参考：
自动驾驶决策规划算法第二章第二节(上) 参考线模块_哔哩哔哩_bilibili
自动驾驶决策规划算法第二章第二节(下) 参考线代码实践_哔哩哔哩_bilibili
QP函数，二次规划的逻辑
function [smooth_path_x,smooth_path_y] = QP(path_x, path_y, w_cost_smooth, w_cost_length, w_cost_ref, x_lb, x_ub, y_lb, y_ub)% 采用二次规划平滑路径
% 0.5X'Hx + f'x = min
% lb <= x <= ub% 输入：
% path_x, path_y: 待平滑路径
% w_cost_smooth, w_cost_length, w_cost_re：平滑代价，紧凑代价，几何相似代价权重系数
% x_lb, x_ub, y_lb, y_ub：允许x, y变化的上下界(约束条件)% 输出：
% smooth_path_x,smooth_path_y：平滑后路径% 分析二次规划的形式
% H = 2 * (w_cost_smooth * A1' * A1 + w_cost_length * A2' * A2 + w_cost_ref * A3' * A3)
% f = -2 * w_cost_ref * path(待优化路径的xy组合)
% A1 = [1, 0, -2, 0, 1, 0
% 1, 0, -2, 0, 1, 0
% 1, 0, -2, 0, 1, 0
% 1, 0, -2, 0, 1, 0
% .............]
% A2 = [1, 0, -1, 0,
% 1, 0, -1, 0,
% 1, 0, -1, 0,
% 1, 0, -1, 0,
% .............]
% A3 为2n*2n的单位矩阵，n为平滑点的个数n = length(path_x);% 初始化平滑路径
smooth_path_x = zeros(n,1);
smooth_path_y = zeros(n,1);% 初始化损失函数相关信息
A1 = zeros(2*n-4, 2*n);
A2 = zeros(2*n-2, 2*n);
A3 = eye(2*n, 2*n);
path = zeros(2*n, 1);% 初始化不等式约束
lb = zeros(2*n, 1);
ub = zeros(2*n, 1);% 初始化等式约束（起点和终点必须一致）
Aeq = zeros(2*n, 2*n);
beq = zeros(2*n, 1);%计算f、lb、ub
for i = 1:npath(2*i-1) = path_x(i);path(2*i) = path_y(i);lb(2*i-1) = path_x(i) - x_lb;lb(2*i) = path_y(i) - y_lb;ub(2*i-1) = path_x(i) + x_ub;ub(2*i) = path_y(i) + y_ub;
end% 计算A1
for i = 1:2*n-5A1(i, i) = 1;A1(i, i+2) = -2;A1(i, i+4) = 1;A1(i+1, i+1) = 1;A1(i+1, i+3) = -2;A1(i+1, i+5) = 1;
end% 计算A2
for i = 1:2*n-3A2(i,i) = 1;A2(i,i+2) = -1;A2(i+1,i+1) = 1;A2(i+1,i+3) = -1;
endH = 2 * (w_cost_smooth * (A1' * A1) + w_cost_length * (A2' * A2) + w_cost_ref * (A3' * A3));
f = -2 * w_cost_ref * path;% 计算Aeq、beq
Aeq(1,1) = 1;
Aeq(2,2) = 1;
Aeq(2*n-1,2*n-1) = 1;
Aeq(2*n,2*n) = 1;
beq(1,1) = path(1,1);
beq(2,1) = path(2,1);
beq(2*n-1,1) = path(2*n-1,1);
beq(2*n,1) = path(2*n,1);%起点
X0 = path;% 进行二次规划
X = quadprog(H,f,[],[],Aeq,beq,lb,ub,X0);for i = 1:nsmooth_path_x(i) = X(2*i-1);smooth_path_y(i) = X(2*i);
endendmain函数，运行QP
clear;
clc;path_x = [0, 1, 2, 4, 5, 5, 5, 6, 7, 8, 9, 10, 11, 12, 12, 13, 14, 14];
path_y = [0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 9, 10, 11, 12, 12, 13, 14, 14];
w_cost_smooth = 3;
w_cost_length = 2;
w_cost_ref = 1;
x_lb = 1;
x_ub = 1;
y_lb = 1;
y_ub = 1;[smooth_path_x, smooth_path_y] = QP(path_x, path_y, w_cost_smooth, w_cost_length, w_cost_ref, x_lb, x_ub, y_lb, y_ub);hold on;
plot(path_x, path_y,'Color','red');
plot(smooth_path_x, smooth_path_y,'Color','blue');
运行结果：蓝色为原路径，红色为平滑路径
