345 lines
23 KiB
Plaintext
345 lines
23 KiB
Plaintext
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{
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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"
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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": 120,
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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": 120,
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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": 121,
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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": 121,
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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": "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": 98,
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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": 110,
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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) #index ox closest wp\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,:], 3, 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"
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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": 111,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"array([-0.12829547, 0.96144073, -1.47313104, -0.47433648])"
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]
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},
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"execution_count": 111,
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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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"coeff"
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]
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},
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{
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"cell_type": "code",
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|
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"execution_count": 113,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"image/png": "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"text/plain": [
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"<Figure size 432x288 with 2 Axes>"
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]
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},
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"metadata": {},
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||
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"output_type": "display_data"
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}
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],
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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(0,3,0.001) #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.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",
|
||
|
|
"plt.axis('equal')\n",
|
||
|
|
"\n",
|
||
|
|
"# MAP REF FRAME\n",
|
||
|
|
"plt.subplot(2,1,2)\n",
|
||
|
|
"plt.scatter(state[0],state[1])\n",
|
||
|
|
"plt.scatter(track[0,:],track[1,:])\n",
|
||
|
|
"plt.scatter(track[0,nn_idx:nn_idx+LOOKAHED],track[1,nn_idx:nn_idx+LOOKAHED])\n",
|
||
|
|
"plt.axis('equal');"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"# COMPUTE ERRORS"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"* **crosstrack error** cte -> desired y-position - y-position of vehicle: this is the bias term of the fitted polynomial (road curve)\n",
|
||
|
|
" \n",
|
||
|
|
"$\n",
|
||
|
|
"f = K_0 * x^3 + K_1 * x^2 + K_2 * x + K_3\n",
|
||
|
|
"$\n",
|
||
|
|
"\n",
|
||
|
|
"Then for the origin cte = K_3\n",
|
||
|
|
" \n",
|
||
|
|
"* **heading error** epsi -> desired heading - heading of vehicle : is the inclination of tangent to the fitted polynomial (road curve)\n",
|
||
|
|
"\n",
|
||
|
|
"The derivative of the fitted poly has the form\n",
|
||
|
|
"\n",
|
||
|
|
"$\n",
|
||
|
|
"f' = 3.0 * K_0 * x^2 + 2.0 * K_1 * x + K_2\n",
|
||
|
|
"$\n",
|
||
|
|
"\n",
|
||
|
|
"Then for the origin the equation of the tangent in the origin is $y=k2$ \n",
|
||
|
|
"\n",
|
||
|
|
"epsi = -atan(K_2)\n",
|
||
|
|
"\n",
|
||
|
|
"in general:\n",
|
||
|
|
"\n",
|
||
|
|
"$\n",
|
||
|
|
"y_{desired} = f(px) \\\\\n",
|
||
|
|
"heading_{desired} = -atan(f`(px))\n",
|
||
|
|
"$"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": 119,
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [
|
||
|
|
{
|
||
|
|
"name": "stdout",
|
||
|
|
"output_type": "stream",
|
||
|
|
"text": [
|
||
|
|
"-0.47433647872956336\n",
|
||
|
|
"55.83031014720696\n"
|
||
|
|
]
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"source": [
|
||
|
|
"#for 0\n",
|
||
|
|
"\n",
|
||
|
|
"cte=coeff[3]\n",
|
||
|
|
"epsi=-np.arctan(coeff[2])\n",
|
||
|
|
"print(cte)\n",
|
||
|
|
"print(np.degrees(epsi))"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"# ADD DELAY (for real time implementation)"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "markdown",
|
||
|
|
"metadata": {},
|
||
|
|
"source": [
|
||
|
|
"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",
|
||
|
|
"\n",
|
||
|
|
"* $x_{delay} = 0.0 + v * dt$\n",
|
||
|
|
"* $y_{delay} = 0.0$\n",
|
||
|
|
"* $psi_{delay} = 0.0 + w * dt$\n",
|
||
|
|
"* $cte_{delay} = cte + v * sin(epsi) * dt$\n",
|
||
|
|
"* $epsi_{delay} = epsi - w * dt$\n",
|
||
|
|
"\n",
|
||
|
|
"Note that the current position and heading is always 0; this is becouse the path is parametrized to vehicle reference frame"
|
||
|
|
]
|
||
|
|
},
|
||
|
|
{
|
||
|
|
"cell_type": "code",
|
||
|
|
"execution_count": null,
|
||
|
|
"metadata": {},
|
||
|
|
"outputs": [],
|
||
|
|
"source": []
|
||
|
|
}
|
||
|
|
],
|
||
|
|
"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": "3.6.9"
|
||
|
|
}
|
||
|
|
},
|
||
|
|
"nbformat": 4,
|
||
|
|
"nbformat_minor": 4
|
||
|
|
}
|