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
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
Fig. 1 : The (perturbed) aSSM A~ε anchored to the slow manifold Lε is a low-dimensional invariant manifold under inputs u(t)=us(εt)+ud(t) , where us is slow and ud is uniformly bounded for all times.
Fig. 2 : The SoPrA arm (left) and its model in the SORS simulator [ 20 ] (right).
Fig. 3 : Observables of SoPrA.
Fig. 4 : Position (a) and orientation (b) of the end effector in task 1. Orientation is plotted as the angle arctan(z5/z4) .
Method
controller weights
control effort
relative ℓ2 error
R
Rdu
task 1
task 2
task 1
task 2
aSSM-MPC
\diag(0.02,0.004⋅I6,2⋅I7)
\diag(10⋅I7,0.1⋅I7)
39.6
33.4
2.6%
2.4%
aSSM-MPC, no reference input
\diag(0.02,0.004⋅I6,2⋅I7)
\diag(10⋅I7,0.1⋅I7)
32.9
25.9
4.3%
5.3%
aSSM-MPC, static approx.
0.1⋅I7
0.5⋅I7
41
35.7
6.6%
6.1%
DMDc-MPC
\diag(0.2,0.04⋅I6)
10⋅I7
41
38.1
14.7%
16.3%
Koopman pregain
0.12⋅I7
−
35.5
21.7
29.9%
35%
TABLE I : Parameters, control effort, and tracking performance of the tested controllers.
Fig. 5 : Position (a) and orientation (b) of the end effector in task 2. Orientation is plotted as the angle arctan(z5/z4) .
Fig. 6 : Instantaneous relative error in tasks 1 (a) and 2 (b).
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.
Paul Leonard Wolff, Hugo Buurmeijer, Luis Pabon +6
Autonomous Systems Lab, Stanford University, USA. · Soft Robotics Lab, ETH Zurich, Switzerland. · Institute of Mechanical Systems, ETH Zurich, Switzerland. +1
Precise control of soft manipulators remains challenging due to the difficulty of developing accurate yet computationally tractable models for model-based estimation and control. Reduced Cosserat-rod models provide a physics-based and control-oriented description of soft-robot dynamics, offering an explicit alternative to purely data-driven input-output representations. In this paper, we propose a moving-horizon estimation (MHE) and nonlinear model predictive control (NMPC) framework for cable-driven soft manipulators based on reduced Cosserat dynamics. A smooth cable-length-driven modeling formulation is developed by approximating the complementarity relationship between cable tension and cable slackness, enabling cable-length control without direct tension sensing. Based on this formulation, an MHE method is introduced to estimate the reduced state and reconstruct the manipulator configuration from end-effector pose measurements and cable-length information. An NMPC controller is then formulated to achieve task-space control under cable-length and cable-rate constraints. The proposed framework is validated through numerical simulations and experiments. Simulation results demonstrate the effectiveness of the estimator and controller for pose and strain-related regulation on a multi-cable soft manipulator. Experimental results on a four-cable prototype further show that the proposed MHE-NMPC scheme can be implemented in real time and enables accurate end-effector position tracking through cable-length control.
Soft and soft-rigid hybrid robots are inherently underactuated and operate under tight actuator limits, making task-space control with stability guarantees challenging. Common nonlinear strategies for soft robots (e.g., those based on PD control) often rely on the assumption of full actuation with no actuator limits. This paper presents a general control framework for task-space regulation and tracking of underactuated soft robots under bounded inputs. The method enforces a rapidly exponentially stabilizing control Lyapunov function as a convex inequality constraint while simultaneously satisfying underactuated full-body dynamics and actuator bounds. We validate the approach in simulation on several platforms spanning increasing underactuation: a simple two link tendon-driven "finger", a trimmed helicoid manipulator, and a highly underactuated spiral robot. We compare against a number of baseline methods from the literature. Results show improved task-space accuracy and consistent Lyapunov convergence under input limits, achieving superior set-point and trajectory-tracking performance.
Huy Pham, Zach J. Patterson
Mechanical and Aerospace Engineering, Case Western Reserve University