Optimal sensor configuration for passive position estimation

Abstract

International audienceThe goal of passive source localization is to acoustically detect objects producing noises by multiple sensors (e.g. microphones, hydrophones) and to estimate their position using only the sound information. While within the last four decades a lot of work was carried out on how to best measure the time delay of arrivals (TDOAs) and on finding an optimal location estimator, relatively little work can be found on how to best place the sensors. However, the performance of such estimators is strongly correlated to the sensor configuration. Therefore, we propose a procedure for an optimal sensor setup minimizing the condition numbers of an analytic linear least-squares (LLS) estimator and an iterative, linearized model (LM) estimator. An advantage of using the condition number as the cost function is that, unlike the Cramer Rao Lower Bound, it defines an upper bound for the estimation error. Further, no assumptions about the disturbance noise need to be made and a robust sensor configuration will be found, which is invariant to rotation and dilatation. The two condition numbers of the presented passive source localization algorithms are independent of the number of sensors. However, it will be shown, that the estimation error decreases proportionally to the inverse of the square-root of the number of sensors. Some analytical forms of optimal sensor configurations will be derived, which attain the global minimum of the condition number of the LLS estimator or which minimize the condition number of the LM estimator. Further, a sensor geometry using a minimum number of sensors is derived, which forces the condition numbers of both estimators equal to one. The interest of such a setup lies in a possible combination of both estimators. The LM estimator might then be initialized by the position estimate found by the LLS estimator. A variety of alternative estimators are closely related to the LLS estimator. Their performances will be compared, and it will be shown, that the o- - ptimal sensor geometry specially derived for the LLS estimator also increases their accuracies

An Ultra-Wideband System for Vehicle Positioning

“…The problem of finding an optimal configuration was reported in several publications e.g. [9], [11]. Similar results were obtained for different positioning algorithms and different cost functions.…”

Underwater Target Positioning with a Single Acoustic Sensor

“…Interesting results in the area of sensor networks for positioning go back to the work of Abel [1990], where the Cramer-Rao bound is used as an indicator of the accuracy of source position estimation and a simple geometric interpretation of this bound is offered. Related work in this area can be found in Levanon [2000], Martinez & Bullo [2006], Zhang [1995], Neering et al [2008], Jourdan & Roy [2008], Bishop et al [2010]. In the marine systems field we can find the works Bahr et al [2009], Moreno-Salinas et al [2011] and Moreno-Salinas et al [2013].…”

Multiple underwater target positioning with optimally placed acoustic surface sensor networks

“…Important work in this field includes also Yang and Scheuing, 6 Yang, 7 Neering et al, 8 Jourdan and Roy, 9 Bishop et al, 10 Isaacs et al, 11 Dogancay and Hmam,12 and Lui and So. 13 In general, the above works search for the conditions that optimize some indicator based on the FIM or the CRB to define the optimal sensor positions in 2D scenarios.…”