A Probabilistic, Rotation-Preserving Kernel Operator Surrogate for Continuum-Robot Design Spaces

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.