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

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(Nd2) 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 of the order of 10–20 μs for the Coriolis matrix and 40–120 μs for the Christoffel symbols on systems with 20-degrees-of-freedom (DoF). The results demonstrate feasibility for the adoption of these algorithms within high-rate (>1 kHz) loops for model-based control applications.