sympy_test.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "####################################################################################################\n",
    "####################################################################################################\n",
    "#########DISCLAIMER: THIS NOTEBOOK DOES NOT CONTAIN THE CORRECT EQUATIONS FOR ANYTHING##############\n",
    "####################################################################################################\n",
    "####################################################################################################\n",
    "####################################################################################################\n",
    "####################################################################################################\n",
    "####################################################################################################\n",
    "####################################################################################################\n",
    "####################################################################################################\n",
    "\n",
    "\n",
    "\n",
    "from sympy import *\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "init_printing()\n",
    "\n",
    "m_cart = Symbol('m_c')\n",
    "m_pole = Symbol('m_p')\n",
    "\n",
    "x, theta, x_dot, theta_dot, x_dot_dot, theta_dot_dot = symbols('x theta xdot thetadot xdotdot thetadotdot', cls=Function)\n",
    "\n",
    "g = Symbol('g')\n",
    "l = Symbol('l')\n",
    "t = Symbol('t')\n",
    "F1 = Symbol('F1')\n",
    "F2 = Symbol('F2')\n",
    "\n",
    "eq_f1 = Eq(-m_pole * l * theta_dot(t) * sin(theta(t)) + (m_cart + m_pole) * x_dot_dot(t) + m_pole * l * theta_dot_dot(t) * cos(theta(t)), F1)\n",
    "eq_f2 = Eq(m_pole * g * l * sin(theta(t)) + m_pole * l * x_dot_dot(t) * cos(theta(t)) + m_pole * l**2 * theta_dot_dot(t), F2)\n",
    "\n",
    "eq_theta_dot_f1 = dsolve(eq_f1, theta_dot(t))\n",
    "#eq_x_dot = dsolve(eq_f1, x_dot(t))\n",
    "#eq_theta_dot_f1\n",
    "#eq_x_dot\n",
    "\n",
    "#eq_theta_dot_dot_subs = eq_theta_dot_dot.subs(x_dot_dot(t), eq_x_dot_dot)\n",
    "#eq_theta_dot_dot_subs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle - l m_{p} \\theta_{2}{\\left(t \\right)} \\sin{\\left(\\theta_{1}{\\left(t \\right)} \\right)} + l m_{p} \\dot{theta2}{\\left(t \\right)} \\cos{\\left(\\theta_{1}{\\left(t \\right)} \\right)} + \\left(m_{c} + m_{p}\\right) \\dot{x2}{\\left(t \\right)} = F_{1}$"
      ],
      "text/plain": [
       "-l⋅mₚ⋅θ₂(t)⋅sin(θ₁(t)) + l⋅mₚ⋅theṫa2(t)⋅cos(θ₁(t)) + (m_c + mₚ)⋅x2̇(t) = F₁"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import *\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "init_printing()\n",
    "\n",
    "###Same thing with expanded state vector in order to convert second order system to first order system\n",
    "\n",
    "m_c = Symbol('m_c')\n",
    "m_p = Symbol('m_p')\n",
    "\n",
    "x1, x2, x1_dot, x2_dot, theta1, theta2, theta1_dot, theta2_dot = symbols('x1 x2 x1dot x2dot theta1 theta2 theta1dot theta2dot', cls=Function)\n",
    "\n",
    "g = Symbol('g')\n",
    "l = Symbol('l')\n",
    "t = Symbol('t')\n",
    "F1 = Symbol('F1')\n",
    "F2 = Symbol('F2')\n",
    "\n",
    "eq_f1 = Eq(-m_p * l * theta2(t) * sin(theta1(t)) + (m_c + m_p) * x2_dot(t) + m_p * l * theta2_dot(t) * cos(theta1(t)), F1)\n",
    "eq_f1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle g l m_{p} \\sin{\\left(\\theta_{1}{\\left(t \\right)} \\right)} + l^{2} m_{p} \\dot{theta2}{\\left(t \\right)} + l m_{p} \\dot{x2}{\\left(t \\right)} \\cos{\\left(\\theta_{1}{\\left(t \\right)} \\right)} = F_{2}$"
      ],
      "text/plain": [
       "                     2                                          \n",
       "g⋅l⋅mₚ⋅sin(θ₁(t)) + l ⋅mₚ⋅theṫa2(t) + l⋅mₚ⋅x2̇(t)⋅cos(θ₁(t)) = F₂"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq_f2 = Eq(m_p * g * l * sin(theta1(t)) + m_p * l * x2_dot(t) * cos(theta1(t)) + m_p * l**2 * theta2_dot(t), F2)\n",
    "eq_f2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\dot{theta2}{\\left(t \\right)} = \\frac{F_{1} + l m_{p} \\theta_{2}{\\left(t \\right)} \\sin{\\left(\\theta_{1}{\\left(t \\right)} \\right)} - m_{c} \\dot{x2}{\\left(t \\right)} - m_{p} \\dot{x2}{\\left(t \\right)}}{l m_{p} \\cos{\\left(\\theta_{1}{\\left(t \\right)} \\right)}}$"
      ],
      "text/plain": [
       "            F₁ + l⋅mₚ⋅θ₂(t)⋅sin(θ₁(t)) - m_c⋅x2̇(t) - mₚ⋅x2̇(t)\n",
       "theṫa2(t) = ─────────────────────────────────────────────────\n",
       "                             l⋅mₚ⋅cos(θ₁(t))                 "
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq_theta2_dot_f1 = dsolve(eq_f1, theta2_dot(t))\n",
    "eq_theta2_dot_f1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\dot{theta2}{\\left(t \\right)} = \\frac{F_{2} - l m_{p} \\left(g \\sin{\\left(\\theta_{1}{\\left(t \\right)} \\right)} + \\dot{x2}{\\left(t \\right)} \\cos{\\left(\\theta_{1}{\\left(t \\right)} \\right)}\\right)}{l^{2} m_{p}}$"
      ],
      "text/plain": [
       "            F₂ - l⋅mₚ⋅(g⋅sin(θ₁(t)) + x2̇(t)⋅cos(θ₁(t)))\n",
       "theṫa2(t) = ───────────────────────────────────────────\n",
       "                                2                      \n",
       "                               l ⋅mₚ                   "
      ]
     },
     "execution_count": 106,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq_theta2_dot_f2 = dsolve(eq_f2, theta2_dot(t))\n",
    "eq_theta2_dot_f2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\dot{x2}{\\left(t \\right)} = \\frac{F_{1} + l m_{p} \\theta_{2}{\\left(t \\right)} \\sin{\\left(\\theta_{1}{\\left(t \\right)} \\right)} - l m_{p} \\dot{theta2}{\\left(t \\right)} \\cos{\\left(\\theta_{1}{\\left(t \\right)} \\right)}}{m_{c} + m_{p}}$"
      ],
      "text/plain": [
       "        F₁ + l⋅mₚ⋅θ₂(t)⋅sin(θ₁(t)) - l⋅mₚ⋅theṫa2(t)⋅cos(θ₁(t))\n",
       "x2̇(t) = ──────────────────────────────────────────────────────\n",
       "                               m_c + mₚ                       "
      ]
     },
     "execution_count": 107,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq_x2_dot_f1 = dsolve(eq_f1, x2_dot(t))\n",
    "eq_x2_dot_f1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\dot{x2}{\\left(t \\right)} = \\frac{F_{2} - l m_{p} \\left(g \\sin{\\left(\\theta_{1}{\\left(t \\right)} \\right)} + l \\dot{theta2}{\\left(t \\right)}\\right)}{l m_{p} \\cos{\\left(\\theta_{1}{\\left(t \\right)} \\right)}}$"
      ],
      "text/plain": [
       "        F₂ - l⋅mₚ⋅(g⋅sin(θ₁(t)) + l⋅theṫa2(t))\n",
       "x2̇(t) = ──────────────────────────────────────\n",
       "                   l⋅mₚ⋅cos(θ₁(t))            "
      ]
     },
     "execution_count": 108,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq_x2_dot_f2 = dsolve(eq_f2, x2_dot(t))\n",
    "eq_x2_dot_f2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\frac{F_{1} - l \\left(g \\sin{\\left(\\theta_{1}{\\left(t \\right)} \\right)} + \\frac{F_{1} + l \\theta_{2}{\\left(t \\right)} \\sin{\\left(\\theta_{1}{\\left(t \\right)} \\right)} - 6 \\dot{x2}{\\left(t \\right)}}{\\cos{\\left(\\theta_{1}{\\left(t \\right)} \\right)}}\\right)}{l \\cos{\\left(\\theta_{1}{\\left(t \\right)} \\right)}} = \\dot{x2}{\\left(t \\right)}$"
      ],
      "text/plain": [
       "       ⎛               F₁ + l⋅θ₂(t)⋅sin(θ₁(t)) - 6⋅x2̇(t)⎞        \n",
       "F₁ - l⋅⎜g⋅sin(θ₁(t)) + ─────────────────────────────────⎟        \n",
       "       ⎝                           cos(θ₁(t))           ⎠        \n",
       "───────────────────────────────────────────────────────── = x2̇(t)\n",
       "                       l⋅cos(θ₁(t))                              "
      ]
     },
     "execution_count": 109,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq_x2_dot_subs = Eq((F1 - l * (g * sin(theta1(t)) + l * ((F1 + l * theta2(t) * sin(theta1(t)) - 6 * x2_dot(t))/ (l * cos(theta1(t))))))/(l * cos(theta1(t))), x2_dot(t))\n",
    "eq_x2_dot_subs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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/Dk+AYO8Gb9xx3Z3mIXDKeLW4r+naMEawz6X2oLFXX5s75pTj0jxE21y99i3Gsi0m5PRkbTpuzVqK0g3A3oC3UjUMoBON8ewOAsKRC0rHwtsQBj+l4TkAX/Z2nIoxkyiTPq8LDssrOt+cdmjb0AZw0vGREl4rZEJmv82HOnh6tomeTcLkpU9m4M95OsnjocEAIu9WB27rqWuBV6z15hS4Ux5PDXXxiib1Hks9WdxLtDl4nRTWBmvkcr+9INKhOTyd4g82i35s/dauc5r2YIGZ0fBTx6tFfS32Ex7UkVujvfpac8yRbjwU2Z6/g86ZH+i73FNlOoY06dYeDCa+Qt16waPXvkVYSuYU9WRtOm699+7dfi8iCGwuwFOF2dssNXTEwxM/E5MNGDW2B5cmPHDdMpGVa8arYRGxX2Pz8CKd9mjbIoUmmKUvxhQG5En36Jx2nyJLZTHYeAOkec/M4ZmAw7OFgHBk3GIi+kBxM3IcmwtAQNeDsXN44Ivn7K/DszCQzo/u1yo72dcGQWeKnNK+VGXVMzkXnSJrDpYtnlkem7QxFo8VYnVBGCwQT8LZTXKXPLjJmZBnueTFx/dH2LxJnSdNHKbDPQnpOFtT6s3YWlZa/x5tS+WdGuf+5ynHnpaOrW9Ou0+RxVNey7tkOs/hMV4P2wiE66Qx67bN4jl7I6DrgReEa/NKcYxPiH5b82xs3deC8DP+nNK+VO2LHbeO8tjoxqBBLAEMN4XigIXrHYt4NDEqjf05XyqsGilKH1nJSkMOX5JlOeFBkzDAeGQSxUomzubWOZ/QFttl0zW3TbrjbmXjaPW+7iG/tN3HyFKZyT40h6fXDs+7Q0A4Mgb2ljkcqjMhoGvDf46NJmKlVz8xoPTF/VplJvvamZo/EntM+6wSlV00Fx0jaw6WPZ5jPTZ04Gy9XkLwrLDOH9ywBgJhTG/+TbrysaBZ+sgmB52zt4dNXvxLb2vvQCrq3sbVfrA51TNwkfhcedsw7rnnZ3ki0wtwRLuPkYVuwwNIKj+Jz+FJ2D1aIqBrydjnRk0JzIWc6/rM+XuGVNut+loq45zxY9oX9L2GcevRkcjyF/O1N6Tw1NC5sehSCpuS0oQizqRQXcJS+isduMmdHIGLQ0D3+q2UwmOTvTK+haJLZUWdut6kOTxbtOU+1SkM8aJ+ojA8eHCuo3wT7z41+d63RddvUb8WP/2/29cuCbSl7TtF96Wy5mA5xXPsUhRGDR032yAnYTz54X3hLZBgqCjEyOFPB3sbF1mGYk8NBtCIlI688k2WEZ8nOAKOgCOwJwIamxgHX+vgAcwIA4dldyZHJ0fAEdgZgWEpSp2Q9UF76vxMcYwMjBI6Lq/R/i6esByksLXnxZ5S0iUlDJ3anhuecnCHISPIUb1scOQfSstlJ8rDO6pHaU6OgCPgCJwLgZ8lmLGTh7qBNIa5UTOg4RFHYF8EgmGjTkjH5P334DFRiIET3kFXnA+Z4VGho5YGh5LuSDwYKhgo8KedmmWmrNNTQjwYKeEfchXiuu3tUUB+cPMqnCTVxR4g9FlC7AGaLWNJxc7rCDgC9xMBjRlL927cTyC8VY7ABSEQDBvpwx6W1JXKshFGin1vhgk/9cLodER4W2rLSdTzdsT9/4Te/hrjojzG1yzSYFNd0ppV2JkcAUfAEXAEHAFH4GoRMMPmlYyB1MvCkhLelJCmsLsEpHyMGvhrHo/HykvrLsHCszL1h48YVRhIZye1cb8vGp69ta6AI+AIOAKOgCNwXQgEw0aTdWl4YGzM8nqoLN+uOSisGTVdNFTmfTFgsLDsdRUknavfPbgK5V1JR8ARcAQcAUfgniNgHpuhmZq4bQNw10tDAfGyMXh4zTGmBc+K8mzpqreM9CSWGWSpHK+Ll4YWXh+rjyJdUvlj9tjw30NsUHZyBBwBR8ARcAQcgStFwDw24c2laFDgrTkoHl7XjnG8Mt/HfJLIxwDiz7ZKTw11pUtLGCQYJjXK9teoLsrCP8iOhTCWSmMnZo0D1TPL2zQu6SmOgCOwJwLqq+FrvZJJH6fv81mH2X19T11dliPgCFwHAjcaRDBk7MujGCTZ20nKD0ZFOtjENMqwWZiBCWJZCQPmU6Wlb09hpLReD6dM8MSoDHG+PpwaRUoKRPnBqxPTPHAEHIErRiCOHfx/nI0BjCm8Pt0aL664ta66I+AI7IUAHptfdGBMYFTgmWE55ivF2RAcvi6suL0dpaRADEAYPGF/zV3S8BsGqeEsvjaenKdRjCIGtvD9HIWpQZTysWT1Ik3wuCPgCJyGgPobffhppX+fVvH80nhoGWeMXinCv5CP/l7FGMpwqg3K54HpmQ7+xbn20FRW2TyfkpUWFC9t+1VhOR4Gtqn8tK5LiF+bvpeAmetwPgSO+vLwUnXVKfgODX+OWe3kvfpUhoGJwa755eJeec9zBB4CAuofPGSwLIyxMnz5u9X22K9+Uph5aFv8W6RLdvg+lcLgjVWI7jZWTHpoxc/Y0GyD8vFGsyzNwxpxxqBy6VzJ0zQlq1aDyvAAyNJaddybyq/VuWWa9OFB8/co40OFPyht2BZwafpuiYXXfd0IPNpJfTrMsfteXqps+lS3k8ouxhG4HgQ06eDtDMvC6WTUaQFLPkdN8p06F2VJT148SA0Y9u1BeJHnULMNqpe6MCzCXxvoHG8NnmiMoWOoKYvKVC9fTMcwSwkvMzq0aCq/VW7VdDDRgXeev77hA6vcS+DEX0WkdBH6pgp53BGoIbCLYaOOwlIW7uWy49d0GtLET+fCVd5aohp4PeIIOAJhb0pqKFQhUX9i6ZcvbVc9CdVC+ySGByDpdTslbkYbzFtS1oXnZhFNyVI+y06j5bPYDr7vVVuyP0zlL1LyNGaw4n5I7x3G6rdptRekb6qWxx2BEQLssdmLcAfTgZa4vuH3V7D3ukIu59oRYC8J+1SmCO/pkn44Vd/J+Zo0MWowAuY+xDTboDow3B4rLPcGomfrDU3yWtSUFQuA5bBkU1TC9fivjla7pvKL6tY9FUYYehzZUr/SW/fHWfVdt/Ve231FYBePDeCpo/DkxN+6h43CU4BGvqv5G/ip9ni+I7AlAuovPGHj4UyfukciI99B4cV4a6QLYwLLIV+PFK4kiI+2HhS22oAh8mNaVLy2zJV5IVKeWnyGLIphGFRxV3nGPb6RBc+IpvJHBdZPAKtb6dHCMpN4Afpm+viJI1BD4KaWuFVa7Dy1p6iRSPHO4hsV9ARH4GEiECZO9ZuW58BQYdmkOgnDoPIYR3hPmJAhztl7MbxRFHnY+5ZuNIVnqFdxDAl0YsLES0I9z5Wevcqtc3iGj3zGc95g6rWj2YZYHsMHb29KAR8lWLvSvF68KivKwZuMLA4+c8FeQPbalN4ZcIF3wEfxlKbyU9614+DC21tcr+c6uKZ4b9iUfYn6SjUnR6CPwE0/23MdAUfgShBg6aA1EaVNgK+c9EO+JjImaDaRvlA8GDJxwvtZYXh1OZ6HJWXFh6d8xTFswjesFGLE/E1htpyh88wjAz98OvivOgwICJ6p5edmG2L5UE8hLxg2SpuDEeWNqrJiPSydoTf7ALO2WuEY8qZX1vaF+QX7OqfSmetk9ETneG8CKY6Bxhtdg0FreQqn2pOwetQR2B8BN2z2x9wlOgJbIMDE/WpGxRgvreUYDBaWTdLJDH7IJkF4+DDnYNSE3Lu3HvksA4YDHhomyvLvUcq3G3nTiHozQ0vlbpXWo14bMJYwwjLjSOf8ee3IC6T0YFwpbBkmPVnoSLlRvWQkBN6GX5I8RKfyB8aNIhikpYec6/ha6Rhv5fU4t74bweDV3hcE3LC5L1fS2/FgEdDEw+TMxJkaJC08MDrKieqgOihPPdkEp3TqDPVGOUz0b3RkpDyMCdJYbuJJn8nvD4VMkPzJLcZQtkSj8w+UfgxV2xArQr9SDu2CBsNPskkzLwplWtSTRRkMyincMQJ7MqbykXOQzlwjjMElhPft11oBpbO3hqzSSCUNzx17n57oKL1cs/RVOSdH4CwIuGFzFthdqCOwKgJMrrM3gDYk28TLMkOLjGdkGCUFzIhgLw37cNCN5SaWpvi/OTMmlLQuqW4mfqg0vJAJPoMBojiTPS8noB/HYorywATDbXOSPHDP9iitIBQjpXc97ZqvIMqrcAT2QeDRPmJciiPgCGyIAMsh4ak6TrY9Ua1lBHtqz177LSoyHjMgiuxwylJWmAwV4rnhK8jvKQfj4ivFjzIiCkGtNhhbOVHjeXhlmQvDniy8GQe1afBoKF7DBq+PYVcTP5VfK7NWGgZez3ip6X1Ofddqt9dzjxFww+YeX1xv2oNBAGPBvAZ4SXrERMXElJEmZIwBJrkwWWeZOlH+FzrIh2+0H0V5n8Yy7Jdhosw+66B8lrjwmPQmUWXPol4bMqNGcvk4HsZWtjw1S8odU1VWLJ/tr5EMNhLX2kdaplcsb8FUvvFtEWLwsRfq/aJyPEN4uQajLck/p76JGh51BOoIPKone6oj4AhcGQJM3kxO9gp2S32Mk9ZyRnh9WfWURglLSZSDPtfxTDxmyIRE/fD5fT7HbxPhy6iP5ROin+Wn6UvjvTb8qMqCbpLPBIynaGSILRDYk0V7gkcjtpWvDxtOqQjw7rV7Kj+ta9V41BeDczCIY1v42OOLhrCz6dvQx5MdgQyBm+zMTxwBR+AaEWDyxijhTx6HV3YbDcGzU76dFFhVFuPoI53wNgyTF/ttPtTxHXkKDwrZJEwey0y3pImY4DlPJ288NBhA5MHHEhf7bEI9ip9CzTaoUtqPHDw1yGQpzPTU6WLqycLgA5tgCCpseYXwgrWMBBSayodnM5Le/J8WmHFfgBUG4ec6rxlp6HFWfVHAyRHoIbDLv3v3FPA8R8AR2BcBTVgYLBhBaxgZ+yofpa3VBtXDMh7GSXNv0SmyVBbDjtfgq/VP5Z8F3I7Qa9O30xTPuscI+FLUPb643jRHoIEAnoYpz06j6MUk79mGU2SxxFP1kEUkp/IvBvAr1ffS8HN9dkDADZsdQHYRjsAlIaCnbpaJ2A/CksNV0optwKPSpWNlqRx1P1VYXaKayu8qdYbMa9P3DBC5yAtBwPfYXMiFcDUcgZ0RYE8ObzCdsrF2Z5VH4o5ugyZpjDq8VixFYeSBxRuFVSNEecfIok7KtWgqv1XuXOnXpu+5cHK5Z0bA99ic+QK4eEfgXAjEyR2PQva14XPpc4zcPduwRJZ42VDM3xFU9zFN5R+DxZZlrk3fLbHwui8fgf8BrQ6xmCfDbGQAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\dot{x2}{\\left(t \\right)} = \\frac{- F_{1} l + F_{1} \\cos{\\left(\\theta_{1}{\\left(t \\right)} \\right)} - \\frac{g l \\sin{\\left(2 \\theta_{1}{\\left(t \\right)} \\right)}}{2} - l^{2} \\theta_{2}{\\left(t \\right)} \\sin{\\left(\\theta_{1}{\\left(t \\right)} \\right)}}{l \\left(\\cos^{2}{\\left(\\theta_{1}{\\left(t \\right)} \\right)} - 6\\right)}$"
      ],
      "text/plain": [
       "                                g⋅l⋅sin(2⋅θ₁(t))    2                 \n",
       "        -F₁⋅l + F₁⋅cos(θ₁(t)) - ──────────────── - l ⋅θ₂(t)⋅sin(θ₁(t))\n",
       "                                       2                              \n",
       "x2̇(t) = ──────────────────────────────────────────────────────────────\n",
       "                               ⎛   2           ⎞                      \n",
       "                             l⋅⎝cos (θ₁(t)) - 6⎠                      "
      ]
     },
     "execution_count": 110,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dsolve(eq_x2_dot_subs, x2_dot(t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\theta_{2}{\\left(t \\right)} = \\frac{- F_{1} + l m_{p} \\dot{theta2}{\\left(t \\right)} \\cos{\\left(\\theta_{1}{\\left(t \\right)} \\right)} + m_{c} \\dot{x2}{\\left(t \\right)} + m_{p} \\dot{x2}{\\left(t \\right)}}{l m_{p} \\sin{\\left(\\theta_{1}{\\left(t \\right)} \\right)}}$"
      ],
      "text/plain": [
       "        -F₁ + l⋅mₚ⋅theṫa2(t)⋅cos(θ₁(t)) + m_c⋅x2̇(t) + mₚ⋅x2̇(t)\n",
       "θ₂(t) = ──────────────────────────────────────────────────────\n",
       "                           l⋅mₚ⋅sin(θ₁(t))                    "
      ]
     },
     "execution_count": 111,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq_theta2_f1 = dsolve(eq_f1, theta2(t))\n",
    "eq_theta2_f1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAUCAYAAACwG3xrAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAWElEQVQoFWOsr68vY2BgCAdiEEgFYhMgTgdiARDNAiSEgbi8oaFhD5AGgXNAPAvI7wRxmEAEPjCqABI6o+EACQdQinoLxDOBKegDkAYlORcgBiVBUJLbDQD2sA/lgdqryAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\right]$"
      ],
      "text/plain": [
       "[]"
      ]
     },
     "execution_count": 121,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x, y, z, u, v = symbols('x y z u v')\n",
    "eq_test = Eq(2 * x + 4 * y, z)\n",
    "eq_subs_part = Eq(5.4 * y - 3.3 * v, u)\n",
    "eq_subs = eq_test.subs(y, eq_subs_part)\n",
    "eq_subs\n",
    "eq_subs_solved = solve(eq_subs, z)\n",
    "eq_subs_solved"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[2, 2, 3],\n",
       "       [4, 5, 6]])"
      ]
     },
     "execution_count": 122,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "test = np.array([[1,2,3], [4,5,6]])\n",
    "test2 = np.array([[1,0,0], [0,0,0]])\n",
    "test + test2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.6.9"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}