Numerical Methods to Compute the Coriolis Matrix and Christoffel Symbols for Rigid-Body Systems

TL;DR

This paper introduces efficient numerical algorithms for computing the Coriolis matrix and Christoffel symbols in rigid-body systems, enabling high-speed calculations suitable for real-time control without symbolic derivatives.

Contribution

The authors develop recursive, purely numerical algorithms for Coriolis and Christoffel computations with optimal complexity, suitable for high-rate control applications.

Findings

Algorithms achieve computation times of 10-20 μs for Coriolis matrix.

Christoffel symbols computed in 40-120 μs for 20-DOF systems.

Feasible for real-time control at over 1kHz update rates.

Abstract

This article presents methods to efficiently compute the Coriolis matrix and underlying Christoffel symbols (of the first kind) for tree-structure rigid-body systems. The algorithms can be executed purely numerically, without requiring partial derivatives as in unscalable symbolic techniques. The computations share a recursive structure in common with classical methods such as the Composite-Rigid-Body Algorithm and are of the lowest possible order: $O(Nd)$ for the Coriolis matrix and $O(Nd^2)$ for the Christoffel symbols, where $N$ is the number of bodies and $d$ is the depth of the kinematic tree. Implementation in C/C++ shows computation times on the order of 10-20 $\mu$s for the Coriolis matrix and 40-120 $\mu$s for the Christoffel symbols on systems with 20 degrees of freedom. The results demonstrate feasibility for the adoption of these algorithms within high-rate ($>$1kHz) loops for model-based control applications.