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

Carina Veil; Moritz Flaschel; Ellen Kuhl; Cosimo Della Santina

cs.ROFeb 20, 2026

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

Authors: Carina Veil, Moritz Flaschel, Ellen Kuhl, Cosimo Della Santina

Organizations: Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA · Institute of Applied Mechanics, Friedrich-Alexander-Universität Erlangen–Nürnberg, 91058 Erlangen, Germany · Cognitive Robotics Department, Delft University of Technology, 2628 Delft, The Netherlands

Abstract

For fully actuated rigid robots, kinematic inversion is a purely geometric problem, efficiently solved by closed-loop inverse kinematics (CLIK) schemes that compute joint velocities to position the robot body in space. For underactuated soft robots, however, not all configurations are attainable through control action, making kinematic inversion extremely ill-posed. Extensions of CLIK 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 formulate CLIK in the infinite-dimensional domain to reason about the entire soft robot shape while solving tasks, thereby skipping intermediate approximation steps. 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 complex soft-robot models are rarely available in closed form, and neither the actuation-to-shape mapping nor its Jacobian can typically be obtained analytically, we propose to learn it using differentiable neural operator networks. 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.

Figures & tables

Fig. 1 : (Neural) Infinite-dimensional closed-loop inverse kinematics (CLIK) : We propose an extension of finite-dimensional CLIK that allows us to 1) skip intermediate approximation steps, and 2) reason on the whole soft robot shape while solving tasks. Since analytical expressions of the Jacobian are difficult to obtain in practice for infinite-dimensional soft robot models, we embed a neural version of the algorithm that uses the neural operator of the robot learned from simulations.

Fig. 2 : Block diagram of the infinite-dimensional closed-loop inverse kinematics ( 23 ) in actuation space $q_{\text{a}}\in\mathbb{R}^{m}$ , with a square actuation-to-task mapping. Trapezoids emphasize the change in dimensionality from input to output.

Fig. 3 : While standard tasks usually focus to position the end-effector $r(\bar{s}=1)$ , the continuum nature of the soft robot invites for just reaching a target point with the closest point $r(s_{*})$ on its backbone, where $s_{*}$ is adapted in each iteration.

Fig. 4 : Constant curvature segment: robot evolution and corresponding discrete CLIK iterations (

1000

steps with

dt=0.001

) of the actuation variable

q

and the task variable

x

for the point-reaching task

\Phi_{\text{pos}}^{\text{fixed}}

(top) and the closest-point task

\Phi_{\text{opt}}^{\text{fixed}}

(bottom).

Fig. 5 : Three-fiber soft robotic arm [ 28 , 26 ] : The orientation of the fibers is inspired by the muscular structure of the elephant trunk, with one straight fiber and a helical fiber pair, enabling continuous deformations and optimal reachability.

Fig. 6 : Three-fiber soft robotic arm. Robot evolution ( ) and time evolutions of the actuation variable ( activations $\gamma_{i}$ ) and the task variable $x$ for the fixed point-reaching task $\Phi_{\text{pos}}^{\text{fixed}}$ (left), closest-point task $\Phi_{\text{pos}}^{\text{opt}}$ (center), and distributed task $\Phi_{\text{cen}}$ (right). For the robot evolution, we plot every 10th shape. The axes of the shape plot are multiplied by 100 for better readability. For the distributed task, the tip position is highlighted in yellow to better track the trajectory

Architecture and Training
Optimizer	Adam
Activation	tanh
Learning rate	Exponential decay
Epochs	500
Batch size	32
Branch layers	[3, 64, 64, 64, 192]

TABLE I : Neural operator hyperparameters and performance.

Symbol	Meaning
$m$	Degrees of actuation
$p$	Task dimension
$X$	Hilbert space of robot shapes
$q_{\text{a}}\in\mathbb{R}^{m}$	Actuation coordinates
$x\in\mathbb{R}^{p}$	Task coordinates
$r(s)\in X$	Robot shape

TABLE II : Notation overview.

Inverse Kinematics

Inverse kinematics maps desired robot or articulated-body motions to joint configurations while handling constraints, redundancy, and multiple solutions.

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LOInK: Learned Optimal Inverse Kinematics via Structured Neural Surrogate Models

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Australian Robotic Inspection and Asset Management (ARIAM) Hub, the Australian Centre for Robotics (ACFR), and the School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Australia

Inverse Kinematics Robot Kinematics Latent Space Optimization

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We introduce Learned Optimal Inverse Kinematics (LOInK), a method to generate approximately optimal solutions to an inverse kinematics problem. When trained on data consisting of sampled configurations and associated task variables and a given cost function, LOInK learns a bi-Lipschitz invertible mapping from configuration space to a decoupled task/latent space, and moreover, the latent space is structured so as to place cost-minimizing solutions at the origin. This enables efficient sampling of cost-minimizing solutions via a network-inversion algorithm based on operator splitting. We demonstrate the proposed approach on three problems: an illustrative three degree-of-freedom manipulator problem; a quadrupedal climbing robot for which LOInK can generate near-optimal solutions on average 31 times faster and up to 100 times faster than a constrained optimization approach; and a simulated soft actuator as a purely data-driven example, in which LOInK can explicitly generate high-quality solutions, unlike existing generative approaches that require diverse sampling and evaluation of candidate solutions.

Abstract

Inverse Kinematics

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