Filter-Type Fault Detection and Exclusion on Multi-Frequency GNSS Receiver

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This thesis is concerned with topics on the problems of satellite fault detection and exclusion (FDE). The purpose of FDE is to detect the presence of unacceptably large positioning error and, further, to exclude the source causing the error, thereby allowing the satellite navigation to continue. To enhance the capability of the existing fault detection and exclusion methods, we propose three type FDE algorithms based on the multi-frequency technique, the auto-regressive moving average (ARMA) filter technique and the Kalman filter technique, respectively.

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Filter-Type Fault Detection and Exclusion on Multi-Frequency GNSS Receiver

  1. 1. 國立臺灣大學電機工程學研究所博士論文 指導教授:張帆人 博士 濾波式故障偵測與排除 於多頻 GNSS 接收機之應用 Filter-Type Fault Detection and Exclusion on Multi-Frequency GNSS Receiver 研究生:蔡宜學 撰 中華民國九十三年十一月
  2. 2. i Acknowledgements At the end of my modest work, I want to express my humble thanks to the people, whom has guided me throughout a long process of studies and research. I am extremely grateful to Prof. Fan-Ren Chang, supervisor of my thesis, for generously helping and directing me in my work. It was my pleasure to work under his guidance. Thanks to his benevolence and patience, I have had many opportunities to make abundant explorations, which have enriched my life and broadened my horizons. In addition, I thank all the members of the Oral Examination Committee for their precious questions and constructive suggestions. Special thanks are due to Dr. Weh-Chieh Yang for his professional enlightenment on the thesis. He was my best research partner in developing the techniques used in the thesis. His scholarly advice often cast light on complicated problems. Moreover, I wish to acknowledge the help given by Dr. Chih-Min Fan in editing the manuscript. In addition, I owe a great debt to Wei-Cheng Lin for his encouragement. When I was experiencing difficulties in my research, he was there to encourage me. I am very fortunate to have the opportunity to work with the other members in the Control and Decision Laboratory. They are He-Sheng Wang, Hsin-Min Peng, Yi-Te Chiang, Kun-Yuan Tu, Ter-Feng Wu, Pu-Sheng Tsai, Tsung-Ching Lin, Shao-I Chu, Jie-Shan Lin, Chia-Lung Ma, Ching-Lun Tsai, etc. Last but not the least, I would like to appreciate my parents, whose endless love and untir- ing support gave me the fortitude to pursue this thesis.
  3. 3. iii Abstract This thesis is concerned with topics on the problems of satellite fault detection and exclu- sion (FDE). The purpose of FDE is to detect the presence of unacceptably large positioning error and, further, to exclude the source causing the error, thereby allowing the satellite naviga- tion to continue. To enhance the capability of the existing fault detection and exclusion meth- ods, we propose three type FDE algorithms based on the multi-frequency technique, the autore- gressive moving average (ARMA) filter technique and the Kalman filter technique, respectively. At the first part of this thesis, algorithms using multi-frequency measurements are proposed for GNSS (GPS + Galileo) positioning and FDE. Conventional algorithms adopt only the sin- gle frequency L1. However, GPS satellites carrying the L2 and L5 signals for civil use will soon be launched in 2005, and the Galileo system will be fully operational in 2008. Since Galileo will be interoperable with GPS, receivers can be designed to simultaneously access both Galileo and GPS systems. Hence, the number of visible GNSS satellites will be significantly increased. Using the multi-frequency technique can eliminate the ionospheric effect because it is highly related to the carrier frequency of the signal. In addition, the new signals can also be regarded as a backup, and this will significantly increase the safety of navigation. Therefore, application of multi-frequency algorithms will improve the positioning accuracy, shorten the failure detection time, and reduce the incorrect exclusion rate (IER). Simulation results show that, in comparison with the conventional single frequency method, the proposed multi-fre- quency algorithms not only possess more accurate positioning results but also demonstrate higher performance in detecting and excluding failures. At the second part of this thesis, we propose an algorithm based on the autoregressive mov- ing average to perform satellite failure detection and exclusion. ARMA filter is widely used in the field of quality control as a tool for fault diagnosis. It uses the historical data as well as the up-to-date information since failure may exist in past measurements before it is detected. The proposed algorithm includes fault detection and fault exclusion. For fault detection, the ARMA-filter is proposed to speed up the detection time by taking the average of the last several sums of the squares of the range residual errors. Speeding up of the failure detection can pro-
  4. 4. iv vide more time for pilots to prevent serious deviations of vehicles from their intended paths. In order to determine the detection threshold under a specified false alarm rate (FAR), the ARMA model is firstly transformed into the state-space model, and the threshold can then be approxi- mated by a “discrete finite-state Markov chain”. Moreover, the alteration of the number of visible satellites will cause problems in data fusion. The probability integral transformation (PIT) method is adopted to solve it. As for fault exclusion, the multivariate ARMA-filter is proposed to reduce the IER by taking the average of the last several parity vectors. Simulation results show that, in comparison with the conventional fault detection methods, the ARMA-filter has higher performance in detecting small failures and however, in detecting large failures, their performances are similar. Moreover, simulation results also verify that the proposed method can reduce the IER in excluding the failed satellite. At the third part of this thesis, we propose an algorithm based on a parallel bank of Kalman filters to perform satellite positioning and FDE. Conventionally, the well known position-ve- locity-acceleration (PVA) model is adopted as the dynamic model of Kalman filter for navigation. However, as a moving vehicle accelerates or slows down furiously, or as the vehicle corners at faster speeds, the conventional PVA model without using extra sensors (such as inertial naviga- tion sensors) can no longer be adequate for describing the motion of the vehicle. Therefore, the positioning result of the vehicle will become less accurate. Moreover, the normalized innova- tion squared (NIS) will deviate from the chi-square distribution and is no longer suitable as the test statistic for FDE. To overcome these problems, the delta range (DR) equation is proposed to accurately model the dynamic behavior of a maneuvering vehicle. Simulation results show that using the proposed DR to replace the PVA model can obtain better positioning and FDE re- sults as the vehicle maneuvers. Furthermore, as a satellite fails at a specified time and if the range measurements associated to the failed one is not yet excluded, the positioning result of the vehicle will become inaccuracy and even unusable. To solve this, an algorithm based on multi- ple model (MM) approach is proposed. Simulation results also present that, compared to the original Kalman filter, the proposed MM can perform positioning well as the satellite is failed.
  5. 5. v 摘要 本論文之議題為探討衛星故障之偵測與排除 (fault detection and exclu- sion; FDE),其目地是為了偵測因衛星故障所引發之嚴重定位誤差,並且進一 步 將 產 生 誤 差 的 來 源 消 除 , 以 使 導 航 能 夠 延 續 ; 在 此 提 出 分 別 以 多 頻 (multi-frequency) 技術、自我迴歸移動平均 (autoregressive moving average; ARMA) 濾波器、卡爾曼濾波器 (Kalman filter) 為基礎之三種演算法,以改 進現有之故障偵測與排除演算法。 本文首先提出使用多頻量測量於 GNSS (GPS + Galileo) 定位以及故障與 排除之演算法;傳統演算法只採用 L1 單頻,然而民用 L2 與 L5 訊號的 GPS 衛 星將在 2005 年發射升空,且 Galileo 系統預計於 2008 年完全運轉;因 Galileo 將與 GPS 相容運轉,接收機可以設計同時接收 Galileo 與 GPS 系統之訊號; 所以對於 GNSS 系統之可視衛星數目將會因此而大幅提升。因電離層效應與 載波頻率息息相關,採用多頻量測量將可以消除該效應;此外新載波可作為 備用之量測量,藉此可提升導航之安全性,所以採用多頻演算法將可以提升 定位精確度、縮短故障偵測時間以及減低故障排除錯誤率 (incorrect exclusion rate; IER)。由模擬結果得知,多頻演算法比起傳統的 L1 單頻演算法不僅有較 精確之定位結果,更能在偵測與排除故障方面有良好成果。 本文接著提出以自我迴歸移動平均 (ARMA) 之演算法來達成衛星系統 故障之偵測與排除,該方法在品管領域已被廣泛採用為故障診斷的工具;因 故障在被偵測到之前可能已經存於量測量之中,該演算法不僅使用到現在的 資料,更用到了先前的資料;其中所提出的演算法包含故障偵測與故障排除 兩部分。在故障偵測上,我們提出以自我迴歸移動平均濾波器為基礎之演算 法,其藉由對現在與先前之虛擬距離殘差平方和 (sums of the squares of the range residual errors) 之資料來加權平均,以縮短故障偵測時間;而提早偵測
  6. 6. vi 到故障可提供駕駛員更多反應時間,以避免載具嚴重偏離預定路徑。然而在 決定某一特定假警報率 (false alarm rate; FAR) 下之偵測臨界值 (detection threshold) 時 , 我 們 乃 先 將 自 我 迴 歸 移 動 平 均 模 型 轉 換 至 狀 態 空 間 (state-space) 模型,再藉由離散有限狀態馬可夫鏈 (Markov chain) 近似法求 得臨界值。再者空中可視衛星數目會隨著時間而改變,做資料匯集時,不能 直接運算,本文採用機率積分轉換 (probability integral transformation; PIT) 的 技巧來解決。至於在故障排除上,我們採用多變量自我迴歸移動平均 (mul- tivariate ARMA) 濾波器,藉由對現在與先前之同位向量 (parity vector) 之資 料加權平均,來減低故障排除錯誤率。由模擬結果得知,相較於原來的方法, 自我迴歸移動平均濾波器對於偵測小故障量有較佳的性能,而對於偵測大故 障量而言其效果差異不大;最後由模擬結果,我們驗證所提出之故障排除演 算法能夠減低故障排除錯誤率。 本文最後提出並列式卡爾曼濾波器 (Kalman filter) 來達成衛星定位以及 故障之偵測與排除。傳統上,卡爾曼濾波器採用常見的位置-速度-加速度 (position- velocity-acceleration; PVA) 模型為載具之動態模型。然而,在缺乏 額外感知器 (如慣性導航感知器) 的情況下,傳統位置-速度-加速度模型將無 法正確描述載具劇烈加減速度亦或高速轉彎之移動狀態。因此載具之定位結 果將變得較不精確。再者,標準化資訊創新平方 (normalized innovation squared; NIS) 將不再是卡方分布 (chi-square distribution),因而無法作為故障偵測與 排除之檢定統計量 (test statistic);而為了解決該問題,我們採用距離差量 (delta range; DR) 方程式來描述速變載具 (maneuvering vehicle) 之動態模 型;由模擬結果得知,在載具速變時,若採用距離差量方程式取代位置-速度- 加速度模型,將對於定位以及處理故障偵測與排除上可得到較佳的結果。此 外,當衛星故障發生且故障尚未被排除之前,載具之定位結果將變的不合理 甚至因此而無法使用。在此,我們乃採用多模型 (multiple model; MM) 演算 法的技巧來解決。從模擬結果顯示,相較於原來的方法,多模型演算法在衛 星故障發生時,亦能正常的執行定位功能。
  7. 7. vii Contents Acknowledgements...............................................................................................................................i Abstract...............................................................................................................................................iii Contents.............................................................................................................................................vii List of Figures ....................................................................................................................................xi List of Tables.....................................................................................................................................xiii List of Abbreviates.............................................................................................................................xv List of Symbols .................................................................................................................................xix Chapter 1 Introduction........................................................................................................................1 1.1 Scope of Thesis ..................................................................................................................2 1.1.1 Snapshot FDE Algorithms: Multi-frequency Technique..................................................... 2 1.1.2 Filter-type FDE Algorithm: ARMA Filter Technique ......................................................... 3 1.1.3 Filter-type FDE Algorithm: Kalman Filter Technique........................................................ 4 1.2 Organization of Thesis......................................................................................................5 Chapter 2 GNSS Architecture, Observables, and Fault Detection and Exclusion Algorithms .......7 2.1 Global Navigation Satellite System (GNSS) Architecture.............................................8 2.1.1 Global Positioning System (GPS)....................................................................................... 8 2.1.2 Global Navigation Satellite System (GLONASS) .............................................................. 9 2.1.3 Galileo............................................................................................................................... 10 2.2 Observables .....................................................................................................................12
  8. 8. viii 2.2.1 Pseudorange Measurement ................................................................................................12 2.2.2 Delta Range (Carrier Phase Difference) Measurement......................................................13 2.2.3 Doppler Shift Measurement...............................................................................................13 2.2.4 Ionospheric Effect and Ionosphere-free Measurements.....................................................14 2.3 Conventional Fault detection and Exclusion Algorithms............................................15 2.3.1 Least-Squares-Residuals....................................................................................................15 2.3.2 Parity Space Method..........................................................................................................18 Chapter 3 Using Multi-Frequency Technique on Fault Detection and Exclusion.........................21 3.1 Linearization of GNSS Pseudorange Measurements...................................................22 3.2 GPS Dual Frequency (L1/L2) Algorithms ....................................................................22 3.2.1 Positioning Algorithm........................................................................................................23 3.2.2 Fault Detection and Exclusion Algorithm..........................................................................24 3.3 GPS Triple Frequency (L1/L2/L5) Algorithms.............................................................25 3.3.1 Positioning Algorithm........................................................................................................26 3.3.2 Fault Detection and Exclusion Algorithm..........................................................................26 3.4 GNSS (L1/L2/E2-L1-E1/E6) Algorithms.......................................................................29 3.4.1 Positioning Algorithm........................................................................................................29 3.4.2 Fault Detection and Exclusion Algorithm..........................................................................30 3.5 Simulation Results and Analysis....................................................................................32 3.5.1 Positioning .........................................................................................................................32 3.5.2 Fault Detection...................................................................................................................35 3.5.3 Fault Exclusion ..................................................................................................................39 Chapter 4 Fault Detection and Exclusion via ARMA-Filter...........................................................41 4.1 Using ARMA-Filter on Fault Detection ........................................................................42 4.2 Determination of the Threshold Value for MA-Filters ................................................42
  9. 9. ix 4.2.1 MA-filter with Window Size 2.......................................................................................... 44 4.2.2 MA-filter with Window Size 3.......................................................................................... 46 4.2.3 MA-filter with Window Size Larger than 3 ...................................................................... 48 4.3 Determination of the Threshold Value for PFARMA-Filters......................................51 4.3.1 PFARMA -filter with Order 1 ........................................................................................... 51 4.3.2 PFARMA-filter with Order 2............................................................................................ 53 4.3.3 PFARMA-filter with Order Larger than 2......................................................................... 56 4.4 Data Adjustment for Different Numbers of Satellites - PIT .......................................58 4.5 Using Multivariate ARMA-filter on Fault Exclusion ..................................................60 4.6 Simulation Results and Analysis....................................................................................61 4.6.1 Fault Detection and Exclusion via MA-filters .................................................................. 61 4.6.2 Fault Detection and Exclusion via PFARMA-filters......................................................... 68 Chapter 5 Fault Detection and Exclusion via Kalman Filter .........................................................75 5.1 Linearization of the Ionosphere-Free Measurements for Kalman Filter ..................76 5.2 Using Conventional PVA Model on Kalman Filter......................................................77 5.2.1 Positioning Algorithm....................................................................................................... 77 5.2.2 Fault Detection and Exclusion Algorithm......................................................................... 78 5.3 Using Delta Range Equation on Kalman Filter for a Maneuvering Vehicle .............80 5.3.1 Positioning Algorithm....................................................................................................... 81 5.3.2 Fault Detection and Exclusion Algorithm......................................................................... 82 5.4 Using Multiple Model Approach on Kalman Filter.....................................................84 5.5 Simulation Results and Analysis....................................................................................87 5.5.1 Using Delta Range Equation on Kalman Filter for a Maneuvering Vehicle..................... 87 5.5.2 Using Multiple Model Approach on Kalman Filter .......................................................... 89 Chapter 6 Conclusions and Future Works.......................................................................................97
  10. 10. x 6.1 Conclusions ......................................................................................................................97 6.1.1 Using Multi-Frequency on FDE ........................................................................................97 6.1.2 Using ARMA Filter on FDE ..............................................................................................98 6.1.3 Using Kalman Filter on FDE .............................................................................................98 6.2 Future Works...................................................................................................................99 6.2.1 Using Multi-Frequency on FDE ........................................................................................99 6.2.2 Using ARMA Filter on FDE ..............................................................................................99 6.2.3 Using Kalman Filter on FDE ...........................................................................................100 Bibliography ....................................................................................................................................101 Appendix A Required Navigation Performance (RNP) .................................................................105 Appendix B History of GPS and GLONASS Satellites..................................................................107 Appendix C Parity Space Method...................................................................................................113 C.1 Maximization of Conditional Probability ...................................................................113 C.2 Existence of the Parity Matrix .....................................................................................114 Appendix D Partition Matrix Inverse.............................................................................................117 Appendix E Simulation Environments...........................................................................................119
  11. 11. xi List of Figures Figure 2-1 Determination of detection threshold Td with eight visible satellites...............................16 Figure 2-2 Parity space plot with six visible satellites.......................................................................19 Figure 3-1 Positioning error...............................................................................................................35 Figure 3-2 ADT under ramp-type failure...........................................................................................37 Figure 3-3 ADT under step-type failure.............................................................................................38 Figure 3-4 Incorrect exclusion rate....................................................................................................40 Figure 4-1 States of Markov chain for MA-filter with window size 2 ..............................................44 Figure 4-2 Transient of Markov chain for MA-filter with window size 2.........................................45 Figure 4-3 Initial states of Markov chain for MA-filter with window size 2 ....................................45 Figure 4-4 States of Markov chain for MA-filter with window size 3 ..............................................46 Figure 4-5 Definition of υ1(i) and υ2(i) for L′ = 3.............................................................................47 Figure 4-6 Transient of Markov chain for MA-filter with window size 3.........................................47 Figure 4-7 Initial states of Markov chain for MA-filter with window size 3 ....................................48 Figure 4-8 Flow chart for the threshold calculation of MA-filter......................................................50 Figure 4-9 States of Markov chain for PFARMA-filter with order 1 ................................................52 Figure 4-10 Transient of Markov chain for PFARMA-filter with order 1.........................................52 Figure 4-11 Initial states of Markov chain for PFARMA-filter with order 1 ....................................53 Figure 4-12 States of Markov chain for PFARMA-filter with order 2 ..............................................54 Figure 4-13 Transient of Markov chain for PFARMA-filter with order 2.........................................54
  12. 12. xii Figure 4-14 Initial states of Markov chain for PFARMA-filter with order 2.....................................55 Figure 4-15 Flow chart for the threshold calculation of PFARMA-filter ..........................................57 Figure 4-16 Illustration of PIT method ..............................................................................................59 Figure 4-17 Ramp-type pseudorange error (MA-filter).....................................................................63 Figure 4-18 Step-type pseudorange error (MA-filter)........................................................................64 Figure 4-19 IER under a ramp-type failure (multivariate MA-filter) ................................................66 Figure 4-20 IER under a step-type failure (multivariate MA-filter) ..................................................67 Figure 4-21 Ramp-type pseudorange error (PFARMA1-filter) .........................................................69 Figure 4-22 Step-type pseudorange error (PFARMA1-filter)............................................................70 Figure 4-23 IER under a ramp-type failure (multivariate PFARMA1-filter).....................................72 Figure 4-24 IER under a step-type failure (multivariate PFARMA1-filter) ......................................73 Figure 5-1 Using PVA model on a parallel bank of Kalman filter for FDE algorithm ......................80 Figure 5-2 Using DR equation on a parallel bank of Kalman filter for FDE algorithm ....................84 Figure 5-3 Using MM approach on Kalman filter for positioning.....................................................86 Figure 5-4 Positioning errors and innovations...................................................................................88 Figure 5-5 Using MM approach on Kalman filter (slope = 0.2 m/s) .................................................91 Figure 5-6 Using MM approach on Kalman filter (slope = 0.5 m/s) .................................................92 Figure 5-7 Using MM approach on Kalman filter (slope = 1 m/s) ....................................................93 Figure 5-8 Using MM approach on Kalman filter (slope = 2 m/s) ....................................................94 Figure 5-9 Using MM approach on Kalman filter (bias = 20 m).......................................................95 Figure 5-10 Using MM approach on Kalman filter (bias = 30 m).....................................................96 Figure A-1 RNP types ......................................................................................................................105
  13. 13. xiii List of Tables Table 2-1 Comparisons of the three GPS L-Band signals ...................................................................9 Table 2-2 Comparison of the four Galileo L-Band signals................................................................11 Table 2-3 The value of detection threshold Td for ( )ks ....................................................................17 Table 3-1 Standard deviation of the positioning error .......................................................................33 Table 3-2 Geographic locations for simulation..................................................................................36 Table 4-1 Threshold value of MA-filter with 6 visible satellites under FAR = 1/15000...................61 Table 4-2 Threshold value of PFARMA1-filter with 6 visible satellites under FAR = 1/15000 .......68 Table 5-1 The value of detection threshold Td for ( )ksPVA ................................................................79 Table 5-2 The value of detection threshold Td for ( )ksDR .................................................................83 Table A-1 RNP values for non precision approach phases of flight ................................................106 Table A-2 RNP values for precision approach phases of flight .......................................................106 Table B-1 History of GPS Satellites and Status...............................................................................107 Table B-2 History of GLONASS Satellites and Status....................................................................109 Table E-1 User-satellite geometry....................................................................................................120 Table E-2 Standard derivations of the pseudorange measurement noise.........................................121 Table E-3 Standard derivations of the ionoshere-free measurement noise......................................122
  14. 14. xv List of Abbreviates ADT average detection time AR autoregressive ARMA autoregressive moving average A-S anti-spoofing ATM air traffic management C/A code coarse/acquisition code cdf cumulative distribution function CDMA code division multiple access CNS communication, navigation and surveillance CTS command tracking stations CUSUM cumulative sum DOD Development of Defense DOF degrees of freedom DOP dilution of precision DF dual frequency DR delta range DT detection time ESA European Space Agency
  15. 15. xvi EWMA exponentially weighted moving average FAA Federal Aviation Administration FANS future air navigation systems FAR false alarm rate FDE fault detection and exclusion FDMA frequency division multiple access FOC full operational capability GCC Galileo control centers GCS ground-based control complex GDOP geometric dilution of precision GLONASS global navigation satellite system GNSS global navigation satellite system GPB generalized pseudo-Bayesian GPS global positioning system GSS Galileo sensor stations GSTB Galileo system test bed HDOP horizontal position dilution of precision HP high precision (navigation signal) ICAO International Civil Aviation Organization IE incorrect exclusion IER incorrect exclusion rate IMM interacting multiple model IOV in-orbit validation
  16. 16. xvii KF Kalman filter LAAS local area augmentation system LRT likelihood ratio test LSR least-squares-residuals MA moving average MEO medium earth orbit MF multi-frequency MM multiple model MTFA mean time to false alarm NIS normalized innovation squared OCS operational control system P code precise code pdf probability density function PDOP position dilution of precision PFARMA parallel-form ARMA PIADT percentage improvements of ADT PIT probability integral transformation PRN pseudorandom noise PVA position-velocity-acceleration RAIM receiver autonomous integrity monitoring RNP required navigation performance SA selective availability SAR search and rescue
  17. 17. xviii SCC system control center SP standard precision (navigation signal) SSE sum of the squares of the range residual errors SVD singular value decomposition TEC total electronic content TF triple frequency TFA time to false alarm TLS total least square US United States UTC universal time coordinated VDOP vertical position dilution of precision WAAS wide area augmentation system
  18. 18. xix List of Symbols DFα 2 2 2 2 2 1 2 1 −−−− += LLLL ff σσ DFα ) 2 6 2 6 2 1 2 1 −−−− += EEEE ff σσ iα coefficients of the ARMA-filter TFα 2 5 2 5 2 2 2 2 2 1 2 1 −−−−−− ++= LLLLLL fff σσσ β the weighting vector of the ARMA-filter and the MA-filter DFβ 4 2 2 2 4 1 2 1 −−−− += LLLL ff σσ DFβ ) 4 6 2 6 4 1 2 1 −−−− += EEEE ff σσ iβ weights of the ARMA-filter and the MA-filter TFβ 4 5 2 5 4 2 2 2 4 1 2 1 −−−−−− ++= LLLLLL fff σσσ ( )νχ2 chi-square distribution with ν degrees of freedom DFδ ( ) 222 2 2 1 DFDFLL αβσσ −+= −− DFδ ) ( ) 222 6 2 1 DFDFEE αβσσ )) −+= −− ( )kRδ receiver clock bias deviation ( )kRδ& ( )( ) ktR dttd == δ TFδ ( ) 222 5 2 2 2 1 TFTFLLL αβσσσ −++= −−− ( )kφ delta range measurement vector ( )kφ true range difference vector ( )kDRΦ transition matrix in Kalman filter using DR equation ( )kGIFφ GNSS ionospheric-free delta range measurement vector ( )kL1φ delta range measurement vector corresponding to the L1 signal
  19. 19. xx ( )kL12φ GPS ionospheric-free delta range measurement vector ( )kL2φ delta range measurement vector corresponding to the L2 signal MAΦ transition matrix of the MA-filter PFARMAΦ transition matrix of the PFARMA-filter ( )kPVAΦ transition matrix in Kalman filter using PVA model ( )kiϕ prior probability that the ith model, iΘ , is correct ( )kγ the satellite clock offset to the GPS time ( )kγ& ( )( ) ktdttd == γ η the weighting vector of the PFARMA-filter iη weights of the PFARMA-filter ( )kiΛ likelihood function of the ith model, iΘ ( )kiκ 40.3 times the TEC associated to the ith satellite ( )kκ the vector with the ith element as ( )kiκ ( )kκ& ( )( ) ktdtkd == κ ( )kDFκˆ estimate of ( )kκ using dual freq. ( )kTFκˆ estimate of ( )kκ using triple freq. λ [ ]T nl λλλ −−−= 111 1 LL iλ parameters of the PFARMA-filter ( )kθ state vector of the ARMA filter, the MA-filter and the PFARMA-filter iΘ model of the ith Kalman filter using MM approach ( )kπ probability vector representing the distribution of ( )kz in transient states ( )kρ pseudorange measurement vector ( )kρ true range vector GIFρ ionospheric-free GNSS pseudorange measurement vector
  20. 20. xxi 1Lρ pseudorange measurement vector corresponding to the L1 signal 12Lρ ionospheric-free GPS pseudorange measurement vector 2Lρ pseudorange measurement vector corresponding to the L1 signal 1Lσ standard derivation of the L1 measurement noises 2Lσ standard derivation of the L2 measurement noises 5Lσ standard derivation of the L5 measurement noises ( )kω delta range measurement noise vector ( )kΩ covariance matrix of the delta range measurement noise vector ( )kω ( )kτ tropospheric delay ( )kτ& ( )( ) ktdtkd == τ ( )kψ Doppler shift measurement vector ( )kψ true Doppler shift vector ( )kGIFψ ionospheric-free GNSS Doppler shift measurement vector 1Lψ Doppler shift measurement vector corresponding to the L1 signal 12Lψ ionospheric-free GPS Doppler shift measurement vector 2Lψ Doppler shift measurement vector corresponding to the L2 signal ( )kiζ the ith column vector of the matrix ( )kS ( )kζii the ith diagonal element of the matrix ( )kS c speed of light 1Lc ( )2 1 222 1 −−− −= LTFTFTFL fαβδσ 2Lc ( )2 2 222 2 −−− −= LTFTFTFL fαβδσ 5Lc ( )2 5 222 5 −−− −= LTFTFTFL fαβδσ ( )kd the predicted range vector based on the reference point ( )krefx ( )kd& the predicted range rate vector based on the reference point ( )krefx
  21. 21. xxii []⋅E expectation operator in,e an n×1 column vector with all elements zeros except the i-th element is one ()⋅νF cdf for the chi-square distribution with degree of freedom ν 1Ef the carrier frequency of the E2-L1-E1 signal 6Ef the carrier frequency of the E6 signal 1Lf the carrier frequency of the L1 signal 2Lf the carrier frequency of the L2 signal 5Lf the carrier frequency of the L5 signal ( )kH observation matrix ( )kH ) Galileo observation matrix ( )kDFH observation matrix for the dual freq. GPS ( )kDFH ) observation matrix for the dual freq. GNSS ( )kTFH the observation matrix for the triple freq. GPS ( )kPVAH measurement matrix for the PVA model nI an n× n identity matrix k current time instant; discrete time ( )kDRK Kalman gain in Kalman filter using DR equation ( )kPVAK Kalman gain in Kalman filter using PVA model ( )kL input gain matrix in Kalman filter using DR equation nl an n×1 column vector with all elements equal to one m number of Galileo satellites in view M window size of the ARMA-filter and the MA-filter n number of satellites in view N the order of the ARMA-filter and the PFARMA-filter
  22. 22. xxiii 0 zero matrix; a matrix with all elements zeros ( )kp parity vector ( )kP parity matrix ( )kDFp parity vector for the dual freq. GPS ( )kDFP parity matrix for the dual freq. GPS ( )kDFp ) parity vector for the dual freq. GNSS ( )kDFP ) parity matrix for the dual freq. GNSS ( )kE iDF 1 ,p ) the i-th channel vector corresponding to the E2-L1-E1 signal for GNSS ( )kE iDF 6 ,p ) the i-th channel vector corresponding to the E2 signal for GNSS ( )kL iDF 1 ,p the i-th channel vector corresponding to the L1 signal for the dual freq. GPS ( )kL iDF 2 ,p the i-th channel vector corresponding to the L2 signal for the dual freq. GPS ( )kL iDF 1 ,p ) the i-th channel vector corresponding to the L1 signal for GNSS ( )kL iDF 2 ,p ) the i-th channel vector corresponding to the L2 signal for GNSS ( )k |kDRP updated state covariance in Kalman filter using DR equation ( )k |klDR,P updated state covariance corresponding to the ith model, iΘ ( )1−k |kDRP state prediction covariance in Kalman filter using DR equation ( )k |kMMP covariance of the combined estimate in MM approach ( )k |kPVAP updated state covariance in Kalman filter using PVA model ( )1−k |kPVAP state prediction covariance in Kalman filter using PVA model ( )kTFp parity vector for the triple freq. GPS ( )kTFP parity matrix for the triple freq. GPS ( )kG iDF 1 ,p ) the i-th channel vector corresponding to the L1/E2-L1-E1 signal ( )kG iDF 2 ,p ) the i-th channel vector corresponding to the L2/E6 signal ( )kL iTF 1 ,p the i-th channel vector corresponding to the L1 signal (triple freq. GPS)
  23. 23. xxiv ( )kL iTF 2 ,p the i-th channel vector corresponding to the L2 signal (triple freq. GPS) ( )kL iTF 5 ,p the i-th channel vector corresponding to the L5 signal (triple freq. GPS) []⋅Pr probability operator ( )kDRQ covariance matrix of the process noise vector ( )kDRv PVAQ covariance matrix of the process noise vector ( )kPVAv ( )kR covariance matrix of the pseudorange measurement noise vector ( )kw ( )kvR covariance matrix of the Doppler shift measurement noise vector ( )kvw ( )kDFR covariance matrix of the measurement noise vector for the dual freq. GPS ( )kDFR ) covariance matrix of the measurement noise vector for the dual freq. GNSS PVAR covariance matrix of the measurement noise vector ( )kPVAw ( )kTFR covariance matrix of the measurement noise vector for the triple freq. GPS ( )ks normalized SSE ( )kS ( ) ( )kkT PP≡ ( )ksDF the test statistic of the failure detection for the dual freq. GPS ( )ksDF ) the test statistic of the failure detection for the dual freq. GNSS ( )ksDR NIS in Kalman filter using DR equation ( )ks iDR, NIS in the ith Kalman filter using DR equation ( )kDRS covariance matrix of the innovation vector ( )kDRy~ ( )ksi normalized SSE of the ith subset ( )kSi the ith transient state of the Markov chain ( )ksPVA NIS in Kalman filter using PVA model ( )ks iPVA, NIS in the ith Kalman filter using PVA model ( )kPVAS covariance matrix of the innovation vector ( )kPVAy~ ( )ksTF the test statistic of the failure detection for the triple freq. GPS
  24. 24. xxv T transition probability matrix dT detection threshold eT exclusion threshold ijT the ijth element of the transition probability matrix T St the sampling time; the time interval between two consecutive data ( )ku linearized ionosphere-free delta range measurement for Kalman filter ( )kDRv process noise vector for the DR equation ( )kPVAv process noise vector for the PVA model ( )kw pseudorange measurement noise vector ( )kvw Doppler shift measurement noise vector ( )kE1w measurement noise vector corresponding to the E2-L1-E1 signal ( )kE6w measurement noise vector corresponding to the E6 signal ( )kL1w measurement noise vector corresponding to the L1 signal ( )kL2w measurement noise vector corresponding to the L2 signal ( )kL5w measurement noise vector corresponding to the L5 signal ( )kPVAw measurement noise vector for the PVA model ( )kx state vector comprised of the true position (deviation) from the nominal position, plus the user clock bias (deviation) ( )kxˆ weighted least-squares estimate of the state vector ( )kax state vector comprised of the true acceleration, plus the change rate of the user clock drift rate ( )kDFxˆ estimate of the state vector for the dual freq. GPS ( )kGNSSDF, ˆx estimate of the state vector for the dual freq. GNSS ( )kkDR |ˆx updated estimate of the state vector in Kalman filter using DR equation ( )kklDR |ˆ ,x updated estimate of the state vector corresponding to the lth model, lΘ
  25. 25. xxvi ( )kkMM |ˆx combination of the model-conditioned estimates in MM approach ( )kposxˆ estimated positioning vector ( )kposx~ estimated positioning error ( )kPVAx state vector of the PVA model ( )kkPVA |ˆx updated estimate of the state vector in Kalman filter using PVA model ( )krefx reference point comprised of the nominal user position plus the nominal user clock drift ( )kTFxˆ estimate of the state vector for the triple freq. GPS ( )ktruex true position vector ( )kvx state vector comprised of the true velocity, plus the user clock drift rate ( )ky pseudorange measurement ( )ky~ ( ) ( )kk yy ˆ−= ; the residual vector in RAIM ( )kDFy measurement vector for the dual freq. GPS ( )kDFy ) measurement vector for the dual freq. GNSS ( )kE1y measurement vector corresponding to the E2-L1-E1 signal ( )kE16y ionospheric-free Galileo measurement vector ( )kE6y measurement vector corresponding to the E6 signal ( )kL1y measurement vector corresponding to the L1 signal ( )kL12y ionospheric-free GPS measurement vector ( )kL125y ( ) ( ) ( )kckckc LLLLLL 552211 yyy ++= ( )kL2y measurement vector corresponding to the L2 signal ( )kL5y measurement vector corresponding to the L5 signal ( )kTFy measurement vector for the triple freq. GPS ( )kz output variable of the ARMA-filter, the MA-filter and the PDARMA-filter ( )kz linearized ionosphere-free pseudorange measurement for Kalman filter
  26. 26. xxvii ( )kvz linearized ionosphere-free Doppler shift measurement for Kalman filter ( )kDRz~ innovation vector in Kalman filter using DR equation ( )kiDR, ~z innovation vector corresponding to the ith model, iΘ ( )kPVAz measurement vector in Kalman filter using PVA model ( )kPVAz~ innovation vector in Kalman filter using PVA model
  27. 27. 1 Chapter 1 Introduction Fault detection and exclusion (FDE) in the global navigation satellite system (GNSS) is a crucial issue in aviation navigation, as satellite failures may result in serious deviations of air- crafts from their intended flight paths. Sometimes the term “receiver autonomous integrity monitoring (RAIM)” is adopted for the similar concern. To ensure the safety of satellite-based navigation, interest in integrity monitoring techniques was spurred on by the Federal Aviation Administration (FAA) in the 1980s. In early years of the 1980s, the International Civil Aviation Organization (ICAO) realized the increasing limitations of the current air navigation systems, and recognized the need for improvement. Therefore, a new committee called the future air navigation systems (FANS) was formed by the ICAO. This committee was asked to make recommendations for the coordinated development of air navigation in the next 25 years. In 1988, this committee proposed the development of new communication, navigation and surveil- lance (CNS) means, and the elaboration of a new method for air traffic management (ATM). This proposition was called the CNS/ATM concept. Moreover, to ensure safe aircraft operation, the ICAO has established the navigation performance requirements for each phase of flight. These requirements can be expressed in terms of accuracy, integrity, availability and continuity of service, which are described in Appendix A. The integrity monitoring for a stand-alone GPS use is performed using the FDE algorithm [16]. For a satellite constellation where geometry can provide enough information redundancy, the FDE algorithm adopts a least squares criterion to check the integrity of the navigation solution. Since the FDE algorithm is so important to aviation navigation, three types of useful methods are proposed in this thesis to improve the ex- isted algorithms. In one of the proposed methods, extra measurements are used to increase the dimension of the information redundancy, and in the others, filters are adopted to process past as well as current uncorrelated information. These can indeed provide enhanced capability of the fault detection and exclusion.
  28. 28. 2 1.1 Scope of Thesis This thesis is concerned with topics on the problems of the fault detection and exclusion. FDE is a crucial issue in aviation navigation because the aircrafts travel at high speeds and can quickly deviate from its intended flight paths if a navigation satellite is failed. According to RTCA SC-159 [16], the purpose of the FDE is to detect the presence of unacceptably large posi- tion error and, further, to exclude the source causing the error, thereby allowing GPS navigation to continue. To achieve this goal, a number of FDE algorithms have been published over the last few decades. One of them is called “snapshot FDE algorithm” since it uses only current data to perform satellite fault detection and exclusion. Others are referred to as “filter-type FDE algorithm” since they process past as well as current uncorrelated information. The main goal of this thesis is to enhance the capability of the fault detection and exclusion, and thus we propose three types of FDE algorithms, i.e. the multi-frequency technique, the autoregressive moving average (ARMA) filter technique and the Kalman filter technique, respectively. Since the probability that two satellites failed at the same time is extremely low and can thus be ig- nored, it is assumed throughout this thesis that at most one satellite failure occurs at a time [16]. 1.1.1 Snapshot FDE Algorithms: Multi-frequency Technique To perform satellite fault detection and exclusion, several useful snapshot algorithms have been published at the ends of the 1980s. First, Parkinson and Axelrad [17] suggested a least-squares residual method for autonomous GPS satellite fault detection and exclusion. Next, Sturza [21] proposed the standard parity space algorithm to detect the satellite failure and further to exclude the range measurement associated to the failed one. Brenner [3] introduced an al- ternate parity space algorithm, in which the parity space coordinates are rotated to lie along with one coordinate axis to perform fault detection and exclusion. Then Kelly [14] proposed a maximum residual algorithm using likelihood ratio test (LRT) to achieve fault detection and ex- clusion. Moreover, he also compared these four methods and proved their equivalence. Per- van et al. [18] used the same algorithm but replaced the code measurement by more accuracy carrier phase measurement. In a later time, Juang and Jang [13] proposed a total least square (TLS) algorithm for the same purpose. In the proposed algorithm, the singular value decompo- sition (SVD) approach is adopted to determine the associated fault matrix. By using this matrix, the satellite failure can be detected and further the failed satellite can be identified. In this thesis, algorithms using multi-frequency measurements are proposed for GNSS (GPS + Galileo) positioning and FDE. The conventional algorithms adopt only the single frequency
  29. 29. 3 L1. However, a GPS modernization policy has been approved by the US government, and the C/A code (coarse/acquisition code) will be modulated onto the L2 and L5 signals for civil use [9]. In addition, the first experimental Galileo satellite will be launched in the second semester of 2005, and more, the operational Galileo satellites will be installed to reach the full operational capability (FOC) in 2008. Since Galileo will be interoperable with GPS, receivers can be de- signed to simultaneously access both Galileo and GPS systems. Hence, the number of visible GNSS satellites will be significantly increased. Therefore, more range measurements will be available, and the dimension of the parity space will be increased under a fixed number of visible satellites. The minimum angle between two channel vectors will thus be increased; as a conse- quence the performance of the failure exclusion will be improved. Furthermore, using the multi-frequency technique can eliminate the ionospheric effect because it is highly related to the carrier frequency of the signal. Monte Carlo simulations are conducted to verify that the pro- posed algorithm. Simulation results show that, in comparison with the conventional single fre- quency method, the proposed multi-frequency algorithms not only possess more accurate posi- tioning results but also demonstrate higher performance in detecting and excluding failures. 1.1.2 Filter-type FDE Algorithm: ARMA Filter Technique To be precise, the algorithms described above are static and depend only on current meas- urements. The statistical approaches such as CUSUM (cumulative sum) algorithm, EWMA (exponential weighted moving average) [20], GMA (geometric moving average) have been used as a tool to improve the capability of the fault detection and exclusion. These well-known al- gorithms have already been applied in the field of quality control. Younes et al. [28] proposed a sequential RAIM algorithm to detect satellite malfunction and also to exclude the failed one. To achieve this goal, the CUSUM algorithm is adopted to detect the occurrence of mean changes in GPS least square residuals by comparing the test statistic from CUSUM with the detection threshold determined through the sequential theory. In a later time, Yang et al. [26] proposed the exponential weighted moving average filter to perform fault detection. The EWMA-filter contains recursive formula, all the past measurement are used with different weighting to detect the possible failures. However, the proposed method only focuses on fault detection. An algorithm based on the autoregressive moving average, which has been widely used in the field of quality control as a tool for fault diagnosis, is proposed in this thesis to perform FDE. It uses the previous data as well as the current information, since failure may exist in past meas- urements before it is detected. The proposed algorithm includes two parts, fault detection and fault exclusion. In the first part, the ARMA-filter is proposed to speed up the fault detection by
  30. 30. 4 taking the average of the last several sums of the squares of the range residual errors. Speeding up of the fault detection might provide more time for pilots to prevent serious deviations of air- crafts from their intended flight paths. In order to calculate the detection threshold under a specified false alarm rate (FAR), the filter is first transformed into the state-space model, and then the threshold can be approximated by a “discrete finite-state Markov chain”. Although the calculation of threshold for ARMA-filter is complex, it is independent of satellite geometry. Therefore, it can be computed off-line and tabulated in computer memory. Because the num- bers of visible satellites may vary with time, the calculated SSE may have different distributions and cannot be directly applied to the ARMA-filter. Thus, the probability integral transforma- tion (PIT) [19] is adopted to deal with this problem. Note that, compared with the snapshot method, the primary computational burden of the ARMA detector lies in the PIT process; how- ever, the calculation of the PIT has been simplified in this thesis. In the second part, the multi- variate ARMA-filter is proposed to reduce the IER by taking the average of the last several par- ity vectors. Since the alteration of the composition of visible satellites will cause a problem in data fusion, a procedure by adjusting the measurements of the remaining satellites is suggested to solve it. Monte Carlo simulations are conducted to verify that the proposed algorithm. Simu- lation results show that, in comparison with the conventional fault detection methods, the ARMA-filter has higher performance in detecting small failures and, in detecting large failures, their performances are similar. Moreover, simulation results also verify that the proposed method has lower IER than the parity space method has. 1.1.3 Filter-type FDE Algorithm: Kalman Filter Technique Besides, Kalman filter has already been widely used in the field of navigation. In order to ascertain normal navigation, several algorithms based on Kalman filter have been published to perform FDE. Brown and Hwang [5] proposed a dynamic algorithm by using a parallel bank of Kalman filters as a multiple hypothesis tester to detect failure. The GPS pseudorange meas- urement is first subtracted by a given failure signal and then applied to a Kalman filter. For each type of failure to be protected, a corresponding Kalman filter is necessary. In order to de- tect all kinds of failures, a set of Kalman filters are required. However, some assumptions on the “signatures” of the failure signal are required, or the approach can not be applied. Da and Lin [7] proposed a dynamic algorithm to detect GPS failure by using two Kalman filters. One filter processes only measurement from a GPS receiver, and the other filter processes measure- ments from the GPS receiver and inertial navigation sensors (INS). By using the estimated states and the associated covariance matrices, a test statistic can be obtained for performing sat-
  31. 31. 5 ellite fault detection. The proposed method, however, only focuses on fault detection. Young and McGraw [29] also proposed a dynamic algorithm to perform failure detection by using two Kalman filters. Measurements from INS are also needed in this algorithm. Moreover, a par- allel bank of Kalman filters is adopted to exclude the failed satellite. It can be achieved by re- moving one of the visible satellites at a time for each Kalman filter in the parallel bank. We propose an alternative algorithm based on a parallel bank of Kalman filters to perform satellite positioning and FDE. The position-velocity-acceleration (PVA) modeled Kalman filter has been widely used in the filed of navigation. Under the hypothesis that no failure is taking place and the vehicle is moving smoothly, the normalized innovation squared (NIS) will be chi-square distributed and thus can be used as a test statistic for FDE. However, as a moving vehicle accelerates or slows down furiously, or as the vehicle corners at faster speeds, the con- ventional PVA model without using extra sensors (such as INS) can no longer be adequate for describing the motion of the vehicle. Therefore, the positioning result of the vehicle will be- come less accurate. Also, the resulting NIS will deviate from the chi-square distribution and thus it is not suitable used as the test statistic for FDE. To overcome these problems, the delta range (DR) information is adopted to accurately describe the dynamic behavior of a maneuvering vehicle. Simulation results show that using the proposed DR to replace the PVA model can produce better positioning and FDE results as the vehicle maneuvers. In addition, as the satel- lite has failed at a specified time and if the range measurements associated to the failed one is not yet excluded, the positioning result of the vehicle becomes inaccuracy and even unusable. To solve this, an algorithm based on multiple model (MM) approach is proposed. MM is a well known approach adopted in target tracking. The basic idea behind MM is to assume a set of models as possible candidates of the true model. The set of candidate models then generates overall estimates by running a bank of Kalman filters. Since the true model is among the set of possible candidates, the correct positioning result will be figured out. From that, the range measurements associated to the failed satellite can be identified. Simulation results also present that, compared to the original Kalman filter, the proposed MM can perform positioning well as the satellite failed. 1.2 Organization of Thesis The remainder of this thesis is organized as follows. Chapter 2 describes the GNSS archi- tecture, observables and conventional FDE algorithms including the least-squares-residuals method and the parity space method. Chapter 3 is devoted to issues of an alternative snapshot FDE algorithm. In this chapter, the FDE algorithms using multi-frequency GNSS receiver are
  32. 32. 6 systematically derived, together with the related simulation results and analysis. Note that this chapter focuses on introducing the snapshot-type method; the filter-type algorithms will be dis- cussed in the following two chapters. Chapter 4 is concerned with the problem on the fault de- tection and exclusion via ARMA filters. In this chapter, we apply the ARMA-filter algorithms to the satellite fault detection and propose the Markov chain approach to determine the threshold of these filters. Moreover, the PIT method is adopted to resolve the problem caused by the al- ternation of the number of satellites in view. Furthermore, we apply the multivariate ARMA-filter algorithms to perform the satellite fault exclusion. In deriving the algorithms, the multi-frequency technique is adopted to eliminate the ionospheric effect. In addition, the re- lated simulation results and analysis are described at the end of this chapter. Chapter 5 shows the scheme of the fault detection and exclusion via Kalman filter. In this chapter, the delta range (DR) equation is proposed to accurately model the dynamic behavior of a maneuvering vehicle, and multiple model (MM) approach is proposed to reduce the positioning error before the range measurements associated to the failed satellite is excluded. Note that the iono- sphere-free measurements are used in the proposed algorithms. In addition, the related simula- tion results and analysis are given at the end of this chapter. Finally the conclusion remarks and future works are discussed in Chapter 6.
  33. 33. 7 Chapter 2 GNSS Architecture, Observables, and Fault Detection and Exclusion Algorithms Global navigation satellite system (GNSS) is the generic name given to the satellite-based navigation systems including GPS (global positioning system), GLONASS (global navigation satellite system), and Galileo. GPS is the first passive one-way ranging satellite system to be- come operational. While GPS was under development by United States (US), the Soviet Union undertook to develop a similar system, called GLONASS. Like GPS, GLONASS was designed primarily for the military, and was also offered for civil use. In a later time, the European Un- ion decided to develop a similar system planed to under civil control. This system is called Galileo, which is now developed by European Space Agency (ESA) and would be full opera- tional in 2008. In GNSS receiver, three different types of observables, the pseudorange meas- urement, the delta range (i.e., the carrier phase difference) measurement, and the Doppler shift measurement, can be obtained from the signals of a satellite. These measurements can be used to determine the position and velocity of an aircraft. In addition, to ensure safe aircraft opera- tion, the navigation performance requirements have been established by the International Civil Aviation Organization (ICAO). These requirements are expressed in terms of accuracy, integ- rity, availability and continuity of service. The integrity monitoring is currently performed us- ing the fault detection and exclusion (FDE) algorithm. The purpose of FDE is to detect the presence of unacceptably large position error and, further, to exclude the source causing the error, thereby allowing GNSS navigation to continue. To achieve this goal, a number of useful FDE algorithms have been developed. In this chapter, the architecture and the developments of GNSS will be introduced in Section 2.1. Moreover, three basic observables, the pseudorange measurement, the delta range measurement, and the Doppler shift measurement, are described in Section 2.2. Finally, two kinds of conventional FDE algorithms, least-squares residuals and parity space method, are discussed in Section 2.3.
  34. 34. 8 2.1 Global Navigation Satellite System (GNSS) Architecture 2.1.1 Global Positioning System (GPS) GPS, a space-based navigation system developed by the DOD (Department of Defense) of US, is available world-wide in all-weather in a common grid system. The GPS is comprised of three segments: the space segment, the control segment, and the user segment [10][32]. The space segment consists of satellites which transmit signals through space to receivers. These satellites have nearly circular orbits with an altitude of about 20200 km above the Earth and a period of 11 hours and 58 minutes. The present constellation consists of 24 operational satel- lites deployed in six equally spaced planes with an inclination of 55° and with four satellites per plane. Normally GPS contains features such as anti-spoofing (A-S) and selective availability (SA), which limit the full accuracy of the service only to authorized users and protection from spoofing. When A-S is active, the precise code (P code) on the L1 and the L2 carrier is re- placed by the Y code (i.e., encrypted P code). Moreover, as SA effect is turn on, the navigation accuracy is degraded by dithering the satellite clock and manipulating the ephemeredes. How- ever, on May 2, 2000 (UTC time), the effect of SA has already been shutdown. The operational control system (OCS), i.e., the control segment, consists of a master control station, five monitor stations and several ground control stations. Monitor stations measure pseudoranges from the satellites in view to compute precise orbital ephemeris data and satellite clock corrections for each satellite. Ground stations collocated with the monitor stations are the communication links to the satellites and mainly consist of the ground antennas. The satellite ephemeris and clock data, computed at the master control station, are uploaded to the satellites via ground stations. The satellites then send subsets of the ephemeris data to GPS receivers over radio signals. The user segment consists of the GPS receivers and the user communities. The PRN (pseudo random number) codes broadcast by the satellites enables a receiver to measure the tran- sit time of the signals and thereby determine the pseudorange between each satellite and the re- ceiver. By using the ephemeris data, the position of each satellite at the time the signals were transmitted can be calculated. Then the users’ own position can be determined. To improve the navigation performance of GPS, some augmentation systems for aviation, such as the wide area augmentation system (WAAS) and the local area augmentation system (LAAS) are devel- oped. The WAAS is a geographically expansive augmentation to the basic GPS service and improves the accuracy, integrity, and function together to supply users with seamless satel-
  35. 35. 9 lite-based navigation for all phases of flight. The LAAS will be used to fulfill existing naviga- tion and landing requirements (such as availability) at locations where the WAAS is unable to meet. GPS is the premier application of code division multiple access (CDMA), where the spread spectrum codes enable the satellites to transmit on the same frequencies simultaneously. Every satellite transmits two signals: the L1 signal, centered at the frequency of 1575.42 MHz, and the L2 signal, centered at the frequency of 1227.60 MHz. The L1 carrier is modulated by two types of PRN ranging codes, the coarse/acquisition code (C/A code) and the encrypted precise code (P code), and the L2 carrier is only modulated by the encrypted P code. Recently, a GPS modernization policy has been approved by the US government to improve the performance of the current navigation system [9]. The C/A code will be modulated onto the L2 and L5 signals for civil use. In addition, the C/A code in the L5 signal, centered at the frequency of 1176.45 MHz, is as precise as the P code in the L1 signal. The comparisons of the three L-Band signals (the L1, L2 and L5 signals) are listed in Table 2-1. The main advantages of the GPS moderni- zation are performance enhancement in positioning accuracy, fault detection and fault exclusion. Moreover, the new signal can be regarded as a backup, and thus will significantly increase the safety of navigation. Table 2-1 Comparisons of the three GPS L-Band signals Civil Signal L1 L2 L5 Frequency (MHz) 1575.42 1227.60 1176.45 Code Length 1023 10230 10230 Code Clock (MHz) 1.023 1.023 10.23 Phases Bi-Phase Bi-Phase Quad-Phase Bit-Rate (bps) 50 25 50 Fully Available (year) Now ~2011 ~2015 Ionospheric Error Ratio 1.00 1.65 1.79 IIR √ × × IIR-M √ √ ×GPS satellite IIF √ √ √ 2.1.2 Global Navigation Satellite System (GLONASS) The global navigation satellite system (GLONASS) is a satellite-base navigation system
  36. 36. 10 developed by the Russian Federation Government and managed by the Russian Space Forces [33][34]. Fully deployed constellation consists of 24 satellites in three orbital planes whose as- cending nodes are 120° apart. Eight satellites are equally spaced in each plane with argument of latitude displacement of 45°. Besides the planes have 15° argument of latitude displacement. Each satellite operates in circular 19100 km orbits at an inclination angle of 64.8° and each satel- lite completes an orbit in approximately 11 hours 15 minutes. The spacing of satellites in orbits is arranged so that a minimum of 5 satellites are in view to users’ world-wide, with adequate geometry. The satellite constellation is operated by ground-based control complex (GCS), consists of the system control center (SCC) and a several command tracking stations (CTS) placed over a wide area of Russia. The CTSs track the satellites in view and accumulate ranging data and te- lemetry from the satellites signals. The information from CTSs is processed at the SCC to de- termine satellite clock and orbit states and to update the navigation message of each satellite. This updated information is transmitted to the satellites via the CTSs. The CTSs ranging data is periodically calibrated using a laser ranging devices at the quantum optical tracking stations which are within GCS. The synchronization of all the processes in the system is done by the central synchronizer within GCS. The onboard time scales of all the satellites are synchronized with the State Etalon UTC (CIS) in Mendeleevo, Moscow region, through the GLONASS sys- tem time scale. Each satellite transmits two types of radiofrequency navigation signals: standard precision (SP) and high precision (HP). The navigation signals contain navigation message for users. SP signal L1 have a frequency division multiple access (FDMA) in L-band with the carrier cen- tered at ( )hf GLONASS hL 5625.01602,1 += MHz, where h denotes the frequency channel number (h = 0, 1, 2, etc). At most two satellites transmit signal on the same frequency. However those sat- ellites have the same frequencies are placed in antipodal slots of orbit planes and thus they will not appear at the same time in user's view. 2.1.3 Galileo Galileo, a Europe’s own satellite-base navigation system, will provide a highly accurate global positioning service under civil control [30]. It will be interoperable with GPS and GLONASS. The fully deployed constellation consists of 30 satellites (27 operational and 3 ac- tive spares), positioned in three circular medium earth orbit (MEO) planes with an altitude of about 23616 km above the Earth, an inclination of the orbital planes of 56° with reference to the equatorial plane and a period of 14 hours 22 minutes. The Galileo navigation signals will pro-
  37. 37. 11 vide a good coverage even at latitudes up to 75°N, which corresponds to the North Cape, and beyond. The large number of satellites together with the optimization of the constellation, and the availability of the three active spare satellites, will ensure that the loss of one satellite has no discernible effect on the user. The first experimental satellite, called Galileo system test bed (GSTB), will be launched in the second semester of 2005. Thereafter up to four operational satellites will be launched in the timeframe 2005-2006 to validate the basic Galileo space and related ground segment. Once this in-orbit validation (IOV) phase has been completed, the re- maining satellites will be installed to reach the full operational capability (FOC) in 2008. To provide for the control of the satellites and to perform the navigation mission manage- ment, two Galileo control centers (GCC) will be implemented on European ground [30][31]. The data provided by a global network of 20 Galileo sensor stations (GSS) will be sent to the GCCs through a redundant communications network. The GCCs will use the data of the GSSs to compute the integrity information and to synchronize the time signal of all satellites and of the ground station clocks. The exchange of the data between the GCCs and the satellites will be performed through so-called uplink stations. The satellites then send data messages to receivers over radio signals. The data messages will include not only satellite clock, orbit ephemeris and constellation almanac information, but also an accuracy signal giving users a prediction of the satellite clock and ephemeris accuracy over time. It will allow receivers to weigh the meas- urements of each satellite and improve their navigation accuracy. All satellites are being de- signed to transmit up to four L-band carriers, and the use of C-band remains under consideration. Table 2-2 Comparison of the four Galileo L-Band signals Civil Signal E2-L1-E1 E6 E5B E5A Frequency (MHz) 1575.42 1278.75 1207.14 1176.45 Sub-carrier A B C A B C I Q I Q Purpose Data Data Pilot Data Data Pilot Data Pilot Data Pilot Secondary Code Length 25 25 4 100 20 100 Code Length NP 8184 NP 5115 10230 10230 Code Clock (MHz) N 2.046 5.115 10.23 10.23 Each satellite will transmit 10 signals: six serve open and safety-of-life services, two are for commercial services and two are for public regulated services. They will be broadcast in the following frequency bands: E2-L1-E1 (1559-1591 MHz), E6 (1260-1300 MHz), E5A-E5B
  38. 38. 12 (1164-1215 MHz), respectively. The comparison of the three Galileo L-Band signals is listed in Table 2-2. E2-L1-E1 is already used by GPS, sharing this band with GPS will be on a nonin- terference basis. It will offer user a simultaneous access to GPS and Galileo at minimal in- creases in terminal cost and complexity. Based on the operational Cospas-Sarsat system, Galileo will provide a global search and rescue (SAR) function as a further feature [30][31]. To do so, each satellite will be equipped with a transponder, which is able to transfer the distress signals from the user transmitters to the Rescue Coordination Centre, which will then initiate the rescue operation. At the same time, the system will provide a signal to the user, informing him that his situation has been detected and that help is under way. This feature is considered a major upgrade compared to the existing system, which does not provide a feedback to the user. 2.2 Observables Three different types of observables, the pseudorange measurement, the delta range (i.e., carrier phase difference) measurement, and the Doppler shift measurement, can be obtained from the signals of a satellite. 2.2.1 Pseudorange Measurement Each navigation satellite will transmit the signals modulated by the PRN code to determine the distance between the satellite and the receiver for positioning purpose. This distance, so-called the pseudorange measurement, is made by replicating the code being generated by the satellite and determining the time offset between the arrival of a particular transition in the code and the same transition in the code replica [10]. Then the pseudorange is obtained by multi- plying this time offset with the speed of light, where the time offset is simply the time interval that the signal takes to propagate from the satellite to the receiver. The pseudorange is biased by several other effects including the ionospheric effect, the tropospheric delay, the receiver clock bias, the multipath effect, the receiver noise, …, etc. The equation for the code pseudo- range measurements can be formulated as the following equation ( ) ( ) ( ) ( )( ) ( ) ( ) ( )kkkfkckckk nR wτκlγρρ +++−+= −2 δ , (2-1) where ( )kρ is the n×1 vector with the ith element as the pseudorange measurement associated to the ith satellite; ( )kρ is the n×1 vector with the ith element as the true range from the receiver to the ith satellite; ( )kγ is the n×1 vector with the ith element as the satellite i clock offset to the
  39. 39. 13 GPS time; ( )kRδ is the receiver clock bias deviation; nl is an n×1 column vector with all ele- ments equal to one; c is the speed of light; f is the carrier frequency of the satellite signal; ( )kκ is an n×1 vector with the ith element as 40.3 times the TEC (total electronic content) asso- ciated to the ith satellite (see Subsection 2.2.4); ( )kτ is an n×1 vector with the ith element as the tropospheric delay associated to the ith satellite; ( )kw is an n×1 zero mean Gaussian noise vec- tor with covariance matrix ( )kR . 2.2.2 Delta Range (Carrier Phase Difference) Measurement To lock the signal of a navigation satellite, the carrier of the received signal must be tracked by the receiver. It provides an alternative measurement from the signals of the satellite. Al- though the carrier generated by the receiver has a nominally constant frequency, the received carrier will still change in frequency due to the Doppler shift induced by the relative motion of the satellite and the receiver [10]. The phase of the received carrier is related to the phase of the carrier at the satellite through the time interval required for the signal to propagate from the satellite to the receiver. The carrier phase observable would be the total number of full carrier cycles and fractional cycles between the antenna of a satellite and a receiver at any instant. Since a receiver has no way of distinguishing one cycle of a carrier from another, only the initial fractional phase plus the changes to the phase, i.e., the delta range, can be measured. The equa- tion for the delta range measurements can be formulated as the following equation ( ) ( ) ( ) ( )( ) ( ) ( )( )[ ] ( ) ( )( ) ( ) ( )( ) ( )kkkkkf ktkkckk nRR ωττκκ lγγ +−−+−−− −−−−−+= − 11 11 2 δδφφ , (2-2) where ( )kφ is the n×1 vector with the ith element as the delta range measurement associated to the ith satellite; ( )kφ is the n×1 vector with the ith element as the true range difference associ- ated to the ith satellite; ( )kγ is the n×1 vector with the ith element as the satellite i clock offset to the GPS time; ( )kRδ is the receiver clock bias deviation; nl is an n×1 column vector with all elements equal to one; c is the speed of light; f is the carrier frequency of the satellite signal; ( )kκ is an n×1 vector with the ith element as 40.3 times the TEC associated to the ith satellite; ( )kτ is an n×1 vector with the ith element as the tropospheric delay associated to the ith satellite; ( )kω is an n×1 zero mean Gaussian noise vector with covariance matrix ( )kΩ . 2.2.3 Doppler Shift Measurement As discussed in the previous section, the Doppler shift observation is induced by the relative motion of a satellite and a receiver [10]. In fact, the raw Doppler shift measurement is linearly
  40. 40. 14 dependent on the radial velocity and thus can be adopted to calculate the velocity of a moving object. The equation for the Doppler shift measurements can be formulated as the following equation ( ) ( ) ( ) ( )[ ] ( ) ( ) ( )kkkfkkckk vnR wτκlγψψ ++−−+= − &&&& 2 δ , (2-3) where ( )kψ is the n×1 vector with the ith element as the Doppler shift measurement associated to the ith satellite; ( )kψ is the n×1 vector with the ith element as the true Doppler shift associ- ated to the ith satellite; ( )kγ& is the n×1 vector defined as ( ) ( )( ) ktdttdk == γγ& ; ( )kRδ& is the n×1 vector defined as ( ) ( )( ) ktRR dttdk == δδ& ; nl is an n×1 column vector with all elements equal to one; c is the speed of light; f is the carrier frequency of the satellite signal; ( )kκ& is the n×1 vector defined as ( ) ( )( ) ktdtkdk == κκ& ; ( )kτ& is the n×1 vector defined as ( ) ( )( ) ktdtkdk == ττ& ; ( )kvw is an n×1 zero mean Gaussian noise vector with covariance matrix ( )kvR . The raw Doppler shift measurement is less accurate than the integrated Doppler shift (i.e., the delta range) measurement. In order to obtain an estimate of the achievable accuracy, the raw Doppler shift should be accumulated over 0.001Hz [10]. This corresponding to 0.3 ms if the Doppler shift is measured in the C/A-code tracking loop. 2.2.4 Ionospheric Effect and Ionosphere-free Measurements The ionosphere is one among the various layers from about 50km to 1000km above earth [10]. It is a dispersive medium with respect to the satellite radio signal. Its refractive index is a function of the frequency, and the group and the phase refractive indices derivate from unity with opposite sign. Therefore, the code measurements are delayed and the carrier phases are advanced. In the order word, the group (code) delay and the phase advance are equal in mag- nitude but opposite in sign. The ionospheric effect in code and phase measurement can be ob- tained as follows ( ) ii fTEC f advancephasedelaygroupcode κ2 2 3.40 − ==−= , (2-4) where f is the center frequency of the carrier, TECi is the total electron content in the path where the signal of the ith satellite go through, and ( )kiκ is the ith element of ( )kκ . Since the iono- spheric effect is highly related to the carrier frequency of the signal, using the dual frequency technique can eliminate the ionospheric effect [10]. Assume ( )kL1ρ and ( )kL2ρ are the L1 and L2 pseudorange measurement vectors at time k, respectively. To obtain the ionosphere-free pseudorange measurement ( )kL12ρ , the following linear combination is adopted
  41. 41. 15 ( ) ( ) ( ) ( )( )kfkfffk LLLLLLL 2 2 11 2 2 12 1 2 212 ρρρ −−−−− −−≡ , (2-5) where 1Lf and 2Lf are carrier frequencies of the L1 and L2 signals, respectively. Furthermore, the ionosphere-free delta range measurement ( )kL12φ and the ionosphere-free Doppler shift measurement ( )kL12ψ can be obtained through the following equations ( ) ( ) ( ) ( )( )kfkfffk LLLLLLL 2 2 11 2 2 12 1 2 212 φφφ −−−−− −−≡ (2-6) and ( ) ( ) ( ) ( )( )kfkfffk LLLLLLL 2 2 11 2 2 12 1 2 212 ψψψ −−−−− −−≡ , (2-7) respectively, where ( )kL1φ and ( )kL2φ denotes the L1 and L2 delta range measurement vectors, respectively, and ( )kL1ψ and ( )kL2ψ denotes the L1 and L2 Doppler shift measurement vectors, respectively. 2.3 Conventional Fault detection and Exclusion Algorithms Two conventional fault detection and exclusion algorithms, least-squares residuals and par- ity space method, are described in this section. Assume the nominal vector, comprised of the nominal user position plus the nominal user clock drift, is set as the reference point ( )krefx with dimension 4×1. Then the linearized GPS measurement equation can be obtained as ( ) ( ) ( ) ( )kkkk wxHy += , (2-8) where ( )ky is the n×1 GPS measurement vector, which is the difference between pseudorange measurement ( )kρ and the predicted range based on the nominal user position; ( )kH is the n×4 observation matrix arrived at by linearizing around the nominal user position and clock bias; ( )kx is the 4×1 state vector comprised of the true position deviation from the nominal position, plus the user clock bias deviation; ( )kw is an n×1 zero mean Gaussian noise vector with co- variance matrix ( )kR ; and n is the number of visible satellites. 2.3.1 Least-Squares-Residuals The least-squares-residuals (LSR) method for fault detection is derived as follows. Ac- cording to [17], the estimate of the state vector is ( ) ( ) ( ) ( )( ) ( ) ( ) ( )kkkkkkk TT yRHHRHx 111 ˆ −−− = , (2-9) Then the estimate of ( )ky is
  42. 42. 16 ( ) ( ) ( )kkk xHy ˆˆ = , (2-10) and the range residual vector is ( ) ( ) ( )kkk yyy ˆ~ −= . (2-11) By defining the normalized sum of the squares of the range residual errors (SSE) as ( ) ( ) ( ) ( )kkkks T yRy ~~ 1− ≡ . (2-12) Parkinson [17] showed that the distribution of ( )ks is ( )42 −nχ , where n is the number of visi- ble GPS satellite and ( )νχ 2 represents the chi-square distribution with ν degrees of freedom. Then ( )ks will be compared with the detection threshold Td to judge whether the system is failed or not. The detection threshold value under a specified false alarm rate (FAR) can be calculated directly through the cumulative distribution function (cdf) of ( )42 −nχ . An exam- ple is given to explain how to determine the threshold value. Suppose there are eight visible satellites, i.e., the statistic ( )ks is ( )42 χ , and the value of FAR is selected as 1/100. Then the pertinent probability density functions (pdf) and key parameters are shown in Figure 2-1. This figure shows that we can choose the detection threshold such that the sum of the shaded area is 0 5 10 15 20 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 false alarm rate = 1/100 normalized SSE (m2 ) probabilitydensityfunction determination of threshold Td Figure 2-1 Determination of detection threshold Td with eight visible satellites
  43. 43. 17 equal to the value of FAR. For aviation navigation, the value of FAR is chosen as 1/15000 [37], and the resulting threshold values are listed in Table 2-3. In order to maintain sufficient redun- dancy, at least five visible satellites are required. Table 2-3 The value of detection threshold Td for ( )ks Number of satellites in view, n Chi-square degrees of freedom Detection Threshold 5 1 15.9032 6 2 19.2316 7 3 21.9546 8 4 24.3914 9 5 26.6521 10 6 28.7899 11 7 30.8356 12 8 32.8089 After the detection of satellite malfunction, the range measurements associated to the failed satellite must be excluded to ensure uninterrupted navigation. According to [17], the subset ex- clusion algorithm for fault exclusion is derived as follows. Subset solutions are formed by re- moving one of the visible satellites at a time. To be precise, the number of subsets is equal to the number of visible satellites. As for the ith subset, the pseudorange measurements are ob- tained from all but the ith visible satellites. Then a corresponding normalized SSE ( )ksi can also be obtained through (2-9) to (2-12). Based on the subset exclusion algorithm [17], a satel- lite is identified as failed if and only if the following equation is satisfied ( ) ( ) ( )⎩ ⎨ ⎧ > =≤ otherwiseTks kniTks ei fei , , , (2-13) where ( )knf denotes the channel number of the failed satellite at time k, Te is a selected thresh- old value for satellite fault exclusion. In order to maintain sufficient redundancy, at least six visible satellites are required. Under the hypothesis that no failure is taking place, the distribu- tion of ( )ksi is ( )52 −nχ . Then the exclusion threshold value under a specified FAR can be calculated directly through the cumulative distribution function of ( )52 −nχ . In order to per- form the fault exclusion, at least six visible satellites are required.
  44. 44. 18 According to [17], the user may perform integrity checking and failed satellite exclusion through following five steps: Step 1: Compute the normalized SSE, ( )ks , using data of all visible satellites from (2-12). Step 2: If ( )ks is larger than the detection threshold Td, then declare detection of a failure, and goto Step 3. Otherwise, all satellites are assumed to operate properly, and the integrity check has been completed. Step 3: Compute the residual parameters, ( )ks1 , …, ( )ksi , …, ( )ksn , using data from all but the ith visible satellites, for i = 1, …, n. Step 4: If one of the residual parameters (such as ( )ks fn ) is less than the exclusion threshold, Te, and all others are larger than Te, identify the satellite omitted from the nfth subset as the failed one. If two or more of the residual parameters are below the threshold, the failed satellite cannot be excluded. (see (2-13)) Step 5: If a failed satellite is detected and excluded, use the navigation solution formed by omitting the failed one. If a failed navigation satellite is detected, but cannot be ex- cluded, use the all-in-view solution if necessary, but recognize that the positioning ac- curacy is degraded. 2.3.2 Parity Space Method The parity space method for fault detection and exclusion is derived as follows. The parity space method can also perform the fault detection, and more, to exclude the failed satellite. According to Appendix C, there exists a parity matrix ( )kP satisfying the following equation ( ) ( ) ( ) ( ) ( )⎩ ⎨ ⎧ = = −4n T kkk kk IPRP 0HP , (2-14) where n is the number of GPS satellites in view at time k. After the parity matrix ( )kP is found, the parity vector can be defined as ( ) ( ) ( )kkk yPp ≡ . (2-15) Brown [4] showed that ( ) ( ) ( )kskkT =pp , (2-16) so, as in the least-squares-residual method, the parity space method can also be used to perform
  45. 45. 19 the fault detection. In order to maintain sufficient redundancy, at least five visible satellites are required. After the detection of satellite malfunction, the failed one must be excluded to ensure uninterrupted navigation. According to (A-10), the algorithm to identify the failed satellite is as follows ( ) ⎟ ⎠ ⎞⎜ ⎝ ⎛ ⎟ ⎠ ⎞⎜ ⎝ ⎛⎟ ⎠ ⎞⎜ ⎝ ⎛ = = k kk kn i i T ni f p pp ,,1 maxarg K , (2-17) where ( )knf is the channel number of the failed satellite at time k and ( )kip is the ith column vector of the parity matrix ( )kP . ( )kip is also called the ith channel vector since it is related to the ith satellite. In order to perform the fault exclusion, at least six visible satellites are re- quired. Figure 2-2 gives the plot of a 2-D parity space, where the number of the visible satel- lites is assumed as six. In this figure, six channel vectors corresponding to the six visible satel- lites are shown, and the parity vector is closest to the channel vector (or its opposite vector) as- sociated with the failed satellite. This coincides with the criteria stated in (2-17). Figure 2-2 Parity space plot with six visible satellites To avoid complex calculation of parity vector, another criteria will be introduced [14][24]. At first, define an n×n symmetric matrix as follows ( ) ( ) ( )kkk T PPS ≡ , (2-18) and Brown [4] showed that ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )kkkkkkkkk TT 11111 −−−−− −= RHHRHHRRS . Then the ith column vector of the matrix ( )kS can be represented as ( ) ( ) ( )kkk i T i pPζ = , (2-19) ith channel vector nfth channel vector (failed satellite) parity vector
  46. 46. 20 and the ith diagonal element of the matrix ( )kS can be represented as ( ) ( ) ( )kkkζ i T iii pp= . (2-20) Substitute both (2-19) and (2-20) into equation (2-17), (2-17) can be rewritten as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )kζ kk kk kkk kn ii i T ni i T i i TT ni f ζy pp pPy ,,1,,1 maxargmaxarg KK == == . (2-21) Since ( )kS matrix defined in equation (2-18) is unique for each ( )kH matrix, the failed satellite in equation (2-21) will be determined no mater how the parity matrix ( )kP is chosen.
  47. 47. 21 Chapter 3 Using Multi-Frequency Technique on Fault Detection and Exclusion To improve the performance of the current navigation system, a GPS modernization policy has been approved by the US government. GPS satellites carrying the L2 and L5 signals will be launched in 2005, and the C/A code will be modulated onto these signals for civil use. The main advantages of multi-frequency GPS are performance enhancement in positioning accuracy and satellite fault detection and exclusion (FDE). In addition, the new signal can be regarded as a backup, and thus will significantly increase the safety of navigation [9]. Besides, the first ex- perimental Galileo satellite will be launched in the second semester of 2005. Furthermore, four operational Galileo satellites will be launched in the 2005-2006, and the remaining satellites will be installed to reach the full operational capability (FOC) in 2008. Since Galileo will be inter- operable with GPS, receivers can be designed to simultaneously access both Galileo and GPS systems. Hence, the number of visible GNSS satellites (24 GPS satellites plus 27 Galileo satel- lites) will be significantly increased, and thus both accuracy and safety of navigation will be greatly raised. Therefore, algorithms using multi-frequency GNSS technique are systematically derived for positioning and FDE in this chapter. These proposed algorithms are expected to improve the positioning accuracy, shorten the failure detection time, and reduce the incorrect ex- clusion rate. Note that these algorithms are snapshot-type method; the filter-type algorithms will be discussed in the following two chapters. First, the linearization of GNSS pseudorange measurement is introduced in Section 3.1. Next, the dual frequency algorithm for GPS posi- tioning and FDE is described in Section 3.2. Moreover, the dual frequency algorithm is further extended to the triple frequency algorithm in Section 3.3. Furthermore, to simultaneously use both Galileo and GPS, the dual frequency algorithm for GNSS is derived in Section 3.4. Simu- lation results and analysis of the proposed algorithms are shown in Section 3.5.
  48. 48. 22 3.1 Linearization of GNSS Pseudorange Measurements The linearization of multi-frequency pseudorange measurement for GNSS, including GPS and Galileo, is described as follows. Assume the reference point ( )krefx is the vector com- prised of the nominal user position plus the nominal user clock drift. Then the linearized GPS and Galileo measurement equations with ionospheric delay can be represented as ( ) ( ) ( ) ( ) ( )kkfkkk LLL wκxHy ++= −2 , (3-1) and ( ) ( ) ( ) ( ) ( )kkfkkk EEE wκxHy ++= − )) 2 , (3-2) where ( )kLy is the n×1 GPS measurement vector, which is the difference between ( )kLρ and the predicted range based on the nominal user position; ( )kEy is the m×1 Galileo measurement vector, which is the difference between ( )kEρ and the predicted range based on the nominal user position; moreover, ( )kLρ and ( )kEρ are the pseudorange measurements corresponding to the GPS and Galileo satellites, respectively; ( )kH and ( )kH ) are the n×4 GPS observation ma- trix and the m×4 Galileo observation matrix, respectively; in addition, ( )kH and ( )kH ) are ma- trices arrived at by linearizing around the nominal user position and clock bias; ( )kx is the 4×1 state vector comprised of the true position deviation from the nominal position, plus the user clock bias deviation; ( )kκ is an n×1 vector with the ith element as 40.3 times the TEC associ- ated to the ith GPS satellite; ( )kκ ) is an m×1 vector with the ith element as 40.3 times the TEC associated to the ith Galileo satellite; Lf and Ef are the carrier frequencies of the GPS and Galileo signals, respectively; ( )kLw is an n×1 zero mean Gaussian noise vector with covariance matrix nLI2 σ ; ( )kEw is an m×1 zero mean Gaussian noise vector with covariance matrix mEI2 σ ; n and m are the number of visible satellites corresponding to the GPS and Galileo systems, re- spectively. 3.2 GPS Dual Frequency (L1/L2) Algorithms The dual frequency algorithm for GPS positioning and FDE is described in this section. According to (3-1), the linearized GPS measurement equations based on dual frequency can be expressed as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )⎥ ⎦ ⎤ ⎢ ⎣ ⎡ +⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ =⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − k k k k fk fk k k L L nL nL L L 2 1 2 2 2 1 2 1 w w κ x IH IH y y , (3-3)
  49. 49. 23 where ( )kL1y and ( )kL2y are n×1 measurement vectors corresponding to the L1 and L2 signals, respectively; ( )kx is the 4×1 state vector comprised of the true position deviation from the nominal position, plus the user clock bias deviation; ( )kκ is an n×1 vector with the ith element as 40.3 times the TEC associated to the ith GPS satellite; ( )kL1w and ( )kL2w are the n×1 zero mean Gaussian noise vectors with covariance matrices nL I2 1σ and nL I2 2σ , respectively; further- more, ( )kL1w and ( )kL2w are assumed to be mutually independent; 1Lf and 2Lf are carrier frequencies of the L1 and L2 signals, respectively. 3.2.1 Positioning Algorithm The dual frequency positioning algorithms are derived as follows. According to Subsec- tion 2.3.1, the estimate of the state vector is ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )kkkkkk kˆ kˆ DFDF T DFDFDF T DF DF DF yRHHRH κ x 111 −−− =⎥⎦ ⎤ ⎢⎣ ⎡ , (3-4) where ( ) ( ) ( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = − − nL nL DF fk fk k IH IH H 2 2 2 1 , ( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = nL nL TF k I0 0I R 2 2 2 1 σ σ , and ( ) ( ) ( )⎥⎦ ⎤ ⎢⎣ ⎡ = k k k L L DF 2 1 y y y . In order to simplify equation (3-4), the right part of (3-4) is split into two terms, ( ) ( ) ( )( ) 11 −− kkk DFDF T DF HRH and ( ) ( ) ( )kkk DFDF T DF yRH 1− . According to (D-5), the first terms can be expressed in partitioned matrix form as ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )( )⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ +− − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + = −−−−− −−−− −−− −− kkkkkkk kkkkk k kkk kkk TT DFDFnDF T DFDF TT DFDF T DFDF nDFDF T DF T LL DFDF T DF HHHHIHHH HHHHH IH HHH HRH 122212 12122 1 2 2 2 2 1 11 δαβδα δαδβ βα ασσ , (3-5) where 2 2 2 2 2 1 2 1 −−−− += LLLLDF ff σσα , 4 2 2 2 4 1 2 1 −−−− += LLLLDF ff σσβ and ( ) 222 2 2 1 DFDFLLDF αβσσδ −+= −− . Moreover, the second terms can be calculated as ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + = −−−− −− − kfkf kkk kkk LLLLLL LLLL T DFDF T DF 2 2 2 2 21 2 1 2 1 2 2 21 2 11 yy yyH yRH σσ σσ . (3-6) Finally, substitute both (3-5) and (3-6) into (3-4), and then the estimated states can be derived as ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )⎪⎩ ⎪ ⎨ ⎧ −= = − − kˆkkkˆ kkkkkˆ DFDFDFLDF L TT DF xHyκ yHHHx 2 12 12 1 βακ , (3-7)
  50. 50. 24 where ( ) ( ) ( ) ( )( )kfkfffk LLLLLLL 2 2 11 2 2 12 1 2 212 yyy −−−−− −−= and ( ) ( )( ( ))kfkfk LLLLLLDFL 2 2 2 2 21 2 1 2 1 2 12 yyy −−−−− += σσβκ . 3.2.2 Fault Detection and Exclusion Algorithm The dual frequency FDE algorithm is derived in this subsection. According to Subsection 2.3.2, there exists a parity matrix ( )kDFP satisfying the following equations ( ) ( ) ( ) ( ) ( )⎩ ⎨ ⎧ = = −4n T DFDFDF DFDF kkk kk IPRP 0HP . (3-8) To derive a simpler form of the parity matrix ( )kDFP , we first represent ( )kDFP as the parti- tioned form ( ) ( )[ ]kk LL 21 PP , where ( )kL1P and ( )kL2P are the (n−4)×n submatrices of ( )kDFP . Substitute the partitioned parity matrix into (3-8), and then (3-8) can be rewritten as the follow- ing equations ( ) ( )( ) ( ) ( ) ( ) ( ) ( )⎩ ⎨ ⎧ =+ =+ −422 2 211 2 1 21 n T LLL T LLL LL kkkk kkk IPPPP 0HPP σσ , (3-9) and ( ) ( ) 0PP =+ −− kfkf LLLL 2 2 21 2 1 , i.e., ( ) ( )kffk LLLL 1 2 1 2 22 PP − −= . (3-10) Furthermore, by using (3-10) to replace the term ( )kL2P in equation (3-9), the following equa- tions can be derived ( ) ( ) ( ) ( ) ( )⎩ ⎨ ⎧ = =− − − 411 4 2 2 2 2 1 2 1 2 1 2 21 n T LLLLLDF LLL kkf kkff IPP 0HP σσβ , i.e., ( ) ( ) ( ) ( )⎩ ⎨ ⎧ = = −4n T kk kk IPP 0HP , (3-11) where ( )kP is the matrix defined as ( ) ( )kfk LLLLDF 1 2 221 PP σσβ= . Therefore, the parity matrix ( )kDFP can be represented as ( ) ( ) ( )[ ] ( ) ( )[ ]kfkfkkk LLLLDFLLDF PPPPP 2 1 2 2 1 2 1 1 1 21 −−−−− −== σσβ , (3-12) where ( )kP is the (n−4)×n matrix satisfying ( ) ( ) 0HP =kk and ( ) ( ) 4−= n T kk IPP . After the parity matrix is found, the parity vector can be expressed as ( ) ( ) ( ) ( ) ( ) ( )kkffkkk LLLLLDFDFDFDF 12 2 1 2 2 1 2 1 1 1 yPyPp −−−−− −== σσβ . (3-13) Moreover, the test statistic for fault detection can be defined as ( ) ( ) ( )kkks DF T DFDF pp≡ . (3-14) Under the hypothesis that no failure is taking place, it can be proved that the distribution of
  51. 51. 25 ( )ksDF is ( )42 −nχ , where ( )νχ 2 represents the chi-square distribution with ν degrees of free- dom. Then ( )ksDF will be compared with a detection threshold Td to judge whether the system is failed or not. The detection threshold value under a specified false alarm rate (FAR) can be calculated directly through the cumulative distribution function of ( )42 −nχ . After the detection of satellite malfunction, the range measurements associated to the failed satellite must be excluded to ensure uninterrupted navigation. Let ( )kDFP be partitioned into columns ( ) ( ) ( ) ( )[ ]kkkk L n,DF L ,DF L n,DF L ,DF 22 1 11 1 pppp LL . From (3-12), ( )kL i,DF 1 p and ( )kL i,DF 2 p can be expressed as ( ) ( )kfk iLLLDF L i,DF pp 2 2 1 2 1 1 11 −−−− = σσβ and ( ) ( )kfk iLLLDF L i,DF pp 2 1 1 2 1 1 12 −−−− −= σσβ , respectively, where ( )kip is the ith column vector of the matrix ( )kP . Based on the standard parity space method, the algorithm to identify the failed satellite can then be derived as follows [23] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ = ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = == k kk maxarg k kk , k kk maxmaxargkn i i T DF n,,i L i,DF L i,DF T DF L i,DF L i,DF T DF n,,i f p pp p pp p pp KK 1 2 2 1 1 1 ,(3-15) where ( )knf denotes the channel number of the failed satellite at time k. The above derivation is not restricted to the case of using the L1/L2 signal only. It can also be applied to other com- binations of frequencies, such as the L1/L5 or L2/L5 signal, as well. 3.3 GPS Triple Frequency (L1/L2/L5) Algorithms In this section, the dual frequency algorithm is extended to the triple frequency algorithm. According to (3-1), the linearized GPS measurement equations based on triple frequency can be expressed as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ +⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − − k k k k k fk fk fk k k k L L L nL nL nL L L L 5 2 1 2 5 2 2 2 1 5 2 1 w w w κ x IH IH IH y y y , (3-16) where ( )kL1y , ( )kL2y , and ( )kL5y are n×1 measurement vectors corresponding to the L1, L2, and L5 signals, respectively; ( )kx is the 4×1 state vector comprised of the true position devia- tion from the nominal position, plus the user clock bias deviation; ( )kκ is an n×1 vector with the ith element as 40.3 times the TEC associated to the ith GPS satellite; 1Lf , 2Lf and 5Lf are car- rier frequencies of the L1, L2 and L5 signals, respectively; ( )kL1w , ( )kL2w and ( )kL5w are n×1 zero mean Gaussian noise vectors with covariance matrices nL I2 1σ , nL I2 2σ and nL I2 5σ , respec- tively; furthermore, ( )kL1w , ( )kL2w and ( )kL5w are assumed to be mutually independent.

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