Mathematical Modelling of The Global Positioning System Tracking Signals


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1 Master Thesis Mathematical Modeling and Simulation Thesis no: May 2008 Mathematical Modelling of The Global Positioning System Tracking Signals Mounchili Mama School of Engineering Blekinge Institute of Technology SE Karlskrona Sweden
2 This thesis is submitted to the School of Engineering at Blekinge Institute of Technology in partial fulfillment of the requirements for the degree of Master of Science in Mathematical Modeling and Simulation. The thesis is 15 credit (15 ECTS ) equivalent to ten weeks of full time studies. Contact Information Author : Mounchili Mama University advisor : Assistant Professor  Claes Jogréus Department of Mathematical Science School of Engineering Blekinge Institute of Technology SE Karlskrona Sweden Internet :
3 Sincere thanks to my Parents and Siblings who always stood by me whenever needed, without whom I couldn t be a complete person. Finally to my wife Fatimatou Zahra embellishing my life, by caring so much about me. I love you Zahra.
4 Abstract Recently, there has been increasing interest within the potential user community of Global Positioning System (GPS) for high precision navigation problems such as aircraft non precision approach, river and harbor navigation, realtime or kinematic surveying. In view of more and more GPS applications, the reliability of GPS is at this issue. The Global Positioning System (GPS) is a spacebased radio navigation system that provides consistent positioning, navigation, and timing services to civilian users on a continuous worldwide basis. The GPS system receiver provides exact location and time information for an unlimited number of users in all weather, day and night, anywhere in the world. The work in this thesis will mainly focuss on how to model a Mathematical expression for tracking GPS Signal using Phase Locked Loop filter receiver. Mathematical formulation of the filter are of two types: the first order and the second order loops are tested successively in order to find out a compromised on which one best provide a zero steady state error that will likely minimize noise bandwidth to tracks frequency modulated signal and returns the phase comparator characteristic to the null point. Then the Ztransform is used to build a phaselocked loop in software for digitized data. Finally, a Numerical Methods approach is developed using either MATLAB or Mathematica containing the package for Gaussian elimination to provide the exact location or the tracking of a GPS in the space for a given a coarse/acquisition (C/A) code. i
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6 Acknowledgements I would like to express my sincere thanks to my supervisor Assistant Professor  Claes Jogréus for his constructive comments as well as useful advice throughout the thesis work. iii
7 Contents Contents List of Figures iv vi 1 Introduction Introduction GPS Overview Motivation Scope and Objective Thesis Structure Modeling of GPS Tracking Signals GPS Signal Structure Basic PhaseLocked Loops FirstOrder PhaseLocked Loop SecondOrder PhaseLocked Loop ZTransform of Continuous Systems Carrier and Code Tracking GPS Tarcking Loop Signals Conclusion Numerical Method of GPS Tracking Signal Introduction Numerical Expression of The Coordinates Numerical Solution Using Mathematica Global Position Systems Applications Introduction GPS for The Utilities Industry iv
8 v 4.3 GPS for Civil Engineering Applications GPS for Land Seismic Surveying GPS for Marine Seismic Surveying GPS for Vehicle Navigation GPS for Transit Systems Bibliography 33 A GPS Source Code for Data Acquisition 37
9 List of Figures 1.1 GPS segments Architecture of Softwarebased GPS Receiver Schematic showing the generation of L1 band GPS signal. The equation is the mathematical representation of C/A code in L1 band A basic PhaseLocked Loop Loop Filter Code and carrier tracking loops Basic idea of GPS positioning Satellite and Observer in the Global Positioning System GPS for utility mapping GPS for construction applications GPS for land seismic surveying GPS for marine seismic surveying GPS for vehicle navigation GPS for transit systems vi
10 Chapter 1 Introduction 1.1 Introduction The Global Positioning System (GPS) is a satellitebased navigation system [1], that was developed by the U.S. Department of Defense (DoD) in the early 1970s. Primarily, GPS was developed as a military system to fulfill U.S. military needs. However, it was later made available to civilians, and is now a dualuse system that can be accessed by both military and civilian users [2]. GPS provides continuous positioning and timing information, anywhere in the world under any weather conditions. Because it serves an unlimited number of users as well as being used for security reasons, GPS is a onewayranging (passive) system [4]. That is, users can only receive the satellite signals. 1.2 GPS Overview The GPS is made up of three parts: satellites orbiting the Earth; control and monitoring stations on Earth; and the GPS receivers owned by users. GPS satellites broadcast signals from space that are picked up and identified by GPS receivers. Each GPS receiver then provides threedimensional location (latitude, longitude, and altitude) plus the time. Furthermore GPS consist of three segments as shown in Figure 1.1: the space segment, the control segment and the user segment. The space segment consists of a nominal constellation of 24 operating satellites that transmit oneway signals that give the current GPS satellite position and time. 1
11 2 CHAPTER 1. INTRODUCTION Figure 1.1: GPS segments The control segment consists of worldwide monitor and control stations that maintain the satellites in their proper orbits through occasional command maneuvers, and adjust the satellite clocks. It tracks the GPS satellites, uploads updated navigational data, and maintains health and status of the satellite constellation. The user segment consists of the GPS receiver equipment, which receives the signals from the GPS satellites and uses the transmitted information to calculate the user s threedimensional position and time. 1.3 Motivation GPS satellites provide service to civilian and military users [8]. The civilian service is freely available to all users on a continuous, worldwide basis. The military service is available to governments and allied armed forces as well as other Government agencies. A variety of GPS augmentation systems and techniques are accessible to improve system performance to meet specific user requirements. The improved signal availability, accuracy, and integrity, allows even better performance than is possible using the basic GPS civilian service.
12 1.4. SCOPE AND OBJECTIVE 3 The outstanding performance of GPS over many years has earned the confidence of millions of civil users worldwide. It has proven its dependability in the past and promises to be of benefit to users, all over the world, far into the future. 1.4 Scope and Objective The goal of this thesis work is primarily to formulate some mathematical models that best describe phase locked loop necessary to determine approximatively the GPS tracking signal. Then the next step will be to make a transform from continuous to discrete systems to build a phaselocked loop in software for digitized data, the continuous system must be changed into a discrete system. Numerical method such as the Gaussian Elimination method will be use to simulation on mathematica package the exact position of a GPS signal receiver in the three dimension space. Finally some GPS applications area will be discussed. 1.5 Thesis Structure Chapter 1 of the book introduces the GPS system and its components. Chapter 2 examines the mathematical modeling or formulation of GPS tracking signal structure, coupled with the GPS modernization, and the key types of the GPS measurements. Another simple way to determine the GPS signal is developed using Gaussian Elimination method package of Mathematical, is presented in Chapter 3. The GPS applications in the various fields are given in Chapter 4, which covers the other satellite navigation systems developed or proposed in different parts of the world that fit with different purpose.
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14 Chapter 2 Modeling of GPS Tracking Signals The main objective in this chapter is to determine how GPS signal can be processed by using acquisition and tracking algorithms to extract the navigation information bits from the raw data. The navigation data bits provide all the necessary information to compute the pseudorange between the receiver and the visible satellites. 2.1 GPS Signal Structure Generally the GPS has two bands, L1 and L2. Where L1 band has coarse / acquisition (C/A) code, Pcode and navigation data and L2 band has only Pcode. The Pcode is encrypted with an encryption code and is named Ycode. While the C/A code is a succession of zeros and ones and is unique for every satellite. The code is relied on Gold Codes. Two strings of gold codes with different phase tapings are considered to generate unique C/A code for each satellite. The C/A code is also called pseudorandom number (PRN) code. PRN code or number is attributed to every satellite depending on which PRN code is attributed to a particular satellite. If PRN (C/A) code of 1 is assigned to a satellite, in that case this satellite is named as PRN1. The particularities of C/A codes are that they have the best crosscorrelation feature. The crosscorrelation among any two codes is much lower than autocorrelation of each one of the codes. The frequency of the C/A code is MHz. Pcode is also a sequence of zeros and ones and is generated using a set of Gold Codes. Pcode has a frequency of MHz. Fig.2.1 depicss a general schematic to generate a L1 band signal as rep 5
15 6 CHAPTER 2. MODELING OF GPS TRACKING SIGNALS Figure 2.1: Architecture of Softwarebased GPS Receiver Figure 2.2: Schematic showing the generation of L1 band GPS signal. The equation is the mathematical representation of C/A code in L1 band
16 2.2. BASIC PHASELOCKED LOOPS 7 resented by the equation shown in the figure for C/A code. In conventional hardwarebased GPS receiver, the lower three blocks in Fig.2.2 are implemented in an IC chip and hence the user cannot access the algorithms built inside the chips. In softwarebased receiver, these blocks are fully implemented using high level programming languages granting the user a complete control over the algorithms. Hence the main difference between the softwarebased GPS receiver and a conventional hardware receiver. 2.2 Basic PhaseLocked Loops During the tracking process, the center frequency of the narrowband filter is fixed, while a locally generated signal follows the frequency of the input signal. Moreover the phases of both signals are compared through a phase comparator of which passes through a narrowband filter. Given that the tracking circuit posses narrow bandwidth, the sensitivity is fairly high in comparison with the acquisition method. In the case of phase shifts in the carrier due to the C/A code, as in a GPS signal, the code has to be stripped off first. The tracking process will follow the signal and acquire the information of the navigation data. In case a GPS receiver is stationary, the desired frequency variation is very slow due to satellite motion. Given this condition, the frequency variation of the locally generated signal is also slow; consequently, the update rate of the tracking loop can be low. Therefore, to track a GPS signal two tracking loops are needed: a loop to track the carrier frequency referred as carrier loop and the other one to track the C/A code known as the code loop. The main purpose of a phaselocked loop is to adjust the frequency of a local oscillator to match the frequency of an input signal, which is sometimes referred to as the reference signal. A basic phaselocked loop is shown in Figure 2.3. Figure 2.3a shows the time domain configuration and Figure 2.3b shows the sdomain configuration, which is obtained from the Laplace transform. The input signal is θ i (t) and the output from the voltagecontrolled oscillator (VCO) is θ f (t). The phase comparator Σ measures the phase difference of these two signals. The amplifier k 0 represents the gain of the phase comparator and the lowpass filter limits the noise in the loop. The input
17 8 CHAPTER 2. MODELING OF GPS TRACKING SIGNALS Figure 2.3: A basic PhaseLocked Loop voltage V o to the VCO controls its output frequency, which can be expressed as: ω 2 (t) = ω 0 + k 1 u(t) (2.1) where ω 0 is the center angular frequency of the VCO, k 1 is the gain of the VCO, and u(t) is a unit step function, which is defined as [6]: u(t) = { 0 for t < 0 1 for t > 0 (2.2) The phase angle of the VCO can be obtained by integrating Equation (2.1) as: Where: t 0 ω 2 (t)dt = ω 0 t + θ f (t) = ω 0 t + θ f (t) = The Laplace transform [7] of θ f (t) is: t 0 t 0 k 1 u(t)dt k 1 u(t)dt (2.3) θ f (s) = k 1 s We can obtain from the Figure 2.1b the following equations: (2.4) V c (s) = k 0 ɛ(s) = k 0 [θ i (s) θ f (s)] (2.5) V 0 (s) = V c (s)f (s) (2.6) Hence result the following: θ f (s) = V 0 (s) k 1 s ɛ(t) = θ i (s) θ f (s) = V c(s) k 0 = V 0(s) k 0 F (s) = sθ f(s) k 0 k 1 F (s) or (2.7)
18 2.2. BASIC PHASELOCKED LOOPS 9 ( ) s θ i (s) = θ f (s) 1 + k 0 k 1 F (s) (2.8) Where ɛ(s) is the error function. The transfer function H(s) of the loop is defined as: H(s) θ f(s) θ i (s) = The error transfer function can be defined as: H e (s) = ɛ(s) θ i (s) = θ i(s) θ f (s) θ i (s) k 0k 1 F (s) s + k 0 k 1 F (s) = 1 H(s) = s s + k 0 k 1 F (s) (2.9) (2.10) The equivalent noise bandwidth is defined as: B n = 0 H(jω) 2 df (2.11) Where ω is the angular frequency and is related to the frequency f by ω = 2πf In order to study the properties of the phaselocked loops, two types of input signals are usually studied. The first type is a unit step function as: θ i (t) = u(t) or θ i (s) = 1 s (2.12) The second type is a frequencymodulated signal θ i (t) = ω(t) or θ i (s) = ω s 2 (2.13) FirstOrder PhaseLocked Loop A firstorder phaselocked loop implies that the denominator of the transfer function H(s) is a firstorder function of s [6]. The order of the phaselocked loop depends on the order of the filter in the loop. The expression of the filter function in the case of the first order phaselocked loop, is: F (s) = 1 (2.14) Hence the simplest expression of phaselocked loop. In the case of a unit step function input, the equivalent transfer function from Equation (2.9) becomes:
19 10 CHAPTER 2. MODELING OF GPS TRACKING SIGNALS Then the noise bandwidth can be expressed as: B n = 0 H(s) = k 0k 1 s + k 0 k 1 (2.15) (k 0 k 1 ) 2 ω 2 + (k 0 k 1 ) df = (k 0k 1 ) 2 dω 2 2π 0 ω 2 + (k 0 k 1 ) 2 ( ) ω B n = (k 0k 1 ) 2 2πk 0 k 1 tan 1 k 0 k 1 0 = k 0k 1 4 (2.16) Given the input signal θ i (s) = 1/s, the error function can be found from Equation (2.10) as: 1 ɛ(s) = θ i (s)h e (s) (2.17) s + k 0 k 1 The steadystate error can be found from the final value theorem [7] of the Laplace transform as follow: lim t y(t) = lim sy (s) (2.18) s 0 It follows from this relation, the final value of ɛ(t) can be found as: lim t ɛ(t) = lim sɛ(s) = lim s 0 s 0 s s + k 0 k 1 = 0 (2.19) Given the input signal θ i (s) = ω/s 2, the error function becomes: ɛ(s) = θ i (s)h e (s) ω s Then the steady state error becomes: 1 s + k 0 k 1 (2.20) ω lim ɛ(t) = lim sɛ(s) = lim = ω (2.21) t 0 s 0 s 0 s + k 0 k 1 k 0 k 1 It results that the steadystate phase error is not equal to zero. Therefore for a large value of k 0 k 1 makes the error small. However, from Equation (2.15) the 3 db bandwidth occurs at s = k 0 k 1. Thus a small value of ɛ(t) also means large bandwidth, that is likely to contain more noise SecondOrder PhaseLocked Loop In the case of a secondorder phaselocked loop means the denominator of the transfer function H(s) is a secondorder function of s. One of the filters
20 2.2. BASIC PHASELOCKED LOOPS 11 to make such a secondorder phaselocked loop is F (s) = sτ sτ 1 (2.22) Replacing this relation into Equation (2.9), the transfer function becomes: k 0 k 1 τ 2 s τ 1 + k 0k 1 τ 1 H(s) = s 2 k 0 k 1 τ 2 s τ 1 + k 0k 1 2ζω ns + ω2 n s 2 + 2ζω τ n s + ω 2 1 n where ω n is the natural frequency, which can be expressed as: (2.23) ω n = k0 k 1 τ 1 (2.24) and ζ the damping factor which can be determined as follow: 2ζω n = k 0k 1 τ 2 or ζ = ω nτ 2 τ 1 2 Therefore the noise bandwidth can be found as: (2.25) B n = 0 H(jω) 2 df = ω n 2π 0 [ 1 ) (2ζ ωωn ( ) ] 2 2 ) dw 2 ω ω n + (2ζ ωωn or = ω n 2π 0 ( 1 + 4ζ 2 ( ) 2 ω ω n ) 4 ω ω n + 2(2ζ2 1) ( ) 2 dw = ω ω n + 1 ω n 2 ( ζ + 1 ) 4ζ (2.26) This integration can be found in the appendix of [6]. Therefore the error transfer function can be obtained from Equation (2.10) as: H e (s) = 1 H(s) = s 2 + 2ζω n s + ωn 2 And whenever θ i (s) = 1/s, the error function becomes: s 2 (2.27) ɛ(s) = s 2 s 2 + 2ζω n s + ω 2 n (2.28) At steady state the error is given by: lim t ɛ(t) = lim sɛ(s) = 0 (2.29) s 0
21 12 CHAPTER 2. MODELING OF GPS TRACKING SIGNALS Figure 2.4: Loop Filter Unlike the firstorder loop, the steadystate error is zero for the frequency modulated signal. Meaning that, the secondorder loop tracks a frequencymodulated signal and returns the phase comparator characteristic to the null point. Therefore, the conventional phaselocked loop in a GPS receiver is usually a secondorder one. 2.3 ZTransform of Continuous Systems In order to build a phaselocked loop in software for digitized data, the continuous system must be changed into a discrete system. The transfer from the continuous sdomain into the discrete zdomain is through bilinear transform as: s = 2 t s 1 z z 1 (2.30) where t s is the sampling interval. Substituting this relation in Equation (2.22) the filter is transformed to: where: F (z) = C 1 + C 2 1 z 1 = (C1 + C2) C 1z 1 1 z 1 (2.31) C 1 = 2τ 2 t s 2τ 1 and C 2 = t s τ 1 (2.32) This filter is shown in Figure 2.4. The VCO in the phaselocked loop is replaced by a direct digital frequency synthesizer and its transfer function N(z) can be used to replace the result in Equation (2.7) with:
22 2.4. CARRIER AND CODE TRACKING 13 N(z) = θ f(z) V 0 (z) k 1z 1 1 z 1 (2.33) Using the same approach as Equation (2.8), the transfer function H(z) can be written as: H(z) = θ f(z) θ i (z) = k 0F (z)n(z) (2.34) 1 + k 0 F (z)n(z) The substituting the results of Equations (2.33) and (2.35) into the above equation, yields: H(z) = k 0 k 1 (C 1 + C 2 )z 1 k 0 k 1 C 1 z [k 0 k 1 (C 1 + C 2 ) 2]z 1 + (1 k 0 k 1 C 1 )z 2 (2.35) By using the bilinear transform in Equation (2.32) to Equation (2.23), the result can be written as: [4ζω n + (ω n t s ) 2 ] + 2(ω n t s ) 2 z 1 + [(ω n t s ) 2 rζω n t s ]z 2 H(z) = [4 + 4ζω n + (ω n t s ) 2 ] + [2(ω n t s ) 2 8]z 1 + [4 4ζω n + (ω n t s ) 2 ]z 2 (2.36) By equating the denominator polynomials in the above two equations, C 1 and C 2 can be determined as: C 1 = 1 8ζω n t s k 0 k ζω n t s + (ω n t s ) 2 C 2 = 1 4ζω n t s (2.37) k 0 k ζω n t s + (ω n t s ) 2 The filter here is implemented in digital format and the result can be used for phaselocked loop designs, but it is not the part of this thesis scope since we are only concerned with mathematical expression. Where the desired noise bandwidth signal from the Equation (2.26) can is expressed as: B n = ω n 2 ( ζ + 1 ) 4ζ 2.4 Carrier and Code Tracking Within a GPS receiver, the input is the GPS signal and a phaselocked loop must follow (or track) this signal. On the other hand, the GPS signal is a biphase coded signal. The carrier and code frequencies change due to the Doppler Effect, which is caused by the motion of the GPS satellite as well
23 14 CHAPTER 2. MODELING OF GPS TRACKING SIGNALS Figure 2.5: Code and carrier tracking loops. as from the motion of the GPS receiver. In order to track the GPS signal, the C/A code information have to be removed. Consequently, it requires two phaselocked loops to track a GPS signal. One loop is to track the C/ A code and the other one is to track the carrier frequency. These two loops must be coupled together are depicted in Figure 3.3. In Figure 2.3, the C/A code loop generates three outputs: an early code, a late code, and a prompt code. The prompt code is applied to the digitized input signal and strips the C/A code from the input signal. Stripping the C/A code means to multiply the C/A code to the input signal with the proper phase. The output will be a continuous wave (cw) signal with phase transition caused only by the navigation data. This signal is applied to the input of the carrier loop. The output from the carrier loop is a cw with the carrier frequency of the input signal. This signal is used to strip the carrier from the digitized input signal, which means using this signal to multiply the input signal. The output is a signal with only a C/A code and no carrier frequency, which is applied to the input of the code loop. Every output passes through a moving average filter and the output of the filter is squared. The two squared outputs are compared to generate a
24 2.5. GPS TARCKING LOOP SIGNALS 15 control signal to adjust the rate of the locally generated C/A code to match the C/A code of the input signal. The locally generated C/A code is the prompt C/A code and this signal is used to strip the C/A code from the digitized input signal. The carrier frequency loop receives a cw signal phase modulated only by the navigation data as the C/A code is stripped off from the input signal. The acquisition program determines the initial value of the carrier frequency. The voltagecontrolled oscillator (VCO) generates a carrier frequency with respect to the value obtained from the acquisition program. This signal is divided into two paths: a direct path and the other one with a 90degree phase shift. These two signals are correlated with the input signal. The outputs of the correlators are filtered and their phases are compared against each other through an arctangent comparator. The arctangent process is insensitive to the phase transition caused by the navigation data and it can be considered as one type of a Costas loop. A Costas loop is a phaselocked loop, which is insensitive to phase transition. The output of the comparator is filtered again and generates a control signal. This control signal is used to tune the oscillator to generate a carrier frequency to follow the input cw signal. This carrier frequency is also used to strip the carrier from the input signal. 2.5 GPS Tarcking Loop Signals The input data to the tracking loop are collected from satellites. Several constants have to be determined such as the noise bandwidth, the gain factors of the phase detector, and the VCO (or the digital frequency synthesizer). These constants are determined through trial and error or guessing and are by no means optimized. This tracking program is applied only on limited data length. Although it generates acceptable results, further study might be needed if it is used in a software GPS receiver designed to track long records of data. The following steps can be applied to both the code loop and the carrier loop: 1. Set the bandwidths and the gain of the code and carrier loops. The loop gain includes the gains of the phase detector and the VCO. The bandwidth of the code loop is narrower than the carrier loop because it tracks the signal for a longer period of time. Select the noise band
25 16 CHAPTER 2. MODELING OF GPS TRACKING SIGNALS width of the code loop to be 1 Hz and the carrier loop to be 20 Hz. This is one set of several probable selections that the tracking program can run. 2. Choose the damping factor in Equation (2.25) to be ζ =.707. The ζ value is often assumed near about its optimum [9]. 3. The natural frequency can be found from Equation (2.26). 4. Select the code loop gain (k 0 k 1 ) to be 50 and the carrier loop gain to be 4π100. These values are also one set of several possible selections. The constants C 1 and C 2 of the filter can be found from Equation (2.39). The above four steps provide the necessary information for the two loops. Once the constants of the loops are known, the phase of the code loop and the phase of the carrier frequency can be adjusted to follow the input signals. 2.6 Conclusion Hence the completed the algorithms for acquisition and tracking. The goal of these algorithms has to be tested on various data sets in different environment using different type of antenna including left hand and right hand polarized antenna. I have found that, it is necessary to tune the parameters of the tracking algorithm and threshold values of acquisition either dynamically or based on some rule for successful acquisition and tracking of GPS signal in standard environment. In some cases, we have found that tracking could not be done for satellites though acquisition is perfect. In this case, a change of parameter values of the Phase Lock Loop manually makes the tracking successful. This type of manual setting shall be automated in the future. The future work consists of extracting the navigation message from the tracking output and to compute the position of the receiver. It is also necessary to make the processing as fast as possible.
26 Chapter 3 Numerical Method of GPS Tracking Signal The idea behind GPS is rather simple. If the distances from a point on the Earth (a GPS receiver) to three GPS satellites are known along with the satellite locations, then the location of the point (or receiver) can be determined by simply applying the wellknown concept of resection [5]. Now arises a question on how we can get the distances to the satellites as well as the satellite locations? 3.1 Introduction As mentioned before, each GPS satellite continuously transmits a microwave radio signal composed of two carriers, two codes, and a navigation message. When a GPS receiver is switched on, it will pick up the GPS signal through the receiver antenna. Once the receiver acquires the GPS signal, it will process it using its builtin software. The partial outcome of the signal processing consists of the distances to the GPS satellites through the digital codes (known as the pseudo ranges) and the satellite coordinates through the navigation message. Theoretically, only three distances to three simultaneously tracked satellites are needed. In this case, the receiver would be located at the intersection of three spheres; each has a radius of one receiversatellite distance and is centered on that particular satellite (Figure 3.1). However from the practical point of view, a fourth satellite needed to account for the receiver clock offset 17
27 18 CHAPTER 3. NUMERICAL METHOD OF GPS TRACKING SIGNAL Figure 3.1: Basic idea of GPS positioning [3]. 3.2 Numerical Expression of The Coordinates The launching over the last three decade of the socalled geosynchronous satellites (i.e., satellites located at a fixed point above the surface of the earth) has made it possible to determine tracking signal position anywhere on earth. this is done by measuring the time it take for a signal to travel between the observer and any satellite and translating it into distance r i between the two. To exact the desired coordinates of the observer from these measurements, we construct a sphere of radius r i about each of four satellites. The equations these spheres are given by: (x x 1 ) 2 + (y y 1 ) 2 + (z z 1 ) 2 = r 2 1 (x x 2 ) 2 + (y y 2 ) 2 + (z z 2 ) 2 = r 2 2 (x x 3 ) 2 + (y y 3 ) 2 + (z z 3 ) 2 = r 2 3 (x x 4 ) 2 + (y y 4 ) 2 + (z z 4 ) 2 = r 2 4 Where x i, y i, z i are the known coordinates of the satellites in the space and x, y, z are the unknown coordinate of the observer on the earth as shown on the Figure 3.2. We now subtract the first of these equations from each of the three. This eliminates quadratic terms in x, y, z and leads to the following set of nonhomogeneous linear equation in x, y, z:
28 3.2. NUMERICAL EXPRESSION OF THE COORDINATES 19 Figure 3.2: Satellite and Observer in the Global Positioning System (S) 2(x 2 x 1 )x + 2(y 2 y 1 )y + 2(z 2 z 1 )z = r 2 1 x 2 1 r x 2 2 2(x 3 x 1 )x + 2(y 3 y 1 )y + 2(z 3 z 1 )z = r 2 1 x 2 1 r x 2 3 2(x 4 x 1 )x + 2(y 4 y 1 )y + 2(z 4 z 1 )z = r 2 1 x 2 1 r x 2 4 This system can be solved by either Mathematica or Matlab containing package for Gaussian elimination. Let us consider an actual numerical example. Assuming that the coordinate of the satellites with respect to a chosen point on the earth are given by the following values (in meters): x 1 = 2, 088, x 3 = 35, 606, y 1 = 11, 757, y 3 = 94, 447, z 1 = 25, 391, z 3 = 9, 101, x 2 = 11, 092, x 4 = 3, 966, y 2 = 14, 198, y 4 = 7, 362, z 2 = 21, 471, z 4 = 26, 388, Suppose that in addition the distances registered by the GPS devices are
29 20 CHAPTER 3. NUMERICAL METHOD OF GPS TRACKING SIGNAL given by: r 1 = 23, 204, r 2 = 21, 585, 835, 37 r 3 = 31, 364, r 4 = 24, 966, Using the Gaussian Elimination to solve the system (S), the following results are obtained: x = m y = m z = m These are the coordinates of the observer with respect to a fixed known origin on earth. Hence we have here the application of simple expressions drawn from the analytical geometry of sphere and the elementary use of Gaussian elimination to arrive at the solution of a problem of some sophistication. The practical implementation of the GPS had to await the deployment of geosynchronous satellite and the hardware needed to produce compact GPS devices, which can now be installed in automobiles. It is worth mentioning here that three equation (i.e., the use of only three satellites in our case given that many can be involved) would have been sufficient to calculate the coordinates x, y, z of the observer. In practice problem when many satellites come into play. Hence the need to develop computer algorithm that could be used to solve the problem in such situation. 3.3 Numerical Solution Using Mathematica The computer outpout obtained in connection with the above problem are as follow: GPS The 3dimensional coordinates of each satellite are the following: x1 := y1 := z1 := x2 := y2 := z2 :=
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