# Optimal Vehicle Lane Change Trajectory Planning in Multi-Vehicle Traffic Environments
## 2 Methods
### Vehicle Lateral Dynamic Stability Constraint  
The 2-DOF vehicle model was built to determine the boundary value of vehicle lateral stability, from which some trajectory clusters that exceed the lateral stability constraints can be eliminated, the 2-DOF vehicle model is shown in Figure 3. The remaining trajectory clusters are lane changing trajectory clusters with lateral stability. According to automobile theory, the relationship between the vehicle yaw rate and the front and rear axle slip angles can be determined using the following formula [24]:  
\[\dot{\beta}=\frac{F_{yr}+F_{yf}\cos\delta_{f}}{m\upsilon_{x}}-\gamma\approx \frac{F_{yr}+F_{yf}}{mu}-\gamma \tag{11}\]  
where \(\dot{\beta}\) is the side-slip angular velocity of the center of mass, \(F_{yf}\) and \(F_{yr}\) are the lateral force of the front axle and the lateral force of the rear axle, respectively, \(\delta_{f}\) is the front wheel angle of the vehicle (the left and right wheel angles are approximately equal), \(m\) is the vehicle mass, and \(\gamma\) is the yaw rate. To keep the vehicle lateral dynamics in a stable state, \(\dot{\beta}\) approaches 0 [25], and the yaw rate limit can be deduced using Formula (12). \[|\gamma_{s}|=\frac{F_{yr}+F_{yf}}{mu}\leq\frac{mg\mu}{mu}=\frac{g\mu}{u} \tag{12}\]  
The brush tire model can be used to calculate the tire lateral force, and the tire slip angle can be approximated according to the vertical force. Then, the brush tire model is referenced to constrain the side slip angle of the vehicle's center of mass, as Formula (13):  
\[F_{yf,r}=\left\{ \begin{array}{l}-C_{xf,r}\tan\alpha_{f,r}+\frac{C_{xf,r}^{2} |\tan\alpha_{f,r}|}{3\mu F_{z}}\tan\alpha_{f,r}-\frac{C_{xf,r}^{3}\tan\alpha_ {f,r}^{3}}{27\mu^{2}F_{z}^{2}},\alpha_{f,r}<\arctan(\frac{3\mu F_{zf,r}}{C_{xf,r}})\\ -\mu F_{z}\text{sgn}\alpha_{f,r},\alpha_{f,r}\geq\arctan(\frac{3\mu F_{zf,r}} {C_{xf,r}})\end{array} \right. \tag{13}\]  
where \(F_{y}\) is tire lateral force, \(\alpha_{f,r}\) is the side slip angle between the rear axle and the front axle, \(F_{zf,r}\) is the vertical force of the rear or front axle of the vehicle, and \(C_{af,r}\) is the cornering  
Figure 3: 2-DOF vehicle dynamic model. stiffness of the rear and front axles. Based on the assumption of a small angle, the range of stability is:  
\[\alpha_{r}\approx\beta-\frac{b\gamma_{s}}{u}\leq\tan^{-1}\left(\frac{3 mg\mu}{C_{ ar}}\cdot\frac{a}{a+b}\right) \tag{14}\]  
where \(a\) and \(b\) are the distances from the front and rear axles to the center of mass of the vehicle, derived from Formulas (13) and (14). The limit of the side slip angle of the center of mass can be obtained as:  
\[|\beta_{s}|\leq\tan^{-1}\left(\frac{3 mg\mu}{C_{ar}}\cdot\frac{a}{l}\right)+ \frac{b\gamma_{s}}{u} \tag{15}\]  
In summary, the stability limits of the vehicle yaw rate and the of sideslip angle of centroid can be determined using Formulas (12) and (14) respectively. From the above vehicle dynamics stability limit, in order to accurately quantify the lateral dynamic stability limit, it is necessary to know the vehicle model and other relevant parameters (such as the parameters of the steering and tire systems. The D-class vehicle in CarSim is used as the research object; it is equipped with a 150 kw engine, a six-speed automatic transmission, an independent suspension, and an ABS (Anti-lock Braking System) system. The vehicle changes lanes at a speed of 72 km/h. Different expected trajectories are calculated by changing the end time of different lane changes. At the same time, the road adhesion coefficient and vehicle speed are constantly changed to complete the lane change. The time of each lane change can be measured. Through data fitting, the following three-dimensional diagram can be obtained:  
It can be found from Figure 4 that within the vehicle dynamic stability limit, \(t_{f}\) becomes larger, the vehicle speed increases, and the adhesion coefficient decreases; when the road adhesion coefficient is larger, x decreases significantly; this is also in line with the driver's daily driving habits.