  We present theory and algorithms for the computation of probability-weighted
"keep-out" sets to assure probabilistically safe navigation in the presence of
multiple rigid body obstacles with stochastic dynamics. Our forward stochastic
reachability-based approach characterizes the stochasticity of the future
obstacle states in a grid-free and recursion-free manner, using Fourier
transforms and computational geometry. We consider discrete-time Markovian
switched systems with affine parameter-varying stochastic subsystems (DMSP) as
the obstacle dynamics, which includes Markov jump affine systems and
discrete-time affine parameter-varying stochastic systems (DPV). We define a
probabilistic occupancy function, to describe the probability that a given
state is occupied by a rigid body obstacle with stochastic dynamics at a given
time; keep-out sets are the super-level sets of this occupancy function. We
provide sufficient conditions that ensure convexity and compactness of these
keep-out sets for DPV obstacle dynamics. We also propose two computationally
efficient algorithms to overapproximate the keep-out sets --- a tight polytopic
approximation using projections, and an overapproximation using Minkowski sum.
For DMSP obstacle dynamics, we compute a union of convex and compact sets that
covers the potentially non-convex keep-out set. Numerical simulations show the
efficacy of the proposed algorithms for a modified version of the classical
unicycle dynamics, modeled as a DMSP.
