#!/usr/bin/env python3 # -*- coding: utf-8 -*- # Software License Agreement (Lesser GPL) # # Copyright (C) 2009-2012 Rosen Diankov # # ikfast is free software: you can redistribute it and/or modify # it under the terms of the GNU Lesser General Public License as published by # the Free Software Foundation, either version 3 of the License, or # at your option) any later version. # # ikfast is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU Lesser General Public License for more details. # # You should have received a copy of the GNU Lesser General Public License # along with this program. If not, see . """ .. _ikfast_compiler: IKFast: The Robot Kinematics Compiler ------------------------------------- .. image:: ../../images/ikfast_robots.jpg :width: 640 IKFast analytically solves robot inverse kinematics equations and generates optimized C++ files. The inverse kinematics equations arise from attemping to place the robot end effector coordinate system in the world while maintaining joint and user-specified constraints. User-specified constraints make up many different `IK Types`_, each of them having advantages depending on the task. IKFast will work with any number of joints arranged in a chain; this is defined by the `Robot.Manipulator`. For chains containing more degrees of freedom (DOF) than the IK type requires, the user can set arbitrary values of a subset of the joints until the number of unknown joints matches the degrees of freedom of the IK type. It is not trivial to create hand-optimized inverse kinematics solutions for arms that can capture all degenerate cases, having closed-form IK speeds up many tasks including planning algorithms, so it really is a must for most robotics researchers. Closed-form solutions are necessary for motion planning due to two reasons: - Numerical inverse kinematics solvers will always be much slower than closed form solutions. Planners require being able to process thousands of configurations per second. The closed-form code generated by ikfast can produce solutions on the order of **~4 microseconds**! As a comparison, most numerical solutions are on the order of 10 milliseconds (assuming good convergence). - The null space of the solution set can be explored because all solutions are computed. Features ======== - Can handle robots with arbitrary joint complexity like non-intersecting axes. - All possible discrete solutions calculated (can be up to 16). - Generated C++ code **independent** of OpenRAVE or any other library. - Automatically detects degenerate cases where 2 or more axes align and cause infinite solutions. - Invalid solutions are detected by checking if square roots are given negative values or arc sines and arc cosines are given inputs exceeding the [-1,1] range. - All divide by zero conditions are automatically checked and handled. .. _ikfast_types: IK Types -------- The following inverse kinematics types are supported: * **Transform6D** - end effector reaches desired 6D transformation * **Rotation3D** - end effector reaches desired 3D rotation * **Translation3D** - end effector origin reaches desired 3D translation * **Direction3D** - direction on end effector coordinate system reaches desired direction * **Ray4D** - ray on end effector coordinate system reaches desired global ray * **Lookat3D** - direction on end effector coordinate system points to desired 3D position * **TranslationDirection5D** - end effector origin and direction reaches desired 3D translation and direction. Can be thought of as Ray IK where the origin of the ray must coincide. * **TranslationXY2D** - end effector origin reaches desired XY translation position, Z is ignored. The coordinate system with relative to the base link. * **TranslationLocalGlobal6D** - local point on end effector origin reaches desired 3D global point. Because both local point and global point can be specified, there are 6 values. * **TranslationXAxisAngle4D**, **TranslationYAxisAngle4D**, **TranslationZAxisAngle4D** - end effector origin reaches desired 3D translation, manipulator direction makes a specific angle with x/y/z-axis (defined in the manipulator base link's coordinate system) * **TranslationXAxisAngleZNorm4D**, **TranslationYAxisAngleXNorm4D**, **TranslationZAxisAngleYNorm4D** - end effector origin reaches desired 3D translation, manipulator direction needs to be orthogonal to z, x, or y axis and be rotated at a certain angle starting from the x, y, or z axis (defined in the manipulator base link's coordinate system) The possible solve methods are defined by `ikfast.IKFastSolver.GetSolvers()` Usage ----- The main file ikfast.py can be used both as a library and as an executable program. For advanced users, it is also possible to use run ikfast.py as a stand-alone program, which makes it mostly independent of the OpenRAVE run-time. **However, the recommended way of using IKFast** is through the OpenRAVE :mod:`.databases.inversekinematics` database generator which directly loads the IK into OpenRAVE as an interface. Stand-alone Executable ====================== To get help and a description of the ikfast arguments type .. code-block:: bash python `openrave-config --python-dir`/openravepy/_openravepy_/ikfast.py --help A simple example to generate IK for setting the 3rd joint free of the Barrett WAM is .. code-block:: bash python `openrave-config --python-dir`/openravepy/_openravepy_/ikfast.py --robot=robots/barrettwam.robot.xml --baselink=0 --eelink=7 --savefile=ik.cpp --freeindex=2 Through Python ============== IKFast can also be used as a library in python. Generating 6D IK for the Barrett WAM while setting the 3rd joint free can be achieved with: .. code-block:: python env = Environment() kinbody = env.ReadRobotXMLFile('robots/barrettwam.robot.xml') env.Add(kinbody) solver = ikfast.IKFastSolver(kinbody=kinbody) chaintree = solver.generateIkSolver(baselink=0,eelink=7,freeindices=[2],solvefn=ikfast.IKFastSolver.solveFullIK_6D) code = solver.writeIkSolver(chaintree) with open('ik.cpp','w') as f: f.write(code) .. _ikfast_generatedcpp: Using Generated IK Files ======================== The common usage is to generate a C++ file that can be compiled into a stand-alone shared object/DLL, an executable program, or linked in statically to a bigger project. For more complex kinematics, LAPACK_ is needed. Here is the header file, which can be found in `share/openrave-X.Y/python/ikfast.h <../../coreapihtml/ikfast_8h.html>`_. Compiling with GCC ~~~~~~~~~~~~~~~~~~ The most basic command is: .. code-block:: bash gcc -lstdc++ -o ik ik.cpp This will generate a small program that outputs all solutions given the end effector with respect to the robot base. Using gcc, this requires "-llapack" to be added. For MSVC++, users will have to compile lapack and link it themselves. Compiling with MSVC ~~~~~~~~~~~~~~~~~~~ `LAPACK For Windows`_ should be installed in order to get complex kinematics linking correctly. Details ------- Terminology: - **solve joints** - the joints to solve for using inverse kinematics - **free joints** - the joints that are specified before the IK is run, these values are known at runtime, but not known at IK generation time. The top level class is `ikfast.IKFastSolver` and generates an Abstract Syntax Tree (AST) using definitions from `ikfast.AST`. The AST is then passed to the language-specific generators defined in `ikfast.CodeGenerators`. Internal symbolic math uses sympy_. Infinite precision fractions are used in order to keep track of linearly independent equations and when they evaluate to 0. The infinite precision fractions are converted to decimals in the generators. .. _LAPACK: http://www.netlib.org/lapack/ .. _`LAPACK For Windows`: http://icl.cs.utk.edu/lapack-for-windows/ .. _sympy: http://code.google.com/p/sympy/ Open Issues ----------- 1. currently ikfast does not handle big decimal numbers well. for example defining the axes or anchors as 1.032513241 will produce very big fractions and make things slow. 2. there are cases when axes align and there are infinite solutions. although ikfast can detect such cases, we need a lot more work in this area. 3. for 6D ik, there are still mechanisms it cannot solve, please send the kinematics model if such a situation is encountered. 4. there are 10 different types of IK, currently ray4d IK needs a lot of work. FAQ --- Q. **ikfast has been running for more than an hour, will it ever finish?** A. Most likely not, usually an iksolver finishes within 10 minutes. ---- """ from __future__ import with_statement # for python 2.5 from sympy import __version__ as sympy_version if sympy_version < '0.7.0': raise ImportError('ikfast needs sympy 0.7.x or greater') sympy_smaller_073 = sympy_version < '0.7.3' from sympy import * from sympy.simplify import cse_main if sympy_version > '0.7.1': _zeros, _ones = zeros, ones zeros = lambda args: _zeros(*args) ones = lambda args: _ones(*args) __author__ = 'Rosen Diankov' __copyright__ = 'Copyright (C) 2009-2012 Rosen Diankov ' __license__ = 'Lesser GPL, Version 3' __version__ = '0x1000004c' # hex of the version, has to be prefixed with 0x. also in ikfast.h # has to specify __version__ after importing as sympy as it will be overwritten by sympy.__version__ import sys, copy, time, math, datetime if sys.version_info[0]<3: import __builtin__ else: import builtins as __builtin__ from optparse import OptionParser try: from openravepy.metaclass import AutoReloader from openravepy import axisAngleFromRotationMatrix except: axisAngleFromRotationMatrix = None class AutoReloader: pass import numpy # required for fast eigenvalue computation try: import mpmath # on some distributions, sympy does not have mpmath in its scope mpmath.mp.eps = 1e-9 # sometimes mpmath.polyroots does not converge well, relax the threshold except ImportError: pass try: import re # for latex cleanup except ImportError: pass try: from math import isinf, isnan except ImportError: # python 2.5 from numpy import isinf as _isinf from numpy import isnan as _isnan def isinf(x): return _isinf(float(x)) def isnan(x): return _isnan(float(x)) from operator import itemgetter from itertools import chain, product try: from itertools import izip except ImportError: izip = zip try: from itertools import combinations, permutations except ImportError: def combinations(items,n): if n == 0: yield[] else: _internal_items=list(items) for i in xrange(len(_internal_items)): for cc in combinations(_internal_items[i+1:],n-1): yield [_internal_items[i]]+cc def permutations(iterable, r=None): # permutations('ABCD', 2) --> AB AC AD BA BC BD CA CB CD DA DB DC # permutations(range(3)) --> 012 021 102 120 201 210 pool = tuple(iterable) n = len(pool) r = n if r is None else r if r > n: return indices = list(range(n)) cycles = list(range(n, n-r, -1)) yield tuple(pool[i] for i in indices[:r]) while n: for i in reversed(range(r)): cycles[i] -= 1 if cycles[i] == 0: indices[i:] = indices[i+1:] + indices[i:i+1] cycles[i] = n - i else: j = cycles[i] indices[i], indices[-j] = indices[-j], indices[i] yield tuple(pool[i] for i in indices[:r]) break else: return import six import logging log = logging.getLogger('openravepy.ikfast') try: # not necessary, just used for testing import swiginac using_swiginac = True except ImportError: using_swiginac = False # from sympy/core/core.py ordering_of_classes = [ # singleton numbers 'Zero', 'One', 'Half', 'Infinity', 'NaN', 'NegativeOne', 'NegativeInfinity', # numbers 'Integer', 'Rational', 'Float', # singleton symbols 'Exp1', 'Pi', 'ImaginaryUnit', # symbols 'Symbol', 'Wild', 'Temporary', # arithmetic operations 'Pow', 'Mul', 'Add', # function values 'Derivative', 'Integral', # defined singleton functions 'Abs', 'Sign', 'Sqrt', 'Floor', 'Ceiling', 'Re', 'Im', 'Arg', 'Conjugate', 'Exp', 'Log', 'Sin', 'Cos', 'Tan', 'Cot', 'ASin', 'ACos', 'ATan', 'ACot', 'Sinh', 'Cosh', 'Tanh', 'Coth', 'ASinh', 'ACosh', 'ATanh', 'ACoth', 'RisingFactorial', 'FallingFactorial', 'factorial', 'binomial', 'Gamma', 'LowerGamma', 'UpperGamma', 'PolyGamma', 'Erf', # special polynomials 'Chebyshev', 'Chebyshev2', # undefined functions 'Function', 'WildFunction', # anonymous functions__l 'Lambda', # Landau O symbol 'Order', # relational operations 'Equality', 'Unequality', 'StrictGreaterThan', 'StrictLessThan', 'GreaterThan', 'LessThan', ] CodeGenerators = {} # try: # import ikfast_generator_vb # CodeGenerators['vb'] = ikfast_generator_vb.CodeGenerator # CodeGenerators['vb6'] = ikfast_generator_vb.CodeGeneratorVB6 # CodeGenerators['vb6special'] = ikfast_generator_vb.CodeGeneratorVB6Special # except ImportError: # pass try: from . import ikfast_generator_cpp CodeGenerators['cpp'] = ikfast_generator_cpp.CodeGenerator IkType = ikfast_generator_cpp.IkType except ImportError: pass # changes to sympy: # core/power.py Pow def Pow_eval_subs(self, old, new): if self == old: return new if old.func is self.func and self.base == old.base: coeff1, terms1 = self.exp.as_coeff_mul() coeff2, terms2 = old.exp.as_coeff_mul() if terms1==terms2: # pow = coeff1/coeff2 # if pow.is_Integer or self.base.is_commutative: # return Pow(new, pow) # (x**(2*y)).subs(x**(3*y),z) -> z**(2/3) # only divide if coeff2 is a divisor of coeff1 if coeff1.is_integer and coeff2.is_integer and (coeff1/coeff2).is_integer: return new ** (coeff1/coeff2) # (x**(2*y)).subs(x**(3*y),z) -> z**(2/3*y) if old.func is C.exp: coeff1, terms1 = old.args[0].as_coeff_mul() coeff2, terms2 = (self.exp*C.log(self.base)).as_coeff_mul() if terms1==terms2: # only divide if coeff2 is a divisor of coeff1 if coeff1.is_integer and coeff2.is_integer and (coeff1/coeff2).is_integer: return new ** (coeff1/coeff2) # (x**(2*y)).subs(exp(3*y*log(x)),z) -> z**(2/3*y) return Pow(self.base._eval_subs(old, new), self.exp._eval_subs(old, new)) if sympy_smaller_073: power.Pow._eval_subs = Pow_eval_subs # simplify/simplify.py def trigsimp_custom(self, **args): """ Default trigsimp (in sympy >= 0.7.3) reduces sum of sin/cos products, for example trigsimp(-sin(x)⋅cos(y) + sin(y)⋅cos(x)) >> -sin(x - y) We have to undo this step, which is what happens here. """ from sympy.simplify import trigsimp as sympy_trigsimp from sympy.simplify.fu import TR10 return TR10(sympy_trigsimp(self, **args)) if not sympy_smaller_073: trigsimp = trigsimp_custom # def custom_trigsimp_nonrecursive(expr, deep=False): # """ # A nonrecursive trig simplifier, used from trigsimp. # # == Usage == # trigsimp_nonrecursive(expr) -> reduces expression by using known trig # identities # # == Notes == # # deep ........ apply trigsimp inside functions # # == Examples == # >>> from sympy import cos, sin, log # >>> from sympy.simplify.simplify import trigsimp, trigsimp_nonrecursive # >>> from sympy.abc import x, y # >>> e = 2*sin(x)**2 + 2*cos(x)**2 # >>> trigsimp(e) # 2 # >>> trigsimp_nonrecursive(log(e)) # log(2*cos(x)**2 + 2*sin(x)**2) # >>> trigsimp_nonrecursive(log(e), deep=True) # log(2) # # """ # from sympy.core.basic import S # sin, cos, tan, cot = C.sin, C.cos, C.tan, C.cot # # if expr.is_Function: # if deep: # return expr.func(trigsimp_nonrecursive(expr.args[0], deep)) # elif expr.is_Mul: # ret = S.One # for x in expr.args: # ret *= trigsimp_nonrecursive(x, deep) # # return ret # elif expr.is_Pow: # return Pow(trigsimp_nonrecursive(expr.base, deep), # trigsimp_nonrecursive(expr.exp, deep)) # elif expr.is_Add: # # TODO this needs to be faster # # # The types of trig functions we are looking for # a,b,c = map(Wild, 'abc') # matchers = ( # (a*sin(b)**2, a - a*cos(b)**2), # (a*tan(b)**2, a*(1/cos(b))**2 - a), # (a*cot(b)**2, a*(1/sin(b))**2 - a) # ) # # # Scan for the terms we need # ret = S.Zero # for term in expr.args: # term = trigsimp_nonrecursive(term, deep) # res = None # for pattern, result in matchers: # res = term.match(pattern) # if res is not None: # ret += result.subs(res) # break # if res is None: # ret += term # # # Reduce any lingering artifacts, such as sin(x)**2 changing # # to 1-cos(x)**2 when sin(x)**2 was "simpler" # artifacts = ( # (a - a*cos(b)**2 + c, a*sin(b)**2 + c, cos), # (a - a*(1/cos(b))**2 + c, -a*tan(b)**2 + c, cos), # (a - a*(1/sin(b))**2 + c, -a*cot(b)**2 + c, sin) # ) # # expr = ret # for pattern, result, ex in artifacts: # # Substitute a new wild that excludes some function(s) # # to help influence a better match. This is because # # sometimes, for example, 'a' would match sec(x)**2 # a_t = Wild('a', exclude=[ex]) # pattern = pattern.subs(a, a_t) # result = result.subs(a, a_t) # if expr.is_number: # continue # try: # m = expr.match(pattern) # except (TypeError): # break # # while m is not None: # if m[a_t] == 0 or -m[a_t] in m[c].args or m[a_t] + m[c] == 0: # break # expr = result.subs(m) # if expr.is_number: # continue # try: # m = expr.match(pattern) # except (TypeError): # break # # # return expr # return expr # # simplify.simplify.trigsimp_nonrecursive = custom_trigsimp_nonrecursive class AST: """Abstarct Syntax Tree class definitions specific for evaluating complex math equations. """ class SolverBase(object): def GetChildrenOfType(self, classinstance): return [] def GetZeroThreshold(self): """returns the threshold to use to check for zeros """ return None class SolverSolution(SolverBase): """Contains equations for evaluating one unknown variable. The variable can have multiple solutions, and the solution is only valid if every equation in checkforzeros is non-zero """ jointname = None jointeval = None jointevalcos = None jointevalsin = None AddPiIfNegativeEq = None isHinge = True checkforzeros = None thresh = None AddHalfTanValue = False dictequations = None presetcheckforzeros = None checkEquationsUsed = True # if True then check equationsused at the end to make sure solution is consistent equationsused = None """Meaning of FeasibleIsZeros: If set to false, then solution is feasible only if all of these equations evalute to non-zero. If set to true, solution is feasible only if all these equations evaluate to zero. """ FeasibleIsZeros = False score = None def __init__(self, jointname, jointeval=None,jointevalcos=None,jointevalsin=None,AddPiIfNegativeEq=None,isHinge=True,thresh=0.000001): self.jointname = jointname self.jointeval = jointeval self.jointevalcos = jointevalcos self.jointevalsin = jointevalsin self.AddPiIfNegativeEq = AddPiIfNegativeEq self.isHinge=isHinge self.thresh = thresh self.presetcheckforzeros = [] self.dictequations = [] self.equationsused = [] assert(self.checkValidSolution()) def subs(self,solsubs): if self.jointeval is not None: self.jointeval = [e.subs(solsubs) for e in self.jointeval] if self.jointevalcos is not None: self.jointevalcos = [e.subs(solsubs) for e in self.jointevalcos] if self.jointevalsin is not None: self.jointevalsin = [e.subs(solsubs) for e in self.jointevalsin] if self.checkforzeros is not None: self.checkforzeros = [e.subs(solsubs) for e in self.checkforzeros] self.dictequations = [(s,v.subs(solsubs)) for s,v in self.dictequations] self.presetcheckforzeros = [e.subs(solsubs) for e in self.presetcheckforzeros] self.equationsused = [e.subs(solsubs) for e in self.equationsused] if not self.checkValidSolution(): raise IKFastSolver.CannotSolveError('substitution produced invalid results') return self def generate(self, generator): assert(self.checkValidSolution()) return generator.generateSolution(self) def end(self, generator): return generator.endSolution(self) def numsolutions(self): n=0 if self.jointeval is not None: n += len(self.jointeval) if self.jointevalcos is not None: n += 2*len(self.jointevalcos) if self.jointevalsin is not None: n += 2*len(self.jointevalsin) return n def checkValidSolution(self): valid=True if self.jointeval is not None: valid &= all([IKFastSolver.isValidSolution(e) for e in self.jointeval]) if self.jointevalsin is not None: valid &= all([IKFastSolver.isValidSolution(e) for e in self.jointevalsin]) if self.jointevalcos is not None: valid &= all([IKFastSolver.isValidSolution(e) for e in self.jointevalcos]) return valid def getPresetCheckForZeros(self): return self.presetcheckforzeros def getEquationsUsed(self): if self.checkEquationsUsed: return self.equationsused return None def GetZeroThreshold(self): return self.thresh class SolverPolynomialRoots(SolverBase): """find all roots of the polynomial and plug it into jointeval. poly should be Poly """ jointname = None poly = None polybackup = None # if poly does not yield and results, this polynomial will be solved instead jointeval = None jointevalcos = None # not used jointevalsin = None # not used checkforzeros = None postcheckforzeros = None # fail if any zero postcheckfornonzeros = None # fail if any nonzero postcheckforNumDenom = None # list of (A,B) pairs where Ax=B was used. Fail if A==0&&B!=0 postcheckforrange = None # checks that value is within [-1,1] dictequations = None postcheckforzerosThresh = 1e-8 # threshold for checking postcheckforzeros. if abs(val) <= postcheckforzerosThresh: skip postcheckfornonzerosThresh = 1e-8 # threshold for checking postcheckfornonzeros. if abs(val) > postcheckfornonzerosThresh: skip postcheckforrangeThresh = 1e-8 # threshold for checking postcheckforrange. if val <= -1-postcheckforrangeThresh || val > 1+postcheckforrangeThresh: skip postcheckforNumDenomThresh = 1e-8 # threshold for checking postcheckforNumDenom: if abs(val[0]) <= postcheckforNumDenomThresh && abs(val[1]) > postcheckforNumDenomThresh: skip isHinge = True FeasibleIsZeros = False AddHalfTanValue = False score = None equationsused = None def __init__(self, jointname, poly=None, jointeval=None,isHinge=True): self.poly = poly assert(self.poly.degree(0)>0) self.jointname=jointname self.jointeval = jointeval self.isHinge = isHinge self.dictequations = [] self.equationsused = [] def numsolutions(self): if self.polybackup is None: return self.poly.degree(0) else: return max(self.poly.degree(0), self.polybackup.degree(0)) def subs(self,solsubs): if self.jointeval is not None: self.jointeval = [e.subs(solsubs) for e in self.jointeval] if self.checkforzeros is not None: self.checkforzeros = [e.subs(solsubs) for e in self.checkforzeros] if self.postcheckforzeros is not None: self.postcheckforzeros = [e.subs(solsubs) for e in self.postcheckforzeros] if self.postcheckfornonzeros is not None: self.postcheckfornonzeros = [e.subs(solsubs) for e in self.postcheckfornonzeros] if self.postcheckforrange is not None: self.postcheckforrange = [e.subs(solsubs) for e in self.postcheckforrange] if self.postcheckforNumDenom is not None: self.postcheckforNumDenom = [e.subs(solsubs) for e in self.postcheckforNumDenom] self.dictequations = [(s,v.subs(solsubs)) for s,v in self.dictequations] self.equationsused = [e.subs(solsubs) for e in self.equationsused] if self.poly is not None: self.poly = Poly(self.poly.subs(solsubs),*self.poly.gens) if self.polybackup is not None: self.polybackup = Poly(self.polybackup.subs(solsubs),*self.polybackup.gens) assert(self.checkValidSolution()) return self def generate(self, generator): return generator.generatePolynomialRoots(self) def end(self, generator): return generator.endPolynomialRoots(self) def checkValidSolution(self): valid = True if self.poly is not None: valid &= IKFastSolver.isValidSolution(self.poly.as_expr()) if self.polybackup is not None: valid &= IKFastSolver.isValidSolution(self.polybackup.as_expr()) if self.jointeval is not None: valid &= all([IKFastSolver.isValidSolution(e) for e in self.jointeval]) return valid def getPresetCheckForZeros(self): # make sure that all the coefficients containing higher-order variables are not 0 zeroeq = S.Zero for monom, coeff in self.poly.terms(): if monom[0] > 0: if len(self.dictequations) > 0: # bug with sympy? zeroeq += abs(coeff.subs(self.dictequations)) else: zeroeq += abs(coeff) if self.polybackup is not None: for monom, coeff in self.polybackup.terms(): if monom[0] > 0: if len(self.dictequations) > 0: # bug with sympy? zeroeq += abs(coeff.subs(self.dictequations)) else: zeroeq += abs(coeff) return [zeroeq]#self.poly.LC()] def getEquationsUsed(self): return self.equationsused def GetZeroThreshold(self): return self.postcheckforzerosThresh # not really sure... class SolverCoeffFunction(SolverBase): """Evaluate a set of coefficients and pass them to a custom function which will then return all possible values of the specified variables in jointnames. """ jointnames = None jointeval = None isHinges = True exportvar = None exportcoeffeqs = None rootmaxdim = None exportfnname = None jointevalcos = None # used for half angles jointevalsin = None # used for half angles checkforzeros = None FeasibleIsZeros = False score = None presetcheckforzeros = None dictequations = None equationsused = None def __init__(self, jointnames, jointeval=None, exportvar=None, exportcoeffeqs=None,exportfnname=None,isHinges=None,rootmaxdim=16,jointevalcos=None,jointevalsin=None): self.jointnames=jointnames self.jointeval = jointeval self.isHinges = isHinges self.exportvar=exportvar self.exportcoeffeqs=exportcoeffeqs self.exportfnname=exportfnname self.rootmaxdim=rootmaxdim self.jointevalsin=jointevalsin self.jointevalcos=jointevalcos self.presetcheckforzeros = [] self.dictequations = [] self.equationsused = [] def numsolutions(self): return self.rootmaxdim def subs(self,solsubs): if self.jointeval is not None: self.jointeval = [e.subs(solsubs) for e in self.jointeval] if self.jointevalcos is not None: self.jointevalcos = [e.subs(solsubs) for e in self.jointevalcos] if self.jointevalsin is not None: self.jointevalsin = [e.subs(solsubs) for e in self.jointevalsin] if self.checkforzeros is not None: self.checkforzeros = [e.subs(solsubs) for e in self.checkforzeros] self.dictequations = [(s,v.subs(solsubs)) for s,v in self.dictequations] self.presetcheckforzeros = [e.subs(solsubs) for e in self.presetcheckforzeros] self.equationsused = [e.subs(solsubs) for e in self.equationsused] #if self.poly is not None: # self.poly = Poly(self.poly.subs(solsubs)...) assert(self.checkValidSolution()) return self def generate(self, generator): return generator.generateCoeffFunction(self) def end(self, generator): return generator.endCoeffFunction(self) def checkValidSolution(self): #if self.poly is not None: # valid = IKFastSolver.isValidSolution(self.poly.as_expr()) if self.jointeval is not None: valid &= all([IKFastSolver.isValidSolution(e) for e in self.jointeval]) if self.jointevalcos is not None: valid &= all([IKFastSolver.isValidSolution(e) for e in self.jointevalcos]) if self.jointevalsin is not None: valid &= all([IKFastSolver.isValidSolution(e) for e in self.jointevalsin]) return valid def getPresetCheckForZeros(self): return self.presetcheckforzeros def getEquationsUsed(self): return self.equationsused class SolverMatrixInverse(SolverBase): """Take the inverse of a large matirx and set the coefficients of the inverse to the symbols in Asymbols. """ A = None Asymbols = None # has to be same size as B checkforzeros = None def __init__(self, A, Asymbols): self.A = A self.Asymbols = Asymbols def subs(self,solsubs): return self def generate(self, generator): return generator.generateMatrixInverse(self) def end(self, generator): return generator.endMatrixInverse(self) def checkValidSolution(self): return True def getsubs(self,psubs): Asub = self.A.subs(psubs) d = Asub.det() if d == S.Zero: raise IKFastSolver.CannotSolveError('determinant for matrix is zero') Anew = Asub.inv() subs = [] for i in range(self.A.shape[0]): for j in range(self.A.shape[1]): if self.Asymbols[i][j] is not None: subs.append((self.Asymbols[i][j],Anew[i,j])) return subs class SolverConditionedSolution(SolverBase): """set solutions based on evaluating equations """ dictequations = None solversolutions = None # a list of solutions. If the solution's checkforzeros evaluates to all zeros, then that solution us used thresh=0.000001 def __init__(self, solversolutions): self.solversolutions = solversolutions self.dictequations = [] def subs(self,solsubs): for s in self.solversolutions: s.subs(solsubs) return self def generate(self, generator): return generator.generateConditionedSolution(self) def end(self, generator): return generator.endConditionedSolution(self) def GetChildrenOfType(self, classinstance): nodes = [] for childnode in self.solversolutions: if isinstance(childnode, classinstance): nodes.append(childnode) nodes += childnode.GetChildrenOfType(classinstance) return nodes def GetZeroThreshold(self): return self.thresh class SolverBranchConds(SolverBase): """ take certain branches depending if a set of equations evaluate to zero. Each branch can also have dictequations """ jointbranches = None # list of (checkzeroequations, branch, dictequations) thresh = 0.000005 # because it is && comparison, have to relax the threshold otherwise SolverCheckZeros checks that check with same thresh can fail on either equation not being 0, and this only passes when all equations are 0. so have to set threshold higher than what is used for SolverCheckZeros def __init__(self, jointbranches): self.jointbranches = jointbranches def generate(self, generator): return generator.generateBranchConds(self) def end(self, generator): return generator.endBranchConds(self) def GetChildrenOfType(self, classinstance): nodes = [] for checkzeroequations, branch, extradictequations in self.jointbranches: for childnode in branch: if isinstance(childnode, classinstance): nodes.append(childnode) nodes += childnode.GetChildrenOfType(classinstance) return nodes def GetZeroThreshold(self): return self.thresh class SolverCheckZeros(SolverBase): jointname = None jointcheckeqs = None # only used for evaluation zerobranch = None nonzerobranch = None anycondition=None dictequations=None thresh=None # a threshold of 1e-6 breaks hiro ik equationsused = None def __init__(self, jointname, jointcheckeqs, zerobranch, nonzerobranch,thresh=None,anycondition=True): self.jointname = jointname self.jointcheckeqs = jointcheckeqs self.zerobranch = zerobranch self.nonzerobranch = nonzerobranch if thresh is None: self.thresh = 0.000001 else: self.thresh = thresh self.anycondition = anycondition self.dictequations = [] def generate(self, generator): return generator.generateCheckZeros(self) def end(self, generator): return generator.endCheckZeros(self) def getPresetCheckForZeros(self): return [] def checkValidSolution(self): for branch in self.nonzerobranch: if not branch.checkValidSolution(): return False for branch in self.zerobranch: if not branch.checkValidSolution(): return False return True def numsolutions(self): return 1 def subs(self,solsubs): for branch in self.nonzerobranch: if hasattr(branch,'subs'): branch.subs(solsubs) for branch in self.zerobranch: if hasattr(branch,'subs'): branch.subs(solsubs) return self def getEquationsUsed(self): return self.equationsused def GetChildrenOfType(self, classinstance): nodes = [] for childnode in self.nonzerobranch + self.zerobranch: if isinstance(childnode, classinstance): nodes.append(childnode) nodes += childnode.GetChildrenOfType(classinstance) return nodes def GetZeroThreshold(self): return self.thresh class SolverFreeParameter(SolverBase): jointname = None jointtree = None def __init__(self, jointname, jointtree): self.jointname = jointname self.jointtree = jointtree def generate(self, generator): return generator.generateFreeParameter(self) def end(self, generator): return generator.endFreeParameter(self) def GetChildrenOfType(self, classinstance): nodes = [] for childnode in self.jointtree: if isinstance(childnode, classinstance): nodes.append(childnode) nodes += childnode.GetChildrenOfType(classinstance) return nodes class SolverRotation(SolverBase): T = None jointtree = None functionid=0 def __init__(self, T, jointtree): self.T = T self.jointtree = jointtree self.dictequations = [] def generate(self, generator): return generator.generateRotation(self) def end(self, generator): return generator.endRotation(self) class SolverFunction(SolverBase): jointtree = None name='innerfn' def __init__(self, name, jointtree): self.name = name self.jointtree = jointtree self.dictequations = [] def generate(self, generator): return generator.generateFunction(self) def end(self, generator): return generator.endFunction(self) def GetChildrenOfType(self, classinstance): nodes = [] for childnode in self.jointtree: if isinstance(childnode, classinstance): nodes.append(childnode) nodes += childnode.GetChildrenOfType(classinstance) return nodes class SolverStoreSolution(SolverBase): """Called when all the unknowns have been solved to add a solution. """ alljointvars = None checkgreaterzero = None # used for final sanity checks to ensure IK solution is consistent thresh = 0 offsetvalues = None isHinge = None def __init__(self, alljointvars,checkgreaterzero=None,isHinge=None): self.alljointvars = alljointvars self.checkgreaterzero = checkgreaterzero self.isHinge=isHinge if isHinge is None: log.warn('SolverStoreSolution.isHinge is not initialized') self.isHinge = [True]*len(self.alljointvars) def generate(self, generator): return generator.generateStoreSolution(self) def end(self, generator): return generator.endStoreSolution(self) class SolverSequence(SolverBase): jointtrees = None def __init__(self, jointtrees): self.jointtrees = jointtrees def generate(self, generator): return generator.generateSequence(self) def end(self, generator): return generator.endSequence(self) def GetChildrenOfType(self, classinstance): nodes = [] for tree in self.jointtrees: for childnode in tree: if isinstance(childnode, classinstance): nodes.append(childnode) nodes += childnode.GetChildrenOfType(classinstance) return nodes class SolverBreak(SolverBase): """Terminates this scope""" comment = None # a comment for the reason of the break varsubs = None # variable substitutions that were valid at the break othersolvedvars = None # the solved variables already solsubs = None # the substitutions of the solved variables endbranchtree = None # a node that points to the end of the tree def __init__(self, comment, varsubs=list(), othersolvedvars=list(), solsubs=list(), globalsymbols=list(), endbranchtree=None): self.comment = comment self.varsubs = list(varsubs) self.othersolvedvars = list(othersolvedvars) self.solsubs = list(solsubs) self.endbranchtree = endbranchtree def generate(self,generator): return generator.generateBreak(self) def end(self,generator): return generator.endBreak(self) def checkValidSolution(self): return True class SolverIKChainTransform6D(SolverBase): solvejointvars = None freejointvars = None jointtree = None Tfk = None Tee = None dictequations = None def __init__(self, solvejointvars, freejointvars, Tee, jointtree,Tfk=None): self.solvejointvars = solvejointvars self.freejointvars = freejointvars self.Tee = Tee self.jointtree = jointtree self.Tfk = Tfk self.dictequations = [] def generate(self, generator): return generator.generateChain(self) def end(self, generator): return generator.endChain(self) def leftmultiply(self,Tleft,Tleftinv): self.Tfk = Tleft*self.Tfk self.Tee = Tleftinv*self.Tee class SolverIKChainRotation3D(SolverBase): solvejointvars = None freejointvars = None Rfk = None Ree = None jointtree = None dictequations = None def __init__(self, solvejointvars, freejointvars, Ree, jointtree,Rfk=None): self.solvejointvars = solvejointvars self.freejointvars = freejointvars self.Ree = Ree self.Rfk=Rfk self.jointtree = jointtree self.dictequations = [] def generate(self, generator): return generator.generateIKChainRotation3D(self) def end(self, generator): return generator.endIKChainRotation3D(self) def leftmultiply(self,Tleft,Tleftinv): self.Rfk = Tleft[0:3,0:3]*self.Rfk self.Ree = Tleftinv[0:3,0:3]*self.Ree class SolverIKChainTranslation3D(SolverBase): solvejointvars = None freejointvars = None jointtree = None Pfk = None Pee = None dictequations = None uselocaltrans = False def __init__(self, solvejointvars, freejointvars, Pee, jointtree,Pfk=None): self.solvejointvars = solvejointvars self.freejointvars = freejointvars self.Pee = Pee self.jointtree = jointtree self.Pfk=Pfk self.dictequations = [] def generate(self, generator): return generator.generateIKChainTranslation3D(self) def end(self, generator): return generator.endIKChainTranslation3D(self) def leftmultiply(self,Tleft,Tleftinv): self.Pfk = Tleft[0:3,0:3]*self.Pfk+Tleft[0:3,3] self.Pee = Tleftinv[0:3,0:3]*self.Pee+Tleftinv[0:3,3] class SolverIKChainTranslationXY2D(SolverBase): solvejointvars = None freejointvars = None jointtree = None Pfk = None Pee = None dictequations = None def __init__(self, solvejointvars, freejointvars, Pee, jointtree,Pfk=None): self.solvejointvars = solvejointvars self.freejointvars = freejointvars self.Pee = Pee self.jointtree = jointtree self.Pfk=Pfk self.dictequations = [] def generate(self, generator): return generator.generateIKChainTranslationXY2D(self) def end(self, generator): return generator.endIKChainTranslationXY2D(self) def leftmultiply(self,Tleft,Tleftinv): self.Pfk = Tleft[0:2,0:2]*self.Pfk+Tleft[0:2,3] self.Pee = Tleftinv[0:2,0:2]*self.Pee+Tleftinv[0:2,3] class SolverIKChainDirection3D(SolverBase): solvejointvars = None freejointvars = None jointtree = None Dfk = None Dee = None dictequations = None def __init__(self, solvejointvars, freejointvars, Dee, jointtree,Dfk=None): self.solvejointvars = solvejointvars self.freejointvars = freejointvars self.Dee = Dee self.jointtree = jointtree self.Dfk=Dfk self.dictequations = [] def generate(self, generator): return generator.generateIKChainDirection3D(self) def end(self, generator): return generator.endIKChainDirection3D(self) def leftmultiply(self,Tleft,Tleftinv): self.Dfk = Tleft[0:3,0:3]*self.Dfk self.Dee = Tleftinv[0:3,0:3]*self.Dee class SolverIKChainRay(SolverBase): solvejointvars = None freejointvars = None jointtree = None Pfk = None Dfk = None Pee = None Dee = None dictequations = None is5dray = False # if True, then full 3D position becomes important and things shouldn't be normalized def __init__(self, solvejointvars, freejointvars, Pee, Dee, jointtree,Pfk=None,Dfk=None,is5dray=False): self.solvejointvars = solvejointvars self.freejointvars = freejointvars self.Pee = Pee self.Dee = Dee self.jointtree = jointtree self.Pfk = Pfk self.Dfk = Dfk self.dictequations = [] self.is5dray=is5dray def generate(self, generator): return generator.generateIKChainRay(self) def end(self, generator): return generator.endIKChainRay(self) def leftmultiply(self,Tleft,Tleftinv): self.Pfk = Tleft[0:3,0:3]*self.Pfk+Tleft[0:3,3] self.Dfk = Tleft[0:3,0:3]*self.Dfk self.Pee = Tleftinv[0:3,0:3]*self.Pee+Tleftinv[0:3,3] self.Dee = Tleftinv[0:3,0:3]*self.Dee class SolverIKChainLookat3D(SolverBase): solvejointvars = None freejointvars = None jointtree = None Pfk = None Dfk = None Pee = None dictequations = None def __init__(self, solvejointvars, freejointvars, Pee, jointtree,Pfk=None,Dfk=None): self.solvejointvars = solvejointvars self.freejointvars = freejointvars self.Pee = Pee self.jointtree = jointtree self.Pfk=Pfk self.Dfk=Dfk self.dictequations = [] def generate(self, generator): return generator.generateIKChainLookat3D(self) def end(self, generator): return generator.endIKChainLookat3D(self) def leftmultiply(self,Tleft,Tleftinv): self.Pfk = Tleft[0:3,0:3]*self.Pfk+Tleft[0:3,3] self.Dfk = Tleft[0:3,0:3]*self.Dfk self.Pee = Tleftinv[0:3,0:3]*self.Pee+Tleftinv[0:3,3] class SolverIKChainAxisAngle(SolverBase): solvejointvars = None freejointvars = None jointtree = None Pfk = None Pee = None dictequations = None angleee=None anglefk=None iktype=None def __init__(self, solvejointvars, freejointvars, Pee, angleee,jointtree,Pfk=None,anglefk=None,iktype=None): self.solvejointvars = solvejointvars self.freejointvars = freejointvars self.Pee = Pee self.anglefk=anglefk self.jointtree = jointtree self.Pfk=Pfk self.angleee=angleee self.dictequations = [] self.iktype=iktype def generate(self, generator): return generator.generateSolverIKChainAxisAngle(self) def end(self, generator): return generator.endSolverIKChainAxisAngle(self) def leftmultiply(self,Tleft,Tleftinv): self.Pfk = Tleft[0:2,0:2]*self.Pfk+Tleft[0:2,3] self.Pee = Tleftinv[0:2,0:2]*self.Pee+Tleftinv[0:2,3] assert(0) # need to change angle from sympy.core import function # for sympy 0.7.1+ class fmod(function.Function): """defines floating-point mod""" nargs = 2 is_real = True is_Function = True class atan2check(atan2): """defines floating-point mod""" nargs = 2 is_real = True is_Function = True class GinacUtils: @staticmethod def ConvertToGinac(eq,localsymbolmap): if eq.is_Add: geq = None for arg in eq.args: geq2 = GinacUtils.ConvertToGinac(arg,localsymbolmap) if geq is None: geq = geq2 else: geq += geq2 return geq if geq is not None else swiginac.numeric(0) elif eq.is_Mul: geq = None for arg in eq.args: geq2 = GinacUtils.ConvertToGinac(arg,localsymbolmap) if geq is None: geq = geq2 else: geq *= geq2 return geq if geq is not None else swiginac.numeric(1) elif eq.is_Pow: gbase = GinacUtils.ConvertToGinac(eq.base,localsymbolmap) if eq.exp == S.One: return gbase elif eq.exp == -S.One: return 1/gbase else: return pow(gbase,GinacUtils.ConvertToGinac(eq.exp,localsymbolmap)) elif eq.is_number: return swiginac.numeric(str(eq)) elif eq.is_Symbol: if str(eq) in localsymbolmap: return localsymbolmap[str(eq)] else: gsym = swiginac.symbol(str(eq)) localsymbolmap[str(eq)] = gsym return gsym raise ValueError('unknown equation %s'%str(eq)) @staticmethod def ConvertFromGinac(geq): if isinstance(geq, swiginac.add): return Add(*[GinacUtils.ConvertFromGinac(geq.op(i)) for i in range(geq.nops())]) elif isinstance(geq, swiginac.mul): return Mul(*[GinacUtils.ConvertFromGinac(geq.op(i)) for i in range(geq.nops())]) elif isinstance(geq, swiginac.power): ebase = GinacUtils.ConvertFromGinac(geq.op(0)) if geq.op(1) == 1: return ebase elif geq.op(1) == -1: return S.One/ebase else: return Pow(ebase,GinacUtils.ConvertFromGinac(geq.op(1)),evaluate=False) elif isinstance(geq, swiginac.numeric): if geq.is_integer(): return Integer(str(geq)) elif geq.is_rational(): return Rational(str(geq)) else: return geq.eval() elif isinstance(geq, swiginac.symbol): return Symbol(str(geq)) else: raise ValueError('unknown equation %s'%str(eq)) @staticmethod def ConvertMatrixToGinac(M,name='M', localsymbolmap={}): gM = swiginac.symbolic_matrix(M.shape[0],M.shape[1],'M') for i in range(M.shape[0]): for j in range(M.shape[1]): gM[i,j] = GinacUtils.ConvertToGinac(M[i,j],localsymbolmap) return gM @staticmethod def GetPolyTermsFromGinac(geq, gothersymbols, othersymbols): """return a dict of monom:coeff items """ terms = {} for i, gothersymbol in enumerate(gothersymbols): for degree in range(geq.ldegree(gothersymbol),geq.degree(gothersymbol)+1): monomprefix = (0,)*i + (degree,) gcoeff = geq.coeff(gothersymbol,degree) if i+1 < len(gothersymbols): newterms = GinacUtils.GetPolyTermsFromGinac(gcoeff,gothersymbols[i+1:],othersymbols[i+1:]) for newmonom, newcoeff in newterms.items(): assert(len(newmonom)==len(gothersymbols)-i-1) terms[monomprefix+newmonom] = newcoeff else: # ConvertFromGinac is very slow terms[monomprefix] = gcoeff#GinacUtils.ConvertFromGinac(gcoeff) return terms @staticmethod def SolveUpperTriangular(gA, gB, name='X'): """solves for gA * X = gB. All parameters have to be ginac objects """ gX = swiginac.symbolic_matrix(gB.rows(),gB.cols(),name) for i in reversed(xrange(gA.rows())): if gA[i, i] == 0: raise ValueError("Matrix must be non-singular.") gX[i, 0] = (gB[i, 0] - sum(gA[i, k] * gX[k, 0] for k in xrange(i+1, gA.rows()))) / gA[i, i] return gX class IKFastSolver(AutoReloader): """Solves the analytical inverse kinematics equations. The symbol naming conventions are as follows: cjX - cos joint angle constX - temporary constant used to simplify computations dummyX - dummy intermediate variables to solve for gconstX - global constant that is also used during ik generation phase htjX - half tan of joint angle jX - joint angle pX - end effector position information rX - end effector rotation information sjX - sin joint angle tconstX - second-level temporary constant tjX - tan of joint angle """ @six.python_2_unicode_compatible class CannotSolveError(Exception): """thrown when ikfast fails to solve a particular set of equations with the given knowns and unknowns """ def __init__(self,value=u''): self.value = value def __str__(self): return u'%s: %s'%(self.__class__.__name__, self.value) def __repr__(self): return '<%s(%r)>'%(self.__class__.__name__, self.value) def __eq__(self, r): return self.value == r.value def __ne__(self, r): return self.value != r.value class IKFeasibilityError(Exception): """thrown when it is not possible to solve the IK due to robot not having enough degrees of freedom. For example, a robot with 5 joints does not have 6D IK """ def __init__(self,equations,checkvars): self.equations=equations self.checkvars=checkvars def __str__(self): s = "Not enough equations to solve variables %s!\nThis means one of several things: not enough constraints to solve all variables, or the manipulator does not span the target IK space. This is not an ikfast failure, it just means the robot kinematics are invalid for this type of IK. Equations that are not uniquely solvable are:\n"%str(self.checkvars) for eq in self.equations: s += str(eq) + '\n' return s class JointAxis: __slots__ = ['joint','iaxis'] class Variable: __slots__ = ['name','var','svar','cvar','tvar','htvar','vars','subs','subsinv'] def __init__(self, var): self.name = var.name self.var = var self.svar = Symbol("s%s"%var.name) self.cvar = Symbol("c%s"%var.name) self.tvar = Symbol("t%s"%var.name) self.htvar = Symbol("ht%s"%var.name) self.vars = [self.var,self.svar,self.cvar,self.tvar,self.htvar] self.subs = [(cos(self.var),self.cvar),(sin(self.var),self.svar),(tan(self.var),self.tvar),(tan(self.var/2),self.htvar)] self.subsinv = [(self.cvar,cos(self.var)),(self.svar, sin(self.var)),(self.tvar,tan(self.var))] def getsubs(self,value): return [(self.var,value)]+[(s,v.subs(self.var,value).evalf()) for v,s in self.subs] class DegenerateCases: def __init__(self): self.handleddegeneratecases = [] def Clone(self): clone=IKFastSolver.DegenerateCases() clone.handleddegeneratecases = self.handleddegeneratecases[:] return clone def AddCasesWithConditions(self,newconds,currentcases): for case in newconds: newcases = set(currentcases) newcases.add(case) assert(not self.CheckCases(newcases)) self.handleddegeneratecases.append(newcases) def AddCases(self,currentcases): if not self.CheckCases(currentcases): self.handleddegeneratecases.append(currentcases) else: log.warn('case already added') # sometimes this can happen, but it isn't a bug, just bad bookkeeping def RemoveCases(self, currentcases): for i, handledcases in enumerate(self.handleddegeneratecases): if handledcases == currentcases: self.handleddegeneratecases.pop(i) return True return False def GetHandledConditions(self,currentcases): handledconds = [] for handledcases in self.handleddegeneratecases: if len(currentcases)+1==len(handledcases) and currentcases < handledcases: handledconds.append((handledcases - currentcases).pop()) return handledconds def CheckCases(self,currentcases): for handledcases in self.handleddegeneratecases: if handledcases == currentcases: return True return False def __init__(self, kinbody=None,kinematicshash='',precision=None, checkpreemptfn=None): """ :param checkpreemptfn: checkpreemptfn(msg, progress) called periodically at various points in ikfast. Takes in two arguments to notify user how far the process has completed. """ self._checkpreemptfn = checkpreemptfn self.usinglapack = False self.useleftmultiply = True self.freevarsubs = [] self.degeneratecases = None self.kinematicshash = kinematicshash self.testconsistentvalues = None self.maxcasedepth = 3 # the maximum depth of special/degenerate cases to process before system gives up self.globalsymbols = [] # global symbols for substitutions self._scopecounter = 0 # a counter for debugging purposes that increaes every time a level changes self._dodebug = False self._ikfastoptions = 0 if precision is None: self.precision=8 else: self.precision=precision self.kinbody = kinbody self._iktype = None # the current iktype processing self.axismap = {} self.axismapinv = {} with self.kinbody: for idof in range(self.kinbody.GetDOF()): axis = IKFastSolver.JointAxis() axis.joint = self.kinbody.GetJointFromDOFIndex(idof) axis.iaxis = idof-axis.joint.GetDOFIndex() name = str('j%d')%idof self.axismap[name] = axis self.axismapinv[idof] = name sys.setrecursionlimit(3000) # needed for non-intersecting axes/5D IK robots def _CheckPreemptFn(self, msg=u'', progress=0.25): """progress is a value from [0,1] where 0 is just starting and 1 is complete """ if self._checkpreemptfn is not None: self._checkpreemptfn(msg, progress=progress) def convertRealToRational(self, x,precision=None): if precision is None: precision=self.precision if Abs(x) < 10**-precision: return S.Zero r0 = Rational(str(round(Float(float(x),30),precision))) if x == 0: return r0 r1 = 1/Rational(str(round(Float(1/float(x),30),precision))) return r0 if len(str(r0)) < len(str(r1)) else r1 def ConvertRealToRationalEquation(self, eq, precision=None): if eq.is_Add: neweq = S.Zero for subeq in eq.args: neweq += self.ConvertRealToRationalEquation(subeq,precision) elif eq.is_Mul: neweq = self.ConvertRealToRationalEquation(eq.args[0],precision) for subeq in eq.args[1:]: neweq *= self.ConvertRealToRationalEquation(subeq,precision) elif eq.is_Function: newargs = [self.ConvertRealToRationalEquation(subeq,precision) for subeq in eq.args] neweq = eq.func(*newargs) elif eq.is_number: if eq.is_irrational: # don't touch it since it could be pi! neweq = eq else: neweq = self.convertRealToRational(eq,precision) else: neweq=eq return neweq def normalizeRotation(self,M): """error from openrave can be on the order of 1e-6 (especially if they are defined diagonal to some axis) """ right = Matrix(3,1,[self.convertRealToRational(x,self.precision-3) for x in M[0,0:3]]) right = right/right.norm() up = Matrix(3,1,[self.convertRealToRational(x,self.precision-3) for x in M[1,0:3]]) up = up - right*right.dot(up) up = up/up.norm() d = right.cross(up) for i in range(3): # don't round the rotational part anymore since it could lead to unnormalized rotations! M[0,i] = right[i] M[1,i] = up[i] M[2,i] = d[i] M[i,3] = self.convertRealToRational(M[i,3]) M[3,i] = S.Zero M[3,3] = S.One return M def GetMatrixFromNumpy(self,T): return Matrix(4,4,[x for x in T.flat]) def RoundMatrix(self, T): """given a sympy matrix, will round the matrix and snap all its values to 15, 30, 45, 60, and 90 degrees. """ if axisAngleFromRotationMatrix is not None: Teval = T.evalf() axisangle = axisAngleFromRotationMatrix([[Teval[0,0], Teval[0,1], Teval[0,2]], [Teval[1,0], Teval[1,1], Teval[1,2]], [Teval[2,0], Teval[2,1], Teval[2,2]]]) angle = sqrt(axisangle[0]**2+axisangle[1]**2+axisangle[2]**2) if abs(angle) < 10**(-self.precision): # rotation is identity M = eye(4) else: axisangle = axisangle/angle log.debug('rotation angle: %f, axis=[%f,%f,%f]', (angle*180/pi).evalf(),axisangle[0],axisangle[1],axisangle[2]) accurateaxisangle = Matrix(3,1,[self.convertRealToRational(x,self.precision-3) for x in axisangle]) accurateaxisangle = accurateaxisangle/accurateaxisangle.norm() # angle is not a multiple of 90, can get long fractions. so check if there's any way to simplify it if abs(angle-3*pi/2) < 10**(-self.precision+2): quat = [-S.One/sqrt(2), accurateaxisangle[0]/sqrt(2), accurateaxisangle[1]/sqrt(2), accurateaxisangle[2]/sqrt(2)] elif abs(angle-pi) < 10**(-self.precision+2): quat = [S.Zero, accurateaxisangle[0], accurateaxisangle[1], accurateaxisangle[2]] elif abs(angle-2*pi/3) < 10**(-self.precision+2): quat = [Rational(1,2), accurateaxisangle[0]*sqrt(3)/2, accurateaxisangle[1]*sqrt(3)/2, accurateaxisangle[2]*sqrt(3)/2] elif abs(angle-pi/2) < 10**(-self.precision+2): quat = [S.One/sqrt(2), accurateaxisangle[0]/sqrt(2), accurateaxisangle[1]/sqrt(2), accurateaxisangle[2]/sqrt(2)] elif abs(angle-pi/3) < 10**(-self.precision+2): quat = [sqrt(3)/2, accurateaxisangle[0]/2, accurateaxisangle[1]/2, accurateaxisangle[2]/2] elif abs(angle-pi/4) < 10**(-self.precision+2): # cos(pi/8) = sqrt(sqrt(2)+2)/2 # sin(pi/8) = sqrt(-sqrt(2)+2)/2 quat = [sqrt(sqrt(2)+2)/2, sqrt(-sqrt(2)+2)/2*accurateaxisangle[0], sqrt(-sqrt(2)+2)/2*accurateaxisangle[1], sqrt(-sqrt(2)+2)/2*accurateaxisangle[2]] elif abs(angle-pi/6) < 10**(-self.precision+2): # cos(pi/12) = sqrt(2)/4+sqrt(6)/4 # sin(pi/12) = -sqrt(2)/4+sqrt(6)/4 quat = [sqrt(2)/4+sqrt(6)/4, (-sqrt(2)/4+sqrt(6)/4)*accurateaxisangle[0], (-sqrt(2)/4+sqrt(6)/4)*accurateaxisangle[1], (-sqrt(2)/4+sqrt(6)/4)*accurateaxisangle[2]] else: # could not simplify further #assert(0) return self.normalizeRotation(T) M = self.GetMatrixFromQuat(quat) for i in range(3): M[i,3] = self.convertRealToRational(T[i,3],self.precision) return M if isinstance(T, Matrix): return self.normalizeRotation(Matrix(4,4,[x for x in T])) else: return self.normalizeRotation(Matrix(4,4,[x for x in T.flat])) def numpyVectorToSympy(self,v,precision=None): return Matrix(len(v),1,[self.convertRealToRational(x,precision) for x in v]) @staticmethod def rodrigues(axis, angle): return IKFastSolver.rodrigues2(axis,cos(angle),sin(angle)) @staticmethod def GetMatrixFromQuat(quat): """quaternion is [cos(angle/2), v*sin(angle/2)] return 4x4 matrix with rotation component set """ M = eye(4) qq1 = 2*quat[1]*quat[1] qq2 = 2*quat[2]*quat[2] qq3 = 2*quat[3]*quat[3] M[0,0] = 1 - qq2 - qq3 M[0,1] = 2*(quat[1]*quat[2] - quat[0]*quat[3]) M[0,2] = 2*(quat[1]*quat[3] + quat[0]*quat[2]) M[1,0] = 2*(quat[1]*quat[2] + quat[0]*quat[3]) M[1,1]= 1 - qq1 - qq3 M[1,2]= 2*(quat[2]*quat[3] - quat[0]*quat[1]) M[2,0] = 2*(quat[1]*quat[3] - quat[0]*quat[2]) M[2,1] = 2*(quat[2]*quat[3] + quat[0]*quat[1]) M[2,2] = 1 - qq1 - qq2 return M @staticmethod def rodrigues2(axis, cosangle, sinangle): skewsymmetric = Matrix(3, 3, [S.Zero,-axis[2],axis[1],axis[2],S.Zero,-axis[0],-axis[1],axis[0],S.Zero]) return eye(3) + sinangle * skewsymmetric + (S.One-cosangle)*skewsymmetric*skewsymmetric @staticmethod def affineInverse(affinematrix): T = eye(4) T[0:3,0:3] = affinematrix[0:3,0:3].transpose() T[0:3,3] = -affinematrix[0:3,0:3].transpose() * affinematrix[0:3,3] return T @staticmethod def affineSimplify(T): return Matrix(T.shape[0],T.shape[1],[trigsimp(x.expand()) for x in T]) @staticmethod def multiplyMatrix(Ts): Tfinal = eye(4) for T in Ts: Tfinal = Tfinal*T return Tfinal @staticmethod def equal(eq0,eq1): if isinstance(eq0, Poly): eq0 = eq0.as_expr() if isinstance(eq1, Poly): eq1 = eq1.as_expr() return expand(eq0-eq1) == S.Zero def chop(self,expr,precision=None): return expr def IsHinge(self,axisname): if axisname[0]!='j' or not axisname in self.axismap: if axisname == 'j100': # always revolute! return True log.info('IsHinge returning false for variable %s'%axisname) return False # dummy joint most likely for angles return self.axismap[axisname].joint.IsRevolute(self.axismap[axisname].iaxis) def IsPrismatic(self,axisname): if axisname[0]!='j' or not axisname in self.axismap: log.info('IsPrismatic returning false for variable %s'%axisname) return False # dummy joint most likely for angles return self.axismap[axisname].joint.IsPrismatic(self.axismap[axisname].iaxis) def forwardKinematicsChain(self, chainlinks, chainjoints): """The first and last matrices returned are always non-symbolic """ with self.kinbody: assert(len(chainjoints)+1==len(chainlinks)) Links = [] Tright = eye(4) jointvars = [] jointinds = [] for i,joint in enumerate(chainjoints): if len(joint.GetName()) == 0: raise self.CannotSolveError('chain %s:%s contains a joint with no name!'%(chainlinks[0].GetName(),chainlinks[-1].GetName())) if chainjoints[i].GetHierarchyParentLink() == chainlinks[i]: TLeftjoint = self.GetMatrixFromNumpy(joint.GetInternalHierarchyLeftTransform()) TRightjoint = self.GetMatrixFromNumpy(joint.GetInternalHierarchyRightTransform()) axissign = S.One else: TLeftjoint = self.affineInverse(self.GetMatrixFromNumpy(joint.GetInternalHierarchyRightTransform())) TRightjoint = self.affineInverse(self.GetMatrixFromNumpy(joint.GetInternalHierarchyLeftTransform())) axissign = -S.One if joint.IsStatic(): Tright = self.affineSimplify(Tright * TLeftjoint * TRightjoint) else: Tjoints = [] for iaxis in range(joint.GetDOF()): var = None if joint.GetDOFIndex() >= 0: var = Symbol(self.axismapinv[joint.GetDOFIndex()]) cosvar = cos(var) sinvar = sin(var) jointvars.append(var) elif joint.IsMimic(iaxis): # get the mimic equation var = joint.GetMimicEquation(iaxis) for itestjoint, testjoint in enumerate(chainjoints): var = var.replace(testjoint.GetName(), 'j%d'%testjoint.GetDOFIndex()) # this needs to be reduced! cosvar = expand_trig(cos(var)) # cos(j1+j2) -> cos(j1)*cos(j2) - sin(j1)*sin(j2) sinvar = expand_trig(sin(var)) elif joint.IsStatic(): # joint doesn't move so assume identity pass else: raise ValueError('cannot solve for mechanism when a non-mimic passive joint %s is in chain'%str(joint)) Tj = eye(4) if var is not None: jaxis = axissign*self.numpyVectorToSympy(joint.GetInternalHierarchyAxis(iaxis)) if joint.IsRevolute(iaxis): Tj[0:3,0:3] = self.rodrigues2(jaxis,cosvar,sinvar) elif joint.IsPrismatic(iaxis): Tj[0:3,3] = jaxis*(var) else: raise ValueError('failed to process joint %s'%joint.GetName()) Tjoints.append(Tj) if axisAngleFromRotationMatrix is not None: axisangle = axisAngleFromRotationMatrix(numpy.array(numpy.array(Tright * TLeftjoint),numpy.float64)) angle = sqrt(axisangle[0]**2+axisangle[1]**2+axisangle[2]**2) if angle > 1e-8: axisangle = axisangle/angle log.debug('rotation angle of Links[%d]: %f, axis=[%f,%f,%f]', len(Links), (angle*180/pi).evalf(),axisangle[0],axisangle[1],axisangle[2]) Links.append(self.RoundMatrix(Tright * TLeftjoint)) for Tj in Tjoints: jointinds.append(len(Links)) Links.append(Tj) Tright = TRightjoint Links.append(self.RoundMatrix(Tright)) # before returning the final links, try to push as much translation components # outwards to both ends. Sometimes these components can get in the way of detecting # intersecting axes if len(jointinds) > 0: iright = jointinds[-1] Ttrans = eye(4) Ttrans[0:3,3] = Links[iright-1][0:3,0:3].transpose() * Links[iright-1][0:3,3] Trot_with_trans = Ttrans * Links[iright] separated_trans = Trot_with_trans[0:3,0:3].transpose() * Trot_with_trans[0:3,3] for j in range(0,3): if separated_trans[j].has(*jointvars): Ttrans[j,3] = Rational(0) else: Ttrans[j,3] = separated_trans[j] Links[iright+1] = Ttrans * Links[iright+1] Links[iright-1] = Links[iright-1] * self.affineInverse(Ttrans) log.info("moved translation %s to right end",Ttrans[0:3,3].transpose()) if len(jointinds) > 1: ileft = jointinds[0] separated_trans = Links[ileft][0:3,0:3] * Links[ileft+1][0:3,3] Ttrans = eye(4) for j in range(0,3): if not separated_trans[j].has(*jointvars): Ttrans[j,3] = separated_trans[j] Links[ileft-1] = Links[ileft-1] * Ttrans Links[ileft+1] = self.affineInverse(Ttrans) * Links[ileft+1] log.info("moved translation %s to left end",Ttrans[0:3,3].transpose()) if len(jointinds) > 3: # last 3 axes always have to be intersecting, move the translation of the first axis to the left ileft = jointinds[-3] separated_trans = Links[ileft][0:3,0:3] * Links[ileft+1][0:3,3] Ttrans = eye(4) for j in range(0,3): if not separated_trans[j].has(*jointvars): Ttrans[j,3] = separated_trans[j] Links[ileft-1] = Links[ileft-1] * Ttrans Links[ileft+1] = self.affineInverse(Ttrans) * Links[ileft+1] log.info("moved translation on intersecting axis %s to left",Ttrans[0:3,3].transpose()) return Links, jointvars def countVariables(self,expr,var): """Counts number of terms variable appears in""" if not expr.is_Add: if expr.has(var): return 1 return 0 num = 0 for term in expr.args: if term.has(var): num += 1 return num @staticmethod def isValidPowers(expr): if expr.is_Pow: if not expr.exp.is_number or expr.exp < 0: return False return IKFastSolver.isValidPowers(expr.base) elif expr.is_Add or expr.is_Mul or expr.is_Function: return all([IKFastSolver.isValidPowers(arg) for arg in expr.args]) else: return True @staticmethod def rotateDirection(sourcedir,targetdir): sourcedir /= sqrt(sourcedir.dot(sourcedir)) targetdir /= sqrt(targetdir.dot(targetdir)) rottodirection = sourcedir.cross(targetdir) fsin = sqrt(rottodirection.dot(rottodirection)) fcos = sourcedir.dot(targetdir) M = eye(4) if fsin > 1e-6: M[0:3,0:3] = IKFastSolver.rodrigues(rottodirection*(1/fsin),atan2(fsin,fcos)) elif fcos < 0: # hand is flipped 180, rotate around x axis rottodirection = Matrix(3,1,[S.One,S.Zero,S.Zero]) rottodirection -= sourcedir * sourcedir.dot(rottodirection) M[0:3,0:3] = IKFastSolver.rodrigues(rottodirection.normalized(), atan2(fsin, fcos)) return M @staticmethod def has(eqs,*sym): return any([eq.has(*sym) for eq in eqs]) if len(sym) > 0 else False def trigsimp(self, eq,trigvars): """recurses the sin**2 = 1-cos**2 equation for every trig var """ trigsubs = [] for v in trigvars: if self.IsHinge(v.name): trigsubs.append((sin(v)**2,1-cos(v)**2)) trigsubs.append((Symbol('s%s'%v.name)**2,1-Symbol('c%s'%v.name)**2)) eq=expand(eq) curcount = eq.count_ops() while True: eq=eq.subs(trigsubs).expand() newcount = eq.count_ops() if IKFastSolver.equal(curcount,newcount): break curcount=newcount return eq def SimplifyAtan2(self, eq, incos=False, insin=False, epsilon=None): """simplifies equations like sin(atan2(y,x)) to y/sqrt(x**2+y**2) Sometimes can get equations like sin(-atan2(-r21, -r20)) cos(-atan2(-r21, -r20) + 3.14159265358979) which means the operations internally have to be carried over """ processed = False # if incos or insin set to True, then this flag specifies whether the function was already taking into account or not. if eq.is_Add: if incos: lefteq = eq.args[1] if len(eq.args) > 2: for ieq in range(2,len(eq.args)): lefteq += eq.args[ieq] neweq = self.SimplifyAtan2(eq.args[0], incos=True) * self.SimplifyAtan2(lefteq, incos=True) - self.SimplifyAtan2(eq.args[0], insin=True) * self.SimplifyAtan2(lefteq, insin=True) processed = True elif insin: lefteq = eq.args[1] if len(eq.args) > 2: for ieq in range(2,len(eq.args)): lefteq += eq.args[ieq] neweq = self.SimplifyAtan2(eq.args[0], incos=True) * self.SimplifyAtan2(lefteq, insin=True) + self.SimplifyAtan2(eq.args[0], insin=True) * self.SimplifyAtan2(lefteq, incos=True) processed = True else: neweq = S.Zero for subeq in eq.args: neweq += self.SimplifyAtan2(subeq) # call simplify in order to take in common terms if self.codeComplexity(neweq) > 80: neweq2 = neweq else: #log.info('complexity: %d', self.codeComplexity(neweq)) neweq2 = simplify(neweq) if neweq2 != neweq: neweq = self.SimplifyAtan2(neweq2) else: try: #print 'simplifying',neweq neweq = self.SimplifyTransform(neweq) except PolynomialError: # ok if neweq is too complicated pass elif eq.is_Mul: if incos and len(eq.args) == 2: num = None if eq.args[0].is_integer: num = eq.args[0] eq2 = eq.args[1] elif eq.args[1].is_integer: num = eq.args[1] eq2 = eq.args[0] if num is not None: if num == S.One: neweq = self.SimplifyAtan2(eq2,incos=True) processed = True if num == -S.One: neweq = self.SimplifyAtan2(eq2,incos=True) processed = True elif insin and len(eq.args) == 2: num = None if eq.args[0].is_integer: num = eq.args[0] eq2 = eq.args[1] elif eq.args[1].is_integer: num = eq.args[1] eq2 = eq.args[0] if num is not None: if num == S.One: neweq = self.SimplifyAtan2(eq2,insin=True) processed = True if num == -S.One: neweq = -self.SimplifyAtan2(eq2,insin=True) processed = True if not processed: neweq = self.SimplifyAtan2(eq.args[0]) for subeq in eq.args[1:]: neweq *= self.SimplifyAtan2(subeq) elif eq.is_Function: if incos and eq.func == atan2: yeq = self.SimplifyTransform(self.SimplifyAtan2(eq.args[0])) xeq = self.SimplifyTransform(self.SimplifyAtan2(eq.args[1])) neweq = xeq / sqrt(self.SimplifyTransform(yeq**2+xeq**2)) processed = True elif insin and eq.func == atan2: yeq = self.SimplifyTransform(self.SimplifyAtan2(eq.args[0])) xeq = self.SimplifyTransform(self.SimplifyAtan2(eq.args[1])) neweq = yeq / sqrt(self.SimplifyTransform(yeq**2+xeq**2)) processed = True elif eq.func == cos: neweq = self.SimplifyAtan2(eq.args[0], incos=True) elif eq.func == sin: neweq = self.SimplifyAtan2(eq.args[0], insin=True) else: newargs = [self.SimplifyAtan2(subeq) for subeq in eq.args] neweq = eq.func(*newargs) elif eq.is_Pow: neweq = None if eq.exp.is_number and eq.exp-0.5 == S.Zero: if eq.base.is_Pow and eq.base.exp.is_number and eq.base.exp-2 == S.Zero: # should be abs(eq.base.base), but that could make other simplifications more difficult? neweq = abs(self.SimplifyAtan2(eq.base.base)) if neweq is None: neweq = self.SimplifyAtan2(eq.base)**self.SimplifyAtan2(eq.exp) elif eq.is_number: if epsilon is None: epsilon = 1e-15 if insin: neweq = sin(eq) elif incos: neweq = cos(eq) else: neweq = eq processed = True if abs(neweq.evalf()) <= epsilon: neweq = S.Zero else: neweq=eq if not processed and insin: return sin(neweq) elif not processed and incos: return cos(neweq) return neweq @staticmethod def codeComplexity(expr): complexity = 1 if expr.is_Add: for term in expr.args: complexity += IKFastSolver.codeComplexity(term) elif expr.is_Mul: for term in expr.args: complexity += IKFastSolver.codeComplexity(term) elif expr.is_Pow: complexity += IKFastSolver.codeComplexity(expr.base)+IKFastSolver.codeComplexity(expr.exp) elif expr.is_Function: complexity += 1 for term in expr.args: complexity += IKFastSolver.codeComplexity(term) return complexity def ComputePolyComplexity(self, peq): """peq is a polynomial """ complexity = 0 for monoms,coeff in peq.terms(): coeffcomplexity = self.codeComplexity(coeff) for m in monoms: if m > 1: complexity += 2 elif m > 0: complexity += 1 complexity += coeffcomplexity + 1 return complexity def sortComplexity(self,exprs): exprs.sort(key=lambda e: self.codeComplexity(e)) return exprs def checkForDivideByZero(self,eq): """returns the equations to check for zero """ checkforzeros = [] try: if eq.is_Function: if eq.func == atan2: # atan2 is only a problem when both numerator and denominator are 0! #checkforzeros.append((eq.args[0]**2+eq.args[1]**2).expand()) # have to re-substitute given the global symbols # args[0] and args[1] are very complicated, then there's no reason to do this check substitutedargs = [] for argeq in eq.args: argeq2 = self._SubstituteGlobalSymbols(argeq) if self.codeComplexity(argeq2) < 200: substitutedargs.append(self.SimplifyAtan2(argeq2)) else: substitutedargs.append(argeq2) # has to be greater than 20 since some const coefficients can be simplified if self.codeComplexity(substitutedargs[0]) < 30 and self.codeComplexity(substitutedargs[1]) < 30: if not substitutedargs[0].is_number or substitutedargs[0] == S.Zero: if not substitutedargs[1].is_number or substitutedargs[1] == S.Zero: sumeq = substitutedargs[0]**2+substitutedargs[1]**2 if self.codeComplexity(sumeq) < 400: testeq = self.SimplifyAtan2((substitutedargs[0]**2+substitutedargs[1]**2).expand()) else: testeq = sumeq testeq2 = abs(substitutedargs[0])+abs(substitutedargs[1]) if self.codeComplexity(testeq) < self.codeComplexity(testeq2): testeqmin = testeq else: testeqmin = testeq2 if testeqmin.is_Mul: checkforzeros += testeqmin.args else: checkforzeros.append(testeqmin) if checkforzeros[-1].evalf() == S.Zero: raise self.CannotSolveError('equation evaluates to 0, so can never be ok') log.info('adding atan2(%r, %r) = %r all zeros check', substitutedargs[0], substitutedargs[1], checkforzeros[-1]) for arg in eq.args: checkforzeros += self.checkForDivideByZero(arg) elif eq.is_Add: for arg in eq.args: checkforzeros += self.checkForDivideByZero(arg) elif eq.is_Mul: for arg in eq.args: checkforzeros += self.checkForDivideByZero(arg) elif eq.is_Pow: for arg in eq.args: checkforzeros += self.checkForDivideByZero(arg) if eq.exp.is_number and eq.exp < 0: checkforzeros.append(eq.base) except AssertionError as e: log.warn('%s',e) if len(checkforzeros) > 0: newcheckforzeros = [] for eqtemp in checkforzeros: # check for abs(x**y), in that case choose x if eqtemp.is_Function and eqtemp.func == Abs: eqtemp = eqtemp.args[0] while eqtemp.is_Pow: eqtemp = eqtemp.base #self.codeComplexity(eqtemp) if self.codeComplexity(eqtemp) < 500: checkeq = self.removecommonexprs(eqtemp,onlygcd=False,onlynumbers=True) if self.CheckExpressionUnique(newcheckforzeros,checkeq): newcheckforzeros.append(checkeq) else: # not even worth checking since the equation is so big... newcheckforzeros.append(eqtemp) return newcheckforzeros return checkforzeros def ComputeSolutionComplexity(self,sol,solvedvars,unsolvedvars): """ for all solutions, check if there is a divide by zero fills checkforzeros for the solution """ sol.checkforzeros = sol.getPresetCheckForZeros() sol.score = 20000*sol.numsolutions() try: # multiby by 400 in order to prioritize equations with less solutions if hasattr(sol,'jointeval') and sol.jointeval is not None: for s in sol.jointeval: sol.score += self.codeComplexity(s) sol.checkforzeros += self.checkForDivideByZero(s.subs(sol.dictequations)) subexprs = sol.jointeval elif hasattr(sol,'jointevalsin') and sol.jointevalsin is not None: for s in sol.jointevalsin: sol.score += self.codeComplexity(s) sol.checkforzeros += self.checkForDivideByZero(s.subs(sol.dictequations)) subexprs = sol.jointevalsin elif hasattr(sol,'jointevalcos') and sol.jointevalcos is not None: for s in sol.jointevalcos: sol.score += self.codeComplexity(s) sol.checkforzeros += self.checkForDivideByZero(s.subs(sol.dictequations)) subexprs = sol.jointevalcos else: return sol.score # have to also check solution dictionary for s,v in sol.dictequations: sol.score += self.codeComplexity(v) sol.checkforzeros += self.checkForDivideByZero(v.subs(sol.dictequations)) def checkpow(expr,sexprs): score = 0 if expr.is_Pow: sexprs.append(expr.base) if expr.base.is_finite is not None and not expr.base.is_finite: return oo # infinity if expr.exp.is_number and expr.exp < 0: # check if exprbase contains any variables that have already been solved containsjointvar = expr.base.has(*solvedvars) cancheckexpr = not expr.base.has(*unsolvedvars) score += 10000 if not cancheckexpr: score += 100000 elif not self.isValidSolution(expr): return oo # infinity return score sexprs = subexprs[:] while len(sexprs) > 0: sexpr = sexprs.pop(0) if sexpr.is_Add: for arg in sexpr.args: if arg.is_Mul: for arg2 in arg.args: sol.score += checkpow(arg2,sexprs) else: sol.score += checkpow(arg,sexprs) elif sexpr.is_Mul: for arg in sexpr.args: sol.score += checkpow(arg,sexprs) elif sexpr.is_Function: sexprs += sexpr.args elif not self.isValidSolution(sexpr): log.warn('not valid: %s',sexpr) sol.score = oo # infinity else: sol.score += checkpow(sexpr,sexprs) except AssertionError as e: log.warn('%s',e) sol.score=1e10 newcheckforzeros = [] for eqtemp in sol.checkforzeros: if self.codeComplexity(eqtemp) < 1000: # if there's a sign, there's an infinite recursion? if len(eqtemp.find(sign)) > 0: newcheckforzeros.append(eqtemp) else: checkeq = self.removecommonexprs(eqtemp,onlygcd=False,onlynumbers=True) if self.CheckExpressionUnique(newcheckforzeros,checkeq): newcheckforzeros.append(checkeq) else: newcheckforzeros.append(eqtemp) sol.checkforzeros = newcheckforzeros return sol.score def checkSolvability(self,AllEquations,checkvars,othervars): pass def checkSolvabilityReal(self,AllEquations,checkvars,othervars): """returns true if there are enough equations to solve for checkvars """ subs = [] checksymbols = [] allsymbols = [] for var in checkvars: subs += self.Variable(var).subs checksymbols += self.Variable(var).vars allsymbols = checksymbols[:] for var in othervars: subs += self.Variable(var).subs allsymbols += self.Variable(var).vars found = False for testconsistentvalue in self.testconsistentvalues: psubvalues = [(s,v) for s,v in testconsistentvalue if not s.has(*checksymbols)] eqs = [eq.subs(self.globalsymbols).subs(subs).subs(psubvalues) for eq in AllEquations] usedsymbols = [s for s in checksymbols if self.has(eqs,s)] eqs = [Poly(eq,*usedsymbols) for eq in eqs if eq != S.Zero] # check if any equations have monos of degree more than 1, if yes, then quit with success since 0.6.7 sympy solver will freeze numhigherpowers = 0 for eq in eqs: for monom in eq.monoms(): if any([m > 1 for m in monom]): numhigherpowers += 1 if numhigherpowers > 0: log.info('checkSolvability has %d higher powers, returning solvable if > 6'%numhigherpowers) if numhigherpowers > 6: found = True break for var in checkvars: varsym = self.Variable(var) if self.IsHinge(var.name): if varsym.cvar in usedsymbols and varsym.svar in usedsymbols: eqs.append(Poly(varsym.cvar**2+varsym.svar**2-1,*usedsymbols)) # have to make sure there are representative symbols of all the checkvars, otherwise degenerate solution setusedsymbols = set(usedsymbols) if any([len(setusedsymbols.intersection(self.Variable(var).vars)) == 0 for var in checkvars]): continue try: sol=solve_poly_system(eqs) if sol is not None and len(sol) > 0 and len(sol[0]) == len(usedsymbols): found = True break except: pass if not found: raise self.IKFeasibilityError(AllEquations,checkvars) def writeIkSolver(self,chaintree,lang=None): """write the ast into a specific langauge, prioritize c++ """ self._CheckPreemptFn(progress=0.5) if lang is None: if 'cpp' in CodeGenerators: lang = 'cpp' else: lang = next(iter(CodeGenerators.keys())) log.info('generating %s code...'%lang) if self._checkpreemptfn is not None: import weakref weakself = weakref.proxy(self) def _CheckPreemtCodeGen(msg, progress): # put the progress in the latter half weakself._checkpreemptfn(u'CodeGen %s'%msg, 0.5+0.5*progress) else: _CheckPreemtCodeGen = None return CodeGenerators[lang](kinematicshash=self.kinematicshash,version=__version__,iktypestr=self._iktype, checkpreemptfn=_CheckPreemtCodeGen).generate(chaintree) def generateIkSolver(self, baselink, eelink, freeindices=None, solvefn=None, ikfastoptions=0): """ :param ikfastoptions: options that control how ikfast. """ self._CheckPreemptFn(progress=0) if solvefn is None: solvefn = IKFastSolver.solveFullIK_6D chainlinks = self.kinbody.GetChain(baselink,eelink,returnjoints=False) chainjoints = self.kinbody.GetChain(baselink,eelink,returnjoints=True) LinksRaw, jointvars = self.forwardKinematicsChain(chainlinks,chainjoints) for T in LinksRaw: log.info('[' + ','.join(['[%s, %s, %s, %s]'%(T[i,0],T[i,1],T[i,2],T[i,3]) for i in range(3)]) + ']') self.degeneratecases = None if freeindices is None: # need to iterate through all combinations of free joints assert(0) isolvejointvars = [] solvejointvars = [] self._ikfastoptions = ikfastoptions self.ifreejointvars = [] self.freevarsubs = [] self.freevarsubsinv = [] self.freevars = [] self.freejointvars = [] self.invsubs = [] for i,v in enumerate(jointvars): var = self.Variable(v) axis = self.axismap[v.name] dofindex = axis.joint.GetDOFIndex()+axis.iaxis if dofindex in freeindices: # convert all free variables to constants self.ifreejointvars.append(i) self.freevarsubs += [(cos(var.var), var.cvar), (sin(var.var), var.svar)] self.freevarsubsinv += [(var.cvar,cos(var.var)), (var.svar,sin(var.var))] self.freevars += [var.cvar,var.svar] self.freejointvars.append(var.var) else: solvejointvars.append(v) isolvejointvars.append(i) self.invsubs += [(var.cvar,cos(v)),(var.svar,sin(v))] self._solvejointvars = solvejointvars self._jointvars = jointvars # set up the destination symbols self.Tee = eye(4) for i in range(0,3): for j in range(0,3): self.Tee[i,j] = Symbol("r%d%d"%(i,j)) self.Tee[0,3] = Symbol("px") self.Tee[1,3] = Symbol("py") self.Tee[2,3] = Symbol("pz") r00,r01,r02,px,r10,r11,r12,py,r20,r21,r22,pz = self.Tee[0:12] self.pp = Symbol('pp') self.ppsubs = [(self.pp,px**2+py**2+pz**2)] self.npxyz = [Symbol('npx'),Symbol('npy'),Symbol('npz')] self.npxyzsubs = [(self.npxyz[i],px*self.Tee[0,i]+py*self.Tee[1,i]+pz*self.Tee[2,i]) for i in range(3)] # cross products between columns of self.Tee self.rxp = [] self.rxpsubs = [] for i in range(3): self.rxp.append([Symbol('rxp%d_%d'%(i,j)) for j in range(3)]) c = self.Tee[0:3,i].cross(self.Tee[0:3,3]) self.rxpsubs += [(self.rxp[-1][j],c[j]) for j in range(3)] # have to include new_rXX self.pvars = self.Tee[0:12]+self.npxyz+[self.pp]+self.rxp[0]+self.rxp[1]+self.rxp[2] + [Symbol('new_r00'), Symbol('new_r01'), Symbol('new_r02'), Symbol('new_r10'), Symbol('new_r11'), Symbol('new_r12'), Symbol('new_r20'), Symbol('new_r21'), Symbol('new_r22')] self._rotsymbols = list(self.Tee[0:3,0:3]) # add positions ip = 9 inp = 12 ipp = 15 irxp = 16 self._rotpossymbols = self._rotsymbols + list(self.Tee[0:3,3])+self.npxyz+[self.pp]+self.rxp[0]+self.rxp[1]+self.rxp[2] # groups of rotation variables that are unit vectors self._rotnormgroups = [] for i in range(3): self._rotnormgroups.append([self.Tee[i,0],self.Tee[i,1],self.Tee[i,2],S.One]) self._rotnormgroups.append([self.Tee[0,i],self.Tee[1,i],self.Tee[2,i],S.One]) self._rotposnormgroups = list(self._rotnormgroups) self._rotposnormgroups.append([self.Tee[0,3],self.Tee[1,3],self.Tee[2,3],self.pp]) # dot product of rotation rows and columns is always 0 self._rotdotgroups = [] for i,j in combinations(range(3),2): self._rotdotgroups.append([[i,j],[i+3,j+3],[i+6,j+6],S.Zero]) self._rotdotgroups.append([[3*i,3*j],[3*i+1,3*j+1],[3*i+2,3*j+2],S.Zero]) self._rotposdotgroups = list(self._rotdotgroups) for i in range(3): self._rotposdotgroups.append([[i,ip],[i+3,ip+1],[i+6,ip+2],self.npxyz[i]]) self._rotposdotgroups.append([[3*i+0,inp],[3*i+1,inp+1],[3*i+2,inp+2],self.Tee[i,3]]) self._rotcrossgroups = [] # cross products of rotation rows and columns always yield the left over vector for i,j,k in [(0,1,2),(0,2,1),(1,2,0)]: # column self._rotcrossgroups.append([[i+3,j+6],[i+6,j+3],k]) self._rotcrossgroups.append([[i+6,j],[i,j+6],k+3]) self._rotcrossgroups.append([[i,j+3],[i+3,j],k+6]) # row self._rotcrossgroups.append([[3*i+1,3*j+2],[3*i+2,3*j+1],3*k]) self._rotcrossgroups.append([[3*i+2,3*j],[3*i,3*j+2],3*k+1]) self._rotcrossgroups.append([[3*i,3*j+1],[3*i+1,3*j],3*k+2]) # swap if sign is negative: if j!=1+i if j!=1+i: for crossgroup in self._rotcrossgroups[-6:]: crossgroup[0],crossgroup[1] = crossgroup[1],crossgroup[0] # add positions self._rotposcrossgroups = list(self._rotcrossgroups) for i in range(3): # column i cross position self._rotposcrossgroups.append([[i+3,ip+2],[i+6,ip+1],irxp+3*i+0]) self._rotposcrossgroups.append([[i+6,ip+0],[i,ip+2],irxp+3*i+1]) self._rotposcrossgroups.append([[i,ip+1],[i+3,ip+0],irxp+3*i+2]) self.Teeinv = self.affineInverse(self.Tee) LinksLeft = [] if self.useleftmultiply: while not self.has(LinksRaw[0],*solvejointvars): LinksLeft.append(LinksRaw.pop(0)) LinksLeftInv = [self.affineInverse(T) for T in LinksLeft] self.testconsistentvalues = None self.gsymbolgen = cse_main.numbered_symbols('gconst') self.globalsymbols = [] self._scopecounter = 0 # before passing to the solver, set big numbers to constant variables, this will greatly reduce computation times # numbersubs = [] # LinksRaw2 = [] # for Torig in LinksRaw: # T = Matrix(Torig) # #print axisAngleFromRotationMatrix(numpy.array(numpy.array(T[0:3,0:3]),numpy.float64)) # for i in range(12): # ti = T[i] # if ti.is_number and len(str(ti)) > 30: # matchnumber = self.MatchSimilarFraction(ti,numbersubs) # if matchnumber is None: # sym = next(self.gsymbolgen) # log.info('adding global symbol %s=%s'%(sym,ti)) # numbersubs.append((sym,ti)) # T[i] = sym # else: # T[i] = matchnumber # LinksRaw2.append(T) # if len(numbersubs) > 10: # log.info('substituting %d global symbols',len(numbersubs)) # LinksRaw = LinksRaw2 # self.globalsymbols += numbersubs self.Teeleftmult = self.multiplyMatrix(LinksLeft) # the raw ee passed to the ik solver function self._CheckPreemptFn(progress=0.01) chaintree = solvefn(self, LinksRaw, jointvars, isolvejointvars) if self.useleftmultiply: chaintree.leftmultiply(Tleft=self.multiplyMatrix(LinksLeft), Tleftinv=self.multiplyMatrix(LinksLeftInv[::-1])) chaintree.dictequations += self.globalsymbols return chaintree def MatchSimilarFraction(self,num,numbersubs,matchlimit = 40): """returns None if no appropriate match found """ for c,v in numbersubs: if self.equal(v,num): return c # nothing matched, so check gcd largestgcd = S.One retnum = None for c,v in numbersubs: curgcd = gcd(v,num) if len(str(curgcd)) > len(str(largestgcd)): newfraction = (num/v) if len(str(newfraction)) <= matchlimit: largestgcd = curgcd retnum = c * newfraction return retnum def ComputeConsistentValues(self,jointvars,T,numsolutions=1,subs=None): """computes a set of substitutions that satisfy the IK equations """ possibleangles = [S.Zero, pi.evalf()/2, asin(3.0/5).evalf(), asin(4.0/5).evalf(), asin(5.0/13).evalf(), asin(12.0/13).evalf()] possibleanglescos = [S.One, S.Zero, Rational(4,5), Rational(3,5), Rational(12,13), Rational(5,13)] possibleanglessin = [S.Zero, S.One, Rational(3,5), Rational(4,5), Rational(5,13), Rational(12,13)] testconsistentvalues = [] varsubs = [] for jointvar in jointvars: varsubs += self.Variable(jointvar).subs for isol in range(numsolutions): inds = [0]*len(jointvars) if isol < numsolutions-1: for j in range(len(jointvars)): inds[j] = (isol+j)%len(possibleangles) valsubs = [] for i,ind in enumerate(inds): v,s,c = possibleangles[ind],possibleanglessin[ind],possibleanglescos[ind] var = self.Variable(jointvars[i]) valsubs += [(var.var,v),(var.cvar,c),(var.svar,s),(var.tvar,s/c),(var.htvar,s/(1+c))] psubs = [] for i in range(12): psubs.append((self.pvars[i],T[i].subs(varsubs).subs(self.globalsymbols+valsubs))) for s,v in self.ppsubs+self.npxyzsubs+self.rxpsubs: psubs.append((s,v.subs(psubs))) allsubs = valsubs+psubs if subs is not None: allsubs += [(dvar,var.subs(varsubs).subs(valsubs)) for dvar,var in subs] testconsistentvalues.append(allsubs) return testconsistentvalues def solveFullIK_Direction3D(self,LinksRaw, jointvars, isolvejointvars, rawmanipdir=Matrix(3,1,[S.Zero,S.Zero,S.One])): """manipdir needs to be filled with a 3elemtn vector of the initial direction to control""" self._iktype = 'direction3d' manipdir = Matrix(3,1,[Float(x,30) for x in rawmanipdir]) manipdir /= sqrt(manipdir[0]*manipdir[0]+manipdir[1]*manipdir[1]+manipdir[2]*manipdir[2]) for i in range(3): manipdir[i] = self.convertRealToRational(manipdir[i]) Links = LinksRaw[:] LinksInv = [self.affineInverse(link) for link in Links] T = self.multiplyMatrix(Links) self.Tfinal = zeros((4,4)) self.Tfinal[0,0:3] = (T[0:3,0:3]*manipdir).transpose() self.testconsistentvalues = self.ComputeConsistentValues(jointvars,self.Tfinal,numsolutions=4) endbranchtree = [AST.SolverStoreSolution(jointvars,isHinge=[self.IsHinge(var.name) for var in jointvars])] solvejointvars = [jointvars[i] for i in isolvejointvars] if len(solvejointvars) != 2: raise self.CannotSolveError('need 2 joints') log.info('ikfast direction3d: %s',solvejointvars) Daccum = self.Tee[0,0:3].transpose() numvarsdone = 2 Ds = [] Dsee = [] for i in range(len(Links)-1): T = self.multiplyMatrix(Links[i:]) D = T[0:3,0:3]*manipdir hasvars = [self.has(D,v) for v in solvejointvars] if __builtin__.sum(hasvars) == numvarsdone: Ds.append(D) Dsee.append(Daccum) numvarsdone -= 1 Tinv = self.affineInverse(Links[i]) Daccum = Tinv[0:3,0:3]*Daccum AllEquations = self.buildEquationsFromTwoSides(Ds,Dsee,jointvars,uselength=False) self.checkSolvability(AllEquations,solvejointvars,self.freejointvars) tree = self.SolveAllEquations(AllEquations,curvars=solvejointvars,othersolvedvars = self.freejointvars[:],solsubs = self.freevarsubs[:],endbranchtree=endbranchtree) tree = self.verifyAllEquations(AllEquations,solvejointvars,self.freevarsubs,tree) return AST.SolverIKChainDirection3D([(jointvars[ijoint],ijoint) for ijoint in isolvejointvars], [(v,i) for v,i in izip(self.freejointvars,self.ifreejointvars)], Dee=self.Tee[0,0:3].transpose().subs(self.freevarsubs), jointtree=tree,Dfk=self.Tfinal[0,0:3].transpose()) def solveFullIK_Lookat3D(self,LinksRaw, jointvars, isolvejointvars,rawmanipdir=Matrix(3,1,[S.Zero,S.Zero,S.One]),rawmanippos=Matrix(3,1,[S.Zero,S.Zero,S.Zero])): """manipdir,manippos needs to be filled with a direction and position of the ray to control the lookat """ self._iktype = 'lookat3d' manipdir = Matrix(3,1,[Float(x,30) for x in rawmanipdir]) manippos = Matrix(3,1,[self.convertRealToRational(x) for x in rawmanippos]) manipdir /= sqrt(manipdir[0]*manipdir[0]+manipdir[1]*manipdir[1]+manipdir[2]*manipdir[2]) for i in range(3): manipdir[i] = self.convertRealToRational(manipdir[i]) manippos = manippos-manipdir*manipdir.dot(manippos) Links = LinksRaw[:] LinksInv = [self.affineInverse(link) for link in Links] T = self.multiplyMatrix(Links) self.Tfinal = zeros((4,4)) self.Tfinal[0,0:3] = (T[0:3,0:3]*manipdir).transpose() self.Tfinal[0:3,3] = T[0:3,0:3]*manippos+T[0:3,3] self.testconsistentvalues = self.ComputeConsistentValues(jointvars,self.Tfinal,numsolutions=4) solvejointvars = [jointvars[i] for i in isolvejointvars] if len(solvejointvars) != 2: raise self.CannotSolveError('need 2 joints') log.info('ikfast lookat3d: %s',solvejointvars) Paccum = self.Tee[0:3,3] numvarsdone = 2 Positions = [] Positionsee = [] for i in range(len(Links)-1): T = self.multiplyMatrix(Links[i:]) P = T[0:3,0:3]*manippos+T[0:3,3] D = T[0:3,0:3]*manipdir hasvars = [self.has(P,v) or self.has(D,v) for v in solvejointvars] if __builtin__.sum(hasvars) == numvarsdone: Positions.append(P.cross(D)) Positionsee.append(Paccum.cross(D)) numvarsdone -= 1 Tinv = self.affineInverse(Links[i]) Paccum = Tinv[0:3,0:3]*Paccum+Tinv[0:3,3] frontcond = (Links[-1][0:3,0:3]*manipdir).dot(Paccum-(Links[-1][0:3,0:3]*manippos+Links[-1][0:3,3])) for v in jointvars: frontcond = frontcond.subs(self.Variable(v).subs) endbranchtree = [AST.SolverStoreSolution (jointvars,checkgreaterzero=[frontcond],isHinge=[self.IsHinge(var.name) for var in jointvars])] AllEquations = self.buildEquationsFromTwoSides(Positions,Positionsee,jointvars,uselength=True) self.checkSolvability(AllEquations,solvejointvars,self.freejointvars) tree = self.SolveAllEquations(AllEquations,curvars=solvejointvars,othersolvedvars = self.freejointvars[:],solsubs = self.freevarsubs[:],endbranchtree=endbranchtree) tree = self.verifyAllEquations(AllEquations,solvejointvars,self.freevarsubs,tree) chaintree = AST.SolverIKChainLookat3D([(jointvars[ijoint],ijoint) for ijoint in isolvejointvars], [(v,i) for v,i in izip(self.freejointvars,self.ifreejointvars)], Pee=self.Tee[0:3,3].subs(self.freevarsubs), jointtree=tree,Dfk=self.Tfinal[0,0:3].transpose(),Pfk=self.Tfinal[0:3,3]) chaintree.dictequations += self.ppsubs return chaintree def solveFullIK_Rotation3D(self,LinksRaw, jointvars, isolvejointvars, Rbaseraw=eye(3)): self._iktype = 'rotation3d' Rbase = eye(4) for i in range(3): for j in range(3): Rbase[i,j] = self.convertRealToRational(Rbaseraw[i,j]) Tfirstright = LinksRaw[-1]*Rbase Links = LinksRaw[:-1] LinksInv = [self.affineInverse(link) for link in Links] self.Tfinal = self.multiplyMatrix(Links) self.testconsistentvalues = self.ComputeConsistentValues(jointvars,self.Tfinal,numsolutions=4) endbranchtree = [AST.SolverStoreSolution (jointvars,isHinge=[self.IsHinge(var.name) for var in jointvars])] solvejointvars = [jointvars[i] for i in isolvejointvars] if len(solvejointvars) != 3: raise self.CannotSolveError('need 3 joints') log.info('ikfast rotation3d: %s',solvejointvars) AllEquations = self.buildEquationsFromRotation(Links,self.Tee[0:3,0:3],solvejointvars,self.freejointvars) self.checkSolvability(AllEquations,solvejointvars,self.freejointvars) tree = self.SolveAllEquations(AllEquations,curvars=solvejointvars[:],othersolvedvars=self.freejointvars,solsubs = self.freevarsubs[:],endbranchtree=endbranchtree) tree = self.verifyAllEquations(AllEquations,solvejointvars,self.freevarsubs,tree) return AST.SolverIKChainRotation3D([(jointvars[ijoint],ijoint) for ijoint in isolvejointvars], [(v,i) for v,i in izip(self.freejointvars,self.ifreejointvars)], (self.Tee[0:3,0:3] * self.affineInverse(Tfirstright)[0:3,0:3]).subs(self.freevarsubs), tree, Rfk = self.Tfinal[0:3,0:3] * Tfirstright[0:3,0:3]) def solveFullIK_TranslationLocalGlobal6D(self,LinksRaw, jointvars, isolvejointvars, Tmanipraw=eye(4)): self._iktype = 'translation3d' Tgripper = eye(4) for i in range(4): for j in range(4): Tgripper[i,j] = self.convertRealToRational(Tmanipraw[i,j]) localpos = Matrix(3,1,[self.Tee[0,0],self.Tee[1,1],self.Tee[2,2]]) chain = self._solveFullIK_Translation3D(LinksRaw,jointvars,isolvejointvars,Tgripper[0:3,3]+Tgripper[0:3,0:3]*localpos,False) chain.uselocaltrans = True return chain def solveFullIK_Translation3D(self,LinksRaw, jointvars, isolvejointvars, rawmanippos=Matrix(3,1,[S.Zero,S.Zero,S.Zero])): self._iktype = 'translation3d' manippos = Matrix(3,1,[self.convertRealToRational(x) for x in rawmanippos]) return self._solveFullIK_Translation3D(LinksRaw,jointvars,isolvejointvars,manippos) def _solveFullIK_Translation3D(self,LinksRaw, jointvars, isolvejointvars, manippos,check=True): Links = LinksRaw[:] LinksInv = [self.affineInverse(link) for link in Links] self.Tfinal = self.multiplyMatrix(Links) self.Tfinal[0:3,3] = self.Tfinal[0:3,0:3]*manippos+self.Tfinal[0:3,3] self.testconsistentvalues = self.ComputeConsistentValues(jointvars,self.Tfinal,numsolutions=4) endbranchtree = [AST.SolverStoreSolution (jointvars,isHinge=[self.IsHinge(var.name) for var in jointvars])] solvejointvars = [jointvars[i] for i in isolvejointvars] if len(solvejointvars) != 3: raise self.CannotSolveError('need 3 joints') log.info('ikfast translation3d: %s',solvejointvars) Tmanipposinv = eye(4) Tmanipposinv[0:3,3] = -manippos T1links = [Tmanipposinv]+LinksInv[::-1]+[self.Tee] T1linksinv = [self.affineInverse(Tmanipposinv)]+Links[::-1]+[self.Teeinv] AllEquations = self.buildEquationsFromPositions(T1links,T1linksinv,solvejointvars,self.freejointvars,uselength=True) if check: self.checkSolvability(AllEquations,solvejointvars,self.freejointvars) transtree = self.SolveAllEquations(AllEquations,curvars=solvejointvars[:],othersolvedvars=self.freejointvars,solsubs = self.freevarsubs[:],endbranchtree=endbranchtree) transtree = self.verifyAllEquations(AllEquations,solvejointvars,self.freevarsubs,transtree) chaintree = AST.SolverIKChainTranslation3D([(jointvars[ijoint],ijoint) for ijoint in isolvejointvars], [(v,i) for v,i in izip(self.freejointvars,self.ifreejointvars)], Pee=self.Tee[0:3,3], jointtree=transtree, Pfk = self.Tfinal[0:3,3]) chaintree.dictequations += self.ppsubs return chaintree def solveFullIK_TranslationXY2D(self,LinksRaw, jointvars, isolvejointvars, rawmanippos=Matrix(2,1,[S.Zero,S.Zero])): self._iktype = 'translationxy2d' self.ppsubs = [] # disable since pz is not valid self.pp = None manippos = Matrix(2,1,[self.convertRealToRational(x) for x in rawmanippos]) Links = LinksRaw[:] LinksInv = [self.affineInverse(link) for link in Links] self.Tfinal = self.multiplyMatrix(Links) self.Tfinal[0:2,3] = self.Tfinal[0:2,0:2]*manippos+self.Tfinal[0:2,3] self.testconsistentvalues = self.ComputeConsistentValues(jointvars,self.Tfinal,numsolutions=4) endbranchtree = [AST.SolverStoreSolution (jointvars,isHinge=[self.IsHinge(var.name) for var in jointvars])] solvejointvars = [jointvars[i] for i in isolvejointvars] if len(solvejointvars) != 2: raise self.CannotSolveError('need 2 joints') log.info('ikfast translationxy2d: %s',solvejointvars) Tmanipposinv = eye(4) Tmanipposinv[2,2] = S.Zero Tmanipposinv[0:2,3] = -manippos Tmanippos = eye(4) Tmanippos[2,2] = S.Zero Tmanippos[0:2,3] = manippos T1links = [Tmanipposinv]+LinksInv[::-1]+[self.Tee] T1linksinv = [Tmanippos]+Links[::-1]+[self.Teeinv] Taccum = eye(4) numvarsdone = 1 Positions = [] Positionsee = [] for i in range(len(T1links)-1): Taccum = T1linksinv[i]*Taccum hasvars = [self.has(Taccum,v) for v in solvejointvars] if __builtin__.sum(hasvars) == numvarsdone: Positions.append(Taccum[0:2,3]) Positionsee.append(self.multiplyMatrix(T1links[(i+1):])[0:2,3]) numvarsdone += 1 if numvarsdone > 2: # more than 2 variables is almost always useless break if len(Positions) == 0: Positions.append(zeros((2,1))) Positionsee.append(self.multiplyMatrix(T1links)[0:2,3]) AllEquations = self.buildEquationsFromTwoSides(Positions,Positionsee,solvejointvars+self.freejointvars,uselength=True) self.checkSolvability(AllEquations,solvejointvars,self.freejointvars) transtree = self.SolveAllEquations(AllEquations,curvars=solvejointvars[:],othersolvedvars=self.freejointvars,solsubs = self.freevarsubs[:],endbranchtree=endbranchtree) transtree = self.verifyAllEquations(AllEquations,solvejointvars,self.freevarsubs,transtree) chaintree = AST.SolverIKChainTranslationXY2D([(jointvars[ijoint],ijoint) for ijoint in isolvejointvars], [(v,i) for v,i in izip(self.freejointvars,self.ifreejointvars)], Pee=self.Tee[0:2,3], jointtree=transtree, Pfk = self.Tfinal[0:2,3]) chaintree.dictequations += self.ppsubs return chaintree def solveFullIK_TranslationXYOrientation3D(self,LinksRaw, jointvars, isolvejointvars, rawmanippos=Matrix(2,1,[S.Zero,S.Zero]), rawangle=S.Zero): self._iktype = 'translationxyorientation3d' raise self.CannotSolveError('TranslationXYOrientation3D not implemented yet') def solveFullIK_Ray4D(self,LinksRaw, jointvars, isolvejointvars, rawmanipdir=Matrix(3,1,[S.Zero,S.Zero,S.One]),rawmanippos=Matrix(3,1,[S.Zero,S.Zero,S.Zero])): """manipdir,manippos needs to be filled with a direction and position of the ray to control""" self._iktype = 'ray4d' manipdir = Matrix(3,1,[Float(x,30) for x in rawmanipdir]) manippos = Matrix(3,1,[self.convertRealToRational(x) for x in rawmanippos]) manipdir /= sqrt(manipdir[0]*manipdir[0]+manipdir[1]*manipdir[1]+manipdir[2]*manipdir[2]) for i in range(3): manipdir[i] = self.convertRealToRational(manipdir[i]) manippos = manippos-manipdir*manipdir.dot(manippos) Links = LinksRaw[:] LinksInv = [self.affineInverse(link) for link in Links] T = self.multiplyMatrix(Links) self.Tfinal = zeros((4,4)) self.Tfinal[0,0:3] = (T[0:3,0:3]*manipdir).transpose() self.Tfinal[0:3,3] = T[0:3,0:3]*manippos+T[0:3,3] self.testconsistentvalues = self.ComputeConsistentValues(jointvars,self.Tfinal,numsolutions=4) endbranchtree = [AST.SolverStoreSolution (jointvars,isHinge=[self.IsHinge(var.name) for var in jointvars])] solvejointvars = [jointvars[i] for i in isolvejointvars] if len(solvejointvars) != 4: raise self.CannotSolveError('need 4 joints') log.info('ikfast ray4d: %s',solvejointvars) Pee = self.Tee[0:3,3] Dee = self.Tee[0,0:3].transpose() numvarsdone = 2 Positions = [] Positionsee = [] for i in range(len(Links)-1): T = self.multiplyMatrix(Links[i:]) P = T[0:3,0:3]*manippos+T[0:3,3] D = T[0:3,0:3]*manipdir hasvars = [self.has(P,v) or self.has(D,v) for v in solvejointvars] if __builtin__.sum(hasvars) == numvarsdone: Positions.append(P.cross(D)) Positionsee.append(Pee.cross(Dee)) Positions.append(D) Positionsee.append(Dee) break Tinv = self.affineInverse(Links[i]) Pee = Tinv[0:3,0:3]*Pee+Tinv[0:3,3] Dee = Tinv[0:3,0:3]*Dee AllEquations = self.buildEquationsFromTwoSides(Positions,Positionsee,jointvars,uselength=True) self.checkSolvability(AllEquations,solvejointvars,self.freejointvars) #try: tree = self.SolveAllEquations(AllEquations,curvars=solvejointvars[:],othersolvedvars = self.freejointvars[:],solsubs = self.freevarsubs[:],endbranchtree=endbranchtree) #except self.CannotSolveError: # build the raghavan/roth equations and solve with higher power methods # pass tree = self.verifyAllEquations(AllEquations,solvejointvars,self.freevarsubs,tree) chaintree = AST.SolverIKChainRay([(jointvars[ijoint],ijoint) for ijoint in isolvejointvars], [(v,i) for v,i in izip(self.freejointvars,self.ifreejointvars)], Pee=self.Tee[0:3,3].subs(self.freevarsubs), Dee=self.Tee[0,0:3].transpose().subs(self.freevarsubs),jointtree=tree,Dfk=self.Tfinal[0,0:3].transpose(),Pfk=self.Tfinal[0:3,3]) chaintree.dictequations += self.ppsubs return chaintree def solveFullIK_TranslationDirection5D(self, LinksRaw, jointvars, isolvejointvars, rawmanipdir=Matrix(3,1,[S.Zero,S.Zero,S.One]),rawmanippos=Matrix(3,1,[S.Zero,S.Zero,S.Zero])): """Solves 3D translation + 3D direction """ self._iktype = 'translationdirection5d' manippos = Matrix(3,1,[self.convertRealToRational(x) for x in rawmanippos]) manipdir = Matrix(3,1,[Float(x,30) for x in rawmanipdir]) manipdir /= sqrt(manipdir[0]*manipdir[0]+manipdir[1]*manipdir[1]+manipdir[2]*manipdir[2]) # try to simplify manipdir based on possible angles for i in range(3): value = None # TODO should restore 12 once we can capture stuff like pi/12+sqrt(12531342/5141414) for num in [3,4,5,6,7,8]:#,12]: if abs((manipdir[i]-cos(pi/num))).evalf() <= (10**-self.precision): value = cos(pi/num) break elif abs((manipdir[i]+cos(pi/num))).evalf() <= (10**-self.precision): value = -cos(pi/num) break elif abs((manipdir[i]-sin(pi/num))).evalf() <= (10**-self.precision): value = sin(pi/num) break elif abs((manipdir[i]+sin(pi/num))).evalf() <= (10**-self.precision): value = -sin(pi/num) break if value is not None: manipdir[i] = value else: manipdir[i] = self.convertRealToRational(manipdir[i],5) manipdirlen2 = trigsimp(manipdir[0]*manipdir[0]+manipdir[1]*manipdir[1]+manipdir[2]*manipdir[2]) # unfortunately have to do it again... manipdir /= sqrt(manipdirlen2) offsetdist = manipdir.dot(manippos) manippos = manippos-manipdir*offsetdist Links = LinksRaw[:] endbranchtree = [AST.SolverStoreSolution (jointvars,isHinge=[self.IsHinge(var.name) for var in jointvars])] numzeros = int(manipdir[0]==S.Zero) + int(manipdir[1]==S.Zero) + int(manipdir[2]==S.Zero) # if numzeros < 2: # try: # log.info('try to rotate the last joint so that numzeros increases') # assert(not self.has(Links[-1],*solvejointvars)) # localdir = Links[-1][0:3,0:3]*manipdir # localpos = Links[-1][0:3,0:3]*manippos+Links[-1][0:3,3] # AllEquations = Links[-2][0:3,0:3]*localdir # tree=self.SolveAllEquations(AllEquations,curvars=solvejointvars[-1:],othersolvedvars = [],solsubs = [],endbranchtree=[]) # offset = tree[0].jointeval[0] # endbranchtree[0].offsetvalues = [S.Zero]*len(solvejointvars) # endbranchtree[0].offsetvalues[-1] = offset # Toffset = Links[-2].subs(solvejointvars[-1],offset).evalf() # localdir2 = Toffset[0:3,0:3]*localdir # localpos2 = Toffset[0:3,0:3]*localpos+Toffset[0:3,3] # Links[-1]=eye(4) # for i in range(3): # manipdir[i] = self.convertRealToRational(localdir2[i]) # manipdir /= sqrt(manipdir[0]*manipdir[0]+manipdir[1]*manipdir[1]+manipdir[2]*manipdir[2]) # unfortunately have to do it again... # manippos = Matrix(3,1,[self.convertRealToRational(x) for x in localpos2]) # except Exception, e: # print 'failed to rotate joint correctly',e LinksInv = [self.affineInverse(link) for link in Links] T = self.multiplyMatrix(Links) self.Tfinal = zeros((4,4)) self.Tfinal[0,0:3] = (T[0:3,0:3]*manipdir).transpose() self.Tfinal[0:3,3] = T[0:3,0:3]*manippos+T[0:3,3] self.testconsistentvalues = self.ComputeConsistentValues(jointvars,self.Tfinal,numsolutions=4) solvejointvars = [jointvars[i] for i in isolvejointvars] if len(solvejointvars) != 5: raise self.CannotSolveError('need 5 joints') log.info('ikfast translation direction 5d: %r, direction=%r', solvejointvars, manipdir) # if last two axes are intersecting, can divide computing position and direction ilinks = [i for i,Tlink in enumerate(Links) if self.has(Tlink,*solvejointvars)] T = self.multiplyMatrix(Links[ilinks[-2]:]) P = T[0:3,0:3]*manippos+T[0:3,3] D = T[0:3,0:3]*manipdir tree = None if not self.has(P,*solvejointvars): Tposinv = eye(4) Tposinv[0:3,3] = -P T0links=[Tposinv]+Links[:ilinks[-2]] try: log.info('last 2 axes are intersecting') tree = self.solve5DIntersectingAxes(T0links,manippos,D,solvejointvars,endbranchtree) except self.CannotSolveError as e: log.warn('%s', e) if tree is None: rawpolyeqs2 = [None]*len(solvejointvars) coupledsolutions = None endbranchtree2 = [] for solvemethod in [self.solveLiWoernleHiller, self.solveKohliOsvatic]:#, self.solveManochaCanny]: if coupledsolutions is not None: break for index in [2,3]: T0links=LinksInv[:ilinks[index]][::-1] T0 = self.multiplyMatrix(T0links) T1links=Links[ilinks[index]:] T1 = self.multiplyMatrix(T1links) p0 = T0[0:3,0:3]*self.Tee[0:3,3]+T0[0:3,3] p1 = T1[0:3,0:3]*manippos+T1[0:3,3] l0 = T0[0:3,0:3]*self.Tee[0,0:3].transpose() l1 = T1[0:3,0:3]*manipdir AllEquations = [] for i in range(3): AllEquations.append(self.SimplifyTransform(p0[i]-p1[i]).expand()) AllEquations.append(self.SimplifyTransform(l0[i]-l1[i]).expand()) # check if all joints in solvejointvars[index:] are revolute and oriented in the same way checkorientationjoints = None leftside = None if len(solvejointvars[:index]) == 3 and all([self.IsHinge(j.name) for j in solvejointvars[:index]]): Taccums = None for T in T0links: if self.has(T, solvejointvars[0]): Taccums = [T] elif Taccums is not None: Taccums.append(T) if self.has(T, solvejointvars[index-1]): break if Taccums is not None: Tcheckorientation = self.multiplyMatrix(Taccums) checkorientationjoints = solvejointvars[:index] leftside = True if len(solvejointvars[index:]) == 3 and all([self.IsHinge(j.name) for j in solvejointvars[index:]]): Taccums = None for T in T1links: if self.has(T, solvejointvars[index]): Taccums = [T] elif Taccums is not None: Taccums.append(T) if self.has(T, solvejointvars[-1]): break if Taccums is not None: Tcheckorientation = self.multiplyMatrix(Taccums) checkorientationjoints = solvejointvars[index:] leftside = False newsolvejointvars = solvejointvars if checkorientationjoints is not None: # TODO, have to consider different signs of the joints cvar3 = cos(checkorientationjoints[0] + checkorientationjoints[1] + checkorientationjoints[2]).expand(trig=True) svar3 = sin(checkorientationjoints[0] + checkorientationjoints[1] + checkorientationjoints[2]).expand(trig=True) # to check for same orientation, see if T's rotation is composed of cvar3 and svar3 sameorientation = True for i in range(3): for j in range(3): if Tcheckorientation[i,j] != S.Zero and not self.equal(Tcheckorientation[i,j], cvar3) and not self.equal(Tcheckorientation[i,j], -cvar3) and not self.equal(Tcheckorientation[i,j], svar3) and not self.equal(Tcheckorientation[i,j], -svar3) and Tcheckorientation[i,j] != S.One: sameorientation = False break if sameorientation: log.info('found joints %r to have same orientation, adding more equations', checkorientationjoints) sumjoint = Symbol('j100') for i in range(3): for j in range(3): if self.equal(Tcheckorientation[i,j], cvar3): Tcheckorientation[i,j] = cos(sumjoint) elif self.equal(Tcheckorientation[i,j], -cvar3): Tcheckorientation[i,j] = -cos(sumjoint) elif self.equal(Tcheckorientation[i,j], svar3): Tcheckorientation[i,j] = sin(sumjoint) elif self.equal(Tcheckorientation[i,j], -svar3): Tcheckorientation[i,j] = -sin(sumjoint) if not leftside: newT1links=[Tcheckorientation] + Links[ilinks[-1]+1:] newT1 = self.multiplyMatrix(newT1links) newp1 = newT1[0:3,0:3]*manippos+newT1[0:3,3] newl1 = newT1[0:3,0:3]*manipdir newp1 = newp1.subs(sin(checkorientationjoints[2]), sin(sumjoint - checkorientationjoints[0] - checkorientationjoints[1]).expand(trig=True)).expand() newl1 = newl1.subs(sin(checkorientationjoints[2]), sin(sumjoint - checkorientationjoints[0] - checkorientationjoints[1]).expand(trig=True)).expand() for i in range(3): newp1[i] = self.trigsimp(newp1[i], [sumjoint, checkorientationjoints[0], checkorientationjoints[1]]) newl1[i] = self.trigsimp(newl1[i], [sumjoint, checkorientationjoints[0], checkorientationjoints[1]]) for i in range(3): AllEquations.append(self.SimplifyTransform(p0[i]-newp1[i]).expand()) AllEquations.append(self.SimplifyTransform(l0[i]-newl1[i]).expand()) AllEquations.append(checkorientationjoints[0] + checkorientationjoints[1] + checkorientationjoints[2] - sumjoint) AllEquations.append((sin(checkorientationjoints[0] + checkorientationjoints[1]) - sin(sumjoint-checkorientationjoints[2])).expand(trig=True)) AllEquations.append((cos(checkorientationjoints[0] + checkorientationjoints[1]) - cos(sumjoint-checkorientationjoints[2])).expand(trig=True)) AllEquations.append((sin(checkorientationjoints[1] + checkorientationjoints[2]) - sin(sumjoint-checkorientationjoints[0])).expand(trig=True)) AllEquations.append((cos(checkorientationjoints[1] + checkorientationjoints[2]) - cos(sumjoint-checkorientationjoints[0])).expand(trig=True)) AllEquations.append((sin(checkorientationjoints[2] + checkorientationjoints[0]) - sin(sumjoint-checkorientationjoints[1])).expand(trig=True)) AllEquations.append((cos(checkorientationjoints[2] + checkorientationjoints[0]) - cos(sumjoint-checkorientationjoints[1])).expand(trig=True)) for consistentvalues in self.testconsistentvalues: var = self.Variable(sumjoint) consistentvalues += var.getsubs((checkorientationjoints[0] + checkorientationjoints[1] + checkorientationjoints[2]).subs(consistentvalues)) newsolvejointvars = solvejointvars + [sumjoint] self.sortComplexity(AllEquations) if rawpolyeqs2[index] is None: rawpolyeqs2[index] = self.buildRaghavanRothEquations(p0,p1,l0,l1,solvejointvars) try: coupledsolutions,usedvars = solvemethod(rawpolyeqs2[index],newsolvejointvars,endbranchtree=[AST.SolverSequence([endbranchtree2])], AllEquationsExtra=AllEquations) break except self.CannotSolveError as e: log.warn('%s', e) continue if coupledsolutions is None: raise self.CannotSolveError('raghavan roth equations too complex') log.info('solved coupled variables: %s',usedvars) if len(usedvars) < len(solvejointvars): curvars=solvejointvars[:] solsubs = self.freevarsubs[:] for var in usedvars: curvars.remove(var) solsubs += self.Variable(var).subs self.checkSolvability(AllEquations,curvars,self.freejointvars+usedvars) localtree = self.SolveAllEquations(AllEquations,curvars=curvars,othersolvedvars = self.freejointvars+usedvars,solsubs = solsubs,endbranchtree=endbranchtree) # make it a function so compiled code is smaller endbranchtree2.append(AST.SolverFunction('innerfn', self.verifyAllEquations(AllEquations,curvars,solsubs,localtree))) tree = coupledsolutions else: endbranchtree2 += endbranchtree tree = coupledsolutions chaintree = AST.SolverIKChainRay([(jointvars[ijoint],ijoint) for ijoint in isolvejointvars], [(v,i) for v,i in izip(self.freejointvars,self.ifreejointvars)], Pee=(self.Tee[0:3,3]-self.Tee[0,0:3].transpose()*offsetdist).subs(self.freevarsubs), Dee=self.Tee[0,0:3].transpose().subs(self.freevarsubs),jointtree=tree,Dfk=self.Tfinal[0,0:3].transpose(),Pfk=self.Tfinal[0:3,3],is5dray=True) chaintree.dictequations += self.ppsubs return chaintree def solve5DIntersectingAxes(self, T0links, manippos, D, solvejointvars, endbranchtree): LinksInv = [self.affineInverse(T) for T in T0links] T0 = self.multiplyMatrix(T0links) Tmanipposinv = eye(4) Tmanipposinv[0:3,3] = -manippos T1links = [Tmanipposinv]+LinksInv[::-1]+[self.Tee] T1linksinv = [self.affineInverse(Tmanipposinv)]+T0links[::-1]+[self.Teeinv] AllEquations = self.buildEquationsFromPositions(T1links,T1linksinv,solvejointvars,self.freejointvars,uselength=True) transvars = [v for v in solvejointvars if self.has(T0,v)] self.checkSolvability(AllEquations,transvars,self.freejointvars) dirtree = [] newendbranchtree = [AST.SolverSequence([dirtree])] transtree = self.SolveAllEquations(AllEquations,curvars=transvars[:],othersolvedvars=self.freejointvars,solsubs = self.freevarsubs[:],endbranchtree=newendbranchtree) transtree = self.verifyAllEquations(AllEquations,solvejointvars,self.freevarsubs,transtree) rotvars = [v for v in solvejointvars if self.has(D,v)] solsubs = self.freevarsubs[:] for v in transvars: solsubs += self.Variable(v).subs AllEquations = self.buildEquationsFromTwoSides([D],[T0[0:3,0:3].transpose()*self.Tee[0,0:3].transpose()],solvejointvars,uselength=False) self.checkSolvability(AllEquations,rotvars,self.freejointvars+transvars) localdirtree = self.SolveAllEquations(AllEquations,curvars=rotvars[:],othersolvedvars = self.freejointvars+transvars,solsubs=solsubs,endbranchtree=endbranchtree) # make it a function so compiled code is smaller dirtree.append(AST.SolverFunction('innerfn', self.verifyAllEquations(AllEquations,rotvars,solsubs,localdirtree))) return transtree def solveFullIK_6D(self, LinksRaw, jointvars, isolvejointvars,Tmanipraw=eye(4)): """Solves the full 6D translatio + rotation IK """ self._iktype = 'transform6d' Tgripper = eye(4) for i in range(4): for j in range(4): Tgripper[i,j] = self.convertRealToRational(Tmanipraw[i,j]) Tfirstright = LinksRaw[-1]*Tgripper Links = LinksRaw[:-1] # if Links[0][0:3,0:3] == eye(3): # # first axis is prismatic, so zero out self.Tee # for i in range(3): # if Links[0][i,3] != S.Zero: # self.Tee[i,3] = S.Zero # self.Teeinv = self.affineInverse(self.Tee) LinksInv = [self.affineInverse(link) for link in Links] self.Tfinal = self.multiplyMatrix(Links) self.testconsistentvalues = self.ComputeConsistentValues(jointvars,self.Tfinal,numsolutions=4) endbranchtree = [AST.SolverStoreSolution (jointvars,isHinge=[self.IsHinge(var.name) for var in jointvars])] solvejointvars = [jointvars[i] for i in isolvejointvars] if len(solvejointvars) > 6 or len(solvejointvars) < 4: raise self.CannotSolveError('need at most 6 joints') log.info('ikfast 6d: %s',solvejointvars) tree = self.TestIntersectingAxes(solvejointvars,Links, LinksInv,endbranchtree) if tree is None: sliderjointvars = [var for var in solvejointvars if not self.IsHinge(var.name)] if len(sliderjointvars) > 0: ZeroMatrix = zeros(4) for i,Tlink in enumerate(Links): if self.has(Tlink,*sliderjointvars): # try sliding left if i > 0: ileftsplit = None for isplit in range(i-1,-1,-1): M = self.multiplyMatrix(Links[isplit:i]) if M*Tlink-Tlink*M != ZeroMatrix: break if self.has(M,*solvejointvars): # surpassed a variable! ileftsplit = isplit if ileftsplit is not None: # try with the new order log.info('rearranging Links[%d] to Links[%d]',i,ileftsplit) NewLinks = list(Links) NewLinks[(ileftsplit+1):(i+1)] = Links[ileftsplit:i] NewLinks[ileftsplit] = Links[i] NewLinksInv = list(LinksInv) NewLinksInv[(ileftsplit+1):(i+1)] = Links[ileftsplit:i] NewLinksInv[ileftsplit] = LinksInv[i] tree = self.TestIntersectingAxes(solvejointvars,NewLinks, NewLinksInv,endbranchtree) if tree is not None: break # try sliding right if i+1 < len(Links): irightsplit = None for isplit in range(i+1,len(Links)): M = self.multiplyMatrix(Links[i+1:(isplit+1)]) if M*Tlink-Tlink*M != ZeroMatrix: break if self.has(M,*solvejointvars): # surpassed a variable! irightsplit = isplit if irightsplit is not None: log.info('rearranging Links[%d] to Links[%d]',i,irightsplit) # try with the new order NewLinks = list(Links) NewLinks[i:irightsplit] = Links[(i+1):(irightsplit+1)] NewLinks[irightsplit] = Links[i] NewLinksInv = list(LinksInv) NewLinksInv[i:irightsplit] = LinksInv[(i+1):(irightsplit+1)] NewLinksInv[irightsplit] = LinksInv[i] tree = self.TestIntersectingAxes(solvejointvars,NewLinks, NewLinksInv,endbranchtree) if tree is not None: break if tree is None: linklist = list(self.iterateThreeNonIntersectingAxes(solvejointvars,Links, LinksInv)) # first try LiWoernleHiller since it is most robust for ilinklist, (T0links, T1links) in enumerate(linklist): log.info('try first group %d/%d', ilinklist, len(linklist)) try: # if T1links[-1] doesn't have any symbols, put it over to T0links. Since T1links has the position unknowns, putting over the coefficients to T0links makes things simpler if not self.has(T1links[-1], *solvejointvars): T0links.append(self.affineInverse(T1links.pop(-1))) tree = self.solveFullIK_6DGeneral(T0links, T1links, solvejointvars, endbranchtree, usesolvers=1) break except (self.CannotSolveError,self.IKFeasibilityError) as e: log.warn('%s',e) if tree is None: log.info('trying the rest of the general ik solvers') for ilinklist, (T0links, T1links) in enumerate(linklist): log.info('try second group %d/%d', ilinklist, len(linklist)) try: # if T1links[-1] doesn't have any symbols, put it over to T0links. Since T1links has the position unknowns, putting over the coefficients to T0links makes things simpler if not self.has(T1links[-1], *solvejointvars): T0links.append(self.affineInverse(T1links.pop(-1))) tree = self.solveFullIK_6DGeneral(T0links, T1links, solvejointvars, endbranchtree, usesolvers=6) break except (self.CannotSolveError,self.IKFeasibilityError) as e: log.warn('%s',e) if tree is None: raise self.CannotSolveError('cannot solve 6D mechanism!') chaintree = AST.SolverIKChainTransform6D([(jointvars[ijoint],ijoint) for ijoint in isolvejointvars], [(v,i) for v,i in izip(self.freejointvars,self.ifreejointvars)], (self.Tee * self.affineInverse(Tfirstright)).subs(self.freevarsubs), tree,Tfk=self.Tfinal*Tfirstright) chaintree.dictequations += self.ppsubs+self.npxyzsubs+self.rxpsubs return chaintree def TestIntersectingAxes(self,solvejointvars,Links,LinksInv,endbranchtree): for T0links,T1links,transvars,rotvars,solveRotationFirst in self.iterateThreeIntersectingAxes(solvejointvars,Links, LinksInv): try: return self.solve6DIntersectingAxes(T0links,T1links,transvars,rotvars,solveRotationFirst=solveRotationFirst, endbranchtree=endbranchtree) except (self.CannotSolveError,self.IKFeasibilityError) as e: log.warn('%s',e) return None def _ExtractTranslationsOutsideOfMatrixMultiplication(self, Links, solvejointvars): """try to extract translations outside of the multiplication (left and right) Tlefttrans * MultiplyMatrix((NewLinks) * Trighttrans = MultiplyMatrix(Links) where Tleftrans and Trighttrans are only translation matrices :return: Tlefttrans, NewLinks, Trighttrans """ NewLinks = list(Links) if len(NewLinks) == 1: return eye(4), NewLinks, eye(4) Trighttrans = eye(4) Trighttrans[0:3,3] = NewLinks[-2][0:3,0:3].transpose() * NewLinks[-2][0:3,3] Trot_with_trans = Trighttrans * NewLinks[-1] separated_trans = Trot_with_trans[0:3,0:3].transpose() * Trot_with_trans[0:3,3] for j in range(0,3): if separated_trans[j].has(*solvejointvars): Trighttrans[j,3] = S.Zero else: Trighttrans[j,3] = separated_trans[j] NewLinks[-2] = NewLinks[-2] * self.affineInverse(Trighttrans) separated_trans = NewLinks[0][0:3,0:3] * NewLinks[1][0:3,3] Tlefttrans = eye(4) for j in range(0,3): if not separated_trans[j].has(*solvejointvars): Tlefttrans[j,3] = separated_trans[j] NewLinks[1] = self.affineInverse(Tlefttrans) * NewLinks[1] return Tlefttrans, NewLinks, Trighttrans def iterateThreeIntersectingAxes(self, solvejointvars, Links, LinksInv): """Search for 3 consectuive intersecting axes. If a robot has this condition, it makes a lot of IK computations simpler. """ TestLinks=Links TestLinksInv=LinksInv ilinks = [i for i,Tlink in enumerate(TestLinks) if self.has(Tlink,*solvejointvars)] hingejointvars = [var for var in solvejointvars if self.IsHinge(var.name)] polysymbols = [] for solvejointvar in solvejointvars: polysymbols += [s[0] for s in self.Variable(solvejointvar).subs] numExpectedRotation = min(3, len(solvejointvars)-3) # the left over are rotation # start backwards since most likely rotation variables are at the end of the chain for i in range(len(ilinks)-(numExpectedRotation-1)-1, -1, -1): startindex = ilinks[i] endindex = ilinks[i+numExpectedRotation-1]+1 Tlefttrans, T0links, Trighttrans = self._ExtractTranslationsOutsideOfMatrixMultiplication(TestLinks[startindex:endindex], solvejointvars) T0 = self.multiplyMatrix(T0links) # count number of variables in T0[0:3,0:3] numVariablesInRotation = sum([self.has(T0[0:3,0:3],solvejointvar) for solvejointvar in solvejointvars]) if numVariablesInRotation < numExpectedRotation: continue solveRotationFirst = None # sometimes the intersecting condition can be there, but is masked by small epsilon errors # so set any coefficients in T0[:3,3] below self.precision precision to zero translationeqs = [self.RoundEquationTerms(eq.expand()) for eq in T0[:3,3]] if not self.has(translationeqs,*hingejointvars): T1links = TestLinksInv[:startindex][::-1] if len(T1links) > 0: T1links[0] = self.affineInverse(Tlefttrans) * T1links[0] else: T1links = [self.affineInverse(Tlefttrans)] T1links.append(self.Tee) T1links += TestLinksInv[endindex:][::-1] T1links[-1] = T1links[-1] * self.affineInverse(Trighttrans) solveRotationFirst = False else: Tlefttrans, T0links, Trighttrans = self._ExtractTranslationsOutsideOfMatrixMultiplication(TestLinksInv[startindex:endindex][::-1], solvejointvars) T0 = self.multiplyMatrix(T0links) translationeqs = [self.RoundEquationTerms(eq.expand()) for eq in T0[:3,3]] if not self.has(translationeqs,*hingejointvars): T1links = TestLinks[endindex:] if len(T1links) > 0: T1links[0] = Trighttrans * T1links[0] else: T1links = [Trighttrans] T1links.append(self.Teeinv) T1links += TestLinks[:startindex] T1links[-1] = T1links[-1] * Tlefttrans solveRotationFirst = False if solveRotationFirst is not None: rotvars = [] transvars = [] for svar in solvejointvars: if self.has(T0[0:3,0:3],svar): rotvars.append(svar) else: transvars.append(svar) if len(rotvars) == numExpectedRotation and len(transvars) == 3: log.info('found 3 consecutive intersecting axes links[%d:%d], rotvars=%s, translationvars=%s',startindex, endindex, rotvars,transvars) yield T0links,T1links,transvars,rotvars,solveRotationFirst def RoundEquationTerms(self,eq,epsilon=None): if eq.is_Add: neweq = S.Zero for subeq in eq.args: neweq += self.RoundEquationTerms(subeq,epsilon) elif eq.is_Mul: neweq = self.RoundEquationTerms(eq.args[0],epsilon) for subeq in eq.args[1:]: neweq *= self.RoundEquationTerms(subeq,epsilon) elif eq.is_Function: newargs = [self.RoundEquationTerms(subeq,epsilon) for subeq in eq.args] neweq = eq.func(*newargs) elif eq.is_number: if epsilon is None: epsilon = 5*(10**-self.precision) if abs(eq.evalf()) <= epsilon: neweq = S.Zero else: neweq = eq else: neweq=eq return neweq def RoundPolynomialTerms(self,peq,epsilon): terms = {} for monom, coeff in peq.terms(): if not coeff.is_number or abs(coeff) > epsilon: terms[monom] = coeff if len(terms) == 0: return Poly(S.Zero,peq.gens) return peq.from_dict(terms, *peq.gens) def iterateThreeNonIntersectingAxes(self, solvejointvars, Links, LinksInv): """check for three consecutive non-intersecting axes. if several points exist, so have to choose one that is least complex? """ ilinks = [i for i,Tlink in enumerate(Links) if self.has(Tlink,*solvejointvars)] usedindices = [] for imode in range(2): for i in range(len(ilinks)-2): if i in usedindices: continue startindex = ilinks[i] endindex = ilinks[i+2]+1 p0 = self.multiplyMatrix(Links[ilinks[i]:ilinks[i+1]])[0:3,3] p1 = self.multiplyMatrix(Links[ilinks[i+1]:ilinks[i+2]])[0:3,3] has0 = self.has(p0,*solvejointvars) has1 = self.has(p1,*solvejointvars) if (imode == 0 and has0 and has1) or (imode == 1 and (has0 or has1)): T0links = Links[startindex:endindex] T1links = LinksInv[:startindex][::-1] T1links.append(self.Tee) T1links += LinksInv[endindex:][::-1] usedindices.append(i) usedvars = [var for var in solvejointvars if any([self.has(T0,var) for T0 in T0links])] log.info('found 3 consecutive non-intersecting axes links[%d:%d], vars=%s',startindex,endindex,str(usedvars)) yield T0links, T1links def solve6DIntersectingAxes(self, T0links, T1links, transvars,rotvars,solveRotationFirst,endbranchtree): """Solve 6D equations using fact that 3 axes are intersecting. The 3 intersecting axes are all part of T0links and will be used to compute the rotation of the robot. The other 3 axes are part of T1links and will be used to first compute the position. """ self._iktype = 'transform6d' assert(len(transvars)==3) # and len(rotvars) == 3) # ok if rotation is 2 T0 = self.multiplyMatrix(T0links) T0posoffset = eye(4) T0posoffset[0:3,3] = -T0[0:3,3] T0links = [T0posoffset] + T0links T1links = [T0posoffset] + T1links T1 = self.multiplyMatrix(T1links) othersolvedvars = rotvars+self.freejointvars if solveRotationFirst else self.freejointvars[:] T1linksinv = [self.affineInverse(T) for T in T1links] AllEquations = self.buildEquationsFromPositions(T1links,T1linksinv,transvars,othersolvedvars,uselength=True) self.checkSolvability(AllEquations,transvars,self.freejointvars) rottree = [] if solveRotationFirst: newendbranchtree = endbranchtree else: newendbranchtree = [AST.SolverSequence([rottree])] curvars = transvars[:] solsubs=self.freevarsubs[:] transtree = self.SolveAllEquations(AllEquations,curvars=curvars,othersolvedvars=othersolvedvars[:],solsubs=solsubs,endbranchtree=newendbranchtree) transtree = self.verifyAllEquations(AllEquations,rotvars if solveRotationFirst else transvars+rotvars,self.freevarsubs[:],transtree) solvertree= [] solvedvarsubs = self.freevarsubs[:] if solveRotationFirst: storesolutiontree = transtree else: solvertree += transtree storesolutiontree = endbranchtree for tvar in transvars: solvedvarsubs += self.Variable(tvar).subs Ree = zeros((3,3)) for i in range(3): for j in range(3): Ree[i,j] = Symbol('new_r%d%d'%(i,j)) try: T1sub = T1.subs(solvedvarsubs) othersolvedvars = self.freejointvars if solveRotationFirst else transvars+self.freejointvars AllEquations = self.buildEquationsFromRotation(T0links,Ree,rotvars,othersolvedvars) self.checkSolvability(AllEquations,rotvars,othersolvedvars) currotvars = rotvars[:] rottree += self.SolveAllEquations(AllEquations,curvars=currotvars,othersolvedvars=othersolvedvars,solsubs=self.freevarsubs[:],endbranchtree=storesolutiontree) if len(rotvars) < 3: # since rotation variables are not enough for 3D rotation, do not check equations used for ss in rottree: ss.checkEquationsUsed = False # has to be after SolveAllEquations...? for i in range(3): for j in range(3): self.globalsymbols.append((Ree[i,j],T1sub[i,j])) if len(rottree) == 0: raise self.CannotSolveError('could not solve for all rotation variables: %s:%s'%(str(freevar),str(freevalue))) if solveRotationFirst: solvertree.append(AST.SolverRotation(T1sub, rottree)) else: rottree[:] = [AST.SolverRotation(T1sub, rottree[:])] return solvertree finally: # remove the Ree global symbols removesymbols = set() for i in range(3): for j in range(3): removesymbols.add(Ree[i,j]) self.globalsymbols = [g for g in self.globalsymbols if not g[0] in removesymbols] def solveFullIK_6DGeneral(self, T0links, T1links, solvejointvars, endbranchtree, usesolvers=7): """Solve 6D equations of a general kinematics structure. This method only works if there exists 3 consecutive joints in that do not always intersect! """ self._iktype = 'transform6d' rawpolyeqs2 = [None,None] coupledsolutions = None leftovervarstree = [] origendbranchtree = endbranchtree solvemethods = [] if usesolvers & 1: solvemethods.append(self.solveLiWoernleHiller) if usesolvers & 2: solvemethods.append(self.solveKohliOsvatic) if usesolvers & 4: solvemethods.append(self.solveManochaCanny) for solvemethod in solvemethods: if coupledsolutions is not None: break complexities = [0,0] for splitindex in [0, 1]: if rawpolyeqs2[splitindex] is None: if splitindex == 0: # invert, this seems to always give simpler solutions, so prioritize it T0 = self.affineSimplify(self.multiplyMatrix([self.affineInverse(T) for T in T0links][::-1])) T1 = self.affineSimplify(self.multiplyMatrix([self.affineInverse(T) for T in T1links][::-1])) else: T0 = self.affineSimplify(self.multiplyMatrix(T0links)) T1 = self.affineSimplify(self.multiplyMatrix(T1links)) rawpolyeqs,numminvars = self.buildRaghavanRothEquationsFromMatrix(T0,T1,solvejointvars,simplify=False) if numminvars <= 5 or len(rawpolyeqs[0][1].gens) <= 6: rawpolyeqs2[splitindex] = rawpolyeqs complexities[splitindex] = sum([self.ComputePolyComplexity(peq0)+self.ComputePolyComplexity(peq1) for peq0, peq1 in rawpolyeqs2[splitindex]]) # try the lowest complexity first and then simplify! sortedindices = sorted(zip(complexities,[0,1])) for complexity, splitindex in sortedindices: for peqs in rawpolyeqs2[splitindex]: c = sum([self.codeComplexity(eq) for eq in peqs[0].coeffs()]) if c < 5000: peqs[0] = self.SimplifyTransformPoly (peqs[0]) else: log.info('skipping simplification since complexity is %d...', c) #self.codeComplexity(poly0.as_expr()) < 2000: c = sum([self.codeComplexity(eq) for eq in peqs[1].coeffs()]) if c < 5000: peqs[1] = self.SimplifyTransformPoly (peqs[1]) else: log.info('skipping simplification since complexity is %d...', c) try: if rawpolyeqs2[splitindex] is not None: rawpolyeqs=rawpolyeqs2[splitindex] endbranchtree=[AST.SolverSequence([leftovervarstree])] unusedsymbols = [] for solvejointvar in solvejointvars: usedinequs = any([var in rawpolyeqs[0][0].gens or var in rawpolyeqs[0][1].gens for var in self.Variable(solvejointvar).vars]) if not usedinequs: unusedsymbols += self.Variable(solvejointvar).vars AllEquationsExtra = [] AllEquationsExtraPruned = [] # prune equations for variables that are not used in rawpolyeqs for i in range(3): for j in range(4): # have to make sure that any variable not in rawpolyeqs[0][0].gens and rawpolyeqs[0][1].gens is not used eq = self.SimplifyTransform(T0[i,j]-T1[i,j]) if not eq.has(*unusedsymbols): AllEquationsExtraPruned.append(eq) AllEquationsExtra.append(eq) self.sortComplexity(AllEquationsExtraPruned) self.sortComplexity(AllEquationsExtra) coupledsolutions,usedvars = solvemethod(rawpolyeqs,solvejointvars,endbranchtree=endbranchtree,AllEquationsExtra=AllEquationsExtraPruned) break except self.CannotSolveError as e: if rawpolyeqs2[splitindex] is not None and len(rawpolyeqs2[splitindex]) > 0: log.warn(u'solving %s: %s', rawpolyeqs2[splitindex][0][0].gens, e) else: log.warn(e) continue if coupledsolutions is None: raise self.CannotSolveError('6D general method failed, raghavan roth equations might be too complex') log.info('solved coupled variables: %s',usedvars) curvars=solvejointvars[:] solsubs = self.freevarsubs[:] for var in usedvars: curvars.remove(var) solsubs += self.Variable(var).subs if len(curvars) > 0: self.sortComplexity(AllEquationsExtra) self.checkSolvability(AllEquationsExtra,curvars,self.freejointvars+usedvars) leftovertree = self.SolveAllEquations(AllEquationsExtra,curvars=curvars,othersolvedvars = self.freejointvars+usedvars,solsubs = solsubs,endbranchtree=origendbranchtree) leftovervarstree.append(AST.SolverFunction('innerfn',leftovertree)) else: leftovervarstree += origendbranchtree return coupledsolutions def solveFullIK_TranslationAxisAngle4D(self, LinksRaw, jointvars, isolvejointvars, rawmanipdir=Matrix(3,1,[S.One,S.Zero,S.Zero]),rawmanippos=Matrix(3,1,[S.Zero,S.Zero,S.Zero]),rawglobaldir=Matrix(3,1,[S.Zero,S.Zero,S.One]), rawglobalnormaldir=None, ignoreaxis=None, rawmanipnormaldir=None, Tmanipraw=None): """Solves 3D translation + Angle with respect to an axis :param rawglobalnormaldir: the axis in the base coordinate system that will be computing a rotation about :param rawglobaldir: the axis normal to rawglobalnormaldir that represents the 0 angle. :param rawmanipnormaldir: the normal dir in the manip coordinate system for emasuring the 0 angle offset. complements rawglobalnormaldir, which shoudl be in the base coordinate system. :param rawmanipdir: the axis in the manip coordinate system measuring the in-plane angle with. :param rawmanippos: the position in manip effector coordinate system for measuring position :param Tmanipraw: extra transform of the manip coordinate system with respect to the end effector """ self._iktype = 'translationaxisangle4d' globaldir = Matrix(3,1,[Float(x,30) for x in rawglobaldir]) globaldir /= sqrt(globaldir[0]*globaldir[0]+globaldir[1]*globaldir[1]+globaldir[2]*globaldir[2]) for i in range(3): globaldir[i] = self.convertRealToRational(globaldir[i],5) iktype = None if rawglobalnormaldir is not None: globalnormaldir = Matrix(3,1,[Float(x,30) for x in rawglobalnormaldir]) binormaldir = globalnormaldir.cross(globaldir).transpose() if globaldir[0] == S.One and globalnormaldir[2] == S.One: if ignoreaxis == 2: iktype = IkType.TranslationXYOrientation3D else: iktype = IkType.TranslationXAxisAngleZNorm4D elif globaldir[1] == S.One and globalnormaldir[0] == S.One: iktype = IkType.TranslationYAxisAngleXNorm4D elif globaldir[2] == S.One and globalnormaldir[1] == S.One: iktype = IkType.TranslationZAxisAngleYNorm4D else: globalnormaldir = None if globaldir[0] == S.One: iktype = IkType.TranslationXAxisAngle4D elif globaldir[1] == S.One: iktype = IkType.TranslationYAxisAngle4D elif globaldir[2] == S.One: iktype = IkType.TranslationZAxisAngle4D if rawmanipnormaldir is None: manipnormaldir = globalnormaldir else: manipnormaldir = Matrix(3,1,[self.convertRealToRational(x) for x in rawmanipnormaldir]) if iktype is None: raise ValueError('currently globaldir can only by one of x,y,z axes') manippos = Matrix(3,1,[self.convertRealToRational(x) for x in rawmanippos]) manipdir = Matrix(3,1,[Float(x,30) for x in rawmanipdir]) L = sqrt(manipdir[0]*manipdir[0]+manipdir[1]*manipdir[1]+manipdir[2]*manipdir[2]) manipdir /= L for i in range(3): manipdir[i] = self.convertRealToRational(manipdir[i],5) manipdir /= sqrt(manipdir[0]*manipdir[0]+manipdir[1]*manipdir[1]+manipdir[2]*manipdir[2]) # unfortunately have to do it again... Links = LinksRaw[:] if Tmanipraw is not None: Links.append(self.RoundMatrix(self.GetMatrixFromNumpy(Tmanipraw))) endbranchtree = [AST.SolverStoreSolution (jointvars,isHinge=[self.IsHinge(var.name) for var in jointvars])] LinksInv = [self.affineInverse(link) for link in Links] Tallmult = self.multiplyMatrix(Links) self.Tfinal = zeros((4,4)) if globalnormaldir is None: self.Tfinal[0,0] = acos(globaldir.dot(Tallmult[0:3,0:3]*manipdir)) else: self.Tfinal[0,0] = atan2(binormaldir.dot(Tallmult[0:3,0:3]*manipdir), globaldir.dot(Tallmult[0:3,0:3]*manipdir)) if self.Tfinal[0,0] == nan: raise self.CannotSolveError('cannot solve 4D axis angle IK. Most likely manipulator direction is aligned with the rotation axis') self.Tfinal[0:3,3] = Tallmult[0:3,0:3]*manippos+Tallmult[0:3,3] self.testconsistentvalues = self.ComputeConsistentValues(jointvars,self.Tfinal,numsolutions=4) solvejointvars = [jointvars[i] for i in isolvejointvars] expecteddof = 4 if ignoreaxis is not None: expecteddof -= 1 if len(solvejointvars) != expecteddof: raise self.CannotSolveError('need %d joints'%expecteddof) log.info('ikfast translation axis %dd, globaldir=%s, manipdir=%s: %s', expecteddof, globaldir, manipdir, solvejointvars) # if last two axes are intersecting, can divide computing position and direction ilinks = [i for i,Tlink in enumerate(Links) if self.has(Tlink,*solvejointvars)] Tmanipposinv = eye(4) Tmanipposinv[0:3,3] = -manippos T1links = [Tmanipposinv]+LinksInv[::-1]+[self.Tee] T1linksinv = [self.affineInverse(Tmanipposinv)]+Links[::-1]+[self.Teeinv] AllEquations = self.buildEquationsFromPositions(T1links,T1linksinv,solvejointvars,self.freejointvars,uselength=True, ignoreaxis=ignoreaxis) if not all([abs(eq.subs(self.testconsistentvalues[0]).evalf())<=1e-10 for eq in AllEquations]): raise self.CannotSolveError('some equations are not consistent with the IK, double check if using correct IK type') for index in range(len(ilinks)): # inv(T0) * T1 * manipdir = globaldir # => T1 * manipdir = T0 * globaldir T0links=LinksInv[:ilinks[index]][::-1] T0 = self.multiplyMatrix(T0links) T1links=Links[ilinks[index]:] T1 = self.multiplyMatrix(T1links) globaldir2 = T0[0:3,0:3]*globaldir manipdir2 = T1[0:3,0:3]*manipdir for i in range(3): if globaldir2[i].is_number: globaldir2[i] = self.convertRealToRational(globaldir2[i]) if manipdir2[i].is_number: manipdir2[i] = self.convertRealToRational(manipdir2[i]) eq = self.SimplifyTransform(self.trigsimp(globaldir2.dot(manipdir2),solvejointvars))-cos(self.Tee[0]) if self.CheckExpressionUnique(AllEquations,eq): AllEquations.append(eq) if globalnormaldir is not None: binormaldir2 = T0[0:3,0:3]*binormaldir for i in range(3): if binormaldir2[i].is_number: binormaldir2[i] = self.convertRealToRational(binormaldir2[i]) eq = self.SimplifyTransform(self.trigsimp(binormaldir2.dot(manipdir2),solvejointvars))-sin(self.Tee[0]) if self.CheckExpressionUnique(AllEquations,eq): AllEquations.append(eq) # check if planar with respect to globalnormaldir extravar = None if globalnormaldir is not None: if Tallmult[0:3,0:3]*manipnormaldir == globalnormaldir: Tnormaltest = self.rodrigues(manipnormaldir,pi/2) # planar, so know that the sum of all hinge joints is equal to the final angle # can use this fact to substitute one angle with the other values angles = [] isanglepositive = [] for solvejoint in solvejointvars: if self.IsHinge(solvejoint.name): Tall0 = Tallmult[0:3,0:3].subs(solvejoint,S.Zero) Tall1 = Tallmult[0:3,0:3].subs(solvejoint,pi/2) if all([f==S.Zero for f in Tall0*Tnormaltest-Tall1]): angles.append(solvejoint) isanglepositive.append(True) else: angles.append(solvejoint) isanglepositive.append(False) Tzero = Tallmult.subs([(a,S.Zero) for a in angles]) for i in range(3): if binormaldir[i].is_number: binormaldir[i] = self.convertRealToRational(binormaldir[i]) if manipdir[i].is_number: manipdir[i] = self.convertRealToRational(manipdir[i]) zeroangle = atan2(binormaldir.dot(Tzero[0:3,0:3]*manipdir), globaldir.dot(Tzero[0:3,0:3]*manipdir)) eqangles = self.Tee[0]-zeroangle for iangle, a in enumerate(angles[:-1]): if isanglepositive[iangle]: eqangles -= a else: eqangles += a if not isanglepositive[-1]: eqangles = -eqangles extravar = (angles[-1],eqangles) coseq = cos(eqangles).expand(trig=True) sineq = sin(eqangles).expand(trig=True) AllEquationsOld = AllEquations AllEquations = [self.trigsimp(eq.subs([(cos(angles[-1]),coseq),(sin(angles[-1]),sineq)]).expand(),solvejointvars) for eq in AllEquationsOld] solvejointvarsold = list(solvejointvars) for var in solvejointvars: if angles[-1].has(var): solvejointvars.remove(var) break self.sortComplexity(AllEquations) endbranchtree = [AST.SolverStoreSolution (jointvars,isHinge=[self.IsHinge(var.name) for var in jointvars])] if extravar is not None: solution=AST.SolverSolution(extravar[0].name, jointeval=[extravar[1]],isHinge=self.IsHinge(extravar[0].name)) endbranchtree.insert(0,solution) try: tree = self.SolveAllEquations(AllEquations,curvars=solvejointvars[:],othersolvedvars=self.freejointvars,solsubs=self.freevarsubs[:],endbranchtree=endbranchtree) tree = self.verifyAllEquations(AllEquations,solvejointvars,self.freevarsubs,tree) except self.CannotSolveError as e: log.debug('failed to solve using SolveAllEquations: %s', e) if 0: solvejointvar0sols = solve(AllEquations[4], solvejointvars[0]) NewEquations = [eq.subs(solvejointvars[0], solvejointvar0sols[0]) for eq in AllEquations] newsolution=AST.SolverSolution(solvejointvars[0].name, jointeval=solvejointvar0sols,isHinge=self.IsHinge(solvejointvars[0].name)) endbranchtree.insert(0,newsolution) tree = self.SolveAllEquations(NewEquations,curvars=solvejointvars[1:],othersolvedvars=self.freejointvars,solsubs=self.freevarsubs[:],endbranchtree=endbranchtree) else: othersolvedvars = self.freejointvars[:] solsubs = self.freevarsubs[:] freevarinvsubs = [(f[1],f[0]) for f in self.freevarsubs] solinvsubs = [(f[1],f[0]) for f in solsubs] # single variable solutions solutions = [] gatheredexceptions = [] for curvar in solvejointvars: othervars = [var for var in solvejointvars if var != curvar] curvarsym = self.Variable(curvar) raweqns = [] for e in AllEquations: if (len(othervars) == 0 or not e.has(*othervars)) and e.has(curvar,curvarsym.htvar,curvarsym.cvar,curvarsym.svar): eq = e.subs(self.freevarsubs+solsubs) if self.CheckExpressionUnique(raweqns,eq): raweqns.append(eq) if len(raweqns) > 0: try: rawsolutions=self.solveSingleVariable(self.sortComplexity(raweqns),curvar,othersolvedvars,unknownvars=solvejointvars) for solution in rawsolutions: self.ComputeSolutionComplexity(solution,othersolvedvars,solvejointvars) solutions.append((solution,curvar)) except self.CannotSolveError as e: gatheredexceptions.append((curvar.name, e)) else: gatheredexceptions.append((curvar.name,None)) if len(solutions) == 0: raise self.CannotSolveError('failed to solve for equations. Possible errors are %s'%gatheredexceptions) firstsolution, firstvar = solutions[0] othersolvedvars.append(firstvar) solsubs += self.Variable(firstvar).subs curvars=solvejointvars[:] curvars.remove(firstvar) trigsubs = [] polysubs = [] polyvars = [] for v in curvars: if self.IsHinge(v.name): var = self.Variable(v) polysubs += [(cos(v),var.cvar),(sin(v),var.svar)] polyvars += [var.cvar,var.svar] trigsubs.append((var.svar**2,1-var.cvar**2)) trigsubs.append((var.svar**3,var.svar*(1-var.cvar**2))) else: polyvars.append(v) polysubsinv = [(b,a) for a,b in polysubs] rawpolyeqs = [Poly(Poly(eq.subs(polysubs),*polyvars).subs(trigsubs),*polyvars) for eq in AllEquations if eq.has(*curvars)] dummys = [] dummysubs = [] dummysubs2 = [] dummyvars = [] for i in range(0,len(polyvars),2): dummy = Symbol('ht%s'%polyvars[i].name[1:]) # [0] - cos, [1] - sin dummys.append(dummy) dummysubs += [(polyvars[i],(1-dummy**2)/(1+dummy**2)),(polyvars[i+1],2*dummy/(1+dummy**2))] var = polyvars[i].subs(self.invsubs).args[0] dummysubs2.append((var,2*atan(dummy))) dummyvars.append((dummy,tan(0.5*var))) newreducedeqs = [] for peq in rawpolyeqs: maxdenom = [0]*(len(polyvars)//2) for monoms in peq.monoms(): for i in range(len(maxdenom)): maxdenom[i] = max(maxdenom[i],monoms[2*i]+monoms[2*i+1]) eqnew = S.Zero for monoms,c in peq.terms(): term = c for i in range(len(polyvars)): num,denom = fraction(dummysubs[i][1]) term *= num**monoms[i] # the denoms for 0,1 and 2,3 are the same for i in range(len(maxdenom)): denom = fraction(dummysubs[2*i][1])[1] term *= denom**(maxdenom[i]-monoms[2*i]-monoms[2*i+1]) eqnew += term newreducedeqs.append(Poly(eqnew,*dummys)) newreducedeqs.sort(key=lambda e: len(e.monoms())) ileftvar = 0 leftvar = dummys[ileftvar] exportcoeffeqs=None for ioffset in range(len(newreducedeqs)): try: exportcoeffeqs,exportmonoms = self.solveDialytically(newreducedeqs[ioffset:],ileftvar) log.info('ioffset %d'%ioffset) break except self.CannotSolveError as e: log.debug('solveDialytically errors: %s',e) if exportcoeffeqs is None: raise self.CannotSolveError('failed to solveDialytically') coupledsolution = AST.SolverCoeffFunction(jointnames=[v.name for v in curvars],jointeval=[v[1] for v in dummysubs2],jointevalcos=[dummysubs[2*i][1] for i in range(len(curvars))],jointevalsin=[dummysubs[2*i+1][1] for i in range(len(curvars))],isHinges=[self.IsHinge(v.name) for v in curvars],exportvar=[v.name for v in dummys],exportcoeffeqs=exportcoeffeqs,exportfnname='solvedialyticpoly12qep',rootmaxdim=16) self.usinglapack = True tree = [firstsolution,coupledsolution]+ endbranchtree # package final solution chaintree = AST.SolverIKChainAxisAngle([(jointvars[ijoint],ijoint) for ijoint in isolvejointvars], [(v,i) for v,i in izip(self.freejointvars,self.ifreejointvars)], Pee=self.Tee[0:3,3].subs(self.freevarsubs), angleee=self.Tee[0,0].subs(self.freevarsubs),jointtree=tree,Pfk=self.Tfinal[0:3,3],anglefk=self.Tfinal[0,0],iktype=iktype) chaintree.dictequations += self.ppsubs return chaintree def buildEquationsFromTwoSides(self,leftside, rightside, usedvars, uselength=True): # try to shift all the constants of each Position expression to one side for i in range(len(leftside)): for j in range(leftside[i].shape[0]): p = leftside[i][j] pee = rightside[i][j] pconstterm = None peeconstterm = None if p.is_Add: pconstterm = [term for term in p.args if term.is_number] elif p.is_number: pconstterm = [p] else: continue if pee.is_Add: peeconstterm = [term for term in pee.args if term.is_number] elif pee.is_number: peeconstterm = [pee] else: continue if len(pconstterm) > 0 and len(peeconstterm) > 0: # shift it to the one that has the least terms for term in peeconstterm if len(p.args) < len(pee.args) else pconstterm: leftside[i][j] -= term rightside[i][j] -= term AllEquations = [] for i in range(len(leftside)): for j in range(leftside[i].shape[0]): e = self.trigsimp(leftside[i][j] - rightside[i][j],usedvars) if self.codeComplexity(e) < 1500: e = self.SimplifyTransform(e) if self.CheckExpressionUnique(AllEquations,e): AllEquations.append(e) if uselength: p2 = S.Zero pe2 = S.Zero for j in range(leftside[i].shape[0]): p2 += leftside[i][j]**2 pe2 += rightside[i][j]**2 if self.codeComplexity(p2) < 1200 and self.codeComplexity(pe2) < 1200: # sympy's trigsimp/customtrigsimp give up too easily e = self.SimplifyTransform(self.trigsimp(p2,usedvars)-self.trigsimp(pe2,usedvars)) if self.CheckExpressionUnique(AllEquations,e): AllEquations.append(e.expand()) else: log.info('length equations too big, skipping %d,%d',self.codeComplexity(p2),self.codeComplexity(pe2)) self.sortComplexity(AllEquations) return AllEquations def buildEquationsFromPositions(self,T1links,T1linksinv,transvars,othersolvedvars,uselength=True,removesmallnumbers=True, ignoreaxis=None): """multiplies out all the matrices and builds up the equations :param ignoreaxis: if not None, can be 0, 1, 2 to specify the axes which should be ignored """ Taccum = eye(4) numvarsdone = 1 Positions = [] Positionsee = [] indices = [0,1,2] if ignoreaxis is not None: indices.remove(ignoreaxis) for i in range(len(T1links)-1): Taccum = T1linksinv[i]*Taccum hasvars = [self.has(Taccum,v) for v in transvars] if __builtin__.sum(hasvars) == numvarsdone: Positions.append(Taccum.extract(indices,[3])) Positionsee.append(self.multiplyMatrix(T1links[(i+1):]).extract(indices,[3])) numvarsdone += 1 if numvarsdone > 2: # more than 2 variables is almost always useless break if len(Positions) == 0: Positions.append(zeros((len(indices),1))) Positionsee.append(self.multiplyMatrix(T1links).extract(indices,[3])) if removesmallnumbers: for i in range(len(Positions)): for j in range(len(indices)): Positions[i][j] = self.RoundEquationTerms(Positions[i][j].expand()) Positionsee[i][j] = self.RoundEquationTerms(Positionsee[i][j].expand()) return self.buildEquationsFromTwoSides(Positions,Positionsee,transvars+othersolvedvars,uselength=uselength) def buildEquationsFromRotation(self,T0links,Ree,rotvars,othersolvedvars): """Ree is a 3x3 matrix """ Raccum = Ree numvarsdone = 0 AllEquations = [] for i in range(len(T0links)): Raccum = T0links[i][0:3,0:3].transpose()*Raccum # transpose is the inverse hasvars = [self.has(Raccum,v) for v in rotvars] if len(AllEquations) > 0 and __builtin__.sum(hasvars) >= len(rotvars): break if __builtin__.sum(hasvars) == numvarsdone: R = self.multiplyMatrix(T0links[(i+1):]) Reqs = [] for i in range(3): Reqs.append([self.trigsimp(Raccum[i,j]-R[i,j],othersolvedvars+rotvars) for j in range(3)]) for i in range(3): for eq in Reqs[i]: AllEquations.append(eq) numvarsdone += 1 # take dot products (equations become unnecessarily complex) # eqdots = [S.Zero, S.Zero, S.Zero] # for i in range(3): # eqdots[0] += Reqs[0][i] * Reqs[1][i] # eqdots[1] += Reqs[1][i] * Reqs[2][i] # eqdots[2] += Reqs[2][i] * Reqs[0][i] # for i in range(3): # AllEquations.append(self.trigsimp(eqdots[i].expand(),othersolvedvars+rotvars)) #AllEquations.append((eqs[0]*eqs[0]+eqs[1]*eqs[1]+eqs[2]*eqs[2]-S.One).expand()) self.sortComplexity(AllEquations) return AllEquations def buildRaghavanRothEquationsFromMatrix(self,T0,T1,solvejointvars, simplify=True, currentcasesubs=None): """Builds the 14 equations using only 5 unknowns. Method explained in [Raghavan1993]_. Basically take the position and one column/row so that the least number of variables are used. .. [Raghavan1993] M Raghavan and B Roth, "Inverse Kinematics of the General 6R Manipulator and related Linkages", Journal of Mechanical Design, Volume 115, Issue 3, 1993. """ p0 = T0[0:3,3] p1 = T1[0:3,3] p=p0-p1 T = T0-T1 numminvars = 100000 for irow in range(3): hasvar = [self.has(T[0:3,irow],var) or self.has(p,var) for var in solvejointvars] numcurvars = __builtin__.sum(hasvar) if numminvars > numcurvars and numcurvars > 0: numminvars = numcurvars l0 = T0[0:3,irow] l1 = T1[0:3,irow] hasvar = [self.has(T[irow,0:3],var) or self.has(p,var) for var in solvejointvars] numcurvars = __builtin__.sum(hasvar) if numminvars > numcurvars and numcurvars > 0: numminvars = numcurvars l0 = T0[irow,0:3].transpose() l1 = T1[irow,0:3].transpose() if currentcasesubs is not None: p0 = p0.subs(currentcasesubs) p1 = p1.subs(currentcasesubs) l0 = l0.subs(currentcasesubs) l1 = l1.subs(currentcasesubs) return self.buildRaghavanRothEquations(p0,p1,l0,l1,solvejointvars,simplify,currentcasesubs),numminvars def CheckEquationForVarying(self, eq): return eq.has('vj0px') or eq.has('vj0py') or eq.has('vj0pz') def buildRaghavanRothEquationsOld(self,p0,p1,l0,l1,solvejointvars): eqs = [] for i in range(3): eqs.append([l0[i],l1[i]]) for i in range(3): eqs.append([p0[i],p1[i]]) l0xp0 = l0.cross(p0) l1xp1 = l1.cross(p1) for i in range(3): eqs.append([l0xp0[i],l1xp1[i]]) ppl0 = p0.dot(p0)*l0 - 2*l0.dot(p0)*p0 ppl1 = p1.dot(p1)*l1 - 2*l1.dot(p1)*p1 for i in range(3): eqs.append([ppl0[i],ppl1[i]]) eqs.append([p0.dot(p0),p1.dot(p1)]) eqs.append([l0.dot(p0),l1.dot(p1)]) # prune any that have varying symbols eqs = [(eq0,eq1) for eq0,eq1 in eqs if not self.CheckEquationForVarying(eq0) and not self.CheckEquationForVarying(eq1)] trigsubs = [] polysubs = [] polyvars = [] for v in solvejointvars: self._CheckPreemptFn(progress=0.05) polyvars.append(v) if self.IsHinge(v.name): var = self.Variable(v) polysubs += [(cos(v),var.cvar),(sin(v),var.svar)] polyvars += [var.cvar,var.svar] trigsubs.append((var.svar**2,1-var.cvar**2)) trigsubs.append((var.svar**3,var.svar*(1-var.cvar**2))) for v in self.freejointvars: if self.IsHinge(v.name): trigsubs.append((sin(v)**2,1-cos(v)**2)) trigsubs.append((sin(v)**3,sin(v)*(1-cos(v)**2))) polysubsinv = [(b,a) for a,b in polysubs] usedvars = [] for j in range(2): usedvars.append([var for var in polyvars if any([eq[j].subs(polysubs).has(var) for eq in eqs])]) polyeqs = [] for i in range(len(eqs)): polyeqs.append([None,None]) for j in range(2): self._CheckPreemptFn(progress=0.05) for i in range(len(eqs)): poly0 = Poly(eqs[i][j].subs(polysubs),*usedvars[j]).subs(trigsubs) poly1 = Poly(poly0.expand().subs(trigsubs),*usedvars[j]) if poly1 == S.Zero: polyeqs[i][j] = poly1 else: polyeqs[i][j] = self.SimplifyTransformPoly(poly1) # remove all fractions? having big integers could blow things up... return polyeqs def buildRaghavanRothEquations(self,p0,p1,l0,l1,solvejointvars,simplify=True,currentcasesubs=None): trigsubs = [] polysubs = [] polyvars = [] for v in solvejointvars: polyvars.append(v) if self.IsHinge(v.name): var = self.Variable(v) polysubs += [(cos(v),var.cvar),(sin(v),var.svar)] polyvars += [var.cvar,var.svar] trigsubs.append((var.svar**2,1-var.cvar**2)) trigsubs.append((var.svar**3,var.svar*(1-var.cvar**2))) for v in self.freejointvars: if self.IsHinge(v.name): trigsubs.append((sin(v)**2,1-cos(v)**2)) trigsubs.append((sin(v)**3,sin(v)*(1-cos(v)**2))) if currentcasesubs is not None: trigsubs += currentcasesubs polysubsinv = [(b,a) for a,b in polysubs] polyeqs = [] for i in range(14): polyeqs.append([None,None]) eqs = [] for i in range(3): eqs.append([l0[i],l1[i]]) for i in range(3): eqs.append([p0[i],p1[i]]) l0xp0 = l0.cross(p0) l1xp1 = l1.cross(p1) for i in range(3): eqs.append([l0xp0[i],l1xp1[i]]) eqs.append([p0.dot(p0),p1.dot(p1)]) eqs.append([l0.dot(p0),l1.dot(p1)]) starttime = time.time() usedvars = [] for j in range(2): usedvars.append([var for var in polyvars if any([eq[j].subs(polysubs).has(var) for eq in eqs])]) for i in range(len(eqs)): self._CheckPreemptFn(progress=0.05) if not self.CheckEquationForVarying(eqs[i][0]) and not self.CheckEquationForVarying(eqs[i][1]): for j in range(2): if polyeqs[i][j] is not None: continue poly0 = Poly(eqs[i][j].subs(polysubs),*usedvars[j]).subs(trigsubs) if self.codeComplexity(poly0.as_expr()) < 5000: poly1 = Poly(poly0.expand().subs(trigsubs),*usedvars[j]) if not simplify or poly1 == S.Zero: polyeqs[i][j] = poly1 else: polyeqs[i][j] = self.SimplifyTransformPoly(poly1) else: polyeqs[i][j] = Poly(poly0.expand().subs(trigsubs),*usedvars[j]) #ppl0 = p0.dot(p0)*l0 - 2*l0.dot(p0)*p0 #ppl1 = p1.dot(p1)*l1 - 2*l1.dot(p1)*p1 ppl0 = polyeqs[9][0].as_expr()*l0 - 2*polyeqs[10][0].as_expr()*p0 # p0.dot(p0)*l0 - 2*l0.dot(p0)*p0 ppl1 = polyeqs[9][1].as_expr()*l1 - 2*polyeqs[10][1].as_expr()*p1 # p1.dot(p1)*l1 - 2*l1.dot(p1)*p1 for i in range(3): eqs.append([ppl0[i],ppl1[i]]) for i in range(11, len(eqs)): if not self.CheckEquationForVarying(eqs[i][0]) and not self.CheckEquationForVarying(eqs[i][1]): for j in range(2): if polyeqs[i][j] is not None: continue poly0 = Poly(eqs[i][j].subs(polysubs),*usedvars[j]).subs(trigsubs) if self.codeComplexity(poly0.as_expr()) < 5000: poly1 = Poly(poly0.expand().subs(trigsubs),*usedvars[j]) if not simplify or poly1 == S.Zero: polyeqs[i][j] = poly1 else: polyeqs[i][j] = self.SimplifyTransformPoly(poly1) else: log.warn('raghavan roth equation (%d,%d) too complex', i, j) polyeqs[i][j] = Poly(poly0.expand().subs(trigsubs),*usedvars[j]) log.info('computed in %fs', time.time()-starttime) # prune any that have varying symbols # remove all fractions? having big integers could blow things up... return [[peq0, peq1] for peq0, peq1 in polyeqs if peq0 is not None and peq1 is not None and not self.CheckEquationForVarying(peq0) and not self.CheckEquationForVarying(peq1)] def reduceBothSides(self,polyeqs): """Reduces a set of equations in 5 unknowns to a set of equations with 3 unknowns by solving for one side with respect to another. The input is usually the output of buildRaghavanRothEquations. """ usedvars = [polyeqs[0][0].gens, polyeqs[0][1].gens] reducedelayed = [] for j in range(2): if len(usedvars[j]) <= 4: leftsideeqs = [polyeq[j] for polyeq in polyeqs if sum(polyeq[j].degree_list()) > 0] rightsideeqs = [polyeq[1-j] for polyeq in polyeqs if sum(polyeq[j].degree_list()) > 0] if all([all(d <= 2 for d in eq.degree_list()) for eq in leftsideeqs]): try: numsymbolcoeffs, _computereducedequations = self.reduceBothSidesSymbolicallyDelayed(leftsideeqs,rightsideeqs) reducedelayed.append([j,leftsideeqs,rightsideeqs,__builtin__.sum(numsymbolcoeffs), _computereducedequations]) except self.CannotSolveError: continue # sort with respect to least number of symbols reducedelayed.sort(key=lambda e: e[3]) reducedeqs = [] tree = [] for j,leftsideeqs,rightsideeqs,numsymbolcoeffs, _computereducedequations in reducedelayed: self._CheckPreemptFn(progress=0.06) try: reducedeqs2 = _computereducedequations() if len(reducedeqs2) == 0: log.info('forcing matrix inverse (might take some time)') reducedeqs2,tree = self.reduceBothSidesInverseMatrix(leftsideeqs,rightsideeqs) if len(reducedeqs2) > 0: # success, add all the reduced equations reducedeqs += [[Poly(eq[0],*usedvars[j]),Poly(eq[1],*usedvars[1-j])] for eq in reducedeqs2] + [[Poly(S.Zero,*polyeq[j].gens),polyeq[1-j]-polyeq[j].as_expr()] for polyeq in polyeqs if sum(polyeq[j].degree_list()) == 0] if len(reducedeqs) > 0: break; except self.CannotSolveError as e: log.warn(e) continue if len(reducedeqs) > 0: # check if any substitutions are needed # for eq in reducedeqs: # for j in range(2): # eq[j] = Poly(eq[j].subs(trigsubs).as_expr().expand(),*eq[j].gens) polyeqs = reducedeqs return [eq for eq in polyeqs if eq[0] != S.Zero or eq[1] != S.Zero],tree def reduceBothSidesInverseMatrix(self,leftsideeqs,rightsideeqs): """solve a linear system inside the program since the matrix cannot be reduced so easily """ allmonomsleft = set() for peq in leftsideeqs: allmonomsleft = allmonomsleft.union(set(peq.monoms())) allmonomsleft = list(allmonomsleft) allmonomsleft.sort() if __builtin__.sum(allmonomsleft[0]) == 0: allmonomsleft.pop(0) if len(leftsideeqs) < len(allmonomsleft): raise self.CannotSolveError('left side has too few equations for the number of variables %d<%d'%(len(leftsideeqs),len(allmonomsleft))) systemcoeffs = [] for ileft,left in enumerate(leftsideeqs): coeffs = [S.Zero]*len(allmonomsleft) rank = 0 for m,c in left.terms(): if __builtin__.sum(m) > 0: if c != S.Zero: rank += 1 coeffs[allmonomsleft.index(m)] = c systemcoeffs.append((rank,ileft,coeffs)) # ideally we want to try all combinations of simple equations first until we arrive to linearly independent ones. # However, in practice most of the first equations are linearly dependent and it takes a lot of time to prune all of them, # so start at the most complex systemcoeffs.sort(key=lambda e: -e[0]) # sort left and right in the same way leftsideeqs = [leftsideeqs[ileft] for rank,ileft,coeffs in systemcoeffs] rightsideeqs = [rightsideeqs[ileft] for rank,ileft,coeffs in systemcoeffs] A = zeros((len(allmonomsleft),len(allmonomsleft))) Asymbols = [] for i in range(A.shape[0]): Asymbols.append([Symbol('gconst%d_%d'%(i,j)) for j in range(A.shape[1])]) solution = None for eqindices in combinations(range(len(leftsideeqs)),len(allmonomsleft)): self._CheckPreemptFn(progress=0.06) for i,index in enumerate(eqindices): for k in range(len(allmonomsleft)): A[i,k] = systemcoeffs[index][2][k] nummatrixsymbols = __builtin__.sum([1 for a in A if not a.is_number]) if nummatrixsymbols > 10: # if too many symbols, evaluate numerically if not self.IsDeterminantNonZeroByEval(A, evalfirst=nummatrixsymbols>60): # pi_robot has 55 symbols and still finishes ok continue log.info('found non-zero determinant by evaluation') else: det = self.det_bareis(A,*self.pvars) if det == S.Zero: continue solution.checkforzeros = [self.removecommonexprs(det,onlygcd=False,onlynumbers=True)] solution = AST.SolverMatrixInverse(A=A,Asymbols=Asymbols) self.usinglapack = True Aadj=A.adjugate() # too big to be useful for now, but can be used to see if any symbols are always 0 break if solution is None: raise self.CannotSolveError('failed to find %d linearly independent equations'%len(allmonomsleft)) reducedeqs = [] for i in range(len(allmonomsleft)): var=S.One for k,kpower in enumerate(allmonomsleft[i]): if kpower != 0: var *= leftsideeqs[0].gens[k]**kpower pright = S.Zero for k in range(len(allmonomsleft)): if Aadj[i,k] != S.Zero: pright += Asymbols[i][k] * (rightsideeqs[eqindices[k]].as_expr()-leftsideeqs[eqindices[k]].TC()) reducedeqs.append([var,pright.expand()]) othereqindices = set(range(len(leftsideeqs))).difference(set(eqindices)) for i in othereqindices: # have to multiply just the constant by the determinant neweq = rightsideeqs[i].as_expr() for m,c in leftsideeqs[i].terms(): if __builtin__.sum(m) > 0: neweq -= c*reducedeqs[allmonomsleft.index(m)][1] else: neweq -= c reducedeqs.append([S.Zero,neweq]) return reducedeqs, [solution] # Adj=M[:,:-1].adjugate() # #D=M[:,:-1].det() # D=M[:,:-1].det() # sols=-Adj*M[:,-1] # solsubs = [] # for i,v in enumerate(newunknowns): # newsol=sols[i].subs(localsymbols) # solsubs.append((v,newsol)) # reducedeqs.append([v.subs(localsymbols)*D,newsol]) # othereqindices = set(range(len(newleftsideeqs))).difference(set(eqindices)) # for i in othereqindices: # # have to multiply just the constant by the determinant # newpoly = S.Zero # for c,m in newleftsideeqs[i].terms(): # monomindices = [index for index in range(len(newunknowns)) if m[index]>0] # if len(monomindices) == 0: # newpoly += c.subs(localsymbols)*D # else: # assert(len(monomindices)==1) # newpoly += c.subs(localsymbols)*solsubs[monomindices[0]][1] # reducedeqs.append([S.Zero,newpoly]) # break # # there are too many symbols, so have to resolve to a little more involved method # P,L,DD,U= M[:,:-1].LUdecompositionFF(*self.pvars) # finalnums = S.One # finaldenoms = S.One # for i in range(len(newunknowns)): # n,d = self.recursiveFraction(L[i,i]*U[i,i]/DD[i,i]) # finalnums *= n # finaldenoms *= d # n,d = self.recursiveFraction(DD[i,i]) # q,r = div(n,d,*pvars) # DD[i,i] = q # assert(r==S.Zero) # det,r = div(finalnums,finaldenoms,*pvars) # assert(r==S.Zero) # b = -P*M[:,-1] # y = [[b[0],L[0,0]]] # for i in range(1,L.shape[0]): # commondenom=y[0][1] # for j in range(1,i): # commondenom=lcm(commondenom,y[j][1],*pvars) # accum = S.Zero # for j in range(i): # accum += L[i,j]*y[j][0]*(commondenom/y[j][1]) # res = (commondenom*b[i]-accum)/(commondenom*L[i,i]) # y.append(self.recursiveFraction(res)) # # ynew = [] # for i in range(L.shape[0]): # q,r=div(y[i][0]*DD[i,i],y[i][1],*pvars) # print 'remainder: ',r # ynew.append(q) # # x = [[ynew[-1],U[-1,-1]]] # for i in range(U.shape[0]-2,-1,-1): # commondenom=x[0][1] # for j in range(i+1,U.shape[0]): # commondenom=lcm(commondenom,x[j][1],*pvars) # accum = S.Zero # for j in range(i+1,U.shape[0]): # accum += U[i,j]*x[j][0]*(commondenom/x[j][1]) # res = (commondenom*b[i]-accum)/(commondenom*U[i,i]) # x.append(self.recursiveFraction(res)) # # print 'ignoring num symbols: ',numsymbols # continue def reduceBothSidesSymbolically(self,*args,**kwargs): numsymbolcoeffs, _computereducedequations = self.reduceBothSidesSymbolicallyDelayed(*args,**kwargs) return _computereducedequations() def reduceBothSidesSymbolicallyDelayed(self,leftsideeqs,rightsideeqs,maxsymbols=10,usesymbols=True): """the left and right side of the equations need to have different variables """ assert(len(leftsideeqs)==len(rightsideeqs)) # first count the number of different monomials, then try to solve for each of them symbolgen = cse_main.numbered_symbols('const') vargen = cse_main.numbered_symbols('tempvar') rightsidedummy = [] localsymbols = [] dividesymbols = [] allmonoms = dict() for left,right in izip(leftsideeqs,rightsideeqs): if right != S.Zero: rightsidedummy.append(next(symbolgen)) localsymbols.append((rightsidedummy[-1],right.as_expr().expand())) else: rightsidedummy.append(S.Zero) for m in left.monoms(): if __builtin__.sum(m) > 0 and not m in allmonoms: newvar = next(vargen) localsymbols.append((newvar,Poly.from_dict({m:S.One},*left.gens).as_expr())) allmonoms[m] = newvar if len(leftsideeqs) < len(allmonoms): raise self.CannotSolveError('left side has too few equations for the number of variables %d<%d'%(len(leftsideeqs),len(allmonoms))) if len(allmonoms) == 0: def _returnequations(): return [[left,right] for left,right in izip(leftsideeqs,rightsideeqs)] return 0, _returnequations unknownvars = leftsideeqs[0].gens newleftsideeqs = [] numsymbolcoeffs = [] for left,right in izip(leftsideeqs,rightsidedummy): left = left - right newleft = Poly(S.Zero,*allmonoms.values()) leftcoeffs = [c for m,c in left.terms() if __builtin__.sum(m) > 0] allnumbers = all([c.is_number for c in leftcoeffs]) if usesymbols and not allnumbers: # check if all the equations are within a constant from each other # This is neceesary since the current linear system solver cannot handle too many symbols. reducedeq0,common0 = self.removecommonexprs(leftcoeffs[0],returncommon=True) commonmults = [S.One] for c in leftcoeffs[1:]: reducedeq1,common1 = self.removecommonexprs(c,returncommon=True) if self.equal(reducedeq1,reducedeq0): commonmults.append(common1/common0) elif self.equal(reducedeq1,-reducedeq0): commonmults.append(-common1/common0) else: break if len(commonmults) == len(leftcoeffs): # divide everything by reducedeq0 index = 0 for m,c in left.terms(): if __builtin__.sum(m) > 0: newleft = newleft + commonmults[index]*allmonoms.get(m) index += 1 else: # look in the dividesymbols for something similar gmult = None for gsym,geq in dividesymbols: greducedeq,gcommon = self.removecommonexprs(S.One/geq,returncommon=True) if self.equal(greducedeq,reducedeq0): gmult = gsym*(gcommon/common0) break elif self.equal(greducedeq,-reducedeq0): gmult = gsym*(-gcommon/common0) break if gmult is None: gmult = next(symbolgen) dividesymbols.append((gmult,S.One/leftcoeffs[0])) newc = (c*gmult).subs(localsymbols).expand() sym = next(symbolgen) localsymbols.append((sym,newc)) newleft = newleft + sym numsymbolcoeffs.append(0) newleftsideeqs.append(newleft) continue numsymbols = 0 for m,c in left.terms(): polyvar = S.One if __builtin__.sum(m) > 0: polyvar = allmonoms.get(m) if not c.is_number: numsymbols += 1 newleft = newleft + c*polyvar numsymbolcoeffs.append(numsymbols) newleftsideeqs.append(newleft) def _computereducedequations(): reducedeqs = [] # order the equations based on the number of terms newleftsideeqs.sort(key=lambda e: len(e.monoms())) newunknowns = newleftsideeqs[0].gens log.info('solving for all pairwise variables in %s, number of symbol coeffs are %s',unknownvars,__builtin__.sum(numsymbolcoeffs)) systemcoeffs = [] for eq in newleftsideeqs: eqdict = eq.as_dict() coeffs = [] for i,var in enumerate(newunknowns): monom = [0]*len(newunknowns) monom[i] = 1 coeffs.append(eqdict.get(tuple(monom),S.Zero)) monom = [0]*len(newunknowns) coeffs.append(-eqdict.get(tuple(monom),S.Zero)) systemcoeffs.append(coeffs) detvars = [s for s,v in localsymbols] + self.pvars for eqindices in combinations(range(len(newleftsideeqs)),len(newunknowns)): # very quick rejection numsymbols = __builtin__.sum([numsymbolcoeffs[i] for i in eqindices]) if numsymbols > maxsymbols: continue M = Matrix([systemcoeffs[i] for i in eqindices]) det = self.det_bareis(M[:,:-1], *detvars) if det == S.Zero: continue try: eqused = [newleftsideeqs[i] for i in eqindices] solution=solve(eqused,newunknowns) except IndexError: # not enough equations? continue if solution is not None and all([self.isValidSolution(value.subs(localsymbols)) for key,value in solution.items()]): # substitute solsubs = [] allvalid = True for key,value in solution.items(): valuesub = value.subs(localsymbols) solsubs.append((key,valuesub)) reducedeqs.append([key.subs(localsymbols),valuesub]) othereqindices = set(range(len(newleftsideeqs))).difference(set(eqindices)) for i in othereqindices: reducedeqs.append([S.Zero,(newleftsideeqs[i].subs(solsubs).subs(localsymbols)).as_expr().expand()]) break # remove the dividesymbols from reducedeqs for sym,ivalue in dividesymbols: value=1/ivalue for i in range(len(reducedeqs)): eq = reducedeqs[i][1] if eq.has(sym): neweq = S.Zero peq = Poly(eq,sym) for m,c in peq.terms(): neweq += c*value**(peq.degree(0) - m[0]) reducedeqs[i][1] = neweq.expand() reducedeqs[i][0] = (reducedeqs[i][0]*value**peq.degree(0)).expand() if len(reducedeqs) > 0: log.info('finished with %d equations',len(reducedeqs)) return reducedeqs return numsymbolcoeffs, _computereducedequations def solveManochaCanny(self,rawpolyeqs,solvejointvars,endbranchtree, AllEquationsExtra=None, currentcases=None, currentcasesubs=None): """Solves the IK equations using eigenvalues/eigenvectors of a 12x12 quadratic eigenvalue problem. Method explained in Dinesh Manocha and J.F. Canny. "Efficient inverse kinematics for general 6R manipulators", IEEE Transactions on Robotics and Automation, Volume 10, Issue 5, Oct 1994. """ log.info('attempting manocha/canny general ik method') PolyEquations, raghavansolutiontree = self.reduceBothSides(rawpolyeqs) # find all equations with zeros on the left side RightEquations = [] for ipeq,peq in enumerate(PolyEquations): if peq[0] == S.Zero: if len(raghavansolutiontree) > 0 or peq[1] == S.Zero: # give up on optimization RightEquations.append(peq[1]) else: RightEquations.append(self.SimplifyTransformPoly(peq[1])) if len(RightEquations) < 6: raise self.CannotSolveError('number of equations %d less than 6'%(len(RightEquations))) # sort with respect to the number of monomials RightEquations.sort(key=lambda e: len(e.monoms())) # substitute with dummy=tan(half angle) symbols = RightEquations[0].gens symbolsubs = [(symbols[i].subs(self.invsubs),symbols[i]) for i in range(len(symbols))] unsolvedsymbols = [] for solvejointvar in solvejointvars: testvars = self.Variable(solvejointvar).vars if not any([v in symbols for v in testvars]): unsolvedsymbols += testvars # check that the coefficients of the reduced equations do not contain any unsolved variables for peq in RightEquations: if peq.has(*unsolvedsymbols): raise self.CannotSolveError('found unsolved symbol being used so ignoring: %s'%peq) log.info('solving simultaneously for symbols: %s',symbols) dummys = [] dummysubs = [] dummysubs2 = [] dummyvars = [] usedvars = [] singlevariables = [] i = 0 while i < len(symbols): dummy = Symbol('ht%s'%symbols[i].name[1:]) var = symbols[i].subs(self.invsubs) if not isinstance(var,Symbol): # [0] - cos, [1] - sin var = var.args[0] dummys.append(dummy) dummysubs += [(symbols[i],(1-dummy**2)/(1+dummy**2)),(symbols[i+1],2*dummy/(1+dummy**2))] dummysubs2.append((var,2*atan(dummy))) dummyvars.append((dummy,tan(0.5*var))) if not var in usedvars: usedvars.append(var) i += 2 else: singlevariables.append(var) # most likely a single variable dummys.append(var) dummysubs += [(var,var)] dummysubs2.append((var,var)) if not var in usedvars: usedvars.append(var) i += 1 newreducedeqs = [] for peq in RightEquations: maxdenom = dict() for monoms in peq.monoms(): i = 0 while i < len(monoms): if peq.gens[i].name[0] == 'j': # single variable maxdenom[peq.gens[i]] = max(maxdenom.get(peq.gens[i],0),monoms[i]) i += 1 else: maxdenom[peq.gens[i]] = max(maxdenom.get(peq.gens[i],0),monoms[i]+monoms[i+1]) i += 2 eqnew = S.Zero for monoms,c in peq.terms(): term = c for i in range(len(dummysubs)): num,denom = fraction(dummysubs[i][1]) term *= num**monoms[i] # the denoms for 0,1 and 2,3 are the same i = 0 while i < len(monoms): if peq.gens[i].name[0] == 'j': denom = fraction(dummysubs[i][1])[1] term *= denom**(maxdenom[peq.gens[i]]-monoms[i]) i += 1 else: denom = fraction(dummysubs[i][1])[1] term *= denom**(maxdenom[peq.gens[i]]-monoms[i]-monoms[i+1]) i += 2 eqnew += term newreducedeqs.append(Poly(eqnew,*dummys)) # check for equations with a single variable if len(singlevariables) > 0: try: AllEquations = [eq.subs(self.invsubs).as_expr() for eq in newreducedeqs] tree = self.SolveAllEquations(AllEquations,curvars=dummys,othersolvedvars=[],solsubs=self.freevarsubs,endbranchtree=endbranchtree, currentcases=currentcases, currentcasesubs=currentcasesubs) return raghavansolutiontree+tree,usedvars except self.CannotSolveError: pass if 0: # try solving for the single variable and substituting for the rest of the equations in order to get a set of equations without the single variable var = singlevariables[0] monomindex = symbols.index(var) singledegreeeqs = [] AllEquations = [] for peq in newreducedeqs: if all([m[monomindex] <= 1 for m in peq.monoms()]): newpeq = Poly(peq,var) if sum(newpeq.degree_list()) > 0: singledegreeeqs.append(newpeq) else: AllEquations.append(peq.subs(self.invsubs).as_expr()) for peq0, peq1 in combinations(singledegreeeqs,2): AllEquations.append(simplify((peq0.TC()*peq1.LC() - peq0.LC()*peq1.TC()).subs(self.invsubs))) log.info(str(AllEquations)) #sol=self.SolvePairVariablesHalfAngle(AllEquations,usedvars[1],usedvars[2],[]) # choose which leftvar can determine the singularity of the following equations! exportcoeffeqs = None getsubs = raghavansolutiontree[0].getsubs if len(raghavansolutiontree) > 0 else None for ileftvar in range(len(dummys)): leftvar = dummys[ileftvar] try: exportcoeffeqs,exportmonoms = self.solveDialytically(newreducedeqs,ileftvar,getsubs=getsubs) break except self.CannotSolveError as e: log.warn('failed with leftvar %s: %s',leftvar,e) if exportcoeffeqs is None: raise self.CannotSolveError('failed to solve dialytically') if ileftvar > 0: raise self.CannotSolveError('solving equations dialytically succeeded with var index %d, unfortunately code generation supports only index 0'%ileftvar) jointevalcos=[d[1] for d in dummysubs if d[0].name[0] == 'c'] jointevalsin=[d[1] for d in dummysubs if d[0].name[0] == 's'] #jointeval=[d[1] for d in dummysubs if d[0].name[0] == 'j'] coupledsolution = AST.SolverCoeffFunction(jointnames=[v.name for v in usedvars],jointeval=[v[1] for v in dummysubs2],jointevalcos=jointevalcos, jointevalsin=jointevalsin, isHinges=[self.IsHinge(v.name) for v in usedvars],exportvar=[v.name for v in dummys],exportcoeffeqs=exportcoeffeqs,exportfnname='solvedialyticpoly12qep',rootmaxdim=16) self.usinglapack = True return raghavansolutiontree+[coupledsolution]+endbranchtree,usedvars def solveLiWoernleHiller(self,rawpolyeqs,solvejointvars,endbranchtree,AllEquationsExtra=[], currentcases=None, currentcasesubs=None): """Li-Woernle-Hiller procedure covered in Jorge Angeles, "Fundamentals of Robotics Mechanical Systems", Springer, 2007. """ log.info('attempting li/woernle/hiller general ik method') if len(rawpolyeqs[0][0].gens) = 8: break log.warn('allowedindex %d found %d equations where coefficients of equations match', allowedindex, len(neweqs)) if len(neweqs) < 8: raise self.CannotSolveError('found %d equations where coefficients of equations match! need at least 8'%len(neweqs)) mysubs = [] badjointvars = [] for solvejointvar in solvejointvars: varsubs = self.Variable(solvejointvar).subs # only choose if varsubs has entry in originalsymbols or othersymbols if len([s for s in varsubs if s[1] in originalsymbols+othersymbols]) > 0: mysubs += varsubs else: badjointvars.append(solvejointvar) AllEquationsExtra = [eq for eq in AllEquationsExtra if not eq.has(*badjointvars)] AllPolyEquationsExtra = [] for eq in AllEquationsExtra: mysubs = [] for solvejointvar in solvejointvars: mysubs += self.Variable(solvejointvar).subs peq = Poly(eq.subs(mysubs), rawpolyeqs[0][0].gens) mixed = False for monom, coeff in peq.terms(): if sum(monom) > 0: # make sure coeff doesn't have any symbols from if coeff.has(*rawpolyeqs[0][1].gens): mixed = True break if not mixed: AllPolyEquationsExtra.append((peq - peq.TC(), Poly(-peq.TC(), rawpolyeqs[0][1].gens))) for polyeq in [polyeqs[ipeq] for ipeq in unusedindices] + AllPolyEquationsExtra: p0 = Poly(polyeq[0],cvar,svar) p1 = polyeq[1] # need to substitute cvar and svar with tvar maxdenom = 0 for monoms in p0.monoms(): maxdenom=max(maxdenom,monoms[0]+monoms[1]) eqnew = S.Zero for monoms,c in p0.terms(): term = c*((1-tvar**2)**monoms[0])*(2*tvar)**monoms[1]*(1+tvar**2)**(maxdenom-monoms[0]-monoms[1]) eqnew += term neweqs.append([Poly(eqnew,*symbols),Poly(p1.as_expr()*(1+tvar**2)**maxdenom,*othersymbols)]) neweqs.append([Poly(eqnew*tvar,*symbols),Poly(p1.as_expr()*tvar*(1+tvar**2)**maxdenom,*othersymbols)]) for ipeq,peq in enumerate(rawpolyeqs): if not peq[0].has(cvar,svar): neweqs.append([Poly(peq[0],*symbols),Poly(peq[1],*othersymbols)]) neweqs.append([Poly(peq[0].as_expr()*tvar,*symbols),Poly(peq[1].as_expr()*tvar,*othersymbols)]) # according to theory, neweqs should have 20 equations, however this isn't always the case # one side should have only numbers, this makes the following inverse operations trivial for peq in neweqs: peq0dict = peq[0].as_dict() peq[1] = peq[1] - tvar*peq0dict.get((0,0,0,0,1),S.Zero)-peq[0].TC() peq[0] = peq[0] - tvar*peq0dict.get((0,0,0,0,1),S.Zero)-peq[0].TC() hasreducedeqs = True while hasreducedeqs: self._CheckPreemptFn(progress=0.08) hasreducedeqs = False for ipeq,peq in enumerate(neweqs): peq0dict = peq[0].as_dict() if len(peq0dict) == 1: monomkey = next(iter(peq0dict.keys())) monomcoeff = peq0dict[monomkey] monomvalue = peq[1].as_expr() if sympy_smaller_073: monomexpr = Monomial(*monomkey).as_expr(*peq[0].gens) else: monomexpr = Monomial(monomkey).as_expr(*peq[0].gens) # for every equation that has this monom, substitute it for ipeq2, peq2 in enumerate(neweqs): if ipeq == ipeq2: continue for monoms,c in peq2[0].terms(): if monoms == monomkey: # have to remove any common expressions between c and monomcoeff, or else equation can get huge num2, denom2 = fraction(cancel(c/monomcoeff)) denom3, num3 = fraction(cancel(monomcoeff/c)) if denom2.is_number and denom3.is_number and abs(denom2.evalf()) > abs(denom3.evalf()): # have to select one with least abs value, or else equation will skyrocket denom2 = denom3 num2 = num3 # have to be careful when multiplying or equation magnitude can get really skewed if denom2.is_number and abs(denom2.evalf()) > 100: peq2[0] = (peq2[0] - c*monomexpr)*monomcoeff peq2[1] = peq2[1]*monomcoeff - c*monomvalue else: peq2[0] = (peq2[0] - c*monomexpr)*denom2 peq2[1] = peq2[1]*denom2 - num2*monomvalue hasreducedeqs = True break # see if there's two equations with two similar monomials on the left-hand side # observed problem: coefficients become extremely huge (100+ digits), need a way to simplify them # for ipeq,peq in enumerate(neweqs): # peq0monoms = peq[0].monoms() # if peq[0] != S.Zero and len(peq0monoms) == 2: # for ipeq2, peq2 in enumerate(neweqs): # if ipeq2 == ipeq: # continue # if peq0monoms == peq2[0].monoms(): # peqdict = peq[0].as_dict() # peq2dict = peq2[0].as_dict() # monom0num, monom0denom = fraction(cancel(peqdict[peq0monoms[0]]/peq2dict[peq0monoms[0]])) # peqdiff = peq2[0]*monom0num - peq[0]*monom0denom # #peqdiff = peq2[0]*peqdict[peq0monoms[0]] - peq[0]*peq2dict[peq0monoms[0]] # if peqdiff != S.Zero: # # there's one monomial left # peqright = (peq2[1]*monom0num - peq[1]*monom0denom) # if peqdiff.LC() != S.Zero: # # check if peqdiff.LC() divides everything cleanly # q0, r0 = div(peqright, peqdiff.LC()) # if r0 == S.Zero: # q1, r1 = div(peqdiff, peqdiff.LC()) # if r1 == S.Zero: # peqright = Poly(q0, *peqright.gens) # peqdiff = Poly(q1, *peqdiff.gens) # # now solve for the other variable # monom1num, monom1denom = fraction(cancel(peqdict[peq0monoms[1]]/peq2dict[peq0monoms[1]])) # peq2diff = peq2[0]*monom1num - peq[0]*monom1denom # peq2right = (peq2[1]*monom1num - peq[1]*monom1denom) # if peq2diff.LC() != S.Zero: # # check if peqdiff.LC() divides everything cleanly # q0, r0 = div(peq2right, peq2diff.LC()) # if r0 == S.Zero: # q1, r1 = div(peq2diff, peq2diff.LC()) # if r1 == S.Zero: # peq2right = Poly(q0, *peq2right.gens) # peq2diff = Poly(q1, *peq2diff.gens) # eqdiff, eqdiffcommon = self.removecommonexprs(peqdiff.as_expr(),returncommon=True, onlygcd=False) # eqright, eqrightcommon = self.removecommonexprs(peqright.as_expr(),returncommon=True, onlygcd=False) # eqdiffmult = cancel(eqdiffcommon/eqrightcommon) # peq[0] = Poly(eqdiff*eqdiffmult, *peqdiff.gens) # peq[1] = Poly(eqright, *peqright.gens) # # eq2diff, eq2diffcommon = self.removecommonexprs(peq2diff.as_expr(),returncommon=True, onlygcd=False) # eq2right, eq2rightcommon = self.removecommonexprs(peq2right.as_expr(),returncommon=True, onlygcd=False) # eq2diffmult = cancel(eq2diffcommon/eq2rightcommon) # peq2[0] = Poly(eq2diff*eq2diffmult, *peq2diff.gens) # peq2[1] = Poly(eq2right, *peq2right.gens) # hasreducedeqs = True # break # else: # # overwrite peq2 in case there are others # peq2[0] = peqdiff # peq2[1] = peq2[1]*monom0num - peq[1]*monom0denom # hasreducedeqs = True neweqs_full = [] reducedeqs = [] # filled with equations where one variable is singled out reducedsinglevars = [None,None,None,None] for ipeq, peq in enumerate(neweqs): peqcomb = Poly(peq[1].as_expr()-peq[0].as_expr(), peq[0].gens[:-1] + peq[1].gens) minimummonom = None for monom in (peqcomb).monoms(): if minimummonom is None: minimummonom = monom else: minimummonom = [min(minimummonom[i], monom[i]) for i in range(len(monom))] if minimummonom is None: continue diveq = None for i in range(len(minimummonom)): if minimummonom[i] > 0: if diveq is None: diveq = peqcomb.gens[i]**minimummonom[i] else: diveq *= peqcomb.gens[i]**minimummonom[i] if diveq is not None: log.info(u'assuming equation %r is non-zero, dividing by %r', diveq, peqcomb) peqcombnum, r = div(peqcomb, diveq) assert(r==S.Zero) peqcombold = peqcomb # save for debugging peqcomb = Poly(peqcombnum, peqcomb.gens)#Poly(peqcomb / diveq, peqcomb.gens) peq0norm, r = div(peq[0], diveq) assert(r==S.Zero) peq1norm, r = div(peq[1], diveq) assert(r==S.Zero) peq = (Poly(peq0norm, *peq[0].gens), Poly(peq1norm, *peq[1].gens)) coeff, factors = peqcomb.factor_list() # check if peq[1] can factor out certain monoms if len(factors) > 1: # if either of the factors evaluate to 0, then we are ok # look for trivial factors that evaluate to 0 or some constant expression and put those into the checkforzeros eq = S.One divisoreq = S.One newzeros = [] for factor, fdegree in factors: if sum(factor.degree_list()) == 1: # actually causes fractions to blow up, so don't use #if factor.as_expr().has(*(originalsymbols+othersymbols)): # eq *= factor.as_expr() # continue log.info(u'assuming equation %r is non-zero', factor) newzeros.append(factor.as_expr()) divisoreq *= factor.as_expr() else: eq *= factor.as_expr() eq = coeff*eq.expand() # have to multiply by the coeff, or otherwise the equation will be weighted different and will be difficult to determine epsilons if peq[0] != S.Zero: peq0norm, r = div(peq[0], divisoreq) assert(r==S.Zero) peq1norm, r = div(peq[1], divisoreq) assert(r==S.Zero) peq0norm = Poly(peq0norm, *peq[0].gens) peq1norm = Poly(peq1norm, *peq[1].gens) peq0dict = peq0norm.as_dict() monom, value = next(iter(peq0dict.items())) if len(peq0dict) == 1 and __builtin__.sum(monom) == 1: indices = [index for index in range(4) if monom[index] == 1] if len(indices) > 0 and indices[0] < 4: reducedsinglevars[indices[0]] = (value, peq1norm.as_expr()) isunique = True for test0, test1 in neweqs_full: if (self.equal(test0,peq0norm) and self.equal(test1,peq1norm)) or (self.equal(test0,-peq0norm) and self.equal(test1,-peq1norm)): isunique = False break if isunique: neweqs_full.append((peq0norm, peq1norm)) else: log.info('not unique: %r', eq) else: eq = eq.subs(self.freevarsubs) if self.CheckExpressionUnique(reducedeqs, eq): reducedeqs.append(eq) else: log.info('factors %d not unique: %r', len(factors), eq) else: if peq[0] != S.Zero: peq0dict = peq[0].as_dict() monom, value = next(iter(peq0dict.items())) if len(peq0dict) == 1 and __builtin__.sum(monom) == 1: indices = [index for index in range(4) if monom[index] == 1] if len(indices) > 0 and indices[0] < 4: reducedsinglevars[indices[0]] = (value,peq[1].as_expr()) isunique = True for test0, test1 in neweqs_full: if (self.equal(test0,peq[0]) and self.equal(test1,peq[1])) or (self.equal(test0,-peq[0]) and self.equal(test1,-peq[1])): isunique = False break if isunique: neweqs_full.append(peq) else: log.info('not unique: %r', peq) else: eq = peq[1].as_expr().subs(self.freevarsubs) if self.CheckExpressionUnique(reducedeqs, eq): reducedeqs.append(eq) else: log.info('factors %d reduced not unique: %r', len(factors), eq) for ivar in range(2): if reducedsinglevars[2*ivar+0] is not None and reducedsinglevars[2*ivar+1] is not None: # a0*cos = b0, a1*sin = b1 a0,b0 = reducedsinglevars[2*ivar+0] a1,b1 = reducedsinglevars[2*ivar+1] reducedeqs.append((b0*a1)**2 + (a0*b1)**2 - (a0*a1)**2) haszeroequations = len(reducedeqs)>0 neweqs_simple = [] neweqs_complex = [] for peq in neweqs_full: hassquare = False for monom in peq[0].monoms(): if any([m > 1 for m in monom]): hassquare = True if not hassquare: neweqs_simple.append(peq) else: neweqs_complex.append(peq) # add more equations by multiplying tvar. this makes it possible to have a fuller matrix neweqs2 = neweqs_simple + [(Poly(peq[0]*tvar, peq[0].gens), Poly(peq[1]*tvar, peq[1].gens)) for peq in neweqs_simple if not peq[0].has(tvar)] # check hacks for 5dof komatsu ik if 1: for itest in range(0,len(neweqs_simple)-1,2): if neweqs_simple[itest][0]*tvar-neweqs_simple[itest+1][0] == S.Zero: eq = (neweqs_simple[itest][1]*tvar-neweqs_simple[itest+1][1]).as_expr() if eq != S.Zero and self.CheckExpressionUnique(reducedeqs, eq): reducedeqs.append(eq) if neweqs_simple[itest+1][0]*tvar-neweqs_simple[itest][0] == S.Zero: eq = (neweqs_simple[itest+1][1]*tvar-neweqs_simple[itest][1]).as_expr() if eq != S.Zero and self.CheckExpressionUnique(reducedeqs, eq): reducedeqs.append(eq) for testrational in [Rational(-103651, 500000), Rational(-413850340369, 2000000000000), Rational(151,500), Rational(301463, 1000000)]: if ((neweqs_simple[0][0]*tvar - neweqs_simple[1][0])*testrational + neweqs_simple[6][0]*tvar - neweqs_simple[7][0]).expand() == S.Zero: if (neweqs_simple[0][0]*tvar - neweqs_simple[1][0]) == S.Zero: eq = (neweqs_simple[6][1]*tvar - neweqs_simple[7][1]).expand().as_expr() else: eq = ((neweqs_simple[0][1]*tvar - neweqs_simple[1][1])*testrational + neweqs_simple[6][1]*tvar - neweqs_simple[7][1]).expand().as_expr() if self.CheckExpressionUnique(reducedeqs, eq): reducedeqs.append(eq) if ((neweqs_simple[0][0]*tvar - neweqs_simple[1][0])*testrational + (neweqs_simple[2][0]*tvar-neweqs_simple[3][0])*sqrt(2)) == S.Zero: if (neweqs_simple[0][0]*tvar - neweqs_simple[1][0]) == S.Zero: eq = ((neweqs_simple[2][1]*tvar-neweqs_simple[3][1])*sqrt(2)).as_expr() else: eq = ((neweqs_simple[0][1]*tvar - neweqs_simple[1][1])*testrational + (neweqs_simple[2][1]*tvar-neweqs_simple[3][1])*sqrt(2)).as_expr() if self.CheckExpressionUnique(reducedeqs, eq): reducedeqs.append(eq) neweqs_test = neweqs2#neweqs_simple allmonoms = set() for ipeq, peq in enumerate(neweqs_test): allmonoms = allmonoms.union(set(peq[0].monoms())) allmonoms = list(allmonoms) allmonoms.sort() if len(allmonoms) > len(neweqs_full) and len(reducedeqs) < 3: raise self.CannotSolveError('new monoms is %d>%d, reducedeqs=%d'%(len(allmonoms), len(neweqs_full), len(reducedeqs))) # the equations are ginac objects getsubs = None dictequations = [] preprocesssolutiontree = [] localsymbolmap = {} AUinv = None if len(allmonoms) < len(neweqs_test): # order with respect to complexity of [0], this is to make the inverse of A faster complexity = [(self.codeComplexity(peq[0].as_expr()),peq) for peq in neweqs_test] complexity.sort(key=itemgetter(0)) neweqs_test = [peq for c,peq in complexity] A = zeros((len(neweqs_test),len(allmonoms))) B = zeros((len(neweqs_test),1)) for ipeq,peq in enumerate(neweqs_test): for m,c in peq[0].terms(): A[ipeq,allmonoms.index(m)] = c.subs(self.freevarsubs) B[ipeq] = peq[1].as_expr().subs(self.freevarsubs) AU = zeros((len(allmonoms),len(allmonoms))) AL = zeros((A.shape[0]-len(allmonoms),len(allmonoms))) BU = zeros((len(allmonoms),1)) BL = zeros((A.shape[0]-len(allmonoms),1)) AUadjugate = None AU = A[:A.shape[1],:] nummatrixsymbols = 0 numcomplexpows = 0 # for non-symbols, how many non-integer pows there are for a in AU: if not a.is_number: nummatrixsymbols += 1 continue hascomplexpow = False for poweq in a.find(Pow): if poweq.exp != S.One and poweq.exp != -S.One: hascomplexpow = True break if hascomplexpow: numcomplexpows += 1 # the 150 threshold is a guess if nummatrixsymbols > 150: log.info('found a non-singular matrix with %d symbols, but most likely there is a better one', nummatrixsymbols) raise self.CannotSolveError('matrix has too many symbols (%d), giving up since most likely will freeze'%nummatrixsymbols) log.info('matrix has %d symbols', nummatrixsymbols) if nummatrixsymbols > 10: # if matrix symbols are great, yield so that other combinations can be tested? pass AUdetmat = None if self.IsDeterminantNonZeroByEval(AU): rows = list(range(A.shape[1])) AUdetmat = AU elif not self.IsDeterminantNonZeroByEval(A.transpose()*A): raise self.CannotSolveError('coefficient matrix is singular') else: # prune the dependent vectors AU = A[0:1,:] rows = [0] for i in range(1,A.shape[0]): self._CheckPreemptFn(progress=0.09) AU2 = AU.col_join(A[i:(i+1),:]) if AU2.shape[0] == AU2.shape[1]: AUdetmat = AU2 else: AUdetmat = AU2*AU2.transpose() # count number of fractions/symbols numausymbols = 0 numaufractions = 0 for f in AUdetmat: if not f.is_number: numausymbols += 1 if f.is_rational and not f.is_integer: numaufractions += 1 # if fraction is really huge, give it more counts (the bigger it is, the slower it takes to compute) flength = len(str(f)) numaufractions += int(flength/20) #d = AUdetmat.det().evalf() #if d == S.Zero: if not self.IsDeterminantNonZeroByEval(AUdetmat, len(rows)>9 and (numaufractions > 120 or numaufractions+numausymbols > 120)): log.info('skipping dependent index %d, numausymbols=%d, numausymbols=%d', i,numaufractions,numausymbols) continue AU = AU2 rows.append(i) if AU.shape[0] == AU.shape[1]: break if AU.shape[0] != AU.shape[1]: raise self.CannotSolveError('could not find non-singular matrix %r'%(AU.shape,)) otherrows = list(range(A.shape[0])) for i,row in enumerate(rows): BU[i] = B[row] otherrows.remove(row) for i,row in enumerate(otherrows): BL[i] = B[row] AL[i,:] = A[row,:] if 0:#self.has(A,*self.freevars): AUinv = AU.inv() AUdet = AUdetmat.det() log.info('AU has symbols, so working with inverse might take some time') AUdet = self.trigsimp(AUdet.subs(self.freevarsubsinv),self.freejointvars).subs(self.freevarsubs) # find the adjugate by simplifying from the inverse AUadjugate = zeros(AUinv.shape) sinsubs = [] for freevar in self.freejointvars: var=self.Variable(freevar) for ideg in range(2,40): if ideg % 2: sinsubs.append((var.cvar**ideg,var.cvar*(1-var.svar**2)**int((ideg-1)/2))) else: sinsubs.append((var.cvar**ideg,(1-var.svar**2)**(ideg/2))) for i in range(AUinv.shape[0]): log.info('replacing row %d', i) for j in range(AUinv.shape[1]): numerator,denominator = self.recursiveFraction(AUinv[i,j]) numerator = self.trigsimp(numerator.subs(self.freevarsubsinv),self.freejointvars).subs(self.freevarsubs) numerator, common = self.removecommonexprs(numerator,onlygcd=True,returncommon=True) denominator = self.trigsimp((denominator/common).subs(self.freevarsubsinv),self.freejointvars).subs(self.freevarsubs) try: q,r=div(numerator*AUdet,denominator,self.freevars) except PolynomialError as e: # 1/(-9000000*cj16 - 9000000) contains an element of the generators set raise self.CannotSolveError('cannot divide for matrix inversion: %s'%e) if r != S.Zero: # sines and cosines can mix things up a lot, so converto to half-tan numerator2, numerator2d, htvarsubsinv = self.ConvertSinCosEquationToHalfTan((AUdet*numerator).subs(sinsubs).expand().subs(sinsubs).expand().subs(sinsubs).expand(), self.freejointvars) denominator2, denominator2d, htvarsubsinv = self.ConvertSinCosEquationToHalfTan(denominator.subs(sinsubs).expand().subs(sinsubs).expand().subs(sinsubs).expand(), self.freejointvars) extranumerator, extradenominator = fraction(numerator2d/denominator2d) htvars = [v for v,eq in htvarsubsinv] q,r=div((numerator2*extradenominator).expand(),(denominator2).expand(),*htvars) if r != S.Zero: log.warn('cannot get rid of denominator for element (%d, %d) in (%s/%s)',i, j, numerator2,denominator2) #raise self.CannotSolveError('cannot get rid of denominator') # convert back to cos/sin in order to get rid of the denominator term? sym = next(self.gsymbolgen) dictequations.append((sym, q / extranumerator)) q = sym #div(q.subs(htvarsubsinv).expand(), extranumerator.subs(htvarsubsinv).expand(), *self.freevars) #newsubs=[(Symbol('htj4'), sin(self.freejointvars[0])/(1+cos(self.freejointvars[0])))] #div(q.extranumerator AUadjugate[i,j] = self.trigsimp(q.subs(self.freevarsubsinv),self.freejointvars).subs(self.freevarsubs) checkforzeros.append(self.removecommonexprs(AUdet,onlygcd=False,onlynumbers=True)) # reason we're multiplying by adjugate instead of inverse is to get rid of the potential divides by (free) parameters BUresult = AUadjugate*BU C = AL*BUresult-BL*AUdet for c in C: reducedeqs.append(c) else: # usually if nummatrixsymbols == 0, we would just solve the inverse of the matrix. however if non-integer powers get in the way, we have to resort to solving the matrix dynamically... if nummatrixsymbols + numcomplexpows/4 > 40: Asymbols = [] for i in range(AU.shape[0]): Asymbols.append([Symbol('gclwh%d_%d'%(i,j)) for j in range(AU.shape[1])]) matrixsolution = AST.SolverMatrixInverse(A=AU,Asymbols=Asymbols) getsubs = matrixsolution.getsubs preprocesssolutiontree.append(matrixsolution) self.usinglapack = True # evaluate the inverse at various solutions and see which entries are always zero isnotzero = zeros((AU.shape[0],AU.shape[1])) epsilon = 1e-15 epsilondet = 1e-30 hasOneNonSingular = False for itest,subs in enumerate(self.testconsistentvalues): AUvalue = AU.subs(subs) isallnumbers = True for f in AUvalue: if not f.is_number: isallnumbers = False break if isallnumbers: # compute more precise determinant AUdetvalue = AUvalue.det() else: AUdetvalue = AUvalue.evalf().det().evalf() if abs(AUdetvalue) > epsilondet:# != S.Zero: hasOneNonSingular = True AUinvvalue = AUvalue.evalf().inv() for i in range(AUinvvalue.shape[0]): for j in range(AUinvvalue.shape[1]): # since making numerical approximations, need a good value for zero if abs(AUinvvalue[i,j]) > epsilon:#!= S.Zero: isnotzero[i,j] = 1 if not hasOneNonSingular: raise self.CannotSolveError('inverse matrix is always singular') AUinv = zeros((AU.shape[0],AU.shape[1])) for i in range(AUinv.shape[0]): for j in range(AUinv.shape[1]): if isnotzero[i,j] == 0: Asymbols[i][j] = None else: AUinv[i,j] = Asymbols[i][j] BUresult = AUinv*BU C = AL*BUresult-BL for c in C: reducedeqs.append(c) elif 0:#nummatrixsymbols > 60: # requires swiginac getsubs = lambda valuesubs: self.SubstituteGinacEquations(dictequations, valuesubs, localsymbolmap) # cannot compute inverse since too many symbols log.info('lu decomposition') # PA = L DD**-1 U P, L, DD, U = self.LUdecompositionFF(AU,*self.pvars) log.info('lower triangular solve') res0 = L.lower_triangular_solve(P*BU) # have to use ginac, since sympy is too slow # there are divides in res0, so have to simplify gres1 = swiginac.symbolic_matrix(len(res0),1,'gres1') for i in range(len(res0)): gres0i = GinacUtils.ConvertToGinac(res0[i],localsymbolmap) gDDi = GinacUtils.ConvertToGinac(DD[i,i],localsymbolmap) gres1[i,0] = gres0i*gDDi gothersymbols = [localsymbolmap[s.name] for s in othersymbols if s.name in localsymbolmap] res2 = [] gres2 = swiginac.symbolic_matrix(len(res0),1,'gres2') for icol in range(len(gres1)): log.info('extracting poly monoms from L solving: %d', icol) polyterms = GinacUtils.GetPolyTermsFromGinac(gres1[icol],gothersymbols,othersymbols) # create a new symbol for every term eq = S.Zero for monom, coeff in polyterms.items(): sym = next(self.gsymbolgen) dictequations.append((sym,coeff)) localsymbolmap[sym.name] = swiginac.symbol(sym.name) if sympy_smaller_073: eq += sym*Monomial(*monom).as_expr(*othersymbols) else: eq += sym*Monomial(monom).as_expr(*othersymbols) res2.append(eq) gres2[icol] = GinacUtils.ConvertToGinac(eq,localsymbolmap) gU = GinacUtils.ConvertMatrixToGinac(U,'U',localsymbolmap) log.info('upper triangular solve') gres3 = GinacUtils.SolveUpperTriangular(gU, gres2, 'gres3') res3 = [] for icol in range(len(gres3)): log.info('extracting poly monoms from U solving: %d', icol) polyterms = GinacUtils.GetPolyTermsFromGinac(gres3[icol],gothersymbols,othersymbols) # create a new symbol for every term eq = S.Zero for monom, coeff in polyterms.items(): sym = next(self.gsymbolgen) dictequations.append((sym,coeff)) localsymbolmap[sym.name] = swiginac.symbol(sym.name) if sympy_smaller_073: eq += sym*Monomial(*monom).as_expr(*othersymbols) else: eq += sym*Monomial(monom).as_expr(*othersymbols) res3.append(eq) BUresult = Matrix(gres3.rows(),gres3.cols(),res3) C = AL*BUresult-BL for c in C: reducedeqs.append(c) else: # if AU has too many fractions, it can prevent computation allzeros = True for b in BU: if b != S.Zero: allzeros = False if not allzeros: try: AUinv = AU.inv() except ValueError as e: raise self.CannotSolveError(u'failed to invert matrix: %e'%e) BUresult = AUinv*BU C = AL*BUresult-BL else: C = -BL for c in C: if c != S.Zero: reducedeqs.append(c) log.info('computed non-singular AU matrix') if len(reducedeqs) == 0: raise self.CannotSolveError('reduced equations are zero') # is now a (len(neweqs)-len(allmonoms))x1 matrix, usually this is 4x1 htvars = [] htvarsubs = [] htvarsubs2 = [] usedvars = [] htvarcossinoffsets = [] nonhtvars = [] for iothersymbol, othersymbol in enumerate(othersymbols): if othersymbol.name[0] == 'c': assert(othersymbols[iothersymbol+1].name[0] == 's') htvarcossinoffsets.append(iothersymbol) name = othersymbol.name[1:] htvar = Symbol('ht%s'%name) htvarsubs += [(othersymbol,(1-htvar**2)/(1+htvar**2)),(othersymbols[iothersymbol+1],2*htvar/(1+htvar**2))] htvars.append(htvar) htvarsubs2.append((Symbol(name),2*atan(htvar))) usedvars.append(Symbol(name)) elif othersymbol.name[0] != 'h' and othersymbol.name[0] != 's': # not half-tan, sin, or cos nonhtvars.append(othersymbol) usedvars.append(othersymbol) htvarsubs += [(cvar,(1-tvar**2)/(1+tvar**2)),(svar,2*tvar/(1+tvar**2))] htvars.append(tvar) htvarsubs2.append((Symbol(varname),2*atan(tvar))) usedvars.append(Symbol(varname)) if haszeroequations: log.info('special structure in equations detected, try to solve through elimination') AllEquations = [eq.subs(self.invsubs) for eq in reducedeqs if self.codeComplexity(eq) < 2000] for curvar in usedvars[:-1]: try: unknownvars = usedvars[:] unknownvars.remove(curvar) jointtrees2=[] curvarsubs=self.Variable(curvar).subs treefirst = self.SolveAllEquations(AllEquations,curvars=[curvar],othersolvedvars=self.freejointvars,solsubs=self.freevarsubs[:],endbranchtree=[AST.SolverSequence([jointtrees2])],unknownvars=unknownvars+[tvar], canguessvars=False, currentcases=currentcases, currentcasesubs=currentcasesubs) # solvable, which means we now have len(AllEquations)-1 with two variables, solve with half angles halfanglesolution=self.SolvePairVariablesHalfAngle(raweqns=[eq.subs(curvarsubs) for eq in AllEquations],var0=unknownvars[0],var1=unknownvars[1],othersolvedvars=self.freejointvars+[curvar])[0] # sometimes halfanglesolution can evaluate to all zeros (katana arm), need to catch this and go to a different branch halfanglesolution.AddHalfTanValue = True jointtrees2.append(halfanglesolution) halfanglevar = unknownvars[0] if halfanglesolution.jointname==unknownvars[0].name else unknownvars[1] unknownvars.remove(halfanglevar) try: # give that two variables are solved, can most likely solve the rest. Solving with the original # equations yields simpler solutions since reducedeqs hold half-tangents curvars = solvejointvars[:] curvars.remove(curvar) curvars.remove(halfanglevar) subsinv = [] for v in solvejointvars: subsinv += self.Variable(v).subsinv AllEquationsOrig = [(peq[0].as_expr()-peq[1].as_expr()).subs(subsinv) for peq in rawpolyeqs] self.sortComplexity(AllEquationsOrig) jointtrees2 += self.SolveAllEquations(AllEquationsOrig,curvars=curvars,othersolvedvars=self.freejointvars+[curvar,halfanglevar],solsubs=self.freevarsubs+curvarsubs+self.Variable(halfanglevar).subs,endbranchtree=endbranchtree, canguessvars=False, currentcases=currentcases, currentcasesubs=currentcasesubs) return preprocesssolutiontree+solutiontree+treefirst,solvejointvars except self.CannotSolveError as e: # try another strategy log.debug(e) # solve all the unknowns now jointtrees3=[] treesecond = self.SolveAllEquations(AllEquations,curvars=unknownvars,othersolvedvars=self.freejointvars+[curvar,halfanglevar],solsubs=self.freevarsubs+curvarsubs+self.Variable(halfanglevar).subs,endbranchtree=[AST.SolverSequence([jointtrees3])], canguessvars=False, currentcases=currentcases, currentcasesubs=currentcasesubs) for t in treesecond: # most likely t is a solution... t.AddHalfTanValue = True if isinstance(t,AST.SolverCheckZeros): for t2 in t.zerobranch: t2.AddHalfTanValue = True for t2 in t.nonzerobranch: t2.AddHalfTanValue = True if len(t.zerobranch) == 0 or isinstance(t.zerobranch[0],AST.SolverBreak): log.info('detected zerobranch with SolverBreak, trying to fix') jointtrees2 += treesecond # using these solutions, can evaluate all monoms and check for consistency, this step is crucial since # AllEquations might not constrain all degrees of freedom (check out katana) indices = [] for i in range(4): monom = [0]*len(symbols) monom[i] = 1 indices.append(allmonoms.index(tuple(monom))) if AUinv is not None: X = AUinv*BU for i in [0,2]: jointname=symbols[i].name[1:] try: # atan2(0,0) produces an invalid solution jointtrees3.append(AST.SolverSolution(jointname,jointeval=[atan2(X[indices[i+1]],X[indices[i]])],isHinge=self.IsHinge(jointname))) usedvars.append(Symbol(jointname)) except Exception as e: log.warn(e) jointcheckeqs = [] for i,monom in enumerate(allmonoms): if not i in indices: eq = S.One for isymbol,ipower in enumerate(monom): eq *= symbols[isymbol]**ipower jointcheckeqs.append(eq-X[i]) # threshold can be a little more loose since just a sanity check jointtrees3.append(AST.SolverCheckZeros('sanitycheck',jointcheckeqs,zerobranch=endbranchtree,nonzerobranch=[AST.SolverBreak('sanitycheck for solveLiWoernleHiller')],anycondition=False,thresh=0.001)) return preprocesssolutiontree+solutiontree+treefirst,usedvars else: log.warn('AUinv not initialized, perhaps missing important equations') except self.CannotSolveError as e: log.info(e) try: log.info('try to solve first two variables pairwise') #solution = self.SolvePairVariables(AllEquations,usedvars[0],usedvars[1],self.freejointvars,maxcomplexity=50) jointtrees=[] unusedvars = [s for s in solvejointvars if not s in usedvars] raweqns=[eq for eq in AllEquations if not eq.has(tvar, *unusedvars)] if len(raweqns) > 1: halfanglesolution = self.SolvePairVariablesHalfAngle(raweqns=raweqns,var0=usedvars[0],var1=usedvars[1],othersolvedvars=self.freejointvars)[0] halfanglevar = usedvars[0] if halfanglesolution.jointname==usedvars[0].name else usedvars[1] unknownvar = usedvars[1] if halfanglesolution.jointname==usedvars[0].name else usedvars[0] nexttree = self.SolveAllEquations(raweqns,curvars=[unknownvar],othersolvedvars=self.freejointvars+[halfanglevar],solsubs=self.freevarsubs+self.Variable(halfanglevar).subs,endbranchtree=[AST.SolverSequence([jointtrees])], canguessvars=False, currentcases=currentcases, currentcasesubs=currentcasesubs) #finalsolution = self.solveSingleVariable(AllEquations,usedvars[2],othersolvedvars=self.freejointvars+usedvars[0:2],maxsolutions=4,maxdegree=4) try: finaltree = self.SolveAllEquations(AllEquations,curvars=usedvars[2:],othersolvedvars=self.freejointvars+usedvars[0:2],solsubs=self.freevarsubs+self.Variable(usedvars[0]).subs+self.Variable(usedvars[1]).subs,endbranchtree=endbranchtree, canguessvars=False, currentcases=currentcases, currentcasesubs=currentcasesubs) jointtrees += finaltree return preprocesssolutiontree+[halfanglesolution]+nexttree,usedvars except self.CannotSolveError as e: log.debug('failed to solve for final variable %s, so returning just two: %s'%(usedvars[2],str(usedvars[0:2]))) jointtrees += endbranchtree # sometimes the last variable cannot be solved, so returned the already solved variables and let the higher function take care of it return preprocesssolutiontree+[halfanglesolution]+nexttree,usedvars[0:2] except self.CannotSolveError as e: log.debug(u'failed solving first two variables pairwise: %s', e) if len(reducedeqs) < 3: raise self.CannotSolveError('have need at least 3 reducedeqs (%d)'%len(reducedeqs)) log.info('reducing %d equations', len(reducedeqs)) newreducedeqs = [] hassinglevariable = False for eq in reducedeqs: self._CheckPreemptFn(progress=0.10) complexity = self.codeComplexity(eq) if complexity > 1500: log.warn('equation way too complex (%d), looking for another solution', complexity) continue if complexity > 1500: log.info('equation way too complex (%d), so try breaking it down', complexity) # don't support this yet... eq2 = eq.expand() assert(eq2.is_Add) log.info('equation has %d additions', len(eq2.args)) indices = list(range(0, len(eq2.args), 100)) indices[-1] = len(eq2.args) testpolyeqs = [] startvalue = 0 for nextvalue in indices[1:]: log.info('computing up to %d', nextvalue) testadd = S.Zero for i in range(startvalue,nextvalue): testadd += eq2.args[i] testpolyeqs.append(Poly(testadd,*othersymbols)) startvalue = nextvalue # convert each poly's coefficients to symbols peq = Poly(S.Zero, *othersymbols) for itest, testpolyeq in enumerate(testpolyeqs): log.info('adding equation %d', itest) newpeq = Poly(S.Zero, *othersymbols) for monom, coeff in newpeq.terms(): sym = next(self.gsymbolgen) dictequations.append((sym,coeff)) if sympy_smaller_073: newpeq += sym*Monomial(*monom).as_expr(*othersymbols) else: newpeq += sym*Monomial(monom).as_expr(*othersymbols) peq += newpeq else: peq = Poly(eq,*othersymbols) maxdenom = [0]*len(htvarcossinoffsets) for monoms in peq.monoms(): for i,ioffset in enumerate(htvarcossinoffsets): maxdenom[i] = max(maxdenom[i],monoms[ioffset]+monoms[ioffset+1]) eqnew = S.Zero for monoms,c in peq.terms(): term = c for i,ioffset in enumerate(htvarcossinoffsets): # for cos num, denom = fraction(htvarsubs[2*i][1]) term *= num**monoms[ioffset] # for sin num, denom = fraction(htvarsubs[2*i+1][1]) term *= num**monoms[ioffset+1] # the denoms for sin/cos of the same joint variable are the same for i,ioffset in enumerate(htvarcossinoffsets): denom = fraction(htvarsubs[2*i][1])[1] term *= denom**(maxdenom[i]-monoms[ioffset]-monoms[ioffset+1]) # multiply the rest of the monoms for imonom, monom in enumerate(monoms): if not imonom in htvarcossinoffsets and not imonom-1 in htvarcossinoffsets: # handle non-sin/cos variables yet term *= othersymbols[imonom]**monom eqnew += term newpeq = Poly(eqnew,htvars+nonhtvars) if newpeq != S.Zero: newreducedeqs.append(newpeq) hassinglevariable |= any([all([__builtin__.sum(monom)==monom[i] for monom in newpeq.monoms()]) for i in range(3)]) if hassinglevariable: log.info('hassinglevariable, trying with raw equations') AllEquations = [] for eq in reducedeqs: peq = Poly(eq,tvar) if sum(peq.degree_list()) == 0: AllEquations.append(peq.TC().subs(self.invsubs).expand()) elif sum(peq.degree_list()) == 1 and peq.TC() == S.Zero: AllEquations.append(peq.LC().subs(self.invsubs).expand()) else: # two substitutions: sin/(1+cos), (1-cos)/sin neweq0 = S.Zero neweq1 = S.Zero for monoms,c in peq.terms(): neweq0 += c*(svar**monoms[0])*((1+cvar)**(peq.degree(0)-monoms[0])) neweq1 += c*((1-cvar)**monoms[0])*(svar**(peq.degree(0)-monoms[0])) AllEquations.append(neweq0.subs(self.invsubs).expand()) AllEquations.append(neweq1.subs(self.invsubs).expand()) unusedvars = [solvejointvar for solvejointvar in solvejointvars if not solvejointvar in usedvars] for eq in AllEquationsExtra: #if eq.has(*usedvars) and not eq.has(*unusedvars): AllEquations.append(eq) self.sortComplexity(AllEquations) # first try to solve all the variables at once try: solutiontree = self.SolveAllEquations(AllEquations,curvars=solvejointvars,othersolvedvars=self.freejointvars[:], solsubs=self.freevarsubs[:], endbranchtree=endbranchtree, canguessvars=False, currentcases=currentcases, currentcasesubs=currentcasesubs) return solutiontree, solvejointvars except self.CannotSolveError as e: log.debug(u'failed solving all variables: %s', e) try: solutiontree = self.SolveAllEquations(AllEquations,curvars=usedvars,othersolvedvars=self.freejointvars[:],solsubs=self.freevarsubs[:], unknownvars=unusedvars, endbranchtree=endbranchtree, canguessvars=False, currentcases=currentcases, currentcasesubs=currentcasesubs) return solutiontree, usedvars except self.CannotSolveError as e: log.debug(u'failed solving used variables: %s', e) for ivar in range(3): try: unknownvars = usedvars[:] unknownvars.pop(ivar) endbranchtree2 = [] if 1: solutiontree = self.SolveAllEquations(AllEquations,curvars=[usedvars[ivar]],othersolvedvars=self.freejointvars[:],solsubs=self.freevarsubs[:],endbranchtree=[AST.SolverSequence([endbranchtree2])],unknownvars=unknownvars+unusedvars, canguessvars=False, currentcases=currentcases, currentcasesubs=currentcasesubs) endbranchtree2 += self.SolveAllEquations(AllEquations,curvars=unknownvars[0:2],othersolvedvars=self.freejointvars[:]+[usedvars[ivar]],solsubs=self.freevarsubs[:]+self.Variable(usedvars[ivar]).subs, unknownvars=unusedvars, endbranchtree=endbranchtree, canguessvars=False, currentcases=currentcases, currentcasesubs=currentcasesubs) return preprocesssolutiontree+solutiontree, usedvars#+unusedvars#[unknownvars[1], usedvars[ivar]]# except self.CannotSolveError as e: log.debug(u'single variable %s failed: %s', usedvars[ivar], e) # try: # testvars = [Symbol(othersymbols[0].name[1:]),Symbol(othersymbols[2].name[1:]),Symbol(varname)] # AllEquations = [(peq[0].as_expr()-peq[1].as_expr()).expand() for peq in polyeqs if not peq[0].has(*symbols)] # coupledsolutions = self.SolveAllEquations(AllEquations,curvars=testvars,othersolvedvars=self.freejointvars[:],solsubs=self.freevarsubs[:],endbranchtree=endbranchtree) # return coupledsolutions,testvars # except self.CannotSolveError: # pass # exportcoeffeqs = None # only support ileftvar=0 for now for ileftvar in [0]:#range(len(htvars)): # always take the equations 4 at a time....? if len(newreducedeqs) == 3: try: exportcoeffeqs,exportmonoms = self.solveDialytically(newreducedeqs,ileftvar,getsubs=getsubs) break except self.CannotSolveError as e: log.warn('failed with leftvar %s: %s',newreducedeqs[0].gens[ileftvar],e) else: for dialyticeqs in combinations(newreducedeqs,3): try: exportcoeffeqs,exportmonoms = self.solveDialytically(dialyticeqs,ileftvar,getsubs=getsubs) break except self.CannotSolveError as e: log.warn('failed with leftvar %s: %s',newreducedeqs[0].gens[ileftvar],e) for dialyticeqs in combinations(newreducedeqs,4): try: exportcoeffeqs,exportmonoms = self.solveDialytically(dialyticeqs,ileftvar,getsubs=getsubs) break except self.CannotSolveError as e: log.warn('failed with leftvar %s: %s',newreducedeqs[0].gens[ileftvar],e) if exportcoeffeqs is None: filteredeqs = [peq for peq in newreducedeqs if peq.degree() <= 2] # has never worked for higher degrees than 2 for dialyticeqs in combinations(filteredeqs,6): try: exportcoeffeqs,exportmonoms = self.solveDialytically(dialyticeqs,ileftvar,getsubs=getsubs) break except self.CannotSolveError as e: log.warn('failed with leftvar %s: %s',newreducedeqs[0].gens[ileftvar],e) if exportcoeffeqs is not None: break self._CheckPreemptFn(progress=0.11) if exportcoeffeqs is None: if len(nonhtvars) > 0 and newreducedeqs[0].degree: log.info('try to solve one variable in terms of the others') doloop = True while doloop: doloop = False # check if there is an equation that can solve for nonhtvars[0] easily solvenonhtvareq = None for peq in newreducedeqs: if peq.degree(len(htvars)) == 1: solvenonhtvareq = peq break if solvenonhtvareq is None: break # nonhtvars[0] index is len(htvars) usedvar0solution = solve(newreducedeqs[0],nonhtvars[0])[0] num,denom = fraction(usedvar0solution) igenoffset = len(htvars) # substitute all instances of the variable processedequations = [] for peq in newreducedeqs[1:]: if self.codeComplexity(peq.as_expr()) > 10000: log.warn('equation too big') continue maxdegree = peq.degree(igenoffset) eqnew = S.Zero for monoms,c in peq.terms(): term = c term *= denom**(maxdegree-monoms[igenoffset]) term *= num**(monoms[igenoffset]) for imonom, monom in enumerate(monoms): if imonom != igenoffset and imonom < len(htvars): term *= htvars[imonom]**monom eqnew += term try: newpeq = Poly(eqnew,htvars) except PolynomialError as e: # most likel uservar0solution was bad... raise self.CannotSolveError('equation %s cannot be represented as a polynomial'%eqnew) if newpeq != S.Zero: processedequations.append(newpeq) if len(processedequations) == 0: break # check if any variables have degree <= 1 for all equations for ihtvar,htvar in enumerate(htvars): leftoverhtvars = list(htvars) leftoverhtvars.pop(ihtvar) freeequations = [] linearequations = [] higherequations = [] for peq in processedequations: if peq.degree(ihtvar) == 0: freeequations.append(peq) elif peq.degree(ihtvar) == 1: linearequations.append(peq) else: higherequations.append(peq) if len(freeequations) > 0: log.info('found a way to solve this! still need to implement it though...') elif len(linearequations) > 0 and len(leftoverhtvars) == 1: # try substituting one into the other equations Ax = B A = S.Zero B = S.Zero for monoms,c in linearequations[0].terms(): term = c for imonom, monom in enumerate(monoms): if imonom != ihtvar: term *= htvars[imonom]**monom if monoms[ihtvar] > 0: A += term else: B -= term Apoly = Poly(A,leftoverhtvars) Bpoly = Poly(B,leftoverhtvars) singlepolyequations = [] useequations = linearequations[1:] if len(useequations) == 0: useequations += higherequations for peq in useequations: complexity = self.codeComplexity(peq.as_expr()) if complexity < 2000: peqnew = Poly(S.Zero,leftoverhtvars) maxhtvardegree = peq.degree(ihtvar) for monoms,c in peq.terms(): term = c for imonom, monom in enumerate(monoms): if imonom != ihtvar: term *= htvars[imonom]**monom termpoly = Poly(term,leftoverhtvars) peqnew += termpoly * (Bpoly**(monoms[ihtvar]) * Apoly**(maxhtvardegree-monoms[ihtvar])) singlepolyequations.append(peqnew) if len(singlepolyequations) > 0: jointsol = 2*atan(leftoverhtvars[0]) jointname = leftoverhtvars[0].name[2:] firstsolution = AST.SolverPolynomialRoots(jointname=jointname,poly=singlepolyequations[0],jointeval=[jointsol],isHinge=self.IsHinge(jointname)) firstsolution.checkforzeros = [] firstsolution.postcheckforzeros = [] firstsolution.postcheckfornonzeros = [] firstsolution.postcheckforrange = [] # in Ax=B, if A is 0 and B is non-zero, then equation is invalid # however if both A and B evaluate to 0, then equation is still valid # therefore equation is invalid only if A==0&&B!=0 firstsolution.postcheckforNumDenom = [(A.as_expr(), B.as_expr())] firstsolution.AddHalfTanValue = True # actually both A and B can evaluate to zero, in which case we have to use a different method to solve them AllEquations = [] for eq in reducedeqs: if self.codeComplexity(eq) > 500: continue peq = Poly(eq, tvar) if sum(peq.degree_list()) == 0: AllEquations.append(peq.TC().subs(self.invsubs).expand()) elif sum(peq.degree_list()) == 1 and peq.TC() == S.Zero: AllEquations.append(peq.LC().subs(self.invsubs).expand()) else: # two substitutions: sin/(1+cos), (1-cos)/sin neweq0 = S.Zero neweq1 = S.Zero for monoms,c in peq.terms(): neweq0 += c*(svar**monoms[0])*((1+cvar)**(peq.degree(0)-monoms[0])) neweq1 += c*((1-cvar)**monoms[0])*(svar**(peq.degree(0)-monoms[0])) if self.codeComplexity(neweq0) > 1000 or self.codeComplexity(neweq1) > 1000: break AllEquations.append(neweq0.subs(self.invsubs).expand()) AllEquations.append(neweq1.subs(self.invsubs).expand()) #oldmaxcasedepth = self.maxcasedepth try: #self.maxcasedepth = min(self.maxcasedepth, 2) solvevar = Symbol(jointname) curvars = list(usedvars) curvars.remove(solvevar) unusedvars = [solvejointvar for solvejointvar in solvejointvars if not solvejointvar in usedvars] solutiontree = self.SolveAllEquations(AllEquations+AllEquationsExtra,curvars=curvars+unusedvars,othersolvedvars=self.freejointvars[:]+[solvevar],solsubs=self.freevarsubs[:]+self.Variable(solvevar).subs,endbranchtree=endbranchtree, canguessvars=False, currentcases=currentcases, currentcasesubs=currentcasesubs) #secondSolutionComplexity = self.codeComplexity(B) + self.codeComplexity(A) #if secondSolutionComplexity > 500: # log.info('solution for %s is too complex, so delaying its solving') #solutiontree = self.SolveAllEquations(AllEquations,curvars=curvars,othersolvedvars=self.freejointvars[:]+[solvevar],solsubs=self.freevarsubs[:]+self.Variable(solvevar).subs,endbranchtree=endbranchtree) return preprocesssolutiontree+[firstsolution]+solutiontree,usedvars+unusedvars except self.CannotSolveError as e: log.debug('could not solve full variables from scratch, so use existing solution: %s', e) secondsolution = AST.SolverSolution(htvar.name[2:], isHinge=self.IsHinge(htvar.name[2:])) secondsolution.jointeval = [2*atan2(B.as_expr(), A.as_expr())] secondsolution.AddHalfTanValue = True thirdsolution = AST.SolverSolution(nonhtvars[0].name, isHinge=self.IsHinge(nonhtvars[0].name)) thirdsolution.jointeval = [usedvar0solution] return preprocesssolutiontree+[firstsolution, secondsolution, thirdsolution]+endbranchtree, usedvars # finally: # self.maxcasedepth = oldmaxcasedepth # try to factor the equations manually deg1index = None for i in range(len(newreducedeqs)): if newreducedeqs[i].degree(2) == 1: if self.codeComplexity(newreducedeqs[i].as_expr()) <= 5000: deg1index = i break else: log.warn('found equation with linear dof, but too complex so skipping') if deg1index is not None: # try to solve one variable in terms of the others if len(htvars) > 2: usedvar0solutions = [solve(newreducedeqs[deg1index],htvars[2])[0]] # check which index in usedvars matches htvars[2] for igenoffset in range(len(usedvars)): if htvars[2].name.find(usedvars[igenoffset].name) >= 0: break polyvars = htvars[0:2] elif len(htvars) > 1: usedvar0solutions = solve(newreducedeqs[deg1index],htvars[1]) igenoffset = 1 polyvars = htvars[0:1] + nonhtvars else: usedvar0solutions = [] processedequations = [] if len(usedvar0solutions) > 0: usedvar0solution = usedvar0solutions[0] num,denom = fraction(usedvar0solution) # substitute all instances of the variable for ipeq, peq in enumerate(newreducedeqs): if ipeq == deg1index: continue newpeq = S.Zero if peq.degree(igenoffset) > 1: # ignore higher powers continue elif peq.degree(igenoffset) == 0: newpeq = Poly(peq,*polyvars) else: maxdegree = peq.degree(igenoffset) eqnew = S.Zero for monoms,c in peq.terms(): term = c*denom**(maxdegree-monoms[igenoffset]) term *= num**(monoms[igenoffset]) for imonom, monom in enumerate(monoms): if imonom != igenoffset: term *= peq.gens[imonom]**monom eqnew += term.expand() try: newpeq = Poly(eqnew,*polyvars) except PolynomialError as e: # most likel uservar0solution was bad raise self.CannotSolveError('equation %s cannot be represented as a polynomial'%eqnew) if newpeq != S.Zero: # normalize by the greatest coefficient in LC, or otherwise determinant will never succeed LC=newpeq.LC() highestcoeff = None if LC.is_Add: for arg in LC.args: coeff = None if arg.is_Mul: coeff = S.One for subarg in arg.args: if subarg.is_number: coeff *= abs(subarg) elif arg.is_number: coeff = abs(arg) if coeff is not None: if coeff > S.One: # round to the nearest integer coeff = int(round(coeff.evalf())) if highestcoeff is None or coeff > highestcoeff: highestcoeff = coeff if highestcoeff == oo: log.warn('an equation has inifinity?!') else: if highestcoeff is not None: processedequations.append(newpeq*(S.One/highestcoeff)) else: processedequations.append(newpeq) else: log.info('equation is zero, so ignoring') for dialyticeqs in combinations(processedequations,3): Mall = None leftvar = None for ileftvar in range(2): # TODO, sometimes this works and sometimes this doesn't try: Mall, allmonoms = self.solveDialytically(dialyticeqs,ileftvar,returnmatrix=True) if Mall is not None: leftvar=processedequations[0].gens[ileftvar] break except self.CannotSolveError as e: log.debug(e) if Mall is None: continue log.info('success in solving sub-coeff matrix!') shape=Mall[0].shape Malltemp = [None]*len(Mall) M = zeros(shape) dictequations2 = list(dictequations) for idegree in range(len(Mall)): Malltemp[idegree] = zeros(shape) for i in range(shape[0]): for j in range(shape[1]): if Mall[idegree][i,j] != S.Zero: sym = next(self.gsymbolgen) Malltemp[idegree][i,j] = sym dictequations2.append((sym,Mall[idegree][i,j])) M += Malltemp[idegree]*leftvar**idegree tempsymbols = [next(self.gsymbolgen) for i in range(len(M))] tempsubs = [] for i in range(len(tempsymbols)): if M[i] != S.Zero: tempsubs.append((tempsymbols[i],Poly(M[i],leftvar))) else: tempsymbols[i] = S.Zero Mtemp = Matrix(M.shape[0],M.shape[1],tempsymbols) dettemp=Mtemp.det() log.info('multiplying all determinant coefficients for solving %s',leftvar) eqadds = [] for arg in dettemp.args: eqmuls = [Poly(arg2.subs(tempsubs),leftvar) for arg2 in arg.args] if sum(eqmuls[0].degree_list()) == 0: eq = eqmuls.pop(0) eqmuls[0] = eqmuls[0]*eq while len(eqmuls) > 1: ioffset = 0 eqmuls2 = [] while ioffset < len(eqmuls)-1: eqmuls2.append(eqmuls[ioffset]*eqmuls[ioffset+1]) ioffset += 2 eqmuls = eqmuls2 eqadds.append(eqmuls[0]) det = Poly(S.Zero,leftvar) for eq in eqadds: det += eq jointsol = 2*atan(leftvar) firstsolution = AST.SolverPolynomialRoots(jointname=usedvars[ileftvar].name,poly=det,jointeval=[jointsol],isHinge=self.IsHinge(usedvars[ileftvar].name)) firstsolution.checkforzeros = [] firstsolution.postcheckforzeros = [] firstsolution.postcheckfornonzeros = [] firstsolution.postcheckforrange = [] firstsolution.dictequations = dictequations2 firstsolution.AddHalfTanValue = True # just solve the lowest degree one complexity = [(eq.degree(1-ileftvar)*100000+self.codeComplexity(eq.as_expr()),eq) for eq in processedequations if eq.degree(1-ileftvar) > 0] complexity.sort(key=itemgetter(0)) orderedequations = [peq for c,peq in complexity] jointsol = 2*atan(htvars[1-ileftvar]) secondsolution = AST.SolverPolynomialRoots(jointname=usedvars[1-ileftvar].name,poly=Poly(orderedequations[0],htvars[1-ileftvar]),jointeval=[jointsol],isHinge=self.IsHinge(usedvars[1-ileftvar].name)) secondsolution.checkforzeros = [] secondsolution.postcheckforzeros = [] secondsolution.postcheckfornonzeros = [] secondsolution.postcheckforrange = [] secondsolution.AddHalfTanValue = True thirdsolution = AST.SolverSolution(usedvars[2].name, isHinge=self.IsHinge(usedvars[2].name)) thirdsolution.jointeval = [usedvar0solution] return preprocesssolutiontree+[firstsolution, secondsolution, thirdsolution]+endbranchtree, usedvars raise self.CannotSolveError('failed to solve dialytically') if 0: # quadratic equations iquadvar = 1 quadpoly0 = Poly(newreducedeqs[0].as_expr(), htvars[iquadvar]) quadpoly1 = Poly(newreducedeqs[2].as_expr(), htvars[iquadvar]) a0, b0, c0 = quadpoly0.coeffs() a1, b1, c1 = quadpoly1.coeffs() quadsolnum = (-a1*c0 + a0*c1).expand() quadsoldenom = (-a1*b0 + a0*b1).expand() if ileftvar > 0: raise self.CannotSolveError('solving equations dialytically succeeded with var index %d, unfortunately code generation supports only index 0'%ileftvar) exportvar = [htvars[ileftvar].name] exportvar += [v.name for i,v in enumerate(htvars) if i != ileftvar] exportfnname = 'solvedialyticpoly12qep' if len(exportmonoms) == 9 else 'solvedialyticpoly8qep' coupledsolution = AST.SolverCoeffFunction(jointnames=[v.name for v in usedvars],jointeval=[v[1] for v in htvarsubs2],jointevalcos=[htvarsubs[2*i][1] for i in range(len(htvars))],jointevalsin=[htvarsubs[2*i+1][1] for i in range(len(htvars))],isHinges=[self.IsHinge(v.name) for v in usedvars],exportvar=exportvar,exportcoeffeqs=exportcoeffeqs,exportfnname=exportfnname, rootmaxdim=16) coupledsolution.presetcheckforzeros = checkforzeros coupledsolution.dictequations = dictequations solutiontree.append(coupledsolution) self.usinglapack = True if 0: if currentcases is None: currentcases = set() if currentcasesubs is None: currentcasesubs = list() rotsymbols = set(self.Tee[:3,:3]) possiblesub = [(self.Tee[1,2], S.Zero)] possiblesub2 = [(self.Tee[2,2], S.Zero)] possiblevar,possiblevalue = possiblesub[0] possiblevar2,possiblevalue2 = possiblesub2[0] cond = Abs(possiblevar-possiblevalue.evalf(n=30)) evalcond = Abs(fmod(possiblevar-possiblevalue+pi,2*pi)-pi)# + evalcond cond2 = Abs(possiblevar2-possiblevalue2.evalf(n=30)) evalcond2 = Abs(fmod(possiblevar2-possiblevalue2+pi,2*pi)-pi)# + evalcond if self._iktype == 'transform6d' and possiblevar in rotsymbols and possiblevalue == S.Zero and possiblevar2 in rotsymbols and possiblevalue2 == S.Zero: checkexpr = [[cond+cond2],evalcond+evalcond2, possiblesub+possiblesub2, []] #flatzerosubstitutioneqs.append(checkexpr) #localsubstitutioneqs.append(checkexpr) #handledconds.append(cond+cond2) row1 = int(possiblevar.name[-2]) col1 = int(possiblevar.name[-1]) row2 = int(possiblevar2.name[-2]) col2 = int(possiblevar2.name[-1]) row3 = 3 - row1 - row2 col3 = 3 - col1 - col2 if row1 == row2: # (row1, col3) is either 1 or -1, but don't know which. # know that (row1+1,col3) and (row1+2,col3) are zero though... checkexpr[2].append((Symbol('%s%d%d'%(possiblevar.name[:-2], (row2+1)%3, col3)), S.Zero)) checkexpr[2].append((Symbol('%s%d%d'%(possiblevar.name[:-2], (row1+2)%3, col3)), S.Zero)) checkexpr[2].append((Symbol('%s%d%d'%(possiblevar.name[:-2], row1, col3))**2, S.One)) # squared in the corner should always be 1 checkexpr[2].append((Symbol('%s%d%d'%(possiblevar.name[:-2], row1, col3))**3, Symbol('%s%d%d'%(possiblevar.name[:-2], row1, col3)))) # squared in the corner should always be 1 # furthermore can defer that the left over 4 values are [cos(ang), sin(ang), cos(ang), -sin(ang)] = abcd if row1 == 1: minrow = 0 maxrow = 2 else: minrow = (row1+1)%3 maxrow = (row1+2)%3 ra = Symbol('%s%d%d'%(possiblevar.name[:-2], minrow, col1)) rb = Symbol('%s%d%d'%(possiblevar.name[:-2], minrow, col2)) rc = Symbol('%s%d%d'%(possiblevar.name[:-2], maxrow, col1)) rd = Symbol('%s%d%d'%(possiblevar.name[:-2], maxrow, col2)) checkexpr[2].append((rb**2, S.One-ra**2)) checkexpr[2].append((rb**3, rb-rb*ra**2)) # need 3rd power since sympy cannot divide out the square checkexpr[2].append((rc**2, S.One-ra**2)) #checkexpr[2].append((rc, -rb)) # not true #checkexpr[2].append((rd, ra)) # not true elif col1 == col2: # (row3, col1) is either 1 or -1, but don't know which. # know that (row3,col1+1) and (row3,col1+2) are zero though... checkexpr[2].append((Symbol('%s%d%d'%(possiblevar.name[:-2], row3, (col1+1)%3)), S.Zero)) checkexpr[2].append((Symbol('%s%d%d'%(possiblevar.name[:-2], row3, (col1+2)%3)), S.Zero)) checkexpr[2].append((Symbol('%s%d%d'%(possiblevar.name[:-2], row3, col1))**2, S.One)) # squared in the corner should always be 1 checkexpr[2].append((Symbol('%s%d%d'%(possiblevar.name[:-2], row3, col1))**3, Symbol('%s%d%d'%(possiblevar.name[:-2], row3, col1)))) # squared in the corner should always be 1 # furthermore can defer that the left over 4 values are [cos(ang), sin(ang), cos(ang), -sin(ang)] = abcd if col1 == 1: mincol = 0 maxcol = 2 else: mincol = (col1+1)%3 maxcol = (col1+2)%3 ra = Symbol('%s%d%d'%(possiblevar.name[:-2], row1, mincol)) rb = Symbol('%s%d%d'%(possiblevar.name[:-2], row2, mincol)) rc = Symbol('%s%d%d'%(possiblevar.name[:-2], row1, maxcol)) rd = Symbol('%s%d%d'%(possiblevar.name[:-2], row2, maxcol)) checkexpr[2].append((rb**2, S.One-ra**2)) checkexpr[2].append((rb**3, rb-rb*ra**2)) # need 3rd power since sympy cannot divide out the square checkexpr[2].append((rc**2, S.One-ra**2)) #checkexpr[2].append((rc, -rb)) # not true #checkexpr[2].append((rd, ra)) # not true return preprocesssolutiontree+solutiontree+endbranchtree,usedvars def ConvertSinCosEquationToHalfTan(self, eq, convertvars): """converts all the sin/cos of variables to half-tangents. Returns two equations (poly, denominator) """ cossinvars = [] htvarsubs = [] htvars = [] htvarsubsinv = [] cossinsubs = [] for varsym in convertvars: var = self.Variable(varsym) cossinvars.append(var.cvar) cossinvars.append(var.svar) htvar = Symbol('ht%s'%varsym.name) htvarsubs += [(var.cvar,(1-htvar**2)/(1+htvar**2)),(var.svar,2*htvar/(1+htvar**2))] htvarsubsinv.append((htvar, (1-var.cvar)/var.svar)) htvars.append(htvar) cossinsubs.append((cos(varsym), var.cvar)) cossinsubs.append((sin(varsym), var.svar)) peq = Poly(eq.subs(cossinsubs),*cossinvars) maxdenom = [0]*len(convertvars) for monoms in peq.monoms(): for i in range(len(convertvars)): maxdenom[i] = max(maxdenom[i],monoms[2*i]+monoms[2*i+1]) eqnew = S.Zero for monoms,c in peq.terms(): term = c for i in range(len(convertvars)): # for cos num, denom = fraction(htvarsubs[2*i][1]) term *= num**monoms[2*i] # for sin num, denom = fraction(htvarsubs[2*i+1][1]) term *= num**monoms[2*i+1] # the denoms for sin/cos of the same joint variable are the same for i in range(len(convertvars)): denom = fraction(htvarsubs[2*i][1])[1] exp = maxdenom[i] - monoms[2*i] - monoms[2*i+1] if exp > 0: term *= denom**exp eqnew += term #newpeq = Poly(eqnew,htvars) othereq = S.One for i in range(len(convertvars)): othereq *= (1+htvars[i]**2)**maxdenom[i] return eqnew, othereq, htvarsubsinv def ConvertHalfTanEquationToSinCos(self, eq, convertvars): """converts all the sin/cos of variables to half-tangents. Returns two equations (poly, denominator) """ assert(0) cossinvars = [] htvarsubs = [] htvars = [] htvarsubsinv = [] for varsym in convertvars: var = self.Variable(varsym) cossinvars.append(var.cvar) cossinvars.append(var.svar) htvar = Symbol('ht%s'%varsym.name) htvarsubs += [(var.cvar,(1-htvar**2)/(1+htvar**2)),(var.svar,2*htvar/(1+htvar**2))] htvarsubsinv.append((htvar, (1-var.cvar)/var.svar)) htvars.append(htvar) peq = Poly(eq,*cossinvars) maxdenom = [0]*len(convertvars) for monoms in peq.monoms(): for i in range(len(convertvars)): maxdenom[i] = max(maxdenom[i],monoms[2*i]+monoms[2*i+1]) eqnew = S.Zero for monoms,c in peq.terms(): term = c for i in range(len(convertvars)): # for cos num, denom = fraction(htvarsubs[2*i][1]) term *= num**monoms[2*i] # for sin num, denom = fraction(htvarsubs[2*i+1][1]) term *= num**monoms[2*i+1] # the denoms for sin/cos of the same joint variable are the same for i in range(len(convertvars)): denom = fraction(htvarsubs[2*i][1])[1] exp = maxdenom[i] - monoms[2*i] - monoms[2*i+1] if exp > 0: term *= denom**exp eqnew += term #newpeq = Poly(eqnew,htvars) othereq = S.One for i in range(len(convertvars)): othereq *= (1+htvars[i]**2)**maxdenom[i] return eqnew, othereq, htvarsubsinv def solveKohliOsvatic(self,rawpolyeqs,solvejointvars,endbranchtree, AllEquationsExtra=None, currentcases=None, currentcasesubs=None): """Find a 16x16 matrix where the entries are linear with respect to the tan half-angle of one of the variables [Kohli1993]_. Takes in the 14 raghavan/roth equations. .. [Kohli1993] Dilip Kohli and M. Osvatic, "Inverse Kinematics of General 6R and 5R,P Serial Manipulators", Journal of Mechanical Design, Volume 115, Issue 4, Dec 1993. """ log.info('attempting kohli/osvatic general ik method') if len(rawpolyeqs[0][0].gens) < len(rawpolyeqs[0][1].gens): for peq in rawpolyeqs: peq[0],peq[1] = peq[1],peq[0] symbols = list(rawpolyeqs[0][0].gens) othersymbols = list(rawpolyeqs[0][1].gens) othersymbolsnames = [] for s in othersymbols: testeq = s.subs(self.invsubs) for solvejointvar in solvejointvars: if testeq.has(solvejointvar): othersymbolsnames.append(solvejointvar) break assert(len(othersymbols)==len(othersymbolsnames)) symbolsubs = [(symbols[i].subs(self.invsubs),symbols[i]) for i in range(len(symbols))] if len(symbols) != 6: raise self.CannotSolveError('Kohli/Osvatic method requires 3 unknown variables') # choose which leftvar can determine the singularity of the following equations! for i in range(0,6,2): eqs = [peq for peq in rawpolyeqs if peq[0].has(symbols[i],symbols[i+1])] if len(eqs) <= 8: break if len(eqs) > 8: raise self.CannotSolveError('need 8 or less equations of one variable, currently have %d'%len(eqs)) cvar = symbols[i] svar = symbols[i+1] tvar = Symbol('t'+cvar.name[1:]) symbols.remove(cvar) symbols.remove(svar) othereqs = [peq for peq in rawpolyeqs if not peq[0].has(cvar,svar)] polyeqs = [[eq[0].as_expr(),eq[1]] for eq in eqs] if len(polyeqs) < 8: raise self.CannotSolveError('solveKohliOsvatic: need 8 or more polyeqs') # solve the othereqs for symbols without the standalone symbols[2] and symbols[3] reducedeqs = [] othersymbolsnamesunique = list(set(othersymbolsnames)) # get the unique names for jother in range(len(othersymbolsnamesunique)): if not self.IsHinge(othersymbolsnamesunique[jother].name): continue othervar=self.Variable(othersymbolsnamesunique[jother]) cosmonom = [0]*len(othersymbols) cosmonom[othersymbols.index(othervar.cvar)] = 1 cosmonom = tuple(cosmonom) sinmonom = [0]*len(othersymbols) sinmonom[othersymbols.index(othervar.svar)] = 1 sinmonom = tuple(sinmonom) leftsideeqs = [] rightsideeqs = [] finaleqsymbols = symbols + [othervar.cvar,othervar.svar] for eq0,eq1 in othereqs: leftsideeq = Poly(eq1,*othersymbols) leftsideeqdict = leftsideeq.as_dict() rightsideeq = Poly(eq0,*finaleqsymbols) coscoeff = leftsideeqdict.get(cosmonom,S.Zero) if coscoeff != S.Zero: rightsideeq = rightsideeq - othervar.cvar*coscoeff leftsideeq = leftsideeq - othervar.cvar*coscoeff sincoeff = leftsideeqdict.get(sinmonom,S.Zero) if sincoeff != S.Zero: rightsideeq = rightsideeq - othervar.svar*sincoeff leftsideeq = leftsideeq - othervar.svar*sincoeff const = leftsideeq.TC() if const != S.Zero: rightsideeq = rightsideeq - const leftsideeq = leftsideeq - const # check that leftsideeq doesn't hold any terms with cosmonom and sinmonom? rightsideeqs.append(rightsideeq) leftsideeqs.append(leftsideeq) # number of symbols for kawada-hiro robot is 16 if len(othersymbols) > 2: reducedeqs = self.reduceBothSidesSymbolically(leftsideeqs,rightsideeqs,usesymbols=False,maxsymbols=18) for peq in reducedeqs: peq[0] = Poly(peq[0],*othersymbols) else: reducedeqs = [[left,right] for left,right in izip(leftsideeqs,rightsideeqs)] if len(reducedeqs) > 0: break if len(reducedeqs) == 0: raise self.CannotSolveError('KohliOsvatic method: could not reduce the equations') finaleqs = [] for peq0,eq1 in reducedeqs: if peq0 == S.Zero: finaleqs.append(Poly(eq1,*finaleqsymbols)) if len(finaleqs) >= 2: # perhaps can solve finaleqs as is? # transfer othersymbols[2*jother:(2+2*jother)] to the leftside try: leftsideeqs = [] rightsideeqs = [] for finaleq in finaleqs: peq=Poly(finaleq,*othersymbols[2*jother:(2+2*jother)]) leftsideeqs.append(peq.sub(peq.TC())) rightsideeqs.append(-peq.TC()) reducedeqs2 = self.reduceBothSidesSymbolically(leftsideeqs,rightsideeqs,usesymbols=False,maxsymbols=18) # find all the equations with left side = to zero usedvars = set() for symbol in symbols: usedvars.add(Symbol(symbol.name[1:])) AllEquations = [] for eq0, eq1 in reducedeqs2: if eq0 == S.Zero: AllEquations.append(eq1.subs(self.invsubs)) if len(AllEquations) > 0: otherjointtrees = [] tree = self.SolveAllEquations(AllEquations,curvars=list(usedvars),othersolvedvars=[],solsubs=self.freevarsubs,endbranchtree=[AST.SolverSequence([otherjointtrees])], canguessvars=False, currentcases=currentcases, currentcasesubs=currentcasesubs) log.info('first SolveAllEquations successful: %s',usedvars) # try: # # although things can be solved at this point, it yields a less optimal solution than if all variables were considered... # solsubs=list(self.freevarsubs) # for usedvar in usedvars: # solsubs += self.Variable(usedvar).subs # # solved, so substitute back into reducedeqs and see if anything new can be solved # otherusedvars = set() # for symbol in othersymbols: # otherusedvars.add(Symbol(symbol.name[1:])) # OtherAllEquations = [] # for peq0,eq1 in reducedeqs: # OtherAllEquations.append((peq0.as_expr()-eq1).subs(self.invsubs).expand()) # otherjointtrees += self.SolveAllEquations(OtherAllEquations,curvars=list(otherusedvars),othersolvedvars=list(usedvars),solsubs=solsubs,endbranchtree=endbranchtree) # return tree, list(usedvars)+list(otherusedvars) # except self.CannotSolveError: # still have the initial solution otherjointtrees += endbranchtree return tree, list(usedvars) except self.CannotSolveError as e: pass log.info('build final equations for symbols: %s',finaleqsymbols) neweqs=[] for i in range(0,8,2): p0 = Poly(polyeqs[i][0],cvar,svar) p0dict = p0.as_dict() p1 = Poly(polyeqs[i+1][0],cvar,svar) p1dict = p1.as_dict() r0 = polyeqs[i][1].as_expr() r1 = polyeqs[i+1][1].as_expr() if self.equal(p0dict.get((1,0),S.Zero),-p1dict.get((0,1),S.Zero)) and self.equal(p0dict.get((0,1),S.Zero),p1dict.get((1,0),S.Zero)): p0,p1 = p1,p0 p0dict,p1dict=p1dict,p0dict r0,r1 = r1,r0 if self.equal(p0dict.get((1,0),S.Zero),p1dict.get((0,1),S.Zero)) and self.equal(p0dict.get((0,1),S.Zero),-p1dict.get((1,0),S.Zero)): # p0+tvar*p1, p1-tvar*p0 # subs: tvar*svar + cvar = 1, svar-tvar*cvar=tvar neweqs.append([Poly(p0dict.get((1,0),S.Zero) + p0dict.get((0,1),S.Zero)*tvar + p0.TC() + tvar*p1.TC(),*symbols), Poly(r0+tvar*r1,*othersymbols)]) neweqs.append([Poly(p0dict.get((1,0),S.Zero)*tvar - p0dict.get((0,1),S.Zero) - p0.TC()*tvar + p1.TC(),*symbols), Poly(r1-tvar*r0,*othersymbols)]) if len(neweqs) != 8: raise self.CannotSolveError('coefficients of equations need to match! only got %d reduced equations'%len(neweqs)) for eq0,eq1 in neweqs: commondenom = Poly(S.One,*self.pvars) hasunknown = False for m,c in eq1.terms(): foundreq = [req[1] for req in reducedeqs if req[0].monoms()[0] == m] if len(foundreq) > 0: n,d = fraction(foundreq[0]) commondenom = Poly(lcm(commondenom,d),*self.pvars) else: if m[2*(1-jother)] > 0 or m[2*(1-jother)+1] > 0: # perhaps there's a way to combine what's in reducedeqs? log.warn('unknown %s',m) hasunknown = True if hasunknown: continue commondenom = self.removecommonexprs(commondenom.as_expr(),onlygcd=True,onlynumbers=True) finaleq = eq0.as_expr()*commondenom for m,c in eq1.terms(): foundreq = [req[1] for req in reducedeqs if req[0].monoms()[0] == m] if len(foundreq) > 0: finaleq = finaleq - c*simplify(foundreq[0]*commondenom) else: finaleq = finaleq - Poly.from_dict({m:c*commondenom},*eq1.gens).as_expr() finaleqs.append(Poly(finaleq.expand(),*finaleqsymbols)) # finally do the half angle substitution with symbols # set: # j=othersymbols[2]*(1+dummys[0]**2)*(1+dummys[1]**2) # k=othersymbols[3]*(1+dummys[0]**2)*(1+dummys[1]**2) dummys = [] dummysubs = [] dummysubs2 = [] dummyvars = [] usedvars = [] dummys.append(tvar) dummyvars.append((tvar,tan(0.5*Symbol(tvar.name[1:])))) usedvars.append(Symbol(cvar.name[1:])) dummysubs2.append((usedvars[-1],2*atan(tvar))) dummysubs += [(cvar,(1-tvar**2)/(1+tvar**2)),(svar,2*tvar/(1+tvar**2))] for i in range(0,len(symbols),2): dummy = Symbol('ht%s'%symbols[i].name[1:]) # [0] - cos, [1] - sin dummys.append(dummy) dummysubs += [(symbols[i],(1-dummy**2)/(1+dummy**2)),(symbols[i+1],2*dummy/(1+dummy**2))] var = symbols[i].subs(self.invsubs).args[0] dummyvars.append((dummy,tan(0.5*var))) dummysubs2.append((var,2*atan(dummy))) if not var in usedvars: usedvars.append(var) commonmult = (1+dummys[1]**2)*(1+dummys[2]**2) usedvars.append(Symbol(othersymbols[2*jother].name[1:])) dummyj = Symbol('dummyj') dummyk = Symbol('dummyk') dummyjk = Symbol('dummyjk') dummys.append(dummyj) dummyvars.append((dummyj,othersymbols[2*jother]*(1+dummyvars[1][1]**2)*(1+dummyvars[2][1]**2))) dummysubs.append((othersymbols[2*jother],cos(dummyjk))) dummys.append(dummyk) dummyvars.append((dummyk,othersymbols[1+2*jother]*(1+dummyvars[1][1]**2)*(1+dummyvars[2][1]**2))) dummysubs.append((othersymbols[1+2*jother],sin(dummyjk))) dummysubs2.append((usedvars[-1],dummyjk)) newreducedeqs = [] for peq in finaleqs: eqnew = S.Zero for monoms,c in peq.terms(): term = S.One for i in range(4): term *= dummysubs[i+2][1]**monoms[i] if monoms[4] == 1: eqnew += c * dummyj elif monoms[5] == 1: eqnew += c * dummyk else: eqnew += c*simplify(term*commonmult) newreducedeqs.append(Poly(eqnew,*dummys)) exportcoeffeqs = None for ileftvar in range(len(dummys)): leftvar = dummys[ileftvar] try: exportcoeffeqs,exportmonoms = self.solveDialytically(newreducedeqs,ileftvar,getsubs=None) break except self.CannotSolveError as e: log.warn('failed with leftvar %s: %s',leftvar,e) if exportcoeffeqs is None: raise self.CannotSolveError('failed to solve dialytically') if ileftvar > 0: raise self.CannotSolveError('solving equations dialytically succeeded with var index %d, unfortunately code generation supports only index 0'%ileftvar) coupledsolution = AST.SolverCoeffFunction(jointnames=[v.name for v in usedvars],jointeval=[v[1] for v in dummysubs2],jointevalcos=[dummysubs[2*i][1] for i in range(len(usedvars))],jointevalsin=[dummysubs[2*i+1][1] for i in range(len(usedvars))],isHinges=[self.IsHinge(v.name) for v in usedvars],exportvar=dummys[0:3]+[dummyjk],exportcoeffeqs=exportcoeffeqs,exportfnname='solvedialyticpoly16lep',rootmaxdim=16) self.usinglapack = True return [coupledsolution]+endbranchtree,usedvars def solveDialytically(self,dialyticeqs,ileftvar,returnmatrix=False,getsubs=None): """ Return the coefficients to solve equations dialytically (Salmon 1885) leaving out variable index ileftvar. Extract the coefficients of 1, leftvar**1, leftvar**2, ... of every equation every len(dialyticeqs)*len(monoms) coefficients specify one degree of all the equations (order of monoms is specified in exportmonomorder there should be len(dialyticeqs)*len(monoms)*maxdegree coefficients Method also checks if the equations are linearly dependent """ self._CheckPreemptFn(progress=0.12) if len(dialyticeqs) == 0: raise self.CannotSolveError('solveDialytically given zero equations') allmonoms = set() origmonoms = set() maxdegree = 0 leftvar = dialyticeqs[0].gens[ileftvar] extradialyticeqs = [] for peq in dialyticeqs: if sum(peq.degree_list()) == 0: log.warn('solveDialytically: polynomial %s degree is 0',peq) continue for m in peq.monoms(): mlist = list(m) maxdegree=max(maxdegree,mlist.pop(ileftvar)) allmonoms.add(tuple(mlist)) origmonoms.add(tuple(mlist)) mlist[0] += 1 allmonoms.add(tuple(mlist)) # check if any monoms are not expressed in this poly, and if so, add another poly with the monom multiplied, will this give bad solutions? for igen in range(len(peq.gens)): if all([m[igen]==0 for m in peq.monoms()]): log.debug('adding extra equation multiplied by %s', peq.gens[igen]) extradialyticeqs.append(peq*peq.gens[igen]) # multiply by peq.gens[igen] for m in peq.monoms(): mlist = list(m) mlist[igen] += 1 maxdegree=max(maxdegree,mlist.pop(ileftvar)) allmonoms.add(tuple(mlist)) origmonoms.add(tuple(mlist)) mlist[0] += 1 allmonoms.add(tuple(mlist)) dialyticeqs = list(dialyticeqs) + extradialyticeqs # dialyticeqs could be a tuple allmonoms = list(allmonoms) allmonoms.sort() origmonoms = list(origmonoms) origmonoms.sort() if len(origmonoms) == 0 or len(allmonoms) == 0: raise self.CannotSolveError('solveDialytically has no equations') if len(allmonoms)<2*len(dialyticeqs): log.warn('solveDialytically equations %d > %d, should be equal...', 2*len(dialyticeqs),len(allmonoms)) # TODO not sure how to select the equations N = len(allmonoms)//2 dialyticeqs = dialyticeqs[:N] if len(allmonoms) == 0 or len(allmonoms)>2*len(dialyticeqs): raise self.CannotSolveError('solveDialytically: more unknowns than equations %d>%d'%(len(allmonoms), 2*len(dialyticeqs))) Mall = [zeros((2*len(dialyticeqs),len(allmonoms))) for i in range(maxdegree+1)] Mallindices = [-ones((2*len(dialyticeqs),len(allmonoms))) for i in range(maxdegree+1)] exportcoeffeqs = [S.Zero]*(len(dialyticeqs)*len(origmonoms)*(maxdegree+1)) for ipeq,peq in enumerate(dialyticeqs): for m,c in peq.terms(): mlist = list(m) degree=mlist.pop(ileftvar) exportindex = degree*len(origmonoms)*len(dialyticeqs) + len(origmonoms)*ipeq+origmonoms.index(tuple(mlist)) assert(exportcoeffeqs[exportindex] == S.Zero) exportcoeffeqs[exportindex] = c Mall[degree][len(dialyticeqs)+ipeq,allmonoms.index(tuple(mlist))] = c Mallindices[degree][len(dialyticeqs)+ipeq,allmonoms.index(tuple(mlist))] = exportindex mlist[0] += 1 Mall[degree][ipeq,allmonoms.index(tuple(mlist))] = c Mallindices[degree][ipeq,allmonoms.index(tuple(mlist))] = exportindex # check if any monoms are not expressed in this poly, and if so, add another poly with the monom multiplied, will this give bad solutions? for igen in range(len(peq.gens)): if all([m[igen]==0 for m in peq.monoms()]): for m,c in peq.terms(): mlist = list(m) mlist[igen] += 1 degree=mlist.pop(ileftvar) if tuple(mlist) not in origmonoms: raise self.CannotSolveError('equations too simple, monom %r is not in %r'%(mlist, origmonoms)) exportindex = degree*len(origmonoms)*len(dialyticeqs) + len(origmonoms)*ipeq+origmonoms.index(tuple(mlist)) assert(exportcoeffeqs[exportindex] == S.Zero) exportcoeffeqs[exportindex] = c Mall[degree][len(dialyticeqs)+ipeq,allmonoms.index(tuple(mlist))] = c Mallindices[degree][len(dialyticeqs)+ipeq,allmonoms.index(tuple(mlist))] = exportindex mlist[0] += 1 Mall[degree][ipeq,allmonoms.index(tuple(mlist))] = c Mallindices[degree][ipeq,allmonoms.index(tuple(mlist))] = exportindex # have to check that the determinant is not zero for several values of ileftvar! It is very common that # some equations are linearly dependent and not solvable through this method. if self.testconsistentvalues is not None: linearlyindependent = False for itest,subs in enumerate(self.testconsistentvalues): if getsubs is not None: # have to explicitly evaluate since testsubs can be very complex subsvals = [(s,v.evalf()) for s,v in subs] try: subs = subsvals+getsubs(subsvals) except self.CannotSolveError as e: # getsubs failed (sometimes it requires solving inverse matrix), so go to next set continue # have to sub at least twice with the global symbols A = Mall[maxdegree].subs(subs) for i in range(A.shape[0]): for j in range(A.shape[1]): A[i,j] = self._SubstituteGlobalSymbols(A[i,j]).subs(subs).evalf() eps = 10**-(self.precision-3) try: Anumpy = numpy.array(numpy.array(A), numpy.float64) except ValueError as e: log.warn(u'could not convert to numpy array: %s', e) continue if numpy.isnan(numpy.sum(Anumpy)): log.info('A has NaNs') break eigenvals = numpy.linalg.eigvals(Anumpy) if all([Abs(f) > eps for f in eigenvals]): try: Ainv = A.inv(method='LU') except ValueError as e: log.error('error when taking inverse: %s', e) continue B = Ainv*Mall[1].subs(subs) for i in range(B.shape[0]): for j in range(B.shape[1]): B[i,j] = self._SubstituteGlobalSymbols(B[i,j]).subs(subs).evalf() C = Ainv*Mall[0].subs(subs).evalf() for i in range(C.shape[0]): for j in range(C.shape[1]): C[i,j] = self._SubstituteGlobalSymbols(C[i,j]).subs(subs).evalf() A2 = zeros((B.shape[0],B.shape[0]*2)) for i in range(B.shape[0]): A2[i,B.shape[0]+i] = S.One A2=A2.col_join((-C).row_join(-B)) eigenvals2,eigenvecs2 = numpy.linalg.eig(numpy.array(numpy.array(A2),numpy.float64)) # check if solutions can actually be extracted # find all the zero eigenvalues roots = [] numrepeating = 0 for ieig,eigenvalue in enumerate(eigenvals2): if abs(numpy.imag(eigenvalue)) < 1e-12: if abs(numpy.real(eigenvalue)) > 1: ev = eigenvecs2[A.shape[0]:,ieig] else: ev = eigenvecs2[:A.shape[0],ieig] if abs(ev[0]) < 1e-14: continue br = ev[1:] / ev[0] dists = abs(numpy.array(roots) - numpy.real(eigenvalue)) if any(dists<1e-7): numrepeating += 1 roots.append(numpy.real(eigenvalue)) if numrepeating > 0: log.info('found %d repeating roots in solveDialytically matrix: %s',numrepeating,roots) # should go on even if there's repeating roots? continue Atotal = None for idegree in range(maxdegree+1): Adegree = Mall[idegree].subs(subs) for i in range(Adegree.shape[0]): for j in range(Adegree.shape[1]): Adegree[i,j] = self._SubstituteGlobalSymbols(Adegree[i,j]).subs(subs).evalf() if Atotal is None: Atotal = Adegree else: Atotal += Adegree*leftvar**idegree # make sure the determinant of Atotal is not-zero for at least several solutions leftvarvalue = leftvar.subs(subs).evalf() hasnonzerodet = False for testvalue in [-10*S.One, -S.One,-0.5*S.One, 0.5*S.One, S.One, 10*S.One]: Atotal2 = Atotal.subs(leftvar,leftvarvalue+testvalue).evalf() detvalue = Atotal2.det() if abs(detvalue) > 1e-10: hasnonzerodet = True if not hasnonzerodet: log.warn('has zero det, so failed') else: linearlyindependent = True break else: log.info('not all abs(eigenvalues) > %e. min is %e', eps, min([Abs(f) for f in eigenvals if Abs(f) < eps])) if not linearlyindependent: raise self.CannotSolveError('equations are not linearly independent') if returnmatrix: return Mall,allmonoms return exportcoeffeqs,origmonoms def SubstituteGinacEquations(self,dictequations, valuesubs, localsymbolmap): gvaluesubs = [] for var, value in valuesubs: if value != oo: if var.name in localsymbolmap: gvaluesubs.append(localsymbolmap[var.name] == GinacUtils.ConvertToGinac(value,localsymbolmap)) retvalues = [] for var, value in dictequations: newvalue = value.subs(gvaluesubs).evalf() if var.name in localsymbolmap: gvaluesubs.append(localsymbolmap[var.name]==newvalue) else: log.warn('%s not in map',var) retvalues.append((var,newvalue)) return retvalues def SimplifyTransformPoly(self,peq): """simplifies the coefficients of the polynomial with simplifyTransform and returns the new polynomial """ if peq == S.Zero: return peq return peq.termwise(lambda m,c: self.SimplifyTransform(c)) def SimplifyTransform(self,eq,othervars=None): """Attemps to simplify an equation given that variables from a rotation matrix have been used. There are 12 constraints that are tested: - lengths of rows and colums are 1 - dot products of combinations of rows/columns are 0 - cross products of combinations of rows/columns yield the left over row/column :param othervars: optional list of the unknown variables inside the equations. Help simplify depending on the terms of these variables """ if othervars is not None: peq = Poly(eq,*othervars) if peq == S.Zero: return S.Zero peqnew = peq.termwise(lambda m,c: self.SimplifyTransform(c)) return peqnew.as_expr() # there can be gobal substitutions like pz=0. get all of them that do not start with gconst transformsubstitutions = [(var,value) for var, value in self.globalsymbols if var.is_Symbol and not var.name.startswith('gconst')] if self._iktype == 'translationdirection5d': # since includes direction, only try to simplify r00**2+r01**2+r02**2=1 simpiter = 0 origeq = eq # first simplify just rotations since they don't add any new variables changed = True while changed and eq.has(*self._rotsymbols): #log.info('simpiter=%d, complexity=%d', simpiter, self.codeComplexity(eq.as_expr() if isinstance(eq,Poly) else eq)) simpiter += 1 changed = False neweq = self._SimplifyRotationNorm(eq, self._rotnormgroups[0:1]) if neweq is not None: eq2 = self._SubstituteGlobalSymbols(neweq, transformsubstitutions) if not self.equal(eq,eq2): eq = eq2 changed = True if isinstance(eq, Poly): eq = eq.as_expr() return eq if self._iktype != 'transform6d': return eq # TODO if there is a divide by self._rotsymbols, then cannot proceed since cannot make Polynomials from them if fraction(eq)[1].has(*self._rotsymbols): log.info('equation %s has rot symbols in denom, so skipping...', eq) return eq if eq.is_Add: for arg in eq.args: if fraction(arg)[1].has(*self._rotsymbols): log.info('equation %s has rot symbols in denom, so skipping...', arg) return eq simpiter = 0 origeq = eq # first simplify just rotations since they don't add any new variables changed = True while changed and eq.has(*self._rotsymbols): #log.info('simpiter=%d, complexity=%d', simpiter, self.codeComplexity(eq.as_expr() if isinstance(eq,Poly) else eq)) simpiter += 1 changed = False neweq = self._SimplifyRotationNorm(eq, self._rotnormgroups) if neweq is not None: eq2 = self._SubstituteGlobalSymbols(neweq, transformsubstitutions) if not self.equal(eq,eq2): eq = eq2 changed = True neweq = self._SimplifyRotationDot(eq, self._rotsymbols, self._rotdotgroups) if neweq is not None: eq2 = self._SubstituteGlobalSymbols(neweq, transformsubstitutions) if not self.equal(eq,eq2): eq = eq2 changed = True neweq = self._SimplifyRotationCross(eq, self._rotsymbols, self._rotcrossgroups) if neweq is not None: eq2 = self._SubstituteGlobalSymbols(neweq, transformsubstitutions) if not self.equal(eq,eq2): eq = eq2 changed = True # check if full 3D position is available if self.pp is not None: changed = True while changed and eq.has(*self._rotpossymbols): changed = False neweq = self._SimplifyRotationNorm(eq, self._rotposnormgroups) if neweq is not None: eq2 = self._SubstituteGlobalSymbols(neweq, transformsubstitutions) if not self.equal(eq,eq2): eq = eq2 changed = True neweq = self._SimplifyRotationDot(eq, self._rotpossymbols, self._rotposdotgroups) if neweq is not None: eq2 = self._SubstituteGlobalSymbols(neweq, transformsubstitutions) if not self.equal(eq,eq2): eq = eq2 changed = True neweq = self._SimplifyRotationCross(eq, self._rotpossymbols, self._rotposcrossgroups) if neweq is not None: eq2 = self._SubstituteGlobalSymbols(neweq, transformsubstitutions) if not self.equal(eq,eq2): eq = eq2 changed = True if isinstance(eq, Poly): eq = eq.as_expr() #log.info("simplify eq:\n%r\n->new eq:\n%r", origeq, eq) return eq def _SimplifyRotationNorm(self, eq, groups): """simplify equation using self._rotnormgroups """ neweq = None for group in groups: try: # not sure about this thresh if self.codeComplexity(eq) > 300: log.warn(u'equation too complex to simplify for rot norm: %s', eq) continue # need to do 1234*group[3] hack in order to get the Poly domain to recognize group[3] (sympy 0.7.1) p = Poly(eq+1234*group[3],group[0],group[1],group[2]) p -= Poly(1234*group[3], *p.gens, domain=p.domain) except (PolynomialError, CoercionFailed, ZeroDivisionError) as e: continue changed = False listterms = list(p.terms()) if len(listterms) == 1: continue usedindices = set() for index0, index1 in combinations(range(len(listterms)),2): if index0 in usedindices or index1 in usedindices: continue m0,c0 = listterms[index0] m1,c1 = listterms[index1] if self.equal(c0,c1): for i,j,k in [(0,1,2),(0,2,1),(1,2,0)]: if m0[k]==m1[k]: if m0[i] >= 2 and m1[j] >= 2 and m0[i] == m1[i]+2 and m0[j]+2 == m1[j]: p = p + Poly(c0*(group[3]-group[0]**2-group[1]**2-group[2]**2)*(group[k]**m0[k])*(group[i]**(m0[i]-2))*(group[j]**(m0[j])), group[0],group[1],group[2]) changed = True if m1[i] >= 2 and m0[j] >= 2 and m1[i] == m0[i]+2 and m1[j]+2 == m0[j]: p = p + Poly(c0*(group[3]-group[0]**2-group[1]**2-group[2]**2)*(group[k]**m1[k])*(group[i]**(m1[i]-2))*(group[j]**(m1[j])), group[0],group[1],group[2]) changed = True #if ((m0[i] == 2 and m1[j] == 2) or (m0[j]==2 and m1[i]==2)) and m0[k]==m1[k]: #p = p + c0*(group[3]-group[0]**2-group[1]**2-group[2]**2)*group[k]**(m0[k])*group[i]**(m0[i] if changed: neweq = p#.as_expr() eq = neweq changed = True usedindices.add(index0) usedindices.add(index1) break elif self.equal(c0,-c1): # x0**4 - x1**4 = (x0**2-x1**2)*(x0**2+x1**2) = (x0**2 - x1**2)*(x3-x2**2) for i,j,k in [(0,1,2),(0,2,1),(1,2,0)]: if m0[k]==m1[k]: if m0[i] == 4 and m1[j] == 4: p = p + Poly(c0*group[k]**m0[k]*((group[3]-group[k]**2)*(group[i]**2-group[j]**2) - group[i]**4 + group[j]**4), group[0],group[1],group[2]) changed = True if m0[j]==4 and m1[i]==4: p = p + Poly(c0*group[k]**m0[k]*((group[3]-group[k]**2)*(group[j]**2-group[i]**2) - group[j]**4 + group[i]**4), group[0],group[1],group[2]) changed = True if changed: neweq = p#.as_expr() eq = neweq changed = True usedindices.add(index0) usedindices.add(index1) break return neweq def _SimplifyRotationDot(self, eq, symbols, groups): """check for dot products between rows and columns """ try: p = Poly(eq,*symbols) except (PolynomialError, CoercionFailed) as e: return None changed = False listterms = list(p.terms()) usedindices = set() for dg in groups: for i,j,k in [(0,1,2),(0,2,1),(1,2,0)]: for index0, index1 in combinations(range(len(listterms)),2): if index0 in usedindices or index1 in usedindices: continue if self.equal(listterms[index0][1], listterms[index1][1]): for (m0,c0),(m1,c1) in [[listterms[index0], listterms[index1]], [listterms[index1], listterms[index0]]]: if m0[dg[i][0]] == 1 and m0[dg[i][1]] == 1 and m1[dg[j][0]] == 1 and m1[dg[j][1]] == 1: # make sure the left over terms are also the same m0l = list(m0); m0l[dg[i][0]] = 0; m0l[dg[i][1]] = 0 m1l = list(m1); m1l[dg[j][0]] = 0; m1l[dg[j][1]] = 0 if tuple(m0l) == tuple(m1l): m2 = list(m0l); m2[dg[k][0]] += 1; m2[dg[k][1]] += 1 # there is a bug in sympy v0.6.7 polynomial adding here! p = p.sub(Poly.from_dict({m0:c0},*p.gens)).sub(Poly.from_dict({m1:c1},*p.gens)).sub(Poly.from_dict({tuple(m2):c0},*p.gens)) if dg[3] != S.Zero: p = p.add(Poly(dg[3],*p.gens)*Poly.from_dict({tuple(m0l):c0},*p.gens)) changed = True usedindices.add(index0) usedindices.add(index1) break return p if changed else None def _SimplifyRotationCross(self, eq, symbols, groups): """simplify rotations using cross products """ changed = False try: p = Poly(eq,*symbols) except (PolynomialError, CoercionFailed) as e: return None listterms = list(p.terms()) usedindices = set() for cg in groups: for index0, index1 in combinations(range(len(listterms)),2): if index0 in usedindices or index1 in usedindices: continue if self.equal(listterms[index0][1],-listterms[index1][1]): for (m0,c0),(m1,c1) in [[listterms[index0], listterms[index1]],[listterms[index1], listterms[index0]]]: if m0[cg[0][0]] == 1 and m0[cg[0][1]] == 1 and m1[cg[1][0]] == 1 and m1[cg[1][1]] == 1: # make sure the left over terms are also the same m0l = list(m0); m0l[cg[0][0]] = 0; m0l[cg[0][1]] = 0 m1l = list(m1); m1l[cg[1][0]] = 0; m1l[cg[1][1]] = 0 if tuple(m0l) == tuple(m1l): m2 = m0l; m2[cg[2]] += 1 # there is a bug in sympy polynomial caching here! (0.6.7) p = p.sub(Poly.from_dict({m0:c0},*p.gens)).sub(Poly.from_dict({m1:c1},*p.gens)).add(Poly.from_dict({tuple(m2):c0},*p.gens)) changed = True usedindices.add(index0) usedindices.add(index1) break return p if changed else None def CheckExpressionUnique(self, exprs, expr, checknegative=True, removecommoncoeff=False): """checks if expr is inside exprs. :param checknegative: if True, then also check if -expr is inside exprs :param removecommoncoeff: if True, removes any coefficients so that self.CheckExpressionUnique can work """ if removecommoncoeff: expr = self.removecommonexprs(expr) for exprtest in exprs: if expr.is_Function != exprtest.is_Function: continue if expr.is_Function: if not expr.is_Function: return False if exprtest.func == sign: # infinite loop for some reason if checking for this return False if expr.func == sign: # infinite loop for some reason if checking for this return False if self.equal(expr,exprtest): return False if checknegative: for exprtest in exprs: if expr.is_Function != exprtest.is_Function: continue if expr.is_Function: if not expr.is_Function: return False if exprtest.func == sign: # infinite loop for some reason if checking for this return False if expr.func == sign: # infinite loop for some reason if checking for this return False if self.equal(-expr,exprtest): return False return True def getCommonExpression(self, exprs, expr): for i,exprtest in enumerate(exprs): if self.equal(expr,exprtest): return i return None def verifyAllEquations(self,AllEquations,unsolvedvars, solsubs, tree=None): extrazerochecks=[] for i in range(len(AllEquations)): expr = AllEquations[i] if not self.isValidSolution(expr): raise self.CannotSolveError('verifyAllEquations: equation is not valid: %s'%(str(expr))) if not expr.has(*unsolvedvars) and self.CheckExpressionUnique(extrazerochecks,expr): extrazerochecks.append(self.removecommonexprs(expr.subs(solsubs).evalf(),onlygcd=False,onlynumbers=True)) if len(extrazerochecks) > 0: return [AST.SolverCheckZeros('verify',extrazerochecks,tree,[AST.SolverBreak('verifyAllEquations')],anycondition=False)] return tree def PropagateSolvedConstants(self, AllEquations, othersolvedvars, unknownvars, constantSymbols=None): """ Sometimes equations can be like "npz", or "pp-1", which means npz=0 and pp=1. Check for these constraints and apply them to the rest of the equations Return a new set of equations :param constantSymbols: the variables to try to propagage, if None will use self.pvars """ if constantSymbols is not None: constantSymbols = list(constantSymbols) else: constantSymbols = list(self.pvars) for othersolvedvar in othersolvedvars: constantSymbols.append(othersolvedvar) if self.IsHinge(othersolvedvar.name): constantSymbols.append(cos(othersolvedvar)) constantSymbols.append(sin(othersolvedvar)) newsubsdict = {} for eq in AllEquations: if not eq.has(*unknownvars) and eq.has(*constantSymbols): try: reducedeq = self.SimplifyTransform(eq) for constantSymbol in constantSymbols: if eq.has(constantSymbol): try: peq = Poly(eq,constantSymbol) if peq.degree(0) == 1: # equation is only degree 1 in the variable, and doesn't have any solvevars multiplied with it newsolution = solve(peq,constantSymbol)[0] if constantSymbol in newsubsdict: if self.codeComplexity(newsolution) < self.codeComplexity(newsubsdict[constantSymbol]): newsubsdict[constantSymbol] = newsolution else: newsubsdict[constantSymbol] = newsolution except PolynomialError: pass except PolynomialError as e: # expected from simplifyTransform if eq is too complex pass # first substitute everything that doesn't have othersolvedvar or unknownvars numberSubstitutions = [] otherSubstitutions = [] for var, value in newsubsdict.items(): if not value.has(*constantSymbols): numberSubstitutions.append((var,value)) else: otherSubstitutions.append((var,value)) NewEquations = [] for ieq, eq in enumerate(AllEquations): if 1:#not eq.has(*unknownvars): neweq = eq.subs(numberSubstitutions).expand() if neweq != S.Zero: # don't expand here since otherSubstitutions could make it very complicated neweq2 = neweq.subs(otherSubstitutions) if self.codeComplexity(neweq2) < self.codeComplexity(neweq)*2: neweq2 = neweq2.expand() if self.codeComplexity(neweq2) < self.codeComplexity(neweq) and neweq2 != S.Zero: NewEquations.append(neweq2) else: NewEquations.append(neweq) else: NewEquations.append(neweq) else: NewEquations.append(eq) return NewEquations def SolveAllEquations(self,AllEquations,curvars,othersolvedvars,solsubs,endbranchtree,currentcases=None,unknownvars=None, currentcasesubs=None, canguessvars=True): """ :param canguessvars: if True, can guess the variables given internal conditions are satisified """ self._CheckPreemptFn(progress=0.15+(0.3-0.3*100/(self._scopecounter+100))) # go from 0.15 - 0.45. usually scope counters go to several hundred if len(curvars) == 0: return endbranchtree if unknownvars is None: unknownvars = [] self._scopecounter+=1 scopecounter = int(self._scopecounter) log.info('depth=%d c=%d, %s %s: cases=%r', len(currentcases) if currentcases is not None else 0, self._scopecounter, othersolvedvars,curvars, currentcases) solsubs = solsubs[:] freevarinvsubs = [(f[1],f[0]) for f in self.freevarsubs] solinvsubs = [(f[1],f[0]) for f in solsubs] # single variable solutions solutions = [] for curvar in curvars: othervars = unknownvars+[var for var in curvars if var != curvar] curvarsym = self.Variable(curvar) raweqns = [] for e in AllEquations: if (len(othervars) == 0 or not e.has(*othervars)) and e.has(curvar,curvarsym.htvar,curvarsym.cvar,curvarsym.svar): eq = e.subs(self.freevarsubs+solsubs) if self.CheckExpressionUnique(raweqns,eq): raweqns.append(eq) if len(raweqns) > 0: try: rawsolutions=self.solveSingleVariable(self.sortComplexity(raweqns),curvar,othersolvedvars, unknownvars=curvars+unknownvars) for solution in rawsolutions: self.ComputeSolutionComplexity(solution,othersolvedvars,curvars) if solution.numsolutions()>0: solutions.append((solution,curvar)) else: log.warn('solution did not have any equations') except self.CannotSolveError: pass # only return here if a solution was found that perfectly determines the unknown # otherwise, the pairwise solver could come up with something.. # There is still a problem with this: (bertold robot) # Sometimes an equation like atan2(y,x) evaluates to atan2(0,0) during runtime. # This cannot be known at compile time, so the equation is selected and any other possibilities are rejected. # In the bertold robot case, the next possibility is a pair-wise solution involving two variables if any([s[0].numsolutions()==1 for s in solutions]): return self.AddSolution(solutions,AllEquations,curvars,othersolvedvars,solsubs,endbranchtree,currentcases=currentcases, currentcasesubs=currentcasesubs, unknownvars=unknownvars) curvarsubssol = [] for var0,var1 in combinations(curvars,2): othervars = unknownvars+[var for var in curvars if var != var0 and var != var1] raweqns = [] complexity = 0 for e in AllEquations: if (len(othervars) == 0 or not e.has(*othervars)) and e.has(var0,var1): eq = e.subs(self.freevarsubs+solsubs) if self.CheckExpressionUnique(raweqns,eq): raweqns.append(eq) complexity += self.codeComplexity(eq) if len(raweqns) > 1: curvarsubssol.append((var0,var1,raweqns,complexity)) curvarsubssol.sort(key=lambda e: e[3]) if len(curvars) == 2 and self.IsHinge(curvars[0].name) and self.IsHinge(curvars[1].name) and len(curvarsubssol) > 0: # there's only two variables left, it might be the case that the axes are aligning and the two variables are dependent on each other # note that the axes's anchors also have to be along the direction! var0,var1,raweqns,complexity = curvarsubssol[0] dummyvar = Symbol('dummy') dummyvalue = var0 + var1 NewEquations = [] NewEquationsAll = [] hasExtraConstraints = False for eq in raweqns: neweq = self.trigsimp(eq.subs(var0,dummyvar-var1).expand(trig=True),curvars) eq = neweq.subs(self.freevarsubs+solsubs) if self.CheckExpressionUnique(NewEquationsAll,eq): NewEquationsAll.append(eq) if neweq.has(dummyvar): if neweq.has(*(othervars+curvars)): hasExtraConstraints = True #break # don't know why breaking here... sometimes equations can be too complex but that doesn't mean variables are not dependent else: eq = neweq.subs(self.freevarsubs+solsubs) if self.CheckExpressionUnique(NewEquations,eq): NewEquations.append(eq) if len(NewEquations) < 2 and hasExtraConstraints: # try subtracting NewEquations = [] NewEquationsAll = [] hasExtraConstraints = False dummyvalue = var0 - var1 for eq in raweqns: neweq = self.trigsimp(eq.subs(var0,dummyvar+var1).expand(trig=True),curvars) eq = neweq.subs(self.freevarsubs+solsubs) if self.CheckExpressionUnique(NewEquationsAll,eq): NewEquationsAll.append(eq) if neweq.has(dummyvar): if neweq.has(*(othervars+curvars)): hasExtraConstraints = True #break # don't know why breaking here... sometimes equations can be too complex but that doesn't mean variables are not dependent else: eq = neweq.subs(self.freevarsubs+solsubs) if self.CheckExpressionUnique(NewEquations,eq): NewEquations.append(eq) if len(NewEquations) >= 2: dummysolutions = [] try: rawsolutions=self.solveSingleVariable(NewEquations,dummyvar,othersolvedvars, unknownvars=curvars+unknownvars) for solution in rawsolutions: self.ComputeSolutionComplexity(solution,othersolvedvars,curvars) dummysolutions.append(solution) except self.CannotSolveError: pass if any([s.numsolutions()==1 for s in dummysolutions]): # two axes are aligning, so modify the solutions to reflect the original variables and add a free variable log.info('found two aligning axes %s: %r',dummyvalue, NewEquations) solutions = [] for dummysolution in dummysolutions: if dummysolution.numsolutions() != 1: continue if dummysolution.jointevalsin is not None or dummysolution.jointevalcos is not None: log.warn('dummy solution should not have sin/cos parts!') sindummyvarsols = [] cosdummyvarsols = [] for eq in NewEquations: sols = solve(eq, sin(dummyvar)) sindummyvarsols += sols sols = solve(eq, cos(dummyvar)) cosdummyvarsols += sols # double check with NewEquationsAll that everything evluates to 0 newsubs = [(value, sin(dummyvar)) for value in sindummyvarsols] + [(value, cos(dummyvar)) for value in cosdummyvarsols] + [(-value, -sin(dummyvar)) for value in sindummyvarsols] + [(-value, -cos(dummyvar)) for value in cosdummyvarsols] allzeros = True for eq in NewEquationsAll: if trigsimp(eq.subs(newsubs)) != S.Zero: allzeros = False break if allzeros: solution=AST.SolverSolution(curvars[0].name, isHinge=self.IsHinge(curvars[0].name)) solution.jointeval = [dummysolution.jointeval[0] - dummyvalue + curvars[0]] self.ComputeSolutionComplexity(solution,othersolvedvars,curvars) solutions.append((solution,curvars[0])) else: log.warn('not all equations zero, so %s vars are not collinear', curvars) if len(solutions) > 0: tree = self.AddSolution(solutions,raweqns,curvars[0:1],othersolvedvars+curvars[1:2],solsubs+self.Variable(curvars[1]).subs,endbranchtree,currentcases=currentcases, currentcasesubs=currentcasesubs, unknownvars=unknownvars) if tree is not None: return [AST.SolverFreeParameter(curvars[1].name, tree)] else: log.warn('almost found two axes but num solutions was: %r', [s.numsolutions()==1 for s in dummysolutions]) for var0,var1,raweqns,complexity in curvarsubssol: try: rawsolutions=self.SolvePrismaticHingePairVariables(raweqns,var0,var1,othersolvedvars,unknownvars=curvars+unknownvars) for solution in rawsolutions: #solution.subs(freevarinvsubs) self.ComputeSolutionComplexity(solution,othersolvedvars,curvars) solutions.append((solution,Symbol(solution.jointname))) if len(rawsolutions) > 0: # solving a pair is rare, so any solution will do break except self.CannotSolveError: pass for var0,var1,raweqns,complexity in curvarsubssol: try: rawsolutions=self.SolvePairVariables(raweqns,var0,var1,othersolvedvars,unknownvars=curvars+unknownvars) except self.CannotSolveError as e: log.debug(e) # try: # rawsolutions=self.SolvePrismaticHingePairVariables(raweqns,var0,var1,othersolvedvars,unknownvars=curvars+unknownvars) # except self.CannotSolveError as e: # log.debug(e) rawsolutions = [] for solution in rawsolutions: #solution.subs(freevarinvsubs) try: self.ComputeSolutionComplexity(solution,othersolvedvars,curvars) solutions.append((solution,Symbol(solution.jointname))) except self.CannotSolveError as e: log.warn(u'equation failed to compute solution complexity: %s', solution.jointeval) if len(rawsolutions) > 0: # solving a pair is rare, so any solution will do break # take the least complex solution and go on if len(solutions) > 0: return self.AddSolution(solutions,AllEquations,curvars,othersolvedvars,solsubs,endbranchtree,currentcases=currentcases, currentcasesubs=currentcasesubs, unknownvars=unknownvars) # test with higher degrees, necessary? for curvar in curvars: othervars = unknownvars+[var for var in curvars if var != curvar] raweqns = [] for e in AllEquations: if (len(othervars) == 0 or not e.has(*othervars)) and e.has(curvar): eq = e.subs(self.freevarsubs+solsubs) if self.CheckExpressionUnique(raweqns,eq): raweqns.append(eq) for raweqn in raweqns: try: log.debug('testing with higher degrees') solution=self.solveHighDegreeEquationsHalfAngle([raweqn],self.Variable(curvar)) self.ComputeSolutionComplexity(solution,othersolvedvars,curvars) solutions.append((solution,curvar)) except self.CannotSolveError: pass if len(solutions) > 0: return self.AddSolution(solutions,AllEquations,curvars,othersolvedvars,solsubs,endbranchtree,currentcases=currentcases, currentcasesubs=currentcasesubs, unknownvars=unknownvars) # solve with all 3 variables together? # htvars = [self.Variable(varsym).htvar for varsym in curvars] # reducedeqs = [] # for eq in AllEquations: # if eq.has(*curvars): # num, denom, htvarsubsinv = self.ConvertSinCosEquationToHalfTan(eq, curvars) # reducedeqs.append(Poly(num, *htvars)) # # only guess if final joint to be solved, or there exists current cases and at least one joint has been solved already. # don't want to start guessing when no joints have been solved yet, this is an indication of bad equations if canguessvars and len(othersolvedvars)+len(curvars) == len(self.freejointvars)+len(self._solvejointvars) and (len(curvars) == 1 or (len(curvars) < len(self._solvejointvars) and currentcases is not None and len(currentcases) > 0)): # only estimate when deep in the hierarchy, do not want the guess to be executed all the time # perhaps there's a degree of freedom that is not trivial to compute? # take the highest hinge variable and set it log.info('trying to guess variable from %r', curvars) return self.GuessValuesAndSolveEquations(AllEquations, curvars, othersolvedvars, solsubs, endbranchtree, currentcases, unknownvars, currentcasesubs) # have got this far, so perhaps two axes are aligned? raise self.CannotSolveError('SolveAllEquations failed to find a variable to solve') def _SubstituteGlobalSymbols(self, eq, globalsymbols=None): if globalsymbols is None: globalsymbols = self.globalsymbols preveq = eq neweq = preveq.subs(globalsymbols) while preveq != neweq: if not self.isValidSolution(neweq): raise self.CannotSolveError('equation %r is not valid'%neweq) preveq = neweq neweq = preveq.subs(globalsymbols) return neweq def _AddToGlobalSymbols(self, var, eq): """adds to the global symbols, returns True if replaced with an existing entry """ for iglobal, gvarexpr in enumerate(self.globalsymbols): if var == gvarexpr[0]: self.globalsymbols[iglobal] = (var, eq) return True self.globalsymbols.append((var, eq)) return False def AddSolution(self,solutions,AllEquations,curvars,othersolvedvars,solsubs,endbranchtree, currentcases=None, currentcasesubs=None, unknownvars=None): """Take the least complex solution of a set of solutions and resume solving """ self._CheckPreemptFn() self._scopecounter += 1 scopecounter = int(self._scopecounter) solutions = [s for s in solutions if s[0].score < oo and s[0].checkValidSolution()] # remove infinite scores if len(solutions) == 0: raise self.CannotSolveError('no valid solutions') if unknownvars is None: unknownvars = [] solutions.sort(key=lambda e: e[0].score) hasonesolution = False for solution in solutions: checkforzeros = solution[0].checkforzeros hasonesolution |= solution[0].numsolutions() == 1 if len(checkforzeros) == 0 and solution[0].numsolutions() == 1: # did find a good solution, so take it. Make sure to check any zero branches var = solution[1] newvars=curvars[:] newvars.remove(var) return [solution[0].subs(solsubs)]+self.SolveAllEquations(AllEquations,curvars=newvars,othersolvedvars=othersolvedvars+[var],solsubs=solsubs+self.Variable(var).subs,endbranchtree=endbranchtree, currentcases=currentcases, currentcasesubs=currentcasesubs, unknownvars=unknownvars) if not hasonesolution: # check again except without the number of solutions requirement for solution in solutions: checkforzeros = solution[0].checkforzeros if len(checkforzeros) == 0: # did find a good solution, so take it. Make sure to check any zero branches var = solution[1] newvars=curvars[:] newvars.remove(var) return [solution[0].subs(solsubs)]+self.SolveAllEquations(AllEquations,curvars=newvars,othersolvedvars=othersolvedvars+[var],solsubs=solsubs+self.Variable(var).subs,endbranchtree=endbranchtree,currentcases=currentcases, currentcasesubs=currentcasesubs, unknownvars=unknownvars) originalGlobalSymbols = self.globalsymbols # all solutions have check for zero equations # choose the variable with the shortest solution and compute (this is a conservative approach) usedsolutions = [] # remove any solutions with similar checkforzero constraints (because they are essentially the same) for solution,var in solutions: solution.subs(solsubs) if len(usedsolutions) == 0: usedsolutions.append((solution,var)) else: match = False for usedsolution,usedvar in usedsolutions: if len(solution.checkforzeros) == len(usedsolution.checkforzeros): if not any([self.CheckExpressionUnique(usedsolution.checkforzeros,eq) for eq in solution.checkforzeros]): match = True break if not match: usedsolutions.append((solution,var)) if len(usedsolutions) >= 3: # don't need more than three alternatives (used to be two, but then lookat barrettwam4 proved that wrong) break nextsolutions = dict() allvars = [] for v in curvars: allvars += self.Variable(v).vars allothersolvedvars = [] for v in othersolvedvars: allothersolvedvars += self.Variable(v).vars lastbranch = [] prevbranch=lastbranch if currentcases is None: currentcases = set() if currentcasesubs is None: currentcasesubs = list() if self.degeneratecases is None: self.degeneratecases = self.DegenerateCases() handledconds = self.degeneratecases.GetHandledConditions(currentcases) # one to one correspondence with usedsolutions and the SolverCheckZeros hierarchies (used for cross product of equations later on) zerosubstitutioneqs = [] # indexed by reverse ordering of usedsolutions (len(usedsolutions)-solutionindex-1) # zerosubstitutioneqs equations flattened for easier checking flatzerosubstitutioneqs = [] hascheckzeros = False addhandleddegeneratecases = [] # for bookkeeping/debugging # iterate in reverse order and put the most recently processed solution at the front. # There is a problem with this algorithm transferring the degenerate cases correctly. # Although the zeros of the first equation are checked, they are not added as conditions # to the later equations, so that the later equations will also use variables as unknowns (even though they are determined to be specific constants). This is most apparent in rotations. for solution,var in usedsolutions[::-1]: # there are divide by zeros, so check if they can be explicitly solved for joint variables checkforzeros = [] localsubstitutioneqs = [] for checkzero in solution.checkforzeros: if checkzero.has(*allvars): log.info('ignoring special check for zero since it has symbols %s: %s',str(allvars),str(checkzero)) continue # bother trying to extract something if too complex (takes a lot of computation time to check and most likely nothing will be extracted). 100 is an arbitrary value checkzeroComplexity = self.codeComplexity(checkzero) if checkzeroComplexity > 120: log.warn('checkforzero too big (%d): %s', checkzeroComplexity, checkzero) # don't even add it if it is too big if checkzeroComplexity < 500: checkforzeros.append(checkzero)#self.removecommonexprs(checkzero.evalf(),onlygcd=False,onlynumbers=True)) else: checkzero2 = self._SubstituteGlobalSymbols(checkzero, originalGlobalSymbols) checkzero2Complexity = self.codeComplexity(checkzero2) if checkzero2Complexity < 2*checkzeroComplexity: # check that with substitutions, things don't get too big checkzero = checkzero2 # fractions could get big, so evaluate directly checkzeroeval = checkzero.evalf() if checkzero2Complexity < self.codeComplexity(checkzeroeval): checkforzeros.append(checkzero) else: checkforzeros.append(checkzero.evalf())#self.removecommonexprs(checkzero.evalf(),onlygcd=False,onlynumbers=True) checksimplezeroexprs = [checkzero] if not checkzero.has(*allothersolvedvars): sumsquaresexprs = self._GetSumSquares(checkzero) if sumsquaresexprs is not None: checksimplezeroexprs += sumsquaresexprs sumsquaresexprstozero = [] for sumsquaresexpr in sumsquaresexprs: if sumsquaresexpr.is_Symbol: sumsquaresexprstozero.append(sumsquaresexpr) elif sumsquaresexpr.is_Mul: for arg in sumsquaresexpr.args: if arg.is_Symbol: sumsquaresexprstozero.append(arg) if len(sumsquaresexprstozero) > 0: localsubstitutioneqs.append([sumsquaresexprstozero,checkzero,[(sumsquaresexpr,S.Zero) for sumsquaresexpr in sumsquaresexprstozero], []]) handledconds += sumsquaresexprstozero for checksimplezeroexpr in checksimplezeroexprs: #if checksimplezeroexpr.has(*othersolvedvars): # cannot do this check since sjX,cjX might be used for othervar in othersolvedvars: sothervar = self.Variable(othervar).svar cothervar = self.Variable(othervar).cvar if checksimplezeroexpr.has(othervar,sothervar,cothervar): # the easiest thing to check first is if the equation evaluates to zero on boundaries 0,pi/2,pi,-pi/2 s = AST.SolverSolution(othervar.name,jointeval=[],isHinge=self.IsHinge(othervar.name)) for value in [S.Zero,pi/2,pi,-pi/2]: try: # doing (1/x).subs(x,0) produces a RuntimeError (infinite recursion...) checkzerosub=checksimplezeroexpr.subs([(othervar,value),(sothervar,sin(value).evalf(n=30)),(cothervar,cos(value).evalf(n=30))]) if self.isValidSolution(checkzerosub) and checkzerosub.evalf(n=30) == S.Zero: if s.jointeval is None: s.jointeval = [] s.jointeval.append(S.One*value) except (RuntimeError, AssertionError) as e: # log.warn('othervar %s=%f: %s',str(othervar),value,e) if s.jointeval is not None and len(s.jointeval) > 0: ss = [s] else: ss = [] try: # checksimplezeroexpr can be simple like -cj4*r21 - r20*sj4 # in which case the solutions would be [-atan2(-r21, -r20), -atan2(-r21, -r20) + 3.14159265358979] ss += self.solveSingleVariable([checksimplezeroexpr.subs([(sothervar,sin(othervar)),(cothervar,cos(othervar))])],othervar,othersolvedvars) except PolynomialError: # checksimplezeroexpr was too complex pass except self.CannotSolveError as e: # this is actually a little tricky, sometimes really good solutions can have a divide that looks like: # ((0.405 + 0.331*cj2)**2 + 0.109561*sj2**2 (manusarm_left) # This will never be 0, but the solution cannot be solved. Instead of rejecting, add a condition to check if checksimplezeroexpr itself is 0 or not pass for s in ss: # can actually simplify Positions and possibly get a new solution! if s.jointeval is not None: for eq in s.jointeval: eq = self._SubstituteGlobalSymbols(eq, originalGlobalSymbols) # why checking for just number? ok to check if solution doesn't contain any other variableS? # if the equation is non-numerical, make sure it isn't deep in the degenerate cases if eq.is_number or (len(currentcases) <= 1 and not eq.has(*allothersolvedvars) and self.codeComplexity(eq) < 100): isimaginary = self.AreAllImaginaryByEval(eq) or eq.evalf().has(I) # TODO should use the fact that eq is imaginary if isimaginary: log.warn('eq %s is imaginary, but currently do not support this', eq) continue dictequations = [] if not eq.is_number and not eq.has(*allothersolvedvars): # not dependent on variables, so it could be in the form of atan(px,py), so convert to a global symbol since it never changes sym = next(self.gsymbolgen) dictequations.append((sym,eq)) #eq = sym sineq = next(self.gsymbolgen) dictequations.append((sineq,self.SimplifyAtan2(sin(eq)))) coseq = next(self.gsymbolgen) dictequations.append((coseq,self.SimplifyAtan2(cos(eq)))) else: sineq = sin(eq).evalf(n=30) coseq = cos(eq).evalf(n=30) cond=Abs(othervar-eq.evalf(n=30)) if self.CheckExpressionUnique(handledconds, cond): if self.IsHinge(othervar.name): evalcond=fmod(cond+pi,2*pi)-pi else: evalcond=cond localsubstitutioneqs.append([[cond],evalcond,[(sothervar,sineq),(sin(othervar),sineq),(cothervar,coseq),(cos(othervar),coseq),(othervar,eq)], dictequations]) handledconds.append(cond) elif s.jointevalsin is not None: for eq in s.jointevalsin: eq = self.SimplifyAtan2(self._SubstituteGlobalSymbols(eq, originalGlobalSymbols)) if eq.is_number or (len(currentcases) <= 1 and not eq.has(*allothersolvedvars) and self.codeComplexity(eq) < 100): dictequations = [] # test when cos(othervar) > 0 # don't use asin(eq)!! since eq = (-pz**2/py**2)**(1/2), which would produce imaginary numbers #cond=othervar-asin(eq).evalf(n=30) # test if eq is imaginary, if yes, then only solution is when sothervar==0 and eq==0 isimaginary = self.AreAllImaginaryByEval(eq) or eq.evalf().has(I) if isimaginary: cond = abs(sothervar) + abs((eq**2).evalf(n=30)) + abs(sign(cothervar)-1) else: if not eq.is_number and not eq.has(*allothersolvedvars): # not dependent on variables, so it could be in the form of atan(px,py), so convert to a global symbol since it never changes sym = next(self.gsymbolgen) dictequations.append((sym,eq)) #eq = sym cond=abs(sothervar-eq.evalf(n=30)) + abs(sign(cothervar)-1) if self.CheckExpressionUnique(handledconds, cond): if self.IsHinge(othervar.name): evalcond=fmod(cond+pi,2*pi)-pi else: evalcond=cond if isimaginary: localsubstitutioneqs.append([[cond],evalcond,[(sothervar,S.Zero),(sin(othervar),S.Zero),(cothervar,S.One),(cos(othervar),S.One),(othervar,S.One)], dictequations]) else: localsubstitutioneqs.append([[cond],evalcond,[(sothervar,eq),(sin(othervar),eq),(cothervar,sqrt(1-eq*eq).evalf(n=30)),(cos(othervar),sqrt(1-eq*eq).evalf(n=30)),(othervar,asin(eq).evalf(n=30))], dictequations]) handledconds.append(cond) # test when cos(othervar) < 0 if isimaginary: cond = abs(sothervar) + abs((eq**2).evalf(n=30)) + abs(sign(cothervar)+1) else: cond=abs(sothervar-eq.evalf(n=30))+abs(sign(cothervar)+1) #cond=othervar-(pi-asin(eq).evalf(n=30)) if self.CheckExpressionUnique(handledconds, cond): if self.IsHinge(othervar.name): evalcond=fmod(cond+pi,2*pi)-pi else: evalcond=cond if isimaginary: localsubstitutioneqs.append([[cond],evalcond,[(sothervar,S.Zero),(sin(othervar),S.Zero),(cothervar,-S.One),(cos(othervar),-S.One),(othervar,pi.evalf(n=30))], dictequations]) else: localsubstitutioneqs.append([[cond],evalcond,[(sothervar,eq),(sin(othervar),eq),(cothervar,-sqrt(1-eq*eq).evalf(n=30)),(cos(othervar),-sqrt(1-eq*eq).evalf(n=30)),(othervar,(pi-asin(eq)).evalf(n=30))], dictequations]) handledconds.append(cond) elif s.jointevalcos is not None: for eq in s.jointevalcos: eq = self.SimplifyAtan2(self._SubstituteGlobalSymbols(eq, originalGlobalSymbols)) if eq.is_number or (len(currentcases) <= 1 and not eq.has(*allothersolvedvars) and self.codeComplexity(eq) < 100): dictequations = [] # test when sin(othervar) > 0 # don't use acos(eq)!! since eq = (-pz**2/px**2)**(1/2), which would produce imaginary numbers # that's why check eq.evalf().has(I) #cond=othervar-acos(eq).evalf(n=30) isimaginary = self.AreAllImaginaryByEval(eq) or eq.evalf().has(I) if isimaginary: cond=abs(cothervar)+abs((eq**2).evalf(n=30)) + abs(sign(sothervar)-1) else: if not eq.is_number and not eq.has(*allothersolvedvars): # not dependent on variables, so it could be in the form of atan(px,py), so convert to a global symbol since it never changes sym = next(self.gsymbolgen) dictequations.append((sym,eq)) eq = sym cond=abs(cothervar-eq.evalf(n=30)) + abs(sign(sothervar)-1) if self.CheckExpressionUnique(handledconds, cond): if self.IsHinge(othervar.name): evalcond=fmod(cond+pi,2*pi)-pi else: evalcond=cond if isimaginary: localsubstitutioneqs.append([[cond],evalcond,[(sothervar,S.One),(sin(othervar),S.One),(cothervar,S.Zero),(cos(othervar),S.Zero),(othervar,(pi/2).evalf(n=30))], dictequations]) else: localsubstitutioneqs.append([[cond],evalcond,[(sothervar,sqrt(1-eq*eq).evalf(n=30)),(sin(othervar),sqrt(1-eq*eq).evalf(n=30)),(cothervar,eq),(cos(othervar),eq),(othervar,acos(eq).evalf(n=30))], dictequations]) handledconds.append(cond) #cond=othervar+acos(eq).evalf(n=30) if isimaginary: cond=abs(cothervar)+abs((eq**2).evalf(n=30)) + abs(sign(sothervar)+1) else: cond=abs(cothervar-eq.evalf(n=30)) + abs(sign(sothervar)+1) if self.CheckExpressionUnique(handledconds, cond): if self.IsHinge(othervar.name): evalcond=fmod(cond+pi,2*pi)-pi else: evalcond=cond if isimaginary: localsubstitutioneqs.append([[cond],evalcond,[(sothervar,-S.One),(sin(othervar),-S.One),(cothervar,S.Zero),(cos(othervar),S.Zero),(othervar,(-pi/2).evalf(n=30))], dictequations]) else: localsubstitutioneqs.append([[cond],evalcond,[(sothervar,-sqrt(1-eq*eq).evalf(n=30)),(sin(othervar),-sqrt(1-eq*eq).evalf(n=30)),(cothervar,eq),(cos(othervar),eq),(othervar,-acos(eq).evalf(n=30))], dictequations]) handledconds.append(cond) flatzerosubstitutioneqs += localsubstitutioneqs zerosubstitutioneqs.append(localsubstitutioneqs) if not var in nextsolutions: try: newvars=curvars[:] newvars.remove(var) # degenreate cases should get restored here since once we go down a particular branch, there's no turning back olddegeneratecases = self.degeneratecases self.degeneratecases = olddegeneratecases.Clone() nextsolutions[var] = self.SolveAllEquations(AllEquations,curvars=newvars,othersolvedvars=othersolvedvars+[var],solsubs=solsubs+self.Variable(var).subs,endbranchtree=endbranchtree,currentcases=currentcases, currentcasesubs=currentcasesubs, unknownvars=unknownvars) finally: addhandleddegeneratecases += olddegeneratecases.handleddegeneratecases self.degeneratecases = olddegeneratecases if len(checkforzeros) > 0: hascheckzeros = True solvercheckzeros = AST.SolverCheckZeros(jointname=var.name,jointcheckeqs=checkforzeros,nonzerobranch=[solution]+nextsolutions[var],zerobranch=prevbranch,anycondition=True,thresh=solution.GetZeroThreshold()) # have to transfer the dictionary! solvercheckzeros.dictequations = originalGlobalSymbols + solution.dictequations solvercheckzeros.equationsused = AllEquations solution.dictequations = [] prevbranch=[solvercheckzeros] else: prevbranch = [solution]+nextsolutions[var] if len(prevbranch) == 0: raise self.CannotSolveError('failed to add solution!') maxlevel2scopecounter = 300 # used to limit how deep the hierarchy goes or otherwise IK can get too big if len(currentcases) >= self.maxcasedepth or (scopecounter > maxlevel2scopecounter and len(currentcases) >= 2): log.warn('c=%d, %d levels deep in checking degenerate cases, skipping. curvars=%r, AllEquations=%r', scopecounter, len(currentcases), curvars, AllEquations) lastbranch.append(AST.SolverBreak('%d cases reached'%self.maxcasedepth, [(var,self.SimplifyAtan2(self._SubstituteGlobalSymbols(eq, originalGlobalSymbols))) for var, eq in currentcasesubs], othersolvedvars, solsubs, originalGlobalSymbols, endbranchtree)) return prevbranch # fill the last branch with all the zero conditions if hascheckzeros: # count the number of rotation symbols seen in the current cases numRotSymbolsInCases = 0 if self._iktype == 'transform6d' or self._iktype == 'rotation3d': rotsymbols = set(self.Tee[:3,:3]).union([Symbol('new_r00'), Symbol('new_r01'), Symbol('new_r02'), Symbol('new_r10'), Symbol('new_r11'), Symbol('new_r12'), Symbol('new_r20'), Symbol('new_r21'), Symbol('new_r22')]) for var, eq in currentcasesubs: if var in rotsymbols: numRotSymbolsInCases += 1 else: rotsymbols = [] # if not equations found, try setting two variables at once # also try setting px, py, or pz to 0 (barrettwam4 lookat) # sometimes can get the following: cj3**2*sj4**2 + cj4**2 threshnumsolutions = 1 # # number of solutions to take usedsolutions[:threshnumsolutions] for the dual values for isolution,(solution,var) in enumerate(usedsolutions[::-1]): if isolution < len(usedsolutions)-threshnumsolutions and len(flatzerosubstitutioneqs) > 0: # have at least one zero condition... continue localsubstitutioneqs = [] for checkzero in solution.checkforzeros: if checkzero.has(*allvars): log.info('ignoring special check for zero 2 since it has symbols %s: %s',str(allvars), str(checkzero)) continue # don't bother trying to extract something if too complex (takes a lot of computation time to check and most likely nothing will be extracted). 120 is an arbitrary value if self.codeComplexity(checkzero) > 120: continue possiblesubs = [] ishinge = [] for preal in self.Tee[:3,3]: if checkzero.has(preal): possiblesubs.append([(preal,S.Zero)]) ishinge.append(False) # have to be very careful with the rotations since they are dependent on each other. For example if r00 and r01 are both 0, then r02 is +- 1, and r12 and r22 are 0. Then r10, r12, r20, r21 is a 2D rotation matrix if numRotSymbolsInCases < 2: for preal in rotsymbols: if checkzero.has(preal): possiblesubs.append([(preal,S.Zero)]) ishinge.append(False) for othervar in othersolvedvars: othervarobj = self.Variable(othervar) if checkzero.has(*othervarobj.vars): if not self.IsHinge(othervar.name): possiblesubs.append([(othervar,S.Zero)]) ishinge.append(False) continue else: sothervar = othervarobj.svar cothervar = othervarobj.cvar for value in [S.Zero,pi/2,pi,-pi/2]: possiblesubs.append([(othervar,value),(sothervar,sin(value).evalf(n=30)),(sin(othervar),sin(value).evalf(n=30)), (cothervar,cos(value).evalf(n=30)), (cos(othervar),cos(value).evalf(n=30))]) ishinge.append(True) # all possiblesubs are present in checkzero for ipossiblesub, possiblesub in enumerate(possiblesubs): try: eq = checkzero.subs(possiblesub).evalf(n=30) except RuntimeError as e: # most likely doing (1/x).subs(x,0) produces a RuntimeError (infinite recursion...) log.warn(e) continue if not self.isValidSolution(eq): continue # only take the first index possiblevar,possiblevalue = possiblesub[0] cond = Abs(possiblevar-possiblevalue.evalf(n=30)) if not self.CheckExpressionUnique(handledconds, cond): # already present, so don't use it for double expressions continue if ishinge[ipossiblesub]: evalcond = Abs(fmod(possiblevar-possiblevalue+pi,2*pi)-pi) else: evalcond = cond if eq == S.Zero: log.info('c=%d, adding case %s=%s in %s', scopecounter, possiblevar, possiblevalue,checkzero) # if the variable is 1 and part of the rotation matrix, can deduce other variables if possiblevar in rotsymbols and (possiblevalue == S.One or possiblevalue == -S.One): row1 = int(possiblevar.name[-2]) col1 = int(possiblevar.name[-1]) possiblesub.append((Symbol('%s%d%d'%(possiblevar.name[:-2], row1, (col1+1)%3)), S.Zero)) possiblesub.append((Symbol('%s%d%d'%(possiblevar.name[:-2], row1, (col1+2)%3)), S.Zero)) possiblesub.append((Symbol('%s%d%d'%(possiblevar.name[:-2], (row1+1)%3, col1)), S.Zero)) possiblesub.append((Symbol('%s%d%d'%(possiblevar.name[:-2], (row1+2)%3, col1)), S.Zero)) checkexpr = [[cond],evalcond,possiblesub, []] flatzerosubstitutioneqs.append(checkexpr) localsubstitutioneqs.append(checkexpr) handledconds.append(cond) continue # try another possiblesub for ipossiblesub2, possiblesub2 in enumerate(possiblesubs[ipossiblesub+1:]): possiblevar2,possiblevalue2 = possiblesub2[0] if possiblevar == possiblevar2: # same var, so skip continue try: eq2 = eq.subs(possiblesub2).evalf(n=30) except RuntimeError as e: # most likely doing (1/x).subs(x,0) produces a RuntimeError (infinite recursion...) log.warn(e) continue if not self.isValidSolution(eq2): continue if eq2 == S.Zero: possiblevar2,possiblevalue2 = possiblesub2[0] cond2 = Abs(possiblevar2-possiblevalue2.evalf(n=30)) if not self.CheckExpressionUnique(handledconds ,cond2): # already present, so don't use it for double expressions continue # don't combine the conditions like cond+cond2, instead test them individually (this reduces the solution tree) if ishinge[ipossiblesub+ipossiblesub2+1]: evalcond2 = Abs(fmod(possiblevar2-possiblevalue2+pi,2*pi)-pi)# + evalcond else: evalcond2 = cond2# + evalcond #cond2 += cond if self.CheckExpressionUnique(handledconds, cond+cond2): # if the variables are both part of the rotation matrix and both zeros, can deduce other rotation variables if self._iktype == 'transform6d' and possiblevar in rotsymbols and possiblevalue == S.Zero and possiblevar2 in rotsymbols and possiblevalue2 == S.Zero: checkexpr = [[cond+cond2],evalcond+evalcond2, possiblesub+possiblesub2, []] flatzerosubstitutioneqs.append(checkexpr) localsubstitutioneqs.append(checkexpr) handledconds.append(cond+cond2) row1 = int(possiblevar.name[-2]) col1 = int(possiblevar.name[-1]) row2 = int(possiblevar2.name[-2]) col2 = int(possiblevar2.name[-1]) row3 = 3 - row1 - row2 col3 = 3 - col1 - col2 if row1 == row2: # (row1, col3) is either 1 or -1, but don't know which. # know that (row1+1,col3) and (row1+2,col3) are zero though... checkexpr[2].append((Symbol('%s%d%d'%(possiblevar.name[:-2], (row2+1)%3, col3)), S.Zero)) checkexpr[2].append((Symbol('%s%d%d'%(possiblevar.name[:-2], (row1+2)%3, col3)), S.Zero)) # furthermore can defer that the left over 4 values are [cos(ang), sin(ang), cos(ang), -sin(ang)] = abcd if row1 == 1: minrow = 0 maxrow = 2 else: minrow = (row1+1)%3 maxrow = (row1+2)%3 ra = Symbol('%s%d%d'%(possiblevar.name[:-2], minrow, col1)) rb = Symbol('%s%d%d'%(possiblevar.name[:-2], minrow, col2)) rc = Symbol('%s%d%d'%(possiblevar.name[:-2], maxrow, col1)) rd = Symbol('%s%d%d'%(possiblevar.name[:-2], maxrow, col2)) checkexpr[2].append((rb**2, S.One-ra**2)) checkexpr[2].append((rb**3, rb-rb*ra**2)) # need 3rd power since sympy cannot divide out the square checkexpr[2].append((rc**2, S.One-ra**2)) #checkexpr[2].append((rc, -rb)) # not true #checkexpr[2].append((rd, ra)) # not true elif col1 == col2: # (row3, col1) is either 1 or -1, but don't know which. # know that (row3,col1+1) and (row3,col1+2) are zero though... checkexpr[2].append((Symbol('%s%d%d'%(possiblevar.name[:-2], row3, (col1+1)%3)), S.Zero)) checkexpr[2].append((Symbol('%s%d%d'%(possiblevar.name[:-2], row3, (col1+2)%3)), S.Zero)) # furthermore can defer that the left over 4 values are [cos(ang), sin(ang), cos(ang), -sin(ang)] = abcd if col1 == 1: mincol = 0 maxcol = 2 else: mincol = (col1+1)%3 maxcol = (col1+2)%3 ra = Symbol('%s%d%d'%(possiblevar.name[:-2], row1, mincol)) rb = Symbol('%s%d%d'%(possiblevar.name[:-2], row2, mincol)) rc = Symbol('%s%d%d'%(possiblevar.name[:-2], row1, maxcol)) rd = Symbol('%s%d%d'%(possiblevar.name[:-2], row2, maxcol)) checkexpr[2].append((rb**2, S.One-ra**2)) checkexpr[2].append((rb**3, rb-rb*ra**2)) # need 3rd power since sympy cannot divide out the square checkexpr[2].append((rc**2, S.One-ra**2)) #checkexpr[2].append((rc, -rb)) # not true #checkexpr[2].append((rd, ra)) # not true log.info('dual constraint %r in %s', checkexpr[2],checkzero) else: # shouldn't have any rotation vars if not possiblevar in rotsymbols and not possiblevar2 in rotsymbols: checkexpr = [[cond+cond2],evalcond+evalcond2, possiblesub+possiblesub2, []] flatzerosubstitutioneqs.append(checkexpr) localsubstitutioneqs.append(checkexpr) handledconds.append(cond+cond2) zerosubstitutioneqs[isolution] += localsubstitutioneqs # test the solutions # PREV: have to take the cross product of all the zerosubstitutioneqs in order to form stronger constraints on the equations because the following condition will be executed only if all SolverCheckZeros evalute to 0 # NEW: not sure why cross product is necessary anymore.... zerobranches = [] accumequations = [] # # since sequence_cross_product requires all lists to be non-empty, insert None for empty lists # for conditioneqs in zerosubstitutioneqs: # if len(conditioneqs) == 0: # conditioneqs.append(None) # for conditioneqs in self.sequence_cross_product(*zerosubstitutioneqs): # validconditioneqs = [c for c in conditioneqs if c is not None] # if len(validconditioneqs) > 1: # # merge the equations, be careful not to merge equations constraining the same variable # cond = [] # evalcond = S.Zero # othervarsubs = [] # dictequations = [] # duplicatesub = False # for subcond, subevalcond, subothervarsubs, subdictequations in validconditioneqs: # cond += subcond # evalcond += abs(subevalcond) # for subothervarsub in subothervarsubs: # if subothervarsub[0] in [sym for sym,value in othervarsubs]: # # variable is duplicated # duplicatesub = True # break # othervarsubs.append(subothervarsub) # if duplicatesub: # break # dictequations += subdictequations # if not duplicatesub: # flatzerosubstitutioneqs.append([cond,evalcond,othervarsubs,dictequations]) if self._iktype == 'transform6d' or self._iktype == 'rotation3d': trysubstitutions = self.ppsubs+self.npxyzsubs+self.rxpsubs else: trysubstitutions = self.ppsubs log.debug('c=%d have %d zero substitutions', scopecounter, len(flatzerosubstitutioneqs)) for iflatzerosubstitutioneqs, (cond, evalcond, othervarsubs, dictequations) in enumerate(flatzerosubstitutioneqs): # have to convert to fractions before substituting! if not all([self.isValidSolution(v) for s,v in othervarsubs]): continue othervarsubs = [(s,self.ConvertRealToRationalEquation(v)) for s,v in othervarsubs] #NewEquations = [eq.subs(self.npxyzsubs + self.rxpsubs).subs(othervarsubs) for eq in AllEquations] NewEquations = [eq.subs(othervarsubs) for eq in AllEquations] NewEquationsClean = self.PropagateSolvedConstants(NewEquations, othersolvedvars, curvars) try: # forcing a value, so have to check if all equations in NewEquations that do not contain # unknown variables are really 0 extrazerochecks=[] for i in range(len(NewEquations)): expr = NewEquations[i] if not self.isValidSolution(expr): log.warn('not valid: %s',expr) extrazerochecks=None break if not expr.has(*allvars) and self.CheckExpressionUnique(extrazerochecks,expr): if expr.is_Symbol: # can set that symbol to zero and create a new set of equations! extrazerochecks.append(expr.subs(solsubs).evalf(n=30)) if extrazerochecks is not None: newcases = set(currentcases) for singlecond in cond: newcases.add(singlecond) if not self.degeneratecases.CheckCases(newcases): log.info('depth=%d, c=%d, iter=%d/%d, starting newcases: %r', len(currentcases), scopecounter, iflatzerosubstitutioneqs, len(flatzerosubstitutioneqs), newcases) if len(NewEquationsClean) > 0: newcasesubs = currentcasesubs+othervarsubs self.globalsymbols = [] for casesub in newcasesubs: self._AddToGlobalSymbols(casesub[0], casesub[1]) extradictequations = [] for s,v in trysubstitutions: neweq = v.subs(newcasesubs) if neweq != v: # should we make sure we're not adding it a second time? newcasesubs.append((s, neweq)) extradictequations.append((s, neweq)) self._AddToGlobalSymbols(s, neweq) for var, eq in chain(originalGlobalSymbols, dictequations): neweq = eq.subs(othervarsubs) if not self.isValidSolution(neweq): raise self.CannotSolveError('equation %s is invalid because of the following substitutions: %s'%(eq, othervarsubs)) if neweq == S.Zero: extradictequations.append((var, S.Zero)) self._AddToGlobalSymbols(var, neweq) if len(extradictequations) > 0: # have to re-substitute since some equations evaluated to zero NewEquationsClean = [eq.subs(extradictequations).expand() for eq in NewEquationsClean] newtree = self.SolveAllEquations(NewEquationsClean,curvars,othersolvedvars,solsubs,endbranchtree,currentcases=newcases, currentcasesubs=newcasesubs, unknownvars=unknownvars) accumequations.append(NewEquationsClean) # store the equations for debugging purposes else: log.info('there are no new equations, so most likely the following variables can be freely determined: %r', curvars) # unfortunately cannot add as a FreeVariable since all the left over variables will have complex dependencies # therefore, iterate a couple of jointevals newtree = [] for curvar in curvars: newtree.append(AST.SolverSolution(curvar.name, jointeval=[S.Zero,pi/2,pi,-pi/2], isHinge=self.IsHinge(curvar.name))) newtree += endbranchtree zerobranches.append(([evalcond]+extrazerochecks,newtree,dictequations)) # what about extradictequations? log.info('depth=%d, c=%d, iter=%d/%d, adding newcases: %r', len(currentcases), scopecounter, iflatzerosubstitutioneqs, len(flatzerosubstitutioneqs), newcases) self.degeneratecases.AddCases(newcases) else: log.warn('already has handled cases %r', newcases) except self.CannotSolveError as e: log.debug(e) continue finally: # restore the global symbols self.globalsymbols = originalGlobalSymbols if len(zerobranches) > 0: branchconds = AST.SolverBranchConds(zerobranches+[(None,[AST.SolverBreak('branch miss %r'%curvars, [(var,self._SubstituteGlobalSymbols(eq, originalGlobalSymbols)) for var, eq in currentcasesubs], othersolvedvars, solsubs, originalGlobalSymbols, endbranchtree)],[])]) branchconds.accumequations = accumequations lastbranch.append(branchconds) else: # add GuessValuesAndSolveEquations? lastbranch.append(AST.SolverBreak('no branches %r'%curvars, [(var,self._SubstituteGlobalSymbols(eq, originalGlobalSymbols)) for var, eq in currentcasesubs], othersolvedvars, solsubs, originalGlobalSymbols, endbranchtree)) return prevbranch def GuessValuesAndSolveEquations(self, AllEquations, curvars, othersolvedvars, solsubs, endbranchtree, currentcases=None, unknownvars=None, currentcasesubs=None): # perhaps there's a degree of freedom that is not trivial to compute? # take the highest hinge variable and set it scopecounter = int(self._scopecounter) hingevariables = [curvar for curvar in sorted(curvars,reverse=True) if self.IsHinge(curvar.name)] if len(hingevariables) > 0 and len(curvars) >= 2: curvar = hingevariables[0] leftovervars = list(curvars) leftovervars.remove(curvar) newtree = [AST.SolverConditionedSolution([])] zerovalues = [] for jointeval in [S.Zero,pi/2,pi,-pi/2]: checkzeroequations = [] NewEquations = [] for eq in AllEquations: neweq = eq.subs(curvar, jointeval) neweqeval = neweq.evalf() if neweq.is_number: # if zero, then can ignore if neweq == S.Zero: continue # if not zero, then a contradiciton, so jointeval is bad NewEquations = None break if neweq.has(*leftovervars): NewEquations.append(neweq) else: checkzeroequations.append(neweq) if NewEquations is None: continue # check to make sure all leftover vars are in scope cansolve = True for leftovervar in leftovervars: if not any([eq.has(leftovervar) for eq in NewEquations]): cansolve = False break if not cansolve: continue if len(checkzeroequations) > 0: solution = AST.SolverSolution(curvar.name, jointeval=[jointeval], isHinge=self.IsHinge(curvar.name)) solution.checkforzeros = checkzeroequations solution.FeasibleIsZeros = True newtree[0].solversolutions.append(solution) else: # one value is enough zerovalues.append(jointeval) if len(zerovalues) > 0: # prioritize these solutions since they don't come with any extra checks solution = AST.SolverSolution(curvar.name, jointeval=zerovalues, isHinge=self.IsHinge(curvar.name)) solution.FeasibleIsZeros = True newtree = [solution] elif len(newtree[0].solversolutions) == 0: # nothing found so remove the condition node newtree = [] if len(newtree) > 0: log.warn('c=%d, think there is a free variable, but cannot solve relationship, so setting variable %s', scopecounter, curvar) newtree += self.SolveAllEquations(AllEquations, leftovervars, othersolvedvars+[curvar], solsubs+self.Variable(curvar).subs, endbranchtree,currentcases=currentcases, currentcasesubs=currentcasesubs, unknownvars=unknownvars) return newtree if len(curvars) == 1: log.info('have only one variable left %r and most likely it is not in equations %r', curvars[0], AllEquations) solution = AST.SolverSolution(curvars[0].name, jointeval=[S.Zero], isHinge=self.IsHinge(curvars[0].name)) solution.FeasibleIsZeros = True return [solution]+endbranchtree raise self.CannotSolveError('cannot find a good variable') def SolvePairVariablesHalfAngle(self,raweqns,var0,var1,othersolvedvars,subs=None): """solves equations of two variables in sin and cos """ varsym0 = self.Variable(var0) varsym1 = self.Variable(var1) varsyms = [varsym0,varsym1] unknownvars=[varsym0.cvar,varsym0.svar,varsym1.cvar,varsym1.svar] varsubs=varsym0.subs+varsym1.subs varsubsinv = varsym0.subsinv+varsym1.subsinv halftansubs = [] for varsym in varsyms: halftansubs += [(varsym.cvar,(1-varsym.htvar**2)/(1+varsym.htvar**2)),(varsym.svar,2*varsym.htvar/(1+varsym.htvar**2))] dummyvars = [] for othervar in othersolvedvars: v = self.Variable(othervar) dummyvars += [v.cvar,v.svar,v.var,v.htvar] polyeqs = [] for eq in raweqns: trigsubs = [(varsym0.svar**2,1-varsym0.cvar**2), (varsym0.svar**3,varsym0.svar*(1-varsym0.cvar**2)), (varsym1.svar**2,1-varsym1.cvar**2), (varsym1.svar**3,varsym1.svar*(1-varsym1.cvar**2))] peq = Poly(eq.subs(varsubs).subs(trigsubs).expand().subs(trigsubs),*unknownvars) if peq.has(varsym0.var) or peq.has(varsym1.var): raise self.CannotSolveError('expecting only sin and cos! %s'%peq) maxmonoms = [0,0,0,0] maxdenom = [0,0] for monoms in peq.monoms(): for i in range(4): maxmonoms[i] = max(maxmonoms[i],monoms[i]) maxdenom[0] = max(maxdenom[0],monoms[0]+monoms[1]) maxdenom[1] = max(maxdenom[1],monoms[2]+monoms[3]) eqnew = S.Zero for monoms,c in peq.terms(): term = c for i in range(4): num,denom = fraction(halftansubs[i][1]) term *= num**monoms[i] # the denoms for 0,1 and 2,3 are the same for i in [0,2]: denom = fraction(halftansubs[i][1])[1] term *= denom**(maxdenom[i//2]-monoms[i]-monoms[i+1]) complexityvalue = self.codeComplexity(term.expand()) if complexityvalue < 450: eqnew += simplify(term) else: # too big, so don't simplify? eqnew += term polyeq = Poly(eqnew,varsym0.htvar,varsym1.htvar) if polyeq.TC() == S.Zero: # might be able to divide out variables? minmonoms = None for monom in polyeq.monoms(): if minmonoms is None: minmonoms = list(monom) else: for i in range(len(minmonoms)): minmonoms[i] = min(minmonoms[i],monom[i]) newpolyeq = Poly(S.Zero,*polyeq.gens) for m,c in polyeq.terms(): newm = list(m) for i in range(len(minmonoms)): newm[i] -= minmonoms[i] newpolyeq = newpolyeq.add(Poly.from_dict({tuple(newm):c},*newpolyeq.gens)) log.warn('converting polyeq "%s" to "%s"'%(polyeq,newpolyeq)) # check if any equations are only in one variable polyeq = newpolyeq polyeqs.append(polyeq) try: return self.solveSingleVariable(self.sortComplexity([e.as_expr() for e in polyeqs if not e.has(varsym1.htvar)]),varsym0.var,othersolvedvars,unknownvars=[]) except self.CannotSolveError: pass try: return self.solveSingleVariable(self.sortComplexity([e.as_expr() for e in polyeqs if not e.has(varsym0.htvar)]),varsym1.var,othersolvedvars,unknownvars=[]) except self.CannotSolveError: pass complexity = [(self.codeComplexity(peq.as_expr()),peq) for peq in polyeqs] complexity.sort(key=itemgetter(0)) polyeqs = [peq[1] for peq in complexity] solutions = [None,None] linearsolution = None for ileftvar in range(2): if linearsolution is not None: break leftvar = varsyms[ileftvar].htvar newpolyeqs = [Poly(eq,varsyms[1-ileftvar].htvar) for eq in polyeqs] mindegree = __builtin__.min([max(peq.degree_list()) for peq in newpolyeqs]) maxdegree = __builtin__.max([max(peq.degree_list()) for peq in newpolyeqs]) for peq in newpolyeqs: if len(peq.monoms()) == 1: possiblefinaleq = self.checkFinalEquation(Poly(peq.LC(),leftvar),subs) if possiblefinaleq is not None: solutions[ileftvar] = [possiblefinaleq] break for degree in range(mindegree,maxdegree+1): if solutions[ileftvar] is not None or linearsolution is not None: break newpolyeqs2 = [peq for peq in newpolyeqs if max(peq.degree_list()) <= degree] if degree+1 <= len(newpolyeqs2): # in order to avoid wrong solutions, have to get resultants for all equations possibilities = [] unusedindices = list(range(len(newpolyeqs2))) for eqsindices in combinations(range(len(newpolyeqs2)),degree+1): Mall = zeros((degree+1,degree+1)) totalcomplexity = 0 for i,eqindex in enumerate(eqsindices): eq = newpolyeqs2[eqindex] for j,c in eq.terms(): totalcomplexity += self.codeComplexity(c.expand()) Mall[i,j[0]] = c if degree >= 4 and totalcomplexity > 5000: # the determinant will never finish otherwise continue # det_bareis freezes when there are huge fractions #det=self.det_bareis(Mall,*(self.pvars+dummyvars+[leftvar])) # for i in range(Mall.shape[0]): # for j in range(Mall.shape[1]): # Mall[i,j] = Poly(Mall[i,j],leftvar) try: Malldet = Mall.berkowitz_det() except Exception as e: log.warn('failed to compute determinant: %s', e) continue complexity = self.codeComplexity(Malldet) if complexity > 1200: log.warn('determinant complexity is too big %d', complexity) continue possiblefinaleq = self.checkFinalEquation(Poly(Malldet,leftvar),subs) if possiblefinaleq is not None: # sometimes +- I are solutions, so remove them q,r = div(possiblefinaleq,leftvar+I) if r == S.Zero: possiblefinaleq = Poly(q,leftvar) q,r = div(possiblefinaleq,leftvar-I) if r == S.Zero: possiblefinaleq = Poly(q,leftvar) possibilities.append(possiblefinaleq) for eqindex in eqsindices: if eqindex in unusedindices: unusedindices.remove(eqindex) if len(unusedindices) == 0: break if len(possibilities) > 0: if len(possibilities) > 1: try: linearsolutions = self.solveVariablesLinearly(possibilities,othersolvedvars) # if can solve for a unique solution linearly, then prioritize this over anything prevsolution = AST.SolverBreak('SolvePairVariablesHalfAngle fail') for divisor,linearsolution in linearsolutions: assert(len(linearsolution)==1) divisorsymbol = next(self.gsymbolgen) solversolution = AST.SolverSolution(varsyms[ileftvar].name,jointeval=[2*atan(linearsolution[0]/divisorsymbol)],isHinge=self.IsHinge(varsyms[ileftvar].name)) prevsolution = AST.SolverCheckZeros(varsyms[ileftvar].name,[divisorsymbol],zerobranch=[prevsolution],nonzerobranch=[solversolution],thresh=1e-6) prevsolution.dictequations = [(divisorsymbol,divisor)] linearsolution = prevsolution break except self.CannotSolveError: pass # sort with respect to degree equationdegrees = [(max(peq.degree_list())*100000+self.codeComplexity(peq.as_expr()),peq) for peq in possibilities] equationdegrees.sort(key=itemgetter(0)) solutions[ileftvar] = [peq[1] for peq in equationdegrees] break if linearsolution is not None: return [linearsolution] # take the solution with the smallest degree pfinals = None ileftvar = None if solutions[0] is not None: if solutions[1] is not None: if max(solutions[1][0].degree_list()) < max(solutions[0][0].degree_list()): pfinals = solutions[1] ileftvar = 1 elif max(solutions[1][0].degree_list()) == max(solutions[0][0].degree_list()) and self.codeComplexity(solutions[1][0].as_expr()) < self.codeComplexity(solutions[0][0].as_expr()): pfinals = solutions[1] ileftvar = 1 else: pfinals = solutions[0] ileftvar = 0 else: pfinals = solutions[0] ileftvar = 0 elif solutions[1] is not None: pfinals = solutions[1] ileftvar = 1 dictequations = [] if pfinals is None: #simplifyfn = self._createSimplifyFn(self.freejointvars,self.freevarsubs,self.freevarsubsinv) for newreducedeqs in combinations(polyeqs,2): try: Mall = None numrepeating = None for ileftvar in range(2): # TODO, sometimes this works and sometimes this doesn't try: Mall, allmonoms = self.solveDialytically(newreducedeqs,ileftvar,returnmatrix=True) if Mall is not None: leftvar=polyeqs[0].gens[ileftvar] break except self.CannotSolveError as e: log.debug(e) if Mall is None: continue shape=Mall[0].shape assert(shape[0] == 4 and shape[1] == 4) Malltemp = [None]*len(Mall) M = zeros(shape) for idegree in range(len(Mall)): Malltemp[idegree] = zeros(shape) for i in range(shape[0]): for j in range(shape[1]): if Mall[idegree][i,j] != S.Zero: if self.codeComplexity(Mall[idegree][i,j])>5: sym = next(self.gsymbolgen) Malltemp[idegree][i,j] = sym dictequations.append((sym,Mall[idegree][i,j])) else: Malltemp[idegree][i,j] = Mall[idegree][i,j] M += Malltemp[idegree]*leftvar**idegree tempsymbols = [next(self.gsymbolgen) for i in range(16)] tempsubs = [] for i in range(16): if M[i] != S.Zero: tempsubs.append((tempsymbols[i],Poly(M[i],leftvar))) else: tempsymbols[i] = S.Zero Mtemp = Matrix(4,4,tempsymbols) dettemp=Mtemp.det() log.info('multiplying all determinant coefficients for solving %s',leftvar) eqadds = [] for arg in dettemp.args: eqmuls = [Poly(arg2.subs(tempsubs),leftvar) for arg2 in arg.args] if sum(eqmuls[0].degree_list()) == 0: eq = eqmuls.pop(0) eqmuls[0] = eqmuls[0]*eq while len(eqmuls) > 1: ioffset = 0 eqmuls2 = [] while ioffset < len(eqmuls)-1: eqmuls2.append(eqmuls[ioffset]*eqmuls[ioffset+1]) ioffset += 2 eqmuls = eqmuls2 eqadds.append(eqmuls[0]) log.info('done multiplying all determinant, now convert to poly') det = Poly(S.Zero,leftvar) for ieq, eq in enumerate(eqadds): log.info('adding to det %d/%d', ieq, len(eqadds)) det += eq if len(Mall) <= 3: # need to simplify further since self.globalsymbols can have important substitutions that can yield the entire determinant to zero log.info('attempting to simplify determinant...') newdet = Poly(S.Zero,leftvar) for m,c in det.terms(): origComplexity = self.codeComplexity(c) # 100 is a guess if origComplexity < 100: neweq = c.subs(dictequations) if self.codeComplexity(neweq) < 100: neweq = self._SubstituteGlobalSymbols(neweq).expand() newComplexity = self.codeComplexity(neweq) if newComplexity < origComplexity: c = neweq newdet += c*leftvar**m[0] det = newdet if det.degree(0) <= 0: continue pfinals = [det] break except self.CannotSolveError as e: log.debug(e) if pfinals is None: raise self.CannotSolveError('SolvePairVariablesHalfAngle: failed to solve dialytically with %d equations'%(len(polyeqs))) jointsol = 2*atan(varsyms[ileftvar].htvar) solution = AST.SolverPolynomialRoots(jointname=varsyms[ileftvar].name,poly=pfinals[0],jointeval=[jointsol],isHinge=self.IsHinge(varsyms[ileftvar].name)) solution.checkforzeros = [] solution.postcheckforzeros = [] if len(pfinals) > 1: # verify with at least one solution solution.postcheckfornonzeros = [peq.as_expr() for peq in pfinals[1:2]] solution.polybackup = pfinals[1] solution.postcheckforrange = [] solution.dictequations = dictequations solution.postcheckfornonzerosThresh = 1e-7 # make threshold a little loose since can be a lot of numbers compounding. depending on the degree, can expect small coefficients to be still valid solution.AddHalfTanValue = True return [solution] def _createSimplifyFn(self,vars,varsubs,varsubsinv): return lambda eq: self.trigsimp(eq.subs(varsubsinv),vars).subs(varsubs) def solveVariablesLinearly(self,polyeqs,othersolvedvars,maxsolvabledegree=4): log.debug('solvevariables=%r, othersolvedvars=%r',polyeqs[0].gens,othersolvedvars) nummonoms = [len(peq.monoms())-int(peq.TC()!=S.Zero) for peq in polyeqs] mindegree = __builtin__.min(nummonoms) maxdegree = min(__builtin__.max(nummonoms),len(polyeqs)) complexity = [(self.codeComplexity(peq.as_expr()),peq) for peq in polyeqs] complexity.sort(key=itemgetter(0)) polyeqs = [peq[1] for peq in complexity] trigsubs = [] trigsubsinv = [] othersolvedvarssyms = [] for othervar in othersolvedvars: v = self.Variable(othervar) othersolvedvarssyms += v.vars trigsubs += v.subs trigsubsinv += v.subsinv symbolscheck = [] for i,solvevar in enumerate(polyeqs[0].gens): monom = [0]*len(polyeqs[0].gens) monom[i] = 1 symbolscheck.append(tuple(monom)) solutions = [] for degree in range(mindegree,maxdegree+1): allindices = [i for i,n in enumerate(nummonoms) if n <= degree] if len(allindices) >= degree: allmonoms = set() for index in allindices: allmonoms = allmonoms.union(set(polyeqs[index].monoms())) allmonoms = list(allmonoms) allmonoms.sort() if __builtin__.sum(allmonoms[0]) == 0: allmonoms.pop(0) # allmonoms has to have symbols as a single variable if not all([check in allmonoms for check in symbolscheck]): continue if len(allmonoms) == degree: if degree > maxsolvabledegree: log.warn('cannot handle linear solving for more than 4 equations') continue systemequations = [] consts = [] for index in allindices: pdict = polyeqs[index].as_dict() systemequations.append([pdict.get(monom,S.Zero) for monom in allmonoms]) consts.append(-polyeqs[index].TC()) # generate at least two solutions in case first's determinant is 0 solutions = [] for startrow in range(len(systemequations)): rows = [startrow] M = Matrix(1,len(allmonoms),systemequations[rows[0]]) for i in range(startrow+1,len(systemequations)): numequationsneeded = M.shape[1] - M.shape[0] if i+numequationsneeded > len(systemequations): # cannot do anything break mergedsystemequations = list(systemequations[i]) for j in range(1,numequationsneeded): mergedsystemequations += systemequations[i+j] M2 = M.col_join(Matrix(numequationsneeded,len(allmonoms),mergedsystemequations)) complexity = 0 for i2 in range(M2.rows): for j2 in range(M2.cols): complexity += self.codeComplexity(M2[i,j]) if self.IsDeterminantNonZeroByEval(M2): if complexity < 5000: Mdet = M2.det() if Mdet != S.Zero: M = M2 for j in range(numequationsneeded): rows.append(i+j) break else: log.warn('found solution, but matrix is too complex and determinant will most likely freeze (%d)', complexity) if M.shape[0] == M.shape[1]: Mdet = self.trigsimp(Mdet.subs(trigsubsinv),othersolvedvars).subs(trigsubs) #Minv = M.inv() B = Matrix(M.shape[0],1,[consts[i] for i in rows]) Madjugate = M.adjugate() solution = [] for check in symbolscheck: value = Madjugate[allmonoms.index(check),:]*B solution.append(self.trigsimp(value[0].subs(trigsubsinv),othersolvedvars).subs(trigsubs)) solutions.append([Mdet,solution]) if len(solutions) >= 2: break if len(solutions) > 0: break if len(solutions) == 0: raise self.CannotSolveError('solveVariablesLinearly failed') return solutions def solveSingleVariableLinearly(self,raweqns,solvevar,othervars,maxnumeqs=2,douniquecheck=True): """tries to linearly solve for one variable treating everything else as constant. need at least 3 equations """ cvar = Symbol('c%s'%solvevar.name) svar = Symbol('s%s'%solvevar.name) varsubs = [(cos(solvevar),cvar),(sin(solvevar),svar)] othervarsubs = [(sin(v)**2,1-cos(v)**2) for v in othervars] eqpolys = [Poly(eq.subs(varsubs),cvar,svar) for eq in raweqns] eqpolys = [eq for eq in eqpolys if sum(eq.degree_list()) == 1 and not eq.TC().has(solvevar)] #eqpolys.sort(key=lambda e: iksolver.codeComplexity(e)) partialsolutions = [] neweqs = [] for p0,p1 in combinations(eqpolys,2): p0dict = p0.as_dict() p1dict = p1.as_dict() M = Matrix(2,3,[p0dict.get((1,0),S.Zero),p0dict.get((0,1),S.Zero),p0.TC(),p1dict.get((1,0),S.Zero),p1dict.get((0,1),S.Zero),p1.TC()]) M = M.subs(othervarsubs).expand() partialsolution = [-M[1,1]*M[0,2]+M[0,1]*M[1,2],M[1,0]*M[0,2]-M[0,0]*M[1,2],M[0,0]*M[1,1]-M[0,1]*M[1,0]] partialsolution = [eq.expand().subs(othervarsubs).expand() for eq in partialsolution] rank = [self.codeComplexity(eq) for eq in partialsolution] partialsolutions.append([rank,partialsolution]) # cos(A)**2 + sin(A)**2 - 1 = 0, useful equation but the squares introduce wrong solutions #neweqs.append(partialsolution[0]**2+partialsolution[1]**2-partialsolution[2]**2) # try to cross partialsolutions.sort(key=lambda e: min(e[0])) for (rank0,ps0),(rank1,ps1) in combinations(partialsolutions,2): if self.equal(ps0[0]*ps1[2]-ps1[0]*ps0[2],S.Zero): continue neweqs.append(ps0[0]*ps1[2]-ps1[0]*ps0[2]) neweqs.append(ps0[1]*ps1[2]-ps1[1]*ps0[2]) # probably a linear combination of the first two #neweqs.append(ps0[0]*ps1[1]-ps1[0]*ps0[1]) # too long #neweqs.append(ps0[0]*ps1[0]+ps0[1]*ps1[1]-ps0[2]*ps1[2]) if len(neweqs) >= maxnumeqs: break; neweqs2 = [eq.expand().subs(othervarsubs).expand() for eq in neweqs] if douniquecheck: reducedeqs = [] i = 0 while i < len(neweqs2): reducedeq = self.removecommonexprs(neweqs2[i]) if neweqs2[i] != S.Zero and self.CheckExpressionUnique(reducedeqs,reducedeq): reducedeqs.append(reducedeq) i += 1 else: eq=neweqs2.pop(i) return neweqs2 def solveHighDegreeEquationsHalfAngle(self,lineareqs,varsym,subs=None): """solve a set of equations in one variable with half-angle substitution """ dummysubs = [(varsym.cvar,(1-varsym.htvar**2)/(1+varsym.htvar**2)),(varsym.svar,2*varsym.htvar/(1+varsym.htvar**2))] polyeqs = [] for eq in lineareqs: trigsubs = [(varsym.svar**2,1-varsym.cvar**2), (varsym.svar**3,varsym.svar*(1-varsym.cvar**2))] try: peq = Poly(eq.subs(varsym.subs).subs(trigsubs),varsym.cvar,varsym.svar) except PolynomialError as e: raise self.CannotSolveError('solveHighDegreeEquationsHalfAngle: poly error (%r)'%eq) if peq.has(varsym.var): raise self.CannotSolveError('solveHighDegreeEquationsHalfAngle: expecting only sin and cos! %s'%peq) if sum(peq.degree_list()) == 0: continue # check if all terms are multiples of cos/sin maxmonoms = [0,0] maxdenom = 0 for monoms in peq.monoms(): for i in range(2): maxmonoms[i] = max(maxmonoms[i],monoms[i]) maxdenom = max(maxdenom,monoms[0]+monoms[1]) eqnew = S.Zero for monoms,c in peq.terms(): if c.evalf() != S.Zero: # big fractions might make this difficult to reduce to 0 term = c for i in range(2): num,denom = fraction(dummysubs[i][1]) term *= num**monoms[i] # the denoms for 0,1 and 2,3 are the same denom = fraction(dummysubs[0][1])[1] term *= denom**(maxdenom-monoms[0]-monoms[1]) eqnew += simplify(term) polyeqs.append(Poly(eqnew,varsym.htvar)) for peq in polyeqs: # do some type of resultants, for now just choose first polynomial finaleq = simplify(peq.as_expr()).expand() pfinal = Poly(self.removecommonexprs(finaleq,onlygcd=False,onlynumbers=True),varsym.htvar) pfinal = self.checkFinalEquation(pfinal,subs) if pfinal is not None and pfinal.degree(0) > 0: jointsol = 2*atan(varsym.htvar) solution = AST.SolverPolynomialRoots(jointname=varsym.name,poly=pfinal,jointeval=[jointsol],isHinge=self.IsHinge(varsym.name)) solution.AddHalfTanValue = True solution.checkforzeros = [] solution.postcheckforzeros = [] solution.postcheckfornonzeros = [] solution.postcheckforrange = [] return solution raise self.CannotSolveError('half-angle substitution for joint %s failed, %d equations examined'%(varsym.var,len(polyeqs))) def checkFinalEquation(self,pfinal,subs=None): """check an equation in one variable for validity """ assert(len(pfinal.gens)==1) if subs is None: subs = [] htvar = pfinal.gens[0] # remove all trivial 0s while sum(pfinal.degree_list()) > 0 and pfinal.TC() == S.Zero: pfinalnew = Poly(S.Zero,htvar) for m,c in pfinal.terms(): if m[0] > 0: pfinalnew += c*htvar**(m[0]-1) pfinal = pfinalnew # check to see that LC is non-zero for at least one solution if pfinal.LC().evalf() == S.Zero or all([pfinal.LC().subs(subs).subs(self.globalsymbols).subs(testconsistentvalue).evalf()==S.Zero for testconsistentvalue in self.testconsistentvalues]): return None # sanity check that polynomial can produce a solution and is not actually very small values found = False LCnormalized, common = self.removecommonexprs(pfinal.LC(),returncommon=True,onlygcd=False,onlynumbers=True) pfinaldict = pfinal.as_dict() for testconsistentvalue in self.testconsistentvalues: coeffs = [] globalsymbols = [(s,v.subs(self.globalsymbols).subs(testconsistentvalue).evalf()) for s,v in self.globalsymbols] for degree in range(pfinal.degree(0),-1,-1): value = pfinaldict.get((degree,),S.Zero).subs(subs).subs(globalsymbols+testconsistentvalue).evalf()/common.evalf() if value.has(I): # check if has imaginary number coeffs = None break coeffs.append(value) # since coeffs[0] is normalized with the LC constant, can compare for precision if len(coeffs) == 1: if ordering_of_classes.index(type(coeffs[0]).__name__) <= ordering_of_classes.index('Float') and Abs(coeffs[0]) < 2*(10.0**-self.precision): coeffs = None break if coeffs is None: continue if not all([c.is_number for c in coeffs]): # cannot evalute log.warn('cannot evalute: %s',coeffs) found = True break realsolution = pfinal.gens[0].subs(subs).subs(self.globalsymbols).subs(testconsistentvalue).evalf() # need to convert to float64 first, X.evalf() is still a sympy object roots = mpmath.polyroots(numpy.array(numpy.array(coeffs),numpy.float64)) for root in roots: if Abs(float(root.imag)) < 10.0**-self.precision and Abs(float(root.real)-realsolution) < 10.0**-(self.precision-2): found = True break if found: break return pfinal if found else None def solveSingleVariable(self,raweqns,var,othersolvedvars,maxsolutions=4,maxdegree=2,subs=None, unknownvars=None): varsym = self.Variable(var) vars = [varsym.cvar,varsym.svar,varsym.htvar,var] othersubs = [] for othersolvedvar in othersolvedvars: othersubs += self.Variable(othersolvedvar).subs # eqns = [] # for eq in raweqns: # if eq.has(*vars): # # for equations that are very complex, make sure at least one set of values yields a non zero equation # testeq = eq.subs(varsym.subs+othersubs) # if any([testeq.subs(testconsistentvalue).evalf()!=S.Zero for testconsistentvalue in self.testconsistentvalues]): # eqns.append(eq) eqns = [eq.expand() for eq in raweqns if eq.has(*vars)] if len(eqns) == 0: raise self.CannotSolveError('not enough equations') # prioritize finding a solution when var is alone returnfirstsolutions = [] for eq in eqns: symbolgen = cse_main.numbered_symbols('const') eqnew, symbols = self.groupTerms(eq.subs(varsym.subs), vars, symbolgen) try: ps = Poly(eqnew,varsym.svar) pc = Poly(eqnew,varsym.cvar) if sum(ps.degree_list()) > 0 or sum(pc.degree_list()) > 0 or ps.TC() == S.Zero or pc.TC() == S.Zero: continue except PolynomialError: continue numvar = self.countVariables(eqnew,var) if numvar >= 1 and numvar <= 2: try: tempsolutions = solve(eqnew,var) jointsolutions = [self.SimplifyTransform(self.trigsimp(s.subs(symbols),othersolvedvars)) for s in tempsolutions] if all([self.isValidSolution(s) and s != S.Zero for s in jointsolutions]) and len(jointsolutions)>0: # check if any solutions don't have divide by zero problems returnfirstsolutions.append(AST.SolverSolution(var.name,jointeval=jointsolutions,isHinge=self.IsHinge(var.name))) hasdividebyzero = any([len(self.checkForDivideByZero(self._SubstituteGlobalSymbols(s)))>0 for s in jointsolutions]) if not hasdividebyzero: return returnfirstsolutions except NotImplementedError as e: # when solve cannot solve an equation log.warn(e) numvar = self.countVariables(eqnew,varsym.htvar) if Poly(eqnew,varsym.htvar).TC() != S.Zero and numvar >= 1 and numvar <= 2: try: tempsolutions = solve(eqnew,varsym.htvar) jointsolutions = [] for s in tempsolutions: s2 = s.subs(symbols) s3 = self.trigsimp(s2,othersolvedvars) s4 = self.SimplifyTransform(s3) try: # check s4 == 0, becasue of a bug in evalf() funcion atan(0, evaluate=False).evalf() fails with NoneTypeError, https://github.com/sympy/sympy/pull/1021. Fixed in sympy version==0.7.2, currently using 0.7.1 if s4 == 0: jointsolutions.append(2*atan(s4, evaluate=True)) else: jointsolutions.append(2*atan(s4, evaluate=False)) # don't evalute since chances if this being a number is very low except RuntimeError as e: log.warn('got runtime error when taking atan: %s', e) if all([self.isValidSolution(s) and s != S.Zero for s in jointsolutions]) and len(jointsolutions)>0: returnfirstsolutions.append(AST.SolverSolution(var.name,jointeval=jointsolutions,isHinge=self.IsHinge(var.name))) hasdividebyzero = any([len(self.checkForDivideByZero(self._SubstituteGlobalSymbols(s)))>0 for s in jointsolutions]) if not hasdividebyzero: return returnfirstsolutions except NotImplementedError as e: # when solve cannot solve an equation log.warn(e) if len(returnfirstsolutions) > 0: # already computed some solutions, so return them, note that this means that all solutions have a divide-by-zero condition return returnfirstsolutions solutions = [] if len(eqns) > 1: neweqns = [] listsymbols = [] symbolgen = cse_main.numbered_symbols('const') for e in eqns: enew, symbols = self.groupTerms(e.subs(varsym.subs),[varsym.cvar,varsym.svar,var], symbolgen) try: # remove coupled equations if any([(m[0]>0)+(m[1]>0)+(m[2]>0)>1 for m in Poly(enew,varsym.cvar,varsym.svar,var).monoms()]): continue except PolynomialError: continue try: # ignore any equations with degree 3 or more if max(Poly(enew,varsym.svar).degree_list()) > maxdegree or max(Poly(enew,varsym.cvar).degree_list()) > maxdegree: log.debug('ignoring equation: %r',enew) continue except PolynomialError: continue try: if Poly(enew,varsym.svar).TC() == S.Zero or Poly(enew,varsym.cvar) == S.Zero or Poly(enew,varsym.var) == S.Zero: log.debug('equation %s is allowing trivial solution for variable %s, ignoring ',e,varsym.name) continue except PolynomialError: continue rank = self.codeComplexity(enew) for s in symbols: rank += self.codeComplexity(s[1]) neweqns.append((rank,enew)) listsymbols += symbols # since we're solving for two variables, we only want to use two equations, so # start trying all the equations starting from the least complicated ones to the most until a solution is found eqcombinations = [] for eqs in combinations(neweqns,2): eqcombinations.append((eqs[0][0]+eqs[1][0],[Eq(e[1],0) for e in eqs])) eqcombinations.sort(key=lambda e: e[0]) hasgoodsolution = False for icomb,comb in enumerate(eqcombinations): # skip if too complex if len(solutions) > 0 and comb[0] > 200: break # try to solve for both sin and cos terms if not self.has(comb[1],varsym.svar) or not self.has(comb[1], varsym.cvar): continue try: s = solve(comb[1],[varsym.svar,varsym.cvar]) except (PolynomialError,CoercionFailed) as e: log.debug('solveSingleVariable: failed: %s',e) continue if s is not None: sollist = None if isinstance(s,dict): if varsym.svar in s and varsym.cvar in s: sollist = [(s[varsym.svar],s[varsym.cvar])] else: sollist = [] else: sollist = s solversolution = AST.SolverSolution(var.name,jointeval=[],isHinge=self.IsHinge(var.name)) goodsolution = 0 for svarsol,cvarsol in sollist: # solutions cannot be trivial soldiff = (svarsol-cvarsol).subs(listsymbols) soldiffComplexity = self.codeComplexity(soldiff) if soldiffComplexity < 1000 and soldiff.expand() == S.Zero: break svarComplexity = self.codeComplexity(svarsol.subs(listsymbols)) cvarComplexity = self.codeComplexity(cvarsol.subs(listsymbols)) if (svarComplexity < 600 and svarsol.subs(listsymbols).expand() == S.Zero) and (cvarComplexity < 600 and Abs(cvarsol.subs(listsymbols).expand()) - S.One != S.Zero): break if (cvarComplexity < 600 and cvarsol.subs(listsymbols).expand() == S.Zero) and (svarComplexity < 600 and Abs(svarsol.subs(listsymbols).expand()) - S.One != S.Zero): break # check the numerator and denominator if solutions are the same or for possible divide by zeros svarfrac=fraction(svarsol) svarfrac = [svarfrac[0].subs(listsymbols), svarfrac[1].subs(listsymbols)] cvarfrac=fraction(cvarsol) cvarfrac = [cvarfrac[0].subs(listsymbols), cvarfrac[1].subs(listsymbols)] if self.equal(svarfrac[0],cvarfrac[0]) and self.equal(svarfrac[1],cvarfrac[1]): break if not self.isValidSolution(svarfrac[0]) or not self.isValidSolution(svarfrac[1]) or not self.isValidSolution(cvarfrac[0]) or not self.isValidSolution(cvarfrac[1]): continue # check if there exists at least one test solution with non-zero denominators if subs is None: testeqs = [svarfrac[1].subs(othersubs),cvarfrac[1].subs(othersubs)] else: testeqs = [svarfrac[1].subs(subs).subs(othersubs),cvarfrac[1].subs(subs).subs(othersubs)] testsuccess = False for testconsistentvalue in self.testconsistentvalues: if all([testeq.subs(self.globalsymbols).subs(testconsistentvalue).evalf()!=S.Zero for testeq in testeqs]): testsuccess = True break if not testsuccess: continue scomplexity = self.codeComplexity(svarfrac[0])+self.codeComplexity(svarfrac[1]) ccomplexity = self.codeComplexity(cvarfrac[0])+self.codeComplexity(cvarfrac[1]) if scomplexity > 1200 or ccomplexity > 1200: log.debug('equation too complex for single variable solution (%d,%d).... (probably wrong?)',scomplexity,ccomplexity) break if scomplexity < 500 and len(str(svarfrac[1])) < 600: # if fractions are too long, then will take a long time to simplify, so also checkout the length fo the entire equation svarfrac[1] = simplify(svarfrac[1]) if self.chop(svarfrac[1])== 0: break if ccomplexity < 500 and len(str(cvarfrac[1])) < 600: cvarfrac[1] = simplify(cvarfrac[1]) if self.chop(cvarfrac[1])== 0: break # sometimes the returned simplest solution makes really gross approximations svarfracsimp_denom = self.SimplifyTransform(self.trigsimp(svarfrac[1],othersolvedvars)) cvarfracsimp_denom = self.SimplifyTransform(self.trigsimp(cvarfrac[1],othersolvedvars)) # self.SimplifyTransform could help in reducing denoms further... denomsequal = False if self.equal(svarfracsimp_denom,cvarfracsimp_denom): denomsequal = True elif self.equal(svarfracsimp_denom,-cvarfracsimp_denom): cvarfrac[0] = -cvarfrac[0] cvarfracsimp_denom = -cvarfracsimp_denom if self.equal(svarfracsimp_denom,cvarfracsimp_denom) and not svarfracsimp_denom.is_number: log.debug('%s solution: denominator is equal %s, doing a global substitution',var.name,svarfracsimp_denom) #denom = next(self.gsymbolgen) #solversolution.dictequations.append((denom,sign(svarfracsimp_denom))) svarsolsimp = self.SimplifyTransform(self.trigsimp(svarfrac[0],othersolvedvars))#*denom) cvarsolsimp = self.SimplifyTransform(self.trigsimp(cvarfrac[0],othersolvedvars))#*denom) solversolution.FeasibleIsZeros = False solversolution.presetcheckforzeros.append(svarfracsimp_denom) # instead of doing atan2(sign(dummy)*s, sign(dummy)*c), do atan2(s,c) + pi/2*(1-1/sign(dummy)) so equations become simpler expandedsol = atan2(svarsolsimp,cvarsolsimp) + pi/2*(-S.One + S.One/sign(svarfracsimp_denom)) else: svarfracsimp_num = self.SimplifyTransform(self.trigsimp(svarfrac[0],othersolvedvars)) cvarfracsimp_num = self.SimplifyTransform(self.trigsimp(cvarfrac[0],othersolvedvars)) svarsolsimp = svarfracsimp_num/svarfracsimp_denom cvarsolsimp = cvarfracsimp_num/cvarfracsimp_denom if svarsolsimp.is_number and cvarsolsimp.is_number: if Abs(svarsolsimp**2+cvarsolsimp**2-S.One).evalf() > 1e-10: log.debug('%s solution: atan2(%s,%s), sin/cos not on circle so ignoring',var.name,svarsolsimp,cvarsolsimp) continue svarsolsimpcomplexity = self.codeComplexity(svarsolsimp) cvarsolsimpcomplexity = self.codeComplexity(cvarsolsimp) if svarsolsimpcomplexity > 3000 or cvarsolsimpcomplexity > 3000: log.warn('new substituted solutions too complex: %d, %d', svarsolsimpcomplexity, cvarsolsimpcomplexity) continue try: expandedsol = atan2check(svarsolsimp,cvarsolsimp) except RuntimeError as e: log.warn(u'most likely got recursion error when calling atan2: %s', e) continue solversolution.FeasibleIsZeros = False log.debug('%s solution: atan2 check for joint',var.name) solversolution.jointeval.append(expandedsol) if unknownvars is not None: unsolvedsymbols = [] for unknownvar in unknownvars: if unknownvar != var: unsolvedsymbols += self.Variable(unknownvar).vars if len(unsolvedsymbols) > 0: solversolution.equationsused = [eq for eq in eqns if not eq.has(*unsolvedsymbols)] else: solversolution.equationsused = eqns if len(solversolution.equationsused) > 0: log.info('%s solution: equations used for atan2: %s',var.name, str(solversolution.equationsused)) if len(self.checkForDivideByZero(expandedsol.subs(solversolution.dictequations))) == 0: goodsolution += 1 if len(solversolution.jointeval) == len(sollist) and len(sollist) > 0: solutions.append(solversolution) if goodsolution > 0: hasgoodsolution = True if len(sollist) == goodsolution and goodsolution == 1 and len(solutions) >= 2: break if len(solutions) >= maxsolutions: # probably more than enough already? break if len(solutions) > 0 or hasgoodsolution: # found a solution without any divides, necessary for pr2 head_torso lookat3d ik return solutions # solve one equation for ieq,eq in enumerate(eqns): symbolgen = cse_main.numbered_symbols('const') eqnew, symbols = self.groupTerms(eq.subs(varsym.subs), [varsym.cvar,varsym.svar,varsym.var], symbolgen) try: # ignore any equations with degree 3 or more ps = Poly(eqnew,varsym.svar) pc = Poly(eqnew,varsym.cvar) if max(ps.degree_list()) > maxdegree or max(pc.degree_list()) > maxdegree: log.debug('cannot solve equation with high degree: %s',str(eqnew)) continue if ps.TC() == S.Zero and len(ps.monoms()) > 0: log.debug('equation %s has trivial solution, ignoring...', ps) continue if pc.TC() == S.Zero and len(pc.monoms()) > 0: log.debug('equation %s has trivial solution, ignoring...', pc) continue except PolynomialError: # might not be a polynomial, so ignore continue equationsused = None if unknownvars is not None: unsolvedsymbols = [] for unknownvar in unknownvars: if unknownvar != var: unsolvedsymbols += self.Variable(unknownvar).vars if len(unsolvedsymbols) > 0: equationsused = [eq2 for ieq2,eq2 in enumerate(eqns) if ieq2!=ieq and not eq2.has(*unsolvedsymbols)] else: equationsused = eqns[:] equationsused.pop(ieq) numcvar = self.countVariables(eqnew,varsym.cvar) numsvar = self.countVariables(eqnew,varsym.svar) if numcvar == 1 and numsvar == 1: a = Wild('a',exclude=[varsym.svar,varsym.cvar]) b = Wild('b',exclude=[varsym.svar,varsym.cvar]) c = Wild('c',exclude=[varsym.svar,varsym.cvar]) m = eqnew.match(a*varsym.cvar+b*varsym.svar+c) if m is not None: symbols += [(varsym.svar,sin(var)),(varsym.cvar,cos(var))] asinsol = trigsimp(asin(-m[c]/Abs(sqrt(m[a]*m[a]+m[b]*m[b]))).subs(symbols),deep=True) # can't use atan2().evalf()... maybe only when m[a] or m[b] is complex? if m[a].has(I) or m[b].has(I): continue constsol = (-atan2(m[a], m[b]).subs(symbols)).evalf() jointsolutions = [constsol+asinsol,constsol+pi.evalf()-asinsol] if not constsol.has(I) and all([self.isValidSolution(s) and self.isValidSolution(s) for s in jointsolutions]) and len(jointsolutions) > 0: #self.checkForDivideByZero(expandedsol) solutions.append(AST.SolverSolution(var.name,jointeval=jointsolutions,isHinge=self.IsHinge(var.name))) solutions[-1].equationsused = equationsused continue if numcvar > 0: try: # substitute cos if self.countVariables(eqnew,varsym.svar) <= 1 or (self.countVariables(eqnew,varsym.cvar) <= 2 and self.countVariables(eqnew,varsym.svar) == 0): # anything more than 1 implies quartic equation tempsolutions = solve(eqnew.subs(varsym.svar,sqrt(1-varsym.cvar**2)).expand(),varsym.cvar) jointsolutions = [] for s in tempsolutions: s2 = self.trigsimp(s.subs(symbols+varsym.subsinv),othersolvedvars) if self.isValidSolution(s2): jointsolutions.append(self.SimplifyTransform(s2)) if len(jointsolutions) > 0 and all([self.isValidSolution(s) and self.isValidSolution(s) for s in jointsolutions]): solutions.append(AST.SolverSolution(var.name,jointevalcos=jointsolutions,isHinge=self.IsHinge(var.name))) solutions[-1].equationsused = equationsused continue except self.CannotSolveError as e: log.debug(e) except NotImplementedError as e: # when solve cannot solve an equation log.warn(e) if numsvar > 0: # substitute sin try: if self.countVariables(eqnew,varsym.svar) <= 1 or (self.countVariables(eqnew,varsym.svar) <= 2 and self.countVariables(eqnew,varsym.cvar) == 0): # anything more than 1 implies quartic equation tempsolutions = solve(eqnew.subs(varsym.cvar,sqrt(1-varsym.svar**2)).expand(),varsym.svar) jointsolutions = [self.SimplifyTransform(self.trigsimp(s.subs(symbols+varsym.subsinv),othersolvedvars)) for s in tempsolutions] if all([self.isValidSolution(s) and self.isValidSolution(s) for s in jointsolutions]) and len(jointsolutions) > 0: solutions.append(AST.SolverSolution(var.name,jointevalsin=jointsolutions,isHinge=self.IsHinge(var.name))) solutions[-1].equationsused = equationsused continue except self.CannotSolveError as e: log.debug(e) except NotImplementedError as e: # when solve cannot solve an equation log.warn(e) if numcvar == 0 and numsvar == 0: try: tempsolutions = solve(eqnew,var) jointsolutions = [] for s in tempsolutions: eqsub = s.subs(symbols) if self.codeComplexity(eqsub) < 2000: eqsub = self.SimplifyTransform(self.trigsimp(eqsub,othersolvedvars)) jointsolutions.append(eqsub) if all([self.isValidSolution(s) and s != S.Zero for s in jointsolutions]) and len(jointsolutions) > 0: solutions.append(AST.SolverSolution(var.name,jointeval=jointsolutions,isHinge=self.IsHinge(var.name))) solutions[-1].equationsused = equationsused except NotImplementedError as e: # when solve cannot solve an equation log.warn(e) continue try: solution = self.solveHighDegreeEquationsHalfAngle([eqnew],varsym,symbols) solutions.append(solution.subs(symbols)) solutions[-1].equationsused = equationsused except self.CannotSolveError as e: log.debug(e) if len(solutions) > 0: return solutions return [self.solveHighDegreeEquationsHalfAngle(eqns,varsym)] def SolvePrismaticHingePairVariables(self, raweqns, var0,var1,othersolvedvars,unknownvars=None): """solves one hinge and one prismatic variable together """ if self.IsPrismatic(var0.name) and self.IsHinge(var1.name): prismaticSymbol = var0 hingeSymbol = var1 elif self.IsHinge(var0.name) and self.IsPrismatic(var1.name): hingeSymbol = var0 prismaticSymbol = var1 else: raise self.CannotSolveError('need to have one hinge and one prismatic variable') prismaticVariable = self.Variable(prismaticSymbol) hingeVariable = self.Variable(hingeSymbol) chingeSymbol,shingeSymbol = hingeVariable.cvar, hingeVariable.svar varsubs=prismaticVariable.subs+hingeVariable.subs varsubsinv = prismaticVariable.subsinv+hingeVariable.subsinv unknownvars=[chingeSymbol,shingeSymbol,prismaticSymbol] reducesubs = [(shingeSymbol**2,1-chingeSymbol**2)] polyeqs = [Poly(eq.subs(varsubs).subs(reducesubs).expand(),unknownvars) for eq in raweqns if eq.has(prismaticSymbol,hingeSymbol)] if len(polyeqs) <= 1: raise self.CannotSolveError('not enough equations') # try to solve one variable in terms of the others solvevariables = [] for polyeq in polyeqs: if polyeq.degree(0) == 1 and polyeq.degree(1) == 0: chingeSolutions = solve(polyeq,chingeSymbol) solvevariables.append((prismaticSymbol,[(chingeSymbol,chingeSolutions[0])])) elif polyeq.degree(0) == 0 and polyeq.degree(1) == 1: shingeSolutions = solve(polyeq,shingeSymbol) solvevariables.append((prismaticSymbol,[(shingeSymbol,shingeSolutions[0])])) elif polyeq.degree(2) == 1: prismaticSolutions = solve(polyeq,prismaticSymbol) solvevariables.append((hingeSymbol,[(prismaticSymbol,prismaticSolutions[0])])) # prioritize solving the hingeSymbol out for solveSymbol in [hingeSymbol,prismaticSymbol]: for solveSymbol2, solvesubs in solvevariables: if solveSymbol == solveSymbol2: # have a solution for one variable, so substitute it in and see if the equations become solvable with one variable reducedeqs = [] for polyeq2 in polyeqs: eqnew = simplify(polyeq2.as_expr().subs(solvesubs)) if eqnew != S.Zero: reducedeqs.append(eqnew) self.sortComplexity(reducedeqs) try: rawsolutions = self.solveSingleVariable(reducedeqs,solveSymbol,othersolvedvars, unknownvars=unknownvars) if len(rawsolutions) > 0: return rawsolutions except self.CannotSolveError: pass raise self.CannotSolveError(u'SolvePrismaticHingePairVariables: failed to find variable with degree 1') def SolvePairVariables(self,raweqns,var0,var1,othersolvedvars,maxcomplexity=50,unknownvars=None): """solves two hinge variables together """ # make sure both variables are hinges if not self.IsHinge(var0.name) or not self.IsHinge(var1.name): raise self.CannotSolveError('pairwise variables only supports hinge joints') varsym0 = self.Variable(var0) varsym1 = self.Variable(var1) cvar0,svar0 = varsym0.cvar, varsym0.svar cvar1,svar1 = varsym1.cvar, varsym1.svar varsubs=varsym0.subs+varsym1.subs varsubsinv = varsym0.subsinv+varsym1.subsinv unknownvars=[cvar0,svar0,cvar1,svar1] reducesubs = [(svar0**2,1-cvar0**2),(svar1**2,1-cvar1**2)] eqns = [eq.subs(varsubs).subs(reducesubs).expand() for eq in raweqns if eq.has(var0,var1)] if len(eqns) <= 1: raise self.CannotSolveError('not enough equations') # group equations with single variables symbolgen = cse_main.numbered_symbols('const') orgeqns = [] allsymbols = [] for eq in eqns: eqnew, symbols = self.groupTerms(eq, unknownvars, symbolgen) allsymbols += symbols orgeqns.append([self.codeComplexity(eq),Poly(eqnew,*unknownvars)]) orgeqns.sort(key=lambda e: e[0]) neweqns = orgeqns[:] pairwisesubs = [(svar0*cvar1,Symbol('s0c1')),(svar0*svar1,Symbol('s0s1')),(cvar0*cvar1,Symbol('c0c1')),(cvar0*svar1,Symbol('c0s1')),(cvar0*svar0,Symbol('s0c0')),(cvar1*svar1,Symbol('c1s1'))] pairwiseinvsubs = [(f[1],f[0]) for f in pairwisesubs] pairwisevars = [f[1] for f in pairwisesubs] reduceeqns = [Poly(eq.as_expr().subs(pairwisesubs),*pairwisevars) for rank,eq in orgeqns if rank < 4*maxcomplexity] for i,eq in enumerate(reduceeqns): if eq.TC != S.Zero and not eq.TC().is_Symbol: n=next(symbolgen) allsymbols.append((n,eq.TC().subs(allsymbols))) reduceeqns[i] += n-eq.TC() # try to at least subtract as much paired variables out eqcombs = [c for c in combinations(reduceeqns,2)] while len(eqcombs) > 0 and len(neweqns) < 20: eq0,eq1 = eqcombs.pop() eq0dict = eq0.as_dict() eq1dict = eq1.as_dict() for i in range(6): monom = [0,0,0,0,0,0] monom[i] = 1 eq0value = eq0dict.get(tuple(monom),S.Zero) eq1value = eq1dict.get(tuple(monom),S.Zero) if eq0value != 0 and eq1value != 0: tempeq = (eq0.as_expr()*eq1value-eq0value*eq1.as_expr()).subs(allsymbols+pairwiseinvsubs).expand() if self.codeComplexity(tempeq) > 200: continue eq = simplify(tempeq) if eq == S.Zero: continue peq = Poly(eq,*pairwisevars) if max(peq.degree_list()) > 0 and self.codeComplexity(eq) > maxcomplexity: # don't need such complex equations continue if not self.CheckExpressionUnique(eqns,eq): continue if eq.has(*unknownvars): # be a little strict about new candidates eqns.append(eq) eqnew, symbols = self.groupTerms(eq, unknownvars, symbolgen) allsymbols += symbols neweqns.append([self.codeComplexity(eq),Poly(eqnew,*unknownvars)]) orgeqns = neweqns[:] # try to solve for all pairwise variables systemofequations = [] for i in range(len(reduceeqns)): if reduceeqns[i].has(pairwisevars[4],pairwisevars[5]): continue if not all([__builtin__.sum(m) <= 1 for m in reduceeqns[i].monoms()]): continue arr = [S.Zero]*5 for m,c in reduceeqns[i].terms(): if __builtin__.sum(m) == 1: arr[list(m).index(1)] = c else: arr[4] = c systemofequations.append(arr) if len(systemofequations) >= 4: singleeqs = None for eqs in combinations(systemofequations,4): M = zeros((4,4)) B = zeros((4,1)) for i,arr in enumerate(eqs): for j in range(4): M[i,j] = arr[j] B[i] = -arr[4] det = self.det_bareis(M,*(self.pvars+unknownvars)).subs(allsymbols) if det.evalf() != S.Zero: X = M.adjugate()*B singleeqs = [] for i in range(4): eq = (pairwisesubs[i][0]*det - X[i]).subs(allsymbols) eqnew, symbols = self.groupTerms(eq, unknownvars, symbolgen) allsymbols += symbols singleeqs.append([self.codeComplexity(eq),Poly(eqnew,*unknownvars)]) break if singleeqs is not None: neweqns += singleeqs neweqns.sort(key=lambda e: e[0]) # check if any equations are at least degree 1 (if not, try to compute some) for ivar in range(2): polyunknown = [] for rank,eq in orgeqns: p = Poly(eq,unknownvars[2*ivar],unknownvars[2*ivar+1]) if sum(p.degree_list()) == 1 and __builtin__.sum(p.LM()) == 1: polyunknown.append((rank,p)) if len(polyunknown) > 0: break if len(polyunknown) == 0: addedeqs = eqns[:] polyeqs = [] for ivar in range(2): polyunknown = [] for rank,eq in orgeqns: p = Poly(eq,unknownvars[2*ivar],unknownvars[2*ivar+1]) polyunknown.append(Poly(p.subs(unknownvars[2*ivar+1]**2,1-unknownvars[2*ivar]**2),unknownvars[2*ivar],unknownvars[2*ivar+1])) if len(polyunknown) >= 2: monomtoremove = [[polyunknown,(2,0)],[polyunknown,(1,1)]] for curiter in range(2): # remove the square polyunknown,monom = monomtoremove[curiter] pbase = [p for p in polyunknown if p.as_dict().get(monom,S.Zero) != S.Zero] if len(pbase) == 0: continue pbase = pbase[0] pbasedict = pbase.as_dict() for i in range(len(polyunknown)): eq = (polyunknown[i]*pbasedict.get(monom,S.Zero)-pbase*polyunknown[i].as_dict().get(monom,S.Zero)).as_expr().subs(allsymbols) if self.codeComplexity(eq) > 4000: # .. way too complex continue eq = eq.expand() if self.codeComplexity(eq) > 10000: # .. way too complex continue if len(addedeqs) > 10 and self.codeComplexity(eq) > 2000: # .. already have enough... continue if eq != S.Zero and self.CheckExpressionUnique(addedeqs,eq): eqnew, symbols = self.groupTerms(eq, unknownvars, symbolgen) allsymbols += symbols p = Poly(eqnew,*pbase.gens) if p.as_dict().get((1,1),S.Zero) != S.Zero and curiter == 0: monomtoremove[1][0].insert(0,p) polyeqs.append([self.codeComplexity(eqnew),Poly(eqnew,*unknownvars)]) addedeqs.append(eq) neweqns += polyeqs neweqns.sort(key=lambda e: e[0]) rawsolutions = [] # try single variable solution, only return if a single solution has been found # returning multiple solutions when only one exists can lead to wrong results. try: rawsolutions += self.solveSingleVariable(self.sortComplexity([e.as_expr().subs(varsubsinv).expand() for score,e in neweqns if not e.has(cvar1,svar1,var1)]),var0,othersolvedvars,subs=allsymbols,unknownvars=unknownvars) except self.CannotSolveError: pass try: rawsolutions += self.solveSingleVariable(self.sortComplexity([e.as_expr().subs(varsubsinv).expand() for score,e in neweqns if not e.has(cvar0,svar0,var0)]),var1,othersolvedvars,subs=allsymbols,unknownvars=unknownvars) except self.CannotSolveError: pass if len(rawsolutions) > 0: solutions = [] for s in rawsolutions: try: solutions.append(s.subs(allsymbols)) except self.CannotSolveError: pass if len(solutions) > 0: return solutions groups=[] for i,unknownvar in enumerate(unknownvars): listeqs = [] listeqscmp = [] for rank,eq in neweqns: # if variable ever appears, it should be alone if all([m[i] == 0 or (__builtin__.sum(m) == m[i] and m[i]>0) for m in eq.monoms()]) and any([m[i] > 0 for m in eq.monoms()]): # make sure there's only one monom that includes other variables othervars = [__builtin__.sum(m) - m[i] > 0 for m in eq.monoms()] if __builtin__.sum(othervars) <= 1: eqcmp = self.removecommonexprs(eq.subs(allsymbols).as_expr(),onlynumbers=False,onlygcd=True) if self.CheckExpressionUnique(listeqscmp,eqcmp): listeqs.append(eq) listeqscmp.append(eqcmp) groups.append(listeqs) # find a group that has two or more equations: useconic=False goodgroup = [(i,g) for i,g in enumerate(groups) if len(g) >= 2] if len(goodgroup) == 0: # might have a set of equations that can be solved with conics # look for equations where the variable and its complement are alone groups=[] for i in [0,2]: unknownvar = unknownvars[i] complementvar = unknownvars[i+1] listeqs = [] listeqscmp = [] for rank,eq in neweqns: # if variable ever appears, it should be alone addeq = False if all([__builtin__.sum(m) == m[i]+m[i+1] for m in eq.monoms()]): addeq = True else: # make sure there's only one monom that includes other variables othervars = 0 for m in eq.monoms(): if __builtin__.sum(m) > m[i]+m[i+1]: if m[i] == 0 and m[i+1]==0: othervars += 1 else: othervars = 10000 if othervars <= 1: addeq = True if addeq: eqcmp = self.removecommonexprs(eq.subs(allsymbols).as_expr(),onlynumbers=False,onlygcd=True) if self.CheckExpressionUnique(listeqscmp,eqcmp): listeqs.append(eq) listeqscmp.append(eqcmp) groups.append(listeqs) groups.append([]) # necessary to get indices correct goodgroup = [(i,g) for i,g in enumerate(groups) if len(g) >= 2] useconic=True if len(goodgroup) == 0: try: return self.SolvePairVariablesHalfAngle(raweqns,var0,var1,othersolvedvars) except self.CannotSolveError as e: log.warn('%s',e) # try a separate approach where the two variables are divided on both sides neweqs = [] for rank,eq in neweqns: p = Poly(eq,unknownvars[0],unknownvars[1]) iscoupled = False for m,c in p.terms(): if __builtin__.sum(m) > 0: if c.has(unknownvars[2],unknownvars[3]): iscoupled = True break if not iscoupled: neweqs.append([p-p.TC(),Poly(-p.TC(),unknownvars[2],unknownvars[3])]) if len(neweqs) > 0: for ivar in range(2): lineareqs = [eq for eq in neweqs if __builtin__.sum(eq[ivar].LM())==1] for paireq0,paireq1 in combinations(lineareqs,2): log.info('solving separated equations with linear terms') eq0 = paireq0[ivar] eq0dict = eq0.as_dict() eq1 = paireq1[ivar] eq1dict = eq1.as_dict() disc = (eq0dict.get((1,0),S.Zero)*eq1dict.get((0,1),S.Zero) - eq0dict.get((0,1),S.Zero)*eq1dict.get((1,0),S.Zero)).subs(allsymbols).expand() if disc == S.Zero: continue othereq0 = paireq0[1-ivar].as_expr() - eq0.TC() othereq1 = paireq1[1-ivar].as_expr() - eq1.TC() csol = - eq1dict.get((0,1),S.Zero) * othereq0 + eq0dict.get((0,1),S.Zero) * othereq1 ssol = eq1dict.get((1,0),S.Zero) * othereq0 - eq0dict.get((1,0),S.Zero) * othereq1 polysymbols = paireq0[1-ivar].gens totaleq = (csol**2+ssol**2-disc**2).subs(allsymbols).expand() if self.codeComplexity(totaleq) < 4000: log.info('simplifying final equation to %d',self.codeComplexity(totaleq)) totaleq = simplify(totaleq) ptotal_cos = Poly(Poly(totaleq,*polysymbols).subs(polysymbols[0]**2,1-polysymbols[1]**2).subs(polysymbols[1]**2,1-polysymbols[0]**2),*polysymbols) ptotal_sin = Poly(S.Zero,*polysymbols) for m,c in ptotal_cos.terms(): if m[1] > 0: assert(m[1] == 1) ptotal_sin = ptotal_sin.sub(Poly.from_dict({(m[0],0):c},*ptotal_sin.gens)) ptotal_cos = ptotal_cos.sub(Poly.from_dict({m:c},*ptotal_cos.gens)) ptotalcomplexity = self.codeComplexity(ptotal_cos.as_expr()) + self.codeComplexity(ptotal_sin.as_expr()) if ptotalcomplexity < 50000: #log.info('ptotal complexity is %d', ptotalcomplexity) finaleq = (ptotal_cos.as_expr()**2 - (1-polysymbols[0]**2)*ptotal_sin.as_expr()**2).expand() # sometimes denominators can accumulate pfinal = Poly(self.removecommonexprs(finaleq,onlygcd=False,onlynumbers=True),polysymbols[0]) pfinal = self.checkFinalEquation(pfinal) if pfinal is not None: jointsol = atan2(ptotal_cos.as_expr()/ptotal_sin.as_expr(), polysymbols[0]) var = var1 if ivar == 0 else var0 solution = AST.SolverPolynomialRoots(jointname=var.name,poly=pfinal,jointeval=[jointsol],isHinge=self.IsHinge(var.name)) solution.postcheckforzeros = [ptotal_sin.as_expr()] solution.postcheckfornonzeros = [] solution.postcheckforrange = [] return [solution] # if maxnumeqs is any less, it will miss linearly independent equations lineareqs = self.solveSingleVariableLinearly(raweqns,var0,[var1],maxnumeqs=len(raweqns)) if len(lineareqs) > 0: try: return [self.solveHighDegreeEquationsHalfAngle(lineareqs,varsym1)] except self.CannotSolveError as e: log.warn('%s',e) raise self.CannotSolveError('cannot cleanly separate pair equations') varindex=goodgroup[0][0] var = var0 if varindex < 2 else var1 varsym = varsym0 if varindex < 2 else varsym1 unknownvar=unknownvars[goodgroup[0][0]] eqs = goodgroup[0][1][0:2] simpleterms = [] complexterms = [] domagicsquare = False for i in range(2): if useconic: terms=[(c,m) for m,c in eqs[i].terms() if __builtin__.sum(m) - m[varindex] - m[varindex+1] > 0] else: terms=[(c,m) for m,c in eqs[i].terms() if __builtin__.sum(m) - m[varindex] > 0] if len(terms) > 0: simpleterms.append(eqs[i].sub(Poly.from_dict({terms[0][1]:terms[0][0]},*eqs[i].gens)).as_expr()/terms[0][0]) # divide by the coeff complexterms.append(Poly({terms[0][1]:S.One},*unknownvars).as_expr()) domagicsquare = True else: simpleterms.append(eqs[i].as_expr()) complexterms.append(S.Zero) finaleq = None checkforzeros = [] if domagicsquare: # here is the magic transformation: finaleq = self.trigsimp(expand(((complexterms[0]**2+complexterms[1]**2) - simpleterms[0]**2 - simpleterms[1]**2).subs(varsubsinv)),othersolvedvars+[var0,var1]).subs(varsubs) denoms = [fraction(simpleterms[0])[1], fraction(simpleterms[1])[1], fraction(complexterms[0])[1], fraction(complexterms[1])[1]] lcmvars = self.pvars+unknownvars for othersolvedvar in othersolvedvars: lcmvars += self.Variable(othersolvedvar).vars denomlcm = Poly(S.One,*lcmvars) for denom in denoms: if denom != S.One: checkforzeros.append(self.removecommonexprs(denom,onlygcd=False,onlynumbers=True)) denomlcm = Poly(lcm(denomlcm,denom),*lcmvars) finaleq = simplify(finaleq*denomlcm.as_expr()**2) complementvarindex = varindex-(varindex%2)+((varindex+1)%2) complementvar = unknownvars[complementvarindex] finaleq = simplify(finaleq.subs(complementvar**2,1-unknownvar**2)).subs(allsymbols).expand() else: # try to reduce finaleq p0 = Poly(simpleterms[0],unknownvars[varindex],unknownvars[varindex+1]) p1 = Poly(simpleterms[1],unknownvars[varindex],unknownvars[varindex+1]) if max(p0.degree_list()) > 1 and max(p1.degree_list()) > 1 and max(p0.degree_list()) == max(p1.degree_list()) and p0.LM() == p1.LM(): finaleq = (p0*p1.LC()-p1*p0.LC()).as_expr() finaleq = expand(simplify(finaleq.subs(allsymbols))) if finaleq == S.Zero: finaleq = expand(p0.as_expr().subs(allsymbols)) if finaleq is None: log.warn('SolvePairVariables: did not compute a final variable. This is a weird condition...') return self.SolvePairVariablesHalfAngle(raweqns,var0,var1,othersolvedvars) if not self.isValidSolution(finaleq): log.warn('failed to solve pairwise equation: %s'%str(finaleq)) return self.SolvePairVariablesHalfAngle(raweqns,var0,var1,othersolvedvars) newunknownvars = unknownvars[:] newunknownvars.remove(unknownvar) if finaleq.has(*newunknownvars): log.warn('equation relies on unsolved variables(%s): %s',newunknownvars,finaleq) return self.SolvePairVariablesHalfAngle(raweqns,var0,var1,othersolvedvars) if not finaleq.has(unknownvar): # somehow removed all variables, so try the general method return self.SolvePairVariablesHalfAngle(raweqns,var0,var1,othersolvedvars) try: if self.codeComplexity(finaleq) > 100000: return self.SolvePairVariablesHalfAngle(raweqns,var0,var1,othersolvedvars) except self.CannotSolveError: pass if useconic: # conic roots solver not as robust as half-angle transform! #return [SolverConicRoots(var.name,[finaleq],isHinge=self.IsHinge(var.name))] solution = self.solveHighDegreeEquationsHalfAngle([finaleq],varsym) solution.checkforzeros += checkforzeros return [solution] # now that everything is with respect to one variable, simplify and solve the equation eqnew, symbols = self.groupTerms(finaleq, unknownvars, symbolgen) allsymbols += symbols solutions=solve(eqnew,unknownvar) log.info('pair solution: %s, %s', eqnew,solutions) if solutions: solversolution=AST.SolverSolution(var.name, isHinge=self.IsHinge(var.name)) processedsolutions = [] for s in solutions: processedsolution = s.subs(allsymbols+varsubsinv).subs(varsubs) # trigsimp probably won't work on long solutions if self.codeComplexity(processedsolution) < 2000: # complexity of 2032 for pi robot freezes log.info('solution complexity: %d', self.codeComplexity(processedsolution)) processedsolution = self.SimplifyTransform(self.trigsimp(processedsolution,othersolvedvars)) processedsolutions.append(processedsolution.subs(varsubs)) if (varindex%2)==0: solversolution.jointevalcos=processedsolutions else: solversolution.jointevalsin=processedsolutions return [solversolution] return self.SolvePairVariablesHalfAngle(raweqns,var0,var1,othersolvedvars) #raise self.CannotSolveError('cannot solve pair equation') ## SymPy helper routines @staticmethod def isValidSolution(expr): """return true if solution does not contain any nan or inf terms""" if expr.is_number: e=expr.evalf() if e.has(I) or isinf(e) or isnan(e): return False return True if expr.is_Mul: # first multiply all numbers number = S.One for arg in expr.args: if arg.is_number: number *= arg elif not IKFastSolver.isValidSolution(arg): return False # finally evalute the multiplied form return IKFastSolver.isValidSolution(number.evalf()) for arg in expr.args: if not IKFastSolver.isValidSolution(arg): return False return True @staticmethod def _GetSumSquares(expr): """if expr is a sum of squares, returns the list of individual expressions that were squared. otherwise returns None """ values = [] if expr.is_Add: for arg in expr.args: if arg.is_Pow and arg.exp.is_number and arg.exp > 0 and (arg.exp%2) == 0: values.append(arg.base) else: return [] elif expr.is_Mul: values = IKFastSolver._GetSumSquares(expr.args[0]) for arg in expr.args[1:]: values2 = IKFastSolver._GetSumSquares(arg) if len(values2) > 0: values = [x*y for x,y in product(values,values2)] else: values = [x*arg for x in values] return values else: if expr.is_Pow and expr.exp.is_number and expr.exp > 0 and (expr.exp%2) == 0: values.append(expr.base) return values @staticmethod def recursiveFraction(expr): """return the numerator and denominator of th eexpression as if it was one fraction """ if expr.is_Add: allpoly = [] finaldenom = S.One for arg in expr.args: n,d = IKFastSolver.recursiveFraction(arg) finaldenom = finaldenom*d allpoly.append([n,d]) finalnum = S.Zero for n,d in allpoly: finalnum += n*(finaldenom/d) return finalnum,finaldenom elif expr.is_Mul: finalnum = S.One finaldenom = S.One for arg in expr.args: n,d = IKFastSolver.recursiveFraction(arg) finalnum = finalnum * n finaldenom = finaldenom * d return finalnum,finaldenom elif expr.is_Pow and expr.exp.is_number: n,d=IKFastSolver.recursiveFraction(expr.base) if expr.exp < 0: exponent = -expr.exp n,d = d,n else: exponent = expr.exp return n**exponent,d**exponent else: return fraction(expr) @staticmethod def groupTerms(expr,vars,symbolgen = None): """Separates all terms that do have var in them""" if symbolgen is None: symbolgen = cse_main.numbered_symbols('const') symbols = [] try: p = Poly(expr,*vars) except PolynomialError: return expr, symbols newexpr = S.Zero for m,c in p.terms(): # make huge numbers into constants too if (c.is_number and len(str(c)) > 40) or (not c.is_number and not c.is_Symbol): # if it is a product of a symbol and a number, then ignore if not c.is_Mul or not all([e.is_number or e.is_Symbol for e in c.args]): sym = next(symbolgen) symbols.append((sym,c)) c = sym if __builtin__.sum(m) == 0: newexpr += c else: for i,degree in enumerate(m): c = c*vars[i]**degree newexpr += c return newexpr,symbols @staticmethod def replaceNumbers(expr,symbolgen = None): """Replaces all numbers with symbols, this is to make gcd faster when fractions get too big""" if symbolgen is None: symbolgen = cse_main.numbered_symbols('const') symbols = [] if expr.is_number: result = next(symbolgen) symbols.append((result,expr)) elif expr.is_Mul: result = S.One for arg in expr.args: newresult, newsymbols = IKFastSolver.replaceNumbers(arg,symbolgen) result *= newresult symbols += newsymbols elif expr.is_Add: result = S.Zero for arg in expr.args: newresult, newsymbols = IKFastSolver.replaceNumbers(arg,symbolgen) result += newresult symbols += newsymbols elif expr.is_Pow: # don't replace the exponent newresult, newsymbols = IKFastSolver.replaceNumbers(expr.base,symbolgen) symbols += newsymbols result = newresult**expr.exp else: result = expr return result,symbols @staticmethod def frontnumbers(eq): if eq.is_Number: return [eq] if eq.is_Mul: n = [] for arg in eq.args: n += IKFastSolver.frontnumbers(arg) return n return [] def IsAnyImaginaryByEval(self, eq): """checks if an equation ever evaluates to an imaginary number """ for testconsistentvalue in self.testconsistentvalues: value = eq.subs(testconsistentvalue).evalf() if value.is_complex and not value.is_real: return True return False def AreAllImaginaryByEval(self, eq): """checks if an equation ever evaluates to an imaginary number """ for testconsistentvalue in self.testconsistentvalues: value = eq.subs(testconsistentvalue).evalf() if not (value.is_complex and not value.is_real): return False return True def IsDeterminantNonZeroByEval(self, A, evalfirst=True): """checks if a determinant is non-zero by evaluating all the possible solutions. :param evalfirst: if True, then call evalf() first before any complicated operation in order to avoid freezes. Set this to false to get more accurate results when A is known to be simple. :return: True if there exist values where det(A) is not zero """ N = A.shape[0] thresh = 0.0003**N # when translationdirection5d is used with direction that is 6+ digits, the determinent gets small... pi_robot requires 0.0003**N if evalfirst: if thresh > 1e-14: # make sure thresh isn't too big... thresh = 1e-14 else: # can have a tighter thresh since evaluating last... if thresh > 1e-40: # make sure thresh isn't too big... thresh = 1e-40 nummatrixsymbols = __builtin__.sum([1 for a in A if not a.is_number]) if nummatrixsymbols == 0: if evalfirst: return abs(A.evalf().det()) > thresh else: return abs(A.det().evalf()) > thresh for testconsistentvalue in self.testconsistentvalues: if evalfirst: detvalue = A.subs(testconsistentvalue).evalf().det() else: detvalue = A.subs(testconsistentvalue).det().evalf() if abs(detvalue) > thresh: return True return False @staticmethod def removecommonexprs(eq,returncommon=False,onlygcd=False,onlynumbers=True): """removes common expressions from a sum. Assumes all the coefficients are rationals. For example: a*c_0 + a*c_1 + a*c_2 = 0 will return in c_0 + c_1 + c_2 = 0 """ eq = eq.expand() # doesn't work otherwise if eq.is_Add: exprs = eq.args totaldenom = S.One common = S.One if onlynumbers: for i in range(len(exprs)): denom = S.One for d in IKFastSolver.frontnumbers(fraction(exprs[i])[1]): denom *= d if denom != S.One: exprs = [expr*denom for expr in exprs] totaldenom *= denom if onlygcd: common = None for i in range(len(exprs)): coeff = S.One for n in IKFastSolver.frontnumbers(exprs[i]): coeff *= n if common == None: common = coeff else: common = igcd(common,coeff) if common == S.One: break else: for i in range(len(exprs)): denom = fraction(exprs[i])[1] if denom != S.One: exprs = [expr*denom for expr in exprs] totaldenom *= denom # there are no fractions, so can start simplifying common = exprs[0]/fraction(cancel(exprs[0]/exprs[1]))[0] for i in range(2,len(exprs)): common = common/fraction(cancel(common/exprs[i]))[0] if common.is_number: common=S.One # find the smallest number and divide by it if not onlygcd: smallestnumber = None for expr in exprs: if expr.is_number: if smallestnumber is None or smallestnumber > Abs(expr): smallestnumber = Abs(expr) elif expr.is_Mul: n = S.One for arg in expr.args: if arg.is_number: n *= arg if smallestnumber is None or smallestnumber > Abs(n): smallestnumber = Abs(n) if smallestnumber is not None: common = common*smallestnumber eq = S.Zero for expr in exprs: eq += expr/common if returncommon: return eq,common/totaldenom elif eq.is_Mul: coeff = S.One for d in IKFastSolver.frontnumbers(eq): coeff *= d if returncommon: return eq/coeff,coeff return eq/coeff if returncommon: return eq,S.One return eq # def det_bareis(M,*vars,**kwargs): # return M.det_bareis() @staticmethod def det_bareis(M,*vars,**kwargs): """Function from sympy with a couple of improvements. Compute matrix determinant using Bareis' fraction-free algorithm which is an extension of the well known Gaussian elimination method. This approach is best suited for dense symbolic matrices and will result in a determinant with minimal number of fractions. It means that less term rewriting is needed on resulting formulae. TODO: Implement algorithm for sparse matrices (SFF). Function from sympy/matrices/matrices.py """ if not M.is_square: raise NonSquareMatrixException() n = M.rows M = M[:,:] # make a copy if n == 1: det = M[0, 0] elif n == 2: det = M[0, 0]*M[1, 1] - M[0, 1]*M[1, 0] else: sign = 1 # track current sign in case of column swap for k in range(n-1): # look for a pivot in the current column # and assume det == 0 if none is found if M[k, k] == 0: for i in range(k+1, n): if M[i, k] != 0: M.row_swap(i, k) sign *= -1 break else: return S.Zero # proceed with Bareis' fraction-free (FF) # form of Gaussian elimination algorithm for i in range(k+1, n): for j in range(k+1, n): D = M[k, k]*M[i, j] - M[i, k]*M[k, j] if k > 0: if len(vars) > 0 and D != S.Zero and not M[k-1, k-1].is_number: try: D,r = div(Poly(D,*vars),M[k-1, k-1]) except UnificationFailed: log.warn('unification failed, trying direct division') D /= M[k-1, k-1] else: D /= M[k-1, k-1] if D.is_Atom: M[i, j] = D else: if len(vars) > 0: M[i, j] = D else: M[i, j] = Poly.cancel(D) det = sign * M[n-1, n-1] return det.expand() @staticmethod def LUdecompositionFF(self,*vars): """ Compute a fraction-free LU decomposition. Returns 4 matrices P, L, D, U such that PA = L D**-1 U. If the elements of the matrix belong to some integral domain I, then all elements of L, D and U are guaranteed to belong to I. **Reference** - W. Zhou & D.J. Jeffrey, "Fraction-free matrix factors: new forms for LU and QR factors". Frontiers in Computer Science in China, Vol 2, no. 1, pp. 67-80, 2008. """ n, m = self.rows, self.cols U, L, P = self[:,:], eye(n), eye(n) DD = zeros(n) # store it smarter since it's just diagonal oldpivot = S.One for k in range(n-1): log.info('row=%d', k) if U[k,k] == 0: for kpivot in range(k+1, n): if U[kpivot, k] != 0: break else: raise ValueError("Matrix is not full rank") U[k, k:], U[kpivot, k:] = U[kpivot, k:], U[k, k:] L[k, :k], L[kpivot, :k] = L[kpivot, :k], L[k, :k] P[k, :], P[kpivot, :] = P[kpivot, :], P[k, :] L[k,k] = Ukk = U[k,k] DD[k,k] = oldpivot * Ukk for i in range(k+1, n): L[i,k] = Uik = U[i,k] for j in range(k+1, m): #U[i,j] = simplify((Ukk * U[i,j] - U[k,j]*Uik) / oldpivot) D = Ukk * U[i,j] - U[k,j]*Uik if len(vars) > 0 and D != S.Zero and not oldpivot.is_number: try: D,r = div(Poly(D,*vars),oldpivot) except UnificationFailed: log.warn('unification failed, trying direct division') D /= oldpivot else: D /= oldpivot # save if D.is_Atom: U[i,j] = D.as_expr() else: if len(vars) > 0: U[i,j] = D.as_expr() else: U[i,j] = D.cancel() U[i,k] = 0 oldpivot = Ukk DD[n-1,n-1] = oldpivot return P, L, DD, U @staticmethod def sequence_cross_product(*sequences): """iterates through the cross product of all items in the sequences""" # visualize an odometer, with "wheels" displaying "digits"...: wheels = list(map(iter, sequences)) digits = [next(it) for it in wheels] while True: yield tuple(digits) for i in range(len(digits)-1, -1, -1): try: digits[i] = next(wheels[i]) break except StopIteration: wheels[i] = iter(sequences[i]) digits[i] = next(wheels[i]) else: break @staticmethod def tolatex(e): s = printing.latex(e) s1 = re.sub('\\\\operatorname\{(sin|cos)\}\\\\left\(j_\{(\d)\}\\\\right\)','\g<1>_\g<2>',s) s2 = re.sub('1\.(0*)([^0-9])','1\g<2>',s1) s3 = re.sub('1 \\\\(sin|cos)','\g<1>',s2) s4 = re.sub('(\d*)\.([0-9]*[1-9])(0*)([^0-9])','\g<1>.\g<2>\g<4>',s3) s5 = re.sub('sj_','s_',s4) s5 = re.sub('cj_','c_',s5) s5 = re.sub('sin','s',s5) s5 = re.sub('cos','c',s5) replacements = [('px','p_x'),('py','p_y'),('pz','p_z'),('r00','r_{00}'),('r01','r_{01}'),('r02','r_{02}'),('r10','r_{10}'),('r11','r_{11}'),('r12','r_{12}'),('r20','r_{20}'),('r21','r_{21}'),('r022','r_{22}')] for old,new in replacements: s5 = re.sub(old,new,s5) return s5 @staticmethod def GetSolvers(): """Returns a dictionary of all the supported solvers and their official identifier names""" return {'transform6d':IKFastSolver.solveFullIK_6D, 'rotation3d':IKFastSolver.solveFullIK_Rotation3D, 'translation3d':IKFastSolver.solveFullIK_Translation3D, 'direction3d':IKFastSolver.solveFullIK_Direction3D, 'ray4d':IKFastSolver.solveFullIK_Ray4D, 'lookat3d':IKFastSolver.solveFullIK_Lookat3D, 'translationdirection5d':IKFastSolver.solveFullIK_TranslationDirection5D, 'translationxy2d':IKFastSolver.solveFullIK_TranslationXY2D, 'translationxyorientation3d':IKFastSolver.solveFullIK_TranslationXYOrientation3D, 'translationxaxisangle4d':IKFastSolver.solveFullIK_TranslationAxisAngle4D, 'translationyaxisangle4d':IKFastSolver.solveFullIK_TranslationAxisAngle4D, 'translationzaxisangle4d':IKFastSolver.solveFullIK_TranslationAxisAngle4D, 'translationxaxisangleznorm4d':IKFastSolver.solveFullIK_TranslationAxisAngle4D, 'translationyaxisanglexnorm4d':IKFastSolver.solveFullIK_TranslationAxisAngle4D, 'translationzaxisangleynorm4d':IKFastSolver.solveFullIK_TranslationAxisAngle4D } if __name__ == '__main__': import openravepy parser = OptionParser(description="""IKFast: The Robot Kinematics Compiler Software License Agreement (Lesser GPL v3). Copyright (C) 2009-2011 Rosen Diankov. IKFast is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details. IKFast is part of OpenRAVE. This program can be used with robots or kinbodies defined and is independent of the OpenRAVE databases. Example usage for 7 DOF Barrett WAM where the 3rd joint is a free parameter: python ikfast.py --robot=robots/barrettwam.robot.xml --baselink=0 --eelink=7 --savefile=ik.cpp --freeindex=2 """,version=__version__) parser.add_option('--robot', action='store', type='string', dest='robot',default=None, help='robot file (COLLADA or OpenRAVE XML)') parser.add_option('--savefile', action='store', type='string', dest='savefile',default='ik.cpp', help='filename where to store the generated c++ code') parser.add_option('--baselink', action='store', type='int', dest='baselink', help='base link index to start extraction of ik chain') parser.add_option('--eelink', action='store', type='int', dest='eelink', help='end effector link index to end extraction of ik chain') parser.add_option('--freeindex', action='append', type='int', dest='freeindices',default=[], help='Optional joint index specifying a free parameter of the manipulator. If not specified, assumes all joints not solving for are free parameters. Can be specified multiple times for multiple free parameters.') parser.add_option('--iktype', action='store', dest='iktype',default='transform6d', help='The iktype to generate the ik for. Possible values are: %s'%(', '.join(name for name,fn in IKFastSolver.GetSolvers().items()))) parser.add_option('--maxcasedepth', action='store', type='int', dest='maxcasedepth',default=3, help='The max depth to go into degenerate cases. If ikfast file is too big, try reducing this, (default=%default).') parser.add_option('--lang', action='store',type='string',dest='lang',default='cpp', help='The language to generate the code in (default=%default), available=('+','.join(name for name,value in CodeGenerators.items())+')') parser.add_option('--debug','-d', action='store', type='int',dest='debug',default=logging.INFO, help='Debug level for python nose (smaller values allow more text).') (options, args) = parser.parse_args() if options.robot is None or options.baselink is None or options.eelink is None: print('Error: Not all arguments specified') sys.exit(1) format = logging.Formatter('%(levelname)s: %(message)s') handler = logging.StreamHandler(sys.stdout) handler.setFormatter(format) log.addHandler(handler) log.setLevel(options.debug) solvefn=IKFastSolver.GetSolvers()[options.iktype.lower()] if options.robot is not None: try: env=openravepy.Environment() kinbody=env.ReadRobotXMLFile(options.robot) env.Add(kinbody) solver = IKFastSolver(kinbody,kinbody) solver.maxcasedepth = options.maxcasedepth chaintree = solver.generateIkSolver(options.baselink,options.eelink,options.freeindices,solvefn=solvefn) code=solver.writeIkSolver(chaintree,lang=options.lang) finally: openravepy.RaveDestroy() if len(code) > 0: with open(options.savefile,'w') as f: f.write(code)