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

Katharina Friedl, Noémie Jaquier, Seungyeon Kim, Jens Lundell, Danica Kragic

TL;DR

This work tackles tracking control for high-dimensional Lagrangian systems with unknown dynamics by learning a structure-preserving reduced-order model (RO-LNN) that preserves Lagrangian structure through a Riemannian projection framework. A latent PD+ controller leverages the RO-LNN to achieve trajectory tracking in the latent space, with stability guarantees characterized as local exponential input-to-state stability (ISS) in the latent domain and extended to the embedded submanifold and full state under bounded disturbances from modeling and projection errors. The approach explicitly accounts for projection misalignment and nullspace dynamics, providing bounds and conditions for convergence, and extends to indirectly-actuated and underactuated systems via learned actuation patterns and lifting. Experimental validation on a simulated 15-DoF augmented pendulum and a real underactuated puppet demonstrates accurate latent tracking, improved regulation/tracking over baselines, and the practical viability of RO-LNN-based control for high-dimensional robotics. The results highlight the potential of physics-informed, non-intrusive ROMs to enable principled control and formal guarantees in complex robotic systems.

Abstract

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.

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

TL;DR

This work tackles tracking control for high-dimensional Lagrangian systems with unknown dynamics by learning a structure-preserving reduced-order model (RO-LNN) that preserves Lagrangian structure through a Riemannian projection framework. A latent PD+ controller leverages the RO-LNN to achieve trajectory tracking in the latent space, with stability guarantees characterized as local exponential input-to-state stability (ISS) in the latent domain and extended to the embedded submanifold and full state under bounded disturbances from modeling and projection errors. The approach explicitly accounts for projection misalignment and nullspace dynamics, providing bounds and conditions for convergence, and extends to indirectly-actuated and underactuated systems via learned actuation patterns and lifting. Experimental validation on a simulated 15-DoF augmented pendulum and a real underactuated puppet demonstrates accurate latent tracking, improved regulation/tracking over baselines, and the practical viability of RO-LNN-based control for high-dimensional robotics. The results highlight the potential of physics-informed, non-intrusive ROMs to enable principled control and formal guarantees in complex robotic systems.

Abstract

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.
Paper Structure (37 sections, 6 theorems, 76 equations, 15 figures)

This paper contains 37 sections, 6 theorems, 76 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Primer on Riemannian Geometry
  4. Riemannian Geometry of Lagrangian Systems
  5. Projection-Based Lagrangian Reduced-Order Models
  6. Reduced-order Lagrangian Neural Networks (RO-LNNs)
  7. Model-based Control via RO-LNNs
  8. A RO-LNN-based Control Law for Trajectory Tracking
  9. Latent Space Analysis of Stability and Convergence
  10. Closed-loop dynamics
  11. Stability and convergence analysis
  12. Stability and convergence of the Original System
  13. Stability and convergence analysis on the embedded submanifold $\varphi_{\mathcal{Q}}(\check{\mathcal{Q}})\!\subseteq\! \mathcal{Q}$
  14. Stability and convergence analysis on $\mathcal{Q}$
  15. Reduced-Order Controllers for Indirectly-actuated and Underactuated Systems
  16. ...and 22 more sections

Key Result

Proposition 1

The latent closed-loop dynamics induced by the control torque $\tilde{\bm{\tau}}_{\text{c}}$eq:torque_reconstruction take the form where $\check{\Delta}_{\theta}$ and $\check{\Delta}_{\perp}$ are disturbances due to dynamic modeling and projection alignment errors, that are bounded in a compact neighborhood $\mathcal{N}\!\subseteq\!\mathcal{T}\mathcal{Q}$ of the training data distribution.

Figures (15)

  • Figure 1: rolnn-based control loop. The AE is depicted in blue and the latent lnn components are depicted within the brown box. The control torques are computed in the learned latent space and lifted through the decoder.
  • Figure 2: Riemannian residual angle $(90°-\alpha)$ between the $\bm{M}$-orthogonal ($\bm P_\perp$) and learned ($\bm P$) projections.
  • Figure 3: Riemannian angle between the image and kernel of the ae projection $\bm P$ at different training epochs, histogram over $5000$ testing points.
  • Figure 4: Regulation error (median and quartiles) for the rolnn (), linear-intrusive (), and model-free pd () controllers over $10$ random initial and desired configurations.
  • Figure 5: Left: Original () and projected () tracking reference, and rolnn-based (), hybrid rolnn-based (), and model-free pd () controllers trajectories. Right: Reference and controllers trajectories on a mesh dof, also depicting the projected reference () and trajectory () of the rolnn-based controller in $\varphi(\mathcal{T}\check{\mathcal{Q}})$, and linear-intrusive controller ().
  • ...and 10 more figures

Theorems & Definitions (16)

  • Proposition 1: Latent dynamics
  • proof : Proof sketch
  • Corollary 1
  • proof
  • Corollary 2
  • proof : Proof Sketch
  • Remark 1
  • Theorem 1: Latent stability and convergence
  • proof : Proof Sketch
  • Proposition 2
  • ...and 6 more