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 (a) explores if and when a control designer has flexibility in this choice, (b) develops a canonical choice related to the Christoffel symbols, and (c) describes how to efficiently perform control computations with it for constrained mechanical systems. 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 the associated factorization. In particular, the factorization based on the Christoffel symbols is linked with a torsion-free property that can limit the twisting of system trajectories during passivity-based control. We then develop a way to induce Coriolis factorizations for constrained mechanisms from unconstrained ones, which provides a pathway to use the theory for efficient control computations with high-dimensional systems such as humanoids and quadruped robots with open- and closed-chain mechanisms. A collection of algorithms is provided (and made available open source) to support the recursive computation of passivity-based control laws, adaptation laws, and regressor matrices in future applications.