作为无人机自主运动的入门路径规划算法，详细表述了路径规划的前端和后端，前端为路径搜索，后端未路径优化。具体参考代码。

一、kinodynamic a_star 路径搜索

这里参考的文章是Path Planning for Autonomous Vehicles in Unknown Semi-structured Environments

总的可以概括为融合动力学约束的A*搜索，通过共轭梯度下降求局部最优。

解析kinodynamic_astar.cpp

Pathnode类(文章中motion Primitives)反向链表

class PathNode { public: /* -------------------- */ Eigen::Vector3i index; //voxel坐标 Eigen::Matrix<double, 6, 1> state; //基元状态,分别为xyz及其一阶导(vel) double g_score, f_score; //代价函数、启发函数 Eigen::Vector3d input; //输入，分别为xyz上的二阶导(acc) double duration; //输入持续时间 double time; // dyn int time_idx; PathNode* parent; //父节点 char node_state; /* -------------------- */ PathNode() { parent = NULL; node_state = NOT_EXPAND; } ~PathNode(){ }; EIGEN_MAKE_ALIGNED_OPERATOR_NEW};typedef PathNode* PathNodePtr;

主函数

/* kinodynamic A*的主函数 parameter： - start_pt：起点位置 - start_v: 起始速度 - start_a: 起始加速度 - end_pt: 终点位置 - end_v: 终点速度 - init： 初始化成功标志位 - dynamic：动静规划标志位 - time_start:起始时间*/int KinodynamicAstar::search(Eigen::Vector3d start_pt, Eigen::Vector3d start_v, Eigen::Vector3d start_a, Eigen::Vector3d end_pt, Eigen::Vector3d end_v, bool init, bool dynamic, double time_start)

读取传入参数

//读取传入参数 start_vel_ = start_v; start_acc_ = start_a; PathNodePtr cur_node = path_node_pool_[0]; //取出第一个路径点赋给当前节点 cur_node->parent = NULL; //父节点 cur_node->state.head(3) = start_pt; cur_node->state.tail(3) = start_v; cur_node->index = posToIndex(start_pt); //获得全局坐标系下的位置索引 cur_node->g_score = 0.0; //记录当前成本代价 Eigen::VectorXd end_state(6); Eigen::Vector3i end_index; double time_to_goal; //路径规划时间 end_state.head(3) = end_pt; end_state.tail(3) = end_v; end_index = posToIndex(end_pt); cur_node->f_score = lambda_heu_ * estimateHeuristic(cur_node->state, end_state, time_to_goal); //计算启发函数 cur_node->node_state = IN_OPEN_SET; //标记为OPEN open_set_.push(cur_node); //将该点放入OPEN集，即论文中的C use_node_num_ += 1; //扩展点数计数

启发函数

该文利用Pontryagins minimum principle计算从Xc到Xg轨迹的启发函数。笔者根据自身能力进行推导，具体参考A Computationally Efficient Motion Primitive for Quadrocopter Trajectory Generation及D. P. Bertsekas的《动态规划和最优控制》。

3D运动的cost function，Jk为投影到三轴上的加加速度

状态： S k = ( p k , v k , a k ) S_k=(p_k,v_k,a_k) Sk​=(pk​,vk​,ak​)

状态方程： S ˙ k = f s ( S k , j k ) = ( v k , a k , j k ) \dot{S}_k=f_s(S_k,j_k)=(v_k,a_k,j_k) S˙k​=fs​(Sk​,jk​)=(vk​,ak​,jk​)

构建Hamiltonian函数和协态方程：

求解costate的微分方程得

最佳输入轨迹J*(t)通过求最小化Hamiltonian函数(PMP原理)获得，即

j*(t)为加加速度的轨迹， S 0 = ( p 0 , v 0 , a 0 ) S_0=(p_0,v_0,a_0) S0​=(p0​,v0​,a0​)通过积分可以获 a ∗ ( t ) , v ∗ ( t ) , p ∗ ( t ) a^*(t),v^*(t),p*(t) a∗(t),v∗(t),p∗(t)

目标状态 S f = ( p f , v f , a f ) S_f=(p_f,v_f,a_f) Sf​=(pf​,vf​,af​)，

解得：

FAST-Planner当中最小化加速度，故与上述最小化加加速度不同，文中给出的公式如下：

启发函数代码(即上述使J取最小的T)：

/* 计算启发函数 parameters： - x1,x2:起点和终点状态 - optimal_time:达到最终点的最优时间 return: - 代价值*/double KinodynamicAstar::estimateHeuristic(Eigen::VectorXd x1, Eigen::VectorXd x2, double&amp; optimal_time) { const Vector3d dp = x2.head(3) - x1.head(3); //位置变化量 const Vector3d v0 = x1.segment(3, 3); //起点速度矩阵 const Vector3d v1 = x2.segment(3, 3); //终点速度矩阵 double c1 = -36 * dp.dot(dp); double c2 = 24 * (v0 + v1).dot(dp); double c3 = -4 * (v0.dot(v0) + v0.dot(v1) + v1.dot(v1)); double c4 = 0; double c5 = w_time_; std::vector&lt;double&gt; ts = quartic(c5, c4, c3, c2, c1); //求解获得极值点的T double v_max = max_vel_ * 0.5; double t_bar = (x1.head(3) - x2.head(3)).lpNorm<Infinity>() / v_max; ts.push_back(t_bar); double cost = 100000000; double t_d = t_bar;//将根带回启发函数求最优Ts for (auto t : ts) { if (t &lt; t_bar) continue; double c = -c1 / (3 * t * t * t) - c2 / (2 * t * t) - c3 / t + w_time_ * t; if (c &lt; cost) { cost = c; t_d = t; } } optimal_time = t_d; return 1.0 * (1 + tie_breaker_) * cost;}

启发函数对T求导获得极值点或者鞍点对应的时间T，然后把多个根代回启发函数获得最优时间T。

搜索

循环搜索第一步判断当前节点是否满足停止搜索的要求

// 判断但前节点是否超出horizon或是离终点较近了 bool reach_horizon = (cur_node->state.head(3) - start_pt).norm() >= horizon_; bool near_end = abs(cur_node->index(0) - end_index(0)) <= tolerance && abs(cur_node->index(1) - end_index(1)) <= tolerance && abs(cur_node->index(2) - end_index(2)) <= tolerance; if (reach_horizon || near_end) { terminate_node = cur_node; //判断其中一个达到了要求，将该节点设为终点 retrievePath(terminate_node);//从终点到起点的反向路径存储到path_nodes_ if (near_end) { // Check whether shot traj exist estimateHeuristic(cur_node->state, end_state, time_to_goal); //计算优化的最小时间 computeShotTraj(cur_node->state, end_state, time_to_goal); //带入优化的时间判断轨迹是否合理 if (init_search) ROS_ERROR("Shot in first search loop!"); } } if (reach_horizon) if (near_end)

其中当满足终点条件时，要重新检查一下轨迹是否合理，即将轨迹带入地图，用到下面这个函数

//将计算得到的最优轨迹带入地图中检查轨迹是否安全，有没有发生碰撞，速度、加速度是否超过限制bool KinodynamicAstar::computeShotTraj(Eigen::VectorXd state1, Eigen::VectorXd state2, double time_to_goal)

当节点不满足终止条件，需要扩展节点，以当前节点为根，离散输入控制量，得到下一状态。

/* ---3）若当前节点没有抵达终点，就要进行节点扩张 ---------- */ //1、在open_list中删除节点，并在close_list中添加节点 open_set_.pop(); //弹出该节点 cur_node->node_state = IN_CLOSE_SET; //将该节点放入close节点当中 iter_num_ += 1; //2、初始化状态传递 double res = 1 / 2.0, time_res = 1 / 1.0, time_res_init = 1 / 20.0; //时间常数 Eigen::Matrix<double, 6, 1> cur_state = cur_node->state; //当前节点状态 Eigen::Matrix<double, 6, 1> pro_state; //下一状态 vector<PathNodePtr> tmp_expand_nodes; //临时扩展节点列表 Eigen::Vector3d um; //控制量，本文是加速度 double pro_t; //扩展节点的时间戳 vector<Eigen::Vector3d> inputs; vector<double> durations; //输入控制量的持续时间 //3、判断节点没有被扩展过，把这个节点存下来 if (init_search) { inputs.push_back(start_acc_); for (double tau = time_res_init * init_max_tau_; tau <= init_max_tau_ + 1e-3; tau += time_res_init * init_max_tau_) durations.push_back(tau); init_search = false; } else { for (double ax = -max_acc_; ax <= max_acc_ + 1e-3; ax += max_acc_ * res) for (double ay = -max_acc_; ay <= max_acc_ + 1e-3; ay += max_acc_ * res) for (double az = -max_acc_; az <= max_acc_ + 1e-3; az += max_acc_ * res) { um << ax, ay, az; inputs.push_back(um); } for (double tau = time_res * max_tau_; tau <= max_tau_; tau += time_res * max_tau_) durations.push_back(tau); } //4、状态传递循环迭代，检查速度约束，检查碰撞，都通过的话，就计算当前节点的g_score以及f_score.并且对重复的扩展节点进行剪枝 // cout << "cur state:" << cur_state.head(3).transpose() << endl; for (int i = 0; i < inputs.size(); ++i) for (int j = 0; j < durations.size(); ++j) { um = inputs[i]; double tau = durations[j]; stateTransit(cur_state, pro_state, um, tau); //状态传递，通过前向积分得到扩展节点的位置和速度 pro_t = cur_node->time + tau; // Check inside map range Eigen::Vector3d pro_pos = pro_state.head(3); if (!edt_environment_->sdf_map_->isInBox(pro_pos)) { if (init_search) std::cout << "box" << std::endl; continue; } // Check if in close set Eigen::Vector3i pro_id = posToIndex(pro_pos); int pro_t_id = timeToIndex(pro_t); PathNodePtr pro_node = dynamic ? expanded_nodes_.find(pro_id, pro_t_id) : expanded_nodes_.find(pro_id); if (pro_node != NULL && pro_node->node_state == IN_CLOSE_SET) { if (init_search) std::cout << "close" << std::endl; continue; } // Check maximal velocity Eigen::Vector3d pro_v = pro_state.tail(3); if (fabs(pro_v(0)) > max_vel_ || fabs(pro_v(1)) > max_vel_ || fabs(pro_v(2)) > max_vel_) { if (init_search) std::cout << "vel" << std::endl; continue; } // Check not in the same voxel Eigen::Vector3i diff = pro_id - cur_node->index; int diff_time = pro_t_id - cur_node->time_idx; if (diff.norm() == 0 && ((!dynamic) || diff_time == 0)) { if (init_search) std::cout << "same" << std::endl; continue; } // Check safety Eigen::Vector3d pos; Eigen::Matrix<double, 6, 1> xt; bool is_occ = false; for (int k = 1; k <= check_num_; ++k) { double dt = tau * double(k) / double(check_num_); stateTransit(cur_state, xt, um, dt); pos = xt.head(3); if (edt_environment_->sdf_map_->getInflateOccupancy(pos) == 1 || !edt_environment_->sdf_map_->isInBox(pos)) { is_occ = true; break; } if (!optimistic_ && edt_environment_->sdf_map_->getOccupancy(pos) == SDFMap::UNKNOWN) { is_occ = true; break; } } if (is_occ) { if (init_search) std::cout << "safe" << std::endl; continue; } //计算代价值 double time_to_goal, tmp_g_score, tmp_f_score; //计算当前节点的真实代价 tmp_g_score = (um.squaredNorm() + w_time_) * tau + cur_node->g_score; //计算启发函数代价 tmp_f_score = tmp_g_score + lambda_heu_ * estimateHeuristic(pro_state, end_state, time_to_goal); /* ----------5)在循环中对比扩展节点,进行节点剪枝---------- */ // Compare nodes expanded from the same parent bool prune = false; //剪枝标志位 for (int j = 0; j < tmp_expand_nodes.size(); ++j) { PathNodePtr expand_node = tmp_expand_nodes[j]; // 首先判断当前临时扩展节点与current node的其他临时扩展节点是否在同一个voxel中，如果是的话就是扩展的节点多余了，就要进行剪枝。 if ((pro_id - expand_node->index).norm() == 0 && ((!dynamic) || pro_t_id == expand_node->time_idx)) { prune = true; if (tmp_f_score < expand_node->f_score) { expand_node->f_score = tmp_f_score; expand_node->g_score = tmp_g_score; expand_node->state = pro_state; expand_node->input = um; expand_node->duration = tau; if (dynamic) expand_node->time = cur_node->time + tau; } break; } } // This node end up in a voxel different from others if (!prune) { if (pro_node == NULL) { pro_node = path_node_pool_[use_node_num_]; pro_node->index = pro_id; pro_node->state = pro_state; pro_node->f_score = tmp_f_score; pro_node->g_score = tmp_g_score; pro_node->input = um; pro_node->duration = tau; pro_node->parent = cur_node; pro_node->node_state = IN_OPEN_SET; if (dynamic) { pro_node->time = cur_node->time + tau; pro_node->time_idx = timeToIndex(pro_node->time); } open_set_.push(pro_node); if (dynamic) expanded_nodes_.insert(pro_id, pro_node->time, pro_node); else expanded_nodes_.insert(pro_id, pro_node); tmp_expand_nodes.push_back(pro_node); use_node_num_ += 1; if (use_node_num_ == allocate_num_) { cout << "run out of memory." << endl; return NO_PATH; } } // 如果不用剪枝的话，则首先判断当前临时扩展节点pro_node是否出现在open集中 else if (pro_node->node_state == IN_OPEN_SET) { if (tmp_g_score < pro_node->g_score) { // pro_node->index = pro_id; pro_node->state = pro_state; pro_node->f_score = tmp_f_score; pro_node->g_score = tmp_g_score; pro_node->input = um; pro_node->duration = tau; pro_node->parent = cur_node; if (dynamic) pro_node->time = cur_node->time + tau; } } // 如果不是存在于open集的话，则可以直接将pro_node加入open集中 else { cout << "error type in searching: " << pro_node->node_state << endl; } } } // init_search = false; } cout << "open set empty, no path!" << endl; cout << "use node num: " << use_node_num_ << endl; cout << "iter num: " << iter_num_ << endl; return NO_PATH;

获得规划得到的路径点

kinodynamic A*的最后一步

/* 作用： 完成路径搜索后，按照预设的时间分辨率delta_t，通过节点回溯和状态前向积分得到分辨率最高的路径点。 param： delta_t:时间分辨率 return： state_list=节点扩张的回溯轨迹+两点边界轨迹*/std::vector<Eigen::Vector3d> KinodynamicAstar::getKinoTraj(double delta_t) { vector<Vector3d> state_list; /* ---------- get traj of searching ---------- */ PathNodePtr node = path_nodes_.back(); Matrix<double, 6, 1> x0, xt; while (node->parent != NULL) { Vector3d ut = node->input; double duration = node->duration; x0 = node->parent->state; for (double t = duration; t >= -1e-5; t -= delta_t) { stateTransit(x0, xt, ut, t); state_list.push_back(xt.head(3)); } node = node->parent; } reverse(state_list.begin(), state_list.end()); /* ---------- get traj of one shot ---------- */ if (is_shot_succ_) { Vector3d coord; VectorXd poly1d, time(4); for (double t = delta_t; t <= t_shot_; t += delta_t) { for (int j = 0; j < 4; j++) time(j) = pow(t, j); for (int dim = 0; dim < 3; dim++) { poly1d = coef_shot_.row(dim); coord(dim) = poly1d.dot(time); } state_list.push_back(coord); } } return state_list;}

第一部分结束，未完待续。。。