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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Iterative Linearization"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The goal is to have a more accurate linearization of the diff equations. For every time step the optimization is iterativelly repeated using he previous optimization results **u_bar** to approximate the vehicle dynamics, instead of a random starting guess and/or the rsult at time t-1.\n",
"\n",
"In previous case the results at t-1 wer used to approimate the dynamics art time t!\n",
"\n",
"This maks the results less correlated but makes the controller slower!"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"from scipy.integrate import odeint\n",
"from scipy.interpolate import interp1d\n",
"import cvxpy as cp\n",
"\n",
"import matplotlib.pyplot as plt\n",
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"\n",
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"plt.style.use(\"ggplot\")\n",
"\n",
"import time"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"\"\"\"\n",
"Control problem statement.\n",
"\"\"\"\n",
"\n",
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"N = 4 # number of state variables\n",
"M = 2 # number of control variables\n",
"T = 20 # Prediction Horizon\n",
"DT = 0.2 # discretization step\n",
"\n",
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"\n",
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"def get_linear_model(x_bar, u_bar):\n",
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" \"\"\"\n",
" Computes the LTI approximated state space model x' = Ax + Bu + C\n",
" \"\"\"\n",
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"\n",
" L = 0.3 # vehicle wheelbase\n",
"\n",
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" x = x_bar[0]\n",
" y = x_bar[1]\n",
" v = x_bar[2]\n",
" theta = x_bar[3]\n",
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"\n",
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" a = u_bar[0]\n",
" delta = u_bar[1]\n",
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"\n",
" A = np.zeros((N, N))\n",
" A[0, 2] = np.cos(theta)\n",
" A[0, 3] = -v * np.sin(theta)\n",
" A[1, 2] = np.sin(theta)\n",
" A[1, 3] = v * np.cos(theta)\n",
" A[3, 2] = v * np.tan(delta) / 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 * np.cos(delta) ** 2)\n",
" B_lin = DT * B\n",
"\n",
" f_xu = np.array(\n",
" [v * np.cos(theta), v * np.sin(theta), a, v * np.tan(delta) / L]\n",
" ).reshape(N, 1)\n",
" C_lin = DT * (\n",
" f_xu - np.dot(A, x_bar.reshape(N, 1)) - np.dot(B, u_bar.reshape(M, 1))\n",
" )\n",
"\n",
" return np.round(A_lin, 4), np.round(B_lin, 4), np.round(C_lin, 4)\n",
"\n",
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"\n",
"\"\"\"\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",
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"\n",
"\n",
"def kinematics_model(x, t, u):\n",
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" \"\"\"\n",
" Returns the set of ODE of the vehicle model.\n",
" \"\"\"\n",
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"\n",
" L = 0.3 # vehicle wheelbase\n",
" dxdt = x[2] * np.cos(x[3])\n",
" dydt = x[2] * np.sin(x[3])\n",
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" dvdt = u[0]\n",
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" dthetadt = x[2] * np.tan(u[1]) / L\n",
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"\n",
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" dqdt = [dxdt, dydt, dvdt, dthetadt]\n",
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"\n",
" return dqdt\n",
"\n",
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"\n",
"def predict(x0, u):\n",
" \"\"\" \"\"\"\n",
"\n",
" x_ = np.zeros((N, T + 1))\n",
"\n",
" x_[:, 0] = x0\n",
"\n",
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" # solve ODE\n",
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" for t in range(1, T + 1):\n",
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"\n",
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" tspan = [0, DT]\n",
" x_next = odeint(kinematics_model, x0, tspan, args=(u[:, t - 1],))\n",
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"\n",
" x0 = x_next[1]\n",
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" x_[:, t] = x_next[1]\n",
"\n",
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" return x_\n",
"\n",
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"\n",
"def compute_path_from_wp(start_xp, start_yp, step=0.1):\n",
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" \"\"\"\n",
" Computes a reference path given a set of waypoints\n",
" \"\"\"\n",
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"\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(\n",
" np.sqrt(\n",
" np.power(np.diff(start_xp[idx : idx + 2]), 2)\n",
" + np.power(np.diff(start_yp[idx : idx + 2]), 2)\n",
" )\n",
" )\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",
" final_xp = np.append(final_xp, fx(interp_range))\n",
" final_yp = np.append(final_yp, fy(interp_range))\n",
"\n",
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" 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",
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" return np.vstack((final_xp, final_yp, theta))\n",
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"\n",
"\n",
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"def get_nn_idx(state, path):\n",
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" \"\"\"\n",
" Computes the index of the waypoint closest to vehicle\n",
" \"\"\"\n",
"\n",
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" dx = state[0] - path[0, :]\n",
" dy = state[1] - path[1, :]\n",
" dist = np.hypot(dx, dy)\n",
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" nn_idx = np.argmin(dist)\n",
"\n",
" try:\n",
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" v = [\n",
" path[0, nn_idx + 1] - path[0, nn_idx],\n",
" path[1, nn_idx + 1] - path[1, nn_idx],\n",
" ]\n",
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" v /= np.linalg.norm(v)\n",
"\n",
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" d = [path[0, nn_idx] - state[0], 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",
" else:\n",
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" target_idx = nn_idx + 1\n",
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"\n",
" except IndexError as e:\n",
" target_idx = nn_idx\n",
"\n",
" return target_idx\n",
"\n",
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"\n",
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"def get_ref_trajectory(state, path, target_v):\n",
" \"\"\"\n",
" Adapted from pythonrobotics\n",
" \"\"\"\n",
" 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",
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" ncourse = path.shape[1]\n",
"\n",
" ind = get_nn_idx(state, path)\n",
"\n",
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" xref[0, 0] = path[0, ind] # X\n",
" xref[1, 0] = path[1, ind] # Y\n",
" xref[2, 0] = target_v # sp[ind] #V\n",
" xref[3, 0] = path[2, ind] # Theta\n",
" dref[0, 0] = 0.0 # steer operational point should be 0\n",
"\n",
" dl = 0.05 # Waypoints spacing [m]\n",
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" travel = 0.0\n",
"\n",
" for i in range(T + 1):\n",
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" travel += abs(target_v) * DT # current V or target V?\n",
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" dind = int(round(travel / dl))\n",
"\n",
" if (ind + dind) < ncourse:\n",
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" xref[0, i] = path[0, ind + dind]\n",
" xref[1, i] = path[1, ind + dind]\n",
" xref[2, i] = target_v # sp[ind + dind]\n",
" xref[3, i] = path[2, ind + dind]\n",
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" dref[0, i] = 0.0\n",
" else:\n",
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" xref[0, i] = path[0, ncourse - 1]\n",
" xref[1, i] = path[1, ncourse - 1]\n",
" xref[2, i] = 0.0 # stop? #sp[ncourse - 1]\n",
" xref[3, i] = path[2, ncourse - 1]\n",
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" dref[0, i] = 0.0\n",
"\n",
" return xref, dref"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"<ipython-input-2-e22409964dd8>:127: RuntimeWarning: invalid value encountered in true_divide\n",
" v /= np.linalg.norm(v)\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"CVXPY Optimization Time: Avrg: 0.5979s Max: 0.8275s Min: 0.2939s\n"
]
}
],
"source": [
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"track = compute_path_from_wp(\n",
" [0, 3, 4, 6, 10, 12, 14, 6, 1, 0], [0, 0, 2, 4, 3, 3, -2, -6, -2, -2], 0.05\n",
")\n",
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"\n",
"# track = compute_path_from_wp([0,10,10,0],\n",
"# [0,0,1,1],0.05)\n",
"\n",
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"sim_duration = 200 # time steps\n",
"opt_time = []\n",
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"\n",
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"x_sim = np.zeros((N, sim_duration))\n",
"u_sim = np.zeros((M, sim_duration - 1))\n",
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"\n",
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"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",
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"\n",
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"REF_VEL = 1.0 # m/s\n",
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"\n",
"# Starting Condition\n",
"x0 = np.zeros(N)\n",
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"x0[0] = 0 # x\n",
"x0[1] = -0.25 # y\n",
"x0[2] = 0.0 # v\n",
"x0[3] = np.radians(-0) # yaw\n",
"\n",
"for sim_time in range(sim_duration - 1):\n",
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"\n",
" iter_start = time.time()\n",
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"\n",
" # starting guess for ctrl\n",
" u_bar = np.zeros((M, T))\n",
" u_bar[0, :] = MAX_ACC / 2 # a\n",
" u_bar[1, :] = 0.0 # delta\n",
"\n",
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" for _ in range(5):\n",
" u_prev = u_bar\n",
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"\n",
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" # dynamics starting state\n",
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" x_bar = np.zeros((N, T + 1))\n",
" x_bar[:, 0] = x_sim[:, sim_time]\n",
"\n",
" # prediction for linearization of costrains\n",
" for t in range(1, T + 1):\n",
" xt = x_bar[:, t - 1].reshape(N, 1)\n",
" ut = u_bar[:, t - 1].reshape(M, 1)\n",
" A, B, C = get_linear_model(xt, ut)\n",
" xt_plus_one = np.squeeze(np.dot(A, xt) + np.dot(B, ut) + C)\n",
" x_bar[:, t] = xt_plus_one\n",
"\n",
" # CVXPY Linear MPC problem statement\n",
" x = cp.Variable((N, T + 1))\n",
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" u = cp.Variable((M, T))\n",
" cost = 0\n",
" constr = []\n",
"\n",
" # Cost Matrices\n",
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" Q = np.diag([20, 20, 10, 0]) # state error cost\n",
" Qf = np.diag([30, 30, 30, 0]) # state final error cost\n",
" R = np.diag([10, 10]) # input cost\n",
" R_ = np.diag([10, 10]) # input rate of change cost\n",
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"\n",
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" # Get Reference_traj\n",
" x_ref, d_ref = get_ref_trajectory(x_bar[:, 0], track, REF_VEL)\n",
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"\n",
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" # Prediction Horizon\n",
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" for t in range(T):\n",
"\n",
" # Tracking Error\n",
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" cost += cp.quad_form(x[:, t] - x_ref[:, t], Q)\n",
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"\n",
" # Actuation effort\n",
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" cost += cp.quad_form(u[:, t], R)\n",
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"\n",
" # Actuation rate of change\n",
" if t < (T - 1):\n",
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" cost += cp.quad_form(u[:, t + 1] - u[:, t], R_)\n",
" constr += [\n",
" cp.abs(u[0, t + 1] - u[0, t]) / DT <= MAX_D_ACC\n",
" ] # max acc rate of change\n",
" constr += [\n",
" cp.abs(u[1, t + 1] - u[1, t]) / DT <= MAX_D_STEER\n",
" ] # max steer rate of change\n",
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"\n",
" # Kinrmatics Constrains (Linearized model)\n",
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" A, B, C = get_linear_model(x_bar[:, t], u_bar[:, t])\n",
" constr += [x[:, t + 1] == A @ x[:, t] + B @ u[:, t] + C.flatten()]\n",
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"\n",
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" # Final Point tracking\n",
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" cost += cp.quad_form(x[:, T] - x_ref[:, T], Qf)\n",
"\n",
" # sums problem objectives and concatenates constraints.\n",
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" constr += [x[:, 0] == x_bar[:, 0]] # starting condition\n",
" constr += [x[2, :] <= MAX_SPEED] # max speed\n",
" constr += [x[2, :] >= 0.0] # min_speed (not really needed)\n",
" constr += [cp.abs(u[0, :]) <= MAX_ACC] # max acc\n",
" constr += [cp.abs(u[1, :]) <= MAX_STEER] # max steer\n",
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"\n",
" # Solve\n",
" prob = cp.Problem(cp.Minimize(cost), constr)\n",
" solution = prob.solve(solver=cp.OSQP, verbose=False)\n",
"\n",
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" # retrieved optimized U and assign to u_bar to linearize in next step\n",
" u_bar = np.vstack(\n",
" (np.array(u.value[0, :]).flatten(), (np.array(u.value[1, :]).flatten()))\n",
" )\n",
"\n",
" # check how this solution differs from previous\n",
" # if the solutions are very\n",
" delta_u = np.sum(np.sum(np.abs(u_bar - u_prev), axis=0), axis=0)\n",
" if delta_u < 0.05:\n",
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" break\n",
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"\n",
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" # select u from best iteration\n",
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" u_sim[:, sim_time] = u_bar[:, 0]\n",
"\n",
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" # Measure elpased time to get results from cvxpy\n",
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" opt_time.append(time.time() - iter_start)\n",
"\n",
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" # move simulation to t+1\n",
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" tspan = [0, DT]\n",
" x_sim[:, sim_time + 1] = odeint(\n",
" kinematics_model, x_sim[:, sim_time], tspan, args=(u_bar[:, 0],)\n",
" )[1]\n",
"\n",
" # reset u_bar? -> this simulates that we don use previous solution!\n",
" u_bar[0, :] = MAX_ACC / 2 # a\n",
" u_bar[1, :] = 0.0 # delta\n",
"\n",
"\n",
"print(\n",
" \"CVXPY Optimization Time: Avrg: {:.4f}s Max: {:.4f}s Min: {:.4f}s\".format(\n",
" np.mean(opt_time), np.max(opt_time), np.min(opt_time)\n",
" )\n",
")"
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]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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
"text/plain": [
"<Figure size 1080x720 with 5 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
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"# plot trajectory\n",
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"grid = plt.GridSpec(4, 5)\n",
"\n",
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"plt.figure(figsize=(15, 10))\n",
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"\n",
"plt.subplot(grid[0:4, 0:4])\n",
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"plt.plot(track[0, :], track[1, :], \"b+\")\n",
"plt.plot(x_sim[0, :], x_sim[1, :])\n",
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"plt.axis(\"equal\")\n",
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"plt.ylabel(\"y\")\n",
"plt.xlabel(\"x\")\n",
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"\n",
"plt.subplot(grid[0, 4])\n",
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"plt.plot(u_sim[0, :])\n",
"plt.ylabel(\"a(t) [m/ss]\")\n",
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"\n",
"plt.subplot(grid[1, 4])\n",
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"plt.plot(x_sim[2, :])\n",
"plt.ylabel(\"v(t) [m/s]\")\n",
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"\n",
"plt.subplot(grid[2, 4])\n",
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"plt.plot(np.degrees(u_sim[1, :]))\n",
"plt.ylabel(\"delta(t) [rad]\")\n",
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"\n",
"plt.subplot(grid[3, 4])\n",
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"plt.plot(x_sim[3, :])\n",
"plt.ylabel(\"theta(t) [rad]\")\n",
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"\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
}
