  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.
