Indirect Optimization for Finite-Thrust Time-Optimal Orbital Maneuver

Abstract

A N IMPORTANTproblem in astronautics is to transfer a satellite between elliptic orbits, which has been widely studied bymany researchers in both the impulsive case and the continuous low-thrust case [1,2]. Recently, much attention has been focused on the computation of the optimal trajectory for the case of the low-thrust orbital transfer, which can be performed by minimizing the cost of the final time or the fuel-consumption under some additional constraints [2–8]. The dynamics of the satellite are usually described by the positionspeed variables or the modified equinoctial elements. Bonnard et al. [3], Caillau and Noailles [4], and Caillau et al. [5] studied the lowthrust time-optimal and minimum fuel-consumption orbital transfer problem using themodified equinoctial elements. The controllability property of the system, the existence of the optimal control and the singularity observed in the problem were also proposed from the geometrical analysis viewpoint. In the numerical experiment of the minimum-time transfer problem, some researchers found that the minimum time tfmin and the magnitude of the maximal thrust Tmax have the relationship that tfmin Tmax c, where c is a constant [5,7]. However, the problem of whether there exists a positive constant c such that tfmin Tmax tends to c as Tmax tends to zero is still open [8]. The missions of the orbital maneuver include the orbital rendezvous and the orbital intercept, which are different from that of the orbital transfer mainly in the terminal constraint conditions. As early as the 1950s and 1960s, the rendezvous and intercept problems had been widely investigated in the impulsive thrust case [9]. As for the rendezvous problem, the relative dynamics of the satellites (for example, the Hill–Clohessy–Wiltshire equations) have also been concerned [10]. The relative dynamics using the position-speed variables have also been used to deal with the orbital intercept problem [11]. However, the optimization problems for these two maneuver cases are not as widely studied as that for the transfer in the mode of continuous low-thrust using the equations described by the modified equinoctial elements. In this paper, the optimal-time orbital maneuver problems for the cases of the transfer, intercept, and rendezvous under a unified framework using the modified equinoctial elements are considered. The trajectory optimization problem is reduced to a two-point boundary-value (TPBV) problem by using the PontryaginMaximum Principle (PMP) [12], and the corresponding terminal conditions for the cases of the orbital transfer, intercept, and rendezvous are studied, respectively. The main contribution of this paper is twofold. First, the indirect optimization method is applied to the discussion of the finite-thrust orbital intercept and rendezvous problems using the modified equinoctial elements, which is quite different from the previous results on these problems obtained in the impulsive thrustmode and/or by using the relative dynamics of the spacecrafts. To the best knowledge of the authors, this is a novel approach to solve these problems. Second, the numerical simulation results show that the product of the minimum flight time and themaximal thrust is also approximately a constant in the orbital rendezvous and intercept. It is also shown that the orbital transfer and the rendezvous share almost the same optimal trajectory for a fixed maximum magnitude of the thrust. In this paper, the following notations are used: h; i indicates the inner product of two vectors, j j is the finite-dimensional Euclidean norm, and the superscript T means the transpose of a matrix.