  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.
