Physics-guided residual inverse-transition modeling for continuum robots with simulation and ordered experimental evaluation

Abstract This study distinguishes two levels that were conflated in the original submission: a tendon-driven mechanical dynamics model used only in simulation, and an ordered one-step inverse-transition estimator used for the recorded experiment. The experimental estimator does not identify force-level inverse dynamics because the supplied dataset contains no verified sampling interval, velocity, acceleration, tendon tension, actuator torque, pneumatic pressure, or contact-force channel. Instead, it predicts the next measured six-channel length-state increment from the present Cartesian state, desired Cartesian increment, and a structured representation of the measured length state. The first-order prior is a global empirical differential map, $$\widehat{{\bf J}}\in \mathbb {R}^{3\times 6}$$ , estimated only from the training block; it has rank 3, condition number 3.49, and singular values 1.425, 1.317, and 0.409. Reviewer-driven ablation further showed that the Clarke-like $$3\times 3$$ transform is a full-rank coordinate change rather than a dimensionality reduction, that appending both raw channels and deterministic modes reweights Gaussian-kernel distances, and that the one-step history term does not improve the held-out experimental prediction. The revised experimental instantiation therefore retains only the four bending coordinates $$(q_{\alpha ,1},q_{\beta ,1},q_{\alpha ,2},q_{\beta ,2})$$ in the nonlinear residual learner and excludes both the common mode and duplicated raw/mode features. On the chronologically held-out 39-transition test block, this bending-mode residual KRR obtains a task-space RMSE of 0.365 mm and a cable/length-state RMSE of 0.745 mm, compared with 0.469 mm for a validation-tuned direct KRR, 0.596 mm for ridge regression, 0.654 mm for the NARX-feature model, and 0.795 mm for the global empirical prior. A four-sample moving-block bootstrap gives a within-trajectory 95% interval of approximately 0.282–0.437 mm for the revised task-space RMSE. A paired sign-randomization test against direct KRR gives $$p\approx 0.060$$ , so the small test block does not justify a claim of statistically conclusive superiority over that strongest nonlinear baseline. The independently learned forward evaluator has a test component RMSE of 0.139 mm and vector RMSE of 0.240 mm, but its larger validation error also reveals chronological distribution shift. Accordingly, the revised claims are restricted to free-space, one-step inverse-transition prediction on the recorded trajectory family; no cross-speed, cross-platform, force/contact, or closed-loop hardware generalization is claimed.