Expectation-maximisation-based localisation using anchors with uncertainties in wireless sensor networks

Localisation in wireless sensor networks (WSNs) has received much attention, where most studies focus on mitigating the effects of measurement noise under the assumption of accurate anchors' positions. However, anchors' positions could be inaccurate for the inevitable errors in practical observations. This paper studies the sensor localisation with both inaccurate anchors' positions and noisy range measurements in WSNs. To solve the intractable integrals in likelihood function, the authors propose to use expectation-maximisation (EM) algorithm to obtain the maximum likelihood (ML) estimation iteratively. The 'a posteriori' distribution of the anchor's position uncertainty is approximated to a circularly symmetric Gaussian distribution by minimising the Kullback-Leibler divergence between them. Building on this, the authors derive the expectation step in a closed-form expression. In the maximisation step, based on the Taylor expansion of the confluent hypergeometric function of the first kind presented in the expectation step, analytical solutions are obtained. Simulation results show that the proposed EM estimator significantly outperforms the approximated ML estimator. The performance gain by using the EM estimator becomes larger as the increase of anchors' position uncertainties. Moreover, the performance of the EM estimator is close to that of the Monte Carlo-based estimator with much less computational complexities.