# A Graph-Based Method for Tactical Planning of Lane-Level Driving Tasks in the Outlook Region
## 5 Modeling of Lane-Following and Lane-Change Maneuvers
A statistically based lane-change duration model proposed by [PERSON] [9] is adopted here to estimate the time cost of lane changes for each lane cell and the cell where the vehicle will be located after the lane change. As shown in Figure 5, the entire lane-change process can be divided into two phases: the first phase involves turning on the turn signal to indicate intent and the second phase involves performing the turning action to enter the adjacent lane. Assuming the starting cell is \(n_{p}\) and the target unit is \(n^{\prime}_{p+q^{\prime}}\) the cost of the lateral edge representing the lane-change maneuver can be represented as follows:  
\[C_{p,\;q}=t_{light}+t_{lcd}, \tag{5}\]  
where \(t_{light}\) denotes the time for turning on the turn signal to indicate intent, which can be regarded as a constant; \(t_{lcd}\) denotes the time for performing the lane-change action, that is, the lane-change duration in [PERSON]'s model. [PERSON]'s lane-change duration model is an empirical model obtained from a large amount of natural driving data statistics and has the following form:  
\[\ln(t_{lcd})=\mathbf{\beta}\cdot\mathbf{X}, \tag{6}\]  
where \(t_{lcd}\) denotes the lane change duration, \(\mathbf{X}\) denotes the vector of explanatory variables, and \(\mathbf{\beta}\) denotes corresponding parameters. Equation (6) can be expanded as follows:  
\[\ln(t_{lcd})=\beta_{0}+\beta_{1}\cdot\overline{p}+\beta_{2}\cdot D +\beta_{3}\cdot min\Big{[}0,\;\Delta V_{front}\Big{]}+\beta_{4}\cdot d_{ front}+\beta_{5}\cdot min\Big{[}0,\;\Delta V_{lag,\;lead}\Big{]} \tag{7}\] \[+\beta_{6}\cdot max\Big{[}0,\;\Delta V_{lag,\;lead}\Big{]}+\beta _{7}\cdot d_{lag,\;lead},\]  
where \(\overline{p}\) denotes the average traffic density, \(D\) denotes the change direction, \(\Delta V_{front}\) denotes the relative speed between the ego vehicle and its preceding vehicle, \(\Delta V_{lag,\;lead}\) denotes  
Figure 5: The modeling of lane-change maneuver. the relative speed between the lead vehicle and the lag vehicle in the adjacent lane, \(d_{front}\) denotes the distance between the ego vehicle and its preceding vehicle, and \(d_{lag,~{}lead}\) denotes the gap of the lead vehicle and the lag vehicle in the adjacent lane. In the traffic flow-based environment model build in this paper, we replaced some variables in Equation (8),  
\[\left\{ \begin{array}{l}\Delta V_{front}~{}\rightarrow~{}\Delta v_{self}\\ \Delta V_{lag,~{}lead}~{}\rightarrow\Delta v_{side}\\ d_{front}~{}\rightarrow~{}1/\rho_{self}\\ d_{lag,~{}lead}~{}\rightarrow~{}1/\rho_{side}\\ \end{array} \right., \tag{8}\]  
where \(\rho_{self}\) denotes the traffic density at current cell, \(\rho_{side}\) denotes the traffic density at adjacent cell, \(\Delta v_{self}\) denotes the traffic flow speed difference between the current lane cell and its preceding cell, and \(\Delta v_{side}\) denotes the traffic flow speed difference between the adjacent cell and its preceding cell. Combining the estimated speed \(v_{est}\) with the duration of the lane-change process at each cell, the cell \(n^{\prime}_{p+q}\) where the ego vehicle located when lane-changing ends can be obtained. This process can be represented with the following equations:  
\[v^{\prime}_{est}=\left\{ \begin{array}{l}v_{est}|n_{p+k},~{}t_{k}\leq t_{ light}+0.5\cdot t_{lcd}\\ \eta_{est}|n^{\prime}_{p+k},~{}t_{k}>t_{light}+0.5\cdot t_{lcd}\\ \Delta t_{i}=\Delta l/v^{\prime}_{est},\\ t_{k}=\sum_{k}\Delta t_{i}\\ \end{array} \right. \tag{9}\]  
where \(v^{\prime}_{est}\) denotes the expected speed of the lane-change progress and \(v_{est}\) is the estimated speed obtained by Equation (4). \(n_{p+k}\) denotes any cell in the current lane, while \(n^{\prime}_{p+k}\) denotes any cell in the adjacent lane, \(\Delta t_{i}\) represents the time for passing through the cell, \(t_{k}\) denotes the time consumed during lane change, and if \(t_{k}\geq C_{p,~{}q}\), \(n^{\prime}_{p+k}=n^{\prime}_{p+q}\).