A Class of Optimal Two-Impulse Rendezvous Using Multiple-Revolution Lambert Solutions
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A Class of Optimal Two-Impulse Rendezvous Using Multiple-Revolution Lambert Solutions 1 John E. Prussing' Abstract In minimum-fuel impulsive spacecraft trajectories, long-duration coast arcs between thrust impulses can occur. If the coast time is long enough to allow one or more complete revolutions of the central body, the solution becomes complicated. Lambert's problem, the determination of the orbit that connects two specified terminal points in a specified time interval, affords multiple solutions. For a transfer time long enough to allow N revolutions of the central body there exist 2N + 1 trajectories that satisfy the boundary value problem. An algorithm based on the classical Lagrange formulation for an elliptic orbit is developed that determines all the trajectories. The procedure is applied to the problem of rendezvous with a target in the same circular orbit as the spacecraft. The minimum-fuel optimality of the two-impulse trajectory is determined using primer vector theory.
Introduction The classical orbit boundary-value problem known as Lambert's problem can be thought of as both an orbit determination problem and a spacecraft targeting problem. Its solution in the two-body problem is the conic orbit that connects two specified terminal points in a specified time interval. Battin [1] provides an detailed analysis of the problem along with historical references to fundamental work by Lagrange (1778) and Gauss (1809). Additional research on methods of solution by Lancaster, Blanchard, and Devaney [2], Sun, Vinh, and Chern [3], Battin and his student Robin Vaughan [4], and Gooding [5] has also been published more recently. The multiple-revolution Lambert's problem is discussed in references [1], [3], and [5], using various formulations of the problem, and has also been analyzed by another of Battin's students, Laura Loechler [6]. The present study of the multiple'Presented as Paper AAS 00-250 at the AAS Richard H. Battin Astrodynamics Symposium, College Station, Texas, March 2000. 2p rofessor, Department of Aeronautical and Astronautical Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801-2835. 131
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revolution case is based on references [7] and [8] which utilize the classical Lagrange formulation of Lambert's problem for an elliptic orbit. This formulation is used because of the ease of describing and displaying the transfer time as a function of semimajor axis, and because the elliptic orbit is the only conic orbit that affords multiple-revolution solutions. The determination of the maximum number of revolutions N for a specified transfer time and all 2N + 1 of the orbit solutions can be readily obtained. This formulation is, however, not the most efficient computationally. The Battin-Vaughan algorithm mentioned above is a highly efficient solution algorithm.
Multiple-Revolution Lambert's Problem Formulation The basic orbital geometry is shown in Fig. 1. The two terminal radius vectors rl and r2 locate the terminal points PI and P2 relative to the focus of the elli
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