Duality of the Existing Geometric Variable Strain Models for the Dynamic Modeling of Continuum Robots

The Cosserat rod theory has become a gold standard for modeling the statics and dynamics of serial and parallel continuum robots. Recently, a weak form of these Cosserat rod models called the geometric variable strain model has been derived where the robot deformations are projected on finite-dimensional basis functions. This model has very interesting features for continuum robotics, such as a Lagrangian form close to classical rigid robots and the ability to tune its performances in terms of computation time and accuracy. Two approaches have been proposed to obtain and compute it. The first is based on the Newton-Euler recursive algorithm and the second, on the projection of the strong form equations using Jacobian matrices. Although these approaches yield identical model forms, their disparate implementations and numerical schemes render each uniquely suited to specific applications. Notably, underlying these disparities lies a profound duality between these models, prompting our quest for a comprehensive overview of this duality along with an analysis of their algorithmic differences. Finally, we discuss perspectives for these two approaches, in particular their hybridization, based on the current knowledge of rigid robotics.