Optimal Guidance of Low-Thrust Trajectories

Abstract

E LECTRIC propulsion has been selected in several interplanetary missions due to its higher efficiency compared with chemical propulsion. Perturbations such as maneuver execution errors, navigation uncertainties, operational constraints, uncertain parameters (e.g. initial state), and disturbance forces, produce deviations from the reference trajectory that the guidance and control functions must cancel at the expense of additional propellant mass. NASA Deep Space 1 mission proved an autonomous low-thrust guidance method [1] based on linearization of the terminal state in terms of a certain parameterization of the thrust profile. The terminal controller [2] was solved via Lagrange multipliers. This method has been refined [3] including a line search during the improvement step in order to assure convergence in case of poorly defined reference trajectory (e.g. sparse nodes, low numerical resolution). An interesting finite-thrust guidance and control method has been applied to small-body proximity operations [4]. It is based on pseudowaypoints generation and linear matrix inequality based feedback control. Waypoint-based, autonomous guidance with lowthrust propulsion was implemented for the approach phase of rendezvous missions to small asteroids [5]. The thrust profile was parameterized using a simple law that reduces the guidance problem to solve a system of nonlinear equations. Another autonomous guidance algorithm [6] used the Pontryaginminimumprinciple [2] to derive the optimal corrections to the nominal thrust profile minimizing the propellant consumption. Receding horizon control has been analyzed [7] using direct transcription to discretize the optimal control problem and an interior-point algorithm to solve numerically the resulting nonlinear programming problem. The optimal guidance and control methods presented in this paper focus on closed-form algorithms as in Deep Space 1 autonomous guidance [1], with special emphasis on the propagation and computation of the required partial derivatives matrices. The guidance function is a terminal controller that aims to reach the next waypoint at a specific time. A receding horizon control maintains the spacecraft in the proximity of the reference trajectory during the thrusting arc. The main differences with the aforementioned methods are that the optimal guidance and control laws are derived on a closed-form solution and accommodate thrust constraints in magnitude and direction. These qualities make them suitable for operational use even in autonomous guidance, navigation and control systems [8].