Time/Fuel-optimal guidance laws for spacecraft rendezvous

Abstract

Spacecraft rendezvous and docking technology is a cornerstone of modern space missions, essential for expanding humanity's activities in space and achieving sustainable space habitation. This technology precisely adjusts the relative position and velocity of two or more spacecraft in orbit to ensure a safe approach and docking. It is crucial in various space missions, including satellite maintenance, space station construction, deep space exploration, and multiple spacecraft operations. This study proposes time- and fuel-optimal guidance laws for spacecraft rendezvous. Theoretically, the solutions to time/fuel-optimal control problems follow Bang-Bang or Bang-Off-Bang control profiles. However, such control profiles involve switching the control input between its maximum and minimum values at specific times, introducing nonlinearity and discontinuity into the system, which makes deriving the optimal solution challenging. This complexity is further aggravated in spacecraft rendezvous problems due to the coupling between x-axis and y-axis motions. This study employs a frequency-domain approach to address these challenges. The Bang-Bang and Bang-Off-Bang profiles can be expressed as a sum of time-delayed step functions, referred to as time-delay filters, in the frequency domain. Doing so can transform the optimal control problem into a parameter optimization problem that identifies switching times, allowing for numerically efficient and fast computation to find the optimal solution. Accurately determining the number of switchings in the control profile is critical for effectively implementing the proposed approach. While optimal control theory provides an upper bound for the maximum number of switchings, the exact number depends on the boundary conditions, adding to the challenge. This study analyzes the characteristics of the optimal solution for time/fuel-optimal rendezvous problems using a direct allocation method to design suitable time-delay filters. It estimates the number of switchings in the control profile. Finally, forms of time-delay filters based on given boundary conditions are proposed. The dissertation then states the process of transforming the time/fuel-optimal rendezvous problem into a parameter optimization problem. Numerical simulations under various boundary conditions are conducted to validate the effectiveness of the proposed approach. The results demonstrate high computational efficiency and robustness to system uncertainties. The findings of this study provide a practical guidance method that simultaneously enhances time and fuel efficiency, aligning with the increasing demands of advanced rendezvous and docking technology and multi-spacecraft missions in the New Space era. The proposed approach’s flexibility to adapt to arbitrary boundary conditions and its robustness to system uncertainties maximize its practical applicability. Furthermore, the time-optimal rendezvous guidance law establishes the lower bound of achievable rendezvous time under given conditions, offering significant insights into mission planning.