-
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)
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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
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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
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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
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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).
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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
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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 =
[Φ̂ΦΦ ∆ΦΦΦ0 ΦΦΦ
]Pek−1|k−1
[Φ̂ΦΦ ∆ΦΦΦ0 ΦΦΦ
]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.
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