LQR轨迹跟踪算法Python/Matlab算法实现_基于matlablqr轨迹跟踪csdn

Python:

"""
Path tracking simulation with LQR steering control and PID speed control.
author Atsushi Sakai (@Atsushi_twi)
"""
import sys
sys.path.append("H:\Project\TrajectoryPlanningModelDesign\Codes\frenet_optimal\frenet_optimal")
import cubic_spline
import numpy as np
import math
import matplotlib.pyplot as plt
import scipy.linalg as la


Kp = 1.0  # speed proportional gain

# LQR parameter
Q = np.eye(4)
R = np.eye(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
#  show_animation = False


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 PIDControl(target, current):
    a = Kp * (target - current)

    return a


def pi_2_pi(angle): # the unit of angle is in rad;
    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 lqr_steering_control(state, cx, cy, cyaw, ck, pe, pth_e):
    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] = dt
    A[1, 2] = v
    A[2, 2] = 1.0
    A[2, 3] = dt
    # print(A)

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

    K, _, _ = dlqr(A, B, Q, R)

    x = np.matrix(np.zeros((4, 1)))

    x[0, 0] = e
    x[1, 0] = (e - pe) / dt
    x[2, 0] = th_e
    x[3, 0] = (th_e - pth_e) / dt

    ff = math.atan2(L * k, 1)
    fb = pi_2_pi((-K * x)[0, 0])

    delta = 2*ff + 1 * fb

    return delta, ind, 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 = [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 = 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 = lqr_steering_control(
            state, cx, cy, cyaw, ck, e, e_th)

        ai = PIDControl(speed_profile[target_ind], state.v)
        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_profile[-1] = 0.0

    #  flg, ax = plt.subplots(1)
    #  plt.plot(speed_profile, "-r")
    #  plt.show()

    return speed_profile


def main():
    print("LQR steering control tracking start!!")
    ax = [0.0, 6.0, 12.5, 10.0, 7.5, 3.0, -1.0]
    ay = [0.0, -3.0, -5.0, 6.5, 3.0, 5.0, -2.0]
    goal = [ax[-1], ay[-1]]

    cx, cy, cyaw, ck, s = cubic_spline.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 show_animation:
        plt.close()
        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()
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然后还要把cubic spline放在同一个project下进行调用， 注意修改路径，我的路径是：

sys.path.append("H:\Project\TrajectoryPlanningModelDesign\Codes\python\tracking_pure_stan")
#自己把它修改成你的路径。然后系统就可以调用了。
#下面就是cubic spline的class,
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import math
import numpy as np
import bisect


class Spline:
    u"""
    Cubic Spline class
    """

    def __init__(self, x, y):
        self.b, self.c, self.d, self.w = [], [], [], []

        self.x = x
        self.y = y

        self.nx = len(x)  # dimension of x
        h = np.diff(x)

        # calc coefficient c
        self.a = [iy for iy in y]

        # calc coefficient c
        A = self.__calc_A(h)
        B = self.__calc_B(h)
        self.c = np.linalg.solve(A, B)
        #  print(self.c1)

        # calc spline coefficient b and d
        for i in range(self.nx - 1):
            self.d.append((self.c[i + 1] - self.c[i]) / (3.0 * h[i]))
            tb = (self.a[i + 1] - self.a[i]) / h[i] - h[i] * \
                (self.c[i + 1] + 2.0 * self.c[i]) / 3.0
            self.b.append(tb)

    def calc(self, t):
        u"""
        Calc position
        if t is outside of the input x, return None
        """

        if t < self.x[0]:
            return None
        elif t > self.x[-1]:
            return None

        i = self.__search_index(t)
        dx = t - self.x[i]
        result = self.a[i] + self.b[i] * dx + \
            self.c[i] * dx ** 2.0 + self.d[i] * dx ** 3.0

        return result

    def calcd(self, t):
        u"""
        Calc first derivative
        if t is outside of the input x, return None
        """

        if t < self.x[0]:
            return None
        elif t > self.x[-1]:
            return None

        i = self.__search_index(t)
        dx = t - self.x[i]
        result = self.b[i] + 2.0 * self.c[i] * dx + 3.0 * self.d[i] * dx ** 2.0
        return result

    def calcdd(self, t):
        u"""
        Calc second derivative
        """

        if t < self.x[0]:
            return None
        elif t > self.x[-1]:
            return None

        i = self.__search_index(t)
        dx = t - self.x[i]
        result = 2.0 * self.c[i] + 6.0 * self.d[i] * dx
        return result

    def __search_index(self, x):
        u"""
        search data segment index
        """
        return bisect.bisect(self.x, x) - 1

    def __calc_A(self, h):
        u"""
        calc matrix A for spline coefficient c
        """
        A = np.zeros((self.nx, self.nx))
        A[0, 0] = 1.0
        for i in range(self.nx - 1):
            if i != (self.nx - 2):
                A[i + 1, i + 1] = 2.0 * (h[i] + h[i + 1])
            A[i + 1, i] = h[i]
            A[i, i + 1] = h[i]

        A[0, 1] = 0.0
        A[self.nx - 1, self.nx - 2] = 0.0
        A[self.nx - 1, self.nx - 1] = 1.0
        #  print(A)
        return A

    def __calc_B(self, h):
        u"""
        calc matrix B for spline coefficient c
        """
        B = np.zeros(self.nx)
        for i in range(self.nx - 2):
            B[i + 1] = 3.0 * (self.a[i + 2] - self.a[i + 1]) / \
                h[i + 1] - 3.0 * (self.a[i + 1] - self.a[i]) / h[i]
        #  print(B)
        return B


class Spline2D:
    u"""
    2D Cubic Spline class
    """

    def __init__(self, x, y):
        self.s = self.__calc_s(x, y)
        self.sx = Spline(self.s, x)
        self.sy = Spline(self.s, y)

    def __calc_s(self, x, y):
        dx = np.diff(x)
        dy = np.diff(y)
        self.ds = [math.sqrt(idx ** 2 + idy ** 2)
                   for (idx, idy) in zip(dx, dy)]
        s = [0]
        s.extend(np.cumsum(self.ds))
        return s

    def calc_position(self, s):
        u"""
        calc position
        """
        x = self.sx.calc(s)
        y = self.sy.calc(s)

        return x, y

    def calc_curvature(self, s):
        u"""
        calc curvature
        """
        dx = self.sx.calcd(s)
        ddx = self.sx.calcdd(s)
        dy = self.sy.calcd(s)
        ddy = self.sy.calcdd(s)
        k = (ddy * dx - ddx * dy) / (dx ** 2 + dy ** 2)
        return k

    def calc_yaw(self, s):
        u"""
        calc yaw
        """
        dx = self.sx.calcd(s)
        dy = self.sy.calcd(s)
        yaw = math.atan2(dy, dx)
        return yaw


def calc_spline_course(x, y, ds=0.1):
    sp = Spline2D(x, y)
    s = list(np.arange(0, sp.s[-1], ds))

    rx, ry, ryaw, rk = [], [], [], []
    for i_s in s:
        ix, iy = sp.calc_position(i_s)
        rx.append(ix)
        ry.append(iy)
        ryaw.append(sp.calc_yaw(i_s))
        rk.append(sp.calc_curvature(i_s))

    return rx, ry, ryaw, rk, s
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Matlab

clc
clear all
Kp = 1.0 ; 
dt = 0.1  ;% [s]
L = 2.9 ;% [m] wheel base of vehicle
Q = eye(4);
R = eye(1);
max_steer =60 * pi/180; % in rad
target_speed = 15.0 / 3.6;

cx = 0:0.1:200; % sampling interception from 0 to 100, with step 0.1
for i = 1:500% here we create a original reference line, which the vehicle should always follow when there is no obstacles;
    cy(i) = -sin(cx(i)/20)*cx(i)/2;
end
for i = 501: length(cx)
    cy(i) = -sin(cx(i)/20)*cx(i)/2; %cy(500);
end

p = [cx', cy'];
 for i = 1:length(cx)-1
 pd(i) = (p(i+1,2)-p(i,2))/(p(i+1,1)-p(i,1));
 end
 pd(end+1) = pd(end);
  %计算一阶导数
 for i =2: length(cx)-1
     pdd(i) = (p(i+1,2)-2*p(i,2) + p(i-1,2))/(0.5*(-p(i-1,1)+p(i+1,1)))^2;
 end
      pdd(1) = pdd(2);
     pdd(length(cx)) = pdd(length(cx)-1);
     
  for i  = 1:length(cx)
     k(i) = (pdd(i))/(1+pd(i)^2)^(1.5);
  end
  
  cx= cx'
  cy =cy'
  cyaw = atan(pd');
 
  ck = k'
%   plot(1:1001, cyaw)
%   plot(1:1001, ck)
%   plot(1:1001, pd)
pe = 0; pth_e = 0;
i = 1;
target_v = 10/3.6;
T = 80;
lastIndex = length(cx);
x = 0; y = -1; yaw = 0; v = 0;
time = 0;
ind =0;
figure

while ind < length(cx)
   
    [delta,ind,e,th_e] =  lqr_steering_control(x,y,v,yaw,cx,cy,cyaw,ck, pe, pth_e,L,Q,R,dt);
    pe =e;
    pth_e = th_e;
    if abs(e)> 4
        fprintf('mayday mayday!\n')
        break;
    end
    delta
    a = PIDcontrol(target_v, v, Kp);
     [x,y,yaw,v] = update(x,y,yaw,v, a, delta, dt,L, max_steer);
     posx(i) = x;
     posy(i)  =y;
     i = i+1;
     plot(cx,cy,'r-')
     hold on
     plot(posx(i-1),posy(i-1),'bo')
      drawnow
     
      hold on
end

% function"Update" updates vehicle states
function [x, y, yaw, v] = update(x, y, yaw, v, a, delta,dt,L,max_steer)
 delta = max(min(max_steer, delta), -max_steer);
    x = x + v * cos(yaw) * dt;
    y = y + v * sin(yaw) * dt;
    yaw = yaw + v / L * tan(delta) * dt;
   v = v + a * dt;
end

function [a] = PIDcontrol(target_v, current_v, Kp)
a = Kp * (target_v - current_v);
end

function [angle] = pipi(angle) % the unit of angle is in rad;

if (angle > pi)
    angle =  angle - 2*pi;
elseif (angle < -pi)
    angle = angle + 2*pi;
else
    angle = angle;
end
end

function [Xn] = solve_DARE(A,B,Q,R)
X = Q;
maxiter = 150;
epsilon = 0.01;
for i = 1:maxiter
    Xn = A' * X * A - A' * X * B * ((R + B' * X * B) \ B') * X * A +Q;
    if abs(Xn - X) <= epsilon
        X = Xn;
        break;
    end
    X = Xn;
end
end

function [K] = dlqr (A,B,Q,R)
X = solve_DARE(A,B,Q,R);
K = (B' * X * B + R) \ (B' * X * A);
end

function [delta,ind,e,th_e] =  lqr_steering_control(x,y,v,yaw,cx,cy,cyaw,ck, pe, pth_e,L,Q,R,dt)
[ind, e] = calc_target_index(x,y,cx,cy,cyaw);
k =ck(ind);
th_e = pipi(yaw -cyaw(ind));
A = zeros(4,4);

A(1,1) = 1; A(1,2) = dt; A(2,3) = v; A(3,3) = 1; A(3,4) = dt;
B= zeros(4,1);
B(4,1) = v/L;
K = dlqr(A,B,Q,R);
x = zeros(4,1);
x(1,1)=e; x(2,1)= (e-pe)/dt; x(3,1) = th_e; x(4,1) = (th_e - pth_e)/dt;
ff = atan(L * (k));
fb = pipi(-K * x);
delta =   1*ff + 1*fb;
ff
fb
end

function [ind, error] = calc_target_index(x,y, cx,cy,cyaw)
N =  length(cx);
Distance = zeros(N,1);
for i = 1:N
Distance(i) =  sqrt((cx(i)-x)^2 + (cy(i)-y)^2);
end
[value, location]= min(Distance);
ind = location

dx1 = cx(ind)- x;
dy1 = cy(ind) -y;
angle = pipi(cyaw(ind)-atan(dy1/dx1));
heading = cyaw(ind)*180/pi
    if y<cy(ind) 
        error = -value;
    else
        error = value;
    end

end

% if cx(ind)> x
%     error = -value;
% else
% error = value;
% end



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15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
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39
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41
42
43
44
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51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
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78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166

效果如图：


效果非常一般，我更加推荐Stanley 和Pure pursuit