Estimation of tire-road contact forces through a model-based approach employing strain measurements

Abstract

The employment of intelligent tires to test the vehicle performances is ever-increasing in the last years. Many research activities have been made to correlate measurements provided by sensors to the tire dynamics. In this paper, a novel tire-road forces estimation technique is presented. The developed estimator is based on an approximation of the well-known Flexible Ring Tire Model (FRTM) and therefore, it is suitable for real-time normal and tangential forces identification. Only the tread band strain measurements are employed in the developed algorithm which can estimate the tire-road forces at every wheel revolution. The tool is validated through both numerical and experimental tests. The results indicate that the developed estimation algorithm, obtained by a mathematical inversion of the FRTM, can be employed as a monitoring tool for tires and vehicles.

1 Introduction

The growing development of autonomous vehicles has as direct consequence the search for intelligent systems and devices adopted to improve safety and ride quality. Among these, smart tires play a fundamental role thanks to their ability to estimate the fundamental characteristics of interaction between vehicle and road [1]. In the last decade, various types of smart tires have been developed. Accelerometers inside the tire have been used in [2, 3] to estimate the contact patch length. In [4], accelerometers inside the tire have been adopted to identify the friction coefficient on the road surfaces. A 3-axial accelerometer mounted on the tire inner linear has been adopted in [5] to estimate tire-road friction. In [6], piezoelectric sensors fixed inside the tire have been used to estimate the tire carcass lateral deformation, Magnets vulcanized into the tire tread have been used to measure the tire deformations [7]. An optical sensor to provide estimations about the vehicle state has been presented in [8, 9]. Different configurations of strain-based smart tires have been presented in [10,11,12,13,14].

As confirmed by the paper review [10], smart tires can be divided into two broad categories referring to two different measures: acceleration or deformation. For both cases, a problem of fundamental importance is the estimation of quantities that are not directly measurable. As a result, many papers have been presented with the aim of proposing data-based estimation techniques and methodologies based on the physics modelling of the smart tire. The estimation procedures, based on signal processing, allow to correlate sensor measurements to the tire operative conditions but, they need long times to analyse data obtained by a large number of experimental tests. [10].

Many published studies indicate that strain sensors could be very effective for smart tire applications [15,16,17,18]. Indeed, strain measurements are less noisy if compared with the acceleration measures and strain sensors are less expensive and bulky than other class of sensors adopted to estimate tire deformations.

In a previous work [19], the authors presented a validation procedure of a flexible-ring model suitable for the reconstruction of tire deformations, therefore in [19] it was demonstrated that a physics-based modelling approach can reproduce the measurements acquired with strain sensors and it can be employed in real-time applications. This methodology does not require initial information on lateral slips, friction, etc., and it is particularly efficient in terms of computational effort than finite element models [20, 21]. The next step consisted in understanding how a classical UKF type estimator based on the physical model could provide some parameters of the tire/road interaction such as normal and tangential forces. In [22], the results of this study have been presented, showing the effectiveness of the proposed method. One of the limitations of the approach presented in [22] is the computational load, since the algorithm must perform a large amount of matrix operations at each time step.

The observation underlying the methodology proposed in this paper is that in common smart tire applications, vehicle and the road working condition can be estimated not necessarily at every time step but at every wheel revolution. Consequently, an estimator based on the physical modelling of the tire with a reduced computational load is developed and validated through numerical and experimental analysis. In particular, the developed estimator is based on the popular tire model developed by Gong [23, 24]. Its principal employment is the estimation of the tire-road contact forces in order to monitor the tire health conditions.

This paper is organized as follows: in Sect. 2, the estimation technique is illustrated; simulation results are presented in Sect. 3; the experimental validation is described in Sect. 4. Finally, conclusions and future developments are drawn in Sect. 5.

2 Estimation technique

The developed estimation algorithm employs the tire treadband circumferential strain measurements, provided by strain-gages mounted on the tire, combined with a physical tire model to estimate the tire-road contact forces.

In particular, the Flexible Ring Tire Model (FRTM) [23, 24] is employed to obtain, analytically, the tire inner liner circumferential strains in a closed-form to reproduce the typical measurements provided by the strain-gages. Furthermore, an approximation of the FRTM is the principal component of the developed estimator. The FRTM is schematically represented in Fig. 1. The stiffness ${k}_{w}$ and $k_{v}$ describe the elastic behaviour of the sidewall structure in normal and tangential directions, respectively.

Fig. 1

FRTM schematization

In this study, the subsequent numerical and experimental analyses are based on the hypotheses of a tire rolling on a flat road at a constant forward speed. Therefore, the rim can be considered as fixed in space and, the wheel can only rotate at a constant angular velocity. The FRTM motion description is given by the employment of space-fixed and rotating coordinate systems, both portrayed in Fig. 2.

Fig. 2

FRTM coordinate system

In particular, the rotating coordinate system is helpful to analyse the contact forces and deformation. The cylindrical coordinates $\left( {r,\phi } \right)$ and $\left( {r,\theta } \right)$ indicate the ring infinitesimal element position, respectively, in the only translational and the rotating coordinate systems.

The centre of the wheel localizes the origins of both coordinate systems. The $\left( {x,z} \right)$ and $\left( {x^{*} ,z^{*} } \right)$ coordinates represent the wheel centre translational displacement, respectively, in the only translational and the rotating coordinate systems.

The two coordinate systems are related by the following equations:

$$\begin{gathered} x = x^{*} \cos \left( {{\Omega }t} \right) - z^{*} {\text{sin}}\left( {{\Omega }t} \right) \hfill \\ z = x^{*} \sin \left( {{\Omega }t} \right) + z^{*} \cos \left( {{\Omega }t} \right) \hfill \\ \phi = \theta + {\Omega }t \hfill \\ \end{gathered}$$

(1)

where ${\Omega }$ is the wheel angular velocity and $t$ is the time. The tread band is assumed as a curved and inextensible bending beam based on the Bernoulli–Euler beam model. Therefore, the normal displacement field $w$ is related to the tangential one $v$ by the following equation:

$$w = - \frac{\partial v}{{\partial \theta }}$$

(2)

The displacement fields $w$ and $v$ are referred to any point on the elastic ring middle surface. From the treadband and wheel equations of motion, described in detail in [23, 24] in terms of the displacement fields $w$ and $v$, the normal and tangential forces $F_{x}^{*}$ and $F_{z}^{*}$ acting at the rim center are expressed as:

$$F_{x}^{*} = R\mathop \int \limits_{0}^{2\pi } \left( {k_{w} w\sin \left( \theta \right) + k_{v} v\cos \left( \theta \right)} \right)d\theta$$

(3)

$$F_{z}^{*} = - R\mathop \int \limits_{0}^{2\pi } \left( {k_{w} w\cos \left( \theta \right) - k_{v} v\sin \left( \theta \right)} \right)d\theta$$

(4)

where $k_{w}$ and $k_{v}$ are the stiffness per unit of length between the rim and the treadband, respectively, in normal and tangential directions and, $R$ is the ring mean radius. The Modal Expansion Method (MEM) is adopted to compute the displacements fields $w$ and $v$ by the partial derivative equations of motion, transforming them in a set of a time-dependent ordinary differential equations [19, 23, 24]. Therefore, $w$ and $v$ can be expressed in the rotating coordinate system through the MEM as:

$$\begin{gathered} v\left( {\theta ,t} \right) = \mathop \sum \limits_{n = 0}^{\infty } \left[ {a_{n} \left( t \right)\cos \left( {n\theta } \right) + b_{n} \left( t \right)\sin \left( {n\theta } \right)} \right] \hfill \\ w\left( {\theta ,t} \right) = - \frac{\partial v}{{\partial \theta }} = \mathop \sum \limits_{n = 0}^{\infty } \left[ {na_{n} \left( t \right)\sin \left( {n\theta } \right) - nb_{n} \left( t \right)\cos \left( {n\theta } \right)} \right] \hfill \\ \end{gathered}$$

(5)

where $a_{n} \left( t \right)$ and $b_{n} \left( t \right)$ are the generalized modal displacements and $n$ is the mode number. The parameters relative to the FRTM are given in Table 1.

Table 1 FRTM parameters

The external line forces on the ring are characterized by the following equations:

$$\begin{gathered} q_{w} \left( {\theta ,t} \right) = Q_{w} \delta \left( {\phi - \phi_{0} } \right) = Q_{w} \delta \left( {\theta - \theta_{0} } \right) = Q_{w} \delta \left( {\theta - \left( {\phi_{0} - {\Omega }t} \right)} \right) \hfill \\ q_{v} \left( {\theta ,t} \right) = Q_{v} \delta \left( {\phi - \phi_{0} } \right) = Q_{v} \delta \left( {\theta - \theta_{0} } \right) = Q_{v} \delta \left( {\theta - \left( {\phi_{0} - {\Omega }t} \right)} \right) \hfill \\ \end{gathered}$$

(6)

where $\delta$ is the Dirac delta function while $Q_{w}$ and $Q_{v}$ are the magnitudes of the normal and tangential line forces referred to a specified stationary angular coordinate $\phi_{0}$. The tire treadband tangential displacement and the corresponding normal one due to the concentrated line forces, in steady-state conditions, can be obtained through the MEM as:

$$v\left( \phi \right) = \mathop \sum \limits_{n = 0}^{\infty } [A_{n1} Q_{v} \cos n\left( {\phi_{0} - \phi + \gamma_{n} } \right) + A_{n2} Q_{w} \sin n\left( {\phi_{0} - \phi + \gamma_{n} } \right)]$$

(7)

where the modal parameters $A_{n1}$, $A_{n2}$ and $\gamma_{n}$ are given in the Appendix. These parameters are dependent on the initial stress $\sigma_{\theta }^{0}$ in the treadband due to the action of the centrifugal force and inflation pressure $p_{0}$. The initial stress $\sigma_{\theta }^{0}$ is related to the wheel angular velocity ${\Omega }$ and the inflation pressure $p_{0}$ by the following expression:

$$\sigma_{\theta }^{0} A = \left( {p_{0} bR + \rho AR^{2} {\Omega }^{2} } \right)$$

(8)

Considering a distributed load on the contact patch bounded between the front and rear angular coordinates $\phi_{f}$ and $\phi_{r}$, respectively, the solution of the FRTM can be obtained applying the superposition principle. Adopting the principle of effects superposition, well described in [19], the normal and tangential pressure distributions $q_{w}$ and $q_{v}$, considering symmetric contact patch $\left( {\phi_{r} = - \phi_{f} } \right)$, are described as follows:

$$\begin{gathered} q_{w} \left( \phi \right) = \frac{3}{4}\frac{{F_{z} }}{{R\phi_{r}^{3} }}\frac{{\left( {\phi_{r}^{2} - \phi^{2} } \right)}}{{{\text{cos}}\left( \phi \right)}} + \tau_{x} \left( \phi \right)\sin \left( \phi \right) \hfill \\ q_{v} \left( \phi \right) = \tau_{x} \left( \phi \right)\cos \left( \phi \right) \hfill \\ \end{gathered}$$

(9)

where $F_{z}$ is the normal load acting on the wheel and $\tau_{x}$ is the tangential force per unit length. The $\tau_{x}$ computation is made through a generalised Coulomb friction model as described in the following equation:

$$\tau_{x} \left( \phi \right) = \left\{ {\begin{array}{*{20}c} {\left( {\phi_{f} - \phi } \right)k_{x} \varepsilon_{x} , \quad \left( {\phi_{f} - \phi } \right)k_{x} \varepsilon_{x} \le \mu_{s} q_{z} \left( \phi \right) } \\ { - \mu_{d} q_{z} \left( \phi \right)sign\left( {\varepsilon_{x} } \right), \quad elsewhere} \\ \end{array} } \right.$$

(10)

where $q_{z} \left( \phi \right) = \frac{3}{4}\frac{{F_{z} }}{{R\phi_{r}^{3} }}\left( {\phi_{r}^{2} - \phi^{2} } \right)$ is the parabolic approximated normal pressure distribution acting on the contact patch, $\varepsilon_{x}$ is the slip ratio, $k_{x}$ is the bristle stiffness, $\mu_{s}$ and $\mu_{d}$ are, respectively, the static and dynamic friction coefficients. Discretizing the contact patch in $N_{\phi }$ equal angular steps $\Delta \phi$, the concentrated forces $Q_{w,i}$ and $Q_{v,i}$ acting at point with angular coordinate $\phi_{0,i}$, can be written as:

$$\begin{gathered} Q_{w,i} = q_{w} \left( {\phi_{0,i} } \right)\Delta \phi \hfill \\ Q_{v,i} = q_{v} \left( {\phi_{0,i} } \right)\Delta \phi \hfill \\ \end{gathered}$$

(11)

Consequently, superimposing the solution, the tangential and normal displacements are:

$$\begin{aligned} v( \phi) & = \mathop \sum \limits_{i = 1}^{{N_{\phi } }} \mathop \sum \limits_{n = 0}^{\infty } \left[ A_{n1} Q_{v,i} \cos \left( {n\left( {\phi_{0,i} - \phi + \gamma_{n} } \right)} \right) \right.\\ &\quad \left. + A_{n2} Q_{w,i} \sin \left( {n\left( {{\phi_{0,i}} - {\phi} + {\gamma}_{n} } \right)} \right) \right] \end{aligned}$$

(12)

$$w\left( \phi \right) = \mathop \sum \limits_{i = 1}^{{N_{\phi } }} \mathop \sum \limits_{n = 0}^{\infty } n\left[ { - A_{n1} Q_{v,i} \sin \left( {n\left( {\phi_{0,i} - \phi + \gamma_{n} } \right)} \right) + A_{n2} Q_{w,i} \cos \left( {n\left( {\phi_{0,i} - \phi + \gamma_{n} } \right)} \right)} \right]$$

(13)

Considering the expression of the treadband circumferential deformation [23,24,25]:

$$\varepsilon_{\theta } = \frac{y}{{R^{2} }}\left( {\frac{\partial v}{{\partial \theta }} - \frac{{\partial^{2} w}}{{\partial \theta^{2} }}} \right)$$

(14)

and substituting the expressions of $v\left( \phi \right)$ and $w\left( \phi \right)$ the circumferential strain is obtained in the MEM form:

$$\begin{aligned} \varepsilon_{\theta} (\phi) a & =\sum\limits_{i = 1}^{N_{\phi}} \sum\limits_{n =0}^{\infty} \frac{y}{{R^{2}}n^{3}} \left[ A_{n1} Q_{v,i} \sin (n(\phi_{0,i} - \phi + \gamma_{n}))\right.\\ & \quad \left.- A_{n2} Q_{w,i} \cos \left(n\left({\phi_{0,i} - \phi + \gamma_{n} } \right) \right) \right]\end{aligned}$$

(15)

The FRTM closed-form solution, in terms of circumferential strains shown in Eq. (15), is employed to generate numerical data for the force’s estimation tool validation. In particular, exploiting the previous hypothesis of constant wheel angular velocity is simple to create time-dependent strains data for multiple wheel turns.

The developed Strain-Based Forces Estimator, briefly called SBFE, considers the strain measurements collected over a wheel revolution locating their two consecutive maximum peaks in a time window of length $\Delta t_{turn}$ and centering it on the first maximum peak as shown in Fig. 3.

Fig. 3

Time window between two maximum strain peaks

The tire-road normal and tangential forces are estimated on the basis of the acquired or computed strains data, adopting a first-order approximation of the Eq. (14):

$$\varepsilon_{\theta } = \frac{y}{{R^{2} }}\left( {\frac{\partial v}{{\partial \theta }}} \right)$$

(16)

Taking into account that the normal displacement field $w = - \frac{\partial v}{{\partial \theta }}$, its approximation can be expressed as follows:

$$w\left( \theta \right) = - \frac{{R^{2} }}{y}\varepsilon_{\theta }$$

(17)

and the tangential displacement field $v$ can be given by:

$$v\left( \theta \right) = \frac{{R^{2} }}{y}\mathop \int \limits_{0}^{2\pi } \varepsilon_{\theta } d\theta$$

(18)

By introducing the integral function $\zeta \left( \theta \right)$ of the treadband circumferential strain, the integral $\mathop \int \limits_{0}^{2\pi } \varepsilon_{\theta } d\theta$ can be computed in discrete form considering the constant angular step $\Delta \theta = \Delta \phi$, numerically equivalent to the differential $d\theta$.

On the basis of the strain data knowledge, the function $\zeta \left( \theta \right)$ is computed employing the Forward Euler Method as follows:

$$\zeta \left( {\theta_{k} } \right) = \zeta \left( {\theta_{k - 1} } \right) + \varepsilon_{\theta } \left( {\theta_{k - 1} } \right) \Delta \theta$$

(19)

where $k = 0, \ldots , 2\pi$ is the angular interval relative to a single wheel revolution, divided in $k - \Delta \theta$ equal angular steps. Therefore, the displacements fields can be expressed in discrete form:

$$\begin{gathered} w\left( {\theta_{k} } \right) = - \frac{{R^{2} }}{y}\varepsilon_{\theta } \left( {\theta_{k} } \right) \hfill \\ v\left( {\theta_{k} } \right) = \frac{{R^{2} }}{y}\zeta \left( {\theta_{k} } \right) \hfill \\ \end{gathered}$$

(20)

where $\varepsilon_{\theta } \left( {\theta_{k} } \right)$ is the actual measured or numerically computed circumferential strain for $k = 0, \ldots , 2\pi$.

Finally, discretizing the equations (a) and (b) of the normal and tangential forces, the estimation of them is obtained as follows substituting at the displacement fields $w$ and $v$ their discrete forms:

$$\begin{gathered} F_{xest} = R\Delta \theta \mathop \sum \limits_{k = 0}^{2\pi } \left( {k_{w} w\left( {\theta_{k} } \right)\sin \left( {\theta_{k} } \right) + k_{v} v\left( {\theta_{k} } \right)\cos \left( {\theta_{k} } \right)} \right) \hfill \\ F_{zest} = - R\Delta \theta \mathop \sum \limits_{k = 0}^{2\pi } \left( {k_{w} w\left( {\theta_{k} } \right)\cos \left( {\theta_{k} } \right) - k_{v} v\left( {\theta_{k} } \right)\sin \left( {\theta_{k} } \right)} \right) \hfill \\ \end{gathered}$$

(21)

The Eq. (21) allows to estimate in real-time the tire-road contact forces in tangential and normal directions, at every wheel revolution, through a model-based approach on an approximated inversion of the FRTM.

Furthermore, based on the hypothesis of constant wheel angular velocity, the time between two consecutive strain maximum peaks $\Delta t_{turn}$ is employed to estimate the peripheral speed:

$$V_{est} = \frac{2\pi R}{{\Delta t_{turn} }}$$

(22)

Therefore, the SBFE (Fig. 4) can be summarized as follows:

• (1) Detect two consecutive maximum peaks of the circumferential strain;

• (2) Select a sample window equal to the distance between two consecutive peaks and center the window on the first maximum peak;

• (3) Compute the estimated wheel peripheral speed through the Eq. (22);

• (4) Compute the estimated normal and tangential forces through the Eq. (21).

Fig. 4

Strain-Based Forces Estimator workflow

3 Estimator testing

The proposed SBFE is validated through both numerical and experimental tests. The theoretical tread band circumferential strains are obtained by computing them employing the FRTM reported in Sect. 2 for different values of the slip ratio, normal loads, tangential loads, and longitudinal speeds. The experimental strain data are provided by acquired signals through strain gauges in different tire load and speed conditions. In Fig. 5, the employed four-strain time-histories to test the SBFE are portrayed.

Fig. 5

Strain datasets for numerical and experimental tests

A description of each dataset is summarized as follows:

• Numerical test 1: the simulated circumferential strains are computed applying an incremental normal load $F_{z}$ from 250 to 1000 N with a variation of 250 N for every ten revolutions of the wheel. Therefore, there are four normal load levels (250 N, 500 N, 750 N, 1000 N). The described FRTM in Sect. 2 includes the brush model for the tangential force $F_{x}$ computation. Therefore, a friction coefficient $\mu_{d} = 0.55$ is assigned to compute it. Finally, the applied longitudinal speed and the longitudinal slip ratio are $V = 10 \;{\text{km/h}}$ and $\varepsilon_{x} = 0.02$, respectively.

• Numerical test 2: the simulated circumferential strains are obtained applying a constant normal load $F_{z} = 500 \;{\text{N}}$. The total number of wheel revolutions is 40. Every ten revolutions a friction coefficient $\mu_{d}$ is assigned ($\mu_{d1} = 0.55$, $\mu_{d2} = 0.25$, $\mu_{d3} = 0.45$, $\mu_{d4} = 0.75$) to compute the tangential force $F_{x}$. The applied longitudinal speed and the longitudinal slip ratio are $V = 20\;{\text{km/h}}$ and $\varepsilon_{x} = 0.2$, respectively.

• Experimental test 1: the acquired strain data are obtained applying a constant normal load $F_{z} = 500 \;{\text{N}}$ and a measured tangential load $F_{x} = 97\; {\text{N}}$. The tire longitudinal speed is $V = 10 \;{\text{km/h}}$.

• Experimental test 2: an incremental normal load $F_{z}$ from 250 to 1000 N is applied on the tire with a gradual transition between 250 to 500 N and 500 N to 750 N. No tangential load $F_{x}$ is applied. The tire longitudinal speed is $V = 20 \;{\text{km/h}}$.

The normal and tangential loads have been estimated by the proposed algorithm exploiting the strain value under the hypothesis of a tire in straight-line free rolling conditions. The maximum peaks of each strain data are shown in Fig. 5 with circular red markers. The identification of this feature allows locating the single wheel turn to estimate $F_{x}$ and $F_{z}$ on it. The employed operative tire inflation pressure for each test is $p_{0} = 1 bar$. The FRTM parameters relative to the numerical tests are listed in Table 2.

Table 2 FRTM parameters values for numerical tests

The SBFE requires the employment of only the $h$, $R$, $k_{w}$ and $k_{v}$, which have been identified to apply the SBFE to the experimental strain data. The parameters employed for the SBFE are listed in Table 3.

Table 3 SBFE parameters values for both numerical and experimental tests

3.1 Numerical results

3.1.1 Numerical test 1

In Fig. 6, the comparison between the computed normal load variation and the estimated one, through the proposed SBFE, is shown, referring to the numerical strain dataset portrayed in Fig. 5.

Fig. 6

Normal load comparison (Numerical test 1)

The proposed algorithm is able to estimate the simulated normal load variation accurately. The deviation from the simulated data due to the first-order approximation of the circumferential strain analytical formulation is small.

Figures 7 and 8 show the mean trends and the mean percentage error (in absolute value) between the simulated and the estimated normal load.

Fig. 7

Normal load mean trend (Numerical test 1)

Fig. 8

Normal load mean percentage error (Numerical test 1)

The maximum mean percentage error of around 0.1% between the estimated normal load mean trend and the simulated one demonstrates the suitability of the developed SBFE.

In Fig. 9, the estimated tangential load is shown. The deviation between the estimated and simulated tangential load values is slightly higher than the results concerning the normal load.

Fig. 9

Tangential load comparison (Numerical test 1)

Figures 10 and 11 show the mean trends and the mean percentage error (in absolute value) between the simulated and the estimated tangential load. The maximum mean percentage error of 0.62% constitutes an indication of the SBFE’s good accuracy regarding the tangential load estimation.

Fig. 10

Tangential load mean trend (Numerical test 1)

Fig. 11

Tangential load mean percentage error (Numerical test 1)

The estimated longitudinal speed shown in Fig. 12 demonstrates the capability of the SBFE to reproduce, faithfully, the trend of the applied constant speed in real-time and for every wheel revolution. The mean errors on the normal and tangential loads are computed considering four windows, one for each $F_{z}$ level, in order to evaluate the mean value for both the simulated and estimated loads to compare them.

Fig. 12

Estimated tire longitudinal speed (Numerical test 1)

The estimated normal and tangential loads represented in Figs. 6 and 9 can be employed to regenerate the circumferential strain data through the FRTM. In Fig. 13, the comparison between the simulated and the estimated circumferential strains is represented for each normal load level. The strain data are shown about a wheel turn bounded from − 180° to 180°.

Fig. 13

Regenerated strain data (Numerical test 1)

The FRTM demonstrates its suitability to reproduce the strain data from the estimated loads through the proposed SBFE.

3.1.2 Numerical test 2

In Fig. 14, the comparison between the computed normal load variation and the estimated one, through the proposed SBFE, is shown, referring to the numerical strain dataset portrayed in Fig. 5.

Fig. 14

Normal load comparison (Numerical test 2)

The estimated constant normal load through the proposed algorithm is in agreement with the computed ones. Figures 15 and 16 show the mean trends and the mean percentage error (in absolute value) between the simulated and the estimated normal load.

Fig. 15

Normal load mean trend (Numerical test 2)

Fig. 16

Normal load mean percentage error (Numerical test 2)

The maximum mean percentage error of around 0.1% between the estimated normal load mean trend and the simulated one demonstrates the suitability of the developed SBFE. In Fig. 17, the estimated tangential load is shown. The deviation between the estimated and simulated tangential load values is slightly higher than the results concerning the normal load. The applied friction coefficient variation, described in paragraph 4 for this test, can be observed in the tangential force, confirming the capability of the SBFE to recognize a load change due to different tire-road conditions.

Fig. 17

Tangential load comparison (Numerical test 2)

Figures 18 and 19 show the mean trends and the mean percentage error (in absolute value) between the simulated and the estimated tangential load. The maximum mean percentage error of 1.1% constitutes an indication of the SBFE’s good accuracy regarding the tangential load estimation.

Fig. 18

Tangential load mean trend (Numerical test 2)

Fig. 19

Tangential load mean percentage error (Numerical test 2)

The estimated longitudinal speed shown in Fig. 20 demonstrates the capability of the SBFE to reproduce, faithfully, the trend of the applied constant speed in real-time and for every wheel revolution.

Fig. 20

Estimated tire longitudinal speed (Numerical test 2)

The regenerated circumferential strains by the FRTM based on the estimated normal and tangential loads are shown in Fig. 21. The quality of the estimated $F_{z}$ and $F_{x}$ is confirmed by the capability of the FRTM to reproduce the strain data. The tangential load variation can be observed, also in the strain data through the asymmetrical behaviour of the two minimum peaks. The asymmetry between the two minimum peaks increases with the tangential load increasing.

Fig. 21

Regenerated strain data (Numerical test 2)

4 Experimental validation

A prototype of intelligent tire based on a commercial slick radial tire DUNLOP SP SPORT 175/505 R13 (tubeless) employed in SAE Formula Student vehicles has been used for testing. In this competition, most of the vehicles have a total weight between 150 and 250 kg, prepared for short and twisty circuits, high accelerations, and velocities between 20 and 60 km/h in about 85% of every lap. In the inner liner of the tire treadband were attached three rosettes of strain gauges. The location is indicated in Fig. 22: two of them in the same cross section and the third one separated by 123.75º of angular rotation. Figure 22 also shows the support installation of the rectangular rosette strain sensors on the tire tread centre line on the tire inner liner surface of the tread. It can be assumed that the stiffness of the strain gauge doesn´t cause local stiffening effect. The strain gauge´s length is 2 mm with gauge resistance 120 Ω. The resolution of the strain measurement is 0.001 µε.

Fig. 22

Strain gauges configuration

The data acquisition system is formed by a portable module set up to a sampling frequency of 1000 Hz. The range of operation of the strain gauge module is from − 5000 to 5000 µε. It is configured in a Wheatstone quarter bridge for each strain sensor. It is equipped with batteries to supply the voltage for the operation of the microprocessor Power/Communication module. In Table 4, the employed data acquisition system with the attached strain gauges is further illustrated.

Table 4 Data acquisition system parts

The prototype of intelligent tire has been tested in a triaxial test rig which allows the speed, normal load and inflation pressure to be varied during the experiments. The test system setup is shown in Fig. 23. The drum’s curved surface has a minimal effect on the results because of its large diameter (2.44 m) which ensures that there is only slight difference between a flat road and the large drum for tire/road contact. The error in contact length is less than 0.1% [26]. The high moment of inertia of the metal drum on which the tire rolls makes it difficult to accelerate. Since the drum acceleration is the only way to apply tangential force to the tire, the low acceleration limits the applied tangential force in the experimental test. The main reason for choosing accelerations at the lower end of the speed range is that the increase in speed is more uniform, resulting in a more progressive behaviour of the measured microstrain in the instrumented tire, which is not the case at higher speeds. Therefore, the capability to accelerate the metal drum of the test rig is limited by its high moment of inertia.

Fig. 23

Indoor tire test rig

4.1 Experimental test 1

Figure 24 shows the comparison between the imposed constant normal load and the estimated one through the proposed SBFE, referring to the experimental strain dataset portrayed in Fig. 5. The imposed normal and tangential loads denoted with black straight lines shown in plots are the assumed ones as reference. Therefore, they are not the experimentally measured loads.

Fig. 24

Normal load comparison (Experimental test 1)

The proposed algorithm is able to estimate the imposed normal load variation with a higher deviation than the numerical tests. In particular, the estimated normal load value is around 550 N compared to the applied normal load of 500 N. The difference is due to the neglected nonlinearities provided by the presence of noise in measurements, causing errors in the acquisition of strains and, therefore, errors in loads prediction. Furthermore, the nonlinear tire mechanical structure is another aspect to consider regarding the deviation between the imposed and the estimated loads. However, despite the FRTM first-order approximation, the SBFE is able to provide an indication of the tire-road forces close to the experimentally applied loads directly from strain data. Figures 25 and 26 show the mean trends and the mean percentage error (in absolute value) between the simulated and the estimated normal load.

Fig. 25

Normal load mean trend (Experimental test 1)

Fig. 26

Normal load mean percentage error (Experimental test 1)

The maximum mean percentage error of around 10.3% between the estimated normal load mean trend and the simulated one demonstrates the suitability of the developed SBFE with the experimental data. A constant discordance can be observed in the test time window.

In Fig. 27, the estimated tangential load is compared to the experimentally applied ones. The SBFE can estimate, with a good estimation quality, the applied tangential load on the tire.

Fig. 27

Tangential load comparison (Experimental test 1)

Figures 28 and 29 show the mean trends and the mean percentage error (in absolute value) between the applied and the estimated tangential load. The maximum mean percentage error of 0.85% constitutes an indication of the SBFE’s good accuracy regarding the tangential load estimation.

Fig. 28

Tangential load mean trend (Experimental test 1)

Fig. 29

Tangential load mean percentage error (Experimental test 1)

The estimated longitudinal speed shown in Fig. 30 demonstrates the capability of the SBFE to reproduce, faithfully, the trend of the applied constant speed in real-time and for every wheel revolution.

Fig. 30

Estimated tire longitudinal speed (Experimental test 1)

As for the numerical tests, the FRTM is employed to regenerate the strain data starting from the estimated $F_{x}$ and $F_{z}$. In Fig. 31, the comparison between the acquired and the estimated circumferential strains are represented for applied constant normal and tangential loads. The strain data are shown about a wheel turn bounded from − 180° to 180°.

Fig. 31

Regenerated strain data (Experimental test 1)

The FRTM demonstrates its suitability to reproduce the experimental strain data from the estimated loads through the proposed SBFE. A higher reproduced strain amplitude than the experimentally acquired is observable in accordance with the slight normal load overestimation. The asymmetry of the two strain minimum peaks is well reproduced following the good tangential load estimation.

4.2 Experimental test 2

In the last test, only the results concerning the normal load estimation are provided, taking into account the non-presence of applied tangential loads. The imposed normal loads denoted with black straight lines shown in plots are the assumed ones as reference. Therefore, they are not the experimentally measured loads. In Fig. 32, the estimated normal load is compared with the applied one, referring to the experimental strain dataset portrayed in Fig. 5.

Fig. 32

Normal load comparison (Experimental test 2)

The SBFE provides a good estimation of the imposed normal load also in the transition between two different load levels. The SBFE provides a reliable estimation of the imposed normal load. The proposed algorithm demonstrates its capability to capture also the transition between two different load levels employing only strain data.

Figures 33 and 34 show the mean trends and the mean percentage error (in absolute value) between the simulated and the estimated normal load.

Fig. 33

Normal load mean trend (Experimental test 2)

Fig. 34

Normal load mean percentage error (Experimental test 2)

The maximum mean percentage error is around 12%, showing a slight disparity between the estimated normal load mean trend and the experimental one, in particular during the load transition from 500 to 750 N and for a 1000 N normal load. This amount of error is due to the approximation at the first order of the tread band circumferential strain analytical formulation, which is fundamental for the development of the proposed SBFE allowing it to invert the FRTM to obtain forces only employing strain data.

Despite this disagreement, the results demonstrate that the SBFE is able to estimate the tire-road forces for different normal loads.

The estimated longitudinal speed shown in Fig. 35 demonstrates the capability of the SBFE to reproduce, correctly, the trend of the applied constant speed in real-time and for every wheel revolution.

Fig. 35

Estimated tire longitudinal speed (Experimental test 2)

The regenerated circumferential strains by the FRTM based on the estimated normal and tangential load are shown in Fig. 36. A small asymmetric behaviour is observable on the two minimum peaks of the strains due to a residual estimated tangential force by the SBFE. Overall, the reproduced strain data demonstrate the quality of the estimated $F_{z}$ and $F_{x}$, confirming the suitability of the SBFE to estimate the forces acting on the tire.

Fig. 36

Regenerated strain data (Experimental test 2)

Histograms for summarizing results referred to vertical load $F_{z}$ and longitudinal speed $V$ referred to experimental tests are shown in Fig. 37. Although the aim of this work is not the estimation of the longitudinal speed, an overall low percentage error is observable, demonstrating the suitability of the SBFE to estimate this vehicle feature for each wheel revolution by the characteristics of the strain signals. Regarding the normal load, for each experimental test (EXP 1 and EXP 2), the overall error is higher than that associated with the longitudinal speed, but it does not exceed the 10%. Furthermore, it can be observed that for the EXP 2 test, the error is around 5.5% because it is referred to the test in which there are four levels of $F_{z}$. Therefore, the global mean value of the percentage error is indicative of an isolated peak of error due to the approximation made for the SBFE development. The previously mentioned approximation allows the employment of only the experimental measurements of strains obtained through strain gauges to estimate at each wheel revolution, both normal and tangential loads. Similar considerations can be made for the global mean percentage error of $F_{z}$, referred to EXP 1, in which the imposed normal load is constant.

Fig. 37

Global mean percentage error of normal load and longitudinal speed referred to the Experimental Tests

The SBFE is a physical-based estimator developed based on the linearization of the tire tread band circumferential strain analytical formulation. The nonlinearities concerning the structural characteristics of the tire coupled with the errors inherent in the measurements of tread band circumferential strains, including the presence of noise, have been neglected in this preliminary study. Therefore, in future works, the nonlinearities mentioned above will be considered for reducing the estimation error and increasing the accuracy of the SBFE. It is important to note that the shown percentage error is acceptable for the proposed application, also considering the potential of the SBFE to capture the trend of normal and tangential loads directly from strain data with overall good accuracy.

The results demonstrate that the SBFE can be employed to obtain a reliable indication of the tire-road forces during the vehicle running to prevent accidents and monitor vehicle health.

5 Conclusions

A model-based tool, briefly called SBFE, to estimate the normal and tangential tire-road forces employing strain measurements has been presented.

The SBFE has been developed on the basis of a first-order approximation of the Flexible Ring Tire Model (FRTM) strain solution. The approach employed to build the SBFE allowed obtaining an estimator based on a physical tire model that provides a closed-form of the tread band displacements and circumferential strains. Furthermore, the SBFE can estimate the tire-road forces at every wheel revolution through the processing of treadband strain measurements.

The SBFE has been validated through both numerical and experimental tests. The tests have been made at different tire longitudinal speeds and with normal load variations in order to ensure the capability of the SBFE to estimate the tire-road forces in various longitudinal dynamics conditions.

Furthermore, the possibility to employ the FRTM for the circumferential strain data regeneration by the estimated forces allows verifying the goodness of fit of the entire estimation process. The results have demonstrated the suitability of the developed algorithm as a real-time tool to estimate the tire-road forces for operating conditions related to SAE Formula Student racing cars. The SBFE can be used in vehicle onboard systems or to develop a smart and portable test-rig for the intelligent tire road tests. Further works will deal with the investigation in different operative conditions of other types of vehicles equipped with different tires, also trying to consider nonlinearities associated with the structure of tires and the strain measurements. The SBFE is employable for various scenarios concerning tires thanks to the possibility of parametrizing it based on the characteristics of the specific tire.

Appendix

Modal parameters:

$$A_{n1} = \frac{1}{{\pi \sqrt {\left( {M_{n} - G_{n} } \right)^{2} } + \left( {C_{n} } \right)^{2} }}$$

$$A_{n2} = nA_{n1}$$

$$n\gamma_{n} = \tan^{ - 1} \left( {\frac{{C_{n} }}{{M_{n} - G_{n} }}} \right)$$

$$M_{n} = k_{n} - \left( {n{\Omega }} \right)^{2} m_{n}$$

$$G_{n} = \left( {n{\Omega }} \right)g_{n}$$

$$C_{n} = \left( {n{\Omega }} \right)c_{n}$$

$$m_{n} = \rho A\left( {1 + n^{2} } \right)$$

$$g_{n} = - 4\rho An{\Omega }$$

$$c_{n} = c_{v} + c_{w} n^{2}$$

$$k_{n} = \left( {\frac{EI}{{R^{4} }}n^{2} + \frac{{\sigma_{\theta }^{0} }}{{R^{2} }}} \right)\left( {1 - n^{2} } \right)^{2} - \frac{{p_{0} b}}{R}\left( {1 - n^{2} } \right) + k_{v} + k_{w} n^{2} - \rho A\left( {1 + n^{2} } \right){\Omega }^{2}$$