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

Paul Leonard Wolff, Hugo Buurmeijer, Luis Pabon, John Irvin Alora, Mark Leone, Roshan S. Kaundinya, Amirhossein Kazemipour, Robert K. Katzschmann, Marco Pavone

Abstract

Continuum robots exhibit high-dimensional, nonlinear dynamics which are often coupled with their actuation mechanism. Spectral submanifold (SSM) reduction has emerged as a leading method for reducing high-dimensional nonlinear dynamical systems to low-dimensional invariant manifolds. Our proposed control-augmented SSMs (caSSMs) extend this methodology by explicitly incorporating control inputs into the state representation, enabling these models to capture nonlinear state-input couplings. Training these models relies solely on controlled decay trajectories of the actuator-augmented state, thereby removing the additional actuation-calibration step commonly needed by prior SSM-for-control methods. We learn a compact caSSM model for a tendon-driven trunk robot, enabling real-time control and reducing open-loop prediction error by 40% compared to existing methods. In closed-loop experiments with model predictive control (MPC), caSSM reduces tracking error by 52%, demonstrating improved performance against Koopman and SSM based MPC and practical deployability on hardware continuum robots.

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

Abstract

Continuum robots exhibit high-dimensional, nonlinear dynamics which are often coupled with their actuation mechanism. Spectral submanifold (SSM) reduction has emerged as a leading method for reducing high-dimensional nonlinear dynamical systems to low-dimensional invariant manifolds. Our proposed control-augmented SSMs (caSSMs) extend this methodology by explicitly incorporating control inputs into the state representation, enabling these models to capture nonlinear state-input couplings. Training these models relies solely on controlled decay trajectories of the actuator-augmented state, thereby removing the additional actuation-calibration step commonly needed by prior SSM-for-control methods. We learn a compact caSSM model for a tendon-driven trunk robot, enabling real-time control and reducing open-loop prediction error by 40% compared to existing methods. In closed-loop experiments with model predictive control (MPC), caSSM reduces tracking error by 52%, demonstrating improved performance against Koopman and SSM based MPC and practical deployability on hardware continuum robots.
Paper Structure (21 sections, 1 theorem, 40 equations, 6 figures, 2 tables)

This paper contains 21 sections, 1 theorem, 40 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background on Spectral Submanifolds
  4. Fitting SSMs from Observational Data
  5. Assumptions and Practical Limitations
  6. Control-Augmented SSMs
  7. Derivation of External Control Contribution
  8. Obtaining Linear Actuator Dynamics
  9. Augmenting with Actuator-Side Feedback
  10. Deploying caSSMs on Hardware
  11. Data Collection
  12. Feature Maps for $w_{\mathrm{nl}}$ and $r_{\mathrm{nl}}$
  13. Control Reference Vector
  14. Low-Dimensional MPC Formulation
  15. Experiments
  16. ...and 6 more sections

Key Result

Theorem 1

Under the augmented dynamics eq:augmented_general and the graph parameterization eq:graph_z–eq:red_dyn, the reduced input term that preserves manifold invariance is

Figures (6)

  • Figure 1: Workflow for deploying SSMs on hardware for closed-loop control: (a) fully automated decay-based data collection, (b) actuator-aware manifold modeling from data, (c) real-time MPC rollout on hardware.
  • Figure 2: Eigenvalue layout: slow system modes are shown in green and actuator modes in blue. The region left of $\gamma \ll 0$ indicates rapidly decaying modes that can safely be neglected. Spectral overlap of modes signals that neglecting actuator dynamics can induce significant modeling errors.
  • Figure 3: RMSE of predictive results across three models for the same ground truth trajectory. caSSM shows lowest RMSE across all segments, while Koopman diverges near training data boundaries.
  • Figure 4: The tendon-driven trunk hardware system, shown in its rest position. The flexible body has a length of 31. Tendons extend from the actuators to rigid clamps attached to the flexible body. Observations were captured using an OptiTrack motion capture system.
  • Figure 5: Closed-loop MPC performance on circular and figure-eight trajectories. Reference paths are shown in gray, with MPC rollouts in lighter gray. The Koopman model fails to track the reference, oSSM struggles on sharp curves while only caSSM tracks consistently across both shapes.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Definition 1: Autonomous SSM
  • Theorem 1
  • proof
  • proof