Reduced-order Control and Geometric Structure of Learned Lagrangian Latent Dynamics

Model-based controllers can offer strong guarantees on stability and convergence by relying on physically accurate dynamic models. However, these are rarely available for high-dimensional mechanical systems such as deformable objects or soft robots. While neural architectures can learn to approximate complex dynamics, they are either limited to low-dimensional systems or provide only limited formal control guarantees due to a lack of embedded physical structure. This paper introduces a latent control framework based on learned structure-preserving reduced-order dynamics for high-dimensional Lagrangian systems. We derive a reduced tracking law for fully actuated systems and adopt a Riemannian perspective on projection-based model-order reduction to study the resulting latent and projected closed-loop dynamics. By quantifying the sources of modeling error, we derive interpretable conditions for stability and convergence. We extend the proposed controller and analysis to underactuated systems by introducing learned actuation patterns. Experimental results on simulated and real-world systems validate our theoretical investigation and the accuracy of our controllers.

💡 Research Summary

This paper tackles the challenge of model‑based control for high‑dimensional Lagrangian systems—such as soft robots and deformable objects—where accurate analytical models are rarely available. The authors propose a latent‑space control framework that learns a structure‑preserving reduced‑order model (ROM) directly from data using Reduced‑Order Lagrangian Neural Networks (RO‑LNNs). RO‑LNNs combine a constrained auto‑encoder (which learns an embedding φ and a reduction ρ that satisfy the idempotent projection property Π = φ ∘ ρ) with a Lagrangian neural network that parameterizes the reduced mass‑inertia matrix, potential energy, and damping as symmetric positive‑definite (SPD) networks. This construction guarantees that the learned ROM respects the original system’s geometric and energetic structure.

For fully actuated systems (B = I), the authors design a tracking controller that augments a standard PD law with a feed‑forward term computed from the latent dynamics: τ_c = M̃ ¨q_d + (C̃ + D̃) · ˙q_d + g̃ − K_P e − K_D · ˙e. By expressing the closed‑loop error dynamics in the latent coordinates, they identify a disturbance term Δ that aggregates modeling errors and projection‑induced errors. Using Riemannian submersion theory, they bound Δ as a linear function of the tracking error plus a constant bias, and prove that the error system is locally exponentially input‑to‑state stable (ISS) provided the ROM is sufficiently accurate.

The framework is then extended to under‑actuated or indirectly actuated systems. A learned actuation pattern (\hat B(\tilde q)) replaces the unknown full‑rank input matrix, and the same ISS analysis applies after accounting for the “virtual under‑actuation” introduced by the reduced model.

Experimental validation includes (1) a simulated 7‑DOF pendulum where the ROM reduces the state dimension from 7 to 3, achieving a three‑fold reduction in tracking error compared with a conventional PD+ controller, and (2) a real‑world task where a humanoid robot manipulates a soft plush puppet. In both cases the controller tracks time‑varying reference trajectories with bounded error, and the stability guarantees hold as long as the ROM error stays below roughly 10 % of the system’s natural dynamics.

Key contributions are: (i) a novel ROM‑based tracking controller for high‑dimensional systems with unknown dynamics, (ii) a rigorous geometric analysis that yields interpretable stability conditions despite model reduction, and (iii) an extension to partially actuated systems via learned actuation mappings. The work demonstrates that preserving physical structure during data‑driven model reduction enables the transfer of classical Lagrangian control theory to latent spaces, opening a pathway for reliable, model‑based control of complex soft and deformable robotic platforms.