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Choosing a kinematic model for a continuum robot typically involves making atradeoff between accuracy and computational complexity. One common modelingapproach is to use the Cosserat rod equations, which have been shown to beaccurate for many types of continuum robots. This approach, however, stillpresents significant computational cost, particularly when many Cosserat rodsare coupled via kinematic constraints. In this work, we propose a numericalmethod that combines orthogonal collocation on the local rod curvature andforward integration of the Cosserat rod kinematic equations via the Magnusexpansion, allowing the equilibrium shape to be written as a product of matrixexponentials. We provide a bound on the maximum step size to guaranteeconvergence of the Magnus expansion for the case of Cosserat rods, compare insimulation against other approaches, and demonstrate the tradeoffs betweenspeed and accuracy for the fourth and sixth order Magnus expansions as well asfor different numbers of collocation points. Our results show that the proposedmethod can find accurate solutions to the Cosserat rod equations and canpotentially be competitive in computation speed.

Solving Cosserat Rod Models via Collocation and the Magnus Expansion