diff --git a/Excalidraw/Drawing 2025-03-27 15.49.37.excalidraw.md b/Excalidraw/Drawing 2025-03-27 15.49.37.excalidraw.md
new file mode 100644
index 0000000..601b965
--- /dev/null
+++ b/Excalidraw/Drawing 2025-03-27 15.49.37.excalidraw.md	
@@ -0,0 +1,600 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+%%
+## Drawing
+```compressed-json
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+diKR6/X5r2zVmjx/C1jx54nNtJ9dE9JDnK4PDHy6VTCPs9pNYKpr38R/1lc8/AP7eYAdkJen3VTVWH4JahPW7zH3HvaIX4G6+JcJ6DYtK+j8mBJ8gNcDrY+gGQaPcvz07rzYAah3Kz1QBv7XHYdJ4g6frAKDRrZc46Cuk+xrN+Yz7ZQ9rnih9bXvR+v3mS4kXOW+1hNR7h3LcnswtCLDrkU0UXBvdO4wgxtNyM3mPd1cDthv2KqOiF/74bVDHTqs
+
+kd54d5jRnukIDI+F3ZBlODCksMTWp5E31ZNP6au/jDVds2D52dhbMy6MTtM7vF/ydLb6XJR+wQs73wWt3ur2zL1uGh9dr2x9dIkvX53jPCDHM2f9XTuoEE9J9d+oLfXvvPKjyR7KLjB+7tok8AyDOor3WiczX6ACcgsXVCgiZ30AgJOBJ7UM2AXQCxORh1WlOQ8Vn69p6zJ7ovzv27VnOxFsmPu66PFo5xtPy+tHnJ8h33J9cht075PkV1ll8Det
+
+ng67KrqZf30A68IpUp/jrMp8YRbG9Hw6e5ZH7S+A70J8WXX45GN2PJwXxweTDLFqBlWzJQPY5frQDI5z1rpzePaC/C15AqePajxePmhb4nL8Zej9GKdP4BIXPg7JT1O5/89EU6OF04XKjpudCnQ1zbLyHoJsaxPfPdumI54tZxh1Ah/PUZ9A3U5apX2R/3hv56WmRvX68V6GwAMezgAFwBRMPS383+nNKF7EHIntR5eqII04XcVdrXSysczcg6S3
+
+79OtDLJ+EX/u6v3Yi8bPwe77HgK6CdMlbxAHSR2hmq6gSVtwpzESjj3/pXfA5qkHP8To0rKe4eOuPdWPfGfWPdPGTqlrBQXm52yd9OvRsth6IZKKdKdN68IWCZ4lrxqr53OJvYbux5XPqVIp9+eUw7envYrtPqI74l48G+svx3eM6m9U58Fmz8hNcV56UThCzXP3eZa7Gw8l32VOutW2/2MZaRxS+u/kzgU/C1EtSqxBNl8vWyWTSIl57diHZNBB
+
+nVoP4zwElWJodY0V8IXE+5AvupMdnLgk83EAHZNPsUNGYrrVoElqgawwDCgD6Cud6F77b7Hk0n2blQn9mag+sh9dx8h8V7La9Ivyh76Pg8P+XGI4K7TXvy3rpVGGPFE9ShNUcD7wF6vjE/aaxh9wBI5+xFiK4EvBi6Ev73LfTSp9is0l9Z7Xg59dzedCn9GMMTJaSG3A9fKjYnPC1NDI2vx57PX4CYdP6ha2vzOr8nyYbEv8vFG3im7wPaw7Iq7h
+
+/X5axMp7Fl9eCyR9NWz1+XnSJ6yPupJyPMQ83nbdiMARuI8gZUU1GQgCDpFAEBzSjivNb5NWlV9KLP+o7d3Abb96KNoodp+/zdQi9qvl+/qvHJ6y3Ax7R1OS/3zbZ4lQw2iTZk08HXanbboqYxiUIRoHPjrkGv7JOx3fat/7Q3es7dM3Ujli7xdd1pOPNLqE33VMO3rN5Ex2pYCPkBKZ9Pp7j9lp63Ppr3X5wCaaN7F31lvp9CDyB4DPBHMLbhwe
+
+U337YMv0BVE3+suF3LN5acJ26iHZp9IW2t4xPRvX6WmAHwAqlouqyao+AFAFaAmAG9E+HmCgoh/znOWWto52Chm/Q0icm12+9WTb4KL6hCN8izWrhGernKN9S330/Rv7J/IvWN65P6h50Hbrfxv6pVng2h+JvJRizL712Oh20M4v1N4WPw56WP/bk5rCWZmnAB61LE6r9VLnbrTlK513JC/v4P19zgs3xeAzyG3zpy6ZSrQmcAYgFIAq4DUOJEvJ
+
+PYstKnDy4FXW1xHbmCqqDVvuczyN6IvqN+6Plo8jbTqb1n6I7qHnU7YTURdVX08g1XPRNCzm3T6GAFH3TVN+DMNN7QZph5Gvl0fYn10aa7V6Y+TBzNVlOM5ORbnpXBtx/l4JnZZ3tquTH9O+4bwtu0TYTMnRoJ6+vat/XPNl8qTEnx0v4fJOZze9VvGkbQqb4Zez0Z4r9Lm4aTfM7rbeyHAAdEFIQ1hY1AaiG4A5YGgAiIAyAswDdoiwAYALzjVH
+
+mjNZA8d2If27LxLIgFfgjVnSAGoEIvRijIfGkA2glD5LwCh5zEdD4ofm4D7KhBpKAcofofWQEYf1D79RrD4Yf7D/4f87aXIgj94f7D7OQ6rXEfXJXYfCCup2Mj8Yf1hiyAXdlYg+AFMmuD+4fbD/SAyj/5dhACyZAnU0f5D6Ef6QAgNmABjgRAEVDswF68UdEUfwj+jgnWFawxdDsf6QFfWxAEcfMeETwccB5ALj+dzf6EGAnQjvgPj+YA+JcZA1
+
+RS8oLbX1AoT7VA7yBIavjb1Xj1FwfxQIMAaD6JgBAAWQ5oAzwvj6kfmOvKAQT9wfQoBIAIwH0f1YFRt/JiKfm4HRME1AKf78P68bACvIr6wcLwQBHE5T/oy2cB/2+AFzgpADLwuAAAAFNNlqAGnwPwEM/Bn9xgAAJR6gCeLKACZCkobp98gfp+T8LfRegJZ+jP4BATPrJ9aPjaAiPhAAIKqADjgNbSfNCeK5gd+FB4DTC1YIRRuvLGC7OA+mEAKp
+
++XrIRRYPnXrCAfqQ5gYLhZPuwAW4xBo8+IRRiu+p8IARp+XPlp/8gPZ+MAev2aBBGC9tuUMUgAwABPvogNLsPQlBEF/PrT8ngvpVDmQcADowdsxeZYAC1gEAC1gIAA==
+```
+%%
\ No newline at end of file
diff --git a/多体+耦合求解器/WindTurbine_Kane推导.ipynb b/多体+耦合求解器/WindTurbine_Kane推导.ipynb
new file mode 100644
index 0000000..939335e
--- /dev/null
+++ b/多体+耦合求解器/WindTurbine_Kane推导.ipynb
@@ -0,0 +1,218 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import sympy as sm\n",
+    "import sympy.physics.mechanics as me\n",
+    "me.init_vprinting(use_latex='mathjax')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# 建立坐标系"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def smll_rot_trans(theta1, theta2, theta3):\n",
+    "    theta11 = theta1 * theta1\n",
+    "    theta22 = theta2 * theta2\n",
+    "    theta33 = theta3 * theta3\n",
+    "    sqrd_sum = theta11 + theta22 + theta33\n",
+    "    sqrt1_sqrd_sum = (1.0 + sqrd_sum)**0.5\n",
+    "    com_denom = sqrd_sum * sqrt1_sqrd_sum\n",
+    "    theta12s = theta1 * theta2 * (sqrt1_sqrd_sum - 1.0)\n",
+    "    theta13s = theta1 * theta3 * (sqrt1_sqrd_sum - 1.0)\n",
+    "    theta23s = theta2 * theta3 * (sqrt1_sqrd_sum - 1.0)\n",
+    "    \n",
+    "    if com_denom == 0.0:\n",
+    "        trans_mat = sm.Matrix([[1.0, 0.0, 0.0],\n",
+    "                        [0.0, 1.0, 0.0],\n",
+    "                        [0.0, 0.0, 1.0]])\n",
+    "        \n",
+    "    else:\n",
+    "        trans_mat = sm.Matrix([[(theta11 * sqrt1_sqrd_sum + theta22 + theta33) / com_denom, (theta3 * sqrd_sum + theta12s) / com_denom, (-theta2 * sqrd_sum + theta13s) / com_denom],\n",
+    "                        [(-theta3 * sqrd_sum + theta12s) / com_denom, (theta11 + theta22 * sqrt1_sqrd_sum + theta33) / com_denom, (theta1 * sqrd_sum + theta23s) / com_denom],\n",
+    "                        [(theta2 * sqrd_sum + theta13s) / com_denom, (-theta1 * sqrd_sum + theta23s) / com_denom, (theta11 + theta22 + theta33 * sqrt1_sqrd_sum) / com_denom]])\n",
+    "    return trans_mat"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 31,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left[\\begin{matrix}\\frac{\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5} \\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}}{\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5}} & \\frac{\\left(\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5} - 1.0\\right) \\theta_{1} \\theta_{2} + \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\theta_{3}}{\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5}} & \\frac{\\left(\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5} - 1.0\\right) \\theta_{1} \\theta_{3} - \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\theta_{2}}{\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5}}\\\\\\frac{\\left(\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5} - 1.0\\right) \\theta_{1} \\theta_{2} - \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\theta_{3}}{\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5}} & \\frac{\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5} \\theta_{2}^{2} + \\theta_{1}^{2} + \\theta_{3}^{2}}{\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5}} & \\frac{\\left(\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5} - 1.0\\right) \\theta_{2} \\theta_{3} + \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\theta_{1}}{\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5}}\\\\\\frac{\\left(\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5} - 1.0\\right) \\theta_{1} \\theta_{3} + \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\theta_{2}}{\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5}} & \\frac{\\left(\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5} - 1.0\\right) \\theta_{2} \\theta_{3} - \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\theta_{1}}{\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5}} & \\frac{\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5} \\theta_{3}^{2} + \\theta_{1}^{2} + \\theta_{2}^{2}}{\\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2}\\right) \\left(\\theta_{1}^{2} + \\theta_{2}^{2} + \\theta_{3}^{2} + 1.0\\right)^{0.5}}\\end{matrix}\\right]$"
+      ],
+      "text/plain": [
+       "⎡          ⎛                       0.5                ⎞                        ↪\n",
+       "⎢          ⎜⎛  2     2     2      ⎞      2     2     2⎟ ⎛  2     2     2       ↪\n",
+       "⎢          ⎝⎝θ₁  + θ₂  + θ₃  + 1.0⎠   ⋅θ₁  + θ₂  + θ₃ ⎠⋅⎝θ₁  + θ₂  + θ₃  + 1.0 ↪\n",
+       "⎢          ─────────────────────────────────────────────────────────────────── ↪\n",
+       "⎢                                        2     2     2                         ↪\n",
+       "⎢                                      θ₁  + θ₂  + θ₃                          ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢⎛⎛                       0.5      ⎞                             ⎞             ↪\n",
+       "⎢⎜⎜⎛  2     2     2      ⎞         ⎟         ⎛  2     2     2⎞   ⎟ ⎛  2     2  ↪\n",
+       "⎢⎝⎝⎝θ₁  + θ₂  + θ₃  + 1.0⎠    - 1.0⎠⋅θ₁⋅θ₂ - ⎝θ₁  + θ₂  + θ₃ ⎠⋅θ₃⎠⋅⎝θ₁  + θ₂   ↪\n",
+       "⎢───────────────────────────────────────────────────────────────────────────── ↪\n",
+       "⎢                                         2     2     2                        ↪\n",
+       "⎢                                       θ₁  + θ₂  + θ₃                         ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢⎛⎛                       0.5      ⎞                             ⎞             ↪\n",
+       "⎢⎜⎜⎛  2     2     2      ⎞         ⎟         ⎛  2     2     2⎞   ⎟ ⎛  2     2  ↪\n",
+       "⎢⎝⎝⎝θ₁  + θ₂  + θ₃  + 1.0⎠    - 1.0⎠⋅θ₁⋅θ₃ + ⎝θ₁  + θ₂  + θ₃ ⎠⋅θ₂⎠⋅⎝θ₁  + θ₂   ↪\n",
+       "⎢───────────────────────────────────────────────────────────────────────────── ↪\n",
+       "⎢                                         2     2     2                        ↪\n",
+       "⎣                                       θ₁  + θ₂  + θ₃                         ↪\n",
+       "\n",
+       "↪  -0.5             ⎛⎛                       0.5      ⎞                        ↪\n",
+       "↪ ⎞                 ⎜⎜⎛  2     2     2      ⎞         ⎟         ⎛  2     2     ↪\n",
+       "↪ ⎠                 ⎝⎝⎝θ₁  + θ₂  + θ₃  + 1.0⎠    - 1.0⎠⋅θ₁⋅θ₂ + ⎝θ₁  + θ₂  + θ ↪\n",
+       "↪ ─────             ────────────────────────────────────────────────────────── ↪\n",
+       "↪                                                            2     2     2     ↪\n",
+       "↪                                                          θ₁  + θ₂  + θ₃      ↪\n",
+       "↪                                                                              ↪\n",
+       "↪             -0.5            ⎛                       0.5                ⎞     ↪\n",
+       "↪     2      ⎞                ⎜⎛  2     2     2      ⎞      2     2     2⎟ ⎛   ↪\n",
+       "↪ + θ₃  + 1.0⎠                ⎝⎝θ₁  + θ₂  + θ₃  + 1.0⎠   ⋅θ₂  + θ₁  + θ₃ ⎠⋅⎝θ₁ ↪\n",
+       "↪ ────────────────            ──────────────────────────────────────────────── ↪\n",
+       "↪                                                           2     2     2      ↪\n",
+       "↪                                                         θ₁  + θ₂  + θ₃       ↪\n",
+       "↪                                                                              ↪\n",
+       "↪             -0.5  ⎛⎛                       0.5      ⎞                        ↪\n",
+       "↪     2      ⎞      ⎜⎜⎛  2     2     2      ⎞         ⎟         ⎛  2     2     ↪\n",
+       "↪ + θ₃  + 1.0⎠      ⎝⎝⎝θ₁  + θ₂  + θ₃  + 1.0⎠    - 1.0⎠⋅θ₂⋅θ₃ - ⎝θ₁  + θ₂  + θ ↪\n",
+       "↪ ────────────────  ────────────────────────────────────────────────────────── ↪\n",
+       "↪                                                            2     2     2     ↪\n",
+       "↪                                                          θ₁  + θ₂  + θ₃      ↪\n",
+       "\n",
+       "↪       ⎞                        -0.5  ⎛⎛                       0.5      ⎞     ↪\n",
+       "↪  2⎞   ⎟ ⎛  2     2     2      ⎞      ⎜⎜⎛  2     2     2      ⎞         ⎟     ↪\n",
+       "↪ ₃ ⎠⋅θ₃⎠⋅⎝θ₁  + θ₂  + θ₃  + 1.0⎠      ⎝⎝⎝θ₁  + θ₂  + θ₃  + 1.0⎠    - 1.0⎠⋅θ₁⋅ ↪\n",
+       "↪ ───────────────────────────────────  ─────────────────────────────────────── ↪\n",
+       "↪                                                                              ↪\n",
+       "↪                                                                              ↪\n",
+       "↪                                                                              ↪\n",
+       "↪                     -0.5             ⎛⎛                       0.5      ⎞     ↪\n",
+       "↪ 2     2     2      ⎞                 ⎜⎜⎛  2     2     2      ⎞         ⎟     ↪\n",
+       "↪   + θ₂  + θ₃  + 1.0⎠                 ⎝⎝⎝θ₁  + θ₂  + θ₃  + 1.0⎠    - 1.0⎠⋅θ₂⋅ ↪\n",
+       "↪ ────────────────────────             ─────────────────────────────────────── ↪\n",
+       "↪                                                                              ↪\n",
+       "↪                                                                              ↪\n",
+       "↪                                                                              ↪\n",
+       "↪       ⎞                        -0.5            ⎛                       0.5   ↪\n",
+       "↪  2⎞   ⎟ ⎛  2     2     2      ⎞                ⎜⎛  2     2     2      ⎞      ↪\n",
+       "↪ ₃ ⎠⋅θ₁⎠⋅⎝θ₁  + θ₂  + θ₃  + 1.0⎠                ⎝⎝θ₁  + θ₂  + θ₃  + 1.0⎠   ⋅θ ↪\n",
+       "↪ ───────────────────────────────────            ───────────────────────────── ↪\n",
+       "↪                                                                              ↪\n",
+       "↪                                                                            θ ↪\n",
+       "\n",
+       "↪                          ⎞                        -0.5⎤\n",
+       "↪      ⎛  2     2     2⎞   ⎟ ⎛  2     2     2      ⎞    ⎥\n",
+       "↪ θ₃ - ⎝θ₁  + θ₂  + θ₃ ⎠⋅θ₂⎠⋅⎝θ₁  + θ₂  + θ₃  + 1.0⎠    ⎥\n",
+       "↪ ──────────────────────────────────────────────────────⎥\n",
+       "↪   2     2     2                                       ⎥\n",
+       "↪ θ₁  + θ₂  + θ₃                                        ⎥\n",
+       "↪                                                       ⎥\n",
+       "↪                          ⎞                        -0.5⎥\n",
+       "↪      ⎛  2     2     2⎞   ⎟ ⎛  2     2     2      ⎞    ⎥\n",
+       "↪ θ₃ + ⎝θ₁  + θ₂  + θ₃ ⎠⋅θ₁⎠⋅⎝θ₁  + θ₂  + θ₃  + 1.0⎠    ⎥\n",
+       "↪ ──────────────────────────────────────────────────────⎥\n",
+       "↪   2     2     2                                       ⎥\n",
+       "↪ θ₁  + θ₂  + θ₃                                        ⎥\n",
+       "↪                                                       ⎥\n",
+       "↪               ⎞                        -0.5           ⎥\n",
+       "↪  2     2     2⎟ ⎛  2     2     2      ⎞               ⎥\n",
+       "↪ ₃  + θ₁  + θ₂ ⎠⋅⎝θ₁  + θ₂  + θ₃  + 1.0⎠               ⎥\n",
+       "↪ ───────────────────────────────────────────           ⎥\n",
+       "↪  2     2     2                                        ⎥\n",
+       "↪ ₁  + θ₂  + θ₃                                         ⎦"
+      ]
+     },
+     "execution_count": 31,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "A = me.ReferenceFrame('A')\n",
+    "B = me.ReferenceFrame('B')\n",
+    "D = me.ReferenceFrame('D')\n",
+    "C = me.ReferenceFrame('C')\n",
+    "G = me.ReferenceFrame('G')\n",
+    "G_prime1 = me.ReferenceFrame('G_prime1')\n",
+    "G_prime2 = me.ReferenceFrame('G_prime2')\n",
+    "G_prime3 = me.ReferenceFrame('G_prime3')\n",
+    "# fast x-1, -y-3, z-2\n",
+    "\n",
+    "# 广义坐标\n",
+    "q1, q2, q3, q4, q_yaw, q_drtr, q_geaz, q_1f1, q_1f2, q_1e1, q_2f1, q_2f2, q_2e1, q_3f1, q_3f2, q_3e1 = me.dynamicsymbols('q1, q2, q3, q4, q_yaw, q_drtr, q_geaz, q_1f1, q_1f2, q_1e1, q_2f1, q_2f2, q_2e1, q_3f1, q_3f2, q_3e1')\n",
+    "# tower top\n",
+    "theta1, theta2, theta3 = me.dynamicsymbols(\"theta1, theta2, theta3\")\n",
+    "dcm = smll_rot_trans(theta1, theta2, theta3)\n",
+    "B.orient_dcm(A, dcm)\n",
+    "\n",
+    "# nacelle\n",
+    "D.orient_axis(B, q_yaw, B.y)\n",
+    "\n",
+    "# shaft C \n",
+    "theta_tilt = sm.symbols(\"theta_tilt\")\n",
+    "C.orient_axis(D, theta_tilt, D.z)\n",
+    "\n",
+    "# Hub G = Azimuth E\n",
+    "Azimuth = q_drtr + q_geaz\n",
+    "G.orient_axis(C, Azimuth, C.x)\n",
+    "\n",
+    "# 到每只叶片\n",
+    "angle = 2*sm.pi/3\n",
+    "G_prime1.orient_axis(G, angle * 0, G.x)\n",
+    "G_prime2.orient_axis(G, angle * 1, G.x)\n",
+    "G_prime3.orient_axis(G, angle * 2, G.x)\n",
+    "\n",
+    "B.dcm(A)\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# 速度"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "MinerU",
+   "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": "3.10.16"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/多体+耦合求解器/风机模型.excalidraw.md b/多体+耦合求解器/风机模型.excalidraw.md
new file mode 100644
index 0000000..2d86d06
--- /dev/null
+++ b/多体+耦合求解器/风机模型.excalidraw.md
@@ -0,0 +1,943 @@
+---
+
+excalidraw-plugin: parsed
+tags: [excalidraw]
+
+---
+==⚠  Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
+
+
+# Excalidraw Data
+
+## Text Elements
+## Embedded Files
+c0166535889251fc00499757c5d32377c8fd6767: [[Pasted Image 20250120103502_252.png]]
+
+%%
+## Drawing
+```compressed-json
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+```
+%%
\ No newline at end of file
diff --git a/工作OKRs/25.2-5 OKR.canvas b/工作OKRs/25.2-5 OKR.canvas
index 6833d1c..2f0e7aa 100644
--- a/工作OKRs/25.2-5 OKR.canvas	
+++ b/工作OKRs/25.2-5 OKR.canvas	
@@ -1,7 +1,7 @@
 {
 	"nodes":[
 		{"id":"b698e88ca5fb9c51","type":"text","text":"状态指标：\n推进OKR的时候也要关注这些事情，它们是完成OKR的保障。\n\n\n效率状态  green","x":-96,"y":80,"width":456,"height":347},
-		{"id":"58be7961ae7275a7","type":"text","text":"# 计划\n这周要做的3~5件重要的事情，这些事情能有效推进实现OKR。\n\nP1 必须做。P2 应该做\n\nP1 多体原理学习 YouTube课程 016\nP1 根据公共mesh模块调整代码 yes\nP1 气动模块联合调试，跑通 no\nP1 稳态工况多体动力学求解方法\n\n","x":-620,"y":-307,"width":450,"height":347},
+		{"id":"58be7961ae7275a7","type":"text","text":"# 计划\n这周要做的3~5件重要的事情，这些事情能有效推进实现OKR。\n\nP1 必须做。P2 应该做\n\nP1 多体原理学习 YouTube课程 018\nP1 气动模块联合调试，跑通 no\nP1 稳态工况多体动力学求解方法\nP1 使用python搭建风电机组多体模型\n\n","x":-620,"y":-307,"width":450,"height":347},
 		{"id":"2b068bfe5df15a72","type":"text","text":"# 目标：多体动力学模块完善\n### 每周盘点一下它们\n\n\n关键结果：建模原理、建模方法掌握    （6.8/10）\n\n关键结果：对标Bladed模块完成  （7.2/10）\n\n关键结果：风机多体动力学文献调研情况完成 （5/10）","x":-96,"y":-307,"width":456,"height":347},
 		{"id":"01ee5c157d0deeae","type":"text","text":"# 推进计划\n未来四周计划推进的重要事情\n\n文献调研启动\n\n建模重新推导\n\n\n","x":-620,"y":80,"width":456,"height":347}
 	],
diff --git a/工作总结/周报/周报70-郭翼泽.docx b/工作总结/周报/周报70-郭翼泽.docx
new file mode 100644
index 0000000..156820c
Binary files /dev/null and b/工作总结/周报/周报70-郭翼泽.docx differ
diff --git a/补课/多体动力学/16_Generalized_Forces.excalidraw.md b/补课/多体动力学/16_Generalized_Forces.excalidraw.md
index 201ba5a..8740375 100644
--- a/补课/多体动力学/16_Generalized_Forces.excalidraw.md
+++ b/补课/多体动力学/16_Generalized_Forces.excalidraw.md
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 ```
 %%
\ No newline at end of file
diff --git a/补课/多体动力学/17_Equations_of_Motions.excalidraw.md b/补课/多体动力学/17_Equations_of_Motions.excalidraw.md
index d5b2a81..61a2dea 100644
--- a/补课/多体动力学/17_Equations_of_Motions.excalidraw.md
+++ b/补课/多体动力学/17_Equations_of_Motions.excalidraw.md
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 ```
 %%
\ No newline at end of file
diff --git a/补课/多体动力学/17_Equations_of_Motions.ipynb b/补课/多体动力学/17_Equations_of_Motions.ipynb
index 44ea833..32c77b5 100644
--- a/补课/多体动力学/17_Equations_of_Motions.ipynb
+++ b/补课/多体动力学/17_Equations_of_Motions.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 2,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -10,17 +10,1085 @@
     "import sympy.physics.mechanics as me\n",
     "me.init_vprinting(use_latex='mathjax')"
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Body Fixed Newton-Euler Equations"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "m, Ixx, Iyy, Izz = sm.symbols('m, I_{xx}, I_{yy}, I_{zz}')\n",
+    "Ixy, Iyz, Ixz = sm.symbols('I_{xy}, I_{yz}, I_{xz}')\n",
+    "Fx, Fy, Fz, Mx, My, Mz = me.dynamicsymbols('F_x, F_y, F_z, M_x, M_y, M_z')\n",
+    "u1, u2, u3, u4, u5, u6 = me.dynamicsymbols('u1, u2, u3, u4, u5, u6')\n",
+    "\n",
+    "A = me.ReferenceFrame('A')\n",
+    "B = me.ReferenceFrame('B')\n",
+    "\n",
+    "Bo = me.Point('Bo')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "A_w_B = u4*B.x + u5*B.y + u6*B.z\n",
+    "B.set_ang_vel(A, A_w_B)\n",
+    "\n",
+    "A_v_Bo = u1*B.x + u2*B.y + u3*B.z\n",
+    "Bo.set_vel(A, A_v_Bo)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left( \\mathbf{\\hat{b}_x}, \\  \\mathbf{\\hat{b}_y}, \\  \\mathbf{\\hat{b}_z}, \\  0, \\  0, \\  0\\right)$"
+      ],
+      "text/plain": [
+       "(b_x, b_y, b_z, 0, 0, 0)"
+      ]
+     },
+     "execution_count": 5,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "v_Bo_1 = A_v_Bo.diff(u1, A, var_in_dcm=False)\n",
+    "v_Bo_2 = A_v_Bo.diff(u2, A, var_in_dcm=False)\n",
+    "v_Bo_3 = A_v_Bo.diff(u3, A, var_in_dcm=False)\n",
+    "v_Bo_4 = A_v_Bo.diff(u4, A, var_in_dcm=False)\n",
+    "v_Bo_5 = A_v_Bo.diff(u5, A, var_in_dcm=False)\n",
+    "v_Bo_6 = A_v_Bo.diff(u6, A, var_in_dcm=False)\n",
+    "\n",
+    "v_Bo_1, v_Bo_2, v_Bo_3, v_Bo_4, v_Bo_5, v_Bo_6"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left( 0, \\  0, \\  0, \\  \\mathbf{\\hat{b}_x}, \\  \\mathbf{\\hat{b}_y}, \\  \\mathbf{\\hat{b}_z}\\right)$"
+      ],
+      "text/plain": [
+       "(0, 0, 0, b_x, b_y, b_z)"
+      ]
+     },
+     "execution_count": 6,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "w_B_1 = A_w_B.diff(u1, A, var_in_dcm=False)\n",
+    "w_B_2 = A_w_B.diff(u2, A, var_in_dcm=False)\n",
+    "w_B_3 = A_w_B.diff(u3, A, var_in_dcm=False)\n",
+    "w_B_4 = A_w_B.diff(u4, A, var_in_dcm=False)\n",
+    "w_B_5 = A_w_B.diff(u5, A, var_in_dcm=False)\n",
+    "w_B_6 = A_w_B.diff(u6, A, var_in_dcm=False)\n",
+    "\n",
+    "w_B_1, w_B_2, w_B_3, w_B_4, w_B_5, w_B_6"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left[ \\left[ \\mathbf{\\hat{b}_x}, \\  \\mathbf{\\hat{b}_y}, \\  \\mathbf{\\hat{b}_z}, \\  0, \\  0, \\  0\\right], \\  \\left[ 0, \\  0, \\  0, \\  \\mathbf{\\hat{b}_x}, \\  \\mathbf{\\hat{b}_y}, \\  \\mathbf{\\hat{b}_z}\\right]\\right]$"
+      ],
+      "text/plain": [
+       "[[b_x, b_y, b_z, 0, 0, 0], [0, 0, 0, b_x, b_y, b_z]]"
+      ]
+     },
+     "execution_count": 7,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "par_vels = me.partial_velocity([A_v_Bo, A_w_B], [u1, u2, u3, u4, u5, u6], A)\n",
+    "\n",
+    "par_vels"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left[\\begin{matrix}F_{x}\\\\F_{y}\\\\F_{z}\\\\M_{x}\\\\M_{x}\\\\M_{z}\\end{matrix}\\right]$"
+      ],
+      "text/plain": [
+       "⎡Fₓ ⎤\n",
+       "⎢   ⎥\n",
+       "⎢F_y⎥\n",
+       "⎢   ⎥\n",
+       "⎢F_z⎥\n",
+       "⎢   ⎥\n",
+       "⎢Mₓ ⎥\n",
+       "⎢   ⎥\n",
+       "⎢Mₓ ⎥\n",
+       "⎢   ⎥\n",
+       "⎣M_z⎦"
+      ]
+     },
+     "execution_count": 8,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "T = Mx*B.x + My*B.y + Mz*B.z\n",
+    "R = Fx*B.x + Fy*B.y + Fz*B.z\n",
+    "\n",
+    "F1 = v_Bo_1.dot(R) + w_B_1.dot(T)\n",
+    "F2 = v_Bo_2.dot(R) + w_B_2.dot(T)\n",
+    "F3 = v_Bo_3.dot(R) + w_B_3.dot(T)\n",
+    "F4 = v_Bo_4.dot(R) + w_B_4.dot(T)\n",
+    "F5 = v_Bo_5.dot(R) + w_B_5.dot(T)\n",
+    "F6 = v_Bo_6.dot(R) + w_B_6.dot(T)\n",
+    "\n",
+    "Fr = sm.Matrix([F1, F2, F3, F4, F4, F6])\n",
+    "Fr"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 12,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle I_{xx}\\mathbf{\\hat{b}_x}\\otimes \\mathbf{\\hat{b}_x} + I_{xy}\\mathbf{\\hat{b}_x}\\otimes \\mathbf{\\hat{b}_y} + I_{xz}\\mathbf{\\hat{b}_x}\\otimes \\mathbf{\\hat{b}_z} + I_{xy}\\mathbf{\\hat{b}_y}\\otimes \\mathbf{\\hat{b}_x} + I_{yy}\\mathbf{\\hat{b}_y}\\otimes \\mathbf{\\hat{b}_y} + I_{yz}\\mathbf{\\hat{b}_y}\\otimes \\mathbf{\\hat{b}_z} + I_{xz}\\mathbf{\\hat{b}_z}\\otimes \\mathbf{\\hat{b}_x} + I_{yz}\\mathbf{\\hat{b}_z}\\otimes \\mathbf{\\hat{b}_y} + I_{zz}\\mathbf{\\hat{b}_z}\\otimes \\mathbf{\\hat{b}_z}$"
+      ],
+      "text/plain": [
+       "I_{xx} b_x⊗b_x + I_{xy} b_x⊗b_y + I_{xz} b_x⊗b_z + I_{xy} b_y⊗b_x + I_{yy} b_y⊗b_y + I_{yz} b_y⊗b_z + I_{xz} b_z⊗b_x + I_{yz} b_z⊗b_y + I_{zz} b_z⊗b_z"
+      ]
+     },
+     "execution_count": 12,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "I = me.inertia(B, Ixx, Iyy, Izz, Ixy, Iyz, Ixz)\n",
+    "I"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left[\\begin{matrix}- m \\left(- u_{2} u_{6} + u_{3} u_{5} + \\dot{u}_{1}\\right)\\\\- m \\left(u_{1} u_{6} - u_{3} u_{4} + \\dot{u}_{2}\\right)\\\\- m \\left(- u_{1} u_{5} + u_{2} u_{4} + \\dot{u}_{3}\\right)\\\\- I_{xx} \\dot{u}_{4} - I_{xy} \\dot{u}_{5} - I_{xz} \\dot{u}_{6} - \\left(- I_{xy} u_{6} + I_{xz} u_{5}\\right) u_{4} - \\left(- I_{yy} u_{6} + I_{yz} u_{5}\\right) u_{5} - \\left(- I_{yz} u_{6} + I_{zz} u_{5}\\right) u_{6}\\\\- I_{xy} \\dot{u}_{4} - I_{yy} \\dot{u}_{5} - I_{yz} \\dot{u}_{6} - \\left(I_{xx} u_{6} - I_{xz} u_{4}\\right) u_{4} - \\left(I_{xy} u_{6} - I_{yz} u_{4}\\right) u_{5} - \\left(I_{xz} u_{6} - I_{zz} u_{4}\\right) u_{6}\\\\- I_{xz} \\dot{u}_{4} - I_{yz} \\dot{u}_{5} - I_{zz} \\dot{u}_{6} - \\left(- I_{xx} u_{5} + I_{xy} u_{4}\\right) u_{4} - \\left(- I_{xy} u_{5} + I_{yy} u_{4}\\right) u_{5} - \\left(- I_{xz} u_{5} + I_{yz} u_{4}\\right) u_{6}\\end{matrix}\\right]$"
+      ],
+      "text/plain": [
+       "⎡                                                  -m⋅(-u₂⋅u₆ + u₃⋅u₅ + u₁̇)    ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                  -m⋅(u₁⋅u₆ - u₃⋅u₄ + u₂̇)     ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                  -m⋅(-u₁⋅u₅ + u₂⋅u₄ + u₃̇)    ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢-I_{xx}⋅u₄̇ - I_{xy}⋅u₅̇ - I_{xz}⋅u₆̇ - (-I_{xy}⋅u₆ + I_{xz}⋅u₅)⋅u₄ - (-I_{yy}⋅u ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢ -I_{xy}⋅u₄̇ - I_{yy}⋅u₅̇ - I_{yz}⋅u₆̇ - (I_{xx}⋅u₆ - I_{xz}⋅u₄)⋅u₄ - (I_{xy}⋅u₆ ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎣-I_{xz}⋅u₄̇ - I_{yz}⋅u₅̇ - I_{zz}⋅u₆̇ - (-I_{xx}⋅u₅ + I_{xy}⋅u₄)⋅u₄ - (-I_{xy}⋅u ↪\n",
+       "\n",
+       "↪                                                ⎤\n",
+       "↪                                                ⎥\n",
+       "↪                                                ⎥\n",
+       "↪                                                ⎥\n",
+       "↪                                                ⎥\n",
+       "↪                                                ⎥\n",
+       "↪ ₆ + I_{yz}⋅u₅)⋅u₅ - (-I_{yz}⋅u₆ + I_{zz}⋅u₅)⋅u₆⎥\n",
+       "↪                                                ⎥\n",
+       "↪  - I_{yz}⋅u₄)⋅u₅ - (I_{xz}⋅u₆ - I_{zz}⋅u₄)⋅u₆  ⎥\n",
+       "↪                                                ⎥\n",
+       "↪ ₅ + I_{yy}⋅u₄)⋅u₅ - (-I_{xz}⋅u₅ + I_{yz}⋅u₄)⋅u₆⎦"
+      ]
+     },
+     "execution_count": 13,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Rs = -m*Bo.acc(A)\n",
+    "Ts = -(B.ang_acc_in(A).dot(I) + me.cross(A_w_B, I).dot(A_w_B))\n",
+    "\n",
+    "F1s = v_Bo_1.dot(Rs) + w_B_1.dot(Ts)\n",
+    "F2s = v_Bo_2.dot(Rs) + w_B_2.dot(Ts)\n",
+    "F3s = v_Bo_3.dot(Rs) + w_B_3.dot(Ts)\n",
+    "F4s = v_Bo_4.dot(Rs) + w_B_4.dot(Ts)\n",
+    "F5s = v_Bo_5.dot(Rs) + w_B_5.dot(Ts)\n",
+    "F6s = v_Bo_6.dot(Rs) + w_B_6.dot(Ts)\n",
+    "\n",
+    "Frs = sm.Matrix([F1s, F2s, F3s, F4s, F5s, F6s])\n",
+    "Frs"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 14,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left[\\begin{matrix}- m \\left(- u_{2} u_{6} + u_{3} u_{5} + \\dot{u}_{1}\\right) + F_{x}\\\\- m \\left(u_{1} u_{6} - u_{3} u_{4} + \\dot{u}_{2}\\right) + F_{y}\\\\- m \\left(- u_{1} u_{5} + u_{2} u_{4} + \\dot{u}_{3}\\right) + F_{z}\\\\- I_{xx} \\dot{u}_{4} - I_{xy} \\dot{u}_{5} - I_{xz} \\dot{u}_{6} - \\left(- I_{xy} u_{6} + I_{xz} u_{5}\\right) u_{4} - \\left(- I_{yy} u_{6} + I_{yz} u_{5}\\right) u_{5} - \\left(- I_{yz} u_{6} + I_{zz} u_{5}\\right) u_{6} + M_{x}\\\\- I_{xy} \\dot{u}_{4} - I_{yy} \\dot{u}_{5} - I_{yz} \\dot{u}_{6} - \\left(I_{xx} u_{6} - I_{xz} u_{4}\\right) u_{4} - \\left(I_{xy} u_{6} - I_{yz} u_{4}\\right) u_{5} - \\left(I_{xz} u_{6} - I_{zz} u_{4}\\right) u_{6} + M_{x}\\\\- I_{xz} \\dot{u}_{4} - I_{yz} \\dot{u}_{5} - I_{zz} \\dot{u}_{6} - \\left(- I_{xx} u_{5} + I_{xy} u_{4}\\right) u_{4} - \\left(- I_{xy} u_{5} + I_{yy} u_{4}\\right) u_{5} - \\left(- I_{xz} u_{5} + I_{yz} u_{4}\\right) u_{6} + M_{z}\\end{matrix}\\right]$"
+      ],
+      "text/plain": [
+       "⎡                                                  -m⋅(-u₂⋅u₆ + u₃⋅u₅ + u₁̇) +  ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                  -m⋅(u₁⋅u₆ - u₃⋅u₄ + u₂̇) + F ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                  -m⋅(-u₁⋅u₅ + u₂⋅u₄ + u₃̇) +  ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢-I_{xx}⋅u₄̇ - I_{xy}⋅u₅̇ - I_{xz}⋅u₆̇ - (-I_{xy}⋅u₆ + I_{xz}⋅u₅)⋅u₄ - (-I_{yy}⋅u ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢  -I_{xy}⋅u₄̇ - I_{yy}⋅u₅̇ - I_{yz}⋅u₆̇ - (I_{xx}⋅u₆ - I_{xz}⋅u₄)⋅u₄ - (I_{xy}⋅u ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎣-I_{xz}⋅u₄̇ - I_{yz}⋅u₅̇ - I_{zz}⋅u₆̇ - (-I_{xx}⋅u₅ + I_{xy}⋅u₄)⋅u₄ - (-I_{xy}⋅u ↪\n",
+       "\n",
+       "↪ Fₓ                                                   ⎤\n",
+       "↪                                                      ⎥\n",
+       "↪ _y                                                   ⎥\n",
+       "↪                                                      ⎥\n",
+       "↪ F_z                                                  ⎥\n",
+       "↪                                                      ⎥\n",
+       "↪ ₆ + I_{yz}⋅u₅)⋅u₅ - (-I_{yz}⋅u₆ + I_{zz}⋅u₅)⋅u₆ + Mₓ ⎥\n",
+       "↪                                                      ⎥\n",
+       "↪ ₆ - I_{yz}⋅u₄)⋅u₅ - (I_{xz}⋅u₆ - I_{zz}⋅u₄)⋅u₆ + Mₓ  ⎥\n",
+       "↪                                                      ⎥\n",
+       "↪ ₅ + I_{yy}⋅u₄)⋅u₅ - (-I_{xz}⋅u₅ + I_{yz}⋅u₄)⋅u₆ + M_z⎦"
+      ]
+     },
+     "execution_count": 14,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Fr + Frs"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 15,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "u = sm.Matrix([u1, u2, u3, u4, u5, u6])\n",
+    "t = me.dynamicsymbols._t\n",
+    "ud = u.diff(t)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 16,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left[\\begin{matrix}m & 0 & 0 & 0 & 0 & 0\\\\0 & m & 0 & 0 & 0 & 0\\\\0 & 0 & m & 0 & 0 & 0\\\\0 & 0 & 0 & I_{xx} & I_{xy} & I_{xz}\\\\0 & 0 & 0 & I_{xy} & I_{yy} & I_{yz}\\\\0 & 0 & 0 & I_{xz} & I_{yz} & I_{zz}\\end{matrix}\\right]$"
+      ],
+      "text/plain": [
+       "⎡m  0  0    0       0       0   ⎤\n",
+       "⎢                               ⎥\n",
+       "⎢0  m  0    0       0       0   ⎥\n",
+       "⎢                               ⎥\n",
+       "⎢0  0  m    0       0       0   ⎥\n",
+       "⎢                               ⎥\n",
+       "⎢0  0  0  I_{xx}  I_{xy}  I_{xz}⎥\n",
+       "⎢                               ⎥\n",
+       "⎢0  0  0  I_{xy}  I_{yy}  I_{yz}⎥\n",
+       "⎢                               ⎥\n",
+       "⎣0  0  0  I_{xz}  I_{yz}  I_{zz}⎦"
+      ]
+     },
+     "execution_count": 16,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "M = -Frs.jacobian(ud)\n",
+    "M"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left[\\begin{matrix}m \\left(- u_{2} u_{6} + u_{3} u_{5}\\right)\\\\m \\left(u_{1} u_{6} - u_{3} u_{4}\\right)\\\\m \\left(- u_{1} u_{5} + u_{2} u_{4}\\right)\\\\\\left(- I_{xy} u_{6} + I_{xz} u_{5}\\right) u_{4} + \\left(- I_{yy} u_{6} + I_{yz} u_{5}\\right) u_{5} + \\left(- I_{yz} u_{6} + I_{zz} u_{5}\\right) u_{6}\\\\\\left(I_{xx} u_{6} - I_{xz} u_{4}\\right) u_{4} + \\left(I_{xy} u_{6} - I_{yz} u_{4}\\right) u_{5} + \\left(I_{xz} u_{6} - I_{zz} u_{4}\\right) u_{6}\\\\\\left(- I_{xx} u_{5} + I_{xy} u_{4}\\right) u_{4} + \\left(- I_{xy} u_{5} + I_{yy} u_{4}\\right) u_{5} + \\left(- I_{xz} u_{5} + I_{yz} u_{4}\\right) u_{6}\\end{matrix}\\right]$"
+      ],
+      "text/plain": [
+       "⎡                                  m⋅(-u₂⋅u₆ + u₃⋅u₅)                          ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                   m⋅(u₁⋅u₆ - u₃⋅u₄)                          ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                  m⋅(-u₁⋅u₅ + u₂⋅u₄)                          ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢(-I_{xy}⋅u₆ + I_{xz}⋅u₅)⋅u₄ + (-I_{yy}⋅u₆ + I_{yz}⋅u₅)⋅u₅ + (-I_{yz}⋅u₆ + I_{ ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢ (I_{xx}⋅u₆ - I_{xz}⋅u₄)⋅u₄ + (I_{xy}⋅u₆ - I_{yz}⋅u₄)⋅u₅ + (I_{xz}⋅u₆ - I_{zz ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎣(-I_{xx}⋅u₅ + I_{xy}⋅u₄)⋅u₄ + (-I_{xy}⋅u₅ + I_{yy}⋅u₄)⋅u₅ + (-I_{xz}⋅u₅ + I_{ ↪\n",
+       "\n",
+       "↪           ⎤\n",
+       "↪           ⎥\n",
+       "↪           ⎥\n",
+       "↪           ⎥\n",
+       "↪           ⎥\n",
+       "↪           ⎥\n",
+       "↪ zz}⋅u₅)⋅u₆⎥\n",
+       "↪           ⎥\n",
+       "↪ }⋅u₄)⋅u₆  ⎥\n",
+       "↪           ⎥\n",
+       "↪ yz}⋅u₄)⋅u₆⎦"
+      ]
+     },
+     "execution_count": 17,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "C = -Frs.xreplace({udi: 0 for udi in ud})\n",
+    "C"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 18,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left[\\begin{matrix}F_{x}\\\\F_{y}\\\\F_{z}\\\\M_{x}\\\\M_{x}\\\\M_{z}\\end{matrix}\\right]$"
+      ],
+      "text/plain": [
+       "⎡Fₓ ⎤\n",
+       "⎢   ⎥\n",
+       "⎢F_y⎥\n",
+       "⎢   ⎥\n",
+       "⎢F_z⎥\n",
+       "⎢   ⎥\n",
+       "⎢Mₓ ⎥\n",
+       "⎢   ⎥\n",
+       "⎢Mₓ ⎥\n",
+       "⎢   ⎥\n",
+       "⎣M_z⎦"
+      ]
+     },
+     "execution_count": 18,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "F = Fr\n",
+    "F"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# 两杆算例"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 19,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left( \\frac{l u_{1}}{2}\\mathbf{\\hat{a}_y}, \\  l u_{1}\\mathbf{\\hat{a}_y}, \\  u_{1}\\mathbf{\\hat{n}_z}, \\  u_{2}\\mathbf{\\hat{a}_x} + u_{1}\\mathbf{\\hat{n}_z}\\right)$"
+      ],
+      "text/plain": [
+       "⎛l⋅u₁                                       ⎞\n",
+       "⎜──── a_y, l⋅u₁ a_y, u₁ n_z, u₂ a_x + u₁ n_z⎟\n",
+       "⎝ 2                                         ⎠"
+      ]
+     },
+     "execution_count": 19,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "m, g, kt, kl, l = sm.symbols('m, g, k_t, k_l, l')\n",
+    "q1, q2, q3 = me.dynamicsymbols('q1, q2, q3')\n",
+    "u1, u2, u3 = me.dynamicsymbols('u1, u2, u3')\n",
+    "\n",
+    "N = me.ReferenceFrame('N')\n",
+    "A = me.ReferenceFrame('A')\n",
+    "B = me.ReferenceFrame('B')\n",
+    "\n",
+    "A.orient_axis(N, q1, N.z)\n",
+    "B.orient_axis(A, q2, A.x)\n",
+    "\n",
+    "A.set_ang_vel(N, u1*N.z)\n",
+    "B.set_ang_vel(A, u2*A.x)\n",
+    "\n",
+    "O = me.Point('O')\n",
+    "Ao = me.Point('A_O')\n",
+    "Bo = me.Point('B_O')\n",
+    "\n",
+    "Ao.set_pos(O, l/2*A.x)\n",
+    "Bo.set_pos(O, l*A.x)\n",
+    "\n",
+    "O.set_vel(N, 0)\n",
+    "Ao.v2pt_theory(O, N, A)\n",
+    "Bo.v2pt_theory(O, N, A)\n",
+    "\n",
+    "Ao.vel(N), Bo.vel(N), A.ang_vel_in(N), B.ang_vel_in(N)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 20,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle -  q_{3} u_{1} \\cos{\\left(q_{2} \\right)}\\mathbf{\\hat{b}_x} + u_{3}\\mathbf{\\hat{b}_y} + q_{3} u_{2}\\mathbf{\\hat{b}_z} + l u_{1}\\mathbf{\\hat{a}_y}$"
+      ],
+      "text/plain": [
+       "-q₃⋅u₁⋅cos(q₂) b_x + u₃ b_y + q₃⋅u₂ b_z + l⋅u₁ a_y"
+      ]
+     },
+     "execution_count": 20,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Q = me.Point('Q')\n",
+    "Q.set_pos(Bo, q3*B.y)\n",
+    "Q.set_vel(B, u3*B.y)\n",
+    "Q.v1pt_theory(Bo, N, B)\n",
+    "\n",
+    "Q.vel(N)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 21,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left( -  \\frac{l u_{1}^{2}}{2}\\mathbf{\\hat{a}_x} + \\frac{l \\dot{u}_{1}}{2}\\mathbf{\\hat{a}_y}, \\  -  l u_{1}^{2}\\mathbf{\\hat{a}_x} + l \\dot{u}_{1}\\mathbf{\\hat{a}_y}, \\  \\dot{u}_{1}\\mathbf{\\hat{n}_z}, \\  \\dot{u}_{2}\\mathbf{\\hat{a}_x} + u_{1} u_{2}\\mathbf{\\hat{a}_y} + \\dot{u}_{1}\\mathbf{\\hat{n}_z}\\right)$"
+      ],
+      "text/plain": [
+       "⎛     2                                                                        ↪\n",
+       "⎜-l⋅u₁         l⋅u₁̇           2                                                ↪\n",
+       "⎜─────── a_x + ──── a_y, -l⋅u₁  a_x + l⋅u₁̇ a_y, u₁̇ n_z, u₂̇ a_x + u₁⋅u₂ a_y + u ↪\n",
+       "⎝   2           2                                                              ↪\n",
+       "\n",
+       "↪      ⎞\n",
+       "↪      ⎟\n",
+       "↪ ₁̇ n_z⎟\n",
+       "↪      ⎠"
+      ]
+     },
+     "execution_count": 21,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Ao.acc(N), Bo.acc(N), A.ang_acc_in(N), B.ang_acc_in(N)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 22,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle (q_{3} u_{1} u_{2} \\sin{\\left(q_{2} \\right)} + q_{3} u_{1} \\sin{\\left(q_{2} \\right)} \\dot{q}_{2} - q_{3} \\cos{\\left(q_{2} \\right)} \\dot{u}_{1} - u_{1} u_{3} \\cos{\\left(q_{2} \\right)} - u_{1} \\cos{\\left(q_{2} \\right)} \\dot{q}_{3})\\mathbf{\\hat{b}_x} + (- q_{3} u_{1}^{2} \\cos^{2}{\\left(q_{2} \\right)} - q_{3} u_{2}^{2} + \\dot{u}_{3})\\mathbf{\\hat{b}_y} + (q_{3} u_{1}^{2} \\sin{\\left(q_{2} \\right)} \\cos{\\left(q_{2} \\right)} + q_{3} \\dot{u}_{2} + u_{2} u_{3} + u_{2} \\dot{q}_{3})\\mathbf{\\hat{b}_z} -  l u_{1}^{2}\\mathbf{\\hat{a}_x} + l \\dot{u}_{1}\\mathbf{\\hat{a}_y}$"
+      ],
+      "text/plain": [
+       "                                                                               ↪\n",
+       "(q₃⋅u₁⋅u₂⋅sin(q₂) + q₃⋅u₁⋅sin(q₂)⋅q₂̇ - q₃⋅cos(q₂)⋅u₁̇ - u₁⋅u₃⋅cos(q₂) - u₁⋅cos( ↪\n",
+       "\n",
+       "↪               ⎛       2    2            2     ⎞       ⎛     2                ↪\n",
+       "↪ q₂)⋅q₃̇) b_x + ⎝- q₃⋅u₁ ⋅cos (q₂) - q₃⋅u₂  + u₃̇⎠ b_y + ⎝q₃⋅u₁ ⋅sin(q₂)⋅cos(q₂ ↪\n",
+       "\n",
+       "↪                          ⎞            2               \n",
+       "↪ ) + q₃⋅u₂̇ + u₂⋅u₃ + u₂⋅q₃̇⎠ b_z + -l⋅u₁  a_x + l⋅u₁̇ a_y"
+      ]
+     },
+     "execution_count": 22,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Q.acc(N)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 23,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle (2 q_{3} u_{1} u_{2} \\sin{\\left(q_{2} \\right)} - q_{3} \\cos{\\left(q_{2} \\right)} \\dot{u}_{1} - 2 u_{1} u_{3} \\cos{\\left(q_{2} \\right)})\\mathbf{\\hat{b}_x} + (- q_{3} u_{1}^{2} \\cos^{2}{\\left(q_{2} \\right)} - q_{3} u_{2}^{2} + \\dot{u}_{3})\\mathbf{\\hat{b}_y} + (q_{3} u_{1}^{2} \\sin{\\left(q_{2} \\right)} \\cos{\\left(q_{2} \\right)} + q_{3} \\dot{u}_{2} + 2 u_{2} u_{3})\\mathbf{\\hat{b}_z} -  l u_{1}^{2}\\mathbf{\\hat{a}_x} + l \\dot{u}_{1}\\mathbf{\\hat{a}_y}$"
+      ],
+      "text/plain": [
+       "                                                             ⎛       2    2    ↪\n",
+       "(2⋅q₃⋅u₁⋅u₂⋅sin(q₂) - q₃⋅cos(q₂)⋅u₁̇ - 2⋅u₁⋅u₃⋅cos(q₂)) b_x + ⎝- q₃⋅u₁ ⋅cos (q₂ ↪\n",
+       "\n",
+       "↪          2     ⎞       ⎛     2                                  ⎞            ↪\n",
+       "↪ ) - q₃⋅u₂  + u₃̇⎠ b_y + ⎝q₃⋅u₁ ⋅sin(q₂)⋅cos(q₂) + q₃⋅u₂̇ + 2⋅u₂⋅u₃⎠ b_z + -l⋅u ↪\n",
+       "\n",
+       "↪  2               \n",
+       "↪ ₁  a_x + l⋅u₁̇ a_y"
+      ]
+     },
+     "execution_count": 23,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "t = me.dynamicsymbols._t\n",
+    "\n",
+    "qdot_repl = {q1.diff(t): u1,\n",
+    "             q2.diff(t): u2,\n",
+    "             q3.diff(t): u3}\n",
+    "\n",
+    "Q.set_acc(N, Q.acc(N).xreplace(qdot_repl))\n",
+    "Q.acc(N)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 24,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "R_Ao = m*g*N.x\n",
+    "R_Bo = m*g*N.x + kl*q3*B.y\n",
+    "R_Q = m/4*g*N.x - kl*q3*B.y\n",
+    "T_A = -kt*q1*N.z + kt*q2*A.x\n",
+    "T_B = -kt*q2*A.x"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 25,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "I = m*l**2/12\n",
+    "I_A_Ao = I*me.outer(A.y, A.y) + I*me.outer(A.z, A.z)\n",
+    "I_B_Bo = I*me.outer(B.x, B.x) + I*me.outer(B.z, B.z)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 26,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "v_Ao_1 = Ao.vel(N).diff(u1, N)\n",
+    "v_Bo_1 = Bo.vel(N).diff(u1, N)\n",
+    "v_Q_1 = Q.vel(N).diff(u1, N)\n",
+    "\n",
+    "v_Ao_2 = Ao.vel(N).diff(u2, N)\n",
+    "v_Bo_2 = Bo.vel(N).diff(u2, N)\n",
+    "v_Q_2 = Q.vel(N).diff(u2, N)\n",
+    "\n",
+    "v_Ao_3 = Ao.vel(N).diff(u3, N)\n",
+    "v_Bo_3 = Bo.vel(N).diff(u3, N)\n",
+    "v_Q_3 = Q.vel(N).diff(u3, N)\n",
+    "\n",
+    "w_A_1 = A.ang_vel_in(N).diff(u1, N)\n",
+    "w_B_1 = B.ang_vel_in(N).diff(u1, N)\n",
+    "\n",
+    "w_A_2 = A.ang_vel_in(N).diff(u2, N)\n",
+    "w_B_2 = B.ang_vel_in(N).diff(u2, N)\n",
+    "\n",
+    "w_A_3 = A.ang_vel_in(N).diff(u3, N)\n",
+    "w_B_3 = B.ang_vel_in(N).diff(u3, N)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 27,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "F1 = v_Ao_1.dot(R_Ao) + v_Bo_1.dot(R_Bo) + v_Q_1.dot(R_Q) + w_A_1.dot(T_A) + w_B_1.dot(T_B)\n",
+    "F2 = v_Ao_2.dot(R_Ao) + v_Bo_2.dot(R_Bo) + v_Q_2.dot(R_Q) + w_A_2.dot(T_A) + w_B_2.dot(T_B)\n",
+    "F3 = v_Ao_3.dot(R_Ao) + v_Bo_3.dot(R_Bo) + v_Q_3.dot(R_Q) + w_A_3.dot(T_A) + w_B_3.dot(T_B)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 28,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left[\\begin{matrix}- \\frac{7 g l m \\sin{\\left(q_{1} \\right)}}{4} - \\frac{g m q_{3} \\cos{\\left(q_{1} \\right)} \\cos{\\left(q_{2} \\right)}}{4} - k_{t} q_{1}\\\\\\frac{g m q_{3} \\sin{\\left(q_{1} \\right)} \\sin{\\left(q_{2} \\right)}}{4} - k_{t} q_{2}\\\\- \\frac{g m \\sin{\\left(q_{1} \\right)} \\cos{\\left(q_{2} \\right)}}{4} - k_{l} q_{3}\\end{matrix}\\right]$"
+      ],
+      "text/plain": [
+       "⎡  7⋅g⋅l⋅m⋅sin(q₁)   g⋅m⋅q₃⋅cos(q₁)⋅cos(q₂)        ⎤\n",
+       "⎢- ─────────────── - ────────────────────── - kₜ⋅q₁⎥\n",
+       "⎢         4                    4                   ⎥\n",
+       "⎢                                                  ⎥\n",
+       "⎢          g⋅m⋅q₃⋅sin(q₁)⋅sin(q₂)                  ⎥\n",
+       "⎢          ────────────────────── - kₜ⋅q₂          ⎥\n",
+       "⎢                    4                             ⎥\n",
+       "⎢                                                  ⎥\n",
+       "⎢            g⋅m⋅sin(q₁)⋅cos(q₂)                   ⎥\n",
+       "⎢          - ─────────────────── - kₗ⋅q₃           ⎥\n",
+       "⎣                     4                            ⎦"
+      ]
+     },
+     "execution_count": 28,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Fr = sm.Matrix([F1, F2, F3])\n",
+    "Fr"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 29,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "TAs = -(A.ang_acc_in(N).dot(I_A_Ao) + me.cross(A.ang_vel_in(N), I_A_Ao).dot(A.ang_vel_in(N)))\n",
+    "TBs = -(B.ang_acc_in(N).dot(I_B_Bo) + me.cross(B.ang_vel_in(N), I_B_Bo).dot(B.ang_vel_in(N)))\n",
+    "\n",
+    "F1s = v_Ao_1.dot(-m*Ao.acc(N)) + v_Bo_1.dot(-m*Bo.acc(N)) + v_Q_1.dot(-m/4*Q.acc(N))\n",
+    "F1s += w_A_1.dot(TAs) + w_B_1.dot(TBs)\n",
+    "\n",
+    "F2s = v_Ao_2.dot(-m*Ao.acc(N)) + v_Bo_2.dot(-m*Bo.acc(N)) + v_Q_2.dot(-m/4*Q.acc(N))\n",
+    "F2s += w_A_2.dot(TAs) + w_B_2.dot(TBs)\n",
+    "\n",
+    "F3s = v_Ao_3.dot(-m*Ao.acc(N)) + v_Bo_3.dot(-m*Bo.acc(N)) + v_Q_3.dot(-m/4*Q.acc(N))\n",
+    "F3s += w_A_3.dot(TAs) + w_B_3.dot(TBs)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 30,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left[\\begin{matrix}- \\frac{19 l^{2} m \\dot{u}_{1}}{12} - \\frac{l m q_{3} u_{1}^{2} \\cos{\\left(q_{2} \\right)}}{4} + l \\left(- \\frac{m \\left(- q_{3} u_{1}^{2} \\cos^{2}{\\left(q_{2} \\right)} - q_{3} u_{2}^{2} + \\dot{u}_{3}\\right) \\cos{\\left(q_{2} \\right)}}{4} + \\frac{m \\left(q_{3} u_{1}^{2} \\sin{\\left(q_{2} \\right)} \\cos{\\left(q_{2} \\right)} + q_{3} \\dot{u}_{2} + 2 u_{2} u_{3}\\right) \\sin{\\left(q_{2} \\right)}}{4}\\right) + \\frac{m \\left(2 q_{3} u_{1} u_{2} \\sin{\\left(q_{2} \\right)} - q_{3} \\cos{\\left(q_{2} \\right)} \\dot{u}_{1} - 2 u_{1} u_{3} \\cos{\\left(q_{2} \\right)}\\right) q_{3} \\cos{\\left(q_{2} \\right)}}{4} + \\left(- \\frac{l^{2} m \\left(- u_{1} u_{2} \\sin{\\left(q_{2} \\right)} + \\cos{\\left(q_{2} \\right)} \\dot{u}_{1}\\right)}{12} + \\frac{l^{2} m u_{1} u_{2} \\sin{\\left(q_{2} \\right)}}{12}\\right) \\cos{\\left(q_{2} \\right)}\\\\- \\frac{l^{2} m u_{1}^{2} \\sin{\\left(q_{2} \\right)} \\cos{\\left(q_{2} \\right)}}{12} - \\frac{l^{2} m \\dot{u}_{2}}{12} + \\frac{l m q_{3} \\sin{\\left(q_{2} \\right)} \\dot{u}_{1}}{4} - \\frac{m \\left(q_{3} u_{1}^{2} \\sin{\\left(q_{2} \\right)} \\cos{\\left(q_{2} \\right)} + q_{3} \\dot{u}_{2} + 2 u_{2} u_{3}\\right) q_{3}}{4}\\\\- \\frac{l m \\cos{\\left(q_{2} \\right)} \\dot{u}_{1}}{4} - \\frac{m \\left(- q_{3} u_{1}^{2} \\cos^{2}{\\left(q_{2} \\right)} - q_{3} u_{2}^{2} + \\dot{u}_{3}\\right)}{4}\\end{matrix}\\right]$"
+      ],
+      "text/plain": [
+       "⎡      2                 2             ⎛    ⎛       2    2            2     ⎞  ↪\n",
+       "⎢  19⋅l ⋅m⋅u₁̇   l⋅m⋅q₃⋅u₁ ⋅cos(q₂)     ⎜  m⋅⎝- q₃⋅u₁ ⋅cos (q₂) - q₃⋅u₂  + u₃̇⎠⋅ ↪\n",
+       "⎢- ────────── - ────────────────── + l⋅⎜- ──────────────────────────────────── ↪\n",
+       "⎢      12               4              ⎝                       4               ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎣                                                                              ↪\n",
+       "\n",
+       "↪             ⎛     2                                  ⎞        ⎞              ↪\n",
+       "↪ cos(q₂)   m⋅⎝q₃⋅u₁ ⋅sin(q₂)⋅cos(q₂) + q₃⋅u₂̇ + 2⋅u₂⋅u₃⎠⋅sin(q₂)⎟   m⋅(2⋅q₃⋅u₁ ↪\n",
+       "↪ ─────── + ────────────────────────────────────────────────────⎟ + ────────── ↪\n",
+       "↪                                    4                          ⎠              ↪\n",
+       "↪                                                                              ↪\n",
+       "↪              2     2                    2                              ⎛     ↪\n",
+       "↪             l ⋅m⋅u₁ ⋅sin(q₂)⋅cos(q₂)   l ⋅m⋅u₂̇   l⋅m⋅q₃⋅sin(q₂)⋅u₁̇   m⋅⎝q₃⋅u ↪\n",
+       "↪           - ──────────────────────── - ─────── + ───────────────── - ─────── ↪\n",
+       "↪                        12                12              4                   ↪\n",
+       "↪                                                                              ↪\n",
+       "↪                                                          ⎛       2    2      ↪\n",
+       "↪                                       l⋅m⋅cos(q₂)⋅u₁̇   m⋅⎝- q₃⋅u₁ ⋅cos (q₂)  ↪\n",
+       "↪                                     - ────────────── - ───────────────────── ↪\n",
+       "↪                                             4                           4    ↪\n",
+       "\n",
+       "↪                                                             ⎛   2            ↪\n",
+       "↪ ⋅u₂⋅sin(q₂) - q₃⋅cos(q₂)⋅u₁̇ - 2⋅u₁⋅u₃⋅cos(q₂))⋅q₃⋅cos(q₂)   ⎜  l ⋅m⋅(-u₁⋅u₂⋅ ↪\n",
+       "↪ ───────────────────────────────────────────────────────── + ⎜- ───────────── ↪\n",
+       "↪                        4                                    ⎝                ↪\n",
+       "↪                                                                              ↪\n",
+       "↪  2                                  ⎞                                        ↪\n",
+       "↪ ₁ ⋅sin(q₂)⋅cos(q₂) + q₃⋅u₂̇ + 2⋅u₂⋅u₃⎠⋅q₃                                     ↪\n",
+       "↪ ────────────────────────────────────────                                     ↪\n",
+       "↪                 4                                                            ↪\n",
+       "↪                                                                              ↪\n",
+       "↪        2     ⎞                                                               ↪\n",
+       "↪ - q₃⋅u₂  + u₃̇⎠                                                               ↪\n",
+       "↪ ──────────────                                                               ↪\n",
+       "↪                                                                              ↪\n",
+       "\n",
+       "↪                          2                ⎞        ⎤\n",
+       "↪ sin(q₂) + cos(q₂)⋅u₁̇)   l ⋅m⋅u₁⋅u₂⋅sin(q₂)⎟        ⎥\n",
+       "↪ ───────────────────── + ──────────────────⎟⋅cos(q₂)⎥\n",
+       "↪    12                           12        ⎠        ⎥\n",
+       "↪                                                    ⎥\n",
+       "↪                                                    ⎥\n",
+       "↪                                                    ⎥\n",
+       "↪                                                    ⎥\n",
+       "↪                                                    ⎥\n",
+       "↪                                                    ⎥\n",
+       "↪                                                    ⎥\n",
+       "↪                                                    ⎥\n",
+       "↪                                                    ⎥\n",
+       "↪                                                    ⎦"
+      ]
+     },
+     "execution_count": 30,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Frs = sm.Matrix([F1s, F2s, F3s])\n",
+    "Frs"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 31,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left\\{q_{1}, q_{2}, q_{3}\\right\\}$"
+      ],
+      "text/plain": [
+       "{q₁, q₂, q₃}"
+      ]
+     },
+     "execution_count": 31,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "me.find_dynamicsymbols(Fr)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 32,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left\\{q_{2}, q_{3}, u_{1}, u_{2}, u_{3}, \\dot{u}_{1}, \\dot{u}_{2}, \\dot{u}_{3}\\right\\}$"
+      ],
+      "text/plain": [
+       "{q₂, q₃, u₁, u₂, u₃, u₁̇, u₂̇, u₃̇}"
+      ]
+     },
+     "execution_count": 32,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "me.find_dynamicsymbols(Frs)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 33,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "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": 33,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "u = sm.Matrix([u1, u2, u3])\n",
+    "ud = u.diff(t)\n",
+    "ud"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 34,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left[\\begin{matrix}- \\frac{l^{2} m \\cos^{2}{\\left(q_{2} \\right)}}{12} - \\frac{19 l^{2} m}{12} - \\frac{m q_{3}^{2} \\cos^{2}{\\left(q_{2} \\right)}}{4} & \\frac{l m q_{3} \\sin{\\left(q_{2} \\right)}}{4} & - \\frac{l m \\cos{\\left(q_{2} \\right)}}{4}\\\\\\frac{l m q_{3} \\sin{\\left(q_{2} \\right)}}{4} & - \\frac{l^{2} m}{12} - \\frac{m q_{3}^{2}}{4} & 0\\\\- \\frac{l m \\cos{\\left(q_{2} \\right)}}{4} & 0 & - \\frac{m}{4}\\end{matrix}\\right]$"
+      ],
+      "text/plain": [
+       "⎡   2      2           2         2    2                                   ⎤\n",
+       "⎢  l ⋅m⋅cos (q₂)   19⋅l ⋅m   m⋅q₃ ⋅cos (q₂)  l⋅m⋅q₃⋅sin(q₂)  -l⋅m⋅cos(q₂) ⎥\n",
+       "⎢- ───────────── - ─────── - ──────────────  ──────────────  ─────────────⎥\n",
+       "⎢       12           12            4               4               4      ⎥\n",
+       "⎢                                                                         ⎥\n",
+       "⎢                                               2         2               ⎥\n",
+       "⎢              l⋅m⋅q₃⋅sin(q₂)                  l ⋅m   m⋅q₃                ⎥\n",
+       "⎢              ──────────────                - ──── - ─────        0      ⎥\n",
+       "⎢                    4                          12      4                 ⎥\n",
+       "⎢                                                                         ⎥\n",
+       "⎢              -l⋅m⋅cos(q₂)                                       -m      ⎥\n",
+       "⎢              ─────────────                       0              ───     ⎥\n",
+       "⎣                    4                                             4      ⎦"
+      ]
+     },
+     "execution_count": 34,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Md = Frs.jacobian(ud)\n",
+    "Md"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 35,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left\\{ \\dot{u}_{1} : 0, \\  \\dot{u}_{2} : 0, \\  \\dot{u}_{3} : 0\\right\\}$"
+      ],
+      "text/plain": [
+       "{u₁̇: 0, u₂̇: 0, u₃̇: 0}"
+      ]
+     },
+     "execution_count": 35,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "ud_zerod = {udr: 0 for udr in ud}\n",
+    "ud_zerod\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 36,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left[\\begin{matrix}- \\frac{7 g l m \\sin{\\left(q_{1} \\right)}}{4} - \\frac{g m q_{3} \\cos{\\left(q_{1} \\right)} \\cos{\\left(q_{2} \\right)}}{4} - k_{t} q_{1} + \\frac{l^{2} m u_{1} u_{2} \\sin{\\left(q_{2} \\right)} \\cos{\\left(q_{2} \\right)}}{6} - \\frac{l m q_{3} u_{1}^{2} \\cos{\\left(q_{2} \\right)}}{4} + l \\left(- \\frac{m \\left(- q_{3} u_{1}^{2} \\cos^{2}{\\left(q_{2} \\right)} - q_{3} u_{2}^{2}\\right) \\cos{\\left(q_{2} \\right)}}{4} + \\frac{m \\left(q_{3} u_{1}^{2} \\sin{\\left(q_{2} \\right)} \\cos{\\left(q_{2} \\right)} + 2 u_{2} u_{3}\\right) \\sin{\\left(q_{2} \\right)}}{4}\\right) + \\frac{m \\left(2 q_{3} u_{1} u_{2} \\sin{\\left(q_{2} \\right)} - 2 u_{1} u_{3} \\cos{\\left(q_{2} \\right)}\\right) q_{3} \\cos{\\left(q_{2} \\right)}}{4}\\\\\\frac{g m q_{3} \\sin{\\left(q_{1} \\right)} \\sin{\\left(q_{2} \\right)}}{4} - k_{t} q_{2} - \\frac{l^{2} m u_{1}^{2} \\sin{\\left(q_{2} \\right)} \\cos{\\left(q_{2} \\right)}}{12} - \\frac{m \\left(q_{3} u_{1}^{2} \\sin{\\left(q_{2} \\right)} \\cos{\\left(q_{2} \\right)} + 2 u_{2} u_{3}\\right) q_{3}}{4}\\\\- \\frac{g m \\sin{\\left(q_{1} \\right)} \\cos{\\left(q_{2} \\right)}}{4} - k_{l} q_{3} - \\frac{m \\left(- q_{3} u_{1}^{2} \\cos^{2}{\\left(q_{2} \\right)} - q_{3} u_{2}^{2}\\right)}{4}\\end{matrix}\\right]$"
+      ],
+      "text/plain": [
+       "⎡                                                      2                       ↪\n",
+       "⎢  7⋅g⋅l⋅m⋅sin(q₁)   g⋅m⋅q₃⋅cos(q₁)⋅cos(q₂)           l ⋅m⋅u₁⋅u₂⋅sin(q₂)⋅cos(q ↪\n",
+       "⎢- ─────────────── - ────────────────────── - kₜ⋅q₁ + ──────────────────────── ↪\n",
+       "⎢         4                    4                                  6            ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                          g⋅m ↪\n",
+       "⎢                                                                          ─── ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎣                                                                              ↪\n",
+       "\n",
+       "↪               2             ⎛    ⎛       2    2            2⎞             ⎛  ↪\n",
+       "↪ ₂)   l⋅m⋅q₃⋅u₁ ⋅cos(q₂)     ⎜  m⋅⎝- q₃⋅u₁ ⋅cos (q₂) - q₃⋅u₂ ⎠⋅cos(q₂)   m⋅⎝q ↪\n",
+       "↪ ── - ────────────────── + l⋅⎜- ────────────────────────────────────── + ──── ↪\n",
+       "↪              4              ⎝                    4                           ↪\n",
+       "↪                                                                              ↪\n",
+       "↪                                2     2                     ⎛     2           ↪\n",
+       "↪ ⋅q₃⋅sin(q₁)⋅sin(q₂)           l ⋅m⋅u₁ ⋅sin(q₂)⋅cos(q₂)   m⋅⎝q₃⋅u₁ ⋅sin(q₂)⋅c ↪\n",
+       "↪ ─────────────────── - kₜ⋅q₂ - ──────────────────────── - ─────────────────── ↪\n",
+       "↪        4                                 12                                  ↪\n",
+       "↪                                                                              ↪\n",
+       "↪                                                   ⎛       2    2             ↪\n",
+       "↪                   g⋅m⋅sin(q₁)⋅cos(q₂)           m⋅⎝- q₃⋅u₁ ⋅cos (q₂) - q₃⋅u₂ ↪\n",
+       "↪                 - ─────────────────── - kₗ⋅q₃ - ──────────────────────────── ↪\n",
+       "↪                            4                                  4              ↪\n",
+       "\n",
+       "↪     2                          ⎞        ⎞                                    ↪\n",
+       "↪ ₃⋅u₁ ⋅sin(q₂)⋅cos(q₂) + 2⋅u₂⋅u₃⎠⋅sin(q₂)⎟   m⋅(2⋅q₃⋅u₁⋅u₂⋅sin(q₂) - 2⋅u₁⋅u₃⋅ ↪\n",
+       "↪ ────────────────────────────────────────⎟ + ──────────────────────────────── ↪\n",
+       "↪                  4                      ⎠                            4       ↪\n",
+       "↪                                                                              ↪\n",
+       "↪                 ⎞                                                            ↪\n",
+       "↪ os(q₂) + 2⋅u₂⋅u₃⎠⋅q₃                                                         ↪\n",
+       "↪ ────────────────────                                                         ↪\n",
+       "↪ 4                                                                            ↪\n",
+       "↪                                                                              ↪\n",
+       "↪ 2⎞                                                                           ↪\n",
+       "↪  ⎠                                                                           ↪\n",
+       "↪ ──                                                                           ↪\n",
+       "↪                                                                              ↪\n",
+       "\n",
+       "↪                    ⎤\n",
+       "↪ cos(q₂))⋅q₃⋅cos(q₂)⎥\n",
+       "↪ ───────────────────⎥\n",
+       "↪                    ⎥\n",
+       "↪                    ⎥\n",
+       "↪                    ⎥\n",
+       "↪                    ⎥\n",
+       "↪                    ⎥\n",
+       "↪                    ⎥\n",
+       "↪                    ⎥\n",
+       "↪                    ⎥\n",
+       "↪                    ⎥\n",
+       "↪                    ⎥\n",
+       "↪                    ⎦"
+      ]
+     },
+     "execution_count": 36,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "gd = Frs.xreplace(ud_zerod) + Fr\n",
+    "gd"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 37,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left\\{q_{2}, q_{3}\\right\\}$"
+      ],
+      "text/plain": [
+       "{q₂, q₃}"
+      ]
+     },
+     "execution_count": 37,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "me.find_dynamicsymbols(Md)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 38,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left\\{q_{1}, q_{2}, q_{3}, u_{1}, u_{2}, u_{3}\\right\\}$"
+      ],
+      "text/plain": [
+       "{q₁, q₂, q₃, u₁, u₂, u₃}"
+      ]
+     },
+     "execution_count": 38,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "me.find_dynamicsymbols(gd)"
+   ]
   }
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Turtle",
+   "display_name": "MinerU",
    "language": "python",
    "name": "python3"
   },
   "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
    "name": "python",
-   "version": "3.11.8"
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.16"
   }
  },
  "nbformat": 4,
diff --git a/补课/多体动力学/18_Simulation_and_Visualization.ipynb b/补课/多体动力学/18_Simulation_and_Visualization.ipynb
new file mode 100644
index 0000000..1b9e38e
--- /dev/null
+++ b/补课/多体动力学/18_Simulation_and_Visualization.ipynb
@@ -0,0 +1,1374 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import sympy as sm\n",
+    "import sympy.physics.mechanics as me\n",
+    "me.init_vprinting(use_latex='mathjax')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left( \\frac{l u_{1}}{2}\\mathbf{\\hat{a}_y}, \\  l u_{1}\\mathbf{\\hat{a}_y}, \\  u_{1}\\mathbf{\\hat{n}_z}, \\  u_{2}\\mathbf{\\hat{a}_x} + u_{1}\\mathbf{\\hat{n}_z}\\right)$"
+      ],
+      "text/plain": [
+       "⎛l⋅u₁                                       ⎞\n",
+       "⎜──── a_y, l⋅u₁ a_y, u₁ n_z, u₂ a_x + u₁ n_z⎟\n",
+       "⎝ 2                                         ⎠"
+      ]
+     },
+     "execution_count": 5,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "m, g, kt, kl, l = sm.symbols('m, g, k_t, k_l, l')\n",
+    "q1, q2, q3 = me.dynamicsymbols('q1, q2, q3')\n",
+    "u1, u2, u3 = me.dynamicsymbols('u1, u2, u3')\n",
+    "\n",
+    "N = me.ReferenceFrame('N')\n",
+    "A = me.ReferenceFrame('A')\n",
+    "B = me.ReferenceFrame('B')\n",
+    "\n",
+    "A.orient_axis(N, q1, N.z)\n",
+    "B.orient_axis(A, q2, A.x)\n",
+    "\n",
+    "A.set_ang_vel(N, u1*N.z)\n",
+    "B.set_ang_vel(A, u2*A.x)\n",
+    "\n",
+    "O = me.Point('O')\n",
+    "Ao = me.Point('A_O')\n",
+    "Bo = me.Point('B_O')\n",
+    "\n",
+    "Ao.set_pos(O, l/2*A.x)\n",
+    "Bo.set_pos(O, l*A.x)\n",
+    "\n",
+    "O.set_vel(N, 0)\n",
+    "Ao.v2pt_theory(O, N, A)\n",
+    "Bo.v2pt_theory(O, N, A)\n",
+    "\n",
+    "Ao.vel(N), Bo.vel(N), A.ang_vel_in(N), B.ang_vel_in(N)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle -  q_{3} u_{1} \\cos{\\left(q_{2} \\right)}\\mathbf{\\hat{b}_x} + u_{3}\\mathbf{\\hat{b}_y} + q_{3} u_{2}\\mathbf{\\hat{b}_z} + l u_{1}\\mathbf{\\hat{a}_y}$"
+      ],
+      "text/plain": [
+       "-q₃⋅u₁⋅cos(q₂) b_x + u₃ b_y + q₃⋅u₂ b_z + l⋅u₁ a_y"
+      ]
+     },
+     "execution_count": 6,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Q = me.Point('Q')\n",
+    "Q.set_pos(Bo, q3*B.y)\n",
+    "Q.set_vel(B, u3*B.y)\n",
+    "Q.v1pt_theory(Bo, N, B)\n",
+    "\n",
+    "Q.vel(N)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left( -  \\frac{l u_{1}^{2}}{2}\\mathbf{\\hat{a}_x} + \\frac{l \\dot{u}_{1}}{2}\\mathbf{\\hat{a}_y}, \\  -  l u_{1}^{2}\\mathbf{\\hat{a}_x} + l \\dot{u}_{1}\\mathbf{\\hat{a}_y}, \\  \\dot{u}_{1}\\mathbf{\\hat{n}_z}, \\  \\dot{u}_{2}\\mathbf{\\hat{a}_x} + u_{1} u_{2}\\mathbf{\\hat{a}_y} + \\dot{u}_{1}\\mathbf{\\hat{n}_z}\\right)$"
+      ],
+      "text/plain": [
+       "⎛     2                                                                        ↪\n",
+       "⎜-l⋅u₁         l⋅u₁̇           2                                                ↪\n",
+       "⎜─────── a_x + ──── a_y, -l⋅u₁  a_x + l⋅u₁̇ a_y, u₁̇ n_z, u₂̇ a_x + u₁⋅u₂ a_y + u ↪\n",
+       "⎝   2           2                                                              ↪\n",
+       "\n",
+       "↪      ⎞\n",
+       "↪      ⎟\n",
+       "↪ ₁̇ n_z⎟\n",
+       "↪      ⎠"
+      ]
+     },
+     "execution_count": 7,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Ao.acc(N), Bo.acc(N), A.ang_acc_in(N), B.ang_acc_in(N)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle (q_{3} u_{1} u_{2} \\sin{\\left(q_{2} \\right)} + q_{3} u_{1} \\sin{\\left(q_{2} \\right)} \\dot{q}_{2} - q_{3} \\cos{\\left(q_{2} \\right)} \\dot{u}_{1} - u_{1} u_{3} \\cos{\\left(q_{2} \\right)} - u_{1} \\cos{\\left(q_{2} \\right)} \\dot{q}_{3})\\mathbf{\\hat{b}_x} + (- q_{3} u_{1}^{2} \\cos^{2}{\\left(q_{2} \\right)} - q_{3} u_{2}^{2} + \\dot{u}_{3})\\mathbf{\\hat{b}_y} + (q_{3} u_{1}^{2} \\sin{\\left(q_{2} \\right)} \\cos{\\left(q_{2} \\right)} + q_{3} \\dot{u}_{2} + u_{2} u_{3} + u_{2} \\dot{q}_{3})\\mathbf{\\hat{b}_z} -  l u_{1}^{2}\\mathbf{\\hat{a}_x} + l \\dot{u}_{1}\\mathbf{\\hat{a}_y}$"
+      ],
+      "text/plain": [
+       "                                                                               ↪\n",
+       "(q₃⋅u₁⋅u₂⋅sin(q₂) + q₃⋅u₁⋅sin(q₂)⋅q₂̇ - q₃⋅cos(q₂)⋅u₁̇ - u₁⋅u₃⋅cos(q₂) - u₁⋅cos( ↪\n",
+       "\n",
+       "↪               ⎛       2    2            2     ⎞       ⎛     2                ↪\n",
+       "↪ q₂)⋅q₃̇) b_x + ⎝- q₃⋅u₁ ⋅cos (q₂) - q₃⋅u₂  + u₃̇⎠ b_y + ⎝q₃⋅u₁ ⋅sin(q₂)⋅cos(q₂ ↪\n",
+       "\n",
+       "↪                          ⎞            2               \n",
+       "↪ ) + q₃⋅u₂̇ + u₂⋅u₃ + u₂⋅q₃̇⎠ b_z + -l⋅u₁  a_x + l⋅u₁̇ a_y"
+      ]
+     },
+     "execution_count": 8,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Q.acc(N)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle (2 q_{3} u_{1} u_{2} \\sin{\\left(q_{2} \\right)} - q_{3} \\cos{\\left(q_{2} \\right)} \\dot{u}_{1} - 2 u_{1} u_{3} \\cos{\\left(q_{2} \\right)})\\mathbf{\\hat{b}_x} + (- q_{3} u_{1}^{2} \\cos^{2}{\\left(q_{2} \\right)} - q_{3} u_{2}^{2} + \\dot{u}_{3})\\mathbf{\\hat{b}_y} + (q_{3} u_{1}^{2} \\sin{\\left(q_{2} \\right)} \\cos{\\left(q_{2} \\right)} + q_{3} \\dot{u}_{2} + 2 u_{2} u_{3})\\mathbf{\\hat{b}_z} -  l u_{1}^{2}\\mathbf{\\hat{a}_x} + l \\dot{u}_{1}\\mathbf{\\hat{a}_y}$"
+      ],
+      "text/plain": [
+       "                                                             ⎛       2    2    ↪\n",
+       "(2⋅q₃⋅u₁⋅u₂⋅sin(q₂) - q₃⋅cos(q₂)⋅u₁̇ - 2⋅u₁⋅u₃⋅cos(q₂)) b_x + ⎝- q₃⋅u₁ ⋅cos (q₂ ↪\n",
+       "\n",
+       "↪          2     ⎞       ⎛     2                                  ⎞            ↪\n",
+       "↪ ) - q₃⋅u₂  + u₃̇⎠ b_y + ⎝q₃⋅u₁ ⋅sin(q₂)⋅cos(q₂) + q₃⋅u₂̇ + 2⋅u₂⋅u₃⎠ b_z + -l⋅u ↪\n",
+       "\n",
+       "↪  2               \n",
+       "↪ ₁  a_x + l⋅u₁̇ a_y"
+      ]
+     },
+     "execution_count": 9,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "t = me.dynamicsymbols._t\n",
+    "\n",
+    "qdot_repl = {q1.diff(t): u1,\n",
+    "             q2.diff(t): u2,\n",
+    "             q3.diff(t): u3}\n",
+    "\n",
+    "Q.set_acc(N, Q.acc(N).xreplace(qdot_repl))\n",
+    "Q.acc(N)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "R_Ao = m*g*N.x\n",
+    "R_Bo = m*g*N.x + kl*q3*B.y\n",
+    "R_Q = m/4*g*N.x - kl*q3*B.y\n",
+    "T_A = -kt*q1*N.z + kt*q2*A.x\n",
+    "T_B = -kt*q2*A.x"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "I = m*l**2/12\n",
+    "I_A_Ao = I*me.outer(A.y, A.y) + I*me.outer(A.z, A.z)\n",
+    "I_B_Bo = I*me.outer(B.x, B.x) + I*me.outer(B.z, B.z)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 12,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "v_Ao_1 = Ao.vel(N).diff(u1, N)\n",
+    "v_Bo_1 = Bo.vel(N).diff(u1, N)\n",
+    "v_Q_1 = Q.vel(N).diff(u1, N)\n",
+    "\n",
+    "v_Ao_2 = Ao.vel(N).diff(u2, N)\n",
+    "v_Bo_2 = Bo.vel(N).diff(u2, N)\n",
+    "v_Q_2 = Q.vel(N).diff(u2, N)\n",
+    "\n",
+    "v_Ao_3 = Ao.vel(N).diff(u3, N)\n",
+    "v_Bo_3 = Bo.vel(N).diff(u3, N)\n",
+    "v_Q_3 = Q.vel(N).diff(u3, N)\n",
+    "\n",
+    "w_A_1 = A.ang_vel_in(N).diff(u1, N)\n",
+    "w_B_1 = B.ang_vel_in(N).diff(u1, N)\n",
+    "\n",
+    "w_A_2 = A.ang_vel_in(N).diff(u2, N)\n",
+    "w_B_2 = B.ang_vel_in(N).diff(u2, N)\n",
+    "\n",
+    "w_A_3 = A.ang_vel_in(N).diff(u3, N)\n",
+    "w_B_3 = B.ang_vel_in(N).diff(u3, N)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "F1 = v_Ao_1.dot(R_Ao) + v_Bo_1.dot(R_Bo) + v_Q_1.dot(R_Q) + w_A_1.dot(T_A) + w_B_1.dot(T_B)\n",
+    "F2 = v_Ao_2.dot(R_Ao) + v_Bo_2.dot(R_Bo) + v_Q_2.dot(R_Q) + w_A_2.dot(T_A) + w_B_2.dot(T_B)\n",
+    "F3 = v_Ao_3.dot(R_Ao) + v_Bo_3.dot(R_Bo) + v_Q_3.dot(R_Q) + w_A_3.dot(T_A) + w_B_3.dot(T_B)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 14,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left[\\begin{matrix}- \\frac{7 g l m \\sin{\\left(q_{1} \\right)}}{4} - \\frac{g m q_{3} \\cos{\\left(q_{1} \\right)} \\cos{\\left(q_{2} \\right)}}{4} - k_{t} q_{1}\\\\\\frac{g m q_{3} \\sin{\\left(q_{1} \\right)} \\sin{\\left(q_{2} \\right)}}{4} - k_{t} q_{2}\\\\- \\frac{g m \\sin{\\left(q_{1} \\right)} \\cos{\\left(q_{2} \\right)}}{4} - k_{l} q_{3}\\end{matrix}\\right]$"
+      ],
+      "text/plain": [
+       "⎡  7⋅g⋅l⋅m⋅sin(q₁)   g⋅m⋅q₃⋅cos(q₁)⋅cos(q₂)        ⎤\n",
+       "⎢- ─────────────── - ────────────────────── - kₜ⋅q₁⎥\n",
+       "⎢         4                    4                   ⎥\n",
+       "⎢                                                  ⎥\n",
+       "⎢          g⋅m⋅q₃⋅sin(q₁)⋅sin(q₂)                  ⎥\n",
+       "⎢          ────────────────────── - kₜ⋅q₂          ⎥\n",
+       "⎢                    4                             ⎥\n",
+       "⎢                                                  ⎥\n",
+       "⎢            g⋅m⋅sin(q₁)⋅cos(q₂)                   ⎥\n",
+       "⎢          - ─────────────────── - kₗ⋅q₃           ⎥\n",
+       "⎣                     4                            ⎦"
+      ]
+     },
+     "execution_count": 14,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Fr = sm.Matrix([F1, F2, F3])\n",
+    "Fr"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 15,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "TAs = -(A.ang_acc_in(N).dot(I_A_Ao) + me.cross(A.ang_vel_in(N), I_A_Ao).dot(A.ang_vel_in(N)))\n",
+    "TBs = -(B.ang_acc_in(N).dot(I_B_Bo) + me.cross(B.ang_vel_in(N), I_B_Bo).dot(B.ang_vel_in(N)))\n",
+    "\n",
+    "F1s = v_Ao_1.dot(-m*Ao.acc(N)) + v_Bo_1.dot(-m*Bo.acc(N)) + v_Q_1.dot(-m/4*Q.acc(N))\n",
+    "F1s += w_A_1.dot(TAs) + w_B_1.dot(TBs)\n",
+    "\n",
+    "F2s = v_Ao_2.dot(-m*Ao.acc(N)) + v_Bo_2.dot(-m*Bo.acc(N)) + v_Q_2.dot(-m/4*Q.acc(N))\n",
+    "F2s += w_A_2.dot(TAs) + w_B_2.dot(TBs)\n",
+    "\n",
+    "F3s = v_Ao_3.dot(-m*Ao.acc(N)) + v_Bo_3.dot(-m*Bo.acc(N)) + v_Q_3.dot(-m/4*Q.acc(N))\n",
+    "F3s += w_A_3.dot(TAs) + w_B_3.dot(TBs)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 16,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left[\\begin{matrix}- \\frac{19 l^{2} m \\dot{u}_{1}}{12} - \\frac{l m q_{3} u_{1}^{2} \\cos{\\left(q_{2} \\right)}}{4} + l \\left(- \\frac{m \\left(- q_{3} u_{1}^{2} \\cos^{2}{\\left(q_{2} \\right)} - q_{3} u_{2}^{2} + \\dot{u}_{3}\\right) \\cos{\\left(q_{2} \\right)}}{4} + \\frac{m \\left(q_{3} u_{1}^{2} \\sin{\\left(q_{2} \\right)} \\cos{\\left(q_{2} \\right)} + q_{3} \\dot{u}_{2} + 2 u_{2} u_{3}\\right) \\sin{\\left(q_{2} \\right)}}{4}\\right) + \\frac{m \\left(2 q_{3} u_{1} u_{2} \\sin{\\left(q_{2} \\right)} - q_{3} \\cos{\\left(q_{2} \\right)} \\dot{u}_{1} - 2 u_{1} u_{3} \\cos{\\left(q_{2} \\right)}\\right) q_{3} \\cos{\\left(q_{2} \\right)}}{4} + \\left(- \\frac{l^{2} m \\left(- u_{1} u_{2} \\sin{\\left(q_{2} \\right)} + \\cos{\\left(q_{2} \\right)} \\dot{u}_{1}\\right)}{12} + \\frac{l^{2} m u_{1} u_{2} \\sin{\\left(q_{2} \\right)}}{12}\\right) \\cos{\\left(q_{2} \\right)}\\\\- \\frac{l^{2} m u_{1}^{2} \\sin{\\left(q_{2} \\right)} \\cos{\\left(q_{2} \\right)}}{12} - \\frac{l^{2} m \\dot{u}_{2}}{12} + \\frac{l m q_{3} \\sin{\\left(q_{2} \\right)} \\dot{u}_{1}}{4} - \\frac{m \\left(q_{3} u_{1}^{2} \\sin{\\left(q_{2} \\right)} \\cos{\\left(q_{2} \\right)} + q_{3} \\dot{u}_{2} + 2 u_{2} u_{3}\\right) q_{3}}{4}\\\\- \\frac{l m \\cos{\\left(q_{2} \\right)} \\dot{u}_{1}}{4} - \\frac{m \\left(- q_{3} u_{1}^{2} \\cos^{2}{\\left(q_{2} \\right)} - q_{3} u_{2}^{2} + \\dot{u}_{3}\\right)}{4}\\end{matrix}\\right]$"
+      ],
+      "text/plain": [
+       "⎡      2                 2             ⎛    ⎛       2    2            2     ⎞  ↪\n",
+       "⎢  19⋅l ⋅m⋅u₁̇   l⋅m⋅q₃⋅u₁ ⋅cos(q₂)     ⎜  m⋅⎝- q₃⋅u₁ ⋅cos (q₂) - q₃⋅u₂  + u₃̇⎠⋅ ↪\n",
+       "⎢- ────────── - ────────────────── + l⋅⎜- ──────────────────────────────────── ↪\n",
+       "⎢      12               4              ⎝                       4               ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎣                                                                              ↪\n",
+       "\n",
+       "↪             ⎛     2                                  ⎞        ⎞              ↪\n",
+       "↪ cos(q₂)   m⋅⎝q₃⋅u₁ ⋅sin(q₂)⋅cos(q₂) + q₃⋅u₂̇ + 2⋅u₂⋅u₃⎠⋅sin(q₂)⎟   m⋅(2⋅q₃⋅u₁ ↪\n",
+       "↪ ─────── + ────────────────────────────────────────────────────⎟ + ────────── ↪\n",
+       "↪                                    4                          ⎠              ↪\n",
+       "↪                                                                              ↪\n",
+       "↪              2     2                    2                              ⎛     ↪\n",
+       "↪             l ⋅m⋅u₁ ⋅sin(q₂)⋅cos(q₂)   l ⋅m⋅u₂̇   l⋅m⋅q₃⋅sin(q₂)⋅u₁̇   m⋅⎝q₃⋅u ↪\n",
+       "↪           - ──────────────────────── - ─────── + ───────────────── - ─────── ↪\n",
+       "↪                        12                12              4                   ↪\n",
+       "↪                                                                              ↪\n",
+       "↪                                                          ⎛       2    2      ↪\n",
+       "↪                                       l⋅m⋅cos(q₂)⋅u₁̇   m⋅⎝- q₃⋅u₁ ⋅cos (q₂)  ↪\n",
+       "↪                                     - ────────────── - ───────────────────── ↪\n",
+       "↪                                             4                           4    ↪\n",
+       "\n",
+       "↪                                                             ⎛   2            ↪\n",
+       "↪ ⋅u₂⋅sin(q₂) - q₃⋅cos(q₂)⋅u₁̇ - 2⋅u₁⋅u₃⋅cos(q₂))⋅q₃⋅cos(q₂)   ⎜  l ⋅m⋅(-u₁⋅u₂⋅ ↪\n",
+       "↪ ───────────────────────────────────────────────────────── + ⎜- ───────────── ↪\n",
+       "↪                        4                                    ⎝                ↪\n",
+       "↪                                                                              ↪\n",
+       "↪  2                                  ⎞                                        ↪\n",
+       "↪ ₁ ⋅sin(q₂)⋅cos(q₂) + q₃⋅u₂̇ + 2⋅u₂⋅u₃⎠⋅q₃                                     ↪\n",
+       "↪ ────────────────────────────────────────                                     ↪\n",
+       "↪                 4                                                            ↪\n",
+       "↪                                                                              ↪\n",
+       "↪        2     ⎞                                                               ↪\n",
+       "↪ - q₃⋅u₂  + u₃̇⎠                                                               ↪\n",
+       "↪ ──────────────                                                               ↪\n",
+       "↪                                                                              ↪\n",
+       "\n",
+       "↪                          2                ⎞        ⎤\n",
+       "↪ sin(q₂) + cos(q₂)⋅u₁̇)   l ⋅m⋅u₁⋅u₂⋅sin(q₂)⎟        ⎥\n",
+       "↪ ───────────────────── + ──────────────────⎟⋅cos(q₂)⎥\n",
+       "↪    12                           12        ⎠        ⎥\n",
+       "↪                                                    ⎥\n",
+       "↪                                                    ⎥\n",
+       "↪                                                    ⎥\n",
+       "↪                                                    ⎥\n",
+       "↪                                                    ⎥\n",
+       "↪                                                    ⎥\n",
+       "↪                                                    ⎥\n",
+       "↪                                                    ⎥\n",
+       "↪                                                    ⎥\n",
+       "↪                                                    ⎦"
+      ]
+     },
+     "execution_count": 16,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Frs = sm.Matrix([F1s, F2s, F3s])\n",
+    "Frs"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "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": 17,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "u = sm.Matrix([u1, u2, u3])\n",
+    "ud = u.diff(t)\n",
+    "ud"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 18,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left[\\begin{matrix}- \\frac{l^{2} m \\cos^{2}{\\left(q_{2} \\right)}}{12} - \\frac{19 l^{2} m}{12} - \\frac{m q_{3}^{2} \\cos^{2}{\\left(q_{2} \\right)}}{4} & \\frac{l m q_{3} \\sin{\\left(q_{2} \\right)}}{4} & - \\frac{l m \\cos{\\left(q_{2} \\right)}}{4}\\\\\\frac{l m q_{3} \\sin{\\left(q_{2} \\right)}}{4} & - \\frac{l^{2} m}{12} - \\frac{m q_{3}^{2}}{4} & 0\\\\- \\frac{l m \\cos{\\left(q_{2} \\right)}}{4} & 0 & - \\frac{m}{4}\\end{matrix}\\right]$"
+      ],
+      "text/plain": [
+       "⎡   2      2           2         2    2                                   ⎤\n",
+       "⎢  l ⋅m⋅cos (q₂)   19⋅l ⋅m   m⋅q₃ ⋅cos (q₂)  l⋅m⋅q₃⋅sin(q₂)  -l⋅m⋅cos(q₂) ⎥\n",
+       "⎢- ───────────── - ─────── - ──────────────  ──────────────  ─────────────⎥\n",
+       "⎢       12           12            4               4               4      ⎥\n",
+       "⎢                                                                         ⎥\n",
+       "⎢                                               2         2               ⎥\n",
+       "⎢              l⋅m⋅q₃⋅sin(q₂)                  l ⋅m   m⋅q₃                ⎥\n",
+       "⎢              ──────────────                - ──── - ─────        0      ⎥\n",
+       "⎢                    4                          12      4                 ⎥\n",
+       "⎢                                                                         ⎥\n",
+       "⎢              -l⋅m⋅cos(q₂)                                       -m      ⎥\n",
+       "⎢              ─────────────                       0              ───     ⎥\n",
+       "⎣                    4                                             4      ⎦"
+      ]
+     },
+     "execution_count": 18,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Md = Frs.jacobian(ud)\n",
+    "Md"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 19,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left\\{ \\dot{u}_{1} : 0, \\  \\dot{u}_{2} : 0, \\  \\dot{u}_{3} : 0\\right\\}$"
+      ],
+      "text/plain": [
+       "{u₁̇: 0, u₂̇: 0, u₃̇: 0}"
+      ]
+     },
+     "execution_count": 19,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "ud_zerod = {udr: 0 for udr in ud}\n",
+    "ud_zerod\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 20,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left[\\begin{matrix}- \\frac{7 g l m \\sin{\\left(q_{1} \\right)}}{4} - \\frac{g m q_{3} \\cos{\\left(q_{1} \\right)} \\cos{\\left(q_{2} \\right)}}{4} - k_{t} q_{1} + \\frac{l^{2} m u_{1} u_{2} \\sin{\\left(q_{2} \\right)} \\cos{\\left(q_{2} \\right)}}{6} - \\frac{l m q_{3} u_{1}^{2} \\cos{\\left(q_{2} \\right)}}{4} + l \\left(- \\frac{m \\left(- q_{3} u_{1}^{2} \\cos^{2}{\\left(q_{2} \\right)} - q_{3} u_{2}^{2}\\right) \\cos{\\left(q_{2} \\right)}}{4} + \\frac{m \\left(q_{3} u_{1}^{2} \\sin{\\left(q_{2} \\right)} \\cos{\\left(q_{2} \\right)} + 2 u_{2} u_{3}\\right) \\sin{\\left(q_{2} \\right)}}{4}\\right) + \\frac{m \\left(2 q_{3} u_{1} u_{2} \\sin{\\left(q_{2} \\right)} - 2 u_{1} u_{3} \\cos{\\left(q_{2} \\right)}\\right) q_{3} \\cos{\\left(q_{2} \\right)}}{4}\\\\\\frac{g m q_{3} \\sin{\\left(q_{1} \\right)} \\sin{\\left(q_{2} \\right)}}{4} - k_{t} q_{2} - \\frac{l^{2} m u_{1}^{2} \\sin{\\left(q_{2} \\right)} \\cos{\\left(q_{2} \\right)}}{12} - \\frac{m \\left(q_{3} u_{1}^{2} \\sin{\\left(q_{2} \\right)} \\cos{\\left(q_{2} \\right)} + 2 u_{2} u_{3}\\right) q_{3}}{4}\\\\- \\frac{g m \\sin{\\left(q_{1} \\right)} \\cos{\\left(q_{2} \\right)}}{4} - k_{l} q_{3} - \\frac{m \\left(- q_{3} u_{1}^{2} \\cos^{2}{\\left(q_{2} \\right)} - q_{3} u_{2}^{2}\\right)}{4}\\end{matrix}\\right]$"
+      ],
+      "text/plain": [
+       "⎡                                                      2                       ↪\n",
+       "⎢  7⋅g⋅l⋅m⋅sin(q₁)   g⋅m⋅q₃⋅cos(q₁)⋅cos(q₂)           l ⋅m⋅u₁⋅u₂⋅sin(q₂)⋅cos(q ↪\n",
+       "⎢- ─────────────── - ────────────────────── - kₜ⋅q₁ + ──────────────────────── ↪\n",
+       "⎢         4                    4                                  6            ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                          g⋅m ↪\n",
+       "⎢                                                                          ─── ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎢                                                                              ↪\n",
+       "⎣                                                                              ↪\n",
+       "\n",
+       "↪               2             ⎛    ⎛       2    2            2⎞             ⎛  ↪\n",
+       "↪ ₂)   l⋅m⋅q₃⋅u₁ ⋅cos(q₂)     ⎜  m⋅⎝- q₃⋅u₁ ⋅cos (q₂) - q₃⋅u₂ ⎠⋅cos(q₂)   m⋅⎝q ↪\n",
+       "↪ ── - ────────────────── + l⋅⎜- ────────────────────────────────────── + ──── ↪\n",
+       "↪              4              ⎝                    4                           ↪\n",
+       "↪                                                                              ↪\n",
+       "↪                                2     2                     ⎛     2           ↪\n",
+       "↪ ⋅q₃⋅sin(q₁)⋅sin(q₂)           l ⋅m⋅u₁ ⋅sin(q₂)⋅cos(q₂)   m⋅⎝q₃⋅u₁ ⋅sin(q₂)⋅c ↪\n",
+       "↪ ─────────────────── - kₜ⋅q₂ - ──────────────────────── - ─────────────────── ↪\n",
+       "↪        4                                 12                                  ↪\n",
+       "↪                                                                              ↪\n",
+       "↪                                                   ⎛       2    2             ↪\n",
+       "↪                   g⋅m⋅sin(q₁)⋅cos(q₂)           m⋅⎝- q₃⋅u₁ ⋅cos (q₂) - q₃⋅u₂ ↪\n",
+       "↪                 - ─────────────────── - kₗ⋅q₃ - ──────────────────────────── ↪\n",
+       "↪                            4                                  4              ↪\n",
+       "\n",
+       "↪     2                          ⎞        ⎞                                    ↪\n",
+       "↪ ₃⋅u₁ ⋅sin(q₂)⋅cos(q₂) + 2⋅u₂⋅u₃⎠⋅sin(q₂)⎟   m⋅(2⋅q₃⋅u₁⋅u₂⋅sin(q₂) - 2⋅u₁⋅u₃⋅ ↪\n",
+       "↪ ────────────────────────────────────────⎟ + ──────────────────────────────── ↪\n",
+       "↪                  4                      ⎠                            4       ↪\n",
+       "↪                                                                              ↪\n",
+       "↪                 ⎞                                                            ↪\n",
+       "↪ os(q₂) + 2⋅u₂⋅u₃⎠⋅q₃                                                         ↪\n",
+       "↪ ────────────────────                                                         ↪\n",
+       "↪ 4                                                                            ↪\n",
+       "↪                                                                              ↪\n",
+       "↪ 2⎞                                                                           ↪\n",
+       "↪  ⎠                                                                           ↪\n",
+       "↪ ──                                                                           ↪\n",
+       "↪                                                                              ↪\n",
+       "\n",
+       "↪                    ⎤\n",
+       "↪ cos(q₂))⋅q₃⋅cos(q₂)⎥\n",
+       "↪ ───────────────────⎥\n",
+       "↪                    ⎥\n",
+       "↪                    ⎥\n",
+       "↪                    ⎥\n",
+       "↪                    ⎥\n",
+       "↪                    ⎥\n",
+       "↪                    ⎥\n",
+       "↪                    ⎥\n",
+       "↪                    ⎥\n",
+       "↪                    ⎥\n",
+       "↪                    ⎥\n",
+       "↪                    ⎦"
+      ]
+     },
+     "execution_count": 20,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "gd = Frs.xreplace(ud_zerod) + Fr\n",
+    "gd"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 22,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left( \\left[\\begin{matrix}q_{1}\\\\q_{2}\\\\q_{3}\\end{matrix}\\right], \\  \\left[\\begin{matrix}u_{1}\\\\u_{2}\\\\u_{3}\\end{matrix}\\right], \\  \\left[\\begin{matrix}\\dot{q}_{1}\\\\\\dot{q}_{2}\\\\\\dot{q}_{3}\\end{matrix}\\right], \\  \\left[\\begin{matrix}\\dot{u}_{1}\\\\\\dot{u}_{2}\\\\\\dot{u}_{3}\\end{matrix}\\right]\\right)$"
+      ],
+      "text/plain": [
+       "⎛⎡q₁⎤  ⎡u₁⎤  ⎡q₁̇⎤  ⎡u₁̇⎤⎞\n",
+       "⎜⎢  ⎥  ⎢  ⎥  ⎢  ⎥  ⎢  ⎥⎟\n",
+       "⎜⎢q₂⎥, ⎢u₂⎥, ⎢q₂̇⎥, ⎢u₂̇⎥⎟\n",
+       "⎜⎢  ⎥  ⎢  ⎥  ⎢  ⎥  ⎢  ⎥⎟\n",
+       "⎝⎣q₃⎦  ⎣u₃⎦  ⎣q₃̇⎦  ⎣u₃̇⎦⎠"
+      ]
+     },
+     "execution_count": 22,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "q = sm.Matrix([q1, q2, q3])\n",
+    "qd = q.diff(t)\n",
+    "q, u, qd, ud"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 24,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left( \\left[\\begin{matrix}-1 & 0 & 0\\\\0 & -1 & 0\\\\0 & 0 & -1\\end{matrix}\\right], \\  \\left[\\begin{matrix}u_{1}\\\\u_{2}\\\\u_{3}\\end{matrix}\\right]\\right)$"
+      ],
+      "text/plain": [
+       "⎛⎡-1  0   0 ⎤  ⎡u₁⎤⎞\n",
+       "⎜⎢          ⎥  ⎢  ⎥⎟\n",
+       "⎜⎢0   -1  0 ⎥, ⎢u₂⎥⎟\n",
+       "⎜⎢          ⎥  ⎢  ⎥⎟\n",
+       "⎝⎣0   0   -1⎦  ⎣u₃⎦⎠"
+      ]
+     },
+     "execution_count": 24,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Mk = -sm.eye(3)\n",
+    "gk = u\n",
+    "Mk, gk"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 25,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left( \\left[\\begin{matrix}- \\frac{l^{2} m \\cos^{2}{\\left(q_{2} \\right)}}{12} - \\frac{19 l^{2} m}{12} - \\frac{m q_{3}^{2} \\cos^{2}{\\left(q_{2} \\right)}}{4} & \\frac{l m q_{3} \\sin{\\left(q_{2} \\right)}}{4} & - \\frac{l m \\cos{\\left(q_{2} \\right)}}{4}\\\\\\frac{l m q_{3} \\sin{\\left(q_{2} \\right)}}{4} & - \\frac{l^{2} m}{12} - \\frac{m q_{3}^{2}}{4} & 0\\\\- \\frac{l m \\cos{\\left(q_{2} \\right)}}{4} & 0 & - \\frac{m}{4}\\end{matrix}\\right], \\  \\left[\\begin{matrix}- \\frac{7 g l m \\sin{\\left(q_{1} \\right)}}{4} - \\frac{g m q_{3} \\cos{\\left(q_{1} \\right)} \\cos{\\left(q_{2} \\right)}}{4} - k_{t} q_{1} + \\frac{l^{2} m u_{1} u_{2} \\sin{\\left(q_{2} \\right)} \\cos{\\left(q_{2} \\right)}}{6} - \\frac{l m q_{3} u_{1}^{2} \\cos{\\left(q_{2} \\right)}}{4} + l \\left(- \\frac{m \\left(- q_{3} u_{1}^{2} \\cos^{2}{\\left(q_{2} \\right)} - q_{3} u_{2}^{2}\\right) \\cos{\\left(q_{2} \\right)}}{4} + \\frac{m \\left(q_{3} u_{1}^{2} \\sin{\\left(q_{2} \\right)} \\cos{\\left(q_{2} \\right)} + 2 u_{2} u_{3}\\right) \\sin{\\left(q_{2} \\right)}}{4}\\right) + \\frac{m \\left(2 q_{3} u_{1} u_{2} \\sin{\\left(q_{2} \\right)} - 2 u_{1} u_{3} \\cos{\\left(q_{2} \\right)}\\right) q_{3} \\cos{\\left(q_{2} \\right)}}{4}\\\\\\frac{g m q_{3} \\sin{\\left(q_{1} \\right)} \\sin{\\left(q_{2} \\right)}}{4} - k_{t} q_{2} - \\frac{l^{2} m u_{1}^{2} \\sin{\\left(q_{2} \\right)} \\cos{\\left(q_{2} \\right)}}{12} - \\frac{m \\left(q_{3} u_{1}^{2} \\sin{\\left(q_{2} \\right)} \\cos{\\left(q_{2} \\right)} + 2 u_{2} u_{3}\\right) q_{3}}{4}\\\\- \\frac{g m \\sin{\\left(q_{1} \\right)} \\cos{\\left(q_{2} \\right)}}{4} - k_{l} q_{3} - \\frac{m \\left(- q_{3} u_{1}^{2} \\cos^{2}{\\left(q_{2} \\right)} - q_{3} u_{2}^{2}\\right)}{4}\\end{matrix}\\right]\\right)$"
+      ],
+      "text/plain": [
+       "⎛                                                                              ↪\n",
+       "⎜⎡   2      2           2         2    2                                   ⎤   ↪\n",
+       "⎜⎢  l ⋅m⋅cos (q₂)   19⋅l ⋅m   m⋅q₃ ⋅cos (q₂)  l⋅m⋅q₃⋅sin(q₂)  -l⋅m⋅cos(q₂) ⎥   ↪\n",
+       "⎜⎢- ───────────── - ─────── - ──────────────  ──────────────  ─────────────⎥   ↪\n",
+       "⎜⎢       12           12            4               4               4      ⎥   ↪\n",
+       "⎜⎢                                                                         ⎥   ↪\n",
+       "⎜⎢                                               2         2               ⎥   ↪\n",
+       "⎜⎢              l⋅m⋅q₃⋅sin(q₂)                  l ⋅m   m⋅q₃                ⎥,  ↪\n",
+       "⎜⎢              ──────────────                - ──── - ─────        0      ⎥   ↪\n",
+       "⎜⎢                    4                          12      4                 ⎥   ↪\n",
+       "⎜⎢                                                                         ⎥   ↪\n",
+       "⎜⎢              -l⋅m⋅cos(q₂)                                       -m      ⎥   ↪\n",
+       "⎜⎢              ─────────────                       0              ───     ⎥   ↪\n",
+       "⎝⎣                    4                                             4      ⎦   ↪\n",
+       "\n",
+       "↪ ⎡                                                      2                     ↪\n",
+       "↪ ⎢  7⋅g⋅l⋅m⋅sin(q₁)   g⋅m⋅q₃⋅cos(q₁)⋅cos(q₂)           l ⋅m⋅u₁⋅u₂⋅sin(q₂)⋅cos ↪\n",
+       "↪ ⎢- ─────────────── - ────────────────────── - kₜ⋅q₁ + ────────────────────── ↪\n",
+       "↪ ⎢         4                    4                                  6          ↪\n",
+       "↪ ⎢                                                                            ↪\n",
+       "↪ ⎢                                                                            ↪\n",
+       "↪ ⎢                                                                          g ↪\n",
+       "↪ ⎢                                                                          ─ ↪\n",
+       "↪ ⎢                                                                            ↪\n",
+       "↪ ⎢                                                                            ↪\n",
+       "↪ ⎢                                                                            ↪\n",
+       "↪ ⎢                                                                            ↪\n",
+       "↪ ⎢                                                                            ↪\n",
+       "↪ ⎣                                                                            ↪\n",
+       "\n",
+       "↪                 2             ⎛    ⎛       2    2            2⎞              ↪\n",
+       "↪ (q₂)   l⋅m⋅q₃⋅u₁ ⋅cos(q₂)     ⎜  m⋅⎝- q₃⋅u₁ ⋅cos (q₂) - q₃⋅u₂ ⎠⋅cos(q₂)   m⋅ ↪\n",
+       "↪ ──── - ────────────────── + l⋅⎜- ────────────────────────────────────── + ── ↪\n",
+       "↪                4              ⎝                    4                         ↪\n",
+       "↪                                                                              ↪\n",
+       "↪                                  2     2                     ⎛     2         ↪\n",
+       "↪ ⋅m⋅q₃⋅sin(q₁)⋅sin(q₂)           l ⋅m⋅u₁ ⋅sin(q₂)⋅cos(q₂)   m⋅⎝q₃⋅u₁ ⋅sin(q₂) ↪\n",
+       "↪ ───────────────────── - kₜ⋅q₂ - ──────────────────────── - ───────────────── ↪\n",
+       "↪          4                                 12                                ↪\n",
+       "↪                                                                              ↪\n",
+       "↪                                                     ⎛       2    2           ↪\n",
+       "↪                     g⋅m⋅sin(q₁)⋅cos(q₂)           m⋅⎝- q₃⋅u₁ ⋅cos (q₂) - q₃⋅ ↪\n",
+       "↪                   - ─────────────────── - kₗ⋅q₃ - ────────────────────────── ↪\n",
+       "↪                              4                                  4            ↪\n",
+       "\n",
+       "↪ ⎛     2                          ⎞        ⎞                                  ↪\n",
+       "↪ ⎝q₃⋅u₁ ⋅sin(q₂)⋅cos(q₂) + 2⋅u₂⋅u₃⎠⋅sin(q₂)⎟   m⋅(2⋅q₃⋅u₁⋅u₂⋅sin(q₂) - 2⋅u₁⋅u ↪\n",
+       "↪ ──────────────────────────────────────────⎟ + ────────────────────────────── ↪\n",
+       "↪                    4                      ⎠                            4     ↪\n",
+       "↪                                                                              ↪\n",
+       "↪                   ⎞                                                          ↪\n",
+       "↪ ⋅cos(q₂) + 2⋅u₂⋅u₃⎠⋅q₃                                                       ↪\n",
+       "↪ ──────────────────────                                                       ↪\n",
+       "↪   4                                                                          ↪\n",
+       "↪                                                                              ↪\n",
+       "↪   2⎞                                                                         ↪\n",
+       "↪ u₂ ⎠                                                                         ↪\n",
+       "↪ ────                                                                         ↪\n",
+       "↪                                                                              ↪\n",
+       "\n",
+       "↪                      ⎤⎞\n",
+       "↪ ₃⋅cos(q₂))⋅q₃⋅cos(q₂)⎥⎟\n",
+       "↪ ─────────────────────⎥⎟\n",
+       "↪                      ⎥⎟\n",
+       "↪                      ⎥⎟\n",
+       "↪                      ⎥⎟\n",
+       "↪                      ⎥⎟\n",
+       "↪                      ⎥⎟\n",
+       "↪                      ⎥⎟\n",
+       "↪                      ⎥⎟\n",
+       "↪                      ⎥⎟\n",
+       "↪                      ⎥⎟\n",
+       "↪                      ⎥⎟\n",
+       "↪                      ⎦⎠"
+      ]
+     },
+     "execution_count": 25,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Md, gd"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 26,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left\\{g, k_{l}, k_{t}, l, m, t\\right\\}$"
+      ],
+      "text/plain": [
+       "{g, kₗ, kₜ, l, m, t}"
+      ]
+     },
+     "execution_count": 26,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Mk.free_symbols | gk.free_symbols | Md.free_symbols | gd.free_symbols"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 27,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left[\\begin{matrix}g\\\\k_{l}\\\\k_{t}\\\\l\\\\m\\end{matrix}\\right]$"
+      ],
+      "text/plain": [
+       "⎡g ⎤\n",
+       "⎢  ⎥\n",
+       "⎢kₗ⎥\n",
+       "⎢  ⎥\n",
+       "⎢kₜ⎥\n",
+       "⎢  ⎥\n",
+       "⎢l ⎥\n",
+       "⎢  ⎥\n",
+       "⎣m ⎦"
+      ]
+     },
+     "execution_count": 27,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "p = sm.Matrix([g, kl, kt, l, m])\n",
+    "p"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 28,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "eval_eom = sm.lambdify((q, u, p), [Mk, gk, Md, gd])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 29,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "(array([0.43633231, 0.08726646, 0.1       ]), numpy.ndarray, (3,))"
+      ]
+     },
+     "execution_count": 29,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "q_vals = np.array([\n",
+    "    np.deg2rad(25.0),  # q1, rad\n",
+    "    np.deg2rad(5.0),  # q2, rad\n",
+    "    0.1,  # q3, m\n",
+    "])\n",
+    "q_vals, type(q_vals), q_vals.shape"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 30,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "(array([0.1, 2.2, 0.3]), numpy.ndarray, (3,))"
+      ]
+     },
+     "execution_count": 30,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "u_vals = np.array([\n",
+    "    0.1,  # u1, rad/s\n",
+    "    2.2,  # u2, rad/s\n",
+    "    0.3,  # u3, m/s\n",
+    "])\n",
+    "u_vals, type(u_vals), u_vals.shape"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 31,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "(array([9.81, 2.  , 0.01, 0.6 , 1.  ]), numpy.ndarray, (5,))"
+      ]
+     },
+     "execution_count": 31,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "p_vals = np.array([\n",
+    "    9.81,  # g, m/s**2\n",
+    "    2.0,  # kl, N/m\n",
+    "    0.01,  # kt, Nm/rad\n",
+    "    0.6,  # l, m\n",
+    "    1.0,  # m, kg\n",
+    "])\n",
+    "p_vals, type(p_vals), p_vals.shape"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 32,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "(array([[-1,  0,  0],\n",
+       "        [ 0, -1,  0],\n",
+       "        [ 0,  0, -1]]),\n",
+       " array([[0.1],\n",
+       "        [2.2],\n",
+       "        [0.3]]),\n",
+       " array([[-0.60225313,  0.00130734, -0.1494292 ],\n",
+       "        [ 0.00130734, -0.0325    ,  0.        ],\n",
+       "        [-0.1494292 ,  0.        , -0.25      ]]),\n",
+       " array([[-4.48963535],\n",
+       "        [-0.02486744],\n",
+       "        [-1.1112791 ]]))"
+      ]
+     },
+     "execution_count": 32,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Mk_vals, gk_vals, Md_vals, gd_vals = eval_eom(q_vals, u_vals, p_vals)\n",
+    "Mk_vals, gk_vals, Md_vals, gd_vals"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 33,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([0.1, 2.2, 0.3])"
+      ]
+     },
+     "execution_count": 33,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "qd_vals = np.linalg.solve(-Mk_vals, np.squeeze(gk_vals))\n",
+    "qd_vals"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 34,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([-7.46056427, -1.06525862,  0.01418834])"
+      ]
+     },
+     "execution_count": 34,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "ud_vals = np.linalg.solve(-Md_vals, np.squeeze(gd_vals))\n",
+    "ud_vals"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 35,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def eval_rhs(t, x, p):\n",
+    "    \"\"\"Return the right hand side of the explicit ordinary differential\n",
+    "    equations which evaluates the time derivative of the state ``x`` at time\n",
+    "    ``t``.\n",
+    "\n",
+    "    Parameters\n",
+    "    ==========\n",
+    "    t : float\n",
+    "       Time in seconds.\n",
+    "    x : array_like, shape(6,)\n",
+    "       State at time t: [q1, q2, q3, u1, u2, u3]\n",
+    "    p : array_like, shape(5,)\n",
+    "       Constant parameters: [g, kl, kt, l, m]\n",
+    "\n",
+    "    Returns\n",
+    "    =======\n",
+    "    xd : ndarray, shape(6,)\n",
+    "        Derivative of the state with respect to time at time ``t``.\n",
+    "\n",
+    "    \"\"\"\n",
+    "\n",
+    "    # unpack the q and u vectors from x\n",
+    "    q = x[:3]\n",
+    "    u = x[3:]\n",
+    "\n",
+    "    # evaluate the equations of motion matrices with the values of q, u, p\n",
+    "    Mk, gk, Md, gd = eval_eom(q, u, p)\n",
+    "\n",
+    "    # solve for q' and u'\n",
+    "    qd = np.linalg.solve(-Mk, np.squeeze(gk))\n",
+    "    ud = np.linalg.solve(-Md, np.squeeze(gd))\n",
+    "\n",
+    "    # pack dq/dt and du/dt into a new state time derivative vector dx/dt\n",
+    "    xd = np.empty_like(x)\n",
+    "    xd[:3] = qd\n",
+    "    xd[3:] = ud\n",
+    "\n",
+    "    return xd"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 36,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def euler_integrate(rhs_func, tspan, x0_vals, p_vals, delt=0.03):\n",
+    "    \"\"\"Returns state trajectory and corresponding values of time resulting\n",
+    "    from integrating the ordinary differential equations with Euler's\n",
+    "    Method.\n",
+    "\n",
+    "    Parameters\n",
+    "    ==========\n",
+    "    rhs_func : function\n",
+    "       Python function that evaluates the derivative of the state and takes\n",
+    "       this form ``dxdt = f(t, x, p)``.\n",
+    "    tspan : 2-tuple of floats\n",
+    "       The initial time and final time values: (t0, tf).\n",
+    "    x0_vals : array_like, shape(2*n,)\n",
+    "       Values of the state x at t0.\n",
+    "    p_vals : array_like, shape(o,)\n",
+    "       Values of constant parameters.\n",
+    "    delt : float\n",
+    "       Integration time step in seconds/step.\n",
+    "\n",
+    "    Returns\n",
+    "    =======\n",
+    "    ts : ndarray(m, )\n",
+    "       Monotonically equally spaced increasing values of time spaced apart\n",
+    "       by ``delt``.\n",
+    "    xs : ndarray(m, 2*n)\n",
+    "       State values at each time in ts.\n",
+    "\n",
+    "    \"\"\"\n",
+    "    # generate monotonically increasing values of time.\n",
+    "    duration = tspan[1] - tspan[0]\n",
+    "    num_samples = round(duration/delt) + 1\n",
+    "    ts = np.arange(tspan[0], tspan[0] + delt*num_samples, delt)\n",
+    "\n",
+    "    # create an empty array to hold the state values.\n",
+    "    x = np.empty((len(ts), len(x0_vals)))\n",
+    "\n",
+    "    # set the initial conditions to the first element.\n",
+    "    x[0, :] = x0_vals\n",
+    "\n",
+    "    # use a for loop to sequentially calculate each new x.\n",
+    "    for i, ti in enumerate(ts[:-1]):\n",
+    "        x[i + 1, :] = x[i, :] + delt*rhs_func(ti, x[i, :], p_vals)\n",
+    "\n",
+    "    return ts, x"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 37,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "x0 = np.empty(6)\n",
+    "x0[:3] = q_vals\n",
+    "x0[3:] = u_vals\n",
+    "\n",
+    "t0 = 0.1"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 38,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([ 0.1       ,  2.2       ,  0.3       , -7.46056427, -1.06525862,\n",
+       "        0.01418834])"
+      ]
+     },
+     "execution_count": 38,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "eval_rhs(t0, x0, p_vals)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 39,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "tf = 2.0\n",
+    "ts, xs = euler_integrate(eval_rhs, (t0, tf), x0, p_vals)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 40,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([0.1 , 0.13, 0.16, 0.19, 0.22, 0.25, 0.28, 0.31, 0.34, 0.37, 0.4 ,\n",
+       "       0.43, 0.46, 0.49, 0.52, 0.55, 0.58, 0.61, 0.64, 0.67, 0.7 , 0.73,\n",
+       "       0.76, 0.79, 0.82, 0.85, 0.88, 0.91, 0.94, 0.97, 1.  , 1.03, 1.06,\n",
+       "       1.09, 1.12, 1.15, 1.18, 1.21, 1.24, 1.27, 1.3 , 1.33, 1.36, 1.39,\n",
+       "       1.42, 1.45, 1.48, 1.51, 1.54, 1.57, 1.6 , 1.63, 1.66, 1.69, 1.72,\n",
+       "       1.75, 1.78, 1.81, 1.84, 1.87, 1.9 , 1.93, 1.96, 1.99])"
+      ]
+     },
+     "execution_count": 40,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "ts"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 41,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([[ 4.36332313e-01,  8.72664626e-02,  1.00000000e-01,\n",
+       "         1.00000000e-01,  2.20000000e+00,  3.00000000e-01],\n",
+       "       [ 4.39332313e-01,  1.53266463e-01,  1.09000000e-01,\n",
+       "        -1.23816928e-01,  2.16804224e+00,  3.00425650e-01],\n",
+       "       [ 4.35617805e-01,  2.18307730e-01,  1.18012770e-01,\n",
+       "        -3.48867554e-01,  2.13303503e+00,  2.99423633e-01],\n",
+       "       [ 4.25151779e-01,  2.82298781e-01,  1.26995479e-01,\n",
+       "        -5.72475739e-01,  2.09463526e+00,  2.97361542e-01],\n",
+       "       [ 4.07977506e-01,  3.45137839e-01,  1.35916325e-01,\n",
+       "        -7.91929289e-01,  2.05197871e+00,  2.94628416e-01],\n",
+       "       [ 3.84219628e-01,  4.06697200e-01,  1.44755177e-01,\n",
+       "        -1.00444789e+00,  2.00382451e+00,  2.91554565e-01],\n",
+       "       [ 3.54086191e-01,  4.66811935e-01,  1.53501814e-01,\n",
+       "        -1.20715867e+00,  1.94877129e+00,  2.88335792e-01],\n",
+       "       [ 3.17871431e-01,  5.25275074e-01,  1.62151888e-01,\n",
+       "        -1.39708735e+00,  1.88551903e+00,  2.84974403e-01],\n",
+       "       [ 2.75958810e-01,  5.81840645e-01,  1.70701120e-01,\n",
+       "        -1.57117194e+00,  1.81314156e+00,  2.81247424e-01],\n",
+       "       [ 2.28823652e-01,  6.36234892e-01,  1.79138543e-01,\n",
+       "        -1.72630399e+00,  1.73132870e+00,  2.76708292e-01],\n",
+       "       [ 1.77034532e-01,  6.88174753e-01,  1.87439792e-01,\n",
+       "        -1.85939863e+00,  1.64055711e+00,  2.70722509e-01],\n",
+       "       [ 1.21252574e-01,  7.37391466e-01,  1.95561467e-01,\n",
+       "        -1.96749077e+00,  1.54215738e+00,  2.62531540e-01],\n",
+       "       [ 6.22278505e-02,  7.83656188e-01,  2.03437413e-01,\n",
+       "        -2.04785035e+00,  1.43826241e+00,  2.51334050e-01],\n",
+       "       [ 7.92340069e-04,  8.26804060e-01,  2.10977435e-01,\n",
+       "        -2.09810628e+00,  1.33164459e+00,  2.36370890e-01],\n",
+       "       [-6.21508484e-02,  8.66753398e-01,  2.18068561e-01,\n",
+       "        -2.11636717e+00,  1.22547077e+00,  2.17000797e-01],\n",
+       "       [-1.25641864e-01,  9.03517521e-01,  2.24578585e-01,\n",
+       "        -2.10132625e+00,  1.12301845e+00,  1.92757262e-01],\n",
+       "       [-1.88681651e-01,  9.37208074e-01,  2.30361303e-01,\n",
+       "        -2.05234005e+00,  1.02739894e+00,  1.63382255e-01],\n",
+       "       [-2.50251853e-01,  9.68030042e-01,  2.35262771e-01,\n",
+       "        -1.96947297e+00,  9.41324864e-01,  1.28837671e-01],\n",
+       "       [-3.09336042e-01,  9.96269788e-01,  2.39127901e-01,\n",
+       "        -1.85350427e+00,  8.66944026e-01,  8.92990821e-02],\n",
+       "       [-3.64941170e-01,  1.02227811e+00,  2.41806873e-01,\n",
+       "        -1.70589822e+00,  8.05745012e-01,  4.51378231e-02],\n",
+       "       [-4.16118116e-01,  1.04645046e+00,  2.43161008e-01,\n",
+       "        -1.52874283e+00,  7.58527111e-01, -3.10326579e-03],\n",
+       "       [-4.61980401e-01,  1.06920627e+00,  2.43067910e-01,\n",
+       "        -1.32466575e+00,  7.25419993e-01, -5.47370444e-02],\n",
+       "       [-5.01720374e-01,  1.09096887e+00,  2.41425799e-01,\n",
+       "        -1.09673819e+00,  7.05937072e-01, -1.08961098e-01],\n",
+       "       [-5.34622520e-01,  1.11214698e+00,  2.38156966e-01,\n",
+       "        -8.48377776e-01,  6.99048783e-01, -1.64889059e-01],\n",
+       "       [-5.60073853e-01,  1.13311845e+00,  2.33210294e-01,\n",
+       "        -5.83259685e-01,  7.03266085e-01, -2.21585204e-01],\n",
+       "       [-5.77571643e-01,  1.15421643e+00,  2.26562738e-01,\n",
+       "        -3.05242994e-01,  7.16728855e-01, -2.78102687e-01],\n",
+       "       [-5.86728933e-01,  1.17571830e+00,  2.18219657e-01,\n",
+       "        -1.83153063e-02,  7.37297421e-01, -3.33524242e-01],\n",
+       "       [-5.87278392e-01,  1.19783722e+00,  2.08213930e-01,\n",
+       "         2.73444426e-01,  7.62647870e-01, -3.87002450e-01],\n",
+       "       [-5.79075060e-01,  1.22071666e+00,  1.96603856e-01,\n",
+       "         5.65888142e-01,  7.90372680e-01, -4.37795193e-01],\n",
+       "       [-5.62098415e-01,  1.24442784e+00,  1.83470001e-01,\n",
+       "         8.54811324e-01,  8.18087698e-01, -4.85291412e-01],\n",
+       "       [-5.36454076e-01,  1.26897047e+00,  1.68911258e-01,\n",
+       "         1.13596271e+00,  8.43544507e-01, -5.29023016e-01],\n",
+       "       [-5.02375194e-01,  1.29427680e+00,  1.53040568e-01,\n",
+       "         1.40505807e+00,  8.64744480e-01, -5.68660838e-01],\n",
+       "       [-4.60223452e-01,  1.32021914e+00,  1.35980743e-01,\n",
+       "         1.65780471e+00,  8.80047808e-01, -6.03995480e-01],\n",
+       "       [-4.10489311e-01,  1.34662057e+00,  1.17860878e-01,\n",
+       "         1.88994245e+00,  8.88268730e-01, -6.34907053e-01],\n",
+       "       [-3.53791037e-01,  1.37326863e+00,  9.88136667e-02,\n",
+       "         2.09730514e+00,  8.88747420e-01, -6.61330213e-01],\n",
+       "       [-2.90871883e-01,  1.39993105e+00,  7.89737603e-02,\n",
+       "         2.27590461e+00,  8.81389870e-01, -6.83221797e-01],\n",
+       "       [-2.22594745e-01,  1.42637275e+00,  5.84771064e-02,\n",
+       "         2.42203585e+00,  8.66668771e-01, -7.00537408e-01],\n",
+       "       [-1.49933669e-01,  1.45237281e+00,  3.74609842e-02,\n",
+       "         2.53239896e+00,  8.45580400e-01, -7.13220652e-01],\n",
+       "       [-7.39617005e-02,  1.47774023e+00,  1.60643646e-02,\n",
+       "         2.60422940e+00,  8.19554682e-01, -7.21205288e-01],\n",
+       "       [ 4.16518154e-03,  1.50232687e+00, -5.57179403e-03,\n",
+       "         2.63542555e+00,  7.90319326e-01, -7.24427319e-01],\n",
+       "       [ 8.32279481e-02,  1.52603645e+00, -2.73046136e-02,\n",
+       "         2.62466048e+00,  7.59725678e-01, -7.22841890e-01],\n",
+       "       [ 1.61967762e-01,  1.54882822e+00, -4.89898703e-02,\n",
+       "         2.57146487e+00,  7.29553925e-01, -7.16439450e-01],\n",
+       "       [ 2.39111708e-01,  1.57071483e+00, -7.04830538e-02,\n",
+       "         2.47627011e+00,  7.01325433e-01, -7.05256667e-01],\n",
+       "       [ 3.13399812e-01,  1.59175460e+00, -9.16407538e-02,\n",
+       "         2.34040397e+00,  6.76154998e-01, -6.89379799e-01],\n",
+       "       [ 3.83611931e-01,  1.61203925e+00, -1.12322148e-01,\n",
+       "         2.16603688e+00,  6.54671168e-01, -6.68940608e-01],\n",
+       "       [ 4.48593037e-01,  1.63167938e+00, -1.32390366e-01,\n",
+       "         1.95608296e+00,  6.37018712e-01, -6.44106831e-01],\n",
+       "       [ 5.07275526e-01,  1.65078994e+00, -1.51713571e-01,\n",
+       "         1.71406620e+00,  6.22938813e-01, -6.15070314e-01],\n",
+       "       [ 5.58697512e-01,  1.66947811e+00, -1.70165680e-01,\n",
+       "         1.44396681e+00,  6.11906749e-01, -5.82036029e-01],\n",
+       "       [ 6.02016516e-01,  1.68783531e+00, -1.87626761e-01,\n",
+       "         1.15006549e+00,  6.03298605e-01, -5.45214730e-01],\n",
+       "       [ 6.36518481e-01,  1.70593427e+00, -2.03983203e-01,\n",
+       "         8.36802201e-01,  5.96558484e-01, -5.04821104e-01],\n",
+       "       [ 6.61622547e-01,  1.72383102e+00, -2.19127836e-01,\n",
+       "         5.08663225e-01,  5.91343485e-01, -4.61078217e-01],\n",
+       "       [ 6.76882444e-01,  1.74157133e+00, -2.32960183e-01,\n",
+       "         1.70104973e-01,  5.87631757e-01, -4.14227908e-01],\n",
+       "       [ 6.81985593e-01,  1.75920028e+00, -2.45387020e-01,\n",
+       "        -1.74482332e-01,  5.85786780e-01, -3.64545604e-01],\n",
+       "       [ 6.76751123e-01,  1.77677388e+00, -2.56323388e-01,\n",
+       "        -5.20773373e-01,  5.86577261e-01, -3.12357072e-01],\n",
+       "       [ 6.61127922e-01,  1.79437120e+00, -2.65694100e-01,\n",
+       "        -8.64475606e-01,  5.91156371e-01, -2.58054046e-01],\n",
+       "       [ 6.35193654e-01,  1.81210589e+00, -2.73435722e-01,\n",
+       "        -1.20127673e+00,  6.01006553e-01, -2.02105739e-01],\n",
+       "       [ 5.99155352e-01,  1.83013609e+00, -2.79498894e-01,\n",
+       "        -1.52678495e+00,  6.17857094e-01, -1.45063860e-01],\n",
+       "       [ 5.53351803e-01,  1.84867180e+00, -2.83850810e-01,\n",
+       "        -1.83647709e+00,  6.43581213e-01, -8.75598764e-02],\n",
+       "       [ 4.98257491e-01,  1.86797924e+00, -2.86477606e-01,\n",
+       "        -2.12567032e+00,  6.80078039e-01, -3.02944186e-02],\n",
+       "       [ 4.34487381e-01,  1.88838158e+00, -2.87386438e-01,\n",
+       "        -2.38953235e+00,  7.29142906e-01,  2.59803906e-02],\n",
+       "       [ 3.62801410e-01,  1.91025587e+00, -2.86607027e-01,\n",
+       "        -2.62314224e+00,  7.92327527e-01,  8.04846656e-02],\n",
+       "       [ 2.84107143e-01,  1.93402569e+00, -2.84192487e-01,\n",
+       "        -2.82160885e+00,  8.70790639e-01,  1.32439931e-01],\n",
+       "       [ 1.99458878e-01,  1.96014941e+00, -2.80219289e-01,\n",
+       "        -2.98024641e+00,  9.65140541e-01,  1.81104004e-01],\n",
+       "       [ 1.10051485e-01,  1.98910363e+00, -2.74786169e-01,\n",
+       "        -3.09479777e+00,  1.07527459e+00,  2.25812638e-01]])"
+      ]
+     },
+     "execution_count": 41,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "xs"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 42,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import matplotlib.pyplot as plt"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 43,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[<matplotlib.lines.Line2D at 0x1e335aa5870>,\n",
+       " <matplotlib.lines.Line2D at 0x1e335aa5960>,\n",
+       " <matplotlib.lines.Line2D at 0x1e335aa5a50>,\n",
+       " <matplotlib.lines.Line2D at 0x1e335aa5b40>,\n",
+       " <matplotlib.lines.Line2D at 0x1e335aa5c30>,\n",
+       " <matplotlib.lines.Line2D at 0x1e335aa5d20>]"
+      ]
+     },
+     "execution_count": 43,
+     "metadata": {},
+     "output_type": "execute_result"
+    },
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 432x288 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "plt.plot(ts, xs)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 44,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from scipy.integrate import solve_ivp"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 45,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "result = solve_ivp(eval_rhs, (t0, tf), x0, args=(p_vals,))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [
+    {
+     "ename": "ValueError",
+     "evalue": "x and y must have same first dimension, but have shapes (10,) and (6, 10)",
+     "output_type": "error",
+     "traceback": [
+      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
+      "\u001b[1;31mValueError\u001b[0m                                Traceback (most recent call last)",
+      "\u001b[1;32m~\\AppData\\Local\\Temp\\ipykernel_38236\\1859627624.py\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0mplt\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mplot\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mresult\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mt\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mresult\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0my\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
+      "\u001b[1;32mc:\\Users\\13063\\.conda\\envs\\MinerU\\lib\\site-packages\\matplotlib\\pyplot.py\u001b[0m in \u001b[0;36mplot\u001b[1;34m(scalex, scaley, data, *args, **kwargs)\u001b[0m\n\u001b[0;32m   3706\u001b[0m     \u001b[1;33m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   3707\u001b[0m ) -> list[Line2D]:\n\u001b[1;32m-> 3708\u001b[1;33m     return gca().plot(\n\u001b[0m\u001b[0;32m   3709\u001b[0m         \u001b[1;33m*\u001b[0m\u001b[0margs\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   3710\u001b[0m         \u001b[0mscalex\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0mscalex\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
+      "\u001b[1;32mc:\\Users\\13063\\.conda\\envs\\MinerU\\lib\\site-packages\\matplotlib\\axes\\_axes.py\u001b[0m in \u001b[0;36mplot\u001b[1;34m(self, scalex, scaley, data, *args, **kwargs)\u001b[0m\n\u001b[0;32m   1777\u001b[0m         \"\"\"\n\u001b[0;32m   1778\u001b[0m         \u001b[0mkwargs\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mcbook\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mnormalize_kwargs\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mmlines\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mLine2D\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m-> 1779\u001b[1;33m         \u001b[0mlines\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;33m[\u001b[0m\u001b[1;33m*\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_get_lines\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mself\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m*\u001b[0m\u001b[0margs\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mdata\u001b[0m\u001b[1;33m=\u001b[0m\u001b[0mdata\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m**\u001b[0m\u001b[0mkwargs\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m   1780\u001b[0m         \u001b[1;32mfor\u001b[0m \u001b[0mline\u001b[0m \u001b[1;32min\u001b[0m \u001b[0mlines\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m   1781\u001b[0m             \u001b[0mself\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0madd_line\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mline\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
+      "\u001b[1;32mc:\\Users\\13063\\.conda\\envs\\MinerU\\lib\\site-packages\\matplotlib\\axes\\_base.py\u001b[0m in \u001b[0;36m__call__\u001b[1;34m(self, axes, data, *args, **kwargs)\u001b[0m\n\u001b[0;32m    294\u001b[0m                 \u001b[0mthis\u001b[0m \u001b[1;33m+=\u001b[0m \u001b[0margs\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    295\u001b[0m                 \u001b[0margs\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0margs\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 296\u001b[1;33m             yield from self._plot_args(\n\u001b[0m\u001b[0;32m    297\u001b[0m                 axes, this, kwargs, ambiguous_fmt_datakey=ambiguous_fmt_datakey)\n\u001b[0;32m    298\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n",
+      "\u001b[1;32mc:\\Users\\13063\\.conda\\envs\\MinerU\\lib\\site-packages\\matplotlib\\axes\\_base.py\u001b[0m in \u001b[0;36m_plot_args\u001b[1;34m(self, axes, tup, kwargs, return_kwargs, ambiguous_fmt_datakey)\u001b[0m\n\u001b[0;32m    484\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    485\u001b[0m         \u001b[1;32mif\u001b[0m \u001b[0mx\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mshape\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m]\u001b[0m \u001b[1;33m!=\u001b[0m \u001b[0my\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mshape\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 486\u001b[1;33m             raise ValueError(f\"x and y must have same first dimension, but \"\n\u001b[0m\u001b[0;32m    487\u001b[0m                              f\"have shapes {x.shape} and {y.shape}\")\n\u001b[0;32m    488\u001b[0m         \u001b[1;32mif\u001b[0m \u001b[0mx\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mndim\u001b[0m \u001b[1;33m>\u001b[0m \u001b[1;36m2\u001b[0m \u001b[1;32mor\u001b[0m \u001b[0my\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mndim\u001b[0m \u001b[1;33m>\u001b[0m \u001b[1;36m2\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
+      "\u001b[1;31mValueError\u001b[0m: x and y must have same first dimension, but have shapes (10,) and (6, 10)"
+     ]
+    },
+    {
+     "data": {
+      "image/png": "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",
+      "text/plain": [
+       "<Figure size 432x288 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "plt.plot(result.t, np.transpose(result.y))"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "MinerU",
+   "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": "3.10.16"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/补课/多体动力学/框架.canvas b/补课/多体动力学/框架.canvas
index 6a069d7..7145c5e 100644
--- a/补课/多体动力学/框架.canvas
+++ b/补课/多体动力学/框架.canvas
@@ -1,11 +1,17 @@
 {
 	"nodes":[
 		{"id":"330ceae2327436f1","type":"text","text":"牛顿-欧拉方程\n\n$$\n\\begin{split}\\bar{F} = & \\frac{{}^N d \\bar{p}}{dt} \\quad \\textrm{ where } \\bar{p} = m_B{}^N\\bar{v}^{B_o} \\\\\\bar{M} = & \\frac{{}^N d\\bar{H}}{dt} \\quad \\textrm{ where }\\bar{H} = \\breve{I}^{B/B_o} \\cdot {}^N\\bar{\\omega}^{B}\\end{split}|","x":-242,"y":-194,"width":542,"height":174},
-		{"id":"04fa17192ff0596c","x":-242,"y":160,"width":250,"height":60,"type":"text","text":"GAF Generalized Active Force 左端项"},
-		{"id":"c574d94bf9b233b4","x":50,"y":160,"width":250,"height":60,"type":"text","text":"GIF Generalized Inertia Force 右端项"}
+		{"id":"04fa17192ff0596c","type":"text","text":"GAF Generalized Active Force 左端项","x":-448,"y":160,"width":250,"height":60},
+		{"id":"3dd1bf823248bf74","type":"text","text":"完整约束(holonomic)\n**约束方程中不包含坐标对时间的导数(不包含运动约束)**，或者约束方程中的微分项可以积分为有限形式的约束（几何约束或可积分的运动约束）\n$f_s(x_k, y_k, z_k;;t)=0$\n$f_s(x_k, y_k, z_k;\\dot{x_k},\\dot{y_k},\\dot{z_k};t)=0$","x":960,"y":-214,"width":340,"height":214},
+		{"id":"3f3febc750ff1af8","type":"text","text":"非完整约束(nonholonomic)\n**约束方程中包含坐标对时间的导数（包含运动约束，对广义速度u1...ur的约束）**，且不可能积分成有限形式的约束（包括积分的运动约束）\n$f_s(x_k, y_k, z_k;\\dot{x_k},\\dot{y_k},\\dot{z_k};t)=0$","x":960,"y":60,"width":340,"height":200},
+		{"id":"c574d94bf9b233b4","type":"text","text":"GIF Generalized Inertia Force 右端项","x":231,"y":160,"width":250,"height":60},
+		{"id":"027a3e957d393870","type":"text","text":"对于holonomic系统中的刚体B，质量$m_B$，质量中心$B_o$，中心inertia dyadic $\\breve{I}^{B/Bo}$。\n$(F_r^*)_B := {}^A\\bar{v}^{B_o}_r \\cdot \\bar{R}^* + {}^A\\bar{\\omega}^B_r \\cdot \\bar{T}^*$\n${}^A\\bar{v}^{B_o}_r$对广义速度$u_r$的偏速度\n${}^A\\bar{\\omega}^B_r$对广义速度$u_r$的偏角速度\n inertia force on the body:\n $\\bar{R}^* := -m_{B} {}^A\\bar{a}^{B_o}$\n ${}^A\\bar{a}^{B_o}$ $B_0$在A的线加速度\n _inertia torque_ on the body:\n $\\bar{T}^* := -\\left({}^A\\bar{\\alpha}^B \\cdot \\breve{I}^{B/Bo} +{}^A\\bar{\\omega}^B \\times\\breve{I}^{B/Bo} \\cdot {}^A\\bar{\\omega}^B\\right)$\n${}^A\\bar{\\alpha}^B$ $B$在A的角加速度\n \n ","x":126,"y":300,"width":460,"height":320},
+		{"id":"4fb6c3b08416426b","x":-537,"y":300,"width":429,"height":220,"type":"text","text":"对于holonomic系统中的刚体B，B上的力表示为一个作用与B上Q点的合力与合力偶。\n$(F_r)_B := {}^A\\bar{v}^Q_r \\cdot \\bar{R} + {}^A\\bar{\\omega}^B_r \\cdot \\bar{T}$\n${}^A\\bar{v}^Q_r$对广义速度$u_r$的偏速度\n${}^A\\bar{\\omega}^B_r$对广义速度$u_r$的偏角速度\n$\\bar{R}$ 作用线通过B上Q点的合力，Q可以是质量中心$B_o$\n$\\bar{T}$ B上的扭矩"}
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