Jupyter notebook to derive Jacobaians of rotational transforms

Author: petercorke

symbolic/rotation-derivative.ipynb

@@ -0,0 +1,381 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Determine derivatives of angular sequence to rotation matrix\n",
+    "\n",
+    "SymPy code to evaluate the mappings from rotation angle sequences to rotation matrix, then compute the derivatives to determine the mapping from angular rates to angular velocity.\n",
+    "\n",
+    "Adjust the next cell to choose the particular angle sequence."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 225,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "angle_names = ('alpha', 'beta', 'gamma')\n",
+    "func = eul2r\n",
+    "\n",
+    "#angle_names =  ('phi', 'theta', 'psi')\n",
+    "#func = lambda Gamma: rpy2r(Gamma, order='xyz')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 226,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from spatialmath.base import *\n",
+    "from sympy import *"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Define a symbol for time"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 227,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "t = symbols('t')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Define arrays of symbols for the angles, angle as function of time, time derivative of angle as a function of time"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 228,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "angle = []  # names of angles, eg. theta\n",
+    "anglet = []  # angles as function of time, eg. theta(t)\n",
+    "angled = []  # derivative of above, eg. d theta(t) / dt\n",
+    "angledn = []  # symbol to represent above, eg. theta_dot\n",
+    "for i in angle_names:\n",
+    "    angle.append(symbols(i))\n",
+    "    anglet.append(Function(i)(t))\n",
+    "    angled.append(anglet[-1].diff(t))\n",
+    "    angledn.append(i + '_dot')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Compute the rotation matrix"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 229,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left[\\begin{matrix}- \\sin{\\left(\\alpha{\\left(t \\right)} \\right)} \\sin{\\left(\\gamma{\\left(t \\right)} \\right)} + \\cos{\\left(\\alpha{\\left(t \\right)} \\right)} \\cos{\\left(\\beta{\\left(t \\right)} \\right)} \\cos{\\left(\\gamma{\\left(t \\right)} \\right)} & - \\sin{\\left(\\alpha{\\left(t \\right)} \\right)} \\cos{\\left(\\gamma{\\left(t \\right)} \\right)} - \\sin{\\left(\\gamma{\\left(t \\right)} \\right)} \\cos{\\left(\\alpha{\\left(t \\right)} \\right)} \\cos{\\left(\\beta{\\left(t \\right)} \\right)} & \\sin{\\left(\\beta{\\left(t \\right)} \\right)} \\cos{\\left(\\alpha{\\left(t \\right)} \\right)}\\\\\\sin{\\left(\\alpha{\\left(t \\right)} \\right)} \\cos{\\left(\\beta{\\left(t \\right)} \\right)} \\cos{\\left(\\gamma{\\left(t \\right)} \\right)} + \\sin{\\left(\\gamma{\\left(t \\right)} \\right)} \\cos{\\left(\\alpha{\\left(t \\right)} \\right)} & - \\sin{\\left(\\alpha{\\left(t \\right)} \\right)} \\sin{\\left(\\gamma{\\left(t \\right)} \\right)} \\cos{\\left(\\beta{\\left(t \\right)} \\right)} + \\cos{\\left(\\alpha{\\left(t \\right)} \\right)} \\cos{\\left(\\gamma{\\left(t \\right)} \\right)} & \\sin{\\left(\\alpha{\\left(t \\right)} \\right)} \\sin{\\left(\\beta{\\left(t \\right)} \\right)}\\\\- \\sin{\\left(\\beta{\\left(t \\right)} \\right)} \\cos{\\left(\\gamma{\\left(t \\right)} \\right)} & \\sin{\\left(\\beta{\\left(t \\right)} \\right)} \\sin{\\left(\\gamma{\\left(t \\right)} \\right)} & \\cos{\\left(\\beta{\\left(t \\right)} \\right)}\\end{matrix}\\right]$"
+      ],
+      "text/plain": [
+       "Matrix([\n",
+       "[-sin(alpha(t))*sin(gamma(t)) + cos(alpha(t))*cos(beta(t))*cos(gamma(t)), -sin(alpha(t))*cos(gamma(t)) - sin(gamma(t))*cos(alpha(t))*cos(beta(t)), sin(beta(t))*cos(alpha(t))],\n",
+       "[ sin(alpha(t))*cos(beta(t))*cos(gamma(t)) + sin(gamma(t))*cos(alpha(t)), -sin(alpha(t))*sin(gamma(t))*cos(beta(t)) + cos(alpha(t))*cos(gamma(t)), sin(alpha(t))*sin(beta(t))],\n",
+       "[                                            -sin(beta(t))*cos(gamma(t)),                                              sin(beta(t))*sin(gamma(t)),               cos(beta(t))]])"
+      ]
+     },
+     "execution_count": 229,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "R = Matrix(func(anglet))\n",
+    "R"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Compute its time derivative"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 230,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left[\\begin{matrix}- \\sin{\\left(\\alpha{\\left(t \\right)} \\right)} \\cos{\\left(\\beta{\\left(t \\right)} \\right)} \\cos{\\left(\\gamma{\\left(t \\right)} \\right)} \\frac{d}{d t} \\alpha{\\left(t \\right)} - \\sin{\\left(\\alpha{\\left(t \\right)} \\right)} \\cos{\\left(\\gamma{\\left(t \\right)} \\right)} \\frac{d}{d t} \\gamma{\\left(t \\right)} - \\sin{\\left(\\beta{\\left(t \\right)} \\right)} \\cos{\\left(\\alpha{\\left(t \\right)} \\right)} \\cos{\\left(\\gamma{\\left(t \\right)} \\right)} \\frac{d}{d t} \\beta{\\left(t \\right)} - \\sin{\\left(\\gamma{\\left(t \\right)} \\right)} \\cos{\\left(\\alpha{\\left(t \\right)} \\right)} \\cos{\\left(\\beta{\\left(t \\right)} \\right)} \\frac{d}{d t} \\gamma{\\left(t \\right)} - \\sin{\\left(\\gamma{\\left(t \\right)} \\right)} \\cos{\\left(\\alpha{\\left(t \\right)} \\right)} \\frac{d}{d t} \\alpha{\\left(t \\right)} & \\sin{\\left(\\alpha{\\left(t \\right)} \\right)} \\sin{\\left(\\gamma{\\left(t \\right)} \\right)} \\cos{\\left(\\beta{\\left(t \\right)} \\right)} \\frac{d}{d t} \\alpha{\\left(t \\right)} + \\sin{\\left(\\alpha{\\left(t \\right)} \\right)} \\sin{\\left(\\gamma{\\left(t \\right)} \\right)} \\frac{d}{d t} \\gamma{\\left(t \\right)} + \\sin{\\left(\\beta{\\left(t \\right)} \\right)} \\sin{\\left(\\gamma{\\left(t \\right)} \\right)} \\cos{\\left(\\alpha{\\left(t \\right)} \\right)} \\frac{d}{d t} \\beta{\\left(t \\right)} - \\cos{\\left(\\alpha{\\left(t \\right)} \\right)} \\cos{\\left(\\beta{\\left(t \\right)} \\right)} \\cos{\\left(\\gamma{\\left(t \\right)} \\right)} \\frac{d}{d t} \\gamma{\\left(t \\right)} - \\cos{\\left(\\alpha{\\left(t \\right)} \\right)} \\cos{\\left(\\gamma{\\left(t \\right)} \\right)} \\frac{d}{d t} \\alpha{\\left(t \\right)} & - \\sin{\\left(\\alpha{\\left(t \\right)} \\right)} \\sin{\\left(\\beta{\\left(t \\right)} \\right)} \\frac{d}{d t} \\alpha{\\left(t \\right)} + \\cos{\\left(\\alpha{\\left(t \\right)} \\right)} \\cos{\\left(\\beta{\\left(t \\right)} \\right)} \\frac{d}{d t} \\beta{\\left(t \\right)}\\\\- \\sin{\\left(\\alpha{\\left(t \\right)} \\right)} \\sin{\\left(\\beta{\\left(t \\right)} \\right)} \\cos{\\left(\\gamma{\\left(t \\right)} \\right)} \\frac{d}{d t} \\beta{\\left(t \\right)} - \\sin{\\left(\\alpha{\\left(t \\right)} \\right)} \\sin{\\left(\\gamma{\\left(t \\right)} \\right)} \\cos{\\left(\\beta{\\left(t \\right)} \\right)} \\frac{d}{d t} \\gamma{\\left(t \\right)} - \\sin{\\left(\\alpha{\\left(t \\right)} \\right)} \\sin{\\left(\\gamma{\\left(t \\right)} \\right)} \\frac{d}{d t} \\alpha{\\left(t \\right)} + \\cos{\\left(\\alpha{\\left(t \\right)} \\right)} \\cos{\\left(\\beta{\\left(t \\right)} \\right)} \\cos{\\left(\\gamma{\\left(t \\right)} \\right)} \\frac{d}{d t} \\alpha{\\left(t \\right)} + \\cos{\\left(\\alpha{\\left(t \\right)} \\right)} \\cos{\\left(\\gamma{\\left(t \\right)} \\right)} \\frac{d}{d t} \\gamma{\\left(t \\right)} & \\sin{\\left(\\alpha{\\left(t \\right)} \\right)} \\sin{\\left(\\beta{\\left(t \\right)} \\right)} \\sin{\\left(\\gamma{\\left(t \\right)} \\right)} \\frac{d}{d t} \\beta{\\left(t \\right)} - \\sin{\\left(\\alpha{\\left(t \\right)} \\right)} \\cos{\\left(\\beta{\\left(t \\right)} \\right)} \\cos{\\left(\\gamma{\\left(t \\right)} \\right)} \\frac{d}{d t} \\gamma{\\left(t \\right)} - \\sin{\\left(\\alpha{\\left(t \\right)} \\right)} \\cos{\\left(\\gamma{\\left(t \\right)} \\right)} \\frac{d}{d t} \\alpha{\\left(t \\right)} - \\sin{\\left(\\gamma{\\left(t \\right)} \\right)} \\cos{\\left(\\alpha{\\left(t \\right)} \\right)} \\cos{\\left(\\beta{\\left(t \\right)} \\right)} \\frac{d}{d t} \\alpha{\\left(t \\right)} - \\sin{\\left(\\gamma{\\left(t \\right)} \\right)} \\cos{\\left(\\alpha{\\left(t \\right)} \\right)} \\frac{d}{d t} \\gamma{\\left(t \\right)} & \\sin{\\left(\\alpha{\\left(t \\right)} \\right)} \\cos{\\left(\\beta{\\left(t \\right)} \\right)} \\frac{d}{d t} \\beta{\\left(t \\right)} + \\sin{\\left(\\beta{\\left(t \\right)} \\right)} \\cos{\\left(\\alpha{\\left(t \\right)} \\right)} \\frac{d}{d t} \\alpha{\\left(t \\right)}\\\\\\sin{\\left(\\beta{\\left(t \\right)} \\right)} \\sin{\\left(\\gamma{\\left(t \\right)} \\right)} \\frac{d}{d t} \\gamma{\\left(t \\right)} - \\cos{\\left(\\beta{\\left(t \\right)} \\right)} \\cos{\\left(\\gamma{\\left(t \\right)} \\right)} \\frac{d}{d t} \\beta{\\left(t \\right)} & \\sin{\\left(\\beta{\\left(t \\right)} \\right)} \\cos{\\left(\\gamma{\\left(t \\right)} \\right)} \\frac{d}{d t} \\gamma{\\left(t \\right)} + \\sin{\\left(\\gamma{\\left(t \\right)} \\right)} \\cos{\\left(\\beta{\\left(t \\right)} \\right)} \\frac{d}{d t} \\beta{\\left(t \\right)} & - \\sin{\\left(\\beta{\\left(t \\right)} \\right)} \\frac{d}{d t} \\beta{\\left(t \\right)}\\end{matrix}\\right]$"
+      ],
+      "text/plain": [
+       "Matrix([\n",
+       "[-sin(alpha(t))*cos(beta(t))*cos(gamma(t))*Derivative(alpha(t), t) - sin(alpha(t))*cos(gamma(t))*Derivative(gamma(t), t) - sin(beta(t))*cos(alpha(t))*cos(gamma(t))*Derivative(beta(t), t) - sin(gamma(t))*cos(alpha(t))*cos(beta(t))*Derivative(gamma(t), t) - sin(gamma(t))*cos(alpha(t))*Derivative(alpha(t), t), sin(alpha(t))*sin(gamma(t))*cos(beta(t))*Derivative(alpha(t), t) + sin(alpha(t))*sin(gamma(t))*Derivative(gamma(t), t) + sin(beta(t))*sin(gamma(t))*cos(alpha(t))*Derivative(beta(t), t) - cos(alpha(t))*cos(beta(t))*cos(gamma(t))*Derivative(gamma(t), t) - cos(alpha(t))*cos(gamma(t))*Derivative(alpha(t), t), -sin(alpha(t))*sin(beta(t))*Derivative(alpha(t), t) + cos(alpha(t))*cos(beta(t))*Derivative(beta(t), t)],\n",
+       "[-sin(alpha(t))*sin(beta(t))*cos(gamma(t))*Derivative(beta(t), t) - sin(alpha(t))*sin(gamma(t))*cos(beta(t))*Derivative(gamma(t), t) - sin(alpha(t))*sin(gamma(t))*Derivative(alpha(t), t) + cos(alpha(t))*cos(beta(t))*cos(gamma(t))*Derivative(alpha(t), t) + cos(alpha(t))*cos(gamma(t))*Derivative(gamma(t), t), sin(alpha(t))*sin(beta(t))*sin(gamma(t))*Derivative(beta(t), t) - sin(alpha(t))*cos(beta(t))*cos(gamma(t))*Derivative(gamma(t), t) - sin(alpha(t))*cos(gamma(t))*Derivative(alpha(t), t) - sin(gamma(t))*cos(alpha(t))*cos(beta(t))*Derivative(alpha(t), t) - sin(gamma(t))*cos(alpha(t))*Derivative(gamma(t), t),  sin(alpha(t))*cos(beta(t))*Derivative(beta(t), t) + sin(beta(t))*cos(alpha(t))*Derivative(alpha(t), t)],\n",
+       "[                                                                                                                                                                                                            sin(beta(t))*sin(gamma(t))*Derivative(gamma(t), t) - cos(beta(t))*cos(gamma(t))*Derivative(beta(t), t),                                                                                                                                                                                                            sin(beta(t))*cos(gamma(t))*Derivative(gamma(t), t) + sin(gamma(t))*cos(beta(t))*Derivative(beta(t), t),                                                                    -sin(beta(t))*Derivative(beta(t), t)]])"
+      ]
+     },
+     "execution_count": 230,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Rdot = Matrix(R).diff(t)\n",
+    "Rdot"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Get angular velocity vector in terms of angles and angle rates"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 231,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "omega = Matrix(vex(Rdot * R.T))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "For each element of this 3x1 matrix get the coefficients of each angle derivative"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 232,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "A = Matrix.zeros(3,3)\n",
+    "for i in range(3):\n",
+    "    e = omega[i,0].expand()\n",
+    "    for j in range(3):\n",
+    "        A[i, j] = e.coeff(angled[j])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The result is a 3x3 matrix. Mapping from angle rates to angular velocity. We subsitute angle as a function of time to plain angle, then simplify."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 233,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "A = trigsimp(A.subs(a for a in zip(anglet, angle)))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Compute the inverse and simplify.  Mapping from angular velocity to angle rates."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 234,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "Ai = trigsimp(A.inv())"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Render as code"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 235,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "'np.array([[0, -math.sin(alpha), math.sin(beta)*math.cos(alpha)], [0, math.cos(alpha), math.sin(alpha)*math.sin(beta)], [1, 0, math.cos(beta)]])'"
+      ]
+     },
+     "execution_count": 235,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "pycode(A).replace('ImmutableDenseMatrix', 'np.array')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 236,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "'np.array([[-math.cos(alpha)/math.tan(beta), -math.sin(alpha)/math.tan(beta), 1], [-math.sin(alpha), math.cos(alpha), 0], [math.cos(alpha)/math.sin(beta), math.sin(alpha)/math.sin(beta), 0]])'"
+      ]
+     },
+     "execution_count": 236,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "pycode(Ai).replace('ImmutableDenseMatrix', 'np.array')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Compute the time derivative of `Ai`, from angular acceleration to angle acceleration"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 237,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "Ai = Ai.subs(a for a in zip(angle, anglet))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 238,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\left[\\begin{matrix}\\frac{\\alpha_{dot} \\sin{\\left(\\alpha \\right)}}{\\tan{\\left(\\beta \\right)}} + \\frac{\\beta_{dot} \\cos{\\left(\\alpha \\right)}}{\\sin^{2}{\\left(\\beta \\right)}} & - \\frac{\\alpha_{dot} \\cos{\\left(\\alpha \\right)}}{\\tan{\\left(\\beta \\right)}} + \\frac{\\beta_{dot} \\sin{\\left(\\alpha \\right)}}{\\sin^{2}{\\left(\\beta \\right)}} & 0\\\\- \\alpha_{dot} \\cos{\\left(\\alpha \\right)} & - \\alpha_{dot} \\sin{\\left(\\alpha \\right)} & 0\\\\- \\frac{\\alpha_{dot} \\sin{\\left(\\alpha \\right)} + \\frac{\\beta_{dot} \\cos{\\left(\\alpha \\right)} \\cos{\\left(\\beta \\right)}}{\\sin{\\left(\\beta \\right)}}}{\\sin{\\left(\\beta \\right)}} & \\frac{\\alpha_{dot} \\cos{\\left(\\alpha \\right)} - \\frac{\\beta_{dot} \\sin{\\left(\\alpha \\right)} \\cos{\\left(\\beta \\right)}}{\\sin{\\left(\\beta \\right)}}}{\\sin{\\left(\\beta \\right)}} & 0\\end{matrix}\\right]$"
+      ],
+      "text/plain": [
+       "Matrix([\n",
+       "[          alpha_dot*sin(alpha)/tan(beta) + beta_dot*cos(alpha)/sin(beta)**2,         -alpha_dot*cos(alpha)/tan(beta) + beta_dot*sin(alpha)/sin(beta)**2, 0],\n",
+       "[                                                      -alpha_dot*cos(alpha),                                                      -alpha_dot*sin(alpha), 0],\n",
+       "[-(alpha_dot*sin(alpha) + beta_dot*cos(alpha)*cos(beta)/sin(beta))/sin(beta), (alpha_dot*cos(alpha) - beta_dot*sin(alpha)*cos(beta)/sin(beta))/sin(beta), 0]])"
+      ]
+     },
+     "execution_count": 238,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Ai_dot = trigsimp(Ai.diff(t).subs(a for a in zip(angled, angledn)).subs(a for a in zip(anglet, angle)))\n",
+    "Ai_dot"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 239,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "'np.array([[alpha_dot*math.sin(alpha)/math.tan(beta) + beta_dot*math.cos(alpha)/math.sin(beta)**2, -alpha_dot*math.cos(alpha)/math.tan(beta) + beta_dot*math.sin(alpha)/math.sin(beta)**2, 0], [-alpha_dot*math.cos(alpha), -alpha_dot*math.sin(alpha), 0], [-(alpha_dot*math.sin(alpha) + beta_dot*math.cos(alpha)*math.cos(beta)/math.sin(beta))/math.sin(beta), (alpha_dot*math.cos(alpha) - beta_dot*math.sin(alpha)*math.cos(beta)/math.sin(beta))/math.sin(beta), 0]])'"
+      ]
+     },
+     "execution_count": 239,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "pycode(Ai_dot).replace('ImmutableDenseMatrix', 'np.array')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
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+}