Localisation optimization with optimal sensor-target geometry

localisation estimation bias reduction

In this thesis, special attention has been paid on two generic factors that influence the accuracy of almost all localisation systems. They are sensor-target geometry and location estimation bias.

2.3.2.1 Localisation with optimal sensor-target geometry

It is well known that relative sensor-target geometry can significantly influence the potential performance of any particular localisation algorithm Gustafsson and Gun- narsson [2005]. The Cramer-Rao bound (CRB) is a function of the relative sensor- target geometry along with the specific measurement technology employed by the sensors. A number of authors have attempted to identify the geometric configura- tions that minimise some measure of the variance lower bound. Popular measures of the optimal design criterion are T-optimality, which maximises the trace of the Fisher Information Matrix; E-optimality, which maximises the minimum eigenvalue of the Fisher Information Matrix and D-optimality which explores to maximize the determinant of the Fisher Information Matrix Ucinski [2004]. The FIM quantifies the amount of information that the observable random measurements carry about the unobservable parameters to be estimated. The CRB matrix is related to the FIM by taking the inverse matrix operation.

Different localisation scenarios have been considered in the design of optimal sensor-target geometry. Most of studies consider static localisation problems with a single stationary target and multiple stationary sensors located in two-dimensional space. Bishop et al. [2010] analyses, in depth, the geometry of the optimal sensor- target angular geometries for range-only and TOA-based localisation. It is shown that the optimal sensor-target angular geometries for range-only and TOA-based localisation are not unique. An infinite number of optimal sensor-target geomet- ric configurations can be found if the number of sensors exceeds a certain small number. In addition, the optimal sensor-target angular geometries for bearing-only localisation is also explored for an arbitrary number of sensors and for fixed, but arbitrary, sensor-target ranges. Unlike range-only and TOA-based localisation which is independent on the sensor-target range, the optimal sensor-target geometry for bearing-only-based localisation is explicitly dependent on the sensor-target ranges and the change of even a single sensor-target range can change the optimal geometry significantly. Again, the optimal sensor-target angular geometries for bearing-only localisation is not unique and an infinite number of optimal configurations can be identified when the number of sensors exceeds a small number. In addition, the optimal sensor-target angular geometries for TDOA-based localisation, is studied in

Meng et al. [2011, 2012]. They study two types of sensor pairing, the centralised sensor pairing and the decentralised sensor pairing. It is shown that the optimal sen- sor pair geometry does not depend on the ranges between the source and sensors. It only depends on sensor-target angular geometries. Furthermore, in the optimal sensor-source geometry setting, any sensor can be chosen as the reference, which results in the same lower bound of the localisation performance. For decentralised sensor pairing, under the condition without sensor sharing in a group of sensors, to achieve the optimal sensor pair geometry, two factors need to be met: 1) large angle between a pair of sensors that subtend to the target and 2) large intersection angle among different sensor pairs. Moreover, the optimal sensor-target geometry for hy- brid localisation are also studied, such as AOA/scan-based localisation Dogançay [2007].

As an extension of the static localisation problem, the optimal sensor-target ge- ometry of mobile localisation with either mobile sensors or mobile targets is also studied. This is known as optimal trajectory. The optimal trajectory is usually ob- tained from the optimal sensor-target geometry at each time instant. Similar to the static case, the measure in the mobile case is generally the same, but instead of tak- ing measurements once in the static case, the measure needs to be implemented at every time instant during the movement of either the sensors or the targets in the mobile localisation problem. The study of optimal trajectory is usually followed by the motion coordination of the sensors to meet the condition of optimal trajectory to reduce the localisation error. If the target is mobile, the extended Kalman filter (EKF) is used to track the target Oshman and Davidson [1999]; MartíNez and Bullo [2006]; Bishop and Pathirana [2008]; Meng et al. [2012]; Kaune and Charlish [2013].

2.3.2.2 Localisation geometric constraints

When conditions of optimal sensor-target geometry or optimal trajectory are met, the localisation accuracy can be improved. Another way to use the geometric infor- mation to improve the localisation accuracy is to implement the constrained optimi- sation during the process of location estimation using the underlying sensor-target geometry, so the localisation problem can be formulated as a constrained optimisa- tion problem. Bishop et al. [2008] derives a constraint for range-difference-of-arrival based localisation of a stationary emitter, which accounts for the relationship among the underlying geometry, the measurements and the nature of the true measure- ment errors. With the constraints, the measurement error which is consistent with the geometrical requirements can be estimated to correct the noisy measurements. Another constrained optimisation algorithm is derived following the similar princi- ple for bearing-only localisation Bishop et al. [2009]. In addition, when more than one emitters need to be localised, the emitter-emitter distance can also be used as a geometric constraint in the constrained optimisation process for emitter localisation Ekanayake et al. [2012].