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import time
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import matplotlib.pyplot as plt
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import numpy as np
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from math import *
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from cvxopt import matrix, solvers
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class MPC:
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def __init__(self):
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self.Np = 60 # 预测步长
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self.Nc = 30 # 控制步长
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self.dt = 0.1 # 时间间隔
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self.Length = 1.0 # 车辆轴距
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max_steer = 30 * pi / 180 # 最大方向盘打角
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max_steer_v = 15 * pi / 180 # 最大方向盘打角速度
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max_v = 8.7 # 最大车速
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max_a = 1.0 # 最大加速度
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# 目标函数相关矩阵
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self.Q =150* np.identity(3 * self.Np) # 位姿权重 100为权重
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print('S',self.Q)
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self.R = 100 * np.identity(2 * self.Nc) # 控制权重
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print('r',self.R)
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self.kesi = np.zeros((5, 1))
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print('kesi',self.kesi)
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self.A = np.identity(5)
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print('A',self.A)
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self.B = np.block([
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[np.zeros((3, 2))],
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[np.identity(2)]
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])
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print('B',self.B)
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self.C = np.block([
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[np.identity(3), np.zeros((3, 2))]
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])
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print(self.C)
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self.PHI = np.zeros((3 * self.Np, 5))
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self.THETA = np.zeros((3 * self.Np, 2 * self.Nc))
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self.CA = (self.Np + 1) * [self.C]
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# print(self.CA)
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self.H = np.zeros((2 * self.Nc, 2 * self.Nc))
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self.f = np.zeros((2 * self.Nc, 1))
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# 不等式约束相关矩阵
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A_t = np.zeros((self.Nc, self.Nc))
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for p in range(self.Nc):
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# print('p',p)
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for q in range(p + 1):
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# print('q', q)
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A_t[p][q] = 1
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A_I = np.kron(A_t, np.identity(2))
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# print(A_I)
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# 控制量约束
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umin = np.array([[-max_v], [-max_steer]])
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print(umin)
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umax = np.array([[max_v], [max_steer]])
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self.Umin = np.kron(np.ones((self.Nc, 1)), umin)
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self.Umax = np.kron(np.ones((self.Nc, 1)), umax)
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# 控制增量约束
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delta_umin = np.array([[-max_a * self.dt], [-max_steer_v * self.dt]])
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delta_umax = np.array([[max_a * self.dt], [max_steer_v * self.dt]])
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delta_Umin = np.kron(np.ones((self.Nc, 1)), delta_umin)
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delta_Umax = np.kron(np.ones((self.Nc, 1)), delta_umax)
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self.A_cons = np.zeros((2 * 2 * self.Nc, 2 * self.Nc))
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self.A_cons[0:2 * self.Nc, 0:2 * self.Nc] = A_I
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self.A_cons[2 * self.Nc:4 * self.Nc, 0:2 * self.Nc] = np.identity(2 * self.Nc)
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self.lb_cons = np.zeros((2 * 2 * self.Nc, 1))
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self.lb_cons[2 * self.Nc:4 * self.Nc, 0:1] = delta_Umin
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self.ub_cons = np.zeros((2 * 2 * self.Nc, 1))
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self.ub_cons[2 * self.Nc:4 * self.Nc, 0:1] = delta_Umax
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def mpcControl(self, x, y, yaw, v, angle, tar_x, tar_y, tar_yaw, tar_v, tar_angle): # mpc优化控制
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T = self.dt
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L = self.Length
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# 更新误差
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self.kesi[0][0] = x - tar_x
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print('1,0',self.kesi[0][0])
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self.kesi[1][0] = y - tar_y
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self.kesi[2][0] = self.normalizeTheta(yaw - tar_yaw)
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self.kesi[3][0] = v - tar_v
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self.kesi[4][0] = angle - tar_angle
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# 更新A矩阵
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self.A[0][2] = -tar_v * sin(tar_yaw) * T
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self.A[0][3] = cos(tar_yaw) * T
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self.A[1][2] = tar_v * cos(tar_yaw) * T
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self.A[1][3] = sin(tar_yaw) * T
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self.A[2][3] = tan(tar_angle) * T / L
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self.A[2][4] = tar_v * T / (L * (cos(tar_angle) ** 2))
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# 更新B矩阵
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self.B[0][0] = cos(tar_yaw) * T
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self.B[1][0] = sin(tar_yaw) * T
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self.B[2][0] = tan(tar_angle) * T / L
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self.B[2][1] = tar_v * T / (L * (cos(tar_angle) ** 2))
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# 更新CA
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for i in range(1, self.Np + 1):
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self.CA[i] = np.dot(self.CA[i - 1], self.A)
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# 更新PHI和THETA
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for j in range(self.Np):
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self.PHI[3 * j:3 * (j + 1), 0:5] = self.CA[j + 1]
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for k in range(min(self.Nc, j + 1)):
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self.THETA[3 * j:3 * (j + 1), 2 * k: 2 * (k + 1)
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] = np.dot(self.CA[j - k], self.B)
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# 更新H
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self.H = np.dot(np.dot(self.THETA.transpose(), self.Q),
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self.THETA) + self.R
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# 更新f
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self.f = 2 * np.dot(np.dot(self.THETA.transpose(), self.Q),
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np.dot(self.PHI, self.kesi))
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# 更新约束
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Ut = np.kron(np.ones((self.Nc, 1)), np.array([[v], [angle]]))
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self.lb_cons[0:2 * self.Nc, 0:1] = self.Umin - Ut
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self.ub_cons[0:2 * self.Nc, 0:1] = self.Umax - Ut
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# 求解QP
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P = matrix(self.H)
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q = matrix(self.f)
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G = matrix(np.block([
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[self.A_cons],
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[-self.A_cons]
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]))
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h = matrix(np.block([
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[self.ub_cons],
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[-self.lb_cons]
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]))
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solvers.options['show_progress'] = False
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sol = solvers.qp(P, q, G, h)
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X = sol['x']
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# 输出结果
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v += X[0]
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angle += X[1]
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return v, angle
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def normalizeTheta(self, angle): # 角度归一化
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while (angle >= pi):
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angle -= 2 * pi
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while (angle < -pi):
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angle += 2 * pi
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return angle
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def findIdx(self, x, y, cx, cy): # 寻找欧式距离最近的点
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min_dis = float('inf')
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idx = 0
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for i in range(len(cx)):
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dx = x - cx[i]
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dy = y - cy[i]
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dis = dx ** 2 + dy ** 2
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if (dis < min_dis):
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min_dis = dis
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idx = i
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return idx
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def update(self, x, y, yaw, v, angle): # 模拟车辆位置
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x += v * cos(yaw) * self.dt
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y += v * sin(yaw) * self.dt
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yaw += v / self.Length * tan(angle) * self.dt
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return x, y, yaw
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if __name__ == '__main__':
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cx = np.linspace(0, 200, 2000)
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cy = np.zeros(len(cx))
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dx = np.zeros(len(cx))
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ddx = np.zeros(len(cy))
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cyaw = np.zeros(len(cx))
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ck = np.zeros(len(cx))
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for i in range(len(cx)):
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cy[i] = cos(cx[i] / 10) * cx[i] / 10
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# 计算一阶导数
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for i in range(len(cx) - 1):
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dx[i] = (cy[i + 1] - cy[i]) / (cx[i + 1] - cx[i])
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dx[len(cx) - 1] = dx[len(cx) - 2]
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# 计算二阶导数
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for i in range(len(cx) - 2):
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ddx[i] = (cy[i + 2] - 2 * cy[i + 1] + cy[i]) / (0.5 * (cx[i + 2] - cx[i])) ** 2
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ddx[len(cx) - 2] = ddx[len(cx) - 3]
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ddx[len(cx) - 1] = ddx[len(cx) - 2]
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# 计算偏航角
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for i in range(len(cx)):
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cyaw[i] = atan(dx[i])
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# 计算曲率
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for i in range(len(cx)):
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ck[i] = ddx[i] / (1 + dx[i] ** 2) ** 1.5
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# 初始状态
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x = 0.0
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y = 5.0
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yaw = 0.0
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v = 0.0
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angle = 0.0
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t = 0
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# 历史状态
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xs = [x]
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ys = [y]
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vs = [v]
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angles = [angle]
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ts = [t]
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# 实例化
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mpc = MPC()
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while (1):
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idx = mpc.findIdx(x, y, cx, cy)
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if (idx == len(cx) - 1):
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break
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tar_v = 80 / 3.6
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tar_angle = atan(mpc.Length * ck[idx])
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(v, angle) = mpc.mpcControl(x, y, yaw, v, angle,
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cx[idx], cy[idx], cyaw[idx], tar_v, tar_angle)
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(x, y, yaw) = mpc.update(x, y, yaw, v, angle)
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# 保存状态
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xs.append(x)
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ys.append(y)
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vs.append(v)
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angles.append(angle)
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t = t + 0.1
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ts.append(t)
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# 显示
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plt.plot(cx, cy)
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plt.scatter(xs, ys, c='r', marker='*')
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plt.pause(0.01) # 暂停0.01秒
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plt.clf()
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# plt.close()
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# plt.subplot(2, 1, 1)
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# plt.plot(ts, vs)
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# plt.subplot(2, 1, 2)
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# plt.plot(ts, angles)
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# plt.show()
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