# Feasible Trajectories Generation for Autonomous Driving Vehicles
## 6 Symmetric Polynomial Trajectory Generation
Other parameters regarding the vehicle angular velocity, \(v_{1}(t)\), the vehicle body angle, \(\theta(t)\), the vehicle body angular velocity, \(\dot{\theta}(t)\), the vehicle steering angle, \(\phi(t)\), and the vehicle steering angular velocity, \(\dot{\phi}(t)\) are calculated using the following Equations:  
\[y(t)=-(\frac{t}{T}-1)^{3}y_{0}+\bigg{(}\frac{t}{T}\bigg{)}^{3}x_{T}+a_{y} \bigg{(}\frac{t}{T}\bigg{)}^{2}(\frac{t}{T}-1)+b_{y}\frac{t}{T}(\frac{t}{T}-1) ^{2} \tag{43}\]2  
and  
\[y(t)=-(\frac{t}{T}-1)^{3}y_{0}+\bigg{(}\frac{t}{T}\bigg{)}^{3}x_{T}+a_{y} \bigg{(}\frac{t}{T}\bigg{)}^{2}(\frac{t}{T}-1)+b_{y}\frac{t}{T}(\frac{t}{T}-1) ^{2} \tag{43}\]3  
and  
\[y(t)=-(\frac{t}{T}-1)^{3}y_{0}+\bigg{(}\frac{t}{T}\bigg{)}^{3}x_{T}+a_{y} \bigg{(}\frac{t}{T}\bigg{)}^{2}(\frac{t}{T}-1)+b_{y}\frac{t}{T}(\frac{t}{T}-1) ^{2} \tag{43}\]4  
and  
\[y(t)=-(\frac{t}{T}-1)^{3}y_{0}+\bigg{(}\frac{t}{T}\bigg{)}^{3}x_{T}+a_{y} \bigg{(}\frac{t}{T}\bigg{)}^{2}(\frac{t}{T}-1)+b_{y}\frac{t}{T}(\frac{t}{T}-1) ^{2} \tag{43}\]5and  
\[y(t)=-(\frac{t}{T}-1)^{3}y_{0}+\bigg{(}\frac{t}{T}\bigg{)}^{3}x_{T}+a_{y} \bigg{(}\frac{t}{T}\bigg{)}^{2}(\frac{t}{T}-1)+b_{y}\frac{t}{T}(\frac{t}{T}-1) ^{2} \tag{43}\]6  
The results of this method were observed to be better than previous methods: The vehicle trajectory is smoother. The vehicle speed also increases smoothly from the starting point and reaches the maximum speed in the middle of the trajectory. Then, the speed is reduced as the vehicle moves to the destination point, as is shown in \[y(t)=-(\frac{t}{T}-1)^{3}y_{0}+\bigg{(}\frac{t}{T}\bigg{)}^{3}x_{T}+a_{y} \bigg{(}\frac{t}{T}\bigg{)}^{2}(\frac{t}{T}-1)+b_{y}\frac{t}{T}(\frac{t}{T}-1) ^{2} \tag{43}\]7  
This method appears to be more realistic and controls the vehicle tracking in its trajectory more easily. The body and steering angle and the angular velocity are shown in \[y(t)=-(\frac{t}{T}-1)^{3}y_{0}+\bigg{(}\frac{t}{T}\bigg{)}^{3}x_{T}+a_{y} \bigg{(}\frac{t}{T}\bigg{)}^{2}(\frac{t}{T}-1)+b_{y}\frac{t}{T}(\frac{t}{T}-1) ^{2} \tag{43}\]8  
\[y(t)=-(\frac{t}{T}-1)^{3}y_{0}+\bigg{(}\frac{t}{T}\bigg{)}^{3}x_{T}+a_{y} \bigg{(}\frac{t}{T}\bigg{)}^{2}(\frac{t}{T}-1)+b_{y}\frac{t}{T}(\frac{t}{T}-1) ^{2} \tag{43}\]9 Body and steer angle. \[\dot{x}(t)=-3(\frac{t}{T}-1)^{2}x_{0}+3\bigg{(}\frac{t}{T}\bigg{)}^{2}x_{T}+a_ {x}2\frac{t}{T}(\frac{t}{T}-1)+a_{x}\bigg{(}\frac{t}{T}\bigg{)}^{2}+b_{x}( \frac{t}{T}-1)^{2}+b_{x}2\frac{t}{T}(\frac{t}{T}-1) \tag{44}\]0 Vehicle trajectory and velocity. Figure 12 shows the feasible trajectory of the symmetric polynomial method is the smoothest, and the velocity of this method is also symmetrical, increasing from 0 and reducing to 0. Therefore, this method is useful in the application of auto-parking systems. Figure 13 shows the comparison of the body and steering angles and angular velocity. This symmetric method also provides the smoothest curve of the body and steering angles and velocity. A vehicle's feasible trajectories are the continuous segments connected to each other along the vehicle's feasible path. All trajectories are required to join each other smoothly and continuously. The control of vehicle tracking along its feasible trajectories is performed by the control of the vehicle velocity and the steering angle. The following are the simulations of a vehicle's auto-parking performance with the symmetric polynomial trajectory generated from a starting point of \(x(0),y(0)\) to a destination point of \(x(-6),y(3)\). Figure 14 shows the vehicle moving in reverse and the steering angle \(\phi\) moving from positive to negative and returning to 0 position. The steering angle, \(-45^{\circ}\leq\phi\leq 45^{\circ}\), satisfied the constraint as per the vehicle dynamic moving in reverse in Equation (2).