{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Kinematic chain in a plane (2D)\n",
    "\n",
    "> Marcos Duarte, Renato Naville Watanabe   \n",
    "> Laboratory of Biomechanics and Motor Control ([http://pesquisa.ufabc.edu.br/bmclab](http://pesquisa.ufabc.edu.br/bmclab))  \n",
    "> Federal University of ABC, Brazil"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "Kinematic chain refers to an assembly of rigid bodies (links) connected by joints that is the mathematical model for a mechanical system which in turn can represent a biological system such as the human arm ([Wikipedia](http://en.wikipedia.org/wiki/Kinematic_chain)). \n",
    "\n",
    "The term chain refers to the fact that the links are constrained by their connections (typically, by a hinge joint which is also called pin joint or revolute joint) to other links. As consequence of this constraint, a kinematic chain in a plane is an example of circular motion of a rigid object.  \n",
    "\n",
    "Chapter 16 of Ruina and Rudra's book is a good formal introduction on the topic of circular motion of a rigid object. However, in this notebook we will not employ the mathematical formalism introduced in that chapter - the concept of a rotating reference frame and the related rotation matrix - we will do that in a future notebook when talking about rigid body transformation. For now we will specify the kinematics in a Cartesian coordinate system using trigonometry and calculus. This approach is simpler and more intuitive but it gets too complicated for a kinematic chain with many links or in the 3D space. For such more complicated problems, it would be recommended using rigid transformations (see for example, Siciliano et al. (2009)). \n",
    "\n",
    "We will deduce the kinematic properties of kinematic chains algebraically using [Sympy](http://sympy.org/), a Python library for symbolic mathematics. With Sympy, we could have used the [mechanics module](http://docs.sympy.org/latest/modules/physics/mechanics/index.html), a specific module for creation of symbolic equations of motion for multibody systems, but let's deduce most of the stuff by ourselves to understand the details."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "## Properties of kinematic chains\n",
    "\n",
    "For a kinematic chain, the base is the extremity (origin) of a kinematic chain which is typically considered attached to the ground, body or fixed. The endpoint is the other extremity (end) of a kinematic chain and typically can move. In robotics, the term end-effector is used and usually refers to a last link (rigid body) in this chain. \n",
    "\n",
    "In topological terms, a kinematic chain is termed open when there is only one sequence of links connecting the two ends of the chain. Otherwise it's termed closed and in this case a sequence of links forms a loop. A kinematic chain can be classified as serial or parallel or a mixed of both. In a serial chain the links are connected in a serial order. A serial chain is an open chain, otherwise it is a parallel chain or a branched chain (e.g., hand and fingers). \n",
    "\n",
    "Another important term to characterize a kinematic chain is <a href=\"https://en.wikipedia.org/wiki/Degrees_of_freedom_(mechanics)\">degree of freedom (DOF)</a>. In mechanics, the degree of freedom of a mechanical system is the number of independent parameters that define its configuration or that determine the state of a physical system. A particle in the 3D space has three DOFs because we need three coordinates to specify its position. A rigid body in the 3D space has six DOFs because we need three coordinates of one point at the body to specify its position and three angles to to specify its orientation in order to completely define the configuration of the rigid body. For a link attached to a fixed body by a hinge joint in a plane, all we need to define the configuration of the link is one angle and then this link has only one DOF. A kinematic chain with two links in a plane has two DOFs, and so on.\n",
    "\n",
    "The mobility of a kinematic chain is its total number of degrees of freedom. The redundancy of a kinematic chain is its mobility minus the number of degrees of freedom of the endpoint."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## The kinematics of one-link system\n",
    "\n",
    "First, let's study the case of a system composed by one planar hinge joint and one link, which technically it's not a chain but it will be useful to review (or introduce) key concepts.  \n",
    "<br>\n",
    "<figure><img src=\"./../images/onelink.gif\" width=350 alt=\"onelink\"/><figcaption><center><i>Figure. One link attached to a fixed body by a hinge joint in a plane.</i></center></figcaption> </figure>\n",
    "\n",
    "First, let's import the necessary libraries from Python and its ecosystem:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "import seaborn as sns\n",
    "sns.set_context(\"notebook\", font_scale=1.2, rc={\"lines.linewidth\": 2,\n",
    "                                                \"lines.markersize\": 10})\n",
    "from IPython.display import display, Math\n",
    "\n",
    "from sympy import Symbol, symbols, Function, Matrix, simplify, lambdify, expand, latex\n",
    "from sympy import diff, cos, sin, sqrt, acos, atan2, atan\n",
    "from sympy.vector import CoordSys3D\n",
    "from sympy.physics.mechanics import dynamicsymbols, mlatex, init_vprinting\n",
    "init_vprinting()\n",
    "\n",
    "import sys\n",
    "sys.path.insert(1, r'./../functions')  # add to pythonpath"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "We need to define a Cartesian coordinate system and the symbolic variables, $t$, $\\ell$, $\\theta$ (and make $\\theta$ a function of time):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "G = CoordSys3D('')\n",
    "t = Symbol('t')\n",
    "l = Symbol('ell', positive=True)\n",
    "theta = dynamicsymbols('theta')  # or Function('theta')(t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Using trigonometry, the endpoint position in terms of the joint angle and link length is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle (\\ell \\cos{\\left(\\theta{\\left(t \\right)} \\right)})\\mathbf{\\hat{i}_{}} + (\\ell \\sin{\\left(\\theta{\\left(t \\right)} \\right)})\\mathbf{\\hat{j}_{}}$"
      ],
      "text/plain": [
       "(ell*cos(theta(t)))*.i + (ell*sin(theta(t)))*.j"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r_p = l*cos(theta)*G.i + l*sin(theta)*G.j\n",
    "r_p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "With the components:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{ \\mathbf{\\hat{i}_{}} : \\ell \\operatorname{cos}\\left(\\theta\\right), \\  \\mathbf{\\hat{j}_{}} : \\ell \\operatorname{sin}\\left(\\theta\\right)\\right\\}$"
      ],
      "text/plain": [
       "{_i: ell⋅cos(θ), _j: ell⋅sin(θ)}"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r_p.components"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Forward and inverse kinematics\n",
    "\n",
    "Computing the configuration of a link or a chain (including the endpoint location) from the joint parameters (joint angles and link lengths) as we have done is called [forward or direct kinematics](https://en.wikipedia.org/wiki/Forward_kinematics).\n",
    "\n",
    "If the linear coordinates of the endpoint position are known (for example, if they are measured with a motion capture system) and one wants to obtain the joint angle(s), this process is known as [inverse kinematics](https://en.wikipedia.org/wiki/Inverse_kinematics). For the one-link system above:\n",
    "\n",
    "$$ \\theta = arctan\\left(\\frac{y_P}{x_P}\\right) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Matrix representation of the kinematics\n",
    "\n",
    "The mathematical manipulation will be easier if we use the matrix formalism (and let's drop the explicit dependence on $t$):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\ell \\operatorname{cos}\\left(\\theta\\right)\\\\\\ell \\operatorname{sin}\\left(\\theta\\right)\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡ell⋅cos(θ)⎤\n",
       "⎢          ⎥\n",
       "⎣ell⋅sin(θ)⎦"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r = Matrix((r_p.dot(G.i), r_p.dot(G.j)))\n",
    "r"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We could have used Sympy for switching to matrix representation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\ell \\operatorname{cos}\\left(\\theta\\right)\\\\\\ell \\operatorname{sin}\\left(\\theta\\right)\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡ell⋅cos(θ)⎤\n",
       "⎢          ⎥\n",
       "⎢ell⋅sin(θ)⎥\n",
       "⎢          ⎥\n",
       "⎣    0     ⎦"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r_p.to_matrix(G)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The third element of the matrix above refers to the $\\hat{\\mathbf{k}}$ component which is zero for the present case."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Differential kinematics\n",
    "\n",
    "Differential kinematics gives the relationship between the joint velocities and the corresponding endpoint linear velocity. This mapping is described by a matrix, termed [Jacobian matrix](http://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant), which depends on the kinematic chain configuration and it is of great use in the study of kinematic chains.  \n",
    "First, let's deduce the endpoint velocity without using the Jacobian and then we will see how to calculate the endpoint velocity using the Jacobian matrix.\n",
    "\n",
    "The velocity of the endpoint can be obtained by the first-order derivative of the position vector. The derivative of a vector is obtained by differentiating each vector component:   \n",
    "\n",
    "$$\n",
    "\\frac{\\mathrm{d}\\overrightarrow{\\mathbf{r}}}{\\mathrm{d}t} = \n",
    "\\large\n",
    "\\begin{bmatrix}\n",
    "\\frac{\\mathrm{d}x_P}{\\mathrm{d}t} \\\\\n",
    "\\frac{\\mathrm{d}y_P}{\\mathrm{d}t} \\\\\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "Note that the derivative is with respect to time but $x_P$ and $y_P$ depend explicitly on $\\theta$ and it's $\\theta$ that depends on $t$ ($x_P$ and $y_P$ depend implicitly on $t$). To calculate this type of derivative we will use the [chain rule](http://en.wikipedia.org/wiki/Chain_rule).  \n",
    "\n",
    "<br />\n",
    "<div style=\"background-color:#FBFBEF;border:1px solid black;padding:10px;\"> \n",
    "<b><a href=\"http://en.wikipedia.org/wiki/Chain_rule\">Chain rule</a></b>   \n",
    "<br />\n",
    "For variable $f$ which is function of variable $g$ which in turn is function of variable $t$, $f(g(t))$ or $(f\\circ g)(t)$, the derivative of $f$  with respect to $t$ is (using <a href=\"http://en.wikipedia.org/wiki/Notation_for_differentiation\">Lagrange's notation</a>):   \n",
    "<br />\n",
    "$$(f\\circ g)^{'}(t) = f'(g(t)) \\cdot g'(t)$$   \n",
    "\n",
    "Or using what is known as <a href=\"http://en.wikipedia.org/wiki/Notation_for_differentiation\">Leibniz's notation</a>:   \n",
    "<br />\n",
    "$$\\frac{\\mathrm{d}f}{\\mathrm{d}t} = \\frac{\\mathrm{d}f}{\\mathrm{d}g} \\cdot \\frac{\\mathrm{d}g}{\\mathrm{d}t}$$   \n",
    "\n",
    "If $f$ is function of two other variables which both are function of $t$, $ f(x(t),y(t))$, the chain rule for this case is:   \n",
    "<br />\n",
    "$$\\frac{\\mathrm{d}f}{\\mathrm{d}t} = \\frac{\\partial f}{\\partial x} \\cdot \\frac{\\mathrm{d}x}{\\mathrm{d}t} + \\frac{\\partial f}{\\partial y} \\cdot \\frac{\\mathrm{d}y}{\\mathrm{d}t}$$   \n",
    "\n",
    "Where $df/dt$ represents the <a href=\"http://en.wikipedia.org/wiki/Total_derivative\">total derivative</a> and $\\partial f / \\partial x$ represents the <a href=\"http://en.wikipedia.org/wiki/Partial_derivative\">partial derivative</a> of a function.   \n",
    "<br />\n",
    "<b><a href=\"http://en.wikipedia.org/wiki/Product_rule\">Product rule</a></b>   \n",
    "<br />\n",
    "The derivative of the product of two functions is:   \n",
    "<br />\n",
    "$$ (f \\cdot g)' = f' \\cdot g + f \\cdot g'  $$\n",
    "\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Linear velocity of the endpoint\n",
    "\n",
    "For the planar one-link case, the linear velocity of the endpoint is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\ell \\operatorname{sin}\\left(\\theta\\right) \\dot{\\theta}\\\\\\ell \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-ell⋅sin(θ)⋅θ̇⎤\n",
       "⎢              ⎥\n",
       "⎣ell⋅cos(θ)⋅θ̇ ⎦"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = r.diff(t)\n",
    "v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Where we used the [Newton's notation](http://en.wikipedia.org/wiki/Notation_for_differentiation) for differentiation. Note that $\\dot{\\theta}$ represents the unknown angular velocity of the joint; this is why the derivative of $\\theta$ is not explicitly solved. \n",
    "The magnitude or [Euclidian norm](http://en.wikipedia.org/wiki/Vector_norm) of the vector $\\overrightarrow{\\mathbf{v}}$ is:\n",
    "\n",
    "$$ ||\\overrightarrow{\\mathbf{v}}||=\\sqrt{v_x^2+v_y^2} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\ell \\sqrt{\\dot{\\theta}^{2}}$"
      ],
      "text/plain": [
       "       _____\n",
       "      ╱   2 \n",
       "ell⋅╲╱  θ̇  "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(sqrt(v[0]**2+v[1]**2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Which is $\\ell\\dot{\\theta}$.  \n",
    "We could have used the function norm of Sympy, v.norm(), but the output does not simplify nicely:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\ell \\sqrt{\\left|{\\operatorname{sin}\\left(\\theta\\right) \\dot{\\theta}}\\right|^{2} + \\left|{\\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta}}\\right|^{2}}$"
      ],
      "text/plain": [
       "       _____________________________\n",
       "      ╱            2              2 \n",
       "ell⋅╲╱  │sin(θ)⋅θ̇│  + │cos(θ)⋅θ̇│  "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(v.norm())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "The direction of $\\overrightarrow{\\mathbf{v}}$ is tangent to the circular trajectory of the endpoint as can be seen in the figure below.\n",
    "\n",
    "<figure><img src=\"./../images/onelink_vel.gif\" width=350 alt=\"onelinkVel\"/><figcaption><center><i>Figure. Endpoint velocity of one link attached to a fixed body by a hinge joint in a plane.</i></center></figcaption></figure>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Linear acceleration of the endpoint\n",
    "\n",
    "The acceleration of the endpoint position can be given by the second-order derivative of the position or by the first-order derivative of the velocity.  \n",
    "Using the chain and product rules for differentiation, the linear acceleration of the endpoint is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\ell \\operatorname{sin}\\left(\\theta\\right) \\ddot{\\theta} - \\ell \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta}^{2}\\\\- \\ell \\operatorname{sin}\\left(\\theta\\right) \\dot{\\theta}^{2} + \\ell \\operatorname{cos}\\left(\\theta\\right) \\ddot{\\theta}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                              2 ⎤\n",
       "⎢-ell⋅sin(θ)⋅θ̈ - ell⋅cos(θ)⋅θ̇  ⎥\n",
       "⎢                                ⎥\n",
       "⎢               2                ⎥\n",
       "⎣- ell⋅sin(θ)⋅θ̇  + ell⋅cos(θ)⋅θ̈⎦"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "acc = v.diff(t, 1)\n",
    "acc"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Examining the terms of the expression for the linear acceleration, we see there are two types of them: the term (in each direction) proportional to the angular acceleration $\\ddot{\\theta}$ and other term proportional to the square of the angular velocity $\\dot{\\theta}^{2}$.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Tangential acceleration\n",
    "\n",
    "The term proportional to angular acceleration, $a_t$, is always tangent to the trajectory of the endpoint (see figure below) and it's magnitude or Euclidean norm is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABcAAAAZCAYAAADaILXQAAAABHNCSVQICAgIfAhkiAAAAb5JREFU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\nTkSuQmCC\n",
      "text/latex": [
       "$$\\ell \\ddot{\\theta}$$"
      ],
      "text/plain": [
       "ell⋅θ̈"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = theta.diff(t, 2)\n",
    "simplify(sqrt(expand(acc[0]).coeff(A)**2 + expand(acc[1]).coeff(A)**2))*A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Centripetal acceleration\n",
    "\n",
    "The term proportional to angular velocity, $a_c$, always points to the joint, the center of the circular motion (see figure below), because of that this term is termed [centripetal acceleration](http://en.wikipedia.org/wiki/Centripetal_acceleration#Tangential_and_centripetal_acceleration). Its magnitude is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB8AAAAaCAYAAABPY4eKAAAABHNCSVQICAgIfAhkiAAAAhNJREFU\nSInt1k2IjlEUB/CfMfROM4Qk0ywoVmIsCQuLKQsJCzWhNCsh2dggIjWlGflYCMV+sqB8lIWFshBi\n4SNZyDSaGU3NGGYxRj4W97x55u1532leGpv51+3U+Z97/ue553Tvw99jWaz/gg+xJo2af1rGNBKO\n4Cm+YAC3sXKqxO+jLQRX4Sb6sWCqCsiiAT+wpejIG7gCrmIQt8okqsMJvMUoetCOWRXE54TeYKUK\nT2MIHfiF2hK+Ea8whi6cweuIvVIhbxdeYGa5gFppONqlgflaws/GM4xgfcbfgG7pWBfn5O1AH5ZX\nKE5LfEEzbuB5CX80+AM5e88Ht6PEfxafsKKScDGwHzNiQ2eGq5Pa0Su/t8dD/GDGd7GScGk/N+IR\n1mAR7mS47ZiHa/iek6sQdizsJezGNmnIiu0YiTVOvB6rpePehfd4mOE3h23CyRzxlrA9YfeFfVAS\ndypv/wbp2FoxjMMlfHfwE62lOYVNiL2x+bI08Q0Zrj64l2X2NkjH3VOGz0X2kmn0Z1o7RV8CTWF7\ny+TZJA3h3WrFC9KUD+NCSdzssN/K5GkLe30y4lkck758aw63MLgnOdxa/MS9aoVr8DiSzA9fwfi+\nv5FusOaMb4n0FzOkykGDQ9K7OypdFo14Jw1hETulrx/AOenx+RzC66oVniu9ta3SpdAXa39O7B7p\nERnFxyigKSduGtPIxW/nY33MspKitAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\ell \\dot{\\theta}^{2}$$"
      ],
      "text/plain": [
       "      2\n",
       "ell⋅θ̇ "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = theta.diff(t)**2\n",
    "simplify(sqrt(expand(acc[0]).coeff(A)**2+expand(acc[1]).coeff(A)**2))*A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "This means that there will be a linear acceleration even if the angular acceleration is zero because although the magnitude of the linear velocity is constant in this case, its direction varies (due to the centripetal acceleration).  \n",
    "<br>\n",
    "<figure><img src=\"./../images/onelink_acc.gif\" width=350 alt=\"onelinkAcc\"/><figcaption><center><i>Figure. Endpoint tangential and centripetal acceleration terms of one link attached to a fixed body by a hinge joint in a plane.</i></center></figcaption> </figure>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "Let's plot some simulated data to have an idea of the one-link kinematics. Consider $\\ell=1\\:m,\\theta_0=0^o,\\theta_f=90^o $, and $1\\:s$ of movement duration, and that it is a [minimum-jerk movement](https://nbviewer.jupyter.org/github/BMClab/bmc/blob/master/notebooks/MinimumJerkHypothesis.ipynb)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "theta0, thetaf, d = 0, np.pi/2, 1\n",
    "ts = np.arange(0.01, 1.01, .01)\n",
    "mjt  = theta0 + (thetaf - theta0)*(10*(t/d)**3 - 15*(t/d)**4 + 6*(t/d)**5)\n",
    "\n",
    "ang  = lambdify(t, mjt, 'numpy'); ang = ang(ts)\n",
    "vang = lambdify(t, mjt.diff(t,1), 'numpy'); vang = vang(ts)\n",
    "aang = lambdify(t, mjt.diff(t,2), 'numpy'); aang = aang(ts)\n",
    "jang = lambdify(t, mjt.diff(t,3), 'numpy'); jang = jang(ts)\n",
    "\n",
    "b, c, d, e = symbols('b c d e')\n",
    "dicti = {l:1, theta:b, theta.diff(t, 1):c, theta.diff(t, 2):d, theta.diff(t, 3):e}\n",
    "\n",
    "r2 = r.subs(dicti);\n",
    "rxfu = lambdify(b, r2[0], modules = 'numpy')\n",
    "ryfu = lambdify(b, r2[1], modules = 'numpy')\n",
    "\n",
    "v2 = v.subs(dicti);\n",
    "vxfu = lambdify((b, c), v2[0], modules = 'numpy')\n",
    "vyfu = lambdify((b, c), v2[1], modules = 'numpy')\n",
    "\n",
    "acc2 = acc.subs(dicti);\n",
    "axfu = lambdify((b, c, d), acc2[0], modules = 'numpy')\n",
    "ayfu = lambdify((b, c, d), acc2[1], modules = 'numpy')\n",
    "\n",
    "jerk = r.diff(t,3)\n",
    "jerk2 = jerk.subs(dicti);\n",
    "jxfu = lambdify((b, c, d, e), jerk2[0], modules = 'numpy')\n",
    "jyfu = lambdify((b, c, d, e), jerk2[1], modules = 'numpy')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Af1DVjXGKWelvTYy2maxvo1GHMbNAo6bxAs504Unc6O5T0ZPXOAKmE+GUrqp2TJJB4MPU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JylfYVNGb0E87ISmTY0lctzU8+aPig8z18HV52ldpxTd9I2ShsKePSOqdgU2JQiNEQREWojIjSE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\nhiPrOwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x23e9f83ed68>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, hax = plt.subplots(2, 4, sharex = True, figsize=(14, 7))\n",
    "hax[0, 0].plot(ts,ang*180/np.pi, linewidth=3)\n",
    "hax[0, 0].set_title('Angular displacement [ $^o$]');\n",
    "hax[0, 0].set_ylabel('Joint')\n",
    "hax[0, 1].plot(ts,vang*180/np.pi, linewidth=3)\n",
    "hax[0, 1].set_title('Angular velocity [ $^o/s$]');\n",
    "hax[0, 2].plot(ts,aang*180/np.pi, linewidth=3)\n",
    "hax[0, 2].set_title('Angular acceleration [ $^o/s^2$]');\n",
    "hax[0, 3].plot(ts,jang*180/np.pi, linewidth=3)\n",
    "hax[0, 3].set_title('Angular jerk [ $^o/s^3$]');\n",
    "hax[1, 0].plot(ts,rxfu(ang), 'r', linewidth=3, label = 'x')\n",
    "hax[1, 0].plot(ts,ryfu(ang), 'k', linewidth=3, label = 'y')\n",
    "hax[1, 0].set_xlabel('Time [s]'); \n",
    "hax[1, 0].set_title('Linear displacement [$m$]');\n",
    "hax[1, 0].legend(loc='best').get_frame().set_alpha(0.8)\n",
    "hax[1, 0].set_ylabel('Endpoint')\n",
    "hax[1, 1].plot(ts,vxfu(ang,vang), 'r', linewidth=3)\n",
    "hax[1, 1].plot(ts,vyfu(ang,vang), 'k', linewidth=3)\n",
    "hax[1, 1].set_xlabel('Time [s]'); \n",
    "hax[1, 1].set_title('Linear velocity [$m/s$]');\n",
    "hax[1, 2].plot(ts,axfu(ang,vang,aang), 'r', linewidth=3)\n",
    "hax[1, 2].plot(ts,ayfu(ang,vang,aang), 'k', linewidth=3)\n",
    "hax[1, 2].set_xlabel('Time [s]'); \n",
    "hax[1, 2].set_title('Linear acceleration [$m/s^2$]');\n",
    "hax[1, 3].plot(ts,jxfu(ang,vang,aang,jang), 'r', linewidth=3)\n",
    "hax[1, 3].plot(ts,jyfu(ang,vang,aang,jang), 'k', linewidth=3)\n",
    "hax[1, 3].set_xlabel('Time [s]'); \n",
    "hax[1, 3].set_title('Linear jerk [$m/s^2$]');\n",
    "fig.suptitle('Minimum jerk trajectory kinematics of one-link system', fontsize=20);\n",
    "for hax2 in hax.flat:\n",
    "        hax2.locator_params(nbins=5)\n",
    "        hax2.grid(True)\n",
    "plt.subplots_adjust(hspace=0.15) #plt.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Jacobian matrix\n",
    "\n",
    "The [Jacobian matrix](http://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant) is the matrix of all first-order partial derivatives of a vector-valued function $F$.\n",
    "\n",
    "$F(q_1,...q_n) = \\begin{bmatrix}F_{1}(q_1,...q_n)\\\\\n",
    "\\vdots\\\\ \n",
    "F_{m}(q_1,...q_n)\\\\ \n",
    "\\end{bmatrix}\n",
    "$\n",
    "\n",
    "\n",
    "In a general form, the Jacobian matrix of the function $F$ is:   \n",
    "<br />\n",
    "$$\n",
    "\\mathbf{J}= \n",
    "\\large\n",
    "\\begin{bmatrix}\n",
    "\\frac{\\partial F_{1}}{\\partial q_{1}} & ... & \\frac{\\partial F_{1}}{\\partial q_{n}} \\\\\n",
    "\\vdots  & \\ddots  & \\vdots \\\\ \n",
    "\\frac{\\partial F_{m}}{\\partial q_{1}} & ... & \\frac{\\partial F_{m}}{\\partial q_{n}} \\\\ \n",
    "\\end{bmatrix}\n",
    "$$    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Derivative of a vector-valued function using the Jacobian matrix\n",
    "\n",
    "The time-derivative of a vector-valued function $F$ can be computed using the Jacobian matrix:\n",
    "\n",
    "$\\frac{dF}{dt} = J\\begin{bmatrix}\\frac{d q_1}{dt}\\\\\n",
    "\\vdots\\\\ \n",
    "\\frac{d q_n}{dt}\\\\ \n",
    "\\end{bmatrix}$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Jacobian matrix in the context of kinematic chains\n",
    "\n",
    "In the context of kinematic chains, the Jacobian is a matrix of all first-order partial derivatives of the linear position vector of the endpoint with respect to the angular position vector.  \n",
    "\n",
    "The Jacobian matrix for a kinematic chain relates differential changes in the joint angle vector with the resulting differential changes in the linear position vector of the endpoint.  \n",
    "\n",
    "For a kinematic chain, the function $F_{i}$ is the expression of the endpoint position in $m$ coordinates and the variable $q_{i}$ is the angle of each $n$ joints. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Jacobian matrix of one-link chain\n",
    "\n",
    "For the planar one-link case, the Jacobian matrix of the position vector of the endpoint with respect to the angular position vector is (<span class=\"notranslate\">\n",
    "$\\mathbf{r}= \\begin{bmatrix}\n",
    "x_P\\\\\n",
    "y_P\n",
    "\\end{bmatrix} = \n",
    "\\begin{bmatrix}\n",
    "l\\cos(\\theta)\\\\\n",
    "l\\sin(\\theta)\n",
    "\\end{bmatrix}\n",
    "$</span> and <span class=\"notranslate\">\n",
    "$q_1=\\theta$</span>):\n",
    "\n",
    "<span class=\"notranslate\">\n",
    "\\begin{equation}\n",
    "\\mathbf{J}= \n",
    "\\large\n",
    "\\begin{bmatrix}\n",
    "\\frac{\\partial \\:x_P}{\\partial \\theta} \\\\\n",
    "\\frac{\\partial \\:y_P}{\\partial \\theta} \\\\\n",
    "\\end{bmatrix}\n",
    "\\end{equation}\n",
    "</span>\n",
    "\n",
    "Which calculates to:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\ell \\operatorname{sin}\\left(\\theta\\right)\\\\\\ell \\operatorname{cos}\\left(\\theta\\right)\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-ell⋅sin(θ)⎤\n",
       "⎢           ⎥\n",
       "⎣ell⋅cos(θ) ⎦"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "J = r.diff(theta)\n",
    "J"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "And Sympy has a function to calculate the Jacobian:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGAAAAAyCAYAAACj3t7EAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAIpklEQVR4Ae2cjVEdOQzHHwwFMFwHXAdAKkjSQS5XQaCDZFLBDXSQ0EGSDpJUwEEH0MEBHXD/n5/l2N61d99+PMjwNGPWtmRZlmRZb9fJ1mKxuFPZVYnh7OHh4UPcsamP08DW1tapOLzPuNxvqQMDvFO5ipC3MsB91N5UR2pABsDJ9yI2B6qf7/iOGyn8JkLOVpUgx2K+q/nOZptkBGPJt6/hryTf5zY2XpFvhcNJv8U0wr1R+6pNl+rDoYNTez6L7ZjB3HVN+snP8Vl1DLESaMy+yt2QsX0mEl+89FNF+a+E/6ryBX6it/XQXHiDnKofI/YGQtCBBsNgtqI58I5L5hCw0DerzqcxbFsYnK46tg+9+F6q7LfRqp+53XlpeLWRhd0c9EZbxa0z7s/ronH81rkDzjXpPyrAkUp85rjOrj9aBNt7S2XyBMHvqn/FuxSK8fwPwocw4uVlVwTw+B/ilx+4gSavzL4DNCHh5k7COU9RfRYPNv5DnpLpWqXk/Yn8xl/07IBja9tTfeyCsF7rj5/Cr3UHnGhCi5vERxa7MhBbVQ4oKw+uDIAvaCmo5P3sOCe/sYlkuLU+e4oPu+RGNMnuMHz8tCwo7pu07heHwiz8sI0xSAP8ohAaReyp4El/i/7Q8+HQA0/28ZcKByG8CW8oESUZjaqLFyoXGn9GowKcTz/a8OLPfPAmBMVAP5CHpGXvkh8ytvI1Ip6zhiDxJxa6w0pPFtp64KsfZX+XssKBRl1wHfepzUH5Ne4zOvV/V3lvONVRHExaQ0tEx7hGKPF8UTw8eMYFvYkkldfawhG2ioexcDjO3Y7+zA2vNYF59J4ETHLnaHIO5iN5HFlF7FVJqieaxpb3PJjjSGOZz4HqhAHqeGtrXu8Il4Yq8UVRHP5ux3l6dh5GqSUS8MOpqlA1AMrQ6J9VDk3kOwkbC4YXohzSzmIoEI7MAaHJ89m2eOW32hjhc2CeNuhSBOEuNnrMA/kTuSUfRgEsrC5b6V9kYWwVqgbQ4hHqsMqhG4kQeFCyiMIw5vqogsfy7oQfNZ81tvXMED4HDDgZaG4z3EXGFHnuJVdpN2fk5eZ2GTUeowWYB3Tm7UarRXFIH6oQO1josXAYZE7oChf57iC+17wfWdlVpR0J3sFsBvAKNc/t45kYi4UFkBGI23iZGTLgJq6gKBSWgOZH8YnytS6SCt6dde1oZE7GJsx9YzYDiD+ZB55PPA8KxDAqZENt8FG43QxBO0/lcppsyMrNK40ohdovwrmYj+yq41ThoFe9BPDL5W6lnTsNJXyQyaA0FN/6K1j9FvfZBXgZT84B3kzq4ZRgKaGlhfBEOXk/fczFQQ4ta2ykrvD1vJk7SXcjHHMgPzK5dRiu9hQt6XJryu3nREbkmvd3QDQZCq3m47UFzY2TbMVXEavOLV4YrdWgxkt4Z4A5Q5DmWIImJQsim+k8lGzMIzzZbZ3JQk+5yOTYLZ2wFgN0SvEECHAQicH5FM6rIWL5M4yw2XVIO/YbA6Ra5tduL89NhyUtzqPkV3OCzRobA0QKkdfeq3kiL07S4YikWvXjTlYJtTtVjs8Q6ZVXe29U1IrGrjxuswOK6lwPYmOA9ei5OMvGAEXVrAexMcB69FycZVIDkAWo8JP9SYNkJN8vZjrC7YJXabyzok9lf6oFTmYACWX586BLV1MtqIuP5HTvdkoZi/C8FyKX5yXcZJev4FWC0S/jxBhPGXXpSgpJvgXP1UZOldZ3Uuq3F2ThspX6eJkX2shFW6X4vbeP7Bo/6bugczG0DxRHqvN698mBvJuw8yiXr2rKGLUDxJhFhUtIqre+bu7jFXPTSLbiG898HSaL+nH5xo0J9bELwrqNvu9TYyfbAXygsHjJ4TTo0pXGzQryfndwSkGlN7KzXb6qLWzUqwi/KCxp4ad46QohRI/X8Nr3nraANneBwsdtT8Pr3P8gEPyhAk34uiQa5uSwtE+J8HEXuPQsAedU4BETiR+8MBCHbwz0AybvsvXrL/x6Xb76NaRZGxyCxIqU0x1SerLA2hcgFshc4VY09L7PHYq+3QgT6ufLlrtwpacZLTm01d/1AQQejVBCyBDYFzWecUFekbQnCMIRfgcdxhrn1j5qB4hJ30tXInUL42N28Hb1ubCgJ0oFWDx3gfIwQXi4lKficXsqfS5wiSwB5ipdDkAZs12+SqRoaYzZAXhr8M6Kp6BgXI1//JB4rrWFQwnQhB1iOJ4e5w541ZmXTje3nq1pZTaedbrvy3F/zttwoq3K48c5GhuzytPzv9tWZQzgVVxQ6vr6Y55eO6CNphRvkZMFA9w4YM49Fc6Ua+0O+yGoZn/QONt9F9kokotJLl9lfJPmYANIcFNYn++oFlL+TGZPG0ZjCkmxyxZ3Pd28MvqqF7gIPzXeueGJ75ZctMlCHw5gcpdoqv2DDOCVgIcApbi6xOqvlMXi+HHGj7QGiB9hBzx0jTs3wpvnc0ZgAJQTQGP7XOBCUSgsAS9bonzNR3Ix2eWrZMKsMcgA4kHMxfM5FJ1HwhfDoEzqLUC61ngJJnpCCMoHXqq8VZ8p3HXqz7kK/3bZ0si+F7hsPE/meLTLV7Egcd3+nfBLLc6UEOOrdSmKPBnFYgzqL7xhVG2C6HfVizJvVTgPyPE5mMM2Fg0GhZ95JWO4VOWU7+dklzAeHHSEtuS3gtoJ+HHM1QiDXi53lmgQeMKbzZ/wiRsax3ul/DZ4TFKsayxO9hOCwVmQhGQ8jAgJnZkI9I9ZJCNGm0RO8cH41d8etbVqLHobnQWh0N/h0pXW6gAv75M0ePLqg1/rgzKvmOvQMyDm8dvU5SyPcvmqpqBnZQCvCM6ssZ5LNgaf0fDsDKBdwOG61stXNSuNfRdU4/1kcTICWdfKl6hYkA9jk63NDEB+Hqddt95TJpvouTPyqe5epAfS7YX9DiClimHzHzbF2pigLgOQgeU3Ru7/BzQdP40d6JG7AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\ell \\operatorname{sin}\\left(\\theta\\right)\\\\\\ell \\operatorname{cos}\\left(\\theta\\right)\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-ell⋅sin(θ)⎤\n",
       "⎢           ⎥\n",
       "⎣ell⋅cos(θ) ⎦"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "J = r.jacobian([theta])\n",
    "J"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "We can recalculate the kinematic expressions using the Jacobian matrix, which can be useful for simplifying the deduction.\n",
    "\n",
    "The linear velocity of the end-effector is given by the product between the Jacobian of the kinematic link and the angular velocity:\n",
    "\n",
    "$$ \\overrightarrow{\\mathbf{v}} = \\mathbf{J} \\cdot \\dot{\\theta}$$\n",
    "\n",
    "Where:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAATCAYAAACp65zuAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA9UlEQVQoFY2SjQ2CQAxGOeMAzIAbOIMjEEdgBJmBEVxBXcURcARgA3xfc700EROafDnbvrv+SFrXtdpjhy0opVSjGb09vwlSZQGY0MfBtLf00W/EU6Xxr2jioZdyP6WBLsSf6CEA/66zUmkX7hnNqA4xraUukBLYiDqHckyJtoAC0ByhAHaxxx7Q+uI0oz+1IpsMzAM0BDRENA0mW2yPgAJaZKtQJptADZasRxwN8f7Tn8Ud1GRDBPHVn02ch2JHIeAwMS26bIHfBbw4lF/QazePeWn9GwUUgEaH7GK+rTJ2m1Nr0nBNBH096nPIwImzB9I3WewLNx3VQid+wMkAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\dot{\\theta}$"
      ],
      "text/plain": [
       "θ̇"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w = theta.diff(t)\n",
    "w"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "The angular velocity is also a vector; it's direction is perpendicular to the plane of rotation and using the [right-hand rule](http://en.wikipedia.org/wiki/Right-hand_rule) this direction is the same as of the versor $\\hat{\\mathbf{k}}$ coming out of the screen (paper).   \n",
    "\n",
    "Then:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\ell \\operatorname{sin}\\left(\\theta\\right) \\dot{\\theta}\\\\\\ell \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-ell⋅sin(θ)⋅θ̇⎤\n",
       "⎢              ⎥\n",
       "⎣ell⋅cos(θ)⋅θ̇ ⎦"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "velJ = J*w\n",
    "velJ"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "And the linear acceleration of the endpoint is given by the derivative of this product:\n",
    " \n",
    "$$ \\overrightarrow{\\mathbf{a}} = \\dot{\\mathbf{J}} \\cdot \\overrightarrow{\\mathbf{\\omega}} + \\mathbf{J} \\cdot \\dot{\\overrightarrow{\\mathbf{\\omega}}} $$\n",
    "\n",
    "Let's calculate this derivative:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\ell \\operatorname{sin}\\left(\\theta\\right) \\ddot{\\theta} - \\ell \\operatorname{cos}\\left(\\theta\\right) \\dot{\\theta}^{2}\\\\- \\ell \\operatorname{sin}\\left(\\theta\\right) \\dot{\\theta}^{2} + \\ell \\operatorname{cos}\\left(\\theta\\right) \\ddot{\\theta}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡                              2 ⎤\n",
       "⎢-ell⋅sin(θ)⋅θ̈ - ell⋅cos(θ)⋅θ̇  ⎥\n",
       "⎢                                ⎥\n",
       "⎢               2                ⎥\n",
       "⎣- ell⋅sin(θ)⋅θ̇  + ell⋅cos(θ)⋅θ̈⎦"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "accJ = J.diff(t)*w + J*w.diff(t)\n",
    "accJ"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "These two expressions derived with the Jacobian are the same as the direct derivatives of the equation for the endpoint position."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## The kinematics of a two-link chain\n",
    "\n",
    "We now will look at the case of a planar kinematic chain with two links, as shown below. The deduction will be similar to the case with one link we just saw.  \n",
    "<br>\n",
    "<figure><img src=\"./../images/twolinks.gif\" width=400 alt=\"twolinks\"/><figcaption><center><i>Figure. Kinematics of a two-link chain with hinge joints in a plane.</i></center></figcaption> </figure>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We need to define a Cartesian coordinate system and the symbolic variables $t,\\:\\ell_1,\\:\\ell_2,\\:\\theta_1,\\:\\theta_2$ (and make $\\theta_1$ and $\\theta_2$ function of time):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "O = CoordSys3D('')\n",
    "t = Symbol('t')\n",
    "l1, l2 = symbols('ell_1 ell_2', positive=True)\n",
    "theta1, theta2 = dynamicsymbols('theta1 theta2')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The position of the endpoint in terms of the joint angles and link lengths is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle (\\ell_{1} \\cos{\\left(\\theta_{1}{\\left(t \\right)} \\right)} + \\ell_{2} \\cos{\\left(\\theta_{1}{\\left(t \\right)} + \\theta_{2}{\\left(t \\right)} \\right)})\\mathbf{\\hat{i}_{}} + (\\ell_{1} \\sin{\\left(\\theta_{1}{\\left(t \\right)} \\right)} + \\ell_{2} \\sin{\\left(\\theta_{1}{\\left(t \\right)} + \\theta_{2}{\\left(t \\right)} \\right)})\\mathbf{\\hat{j}_{}}$"
      ],
      "text/plain": [
       "(ell_1*cos(theta1(t)) + ell_2*cos(theta1(t) + theta2(t)))*.i + (ell_1*sin(theta1(t)) + ell_2*sin(theta1(t) + theta2(t)))*.j"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r2_p = (l1*cos(theta1) + l2*cos(theta1 + theta2))*O.i + (l1*sin(theta1) + l2*sin(theta1 + theta2))*O.j\n",
    "r2_p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "With the components:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{ \\mathbf{\\hat{i}_{}} : \\ell_{1} \\operatorname{cos}\\left(\\theta_{1}\\right) + \\ell_{2} \\operatorname{cos}\\left(\\theta_{1} + \\theta_{2}\\right), \\  \\mathbf{\\hat{j}_{}} : \\ell_{1} \\operatorname{sin}\\left(\\theta_{1}\\right) + \\ell_{2} \\operatorname{sin}\\left(\\theta_{1} + \\theta_{2}\\right)\\right\\}$"
      ],
      "text/plain": [
       "{_i: ell₁⋅cos(θ₁) + ell₂⋅cos(θ₁ + θ₂), _j: ell₁⋅sin(θ₁) + ell₂⋅sin(θ₁ + θ₂)}"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r2_p.components"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "And in matrix form:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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CtjHIPX9mRQOe5AuvLX8/7hrYNXCiGlAfZ/l7FbDvnvuxSIDCEF5qEJj6mGxIbz/fNbBrYGMNaBDgrgrPv4Rw/S9Wutd+bViqgAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\ell_{1} \\operatorname{cos}\\left(\\theta_{1}\\right) + \\ell_{2} \\operatorname{cos}\\left(\\theta_{1} + \\theta_{2}\\right)\\\\\\ell_{1} \\operatorname{sin}\\left(\\theta_{1}\\right) + \\ell_{2} \\operatorname{sin}\\left(\\theta_{1} + \\theta_{2}\\right)\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡ell₁⋅cos(θ₁) + ell₂⋅cos(θ₁ + θ₂)⎤\n",
       "⎢                                ⎥\n",
       "⎣ell₁⋅sin(θ₁) + ell₂⋅sin(θ₁ + θ₂)⎦"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r2 = Matrix((r2_p.dot(O.i), r2_p.dot(O.j)))\n",
    "r2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Joint and segment angles\n",
    "\n",
    "Note that $\\theta_2$ is a joint angle (referred as measured in the joint space); the angle of the segment 2 with respect to the horizontal is $\\theta_1+\\theta_2$ and is referred as an angle in the segmental space. A joint and segment angles are also referred as relative and absolute angles, respectively."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Inverse kinematics \n",
    "\n",
    "Using the [cosine rule](http://en.wikipedia.org/wiki/Law_of_cosines), in terms of the endpoint position, the angle $\\theta_2$ is:\n",
    "\n",
    "<span class=\"notranslate\">\n",
    "\\begin{equation}\n",
    "x_P^2 + y_P^2 = \\ell_1^2+\\ell_2^2 - 2\\ell_1 \\ell_2 cos(\\pi-\\theta_2)\n",
    "\\end{equation}\n",
    "</span>\n",
    "\n",
    "<span class=\"notranslate\">\n",
    "\\begin{equation}\n",
    "\\theta_2 = \\arccos\\left(\\frac{x_P^2 + y_P^2 - \\ell_1^2 - \\ell_2^2}{2\\ell_1 \\ell_2}\\;\\;\\right)\n",
    "\\end{equation}\n",
    "</span>\n",
    "\n",
    "To find the angle $\\theta_1$, if we now look at the triangle in red in the figure below, its angle $\\phi$ is:\n",
    "\n",
    "<span class=\"notranslate\">\n",
    "\\begin{equation}\n",
    "\\phi = \\arctan\\left(\\frac{\\ell_2 \\sin(\\theta_2)}{\\ell_1 + \\ell_2 \\cos(\\theta_2)}\\right)\n",
    "\\end{equation}\n",
    "</span>\n",
    "\n",
    "And the angle of its hypotenuse with the horizontal is:\n",
    "\n",
    "<span class=\"notranslate\">\n",
    "\\begin{equation}\n",
    "\\theta_1 + \\phi = \\arctan\\left(\\frac{y_P}{x_P}\\right)\n",
    "\\end{equation}\n",
    "</span>\n",
    "\n",
    "Then, the angle $\\theta_1$ is:\n",
    "\n",
    "<span class=\"notranslate\">\n",
    "\\begin{equation}\n",
    "\\theta_1 = \\arctan\\left(\\frac{y_P}{x_P}\\right) - \\arctan\\left(\\frac{\\ell_2 \\sin(\\theta_2)}{\\ell_1+\\ell_2 \\cos(\\theta_2)}\\right)\n",
    "\\end{equation}\n",
    "</span>\n",
    "\n",
    "Note that there are two possible sets of $(\\theta_1, \\theta_2)$ angles for the same $(x_P, y_P)$ coordinate that satisfy the equations above. The figure below shows in orange another possible configuration of the kinematic chain with the same endpoint coordinate. The other solution is $\\theta_2'=2\\pi - \\theta_2$, but <span class=\"notranslate\">$\\sin(\\theta_2')=-sin(\\theta_{2})$</span> and then the $arctan()$ term in the last equation becomes negative.   \n",
    "Even for a simple two-link chain we already have a problem of redundancy, there is more than one joint configuration for the same endpoint position; this will be much more problematic for chains with more links (more degrees of freedom).  \n",
    "<br>\n",
    "<figure><img src=\"./../images/twolinks_ik.gif\" width=350 alt=\"twolinks_ik\"/><figcaption><center><i>Figure. Indetermination in the inverse kinematics approach to determine one of the joint angles for a two-link chain with hinge joints in a plane.</i></center></figcaption> </figure>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Differential kinematics \n",
    "\n",
    "The linear velocity of the endpoint is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\ell_{1} \\operatorname{sin}\\left(\\theta_{1}\\right) \\dot{\\theta}_{1} - \\ell_{2} \\left(\\dot{\\theta}_{1} + \\dot{\\theta}_{2}\\right) \\operatorname{sin}\\left(\\theta_{1} + \\theta_{2}\\right)\\\\\\ell_{1} \\operatorname{cos}\\left(\\theta_{1}\\right) \\dot{\\theta}_{1} + \\ell_{2} \\left(\\dot{\\theta}_{1} + \\dot{\\theta}_{2}\\right) \\operatorname{cos}\\left(\\theta_{1} + \\theta_{2}\\right)\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-ell₁⋅sin(θ₁)⋅θ₁̇ - ell₂⋅(θ₁̇ + θ₂̇)⋅sin(θ₁ + θ₂)⎤\n",
       "⎢                                                 ⎥\n",
       "⎣ell₁⋅cos(θ₁)⋅θ₁̇ + ell₂⋅(θ₁̇ + θ₂̇)⋅cos(θ₁ + θ₂) ⎦"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vel2 = r2.diff(t)\n",
    "vel2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "The linear velocity of the endpoint is the sum of the velocities at each joint, i.e., it is the velocity of the endpoint in relation to joint 2, for instance, <span class=\"notranslate\">\n",
    "$\\ell_2cos(\\theta_1 + \\theta_2)\\dot{\\theta}_1$</span>, plus the velocity of joint 2 in relation to joint 1, for instance, <span class=\"notranslate\">$\\ell_1\\dot{\\theta}_1 cos(\\theta_1)$</span>, and this last term we already saw for the one-link example. In classical mechanics this is known as [relative velocity](http://en.wikipedia.org/wiki/Relative_velocity), an example of [Galilean transformation](http://en.wikipedia.org/wiki/Galilean_transformation)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "The linear acceleration of the endpoint is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- (\\ell_{1} \\operatorname{sin}\\left(\\theta_{1}\\right) \\ddot{\\theta}_{1} + \\ell_{1} \\operatorname{cos}\\left(\\theta_{1}\\right) \\dot{\\theta}_{1}^{2} + \\ell_{2} \\left(\\dot{\\theta}_{1} + \\dot{\\theta}_{2}\\right)^{2} \\operatorname{cos}\\left(\\theta_{1} + \\theta_{2}\\right) + \\ell_{2} \\left(\\ddot{\\theta}_{1} + \\ddot{\\theta}_{2}\\right) \\operatorname{sin}\\left(\\theta_{1} + \\theta_{2}\\right))\\\\- \\ell_{1} \\operatorname{sin}\\left(\\theta_{1}\\right) \\dot{\\theta}_{1}^{2} + \\ell_{1} \\operatorname{cos}\\left(\\theta_{1}\\right) \\ddot{\\theta}_{1} - \\ell_{2} \\left(\\dot{\\theta}_{1} + \\dot{\\theta}_{2}\\right)^{2} \\operatorname{sin}\\left(\\theta_{1} + \\theta_{2}\\right) + \\ell_{2} \\left(\\ddot{\\theta}_{1} + \\ddot{\\theta}_{2}\\right) \\operatorname{cos}\\left(\\theta_{1} + \\theta_{2}\\right)\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡ ⎛                                   2                   2                   \n",
       "⎢-⎝ell₁⋅sin(θ₁)⋅θ₁̈ + ell₁⋅cos(θ₁)⋅θ₁̇  + ell₂⋅(θ₁̇ + θ₂̇) ⋅cos(θ₁ + θ₂) + ell\n",
       "⎢                                                                             \n",
       "⎢                  2                                      2                   \n",
       "⎣- ell₁⋅sin(θ₁)⋅θ₁̇  + ell₁⋅cos(θ₁)⋅θ₁̈ - ell₂⋅(θ₁̇ + θ₂̇) ⋅sin(θ₁ + θ₂) + ell\n",
       "\n",
       "                          ⎞⎤\n",
       "₂⋅(θ₁̈ + θ₂̈)⋅sin(θ₁ + θ₂)⎠⎥\n",
       "                           ⎥\n",
       "                           ⎥\n",
       "₂⋅(θ₁̈ + θ₂̈)⋅cos(θ₁ + θ₂) ⎦"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "acc2 = r2.diff(t, 2)\n",
    "acc2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "We can separate the equation above for the linear acceleration in three types of terms: proportional to $\\ddot{\\theta}$ and to $\\dot{\\theta}^2$, as we already saw for the one-link case, and a new term, proportional to $\\dot{\\theta}_1\\dot{\\theta}_2$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle Tangential:\\:\\left[\\begin{matrix}- \\ell_{2} \\operatorname{sin}\\left(\\theta_{1} + \\theta_{2}\\right) \\ddot{\\theta}_{2} + \\left(- \\ell_{1} \\operatorname{sin}\\left(\\theta_{1}\\right) - \\ell_{2} \\operatorname{sin}\\left(\\theta_{1} + \\theta_{2}\\right)\\right) \\ddot{\\theta}_{1}\\\\\\ell_{2} \\operatorname{cos}\\left(\\theta_{1} + \\theta_{2}\\right) \\ddot{\\theta}_{2} + \\left(\\ell_{1} \\operatorname{cos}\\left(\\theta_{1}\\right) + \\ell_{2} \\operatorname{cos}\\left(\\theta_{1} + \\theta_{2}\\right)\\right) \\ddot{\\theta}_{1}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle Centripetal:\\left[\\begin{matrix}- \\ell_{2} \\operatorname{cos}\\left(\\theta_{1} + \\theta_{2}\\right) \\dot{\\theta}_{2}^{2} + \\left(- \\ell_{1} \\operatorname{cos}\\left(\\theta_{1}\\right) - \\ell_{2} \\operatorname{cos}\\left(\\theta_{1} + \\theta_{2}\\right)\\right) \\dot{\\theta}_{1}^{2}\\\\- \\ell_{2} \\operatorname{sin}\\left(\\theta_{1} + \\theta_{2}\\right) \\dot{\\theta}_{2}^{2} + \\left(- \\ell_{1} \\operatorname{sin}\\left(\\theta_{1}\\right) - \\ell_{2} \\operatorname{sin}\\left(\\theta_{1} + \\theta_{2}\\right)\\right) \\dot{\\theta}_{1}^{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle Coriolis:\\;\\;\\;\\;\\:\\left[\\begin{matrix}- 2 \\ell_{2} \\operatorname{cos}\\left(\\theta_{1} + \\theta_{2}\\right) \\dot{\\theta}_{1} \\dot{\\theta}_{2}\\\\- 2 \\ell_{2} \\operatorname{sin}\\left(\\theta_{1} + \\theta_{2}\\right) \\dot{\\theta}_{1} \\dot{\\theta}_{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "acc2 = acc2.expand()\n",
    "A = theta1.diff(t, 2)\n",
    "B = theta2.diff(t, 2)\n",
    "tg = A*Matrix((acc2[0].coeff(A),acc2[1].coeff(A)))+B*Matrix((acc2[0].coeff(B),acc2[1].coeff(B)))\n",
    "\n",
    "A = theta1.diff(t)**2\n",
    "B = theta2.diff(t)**2\n",
    "ct = A*Matrix((acc2[0].coeff(A),acc2[1].coeff(A)))+B*Matrix((acc2[0].coeff(B),acc2[1].coeff(B)))\n",
    "\n",
    "A = theta1.diff(t)*theta2.diff(t)\n",
    "co = A*Matrix((acc2[0].coeff(A),acc2[1].coeff(A)))\n",
    "\n",
    "display(Math(mlatex(r'Tangential:\\:') + mlatex(tg)))\n",
    "display(Math(mlatex(r'Centripetal:') + mlatex(ct)))\n",
    "display(Math(mlatex(r'Coriolis:\\;\\;\\;\\;\\:') + mlatex(co)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "This new term is called the [Coriolis acceleration](http://en.wikipedia.org/wiki/Coriolis_effect); it is 'felt' by the endpoint when its distance to the instantaneous center of rotation varies, due to the links' constraints, and as consequence the endpoint motion is deflected (its direction is perpendicular to the relative linear velocity of the endpoint with respect to the linear velocity at the second joint, $\\mathbf{v} - \\mathbf{v}_{joint2}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Let's now deduce the Jacobian for this planar two-link chain:\n",
    "\n",
    "$$\n",
    "\\mathbf{J} = \n",
    "\\large\n",
    "\\begin{bmatrix}\n",
    "\\frac{\\partial x_P}{\\partial \\theta_{1}} & \\frac{\\partial x_P}{\\partial \\theta_{2}} \\\\\n",
    "\\frac{\\partial y_P}{\\partial \\theta_{1}} & \\frac{\\partial y_P}{\\partial \\theta_{2}} \\\\\n",
    "\\end{bmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We could manually run:  \n",
    "```python\n",
    "J = Matrix([[r2[0].diff(theta1), r2[0].diff(theta2)], [r2[1].diff(theta1), r2[1].diff(theta2)]])\n",
    "```\n",
    "But it's shorter with the Sympy Jacobian function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\ell_{1} \\operatorname{sin}\\left(\\theta_{1}\\right) - \\ell_{2} \\operatorname{sin}\\left(\\theta_{1} + \\theta_{2}\\right) & - \\ell_{2} \\operatorname{sin}\\left(\\theta_{1} + \\theta_{2}\\right)\\\\\\ell_{1} \\operatorname{cos}\\left(\\theta_{1}\\right) + \\ell_{2} \\operatorname{cos}\\left(\\theta_{1} + \\theta_{2}\\right) & \\ell_{2} \\operatorname{cos}\\left(\\theta_{1} + \\theta_{2}\\right)\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-ell₁⋅sin(θ₁) - ell₂⋅sin(θ₁ + θ₂)  -ell₂⋅sin(θ₁ + θ₂)⎤\n",
       "⎢                                                     ⎥\n",
       "⎣ell₁⋅cos(θ₁) + ell₂⋅cos(θ₁ + θ₂)   ell₂⋅cos(θ₁ + θ₂) ⎦"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "J2 = r2.jacobian([theta1, theta2])\n",
    "J2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Using the Jacobian, the linear velocity of the endpoint is:   \n",
    "\n",
    "$$ \\mathbf{v_J} = \\mathbf{J} \\cdot \\begin{bmatrix}\\dot{\\theta_1}\\\\\n",
    "\\dot{\\theta_2}\\\\ \n",
    "\\end{bmatrix} $$ \n",
    "\n",
    "Where:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\dot{\\theta}_{1}\\\\\\dot{\\theta}_{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡θ₁̇⎤\n",
       "⎢   ⎥\n",
       "⎣θ₂̇⎦"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w2 = Matrix((theta1, theta2)).diff(t)\n",
    "w2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Then:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\ell_{2} \\operatorname{sin}\\left(\\theta_{1} + \\theta_{2}\\right) \\dot{\\theta}_{2} + \\left(- \\ell_{1} \\operatorname{sin}\\left(\\theta_{1}\\right) - \\ell_{2} \\operatorname{sin}\\left(\\theta_{1} + \\theta_{2}\\right)\\right) \\dot{\\theta}_{1}\\\\\\ell_{2} \\operatorname{cos}\\left(\\theta_{1} + \\theta_{2}\\right) \\dot{\\theta}_{2} + \\left(\\ell_{1} \\operatorname{cos}\\left(\\theta_{1}\\right) + \\ell_{2} \\operatorname{cos}\\left(\\theta_{1} + \\theta_{2}\\right)\\right) \\dot{\\theta}_{1}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-ell₂⋅sin(θ₁ + θ₂)⋅θ₂̇ + (-ell₁⋅sin(θ₁) - ell₂⋅sin(θ₁ + θ₂))⋅θ₁̇⎤\n",
       "⎢                                                                ⎥\n",
       "⎣ ell₂⋅cos(θ₁ + θ₂)⋅θ₂̇ + (ell₁⋅cos(θ₁) + ell₂⋅cos(θ₁ + θ₂))⋅θ₁̇ ⎦"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vel2J = J2*w2\n",
    "vel2J"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "This expression derived with the Jacobian is the same as the first-order derivative  of the equation for the endpoint position. We can show this equality by comparing the two expressions with Sympy:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vel2.expand() == vel2J.expand()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Once again, the linear acceleration of the endpoint is given by the derivative of the product between the Jacobian and the angular velocity:\n",
    "\n",
    "<span class=\"notranslate\">\n",
    "\\begin{equation}\n",
    "\\mathbf{a} = \\dot{\\mathbf{J}} \\cdot \\mathbf{\\omega} + \\mathbf{J} \\cdot \\dot{\\mathbf{\\omega}}\n",
    "\\end{equation}\n",
    "</span>\n",
    "\n",
    "Let's calculate this derivative:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\ell_{2} \\left(\\dot{\\theta}_{1} + \\dot{\\theta}_{2}\\right) \\operatorname{cos}\\left(\\theta_{1} + \\theta_{2}\\right) \\dot{\\theta}_{2} - \\ell_{2} \\operatorname{sin}\\left(\\theta_{1} + \\theta_{2}\\right) \\ddot{\\theta}_{2} + \\left(- \\ell_{1} \\operatorname{sin}\\left(\\theta_{1}\\right) - \\ell_{2} \\operatorname{sin}\\left(\\theta_{1} + \\theta_{2}\\right)\\right) \\ddot{\\theta}_{1} + \\left(- \\ell_{1} \\operatorname{cos}\\left(\\theta_{1}\\right) \\dot{\\theta}_{1} - \\ell_{2} \\left(\\dot{\\theta}_{1} + \\dot{\\theta}_{2}\\right) \\operatorname{cos}\\left(\\theta_{1} + \\theta_{2}\\right)\\right) \\dot{\\theta}_{1}\\\\- \\ell_{2} \\left(\\dot{\\theta}_{1} + \\dot{\\theta}_{2}\\right) \\operatorname{sin}\\left(\\theta_{1} + \\theta_{2}\\right) \\dot{\\theta}_{2} + \\ell_{2} \\operatorname{cos}\\left(\\theta_{1} + \\theta_{2}\\right) \\ddot{\\theta}_{2} + \\left(\\ell_{1} \\operatorname{cos}\\left(\\theta_{1}\\right) + \\ell_{2} \\operatorname{cos}\\left(\\theta_{1} + \\theta_{2}\\right)\\right) \\ddot{\\theta}_{1} + \\left(- \\ell_{1} \\operatorname{sin}\\left(\\theta_{1}\\right) \\dot{\\theta}_{1} - \\ell_{2} \\left(\\dot{\\theta}_{1} + \\dot{\\theta}_{2}\\right) \\operatorname{sin}\\left(\\theta_{1} + \\theta_{2}\\right)\\right) \\dot{\\theta}_{1}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-ell₂⋅(θ₁̇ + θ₂̇)⋅cos(θ₁ + θ₂)⋅θ₂̇ - ell₂⋅sin(θ₁ + θ₂)⋅θ₂̈ + (-ell₁⋅sin(θ₁) -\n",
       "⎢                                                                             \n",
       "⎣-ell₂⋅(θ₁̇ + θ₂̇)⋅sin(θ₁ + θ₂)⋅θ₂̇ + ell₂⋅cos(θ₁ + θ₂)⋅θ₂̈ + (ell₁⋅cos(θ₁) + \n",
       "\n",
       " ell₂⋅sin(θ₁ + θ₂))⋅θ₁̈ + (-ell₁⋅cos(θ₁)⋅θ₁̇ - ell₂⋅(θ₁̇ + θ₂̇)⋅cos(θ₁ + θ₂))⋅\n",
       "                                                                              \n",
       "ell₂⋅cos(θ₁ + θ₂))⋅θ₁̈ + (-ell₁⋅sin(θ₁)⋅θ₁̇ - ell₂⋅(θ₁̇ + θ₂̇)⋅sin(θ₁ + θ₂))⋅θ\n",
       "\n",
       "θ₁̇⎤\n",
       "   ⎥\n",
       "₁̇ ⎦"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "acc2J = J2.diff(t)*w2 + J2*w2.diff(t)\n",
    "acc2J"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Once again, the expression above is the same as the second-order derivative of the equation for the endpoint position:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "acc2.expand() == acc2J.expand()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "Let's plot some simulated data to have an idea of the two-link kinematics.  \n",
    "Consider $\\ell_1=\\ell_2=0.5m, \\theta_1(0)=\\theta_2(0)=0$, 1 s of movement duration, $\\theta_1(1)=\\theta_2(1)=90^o$, and that it is a [minimum-jerk movement](https://nbviewer.jupyter.org/github/BMClab/bmc/blob/master/notebooks/MinimumJerkHypothesis.ipynb) of the endpoint.   \n",
    "\n",
    "First, the simulated trajectories:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "t, p0, pf, d, rx, ry = symbols('t p0 pf d rx ry')\n",
    "\n",
    "# minimum jerk kinematics\n",
    "mjt = p0 + (pf - p0)*(10*(t/d)**3 - 15*(t/d)**4 + 6*(t/d)**5)\n",
    "rfu = lambdify((t, p0, pf, d), mjt, 'numpy')\n",
    "vfu = lambdify((t, p0, pf, d), diff(mjt, t, 1), 'numpy')\n",
    "afu = lambdify((t, p0, pf, d), diff(mjt, t, 2), 'numpy')\n",
    "jfu = lambdify((t, p0, pf, d), diff(mjt, t, 3), 'numpy')\n",
    "\n",
    "#initial values:\n",
    "p0, pf, d, L1, L2 = [1, 0], [0.5, .5], 1, .5, .5\n",
    "ts = np.arange(0.01, 1.01, .01)\n",
    "\n",
    "#endpoint kinematics\n",
    "x  = rfu(ts, p0[0], pf[0], d)\n",
    "y  = rfu(ts, p0[1], pf[1], d)\n",
    "vx = vfu(ts, p0[0], pf[0], d)\n",
    "vy = vfu(ts, p0[1], pf[1], d)\n",
    "ax = afu(ts, p0[0], pf[0], d)\n",
    "ay = afu(ts, p0[1], pf[1], d)\n",
    "jx = jfu(ts, p0[0], pf[0], d)\n",
    "jy = jfu(ts, p0[1], pf[1], d)\n",
    "\n",
    "#inverse kinematics\n",
    "ang2b = np.arccos((x**2 + y**2 - L1**2 - L2**2)/(2*L1*L2))\n",
    "ang1b = np.arctan2(y, x) - (np.arctan2(L2*np.sin(ang2b), (L1+L2*np.cos(ang2b))))\n",
    "\n",
    "ang2 = acos((rx**2 + ry**2 - l1**2 - l2**2)/(2*l1*l2))\n",
    "ang2fu = lambdify((rx ,ry, l1, l2), ang2, 'numpy');\n",
    "ang2 = ang2fu(x, y, L1, L2)\n",
    "ang1 = atan2(ry, rx) - (atan(l2*sin(acos((rx**2 + ry**2 - l1**2 - l2**2)/(2*l1*l2)))/ \\\n",
    "                            (l1+l2*cos(acos((rx**2 + ry**2 - l1**2 - l2**2)/(2*l1*l2))))))\n",
    "ang1fu = lambdify((rx, ry, l1, l2), ang1, 'numpy');\n",
    "ang1 = ang1fu(x, y, L1, L2)\n",
    "\n",
    "rx = rx(t)\n",
    "ry = ry(t)\n",
    "ang2b = acos((rx**2 + ry**2 - l1**2 - l2**2)/(2*l1*l2))\n",
    "ang1b = atan2(ry, rx) - (atan(l2*sin(acos((rx**2 + ry**2 - l1**2 - l2**2)/(2*l1*l2)))/ \\\n",
    "                            (l1 + l2*cos(acos((rx**2 + ry**2-l1**2 - l2**2)/(2*l1*l2))))))\n",
    "X, Y, Xd, Yd, Xdd, Ydd, Xddd, Yddd = symbols('X Y Xd Yd Xdd Ydd Xddd Yddd')\n",
    "dicti = {rx:X, ry:Y, rx.diff(t, 1):Xd, ry.diff(t, 1):Yd, \\\n",
    "        rx.diff(t, 2):Xdd, ry.diff(t, 2):Ydd, rx.diff(t, 3):Xddd, ry.diff(t, 3):Yddd, l1:L1, l2:L2}\n",
    "vang1 = diff(ang1b, t, 1)\n",
    "vang1 = vang1.subs(dicti)\n",
    "vang1fu = lambdify((X, Y, Xd, Yd, l1, l2), vang1, 'numpy')\n",
    "vang1 = vang1fu(x, y, vx, vy, L1, L2)\n",
    "vang2 = diff(ang2b, t, 1)\n",
    "vang2 = vang2.subs(dicti)\n",
    "vang2fu = lambdify((X, Y, Xd, Yd, l1, l2), vang2, 'numpy')\n",
    "vang2 = vang2fu(x, y, vx, vy, L1, L2)\n",
    "\n",
    "aang1 = diff(ang1b, t, 2)\n",
    "aang1 = aang1.subs(dicti)\n",
    "aang1fu = lambdify((X, Y, Xd, Yd, Xdd, Ydd, l1, l2), aang1, 'numpy')\n",
    "aang1 = aang1fu(x, y, vx, vy, ax, ay, L1, L2)\n",
    "aang2 = diff(ang2b, t, 2)\n",
    "aang2 = aang2.subs(dicti)\n",
    "aang2fu = lambdify((X, Y, Xd, Yd, Xdd, Ydd, l1, l2), aang2, 'numpy')\n",
    "aang2 = aang2fu(x, y, vx, vy, ax, ay, L1, L2)\n",
    "\n",
    "jang1 = diff(ang1b, t, 3)\n",
    "jang1 = jang1.subs(dicti)\n",
    "jang1fu = lambdify((X, Y, Xd, Yd, Xdd, Ydd, Xddd, Yddd, l1, l2), jang1, 'numpy')\n",
    "jang1 = jang1fu(x, y, vx, vy, ax, ay, jx, jy, L1, L2)\n",
    "jang2 = diff(ang2b, t, 3)\n",
    "jang2 = jang2.subs(dicti)\n",
    "jang2fu = lambdify((X, Y, Xd, Yd, Xdd, Ydd, Xddd, Yddd, l1, l2), jang2, 'numpy')\n",
    "jang2 = jang2fu(x, y, vx, vy, ax, ay, jx, jy, L1, L2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "And the plots for the trajectories:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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24XoCm1zuCzBWLS90+d3n+C0DHgHKZCKfE6L82o2Gg/Ojj3sr0NA5rowZJZ4i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8tbVK/ucLC+/97/HKzoaPXZsvPOO45eNV6kefPm9O3blxs3bnDkyBHWr1/v1F/IH9i8eTO3bY5q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l7lzVqqXU4lavVo7Y5csquvXpp9C3LzRtCo88knHnKjeT/EHDwYMQFWWcLVkhPFz9jJNo3x66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\npUQNrG7ofe8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SERERERFJiYYIioiIiIiIpEQNLBERERERkZSogSUiIiIiIpISNbBERERE\nRERSogaWiIiIiIhIStTAEhERERERSYkaWCIiIiIiIin5f/bTAw/nFWNxAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x23ea06db400>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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MdJwiqV7d8p2ff96K84KN3v362TrBL7+Ea5+zu6gDjFDVpP1XLSLV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\nCIIgCIKgCtFYCoIgCIIgqML/Ayn4oNwDQdH8AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x23ea01190b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, hax = plt.subplots(2, 4, sharex = True, figsize=(14, 7))\n",
    "hax[0, 0].plot(ts, x, 'r', linewidth=3, label = 'x')\n",
    "hax[0, 0].plot(ts, y, 'k', linewidth=3, label = 'y')\n",
    "hax[0, 0].set_title('Linear displacement [$m$]')\n",
    "hax[0, 0].legend(loc='best').get_frame().set_alpha(0.8)\n",
    "hax[0, 0].set_ylabel('Endpoint')\n",
    "hax[0, 1].plot(ts, vx, 'r', linewidth=3)\n",
    "hax[0, 1].plot(ts, vy, 'k', linewidth=3)\n",
    "hax[0, 1].set_title('Linear velocity [$m/s$]')\n",
    "hax[0, 2].plot(ts, ax, 'r', linewidth=3)\n",
    "hax[0, 2].plot(ts, ay, 'k', linewidth=3)\n",
    "hax[0, 2].set_title('Linear acceleration [$m/s^2$]')\n",
    "hax[0, 3].plot(ts, jx, 'r', linewidth=3)\n",
    "hax[0, 3].plot(ts, jy, 'k', linewidth=3)\n",
    "hax[0, 3].set_title('Linear jerk [$m/s^3$]')\n",
    "hax[1, 0].plot(ts,ang1*180/np.pi, 'b', linewidth=3, label = 'Ang1')\n",
    "hax[1, 0].plot(ts,ang2*180/np.pi, 'g', linewidth=3, label = 'Ang2')\n",
    "hax[1, 0].set_xlabel('Time [$s$]')\n",
    "hax[1, 0].set_title('Angular displacement [ $^o$]')\n",
    "hax[1, 0].legend(loc='best').get_frame().set_alpha(0.8)\n",
    "hax[1, 0].set_ylabel('Joint')\n",
    "hax[1, 1].plot(ts,vang1*180/np.pi, 'b', linewidth=3)\n",
    "hax[1, 1].plot(ts,vang2*180/np.pi, 'g', linewidth=3)\n",
    "hax[1, 1].set_xlabel('Time [$s$]')\n",
    "hax[1, 1].set_title('Angular velocity [ $^o/s$]')\n",
    "hax[1, 2].plot(ts,aang1*180/np.pi, 'b', linewidth=3)\n",
    "hax[1, 2].plot(ts,aang2*180/np.pi, 'g', linewidth=3)\n",
    "hax[1, 2].set_xlabel('Time [$s$]')\n",
    "hax[1, 2].set_title('Angular acceleration [ $^o/s^2$]')\n",
    "hax[1, 3].plot(ts,jang1*180/np.pi, 'b', linewidth=3)\n",
    "hax[1, 3].plot(ts,jang2*180/np.pi, 'g', linewidth=3)\n",
    "hax[1, 3].set_xlabel('Time [$s$]')\n",
    "hax[1, 3].set_title('Angular jerk [ $^o/s^3$]')\n",
    "tit = fig.suptitle('Minimum jerk trajectory kinematics of a two-link chain', fontsize=20)\n",
    "for hax2 in hax.flat:\n",
    "    hax2.locator_params(nbins=5)\n",
    "    hax2.grid(True)\n",
    "plt.subplots_adjust(hspace=0.15, wspace=0.25) #plt.tight_layout()\n",
    "\n",
    "fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 3))\n",
    "ax1.plot(x, y, 'r', linewidth=3)\n",
    "ax1.set_xlabel('Displacement in x [$m$]')\n",
    "ax1.set_ylabel('Displacement in y [$m$]')\n",
    "ax1.set_title('Endpoint space', fontsize=14)\n",
    "ax1.grid(True)\n",
    "ax2.plot(ang1*180/np.pi, ang2*180/np.pi, 'b', linewidth=3)\n",
    "ax2.set_xlabel('Displacement in joint 1 [ $^o$]')\n",
    "ax2.set_ylabel('Displacement in joint 2 [ $^o$]')\n",
    "ax2.set_title('Joint sapace', fontsize=14)\n",
    "plt.subplots_adjust(left=0.3, wspace=0.3)\n",
    "ax2.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Problems\n",
    "\n",
    "1. For the numerical example of the two-link chain plotted above, calculate and plot the values for the each type of acceleration (tangential, centripetal and Coriolis). See solution below.\n",
    "\n",
    "2. For the two-link chain, calculate and interpret the Jacobian and the expressions for the position, velocity, and acceleration of the endpoint for the following cases:   \n",
    " a) When the first joint (the joint at the base) is fixed at $0^o$.   \n",
    " b) When the second joint is fixed at $0^o$.  \n",
    "\n",
    "3. For the two-link chain, a special case of movement occurs when the endpoint moves along a line passing through the first joint (the joint at the base). A system with this behavior is known as a polar manipulator (Mussa-Ivaldi, 1986). For simplicity, consider that the lengths of the two links are equal to $\\ell$. In this case, the two joint angles are related by: $2\\theta_1+\\theta_2=\\pi$.  \n",
    " a) Calculate the Jacobian for this polar manipulator and compare it with the Jacobian for the  standard two-link chain. Note the difference between the off-diagonal terms.  \n",
    " b) Calculate the expressions for the endpoint position, velocity, and acceleration.   \n",
    " c) For the endpoint acceleration of the polar manipulator, identify the tangential, centrifugal, and Coriolis components and compare them with the expressions for the standard two-link chain.  \n",
    " \n",
    "4. Deduce the equations for the kinematics of a two-link pendulum with the angles in relation to the vertical.  \n",
    "\n",
    "5. Deduce the equations for the kinematics of a two-link system using segment angles and compare with the deduction employing joint angles.\n",
    "\n",
    "6. Calculate the Jacobian matrix for the following function:\n",
    "\n",
    "<span class=\"notranslate\">\n",
    "\\begin{equation}\n",
    "f(x, y) = \\begin{bmatrix}\n",
    "x^2 y \\\\\n",
    "5 x + \\sin y \\end{bmatrix}\n",
    "\\end{equation}\n",
    "</span>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Calculation of each type of acceleration of the endpoint for the numerical example of the two-link system"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [],
   "source": [
    "# tangential acceleration\n",
    "A1, A2, A1d, A2d, A1dd, A2dd = symbols('A1 A2 A1d A2d A1dd A2dd')\n",
    "dicti = {theta1:A1, theta2:A2, theta1.diff(t,1):A1d, theta2.diff(t,1):A2d, \\\n",
    "         theta1.diff(t,2):A1dd, theta2.diff(t,2):A2dd, l1:L1, l2:L2}\n",
    "tg2 = tg.subs(dicti)\n",
    "tg2fu = lambdify((A1, A2, A1dd, A2dd), tg2, 'numpy');\n",
    "tg2n = tg2fu(ang1, ang2, aang1, aang2)\n",
    "tg2n = tg2n.reshape((2, 100)).T\n",
    "# centripetal acceleration\n",
    "ct2 = ct.subs(dicti)\n",
    "ct2fu = lambdify((A1, A2, A1d, A2d), ct2, 'numpy');\n",
    "ct2n = ct2fu(ang1, ang2, vang1, vang2)\n",
    "ct2n = ct2n.reshape((2, 100)).T\n",
    "# coriolis acceleration\n",
    "co2 = co.subs(dicti)\n",
    "co2fu = lambdify((A1, A2, A1d, A2d), co2, 'numpy');\n",
    "co2n = co2fu(ang1, ang2, vang1, vang2)\n",
    "co2n = co2n.reshape((2, 100)).T\n",
    "# total acceleration (it has to be the same calculated before)\n",
    "acc_tot = tg2n + ct2n + co2n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### And the corresponding plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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lJgrapr9PQ2uzVub6oaHKbNtr/WJlrj9iUe5wt1nrNmOxWCwWi8ViscQIVnm3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pOcnByT872Xl1erOrk1O3hXFKU/olHTi7U3Q/R6F/Cm5YZmIziVgaYhC6r9/cFMf3ZFUaQnl7y8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\ndOnCa6+95ovwhegQJk6cuExrPTrYcbQUyDYbt2sXZ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      "text/plain": [
       "<matplotlib.figure.Figure at 0x23ea1b6e2e8>"
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   "source": [
    "fig, hax = plt.subplots(1, 3, sharex = True, sharey = True, figsize=(12, 5))\n",
    "hax[0].plot(ts, acc_tot[:, 0], color=(1,0,0,.3), linewidth=5, label = 'x total')\n",
    "hax[0].plot(ts, acc_tot[:, 1], color=(0,0,0,.3), linewidth=5, label = 'y total')\n",
    "hax[0].plot(ts, tg2n[:, 0], 'r', linewidth=2, label = 'x')\n",
    "hax[0].plot(ts, tg2n[:, 1], 'k', linewidth=2, label = 'y')\n",
    "hax[0].set_title('Tangential')\n",
    "hax[0].set_ylabel('Endpoint acceleration [$m/s^2$]')\n",
    "hax[0].set_xlabel('Time [$s$]')\n",
    "hax[1].plot(ts, acc_tot[:, 0], color=(1,0,0,.3), linewidth=5, label = 'x total')\n",
    "hax[1].plot(ts, acc_tot[:, 1], color=(0,0,0,.3), linewidth=5, label = 'y total')\n",
    "hax[1].plot(ts, ct2n[:, 0], 'r', linewidth=2, label = 'x')\n",
    "hax[1].plot(ts, ct2n[:, 1], 'k', linewidth=2, label = 'y')\n",
    "hax[1].set_title('Centripetal')\n",
    "hax[1].set_xlabel('Time [$s$]')\n",
    "hax[1].legend(loc='best').get_frame().set_alpha(0.8)\n",
    "hax[2].plot(ts, acc_tot[:, 0], color=(1,0,0,.3), linewidth=5, label = 'x total')\n",
    "hax[2].plot(ts, acc_tot[:, 1], color=(0,0,0,.3), linewidth=5, label = 'y total')\n",
    "hax[2].plot(ts, co2n[:, 0], 'r', linewidth=2, label = 'x')\n",
    "hax[2].plot(ts, co2n[:, 1], 'k', linewidth=2, label = 'y')\n",
    "hax[2].set_title('Coriolis')\n",
    "hax[2].set_xlabel('Time [$s$]')\n",
    "tit = fig.suptitle('Acceleration terms for the minimum jerk trajectory of a two-link chain', fontsize=16)\n",
    "for hax2 in hax:\n",
    "    hax2.locator_params(nbins=5)\n",
    "    hax2.grid(True)\n",
    "# plt.subplots_adjust(hspace=0.15, wspace=0.25) #plt.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "## References\n",
    "\n",
    "- Mussa-Ivaldi FA (1986) Compliance.  In: Morasso P Tagliasco V (eds), [Human Movement Understanding: from computational geometry to artificial Intelligence](http://books.google.com.br/books?id=ZlZyLKNoAtEC). North-Holland, Amsterdam.  \n",
    "- Ruina A, Rudra P (2015) [Introduction to Statics and Dynamics](http://ruina.tam.cornell.edu/Book/index.html). Oxford University Press.  \n",
    "- Siciliano B et al. (2009) [Robotics - Modelling, Planning and Control](http://books.google.com.br/books/about/Robotics.html?hl=pt-BR&id=jPCAFmE-logC). Springer-Verlag London.\n",
    "- Zatsiorsky VM (1998) [Kinematics of Human Motions](http://books.google.com.br/books/about/Kinematics_of_Human_Motion.html?id=Pql_xXdbrMcC&redir_esc=y). Champaign, Human Kinetics."
   ]
  }
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