464 lines
12 KiB
Plaintext
464 lines
12 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# STATE SPACE MODEL MATRICES\n",
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"\n",
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"### Diff drive"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/latex": [
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"$\\displaystyle \\left[\\begin{matrix}0 & 0 & - v \\sin{\\left(\\theta \\right)} & 0 & 0\\\\0 & 0 & v \\cos{\\left(\\theta \\right)} & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & v \\cos{\\left(\\psi \\right)} & 0\\end{matrix}\\right]$"
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],
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"text/plain": [
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"Matrix([\n",
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"[0, 0, -v*sin(theta), 0, 0],\n",
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"[0, 0, v*cos(theta), 0, 0],\n",
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"[0, 0, 0, 0, 0],\n",
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"[0, 0, 0, 0, 0],\n",
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"[0, 0, 0, v*cos(psi), 0]])"
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]
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},
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"execution_count": 1,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"import sympy as sp\n",
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"\n",
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"x,y,theta,psi,cte,v,w = sp.symbols(\"x y theta psi cte v w\")\n",
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"\n",
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"gs = sp.Matrix([[ sp.cos(theta)*v],\n",
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" [ sp.sin(theta)*v],\n",
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" [w],\n",
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" [-w],\n",
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" [ v*sp.sin(psi)]])\n",
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"\n",
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"state = sp.Matrix([x,y,theta,psi,cte])\n",
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"\n",
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"#A\n",
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"gs.jacobian(state)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/latex": [
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"$\\displaystyle \\left[\\begin{matrix}\\cos{\\left(\\theta \\right)} & 0\\\\\\sin{\\left(\\theta \\right)} & 0\\\\0 & 1\\\\0 & -1\\\\\\sin{\\left(\\psi \\right)} & 0\\end{matrix}\\right]$"
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],
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"text/plain": [
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"Matrix([\n",
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"[cos(theta), 0],\n",
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"[sin(theta), 0],\n",
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"[ 0, 1],\n",
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"[ 0, -1],\n",
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"[ sin(psi), 0]])"
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]
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},
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"execution_count": 2,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"state = sp.Matrix([v,w])\n",
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"\n",
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"#B\n",
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"gs.jacobian(state)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/latex": [
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"$\\displaystyle \\left[\\begin{matrix}1 & 0 & - dt v \\sin{\\left(\\theta \\right)}\\\\0 & 1 & dt v \\cos{\\left(\\theta \\right)}\\\\0 & 0 & 1\\end{matrix}\\right]$"
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],
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"text/plain": [
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"Matrix([\n",
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"[1, 0, -dt*v*sin(theta)],\n",
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"[0, 1, dt*v*cos(theta)],\n",
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"[0, 0, 1]])"
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]
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},
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"execution_count": 3,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"import sympy as sp\n",
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"\n",
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"x,y,theta,psi,cte,v,w ,dt= sp.symbols(\"x y theta psi cte v w dt\")\n",
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"\n",
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"gs = sp.Matrix([[x + sp.cos(theta)*v*dt],\n",
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" [y+ sp.sin(theta)*v*dt],\n",
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" [theta + w*dt]])\n",
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"\n",
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"state = sp.Matrix([x,y,theta])\n",
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"\n",
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"#A\n",
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"gs.jacobian(state)#.subs({x:0,y:0,theta:0})"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/latex": [
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"$\\displaystyle \\left[\\begin{matrix}dt \\cos{\\left(\\theta \\right)} & 0\\\\dt \\sin{\\left(\\theta \\right)} & 0\\\\0 & dt\\end{matrix}\\right]$"
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],
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"text/plain": [
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"Matrix([\n",
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"[dt*cos(theta), 0],\n",
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"[dt*sin(theta), 0],\n",
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"[ 0, dt]])"
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]
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},
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"state = sp.Matrix([v,w])\n",
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"\n",
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"#B\n",
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"gs.jacobian(state)#.subs({x:0,y:0,theta:0})"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Ackermann"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/latex": [
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"$\\displaystyle \\left[\\begin{matrix}0 & 0 & \\cos{\\left(\\theta \\right)} & - v \\sin{\\left(\\theta \\right)}\\\\0 & 0 & \\sin{\\left(\\theta \\right)} & v \\cos{\\left(\\theta \\right)}\\\\0 & 0 & 0 & 0\\\\0 & 0 & \\frac{\\tan{\\left(\\delta \\right)}}{L} & 0\\end{matrix}\\right]$"
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],
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"text/plain": [
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"Matrix([\n",
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"[0, 0, cos(theta), -v*sin(theta)],\n",
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"[0, 0, sin(theta), v*cos(theta)],\n",
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"[0, 0, 0, 0],\n",
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"[0, 0, tan(delta)/L, 0]])"
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]
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},
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"execution_count": 8,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"x,y,theta,v,delta,L,a = sp.symbols(\"x y theta v delta L a\")\n",
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"\n",
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"gs = sp.Matrix([[ sp.cos(theta)*v],\n",
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" [ sp.sin(theta)*v],\n",
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" [a],\n",
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" [ v*sp.tan(delta)/L]])\n",
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"\n",
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"state = sp.Matrix([x,y,v,theta])\n",
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"\n",
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"#A\n",
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"gs.jacobian(state)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/latex": [
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"$\\displaystyle \\left[\\begin{matrix}0 & 0\\\\0 & 0\\\\1 & 0\\\\0 & \\frac{v \\left(\\tan^{2}{\\left(\\delta \\right)} + 1\\right)}{L}\\end{matrix}\\right]$"
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],
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"text/plain": [
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"Matrix([\n",
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"[0, 0],\n",
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"[0, 0],\n",
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"[1, 0],\n",
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"[0, v*(tan(delta)**2 + 1)/L]])"
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]
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},
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"execution_count": 7,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"state = sp.Matrix([a,delta])\n",
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"\n",
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"#B\n",
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"gs.jacobian(state)#.subs({x:0,y:0,theta:0})"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# PATH WAYPOINTS AS PARAMETRIZED CURVE"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"from scipy.interpolate import interp1d\n",
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"\n",
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"def compute_path_from_wp(start_xp, start_yp, step = 0.1):\n",
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" final_xp=[]\n",
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" final_yp=[]\n",
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" delta = step #[m]\n",
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"\n",
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" for idx in range(len(start_xp)-1):\n",
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" section_len = np.sum(np.sqrt(np.power(np.diff(start_xp[idx:idx+2]),2)+np.power(np.diff(start_yp[idx:idx+2]),2)))\n",
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"\n",
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" interp_range = np.linspace(0,1,np.floor(section_len/delta).astype(int))\n",
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" \n",
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" fx=interp1d(np.linspace(0,1,2),start_xp[idx:idx+2],kind=1)\n",
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" fy=interp1d(np.linspace(0,1,2),start_yp[idx:idx+2],kind=1)\n",
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" \n",
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" final_xp=np.append(final_xp,fx(interp_range))\n",
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" final_yp=np.append(final_yp,fy(interp_range))\n",
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"\n",
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" return np.vstack((final_xp,final_yp))\n",
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"\n",
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"def get_nn_idx(state,path):\n",
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"\n",
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" dx = state[0]-path[0,:]\n",
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" dy = state[1]-path[1,:]\n",
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" dist = np.sqrt(dx**2 + dy**2)\n",
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" nn_idx = np.argmin(dist)\n",
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"\n",
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" try:\n",
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" v = [path[0,nn_idx+1] - path[0,nn_idx],\n",
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" path[1,nn_idx+1] - path[1,nn_idx]] \n",
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" v /= np.linalg.norm(v)\n",
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"\n",
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" d = [path[0,nn_idx] - state[0],\n",
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" path[1,nn_idx] - state[1]]\n",
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"\n",
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" if np.dot(d,v) > 0:\n",
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" target_idx = nn_idx\n",
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" else:\n",
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" target_idx = nn_idx+1\n",
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"\n",
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" except IndexError as e:\n",
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" target_idx = nn_idx\n",
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"\n",
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" return target_idx"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"#define track\n",
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"wp=np.array([0,5,6,10,11,15, 0,0,2,2,0,4]).reshape(2,-1)\n",
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"track = compute_path_from_wp(wp[0,:],wp[1,:],step=0.5)\n",
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"\n",
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"#vehicle state\n",
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"state=[3.5,0.5,np.radians(30)]\n",
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"\n",
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"#given vehicle pos find lookahead waypoints\n",
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"nn_idx=get_nn_idx(state,track)-1 #index ox closest wp, take the previous to have a straighter line\n",
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"LOOKAHED=6\n",
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"lk_wp=track[:,nn_idx:nn_idx+LOOKAHED]\n",
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"\n",
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"#trasform lookahead waypoints to vehicle ref frame\n",
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"dx = lk_wp[0,:] - state[0]\n",
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"dy = lk_wp[1,:] - state[1]\n",
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"\n",
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"wp_vehicle_frame = np.vstack(( dx * np.cos(-state[2]) - dy * np.sin(-state[2]),\n",
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" dy * np.cos(-state[2]) + dx * np.sin(-state[2]) ))\n",
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"\n",
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"#fit poly\n",
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"coeff=np.polyfit(wp_vehicle_frame[0,:], wp_vehicle_frame[1,:], 5, rcond=None, full=False, w=None, cov=False)\n",
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"\n",
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"#def f(x,coeff):\n",
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"# return coeff[0]*x**3+coeff[1]*x**2+coeff[2]*x**1+coeff[3]*x**0\n",
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"def f(x,coeff):\n",
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" return coeff[0]*x**5+coeff[1]*x**4+coeff[2]*x**3+coeff[3]*x**2+coeff[4]*x**1+coeff[5]*x**0"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"coeff"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"import matplotlib.pyplot as plt\n",
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"plt.style.use(\"ggplot\")\n",
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"\n",
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"x=np.arange(-1,2,0.001) #interp range of curve \n",
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"\n",
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"# VEHICLE REF FRAME\n",
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"plt.subplot(2,1,1)\n",
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"plt.title('parametrized curve, vehicle ref frame')\n",
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"plt.scatter(0,0)\n",
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"plt.scatter(wp_vehicle_frame[0,:],wp_vehicle_frame[1,:])\n",
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"plt.plot(x,[f(xs,coeff) for xs in x])\n",
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"plt.axis('equal')\n",
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"\n",
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"# MAP REF FRAME\n",
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"plt.subplot(2,1,2)\n",
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"plt.title('waypoints, map ref frame')\n",
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"plt.scatter(state[0],state[1])\n",
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"plt.scatter(track[0,:],track[1,:])\n",
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"plt.scatter(track[0,nn_idx:nn_idx+LOOKAHED],track[1,nn_idx:nn_idx+LOOKAHED])\n",
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"plt.axis('equal')\n",
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"\n",
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"plt.tight_layout()\n",
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"plt.show()\n",
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"#plt.savefig(\"fitted_poly\")"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# COMPUTE ERRORS"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"* **crosstrack error** cte -> desired y-position - y-position of vehicle: this is the value of the fitted polynomial (road curve)\n",
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" \n",
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"$\n",
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"f = K_0 * x^3 + K_1 * x^2 + K_2 * x + K_3\n",
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"$\n",
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"\n",
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"Then for the origin cte = K_3\n",
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" \n",
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"* **heading error** epsi -> desired heading - heading of vehicle : is the inclination of tangent to the fitted polynomial (road curve)\n",
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"\n",
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"The derivative of the fitted poly has the form\n",
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"\n",
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"$\n",
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"f' = 3.0 * K_0 * x^2 + 2.0 * K_1 * x + K_2\n",
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"$\n",
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"\n",
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"Then for the origin the equation of the tangent in the origin is $y=k2$ \n",
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"\n",
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"epsi = -atan(K_2)\n",
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"\n",
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"in general:\n",
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"\n",
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"$\n",
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"y_{desired} = f(px) \\\\\n",
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"heading_{desired} = -atan(f`(px))\n",
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"$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"#for 0\n",
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"\n",
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"cte=coeff[3]\n",
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"epsi=-np.arctan(coeff[2])\n",
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"print(cte)\n",
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"print(np.degrees(epsi))"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# ADD DELAY (for real time implementation)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"It is necessary to take *actuation latency* into account: so instead of using the actual state as estimated, the delay factored in using the kinematic model\n",
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"\n",
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"* $x_{delay} = 0.0 + v * dt$\n",
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"* $y_{delay} = 0.0$\n",
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"* $psi_{delay} = 0.0 + w * dt$\n",
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"* $cte_{delay} = cte + v * sin(epsi) * dt$\n",
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"* $epsi_{delay} = epsi - w * dt$\n",
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"\n",
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"Note that the starting position and heading is always 0; this is becouse the path is parametrized to **vehicle reference frame**"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.8.3"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 4
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}
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