  Planning robust executions under uncertainty is a fundamental challenge for
building autonomous robots. Partially Observable Markov Decision Processes
(POMDPs) provide a standard framework for modeling uncertainty in many
applications. In this work, we study POMDPs with safe-reachability objectives,
which require that with a probability above some threshold, a goal state is
eventually reached while keeping the probability of visiting unsafe states
below some threshold. This POMDP formulation is different from the traditional
POMDP models with optimality objectives and we show that in some cases, POMDPs
with safe-reachability objectives can provide a better guarantee of both safety
and reachability than the existing POMDP models through an example. A key
algorithmic problem for POMDPs is policy synthesis, which requires reasoning
over a vast space of beliefs (probability distributions). To address this
challenge, we introduce the notion of a goal-constrained belief space, which
only contains beliefs reachable from the initial belief under desired
executions that can achieve the given safe-reachability objective. Our method
compactly represents this space over a bounded horizon using symbolic
constraints, and employs an incremental Satisfiability Modulo Theories (SMT)
solver to efficiently search for a valid policy over it. We evaluate our method
using a case study involving a partially observable robotic domain with
uncertain obstacles. The results show that our method can synthesize policies
over large belief spaces with a small number of SMT solver calls by focusing on
the goal-constrained belief space.
