## 6 Point-of-Conflict Prediction
### Pedestrian Obstacle  
Pedestrian obstacles are usually the most common type of obstacle that could interact with the pedestrian agent. However, the definition of pedestrian obstacle in our study does not include a pedes-trian crossing the environment, as it is considered as a single diagonal movement ob-stacle discussed above. To predict the position of a POC, the agent needs to specify the navigation path that the obstacle might take. While the model for single diagonal movement obstacle could also be adopted in this case, its result would be fairly inaccurate, and more importantly, does not conform to the human predictive system.  
For that reason, we have proposed a unique method of predicting the POC for a pedestrian obstacle. Firstly, to define the predicted navigation path of the obstacle, we utilized our existing reinforcement learning path-planning model. By doing this, the predicted navigation path would have the same advantage as our reinforcement learning model and, therefore, could replicate a realistic navigation path. Subsequently, the POC will be specified on that navigation path, using the velocity of the agent and the obstacle. Figure 6 represents the POC's prediction in the case of a pedestrian obstacle.  
Figure 5: Point-of-conflict of a single diagonal movement obstacle.  
Before the obstacle's navigation path could be constructed, its estimated destination needs to be determined. This could be achieved by projecting the obstacle's orientation to the end of its navigation environment (separate from the agent's environment). The projected destination \(\left(x_{D}^{(obs)},y_{D}^{(obs)}\right)\) could be formulated as follows:  
\[\left(x_{D}^{(obs)},y_{D}^{(obs)}\right)=\left(x_{obs}-\frac{\lambda\ L\ v_{x}}{y_{y}},y_{ obs}-\lambda\ L\right), \tag{15}\]  
where \(\left(x_{obs},y_{obs}\right)\) is the obstacle's position, \(\left(v_{x},v_{y}\right)\) is the orientation vector of the obstacle, and \(L\) is the length of the obstacle's environment. In our proposed model's environment, \(L\) has a length of 22 m.  
The observations of the obstacle consist of the obstacle's position and its projected destination. The observations do not include the observation of an obstacle (i.e., the pedestrian agent in the obstacle's environment) for two reasons. The first reason is that the POC prediction happens before the path-planning process; therefore, the obstacle cannot specify the agent's path. Trying to specify the paths of the agent and the obstacle at the same time would certainly cause conflict. Another reason is related to the process of human thinking in real life. When a pedestrian is predicting the navigation path of the obstacle, he would not consider himself as an obstacle, but rather trying to navigate in a way that could avoid a collision.  
The RL model used in our obstacle's path-planning process is the same one used by the pedestrian agent. The reason is that usually a person often thinks other people would act the same way, for example, navigating the same way as he would do. Alternatively, the obstacle could use the mean RL model from multiple training.  
The predicted position of the POC could be subsequently determined using the scale between the velocities of the agent and the obstacle. The calculation of the POC's y coordinate is formulated as follows:  
\[y_{POC}=y_{obs}-\delta\ \frac{v_{obs}}{v_{agent}+v_{obs}}, \tag{16}\]  
Figure 6: Point-of-conflict of a pedestrian obstacle.  
where \(v_{agent}\) and \(v_{obs}\) are the velocity of the agent and the obstacle, respectively; \(\delta\) is the difference in the y axis between the agent and the obstacle.  
Finally, the location of the predicted POC is specified by the point on the pedestrian's navigation path at the \(y_{POC}\) value in the y axis.