""" Path tracking simulation with LQR speed and steering control author Atsushi Sakai (@Atsushi_twi) """ import math import sys import matplotlib.pyplot as plt import numpy as np import scipy.linalg as la import pathlib sys.path.append(str(pathlib.Path(__file__).parent.parent.parent)) from PathPlanning.CubicSpline import cubic_spline_planner # === Parameters ===== # LQR parameter lqr_Q = np.eye(5) lqr_R = np.eye(2) dt = 0.1 # time tick[s] L = 0.5 # Wheel base of the vehicle [m] max_steer = np.deg2rad(45.0) # maximum steering angle[rad] show_animation = True class State: def __init__(self, x=0.0, y=0.0, yaw=0.0, v=0.0): self.x = x self.y = y self.yaw = yaw self.v = v def update(state, a, delta): if delta >= max_steer: delta = max_steer if delta <= - max_steer: delta = - max_steer state.x = state.x + state.v * math.cos(state.yaw) * dt state.y = state.y + state.v * math.sin(state.yaw) * dt state.yaw = state.yaw + state.v / L * math.tan(delta) * dt state.v = state.v + a * dt return state def pi_2_pi(angle): return (angle + math.pi) % (2 * math.pi) - math.pi def solve_dare(A, B, Q, R): """ solve a discrete time_Algebraic Riccati equation (DARE) """ x = Q x_next = Q max_iter = 150 eps = 0.01 for i in range(max_iter): x_next = A.T @ x @ A - A.T @ x @ B @ \ la.inv(R + B.T @ x @ B) @ B.T @ x @ A + Q if (abs(x_next - x)).max() < eps: break x = x_next return x_next def dlqr(A, B, Q, R): """Solve the discrete time lqr controller. x[k+1] = A x[k] + B u[k] cost = sum x[k].T*Q*x[k] + u[k].T*R*u[k] # ref Bertsekas, p.151 """ # first, try to solve the ricatti equation X = solve_dare(A, B, Q, R) # compute the LQR gain K = la.inv(B.T @ X @ B + R) @ (B.T @ X @ A) eig_result = la.eig(A - B @ K) return K, X, eig_result[0] def lqr_speed_steering_control(state, cx, cy, cyaw, ck, pe, pth_e, sp, Q, R): ind, e = calc_nearest_index(state, cx, cy, cyaw) tv = sp[ind] k = ck[ind] v = state.v th_e = pi_2_pi(state.yaw - cyaw[ind]) # A = [1.0, dt, 0.0, 0.0, 0.0 # 0.0, 0.0, v, 0.0, 0.0] # 0.0, 0.0, 1.0, dt, 0.0] # 0.0, 0.0, 0.0, 0.0, 0.0] # 0.0, 0.0, 0.0, 0.0, 1.0] A = np.zeros((5, 5)) A[0, 0] = 1.0 A[0, 1] = dt A[1, 2] = v A[2, 2] = 1.0 A[2, 3] = dt A[4, 4] = 1.0 # B = [0.0, 0.0 # 0.0, 0.0 # 0.0, 0.0 # v/L, 0.0 # 0.0, dt] B = np.zeros((5, 2)) B[3, 0] = v / L B[4, 1] = dt K, _, _ = dlqr(A, B, Q, R) # state vector # x = [e, dot_e, th_e, dot_th_e, delta_v] # e: lateral distance to the path # dot_e: derivative of e # th_e: angle difference to the path # dot_th_e: derivative of th_e # delta_v: difference between current speed and target speed x = np.zeros((5, 1)) x[0, 0] = e x[1, 0] = (e - pe) / dt x[2, 0] = th_e x[3, 0] = (th_e - pth_e) / dt x[4, 0] = v - tv # input vector # u = [delta, accel] # delta: steering angle # accel: acceleration ustar = -K @ x # calc steering input ff = math.atan2(L * k, 1) # feedforward steering angle fb = pi_2_pi(ustar[0, 0]) # feedback steering angle delta = ff + fb # calc accel input accel = ustar[1, 0] return delta, ind, e, th_e, accel def calc_nearest_index(state, cx, cy, cyaw): dx = [state.x - icx for icx in cx] dy = [state.y - icy for icy in cy] d = [idx ** 2 + idy ** 2 for (idx, idy) in zip(dx, dy)] mind = min(d) ind = d.index(mind) mind = math.sqrt(mind) dxl = cx[ind] - state.x dyl = cy[ind] - state.y angle = pi_2_pi(cyaw[ind] - math.atan2(dyl, dxl)) if angle < 0: mind *= -1 return ind, mind def do_simulation(cx, cy, cyaw, ck, speed_profile, goal): T = 500.0 # max simulation time goal_dis = 0.3 stop_speed = 0.05 state = State(x=-0.0, y=-0.0, yaw=0.0, v=0.0) time = 0.0 x = [state.x] y = [state.y] yaw = [state.yaw] v = [state.v] t = [0.0] e, e_th = 0.0, 0.0 while T >= time: dl, target_ind, e, e_th, ai = lqr_speed_steering_control( state, cx, cy, cyaw, ck, e, e_th, speed_profile, lqr_Q, lqr_R) state = update(state, ai, dl) if abs(state.v) <= stop_speed: target_ind += 1 time = time + dt # check goal dx = state.x - goal[0] dy = state.y - goal[1] if math.hypot(dx, dy) <= goal_dis: print("Goal") break x.append(state.x) y.append(state.y) yaw.append(state.yaw) v.append(state.v) t.append(time) if target_ind % 1 == 0 and show_animation: plt.cla() # for stopping simulation with the esc key. plt.gcf().canvas.mpl_connect('key_release_event', lambda event: [exit(0) if event.key == 'escape' else None]) plt.plot(cx, cy, "-r", label="course") plt.plot(x, y, "ob", label="trajectory") plt.plot(cx[target_ind], cy[target_ind], "xg", label="target") plt.axis("equal") plt.grid(True) plt.title("speed[km/h]:" + str(round(state.v * 3.6, 2)) + ",target index:" + str(target_ind)) plt.pause(0.0001) return t, x, y, yaw, v def calc_speed_profile(cyaw, target_speed): speed_profile = [target_speed] * len(cyaw) direction = 1.0 # Set stop point for i in range(len(cyaw) - 1): dyaw = abs(cyaw[i + 1] - cyaw[i]) switch = math.pi / 4.0 <= dyaw < math.pi / 2.0 if switch: direction *= -1 if direction != 1.0: speed_profile[i] = - target_speed else: speed_profile[i] = target_speed if switch: speed_profile[i] = 0.0 # speed down for i in range(40): speed_profile[-i] = target_speed / (50 - i) if speed_profile[-i] <= 1.0 / 3.6: speed_profile[-i] = 1.0 / 3.6 return speed_profile def main(): print("LQR steering control tracking start!!") ax = [0.0, 6.0, 12.5, 10.0, 17.5, 20.0, 25.0] ay = [0.0, -3.0, -5.0, 6.5, 3.0, 0.0, 0.0] goal = [ax[-1], ay[-1]] cx, cy, cyaw, ck, s = cubic_spline_planner.calc_spline_course( ax, ay, ds=0.1) target_speed = 10.0 / 3.6 # simulation parameter km/h -> m/s sp = calc_speed_profile(cyaw, target_speed) t, x, y, yaw, v = do_simulation(cx, cy, cyaw, ck, sp, goal) if show_animation: # pragma: no cover plt.close() plt.subplots(1) plt.plot(ax, ay, "xb", label="waypoints") plt.plot(cx, cy, "-r", label="target course") plt.plot(x, y, "-g", label="tracking") plt.grid(True) plt.axis("equal") plt.xlabel("x[m]") plt.ylabel("y[m]") plt.legend() plt.subplots(1) plt.plot(s, [np.rad2deg(iyaw) for iyaw in cyaw], "-r", label="yaw") plt.grid(True) plt.legend() plt.xlabel("line length[m]") plt.ylabel("yaw angle[deg]") plt.subplots(1) plt.plot(s, ck, "-r", label="curvature") plt.grid(True) plt.legend() plt.xlabel("line length[m]") plt.ylabel("curvature [1/m]") plt.show() if __name__ == '__main__': main()