A Fused Approach of Lie Group SE2(3) and Hill-Clohessy-Wiltshire Optimization for Relative Navigation of a Group of Small Objects.

This research presents a performance analysis of a hybrid navigation approach combining Lie group theory and the Hill Clohessy Wiltshire (HCW) equations for managing a swarm of small spatial objects. Our study addresses the challenges of precise relative navigation by leveraging the geometric structure of the SE₂(3) Lie group coupled with the linearized orbital dynamics of the Hill Clohessy Wiltshire model. Using the MATLAB environment, we developed a unified mathematical model that integrates complex rigid transformations and orbital proximity constraints, providing a robust framework for trajectory tracking and collision avoidance between objects.
The research methodology in this work was structured around a rigorous three-step approach. First, we formulated the navigation state on SE₂(3) to naturally capture the coupling between rotation and translation of objects. Secondly, we merged this framework with Hill Clohessy Wiltshire optimization to linearize relative perturbations around a reference orbit, thus enabling optimal state estimation. Finally, this merged approach was validated by a high-fidelity simulation involving a multi-object deployment, taking into account sensor measurement errors and environmental perturbations. Our results demonstrate that while the traditional Hill Clohessy Wiltshire model excels in simple proximity calculations, merging it with optimization on SE₂(3) significantly reduces singularity errors and improves navigation convergence during wide angle maneuvers.