{
"cells": [
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Mon Nov 20 15:21:48 2017\n"
]
}
],
"source": [
"import time\n",
"print(time.ctime())"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"<sub>Bei diesem IPython-Notebook handelt es sich um eine Beispiel-Rechnung als Ergänzung zur Dissertationsschrift von C. Knoll. Allgemeine Hinweise zu den Beispielen sind in der [readme.md-Datei des entsprechenden Repositoriums](https://github.com/cknoll/beispiele/blob/master/readme.md) zu finden.</sub>\n",
"\n",
"---"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Betrachtetes System**: aktuiertes Pendel, dessen Aufhängepunkt an einem horizontal verschieblichen Wagen hängt, der elastisch mit einem aktuierten Wagen verbunden ist.\n",
"\n",
"\n",
"Bild stimmt nicht mehr!!\n",
"\n",
"<img src=\"files/images/pendel_wagen_feder_wagen.png\">\n",
"\n",
"**Betrachtete Aspekte**:\n",
"\n",
"* Ist das System statisch Eingangs-Zustands-Linearisierbar.\n",
" * Wenn ja, wie sieht ein entsprechender linearisierender Ausgang aus?\n",
"\n",
"* Wie ist die Situation, wenn ein \"intuitiver\" flacher Ausgang (Absolutkoordinaten des Pendelschwerpunkts) untersucht wird?\n",
"\n",
"* Flachheitsbasierte Trajektorienplanung\n",
"\n",
"\n",
"[//]: # (custom_config_start)\n",
"$$\n",
"\\newcommand{\\col}{\\mathrm{col}}\n",
"\\newcommand{\\bs}{\\boldsymbol}\n",
"\\newcommand{\\y}{\\bs y}\n",
"\\newcommand{\\nx}{n_{\\boldsymbol x}}\n",
"\\newcommand{\\Ddt}{\\left(\\tfrac{d}{dt}\\right)}\n",
"$$\n",
"[//]: # (custom_config_end)\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [],
"source": [
"%load_ext ipydex.displaytools\n",
"%matplotlib inline\n",
"\n",
"import sympy as sp\n",
"from sympy import sin, cos, pi\n",
"from sympy.interactive import printing\n",
"import symbtools as st\n",
"import scipy.integrate as sc_integrate\n",
"import matplotlib.pyplot as pl\n",
"\n",
"import symbtools.modeltools as mt\n",
"\n",
"\n",
"printing.init_printing(1)"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAEQAAAAZBAMAAAB6LZWoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu90iiWZ2EETvq82Z\nVDJF77rnAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABC0lEQVQoFWMQUjJgwAM4lFQYFPDIg6WshqkS\nk7CyBjS/M7umZ0CEwJ7mmMywvgBNSacB13ckJZUBDKsZ+DYgK2KfysC1geXgcaAY2JTzDAxT+E4p\nICvh+cDA6eAM1AlVMpGB+QcDL4oSTgEG+wZlhvfA+AOZwv6FgfUDmpJ6B4Z4Bg2gMqiSvwyMG9CV\nXGCQBlpyH+gLsFu2McQ7oCnhTmD9AlSiBsRgJVVp9xvQlLAvy/3BwMByAKaEAWgvmhIGkPMYYoAq\nIKYwgDyH6iMGYLAwcDhwPIArmQJUMgGkBQE4HRhe370Icy6D96cDzId+JSDkgc7Yr8Mw//9/oBDY\nuchSmGxqKVEilI+0AMtTPuYh/ishAAAAAElFTkSuQmCC\n",
"text/latex": [
"$$\\texttt{ttheta.T} := \\left[\\begin{matrix}q_{1} & q_{2}\\end{matrix}\\right]$$"
],
"text/plain": [
"ttheta.T := [q₁ q₂]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"---\n"
]
},
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAEQAAAAZBAMAAAB6LZWoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu90iEM2riWZ2RO+Z\nVDKhftUhAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABJElEQVQoFWMQUjJgwAO4lNQZFPDIg6XMqKnE\nPwDDOpavUCGoRUwJGEoY1FCVYCpAiBDtXPNOzwUIbWAWc8WUWRAhsClcmxnyHdCUrDZg+4KkxLuB\nIZWBD8VTLDsY2ALYDx4BqgKbcp6BYTvfMQVkc3g+MHAXFAN1QpVsZGD+xcCLooRbgMF+gTLDe2D8\ngUxh+c7A8QFNiX8BQz+DFlAZVMlfBsYAdCUXGESBltx3gFoUydBfgKaEdQLHd6ASTSAGO9dt5v0F\naEpYMuf+YmBgPwBTwgC0F00JA8h5DD1AFRBTGECeQ/URAzBYGLgKuB7AlWwHKtkA0oIA3AUMT+9e\ndIApKf10gPnQtwkIeaAz4nUY9v//DxQCOxdZCpNNLSVKhPKRGgBuSkDnWrJ6RAAAAABJRU5ErkJg\ngg==\n",
"text/latex": [
"$$\\texttt{tthetad.T} := \\left[\\begin{matrix}\\dot{q}_{1} & \\dot{q}_{2}\\end{matrix}\\right]$$"
],
"text/plain": [
"tthetad.T := [q̇₁ q̇₂]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"---\n"
]
},
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAEQAAAAZBAMAAAB6LZWoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu90imYlmdhBE76vN\nVDLrfj8uAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABHElEQVQoFWMQUjJgwAM4lVQZFPDIg6WsqajE\nicEJwzq4GMQiJ4YgLEqgYtR0i2l6+wQ0m5jDKiohQmCLOBcz7G9AUzLLgPsHkpKuBIY9DHwHkBVx\nLGPgPsDq6AEUA5viz8CwlM9bAVkJywcGroBgoE6okoUMzD8ZeFGUcAkw2E9QZngPjD+QKRxfGdg+\noCnpD2DIZ9AEKoMq+cvAeABdyQUGGaAl9xugFh1nyA9AU8JTwPYVqEQdiMHO7S6/PwFNCcf22p8M\nDKwOMCUMQHvRlDCAnMeQAlQBMYUB5DlUHzEAg4WBM4DzAVzJUqCSBSAtCMAVwPDm7sUGmJKYTw7M\nTr8LEPJAZ5xXYVj//z9QCOxcZClMNrWUKBHKR9oA6QpA/GlcEr0AAAAASUVORK5CYII=\n",
"text/latex": [
"$$\\texttt{tthetadd.T} := \\left[\\begin{matrix}\\ddot{q}_{1} & \\ddot{q}_{2}\\end{matrix}\\right]$$"
],
"text/plain": [
"tthetadd.T := [q̈₁ q̈₂]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"---\n"
]
}
],
"source": [
"t = sp.Symbol('t')\n",
"np = 0\n",
"nq = 2\n",
"n = np + nq\n",
"pp = st.symb_vector(\"p1:{0}\".format(np+1))\n",
"qq = st.symb_vector(\"q1:{0}\".format(nq+1))\n",
"\n",
"aa = st.symb_vector(\"a1:{0}\".format(nq+1))\n",
"\n",
"ttheta = st.row_stack(pp, qq) ##:T\n",
"tthetad = st.time_deriv(ttheta, ttheta) ##:T\n",
"tthetadd = st.time_deriv(ttheta, ttheta, order=2) ##:T\n",
"st.make_global(ttheta, tthetad, tthetadd)"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"params = sp.symbols('s3, m2, m3, g')\n",
"st.make_global(params)\n",
"\n",
"tau1, tau2 = ttau = st.symb_vector(\"tau1, tau2\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Festlegung der Geometrie des mechanischen Systemes"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAI4AAAAyBAMAAAB42/41AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nggkSb55v1HYAAAAASUVORK5CYII=\n",
"text/latex": [
"$$\\texttt{S3} := \\left[\\begin{matrix}q_{1} - s_{3} \\sin{\\left (q_{2} \\right )}\\\\s_{3} \\cos{\\left (q_{2} \\right )}\\end{matrix}\\right]$$"
],
"text/plain": [
"S3 := ⎡q₁ - s₃⋅sin(q₂)⎤\n",
"⎢ ⎥\n",
"⎣ s₃⋅cos(q₂) ⎦"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"---\n"
]
},
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAB8AAAAyBAMAAABWsh3iAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhDN3asiZnZE\nmbs5ao2tAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABCklEQVQoFWOQ//+JAQaY/v8XYBB2cYXxGVhd\nnAUYROBcEIMFlwDrN6g6uIoKdAEoH6cZnFpb90DUQM04OIHjK4MbSAgiwCrFwJHg244Q4P3AwKjA\nYI4QYHzAMP8AskC8AsN6BhQBA4ZnKAJsG7iAQYBkBmv35j8oAgwMXB/QBDgSGBgMEdYyMDAqMMyt\n7waKQJ3OlF8LkgYCeHhAuHQXYL20yAFkNdwdbAaslSgCCxkY1FAEXjAw2Acga/nBwHDeAEmA9S9Q\nYAOSACcwCP0VkAWAKlAEMLQwAA21RzaU4R0Dw3oUa4EOuwY0E+F0dgNWcHKE+4VVe4cDigoQBwTg\nKiBcqglgZA/0DAQAA4w9aU6O4DgAAAAASUVORK5CYII=\n",
"text/latex": [
"$$\\left[\\begin{matrix}\\dot{q}_{1}\\\\0\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡q̇₁⎤\n",
"⎢ ⎥\n",
"⎣ 0 ⎦"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAKMAAAAyBAMAAADCeHZVAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAElFTkSuQmCC\n",
"text/latex": [
"$$\\left[\\begin{matrix}\\dot{q}_{1} - \\dot{q}_{2} s_{3} \\cos{\\left (q_{2} \\right )}\\\\- \\dot{q}_{2} s_{3} \\sin{\\left (q_{2} \\right )}\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡q̇₁ - q̇₂⋅s₃⋅cos(q₂)⎤\n",
"⎢ ⎥\n",
"⎣ -q̇₂⋅s₃⋅sin(q₂) ⎦"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"---\n"
]
}
],
"source": [
"#Einheitsvektoren\n",
"\n",
"ex = sp.Matrix([1,0])\n",
"ey = sp.Matrix([0,1])\n",
"\n",
"\n",
"# Koordinaten der Schwerpunkte und Gelenke\n",
"S2 = ex*q1 # Schwerpunkt K2\n",
"G3 = S2 # Gelenk\n",
"\n",
"# Schwerpunkt des Pendels #zeigt nach oben\n",
"S3 = G3 + mt.Rz(q2)*ey*s3 ##:\n",
"\n",
"# Zeitableitungen der Schwerpunktskoordinaten\n",
"Sd2, Sd3 = st.col_split(st.time_deriv(st.col_stack(S2, S3), ttheta)) ##"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
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"text/latex": [
"$$\\texttt{T} := \\frac{m_{2} \\dot{q}_{1}^{2}}{2} + \\frac{m_{3} \\dot{q}_{2}^{2}}{2} s_{3}^{2} \\sin^{2}{\\left (q_{2} \\right )} + \\frac{m_{3}}{2} \\left(\\dot{q}_{1} - \\dot{q}_{2} s_{3} \\cos{\\left (q_{2} \\right )}\\right)^{2}$$"
],
"text/plain": [
"T := 2 2 2 2 2\n",
"m₂⋅q̇₁ m₃⋅q̇₂ ⋅s₃ ⋅sin (q₂) m₃⋅(q̇₁ - q̇₂⋅s₃⋅cos(q₂)) \n",
"─────── + ──────────────────── + ──────────────────────────\n",
" 2 2 2 "
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"---\n"
]
},
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAH0AAAAUBAMAAABSVS+EAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nx60r//l//QGEl5pF9StaXAAAAABJRU5ErkJggg==\n",
"text/latex": [
"$$\\texttt{V} := g m_{3} s_{3} \\cos{\\left (q_{2} \\right )}$$"
],
"text/plain": [
"V := g⋅m₃⋅s₃⋅cos(q₂)"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"---\n"
]
}
],
"source": [
"# Energie\n",
"\n",
"T_rot = 0 # Pendel as mathematisches Pendel angenommen\n",
"T_trans = ( m2*Sd2.T*Sd2 + m3*Sd3.T*Sd3 )/2\n",
"\n",
"T = T_rot + T_trans[0] ##:\n",
"V = m3*g*S3[1] ##:"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"external_forces = [tau1, tau2]\n",
"assert not any(external_forces[:np])\n",
"mod = mt.generate_symbolic_model(T, V, ttheta, external_forces)"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAASkAAAAyBAMAAADy0TBdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\neX9HrBrPTQMelKvW/E9cS/7X8j+i+td6pc4aqQAAAABJRU5ErkJggg==\n",
"text/latex": [
"$$\\left[\\begin{matrix}m_{2} + m_{3} & - m_{3} s_{3} \\cos{\\left (q_{2} \\right )}\\\\- m_{3} s_{3} \\cos{\\left (q_{2} \\right )} & m_{3} s_{3}^{2}\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡ m₂ + m₃ -m₃⋅s₃⋅cos(q₂)⎤\n",
"⎢ ⎥\n",
"⎢ 2 ⎥\n",
"⎣-m₃⋅s₃⋅cos(q₂) m₃⋅s₃ ⎦"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Massenmatrix\n",
"mod.MM"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"data": {
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"text/latex": [
"$$\\left[\\begin{matrix}m_{2} \\ddot{q}_{1} + m_{3} \\left(\\ddot{q}_{1} - \\ddot{q}_{2} s_{3} \\cos{\\left (q_{2} \\right )} + \\dot{q}_{2}^{2} s_{3} \\sin{\\left (q_{2} \\right )}\\right) - \\tau_{1}\\\\- g m_{3} s_{3} \\sin{\\left (q_{2} \\right )} - m_{3} \\ddot{q}_{1} s_{3} \\cos{\\left (q_{2} \\right )} + m_{3} \\ddot{q}_{2} s_{3}^{2} - \\tau_{2}\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡ ⎛ 2 ⎞ ⎤\n",
"⎢m₂⋅q̈₁ + m₃⋅⎝q̈₁ - q̈₂⋅s₃⋅cos(q₂) + q̇₂ ⋅s₃⋅sin(q₂)⎠ - τ₁⎥\n",
"⎢ ⎥\n",
"⎢ 2 ⎥\n",
"⎣ -g⋅m₃⋅s₃⋅sin(q₂) - m₃⋅q̈₁⋅s₃⋅cos(q₂) + m₃⋅q̈₂⋅s₃ - τ₂ ⎦"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mod.eqns"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"mod.calc_state_eq()"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
"data": {
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"text/latex": [
"$$\\texttt{mod.f } := \\left[\\begin{matrix}\\dot{q}_{1}\\\\\\dot{q}_{2}\\\\\\frac{m_{3} \\left(g \\cos{\\left (q_{2} \\right )} - \\dot{q}_{2}^{2} s_{3}\\right) \\sin{\\left (q_{2} \\right )}}{m_{2} + m_{3} \\sin^{2}{\\left (q_{2} \\right )}}\\\\\\frac{\\left(g \\left(m_{2} + m_{3}\\right) - m_{3} \\dot{q}_{2}^{2} s_{3} \\cos{\\left (q_{2} \\right )}\\right) \\sin{\\left (q_{2} \\right )}}{s_{3} \\left(m_{2} + m_{3} \\sin^{2}{\\left (q_{2} \\right )}\\right)}\\end{matrix}\\right]$$"
],
"text/plain": [
"mod.f := ⎡ q̇₁ ⎤\n",
"⎢ ⎥\n",
"⎢ q̇₂ ⎥\n",
"⎢ ⎥\n",
"⎢ ⎛ 2 ⎞ ⎥\n",
"⎢ m₃⋅⎝g⋅cos(q₂) - q̇₂ ⋅s₃⎠⋅sin(q₂) ⎥\n",
"⎢ ──────────────────────────────── ⎥\n",
"⎢ 2 ⎥\n",
"⎢ m₂ + m₃⋅sin (q₂) ⎥\n",
"⎢ ⎥\n",
"⎢⎛ 2 ⎞ ⎥\n",
"⎢⎝g⋅(m₂ + m₃) - m₃⋅q̇₂ ⋅s₃⋅cos(q₂)⎠⋅sin(q₂)⎥\n",
"⎢──────────────────────────────────────────⎥\n",
"⎢ ⎛ 2 ⎞ ⎥\n",
"⎣ s₃⋅⎝m₂ + m₃⋅sin (q₂)⎠ ⎦"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"___\n"
]
},
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAXIAAAByCAMAAAB5jaqPAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADFBd4eAAAAFHRS\nTlMAMquZdlQQQO0wRIlmzd0i77t8bBwggJIAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAqQSURBVHgB\n7V2BlqMqDMVq3X3VGdv3/P9vfQGCgiRqhdB2Rs/ZaVUkuVeKkFxZVY1mu6hzE2bg2zKtVDXWDWxX\nYXtn9arXPF9GTXmVmY627voud6WZfWSqk3e9F6H8C34x7ffAoHrrw/Kui1De3zWr3ddbc0s7V8B1\nEcof5kl8HVsa1jsfLeC6COWjofw2fuADuYDrEpS3Y60b8m1s3rk9k76VcF2C8mHsNJ7KfpDQ3vVg\nCddlKDet/DMpl3ddgvISv06hn0kJ1yUoV/YZdP3cx6eo6yKUPx66EfYfOUiUd12EcjufqD93KiTq\nugjl6ltP+O8fGWSRd12G8raDgNlHMq7kXZehXGg88TOqPSkvfh9Pyk/KizNQ3ODZyk/KizNQ3ODZ\nyk/KizNQ3ODZyk/KizNQ3ODZyn8I5fL6GzGi5F2XaeXy+hsxyuVdF6G8gP5GivICrotQXkB/I0V5\nAddFKC+gv5GivIDrEpSXSJMLUV7CdQnKS+hvhCgv4boM5fL6GzHK5V2XoLzEr1OI8hKuS1B+SodW\nG4SmvL1mFoKf0qE1zgdQnGR/V+iUDq1RLtKxnNKh8pTL62/WMCWdk3ddppUngf7pF5+UF7/DJ+Un\n5cUZKG7wbOUn5cUZKG7wbOUn5cUZoA3GSz5ke0PhbOUk5eZd4fDM9RbuH97TlP8Z/xy+nruw+sR3\nsyYwl7iRK1VTB6dL9n/5V2Q9lqq+fO/34e1KVkQjh4BrppWwpDqW6ydTzrxZ9pWnmf9myruuq+Gh\nCMtddZAwaJq+w6U1bHNpYcWxm7++SdNn+T3+Ysov0H3Av4d+R/Vb3WBvsB3KYChvvyvIJAyq6rqH\nbd7XPD3L76V8sO+7V/AwU+rRDGN9bW127GZeKq8h8VzBilWwc7PN3h5Pbum/l/Kb4VrZDNalVrfH\nONpx4M205ju0/ga+wdCrxbuTZxT2eykfoNOAzbXyCvbwmV/pVt7q0w/bueNxeyvOVn6cAWjZqm3U\nAx6Kw729agGL6VF0zw4bLEGASeEWhyo9OXZ82gGxVm7W7XvanaIXdHWjSdQDl0FdYdmBzj4m1bfu\n02+XpjN9T3tBAUSdZ/4pQ/lQf4+Xz1vMzN1wtxZtrzvvth5a27zzdOWwTo2AqMK5/rGftoPRQ0il\n9NLApqe55hmWG8rbBn85H0tRbsctu/X45YUPW6NWzGBpgJ9/dulQBr9eXMXV4xpdydYuVzsWu9r2\nK/4eYPwVbu6wGSNZpTwunnpknsqZmnCE4NUaty7v5PNfYwPP17G8YicGFok85b2dLVvH56mc3ifG\nudkSAcYeYcD68dzfAII3HdW1ECYsBhaJPOXLOdsc1hVNBGg6SAP6xJPbEoKbpkI1pAmbzOBSGscp\nPxhWc1M5eGoTDQSGwQfrJVikDXgFj5rah4FDcpxyHLt6COivbdCpTVM5aCDBieniTIkAqI8xMFnC\n+f28z30LIUCrmAbVjAmLgUGyTnlf91XT1BC+j+t2lLNlbMy/fcBka/i6V00HjWqeysEcwyJcZgIS\nEgEwZ/dTCsYA6x5YPwSBwLCEoCwGBskG5bcRplx3uKs6jOlF631/F2X6RqdaIEYB/YaO+bc6UDHo\n9W/hXe1pKgeHLOVRJuB4IgBiIDoIWKEH1kC/cG9KOByFEGOIICiLgUGySvnQNpoXHUaudccLeDBa\nP/sblbmpqwlGuJg/Ug6/xTDqggH/KBNwPBHweJgkwwM9MBVF7uWEADF2IEJFEPA4g2SVcsuSTo2Y\n9yKm4dHlC7a7/qNDV5rJuQxQ29nnIsb8kXIsCB+44TAgygQcl2NUl1H/dz3OAzQQuudGqVkgQLRR\ng4kgKIuBQbJBuV63toFbOYzKPuzmIZ65wcZiWKbq7BkX8+coN4kAIhMQjcjwFm1+XGHeY35I6IE1\noOPeUhBMP0ZAwFvBINGU//n7D4MHmLYw+ocd88/Do+nZE5VRN9OxuJg/R7lNBMSZgMOJAEgPK/sY\nsR5YA5F7GSFsJDMYJP/9hV6B/d+zzH3SY53honUH/vBoonxRRnc1JsWFMf/2Av8ZCAxbLqq561WT\n580kAuJMwOFEQN/3jU4iTB5gpgEOSUFYT2YwSDY6lpkg/c0f4k2UL4pAx2869vAwtTcN6MJMgPmJ\nUOV3HmsnDyYD3pV5IShnIoSgH3qwMUieotwb4s2DWg+P/gr6m3gMvyiDu+5pEGQC0hMBswfOgGc+\nMwTX8AIIymLgkDxFuec69DXB3pEd9CnIBGRLBGiHONDO2XQIaCKAAJ2BNsAiOU6583vlM4Zkhz3u\nEslEgLFBGHC2c30SJmwyg01pSFJuh+cBNjaiGZT62TuClK/ENX82pxvo5CinQ6dcRHPDzZ90Wo5y\nZtzCRDTfidOjYfSdGFIpZzXaGJtl4po7vXtNMWJsmdORRMr1eBT+xRptjM1ycc2cEA7VlRRGP2Rx\nuiiNcl6jvRHXnOy/6ktKGD3R5zTKeY32Rlwz0evky5PC6InW0yjnNdobcc1ErzNcnhJGTzOfRrli\nNdobcc00p3NcnRBGTzSfSLliNdrrcc1Er9MvTwmjJ1pPpZw3vxrX5C8rdGYR5X8qE5Doohzla3HN\nRKfzX545jL7qoCDlNnRKxTVXPXrJydxh9DUQgpQrNq655tAbnItjzlmdCinfoZd+RREW8SucOW4T\nYYSUk9gkbjqEMWYREpVhwlxGmNIg3Vs76JuRgDG/bG68iE1QMLYpJxINayjZc4FIGzKx+MqwLk+Y\nwFxGYkrDM0PYYF1dOxHA8G/pbhiblJOJhjWfuHNLIc0sQiJNoDqbE2lzVqLjaIa0ERXecSCA4d1S\nCO/FjXxaN8eHsUU5nWjwXDsaXJ4VPLQJzGWkpjTQDG0jA4xZ3U+bIGBsUc4kGmZndweXQ5H2Xo22\nUf3M1p795sxIwVBzy2FMYErGy8xElFMabbUSXXaU80XSdOaMSBu5Z61aGNMMJ11qjkvkhHJ5f9Jq\ntdugpQgXzkH/PRhLykmNtlqJLk+UL3TcTmaeqjNnRNqOcsYqwnAznHSpOQ1juqUw6FqXy3swlpRT\nGm21Fl1GyqMiTmYOol27tsxBnTkj0raUs1YRhuuH0qXmNAx3S8EO+hlpzfG4B2NJOa3RXmjI8UG9\nT6MNzpi1ZTgFLg4BIpE2qrMZkbYjMwx7O2m5QhiuFNoIC7vhRmkYC8oZjfaKSNt1LMsiTmauEnXm\nwaDMUTh/LsLeaNXBcO+wpEvNM8JYUE5rtLWin9OZO8qjIigzV4k6c0akjaQzVh0M9w5LutQ8I4wF\n5bRG2yYyaZG2o9y0xrnIJPJ2a8sc1ZkzIm2knLHqYEwdTbLUPCOMBeWIZF2jHejMHeV4ZVRBeJza\nc7kMFYq0oavVG37YnY2/s9uuoOveJhvuBHx6ww1Ouh1X6F2/+DqZ2IJBU+6pxClKvQe1S0Qs7HsV\nLM/E+85EINJGofKWXjmsbZaWT8ftWzSUmz4K6ryugqhwqnn5ZS8MS7kJSNJT9y3MVGBh6czGPpoI\nkhmozmZF2htV4mmve9tYMqgMDH2XYYNfGCxBChuzWBeRaNiHd38pwgSqs1mR9q7K/X6BsLGrjicK\nESYWMHrL9BN1flzRZ/qFkuD+BxCkvs6S+KNkAAAAAElFTkSuQmCC\n",
"text/latex": [
"$$\\texttt{mod.g } := \\left[\\begin{matrix}0 & 0\\\\0 & 0\\\\\\frac{1}{m_{2} + m_{3} \\sin^{2}{\\left (q_{2} \\right )}} & \\frac{\\cos{\\left (q_{2} \\right )}}{s_{3} \\left(m_{2} + m_{3} \\sin^{2}{\\left (q_{2} \\right )}\\right)}\\\\\\frac{\\cos{\\left (q_{2} \\right )}}{s_{3} \\left(m_{2} + m_{3} \\sin^{2}{\\left (q_{2} \\right )}\\right)} & \\frac{m_{2} + m_{3}}{m_{3} s_{3}^{2} \\left(m_{2} + m_{3} \\sin^{2}{\\left (q_{2} \\right )}\\right)}\\end{matrix}\\right]$$"
],
"text/plain": [
"mod.g := ⎡ 0 0 ⎤\n",
"⎢ ⎥\n",
"⎢ 0 0 ⎥\n",
"⎢ ⎥\n",
"⎢ 1 cos(q₂) ⎥\n",
"⎢ ──────────────── ───────────────────── ⎥\n",
"⎢ 2 ⎛ 2 ⎞ ⎥\n",
"⎢ m₂ + m₃⋅sin (q₂) s₃⋅⎝m₂ + m₃⋅sin (q₂)⎠ ⎥\n",
"⎢ ⎥\n",
"⎢ cos(q₂) m₂ + m₃ ⎥\n",
"⎢───────────────────── ─────────────────────────⎥\n",
"⎢ ⎛ 2 ⎞ 2 ⎛ 2 ⎞⎥\n",
"⎣s₃⋅⎝m₂ + m₃⋅sin (q₂)⎠ m₃⋅s₃ ⋅⎝m₂ + m₃⋅sin (q₂)⎠⎦"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"___\n"
]
}
],
"source": [
"mod.f ##:\n",
"\n",
"mod.g ##:"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAACIAAABkCAMAAADUmBKBAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\n",
"text/latex": [
"$$\\texttt{f} := \\left[\\begin{matrix}\\dot{q}_{1}\\\\\\dot{q}_{2}\\\\0\\\\0\\end{matrix}\\right]$$"
],
"text/plain": [
"f := ⎡q̇₁⎤\n",
"⎢ ⎥\n",
"⎢q̇₂⎥\n",
"⎢ ⎥\n",
"⎢ 0 ⎥\n",
"⎢ ⎥\n",
"⎣ 0 ⎦"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"---\n"
]
},
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAADkAAABkCAMAAAAL1mieAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nJ2BrL6+ybPryzdha8wvC6SxOy90ZPwAAAABJRU5ErkJggg==\n",
"text/latex": [
"$$\\texttt{G} := \\left[\\begin{matrix}0 & 0\\\\0 & 0\\\\1 & 0\\\\0 & 1\\end{matrix}\\right]$$"
],
"text/plain": [
"G := ⎡0 0⎤\n",
"⎢ ⎥\n",
"⎢0 0⎥\n",
"⎢ ⎥\n",
"⎢1 0⎥\n",
"⎢ ⎥\n",
"⎣0 1⎦"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"---\n"
]
},
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAJIAAAAZBAMAAAAoImqmAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
"text/latex": [
"$$\\texttt{xx.T} := \\left[\\begin{matrix}q_{1} & q_{2} & \\dot{q}_{1} & \\dot{q}_{2}\\end{matrix}\\right]$$"
],
"text/plain": [
"xx.T := [q₁ q₂ q̇₁ q̇₂]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"---\n"
]
}
],
"source": [
"# partiell linearisierte Gleichungen (hier: partiell=vollständig)\n",
"mod.calc_coll_part_lin_state_eq()\n",
"\n",
"f = mod.ff ##:\n",
"G = mod.gg ##:\n",
"g1, g2 = st.col_split(G)\n",
"\n",
"xx = mod.x ##:T"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Überprüfung auf statische Eingangs-Zustands-Linearisierbarkeit\n",
"\n",
"Konstrunktion der Distributionen $\\Delta_k$"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"nx = len(xx)\n",
"\n",
"# Liste der Distributionen\n",
"# (mit Eingangsvektorfeldern starten (Delta_0 = G) )\n",
"Delta_list = [mod.g]\n",
"\n",
"for k in range(1, nx):\n",
" Delta_j_full = Delta_list[-1]\n",
" Delta_j = Delta_j_full[:, -nq:]# nur die letzten nq Spalten für Lieklammer relevant\n",
" Delta_k = []\n",
" for vf in st.col_split(Delta_j):\n",
" Delta_k.append(st.lie_bracket(mod.f, vf, xx))\n",
" \n",
" # neu berechnete VF zusammen mit bisherigen in Matrix konvertieren\n",
" Delta_k = st.col_stack(Delta_j_full, *Delta_k)\n",
" Delta_list.append(Delta_k)"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"dim(Delta_0) = 2\n",
"Delta_0 involutiv? True\n",
"\n",
"dim(Delta_1) = 4\n",
"Delta_1 involutiv? True\n",
"\n",
"dim(Delta_2) = 4\n",
"Delta_2 involutiv? True\n",
"\n",
"dim(Delta_3) = 4\n",
"Delta_3 involutiv? True\n",
"\n"
]
}
],
"source": [
"for i, D in enumerate(Delta_list):\n",
" print(\"dim(Delta_%i) =\" %i, st.rnd_number_rank(D))\n",
" \n",
" # failing_vf enthält die ersten beiden VF, deren Lie-Klammer nicht in der Distribution liegen\n",
" inv_flag, failing_vf = st.involutivity_test(D, xx)\n",
" print(\"Delta_%i involutiv? %s\\n\" %(i, inv_flag))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Fazit:\n",
"Das System ist also statisch Eingangs-Zustands-Linearisierbar.\n",
"\n",
"---\n",
"\n",
"Überprüfung: wird durch die Absolutkoordinaten des Pendelschwerpunkts ein in statisch linearisierender Ausgang gebildet (d.h. hat dieser Ausgang einen wohldefinierten und vollständigen relativen Grad?):"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"h1, h2 = S3"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Funktion definieren, die $L_{\\bs g_i}L_{\\bs f}^k h(\\bs x)$ in Abhängigkeit von $h, i$ und $k$ berechnet"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def LgLf(h, i, k):\n",
" \"\"\"\n",
" h: Skalarfeld\n",
" i: Index für Eingangsvektorfeld\n",
" k: Ordnung der Lieableitung bezüglich k\n",
" \"\"\"\n",
" Lf = st.lie_deriv(h, mod.f, xx, n=k)\n",
" res = st.lie_deriv(Lf, mod.g[:, i], xx)\n",
" return res.simplify()"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {},
"outputs": [
{
"data": {
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"text/latex": [
"$$\\left[\\begin{matrix}\\frac{\\sin^{2}{\\left (q_{2} \\right )}}{m_{2} + m_{3} \\sin^{2}{\\left (q_{2} \\right )}} & - \\frac{m_{2} \\cos{\\left (q_{2} \\right )}}{m_{3} s_{3} \\left(m_{2} + m_{3} \\sin^{2}{\\left (q_{2} \\right )}\\right)}\\\\- \\frac{\\sin{\\left (2 q_{2} \\right )}}{2 m_{2} - m_{3} \\cos{\\left (2 q_{2} \\right )} + m_{3}} & - \\frac{\\left(m_{2} + m_{3}\\right) \\sin{\\left (q_{2} \\right )}}{m_{3} s_{3} \\left(m_{2} + m_{3} \\sin^{2}{\\left (q_{2} \\right )}\\right)}\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡ 2 ⎤\n",
"⎢ sin (q₂) -m₂⋅cos(q₂) ⎥\n",
"⎢ ──────────────── ────────────────────────⎥\n",
"⎢ 2 ⎛ 2 ⎞⎥\n",
"⎢ m₂ + m₃⋅sin (q₂) m₃⋅s₃⋅⎝m₂ + m₃⋅sin (q₂)⎠⎥\n",
"⎢ ⎥\n",
"⎢ -sin(2⋅q₂) -(m₂ + m₃)⋅sin(q₂) ⎥\n",
"⎢──────────────────────── ────────────────────────⎥\n",
"⎢2⋅m₂ - m₃⋅cos(2⋅q₂) + m₃ ⎛ 2 ⎞⎥\n",
"⎣ m₃⋅s₃⋅⎝m₂ + m₃⋅sin (q₂)⎠⎦"
]
},
"execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sp.Matrix([[LgLf(h1, i=0, k=1), LgLf(h1, i=1, k=1)],\n",
" [LgLf(h2, i=0, k=1), LgLf(h2, i=1, k=1)]])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Für $q_2 \\neq 0$ usw. ist diese Matrix regulär. D.h. $\\y$ hat bis auf Singularitäten den vektoriellen relativen Grad $(2, 2)^T$"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": []
}
],
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"language": "python",
"name": "python3"
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