# Optimal Vehicle Lane Change Trajectory Planning in Multi-Vehicle Traffic Environments
## 2 Methods
### TOPSIS Algorithm for Solving Multiple Indexes  
Clusters of safe trajectories are multi-objectively optimised to solve for optimal lane change trajectories. Autonomous vehicles change lanes not only for dynamic stability and safety, but also for comfort and lane change efficiency. Four evaluation indicators are proposed for this question, including Index of trajectory deviation, Index of rollover hazard, Index of Side Sliping and Index of Efficiency [28,29].  
1. Index of trajectory deviation  
Figure 8: Schematic diagram of the TOPSIS algorithm.  
Figure 7: (a) Collision analysis scenarios; (b) Collision-free analysis scenarios.  
\[J_{T}=\sqrt{\left[\left(\int_{0}^{\tau}\left[\frac{f(t)-y(t)}{\overset{\wedge}{E}} \right]^{2}dt\right)^{2}+\left(\int_{0}^{\tau}\left[\frac{u\overset{\wedge}{- \beta(t)}}{\overset{\wedge}{\beta}}\right]^{2}dt\right)^{2}\right]}\bigg{/}2 \tag{23}\]  
where \(f(t)\) is the reference trajectory, \(y(t)\) is the actual vehicle trajectory, \(\dot{\beta}(t)\) is the mass-side eccentricity speed, \(\overset{\wedge}{E}\) is the trajectory deviation threshold and \(\overset{\wedge}{\beta}\) is the mass-side eccentricity threshold.  
2. Index of rollover hazard  
\[J_{R}=\sqrt{\left[\left(\int_{0}^{\tau}\left[\frac{a_{y}(t)}{\overset{ \wedge}{a_{y}}}\right]^{2}dt\right)^{2}+\left(\int_{0}^{\tau}\left[\frac{\phi(t )}{\overset{\wedge}{\phi}}\right]^{2}dt\right)^{2}\right]}\bigg{/}2 \tag{24}\]  
where \(\overset{\wedge}{a_{y}}\) is the lateral acceleration threshold, \(a_{y}(t)\) is the vehicle lateral acceleration, \(\phi(t)\) is the vehicle side camber and \(\overset{\wedge}{\phi}\) is the vehicle side camber threshold.  
3. Index of Side Slip  
\[J_{s}=\max\left(\int_{0}^{\tau}\left[\frac{F_{yf,r}(t)/F_{zf,r}(t)}{\overset{ \wedge}{\mu}}\right]^{2}dt\right) \tag{25}\]  
where \(F_{yf,r}(t)\) is the lateral force on the front and rear axles respectively, \(F_{zf,r}(t)\) is the vertical force on the front and rear axles respectively and \(\overset{\wedge}{\mu}\) is the road friction coefficient threshold value.  
4. Index of Efficiency  
\[J_{E}=t_{f} \tag{26}\]  
The less time it takes to change lanes, the higher the efficiency of changing lanes [30]. The attributes of each evaluation index are defined as:  
\[N=\left[1/J_{T}\quad 1/J_{R}\quad 1/J_{S}\quad J_{E}\right] \tag{27}\]  
Benefit indicators include Index of trajectory deviation, Index of rollover hazard and Index of Side Sliping, the cost index is Index of Efficiency. Then the vector \(N\) is expanded into a decision matrix, which is the first step of the TOPSIS algorithm.  
\[M_{1}=\begin{bmatrix}(1/J_{S})_{11}&(1/J_{T})_{12}&(1/J_{R})_{13}&(J_{E})_{14 }\\ \bullet&\bullet&\bullet&\bullet\\ \bullet&\bullet&\bullet&\bullet\\ (1/J_{S})_{q1}&(1/J_{T})_{q2}&(1/J_{R})_{q3}&(J_{E})_{q4}\end{bmatrix} \tag{28}\]  
Among them, the number of rows of the matrix is calculated as Formula (28).  
\[q=\frac{t_{\text{max}}-t_{\text{fmin}}(u)}{\Delta t}+1 \tag{29}\]  
where \(q\) is the number of rows in the decision matrix, \(t_{\text{fmax}}\) represents the maximum lane change time, which is 9 s, \(t_{\text{fmin}}(u)\) is the minimum lane-changing time. Previous studies have shown that the minimum lane change time can be calculated when the longitudinal