  This paper presents a computationally efficient optimization algorithm for
solving nonconvex optimal control problems that involve discrete logic
constraints. Traditional solution methods for these constraints require binary
variables and mixed-integer programming, which is prohibitively slow and
computationally expensive. This paper targets a fast solution that is capable
of real-time implementation onboard spacecraft. To do so, a novel algorithm is
developed that blends sequential convex programming and numerical continuation
into a single iterative solution process. Inside the algorithm, discrete logic
constraints are approximated by smooth functions, and a homotopy parameter
governs the accuracy of this approximation. As the algorithm converges, the
homotopy parameter is updated such that the smooth approximations enforce the
exact discrete logic. The effectiveness of this approach is numerically
demonstrated for a realistic rendezvous scenario inspired by the Apollo
Transposition and Docking maneuver. In under 15 seconds of cumulative solver
time, the algorithm is able to reliably find difficult fuel-optimal
trajectories that obey the following discrete logic constraints: thruster
minimum impulse-bit, range-triggered approach cone, and range-triggered plume
impingement. The optimized trajectory uses significantly less fuel than
reported NASA design targets.
