## 3 Obstacle Avoidance Motion Planning Framework
### Compliance-Oriented Potential Field
#### 4.2.2 Stop-Line Potential Field  
Types of stop lines include marked stop lines and virtual stop lines. The virtual stop line is adopted as the basis for the right-of-way yielding line and the safety following line. When the trajectory of the ego vehicle intersects with that of the vehicle possessing the right-of-way, a virtual yield line is created perpendicular to the ego vehicle's direction at the point of intersection. Subsequently, a potential field for the yield stop line is established based on this virtual line.  
The distance between the foremost part of a vehicle and the stop line is used as the independent variable of the stop line PF, which is determined as:  
\[s_{S}^{(t+k,t)}=s_{f}^{(t+k,t)}-s_{s} \tag{21}\]  
where \(s_{f}\) and \(s_{s}\) are the longitudinal position of the foremost part of the vehicle and the longitudinal position of the stop line, respectively. The stop-line potential field is also  
Figure 4: Crossable lane-marking potential field.  
Figure 3: Non-crossable lane-marking potential field.  
regarded as non-crossable. The inverse proportional function and power function are utilized. The stop line PF is designed as:  
\[U_{\text{SL}}^{(t+k,t)}=\begin{cases}A_{\text{SL}},&s_{S}^{(t+k,t)}\leq s_{ \text{m,SL}}\\ \frac{a_{\text{SL}}}{(s_{S}^{(t+k,t)})_{\text{SL}}}-e_{\text{s,SL}},&s_{\text{ m,SL}}<s_{S}^{(t+k,t)}<s_{\text{in,SL}}\\ 0,&s_{\text{in,SL}}\leq s_{S}^{(t+k,t)}\end{cases} \tag{22}\]  
where \(a_{\text{SL}}\) and \(b_{\text{SL}}\) are the intensity and shape parameters of the PF; \(s_{\text{m,SL}}\) is the minimal distance and is set to 0.1 m; the influence distance \(s_{\text{in,SL}}\) is defined according to the length of the guide lane. The smooth parameter \(e_{\text{s,NC}}\) and the maximum amplitude \(A_{\text{NC}}\) are defined as:  
\[e_{\text{s,SL}}=\frac{a_{\text{SL}}}{s_{\text{in,SL}}^{\text{n,SL}}} \tag{23}\]  
\[A_{\text{SL}}=\frac{a_{\text{SL}}}{s_{\text{m,SL}}^{\text{n,SL}}}-e_{\text{s, SL}} \tag{24}\]  
The visualization of the stop-line potential field is shown in Figure 5, where the intensity and shape parameters of the stop line PF are set to 0.2 and 2.0, respectively.  
When driving in a queue sequence, it is necessary to maintain a safe following distance with the vehicle in front. The independent variable of the car-following PF is defined as:  
\[s_{F}^{(t+k,t)}=s_{f}^{(t+k,t)}-s_{\text{m,CF}} \tag{25}\]  
where \(s_{f}\) is the longitudinal position of the foremost part of the ego vehicle; \(s_{\text{m,CF}}\) is the longitudinal position of the safe following line.  
The foundation of car-following PF is the Yukawa potential, with a slower rate of rise at a moderate distance. This property allows for a smoother transition between different driving regimes, such as free flow, synchronized flow, and congested flow. The car-following PF is defined as:  
\[U_{\text{CF}}^{(t+k,t)}=A_{\text{CF}}\frac{e^{-4\text{CF}_{\text{F}}^{(t+k,t) }}}{s_{F}^{(t+k,t)}} \tag{26}\]  
where \(A_{\text{CF}}\) and \(a_{\text{CF}}\) are the maximum amplitude and the scale factor of the car-following PF. The obstacle PF employed in this study is based on [37]. The visualization of the car and car-following potential field is shown in Figure 6, where \(A_{\text{CF}}\) and \(a_{\text{CF}}\) are set to 2.0 and 0.5, respectively.  
Figure 5: Stop-line potential field.