⚡ electrical engineering

Orientation Control of Soft Robots via Adiabatic Spectral Submanifolds

This paper presents an enhanced data-driven model predictive control scheme for soft robots that utilizes adiabatic spectral submanifolds to accurately capture infinite-dimensional nonlinear dynamics, achieving over a 60% reduction in position and orientation tracking errors compared to existing baselines.

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

Published 2026-09-15

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CC BY 4.0

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

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Soft robots are the gentle giants of the engineering world, built from materials that bend, stretch, and squish rather than snap and grind like traditional metal machines. Because they are made of compliant, continuously deformable substances, they are ideal for navigating delicate environments where a human hand might be too clumsy or a metal gripper too dangerous. Think of them as the robotic equivalent of an octopus or an elephant's trunk, capable of slipping into tight spaces and interacting safely with fragile objects. However, this very flexibility makes them incredibly difficult to control. Unlike a rigid robot arm that moves in predictable, straight lines, a soft robot has an infinite number of ways it can wiggle and deform. To steer one accurately, a computer needs a perfect map of its internal physics, but these maps are often too complex to calculate in real-time, and the exact laws governing the material's behavior are frequently unknown.

For years, engineers have tried to solve this by either simplifying the robot's physics into rough approximations or by teaching computers to guess the right moves through trial and error. The first approach often fails because it ignores the subtle, non-linear ways soft materials behave, while the second requires massive amounts of data and powerful simulators that don't always translate well to the real world. The challenge has been finding a middle ground: a way to create a simple, fast, and accurate model of a soft robot's movement that a computer can use to make split-second decisions, without needing to know every single detail of the robot's internal structure.

A team of researchers has now developed a new method to bridge this gap, allowing them to control both the position and the orientation of a soft robot arm with unprecedented precision. Their approach relies on a mathematical concept called an adiabatic spectral submanifold, which can be thought of as a hidden, low-dimensional "skeleton" that captures the most important movements of the robot while ignoring the chaotic noise. In the physical world, soft robots are highly dissipative, meaning they lose energy quickly and settle down fast after being disturbed. The researchers realized that while the robot's internal vibrations happen rapidly, the tasks we want it to perform—like moving a tool to a specific spot—happen much more slowly. By separating these two speeds, they could identify a simplified surface that the robot's motion naturally follows, effectively reducing the infinite complexity of the soft arm down to a manageable set of rules.

To test this idea, the team built a high-fidelity simulation of a soft arm named SoPrA, which is twenty-eight centimeters long and powered by six internal air chambers. They also added a motor at the base to rotate the entire arm, allowing them to control not just where the tip of the arm was, but also which way it was pointing. This orientation control is crucial for tasks like inserting a tool or inspecting a surface, yet it had proven difficult to achieve with previous data-driven methods. The researchers fed the simulation thousands of random movements to generate a dataset, then used their new algorithm to learn the robot's underlying "skeleton" directly from the data, without needing to know the specific stiffness or weight of the materials.

The result was a new control system that could predict the robot's future path with remarkable accuracy. Unlike older methods that assumed the robot would stay in a fixed, static position while making tiny adjustments, this new system allowed the robot to move freely along its natural path while simultaneously adjusting for small errors. In their tests, the new controller reduced the error in tracking both position and orientation by more than sixty percent compared to existing data-driven techniques. It performed significantly better than linear models, which failed to capture the robot's complex curves, and even outperformed previous versions of their own method that relied on static approximations. The system was fast enough to run in real time, calculating the necessary adjustments in less than five milliseconds on a standard laptop.

Perhaps most importantly, the researchers found that their method worked well even when the robot was moving faster than the theory strictly predicted it should. This suggests that the approach is robust enough for practical use, even when the conditions are not perfectly ideal. By combining a deep understanding of the robot's natural dynamics with a flexible, data-driven learning process, the team has created a tool that could eventually allow soft robots to perform delicate, complex tasks in the real world, from surgical procedures to search-and-rescue operations, with a level of grace and accuracy that was previously out of reach.

Technical Summary: Orientation Control of Soft Robots via Adiabatic Spectral Submanifolds

Problem Statement
Soft robots, constructed from compliant materials, are increasingly vital for safe interactions in delicate environments. However, controlling their position and orientation is challenging due to their infinite-dimensional, underactuated, and intrinsically nonlinear dynamics. While Model Predictive Control (MPC) is a promising framework, it requires accurate, computationally efficient models of these dynamics. Existing high-fidelity finite-element models are too computationally expensive for real-time closed-loop control. Conversely, data-driven approaches often rely on linear approximations (e.g., Dynamic Mode Decomposition with Control, DMDc) which lack predictive power for nonlinear systems, or black-box methods like reinforcement learning that require massive datasets and accurate simulators.

Recent work introduced Adiabatic Spectral Submanifolds (aSSMs) to reduce the dimensionality of soft robot dynamics by exploiting the time-scale separation between fast internal dissipation and slow task execution. However, previous aSSM-based MPC implementations fixed the slow input and the SSM approximation for the entire planning horizon, optimizing only over small, fast deviations. This limitation restricted control authority and predictive accuracy, particularly for complex tasks involving orientation changes.

Methodology
The authors propose an upgraded aSSM-MPC scheme that addresses these limitations through three main technical advancements:

Full Time-Varying Geometry: Unlike prior methods that fix the slow input sequence, the proposed scheme utilizes the full, time-varying geometry of the aSSM across the entire prediction horizon. This allows the controller to optimize over both the slow input component ( $u_s$ ) and the fast deviation component ( $u_d$ ) independently.
Orientation Observables: To enable orientation control, the authors introduce a specific set of observables. Instead of using Euler angles, which require modular arithmetic, they define the orientation using a vector $r_2$ pointing from the end-effector center to a point on its edge, alongside the position vector $r_1$ . This approach leverages the expressivity of Lie group $SO(3)$ descriptions without the computational overhead of modular constraints in data-driven modeling.
Data-Driven Model Construction: The reduced-order model is constructed by:
- Learning a critical manifold $L_0$ mapping static inputs to steady states.
- Identifying a global spectral subspace ( $E_{global}$ ) from decaying trajectories.
- Fitting polynomial parametrizations for the spectral submanifolds ( $W$ ) and reduced dynamics ( $R$ ) as functions of the reduced coordinates and the static input.
- Learning a linear control matrix ( $B$ ) to account for fast input deviations.
- The entire pipeline is implemented in a new Python library, aSSMPy.

The control loop employs a sequential convex programming technique (GuSTO) to solve the MPC optimization problem, minimizing the deviation of workspace coordinates from a target trajectory while respecting input constraints and slew rates.

Key Contributions
The paper makes the following specific contributions:

Extended aSSM-MPC Scheme: An algorithm that optimizes over both slow and fast input components using the full aSSM geometry, significantly improving predictive accuracy and control authority compared to static approximations.
Orientation Control Framework: A novel set of observables and control formulation that successfully applies aSSM-reduction to the combined position and orientation control of a soft continuum arm.
Open-Source Implementation: The release of aSSMPy, a general-purpose Python implementation for data-driven aSSM-reduction.

Experimental Results
The method was evaluated on a high-fidelity finite-element simulation of "SoPrA," a 28 cm long, pressure-actuated soft arm with six chambers and a rotating base. The controller was tested on two dynamic tracking tasks involving complex position and orientation trajectories.

Performance: The proposed aSSM-MPC reduced position and orientation tracking errors by more than 60% compared to existing linear data-driven baselines (DMDc-MPC and Koopman pregain).
Comparison to Static Approximation: Even when compared to the previous aSSM-MPC method (which uses static approximations), the new scheme showed significant improvements, particularly in orientation tracking and when operating far from the unforced equilibrium.
Real-Time Feasibility: On a MacBook Pro with an M5 Pro chip, the average computation time per MPC iteration was 4.85 ms (95th percentile at 7.38 ms), enabling real-time control at a 100 Hz discretization rate.
Robustness: The method remained effective even when the "slowness metric" ( $r_s$ ) of the tasks exceeded the theoretical limit of $r_s \ll 1$ , with experimental values reaching $r_s \approx 2.7$ .

Significance and Claims
The authors claim that their approach provides a rigorous, data-driven pathway to control soft robots with high accuracy without requiring explicit knowledge of the governing physical equations. By extending aSSM theory to optimize over full time-varying inputs and orientation, the method overcomes the limitations of linear approximations and static manifold assumptions.

The paper emphasizes that because the method is entirely data-driven and requires only a modest training dataset, it is well-suited for direct deployment on physical hardware. The authors note that while this work validates the approach in simulation, the immediate next step is physical validation on hardware, citing ongoing successful work with similar methods on tendon-driven robots. They also identify the derivation of rigorous stability guarantees for aSSM-MPC as an important direction for future research.

Technical Summary: Orientation Control of Soft Robots via Adiabatic Spectral Submanifolds

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