Variational Estimation for Mechanical Systems on Lie Groups based on Geometric Mechanics

Geometric mechanics analyzes mechanical systems in the framework of variational mechanics, while accounting for the geometry of the configuration space. From the late 1970s, developments in this area produced several schemes for geometric control of mechanical systems in continuous time and discrete time. In the mid to late 2000s, geometric mechanics was first applied to state estimation of mechanical systems, particularly systems evolving on Lie groups as configuration manifolds, like rigid body systems. Much of the existing work on geometric mechanics-based estimation has been in continuous time, using deterministic, semi-stochastic and stochastic approaches. While the body of existing literature on discrete-time estimation schemes on Lie groups is not as extensive, the literature on this topic is contemporaneous with continuous-time schemes. This work describes some recent and ongoing research on geometric mechanics-based estimation schemes in continuous and discrete time from the mid-2010s, which were developed using the Lagrange-d’Alembert principle applied to rigid body systems. This approach gives (deterministic) observer designs with strong stability and robustness properties. This work concludes with potential extensions of this approach to mechanical systems with principal fiber bundles as configuration manifolds.