[Paper Review] A Minimum-propellant Pontryagin-based Nonlinear MPC for Spacecraft Rendezvous in Lunar Orbit: the Extended Version

Michele Pagone, Giordana Bucchioni+2 more|arXiv (Cornell University)|Sep 11, 2023

Astro and Planetary SciencePhysics and Astronomy3 citations

TL;DR

This paper proposes a Pontryagin Minimum Principle-based Nonlinear Model Predictive Control (NMPC) strategy for minimum-propellant spacecraft rendezvous in lunar orbit, leveraging the Circular Restricted Three-Body Problem (CR3BP) dynamics. The controller achieves bang-bang thrust behavior, ensuring fuel efficiency while maintaining high tracking accuracy with sub-meter position error and centimeter-per-second velocity error over a 4-hour maneuver.

ABSTRACT

We propose a Nonlinear Model Predictive Control approach to spacecraft rendezvous in non-Keplerian Lunar orbits. The approach is based on the Pontryagin Minimum Principle and allows the accomplishment of minimum-propellant maneuvers. The relative motion between the chaser and the target is described by the nonlinear and unstable dynamics of the circular restricted three body-problem. In the proposed formulation, we design a minimum-propellant controller, which leads to a bang-bang behavior of the control signal. Under suitable assumptions, simplified dynamics is employed as prediction model, in order to reduce the complexity of the controller algorithm but, at the same time, without penalizing the controller tracking performance. The proposed approach's effectiveness is validated by a simulation example.

Motivation & Objective

• To develop a fuel-optimal control strategy for the final phase of spacecraft rendezvous in non-Keplerian lunar orbits.
• To reduce computational complexity in NMPC by using simplified prediction dynamics without degrading tracking performance.
• To enforce minimum-propellant control via Pontryagin's Minimum Principle, yielding explicit state and costate-dependent control laws.
• To achieve high-precision relative state tracking under realistic thrust constraints and nonlinear dynamics.
• To validate the controller’s robustness and fuel efficiency through simulation in a lunar CR3BP environment.

Proposed method

• Formulates the relative chaser-target dynamics using the nonlinear and unstable CR3BP equations in a co-rotating synodic frame.
• Applies the Pontryagin Minimum Principle to derive an explicit optimal control law dependent on state and costate variables.
• Uses a simplified version of the CR3BP dynamics as the prediction model to reduce algorithmic complexity.
• Solves the resulting Two-Point Boundary Value Problem (TPBVP) via the bvp5c solver in MATLAB to compute optimal control inputs.
• Imposes thrust magnitude constraints with ||u||₂ ≤ 0.02 m/s², corresponding to ~10 N maximum thrust.
• Employs a receding horizon NMPC framework with sampling time Ts = 2 s and prediction horizon Tp = 90 s.

Experimental results

Research questions

• RQ1 Can a Pontryagin-based NMPC achieve minimum-propellant control in lunar rendezvous under CR3BP dynamics?
• RQ2 How does using simplified prediction dynamics affect tracking performance compared to full dynamics?
• RQ3 Does the controller exhibit bang-bang thrust behavior, indicating fuel-optimal control?
• RQ4 What level of tracking accuracy can be achieved for position and velocity in the final rendezvous phase?
• RQ5 How does the switching function behavior correlate with engine on/off states in the optimal control solution?

Key findings

• The controller achieves a final position error of less than 1 m on the x-axis and sub-millimeter errors on the y and z axes.
• Final velocity errors are below 1 cm/s across all components, indicating high-precision relative state convergence.
• The total impulse delivered by the thrusters is 28 m/s²·s, comparable to results from SDRE-based controllers.
• The thrust magnitude exhibits clear bang-bang behavior, with the switching function Υ crossing zero at engine on/off transitions.
• The simplified prediction model maintains high tracking accuracy despite reduced computational complexity.
• The controller successfully maintains thrust within the 0.02 m/s² limit, with individual components varying freely within the constraint set.