A 3D Discrete Elastic Rod Model and Observer for Continuum Robots

Soft and continuum robots have unique advantages and capabilities useful in medical applications and confined environments due to their highly flexible structure. However, their underactuation and theoretically infinite degrees-of-freedom present significant challenges, particularly in control and state estimation. Prior work showed that a 2D discrete rod mechanics model, derived from the continuous partial differential equations of a Kirchhoff rod and expressed in maximal coordinates, can facilitate continuum robot state estimation and feedback-linearization-based control without conversion to a classical minimal robot dynamics form. We here extend that maximal-coordinate modeling approach to 3D, in a symmetric, constrained-Lagrangian form, using quaternions and Baumgarte constraint stabilization. We also formulate a physically intuitive, model-based observer to estimate the full state of a continuum robot by introducing virtual forces and moments. We validate both the open-loop model and the observer through optical tracking in experiments on a large tendon-driven continuum robot prototype, demonstrating the ability to accurately estimate the robot’s dynamic state during actuation.