Measurement Error and Fault Models for Multi-Constellation Navigation Systems. Mathieu Joerger Illinois Institute of Technology

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1 Measurement Error and Fault Models for Multi-Constellation Navigation Systems Mathieu Joerger Illinois Institute of Technology Colloquium on Satellite Navigation at TU München May 16,

2 Multi-Constellation Navigation System In this research, we explore the potential of multi-constellation navigation systems to achieve high-integrity, precision positioning over wide areas. GPS, Galileo and Iridium Orbits Signals from GPS and Galileo are combined with ranging measurements from Iridium low-earth-orbiting (LEO) telecommunication satellite. GPS Galileo Iridium 2

3 Geometric diversity Motivation for LEO Augmentation large angular variations of line-of-sight (LOS) for Iridium fast floating cycle ambiguity estimation Satellite redundancy guaranteed with GPS/Galileo/Iridium fault detection using receiver autonomous integrity monitoring (RAIM) Measurement errors Azimuth-Elevation Sky Plot (over 10min) W N E eliminated using local differential corrections [Rabinowitz] consider occasional wide-area corrections, together with robust error models, which must be validated over the duration of cycle ambiguity estimation S GPS Iridium 150 Rabinowitz, M. A Differential Carrier-Phase Navigation System Combining GPS with Low Earth Orbit Satellites for Rapid Resolution of Integer Cycle Ambiguities. PhD Thesis. Stanford, CA (2000). 3

4 Conceptual System Design In this research, the system is designed for civilian air and land transportation applications. 1. User segment: Single-frequency Iridium and GPS signal receivers Dual-frequency (DF) GPS/Galileo by 2025 DF Iridium also considered in the analysis 60 N 2. Example ground segment: a network of ground stations similar to WAAS (performance results documented since 2003) aims at computing Iridium clock and orbit information WAAS-like corrections for GPS clock and orbit ephemeris and for iono errors 40 N 140 W 120 W 100 W Wide Area Augmentation System (WAAS) 80 W 4

5 Conceptual System Design 3. Space segment: models for existing constellations Nominal 24 GPS satellite constellation Nominal 27 Galileo satellite constellation 66 Iridium SVs arranged in 6 orbital planes Near-polar orbits impact the positioning performance. Higher satellite density near the poles Directionality of the satellite motion Iridium Satellite Coverage 5

6 Measurement Error Models To exploit changes in geometry, measurements are filtered over time (over a filtering period T F, lower than 10 min). Therefore, ranging error models must account for absolute error at filter initiation relative error over filtering period T F, and the fact that Iridium satellites cross large sections of the atmosphere within the T F period. Existing measurement error models (e.g., used in WAAS and in the Local Area Augmentation System) are insufficient. Derived new models, as illustrated (next slides) with the model for the ionosphere (similar methods for troposphere and SV orbit and clock ephemeris errors). 6

7 Ionospheric Error Modeling For near-term future single-frequency (SF) implementations, the ionosphere is the largest source of measurement error. In this work, the new models are employed in two distinct ways first, we evaluate the fidelity of the models to DF GPS data then, we use these stochastic models to compute probability bounds on position estimates Limits of measurement error modeling: models must be robust, but not overly-conservative must limit the number of model parameters to avoid unrealistic bounds on the position estimate 7

8 Nominal Ionospheric Error Model Ionospheric pierce point (IPP) intersection of the LOS with thin shell (highest density of e - ) 350km IPP displacement (d IPP ) due to earth, user & SV motion computed in a sun-fixed frame Main assumption vertical ionospheric delay varies linearly with d IPP, in a sun-fixed frame over limited d IPP (piecewise linear model) model used with GPS and LEO, no local ref. station assumption b VI I d IPP IPP c b d g I, k OI, k VI IPP, k VI g VI Instantaneous Over time (d IPP ) d IPP,MAX = 750km d IPP 8

9 Experimental Evaluation Dual-frequency carrier phase GPS data (at f L1 and f L2 ) provide a biased and noisy measurement of the ionospheric delay at the L1 frequency: I, k OI, k VI IPP, k VI f f 2 L2 2 2 L1 fl2 Equivalent Ionospheric Delay on f L1 c b d g b v L1, k L2, k I, k I I, k Unknown bias (differenced cycle ambiguities) vi, k ~ 0, I 2.3cm 2 I Processed GPS data from CORS at seven sites across the United States, from January to August 2007 During a period of low activity in the 11-year-long solar cycle Considered quiet days (91 quiet days) at mid-latitude region 9

10 Preliminary Model Validation Preliminary model evaluation assuming a satellite elevation mask of 50 deg. Fit the ionospheric error model to the data and compute the residual error (due to mis-modeling) over increasing lengths of the fit interval (d IPP,MAX ). 750 km d IPP correspond to: about 30 min of GPS SV motion about 3 min of Iridium SV motion Residual (cm) Fidelity of the Model to the Data vs. d IPP,MAX nominal 750km meas. noise level 1-sigma d (km) IPP,MAX Percentage of occurr. per d IPP,MAX bin

11 Example Experimental Result (1) At low elevations, in the large majority of cases, the nominal model fits the data well: residual errors on the order of the expected measurement noise few cases of wave-like structures Ionospheric Delay (m) PRN 1 on 07/28/2007 at 2:50pm local time in Los Angeles, California Measured Estimated d IPP (km) Ionospheric Delay (m) PRN 13 on 01/08/2007 at 2:26pm local time in Battle Creek, Michigan Measured Estimated d IPP (km) Detrended Delay (m) d IPP (km) Detrended Delay (m) d IPP (km) 11

12 Traveling Ionospheric Disturbance (TID) These temporary, localized, moving wave-like structures are in fact TIDs. TIDs have been studied over decades, and have been observed at all latitudes, longitudes, times of day, season, and solar cycle [Hunsucker 1982]. TIDs are generally thought to be caused by the propagation of atmospheric gravity waves through the neutral atmosphere, which in turn perturbs the ionospheric electron density distribution. We anticipate that TIDs will significantly impact the SF carrier phase filtering process, which is sensitive even to low-magnitude error variations (in contrast to other GPS-based implementations). Hunsucker, R. D. (1982), Atmospheric Gravity Waves Generated in the High-Latitude Ionosphere: A Review, Rev. Geophys. and Space Phys., 20, 2,

13 New Ionospheric Error Model TID properties that we observed using CORS data are consistent with values found in the literature wavelength: ~ 300 km amplitude: ~ cm and higher (up to 60 cm [Ding 2008]) We determined that in our 8-month long set of data, about 1% of GPS satellite passes were affected by decimeter-level TIDs. TID parameters are therefore included in the new fault-free error model. sin I, k coi, k bvi dipp, k gvi c OI, k ac cos IdIPP, k as Id IPP, k Nominal model Amplitude of the TID sin and cos terms TID frequency (rad/m) Ding, F., W. Wan, L. Liu, E. L. Afraimovich, S. V. Voeykov, and N. P. Perevalova (2008), A statistical study of large-scale traveling ionospheric disturbances observed by GPS TEC during major magnetic storms over the years , J. Geophys. Res. 13

14 Example Experimental Result (2) The new non-linear model is fit to the data using an iterative Newton-Raphson process residual errors (after removing the model from the data) are lower than with the nominal model TIDs with non-sinusoidal profiles still cause large residual errors PRN 20 on 01/08/2007 at 10:14am local time in Cleveland, Ohio PRN 10 on 02/10/2007 at 10:54am local time in Holland, Michigan Detrended Delay (m) Nominal Model New Model Detrended Delay (m) Nominal Model New Model d IPP (km) d IPP (km) 14

15 Result for the Entire Data Set Folded CDF of residual errors using the nominal and new models: data collected over 8 months (91 days) for all visible SVs at 7 locations 57,478 segments of 750 km long SV passes (3,777,406 data points) Cumulative Probability Residual Error (m) data (nominal) over-bound (nominal) data (new) over-bound (new) DeCleene, B. Defining Pseudorange Integrity - Overbounding. Proceedings of the Institute of Navigation GPS Conference. Salt Lake City, UT. (2000):

16 Summary of Ionospheric Error Modeling Nominal Model New Model Measurement Noise Sigma over-bound 8.2 cm 5.7 cm 2.3 cm Max residual error 40 cm 27 cm The new model brings about some improvement, but substantial residual mis-modeling errors remain. It is extremely challenging to capture the wide variety of TID profiles through simple modeling. In the next slides, a different approach to evaluate the impact of TIDs on the positioning performance is established (using a position estimate error bounding analysis). 16

17 Complete Measurement Equation satellite s epoch k u k x s s T k s s s s s s s L, k ek 1 N I, k O, k C, k T, k M, k RN, k k Ionospheric Delay SV Orbit Error Tropospheric Delay SV Clock Error Multipath & Rx Noise c b d g s s s s I, k OI, k VI IPP, k VI Similar methods for orbit ephemeris and tropospheric error models Ranging error model parameters (absolute and relative over time): are assumed constant over short time periods their probability distribution can be over-bounded by a Gaussian bound on uncertainty is the prior knowledge (PK) used in the estimation algorithm 17

18 Batch Least-Squares Estimation Batch of current-time and stored past-time measurements: stacked over time over T F = 10 min for all satellites Batch measurement equation: s s s φ 0 n 1 1 T ns φ φ φ P T T T φ z Hx v ρ v ~ 0, V x u u N x P T T T T 0 n 1 ERR Error States (incl. s b VI and s g VI for each satellite, T assumed constant) Weighted least-squares state estimate (with prior knowledge): xˆ S z S P H V x T 1 P x P 1 PK T 1 H V H 1 T Px T 2 T U U U 18

19 Nominal VPL Equation For integrity analysis under fault-free conditions, protection levels are computed; focus is on vertical protection level VPL because of usually worst position estimation performance in vertical direction. We first consider a nominal VPL equation, assuming no modeling error caused by TIDs: VPL b FF U specified integrity multiplier (corresponding to ) 19

20 Benchmark Application Aircraft Precision Approach Up (km) m/s Distance to TD (km) Filtering Period: T F 5 0 PL ellipses (x100) Iridium/GPS GPS-WAAS TD -5 2 VAL (x100) North North (km) (km) The fault-free (FF) availability criterion: VPL : Computed fault-free Vertical Protection Level VAL = 10m : Specified Vertical Alert Limit VPL VAL 20

21 Availability Analysis Methodology Approaches are simulated at regular 30s intervals over 3 days (10 days with Galileo), to simulate satellite geometries (representative of the entire constellation). 30s intervals over 3days VPL at TD VPL at TD (m) s VAL 5 10 Time (hrs) The percentage of approaches that meet the VAL requirement over the total number of simulated approaches is the measure of fault-free performance (FF availability). 21

22 Fault-Free Integrity Analysis FF Availability = 100% % 100% 10 min filtering period T F, assuming NO TIDs at MIAMI VPL at TD (m) 10 5 VAL VPL VAL Time (hours) 22

23 VPL Equation Accounting for TIDs Updated VPL equation accounting for mis-modeling errors. VPL b FF U The extra term b extra term accounting for mis-modeling TIDs assumes single-satellite TIDs (the worst-case SV is selected) is computed using Hölder s inequality (details in [Joerger 10]) is a conservative upper bound of the impact of mis-modeling errors on the position estimate (infinity norm) the input parameter is maximum mis-modeling error [Joerger 10] Joerger, M., Neale, J., Datta-Barua, S., and Pervan, B., Ionospheric Error Modeling for Carrier Phase-Based Multi- Constellation Navigation Systems, IEEE Transactions on Aerospace and Electronic Systems, under review 23

24 Fault-Free Integrity Analysis Iridium/GPS, 10cm TIDs Iridium/GPS, NO TIDs Availability 51% 100% VPL at TD (m) 10 VAL VPL VAL consider dual-frequency (or differential implementations) Time (hours) 24

25 Batch Least-Squares RAIM Residual For high-integrity applications, the system must be protected against rare event faults such as satellite failures. Perform residual-based RAIM for a sequence of carrier phase measurements: IMPACT OF MEAS. FAULT 1. on vert. pos. estimate error z Hx v f 2 x U ~ T U S f, U 2. on detection test statistic (norm of the RAIM residual weighted by the r r z - Hxˆ W T 1 r V r meas. noise covariance) 2 2 T 1 r ~ dof, f V ( I HS) f W Single satellite faults: f T f Z NZ 25

26 Detection Algorithm: RAIM HAZARDOUS x U VAL Missed-Detection Area 1. the vertical position estimate error is compared to a specified vertical alert limit VAL 2. the detection test statistic, is compared to a threshold R C set according to a specified continuity risk MISLEADING r W R C 26

27 Single-Satellite Fault Integrity Analysis In previous work, we derived realistic single-satellite faults (SSF), which included impulses, steps and ramps of all magnitudes and start times. more than 7000 faults per aircraft approach In this work, we directly address worst-case faults: that maximize the ratio of the mean of the position estimate error over the non-centrality parameter of the detection test-statistic T T 2 NZ U U NZ FM T T fnzmzmzfnz g f M M f M T ST U U Z 1 T 1 2 M T V I HS T Z Z Z after a few steps (paper, [Angus 2007]) where v MAX is the eigenvector corresponding to the max eigenvalue of f T M v 1 WORST Z Z MAX M M M M 1 T 1 Z U U Z Angus, J. E., (2007) RAIM with Multiple Faults, NAVIGATION: The Journal of the Institute of Navigation, Vol. 53, No

28 Detection Algorithm: RAIM x U VAL Missed-Detection Area failure mode s U k, F f VAL T 1 f V I HS f R C r W R C 28

29 x U VAL Probability of Missed Detection Missed-Detection Area Probability of missed-detection P MD s U k, F f VAL T 1 f V I HS f R C 25 Lines of constant probability P (-10 log P) r W R C 29

30 Overall System Performance An overall integrity requirement is set [MASPS 2004], and is allocated as follows: Simplified Integrity Allocation Tree Total System Integrity FF Integrity SSF Integrity Multiple Simult. ~ ~ Faults ~ requirement on VPL (estimation) requirement on P MD (detection) probability of occurrence computed combined FF and SSF availability criterion RTCA Special Committee 159. (2004) Minimum Aviation System Performance Standards for the Local Area Augmentation System (LAAS). Document No. RTCA/DO-245. Washington, DC. 30

31 Worst-Case Dual-Frequency Performance Evaluation Worst-case performance is established for a nominal configuration dual-frequency code and carrier phase signals, over T F = 10 min low-rate corrections from widearea ground stations Combined FF-SSF Availability N 40 N 50 N Iridium/GPS GPS/Galileo performance for the assumed 10 m VAL is extremely poor (not represented) 120 W 50 N 110 W 100 W 90 W 80 W 70 W Iridium/GPS/Galileo 40 N An extended-window RAIM algorithm was also derived, which provides 100% availability at all locations. 30 N 120 W 110 W 100 W 90 W 80 W 70 W 31

32 Conclusion Single-frequency implementation: designed and experimentally evaluated two ionospheric error models developed a conservative approach to account for mis-modeling errors Traveling ionospheric disturbances: are extremely challenging to model and affect positioning performance other strategies OR dual-frequency may be needed for high-integrity Worst-case dual-frequency performance evaluation derived an analytical expression of the worst-case fault mode dual-frequency Iridium/GPS and Iridium/GPS/Galileo can potentially provide high-integrity positioning over wide areas Other measurement error models (SV clock and orbit, troposphere) require experimental validation 32

33 Acknowledgment Sponsors The Boeing Company and the Naval Research Laboratory Colleagues Boris Pervan (advisor) Jason Neale Seebany Datta-Barua 33

34 END 34