A New Approach to Impulsive Rendezvous near Circular Orbit

TL;DR

This paper introduces a novel closed-form method for optimal impulsive spacecraft rendezvous near circular orbits, simplifying calculations and providing explicit conditions for the number of impulses needed.

Contribution

It develops a new optimization approach using a characteristic-value function and primer-vector theory, enabling easier computation of optimal velocity increments.

Findings

Closed-form solution for optimal impulsive rendezvous

Explicit conditions for three-impulse solutions

Practical example demonstrating the method

Abstract

A new approach is presented for the problem of optimal impulsive rendezvous of a spacecraft in an inertial frame near a circular orbit in a Newtonian gravitational field. The total characteristic velocity to be minimized is replaced by a related characteristic-value function and this related optimization problem can be solved in closed form. The solution of this problem is shown to approach the solution of the original problem in the limit as the boundary conditions approach those of a circular orbit. Using a form of primer-vector theory the problem is formulated in a way that leads to relatively easy calculation of the optimal velocity increments. A certain vector that can easily be calculated from the boundary conditions determines the number of impulses required for solution of the optimization problem and also is useful in the computation of these velocity increments. Necessary and sufficient conditions for boundary conditions to require exactly three nonsingular non-degenerate impulses for solution of the related optimal rendezvous problem, and a means of calculating these velocity increments are presented. If necessary these velocity increments could be calculated from a hand calculator containing trigonometric functions. A simple example of a three-impulse rendezvous problem is solved and the resulting trajectory is depicted.