## 2 Materials and Methods
### The Voxel Grid and the Approach to Capture Data
Figure 2:2

where

\(S\) is a set of voxels within the voxel grid;

\(k\) is the number of unique objects within the environment the agents has mapped;

\(x\) is an observation data point;

\(\mu_{i}\) is the mean of points in \(S_{i}\)

\(d(x,\mu_{i})\) is some arbitrary function that measures the distance between points in any given unit. In this case, the Euclidean distance function will be used, given by

Figure 2:3

Figure 2:4 Figure showing centre point of a detected object with real data. Figure 2:5 Figure showing centre point of a detected object. The below images show data taken from a single frame with all voxels removed that did not contain an object. The data show a frame containing two objects. ### The Frame-to-Frame Alignment

To align from two frames, the agent considers two key aspects of information. The first is the internal odometer and the second is the holographic difference between the two frames. The odometer would, in principle, be the only information required, as by considering how much the agent has moved in the x, y, and z plane, while also considering the rotation, an updated position of the agent could be determined in the global world space. However, outside of the simulation, there is no perfect odometer information and an error will also occur. This error will have a cumulative effort on the perceived position of the agent, and unless the error is near-perfect Gaussian, it will lead to an open-loop mapping. In SLAM, this problem is solved using an extended Kalman filter, and in the proposed approach, that is also what is being utilised, where the frame-to-frame alignment is a prediction based on the odometric reading and the discrepancy of features detected in the frame. Figure 2:6

Each time the agent moves, the odometer of the agent is merged with the frame alignment process. Note that this fusion could be complemented with other forms of localisation such as GPS. This will be further covered in the section relating to heterogeneous agent design. Let us break down the frame alignment technique into steps and walk through the processes. Figure 13 defines the scenario for the subsequent explanation of frame-to-frame alignment. _Step 1_--is to find all image features; these are our key points to connect between one frame and another. \(f_{t}=x\mid x\) SURF Feature. This can be seen in Figure 2:7

_Step 2_--The agent moves; this movement can be in any form of translation or rotation, but for this example, the movement will be limited to a linear transform in the X plane. This can be seen in Figure 2:8

Figure 2:9 Plot of K-means clustered data of two separate objects. Figure 4:0 Figure showing features detected in frame. Figure 4:1 Figure showing key for agent alignment images. Figure 4:2 Figure showing features detected from agent’s view. _Step 3_--Compare the difference between the original image frame and the agent's new location image frame with respect to the matched features detected. \(h:f_{t-1}-f_{t}\). This can be seen in Figure 4:3

_Step 4_--Compare the agent's odometric translation with respect to the matched features. \(o:f_{t-1}+CurrentOdometrereading\). This can be seen in Figure 17 and further explained in Figure 4:4

_Step 5_--Take a final weighted value from the two comparison figures. This can be seen in Figure 19 with the full algothem broken down in Algorithm 2. ### Multiagent Alignment of Maps

Multiagent environment mapping--Multiagent environment mapping is a critical area of research in robotics that focuses on utilising multiple robotic agents to collaboratively map, assess, and interact with dynamic and complex environments. This approach is particularly valuable in situations where human intervention is risky or impractical, such as during disaster response or exploration missions. Key aspects of multiagent mapping include effective communication among agents, decentralised data fusion, and consensus algorithms. A good example can be found at [27].