# Research on Tire/Road Peak Friction Coefficient Estimation Considering Effective Contact Characteristics between Tire and Three-Dimensional Road Surface

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## Abstract

**:**

## 1. Introduction

## 2. Overall Estimation Strategy

## 3. Establishment of Vehicle Dynamics Model

## 4. Modified Tire/Road Contact Model

#### 4.1. Vertical Dynamics Model

#### 4.2. Construction of 3D Road

#### 4.3. Modified LuGre Tire Model

#### 4.4. Tire Model Normalization

#### 4.5. Establishment of System Equations

## 5. Unscented Kalman Filter

## 6. Simulation Verification Analysis

#### 6.1. Straight-Line Braking Condition

^{2}and that the TRPFC converged to 0.8 before 0.8 s and then fluctuated around 0.85, but the overall error was maintained in [−0.05, 0.05].

#### 6.2. Combined Turning and Braking Condition

^{2}, the maximum lateral acceleration could reach 9 m/s

^{2}, and the TRPFC converged to 0.9 after 0.2 s. After that, it fluctuated around 0.85 and was generally stable.

## 7. Real Vehicle Test

#### 7.1. Real Vehicle Test Platform

#### 7.2. Straight-Line Test

^{2}and that the TRPFC converged to 0.8 at about 0.3 s, then fluctuated steadily at about 0.86.

#### 7.3. Curved Test

^{2}, the lateral acceleration fluctuated between −2.2 m/s

^{2}and −1.4 m/s

^{2}, and the steering wheel angle was up to 72 degrees. The TRPFC converged to 0.82 before 0.2 s and rose to about 0.87 after 0.6 s, then basically remained stable.

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- National Data Traffic Accident Statistics. China National Bureau of Statistics. 2021. Available online: https://data.stats.gov.cn/easyquery.htm?cn=C01&zb=A0S0D01&sj=2021 (accessed on 25 February 2022).
- World Health Organization. Global Status Report on Road Safety 2015; World Health Organization: Geneva, Switzerland, 2015. [Google Scholar]
- Chen, B.; Zhang, X.; Yu, J.; Wang, Y. Impact of contact stress distribution on skid resistance of asphalt roads. Constr. Build. Mater.
**2017**, 133, 330–339. [Google Scholar] [CrossRef] - Zhu, S.; Liu, X.; Cao, Q.; Huang, X. Numerical Study of Tire Hydroplaning Based on Power Spectrum of Asphalt Road and Kinetic Friction Coefficient. Adv. Mater. Sci. Eng.
**2017**, 2017, 1–11. [Google Scholar] - Wang, Y.; Hu, J.; Wang, F.; Dong, H.; Yan, Y.; Ren, Y.; Zhou, C.; Yin, G. Tire Road Friction Coefficient Estimation: Review and Research Perspectives. Chin. J. Mech. Eng.
**2022**, 35, 1–11. [Google Scholar] [CrossRef] - Guo, F.; Pei, J.; Zhang, J.; Li, R.; Zhou, B.; Chen, Z. Study on the skid resistance of asphalt pavement: A state-of-the-art review and future prospective. Constr. Build. Mater.
**2021**, 303, 124411. [Google Scholar] [CrossRef] - Mendoza-Petit, M.F.; Garcia-Pozuelo, D.; Diaz, V.; Garrosa, M. Characterization of the loss of grip condition in the Strain-Based Intelligent Tire at severe maneuvers. Mech. Syst. Signal Processing
**2021**, 168, 108586. [Google Scholar] [CrossRef] - Matsuzaki, R.; Kamai, K.; Seki, R. Intelligent tires for identifying coefficient of friction of tire/road contact surfaces using three-axis accelerometer. Smart Mater. Struct.
**2015**, 24, 025010. [Google Scholar] [CrossRef] - Kim, M.-H.; Park, J.; Choi, S. Road Type Identification Ahead of the Tire Using D-CNN and Reflected Ultrasonic Signals. Int. J. Automot. Technol.
**2021**, 22, 47–54. [Google Scholar] [CrossRef] - Yoon, J.-H.; Li, S.E.; Ahn, C. Estimation of vehicle sideslip angle and tire-road friction coefficient based on magnetometer with GPS. Int. J. Automot. Technol.
**2016**, 17, 427–435. [Google Scholar] [CrossRef] - Khaleghian, S.; Emami, A.; Taheri, S. A technical survey on tire-road friction estimation. Friction
**2017**, 5, 123–146. [Google Scholar] [CrossRef] [Green Version] - Zhang, Z.; Zheng, L.; Wu, H.; Zhang, Z.; Li, Y.; Liang, Y. An estimation scheme of road friction coefficient based on novel tyre model and improved SCKF. Veh. Syst. Dyn.
**2021**, 60, 2775–2804. [Google Scholar] [CrossRef] - Gustafsson, F. Slip-based tire-road friction estimation. Automatica
**1997**, 33, 1087–1099. [Google Scholar] [CrossRef] - Gustafsson, F. Monitoring tire-road friction using the wheel slip. IEEE Control Syst.
**1998**, 18, 42–49. [Google Scholar] - Wang, J.; Alexander, L.; Rajamani, R. Friction Estimation on Highway Vehicles Using Longitudinal Measurements. J. Dyn. Syst. Meas. Control
**2004**, 126, 265–275. [Google Scholar] [CrossRef] - Germann, S.; Wurtenberger, M.; Daiss, A. Monitoring of the friction coefficient between tire and road surface. In Proceedings of the IEEE Conference on Control Applications, Scotland, UK, 24–26 August 1994; IEEE: Piscataway, NJ, USA, 1994; pp. 613–618. [Google Scholar]
- de Castro, R.; Araujo, R.E.; Cardoso, J.S.; Freitas, D. A new linear parametrization for peak friction coefficient estimation in real time. In Proceedings of the Vehicle Power and Propulsion Conference, Lille, France, 1–3 September 2010; IEEE: Piscataway, NJ, USA, 2010; pp. 1–6. [Google Scholar]
- Wang, B.; Guan, H.; Lu, P.; Zhang, A. Road surface condition identification approach based on road characteristic value. J. Terramechanics
**2014**, 56, 103–117. [Google Scholar] [CrossRef] - Xin, W.; Liang, G.; Mingming, D.; Xiaolei, L. State estimation of tire-road friction and suspension system coupling dynamic in braking process and change detection of road adhesive ability. Proc. Inst. Mech. Eng. Part D J. Automob. Eng.
**2021**, 236, 1170–1187. [Google Scholar] [CrossRef] - Li, G.; Fan, D.S.; Wang, Y.; Xie, R.C. Study on vehicle driving state and parameters estimation based on triple cubature Kalman filter. Int. J. Heavy Veh. Syst.
**2020**, 27, 126–144. [Google Scholar] [CrossRef] - Sharifzadeh, M.; Senatore, A.; Farnam, A.; Akbari, A.; Timpone, F. A real-time approach to robust identification of tire–road friction characteristics on mixed-μ roads. Veh. Syst. Dyn.
**2018**, 57, 1338–1362. [Google Scholar] [CrossRef] - Gao, L.; Xiong, L.; Lin, X.; Xia, X.; Liu, W.; Lu, Y.; Yu, Z. Multi-sensor Fusion Road Friction Coefficient Estimation During Steering with Lyapunov Method. Sensors
**2019**, 19, 3816. [Google Scholar] [CrossRef] [Green Version] - Chu, L.; Cui, X.; Zhang, K.; Fwa, T.F.; Han, S. Directional Skid Resistance Characteristics of Road Pavement: Implications for Friction Measurements by British Pendulum Tester and Dynamic Friction Tester. Transp. Res. Rec. J. Transp. Res. Board
**2019**, 2673, 793–803. [Google Scholar] [CrossRef] - Uz, V.E.; Gökalp, I. Comparative laboratory evaluation of macro texture depth of surface coatings with standard volumetric test methods. Constr. Build. Mater.
**2017**, 139, 267–276. [Google Scholar] [CrossRef] - Bitelli, G.; Simone, A.; Girardi, F.; Lantieri, C. Laser Scanning on Road Pavements: A New Approach for Characterizing Surface Texture. Sensors
**2012**, 12, 9110–9128. [Google Scholar] [CrossRef] [PubMed] - Vollor, T.W.; Hanson, D.I. Development of Laboratory Procedure for Measuring Friction of HMA Mixtures—Phase I. Final Report of NCAT 06–06 (2006); National Center for Asphalt Technology, Auburn, Alabama, 2006.
- Xu, H.; Prozzi, J.A. Correlation of aggregate form properties with fourier frequencies. Transp. Res. Rec. J. Transp. Res. Board
**2020**, 2674, 405–419. [Google Scholar] [CrossRef] - Wang, D.; Liu, P.; Xu, H.; Kollmann, J.; Oeser, M. Evaluation of the polishing resistance characteristics of fine and coarse aggregate for asphalt road using wehner/schulze test. Constr. Build. Mater.
**2018**, 163, 742–750. [Google Scholar] [CrossRef] - Hu, L.; Yun, D.; Gao, J.; Tang, C. Monitoring and optimizing the surface roughness of high friction exposed aggregate cement concrete in exposure process. Constr. Build. Mater.
**2020**, 230, 117005. [Google Scholar] [CrossRef] - Hao, X.; Sha, A.; Sun, Z.; Li, W.; Zhao, H. Evaluation and comparison of real-time laser and electric sand-patch road texture-depth measurement methods. J. Transp. Eng.
**2016**, 142, 04016022. [Google Scholar] [CrossRef] - Gökalp, I.; Uz, V.E.; Saltan, M. Comparative laboratory evaluation of macro texture depth of chip seal samples using sand patch and outflow meter test methods. In Bearing Capacity of Roads, Railways and Airfields; CRC Press: Boca Raton, FL, USA, 2017; pp. 915–920. [Google Scholar] [CrossRef]
- Zhu, H.; Liao, Y. Present situations of research on anti-skid property of asphalt road. Highway
**2018**, 63, 35–46. [Google Scholar] - Grosch, K.A. Visco-Elastic Properties and the Friction of Solids: Relation between the Friction and Visco-elastic Properties of Rubber. Nature
**1963**, 197, 858–859. [Google Scholar] [CrossRef] - Adam, C.S.; Piotrowski, M. Use of the unified theory of rubber friction for slip-resistance analysis in the testing of footwear outsoles and outsole compounds. Footwear Sci.
**2012**, 4, 23–35. [Google Scholar] [CrossRef] - Mazzola, L.; Galderisi, A.; Fortunato, G.; Ciaravola, V.; Giustiniano, M. Influence of Surface Free Energy and Wettability on Friction Coefficient between Tire and Road Surface in Wet Conditions. Adv. Contact Angle Wettability Adhes.
**2013**, 1, 389–410. [Google Scholar] [CrossRef] - Persson, B.N. Rubber friction: Role of the flash temperature. J. Phys. Condens. Matter
**2006**, 18, 7789–7823. [Google Scholar] [CrossRef] - Ueckermann, D.; Wang, M.; Oeser, B. Steinauer, Calculation of skid resistance from texture measurements. J. Traffic Trans. Eng.
**2015**, 2, 3–16. [Google Scholar] - Hertz, H. On the contact of elastic solids. J. Die Reine Angew. Math.
**1882**, 92, 156–171. [Google Scholar] [CrossRef] - Greenwood, J.A.; Williamson, J. Contact of nominally flat surfaces. Proc. R. Soc. Lond.
**1966**, 295, 300–319. [Google Scholar] - Carbone, G. A slightly corrected Greenwood and Williamson model predicts asymptotic linearity between contact area and load. J. Mech. Phys. Solids
**2009**, 57, 1093–1102. [Google Scholar] [CrossRef] - Persson, B.N.J. Theory of rubber friction and contact mechanics. J. Chem. Phys.
**2001**, 115, 3840–3861. [Google Scholar] [CrossRef] [Green Version] - Lu, Y.; Zhang, J.; Yang, S.; Li, Z. Study on improvement of LuGre dynamical model and its application in vehicle handling dynamics. J. Mech. Sci. Technol.
**2019**, 33, 545–558. [Google Scholar] [CrossRef] - Yongjie, L.; Junning, Z.; Haoyu, L.; Zhizhe, M. Research on tire-road system coupling dynamics based on non-uniform contact. J. Mech. Eng.
**2021**, 57, 87–98. (In Chinese) [Google Scholar] - Yongjie, L.; Wenqing, H.; Junning, Z. Construction of Three-Dimensional Road Surface and Application on Interaction between Vehicle and Road. Shock Vib.
**2018**, 2018, 1–14. [Google Scholar] [CrossRef] [Green Version] - Han, Y.; Lu, Y.; Chen, N.; Wang, H. Research on the Identification of Tire Road Peak Friction Coefficient under Full Slip Rate Range Based on Normalized Tire Model. Actuators
**2022**, 11, 59. [Google Scholar] [CrossRef] - Kiencke, U.; Nielsen, L. Automotive Control Systems: For Engine, Driveline, and Vehicle; Springer: Berlin/Heidelberg, Germany, 2005. [Google Scholar]
- Rajamani, R. Vehicle Dynamics and Control, 2nd ed.; Springer Science: London, UK, 2012. [Google Scholar]
- Mechanical Vibration-Road Surface Profiles-Reporting of Measured Data; Standardization Administration: Beijing, China, 2006.
- Standard’s ISO 21994-2007; Standard’s Passenger Cars-Stopping Distance at Straight-Line Braking with ABS-Openloop Test Method. ISO: Geneva, Switzerland, 2007.
- Standard’s ISO 4138:2004; Standard’s Circular Driving Behaviour—Open-Loop Test Methods. ISO. ISO: Geneva, Switzerland, 2012.

Symbol | Value and Unit | Parameter Name |
---|---|---|

m_{z} | 880 kg | Total vehicle mass |

m_{b} | 788 kg | Sprung mass |

L | 2.040 m | Wheel base |

a | 1.145 m | Distance from centroid to front axle |

b | 0.895 m | Distance from centroid to rear axle |

h_{g} | 0.54 m | Centroid height |

T | 1.3 m | Wheel track width |

I_{z} | 832.3 kg · m^{2} | Moment of inertia about the z-axis |

K_{ψ} | 25,041 N/rad | Tire cornering stiffness |

${K}_{b}$ | $19.6\mathrm{kN}/\mathrm{m}$ | Stiffness coefficient of suspension system |

${c}_{b}$ | $1450\mathrm{N}\cdot \mathrm{s}/\mathrm{m}$ | Damping Constant of Suspension Buffer |

${K}_{w}$ | $250\mathrm{kN}/\mathrm{m}$ | Tire stiffness coefficient |

${C}_{w}$ | $375\mathrm{N}\cdot \mathrm{s}/\mathrm{m}$ | Tire damping coefficient |

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**MDPI and ACS Style**

Han, Y.; Lu, Y.; Liu, J.; Zhang, J.
Research on Tire/Road Peak Friction Coefficient Estimation Considering Effective Contact Characteristics between Tire and Three-Dimensional Road Surface. *Machines* **2022**, *10*, 614.
https://doi.org/10.3390/machines10080614

**AMA Style**

Han Y, Lu Y, Liu J, Zhang J.
Research on Tire/Road Peak Friction Coefficient Estimation Considering Effective Contact Characteristics between Tire and Three-Dimensional Road Surface. *Machines*. 2022; 10(8):614.
https://doi.org/10.3390/machines10080614

**Chicago/Turabian Style**

Han, Yinfeng, Yongjie Lu, Jingxv Liu, and Junning Zhang.
2022. "Research on Tire/Road Peak Friction Coefficient Estimation Considering Effective Contact Characteristics between Tire and Three-Dimensional Road Surface" *Machines* 10, no. 8: 614.
https://doi.org/10.3390/machines10080614