An extended LuGre-brush tyre model for large camber angles and turning speeds

ABSTRACT This paper presents a novel tyre model which combines the LuGre formulation with the exact brush theory recently developed by the authors, and which accounts for large camber angles and turning speeds. Closed-form solutions for the frictional state at the tyre-road interface are provided for the case of constant slip inputs, considering rectangular and elliptical contact patches. The steady-state tyre characteristics resulting from the proposed approach are compared to those obtained by employing the standard formulation of the LuGre-brush tyre models and the exact brush theory for large camber angles. Then, to cope with the general situation of time-varying slips and spins, two approximated lumped models are developed that describe the aggregate dynamics of the tyre forces and moment. In particular, it is found that the transient evolution of the tangential forces may be approximated by a system of two coupled ordinary differential equations (ODEs), whilst the dynamics of the self-aligning moment may be described by combining two systems of two coupled ODEs. Given its stability properties and ease of implementation, the lumped one may be effectively employed for vehicle state estimation and control purposes.


Introduction
The transient behaviour of the tyre is an extremely complicated and fascinating subject [1][2][3], which involves the interaction of several interconnected phenomena. These may include, for example, continuous variations in the slip inputs [4], camber angles and vertical load acting on the tyre [5], nonstationary effects due to the compliance of the tyre tread and carcass, and even abrupt discontinuities in the available friction at the tyre-road interface. An exhaustive understanding of all these phenomena, and their influence upon the transient mechanism of tyre forces and moment generation, is an essential prerequisite when it comes to the design and implementation of algorithms for vehicle state estimation and control [2,6,7].
Whilst the standard brush models [1,2,8] have been relegated for a long time to the study of the steady-state tyre characteristics for pedagogical purposes, and more recently used to enhance test modalities of tyre grip [9][10][11][12], other formulations have successfully been employed to investigate the nonstationary response of the tyre to time-varying slip and spin inputs. In this context, the first analytical description is due to Schlippe and Segel [13,14], and approximates the behaviour of the tyre carcass with that of a stretched string of infinite length. Over the years, the original formulation has undergone a number of extensions, aimed at improving its accuracy by adding, for example, bristle elements representing the tread, or taking into account the compliance of the tyre in the longitudinal direction [15][16][17]. Recent contributions have also considered the exact coupled nonlinear kinematics of the rubber particles contacting the ground and the vehicle motion [18][19][20][21][22][23][24], and the presence of limited friction inside the tyre contact patch [25][26][27] according to the classic Coulomb-Amontons assumption [9,10,[28][29][30][31][32][33][34]. The inherent complexity of modelling the tyre carcass as a distributed system has legitimated the adoption of approximated formulations derived directly from the stretched string models. Amongst these, the single contact point models [35][36][37][38][39][40][41][42][43] constitute a standard approach when it comes to full vehicle dynamics simulations, since they can be easily integrated with Pacejka's Magic Formula (MF) [1,44] or other empirical steady-state tyre formulae. The basic assumption of the single contact point formulation is that the tyre dynamics may be approximated as the one of a linear system, whose main parameter is the so-called relaxation length. This is identified as the distance that the tyre needs to travel to develop 63% of the steady-state forces. In this approach, the tyre carcass is held solely responsible for the whole nonstationary process of transient generation of tyre forces and moments. In this way, the dual nature of the tyre is mimicked by a series system that behaves as a spring at low rolling speed and as a damper at high speeds. This pragmatic approach leads to a very straightforward model, which generally shows a good agreement with experimental evidence and -combined with MF, which is currently able to take into account physical phenomena connected to tyre inflation pressure, temperature and wear [45][46][47] -can also handle the presence of large camber angles and steering speeds. However, two major disadvantages of the single contact point models consist in that they neglect nonstationary phenomena connected to the tread rubber flow inside the contact patch, i.e. the transient of the bristles, and do not come with important dynamical properties, like, e.g. asymptotic and input-to-state stability, that are appetible for control applications.
An alternative approach that is also grounded on the brush theory but accounts for the dynamics of the bristles has been recently developed by the authors and renamed two-regime [48]. As opposed to the single contact point formulation, the two-regime models describe the tyre dynamics by means of a nonlinear system of ordinary differential equations directly in terms of forces and moment, and has been proven to be input-tostate stable even in combined slip conditions. On the other hand, an intrinsic difficulty connected to this formulation is the need for local inversion of the steady-state tyre characteristics.
However, being grounded on the stretched string and brush theories, respectively, both the single contact point and two-regime descriptions inherit the drawbacks originating from the Coulomb-Amontons friction model, which may pose some difficulties in the derivation of control-oriented tyre models [49]. Indeed, Coulomb-like friction models imply nonsmooth relationships between the shear stresses acting on the rubber particles contacting the ground and their sliding velocity. In particular, a major disadvantage connects to the possible presence of multiple adhesion and sliding zones inside the contact patch, whose number is unknown a priori.
Some of these limitations may be overcome by the LuGre formulation [50], which proposes an alternative friction model to the Coulomb-Amontons one. Indeed, one of the strengths of the LuGre-brush theory resides in that the calculation of the tangential forces generated at the tyre-road interface is carried out by multiplying the normal pressure distribution by a frictional state variable, which may be eventually interpreted as the deformation of a bristle travelling inside the contact patch. This eliminates the need for distinguishing between multiple adhesion and sliding zones. Additional advantages connected to the LuGre friction model are that it predicts correctly not only the Stribeck and pre-sliding displacement effects [50], but also variable breakaway forces and frictional lag phenomena, whereas the traditional brush models notoriously fail. Moreover, being based on linear transport partial differential equations (PDEs), the LuGre formulation (in its standard form) leads to a nicely-behaving dynamical system. In this context, extensive research has already been devoted to incorporating the LuGre model within the brush theory, leading to enhanced tyre models, capable of capturing dynamic effects with high accuracy and extremely suited for control applications [51][52][53][54][55][56][57].
With a few exceptions [57], these investigations are however limited to the case of combined translational slips, and systematically disregard the transient effects of the spin, which may play an important role in conditions of large camber angles and steering speeds that are typical of motorsport applications. On the other hand, these effects have been recently investigated by Romano et al. [58][59][60] taking advantage of the theoretical framework provided by the brush models [1,2,8], under the assumption of vanishing sliding (i.e. virtually infinite friction available inside the contact patch). Specifically, in [58,59], a general procedure to derive the exact analytical solution in the case of time-varying slip inputs was outlined. Then, explicit expressions for the deflection of the bristle were provided for the cases of rectangular and elliptical contact patches. It was also found in [58,59] that vanishing sliding conditions may be treated analytically for many cases of practical interests, whereas, once again, the more stringent situation of limited friction appeared to be a source of further complications when attacking the transient problem, conflicting with the need for simplified nonstationary models to be used in simulation and/or for control purposes. Whilst vanishing sliding conditions are undoubtedly interesting to explore from a theoretical perspective, they are not representative of real tyre operating conditions, therefore limiting the applicability of the results obtained in [58,59].
However, since the structure of the governing PDEs of the standard brush models is similar to that of the LuGre-brush formulation, the intuition and the results obtained in [58,59] may be effectively exploited in the development of an enhanced model for control purposes, and capable of taking correctly into account the individual contributions of the camber and turn spin slips. Therefore, the main scope of this paper is to combine the LuGre friction model with the exact brush theory developed in [58,59]. In this process, the tyre characteristics are obtained as a function of an internal frictional variable, whose dynamics is described using two coupled PDEs. These enjoy important dynamical properties that constitute an essential prerequisite when it comes to designing control algorithms and estimators for distributed dynamical systems. Moreover, although the original formulation in terms of PDEs may eventually be used directly for control-oriented applications [61,62], an approximated description would better serve the purpose. Therefore, starting from the original set of equations, a lumped model is derived that represents the transient tyre forces and moment using multiple sets of simpler ODEs. Similar approximations are well documented in the literature [51][52][53][54][55][56][57] and describe the tyre dynamics using aggregate frictional states, which may be interpreted as global or averaged variables inside the contact patch.
Whilst possible disadvantages and improvements of the LuGre model have been discussed in [63][64][65][66][67], where enhanced descriptions have also been proposed based on refined friction models [65,68], the adoption of the LuGre-brush formulation in this paper is justified by its almost ubiquitous presence in the vast control-oriented literature in the field [69][70][71][72][73][74], together with its nice properties and relative simplicity, which allow for natural extensions in several research directions. In this context, it should be also observed that the present work is mainly theoretical in nature and should be intended as a starting point for future investigations.
The remainder of the manuscript is organised as follows: Section 2 introduces the governing equations of the LuGre-brush tyre models, and states the corresponding boundary (BCs) and initial conditions (ICs); Section 3 derives the closed-form solutions for the time-varying frictional state in case of constant slip inputs and compares the resulting steady-state tyre characteristics against the standard version of the LuGre tyre model and the exact brush theory for large camber angles presented in [58,59]. The development of the approximated lumped models for the tyre characteristics is instead carried out in Section 5. Finally, Section 6 summarises the main findings of the paper and outlines future directions for research.

The LuGre-brush tyre model
In this paper, the governing equations of the brush models according to the LuGre formulation are divided in three sets: the kinematic equations, the friction relationship and the equilibrium equations.

Kinematic equations
The kinematic equations relate the time derivative of the internal frictional state z(x, t) to the local sliding velocity 1 v μ (x, t) between the tyre and the road: where • P denotes the interior of the contact patch P, modelled as a compact subset of the road plane and assumed fixed for simplicity, v μ (z(x, t), x, t) v μ (z(x, t), x, t) and is a constant matrix, often assumed to be positive-definite or even diagonal. It may be noticed that the entries of the matrix C 0 have the same dimension of a curvature. The sliding function g(v μ (z(x, t), x, t)) in Equation (1) is usually postulated in the form where μ s and μ d are the static and dynamic friction coefficients, clearly with μ d ≤ μ s , v δ is the Stribeck velocity, and δ is the Stribeck exponent. It should be observed that μ d and μ s are global quantities in the LuGre friction model, meaning that they are not function of the position inside the contact patch P.
In particular, from the exact formulation of the brush theory, the sliding velocity v μ (x, t) = V r (t)v μ (x, t) and the tangential velocity andv where V r (t) = (t)R r (t) is the so-called rolling speed of the tyre, (t) is the angular speed of the wheel hub around its axis, R r (t) is the so-called rolling radius, is the vector of translational slips, and the spin and camber tensors A ϕ (t), A ϕ γ (t) write respectively as The spin variable ϕ(t) appearing in the spin tensor accounts for both the turning speed of the tyreψ(t) around its axis and the vertical component of the rolling velocity due to camber γ (t). Hence, it may be generally decomposed as The variables ϕ γ (t) and ϕ ψ (t) are called camber and turn spin, respectively. They may be interpreted as two different signed curvatures ϕ γ = 1/R γ and ϕ ψ = −1/R ψ ; the actual curvature of the contact patch centre is thus given by the difference ϕ = 1/R γ − 1/R ψ . The camber and turn spin slips may be conveniently expressed as ratios of the total spin, that is ϕ γ = χ γ ϕ and ϕ ψ = χ ψ ϕ, with χ γ + χ ψ = 1. The coefficients χ γ and χ ψ have been introduced by Romano et al. [58] and are called camber and turn ratio, respectively.
Finally, the quantity ε γ in Equation (7a) is known as camber reduction factor and incorporates additional curvature effects due to ply-steer and conicity. For car and truck tyres, this parameter ranges between 0.4 and 0.7, whilst motorcycle tyres have camber reduction factor ε γ ≈ 0 [1,2,8,75].
Equations (1), (4), (6a) together yield the general form for the governing PDEs of the LuGre-brush tyre model for large camber angles and turning speeds. However, in this formulation, they do not admit analytical solution for the frictional state z(x, t), as also reported by Deur [57]. This is due to the fact that, according to Equations (3) and (4), the sliding function g(·) and the total velocity v μ (z(x, t), x, t) depend not only on the vector position inside the contact patch but also on the frictional state itself. Therefore, to derive a closed-form expression for z(x, t), at least in steady-state conditions, this paper approximates the original PDEs as (4), and considering average values for the longitudinal and lateral components of v μ (x, t). It is also worth observing that, for a symmetrical contact patch, it isv . Some qualitative results for the more general case in whichv μ (x, t) is assumed to vary with the coordinate x, but not with z(x, t), are instead presented in Appendix 1 (see Remark A.2).
The vector Equation (8) is composed of two linear transport PDEs and comes equipped with BC and IC given respectively by possibly for some z 0 (x) ∈ C 1 ( The IC (9b) specifies an initial distribution z 0 (x) for the frictional state, whilst the BC (9a) prescribes that the frictional state must vanish on the leading edge L . The notion of leading edge strictly connects to those of neutral edge and trailing edge, according to the following Definition 2.1.

Definition 2.1 (Leading edge, neutral edge and trailing edge for fixed contact patch):
Assuming a fixed contact shape, the leading, neutral and trailing edges L , N and T are defined respectively by where ∂P is the boundary of the contact patch,ν ∂P (x) is the outward-pointing unit normal to the boundary ∂P of the contact patch.
Definition 2.1 may be extended to the more general case of time-varying contact geometry as in Ref. [59].

Friction relationship
In the LuGre friction model, the local friction coefficient μ( inside the contact patch depends on the internal state z(x, t) according to the following relationship: where C 1 , C 2 are two matrices of constant coefficients: usually assumed to be positive definite and even diagonal. Their entries are expressed in s m −1 .

Equilibrium equations
In the LuGre formulation, the tangential forces exerted at the tyre-road interface are given by the product between the local friction coefficient μ(x, t) and the normal pressure distribution q z (x, t), expressed in N m −2 . Integrating over the contact patch P yields the following expressions for the tangential force vector F t (t) = F x (t) F y (t) T and the self-aligning moment M z (t): Equations (13a) are known as equilibrium equations. It should be observed that, according to Equation (13b), the self-aligning moment is computed on the undeformed configuration.

Analytical solutions for constant slip inputs
In the general case of time-varying slip inputs and contact patch, Equation (8) admits integral solutions in parametric form. These may be easily found by resorting to the classic method of characteristic lines for partial differential equations, as discussed in detail in [59]. However, closed-form expressions for the frictional state z(x, t) are usually only possible when the spin slips are constant and the shape of the contact patch admits simple analytical representations. Therefore, to limit the scope of this paper, only the case where the slip inputs and rolling speed are constant over time is considered. Moreover, the rolling speed is assumed to be always positive, so that the mapping between the travelled distance s = t 0 V r (t ) dt and the time is one-to-one. Owing to these premises, replacing the time variable t with the travelled distance s turns Equation (8) into Renaming and assuming parametrisations of the type , for constant slip inputs and rolling speeds the original PDEs may be restated as where the modified turn spin tensor has been defined as The solution to the above system (16a) may be derived as explained in Romano et al. [59] and reads whilst the vector denotes the position of the cambering centre C γ . Equation (18c) describes instead the dynamics of the transformed internal state ζ (ρ, ς). Specifically, the entries of the modified turning transition matrix write as follows: with the functions φ 1 (·), φ 2 (·) reading It may be easily verified that the transition matrix˜ ϕ ψ (ς , 0) in Equation (21) reduces to a rotation matrix when ϕ xx = ϕ xy = ϕ yx = ϕ yy = 0, yielding exactly the same result of the exact brush theory presented in [58,59]. The functionζ (·, ·) appearing in Equation (18c) may be expressed in matrix form as where the coefficients α xx , α xy , α yx , α yy , β x and β y depend in turn upon σ , ϕ γ , ϕ ψ and V r . Their expressions are quite lengthy and reported in Appendix A.1. Finally,ζ 0 (ρ) ζ (ρ, 0) in Equation (18c).
To derive the analytical expression for the frictional state z(x, s), a mapping between the parametric coordinates (ρ, ς) and the initial variables (x, s) should be found so to transform back the function ζ (ρ, ς) [76][77][78]. This may be done prescribing in turn the BC (9a) and IC (9b), respectively, which guarantee the uniqueness of a diffeomorphism between the coordinates (ρ, ς) and (x, s), and thus also the uniqueness of the solution for the problem at hand, at least locally. Similar considerations about the importance of the inflow BC may be drawn from an application of the energy method, as reported in Appendix 1, where the inflow BC plays again a fundamental role.
In particular, enforcing (9a) and (9b) yields two different solutions for z(x, s), corresponding to the steady-state and transient one, denoted by z − (x) and z + (x, s), and defined respectively on the subdomains P − and P + of the contact patch. The global solution over the contact patch P = P − ∪ P + may be then formally constructed as Sections 3.1 and 3.2 are dedicated to the derivation of the steady-state and transient solutions z − (x) and z + (x, s) for the frictional state.

Steady-state solution
The steady-state solution for the frictional state z − (x, s) should be sought by enforcing the BC (9a), which imposes z(x, s) = 0 on the leading edge L . Compatibility conditions imply therefore ζ 0 (ρ) = 0. By noticing that and manipulating Equations (18a), (18b) as the following representation formula may be obtained for the frictional state in the steadystate region P − of the contact patch: In particular, it should be noticed that, owing to the assumption of constant slips and fixed contact patch, the initial conditions for the independent variables s and x may be parametrised independently as s 0 (ρ) = ρ 1 and x 0 (ρ) = x 0 (ρ 2 ), respectively. Accordingly, the functions (·) and (·) appearing in Equation (29) are as follows: where the expressions for x 0 (ρ) = x 0 (ρ 2 ) should be derived starting from an analytical representation of the leading edge. In particular, the exact functions for x 0 (ρ) = x 0 (ρ 2 ) have been derived by Romano et al. [59] for a rectangular and elliptical contact patch, and are reported in Appendix A.2 for completeness. From Equation (29), it may also be observed that z − (x) is independent of the travelled distance, as it happens in the basic formulation of the brush models. Finally, it is worth mentioning that, in Equation (29), the domain P − is defined from the requirement s 0 (ρ 1 (x, s) Therefore, the subset P − of P, describing the stationary region of the contact patch, may be defined mathematically as In particular, the curve parametrised by γ (x, s) = 0 is referred to as transient or travelling edge in [59].

Transient solution
The transient solution z + (x, s) applies to the transient region of the contact patch: The analytical expression for z + (x, s) may be obtained by parametrising x 0 (ρ) = ρ for s 0 (ρ) = 0. Specifically, it follows again from compatibility that ζ 0 (ρ(x, s)) = z 0 (x 0 (x, s)), and therefore the transient solution is given by wherez 0 (x, s) z(x 0 (x, s)) and The solution constructed by patching the steady-state and transient states according to Equation (26) is clearly continuous on the travelling edge, that is z + (x, (x)) ≡ z − (x).

Steady-state tyre characteristics
Starting from the closed-form solutions for the internal frictional state, the steady-state tyre characteristics may be obtained integrating numerically over the contact patch according to Equation (13a). In this section, the tyre forces and moment are compared to those obtained using the classic LuGre-brush formulation and the exact brush theory recently developed by the authors in [58,59]. Figure 1 compares the stationary tyre forces and moment for a rectangular contact patch for different translational slip inputs σ and a fixed spin ϕ = 3.33 m −1 (with χ γ = 0.9) according to both the LuGre-brush theory for large camber angles presented in this paper and the classic one (Appendix 3). Specifically, Figure 1 was produced considering a contact patch with semilength and semiwidth of a = 0.075 and b = 0.05 m, respectively. The vertical force acting upon the tyre was set to F z = 3000 N, whilst the matrix C 0 was assumed to be diagonal with c 0xx = c 0yy = 240 m −1 . Generally speaking, a good qualitative agreement between the two formulations may be observed, which would eventually confirm the accuracy of the LuGre-brush tyre model even in its standard version. However, the novel formulation tends apparently to predict slightly larger forces. In this context, it is also important to mention that, whilst the classic LuGre-brush theory yields a unique steady-

Comparison with the classic LuGre-brush tyre models
over the entire contact patch, the model for large camber angles predicts the existence of three different analytical functions describing the internal frictional state inside as many subdomains of P, as better explained in Appendix A.2.
Similar considerations hold also true for the tyre characteristics of an elliptical contact patch illustrated in Figure 2. In generating the curves in Figure 2, the same tyre parameters were assumed as for Figure 1, except c 0xx = c 0yy = 320 m −1 . Perhaps, a substantial difference from the trends plotted in Figure 1 is that the self-aligning moment exhibits a local maximum for positive values of the lateral slip σ y . This phenomenon is a minor effect linked to the aspect ratio a/b of the contact ellipse. In particular, the fulfilment of the con- defined on the entire contact patch.

Comparison with the exact brush tyre models
A comparison between the tyre characteristics predicted by the LuGre-brush theory for large camber angles and the exact brush models presented in [58,59] is shown in Figure 3 for a rectangular contact patch, where the trend of the forces and moment is plotted versus the longitudinal slip σ x , for several fixed values of the lateral slip input σ y . Again, in generating Figure 3, a total spin slip of ϕ = 3.33 m −1 was considered. The stiffnesses and the geometrical parameters were also set as in Section 4.1. The stiffness of the bristle according to the exact brush theory was instead chosen as 2.67 · 10 7 N m −3 . Specifically, this value was slightly adjusted starting from the one that, in pure slip conditions, ensures that the derivative of the tyre forces at the origin, i.e. the slip stiffness, is the same for both formulation . 2 To provide a fair comparison, the static and friction coefficients were also assumed to be equal, that is μ s ≡ μ d in Equation (3).
Opposed to what observed from the comparison between the enhanced and classic LuGre formulations, the discrepancy with the exact brush theory (solid lines) is particularly significant, especially at low values of the lateral slip σ y , where the LuGre-brush model (dashed lines) for large camber angles tends to predict much higher lateral forces F y and self-aligning moments M z . In respect to the lateral force, this phenomenon may be easily explained by noticing that, in the standard brush tyre models, the shear stresses acting upon the bristles are always constrained below the traction bound 3 μ s q z (x), and therefore   the magnitude of the total tangential force can never exceed the friction limit μ s F z . This is not necessarily true in the LuGre-brush theory, which might yield relatively large values for the frictional state z − (x) 4 . As a result, in the example of Figure 3(a), the combined effect of the translational slip σ and spin ϕ generates a lateral force F y that in some cases is even higher than the vertical load F z acting upon the tyre. The less pronounced mismatch between the two trends for the self-aligning moment in Figure 3(b) may be explained using a similar rationale.
Analogous considerations also hold for the case of an elliptical contact patch, which is not discussed in this paper for brevity.

Aggregate model for tyre forces and moment
The exact formulation of the LuGre-brush tyre model developed in Section 3 represents the tyre dynamics in terms of PDEs. For simulation and control purposes, however, it may be beneficial to derive a simplified description based on systems of ordinary differential equations (ODEs). In this context, several contributions may be found in the literature that develop approximated formulations considering the aggregate dynamics of the frictional state z(x, t). Following the approaches outlined in [51][52][53][54][55][56][57], the present section extends this class of models to account for large camber angles and two-dimensional contact geometries.
The analysis is conducted under the following Assumption 5.1.

Assumption 5.1:
The vertical pressure distribution q z ∈ C 1 (P; R) vanishes on the trailing edge, i.e. q z (x) = 0 for x ∈ T .

Aggregate model for tyre force
The time-varying tyre forces may be described in terms of an aggregate state, which may be interpreted as an average tangential stress acting inside the contact patch. In this context, it is customary to defineẑ so that Equation (13a) may be restated as Equation (37) should be complemented with another relationship for the dynamics of the aggregate stateẑ(t). To this end, differentiating Equation (36) with respect to time giveṡ where it has been used the fact that the velocity fieldv t (x, t) is solenoidal, i.e. ∇ t ·v t (x, t) = 0. In deriving Equation (38), it should be noticed that the frictional state z(x, t) must always be continuous on a travelling edge. Discontinuities may only take place on the characteristic lines separating the subdomains of the contact patch where different analytical solutions are obtained depending on the boundary prescription. Therefore, on these lines the boundary terms vanish, as it happens also for the last integral in Equation (38) if Assumption 5.1 is satisfied. The matrix K(t) appearing in Equation (38) is diagonal and time-varying: with the entries κ x (t) and κ y (t) given by It should be observed that the functions κ x (t) and κ y (t) defined in the above Equation (40) depend upon the frictional state z(x, t), which is however unknown. Different solutions have been proposed in the literature to approximate them. In particular, a common approach consists in replacing the expression for the time-varying frictional state z(x, t) with the corresponding stationary solution z − (x). This guarantees that the lumped model and the original one yield the same result in steady-state conditions. Another option is to postulate a priori an analytical function to describe the quantity z(x, t). A more exhaustive discussion may be found in dedicated works [51][52][53][54][55][56][57], whereas the present paper privileges the first method for purposes of simplicity. A comparison between the transient response of the tyre forces to a constant slip input σ according to the exact and aggregate formulations is illustrated in Figure 4 for different combinations of σ x , σ y and a fixed value of ϕ = 3.33 m −1 (again with χ γ = 0.9). The evolution of the tyre characteristics is shown versus the travelled distance s instead of the time t (since it is assumed C 1 = C 2 = 0 for simplicity, the problem is indeed independent of the tyre speed). In general, a very encouraging agreement may be noticed between the two models, especially concerning the longitudinal force. In any case, the trends predicted by the lumped formulation converge to the exact ones approximately within s = 0.15 m.

Aggregate model for self-aligning moment
An approximated model for the self-aligning moment with aggregate dynamics may be derived similarly to what done for the tangential forces. In particular, the following states may be defined: where a and b are two characteristic geometrical parameters in longitudinal and lateral direction, respectively (intuitively, for a symmetrical contact patch, they would correspond to the semilength and semiwidth). It may be easily inferred that the aggregate states in Equation (41) represent moments averaged over the contact patch. Then the dynamic equation for the self-aligning moment becomes being the dynamics of the statesẑ xx (t),ẑ xy (t),ẑ yx (t) andẑ yy (t) described by the two following systems of coupled ODEs: andż Once again, it is worth noticing that the boundary terms in Equations (43a) and (44a) vanish if Assumption 5.1 is satisfied. The functions κ xx (t), κ xy (t), κ yx (t) and κ yy (t) appearing in Equations (43a) and (44a) read and may be again approximated using the steady-state solution z − (x) in place of the general expressions for z(x, t).
As already done for the longitudinal and lateral tyre characteristics, some possible transient evolutions of the self-aligning moment according to both the exact and lumped formulations are shown in Figure 5 for different lateral slip inputs σ y and a fixed spin ϕ = 3.33 m −1 . In this case, the response from the aggregate dynamic equations converges still relatively fast to that of the exact formulation, but the discrepancy between the two models is more evident. In particular, whilst the exact formulation predicts a transient peak, the lumped model does not. This should however be expected, since the aggregate formulation consists of two linear systems [79,80].

Conclusion
This paper presented an extended version of the LuGre-brush tyre model to handle large camber angles and turning speeds. In this context, the first main contribution of the manuscript consisted of the analytical derivation of the steady-state and transient solutions for the frictional state variable when considering constant slip inputs. Specifically, closedform solutions were deduced for a rectangular and elliptical contact patch by resorting to classic results for the theory of linear PDEs. A comparison against the standard version of the LuGre-brush tyre model highlighted minor discrepancies between the steady-state tyre characteristics predicted by the refined theory and the standard one. A possible explanation to this may be sought in that, in order to derive closed-form solutions, in this paper, the nonlinear terms involving the sliding velocity were approximated not only by neglecting the contribution due to the internal frictional state but also the variability with the coordinate inside the contact patch. In this context, a possible extension, which would however require numerical methods for hyperbolic systems of PDEs [82], could be to also taken into account the dependence upon the space variables.
On the other hand, significant differences could be observed when comparing the LuGre-brush theory for large camber angles to the exact formulation of the brush models presented in [58,59], especially at low values of the longitudinal slip. The enhanced, physical model presented in this paper may thus have the potential to become a viable alternative to Pacejka's MF, which currently appears to be the only analytical model capable of dealing effectively with large camber angles and steering speeds.
In the second part of the manuscript, approximated models for the transient tangential forces and moment were developed based on the aggregate dynamics of averaged frictional states over the contact patch. This approach is well established in the literature and leads to the derivation of simplified models describing the time-varying tyre characteristics using linear systems of ODEs. As opposed to PDEs, linear dynamical systems of ODEs are in fact easier to implement when it comes to control applications and vehicle dynamics studies. The transient response of the tyre forces and moment predicted using the aggregate model was validated against the exact formulation in terms of PDEs, showing an encouragingly good agreement.
A major disadvantage of the extended formulation presented in the paper is that, owing to the complexity of the analytical solutions, some functions that play a fundamental role in the derivation of the approximated models need to be evaluated numerically. Therefore, future research efforts should be devoted to building look-up tables for these functions depending on different tyre structural parameters and operating conditions. The validation of the model should also be conducted to assess its ability to correctly replicate the steady-state tyre characteristics at large spin slips. This will require ad-hoc experiments or procedures [46], as well as a more detailed model for the contact patch shape as a function of the camber angle. In this context, it would be also interesting to perform a comparison against the forces and moment predicted using a full version of Pacejka's MF.  z(x, t) is interpreted as a real deflection. In the brush models, such a sliding velocity is denoted by

Nomenclature
2. For an isotropic tyre, straightforward calculations yield c 0xx = c 0yy = 4kab/F z , where k is the bristle stiffness according to the exact brush theory. 3. In this regard, it is important to clarify that the term related to the partial derivatives of the bristle deflection was neglected in the calculation of the direction for the shear stress acting inside the sliding zone of the contact patch. This is usually a reasonable approximation [1]. 4. In this context, however, it is worth observing that the problem described by the linear PDEs (8) with BC (9a) fulfils the noncharacteristic condition [59], and therefore the steady-state solution z − (x) for the internal frictional state should not be expected to diverge for any x ∈ P. Further details on the purely mathematical aspects may be found in [76,77].

Appendix 1. Energy estimate for the bristle deflection
This section gives upper bounds on the internal frictional state resulting from the solution of the PDEs (8). For the sake of notation, the time variable t is again replaced by the travelled distance s. The analysis is conducted with respect to the modified formulation described by Equation (8). However, it should be observed that, as opposed to Section 3, the slip inputs and the tangential velocity field are not assumed to be constant over s.
For what follows, it is also beneficial to introduce the notion of L p -norm · L p . In particular, consider a domain ⊆ R 2 ; then, the L p -norm · L p ( ) (1 ≤ p < ∞) is defined as [76,77] A generic measurable function f (·) is said to belong to the space since either z(x, s) = 0 on L orv t (x, s) ·ν ∂P (x) = 0 on N . In this context, it should be also observed that the boundary term on the right-hand side of Equation (A5) is always positive, being by definitionv t (x, s) ·ν ∂P (x) > 0 on T . Using the fact that C 0 is positive definite, which gives z(x, s) T C 0 z(x, s) ≥ λ min (C 0 )z 2 (x, s), the following inequality may hence be deduced: .
( A 6 ) Using Hölder's and then Young's inequality for products yields .
Remark A.1: Proposition A.1 has been proved under the assumption of fixed contact patch. The proof may , however, be conducted similarly even for the case of time-varying contact patch, yielding exactly the same result as in Equation (A2) (see also [81]).

Remark A.2:
If the approximated sliding velocityv μ (x, s) is assumed to also vary with x, but independently of z(x, s), by observing thatv μ (x, s) and the function g(·) are always nonnegative, it holds that and therefore a simpler energy estimate for the total internal frictional state z(x, s) may be derived exactly as for the classic brush tyre models, for example as done in [81].

Appendix 2. Analytical solutions
This appendix yields some analytical solutions for the results presented in Section 3. P 2 stems directly from the fact that the value assumed by the bristle deflection at x = x L 1 (y) = a is the same for both Equations (A19a) and (A19b). In contrast, the continuity of the solution is not preserved on C 3 (x).

A.2.2 Elliptical contact patch
An elliptical contact patch is described as the set P x ∈ x 2 a 2 + with the leading edge parametrised by Setting y 0 (ρ) = ρ 2 and x 0 (ρ) = x L (ρ 2 ) yields the following expression for the composite function y 0 (ρ(x)): where (·) is defined as in Equation (A16). Accordingly, the initial condition for the longitudinal coordinate clearly becomes x 0 (ρ(x)) = x L • y 0 (ρ(x)), with x L (·) reading as in Equation (A23). It is worth mentioning that Equation (A24) is valid under the assumption that a 2 ≤ b(b + 1/ ϕ γ | [59].