Combining the Inherent Distance Geometry Constraints of CMD and Triangulation Constraints to Improve Positioning Accuracy

In wireless sensor networks (WSN), sources are usually localized with the help of pre-deployed base stations (BS). Due to the unknown environment, the main challenge in source localization is the non-line-of-sight (NLOS) error. This paper focuses on two approaches, such as triangulation constraints based on the relationships between each BS, and distance geometry constraints of Cayley-Menger determinants (CMD). However, there is no literature using them jointly. Therefore, this study aims to combine triangulation constraints and Distance geometry constraints of CMD to improve localization performance. First, we obtain the weight coefficients for all combinations of BSs by solving the Karush-Kuhn-Tucker (KKT) conditions for the triangular constraints and the distance geometry constraints of CMD. Since various combinations of BSs have different weighting coefficients, leading to uncertainty. Then, based on the uncertainty of the weight coefficients, we formulate the source localization problem as a robust least squares (RLS) problem. Finally, we eliminate the uncertainty by continuously relaxing the constraints through some mathematical methods, such as Cauchy-Schwarz inequality and norm compatibility. In addition, excessively relaxing the constraints may lead to low estimation accuracy, so we add some inherent constraints. Simulations and experiments demonstrate that the proposed method can perform well in different environments.