lqr_speed_steer_control.py

# -*- coding: utf-8 -*-
"""

Path tracking simulation with LQR speed and steering control

author Atsushi Sakai (@Atsushi_twi)

"""
import numpy as np
import math
import matplotlib.pyplot as plt
import scipy.linalg as la

# LQR parameter
Q = np.diag([3,3,4,4,1])
R = np.diag([3,1])

# parameters
dt = 0.1  # time tick[s]
L = 0.5  # Wheel base of the vehicle [m]
max_steer = math.radians(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  #将角度 从（0,2pi） 转换为  (-pi,pi)


def solve_DARE(A, B, Q, R):
    """
    solve a discrete time_Algebraic Riccati equation (DARE)
    """
    X = Q
    maxiter = 150
    eps = 0.01

    for i in range(maxiter):
        Xn = A.T * X * A - A.T * X * B * \
            la.inv(R + B.T * X * B) * B.T * X * A + Q
        if (abs(Xn - X)).max() < eps:
            X = Xn
            break
        X = Xn

    return Xn


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 = np.matrix(la.inv(B.T * X * B + R) * (B.T * X * A))

    eigVals, eigVecs = la.eig(A - B * K)

    return K, X, eigVals


def lqr_steering_control(cx, cy, cyaw, e, th_e, pe, pth_e, sp):

    #ind, e = calc_nearest_index(state, cx, cy, cyaw)# ind是得到的最近点的下标，e是最近点与当前点的距离误差

    tv = sp  #不修改速度，这里设置当前速度和目标速度一致
    v = sp
    dt = 0.1 # 单位时间 0.1s
    L = 1  # 车长 [m]
    th_e = pi_2_pi(th_e)   #th_e 是当前车的航向角误差，然后转换到了（-pi,pi）范围内

    # LQR parameter
    Q = np.eye(5)
    R = np.eye(2)

    A = np.matrix(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 = np.matrix(np.zeros((5, 2)))
    B[3, 0] = v / L
    B[4, 1] = dt

    K, _, _ = dlqr(A, B, Q, R)      #求解K矩阵

    x = np.matrix(np.zeros((5, 1)))   #状态变量参数定义
    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

    ustar = -K * x             #利用求好的K矩阵，通过状态变量得到控制变量

    # calc steering input
    #ff = math.atan2(L * k, 1)
    fb = pi_2_pi(ustar[0, 0])
    #delta = ff + fb
    delta = fb
    
    # calc accel input
    a = ustar[1, 0]

    return delta, a, e, th_e


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 closed_loop_prediction(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]
    target_ind = calc_nearest_index(state, cx, cy, cyaw)
    e, e_th = 0.0, 0.0

    while T >= time:
        dl, target_ind, e, e_th, ai = lqr_steering_control(state, cx, cy, cyaw, ck, e, e_th, speed_profile)

        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.sqrt(dx ** 2 + dy ** 2) <= 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()
            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(cx, cy, cyaw, target_speed):
    speed_profile = [target_speed] * len(cx)

    direction = 1.0

    # Set stop point
    for i in range(len(cx) - 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(cx, cy, cyaw, target_speed)

    t, x, y, yaw, v = closed_loop_prediction(cx, cy, cyaw, ck, sp, goal)

if __name__ == '__main__':
    main()