Almost Global Stabilization of Curvature-Nullifiable Mechanical Systems via Pure Mechanical Feedback

Mechanical control systems structure, derived from Euler–Lagrange dynamics, is directly tied to physically meaningful coordinates such as joint angles, positions, and velocities. This work investigates when a mechanical control system can be transformed, without changing its physical coordinates, into an equivalent form whose Christoffel symbols vanish, thereby eliminating the configuration-dependent coupling terms in the inertia matrix. We establish a necessary and sufficient condition under which a mechanical control system can, via pure mechanical feedback, be transformed into an equivalent system with zero Christoffel symbols. For three representative examples of mechanical systems, we extensively discuss the global stabilization problem. These case studies demonstrate, respectively, global linearization; local linearization with singularities that can be globalized through an appropriate switching control strategy; and partial linearization, where eliminating the Christoffel symbols enables the design of a globally stabilizing nonlinear controller for a system that is not fully feedback linearizable. These findings demonstrate that achieving vanishing Christoffel symbols, while preserving physically meaningful coordinates, provides a powerful and broadly applicable tool for addressing complex control problems.