# Model Reference Adaptive Control of Vehicle Slip Ratio Based on Speed Tracking
## 2 Problem Formulation  
### Vehicle Slip Ratio Model  
In this section, a quarter vehicle model involving the longitudinal vehicle speed dynamics and the wheel rotational dynamics represents the vehicle fundamental longitudinal dynamics [11; 21]. As shown in Figure 1, the model is obtained from single-wheel vehicle dynamics attached to a mass \(m\). The braking force \(F_{d}\) is  
\[F_{d}=\mu(\lambda)mg \tag{1}\]  
where, \(\lambda\) is the longitudinal slip ratio, \(\mu\) is the road adhesion coefficient, \(m\) is the mass of the vehicle supported by the wheel.  
Obviously, braking force \(F_{d}\) depends on the road adhesion coefficient \(\mu\) which is a nonlinear function of the longitudinal slip ratio \(\lambda\). Among the uncertainties, the road condition is one of the most complicated variables, which is difficult to be measured in real time but affects the control performance  
Figure 1: Quarter-vehicle braking model.  
directly. Some researches [22; 23; 24] adopted empirically parameterized tire models to represent the relationship, such as the Magic Tire Formula [25], the Burckhardt [26; 27; 28] and the LuGre [27]. However, it is difficult to describe the varying road conditions accurately, which is the research focus of this study.  
Considering longitudinal velocity and angular velocity of the wheel as the degrees of freedom during the braking process, the wheel nonlinear dynamic can be described [21]:  
\[n_{w}m\frac{dv}{dt}=n_{w}\mu(\lambda)mg-c_{v}v^{2} \tag{2}\]  
\[J_{w}\frac{d\omega_{w}}{dt}=T_{b}-r_{w}(\mu(\lambda)mg+f_{w}) \tag{3}\]  
where, \(f_{w}=c_{f}\omega_{w}\) denotes the various rolling friction forces and \(c_{f}\) is the friction coefficient, \(r_{w}\) is the wheel radius, \(J_{w}\) is the wheel inertia, \(n_{w}\) is the number of wheels, \(c_{v}\) is the air resistance coefficient.  
The longitudinal slip ratio [21] is defined as follows  
\[\lambda=\frac{v_{w}-v}{\max(v,v_{w})} \tag{4}\]  
where \(v_{w}=\omega_{w}r_{w}\). It describes the normalized difference between the vehicle speed and the speed of the wheel perimeter. If \(\lambda=0\), the wheel is free motion and \(v_{w}=v\). while \(\lambda=-1\) means that the wheel is locked. The relationship of the typical road adhesion coefficient with the wheel slip developed by Burckhardt model is shown in Figure 2[28; 29].  
It can be seen that the slip ratio is different on the different road conditions and the difference between the vehicle speed and the speed of the wheel varies with the conditions too. However, the optimal slip ratio is controlled within a certain scope, which is considered as stable regions. Thus, the desired value can be set in the range or a desired slip ratio trajectory. The control problem is to regulate the slip ratio in accordance with the desired slip ratio and can be solved from the point of view of speed tracking \(v_{w}=(1+\lambda^{*})v\).  
**Remark 1**.: _For the situation of vehicle longitudinal slip ratio control, the braking torque is controlled according to the wheel that first appears braking lock or the road surface with low adhesion coefficient. Therefore, a quarter vehicle model involving the longitudinal vehicle speed dynamics is available for research and it can be further extended to more complex mechanical structures, e.g., full-wheel-drive vehicles. Of course, during the turning process, the lateral vehicle speed, yaw rate, and the lateral slip ratio should be considered which will be the next research._  
Figure 2: Quarter-vehicle braking model.