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In this paper, a distributed cooperative filtering strategy for state estimation has been developed for mobile sensor networks in a spatial–temporal varying field modeled by the advection–diffusion equation. Sensors are organized into distributed cells that resemble a mesh grid covering a spatial area, and estimation of the field value and gradient information at each cell center is obtained by running a constrained cooperative Kalman filter while incorporating the sensor measurements and information from neighboring cells. Within each cell, the finite volume method is applied to discretize and approximate the advection–diffusion equation. These approximations build the weakly coupled relationships between neighboring cells and define the constraints that the cooperative Kalman filters are subjected to. With the estimated information, a gradient-based formation control law has been developed that enables the sensor network to adjust formation size by utilizing the estimated gradient information. Convergence analysis has been conducted for both the distributed constrained cooperative Kalman filter and the formation control. Simulation results with a 9-cell 12-sensor network validate the proposed distributed filtering method and control law. 

This paper proposes cooperative Kalman filters for distributed mobile sensor networks where the mobile sensors are organized into cells that resemble a mesh grid to cover a spatial area. The mobile sensor networks are deployed to map an underlying spatial-temporal field modeled by the Poisson equation. After discretizing the Poisson equation with finite volume method, we found that the cooperative Kalman filters for the cells are subjected to a set of distributed constraints. The field value and gradient information at each cell center can be estimated by the constrained cooperative Kalman filter using measurements within each cell and information from neighboring cells. We also provide convergence analysis for the distributed constrained cooperative Kalman filter. Simulation results with a five cell network validates the proposed distributed filtering method. 

In this paper, a constrained cooperative Kalman filter is developed to estimate field values and gradients along trajectories of mobile robots collecting measurements. We assume the underlying field is generated by a polynomial partial differential equation with unknown time-varying parameters. A long short-term memory (LSTM) based Kalman filter, is applied for the parameter estimation leveraging the updated state estimates from the constrained cooperative Kalman filter. Convergence for the constrained cooperative Kalman filter has been justified. Simulation results in a 2-dimensional field are provided to validate the proposed method. 

This article presents an online parameter identification scheme for advection-diffusion processes using data collected by a mobile sensor network. The advection-diffusion equation is incorporated into the information dynamics associated with the trajectories of the mobile sensors. A constrained cooperative Kalman filter is developed to provide estimates of the field values and gradients along the trajectories of the mobile sensors so that the temporal variations in the field values can be estimated. This leads to a co-design scheme for state estimation and parameter identification for advection-diffusion processes that is different from comparable schemes using sensors installed at fixed spatial locations. Using state estimates from the constrained cooperative Kalman filter, a recursive least-square (RLS) algorithm is designed to estimate unknown model parameters of the advection-diffusion processes. Theoretical justifications are provided for the convergence of the proposed cooperative Kalman filter by deriving a set of sufficient conditions regarding the formation shape and the motion of the mobile sensor network. Simulation and experimental results show satisfactory performance and demonstrate the robustness of the algorithm under realistic uncertainties and disturbances. 

Recursive neural networks can be trained to serve as a memory for robots to perform intelligent behaviors when localization is not available. This paper develops an approach to convert a spatial map, represented as a scalar field, into a trained memory represented by the long short-term memory (LSTM) neural network. The trained memory can be retrieved through sensor measurements collected by robots to achieve intelligent behaviors, such as tracking level curves in the map. Memory retrieval does not require robot locations. The retrieved information is combined with sensor measurements through a Kalman filter enabled by the LSTM (LSTM-KF). Furthermore, a level curve tracking control law is designed. Simulation results show that the LSTM-KF and the control law are effective to generate level curve tracking behaviors for single-robot and multi-robot teams. 

In this paper, we present a Long Short-Term Memory (LSTM)-based Kalman Filter for data assimilation of a 2D spatio-temporally varying depth-averaged ocean flow field for underwater glider path planning. The data source to the filter combines both the Eulerian flow map with the Lagrangian mobile sensor data stream. The depth-averaged flow is modeled as two components, the tidal and the non-tidal flow component. The tidal flow is modeled with ADCIRC (Advanced Three-Dimensional Circulation Model), while the non-tidal flow field is modeled by a set of spatial basis functions and their time series coefficients. The spatial basis functions are the principal modes derived by performing EOF (Empirical Orthogonal Functions) analysis on the historical surface flow field measured by high frequency radar (HFR), and the temporal coefficients of the spatial basis function are modeled by an LSTM neural network. The Kalman Filter is performed to combine the dynamics derived from the LSTM network, and the observations from the glider flow estimation data. Simulation results demonstrate that the proposed data assimilation method can give flow field prediction of reasonable accuracy.