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