## 3 Performance Measures for Path Tracking Control  
In the domain of PTC, \(e_{y}\) and \(e_{\varphi}\) have been widely used as indicators for assessing path tracking performance. However, different from conventional PTC literature, this paper adopts a double lane change maneuver as the target path for collision avoidance [13; 16; 17; 20; 21; 31; 32]. This maneuver was employed to evaluate the reachability and agility of a path tracking controller [31; 32]. Figure 5 shows both the target path and the trajectory of a vehicle. In Figure 5, the points \(A\), \(B\) and \(C\) are on the target path, which represent the peak, zero-crossing and convergence points, respectively. The points \(D\), \(E\), \(F\) and \(G\) are on the vehicle trajectory, which represent the peak, zero-crossing, over-shoot and convergence points, respectively. Five metrics, \(M_{X}\), \(M_{Y}\), \(M_{OS}\), \(M_{DX}\) and \(M_{SX}\), representing path tracking performance, are defined as Equation (37), based on the points along the target path and vehicle trajectory in Figure 5. In Equation (37), the subscripts X and Y correspond to the x- and y-positions of the corresponding point, respectively.  
\[\left\{\begin{array}{l}M_{X}=D_{X}-A_{X}=D_{X}-73.20\\ M_{Y}=D_{Y}-A_{Y}=D_{Y}-3.53\\ M_{DX}=E_{X}-B_{X}=E_{X}-91.50\\ M_{SX}=G_{X}-C_{X}=G_{X}-190.00\\ M_{OS}=\frac{|F_{y}|-1.65}{1.65+3.55}\times 100\end{array}\right. \tag{37}\]  
\(M_{X}\) and \(M_{Y}\) represent the peak's center and lateral offsets, respectively. In this study, agility and reachability are represented by \(M_{X}\) and \(M_{Y}\), respectively. If \(M_{Y}\) exceeds \(-0.02\,\text{m}\), the controller's path tracking performance can be regarded as satisfactory. \(M_{OS}\) represents the percentage of overshoot, which is identical to that of automatic control theory. This indicates lateral damping, reflecting agility. In this paper, a satisfactory performance is achieved if \(M_{OS}\) remains below 16%, which is equivalent to an overshoot of 0.85 m. \(M_{DX}\) and \(M_{SX}\) represent the response delay and the settling delay, respectively. \(M_{DX}\) corresponds to a delay, standing for the agility of longitudinal motion. \(M_{DX}\) highly depends on \(\mu\). \(M_{SX}\) can be interpreted as the settling time, indicating the convergence speed of the vehicle's lateral motion toward a target y-position. \(M_{SX}\) is closely related to \(M_{OS}\). In essence, smaller absolute values of these metrics correspond to improved path tracking performance.  
As mentioned previously, VSC and LSC aim for yaw rate tracking and lateral stability, respectively. Typically, yaw rate tracking performance is assessed by the difference between real and reference yaw rates, while lateral stability performance is evaluated by the  
Figure 5: Measures used to tune PTC.  
magnitude of \(\beta\). It has been considered satisfactory if \(\beta\) is less than 3 degrees [51]. Let the maximum absolute value of \(\beta\) be referred to as MASSA. The side-slip angular rate, when combined with \(\beta\), has been also employed as a measure of lateral stability [30]. Let this denote the maximum absolute value of side-slip angular rate as MASSAR.  
In this paper, the tuning parameters are the maximum allowable value \(\xi_{i}\) given in Equation (14) and the velocity gain \(k_{v}\) given in Equation (5). There are three tuning objectives when tuning these parameters in this paper. The first objective is collision avoidance, which is achieved by tuning parameters such that \(M_{Y}\) is larger than \(-0.02\) m. Generally, the best \(M_{SX}\) is attained when \(M_{OS}\) is around 0.9, provided that \(M_{Y}\) exceeds \(-0.02\) m. The second objective is path tracking, which is achieved by tuning parameters such that \(M_{X}\), \(M_{DX}\) and \(M_{SX}\) are maintained as small as possible. Among them, the minimum of \(M_{DX}\) clearly depends on \(\mu\). The third objective is lateral stability, which is achieved by tuning parameters such that MASSA is less than \(2^{\circ}\).