Techniques for safety critical geometric control

Geometric control provides a natural framework for the analysis and regulation of mechanical systems evolving on nonlinear configuration spaces such as Lie groups. While significant progress has been made in geometric tracking control, the systematic integration of safety constraints in this setting remains comparatively underexplored. In particular, existing Control Barrier Function (CBF) formulations are largely developed for Euclidean systems and often fail to fully exploit the intrinsic geometric structure of rigid-body dynamics. This thesis develops a geometric safety framework for fully actuated mechanical systems evolving on Lie groups, with a particular focus on rigid-body motion on SE(3). After establishing a rigorous differential-geometric foundation and revisiting geometric tracking control laws inspired by F.Bullo’s framework, we introduce three classes of Zeroing Control Barrier Functions (ZCBFs) tailored to Lie-group dynamics. The first class addresses configuration-level constraints and enables obstacle avoidance directly on the manifold. The second class, which constitutes the most novel contribution of this work, introduces directional kinetic-energy constraints expressed in inertial world-frame coordinates, allowing the enforcement of energy bounds along prescribed spatial directions. The third class, formulates directional kinetic-energy constraints in body-frame coordinates. This approach enables the regulation of internal motion characteristics, providing a mechanism for actuator protection and self-preservation through bounded body twists. All proposed safety filters are validated through simulations. Furthermore, the framework is extended to the Riemannian setting and compared with existing geometric CBF constructions, particularly the approach proposed by G.Wu and K.Sreenath. We show that the proposed energy-based filter exhibits passivity properties that are not retained in general by other formulations.