  This paper presents an enhanced direct-method-based approach for the
real-time solution of optimal control problems to handle path constraints, such
as obstacles. The principal contributions of this work are twofold: first, the
existing methods for constructing reachability sets in the literature are
extended to derive the envelope of these sets, which determines the region
swept by all feasible trajectories between adjacent sample points. Second, we
propose a novel method to guarantee constraint violation-free between discrete
states in two dimensions through mesh refinement approach. To illustrate the
effectiveness of the proposed methodology, numerical simulations are conducted
on real-time path planning for fixed-wing unmanned aerial vehicles.
