# Global-Entropy Driven Exploration with Distributed Models under Sparsity Constraints
## 4 Simulations
### Evaluation of Entropy-Driven Exploration with Real Data  
We are now ready to evaluate the proposed exploration strategies. As data, for exploration, we used the magnetic field intensity on the ground in our laboratory. These intensities have been measured with a magnetic field sensor module integrated in the Xsens MTx Package [14]. To evaluate the proposed entropy-driven trajectory planner, we created a simulation environment that uses a part of the magnetic field data of our laboratory. Figure 3 shows the data. The data are sampled on a grid at the positions \(\mathbf{X}=[\mathbf{x}_{0},\dots,\mathbf{x}_{|\mathcal{X}^{\prime}|}]^{T}\), where \(\mathbf{x}\in\mathcal{X}^{\prime}\subset\mathbb{R}^{2}\) and \(|\mathcal{X}^{\prime}|=4128\). The magnetic field clearly shows a spatial correlation and therefore is a good starting point to evaluate the algorithm. As features, we use a two-dimensional Gaussian kernel defined by a kernel width \(\sigma\) and a feature position \(\mathbf{\mu}\in\mathbb{R}^{2}\) as:  
\[\varphi(\mathbf{x}[i],\mathbf{\mu})=\exp\left(-\frac{1}{2\sigma^{2}}\|\mathbf{x}[i]-\mathbf{ \mu}\|^{2}\right),\mathbf{x}[i]\in\mathcal{X}^{\prime}.\]  
In this paper, we set the feature width \(\sigma=30\,\mathrm{cm}\) and the number of agents to \(K=10\). The agent's locations are initialized uniformly distributed over \(\mathcal{X}^{\prime}\). The predefined feature locations for SOE are all discretization points in \(\mathcal{X}^{\prime}\). The first measurements are then taken at their initial positions. We consider each agent in this simulation as a single point, and they can move everywhere within an instant. Therefore, we consider no collision avoidance in this simulation. To benchmark our exploration strategy, we will compare three different types of movement strategies: entropy, meander and random walk strategies. Entropy is the proposed strategy, whereas meander and random walk are used as benchmark algorithms. They are explained in the following. Entropy: After the distributed estimation, the agents hierarchically evaluate the exploration criterion. For SOF, agent \(k\) distributes all possible measurement locations \(\mathcal{X}_{k}\) to its neighbors \(\mathcal{N}_{k}\) and receives each agent's consensus part, cf. Section 3.3. For SOE, the possible measurement locations do not need to be distributed, cf. Section 3.2. Then, for both paradigms, the exploration criterion is evaluated. Afterward, agent \(k\) chooses the location, which yields the best criterion according to (39) for SOF or (28) for SOE. Both schemes are also shown in Algorithms 1 and 2 for SOE and SOF, respectively. Meander: In this exploration strategy, the agents start at an individual predefined position, somewhere at the left border of the environment, i.e., at coordinates with \(\mathbf{x}_{k}[0]=[0,\frac{4.3}{k}]^{T}\), where 4.3 is the maximum of the environment in the y-direction. After estimation, they continue to move towards the opposing border on the right, i.e., they move in the \(x\)-direction with a predefined step size. In this  
Figure 3: A part of the magnetic field on the ground of our laboratory. This plot shows the normalized magnetic field intensity in color coding. Red corresponds to a high magnetic field intensity, and blue corresponds to a low magnetic field intensity. paper, the step size is equal to the kernel width \(\sigma\). On the other side of the border, they move upwards and move back to the other side of the environment, generating a zigzag pattern. Random walk: This is the simplest approach, where the agents choose a new measurement location by random uniform sampling of the whole \(\mathcal{X}^{\prime}\). For meander and random walk, we do not evaluate the exploration criterion for the agents' movements. Consequently, the agents do not exchange the information of their already traversed paths, their proposed measurement positions \(\mathcal{X}_{k}\), nor consensus parts for evaluation of the exploration criterion.