New Approach to Satellite Formation-Keeping: Exact Solution to the Full Nonlinear Problem

Abstract

This paper presents a new, simple, and exact solution to the formation keeping of satellites when the relative distance between the satellites is so large that the linearized relative equations of motion no longer hold. We employ a recently proposed approach, the Udwadia-Kalaba approach, which makes it possible to explicitly obtain the desired control function without making any approximations related to the nonlinearities in the underlying dynamics. We use an inertial frame of reference to describe the motion of a satellite and since no approximations are made, the results obtained apply to situations even when the distance between the satellites is arbitrarily large. The paper deals with a projected circular formation, but the methodology in this paper can be applied to any desired configuration or orbital requirements. Numerical simulations confirm the brevity and the accuracy of the analytical solution to the dynamical control problem developed herein.

Methodology for Satellite Formation-Keeping in the Presence of System Uncertainties

“…It is also assumed that the position vector of the center of mass of the ith follower satellite in the Hill frame [17] is given by x i y i z i T , and its orientation is described by the quaternion…”

“…Because of its remarkable simplicity and applicability, several researchers have used it in fields like robotics [14], modeling of complex multibody systems [15,16], and also in solving the formationkeeping problem. Cho and Yu [17] obtained an analytical solution for formation keeping when the leader satellite is in an unperturbed circular orbit. They did not, however, consider any attitude dynamics.…”

Explicit solution to the full nonlinear problem for satellite formation-keeping

“…For example, in this paper we have so far used two (that is, p =2) constraints given by Eqs. (25) and (26) …”

“…Here, in order to stress the strength of this paper we set more complex and specific constraints instead of the ones in Eqs. (25) and (26) so that the followers execute a circular orbit of radius r=50 km around the leader with a prescribed angular frequency o in the Hill frame. In addition, we add one more constraint: as seen in the Hill frame at all times, the two followers must be located diametrically opposite ARTICLE IN PRESS H. Cho, F.E.…”

“…The initial conditions used are the ones that exactly satisfy the given configuration constraints and the problem is solved in the inertial coordinate frame of reference. Cho and Yu [26] obtain a solution to the formation-keeping problem of multiple satellites circling a spherical Earth. They constrain the relative distances of the formation in the Hill frame, and also use the Udwadia-Kalaba approach.…”