## 5 Simulation Example  
In this subsection, a traffic situation is solved using the graph-based motion-planning algorithm. In Figure 10, the points are presented, where the algorithm is evaluated. During the graph generation the value of \(y_{p,b}\) is set to \(0.1\,\mathrm{m}\).  
First, the motion of the vehicle is predicted, and then the probability values and the weights are computed. The shortest path of the proposed graph can be found using [PERSON]'s algorithm, which is a greedy algorithm. Using the reference vector, the neural network trajectory planning algorithm is evaluated, and finally, the trajectory is tracked by a local controller, where one nonlinear bicycle model is used. The velocity of the ego vehicles are set to \(v_{x}=$20\,\mathrm{m}\mathrm{/}\mathrm{s}$,\mathrm{v}_{\mathrm{other}}=$15\, \mathrm{m}\mathrm{/}\mathrm{s}$\) The distance between them is \(d=$16\,\mathrm{m}$\), and the acceleration value on the first time step is \(a_{other}=$-1\,\mathrm{m}\mathrm{/}\mathrm{s}$^{2}\). The lateral position of the vehicle in the front is \(y_{other}=$0\,\mathrm{m}$\). Finally, the width of the vehicle is set to \(w_{other}=$2.2\,\mathrm{m}$\). In Figure 11 the results for the 1 st point are presented. It can be seen, that the vehicle can track the reference trajectory, which is computed by the neural network.  
It can be said, that the result of the simulations shows, that the algorithm decided to overtake, and the smooth trajectory is planned by the neural network. The local controller of the ego vehicle was able to track the reference trajectory, and the lateral accelerations during the simulation were nearly at the comfort level which is \(1.6\,\mathrm{m}\mathrm{/}\mathrm{s}^{2}\). During the second part of the simulations, only the planned trajectories are presented. Since the goal of this paper is to introduce a motion-planning algorithm,  
Figure 11: Results of the simulation.  
Figure 10: The traffic scenario.  
in these cases, the planned trajectory is not tracked by the controlled vehicle. In Figures 12 and 13 the planned trajectories, and the results of the graph-based algorithm are presented.  
As it can be seen, the algorithm makes the correct decision in different cases, and using the results, the neural network-based algorithm planned the feasible trajectory. In Figure 12 the result of the algorithm can be seen for the second and third cases. The ego vehicle must get to the another lane to avoid the collision with the other vehicle. Finally, in Figure 13, the vehicle gets back to the original lane after overtaking the vehicle, which was in front of the controlled vehicle. In Table 1, the differences between the planned trajectory, and the reference vector are showed.  
Table 1 shows the maximum value of the differences between the reference vector, provided by the graph-based algorithm, and the computed feasible trajectory. The maximum value of the error is 0.435 m. It can be said that the maximum values of the errors are smaller than the maximum value during the generation of the trajectories (see: Appendix A.3). It can be said, that the graph-based algorithm can compute the collision-free trajectory. Using a machine learning-based approach, the feasible trajectory is planned using the results of the graph-based algorithm.