Polynomial-Based Solutions to Targeting Problems for Onboard Applications

Adam Evans, Alberto Fossa, Roberto Armellin, Didier Henrion, Renato Zanetti

This paper solves the targeting problem focusing on accuracy, computational efficiency, and reliability. The trajectory optimization problem is first recast as a polynomial optimization problem (POP) by leveraging differential algebra to compute high-order Taylor expansions of the nonlinear dynamics and constraints. Moment-sum-of-squares (SOS) optimization is then utilized to solve this POP. A convex formulation based on a second-order expansion of the dynamics is also proposed. For impulsive targeting, the moment-SOS and convex approaches are compared against traditional nonlinear programming (NLP) solvers and map inversion techniques. Results indicate that the moment-SOS approach provides solutions as accurate as traditional NLP, but with the critical advantage of guaranteeing convergence to the global optimum under mild assumptions. Furthermore, the method excels at handling large maneuvers and long propagation times, conditions in which standard linear approximations rapidly degrade. To demonstrate its versatility, the methodology is extended to a continuous low-thrust station keeping (SK) scenario in the Earth-Moon Circular Restricted Three-Body Problem. The algorithm's performance is then evaluated in the presence of significant state errors. The ability to directly handle non-convex constraints and recast complex, nonlinear dynamics into formulations with reliable convergence properties makes the moment-SOS approach suitable for autonomous onboard applications.