Fast Homotopy for Spacecraft Rendezvous Trajectory Optimization with Discrete Logic

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. Nomenclature 𝜃 appch = approach cone half-angle, rad Δ𝑡 max = maximum pulse duration, s Δ𝑡 min = minimum pulse duration, s Δ𝑡 𝑖𝑘 = pulse duration of 𝑖-th thruster at 𝑘-th control interval, s Δ𝑡 db = buffer zone around Δ𝑡 min for the wall avoidance constraint, s f𝑖 = thrust direction vector for the 𝑖-th thruster F B = body frame centered at the chaser COM F L = LVLH frame centered at the target COM 𝐹 rcs = constant thrust level generated by a thruster, N

Trajectory Optimization with Complex Obstacle Avoidance Constraints via Homotopy Network Sequential Convex Programming

“…The general idea of the homotopy technique is to solve a series of simple transition problems so that the solution of the transition problem gradually approaches the optimal solution of the primal problem. This method reduces the problem's difficulty, thereby significantly improving the success rate of trajectory planning [4,[23][24][25][26][27].…”

“…Taheri [23,24] and Saranathan [25] first used the homotopy method to convert a multipoint boundary value problem into an easier-to-solve two-point boundary value problem, thereby dealing with multiple propulsion modes and dynamic environments. Malyuta [26] combined the homotopy method with the SCP algorithm to solve trajectory planning in a rendezvous maneuver under the constraints of discrete logic.…”