Coriolis Factorizations and their Connections to Riemannian Geometry

Many energy-based control strategies for mechanical systems require the choice of a Coriolis factorization satisfying a skew-symmetry property. This paper explores (a) if and when a control designer has flexibility in this choice, (b) what choice should be made, and (c) how to efficiently perform control computations with it. We link the choice of a Coriolis factorization to the notion of an affine connection on the configuration manifold and show how properties of the connection relate with ones of the associated factorization. Out of the choices available, the factorization based on the Christoffel symbols is linked with a torsion-free property that limits the twisting of system trajectories during passivity-based control. The machinery of Riemannian geometry also offers a natural way to induce Coriolis factorizations for constrained mechanisms from unconstrained ones, and this result provides a pathway to use the theory for efficient control computations with high-dimensional systems such as humanoids and quadruped robots.