  The paper presents Maximal Covariance Backward Reachable Trees (MAXCOVAR
BRT), which is a multi-query algorithm for planning of dynamic systems under
stochastic motion uncertainty and constraints on the control input with
explicit coverage guarantees. In contrast to existing roadmap-based
probabilistic planning methods that sample belief nodes randomly and draw edges
between them \cite{csbrm_tro2024}, under control constraints, the reachability
of belief nodes needs to be explicitly established and is determined by
checking the feasibility of a non-convex program. Moreover, there is no
explicit consideration of coverage of the roadmap while adding nodes and edges
during the construction procedure for the existing methods. Our contribution is
a novel optimization formulation to add nodes and construct the corresponding
edge controllers such that the generated roadmap results in provably maximal
coverage under control constraints as compared to any other method of adding
nodes and edges. We characterize formally the notion of coverage of a roadmap
in this stochastic domain via introduction of the h-$\operatorname{BRS}$
(Backward Reachable Set of Distributions) of a tree of distributions under
control constraints, and also support our method with extensive simulations on
a 6 DoF model.
