This paper proposes a stable sparse rapidly-exploring random trees (SST) algorithm to solve the optimal motion planning problem for hybrid systems. At each iteration, the proposed algorithm, called HySST, selects a vertex with the lowest cost among all the vertices within the neighborhood of a randomly selected sample and then extends the search tree by flow or jump, which is also chosen randomly when both regimes are possible. In addition, HySST maintains a static set of witness points such that all the vertices within the neighborhood of each witness are pruned except the vertex with the lowest cost. Through a definition of concatenation of functions defined on hybrid time domains, we show that HySST is asymptotically near optimal, namely, the probability of failing to find a motion plan such that its cost is close to the optimal cost approaches zero as the number of iterations of the algorithm increases to infinity. This property is guaranteed under mild conditions on the data defining the motion plan, which include a relaxation of the usual positive clearance assumption imposed in the literature of classical systems. The proposed algorithm is applied to an actuated bouncing ball system and a collision-resilient tensegrity multicopter system so as to highlight its generality and computational features.

翻译：本文提出一种稳定的稀疏快速探索随机树（SST）算法，用于解决混合系统的最优运动规划问题。所提出的算法称为HySST，每次迭代时，在随机选取样本的邻域内所有顶点中选择成本最低的顶点，并通过流或跳跃扩展搜索树；当两种模式均可行时，扩展方式亦随机选择。此外，HySST维护一个静态的见证点集合，使得每个见证点邻域内的所有顶点中仅保留成本最低的顶点，其余均被剪枝。通过定义混合时间域上函数的串联操作，我们证明HySST具有渐近接近最优性，即随着算法迭代次数趋于无穷，未能找到成本接近最优成本的运动规划的概率趋近于零。该性质在定义运动规划数据的温和条件下得到保证，其中包括对经典系统文献中常规正间隙假设的放宽。将所提算法应用于执行器驱动弹跳球系统和抗碰撞张拉整体多旋翼系统，以凸显其通用性与计算特性。