This paper focuses on the motion planning problem for the systems exhibiting both continuous and discrete behaviors, which we refer to as hybrid dynamical systems. Firstly, the motion planning problem for hybrid systems is formulated using the hybrid equation framework, which is general to capture most hybrid systems. Secondly, a propagation algorithm template is proposed that describes a general framework to solve the motion planning problem for hybrid systems. Thirdly, a rapidly-exploring random trees (RRT) implementation of the proposed algorithm template is designed to solve the motion planning problem for hybrid systems. At each iteration, the proposed algorithm, called HyRRT, randomly picks a state sample and extends the search tree by flow or jump, which is also chosen randomly when both regimes are possible. Through a definition of concatenation of functions defined on hybrid time domains, we show that HyRRT is probabilistically complete, namely, the probability of failing to find a motion plan approaches zero as the number of iterations of the algorithm increases. 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 motion plan is computed through the solution of two optimization problems, one associated with the flow and the other with the jumps of the system. The proposed algorithm is applied to an actuated bouncing ball system and a walking robot system so as to highlight its generality and computational features.

翻译：本文聚焦于同时呈现连续与离散行为的系统运动规划问题，此类系统被称为混合动力系统。首先，采用混合方程框架对混合系统运动规划问题进行形式化描述，该框架具有普适性，能够涵盖大多数混合系统。其次，提出传播算法模板，为求解混合系统运动规划问题提供了通用框架。第三，设计了一种基于快速扩展随机树（RRT）的算法模板实现方案，用于求解混合系统运动规划问题。所提出的HyRRT算法在每次迭代中随机选取状态样本，并通过流或跳转扩展搜索树——当两种模式均可行时，扩展模式也通过随机选择确定。通过定义混合时间域上函数的连接操作，我们证明HyRRT具有概率完备性，即随着算法迭代次数增加，无法找到运动规划方案的概率趋近于零。该性质在定义运动规划数据的温和条件下得以保证，这些条件包含对经典系统文献中通常要求的正间隙假设的松弛处理。运动规划方案通过求解两个优化问题获得：一个关联系统的流行为，另一个关联系统的跳转行为。将所提算法应用于驱动弹球系统与步行机器人系统，以突显其通用性与计算特性。