Beyond Black Boxes: Learning Robot Dynamics with Interpretable Models

Author: Denis Avetisyan

New research demonstrates that symbolic regression and sparse identification techniques can unlock clear, physically-meaningful insights into how robots move, offering a powerful alternative to complex neural networks.

Evaluations demonstrate the trained model’s performance using scene reconstruction on real-world data gathered from a 7-degree-of-freedom WAM robot arm, as detailed in Sousa et al.’s work [sousa2014].

This review explores data-driven methods for learning interpretable hybrid robot dynamics, including force estimation and residual learning for improved model-based control.

Accurate robot dynamics modeling remains a challenge due to system complexity and unmodeled effects. This is addressed in ‘Data-driven Interpretable Hybrid Robot Dynamics’, which introduces a framework for learning residual dynamics using symbolic regression and sparse identification of nonlinear dynamics (SINDy). The authors demonstrate that these interpretable models not only achieve high accuracy in predicting robot torques-outperforming neural network baselines-but also recover physically meaningful expressions for previously unknown dynamics. Could this approach unlock more robust and adaptable robot control strategies by bridging the gap between data-driven learning and traditional model-based techniques?

Unveiling Robot Dynamics: The Foundation of Precise Control

Achieving precise control of robotic systems necessitates a thorough understanding and accurate representation of the forces governing their motion, yet conventional approaches frequently encounter limitations. While theoretical models attempt to capture these dynamics, the inherent complexity of robotic mechanisms – coupled with unmodeled effects such as flexible joints or unpredictable external disturbances – introduces significant challenges. These discrepancies between predicted and actual behavior can lead to instability or degraded performance, particularly in high-speed or precision tasks. Consequently, researchers continually strive to refine modeling techniques and develop adaptive control strategies that can effectively mitigate the impact of these unmodeled dynamics, moving beyond purely analytical approaches to incorporate data-driven methods and real-time compensation schemes.

The foundation of predictable robot movement lies in the Rigid Body Dynamics Model, a mathematical representation of how forces affect a robot’s structure. This model doesn’t treat the robot as a simple collection of parts, but as interconnected rigid bodies, each characterized by its Inertia Matrix – a tensor defining resistance to rotational and translational acceleration. Crucially, it accounts for the Coriolis Effect, an apparent force arising from the robot’s rotation that impacts the forces experienced by moving parts. Furthermore, effective control necessitates Gravity Compensation, a calculation that counteracts the constant pull of gravity on the robot’s links, preventing unwanted movements and ensuring accurate positioning. By precisely defining these parameters and incorporating these effects into the model, engineers can predict and control the robot’s behavior with greater accuracy, forming the basis for complex maneuvers and delicate operations.

Despite the sophistication of rigid body dynamic models, achieving truly accurate robot control necessitates addressing the persistent challenge of residual dynamics. These discrepancies arise from a multitude of unmodeled factors, most notably viscous friction within the robot’s joints and actuators. While idealized models assume frictionless movement, real-world robots experience internal resistance proportional to velocity, creating forces that deviate from theoretical predictions. Furthermore, unmodeled elasticity in the transmission systems, external disturbances, and even imperfections in the robot’s geometry contribute to these residual errors. Consequently, advanced control strategies often incorporate adaptive techniques or learning algorithms to estimate and compensate for these unmodeled dynamics, effectively minimizing the impact of $f_{residual}$ and improving overall system performance.

A trajectory was generated in simulation using a 7-DoF Franka robot arm and controlled with a PID controller incorporating gravity compensation.

System Identification: Learning Robot Dynamics from Data

System identification addresses residual dynamics – unmodeled behaviors remaining after accounting for known kinematic and dynamic effects – by directly learning the underlying relationships from robot data. This is achieved by utilizing joint space features, which represent the robot’s configuration in terms of joint angles and velocities, as input to the identification process. These features serve as a concise and informative representation of the robot’s state, enabling the learning algorithms to establish a mapping between the current state and the resulting changes in joint space. The resulting models can then be used to predict future behavior or to improve control performance by compensating for the previously unmodeled dynamics.

Multiple techniques are utilized to model complex robot dynamics from data. Symbolic Regression aims to identify mathematical expressions that best fit observed data, producing interpretable, albeit potentially complex, models. Sparse Identification of Nonlinear Dynamics focuses on identifying a limited number of significant terms within a potentially high-dimensional nonlinear function, improving generalization and reducing computational cost. Neural Networks, conversely, employ interconnected nodes arranged in layers to learn hierarchical representations of the data, offering high flexibility but often at the expense of interpretability; different network architectures, such as recurrent or convolutional networks, can be chosen based on the nature of the dynamics being modeled.

Total Variation Regularization (TVR) is a critical data preprocessing step for system identification, employed to reduce the impact of noise on model learning. TVR operates by penalizing rapid changes in the data, effectively smoothing the input features without blurring significant dynamic events. This process improves the quality of the learned dynamics model, leading to increased accuracy as quantified by Root Mean Squared Error (RMSE). In simulation, implementations utilizing sparse regression and hybrid modeling techniques, combined with TVR, consistently achieve relative RMSEs on the order of $10^{-3}$. This level of accuracy demonstrates the efficacy of TVR in preparing noisy robot data for robust system identification.

Trajectories from Sousa and Cortesão [sousa2014] were analyzed using the numerical derivative estimation method of Chartrand [chartrand2011numerical] to determine corresponding velocities, accelerations, and jerk.

Validation and Performance on Physical Robotic Platforms

The Franka Emika Panda robot and the WAM robot are utilized as physical platforms to validate the accuracy of dynamically identified models. These robots provide a means to assess the fidelity of identified parameters and their ability to predict robot behavior in real-world scenarios. Validation involves comparing predicted kinematic and dynamic responses-such as joint positions, velocities, and accelerations-with measured data obtained from the physical robots executing defined trajectories. This comparison allows for quantitative evaluation of model performance, typically expressed through metrics like Root Mean Squared Error (RMSE), and helps determine the suitability of identified models for control design and simulation purposes. The use of multiple robotic platforms ensures robustness and generalization of the validation process across different robot characteristics and operational conditions.

Quantification of system identification technique accuracy is achieved through direct comparison of predicted and observed robot behavior. This involves utilizing data from physical robot executions to generate a set of ground truth values against which model predictions are assessed. Common metrics employed for this comparison include the Root Mean Squared Error (RMSE), which provides a measure of the average magnitude of error between predicted and actual joint positions, velocities, or accelerations. Lower RMSE values indicate a higher degree of fidelity between the identified model and the physical system. This comparative analysis allows for objective ranking of different system identification techniques based on their ability to accurately represent the robot’s dynamics.

System identification techniques, particularly sparse regression models (SR and SINDy), have proven effective in reducing residual dynamics, thereby improving both robot control and simulation fidelity. Quantitative results indicate that these sparse regression models achieve relative Root Mean Squared Errors (RMSE) on the order of $10^{-3}$ in simulation. On a 7-DoF WAM arm utilizing real-world data, sparse regression models demonstrably outperformed neural networks, as evidenced by lower test set RMSE values and improved generalization performance. This reduction in residual dynamics contributes to more accurate dynamic models and, consequently, enhanced control strategies and more realistic simulations.

The 7-DoF Franka Robot arm is visualized using PyBullet for simulation display, though all control logic is implemented within our custom simulator.

The pursuit of interpretable models, as demonstrated in this work on data-driven hybrid robot dynamics, echoes a fundamental principle of robust system design. If a system survives on duct tape, it’s probably overengineered – a sentiment applicable to complex, black-box approaches. This research champions a return to clarity, favoring symbolic regression and sparse identification to reveal the underlying mechanics governing robot behavior. As Claude Shannon wisely stated, “The most important thing is to get the message across.” Here, the ‘message’ isn’t simply data transmission, but a clear, understandable articulation of the robot’s dynamics-a message lost in the noise of purely data-driven, opaque models. The success of these techniques highlights that structure dictates behavior, and a well-defined, interpretable structure leads to predictable and controllable systems.

The Road Ahead

The pursuit of interpretable models for robot dynamics, as demonstrated by this work, is not merely a quest for accuracy, but a recognition that understanding why a system behaves as it does is fundamental to robust control. The success of symbolic regression and sparse identification in capturing residual dynamics suggests a path beyond the black-box approach of deep learning – a welcome shift, though not a panacea. The limitations remain clear: scaling these methods to systems of significantly higher complexity, or to scenarios involving substantial environmental uncertainty, will demand innovation.

A critical, often overlooked, aspect is the inherent trade-off between model complexity and fidelity. The elegance of a sparse, interpretable model is diminished if it glosses over crucial, albeit subtle, physical phenomena. Future research should explore methods for systematically quantifying this trade-off, and for incorporating prior knowledge – not as rigid constraints, but as guiding principles – into the model identification process. This necessitates a move beyond purely data-driven approaches, towards a synthesis of learning and reasoning.

Ultimately, the true test lies not in achieving incrementally better predictions, but in building robots that can adapt, generalize, and even explain their actions. The development of truly intelligent machines demands models that are not only accurate and interpretable, but also – crucially – understandable. The path forward requires a humility regarding the limits of data, and a renewed appreciation for the power of physical insight.

Original article: https://arxiv.org/pdf/2512.11900.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

Unveiling Robot Dynamics: The Foundation of Precise Control

System Identification: Learning Robot Dynamics from Data

Validation and Performance on Physical Robotic Platforms

The Road Ahead

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