cart-pole.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\newcommand{\\mb}[1]{\\mathbf{ #1 }}$\n",
    "$\\newcommand{\\bs}[1]{\\boldsymbol{ #1 }}$\n",
    "$\\newcommand{\\bb}[1]{\\mathbb{ #1 }}$\n",
    "\n",
    "$\\newcommand{\\R}{\\bb{R}}$\n",
    "\n",
    "$\\newcommand{\\ip}[2]{\\left\\langle #1, #2 \\right\\rangle}$\n",
    "$\\newcommand{\\norm}[1]{\\left\\Vert #1 \\right\\Vert}$\n",
    "\n",
    "$\\newcommand{\\der}[2]{\\frac{\\mathrm{d} #1 }{\\mathrm{d} #2 }}$\n",
    "$\\newcommand{\\derp}[2]{\\frac{\\partial #1 }{\\partial #2 }}$\n",
    "\n",
    "# Cart Pole"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider a cart on a frictionless track. Suppose a pendulum is attached to the cart by a frictionless joint. The cart is modeled as a point mass $m_c$ and the pendulum is modeled as a massless rigid link with point mass $m_p$ a distance $l$ away from the cart."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $\\mathcal{I} = (\\mb{i}^1, \\mb{i}^2, \\mb{i}^3)$ denote an inertial frame. Suppose the position of the cart is resolved in the inertial frame as $\\mb{r}_{co}^{\\mathcal{I}} = (x, 0, 0)$. Additionally, suppose the gravitational force acting on the pendulum is resolved in the inertial frame as $\\mb{f}_g^{\\mathcal{I}} = (0, 0, -m_p g)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $\\mathcal{B} = (\\mb{b}^1, \\mb{b}^2, \\mb{b}^3)$ denote a body reference frame, with $\\mb{b}^2 = \\mb{i}^2$. The position of the pendulum mass relative to the cart is resolved in the body frame as $\\mb{r}_{pc}^\\mathcal{B} = (0, 0, l)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The kinetic energy of the system is:\n",
    "\n",
    "\\begin{equation}\n",
    "    \\frac{1}{2} m_c \\norm{\\dot{\\mb{r}}_{co}^\\mathcal{I}}_2^2 + \\frac{1}{2} m_p \\norm{\\dot{\\mb{r}}_{po}^\\mathcal{I}}_2^2\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, note that $\\dot{\\mb{r}}_{co}^{\\mathcal{I}} = (\\dot{x}, 0, 0)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, note that $\\mb{r}_{po}^\\mathcal{I} = \\mb{r}_{pc}^\\mathcal{I} + \\mb{r}_{co}^\\mathcal{I} = \\mb{C}_{\\mathcal{I}\\mathcal{B}}\\mb{r}_{pc}^\\mathcal{B} + \\mb{r}_{co}^\\mathcal{I}$, where $\\mb{C}_{\\mathcal{I}\\mathcal{B}}$ is the direction cosine matrix (DCM) satisfying:\n",
    "\n",
    "\\begin{equation}\n",
    "    \\mb{C}_{\\mathcal{I}\\mathcal{B}} = \\begin{bmatrix} \\ip{\\mb{i}_1}{\\mb{b}_1} & \\ip{\\mb{i}_1}{\\mb{b}_2} & \\ip{\\mb{i}_1}{\\mb{b}_3} \\\\ \\ip{\\mb{i}_2}{\\mb{b}_1} & \\ip{\\mb{i}_2}{\\mb{b}_2} & \\ip{\\mb{i}_2}{\\mb{b}_3} \\\\ \\ip{\\mb{i}_3}{\\mb{b}_1} & \\ip{\\mb{i}_3}{\\mb{b}_2} & \\ip{\\mb{i}_3}{\\mb{b}_3} \\end{bmatrix}.\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We parameterize the DCM using $\\theta$, measuring the clockwise angle of the pendulum from upright in radians. In this case, the DCM is:\n",
    "\n",
    "\\begin{equation}\n",
    "    \\mb{C}_{\\mathcal{I}\\mathcal{B}} = \\begin{bmatrix} \\cos{\\theta} & 0 & \\sin{\\theta} \\\\ 0 & 1 & 0 \\\\ -\\sin{\\theta} & 0 & \\cos{\\theta} \\end{bmatrix},\n",
    "\\end{equation}\n",
    "\n",
    "following from $\\cos{\\left( \\frac{\\pi}{2} - \\theta \\right)} = \\sin{\\theta}$. Therefore:\n",
    "\n",
    "\\begin{equation}\n",
    "    \\mb{r}_{po}^\\mathcal{I} = \\begin{bmatrix} x + l\\sin{\\theta} \\\\ 0 \\\\ l\\cos{\\theta} \\end{bmatrix}\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have $\\dot{\\mb{r}}_{po}^\\mathcal{I} = \\dot{\\mb{C}}_{\\mathcal{I}\\mathcal{B}} \\mb{r}_{pc}^\\mathcal{B} + \\dot{\\mb{r}}_{co}^\\mathcal{I}$, following from $\\dot{\\mb{r}}_{pc}^\\mathcal{B} = \\mb{0}_3$ since the pendulum is rigid. The derivative of the DCM is:\n",
    "\n",
    "\\begin{equation}\n",
    "    \\der{{\\mb{C}}_{\\mathcal{I}\\mathcal{B}}}{\\theta} = \\begin{bmatrix} -\\sin{\\theta} & 0 & \\cos{\\theta} \\\\ 0 & 0 & 0 \\\\ -\\cos{\\theta} & 0 & -\\sin{\\theta} \\end{bmatrix},\n",
    "\\end{equation}\n",
    "\n",
    "finally yielding:\n",
    "\n",
    "\\begin{equation}\n",
    "    \\dot{\\mb{r}}_{po}^\\mathcal{I} = \\dot{\\theta}  \\der{\\mb{C}_{\\mathcal{I}\\mathcal{B}}}{\\theta} \\mb{r}^{\\mathcal{B}}_{pc} + \\dot{\\mb{r}}_{co}^\\mathcal{I} = \\begin{bmatrix} l\\dot{\\theta}\\cos{\\theta} + \\dot{x} \\\\ 0 \\\\ -l\\dot{\\theta}\\sin{\\theta} \\end{bmatrix}\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define generalized coordinates $\\mb{q} = (x, \\theta)$ with configuration space $\\mathcal{Q} = \\R \\times \\bb{S}^1$, where $\\bb{S}^1$ denotes the $1$-sphere. The kinetic energy can then be expressed as:\n",
    "\n",
    "\\begin{align}\n",
    "    T(\\mb{q}, \\dot{\\mb{q}}) &= \\frac{1}{2} m_c \\begin{bmatrix} \\dot{x} \\\\ \\dot{\\theta} \\end{bmatrix}^\\top \\begin{bmatrix} 1 & 0 \\\\ 0 & 0 \\end{bmatrix} \\begin{bmatrix} \\dot{x} \\\\ \\dot{\\theta} \\end{bmatrix} + \\frac{1}{2} m_p \\begin{bmatrix} \\dot{x} \\\\ \\dot{\\theta} \\end{bmatrix}^\\top \\begin{bmatrix} 1 & l \\cos{\\theta} \\\\ l \\cos{\\theta} & l^2 \\end{bmatrix} \\begin{bmatrix} \\dot{x} \\\\ \\dot{\\theta} \\end{bmatrix}\\\\\n",
    "    &= \\frac{1}{2} \\dot{\\mb{q}}^\\top\\mb{D}(\\mb{q})\\dot{\\mb{q}},\n",
    "\\end{align}\n",
    "\n",
    "where inertia matrix function $\\mb{D}: \\mathcal{Q} \\to \\bb{S}^2_{++}$ is defined as:\n",
    "\n",
    "\\begin{equation}\n",
    "    \\mb{D}(\\mb{q}) = \\begin{bmatrix} m_c + m_p & m_p l \\cos{\\theta} \\\\ m_p l \\cos{\\theta} & m_pl^2 \\end{bmatrix}.\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that:\n",
    "\n",
    "\\begin{equation}\n",
    "    \\derp{\\mb{D}}{x} = \\mb{0}_{2 \\times 2},\n",
    "\\end{equation}\n",
    "\n",
    "and:\n",
    "\n",
    "\\begin{equation}\n",
    "    \\derp{\\mb{D}}{\\theta} = \\begin{bmatrix} 0 & -m_p l \\sin{\\theta} \\\\ -m_p l \\sin{\\theta} & 0 \\end{bmatrix},\n",
    "\\end{equation}\n",
    "\n",
    "so we can express:\n",
    "\n",
    "\\begin{equation}\n",
    "    \\derp{\\mb{D}}{\\mb{q}} = -m_p l \\sin{\\theta} (\\mb{e}_1 \\otimes \\mb{e}_2 \\otimes \\mb{e}_2 + \\mb{e}_2 \\otimes \\mb{e}_1 \\otimes \\mb{e}_2).\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The potential energy of the system is $U: \\mathcal{Q} \\to \\R$ defined as:\n",
    "\n",
    "\\begin{equation}\n",
    "    U(\\mb{q}) = -\\ip{\\mb{f}_g^\\mathcal{I}}{\\mb{r}^{\\mathcal{I}}_{po}} = m_p g l \\cos{\\theta}.\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define $\\mb{G}: \\mathcal{Q} \\to \\R^2$ as:\n",
    "    \n",
    "\\begin{equation}\n",
    "    \\mb{G}(\\mb{q}) = \\left(\\derp{U}{\\mb{q}}\\right)^\\top = \\begin{bmatrix} 0 \\\\ -m_p g l \\sin{\\theta} \\end{bmatrix}.\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assume a force $(F, 0, 0)$ (resolved in the inertial frame) can be applied to the cart. The Euler-Lagrange equation yields:\n",
    "\n",
    "\\begin{align}\n",
    "    \\der{}{t} \\left( \\derp{T}{\\dot{\\mb{q}}} \\right)^\\top - \\left( \\derp{T}{\\mb{q}} - \\derp{U}{\\mb{q}} \\right)^\\top &= \\der{}{t} \\left( \\mb{D}(\\mb{q})\\dot{\\mb{q}} \\right) - \\frac{1}{2}\\derp{\\mb{D}}{\\mb{q}}(\\dot{\\mb{q}}, \\dot{\\mb{q}}, \\cdot) + \\mb{G}(\\mb{q})\\\\\n",
    "    &= \\mb{D}(\\mb{q})\\ddot{\\mb{q}} + \\derp{\\mb{D}}{\\mb{q}}(\\cdot, \\dot{\\mb{q}}, \\dot{\\mb{q}}) - \\frac{1}{2}\\derp{\\mb{D}}{\\mb{q}}(\\dot{\\mb{q}}, \\dot{\\mb{q}}, \\cdot) + \\mb{G}(\\mb{q})\\\\\n",
    "    &= \\mb{B} F,\n",
    "\\end{align}\n",
    "\n",
    "with static actuation matrix:\n",
    "\n",
    "\\begin{equation}\n",
    "    \\mb{B} = \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}.\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that:\n",
    "\n",
    "\\begin{align}\n",
    "    \\derp{\\mb{D}}{\\mb{q}}(\\cdot, \\dot{\\mb{q}}, \\dot{\\mb{q}}) - \\frac{1}{2}\\derp{\\mb{D}}{\\mb{q}}(\\dot{\\mb{q}}, \\dot{\\mb{q}}, \\cdot) &= -m_p l \\sin{\\theta} (\\mb{e}_1 \\dot{\\theta}\\dot{\\theta} + \\mb{e}_2\\dot{x}\\dot{\\theta}) + \\frac{1}{2} m_p l \\sin{\\theta} (\\dot{x}\\dot{\\theta} \\mb{e}_2 + \\dot{\\theta}\\dot{x} \\mb{e}_2)\\\\\n",
    "    &= \\begin{bmatrix} -m_p l \\dot{\\theta}^2 \\sin{\\theta} \\\\ 0 \\end{bmatrix}\\\\\n",
    "    &= \\mb{C}(\\mb{q}, \\dot{\\mb{q}})\\dot{\\mb{q}},\n",
    "\\end{align}\n",
    "\n",
    "with Coriolis terms defined as:\n",
    "\n",
    "\\begin{equation}\n",
    "    \\mb{C}(\\mb{q}, \\dot{\\mb{q}}) = \\begin{bmatrix} 0 & -m_p l \\sin{\\theta} \\\\ 0 & 0 \\end{bmatrix}.\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we have:\n",
    "\n",
    "\\begin{equation}\n",
    "    \\mb{D}(\\mb{q})\\ddot{\\mb{q}} + \\mb{C}(\\mb{q}, \\dot{\\mb{q}})\\dot{\\mb{q}} + \\mb{G}(\\mb{q}) = \\mb{B}F\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from numpy import array, concatenate, cos, dot, reshape, sin, zeros\n",
    "\n",
    "from core.dynamics import RoboticDynamics\n",
    "\n",
    "class CartPole(RoboticDynamics):\n",
    "    def __init__(self, m_c, m_p, l, g=9.81):\n",
    "        RoboticDynamics.__init__(self, 2, 1)\n",
    "        self.params = m_c, m_p, l, g\n",
    "    \n",
    "    def D(self, q):\n",
    "        m_c, m_p, l, _ = self.params\n",
    "        _, theta = q\n",
    "        return array([[m_c + m_p, m_p * l * cos(theta)], [m_p * l * cos(theta), m_p * (l ** 2)]])\n",
    "    \n",
    "    def C(self, q, q_dot):\n",
    "        _, m_p, l, _ = self.params\n",
    "        _, theta = q\n",
    "        _, theta_dot = q_dot\n",
    "        return array([[0, -m_p * l * theta_dot * sin(theta)], [0, 0]])\n",
    "    \n",
    "    def U(self, q):\n",
    "        _, m_p, l, g = self.params\n",
    "        _, theta = q\n",
    "        return m_p * g * l * cos(theta)\n",
    "    \n",
    "    def G(self, q):\n",
    "        _, m_p, l, g = self.params\n",
    "        _, theta = q\n",
    "        return array([0, -m_p * g * l * sin(theta)])\n",
    "    \n",
    "    def B(self, q):\n",
    "        return array([[1], [0]])\n",
    "\n",
    "m_c = 0.5\n",
    "m_p = 0.25\n",
    "l = 0.5\n",
    "cart_pole = CartPole(m_c, m_p, l)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We attempt to stabilize the pendulum upright, that is, drive $\\theta$ to $0$. We'll use the normal form transformation:\n",
    "\n",
    "\\begin{equation}\n",
    "    \\bs{\\Phi}(\\mb{q}, \\dot{\\mb{q}}) = \\begin{bmatrix} \\bs{\\eta}(\\mb{q}, \\dot{\\mb{q}}) \\\\ \\mb{z}(\\mb{q}, \\dot{\\mb{q}}) \\end{bmatrix} = \\begin{bmatrix} \\theta \\\\ \\dot{\\theta} \\\\ x \\\\ m_p l \\dot{x} \\cos{\\theta} + m_p l^2 \\dot{\\theta} \\end{bmatrix}.\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "from core.dynamics import ConfigurationDynamics\n",
    "\n",
    "class CartPoleOutput(ConfigurationDynamics):\n",
    "    def __init__(self, cart_pole):\n",
    "        ConfigurationDynamics.__init__(self, cart_pole, 1)\n",
    "        self.cart_pole = cart_pole\n",
    "    \n",
    "    def y(self, q):\n",
    "        return q[1:]\n",
    "        \n",
    "    def dydq(self, q):\n",
    "        return array([[0, 1]])\n",
    "    \n",
    "    def d2ydq2(self, q):\n",
    "        return zeros((1, 2, 2))\n",
    "    \n",
    "output = CartPoleOutput(cart_pole)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "from numpy import identity\n",
    "\n",
    "from core.controllers import FBLinController, LQRController\n",
    "\n",
    "Q = 10 * identity(2)\n",
    "R = identity(1)\n",
    "lqr = LQRController.build(output, Q, R)\n",
    "fb_lin = FBLinController(output, lqr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "from numpy import linspace, pi\n",
    "\n",
    "x_0 = array([0, pi / 4, 0, 0])\n",
    "ts = linspace(0, 10, 1000 + 1)\n",
    "\n",
    "xs, us = cart_pole.simulate(x_0, fb_lin, ts)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "from matplotlib.pyplot import subplots, show, tight_layout"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x576 with 4 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "_, axs = subplots(2, 2, figsize=(8, 8))\n",
    "ylabels = ['$x$ (m)', '$\\\\theta$ (rad)', '$\\\\dot{x}$ (m / sec)', '$\\\\dot{\\\\theta}$ (rad / sec)']\n",
    "\n",
    "for ax, data, ylabel in zip(axs.flatten(), xs.T, ylabels):\n",
    "    ax.plot(ts, data, linewidth=3)\n",
    "    ax.set_ylabel(ylabel, fontsize=16)\n",
    "    ax.grid()\n",
    "    \n",
    "for ax in axs[-1]:\n",
    "    ax.set_xlabel('$t$ (sec)', fontsize=16)\n",
    "    \n",
    "tight_layout()\n",
    "show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 288x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "_, ax = subplots(figsize=(4, 4))\n",
    "\n",
    "ax.plot(ts[:-1], us, linewidth=3)\n",
    "ax.grid()\n",
    "ax.set_xlabel('$t$ (sec)', fontsize=16)\n",
    "ax.set_ylabel('$F$ (N)', fontsize=16)\n",
    "\n",
    "show()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}