Research Article Open Access
Non-Synchronous Sensor Network Positioning Method Based On RSS and TOA Hybrid Measurement
Lu Kangli1*, Wu Xiaoping2, LvYangmin3 and Hua Yuting4
1School of Information Engineering, Zhejiang Agriculture and Forestry University, Lin’an 311300, China
2Zhejiang Key Laboratory of Forestry Intelligent Monitoring and Information Technology, Lin’an 311300, China
3School of Information Engineering, Zhejiang Agriculture and Forestry University, Lin’an 311300, China
4School of Information Engineering, Zhejiang Agriculture and Forestry University, Lin’an 311300, China
*Corresponding author: Lu Kangli, School of Information Engineering, Zhejiang Agriculture and Forestry University, Lin’an 311300, China, Email: @
Received: August 21, 2018; Accepted: September 07, 2018; Published: September 21, 2018
Citation: Kangli Lu, Xiaoping Wu, Yangmin Lv, Hua Y (2018) Non-Synchronous Sensor Network Positioning Method Based On RSS and TOA Hybrid Measurement. J Comp Sci Appl Inform Technol. 3(3): 1-9. DOI: 10.15226/2474-9257/3/3/00135
Abstract
The method of single range-based measurement will lead to an increase in the uncertainty of positioning results and a lower positioning accuracy. To deal with the positioning problem in three dimensional non-synchronous sensor networks, an accurate positioning method is proposed by using the hybrid Received Signal Strength (RSS) and Time-of-Arrival (TOA) measurements. The Unconstrained Linear Least Squares (ULLS) and Constrained Linear Squares (CLLS) methods are put forward to obtain the node position coordinates by converting the non-linear optimization model established by hybrid measurements into linear equations. The accuracy of the designed algorithm was tested by simulation, and the influence of different measurement noise on the estimation error of the hybrid measurement method was analyzed. The results show that the hybrid measurement positioning method has less error than the single measurement method. The constrained CLLS is more accurate than the unconstrained ULLS. The CLLS and ULLS linear estimation methods have high stability and positioning accuracy under low noise conditions.

Keywords: Positioning; Time synchronization; Received signal strength; Hybrid measurement;
Introduction
The Wireless Sensor Networks (WSN) use a large number of static or mobile wireless transmission nodes for sensing and collection. They have broad application prospects in the fields of environmental monitoring, target tracking, military and intelligent transportation systems, and have been concerned and explored by many researchers [1,2]. The implementation of functions such as target tracking, geographical location routing, and fault alarm in sensor network applications is based on node location [3,4]. Node location technology is one of the basic support technologies of wireless sensor networks [5,6]. Sensor network positioning technology has become a key factor in its application promotion, and the node positioning technology with superior development performance has become an important research hotspot of sensor network positioning content. In a real scenario, the sensor network area deployed by nodes is usually in a 3D environment. However, most of the existing localization algorithms are researched and optimized based on 2D space, which is different from the real environment, resulting in the inability to locate specific locations with high precision. Therefore, the research of 3D positioning algorithm is one of the key contents to solve the problem of wireless sensor networks node positioning in the future [7].

The use of known location beacon nodes to estimate the unknown node position coordinates is a common sensor network positioning method, which relies on some ranging methods between nodes, such as Time of Arrival (TOA) and Arrival Time Difference (TDOA), Angle of Arrival (AOA) and Signal Reception Strength (RSS) [8-11]. Since each node uses a separate clock module for timing, the clock module timing will drift with time and environmental parameters, so the timing of each clock module is not synchronized, which is called the asynchronous sensor network [12]. The single-time measurement method can solve the node position estimation of the asynchronous sensor network, but the single measurement method leads to the increase of uncertainty of the positioning result, the unreliable positioning result, and the low positioning accuracy, it is more difficult to apply to the positioning problem of the 3D sensor network.

The time measurement implementation principle is relatively simple. For example, the literature [13] uses multiple antenna receivers, and proposes a joint estimation method of node clock deviation and target position, which realizes the joint calculation of time synchronization and positioning. And the literature [14] introduced a hybrid measurement method combining TOA and AOA. The RSS measurement does not require additional hardware facilities, and the measurement cost is low, there is also a hybrid measurement method of RSS and TDOA proposed in literature [15]. Among the existing three-dimensional positioning methods, some are derived from three-dimensional positioning methods in three-dimensional space, and some propose new positioning algorithms. For example, the literature [16] solving the problem of 3D location based on RSS and AOA measurements. Therefore, this paper introduces a hybrid RSS and TOA measurement method to solve the positioning problem in the 3D scene in the asynchronous sensor network.

With the increasing demand for location service applications, the positioning accuracy requirements are getting higher and higher. How to reduce the positioning error of positioning algorithm design has become a research hotspot in this field. By establishing a distance-constrained optimization model between nodes, researchers have continuously proposed positioning algorithms for sensor networks, such as Maximum Likelihood (ML) estimation, linear algebra method and convex optimization, and so on [17-19]. The numerical calculation method of ML estimation method relies on the selection of the initial solution, and has better precision in weak noise environment, but it may fall into local optimum. For this reason, linear algebra method and convex optimization method are proposed. The convex optimization method (including semi-definite programming, quadratic cone planning, etc.) relaxes the optimization model into a convex optimization problem, which is a popular method in the current sensor network positioning method, but the convex optimization function has more variables and equality constraint, and the computational complexity is high. Because of the relaxation, the estimation result is not optimal. In order to reduce the computational complexity, the linear algebra method directly represents the calculation result as an algebraic solution, which is faster in calculation and avoids the trouble caused by the initial value selection.

The TOA ranging method is used to establish the distance relationship between nodes by measuring the arrival time of the signal. The accuracy of the ranging depends on the accurate timing ability of the node clock. However, due to the change of environmental factors, may lead to the asynchrony of the timing clock of the node. In this paper, based on the chronograph clock model of signal transmission between nodes, the relationship between observation time and actual time is derived, and the joint estimation equation of time synchronization and positioning is established. Based on the node clock drift and deviation model, a joint linear estimation method of time synchronization and node localization is proposed to realize simultaneous estimation of node clock drift rate, deviation and position coordinates. The nodes communicate with each other in the form of electromagnetic waves, and the Received Signal Strength (RSS) is attenuated with the extension of the transmission path. The RSS method does not need additional hardware, is simple to implement, it has the characteristics of low power consumption, low cost, and the like, and is widely used. The sensor network node positioning is realized by the RSS and TOA hybrid measurement between the nodes, which can make the positioning result more reliable and the positioning accuracy higher. Since the unknown parameters of the 3D space are more than the 2D plane, the difficulty is also increased. The positioning method using the hybrid measurement is more reliable and accurate than the single ranging method, and is more suitable for the positioning problem of the asynchronous sensor network. Based on the RSS and TOA hybrid measurement technique between nodes, this paper proposes a precise positioning method for the wireless signal strength (RSS) and Time of Arrival (TOA) hybrid measurement of the 3D asynchronous sensor network. The nonlinear optimization model established by the hybrid measurement is transformed into a linear equation, and the Unconstrained Linear Least Squares (ULLS) and Constrained Linear Singularity (CLLS) methods for node position coordinate estimation are proposed respectively, and compared with the model of the Carmer-Rao Lower Bound( CRLB) values were compared. The design of hybrid measurement method for 3D spatial positioning depends on less number of beacon nodes and high positioning accuracy.

The first part of this paper begins with introduced the problem description of hybrid positioning of RSS and TOA; the second part deduces the calculation method of ULLS and CLLS; the third part deduces the lower bound of carmer-Rao (CRLB) of model; the fourth part is simulation and analysis; the last part is the conclusion.
Problem Description
The coordinates of unknown nodes are calculated in twodimensional plane. Suppose that there are N beacon nodes with known position coordinates, and their coordinate positions are respectively x i = [ x i y i ] T  (i=1,2,,N) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaWGPbaabeaakiabg2da9iaacUfafaqabeqacaaabaGaamiE amaaBaaaleaacaWGPbaabeaaaOqaaiaadMhadaWgaaWcbaGaamyAaa qabaaaaOGaaiyxamaaCaaaleqabaGaamivaaaakiaabccacaGGOaGa amyAaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacqWIVlctcaGGSa GaamOtaiaacMcaaaa@4A79@ . At the same time, there is an unknown node to be located in the area, and the position coordinate is assumed to be x=[ x y ] T MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEaiaab2 dacaqGBbqbaeqabeGaaaqaaiaadIhaaeaacaWG5baaaiaab2fadaah aaWcbeqaaiaadsfaaaaaaa@3C82@ . Using the ranging method to obtain the coordinates of the unknown node, it is obvious that this ignores the gradient difference between the unknown node and the beacon node, and it is unrealistic. The coordinate parameters of the spatial position of the 3D sensor network space are increased compared with the 2D plane, including three directions of x, y, and z. Increasing the gradient of Z axis can make more accurate calculation. Assuming that the coordinate positions of the beacon a node of the N known position coordinates are respectively x i = [ x i y i z i ] T (i=1,2,,N) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaWGPbaabeaakiabg2da9iaacUfafaqabeqadaaabaGaamiE amaaBaaaleaacaWGPbaabeaaaOqaaiaadMhadaWgaaWcbaGaamyAaa qabaaakeaacaWG6bWaaSbaaSqaaiaadMgaaeqaaaaakiaac2fadaah aaWcbeqaaiaadsfaaaGccaGGOaGaamyAaiabg2da9iaaigdacaGGSa GaaGOmaiaacYcacqWIVlctcaGGSaGaamOtaiaacMcaaaa@4BFB@ . At the same time, the location coordinates of the unknown node to be located in the area is assumed to be x=[ x y z ] T MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEaiaab2 dacaqGBbqbaeqabeWaaaqaaiaadIhaaeaacaWG5baabaGaamOEaaaa caqGDbWaaWbaaSqabeaacaWGubaaaaaa@3D83@ . (Figure 1)
Figure 1: Schematic diagram of unknown node and beacon node model
To estimate the location of an unknown node, the Time of Arrival (TOA) of the signal is measured to represent the spatial distance between the unknown node and the beacon nodes. However, in the non-synchronous sensor network, the node timing clock module is not synchronized due to environmental impact and other reasons, resulting in the node observation time being inconsistent with the actual real time. In order to improve the accuracy of the measurement results, this paper uses the drift and deviation clock timing model to represent the relationship between observation time and actual real time. ω MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdChaaa@37C2@ and θ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AB@ respectively represent the drift rate and the deviation amount of the clock, and represent the time change rate and the time difference between the observation time and the actual real time on the unknown node. The observation time T and the actual real time t of the unknown node are expressed as a relation:
T=ωt+θ       (1) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaiabg2 da9iabeM8a3jaadshacqGHRaWkcqaH4oqCcaqGGaGaaeiiaiaabcca caqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabgdacaqGPaaaaa@43B2@
The clock drift rate and the deviation amount of the beacon nodes i are assumed to be ω i ,  θ i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadMgaaeqaaOGaaiilaiaabccacqaH4oqCdaWgaaWcbaGa amyAaaqabaaaaa@3D09@ . According to the same principle, the observation time and the real time of the beacon nodes i also have the same relationship as equation (1). In this model, it is assumed that the clock parameters of the beacon nodes, including the drift rate ω i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadMgaaeqaaaaa@38DC@ and the deviation amount θ i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUde3aaS baaSqaaiaadMgaaeqaaaaa@38C5@ are calibrated, and are known parameters, while the unknown node clock parameters ω MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdChaaa@37C2@ and θ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AB@ are unknown parameters.

The unknown node sends a signal to the beacon nodes i at T 1,i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaaIXaGaaiilaiaadMgaaeqaaaaa@3953@ time (the observation time of the unknown node clock at T 1,i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaaIXaGaaiilaiaadMgaaeqaaaaa@3953@ ) and the beacon nodes i receive the signal of the unknown node at R 1,i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaaIXaGaaiilaiaadMgaaeqaaaaa@3951@ time (the observation time of the beacon nodes i clock at R 1,i MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaaIXaGaaiilaiaadMgaaeqaaaaa@3952@ ), According to equation (1) and the signal transmission process has the following relationship:
R 1,i ω i T 1,i ω + θ ω θ i ω i = t i + m i      (2) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGsbWaaSbaaSqaaiaaigdacaGGSaGaamyAaaqabaaakeaacqaHjpWD daWgaaWcbaGaamyAaaqabaaaaOGaeyOeI0YaaSaaaeaacaWGubWaaS baaSqaaiaaigdacaGGSaGaamyAaaqabaaakeaacqaHjpWDaaGaey4k aSYaaSaaaeaacqaH4oqCaeaacqaHjpWDaaGaeyOeI0YaaSaaaeaacq aH4oqCdaWgaaWcbaGaamyAaaqabaaakeaacqaHjpWDdaWgaaWcbaGa amyAaaqabaaaaOGaeyypa0JaamiDamaaBaaaleaacaWGPbaabeaaki abgUcaRiaad2gadaWgaaWcbaGaamyAaaqabaGccaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabIcacaqGYaGaaeykaaaa@5921@
In equation (2) t i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaBa aaleaacaWGPbaabeaaaaa@3808@ , is the Signal arrival time between nodes t i = d i /c,  d i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaBa aaleaacaWGPbaabeaakiabg2da9iaadsgadaWgaaWcbaGaamyAaaqa baGccaGGVaGaam4yaiaacYcacaqGGaGaamizamaaBaaaleaacaWGPb aabeaaaaa@4016@ is the spatial distance between the unknown node and the beacon nodes i, c is the propagation speed of the signal, c=3× 10 8 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiabg2 da9iaaiodacqGHxdaTcaaIXaGaaGimamaaCaaaleqabaGaaGioaaaa aaa@3D1B@ m/s m i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaWGPbaabeaaaaa@3801@ represents the time measurement noise between the unknown node and the beacon nodes i. It can be assumed that m i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaWGPbaabeaaaaa@3801@ obeys the Gaussian distribution with a mean of zero, and a variance of δ i,m 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aa0 baaSqaaiaadMgacaGGSaGaamyBaaqaaiaaikdaaaaaaa@3B13@ , which is denoted as m i N(0, δ i,m 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaWGPbaabeaakiabgIGiolaad6eacaGGOaGaaGimaiaacYca cqaH0oazdaqhaaWcbaGaamyAaiaacYcacaWGTbaabaGaaGOmaaaaki aacMcaaaa@424D@ .

When the beacon nodes i receive the signal of the unknown node, the beacon nodes i send a signal to the unknown node at T 2,i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaaIYaGaaiilaiaadMgaaeqaaaaa@3954@ time ( T 2,i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivamaaBa aaleaacaaIYaGaaiilaiaadMgaaeqaaaaa@3954@ is the observation time of the beacon nodes i clock), and the unknown node at R 2,i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaaIYaGaaiilaiaadMgaaeqaaaaa@3952@ time ( R 2,i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaaIYaGaaiilaiaadMgaaeqaaaaa@3952@ is the observation time of the unknown node clock) receives the signal of the beacon nodes, a similar derivation process of the equation (2), there is:
R 2,i ω T 2,i ω i + θ i ω i θ ω = t i + n i       (3) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGsbWaaSbaaSqaaiaaikdacaGGSaGaamyAaaqabaaakeaacqaHjpWD aaGaeyOeI0YaaSaaaeaacaWGubWaaSbaaSqaaiaaikdacaGGSaGaam yAaaqabaaakeaacqaHjpWDdaWgaaWcbaGaamyAaaqabaaaaOGaey4k aSYaaSaaaeaacqaH4oqCdaWgaaWcbaGaamyAaaqabaaakeaacqaHjp WDdaWgaaWcbaGaamyAaaqabaaaaOGaeyOeI0YaaSaaaeaacqaH4oqC aeaacqaHjpWDaaGaeyypa0JaamiDamaaBaaaleaacaWGPbaabeaaki abgUcaRiaad6gadaWgaaWcbaGaamyAaaqabaGccaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGOaGaae4maiaabMcaaaa@59C8@
In the formula, the definition of t i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaBa aaleaacaWGPbaabeaaaaa@3808@ is the same as equation (2), and n i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGPbaabeaaaaa@3802@ is also the noise part. It is assumed that n i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGPbaabeaaaaa@3802@ obeys the mean of zero, and the Gaussian distribution of variance δ i,n 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aa0 baaSqaaiaadMgacaGGSaGaamOBaaqaaiaaikdaaaaaaa@3B14@ is , recorded as n i N(0, δ i,n 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGPbaabeaakiabgIGiolaad6eacaGGOaGaaGimaiaacYca cqaH0oazdaqhaaWcbaGaamyAaiaacYcacaWGUbaabaGaaGOmaaaaki aacMcaaaa@424F@ .

It can be seen from equations (2) and (3) that the clock parameters and the position parameters are intertwined, and the clock parameter estimation and positioning have commonality in time characteristics. For this reason, the clock parameter estimation and the position parameters can be jointly estimated, and carried out to realize both the clock parameters estimation and and the position estimation. Since the positional parameter in the 3D sensor network space has more z-direction than the 2D plane, the method of estimating the position coordinates of the unknown node only by measuring the arrival time of the signal between the unknown node and the beacon nodes, and the measurement result is not accurate. So the (RSS) positioning method is added to measure the signal strength of unknown node and beacon nodes. Express the RSS measurement value between the unknown node and the beacon node i as. p i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaWGPbaabeaaaaa@3804@ According to the logarithmic attenuation model of RSS ranging, there is the following RSS relation expression:
p i = p 0 10βlg d i + ε i          (4) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaWGPbaabeaakiabg2da9iaadchadaWgaaWcbaGaaGimaaqa baGccqGHsislcaaIXaGaaGimaiabek7aIjGacYgacaGGNbGaamizam aaBaaaleaacaWGPbaabeaakiabgUcaRiabew7aLnaaBaaaleaacaWG PbaabeaakiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabIcacaqG0aGaaeykaaaa@4E5C@
In equation (4), β MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3796@ is the path attenuation index, The β MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3796@ value can be obtained through experimental analysis in advance, and the value varies with the change of the propagation medium, generally between 2 and 5. p o MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaWGVbaabeaaaaa@380A@ is the signal emission intensity of the unknown node, which is related to the antenna gain of the node and the battery supply. p i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaWGPbaabeaaaaa@3804@ is the signal reception strength of the beacon nodes i. d i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGPbaabeaaaaa@37F8@ is the spatial linear distance between the unknown node to be located and the beacon nodes i, d i = x x i ,  ε i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGPbaabeaakiabg2da9maafeaabaGaaCiEaiabgkHiTmaa fiaabaGaaCiEamaaBaaaleaacaWGPbaabeaaaOGaayPcSdaacaGLjW oacaGGSaGaaeiiaiabew7aLnaaBaaaleaacaWGPbaabeaaaaa@445F@ represents the measurement noise between the unknown node and the beacon nodes i, and it can be assumed that ϕ i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaS baaSqaaiaadMgaaeqaaaaa@38D7@ obeys the Gaussian distribution with a mean of zero, and variance is δ i,ε 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aa0 baaSqaaiaadMgacaGGSaGaeqyTdugabaGaaGOmaaaaaaa@3BC8@ , which is recorded as ε i N(0, δ i,ε 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgaaeqaaOGaeyicI4SaamOtaiaacIcacaaIWaGaaiil aiabes7aKnaaDaaaleaacaWGPbGaaiilaiabew7aLbqaaiaaikdaaa GccaGGPaaaaa@43B7@ .

To estimate the unknown node position coordinate x MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEaaaa@36F6@ with TOA and RSS hybrid measurements as known values, and the maximum likelihood (ML) estimates by optimizing the following expressions:
arg x min i=1 N ( 1 δ i,m 2 r i,m 2 + 1 δ i,n 2 r i,n 2 + 1 δ i,ε 2 r i,ε 2 )       (5) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGHbGaaiOCaiaacEgaaSqaaiaahIhaaeqaaOGaciyBaiaacMgacaGG UbWaaabCaeaacaGGOaWaaSaaaeaacaaIXaaabaGaeqiTdq2aa0baaS qaaiaadMgacaGGSaGaamyBaaqaaiaaikdaaaaaaaqaaiaadMgacqGH 9aqpcaaIXaaabaGaamOtaaqdcqGHris5aOGaamOCamaaDaaaleaaca WGPbGaaiilaiaad2gaaeaacaaIYaaaaOGaey4kaSYaaSaaaeaacaaI XaaabaGaeqiTdq2aa0baaSqaaiaadMgacaGGSaGaamOBaaqaaiaaik daaaaaaOGaamOCamaaDaaaleaacaWGPbGaaiilaiaad6gaaeaacaaI YaaaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaeqiTdq2aa0baaSqaai aadMgacaGGSaGaeqyTdugabaGaaGOmaaaaaaGccaWGYbWaa0baaSqa aiaadMgacaGGSaGaeqyTdugabaGaaGOmaaaakiaacMcacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabwdacaqG Paaaaa@6CE0@
In equation (5), r i,m ,  r i,n ,  r i,ε MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGPbGaaiilaiaad2gaaeqaaOGaaiilaiaabccacaWGYbWa aSbaaSqaaiaadMgacaGGSaGaamOBaaqabaGccaGGSaGaaeiiaiaadk hadaWgaaWcbaGaamyAaiaacYcacqaH1oqzaeqaaaaa@447E@ respectively represent the error items of the two arrival time measurements and the wireless received signal strength measurement. r i,m = R 1,i ω i T 1,i ω + θ ω θ i ω i t i ,  r i,n = R 2,i ω T 2,i ω i + θ i ω i θ ω t i , MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGPbGaaiilaiaad2gaaeqaaOGaeyypa0ZaaSaaaeaacaWG sbWaaSbaaSqaaiaaigdacaGGSaGaamyAaaqabaaakeaacqaHjpWDda WgaaWcbaGaamyAaaqabaaaaOGaeyOeI0YaaSaaaeaacaWGubWaaSba aSqaaiaaigdacaGGSaGaamyAaaqabaaakeaacqaHjpWDaaGaey4kaS YaaSaaaeaacqaH4oqCaeaacqaHjpWDaaGaeyOeI0YaaSaaaeaacqaH 4oqCdaWgaaWcbaGaamyAaaqabaaakeaacqaHjpWDdaWgaaWcbaGaam yAaaqabaaaaOGaeyOeI0IaamiDamaaBaaaleaacaWGPbaabeaakiaa cYcacaqGGaGaamOCamaaBaaaleaacaWGPbGaaiilaiaad6gaaeqaaO Gaeyypa0ZaaSaaaeaacaWGsbWaaSbaaSqaaiaaikdacaGGSaGaamyA aaqabaaakeaacqaHjpWDaaGaeyOeI0YaaSaaaeaacaWGubWaaSbaaS qaaiaaikdacaGGSaGaamyAaaqabaaakeaacqaHjpWDdaWgaaWcbaGa amyAaaqabaaaaOGaey4kaSYaaSaaaeaacqaH4oqCdaWgaaWcbaGaam yAaaqabaaakeaacqaHjpWDdaWgaaWcbaGaamyAaaqabaaaaOGaeyOe I0YaaSaaaeaacqaH4oqCaeaacqaHjpWDaaGaeyOeI0IaamiDamaaBa aaleaacaWGPbaabeaakiaacYcaaaa@7741@ r i,ε = p i p o +10βlg d i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGPbGaaiilaiabew7aLbqabaGccqGH9aqpcaWGWbWaaSba aSqaaiaadMgaaeqaaOGaeyOeI0IaamiCamaaBaaaleaacaWGVbaabe aakiabgUcaRiaaigdacaaIWaGaeqOSdiMaciiBaiaacEgacaWGKbWa aSbaaSqaaiaadMgaaeqaaaaa@486A@ The optimization function established by equation (5) is a nonlinear equation, and its numerical calculation method depends on the selection of the initial value. If the initial value selection is not suitable, it may fall into local optimum, which causes serious deviation of the positioning result. Therefore, this paper converts the nonlinear optimization equation is transformed into a linear equation, and the two-step calculation method is used to accurately calculate the position coordinates and clock parameters of the unknown node.
Linear Least Squares Estimation Method
Considering the approximate linearization of the measurement equation under small noise conditions, the nonlinear optimization model described in equation (5) is transformed into a linear equation. The designed calculation method is divided into two steps: the Unconstrained Linear Least Squares (ULLS) method and the Constrained Linear Least Squares (CLLS) method.
Unconstrained Least Squares Method
Adding equations (2) and (3) to eliminate common items:
R 2,i T 1,i ω + R 1,i T 2,i ω i =2 t i + m i + n i        (6) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGsbWaaSbaaSqaaiaaikdacaGGSaGaamyAaaqabaGccqGHsislcaWG ubWaaSbaaSqaaiaaigdacaGGSaGaamyAaaqabaaakeaacqaHjpWDaa Gaey4kaSYaaSaaaeaacaWGsbWaaSbaaSqaaiaaigdacaGGSaGaamyA aaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaikdacaGGSaGaamyAaa qabaaakeaacqaHjpWDdaWgaaWcbaGaamyAaaqabaaaaOGaeyypa0Ja aGOmaiaadshadaWgaaWcbaGaamyAaaqabaGccqGHRaWkcaWGTbWaaS baaSqaaiaadMgaaeqaaOGaey4kaSIaamOBamaaBaaaleaacaWGPbaa beaakiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGOaGaaeOnaiaabMcaaaa@5B82@
Since ω MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdChaaa@37C2@ is very close to 1, it can be assumed that ω= 1 1+δ , δ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdCNaey ypa0ZaaSaaaeaacaaIXaaabaGaaGymaiabgUcaRiabes7aKbaacaGG SaGaaeiiaiabes7aKbaa@3FCD@ is a variable close to zero, so equation (6) can be rewritten as:
μ i δ+ λ i =2 t i + m i + n i        (7) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaadMgaaeqaaOGaeqiTdqMaey4kaSIaeq4UdW2aaSbaaSqa aiaadMgaaeqaaOGaeyypa0JaaGOmaiaadshadaWgaaWcbaGaamyAaa qabaGccqGHRaWkcaWGTbWaaSbaaSqaaiaadMgaaeqaaOGaey4kaSIa amOBamaaBaaaleaacaWGPbaabeaakiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGOaGaae4naiaabMcaaaa@4E84@
In equation (7), λ i = R 1,i T 2,i ω i + R 2,i T 1,i ,  μ i = R 2,i T 1,i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaS baaSqaaiaadMgaaeqaaOGaeyypa0ZaaSaaaeaacaWGsbWaaSbaaSqa aiaaigdacaGGSaGaamyAaaqabaGccqGHsislcaWGubWaaSbaaSqaai aaikdacaGGSaGaamyAaaqabaaakeaacqaHjpWDdaWgaaWcbaGaamyA aaqabaaaaOGaey4kaSIaamOuamaaBaaaleaacaaIYaGaaiilaiaadM gaaeqaaOGaeyOeI0IaamivamaaBaaaleaacaaIXaGaaiilaiaadMga aeqaaOGaaiilaiaabccacqaH8oqBdaWgaaWcbaGaamyAaaqabaGccq GH9aqpcaWGsbWaaSbaaSqaaiaaikdacaGGSaGaamyAaaqabaGccqGH sislcaWGubWaaSbaaSqaaiaaigdacaGGSaGaamyAaaqabaaaaa@5A13@ it can be seeing that λ i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaS baaSqaaiaadMgaaeqaaaaa@38C3@ and μ i MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaadMgaaeqaaaaa@38C5@ are known parameters, since t i = d i /c MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaBa aaleaacaWGPbaabeaakiabg2da9iaadsgadaWgaaWcbaGaamyAaaqa baGccaGGVaGaam4yaaaa@3CC0@ and d i =||x x i || MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGPbaabeaakiabg2da9iaacYhacaGG8bGaaCiEaiabgkHi TiaahIhadaWgaaWcbaGaamyAaaqabaGccaGG8bGaaiiFaaaa@411B@ , square sides of the equation (7), ignoring the second high-order term, and sorting the expression:
8 x i x4 x T x+2 c 2 μ i λ i δ=4 x i T x i c 2 λ i 2 + c 2 d i ( m i + n i )    (8) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioaiaahI hadaWgaaWcbaGaamyAaaqabaGccaWH4bGaeyOeI0IaaGinaiaahIha daahaaWcbeqaaiaadsfaaaGccaWH4bGaey4kaSIaaGOmaiaadogada ahaaWcbeqaaiaaikdaaaGccqaH8oqBdaWgaaWcbaGaamyAaaqabaGc cqaH7oaBdaWgaaWcbaGaamyAaaqabaGccqaH0oazcqGH9aqpcaaI0a GaaCiEamaaDaaaleaacaWGPbaabaGaamivaaaakiaahIhadaWgaaWc baGaamyAaaqabaGccqGHsislcaWGJbWaaWbaaSqabeaacaaIYaaaaO Gaeq4UdW2aa0baaSqaaiaadMgaaeaacaaIYaaaaOGaey4kaSIaam4y amaaCaaaleqabaGaaGOmaaaakiaadsgadaWgaaWcbaGaamyAaaqaba GccaGGOaGaamyBamaaBaaaleaacaWGPbaabeaakiabgUcaRiaad6ga daWgaaWcbaGaamyAaaqabaGccaGGPaGaaeiiaiaabccacaqGGaGaae iiaiaabIcacaqG4aGaaeykaaaa@6680@
Equation (8) shows the transformed TOA measurement equation, i=1,2,,N MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabg2 da9iaaigdacaGGSaGaaGOmaiaacYcacqWIVlctcaGGSaGaamOtaaaa @3E31@ . Let z= [ x x T x δ ] T MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEaiabg2 da9maadmaabaqbaeqabeGaaaqaauaabeqabiaaaeaacaWH4baabaGa aCiEamaaCaaaleqabaGaamivaaaakiaahIhaaaaabaGaeqiTdqgaaa Gaay5waiaaw2faamaaCaaaleqabaGaamivaaaaaaa@40C8@ , equation (8) can be written as a linear matrix:
A 1 z= b 1 + γ 1      (9) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyqamaaBa aaleaacaaIXaaabeaakiaahQhacqGH9aqpcaWHIbWaaSbaaSqaaiaa igdaaeqaaOGaey4kaSIaaC4SdmaaBaaaleaacaaIXaaabeaakiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabMdacaqGPaaaaa@43E9@
In equation (9), the row vectors of matrix A 1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyqamaaBa aaleaacaaIXaaabeaaaaa@37A6@ of dimension N×5 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiabgE na0kaaiwdaaaa@399E@ is [ 8 x i 4 2 c 2 μ i λ i ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4wauaabe qabmaaaeaacaaI4aGaaCiEamaaBaaaleaacaWGPbaabeaaaOqaaiab gkHiTiaaisdaaeaacaaIYaGaam4yamaaCaaaleqabaGaaGOmaaaaki abeY7aTnaaBaaaleaacaWGPbaabeaakiabeU7aSnaaBaaaleaacaWG PbaabeaaaaGccaGGDbaaaa@449F@ , the column vector b 1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOyamaaBa aaleaacaaIXaaabeaaaaa@37C7@ and the row element values of γ 1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4SdmaaBa aaleaacaaIXaaabeaaaaa@381B@ of N row are [4 x i T x i c 2 λ i 2 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaais dacaWH4bWaa0baaSqaaiaadMgaaeaacaWGubaaaOGaaCiEamaaBaaa leaacaWGPbaabeaakiabgkHiTiaadogadaahaaWcbeqaaiaaikdaaa GccqaH7oaBdaqhaaWcbaGaamyAaaqaaiaaikdaaaGccaGGDbaaaa@43F4@ and [ c 2 d i ( m i + n i )] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaado gadaahaaWcbeqaaiaaikdaaaGccaWGKbWaaSbaaSqaaiaadMgaaeqa aOGaaiikaiaad2gadaWgaaWcbaGaamyAaaqabaGccqGHRaWkcaWGUb WaaSbaaSqaaiaadMgaaeqaaOGaaiykaiaac2faaaa@4205@ respectively. Equivalent transformation of the RSS measurement equation, that is, equivalent shift transformation of equation (4), rewriting it to:
d i = 10 p 0 p i + ε i 10β          (10) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGPbaabeaakiabg2da9iaaigdacaaIWaWaaWbaaSqabeaa daWcaaqaaiaadchadaWgaaadbaGaaGimaaqabaWccqGHsislcaWGWb WaaSbaaWqaaiaadMgaaeqaaSGaey4kaSIaeqyTdu2aaSbaaWqaaiaa dMgaaeqaaaWcbaGaaGymaiaaicdacqaHYoGyaaaaaOGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeik aiaabgdacaqGWaGaaeykaaaa@4EF1@
Considering that in the smaller noise range, the Taylor series expansion is applied to the right side of equation (10), ignore the high order term, and equation (10) is transformed into:
d i = τ i + τ i ln10 10β ε i       (11) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGPbaabeaakiabg2da9iabes8a0naaBaaaleaacaWGPbaa beaakiabgUcaRmaalaaabaGaeqiXdq3aaSbaaSqaaiaadMgaaeqaaO GaciiBaiaac6gacaaIXaGaaGimaaqaaiaaigdacaaIWaGaeqOSdiga aiabew7aLnaaBaaaleaacaWGPbaabeaakiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabIcacaqGXaGaaeymaiaabMcaaaa@4F97@
In equation (11), i=1,2,,N MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabg2 da9iaaigdacaGGSaGaaGOmaiaacYcacqWIVlctcaGGSaGaamOtaaaa @3E31@ , τ i = 10 p 0 p i 10β MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS baaSqaaiaadMgaaeqaaOGaeyypa0JaaGymaiaaicdadaahaaWcbeqa amaalaaabaGaamiCamaaBaaameaacaaIWaaabeaaliabgkHiTiaadc hadaWgaaadbaGaamyAaaqabaaaleaacaaIXaGaaGimaiabek7aIbaa aaaaaa@439B@ Square the two sides of the equation (11), omitting the high-order terms, it can get:
d i 2 = τ i 2 + τ i 2 ln10 5β ε i      (12) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaDa aaleaacaWGPbaabaGaaGOmaaaakiabg2da9iabes8a0naaDaaaleaa caWGPbaabaGaaGOmaaaakiabgUcaRmaalaaabaGaeqiXdq3aa0baaS qaaiaadMgaaeaacaaIYaaaaOGaciiBaiaac6gacaaIXaGaaGimaaqa aiaaiwdacqaHYoGyaaGaeqyTdu2aaSbaaSqaaiaadMgaaeqaaOGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeymaiaabkdacaqG Paaaaa@5076@
Since d i =||x x i || MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGPbaabeaakiabg2da9iaacYhacaGG8bGaaCiEaiabgkHi TiaahIhadaWgaaWcbaGaamyAaaqabaGccaGG8bGaaiiFaaaa@411B@ , there is an expression for expansion of equation (12):
2 x i T x+ x T x= x i T x i + τ i 2 + τ i 2 ln10 5β ε i        (13) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG OmaiaahIhadaWgaaWcbaGaamyAaaqabaGcdaahaaWcbeqaaiaadsfa aaGccaWH4bGaey4kaSIaaCiEamaaCaaaleqabaGaamivaaaakiaahI hacqGH9aqpcqGHsislcaWH4bWaaSbaaSqaaiaadMgaaeqaaOWaaWba aSqabeaacaWGubaaaOGaaCiEamaaBaaaleaacaWGPbaabeaakiabgU caRiabes8a0naaBaaaleaacaWGPbaabeaakmaaCaaaleqabaGaaGOm aaaakiabgUcaRmaalaaabaGaeqiXdq3aaSbaaSqaaiaadMgaaeqaaO WaaWbaaSqabeaacaaIYaaaaOGaciiBaiaac6gacaaIXaGaaGimaaqa aiaaiwdacqaHYoGyaaGaeqyTdu2aaSbaaSqaaiaadMgaaeqaaOGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqG XaGaae4maiaabMcaaaa@605B@
Equation (13) shows the transformed RSS measurement equation, i=1,2,,N MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabg2 da9iaaigdacaGGSaGaaGOmaiaacYcacqWIVlctcaGGSaGaamOtaaaa @3E31@ , and the same equation (13) can be written as a linear matrix form:
A 2 z= b 2 + γ 2       (14) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyqamaaBa aaleaacaaIYaaabeaakiaahQhacqGH9aqpcaWHIbWaaSbaaSqaaiaa ikdaaeqaaOGaey4kaSIaaC4SdmaaBaaaleaacaaIYaaabeaakiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqGXaGaaein aiaabMcaaaa@453E@
In equation (14), the row vector of matrix A 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyqamaaBa aaleaacaaIYaaabeaaaaa@37A7@ of dimension N×5 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiabgE na0kaaiwdaaaa@399E@ is [ 2 x i T 1 0 ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4wauaabe qabiaaaeaacqGHsislcaaIYaGaaCiEamaaBaaaleaacaWGPbaabeaa kmaaCaaaleqabaGaamivaaaaaOqaauaabeqabiaaaeaacaaIXaaaba GaaGimaaaaaaGaaiyxaaaa@3E22@ . The column vector b 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOyamaaBa aaleaacaaIYaaabeaaaaa@37C8@ and the row element values of γ 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4SdmaaBa aaleaacaaIYaaabeaaaaa@381C@ of N row are [ x i T x i + τ i 2 ], [ τ i 2 ln10 5β ε i ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiabgk HiTiaahIhadaqhaaWcbaGaamyAaaqaaiaadsfaaaGccaWH4bWaaSba aSqaaiaadMgaaeqaaOGaey4kaSIaeqiXdq3aa0baaSqaaiaadMgaae aacaaIYaaaaOGaaiyxaiaacYcacaqGGaGaai4wamaalaaabaGaeqiX dq3aa0baaSqaaiaadMgaaeaacaaIYaaaaOGaciiBaiaac6gacaaIXa GaaGimaaqaaiaaiwdacqaHYoGyaaGaeqyTdu2aaSbaaSqaaiaadMga aeqaaOGaaiyxaaaa@519B@ Establish a unified matrix form by combining equations (9) and (14):
Az=b+γ     (15) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyqaiaahQ hacqGH9aqpcaWHIbGaey4kaSIaaC4SdiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeikaiaabgdacaqG1aGaaeykaaaa@41C6@
In equation (15), A= [ A 1 T A 2 T ] T , b= [ b 1 T b 2 T ] T ,γ= [ γ 1 T γ 2 T ] T MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyqaiabg2 da9iaacUfafaqabeqacaaabaGaaCyqamaaDaaaleaacaaIXaaabaGa amivaaaaaOqaaiaahgeadaqhaaWcbaGaaGOmaaqaaiaadsfaaaaaaO GaaiyxamaaCaaaleqabaGaamivaaaakiaacYcacaqGGaGaaCOyaiab g2da9iaacUfafaqabeqacaaabaGaaCOyamaaDaaaleaacaaIXaaaba GaamivaaaaaOqaaiaahkgadaqhaaWcbaGaaGOmaaqaaiaadsfaaaaa aOGaaiyxamaaCaaaleqabaGaamivaaaakiaacYcacaaMb8UaaC4Sdi abg2da9iaacUfafaqabeqacaaabaGaaC4SdmaaDaaaleaacaaIXaaa baGaamivaaaaaOqaaiaaho7adaqhaaWcbaGaaGOmaaqaaiaadsfaaa aaaOGaaiyxamaaCaaaleqabaGaamivaaaaaaa@58C2@ and A R 2N×5 , b R 2N , v R 2N MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyqaiabgI GiolaadkfadaahaaWcbeqaaiaaikdacaWGobGaey41aqRaaGynaaaa kiaacYcacaqGGaGaaCOyaiabgIGiolaadkfadaahaaWcbeqaaiaaik dacaWGobaaaOGaaiilaiaabccacaWH2bGaeyicI4SaamOuamaaCaaa leqabaGaaGOmaiaad6eaaaaaaa@4A7E@ .

According to the principle of least squares, the estimated value of parameters z MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEaaaa@36F8@ is:
z= ( A T Σ γ 1 A) 1 A T Σ γ 1 b       (16) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEaiabg2 da9iaacIcacaWHbbWaaWbaaSqabeaacaWGubaaaOGaaC4OdmaaDaaa leaacqaHZoWzaeaacqGHsislcaaIXaaaaOGaaCyqaiaacMcadaahaa WcbeqaaiabgkHiTiaaigdaaaGccaWHbbWaaWbaaSqabeaacaWGubaa aOGaaC4OdmaaDaaaleaacqaHZoWzaeaacqGHsislcaaIXaaaaOGaaC OyaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG OaGaaeymaiaabAdacaqGPaaaaa@5142@
In equation (16), the matrix γ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyeIu+aaS baaSqaaiabeo7aNbqabaaaaa@396C@ with a dimension of 2N×2N MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaad6 eacqGHxdaTcaaIYaGaamOtaaaa@3B2A@ is a weight matrix, whose value is γ =Σ( γ T γ ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyeIu+aaS baaSqaaiabeo7aNbqabaGccqGH9aqpcqqHJoWudaqadaqaaiaaho7a daahaaWcbeqaaiaadsfaaaGccaWHZoaacaGLOaGaayzkaaaaaa@4117@ . Representing the variance of γ T γ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4SdmaaCa aaleqabaGaamivaaaakiaaho7aaaa@3983@ and its value is further expressed as:
γ =[ Σ( γ 1 T γ 1 ) Σ( γ 1 T γ 2 ) Σ( γ 2 T γ 1 ) Σ( γ 2 T γ 2 ) ]           (17) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyeIu+aaS baaSqaaiabeo7aNbqabaGccqGH9aqpdaWadaqaauaabeqaciaaaeaa cqqHJoWudaqadaqaaiaaho7adaqhaaWcbaGaaGymaaqaaiaadsfaaa GccaWHZoWaaSbaaSqaaiaaigdaaeqaaaGccaGLOaGaayzkaaaabaGa eu4Odm1aaeWaaeaacaWHZoWaa0baaSqaaiaaigdaaeaacaWGubaaaO GaaC4SdmaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaaaqaaiab fo6atnaabmaabaGaaC4SdmaaDaaaleaacaaIYaaabaGaamivaaaaki aaho7adaWgaaWcbaGaaGymaaqabaaakiaawIcacaGLPaaaaeaacqqH JoWudaqadaqaaiaaho7adaqhaaWcbaGaaGOmaaqaaiaadsfaaaGcca WHZoWaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaaaaaGaay5w aiaaw2faaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabgdacaqG3aGaaeyk aaaa@6764@
Because γ 1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4SdmaaBa aaleaacaaIXaaabeaaaaa@381B@ and γ 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4SdmaaBa aaleaacaaIYaaabeaaaaa@381C@ are not related, there are Σ( γ 1 T γ 2 ), Σ( γ 2 T γ 1 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4Odm1aae WaaeaacaWHZoWaa0baaSqaaiaaigdaaeaacaWGubaaaOGaaC4Sdmaa BaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaaiaacYcacaqGGaGaeu 4Odm1aaeWaaeaacaWHZoWaa0baaSqaaiaaikdaaeaacaWGubaaaOGa aC4SdmaaBaaaleaacaaIXaaabeaaaOGaayjkaiaawMcaaaaa@47D8@ equal to zero, will γ 1 ,  γ 2 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4SdmaaBa aaleaacaaIXaaabeaakiaacYcacaqGGaGaaC4SdmaaBaaaleaacaaI Yaaabeaaaaa@3B9F@ are substituted into each equation, the other values of are as follows γ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyeIu+aaS baaSqaaiabeo7aNbqabaaaaa@396C@
{ Σ( γ 1 T γ 1 )=diag{ τ i 4 4.7 β 2 δ i,ε 2 } Σ( γ 2 T γ 2 )F( ν 2 T ν 2 )=diag{ c 4 d i 2 ( δ i,m + δ i,n ) 2 }          (18) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaeaafa qabeGabaaabaGaeu4Odm1aaeWaaeaacaWHZoWaa0baaSqaaiaaigda aeaacaWGubaaaOGaaC4SdmaaBaaaleaacaaIXaaabeaaaOGaayjkai aawMcaaiabg2da9iaabsgacaqGPbGaaeyyaiaabEgadaGadaqaamaa laaabaGaeqiXdq3aa0baaSqaaiaadMgaaeaacaaI0aaaaaGcbaGaaG inaiaac6cacaaI3aGaeqOSdi2aaWbaaSqabeaacaaIYaaaaaaakiab es7aKnaaDaaaleaacaWGPbGaaiilaiabew7aLbqaaiaaikdaaaaaki aawUhacaGL9baaaeaacqqHJoWudaqadaqaaiaaho7adaqhaaWcbaGa aGOmaaqaaiaadsfaaaGccaWHZoWaaSbaaSqaaiaaikdaaeqaaaGcca GLOaGaayzkaaGaamOramaabmaabaGaeqyVd42aa0baaSqaaiaaikda aeaacaWGubaaaOGaeqyVd42aaSbaaSqaaiaaikdaaeqaaaGccaGLOa GaayzkaaGaeyypa0JaaeizaiaabMgacaqGHbGaae4zamaacmaabaGa am4yamaaCaaaleqabaGaaGinaaaakiaadsgadaqhaaWcbaGaamyAaa qaaiaaikdaaaGcdaqadaqaaiabes7aKnaaBaaaleaacaWGPbGaaiil aiaad2gaaeqaaOGaey4kaSIaeqiTdq2aaSbaaSqaaiaadMgacaGGSa GaamOBaaqabaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaa kiaawUhacaGL9baaaaaacaGL7baacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeymaiaabIda caqGPaaaaa@8595@
Assume the estimated error Δz MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaaC OEaaaa@385E@ of the parameter z MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEaaaa@36F8@ , the value is:
Δz= ( A T Σ γ 1 A) 1 A T Σ γ 1 γ      (19) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaaC OEaiabg2da9iaacIcacaWHbbWaaWbaaSqabeaacaWGubaaaOGaaC4O dmaaBaaaleaacqaHZoWzaeqaaOWaaWbaaSqabeaacqGHsislcaaIXa aaaOGaaCyqaiaacMcadaahaaWcbeqaaiabgkHiTiaaigdaaaGccaWH bbWaaWbaaSqabeaacaWGubaaaOGaaC4OdmaaBaaaleaacqaHZoWzae qaaOWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaaC4SdiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqGXaGaaeyoaiaabM caaaa@52C8@
The variance of the estimation error Δz MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaaC OEaaaa@385E@ is expressed as:
cov(Δz)= ( A T γ 1 A) 1       (20) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGG2bGaaiikaiabfs5aejaahQhacaGGPaGaeyypa0Jaaiikaiaa hgeadaahaaWcbeqaaiaadsfaaaGccqGHris5daqhaaWcbaGaeq4SdC gabaGaeyOeI0IaaGymaaaakiaahgeacaGGPaWaaWbaaSqabeaacqGH sislcaaIXaaaaOGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeikaiaabkdacaqGWaGaaeykaaaa@4F2A@
Extracting z(1:3) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEaiaacI cacaaIXaGaaiOoaiaaiodacaGGPaaaaa@3A87@ slave from the parameter z MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEaaaa@36F8@ is the position coordinates of the unknown node being located. The above solution process is not considered the mutually constrained relationship between the fourth element and the first three element values in z= [ x x T x δ ] T MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEaiabg2 da9maadmaabaqbaeqabeGaaaqaauaabeqabiaaaeaacaWH4baabaGa aCiEamaaCaaaleqabaGaamivaaaakiaahIhaaaaabaGaeqiTdqgaaa Gaay5waiaaw2faamaaCaaaleqabaGaamivaaaaaaa@40C8@ , so the calculation method is called the unconstrained least squares (ULLS) method of the RSS and TOA hybrid positioning problem. Equation (16) obtains an approximate estimate of the coordinates of the unknown node being located, and the exact value of the unknown coordinates of the unknown node can be calculated using the mutual constraint relation between the vector z= [ x x T x δ ] T MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEaiabg2 da9maadmaabaqbaeqabeGaaaqaauaabeqabiaaaeaacaWH4baabaGa aCiEamaaCaaaleqabaGaamivaaaakiaahIhaaaaabaGaeqiTdqgaaa Gaay5waiaaw2faamaaCaaaleqabaGaamivaaaaaaa@40C8@ elements. Since the fifth element and the first four elements are not constrained, this article only selects the constraint relationship of the first four elements. The clock drift rate ω MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdChaaa@37C3@ can be further solved by ω= 1 1+δ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdCNaey ypa0ZaaSaaaeaacaaIXaaabaGaaGymaiabgUcaRiabes7aKbaaaaa@3CD5@ , the equation (2) subtract the equation (3) can further calculate the clock offset θ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AB@ .
Constrained Least Squares
The results obtained by the ULLS method are further optimized by using the constraint relationship of each element value in the vector z MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEaaaa@36F8@ . Assume that the true value of the coordinate position of the unknown node is x o = [ x o y o z o ] T MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaCa aaleqabaGaam4Baaaakiabg2da9iaacUfafaqabeqadaaabaGaamiE amaaCaaaleqabaGaam4BaaaaaOqaaiaadMhadaahaaWcbeqaaiaad+ gaaaaakeaacaWG6bWaaWbaaSqabeaacaWGVbaaaaaakiaac2fadaah aaWcbeqaaiaadsfaaaaaaa@4277@ , there are the following relationships:
{ x o2 = [z(1)+Δz(1)] 2 z (1) 2 +2z(1)Δz(1) y o2 = [z(2)+Δz(2)] 2 z (2) 2 +2z(2)Δz(2) z o2 = [z(3)+Δz(3)] 2 z (3) 2 +2z(3)Δz(3) x o2 + y o2 + z o2 =z(4)+Δz(4)           (21) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaeaafa qabeabbaaaaeaacaWG4bWaaWbaaSqabeaacaWGVbGaaGOmaaaakiab g2da9iaacUfacaWH6bGaaiikaiaaigdacaGGPaGaey4kaSIaeuiLdq KaaCOEaiaacIcacaaIXaGaaiykaiaac2fadaahaaWcbeqaaiaaikda aaGccqGHijYUcaWH6bGaaiikaiaaigdacaGGPaWaaWbaaSqabeaaca aIYaaaaOGaey4kaSIaaGOmaiaahQhacaGGOaGaaGymaiaacMcacqqH uoarcaWH6bGaaiikaiaaigdacaGGPaaabaGaamyEamaaCaaaleqaba Gaam4BaiaaikdaaaGccqGH9aqpcaGGBbGaaCOEaiaacIcacaaIYaGa aiykaiabgUcaRiabfs5aejaahQhacaGGOaGaaGOmaiaacMcacaGGDb WaaWbaaSqabeaacaaIYaaaaOGaeyisISRaaCOEaiaacIcacaaIYaGa aiykamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaikdacaWH6bGaai ikaiaaikdacaGGPaGaeuiLdqKaaCOEaiaacIcacaaIYaGaaiykaaqa aiaadQhadaahaaWcbeqaaiaad+gacaaIYaaaaOGaeyypa0Jaai4wai aahQhacaGGOaGaaG4maiaacMcacqGHRaWkcqqHuoarcaWH6bGaaiik aiaaiodacaGGPaGaaiyxamaaCaaaleqabaGaaGOmaaaakiabgIKi7k aahQhacaGGOaGaaG4maiaacMcadaahaaWcbeqaaiaaikdaaaGccqGH RaWkcaaIYaGaaCOEaiaacIcacaaIZaGaaiykaiabfs5aejaahQhaca GGOaGaaG4maiaacMcaaeaacaWG4bWaaWbaaSqabeaacaWGVbGaaGOm aaaakiabgUcaRiaadMhadaahaaWcbeqaaiaad+gacaaIYaaaaOGaey 4kaSIaamOEamaaCaaaleqabaGaam4BaiaaikdaaaGccqGH9aqpcaWH 6bGaaiikaiaaisdacaGGPaGaey4kaSIaeuiLdqKaaCOEaiaacIcaca aI0aGaaiykaaaaaiaawUhaaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeOmaiaabg dacaqGPaaaaa@AE45@
In equation (21), z(k), Δz(k) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEaiaacI cacaWGRbGaaiykaiaacYcacaqGGaGaeuiLdqKaaCOEaiaacIcacaWG RbGaaiykaaaa@3F46@ represent the k-th element of the vector z, Δz, k=1,2,3,4 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOEaiaacY cacaqGGaGaeuiLdqKaaCOEaiaacYcacaqGGaGaam4Aaiabg2da9iaa igdacaGGSaGaaGOmaiaacYcacaaIZaGaaiilaiaaisdaaaa@42FF@ . Express equation (21) as a linear matrix form:
G μ o =h+η     (22) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4raiaahY 7adaahaaWcbeqaaiaad+gaaaGccqGH9aqpcaWHObGaey4kaSIaaC4T diaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabkdacaqGYa Gaaeykaaaa@4344@
In equation h= [ z (1) 2 z (2) 2 z (3) 2 z (4) 2 ] T ,  μ o = [ x o2 y o2 z o2 ] T , η=LΔz MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiAaiabg2 da9iaacUfafaqabeqaeaaaaeaacaWH6bGaaiikaiaaigdacaGGPaWa aWbaaSqabeaacaaIYaaaaaGcbaGaaCOEaiaacIcacaaIYaGaaiykam aaCaaaleqabaGaaGOmaaaaaOqaaiaahQhacaGGOaGaaG4maiaacMca daahaaWcbeqaaiaaikdaaaaakeaacaWH6bGaaiikaiaaisdacaGGPa WaaWbaaSqabeaacaaIYaaaaaaakiaac2fadaahaaWcbeqaaiaadsfa aaGccaGGSaGaaeiiaiaahY7adaahaaWcbeqaaiaad+gaaaGccqGH9a qpcaGGBbqbaeqabeWaaaqaaiaadIhadaahaaWcbeqaaiaad+gacaaI YaaaaaGcbaGaamyEamaaCaaaleqabaGaam4Baiaaikdaaaaakeaaca WG6bWaaWbaaSqabeaacaWGVbGaaGOmaaaaaaGccaGGDbWaaWbaaSqa beaacaWGubaaaOGaaiilaiaabccacaWH3oGaeyypa0JaaCitaiabfs 5aejaadQhaaaa@622B@ ,
L=diag{ 2z(1) 2z(2) 2z(3) 1 } MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCitaiabg2 da9iaabsgacaqGPbGaaeyyaiaabEgacaGG7bqbaeqabeabaaaabaGa aGOmaiaahQhacaGGOaGaaGymaiaacMcaaeaacaaIYaGaaCOEaiaacI cacaaIYaGaaiykaaqaaiaaikdacaWH6bGaaiikaiaaiodacaGGPaaa baGaaGymaaaacaGG9baaaa@49B9@
G= [ 1 0 0 1 0 1 0 1 0 0 1 1 ] T MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4raiabg2 da9maadmaabaqbaeqabmabaaaabaGaaGymaaqaaiaaicdaaeaacaaI WaaabaGaaGymaaqaaiaaicdaaeaacaaIXaaabaGaaGimaaqaaiaaig daaeaacaaIWaaabaGaaGimaaqaaiaaigdaaeaacaaIXaaaaaGaay5w aiaaw2faamaaCaaaleqabaGaamivaaaaaaa@439C@
According to the principle of linear least squares, the unbiased estimate of the vector μ o MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiVdmaaCa aaleqabaGaam4Baaaaaaa@385E@ is:
μ o = ( G 1 Σ η 1 G) 1 G Σ η 1 h     (23) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiVdmaaCa aaleqabaGaam4Baaaakiabg2da9iaacIcacaWHhbWaaWbaaSqabeaa cqGHsislcaaIXaaaaOGaaC4OdmaaDaaaleaacqaH3oaAaeaacqGHsi slcaaIXaaaaOGaaC4raiaacMcadaahaaWcbeqaaiabgkHiTiaaigda aaGccaWHhbGaaC4OdmaaDaaaleaacqaH3oaAaeaacqGHsislcaaIXa aaaOGaaCiAaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaa bkdacaqGZaGaaeykaaaa@514B@
The weight matrix Σ η MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4OdmaaBa aaleaacqaH3oaAaeqaaaaa@38FC@ value of the dimension 4×4 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiabgE na0kaaisdaaaa@3988@ in equation (23) is:
Σ η =Σ( η T η)= L T cov(Δz)L= L T ( A T γ 1 A) 1 L    (24) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4OdmaaBa aaleaacqaH3oaAaeqaaOGaeyypa0JaaC4OdiaacIcacaWH3oWaaWba aSqabeaacaWGubaaaOGaaC4TdiaacMcacqGH9aqpcaWHmbWaaWbaaS qabeaacaWGubaaaOGaci4yaiaac+gacaGG2bGaaiikaiabfs5aejaa hQhacaGGPaGaaCitaiabg2da9iaadYeadaahaaWcbeqaaiaadsfaaa GccaGGOaGaaCyqamaaCaaaleqabaGaamivaaaakiabggHiLpaaDaaa leaacqaHZoWzaeaacqGHsislcaaIXaaaaOGaaCyqaiaacMcadaahaa WcbeqaaiabgkHiTiaaigdaaaGccaWGmbGaaeiiaiaabccacaqGGaGa aeiiaiaabIcacaqGYaGaaeinaiaabMcaaaa@5E8F@
Due to then the exact estimated value of the coordinates of the location node being located is
x=sign{ diag{ z( 1:3 ) } } μ o       (25) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEaiabg2 da9iaabohacaqGPbGaae4zaiaab6gadaGadaqaaiaabsgacaqGPbGa aeyyaiaabEgadaGadaqaaiaahQhadaqadaqaaiaaigdacaGG6aGaaG 4maaGaayjkaiaawMcaaaGaay5Eaiaaw2haaaGaay5Eaiaaw2haamaa kaaabaGaaCiVdmaaCaaaleqabaGaam4BaaaaaeqaaOGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabkdacaqG1aGaaeyk aaaa@5197@
The calculation process represented by the equation (25) takes into account the mutual constraint relationship between the elements in the parameter z, and obtains accurate positioning results. This calculation method is called the constrained linear least squares (CLLS) method of the asynchronous sensor network positioning problem of the RSS and TOA hybrid measurement methods.
Model of the Cramer-Rao Lower Bound (CRLB)
The CRLB value provides the lower bound of the error variance for the unbiased estimation of the model to be estimated, and the unknown parameter of the model is assumed to be ρ=[ x ω θ ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyWdiabg2 da9maadmaabaqbaeqabeWaaaqaaiaahIhaaeaacqaHjpWDaeaacqaH 4oqCaaaacaGLBbGaayzxaaaaaa@3ECD@ . Assuming that the estimated error variance of unknown parameters ρ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyWdaaa@3742@ is cov(ρ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGG2bGaaiikaiaahg8acaGGPaaaaa@3B71@ . Then according to Cramer-Rao lower bound theory, there is a relationship cov(ρ) F 1 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGG2bGaaiikaiaahg8acaGGPaGaeyyzImRaaCOramaaCaaaleqa baGaeyOeI0IaaGymaaaaaaa@3FDB@ , among them F MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOraaaa@36C4@ is the representation of the FIM (Fisher Information Matrix) of parameter to be evaluated ρ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyWdaaa@3742@ , the matrix F MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOraaaa@36C4@ expressed as:
F= 2 lnP(p|ρ) ρ T ρ       (26) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOraiabg2 da9iabgkHiTmaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGa ciiBaiaac6gacaWGqbGaaiikaiaabchacaGG8bGaaCyWdiaacMcaae aacqGHciITcaWHbpWaaWbaaSqabeaacaWGubaaaOGaaCyWdaaacaqG GaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeOmaiaabA dacaqGPaaaaa@4E19@
The model is solved by RSS, TOA measures parameters, and maximizing the matrix F MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOraaaa@36C4@ . The probability density function lnP(p|ρ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6 gacaWGqbGaaiikaiaabchacaGG8bGaaCyWdiaacMcaaaa@3D47@ can be expressed as:
P(p|ρ)= i=1 N 1 2π δ i,m exp( r i,m 2 2 δ i,m 2 ) 1 2π δ i,n exp( r i,n 2 2 δ i,n 2 ) 1 2π δ i,ε exp( r i,ε 2 2 δ i,ε 2 )   (27) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaacI cacaqGWbGaaeiFaiaahg8acaGGPaGaeyypa0ZaaebCaeaadaWcaaqa aiaaigdaaeaadaGcaaqaaiaaikdacqaHapaCaSqabaGccqaH0oazda WgaaWcbaGaamyAaiaacYcacaWGTbaabeaaaaaabaGaamyAaiabg2da 9iaaigdaaeaacaWGobaaniabg+GivdGcciGGLbGaaiiEaiaacchaca GGOaGaeyOeI0YaaSaaaeaacaWGYbWaa0baaSqaaiaadMgacaGGSaGa amyBaaqaaiaaikdaaaaakeaacaaIYaGaeqiTdq2aa0baaSqaaiaadM gacaGGSaGaamyBaaqaaiaaikdaaaaaaOGaaiykamaalaaabaGaaGym aaqaamaakaaabaGaaGOmaiabec8aWbWcbeaakiabes7aKnaaBaaale aacaWGPbGaaiilaiaad6gaaeqaaaaakiGacwgacaGG4bGaaiiCaiaa cIcacqGHsisldaWcaaqaaiaadkhadaqhaaWcbaGaamyAaiaacYcaca WGUbaabaGaaGOmaaaaaOqaaiaaikdacqaH0oazdaqhaaWcbaGaamyA aiaacYcacaWGUbaabaGaaGOmaaaaaaGccaGGPaWaaSaaaeaacaaIXa aabaWaaOaaaeaacaaIYaGaeqiWdahaleqaaOGaeqiTdq2aaSbaaSqa aiaadMgacaGGSaGaeqyTdugabeaaaaGcciGGLbGaaiiEaiaacchaca GGOaGaeyOeI0YaaSaaaeaacaWGYbWaa0baaSqaaiaadMgacaGGSaGa eqyTdugabaGaaGOmaaaaaOqaaiaaikdacqaH0oazdaqhaaWcbaGaam yAaiaacYcacqaH1oqzaeaacaaIYaaaaaaakiaacMcacaqGGaGaaeii aiaabccacaqGOaGaaeOmaiaabEdacaqGPaaaaa@8F0B@
Take the logarithm of both sides of equation (27) and define the vector r m ,  r n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOCamaaBa aaleaacaWGTbaabeaakiaacYcacaqGGaGaaCOCamaaBaaaleaacaWG Ubaabeaaaaa@3B85@ , r ε , r m =[ r 1,m r 2,m r N,m ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOCamaaBa aaleaacqaH1oqzaeqaaOGaaiilaiaaykW7caWHYbWaaSbaaSqaaiaa d2gaaeqaaOGaeyypa0Jaai4wauaabeqabqaaaaqaaiaadkhadaWgaa WcbaGaaGymaiaacYcacaWGTbaabeaaaOqaaiaadkhadaWgaaWcbaGa aGOmaiaacYcacaWGTbaabeaaaOqaaiabl+UimbqaaiaadkhadaWgaa WcbaGaamOtaiaacYcacaWGTbaabeaaaaGccaGGDbaaaa@4CA7@ , r n =[ r 1,n r 2,n r N,m ], r ε =[ r 1,ε r 2,ε r N,ε ] MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOCamaaBa aaleaacaWGUbaabeaakiabg2da9iaacUfafaqabeqaeaaaaeaacaWG YbWaaSbaaSqaaiaaigdacaGGSaGaamOBaaqabaaakeaacaWGYbWaaS baaSqaaiaaikdacaGGSaGaamOBaaqabaaakeaacqWIVlctaeaacaWG YbWaaSbaaSqaaiaad6eacaGGSaGaamyBaaqabaaaaOGaaiyxaiaacY cacaaMe8UaaCOCamaaBaaaleaacqaH1oqzaeqaaOGaeyypa0Jaai4w auaabeqabqaaaaqaaiaadkhadaWgaaWcbaGaaGymaiaacYcacqaH1o qzaeqaaaGcbaGaamOCamaaBaaaleaacaaIYaGaaiilaiabew7aLbqa baaakeaacqWIVlctaeaacaWGYbWaaSbaaSqaaiaad6eacaGGSaGaeq yTdugabeaaaaGccaGGDbaaaa@5E47@ with the following expression:
lnP( p|ρ )=e( r m T Σ m 1 r m + r n T Σ n 1 r n + r ε T Σ ε 1 r ε )    (28) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6 gacaWGqbWaaeWaaeaacaWHWbGaaiiFaiaahg8aaiaawIcacaGLPaaa cqGH9aqpcaWGLbGaeyOeI0IaaiikaiaahkhadaqhaaWcbaGaamyBaa qaaiaadsfaaaGccaWHJoWaa0baaSqaaiaad2gaaeaacqGHsislcaaI XaaaaOGaaCOCamaaBaaaleaacaWGTbaabeaakiabgUcaRiaahkhada qhaaWcbaGaamOBaaqaaiaadsfaaaGccaWHJoWaa0baaSqaaiaad6ga aeaacqGHsislcaaIXaaaaOGaaCOCamaaBaaaleaacaWGUbaabeaaki abgUcaRiaahkhadaqhaaWcbaGaeqyTdugabaGaamivaaaakiaaho6a daqhaaWcbaGaeqyTdugabaGaeyOeI0IaaGymaaaakiaahkhadaWgaa WcbaGaeqyTdugabeaakiaacMcacaqGGaGaaeiiaiaabccacaqGGaGa aeikaiaabkdacaqG4aGaaeykaaaa@664D@
In equation (28), Σ m =diag{ δ i,m 2 },  Σ n =diag{ δ i,n 2 }, MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4OdmaaBa aaleaacaWGTbaabeaakiabg2da9iaabsgacaqGPbGaaeyyaiaabEga caGG7bGaeqiTdq2aa0baaSqaaiaadMgacaGGSaGaamyBaaqaaiaaik daaaGccaGG9bGaaiilaiaabccacaWHJoWaaSbaaSqaaiaad6gaaeqa aOGaeyypa0JaaeizaiaabMgacaqGHbGaae4zaiaacUhacqaH0oazda qhaaWcbaGaamyAaiaacYcacaWGUbaabaGaaGOmaaaakiaac2hacaGG Saaaaa@5446@ Σ ε =diag{ δ i,ε 2 } i=1,2,,N MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4OdmaaBa aaleaacqaH1oqzaeqaaOGaeyypa0JaaeizaiaabMgacaqGHbGaae4z aiaacUhacqaH0oazdaqhaaWcbaGaamyAaiaacYcacqaH1oqzaeaaca aIYaaaaOGaaiyFaiaabccacaWGPbGaeyypa0JaaGymaiaacYcacaaI YaGaaiilaiabl+UimjaacYcacaWGobaaaa@4E64@ e is the constant part. Substituting equation (28) into equation (26), and the matrix F MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOraaaa@36C4@ can be expressed as:
F= ( r m ρ ) T m 1 r m ρ + ( r n ρ ) T n 1 r n ρ + ( r ε ρ ) T ε 1 r ε ρ     (29) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOraiabg2 da9iaacIcadaWcaaqaaiabgkGi2kaahkhadaWgaaWcbaGaamyBaaqa baaakeaacqGHciITcaWHbpaaaiaacMcadaahaaWcbeqaaiaadsfaaa GcdaaeWaqaamaalaaabaGaeyOaIyRaaCOCamaaBaaaleaacaWGTbaa beaaaOqaaiabgkGi2kaahg8aaaaaleaacaWGTbaabaGaeyOeI0IaaG ymaaqdcqGHris5aOGaey4kaSIaaiikamaalaaabaGaeyOaIyRaaCOC amaaBaaaleaacaWGUbaabeaaaOqaaiabgkGi2kaahg8aaaGaaiykam aaCaaaleqabaGaamivaaaakmaaqadabaWaaSaaaeaacqGHciITcaWH YbWaaSbaaSqaaiaad6gaaeqaaaGcbaGaeyOaIyRaaCyWdaaaaSqaai aad6gaaeaacqGHsislcaaIXaaaniabggHiLdGccqGHRaWkcaGGOaWa aSaaaeaacqGHciITcaWHYbWaaSbaaSqaaiabew7aLbqabaaakeaacq GHciITcaWHbpaaaiaacMcadaahaaWcbeqaaiaadsfaaaGcdaaeWaqa amaalaaabaGaeyOaIyRaaCOCamaaBaaaleaacqaH1oqzaeqaaaGcba GaeyOaIyRaaCyWdaaaaSqaaiabew7aLbqaaiabgkHiTiaaigdaa0Ga eyyeIuoakiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeOmaiaabM dacaqGPaaaaa@7BF9@
In equation (29), r m ρ ,  r n ρ ,  r ε ρ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHciITcaWHYbWaaSbaaSqaaiaad2gaaeqaaaGcbaGaeyOaIyRaaCyW daaacaGGSaGaaeiiamaalaaabaGaeyOaIyRaaCOCamaaBaaaleaaca WGUbaabeaaaOqaaiabgkGi2kaahg8aaaGaaiilaiaabccadaWcaaqa aiabgkGi2kaahkhadaWgaaWcbaGaeqyTdugabeaaaOqaaiabgkGi2k aahg8aaaaaaa@4C35@ are respectively the differential of each vector to the unknown parameter ρ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyWdaaa@3742@ , which can be written as follows:
{ r m ρ = [ r 1,m ρ r 2,m ρ r N,m ρ ] T r n ρ = [ r 1,n ρ r 2,n ρ r N,n ρ ] T r ε ρ = [ r 1,ε ρ r 2,ε ρ r N,ε ρ ] T         (30) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaeaafa qabeWabaaabaWaaSaaaeaacqGHciITcaWHYbWaaSbaaSqaaiaad2ga aeqaaaGcbaGaeyOaIyRaaCyWdaaacqGH9aqpdaWadaqaauaabeqabq aaaaqaaiaahkhadaqhaaWcbaGaaGymaiaacYcacaWGTbaabaGaeqyW dihaaaGcbaGaaCOCamaaDaaaleaacaaIYaGaaiilaiaad2gaaeaacq aHbpGCaaaakeaacqWIVlctaeaacaWHYbWaa0baaSqaaiaad6eacaGG SaGaamyBaaqaaiabeg8aYbaaaaaakiaawUfacaGLDbaadaahaaWcbe qaaiaadsfaaaaakeaadaWcaaqaaiabgkGi2kaahkhadaWgaaWcbaGa amOBaaqabaaakeaacqGHciITcaWHbpaaaiabg2da9maadmaabaqbae qabeabaaaabaGaaCOCamaaDaaaleaacaaIXaGaaiilaiaad6gaaeaa cqaHbpGCaaaakeaacaWHYbWaa0baaSqaaiaaikdacaGGSaGaamOBaa qaaiabeg8aYbaaaOqaaiabl+UimbqaaiaahkhadaqhaaWcbaGaamOt aiaacYcacaWGUbaabaGaeqyWdihaaaaaaOGaay5waiaaw2faamaaCa aaleqabaGaamivaaaaaOqaamaalaaabaGaeyOaIyRaaCOCamaaBaaa leaacqaH1oqzaeqaaaGcbaGaeyOaIyRaaCyWdaaacqGH9aqpdaWada qaauaabeqabqaaaaqaaiaahkhadaqhaaWcbaGaaGymaiaacYcacqaH 1oqzaeaacqaHbpGCaaaakeaacaWHYbWaa0baaSqaaiaaikdacaGGSa GaeqyTdugabaGaeqyWdihaaaGcbaGaeS47IWeabaGaaCOCamaaDaaa leaacaWGobGaaiilaiabew7aLbqaaiabeg8aYbaaaaaakiaawUfaca GLDbaadaahaaWcbeqaaiaadsfaaaaaaaGccaGL7baacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqGZa GaaeimaiaabMcaaaa@96D9@
In equation (30), r i,m ρ ,  r i,n ρ ,  r i,ε ρ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOCamaaDa aaleaacaWGPbGaaiilaiaad2gaaeaacqaHbpGCaaGccaGGSaGaaeii aiaahkhadaqhaaWcbaGaamyAaiaacYcacaWGUbaabaGaeqyWdihaaO GaaiilaiaabccacaWHYbWaa0baaSqaaiaadMgacaGGSaGaeqyTduga baGaeqyWdihaaaaa@49CD@ further expansion of the solution is expressed :
{ r i,m ρ = r i,m ρ =[ x i x c d i T 1,i θ ω 2 1 ω ] r i,n ρ = r i,n ρ =[ x i x c d i θ R 2,i ω 2 1 ω ] r i,ε ρ = r i,ε ρ =[ 4.34β (x x i ) T d i 2 0 0 ]            (31) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaeaafa qabeWabaaabaGaaCOCamaaDaaaleaacaWGPbGaaiilaiaad2gaaeaa cqaHbpGCaaGccqGH9aqpdaWcaaqaaiabgkGi2kaadkhadaWgaaWcba GaamyAaiaacYcacaWGTbaabeaaaOqaaiabgkGi2kaahg8aaaGaeyyp a0ZaamWaaeaafaqabeqadaaabaWaaSaaaeaacaWH4bWaaSbaaSqaai aadMgaaeqaaOGaeyOeI0IaaCiEaaqaaiaadogacaWGKbWaaSbaaSqa aiaadMgaaeqaaaaaaOqaamaalaaabaGaamivamaaBaaaleaacaaIXa GaaiilaiaadMgaaeqaaOGaeyOeI0IaeqiUdehabaGaeqyYdC3aaWba aSqabeaacaaIYaaaaaaaaOqaamaalaaabaGaaGymaaqaaiabeM8a3b aaaaaacaGLBbGaayzxaaaabaGaaCOCamaaDaaaleaacaWGPbGaaiil aiaad6gaaeaacqaHbpGCaaGccqGH9aqpdaWcaaqaaiabgkGi2kaadk hadaWgaaWcbaGaamyAaiaacYcacaWGUbaabeaaaOqaaiabgkGi2kaa hg8aaaGaeyypa0ZaamWaaeaafaqabeqadaaabaWaaSaaaeaacaWH4b WaaSbaaSqaaiaadMgaaeqaaOGaeyOeI0IaaCiEaaqaaiaadogacaWG KbWaaSbaaSqaaiaadMgaaeqaaaaaaOqaamaalaaabaGaeqiUdeNaey OeI0IaamOuamaaBaaaleaacaaIYaGaaiilaiaadMgaaeqaaaGcbaGa eqyYdC3aaWbaaSqabeaacaaIYaaaaaaaaOqaaiabgkHiTmaalaaaba GaaGymaaqaaiabeM8a3baaaaaacaGLBbGaayzxaaaabaGaaCOCamaa DaaaleaacaWGPbGaaiilaiabew7aLbqaaiabeg8aYbaakiabg2da9m aalaaabaGaeyOaIyRaamOCamaaBaaaleaacaWGPbGaaiilaiabew7a LbqabaaakeaacqGHciITcaWHbpaaaiabg2da9maadmaabaqbaeqabe WaaaqaamaalaaabaGaaGinaiaac6cacaaIZaGaaGinaiabek7aIjaa cIcacaWH4bGaeyOeI0IaaCiEamaaBaaaleaacaWGPbaabeaakiaacM cadaahaaWcbeqaaiaadsfaaaaakeaacaWGKbWaa0baaSqaaiaadMga aeaacaaIYaaaaaaaaOqaaiaaicdaaeaacaaIWaaaaaGaay5waiaaw2 faaaaaaiaawUhaaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabodacaqGXa Gaaeykaaaa@AC03@
Then according to the CRLB unbiased estimate lower bound theory has
CRLB( [ρ] p )= [ F 1 ] p,p        (32) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4qaiaabk facaqGmbGaaeOqaiaacIcacaGGBbGaaCyWdiaac2fadaWgaaWcbaGa amiCaaqabaGccaGGPaGaeyypa0Jaai4waiaahAeadaahaaWcbeqaai abgkHiTiaaigdaaaGccaGGDbWaaSbaaSqaaiaadchacaGGSaGaamiC aaqabaGccaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeikaiaabodacaqGYaGaaeykaaaa@4E30@
In the equation (32) p=1,2,,5,  [ F 1 ] p,p MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2 da9iaaigdacaGGSaGaaGOmaiaacYcacqWIVlctcaGGSaGaaGynaiaa cYcacaqGGaGaai4waiaahAeadaahaaWcbeqaaiabgkHiTiaaigdaaa GccaGGDbWaaSbaaSqaaiaadchacaGGSaGaamiCaaqabaaaaa@46AB@ represents the p MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaaaa@36EA@ -th row and the p MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaaaa@36EA@ -th column element value of the inverse matrix of F; CRLB( [ρ] p ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4qaiaabk facaqGmbGaaeOqaiaacIcacaGGBbGaaCyWdiaac2fadaWgaaWcbaGa amiCaaqabaGccaGGPaaaaa@3EB5@ indicates the lower bound of the CRLB unbiased to estimate of the p MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaaaa@36EB@ row element of unknown parameters ρ MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyWdaaa@3742@ .
Simulation Analysis
In this section, numerical results will be provided to verify and compare the feasibility of the proposed non-synchronous sensor network positioning method based on RSS and TOA hybrid measurements. According to the calculation method of the above design, the hybrid positioning algorithm is simulated by MATLAB software, and the simulation results are more intuitively displayed and analyzed. Assume that the clock of the set beacon nodes are synchronized, that are ω i =1,  θ i =0, i=1,2,,N MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaadMgaaeqaaOGaeyypa0JaaGymaiaacYcacaqGGaGaeqiU de3aaSbaaSqaaiaadMgaaeqaaOGaeyypa0JaaGimaiaacYcacaqGGa GaamyAaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacqWIVlctcaGG SaGaamOtaaaa@4A23@ , while the unknown node has clock drift and deviation. The time measurement noise variance δ i,m MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadMgacaGGSaGaamyBaaqabaaaaa@3A56@ with δ i,n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadMgacaGGSaGaamOBaaqabaaaaa@3A57@ , RSS noise variance δ i,ε MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadMgacaGGSaGaeqyTdugabeaaaaa@3B0C@ , signal emission intensity p 0 =45dB MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaaIWaaabeaakiabg2da9iabgkHiTiaaisdacaaI1aGaamiz aiaadkeaaaa@3CFA@ , path attenuation index β=4 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaey ypa0JaaGinaaaa@395A@ are set in advance between the unknown nodes and beacon nodes i. The accuracy of the positioning error of the designed algorithm is based on the Root Mean Square Error (RMSE) analysis. The simulation runs 1000 times, and the average value of the results is used for error analysis.
The influence of the number of beacon nodes on the positioning error
Assuming that the simulation environment is in a 3D region of 100 m × 100 m × 100 m, the coordinates of the unknown node is set to (50, 50, 50) in advance. Because the quantity of beacon nodes is too few lead to the positioning accuracy is too low, so the number of beacon nodes is set as N=4,5,,8 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiabg2 da9iaaisdacaGGSaGaaGynaiaacYcacqWIVlctcaGGSaGaaGioaaaa @3E0B@ . Table 1 compares the Unconstrained Linear Least Squares (ULLS) and constrained least squares (CLLS) positioning accuracy of the three different methods of RSS, TOA, RSS/TOA, and the CRLB value is compared as a system benchmark.
Table 1: Comparison of the influence of the number of beacon nodes on the positioning error

4

5

6

7

8

ULLS

RSS

0.873

0.628

0.531

0.456

0.377

TOA

0.718

0.573

0.359

0.272

0.185

RSS/TOA

0.141

0.113

0.109

0.089

0.075

CLLS

RSS

0.591

0.504

0.438

0.356

0.271

TOA

0.405

0.311

0.237

0.188

0.139

RSS/TOA

0.063

0.039

0.033

0.028

0.025

CRLB

RSS

0.512

0.425

0.351

0.288

0.204

TOA

0.323

0.255

0.189

0.132

0.093

RSS/TOA

0.047

0.024

0.021

0.019

0.018

Table 1 lists 1000 simulations under the premise that the time measurement noise variance δ i,m ,  δ i,n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadMgacaGGSaGaamyBaaqabaGccaGGSaGaaeiiaiabes7a KnaaBaaaleaacaWGPbGaaiilaiaad6gaaeqaaaaa@4015@ and RSS noise variance δ i,ε MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadMgacaGGSaGaeqyTdugabeaaaaa@3B0B@ are all 0.2, and compares the positioning errors of the ULLS and CLLS calculation methods mentioned above in the paper and compared them with CRLB. It can be seen from Table 1 that with the increase of the number of beacon nodes, the measurement connection relationship between nodes increases, and both algorithms and CRLB values decrease accordingly. For example, when the number of beacon nodes is 5, the positioning error of the single RSS in the ULLS method is 0.628 m, and that in the CLLS method is 0.504 m; the positioning error of the single TOA in the ULLS method is 0.573 m, and that in the CLLS method is 0.311 m; The positioning error of the hybrid measured RSS and TOA in the ULLS method is 0.425 m, and that in the CLLS method is 0.255 m. This shows that in the case of the same number of beacon nodes, whether in the unconstrained linear least squares or constrained least squares method, the error of the hybrid measurement method is smaller than the single measurement method (RSS or TOA), which indicates the hybrid measurement method is more accurate. It can also be seen from Table 1 that the proposed hybrid method yield results very close to CRLB, which confirms the feasibility and accuracy of the hybrid measurement method, proposed in this paper.
Influence of noise on positioning error
In the linear estimation method proposed in this paper, each noise of time measurement and signal strength measurement has independence, which leads to a decrease in the accuracy of the positioning method. In order to test the estimation error of the ULLS and CLLS calculation methods proposed in this paper, the positioning accuracy of the algorithm is compared with CRLB. And the results are compared with the SDP method introduced in literature [16]. In order to make the simulation results more representative, the number of beacon nodes is selected to be 5, and keeping the time measurement noise variance δ i,m MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadMgacaGGSaGaamyBaaqabaaaaa@3A56@ with δ i,n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadMgacaGGSaGaamOBaaqabaaaaa@3A57@ both are 0.2 ns, while adjusting RSS measurement noise δ i,ε MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadMgacaGGSaGaeqyTdugabeaaaaa@3B0B@ varies from 0.2 dB to 2 dB. Figure 2 depicts the relation of RMSE positioning error with RSS measurement noise under different algorithms. As shown in figure 2, the value of the RSS measurement noise has a great influence on the positioning accuracy of the algorithm. With the increases of the RSS measurement noise, the positioning error also increases. When the RSS measurement noise is equal to 0.2 dB, the positioning error RMSE of the designed ULLS method is 0.131 m, and the positioning error of the CLLS method is only 0.039 m. The constrained CLLS method is more accurate than the ULLS method, and its positioning result is closer to the CRLB value. Compared with the SDP method, the positioning error of the ULLS method is larger, and the positioning error of the CLLS method is smaller than that of the SDP method. When the RSS measurement noise is equal to 0.2 dB, the positioning error of SDP method is 0.075 m, which is between the positioning error of ULLS and CLLS method.
Figure 2: The effect of RSS measurement noise on positioning error
Similarly, keep adjusting RSS measurement noise δ i,n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadMgacaGGSaGaamOBaaqabaaaaa@3A57@ to 0.2 dB, the time measurement noise variance δ i,m MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadMgacaGGSaGaamyBaaqabaaaaa@3A56@ with δ i,n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadMgacaGGSaGaamOBaaqabaaaaa@3A57@ both vary from 0.2 ns to 2 ns at the same time. It can be seen from figure 3 that the RMSE positioning error under different algorithms increases as the TOA measurement noise increases. Compared with the influence of RSS measurement noise on positioning error, TOA measurement noise has less influence on positioning error. When the TOA measurement noise is equal to 0.2 ns, the positioning error RMSE of the designed ULLS method is 0.107 m, the positioning error of the SDP method is 0.058, and the positioning error of the CLLS method is only 0.031 m. The constrained CLLS method is closer to the CRLB than the ULLS method. The same analysis results as in Fig. 2 show that the positioning error using the constrained CLLS method is smaller than that of the ULLS and SDP methods.
Figure 3: The effect of TOA measurement noise on positioning error
Influence of path attenuation index on positioning error
In order to make the simulation results more representative, this paper compares the influence of path attenuation index on positioning error and sets the RSS measurement noise δ i,ε MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadMgacaGGSaGaeqyTdugabeaaaaa@3B0B@ to 0.2 dB, the time measurement noise variance δ i,m MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadMgacaGGSaGaamyBaaqabaaaaa@3A56@ and δ i,n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadMgacaGGSaGaamOBaaqabaaaaa@3A57@ are set to 0.2 ns, the path attenuation index β MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3796@ value is obtained by experimental analysis in advance, and its value varies with the change of the propagation medium, generally between the typical value of 2 and 5, and adjusts β MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3796@ value changes within the interval.
Figure 4: Influence of path attenuation index on positioning error
Figure 4 plots the variation of the RMSE positioning error with the path attenuation index under different algorithms. It can be seen from the figure that the positioning error gradually decreases with the path attenuation index β MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3796@ value increases. When the path attenuation index β MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3796@ value is equal to 2, the positioning error RMSE of the designed ULLS method is 0.151 m, the positioning error of the SDP method is 0.114m, the positioning error of the constrained CLLS method is 0.091 m, and the CRLB value is 0.079 m; when the path attenuation index β MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3796@ value is equal to 5, the positioning error RMSE of the designed ULLS method is 0.082 m, the positioning error of the SDP method is 0.061m, the positioning error of the constrained CLLS method is 0.039 m, and the CRLB value is 0.032 m.
Time synchronization parameter estimation error
The model designed in this paper assumes that the clock parameters of the beacon nodes, including the drift rate, deviation amount are calibrated, while the unknown node clock synchronization parameter is unknown parameter. The time synchronization parameters of the unknown node including the drift rate, the deviation amount, and the position coordinates of the unknown node are simultaneously estimated. Because the noise error change of RSS in the hybrid algorithm has little effect on the clock drift rate and deviation amount, the fixed RSS noise error value is selected to be 0.2 dB, which makes the TOA noise error change from 0.2 to 2 ns. The clock drift rate and deviation amount of the unknown node are selected to analyze the estimation error. The results are shown in Figure 5 and Figure 6.
Figure 5: Clock drift rate estimation error
Figure 6: Clock deviation amount t estimation error
Observing the change curve of the clock drift rate and deviation amount of Figure 5 and Figure 6 with noise δ i,m MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadMgacaGGSaGaamyBaaqabaaaaa@3A56@ and δ i,n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadMgacaGGSaGaamOBaaqabaaaaa@3A57@ ,it is not difficult to find that the variation law of the algorithm is similar to the result reflected in Figure 3. The general trend is that the clock drift rate increases with the increase of noise, which is an upward trend. As when δ i,m MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadMgacaGGSaGaamyBaaqabaaaaa@3A56@ and δ i,n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadMgacaGGSaGaamOBaaqabaaaaa@3A57@ are 0.2 ns, the clock drift rate RMSE of the ULLS is 0.008, the clock drift rate RMSE of the SDP is 0.007, while the clock drift rate RMSE of the CLLS is 0.006; when δ i,m MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadMgacaGGSaGaamyBaaqabaaaaa@3A56@ and δ i,n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadMgacaGGSaGaamOBaaqabaaaaa@3A57@ are 2 ns, the clock drift rate RMSE of the ULLS increases to 0.172, the clock drift rate RMSE of the SDP increases to 0.131, while the CLLS clock drift rate RMSE increases to 0.125.

Similarly, the clock deviation quantity estimation error is similar to the above-mentioned result, and the overall trend is also increasing with the increase of noise. As when δ i,m MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadMgacaGGSaGaamyBaaqabaaaaa@3A56@ and δ i,n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadMgacaGGSaGaamOBaaqabaaaaa@3A57@ are 0.2 ns, the clock deviation quantity RMSE of the ULLS is 0.36, the clock deviation quantity RMSE of the SDP is 0.29, while the clock deviation quantity RMSE of the CLLS is 0.28 ; and when δ i,m MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadMgacaGGSaGaamyBaaqabaaaaa@3A56@ and δ i,n MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadMgacaGGSaGaamOBaaqabaaaaa@3A57@ are 2 ns, the clock deviation quantity RMSE of ULLS increases to 4.18 , the clock deviation quantity RMSE of SDP increases to 3.71, while the CLLS clock deviation quantity RMSE increases to 3.17.
Conclusion
This paper proposes a 3D asynchronous sensor network positioning method based on hybrid measurement of RSS and TOA. Using the clock drift and deviation TOA ranging model, the time synchronization parameters and the node position coordinates are simultaneously estimated, which realizes both time synchronization and node position coordinate estimation. Converting the nonlinear optimization model of RSS and TOA hybrid measurement into linear equation, obtains the Unconstrained Linear Least Squares linear method (ULLS) of the unknown node position coordinates, and optimizes the positioning result of the ULLS. And convert it to a more precise Constrained Least Squares (CLLS) result. Compared with the SDP method introduced in the literature, the proposed ULLS and CLLS linear estimation method calculation does not depend on the choice of the initial solution, with low complexity and fast operation speed. The simulation results show that the performance of the proposed hybrid positioning algorithm is superior to the traditional single RSS or TOA method. Compared with the SDP method, the error of the constrained CLLS method is smaller, which is closer to the CRLB lower bound value of the positioning result. However, the ULLS and CLLS obtained by the hybrid positioning method proposed in this paper have certain limitations, and the estimation error has not reached the CRLB, which needs further optimization.
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