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iamareebjamal/Inertia.ipynb

Last active September 25, 2018 16:24
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Rotation Matrices
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Inertia.ipynb
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Rotation.ipynb
{
"cells": [
{
"cell_type": "code",
"execution_count": 133,
"metadata": {},
"outputs": [],
"source": [
"from sympy import *\n",
"init_printing()"
]
},
{
"cell_type": "code",
"execution_count": 134,
"metadata": {},
"outputs": [],
"source": [
"x, y, z, t = symbols('x y z t')"
]
},
{
"cell_type": "code",
"execution_count": 135,
"metadata": {},
"outputs": [],
"source": [
"theta1 = Function('f1')(t)\n",
"theta2 = Function('f2')(t)\n",
"theta3 = Function('f3')(t)"
]
},
{
"cell_type": "code",
"execution_count": 136,
"metadata": {},
"outputs": [],
"source": [
"S1, S2, S3 = sin(theta1), sin(theta2), sin(theta3)\n",
"C1, C2, C3 = cos(theta1), cos(theta2), cos(theta3)"
]
},
{
"cell_type": "code",
"execution_count": 137,
"metadata": {},
"outputs": [],
"source": [
"M1 = Matrix([[C1, -S1, 0], [S1, C1, 0], [0, 0, 1]])\n",
"M2 = Matrix([[1, 0, 0], [0, C2, -S2], [0, S2, C2]])\n",
"M3 = Matrix([[C3, 0, S3], [0, 1, 0], [-S3, 0, C3]])"
]
},
{
"cell_type": "code",
"execution_count": 138,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left ( \\left[\\begin{matrix}\\cos{\\left (\\operatorname{f_{1}}{\\left (t \\right )} \\right )} & - \\sin{\\left (\\operatorname{f_{1}}{\\left (t \\right )} \\right )} & 0\\\\\\sin{\\left (\\operatorname{f_{1}}{\\left (t \\right )} \\right )} & \\cos{\\left (\\operatorname{f_{1}}{\\left (t \\right )} \\right )} & 0\\\\0 & 0 & 1\\end{matrix}\\right], \\quad \\left[\\begin{matrix}1 & 0 & 0\\\\0 & \\cos{\\left (\\operatorname{f_{2}}{\\left (t \\right )} \\right )} & - \\sin{\\left (\\operatorname{f_{2}}{\\left (t \\right )} \\right )}\\\\0 & \\sin{\\left (\\operatorname{f_{2}}{\\left (t \\right )} \\right )} & \\cos{\\left (\\operatorname{f_{2}}{\\left (t \\right )} \\right )}\\end{matrix}\\right], \\quad \\left[\\begin{matrix}\\cos{\\left (\\operatorname{f_{3}}{\\left (t \\right )} \\right )} & 0 & \\sin{\\left (\\operatorname{f_{3}}{\\left (t \\right )} \\right )}\\\\0 & 1 & 0\\\\- \\sin{\\left (\\operatorname{f_{3}}{\\left (t \\right )} \\right )} & 0 & \\cos{\\left (\\operatorname{f_{3}}{\\left (t \\right )} \\right )}\\end{matrix}\\right]\\right )$$"
],
"text/plain": [
"⎛⎡cos(f₁(t)) -sin(f₁(t)) 0⎤ ⎡1 0 0 ⎤ ⎡cos(f₃(t)) 0 \n",
"⎜⎢ ⎥ ⎢ ⎥ ⎢ \n",
"⎜⎢sin(f₁(t)) cos(f₁(t)) 0⎥, ⎢0 cos(f₂(t)) -sin(f₂(t))⎥, ⎢ 0 1 \n",
"⎜⎢ ⎥ ⎢ ⎥ ⎢ \n",
"⎝⎣ 0 0 1⎦ ⎣0 sin(f₂(t)) cos(f₂(t)) ⎦ ⎣-sin(f₃(t)) 0 \n",
"\n",
"sin(f₃(t))⎤⎞\n",
" ⎥⎟\n",
" 0 ⎥⎟\n",
" ⎥⎟\n",
"cos(f₃(t))⎦⎠"
]
},
"execution_count": 138,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"M1, M2, M3"
]
},
{
"cell_type": "code",
"execution_count": 139,
"metadata": {},
"outputs": [],
"source": [
"R = M1*M2*M3"
]
},
{
"cell_type": "code",
"execution_count": 140,
"metadata": {},
"outputs": [],
"source": [
"Rdot = diff(R, t)"
]
},
{
"cell_type": "code",
"execution_count": 141,
"metadata": {},
"outputs": [],
"source": [
"result = Rdot*(R.T)"
]
},
{
"cell_type": "code",
"execution_count": 142,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$$\\left[\\begin{matrix}0 & - \\sin{\\left (\\operatorname{f_{2}}{\\left (t \\right )} \\right )} \\frac{d}{d t} \\operatorname{f_{3}}{\\left (t \\right )} - \\frac{d}{d t} \\operatorname{f_{1}}{\\left (t \\right )} & \\sin{\\left (\\operatorname{f_{1}}{\\left (t \\right )} \\right )} \\frac{d}{d t} \\operatorname{f_{2}}{\\left (t \\right )} + \\cos{\\left (\\operatorname{f_{1}}{\\left (t \\right )} \\right )} \\cos{\\left (\\operatorname{f_{2}}{\\left (t \\right )} \\right )} \\frac{d}{d t} \\operatorname{f_{3}}{\\left (t \\right )}\\\\\\sin{\\left (\\operatorname{f_{2}}{\\left (t \\right )} \\right )} \\frac{d}{d t} \\operatorname{f_{3}}{\\left (t \\right )} + \\frac{d}{d t} \\operatorname{f_{1}}{\\left (t \\right )} & 0 & \\sin{\\left (\\operatorname{f_{1}}{\\left (t \\right )} \\right )} \\cos{\\left (\\operatorname{f_{2}}{\\left (t \\right )} \\right )} \\frac{d}{d t} \\operatorname{f_{3}}{\\left (t \\right )} - \\cos{\\left (\\operatorname{f_{1}}{\\left (t \\right )} \\right )} \\frac{d}{d t} \\operatorname{f_{2}}{\\left (t \\right )}\\\\- \\sin{\\left (\\operatorname{f_{1}}{\\left (t \\right )} \\right )} \\frac{d}{d t} \\operatorname{f_{2}}{\\left (t \\right )} - \\cos{\\left (\\operatorname{f_{1}}{\\left (t \\right )} \\right )} \\cos{\\left (\\operatorname{f_{2}}{\\left (t \\right )} \\right )} \\frac{d}{d t} \\operatorname{f_{3}}{\\left (t \\right )} & - \\sin{\\left (\\operatorname{f_{1}}{\\left (t \\right )} \\right )} \\cos{\\left (\\operatorname{f_{2}}{\\left (t \\right )} \\right )} \\frac{d}{d t} \\operatorname{f_{3}}{\\left (t \\right )} + \\cos{\\left (\\operatorname{f_{1}}{\\left (t \\right )} \\right )} \\frac{d}{d t} \\operatorname{f_{2}}{\\left (t \\right )} & 0\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡ \n",
"⎢ 0 - sin(f₂\n",
"⎢ \n",
"⎢ \n",
"⎢ d d \n",
"⎢ sin(f₂(t))⋅──(f₃(t)) + ──(f₁(t)) \n",
"⎢ dt dt \n",
"⎢ \n",
"⎢ d d \n",
"⎢- sin(f₁(t))⋅──(f₂(t)) - cos(f₁(t))⋅cos(f₂(t))⋅──(f₃(t)) - sin(f₁(t))⋅cos(f₂\n",
"⎣ dt dt \n",
"\n",
" d d d \n",
"(t))⋅──(f₃(t)) - ──(f₁(t)) sin(f₁(t))⋅──(f₂(t)) + cos(f₁(t))⋅cos(f\n",
" dt dt dt \n",
" \n",
" d \n",
" 0 sin(f₁(t))⋅cos(f₂(t))⋅──(f₃(t)) - cos(f\n",
" dt \n",
" \n",
" d d \n",
"(t))⋅──(f₃(t)) + cos(f₁(t))⋅──(f₂(t)) 0 \n",
" dt dt \n",
"\n",
" d ⎤\n",
"₂(t))⋅──(f₃(t))⎥\n",
" dt ⎥\n",
" ⎥\n",
" d ⎥\n",
"₁(t))⋅──(f₂(t))⎥\n",
" dt ⎥\n",
" ⎥\n",
" ⎥\n",
" ⎥\n",
" ⎦"
]
},
"execution_count": 142,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"simplify(result)"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.6.4"
}
},
"nbformat": 4,
"nbformat_minor": 2
}
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