# Central Non-Linear Model-Based Predictive Vehicle Dynamics Control
## 3 Central Predictive Control
### Prediction
#### 3.1.2 Self-Steering Behavior  
The basis for the prediction of the self-steering behavior is the single-track model [24]. Here, the wheels of an axle are virtually combined for modeling. The single-track model can be used to describe and predict the self-steering behavior and, thus, the response of the vehicle to steering movements [25]. Within the control system, the self-steering gradient \(SSG\) is used as the characteristic variable:  
\[SSG(k)=\frac{(\alpha_{\text{f}}(k)-\alpha_{\text{r}}(k))}{a_{y}} \tag{4}\]  
The variables \(\alpha_{\text{f}}\) and \(\alpha_{\text{r}}\) are the slip angles at the front and rear axles, respectively. The slip angles are dependent of the yaw rate \(\dot{\psi}\), the velocity \(v\) and the side-slip angle \(\beta\). In addition, the steering angle \(\delta\) affects the slip angle at the front axle \(\alpha_{\text{f}}\). The parameters\(l_{\text{f}}\) and \(l_{\text{r}}\) represent the distance from the center of gravity to the front axle and the rear axle, respectively. \[\alpha_{\text{f}}(k) =\delta-\arctan\left(\frac{l_{\text{f}}\dot{\psi}(k)+v\sin\beta(k)}{ v\cos\beta(k)}\right) \tag{5}\] \[\alpha_{\text{r}}(k) =-\arctan\left(\frac{-l_{\text{r}}\dot{\psi}(k)+v\sin\beta(k)}{v \cos\beta(k)}\right) \tag{6}\]  
Whereas the steering angle and the velocity are kept constant within the prediction, the yaw rate and the side-slip angle are predicted. For this purpose, both [PERSON]'s principle in the lateral direction and the principle of angular momentum in the \(x-y\) plane are applied:  
\[\frac{\left(\cos\delta\big{(}F_{y,\text{fl}}(k)+F_{y,\text{fr}}(k)\big{)}+ \big{(}F_{y,\text{rl}}(k)+F_{y,\text{rr}}(k)\big{)}\right)}{mv\cos\beta(k)}- \dot{\psi}(k)=\dot{\beta}(k), \tag{7}\]  
\[\frac{1}{l_{zz}}\big{(}l_{\text{f}}\cos\delta\big{(}F_{y,\text{fl}}(k)+F_{y, \text{fr}}(k)\big{)}-l_{\text{r}}\big{(}F_{y,\text{rl}}(k)+F_{y,\text{rr}}(k) \big{)}\big{)}=\ddot{\psi}(k). \tag{8}\]  
These two equations are solved for \(\hat{\beta}\) and \(\ddot{\psi}\), respectively. \(F_{y,\text{fl}}\), \(F_{y,\text{fr}}\), \(F_{y,\text{rl}}\) and \(F_{y,\text{rr}}\) represent the lateral forces at the tire front left, front right, rear left and rear right, respectively. The moment of inertia about the vertical axis is denoted as \(l_{zz}\). By applying the scheme of the explicit Euler integration method [26], the time derivative of the side-slip angle \(\dot{\beta}(k)\) and the yaw acceleration \(\ddot{\psi}(k)\) are then used to predict the side-slip angle \(\dot{\beta}(k+1)\) and the yaw rate \(\dot{\psi}(k+1)\), respectively:  
\[\beta(k+1) =\beta(k)+\dot{\beta}(k)l_{\text{S}}, \tag{9}\] \[\dot{\psi}(k+1) =\dot{\psi}(k)+\ddot{\psi}(k)l_{\text{S}}. \tag{10}\]  
The influence of the chassis elements and, thus, the actuators on the self-steering behavior is exerted indirectly via the lateral tire forces \(F_{y,ij}\). Here, the index \(j\) denotes the vehicle side. The lateral tire forces \(F_{y,ij}\) correspond to the product of the slip angles \(\alpha_{i}\) and the respective cornering stiffnesses \(c_{a,ij}\):  
\[F_{y,ij}(k)=c_{a,ij}(k)\alpha_{i}(k). \tag{11}\]  
The cornering stiffness \(c_{a,ij}\) depends on the current wheel load \(F_{z,ij}\). The dependency features a degressive characteristic. This characteristic is illustrated in Figure 2.  
Figure 2: Degressive Characteristic of the Cornering Stiffness Regarding the Wheel Load. This relationship is modeled using a semi-empirical approach according to [27]. The degressive characteristic is described by a mathematical model, which is parameterized by empirical measurements:  
\[\alpha_{\text{f}}(k) =\delta-\arctan\left(\frac{l_{\text{f}}\dot{\psi}(k)+v\sin\beta(k)}{ v\cos\beta(k)}\right) \tag{5}\]0  
The basis for the parameter identification is the tire model used in IPG CarMaker. This results in the parameters \(c_{1}\) and \(c_{2}\) of the semi-empirical tire model. Furthermore, \(F_{20,ij}\) is the nominal wheel load, which is present under static conditions.