Abstract
Primer vector theory is used in analysing a set of linear relative‐motion equations—the Clohessy‐Wiltshire (C/W) equations—to determine the criteria and necessary conditions for an optimal N‐impulse trajectory. Since the state vector for these equations is defined in terms of a linear system of ordinary differential equations, all fundamental relations defining the solution of the state and costate equations and the necessary conditions for optimality can be expressed in terms of elementary functions. The analysis develops the analytical criteria for improving a solution by (1) moving any dependent or independent variable in the initial and/or final orbit, and (2) adding intermediate impulses. If these criteria are violated, the theory establishes a sufficient number of analytical equations. The subsequent satisfaction of these equations will result in the optimal position vectors and times of an N‐impulse trajectory.
The solution is examined for the specific boundary conditions of (1) fixed‐end conditions, two‐impulse, and time‐open transfer; (2) an orbit‐to‐orbit transfer; and (3) a generalized rendezvous problem. A sequence of rendezvous problems is solved to illustrate the analysis and the computational procedure.