## 2 Materials and Methods
### The Voxel Grid and the Approach to Capture Data
The voxel grid also provides wider net data points, as, rather than captured visual data being points x,y,z in a space with no determined size, they fall into a discrete grid of voxels with a defined size. This in turn means that when an agent is scanning for an object it has an increased probability of catching a feature, an example of this is shown in Figure 6

Figure 5: Figure showing how data of a single voxel are an array based on time. with the real data comparison shown in Figure 7. It should be noted, however, that this comes at the cost of the resolution of the visual data. ### Loop Closure and Object Probability over Time

The closed-loop problem that is often present in SLAM is not realised in the same way as traditional methods. As the voxel grid method is already a fixed set of states, or voxels, then there is as such no loop to close; however, the same issue, in principle, can be seen, whereby the voxels that contain object information will drift in a direction overtime and voxels will not link together to create complete environments. In fact, it could be argued that this drift in object detection would be significantly more of an issue as the objects would appear to be dynamic in nature, due to the recording of drift in the voxel information over time. In the example of a car driving a set distance, then the problem can be explained in more simple terms. In this example, the car has instructions to drive 10 metres, then turn 180 degrees over another 20 metres, and, finally, repeat. This creates a \"loop\" whereby the car returns to the position it started in, this is shown in Figure 8. In the first image, this can be seen in effect. The image shows the car with perfect odometry and therefore returns to the same starting position. However, in the second image, we can see that the car has an error on the internal distance and rotation reading and, therefore, even though the car believes it is back to the same position it started at, the ground truth shows a different story and, in fact, the car has not returned to the starting position, leaving an \"open\" loop circuit. Figure 6: Figure showing the voxel approach to collecting data points. Figure 7: Figure showing point cloud and voxel grid. Below, it can be seen what happens when a small error is applied to the reading. The error propagates throughout the processes and the misalignment becomes significantly worse over time. With the voxel grid approach described in this paper, there is not a closed-loop detection, as such. When the agent returns to a position it could consider repeated or closed, the mapping data are not aligned, shown in Figure 9.

### Centre Point of Object, Clustering

_Object centroid_--To utilize the known objects in the world for aligning and scaling the voxel grid, a fixed center point is required for each object. This center point serves as a reference point for applying transformations, assuming that we have at least two identifiable objects. The process of generating identifiable objects is an area of ongoing development and will be explored in future work for more effective methodologies. Figure 10 shows an image of the concept. Figures 11 and 12 shows a real data and graphical plot respectively. Figure 2:0 Figure showing the perfect closed-loop system. Figure 2:1 Figure showing the imperfect closed-loop system. ClusteringTo determine the center point, the object data is clustered based on the voxels containing each object. This helps to identify the center of the voxel cluster belonging to each object. We use a simple K-means clustering algorithm, with K equal to the number of objects present in the environment. Any voxels identified as outliers are disregarded during the clustering process. The below defines the clustering of objects within the agent's voxel grid.