Optimal Fiducial Marker Placement for Satellite Proximity Operations Using Observability Gramians

TL;DR

This work addresses optimal fiducial marker placement on a target satellite to enhance relative state estimation during proximity operations. It employs a dual-quaternion pose model and an empirical observability Gramian to quantify how marker configurations influence observability under nonlinear dynamics, solving a relaxed semidefinite program to select markers from 54 candidates. In a geostationary flyby simulation, two marker sets (5 and 10) are optimized, revealing that placing markers to maximize inter-marker distance and prioritizing the $+\bar{i}_D$ face improves observability even when those locations are visible for shorter durations. The results demonstrate the practical value of observability-driven sensor placement for robust autonomous relative navigation in satellite teams, with implications for docking, inspection, and servicing missions, and point to future work validating performance via Kalman filtering and real-world sensor effects.

Abstract

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.

Figures (6)

• Figure 1: Model of relative spacecraft system.
• Figure 2: AprilTag fiducial marker and tag coordinate frame $T$Honigmann2020.
• Figure 3: Nominal flyby trajectory from +R/+I to -R/+I.
• Figure 4: Feasible fiducial marker set on the desired spacecraft.
• Figure 5: Visible fiducial marker set on the desired spacecraft.