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Calibration of Multilateration Positioning Systems Via Nonlinear Optimization

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Calibration of Multilateration Positioning Systems Via Nonlinear Optimization DEGREE PROJECT, IN OPTIMIZATION AND SYSTEMS THEORY , SECOND LEVEL STOCKHOLM, SWEDEN 2015 Calibration of Multilateration Positioning Systems via Nonlinear Optimization SEBASTIAN BREMBERG KTH ROYAL INSTITUTE OF TECHNOLOGY SCI SCHOOL OF ENGINEERING SCIENCES Calibration of Multilateration Positioning Systems via Nonlinear Optimization SEBASTIAN BREMBERG Master’s Thesis in Optimization and Systems Theory (30 ECTS credits) Master Programme in Applied and Computational Mathematics (120 credits) Royal Institute of Technology year 2015 Supervisor at Ericsson: Daniel Henriksson Supervisor at KTH: Johan Karlsson Examiner: Johan Karlsson TRITA-MAT-E 2015:62 ISRN-KTH/MAT/E--15/62--SE Royal Institute of Technology SCI School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci Kalibrering av System f¨or Multilaterations Positionerssystem genom Icke-linj¨ar Optimering ” Sammanfattning I denna masteruppsats utv¨arderas en metod syftande till att f¨orb¨attra noggran- nheten i den funktion som positionerar sensorer i ett tr˚adl¨ost transmissionsn¨atverk. Den positioneringsmetod som har legat till grund f¨or analysen ¨ar TDOA (Time Dif- ference of Arrival), en multilaterations-teknik som baseras p˚am¨atning av tidsskillnaden av en radiosignal fr˚an tv˚arumsligt separerade och synkrona transmittorer till en mottagande sensor. Metoden syftar till att reducera positioneringsfel som orsakats av att de ursprungliga positionsangivelserna varit felaktiga samt synkroniserings- fel i n¨atet. F¨or rekalibrering av transmissionsn¨atet anv¨ands redan k¨anda sensor- positioner. Detta uppn˚as genom minimering av skillnaden mellan signalbaserade TDOA-m¨atningar fr˚an systemet och uppskattade TDOA-m˚att vilka erh˚allits genom ber¨akningar av en given sensorposition baserat p˚aoptimering via en ickelinj¨ar minstakvadratanpassning. Genom ett antal simuleringar testas sedan den f¨oreslagna metoden med olika grundinst¨allningar och olika grad av m¨atbrus samt ett varier- ande antal sensorer och transmittorer. Denna metod ger tydlig f¨orb¨attring f¨or estimering av systemparametrar och klarar ¨aven av att hantera multipla felk¨allor f¨orutsatt att antalet m¨atningar ¨ar tillr¨ackligt stort. 2 Abstract This master thesis presents an evaluation of a method for improving performance of sensor positioning in a network of emitters. The positioning method used for the analysis is Time Di↵erence of Arrival, TDOA, a multilateration technique based on measurements of di↵erences in signal travel time between a pair of synchronous and spatially separated pairs of emitters and a sensor. The method in question aims at reducing positioning errors caused by errors in initially reported emitter positions as well as network synchronization errors by using already known sensor positions to re-calibrate the network of emitters. This is done by minimizing the di↵erence between signal based TDOA measurements from the system and estimated TDOA measurements made by calculations based on given sensor positions by means of nonlinear least squares optimization. Alterations of the method with di↵erent settings and error contributions and with varying amount of sensors and emitters are tested throughout several simulations. The proposed method shows apparent results of improving the system parameters and also copes well with contributing errors provided that the amount of measurements is sufficiently large. Acknowledgements I would like to extend my greatest gratitude to Daniel Henriksson at Ericsson, who with his knowledge and enthusiasm has given invaluable support and guidance throughout this project. I would also like to thank my supervisor at KTH, Johan Karlsson, who has with his experience been a much appreciated support and dedicated advisor throughout the project. Contents 1 Introduction 2 1.1 Signal Based Positioning . 2 1.2 ThesisOutline ................................ 3 2 Time Di↵erence of Arrival, TDOA 4 2.1 Method .................................... 4 2.2 Factors influencing accuracy . 7 2.2.1 Synchronisation errors and time delays . 8 2.2.2 Emitter position errors . 8 2.2.3 Geometry............................... 9 2.2.4 Altitude................................ 9 2.2.5 Multipath . 10 3 Optimization and Method of Estimation 11 3.1 Unconstrained Optimization . 11 3.2 Nonlinear Least Squares . 12 3.2.1 Trust-Region-Reflective Least Squares Algorithm . 12 3.2.2 Nonlinear Least Square in Larger Scale . 14 3.2.3 Weighted Nonlinear Least Squares Minimisation . 15 3.3 Cram´er-Rao Lower Bound . 15 4 Modelling and Optimizing Antenna Position Errors in Cell Network 19 4.1 Formulation . 20 4.1.1 System of InsufficientRank ..................... 21 4.2 Determination of Weights . 22 4.3 Cram´er Rao Lower Bound Analysis . 23 5 Simulations 26 5.1 Method .................................... 26 6 Results 28 6.0.1 Weighted and Non-Weighted Least Squares . 28 6.0.2 No error in sensor positions . 29 6.0.3 Small error in sensor positions . 30 6.0.4 Large error in sensor positions . 31 7 Discussion 32 7.1 Futurework.................................. 33 5 8 Conclusion 35 Appendix A TDOA Positioning of Sensors 36 A.1 TDOA Positioning by Sum of Least Squares . 36 Appendix B Positioning with height parameter 37 Glossary TDOA Time Di↵erence of Arrival ETDOA Expected Time Di↵erence of Arrival GPS Global Positioning System NLLS Nonlinear Least Squares TTFF Time to First Fix TOA Time of Arrival CRLB Cram´er-Rao Lower Bound GDOP Geometric Dilution of Precision FIM Fischer Information Matrix Nomenclature e Emitter position eest Estimated emitter position s Sensor position sest Estimated sensor position ˜s Estimated sensor position g Emitter position error ✏ TDOA Measurement noise δ Time synchronization delay ⌧ TDOA value 1 1 Introduction This aim of this thesis is to evaluate an algorithm for improving positioning accuracy in wireless networks. Primarily, aspects of improving Time Di↵erence of Arrival (TDOA) positioning are considered. However, some of the methods used are also applicable to other positioning methods in wireless networks and other similar signal based positioning methods. More specifically, the algorithm to be evaluated will address certain common error contributions to see whether an adjusting calibration can decrease systematical errors and their impact on positioning accuracy. 1.1 Signal Based Positioning There are many examples of wireless and signal based systems that feature the possibil- ity of locating the position of an emitting or receiving unit. Global Positioning System, GPS, is a widely adopted positioning method using several time synchronized satellites making signal time measurements and multilateration to locate a receiving GPS signal unit [1]. It has become increasingly important to be able to position a mobile device accur- ately in a cellular network. Positioning a mobile device in a mobile network can ,e.g., enable localization of emergency calls, advanced location based services and possible also aid network traffic optimization [2]. As mobile network coverage is increasing and the technology is becoming increasingly advanced, the methods of positioning are under constant improvement [2]. Both GPS positioning methods and cellular network positioning methods are based on measurements on signals from distant satellites and antennas respectively. An obstacle free environment, where signals can travel without disturbance between the transmitting and receiving unit is to be preferred, however that is in most environments of applica- tion not possible. For example, these disturbances lead to multipath errors which will be described further in Section 2.2. These methods are not ideal for ,e.g., urban and indoor environments. A more recent and evolving positioning feature is that of locating units with local wireless networks, such as WiFi or Wireless Sensor Networks, WSN [4][5]. An example of application is indoor robot localisation, as described by Cheng et al. [4]. 2 As the computational capacity of network related equipment increases, it is possible to combine several positioning methods to increase accuracy. These are known as Hybrid Positioning systems. By combining results from multiple positioning methods, knowing the limitations and sources of errors for each method, it is possible to achieve a higher accuracy than when applying one single method [7]. A common method of combining di↵erent methods is for a GPS to achieve assistance data from a cellular networks to improve start up speed and decrease the Time-to-First-Fix, TTFF, known as the time required for a GPS receiver to acquire necessary GPS signals and fix a first position [7]. The assistance date sent from the cellular network could for example be in what cell sector the unit is situated whereby the GPS can limit its scope of search significantly [7]. The above mentioned positioning methods are examples of methods using known in- formation from emitters to locate unknown positions of sensors. However, the opposite can also be true. An example is when trying to locate from where radar signals are sent from. By measuring the signal and their characteristics from di↵erent known locations it is possible to estimate where the signals are transmitted from [6] [3]. One such method is known as Passive Radar Localization where positioning by means of TDOA is one of the most fundamental and popular localization methods [6]. This method will be described in more detail later in this thesis. However, the emphasis in this thesis will be on sensor positioning in a network of emitters. 1.2 Thesis Outline The
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