## 2 Design of the Revised Braking Regenerative Control Strategy
### Realization of Braking Torque Based on OSR  
As shown in Figure 3a, the motor reaches the allowable braking force upper limit \(T_{mot\_max}\) at point \(A\). The braking strength corresponding to point A is defined as the threshold value of low brake strength \(Z_{a\_J}\), and the braking strength corresponding to point \(B\) is defined as the threshold value of high brake strength \(Z_{b\_J}\). The braking force distribution should follow the ideal braking force distribution curve (\(I\) curve), so that the front and rear wheels are locked at the same time to fully use the road surface adhesion. The \(I\) curve is related to the vehicle's actual mass and static axle load, which are variable in daily driving. The ideal braking force distribution is obtained by adjusting the slip ratio of the front and rear wheels. To realize the braking torque control, the relationship between the front and rear axle braking force and the slip ratio is derived as shown in Figure 3b. According to the \(ECE\)\(R13\)-\(H\) braking regulations, the rear wheels must have a certain braking force to maintain vehicle's stability and braking efficiency when the front and rear wheels are locked. The relationship between the rear wheels' minimum braking force and the front wheels' braking force is the \(M\) curve, as shown in the Figure 3a. \(W\) is the correction factor with the tire-road adhesion condition consideration. Together with \(Z_{a\_J}\) and \(Z_{b\_J}\), they can be calculated as follows:  
\[Z_{a\_J}=\frac{WT_{\max\_mod}i}{mgR} \tag{1}\]  
\[Z_{b\_J}=\frac{\sqrt{m_{f}^{2}+4Z_{a\_J}h_{g}/l}-m_{f}}{2h_{g}/l} \tag{2}\]  
\[W=\left\{ \begin{array}{c}0,0\leq\mu_{\max}\leq\mu_{w\_J}\\ \frac{(\mu_{\max}-\mu_{w\_J})}{(\mu_{w\_J}-h^{\prime}\mu_{w\_J})},\mu_{w\_J} \leq\mu_{\max}\leq\mu_{w\_J}\\ 1,\mu_{w\_J}\leq\mu_{\max}\end{array} \right. \tag{3}\]  
where \(T_{mot\_max}\) is the motor allowable braking force upper limit; \(m\) is the total mass of vehicle; \(R\) is the wheelbase; \(i\) is the transmission ratio; \(\mu_{max}\) is the peak tire-road adhesion coefficient (PTAC); \(\mu_{w\_J}\) is the threshold value of the low tire-road adhesion coefficient (TAC); \(\mu_{w\_h}\) is the threshold value of the high TAC; \(m_{f}\) is the front axle mass distribution coefficient; \(l\) is the front and rear axle wheelbase; and \(h_{g}\) is the centroid height. The distribution values of the slip ratio in three braking stages are described as follows:  
1. Full regenerative braking of the \"\(OA\)\" stage \[S_{f}=\varphi_{f}^{-1}[\frac{Z_{a\_J}}{m_{f}+Z_{\_}{\_}{\text{rd}}h_{g}/l}], S_{r}=0\] (4)  
Figure 3: Coordinate system conversion: (**a**) wheel braking force distribution and (**b**) slip ratio distribution. 2. Hydraulic braking compensation of the \"\(AB\)\" stage \[S_{f}=\varphi_{f}^{-1}[\frac{Z_{a,J}}{m_{f}+Z_{\textit{req}}h_{g}/l}],S_{r}= \varphi_{r}^{-1}[\frac{Z_{\textit{req}}-Z_{a,J}}{m_{f}-Z_{\textit{req}}h_{g}/l}]\] (5) 3. Regenerative braking and hydraulic braking coordination of the \"\(BC\)\" stage \[S_{f}=\varphi_{f}^{-1}[Z_{\textit{req}}],\quad S_{r}=\varphi_{r}^{-1}[Z_{ \textit{req}}]\] (6)  
Considering the difference of the braking torque between the front and rear wheels, the DFLC for the front and rear wheels is designed to ensure OSR tracking control target values effectively. As shown in Figure 4, the demand slip ratio \(s_{\textit{req}}(t)\) of the front and rear wheels can be obtained from the demand braking strength calculation and \(s_{\textit{ost}}(t)\) can be output in real time. If \(s_{\textit{req}}(t)<s_{\textit{ost}}(t)\), the OSR control target \(s_{\textit{csr}}(t)\) is defined as \(s_{\textit{csr}}(t)=s_{\textit{req}}(t)\). Otherwise, \(s_{\textit{csr}}(t)\) is defined as \(s_{\textit{csr}}(t)=s_{\textit{ost}}(t)\), and it generally occurs on the low tire-road adhesion condition or the braking strength demand is too large.