Orientation Control of Soft Robots via Adiabatic Spectral Submanifolds

Aron Karakai; Roshan S. Kaundinya; Mike Yan Michelis; Robert Katzschmann; George Haller

cs.ROSep 13, 2026

Orientation Control of Soft Robots via Adiabatic Spectral Submanifolds

Authors: Aron Karakai, Roshan S. Kaundinya, Mike Yan Michelis, Robert Katzschmann, George Haller

Organizations: Automatic Control Laboratory, ETH Zurich, Switzerland · Institute for Mechanical Systems, ETH Zurich, Switzerland · Soft Robotics Laboratory and the ETH AI Center, ETH Zurich, Switzerland

Abstract

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.

Figures & tables

Panel of Figure 1 — Fig. 1 : The (perturbed) aSSM $\tilde{\mathcal{A}}_{\varepsilon}$ anchored to the slow manifold $\mathcal{L}_{\varepsilon}$ is a low-dimensional invariant manifold under inputs $u(t)=u^{s}(\varepsilon t)+u^{d}(t)$ , where $u^{s}$ is slow and $u^{d}$ is uniformly bounded for all times.

Panel of Figure 2 — Fig. 2 : The SoPrA arm (left) and its model in the SORS simulator [ 20 ] (right).

Panel of Figure 3 — Fig. 3 : Observables of SoPrA.

Panel of Figure 4 — Fig. 4 : Position (a) and orientation (b) of the end effector in task 1. Orientation is plotted as the angle $\arctan(z_{5}/z_{4})$ .

Method	controller weights		control effort		relative $\ell_{2}$ error
	$R$	$R_{du}$	task 1	task 2	task 1	task 2
aSSM-MPC	$\diag(0.02,0.004\cdot I_{6},2\cdot I_{7})$	$\diag(10\cdot I_{7},0.1\cdot I_{7})$	39.6	33.4	$\mathbf{2.6\%}$	$\mathbf{2.4\%}$
aSSM-MPC, no reference input	$\diag(0.02,0.004\cdot I_{6},2\cdot I_{7})$	$\diag(10\cdot I_{7},0.1\cdot I_{7})$	32.9	25.9	$4.3\%$	$5.3\%$
aSSM-MPC, static approx.	$0.1\cdot I_{7}$	$0.5\cdot I_{7}$	41	35.7	$6.6\%$	$6.1\%$
DMDc-MPC	$\diag(0.2,0.04\cdot I_{6})$	$10\cdot I_{7}$	$41$	$38.1$	$14.7\%$	$16.3\%$
Koopman pregain	$0.12\cdot I_{7}$	$-$	$35.5$	$21.7$	$29.9\%$	$35\%$

TABLE I : Parameters, control effort, and tracking performance of the tested controllers.

Panel of Figure 6 — Fig. 5 : Position (a) and orientation (b) of the end effector in task 2. Orientation is plotted as the angle $\arctan(z_{5}/z_{4})$ .

Panel of Figure 7 — Fig. 6 : Instantaneous relative error in tasks 1 (a) and 2 (b).

Robotic Control

Robotic control studies how machines use sensing and computation to produce reliable movement and interaction in changing environments.

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

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Continuum Robot Control Robot Dynamics Learning Dimensionality Reduction

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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.

Abstract

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