# Global-Entropy Driven Exploration with Distributed Models under Sparsity Constraints
## 1 Introduction  
Multi-agent systems or swarms have proven to be advantageous in hazardous and inhospitable environments, such as emergency scenarios, natural disasters and extraterrestrial scenarios [1]. Humans operate these robots from a distance and are not put into danger while getting valuable information in a reconnaissance mission or the search for extraterrestrial life. Swarms precipitate in the information gathering process by their spatial distribution and by their larger sensor aperture [2; 3]. Especially for coverage problems, swarms are very useful [4; 5; 6] and have proven to be advantageous. It is obvious to see that by cooperation, swarms can speed up their mission task significantly. The gain in speed can either be due to performing parallel measurements or due to splitting the computational load among the agents [7]. Additionally, cooperation can open a way for more autonomy in such tasks and as a consequence reduce the mission costs [3]. To process the distributed data cooperatively, two different distribution strategies can be distinguished. The first strategy follows a homogeneous distribution paradigm [8; 9], where the swarm members have their own measurements and agree on a commonly-known model, which is then cooperatively estimated. This paradigm is also referred to as SOE, because every agent has an individual example of a common estimate.The second strategy uses a heterogeneous distribution paradigm [8; 9], also called SOF. In SOF, every agent has the knowledge of all measurements, but only processes parts of the global model. Thus, it can be seen as a splitting into different feature sets. Both approaches have been studied for estimation tasks [8; 9; 10]. However, a swarm can also cooperate by using the obtained information about their task for a cooperative navigation. Nowadays, cooperative information-driven navigation with swarms is the next key step for more autonomy in swarm systems, especially for SLAM algorithms, but also for information-gathering and exploration tasks. In [11], the authors set up a swarm of three mobile robots. One robot was able to manipulate objects in the environment with a robotic arm. The other two agents explored the a priori unknown environment and detected the positions of desired objects for the robot equipped with the manipulator. The two exploring robots explored the environment faster than just a single robot. A decision-theoretic approach, which takes uncertainty information of the agent's sensors into account, for exploring maps was presented in [12]. The authors estimated the map with a Rao-Blackwellized particle filter and performed exploration by maximizing the information gain, when a specific action was performed, i.e., the robot's movement. To estimate the entropy, the authors used a weighted sum of the posterior of their estimated particles. This approach was also tested in experiments, and the authors showed that this approach was more effective compared to non-information-driven approaches. A similar approach was described in [13], where the authors used a combination of the joint entropy of the map and the gained entropy along a path as a metric for exploration. All these methods focused on entropy-driven approaches for estimating a map. The authors did not consider other processes behind this, such as magnetic fields, gas concentrations or temperatures. Thus, we focus in this paper more on the exploration of such processes instead of the exploration of maps, and consequently, we assume in the following that an obstacle map is known or has been estimated before. Furthermore, we focus on distributed systems, which exchange their information to explore cooperatively. In this paper, we aim to estimate the spatial distribution of a static process such as a height profile or a magnetic field intensity in an area [14]. Static means that this process possesses no variation in time. The exploration of dynamic processes is therefore not considered in this work.