Robotics: Science and Systems 2026

Stabilizing 3D Continuum-Arm Rollouts via Equilibrium Anchoring and Feature-Lifted Residual Learning

A hybrid predictor that separates steady-state anchoring from transient correction to keep long-horizon rollouts stable under actuation shift.

Ahsan Tanveer1,2, Rahdar Hussain Afridi1,2,3,*, Waqar Hussain Afridi1,2,*,
Feitian Zhang2,4, Guangming Xie1,2,5,†

1Intelligent Biomimetic Design Lab, Peking University  ·  2State Key Laboratory of Turbulence and Complex Systems, Peking University
3FAW Robotics  ·  4Robotics and Control Lab, Peking University  ·  5Institute of Ocean Research, Peking University
*Equal contribution  ·  †Corresponding author

Paper (PDF coming soon) Code & Data Video
Graphical abstract comparing Equilibrium Prior, Residual, and Hybrid predictors under actuation shift

Recursive rollout comparison for a tendon-driven continuum arm. The equilibrium prior stays stable but drifts in steady state; the residual captures transients but can blow up; the hybrid suppresses drift while tracking the ground truth.

Ground Truth Equilibrium Prior Residual Hybrid
26%
Backbone RMSE reduction vs. equilibrium prior
27%
Tip position RMSE reduction
200
Stable rollout steps (6 s) under OOD actuation
0%
Catastrophic failure on Van der Pol benchmark

Abstract

Multi-step motion prediction for continuum robots is difficult, especially under actuation distribution shift, where error accumulation can distort the predicted steady response and destabilize rollouts. This paper introduces a hybrid equilibrium-anchored residual-learning framework for a tendon-driven 3D continuum arm that makes steady behavior explicit. An equilibrium prior is learned from inexpensive static equilibrium data and used in a contractive update that continuously pulls predictions toward the equilibrium estimate, improving rollout stability. A lightweight feature-lifted residual model, linear in parameters, learns the remaining one-step mismatch from dynamic trajectory data, recovering transient dynamics. The approach is validated on 200-step simulation rollouts under stronger and faster actuation than in training, with an additional soft-tail hardware test under actuation-frequency shift. The Hybrid method reduces backbone position RMSE by 26% and tip position RMSE by 27%, producing consistent accuracy gains over prior-only and residual-only predictors while remaining stable across all tested trajectories. The same proposed model also improves robustness on standard nonlinear benchmarks against a combined Koopman baseline under matched evaluation protocols.

Key Contributions

Unlike additive residual-Koopman models, our method explicitly separates steady-state anchoring from transient correction at every rollout step.

1

Equilibrium prior from static data

An actuation-conditioned equilibrium map is learned from inexpensive static equilibrium pairs and used in a contractive anchor update that pulls predictions toward physically plausible steady configurations, suppressing long-horizon drift.

2

Prior-consistent residual learning

A feature-lifted, linear-in-parameters residual is trained on anchored states—matching the distribution seen during open-loop rollout—to recover transient dynamics while retaining a control-friendly structure amenable to stability analysis.

Method

Two datasets, two roles: static equilibrium data D2 for the anchor; dynamic trajectories D1 for the residual.

Step 1 — Equilibrium anchor

Fit a ridge-regression equilibrium map P(u) from 4,520 static equilibrium samples. At each rollout step, contract the predicted state toward P(uk) with strength β = 0.5.

Step 2 — Residual correction

Train a ridge-fit residual R(z) on anchored transitions to capture the one-step mismatch rk☆ = xk+1 − x̄k+1, recovering transient dynamics the anchor alone cannot model.

x̄k+1 = x̂k + β(P(uk) − x̂k)   →   x̂k+1 = x̄k+1 + R([x̄k+1; uk])
Training and rollout framework for equilibrium-anchored residual learning

Framework overview. (a) Training: learn equilibrium prior P from static dataset D2, form anchored states, then fit residual R on dynamic dataset D1. (b) OOD rollout: iterate the anchored update plus residual correction for multi-step prediction.

Experimental Setup

A tendon-driven 3D continuum arm simulated with a Cosserat-rod model. Training uses moderate actuation; OOD testing uses larger amplitudes and higher dominant frequencies.

Tendon-driven continuum arm model and notation

Continuum arm model. Three routing tendons actuate a flexible backbone with spacer disks. The 144-dimensional state captures configuration at six backbone sampling locations.

Results

Five predictors evaluated on identical 6 s (200-step) OOD rollouts with shifted actuation. Hybrid and Neural Hybrid achieve the lowest errors while remaining stable on every trajectory.

Model State RMSE Backbone RMSE (cm) Tip RMSE (cm) Tip Orient. RMSE (°)
Equilibrium Prior 2.513 15.32 26.10 41.63
Neural Prior 2.515 16.08 27.51 34.44
Residual 1.943 12.21 20.62 23.06
Hybrid 1.914 11.34 19.15 21.97
Neural Hybrid 1.900 11.29 19.06 24.20

OOD rollout RMSE over T = 200 steps on 19 test trajectories.

Rollout error over time and boxplots

Rollout error. Median shape RMSE over the 6 s horizon and per-trajectory distribution. Hybrid achieves the lowest median with a compact interquartile range.

Backbone configuration comparison under actuation shift

Qualitative comparison. Over 6 s, Hybrid stays close to ground truth while Residual can diverge to unphysical configurations under actuation shift.

Hardware & Benchmarks

Validated on a physical PneuNet soft tail under actuation-frequency shift, and on standard Van der Pol and pendulum-cart benchmarks against a residual-Koopman baseline.

Soft-tail hardware validation and Van der Pol benchmark

Beyond simulation. (a–h) PneuNet soft-tail experiments: Hybrid reduces mean RMSE by ≥51% vs. equilibrium prior and ≥74% vs. residual under frequency shift. (i–k) Van der Pol benchmark: Hybrid maintains 0% catastrophic rollout rate while the Koopman baseline fails on harder initial conditions.

Video

Overview of the method, simulation results, and hardware validation.

BibTeX

@inproceedings{tanveer2026stabilizing,
  title={Stabilizing 3D Continuum-Arm Rollouts via Equilibrium Anchoring and Feature-Lifted Residual Learning},
  author={Tanveer, Ahsan and Afridi, Rahdar Hussain and Afridi, Waqar Hussain and Zhang, Feitian and Xie, Guangming},
  booktitle={Robotics: Science and Systems},
  year={2026}
}