## 3 Emergency Braking Control Strategy Design
### Rule-Based Braking Torque Allocating Method  
There are three independent braking actuators for each half-shaft, and how to allocate the demanded braking torque derived in Equation (20) to them is the key to better follow the change of it and get a better braking performance. These three braking actuators have their own advantages and drawbacks in different aspects. The mechanical brake can offer a large amount of braking torque, but its dynamic response is quite poor. The hydraulic retarder has a better response than the mechanical brake and can provide a stable braking torque in a rather high revolution speed, but its braking torque is nonadjustable at a certain revolution speed; what is more, its braking torque declines dramatically with the reduction of speed when the half-shaft rotates under a certain speed. As for the PMSM, its torque response is the best among these actuators and even can provide a positive torque in the braking process, which can help to improve the system response. However, according to its model, the maximum torque of PMSM is related to its revolution speed, so the maximum motor torque declines when exceeding the motor'srated speed, even though the rated torque of it is already much smaller than the mechanical brake can provide; what is more, the accuracy of its torque control and the efficiency of the motor drive system also become worse when the revolution speed of the half-shaft is too low. Therefore, several basic rules can be set according to the above characteristics of these braking actuators. Firstly, the total demanded braking torque can be divided into a steady part that changes less in the braking process and a dynamic part that is associated with the change of \(s_{\mathrm{s}}\). Secondly, the steady part is provided by the mechanical brake and the hydraulic retarder. If the steady part of demanded torque is larger than the maximum torque of the hydraulic retarder at the current speed, it will be started up and generate torque; otherwise, it will be shut down. The difference of the steady part of demanded torque and the hydraulic retarder torque will be compensated by the mechanical brake. Thirdly, the dynamic part is provided by the PMSM. Finally, the hydraulic retarder exits when the speed of the vehicle is less than 12 km/h, and the PMSM exits under the point of 5 km/h to suit its own features. The braking torque allocation method is shown in Figure 5.  
Substituting the definition of the sliding surface into Equation (20), we can get that:  
\[T_{hdem}=r\hat{F}_{x}+\frac{1-\lambda_{d}}{r}I_{ori}\dot{v}_{x}+\frac{\omega I _{ori}}{1-\lambda}(\varepsilon+F_{d})sign(s_{\mathrm{s}})+\left(-\frac{1}{r}I _{ori}\dot{v}_{x}+\frac{\omega I_{ori}}{1-\lambda}k\right)_{\mathrm{s}} \tag{21}\]  
In this equation, we can find that although \(F_{x}\), \(I_{ori}\), and \(\dot{v}_{x}\) are functions of slip ratio \(\lambda\), they change slowly when the slip ratio changes around the point \(\lambda_{d}=0.2\); however, the value of the last two items in this equation changes greatly with the fluctuation of \(s_{\mathrm{s}}\), and the value may be positive, which can only be provided by the PMSM. Thus, the steady and dynamic part of total braking torque can be written as follows:  
\[\left\{ \begin{array}{l}T_{\mathrm{\textit{bdem}}}=r\hat{F}_{x}+\frac{1- \lambda_{d}}{r}J_{ori}\dot{v}_{x}\\ T_{\mathrm{\textit{bdem}}}=\frac{\omega I_{ori}}{1-\lambda}(\varepsilon+F_{d} )sign(s_{\mathrm{s}})+\left(-\frac{1}{r}I_{ori}\dot{v}_{x}+\frac{\omega I_{ ori}}{1-\lambda}k\right)_{\mathrm{s}}\end{array} \right. \tag{22}\]  
However, the dynamic part of total demanded braking torque \(T_{\mathrm{\textit{bdem}}}\) may exceed the ability of the PMSM greatly when the deviation of the real slip ratio \(\lambda\) and demanded slip ratio \(\lambda_{d}\) is large, so it is better to use the mechanical brake compensating the difference of them to improve the control effect. Figure 5: Schematic diagram of the braking torque allocation method.