Satellite Orbit Transfer and Formation Reconfiguration via an Attitude Control Analogy

A control law applicable to orbit transfer as well as formation reconfiguration has been developed via a Lyapunov function similar to that used for spacecraft attitude control. It uses feedback of kinematic and kinetic error states between the current and desired orbital frames, which are parameterized by Euler parameters, orbital radius and its time derivative, and orbital angular velocities. The efficacy of the control law is demonstrated with examples, including orbit transfers and formation reconfiguration, both with low-eccentricity and high-eccentricity reference orbits, in the presence of J2 perturbations. Further simplifications in the control law are introduced and it is shown that stability is not affected. A methodology is proposed to reduce control requirement by appropriate gain selection and control initiation at predefined locations. The equivalent impulse requirements obtained for the examples compare very favorably with analytically and numerically optimized figures presented in other published works. HE problem of orbital transfers of satellites by the use of con- tinuous, low-thrust control has been studied in great detail. Re- cent advances in the field of electric propulsion have made low-thrust propulsion an operational reality. Formation flying of spacecraft is also a relatively new area of research wherein control of relative mo- tion is a key element. Control of relative motion requires controlling each satellite's orbit precisely, with high regard for intersatellite dis- tances, collision hazards, and other operational constraints. The primary purpose of this paper is to develop a Lyapunov-based control law that has a large range of applicability, free from the influ- ence of singularities that arise from the more common descriptions of a satellite's orbit, such as orbital elements. Previous studies in the literature have addressed the problem of continuous-thrust orbit transfer using Lyapunov function-based feedback controllers. Ilgen 1 uses orbital element feedback for orbital transfers, using the clas- sical as well as equinoctial elements. Ilgen's quadratic Lyapunov function is based on only five of the orbital elements. Gurfil 2 uses classical orbital element feedback for orbital transfer and explores nonlinear controllability issues of the problem, and is able to prove by using Gauss variational equations that a spacecraft's orbital el- ements can be controlled by continuous control, except when the desired orbit is parabolic. Chang et al. 3 approach the problem of orbital transfer by feeding back the angular momentum and eccen- tricity vectors. References 1-3 provide excellent bases for the de- velopment of Lyapunov-function-based controllers for formation reconfiguration using orbital elements. Schaub and Alfriend 4 use the local Cartesian coordinates of the Chief/leader satellite and the differential orbital elements between the Deputy/follower satellite and the Chief as feedback. Schaub et al. 5 present two control laws, one in terms of mean elements, and another in terms of inertial co- ordinates, for formation control, with the added simplification of using mean elements in the presence of J2 perturbations arising due