Taming High-Dimensional Dynamics: Learning Optimal Projections onto Spectral Submanifolds

High-dimensional nonlinear systems pose considerable challenges for modeling and control across many domains, from fluid mechanics to advanced robotics. Such systems are typically approximated with reduced order models, which often rely on orthogonal projections, a simplification that may lead to large prediction errors. In this work, we derive optimality of fiber-aligned projections onto spectral submanifolds, preserving the nonlinear geometric structure and minimizing long-term prediction error. We propose a computationally tractable procedure to approximate these projections from data, and show how the effect of control can be incorporated. For a 180-dimensional robotic system, we demonstrate that our reduced-order models outperform previous state-of-the-art approaches by up to fivefold in trajectory tracking accuracy under model predictive control.

Data-Driven Soft Robot Control via Adiabatic Spectral Submanifolds

The mechanical complexity of soft robots creates significant challenges for their model-based control. Specifically, linear data-driven models have struggled to control soft robots on complex, spatially extended paths that explore regions with significant nonlinear behavior. To account for these nonlinearities, we develop here a model-predictive control strategy based on the recent theory of adiabatic spectral submanifolds (aSSMs). This theory is applicable because the internal vibrations of heavily overdamped robots decay at a speed that is much faster than the desired speed of the robot along its intended path. In that case, low-dimensional attracting invariant manifolds (aSSMs) emanate from the path and carry the dominant dynamics of the robot. Aided by this recent theory, we devise an aSSM-based model-predictive control scheme purely from data. We demonstrate the effectiveness of this data-driven model on various dynamic trajectory tracking tasks on a high-fidelity and high-dimensional finite-element model of a soft trunk robot. Notably, we find that four- or five-dimensional aSSM-reduced models outperform the tracking performance of other data-driven modeling methods by a factor up to 10 across all closed-loop control tasks.