## Appendix A Path Representation  
This section and the path following control problem considered in this work are based on [12; 23]. The vehicle should robustly follow a path which is characterized by a motion demand \(\mathbf{\lambda}(s)\) and parameterized by the arc length \(s\) and is defined by  
\[\mathbf{\lambda}(s)=\left(x_{\text{P}}^{\text{I}}(s),y_{\text{P}}^{\text{I}}(s), \psi_{\text{P}}(s),\kappa_{\text{P}}(s),v_{\text{P},x}^{\text{P}}(s)\right) \tag{1}\]  
The superscripts I and P denote that the individual values are considered in the inertial and path reference frame, respectively. Furthermore, the subscript P expresses that the respective value describes a property of the path. Figure 11 shows a graphical depiction of \(\mathbf{\lambda}(s)\) and the values introduced in Equation (1) at the point \(s_{i}\). Here, \(p_{\text{P}}^{\text{I}}(s_{i})=\left[x_{\text{P}}^{\text{I}}(s_{i}),\ y_{ \text{P}}^{\text{I}}(s_{i})\right]\) denotes the path reference point in the inertial coordinate system and \(\psi_{\text{P}}(s_{i})\) represents the path orientation. Furthermore, \(\kappa_{\text{P}}(s_{i})\) expresses the path curvature and \(v_{\text{P}}^{\text{P}}(s_{i})\) depicts the longitudinal velocity tangential to the path in the direction of the tangent vector \(t^{\text{P}}\) at the path reference point \(p_{\text{P}}^{\text{I}}(s_{i})\). Given the current vehicle position \(\mathbf{p}_{\text{C}}^{\text{I}}=\left[x_{\text{C}}^{\text{I}},y_{\text{C}}^{\text{I}}\right]\), the closest point on the reference path \(\mathbf{p}_{\text{P}}^{\text{I}}(s^{*})\) is chosen as reference point on the path [23]. The subscript C represents that properties of the car are regarded. This reference point then provides the current motion demand \(\lambda(s^{*})\) which the vehicle should follow. The reference point is calculated by finding the arc length \(s^{*}\) which minimizes the distance \(e(s)\) between the vehicle position \(\mathbf{p}_{\text{C}}^{\text{I}}\) and the reference path \(\mathbf{p}_{\text{P}}^{\text{I}}(s)\), i.e., \(e(s)=\mathbf{p}_{\text{P}}^{\text{I}}(s)-\mathbf{p}_{\text{C}}^{\text{I}}\). This optimization problem is denoted by  
\[s^{*}=\arg\min_{s}||e(s)||_{2}=\arg\min_{s}||\mathbf{p}_{\text{P}}^{\text{I}}(s)- \mathbf{p}_{\text{C}}^{\text{I}}||_{2}. \tag{10}\]  
Figure 1 illustrates this optimization problem. It can be seen that the optimal solution is the arc length \(s^{*}\) for which the longitudinal position error of the vehicle is zero after transforming it into the path frame centered at \(\mathbf{p}_{\text{P}}^{\text{I}}(s^{*})\), i.e., \(e_{x}^{\text{P}}(s^{*})=0\). In order to obtain \(s^{*}\) through the minimization problem in Equation (10), the time independent path interpolation (TIPI) [12; 23] is applied. To allow for a successful path following control according to [14], the controller needs to minimize certain errors between the vehicle and the path, more specifically the lateral offset \(e_{y}^{\text{P}}\), the velocity error \(e_{y_{x}}^{\text{P}}\) in longitudinal direction of the path and the orientation error \(e_{\varphi}\). The lateral offset of the vehicle with respect to the path is denoted by  
\[e_{y}^{\text{P}}=y_{\text{P}}^{\text{P}}-y_{\text{C}}^{\text{P}}, \tag{11}\]  
with \(y_{\text{P}}^{\text{P}}\) being the desired lateral position and \(y_{\text{C}}^{\text{P}}\) being the lateral position of the car, both referenced in the path frame. Since no lateral offset of the vehicle is desired, the desired lateral position is set to zero in Equation (11), i.e., \(y_{\text{P}}^{\text{P}}=0\). Furthermore, \(e_{v_{x}}^{\rm P}\) denotes the velocity error between the desired velocity \(v_{{\rm P},x}^{\rm P}(s)\) tangential to the path and the longitudinal velocity \(v_{{\rm C},x}^{\rm P}\) of the car in the path frame.