Coriolis Factorizations and their Connections to Riemannian Geometry

TL;DR

The work develops a geometric framework that links Coriolis factorizations in mechanical systems to affine connections on the configuration manifold, showing that a canonical, Christoffel-symbol-based factorization yields a torsion-free property beneficial for passivity-based control. It proves that, while many factorizations exist for $n\ge3$, the Christoffel-consistent choice provides desirable structure, and it demonstrates how to induce factorizations for constrained systems from unconstrained ones via a projection formula $\overline{C}=A^\top C A + A^\top H \dot A$. The paper then provides practical, open-source algorithms for computing these factorizations efficiently in high-DoF systems, including open-chain and closed-chain mechanisms, with complexity analyses like $\mathcal O(N d)$ for open chains and $\mathcal O(N d^2)$ for constrained/embedded cases. Practically, these results enable scalable, geometry-preserving control computations for complex robots (e.g., humanoids, quadrupeds) while preserving passivity properties essential for robust interaction and adaptation.

Abstract

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.

Figures (8)

• Figure 1: Roadmap. For a mechanical system on a Riemannian manifold $\mathcal{Q}$ with kinetic energy metric $\mathbb{G}$ we study how the properties of any chosen affine connection $\nabla$ relate with those of a corresponding Coriolis matrix $C$ and mass matrix $H$ w.r.t. a choice of generalized velocities $v$. We show how these properties for the system on $\mathcal{Q}$ can be related to a constrained system that evolves on a submanifold $\overline{\mathcal{Q}}$ where the dynamics are described with generalized velocities $\overline{v}$ that are related to $v$ by $v = A \overline{v}$. Bolded results indicate the main contributions.
• Figure 2: Motion of a particle on a frictionless surface in the absence of gravity. Equivalently, a geodesic on the surface. The particle acceleration is always normal to the surface (since constraint forces are normal to it). Equivalently, the covariant derivative of its velocity vector w.r.t. the Riemannian connection is zero along the curve.
• Figure 3: We consider parallel transport along the geodesics of four different affine connections. In each case, three vectors are transported -- they start mutually orthogonal, and the geodesic is considered along the initial red vector. The connection is compatible with the Euclidean metric (left column) if the coordinate system remains orthonormal throughout. This notion of being compatible with the metric is distinct from the geodesics of the connection being the same as those of the Euclidean metric (top row). The upper left corner shows a connection with torsion. In this case, the unique torsion-free metric-compatible connection would instead keep the coordinate system at a constant orientation.
• Figure 4: Point mass state trajectories, torsion-free.
• Figure 5: Point mass state trajectories, non-torsion-free.

Theorems & Definitions (29)

• Remark 1
• Proposition 3.1: Metric-Compatible Connections
• proof
• Theorem 3.1: Riemannian Connection, do1992riemannian
• Remark 2
• Proposition 3.2: Geodesic Agreement
• Theorem 3.2
• proof
• Corollary 3.1
• Corollary 3.2