Carrier Phase Integer Ambiguity Resolution Recent results and open issues

Carrier Phase Integer Ambiguity Resolution Recent results and open issues Sandra Verhagen DEOS, Delft University of Technology, The Netherlands Kolloquium Satellitennavigation, TU München, 9 February 2010 Delft University of Technology Challenge the future

Outline Carrier phase ambiguity resolution Background State-of-the-art Performance prospects High-precision applications: examples Carrier phase ambiguity resolution with constraints Background Recent developments and results 2 76

3 76 : number of times (= known) fits into ( = unknown integer) A Simple Ambiguity Resolution Example = = = a p D a p E p, 0 0, 0 1 1 2 2 σ σ ϕ ρ λ ϕ ϕ Model: integer p λ 1 λ 2 λ 3 λ ( 2) a λ ( 1) a λ a ϕ A B ϕ a λ p : measured fraction distance AB ( = 2 mm) σ ϕ : measured distance AB ( = 20 cm) σ p ρ ρ a : unknown distance AB

Relative Positioning: Double Differencing elimination of receiver clock errors elimination of initial receiver phase offsets DD phase ambiguity is an integer number! 1 2 DD code observation: Δp 12 2 1 12 T 12 12 12 12, j = Δp12, j Δp12, j = ( e2 ) Δx12 + μ jδi12, j + ΔT12 + 12, j x 12 DD phase observation: relative receiver position 12 2 1 12 T 12 12 12 12, j = Δϕ12, j Δϕ12, j = ( e2 ) Δx12 μ jδi12, j + ΔT12 + λ jδn12, j ε12, j Δϕ + relative receiver position integer DD ambiguity 4 76

Resolution of the DD ambiguities code observation: dm precision phase observation: mm precision, but: receiver-satellite geometry has to change considerably (long observation time) to solve position with mm-cm accuracy if DD ambiguities are resolved to integers within a short time (or instantaneously), positions (and other parameters) can be solved with mm-cm accuracy 5 76

100 10 Precision of relative GPS positioning code observations N E U 100 10 phase observations float N E U st.dev. [m] 1 st.dev. [m] 1 0.1 0.1 0.01 0.01 fixed 0.001 0.1 1 10 50 time [min] 0.001 two epochs of data: varying time span 0.1 1 10 50 time [min] 6 76

Precision code vs. phase observations code observations phase observations U U E N 1 m E N 0.01 m both RELATIVE positioning phase: provided that the integer ambiguity is KNOWN 7 76

Short vs. long baselines DELAY_1 DELAY_2 atmospheric layer short baselines (few km): signals travel through same part of atmosphere differential atmospheric delays neglected x 12 12 12 I12 = 0 T12 = 0 beneficial for ambiguity resolution (less parameters) long baselines (> few km): differential atmospheric delays need to be modeled (otherwise: wrong ambiguities and biased position) 8 76

Ionosphere-fixed, -float, -weighted model Ionosphere-fixed model: no differential ionospheric delay parameters observations may be corrected a priori for ionosphere for short baselines only can already be based on single-frequency data Ionosphere-float model: parameterization of differential ionospheric delays for long baselines based on at least dual-frequency data Ionosphere-weighted model: Ionosphere corrections from network RTK subtracted for medium to long baselines 9 76

GNSS model observation equations: y = Aa+ Bb+ e, a Z n ; Q yy y a b Q yy data vector ambiguities baseline coordinates & other unknowns variance-covariance matrix of data 10 76

SUCCESSFUL INTEGER AMBIGUITY RESOLUTION is the key to FAST and PRECISE GNSS parameter estimation (baseline coordinates, attitude angles, orbit parameters, atmospheric delays) 11 76

Integer estimation float solution estimate position and carrier ambiguities integer map estimate integer ambiguities fixed solution estimate position (ambiguities fixed) validate float solution validate integer ambiguities 12 76

Float and fixed solution Ambiguities not fixed Ambiguities fixed 1.0 0.010 0.5 0.005 Up (meters) 0.0 Up (meters) 0.000 0.5 0.005 1.0 0.010 1.0 0.0 North (meters) 1.0 1.0 0.5 0.0 East (meters) 0.5 1.0 0.010 0.000 North (meters) 0.010 0.010 0.005 0.000 East (meters) 0.005 0.010 13 76

Integer estimation integer map aˆ R n S( aˆ) = a Z n no holes & no overlap translation invariant ('linearity') 14 76

Different choices of integer estimators 2 integer rounding integer bootstrapping integer least-squares 2 2 1 1 1 0 0 0 1 1 1 2 2 1 0 1 2 2 2 1 0 1 2 2 2 1 0 1 2 after their pull-in region 15 76

Ambiguity resolution Integer ambiguities are derived from stochastic observations Integer ambiguities are not deterministic but stochastic input (stochastic) a ( a) S R n Z n = S ˆ : output (stochastic) 16 76

Integer estimation Optimal integer estimator: integer least-squares 2 1 a = arg min aˆ z 2 z Z n Q aˆ 0 1 2 2 1 0 1 2 17 76

Integer estimation PDF f a ˆ ( x) PMF P( a LS = z) success rate: P ( a = = LS a) faˆ ( x) dx S a 18 76

LAMBDA method Integer estimation: optimal : maximum success rate efficient : (near) real-time Applicable to wide variety of models With or without relative satellite-receiver geometry Stationary or moving receivers With or without atmospheric delays Single- or multi-baseline One, two, three or more frequencies (any GNSS) LAMBDA LAMBDA 19 76

Ambiguity Resolution Methods Search in the 3-dimensional position space (e.g. ambiguity function method); Now deprecated Linear combination of code and phase (using widelane/narrowlane combinations) performance worse with AS has been improved by LAMBDA: 2-dimensional ambiguity resolution/search problem Geometry-free model Search in the m-dimensional ambiguity space Geometry-based model 20 76

Integer validation: ratio test Integer validation:? actually discrimination test: is integer estimate sufficiently more likely than any other integer candidate? in practice: ratio test 21 76

22 76 2 1 0 1 2 2 1 0 1 2 a 2 a â 1 ˆ ˆ 2 2 2 ˆ ˆ a a Q Q a a a a 2 2 2 ˆ ˆ ˆ ˆ a Q a Q a a a a Integer validation: ratio test

Integer validation: ratio test Ratio test: aˆ a aˆ a Problem: how to choose μ? Practice: fixed value of e.g. μ = 0.5 2 2 Q 2 Q aˆ aˆ New theory: choose μ based on the condition that: P (failure) β user-defined μ > μ use a use aˆ Note: reciprocal of ratio test used in practice Look-up table to determine μ for a given ILS failure rate and number of ambiguities 23 76

Integer validation: ratio test white: rejected red: accepted but wrong green: accepted and correct P (float) = undecided rate P (failure) = failure rate < β P (success) = success rate 24 76

On the prospects of GPS+Galileo ambiguity resolution GPS/Galileo 25 76

Number of satellites during a day at three different locations for GPS, Galileo and GPS+Galileo 26 76

Single epoch success-rates 15 km, 30 S, 24 hrs Min&Max (bars),mean ( ) GPS Galileo GPS+Galileo 27 76

Single epoch success-rates 30 km, 30 S, 24 hrs Min&Max (bars),mean ( ) GPS Galileo GPS+Galileo 28 76

Single epoch success-rates 100 km, 30 S, 24 hrs Min&Max (bars),mean ( ) GPS Galileo GPS+Galileo 29 76

Single-Frequency, Single-Epoch, 24H Mean Success Rates 1 45 o North 1 75 o North 0.8 0.8 mean success rate 0.6 0.4 mean success rate 0.6 0.4 <5 km 10 km 20 km 0.2 0.2 0 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 number of satellites 0 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 number of satellites 30 76

Wide Area RTK Multi-frequency double-differenced phase & code observables Each GNSS has its own reference satellite Unknowns: coordinates + ZTD, DD ambiguities, DD ionosphere No predefined linear combinations of observations formed Ionosphere-weighted: to account for uncertainty in WARTK ionospheric corrections 31 76

WARTK performance DUNK 257km σ I = 1cm DELF 413km σ I = 2cm GPS only dual-freq GAL only dual-freq GAL only triple-freq GPS+GAL dual-freq 92.3% 98.0% 98.6% 100% 52.0% 62.1% 83.0% 99.8% 32 76

WARTK performance 0.5 0.4 Horizontal Position Errors of DELF GPS ONLY FLT HPE (95%): 0.67501 FIX HPE (95%): 0.012231 HPx [m] 0.3 0.2 0.1 0 2 1.5 1000 2000 3000 4000 5000 6000 7000 epoch [1 sec] Vertical Position Errors of DELF GPS ONLY FLT VPE (95%): 5.2199 FIX VPE (95%): 0.14579 VPx [m] 1 0.5 0 1000 2000 3000 4000 5000 6000 7000 epoch [1 sec] 33 76

WARTK performance 0.5 0.4 Horizontal Position Errors of DELF GPS & GAL FLT HPE (95%): 0.16621 FIX HPE (95%): 0.0054245 HPx [m] 0.3 0.2 0.1 0 2 1.5 1000 2000 3000 4000 5000 6000 7000 epoch [1 sec] Vertical Position Errors of DELF GPS & GAL FLT VPE (95%): 0.80056 FIX VPE (95%): 0.027094 VPx [m] 1 0.5 0 1000 2000 3000 4000 5000 6000 7000 epoch [1 sec] 34 76

Outline Carrier phase ambiguity resolution Background State-of-the-art Outlook on performance High-precision applications: examples Carrier phase ambiguity resolution with constraints Background Recent developments and results 35 76

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Example Machine Guidance 37 76

JPALS: Joint Precision Approach and Landing System All weather, all mission, all user landing system, based on DGPS GPS to provide accurate and reliable landing guidance Supports fixed base, tactical and shipboard applications First ever fully automated carrier landing at sea on April 23, 2004 38 76

Satellite formation flying dual-frequency GPS relative positioning (incl. LAMBDA ambiguity resolution) JPL BlackJack receiver dual frequency GRACE A and B geo-science twin-sat mission NASA-JPL EarthObservatory.nasa.gov 39 76

Furuno: Satellite Compass 40 76

Outline Carrier phase ambiguity resolution Background State-of-the-art Outlook on performance High-precision applications: examples Carrier phase ambiguity resolution with constraints Background Recent developments and results Open issues 41 76

Multivariate Constrained LAMBDA method Based on presentation by G. Giorgi, P.J.G. Teunissen and S. Verhagen at IAG Scientific Assembly, Buenos Aires, 2009 The observation model: functional and stochastic model of the GNSS observables The set of nonlinear geometrical constraints Constrained integer least-squares Numerical and experimental results 42 76

Modeling the GNSS observations 2 antennae tracking the same n+1 satellites Double Differences Gauss-Markov model: ( ) ( ) n E y = Az+ Gb ; D y = Q ; z Z, b R y p E(.) D(.) Expectation operator Dispersion operator y Vector of phase and code observables (2n) A Matrix of carrier wavelength (2n n) z Vector of integer phase ambiguities (n) G Matrix of Line-of-sight vectors (2n 3) b Vector of real-valued unknowns (p) Q y Variance-covariance matrix (2n 2n) 43 76

Modeling the GNSS observations Multibaseline observations: m+1 antennae Multivariate Gauss-Markov model: 3 ( ) = + ; ( ( )) = ; Z n m, R E Y AZ GB D vec Y Q Z B Y m b m b 1 b 2 [ ] Y = y1, y2,, ym Solvable applying the ILS principle Z = z1, z2,, zm The LAMBDA method [ ] [ ] B= b1, b2,, bm 44 76

Fixed frame of antennae Assumption: set of antennae mounted on a rigid platform Introduction of a local frame of reference u 1, u 2, u 3 x3 u3 u2 [ ] B= b1, b2,, bm x2 x1 [ ] F = f1, f2,, fm u1 45 76

Fixed frame of antennae Parameterization of the baseline coordinates via a rotation matrix R x3 u3 u2 [ ] B= b1, b2,, bm f2 =[f21, f22,0] T x2 B = RF x1 f1 =[f11,0,0] T u1 [ ] F = f1, f2,, fm KNOWN 46 76

Modeling the constrained set of GNSS observations 3 ( ) = + ; ( ( )) = ; Z n m, E Y AZ GB D vec Y Q Z B Additional constraint: the orthogonality of R ( ) ( ( )) D vec Y E Y = AZ + GRF ; Z Z, R O = Q Y Y m n m 33 Unknowns to be determined: Ambiguity matrix Z, in the class of integer matrices nxm Rotation matrix R, in the class of orthogonal matrices 3x3 47 76

Constrained Integer Least-Squares 33 ( ) ; n m = + Z, O E Y AZ GRF Z R Model in vectorial form: ( ( )) ( ) ( ) vec Z ( ( )) T E vec Y = Im A+ F G ; Z Z, R O vec R D vec Y = Q Y n m 33 48 76

Constrained Integer Least-Squares Minimizing the weighted squared norm of the least-squares residuals: min nm Z Z R O 33 vec Y I A vec Z F G vec R = ( ) ( ) ( ) ( T ) ( ) m 2 Q Y Sum of squares decomposition 2 ˆ = E + min vec Z Zˆ vec Rˆ + ( Z) R 2 2 ( ) ( ) n m Y Z ˆ 33 Z Rˆ( Z ) Q Z Q Q R O 49 76

Constrained Integer Least-Squares Objective function Z = arg min vec Z Z + min vec R( Z) R Z Z ( ˆ) ( ˆ ) nm 33 Q Zˆ R O Q Rˆ( Z ) Matrix of float ambiguities Matrix of l-s residuals 2 2 Conditional rotation matrix solution CZ ( ) 2 ˆ E + min vec Z Zˆ vec Rˆ + ( Z) R 2 2 ( ) ( ) n m Y Z ˆ 33 Z Rˆ( Z ) Q Z Q Q R O 50 76

Constrained Integer Least-Squares No closed-form solutions available Direct search in a set of admissible integer candidates Search space no longer ellipsoidal { n m Z Z C Z χ } Ω ( χ ) = ( ) 2 2 ISSUE: The search might be rather inefficient if one cannot limit the size of the search space Modification of the LAMBDA method 51 76

Constrained Integer Least-Squares Z = arg min vec Z Z + min vec R( Z) R Z Z 2 2 ( ˆ) ( ˆ ) nm 33 Q Zˆ R O Q Rˆ( Z ) THE THREE STEP SOLUTION The float estimators The search for the integer minimizer Search and Shrink approach Expansion approach The solution of the nonlinear least-squares problem 52 76

Simulations results SIMULATIONS SET-UP Date and time 22 Jan 2008 00:00 Location Latitude:50º, Longitude: 3º GPS Week 439 Scenario Single baseline, stationary Frequency L1 Baseline length 2 m Samples simulated 10 5 53 76

Simulations results σ φ σ p [ mm] [ cm] N 3 1 30 15 5 30 15 5 LAMBDA method Constrained LAMBDA Single-frequency Single-epoch SUCCESS RATE [%] 5 6 7 8 0.17 5.69 83.87 0.55 10.18 95.48 99.60 99.94 100 100 100 100 9.81 56.89 96.83 30.12 81.35 99.99 99.99 100 100 100 100 100 31.97 73.07 99.74 61.16 91.33 100 100 100 100 100 100 100 81.72 93.12 99.99 99.99 100 100 100 100 100 100 100 100 54 76

Experimental results EXPERIMENT SET-UP 1: choke-ring antenna connected to the Ashtech reciver ANTENNAE 2: antenna connected to the Leica SR530 receiver 3: antenna connected to the Novatel OEM3 receiver 55 76

Experimental results Single-frequency / Single-epoch SUCCESS RATE [%] 56 76

Experimental results Full Attitude Solution: HEADING 57 76

Experimental results Full Attitude Solution: ELEVATION 58 76

Experimental results Full Attitude Solution: BANK 59 76

Experimental results GAIN project Gravimetry using Airborne Inertial Navigation 1 November 2007 flight Cessna Citation II 60 76

Experimental results 61 76

Experimental results Attitude accuracy Heading Elevation Bank σ ψψ INS σ θθins σ φφins [deg] [deg] [deg] 0.065 0.202 0.124 62 76

Formation flying: relative positioning 63 76

Summary and Conclusions Constrained model for the set of GNSS observations The least-squares minimization with constraints: integerness of the ambiguities orthogonality of the rotation matrix Computationally efficient search method based on a modification of LAMBDA The Constrained LAMBDA method is capable of providing extremely reliable single-epoch / single-frequency solutions Also useful for relative positioning of multi-antenna platforms 64 76