## 3 Lane Change Trajectory Planning Subheadings  
When the smart car makes a lane-changing decision, it needs to complete the lane-changing trajectory planning according to the current conditions. Because of the simplicity and mall amount of calculation of polynomial curve fitting, the continuous third-order differentiability of the fifth-order polynomial, and the smoothness of curvature, we make the longitudinal and lateral decoupling of the lane-changing according to the lane-changing characteristics and select a quartic polynomial and a quintic polynomial in the x-direction and y-direction, respectively, for planning.  
The lane-changing trajectory function is shown in Equation (4).  
\[\left\{\begin{array}{l}x(t)=a_{0}+a_{1}t+a_{2}t^{2}+a_{3}t^{3}+a_{4}t^{4}\\ y(t)=b_{0}+b_{1}t+b_{2}t^{2}+b_{3}t^{3}+b_{4}t^{4}+b_{5}t^{5}\end{array}\right. \tag{4}\]  
where \(a_{0}\sim a_{4}\) and \(b_{0}\sim b_{5}\) are the required coefficients for the polynomials \(x(t)\) and \(y(t)\). The solution of the polynomial needs to rely on the initial and end conditions of the vehicle lane change. Let the initial time of vehicle lane change be \(t_{0}\), and the end time of lane change be \(t_{f}\). The initial state can be expressed by \(S_{in}=\left[x_{0},\dot{x}_{0},\ddot{x}_{0},y_{0},\dot{y}_{0},\ddot{y}_{0} \right]\), where the six components represent the longitudinal and lateral displacement, velocity, and acceleration of the car at the initial moment, respectively. Here, the speed at the beginning of the lane change is \(v_{0}\), the speed at the end of the lane change is \(v_{f}\), and the longitudinal position \(w\) is one lane width with a value of 3.75 m. The boundary states of the car are shown in Equations (5) and (6).  
\[\left\{\begin{array}{l}x(t_{0})=0,\dot{x}(t_{0})=v_{0},\ddot{x}(t_{0})=0\\ y(t_{0})=0,\ddot{y}(t_{0})=0,\ddot{y}(t_{0})=0\end{array}\right. \tag{5}\]  
\[\left\{\begin{array}{l}\dot{x}\Big{(}t_{f}\Big{)}=v_{f},\ddot{x}\Big{(}t_{f} \Big{)}=0\\ y\Big{(}t_{f}\Big{)}=w,\ddot{y}\Big{(}t_{f}\Big{)}=0,\ddot{y}\Big{(}t_{f} \Big{)}=0\end{array}\right. \tag{6}\]  
We define the time parameter matrix as follows; see Equations (7) and (8).  
\[T_{5\times 5}=\left[\begin{array}{cccc}1&t_{0}&{t_{0}}^{2}&{t_{0}}^{3}&{t_ {0}}^{4}\\ 0&1&2t_{0}&3{t_{0}}^{2}&4{t_{0}}^{3}\\ 0&0&0&6t_{0}&12{t_{0}}^{3}\\ 0&1&2t_{f}&3{t_{f}}^{2}&4{t_{f}}^{3}\\ 0&0&2&6t_{f}&12{t_{f}}^{2}\end{array}\right] \tag{7}\]\[T_{6\times 6}=\begin{bmatrix}1&t_{0}&t_{0}^{2}&t_{0}^{3}&t_{0}^{4}&t_{0}^{5}\\ 0&1&2t_{0}&3t_{0}^{2}&4t_{0}^{3}&5t_{0}^{4}\\ 0&0&2&6t_{0}&12t_{0}^{2}&30t_{0}^{3}\\ 1&t_{f}&t_{f}^{2}&t_{f}^{3}&t_{f}^{4}&t_{f}^{5}\\ 0&1&2t_{f}&3t_{f}^{2}&4t_{f}^{3}&5t_{f}^{4}\\ 0&0&2&6t_{f}&12t_{f}^{2}&20t_{f}^{3}\end{bmatrix} \tag{8}\]  
We define the following Equations (9) and (10) as the coefficient matrix and the boundary matrix, respectively.  
\[\left\{\begin{array}{l}\overset{\rightarrow}{a}=[a_{4},a_{3},a_{2},a_{1},a _{0}]\\ \overset{\rightarrow}{b}=[b_{5},b_{4},b_{3},b_{2},b_{1},b_{0}]\end{array}\right. \tag{9}\]  
\[\left\{\begin{array}{l}q_{x}=[0,v_{0},0,v_{d},0]\\ q_{y}=[0,0,0,w,0,0]\end{array}\right. \tag{10}\]  
Then, the system of equations can be expressed as below; see Equation (11).  
\[\left\{\begin{array}{l}\overset{\rightarrow}{d}=T_{5\times 5}^{-1}q_{x}\\ \overset{\rightarrow}{b}=T_{6\times 6}^{-1}q_{y}\end{array}\right. \tag{11}\]  
We can obtain the general form of Equation (11) by using the lane-changing boundary conditions, obtain the optimal lane-changing time in different lane-changing scenarios, and then determine the final lane-changing trajectory.