State Transition Matrices for Terminal Rendezvous Studies: Brief Survey and New Example

Abriefsurvey and classie cation of much of the published material on linearized rendezvous is presented. Thisis followed by a new form of solution of the terminal rendezvous problem that is valid in a general central force e eld. This solution and the solution of the related adjoint system are used to construct a general state transition matrix. Because of the generality of the assumptions, this state transition matrix is very concise and e exible. Finally, the work is applied to the problem of terminal rendezvous near any Keplerian orbit in a Newtonian gravitational e eld using the Tschauner ‐Hempel equations. Because this solution is presented in terms of the true anomaly, considerable care is taken to avoid the types of singularities that are typical in this kind of problem. The result is a state-transition matrix for linearized rendezvous studies that is thought to be simpler and more convenient than other versions found in the literature. I. Introduction L INEARIZED equations of motion are useful in describing the terminal rendezvous phase of a mission or in satellite station keeping. These areas of astrodynamics are rich in the variety of linearizations available to investigators and in the resulting mathematical analysis and computations that follow. This paper presents a brief survey of the types of linear models found in rendezvous studies, followed by a new general model that incorporates much previous work as special cases. The work combines some ideas in 19th century celestial mechanics with some recent discoveries. Because this new model assumes a general central force gravitational e eld,muchofthecomplexitiesfoundinspecie ccasesareavoidedin designingarelativelysimplestatetransitionmatrixthatisapplicable to a variety of problems. The work is then applied to the important special case of linearized rendezvous in a gravitational e eld dee ned by an inverse square law using the Tschauner ‐Hempel equations. In fact, it was this problem that motivated the study. The search for a solution devoid of singularities, valid for any Keplerian orbit, that avoids universal functions can lead to a cluttered set of equations. The new general model avoids much of this clutter and presents the results in a relatively concise form.