Approximate Minimum Energy Control Laws for Low-Thrust Formation Reconfiguration

T HERE are a vast number of papers on optimal space trajectory problems. The propulsion systems and the models of satellite motion are major concerns in these papers. The thrust types include impulsive and continuous thrust. According to optimal control theory, Lawden’s [1] pioneering work set up fundamental theories for optimal impulsive space trajectories using the primer vector. In his work, the primer vector determines the time and direction of each impulse. This is in sharp contrast to using optimization techniques to numerically search for the impulses. Pontryagin’s minimum principle is a powerful tool for the investigation of optimal continuous-thrust space trajectories. Edelbaum [2], Lawden [1], Prussing [3], Carter [4], and Kechichian [5] also contributed significantly to the topic. Neustadt [6] showed that the impulsive solution is a limit of the bounded continuous control when the bound becomes arbitrarily large. Because the relative motion of satellite formations is very complicated, much effort has been devoted to simplifying the relative-motion dynamic models to better understand and control relative motion. The first published study in the U.S. for the relative motion of close or neighboring satellites was performed by Clohessy and Wiltshire [7], hence the often-used name: the Clohessy– Wiltshire (C–W) equations. These equations assume that motion is about a spherical Earth, the reference orbit or target is in a circular orbit, and the distance between the satellites is small compared with the orbit radius so that the equations of motion can be linearized. Tschauner andHempel [8] obtained a solution for the relativemotion that included the reference orbit eccentricity. Incorporation of the eccentricity for the reference orbit was also obtained by Lawden [1]. Inalhan et al. [9] and Sengupta et al. [10] investigated the effects of neglecting the reference orbit eccentricity when establishing the relative-motion initial conditions. As a first step to including the nonspherical Earth effects, Gim andAlfriend [11] obtained the statetransition matrix (STM) for the relative motion that includes the absolute and differential gravity perturbation J2 effects. Approximate theories of relative motion were compared by Alfriend and Yan [12]. This note focuses on an optimal low-thrust control law that is based on the state-transition matrix for linearized equations of motion and its applications to satellite formations. The optimal control law of this problem is known in linear control theories [4]. The control law includes an integral that involves a multiplication of the state-transition matrix and its transpose. It is difficult to obtain an analytical solution of the integral, except for simple models such as the C–W equations [13]. Numerical integration is usually needed to evaluate this integral for a complicated state-transition matrix [4]. In thisNote, we use a pseudospectral or directmethod to solve the linear optimal control problem. An approximate analytical low-thrust control lawwithout numerical evaluation of the integral is derived for linear systems and applied to the reconfiguration of a satellite formation. This control law can be efficiently implemented in real time. Although Kechichian [5] included the first-order nonspherical Earth effects in optimal orbital transfers, the solutions are obtained from numerical iteration methods. We use the Gim–Alfriend statetransition matrix as the relative-motion model, which can be implemented in real time in an optimal low-thrust reconfiguration with gravity perturbation J2. The numerical results show that the control law works well for the optimal reconfiguration of a satellite formation.