Mobile robots are being increasingly used as sensor-carrying agents to perform sampling missions, such as searching for harmful biological and chemical agents, search and rescue in disaster areas, and environmental mapping and monitoring. One of the objectives of these sampling missions is ‘Field Estimation’. Field estimation is the construction of an estimate of how a certain parameter varies in space and time, i.e., an estimate of its spatio-temporal distribution, based on observed or sampled data. As the field of interest is spread over a wide area, using a dense and fixed sampling scheme for an efficient field mapping would simply be too costly and will involve a possibly prohibitive computational load. Instead, it is far more interesting to use a mobile sampling scheme that would collect samples at few judiciously selected locations, in a way that would enable it to gain enough information about the field to be able to infer, with significant accuracy, the value of the parameter of interest at the unsampled locations. A multitude of research groups have published results on sampling using mobile robots for chemical plume source localization[1, 2], soil–moisture mapping for crop monitoring[3], ocean sampling[4, 5], forest-fire mapping[6], etc.

The sensor fusion schemes for sampling missions can broadly be classified into three categories based on (i) physical parametric models, (ii) feature-based inference techniques such as clustering algorithms, neural networks, etc., which are generally non-parametric in nature but can lead to black or grey box parametric representation of the process, and (iii) cognitive-based models, which use the inference processes of humans and animals and which are based on fuzzy logic rules, search techniques, information-theoretic approaches, etc. Models acquired using these three broad classes of approaches can be either purely deterministic or purely stochastic. In many cases, deterministic models affected by some random noise can also be assumed.

In the area of physical deterministic parametric modeling representing the first category of sampling missions, Christopoulos and Roumeliotis[2] presented an approach for estimating the parameters of the diffusion equation that describes the propagation of an instantaneously released gas. Cannell and Stilwell[4] presented two approaches for adaptive sampling (AS) of underwater processes using AUVs. The first one assumes a parametric model, while the second one uses an information-theoretic approach. A number of strategies for non-parametric AS can also be found in the literature. A solution for non-parametric ocean sampling is proposed in[7] based on a classification of the sampling area. The multi-robot path planning problem is addressed in[8] using the mutual information collected using different paths. The study of[5] is also similar to that of[8] in the sense that both deal with generating optimal trajectories for multiple underwater vehicles for sampling purposes. Rule-based non-parametric approaches are also used widely in chemical plume tracing on land and in water, odor sensing[2], mine detection, etc.

Forest fires, chemical source leaks, and temperature variations in oceans are examples of complex natural phenomena for which the exact nonlinear model descriptions are unattainable due to the high-level of complexity involved. Demetriou and Hussein[9] present a solution to the problem of estimating a spatial distribution when the process is described by a partial differential equation. In[10], a non-parametric model is considered, and a distributed scheme for field estimation is developed using a Kalman filter-like recursive scheme.

In geostatistics, spatial processes are generally modeled as random fields, and estimation is performed using Kriging Interpolation techniques[11, 12]. Kriging is termed *“simple”* if the mean of the distribution is also known, and *“universal”* if the mean is treated as an unknown linear combination of known basis functions. In[13], a distributed algorithm is presented for spatial estimation using the Kriged Kalman filter. Graham and Cortes[14] proposed a Kriged Kalman filter-based approach for a spatiotemporal field where the discrete-time evolution of the state is governed by the Kalman filter used. In[15], the authors represent the time-varying field with a random process with a covariance known up to a scaling parameter. They proposed gradient descent algorithm which can run in a distributed fashion on multiple robots. Olfati-Saber[16, 17] developed a distributed Kalman filter approach along with consensus filters to estimate the state of a process and reach consensus of all nodes.

Due to the time and energy-critical nature of some of these sampling scenarios, simply requiring the robots to perform a raster scan or randomly sample the field of interest would clearly be a sub-optimal and highly inefficient sampling strategy. Moreover, many time-varying distributions of interest encompass a wide area, and must therefore be observed with sensors having variables characteristics such as multiple size scales, rates, and accuracies[18]. For example, a forest fire is monitored using satellite images which provide a large spatial field-of-view (FOV) but a low-resolution or fidelity. On the other hand, a plane flying at low altitude would provide a low-spatial FOV but high-fidelity information.

In order to effectively fuse these different types of measurements, we proposed a *Multi-scale Multi-rate Adaptive Sampling* approach with a parametric description of the field[6]. In this approach, sampling strategies continuously adapt in response to real-time measurements from sensors of different scales. This scheme relies on building parametric models of the field using spatial sensor measurements collected from a high-altitude, and which are thus less accurate, and then improving the models by using more accurate spot measurements. The extended Kalman filter (EKF) is used to derive a quantitative information measure that is needed for the selection of sampling locations that are mostly likely to yield optimal information. In this approach, the existing low-resolution information of the field is first used to acquire an initial parametric representation of the field whose parameters have a higher initial error covariance which gradually reduces as high-resolution samples are taken and processed.

In our previous work[6], we presented a framework that extends our estimation of a simple parametric field to that of complex time-varying (e.g., forest fires[6]) by representing these with sums of overlapping Gaussians. The resulting algorithm was called EKF–NN–GAS, and is based on (a) a Radial Basis Function (RBF) neural network (NN) for the parameterization of the non-parametric field, (b) an EKF for parameter estimation, and (c) a heuristic search scheme called ‘Greedy Adaptive Sampling’ (GAS).

A further investigation of the AS algorithm using multiple robots is presented in this article. For widespread fields, it may be impractical and certainly inefficient for a single-robot to map the entire field by navigating to different sampling locations, even when guided by an efficient sampling algorithm. However, when using multiple robots, the sampling area is first divided into smaller regions, and then each sampling instance in a particular region gains information about the parameters which have a dominant effect in that region. Therefore, in order to distribute computations, we need to be able to fuse the parameter estimates in order to construct the map of the field density distribution.

This problem is similar to reformulating the algorithm originally designed for a conventional single-sensor single-processor system to work on a more general multi-sensor, multi-processor system. Distributed algorithms have been used before in many applications, and the degree of parallelism used in them varies from one algorithm to another, depending on the application at hand. An example of distributing processing includes target location estimation using several sensors for data collection, and then fusing together the collected measurements either at the central station or at each sensor in a multi-sensor fusion algorithm[19–21].

Since complex fields are represented by hundreds of parameters[6], it is computationally cumbersome for a single-robot to compute and store all parameter estimates and the uncertainty measures. It also quickly becomes unfeasible for individual robots to run a large AS algorithm, and share large covariance matrices wirelessly. Furthermore, with multi-robot sampling, the resources can be allocated efficiently if some resources are either busy or not available.

If the filter computation can be distributed among multiple robots, the number of computations performed by all the robots, i.e., the overall computational efficiency would be greater than the processing carried out by a single-robot having to carry-out both the sampling and computational tasks. Moreover, we expect that the concomitant advantages such as the flexible degree of parallelism, speed of convergence, and reduction in complexity that will be thus gained would be significant. With a single-robot, the total field estimation time includes the time necessary for navigation, sensing, and computation of the estimate (as there is no communication involved in this case). With multiple robots, the field estimation time includes the time taken for sensing, computation, communication, and final fusion to recover the field density distribution. We expect that the speed of convergence would increase by using multiple robots simply because of the sampling being done in parallel, and that the navigation time would be reduced significantly at the cost of modest increases in computation, communication, and fusion.

The rest of the article is organized as follows: in Section 2, we present the general formulation of the AS problem; Section 3 summarizes the existing centralized and decentralized filters, and their application to sensor network for field estimation; in Section 4, we present the novel federated distributed KF; Section 5 presents the simulation results for the proposed algorithm, and their discussion; finally Section 6 concludes the article.