{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sm\n",
    "import sympy.physics.mechanics as me\n",
    "me.init_vprinting()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem Description\n",
    "\n",
    "![](https://objects-us-east-1.dream.io/mae223/2019f/mae223-l17-example-fig.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Generalized coordinates"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "q1, q2, q3 = me.dynamicsymbols('q1, q2, q3')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Generalized speeds"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "u1, u2, u3 = me.dynamicsymbols('u1, u2, u3')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Specified Inputs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "F, T = me.dynamicsymbols('F, T')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Constants"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( k, \\  c, \\  m_{a}, \\  m_{b}, \\  m_{c}, \\  I_{B_bo}, \\  l, \\  k_{T}, \\  g\\right)$"
      ],
      "text/plain": [
       "(k, c, mₐ, m_b, m_c, I_{B_bo}, l, k_T, g)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k, c, ma, mb, mc, IB_bo, l, kT, g = sm.symbols('k, c, m_a, m_b, m_c, I_{B_bo}, l, k_T, g')\n",
    "k, c, ma, mb, mc, IB_bo, l, kT, g"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Reference Frames"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = me.ReferenceFrame('N')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "B = N.orientnew('B', 'Axis', (q2, N.z))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "C = B.orientnew('C', 'Axis', (q3, N.z))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Points"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "O = me.Point('O')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "Pab = O.locatenew('P_{ab}', q1 * N.x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "Bo = Pab.locatenew('B_o', - 2 * l / 3 * B.y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "Pbc = Pab.locatenew('P_{bc}', -l * B.y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "Pc = Pbc.locatenew('P_c', -l * C.y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -  l\\mathbf{\\hat{c}_y} -  l\\mathbf{\\hat{b}_y} + q_{1}\\mathbf{\\hat{n}_x}$"
      ],
      "text/plain": [
       "-l c_y + -l b_y + q₁ n_x"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Pc.pos_from(O)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Linear Velocities"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\dot{q}_{1}\\mathbf{\\hat{n}_x}$"
      ],
      "text/plain": [
       "q₁̇ n_x"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Pab.set_vel(N, Pab.pos_from(O).dt(N))\n",
    "Pab.vel(N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\dot{q}_{1}\\mathbf{\\hat{n}_x} + \\frac{2 l \\dot{q}_{2}}{3}\\mathbf{\\hat{b}_x}$"
      ],
      "text/plain": [
       "          2⋅l⋅q₂̇\n",
       "q₁̇ n_x + ─────── b_x\n",
       "             3"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Bo.v2pt_theory(Pab, N, B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\dot{q}_{1}\\mathbf{\\hat{n}_x} + l \\dot{q}_{2}\\mathbf{\\hat{b}_x}$"
      ],
      "text/plain": [
       "q₁̇ n_x + l⋅q₂̇ b_x"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Pbc.v2pt_theory(Pab, N, B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\dot{q}_{1}\\mathbf{\\hat{n}_x} + l \\dot{q}_{2}\\mathbf{\\hat{b}_x} + l \\left(\\dot{q}_{2} + \\dot{q}_{3}\\right)\\mathbf{\\hat{c}_x}$"
      ],
      "text/plain": [
       "q₁̇ n_x + l⋅q₂̇ b_x + l⋅(q₂̇ + q₃̇) c_x"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Pc.v2pt_theory(Pbc, N, C)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Kinematic Differential Equations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "u1_eq = sm.Eq(u1, Pab.vel(N).dot(N.x))\n",
    "u2_eq = sm.Eq(u2, Bo.vel(N).dot(B.x))\n",
    "u3_eq = sm.Eq(u3, C.ang_vel_in(B).dot(B.z))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{ \\dot{q}_{1} : u_{1}, \\  \\dot{q}_{2} : \\frac{3 \\left(- u_{1} \\operatorname{cos}\\left(q_{2}\\right) + u_{2}\\right)}{2 l}, \\  \\dot{q}_{3} : u_{3}\\right\\}$"
      ],
      "text/plain": [
       "⎧              3⋅(-u₁⋅cos(q₂) + u₂)         ⎫\n",
       "⎨q₁̇: u₁, q₂̇: ────────────────────, q₃̇: u₃⎬\n",
       "⎩                      2⋅l                  ⎭"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "qdots = sm.solve([u1_eq, u2_eq, u3_eq], q1.diff(), q2.diff(), q3.diff())\n",
    "qdots"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Substitute expressions for the $\\dot{q}$'s."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle u_{1}\\mathbf{\\hat{n}_x}$"
      ],
      "text/plain": [
       "u₁ n_x"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Pab.set_vel(N, Pab.vel(N).subs(qdots).simplify())\n",
    "Pab.vel(N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle u_{2}\\mathbf{\\hat{b}_x} -  u_{1} \\operatorname{sin}\\left(q_{2}\\right)\\mathbf{\\hat{b}_y}$"
      ],
      "text/plain": [
       "u₂ b_x + -u₁⋅sin(q₂) b_y"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Bo.set_vel(N, Bo.vel(N).subs(qdots).express(B).simplify())\n",
    "Bo.vel(N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle u_{1}\\mathbf{\\hat{n}_x} + (- \\frac{3 u_{1} \\operatorname{cos}\\left(q_{2}\\right)}{2} + \\frac{3 u_{2}}{2})\\mathbf{\\hat{b}_x} + (l u_{3} - \\frac{3 u_{1} \\operatorname{cos}\\left(q_{2}\\right)}{2} + \\frac{3 u_{2}}{2})\\mathbf{\\hat{c}_x}$"
      ],
      "text/plain": [
       "         ⎛  3⋅u₁⋅cos(q₂)   3⋅u₂⎞       ⎛       3⋅u₁⋅cos(q₂)   3⋅u₂⎞\n",
       "u₁ n_x + ⎜- ──────────── + ────⎟ b_x + ⎜l⋅u₃ - ──────────── + ────⎟ c_x\n",
       "         ⎝       2          2  ⎠       ⎝            2          2  ⎠"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Pc.set_vel(N, Pc.vel(N).subs(qdots).simplify())\n",
    "Pc.vel(N)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Angular Velocities"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{3 \\left(- u_{1} \\operatorname{cos}\\left(q_{2}\\right) + u_{2}\\right)}{2 l}\\mathbf{\\hat{n}_z}$"
      ],
      "text/plain": [
       "3⋅(-u₁⋅cos(q₂) + u₂)\n",
       "──────────────────── n_z\n",
       "        2⋅l"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B.set_ang_vel(N, B.ang_vel_in(N).subs(qdots).simplify())\n",
    "B.ang_vel_in(N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "C.set_ang_vel(B, u3 * N.z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Partial Velocities"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Each partial velocity can be calculated manually like so:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\mathbf{\\hat{n}_x}$"
      ],
      "text/plain": [
       "n_x"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Pab.vel(N).diff(u1, N)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Or you can use this convenience function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\mathbf{\\hat{n}_x}$"
      ],
      "text/plain": [
       "n_x"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Pab.partial_velocity(N, u1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "All partial velocities can be calculated at once with:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\mathbf{\\hat{n}_x}, \\  0, \\  0\\right)$"
      ],
      "text/plain": [
       "(n_x, 0, 0)"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Pab.partial_velocity(N, u1, u2, u3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( -  \\operatorname{sin}\\left(q_{2}\\right)\\mathbf{\\hat{b}_y}, \\  \\mathbf{\\hat{b}_x}, \\  0\\right)$"
      ],
      "text/plain": [
       "(-sin(q₂) b_y, b_x, 0)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Bo.partial_velocity(N, u1, u2, u3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\mathbf{\\hat{n}_x} -  \\frac{3 \\operatorname{cos}\\left(q_{2}\\right)}{2}\\mathbf{\\hat{b}_x} -  \\frac{3 \\operatorname{cos}\\left(q_{2}\\right)}{2}\\mathbf{\\hat{c}_x}, \\  \\frac{3}{2}\\mathbf{\\hat{b}_x} + \\frac{3}{2}\\mathbf{\\hat{c}_x}, \\  l\\mathbf{\\hat{c}_x}\\right)$"
      ],
      "text/plain": [
       "⎛         -3⋅cos(q₂)        -3⋅cos(q₂)  , 3/2 b_x + 3/2 c_x, l c_x⎞\n",
       "⎜n_x + ─────────── b_x + ─────────── c_x                          ⎟\n",
       "⎝                2                 2                              ⎠"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Pc.partial_velocity(N, u1, u2, u3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This also works for angular partial velocities:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( -  \\frac{3 \\operatorname{cos}\\left(q_{2}\\right)}{2 l}\\mathbf{\\hat{n}_z}, \\  \\frac{3}{2 l}\\mathbf{\\hat{n}_z}, \\  0\\right)$"
      ],
      "text/plain": [
       "⎛   -3⋅cos(q₂)  ,     3  , 0⎞\n",
       "⎜─────────── n_z  ─── n_z   ⎟\n",
       "⎝        2⋅l        2⋅l     ⎠"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B.partial_velocity(N, u1, u2, u3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The convenience function is doing this behind the scenes:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ -  \\frac{3 \\operatorname{cos}\\left(q_{2}\\right)}{2 l}\\mathbf{\\hat{n}_z}, \\  \\frac{3}{2 l}\\mathbf{\\hat{n}_z}, \\  0\\right]$"
      ],
      "text/plain": [
       "⎡   -3⋅cos(q₂)  ,     3  , 0⎤\n",
       "⎢─────────── n_z  ─── n_z   ⎥\n",
       "⎣        2⋅l        2⋅l     ⎦"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[B.ang_vel_in(N).diff(ur, N) for ur in [u1, u2, u3]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Angular Accelerations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -  \\frac{6 l \\left(\\operatorname{cos}\\left(q_{2}\\right) \\dot{u}_{1} - \\dot{u}_{2}\\right) + 9 \\left(u_{1} \\operatorname{cos}\\left(q_{2}\\right) - u_{2}\\right) u_{1} \\operatorname{sin}\\left(q_{2}\\right)}{4 l^{2}}\\mathbf{\\hat{n}_z}$"
      ],
      "text/plain": [
       "-(6⋅l⋅(cos(q₂)⋅u₁̇ - u₂̇) + 9⋅(u₁⋅cos(q₂) - u₂)⋅u₁⋅sin(q₂))\n",
       "──────────────────────────────────────────────────────────── n_z\n",
       "                               2\n",
       "                            4⋅l"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "alpha_B_N = B.ang_acc_in(N).subs(qdots).simplify()\n",
    "alpha_B_N"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\dot{u}_{1}\\mathbf{\\hat{n}_x}$"
      ],
      "text/plain": [
       "u₁̇ n_x"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a_Pab_N = Pab.acc(N).subs(qdots).simplify()\n",
    "a_Pab_N"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\frac{l \\dot{u}_{2} - \\frac{3 \\left(u_{1} \\operatorname{cos}\\left(q_{2}\\right) - u_{2}\\right) u_{1} \\operatorname{sin}\\left(q_{2}\\right)}{2}}{l}\\mathbf{\\hat{b}_x} -  \\frac{l \\left(u_{1} \\operatorname{cos}\\left(q_{2}\\right) \\dot{q}_{2} + \\operatorname{sin}\\left(q_{2}\\right) \\dot{u}_{1}\\right) + \\frac{3 \\left(u_{1} \\operatorname{cos}\\left(q_{2}\\right) - u_{2}\\right) u_{2}}{2}}{l}\\mathbf{\\hat{b}_y}$"
      ],
      "text/plain": [
       "        3⋅(u₁⋅cos(q₂) - u₂)⋅u₁⋅sin(q₂)        ⎛\n",
       "l⋅u₂̇ - ──────────────────────────────       -⎜l⋅(u₁⋅cos(q₂)⋅q₂̇ + sin(q₂)⋅u₁̇\n",
       "                      2                       ⎝\n",
       "────────────────────────────────────── b_x + ─────────────────────────────────\n",
       "                  l                                                        l\n",
       "\n",
       "    3⋅(u₁⋅cos(q₂) - u₂)⋅u₂⎞\n",
       ") + ──────────────────────⎟\n",
       "              2           ⎠\n",
       "──────────────────────────── b_y\n"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a_Bo_N = Bo.acc(N).express(B).simplify()\n",
    "a_Bo_N"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\frac{2 l \\left(2 l \\dot{u}_{3} + 2 \\operatorname{cos}\\left(q_{2} + q_{3}\\right) \\dot{u}_{1} - 3 \\operatorname{cos}\\left(q_{2}\\right) \\dot{u}_{1} + 3 \\dot{u}_{2}\\right) - 3 \\left(2 l \\left(\\operatorname{cos}\\left(q_{2}\\right) \\dot{u}_{1} - \\dot{u}_{2}\\right) + 3 \\left(u_{1} \\operatorname{cos}\\left(q_{2}\\right) - u_{2}\\right) u_{1} \\operatorname{sin}\\left(q_{2}\\right)\\right) \\operatorname{cos}\\left(q_{3}\\right) + 9 \\left(u_{1} \\operatorname{cos}\\left(q_{2}\\right) - u_{2}\\right)^{2} \\operatorname{sin}\\left(q_{3}\\right) - 9 \\left(u_{1} \\operatorname{cos}\\left(q_{2}\\right) - u_{2}\\right) u_{1} \\operatorname{sin}\\left(q_{2}\\right)}{4 l}\\mathbf{\\hat{c}_x} + \\frac{- 4 l \\operatorname{sin}\\left(q_{2} + q_{3}\\right) \\dot{u}_{1} + 3 \\left(2 l \\left(\\operatorname{cos}\\left(q_{2}\\right) \\dot{u}_{1} - \\dot{u}_{2}\\right) + 3 \\left(u_{1} \\operatorname{cos}\\left(q_{2}\\right) - u_{2}\\right) u_{1} \\operatorname{sin}\\left(q_{2}\\right)\\right) \\operatorname{sin}\\left(q_{3}\\right) + 9 \\left(u_{1} \\operatorname{cos}\\left(q_{2}\\right) - u_{2}\\right)^{2} \\operatorname{cos}\\left(q_{3}\\right) + \\left(2 l u_{3} - 3 u_{1} \\operatorname{cos}\\left(q_{2}\\right) + 3 u_{2}\\right)^{2}}{4 l}\\mathbf{\\hat{c}_y}$"
      ],
      "text/plain": [
       "\n",
       "2⋅l⋅(2⋅l⋅u₃̇ + 2⋅cos(q₂ + q₃)⋅u₁̇ - 3⋅cos(q₂)⋅u₁̇ + 3⋅u₂̇) - 3⋅(2⋅l⋅(cos(q₂)⋅u\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "\n",
       "\n",
       "                                                                         2\n",
       "₁̇ - u₂̇) + 3⋅(u₁⋅cos(q₂) - u₂)⋅u₁⋅sin(q₂))⋅cos(q₃) + 9⋅(u₁⋅cos(q₂) - u₂) ⋅sin\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                 4⋅l\n",
       "\n",
       "\n",
       "(q₃) - 9⋅(u₁⋅cos(q₂) - u₂)⋅u₁⋅sin(q₂)       -4⋅l⋅sin(q₂ + q₃)⋅u₁̇ + 3⋅(2⋅l⋅(co\n",
       "───────────────────────────────────── c_x + ──────────────────────────────────\n",
       "\n",
       "\n",
       "\n",
       "s(q₂)⋅u₁̇ - u₂̇) + 3⋅(u₁⋅cos(q₂) - u₂)⋅u₁⋅sin(q₂))⋅sin(q₃) + 9⋅(u₁⋅cos(q₂) - u\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                           4⋅l\n",
       "\n",
       "  2                                         2\n",
       "₂) ⋅cos(q₃) + (2⋅l⋅u₃ - 3⋅u₁⋅cos(q₂) + 3⋅u₂)\n",
       "───────────────────────────────────────────── c_y\n"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a_Pc_N = Pc.acc(N).express(C).subs(qdots).simplify()\n",
    "a_Pc_N"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Mass and Inertia"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( m_{a}, \\  m_{c}\\right)$"
      ],
      "text/plain": [
       "(mₐ, m_c)"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ma, mc"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGwAAAAbCAYAAACZQWq0AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAEwUlEQVRoBeWaj3FUNxCHfRkXwJAOXIKhgzgdQAukA1MD6YB0kHE6IB0Q6ACowMEdmO8T2jd6ks7vne/d+W5OM0LSarX721392XdmdX9/f3ZMZbVaPQPz3TFhLrFui/+XUtih9zH2AoyfaK3PDh1vjW8J/EcTMIy9MlhUT5fB+pYdQPfwy2L4vRIPvRIOA+TdfR1Y6b+jfo/xIbfgXAz/SkOPobBDm7erRztUW3pYe7Qp/EcTsClDTmX+aN6wUwnIlJ0HHTCujEvqfVHfTBl0SPO7wN8EDCXX1JvCSabQ76kX+3YG7+tndL6gmhkeXdkF/vPaCyj5U5oBo/nMWIc9WdFosHwFwOWTgdhC8dL4m4CJDQf5zWP592dzev/iA1Nxr+BfK+tvY1NX9L0MuwFD8+9Z+4e9oDgwJXnDvgbWW4Izuo4NJPWaOW+fvW/odQFLJ+wpAE3ETmf5wRw3gOz/URvHOvGYgnzfanW8roOlPGnw/EXXd95vw3+kzyzb40dh80sHyn2/PvTmnoIGFn+SEpP1TWCg77sm7Tv1IujbtMh573rbnkxoXpUlj0FrfFjS4F8Mf6MI4e5eEQ8/A1XKnfeqlEej3I22N1Tpiziu0hkGJ0dVc+oXy6eS/pg+MnyzBvz0R0FjPARL+Xnc9VOpH77F8PcCFg64LJWWfQBoWPM7HjTXNvRy7WP6yAyD39XrmXtF1XvWwdk135wx63sbIgWNuVGwQl5vTcxFC89i+JvvMITH++U30Lpiqt97cD9C957WuH0VU/4o26b+dyEoWpz+B/23VDeL/bo0a2qGifFG+HsB0+heMEq9BtXg1MXs0uxpWyNqufsaNxut3Hz0ez8eNGt2Cfa8FA6gdLqgrU3nswECHwUVuqnuS+pv1NmFdV6jt9T0vUOw3c2blOcFc7oVkJkeGOiO/6eGXWZ+s7O6bOtwssRK9dosT0Wh/lHdjfCP3jDUzXm/NF6HGCDfMlsDfIMhtTx5zeC67yF07/arWEffxGXIAgt6vAG9NyYwp6yW9e744T2jP5oPmb0WXvEmPLTKWacvvZXMexs1eGvZ8CyGv3bwF4Sjb+z4csy8DmgyMmgG7kvJa5/STUKgp8yy5Iem7F7gw+BR8OHVYdKGtJ7+4Gj6kZBo12T6nfFGktEEK7AiS5zqbrAGT9nCtxh+ZCUD3dkhVC87bjKybJCnqZmD5jXp2lcBNtOa4GY5OnngnZAtNh2ZgkErPnFIb7BkWYEnnTb4XDsZtMyn3Ad5mZdnVlaaeRfBPzph4eiHWpSPghK80OOqHIyA5g7XME+fgIerkX5yZLFeh0obeGJu0xYZykq3BW3aFLSeislvJnVRYmOMNlSe06Z0CjfFNZc/6+/i3yhgCIpdO9p9me6JGe12xhqW7vjMM/x6wthADsGhv/ZUzzU0+JDl6dPzCQ9tBLAJQKzptazz2jPQZR3e3N6aJWjoW4v/nMlZxQwJRneXxWzpZ+/s7DkdHWIGNsocob2kus6i8WXxx1XlfKQ1Q/yb9bMzuFJQ2UdeShwy7Ypx2hh5fFfyTvXBY5aZMs8p3qXmp/Dv9P906CyM9iPbP9l4gnzI66AuZetJyNl1wNztfkzfUr8ucYJOIioPGPkDd69brAR9NrIAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle I_{B_bo}\\mathbf{\\hat{b}_z}\\otimes \\mathbf{\\hat{b}_z}$"
      ],
      "text/plain": [
       "I_{B_bo} b_z⊗b_z"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "IB = me.inertia(B, 0, 0, IB_bo)\n",
    "IB"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Loads (forces and torques)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle (- c u_{1} - k q_{1} + F)\\mathbf{\\hat{n}_x}$"
      ],
      "text/plain": [
       "(-c⋅u₁ - k⋅q₁ + F) n_x"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rab = (F - k*q1 - c*qdots[q1.diff()]) * N.x\n",
    "Rab"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -  g m_{b}\\mathbf{\\hat{n}_y}$"
      ],
      "text/plain": [
       "-g⋅m_b n_y"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rbo = -(mb*g)*N.y\n",
    "Rbo"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -  g m_{c}\\mathbf{\\hat{n}_y}$"
      ],
      "text/plain": [
       "-g⋅m_c n_y"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rc = -(mc*g)*N.y\n",
    "Rc"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle (k_{T} q_{3} + T)\\mathbf{\\hat{n}_z}$"
      ],
      "text/plain": [
       "(k_T⋅q₃ + T) n_z"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TB = (T + kT*q3)*N.z\n",
    "TB"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Generalizd Active Forces"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - c u_{1} - k q_{1} + F$"
      ],
      "text/plain": [
       "-c⋅u₁ - k⋅q₁ + F"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Pab.partial_velocity(N, u1).dot(Rab)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOCAYAAAAWo42rAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAzklEQVQoFW1SiRGCMBAkGQtgtAPswKcCKYEa7MESLMWhBGkBOtAOEDuA3XDL3Cg3c9lkby+bEMI4jpkihHDHvEd+kXvkA/UOmGUUmrjFsnLrHOsXskh1E11BDBIJwdGh9cIniFoCIbgKybPlEQOjnOFv5FkZZcQFeBbGZ4bVseCOWyup2yuXZlmzqJ29UPMdhUuXWIdy6yNuKEuRTre4vGXdoLpmreZGwhrCk9/K5nzGLrm6j8vn+n3CAdyBmo11Eo7IG75rYdwZeIEo/RQTNRRqzYwqZmYAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Pab.partial_velocity(N, u2).dot(Rab)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - g m_{b} \\operatorname{sin}\\left(q_{2}\\right) + \\frac{3 \\left(k_{T} q_{3} + T\\right)}{2 l}$"
      ],
      "text/plain": [
       "                 3⋅(k_T⋅q₃ + T)\n",
       "-g⋅m_b⋅sin(q₂) + ──────────────\n",
       "                      2⋅l      "
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Bo.partial_velocity(N, u2).dot(Rbo) + B.partial_velocity(N, u2).dot(TB)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [],
   "source": [
    "u = [u1, u2, u3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[ - c u_{1} + g m_{b} \\operatorname{sin}\\left(q_{2}\\right) \\operatorname{cos}\\left(q_{2}\\right) + \\frac{3 g m_{c} \\left(\\operatorname{sin}\\left(q_{2}\\right) \\operatorname{cos}\\left(q_{3}\\right) + \\operatorname{sin}\\left(q_{3}\\right) \\operatorname{cos}\\left(q_{2}\\right)\\right) \\operatorname{cos}\\left(q_{2}\\right)}{2} + \\frac{3 g m_{c} \\operatorname{sin}\\left(q_{2}\\right) \\operatorname{cos}\\left(q_{2}\\right)}{2} - k q_{1} + F - \\frac{3 \\left(k_{T} q_{3} + T\\right) \\operatorname{cos}\\left(q_{2}\\right)}{2 l}, \\  - g m_{b} \\operatorname{sin}\\left(q_{2}\\right) - \\frac{3 g m_{c} \\left(\\operatorname{sin}\\left(q_{2}\\right) \\operatorname{cos}\\left(q_{3}\\right) + \\operatorname{sin}\\left(q_{3}\\right) \\operatorname{cos}\\left(q_{2}\\right)\\right)}{2} - \\frac{3 g m_{c} \\operatorname{sin}\\left(q_{2}\\right)}{2} + \\frac{3 \\left(k_{T} q_{3} + T\\right)}{2 l}, \\  - g l m_{c} \\left(\\operatorname{sin}\\left(q_{2}\\right) \\operatorname{cos}\\left(q_{3}\\right) + \\operatorname{sin}\\left(q_{3}\\right) \\operatorname{cos}\\left(q_{2}\\right)\\right)\\right]$"
      ],
      "text/plain": [
       "⎡                                3⋅g⋅m_c⋅(sin(q₂)⋅cos(q₃) + sin(q₃)⋅cos(q₂))⋅c\n",
       "⎢-c⋅u₁ + g⋅m_b⋅sin(q₂)⋅cos(q₂) + ─────────────────────────────────────────────\n",
       "⎣                                                         2                   \n",
       "\n",
       "os(q₂)   3⋅g⋅m_c⋅sin(q₂)⋅cos(q₂)              3⋅(k_T⋅q₃ + T)⋅cos(q₂)          \n",
       "────── + ─────────────────────── - k⋅q₁ + F - ──────────────────────, -g⋅m_b⋅s\n",
       "                    2                                  2⋅l                    \n",
       "\n",
       "         3⋅g⋅m_c⋅(sin(q₂)⋅cos(q₃) + sin(q₃)⋅cos(q₂))   3⋅g⋅m_c⋅sin(q₂)   3⋅(k_\n",
       "in(q₂) - ─────────────────────────────────────────── - ─────────────── + ─────\n",
       "                              2                               2               \n",
       "\n",
       "T⋅q₃ + T)                                              ⎤\n",
       "─────────, -g⋅l⋅m_c⋅(sin(q₂)⋅cos(q₃) + sin(q₃)⋅cos(q₂))⎥\n",
       "2⋅l                                                    ⎦"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fr = []\n",
    "\n",
    "for ur in u:\n",
    "    Fr.append(Pab.partial_velocity(N, ur).dot(Rab) + # particle\n",
    "              Pc.partial_velocity(N, ur).dot(Rc) + # particle\n",
    "              Bo.partial_velocity(N, ur).dot(Rbo) + B.partial_velocity(N, ur).dot(TB)) # rigid body\n",
    "    \n",
    "Fr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- c u_{1} + \\frac{g m_{b} \\operatorname{sin}\\left(2 q_{2}\\right)}{2} + \\frac{3 g m_{c} \\operatorname{sin}\\left(q_{2} + q_{3}\\right) \\operatorname{cos}\\left(q_{2}\\right)}{2} + \\frac{3 g m_{c} \\operatorname{sin}\\left(2 q_{2}\\right)}{4} - k q_{1} + F - \\frac{3 \\left(k_{T} q_{3} + T\\right) \\operatorname{cos}\\left(q_{2}\\right)}{2 l}\\\\- g m_{b} \\operatorname{sin}\\left(q_{2}\\right) - \\frac{3 g m_{c} \\operatorname{sin}\\left(q_{2} + q_{3}\\right)}{2} - \\frac{3 g m_{c} \\operatorname{sin}\\left(q_{2}\\right)}{2} + \\frac{3 \\left(k_{T} q_{3} + T\\right)}{2 l}\\\\- g l m_{c} \\operatorname{sin}\\left(q_{2} + q_{3}\\right)\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡        g⋅m_b⋅sin(2⋅q₂)   3⋅g⋅m_c⋅sin(q₂ + q₃)⋅cos(q₂)   3⋅g⋅m_c⋅sin(2⋅q₂)   \n",
       "⎢-c⋅u₁ + ─────────────── + ──────────────────────────── + ───────────────── - \n",
       "⎢               2                       2                         4           \n",
       "⎢                                                                             \n",
       "⎢                                    3⋅g⋅m_c⋅sin(q₂ + q₃)   3⋅g⋅m_c⋅sin(q₂)   \n",
       "⎢                   -g⋅m_b⋅sin(q₂) - ──────────────────── - ─────────────── + \n",
       "⎢                                             2                    2          \n",
       "⎢                                                                             \n",
       "⎣                                            -g⋅l⋅m_c⋅sin(q₂ + q₃)            \n",
       "\n",
       "           3⋅(k_T⋅q₃ + T)⋅cos(q₂)⎤\n",
       "k⋅q₁ + F - ──────────────────────⎥\n",
       "                    2⋅l          ⎥\n",
       "                                 ⎥\n",
       "3⋅(k_T⋅q₃ + T)                   ⎥\n",
       "──────────────                   ⎥\n",
       "     2⋅l                         ⎥\n",
       "                                 ⎥\n",
       "                                 ⎦"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fr = sm.trigsimp(sm.Matrix(Fr))\n",
    "Fr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Generalized Inertia Forces"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [],
   "source": [
    "Frstar = []\n",
    "\n",
    "for ur in u:\n",
    "    Frstar.append(Pab.partial_velocity(N, ur).dot(-ma*a_Pab_N) +\n",
    "                  Pc.partial_velocity(N, ur).dot(-mc*a_Pc_N) +\n",
    "                  Bo.partial_velocity(N, ur).dot(-mb*a_Bo_N) +\n",
    "                  B.partial_velocity(N, ur).dot(-alpha_B_N.dot(IB) -\n",
    "                                                B.ang_vel_in(N).cross(IB.dot(B.ang_vel_in(N)))))\n",
    "    \n",
    "Frstar = sm.Matrix(Frstar)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "Frstar = sm.trigsimp(Frstar)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\frac{9 I_{B_bo} \\operatorname{cos}^{2}\\left(q_{2}\\right) \\dot{u}_{1}}{4 l^{2}} + \\frac{9 I_{B_bo} \\operatorname{cos}\\left(q_{2}\\right) \\dot{u}_{2}}{4 l^{2}} + \\frac{27 I_{B_bo} u^{2}_{1} \\operatorname{sin}^{3}\\left(q_{2}\\right)}{8 l^{3}} - \\frac{27 I_{B_bo} u^{2}_{1} \\operatorname{sin}\\left(q_{2}\\right)}{8 l^{3}} + \\frac{27 I_{B_bo} u_{1} u_{2} \\operatorname{sin}\\left(q_{2}\\right) \\operatorname{cos}\\left(q_{2}\\right)}{8 l^{3}} + l m_{c} u^{2}_{3} \\operatorname{sin}\\left(q_{2}\\right) \\operatorname{cos}\\left(q_{3}\\right) - \\frac{l m_{c} u^{2}_{3} \\operatorname{sin}\\left(q_{3}\\right) \\operatorname{cos}\\left(q_{2}\\right)}{2} + l m_{c} \\operatorname{sin}\\left(q_{2}\\right) \\operatorname{sin}\\left(q_{3}\\right) \\dot{u}_{3} + \\frac{l m_{c} \\operatorname{cos}\\left(q_{2}\\right) \\operatorname{cos}\\left(q_{3}\\right) \\dot{u}_{3}}{2} + \\frac{3 l m_{c} \\operatorname{cos}\\left(q_{2}\\right) \\dot{u}_{3}}{2} - m_{a} \\dot{u}_{1} - m_{b} u_{1} \\operatorname{sin}\\left(q_{2}\\right) \\operatorname{cos}\\left(q_{2}\\right) \\dot{q}_{2} + m_{b} \\operatorname{cos}^{2}\\left(q_{2}\\right) \\dot{u}_{1} - m_{b} \\dot{u}_{1} - 3 m_{c} u_{1} u_{3} \\operatorname{sin}\\left(q_{2} + q_{3}\\right) \\operatorname{cos}\\left(q_{2}\\right) + \\frac{9 m_{c} u_{1} u_{3} \\operatorname{sin}\\left(q_{3}\\right) \\operatorname{cos}^{2}\\left(q_{2}\\right)}{2} + 3 m_{c} u_{2} u_{3} \\operatorname{sin}\\left(q_{2}\\right) \\operatorname{cos}\\left(q_{3}\\right) - \\frac{3 m_{c} u_{2} u_{3} \\operatorname{sin}\\left(q_{3}\\right) \\operatorname{cos}\\left(q_{2}\\right)}{2} + \\frac{3 m_{c} \\operatorname{sin}\\left(q_{2}\\right) \\operatorname{sin}\\left(q_{3}\\right) \\dot{u}_{2}}{2} + 3 m_{c} \\operatorname{cos}\\left(q_{2} + q_{3}\\right) \\operatorname{cos}\\left(q_{2}\\right) \\dot{u}_{1} - \\frac{9 m_{c} \\operatorname{cos}^{2}\\left(q_{2}\\right) \\operatorname{cos}\\left(q_{3}\\right) \\dot{u}_{1}}{2} - \\frac{3 m_{c} \\operatorname{cos}^{2}\\left(q_{2}\\right) \\dot{u}_{1}}{2} + 3 m_{c} \\operatorname{cos}\\left(q_{2}\\right) \\operatorname{cos}\\left(q_{3}\\right) \\dot{u}_{2} + 3 m_{c} \\operatorname{cos}\\left(q_{2}\\right) \\dot{u}_{2} - m_{c} \\dot{u}_{1} - \\frac{3 m_{b} u_{1} u_{2} \\operatorname{sin}\\left(q_{2}\\right) \\operatorname{cos}\\left(q_{2}\\right)}{2 l} + \\frac{3 m_{b} u^{2}_{2} \\operatorname{sin}\\left(q_{2}\\right)}{2 l} - \\frac{9 m_{c} u^{2}_{1} \\operatorname{sin}\\left(q_{2} - q_{3}\\right) \\operatorname{cos}^{2}\\left(q_{2}\\right)}{8 l} + \\frac{9 m_{c} u^{2}_{1} \\operatorname{sin}\\left(q_{2} - q_{3}\\right)}{8 l} - \\frac{9 m_{c} u^{2}_{1} \\operatorname{sin}\\left(q_{2} + q_{3}\\right)}{8 l} + \\frac{9 m_{c} u^{2}_{1} \\operatorname{sin}^{3}\\left(q_{2}\\right)}{2 l} - \\frac{9 m_{c} u^{2}_{1} \\operatorname{sin}\\left(q_{2}\\right) \\operatorname{cos}^{2}\\left(q_{2}\\right) \\operatorname{cos}\\left(q_{3}\\right)}{8 l} + \\frac{9 m_{c} u^{2}_{1} \\operatorname{sin}\\left(q_{2}\\right) \\operatorname{cos}^{2}\\left(q_{2}\\right)}{4 l} - \\frac{9 m_{c} u^{2}_{1} \\operatorname{sin}\\left(q_{2}\\right)}{2 l} + \\frac{27 m_{c} u^{2}_{1} \\operatorname{sin}\\left(q_{3}\\right) \\operatorname{cos}^{3}\\left(q_{2}\\right)}{8 l} - \\frac{27 m_{c} u_{1} u_{2} \\operatorname{sin}\\left(q_{3}\\right) \\operatorname{cos}^{2}\\left(q_{2}\\right)}{4 l} + \\frac{9 m_{c} u_{1} u_{2} \\operatorname{sin}\\left(q_{3}\\right)}{4 l} + \\frac{9 m_{c} u^{2}_{2} \\operatorname{sin}\\left(q_{2} + q_{3}\\right)}{4 l} + \\frac{9 m_{c} u^{2}_{2} \\operatorname{sin}\\left(q_{2}\\right)}{4 l}\\\\\\frac{9 I_{B_bo} \\operatorname{cos}\\left(q_{2}\\right) \\dot{u}_{1}}{4 l^{2}} - \\frac{9 I_{B_bo} \\dot{u}_{2}}{4 l^{2}} + \\frac{27 I_{B_bo} u^{2}_{1} \\operatorname{sin}\\left(2 q_{2}\\right)}{16 l^{3}} - \\frac{27 I_{B_bo} u_{1} u_{2} \\operatorname{sin}\\left(q_{2}\\right)}{8 l^{3}} + \\frac{3 l m_{c} u^{2}_{3} \\operatorname{sin}\\left(q_{3}\\right)}{2} - \\frac{3 l m_{c} \\operatorname{cos}\\left(q_{3}\\right) \\dot{u}_{3}}{2} - \\frac{3 l m_{c} \\dot{u}_{3}}{2} - m_{b} \\dot{u}_{2} + \\frac{9 m_{c} u_{1} u_{3} \\operatorname{sin}\\left(q_{2} - q_{3}\\right)}{4} - \\frac{9 m_{c} u_{1} u_{3} \\operatorname{sin}\\left(q_{2} + q_{3}\\right)}{4} + \\frac{9 m_{c} u_{2} u_{3} \\operatorname{sin}\\left(q_{3}\\right)}{2} + \\frac{9 m_{c} \\operatorname{cos}\\left(q_{2} - q_{3}\\right) \\dot{u}_{1}}{4} + \\frac{3 m_{c} \\operatorname{cos}\\left(q_{2} + q_{3}\\right) \\dot{u}_{1}}{4} + 3 m_{c} \\operatorname{cos}\\left(q_{2}\\right) \\dot{u}_{1} - \\frac{9 m_{c} \\operatorname{cos}\\left(q_{3}\\right) \\dot{u}_{2}}{2} - \\frac{9 m_{c} \\dot{u}_{2}}{2} + \\frac{3 m_{b} u^{2}_{1} \\operatorname{sin}\\left(2 q_{2}\\right)}{4 l} - \\frac{3 m_{b} u_{1} u_{2} \\operatorname{sin}\\left(q_{2}\\right)}{2 l} + \\frac{27 m_{c} u^{2}_{1} \\operatorname{sin}\\left(2 q_{2} - q_{3}\\right)}{16 l} + \\frac{27 m_{c} u^{2}_{1} \\operatorname{sin}\\left(2 q_{2} + q_{3}\\right)}{16 l} + \\frac{27 m_{c} u^{2}_{1} \\operatorname{sin}\\left(2 q_{2}\\right)}{8 l} - \\frac{27 m_{c} u_{1} u_{2} \\operatorname{sin}\\left(q_{2} - q_{3}\\right)}{8 l} - \\frac{27 m_{c} u_{1} u_{2} \\operatorname{sin}\\left(q_{2} + q_{3}\\right)}{8 l} - \\frac{27 m_{c} u_{1} u_{2} \\operatorname{sin}\\left(q_{2}\\right)}{4 l}\\\\- \\frac{m_{c} \\left(2 l \\left(2 l \\dot{u}_{3} + 2 \\operatorname{cos}\\left(q_{2} + q_{3}\\right) \\dot{u}_{1} - 3 \\operatorname{cos}\\left(q_{2}\\right) \\dot{u}_{1} + 3 \\dot{u}_{2}\\right) - 3 \\left(2 l \\left(\\operatorname{cos}\\left(q_{2}\\right) \\dot{u}_{1} - \\dot{u}_{2}\\right) + 3 \\left(u_{1} \\operatorname{cos}\\left(q_{2}\\right) - u_{2}\\right) u_{1} \\operatorname{sin}\\left(q_{2}\\right)\\right) \\operatorname{cos}\\left(q_{3}\\right) + 9 \\left(u_{1} \\operatorname{cos}\\left(q_{2}\\right) - u_{2}\\right)^{2} \\operatorname{sin}\\left(q_{3}\\right) - 9 \\left(u_{1} \\operatorname{cos}\\left(q_{2}\\right) - u_{2}\\right) u_{1} \\operatorname{sin}\\left(q_{2}\\right)\\right)}{4}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡                2                                                  2    3    \n",
       "⎢  9⋅I_{B_bo}⋅cos (q₂)⋅u₁̇   9⋅I_{B_bo}⋅cos(q₂)⋅u₂̇   27⋅I_{B_bo}⋅u₁ ⋅sin (q₂)\n",
       "⎢- ─────────────────────── + ────────────────────── + ────────────────────────\n",
       "⎢               2                        2                         3          \n",
       "⎢            4⋅l                      4⋅l                       8⋅l           \n",
       "⎢                                                                             \n",
       "⎢                                                                             \n",
       "⎢                                                                             \n",
       "⎢                                                                             \n",
       "⎢                                                                             \n",
       "⎢                                                                             \n",
       "⎢                                                                             \n",
       "⎢                                                                             \n",
       "⎢                                                                             \n",
       "⎢                                                                             \n",
       "⎣                                                                             \n",
       "\n",
       "                 2                                                            \n",
       "   27⋅I_{B_bo}⋅u₁ ⋅sin(q₂)   27⋅I_{B_bo}⋅u₁⋅u₂⋅sin(q₂)⋅cos(q₂)           2    \n",
       " - ─────────────────────── + ───────────────────────────────── + l⋅m_c⋅u₃ ⋅sin\n",
       "                3                              3                              \n",
       "             8⋅l                            8⋅l                               \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                       2                                                      \n",
       "               l⋅m_c⋅u₃ ⋅sin(q₃)⋅cos(q₂)                               l⋅m_c⋅c\n",
       "(q₂)⋅cos(q₃) - ───────────────────────── + l⋅m_c⋅sin(q₂)⋅sin(q₃)⋅u₃̇ + ───────\n",
       "                           2                                                  \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                            9⋅\n",
       "                                                                            ──\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "os(q₂)⋅cos(q₃)⋅u₃̇   3⋅l⋅m_c⋅cos(q₂)⋅u₃̇                                      \n",
       "────────────────── + ─────────────────── - mₐ⋅u₁̇ - m_b⋅u₁⋅sin(q₂)⋅cos(q₂)⋅q₂̇\n",
       "     2                        2                                               \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                      2                       \n",
       "I_{B_bo}⋅cos(q₂)⋅u₁̇   9⋅I_{B_bo}⋅u₂̇   27⋅I_{B_bo}⋅u₁ ⋅sin(2⋅q₂)   27⋅I_{B_bo\n",
       "──────────────────── - ────────────── + ───────────────────────── - ──────────\n",
       "          2                    2                      3                       \n",
       "       4⋅l                  4⋅l                   16⋅l                        \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "          2                                                        9⋅m_c⋅u₁⋅u₃\n",
       " + m_b⋅cos (q₂)⋅u₁̇ - m_b⋅u₁̇ - 3⋅m_c⋅u₁⋅u₃⋅sin(q₂ + q₃)⋅cos(q₂) + ───────────\n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                            2                                                 \n",
       "}⋅u₁⋅u₂⋅sin(q₂)   3⋅l⋅m_c⋅u₃ ⋅sin(q₃)   3⋅l⋅m_c⋅cos(q₃)⋅u₃̇   3⋅l⋅m_c⋅u₃̇     \n",
       "─────────────── + ─────────────────── - ─────────────────── - ─────────── - m_\n",
       "    3                      2                     2                 2          \n",
       " 8⋅l                                                                          \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "            2                                                                 \n",
       "⋅sin(q₃)⋅cos (q₂)                                 3⋅m_c⋅u₂⋅u₃⋅sin(q₃)⋅cos(q₂) \n",
       "───────────────── + 3⋅m_c⋅u₂⋅u₃⋅sin(q₂)⋅cos(q₃) - ─────────────────────────── \n",
       "  2                                                            2              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "        9⋅m_c⋅u₁⋅u₃⋅sin(q₂ - q₃)   9⋅m_c⋅u₁⋅u₃⋅sin(q₂ + q₃)   9⋅m_c⋅u₂⋅u₃⋅sin(\n",
       "b⋅u₂̇ + ──────────────────────── - ──────────────────────── + ────────────────\n",
       "                   4                          4                        2      \n",
       "                                                                              \n",
       "                                                                              \n",
       "                  ⎛                                                           \n",
       "             -m_c⋅⎝2⋅l⋅(2⋅l⋅u₃̇ + 2⋅cos(q₂ + q₃)⋅u₁̇ - 3⋅cos(q₂)⋅u₁̇ + 3⋅u₂̇) \n",
       "             ─────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "\n",
       "                                                                        2     \n",
       "  3⋅m_c⋅sin(q₂)⋅sin(q₃)⋅u₂̇                                    9⋅m_c⋅cos (q₂)⋅\n",
       "+ ───────────────────────── + 3⋅m_c⋅cos(q₂ + q₃)⋅cos(q₂)⋅u₁̇ - ───────────────\n",
       "              2                                                            2  \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "q₃)   9⋅m_c⋅cos(q₂ - q₃)⋅u₁̇   3⋅m_c⋅cos(q₂ + q₃)⋅u₁̇                       9⋅\n",
       "─── + ────────────────────── + ────────────────────── + 3⋅m_c⋅cos(q₂)⋅u₁̇ - ──\n",
       "                4                        4                                    \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "- 3⋅(2⋅l⋅(cos(q₂)⋅u₁̇ - u₂̇) + 3⋅(u₁⋅cos(q₂) - u₂)⋅u₁⋅sin(q₂))⋅cos(q₃) + 9⋅(u₁\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                   4                                          \n",
       "\n",
       "                       2                                                      \n",
       "cos(q₃)⋅u₁̇   3⋅m_c⋅cos (q₂)⋅u₁̇                                              \n",
       "─────────── - ────────────────── + 3⋅m_c⋅cos(q₂)⋅cos(q₃)⋅u₂̇ + 3⋅m_c⋅cos(q₂)⋅u\n",
       "                      2                                                       \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                      2                                       \n",
       "m_c⋅cos(q₃)⋅u₂̇   9⋅m_c⋅u₂̇   3⋅m_b⋅u₁ ⋅sin(2⋅q₂)   3⋅m_b⋅u₁⋅u₂⋅sin(q₂)   27⋅m\n",
       "─────────────── - ───────── + ─────────────────── - ─────────────────── + ────\n",
       "      2               2               4⋅l                   2⋅l               \n",
       "                                                                              \n",
       "                                                                              \n",
       "              2                                         ⎞                     \n",
       "⋅cos(q₂) - u₂) ⋅sin(q₃) - 9⋅(u₁⋅cos(q₂) - u₂)⋅u₁⋅sin(q₂)⎠                     \n",
       "──────────────────────────────────────────────────────────                    \n",
       "                                                                              \n",
       "\n",
       "                                                     2                   2    \n",
       "               3⋅m_b⋅u₁⋅u₂⋅sin(q₂)⋅cos(q₂)   3⋅m_b⋅u₂ ⋅sin(q₂)   9⋅m_c⋅u₁ ⋅sin\n",
       "₂̇ - m_c⋅u₁̇ - ─────────────────────────── + ───────────────── - ─────────────\n",
       "                           2⋅l                      2⋅l                       \n",
       "                                                                              \n",
       "                                                                              \n",
       "     2                           2                           2                \n",
       "_c⋅u₁ ⋅sin(2⋅q₂ - q₃)   27⋅m_c⋅u₁ ⋅sin(2⋅q₂ + q₃)   27⋅m_c⋅u₁ ⋅sin(2⋅q₂)   27⋅\n",
       "───────────────────── + ───────────────────────── + ──────────────────── - ───\n",
       "       16⋅l                        16⋅l                     8⋅l               \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "             2               2                        2                       \n",
       "(q₂ - q₃)⋅cos (q₂)   9⋅m_c⋅u₁ ⋅sin(q₂ - q₃)   9⋅m_c⋅u₁ ⋅sin(q₂ + q₃)   9⋅m_c⋅u\n",
       "────────────────── + ────────────────────── - ────────────────────── + ───────\n",
       " 8⋅l                          8⋅l                      8⋅l                    \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "m_c⋅u₁⋅u₂⋅sin(q₂ - q₃)   27⋅m_c⋅u₁⋅u₂⋅sin(q₂ + q₃)   27⋅m_c⋅u₁⋅u₂⋅sin(q₂)     \n",
       "────────────────────── - ───────────────────────── - ────────────────────     \n",
       "        8⋅l                         8⋅l                      4⋅l              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       " 2    3               2            2                       2            2     \n",
       "₁ ⋅sin (q₂)   9⋅m_c⋅u₁ ⋅sin(q₂)⋅cos (q₂)⋅cos(q₃)   9⋅m_c⋅u₁ ⋅sin(q₂)⋅cos (q₂) \n",
       "─────────── - ────────────────────────────────── + ────────────────────────── \n",
       "2⋅l                          8⋅l                              4⋅l             \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "          2                    2            3                               2 \n",
       "  9⋅m_c⋅u₁ ⋅sin(q₂)   27⋅m_c⋅u₁ ⋅sin(q₃)⋅cos (q₂)   27⋅m_c⋅u₁⋅u₂⋅sin(q₃)⋅cos (\n",
       "- ───────────────── + ─────────────────────────── - ──────────────────────────\n",
       "         2⋅l                      8⋅l                            4⋅l          \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "                                                                              \n",
       "\n",
       "                                    2                        2        ⎤\n",
       "q₂)   9⋅m_c⋅u₁⋅u₂⋅sin(q₃)   9⋅m_c⋅u₂ ⋅sin(q₂ + q₃)   9⋅m_c⋅u₂ ⋅sin(q₂)⎥\n",
       "─── + ─────────────────── + ────────────────────── + ─────────────────⎥\n",
       "              4⋅l                    4⋅l                    4⋅l       ⎥\n",
       "                                                                      ⎥\n",
       "                                                                      ⎥\n",
       "                                                                      ⎥\n",
       "                                                                      ⎥\n",
       "                                                                      ⎥\n",
       "                                                                      ⎥\n",
       "                                                                      ⎥\n",
       "                                                                      ⎥\n",
       "                                                                      ⎥\n",
       "                                                                      ⎥\n",
       "                                                                      ⎥\n",
       "                                                                      ⎦"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Frstar"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = Fr + Frstar"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{F, T, q_{1}, q_{2}, q_{3}, u_{1}, u_{2}, u_{3}, \\dot{q}_{2}, \\dot{u}_{1}, \\dot{u}_{2}, \\dot{u}_{3}\\right\\}$"
      ],
      "text/plain": [
       "{F, T, q₁, q₂, q₃, u₁, u₂, u₃, q₂̇, u₁̇, u₂̇, u₃̇}"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "me.find_dynamicsymbols(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{I_{B_bo}, c, g, k, k_{T}, l, m_{a}, m_{b}, m_{c}, t\\right\\}$"
      ],
      "text/plain": [
       "{I_{B_bo}, c, g, k, k_T, l, mₐ, m_b, m_c, t}"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.free_symbols"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Put in first order form"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}u_{1}\\\\u_{2}\\\\u_{3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡u₁⎤\n",
       "⎢  ⎥\n",
       "⎢u₂⎥\n",
       "⎢  ⎥\n",
       "⎣u₃⎦"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u = sm.Matrix(u)\n",
    "u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\dot{u}_{1}\\\\\\dot{u}_{2}\\\\\\dot{u}_{3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡u₁̇⎤\n",
       "⎢   ⎥\n",
       "⎢u₂̇⎥\n",
       "⎢   ⎥\n",
       "⎣u₃̇⎦"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u.diff()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\frac{9 I_{B_bo} \\operatorname{cos}^{2}\\left(q_{2}\\right)}{4 l^{2}} - m_{a} + m_{b} \\operatorname{cos}^{2}\\left(q_{2}\\right) - m_{b} + 3 m_{c} \\operatorname{cos}\\left(q_{2} + q_{3}\\right) \\operatorname{cos}\\left(q_{2}\\right) - \\frac{9 m_{c} \\operatorname{cos}^{2}\\left(q_{2}\\right) \\operatorname{cos}\\left(q_{3}\\right)}{2} - \\frac{3 m_{c} \\operatorname{cos}^{2}\\left(q_{2}\\right)}{2} - m_{c} & \\frac{9 I_{B_bo} \\operatorname{cos}\\left(q_{2}\\right)}{4 l^{2}} + \\frac{3 m_{c} \\operatorname{sin}\\left(q_{2}\\right) \\operatorname{sin}\\left(q_{3}\\right)}{2} + 3 m_{c} \\operatorname{cos}\\left(q_{2}\\right) \\operatorname{cos}\\left(q_{3}\\right) + 3 m_{c} \\operatorname{cos}\\left(q_{2}\\right) & \\frac{l m_{c} \\left(3 \\operatorname{cos}\\left(q_{2} - q_{3}\\right) - \\operatorname{cos}\\left(q_{2} + q_{3}\\right) + 6 \\operatorname{cos}\\left(q_{2}\\right)\\right)}{4}\\\\\\frac{9 I_{B_bo} \\operatorname{cos}\\left(q_{2}\\right)}{4 l^{2}} + \\frac{9 m_{c} \\operatorname{cos}\\left(q_{2} - q_{3}\\right)}{4} + \\frac{3 m_{c} \\operatorname{cos}\\left(q_{2} + q_{3}\\right)}{4} + 3 m_{c} \\operatorname{cos}\\left(q_{2}\\right) & - \\frac{9 I_{B_bo}}{4 l^{2}} - m_{b} - \\frac{9 m_{c} \\operatorname{cos}\\left(q_{3}\\right)}{2} - \\frac{9 m_{c}}{2} & - \\frac{3 l m_{c} \\left(\\operatorname{cos}\\left(q_{3}\\right) + 1\\right)}{2}\\\\- \\frac{l m_{c} \\left(- 3 \\operatorname{cos}\\left(q_{2} - q_{3}\\right) + \\operatorname{cos}\\left(q_{2} + q_{3}\\right) - 6 \\operatorname{cos}\\left(q_{2}\\right)\\right)}{4} & - \\frac{3 l m_{c} \\left(\\operatorname{cos}\\left(q_{3}\\right) + 1\\right)}{2} & - l^{2} m_{c}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡                2                                                            \n",
       "⎢  9⋅I_{B_bo}⋅cos (q₂)               2                                        \n",
       "⎢- ─────────────────── - mₐ + m_b⋅cos (q₂) - m_b + 3⋅m_c⋅cos(q₂ + q₃)⋅cos(q₂) \n",
       "⎢             2                                                               \n",
       "⎢          4⋅l                                                                \n",
       "⎢                                                                             \n",
       "⎢                        9⋅I_{B_bo}⋅cos(q₂)   9⋅m_c⋅cos(q₂ - q₃)   3⋅m_c⋅cos(q\n",
       "⎢                        ────────────────── + ────────────────── + ───────────\n",
       "⎢                                  2                  4                    4  \n",
       "⎢                               4⋅l                                           \n",
       "⎢                                                                             \n",
       "⎢                                    -l⋅m_c⋅(-3⋅cos(q₂ - q₃) + cos(q₂ + q₃) - \n",
       "⎢                                    ─────────────────────────────────────────\n",
       "⎣                                                             4               \n",
       "\n",
       "           2                        2                                         \n",
       "  9⋅m_c⋅cos (q₂)⋅cos(q₃)   3⋅m_c⋅cos (q₂)        9⋅I_{B_bo}⋅cos(q₂)   3⋅m_c⋅si\n",
       "- ────────────────────── - ────────────── - m_c  ────────────────── + ────────\n",
       "            2                    2                         2                  \n",
       "                                                        4⋅l                   \n",
       "                                                                              \n",
       "₂ + q₃)                                                                9⋅I_{B_\n",
       "─────── + 3⋅m_c⋅cos(q₂)                                              - ───────\n",
       "                                                                             2\n",
       "                                                                          4⋅l \n",
       "                                                                              \n",
       "6⋅cos(q₂))                                                                    \n",
       "───────────                                                                   \n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "n(q₂)⋅sin(q₃)                                          l⋅m_c⋅(3⋅cos(q₂ - q₃) -\n",
       "───────────── + 3⋅m_c⋅cos(q₂)⋅cos(q₃) + 3⋅m_c⋅cos(q₂)  ───────────────────────\n",
       "  2                                                                           \n",
       "                                                                              \n",
       "                                                                              \n",
       "bo}         9⋅m_c⋅cos(q₃)   9⋅m_c                                   -3⋅l⋅m_c⋅(\n",
       "─── - m_b - ───────────── - ─────                                   ──────────\n",
       "                  2           2                                               \n",
       "                                                                              \n",
       "                                                                              \n",
       "-3⋅l⋅m_c⋅(cos(q₃) + 1)                                                        \n",
       "───────────────────────                                                     -l\n",
       "           2                                                                  \n",
       "\n",
       "                          ⎤\n",
       " cos(q₂ + q₃) + 6⋅cos(q₂))⎥\n",
       "──────────────────────────⎥\n",
       " 4                        ⎥\n",
       "                          ⎥\n",
       "                          ⎥\n",
       "cos(q₃) + 1)              ⎥\n",
       "─────────────             ⎥\n",
       " 2                        ⎥\n",
       "                          ⎥\n",
       "                          ⎥\n",
       "2                         ⎥\n",
       " ⋅m_c                     ⎥\n",
       "                          ⎦"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = f.jacobian(u.diff())\n",
    "M = sm.trigsimp(M)\n",
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [],
   "source": [
    "G = -(f - M * u.diff())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [],
   "source": [
    "udots = M.LUsolve(G)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [],
   "source": [
    "# udots  # no need to print, too long, may eat up your computer's memory"
   ]
  }
 ],
 "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": "3.6.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}