## 6 Individual Brake Force Estimation
``` Input \(\omega_{i}\), \(r_{eff}\), \(V_{wx,i}\), \(\mu_{0}\), \(a_{x}\), \(a_{y}\), \(\hat{\mu}_{i}\), \(\hat{m}_{y}\), \(t_{f}\), \(t_{r}\), \(l_{f}\), \(l_{r}\), \(g\), \(h\) Output \(\hat{F}_{x:\text{FL}}\), \(\hat{F}_{x:\text{RR}}\), \(\hat{F}_{x:\text{RL}}\), \(\hat{F}_{x:\text{RR}}\)
1:\(\sigma_{xi}=\left(r_{eff}\omega_{i}-V_{wx,i}\right)/\text{max}\left(r_{eff} \omega_{i},V_{wx,i}\right)\)
2:\(F_{z:\text{FL}}=m_{y}\left[g\frac{l_{f}}{h_{f}}-a_{x}\frac{l_{f}}{h_{f}}, \frac{l_{f}}{l_{f}}\right]\)
3:\(F_{z:\text{FR}}=m_{y}\left[g\frac{l_{f}}{h_{f}}-a_{x}\frac{l_{f}}{h_{f}}+a_{y} \frac{l_{f}}{h_{f}l_{f}}\right]\)
4:\(F_{z:\text{RL}}=m_{y}\left[g\frac{l_{f}}{h_{f}}+a_{x}\frac{l_{f}}{h_{f}}-a_{y} \frac{l_{f}}{h_{f}l_{f}}\right]\)
5:\(F_{z:\text{RR}}=m_{y}\left[g\frac{l_{f}}{h_{f}}+a_{x}\frac{l_{f}}{h_{f}}+a_{y} \frac{l_{f}}{h_{f}l_{f}}\right]\)
6:\(p_{i}=\text{abs}(\sigma_{xi}*\mu_{0}/\hat{\mu}_{i})\)
7:\(p_{i}(p_{i}>1.02)=1.02\) %Set the upper limit of value
8:\(p_{i}(p_{i}<0)=0\) %Set the lower limit of value
9: Fori = 1:1:4 % i = 1 (FL)/i = 2 (FR)/i = 3 (RL)/i = 4 (RR),
10:\(\hat{F}_{xi}=-\hat{\mu}_{i}/\mu_{0}*\left(\text{interp}2\left(F_{x0},F_{z0}, \sigma_{0},F_{zi},p_{i},\text{linea}^{\prime}\right)\right)\) % The above \(F_{x0},F_{z0}\) and \(\sigma_{0}\) are reference values of 3D mapped longitudinal forces for the vertical force and slip ratio in Figure 6
11: End

Figure 7: Reference curves of tire force in longitudinal direction. ## 7 Integrated Estimation System

When vehicle velocity is less than 10 km/h, the mass estimation is enabled to obtain and update the total vehicle mass. This mass value will be maintained at the specific value before the suspension of estimation if the estimation condition is not satisfied. The estimated total mass is used to calculate the individual vertical forces of tires via (7) through (10) for the brake force estimation. During vehicle operation, if the brake is applied and the steering wheel angle is smaller than 20 degs., the rest of the integrated estimation (road friction and brake force estimations) will be activated. Based on (15) and (16), the averaged input data of the trained ANN classifier are calculated by vehicle velocity and wheel speed. Then, based on the averaged data, the estimated road friction is produced by the ANN classifier. Finally, with the current estimates of road frictions on each wheel and estimated mass, the individual brake forces are identified by (26) together with (27) and (28). ## 8 Simulation Study

### The Data Preparation and Training Phase

Here, we presented the training data set for the ANN classifier, which is coming from a co-simulation between CarSim and Simulink. A sedan C-class with four-wheel drive, specified in Table 1, is selected from CarSim. The data obtained from the co-simulations are the vehicle velocity, acceleration, wheel speed, brake control signal, and steering angle. Using an Algorithm 2 based on (15) and (16), the data shown in Figure 8 have been generated. The vehicle model with different masses of 1642 kg (Group1), 1842 kg (Group2), and 2042 kg (Group3) runs at three random target speeds, 25 km/h, 79 km/h, and 107 km/h and then the brake is applied until the vehicle is completely stopped. Each brake cycle is performed at different road friction ranging from 0.1-1. Hence, the training data set is constructed from ninety simulations. The label data (true references) for ANN training is presented in Figure 9. The ANN training strategy follows the diagram described in Figure 10 thus the training will be continued until the acceptable error between true references and estimates (outputs of trained ANN) is achieved.