Infinite-Dimensional Closed-Loop Inverse Kinematics for Soft Robots via Neural Operators

Abstract: While kinematic inversion is a purely geometric problem for fully actuated rigid robots, it becomes extremely challenging for underactuated soft robots with infinitely many degrees of freedom. Closed-loop inverse kinematics (CLIK) schemes address this by introducing end-to-end mappings from actuation to task space for the controller to operate on, but typically assume finite dimensions of the underlying virtual configuration space. In this work, we extend CLIK to the infinite-dimensional domain to reason about the entire soft robot shape while solving tasks. We do this by composing an actuation-to-shape map with a shape-to-task map, deriving the differential end-to-end kinematics via an infinite-dimensional chain rule, and thereby obtaining a Jacobian-based CLIK algorithm. Since the actuation-to-shape mapping is rarely available in closed form, we propose to learn it from simulation data using neural operator networks, which are differentiable. We first present an analytical study on a constant-curvature segment, and then apply the neural version of the algorithm to a three-fiber soft robotic arm whose underlying model relies on morphoelasticity and active filament theory. This opens new possibilities for differentiable control of soft robots by exploiting full-body shape information in a continuous, infinite-dimensional framework.