# Robust Estimation of Vehicle Motion States Utilizing an Extended Set-Membership Filter
## 4 Calculation of Tire Forces
### Evaluation of Tire Force Estimator  
Numerical experiment is implemented via CarSim (a professional software widely used in automotive industry) and Matlab/Simulink co-simulation to testify the performance of the scheme chosen for the estimation of the tire forces.  
Figure 5 shows the structure for the calculation of tire forces, where the output signals of CarSim are taken as the inputs of the tire force estimators at each sampling instant. The longitudinal and lateral tire force estimators are built by use of the \"S-Function\" module in Simulink, where the corresponding algorithm code are integrated.  
A front-wheel driven vehicle is selected and set to move at a constant speed of 120 km/h along the route depicted in Figure 6. The corresponding results obtained from these two estimators are used to make comparison with the reference values of the tire forces output by CarSim.  
Figure 5: Structure for the calculation of tire forces.  
Figure 6: Driving route.  
Noticing that the vehicle is driven by front wheels, some representative estimation results are presented in Figure 7 for the convenience of analysis, where the longitudinal forces of two front tires and the lateral forces of two left tires are included.  
It can be seen from Figure 7 that the estimated longitudinal forces for left and right front tires match the reference curves well, where the maximum errors of about 7.5% and 8.9% occur at around time points 3 s and 4 s, respectively.  
For the estimated lateral force of left front tire in Figure 7c, deviation appears at the first few seconds of vehicle motion. The reason for this is the insufficient lateral excitation from ground. After the steering wheel angle starts working, the estimated results almost match the reference values, except for some moments when the vehicle is just in high maneuvers. For example, around time points 2.6 s, 3.5 s, 4.4 s and 5.3 s. The reason for the deviations at these moments can be ascribed to the neglected role of the suspension in vehicle dynamic model.  
The performance of the adopted scheme can be further evaluated by the normalized root-mean-square (NRMS) of the corresponding estimated result [28], which is calculated as follows:  
\[\text{NRMS}=\frac{\sqrt{\sum\limits_{i=1}^{n}\frac{(F_{\text{reference}}-\hat{F }_{\text{estimated}})^{2}}{n}}}{\text{max}(|F_{\text{reference}}|)} \tag{27}\]  
Figure 7: Estimation results of tire forces. (**a**) Longitudinal force of left front tire; (**b**) Longitudinal force of right front tire; (**c**) Lateral force of left front tire; (**d**) Lateral force of left rear tire.  
where \(F_{\text{reference}}\) is the reference value of tire force, \(\hat{F}_{\text{estimated}}\) is the estimated value, and \(n\) is the sampling number. For the estimation results in Figure 7, the NRMSs of \(F_{x1}\), \(F_{x2}\), \(F_{y1}\) and \(F_{y3}\) are 3.44%, 3.56%, 4.10% and 2.76%, respectively, which shows that the accuracy of the estimated tire force is acceptable.  
Based on the above scheme with simple structure, the calculation of tire forces can be provided with good accuracy, where no knowledge of road adhesion coefficient is needed. Therefore, it is likely to be adopted in some practical applications. In the following part of this paper, the output results of the tire force estimators are taken as one portion of the input of the proposed robust estimation strategy.