A Probabilistic, Rotation-Preserving Kernel Operator Surrogate for Continuum-Robot Design Spaces
Authors: Branden Frieden, Alan Kuntz, Varun Shankar
Organizations: The Kahlert School of Computing at the University of Utah, Salt Lake City, UT 84112, USA. · The Departments of Computer Science and Electrical and Computer Engineering at Vanderbilt University, Nashville, TN, 37203, USA.
We develop an uncertainty-aware, operator-learning-based surrogate for tendon-driven continuum robots (TDCRs) that generalizes across robot design and actuation while predicting the complete equilibrium configuration. Our central idea is a nonstationary Gaussian-process (GP) kernel whose metric is learned by a shallow neural network from mechanics-aware design coordinates. The network learns how similarity changes across design space, while the resulting kernel defines the design-to-pose operator, its posterior uncertainty, and a criterion for sequential simulator acquisition. We represent backbone position and rotational strain with spatial kernels and reconstruct orientation directly on SO(3), preserving valid material frames and tendon geometry. We use ASkotch with Nystrom preconditioning for scalable kernel operator solves and a separate Nystrom approximation for posterior covariance. Experiments show that kernel-coverage acquisition reduces mean trajectory error by approximately 15% relative to IID sampling, while posterior variance increases for increasingly unsupported designs. Compared with prior neural-operator TDCR models, the method is 30-50% more accurate using approximately 5x fewer training designs while additionally providing posterior uncertainty estimates.
Figures & tables
Fig. 1: Four-tendon TDCR and the continuous configuration learned by the surrogate. The input contains robot-design and actuation parameters; the output is the backbone/material-frame pose and tendon trajectories.
Fig. 2: Method overview. Mechanics-aware coordinates feed a learned nonstationary kernel metric; the kernel operator learns d↦z(d) , after which structured reconstruction recovers the TDCR pose and uncertainty. The same kernel geometry drives sequential acquisition.
Parameter
Symbol
Range
Tendon load
τi
[ 0, 5 ] N
Robot length
L
[ 0.10, 0.35 ] m
Routing twist rate
ci
[ −20 , 20 ] rad m -1
Routing radius
D
[ 0.005, 0.020 ] m
Young’s modulus
E
[ 15.5, 46.5 ] GPa
Backbone radius
ro
[ 0.0005, 0.0015 ] m
TABLE I: Admissible TDCR design and actuation space used to generate the simulation data.
Fig. 3: Prediction accuracy for ASkotch under different target residuals, with the direct Cholesky solve as the baseline.
Fig. 4: Training time required to reach different ASkotch residual targets as the number of training functions increases. Note that the Cholesky solve was unable to be used past 10,000 training functions due to memory limitations.
Fig. 5: Prediction accuracy for sequential kernel-coverage acquisition compared with IID sampling as the number of training functions increases.
Fig. 6: Design-space posterior variance for different Nyström ranks as the number of training functions increases, compared with the direct Cholesky solve baseline. These ranks were not used in training, only for estimation of the posterior variance
Fig. 7: Design-space posterior variance as the number of training functions increases.
Fig. 8: Design-space posterior variance as a function of the landmark-based OOD score.
Fig. 9: Base-anchored physical-space uncertainty for an example design, showing covariance tubes around the backbone and four tendons and the corresponding covariance ellipsoid at the robot tip.
Continuum robots enable dexterous manipulation in constrained environments, but require accurate and efficient models for real-time manipulation and control. Traditional physics-based models can be computationally expensive and may suffer from inaccuracies due to unmodeled effects, while current learning-based methods often generalize poorly beyond the specific robot on which they are trained. We present a formulation of surrogate modeling for tendon-driven continuum robots as an operator learning problem that maps robot design parameters and tendon actuation inputs to resulting configurations. This formulation enables a single trained model to generalize across a large class of robot designs. We develop four novel neural operator architectures--two based on Deep Operator Networks (DeepONets) and two based on Fourier Neural Operators (FNOs)--and train them on simulation data to predict robot configurations. All architectures achieve good accuracy while allowing for fast and accurate generalization across designs. Our results demonstrate that operator learning provides an effective and generalizable surrogate for continuum robot mechanics in the design space, enabling fast modeling for control, planning, and design optimization in surgical and industrial applications.
Branden Frieden, James M. Ferguson, Alan Kuntz +1
The Robotics Center and the Kahlert School of Computing at the University of Utah, Salt Lake City, UT 84112, USA. · The Departments of Computer Science and Electrical and Computer Engineering at Vanderbilt University, Nashville, TN, 37203, USA.
Many planning and control tasks for tendon-driven continuum robots (TDCRs) require the complete Cartesian backbone geometry. We present a reduced Cartesian framework for planar, axially compressible TDCRs that propagates equilibrium configurations along prescribed tendon-force and tendon-displacement trajectories. The backbone is represented by two global position fields. Following exact variation, a Taylor-Galerkin reduction condenses prescribed spatial properties and distributed loads into offline moment vectors, yielding analytic reduced residuals and Jacobians without online spatial quadrature or numerical differentiation. Analytical differentiation and residual correction yield first-order rate systems requiring one fixed-dimensional linear solve per rate evaluation after initial equilibrium alignment on a regular branch. Across four simulated cases covering variable tendon routing, nonuniform geometry, axial compression, and their combined effects, the propagated Cartesian shapes and distributed strains closely match pointwise geometrically variable-strain (GVS) equilibrium solutions. Residual correction suppresses propagation drift across the tested step sizes while adding only about 0.98% to the mean update time of uncorrected Euler. The proposed method requires 0.508 ms per update on average, approximately 11 times faster than pointwise GVS solves. Displacement-driven experiments yield a maximum normalized mean backbone position error of 1.02% and a maximum end-effector position error of 0.28%. These results support efficient and accurate Cartesian full-shape prediction along prescribed actuation paths.
Predicting the shape of tendon driven continuum robots (TDCRs) at steady state from actuation remains challenging due to continuous deformation, complex tendon routing, compliance, friction, and fabrication variability. In this paper, we address this problem as kinematic self modeling conditioned on action. We present a lightweight 3D printed TDCR hardware platform and an RGB-D data collection pipeline with multiple cameras, and we learn a point cloud flow matching model that maps motor actuation states to the robot's settled 3D geometry. The model is trained from randomly sampled quasi static configurations and evaluated on test motor commands within the same TDCR design family and actuation range. We compare against prior 3D deformable object and robot self modeling approaches in both MuJoCo simulation and real hardware experiments. Experiments on simulated 2-, 3-, and 5-module TDCRs and real 2- and 3-module robots show improved shape prediction accuracy under CD and EMD metrics. We further show in simulation that the same conditional formulation generalizes to tip payload as a conditioning input, enabling payload conditioned steady-state shape prediction. These results demonstrate a data driven self modeling framework for quasi static TDCR geometry prediction.
Jiong Lin, Jinchen Ruan, Hod Lipson
Department of Mechanical Engineering, Columbia University, New York, NY, USA