## 5 Tire-Road Friction Estimation to Improve Predictive Control
### Tire-Road Friction Estimation System  
The friction-estimation circles are presented using three lines. The exact estimation, under-estimation, and over-estimation cases are presented in green, red, and blue, respectively. There are several methods for estimating tire-road friction. For the ELSD system to provide optimal control at the time of wheel slip, the friction value is estimated by real-time monitoring of the driving torque when wheel slip occurs, as shown in Figures 11 and 12, and the following equations.  
Figure 9: Simulation studies. (**a**) Steering wheel angle; (**b**) longitudinal speed; and (**c**) lateral acceleration; (**d**) side slip angle.  
The road-friction coefficient estimation system uses information from the controller area network communication of the vehicle to calculate longitudinal, lateral, and vertical forces, thereby estimating the primary friction coefficient. To update the initially estimated friction coefficient, the system checks the feasibility of calculating wheel slips based on the current road conditions. It assesses the driving state of the vehicle by examining the speed difference between the vehicle and the driven wheels, as well as the speed difference between the front and rear wheels.  
In unstable conditions, such as significant wheel slip where tire grip exceeds the road-friction limit, the system employs values close to the road-friction coefficient. In stable conditions with minimal wheel slip, it selects the larger value between the calculated and the existing friction coefficient values, facilitating a rapid convergence to the actual friction coefficient. While adjusting the cutoff frequency of the low-pass filter according  
Figure 11: Tire–road friction estimation system architecture.  
Figure 12: Control logic diagram for tire–road friction estimation.  
Figure 10: The concept of ELSD intervention time via friction estimation.  
to the driving state of the vehicle, the system ultimately filters the estimated road-friction coefficient for use in ELSD predictive control.  
\[\mu=\frac{\sqrt{\mathbf{{F_{x}}^{2}+\mathbf{{F_{y}}^{2}}}}}{\mathbf{{F_{z}}}} \tag{7}\]  
\[\mathbf{{F_{x}}}\mathbf{=}\frac{\mathbf{{T_{drive}}-\mathbf{{I_{drive}}}\cdot \theta_{drive}^{\cdot\cdot}}}{\mathbf{{R_{tire}}}} \tag{8}\]  
\[\mathbf{{F_{y}}}\mathbf{=}\mathbf{m\cdot}\mathbf{g} \tag{9}\]  
\[\mathbf{{F_{z}}}\mathbf{=}\mathbf{m\cdot}\mathbf{g} \tag{10}\]  
The road-friction coefficient estimation considers not only longitudinal forces based on driving torque (\(\mathbf{{T_{drive}}}\)), driveline rotational inertia (\(\mathbf{{I_{Drive}}}\)), and rotational acceleration (\(\theta_{drive}^{\cdot\cdot}\)) but also lateral forces generated by steering maneuvers.  
Because the electronic stability control (ESC) system operates in the region of the grip limit, the friction coefficient calculated at this time is close to the actual friction value. However, it is necessary to estimate the friction coefficient for ELSD at all times while in operation. Therefore, in the region of the grip limit (unstable wheel slip region that is above the wheel slip criterion value), the instantaneous calculated value is estimated to be the real-time friction coefficient, as in ESC shown in Table 1.  
M(n \(-\) 1): Friction coefficient calculated just before in other regions, minor speed differences occur between the driving and non-driving wheels during moderate acceleration, and the corresponding engine torque is low. Therefore, the calculated friction values are continuously low. Hence, this was defined as the stable region (stable wheel slip region, below the wheel slip criterion value). In this region, the friction coefficient is estimated to be the larger of the current friction coefficient and the previous friction coefficient.