Learning Actuator-Aware Spectral Submanifolds for Precise Control of Continuum Robots

Soft robots are commonly sought for safety-critical interactions in delicate environments, where accurate position and orientation control is imperative. Model predictive control (MPC) offers a solution, but it requires a model of the robot's infinite-dimensional nonlinear dynamics that is at once accurate and computationally cheap. Recent theory on adiabatic spectral submanifolds (aSSMs) and their applications to soft robots provide data-driven model-reduction methods to construct such models. Here, we extend these methods to identify aSSMs from enlarged observable datasets and upgrade the currently available aSSM-MPC schemes. Evaluated on a high-fidelity finite-element simulation of a pressure-actuated soft arm, our controller reduces position and orientation tracking error by more than 60% compared to existing data-driven baselines.

Spectral submanifold (SSM) reduction has emerged as a mathematically principled route to reliable nonlinear reduced-order models, capturing dynamics beyond the reach of linear techniques such as Dynamic Mode Decomposition (DMD). The computation of SSMs, however, remains computationally expensive, particularly for high-dimensional systems. In this work, we introduce equivariant spectral submanifold (eSSM) reduction, a novel extension of the SSM framework that explicitly incorporates symmetries of the full-order model into the reduction process. We establish the mathematical foundations of this approach by showing that SSMs are naturally equivariant submanifolds and that the associated charts and reduced dynamics inherit the appropriate induced group actions. Building on this framework, we develop a novel equivariant SSM reduction algorithm that exploits these symmetries to achieve substantially faster computations while also improving model robustness. We demonstrate the advantages of this approach on several benchmark problems including a test from the Common Task Framework for Science.

Tendon-Driven Continuum Robots (TDCRs) pose significant modeling and control challenges due to complex nonlinearities, such as frictional hysteresis and transmission compliance. This paper proposes a differentiable learning framework that integrates high-fidelity dynamics modeling with robust neural control. We develop a GRU-based dynamics model featuring bidirectional multi-channel connectivity and residual prediction to effectively suppress compounding errors during long-horizon auto-regressive prediction. By treating this model as a gradient bridge, an end-to-end neural control policy is optimized through backpropagation, allowing it to implicitly internalize compensation for intricate nonlinearities. Experimental validation on a physical three-section TDCR demonstrates that our framework achieves accurate tracking and superior robustness against unseen payloads, outperforming Jacobian-based methods by eliminating self-excited oscillations. For implementation details and source code, please refer to https://github.com/ZiqingZou/ContinuumControl.