Formation Flying: Accommodating Nonlinearity and Eccentricity Perturbations

Abstract

Hill-Clohessy-Wiltshire equations describe the relative motion of one satellite with respect to another in a circular reference orbit. Initial conditions that generate periodic solutions to these equations have to be corrected to obtain bounded solutions in the presence of nonlinearity of the differential gravitational acceleration model and eccentricity of the reference orbit. The corrections to the initial conditions due to quadratic terms in the differential gravitational acceleration for circular reference orbits are established e rst by using a perturbation approach. These corrections are related to the period-matching constraint required for bounded relative motion. Next, the solution to the linear problem including the effect of eccentricity is determined, and a procedure for correcting the along-track bias is presented. The two solutions obtained are combined to produce an asymptotic solution for the quadratic eccentricity problem. The effects of nonlinearity and eccentricity on the relative orbits are characterized as functions of their initial position in the formation. HE problem of relative motion dynamics of satellites has been of interest since the 1960s. Much of the work has been per- formed in the context of the rendezvous problem. Accurate mod- eling of the relative motion dynamics for initial conditions close to the target is important for the rendezvous problem. Formation e ying requires bounded relative motion. Therefore, the solutions of interest are restricted to a certain set of initial conditions that lead to bounded relative motion. One particular formation of interest is the relative orbit that is circular when projected on the local hori- zontal plane. This solution is an exact solution to Hill- Clohessy- Wiltshire(HCW)equationsthatmodeltherelativemotiondynamics under the assumption of a circular reference orbit, spherical Earth, and linearized differential gravitational acceleration. Nonlinearity of the differential gravitational acceleration, eccentricity of the ref- erence orbit,and the Earth' s oblateness are the three most important perturbations that breakdown the circular orbit solutions to HCW equations. In this paper, we study the effects of nonlinearity and the eccentricity perturbations; the effects of J2 are ignored. The developments in this paper draw on several previous studies. Melton 1 developed a state transition matrix solution for the lin- earized relative motion dynamics by incorporating the effect of ec- centricity.Inalhan etal. 2 obtainedthe conditionforboundedrelative orbit solutions to the linearized problem with nonzero eccentricity. The effects of including quadratic gravitational acceleration terms were studied in Refs. 3- 6. Karlgaard and Lutze 7 used the method of multiple timescalesto obtain aperturbationsolutiontothe quadratic problem formulated using spherical coordinates. Alfriend et al. 8 used a geometric approach to map relative motion coordinates to orbital element differences. Mitchell and Richardson 9 developed an active nonlinear controller to accommodate quadratic nonlinearities in the zero-eccentricity problem. Broucke 10 has presented an exact state transition matrix solution for the linearized elliptic rendezvous problem. The solution is obtained by taking partial derivatives with