## 3 Emergency Braking Control Strategy Design
### Sliding Mode Slip Ratio Controller  
According to Equation (6), the derivation of the slip ratio with respect to time is:  
\[\dot{\lambda}=\frac{\dot{v}_{x}\omega r-\dot{\omega}rv_{x}}{v_{x}^{2}} \tag{13}\]  
Substitute \(\omega\) as a function of \(\lambda\) from Equation (6) and \(\dot{\omega}\) defined in Equation (8) into Equation (13); we can get that:  
\[\dot{\lambda}=f-\frac{(1-\lambda)}{\omega r_{\rm ori}}T_{b} \tag{14}\]  
where:  
\[f=\frac{(1-\lambda)^{2}J_{\rm ori}\dot{v}_{x}+(1-\lambda)F_{x}r^{2}}{\omega r _{\rm ori}} \tag{15}\]  
Figure 4: Schematic diagram of the braking system.  
Here, \(f\) is a function of vehicle deceleration \(\dot{v}_{x}\), drive wheel angular velocity \(\omega\), and track-ground resultant force \(F_{x}\). Since \(\dot{v}_{x}\) and \(\omega\) are directly measured by the accelerometer and encoder, the errors of them are relatively small, so the error of estimating \(f\) is mainly the result of the error estimating \(F_{x}\). Assuming that the estimation of \(F_{x}\) is bounded, then the error of \(f\) can be expressed as:  
\[\widetilde{f}=\hat{f}-f=\frac{(1-\lambda)}{\omega\varepsilon_{ri}}\Big{(}\hat {F}_{x}-F_{x}\Big{)} \tag{16}\]  
and its boundary can be represented as:  
\[F_{d}=\frac{(1-\lambda)r}{\omega\varepsilon_{ri}}\widetilde{F}_{x,max} \tag{17}\]  
where \(\widetilde{F}_{x,max}\) is the maximum error of estimating \(F_{x}\).  
A zero-order sliding surface can be chosen to track the demanded slip ratio [8; 13; 29]; the sliding surface and its derivation with respect to time are:  
\[\left\{\begin{array}{l}s_{s}=\lambda-\lambda_{d}\\ \dot{s}_{s}=\dot{\lambda}-\dot{\lambda}_{d}\end{array}\right. \tag{18}\]  
Although the \(\mu-\lambda\) curves vary widely according to different track-ground conditions, it is fortunate that the adhesion coefficient always achieves its largest value when the slip ratio nears the point of 0.2, as shown in Figure 3, so the demanded slip ratio can be set as 0.2 for convenience, no matter at what track-ground conditions. Thus, the derivation of the sliding surface equals that of the slip ratio.  
In order to converge as quickly as possible when \(s\) is large and reduce chattering when \(s\) is small, a sliding mode robust control method using the exponential approaching law is suitable for a plant with bounded disturbance [38]. Thus, regarding the error of estimating \(f\) as a disturbance, the derivation of the sliding surface can be rewritten as:  
\[\dot{s}_{s}=-\varepsilon sign(s_{s})-ks_{s}=\hat{f}-\frac{(1-\lambda)}{\omega \varepsilon_{ri}}T_{b}-\widetilde{f},\varepsilon>0,k>0 \tag{19}\]  
where \(sign(s_{s})\) is a sign function of \(s_{s}\), \(k\) is suggested to be designed relatively large to realize a high approaching speed, and \(\varepsilon\) is suggested to be designed relatively small to reduce the chattering near the sliding surface. According to Equations (15), (17), and (19), the total demanded braking torque converted to the drive wheel can be derived as (detailed in Appendix C):  
\[T_{bdem}=r\hat{F}_{x}+\frac{1-\lambda}{r}J_{\sigma i}\dot{v}_{x}+\frac{\omega j _{\sigma i}}{1-\lambda}[(\varepsilon+F_{d})sign(s_{s})+ks_{s}] \tag{20}\]