Technical Summary: Physics-Aware Sparse Learning and Selective Online Adaptation for Euler–Lagrange Robot Dynamics

1. Problem Statement

Accurate dynamics models are fundamental to model-based robotic control, typically relying on Euler–Lagrange (EL) formulations. However, nominal EL models often fail in real-world scenarios due to unmodeled effects such as payload variations, intra-system coupling, friction, aerodynamic drag, and changing operating conditions.

Existing learning-based correction methods generally address this by introducing a single additive residual term to the nominal model. While effective at reducing prediction error, these approaches often treat the mismatch as an unstructured, black-box function. This neglects the internal mechanical structure of EL systems, violating critical physical properties such as symmetry, positive-definiteness of the inertia matrix, and the specific coupling between inertia and velocity-dependent (Coriolis/centrifugal) terms. Consequently, such models may yield physically inconsistent predictions and reduced reliability when embedded in controllers.

Furthermore, real-world systems exhibit time-varying coupling where many state interactions remain weak, suggesting that residual dynamics are partially sparse. Existing methods often fail to leverage this sparsity or adapt efficiently to non-stationary conditions without incurring high computational costs or data inefficiency. The core challenge is to develop a learning framework that preserves the structured mechanical form of EL systems while enabling compact, selective online adaptation to changing disturbances.

2. Methodology

The authors propose a structure-preserving residual learning framework that decomposes model mismatch into distinct components, separating mechanically coupled terms from disturbance-sensitive forces. The methodology consists of four main stages:

A. Structured Residual Decomposition

Instead of learning a single additive residual $\delta$, the model decomposes the correction into three parts based on the EL equation:
$$\delta_t = \Delta M_t \ddot{\chi}_t + \Delta C_t \dot{\chi}_t + \Delta f_t$$

1. Inertia Correction ($\Delta M$): A structured correction to the inertia matrix.
2. Induced Coriolis Term ($\Delta C$): Derived directly from the corrected inertia to preserve the skew-symmetry property ($\dot{M} - 2C$).
3. Generalized Force Residual ($\Delta f$): Captures non-conservative forces (friction, drag, external disturbances).

B. Sparse Latent Interaction Modeling

To handle history-dependent dynamics and exploit sparsity, the system encodes a finite history window of observations ($H_t$) into a latent representation $\tilde{z}_t$ using a temporal convolutional network. This representation is then split into two branches via soft-thresholding (a sparse coding mechanism):

• Structural Branch ($z^M_t$): A sparse code driving the inertia correction.
• Adaptive Branch ($z^f_t$): A sparse code driving the generalized force correction.

This ensures that only a small subset of "active" interaction modes is utilized at any given time, promoting parsimony.

C. Physical Constraints and Parameterization

• Inertia Matrix: The corrected inertia $\hat{M}$ is parameterized as the sum of the nominal inertia and a learned correction. The correction is constructed as a low-rank term plus a strictly positive diagonal term (using a softplus activation), guaranteeing that $\hat{M}$ remains Symmetric Positive Definite (SPD) by construction.
• Coriolis Matrix: The corrected Coriolis matrix $\hat{C}$ is not learned independently but is analytically derived from the learned inertia $\hat{M}$ using Christoffel symbols. This enforces the mechanical coupling between inertia and velocity terms.
• Generalized Force: Modeled as a linear function of the sparse adaptive latent code ($\Delta f = \Theta z^f_t$).

D. Selective Online Adaptation

To handle time-varying conditions (e.g., payload changes), the framework employs selective online adaptation:

• The mechanically structured components ($\Delta M$ and $\Delta C$) are kept fixed after offline training.
• Only the generalized force decoder ($\Theta$) is updated online.
• Updates are performed using Bayesian Linear Regression within the active latent subspace defined by the non-zero elements of $z^f_t$. This allows for rapid, sample-efficient adaptation to disturbances without retraining the entire model or violating the learned mechanical structure.

3. Key Contributions

The paper highlights three primary contributions:

1. Structure-Preserving Residual Decomposition: The formulation of residual learning for EL systems by explicitly separating inertia corrections, induced Coriolis terms, and generalized forces. This preserves key mechanical properties (symmetry, positive-definiteness) rather than absorbing all discrepancies into a generic black-box term.
2. Parsimonious History-Dependent Interaction Modeling: The introduction of a sparse latent representation that captures a limited number of active interaction patterns. This enables compact modeling of both structured mechanical mismatches and disturbance effects, leveraging the observation that many state interactions are consistently weak.
3. Selective Bayesian Online Adaptation: The development of an adaptation mechanism that updates only the disturbance-sensitive generalized-force component within the active latent subspace. This maintains the integrity of the learned mechanical structure while ensuring responsiveness to non-stationary dynamics.

4. Experimental Results

The method was evaluated across five robotic platforms: a differential-drive mobile robot, a 6-DoF manipulator, a quadrotor, a mobile manipulator, and an aerial manipulator. Experiments involved tracking complex figure-eight trajectories under varying payloads and operating conditions.

• Prediction Accuracy: The proposed method achieved the lowest normalized prediction error compared to baselines including Sparse System Identification (SysID), Physics-Inspired TCNs (PI-TCN), and Deep Model Reference Adaptive Control (DMRAC). Baselines struggled with configuration-dependent coupling in mobile and aerial manipulators, whereas the structured approach excelled.
• Trajectory Tracking: In closed-loop control using a computed torque controller, the proposed method demonstrated superior tracking performance (lowest RMSE) compared to offline deep learning models (DNN, Diffusion), meta-learning approaches, and active MLPs. The results indicate that combining structured modeling with recursive, uncertainty-aware updates is critical for robust tracking under non-stationary dynamics.
• Ablation Studies:
  • Sparsity: Moderate sparsity levels yielded the best balance between expressiveness and robustness, improving data efficiency in low-data regimes and enabling faster convergence.
  • Adaptation: The sparse model showed faster error recovery and smoother convergence following transient disturbances (e.g., payload drops) compared to dense models. The active latent dimension dynamically increased during transients and returned to nominal levels, demonstrating adaptive complexity.

5. Significance and Claims

The paper argues that accurate real-world model-based control requires more than just reducing prediction error; it demands physical consistency and selective adaptability.

• Physical Consistency: By preserving the EL structure (SPD inertia, coupled Coriolis terms), the model remains interpretable and reliable, avoiding the pitfalls of unstructured black-box corrections that can lead to unstable control behavior.
• Efficiency and Robustness: The combination of sparse latent representations and selective online adaptation allows the system to handle time-varying conditions efficiently. It avoids the computational burden of full-model retraining and the data inefficiency of adapting large parameter sets.
• Practical Deployment: The framework is designed for real-time deployment on embedded hardware (NVIDIA Jetson Orin NX), demonstrating that structured residual learning can be effectively integrated into model-based controllers for diverse robotic systems.

The authors conclude that while their approach significantly improves performance, a limitation remains: the mechanically structured component is fixed during deployment. They suggest future work should investigate joint adaptation of structured mechanical terms for scenarios involving very large or persistent changes in inertial properties.