Fast Homotopy for Spacecraft Rendezvous Trajectory Optimization with Discrete Logic

TL;DR

This paper introduces a fast, real-time capable optimization algorithm for spacecraft rendezvous that efficiently handles discrete logic constraints by blending sequential convex programming, numerical continuation, and smooth approximations.

Contribution

A novel algorithm combining sequential convex programming and homotopy methods to solve nonconvex optimal control problems with discrete logic constraints efficiently.

Findings

Successfully computed fuel-optimal rendezvous trajectories in under 15 seconds.

The algorithm reliably enforces discrete logic constraints like thruster and plume impingement.

Achieved trajectories with significantly less fuel than NASA design targets.

Abstract

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.