## 3 Obstacle Avoidance Motion Planning Framework
### Compliance-Oriented Potential Field
#### 4.2.1 Lane-Marking Potential Field  
Types of lane markings include solid lines, broken lines, and virtual lane lines. According to [34], solid lines are non-crossable, and broken lines allow vehicles to cross for short periods when appropriate. Virtual lane lines are used to guide vehicles along the optimal path during a turn at an intersection, according to [33]. Optimal turning path information can be provided by high-definition maps [35; 36]. In order to handle the driving constraints of different types of lane markings in an autonomous driving system, a non-crossable potential field is used to model the solid line, and a crossable potential field is used to model the broken line and virtual lane line.  
The non-crossable potential field ensures that the vehicle stays within the lane. The distance from the front tries to the solid line is used as the independent variable of the PF. The distance can be determined as:  
\[d_{L}^{(t+k,t)}=\begin{cases}d_{\text{NC},1}-d_{\text{fl}}^{(t+k,t)},&\text{ left}\\ d_{\text{fr}}^{(t+k,t)}-d_{\text{NC},r},&\text{right}\end{cases} \tag{16}\]  
where \((t+k,t)\) denotes the \(k\)th time step ahead of the current time \(t\) in the prediction horizon; \"left\" and \"right\" indicate the lane markings on the left and right of the vehicle; \(d_{\text{fl}}\) and \(d_{\text{fr}}\) are the lateral positions of the left and right front wheels, respectively; \(d_{\text{NC},1}\) and \(d_{\text{NC},r}\) are the lateral positions of the left and right non-crossable lane markings, respectively. When \(d_{\text{L}}\) decreases to a certain extent, the cost should increase rapidly with the decreasing distance. Therefore, the inverse proportional function and power function are utilized to design the PFs for the non-crossable lane markings. The design of the non-crossable lane-marking potential field is based on the method proposed in [26]. The PF of the non-crossable lane marking is as follows:  
\[U_{\text{NC}_{i}}^{(t+k,t)}=\begin{cases}A_{\text{NC}},&d_{L_{i}}^{(t+k,t)} \leq d_{\text{m,NC}}\\ \frac{a_{\text{NC}}}{(d_{L_{i}}^{(t+k,t)})_{\text{NC}}}-e_{\text{s,NC}},&d_{ \text{m,NC}}<d_{L_{i}}^{(t+k,t)}<d_{\text{in,NC}}\,,\\ 0,&d_{\text{in,NC}}\leq d_{L_{i}}^{(t+k,t)}\end{cases} \tag{17}\]  
where \(d_{\text{m,NC}}\) and \(d_{\text{in,NC}}\) are the minimal distance and the influence distance; \(a_{\text{NC}}\) and \(b_{\text{NC}}\) are the intensity and shape parameters of the PF. The smooth parameters guarantee the continuity of the PF value when \(d_{L_{i}}^{(t+k,t)}\) switches to another condition. The smooth parameter \(e_{\text{s,NC}}\) and the maximum amplitude \(A_{\text{NC}}\) are defined as:  
\[e_{\text{s,NC}}=\frac{a_{\text{NC}}}{D_{\text{in,NC}}^{b_{\text{NC}}}} \tag{18}\]  
\[A_{\text{NC}}=\frac{a_{\text{NC}}}{D_{\text{m,NC}}^{b_{\text{NC}}}}-e_{\text{ s,NC}} \tag{19}\]  
The visualization of the non-crossable lane-marking potential field is shown in Figure 3, where the minimal and influence distance is set to 0.1 m and 1.5 m, respectively. The intensity and shape parameters of the solid line PF are set to 0.2 and 2.0, respectively.  
The crossable lane-marking potential field is used to constrain the vehicle to travel within the lane and, at the same time, is crossable for the vehicle to change lanes. Therefore, the crossable lane-marking potential field is designed based on Gaussian-like functions. The PF of the crossable lane marking is as follows:  
\[U_{C,i}^{(t+k,t)}=A_{C}exp\left(-\frac{\left(d_{L}^{(t+k,t)}-d_{C,i}\right)}{ 2\sigma^{2}}\right) \tag{20}\]  
where \(A_{C}\) is the maximum amplitude of the crossable lane markings PF; \(d_{C}\) is the lateral position of the lane markings; \(\sigma\) is the rate of rise that is related to the lane width. The visualization of the crossable lane-marking potential field is shown in Figure 4, where \(A_{C}\) is set to 2 and \(\sigma\) is set to 0.3 times the lane width.