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

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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  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.
  • 52. 26 3.3.1 Positioning Algorithm In this subsection, the positioning algorithm is systematically derived as follows. Accord- ing to Subsection 2.3.1, the estimate of the state vector is ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )kkkkkk k k TFTF T TFTFTF T TF TF TF yRHHRH κ x 111 ˆ ˆ −−− =⎥ ⎦ ⎤ ⎢ ⎣ ⎡ , (3-17) where ( ) ( ) ( ) ( ) ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = − − − nL nL nL TF fk fk fk k IH IH IH H 2 5 2 2 2 1 , ( ) ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = nL nL nL TF k I00 0I0 00I R 2 5 2 2 2 1 σ σ σ and ( ) ( ) ( ) ( )⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = k k k k L L L TF 5 2 1 y y y y . In order to simplify equation (3-17), the right part of (3-17) is split into two terms, ( ) ( ) ( )( ) 11 −− kkk TFTF T TF HRH and ( ) ( ) ( )kkk TFTF T TF yRH 1− . According to (D-5), the first terms can be derived in the partitioned matrix form as ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )( )⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ +− − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ++ = −−−−− −−−− −−−− −− kkkkkkk kkkkk k kkk kkk TT TFTFn T TFTF TT TFTF T TFTF nTFTF T TF T LLL TFTF T TF HHHHIHHH HHHHH IH HHH HRH 122212 12122 1 2 2 5 2 2 2 1 11 δαβδα δαδβ βα ασσσ , (3-18) where TFα , TFβ and TFδ are the parameters defined as 2 5 2 5 2 2 2 2 2 1 2 1 −−−−−− ++= LLLLLLTF fff σσσα , 4 5 2 5 4 2 2 2 4 1 2 1 −−−−−− ++= LLLLLLTF fff σσσβ and ( ) 222 5 2 2 2 1 TFTFLLLTF αβσσσδ −++= −−− , respectively. More- over, the second terms can be calculated as ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ++ ++ = −−−−−− −−− − kfkfkf kkkk kkk LLLLLLLLL LLLLLL T TFTF T TF 5 2 5 2 52 2 2 2 21 2 1 2 1 5 2 52 2 21 2 11 yyy yyyH yRH σσσ σσσ . (3-19) Substitute both (3-18) and (3-19) into (3-17), and then the estimated states can be derived as ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )⎪⎩ ⎪ ⎨ ⎧ −= = − − kˆkkkˆ kkkkkˆ TFTFLTF L TT TF xHyκ yHHHx 2 125 125 1 βακ , (3-20) where ( ) ( ) ( ) ( )kfkfkfk LLLTFLLLTFLLLTFL 5 2 5 2 5 2 2 2 2 2 2 2 1 2 1 2 1 2 125 yyyy −−−−−−−−− ++= σβσβσβκ ; in addition, ( )=kL125y ( ) ( ) ( )kckckc LLLLLL 552211 yyy ++ with ( )2 1 222 11 −−− −= LTFTFTFLL fc αβδσ , ( )2 2 222 22 −−− −= LTFTFTFLL fc αβδσ and ( )2 5 222 55 −−− −= LTFTFTFLL fc αβδσ . 3.3.2 Fault Detection and Exclusion Algorithm The triple frequency FDE algorithm is systematically derived as follows. According to Subsection 2.3.2, there exists a parity matrix ( )kTFP satisfying the following equation
  • 53. 27 ( ) ( ) ( ) ( ) ( )⎩ ⎨ ⎧ = = −42n T TFTFTF TFTF kkk kk IPRP 0HP (3-21) As in the case of the dual frequency FDE algorithm, we can represent the parity matrix ( )kTFP as the partitioned matrix ( ) ( ) ( ) ( ) ( ) ( ) ( )⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = kkk kkk k lLlLlL uLuLuL TF 551 521 PPP PPP P , where ( )kuL1P , ( )kuL2P and ( )kuL5P are (n−4)×n submatrices, and ( )klL1P , ( )klL2P and ( )klL5P are n×n submatrices. Then (3-21) can be rewritten as the following equations ( ) ( ) ( )( ) ( ) 0HPPP =++ kkkk uLuLuL 521 , (3-22) ( ) ( ) ( )( ) ( ) 0HPPP =++ kkkk lLlLlL 521 , (3-23) ( ) ( ) ( ) 0PPP =++ −−− kfkfkf uLLuLLuLL 5 2 52 2 21 2 1 , (3-24) ( ) ( ) ( ) 0PPP =++ −−− kfkfkf lLLlLLlLL 5 2 52 2 21 2 1 , i.e., ( ) ( ) ( )( )kfkffk lLLlLLLlL 2 2 21 2 1 2 55 PPP −− +−= ,(3-25) ( ) ( ) ( ) ( ) ( ) ( ) 455 2 522 2 211 2 1 −=++ n T uLuLL T uLuLL T uLuLL kkkkkk IPPPPPP σσσ , (3-26) ( ) ( ) ( ) ( ) ( ) ( ) n T lLlLL T lLlLL T lLlLL kkkkkk IPPPPPP =++ 55 2 522 2 211 2 1 σσσ , (3-27) and ( ) ( ) ( ) ( ) ( ) ( ) 0PPPPPP =++ kkkkkk T lLuLL T lLuLL T lLuLL 55 2 522 2 211 2 1 σσσ . (3-28) Moreover, substitute (3-25) into (3-23), and then the following equation can be obtained ( ) ( ) ( ) ( )( ) ( ) 0HPP =−+− −− kkffkff lLLLlLLL 2 2 2 2 51 2 1 2 5 11 . As a result, one of the possible solutions sat- isfied the previous equation is ( ) ( ) 1 2 1 2 5 12 5 2 22 lLLLLLlL ffff PP −−−−− −−= . Replacing the term 2lLP in (3-25), we then have ( ) ( ) 1 2 2 2 1 12 5 2 25 lLLLLLlL ffff PP −−−−− −−= . Furthermore, substitute both 2lLP and 5lLP into (3-27), and it results that ( ) ( ) ( ) n T lLlLLLLTFLL kkff IPP =− −−− 11 2 5 2 2 2 1 222 5 2 2 σσσδ . To give a sim- pler form of ( )klL1P , we can select ( )klL1P as ( ) ( ) nLLLLLTFlL ffk IP 2 5 2 2 1 5 1 2 1 1 1 1 −−−−−− −= σσσδ . Then 1lLP , 2lLP and 5lLP can be expressed as ( ) ( ) ( ) ( ) ( ) ( )⎪ ⎩ ⎪ ⎨ ⎧ −= −= −= −−−−−− −−−−−− −−−−−− nLLLLLTFlL nLLLLLTFlL nLLLLLTFlL ffk ffk ffk IP IP IP 2 2 2 1 1 5 1 2 1 1 1 5 2 1 2 5 1 5 1 2 1 1 1 2 2 5 2 2 1 5 1 2 1 1 1 1 σσσδ σσσδ σσσδ . (3-29) Moreover, substitute 1lLP , 2lLP and 5lLP in (3-29) into (3-28), and we have the following equation ( ) ( ) ( ) 0PPP =−+−+− −−−−−− 5 2 2 2 1 2 52 2 1 2 5 2 21 2 5 2 2 2 1 uLLLLuLLLLuLLLL ffffff σσσ . Then combine this equation with (3-24), and we can derive that ( ) ( )kcck uLLLuL 12 1 12 PP − = and ( ) ( )kcck uLLLuL 15 1 15 PP − = ; furthermore, substitute ( )kuL2P and ( )kuL5P into (3-28), and the following equations can be ob- tained
  • 54. 28 ( ) ( ) ( ) ( ) ( )⎩ ⎨ ⎧ = =++ − −− −− 411 2 1 22 15 1 12 1 11 n T uLuLLTFTF uLLLLL kkc kkcccc IPP 0HP βδ , i.e., ( ) ( ) ( ) ( )⎩ ⎨ ⎧ = = −4n T kk kk IPP 0HP , (3-30) where ( )kP is the matrix defined as ( ) ( )kck uLLTFTF 1 1 1 1 PP −− = βδ . Note that in the previous equa- tion, 22 − TFTF βδ is equal to 2 5 2 5 2 2 2 2 2 1 2 1 LLLLLL ccc σσσ ++ . Therefore, the parity matrix ( )kDFP can be represented as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ −−− = −−−−−−−−−−−−−−− − nLLLLLnLLLLLnLLLLL LTFLTFLTF TFTF ffffff kckckc k III PPP P 2 2 2 1 1 5 1 2 1 1 2 1 2 5 1 5 1 2 1 1 2 5 2 2 1 5 1 2 1 1 5211 σσσσσσσσσ βββ δ ,(3-31) where ( )kP is an (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 ( ) ( ) ( ) ( ) ( ) ( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ == − k kk kkk L LTFTF TFTFTF 125 125 1 y yP yPp βδ , (3-32) where ( ) ( ) ( ) ( ) ( ) ( ) ( )( )kffkffkffk LLLLLLLLLLLLTFL 5 2 2 2 12 2 1 2 51 2 5 2 2 1 5 1 2 1 1 1 125 yyyy −−−−−−−−−− −+−+−= σσσδ . Fur- thermore, the test statistic for fault detection can be defined as ( ) ( ) ( )kkks TF T TFTF pp≡ . (3-33) Under the hypothesis that no failure is taking place, it can be proved that the distribution of ( )ksTF is ( )422 −nχ . The detection threshold Td under a specified FAR can then be calculated directly through the cumulative distribution function of ( )422 −nχ . To judge whether the sys- tem is failed or not, ( )ksTF will be compared with the detection threshold. After the detection of satellite malfunction, the range measurements associated to the failed satellite must be excluded to ensure uninterrupted navigation. Let ( )kTFP is partitioned into columns ( ) ( ) ( ) ( ) ( ) ( )[ ]kkkkkk L n,TF L ,TF L n,TF L ,TF L n,TF L ,TF 55 1 22 1 11 1 pppppp LLL . From equation (3-31), ( )kL i,TF 1 p , ( )kL i,TF 2 p and ( )kL i,TF 5 p can be expressed as ( ) ( ) ( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = −−−−− − i,nLLLLL iLTF TF L i,TF ff kc k e p p 2 5 2 2 1 5 1 2 1 1 111 σσσ β δ , ( ) ( ) ( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = −−−−− − i,nLLLLL iLTF TF L i,TF ff kc k e p p 2 1 2 5 1 5 1 2 1 1 212 σσσ β δ and ( ) ( ) ( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = −−−−− − i,nLLLLL iLTF TF L i,TF ff kc k e p p 2 2 2 1 1 5 1 2 1 1 515 σσσ β δ , respectively, where ( )kip is the ith column vector of the matrix ( )kP and in,e is an n×1 column vector with all elements zeros except that the ith element is one. Based on the standard parity space method, the algorithm to identify the failed satellite can be derived as follows [23] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = = k kk k kk k kk kn L iTF L iTF T TF L iTF L iTF T TF L iTF L iTF T TF ni f 5 , 5 , 2 , 2 , 1 , 1 , ,,1 ,,maxmaxarg p pp p pp p pp K , (3-34)
  • 55. 29 where ( )knf denotes the channel number of the failed satellite at time k. 3.4 GNSS (L1/L2/E2-L1-E1/E6) Algorithms To simultaneously use the measurements of both Galileo and GPS system, the dual fre- quency algorithm for GNSS is derived in this section. According to (3-1) and (3-2), the esti- mate of the state vector can be expressed as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ + ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − − − k k k k k k k fk fk fk fk k k k k E L E L mE nL mE nL E L E L 6 2 1 1 2 6 2 2 2 1 2 1 6 2 1 1 w w w w κ κ x I0H 0IH I0H 0IH y y y y ) ) ) , (3-35) where ( )kL1y and ( )kL2y are n×1 measurement vectors corresponding to the L1 and L2 signals, respectively; ( )kE1y and ( )kE6y are m×1 measurement vectors corresponding to the E2-L1-E1 and E6 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; ( )kκ ) is an m×1 vector with the ith element as 40.3 times the TEC associated to the ith Galileo satellite; 1Lf , 2Lf , 1Ef and 6Ef are carrier frequencies of the L1, L2, E2-L1-E1 and E6 signals, respectively; ( )kL1w and ( )kL2w are n×1 zero mean Gaussian noise vectors with covariance matrices nL I2 1σ and nL I2 2σ , respectively; ( )kE1w and ( )kE6w are m×1 zero mean Gaussian noise vectors with covariance matrices nE I2 1σ and nE I2 6σ , respectively; furthermore, ( )kL1w , ( )kL2w , ( )kE1w and ( )kE6w are assumed to be mutually independent. 3.4.1 Positioning Algorithm The positioning algorithm is systematically derived as follows. According to Subsection 2.3.1, the estimate of the state vector is ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )kkkkkk k k k DFDF T DFDFDF T DF GalileoDF GPSDF GNSSDF yRHHRH κ κ x )))))) 111 , , . ˆ ˆ ˆ −−− = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ , (3-36) where ( ) ( ) ( ) ( ) ( ) ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = − − − − mE nL mE nL DF fk fk fk fk k I0H 0IH I0H 0IH H 2 6 2 2 2 1 2 1 ) ) ) , ( ) ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = mE nL mE nL DF k I000 0I00 00I0 000I R 2 6 2 2 2 1 2 1 σ σ σ σ ) and ( )=kDFy )
  • 56. 30 ( ) ( ) ( ) ( )[ ]TT E T L T E T L kkkk 6211 yyyy . In order to simplify equation (3-36), the right part of (3-36) is split into two terms, ( ) ( ) ( )( ) 11 −− kkk DFDF T DF HRH ))) and ( ) ( ) ( )kkk DFDF T DF yRH ))) 1− . According to (D-8), the first term can be derived as ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ +− +− −− = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ +++ = −−−−−−−− −−−−−−−− −−−−− − −−−− −− kkkkk kkkkk kk k k kkkkkk kkk T DFDFDFmDF T DFDFDFDFDFDFDFDF T DFDFDFDFDF T DFDFDFnDFDFDFDF T DFDFDF T DFDFDFDF nDFDF nDFDF T DF T DF T EE T LL DFDF T DF H∆HIH∆H∆H H∆HH∆HI∆H H∆H∆∆ I0H 0IH HHHHHH HRH ))))))))))) ))) ))) ))) )))) ))) 124212212 122124212 12121 1 2 2 2 6 2 1 2 2 2 1 11 αβββαβαβα βαβααβββα βαβα βα βα αασσσσ ,(3-37) where DF∆ is a matrix defined as ( ) ( ) ( ) ( )kkkk T DFDF T DFDFDF HHHH∆ )))) 2222 δβδβ −− += ; DFα , DFβ and DFδ are parameters defined as in Subsection 3.2.1; DFα ) , DFβ ) and DFδ ) are parameters de- fined as 2 6 2 6 2 1 2 1 −−−− += EEEEDF ff σσα ) , 4 6 2 6 4 1 2 1 −−−− += EEEEDF ff σσβ ) , ( ) 222 6 2 1 DFDFEEDF αβσσδ ))) −+= −− , re- spectively. Moreover, the second terms can be calculated as ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ + + +++ = −−−− −−−− −−−− − kfkf kfkf kkkkkk kkk EEEEEE LLLLLL EEEE T LLLL T DFDF T DF 6 2 6 2 61 2 1 2 1 2 2 2 2 21 2 1 2 1 6 2 61 2 12 2 21 2 1 1 yy yy yyHyyH yRH σσ σσ σσσσ ) ))) . (3-38) After substituting (3-37) and (3-38) into (3-36), we can express the estimated states as ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )⎪ ⎪ ⎩ ⎪⎪ ⎨ ⎧ −= −= += − − −−− kkkk kkkk kkkkk GNSSDFDFDFEGalileoDF GNSSDFDFDFLGPSDF E T DFDFL T DFDFDFGNSSDF , 2 126, , 2 12, 16 22 12 221 , ˆˆ ˆˆ ˆ xHyκ xHyκ yHyH∆x ))) ))) βα βα δβδβ κ κ , (3-39) where both ( )kL12y and ( )kL κ 12y are the n×1 vectors defined in Subsection 3.2.1; ( )=kE16y ( ) ( ) ( )( )kfkfff EEEEEE 6 2 11 2 6 12 1 2 6 yy −−−−− −− and ( ) ( ) ( )( )kfkfk EEEEEEDFE 6 2 6 2 61 2 1 2 1 2 16 yyy −−−−− += σσβκ ) . 3.4.2 Fault Detection and Exclusion Algorithm In this subsection, the FDE algorithm is systematically derived as follows. According to Subsection 2.3.2, there exists a parity matrix, ( )kDFP ) , satisfying the following equation ( ) ( ) ( ) ( ) ( )⎩ ⎨ ⎧ = = −+ 4mn T DFDFDF DFDF kkk kk IPRP 0HP ))) )) . (3-40)
  • 57. 31 As in the case of the dual frequency FDE algorithm, we can represent the parity matrix ( )kDFP ) as the partitioned matrix ( ) ( ) ( ) ( ) ( )[ ]kkkkk ELELDF 6211 PPPPP ))))) = , where ( )kL1P ) and ( )kL2P ) are the (n+m−4)×n submatrices, and ( )kE12P ) and ( )kE6P ) are the (n+m−4)×m submatrices. Then (3-40) can be rewritten as the following equations ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )⎩ ⎨ ⎧ =+++ =+++ −+ 466 2 611 2 122 2 211 2 1 6121 mn T EEE T EEE T LLL T LLL EELL kkkkkkkk kkkkkk IPPPPPPPP 0HPPHPP )))))))) ))))) σσσσ , (3-41) and ( ) ( ) ( ) ( )⎩ ⎨ ⎧ =+ =+ −− −− 0PP 0PP kfkf kfkf EEEE LLLL 6 2 61 2 1 2 2 21 2 1 )) )) , i.e., ( ) ( ) ( ) ( )⎩ ⎨ ⎧ −= −= − − kffk kffk EEEE LLLL 1 2 1 2 66 1 2 1 2 22 PP PP )) )) . (3-42) Furthermore, substitute (3-42) into (3-41), and we have the following equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )⎩ ⎨ ⎧ =+ =−+− −+ −− 411 4 6 2 6 2 1 2 11 4 2 2 2 2 1 2 1 2 1 2 61 2 1 2 2 11 mn T EEEEEDF T LLLLLDF LEELLL kkfkkf kkffkkff IPPPP 0HPHP σσβσσβ ) ) . (3-43) Assume that ( ) ( )kfk LLLLDFL 1 2 221 PP σσβ= and ( ) ( )kfk EEEEDFE 1 2 661 PP σσβ ) = . Then equation (3-43) can be rewritten as ( ) ( )[ ] ( ) 0HPP =kkk GEL , ( ) ( )[ ] ( ) ( )[ ] 4−+= mn T ELEL kkkk IPPPP , where ( ) ( ) ( ) ( ) ( )[ ]TT EEEEDF T LLLLDFG kffkffk HHH )) 2 1 2 6 1 6 1 1 12 1 2 2 1 2 1 1 1 −−−−−−−−−− −−= σσβσσβ . Therefore, the parity ma- trix ( )kDFP ) can be represented as ( ) ( ) ( ) ( ) ( )[ ]kfkfkfkfk EEEEDFLLLLDFEEEEDFLLLLDFDF PPPPP 2 1 1 6 1 1 12 1 1 2 1 1 12 6 1 6 1 1 12 2 1 2 1 1 1 −−−−−−−−−−−−−−−− −−= σσβσσβσσβσσβ ))) ,(3-44) where ( )kLP and ( )kEP are the matrices satisfying ( ) ( )[ ] ( ) ( )[ ] 4−+= mn T ELEL Ikkkk PPPP and ( ) ( )[ ] ( ) 0HPP =kkk GEL . After the parity matrix is found, the parity vector can be expressed as ( ) ( ) ( )kkk DFDFDF yPp ))) = , (3-45) Furthermore, the test statistic for fault detection can be defined as ( ) ( ) ( )kkks DF T DFDF pp ))) = . (3-46) Under the hypothesis that no failure is taking place, it can be proved that the distribution of ( )ksDF ) is ( )42 −+ mnχ . Then ( )ksDF ) will be compared with the detection threshold to judge whether the system is failed or not. The detection threshold Td under a specified FAR can be calculated directly through the cumulative distribution function of ( )42 −+ mnχ . After the detection of satellite malfunction, the range measurements associated to the failed one must be excluded to ensure uninterrupted navigation. As in the case of the dual frequency algorithms, ( )kDFp ) is partitioned into columns ( ) ( ) ( ) ( )[ ]kkkk G mn,DF G ,DF G mn,DF G ,DF 22 1 11 1 ++ pppp ) L )) L ) .
  • 58. 32 From equation (3-44), ( )kG i,DF 1 p ) and ( )kG i,DF 2 p ) can be expressed as ( ) ( ) ( )( ) ( ) ( ) ( )( )⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ ⎩ ⎨ ⎧ ++=− =− = ⎩ ⎨ ⎧ ++= = = − − mn,,ni,kf n,,i,kf k mn,,ni,kf n,,i,kf k ni,EEEEDF i,LLLLDFG i,DF ni,EEEEDF i,LLLLDFG i,DF K ) K) K ) K) 1for 1for 1for 1for 2 161 2 1212 2 661 2 2211 p p p p p p σσβ σσβ σσβ σσβ , (3-47) where ( )kiL,p is the ith column vector of the matrix ( )kLP and ( )kiE,p is the ith column vector of the matrix ( )kEP . Based on the standard parity space method, the algorithm to identify the failed satellite can then be derived as follows [23] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = = k kk k kk kn G iDF G iDF T DF G iDF G iDF T DF ni f 2 , 2 , 1 , 1 , ,,1 ,maxmaxarg p pp p pp ) )) ) )) K , (3-48) where ( )knf denotes the channel number of the failed satellite at time k. The above derivation can also be applied to other combinations of frequencies as well. 3.5 Simulation Results and Analysis The software package “Satellite Navigation ToolBox 2.0 for Matlab,” by GPSoft LLC is adopted in the simulation. It assumed a 24-satellite constellation with perfectly circular orbits. For each satellite, the ionospheric delay, the tropospheric delay, and the thermal noise are added to the pseudorange measurement. In addition, the receiver mask angle is set as 7.5°. More- over, the standard deviations of measurement noises and the user-satellite geometry are listed in Appendix E. Three simulations are given to verify that the application of multi-frequency algo- rithms will improve the positioning accuracy, shorten the failure detection time, and reduce the incorrect exclusion rate. Due to the limitation of the software package, only the GPS satellite constellation is applied in the simulations. 3.5.1 Positioning Monte Carlo simulations are conducted to verify that the proposed algorithm can improve the positioning performance. The simulation time is every minute for 24 hours starting at mid- night at the beginning of the GPS week, and the location is selected at London (the latitude 52°N and the longitude 0°E). A total of 1440 (60×24) sample points were produced according to the selected user locations and simulation times. Then a result of the estimated positioning vector can be obtained for each sample point, and furthermore, the estimated positioning error ( )kposx~
  • 59. 33 can be obtained through the following equation ( ) ( ) ( )kkk truepospos xxx −= ˆ~ , (3-49) where ( )kposxˆ is the 3×1 estimated position vector and ( )ktruex is the 3×1 true position vector. The procedure of using the multi-frequency technique on positioning is summarized as the fol- lowing three steps Step 1: Select a specified positioning algorithm (conventional single frequency or dual fre- quency or triple frequency algorithm). Step 2: Calculate the estimated state ( )kxˆ (or ( )kDFxˆ or ( )kTFxˆ ) from the user-satellite geometry through (2-9) (or (3-7) or (3-20)), and then find out the corresponding esti- mated position. Step 3: Determine the estimated positioning error ( )kposx~ through (3-49). Based on the simulation environments described above, an estimated positioning error can be obtained for each sample point. The standard deviations of the estimated positioning error are listed in Table 3-1. Furthermore, the east and north errors (i.e., the first and the second ele- ment of the estimated positioning error) are plotted in Figure 3-1(a), Figure 3-1(b) and Figure 3-1(c), respectively. The simulation results show that, in comparison with the conventional single frequency method, the proposed multi-frequency algorithms possess more accurate posi- tioning results. The estimated positioning errors of multi-frequency method are concentrated around the origin due to the reduction of measurement error since the ionospheric delay is re- moved by adopting the multi-frequency (dual frequency or triple frequency). Furthermore, the reductions of the estimated positioning errors in vertical axis are especially significant. Table 3-1 Standard deviation of the positioning error Horizontal error (m) Standard deviation East error North error Vertical error (m) Single frequency 5.5499 3.9982 21.2627 Double frequency 2.2234 3.3094 6.1387 Triple frequency 1.8737 2.8852 5.4481
  • 60. 34 -20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 Positioning error (L1) east error (m) northerror(m) (a) Single frequency -20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 Positioning error (L1/L2) east error (m) northerror(m) (b) Dual frequency
  • 61. 35 -20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 Positioning error (L1/L2/L5) east error (m) northerror(m) (c) Triple frequency Figure 3-1 Positioning error 3.5.2 Fault Detection Monte Carlo simulations are conducted to verify that the proposed algorithm can reduce the detection time. A total of 1152 (24×48) space-time sample points were produced according to the user locations and simulation times mentioned in [36]. The user locations covering the 24 geographic locations are listed in Table 3-2, and the simulation time is every half hour for 24 hours starting at midnight at the beginning of the GPS week. Both ramp-type and step-type failures are used to simulate satellite malfunction. The ramp-type failure refers to a failure growing linearly with time, and the step-type failure refers to a constant bias happening at and remaining after a specified time. The procedure of using the multi-frequency technique on sat- ellite failure for each point is summarized as the following three steps. Step 1: Select a specified fault detection algorithm (conventional single frequency or dual fre- quency or triple frequency algorithm) and set the time index as one (k = 1). Step 2: Calculate the test statistic ( )ks (or ( )ksDF or ( )ksTF ) from the user-satellite geome- try through (2-12) (or (3-14) or (3-33)). Step 3: Compare the test statistic ( )ks (or ( )ksDF or ( )ksTF ) with the threshold Td under the
  • 62. 36 specified fault detection algorithm. If the test statistic exceeds Td, then the detection time will be recorded. Otherwise the time index will be increased by one (k = k + 1), and then repeat step 2 and step 3. Table 3-2 Geographic locations for simulation Name Latitude Longitude London 52°N 0°W Liberia 7°N 10°W South Atlantic 30°S 15°W Iceland 65°N 22°W St. Johns 49°N 52°W Buenos Aires 30°S 58°W Ecuador 3°S 80°W New Orleans 30°N 90°W Winnipeg 50°N 95°W Easter Island 27°S 112°W Los Angeles 34°N 118°W Central Pacific 5°S 135°W North Alaska 70°N 150°W Honolulu 22°N 158°W Ross Sea 75°S 180°W New Zealand 40°S 175°E Marshall Islands 8°N 170°E Tokyo 36°N 140°E Perth 32°S 115°E Singapore 2°N 104°E Indian Ocean 45°S 75°E Aral Sea 45°N 60°E Madagascar 15°S 50°E Cape Town 35°S 18°E Based on the simulation environments described above, a detection time (DT) can be ob- tained for each point. DT is defined as the time needed from the onset of the failure to the an- nunciation of an alarm signal. The average detection time (ADT) is the sample mean of the 1152 values of the DT. The simulation result of ADT for ramp-type pseudorange failure is plotted in Figure 3-2. This figure shows that the multi-frequency detection algorithm has
  • 63. 37 shorter ADT than the conventional one (single frequency) under all ramp-type failures (0.5, 1, 2, 5, 10, and 20m/s). The resulting ADT are transferred into the percentage improvements of ADT (PIADT) through the following equation ( ) ( ) ( ) 100% SFADT MFADTSFADT PIADT × − = , (3-50) where ADT(SF) is the ADT when the single frequency is applied, ADT(MF) is the ADT when the multi-frequency (dual frequency or triple frequency) is applied. The best improvement per- centage for dual frequency and triple frequency are 48.3% and 55.9%, respectively. The simu- lation result of ADT for step-type pseudorange failure is plotted in Figure 3-3(a) and (b), respec- tively. This figure shows that the multi-frequency detection algorithm has shorter ADT than the conventional one under all step-type failures (ranges from 15 to 80m). The percentage of im- provement can be obtained through (3-50). The best improvement percentage for dual fre- quency and triple frequency are 99.1% and 99.2%, respectively. To sum up, in comparison with the conventional single frequency method, the multi-frequency algorithms demonstrates higher performance in detecting small failures and, in detecting large failures, demonstrates a similar level of performance. 0.5 1 2 5 10 20 1 2 5 10 20 50 80 Slope (m/s) AverageDetectionTime(sec) Average Detection Time Single frequency Dual frequency Triple frequency Figure 3-2 ADT under ramp-type failure
  • 64. 38 40 45 50 60 70 80 1 2 5 10 20 50 100 200 Bias (m) AverageDetectionTime(sec) Average Detection Time Single frequency Dual frequency Triple frequency (a) Single frequency v.s. Multi-frequency 15 20 25 30 35 40 45 1 2 5 10 20 50 80 Bias (m) AverageDetectionTime(sec) Average Detection Time Dual frequency Triple frequency (b) Dual frequency v.s. Triple frequency Figure 3-3 ADT under step-type failure
  • 65. 39 3.5.3 Fault Exclusion Furthermore, to show that the proposed algorithm can reduce the incorrect exclusion rate (IER), the step-type failures is used to simulate satellite malfunction. A total of 1152 (24×48) space-time sample points were produced according to the user locations and simulation times mentioned in [36]. The user locations covering the 24 geographic locations are listed in Table 3-2, and the simulation time is every half hour for 24 hours starting at midnight at the beginning of the GPS week. The procedure of using the multi-frequency technique on satellite exclusion is summarized as the following four steps. Step 1: Select a specified fault exclusion algorithm (conventional single frequency or dual fre- quency or triple frequency algorithm). Step 2: Calculate parity vector ( )kp (or ( )kDFp or ( )kTFp ) from the user-satellite geometry through (2-15) (or (3-13) or (3-32)). Step 3: Identify the channel number of the failed satellite, ( )knf , from (2-17) (or (3-15) or (3-34)). Step 4: Comparing ( )knf with the channel number of the true failed satellite, if they did not match each other, then an incorrect exclusion (IE) is recorded. An IE occurs when the receiver performs a valid detection, but the failed satellite remains in the solution after the exclusion operation [16]. The incorrect exclusion rate (IER), which is used as a performance index, is defined as 100% exclusiontotalofno. exclusionincorrectofno. IER ×= . (3-51) IER can be used to verify the superior exclusion capability of the proposed algorithms. Since a total of 1152 space-time sample points were produced in this simulation, the number of total ex- clusion is 1152. The bias value of step ranges from 15m to 50m with a 5m incremental. Simulation results are given in Figure 3-4. These figures show that the IER obtained through the dual frequency is about 5% lower than the one through conventional single frequency under the best scenario (bias = 20 m). The IER obtained through the triple frequency under the best scenario (bias = 15 m) is about 9% (12%) lower than the one through the dual (single) frequency. In conclusion, the multi-frequency can reduce the IER in excluding the failed satellite.
  • 66. 40 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 Bias (m) IncorrectExclusionRate(%) Incorrect Exclusion Rate Single frequency Dual frequency Triple frequency Figure 3-4 Incorrect exclusion rate
  • 67. 41 Chapter 4 Fault Detection and Exclusion via ARMA-Filter Autoregressive moving average (ARMA) filter has been widely used as a tool for fault di- agnosis in the field of quality control. It uses the historical data as well as the up-to-date infor- mation since failure may already exist in past measurements before it is detected. Therefore, an algorithm based on the ARMA is proposed in this chapter to perform the satellite fault detection and exclusion. The multi-frequency technique is adopted to eliminate the ionospheric delay. The proposed algorithm includes two parts, fault detection and fault exclusion. In the first part, the ARMA-filter is applied to speed up the failure detection. Speeding up of the failure detec- tion might provide more time for pilots to prevent serious deviations of aircrafts from their in- tended flight paths. In order to calculate the detection threshold under a specified false alarm rate (FAR), the ARMA model is first transformed into the state-space model, and then the threshold can be approximated by a “discrete finite-state Markov chain”. In this chapter, we determine the detection threshold for two special cases of the ARMA-filter, the moving average (MA) filter and the parallel-form autoregressive moving average (PFARMA) filter. In the sec- ond part, the multivariate ARMA-filter is proposed to reduce the incorrect exclusion rate (IER) in fault exclusion by taking the average of the last several parity vectors. In this chapter, failure detectors based the ARMA-filter will be introduced in Section 4.1. The procedures to deter- mine the threshold value for the MA-filter under a given FAR is given in Section 4.2. The pro- cedures to determine the threshold value for the PFARMA-filter under a given FAR is discussed in Section 4.3. Since the numbers of visible satellites may vary with time, the calculated nor- malized SSE may have different distributions and cannot be directly accumulated through ARMA-filter. Therefore, the probability integral transformation (PIT) method is proposed to resolve this in Section 4.4. Furthermore, in Section 4.5, we apply the multivariate ARMA-filter
  • 68. 42 to satellite fault exclusion. Section 4.6 presents simulation results to verify the proposed algo- rithm. 4.1 Using ARMA-Filter on Fault Detection In this section, the ARMA-filter, which has been used in detection of abrupt change [2], is proposed to speed up the satellite failure detection. The number of satellites in view is assumed to be constant at the moment. The case when the number of visible satellites changes will be described later. The ARMA scheme is based on the statistic ( ) ( ) ( )∑∑ == +−+−= M i i N i i iksikzkz 11 1βα , (4-1) where k is the running time index, N is the order of the filter, M is the window size of the filter, ()⋅s is as defined in (2-12), iα is the coefficient of the filter, and iβ is the weight of the filter; furthermore, iα and iβ satisfies 111 =+ ∑∑ == M i i N i i βα . Under the hypothesis that no failure is taking place, ()⋅s is chi-square distributed with ν degrees of freedom, and then the expectation value of ()⋅s will equal to the associated degrees of freedom, ν. Therefore, the initial condi- tions of (4-1) are set as ( ) ( ) ν==−= L10 zz and ( ) ( ) ν==−= L10 ss . Take expectation on both sides of (4-1), and it can be obtained that the expectation value of ()⋅z will equal to the ex- pectation value of ()⋅s as no satellite failure occurs. For the special case of N = 0, the ARMA-filter is exactly the same as an MA-filter. To judge whether the system is failed or not, the test statistic ( )kz will be compared with the detection threshold, Td. In order to calculate the value of Td under a specified FAR, the filter is first transformed into the state-space model, and then the value of Td can be approximated by a “discrete finite-state Markov chain”. In this chapter, we only discuss two special cases of the ARMA-filter, the MA-filter and the PFARMA- filter. The procedures to determine the threshold value for theses two filters are described in the following two sections. 4.2 Determination of the Threshold Value for MA-Filters The procedures to determine the threshold value for MA-filters are introduced in this sec- tion. The scheme of the MA-filter is based on the following statistic ( ) ( )∑= +−= M i i ikskz 1 1β , (4-2) where k is the running time index, M is the window size, ()⋅s is as defined in (2-12), and iβ is the weight of the filter with nonnegative value and satisfies 11 =∑= M i iβ . The initial conditions
  • 69. 43 of (4-2) are set as ( ) ( ) ν==−= L10 ss . Take expectation on both sides of (4-2), and it can be obtained that the expectation value of ()⋅z will equal to the expectation value of ()⋅s as no satel- lite failure occurs. If a satellite failure is detected at time dk , then the past data ( )1−dks , …, ( )mks d − might contain a bias already. Thus, the past data cannot be applied to the MA-filter any more, and the filter should be reset. The process of reset is based on the initial conditions of (4-2) as ( ) ( ) ν=−==− mksks dd L1 . As a result, the cumulative distribution function (cdf) of ()⋅z cannot be directly obtained through algebraic calculation. Therefore, the threshold can not be calculated in the traditional way as in Subsection 2.3.1. The MA filter can be converted into a state space model, which describes the relationship between the current data and the previous one and thus matches the properties of Markov chain. Therefore, the Markov chain approach is proposed to obtain the threshold value of the MA-filter. At first, (4-2) is transformed to a state space model [25]. Set ( ) ( )1+−= ikskiθ , for i = 1, …, M, and define the state vector ( ) ( ) ( ) ( )[ ]T Mi kkkk θθθ LL1=θ . Then ( )kθ can be represented in a dynamic equation as ( ) ( ) ( )kskk MMA 1,1 eθΦθ +−= , (4-3) where MAΦ is an M×M transition matrix with the ijth element as ⎩ ⎨ ⎧ += =Φ othewise,0 1if,1 jiMA ij , and [ ]T M 0011, L=e is an M×1 column vector. The initial conditions of (4-3) is set as ( ) Mlθ ν=0 , where Ml is an M×1 column vector with all elements equal to one. Then (4-2) can be repre- sented as ( ) ( )kkz T θβ= , (4-4) where [ ]T Mββ L1=β is the weighting vector satisfying 1=βlT M . Since ( )kz is related to ( )kθ through (4-4), the calculation of the threshold of ( )kz can be performed on ( )kθ . Because the state vector ( )kθ in equation (4-3) is a Markov process, the calculation of the threshold under a specific false alarm rate (FAR) can be approximated by modeling the process ( )kθ as a discrete Markov chain with a stationary transition probability matrix. There are two types of Markov states: transient states and a terminating state. When ( )kz is less than or equal to the threshold value, Td, the process belongs to transient states. Otherwise, it belongs to the terminating state. Once the state becomes terminating, it will not go back to the transient state. The case of window size equal to 2 is discussed in Subsection 4.2.1; then the case of window size equal to 3 is described in Subsection 4.2.2; finally the general case will be shown in Subsec- tion 4.2.3.
  • 70. 44 4.2.1 MA-filter with Window Size 2 For the case of window size equal to 2, the corresponding state equation is as follows ( ) ( ) ( )kskk ⎥⎦ ⎤ ⎢⎣ ⎡+− ⎥⎦ ⎤ ⎢⎣ ⎡= 0 11 01 00 θθ , (4-5) where ( ) ( ) ( )[ ]T kkk 21 θθ=θ ; both ( )k1θ and ( )k2θ are nonnegative because ( )ks is nonnegative. The process is in transient states at time k if and only if ( ) dTkz ≤ , i.e., ( ) ( ) dTkk ≤+ 2211 θβθβ , where 1β and 2β are nonnegative weights satisfying 121 =+ ββ . Therefore, the transient states are bounded by the triangle area in Figure 4-1. To calculate the transition probability matrix, this area is divided into L subareas, S1, …, Si, …, SL, each of which represents a transient state. The division is performed along θ1-axis with the same width, LTd 1 12 − = βδ . The center of the ith subarea, Si, is represented as ( )ii ,ζµ , where =iµ ( )δ12 −i and ( ) 21 1 2 idi T µββζ −= − . Figure 4-1 States of Markov chain for MA-filter with window size 2 Then, ( )kθ belongs to transient state Si if and only if ( ) [ ] ( ) ( )⎩ ⎨ ⎧ ≤+≤ +−∈ d ii Tkk k 2211 1 0 , θβθβ δµδµθ . (4-6) Assume the probability distribution of ( )kθ can be described by the probability vector ( ) =kπ ( ) ( ) ( )[ ]T Li kkk πππ LL1 , where ( )kiπ represents the probability of ( )kθ being in state Si. Note that ( )kT L πl is the probability that no alarm is activated and the satellites are all judged as normal at time k, where Ll is an L×1 column vector with all elements equal to one. Then the transition of ( )kπ can be represented by ( ) ( )1−= kk Tππ , (4-7) where T is the transition probability matrix among transient states [12]. T is defined as dT1 1 − β 1S LS ()⋅2θ ()⋅1θ 0 dT1 2 − β () () dT=⋅+⋅ 2211 θβθβ iS ( )ii ,ζµ
  • 71. 45 ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = L,LL, ,L, TT TT L MOM L 1 111 T , (4-8) where ( ) ( )[ ]jiij SstateinwaskSstatetogoeskT 1|Pr −≡ θθ . As shown in Figure 4-2, Tij can be approximated as ( ){ } ( ) ( ){ } ( ) ( )[ ]jjdiiij kkTkkkT ζθµθθβθβδµθδµ =−∩=−≤+≤∩+≤≤−= 11|0Pr 2122111 .(4-9) In the previous equation, the elements of ( )1−kθ being in state Sj are approximated by setting ( ) jk µθ =−11 and ( ) jk ζθ =−12 . Substitute ( ) ( )ksk =1θ and ( ) ( ) jkk µθθ =−= 112 into (4-9), and we have ( ){ } ( ){ }[ ] ( ){ } ( ) ( ){ }[ ] ( )[ ] , ,0 ,Pr 0Pr 0Pr 2 1 1 21 ⎩ ⎨ ⎧ ≤≤≤ = −≤≤∩+≤≤−= ≤+≤∩+≤≤−= − otherwise BBifBksB Tksks TksksT ijijijij jdii djiiij µββδµδµ µββδµδµ (4-10) where δµ −= iijB and ( )( )jdiij TB µββδµ 2 1 1,min −+= − . The division is performed only on the θ1-axis because ( ) jk ζθ =−12 will not appear in (4-10) during the transition. Figure 4-2 Transient of Markov chain for MA-filter with window size 2 Figure 4-3 Initial states of Markov chain for MA-filter with window size 2 dT1 2 − β 1S LS 1S LS iS jS ( )k2θ ( )k1θ dT1 1 − βdT1 1 − β ( )12 −kθ dT1 2 − β ( )11 −kθ 0 0 ijT dT1 1 − β 1S LS iniS ()⋅2θ ()⋅1θ dT1 2 − β () () dT=⋅+⋅ 2211 θβθβ ( )νν, 0
  • 72. 46 The initial condition of the probability vector, ( )0π , is set as a vector with all elements equal to zero, except that πini(0) is set to 1, where πini(0) is corresponding to state Sini containing ( ) [ ]T νν=0θ . Figure 4-3 depicts the above. Time to false alarm (TFA) is defined as the time needed from the beginning of detection to the declaration of a false alarm, and the mean of which is called mean time to false alarm (MTFA) [27]. MTFA can be represented as ( ) ( )0 1 πTΙl − −= L T LFAM , (4-11) where IL is a L×L identity matrix. In fact, FAR is equal to the inverse of MTFA. The proce- dures to obtain the threshold value under a specified FAR will be discussed at the end of this sec- tion. 4.2.2 MA-filter with Window Size 3 For the case of window size equal to 3, the corresponding state equation is as follows ( ) ( ) ( )kskk ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ +− ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ = 0 0 1 1 010 001 000 θθ , (4-12) where ( ) ( ) ( ) ( )[ ]T kkkk 321 θθθ=θ ; all elements ( )k1θ , ( )k2θ and ( )k3θ are nonnegative be- cause ( )ks is nonnegative. The process is in transient states at time k if and only if ( ) dTkz ≤ , i.e., ( ) ( ) ( ) dTkkk ≤++ 332211 θβθβθβ , where 1β , 2β , and 3β are nonnegative weights satisfying 1321 =++ βββ . Therefore, the transient states are bounded by the tetrahedron in Figure 4-4. As in the case of window size equal to 2, the divisions can be performed along θ1-axis and θ2-axis only to calculate the transition probability matrix. The division performed on θ1-axis are with the same width, LTd ′= −1 112 βδ , and on θ2-axis with the same width, LTd ′= −1 222 βδ , where L′ is the number of division on both axes. This tetrahedron is divided into L subspace, (a) whole picture (b) top view Figure 4-4 States of Markov chain for MA-filter with window size 3 LS dT1 1 − β ()⋅2θ ()⋅1θ 0 dT1 2 − β () () dT=⋅+⋅ 2211 θβθβ ( )iiiiS ζµµ ,, ,2,1 1S dT1 1 − β ()⋅3θ ()⋅1θ 0 dT1 3 − β ()⋅2θ dT1 2 − β () di ii T=⋅∑= 3 1 θβ
  • 73. 47 S1, …, Si, …, SL, where L is equal to L′(L′+1)/2. The center of the ith subspace, Si, is represented as ( )ii,i, ,, ζµµ 21 , where ( )( ) 11,1 12 δυµ −= ii , ( )( ) 22,2 12 δυµ −= ii and ( )iidi T ,22,11 1 32 1 µβµββζ −−= − . The functions υ1(i) and υ2(i) will map the index of states to the corresponding indexes in θ1-axis and θ2-axis. Figure 4-5 gives an example for the case of L′ = 3. The value in the top of each state is the state number, and the values inside the parenthesis are the corresponding values of (υ1(i), υ2(i)). 3 (1,3) 2 (1,2) 5 (2,2) 1 (1,1) 4 (2,1) 6 (3,1) Figure 4-5 Definition of υ1(i) and υ2(i) for L′ = 3 Figure 4-6 Transient of Markov chain for MA-filter with window size 3 Then, ( )kθ belongs to transient state Si if and only if ( ) [ ] ( ) [ ] ( ) ( ) ( )⎪ ⎩ ⎪ ⎨ ⎧ ≤++≤ +−∈ +−∈ d i,i, i,i, Tkkk ,k ,k 332211 22222 11111 0 θβθβθβ δµδµθ δµδµθ . (4-13) The transition probability matrix among transient states is given in (4-8) with the ijth element as ( ) ( )[ ]jiij SstateinwaskSstatetogoeskT 1|Pr −≡ θθ . As shown in Figure 4-6, Tij can be ap- proximated as ijT 1S LS dT1 1 − β ( )k2θ ( )k1θ 0 dT1 2 − β LS dT1 1 − β ( )12 −kθ ( )11 −kθ 0 dT1 2 − β 1S jS iS
  • 74. 48 ( ){ } ( ){ }[ ( ) ( ) ( ){ } ( ){ } ( ){ } ( ){ }].111| 0 Pr 3,22,11 332211 2,222,21,111,1 jjj d iiiiij kkk Tkkk kkT ζθµθµθ θβθβθβ δµθδµδµθδµ =−∩=−∩=− ≤++≤∩ +≤≤−∩+≤≤−= (4-14) In the previous equation, the elements of ( )1−kθ being in state Sj are approximated by setting ( ) jk ,11 1 µθ =− , ( ) jk ,22 1 µθ =− and ( ) jk ζθ =−13 . Substitute ( ) ( )ksk =1θ , ( ) ( ) jkk ,112 1 µθθ =−= , and ( ) ( ) jkk ,223 1 µθθ =−= into (4-14), and we have ( ){ } { }[ ( ){ } ( )[ ] ( ) ( )( )[ ]{ } { }[ ] ( ){ }[ ] ( )[ ] , otherwise,0 andif,Pr Pr 0Pr 0 Pr 2,2,1 2,2,1 2,2,1,23,12 1 11,11,1 ,23,121 2,2,12,21,11,1 ⎩ ⎨ ⎧ ≤−≤≤≤ = ≤−∩≤≤= ≤−∩+−≤≤∩+≤≤−= ≤++≤∩ +≤≤−∩+≤≤−= − δµµ δµµ δµµµβµββδµδµ µβµββ δµµδµδµδµ ijijijijij ijijij ijjjdii djj ijiiiij BBBksB BksB Tksks Tks ksT (4-15) where 1,1 δµ −= iijB and ( )( )( )jjdiij TB ,23,12 1 11,1 ,min µβµββδµ +−+= − . The initial condition of the probability vector, ( )0π , is set as a vector with all elements equal to zero, except that πini(0) is set to 1, where πini(0) is corresponding to state Sini containing ( ) [ ]T ννν=0θ . Figure 4-7 de- picts the above. The MTFA can be obtained through (4-11), and the procedures to obtain the threshold value will be discussed at the end of this section. Figure 4-7 Initial states of Markov chain for MA-filter with window size 3 4.2.3 MA-filter with Window Size Larger than 3 For the case of window size larger than 3, the corresponding state equation is described in (4-3), where all elements ( )kiθ are nonnegative because ( )ks is nonnegative. The process is in transient states at time k if and only if ( ) dTkz ≤ , i.e., ( ) d T Tk ≤θβ , where β is the weighting vector satisfying 1=βlT M . Therefore, the transient states are bounded by a hyperspace. As in LS dT1 1 − β ()⋅2θ ()⋅1θ 0 dT1 2 − β () () dT=⋅+⋅ 2211 θβθβ ( )ννν ,,iniS 1S
  • 75. 49 the case of window size equal to 2, the divisions can be performed along all axes except θM-axis. The division is performed on θl-axis with the same width, LTw dll ′= −1 2δ for l = 1, …, M−1, where L′ is the number of division on each axis except θM-axis. This hyperspace is divided into L subspaces, S1, …, Si, …, SL, each of which represents a transient state. Represent the center of the ith subspace, Si, as ( )iiMili ζµµµ ,,,,, ,1,,1 −KK . Then, ( )kθ belongs to transient state Si if and only if ( ) [ ] ( )⎪⎩ ⎪ ⎨ ⎧ ≤≤ −=+−∈ ∑= d M l ll lillill Tk Mlk 1 ,, 0 1,,1for,, θβ δµδµθ K . (4-16) The transition probability matrix among transient states is given in (4-8) with the ijth element as ( ) ( )[ ]jiij SstateinwaskSstatetogoeskT 1|Pr −≡ θθ . As in the case of window size equal to 3, Tij can be approximated as ( ){ } ( ){ }[ ( ){ } ( ){ } ( ){ } ( ){ } ( ){ }].111 0 Pr ,11,11 11,111,1 2,222,21,111,1 jMjMMj d M l llMiMMMiM iiiiij kkk Tkk kkT ζθµθµθ θβδµθδµ δµθδµδµθδµ =−∩=−∩∩=− ≤≤∩+≤≤−∩ ∩+≤≤−∩+≤≤−= −− =−−−−− ∑ L L (4-17) In the previous equation, the elements of ( )1−kθ being in state Sj are approximated by setting ( ) jll k ,1 µθ =− for l = 1, …, M−1, and ( ) jM k ζθ =−1 . Substitute ( ) ( )ksk =1θ , and ( )=klθ ( ) jll k ,11 1 −− =− µθ for l = 2, …, M into (4-18), and we have ( ){ }[ { } { } ( ){ }] ( ){ }[ { } { } ( ) ( ){ }] ( )[ ] ( ) ( )[ ]{ }[ { } { }] ( ){ } { } { }[ ],Pr 0Pr 0 Pr 0 Pr 1,1,22,2,1 1,1,22,2,1 2 ,1 1 11,11,1 2 ,1 1 1 1,1,22,2,1 1,11,1 2 ,11 1,1,21,12,2,12,2 1,11,1 −−− −−− = − − = − − −−− = − −−−−− ≤−∩∩≤−∩≤≤= ≤−∩∩≤−∩ −≤≤∩+≤≤−= −≤≤∩ ≤−∩∩≤−∩ +≤≤−= ≤+≤∩ +≤≤−∩∩+≤≤−∩ +≤≤−= ∑ ∑ ∑ miMjMijijij MiMjMij M l illdii M l illd MiMjMij ii d M l ill MiMjMMiMiji iiij BksB Tksks Tks ks Tks ksT δµµδµµ δµµδµµ µββδµδµ µββ δµµδµµ δµδµ µββ δµµδµδµµδµ δµδµ L L L L (4-18) where 1,1 δµ −= iijB and ( )( )∑ = − − −+= M l illdiij TB 2 ,1 1 11,1 ,min µββδµ . In the last line of (4-18), { } { }112221 −−− ≤−∩∩≤− mi,Mj,Mi,j, δµµδµµ L can be viewed as constraints since jδ and j,lµ for l = 1, …, M−1 are deterministic. Thus (4-18) can be simplified as follows
  • 76. 50 ( ){ } { } { }[ ] ( )[ ] . otherwise,0 and,,,if,Pr Pr 1,1,22,2,1 1,1,22,2,1 ⎩ ⎨ ⎧ ≤−≤−≤≤≤ = ≤−∩∩≤−∩≤≤= −−− −−− miMjMijijijijij miMjMijijijij BBBksB BksBT δµµδµµ δµµδµµ L L (4-19) The initial condition of the probability vector, ( )0π , is set as a vector with all elements equal to zero, except πini(0) is set to 1, where πini(0) is corresponding to state Sini containing ( ) 1,0 Meθ ν= . MTFA can be obtained through equation (4-11), and FAR is equal to the inverse of MTFA, i.e., FARM FA 1* = . The procedures to obtain * dT under a specific window size * M and weight * iβ for i = 1, …, M are summarized in the following four steps. The procedures are also depicted in Figure 4-8. Step 1: Set * MM = , * ii ββ = for i = 1, …, M, degrees of freedom ν = 2 and FARM FA 1* = . Step 2: Make an initial guess dTˆ and calculate the corresponding FAMˆ . Step 3: If FAMˆ is larger than * FAM , decrease dTˆ ; otherwise, increase dTˆ . Step 4: If .ˆ * tolMM FAFA ≤− , then the * dT corresponding to * M and * iβ is obtained. The process is repeated until adequate resolution is attained. Figure 4-8 Flow chart for the threshold calculation of MA-filter Set * MM = , * ii ββ = for i = 1, …, M ν = 2 and calculate FARM FA 1* = Make an initial guess dTˆ Calculate the FAMˆ Decrease dTˆ The * dT corresponding to * M and * iβ is obtained Increase dTˆ .ˆ * tolMM FAFA ≤− Yes No No Yes *ˆ FAFA MM >
  • 77. 51 4.3 Determination of the Threshold Value for PFARMA-Filters The procedures to determine the threshold value for PFARMA-filter is introduced in this section. The PFARMA-filter, which is the ARMA filter in the parallel-form structure, is ob- tained by performing a partial fraction expansion. In addition, we assume that each root in the PFARMA filter is real with value between zero and one. The scheme of the PFARMA- filter is based on the following statistic ( ) ( )kkz T θη= , (4-20) where k is the running time index, N is the order of the filter, [ ]T Nl ηηη LL1=η is an N×1 weighting vector satisfying 1=ηlT N and 0≥lη for l = 1, …, N, Nl is an N×1 column vector with all elements equal to one, and ( ) ( ) ( ) ( )[ ]T Ni kkkk θθθ LL1=θ is an N×1 state vector. In ad- dition, the state vector, ( )kθ , can be represented in a dynamic equation as ( ) ( ) ( )kskk PFARMA λθΦθ +−= 1 , (4-21) where { }nlPFARMA diag λλλ ,,,,1 LL=Φ is an N×N transition matrix, lλ is the parameter of the filter satisfying 10 <≤ lλ for l = 1, …, N, [ ]T nl λλλ −−−= 111 1 LLλ is an N×1 column vector, and ()⋅s is as defined in (2-12). Under the hypothesis that no failure is taking place, ()⋅s is chi-square distributed with ν degrees of freedom, and then the expectation value of ()⋅s will equal to the associated degrees of freedom, ν. Therefore, the initial condition of (4-21) is set as ( ) Nlθ ν=0 . Take expectation on both sides of (4-20) and (4-21), and it can then be ob- tained that the expectation value of ()⋅z will equal to the expectation value of ()⋅s as no satellite failure occurs. If a satellite failure is detected at time dk , then the past data ( )1−dkθ might contain a bias already. Therefore, the past data cannot be applied to the PFARMA-filter any more, and the filter should be reset. The process of reset is based on the initial condition of (4-21) as ( ) Nk lθ ν= . As a result, the cumulative distribution function of ()⋅z cannot be di- rectly obtained through algebraic calculation. Because the state vector ( )kθ in equation (4-21) is a Markov process, the calculation of the threshold under a specific FAR can also be approxi- mated by modeling the process ( )kθ as a discrete Markov chain with a stationary transition probability matrix. The case of order equal to 1 is discussed in Subsection 4.3.1; then the case of order equal to 2 is discussed in Subsection 4.3.2; finally the general case will be shown in Subsection 4.3.3. 4.3.1 PFARMA -filter with Order 1 For the case of order equal to 1, the corresponding state equation is as follows [6][11]
  • 78. 52 ( ) ( ) ( ) ( ) ( )kskkkz 1111 11 λθλθ −+−== , (4-22) where ( )k1θ is nonnegative because ( )ks is nonnegative. The process is in transient states at time k if and only if ( ) dTkz ≤ , i.e., ( ) dTk ≤1θ . Therefore, the transient states are bounded by the interval in Figure 4-9. To calculate the transition probability matrix, this segment is divided into L subintervals, S1, …, Si, …, SL, each of which represents a transient state. The division is performed along θ1-axis with the same width, LTd=δ2 . The center of the ith subinterval, Si, is represented as µi, where ( )δµ 12 −= ii . Figure 4-9 States of Markov chain for PFARMA-filter with order 1 Figure 4-10 Transient of Markov chain for PFARMA-filter with order 1 Then ( )kθ belongs to transient state Si if and only if ( ) [ ]δµδµθ +−∈ iik ,1 . (4-23) Assume the probability distribution of ( )kθ can be described by the probability vector ( )=kπ ( ) ( ) ( )[ ]T Li kkk πππ LL1 , where ( )kiπ represents the probability of ( )kθ being in state Si. Note that ( )kT L πl is the probability that no alarm is activated and the satellites are all judged as normal at time k, Ll is an L×1 column vector with all elements equal to one. Then the transi- tion of ( )kπ can be represented by ( ) ( )1−= kk Tππ , (4-24) where T is the transition probability matrix among transient states [12]. T is defined as ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = L,LL, ,L, TT TT L MOM L 1 111 T , (4-25) where ( ) ( )[ ]jiij SstateinwaskSstatetogoeskT 1|Pr −≡ θθ . ijT LS dT 1S ( )11 −kθ 0 jS LS dT 1S ( )k1θ 0 iS iµ LS dT 1S ()⋅1θ 0 iS
  • 79. 53 As shown in Figure 4-10, Tij can be approximated as ( ) ( )[ ]jiiij kkT µθδµθδµ =−+≤≤−= 1|Pr 11 . (4-26) In the previous equation, the elements of ( )1−kθ being in state Sj are approximated by setting ( ) jk µθ =−11 . Substitute ( ) ( ) ( ) ( ) ( ) ( )kskskk j 111111 111 λµλλθλθ −+=−+−= into (4-26), and we have ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ] ( )[ ] , otherwise,0 if,Pr 11Pr 1Pr 1 1 11 1 1 11 ⎩ ⎨ ⎧ ≤≤≤ = −+−≤≤−−−= +≤−+≤−= −− ijijijij jiji ijiij BBBksB ks ksT µλδµλµλδµλ δµλµλδµ (4-27) where ( ) ( )( )0,1max 1 1 1 jiijB µλδµλ −−−= − and ( ) ( )jiijB µλδµλ 1 1 11 −+−= − . The initial condition of the probability vector, ( )0π , is set as a vector with all elements equal to zero, except πini(0) is set to 1, where πini(0) is corresponding to state Sini containing ( ) νθ =01 . Figure 4-11 depicts the above. According to Subsection 4.2.1, MTFA can be represented as ( ) ( )0 1 πTΙl − −= L T LFAM . (4-28) Moreover, FAR is equal to the inverse of MTFA, and the procedures to obtain the threshold value under a specified FAR will be discussed at the end of this section. Figure 4-11 Initial states of Markov chain for PFARMA-filter with order 1 4.3.2 PFARMA-filter with Order 2 For the case of order equal to 2, the corresponding state equation is as follows ( ) ( ) ( )kskk ⎥⎦ ⎤ ⎢⎣ ⎡ − − +−⎥⎦ ⎤ ⎢⎣ ⎡ = 2 1 2 1 1 1 1 0 0 λ λ λ λ θθ , (4-29) where ( ) ( ) ( )[ ]T kkk 21 θθ=θ ; both ( )k1θ and ( )k2θ are nonnegative because ( )ks is nonnegative. The process is in transient states at time k if and only if ( ) dTkz ≤ , i.e., ( ) ( ) dTkk ≤+ 2211 θηθη , where 1η and 2η are weights satisfying 121 =+ηη . Therefore, the transient states are bounded by the triangle area in Figure 4-12. To calculate the transition probability matrix, the divisions LS dT 1S ()⋅1θ 0 iniS ν
  • 80. 54 is performed along both θ1-axis and θ2-axis. The division performed on θ1-axis with the same width, LTd ′= −1 112 ηδ , and on θ2-axis with the same width, LTd ′= −1 222 ηδ , where L′ is the number of division on both axes. This triangle area is divided into L subareas, S1, …, Si, …, SL, where L is equal to ( ) 21+′′ LL . The center of the ith subarea, Si, is represented as ( )ii ,2,1 ,µµ , where ( )( ) 11,1 12 δυµ −= ii and ( )( ) 22,2 12 δυµ −= ii . In addition, the functions υ1(i) and υ2(i) will map the index of states to the corresponding indexes in θ1-axis and θ2-axis (see Subsection 4.2.2). Figure 4-12 States of Markov chain for PFARMA-filter with order 2 Figure 4-13 Transient of Markov chain for PFARMA-filter with order 2 Then ( )kθ belongs to transient state Si if and only if ( ) [ ] ( ) [ ] ( ) ( )⎪ ⎩ ⎪ ⎨ ⎧ ≤+≤ +−∈ +−∈ d ii ii Tkk k k 2211 2,22,22 1,11,11 0 , , θηθη δµδµθ δµδµθ . (4-30) The transition probability matrix among transient states is given in (4-25) with the ijth element as ( ) ( )[ ]jiij SstateinwaskSstatetogoeskT 1|Pr −≡ θθ . As shown in Figure 4-13, Tij can be ijT 1S LS dT1 1 − η ( )k2θ ( )k1θ 0 dT1 2 − η LS dT1 1 − η ( )12 −kθ ( )11 −kθ 0 dT1 2 − η 1S jS iS LS dT1 1 − η ()⋅2θ ()⋅1θ 0 dT1 2 − η () () dT=⋅+⋅ 2211 θηθη ( )iiiS ,2,1 ,µµ 1S
  • 81. 55 approximated as ( ){ } ( ){ }[ ( ) ( ){ } ( ){ } ( ){ }].11|0 Pr ,22,112211 2,222,21,111,1 jjd iiiiij kkTkk kkT µθµθθηθη δµθδµδµθδµ =−∩=−≤+≤∩ +≤≤−∩+≤≤−= (4-31) In the previous equation, the elements of ( )1−kθ being in state Sj are approximated by setting ( ) jll k ,1 µθ =− for l = 1, 2. Substitute ( ) ( ) ( ) ( ) ( ) ( )kskskk ljllllll λµλλθλθ −+=−+−= 111 , for l = 1, 2 into (4-31), and we have ( ) ( ){ }[ ( ) ( ){ } ( ) ( )( ) ( ){ }] ( ) ( ) ( ) ( ) ( ){ }[ ( ) ( ) ( ) ( ) ( ){ } ( ) ( ) ( )( ) ( ){ }] ( )[ ] , otherwise,0 if,Pr 110 11 11Pr 110 1 1Pr ,222,111 1 2211 ,222,2 1 2,222,2 1 2 ,111,1 1 1,111,1 1 1 1211,222,111 2,22,222,2 1,11,111,1 ⎩ ⎨ ⎧ ≤≤≤ = −−−+−≤≤∩ −+−≤≤−−−∩ −+−≤≤−−−= ≤−+−++≤∩ +≤−+≤−∩ +≤−+≤−= − −− −− ijijijij jjd jiji jiji djj iji ijiij BBBksB Tks ks ks Tks ks ksT µληµληληλη µλδµλµλδµλ µλδµλµλδµλ ληληµληµλη δµλµλδµ δµλµλδµ (4-32) where ( )0,,max ,2,1 ijijij BBB = and ( ) ( ) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −+− −− = 2211 ,222,111 ,2,1 11 ,,min ληλη µληµλη jjd ijijij T BBB with =ijlB , ( ) ( )jlllill ,, 1 1 µλδµλ −−− − and ( ) ( )jlllillijlB ,, 1 , 1 µλδµλ −+−= − for l = 1, 2. Figure 4-14 Initial states of Markov chain for PFARMA-filter with order 2 The initial condition of the probability vector, ( )0π , is set as a vector with all elements equal to zero, except that πini(0) is set to 1, where πini(0) is corresponding to state Sini containing ( ) [ ]T νν=0θ . Figure 4-14 depicts the above. Then MTFA can be obtained through (4-28), and the procedures to obtain the threshold value will be discussed at the end of this section. LS dT1 1 − η ()⋅2θ ()⋅1θ 0 dT1 2 − η () () dT=⋅+⋅ 2211 θηθη ( )νν ,Sini 1S
  • 82. 56 4.3.3 PFARMA-filter with Order Larger than 2 For the case of order larger than 2, the corresponding state equation is described in (4-21), where all elements ( )kiθ are nonnegative because ( )ks is nonnegative. The process is in tran- sient states at time k if and only if ( ) dTkz ≤ , i.e., ( ) d T Tk ≤θη , where η is the weighting vector satisfying 1=ηlT N . Therefore, the transient states are bounded by a hyperspace. To calculate the transition probability matrix, the divisions is performed along all axes. The division is per- formed on θl-axis with the same width LTw dll ′= −1 2δ for l = 1, …, N, where L′ denotes the number of division on each axis. Thus, the hyperspace is divided into L subspaces, S1, …, Si, …, SL, each of which represents a transient state. Represent the center of the ith subspace, Si, as ( )iNili ,,,1 ,,,, µµµ KK . Then ( )kθ belongs to transient state Si if and only if ( ) [ ] ( )⎪⎩ ⎪ ⎨ ⎧ ≤≤ =+−∈ ∑= d N i ii lillill Tk Nlk 1 ,, 0 ,,1for,, θη δµδµθ K . (4-33) The transition probability matrix among transient states is given in (4-25) with the ijth element as ( ) ( )[ ]jiij SstateinwaskSstatetogoeskT 1|Pr −≡ θθ . Furthermore, Tij can be approximated as ( ){ } ( ){ }[ ( ){ } ( ){ } ( ){ }].11|0 Pr ,,111 ,,1,111,1 jNNjd N l ll NiNNNiNiiij kkTk kkT µθµθθη δµθδµδµθδµ =−∩∩=−≤≤∩ +≤≤−∩∩+≤≤−= ∑ = L L (4-34) In the previous equation, the elements of ( )1−kθ being in state Sj are approximated by setting ( ) jll k ,1 µθ =− , for l = 1, …, N. Substitute ( ) ( ) ( ) ( ) ( ) ( )kskskk ljllllll λµλλθλθ −+=−+−= 111 , , for l = 1, …, N into (4-34), and we have ( ) ( ){ }[ ( ) ( ){ } ( )( ) ( ){ }] ( ) ( ) ( ) ( ) ( ){ }[ ( ) ( ) ( ) ( ) ( ){ } ( ) ( )( ) ( ) ( )[ ] , otherwise,0 if,Pr 10 11 11Pr 10 1 1Pr 1 , 1 1 ,, 1 ,, 1 ,111,1 1 1,111,1 1 1 11 , ,,, 1,11,111,1 ⎪⎩ ⎪ ⎨ ⎧ ≤≤≤ = ⎥⎦ ⎤ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ −−≤≤∩ −+−≤≤−−−∩ ∩−+−≤≤−−−= ≤−+≤∩ +≤−+≤−∩ ∩+≤−+≤−= ∑∑ ∑∑ = − = −− −− == ijijijij N l jllld N l ll jNNNiNNjNNNiNN jiji d N l ll N l jlll NiNNjNNNiN ijiij BBBksB Tks ks ks Tks ks ksT µληλη µλδµλµλδµλ µλδµλµλδµλ ληµλη δµλµλδµ δµλµλδµ L L (4-35)
  • 83. 57 where ( )01 ,B,,BmaxB ji,Nji,ij L= and ( ) ( )( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − = ∑ ∑ = = N l ll N l jllld ijNijij T BBB 1 1 , ,,1 1 ,,,min λη µλη L with =ijlB , ( ) ( )jlllill ,, 1 1 µλδµλ −−− − and ( ) ( )jlllillijlB ,, 1 , 1 µλδµλ −+−= − for l = 1, …, N. The initial condition of the probability vector, ( )0π , is set as a vector with all elements equal to zero, except πini(0) is set to 1, where πini(0) is corresponding to state Sini containing ( ) Nlθ ν=0 . MTFA can then be obtained through (4-28), and FAR is equal to the inverse of MTFA, i.e., FARM FA 1* = . The procedures to obtain * dT under a specific order * N , coeffi- cient * iλ , and weight * iη for i = 1, …, N are summarized in the following four steps. The pro- cedures are also depicted in Figure 4-15. Step 1: Set * NN = , * ii λλ = and * ii ηη = for i = 1, …, N, degrees of freedom ν = 2 and FARM FA 1* = . Step 2: Make an initial guess dTˆ and calculate the corresponding FAMˆ . Step 3: If FAMˆ is larger than * FAM , decrease dTˆ ; otherwise, increase dTˆ . Step 4: If .ˆ * tolMM FAFA ≤− , then the * dT corresponding to * N , * iλ and * iη is obtained. The process is repeated until adequate resolution is attained. Figure 4-15 Flow chart for the threshold calculation of PFARMA-filter Set * NN = , * ii λλ = , * ii ηη = for i = 1, …, N, ν = 2 and calculate FARM FA 1* = Make an initial guess dTˆ Calculate the FAMˆ Decrease dTˆ The * dT corresponding to * N , * iλ and * iη is obtained Increase dTˆ .ˆ * tolMM FAFA ≤− Yes No No Yes *ˆ FAFA MM >
  • 84. 58 4.4 Data Adjustment for Different Numbers of Satellites - PIT Thus far, the number of satellites in view is supposed to be constant. But in real situation the number of visible satellites may change with time. Because the least-squares-residual method is a snapshot-type method, the determination of thresholds depends only on the current number of the visible satellites. However, the ARMA-filter takes average over the last several data, which may be produced from different numbers of visible satellites. These data cannot be directly applied to the ARMA-filter because they may have different distributions. Thus, the Probability Integral Transformation (PIT) [19] is proposed to preprocess the data into a useable form. The procedure of PIT is to transform a random variable with a specific continuous dis- tribution into another one with different distribution that retains the same cumulative probability value. Because the test statistic is chi-square distributed, we will focus on this distribution. Suppose that X and Y are two random variables, having chi-square distributions with µ and ν de- grees of freedom, respectively. The associated cumulative distribution functions are with the cdf ()⋅µF and ()⋅νF . Assume that x and y are realizations of X and Y, and let ()⋅−1 µF be the in- verse function of ()⋅µF . Define z = Fν(y), and set ( )zFx 1− = µ . The transformation from y into x can be condensed into a single formula as ( )( )yFFx νµ 1− = . (4-36) During the transformation, the cumulative probabilities of x and y are equal since ( ) ( )yFxF νµ = . From “table of the χ 2 distribution” in [15], the chi-square cdf with ν degrees of freedom is ( ) ( ) ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = ∑ ∑ − = − − = + − evenis, k Y e oddis,Y k kY erfcxe YF k k kY k kkY ν ν π ν ν ν 2 2 0 2 2 3 0 2 12 2 !2 1 !12 !22 2 1 , (4-37) where ( ) ∫ ∞ −− = x ux dueexerfcx 222 π is the scaled complementary error function. In (4-37), it is difficult to find a closed form of the inverse chi-square cdf in most cases. However, since the cdf of χ 2 (2) is ( ) 2 2 1 Y eYF − −= , the inverse function can be obtained as ( ) ( )ZZF −−=− 1log21 2 . (4-38)
  • 85. 59 Substitute (4-37) and (4-38) into (4-36), and then the formula of PIT can be represented as [22] ( )( ) ( )( ) ( ) ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + +⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = −−= = ∑ ∑ − = − = + − evenis k Y Y oddisY k kY erfcxY YF YFFX k k k k kk ν ν π ν ν ν ν , !2 log2 , !12 !22 2 log2 1log2 2 2 0 2 3 0 2 12 1 2 , (4-39) where X and Y are random variables, having chi-square distribution with 2 and ν degrees of freedom respectively. From (4-39), the PIT can be represented in a closed form if ν is even, and an extra function erfcx(x) is needed in the formula if ν is odd. 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 random variable cdfvalue Probability Integral Transformation 0.9 4.6 10.6 χ2 (2) χ2 (6) F -1 F 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 random variable cdfvalue Probability Integral Transformation 0.9 4.6 10.6 χ2 (2) χ2 (6) F -1 F Figure 4-16 Illustration of PIT method
  • 86. 60 In order to help in explaining how PIT works, a simple example is demonstrated here. Suppose there are 10 satellites in view, and the statistic SSE is consequently χ2 (6). If an SSE = 10.6 is obtained, then the datum 10.6 is discounted as 4.6, as if there are 6 satellites in view. Since ( ) 9.06.106 =F , the value 0.9 is transferred to 4.6 via ( ) 6.49.01 2 =− F . Figure 4-16 illus- trates the above procedures. 4.5 Using Multivariate ARMA-filter on Fault Exclusion In this section, a parity-space-based fault exclusion algorithm using the multivariate ARMA (MARMA) filter is introduced to reduce the incorrect exclusion rate. The MARMA-filter is defined as ( ) ( ) ( )∑∑ == +−+−= M i i N i i ikikk 11 1pqq βα , (4-40) where k is the running time index, N is the order of the filter, M is the window size, ( )kp is the (n−4)×1 parity vector defined in (2-15), and iα is the coefficient of the filter, iβ is the weight of the filter; furthermore, iα and iβ satisfies 111 =+ ∑∑ == M i i N i i βα . The initial conditions of (4-40) are set as ( ) ( ) 0qq ==−= L10 and ( ) ( ) 0pp ==−= L10 . The parity matrix ( )kP can be regarded as a constant matrix since the observation matrix, ( )kH , changes slowly with time. Therefore, (4-40) can be simplified as ( ) ( ) ( )kkk ARMAzPq = , (4-41) where ( ) ( ) ( )∑∑ == +−+−= M i i N i ARMAiARMA ikikk 11 1yzz βα . According to (2-17), the exclusion algorithm based on (4-41) can be defined as ( ) ⎟ ⎠ ⎞⎜ ⎝ ⎛ ⎟ ⎠ ⎞⎜ ⎝ ⎛⎟ ⎠ ⎞⎜ ⎝ ⎛ = = k kk kn i i T ni f p pq ,,1 maxarg K , (4-42) where ( )knf denotes the channel of the failed satellite at time k, and ( )kip is the i-th column vector of the (n−4)×n parity matrix ( )kP . As the composition of visible satellites changes, it will cause problem in data fusion. To solve this, the filter should be reset, and the process of reset is described as follows. If any satellite in view passes below the mask angle, then only the previous measurements of the remaining satellites are reserved to perform the fault exclusion. However, if an extra satellite passes above the mask angle, then the past data cannot be directly applied to the filter, and thus the MARMA-filter will be reset.
  • 87. 61 4.6 Simulation Results and Analysis Monte Carlo simulations are conducted to verify the proposed FDE algorithm. The soft- ware package “Satellite Navigation ToolBox 2.0 for Matlab,” by GPSoft LLC is adopted in the simulation. The dual frequency is used to eliminate the ionospheric delay. Therefore, only the thermal noise and tropospheric delay are left in the ionosphere-free pseudorange measurements. A total of 1152 (24×48) space-time sample points were produced according to the user locations and simulation times mentioned in [36]. The user locations covering the 24 geographic loca- tions are listed in Table 3-2, and the simulation time is every half hour for 24 hours starting at midnight at the beginning of the GPS week. In addition, the receiver mask angle is set as 7.5°. Moreover, the standard deviations of measurement noises and the user-satellite geometry are listed in Appendix E. Two simulations were conducted to verify the proposed algorithms. One is to verify the proposed MA-filter, and the other to verify the proposed PFARMA-filter. 4.6.1 Fault Detection and Exclusion via MA-filters To show that the proposed MA-filter can speed up the fault detection, the ramp-type and step-type pseudorange failures were used to simulate the satellite malfunction. The ramp-type failure refers to a failure growing linearly with time, and the step-type failure refers to a constant bias happening at and remaining after a specified time. The MA-filter with equal weights was adopted in the simulation of fault detection and exclusion because the obtained test statistics has minimum variance when no satellite failure occurs. The corresponding detection threshold values can be found in Table 4-1. Table 4-1 Threshold value of MA-filter with 6 visible satellites under FAR = 1/15000 Window Size Threshold 1 19.2316 2 12.0159 3 9.3713 4 7.9669 5 7.0898 The procedure of using the MA-filter to detect the satellite failure for each point is summa- rized as follows. Step 1: Set a specified window size M (ranging from 1 to 5) and the time index as one (k = 1). Step 2: Calculate s(k) from the user-satellite geometry through (2-12).
  • 88. 62 Step 3: Check the number of satellites in view. If it is not equal to the specified value 6, then the PIT method (4-39) is applied. Step 4: Calculate z(k) from (4-2). Step 5: Compare the test statistic z(k) with the detection threshold Td under the specified window size. If the test statistic exceeds Td, then the detection time is recorded. Otherwise the time index is increased by one (k=k+1), and repeat step 2 through step 5. Based on the simulation environments described above, a detection time (DT) can be ob- tained for each point. DT is defined as the time needed from the onset of the failure to the an- nunciation of an alarm signal. The average detection time (ADT) is the sample mean of the 1152 values of the DT. The simulation result of the ADT for ramp-type pseudorange failure is plotted in Figure 4-17(a). Averagely speaking, MA-filters with window size larger than 1 have shorter DT than the conventional least-squares-residual (i.e., MA-filter with window size 1). The resulting ADT are transferred into the percentage improvements of ADT (PIADT) through the following equation ( ) ( ) ( ) 100% LSRADT MAADTLSRADT PIADT × − = , (4-43) where ADT(MA) is the ADT when MA-filters is applied and ADT(LSR) is the ADT when the conventional least-squares-residual is applied. The result is depicted in Figure 4-17(b). It can be seen that the best improvement percentage is 26%, i.e., the detection is speed up by 26%. Under ramp-type failures with slopes 0.2, 0.5 and 1 m/s, the PIADT will increase as the window size increase. While under ramp-type failures with slopes 5, 10 and 15, the resulting ADT is within 7 seconds for all detectors; this means that the window size has little influence on the PIADT. The simulation result of the ADT for step-type pseudorange failures is plotted in Figure 4-18(a). The data in the figure are also transferred into PIADT and are plotted in Figure 4-18(b). In this figure, the best improvement percentage is 77%. Under step-type failures with biases 20, 25 and 30 m, the PIADT will also increase as the window size increase. How- ever, under step-type failure with bias 40 m, their performances are also similar. To sum up, in comparison with the conventional fault detection methods, the MA-filter has higher performance in detecting small failures and, in detecting large failures, their performances are similar. Al- though the calculation of threshold for MA-filter is complex, it is independent of satellite ge- ometry. Therefore, it can be computed off-line and tabulated in computer memory. Note that, in comparison with the snapshot method, extra computational burdens for on-line operation are the calculation of the “ARMA-filter” and the “PIT”. In fact, the primary computational burden
  • 89. 63 1 2 3 4 5 2 5 10 20 50 100 120 Average Detection Time Window Size AverageDetectionTime(sec) slope = 0.2 slope = 0.5 slope = 1 slope = 5 slope = 10 slope = 15 (a) ADT 1 2 3 4 5 -5 0 5 10 15 20 25 30 The Percentage Improvements of ADT Window Size ImprovementsofADT(%) slope = 0.2 slope = 0.5 slope = 1 slope = 5 slope = 10 slope = 15 (b) PIADT Figure 4-17 Ramp-type pseudorange error (MA-filter)
  • 90. 64 1 2 3 4 5 1 2 5 10 20 50 90 Average Detection Time Window Size AverageDetectionTime(sec) step = 20 step = 25 step = 30 step = 40 (a) ADT 1 2 3 4 5 0 10 20 30 40 50 60 70 80 The Percentage Improvements of ADT Window Size ImprovementsofADT(%) step = 20 step = 25 step = 30 step = 40 (b) PIADT Figure 4-18 Step-type pseudorange error (MA-filter)
  • 91. 65 of the ARMA detector lies in the PIT process. Different combinations of coefficients will result in different performances for failure de- tection. In fact, the determination of the best coefficients will depend on the type and magni- tude of failure. For example, one set of coefficients might have good performance for one type of failure but it might not be true for another type of failure. Under the case of the ramp-type failure, the magnitude of failure will increase with time. Instinctively, if larger weights are given to the coefficients of the MA-filter for closer data, then it will be more sensitive to the ramp-type failure than the one with equal weights. However, the magnitudes of the step-type failure are equal all the time and thus all combinations of weights will have the same sensitivity on fault detection. However, take the variance into account, equal weights will have the best performance. In addition, magnitudes of ramp-type failure with small slopes (e.g., 0.2, 0.5 m/s) are almost equal within a short period of time (e.g. 5 second in the case of windows size equal to 5) and thus equal weights should be also acceptable. To deal with all types of failures, only the case of equal weights is adopted. Furthermore, to show that the proposed multivariate MA-filter can reduce the incorrect ex- clusion rate, the ramp-type and the step-type failures are also used to simulate satellite malfunc- tion. The procedure using the multivariate MA-filter to exclude the failed satellite is summa- rized as follows. Step 1: Set a specified window size M (ranging from 1 to 5). Step 2: Calculate parity matrix from the user-satellite geometry through (2-14). Step 3: Calculate ( )kq from (4-41) under the specified window size. Step 4: Identify the channel number of the failed satellite, ( )knf , from (4-42). Step 5: Comparing ( )knf with the channel number of the true failed satellite, if they did not match, then an incorrect exclusion (IE) is recorded. An IE occurs when the receiver performs a valid detection, but the failed satellite remains in the solution after the exclusion operation [16]. The incorrect exclusion rate (IER), which is used as a performance index, is defined as 100% exclusiontotalofno. exclusionincorrectofno. IER ×= . (4-44) IER can be used to verify the superior exclusion capability of the proposed multivariate MA filter. In the simulation, the number of total exclusion is chosen as 1152. Simulation results for the
  • 92. 66 80 85 90 95 100 105 110 115 5 10 15 20 25 30 35 40 45 50 time (sec) IncorrectExclusionRate(%) Incorrect Exclusion Rate Window Size = 1 Window Size = 2 Window Size = 3 Window Size = 4 Window Size = 5 35 40 45 50 55 0 5 10 15 20 25 30 35 40 45 time (sec) IncorrectExclusionRate(%) Incorrect Exclusion Rate Window Size = 1 Window Size = 2 Window Size = 3 Window Size = 4 Window Size = 5 (a) slope = 0.2 m/s (b) slope = 0.5 m/s 20 25 30 35 0 5 10 15 20 25 30 35 40 time (sec) IncorrectExclusionRate(%) Incorrect Exclusion Rate Window Size = 1 Window Size = 2 Window Size = 3 Window Size = 4 Window Size = 5 6 7 8 9 0 2 4 6 8 10 12 14 16 18 time (sec) IncorrectExclusionRate(%) Incorrect Exclusion Rate Window Size = 1 Window Size = 2 Window Size = 3 Window Size = 4 Window Size = 5 (c) slope = 1 m/s (d) slope = 5 m/s 3 4 5 6 0 2 4 6 8 10 12 14 16 18 time (sec) IncorrectExclusionRate(%) Incorrect Exclusion Rate Window Size = 1 Window Size = 2 Window Size = 3 Window Size = 4 Window Size = 5 2 3 4 5 0 2 4 6 8 10 12 14 16 18 time (sec) IncorrectExclusionRate(%) Incorrect Exclusion Rate Window Size = 1 Window Size = 2 Window Size = 3 Window Size = 4 Window Size = 5 (e) slope = 10 m/s (f) slope = 15 m/s Figure 4-19 IER under a ramp-type failure (multivariate MA-filter)
  • 93. 67 ramp-type failures with slope 0.2, 0.5, 1, 5, 10 and 15 m/s are given in Figure 4-19 (a), (b), (c), (d), (e) and (f), respectively. The values of records on the abscissa denote the time duration from the onset of satellite failure. It can be seen that the IER will reduce as the window size increase under ramp-type failures with slopes 0.2, 0.5, 1 and 5 m/s. While under ramp-type failures with slopes 10 and 15 m/s, the window size has little influence on the IER. Averagely speaking, the proposed multivariate MA-filter with window size larger than 1 has lower IER than the original parity space method (multivariate MA-filter with window size equal to 1) has. Furthermore, simulation results for the step-type failures with bias 20, 25, 30 and 40 are shown in Figure 4-20 (a), (b), (c) and (d), respectively. These figures also verify that the proposed method has lower IER than the original parity space method has. In summary, the multivariate MA-filters can reduce the IER in excluding the failed satellite. 15 20 30 40 50 60 70 80 85 8 10 15 20 25 30 35 39 time (sec) IncorrectExclusionRate(%) Incorrect Exclusion Rate Window Size = 1 Window Size = 2 Window Size = 3 Window Size = 4 Window Size = 5 5 10 15 20 25 30 5 10 15 20 25 27 time (sec) IncorrectExclusionRate(%) Incorrect Exclusion Rate Window Size = 1 Window Size = 2 Window Size = 3 Window Size = 4 Window Size = 5 (a) bias = 20 m (b) bias = 25 m 3 5 7 9 11 13 3 5 7 9 11 13 15 17 19 time (sec) IncorrectExclusionRate(%) Incorrect Exclusion Rate Window Size = 1 Window Size = 2 Window Size = 3 Window Size = 4 Window Size = 5 2 3 4 5 3 4 5 6 7 8 9 10 11 time (sec) IncorrectExclusionRate(%) Incorrect Exclusion Rate Window Size = 1 Window Size = 2 Window Size = 3 Window Size = 4 Window Size = 5 (c) bias = 30 m (d) bias = 40 m Figure 4-20 IER under a step-type failure (multivariate MA-filter)
  • 94. 68 4.6.2 Fault Detection and Exclusion via PFARMA-filters To show that the proposed PFARMA-filters can also speed up the fault detection, the ramp-type and step-type pseudorange failures were used to simulate the satellite malfunction. In the simulation of failure detection, only the case of the PFARMA-filters with order 1 (PFARMA1-filters) was presented. The corresponding detection threshold values can be found in Table 4-2. The procedure of using the PFARMA1-filter to detect the satellite failure for each point is summarized as follows. Step 1: Set a specified parameter λ1 (ranging from 0 to 0.9 with a 0.1 incremental) and the time index as one (k = 1). Step 2: Calculate s(k) from the user-satellite geometry through (2-12). Step 3: Check the number of satellites in view. If it is not equal to the specified value 6, then the PIT method (4-39) is applied. Step 4: Calculate z(k) from (4-22). Step 5: Compare the test statistic z(k) with the detection threshold Td under the specified parame- ter λ1. If the test statistic exceeds Td, then the detection time is recorded. Otherwise the time index is increased by one (k=k+1), and repeat step 2 through step 5. Table 4-2 Threshold value of PFARMA1-filter with 6 visible satellites under FAR = 1/15000 λ1 Threshold 0.0 19.2316 0.1 17.5180 0.2 15.8230 0.3 14.1464 0.4 12.4881 0.5 10.8482 0.6 9.2260 0.7 7.6169 0.8 6.0065 0.9 4.3426 Based on the simulation environments described above, a detection time (DT) can be ob- tained for each point. The simulation result of the ADT for ramp-type pseudorange failure is
  • 95. 69 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2 5 10 20 50 100 120 Average Detection Time lambda AverageDetectionTime(sec) slope = 0.2 (m/s) slope = 0.5 (m/s) slope = 1 (m/s) slope = 5 (m/s) slope = 10 (m/s) slope = 15 (m/s) (a) ADT 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 -5 0 5 10 15 20 25 30 35 40 Improvement of ADT lambda Improvement(%) slope = 0.2 (m/s) slope = 0.5 (m/s) slope = 1 (m/s) slope = 5 (m/s) slope = 10 (m/s) slope = 15 (m/s) (b) PIADT Figure 4-21 Ramp-type pseudorange error (PFARMA1-filter)
  • 96. 70 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 5 10 20 50 90 Average Detection Time lambda AverageDetectionTime(sec) step = 20 (m) step = 25 (m) step = 30 (m) step = 40 (m) (a) ADT 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 10 20 30 40 50 60 70 80 90 Improvement of ADT lambda Improvement(%) step = 20 (m) step = 25 (m) step = 30 (m) step = 40 (m) (b) PIADT Figure 4-22 Step-type pseudorange error (PFARMA1-filter)
  • 97. 71 plotted in Figure 4-21(a). On the average, the PFARMA1-filter with λ1 larger than zero has shorter DT than the conventional least-squares-residual (i.e., PFARMA1-filter with λ1 equal to zero). The resulting ADT are transferred into the PIADT through the following equation ( ) ( ) ( ) 100% LSRADT EWMAADTLSRADT PIADT × − = , (4-45) where ADT(EWMA) is the ADT when PFARMA1-filters is applied and ADT(LSR) is the ADT when the conventional least-squares-residual is applied. The result is depicted in Figure 4-21(b). It can be seen that the best improvement percentage is 36%, i.e., the detection is speed up by 36%. Under ramp-type failures with slopes 0.2, 0.5 and 1 m/s, the PIADT will increase as the value of λ1 increase. While under ramp-type failures with slopes 5, 10 and 15 m/s, the resulting ADT is within 7 seconds for all detectors; this means that the parameter λ1 has little in- fluence on the PIADT. The simulation result of the ADT for step-type pseudorange failures is plotted in Figure 4-18 (a). The data in the figure are also transferred into PIADT and are plot- ted in Figure 4-18 (b). In this figure, the best improvement percentage is 85%. Under step-type failures with biases 20, 25 and 30 m, the PIADT will also increase as the value of λ1 increase. However, under step-type failures with bias 40 m, their performances are also similar. To sum up, in comparison with the conventional fault detection methods, the PFARMA1-filter has higher performance in detecting small failures and, in detecting large failures, their perform- ances are similar. Furthermore, to show that the proposed multivariate PFARMA-filter can reduce the incor- rect exclusion rate, the ramp-type and step-type pseudorange failures are also used to simulate satellite malfunction. Moreover, only the case of multivariate PFARMA-filters with Order 1 (multivariate PFARMA1-filters) was presented in the simulation. The procedure using the mul- tivariate PFARMA1-filter to exclude the failed satellite is summarized as follows. Step 1: Set a specified parameters λ1 (ranging from 0 to 0.9 with a 0.1 incremental). Step 2: Calculate parity matrix from the user-satellite geometry through (2-14). Step 3: Calculate ( )kq from (4-41) under the specified window size. Step 4: Identify the channel number of the failed satellite, ( )knf , from (4-42). Step 5: Comparing ( )knf with the channel number of the true failed satellite, if they did not match, then an incorrect exclusion (IE) is recorded.
  • 98. 72 70 75 80 85 90 95 100 105 110 115 0 5 10 15 20 25 30 35 40 45 50 55 time (sec) IncorrectExclusionRate(%) Incorrect Exclusion Rate lambda = 0.0 lambda = 0.2 lambda = 0.4 lambda = 0.6 lambda = 0.8 35 40 45 50 55 0 5 10 15 20 25 30 35 40 45 time (sec) IncorrectExclusionRate(%) Incorrect Exclusion Rate lambda = 0.0 lambda = 0.2 lambda = 0.4 lambda = 0.6 lambda = 0.8 (a) slope = 0.2 m/s (b) slope = 0.5 m/s 20 25 30 35 0 5 10 15 20 25 30 35 40 time (sec) IncorrectExclusionRate(%) Incorrect Exclusion Rate lambda = 0.0 lambda = 0.2 lambda = 0.4 lambda = 0.6 lambda = 0.8 6 7 8 9 0 2 4 6 8 10 12 14 16 18 20 time (sec) IncorrectExclusionRate(%) Incorrect Exclusion Rate lambda = 0.0 lambda = 0.2 lambda = 0.4 lambda = 0.6 lambda = 0.8 (c) slope = 1 m/s (d) slope = 5 m/s 3 4 5 6 0 2 4 6 8 10 12 14 16 18 time (sec) IncorrectExclusionRate(%) Incorrect Exclusion Rate lambda = 0.0 lambda = 0.2 lambda = 0.4 lambda = 0.6 lambda = 0.8 2 3 4 5 0 2 4 6 8 10 12 14 16 18 time (sec) IncorrectExclusionRate(%) Incorrect Exclusion Rate lambda = 0.0 lambda = 0.2 lambda = 0.4 lambda = 0.6 lambda = 0.8 (e) slope = 10 m/s (f) slope = 15 m/s Figure 4-23 IER under a ramp-type failure (multivariate PFARMA1-filter)
  • 99. 73 The IER, which is used as a performance index, is defined as (4-44). IER can be used to verify the superior exclusion capability of the proposed multivariate PFARMA1-filter. Simula- tion results for the ramp-type failure with slope 0.2, 0.5, 1, 5, 10 and 15 m/s are given in Figure 4-23 (a), (b), (c), (d), (e) and (f), respectively. The values of records on the abscissa denote the time duration from the onset of satellite failure. These figures show that the proposed multi- variate PFARMA1-filter with λ1 larger than zero has lower IER than the original parity space method (multivariate PFARMA1-filter with λ1 equal to zero) has. Simulation results for the step-type failure with bias 20, 25, 30 and 40 are given in Figure 4-24 (a), (b), (c) and (d), respec- tively. These Figures also verify that the proposed method has lower IER than the original par- ity space method. In conclusion, the multivariate PFARMA1-filters can reduce the IER in ex- cluding the failed satellite. 10 20 30 40 50 60 70 80 5 10 15 20 25 30 35 40 time (sec) IncorrectExclusionRate(%) Incorrect Exclusion Rate lambda = 0.0 lambda = 0.2 lambda = 0.4 lambda = 0.6 lambda = 0.8 5 10 15 20 25 30 4 5 10 15 20 25 27 time (sec) IncorrectExclusionRate(%) Incorrect Exclusion Rate lambda = 0.0 lambda = 0.2 lambda = 0.4 lambda = 0.6 lambda = 0.8 (a) bias = 20 m (b) bias = 25 m 3 5 7 9 11 13 3 5 7 9 11 13 15 17 19 time (sec) IncorrectExclusionRate(%) Incorrect Exclusion Rate lambda = 0.0 lambda = 0.2 lambda = 0.4 lambda = 0.6 lambda = 0.8 2 3 4 5 3 4 5 6 7 8 9 10 11 time (sec) IncorrectExclusionRate(%) Incorrect Exclusion Rate lambda = 0.0 lambda = 0.2 lambda = 0.4 lambda = 0.6 lambda = 0.8 (c) bias = 30 m (d) bias = 40 m Figure 4-24 IER under a step-type failure (multivariate PFARMA1-filter)
  • 100. 75 Chapter 5 Fault Detection and Exclusion via Kalman Filter Kalman filter has been widely used in the filed of navigation, and conventionally, the well known position-velocity-acceleration (PVA) model is adopted as the dynamic model. Under the hypothesis that no failure is taking place and the vehicle is moving smoothly, the normalized innovation squared 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 conventional PVA model without extra sensors (such as inertial navigation sensors) can no longer be adequate for describing the motion of the vehicle. Therefore, the po- sitioning result of the vehicle will become 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. In addition, as a satellite has failed at a specified time and if the range measurements associated to the failed satellite is not yet excluded, the posi- tioning result of the vehicle will become inaccuracy and even unusable. To solve this, an algo- rithm 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 the overall estimates by running a bank of Kalman filters. Since the true model is among the set of possible candi- dates, the correct positioning result will be figured out. From that, the range measurements as- sociated to the failed satellite can be identified. In this chapter, the linearization of the iono- sphere-free pseudorange, delta range, and Doppler shift measurements for Kalman filter will be introduced in Section 5.1. Next, the positioning and FDE algorithm using Kalman filter with the conventional PVA model is described in Section 5.2. Moreover, to accurately model the
  • 101. 76 dynamic behavior of a maneuvering vehicle, the DR equation is adopted as a dynamic equation for Kalman filter in Section 5.3. Furthermore, the positioning algorithm based on multiple model (MM) approach is discussed in Section 5.4. Simulation results and analysis of the pro- posed algorithm are presented in Section 5.5. 5.1 Linearization of the Ionosphere-Free Measurements for Kalman Filter The linearization of three ionosphere-free GNSS measurements for Kalman filter is derived in this section. According to Subsection 3.1, the modernized GNSS receiver can provide three different types of ionosphere-free measurements: the pseudorange measurement ( )kGIFρ , the delta range measurement ( )kGIFφ , and the Doppler shift measurement ( )kGIFψ . Assume the reference point ( )krefx is the vector comprised of the nominal user position plus the nominal user clock drift. Then the linearization of ( )kGIFρ , ( )kGIFφ and ( )kGIFψ can be obtained as ( ) ( ) ( ) ( ) ( )[ ] ( )kkkkkk refGIF wxxHdρ +−+= , (5-1) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ( )kkkkkkkkkk refrefGIF ωxxHxxHdd +−−−−−−+−−= 1111φ , (5-2) and ( ) ( ) ( ) ( ) ( )kkkkk vvGIF wxHdψ ++= & , (5-3) where ( )kd is the n×1 predicted range vector based on the reference point ( )krefx ; ( )kd& is the n×1 predicted range rate vector based on the reference point ( )krefx ; ( )kH is the n×4 observa- tion matrix arrived at by linearizing around ( )krefx ; ( )kx is the 4×1 state vector comprised of the true position, plus the user clock drift; ( )kvx is the 4×1 state vector comprised of the true velocity, plus the user clock drift rate; ( )kw is the n×1 pseudorange measurement noises, and is assumed to be zero mean white Gaussian with the n×n covariance matrix ( )kR ; ( )kω is the n×1 delta range measurement noises, and is assumed to be zero mean white Gaussian with the n×n covariance matrix ( )kΩ ; ( )kvw is the n×1 Doppler shift measurement noises, and is assumed to be zero mean white Gaussian with the n×n covariance matrix ( )kvR ; n is the number of visible satellites. Define ( ) ( ) ( ) ( ) ( )kkkkk refGIF xHdρz +−≡ and substitute ( )kz into (5-1); then the lin- earized pseudorange measurement equation becomes ( ) ( ) ( ) ( )kkkk wxHz += . (5-4)
  • 102. 77 In addition, define ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )111 −−−+−+−≡ kkkkkkkk refrefGIF xHxHddu φ and substi- tute ( )ku into (5-2); then the linearized delta range measurement equation becomes ( ) ( ) ( ) ( ) ( ) ( )kkkkkk ωxHxHu +−−−= 11 . (5-5) Moreover, define ( ) ( ) ( )kkk GIFv dψz &−≡ and substitute ( )kvz into (5-3); then the linearized Dop- pler shift measurement equation becomes ( ) ( ) ( ) ( )kkkk vvv wxHz += . (5-6) 5.2 Using Conventional PVA Model on Kalman Filter Traditionally, Kalman filter is applied for navigation to obtain more precise positioning re- sult. Moreover, to ascertain normal navigation, the satellite fault detection and exclusion algo- rithm based on Kalman filter has been proposed. In detecting the failed satellite, the resulting NIS in Kalman filter is compared with the associated threshold under a given false alarm rate. Furthermore, in excluding the failed satellite, a parallel bank of Kalman filters is adopted. 5.2.1 Positioning Algorithm Conventionally, the well known position-velocity-acceleration (PVA) model is adopted as the dynamic model of Kalman filter for navigation. As such, the dynamic and measurement equations can be obtained as ( ) ( ) ( ) ( ) ( ) ( ) ( )⎩ ⎨ ⎧ += +−= kkkk kkk PVAPVAPVAPVA PVAPVAPVAPVA wxHz vxΦx 1 , (5-7) where ( ) ( ) ( ) ( )[ ]TT a T v T PVA kkkk xxxx = is the 12×1 state vector; ( )kax is the 4×1 vector com- prised of the true acceleration, plus the change rate of the user clock drift rate; ( ) ( ) ( )⎥⎦ ⎤ ⎢⎣ ⎡= k k k v PVA z z z is the 2n×1 measurement vector; ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ = 4 44 4 2 2 1 44 I00 II0 III Φ S SS PVA t tt is the 12×12 transition matrix; 4I is the 4×4 identity matrix; ( ) ( ) ( ) ⎥⎦ ⎤ ⎢⎣ ⎡= 0H0 00H H k k kPVA is the 2n×12 measurement matrix; ( )kPVAv is the 12×1 process noise, and is assumed to be zero mean white Gaussian with the covariance ma- trix ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = 44 2 2 1 4 3 6 1 4 2 2 1 4 3 3 1 4 4 8 1 4 3 6 1 4 4 8 1 4 5 20 1 III III III Q SSS SSS SSS PVAPVA ttt ttt ttt q ; ( )kPVAw is the 2n×1 measurement noise, and is assumed
  • 103. 78 to be zero mean white Gaussian with the covariance matrix ( ) ( ) ( ){ }k,kdiagk vPVA RRR = ; fur- thermore, ( )kPVAw and ( )kPVAv are assumed to be mutually independent; and tS is the sampling time. According to [1], the 12×1 updated state estimate can be obtained as ( ) ( ) ( ) ( )kkkkkk PVAPVAPVAPVAPVA zKxΦx ~1|1ˆ|ˆ +−−= (5-8) with the 12×12 updated state covariance matrix ( ) ( ) ( ) ( ) ( )[ ]11 1|| −− +−= kkkkkkk PVAPVA T PVAPVAPVA HRHPP , (5-9) where ( ) ( ) PVA T PVAPVAPVAPVA kkkk QΦPΦP +−−=− 1|11| is the 12×12 state prediction covariance matrix, ( ) ( ) ( ) ( )kkkkk PVA T PVAPVAPVA 1 | − = RHPK is the 12×2n Kalman gain, and ( ) ( ) ( ) ( )1|1ˆ~ −−−= kkkkk PVAPVAPVAPVAPVA xΦHzz (5-10) is the innovation vector with the 2n×2n covariance matrix ( ) ( ) ( ) ( ) ( )kkkkkk PVA T PVAPVAPVAPVA RHPHS +−= 1| . (5-11) 5.2.2 Fault Detection and Exclusion Algorithm The derivation of the FDE algorithm via Kalman filter using PVA model is described as follows. In terms of the innovation vector and the associated covariance matrix, the normalized innovation squared (NIS) can be defined as ( ) ( ) ( ) ( )kkkks PVAPVA T PVAPVA zSz ~~ 1− ≡ . (5-12) Under the hypothesis that no failure is taking place and the vehicle is moving smoothly, it can then be proved that the distribution of ( )ksPVA is ( )n22 χ , where ( )νχ 2 denotes the chi-square distribution with ν degrees of freedom and n is the number of visible satellites. Then ( )ksPVA will be compared with a detection threshold dT to judge whether the system is failed or not. The threshold value under a specified false alarm rate (FAR) can be calculated directly through the cumulative distribution function (cdf) of ( )n22 χ . The resulting threshold values under FAR equal to 1/15000 [37] are listed in Table 5-1. After the detection of satellite malfunction, the pseudorange and Doppler shift measure- ments associated to the failed satellite must be excluded to ensure uninterrupted navigation. According to [29], the fault exclusion algorithm based on a parallel bank of Kalman filter is de- rived as follows. The number of Kalman filters in this parallel bank is equal to the number of
  • 104. 79 Table 5-1 The value of detection threshold Td for ( )ksPVA Number of satellites in view, n Chi-square degrees of freedom Detection Threshold 4 8 32.8089 5 10 36.5882 6 12 40.1979 7 14 43.6794 8 16 47.0593 9 18 50.3562 10 20 53.5837 11 22 56.7518 12 24 59.8684 visible satellites. As for the ith Kalman filter, the pseudorange and Doppler shift measurements are obtained from all but the ith visible satellites. Then a residual test statistic, ( )ks iPVA, , cor- responding to the ith Kalman filter can be continuously calculated, and a satellite will be identi- fied as failed if and only if the following equation is satisfied ( ) ( ) ( )⎩ ⎨ ⎧ > =≤ otherwiseTks kniTks eiPVA feiPVA , , , , , (5-13) where ( )knf denotes the channel number of the failed satellite at time k, and eT is a selected exclusion threshold value for satellite fault exclusion. Figure 5-1 depicts the system diagram of using PVA model on a parallel bank of Kalman filter for the satellite fault detection and exclu- sion algorithm. According to Subsection 2.3.1, the user may perform integrity checking and failed satellite exclusion through following five steps: Step 1: Compute the normalized innovation squared, ( )ksPVA , using data of all visible satellites from (5-12). Step 2: If ( )ksPVA is larger than Td, then declare detection of a failure, and goto Step 3. Oth- erwise, all satellites are assumed to operate properly, and the integrity check has been completed Step 3: Compute the residual test statistic, ( )ks iPVA, , using data from all but the ith visible sat-
  • 105. 80 ellites, for i = 1, …, n. Step 4: If one of the residual test statistics (such as ( )ks fnPVA, ) is less than the exclusion thresh- old, Te, and all others are larger than Te, identify the satellite omitted from the nfth Kalman filter as the failed one. If two or more of the residual test statistics are below the threshold, the failed satellite cannot be excluded. (see (5-13)) Step 5: If a failed satellite is detected and excluded, use the navigation solution formed by omitting the failed one. If a failed satellites is detected, but cannot be excluded, use the all-in-view solution if necessary, but recognize that the positioning accuracy is de- graded. Figure 5-1 Using PVA model on a parallel bank of Kalman filter for FDE algorithm 5.3 Using Delta Range Equation on Kalman Filter for a Maneu- vering Vehicle Vehicle maneuvers, referring to unpredictable changes in the vehicle course, may cause se- rious inaccuracies in modeling the system. As a moving vehicle accelerates or slows down fu- riously, or as the vehicle corners at faster speeds, the conventional PVA model without using ex- tra 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 will deviate from the chi-square distribution and is no longer suitable as the test sta- Kalman Filter (PVA model) Kalman Filter 1 (PVA model) Kalman Filter n (PVA model) Pseudorange Doppler shift Satellite Fault Detection Satellite Fault Exclusion All Measurements Exclude Meas. 1 Exclude Meas. n
  • 106. 81 tistic for FDE. To overcome this problem, the delta range (DR) equation is adopted as the dy- namic equation for the Kalman filter. 5.3.1 Positioning Algorithm According to (5-5), the delta range equation can be described as ( ) ( ) ( ) ( ) ( ) ( )kkkkkk ωxHxHu +−−−= 11 , (5-14) where ( )kx is the 4×1 state vector comprised of the true position, plus the user clock drift; ( )ku is a vector defined as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )111 −−−+−+−= kkkkkkkk refrefGIF xHxHddu φ ; ( )kω is assumed to be zero mean white Gaussian with covariance matrix ( )kΩ . Suppose that there are at least four visible satellites, i.e., the rank of ( )kH is equal to its column dimension. Next, assume ( )kL is a 4×n matrix satisfying that ( ) ( ) 4IHL =kk , where 4I is the 4×4 identity matrix. Multiply ( )kL on both sides of (5-14), and then the following equation can be derived ( ) ( ) ( ) ( ) ( ) ( )kkkkkk DRDR vxΦxuL −−−= 1 , (5-15) where ( ) ( ) ( )1−= kkkDR HLΦ is the 4×4 transition matrix, and ( ) ( ) ( )kkkDR ωLv −= is the 4×1 noise vector, assumed to be zero mean white Gaussian with the covariance matrix ( )=kDRQ ( ) ( ) ( )kkk T LΩL . To give a simple form of ( )kDRQ , we can select ( )kL as ( ) ( ) ( ) ( )( ) ( ) ( )kkkkkk TT 111 −−− = ΩHHΩHL . (5-16) Then the covariance matrix ( )kDRQ can be derived as ( ) ( ) ( ) ( )( ) 11 −− = kkkk T DR HΩHQ . As a result, ( )kDRQ will have a simpler form if ( )kL is chosen as in (5-16). By applying (5-4) and (5-15) to the Kalman filter, a moving vehicle can be described by the following dynamic and measurement equations ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )⎩ ⎨ ⎧ += ++−= kkkk kkkkkk DRDR wxHz vuLxΦx 1 , (5-17) where ( )kx is the 4×1 state vector; ( )kDRΦ is the 4×4 transition matrix; ( )kz is the n×1 meas- urement vectors; ( )ku is the n×1 input vectors; ( )kL is the 4×n input gain matrix; ( )kDRv is the 4×1 process noise, and is assumed to be zero mean white Gaussian with the 4×4 covariance matrix ( )kDRQ ; ( )kw is the n×1 measurement noise, and is assumed to be zero mean white Gaussian with the n×n covariance matrix ( )kR ; furthermore, ( )kDRv and ( )kw are assumed to be mutually independent. According to [1], the 4×1 updated state estimate can be obtained as
  • 107. 82 ( ) ( ) ( ) ( ) ( ) ( ) ( )kkkkkkkkk DRDRDRDR zKuLxΦx ~1|1ˆ|ˆ ++−−= (5-18) with the 4×4 updated state covariance matrix ( ) ( ) ( ) ( ) ( )[ ]11 1|| −− +−= kkkkkkk T DRDR HRHPP , (5-19) where ( ) ( ) ( ) ( ) ( )kkkkkkk DR T DRDRDRDR QΦPΦP +−−=− 1|11| is the 4×4 state prediction covari- ance matrix, ( ) ( ) ( ) ( )kkkkk T DRDR 1 | − = RHPK is the 4×n Kalman Gain, and ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]kkkkkkkk DRDRDR uLxΦHzz +−−−= 1|1ˆ~ (5-20) is the innovation vector with the n×n covariance matrix ( ) ( ) ( ) ( ) ( )kkkkkk T DRDR RHPHS +−= 1| . (5-21) 5.3.2 Fault Detection and Exclusion Algorithm The derivation of the FDE algorithm via Kalman filter using DR equation is described as follows. In terms of the innovation vector and the associated covariance matrix, the normalized innovation squared can be defined as ( ) ( ) ( ) ( )kkkks DRDR T DRDR zSz ~~ 1− ≡ . (5-22) Under the hypothesis that no failure is taking place, it can then be proved that the distribution of ( )ksDR is ( )n2 χ . The obtained ( )ksDR will be compared with a detection threshold Td to judge whether the system is failed or not. The threshold value under a specified FAR can be calcu- lated directly through the cumulative distribution function of ( )n2 χ . The resulting threshold values under FAR equal to 1/15000 [37] are listed in Table 5-2. After the detection of satellite malfunction, the pseudorange and delta range measurements associated to the failed satellite must be excluded to ensure uninterrupted navigation. Accord- ing to Subsection 5.2.2, the fault exclusion algorithm based on a parallel bank of Kalman filter is derived as follows. The number of Kalman filters in this parallel bank is equal to the number of visible satellites. As for the ith Kalman filter, the pseudorange and delta range measurements are obtained from all but the ith visible satellites. Then a residual test statistic ( )ks iDR, corre- sponding to the ith Kalman filter can be continuously calculated, and a satellite will be identified as failed if and only if the following equation is satisfied ( ) ( ) ( )⎩ ⎨ ⎧ > =≤ otherwiseTks kniTks eiDR feiDR , , , , , (5-23)
  • 108. 83 where ( )knf denotes the channel number of the failed satellite at time k, and eT is a selected ex- clusion threshold value for satellite fault exclusion. Figure 5-2 depicts the system diagram of using DR equation on a parallel bank of Kalman filters for the satellite fault detection and exclu- sion algorithm. Table 5-2 The value of detection threshold Td for ( )ksDR Number of satellites in view, n Chi-square degrees of freedom Detection Threshold 4 4 24.3914 5 5 26.6521 6 6 28.7899 7 7 30.8356 8 8 32.8089 9 9 34.7232 10 10 36.5882 11 11 38.4112 12 12 40.1979 According to Subsection 5.2.2, the user may perform integrity checking and failed satellite exclusion through following five steps: Step 1: Compute the normalized innovation squared, ( )ksDR , using data of all visible satellites from (5-22). Step 2: If ( )ksDR is larger than Td, then declare detection of a failure, and goto Step 3. Oth- erwise, all satellites are assumed to operate properly, and the integrity check has been completed. Step 3: Compute the residual test statistic, ( )ks iDR, , using data from all but the ith visible satel- lites, for i = 1, …, n. Step 4: If one of the residual test statistic (such as ( )ks fnDR, ) is less than the exclusion thresh- old, Te, and all others are larger than Te, identify the satellite omitted from the nfth Kalman filter as the failed one. If two or more of the residual test statistic are below the threshold, the failed satellite cannot be excluded. (see (5-23))
  • 109. 84 Step 5: If a failed satellite is detected and excluded, use the navigation solution formed by omitting the failed one. If a failed satellites is detected, but cannot be excluded, use the all-in-view solution if necessary, but recognize that the positioning accuracy is de- graded. Figure 5-2 Using DR equation on a parallel bank of Kalman filter for FDE algorithm 5.4 Using Multiple Model Approach on Kalman Filter As a satellite is failed at and remained after a specified time, and if the range measurements associated to the failed one is not yet excluded, the accuracy of positioning result will become degraded. To solve this, a positioning algorithm based on multiple model (MM) approach is proposed in this section. MM is a well known approach adopted in target tracking [1]. The basic idea behind MM is to assume a set of models as possible candidates of the true model, and the set of the candidate models then generates the overall estimates by running a bank of Kalman filters. Since it assumes that the correct model is among the set of models, the correct posi- tioning result will be figured out. From that, the range measurements associated to the failed satellite can be identified. According to [1], the corresponding posterior probabilities can be obtained as the Bayesian framework is used, starting with prior probabilities of each model being correct. In addition, during the estimation process, the assumption that no switching occurs from one model to an- Kalman Filter (DR equation) Kalman Filter 1 (DR equation) Kalman Filter n (DR equation) Pseudorange Delta range Satellite Fault Detection Satellite Fault Exclusion All Measurements Exclude Meas. 1 Exclude Meas. n
  • 110. 85 other is made. The model, assumed to be in effect throughout the process, is one of n+1 possi- ble candidates { }n ii 0= Θ∈Θ , (5-24) where n is the number of visible satellites and iΘ represents the model associated to the ith Kalman filter. To be precise, 0Θ represents the model that uses the pseudorange and delta range measurements from all visible satellites. As for the model corresponding to the ith Kal- man filter, iΘ for i = 1, …, n, only the pseudorange and delta range measurements (i.e., ( )kz and ( )ku ) from all but the ith visible satellites will be used. The prior probability that the ith model, iΘ , is correct can be represented as ( ) ( )[ ]kiii k ,|Pr ZΘ≡ϕ , (5-25) where ( )kZ denotes the prior information representing the measurements: ( )1z , …, ( )jz , …, ( )kz ; moreover, ( )kz denotes the pseudorange and delta range measurements observed at time k. Note that ( ) 10 =∑= n l l kϕ since the correct model is assumed to be among the n+1 possible can- didates. By applying Bayes’ formula [1], the posterior probability that the ith model is correct can be obtained recursively as ( ) ( )[ ] ( ) ( )[ ] ( ) ( )[ ] ( )[ ] ( ) ( )[ ] ( ) ( )[ ] ( )[ ] ( ) ( )[ ] ( )[ ] ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )∑ ∑ ∑ = = − − = −− −− − −− − −Λ −Λ = Θ Θ = ΘΘ ΘΘ = ΘΘ = Θ=Θ= n l ll ii n l llkk iikk n l kllkk kiikk kk kiikk kkikii kk kk kz kz z z z z zk 0 0 1 1 0 11 11 1 11 1 1 1 ,|Pr ,|Pr |Pr,|Pr |Pr,|Pr |Pr |Pr,|Pr ,|Pr|Pr ϕ ϕ ϕ ϕ ϕ Z Z ZZ ZZ Z ZZ ZZ . (5-26) In the above equation, ( )kiΛ denotes the likelihood function of the ith model at time k and can be expressed as ( ) ( ) ( )[ ] ( )[ ]kzk iDRikki ,1 ~Pr,|Pr zZ =Θ≡Λ − , (5-27) where ( )kiDR, ~z is the innovation vector corresponding to the ith model, and is assumed to be zero mean white Gaussian with the covariance matrix ( )kiDR,S .
  • 111. 86 The output of the ith Kalman filter is the ith model-conditioned state estimate, ( )kkiDR |ˆ ,x . Moreover, the likelihood function defined in (5-27) will be used to update the model probabili- ties [1]. Therefore, the combination of the model-conditioned estimates can be expressed as follows ( ) ( ) ( )∑= = n l lDRlMM kkkkk 0 , |ˆ|ˆ xx ϕ , (5-28) and the covariance matrix of the combined estimate is ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( )[ ]∑= −−+= n l T MMlDRMMlDRlDRlMM kkkkkkkkkkkkk 0 ,,, |ˆ|ˆ|ˆ|ˆ|| xxxxPP ϕ . (5-29) Since the correct model is among the set of candidate models, the correct positioning result will be obtained before the range measurements associated to the failed satellite is excluded. Hence, the error in the estimated state ( )kkMM |ˆx should be within a reasonable range. Figure 5-3 Using MM approach on Kalman filter for positioning According to Subsection 5.3.2, under the hypothesis that no failure is taking place, the dis- tribution of ( )ksDR 0, , i.e., ( )ksDR , is ( )n2 χ . Then ( )ksDR 0, will be compared with a threshold Td to judge whether the system is failed or not. The threshold value under a specified FAR can be calculated directly through the cumulative distribution function of ( )n2 χ . The resulting threshold values under FAR equal to 1/15000 are listed in Table 5-2. According to (5-26), each Kalman Filter 0 Kalman Filter 1 Kalman Filter n Multiple Model Ap- proach Pseudorange Delta range Satellite Fault detec- tion Correct Model Identification Positioning Result
  • 112. 87 satellite has a corresponding prior probability ( )kiϕ . Since the prior probability corresponding to the correct model should have maximal value, the correct model can be identified through the following equation ( ) ( )kkn i ni KF ϕ ,,1 maxarg K= = , (5-30) where ( )knKF denotes the index number associated to the correct model at time k. Therefore, the range measurements associated to the failed satellite can be identified. The system diagram of applying the MM approach on Kalman filter is illustrated in Figure 5-3. 5.5 Simulation Results and Analysis Two simulations are conducted to verify the proposed algorithms. In the first simulation, it is assumed that the vehicle will first complete a half circle at a specified time and then slows down rapidly at another time. The goal of this simulation is to verify that the proposed DR can still perform both positioning and FDE well as a vehicle maneuvers. In the second simulation, it is assumed that one of the visible satellites is failed at and remained after a specified time. The goal of this simulation is to verify that the proposed MM can reduce the positioning error to a reasonable range as a satellite is failed. In these two simulations, the software package “Sat- ellite Navigation ToolBox 2.0 for Matlab,” by GPSoft LLC is adopted. It assumed a 24-satellite constellation with perfectly circular orbits. For each satellite, the dual frequency is used to eliminate the ionospheric delay. In addition, the receiver mask angle is set as 7.5°. Moreover, the standard deviations of ionosphere-free measurement noises and the user-satellite geometry are listed in Appendix E. 5.5.1 Using Delta Range Equation on Kalman Filter for a Maneuvering Vehicle Monte Carlo simulations are conducted to verify that the proposed DR can still perform po- sitioning and FDE as the vehicle maneuvers. In this simulation, both algorithms, PVA and DR, are applied to a vehicle which followed a constant course with a constant velocity 250m/s along the x-axis until time k = 100 seconds, at which point it began to maneuver in uniform circular motion with a radius of 12.5 kilometers and a centripetal acceleration of 5m/s2 until time k = 257.0796 seconds, at which point it completed a half circle. Then the vehicle followed a con- stant course with a constant velocity -250m/s along the x-axis until time k = 400 seconds, at which point it began to accelerate with a constant acceleration -2.5m/s2 along the z-axis until
  • 113. 88 50 100 150 200 250 300 350 400 450 500 0 6 12 18 24 30 positioningerror(m) PVA 50 100 150 200 250 300 350 400 450 500 0.0 25.0 50.3 90.0 125.0 160.0 Innovation(m2 ) time (sec) 50 100 150 200 250 300 350 400 450 500 0 6 12 18 24 30 positioningerror(m) DR 50 100 150 200 250 300 350 400 450 500 0.0 34.7 60.0 90.0 125.0 160.0 Innovation(m2 ) time (sec) detection threshold detection threshold Figure 5-4 Positioning errors and innovations
  • 114. 89 time k = 440 seconds, at which point it began to course with a constant velocity -250m/s along the x-axis and -100m/s along the z-axis. According to the vehicle course described above, a total of 500 sample points were produced. Moreover, for each sample point, an estimated posi- tioning result can be obtained. Furthermore, the norm of estimated positioning error ( )kposx~ can be calculated through the following equation ( ) ( ) ( )kkˆk~ truepospos xxx −= , (5-31) where ( )kposxˆ is the 3×1 estimated position vector and ( )ktruex is the 3×1 true position vector. The procedure of simulation is summarized as the following four steps Step 1: Set a specified dynamic model of Kalman filter (PVA model or DR equation). Step 2: Calculate the estimated state ( )kkPVA |ˆx from (5-8) (or ( )kkDR |ˆx from (5-18)), and then find out the corresponding estimated positioning result. Step 3: Determine the estimated positioning error ( )kposx~ from (5-31). Step 4: Calculate the normalized innovation squared ( )ksPVA through (5-12) (or ( )ksDR through (5-22)). Based on the simulation environments described above, the norm of the estimated position- ing error and the normalized innovation squared can be obtained for each point. The simulation results of applying the PVA and DR are plotted in Figure 5-4. It shows that, in comparison with the PVA model, the proposed DR equation possesses more accurate positioning results as a moving vehicle accelerates or slows down furiously, or as the vehicle corners at faster speeds. In addition, the NIS from the DR equation will still be chi-square distributed. Therefore, in comparison with the PVA model, the proposed DR can perform both positioning and FDE well as the vehicle maneuvers. 5.5.2 Using Multiple Model Approach on Kalman Filter To illustrate that the multiple model approach can reduce the positioning error to a reason- able range, both the ramp-type and the step-type pseudorange failures were used to simulate the satellite malfunction. The ramp-type failure refers to a failure growing linearly with time, and the step-type failure refers to a constant bias happening at and remaining after a specified time. The simulation time starts at midnight at the beginning of the GPS week, and the location is se- lected at London (the latitude 52°N and the longitude 0°E). Then a result of the estimated posi- tioning vector can be obtained. Furthermore, the norm of estimated positioning error, ( )kposx~ ,
  • 115. 90 can be calculated through (5-31). The procedure of using the MM approach on Kalman filter is summarized as the following five steps Step 1: Calculate the likelihood function of the ith candidate model, ( )kiΛ , from (5-27) Step 2: Determine the posterior probability of the ith candidate model being correct, ( )kiϕ , from (5-26). Step 3: Calculate the estimated state ( )kkDR |ˆx and ( )kkMM |ˆx from (5-18) and (5-28)), respec- tively. Then find out the corresponding estimated positioning result. Step 4: Determine the estimated positioning error ( )kposx~ through (5-31). Step 5: Identify the index number of the correct model, ( )knKF , from (5-30). To simulate the ramp-type failure, different ramp failures (with slope = 0.2, 0.5, 1 and 2 m/s) were added to the measurements of the first satellite at time 50s. The simulation results are then plotted in Figure 5-5, Figure 5-6, Figure 5-7, and Figure 5-8, respectively. These figures show that the positioning error of the original Kalman filter will be increased with time; however, that of the proposed MM will first grow with time and then decay back to the normal range. Therefore, in comparison with the original Kalman filter, the positioning error of the proposed MM is significantly reduced before the failure is detected. These figures also present that the posterior probability associated to the correct model will be increased with time and will finally approach one, and thus the correct model will be identified. Moreover, to simulate the step-type failure, different step failures (with bias = 20 and 30 m) were added to the measurements of the first satellite at time 50s. The simulation results are then plotted in Figure 5-9 and Figure 5-10, respectively. These figures show that the position- ing error of the original Kalman filter will be increased instantly and remain inside a certain level afterwards; however, that of the proposed MM will also instantly grow and then decay back to the normal range rapidly. Therefore, in comparison with the original Kalman filter, the pro- posed MM can also provide more accurate positioning result as the satellite is failed. In addi- tion, the posterior probability associated to the correct model will rapidly approach one, and thus the correct model will be identified. In summary, in comparison with the original Kalman filter, the proposed MM can perform positioning well as the satellite failed. As a consequence, after the application of the proposed MM, the navigation can be still continued even when a satellite is failed.
  • 116. 91 0 25 50 75 100 125 150 175 200 0 20 40 60 detection threshold Innovation innovation(m2 ) 0 25 50 75 100 125 150 175 200 0 12.5 25 37.5 50 Delta Range Model positioningerror(m) 0 25 50 75 100 125 150 175 200 0 12.5 25 37.5 50 Multiple Model Approach positioningerror(m) time (sec) (a) Positioning errors and innovation 0 50 100 150 200 0 0.5 1 φ0 (k) 0 50 100 150 200 0 0.5 1 φ1 (k) 0 50 100 150 200 0 0.5 1 φ2 (k) 0 50 100 150 200 0 0.5 1 φ3 (k) 0 50 100 150 200 0 0.5 1 φ4 (k) 0 50 100 150 200 0 0.5 1 φ5 (k) time (sec) 0 50 100 150 200 0 0.5 1 φ6 (k) time (sec) 0 50 100 150 200 0 0.5 1 φ7 (k) time (sec) (b) Prior probability Figure 5-5 Using MM approach on Kalman filter (slope = 0.2 m/s)
  • 117. 92 0 25 50 75 100 125 150 0 20 40 60 detection threshold Innovation innovation(m2 ) 0 25 50 75 100 125 150 0 12.5 25 37.5 50 Delta Range Model positioningerror(m) 0 25 50 75 100 125 150 0 12.5 25 37.5 50 Multiple Model Approach positioningerror(m) time (sec) (a) Positioning errors and innovation 0 50 100 150 0 0.5 1 φ0 (k) 0 50 100 150 0 0.5 1 φ1 (k) 0 50 100 150 0 0.5 1 φ2 (k) 0 50 100 150 0 0.5 1 φ3 (k) 0 50 100 150 0 0.5 1 φ4 (k) 0 50 100 150 0 0.5 1 φ5 (k) time (sec) 0 50 100 150 0 0.5 1 φ6 (k) time (sec) 0 50 100 150 0 0.5 1 φ7 (k) time (sec) (b) Prior probability Figure 5-6 Using MM approach on Kalman filter (slope = 0.5 m/s)
  • 118. 93 0 10 20 30 40 50 60 70 80 90 100 0 20 40 60 detection threshold Innovation innovation(m2 ) 0 10 20 30 40 50 60 70 80 90 100 0 12.5 25 37.5 50 Delta Range Model positioningerror(m) 0 10 20 30 40 50 60 70 80 90 100 0 12.5 25 37.5 50 Multiple Model Approach positioningerror(m) time (sec) (a) Positioning errors and innovation 0 25 50 75 100 0 0.5 1 φ0 (k) 0 25 50 75 100 0 0.5 1 φ1 (k) 0 25 50 75 100 0 0.5 1 φ2 (k) 0 25 50 75 100 0 0.5 1 φ3 (k) 0 25 50 75 100 0 0.5 1 φ4 (k) 0 25 50 75 100 0 0.5 1 φ5 (k) 0 25 50 75 100 0 0.5 1 φ6 (k) time (sec) 0 25 50 75 100 0 0.5 1 φ7 (k) time (sec) (b) Prior probability Figure 5-7 Using MM approach on Kalman filter (slope = 1 m/s)
  • 119. 94 0 10 20 30 40 50 60 70 0 20 40 60 detection threshold Innovation innovation(m2 ) 0 10 20 30 40 50 60 70 0 12.5 25 37.5 50 Delta Range Model positioningerror(m) 0 10 20 30 40 50 60 70 0 12.5 25 37.5 50 Multiple Model Approach positioningerror(m) time (sec) (a) Positioning errors and innovation 0 25 50 70 0 0.5 1 φ0 (k) 0 25 50 70 0 0.5 1 φ1 (k) 0 25 50 70 0 0.5 1 φ2 (k) 0 25 50 70 0 0.5 1 φ3 (k) 0 25 50 70 0 0.5 1 φ4 (k) 0 25 50 70 0 0.5 1 φ5 (k) 0 25 50 70 0 0.5 1 φ6 (k) time (sec) 0 25 50 70 0 0.5 1 φ7 (k) time (sec) (b) Prior probability Figure 5-8 Using MM approach on Kalman filter (slope = 2 m/s)
  • 120. 95 0 25 50 75 100 125 150 175 200 0 20 40 60 detection threshold Innovation innovation(m2 ) 0 25 50 75 100 125 150 175 200 0 12.5 25 37.5 50 Delta Range Model positioningerror(m) 0 25 50 75 100 125 150 175 200 0 12.5 25 37.5 50 Multiple Model Approach positioningerror(m) time (sec) (a) Positioning errors and innovation 0 50 100 150 200 0 0.5 1 φ0 (k) 0 50 100 150 200 0 0.5 1 φ1 (k) 0 50 100 150 200 0 0.5 1 φ2 (k) 0 50 100 150 200 0 0.5 1 φ3 (k) 0 50 100 150 200 0 0.5 1 φ4 (k) 0 50 100 150 200 0 0.5 1 φ5 (k) 0 50 100 150 200 0 0.5 1 φ6 (k) time (sec) 0 50 100 150 200 0 0.5 1 φ7 (k) time (sec) (b) Prior probability Figure 5-9 Using MM approach on Kalman filter (bias = 20 m)
  • 121. 96 0 10 20 30 40 50 60 70 0 20 40 60 detection threshold Innovation innovation(m2 ) 0 10 20 30 40 50 60 70 0 12.5 25 37.5 50 Delta Range Model positioningerror(m) 0 10 20 30 40 50 60 70 0 12.5 25 37.5 50 Multiple Model Approach positioningerror(m) time (sec) (a) Positioning errors and innovation 0 25 50 70 0 0.5 1 φ0 (k) 0 25 50 70 0 0.5 1 φ1 (k) 0 25 50 70 0 0.5 1 φ2 (k) 0 25 50 70 0 0.5 1 φ3 (k) 0 25 50 70 0 0.5 1 φ4 (k) 0 25 50 70 0 0.5 1 φ5 (k) 0 25 50 70 0 0.5 1 φ6 (k) time (sec) 0 25 50 70 0 0.5 1 φ7 (k) time (sec) (b) Prior probability Figure 5-10 Using MM approach on Kalman filter (bias = 30 m)
  • 122. 97 Chapter 6 Conclusions and Future Works 6.1 Conclusions This thesis is concerned with topics on the problems of GNSS 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. To enhance the capability of the existing fault detection and exclusion methods, we propose here three types of FDE algorithms based on the multi-frequency technique, the ARMA-filter technique and the Kalman filter technique, respectively. 6.1.1 Using Multi-Frequency on FDE First, algorithms using multi-frequency measurements are proposed for GNSS positioning and FDE. The conventional algorithms adopt only the single frequency L1. However, GPS satellites carrying the L2 and L5 signals 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. Monte Carlo simulations are conducted to verify that the proposed algorithm. Simulation results show that, in comparison with the conventional single frequency method, the estimated positioning errors of multi-frequency method are concentrated around the origin due to the reduction of measurement errors, and the reduction of the estimated positioning errors in vertical axis are especially significant. Moreover, the simulation results of the ramp- type failure detection show that the best improvement percentage for the dual fre-
  • 123. 98 quency method and the triple frequency method are 48.3% and 55.9%, respectively. As for the case of the step-type failure detection, the best improvement percentage for the dual frequency method and the triple frequency method are 99.1% and 99.2%, respectively. Furthermore, the simulation result also present that the IER obtained through the dual frequency method is about 5% lower than the one through the conventional single frequency method under the best scenario. In addition, the IER obtained through the triple frequency method under the best scenario is about 9% (12%) lower than the one through the dual (single) frequency method. 6.1.2 Using ARMA Filter on FDE Next, an approach 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 fault detection and exclusion. By taking the average of the last several sums of the squares of the range residual errors, the ARMA-filter can speed up the detection time. The detection threshold can be obtained via the Markov chain approach under a specific false alarm rate. Al- though 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. More- over, 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. Note that, in compari- son with the snapshot method, extra computational burdens for on-line operation are the calcula- tion of the “ARMA-filter” and the “PIT”. In fact, the primary computational burden of the ARMA detector lies in the PIT process. However, the calculation of the PIT has been simpli- fied in this thesis. After a satellite failure is detected, the multivariate ARMA-filter is used to reduce the incorrect exclusion rate by taking average over the last several parity vectors. Monte Carlo simulations are conducted to verify that the proposed algorithm using ARMA-filter. To simulate the satellite malfunction, ramp-type and step-type pseudorange failures are applied. Simulation 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. In summary, the proposed multivari- ate ARMA-filters can reduce the IER in excluding the failed satellite. 6.1.3 Using Kalman Filter on FDE Finally, an algorithm based on a parallel bank of Kalman filters to perform satellite posi- tioning and FDE is proposed. Conventionally, the well known PVA model is adopted as the
  • 124. 99 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 can no longer be adequate for describing the motion of the vehicle. To overcome these problems, the delta range (DR) equation is proposed to accurately model the dynamic behavior of a maneuvering vehicle. In the simulation, it is assumed that the vehicle will first complete a half circle at a specified time and then slow down rapidly at another time. 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 satellite 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 to reduce the positioning error. In the simulation, it is assumed that one of the visible satellites is failed at and remained after a specified time. Simulation results also present that, compared to the original Kalman filter, the proposed MM can perform positioning well as the satellite failed. In addition, the posterior probability associated to the correct model will finally approach one, and thus the correct model will be identified. As a consequence, after the application of the proposed MM, the navigation can be still continued even when a satellite has failed. 6.2 Future Works 6.2.1 Using Multi-Frequency on FDE In applying multiple frequency technique to GNSS FDE, two directions can be further pur- sued. The first direction is on the development of the fault exclusion algorithms to simultane- ously identify two or more failed satellites. The proposed parity-space-based fault exclusion algorithms can perform exclusion under the assumption that only one satellite is failure. The concept of the subset method might be useful to achieve this goal. The second direction is on the development of the positioning and FDE algorithms using the carrier phase (i.e., accumulated delta ranges) measurement. Currently, the proposed algorithms use only the pseudorange meas- urement; however, the carrier phase integrity ambiguity can be resolved quickly by using dual frequency phase data. We expect that using carrier phase to replace pseudorange can obtain better positioning and FDE results since carrier phase measurement is more accurate. 6.2.2 Using ARMA Filter on FDE In using the ARMA-filter on GNSS FDE, five directions can be further pursued. The first
  • 125. 100 direction is on the development of a procedure to determine the detection threshold for ARMA- filter with all possible coefficients. Currently we only focus on special cases of ARMA-filter, such as the MA-filter and the PFARMA-filter. The second direction is to develop a systematic procedure on the determination of the optimal coefficients of ARMA-filter for failure detection and exclusion. To figure out the optimal solutions, however, the “signatures” of the failure (e.g., ramp-type failure with slope 1m/s) should be specified. The third direction is to extend the ARMA-filter to the multivariate form for failure detection. However, the alternation of the number of visible satellites will result in diverse dimensions of the residual vectors. As a con- sequence, these vectors cannot be directly accumulated. In addition, the determination of the threshold is still an issue to be solved. The fourth direction is to find the relationship between the sampling time (i.e., the time interval between two consecutive data) and the detection time using ARMA-filter. Currently, due to the limitation of the software package, the sampling time is limited to be at least one. The last direction is to figure out the missed detection probability for the proposed algorithms. In using the snapshot FDE, the missed detection probability can be calculated directly, but, in using the ARMA-filter, it is difficult to figure out it. 6.2.3 Using Kalman Filter on FDE In using the Kalman filter on GNSS FDE, five directions can be further pursued. The first direction is to find the relationship between the navigation horizontal radial position error and the test statistic. The upper bound of the allowable radial position error is referred to as the alarm limit. If the horizontal error is found to exceed the alarm limit, a timely alarm will be is- sued. However, the horizontal error is unknown to the users. For this reason, a test statistic based on the pseudorange error must be created to perform failure detection. In using the snapshot FDE, the relationship between the horizontal error and the test statistic will follow a straight line with a certain slope, but, in using the Kalman-filter, it is difficult to derive the rela- tionships between them. The second direction is to reduce the computing burden by simplify- ing the algorithm of the MM approach. The concept of the generalized pseudo-Bayesian (GPB) approach [1] and the interacting multiple model (IMM) algorithm [1] might be useful to achieve this goal. The third direction is to find the relationship between the sampling time and the es- timated positioning result when both PVA and DR are applied. Currently, due to the limitation of the software package, the sampling time is only set as one. The fourth direction is on the development of an algorithm based on the adaptive Kalman filter. The last direction is to define an alternative test statistic for failure detection and exclusion using MM approach.
  • 126. 101 Bibliography [1] Bar-Shalom, Y. and Fortman, T. E., Estimation and Tracking: Principles, Techniqus and Software, Artech House, Inc., Boston, London, 1993 [2] Basseville, M. and Nikiforov, I. V., Detection of Abrupt Changes: Theory and Application, Prentice Hall, Inc., New Jersey, 1993. [3] Brenner, M., “Implementation of a RAIM Monitor in a GPS receiver and an Integrated GPS/INS”, Proceedings of ION GPS-90, 1990. [4] Brown, R. G., “A Baseline GPS RAIM Scheme and a Note on the Equivalence of Three RAIM Methods”, Navigation, Journal of the Institute of Navigation, Vol. 39, No.3, 1992, pp. 301-316. [5] Brown, R. G. and Hwang, P., “GPS Fault detection by Autonomous Means within the Cock- pit,” Proceedings of the Forty-Two Annual Meeting of the Institute of Navigation, June, 1986, pp. 5-12. [6] Crowder, S. V., “Design of Exponentially Weighted Moving Average Schemes,” Journal of Quality Technology, Vol. 21, No.3, July, 1989, pp. 155-162. [7] Da, R. and Lin, C.-F., “A New Fault detection Approach and Its Application to GPS Autonomous Integrity Monitoring,” IEEE transactions on Aerospace and Electronic Systems, Vol. 31, No. 1, 1995, pp. 499 ~ pp. 506 [8] Diggelen, V. F. and Brown, A., “Mathematical Aspects of GPS RAIM,” Position Location and Navigation Symposium, IEEE, 1994, pp.733-738. [9] Fontana, D. R., Cheung, W., and Stansell T., “The Modernized L2 Civil Signal: Leaping Forward in the 21st Century”, GPS World, September, 2001, pp.28-34. (http://www.gpsworld.com) [10] Hofmann-Wellenhof, B., Lichtenegger, H, and Collins, J., Global Positioning System: The- ory and Practice, (4th revised edition) Springer Wien New Work, 1997. [11] Hunter, J. S., “The Exponentially Weighted Moving Average,” Journal of Quality Technology, Vol. 18, No.4, October, 1986, pp. 203-210. [12] Isaacson, D. L., Markov Chains: Theory and Applications, John Wiley & Sons, 1976. [13] Juang, J.-C. and Jang, C.-W., “Fault detection approach applying to GPS autonomous integ- rity monitoring,” IEE Proceedings: Radar, Sonar and Navigation, Vol. 145, No. 6, 1998, pp. 342-346. [14] Kelly, J. R., “The Linear Model, RNP, and the Near-Optimum Fault Detection and Exclusion Algorithm”, Global Positioning System, Vol. 5, The Institute of Navigation, 1998, pp. 227-259
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  • 128. 103 [30] Galileo and EGNOS Office Site: http://www.esa.int/export/esaNA/index.html [31] Galileo: The European Programme for Global Navigation Services, ESA, May, 2002. [32] Global Positioning System Standard Positioning Service Signal Specification, GPS NAVSTAR Global Positioning System, 2nd Edition, 1995. [33] Global Satellite Navigation System; Interface Control Document (ICD), Coordination Sci- entific Information Center, 1998 [34] GLONASS Office Site: http://www.glonass-center.ru/ [35] ICAO All Weather Operations Panel. Draft Manual on RNP for Approach, Landing and Departure Operations, AWOP. [36] Minimum Operational Performance Standards for Airborne Supplemental Navigation Equipment Using Global Positioning System (GPS). Document RTCA/DO-208, Radio Technical Commission for Aeronautics, 1991. [37] Minimum Operational Performance Standards for Airborne Equipment Using Global Posi- tioning System/Wide Area Augmentation System. Document RTCA/DO-229, Radio Technical Commission for Aeronautics, 1997.
  • 129. 105 Appendix A Required Navigation Performance (RNP) The growing demand in air transport domain and the inducing air system saturation have led in the late 1980s the ICAO to recommend improvements of the existing conventional naviga- tion systems. [36][37] The ICAO has defined the CNS/ATM concept to develop satellite-based systems, capable to resolve the arising navigation problems. Furthermore, to qualify the future air navigation systems, 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. In order to extend the class of specific normalized navi- gation systems to systems based on new technology and modern methods, the concept of re- quired navigation performance (RNP) capability was introduced. The RNP concept serves to qualify the aircraft capability: a RNP type represents an aircraft certification level and corre- sponds to a navigation area type or to a given phase of flight. Figure A-1 illustrates this. Figure A-1 RNP types For a given RNP type (or phase of flight), the navigation system can be qualified as: (1) a sole means if it meets all the requirements (on accuracy, integrity, continuity and availability), (2) a primary means if it meets requirements on accuracy and integrity, (3) a supplemental means if it achieves the performance of a present sole means on accuracy and integrity. For a navigation system such as GNSS to be declared as a sole means of navigation, it must fulfill all normalized requirements stated for a given phase of flight. Recently proposed figures on performance re- quirements in each phase of flight are listed in Table A-1 and Table A-2. [35] The accuracy
  • 130. 106 represents the total system error limit as the position measure or estimate can experience on the true position in 95% of the time. Vertical accuracy is provided for precision approaches. The integrity represents the trust which can be placed in the correctness of the information supplied by the total system. This measure includes the ability of the system to provide timely and valid warnings (or alerts) to the user when the system should not be used for the intended operation. To ensure the system integrity in the failure case for a given phase of flight the alert must be an- nunciated in a time interval lower than the time to alert. The continuity is the capability of the system to perform its function without non-scheduled interruptions during the phase of flight. The availability is the portion of time during which the system provides reliable information and is to be used for navigation. Table A-1 RNP values for non precision approach phases of flight En Route En Route (Terminal) Initial Approach (NPA, Departure)Operation RNP 20 to 10 RNP 5 to 1 RNP 0.5 to 0.3 Accuracy 95% (Lateral / Vertical) 2.0 NM / - 0.4 NM / - 220m / - Alarm Limit (Lateral / Vertical) 4NM / - 1 NM/ - 555m / - Integrity 1−10-7 per hour Time to Alert 5 min 15 s 10 s Continuity 1−10-4 ~ 1−10-8 per hour Availability 0.99 ~ 0.99999 0.999 ~ 0.99999 0.99 ~ 0.99999 Table A-2 RNP values for precision approach phases of flight IPV Cat. I Cat. II Cat. III Operation RNP 0.3/125 RNP 0.03/50 to 0.02/40 RNP0.01/15 RNP0.003 Accuracy 95% (Lateral / Vertical) 220m / 9.1m 16.0m / 7.7~4m 6.5m / 1.7m 3.9m / 0.8m Alarm Limit (Lateral / Vertical) 600m / 50m 40m / 10 to 15m 17.3m / 5.3m 15.5m / 5.3m Integrity 1−2×10-7 per approach 1−10-9 per approach Time to Alert 10 s 10 s 6 s 1 s Continuity (any 15 s) 1−8×10-6 any 15 s 1−4×10-6 1−2×10-6 Availability 0.99 to 0.99999
  • 131. 107 Appendix B History of GPS and GLONASS Satellites Table B-1 History of GPS Satellites and Status Launch Order PRN SVN Launch Date PLANE Block I-1 - 01 Feb. 22, 1978 no longer in service Block I-2 - 02 May 13, 1978 no longer in service Block I-3 - 03 Oct. 06, 1978 no longer in service Block I-4 - 04 Dec. 10, 1978 no longer in service Block I-5 - 05 Feb. 9, 1980 no longer in service Block I-6 - 06 April 26, 1980 no longer in service Block I-7 - 07 Dec. 18, 1981 unsuccessful launch Block I-8 - 08 July 14, 1983 no longer in service Block I-9 - 09 June 13, 1984 no longer in service Block I-10 - 10 Sep. 8, 1984 no longer in service Block I-11 - 11 Oct. 9, 1985 no longer in service Block II-1 - 14 Feb. 14, 1989 no longer in service Block II-2 - 13 June 10, 1989 no longer in service Block II-3 - 16 Aug. 18, 1989 no longer in service Block II-4 - 19 Oct. 21, 1989 no longer in service Block II-5 17 17 Dec. 11, 1989 D6 Block II-6 - 18 Jan. 24, 1990 no longer in service Block II-7 - 20 March 26, 1990 no longer in service Block II-8 - 21 Aug. 2, 1990 no longer in service Block II-9 15 15 Oct. 1, 1990 D5 Block IIA-10 - 23 Nov. 26, 1990 no longer in service
  • 132. 108 Block IIA-11 24 24 July 4, 1991 D1 Block IIA-12 25 25 Feb. 23, 1992 A2 Block IIA-13 - 28 April 10, 1992 no longer in service Block IIA-14 26 26 July 7, 1992 F2 Block IIA-15 27 27 Sept. 9, 1992 A4 Block IIA-16 01 32 Nov. 22, 1992 F6 Block IIA-17 29 29 Dec. 18, 1992 F5 Block IIA-18 - 22 Feb. 3, 1993 no longer in service Block IIA-19 31 31 March 30, 1993 C5 Block IIA-20 07 37 May 13, 1993 C4 Block IIA-21 09 39 June 26, 1993 A1 Block IIA-22 05 35 Aug. 30, 1993 B4 Block IIA-23 04 34 Oct. 26, 1993 D4 Block IIA-24 06 36 March 10, 1994 C1 Block IIA-25 03 33 March 28, 1996 C2 Block IIA-26 10 40 July 16, 1996 E3 Block IIA-27 30 30 Sept. 12, 1996 B2 Block IIA-28 08 38 Nov. 6, 1997 A3 Block IIR-1 - 42 Jan. 17, 1997 unsuccessful launch Block IIR-2 13 43 July 23, 1997 F3 Block IIR-3 11 46 Oct 7, 1999 D2 Block IIR-4 20 51 May 11, 2000 E1 Block IIR-5 28 44 July 16, 2000 B3 Block IIR-6 14 41 Nov. 10, 2000 F1 Block IIR-7 18 54 Jan. 30, 2001 E4 Block IIR-8 16 56 Jan 29, 2003 B1 Block IIR-9 21 45 March 31, 2003 D3 Block IIR-10 22 47 Dec. 21, 2003 E2 Block IIR-11 19 59 March 20, 2004 C3 Block IIR-12 23 60 June 23, 2004 F4 Block IIR-13 02 61 Nov. 06, 2004 The current GPS constellation consists of 30 Block II/IIA/IIR satellites. The first operational, Block II, satellite was launched in February 1989.
  • 133. 109 Table B-2 History of GLONASS Satellites and Status Block No. GLONASS No. (slot/frequency) Cosmos No. Launch date Put into operation End of Operation (Withdrawn) 1 224 (01/--) 1413 12.10.82 15.10.82 12.01.84(16.04.84) 2 222 (03/--) 1490 10.08.83 03.09.83 05.07.84(31.10.85) 2 223 (02/--) 1491 10.08.83 31.08.83 27.09.84(09.06.88) 3 220 (18/--) 1519 29.12.83 07.01.84 27.09.84 28.01.88) 3 219 (17/--) 1520 29.12.83 15.01.84 30.06.86(16.09.86) 4 218 (19/--) 1554 19.05.84 13.06.84 16.08.85(16.09.86) 4 217 (18/--) 1555 19.05.84 18.06.84 25.10.85(17.09.87) 5 216 (02/10) 1593 04.09.84 22.09.84 28.11.85(19.05.88) 5 215 (03/--) 1594 04.09.84 28.09.84 04.09.86(16.09.86) 6 224 (01/--) 1650 18.05.85 28.06.85 08.11.85(29.11.85) 6 221 (01/07) 1651 18.05.85 14.06.85 09.08.87(17.09.87) 7 209 (18/04) 1710 25.12.85 24.01.86 28.02.87(16.03.89) 7 210 (17/19) 1711 25.12.85 24.01.86 16.05.87(16.09.87) 8 203 (02/11) 1778 16.09.86 19.10.86 20.02.87(13.07.89) 8 202 (03/20) 1779 16.09.86 19.10.86 15.07.88(24.10.88) 8 201 (08/22) 1780 16.09.86 19.10.86 15.06.88(10.10.88) 9 - 1838 24.04.87 Failed launch 9 - 1839 24.04.87 Failed launch 9 - 1840 24.04.87 Failed launch 10 229 (--/--) 1883 16.09.87 12.10.87 06.06.87(03.07.89) 10 228 (--/--) 1884 16.09.87 12.10.87 30.08.88(15.12.88) 10 227 (17/--) 1885 16.09.87 07.10.87 01.02.89(09.03.89) 11 - 1917 17.02.88 Failed launch 11 - 1918 17.02.88 Failed launch 11 - 1919 17.02.88 Failed launch 12 235 (07/--) 1946 21.05.88 15.06.88 10.05.90(22.10.90) 12 234 (08/--) 1947 21.05.88 15.06.88 19.03.91(18.09.91) 12 233 (01/--) 1948 21.05.88 15.06.88 11.06.91(18.09.91) 13 238 (17/--) 1970 16.09.88 11.10.88 21.05.90 (22.10.90) 13 237 (18/--) (1) 1971 16.09.88 11.10.88 31.08.89(30.11.89) 13 236 (19/--) (2) 1972 16.09.88 11.10.88 01.11.91(12.08.92)
  • 134. 110 14 239 (02/09) 1987 10.01.89 01.02.89 14.03.93(03.02.94) 14 240 (03/06) 1988 10.01.89 01.02.89 16.02.92(02.06.92) 14 241 1989 10.01.89 Geodetic reference satellite 15 231 (24/--) 2022 31.05.89 04.07.89 25.01.90(13.03.90) 15 230 (19/--) 2023 31.05.89 17.06.89 18.11.89(13.03.90) 15 232 2024 31.05.89 Geodetic reference satellite 16 242 (17/21) 2079 19.05.90 20.06.90 23.04.94(17.08.94) 16 228 (19/03) 2080 19.05.90 17.06.90 27.07.94(27.08.94) 16 229 (20/15) 2081 19.05.90 11.06.90 18.08.92(20.01.93) 17 247 (07/13) 2109 08.12.90 01.01.91 17.03.94(10.06.94) 17 248 (04/14) 2110 08.12.90 29.12.90 29.10.93(20.01.94) 17 249 (05/19) (4) 2111 08.12.90 28.12.90 09.06.96(15.08.96) 18 750 (22/11) 2139 04.04.91 28.04.91 29.09.94(14.11.94) 18 753 (21/20) 2140 04.04.91 28.04.91 06.01.92(04.06.93) 18 754 (24/14) 2141 04.04.91 04.05.91 26.02.92(16.06.92) 19 768 (03/22) 2177 30.01.92 24.02.92 09.01.93(29.06.93) 19 769 (08/02) 2178 30.01.92 22.02.92 23.05.97(24.06.97) 19 771 (01/17) (3) 2179 30.01.92 18.02.92 25.10.86(21.12.96) 20 756 (18/24) (6) 2204 30.07.92 19.08.92 27.06.97(05.08.97) 20 772 (21/08) 2205 30.07.92 29.08.92 29.06.94(27.08.94) 20 774 (24/01) 2206 30.07.92 25.08.92 18.05.96(26.08.96) 21 773 (02/05) 2234 17.02.93 14.03.93 09.03.94(17.08.94) 21 759 (06/23) (5) 2235 17.02.93 25.08.93 30.06.97(05.08.97) 21 757 (03/12) 2236 17.02.93 14.03.93 27.07.97(23.08.97) 22 758 (18/10) 2275 11.04.94 04.09.94 05.03.99(15.01.00) 22 760 (17/24) 2276 11.04.94 18.05.94 30.07.99(09.09.99) 22 761 (23/03) 2277 11.04.94 16.05.94 24.07.97(29.08.97) 23 767 (12/22) (7) 2287 11.08.94 07.09.94 05.11.98(03.02.99) 23 770 (14/09) 2288 11.08.94 04.09.94 24.08.99(15.01.00) 23 775 (16/22) 2289 11.08.94 07.09.94 13.08.00(28.09.00) 24 762 (04/12) (7) 2294 20.11.94 11.12.94 04.09.99(19.11.99) 24 763 (03/21) 2295 20.11.94 15.12.94 27.07.99(05.10.99) 24 764 (06/13) 2296 20.11.94 16.12.94 27.10.99(30.11.99) 25 765 (20/01) 2307 07.03..95 30.03.95 10.09.99(19.11.99)
  • 135. 111 25 766 (22/10) 2308 07.03.95 05.04.95 21.11.00(05.02.01) 25 777 (19/03) 2309 07.03.95 05.04.95 17.07.97(26.12.97) 26 780 (15/04) 2316 24.07.95 26.08.95 03.12.98(06.04.99) 26 781 (10/09) 2317 24.07.95 22.08.95 24.01.01(15.10.01) 26 785 (11/04) 2318 24.07.95 22.08.95 03.02.01(06.04.01) 27 776 (09/06) 2323 14.12.95 07.01.96 13.08.00(28.09.00) 27 778 (15/11) (8) 2324 14.12.95 26.04.99 29.01.01(30.12.01) 27 782 (13/06) 2325 14.12.95 18.01.96 23.07.01(15.10.01) 28 779 (01/02) 2364 30.12.98 18.02.99 31.01.02(08.07.02) 28 784 (08/08) 2363 30.12.98 29.01.99 19.12.03 28 786 (07/07) 2362 30.12.98 29.01.99 17.10.03 29 783 (18/10) 2374 13.10.00 05.01.01 15.10.04 29 787 (17/05) 2375 13.10.00 04.11.00 11.11.04 29 788 (24/03) 2376 13.10.00 21.11.00 07.11.04 30 789 (03/12) 2381 01.12.01 04.01.02 29.10.04 30 790 (06/09) 2380 01.12.01 04.01.02 19.12.03 30 711 (05/02) 2382 01.12.01 15.04.03 Operational 31 792 (21/05) 2395 25.12.02 31.01.03 Operational 31 791 (22/10) 2394 25.12.02 10.02.03 Operational 31 793 (23/11) 2396 25.12.02 31.01.03 Operational 32 701 (06/-) 2404 10.12.03 32 794 (2/04) 2402 10.12.03 02.02.04 Operational 32 795 (04/06) 2403 10.12.03 02.02.04 Operational Notes: (1) On 6 August 1989 SV 237 had been moved from slot 18 to slot 20 (2) On 5 August 1989 SV 236 had been moved from slot 19 to slot 18 (3) On 2 September 1992 frequency channel of SV 771 had been changed from 17 to 23 (4) On 2 September 1992 frequency channel of SV 249 had been changed from 19 to 23 (5) On December 1994 SV 759 had been moved from slot 6 to slot 7and on 2 September 1993 frequency channel of SV 771 had been changed from 23 to 21 (6) SV 756 had been moved from slot 18 to slot 21 (7) On 27 September 1994 frequency channels of both SV 767 and SV 775 had been changed from 21 to 22 (8) On April 1999 SV 778 had been moved from slot 09 to slot 15
  • 136. 113 Appendix C Parity Space Method C.1 Maximization of Conditional Probability Assume the i-th satellite is failed with a bias ( )kib , and then the linearized pseudorange measurement equation becomes ( ) ( ) ( ) ( ) ( )kkkkk i ebxHy ++= , (C-1) where ( )ky is the n×1 measurement vector; ( )kH is the n×4 observation matrix; ( )kx is the 4×1 state vector; ( )kw is an n×1 zero mean Gaussian noise vector with covariance matrix ( )kR ; and n is the number of visible satellites; ( ) ( ) ini kbk ,ub = with in,u is an n×1 column vector with all elements zeros except the i-th element is one. Substitute (C-1) into (2-15) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )kkkkkkkkkk ii ePbPebxHPp +=++= , (C-2) where ( ) ( )kk eP is an (n-4)×1 zero mean Gaussian vector with covariance matrix 4−nI . (Note that ( ) ( ) ( ) 4−= n T kkk IPRP ) Thus, the parity vector has multivariate normal density with mean ( ) ( )kk ibP and covariance In-4 [8][21]. Therefore, the conditional probability density function of parity vector can be described as follows ( ) ( )( ) ( ) ( ) ( ) ( )( )2 2 12 4 exp2|Pr kkkkk i n i bPpbp −−= − π . (C-3) The failed satellite should be associated to the maxima of (C-3), and the failed one can be identified through the following equation ( ) ( )( ) ( ) ( ) ( )( )2 ,,1,,1 maxarg|Prmaxarg kkkkk i ni i ni bPpbp −−= == KK . (C-4) Substitute ( ) ( ) ini kbk ,ub = into (C-4), (C-4) becomes
  • 137. 114 ( ) ( )( ) ( ) ( ) ( )( )2 ,,1,,1 maxarg|Prmaxarg kkbkkk i ni i ni ppbp −−= == KK . (C-5) where pi (k) is the i-th column vector of the parity matrix P(k). pi (k) is also called the i-th chan- nel vector since it is related to the i-th satellite. Since ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2222 2 kkbkkkbkkkbk i T ii pppppp +−=− , (C-6) and ( )kp is independent of i, equation (C-5) can be rewritten as ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( )22 ,,1 2 ,,1 2maxargmaxarg kkbkkkbkkbk i T i ni i ni ppppp −=−− == KK . (C-7) Since ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 02 2 =−−=− ∂ ∂ kkbkkkkbk kb i T ii ppppp , (C-8) ( ) ( ) ( ) ( )2 k kk kb i T i p pp = maximizes the term ( ) ( ) ( )( )2 kkbk ipp −− , for i = 1, …, n. Thus (C-7) can be rewritten as ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =−− === k kk k kk kkbk i T i ni i T i ni i ni p pp p pp pp ,,1 2 ,,1 2 ,,1 maxargmaxargmaxarg KKK . (C-9) Therefore, the algorithm to identify the failed satellite is as follows ( ) ( ) ( )( ) ⎟ ⎠ ⎞⎜ ⎝ ⎛ ⎟ ⎠ ⎞⎜ ⎝ ⎛⎟ ⎠ ⎞⎜ ⎝ ⎛ == == k kk kkkn i i T ni i ni f p pp bp ,,1,,1 maxarg|Prmaxarg KK , (C-10) where ( )knf denotes the channel number of the failed satellite at time k. C.2 Existence of the Parity Matrix In this subsection, we will prove that there exists a parity matrix ( )kP satisfying the fol- lowing equations ( ) ( ) 0HP =kk . (C-11) and ( ) ( ) ( ) 4−= n T kkk IPRP . (C-12)
  • 138. 115 Suppose that there are at least six visible satellites, i.e., n ≥ 6. Since the covariance matrix of the measurement noise, ( )kR , is symmetric and positive definite, there exists a symmetric and positive definite matrix ( )kR ) such that ( ) ( )kk RR =2 ) . Therefore, ( )kR ) is invertible and symmetric and thus ( ) ( )kk HR 1− ) can be calculated. Represent the singular value decomposi- tion of ( ) ( )kk HR 1− ) as ( ) ( ) ( ) ( ) ( )kkkkk T VΣUHR =−1 ) , (C-13) where ( )kU is an n×n orthogonal matrix; ( )kV is a 4×4 orthogonal matrix; ( ) ( )[ ]TT kk 0DΣ = is an n×(n-4) matrix; in addition, ( )kD is a 4×4 diagonal matrix. Let ( )kU be partitioned into submatrices ( ) ( )[ ]kk UU )~ , where ( )kU ~ is an n×4 matrix and ( )kU ) is an n×(n-4) matrix. De- fine ( ) ( ) ( )kkk T 1− = RUP )) , and substitute ( )kP into equation (C-11); then (C-11) becomes ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) 0 VDUU V 0 D UUU VΣUU HRU HP = = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = = = − kkkk k k kkk kkkk kkk kk TT TT TT T ~ ~ 1 ) )) ) )) . (C-14) In the previous equation, ( ) ( ) 0UU =kkT ~) since ( ) ( )[ ]kk UU )~ is an orthogonal matrix. Further- more, substitute ( )kP into equation (C-12); then (C-12) becomes ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 4 121 1 − −− −− = = = = n TT TT TTT T kk kkkkk kkkkk kkk I UU URRRU URRRU PRP )) ))))) )))) . (C-15) In the previous equation, ( ) ( ) 4−= n T kk IUU )) since ( ) ( )[ ]kk UU )~ is an orthogonal matrix. This proves the existence of a parity matrix ( )kP satisfying ( ) ( ) 0HP =kk and ( ) ( ) ( ) 4−= n T kkk IPRP .
  • 139. 117 Appendix D Partition Matrix Inverse For a nonsingular partition matrix ⎥⎦ ⎤ ⎢⎣ ⎡ 2221 1211 MM MM , there are two kind of the representation of the partition matrix inverse. If both 11M and 12 1 11212222 MMMM∆ − −= are nonsingular ma- trices, then the inverse can be represented as follows ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − −+ =⎥⎦ ⎤ ⎢⎣ ⎡ −−− −−−−−−− 1 22 1 1121 1 22 1 2212 1 11 1 1121 1 2212 1 11 1 11 1 2221 1211 ∆MM∆ ∆MMMM∆MMM MM MM . (D-1) Moreover, if both 22M and 21 1 22121111 MMMM∆ − −= are nonsingular matrices, then the in- verse can be expressed as the following equation ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ +− − =⎥⎦ ⎤ ⎢⎣ ⎡ −−−−−− −−−− 1 2212 1 1121 1 22 1 22 1 1121 1 22 1 2212 1 11 1 11 1 2221 1211 MM∆MMM∆MM MM∆∆ MM MM . (D-2) For a given nonsingular partition matrix (see (3-5) and (3-18)) ( ) ( ) ( ) ( ) ( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = n TT aka kakka k IH HHH M 2212 1211 . (D-3) Suppose that 2 12 1 221111 aaa − −≡δ is a nonzero scalar, and then the following sub-matrix ( ) ( ) ( )( )( ) ( )( ) ( ) ( )kkkaakakka T n TT HHHIHHH∆ 1112 1 22121111 δ=−= − (D-4) is also nonsingular; moreover, the sub-matrix na I22 in (D-3) is nonsingular. According to (D-2), the inverse of (D-3) can then be derived as ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ +− − = −−−−−−− −−−−− − kkkkaaakkkaa kkkaakk k TT n T TTT HHHHIHHH HHHHH M 11 11 2 12 2 22 1 22 11 1112 1 22 11 1112 1 22 11 111 δδ δδ . (D-5)
  • 140. 118 For another nonsingular partition matrix (see (3-37)) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ + = m n TTTT aka aka kakakkakka k I0H 0IH HHHHHH M 2212 2212 12121111 ))) ))))) ) . (D-6) Suppose that 2 12 1 221111 aaa − −=δ and 2 12 1 221111 aaa )))) − −=δ are nonzero scalars, and then the follow- ing sub-matrix ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( )kkkk ka ka a a kakakkakka TT m nTTTT HHHH H H I0 0I HHHHHH∆ ))) ))) ))))) 1111 12 12 1 22 22 1212111111 δδ += ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ −+= − (D-7) is also nonsingular; moreover, the sub-matrix ⎥⎦ ⎤ ⎢⎣ ⎡ m n a a I0 0I 22 22 ) in (D-6) is nonsingular. Accord- ing to (D-2), the inverse of (D-7) can be derived as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ +− +− −− = −−−−−−−− −−−−−−−− −−−−− − kkaaakkaaaakaa kkaaaakkaaakaa kaakaa k T m T TT n TT H∆HIH∆H∆H H∆HH∆HI∆H H∆H∆∆ M ))))))))))) ))) ))) ) 1 11 2 12 2 22 1 22 1 1112 1 2212 1 22 1 1112 1 22 1 1112 1 2212 1 22 1 11 2 12 2 22 1 22 1 1112 1 22 1 1112 1 22 1 1112 1 22 1 11 1 . (D-8)
  • 141. 119 Appendix E Simulation Environments In this appendix, the standard deviation of measurement noises and the user-satellite ge- ometry in the simulation will be calculated. Throughout this thesis, the software package “Sat- ellite Navigation ToolBox 2.0 for Matlab,” by GPSoft LLC is adopted for the simulation. It assumed a 24-satellite constellation with perfectly circular orbits. The receiver mask angle is set as 7.5°. Due to the limitation of the software package, only the GPS satellite constellation is applied in the simulation. The simulation time is every minute for 24 hours starting at midnight at the beginning of the GPS week. The user locations covering the 24 geographic locations are listed in Table 3-2. For each selected user location, a total of 1440 (60×24) sample points were produced according to the simulation time. For each sample point, three different types of measurement noises can be obtained and the satellite geometry can be determined. Further- more, the standard deviations of the measurement noise can be evaluated and the estimated user- satellite geometry will be calculated. A measure of the user-satellite geometry can be expressed in the dilution of precision (DOP) factors [10], including the geometric dilution of precision (GDOP), the position dilution of pre- cision (PDOP), the horizontal position dilution of precision (HDOP) and the vertical position di- lution of precision (VDOP). Based on the simulation environments described above, these fac- tors are calculated and listed in Table 2-1. The standard derivations of pseudorange measure- ment noises (ionospheric effect and other error) can be obtained. The standard deviations of the L1, L2 and L5 pseudorange noises are shown in Table E-2. Moreover, the standard deviations of the ionosphere-free measurement noises (pseudorange, delta range and Doppler shift noises) are listed in Table E-3.
  • 142. 120 Table E-1 User-satellite geometry DOP Location GDOP PDOP HDOP VDOP London 2.251401 1.995762 1.101830 1.658317 Liberia 2.172715 1.941092 0.933508 1.698467 South Atlantic 2.356793 2.067401 1.103095 1.743639 Iceland 2.396141 2.132540 1.019245 1.868487 St. Johns 2.278765 2.008823 1.123569 1.660462 Buenos Aires 2.333322 2.054442 1.084769 1.739017 Ecuador 2.044956 1.839035 0.882633 1.610462 New Orleans 2.361366 2.072272 1.091680 1.755191 Winnipeg 2.265605 1.999297 1.118498 1.652073 Easter Island 2.385146 2.093649 1.087978 1.782232 Los Angeles 2.328920 2.044347 1.114639 1.710732 Central Pacific 2.084720 1.868515 0.914881 1.626702 North Alaska 2.547509 2.265791 0.984441 2.036552 Honolulu 2.302582 2.036485 1.013291 1.760576 Ross Sea 2.581435 2.298955 0.947147 2.091483 New Zealand 2.384102 2.088082 1.147623 1.740901 Marshall Islands 2.174647 1.942334 0.935249 1.698735 Tokyo 2.390080 2.090904 1.135026 1.751723 Perth 2.291242 2.020350 1.077606 1.705022 Singapore 2.064319 1.856297 0.881943 1.630208 Indian Ocean 2.298649 2.025216 1.126307 1.679788 Aral Sea 2.303847 2.026377 1.128356 1.679415 Madagascar 2.254428 1.998926 0.977338 1.737260 Cape Town 2.363300 2.072462 1.123003 1.735856
  • 143. 121 Table E-2 Standard derivations of the pseudorange measurement noise Ionospheric effectNoise Location L1 L2 L5 Other error London 20.894603 34.412250 37.469671 3.795735 Liberia 20.488931 33.744132 36.742193 3.554357 South Atlantic 20.429332 33.645975 36.635315 3.723212 Iceland 20.966592 34.530812 37.598767 3.682614 St. Johns 20.520068 33.795412 36.798029 3.858764 Buenos Aires 19.621383 32.315329 35.186445 3.663732 Ecuador 20.547969 33.841363 36.848063 3.710065 New Orleans 19.413097 31.972292 34.812931 3.725367 Winnipeg 20.418592 33.628286 36.616054 3.834111 Easter Island 20.574262 33.884666 36.895213 3.757591 Los Angeles 20.184153 33.242178 36.195642 3.768753 Central Pacific 21.633784 35.629641 38.795224 3.611902 North Alaska 20.823931 34.295858 37.342938 3.521314 Honolulu 21.023866 34.625140 37.701476 3.744739 Ross Sea 20.528958 33.810053 36.813970 3.443529 New Zealand 20.666706 34.036917 37.060991 3.724554 Marshall Islands 20.508326 33.776073 36.776972 3.616821 Tokyo 19.826788 32.653618 35.554791 3.747443 Perth 19.497355 32.111060 34.964028 3.786272 Singapore 20.603029 33.932044 36.946801 3.510792 Indian Ocean 20.144384 33.176682 36.124326 3.822272 Aral Sea 20.581804 33.897087 36.908737 3.829642 Madagascar 21.452436 35.330970 38.470016 3.792139 Cape Town 20.949639 34.502892 37.568366 3.697004
  • 144. 122 Table E-3 Standard derivations of the ionoshere-free measurement noise Noise Location Pseudorange noise Delta range noise Doppler shift noise London 6.032272 0.068824 0.074844 Liberia 5.889307 0.069002 0.074123 South Atlantic 5.914091 0.067500 0.074910 Iceland 5.953343 0.068527 0.074389 St. Johns 5.980977 0.068940 0.073346 Buenos Aires 5.895626 0.068854 0.074375 Ecuador 5.973634 0.068232 0.074220 New Orleans 5.945720 0.067915 0.073951 Winnipeg 6.058125 0.068344 0.073655 Easter Island 5.919100 0.069299 0.073773 Los Angeles 5.963550 0.068936 0.074208 Central Pacific 5.876772 0.068806 0.074559 North Alaska 5.836831 0.068295 0.073434 Honolulu 5.976672 0.067897 0.073894 Ross Sea 5.767519 0.068964 0.074583 New Zealand 5.935914 0.069638 0.074678 Marshall Islands 5.901946 0.068759 0.073971 Tokyo 5.939646 0.069082 0.073630 Perth 5.931471 0.068549 0.074345 Singapore 5.776020 0.068418 0.074191 Indian Ocean 5.976338 0.069047 0.073999 Aral Sea 5.946073 0.068186 0.073827 Madagascar 5.926254 0.069011 0.074201 Cape Town 5.925371 0.068520 0.074346

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