Simulation of a four bar linkage using sympy.physics.mechanics

fourbar.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Four-bar Linkage Simulation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introduction"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy.physics.mechanics import dynamicsymbols, ReferenceFrame, Point, RigidBody, inertia, KanesMethod, mechanics_printing, mlatex\n",
    "from sympy import symbols, diff, simplify, lambdify, solve\n",
    "from numpy import array, linspace, rad2deg, deg2rad, append, interp\n",
    "from numpy.linalg import pinv, inv\n",
    "from scipy.integrate import odeint, solve_ivp\n",
    "from scipy.optimize import fmin\n",
    "import matplotlib.pyplot as plt\n",
    "from matplotlib.animation import FuncAnimation\n",
    "from IPython.display import display, Latex"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "mechanics_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def dtex(*args):\n",
    "    tex = ''\n",
    "    for arg in args:\n",
    "        if isinstance(arg, str):\n",
    "            # Handle string arguments\n",
    "            tex += f'${arg}$ $\\, = \\,$ '\n",
    "        else:\n",
    "            # Handle SymPy arguments\n",
    "            tex += f'${mlatex(arg)}$ $\\, = \\,$ '\n",
    "    \n",
    "    # Trim off the last equals sign\n",
    "    tex = tex[:-10]  \n",
    "    display(Latex(tex))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Symbolic Variables\n",
    "\n",
    "Symbolic variables are defined using a subscript convention representing the number of the angular coordinate or speed, and corresponding bar length, mass, or inertia.\n",
    "Angles are represented as generalized coordinates $\\boldsymbol{q} = (\\theta_0, \\theta_1, \\theta_2)$, angular rates as generalized speeds $\\boldsymbol{s} = (\\omega_0, \\omega_1, \\omega_2)$, where the time rate of change of the $k^{th}$ coordinate, $\\dot{\\theta}_k = \\omega_k$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "\"\"\"\n",
    "Time-varying symbols are created using the dynamicsymbols function, where the \n",
    "argument is a string containing one or more output variables, comma separated.\n",
    "The kinematic equations are created as the variable named kd, in this case with \n",
    "each derivative of a generalized coordinate being subtracted by the \n",
    "corresponding generalized speed. Convention for the kinematic equations is that\n",
    "each item in the list is assumed to be equal to zero, thus the sk - diff(qk) can\n",
    "be assume to be analogous to sk == diff(qk) from the perspective of the equation \n",
    "of motion generation step to follow.\n",
    "\"\"\"\n",
    "q = θ0, θ1, θ2 = dynamicsymbols('θ_0, θ_1, θ_2')\n",
    "s = ω0, ω1, ω2 = dynamicsymbols('ω_0, ω_1, ω_2')\n",
    "kd = [sk - diff(qk) for qk, sk in zip(q, s)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Constants include the bar lengths, represented by the symbols $(l_0, l_1, l_2, l_3)$, bar masses with the symbols $(m_0, m_1, m_2)$, and bar inertias with the symbols $(I_0, I_1, I_2)$.\n",
    "Gravitational acceleration and time are represented by $g$ and $t$, respectively."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "\"\"\"\n",
    "Non-time-varying symbols are created using the symbols function, with the same\n",
    "argument and output formatting as dynamicsymbols. These, however, will be \n",
    "replaced with a constant, numeric value in the simulation step.\n",
    "\"\"\"\n",
    "lengths = l0, l1, l2, l3 = symbols('l_0, l_1, l_2, l_3')\n",
    "masses = m0, m1, m2 = symbols('m_0, m_1, m_2')\n",
    "inertias = I0, I1, I2 = symbols('I_0, I_1, I_2')\n",
    "g = symbols('g')\n",
    "t = symbols('t')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Kinematics\n",
    "\n",
    "The relative positioning of the links is defined first using frames to define orientation, then points for each of the link intersections, and the link centers of gravity (CG's)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Frames\n",
    "\n",
    "The inertial frame is considered to be stationary and non-rotating, and is denoted $R$. Notably, the unit vector $\\hat{\\boldsymbol{r}}_x$ is aligned horizontally (tangent to the Earth's surface), the unit vector $\\hat{\\boldsymbol{r}}_y$ is aligned upward (opposite of gravitational acceleration), and the unit vector $\\hat{\\boldsymbol{r}}_z$ is orthogonal to $\\hat{\\boldsymbol{r}}_x$ and $\\hat{\\boldsymbol{r}}_y$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "\"\"\"\n",
    "Frames are represented by the ReferenceFrame object, which takes a string\n",
    "formatted name as its first argument. If no additional arguments are provided, \n",
    "then it can be assumed to be stationary.\n",
    "\"\"\"\n",
    "inertial = ReferenceFrame('R')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Three additional frames are defined relative to the inertial frame, with the aforementioned angles previously defined with the generalized coordinate vector $\\boldsymbol{q}$. In each case, the orientation is defined with a rotation about the $\\hat{\\boldsymbol{r}}_z$ axis. Angular velocity vectors for the $k^{th}$ frame are defined with the $k^{th}$ generalized speed, or $\\omega_k \\hat{\\boldsymbol{r}}_z$. The set of orthogonal unit vectors attached to the $k^{th}$ rotating frame is denoted $(\\hat{\\boldsymbol{e}}_{kx}, \\hat{\\boldsymbol{e}}_{ky}, \\hat{\\boldsymbol{e}}_{kz})$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Rotation matrix for the 0th frame\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{R}_0$ $\\, = \\,$ $\\left[\\begin{matrix}\\operatorname{cos}\\left(θ_{0}\\right) & \\operatorname{sin}\\left(θ_{0}\\right) & 0\\\\- \\operatorname{sin}\\left(θ_{0}\\right) & \\operatorname{cos}\\left(θ_{0}\\right) & 0\\\\0 & 0 & 1\\end{matrix}\\right]$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Angular rate for the 0th frame\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{\\omega}_0$ $\\, = \\,$ $ω_{0}\\mathbf{\\hat{r}_z}$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Rotation matrix for the 1th frame\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{R}_1$ $\\, = \\,$ $\\left[\\begin{matrix}\\operatorname{cos}\\left(θ_{1}\\right) & \\operatorname{sin}\\left(θ_{1}\\right) & 0\\\\- \\operatorname{sin}\\left(θ_{1}\\right) & \\operatorname{cos}\\left(θ_{1}\\right) & 0\\\\0 & 0 & 1\\end{matrix}\\right]$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Angular rate for the 1th frame\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{\\omega}_1$ $\\, = \\,$ $ω_{1}\\mathbf{\\hat{r}_z}$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Rotation matrix for the 2th frame\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{R}_2$ $\\, = \\,$ $\\left[\\begin{matrix}\\operatorname{cos}\\left(θ_{2}\\right) & \\operatorname{sin}\\left(θ_{2}\\right) & 0\\\\- \\operatorname{sin}\\left(θ_{2}\\right) & \\operatorname{cos}\\left(θ_{2}\\right) & 0\\\\0 & 0 & 1\\end{matrix}\\right]$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Angular rate for the 2th frame\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{\\omega}_2$ $\\, = \\,$ $ω_{2}\\mathbf{\\hat{r}_z}$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\"\"\"\n",
    "With the inertial frame defined, additional non-stationary frames can be\n",
    "created using the .orientnew method, which takes a string formatted name as its \n",
    "first argument, followed by arguments defining its orientation. The second\n",
    "argument is the orientation method, where 'Axis' specifies a rotation about a\n",
    "single axis in a specified frame. The next argument is the amounts of rotation, \n",
    "which for an 'Axis' definition, is a tuple containing the symbol representating \n",
    "the rotation angle, and the axis that the rotation is defined about.\n",
    "\"\"\"\n",
    "frames = [inertial.orientnew(f'E_{k}', 'Axis', (q[k], inertial.z)) for k in range(3)]\n",
    "for k in range(3):\n",
    "  frames[k].set_ang_vel(inertial, s[k]*inertial.z)\n",
    "\n",
    "  print(f'Rotation matrix for the {k}th frame')\n",
    "  dtex('\\mathbf{R}_' + str(k), frames[k].dcm(inertial))\n",
    "\n",
    "  print(f'Angular rate for the {k}th frame')\n",
    "  dtex('\\mathbf{\\omega}_' + str(k), frames[k].ang_vel_in(inertial))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Points\n",
    "\n",
    "The position of each point is defined relative to the origin, which is stationary and denoted $O$. For indexing purposes, we will consider $O$ as the $0^{th}$ point. Supposing we denote the $k^{th}$ point as $p_k$, and its position vector as $\\boldsymbol{r}_k$, then for each $k=0, 1 ,2$, the subsequent position vector \n",
    "\n",
    "$\\boldsymbol{r}_{k+1} = \\boldsymbol{r}_k + l_k \\boldsymbol{e}_{kx}$.\n",
    "\n",
    "Points representing the CG of each link are denoted $g_k$, where for each $k=0, 1, 2$, the CG position is \n",
    "\n",
    "$r_{gk} = r_{k} + \\frac{1}{2} l_k \\boldsymbol{e}_{kx}$. \n",
    "\n",
    "Velocity vectors for the $0^{th}$ and $3^{rd}$ point are set to zero, while velocity of the $1^{st}$ point is \n",
    "\n",
    "$\\boldsymbol{v}_1 = l_0 \\omega_0 \\hat{\\boldsymbol{e}}_{0y}$, \n",
    "\n",
    "and of the $2^{nd}$ point is \n",
    "\n",
    "$\\boldsymbol{v}_2 = \\boldsymbol{v}_1 +  l_1 \\omega_1 \\hat{\\boldsymbol{e}}_{1y}$. \n",
    "\n",
    "Velocity vectors for the CG's of each link are \n",
    "\n",
    "$\\boldsymbol{v}_{gk} = \\boldsymbol{v}_{k}+\\frac{1}{2}l_k \\omega_k \\hat{\\boldsymbol{e}}_{ky}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Position for the 1th point\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{r}_1$ $\\, = \\,$ $l_{0}\\mathbf{\\hat{e_0}_x}$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Velocity for the 1th point\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{v}_1$ $\\, = \\,$ $l_{0} ω_{0}\\mathbf{\\hat{e_0}_y}$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Position for the CG of the 0th link\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{r}_{g0}$ $\\, = \\,$ $0.5 l_{0}\\mathbf{\\hat{e_0}_x}$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Velocity for the CG of the 0th link\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{v}_{g0}$ $\\, = \\,$ $0.5 l_{0} ω_{0}\\mathbf{\\hat{e_0}_y}$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Position for the 2th point\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{r}_2$ $\\, = \\,$ $l_{1}\\mathbf{\\hat{e_1}_x} + l_{0}\\mathbf{\\hat{e_0}_x}$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Velocity for the 2th point\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{v}_2$ $\\, = \\,$ $l_{0} ω_{0}\\mathbf{\\hat{e_0}_y} + l_{1} ω_{1}\\mathbf{\\hat{e_1}_y}$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Position for the CG of the 1th link\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{r}_{g1}$ $\\, = \\,$ $0.5 l_{1}\\mathbf{\\hat{e_1}_x} + l_{0}\\mathbf{\\hat{e_0}_x}$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Velocity for the CG of the 1th link\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{v}_{g1}$ $\\, = \\,$ $l_{0} ω_{0}\\mathbf{\\hat{e_0}_y} + 0.5 l_{1} ω_{1}\\mathbf{\\hat{e_1}_y}$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Position for the 3th point\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{r}_3$ $\\, = \\,$ $l_{2}\\mathbf{\\hat{e_2}_x} + l_{1}\\mathbf{\\hat{e_1}_x} + l_{0}\\mathbf{\\hat{e_0}_x}$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Velocity for the 3th point\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{v}_3$ $\\, = \\,$ $0$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Position for the CG of the 2th link\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{r}_{g2}$ $\\, = \\,$ $0.5 l_{2}\\mathbf{\\hat{e_2}_x} + l_{1}\\mathbf{\\hat{e_1}_x} + l_{0}\\mathbf{\\hat{e_0}_x}$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Velocity for the CG of the 2th link\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{v}_{g2}$ $\\, = \\,$ $-  0.5 l_{2} ω_{2}\\mathbf{\\hat{e_2}_y}$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\"\"\"\n",
    "The variables points, and cms are both created as Python lists of Point objects.\n",
    "The origin (0th point), and end of the third link (3rd) point, are each \n",
    "specified to have zero velocity, using the .set_vel() method. The locations of \n",
    "each link intersection is specified using the .locatenew() method from the \n",
    "point that precedes it in the list, by indexing the lengths list and the frames.\n",
    "Velocity for the moving points is specified using the .v2pt_theory() method, \n",
    "which calculates velocity using the transport equation.\n",
    "\"\"\"\n",
    "# Create the origin, and specify that it is stationary\n",
    "points = [Point('O')]\n",
    "points[0].set_vel(inertial, 0)\n",
    "\n",
    "# Create the points corresponding to the intersection of links\n",
    "for k in range(3):\n",
    "  points += [points[k].locatenew(f'p_{k}', lengths[k] * frames[k].x)]\n",
    "for k in (1, 2):\n",
    "  points[k].v2pt_theory(points[k-1], inertial, frames[k-1])\n",
    "points[3].set_vel(inertial, 0)\n",
    "\n",
    "# Create the points corresponding to centers of mass\n",
    "cms = []\n",
    "cms = [points[k].locatenew(f'g_{k}', 0.5 * lengths[k] * frames[k].x) for k in range(3)]\n",
    "for k in range(3):\n",
    "  cms[k].v2pt_theory(points[k if k < 2 else k+1], inertial, frames[k])\n",
    "\n",
    "for k in range(3):\n",
    "  print(f'Position for the {k+1}th point')\n",
    "  dtex('\\mathbf{r}_' + str(k+1), points[k+1].pos_from(points[0]))\n",
    "  print(f'Velocity for the {k+1}th point')\n",
    "  dtex('\\mathbf{v}_' + str(k+1), points[k+1].vel(inertial))\n",
    "  print(f'Position for the CG of the {k}th link')\n",
    "  dtex('\\mathbf{r}_{g' + str(k) + '}', cms[k].pos_from(points[0]))\n",
    "  print(f'Velocity for the CG of the {k}th link')\n",
    "  dtex('\\mathbf{v}_{g' + str(k) + '}', cms[k].vel(inertial))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Constraints\n",
    "\n",
    "A constraint is enforced such that the location of the $2^{nd}$ point is fixed to a constant location $\\boldsymbol{r}_2 = l_3 \\hat{\\boldsymbol{r}}_x$. Using the position defined above, this gives the constraint equation \n",
    "\n",
    "$f_c(\\boldsymbol{q}) = l_2 \\hat{\\boldsymbol{e}}_{2x} + l_1 \\hat{\\boldsymbol{e}}_{1x} + l_0 \\hat{\\boldsymbol{e}}_{0x} - l_3 \\hat{\\boldsymbol{r}}_x = 0$. \n",
    "\n",
    "Similarly, a velocity constraint can exists, which is the derivative of $f_c(\\boldsymbol{q})$ with respect to time, or \n",
    "\n",
    "$f_v(\\boldsymbol{q}, \\boldsymbol{s}) = \\frac{\\partial f_c(\\boldsymbol{q})}{\\partial t} = \\sum_{k=0}^2 l_k \\omega_k \\hat{\\boldsymbol{e}}_{kx}$.\n",
    "\n",
    "Implementation of these constraints involves transforming the vector form of the constraint equations into a vector form, evaluated in the inertial frame. Upon this transformation, the positional constraint equation in the $\\hat{\\boldsymbol{r}}_x$ direction is \n",
    "\n",
    "$-l_3 + \\sum_{k=0}^2 l_k \\cos \\theta_k = 0$, \n",
    "\n",
    "and the positional constraint equation in the $\\hat{\\boldsymbol{r}}_y$ direction is \n",
    "\n",
    "$\\sum_{k=0}^2 l_k \\sin \\theta_k = 0$.\n",
    "\n",
    "The corresponding velocity constraint equation in the $\\hat{\\boldsymbol{r}}_x$ direction is \n",
    "\n",
    "$\\sum_{k=0}^2 -l_k \\omega_k \\sin \\theta_0 = 0$,\n",
    "\n",
    "and the velocity constraint equation in the $\\hat{\\boldsymbol{r}}_y$ direction is \n",
    "\n",
    "$\\sum_{k=0}^2 l_k \\omega_k \\cos \\theta_0 = 0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Position constraint equation\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle l_{2}\\mathbf{\\hat{e_2}_x} + l_{1}\\mathbf{\\hat{e_1}_x} + l_{0}\\mathbf{\\hat{e_0}_x} -  l_{3}\\mathbf{\\hat{r}_x}$"
      ],
      "text/plain": [
       "l₂ e_2_x + l₁ e_1_x + l₀ e_0_x + -l₃ r_x"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "'0'"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Position contraint equations in inertial frame\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}l_{0} \\operatorname{cos}\\left(θ_{0}\\right) + l_{1} \\operatorname{cos}\\left(θ_{1}\\right) + l_{2} \\operatorname{cos}\\left(θ_{2}\\right) - l_{3}\\\\l_{0} \\operatorname{sin}\\left(θ_{0}\\right) + l_{1} \\operatorname{sin}\\left(θ_{1}\\right) + l_{2} \\operatorname{sin}\\left(θ_{2}\\right)\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡l₀⋅cos(θ₀) + l₁⋅cos(θ₁) + l₂⋅cos(θ₂) - l₃⎤\n",
       "⎢                                         ⎥\n",
       "⎢  l₀⋅sin(θ₀) + l₁⋅sin(θ₁) + l₂⋅sin(θ₂)   ⎥\n",
       "⎢                                         ⎥\n",
       "⎣                    0                    ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Velocity constraint equation\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle l_{2} ω_{2}\\mathbf{\\hat{e_2}_y} + l_{1} ω_{1}\\mathbf{\\hat{e_1}_y} + l_{0} ω_{0}\\mathbf{\\hat{e_0}_y}$"
      ],
      "text/plain": [
       "l₂⋅ω₂ e_2_y + l₁⋅ω₁ e_1_y + l₀⋅ω₀ e_0_y"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "'0'"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Velocity constraint equations in inertial frame\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- l_{0} ω_{0} \\operatorname{sin}\\left(θ_{0}\\right) - l_{1} ω_{1} \\operatorname{sin}\\left(θ_{1}\\right) - l_{2} ω_{2} \\operatorname{sin}\\left(θ_{2}\\right)\\\\l_{0} ω_{0} \\operatorname{cos}\\left(θ_{0}\\right) + l_{1} ω_{1} \\operatorname{cos}\\left(θ_{1}\\right) + l_{2} ω_{2} \\operatorname{cos}\\left(θ_{2}\\right)\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-l₀⋅ω₀⋅sin(θ₀) - l₁⋅ω₁⋅sin(θ₁) - l₂⋅ω₂⋅sin(θ₂)⎤\n",
       "⎢                                              ⎥\n",
       "⎢l₀⋅ω₀⋅cos(θ₀) + l₁⋅ω₁⋅cos(θ₁) + l₂⋅ω₂⋅cos(θ₂) ⎥\n",
       "⎢                                              ⎥\n",
       "⎣                      0                       ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\"\"\"\n",
    "Positional constraints are determined by obtaining the position of the 3rd point\n",
    "relative to the origin (0th point), using the .pos_from() method included in the\n",
    "Point class, and subtracting the specified location of the constraint. The\n",
    "velocity constraint equation is determined through calling the .dt() method to\n",
    "calculate the time derivative of the position equation. Lastly, the to_matrix() \n",
    "method converts the vector form of the equations to a sympy Matrix object, with \n",
    "the first two elements of the matrix corresponding to the equations evaluated in\n",
    "the rx and ry directions of the inertial frame, the third element is zero\n",
    "because the motion is planar.\n",
    "\"\"\"\n",
    "# Constraint equations\n",
    "fc = points[3].pos_from(points[0]) - lengths[3] * inertial.x\n",
    "fv = fc.dt(inertial)\n",
    "\n",
    "# Constraint matrix equations in inertial frame\n",
    "position_constraint = fc.to_matrix(inertial)\n",
    "velocity_constraint = fv.to_matrix(inertial)\n",
    "\n",
    "print('Position constraint equation')\n",
    "display(fc, '0')\n",
    "\n",
    "print('Position contraint equations in inertial frame')\n",
    "display(position_constraint)\n",
    "\n",
    "print('Velocity constraint equation')\n",
    "display(fv, '0')\n",
    "\n",
    "print('Velocity constraint equations in inertial frame')\n",
    "display(velocity_constraint)\n",
    "\n",
    "# Define a function that will be used to solve for initial conditions\n",
    "def solve_ics(theta_0, subs_dict):\n",
    "  err = position_constraint.subs(q[0], theta_0).subs(subs_dict)\n",
    "  err2 = lambdify(q[1:], err[0]**2 + err[1]**2)\n",
    "  def loss(y):\n",
    "    return err2(y[0], y[1])\n",
    "  min_x = fmin(func=loss, x0=[0.0, 0.0]);\n",
    "  return [theta_0, min_x[0], min_x[1], 0.0, 0.0, 0.0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Dynamics\n",
    "\n",
    "### Rigid Bodies\n",
    "\n",
    "Each of the three links are defined with a mass at its CG in the center of the link, and a moment of inertia $I_k$ relative to the CG and about the $\\hat{\\boldsymbol{e}}_{kz} = \\hat{\\boldsymbol{r}}_z$ axis. Linear momentum for each body in the inertial frame is \n",
    "\n",
    "$\\mathbf{P}_k = m_k \\mathbf{v}_{gk}$\n",
    "\n",
    "and angular momentum for each body, relative to the origin and in the inertial frame, is\n",
    "\n",
    "$\\mathbf{H}_k = I_k \\omega_k \\hat{\\boldsymbol{e}}_{kz} + \\mathbf{r}_{gk} \\times m_k \\mathbf{v}_{gk}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Linear momentum for the 0th link in the inertial frame\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{P}_0$ $\\, = \\,$ $0.5 l_{0} m_{0} ω_{0}\\mathbf{\\hat{e_0}_y}$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Angular momentum for the 0th link relative to the origin, in the inertial frame\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{H}_0$ $\\, = \\,$ $(I_{0} ω_{0} + 0.25 l_{0}^{2} m_{0} ω_{0})\\mathbf{\\hat{e_0}_z}$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Linear momentum for the 1th link in the inertial frame\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{P}_1$ $\\, = \\,$ $l_{0} m_{1} ω_{0}\\mathbf{\\hat{e_0}_y} + 0.5 l_{1} m_{1} ω_{1}\\mathbf{\\hat{e_1}_y}$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Angular momentum for the 1th link relative to the origin, in the inertial frame\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{H}_1$ $\\, = \\,$ $(I_{1} ω_{1} + 0.5 l_{1} m_{1} \\left(l_{0} \\left(\\operatorname{sin}\\left(θ_{0}\\right) \\operatorname{sin}\\left(θ_{1}\\right) + \\operatorname{cos}\\left(θ_{0}\\right) \\operatorname{cos}\\left(θ_{1}\\right)\\right) + 0.5 l_{1}\\right) ω_{1})\\mathbf{\\hat{e_1}_z} + l_{0} m_{1} \\left(l_{0} + 0.5 l_{1} \\left(\\operatorname{sin}\\left(θ_{0}\\right) \\operatorname{sin}\\left(θ_{1}\\right) + \\operatorname{cos}\\left(θ_{0}\\right) \\operatorname{cos}\\left(θ_{1}\\right)\\right)\\right) ω_{0}\\mathbf{\\hat{e_0}_z}$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Linear momentum for the 2th link in the inertial frame\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{P}_2$ $\\, = \\,$ $-  0.5 l_{2} m_{2} ω_{2}\\mathbf{\\hat{e_2}_y}$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Angular momentum for the 2th link relative to the origin, in the inertial frame\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{H}_2$ $\\, = \\,$ $(I_{2} ω_{2} - 0.5 l_{2} m_{2} \\left(l_{0} \\left(\\operatorname{sin}\\left(θ_{0}\\right) \\operatorname{sin}\\left(θ_{2}\\right) + \\operatorname{cos}\\left(θ_{0}\\right) \\operatorname{cos}\\left(θ_{2}\\right)\\right) + l_{1} \\left(\\operatorname{sin}\\left(θ_{1}\\right) \\operatorname{sin}\\left(θ_{2}\\right) + \\operatorname{cos}\\left(θ_{1}\\right) \\operatorname{cos}\\left(θ_{2}\\right)\\right) + 0.5 l_{2}\\right) ω_{2})\\mathbf{\\hat{e_2}_z}$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\"\"\"\n",
    "Bodies are instantiated using the RigidBody class, which takes as arguments a \n",
    "string-formatted name, vector corresponding to the CG position, frame in which\n",
    "its inertia tensor components are defined, mass of the body, and an inertia\n",
    "tuple. The inertia tuple includes an inertia dyadic, and a CG position. For \n",
    "convenience, the inertia dyadic is created using the inertia function, which\n",
    "takes as arguments the frame in which the inertia tensor components are defined,\n",
    "three required inertia terms (ixx, iyy, izz), and optional product of inertia\n",
    "terms (ixy, izx, iyz) as keyword arguments.\n",
    "\"\"\"\n",
    "bodies = [RigidBody(f'Bar {k}', cms[k], frames[k], masses[k], \n",
    "                    (inertia(frames[k], 0, 0, inertias[k]), cms[k])) \n",
    "          for k in range(3)]\n",
    "\n",
    "for k in range(3):\n",
    "  print(f'Linear momentum for the {k}th link in the inertial frame')\n",
    "  dtex('\\mathbf{P}_' + str(k), bodies[k].linear_momentum(inertial))\n",
    "  print(f'Angular momentum for the {k}th link relative to the origin, in the inertial frame')\n",
    "  dtex('\\mathbf{H}_' + str(k), bodies[k].angular_momentum(points[0], inertial))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Loads\n",
    "\n",
    "The external forces on the three links are from gravitational acceleration, which is assumed to act in the $-\\hat{\\boldsymbol{r}}_z$ direction, such that the force at the $k^{th}$ link CG is\n",
    "\n",
    "$\\mathbf{F}_k = -g m_k \\hat{\\boldsymbol{r}}_z$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Force applied at the point g_0\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{F}_2$ $\\, = \\,$ $-  g m_{0}\\mathbf{\\hat{r}_y}$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Force applied at the point g_1\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{F}_2$ $\\, = \\,$ $-  g m_{1}\\mathbf{\\hat{r}_y}$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Force applied at the point g_2\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{F}_2$ $\\, = \\,$ $-  g m_{2}\\mathbf{\\hat{r}_y}$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\"\"\"\n",
    "Loads are created as a list of tuples, where the first element of the tuple is \n",
    "the point at which the force is applied, and the second element is the force \n",
    "vector.\n",
    "\"\"\"\n",
    "loads = [(cms[k], -masses[k] * g * inertial.y) for k in range(3)]\n",
    "for load in loads:\n",
    "  print(f'Force applied at the point {load[0]}')\n",
    "  dtex('\\mathbf{F}_' + str(k), load[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Equations of motion\n",
    "\n",
    "Kane's method is used to derive equations of motion given an inertial reference frame, the set of independent and dependent generalized coordinates and speeds, the set of configuration and velocity constraints, and the kinematic differential equations defining the relationship between generalized coordinates and speeds. The form of the equations is \n",
    "\n",
    "$\\mathbf{F}_r + \\mathbf{F}_r^* = \\mathbf{0}$\n",
    "\n",
    "where if we assume the only independent generalized speed is $\\omega_0$, the applied, or active force terms\n",
    "\n",
    "$\n",
    "\\mathbf{F}_r = \\sum_{k=0}^2 \n",
    "\\frac{\\partial \\mathbf{v}_{gk}}{\\partial \\omega_0} \n",
    "\\cdot \\mathbf{F}_k\n",
    "$\n",
    "\n",
    "and the body, or inertial force terms\n",
    "\n",
    "$\n",
    "\\mathbf{F}_r^* = \\sum_{k=0}^2\n",
    "\\left(\n",
    "\\frac{\\partial \\mathbf{v}_{gk}}{\\partial \\omega_0} \n",
    "\\cdot \\dot{\\mathbf{P}}_k\n",
    "+\n",
    "\\frac{\\partial \\mathbf{\\omega}_{k}}{\\partial \\omega_0} \n",
    "\\cdot \\dot{\\mathbf{H}}_k\n",
    "\\right)\n",
    "$\n",
    "\n",
    "These equations are expanded to a mass matrix and forcing vector form, where\n",
    "\n",
    "$\\mathbf{M}(\\boldsymbol{q}) \\dot{\\mathbf{x}} = \\boldsymbol{F}\\,(\\boldsymbol{q}, \\boldsymbol{s})$\n",
    "\n",
    "where $\\boldsymbol{x}=\\left(\\boldsymbol{q}, \\boldsymbol{s}\\right)$ is the combined state vector containing generalized coordinates and speeds."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Applied Forces\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{F}_r$ $\\, = \\,$ $\\left[\\begin{matrix}\\frac{1.0 g l_{0} \\left(- 0.5 m_{0} \\operatorname{sin}\\left(θ_{1} - θ_{2}\\right) \\operatorname{cos}\\left(θ_{0}\\right) + 0.5 m_{1} \\operatorname{sin}\\left(θ_{0} + θ_{2}\\right) \\operatorname{cos}\\left(θ_{1}\\right) - 1.0 m_{1} \\operatorname{sin}\\left(θ_{1}\\right) \\operatorname{cos}\\left(θ_{0}\\right) \\operatorname{cos}\\left(θ_{2}\\right) + 0.5 m_{2} \\operatorname{sin}\\left(θ_{0} - θ_{1}\\right) \\operatorname{cos}\\left(θ_{2}\\right)\\right)}{\\operatorname{sin}\\left(θ_{1} - θ_{2}\\right)}\\end{matrix}\\right]$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "'Internal Forces'"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{F}_r^*$ $\\, = \\,$ $\\left[\\begin{matrix}\\frac{- l_{0} l_{1} \\left(I_{2} + 0.25 l_{2}^{2} m_{2}\\right) \\operatorname{sin}\\left(θ_{0} - θ_{1}\\right) \\dot{ω}_{2} + l_{0} l_{2} \\left(- 0.5 l_{0} l_{1} m_{1} ω^{2}_{0} \\operatorname{sin}\\left(θ_{0} - θ_{1}\\right) + 0.5 l_{0} l_{1} m_{1} \\operatorname{cos}\\left(θ_{0} - θ_{1}\\right) \\dot{ω}_{0} + \\left(I_{1} + 0.25 l_{1}^{2} m_{1}\\right) \\dot{ω}_{1}\\right) \\operatorname{sin}\\left(θ_{0} - θ_{2}\\right) - l_{1} l_{2} \\left(0.5 l_{0} l_{1} m_{1} ω^{2}_{1} \\operatorname{sin}\\left(θ_{0} - θ_{1}\\right) + 0.5 l_{0} l_{1} m_{1} \\operatorname{cos}\\left(θ_{0} - θ_{1}\\right) \\dot{ω}_{1} + \\left(I_{0} + 0.25 l_{0}^{2} m_{0} + l_{0}^{2} m_{1}\\right) \\dot{ω}_{0}\\right) \\operatorname{sin}\\left(θ_{1} - θ_{2}\\right)}{l_{1} l_{2} \\operatorname{sin}\\left(θ_{1} - θ_{2}\\right)}\\end{matrix}\\right]$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "'Mass Matrix'"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{M}$ $\\, = \\,$ $\\left[\\begin{matrix}1 & 0 & 0 & 0 & 0 & 0\\\\0 & 1 & 0 & 0 & 0 & 0\\\\0 & 0 & 1 & 0 & 0 & 0\\\\0 & 0 & 0 & 1.0 I_{0} + 0.25 l_{0}^{2} m_{0} - \\frac{0.5 l_{0}^{2} m_{1} \\operatorname{sin}\\left(θ_{0} - θ_{2}\\right) \\operatorname{cos}\\left(θ_{0} - θ_{1}\\right)}{\\operatorname{sin}\\left(θ_{1} - θ_{2}\\right)} + 1.0 l_{0}^{2} m_{1} & \\frac{l_{0} \\left(- 1.0 I_{1} \\operatorname{sin}\\left(θ_{0} - θ_{2}\\right) - \\frac{0.125 l_{1}^{2} m_{1} \\operatorname{cos}\\left(θ_{0} - 3 θ_{1} + θ_{2}\\right)}{\\operatorname{sin}\\left(θ_{1}\\right)} + \\frac{0.125 l_{1}^{2} m_{1} \\operatorname{cos}\\left(θ_{0} - θ_{1} + θ_{2}\\right)}{\\operatorname{sin}\\left(θ_{1}\\right)}\\right)}{l_{1} \\operatorname{sin}\\left(θ_{1} - θ_{2}\\right)} & \\frac{l_{0} \\left(I_{2} + 0.25 l_{2}^{2} m_{2}\\right) \\operatorname{sin}\\left(θ_{0} - θ_{1}\\right)}{l_{2} \\operatorname{sin}\\left(θ_{1} - θ_{2}\\right)}\\\\0 & 0 & 0 & - l_{0} \\operatorname{sin}\\left(θ_{0}\\right) & - l_{1} \\operatorname{sin}\\left(θ_{1}\\right) & - l_{2} \\operatorname{sin}\\left(θ_{2}\\right)\\\\0 & 0 & 0 & l_{0} \\operatorname{cos}\\left(θ_{0}\\right) & l_{1} \\operatorname{cos}\\left(θ_{1}\\right) & l_{2} \\operatorname{cos}\\left(θ_{2}\\right)\\end{matrix}\\right]$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "'Forcing'"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\mathbf{F}$ $\\, = \\,$ $\\left[\\begin{matrix}ω_{0}\\\\ω_{1}\\\\ω_{2}\\\\\\frac{l_{0} \\left(0.5 g m_{2} \\operatorname{sin}\\left(θ_{0} - θ_{1}\\right) \\operatorname{cos}\\left(θ_{2}\\right) + 0.5 m_{1} \\left(g \\operatorname{cos}\\left(θ_{1}\\right) - l_{0} ω^{2}_{0} \\operatorname{sin}\\left(θ_{0} - θ_{1}\\right)\\right) \\operatorname{sin}\\left(θ_{0} - θ_{2}\\right) - \\left(0.5 g m_{0} \\operatorname{cos}\\left(θ_{0}\\right) + 1.0 g m_{1} \\operatorname{cos}\\left(θ_{0}\\right) + 0.5 l_{1} m_{1} ω^{2}_{1} \\operatorname{sin}\\left(θ_{0} - θ_{1}\\right)\\right) \\operatorname{sin}\\left(θ_{1} - θ_{2}\\right)\\right)}{\\operatorname{sin}\\left(θ_{1} - θ_{2}\\right)}\\\\l_{0} ω^{2}_{0} \\operatorname{cos}\\left(θ_{0}\\right) + l_{1} ω^{2}_{1} \\operatorname{cos}\\left(θ_{1}\\right) + l_{2} ω^{2}_{2} \\operatorname{cos}\\left(θ_{2}\\right)\\\\l_{0} ω^{2}_{0} \\operatorname{sin}\\left(θ_{0}\\right) + l_{1} ω^{2}_{1} \\operatorname{sin}\\left(θ_{1}\\right) + l_{2} ω^{2}_{2} \\operatorname{sin}\\left(θ_{2}\\right)\\end{matrix}\\right]$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\"\"\"\n",
    "The KanesMethod object takes the inertial frame as its sole required argument. \n",
    "Additionally, the keyword arguments q_ind, and u_ind define the independent\n",
    "generalized coordinates and speeds, in this case the first entry in the q and s\n",
    "lists, which are surrounded by brackets so that the argument is in the required\n",
    "list form. the q_dependent and u_dependent keyword arguments are the remaining \n",
    "entries in q and s. The configuration_constraints and velocity_constraints \n",
    "keyword arguments are taken from the matrix form of the contraint equations, in\n",
    "the first two, non-zero directions. The kd_eqs keyword argument are kinematic \n",
    "equations defining the relation between generalized coordinates and speeds. The\n",
    ".kanes_equations() method is called with the lists of bodies and loads as\n",
    "arguments. After this is called, the equations can be accessed in various forms,\n",
    "Notably in the mass matrix and forcing vector form that will be integrated here.\n",
    "\"\"\"\n",
    "kane = KanesMethod(inertial, q_ind=[q[0]], u_ind=[s[0]], \n",
    "                   q_dependent=q[1:], u_dependent=s[1:], \n",
    "                   configuration_constraints=position_constraint[:2], \n",
    "                   velocity_constraints=velocity_constraint[:2], \n",
    "                   kd_eqs=kd)\n",
    "fr, frstar = kane.kanes_equations(bodies, loads)\n",
    "print('Applied Forces')\n",
    "dtex('\\mathbf{F}_r', simplify(fr))\n",
    "display('Internal Forces')\n",
    "dtex('\\mathbf{F}_r^*', simplify(frstar))\n",
    "display('Mass Matrix')\n",
    "M = simplify(kane.mass_matrix_full)\n",
    "dtex('\\mathbf{M}', M)\n",
    "display('Forcing')\n",
    "F = simplify(kane.forcing_full).subs(kane.kindiffdict())\n",
    "dtex('\\mathbf{F}', F)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Analysis\n",
    "\n",
    "### Simulation\n",
    "\n",
    "To obtain a callable ordinary differential equation (ODE) function, first we create a dictionary containing numeric values for the previously defined symbolic quantities. Assuming units of kilograms ($kg$), meters ($m$), Newtons ($N$), and seconds ($s$), the gravitational acceleration is specified to be 9.81 $m/s^2$. For this model, we will assume links of length 1, 2, 3, and 4 $m$, respectively, and with mass value of 1, 2, and 3 $kg$ for the first 3 links (the fourth link is fixed to the ground, it is not assigned mass). Moment of inertia values are assumed for a thin rod, where\n",
    "\n",
    "$I_k = \\frac{1}{12} m_k l_k^2$\n",
    "\n",
    "These values are substituted into the equations for mass matrix and forcing prior to creation of the ODE function. The `scipy.integrate.solve_ivp` function is used to integrate the initial value problem, and requires and ode function of the form $\\dot{\\boldsymbol{x}} = f(t, \\boldsymbol{x})$. In this case, $f(t, \\boldsymbol{x}) = \\mathbf{M}^{-1} \\boldsymbol{F}$. Since $\\mathbf{M}$ and $\\boldsymbol{F}$ are state-dependent, these are calculated (and the mass matrix inverted) at each time step in the simulation. Initial states are estimated using an optimization function to minimize the positional constraint equations given a specified $\\theta_0(0) = 85^\\circ$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 0.000000\n",
      "         Iterations: 70\n",
      "         Function evaluations: 134\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\theta_0(0)$ $\\, = \\,$ $85.0$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\theta_1(0)$ $\\, = \\,$ $31.301991721073403$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\theta_2(0)$ $\\, = \\,$ $-42.722211361919406$ "
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Create a dictionary of constants to be substituted into the symbolic equations\n",
    "subs_dict = {g: 9.81, \n",
    "             m0: 1.0, m1: 2.0, m2: 3.0, \n",
    "             l0: 1.0, l1: 2.0, l2: 3.0, l3: 4.0}\n",
    "for I, m, l in zip(inertias, masses, lengths[:3]):\n",
    "  subs_dict[I] = 1.0 / 12.0 * subs_dict[m] * subs_dict[l]**2\n",
    "\n",
    "# Use lambdify function to create anonymous functions for the mass matrix and \n",
    "# forcing, which will yield speed-optimized function given the state arguments,\n",
    "# and will output a numpy array object   \n",
    "mass_fcn = lambdify([t]+q+s, M.subs(subs_dict))\n",
    "forcing_fcn = lambdify([t]+q+s, F.subs(subs_dict))\n",
    "\n",
    "# Create a callable ordinary differential equation of time and state\n",
    "def ode(t, x):\n",
    "  args = append([t], x)\n",
    "  dx = inv(mass_fcn(*args)) @ forcing_fcn(*args)\n",
    "  return dx[:, 0]\n",
    "\n",
    "# Determine initial conditions, and simulate\n",
    "x0 = solve_ics(deg2rad(85), subs_dict)\n",
    "sol = solve_ivp(ode, [0, 20], x0, rtol=1e-6)\n",
    "for k in range(3):\n",
    "  dtex(f'\\\\theta_{k}(0)', str(rad2deg(x0[k])))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plotting\n",
    "\n",
    "Generalized coordinates derived from simulating the equations of motion giving the specified constants and initial values are shown. Notable features include the periodicity of the $\\theta_0$ coordinate, which oscillates between its initial $85^\\circ$, and $405^\\circ$. Since the only force acting on the system is gravity, there are no initial rates, and the constraint forces are conservative it does not complete one full rotation. Also, at instants where $\\theta_0(t)=180^\\circ$, $\\theta_1(t) = \\theta_2(t) = 0$ because $l_0 + l_3 = l_1 + l_2$, meaning at these times all four links will be in alignment. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "\"\"\"\n",
    "The sol structure contains the time variable t, and integrated states y. The\n",
    "first three indices in y are the generalized coordinates, which are converted\n",
    "from radians to degrees for plotting. Plotting functions are taken from the \n",
    "matplotlib.pyplot which was imported as plt.\n",
    "\"\"\"\n",
    "for k in range(3):\n",
    "  plt.plot(sol.t, rad2deg(sol.y[k, :]))\n",
    "plt.xlabel('Time (sec)')\n",
    "plt.ylabel('Angle (deg)')\n",
    "plt.legend(q);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Generalized speeds are similarly plotted. These are distinctly \"spiky,\" with positive and negative peaks corresponding to the times at which $\\theta_0(t) = 180^\\circ$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "\"\"\"\n",
    "The remaining indices in the sol structure are the generalized speeds, which are\n",
    "plotting in a similar manner to the coordinates, including conversion to \n",
    "degrees.\n",
    "\"\"\"\n",
    "for k in range(3):\n",
    "  plt.plot(sol.t, rad2deg(sol.y[k+3, :]))\n",
    "plt.xlabel('Time (sec)')\n",
    "plt.ylabel('Angular Rate (deg)')\n",
    "plt.legend(s);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "An animation of the positions of each of the link interface is generated to visualize the spatial relations between points as a function of time. Interestingly, we can see the time at which $\\theta_0=180^\\circ$ corresponds to an apparrent \"bounce\" of the third link. This occurs because it is kinematically impossible given the link geometries for this link to travel past the horizontal, and abruptly reverses direction at this point."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<link rel=\"stylesheet\"\n",
       "href=\"https://maxcdn.bootstrapcdn.com/font-awesome/4.4.0/\n",
       "css/font-awesome.min.css\">\n",
       "<script language=\"javascript\">\n",
       "  function isInternetExplorer() {\n",
       "    ua = navigator.userAgent;\n",
       "    /* MSIE used to detect old browsers and Trident used to newer ones*/\n",
       "    return ua.indexOf(\"MSIE \") > -1 || ua.indexOf(\"Trident/\") > -1;\n",
       "  }\n",
       "\n",
       "  /* Define the Animation class */\n",
       "  function Animation(frames, img_id, slider_id, interval, loop_select_id){\n",
       "    this.img_id = img_id;\n",
       "    this.slider_id = slider_id;\n",
       "    this.loop_select_id = loop_select_id;\n",
       "    this.interval = interval;\n",
       "    this.current_frame = 0;\n",
       "    this.direction = 0;\n",
       "    this.timer = null;\n",
       "    this.frames = new Array(frames.length);\n",
       "\n",
       "    for (var i=0; i<frames.length; i++)\n",
       "    {\n",
       "     this.frames[i] = new Image();\n",
       "     this.frames[i].src = frames[i];\n",
       "    }\n",
       "    var slider = document.getElementById(this.slider_id);\n",
       "    slider.max = this.frames.length - 1;\n",
       "    if (isInternetExplorer()) {\n",
       "        // switch from oninput to onchange because IE <= 11 does not conform\n",
       "        // with W3C specification. It ignores oninput and onchange behaves\n",
       "        // like oninput. In contrast, Mircosoft Edge behaves correctly.\n",
       "        slider.setAttribute('onchange', slider.getAttribute('oninput'));\n",
       "        slider.setAttribute('oninput', null);\n",
       "    }\n",
       "    this.set_frame(this.current_frame);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.get_loop_state = function(){\n",
       "    var button_group = document[this.loop_select_id].state;\n",
       "    for (var i = 0; i < button_group.length; i++) {\n",
       "        var button = button_group[i];\n",
       "        if (button.checked) {\n",
       "            return button.value;\n",
       "        }\n",
       "    }\n",
       "    return undefined;\n",
       "  }\n",
       "\n",
       "  Animation.prototype.set_frame = function(frame){\n",
       "    this.current_frame = frame;\n",
       "    document.getElementById(this.img_id).src =\n",
       "            this.frames[this.current_frame].src;\n",
       "    document.getElementById(this.slider_id).value = this.current_frame;\n",
       "  }\n",
       "\n",
       "  Animation.prototype.next_frame = function()\n",
       "  {\n",
       "    this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));\n",
       "  }\n",
       "\n",
       "  Animation.prototype.previous_frame = function()\n",
       "  {\n",
       "    this.set_frame(Math.max(0, this.current_frame - 1));\n",
       "  }\n",
       "\n",
       "  Animation.prototype.first_frame = function()\n",
       "  {\n",
       "    this.set_frame(0);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.last_frame = function()\n",
       "  {\n",
       "    this.set_frame(this.frames.length - 1);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.slower = function()\n",
       "  {\n",
       "    this.interval /= 0.7;\n",
       "    if(this.direction > 0){this.play_animation();}\n",
       "    else if(this.direction < 0){this.reverse_animation();}\n",
       "  }\n",
       "\n",
       "  Animation.prototype.faster = function()\n",
       "  {\n",
       "    this.interval *= 0.7;\n",
       "    if(this.direction > 0){this.play_animation();}\n",
       "    else if(this.direction < 0){this.reverse_animation();}\n",
       "  }\n",
       "\n",
       "  Animation.prototype.anim_step_forward = function()\n",
       "  {\n",
       "    this.current_frame += 1;\n",
       "    if(this.current_frame < this.frames.length){\n",
       "      this.set_frame(this.current_frame);\n",
       "    }else{\n",
       "      var loop_state = this.get_loop_state();\n",
       "      if(loop_state == \"loop\"){\n",
       "        this.first_frame();\n",
       "      }else if(loop_state == \"reflect\"){\n",
       "        this.last_frame();\n",
       "        this.reverse_animation();\n",
       "      }else{\n",
       "        this.pause_animation();\n",
       "        this.last_frame();\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.anim_step_reverse = function()\n",
       "  {\n",
       "    this.current_frame -= 1;\n",
       "    if(this.current_frame >= 0){\n",
       "      this.set_frame(this.current_frame);\n",
       "    }else{\n",
       "      var loop_state = this.get_loop_state();\n",
       "      if(loop_state == \"loop\"){\n",
       "        this.last_frame();\n",
       "      }else if(loop_state == \"reflect\"){\n",
       "        this.first_frame();\n",
       "        this.play_animation();\n",
       "      }else{\n",
       "        this.pause_animation();\n",
       "        this.first_frame();\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.pause_animation = function()\n",
       "  {\n",
       "    this.direction = 0;\n",
       "    if (this.timer){\n",
       "      clearInterval(this.timer);\n",
       "      this.timer = null;\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.play_animation = function()\n",
       "  {\n",
       "    this.pause_animation();\n",
       "    this.direction = 1;\n",
       "    var t = this;\n",
       "    if (!this.timer) this.timer = setInterval(function() {\n",
       "        t.anim_step_forward();\n",
       "    }, this.interval);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.reverse_animation = function()\n",
       "  {\n",
       "    this.pause_animation();\n",
       "    this.direction = -1;\n",
       "    var t = this;\n",
       "    if (!this.timer) this.timer = setInterval(function() {\n",
       "        t.anim_step_reverse();\n",
       "    }, this.interval);\n",
       "  }\n",
       "</script>\n",
       "\n",
       "<style>\n",
       ".animation {\n",
       "    display: inline-block;\n",
       "    text-align: center;\n",
       "}\n",
       "input[type=range].anim-slider {\n",
       "    width: 374px;\n",
       "    margin-left: auto;\n",
       "    margin-right: auto;\n",
       "}\n",
       ".anim-buttons {\n",
       "    margin: 8px 0px;\n",
       "}\n",
       ".anim-buttons button {\n",
       "    padding: 0;\n",
       "    width: 36px;\n",
       "}\n",
       ".anim-state label {\n",
       "    margin-right: 8px;\n",
       "}\n",
       ".anim-state input {\n",
       "    margin: 0;\n",
       "    vertical-align: middle;\n",
       "}\n",
       "</style>\n",
       "\n",
       "<div class=\"animation\">\n",
       "  <img id=\"_anim_img9463b03b9b2b4700819ef823cd6fa729\">\n",
       "  <div class=\"anim-controls\">\n",
       "    <input id=\"_anim_slider9463b03b9b2b4700819ef823cd6fa729\" type=\"range\" class=\"anim-slider\"\n",
       "           name=\"points\" min=\"0\" max=\"1\" step=\"1\" value=\"0\"\n",
       "           oninput=\"anim9463b03b9b2b4700819ef823cd6fa729.set_frame(parseInt(this.value));\"></input>\n",
       "    <div class=\"anim-buttons\">\n",
       "      <button onclick=\"anim9463b03b9b2b4700819ef823cd6fa729.slower()\"><i class=\"fa fa-minus\"></i></button>\n",
       "      <button onclick=\"anim9463b03b9b2b4700819ef823cd6fa729.first_frame()\"><i class=\"fa fa-fast-backward\">\n",
       "          </i></button>\n",
       "      <button onclick=\"anim9463b03b9b2b4700819ef823cd6fa729.previous_frame()\">\n",
       "          <i class=\"fa fa-step-backward\"></i></button>\n",
       "      <button onclick=\"anim9463b03b9b2b4700819ef823cd6fa729.reverse_animation()\">\n",
       "          <i class=\"fa fa-play fa-flip-horizontal\"></i></button>\n",
       "      <button onclick=\"anim9463b03b9b2b4700819ef823cd6fa729.pause_animation()\"><i class=\"fa fa-pause\">\n",
       "          </i></button>\n",
       "      <button onclick=\"anim9463b03b9b2b4700819ef823cd6fa729.play_animation()\"><i class=\"fa fa-play\"></i>\n",
       "          </button>\n",
       "      <button onclick=\"anim9463b03b9b2b4700819ef823cd6fa729.next_frame()\"><i class=\"fa fa-step-forward\">\n",
       "          </i></button>\n",
       "      <button onclick=\"anim9463b03b9b2b4700819ef823cd6fa729.last_frame()\"><i class=\"fa fa-fast-forward\">\n",
       "          </i></button>\n",
       "      <button onclick=\"anim9463b03b9b2b4700819ef823cd6fa729.faster()\"><i class=\"fa fa-plus\"></i></button>\n",
       "    </div>\n",
       "    <form action=\"#n\" name=\"_anim_loop_select9463b03b9b2b4700819ef823cd6fa729\" class=\"anim-state\">\n",
       "      <input type=\"radio\" name=\"state\" value=\"once\" id=\"_anim_radio1_9463b03b9b2b4700819ef823cd6fa729\"\n",
       "             >\n",
       "      <label for=\"_anim_radio1_9463b03b9b2b4700819ef823cd6fa729\">Once</label>\n",
       "      <input type=\"radio\" name=\"state\" value=\"loop\" id=\"_anim_radio2_9463b03b9b2b4700819ef823cd6fa729\"\n",
       "             checked>\n",
       "      <label for=\"_anim_radio2_9463b03b9b2b4700819ef823cd6fa729\">Loop</label>\n",
       "      <input type=\"radio\" name=\"state\" value=\"reflect\" id=\"_anim_radio3_9463b03b9b2b4700819ef823cd6fa729\"\n",
       "             >\n",
       "      <label for=\"_anim_radio3_9463b03b9b2b4700819ef823cd6fa729\">Reflect</label>\n",
       "    </form>\n",
       "  </div>\n",
       "</div>\n",
       "\n",
       "\n",
       "<script language=\"javascript\">\n",
       "  /* Instantiate the Animation class. */\n",
       "  /* The IDs given should match those used in the template above. */\n",
       "  (function() {\n",
       "    var img_id = \"_anim_img9463b03b9b2b4700819ef823cd6fa729\";\n",
       "    var slider_id = \"_anim_slider9463b03b9b2b4700819ef823cd6fa729\";\n",
       "    var loop_select_id = \"_anim_loop_select9463b03b9b2b4700819ef823cd6fa729\";\n",
       "    var frames = new Array(93);\n",
       "    \n",
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