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

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1. Introduction ...

1. Introduction
2. GNSS Architecture, Observables, and Covenantal Fault
3. Detection and Exclusion
4. Using Multi-Frequency Tech. on FDE
5. Using ARMA-Filter on FDE
6. Using Kalman Filter on FDE
7. Conclusions and Future Works

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    Filter-Type Fault Detection and Exclusion (FDE) on Multi-Frequency GNSS Receiver Filter-Type Fault Detection and Exclusion (FDE) on Multi-Frequency GNSS Receiver Presentation Transcript

    • Filter-Type Fault Detection and Exclusion (FDE) on Multi- Frequency GNSS Receiver Advisor: Prof. Chang, Fan-Ren Presenters: Tsai, Yi-Hsueh
    • 2 Outline 1. Introduction 2. GNSS Architecture, Observables, and Covenantal Fault Detection and Exclusion 3. Using Multi-Frequency Tech. on FDE 4. Using ARMA-Filter on FDE 5. Using Kalman Filter on FDE 6. Conclusions and Future Works
    • 3 Introduction (1/2) accuracy FDE algorithm (stand-alone GPS use) integrity availability continuity ICAO (International Civil Aviation Organization) navigation performance requirements
    • 4 Introduction (2/2) current data time process Filter-Type FDE Algorithm (use current & past data) Snapshot FDE Algorithm (use current data) past data
    • 5 GNSS Architecture System GPS GLONASS Galileo Developed by United State Russian European Sat. No. / planes 24 / 6 24 / 3 27 / 3 Inclination 55 64.8 56 Altitude 20200 km 19100 km 23616 km Period 11 hr 58 min 11 hr 15 min 14 hr 22 min Modulation CDMA FDMA CDMA L1*, E1-L1-E2** 1575.42 MHz* 1602+0.5625n MHz* 1575.42 MHz** L2*, E6** 1227.60 MHz* 1246+0.4375n MHz* 1278.75 MHz** E5B** 1207.14 MHz** L5*, E5A** 1176.45 MHz* 1176.45 MHz**
    • 6 Observables ionosphere    2 frequencycarrier 1   advancephase delaygroupcode Doppler shift delta range
    • 7 Least-Squares-Residuals (1/3)               kkkkkkk TT yRHHRHx 111 ˆ        kkk xHy ˆˆ  The estimate of the state vector is and the range residual vector is The estimate of y(k) is The linearized GPS measurement equation is      kkk yyy ˆ~         ,kkkk wxHy      kNk R0w ,~
    • 8 Least-Squares-Residuals (2/3) The test statistic        kkkks T yRy ~~ 1  0 5 10 15 20 0 0.05 0.1 0.15 0.2 false alarm rate = 1/100 normalized SSE (m2 ) probabilitydensityfunction determination of threshold Td 2(4)Parkinson showed that the distribution of s(k) is chi-square distributed with degrees of freedom as n4.
    • 9 Least-Squares-Residuals (3/3) detection threshold Td under FAR = 1/15000 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
    • 10 > Te Subset Method Subset solutions are formed by removing one of the visible satellites at a time. s1(k) si (k) sn(k) snf (k)  Te
    • 11 satisfying and Parity Space Method (1/2) The parity matrix The parity vector     0HP kk      kkk yPp          kkkk ni pppP 1      kkk wxH    nfkb e 0    kkb nfp         kkkkk wPxHP        4 n T kkk IPRP nfth channel vector
    • 12 Parity Space Method (2/2) ith channel vector nfth channel vector (failed satellite) parity vector                k kk maxarg i i T n,,i p pp 1 No of visible satellite = 6
    • Using Multi-Frequency Technique on Failure Detection and Exclusion
    • 14 Dual Frequency GPS (1/2)                                          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 The estimated positioning result The linearized measurement equation           kkkkk L TT DF 12 1 ˆ yHHHx           kfkfffk LLLLLLL 2 2 11 2 2 12 1 2 212 yyy  where      kkk DFDFDF wHy
    • 15 Dual Frequency GPS (2/2) The parity matrix The parity vector       kfkfk LLLLDFDF PPP 2 1 2 2 1 2 1 1 1               kkff kkk LLLLLDF DFDFDF 12 2 1 2 2 1 2 1 1 1 yP yPp     The algorithm to identify the failed satellite      kkks DF T DFDF ppThe test statistic                              k kk k kk L iDF L iDF T DF L iDF L iDF T DF ni 2 , 2 , 1 , 1 , ,,1 ,maxmaxarg p pp p pp 
    • 16 GPS Triple Frequency (1/2) The estimated positioning result Extend to the triple frequency           kkkkk L TT TF 125 1 ˆ yHHHx                                                                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        kckckck LLLLLLL 552211125 yyyy where      kkk TFTFTF wHy
    • 17 GPS Triple Frequency (2/2) The algorithm to identify the failed satellite The test statistic      kkks TF T TFTF pp                                    k kk , k kk , k kk maxmaxarg L i,TF L i,TF T TF L i,TF L i,TF T TF L i,TF L i,TF T TF n,,i 5 5 2 2 1 1 1 p pp p pp p pp  The parity vector      kkk TFTFTF yPp  The parity matrix                         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   
    • 18 GNSS (GPS + Galileo) To simultaneously use the measurements of both Galileo and GPS system                                                                                    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         kkk DFDFDF wHy 
    • 19 Simulation Results: Positioning (1/2) Dual freq. L1/L2 Single freq. L1 Triple freq. L1/L2/L5 -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) -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) -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)
    • 20 Simulation Results: Positioning (2/2) Standard Derivation Horizontal error (m) Vertical error (m) East error North error Single Freq. 5.5499 3.9982 21.2627 Double Freq. 2.2234 3.3094 6.1387 Triple Freq. 1.8737 2.8852 5.4481
    • 21 Simulation Results: Failure Detection (1/2) sound an alarm begin failure    11521152 1 timedetectionaverage DTADT detection time (DT) 24 space 48 time (RTCA DO-208)
    • 22 Simulation Results: Failure Detection (2/2) 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  The best improvement percentage for dual freq. and triple freq. are 48.3% and 55.9%, respectively.  Application of multi- frequency algorithms will shorten the failure detection time.
    • 23 Simulation Results: Failure Exclusion (1/2) %100 exclusiontotalofno. exclusionincorrectofno. rate)exclusion(incorrect  IER failed excluded
    • 24 Simulation Results: Failure Exclusion (2/2) 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  The IER obtained through the dual (triple) freq. is about 5% (12%) lower than the one through single freq.  Application of multi- frequency algorithms will reduce the IER.
    • Failure Detection and Exclusion via ARMA-Filter
    • 26 Failure Detection via ARMA-filter         M i i N i i iksikzkz 11 1 The scheme of the ARMA-filter (Autoregressive Moving Average filter) Failure z(k) > Td Normal no yes ARMA-filters(k) z(k)
    • 27 Detection Threshold (1/3) FAMMean time to false alarm (MTFA):   FAM FAR 1 RateAlarmFalse  sound a false alarm begin detection time to false alarm (TFA)
    • 28 Detection Threshold (2/3) Determining Td via the Markov Chain Approach L 1 1 L j i k-1 L(k) i(k) 1(k) terminating state transient state k Tij z(k)  Td z(k) > Td
    • 29 Detection Threshold (3/3)    1 kk Tππ where T is the transition probability matrix The transition of (k) can be represented as    0 1 πTΙl   L T LFAMMTFA: Threshold Td FAM FAR 1 
    • 30 MA-filter               kkz kskk T MMA θβ eθΦθ 1,1 The dynamic equation of the MA-filter state space model where i is the weight satisfies The scheme of the MA-filter      M i i ikskz 1 1 11  M i i Markov Chain
    • 31 MA-filter: Window Size 2 (1/2) MA-filter with window size 2:      121  kskskz  dT1 1   1S LS 2 1 0 dT1 2     dT 2211  iS  ii  ,                                                kkkz ks k k k k 2211 2 1 2 1 0 1 1 1 01 00      state space model
    • 32 MA-filter: Window Size 2 (2/2) dT1 2   1S LS 1S LS iSjS  k2  k1 dT1 1  dT1 1    12 k dT1 2    11 k 0 0 ijT   dT 2211   ii  ,     jiij SstateinwaskSstatetogoeskT 1|Pr  θθ
    • 33 PFARMA-filter The scheme of the PFARMA-filter (parallel–form ARMA)               kkz kskk T PFARMA θη λθΦθ 1 parallel-form structure         M i i N i i iksikzkz 11 1 The scheme of the ARMA-filter Markov Chain
    • 34 Probability Integral Transformation (PIT) 0 5 10 15 20 0 0.2 0.4 0.6 0.8 0.9 1 random variable cdfvalue Probability Integral Transformation 4.6 10.6 2 (2)  2 (6) F -1 F 10 Sat. 2(6) 9 Sat. 2(5) 6 Sat. 2(2) Visible Satellite    ZZF  1log21 2
    • 35 Failure Exclusion via Multivariate ARMA The scheme of the multivariate ARMA-filter         M i i N i i ikikk 11 1pqq                 k kk i i T ni p pq ,,1 maxarg  Multivariate ARMA-filterp(k) q(k)
    • 36 Simulation Results (1/5) Window Size Threshold 1 19.2316 2 12.0159 3 9.3713 4 7.9669 5 7.0898 MA-filter with equal weights
    • 37 Simulation Results (2/5) 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  Under small ramp-type failures (slope = 0.2, 0.5 and 1 m/s), the ADT will decrease as the window size increase.  Under large ramp-type failures (slope = 5, 10 and 15 m/s), the window size has little influence on the ADT.
    • 38 Simulation Results (3/5)  Under small step-type failures (bias = 20, 25 and 30 m), the ADT will decrease as the window size increase.  Under large step-type failures (bias = 40 m), the window size has little influence on the ADT. 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
    • 39 Simulation Results (4/5) 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  The IER will reduce as the window size increase under small ramp-type failures.  The window size has little influence on the IER under large ramp-type failures. 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 slope=0.5m/s slope=10m/s
    • 40 Simulation Results (5/5)  The IER will reduce as the window size increase under small step-type failures.  The window size has little influence on the IER under large step-type failures. 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 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 bias=25m bias=40m
    • Failure Detection and Exclusion via Kalman Filter
    • 42 PVA Model (1/2)               , 1      kkkk kkk PVAPVAPVAPVA PVAPVAPVAPVA wxHz vxΦx The dynamic and measurement equations          4 44 4 2 2 1 44 I00 II0 III Φ S SS PVA t tt         0H0 00H H k k PVAwhere        kkkkkk PVAPVAPVAPVAPVA zKxΦx ~1|1ˆ|ˆ  The updated state estimate where        kkkkk PVA T PVAPVAPVA 1 |   RHPK             kNk kNk PVAPVA PVAPVA Rw Qv ,0~ ,0~
    • 43 is the innovation vector with the covariance matrix PVA Model (2/2) The normalized innovation squared (NIS)        kkkks PVAPVA T PVAPVA zSz ~~ 1         1|1ˆ~  kkkkk PVAPVAPVAPVAPVA xΦHzz          kkkkkk PVA T PVAPVAPVAPVA RHPHS  1| Chi-square distributed
    • 44 FDE via Kalman filter Kalman Filter Kalman Filter 1 Kalman Filter n Pseudorange Delta range Doppler shift Satellite Failure Exclusion Satellite Failure Detection All Measurements Exclude Meas. n Exclude Meas. 1
    • 45 Maneuvering Vehicle delta range information
    • 46 Multiply L(k) on both side (L(k)H(k) = I4) Delta Range Equation (1/3)            kkkkkk ωxHxHu  11 The linearized delta range measurement L(k)H(k-1)            kkkkkk DRDR vuLxΦx  1 -L(k)(k) The dynamic equation
    • 47 Delta Range Equation (2/3) The dynamic and measurement equations                         , 1 kkkk kkkkkk DRDR wxHz vuLxΦx The updated state estimate              kkkkkkkkk DRDRDRDR zKuLxΦx ~1|1ˆ|ˆ  where        kkkkk T DRDR 1 |   RHPK             kNk kNk DRDR Rw Qv ,0~ ,0~
    • 48 Delta Range Equation (3/3) The normalized innovation squared (NIS)          kkkkkk T DRDR RHPHS  1|        kkkks DRDR T DRDR zSz ~~ 1                kkkkkkkk DRDRDR uLxΦHzz  1|1ˆ~ is the innovation vector with the covariance matrix
    • 49 Simulation Results 50 100 150 200 250 300 350 400 450 500 0 10 20 30 positioning error(m) PVA 50 100 150 200 250 300 350 400 450 500 0.0 50.3 80.0 120.0 160.0 Innovation time (sec) 50 100 150 200 250 300 350 400 450 500 0 10 20 30 positioning error(m) DR 50 100 150 200 250 300 350 400 450 500 0.0 34.7 80.0 120.0 160.0 Innovation time (sec) detection threshold detection threshold
    • 50 Multiple Model Approach (1/2) Kalman Filter 0 Kalman Filter 1 Kalman Filter n Pseudorange Delta range Positioning Result+        n i iDRi MM kkk kk 0 |ˆ |ˆ ,x x   k0  k1  kn prior prob.
    • 51 Multiple Model Approach (2/2)                 n l ll ii kii kk kk k 0 1 1 |Pr    Z          kzk iDRikki ,11 ~Pr,|Pr zZ   The prior probability where i(k): the likelihood function of the ith model
    • 52 0 25 50 75 100 125 150 175 200 0 20 40 60 detection threshold Innovation innovation 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) Simulation Results 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 1φ4 (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) Ramp-Type Failure (slope = 0.2m/s)
    • 53 0 25 50 75 100 125 150 175 200 0 20 40 60 detection threshold Innovation innovation 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) Simulation Results (2/2) 0 50 100 150 200 0 0.5 1 φ 0 (k) 0 50 100 150 200 0 0.5 1 φ 1 (k) 0 50 100 150 200 0 0.5 1 φ 2 (k) 0 50 100 150 200 0 0.5 1 φ 3 (k) 0 50 100 150 200 0 0.5 1 φ 4 (k) 0 50 100 150 200 0 0.5 1 φ 5 (k) 0 50 100 150 200 0 0.5 1 φ 6 (k) time (sec) 0 50 100 150 200 0 0.5 1 φ 7 (k) time (sec) Step-Type Failure (bias = 20 m)
    • 54 Conclusions (1/2)  In Multi-frequency  propose the dual frequency alogrithm  extend to the triple frequency algorithm  to simultaneously use the measurements of both GPS and Galileo system  In ARMA-filter  propose the ARMA filter for fast failure detection  use the Markov chain approach for calculating the threshold of ARMA
    • 55 Conclusions (2/2)  In ARMA-filter (cont.)  propose the PIT to solve the problem caused by change of number of visible satellites  proposed the multivariate ARMA to reduce the IER  In Kalman filter  use delta range information to accurately describe the dynamic behavior of a maneuvering vehicle  propose multiple model approach to reduce the positioning error
    • 56 Future Work  Multi-frequency  using delta range and/or Doppler shift measurements  ARMA-filter  to extend the ARMA detector to multivariate case  to determine the optimal coefficients for FDE  Kalman filter  to reduce the computing burden  Generalized pseudo-Bayesian (GPB) approach  Interacting multiple model (IMM) algorithm.