模型预测控制路径跟踪python语言实现
模型预测控制路径跟踪python语言实现，参考《无人驾驶车辆模型预测控制》以及网上其他代码。
import matplotlib.pyplot as plt
import numpy as np
from math import *
from cvxopt import matrix, solvers
class MPC:
def __init__(self):
self.Np = 60 # 预测步长
self.Nc = 60 # 控制步长
self.dt = 0.1 # 时间间隔
self.Length = 1.0 # 车辆轴距
max_steer = 30 * pi / 180 # 最大方向盘打角
max_steer_v = 15 * pi / 180 # 最大方向盘打角速度
max_v = 8.7 # 最大车速
max_a = 1.0 # 最大加速度
# 目标函数相关矩阵
self.Q = 50 * np.identity(3*self.Np) # 位姿权重
self.R = 100 * np.identity(2*self.Nc) # 控制权重
self.kesi = np.zeros((5, 1))
self.A = np.identity(5)
self.B = np.block([
[np.zeros((3, 2))],
[np.identity(2)]
])
self.C = np.block([
[np.identity(3), np.zeros((3, 2))]
])
self.PHI = np.zeros((3*self.Np, 5))
self.THETA = np.zeros((3*self.Np, 2*self.Nc))
self.CA = (self.Np+1) * [self.C]
self.H = np.zeros((2*self.Nc, 2*self.Nc))
self.f = np.zeros((2*self.Nc, 1))
# 不等式约束相关矩阵
A_t = np.zeros((self.Nc, self.Nc))
for p in range(self.Nc):
for q in range(p+1):
A_t[p][q] = 1
A_I = np.kron(A_t, np.identity(2))
# 控制量约束
umin = np.array([[-max_v], [-max_steer]])
umax = np.array([[max_v], [max_steer]])
self.Umin = np.kron(np.ones((self.Nc, 1)), umin)
self.Umax = np.kron(np.ones((self.Nc, 1)), umax)
# 控制增量约束
delta_umin = np.array([[-max_a * self.dt], [-max_steer_v * self.dt]])
delta_umax = np.array([[max_a * self.dt], [max_steer_v * self.dt]])
delta_Umin = np.kron(np.ones((self.Nc, 1)), delta_umin)
delta_Umax = np.kron(np.ones((self.Nc, 1)), delta_umax)
self.A_cons = np.zeros((2 * 2*self.Nc, 2*self.Nc))
self.A_cons[0:2*self.Nc, 0:2*self.Nc] = A_I
self.A_cons[2*self.Nc:4*self.Nc, 0:2*self.Nc] = np.identity(2*self.Nc)
self.lb_cons = np.zeros((2 * 2*self.Nc, 1))
self.lb_cons[2*self.Nc:4*self.Nc, 0:1] = delta_Umin
self.ub_cons = np.zeros((2 * 2*self.Nc, 1))
self.ub_cons[2*self.Nc:4*self.Nc, 0:1] = delta_Umax
def mpcControl(self, x, y, yaw, v, angle, tar_x, tar_y, tar_yaw, tar_v, tar_angle): # mpc优化控制
T = self.dt
L = self.Length
# 更新误差
self.kesi[0][0] = x-tar_x
self.kesi[1][0] = y-tar_y
self.kesi[2][0] = self.normalizeTheta(yaw - tar_yaw)
self.kesi[3][0] = v - tar_v
self.kesi[4][0] = angle - tar_angle
# 更新A矩阵
self.A[0][2] = -tar_v * sin(tar_yaw) * T
self.A[0][3] = cos(tar_yaw) * T
self.A[1][2] = tar_v * cos(tar_yaw) * T
self.A[1][3] = sin(tar_yaw) * T
self.A[2][3] = tan(tar_angle) * T / L
self.A[2][4] = tar_v * T / (L * (cos(tar_angle)**2))
# 更新B矩阵
self.B[0][0] = cos(tar_yaw) * T
self.B[1][0] = sin(tar_yaw) * T
self.B[2][0] = tan(tar_angle) * T / L
self.B[2][1] = tar_v * T / (L * (cos(tar_angle)**2))
# 更新CA
for i in range(1, self.Np+1):
self.CA[i] = np.dot(self.CA[i-1], self.A)
# 更新PHI和THETA
for j in range(self.Np):
self.PHI[3*j:3*(j+1), 0:5] = self.CA[j+1]
for k in range(min(self.Nc, j+1)):
self.THETA[3*j:3*(j+1), 2*k: 2*(k+1)
] = np.dot(self.CA[j-k], self.B)
# 更新H
self.H = np.dot(np.dot(self.THETA.transpose(), self.Q),
self.THETA) + self.R
# 更新f
self.f = 2 * np.dot(np.dot(self.THETA.transpose(), self.Q),
np.dot(self.PHI, self.kesi))
# 更新约束
Ut = np.kron(np.ones((self.Nc, 1)), np.array([[v], [angle]]))
self.lb_cons[0:2*self.Nc, 0:1] = self.Umin-Ut
self.ub_cons[0:2*self.Nc, 0:1] = self.Umax-Ut
# 求解QP
P = matrix(self.H)
q = matrix(self.f)
G = matrix(np.block([
[self.A_cons],
[-self.A_cons]
]))
h = matrix(np.block([
[self.ub_cons],
[-self.lb_cons]
]))
solvers.options['show_progress'] = False
sol = solvers.qp(P, q, G, h)
X = sol['x']
# 输出结果
v += X[0]
angle += X[1]
return v, angle
def normalizeTheta(self, angle): # 角度归一化
while(angle >= pi):
angle -= 2*pi
while(angle < -pi):
angle += 2*pi
return angle
def findIdx(self, x, y, cx, cy): # 寻找欧式距离最近的点
min_dis = float('inf')
idx = 0
for i in range(len(cx)):
dx = x - cx[i]
dy = y - cy[i]
dis = dx**2 + dy**2
if(dis < min_dis):
min_dis = dis
idx = i
return idx
def update(self, x, y, yaw, v, angle): # 模拟车辆位置
x += v * cos(yaw) * self.dt
y += v * sin(yaw) * self.dt
yaw += v / self.Length * tan(angle) * self.dt
return x, y, yaw
if __name__ == '__main__':
cx = np.linspace(0, 200, 2000)
cy = np.zeros(len(cx))
dx = np.zeros(len(cx))
ddx = np.zeros(len(cy))
cyaw = np.zeros(len(cx))
ck = np.zeros(len(cx))
for i in range(len(cx)):
cy[i] = cos(cx[i]/10)*cx[i]/10
# 计算一阶导数
for i in range(len(cx)-1):
dx[i] = (cy[i+1] - cy[i])/(cx[i+1] - cx[i])
dx[len(cx)-1] = dx[len(cx)-2]
# 计算二阶导数
for i in range(len(cx)-2):
ddx[i] = (cy[i+2] - 2*cy[i+1] + cy[i]) / (0.5 * (cx[i+2] - cx[i]))**2
ddx[len(cx)-2] = ddx[len(cx)-3]
ddx[len(cx)-1] = ddx[len(cx)-2]
# 计算偏航角
for i in range(len(cx)):
cyaw[i] = atan(dx[i])
# 计算曲率
for i in range(len(cx)):
ck[i] = ddx[i] / (1 + dx[i]**2)**1.5
# 初始状态
x = 0.0
y = 5.0
yaw = 0.0
v = 0.0
angle = 0.0
t = 0
# 历史状态
xs = [x]
ys = [y]
vs = [v]
angles = [angle]
ts = [t]
# 实例化
mpc = MPC()
while(1):
idx = mpc.findIdx(x, y, cx, cy)
if(idx == len(cx)-1):
break
tar_v = 30.0/3.6
tar_angle = atan(mpc.Length * ck[idx])
(v, angle) = mpc.mpcControl(x, y, yaw, v, angle,
cx[idx], cy[idx], cyaw[idx], tar_v, tar_angle)
(x, y, yaw) = mpc.update(x, y, yaw, v, angle)
# 保存状态
xs.append(x)
ys.append(y)
vs.append(v)
angles.append(angle)
t = t+0.1
ts.append(t)
# 显示
plt.plot(cx, cy)
plt.scatter(xs, ys, c='r', marker='*')
plt.pause(0.01) # 暂停0.01秒
plt.clf()
plt.close()
plt.subplot(2, 1, 1)
plt.plot(ts, vs)
plt.subplot(2, 1, 2)
plt.plot(ts, angles)
plt.show()