  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.
