Second-Order Equations for Rendezvous in a Circular Orbit

Abstract

Introduction T HIS Note develops an approximate analytical solution of the two-body orbital boundary-value problem, known as Lambert’s problem, using a time-explicit analytical solution of the relative motion problem. Explicitly stated, the problem is to determine the initial velocity of a satellite, given two position vectors and a time of flight between them. Although the classical origin of the problem related to the determination of planetary orbits, present-day applications are in the area of intercept and rendezvous guidance, where the initial velocity of a maneuvering spacecraft for which its position will coincide with a target position at a specified time must be determined. A short history of Lambert’s problem is given in Ref. 1. Numerical solutions of the problem have been developed by Battin and Vaughan1,2 and Gooding3 for example; however, for small separation distances, such as those encountered in many intercept and rendezvous problems, analytical solutions may be determined using a relative motion formulation. A closed-form solution of Lambert’s problem has been developed previously by Clohessy and Wiltshire in Ref. 4 and has been applied to a variety of intercept and rendezvous problems, as in Refs. 5–10, for example. This solution results from a linear approximation of the equations of motion, and thus the validity range is limited. A higher-order approximation of the equations of motion may extend the range of applicability and lead to an increased level of accuracy. There have been several attempts at developing relative motion equations that take nonlinear dynamics into account.11−14 The solutions presented in Refs. 11–14 result from a straightforward expansion, a method that constructs a truncated Taylor series representation of the exact solution. Anthony and Sasaki,12 Kechichian,13 and Kelly14 show how the nonlinear relative motion equations may be used to solve Lambert’s problem. While these solutions are more accurate than the solutions resulting from the linearized model, the validity region is limited to short periods of time due to the presence of secular terms. For a fixed time, solutions derived from the straightforward expansion will converge to the value of the exact solution as the number of terms in the expansion grows. For a fixed number of terms, however, the accuracy is diminished as time increases and the secular terms begin to dominate the solution. Karlgaard and Lutze15 used a multiple-scale perturbation method to determine a solution of the relative motion problem that included the effects of second-order terms kept in the expansion of the Keplerian equations of motion about a circular reference orbit. The multiple-scale approach allowed elimination of the secular terms by determination of the appropriate differential equations that