  This paper proposes a rapidly-exploring random trees (RRT) algorithm 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 a walking robot so as to highlight
its generality and computational features.
