Lugre

LuGre is a dynamic friction model used in control, robotics, and mechanical engineering to describe the friction force at a contact interface as a function of relative motion. It improves on simple static or kinetic models by introducing an internal state that represents the average deflection of microscopic bristles at the contact. The model captures key friction phenomena, including static sticking, pre-sliding displacement, velocity-weakening behavior (the Stribeck effect), and hysteresis between different motion directions.

The LuGre model uses an internal state z, representing bristle deflection, that evolves with the relative sliding velocity v. A common formulation for the state dynamics is dz/dt = v − (|v|/g) z, where g is a parameter related to bristle geometry and the transition region. The friction force F is modeled as a sum of elastic, viscous, and damping terms: F = sigma0 z + sigma1 dz/dt + sigma2 v. Here sigma0 is the bristle stiffness, sigma1 the bristle damping, and sigma2 the viscous friction coefficient. This structure enables a smooth transition from static to kinetic friction as velocity changes.

Applications of LuGre include modeling and compensating friction in servo actuators, precision motion systems, and robotic platforms. Its ability to reproduce pre-slip behavior and to be fitted to experimental data makes it a popular choice for friction-aware control. While effective, parameter identification can be challenging, and the model remains a phenomenological representation rather than a first-principles description.