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This paper examines the local exponential stability (LES) of trajectories fornonlinear systems on Riemannian manifolds. We present necessary and sufficientconditions for LES of a trajectory on a Riemannian manifold by analyzing thecomplete lift of the system along the given trajectory. These conditions arecoordinate-free which reveal fundamental relationships between exponentialstability and incremental stability in a local sense. We then apply theseresults to design tracking controllers and observers for Euler-Lagrangiansystems on manifolds; a notable advantage of our design is that it visiblyreveals the effect of curvature on system dynamics and hence suggestscompensation terms in the controller and observer. Additionally, we revisitsome well-known intrinsic observer problems using our proposed method, whichlargely simplifies the analysis compared to existing results.

Stability Analysis of Trajectories on Manifolds with Applications to Observer and Controller Design