Kinematics.ipynb

{
 "metadata": {
  "name": "",
  "signature": "sha256:446bed56e98f63afe6287b9050f148a7b7e5efcf1fc1b8d572afaf8d4af731f4"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import sympy\n",
      "import sympy.physics.mechanics as mech\n",
      "import copy\n",
      "import os\n",
      "import scipy.integrate\n",
      "import collections\n",
      "import matplotlib.pyplot as plt\n",
      "import sympy_utils\n",
      "import numpy as np\n",
      "%matplotlib inline\n",
      "sympy.init_printing()\n",
      "sympy.physics.vector.init_vprinting()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x, y, z, phi, theta, psi, t, J_x, J_y, J_z, Jxz, L, D, Y, M, \\\n",
      "    alpha, beta, V, h, g, P, Q, R, l, m, n = \\\n",
      "    sympy.symbols('x, y, z, phi, theta, psi, t, J_x, J_y, J_z, Jxz, L, D, Y, M, alpha, beta, V, h, g, P, Q, R, l, m, n')\n",
      "frame_i = mech.ReferenceFrame('i')\n",
      "frame_b = frame_i.orientnew('b', 'Body', (phi(t), theta(t), psi(t)), '321')\n",
      "frame_b2 = copy.copy(frame_b)\n",
      "frame_b.set_ang_vel(frame_i, P(t)*frame_b.x + Q(t)*frame_b.y + R(t)*frame_b.z)\n",
      "frame_w = frame_b.orientnew('w', 'Body', (beta(t), -alpha(t), 0), '321')\n",
      "point_cm = mech.Point('cm')\n",
      "point_cm.set_vel(frame_b, 0)\n",
      "point_cm.set_vel(frame_i, V(t)*frame_w.x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "w_ib = frame_i.orientnew('b', 'Body', (phi(t), theta(t), psi(t)), '321').ang_vel_in(frame_i)\n",
      "w_ib"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$(- \\operatorname{sin}\\left(\\theta\\right) \\dot{\\phi} + \\dot{\\psi})\\mathbf{\\hat{b}_x} + (\\operatorname{sin}\\left(\\psi\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\phi} + \\operatorname{cos}\\left(\\psi\\right) \\dot{\\theta})\\mathbf{\\hat{b}_y} + (- \\operatorname{sin}\\left(\\psi\\right) \\dot{\\theta} + \\operatorname{cos}\\left(\\psi\\right) \\operatorname{cos}\\left(\\theta\\right) \\dot{\\phi})\\mathbf{\\hat{b}_z}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
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       "prompt_number": 20,
       "text": [
        "(-sin(\u03b8)\u22c5\u03c6\u0307 + \u03c8\u0307) \u001b[94m\u001b[1mb_x\u001b[0;0m\u001b[0;0m + (sin(\u03c8)\u22c5cos(\u03b8)\u22c5\u03c6\u0307 + cos(\u03c8)\u22c5\u03b8\u0307) \u001b[94m\u001b[1mb_y\u001b[0;0m\u001b[0;0m + (-sin(\u03c8)\u22c5\u03b8\u0307 + cos(\u03c8)\u22c5cos(\u03b8)\u22c5\u03c6\u0307) \u001b[94m\u001b[1mb_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "P_eq = w_ib.dot(frame_b.x).simplify()\n",
      "Q_eq = w_ib.dot(frame_b.y).simplify()\n",
      "R_eq = w_ib.dot(frame_b.z).simplify()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 21
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "euler_sol = sympy.solve([P(t) - P_eq, Q(t) - Q_eq, R(t) - R_eq],\n",
      "                        [phi(t).diff(t), theta(t).diff(t), psi(t).diff(t)])\n",
      "euler_sol"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{Bmatrix}\\dot{\\phi} : \\frac{1}{\\operatorname{cos}\\left(\\theta\\right)} \\left(Q \\operatorname{sin}\\left(\\psi\\right) + R \\operatorname{cos}\\left(\\psi\\right)\\right), & \\dot{\\psi} : P + Q \\operatorname{sin}\\left(\\psi\\right) \\operatorname{tan}\\left(\\theta\\right) + R \\operatorname{cos}\\left(\\psi\\right) \\operatorname{tan}\\left(\\theta\\right), & \\dot{\\theta} : Q \\operatorname{cos}\\left(\\psi\\right) - R \\operatorname{sin}\\left(\\psi\\right)\\end{Bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAA8YAAAAmBAMAAAD0E5/tAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nEy60bN5rO9PnQywDx8l6/g/J9UUqxVSUBQAAAABJRU5ErkJggg==\n",
       "prompt_number": 22,
       "text": [
        "\u23a7    Q\u22c5sin(\u03c8) + R\u22c5cos(\u03c8)                                                      \n",
        "\u23a8\u03c6\u0307: \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, \u03c8\u0307: P + Q\u22c5sin(\u03c8)\u22c5tan(\u03b8) + R\u22c5cos(\u03c8)\u22c5tan(\u03b8), \u03b8\u0307: Q\u22c5cos\n",
        "\u23a9           cos(\u03b8)                                                            \n",
        "\n",
        "              \u23ab\n",
        "(\u03c8) - R\u22c5sin(\u03c8)\u23ac\n",
        "              \u23ad"
       ]
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "aircraft = mech.RigidBody('aircraft', point_cm, frame_b, M, (mech.inertia(frame_b, J_x, J_y, J_z, 0, 0, Jxz), point_cm))\n",
      "aircraft.set_potential_energy(m*g*h(t))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 23
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "F_A = -D(t)*frame_b.x -Y(t)*frame_b.y -L(t)*frame_b.z\n",
      "M_A = l(t)*frame_b.x + m(t)*frame_b.y + n(t)*frame_b.z"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 24
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "F_T = F_A + M*g*frame_i.z\n",
      "M_T = M_A"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 25
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "H_cm_i = aircraft.angular_momentum(point_cm, frame_i)\n",
      "di_H_cm_i = H_cm_i.diff(t, frame_b) + frame_b.ang_vel_in(frame_i).cross(H_cm_i)\n",
      "eom_rot = di_H_cm_i - M_T\n",
      "eom_rot"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$(J_{x} \\dot{P} - J_{y} Q R + Jxz \\dot{R} + \\left(J_{z} R + Jxz P\\right) Q - l)\\mathbf{\\hat{b}_x} + (J_{y} \\dot{Q} + \\left(J_{x} P + Jxz R\\right) R - \\left(J_{z} R + Jxz P\\right) P - m)\\mathbf{\\hat{b}_y} + (J_{y} P Q + J_{z} \\dot{R} + Jxz \\dot{P} - \\left(J_{x} P + Jxz R\\right) Q - n)\\mathbf{\\hat{b}_z}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAABQoAAAAcBAMAAADbxJ9KAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nsrKimQYt6fnLte4UUddsTAYuf5r/BUXJ2+qypcsmAAAAAElFTkSuQmCC\n",
       "prompt_number": 26,
       "text": [
        "(J\u2093\u22c5P\u0307 - J_y\u22c5Q\u22c5R + Jxz\u22c5R\u0307 + (J_z\u22c5R + Jxz\u22c5P)\u22c5Q - l) \u001b[94m\u001b[1mb_x\u001b[0;0m\u001b[0;0m + (J_y\u22c5Q\u0307 + (J\u2093\u22c5P + Jxz\u22c5R)\u22c5R - (J_z\u22c5R + Jxz\u22c5P)\u22c5P - m) \u001b[94m\u001b[1mb_y\u001b[0;0m\u001b[0;0m + (J_y\u22c5P\u22c5Q + J_z\u22c5R\u0307 + Jxz\u22c5P\u0307 - (J\u2093\u22c5P + Jxz\u22c5R)\u22c5Q - n) \u001b[94m\u001b[1mb_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 26
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "rot_sol = sympy.solve([eom_rot.dot(frame_b.x) - P(t).diff(t),\n",
      "             eom_rot.dot(frame_b.y) - Q(t).diff(t),\n",
      "             eom_rot.dot(frame_b.z) - R(t).diff(t)], [P(t).diff(t), Q(t).diff(t), R(t).diff(t)])\n",
      "rot_sol"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{Bmatrix}\\dot{P} : \\frac{1}{Jxz^{2} - \\left(J_{x} - 1\\right) \\left(J_{z} - 1\\right)} \\left(Jxz \\left(J_{x} P Q - J_{y} P Q + Jxz Q R + n\\right) - \\left(J_{z} - 1\\right) \\left(J_{y} Q R - J_{z} Q R - Jxz P Q + l\\right)\\right), & \\dot{Q} : \\frac{1}{J_{y} - 1} \\left(- J_{x} P R + J_{z} P R + Jxz P^{2} - Jxz R^{2} + m\\right), & \\dot{R} : \\frac{1}{Jxz^{2} - \\left(J_{x} - 1\\right) \\left(J_{z} - 1\\right)} \\left(Jxz \\left(J_{y} Q R - J_{z} Q R - Jxz P Q + l\\right) - \\left(J_{x} - 1\\right) \\left(J_{x} P Q - J_{y} P Q + Jxz Q R + n\\right)\\right)\\end{Bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": 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       "prompt_number": 27,
       "text": [
        "\u23a7                                                                             \n",
        "\u23aa    Jxz\u22c5(J\u2093\u22c5P\u22c5Q - J_y\u22c5P\u22c5Q + Jxz\u22c5Q\u22c5R + n) - (J_z - 1)\u22c5(J_y\u22c5Q\u22c5R - J_z\u22c5Q\u22c5R - Jxz\n",
        "\u23a8P\u0307: \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "\u23aa                                   2                                         \n",
        "\u23a9                                Jxz  - (J\u2093 - 1)\u22c5(J_z - 1)                    \n",
        "\n",
        "                                        2        2                            \n",
        "\u22c5P\u22c5Q + l)      -J\u2093\u22c5P\u22c5R + J_z\u22c5P\u22c5R + Jxz\u22c5P  - Jxz\u22c5R  + m      Jxz\u22c5(J_y\u22c5Q\u22c5R - J_z\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, Q\u0307: \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, R\u0307: \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                               J_y - 1                                        \n",
        "                                                                              \n",
        "\n",
        "                                                               \u23ab\n",
        "\u22c5Q\u22c5R - Jxz\u22c5P\u22c5Q + l) - (J\u2093 - 1)\u22c5(J\u2093\u22c5P\u22c5Q - J_y\u22c5P\u22c5Q + Jxz\u22c5Q\u22c5R + n)\u23aa\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23ac\n",
        "             2                                                 \u23aa\n",
        "          Jxz  - (J\u2093 - 1)\u22c5(J_z - 1)                            \u23ad"
       ]
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "L_i = aircraft.linear_momentum(frame_i)\n",
      "di_L_i = L_i.diff(t, frame_w) + frame_w.ang_vel_in(frame_i).cross(L_i)\n",
      "eom_trans = di_L_i - F_T\n",
      "eom_trans"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$M \\dot{V}\\mathbf{\\hat{w}_x} + M \\left(- P \\operatorname{sin}\\left(\\alpha\\right) \\operatorname{cos}\\left(\\beta\\right) - Q \\operatorname{sin}\\left(\\alpha\\right) \\operatorname{sin}\\left(\\beta\\right) + R \\operatorname{cos}\\left(\\alpha\\right) + \\operatorname{cos}\\left(\\alpha\\right) \\dot{\\beta}\\right) V\\mathbf{\\hat{w}_y} -  M \\left(- P \\operatorname{sin}\\left(\\beta\\right) + Q \\operatorname{cos}\\left(\\beta\\right) - \\dot{\\alpha}\\right) V\\mathbf{\\hat{w}_z} + D\\mathbf{\\hat{b}_x} + Y\\mathbf{\\hat{b}_y} + L\\mathbf{\\hat{b}_z} -  M g\\mathbf{\\hat{i}_z}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": 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       "prompt_number": 28,
       "text": [
        "M\u22c5V\u0307 \u001b[94m\u001b[1mw_x\u001b[0;0m\u001b[0;0m + M\u22c5(-P\u22c5sin(\u03b1)\u22c5cos(\u03b2) - Q\u22c5sin(\u03b1)\u22c5sin(\u03b2) + R\u22c5cos(\u03b1) + cos(\u03b1)\u22c5\u03b2\u0307)\u22c5V \u001b[94m\u001b[1mw_y\u001b[0;0m\u001b[0;0m - M\u22c5(-P\u22c5sin(\u03b2) + Q\u22c5cos(\u03b2) - \u03b1\u0307)\u22c5V \u001b[94m\u001b[1mw_z\u001b[0;0m\u001b[0;0m + D \u001b[94m\u001b[1mb_x\u001b[0;0m\u001b[0;0m + Y \u001b[94m\u001b[1mb_y\u001b[0;0m\u001b[0;0m + L \u001b[94m\u001b[1mb_z\u001b[0;0m\u001b[0;0m - M\u22c5g \u001b[94m\u001b[1mi_z\u001b[0;0m\u001b[0;0m"
       ]
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Lagrangian = mech.Lagrangian(frame_i, aircraft)\n",
      "Lagrangian"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{J_{y} Q^{2}}{2} + \\frac{M V^{2}}{2} - g m h + \\left(\\frac{J_{x} P}{2} + \\frac{Jxz R}{2}\\right) P + \\left(\\frac{J_{z} R}{2} + \\frac{Jxz P}{2}\\right) R$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAlQAAAA0BAMAAAC+3btpAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nszGZhY41AAAAAElFTkSuQmCC\n",
       "prompt_number": 29,
       "text": [
        "     2      2                                               \n",
        "J_y\u22c5Q    M\u22c5V            \u239bJ\u2093\u22c5P   Jxz\u22c5R\u239e     \u239bJ_z\u22c5R   Jxz\u22c5P\u239e  \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500 - g\u22c5m\u22c5h + \u239c\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u239f\u22c5P + \u239c\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u239f\u22c5R\n",
        "  2       2             \u239d 2       2  \u23a0     \u239d  2       2  \u23a0  "
       ]
      }
     ],
     "prompt_number": 29
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "v_cm_i = point_cm.vel(frame_i).express(frame_i)\n",
      "v_cm_i.dot(frame_i.x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(\\left(\\operatorname{sin}\\left(\\phi\\right) \\operatorname{sin}\\left(\\psi\\right) + \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\phi\\right) \\operatorname{cos}\\left(\\psi\\right)\\right) \\operatorname{sin}\\left(\\alpha\\right) + \\left(- \\operatorname{sin}\\left(\\phi\\right) \\operatorname{cos}\\left(\\psi\\right) + \\operatorname{sin}\\left(\\psi\\right) \\operatorname{sin}\\left(\\theta\\right) \\operatorname{cos}\\left(\\phi\\right)\\right) \\operatorname{sin}\\left(\\beta\\right) \\operatorname{cos}\\left(\\alpha\\right) + \\operatorname{cos}\\left(\\alpha\\right) \\operatorname{cos}\\left(\\beta\\right) \\operatorname{cos}\\left(\\phi\\right) \\operatorname{cos}\\left(\\theta\\right)\\right) V$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAABM8AAAAVBAMAAACj09SnAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMmYiu80QdonvRN2Z\nVKvu110NAAAACXBIWXMAAA7EAAAOxAGVKw4bAAALl0lEQVRoBe1abYxcZRU+M3dndnZ2ZnfBGFIR\nGFsVNCaMtX8abVhbtTEhskbWLzSMH4SaoB012iaAXCXYFbFdbaxCUEfDD0ljupGY+IFxFHUhSJxE\nEvwg6eIPjca0hVYom354nvN+3PveOffe/vIXb7j3vu97znnOc573nfuxlOil9pIC/xcFKh0iHJl2\nMcYR0asy82ZY6eBaIarOoVPQKh2i+sY8HIKZqJuLIzSIk4BHERfdbMKRYrwVV0iVDkK4xJy0sNoW\nkHSTybXSEQnypErz0H0AQNHGbh4TsbNLnh0m2yod29G1JLGXpiper492hEg2wxbOjINF/ZZc5BTN\n49IgmvE7aUNfLHIyAWymZ5JJ38t6PvDJYQrHu6FjgGIdh9WdhxPFhkcRF/ZKzIhBs+FmwOeSCr0f\nOkmJ47gakiNpQDQJNKkcR6e098kC0O31tyoVSjZDlbsaU8PHnctcxV6aKi5ar+Y7nhYihsyt5z9M\ndOmLQ6L3MgccLN5pucip1sMFO2kBHbQrYrnIyQTAXEsmfS/jWTlEz6dwvBs6BijWcXi2ByesofAo\n4AI3b8YAzYabAZ9LKvR+6CQljuNqSJ6koGgSaFI5jk5p75MBoOocPaBUKMkMVXTHFBB76lTmCnt5\nqrhovTb0ZntCRMjU/8uI7TWiSQ7CkWl7ZXwzn00vNNuAzTzrYz8WutgRzNMdOqvj2Oh6P4UTwJjk\nUYcn0S3ggjBnrncx4uYmzCg8G1teha4ulOhh9BKt2ZMM05jKRAIvVeAR8tB8ZI6J/C1hEiCkYlzB\nXoHAMVDZuQYeF5aqeL1+TLMj0cRkuJcTzPAxxYuCI9MOyvj7fP5ExoKhDdjE3fqytT9lr+EFnq8j\nek7HsUDNuRROEG9oVMEPPAq4IMyZ232MuJlw08+eiytMl+hx9RJtFk8ym8hL4KUKPEIemo+ozSUd\nSyoMEOxqYG5MgcDRL1zaNfC4sFSF61VdptmOEDFkbuQEn+bjsD34El3EJ9OaK3KN+VwbmSl645zt\nmCAexJjgR7C01CpkPB+mCt/RPI71lwuymye0x5EZe7I08FyReI+RyWDdndltNBturXwpqTBxDEt0\nuKkSx5E8SUHJEDQSOKnSebJKO58MAPWpyVvSMUkjyBLaCWd3CgR+PDB6y6xzDVzEXppKSnVE0/Eo\nZnpIV4xSq3W8S1HMTveYo3L7noUt5+jl91/yhT7PNIZ8qsw98ifbveSpz0XXrVU+8Os9/2QDgur3\n9CmeZGAAoNlVGPdsrtPkusW5ZelrVN+3d0DshiAJvqi1rWu7GTto7FrlV8VbTbzQGs/AwbcNGE3M\nfHUyuzFPcTXFFdYfXRrRrj13ix+7p0t0OLZEBSlNErVlxLISSLVKiWml4TMOQM2Xrd4wVySiroBS\nVSCWZi9NVbxeRwZ0x0CWwsh2YEAt6L/dHB8nWqF3EV3fj07yzESHT09EnRp/yC7wrpqnaZpYo9pO\nuob3BIIa7RPRaPJ5otfwGM2sguI5c+7Yv07wKybjRN+hDd1Lu7RT3DgI2VujzfQeg5O1g0Z8lLbR\nHT2Jz+FCbx7UD0YjcWFAv9GkCkygFVdIW0eVg5WHaKIvfmGJSItmN5qClCYptYViWQkglVZiSmnI\nqQDQZH3psbkCEXMUUKoKxNLsZalK1uv4sWPfM0SNbFf3zV9dfsnF8/Fgj2JstHfK+xRNj7DZqt3G\nGrVXiJqH+K7C2k3I81cCtrVWqj1alldULIJdBcVzeoFmF3j1Gae2QEfnfsofsK8AIDdk306baL/B\nydqZRjR6nF5LUyY+hwt9m+jayZ6kAKq7o6EK34orpEVqPzsdU+VZ8QtLRFo0u9HGkQKS+RLw2/yY\nBBml8cavAECtySIRcxRQqgrE0uxlqYrXi35CxB+akMzIdmQoT066j4vnY/rUZwkb7e1EL/DM1ID4\ncdWg1gpVTvD4Vzu7stF4z/RNQG8qZkZD3jQ8RrOrMO45G+OhLTi7h3w9SzQbw40bZ+bHb0yPG5ys\nnWnUWZgYN1/wyOHS4hvr75iNmBnVbTRUcdMv0N5dVmH9DMeB5mlRQpj5Eh2uLXFcq4CkIpaVAFIp\nJQZKw2dcQ5rk/WcFyCJAxBwFtKrSrpq9LFXJetFpWWG/WrR7foZLEkHBtH7X+SE22tuSjfYwb7Ta\nvFm9XTeekY027zcaHek2CN+5stHuWlz82eLihxhv3HP3iB5jA1IfYf/2Om+0ebiZ7PzyGNP1Bidr\nx05pnmT7zSZeVnw8w9Q80Ve+bDdaY3Hxhm8uLg4ZHeG+FVfYxM/r6IDolPgZXVyJkjYpUUFKk8yX\nAFJpJaaVhs94hcTf/9XlfBFzFNCqSrtq9rJUJetVP0stS1Rko9mV13NJ5uHFD7A3UXU9tdFwQ3+O\nNtM1I8KXRNSlqwd4dNqNhifeV/nVqdFLPmTMz13x3D2QmyRw/I/xEQCa7LOdqBPxXe4wD7N20Gis\n8NsX7yLE53DZHRMdYFeY0dwdDeG+FVfof9tnxM/o4kp0uPaOpiClSeZLkFdiSmnIoADwb41m1kyF\nWZHw/pGjgFZV2lWzl6UqWa/6utyL3Grxk+TkUNbAfgz8mejvqY2G1+j38bvTtea1rh3TTCe10RD0\nW351+itf8YaLZlZB8ZwaNLAB8GrI72A0up9/1j8CIE8yUK1f7U4MDU7WDhqt+WqvPUheL5UMs+x2\ngIwLX/xGQ7hvxRXyOxq1p4bUXBc/85niSsx8DChIaZIKQSsBpNJKTCkNORUAYuG2dvNFzFNAqSpw\nVexlqcrW6z6+BSWrxa/6p3qyBvv5zMdferQDz03+7wWeaazhFhNjA7T4aB+iiS7fztwdDUEPUr86\n4Osf+ECzG23cs9WHRIITfYMa3Q1z9H4B5EkGaq40sJ8FJ2sHjebyJOHmCx45XFprVP1jpSdm9vQb\nDeG+FVdIWzt0Z+XrVIvFT5j5EpEWzZSoaZUmqYhlJYBUWokppSGDAsC/yMrPORpMsggQMU8BparA\nVbGXpSpbr9XooCVqZGut85DbYXP88MqrBpef/8fl55958sU1c4uu/Gb7F9lYG/DwS5+5u3L83EeO\nn7vsuu/yHILaT2zn7weiD+LEzayC4llf/T3stQGfPrV6J0WP7h/AjYcC9MpXXzXHXcHJ2OWZ9fST\nV2Kj1gb5XJY2dlr7bAp2dY9O98jjKaLiCqn+n71desvSv8WP3dMl1gY8wc1uNAWJUiTzJcgrMaU0\nZFAAmhfvW+oxg9qATxmRhGqOAlpVaVfFXpqqZL1uua3HHGsDtxzVNZI2zYuII9Me4nHUweRWnDJN\nAiojzEbL1mZXQfOUOQ3HZJb/n+hxAgDQoMt8vIaRIu/MbqORhAeAyaC4QsPMlOgl0Es0WRKSSQrT\n8wTzS0yU1nwYgD+ppbkK7dBcfIJksbwCgWOw0nlQ5anK1gs5gR5mqA75gcpHpr2Bx1XZf3ynGGsS\nMNnDfKNvrT+w1/CSQGs4JvMmhHicIB406PMyhXgNI8ngzc2ORBBJuO1nL8UVGmamRI+rl2iyJCSz\niTzB/BITpTUfBnCrrwmQXj5n9wpkyHguupaAKk9Vtl7IqazWDp7GEbYJ3mSSsr4QGswIAYbRTZo5\nNeegdRzJHMNdxwENEjvidQyXQTFLONCVVlKhMDMl6mkTyJBkMu96jmB+iYnSqs8O/HkHLYeJS5Bn\nN8FyLnO9kFQxkFSi/CYvWYRIliz+qZv/53DGjc/RPP8pFSMXin7SECBmZY8mXug5aB1HzH34eQUw\n8A00SOyI1zFcBsUs4R4s7JRUKMxMiXraBC0kmcy7niOYX2KitOqzBX/bQsth4hLk2U2wnMtcLyRV\n2XpZolmyeD8w7wgpPsR/QzNtUzDrBj7A3PTdtHJ1njqOz5yH42jwn1pIjvEULoNm9uHjYSUVemZ5\naVOIPotepCVYViKCdZ+iCkGjzJ6iWuZ6oXadqN8246v1Py9bGTXbhgtuAAAAAElFTkSuQmCC\n",
       "prompt_number": 30,
       "text": [
        "((sin(\u03c6)\u22c5sin(\u03c8) + sin(\u03b8)\u22c5cos(\u03c6)\u22c5cos(\u03c8))\u22c5sin(\u03b1) + (-sin(\u03c6)\u22c5cos(\u03c8) + sin(\u03c8)\u22c5sin(\n",
        "\u03b8)\u22c5cos(\u03c6))\u22c5sin(\u03b2)\u22c5cos(\u03b1) + cos(\u03b1)\u22c5cos(\u03b2)\u22c5cos(\u03c6)\u22c5cos(\u03b8))\u22c5V"
       ]
      }
     ],
     "prompt_number": 30
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x_vect = sympy.Matrix([P, Q, R, phi, theta, psi, V, alpha, beta, x, y, z])\n",
      "u_vect = sympy.Matrix([L, D, Y, l, m, n])\n",
      "\n",
      "State = collections.namedtuple('State', ['phi', 'theta', 'psi', 'P', 'Q', 'R', 'V', 'alpha', 'beta', 'x', 'y', 'z'])\n",
      "Input = collections.namedtuple('Input', ['L', 'D', 'Y', 'l', 'm', 'n'])\n",
      "\n",
      "remove_t_sub = {var[0](t):var[0] for var in x_vect.tolist() + u_vect.tolist()}\n",
      "\n",
      "rhs = sympy.Matrix([\n",
      "    rot_sol[P(t).diff(t)],\n",
      "    rot_sol[Q(t).diff(t)],\n",
      "    rot_sol[R(t).diff(t)],\n",
      "    euler_sol[phi(t).diff(t)],\n",
      "    euler_sol[theta(t).diff(t)],\n",
      "    euler_sol[psi(t).diff(t)],\n",
      "    sympy.solve(eom_trans.dot(frame_w.x), V(t).diff(t))[0],\n",
      "    sympy.solve(eom_trans.dot(frame_w.z), alpha(t).diff(t))[0],\n",
      "    sympy.solve(eom_trans.dot(frame_w.y), beta(t).diff(t))[0],\n",
      "    v_cm_i.dot(frame_i.x),\n",
      "    v_cm_i.dot(frame_i.y),\n",
      "    v_cm_i.dot(frame_i.z),\n",
      "    ]).subs(remove_t_sub)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 31
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print rhs"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Matrix([[(Jxz*(J_x*P*Q - J_y*P*Q + Jxz*Q*R + n) - (J_z - 1)*(J_y*Q*R - J_z*Q*R - Jxz*P*Q + l))/(Jxz**2 - (J_x - 1)*(J_z - 1))], [(-J_x*P*R + J_z*P*R + Jxz*P**2 - Jxz*R**2 + m)/(J_y - 1)], [(Jxz*(J_y*Q*R - J_z*Q*R - Jxz*P*Q + l) - (J_x - 1)*(J_x*P*Q - J_y*P*Q + Jxz*Q*R + n))/(Jxz**2 - (J_x - 1)*(J_z - 1))], [(Q*sin(psi) + R*cos(psi))/cos(theta)], [Q*cos(psi) - R*sin(psi)], [P + Q*sin(psi)*tan(theta) + R*cos(psi)*tan(theta)], [(-D*cos(alpha)*cos(beta) - L*sin(alpha) + M*g*(sin(alpha)*cos(psi)*cos(theta) + sin(beta)*sin(psi)*cos(alpha)*cos(theta) - sin(theta)*cos(alpha)*cos(beta)) - Y*sin(beta)*cos(alpha))/M], [(D*sin(alpha)*cos(beta) - L*cos(alpha) + M*V*(-P*sin(beta) + Q*cos(beta)) + M*g*(-sin(alpha)*sin(beta)*sin(psi)*cos(theta) + sin(alpha)*sin(theta)*cos(beta) + cos(alpha)*cos(psi)*cos(theta)) + Y*sin(alpha)*sin(beta))/(M*V)], [(D*sin(beta) - M*R*V*cos(alpha) + M*V*(P*cos(beta) + Q*sin(beta))*sin(alpha) + M*g*(sin(beta)*sin(theta) + sin(psi)*cos(beta)*cos(theta)) - Y*cos(beta))/(M*V*cos(alpha))], [V*((sin(phi)*sin(psi) + sin(theta)*cos(phi)*cos(psi))*sin(alpha) + (-sin(phi)*cos(psi) + sin(psi)*sin(theta)*cos(phi))*sin(beta)*cos(alpha) + cos(alpha)*cos(beta)*cos(phi)*cos(theta))], [V*((sin(phi)*sin(psi)*sin(theta) + cos(phi)*cos(psi))*sin(beta)*cos(alpha) + (sin(phi)*sin(theta)*cos(psi) - sin(psi)*cos(phi))*sin(alpha) + sin(phi)*cos(alpha)*cos(beta)*cos(theta))], [V*(sin(alpha)*cos(psi)*cos(theta) + sin(beta)*sin(psi)*cos(alpha)*cos(theta) - sin(theta)*cos(alpha)*cos(beta))]])\n"
       ]
      }
     ],
     "prompt_number": 45
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "const = {J_x:0.1, J_y:0.1, J_z:0.1, Jxz:0.1, M:1, g:9.8}\n",
      "x0 = [0,0,0,0,0,0,10,0,0,0,0,0]\n",
      "u0 = [0,0,0,0,0,0]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 56
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sympy_utils.save_sympy_expr(\n",
      "    {'rhs': rhs, 't': t, 'x_vect': x_vect, 'u_vect': u_vect,\n",
      "     'constants': const},\n",
      "    os.path.join('save', 'wind_kinematics.sympy'))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 92
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "wind_kinematics = sympy_utils.load_sympy_expr(\n",
      "    os.path.join('save', 'wind_kinematics.sympy'))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 93
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f_sim, f_jac = sympy_utils.rhs_to_scipy_ode(**wind_kinematics)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 94
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%timeit\n",
      "f_sim(1, x0, u0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "10000 loops, best of 3: 21.4 \u00b5s per loop\n"
       ]
      }
     ],
     "prompt_number": 95
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "ode = scipy.integrate.ode(f_sim, f_jac)\n",
      "ode.set_initial_value(x0)\n",
      "\n",
      "dt = 0.01\n",
      "Data = collections.namedtuple('Data', ['t', 'x'])\n",
      "\n",
      "data = Data([], [])\n",
      "u = u0\n",
      "\n",
      "while ode.t < 10:\n",
      "    data.x.append(ode.y)\n",
      "    data.t.append(ode.t)\n",
      "    state = State(*ode.y)\n",
      "    state.P\n",
      "    u = [9.7,1,1,1,1,10]\n",
      "    ode.set_f_params(u)\n",
      "    ode.set_jac_params(u)\n",
      "    ode.integrate(ode.t + dt)\n",
      "\n",
      "hist = State(*np.array(data.x).T)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 96
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(data.t, np.array([hist.x, hist.y, hist.z]).T);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x6b59b10>"
       ]
      }
     ],
     "prompt_number": 97
    }
   ],
   "metadata": {}
  }
 ]
}