# Research on Multi-Mode Braking Energy Recovery Control Strategy for Battery Electric Vehicles
## 2 Brake Energy Recovery Control Strategy
### Conventional Brake Energy Recovery Control Strategy
#### 2.3.1 Front and Rear Axle Brake Force Distribution  
(1) Ideal front and rear axle brake force distribution I-curve:  
In battery electric vehicles, in the braking dead process, the vehicle's front and rear wheels will appear in three situations: (1) the front wheels first hold dead drag slip, followed by the rear wheels, and then hold dead; (2) the rear wheels first hold dead drag slip, followed by the front wheels, and then hold dead; (3) the front and rear wheels at the same time hold dead. When the wheels appear, the front wheels first hold dead slippery; although the vehicle's front wheels hold dead and lose steering ability, this situation can still ensure the stability of the braking; when the rear wheels first hold dead slippery, although the front wheels do not hold dead, they can still steer but are prone to the tailgate out of control phenomenon; when the vehicle's braking force and its adhesion are equal to the wheel on the road surface, the coefficient of adhesion of the utilization of the maximum at this point. Not only can it ensure the steering ability of the vehicle, but it can also ensure the safety and stability of braking. During the braking of the vehicle, the vehicle is subjected to the force of gravity and the vertical reaction force of the front and rear axles, and the force balance equation in the vertical direction is as follows:  
\[mg=F_{zf}+F_{zr} \tag{10}\] \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**k**} & \multicolumn{4}{c}{**AP\_t**} \\ \cline{2-5}  & & **L** & **M** & **H** \\ \hline \multirow{3}{*}{**v**} & L & M & M & L \\  & M & H & M & L \\ \cline{1-1}  & H & H & M & L \\ \hline \end{tabular}
\end{table} 
Table 1: Single-pedal mode fuzzy rule table. where \(F_{zf}\) is the ground normal reaction force on the front wheels, \(F_{zr}\) is the ground normal reaction force on the rear wheels, \(m\) is the mass of the vehicle, and \(g\) is the acceleration of gravity. This can be obtained by analyzing the forces at the tangent points of the front and rear wheels to the ground:  
\[F_{zr}L=mga-m\frac{dv}{dt}h \tag{11}\]  
\[F_{zr}L=mgb+m\frac{dv}{dt}h\]  
where \(a\) is the distance from the center of mass to the front axle of the vehicle; \(b\) is the distance from the center of mass to the rear axle of the vehicle; \(d_{v}/d_{t}\) is the acceleration of the vehicle; and \(h\) is the height of the center of mass of the vehicle. The braking strength \(z\) is as follows:  
\[zg=\frac{dv}{dt} \tag{12}\]  
The ground force on the front and rear wheels of the vehicle during braking can be obtained after association as follows:  
\[\left\{ \begin{array}{l}F_{zf}=\frac{mg(b+zh)}{L}\\ F_{zr}=\frac{mg(a-zh)}{L}\end{array} \right. \tag{13}\]  
where \(F_{zf}\) is the force of the ground on the front wheel and \(F_{zr}\) is the force of the ground on the rear wheel. \(z\) is the braking strength. In the actual driving process of the vehicle, the braking of the vehicle is also related to the adhesion coefficient of the driving surface. In different adhesion coefficients of the road surface, the vehicle's braking force on the front and rear wheels should meet the following formula:  
\[\left\{ \begin{array}{l}F_{zf}=\frac{mg}{L}(b+\varphi)\\ F_{zr}=\frac{mg}{L}(a-\varphi h)\end{array} \right. \tag{14}\]  
where \(\varphi\) is the pavement adhesion coefficient. In vehicles in the braking process, when the front and rear wheels of the vehicle are simultaneously on hold on the road surface, adhesion coefficient utilization is the largest, which can ensure the stability and safety of the vehicle when braking. Accordingly, it can be obtained as follows:  
\[\left\{ \begin{array}{l}F_{uf}+F_{ur}=F_{z}\varphi\\ \frac{F_{uf}}{F_{ur}}=\frac{b+gh}{d-\varphi h}\end{array} \right. \tag{15}\]  
In the above equation, \(F_{uf}\) and \(F_{ur}\) are the braking forces for the front and rear axles of the vehicle, respectively.