A Fast Geometric Framework for Dynamic Cosserat Rods with Discrete Actuated Joints

Current dynamical models of Cosserat rods often use the finite element method limited by computational efficiency or the finite difference method in a Cartesian framework with a compromise to accuracy. We employ the finite difference method in a geometric framework to develop solutions that are both computationally efficient and accurate. A numerical study is conducted on various backward-differentiation discretization and Runge-Kutta-Munthe-Kaas integration schemes, focusing on their accuracy and computational efficiency. Case studies are conducted on a single-degree-of-freedom joint actuated Cosserat rod to mitigate additional sources of undesired error from the numerical analysis, e.g. multi-body interactions, moving base dynamics, etc. The proposed geometric integrators are demonstrated to improve solution accuracy compared to the published finite difference models. The presented solution is parameterization-free and also computationally efficient with the potential for use in real-time applications, e.g., model-based control of soft manipulators.