First-order friction models with bristle dynamics: lumped and distributed formulations

Abstract: Dynamic models, particularly rate-dependent models, have proven effective in capturing the key phenomenological features of frictional processes, whilst also possessing important mathematical properties that facilitate the design of control and estimation algorithms. However, many rate-dependent formulations are built on empirical considerations, whereas physical derivations may offer greater interpretability. In this context, starting from fundamental physical principles, this paper introduces a novel class of first-order dynamic friction models that approximate the dynamics of a bristle element by inverting the friction characteristic. Amongst the developed models, a specific formulation closely resembling the LuGre model is derived using a simple rheological equation for the bristle element. This model is rigorously analyzed in terms of stability and passivity -- important properties that support the synthesis of observers and controllers. Furthermore, a distributed version, formulated as a hyperbolic partial differential equation (PDE), is presented, which enables the modeling of frictional processes commonly encountered in rolling contact phenomena. The tribological behavior of the proposed description is evaluated through classical experiments and validated against the response predicted by the LuGre model, revealing both notable similarities and key differences.

• The paper derives Friction with Bristle Dynamics models by inverting a bristle friction characteristic, showing that LuGre emerges as a special case while adding a damping-dependent denominator term.
• The lumped model offers global well-posedness and passivity with constant micro-stiffness, while the distributed PDE model provides input-to-state and input-to-output stability under explicit spatial conditions.
• Experiments on a diaphragm valve show prediction errors comparable to LuGre, positioning FrBD as a physically interpretable alternative whose main benefits are structural guarantees rather than improved accuracy.

Motivation and contribution

Rate-dependent dynamic friction models dominate the control-oriented modeling of mechanical systems because they capture pre-sliding hysteresis, frictional lag, and stick-slip behavior, yet most such models — including the widely used LuGre model (2602.09429) — are constructed from empirical considerations rather than physical derivations. This paper addresses that gap by introducing a class of first-order dynamic friction models, termed Friction with Bristle Dynamics (FrBD), obtained through a systematic inversion of the friction characteristic applied to a rheological model of a deformable bristle element. The authors' stated contributions are threefold: a general derivation procedure for lumped and distributed first-order friction models from bristle constitutive laws; the development and rigorous analysis (stability, dissipativity, passivity) of a specific LuGre-inspired variant in both lumped and distributed forms; and experimental validation of the lumped model on a diaphragm valve system.

The paper's central structural claim is that the LuGre model itself can be recovered as a special case of this inversion procedure, thereby providing formal justification for arguments previously made only informally in the literature. Notably, the resulting model family differs from LuGre in one substantive respect: an additional damping-related term $\sigma_1(v)|v|_\varepsilon$ appears in the denominator $g(v)$ of the bristle ODE, which the authors show has significant consequences for well-posedness and passivity.

Derivation via friction-curve inversion

The general rate-dependent structure postulates $F_b = f(\dot z, z, v)$ together with a state ODE $\dot z = h(z,v)$. The proposed derivation equates the bristle force with a regularized generalized Coulomb/Stribeck friction law $F_r(v_s)$ acting at the bristle tip, where $v_s = v - \dot z$ is the sliding velocity. Inverting this implicit relation via successive approximations (an application of the implicit function theorem), truncated at first order under the assumption $|\dot z| \ll |v|$, yields

$$\dot z = -\frac{|v|_\varepsilon}{g(v)}\bigl(f(0,z,v) - \mu(v)\operatorname{sgn}_\varepsilon(v)\bigr), \qquad g(v) = \frac{\partial f}{\partial \dot z}\Big|_0 |v|_\varepsilon + \mu(v).$$

Two caveats are stated explicitly by the authors. First, the approximation is valid only outside the sticking regime ($|\dot z| \ll |v|$); consequently the model describes average bristle behavior and cannot capture instantaneous sticking effects — a limitation consistent with known properties of the LuGre model. Second, the truncation is described as a formal rather than mathematically rigorous justification.

The lumped LuGre-like FrBD model

Adopting a linear viscoelastic constitutive law $F_b = (\sigma_0(v) z + \sigma_1(v)\dot z)p$ for the bristle element produces the scalar ODE

$g(v)$0

Existence and uniqueness of solutions follow from standard ODE arguments, and global solutions are guaranteed for bounded continuous velocity inputs. A remark worth emphasizing: for constant $g(v)$1 and $g(v)$2, the term $g(v)$3 is uniformly bounded, which can ensure well-posedness in interconnections where the standard LuGre formulation would fail. The stationary solution $g(v)$4 coincides formally with that of LuGre, but here it is obtained a posteriori from a physically grounded derivation rather than imposed ad hoc.

Passivity without artificial parametrization. The strongest analytical result of the lumped analysis is Lemma 2: provided only that the micro-stiffness coefficient $g(v)$5 is constant, the FrBD model is passive with storage function $g(v)$6. This holds "virtually for every combination of model parameters," in contrast to the LuGre model, which requires postulating a velocity-dependent micro-damping coefficient to achieve passivity — a modification the authors characterize as artificial. Since passivity underpins observer and controller synthesis (passivity-based control, energy-balancing observers), this result directly widens the admissible parameter space for such designs.

A linearized variant around any stationary operating point is also derived, with stability evident by inspection.

The distributed FrBD model

For rolling contact applications (tire–road, wheel–rail, bearings), the lumped ODE is extended to a hyperbolic semilinear PDE on the contact patch by replacing the Lagrangian derivative with the Eulerian one:

$g(v)$7

with boundary condition $g(v)$8 expressing that bristles enter the contact undeformed, and output given by spatial integration against the pressure distribution $g(v)$9.

Well-posedness is established in two settings: classical solutions in $F_b = f(\dot z, z, v)$0 for smooth inputs and $F_b = f(\dot z, z, v)$1 initial data compatible with the boundary condition, and mild solutions in $F_b = f(\dot z, z, v)$2 for merely continuous inputs. Closed-form solutions along characteristic lines are derived, and steady-state profiles reproduce those of the distributed LuGre model when parameters coincide. Under a constant pressure distribution and constant coefficients, the normalized steady-state force–slip curve matches experimental tire data collected in a low-friction environment closely, for both constant and exponential pressure profiles.

Stability and a critical modeling choice. The PDE is shown to be uniformly input-to-state stable in the spatial $F_b = f(\dot z, z, v)$3 norm, which implies input-to-output stability of the integrated force via a straightforward bound on the output operator. The authors stress that these estimates require defining the virtual friction coefficient using the total time derivative of the distributed state; replacing it with the partial derivative, as done by some prior work on distributed LuGre, makes the output operator unbounded and precludes the IOS result. Distributed passivity (Lemma 5) holds under condition

$F_b = f(\dot z, z, v)$4

identical to the corresponding constraint derived for the distributed LuGre model. Constant and exponentially decaying pressure distributions satisfy it (the latter strictly), but no choice of $F_b = f(\dot z, z, v)$5 satisfies it for a parabolic pressure profile $F_b = f(\dot z, z, v)$6 — a concrete limitation for contact patches whose pressure distributions are parabolic. All ISS/IOS and passivity results are proven for classical solutions with strictly positive regularization $F_b = f(\dot z, z, v)$7; extension to mild solutions and unregularized friction is left open, acknowledged as technically involved.

Tribological behavior and experimental validation

Numerical experiments replicate three canonical tests. Pre-sliding displacement: under sinusoidal forcing at 90% of breakaway, force–displacement loops qualitatively match the classical measurements of Courtney-Pratt and Eisner and resemble LuGre responses. Frictional lag: for nonzero $F_b = f(\dot z, z, v)$8 and sufficiently large Stribeck velocity, the hysteresis loop reverses orientation relative to Hess–Soom's observations, recovering normal behavior at low $F_b = f(\dot z, z, v)$9; loop width shrinks with excitation frequency. Stick-slip: in a mass–spring–slider benchmark, FrBD and LuGre produce similar displacement and velocity trajectories, but FrBD predicts a bristle force that follows the deflection state more closely due to the stronger influence of micro-damping on $\dot z = h(z,v)$0 dynamics — a behavioral difference attributable to the modified $\dot z = h(z,v)$1.

Experimental validation uses an open-access diaphragm valve dataset with ramp (Test 1) and sinusoidal (Test 2) inputs, retaining published parameters except for $\dot z = h(z,v)$2, recalibrated by genetic algorithm. Both models agree well with measurements; FrBD attains marginally lower prediction error in Test 1 (RMSE $\dot z = h(z,v)$3 vs. $\dot z = h(z,v)$4 m) while LuGre performs slightly better in Test 2 (RMSE $\dot z = h(z,v)$5 vs. $\dot z = h(z,v)$6 m). The differences are small, so validation supports equivalence of tribological fidelity rather than superiority — the paper's case for FrBD rests primarily on interpretability and mathematical properties, not accuracy gains. Validation of the distributed variant is not performed experimentally.

Limitations and open questions

Several restrictions are conceded within the analysis. The inversion-based derivation excludes the sticking regime and captures only averaged bristle behavior, so nonlocal memory effects in pre-sliding remain outside the scope of the single-state formulation (the paper does not consider vector-valued states). Passivity of the lumped model requires constant $\dot z = h(z,v)$7; the distributed passivity condition fails entirely for parabolic pressure profiles. Stability and dissipativity results are established only for classical solutions of the regularized problem. Experimentally, only the lumped model was validated, on a single valve system, with one recalibrated parameter; whether the claimed passivity advantages translate into improved observer or controller performance remains untested. The authors themselves identify the natural open questions: alternative bristle rheologies and friction-coefficient laws, experimental validation of the distributed formulation on rolling-contact hardware, and exploitation of the model's passive structure in control and observer design.

Conclusion

This paper supplies a physically grounded derivation route for first-order dynamic friction models based on inversion of the friction characteristic applied to bristle constitutive laws. The resulting LuGre-like FrBD formulation retains LuGre's steady-state and phenomenological behavior while offering two distinct analytical advantages: uniform boundedness properties that can secure well-posedness where LuGre fails, and passivity for essentially arbitrary parametrizations without velocity-dependent damping assumptions. The hyperbolic-PDE extension carries these properties to rolling contact under explicit constraints on the spatially varying stiffness–pressure product, with the notable exception of parabolic pressure profiles. Numerical and experimental evidence indicates tribological parity with LuGre, positioning the contribution primarily as a matter of interpretability and structural guarantees rather than predictive improvement.