Stochastic Modelling of Dry-Clutch Coefficient of Friction for a Wide Range of Operating Conditions

Abstract

This paper presents a stochastic regression model for predicting the coefficient of friction (COF) in automotive dry clutches with organic linings. The influence of temperature, normal load, and slip speed on COF behaviour is investigated based on a large set of clutch wear-characterization data, collected using a custom-designed disc-on-disc tribometer that replicates realistic clutch-engagement cycles. The proposed model calculates both the expected value and standard deviation of the COF. The COF expectation model takes temperature, normal load, and slip speed as inputs, and it has a cubic polynomial form selected through a feature-selection method. The COF standard deviation model is fed by the same three inputs or alternatively the COF expectation input, and it is parameterized using the maximum likelihood method. The overall model is validated on an independent characterization dataset and an additional dataset gained through separate experiments designed to mimic real driving conditions.

1. Introduction

With the motivation to improve efficiency in automotive powertrain systems, dry clutches are used not only in manual transmissions, but also in Automated Manual Transmissions (AMTs) [1], Dual-Clutch Transmissions (DCTs) [2] and Continuously Variable Transmissions (CVTs) [3], as well as in electrified powertrains where dedicated transmissions can be used to improve vehicle range and performance [4]. Apart from their better efficiency due to the lack of cooling-oil-related viscous friction losses, dry-clutch systems are often less expensive and simpler than their wet counterparts. On the other hand, a downside of dry-clutch systems is lower controllability of the transmitted clutch torque than in wet-clutch systems, because of wider coefficient of friction (COF) variations, partly due to a broader operating temperature range [5,6]. Also, the dry-clutch materials are more prone to negative COF vs. slip speed gradient, causing shudder vibrations in specific operating conditions [7]. To gain insights into the engagement process of dry clutches, which involves complex interactions between mechanical, thermal, and tribological factors [8], sophisticated mathematical modelling is necessary [9].

Friction linings for dry automobile clutches are typically manufactured from composites containing glass and copper fibres bound by phenolic resin [10,11], commonly referred to as organic friction materials. When in contact with the steel pressure plate during clutch operation, the lining exhibits friction which is, apart from the applied normal force and clutch geometry parameters, dependent on the COF, which can vary significantly depending on several operating parameters such as slip speed [12], normal force [13], and temperature [10]. These dependencies are not independent, and to predict friction behaviour accurately, one must consider their interaction [14]. In the early stage of a dry clutch plate’s life, the COF undergoes fluctuations accompanied by elevated wear rates, a period known as the running-in phase [15]. After this phase, the COF generally stabilizes under constant operating conditions, while under severe driving conditions it may reduce by up to 30% [16]. The COF reduction relates to an effect called fading, which occurs because of gas formed on the friction surface by thermally induced decomposition of organic clutch linings [17]. During fading, the COF decreases and the wear rate increases significantly; however, both parameters return to their typical values once the fading conditions cease [18]. Mitigation measures include the use of high-permeability friction materials, allowing the gases to escape from the friction surface [19]. COF experimental characterization is mostly performed on pin-on-disc tribometers, where a small sample of friction material slides against the steel disc [20,21]. The temperature increase of the sliding contact is the result of heat generated by friction itself. Because of its significant influence on COF and wear, extensive research has been conducted to determine the temperature fields for a pin-on-disc system [22] and for a full-friction clutch with different groove patterns [23]. The distributions of the contact pressure under different operating conditions is modelled in [24] using a finite element method, while the influence of pressure distribution on the friction surface is investigated in [25]. The COF is also affected by the steel disc surface roughness, where higher roughness leads to more wear debris and decreased COF [26].

This paper deals with experimental characterization and data-driven modelling of COF for a dry-clutch organic friction material pressed against a steel pressure plate. A large amount of experimental data was acquired from a custom-designed disc-on-disc tribometer with real clutch friction and pressure plate materials under realistic clutch closing operating conditions and long wear-related tests. The COF modelling is formulated as a regression problem with three parameter inputs, including clutch temperature, slip speed, and normal force. The obtained regression model, providing the COF expectation, is accompanied by a COF variability model, which delivers a confidence interval of COF estimate, thus accounting for inherent uncertainties. The overall model is finally validated on an independent dataset obtained through regular characterization experiments, as well as on a separate dataset gained through experiments involving frequent random variations of operating parameters to mimic real driving conditions.

The main contributions of the paper can be summarized as follows: (i) the experimental characterization of dry-clutch COF for a wide range of influential operating parameters throughout the full wear depth, and (ii) the development and validation of a stochastic multi-input COF regression model accounting for COF expectation dependence on operating parameters and its inherent uncertainties seen in data.

The remaining part of the paper is organized as follows. Section 2 outlines the tribometer rig, including its mechanical and control subsystems. Section 3 describes the design of experiments used for COF and original wear characterization. Section 4 presents and analyzes the COF characterization results. Section 5 and Section 6 elaborate on the proposed COF expectation and variability models, respectively. Section 7 presents the experimental model validation results, while Section 8 draws concluding remarks.

2. Experimental Setup

2.1. Mechanical Subsystem

The disc-on-disc tribometer (Figure 1; [27]) has two degrees of freedom corresponding to rotational and vertical motion. These two degrees of freedom define the machine’s rotational and vertical axes. The rotational axis includes a water-cooled rotating table driven by an electric servomotor. The rotating table carries the original dry-clutch friction plate. The vertical axis holds a pressure plate cut from the original clutch flywheel (Figure 1a), a water-cooled disc, a leaf-spring suspension, three tri-axial piezoelectric force sensors, and an electric servomotor with a spindle that provides the normal force. The pressure plate is equipped with three circumferentially equidistant temperature sensors positioned 4 mm above the friction surface (Figure 1b). The upper bell of the vertical axis is supported by three linear guides, and three three-axial force sensors are positioned between the lower and upper bells to prevent parasitic force transfer, thereby ensuring high precision of the normal-force and torque measurement system. Since only one side of the friction plate is in contact with the pressure plate, the resulting torque is half of that generated in the real clutch under the same normal force.

2.2. Control Subsystem

The experiments are designed to replicate realistic clutch closing cycles (Figure 2a). Each experiment is defined by four parameter inputs corresponding to the closing cycle conditions: clutch temperature ($T_d$), initial slip speed (${\omega }_0$), clutch torque ($M_z$), and closing time ($t_2$). Each cycle consists of five distinctive phases, as illustrated in Figure 2b [28]. In Phase 1, the slip speed is ramped up to a target initial value ${\omega }_{0R}$ under idling conditions. During Phase 2, the vertical axis is lowered until the friction contact is established. At the beginning of Phase 3, the closed-loop position control of the vertical axis switches to closed-loop force control, and the normal force is then ramped up over the time interval $t_1$. In Phase 4, the normal force is maintained at the reached target level $F_{zR}$ until the clutch stops during the interval $t_2$. Before the next closing cycle, in Phase 5, the cooling delay of a variable length $t_d$ is introduced as the main mechanism of clutch temperature control.

The parameter inputs are accurately controlled via feedback controllers acting on the vertical- and rotational-axis servo motors and the water-cooling system [28]. The clutch is heated by friction itself, where a certain number of closing cycles are required to reach the target temperature $T_{dR}$ (see Figure 2a). Apart from manipulating the cooling delay $t_d$, the temperature control involves adjustment of the water pump speed and an on/off valve that bypasses the cooling disc. The torque is controlled by an integral (I) controller that commands the target normal force ($F_{zR}$) to a cascade force feedback control subsystem. For the purpose of closing time ($t_2$) control, an effect of electric inertia is introduced by manipulating the rotational-axis servomotor torque in proportion to the measured friction torque, where the torque ratio is commanded by an I controller to achieve the target closing time $t_{2R}$. Several closing cycles are generally required for the torque and closing time to stabilize at their target values (see Figure 2a).

3. Design of Experiments

3.1. Characterization Experiments

The characterization experiments were originally designed to characterize clutch wear over a wide range of input parameters [29], while simultaneously collecting a large amount of high-sampling-rate (500 Hz) data for COF characterization. The wear-characterization experiments were performed by repeating clutch closing cycles with fixed target values of parameter inputs (see illustration in Figure 2a) until a sufficient mass of friction material had been worn down (0.3 g, for the sake of precise worn-mass measurement) and at least 750 closing cycles had been recorded (for suppression of wear-rate “noise”) [29]. The parameter input sets involved all combinations of three distinctive levels of each input: $T_{dR}\in \left\{120, 170, 240\right\} ℃$, ${\omega }_{0R}\in \left\{1200, 1700, 2800\right\} \mathrm{r}\mathrm{p}\mathrm{m}$, $t_{2R}\in \left\{0.9, 2.15, 3.4\right\} \mathrm{s}$, and $M_{zR}\in \left\{25, 50, 75\right\} \mathrm{N}\mathrm{m}$, resulting in 3⁴ = 81 experimental points (blue dots in Figure 3). Because of machine limitations in parameter-space edge regions, the parameter inputs were modified for several of those 81 points (empty blue circles). The wear-model training dataset includes an additional 20 extrapolation points, represented by red dots in Figure 3. Finally, there are 20 additional interpolation points used for wear-model validation (green circles in Figure 3). In total, 13 friction plates were worn down to record all 121 wear-characterization points. Note that the recorded COF dataset is very large because there are at least 750 clutch closing cycles per wear-characterization test and many COF sampling points per closing cycle (see Figure 2).

In the early, so-called run-in stage of friction plate life, the friction material experiences increased wear rate and reduced COF [30]. To avoid the influence of run-in transients on the above-described stationary characterization tests, each plate was first subjected to wear under fixed input parameters until wear-rate stabilization was detected (so-called run-in experiments). These run-in recordings are not included in the COF dataset.

Although the wear-characterization experiments correspond to the fixed target values of parameter inputs for hundreds of clutch closing cycles, their actual values typically fluctuate around the target values due to the transient nature of the tests and limited bandwidths of the closed-loop control systems. For instance, the slip speed falls from ${\omega }_{0R}$ to zero during the closing, the temperature exhibits slow transient behaviour at the beginning of each characterization and quasi-stationary fluctuations within a closing cycle, and the normal force fluctuates in response to torque control under the conditions of COF variations (Figure 2). The fluctuations of input parameters are actually beneficial from the perspective of COF characterization and modelling, as they enrich the input parameter dataset compared to the discrete set of target values in Figure 3.

3.2. Cycle-Wise Validation Experiments

In addition to the 20 validation experiments, executed for the fixed operating conditions (Section 3.1), a second set of validation experiments correspond to varying-parameter conditions to emulate real-world clutch operation [29]. These experiments were recorded over the full wear depth of two additional friction plates. The tests for Plate I concerned a constant target temperature of 170 °C, while Plate II experienced floating-temperature. The remaining mechanical input parameters were selected randomly from the set of characterization points (blue points in Figure 3) and changed after every 10 closing cycles. The normal force was restricted to values lower than 1500 N to ensure reliable operation under more dynamic conditions. To maintain the floating temperature response within regular bounds in the case of Plate II, the bypass valve was automatically controlled. Its action was aimed to correct the temperature value if it exceeded the limits set to [100, 170] °C for low-to-mid temperature conditions and [150, 240] °C for high-temperature conditions, where the former was set to occur twice as often as the latter [29].

4. COF Characterization

4.1. Data Preprocessing

COF values were reconstructed from the clutch torque ($M_z$) and normal-force ($F_z$) data, acquired from the three three-axial force sensors (Figure 2c, [27]), as:

$$COF={\displaystyle \frac{1}{r_{eff}}}·{\displaystyle \frac{M_z}{F_z}},$$

(1)

where $r_{eff}={\displaystyle \frac{2}{3}}·{\displaystyle \frac{r_{out}^3 - r_{in}^3}{r_{out}^2 - r_{in}^2}}$ is the friction plate effective radius determined from the plate inner ($r_{in}$) and outer radii ($r_{out}$), respectively. COF values are shown in Figure 2c and throughout the paper in a normalized (per unit, p.u.) form.

The COF data were extracted from all 121 wear-characterization points. To remove high-frequency noise from the data recordings (particularly the torque signal; see Figure 2b), a zero-phase Butterworth filter (Matlab function filtfilt (.)) with a cut-off frequency of 5 Hz was first applied offline to clutch torque, normal force, slip speed, and clutch temperature signals. The cut-off frequency was selected based on the torque signal amplitude-to-frequency characteristic, showing that the relevant, low-frequency content of the signal resided well below 5 Hz. Finally, the filtered data were down-sampled from the original sampling rate of 500 Hz to 10 Hz, thus reducing the input dataset size from 37 GB to less than 1 GB. This resampling rate of 10 Hz was chosen in accordance with the Shannon sampling theorem (a double value of the signal cut-off frequency), guaranteeing that the signal preserved its features after resampling.

The COF can be reconstructed during both clutch-engagement phases ($t_1$ and $t_2$ in Figure 2). The phase $t_1$ is characterized by fast transient of normal force, and the COF reconstruction can be affected by dynamic effects. Thus, the COF reconstruction was narrowed to the longer (and more relevant) quasi-stationary phase $t_2$ only, where the following data-selection criterion was applied to filtered and resampled data (see Figure 2c for illustration):

$${\omega }_{r,k}>25{\displaystyle \frac{\mathrm{r}\mathrm{a}\mathrm{d}}{\mathrm{s}}} \wedge M_{z,k}>10 \mathrm{N}\mathrm{m} \wedge F_{z,k}>0.99·\underset{k}{\max } F_{z,k}, \forall k$$

(2)

to omit the low slip speed/torque and unsettled normal-force data, with k denoting the sampling step.

The parameter inputs of the finally selected COF data are distributed according to the histograms shown in Figure 4. The clutch temperature points group around the target values 80, 120, 170, 240, and 270 °C (cf. Figure 3). The slip speed distribution is continuous due to the continuous, wide-range nature of slip speed time response (cf. Figure 2), with most of the points lying in the low-slip range as it is present in all responses. The normal force is also well distributed (Figure 4c) despite the fixed torque target values, because of COF variations with input parameters and throughout wear depth. The number of selected data points is very large (exceeding three million; see y-axis of plots in Figure 4 and related elaboration in Section 3).

4.2. Analysis of Influence of Individual Operating Parameters on COF

The COF data plotted individually with respect to three input parameters are presented in Figure 5 in the form of heatmaps. Figure 5a reveals that the COF exhibits a positive correlation with temperature. The corresponding correlation coefficient calculated using the Matlab function corr (.) equals 0.483 (

Table 1. Cross-correlation indices of COF and individual parameter inputs, as well as between input parameters themselves, including related VIFs.

Correlation	COF	Normal Force	Slip Speed	Temperature
COF	-	−0.377	0.385	0.483
Normal force	−0.377	-	−0.102	0.025
Slip speed	0.385	−0.102	-	0.234
Temperature	0.483	0.025	0.234	-
VIF	-	1.013	1.071	1.060

; note that the characteristic values are 1, 0, and −1 for perfect correlation, no correlation, and perfect anti-correlation, respectively). When related to normal force, the COF shows negative correlation (Figure 5b), which is somewhat less emphasized (in absolute terms) than in the case of temperature input, but still significant (Table 1). The dense (yellow) areas of the COF vs. normal force heatmap in Figure 5 relate to hyperbolic curves resulting from Equation (1) and corresponding to the fixed target torque levels of 25, 37.5, 50, 62.5, and 75 Nm (see Figure 3; note that COF significantly varies with input parameters and wear even though the target torque is constant). Finally, the correlation of COF with respect to slip speed is positive (Figure 5c) and comparable to that of normal force in terms of absolute value of the correlation coefficient (Table 1). The high density of points in the low-speed region relates to discussed low-speed specifics of the histogram shown in Figure 4b. The same applies to the connection between the dense areas in Figure 5a and the peaks of the histogram in Figure 4a.

From the perspective of modelling, it is important to check that the individual parameter inputs, i.e., predictors, are not correlated with each other. The results given in Table 1 confirm that these correlations are weak (near-zero correlation coefficients). Exceptionally, the correlation between slip speed and temperature may be regarded as considerable (correlation coefficient of 0.234). However, this correlation occurs because of the specific design of experiments for extrapolation points, which are usually (feasibly) recorded at high-temperature–high-energy/slip-speed conditions and low-temperature–low-energy/slip-speed conditions (Section 2). To check the potential multicollinearity between the predictor variables in a more rigorous way, the Variance Inflation Factor (VIF) is evaluated. Table 1 indicates that all predictors exhibit VIF values that are very close to the perfect no-multicollinearity value of 1, thus demonstrating that the characterization dataset is appropriate for regression analysis.

Furthermore, a cluster analysis is performed for three characteristic ranges of input parameters in order to further analyze interactions between the model inputs (predictors) and the COF and to visualize the scattering of COF values under various operating conditions. According to the results shown in Figure 6a, and similarly in Figure 6c, the COF variation/spread increases with temperature in the low-to-mid normal-force region. The COF spread is largely unaffected by the slip speed (Figure 6b). Hence, the COF variations are most emphasized in the mid-to-high-temperature and low-to-mid-normal-force regions.

Figure 7 further illustrates the COF distributions through comparison of probability density functions (PDFs) for narrow ranges of parameter inputs against the PDFs of all data points. At low temperatures (the first row of graphs), the PDFs are very narrow, indicating a stable and consistent COF (i.e., low inherent variability). At mid temperatures, the COF remains consistent only under high normal forces, whereas at high temperatures the COF scatters for all normal forces. These findings correlate well with those shown in Figure 6.

4.3. Specific Effects of COF Behaviour

When analyzing COF data scattering in further detail, some specific, relatively rare effects were observed. These effects are illustrated in Figure 8, while their statistics are summarized in

Table 2. Statistics of specific COF behaviour effects observed in experimental data for three friction materials.

Specific COF Behaviour Effects	Number of Wear-Characterization Points in Which Specific Effect Occurs out of 121 Points from Figure 3
	Material A	Material B	Material C
COF creep	14 (11%)	32 (26%)	26 (21%)
Mid-speed COF drop and scattering	5 (4%)	27 (22%)	9 (7%)
Negative COF vs. slip speed correlation	17 (14%) correlation < −0.2: 6 (5%)	34 (28%) correlation < −0.2: 14 (11%)	27 (22%) correlation < −0.2: 8 (7%)

. Firstly, the COF sometimes slowly varies through consecutive clutch closing cycles (so-called COF creep effect; Figure 8a). The COF creep cannot be explained by input parameters because they are well controlled around the fixed target values for the whole experiment. It may be explained by (i) slowly varying friction lining properties through the depth of friction material due to heterogeneity of the composite friction material [31] (see [32] for a basic analysis of material properties), and (ii) variation in contact pressure, and thus temperature, due to non-ideal contact surface geometry. Secondly, although the COF vs. slip speed curve gradient is generally positive (see Figure 5c), it may sometimes be negative (Figure 8b), i.e., shudder-sensitive [7,28]. Thirdly, the COF response occasionally exhibits characteristic drops and increased variability at around mid speeds, as illustrated in Figure 8c.

Table 2 indicates that the above specific COF effects are quite rare, i.e., for the given friction material (Material A) they occur in around 10% of wear-characterization points from Figure 3. Nevertheless, they contribute to the COF variations observed in Figure 5 and would, accordingly, affect the regression model’s accuracy. The table also presents the specific effect statistics for two additional friction materials tested (see Appendix A). The effects are more emphasized/frequent for the additional materials, but they still occur in a minority of experiments.

4.4. COF Variability Among Different Friction Plates

An additional source of COF variability is due to production deviations of friction plates (so-called plate-to-plate variability). To characterize this variability, the COF run-in responses are used, since they correspond to the same (run-in) operating conditions. Figure 9a shows these responses for 13 characterization friction plates, where the COF values designated represent the average COF over the active clutch closing ($t_2$) phase and for all closing cycles of a single batch of wear-characterization cycles (see Section 2 and [29] for details of experimental procedure). In general, the COF increases during the run-in phase, which is in correlation with simultaneous wear-rate decrease (Section 3, [18,29]). Figure 9b presents a histogram of the average COF values during the stabilized phase for each friction plate, revealing quite significant variability of COF (and also wear rate [29]) caused by production deviations. The histogram follows the normal distribution well (the p-value of 0.28 is higher than the normality threshold of 0.05).

5. Modelling of COF Expectation

5.1. Modelling Methodology

Similarly to wear-rate modelling [29], a linear regression is employed for COF expectation modelling. In accordance with the COF characterization results from Section 4, the model inputs include clutch temperature, slip speed, and normal force. The full cubic polynomial function is selected as a basis for modelling, which is justified by apparently simple individual input–output dependencies, illustrated through data fits in Figure 5. Unlike in the case of wear-rate modelling, the exhaustive, globally optimal best subset modelling approach is not feasible here due to the large input–output dataset. Therefore, a proper, ordered, nearly optimal COF model structure is found using a limited-size (step-wise) search over a number of sub-models of the full cubic model:

$$\begin{array}{l}\widehat{y}={\beta }_1+{\beta }_2{x}_1+{\beta }_3{x}_2+{\beta }_4{x}_3+{\beta }_5{x}_1{x}_2+{\beta }_6{x}_1{x}_3+{\beta }_7{x}_2{x}_3+{\beta }_8{x}_1^2+{\beta }_9{x}_2^2+{\beta }_{10}{x}_3^2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +{\beta }_{11}{x}_1{x}_2{x}_3+{\beta }_{12}{x}_1^2{x}_2+{\beta }_{13}{x}_1^2{x}_3+{\beta }_{14}{x}_1{x}_2^2+{\beta }_{15}{x}_1{x}_3^2+{\beta }_{16}{x}_2^2{x}_3\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +{\beta }_{17}{x}_2{x}_3^2+{\beta }_{18}{x}_1^3+{\beta }_{19}{x}_2^3+{\beta }_{20}{x}_3^3,\end{array}$$

(3)

where $\widehat{y}$ is a dependent response variable; $x_1, x_2$ and $x_3$ are independent predictor variables; and ${\beta }_i$ are model parameters. This model has 20 free parameters and a total number of 2²⁰ − 1 = 1,048,575 possible sub-models. In the context of COF modelling, the predictor variables from (3) correspond to normal force ($x_1=F_z$), slip speed ($x_2={\omega }_r$), and clutch temperature ($x_3=T_d$), while the response variable represents the COF expectation: $\widehat{y}={CoF}_{\mu }$. The predictor variables form the input features $z_i, i\in \left\{\mathrm{1,2},\dots ,20\right\}$ (i.e., $z_1=1$, $z_2=x_1$, …, $z_{20}=x_3^3$), which are normalized to the interval [0, 1] for better numerical conditioning:

$$z_{j,i,norm}={\displaystyle \frac{z_{j,i}-z_{i,\mathrm{m}\mathrm{i}\mathrm{n}}}{z_{i,\mathrm{m}\mathrm{a}\mathrm{x}}-z_{i,\mathrm{m}\mathrm{i}\mathrm{n}}}},$$

(4)

where $z_{j,i,norm}$ and $z_{j,i}$ are the normalized and true values of the ith input feature and the jth modelling point, respectively; $j\in \left\{\mathrm{1,2},\dots ,N\right\}$, $z_{i,\mathrm{m}\mathrm{i}\mathrm{n}}$, and $z_{i,\mathrm{m}\mathrm{a}\mathrm{x}}$ are the minimal and maximal values of the feature $z_i$ across all (N) modelling points.

The assessment of different sub-model candidates and the selection of the optimal one are based on the coefficient of determination R² [33]:

$$R^2=1-{\displaystyle \frac{\sum _{j = 1}^N{\left({COF}_j-{COF}_{\mu ,j}\right)}^2}{\sum _{j = 1}^N{\left({COF}_j-\overline{COF}\right)}^2}},$$

(5)

where $\overline{COF}$ is the mean COF among N recorded modelling points, ${COF}_j$ is the actual recorded COF, and ${COF}_{\mu ,j}$ is the model prediction. The coefficient R² can be interpreted as a proportion of variation in the dependent response variable, here COF, which is predictable from (i.e., explainable by) the independent predictor variables. It takes values in the range [0, 1], where the maximum value of 1 corresponds to the idealized, zero-error model case.

The overall modelling procedure is depicted by the block diagram shown in Figure 10. The points from the modelling dataset are used to form features for the full cubic model (3), which are then normalized according to Equation (4). The following feature-selection methods were considered to determine the optimal model structure: (1) Forward sequential feature selection without sorting, (2) Forward sequential feature selection with sorting, and (3) Least absolute shrinkage and selector operator (LASSO) [34]. Finally, the model-testing procedure leads to the final model selected.

The forward sequential feature selection without sorting introduces the features sequentially in the same order as they appear in Equation (3) (i.e., from $z_1$ to $z_{20}$). After adding a new feature into the model, its contribution to the model performance is evaluated. Although this method provides good insights into how model performance evolves as higher-order features are incrementally introduced, the solution may be quite distant from the optimal one since the features are added in a fixed (unsorted) order.

When the forward sequential feature-selection method involves sorting, the full cubic model is first fitted to the data using the least squares method. The model features are then sorted according to the absolute values of corresponding regression coefficients, ${|\beta }_i|$ $i\in \left\{\mathrm{2,3},\dots ,20\right\}$ (the intercept ${\beta }_1$ is not sorted, but added in all sub-models). The forward selection process then sequentially adds the most influential features (i.e., those with the largest coefficients ${|\beta }_i|$) into the model. By prioritizing the inclusion of features that exhibit greater estimated importance based on the full model, this method aims to construct a reduced-order model that captures the dominant relationships more efficiently than the unsorted method.

The LASSO method penalizes the magnitude of absolute value of regression coefficients, $\left|{\beta }_2\right|,\dots ,\left|{\beta }_p\right|$, within the least squares model parameter-optimization procedure [34]:

$${\widehat{\beta }}^{\mathrm{L}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{o}}=\underset{\beta }{\mathrm{argmin}}\left\{{\displaystyle \frac{1}{2}}\sum _{j =1}^N{\left({COF}_j-\underset{{\widehat{y}}_j}{\underbrace{{\beta }_1-\sum _{i =2}^p{z}_{ij}{\beta }_i}}\right)}^2+\lambda \sum _{i =2}^p\left|{\beta }_i\right|\right\},$$

(6)

where p = 20 is the number of model features (the intercept ${\beta }_1$ is omitted from the penalization term, as it does not affect feature importance), and $\lambda$ ≥ 0 is the regularization parameter. The coefficients which are not significant for modelling accuracy then shrink to near-zero values when conducting the minimization in Equation (6). If $\lambda$ is set to 0, the method reduces to the standard least squares method. As $\lambda$ increases, more model parameters shrink to zero, and with a sufficiently large $\lambda$ all but the intercept parameter ${\beta }_1$ would cease. In this way, a favourable trade-off between model complexity (expressed in the number of features controlled by $\lambda$) and the model accuracy (evaluated through Equation (5)) can be established. The modelling procedure was performed for 100 different values of parameter $\lambda$, ranging from 0 to the value resulting in shrinkage to the intercept parameter only ($\widehat{y}={\beta }_1$).

Parametrization of the sub-models of different complexities, obtained by the considered feature-selection methods, follows the least squares method [34]:

$$\mathbf{\beta }=\underset{\mathbf{\beta }}{\mathrm{argmin}}{\displaystyle \frac{1}{N}}\sum _{j=1}^N{\left({COF}_{\mu ,j}\left(\mathbf{\beta }\right) - {COF}_j\right)}^2.$$

(7)

with the solution

$$\mathbf{\beta }={\left({\mathbf{X}}^{\mathrm{T}}\mathbf{X}\right)}^{-1}{\mathbf{X}}^{\mathrm{T}} \mathbf{C}\mathbf{O}\mathbf{F},$$

(8)

where COF_(Nx1) is the column vector containing the recorded ${COF}_j$ points; X_(Nxp) is the design matrix containing the corresponding selected features $z_{j,i,norm}$ calculated from the recorded input parameters (see Equation (4)).

5.2. Model Selection

The prepared characterization dataset (Section 3) was randomly divided into (i) modelling (20%) and (ii) testing datasets (80% of the total number of points). Only 20% of data points were selected for modelling to computationally unburden the model parametrization (N in Equations (7) and (8) is reduced by five times), without sacrificing the accuracy, since the entire dataset is overwhelming (around 3 million points) and the number of model parameters to be determined is low (20 at maximum). Confirming that the PDFs of COF and related model inputs are comparable for the modelling and full datasets (Figure 11) justifies the data-selection approach.

Figure 12 shows the R² metrics of models obtained by different feature-selection methods, where the full lines (and corresponding circles) and the dashed lines correspond to modelling and testing datasets, respectively. Evidently, the R² profiles obtained for the testing and modelling datasets closely match each other, thus clearly indicating the absence of overfitting. This outcome can be attributed to the massive number of modelling dataset points compared to only up to 20 model parameters. Thus, an intervention to avoid overfitting is unnecessary, unlike in the case of the wear model [29], where for the same number of model parameters there were only 101 modelling points, and where a leave-one-out cross-validation was employed to mitigate overfitting.

Figure 12 further indicates that all the three methods reach a R² value of around 0.45 for five model features, after which the model performance largely saturates (Figure 12a). The zoom-in detail shown in Figure 12b reveals that the forward sequential feature selection without sorting, regarded as a greedy approach, expectedly shows the worst performance, while consistently increasing the performance as more complex features are added. This reveals that there are non-negligible COF dependencies reflected through high-order interactions of model inputs. For low-order models (with four or five features), the LASSO method achieves the best performance, while for high-order models the forward sequential feature selection with sorting clearly dominates other approaches (although the differences in performance are relatively modest).

The model with seven parameters obtained by the forward sequential feature selection with sorting (Figure 12b) was ultimately selected as a good compromise between model complexity and accuracy (the knee point of the performance plot in Figure 12b). It should be noted that selecting a low-order model facilitates its real-time implementation and enhances interpretability of input–output dependencies while obtaining the performance close to that of the full cubic model. Moreover, low-order models are often less sensitive to unseen data in extrapolation regions, which can enhance their robustness when applied beyond the training domain.

The final model structure is given by

$$C\widehat{O}F={\beta }_1+{\beta }_2{F}_z+{\beta }_6{F}_z{T}_d+{\beta }_7{\omega }_r{T}_d+{\beta }_8{F}_z^2+{\beta }_{13}{F}_z^2{T}_d+{\beta }_{18}{F}_z^3.$$

(9)

The only parameter input which appears individually (in all linear, quadratic, and cubic features) is the normal force. The normal force is also present in interaction features with the clutch temperature, while the slip speed appears exclusively in an interaction feature with the clutch temperature. An analysis of the model parameter values showed that (i) the most significant model feature (the one with the highest associated parameter) is the squared normal force, (ii) all features involving the normal force exhibit relatively high parameter values indicating their relevance, and (iii) the least significant model feature is the one involving interaction between slip speed and temperature. The parameters associated with features $F_z$, $F_z^2{T}_d$, and $F_z^3$ are negative (reflecting a falling COF vs. normal-force dependence, Figure 5), while the other parameters are positive.

5.3. Analysis of Selected Model

Table 3. Performance evaluation of different COF expectation models.

		Modelling Dataset		Testing Dataset
Model	Number of Model Features	RMSE [−]	R2 [−]	RMSE [−]	R2 [−]
Basic linear model	4	0.197	0.442	0.198	0.441
Linear model with interaction features	7	0.196	0.445	0.197	0.445
Full cubic model	20	0.191	0.477	0.191	0.476
Selected model	7	0.194	0.461	0.194	0.460
Selected model, full dataset	7	0.194	0.460	-	-

provides a comparative performance evaluation of the selected model and the following three baseline models: (1) the basic linear model containing only the first four features of Equation (3); (2) the linear model containing the basic linear model and the interaction features associated with the parameters ${\beta }_5$, ${\beta }_6$, and ${\beta }_7$; and (3) the full cubic model (3). The model performance is evaluated based on the root mean square error (RMSE) of COF prediction and the R² metrics given by Equation (5), separately for the modelling and testing datasets. The results presented in Table 3 indicate that there are no notable differences between the modelling and testing dataset performance metrics, which is in agreement with the results presented in Figure 12. The selected model has an around 2% and 1.5% higher R² value than the basic linear model and the linear model with interaction features, respectively. On the other hand, it has an around 1.5% lower R² value than the full cubic model, at the advantage of a significantly lower number of parameters (7 compared to 20). Similar comparative results apply when considering the RMSE metrics in Table 3. The competitive performance of the basic linear model suggests that the relationship between COF and the considered input variables is dominated by linear effects.

The relatively low R² value of around 0.45 indicates that a significant portion of the COF variance remains unexplained by the model. This can predominantly be attributed to inherent COF variability, which cannot be explained by the given model inputs, and mainly comes from (i) the specific effects such as the creep of COF observed for the constant inputs and related to varying composite material properties through the wear depth (see Section 4.3) and (ii) plate-to-plate COF variability (see Section 4.4).

The selected model was re-parameterized using the full modelling data set. The results given in the last row of Table 3 show that the performance indices remain very similar to those of the original approach, thereby justifying the use of 20% of the data points for model training (Section 5.2).

The same feature-selection methodology was employed for modelling the COF for two additional dry-clutch friction materials considered in the wear-characterization and -modelling study [29]. The comparative results are presented and discussed in Appendix A. They indicate that the two additional materials have higher overall COF variation, but a higher proportion of that variation can be explained by the model (R² values exceed 0.6).

The selected model response with respect to individual inputs is visualized in Figure 13, while Figure 14 shows the corresponding 4D plots. For the fixed slip speed, the model predicts linear COF dependence on the clutch temperature (see Figure 13a and Figure 14a and Equation (9)), where the dependence gradient is higher for lower normal forces and may be slightly negative at high normal forces. According to Figure 13b and Figure 14b, the COF decreases with normal force’s increase, particularly at high normal forces and temperatures. Finally, the COF dependence on slip speed is linear, with the corresponding gradient being higher for higher temperatures (see Figure 13c and Figure 14b and Equation (9)). The simple linear models can capture the observed linear behaviour with respect to temperature and slip speed, particularly in the version with interaction features (to account for the gradient dependencies). It cannot capture the nonlinear behaviour with respect to normal force, but according to Table 3, this only has a modest influence on the modelling accuracy.

Figure 15a shows the distribution of model residuals plotted against predicted COF values, where the model residuals are calculated as

$$e_j={COF}_j-{C\widehat{O}F}_j.$$

(10)

While most of the points fall around the ideal-fit line (yellow region), a considerable number of points scatter, particularly in the region of high COFs (cf. Figure 13). Specifically, there are 0.4% of points with residuals larger than 0.6 p.u. and 0.1% with residuals smaller than −0.6 p.u. Most of them occur under high-temperature and low-normal-force conditions. Also, some of them likely correspond to the specific, model-input-uncorrelated COF behaviour effects discussed in Section 4 and presented in Table 2. Figure 15b shows the PDFs of recorded COF values (after subtracting the COF mean) and COF model residuals. The model residuals have narrower PDF, showing that the model explains some of the variability in the recorded COF values (around 45% according to the R² coefficient value in Table 3).

6. COF Variability Modelling

Results from Section 4 and Section 5 showed that the recorded COF values exhibit significant variability that cannot be explained by the COF expectation model. This section introduces a model that quantifies this variability in the form of a COF variability model, which provides a confidence interval around the value predicted by the COF expectation model.

6.1. Methodology

The COF variability model is developed based on the maximum likelihood method [29,35]. The COF is assumed to be a random variable following a normal probability distribution, with the mean value provided by the COF expectation model (Section 5) and the standard deviation predicted by the variability model. The normal distribution assumption is found to be reasonable since the distribution of residuals is predominantly unimodal and nearly symmetric (with only a mild skewness on the side of positive values, see Figure 15b). To account for the distribution skewness, application of the Weibull probability distribution is also investigated in Appendix B.

The variability model is derived by maximizing the product of likelihood values of N modelling points, defined as [35]

$$L^{\prime }\left(\mathbf{\theta }\right)=f\left({\mathbf{x}}_1,{COF}_1,\mathbf{\theta }\right)·f\left({\mathbf{x}}_2,{COF}_2,\mathbf{\theta }\right)·\cdots ·f\left({\mathbf{x}}_N,{COF}_N,\mathbf{\theta }\right).$$

(11)

where $\mathbf{\theta }$ is the vector of model parameters to be optimized, ${CoF}_j$ is the observed output value of jth $\in \left\{\mathrm{1,2},\dots ,N\right\}$ random variable (i.e., the recorded COF for jth characterization point) accompanied with the model input vector ${\mathbf{x}}_j$, and $f$ is the probability density function (PDF) providing the likelihood of occurrence of each pair $\left({\mathbf{x}}_j,{CoF}_j\right)$ dependent on the parameters $\mathbf{\theta }$.

For better numerical conditioning of the parameter optimization, the multiplication of PDF values in Equation (11) is typically transformed into a sum by taking its logarithm. Furthermore, the cost function sign is inverted to convert the problem from maximization to minimization. This results in the negative log likelihood (NLL) cost function to be minimized [34]:

$$L\left(\mathbf{\theta }\right)=\sum _{j = 1}^N-ln\left(f\left({\mathbf{x}}_j,{COF}_j,\mathbf{\theta }\right)\right).$$

(12)

The PDF of normal distribution reads

$$f\left({\mathbf{x}}_j, {COF}_j,\mathbf{\theta }\right)={\displaystyle \frac{1}{{COF}_{\sigma }\left({\mathbf{x}}_j,\mathbf{\theta }\right)\sqrt{2\pi }}}{e}^{-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{y_j-{COF}_{\mu }\left({\mathbf{x}}_j\right)}{{COF}_{\sigma }\left({\mathbf{x}}_j,\mathbf{\theta }\right)}}\right)}^2},$$

(13)

where ${COF}_{\mu }$ and ${COF}_{\sigma }$ are its expectation and standard deviation values, respectively. The expectation ${COF}_{\mu }$ is provided by the COF expectation model (9).

The standard deviation ${CoF}_{\sigma ,j}$ is modelled with dependence on (i) the expected value of COF or (ii) the three expectation-model inputs. To this end, four model candidates are nominated, where the first three models involve exponential, linear, and quadratic dependence on COF expectation ${CoF}_{\mu ,j}$, while the fourth model is linear in the three COF model inputs:

$${COF}_{\sigma ,j}=p_1{e}^{p_2 {COF}_{\mu ,j}}$$

(14)

$${COF}_{\sigma ,j}=p_1+p_2 {COF}_{\mu ,j}$$

(15)

$${COF}_{\sigma ,j}=p_1+p_2 {COF}_{\mu ,j}+p_3 {{COF}_{\mu ,j}}^2$$

(16)

$${COF}_{\sigma ,j}=p_1+p_2 T_{d,j}+p_3 {\omega }_{r,j}+p_4 F_{z,j}$$

(17)

where $p_1,\dots ,p_4$ are the variability-model parameters (i.e., the elements of vector $\mathbf{\theta }$).

The procedure of optimizing the model parameters is elaborated in [29] and is outlined in what follows. Firstly, the model parameters $\mathbf{\theta }$ are initialized. The standard deviation ${CoF}_{\sigma ,i}$ is then determined for all modelling points j = 1, …, N based on one of the above models and used to calculate the NLL function (12). The NLL value is fed into the parameter-optimization algorithm (Matlab function fminsearch (.)) to provide a new set of model parameters aiming at NLL cost minimization. The above steps are repeated in a loop until a termination criterion is satisfied. Finally, the 95% CI is calculated from the obtained expectation and standard deviation as ${COF}_{\mu ,j}\pm 2{COF}_{\sigma ,j}$.

Table 4. Performance of COF variability models of different structures.

Model	L(θ)
Exponential, Equation (13)	−39,593 (0.0%)
Linear, Equation (14)	−39,698 (−0.27%)
Quadratic, Equation (15)	−39,702 (−0.28%)
Three-input linear model, Equation (16)	−39,864 (−0.68%)

gives the NNL cost function values reached after the optimization execution. All the considered models have similar performance, with the three-input linear model performing best, which can be explained by the richer input set.

The results for the case of Weibull distribution approximation are presented in Appendix B, showing that this modification only slightly reduces the settled value of the NLL cost function. This indicates that the deviation from normality observed in Figure 15b has limited practical impact on the predictive performance of the model.

6.2. Analysis of Selected Model

Figure 16 shows the expectation-model residuals with respect to individual inputs, along with the variability-model-predicted 95% CIs. For the sake of brevity, the results are given for a single pair of the remaining two inputs, as denoted in the title of each plot. All the four COF variability models show similar 95% CI predictions, with a tendency of the three-input model to provide somewhat-higher accuracy. The 95% CIs increases with normal force’s decrease (Figure 16a), indicating higher COF variability at lower normal loads (cf. Figure 13b). An opposite trend of growing COF variability with the increase in temperature input and, to a lower extent, slip speed input can be observed in Figure 16b,c (cf. Figure 13a,c).

Figure 17 shows a heatmap of recorded versus expectation-model-predicted COF values, where a randomly selected 1% of all recorded points is considered to facilitate plotting. The three-input COF variability model is used to determine the 95% CI, which is employed to extract the points falling outside of this interval. Evidently, the 95% CI becomes wider for higher COFs, i.e., the higher the COF, the higher is its variability (cf. Figure 13). From the 30,289 points presented in the graph, only 1395 points fall outside of the 95% CI, corresponding to a share of 4.6%, which is close to the theoretical expectation of 5%, thereby further confirming the validity of the COF variability model.

7. Model Validation

The developed COF model was validated against (i) the testing dataset extracted from the characterization data (see Section 5) and (ii) the validation dataset recorded for two additional friction plates (see Section 3). Figure 18 shows five characteristic samples of COF time responses recorded on the additional validation plate that experienced floating temperature (Plate II, Section 3). The recorded COF responses are accompanied by the corresponding predictions of COF expectation and variability models. The portion of the clutch closing interval $t_2$, to which the COF characterization data-selection condition (2) applies, is designated by green symbols and labelled by ${t_2}^{*}$. The title of each subplot includes the reference values of initial slip speed, torque, and closing time (${\omega }_{0R}$, $M_{zR}$, and $t_{2R}$), as well as the average value and standard deviation of the actual temperature response (${\overline{T}}_d, {\sigma }_{T_d}$).

Figure 18a illustrates that the predicted COF can be accurate outside of the model-training interval $t_2$. The error is expectedly highest in the initial normal-force ramping-up interval $t_1$ due to the transient effects unseen by the model. Nevertheless, the predicted COF response remains within the predicted 95% CI throughout the response. The model consistently predicts the distinctively highest 95% CI in the initial phase $t_1$ (applies to all responses in Figure 18). Namely, it recognizes this transient interval as the one in which the confidence in COF expectation prediction is lowest, although the data from the interval $t_1$ were unseen by the COF model (i.e., not used for its training). Figure 18b indicates that the model can be very accurate during the nominal (training) phase ${t_2}^{*}$, but rather imprecise in the initial $t_1$ phase (both quantitatively and qualitatively). Figure 18c shows that the COF expectation-modelling accuracy can be very good for some closing cycles, but worse for others, although the input parameters are same. This is because of inherent COF variations through the wear depth (Section 4). Similarly, Figure 18d illustrates that, although the expectation model is quite accurate in general, it cannot predict the sporadic mid-speed COF drops discussed in Section 4. Finally, Figure 18e illustrates that the expectation model’s prediction can be quite inaccurate throughout the response. However, it remains within (or very close to) the 95% CI (applies to all plots in Figure 18).

The left-hand side of

Table 5. Comparative assessment of model performance for different datasets.

	Characterization Plates		Validation Plates
Dataset	Modelling	Testing	I	II	Both
Residuals mean	0.005	0.004	−0.054	0.063	−0.003
RMSE	0.194	0.194	0.220	0.189	0.214
Normalized residuals mean	0.029	0.030	−0.200	0.368	0.042
Normalized residual st. deviation	1.122	1.124	0.989	0.890	0.989
St. deviation of recorded COF	0.264	0.264	0.280	0.231	0.268
R2	0.461	0.460	0.345	0.263	0.356
Points outside of 95% CI	4.64%	4.67%	5.86%	4.48%	5.28%

shows the COF model validation results obtained from the testing portion of the characterization dataset (80% of the full dataset) and given alongside the corresponding modelling/training results. As already discussed with Table 3 and Figure 17, the R² and RMSE metrics are very similar for the modelling and testing datasets, and nearly 5% of points lie outside of the 95% interval for both datasets, thus confirming the COF model validity. In addition, the model is unbiased on both datasets (nearly zero mean), and when given in the normalized sense (see [24] for details), the mean and RMSE approach 0 and 1, respectively. Finally, the RMSE is considerably lower than the standard deviation of recorded COF data.

The right-hand side of Table 5 includes the validation metrics related to datasets collected through experiments on additional validation plates, subject to data-selection condition (2). It may be noted that the individual validation plate metrics are worse in terms of R² when compared to characterization-point testing metrics. This can be explained by the fact that although the validation experiments are richer in terms of operating transients, they span a narrower range of input parameters than the characterization experiments (Section 3). The narrower range of inputs excites lower COF variations (explainable by the inputs), which, in combination with inherent COF variations (not explainable by the inputs; due to specific effects and plate-to-plate variations), expectedly brings R² to lower values. Nevertheless, similar model performance on the validation and modelling datasets is indicated by comparative RMSE values, with the validation Plate II even exhibiting a lower RMSE on the validation dataset (despite having a lower R² value). Furthermore, one may suppose that if there were more additional validation plates (closer to the number of characterization plates), i.e., if the sizes of validation and characterization datasets were comparable, the two sets of metrics would approach each other more closely.

Figure 19 shows the heatmap plot of recorded COF vs. predicted COF expectation for a randomly selected 10% of the total recorded points. Similarly to the results in Figure 17, the majority of points lie close to the ideal-fit line. A narrower range of COF values compared to that in Figure 17 is due to the narrower range of parameter inputs and lower size of the dataset. Consistently, around 5% of points fall outside of the 95% CI (see Table 5).

8. Conclusions

This paper firstly presented the experimental characterization of the coefficient of friction (COF) for an organic dry-clutch friction lining against a steel pressure plate. The characterization was based on around 3 million data points, obtained after filtering the time responses recorded on a custom disc-on-disc tribometer for the purpose of friction lining wear characterization. Among the investigated input parameters, the friction interface temperature was identified as the most influential parameter, exhibiting a positive correlation coefficient with COF of 0.485. The slip speed was found to have a somewhat lower, but still positive, correlation coefficient of 0.385, whereas the normal force exhibited a comparable, but negative/anti-correlation, coefficient of −0.377.

Secondly, a stochastic COF prediction model was proposed, which consists of expectation and variability sub-models. An optimal structure of the COF expectation model, with the three aforementioned input parameters, was identified from the full cubic polynomial model by means of a sequential feature-selection method. The resulting optimal model has 7 free parameters (out of 20 parameters of the full cubic model), which are estimated via the least squares method. The model can explain 46% of the variability in COF data from both the training and testing datasets (the R² coefficient equals 0.46). This coefficient is relatively modest due to inherent COF variability, i.e., the one that cannot be explained by the three model inputs (e.g., slow COF variation through the wear depth for the constant inputs and friction plate-to-plate variations due to production deviations). The COF model behaviour is predominantly linear with respect to temperature and slip speed inputs, while a more complex, nonlinear dependence on normal force appears only for high-normal-force conditions, particularly at higher temperatures. Therefore, linear models, particularly those with interaction terms, can be employed for COF expectation modelling without a major deterioration of modelling accuracy compared to the selected nonlinear model. The modelling methodology has also been demonstrated for two other friction materials, resulting in six and nine model parameters, and accounting for more than 60% of COF variability within respective datasets, but also with higher COF data variation.

The COF variability model was developed using the maximum likelihood method, based on the assumption of COF distributing according to the normal distribution under fixed input parameters. Several COF standard deviation model candidates were nominated, including exponential, linear, and quadratic models inputted by the COF expectation value and a linear model fed by the three expectation-model input parameters. The three-input model was found to have marginally better performance than the three single-input models. The COF variability-model-predicted 95% confidence interval (CI) captures the testing dataset well, with 4.6% of recorded points falling outside of the 95% CI.

The overall model was validated on an independent validation dataset recorded on two separate friction plates, with the experiments designed to mimic realistic clutch operation. The expectation model explains 36% of the COF variation in the validation dataset (the R² coefficient falls to 0.36 for the limited set of validation plates and narrow operating conditions), while the variability model successfully predicts the 95% confidence interval. The model was experimentally validated for the clutch transient conditions, as well, with the following main findings: (i) the expectation-model-predicted COF time response generally follows the recorded response quite well during the clutch closing interval for which the model is trained; (ii) the model response can be inaccurate due to inherent COF variability; (iii) the predicted 95% CI captures well the recorded COF variations around the predicted expectation throughout the response (including the initial normal-force ramping-up phase, for which the model was not trained); (iv) the 95% CI is widest for the initial phase, which is consistent with the observation that the COF response inaccuracy is found to be largest in this phase.

Future research may focus on improving the modelling accuracy by employing more advanced modelling approaches (such as deep neural networks, time convolutional networks, random forests, transformers, and similar), potentially augmented with additional explanatory variables including wear depth, as well as a history of already considered inputs. Further experimental campaigns could be designed to explicitly measure or estimate currently omitted material-specific or environmental factors, in order to add them into the model input set aimed at reducing the unexplained variability to some extent. For transient behaviours (e.g., those related to initial phase) a regime-detection method (based on, e.g., classifiers, hidden Markov models, or recurrent neural networks) could be employed, followed by deployment of regime-specific predictive models (e.g., time-series/state-space formulations applied to full engagement cycles). Finally, applying the developed models to compensate for COF variations in applications of clutch torque control in automated transmissions represents a particularly relevant and valuable direction of future work.