An Efficient Adaptive Sampling Approach for Mobile Robotic Sensor Networks using Proximal ADMM Viet-Anh Le 1 , Linh Nguyen 2 and Truong X. Nghiem 1 Abstract— Adaptive sampling in a resource-constrained mo- bile robotic sensor network for monitoring a spatial phe- nomenon is a fundamental but challenging problem. In ap- plications where a Gaussian Process is employed to model a spatial field and then to predict the field at unobserved locations, the adaptive sampling problem can be formulated as minimizing the negative log determinant of a predicted covariance matrix, which is a non-convex and highly complex function. Consequently, this optimization problem is typically addressed in a grid-based discrete domain, although it is combinatorial NP-hard and only a near-optimal solution can be obtained. To overcome this challenge, we propose using a proximal alternating direction method of multipliers (Px- ADMM) technique to solve the adaptive sampling optimization problem in a continuous domain. Numerical simulations using a real-world dataset demonstrate that the proposed PxADMM- based method outperforms a commonly used grid-based greedy method in the final model accuracy. I. I NRODUCTION A network of mobile robotic sensors is frequently employed in applications of monitoring environmental spatial fields, such as exploring ecosystems on land and in ocean, observing chemical concentration, and monitoring air pollutants and indoor climates [1]. Nevertheless, the fundamental problem of how to optimally drive the resource-constrained mobile sensors on sampling paths so that the information gained by the sensor measurements is maximal while the prediction uncertainty is minimal, known as adaptive sampling, is still practically challenging. In such a problem setting, while the mobile robots navigate through the environment, their on-board sensors take measurements of the spatial field and continually update a model of the field. The collective measurements and the corresponding model are then used to decide optimally where the mobile sensors should move at every moving step. The model used to represent the environmental spatial field in adaptive sampling can be either parametric or non- parametric [2], [3]. A parametric model requires a priori assumptions about the model structure and initial conditions, which are not easily obtained in practice. In contrast, a non- parametric model can be learned purely based on the collected data, resulting in its wide adoption in the literature [2], [3]. One such non-parametric model is the Gaussian Process (GP) [4]. A GP model can effectively learn an unknown spatial 1 V-A. Le and T. Nghiem are with the School of Informatics, Computing, and Cyber Systems, Northern Arizona University, Flagstaff, AZ 86011, USA {vl385,truong.nghiem}@nau.edu 2 L. Nguyen is with the School of Engineering, Information Technology and Physical Sciences, Federation University Australia, Churchill, VIC 3842, Australia l.nguyen@federation.edu.au phenomenon from a limited number of sensor measurements and then predict it at any unobserved locations of interest. Finding the optimal sampling locations for the robotic sensors at each time step can be formulated as minimizing the prediction uncertainty in the regions of interest. In the literature, various adaptive sampling optimization formula- tions have been considered. For instance, Xu et al. in [5] proposed to minimize the Fisher information matrix based objective function to find the optimal sampling locations for a MRSN. In [6], the authors considered a minimization problem where the cost function is derived from the average of the prediction variances over prespecified target points. In [7], the maximum a posterior estimation was exploited to design an adaptive sampling strategy for an MRSN by minimizing the prediction error variances. The adaptive sampling optimization problems in the abovementioned works [5]–[7] were solved by the gradient descent method. However, the robot dynamics and constraints on the physical mobility of the robots were not considered. With expectation of a trade-off between exploration and exploitation, Marchant et al. in [8] employed a Bayesian optimization approach to derive a sampling strategy for a MRSN, where the travelled distances of mobile robots were taken into account. More interestingly, some works used concepts from information theory to formulate the adaptive sampling optimization problem. For instance, in both [9] and [10], the authors utilized conditional entropy in a sampling optimality criterion to represent prediction uncertainty. Xu et al. in [10] aimed to minimize not only the prediction errors but also the hyperparameter uncertainty in the sampling optimization problem. In our previous works [11], [12], we employed both the conditional entropy and the posterior variances to design an adaptive sampling strategy for an MRSN. The dynamic constraints in the network along with a collision avoidance scheme were also incorporated into the optimization problem. To the best of our knowledge, most of the works that employed continuous optimization algorithms to solve the adaptive sampling problem did not take into account the dynamics and the mobility constraints of the robots. On the other hand, the adaptive sampling optimization problems in the works that considered the robot dynamics and re- source constraints were usually solved by grid-based search algorithms. However, these search algorithms suffer from the combinatorial NP-hard complexity of the grid-based approach and can only provide near-optimal solutions. To address the shortcomings of the above two approaches in the literature, in this work we propose using a proximal alternating direction method of multipliers (PxADMM) method [13]