3- Time and Motion.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercises\n",
    "\n",
    "### Ex1 \n",
    "\n",
    "A single-link robot with a rotary joint is motionless at  = -7˚. It is desired to move the joint in a smooth manner to  = 62° in 5 seconds. Find the coefficients of a cubic which accomplishes this motion and brings the arm to rest at the goal. Write expressions for the position, velocity, and acceleration of the joint as a function of time."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**\n",
    "\n",
    "$$\n",
    "\\theta(t) = a_0 + a_1 t + a_2 t^2 + a_3 t^3\n",
    "$$\n",
    "$$\n",
    "\\dot{\\theta}(t) = a_1 +2 a_2 t + 3 a_3 t^2\n",
    "$$\n",
    "$$\n",
    "\\ddot{\\theta}(t) = 2 a_2 + 6 a_3 t\n",
    "$$\n",
    "\n",
    "With Initial conditions \n",
    "\n",
    "$\\theta_0 = -7$ and intial velocity $\\dot{\\theta} = 0$ \n",
    "\n",
    "And final angle $\\theta_f = 62$ and final time $t_f = 5$\n",
    "\n",
    "So:\n",
    "$$\n",
    " a_0 = \\theta_0\n",
    "$$\n",
    "$$\n",
    " a_1 = 0\n",
    "$$\n",
    "\n",
    "$$\n",
    " a_2 = \\frac{3}{t_f^2} (\\theta_f - \\theta_0)\n",
    "$$\n",
    "\n",
    "$$\n",
    "a_3 = \\frac{-2}{t_f^3} (\\theta_f - \\theta_0)\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Theta  [-7, 0, 8.28, -1.104]\n",
      "Theta dot  [0, 16.56, -3.3120000000000003]\n",
      "Theta ddot  [16.56, -6.6240000000000006]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np \n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "th_f = 62\n",
    "th_0 = -7\n",
    "t_f = 5\n",
    "\n",
    "# Coeffs\n",
    "a0 = -7\n",
    "a1 = 0\n",
    "a2 = (3/(t_f**2))*(th_f-th_0)\n",
    "a3 = (-2/(t_f**3))*(th_f-th_0)\n",
    "th_eq = [a0, a1, a2, a3]\n",
    "thdot_eq = [a1, 2*a2, 3*a3]\n",
    "thddot_eq = [2*a2, 6*a3]\n",
    "print(\"Theta \", th_eq)\n",
    "print(\"Theta dot \", thdot_eq)\n",
    "print(\"Theta ddot \", thddot_eq)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So \n",
    "$$\n",
    "\\theta(t) = -7 +  8.28 t^2 -1.104 t^3\n",
    "$$\n",
    "$$\n",
    "\\dot{\\theta}(t) = 16.56 t -3.31 t^2\n",
    "$$\n",
    "$$\n",
    "\\ddot{\\theta}(t) = 16.56-6.62 t\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Ex2\n",
    "\n",
    "A single-link robot with a rotary joint is motionless at Ɵ = 15 degrees. It is desired to move the joint in a smooth manner to Ɵ = 75 degrees in 3 seconds. Find the coefficients of a cubic that accomplishes this motion and brings the manipulator to rest at the goal. Plot the position, velocity, and acceleration of the joint as a function of time.\n",
    "\n",
    "**Solution**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**\n",
    "\n",
    "$$\n",
    "\\theta(t) = a_0 + a_1 t + a_2 t^2 + a_3 t^3\n",
    "$$\n",
    "$$\n",
    "\\dot{\\theta}(t) = a_1 +2 a_2 t + 3 a_3 t^2\n",
    "$$\n",
    "$$\n",
    "\\ddot{\\theta}(t) = 2 a_2 + 6 a_3 t\n",
    "$$\n",
    "\n",
    "With Initial conditions \n",
    "\n",
    "$\\theta_0 = 15$ and intial velocity $\\dot{\\theta} = 0$ \n",
    "\n",
    "And final angle $\\theta_f = 75$ and final time $t_f = 3$\n",
    "\n",
    "So:\n",
    "$$\n",
    " a_0 = \\theta_0\n",
    "$$\n",
    "$$\n",
    " a_1 = \\dot{\\theta_0} = 0\n",
    "$$\n",
    "\n",
    "$$\n",
    " a_2 = \\frac{3}{t_f^2} (\\theta_f - \\theta_0)\n",
    "$$\n",
    "\n",
    "$$\n",
    "a_3 = \\frac{-2}{t_f^3} (\\theta_f - \\theta_0)\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Theta  [15, 0, 20.0, -4.444444444444445]\n",
      "Theta dot  [0, 40.0, -13.333333333333334]\n",
      "Theta ddot  [40.0, -26.666666666666668]\n"
     ]
    }
   ],
   "source": [
    "th_f = 75\n",
    "th_0 = 15\n",
    "t_f = 3\n",
    "thdot_0 = 0\n",
    "\n",
    "# Coeffs\n",
    "a0 = th_0\n",
    "a1 = thdot_0\n",
    "a2 = (3/(t_f**2))*(th_f-th_0)\n",
    "a3 = (-2/(t_f**3))*(th_f-th_0)\n",
    "th_eq = [a0, a1, a2, a3]\n",
    "thdot_eq = [a1, 2*a2, 3*a3]\n",
    "thddot_eq = [2*a2, 6*a3]\n",
    "print(\"Theta \", th_eq)\n",
    "print(\"Theta dot \", thdot_eq)\n",
    "print(\"Theta ddot \", thddot_eq)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So \n",
    "$$\n",
    "\\theta(t) = 15 +  20 t^2 -4.44 t^3\n",
    "$$\n",
    "$$\n",
    "\\dot{\\theta}(t) = 40 t - 13.33 t^2\n",
    "$$\n",
    "$$\n",
    "\\ddot{\\theta}(t) = 40-26.67 t\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Ex4\n",
    "\n",
    "A single-link robot with a rotary joint is motionless at Ɵ = 15 degrees. It is desired to move the joint in a smooth manner to Ɵ = 75 degrees in 3 seconds. Find the coefficients of a cubic with a parabolic blends that accomplishes this motion and brings the manipulator to rest at the goal. Plot the position, velocity, and acceleration of the joint as a function of time.\n",
    "\n",
    "**Solution**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Theta  [15, 0, 20.0, -4.444444444444445]\n",
      "Theta dot  [0, 40.0, -13.333333333333334]\n",
      "Theta ddot  [40.0, -26.666666666666668]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "th_f = 75\n",
    "th_0 = 15\n",
    "t_f = 3\n",
    "thdot_0 = 0\n",
    "\n",
    "# Coeffs\n",
    "a0 = th_0\n",
    "a1 = thdot_0\n",
    "a2 = (3/(t_f**2))*(th_f-th_0)\n",
    "a3 = (-2/(t_f**3))*(th_f-th_0)\n",
    "th_eq = [a0, a1, a2, a3]\n",
    "thdot_eq = [a1, 2*a2, 3*a3]\n",
    "thddot_eq = [2*a2, 6*a3]\n",
    "theta_func = lambda t: a0+a1*t + a2*(t**2) + a3 * (t**3)\n",
    "dotTheta_func = lambda t: a1 + 2*a2*t + 3*a3 * (t**2)\n",
    "ddotTheta_func = lambda t:  2*a2 + 6* a3 * t\n",
    "t = list(np.arange(0, t_f+0.1, 0.1))\n",
    "theta = [theta_func(i) for i in t]\n",
    "dot_theta = [dotTheta_func(i) for i in t]\n",
    "ddot_theta = [ddotTheta_func(i) for i in t]\n",
    "print(\"Theta \", th_eq)\n",
    "print(\"Theta dot \", thdot_eq)\n",
    "print(\"Theta ddot \", thddot_eq)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Angle (Deg)')"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "plt.plot(t, theta)\n",
    "plt.xlabel('Time (Sec)')\n",
    "plt.ylabel('Angle (Deg)')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Velocity (Deg/Sec)')"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "plt.plot(t, dot_theta)\n",
    "plt.xlabel('Time (Sec)')\n",
    "plt.ylabel('Velocity (Deg/Sec)')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Acceleration (Deg/Sec^2)')"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "plt.plot(t, ddot_theta)\n",
    "plt.xlabel('Time (Sec)')\n",
    "plt.ylabel('Acceleration (Deg/Sec^2)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Multi-Segment trajectories"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![](images/Selection_021.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Ex3\n",
    "\n",
    "Solve for the coefficients of two cubic equation that are connected in a two-segment spline with continuous acceleration at the intermediate via point. The initial angle Ɵ0 is the via point is Ɵv and the goal point is Ɵg. where Ɵ0=15 , Ɵv=30 , Ɵg=75 , TF1=3 and TF2=2.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**\n",
    "\n",
    "**For first segment** \n",
    "$$\n",
    "\\theta_1(t) = a_{10} + a_{11} t + a_{12} t^2 + a_{13} t^3\n",
    "$$\n",
    "\n",
    "With Initial conditions \n",
    "\n",
    "$\\theta_{1} = 15$ and intial velocity $\\dot{\\theta}_1 = 0$ \n",
    "\n",
    "And final angle $\\theta_1 = \\theta_v = 30$ and final time $t_f=3$\n",
    "\n",
    "\n",
    "**And for second segment**\n",
    "\n",
    "$$\n",
    "\\theta_2(t) = a_{20} + a_{21} t + a_{22} t^2 + a_{23} t^3\n",
    "$$\n",
    "\n",
    "With Initial conditions \n",
    "\n",
    "$\\theta_{2} = \\theta_v = 30$ and intial velocity $\\dot{\\theta}_2 = \\dot{\\theta_{f1}}$ \n",
    "\n",
    "And final angle $\\theta_2 = \\theta_g = 75$ and final time $t_f=2$ and final velocity $\\dot{\\theta_{f2}} = 0$\n",
    "\n",
    "\n",
    "\n",
    "So:\n",
    "$$\n",
    " a_{10} = \\theta_0\n",
    "$$\n",
    "$$\n",
    " a_{11} = \\dot{ \\theta_0} = 0\n",
    "$$\n",
    "$$\n",
    "a_{12} = \\frac{12\\theta_v-3\\theta_g-9\\theta_0}{4t_{f1}^2}\n",
    "$$\n",
    "\n",
    "$$\n",
    "a_{13} = \\frac{-8\\theta_v+3\\theta_g+5\\theta_0}{4t_{f1}^3}\n",
    "$$\n",
    "\n",
    "$$\n",
    " a_{20} = \\theta_v\n",
    "$$\n",
    "$$\n",
    " a_{21} = \\frac{3\\theta_g - 3\\theta_0}{4t_{f2}}\n",
    "$$\n",
    "$$\n",
    "a_{22} = \\frac{-12\\theta_v+6\\theta_g+6\\theta_0}{4t_{f2}^2}\n",
    "$$\n",
    "\n",
    "$$\n",
    "a_{23} = \\frac{8\\theta_v-5\\theta_g-3\\theta_0}{4t_{f2}^3}\n",
    "$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[15, 0, 0.0, 0.5555555555555556]\n",
      "[30, 22.5, 11.25, -5.625]\n"
     ]
    }
   ],
   "source": [
    "th_0 = 15\n",
    "th_v = 30\n",
    "th_g = 75\n",
    "\n",
    "t_f1 = 3\n",
    "t_f2 = 2\n",
    "\n",
    "# Coeffs\n",
    "a10 = th_0\n",
    "a11 = 0\n",
    "a12 = (12*th_v - 3*th_g - 9*th_0)/(4*t_f1**2)\n",
    "a13 = (-8*th_v + 3*th_g + 5*th_0)/(4*t_f1**3)\n",
    "\n",
    "a20 = th_v\n",
    "a21 = (3*th_g - 3*th_0)/(4*t_f2)\n",
    "a22 = (-12*th_v + 6*th_g + 6*th_0)/(4*t_f2**2)\n",
    "a23 = (8*th_v - 5*th_g - 3*th_0)/(4*t_f2**3)\n",
    "\n",
    "theta1_coeff = [a10, a11, a12, a13]\n",
    "theta2_coeff = [a20, a21, a22, a23]\n",
    "print(theta1_coeff)\n",
    "print(theta2_coeff)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "t1 = np.arange(0, 3.01, 0.1)\n",
    "t2 = np.arange(0, 2.01, 0.1)\n",
    "theta1_func = lambda t: a10+a11*t + a12*(t**2) + a13 * (t**3)\n",
    "theta2_func = lambda t: a20+a21*t + a22*(t**2) + a23 * (t**3)\n",
    "theta1 = [theta1_func(i) for i in t1] \n",
    "theta2 = [theta2_func(i) for i in t2] "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f2fb7acd210>]"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "plt.plot(t1, theta1)\n",
    "plt.plot(t_f1+t2, theta2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So \n",
    "\n",
    "$$\n",
    "\\theta_1(t) = 15 + 0.55t^3\n",
    "$$\n",
    "\n",
    "And \n",
    "\n",
    "$$\n",
    "\\theta_2(t) = 30 + 22.5 t + 11.25 t^2 - 5.625 t^3\n",
    "$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**What is the formula of ?**\n",
    "\n",
    "* $\\dot\\theta_1$ \n",
    "* $\\ddot\\theta_1$ \n",
    "* $\\dot\\theta_2$ \n",
    "* $\\ddot\\theta_2$ \n"
   ]
  }
 ],
 "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": "3.7.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}