Deep Geometric Potential Functions for Tracking on Manifolds

In this paper, we introduce a novel approach for designing invariant control laws through potential functions for fully actuated dynamical systems evolving on manifolds by leveraging the power of neural networks. The geometry and non-linearity inherent to manifold-based dynamical systems pose challenges for traditional control law design, necessitating techniques with the interplay of differential geometry and dynamical systems for ensuring stability. Apart from stability, performance and optimality are other challenging areas to address for dynamical systems evolving on manifolds. On top of these, the concept of invariance helps us improve learning transferability skills from one scene to another scene. We propose invariant potential functions on manifolds defined by neural networks that can be used to generate elastic forces for asymptotic tracking of trajectories. The weights of the potential function can be tuned to shape the potential functions according to the performance requirements through minimizing a loss function.