Curvature shaping control of nonlinear mechanical systems

This paper introduces a curvature-shaping approach for stabilizing nonlinear mechanical systems. The procedure involves modifying the metric to change curvature, thus making the geodesics corresponding to the closed-loop system confined within a prescribed region. The control laws realizing curvature shaping are derived for fully actuated and underactuated systems, respectively. First, a criterion for Jacobi stability is given in terms of the sectional curvature. The feedback control law is then obtained for fully actuated systems on locally conformally flat Riemannian manifolds. The control law manifests as a physically realizable potential field and generates a “curvature well” with a boundary possessing a curvature singularity. Next, a control law for an underactuated system is constructed to make the controlled horizontal metric negative definite and the curvature at the equilibrium point positive. Specific matching conditions are needed to realize the control. Finally, the effectiveness of the proposed approach is demonstrated through the stabilization of the cart-pendulum system. The curvature shaping method reduces stability verification to a scalar sectional curvature, facilitates intuitive parameter tuning, and suggests a promising geometric framework for robust control design in robotics and aerospace systems.