Under-actuated Controllability for Spacecraft Rendezvous

In this report, we examine the controllability of a particular form of the equations of motion for spacecraft formation flying. These equations, the Tschauner-Hempel equations, rescale the formation flying equations to a domain in which the true anomaly is the independent variable. Using this form, we are able to compute an explicit, closed-form Gramian matrix for the period of one full orbit at arbitrary eccentricity. We do this for two cases: 1) the case in which there are three inputs to the system as well as 2) the restricted case where authority only exists in the in-track and cross-track directions. This Gramian is invertible and as a result the system is controllable for both cases. Since the transformation between the time-domain, linear equations of motion and the Tschauner-Hempel equations is bijective, we conclude that the linear equations of motion are also controllable.