Corrected Initial Conditions Accommodating Nonlinearity of Differential Gravitational Acceleration and Eccentricity for Spacecraft Formation Flying

Paper number

IAC-09.C1.5.10

Author

Prof. Zhang Chen guang, Nanjing University of Aeronautics and Astronautics, China

Coauthor

Prof. Xu Bo, Nanjing University of Aeronautics and Astronautics, China

Year

2009

Abstract

Hill-Clohessy-Wiltshire (HCW) equations describe the relative motion of one spacecraft with respect to another under the assumption of linearized differential gravitational acceleration, a circular reference orbit, and spherical Earth. For spacecraft formation flying missions, the modeling errors of HCW equations would build up over time and break down the boundedness of the relative orbit. Thus, initial conditions generating periodic solutions to HCW equations have to be modified to obtain bounded solutions in the presence of the perturbations such as nonlinearity of the differential gravitational acceleration, eccentricity of the reference orbit and the oblateness of the Earth.

In this paper, assuming a spherical earth, we present two new sets of corrected initial conditions leading to bounded relative motion. The first set accommodates the cubic terms in the differential gravitational acceleration for circular reference orbit, while for reference orbit with small eccentricity, the second set captures the quadratic terms.

Firstly, nonlinear equations of relative motion accounting for full nonlinear gravitational acceleration and eccentricity of the reference orbit are established. Then, the cubic terms in the differential gravitational acceleration are retained for circular reference orbit. As for elliptical reference orbit, its true anomaly is expanded successively as functions of the mean anomaly with abandonment of higher order eccentricity items and the quadratic terms in the differential gravitational acceleration are kept. Both of these manipulations lead to two sets of perturbed HCW equations with small perturbation parameters. Using the perturbation method, the approximately analytical perturbation solutions to the perturbed differential equations are obtained. By substituting separately the analytical solutions of HCW equations into the perturbed differential equations, two sets of perturbed analytical solutions for these perturbed HCW equations are obtained. The secular terms in these solutions would create unbounded relative motion and they are parts of the source of HCW equations’ modeling errors. By proper algebraic manipulation, two criterions eliminating these secular terms are derived and further reduced to two sets of perturbed initial conditions. 

Finally, combining separately the two sets of perturbed initial conditions with the initial conditions derived from HCW equations, two sets of corrected initial conditions which lead to bounded relative motion with little drift are presented. These new initial conditions obtained are the generalization of the usual ones obtained from the linearized equations and also of the second order ones computed by S.S.Vaddi etc. Numeric simulations of modeling errors of these initial conditions are performed.

Abstract document

IAC-09.C1.5.10.pdf

Manuscript document

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