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Linear viscoelastic rheological FrBD models

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

arXiv:2601.13799v3eess.SYcs.SY

TL;DR

Dynamic friction models are needed to represent behaviors such as pre-sliding displacement, hysteresis, and memory, while LuGre has passivity and parameter-interpretation limitations. This letter develops FrBD formulations using Generalized Maxwell and Generalized Kelvin-Voigt elements. The models satisfy boundedness and passivity, and a robotic example illustrates passivity-based output-feedback control design.

  • Problem

    Dynamic friction models capture pre-sliding displacement, hysteresis, velocity effects, and memory, while LuGre lacks passivity for general parameter choices and has parameters whose physical interpretation is not straightforward.

    Source: §I · p. 1

  • Method

    The letter extends FrBD with formulations based on Generalized Maxwell and Generalized Kelvin-Voigt elements and analyzes their mathematical properties.

    Source: §I · p. 1, §II · p. 2, §V · p. 6

  • Results

    The proposed models satisfy boundedness and passivity, reproduce frictional hysteresis and stress relaxation, and are illustrated in a robotic output-feedback control example.

    Source: B. Mathematical properties · p. 4, 2) Passivity · p. 4, §V · p. 6

  • Takeaways & Limitations

    The findings unify linear viscoelasticity with rate-and-state friction models within a physically grounded, control-oriented approach.

    Source: §V · p. 6

Abstract

from arXiv · show

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.

I. INTRODUCTION

Dynamic friction models capture behaviors that static or velocity-only laws cannot, but LuGre has passivity and parameter-interpretation limitations. This section extends FrBD with Generalized Maxwell and Generalized Kelvin-Voigt formulations, alongside the notation and model setup used to derive them.

I. INTRODUCTION

  • LuGre captures pre-sliding displacement and frictional lag, but general parameter choices and operating conditions do not guarantee passivity, and some parameters lack straightforward physical interpretation.FrBD was introduced to combine nonlinear friction laws with constitutive equations for internal bristle dynamics [1].Source: §I · p. 1
  • The letter develops FrBD formulations based on Generalized Maxwell and Generalized Kelvin-Voigt representations and investigates their solution properties, friction behavior, and use in robotic output-feedback control.The models are motivated in part by applications involving highly viscoelastic materials, including polymers and rubbers.Source: §I · p. 1

Notation and preliminaries

  • The notation section defines real and integer sets, function spaces, and comparison-function classes used in the analysis.It also identifies the implicit function theorem as a guide for deriving the friction models.Source: Notation and preliminaries · pp. 1–2

II. MODEL DERIVATION

  • The derivation represents bristle dynamics with GM or GKV constitutive equations, yielding FrBD models of order n+1 that generalize the Dahl, LuGre, and FrBD1-KV models [1].GM uses internal branch forces and relaxation times; GKV uses internal branch deformations and normalized micro-damping coefficients.Source: §II · pp. 2–3

III. MODEL PROPERTIES

This section analyzes the FrBDn+1 models for well-posedness, boundedness, and passivity.

III. MODEL PROPERTIES

  • The FrBDn+1 models are analyzed for well-posedness, boundedness, and passivity.Source: §III · p. 3

A. Well-posedness and steady-state solution

The FrBDn+1-GM and FrBDn+1-GKV models are well-posed and admit unique solutions, while their steady-state relationships extend earlier friction models and can reproduce arbitrary steady-state friction characteristics in solids.

A. Well-posedness and steady-state solution

  • The state vectors collect bristle deformation and internal states in R^(n+1), with the listed internal-state variables differing between the GM and GKV variants.The order n+1 coincides with the number of branches in the rheological element.Source: A. Well-posedness and steady-state… · p. 3
  • For continuous inputs and any initial condition, both FrBDn+1 variants admit a unique continuously differentiable solution.For large n, closed-form transient results are unavailable, making the steady-state solution of interest.Source: A. Well-posedness and steady-state… · p. 3
  • The GM steady-state solution coincides with the LuGre and FrBD1-KV solution when σ0 = k̄0.The bristle deformation and internal-state expressions differ between the GM and GKV variants.Source: A. Well-posedness and steady-state… · p. 3
  • The relationships in (14)–(16) show that the models can reproduce arbitrary steady-state friction characteristics in solids.The GKV relationships are consistent with its parallel structure.Source: A. Well-posedness and steady-state… · p. 3

B. Mathematical properties

The FrBDn+1 models are analyzed through Lyapunov functions, and boundedness and passivity are guaranteed for both variants.

B. Mathematical properties

  • Passivity expresses friction’s dissipative character through nonnegative exchanged power, p f(t)v(t) ≥ 0.The paper states that both boundedness and passivity are guaranteed for the presented models.Source: B. Mathematical properties · pp. 3–4
  • The GM analysis considers a Lyapunov function, while the GKV analysis introduces z0(t) = z(t) − Σz_i(t) and a corresponding candidate.These candidates are used to prove boundedness and passivity.Source: B. Mathematical properties · p. 4
  • The candidate functions are differentiated along each model’s dynamics using the corresponding constitutive relationship.The resulting relations are used in the proofs of boundedness and passivity.Source: B. Mathematical properties · p. 4

1) Boundedness:

Under the stated input and parameter condition, solutions of both FrBDn+1 variants remain bounded for all times.

1) Boundedness:

  • Solutions of both FrBDn+1 models are bounded for all times when the stated conditions hold.The result applies to all initial conditions and continuous, bounded inputs.Source: 1) Boundedness · p. 4
  • The boundedness result assumes continuous inputs that are bounded over the nonnegative time axis.It applies to every initial state x0 in R^(n+1).Source: 1) Boundedness · p. 4
  • The lemma requires |y|ε ≥ |y| for every real y; the GKV proof follows the GM argument.The stated input domain is C0(R≥0; R) ∩ L∞(R≥0; R).Source: 1) Boundedness · p. 4

2) Passivity:

For either FrBDn+1 variant, the input-to-output mapping from velocity to friction force is passive, with the corresponding storage function.

2) Passivity:

  • The mapping from input v to output p f is passive for both FrBDn+1 variants.The input and output are continuous functions on [0, T].Source: 2) Passivity · p. 4
  • The GM and GKV variants use storage functions (17) and (20), respectively.These are the storage functions associated with the passivity result.Source: 2) Passivity · p. 4
  • The passivity proof rearranges the model inequalities and integrates them over time.The storage term is evaluated as V(x(t)), using the respective functions in (17) and (20).Source: 2) Passivity · p. 4

IV. DYNAMICAL BEHAVIOR AND CONTROL APPLICATIONS

The section examines the FrBD models’ qualitative dynamics and demonstrates a robotic-manipulator control application.

IV. DYNAMICAL BEHAVIOR AND CONTROL APPLICATIONS

  • The FrBD models’ dynamics are examined through frictional lag and relaxation.Source: §IV · p. 4
  • A robotic-manipulator application illustrates feedback control using the FrBD models.Source: §IV · p. 4

A. Dynamical behavior

The dynamical analysis examines frictional lag and relaxation in FrBD models. It reports frequency-dependent hysteresis behavior and relaxation dynamics that vary with rheological order.

1) Frictional lag:

  • The lag simulations use a friction-coefficient law parameterized by dynamic and static friction coefficients, Stribeck velocity, and Stribeck exponent.Source: 1) Frictional lag · p. 4
  • As excitation frequency rises from 25 to 50 to 100 Hz, peak friction force decreases and hysteresis loops broaden.The reported curves use the GM parametrization; at sufficiently high damping, large Stribeck velocity can invert the loop, while sufficiently small velocity recovers conventional hysteresis.Source: 1) Frictional lag · p. 5, Figure 3 · p. 5

2) Relaxation:

  • FrBDn+1-GM variants with n = 1, 2, and 3 produce the same steady-state response, but their relaxation dynamics depend on rheological order.The illustrated case uses τ1 = 1, τ2 = 0.1, τ3 = 0.01 s and v = 0.01 m s−1.Source: 2) Relaxation · p. 5, Figure 4 · p. 5

B. Feedback control of a robotic arm

The control example considers reference-position tracking for a one-degree-of-freedom robot arm with FrBD friction. An observer-based controller yields asymptotic tracking-error convergence for both GM and GKV models.

B. Feedback control of a robotic arm

V. CONCLUSION

The conclusion presents GM- and GKV-based linear viscoelastic friction models within FrBD. It reports well-posedness, boundedness, and passivity, and identifies experimental validation and model extensions as future work.

V. CONCLUSION

  • The proposed GM- and GKV-based FrBD models are well-posed and satisfy boundedness and passivity for all parametrizations.They are formulated as systems of n+1 nonlinear ODEs and analyzed using an energy-based Lyapunov interpretation.Source: §V · p. 6
  • The models reproduce frictional hysteresis and stress relaxation in ways described as consistent with experimental observations and established dynamic models.Source: §V · p. 6
  • The letter presents the findings as a first unification of linear viscoelasticity and rate-and-state friction models, while experimental validation and nonlinear rheological extensions remain future work.Source: §V · p. 6
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