可以帮我发一下Frenet坐标系下的动态轨迹规划的matlab代码吗，十分感谢。
参考GPT和自己的思路，Frenet坐标系下的动态轨迹规划需要根据具体的问题进行实现，这里提供一个基础的代码框架，供参考：
% Frenet Trajectory Planning
% Define the reference path
refPath = [0 0; 20 20; 40 10; 60 25; 80 10; 100 0];
s = linspace(0, 1, size(refPath, 1));
splineX = spline(s, refPath(:, 1));
splineY = spline(s, refPath(:, 2));
% Define the initial conditions
x0 = [0, 0, 0, 0, 0, 0];
tspan = [0, 5];
% Define the cost function
costFunction = @(x) frenetCost(x, splineX, splineY, s);
% Define the constraints
constraintFunction = @(x) frenetConstraints(x, splineX, splineY, s);
% Solve the optimization problem
options = optimoptions(@fmincon, 'Algorithm', 'interior-point', 'Display', 'iter');
[x, fval, exitflag] = fmincon(costFunction, x0, [], [], [], [], [], [], constraintFunction, options);
% Plot the results
plotFrenetTrajectory(x, splineX, splineY, s);
其中，refPath表示参考路径，splineX和splineY表示参考路径的样条插值函数。x0是初始状态，tspan表示时间范围。costFunction和constraintFunction分别表示代价函数和约束条件。使用fmincon函数求解优化问题，得到最优解x。最后使用plotFrenetTrajectory函数绘制Frenet轨迹。
具体实现时，需要根据问题进行修改和扩展。需要注意的是，Frenet坐标系下的动态轨迹规划需要考虑多种因素，如车辆动力学模型、障碍物避障、速度规划等，因此实现较为复杂。建议先从基础的静态轨迹规划开始学习，逐步增加复杂度。
该回答引用GPTᴼᴾᴱᴺᴬᴵ
这里提供一个基于Frenet坐标系的路径规划MATLAB代码示例，代码包括了路径生成、平滑和路径跟踪三个模块。具体实现过程中用到了cubic-spline插值和LQR控制器。以下是代码：
% Frenet-based dynamic trajectory planning and tracking
clc;
close all;
clear;
%% Path Generation
% Define reference path in Cartesian coordinate system
waypoints = [0 0; 30 20; 60 20; 70 10; 70 -10; 60 -20; 30 -20; 0 0];
% Convert reference path to Frenet coordinate system
[FrenetPath, FrenetFrame] = Cartesian2Frenet(waypoints);
% Generate path with desired speed profile
numPoints = 400;
maxSpeed = 10;
path = GeneratePath(FrenetPath, numPoints, maxSpeed);
%% Path Smoothing
% Cubic spline interpolation
path = CubicSpline(path);
%% Path Tracking
% LQR controller parameters
Q = diag([1, 1, 1, 1, 1]);
R = 0.1;
% Define initial state
x0 = [path(1,1); 0; 0; path(1,2); 0];
% Define time step
dt = 0.1;
% Define simulation time
simTime = 20;
% Path tracking simulation
[x, u] = PathTracking(path, x0, simTime, dt, Q, R);
%% Plot Results
% Plot reference path
figure;
plot(waypoints(:,1), waypoints(:,2), 'k--');
hold on;
% Plot smoothed path
plot(path(:,1), path(:,2), 'b');
% Plot path tracking simulation
plot(x(:,1), x(:,4), 'r');
legend('Reference Path', 'Smoothed Path', 'Path Tracking');
xlabel('X [m]');
ylabel('Y [m]');
其中 Cartesian2Frenet 函数将笛卡尔坐标系的路径转换为 Frenet 坐标系的路径，GeneratePath 函数在 Frenet 坐标系下生成速度剖面匹配的路径，CubicSpline 函数对路径进行三次样条插值，PathTracking 函数利用 LQR 控制器实现路径跟踪。注意代码中的各个函数的实现需要自己编写。
该回答引用ChatGPT
当涉及到动态轨迹规划时，常常使用Frenet坐标系。下面是一个简单的Matlab代码实现，用于计算和可视化Frenet坐标系下的轨迹规划：
% 定义路线的参数化函数
s = linspace(0, 10*pi, 1000);
x = s.*cos(s);
y = s.*sin(s);
% 计算Frenet坐标系下的参数化函数
[kappa, s] = curvature(x, y);
theta = atan2(y', x');
d = zeros(size(s));
for i = 2:length(s)
dx = x(i) - x(i-1);
dy = y(i) - y(i-1);
d(i) = d(i-1) + norm([dx, dy]);
end
% 定义起点和终点
start = [x(1); y(1); theta(1); kappa(1); d(1)];
goal = [x(end); y(end); theta(end); kappa(end); d(end)];
% 使用Dijkstra算法进行路径规划
path = dijkstra(start, goal, @cost);
% 绘制结果
figure
plot(x, y, 'k', 'LineWidth', 2);
hold on
plot(path(:,1), path(:,2), 'r', 'LineWidth', 2);
axis equal
xlabel('X位置');
ylabel('Y位置');
title('Frenet坐标系下的轨迹规划');
% 定义成本函数
function c = cost(x1, x2)
c = norm(x1(5) - x2(5)) + norm(x1(4) - x2(4))*10;
end
% 计算曲率
function [kappa, s] = curvature(x, y)
dx = gradient(x);
ddx = gradient(dx);
dy = gradient(y);
ddy = gradient(dy);
kappa = (dx.*ddy - dy.*ddx)./((dx.^2 + dy.^2).^(3/2));
s = zeros(size(x));
for i = 2:length(x)
ds = norm([dx(i); dy(i)]);
s(i) = s(i-1) + ds;
end
end
% Dijkstra算法
function path = dijkstra(start, goal, costfun)
openSet = PriorityQueue();
openSet.insert(start, 0);
cameFrom = containers.Map();
gScore = containers.Map();
gScore(mat2str(start)) = 0;
while ~openSet.isEmpty()
current = openSet.remove();
if isequal(current, goal)
path = reconstructPath(cameFrom, current);
return
end
neighbors = getNeighbors(current);
for i = 1:size(neighbors, 1)
neighbor = neighbors(i, :);
tentativeGScore = gScore(mat2str(current)) + costfun(current, neighbor);
if ~gScore.isKey(mat2str(neighbor)) || tentativeGScore < gScore(mat2str(neighbor))
cameFrom(mat2str(neighbor)) = current;
gScore(mat2str(neighbor)) = tentativeGScore;
fScore = tentativeGScore;
if ~openSet.contains(neighbor)
openSet.insert(neighbor, fScore);
else
openSet.updatePriority(neighbor, fScore);
end
end
end
end
end
% 重构路径
function path = reconstructPath(cameFrom, current)
path = current;
while
cameFrom.isKey(mat2str(current))
current = cameFrom(mat2str(current));
path = [current; path];
end
end
% 获取邻居节点
function neighbors = getNeighbors(node)
x = node(1);
y = node(2);
theta = node(3);
kappa = node(4);
d = node(5);
L = 2;
delta = linspace(-pi/4, pi/4, 5);
neighbors = zeros(length(delta), 5);
for i = 1:length(delta)
deltaTheta = delta(i);
nextTheta = theta + kappaLtan(deltaTheta);
nextKappa = kappa + deltaTheta/L;
nextD = d + L/cos(deltaTheta);
neighbors(i,:) = [x+Lcos(nextTheta); y+Lsin(nextTheta); nextTheta; nextKappa; nextD];
end
end
% 优先队列
classdef PriorityQueue < handle
properties (Access = private)
heap
priority
count
end
methods
function obj = PriorityQueue()
obj.heap = cell(16, 1);
obj.priority = zeros(16, 1);
obj.count = 0;
end
function insert(obj, element, priority)
obj.count = obj.count + 1;
if obj.count > length(obj.heap)
obj.heap = [obj.heap; cell(16, 1)];
obj.priority = [obj.priority; zeros(16, 1)];
end
obj.heap{obj.count} = element;
obj.priority(obj.count) = priority;
obj.bubbleUp(obj.count);
end
function element = remove(obj)
element = obj.heap{1};
obj.heap{1} = obj.heap{obj.count};
obj.priority(1) = obj.priority(obj.count);
obj.count = obj.count - 1;
obj.bubbleDown(1);
end
function empty = isEmpty(obj)
empty = (obj.count == 0);
end
function updatePriority(obj, element, priority)
index = find(cellfun(@(x)isequal(x, element), obj.heap), 1);
obj.priority(index) = priority;
obj.bubbleUp(index);
end
function containsElement = contains(obj, element)
containsElement = any(cellfun(@(x)isequal(x, element), obj.heap));
end
end
methods (Access = private)
function bubbleUp(obj, index)
if index == 1
return
end
parent = floor(index/2);
if obj.priority(index) < obj.priority(parent)
temp = obj.heap{index};
obj.heap{index} = obj.heap{parent};
obj.heap{parent} = temp;
temp = obj.priority(index);
obj.priority(index) = obj.priority(parent);
obj.priority(parent) = temp;
obj.bubbleUp(parent);
end
end
function bubbleDown(obj, index)
left = 2*index;
right = 2*index + 1;
minIndex = index;
if left <= obj.count && obj.priority(left) < obj.priority(minIndex)
minIndex = left;
end
if right <= obj.count && obj.priority(right) < obj.priority(minIndex)
minIndex = right;
end
if minIndex ~= index
temp = obj.heap(index);
obj.heap{index} = obj.heap{minIndex};
obj.heap{minIndex} = temp;
temp = obj.priority(index);
obj.priority(index) = obj.priority(minIndex);
obj.priority(minIndex) = temp;
obj.bubbleDown(minIndex);
end
end
end
end