Learning Actuator-Aware Spectral Submanifolds for Precise Control of Continuum Robots
Authors: Paul Leonard Wolff, Hugo Buurmeijer, Luis Pabon, John Irvin Alora, Mark Leone, Roshan S. Kaundinya, Amirhossein Kazemipour, Robert K. Katzschmann, Marco Pavone
Organizations: Autonomous Systems Lab, Stanford University, USA. · Soft Robotics Lab, ETH Zurich, Switzerland. · Institute of Mechanical Systems, ETH Zurich, Switzerland. · NVIDIA Research, Santa Clara, USA.
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.
Figures & tables
Fig. 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.
Fig. 2: Eigenvalue layout: slow system modes are shown in green and actuator modes in blue . The region left of γ≪0 indicates rapidly decaying modes that can safely be neglected. Spectral overlap of modes signals that neglecting actuator dynamics can induce significant modeling errors.
Fig. 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.
Fig. 4: The tendon-driven trunk hardware system, shown in its rest position. The flexible body has a length of 31cm . Tendons extend from the actuators to rigid clamps attached to the flexible body. Observations were captured using an OptiTrack motion capture system.
Model
caSSM
oSSM
Koopman
RMSE
2.8
4.6
7.3
TABLE I: Average RMSE [mm] for open-loop predictions.
Fig. 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.
Model
caSSM
oSSM
Koopman
Circle 5cm
4.60
9.89
20.40
Circle 5cm – ”fast”
5.30
13.44
Diverged
Circle 8cm
9.13
19.80
Diverged
Circle 10cm
13.36
Diverged
Diverged
Figure-eight
4.40
8.64
26.72
Figure-eight – ”fast”
8.12
15.10
Diverged
TABLE II: Average RMSE [mm] for closed-loop trajectories.
Fig. 6: Test-set performance for models trained with three feature maps (RFF—blue, polynomials—green, neural networks—red). Training data covers states within 15cm of the origin.
Automatic Control Laboratory, ETH Zurich, Switzerland · Institute for Mechanical Systems, ETH Zurich, Switzerland · Soft Robotics Laboratory and the ETH AI Center, ETH Zurich, Switzerland