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Linear viscoelastic rheological FrBD models

Luigi Romano, Ole Morten Aamo, Jan Åslund, Erik Frisk

TL;DR

This paper extends the Friction with Bristle Dynamics (FrBD) framework by introducing two linear viscoelastic formulations based on Generalized Maxwell ($GM$) and Generalized Kelvin-Voigt ($GKV$) elements, tying rate-and-state friction to classical rheology. It proves well-posedness, boundedness, and passivity for the resulting $n{+}1$ state nonlinear ODEs and shows that the models can reproduce key friction phenomena such as pre-sliding displacement and frictional hysteresis; a passivity-based control example with a robotic arm demonstrates practical stabilization and tracking. The work provides a physically grounded, control-oriented integration of linear viscoelasticity with rate-and-state friction, yielding a unified framework that is amenable to analysis and synthesis, and offering a pathway to extend to nonlinear rheologies in future work.

Abstract

In [1], a new modeling paradigm for developing rate-and-state-dependent, control-oriented friction models was introduced. The framework, termed Friction with Bristle Dynamics (FrBD), combines nonlinear analytical expressions for the friction coefficient with constitutive equations for bristle-like elements. Within the FrBD framework, this letter introduces two novel formulations based on the two most general linear viscoelastic models for solids: the Generalized Maxwell (GM) and Generalized Kelvin-Voigt (GKV) elements. Both are analyzed in terms of boundedness and passivity, revealing that these properties are satisfied for any physically meaningful parametrization. An application of passivity for control design is also illustrated, considering an example from robotics. The findings of this letter systematically integrate rate-and-state dynamic friction models with linear viscoelasticity.

Linear viscoelastic rheological FrBD models

TL;DR

This paper extends the Friction with Bristle Dynamics (FrBD) framework by introducing two linear viscoelastic formulations based on Generalized Maxwell () and Generalized Kelvin-Voigt () elements, tying rate-and-state friction to classical rheology. It proves well-posedness, boundedness, and passivity for the resulting state nonlinear ODEs and shows that the models can reproduce key friction phenomena such as pre-sliding displacement and frictional hysteresis; a passivity-based control example with a robotic arm demonstrates practical stabilization and tracking. The work provides a physically grounded, control-oriented integration of linear viscoelasticity with rate-and-state friction, yielding a unified framework that is amenable to analysis and synthesis, and offering a pathway to extend to nonlinear rheologies in future work.

Abstract

In [1], a new modeling paradigm for developing rate-and-state-dependent, control-oriented friction models was introduced. The framework, termed Friction with Bristle Dynamics (FrBD), combines nonlinear analytical expressions for the friction coefficient with constitutive equations for bristle-like elements. Within the FrBD framework, this letter introduces two novel formulations based on the two most general linear viscoelastic models for solids: the Generalized Maxwell (GM) and Generalized Kelvin-Voigt (GKV) elements. Both are analyzed in terms of boundedness and passivity, revealing that these properties are satisfied for any physically meaningful parametrization. An application of passivity for control design is also illustrated, considering an example from robotics. The findings of this letter systematically integrate rate-and-state dynamic friction models with linear viscoelasticity.
Paper Structure (13 sections, 5 theorems, 31 equations, 3 figures, 1 table)

This paper contains 13 sections, 5 theorems, 31 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model derivation
  3. Model properties
  4. Well-posedness and steady-state solution
  5. Mathematical properties
  6. Boundedness
  7. Passivity
  8. Dynamical behavior and control applications
  9. Dynamical behavior
  10. Pre-sliding displacement
  11. Frictional lag
  12. Feedback control of a robotic arm
  13. Conclusion

Key Result

Theorem I.1

Suppose that the mapping $H : \mathbb{R}^{m+n}\mapsto \mathbb{R}^n$ is $C^1$ in a neighbourhood of a point $(x^\star,y^\star)$, where $H(x^\star,y^\star) = 0$. If the Jacobian matrix $\nabla_{y}H(x^\star,y^\star)^{\mathrm{T}}$ is nonsingular, there exist a neighbourhood $\mathcal{X}$ of $x^\star$ in for $x\in \mathcal{X}$.

Figures (3)

  • Figure 1: A schematic representation of the friction model
  • Figure 2: Linear viscoelastic rheological models for solid elements: (a) Generalized Maxwell (GM); (b) Generalized Kelvin-Voigt (GKV).
  • Figure 3: Dynamical behavior of the FrBD$_2$ models: (a) Pre-sliding displacement hysteresis; (b) frictional lag hysteresis

Theorems & Definitions (7)

  • Theorem I.1: Edwards Edwards
  • Theorem III.1: Existence and uniqueness of solutions
  • Lemma III.2: Boundedness
  • proof
  • Lemma III.3: Passivity
  • proof
  • Theorem IV.1