theta = atan2((x_rand(2) - x_near(2)),(x_rand(1) - x_near(1))); x_new(1) = x_near(1) + cos(theta) * Delta; x_new(2) = x_near(2) + sin(theta) * Delta;

以下是RRT-Connect算法的C++实现示例。 ```cpp #include <iostream> #include <cmath> #include <vector> #include <cstdlib> #include <ctime> #include <chrono> using namespace std; // 定义状态结构体 struct State { double x; double y; }; // 定义节点结构体 struct Node { State state; Node* parent; }; // 定义RRT-Connect类 class RRTConnect { public: RRTConnect(State start, State goal, double map_width, double map_height, double step_size); void planning(); vector<State> getPath(); private: Node* getRandomNode(); Node* getNearestNode(Node* node); bool isCollide(State state); bool isReach(State s1, State s2); void addVertex(Node* parent, Node* child); State generateRandomState(); vector<Node*> tree1; vector<Node*> tree2; State start_; State goal_; double map_width_; double map_height_; double step_size_; double goal_radius_ = 0.5; int max_iterations_ = 5000; }; // RRT-Connect算法类构造函数 RRTConnect::RRTConnect(State start, State goal, double map_width, double map_height, double step_size) : start_(start), goal_(goal), map_width_(map_width), map_height_(map_height), step_size_(step_size) { Node* start_node = new Node{ start_, nullptr }; Node* goal_node = new Node{ goal_, nullptr }; tree1.push_back(start_node); tree2.push_back(goal_node); srand(time(nullptr)); // 初始化随机数种子 } // 获取随机节点 Node* RRTConnect::getRandomNode() { double x = (double)rand() / RAND_MAX * map_width_; double y = (double)rand() / RAND_MAX * map_height_; return new Node{ State{x, y}, nullptr }; } // 获取最近节点 Node* RRTConnect::getNearestNode(Node* node) { Node* nearest_node = tree1.front(); double min_dist = hypot(nearest_node->state.x - node->state.x, nearest_node->state.y - node->state.y); for (auto n : tree1) { double dist = hypot(n->state.x - node->state.x, n->state.y - node->state.y); if (dist < min_dist) { nearest_node = n; min_dist = dist; } } return nearest_node; } // 判断状态是否碰撞 bool RRTConnect::isCollide(State state) { double obs_x = map_width_ / 2; double obs_y = map_height_ / 2; double obs_r = 2.0; double dist = hypot(state.x - obs_x, state.y - obs_y); return dist < obs_r; } // 判断两个状态是否可达 bool RRTConnect::isReach(State s1, State s2) { double dist = hypot(s1.x - s2.x, s1.y - s2.y); return dist < step_size_; } // 添加节点 void RRTConnect::addVertex(Node* parent, Node* child) { child->parent = parent; tree1.push_back(child); } // 生成随机状态 State RRTConnect::generateRandomState() { double x = (double)rand() / RAND_MAX * map_width_; double y = (double)rand() / RAND_MAX * map_height_; return State{ x, y }; } // RRT-Connect算法主函数 void RRTConnect::planning() { for (int i = 0; i < max_iterations_; i++) { Node* q_rand = getRandomNode(); Node* q_near = getNearestNode(q_rand); State new_state; if (isReach(q_near->state, q_rand->state)) { new_state = q_rand->state; } else { double theta = atan2(q_rand->state.y - q_near->state.y, q_rand->state.x - q_near->state.x); new_state = State{ q_near->state.x + step_size_ * cos(theta), q_near->state.y + step_size_ * sin(theta) }; } if (isCollide(new_state)) { continue; } Node* new_node = new Node{ new_state, nullptr }; addVertex(q_near, new_node); if (isReach(new_state, goal_)) { addVertex(tree2.front(), new Node{ goal_, nullptr }); break; } swap(tree1, tree2); } } // 获取路径 vector<State> RRTConnect::getPath() { vector<State> path; Node* node = tree1.back(); while (node) { path.push_back(node->state); node = node->parent; } reverse(path.begin(), path.end()); node = tree2.front()->parent; while (node) { path.push_back(node->state); node = node->parent; } return path; } int main() { State start{ 1.0, 1.0 }; State goal{ 9.0, 9.0 }; RRTConnect rrt(start, goal, 10.0, 10.0, 0.5); auto start_time = chrono::high_resolution_clock::now(); rrt.planning(); auto end_time = chrono::high_resolution_clock::now(); auto duration = chrono::duration_cast<chrono::milliseconds>(end_time - start_time); cout << "Time: " << duration.count() << "ms" << endl; vector<State> path = rrt.getPath(); for (auto state : path) { cout << "(" << state.x << ", " << state.y << ")" << endl; } return 0; } ``` 这个示例中，我们使用了两个树来进行RRT-Connect算法的搜索。其中，tree1是从起点出发的树，tree2是从终点出发的树。在每次迭代中，我们随机生成一个节点q_rand，并在tree1中寻找距离q_rand最近的节点q_near。然后，我们通过扩展q_near，生成新的节点new_node，并将其添加到tree1中。如果new_node离终点goal足够近，我们将生成一个新的终点节点，并结束搜索。在添加完new_node后，我们交换tree1和tree2，以便下一次迭代时，tree1成为从终点开始的树，tree2成为从起点开始的树。 值得注意的是，这个示例中我们假设地图中只有一个障碍物，障碍物的位置和大小都是固定的。在实际应用中，我们需要根据实际场景来确定障碍物的位置和大小。

Informed-RRT*算法是一种用于路径规划的算法，它可以在高维空间中找到一条最短路径。下面是使用C语言实现Informed-RRT*算法的示例代码。 ```c #include <stdio.h> #include <stdlib.h> #include <math.h> #include <time.h> #define WIDTH 500 // 定义环境宽度 #define HEIGHT 500 // 定义环境高度 typedef struct node{ // 定义树节点 int x, y; float cost; struct node *parent; } Node; typedef struct{ // 定义目标点 int x, y; } Goal; float euclidean_distance(Node *node1, Node *node2){ // 计算欧几里得距离 float dx = node1->x - node2->x; float dy = node1->y - node2->y; return sqrt(dx*dx + dy*dy); } Node *generate_random_node(){ // 随机生成节点 Node *node = (Node *)malloc(sizeof(Node)); node->x = rand() % WIDTH; node->y = rand() % HEIGHT; node->cost = 0.0; node->parent = NULL; return node; } Node *nearest_vertex(Node *node, Node **tree, int tree_size){ // 找到最近的节点 float min_distance = 999999.0; Node *nearest_node = NULL; for(int i=0; i<tree_size; i++){ float distance = euclidean_distance(node, tree[i]); if(distance < min_distance){ min_distance = distance; nearest_node = tree[i]; } } return nearest_node; } Node *new_node(Node *nearest_node, Node *random_node, float step_size){ // 创建新节点 Node *new_node = (Node *)malloc(sizeof(Node)); float distance = euclidean_distance(nearest_node, random_node); if(distance < step_size){ // 如果距离小于步长，直接连接两个节点 new_node->x = random_node->x; new_node->y = random_node->y; } else{ // 否则在两个节点之间插值 float theta = atan2(random_node->y - nearest_node->y, random_node->x - nearest_node->x); new_node->x = nearest_node->x + step_size * cos(theta); new_node->y = nearest_node->y + step_size * sin(theta); } new_node->cost = nearest_node->cost + euclidean_distance(nearest_node, new_node); new_node->parent = nearest_node; return new_node; } void add_node_to_tree(Node *node, Node **tree, int *tree_size){ // 将节点添加到树中 tree[*tree_size] = node; (*tree_size)++; } Goal *generate_random_goal(){ // 随机生成目标点 Goal *goal = (Goal *)malloc(sizeof(Goal)); goal->x = rand() % WIDTH; goal->y = rand() % HEIGHT; return goal; } Node *find_best_goal_node(Node **tree, int tree_size, Goal *goal){ // 找到最优的目标节点 float min_distance = 999999.0; Node *best_node = NULL; for(int i=0; i<tree_size; i++){ float distance = euclidean_distance(tree[i], (Node *)goal); if(distance < min_distance){ min_distance = distance; best_node = tree[i]; } } return best_node; } void rewire(Node *node, Node **tree, int tree_size, float step_size){ // 重新连接节点 for(int i=0; i<tree_size; i++){ Node *near_node = tree[i]; float distance = euclidean_distance(node, near_node); float new_cost = node->cost + distance; if(new_cost < near_node->cost){ // 如果新的路径比原来的路径更短，则重新连接节点 float theta = atan2(node->y - near_node->y, node->x - near_node->x); int new_x = near_node->x + step_size * cos(theta); int new_y = near_node->y + step_size * sin(theta); if(new_x > 0 && new_x < WIDTH && new_y > 0 && new_y < HEIGHT){ // 确保新节点在环境内 near_node->parent = node; near_node->cost = new_cost; } } } } void print_path(Node *node){ // 打印路径 while(node != NULL){ printf("(%d,%d) ", node->x, node->y); node = node->parent; } printf("\n"); } void informed_rrt_star(int max_iterations, float step_size){ srand((unsigned int)time(NULL)); // 设置随机种子 Node *start_node = generate_random_node(); // 随机生成起始节点 Node *tree[10000]; int tree_size = 1; tree[0] = start_node; Goal *goal = generate_random_goal(); // 随机生成目标点 for(int i=0; i<max_iterations; i++){ // 执行迭代 Node *random_node = generate_random_node(); // 随机生成节点 Node *nearest_node = nearest_vertex(random_node, tree, tree_size); // 找到最近的节点 Node *new_node = new_node(nearest_node, random_node, step_size); // 创建新节点 if(new_node->x == nearest_node->x && new_node->y == nearest_node->y){ // 如果新节点和最近节点重合，则跳过 continue; } if(new_node->x == goal->x && new_node->y == goal->y){ // 如果新节点是目标节点，则返回路径 print_path(new_node); return; } if(new_node->x < 0 || new_node->x > WIDTH || new_node->y < 0 || new_node->y > HEIGHT){ // 如果新节点不在环境内，则跳过 continue; } add_node_to_tree(new_node, tree, &tree_size); // 将新节点添加到树中 rewire(new_node, tree, tree_size, step_size); // 重新连接节点 } Node *best_goal_node = find_best_goal_node(tree, tree_size, goal); // 找到最优的目标节点 print_path(best_goal_node); // 打印路径 } int main(){ informed_rrt_star(1000, 10.0); return 0; } ``` 上述代码使用了一个简单的环境，节点和目标点都是二维坐标。算法首先随机生成一个起始节点和目标点，然后在环境中随机生成节点，并将其与树中的最近节点连接。如果新节点和最近节点重合，则跳过。如果新节点是目标节点，则返回路径。如果新节点不在环境内，则跳过。然后将新节点添加到树中，并重新连接节点。最后找到最优的目标节点，并返回路径。