Solving Cosserat Rod Models via Collocation and the Magnus Expansion

TL;DR

This paper introduces a novel numerical method combining collocation and the Magnus expansion to efficiently solve Cosserat rod models, balancing accuracy and computational speed for continuum robot simulations.

Contribution

It proposes a new approach that integrates orthogonal collocation with the Magnus expansion for solving Cosserat rod equations more efficiently.

Findings

The method guarantees convergence with a maximum step size bound.

Simulation results show improved speed and accuracy tradeoffs.

The approach is competitive with existing methods in computational efficiency.

Abstract

Choosing a kinematic model for a continuum robot typically involves making a tradeoff between accuracy and computational complexity. One common modeling approach is to use the Cosserat rod equations, which have been shown to be accurate for many types of continuum robots. This approach, however, still presents significant computational cost, particularly when many Cosserat rods are coupled via kinematic constraints. In this work, we propose a numerical method that combines orthogonal collocation on the local rod curvature and forward integration of the Cosserat rod kinematic equations via the Magnus expansion, allowing the equilibrium shape to be written as a product of matrix exponentials. We provide a bound on the maximum step size to guarantee convergence of the Magnus expansion for the case of Cosserat rods, compare in simulation against other approaches, and demonstrate the tradeoffs between speed and accuracy for the fourth and sixth order Magnus expansions as well as for different numbers of collocation points. Our results show that the proposed method can find accurate solutions to the Cosserat rod equations and can potentially be competitive in computation speed.