The ability to sense airborne pollutants with mobile robots provides a valuable asset for domains such as industrial safety and environmental monitoring. Oftentimes, this involves detecting how certain gases are spread out in the environment, commonly referred to as a gas distribution map, to subsequently take actions that depend on the collected information. Since the majority of gas transducers require physical contact with the analyte to sense it, the generation of such a map usually involves slow and laborious data collection from all key locations.
Monitoring and enhancing indoor environmental standards, such as air quality, thermal comfort, or acoustics, are of great importance for improving human health, comfort, and productivity 
. Due to the complexity of indoor structures typical of human-centered environments, airborne pollutants are not distributed homogeneously, resulting in significant spatio-temporal variations.
Mobile robotic olfaction, the discipline that combines artificial olfaction with mobile robotics, often involves applications where a robot has to detect the contour of a gas distribution and measure its intensity 
. In these situations, the robot’s role is to explore an environment to obtain a map of the gas distribution. These maps could then be an asset, for example, to tailor an automated fire evacuation response depending on how smoke is spreading through a building 
, to determine the origin of a volatile substance during robotic Gas Source Localization (GSL) 
, or as an intermediate tool to investigate urban air pollution 
Acquiring the necessary data to generate such gas distribution maps is not a trivial task. The phenomena of diffusion and advection often lead gases to spread in a highly stochastic fashion 
, making it very hard or even impossible to accurately predict how gas particles move with dispersion models alone. Moreover, limitations in current gas sensing technologies have to be taken into account. To date, no gas sensor allows capturing a complete gas distribution with a single snapshot 
, and visiting every possible location to take a gas sample with an electronic nose (e-nose), for instance, is either very time-consuming or, quite often, intractable if there are places that the robot cannot physically reach. Hence, the prevailing approach for olfaction-enabled mobile robots is to perform a partial exploration of the environment and then extrapolate the gas concentration to non-visited locations. This is accomplished with Gas Distribution Modeling (GDM) 
, the process that estimates the shape of a gas distribution from a set of sparse measurements that are composed of gas observations (e.g., from the robot’s e-nose 
) and, in some cases, wind data from an anemometer 
. For a detailed review of gas sensing technologies and e-noses commonly used in mobile robotics, please refer to 
Multiple GDM approaches have been proposed for mobile robotics. Some techniques extrapolate the robot measurements to nearby locations with Gaussian
, others rely on Source Term Estimation (STE) 
to predict the parameters that govern the shape and origin of a gas plume 
, and others constrain the gas dispersion to the presence of obstacles 
. Regardless of the mathematical formulation, all GDM techniques for mobile robotics aim to fulfil one requirement: their computation must happen in real time, or as close as possible to it, to provide the robot with a continuously updated gas map that accounts for the latest gas measurements. This is paramount for applications such as industrial safety 
, where fast decision making is necessary and where more accurate Computational Fluid Dynamics (CFD) simulations would be too slow to perform.
One such real-time GDM algorithm is Gas-Wind Gaussian Markov Random Field (GW-GMRF) distribution modeling 
. This algorithm treats the environment as a lattice and models the interaction between adjacent cells as a probability distribution to estimate the most likely gas concentration at remote locations. The main advantage of GW-GMRF over similar techniques is that it also accounts for the obstacles in the environment and for the local wind measurements captured by the robot to reduce the total number of samples it has to gather. Moreover, because GW-GMRF combines all gas and wind samples under a probabilistic approach, it can assign an uncertainty level to each location of the output map. This provides valuable feedback to determine how trustworthy the estimates are, which, in turn, allows GW-GMRF to deal with challenging situations, such as physically separated areas (e.g., walls), eddies near obstacles, or environments that are only partially explored, without impairing its overall reliability.
Similar to most GDM techniques, GW-GMRF has one important limitation: it is an open-loop method that provides no feedback on where the robot should explore next; it only takes a set of input samples and computes the gas distribution map that explains them the best. However, the limited time and power available to explore the environment with a robot imply that it is critical to plan sampling locations that yield the most informative sensor readings. The main difficulty herein lies in the lack of a priori knowledge about how the gas is distributed, which prevents the robot from computing the optimal exploration path in advance. A conventional sensing strategy that samples along a predefined path does not take into account the distribution and valuable resources might be wasted at locations of little interest 
. Certainly, a better strategy would be one where the robot can dynamically adapt to the latest gas distribution estimates and plan the remainder of its exploration accordingly. In the case of GW-GMRF, this can be achieved by resorting to an uncertainty map, whose function is precisely to highlight where the estimate is deficient and more data are needed. However, just visiting the locations with the highest uncertainty in sequence might still lead to erratic and sub-optimal behavior. For example, the robot could end up in an unintended loop where it repeatedly transverses the same areas to reach locations with high uncertainty at opposite ends of the environment.
For situations where a mobile robot needs to acquire information about the environment yet is unable to plan an optimal strategy without this very information 
, a Partially Observable Markov Decision Process (POMDP) can be a possible solution. The POMDP framework was originally conceived for robotic control, dealing with the uncertainty in the robot measurements and the outcome of its actions 
. In particular, it is useful for non-deterministic scenarios where a robot has to maximize a goal (e.g., get close to a target or collect objects) whilst saving on resources (e.g., energy and execution time). In these cases, the robot is provided with a complete representation of the environment, which is subsequently leveraged to predict the likely outcomes of an action. After implementing some slight modifications, the POMDP framework can also be applied for information gathering tasks 
. For example, when it is applied to GDM, the robot might know the shape of the environment, but not how the gas is distributed within it.
2. Information-Driven Gas Distribution Mapping for Autonomous Mobile Robots
Autonomous mapping, understood as an Informative Path Planning (IPP) problem, is an active research topic that centers on optimizing a robot’s exploration path to gather information about a physical phenomenon with the objective of reconstructing its spatial distribution.
Researchers' work draws inspiration from the work on infotaxis 
, a strategy that searches for gases by maximizing the expected information gain. Infotaxis
relies on an underlying gas dispersion model to compute a prior probability distribution over the environment from past gas measurements. For each incremental step of the exploration, this prior distribution enables the robot to determine the travel direction that should, hypothetically, yield the most informative and valuable samples. The behavior and capabilities of infotaxis
are, consequently, not static, but determined by the employed gas dispersion model and its fitness for the task at hand. The first version of this strategy 
and its multi-robot variant 
, for instance, were exclusively aimed at GSL, finding the release point of a target gas. These employed a simple yet effective model of how frequently a robot is expected to encounter gas patches given its distance to the source. This model worked well to track sparse gas plumes, but not to map their spatial distribution.
Another infotactic approach, based on STE 
, models the gas distribution as a pseudo-Gaussian plume whose origin coincides with the source and whose concentration can be computed for every point in space (also in 3D). However, it aims to track relatively straight gas plumes under stable wind conditions, and thus may fail to provide reliable gas maps in the presence of obstacles. Researchers' work, on the other hand, takes full advantage of GW-GMRFs ability to deal with complex indoor environments, albeit oriented purely towards infotactic GDM rather than GSL.
An important aspect of IPP approaches is how to determine the informative content and overall importance of each potential sampling site. In applications that involve gases 
, this is usually accomplished by computing the Kullback–Leibler Divergence (KLD) between the prior distribution and the later one that would be obtained after taking measurements at each candidate goal position. However, just choosing the sampling sites that yield the most information alone would not lead to a sensible exploration of the environment. The robot must also take into account the time and effort it takes to reach them, and if there are multiple sites of interest, in what order they should be visited. One possible tool to determine such a (pseudo) optimal path could be a POMDP formulation (more details in Section 4
). A POMDP simulates the outcome of every possible movement and then picks the one with the best score according to a given reward function, so that it may also account for other relevant parameters (travel time, distance, and energy cost) aside from the KLD gain 
Regarding computational efficiency, POMDPs and similar IPP methods are usually constrained by the fact that the robot may only choose from a finite set of actions, which have to be checked individually. Efficient alternatives have been proposed for Gaussian Markov random field (GMRF)-based IPP 
exploiting their closed-form solution. However, GW-GMRF introduces a dependency between a gas map (a scalar field) and a wind map (a vector field) that must be solved numerically, and avoids such implementations.