## 5 Results and Discussion  
The results of the point cloud mapping before and after being discretized to 2D grid cells are shown in Figures 10 and 11, respectively, with the ground and ceiling's information being filtered out. During the process, the pixel size information and the minimum and maximum value of \(x\) and \(y\) of the 3D point cloud map is recorded. This will be used during the conversion from the transformed orientation back into the original orientation. Figure 12 illustrates the colorbar that is used as the indication for the potential value at each cell. The blue region shows the area with low potential value and the red region signifies as the high-risk area. Figure 11: Point cloud map centered to 2D grid cells. Figure 12: Potential value’s colorbar for indication. Figure 13 shows the original map without any discretization and transformation applied. For this scenario, the starting position is not blocked by any walls or obstacle, thus the planning is straightforward. Figure 14 shows the altered potential map of Figure 13, discretized to 0.25 m \(\times\) 0.25 m. The attractive force creates a low potential while the obstacles are encapsulated by high potential charges. The start position is located at (228, 143), and the goal location is at the end of the corridor at (102, 44). Similarly, the planning is also straightforward in this case. The path of before and after smoothing is shown in Figure 15. It can be observed that, before smoothing, the path exhibits a zig-zag pattern. Now, the starting position is shifted to position (36.3, 23.2) in the original map. In this case, the planner fails to find a path, as shown in Figure 16. This is due to the planner suffering from an  
Figure 14: A feasible path is found from (228, 143) to (102, 44). Figure 13: Non-discretized potential map. Figure 15: Path before and after filtering for Figure 14.  
infinite loop between two walls. A point to note is that the planner is heading straight to the wall. This is the nature of the APF algorithm; as it searches for the feasible path, it will stay as close as possible to the boundaries of the potentials near the wall. The reason is due the algorithm attempt to minimize the distance between the vehicle and the obstacle. When discretized to 0.25 m \(\times\) 0.25 m grid, as shown in Figure 17, the starting position is now at (242, 135); however, the planner still fails to find a path. This is because the planner also suffers from an infinite loop, as it is stuck between the wall and the pillar at the corner of the area. As the 2D map is a downscaled version of the 3D point cloud map, the planner has added a tolerance value of plus-minus one-pixel value to the starting position--for example, (242, 136), (243, 135) and so on. However, even with the tolerance, the planner still fails, as shown in Figure 17.  
Figure 18 shows the transformed Figure 17. The transformation matrix is obtained by the brute force method, which is rotating the map by 1-degree increment until the path is found. The matrix \(\mathbf{T}\), based on Equation (4), is used to recalculate the transformed starting and goal position. The value depends on the increment of rotation. However, the result did not yield a round value. Thus, the number will first be rounded down. For the first trial, the starting value is at (196, 133) and the goal is at (40, 158). As shown in the same figure, the planner is still stuck in the similar area to that of the previous scenario before the map is rotated. Figure 12:0 Non-discretized potential map—failed scenario. Figure 12:1 Failed scenario at (242, 135), before the map is rotated. The next trial involves rounding up the transformed value. The \(x\)-coordinate starting value is rounded up to 197. Thus, the next starting position is at (197, 133). The goal remains unchanged. The result is illustrated in Figure 12:2  
It can be observed that a feasible path is found. The reason is that the 2D grid cell map is the scaled-down integer representation of the original 3D point cloud map.