Unlocking Robotic Precision: Machine Learning Discovers the Laws of Friction

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Author: Denis Avetisyan

A new approach leverages the power of artificial intelligence to automatically derive accurate and understandable models of static friction for robotic systems.

A systematic refinement process—beginning with robotic data acquisition and iterative model fitting using a Kernelized Artificially-constrained Network (KAN)—culminates in a parsimonious friction model derived through pruning, symbolic regression, and subsequent refinement, effectively distilling complex physical behavior into a concise, analytically tractable form.
A systematic refinement process—beginning with robotic data acquisition and iterative model fitting using a Kernelized Artificially-constrained Network (KAN)—culminates in a parsimonious friction model derived through pruning, symbolic regression, and subsequent refinement, effectively distilling complex physical behavior into a concise, analytically tractable form.

This work introduces a physics-informed machine learning framework based on Kolmogorov–Arnold Networks for interpretable static friction modeling in robotics.

Achieving high-precision motion control in robotics is often hampered by the difficulty of accurately modeling complex frictional forces. This paper, ‘Physics-informed Machine Learning for Static Friction Modeling in Robotic Manipulators Based on Kolmogorov-Arnold Networks’, introduces a novel approach leveraging Kolmogorov-Arnold Networks to automatically discover interpretable static friction models directly from experimental data. The method achieves both high predictive accuracy—exceeding 0.95 coefficient of determination in experiments—and extracts concise, physically meaningful expressions for friction. Could this data-driven, physics-informed framework unlock a new era of adaptable and robust robotic control systems?

Deconstructing Friction: The Illusion of Simplicity

Effective control of robot manipulators hinges on accurately predicting frictional forces during interaction with the environment. However, achieving this precision proves remarkably difficult; traditional friction models often struggle with both computational complexity and sensitivity to external factors. While a seemingly straightforward concept, friction is influenced by a multitude of variables – including surface properties, velocity, and normal force – making its behavior highly nonlinear. Consequently, even relatively simple models require extensive parameter tuning, and these parameters can shift dramatically due to minor changes in conditions like temperature or surface contamination. More sophisticated dynamic models, though theoretically more accurate, demand significant computational power, potentially exceeding the real-time processing capabilities of many robotic systems. This inherent trade-off between accuracy, complexity, and sensitivity presents a persistent challenge in robotics, limiting a robot’s ability to perform delicate or precise manipulation tasks.

The Coulomb friction model, a foundational concept in robotics and physics, posits that frictional force is directly proportional to the normal force pressing two surfaces together – a simplification often expressed as $F = \mu N$, where $\mu$ represents the coefficient of static friction. While intuitively appealing and computationally inexpensive, this model struggles to represent the complexities of real-world contact. Actual friction isn’t merely a constant value overcome at the point of motion; it’s influenced by factors like surface roughness, velocity, dwell time, and material properties. These nuances result in phenomena such as stick-slip motion, hysteresis, and varying coefficients of friction depending on conditions – behaviors entirely absent in the static Coulomb representation. Consequently, relying solely on this model can lead to inaccuracies in robot control, particularly in precision tasks or when dealing with delicate manipulations, highlighting the need for more sophisticated approaches.

While static friction models like Coulomb’s law offer a foundational understanding of contact, accurately representing real-world interactions often necessitates the use of dynamic friction models. These models account for velocity-dependent effects, such as the Stribeck curve – a phenomenon where friction initially decreases with increasing velocity before rising again – and offer a significantly improved fidelity to observed behavior. However, this increased accuracy comes at a cost. Effectively identifying the parameters within these complex models requires extensive experimentation and substantial computational resources, as the models frequently involve numerous variables and nonlinear relationships. The process of parameter estimation can be particularly challenging due to the inherent noise in friction measurements and the difficulty of isolating the specific factors influencing frictional forces, hindering their practical implementation in real-time control systems despite their theoretical advantages.

Various static friction models—including the classical Coulomb model and more complex variations like the Stribeck and Bengisu–Akay models—describe the relationship between tangential relative motion and frictional force, with the Awrejcewicz envelope model defining a range of possible behavior.
Various static friction models—including the classical Coulomb model and more complex variations like the Stribeck and Bengisu–Akay models—describe the relationship between tangential relative motion and frictional force, with the Awrejcewicz envelope model defining a range of possible behavior.

Unveiling the System: KAN – A Symbolic Regression Architecture

The KAN architecture utilizes symbolic regression, a technique that searches for mathematical expressions that best fit a given dataset, to directly generate friction models from experimental data without requiring pre-defined model structures. This process involves evolving a population of candidate equations – composed of mathematical operators and variables representing physical parameters – and selecting those that minimize the error between the predicted and observed friction forces. By automating model discovery, KAN circumvents the need for manual model identification and parameter tuning, providing interpretable models expressed as explicit equations, such as $f = \mu N + Kv$, where $f$ represents friction force, $\mu$ is the coefficient of friction, $N$ is the normal force, $K$ is the viscous friction coefficient, and $v$ is the sliding velocity.

KAN utilizes B-Spline activation functions within its symbolic regression framework to ensure generated friction models possess both smoothness and differentiability. B-Splines are piecewise polynomial functions defined over a set of knots, allowing for local control of curve shape while maintaining global smoothness. This characteristic is critical for control applications, as it enables the calculation of derivatives – specifically, the friction force and its rate of change – necessary for dynamics modeling and controller design. The differentiability of B-Spline-based models avoids the discontinuities that can arise with other activation functions, leading to more stable and predictable control performance. Furthermore, the parameterization of B-Splines allows KAN to efficiently explore the space of possible friction models and converge on solutions that accurately represent the observed experimental data.

Network pruning within the KAN architecture systematically reduces model complexity by removing redundant or inconsequential connections and parameters. This process is achieved post-training through iterative removal and retraining cycles, guided by a sensitivity analysis that assesses the impact of each parameter on the model’s overall error. By decreasing the number of parameters, pruning mitigates overfitting, thereby enhancing the model’s ability to generalize to unseen data. The resulting friction models exhibit improved robustness and computational efficiency, requiring fewer resources for both storage and real-time application in control systems. Pruned networks also demonstrate faster inference times without significant performance degradation, making them suitable for resource-constrained environments.

Symbolic identification reveals that the KAN framework effectively learns friction models for Joints 1 and 2, refining its computational graph through pruning and automatic symbolic selection to highlight key activations (red nodes) based on attribution scores.
Symbolic identification reveals that the KAN framework effectively learns friction models for Joints 1 and 2, refining its computational graph through pruning and automatic symbolic selection to highlight key activations (red nodes) based on attribution scores.

The Experiment: Friction Identification with KAN and L-BFGS

The Limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) optimizer is utilized within the KAN framework to train the friction models. L-BFGS is a quasi-Newton method that approximates the Hessian matrix, enabling efficient minimization of a cost function representing the error between predicted and observed friction forces. This approach avoids the computational expense of calculating and storing the full Hessian, making it suitable for high-dimensional parameter spaces often encountered in robotic systems. The optimization process iteratively adjusts model parameters to reduce the root mean squared error (RMSE) between predicted and actual friction, ultimately yielding a model that accurately represents the frictional behavior of the robot’s joints.

KAN’s friction identification process determines the mathematical structure of friction models based on observed data. These models are not pre-defined; instead, KAN discovers the necessary functional form to accurately represent the friction characteristics of a robotic system. The resulting models can range from simple static friction – where force is proportional to normal load – to more complex dynamic representations that incorporate velocity-dependent terms, hysteresis, and Stribeck effects. This adaptability allows KAN to capture a broad spectrum of frictional behaviors observed in real-world robotic applications, moving beyond the limitations of pre-defined models and accommodating varying levels of complexity in the system’s frictional dynamics.

Experimental validation of the Kinetic and Neural network (KAN) friction identification approach using real robotic data demonstrates a high degree of accuracy, as quantified by a coefficient of determination ($R^2$) consistently exceeding 0.95. This $R^2$ value indicates that the model explains at least 95% of the variance in the observed friction forces, signifying a strong correlation between predicted and actual behavior. The consistent achievement of this threshold across multiple robotic experiments validates KAN’s capability to accurately model friction dynamics and provides a reliable basis for downstream control and planning applications.

Analysis of robotic experiments demonstrates a strong linear relationship between friction forces and resulting joint torques when utilizing the KAN model. Specifically, single-axis motions exhibit a correlation coefficient of 0.9196, indicating a high degree of association between these variables. Multi-axis motions further strengthen this correlation, achieving a coefficient of 0.9306. These values suggest that a substantial portion of the variance in joint torque can be explained by variations in friction, and vice versa, providing a reliable basis for friction compensation and dynamic modeling.

Kernelized Alignment Networks (KAN) effectively fit friction models with unknown functional forms even when data is noisy.
Kernel-based adaptive Newton (KAN) fitting successfully identified the friction model parameters for all six joints despite the unknown functional form.

Beyond Precision: Implications for Advanced Robot Control

The fidelity of robotic movement hinges critically on understanding and accurately modeling friction – a force often simplified in traditional control systems. Recent work leveraging the Kinematic Analysis Network (KAN) has yielded friction models with unprecedented detail, directly translating to improvements in both the precision and responsiveness of robot manipulators. By capturing nuanced frictional behaviors at contact surfaces, KAN enables controllers to predict and compensate for forces that would otherwise introduce errors or slow down movements. This means robots can execute complex tasks with greater accuracy, adapt more quickly to changing conditions, and achieve a level of dexterity previously unattainable – paving the way for more reliable and efficient automation in diverse fields.

The fidelity of robotic simulations and the efficiency of trajectory planning are fundamentally limited by how accurately frictional forces are represented. Traditional models often simplify these interactions, leading to discrepancies between virtual environments and real-world performance. However, capturing the nuanced complexities of friction – including variations arising from surface texture, velocity, and material properties – allows for the creation of simulations that closely mirror actual robotic behavior. This enhanced realism enables developers to refine control algorithms and optimize movement paths before implementation, minimizing errors and maximizing precision. Consequently, robots can navigate and manipulate objects with greater accuracy and adaptability, particularly in scenarios demanding intricate maneuvers or precise contact forces. The ability to predict frictional behavior not only streamlines the design process but also paves the way for more robust and reliable robotic systems capable of operating in unpredictable environments.

The refined accuracy afforded by KAN-identified friction models extends beyond theoretical improvements, promising substantial gains in fields demanding unwavering precision and consistent performance. Applications such as delicate assembly of microelectronics, where minute errors are unacceptable, stand to benefit significantly from the enhanced control offered by these models. Similarly, the intricacies of surgical robotics, requiring sub-millimeter accuracy for tissue manipulation, could be revolutionized, potentially enabling less invasive procedures and improved patient outcomes. Furthermore, the ability to reliably predict and compensate for friction is crucial for robotic exploration in unpredictable environments – from the precise maneuvering required for sample collection on another planet to the stable operation of robots navigating challenging terrains. This advancement therefore represents a pivotal step towards realizing truly autonomous and capable robotic systems across a diverse range of critical applications.

Kernel-based adaptive Newton (KAN) fitting successfully identified the friction model parameters for all six joints despite the unknown functional form.
Kernel-based adaptive Newton (KAN) fitting successfully identified the friction model parameters for all six joints despite the unknown functional form.

The pursuit of physically interpretable models, as demonstrated by this work on static friction, echoes a fundamental tenet of understanding any system: its deconstruction to reveal underlying principles. This research leverages Kolmogorov–Arnold Networks to essentially reverse-engineer the forces governing robotic manipulation, moving beyond mere predictive accuracy toward genuine comprehension. As Bertrand Russell observed, “The point of contact between two disciplines is always a source of illumination.” The illumination here stems from the convergence of machine learning and physics, yielding models that aren’t black boxes, but transparent representations of reality – a testament to the power of probing a system’s boundaries to truly know it.

Pushing Beyond the Surface

The demonstrated success in reverse-engineering static friction models via Kolmogorov–Arnold Networks begs the question: how much of robotic control is currently built upon comfortably inaccurate assumptions? The field readily accepts approximations, largely because exhaustive, high-fidelity modeling proves intractable. Yet, this work suggests a path toward dismantling that compromise. The true test lies not in achieving marginally better performance on benchmark datasets, but in deliberately stressing the discovered models – pushing them into regimes of unexpected behavior to reveal the limits of the learned physics.

Future investigations should not shy away from deliberately incorrect data. Injecting noise, systematic errors, or even fabricated data points allows a rigorous assessment of the model’s robustness and its ability to extrapolate beyond the training domain. A truly insightful system will not simply predict accurately; it will flag anomalies and inconsistencies, effectively questioning the validity of the experimental setup itself. This demands a shift from validation against ground truth to validation against physical plausibility.

Ultimately, the goal isn’t merely to build a better friction model, but to build a system that understands why friction behaves as it does, and can intelligently reason about its uncertainties. The current work is a promising first step, but the real challenge remains: to create robotic systems that aren’t simply programmed to perform tasks, but to genuinely understand the physical world they inhabit.

Original article: https://arxiv.org/pdf/2511.10079.pdf

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

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2025-11-15 19:54