Optimal Fiducial Marker Placement for Satellite Proximity Operations Using Observability Gramians

This paper investigates optimal fiducial marker placement on the surface of a satellite performing relative proximity operations with an observer satellite. The absolute and relative translation and attitude equations of motion for the satellite pair are modeled using dual quaternions. The observability of the relative dual quaternion system is analyzed using empirical observability Gramian methods. The optimal placement of a fiducial marker set, in which each marker gives simultaneous optical range and attitude measurements, is determined for the pair of satellites. A geostationary flyby between the observing body (chaser) and desired (target) satellites is numerically simulated and the optimal fiducial placement sets of five and ten on the surface of the desired satellite are solved. It is shown that the optimal solution maximizes the distance between fiducial markers and selects marker locations that are most sensitive to measuring changes in the state during the nonlinear trajectory, despite being visible for less time than other candidate marker locations. Definitions and properties of quaternions and dual quaternions, and parallels between the two, are presented alongside the relative motion model.