# Road Friction Virtual Sensing: A Review of Estimation Techniques with Emphasis on Low Excitation Approaches

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## Abstract

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## 1. Introduction

## 2. Background

#### 2.1. Vehicle Modelling

#### 2.2. Road-Rubber Friction

#### 2.2.1. Tyre Properties as a Function of Temperature and Stress Frequency

#### 2.2.2. Tyre Friction Force as a Function of Stress Frequency

- Molecular adhesion: The first mechanism is the adhesion [57]. The grip derived from the adhesion between the rubber and the road is the result of the Van der Waals bonding phenomena. The rubber’s molecular chains form, stretch and break, following a cycle of stretching and breaking, and generating visco-elastic work. This adhesion mechanism occurs in a range of stress frequencies between ${10}^{6}$ Hz and ${10}^{9}$ Hz, and requires the separation distance between the road and the rubber to be below ${10}^{-6}$ mm [2]. The bonding phenomena can be explained in a simplified manner by three steps, Figure 5b. In the first step the bond is created. After that, in step 2, the molecular chain is stretched, and a friction force which opposes the tyre skidding is generated. Finally, in the last step, the bond breaks and new bonds form again successively.
- Road roughness effect: The road roughness effect (also denoted as indentation) is primarily caused by the road irregularities [58,59] and the hysteresis of the rubber [2]. The road texture (with rough spots that vary from 1 centimetre to 1 micron) induce a high-frequency excitation on the rubber (with frequencies ranging from ${10}^{2}$ to ${10}^{6}$ Hz [2]), which is distorted and undergoes several compression-relaxation cycles. As the rubber presents an inherent hysteresis, the rubber does not return immediately to its initial position, but exhibits an asymmetrical movement (and therefore an energy loss). Such asymmetrical movement of the rubber block around the rough spot results in a force field, with a tangential component which opposes the slippage and is seen as the tyre force [2].

#### 2.2.3. Tyre Models as a Function of the Friction Coefficient

- Magic FormulaThe Magic Formula [67] consists of a nonlinear formulation based on arctan functions. The coefficients ${D}_{j},{C}_{j},{B}_{j},{E}_{j}$, with $j\in \{x,y\}$, are determined empirically, based on experimental data. A simplified Magic formula formulation is presented here [8,31]. More sophisticaped and complex formulations can be consulted in Hans B. Pacejka [67]. Firstly, the one-directional normalised tyre forces (${\mu}_{x0},{\mu}_{y0}$) are computed from expressions (16) and (17).$$\begin{array}{c}\hfill {\mu}_{x0}\left(\lambda \right)={D}_{x}\mathrm{sin}\left({C}_{x}\mathrm{arctan}({B}_{x}\lambda -{E}_{x}({B}_{x}\lambda -\mathrm{arctan}\left({B}_{x}\lambda \right)))\right)\end{array}$$$$\begin{array}{c}\hfill {\mu}_{y0}\left(\alpha \right)={D}_{y}\mathrm{sin}\left({C}_{y}\mathrm{arctan}({B}_{y}\alpha -{E}_{y}({B}_{y}\alpha -\mathrm{arctan}\left({B}_{y}\alpha \right)))\right)\end{array}$$After that, in order to handle combined efforts, the weighting functions (${G}_{x\alpha},{G}_{y\lambda}$) can be defined in the following manner [8]:$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {G}_{x\alpha}& =\mathrm{cos}\left({C}_{x\alpha}\mathrm{arctan}({B}_{x\alpha}\alpha -{E}_{x\alpha}({B}_{x\alpha}\alpha -\mathrm{arctan}\left(\alpha \right)))\right)\hfill \end{array}\end{array}$$$$\begin{array}{c}\hfill {G}_{y\lambda}=\frac{\mathrm{cos}\left({C}_{y\lambda}\mathrm{arctan}\left({B}_{y\lambda}(\lambda +{S}_{Hy\lambda})\right)\right)}{\mathrm{cos}\left({C}_{y\lambda}\mathrm{arctan}\left({B}_{y\lambda}{S}_{Hy\lambda}\right)\right)}\end{array}$$Finally, the one-directional friction coefficients are obtained from the product of these weighting functions and the normalised forces calculated previously, expression (20).$$\begin{array}{c}\hfill {\mu}_{x}={G}_{x\alpha}{\mu}_{x0},\phantom{\rule{1.em}{0ex}}{\mu}_{y}={G}_{y\lambda}{\mu}_{y0}\end{array}$$If load sensitivity effects are disregarded, the tyre planar forces can be obtained from the vertical forces as (${F}_{j}={\mu}_{j}{F}_{z}$), [8].
- Brush modelIn this formulation the pure longitudinal force is obtained from expressions (21)–(24). During gentle driving conditions, the tyre longitudinal slip $\lambda $ keeps below the critical slip ${\lambda}_{sl}$ and the first expression is used. Under strong braking or acceleration events ($\lambda >{\lambda}_{sl}$) the entire contact patch is sliding, and the second expression is used.$$\begin{array}{cc}\hfill {\mu}_{x}(\lambda ,{\mu}_{\mathrm{max}})& =\left\{\begin{array}{c}3{\mu}_{\mathrm{max}}{\theta}_{x}{\sigma}_{x}\{1-|{\theta}_{x}{\sigma}_{x}|+\frac{1}{3}|{\theta}_{x}{\sigma}_{x}{|}^{2}\}\hfill \\ {\mu}_{\mathrm{max}}sign\left(\lambda \right)\hfill \end{array}\right.\hfill \end{array}$$$$\begin{array}{cc}\hfill {\theta}_{x}& =2\frac{{c}_{p}{l}^{2}}{\left(3{\mu}_{\mathrm{max}}{F}_{z}\right)}\hfill \end{array}$$$$\begin{array}{cc}\hfill {\sigma}_{x}& =\frac{\lambda}{(\lambda +1)}\hfill \end{array}$$$$\begin{array}{cc}\hfill {\lambda}_{sl}& =\frac{1}{({\theta}_{x}-1)}\hfill \end{array}$$In this case the normalised tyre forces are presented (${\mu}_{j}={F}_{j}/{F}_{z}$), the tyre tread stiffness is denoted by ${c}_{p}$, and the longitudinal slip at which the full-sliding condition starts is ${\lambda}_{sl}$. The derivation of the friction model in the lateral direction is straightforward from expressions (25)–(28) [17].$$\begin{array}{cc}\hfill {\mu}_{y}(\alpha ,{\mu}_{\mathrm{max}})& =\left\{\begin{array}{c}-3{\mu}_{\mathrm{max}}{\theta}_{y}{\sigma}_{y}\{1-|{\theta}_{y}{\sigma}_{y}|+\frac{1}{3}|{\theta}_{y}{\sigma}_{y}{|}^{2}\}\hfill \\ -{\mu}_{\mathrm{max}}sign\left(\alpha \right)\hfill \end{array}\right.\hfill \end{array}$$$$\begin{array}{cc}\hfill {\theta}_{y}& =2\frac{{c}_{p}{l}^{2}}{\left(3{\mu}_{\mathrm{max}}{F}_{z}\right)}\hfill \end{array}$$$$\begin{array}{cc}\hfill {\sigma}_{y}& =\mathrm{tan}\left(\alpha \right)\hfill \end{array}$$$$\begin{array}{cc}\hfill {\alpha}_{sl}& =\frac{1}{\mathrm{tan}(1/{\theta}_{y})}\hfill \end{array}$$Once again, during gentle cornering, the lateral slip ($\alpha $) remains far below the nonlinear region ($\alpha <{\alpha}_{sl}$), and the lateral friction is approximated by the first expression. Otherwise, during limit cornering and full sliding conditions, the second expression is used. Finally, the tyre self-alignment torque (SAT) ${\tau}_{a}$ can be calculated with the expression (29) [17].$$\begin{array}{c}\hfill {\tau}_{a}(\alpha ,{\mu}_{\mathrm{max}})=\left\{\begin{array}{c}{\mu}_{\mathrm{max}}{F}_{z}l{\theta}_{y}{\sigma}_{y}(1-|{\theta}_{y}{\sigma}_{y}{\left|\right)}^{3},\phantom{\rule{5.69046pt}{0ex}}\left|\alpha \right|\le |{\alpha}_{sl}|\hfill \\ 0,\phantom{\rule{95.3169pt}{0ex}}\left|\alpha \right|>|{\alpha}_{sl}|\hfill \end{array}\right.\end{array}$$As occurs in the Magic Formula model, in the event of simultaneous efforts in the longitudinal and lateral directions, a resultant friction $\mu $ is calculated (30) based on the total slip $\sigma $ [17].$$\begin{array}{c}\hfill \mu (\alpha ,\lambda ,{\mu}_{\mathrm{max}})=\left\{\begin{array}{c}{\mu}_{\mathrm{max}}(1-{\rho}^{3})\phantom{\rule{14.22636pt}{0ex}}for\phantom{\rule{14.22636pt}{0ex}}\left|\sigma \right|\le |{\sigma}_{sl}|\hfill \\ {\mu}_{\mathrm{max}}sign\left(\alpha \right)\phantom{\rule{14.22636pt}{0ex}}for\phantom{\rule{14.22636pt}{0ex}}\left|\sigma \right|>|{\sigma}_{sl}|\hfill \end{array}\right.\end{array}$$The resultant friction is then projected in the lateral and longitudinal directions assuming the slip-proportionality principle.$$\begin{array}{cc}\hfill {\mu}_{x}& =\mu \frac{{\sigma}_{x}}{\sigma},\phantom{\rule{1.em}{0ex}}{\mu}_{y}=\mu \frac{{\sigma}_{y}}{\sigma}\hfill \end{array}$$$$\begin{array}{cc}\hfill \rho & =1-\theta \sigma \hfill \end{array}$$$$\begin{array}{cc}\hfill \theta & =2\frac{{c}_{p}{l}^{2}}{\left(3{\mu}_{\mathrm{max}}{F}_{z}\right)}\hfill \end{array}$$$$\begin{array}{cc}\hfill \sigma & =\sqrt{{\sigma}_{x}^{2}+{\sigma}_{y}^{2}}\hfill \end{array}$$$$\begin{array}{cc}\hfill {\sigma}_{sl}& =\frac{1}{\theta}\hfill \end{array}$$Finally, the tyre SAT is computed from the projected lateral force as$$\begin{array}{c}\hfill {\tau}_{comb}=-{t}_{p}\left(\sigma \right){F}_{y}\end{array}$$$$\begin{array}{c}\hfill {t}_{p}\left(\sigma \right)=\frac{l{(1-|\theta \sigma \left|\right)}^{3}}{3-3\left|\theta \sigma \right|+{\left|\theta \sigma \right|}^{2}}\end{array}$$
- Dugoff modelThe Dugoff model [75] derives from the research described in Fiala [76]. A uniform rectangular pressure distribution and a rigid tyre carcass are assumed in this model. In addition, conversely to the Brush model described previously, individual tread stiffnesses (${C}_{\lambda}$, ${C}_{\alpha}$) are considered. In first place, a coefficient $\xi $ is defined to account for the coupling between lateral and longitudinal forces, (38) [14,72].$$\begin{array}{c}\hfill \xi =\frac{{\mu}_{\mathrm{max}}{F}_{z}(1+{s}_{l})}{2\sqrt{{\left({C}_{\lambda}{s}_{l}\right)}^{2}+{\left({C}_{\alpha}\mathrm{tan}\left({s}_{s}\right)\right)}^{2}}}\end{array}$$In the formulation presented here, the PRAXIS slip notation [54] is adopted, and the lateral (${s}_{s}$) and longitudinal (${s}_{l}$) slips are computed differently depending on the driving situation:
- −
- Driving:$$\begin{array}{c}\hfill {s}_{s}=\frac{{v}_{y}}{\omega {r}_{e}},\phantom{\rule{1.em}{0ex}}{s}_{l}=\frac{\omega {r}_{e}-{v}_{x}}{\omega {r}_{e}}\end{array}$$
- −
- Braking:$$\begin{array}{c}\hfill {s}_{s}=\frac{{v}_{y}}{{v}_{x}},\phantom{\rule{1.em}{0ex}}{s}_{l}=\frac{\omega {r}_{e}-{v}_{x}}{{v}_{x}}\end{array}$$

After that, the function $f\left(\xi \right)$ is computed from the saturation level of the tyre, (41).$$\begin{array}{c}\hfill f\left(\xi \right)=\left\{\begin{array}{c}(2-\xi )\xi ,\xi <1\hfill \\ 1,\xi \ge 1\hfill \end{array}\right.\end{array}$$Finally, longitudinal and lateral forces are obtained from expression (42) [14,72],$$\begin{array}{c}\hfill {F}_{x}={C}_{\lambda}\frac{{s}_{l}}{1+{s}_{l}}f\left(\xi \right),\phantom{\rule{1.em}{0ex}}{F}_{y}={C}_{\alpha}\frac{\mathrm{tan}\left({s}_{s}\right)}{1+{s}_{l}}f\left(\xi \right)\end{array}$$

#### 2.3. Vehicle Systems: The Role of Road Friction

## 3. Slip-Based Road Friction Monitoring

#### 3.1. Longitudinal Dynamics

#### 3.2. Lateral Dynamics

#### 3.3. Tyre Self-Alignment Torque

#### 3.4. Conclusions for Slip-Based Friction Estimation Approaches

- Overall, tyre model-based approaches dominate the slip-based friction estimation problem. Specifically, the simplified analytical Brush tyre model has been widely employed in the literature.
- Depending on the methodology employed (longitudinal, lateral or SAT-based) slip-based friction monitoring requires the computation of the tyre forces, tyre SAT and tyre longitudinal or lateral slips:
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- Tyre forces might be estimated from a state estimator (e.g., UKF) using a random-walk approach.
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- Tyre SAT is often estimated using a linear disturbance observer and EPS current measurements.
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- Regarding tyre longitudinal slips, the major difficulty resides on estimating the reference velocity during braking events in which the four wheels present large slips.
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- A single-track yaw-body slip observer is often employed to estimate the axle lateral slips.

- Slip-based approaches can accurately predict the friction potential only when a certain level of longitudinal or lateral excitation is present:
- −
- Most promising approaches have been found during the tyre SAT-based survey. In particular, tyre model-less approaches such as the TGM methods are inherently attractive due to its implementation easiness.
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- Regarding slip slope approaches, a clearer treatment between loose surfaces and rigid surfaces should be provided. There is still a lack of understanding in what concerns the slip slope changes when the soft material is present in the road-rubber interface.

- Combined approaches based on friction fusion strategies can speed up the friction potential convergence (fusion of information from tyre SAT, braking manoeuvres and limit cornering manoeuvres).
- A friction recognition module relying exclusively on slip-based approaches is seen insufficient for ADAS functions such as ACC or AEB. Additional methodologies capable of sensing the friction potential during free-rolling events (at least capable of providing an initial rough estimate) should be incorporated.

## 4. Vibration-Based Road Friction Monitoring

#### 4.1. Low Frequency Vibrations

#### 4.2. High Frequency Vibrations

#### 4.3. Conclusions of Vibration-Based Friction Estimation Approaches

- Tyre vibration can be used to estimate the friction coefficient. Different approaches have been tried up to now.
- There are mainly two approaches. One approach focuses on low frequencies and the second approach on high frequencies.
- Regarding the low frequency approaches most of them are based on the assumption that longitudinal and lateral slip stiffness depend on the friction coefficient. However, there are contradicting studies and it seems that researchers cannot agree on this.
- There is more confidence on the influence of micro roughness (high frequencies) on the friction coefficient. However, there is no model that can describe the mechanism and mostly data-based approaches have been applied.
- The field trials in the different studies are not consistent or standardised. Various parameters, such as temperature, tread depth and water level, that influence the developed tyre-road friction coefficient are not always measured.
- The influence of tread depth and pattern has not been considered in greater detail. The same tyre with different tread depth will behave very differently for different water levels. It is possible to estimate tread depth by monitoring the longitudinal slip stiffness over longer periods of time.

## 5. Summary of Presented Approaches

- Slip slope solutions present high variability and still lack of robustness. Further investigations on a wider range of rigid road surfaces are required to corroborate the proposed slip slope versus friction models.
- Lateral or longitudinal model-based approaches require high excitation levels. These can produce accurate estimates during emergency braking manoeuvres or limit cornering events, but cannot anticipate the road grip potential during constant speed conditions. Therefore, such approaches are limited to vehicle systems, such as ABS or ESP, and are not suitable for ADAS functions, such as ACC or AEB.
- Active force excitation is a promising and novel methodology with which accurate friction estimates can be obtained during constant speed driving. Nevertheless, this solution might require the use of additional sensors (e.g., reference velocity from GPS). Moreover, additional studies assessing “when” the active excitation has to be performed are still necessary (e.g., tyre/brake pad wear versus frequency of active excitations to detect sudden changes in the road friction).
- Data-based solutions require the collection of experimental data and the subsequent training with these data. In addition, extrapolation issues might arise if the training data do not cover a wide range of road surfaces. Moreover, a certain level of dynamic excitation is required.
- SAT-based approaches are limited to vehicles incorporating an Electric Power Steering (EPS) system. In case of electrohydraulic steering system, additional measurements (e.g., tie rod axial force) are required. Model-based approaches that monitor the pneumatic trail can be affected by the accuracy of the suspension or steering kinematic model employed. Tyre model-less approaches such as Lateral Grip Margin might be more robust to kinematic model uncertainties. In any case, lateral excitation levels of at least 0.3 G are reported in the literature to achieve an accurate friction estimation.
- Low-frequency vibration-based approaches rely on the slip slope principle. Therefore, their applicability is subjected to the validity of this principle. As mentioned in the first point, the relationship between the tyre stiffness and the road friction potential requires further analyses.
- High-frequency vibration-based or noise-based approaches require establishing a mapping between the feature vector and the road grip potential. Selecting a suitable feature vector is not trivial. Moreover, a more costly and complicated instrumentation is required. Overall, these approaches seem most promising for future applications requiring identification of reduced friction surfaces during constant speed driving. Nevertheless, there is still a long way until these solutions can be fully implemented robustly and generalised to a wide range of vehicles, tyres, and roads.

## 6. Conclusions and Future Challenges

#### 6.1. Conclusions

#### 6.2. Future Challenges

- Friction fusion (integration of slip-based and vibration-based approaches): It seems clear that a combination of different approaches will be needed in order to have a continuous estimation of the friction potential in spite of the driving situations [16,104]. The precise definition of the driving states and the timely identification of these will be key aspects for the correct implementation of friction-fusion strategies.
- Robustness of tyre SAT-based approaches: Despite SAT-based approaches seem most promising among the slip-based group, there are several aspects to consider for future investigations. In particular, a more detailed evaluation during coupled dynamics (lateral and longitudinal forces) is still missing from the literature.
- Robustness of noise-based approaches: At the moment, high-frequency vibration-based approaches have been tested in a reduced number of scenarios. Additional investigations are still needed in order to find clear patterns that could facilitate the extraction of metrics to generalise the problem to a wider range of roads, tyres, and vehicles.
- Integration of ADAS with grip estimation approaches: Finally, the ultimate goal of the investigation on grip recognition approaches is to provide autonomous vehicles with the ability to detect low mu situations in a wide range of scenarios. Integrating the grip monitoring solutions on the ADAS and evaluating the impact of the uncertainty associated with the friction potential estimates on the ADAS performance will be of extreme importance in the future.

## Acknowledgments

## Conflicts of Interest

## Abbreviations

TRFC | Tyre Road Friction Coefficient |

ABS | Anti-lock Braking System |

TCS | Traction Control System |

ESP | Electronic Stability Program |

ADAS | Advanced Driver Assistance System |

ACC | Autonomous Cruise Control |

ADC | Autonomous Drift Control |

SAT | Self-Aligning Torque |

AFS | Active Front Steering |

ANFIS | Adaptive Neuro-Fuzzy Inference System |

CAN | Controller Area Networks |

NN | Neural Networks |

MF | Magic Formula |

RLS | Recursive Least Squares |

NLLS | Non Linear Least Squares |

SMC | Sliding Mode Control |

LRLS | Linearised Recursive Least Squares |

PID | Proportional Integral Derivative |

UKF | Unscented Kalman Filter |

EKF | Extended Kalman Filter |

MSE | Mean Squared Error |

VFF | Vehicle Feature Fusion |

DGPS | Differential Global Positioning System |

EKBF | Extended Kalman Bucy Filter |

GRNN | General Regression Neural Network |

BP | Back Propagation |

GA | Genetic Algorithm |

FWIA | Four Wheel Independently Actuated |

LS | Least Squares |

WFT | Wheel Force Transducers |

SQP | Sequential Quadratic Programming |

APF-IEKF | Auxiliary Particle Filter—Iterated Extended Kalman Filter |

ISS | Input State Stable |

RTLS | Recursive Total Least Squares |

IVSS | Intelligent Vehicle Safety System |

EPS | Electric Power Steering |

LGM | Lateral Grip Margin |

TGM | Tyre Grip Margin |

LSB | Load Sensing Bearing |

HHT | Hilbert Huang Transform |

IMF | Intrinsic Mode Function |

EV | Electric Vehicle |

ICE | Internal Combustion Engine |

LPC | Linear Predictive Coefficient |

PSC | Power Spectrum Coefficients |

SRTT | Standard Reference Test Tyre |

UHP | Ultra High Performance |

PSD | Power Spectral Density |

GPS | Global Positioning System |

SVM | Support Vector Machine |

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**Figure 1.**Overall picture of the effect-based road grip recognition approaches treated in this work.

**Figure 3.**(

**a**) The tyre as a viscoelastic material can be approximated as a spring K connected in parallel to a damper $\eta $; (

**b**) The influence of temperature on the tyre as a viscoelastic material. The modulus of elasticity is maximum for temperatures lower to the glass transition temperature. Hysteresis (energy loss) is maximum at the glass transition temperature [2].

**Figure 4.**The influence of stress frequency on the tyre as a viscoelastic material. The modulus of elasticity becomes maximum above a stress frequency threshold. Hysteresis (energy loss) is maximum at this threshold and reduces at frequencies different to that [2].

**Figure 5.**(

**a**) Road roughness (indentation) friction mechanism; (

**b**) Molecular adhesion friction mechanism.

**Figure 7.**Friction versus longitudinal slip curves representative of (

**a**) high mu asphalt and (

**b**) gravel surface. Curves generated with the tyre parameters presented in [31].

**Figure 8.**Benefit of the collection and distribution of the friction information on several applications. Diagram adapted by the authors from Koskinen [23].

**Figure 9.**Impact of the road friction potential estimation ${\mu}_{\mathrm{max}}$ on different generations of vehicle systems. Current vehicle systems (e.g., ABS) are less dependent on the a priori road friction information than current (e.g., AEB) or expert ADAS (e.g., ADC).

**Figure 11.**(

**a**) Longitudinal forces versus slip ratio $\lambda (\%)$ obtained for different ${\mu}_{\mathrm{max}}$ values from a MF 6.1 205/65 R16 tyre model [3]; (

**b**) Tyre SAT ${\tau}_{a}$ and lateral force ${F}_{y}$ versus lateral wheel slip $\alpha $. Tyre SAT-based methods rely on the earlier saturation of the tyre self-alignment torque.

**Figure 12.**Classification followed on the discussion of slip-based road friction monitoring approaches.

**Figure 13.**Active force excitation strategy proposed by Albinsson et al. [96].

**Figure 17.**(

**a**) Normalised tyre lateral forces; (

**b**) normalised tyre pneumatic trail for different road friction potential coefficients. Figure adapted by the authors from Hsu et al. [40].

**Figure 21.**Classification followed on the discussion of vibration-based road friction monitoring approaches.

**Figure 22.**Estimation structure presented by Chen et al. [4].

**Table 1.**Friction coefficient for four different broad categories of wet road surfaces [2].

Road Category | Friction Coefficient Range ${\mathit{\mu}}_{\mathrm{max}}$ |
---|---|

Macro rough/micro rough surface (draining mixes, bituminous concretes) | 0.5–0.9 |

Macro smooth/micro rough surface (fine mixes) | 0.4–0.8 |

Macro rough/micro smooth surface (rolled aggregates) | 0.2–0.3 |

Macro smooth/micro smooth surface (flushing asphalt) | 0.1–0.2 |

Road | ${\mathit{\kappa}}_{\mathrm{min}}$ | ${\mathit{\kappa}}_{\mathrm{max}}$ | ${\mathit{\mu}}_{\mathrm{max}}$ |
---|---|---|---|

Dry asphalt | 23 | 40 | [0.85,1.15] |

Soapy asphalt | 17 | 28.2 | [0.45,0.75] |

Tyre Parameter | Concrete | Snow | Ice |
---|---|---|---|

${\mu}_{\mathrm{max}}$ | [0.85–1] | [0.35–0.4] | [0.15–0.2] |

${C}_{\lambda}$ | $16\times {10}^{4}$ | $6.6\times {10}^{4}$ | $1.8\times {10}^{4}$ |

**Table 4.**Longitudinal stiffness changes for different tyre temperature, inflation pressure, tread depth, normal load, and road surface wetness [129].

Longitudinal Stiffness Variation $\mathbf{\Delta}{\mathit{C}}_{\mathit{\lambda}}$ (%) | ||
---|---|---|

Test | Tyre 1 | Tyre 2 |

Cold to steady-state temperature | −17% | −21% |

−10% pressure | 17% | 15% |

−20% pressure | 29% | 28% |

Reduced tread (2.5 mm) | 34% | 91% |

+200 kg (normal load ) | 13% | 7% |

+400 kg (normal load ) | 60% | 42% |

Wet road | 4% | −2% |

**Table 5.**Values of the peak frequency ${f}_{p}$, undamped natural frequency ${f}_{n}$, and damping ratio $\xi $ of the first mode of the tyre for different longitudinal slip stiffness scaling factors LKX, [50].

LKX | ${\mathit{f}}_{\mathit{n}}$ | ${\mathit{f}}_{\mathit{p}}$ | $\mathit{\xi}$ |
---|---|---|---|

1 | 35.9 Hz | 34.8 Hz | 0.17 |

0.5 | 34.1 Hz | 32.0 Hz | 0.24 |

0.2 | 29.9 Hz | 23.2 Hz | 0.44 |

Reference | Hypothesis | Method | Frequency Range |
---|---|---|---|

Umeno et al. [51] | Friction influences damping of tyre vibrations | Transfer function—Tyre rotational vibrations | 20–60 Hz |

Chen et al. [4] | Longitudinal slip stiffness is linearly dependent on friction | Resonance frequency of the driveline of an in-wheel motor drive system | 15–22 Hz |

Schmeitz and Alirezaei [50] | normalised slip slope factor LKX is dependent on friction coefficient | Simulations using MF-Swift tyre model | 23–36 Hz |

Chen et al. [130] | tyre self-aligning torque (SAT) stiffness varies linearly with the road friction potential | Resonance frequency of the electric power steering system | 0.5–1.6 Hz |

Reference | Hypothesis | Method | Frequency Range |
---|---|---|---|

Boyraz [46,131] | Features of the acoustic signal depend on the friction coefficient | Linear predictive coefficients, mel-frequency cepstrum coefficients and power spectrum coefficients, Artificial Neural Networks | Not specified |

Alonso et al. [49] | Features of the acoustic signal depend on the friction coefficient | Audio samples 20 Hz-20 kHz, Support Vector Machines | 500 Hz to 8 kHz |

Masino et al. [47,132] | acoustic signal in the tyre cavity | Power Spectral Density | >5000 Hz |

Singh et al. [77] | tyre circumferential acceleration and the road friction potential | In-tyre accelerometer measures vibrations, ratio of vibrations at low and high frequency range | 10–500 Hz and 600–2500 Hz |

Method | Excitation Direction | Description | Advantages | Disadvantages |
---|---|---|---|---|

Slip-based | Longitudinal | Slip slope | Approach easiness | Contradictory results High variability |

Slip-based | Longitudinal | Friction Model-based | Accuracy | High excitation required e.g., ABS braking |

Slip-based | Longitudinal | Active force excitation | Cons. speed conditions | “when”? Blind to mu transitions |

Slip-based | Lateral | Friction Model-based | Accuracy | High excitation required Nonlinear cornering region |

Slip-based | Lateral | Data-based | No model required | Experimental data needed Extrapolation issues |

Slip-based | Tyre SAT | Friction Model-based | Accuracy Lowest excitation among slip-based | Pneumatic trail estimation EPS requirement Mech. trail compensation |

Slip-based | Tyre SAT | Data-based | No model required | Experimental data needed Extrapolation issues |

Slip-based | Tyre SAT | LGM | Easiness Low excitation | Lack of detailed analysis Existing studies focused on stability controller validation |

Method | Frequency band | Description | Advantages | Disadvantages |

Vibration-based | Low | Resonance frequency | Reduced excitation required | Contradictory results High variability |

Vibration-based | High | Noise | Reduced excitation required | Based on empirical results Selection of feature vector |

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**MDPI and ACS Style**

Acosta, M.; Kanarachos, S.; Blundell, M. Road Friction Virtual Sensing: A Review of Estimation Techniques with Emphasis on Low Excitation Approaches. *Appl. Sci.* **2017**, *7*, 1230.
https://doi.org/10.3390/app7121230

**AMA Style**

Acosta M, Kanarachos S, Blundell M. Road Friction Virtual Sensing: A Review of Estimation Techniques with Emphasis on Low Excitation Approaches. *Applied Sciences*. 2017; 7(12):1230.
https://doi.org/10.3390/app7121230

**Chicago/Turabian Style**

Acosta, Manuel, Stratis Kanarachos, and Mike Blundell. 2017. "Road Friction Virtual Sensing: A Review of Estimation Techniques with Emphasis on Low Excitation Approaches" *Applied Sciences* 7, no. 12: 1230.
https://doi.org/10.3390/app7121230