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
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 qa∈Rm , 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(sˉ=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 Φposfixed (top) and the closest-point task Φoptfixed (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 γi ) and the task variable x for the fixed point-reaching task Φposfixed (left), closest-point task Φposopt (center), and distributed task Φcen (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.
Inverse kinematics maps desired robot or articulated-body motions to joint configurations while handling constraints, redundancy, and multiple solutions.
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.
Michael Somerfield, Damian Abood, Ruigang Wang +1
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
Neural models can learn to generate various solutions to the inverse kinematics problem from data, but are usually limited to a single robot. We present MorphIK, a flow-matching model that solves inverse kinematics for revolute-joint-based kinematic chains it has never seen during training. The model uses a transformer architecture to encode the robot's morphology along with the target pose. This encoding then conditions a flow-matching head that generates poses from noise. Trained on purely synthetic data from procedurally generated robots, the model reaches a precision of about 5 cm on unseen real-world robots with 6 to 9 Degrees of Freedom. For higher precision, the model serves as an excellent Prior for further optimization algorithms, reducing error to less than 1 cm after a single step of Damped Least Squares optimization and to sub-1 mm error after 3 steps in most cases. Building on flow matching's generative capabilities to produce highly diverse outputs, our model can efficiently sample the robot's null space, providing a wide variety of configurations for the same pose. Thus, overall, MorphIK allows learning and generalizing neural inverse kinematics for a multitude of known and unknown robots.
Lennart Clasmeier, Jan Gerrit Habekost, Cornelius Weber +1
Knowledge Technology Group, Department of Informatics, University of Hamburg, Hamburg, Germany
Learning the inverse dynamics of soft continuum robots remains challenging due to high-dimensional nonlinearities and complex actuation coupling. Conventional feedback-based controllers often suffer from control chattering due to corrective oscillations, while deterministic regression-based learners struggle to capture the complex nonlinear mappings required for accurate dynamic tracking. Motivated by these limitations, we propose an inverse-dynamics framework for open-loop feedforward control that learns the system's differential dynamics as a generative transport map. Specifically, inverse dynamics is reformulated as a conditional flow-matching problem, and Rectified Flow (RF) is adopted as a lightweight instance to generate physically consistent control inputs rather than conditional averages. Two variants are introduced to further enhance physical consistency: RF-Physical, utilizing a physics-based prior for residual modeling; and RF-FWD, integrating a forward-dynamics consistency loss during flow matching. Extensive evaluations demonstrate that our framework reduces trajectory tracking RMSE by over 50% compared to standard regression baselines (MLP, LSTM, Transformer). The system sustains stable open-loop execution at a peak end-effector velocity of 1.14 m/s with sub-millisecond inference latency (0.995 ms). This work demonstrates flow matching as a robust, high-performance paradigm for learning differential inverse dynamics in soft robotic systems.