Rigid body equations of motion for modeling and control of spacecraft formations. Part 1: Absolute equations of motion

In this paper, we present a tensorial (i.e., coordinate-free) derivation of the equations of motion of a formation consisting of N spacecraft each modeled as a rigid body. Specifically, using spatial velocities and spatial forces we demonstrate that the equations of motion for a single free rigid body (i.e., a single spacecraft) can be naturally expressed in four fundamental forms. The four forms of the dynamic equations include (1) motion about the system center-of-mass in terms of absolute rates-of-change, (2) motion about the system center-of-mass in terms of body rates of change, (3) motion about an arbitrary point fixed on the rigid body in terms of absolute rates-of-change, and (4) motion about an arbitrary point fixed on the rigid body in terms of body rates-of-change. We then introduce the spatial Coriolis dyadic and discuss how a proper choice of this non-unique tensor leads to dynamic models of formations satisfying the skew-symmetry property required by an important class of nonlinear tracking control laws. Next, we demonstrate that the equations of motion of the entire formation have the same structure as the equations of motion of an individual spacecraft. The results presented in this paper form the cornerstone of a coordinate-free modeling environment for developing dynamic models for various formation flying applications.