Overbounding GNSS/INS Integration with Uncertain GNSS ......of GNSS and Inertial Navigation Systems (INS) has been used since the 1990s in avionic systems as part of the Aircraft-Based

Overbounding GNSS/INS Integration withUncertain GNSS Gauss-Markov Error Parameters

Omar Garcı́a Crespillo1, Mathieu Joerger2, Steve Langel31Institute of Communications and Navigation, German Aerospace Center (DLR)

2Department of Aerospace and Ocean Engineering, Virginia Tech3Department of Communications, SIGINT and PNT, The MITRE Corporation

Abstract—The integration of GNSS with Inertial NavigationSystems (INS) has the potential to achieve high levels of continuityand availability as compared to standalone GNSS and thereforeto satisfy stringent navigation requirements. However, robustlyaccounting for time-correlated measurement errors is a challengewhen designing the Kalman filter (KF) used for GNSS/INS cou-pling. In particular, if the error processes are not fully known, theKF estimation error covariance can be misleading, which is prob-lematic in safety-critical applications. In this paper, we designa GNSS/INS integration scheme that guarantees upper boundson the estimation error variance assuming that measurementerrors are first-order Gauss-Markov processes with parametersonly known to reside within pre-established bounds. We evaluatethe filter performance and guaranteed estimation by covarianceanalysis for a simulated precision approach procedure.

Index Terms—Overbounding, Kalman filtering, GNSS, InertialSystems, Guaranteed estimation, Precision Approach, GaussMarkov Process, Colored Noise, ARAIM

I. INTRODUCTION

The demand for increased levels of autonomy in safety-and liability-critical transportation applications motivates theneed for high-integrity navigation solutions. The combinationof GNSS and Inertial Navigation Systems (INS) has beenused since the 1990s in avionic systems as part of theAircraft-Based Augmentation Systems (ABAS). It has alsobecome a baseline approach for applications in challengingland, air, and sea environments, for example in autonomousground vehicles and commercial quadcopters. GNSS/INS canpotentially achieve high levels of continuity and availability ascompared to standalone GNSS. However, providing a rigorousassessment of GNSS/INS integrity still poses unansweredchallenges.

Fault detection algorithms have been developed to addresssensor faults in integrated GNSS/INS schemes using a Kalmanfilter (KF) [1, 2, 3, 4]. However, there is no widely adoptedapproach to account for time-varying sensor errors whenthe structure of the error time correlation is uncertain. Forexample, assuming that sensor errors follow a Gauss-MarkovProcess (GMP), how should one account for an unknown GMPtime constant? Robust estimators are a promising solutionwhen dealing with mis-modeled errors in GNSS [5] andGNSS/INS positioning [6, 7], but no rigorous quantification oftheir estimation error is currently available and therefore theycannot readily be implemented in safety critical applications.Existing error overbounding techniques used for snapshot

GNSS positioning [8] do not account for measurement errorcorrelation and therefore are not guaranteed to overboundthe positioning error distribution when used in a sequentialestimation context.

Correlated error processes affect GNSS pseudoranges, IMUspecific force, and IMU angular rate measurements. Theseerrors have a substantial impact on the performance of theKF, and state estimate variance can be misleading if theseerrors are not properly modelled. Proper characterization ofcorrelated errors can be a difficult task. For instance, inGNSS, the pseudorange error is an accumulation of multipleerror sources including orbit, clock, atmospheric (ionosphericand tropospheric) and multipath, whose stochastic behaviordepends upon complex system processes and a changingenvironment.

In [9, 10], we provided a solution for overbounding sequen-tial estimation errors in the presence of Gauss-Markov (GM)error processes with unknown but bounded time constants anddriving noises. In this paper, we apply the methodology in [9]to design a GNSS/INS Kalman filter so that the navigationestimation error is properly overbounded by the KF covariancefor the states of interest when the error model parameters arenot fully known. First, we revisit the overbounding processin [9]. The different GNSS error sources are then modeledusing different ranges of correlation time constants. We de-scribe the error models provided in aviation standards, inthe ARAIM Working Group C, and in recent publicationsthat address sequential estimation for high-integrity navigation.We then design a KF that includes augmented states foreach of the time-correlated GNSS and IMU measurements.This GNSS/INS Kalman filter is implemented in a realisticsimulated aircraft precision approach trajectory to analyze thetightness of the proposed estimation error variance bound.

II. OVERBOUNDING KALMAN FILTER WITH UNKNOWNGAUSS-MARKOV PROCESSES

Let P̂ be the Kalman filter (KF) estimate error covariancematrix and P be the true estimate error covariance matrix. Theinequality P̂ ≥ P means that the predicted variance αT P̂α isgreater than or equal to the true variance αTPα for any realvector α.

Suppose that the measurement and process noise com-ponents are known to be first-order GMP. These processesare completely specified by a time constant τ , steady-state

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variance σ2 and initial variance σ20 , and propagate accordingto the difference equation

ak+1 = e−∆t/τak +

√σ2(1− e−2∆t/τ

)wk ,

wk ∼WGN (0, 1) and a0 ∼ N(0, σ20)(1)

where k is an arbitrary time index and ∆t = tk+1 − tk isthe discrete-time sampling interval. The notation WGN(0, 1)indicates zero-mean white Gaussian noise with unit varianceand N(0, σ20) denotes a zero-mean normal random variablewith variance σ20 .

It was proved in [9] that when τ and σ2 are only knownto reside in the intervals [τmin, τmax] and [0, σ2max], a state-augmented KF that models ak with

τ̂ = τmax

σ̂2 = σ2max(τmax/τmin)

σ̂20 ≥2σ2max

1 + (τmin/τmax)

(2)

is guaranteed to produce a covariance matrix P̂ ≥ P.A direct interpretation is that the design of the GMP that

produces an overbounded covariance estimation must considerthe maximum time constant from within the possible rangeand must inflate the steady-state maximum variance by afactor which is the ratio between the maximum and minimumtime constant. Please note that when σ20 = σ

2max(τmax/τmin),

the overbounding augmented state error model is stationary.When using the lower bound for σ20 in Equation (2), the GMPvariance will have a transition phase until it converges to itssteady state variance. Appendix A provides further insightabout the nature of this bound with respect to the uncertainGMPs by representing it in different domains.

III. ERROR MODEL IMPLEMENTATION

In the following section, we describe the error models forthe GNSS and IMU measurements. Particularly relevant forthis paper is the consideration of uncertain time constants forthe time correlated errors in GNSS, which we assume to residewithin a known range of values.

A. GNSS

The linearized iono-free code and carrier measurement canbe expressed as:

ρi,jk − ρ̃i,jk (x0,k) =

ui,jT

k ∆xk + bjk + ∆S

i,jk + T

i,jk +mp

i,jρ,k + �

i,jρ,k (3)

φi,jk − φ̃i,jk (x0,k) =

ui,jT

k ∆xk + bjk + ∆S

i,jk + T

i,jk +N

i,jφ +mp

i,jφ,k + �

i,jφ,k (4)

where ρi,jk is the code measurement of satellite i of constel-lation j at time epoch k. ui,j

T

k is a unit line of sight vectoruser to satellite, ∆xk is the user position with respect to the

linearization point. The receiver clock offset with respect toconstellation j is bjk. ∆S

i,jk is the residual satellite clock and

ephemeris error after correction based on broadcast ephemeris,T i,jk is the tropospheric error, mp

i,jρ,k and mp

i,jφ,k the multipath

error in code and carrier respectively, �i,jk the receiver noiseand N i,jφ is the float iono-free integer ambiguity.

1) Satellite Clock and Orbit Errors: According to [11], wecan model the residual satellite clock and ephemeris error fora satellite i as a Gauss-Markov process expressed as:

∆Si,GPS ∼ GM(σ2URA , τ ∈ [4, 50] hours), (5)∆Si,GAL ∼ GM(σ2URA , τ ∈ [2, 38] hours). (6)

The user range accuracy (URA) is nominally chosen to beσURA = 1m [12].

2) Tropospheric Errors: The largest part of the troposphericerror caused by the dry component can be removed byapplying standard models [13]. The remaining wet componentof the troposphere is more unpredictable and it is typicallymodeled as:

∆T i,jk = mtropo(θi,j) · ηtropo,k, (7)

where m(θi,j) is a mapping function that depends on thesatellite elevation θi,j [13]:

m(θi,j) =1.001√

0.002001 + sin(θi,j)2. (8)

The uncertain component ηtropo,k at the zenit is specified in[13] to be overbounded by a zero mean Gaussian distributionwith variance σ2tropo = (0.12)

2m2. In this work, we modelthe random component ηtropo,k as a first-order Gauss-Markovprocess with the range of parameters specified in Table I.

3) Multipath and antenna group delay: For 100 secondscarrier-smoothed code, the following expression is given forthe multipath error standard deviation [12]:

σi,jmp,ρ,sm =

√(f2L1 − f2L5)2f4L1 + f

4L5

(0.13 + 0.53e−θ

i,jdeg /10

), (9)

where fL1 and fL5 are the GNSS carrier frequencies for L1and L5 signals respectively.

We want to consider unsmoothed measurements in order toavoid the artificial correlation that the smoothing would causebetween the measurements in the filter. It is found in [14, 15],that the unsmoothed code and carrier phase standard deviationdue to multipath can be obtained with the following scalingof the smoothed ones:

σmp,ρ = 1.5 · σmp,ρ,sm,σmp,φ = 0.015 · σmp,ρ,sm.

(10)

In this work, we model the total multipath for code and carrieras:

mpi,jρ,k = σρ(θi,jk ) · η

i,jmpρ,k

, (11)

mpi,jφ,k = σφ(θi,jk ) · η

i,jmpφ,k

. (12)

482


The first term will follow Equation (10) and it is used as amapping or scaled parameter. The stochastic process ηi,jmpρ,kand ηi,jmpφ,k are modeled as first-order Gauss-Markov processeswith unit variance and time-correlations in the ranges specifiedin Table I. Airborne multipath models for new signals andconstellations are currently under development [16] and willbe included in future publications.

4) Receiver Noise: The receiver noise component in thecode and carrier measurement is modeled as a zero mean whiteGaussian noise. Receiver code and carrier phase standarddeviations can be expressed as [15]:

σi,j�ρ = 19.6 · σi.j�ρ,sm,

σi,j�φ = 0.196 · σi.j�ρ,sm,

(13)

where σi.j�ρ,sm is the iono-free scaled carrier smoothed codenoise standard deviation which is dependent on elevation [12]:

σi.j�ρ,sm =

√(f2L1 − f2L5)2f4L1 + f

4L5

(0.15 + 0.43e−θ

i,jdeg /6.9

). (14)

5) Receiver Clock: GNSS receiver clocks are typicallyquartz oscillators; their offset with respect to GPS time isoften treated as a parameter to be estimated. In this paper, weconservatively assume that at any particular time, we do nothave any prior knowledge of the clock bias from previous timeinstants. This is modeled as an KF state parameter followinga random walk with infinite (very high) variance.

TABLE I: GNSS Error Model Parameters.

Error Model ParametersError source Mapping Variance τmin τmax

Clock and Eph. (GPS) - 1m2 4 h. 50 h.Clock and Eph. (Gal) - 1m2 2 h. 38 h.

Tropospheric Eq.(8) (0.12 m)2 900 s 2700 sRaw Code Multipath Eqs.(9,10) 1m2 10 s 900 s

Raw Carrier Multipath Eqs.(9,10) 1m2 10 s 900 sReceiver Code Noise - Eqs.(13,14) - -

Receiver Carrier Noise - Eqs.(13,14) - -

B. Inertial Measurement Unit (IMU)

The inertial measurements coming from gyroscopes andaccelerometers are typically modeled as a combination of errorsources and processes. First, we consider deterministic errorsincluding misalignment of the sensor axis, scaling factors andconstant biases. In this paper, we assume that these errors canbe estimated and compensated for using an offline calibrationprocedure. Second, we consider stochastic errors that cannotbe compensated for. A widely-used approach is to modelstochastic errors of the IMU as the sum of a random constantturn-on bias, a time-correlated process and a white-Gaussiannoise. We can therefore express the turn rate and specific forcemeasurements as:

w̃b = wb + bw,0 + bw + ηηηw, (15)

f̃ b = f b + bf,0 + bf + ηηηf , (16)

where w̃b and f̃ b are the 3-axis measured turn rates andspecific forces in a body frame b, respectively; similar, wb andf b are their true values, b?,0 is the turn-on random constantbias with ? referring either to the turn rates w or to the specificforces f . Last, b? is the time-correlated bias and ηηη? are thewhite Gaussian noise vectors of the associated measurements.

The turn-on biases can initially roughly be estimated bya coarse alignment process and further improved using a finealignment process [17]. For low cost sensors in particular, finalestimation of these random constant biases is often performedwhile in operation and thanks to the dynamics of the vehicle.In Section ”Analysis of Overbounded GPS/INS”, we willassume that some previous estimation process was availablethat provided initial values for these biases.

The most widely used model for the time-correlated bias ofIMU measurements is based on a GM approximation, in partbecause it can easily by incorporated in a Kalman filter bystate augmentation. This model is also adopted in this paper.Typical GM model parameter values for two sensor gradesare listed in Table II and Table III for the GM bias over time(including a time constant and driving noise specifications)and for the measurement white noise.

TABLE II: IMU Accelerometer Error Parameters.

Grade Noise Bias Noise τ[µg/

√Hz] [µg] [s]

Navigation 15 20 3000Tactical 50 160 3000

TABLE III: IMU Gyroscope Error Parameters.

Grade Noise Bias Noise τ[°/h/

√Hz] [°h−1] [s]

Navigation 0.01 0.005 12000Tactical 2 0.5 10000

In this paper, we assume that the IMU error parameters areknown. Future work will include the possibility to account foruncertainty in parameter values of the stochastic error models.This will capture the fact that error processes are not alwaysrepeatable, and that error behavior becomes unpredictable inthe presence of variations in temperature and vibrations.

IV. GNSS/INS KALMAN FILTER DESIGN

We consider a tightly coupled integration between GNSSand Inertial Navigation System (INS) where we use an error-state Kalman Filter. A general architecture of the systemdesign is depicted in Figure 1. Note that the INS system is runoutside the KF and it is calibrated with parameters estimatedusing the KF whenever an update step occurs.

A. State Selection

Kalman filter state parameters include position, velocity andattitude errors in a local navigation frame. In order to accountfor the time correlated errors present in IMU measurements we

483


Fig. 1: GNSS/INS Kalman filter Architecture.

include the augmented states bf and bw. The total number oferror states related to the INS system are therefore NINS = 15:

xINS =(δψψψT δvT δpT bTf b

Tw

)T(17)

where δψψψ, δv and δp are the 3D error in attitude, velocityand position of the INS system respectively. The filter statesspecific to GNSS first include the receiver clock biases withrespect to each constellation in use. In order to account for thetime correlated errors, additional augmented states are addedto the state vector to capture the satellite clock and ephemeris,tropospheric and multipath error of each satellite. Finally, weadd the integer ambiguities to each of the satellites in view.The GNSS-specific state vector component is therefore:

xGNSS =(

bclkT ∆ST mtropo

T mpTρ mpTφ Nφ

T)T

(18)where bclk are the user clock biases for each GNSS con-stellation, ∆S the satellite ephemeris and clock error, mtropothe tropospheric error at zenith, mpρ and mpφ the codeand carrier multipath respectively and Nφ the float iono-freeinteger ambiguities. The KF state vector therefore contains

NKF = 15 + Nj + 5Ni parameters, where Nj is the numberof constellations and Ni the number of satellites in view:

xKF =(

xTINS

xTGNSS

)T. (19)

B. KF Prediction

The KF time-update or prediction step propagates the meanand covariance of the state estimates as follows:

x̂k|k−1 = Φ̂ΦΦkx̂k−1|k−1, (20)

P̂k|k−1 = Φ̂ΦΦkP̂k−1|k−1Φ̂ΦΦT

k + GkQ̂kGTk , (21)

where x̂k|k−1 and P̂k|k−1 are the predicted states and covari-ance respectively, Φ̂ΦΦk is the time propagation matrix, Q̂k isthe covariance of the process noise and Gk maps the processnoise vector to the relevant states. The (ˆ) notation on Φ̂ΦΦk andQ̂k indicates that a filter design choice is made: we want toset Φ̂ΦΦk and Q̂k guaranteeing that the computed estimate errorcovariance overbounds the actual estimation uncertainty. Thediscrete propagation matrix Φ̂ΦΦk for the GNSS/INS design canbe expressed as:

Φ̂ΦΦk =

[ΦΦΦk,INS 0

0 Φ̂ΦΦk,GNSS

], (22)

whereΦΦΦk,INS = e

Fk,INS∆t ≈ I + Fk,INS∆t. (23)

The Jacobian matrix Fk,INS can be obtained by differentiatingthe strapdown inertial differential equations at time k. Thismatrix is well known and can be found in textbooks on inertialintegration (for instance, in [17]). It is worth noticing that ΦΦΦINSand FINS do not have the (ˆ) notation because, in this paper,the IMU error process parameters are assumed to be known.This assumption will be relaxed in future work.

The propagation design matrix Φ̂ΦΦk,GNSS is a diagonal matrixexpressed as:

Φ̂ΦΦk,GNSS =

INj×Nj

e−∆t/τ̂∆SINi×Ni

e−∆t/τ̂tropoINi×Ni

e−∆t/τ̂mp,ρINi×Ni

e−∆t/τ̂mp,φINi×Ni

INi×Ni

. (24)

In Equation (21), the Q̂k matrix can also be split into con-tributions from the IMU and GNSS error processes using thefollowing definitions:

Q̂k =

[QIMU 0

0 Q̂k,GNSS

]. (25)

The covariance QIMU contains the IMU noise and GM vari-ances which are not changing at different epochs.

The matrix Q̂k,GNSS is a diagonal matrix expressed as:

Q̂k,GNSS = diag

σ2clk∆t1Nj×1

σ̂2∆S

(1− e−

2∆tτ̂∆S

)1Ni×1

σ̂2tropo

(1− e−

2∆tτ̂tropo

)1Ni×1

σ̂2mp,ρ

(1− e−

2∆tτ̂mp,ρ

)1Ni×1

σ̂2mp,φ

(1− e−

2∆tτ̂mp,φ

)1Ni×1

0Ni×1

T.

(26)484


The notation 1a×b and 0a×b indicates a matrix (or vector) ofsize a × b filled with ones or zeros respectively. The valuesof the σ̂2 and τ̂ parameters are computed using Equation (2)and parameter range limits in Section III to ensure that the KFestimation is overbounded. Finally, matrix Gk can be writtenas:

Gk =

[Gk,IMU 0

NINS×NGNSS

0NGNSS×NINS INGNSS×NGNSS

], (27)

where NINS = 15 and NGNSS = Nj + 5Ni. Matrix Gk,IMU isgiven in Appendix in [18].

C. KF Update

The Kalman filter measurement update is performed usingthe following equations:

x̂k|k = x̂k|k−1 + K̂k(zk −Hkx̂k|k−1

), (28)

P̂k|k =(I− K̂kHk

)P̂k|k−1, (29)

where K̂ is the Kalman filter gain obtained using:

K̂k = P̂k|k−1HTk

(HkP̂k|k−1H

Tk + Rk

)−1. (30)

The linearized KF measurement vector z is made of thedifferences between the iono-free code and carrier phasemeasurements and their predicted values computed using theINS current position:

zk =

ρ1k − ρ1k,INS...

ρNik − ρNik,INS

φ1k − φ1k,INS...

φNik − φNik,INS

. (31)

The measurement matrix H projects the states into measure-ment space and is expressed as:

Hk =

06×Ni 06×Ni

UTk UTk

06×Ni 06×Ni

1Nj×Ni{si∈cj} 1

Nj×Ni{si∈cj}

INi×Ni INi×Ni

Mtropo MtropoMmp,ρ 0

Ni×Ni

0Ni×Ni Mmp,φ0Ni×Ni INi×Ni

T

, (32)

where Uk ∈ RNi×3 contains the line-of-sight vectors relatedto each of the satellites in view. The diagonal matrix Mtropo ∈RNi×Ni contains the tropospheric mapping function for eachof the satellites depending on their elevation as in Equation (8).And the diagonal matrix Mmp,ρ,Mmp,φ ∈ RNi×Ni containsthe scaling of the multipath time-correlated errors accordingto their elevation as in Equation (10) for the code andcarrier phase measurement respectively. Each element (j, i)of the matrix 1Nj×Ni{si∈cj} is one if the satellite s

i belongs to

constellation cj and zero otherwise. Finally, Rk is a diagonalmatrix containing the code and carrier receiver noise variances(Equation (13)):

Rk =

[σ2�ρI

Ni×Ni 0

0 σ2�φINi×Ni

]. (33)

Note that in the case of loss of satellites in view, the size ofstate vector and covariance matrix must be reduced accord-ingly. Similarly, new states must be created and initializedif new satellites are available during the filter runtime. Thechange of satellites in view also affect the size of Φ̂ΦΦk,GNSS,Q̂k,GNSS, Gk, Hk and Rk.

V. ANALYSIS OF OVERBOUNDED GPS/INS

A. Precision Approach Simulation

In order to evaluate the behavior of a GPS/INS systemin a realistic operational scenario, we consider the simulatedprecision approach procedure shown in Figure 2. We considera half race-track procedure starting from a holding position at7000ft with 200 knots speed. For the turn, the bank angle isat its maximum of 25 degrees as specified in [19], which isa worst case trajectory (i.e., high likelihood of losing visiblelow-elevation satellites due to banking). More realistic proce-dures could be defined based on true airspeed and tailwind,but these are beyond the scope of this work.

0 0.5 1 1.5 2

Easting [m] 104

0

2000

4000No

rth

ing

[m

]

Start

End

(a) Horizontal Profile

0 0.5 1 1.5 2

Easting [m] 104

0

1000

2000

He

igh

t [m

]

(b) Height profile

Fig. 2: Simulated Approach Trajectory.

We consider the integration of navigation grade IMU mea-surements at 100 Hz frequency with L1/L5 GPS code andcarrier measurements at 1 Hz.

B. Linearized GPS/INS and KF Initialization

In order to evaluate the overbounding capability of theproposed solution, we implement a Linearized Kalman Filter(LKF) [17], linearized about the true trajectory. This helpsisolate and properly evaluate the impact of the computedKF covariance as compared to the true covariance withoutincluding linearization errors of the Extended Kalman Filter(EKF). We can assume that when a simulated approach inFigure 2 is initiated, the navigation filter must have been

485


running for a significant period of time. Finding realistic valuesfor the initialized KF while limiting computation load is nottrivial. We have selected the following values for the initialcovariance matrix P0 related to the INS states:

P0,δψψψ = diag(0.012, 0.012, 0.032) deg2,

P0,δv = diag(0.0052, 0.0052, 0.0052) (m/s)2,

P0,δp = diag(1.22, 1.22, 1.22) m2,

P0,δbf = diag((10−3)2, (10−3)2, (10−4)2) (m/s/s)2,

P0,δbw = diag((10−5)2, (10−5)2, (10−4)2) (deg/s)2

and for the GNSS states:

σ20,clk = 1 m2,

σ20,∆S = 12.5 m2,

σ20,mtropo = 0.0432 m2,

σ20,mpρ = 90 m2,

σ20,mpφ = 90 m2,

σ20,Nφ = 0.62m2.

The initial variance for the augmented states were selectedaccording to the stationary bound variance from Equation (2).Notice that when we acquire new satellites during the simu-lation, we need to incorporate new augmented states for thecorrelated errors related to each satellites and the initializationis here performed with the non-stationary GM bound conditionto minimize the uncertainty.

C. Covariance Analysis Results

The position covariance estimated using the proposedGPS/INS filter are depicted over time in Figure 3.

0 50 100 150 200 250 300 350 400

Time [s]

0

0.5

1

1.5

2

K

F [m

]

Horizontal

Vertical

Fig. 3: Horizontal and vertical standard deviation estimated bythe overbounding GPS/INS filter.

If the true time correlation of the GNSS errors is known,it is possible to compute the actual true KF estimation errorcovariance in addition to the computed covariance. AppendixB describes a method to derive the true error covariance fora generic discrete-time KF, which also applies to this aircraftnavigation problem. In Figure 4, we compare the estimatedpositioning standard deviation with the true error standarddeviation, assuming that the true values of the error correlationtime constants are in the middle of the range of possible timeconstant values.

0 50 100 150 200 250 300 350 400

Time [s]

0

0.2

0.4

0.6

0.8

K

F -

tr

ue [m

]

Horizontal

Vertical

Fig. 4: Estimated KF standard deviation minus true errorstandard deviation. Here it is assumed that the true timecorrelations for each of the augmented states are in the middleof the range provided in Table I.

In Figure 4, the fact that the curves are positive for allsimulated time epochs shows that the designed KF achievesbounding estimation covariance. The proof that this holds truefor any values of the correlation time constants is provided in[9]. Figure 4 illustrates that this theoretical result holds true forpractical GPS/INS integration applications. The results fromFigure 3 and Figure 4 show that, in this example, the proposedfilter produces estimated positioning standard deviations upto 70 cm larger than the true values in both horizontal andvertical directions. Although this value is a significant fractionof the actual true error standard deviation (also sub-meterlevel), the estimated total uncertainty is kept at the meter level.This has therefore the potential to support the computation ofprotection levels that satisfy stringent integrity and availabilityrequirements for precision approach.

VI. CONCLUSIONS

The concept of overbounding has been developed and is inuse in the context of civil aviation for snapshot estimators.The use of sequential estimators like the Kalman filter canpotentially provide better performance in accuracy, continuityand availability than snapshot algorithms, especially whenintegrating information from multiple sensors. However, ex-tending the concept of overbounding to GNSS/INS is chal-lenging because time-correlated errors must be rigorouslyaccounted for. In this paper, we provided a first solutionto overbounding GNSS/INS estimation error variance in thepresence of uncertain GNSS measurement errors behavingas Gauss-Markov processes (GMP). Our proposed integrationscheme only causes minor changes in algorithms that alreadyincorporated GMP error models but, in addition, producesguaranteed estimation error bounds which are essential forsafety-critical applications.

APPENDIX AGAUSS-MARKOV BOUND INTERPRETATION

The methodology to bound the Kalman filter under uncer-tain GM errors consists on designing the augmented timecorrelated states with another GM process as specified inEquation (2).

486


There are different mathematical tools that are typically usedto model the errors in sensor measurements. Depending onthe type of sensor, the nature of the measurements and thespecific parameters and time correlation that they have, oneor another tool would be more suitable to perform the errormodeling. In this appendix we include the representation ofthe proposed bound in [9] and Equation (2) in the followingdomains: Autocorrelation, Power Spectral Density (PSD) andAllan Variance.

A. Autocorrelation

The autocorrelation of a given stationary GM process withvariance σ2 and time correlation τ is expressed as:

R(∆t) = σ2e−|∆t|τ , (34)

where ∆t is the sampling interval. In Figure 5, we can seesome examples of different GM processes in the autocorrela-tion domain that have σ2 = 1 and τ ∈ [1, 10]s. In Figure 5, the

0 5 10 15 20 25 30 35 40

t

0

0.5

1

1.5

2

2.5

3

3.5

4

R(

t)

GM bound

GM with min

= 1

GM with max

=10

Fig. 5: Autocorrelation of GM processes with σ2 = 1 andτ ∈ [1, 10]s and the GM process bound.

value of the autocorrelation of the bound at zero is expressedas R(0) = σ2 τmaxτmin and its value is therefore 10 for this example.Please note that the bound appears to be quite conservativefor small sample intervals. This can be partially improved byusing the non-stationary bound with a tighter initial varianceas described in Section II.

B. Power Spectral Density (PSD)

The spectral density of a GM process for a given frequencyf can be expressed as [20]:

S(f) =2σ2/τ

(2πf)2 + (1/τ)2. (35)

In Figure 6, the GMPs and the proposed bound are representedin the PSD domain. Please note that the GMPs with the uncer-tain time-correlation constant crosses at different frequencies,which supports the reason why choosing the GMPs with thehighest time constant does not guaranteed an estimation bound.The PSD offers a powerful insight of the underlying errorprocess in the frequency domain. In [21], the authors presenta general overbounding methodology in the frequency domain

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

f [Hz]

-5

0

5

10

log

(S(f

))

GMP Bound

GM with min

= 1

GM with max

=10

Fig. 6: Power Spectral Density (PSD) of GM processes withwith σ2 = 1 and τ ∈ [1, 10]s and the GM process bound.

for time-correlated error processes that goes beyond the Gauss-Markov error structure.

C. Allan Variance

The Allan variance (AV) is a statistical analysis tool toidentify and characterize stochastic processes by observingtheir behaviour over different time periods [22]. The GMPin the AV domain has the following expression [22]:

σ2AV(∆t) =2σ2τ

∆t

[1− τ

2∆t

(3− 4e

−∆tτ + e

−2∆tτ

)](36)

where σ2 is the GM steady state variance, τ is the timeconstant and ∆t is the time interval under consideration.

Figure 7 represents the Allan deviation in log-log scale ofGMPs with different time constants as well as the representa-tion of the Equation (2) bound.

10-1 100 101 102 103

log( t)

10-1

100

log

(A

V)

GMP Bound

GM with min

= 1

GM with max

=10

Fig. 7: Different Gauss-Markov Process with different timeconstant within τ ∈ [1, 10]s in Allan Variance.

We can see that in fact the designed GMP is bounding theunknown process for any time interval for any possible timeconstant also in the AV domain. The maximum peak valueof the bound is aligned with the one of the GMP with themaximum time constant, but its variance level is increased.This is consistent with the interpretation we did in Section II.This graphical representation also suggest that there is a tighter

487


bound. Graphically this bound can be found by fitting a GMPthat bounds in the AV domain both the ascending slope of theGMP with τmin and the descending slope of the GMP withτmax. This new process is provided graphically in Figure 8.

10-1 100 101 102 103

log( t)

10-1

100

log

(A

V)

GMP Bound

GM with min

= 1

GM with max

=10

Tight Bound

Fig. 8: New bound representation in Allan Variance Domain.

The new tighter bound can be found to have a time constantthat is a mean value between the extremes of the range ofpossible time constants in the logarithmic scale:

τb = 10log(τmin)+log(τmax)

2 = 10log(τminτmax)

2 (37)

and the variance of the process must be inflated as:

σ2b = σ2max

τbτmin

. (38)

This new bound is a candidate to provide a tighter bound overuncertain GMP and is first presented here only as a conjectureuntil the formal proof is published [23].

APPENDIX BDISCRETE KF TRUE ERROR COVARIANCE

The general discrete linear system we are working with canbe described as follows:

xk = ΦΦΦxk−1 + Gkwk, (39)zk = Hkxk + νννk. (40)

A Kalman filter estimator that designs imperfectly Φ̂ΦΦ, and Q̂with E[ŵŵT ] = Q̂, can be written as:

x̂k|k−1 = Φ̂ΦΦx̂k−1|k−1, (41)

P̂k|k−1 = Φ̂ΦΦkP̂k−1|k−1Φ̂ΦΦT

k + GkQ̂kGTk , (42)

K̂k = P̂k|k−1HTk

(HkP̂k|k−1H

Tk + Rk

)−1, (43)

x̂k|k = x̂k|k−1 + K̂k(zk −Hkx̂k|k−1

), (44)

P̂k|k =(I− K̂kHk

)P̂k|k−1. (45)

Notice that because the filter is imperfectly designed, thecovariance matrix P̂k|k is not guaranteed to bound the actualtrue error present in x̂k|k. In [24] a methodology is proposedto study the sensitivity of the imperfect modeling for acontinuous Kalman filter. In [10] similar expressions can befound for the hybrid Kalman filter. In here we follow the same

approach to derive the true covariance error for the completediscrete Kalman filter.

Let’s consider the error vector e =̂ x̂ − x. Using it andEquation (39), for the prediction step we can write:

ek|k−1 = x̂k|k−1 − xk= Φ̂ΦΦx̂k−1|k−1 −ΦΦΦxk−1 −Gkwk. (46)

Defining ∆ΦΦΦ =̂Φ̂ΦΦ−ΦΦΦ, Equation (46) can be written as:

ek|k−1 = Φ̂ΦΦek−1|k−1 + ∆ΦΦΦxk−1 −Gwk. (47)

In order to propagate this error over time, we can consider thetime propagation of the extended vector xe =

[e x

]Tas:[

ek|k−1xk

]=

[Φ̂ΦΦ ∆ΦΦΦ0 ΦΦΦ

] [ek−1|k−1

xk−1

]+

[−GkwkGkwk

],

(48)

whose associated covariance is:

Pek|k−1 =


]Pek−1|k−1


]T+

[GkQkG

Tk −GkQkGTk

−GkQkGTk GkQkGTk

]. (49)

For the update step we proceed in a similar way:

ek|k = x̂k|k − xk =(I− K̂kHk

)x̂k|k−1 + K̂kzk − xk

=(I− K̂kHk

)ek|k−1 − K̂kνννk, (50)

which leads to the extended update expression:[ek|kxk

]=

[ (I− K̂kHk

)0

0 I

] [ek|k−1

xk

]+

[−K̂kνννk

0

](51)

and whose associated covariance is now:

Pek|k =

[ (I− K̂kHk

)0

0 I

]Pek|k−1

[ (I− K̂kHk

)0

0 I

]T+

[K̂kRkK̂

Tk 0

0 0

]. (52)

The true error covariance of the states of interest, i.e.,P = E[eeT ] can be extracted from the upper block of thecovariance matrix Pe.

Notice also that Equation (52) cannot be written in asimplified fashion as in Equation (45) because the covarianceof the prediction of ek|k−1 is different from the one that is usedto compute the Kalman gain, which is based on the imperfectlydesigned Kalman filter estimator.

Finally, the initial state covariance P̂0 must be also set ac-cording to our design. Assuming an error state implementationof the filter where E[x0] = 0, the initialization of the extendedmatrix we used for the sensitivity analysis is described as:

Pe0 =

[P0 −P0−P0 P

]. (53)

Using Equation (53), Equation (49) and Equation (52) it ispossible to obtain recursively the true error KF covariancematrix over time.

488


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