try implementing

PathTracking/lqr/lqr_tracking.py

@@ -0,0 +1,295 @@
+#! /usr/bin/python
+"""
+
+Path tracking simulation with rear wheel feedback steering control and PID speed control.
+
+author: Atsushi Sakai
+
+"""
+import numpy as np
+import math
+import matplotlib.pyplot as plt
+import unicycle_model
+from pycubicspline import pycubicspline
+from matplotrecorder import matplotrecorder
+import scipy.linalg as la
+
+Kp = 1.0  # speed propotional gain
+# steering control parameter
+KTH = 1.0
+KE = 0.5
+
+Q = np.eye(4)
+R = np.eye(1)
+
+animation = True
+# animation = False
+
+matplotrecorder.donothing = True
+
+
+def PIDControl(target, current):
+    a = Kp * (target - current)
+
+    return a
+
+
+def pi_2_pi(angle):
+    while (angle > math.pi):
+        angle = angle - 2.0 * math.pi
+
+    while (angle < -math.pi):
+        angle = angle + 2.0 * math.pi
+
+    return angle
+
+
+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 rear_wheel_feedback_control(state, cx, cy, cyaw, ck, preind):
+    ind, e = calc_nearest_index(state, cx, cy, cyaw)
+
+    k = ck[ind]
+    v = state.v
+    th_e = pi_2_pi(state.yaw - cyaw[ind])
+
+    omega = v * k * math.cos(th_e) / (1.0 - k * e) - \
+            KTH * abs(v) * th_e - KE * v * math.sin(th_e) * e / th_e
+
+    if th_e == 0.0 or omega == 0.0:
+        return 0.0, ind
+
+    delta = math.atan2(unicycle_model.L * omega / v, 1.0)
+    #  print(k, v, e, th_e, omega, delta)
+
+    return delta, ind
+
+
+def lqr_steering_control(state, cx, cy, cyaw, ck, target_ind):
+    ind, e = calc_nearest_index(state, cx, cy, cyaw)
+
+    k = ck[ind]
+    v = state.v
+    th_e = pi_2_pi(state.yaw - cyaw[ind])
+
+    A = np.matrix(np.zeros((4, 4)))
+    A[0, 0] = 1.0
+    A[0, 1] = unicycle_model.dt
+    A[1, 2] = v
+    A[2, 2] = 1.0
+    A[2, 3] = unicycle_model.dt
+    # print(A)
+
+    B = np.matrix(np.zeros((4, 1)))
+    B[3, 0] = v / unicycle_model.L
+
+    K, _, _ = dlqr(A, B, Q, R)
+
+    x = np.matrix(np.zeros((4, 1)))
+
+    x[0, 0] = e
+    x[1, 0] = 0.0
+    x[2, 0] = th_e
+    x[3, 0] = 0.0
+
+    ff = math.atan2(unicycle_model.L * k, 1)
+    fb = pi_2_pi((-K * x)[0, 0])
+    print(math.degrees(th_e))
+    # print(K, x)
+    print(math.degrees(ff), math.degrees(fb))
+
+    delta = ff + fb
+    # print(delta)
+    return delta
+
+
+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 = [abs(math.sqrt(idx ** 2 + idy ** 2)) for (idx, idy) in zip(dx, dy)]
+
+    mind = min(d)
+
+    ind = d.index(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 = unicycle_model.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)
+
+    while T >= time:
+        di, target_ind = rear_wheel_feedback_control(
+            state, cx, cy, cyaw, ck, target_ind)
+
+        dl = lqr_steering_control(state, cx, cy, cyaw, ck, target_ind)
+        # print(di, dl)
+
+        ai = PIDControl(speed_profile[target_ind], state.v)
+        # state = unicycle_model.update(state, ai, di)
+        state = unicycle_model.update(state, ai, dl)
+
+        if abs(state.v) <= stop_speed:
+            target_ind += 1
+
+        time = time + unicycle_model.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 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)
+            matplotrecorder.save_frame()  # save each frame
+
+    plt.close()
+    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 = 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_profile[-1] = 0.0
+
+    #  flg, ax = plt.subplots(1)
+    #  plt.plot(speed_profile, "-r")
+    #  plt.show()
+
+    return speed_profile
+
+
+def main():
+    print("rear wheel feedback tracking start!!")
+    ax = [0.0, 6.0, 12.5, 5.0, 7.5, 3.0, -1.0]
+    ay = [0.0, 0.0, 5.0, 6.5, 3.0, 5.0, -2.0]
+    goal = [ax[-1], ay[-1]]
+
+    cx, cy, cyaw, ck, s = pycubicspline.calc_spline_course(ax, ay, ds=0.1)
+    target_speed = 10.0 / 3.6
+
+    sp = calc_speed_profile(cx, cy, cyaw, target_speed)
+
+    t, x, y, yaw, v = closed_loop_prediction(cx, cy, cyaw, ck, sp, goal)
+
+    if animation:
+        matplotrecorder.save_movie("animation.gif", 0.1)  # gif is ok.
+
+    flg, _ = plt.subplots(1)
+    plt.plot(ax, ay, "xb", label="input")
+    plt.plot(cx, cy, "-r", label="spline")
+    plt.plot(x, y, "-g", label="tracking")
+    plt.grid(True)
+    plt.axis("equal")
+    plt.xlabel("x[m]")
+    plt.ylabel("y[m]")
+    plt.legend()
+
+    flg, ax = plt.subplots(1)
+    plt.plot(s, [math.degrees(iyaw) for iyaw in cyaw], "-r", label="yaw")
+    plt.grid(True)
+    plt.legend()
+    plt.xlabel("line length[m]")
+    plt.ylabel("yaw angle[deg]")
+
+    flg, ax = 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()

PathTracking/lqr/unicycle_model.py

@@ -0,0 +1,66 @@
+#! /usr/bin/python
+# -*- coding: utf-8 -*-
+"""
+
+
+author Atsushi Sakai
+"""
+
+import math
+
+dt = 0.1  # [s]
+L = 2.9  # [m]
+
+
+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):
+    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
+
+
+if __name__ == '__main__':
+    print("start unicycle simulation")
+    import matplotlib.pyplot as plt
+
+    T = 100
+    a = [1.0] * T
+    delta = [math.radians(1.0)] * T
+    #  print(delta)
+    #  print(a, delta)
+
+    state = State()
+
+    x = []
+    y = []
+    yaw = []
+    v = []
+
+    for (ai, di) in zip(a, delta):
+        state = update(state, ai, di)
+
+        x.append(state.x)
+        y.append(state.y)
+        yaw.append(state.yaw)
+        v.append(state.v)
+
+    flg, ax = plt.subplots(1)
+    plt.plot(x, y)
+    plt.axis("equal")
+    plt.grid(True)
+
+    flg, ax = plt.subplots(1)
+    plt.plot(v)
+    plt.grid(True)
+
+    plt.show()