{ "metadata": { "name": "n-pendulum-control" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "We've been working on a [conference paper](https://github.com/gilbertgede/idetc-2013-paper) to demonstrate the ability to do multibody dynamics with Python. We've been calling this work flow [PyDy](http://pydy.org), short for Python Dynamics. Several pieces of the puzzle have come together lately to really demonstrate the power of the scientific python software packages to handle complex dynamic and controls problems (i.e. IPython notebooks, matplotlib animations, python-control, and our software package mechanics which is a part of SymPy). After writing the draft of our paper, which uses a general n-link pendulum as it's main example, I came across this [blog post by Wolfram](http://blog.wolfram.com/2011/03/01/stabilized-n-link-pendulum/) demonstrating their ability to symbolically derive the equations of motion for the n-link pendulum and stabilize it with an LQR controller. It inspired me to replicate the example as I realized that it was relatively easy to do with all free and open source software!\n", "\n", "In this example problem we will derive the equations of motion of an n-link pendulum on a laterally sliding cart and then develop a controller to stabilize it. Balancing a single inverted pendulum is a classic problem that is many times a student's first experience with non-linear dynamics and control. The problem here is extended to a general n-link pendulum and as we will see the equations of motion quickly get messy with greater than 2 links.\n", "\n", "The diagram below shows the general description of the problem." ] }, { "cell_type": "code", "collapsed": false, "input": [ "from IPython.display import SVG\n", "SVG(filename='n-pendulum-with-cart.svg')" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "pyout", "prompt_number": 1, "svg": [ "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " image/svg+xml\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "" ], "text": [ "" ] } ], "prompt_number": 1 }, { "cell_type": "markdown", "metadata": {}, "source": [ "I used these software versions for the following computations:\n", "\n", "- IPython: 0.13.1.rc2\n", "- matplotlib: 1.1.1\n", "- NumPy: 1.6.2\n", "- SciPy: 0.10.1\n", "- SymPy: 0.7.2\n", "- python-control: 0.6d \n", "\n", "Equations of Motion\n", "===================\n", "\n", "We'll start by generating the equations of motion for the system with SymPy **[mechanics](http://docs.sympy.org/dev/modules/physics/mechanics/index.html)**. The functionality that mechanics provides is much more in depth than Mathematica's functionality. In the Mathematica example, Lagrangian mechanics were implemented manually with Mathematica's symbolic functionality. **mechanics** provides an assortment of functions and classes to derive the equations of motion for arbitrarily complex (i.e. configuration constraints, nonholonomic motion constraints, etc) multibody systems in a very natural way. First we import the necessary functionality from SymPy." ] }, { "cell_type": "code", "collapsed": false, "input": [ "from sympy import symbols\n", "from sympy.physics.mechanics import *" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 2 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now specify the number of links, $n$. I'll start with 5 since the Wolfram folks only showed four." ] }, { "cell_type": "code", "collapsed": false, "input": [ "n = 5" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 3 }, { "cell_type": "markdown", "metadata": {}, "source": [ "**mechanics** will need the generalized coordinates, generalized speeds, and the input force which are all time dependent variables and the bob masses, link lengths, and acceleration due to gravity which are all constants. Time, $t$, is also made available because we will need to differentiate with respect to time." ] }, { "cell_type": "code", "collapsed": false, "input": [ "q = dynamicsymbols('q:' + str(n + 1)) # Generalized coordinates\n", "u = dynamicsymbols('u:' + str(n + 1)) # Generalized speeds\n", "f = dynamicsymbols('f') # Force applied to the cart\n", " \n", "m = symbols('m:' + str(n + 1)) # Mass of each bob\n", "l = symbols('l:' + str(n)) # Length of each link\n", "g, t = symbols('g t') # Gravity and time" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 4 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we can create and inertial reference frame $I$ and define the point, $O$, as the origin." ] }, { "cell_type": "code", "collapsed": false, "input": [ "I = ReferenceFrame('I') # Inertial reference frame\n", "O = Point('O') # Origin point\n", "O.set_vel(I, 0) # Origin's velocity is zero" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 5 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Secondly, we define the define the first point of the pendulum as a particle which has mass. This point can only move laterally and represents the motion of the \"cart\"." ] }, { "cell_type": "code", "collapsed": false, "input": [ "P0 = Point('P0') # Hinge point of top link\n", "P0.set_pos(O, q[0] * I.x) # Set the position of P0 \n", "P0.set_vel(I, u[0] * I.x) # Set the velocity of P0\n", "Pa0 = Particle('Pa0', P0, m[0]) # Define a particle at P0" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 6 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we can define the $n$ reference frames, particles, gravitational forces, and kinematical differential equations for each of the pendulum links. This is easily done with a loop." ] }, { "cell_type": "code", "collapsed": false, "input": [ "frames = [I] # List to hold the n + 1 frames\n", "points = [P0] # List to hold the n + 1 points\n", "particles = [Pa0] # List to hold the n + 1 particles\n", "forces = [(P0, f * I.x - m[0] * g * I.y)] # List to hold the n + 1 applied forces, including the input force, f\n", "kindiffs = [q[0].diff(t) - u[0]] # List to hold kinematic ODE's\n", "\n", "for i in range(n):\n", " Bi = I.orientnew('B' + str(i), 'Axis', [q[i + 1], I.z]) # Create a new frame\n", " Bi.set_ang_vel(I, u[i + 1] * I.z) # Set angular velocity\n", " frames.append(Bi) # Add it to the frames list\n", "\n", " Pi = points[-1].locatenew('P' + str(i + 1), l[i] * Bi.x) # Create a new point\n", " Pi.v2pt_theory(points[-1], I, Bi) # Set the velocity\n", " points.append(Pi) # Add it to the points list\n", " \n", " Pai = Particle('Pa' + str(i + 1), Pi, m[i + 1]) # Create a new particle\n", " particles.append(Pai) # Add it to the particles list\n", "\n", " forces.append((Pi, -m[i + 1] * g * I.y)) # Set the force applied at the point\n", " \n", " kindiffs.append(q[i + 1].diff(t) - u[i + 1]) # Define the kinematic ODE: dq_i / dt - u_i = 0" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 7 }, { "cell_type": "markdown", "metadata": {}, "source": [ "With all of the necessary point velocities and particle masses defined, the `KanesMethod` class can be used to derive the equations of motion of the system automatically." ] }, { "cell_type": "code", "collapsed": false, "input": [ "kane = KanesMethod(I, q_ind=q, u_ind=u, kd_eqs=kindiffs) # Initialize the object\n", "fr, frstar = kane.kanes_equations(forces, particles) # Generate EoM's fr + frstar = 0" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 8 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The equations of motion are quite long as can been seen below. This is the general nature of most non-simple mutlibody problems. That is why a SymPy is so useful; no more mistakes in algegra, differentiation, or copying in hand written equations. " ] }, { "cell_type": "code", "collapsed": false, "input": [ "fr" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "pyout", "prompt_number": 9, "text": [ "[ -f(t)]\n", "[g*l0*m1*cos(q1(t)) + g*l0*m2*cos(q1(t)) + g*l0*m3*cos(q1(t)) + g*l0*m4*cos(q1(t)) + g*l0*m5*cos(q1(t))]\n", "[ g*l1*m2*cos(q2(t)) + g*l1*m3*cos(q2(t)) + g*l1*m4*cos(q2(t)) + g*l1*m5*cos(q2(t))]\n", "[ g*l2*m3*cos(q3(t)) + g*l2*m4*cos(q3(t)) + g*l2*m5*cos(q3(t))]\n", "[ g*l3*m4*cos(q4(t)) + g*l3*m5*cos(q4(t))]\n", "[ g*l4*m5*cos(q5(t))]" ] } ], "prompt_number": 9 }, { "cell_type": "code", "collapsed": false, "input": [ "frstar" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "pyout", "prompt_number": 10, "text": [ "[ -l0*m1*u1(t)**2*cos(q1(t)) - l0*m2*u1(t)**2*cos(q1(t)) - l0*m3*u1(t)**2*cos(q1(t)) - l0*m4*u1(t)**2*cos(q1(t)) - l0*m5*u1(t)**2*cos(q1(t)) - l1*m2*u2(t)**2*cos(q2(t)) - l1*m3*u2(t)**2*cos(q2(t)) - l1*m4*u2(t)**2*cos(q2(t)) - l1*m5*u2(t)**2*cos(q2(t)) - l2*m3*u3(t)**2*cos(q3(t)) - l2*m4*u3(t)**2*cos(q3(t)) - l2*m5*u3(t)**2*cos(q3(t)) - l3*m4*u4(t)**2*cos(q4(t)) - l3*m5*u4(t)**2*cos(q4(t)) - l4*m5*u5(t)**2*cos(q5(t)) - l4*m5*sin(q5(t))*Derivative(u5(t), t) + (-l3*m4*sin(q4(t)) - l3*m5*sin(q4(t)))*Derivative(u4(t), t) + (-l2*m3*sin(q3(t)) - l2*m4*sin(q3(t)) - l2*m5*sin(q3(t)))*Derivative(u3(t), t) + (-l1*m2*sin(q2(t)) - l1*m3*sin(q2(t)) - l1*m4*sin(q2(t)) - l1*m5*sin(q2(t)))*Derivative(u2(t), t) + (-l0*m1*sin(q1(t)) - l0*m2*sin(q1(t)) - l0*m3*sin(q1(t)) - l0*m4*sin(q1(t)) - l0*m5*sin(q1(t)))*Derivative(u1(t), t) + (m0 + m1 + m2 + m3 + m4 + m5)*Derivative(u0(t), t)]\n", "[-l0*l1*m2*(-sin(q1(t))*cos(q2(t)) + sin(q2(t))*cos(q1(t)))*u2(t)**2 - l0*l1*m3*(-sin(q1(t))*cos(q2(t)) + sin(q2(t))*cos(q1(t)))*u2(t)**2 - l0*l1*m4*(-sin(q1(t))*cos(q2(t)) + sin(q2(t))*cos(q1(t)))*u2(t)**2 - l0*l1*m5*(-sin(q1(t))*cos(q2(t)) + sin(q2(t))*cos(q1(t)))*u2(t)**2 - l0*l2*m3*(-sin(q1(t))*cos(q3(t)) + sin(q3(t))*cos(q1(t)))*u3(t)**2 - l0*l2*m4*(-sin(q1(t))*cos(q3(t)) + sin(q3(t))*cos(q1(t)))*u3(t)**2 - l0*l2*m5*(-sin(q1(t))*cos(q3(t)) + sin(q3(t))*cos(q1(t)))*u3(t)**2 - l0*l3*m4*(-sin(q1(t))*cos(q4(t)) + sin(q4(t))*cos(q1(t)))*u4(t)**2 - l0*l3*m5*(-sin(q1(t))*cos(q4(t)) + sin(q4(t))*cos(q1(t)))*u4(t)**2 + l0*l4*m5*(sin(q1(t))*sin(q5(t)) + cos(q1(t))*cos(q5(t)))*Derivative(u5(t), t) - l0*l4*m5*(-sin(q1(t))*cos(q5(t)) + sin(q5(t))*cos(q1(t)))*u5(t)**2 + (l0*l3*m4*(sin(q1(t))*sin(q4(t)) + cos(q1(t))*cos(q4(t))) + l0*l3*m5*(sin(q1(t))*sin(q4(t)) + cos(q1(t))*cos(q4(t))))*Derivative(u4(t), t) + (l0*l2*m3*(sin(q1(t))*sin(q3(t)) + cos(q1(t))*cos(q3(t))) + l0*l2*m4*(sin(q1(t))*sin(q3(t)) + cos(q1(t))*cos(q3(t))) + l0*l2*m5*(sin(q1(t))*sin(q3(t)) + cos(q1(t))*cos(q3(t))))*Derivative(u3(t), t) + (l0*l1*m2*(sin(q1(t))*sin(q2(t)) + cos(q1(t))*cos(q2(t))) + l0*l1*m3*(sin(q1(t))*sin(q2(t)) + cos(q1(t))*cos(q2(t))) + l0*l1*m4*(sin(q1(t))*sin(q2(t)) + cos(q1(t))*cos(q2(t))) + l0*l1*m5*(sin(q1(t))*sin(q2(t)) + cos(q1(t))*cos(q2(t))))*Derivative(u2(t), t) + (l0**2*m1 + l0**2*m2 + l0**2*m3 + l0**2*m4 + l0**2*m5)*Derivative(u1(t), t) + (-l0*m1*sin(q1(t)) - l0*m2*sin(q1(t)) - l0*m3*sin(q1(t)) - l0*m4*sin(q1(t)) - l0*m5*sin(q1(t)))*Derivative(u0(t), t)]\n", "[ -l0*l1*m2*(sin(q1(t))*cos(q2(t)) - sin(q2(t))*cos(q1(t)))*u1(t)**2 - l0*l1*m3*(sin(q1(t))*cos(q2(t)) - sin(q2(t))*cos(q1(t)))*u1(t)**2 - l0*l1*m4*(sin(q1(t))*cos(q2(t)) - sin(q2(t))*cos(q1(t)))*u1(t)**2 - l0*l1*m5*(sin(q1(t))*cos(q2(t)) - sin(q2(t))*cos(q1(t)))*u1(t)**2 - l1*l2*m3*(-sin(q2(t))*cos(q3(t)) + sin(q3(t))*cos(q2(t)))*u3(t)**2 - l1*l2*m4*(-sin(q2(t))*cos(q3(t)) + sin(q3(t))*cos(q2(t)))*u3(t)**2 - l1*l2*m5*(-sin(q2(t))*cos(q3(t)) + sin(q3(t))*cos(q2(t)))*u3(t)**2 - l1*l3*m4*(-sin(q2(t))*cos(q4(t)) + sin(q4(t))*cos(q2(t)))*u4(t)**2 - l1*l3*m5*(-sin(q2(t))*cos(q4(t)) + sin(q4(t))*cos(q2(t)))*u4(t)**2 + l1*l4*m5*(sin(q2(t))*sin(q5(t)) + cos(q2(t))*cos(q5(t)))*Derivative(u5(t), t) - l1*l4*m5*(-sin(q2(t))*cos(q5(t)) + sin(q5(t))*cos(q2(t)))*u5(t)**2 + (l1*l3*m4*(sin(q2(t))*sin(q4(t)) + cos(q2(t))*cos(q4(t))) + l1*l3*m5*(sin(q2(t))*sin(q4(t)) + cos(q2(t))*cos(q4(t))))*Derivative(u4(t), t) + (l1*l2*m3*(sin(q2(t))*sin(q3(t)) + cos(q2(t))*cos(q3(t))) + l1*l2*m4*(sin(q2(t))*sin(q3(t)) + cos(q2(t))*cos(q3(t))) + l1*l2*m5*(sin(q2(t))*sin(q3(t)) + cos(q2(t))*cos(q3(t))))*Derivative(u3(t), t) + (l1**2*m2 + l1**2*m3 + l1**2*m4 + l1**2*m5)*Derivative(u2(t), t) + (-l1*m2*sin(q2(t)) - l1*m3*sin(q2(t)) - l1*m4*sin(q2(t)) - l1*m5*sin(q2(t)))*Derivative(u0(t), t) + (l0*l1*m2*(sin(q1(t))*sin(q2(t)) + cos(q1(t))*cos(q2(t))) + l0*l1*m3*(sin(q1(t))*sin(q2(t)) + cos(q1(t))*cos(q2(t))) + l0*l1*m4*(sin(q1(t))*sin(q2(t)) + cos(q1(t))*cos(q2(t))) + l0*l1*m5*(sin(q1(t))*sin(q2(t)) + cos(q1(t))*cos(q2(t))))*Derivative(u1(t), t)]\n", "[ -l0*l2*m3*(sin(q1(t))*cos(q3(t)) - sin(q3(t))*cos(q1(t)))*u1(t)**2 - l0*l2*m4*(sin(q1(t))*cos(q3(t)) - sin(q3(t))*cos(q1(t)))*u1(t)**2 - l0*l2*m5*(sin(q1(t))*cos(q3(t)) - sin(q3(t))*cos(q1(t)))*u1(t)**2 - l1*l2*m3*(sin(q2(t))*cos(q3(t)) - sin(q3(t))*cos(q2(t)))*u2(t)**2 - l1*l2*m4*(sin(q2(t))*cos(q3(t)) - sin(q3(t))*cos(q2(t)))*u2(t)**2 - l1*l2*m5*(sin(q2(t))*cos(q3(t)) - sin(q3(t))*cos(q2(t)))*u2(t)**2 - l2*l3*m4*(-sin(q3(t))*cos(q4(t)) + sin(q4(t))*cos(q3(t)))*u4(t)**2 - l2*l3*m5*(-sin(q3(t))*cos(q4(t)) + sin(q4(t))*cos(q3(t)))*u4(t)**2 + l2*l4*m5*(sin(q3(t))*sin(q5(t)) + cos(q3(t))*cos(q5(t)))*Derivative(u5(t), t) - l2*l4*m5*(-sin(q3(t))*cos(q5(t)) + sin(q5(t))*cos(q3(t)))*u5(t)**2 + (l2*l3*m4*(sin(q3(t))*sin(q4(t)) + cos(q3(t))*cos(q4(t))) + l2*l3*m5*(sin(q3(t))*sin(q4(t)) + cos(q3(t))*cos(q4(t))))*Derivative(u4(t), t) + (l2**2*m3 + l2**2*m4 + l2**2*m5)*Derivative(u3(t), t) + (-l2*m3*sin(q3(t)) - l2*m4*sin(q3(t)) - l2*m5*sin(q3(t)))*Derivative(u0(t), t) + (l0*l2*m3*(sin(q1(t))*sin(q3(t)) + cos(q1(t))*cos(q3(t))) + l0*l2*m4*(sin(q1(t))*sin(q3(t)) + cos(q1(t))*cos(q3(t))) + l0*l2*m5*(sin(q1(t))*sin(q3(t)) + cos(q1(t))*cos(q3(t))))*Derivative(u1(t), t) + (l1*l2*m3*(sin(q2(t))*sin(q3(t)) + cos(q2(t))*cos(q3(t))) + l1*l2*m4*(sin(q2(t))*sin(q3(t)) + cos(q2(t))*cos(q3(t))) + l1*l2*m5*(sin(q2(t))*sin(q3(t)) + cos(q2(t))*cos(q3(t))))*Derivative(u2(t), t)]\n", "[ -l0*l3*m4*(sin(q1(t))*cos(q4(t)) - sin(q4(t))*cos(q1(t)))*u1(t)**2 - l0*l3*m5*(sin(q1(t))*cos(q4(t)) - sin(q4(t))*cos(q1(t)))*u1(t)**2 - l1*l3*m4*(sin(q2(t))*cos(q4(t)) - sin(q4(t))*cos(q2(t)))*u2(t)**2 - l1*l3*m5*(sin(q2(t))*cos(q4(t)) - sin(q4(t))*cos(q2(t)))*u2(t)**2 - l2*l3*m4*(sin(q3(t))*cos(q4(t)) - sin(q4(t))*cos(q3(t)))*u3(t)**2 - l2*l3*m5*(sin(q3(t))*cos(q4(t)) - sin(q4(t))*cos(q3(t)))*u3(t)**2 + l3*l4*m5*(sin(q4(t))*sin(q5(t)) + cos(q4(t))*cos(q5(t)))*Derivative(u5(t), t) - l3*l4*m5*(-sin(q4(t))*cos(q5(t)) + sin(q5(t))*cos(q4(t)))*u5(t)**2 + (l3**2*m4 + l3**2*m5)*Derivative(u4(t), t) + (-l3*m4*sin(q4(t)) - l3*m5*sin(q4(t)))*Derivative(u0(t), t) + (l0*l3*m4*(sin(q1(t))*sin(q4(t)) + cos(q1(t))*cos(q4(t))) + l0*l3*m5*(sin(q1(t))*sin(q4(t)) + cos(q1(t))*cos(q4(t))))*Derivative(u1(t), t) + (l1*l3*m4*(sin(q2(t))*sin(q4(t)) + cos(q2(t))*cos(q4(t))) + l1*l3*m5*(sin(q2(t))*sin(q4(t)) + cos(q2(t))*cos(q4(t))))*Derivative(u2(t), t) + (l2*l3*m4*(sin(q3(t))*sin(q4(t)) + cos(q3(t))*cos(q4(t))) + l2*l3*m5*(sin(q3(t))*sin(q4(t)) + cos(q3(t))*cos(q4(t))))*Derivative(u3(t), t)]\n", "[ l0*l4*m5*(sin(q1(t))*sin(q5(t)) + cos(q1(t))*cos(q5(t)))*Derivative(u1(t), t) - l0*l4*m5*(sin(q1(t))*cos(q5(t)) - sin(q5(t))*cos(q1(t)))*u1(t)**2 + l1*l4*m5*(sin(q2(t))*sin(q5(t)) + cos(q2(t))*cos(q5(t)))*Derivative(u2(t), t) - l1*l4*m5*(sin(q2(t))*cos(q5(t)) - sin(q5(t))*cos(q2(t)))*u2(t)**2 + l2*l4*m5*(sin(q3(t))*sin(q5(t)) + cos(q3(t))*cos(q5(t)))*Derivative(u3(t), t) - l2*l4*m5*(sin(q3(t))*cos(q5(t)) - sin(q5(t))*cos(q3(t)))*u3(t)**2 + l3*l4*m5*(sin(q4(t))*sin(q5(t)) + cos(q4(t))*cos(q5(t)))*Derivative(u4(t), t) - l3*l4*m5*(sin(q4(t))*cos(q5(t)) - sin(q5(t))*cos(q4(t)))*u4(t)**2 + l4**2*m5*Derivative(u5(t), t) - l4*m5*sin(q5(t))*Derivative(u0(t), t)]" ] } ], "prompt_number": 10 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Simulation\n", "==========\n", "\n", "Now that the symbolic equations of motion are available we can simulate the pendulum's motion. We will need some more SymPy functionality and several NumPy functions, and most importantly the integration function from SciPy, `odeint`." ] }, { "cell_type": "code", "collapsed": false, "input": [ "from sympy import Dummy, lambdify\n", "from numpy import array, hstack, zeros, linspace, pi\n", "from numpy.linalg import solve\n", "from scipy.integrate import odeint" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 11 }, { "cell_type": "markdown", "metadata": {}, "source": [ "First, define some numeric values for all of the constant parameters in the problem." ] }, { "cell_type": "code", "collapsed": false, "input": [ "arm_length = 1. / n # The maximum length of the pendulum is 1 meter\n", "bob_mass = 0.01 / n # The maximum mass of the bobs is 10 grams\n", "parameters = [g, m[0]] # Parameter definitions starting with gravity and the first bob\n", "parameter_vals = [9.81, 0.01 / n] # Numerical values for the first two\n", "for i in range(n): # Then each mass and length\n", " parameters += [l[i], m[i + 1]] \n", " parameter_vals += [arm_length, bob_mass]" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 12 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Mathematica has a really nice `NDSolve` function for quickly integrating their symbolic differential equations. We have plans to develop something similar for SymPy but haven't found the development time yet to do it properly. So the next bit isn't as clean as we'd like but you can make use of SymPy's lambdify function to create functions that will evaluate the mass matrix, $M$, and forcing vector, $\\bar{f}$ from $M\\dot{u} = \\bar{f}(q, \\dot{q}, u, t)$ as a NumPy function. We make use of dummy symbols to replace the time varying functions in the SymPy equations a simple dummy symbol. " ] }, { "cell_type": "code", "collapsed": false, "input": [ "dynamic = q + u # Make a list of the states\n", "dynamic.append(f) # Add the input force\n", "dummy_symbols = [Dummy() for i in dynamic] # Create a dummy symbol for each variable\n", "dummy_dict = dict(zip(dynamic, dummy_symbols)) \n", "kindiff_dict = kane.kindiffdict() # Get the solved kinematical differential equations\n", "M = kane.mass_matrix_full.subs(kindiff_dict).subs(dummy_dict) # Substitute into the mass matrix \n", "F = kane.forcing_full.subs(kindiff_dict).subs(dummy_dict) # Substitute into the forcing vector\n", "M_func = lambdify(dummy_symbols + parameters, M) # Create a callable function to evaluate the mass matrix \n", "F_func = lambdify(dummy_symbols + parameters, F) # Create a callable function to evaluate the forcing vector " ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 13 }, { "cell_type": "markdown", "metadata": {}, "source": [ "To integrate the ODE's we need to define a function that returns the derivatives of the states given the current state and time." ] }, { "cell_type": "code", "collapsed": false, "input": [ "def right_hand_side(x, t, args):\n", " \"\"\"Returns the derivatives of the states.\n", "\n", " Parameters\n", " ----------\n", " x : ndarray, shape(2 * (n + 1))\n", " The current state vector.\n", " t : float\n", " The current time.\n", " args : ndarray\n", " The constants.\n", "\n", " Returns\n", " -------\n", " dx : ndarray, shape(2 * (n + 1))\n", " The derivative of the state.\n", " \n", " \"\"\"\n", " u = 0.0 # The input force is always zero \n", " arguments = hstack((x, u, args)) # States, input, and parameters\n", " dx = array(solve(M_func(*arguments), # Solving for the derivatives\n", " F_func(*arguments))).T[0]\n", " \n", " return dx" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 14 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now that we have the right hand side function, the initial conditions are set such that the pendulum is in the vertical equilibrium and a slight initial rate is set for each speed to ensure the pendulum falls. The equations can then be integrated with SciPy's `odeint` function given a time series." ] }, { "cell_type": "code", "collapsed": false, "input": [ "x0 = hstack(( 0, pi / 2 * ones(len(q) - 1), 1e-3 * ones(len(u)) )) # Initial conditions, q and u\n", "t = linspace(0, 10, 1000) # Time vector\n", "y = odeint(right_hand_side, x0, t, args=(parameter_vals,)) # Actual integration" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 15 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Plotting\n", "========\n", "\n", "The results of the simulation can be plotted with matplotlib." ] }, { "cell_type": "code", "collapsed": false, "input": [ "lines = plot(t, y[:, :y.shape[1] / 2])\n", "lab = xlabel('Time [sec]')\n", "leg = legend(dynamic[:y.shape[1] / 2])" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "display_data", "png": 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wcIW19WIMGfIDxmpZ4IXEREyJjMTfnp6w0W3ZqqijpNXU4M2UFPh5efU6BQ8w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\nAAAASUVORK5CYII=\n" } ], "prompt_number": 16 }, { "cell_type": "code", "collapsed": false, "input": [ "lines = plot(t, y[:, y.shape[1] / 2:])\n", "lab = xlabel('Time [sec]')\n", "leg = legend(dynamic[y.shape[1] / 2:])" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "display_data", "png": 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Q5npa3r17Nx/5yEf40z/90+Z7x1P7v5OD5JML7s97exnL5Zp2jR3ZTKhVRGMK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/CHf4g8fQw7sLT6GbJpiHZE6MPn4NxWoAR5IZAh6Ez6dlOlcHnBhw9fw4X3vOdZMflL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\nE/Ca1yz+/eDyvp/aljpoEpLXWU+1+vhZe/LSklz59SvTWvM2J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} ], "prompt_number": 17 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Animation\n", "=========\n", "\n", "matplotlib now includes very nice animation functions for animating matplotlib plots. First we import the necessary functions for creating the animation." ] }, { "cell_type": "code", "collapsed": false, "input": [ "from numpy import zeros, cos, sin, arange, around\n", "from matplotlib import pyplot as plt\n", "from matplotlib import animation\n", "from matplotlib.patches import Rectangle" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 18 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The following function was modeled from Jake Vanderplas's [post on matplotlib animations](http://jakevdp.github.com/blog/2012/08/18/matplotlib-animation-tutorial/)." ] }, { "cell_type": "code", "collapsed": false, "input": [ "def animate_pendulum(t, states, length, filename=None):\n", " \"\"\"Animates the n-pendulum and optionally saves it to file.\n", "\n", " Parameters\n", " ----------\n", " t : ndarray, shape(m)\n", " Time array.\n", " states: ndarray, shape(m,p)\n", " State time history.\n", " length: float\n", " The length of the pendulum links.\n", " filename: string or None, optional\n", " If true a movie file will be saved of the animation. This may take some time.\n", "\n", " Returns\n", " -------\n", " fig : matplotlib.Figure\n", " The figure.\n", " anim : matplotlib.FuncAnimation\n", " The animation.\n", "\n", " \"\"\"\n", " # the number of pendulum bobs\n", " numpoints = states.shape[1] / 2\n", "\n", " # first set up the figure, the axis, and the plot elements we want to animate\n", " fig = plt.figure()\n", " \n", " # some dimesions\n", " cart_width = 0.4\n", " cart_height = 0.2\n", " \n", " # set the limits based on the motion\n", " xmin = around(states[:, 0].min() - cart_width / 2.0, 1)\n", " xmax = around(states[:, 0].max() + cart_width / 2.0, 1)\n", " \n", " # create the axes\n", " ax = plt.axes(xlim=(xmin, xmax), ylim=(-1.1, 1.1), aspect='equal')\n", " \n", " # display the current time\n", " time_text = ax.text(0.04, 0.9, '', transform=ax.transAxes)\n", " \n", " # create a rectangular cart\n", " rect = Rectangle([states[0, 0] - cart_width / 2.0, -cart_height / 2],\n", " cart_width, cart_height, fill=True, color='red', ec='black')\n", " ax.add_patch(rect)\n", " \n", " # blank line for the pendulum\n", " line, = ax.plot([], [], lw=2, marker='o', markersize=6)\n", "\n", " # initialization function: plot the background of each frame\n", " def init():\n", " time_text.set_text('')\n", " rect.set_xy((0.0, 0.0))\n", " line.set_data([], [])\n", " return time_text, rect, line,\n", "\n", " # animation function: update the objects\n", " def animate(i):\n", " time_text.set_text('time = {:2.2f}'.format(t[i]))\n", " rect.set_xy((states[i, 0] - cart_width / 2.0, -cart_height / 2))\n", " x = hstack((states[i, 0], zeros((numpoints - 1))))\n", " y = zeros((numpoints))\n", " for j in arange(1, numpoints):\n", " x[j] = x[j - 1] + length * cos(states[i, j])\n", " y[j] = y[j - 1] + length * sin(states[i, j])\n", " line.set_data(x, y)\n", " return time_text, rect, line,\n", "\n", " # call the animator function\n", " anim = animation.FuncAnimation(fig, animate, frames=len(t), init_func=init,\n", " interval=t[-1] / len(t) * 1000, blit=True, repeat=False)\n", " \n", " # save the animation if a filename is given\n", " if filename is not None:\n", " anim.save(filename, fps=30, codec='libx264')" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 19 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we can create the animation of the pendulum. This animation will show the open loop dynamics." ] }, { "cell_type": "code", "collapsed": false, "input": [ "animate_pendulum(t, y, arm_length, filename=\"open-loop.ogv\")\n", "animate_pendulum(t, y, arm_length, filename=\"open-loop.mp4\")" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "display_data", "png": 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} ], "prompt_number": 20 }, { "cell_type": "code", "collapsed": false, "input": [ "from IPython.display import HTML\n", "h = \\\n", "\"\"\"\n", "\n", "\"\"\"\n", "HTML(h)" ], "language": "python", "metadata": {}, "outputs": [ { "html": [ "\n", "\n" ], "output_type": "pyout", "prompt_number": 21, "text": [ "" ] } ], "prompt_number": 21 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Controller Design\n", "=================\n", "\n", "The n-link pendulum can be balanced such that all of the links are inverted above the cart by applying the correct lateral force to the cart. We can design a full state feedback controller based off of a linear model of the pendulum about its upright equilibrium point. We'll start by specifying the equilibrium point and parameters in dictionaries." ] }, { "cell_type": "code", "collapsed": false, "input": [ "equilibrium_point = hstack(( 0, pi / 2 * ones(len(q) - 1), zeros(len(u)) ))\n", "equilibrium_dict = dict(zip(q + u, equilibrium_point))\n", "parameter_dict = dict(zip(parameters, parameter_vals))" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 22 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The `KanesMethod` class has method that linearizes the forcing vector about generic state and input perturbation vectors. The equilibrium point and numerical constants can then be substituted in to give the linear system in this form: $M\\dot{x}=F_Ax+F_Bu$. The state and input matrices, $A$ and $B$, can then be computed by left side multiplication by the inverse of the mass matrix: $A=M^{-1}F_A$ and $B=M^{-1}F_B$." ] }, { "cell_type": "code", "collapsed": false, "input": [ "# symbolically linearize about arbitrary equilibrium\n", "linear_state_matrix, linear_input_matrix, inputs = kane.linearize()\n", "# sub in the equilibrium point and the parameters\n", "f_A_lin = linear_state_matrix.subs(parameter_dict).subs(equilibrium_dict)\n", "f_B_lin = linear_input_matrix.subs(parameter_dict).subs(equilibrium_dict)\n", "m_mat = kane.mass_matrix_full.subs(parameter_dict).subs(equilibrium_dict)\n", "# compute A and B\n", "from numpy import matrix\n", "A = matrix(m_mat.inv() * f_A_lin)\n", "B = matrix(m_mat.inv() * f_B_lin)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 23 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now that we have a linear system, the python-control package can be used to design an optimal controller for the system." ] }, { "cell_type": "code", "collapsed": false, "input": [ "import control\n", "from numpy import dot, rank\n", "from numpy.linalg import matrix_rank" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 24 }, { "cell_type": "markdown", "metadata": {}, "source": [ "First we can check to see if the system is, in fact, controllable." ] }, { "cell_type": "code", "collapsed": false, "input": [ "assert matrix_rank(control.ctrb(A, B)) == A.shape[0]" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 25 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The control matrix is full rank, so now we can compute the optimal gains with a linear quadratic regulator. I chose identity matrices for the weightings for simplicity." ] }, { "cell_type": "code", "collapsed": false, "input": [ "K, X, E = control.lqr(A, B, ones(A.shape), 1);" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 26 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The gains can now be used to define the required input during simulation to stabilize the system. The input $u$ is simply the gain vector multiplied by the error in the state vector from the equilibrium point, $u(t)=K(x_{eq} - x(t))$." ] }, { "cell_type": "code", "collapsed": false, "input": [ "def right_hand_side(x, t, args):\n", " \"\"\"Returns the derivatives of the states.\n", "\n", " Parameters\n", " ----------\n", " x : ndarray, shape(2 * (n + 1))\n", " The current state vector.\n", " t : float\n", " The current time.\n", " args : ndarray\n", " The constants.\n", "\n", " Returns\n", " -------\n", " dx : ndarray, shape(2 * (n + 1))\n", " The derivative of the state.\n", " \n", " \"\"\"\n", " u = dot(K, equilibrium_point - x) # The controller \n", " arguments = hstack((x, u, args)) # States, input, and parameters\n", " dx = array(solve(M_func(*arguments), # Solving for the derivatives\n", " F_func(*arguments))).T[0]\n", " \n", " return dx" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 27 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we can simulate and animate the system to see if the controller works." ] }, { "cell_type": "code", "collapsed": false, "input": [ "x0 = hstack(( 0, pi / 2 * ones(len(q) - 1), 1 * ones(len(u)) )) # Initial conditions, q and u\n", "t = linspace(0, 10, 1000) # Time vector\n", "y = odeint(right_hand_side, x0, t, args=(parameter_vals,)) # Actual integration" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 28 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The plots show that we seem to have a stable system." ] }, { "cell_type": "code", "collapsed": false, "input": [ "lines = plot(t, y[:, :y.shape[1] / 2])\n", "lab = xlabel('Time [sec]')\n", "leg = legend(dynamic[:y.shape[1] / 2])" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "display_data", "png": 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56gawaNfu61qx17Srr13dptjz2tX7hV+ahV+y+tUvyCL61S/IEo+5+t+8vP/HVo7CsnEv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\nNGgQc+bMYdSoUTZ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} ], "prompt_number": 29 }, { "cell_type": "code", "collapsed": false, "input": [ "lines = plot(t, y[:, y.shape[1] / 2:])\n", "lab = xlabel('Time [sec]')\n", "leg = legend(dynamic[y.shape[1] / 2:])" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "display_data", "png": 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} ], "prompt_number": 30 }, { "cell_type": "code", "collapsed": false, "input": [ "animate_pendulum(t, y, arm_length, filename=\"closed-loop.ogv\")\n", "animate_pendulum(t, y, arm_length, filename=\"closed-loop.mp4\")" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "display_data", "png": 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} ], "prompt_number": 31 }, { "cell_type": "code", "collapsed": false, "input": [ "from IPython.display import HTML\n", "h = \\\n", "\"\"\"\n", "\n", "\"\"\"\n", "HTML(h)" ], "language": "python", "metadata": {}, "outputs": [ { "html": [ "\n", "\n" ], "output_type": "pyout", "prompt_number": 32, "text": [ "" ] } ], "prompt_number": 32 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The video clearly shows that our controller can balance all $n$ of the pendulum links. The weightings in the lqr design can be tweaked to give different performance if needed.\n", "\n", "This example shows that the free and open source scientific Python tools for dynamics are easily comparable in ability and quality a commercial package such as Mathematica. Besides the current installation hurdles for Python, I'd like to claim that it may better than commercial packages, due to our much more robust SymPy **mechanics** package and the fact that all of the code is liberally licensed for reuse and hacking.\n", "\n", "The IPython notebook for this example can be downloaded from https://github.com/gilbertgede/idetc-2013-paper/blob/master/n-pendulum-control.ipynb. Yo ucan try out different $n$ values. I've gotten the equations of motion to compute and an open loop simulation of 10 links. My computer ran out of memory when I tried to compute for $n=50$. The controller weightings and initial conditions will probably have to be adjusted for better performance for $n>5$, but it should work. Let me know the resuls if you play with it." ] } ], "metadata": {} } ] }