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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Simplified MPC\n",
"\n",
"Checking the implementation from [Reuben Ferrante](https://github.com/arex18/rocket-lander) \n",
"\n",
"Here the system dynamics matrices are evaluated only once given the current state, input -> So no more need to keep track of **x_bar** and **u_bar**.\n",
"\n",
"This should give less precise results but the computation time should be gratly reduced and the overall code is slimmer, worth checking out!"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import cvxpy as opt\n",
"import time\n",
"from scipy.integrate import odeint\n",
"from scipy.interpolate import interp1d\n",
"import matplotlib.pyplot as plt\n",
"plt.style.use(\"ggplot\")\n",
"\n",
"N = 4 #number of state variables\n",
"M = 2 #number of control variables\n",
"T = 10 #Prediction Horizon\n",
"DT = 0.2 #discretization step\n",
"\n",
"L = 0.3 #vehicle wheelbase\n",
"MAX_SPEED = 1.5 #m/s\n",
"MAX_ACC = 1.0 #m/ss\n",
"MAX_D_ACC = 1.0 #m/sss\n",
"MAX_STEER = np.radians(30) #rad\n",
"MAX_D_STEER = np.radians(30) #rad/s\n",
"\n",
"def get_linear_model_matrices(x_bar,u_bar):\n",
" \"\"\"\n",
" Computes the LTI approximated state space model x' = Ax + Bu + C\n",
" \"\"\"\n",
" \n",
" x = x_bar[0]\n",
" y = x_bar[1]\n",
" v = x_bar[2]\n",
" theta = x_bar[3]\n",
" \n",
" a = u_bar[0]\n",
" delta = u_bar[1]\n",
" \n",
" ct = np.cos(theta)\n",
" st = np.sin(theta)\n",
" cd = np.cos(delta)\n",
" td = np.tan(delta)\n",
" \n",
" A = np.zeros((N,N))\n",
" A[0,2] = ct\n",
" A[0,3] = -v*st\n",
" A[1,2] = st\n",
" A[1,3] = v*ct\n",
" A[3,2] = v*td/L\n",
" A_lin = np.eye(N)+DT*A\n",
" \n",
" B = np.zeros((N,M))\n",
" B[2,0]=1\n",
" B[3,1]=v/(L*cd**2)\n",
" B_lin=DT*B\n",
" \n",
" f_xu=np.array([v*ct, v*st, a, v*td/L]).reshape(N,1)\n",
" C_lin = DT*(f_xu - np.dot(A, x_bar.reshape(N,1)) - np.dot(B, u_bar.reshape(M,1)))#.flatten()\n",
" \n",
" #return np.round(A_lin,6), np.round(B_lin,6), np.round(C_lin,6)\n",
" return A_lin, B_lin, C_lin\n",
"\n",
"class MPC():\n",
" \n",
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" def __init__(self, N, M, Q, R):\n",
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" \"\"\"\n",
" \"\"\"\n",
" self.state_len = N\n",
" self.action_len = M\n",
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" self.state_cost = Q\n",
" self.action_cost = R\n",
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" \n",
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" def optimize_linearized_model(self, A, B, C, initial_state, target, time_horizon=10, Q=None, R=None, verbose=False):\n",
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" \"\"\"\n",
" Optimisation problem defined for the linearised model,\n",
" :param A: \n",
" :param B:\n",
" :param C: \n",
" :param initial_state:\n",
" :param Q:\n",
" :param R:\n",
" :param target:\n",
" :param time_horizon:\n",
" :param verbose:\n",
" :return:\n",
" \"\"\"\n",
" \n",
" assert len(initial_state) == self.state_len\n",
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" \n",
" if (Q == None or R==None):\n",
" Q = self.state_cost\n",
" R = self.action_cost\n",
" \n",
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" # Create variables\n",
" x = opt.Variable((self.state_len, time_horizon + 1), name='states')\n",
" u = opt.Variable((self.action_len, time_horizon), name='actions')\n",
"\n",
" # Loop through the entire time_horizon and append costs\n",
" cost_function = []\n",
"\n",
" for t in range(time_horizon):\n",
"\n",
" _cost = opt.quad_form(target[:, t + 1] - x[:, t + 1], Q) +\\\n",
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" opt.quad_form(u[:, t], R)\n",
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" \n",
" _constraints = [x[:, t + 1] == A @ x[:, t] + B @ u[:, t] + C,\n",
" u[0, t] >= -MAX_ACC, u[0, t] <= MAX_ACC,\n",
" u[1, t] >= -MAX_STEER, u[1, t] <= MAX_STEER]\n",
" #opt.norm(target[:, t + 1] - x[:, t + 1], 1) <= 0.1]\n",
" \n",
" # Actuation rate of change\n",
" if t < (time_horizon - 1):\n",
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" _cost += opt.quad_form(u[:,t + 1] - u[:,t], R * 1)\n",
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" _constraints += [opt.abs(u[0, t + 1] - u[0, t])/DT <= MAX_D_ACC]\n",
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" _constraints += [opt.abs(u[1, t + 1] - u[1, t])/DT <= MAX_D_STEER]\n",
" \n",
" \n",
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" if t == 0:\n",
" #_constraints += [opt.norm(target[:, time_horizon] - x[:, time_horizon], 1) <= 0.01,\n",
" # x[:, 0] == initial_state]\n",
" _constraints += [x[:, 0] == initial_state]\n",
" \n",
" cost_function.append(opt.Problem(opt.Minimize(_cost), constraints=_constraints))\n",
"\n",
" # Add final cost\n",
" problem = sum(cost_function)\n",
" \n",
" # Minimize Problem\n",
" problem.solve(verbose=verbose, solver=opt.OSQP)\n",
" return x, u"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"jupyter": {
"source_hidden": true
}
},
"outputs": [],
"source": [
"\"\"\"\n",
"the ODE is used to update the simulation given the mpc results\n",
"I use this insted of using the LTI twice\n",
"\"\"\"\n",
"def kinematics_model(x,t,u):\n",
" \"\"\"\n",
" Returns the set of ODE of the vehicle model.\n",
" \"\"\"\n",
" \n",
" L=0.3 #vehicle wheelbase\n",
" dxdt = x[2]*np.cos(x[3])\n",
" dydt = x[2]*np.sin(x[3])\n",
" dvdt = u[0]\n",
" dthetadt = x[2]*np.tan(u[1])/L\n",
"\n",
" dqdt = [dxdt,\n",
" dydt,\n",
" dvdt,\n",
" dthetadt]\n",
"\n",
" return dqdt\n",
"\n",
"def predict(x0,u):\n",
" \"\"\"\n",
" \"\"\"\n",
" \n",
" x_ = np.zeros((N,T+1))\n",
" \n",
" x_[:,0] = x0\n",
" \n",
" # solve ODE\n",
" for t in range(1,T+1):\n",
"\n",
" tspan = [0,DT]\n",
" x_next = odeint(kinematics_model,\n",
" x0,\n",
" tspan,\n",
" args=(u[:,t-1],))\n",
"\n",
" x0 = x_next[1]\n",
" x_[:,t]=x_next[1]\n",
" \n",
" return x_\n",
"\n",
"\n",
"\"\"\"\n",
"MODIFIED TO INCLUDE FRAME TRANSFORMATION\n",
"\"\"\"\n",
"def compute_path_from_wp(start_xp, start_yp, step = 0.1):\n",
" \"\"\"\n",
" Computes a reference path given a set of waypoints\n",
" \"\"\"\n",
" \n",
" final_xp=[]\n",
" final_yp=[]\n",
" delta = step #[m]\n",
"\n",
" for idx in range(len(start_xp)-1):\n",
" 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",
"\n",
" interp_range = np.linspace(0,1,np.floor(section_len/delta).astype(int))\n",
" \n",
" fx=interp1d(np.linspace(0,1,2),start_xp[idx:idx+2],kind=1)\n",
" fy=interp1d(np.linspace(0,1,2),start_yp[idx:idx+2],kind=1)\n",
" \n",
" # watch out to duplicate points!\n",
" final_xp=np.append(final_xp,fx(interp_range)[1:])\n",
" final_yp=np.append(final_yp,fy(interp_range)[1:])\n",
" \n",
" dx = np.append(0, np.diff(final_xp))\n",
" dy = np.append(0, np.diff(final_yp))\n",
" theta = np.arctan2(dy, dx)\n",
"\n",
" return np.vstack((final_xp,final_yp,theta))\n",
"\n",
"\n",
"def get_nn_idx(state,path):\n",
" \"\"\"\n",
" Computes the index of the waypoint closest to vehicle\n",
" \"\"\"\n",
"\n",
" dx = state[0]-path[0,:]\n",
" dy = state[1]-path[1,:]\n",
" dist = np.hypot(dx,dy)\n",
" nn_idx = np.argmin(dist)\n",
"\n",
" try:\n",
" v = [path[0,nn_idx+1] - path[0,nn_idx],\n",
" path[1,nn_idx+1] - path[1,nn_idx]] \n",
" v /= np.linalg.norm(v)\n",
"\n",
" d = [path[0,nn_idx] - state[0],\n",
" path[1,nn_idx] - state[1]]\n",
"\n",
" if np.dot(d,v) > 0:\n",
" target_idx = nn_idx\n",
" else:\n",
" target_idx = nn_idx+1\n",
"\n",
" except IndexError as e:\n",
" target_idx = nn_idx\n",
"\n",
" return target_idx\n",
"\n",
"def normalize_angle(angle):\n",
" \"\"\"\n",
" Normalize an angle to [-pi, pi]\n",
" \"\"\"\n",
" while angle > np.pi:\n",
" angle -= 2.0 * np.pi\n",
"\n",
" while angle < -np.pi:\n",
" angle += 2.0 * np.pi\n",
"\n",
" return angle\n",
"\n",
"def get_ref_trajectory(state, path, target_v):\n",
" \"\"\"\n",
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" For each step in the time horizon\n",
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" modified reference in robot frame\n",
" \"\"\"\n",
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" xref = np.zeros((N, T+1))\n",
" dref = np.zeros((1, T+1))\n",
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" \n",
" #sp = np.ones((1,T +1))*target_v #speed profile\n",
" \n",
" ncourse = path.shape[1]\n",
"\n",
" ind = get_nn_idx(state, path)\n",
" dx=path[0,ind] - state[0]\n",
" dy=path[1,ind] - state[1]\n",
" \n",
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" \n",
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" xref[0, 0] = dx * np.cos(-state[3]) - dy * np.sin(-state[3]) #X\n",
" xref[1, 0] = dy * np.cos(-state[3]) + dx * np.sin(-state[3]) #Y\n",
" xref[2, 0] = target_v #V\n",
" xref[3, 0] = normalize_angle(path[2,ind]- state[3]) #Theta\n",
" dref[0, 0] = 0.0 # steer operational point should be 0\n",
" \n",
" dl = 0.05 # Waypoints spacing [m]\n",
" travel = 0.0\n",
" \n",
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" for i in range(1, T+1):\n",
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" travel += abs(target_v) * DT #current V or target V?\n",
" dind = int(round(travel / dl))\n",
" \n",
" if (ind + dind) < ncourse:\n",
" dx=path[0,ind + dind] - state[0]\n",
" dy=path[1,ind + dind] - state[1]\n",
" \n",
" xref[0, i] = dx * np.cos(-state[3]) - dy * np.sin(-state[3])\n",
" xref[1, i] = dy * np.cos(-state[3]) + dx * np.sin(-state[3])\n",
" xref[2, i] = target_v #sp[ind + dind]\n",
" xref[3, i] = normalize_angle(path[2,ind + dind] - state[3])\n",
" dref[0, i] = 0.0\n",
" else:\n",
" dx=path[0,ncourse - 1] - state[0]\n",
" dy=path[1,ncourse - 1] - state[1]\n",
" \n",
" xref[0, i] = dx * np.cos(-state[3]) - dy * np.sin(-state[3])\n",
" xref[1, i] = dy * np.cos(-state[3]) + dx * np.sin(-state[3])\n",
" xref[2, i] = 0.0 #stop? #sp[ncourse - 1]\n",
" xref[3, i] = normalize_angle(path[2,ncourse - 1] - state[3])\n",
" dref[0, i] = 0.0\n",
"\n",
" return xref, dref"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
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"CVXPY Optimization Time: Avrg: 0.0908s Max: 0.1129s Min: 0.0846s\n"
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]
}
],
"source": [
"track = compute_path_from_wp([0,3,4,6,10,12,14,6,1,0],\n",
" [0,0,2,4,3,3,-2,-6,-2,-2],0.05)\n",
"\n",
"# track = compute_path_from_wp([0,10,10,0],\n",
"# [0,0,1,1],0.05)\n",
"\n",
"sim_duration = 200 #time steps\n",
"opt_time=[]\n",
"x_sim = np.zeros((N,sim_duration))\n",
"u_sim = np.zeros((M,sim_duration-1))\n",
"\n",
"# Starting Condition\n",
"x0 = np.zeros(N)\n",
"x0[0] = 0 #x\n",
"x0[1] = -0.25 #y\n",
"x0[2] = 0.0 #v\n",
"x0[3] = np.radians(-0) #yaw\n",
"x_sim[:,0] = x0 #simulation_starting conditions\n",
"\n",
"#starting guess\n",
"action = np.zeros(M)\n",
"action[0] = MAX_ACC/2 #a\n",
"action[1] = 0.0 #delta\n",
"u_sim[:,0] = action\n",
"\n",
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"# Cost Matrices\n",
"Q = np.diag([20,20,10,20]) #state error cost\n",
"Qf = np.diag([30,30,30,30]) #state final error cost\n",
"R = np.diag([10,10]) #input cost\n",
"R_ = np.diag([10,10]) #input rate of change cost\n",
"\n",
"mpc = MPC(N,M,Q,R)\n",
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"REF_VEL = 1.0\n",
"\n",
"for sim_time in range(sim_duration-1):\n",
" \n",
" iter_start = time.time()\n",
" \n",
" # dynamics starting state w.r.t. robot are always null except vel \n",
" start_state = np.array([0, 0, x_sim[2,sim_time], 0])\n",
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" \n",
" # OPTIONAL: Add time delay to starting State- y\n",
" \n",
" current_action = np.array([u_sim[0,sim_time], u_sim[1,sim_time]])\n",
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" \n",
" # State Matrices\n",
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" A,B,C = get_linear_model_matrices(start_state, current_action)\n",
" \n",
" #Get Reference_traj -> inputs are in worldframe\n",
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" target, _ = get_ref_trajectory(x_sim[:,sim_time] ,track, REF_VEL)\n",
" \n",
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" x_mpc, u_mpc = mpc.optimize_linearized_model(A, B, C, start_state, target, time_horizon=T, verbose=False)\n",
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" \n",
" #retrieved optimized U and assign to u_bar to linearize in next step\n",
" u_bar = np.vstack((np.array(u_mpc.value[0,:]).flatten(),\n",
" (np.array(u_mpc.value[1,:]).flatten())))\n",
" \n",
" u_sim[:,sim_time] = u_bar[:,0]\n",
" \n",
" # Measure elpased time to get results from cvxpy\n",
" opt_time.append(time.time()-iter_start)\n",
" \n",
" # move simulation to t+1\n",
" tspan = [0,DT]\n",
" x_sim[:,sim_time+1] = odeint(kinematics_model,\n",
" x_sim[:,sim_time],\n",
" tspan,\n",
" args=(u_bar[:,0],))[1]\n",
" \n",
"print(\"CVXPY Optimization Time: Avrg: {:.4f}s Max: {:.4f}s Min: {:.4f}s\".format(np.mean(opt_time),\n",
" np.max(opt_time),\n",
" np.min(opt_time))) "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### RESUTS\n",
"\n",
"SCS -> Optimization Time: Avrg: 0.2139s Max: 0.3517s Min: 0.1913s\n",
"\n",
"OSQP -> Optimization Time: Avrg: 0.1035s Max: 0.1311s Min: 0.0959s\n",
"\n",
"ECOS -> Avrg: 0.2024s Max: 0.2313s Min: 0.1904s\n",
"\n",
"**Qualitative result** aka \"it drives?\" seems the same..."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
2021-04-19 23:18:08 +08:00
"image/png": "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
2021-04-15 21:46:12 +08:00
"text/plain": [
"<Figure size 1080x720 with 5 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"#plot trajectory\n",
"grid = plt.GridSpec(4, 5)\n",
"\n",
"plt.figure(figsize=(15,10))\n",
"\n",
"plt.subplot(grid[0:4, 0:4])\n",
"plt.plot(track[0,:],track[1,:],\"b+\")\n",
"plt.plot(x_sim[0,:],x_sim[1,:])\n",
"plt.axis(\"equal\")\n",
"plt.ylabel('y')\n",
"plt.xlabel('x')\n",
"\n",
"plt.subplot(grid[0, 4])\n",
"plt.plot(u_sim[0,:])\n",
"plt.ylabel('a(t) [m/ss]')\n",
"\n",
"plt.subplot(grid[1, 4])\n",
"plt.plot(x_sim[2,:])\n",
"plt.ylabel('v(t) [m/s]')\n",
"\n",
"plt.subplot(grid[2, 4])\n",
"plt.plot(np.degrees(u_sim[1,:]))\n",
"plt.ylabel('delta(t) [rad]')\n",
"\n",
"plt.subplot(grid[3, 4])\n",
"plt.plot(x_sim[3,:])\n",
"plt.ylabel('theta(t) [rad]')\n",
"\n",
"plt.tight_layout()\n",
"plt.show()"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python [conda env:.conda-jupyter] *",
"language": "python",
"name": "conda-env-.conda-jupyter-py"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.8.5"
}
},
"nbformat": 4,
"nbformat_minor": 4
}
