Analysis and Modeling of Inertial Sensors Using Allan Variance

Abstract

It is well known that inertial navigation systems can provide high-accuracy position, velocity, and attitude information over short time periods. However, their accuracy rapidly degrades with time. The requirements for an accurate estimation of navigation information necessitate the modeling of the sensors' error components. Several variance techniques have been devised for stochastic modeling of the error of inertial sensors. They are basically very similar and primarily differ in that various signal processings, by way of weighting functions, window functions, etc., are incorporated into the analysis algorithms in order to achieve a particular desired result for improving the model characterizations. The simplest is the Allan variance. The Allan variance is a method of representing the root means square (RMS) random-drift error as a function of averaging time. It is simple to compute and relatively simple to interpret and understand. The Allan variance method can be used to determine the characteristics of the underlying random processes that give rise to the data noise. This technique can be used to characterize various types of error terms in the inertial-sensor data by performing certain operations on the entire length of data. In this paper, the Allan variance technique will be used in analyzing and modeling the error of the inertial sensors used in different grades of the inertial measurement units. By performing a simple operation on the entire length of data, a characteristic curve is obtained whose inspection provides a systematic characterization of various random errors contained in the inertial-sensor output data. Being a directly measurable quantity, the Allan variance can provide information on the types and magnitude of the various error terms. This paper covers both the theoretical basis for the Allan variance for modeling the inertial sensors' error terms and its implementation in modeling different grades of inertial sensors.

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140IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 57, NO. 1, JANUARY 2008
Analysis and Modeling of Inertial
Sensors Using Allan Variance
Naser El-Sheimy, Haiying Hou, and Xiaoji Niu
Abstract—It is well known that inertial navigation systems can
provide high-accuracy position, velocity, and attitude information
over short time periods. However, their accuracy rapidly degrades
with time. The requirements for an accurate estimation of navi-
gation information necessitate the modeling of the sensors’ error
components. Several variance techniques have been devised for
stochastic modeling of the error of inertial sensors. They are
basically very similar and primarily differ in that various signal
processings, by way of weighting functions, window functions, etc.,
are incorporated into the analysis algorithms in order to achieve
a particular desired result for improving the model characteriza-
tions. The simplest is the Allan variance. The Allan variance is
a method of representing the root means square (RMS) random-
drift error as a function of averaging time. It is simple to compute
and relatively simple to interpret and understand. The Allan vari-
ance method can be used to determine the characteristics of the
underlying random processes that give rise to the data noise. This
technique can be used to characterize various types of error terms
in the inertial-sensor data by performing certain operations on the
entire length of data. In this paper, the Allan variance technique
will be used in analyzing and modeling the error of the inertial
sensors used in different grades of the inertial measurement units.
By performing a simple operation on the entire length of data,
a characteristic curve is obtained whose inspection provides a
systematic characterization of various random errors contained
in the inertial-sensor output data. Being a directly measurable
quantity, the Allan variance can provide information on the types
and magnitude of the various error terms. This paper covers
both the theoretical basis for the Allan variance for modeling the
inertial sensors’ error terms and its implementation in modeling
different grades of inertial sensors.
Index Terms—Allan variance, error analysis, gyroscopes, iner-
tial navigation, inertial sensors.
I. INTRODUCTION
A
which are then integrated to obtain the vehicle’s position,
velocity, and attitude. The IMU measurements are usually
corrupted by different types of error sources, such as sensor
N INERTIAL measurement unit (IMU) typically outputs
the vehicle’s (e.g., aircraft) acceleration and angular rate,
Manuscript received June 30, 2005; revised September 3, 2007. This work
was supported in part by the Geomatics for Informed Decision Network
Centre of Excellence (GEOIDE NCE) and in part by the Natural Sciences and
Engineering Research Council (NSERC) of Canada.
N. El-Sheimy is with the Mobile Multisensor Research Group, Department
of Geomatics Engineering, University of Calgary, Calgary, AB T2N 1N4,
Canada (e-mail: naser@geomatics.ucalgary.ca).
H. Hou is with Schlumberger Drilling & Measurement, Calgary, AB T2C
4R7, Canada (e-mail: haiying407@gmail.com).
X.NiuiswithShanghaiSiRFTechnologyCo.,Ltd.,Shanghai200000,China
(e-mail: xjniu@sirf.com).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIM.2007.908635
noises, scale factor, and bias variations with temperature (non-
linear, difficult to characterize), etc. By integrating the IMU
measurements in the navigation algorithm, these errors will
be accumulated, leading to a significant drift in the position
and velocity outputs. A standalone IMU by itself is seldom
useful since the inertial-sensor biases and the fixed-step in-
tegration errors will cause the navigation solution to quickly
diverge. Inertial systems design and performance prediction
depends on the accurate knowledge of the sensors’ noise
model. The requirements for an accurate estimation of naviga-
tion information necessitate the modeling of the sensors’ noise
components.
The frequency-domain approach for modeling noise by us-
ing the power spectral density (PSD) to estimate the transfer
functions is straightforward but difficult for nonsystem analysts
to understand. Several time-domain methods have been devised
for stochastic modeling. The correlation-function approach is
the dual of the PSD approach, which is being related as the
Fourier transform pair. This is analogous to expressing the fre-
quency response function in terms of the partial fraction expan-
sion. Another correlation method relates the autocovariance to
the coefficients of a difference equation, which is expressed as
an autoregressive moving-average process. Correlation meth-
ods are very model-sensitive and not well suited when deal-
ing with odd power-law processes, higher order processes, or
wide dynamic range. They work best with a priori knowledge
based on a model of few terms [1]. Yet, several time-domain
methods have been devised. They are basically very similar and
primarily differ in that various signal processings, by way of
weighting functions, window functions, etc., are incorporated
into the analysis algorithms in order to achieve a particular
desired result of improving the model characterizations. The
simplest is the Allan variance.
The Allan variance is a time-domain-analysis technique orig-
inally developed in the mid-1960s to study the frequency stabil-
ity of precision oscillators [2]–[7]. Being a directly measurable
quantity, it can provide information on the types and magnitude
of various noise terms. Because of the close analogies to
inertial sensors, this method has been adapted to random-drift
characterization of a variety of devices [1], [8]–[12].
Put simply, the Allan variance is a method of representing
the root mean square (RMS) random-drift errors as a function
of averaging times. It is simple to compute and relatively simple
to interpret and understand. The Allan variance method can be
used to determine the characteristics of the underlying random
processes that give rise to the data noise. In this paper, this
technique is used to characterize various types of noise terms
in different inertial-sensor data.
0018-9456/$25.00 © 2008 IEEE
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EL-SHEIMY et al.: ANALYSIS AND MODELING OF INERTIAL SENSORS USING ALLAN VARIANCE141
Although the Allan variance statistic remains useful for
revealing broad spectral trends, the Allan variance does not
always determine a unique noise spectrum because the mapping
from the spectrum to the Allan variance is not one-to-one. This
puts a fundamental limitation on what can be learned about a
noise process from the examination of its Allan variance.
In the following, the mathematical definition of the Allan
variance is given, and the relationship between the Allan vari-
ance and the noise PSD is established. Using this relationship,
the behavior of the characteristic curve for a number of promi-
nent noise terms can be determined.
II. METHODOLOGY
In stochastic modeling, there may be no direct access to an
input. A model is hypothesized which, although excited by
white noise, has the same output characteristics as the unit
under test. The phase information is uniquely determined from
the magnitude response. Thus, for a linear time-invariant sys-
tem, by having a knowledge of the output only, and assuming
a white-noise input, it is possible to characterize the unknown
model [14]. Such models are not generally unique; thus, certain
canonical forms are usually used.
A. Power Spectral Density (PSD)
The PSD is the most commonly used representation of the
spectral decomposition of a time series. It is a powerful tool for
analyzing or characterizing data and for stochastic modeling.
ThePSD,orspectrumanalysis,isalsobettersuitedtoanalyzing
periodic or nonperiodic signals than the other methods [1].
The basic relationship for stationary processes between the
two-sided PSD S(ω) and the covariance K(τ)—the Fourier
transform pair—is expressed by
S(ω) =
∞
?
−∞
e−jωτK(τ)dτ.
(1)
The transfer-function form of the stochastic model may be
directly estimated from the PSD of the output data (on the
assumption of an equivalent white-noise driving function).
For linear systems, the output PSD is the product of the
input PSD and the magnitude square of the system transfer
function. If state-space methods are used, the PSD matrices of
the input and output are related to the system-transfer-function
matrix by
Soutput(ω) = H(jω)Sinput(ω)H∗T(jω)
(2)
where
H
H∗T
Soutput
Sinput
Thus, for the special case of the white-noise input, (Sinputis
equal to some constant value, i.e., N2
gives the system transfer function.
system-transfer-function matrix;
complex conjugate transpose of H;
output PSD;
input PSD.
i), the output PSD directly
B. Allan Variance
For the Allan variance, the idea is that one or more white-
noise sources of strength N2
tion(s), resulting in the same statistical and spectral properties
as the actual device.
In this paper, Allan’s definition and results are related to five
basic noise terms and are expressed in a notation appropriate
for inertial-sensor data reduction. The five basic noise terms
are quantization noise, angle random walk, bias instability, rate
random walk, and rate ramp.
AssumethatthereareN consecutive datapoints,eachhaving
a sample time of t0. Forming a group of n consecutive data
points (with n < N/2), each member of the group is a cluster.
Associated with each cluster is a time T, which is equal to nt0.
If the instantaneous output rate of the inertial sensor is Ω(t), the
cluster average is defined as
idrive the canonical transfer func-
Ωk(T) =1
T
tk+T
?
tK
Ω(t)dt
(3)
where Ωk(t) represents the cluster average of the output rate
for a cluster which starts from the kth data point and contains
the n data points. The definition of the subsequent cluster
average is
Ωnext(T) =1
T
tk+1+T
?
tk+1
Ω(t)dt
(4)
where tk+1= tk+ T.
Performing the average operation for each of the two adja-
cent clusters can form the difference
ξk+1,k= Ωnext(T) − Ωk(T).
For each cluster time T, the ensemble of ξs defined by (5)
forms a set of random variables. The quantity of interest is the
variance of ξs over all the clusters of the same size that can be
formed from the entire data.
Thus, the Allan variance of length T is defined as [8]
(5)
σ2(T) =
1
2(N − 2n)
N−2n
?
k=1
?Ωnext(T) − Ωk(T)?2.
(6)
Obviously, for any finite number of data points (N), a finite
number of clusters of a fixed length (T) can be formed. Hence,
(6) represents an estimation of the quantity σ2(T) whose qual-
ity of estimate depends on the number of independent clusters
of a fixed length that can be formed. The Allan variance can
also be defined in terms of the output angle or velocity as
θ(t) =
t ?
Ω(t)dt.
(7)
The lower integration limit is not specified since only the
angle or velocity differences are employed in the definitions.
The angle or velocity measurements are made at discrete times
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142IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 57, NO. 1, JANUARY 2008
given by t = kt0, k = 1,2,3,...,N. Accordingly, the notation
is simplified by writing θk= θ(kt0).
Equations (3) and (4) can then be redefined by
Ωk(T) =θk+n− θk
T
(8)
and
Ωnext(T) =θk+2n− θk+n
T
.
(9)
According to (6), the Allan variance is estimated as follows:
σ2(T)=
1
2T2(N−2n)
N−2n
?
k=1
(θk+2n−2θk+n+θk)2.
(10)
There is a unique relationship that exists between σ2(T) and
the PSD of the intrinsic random processes. This relationship is
σ2(T) = 4
∞
?
0
df · SΩ(f) ·sin4(πfT)
(πfT)2
(11)
where SΩ(f) is the PSD of the random process Ω(T).
In the derivation of (11), it is assumed that the random
process Ω(T) is stationary in time. This assures that the auto-
correlation function of Ω(T) is not dependent on time, and the
autocorrelation function is even, which is a necessary condition
in the derivation of (11). The detailed derivations can be found
in [8] and [17, Sec. 4.2].
Equation (11) states that the Allan variance is propor-
tional to the total power output of the random process when
passed through a filter with the transfer function of the form
sin4(x)/(x)2. This particular transfer function is the result of
the method used to create and operate on the clusters.
Equation(11)isthefocalpointoftheAllan-variancemethod.
This equation will be used to calculate the Allan variance from
the rate-noise PSD. The PSD of any physically meaningful
random process can be substituted in the integral, and an
expression for the Allan variance σ2(T) as a function of cluster
lengthcanbeobtained.Conversely,sinceσ2(T)isameasurable
quantity, a log–log plot of σ(T) versus T provides a direct
indication of the types of random processes, which exist in
the inertial-sensor data. The corresponding Allan variance of
a stochastic process may be uniquely derived from its PSD;
however, there is no general inversion formula because there
is no one-to-one relation [8].
It is evident from (11) and the previous interpretation that
the filter bandwidth depends on T. This suggests that different
types of random processes can be examined by adjusting the
filter bandwidth, namely, by varying T. Thus, the Allan vari-
ance provides a means of identifying and quantifying various
noise terms that exist in the data. It is normally plotted as the
square root of the Allan variance σ(T) versus T on a log–log
plot. To estimate the amplitude of different noise components,
it is convenient to let n = 2j, j = 0,1,2,... [5].
Fig. 1.
σ(T) plot for quantization noise.
C. Representation of Noise Terms in Allan Variance
The following subsections will show the integral solution
for a number of specific noise terms, which are either known
to exist in the inertial sensor or are suspected to influence the
data. The definition is defined in [1] and [11], and the detailed
derivations are given in [8].
1) Quantization Noise: The quantization noise is one of the
errors introduced into an analog signal by encoding it in digital
form. That noise is caused by the small differences between
the actual amplitudes of the points being sampled and the bit
resolution of the analog-to-digital converter [13].
For a gyro output, for example, the angle PSD for such a
process, as given in [14], is
Sθ(f) = TsQ2
z
?sin2(πfT)
(πfT)2
?
≈ TsQ2
z,f <
1
2Ts
(12)
where
Qz
Ts
The theoretical limit for Qzis equal to S/121/2, where S is
the gyro scaling coefficient for the tests with fixed and uniform
sampling times. The gyro rate PSD, on the other hand, is related
to the angle PSD through the following relationship:
quantization-noise coefficient;
sample interval.
SΩ(2πf) = (2πf)2Sθ(2πf)
(13)
and is
SΩ(f) =4Q2
z
Ts
sin2(πfTs) ≈ (2πf)2TsQ2
z,f <
1
2Ts.
(14)
Substituting (14) into (11) and performing the integration
yield
σ2(T) =3Q2
z
T2.
(15)
This indicates that the quantization noise is represented by a
slope of −1 in a log–log plot of σ(T) versus T, as shown in
Fig. 1. Here, the unit of time axis in the figure is the relative
quantity of the sample time t0. The magnitude of this noise can
be read off as the slope of the line at T = 31/2.
It should be noted that there are other noise terms with
different spectral characteristics, such as the flicker angle noise
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EL-SHEIMY et al.: ANALYSIS AND MODELING OF INERTIAL SENSORS USING ALLAN VARIANCE143
Fig. 2.
σ(T) plot for angle (velocity) random walk.
and the white angle noise, which lead to the same Allan-
variance T dependence [1].
2) Angle (Velocity) Random Walk: The high-frequency
noise terms that have correlation time much shorter than the
sample time can contribute to the gyro angle (or accelerometer
velocity) random walk. However, most of these sources can be
eliminated by design [1]. These noise terms are all character-
ized by a white-noise spectrum on the gyro (or accelerometer)
rate output. The associated rate noise PSD is represented by
SΩ(f) = Q2
(16)
where Q is the angle (velocity) random-walk coefficient.
Substituting (16) into (11) and performing the integration
yield
σ2(T) =Q2
T.
(17)
As shown in Fig. 2, (17) indicates that a log–log plot of σ(T)
versus T has a slope of −1/2. Furthermore, the numerical value
ofQcan bedirectlyobtained byreadingtheslopelineatT = 1.
3) Bias Instability: The origin of this noise is the electronics
or other components that are susceptible to random flickering.
Because of its low-frequency nature, it is indicated as the bias
fluctuations in the data [15]. The rate PSD associated with this
noise is
SΩ(f) =



?
0,
B2
2π
?
1
f,f ≤ f0
f > f0
(18)
where
B
f0
Substituting (18) into (11) and performing the integration
yield
bias instability coefficient;
cutoff frequency.
σ2(T)=2B2
π
×
?
ln2−sin3x
2x2(sinx + 4xcosx)+Ci(2x)−Ci(4x)
?
(19)
Fig. 3.
figure is the result of?
σ(T) plot for bias instability (for f0= 1) [4] (the value 0.664 in the
2ln2/π).
Fig. 4.
σ(T) plot for rate random walk.
where
x
Ci
Fig. 3 shows a log–log plot of (19) that shows that the Allan
variance for bias instability reaches a plateau for T much longer
than the inverse cutoff frequency. Thus, the flat region of the
plot can be examined to estimate the limit of the bias instability.
4) Rate Random Walk: This is a random process of un-
certain origin, possibly a limiting case of an exponentially
correlated noise with a very long correlation time. The rate PSD
associated with this noise is
πf0T;
cosine-integral function.
SΩ(f) =
?K
2π
?21
f2
(20)
where K is the rate random-walk coefficient.
Substituting (20) into (11) and performing the integration
yield
σ2(T) =K2T
3
.
(21)
This indicates that the rate random walk is represented by a
slope of +1/2 on a log–log plot of σ(T) versus T, as shown in
Fig. 4. The magnitude of this noise, K, can be read off as the
slope line at T = 3.
5) Drift Rate Ramp: The error terms considered so far are
of random character. It is, however, useful to determine the
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144 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 57, NO. 1, JANUARY 2008
Fig. 5.
σ(T) plot for drift rate ramp.
behavior of σ(T) under systematic (deterministic) errors. One
such error is the drift rate ramp defined as
Ω = Rt
(22)
where R is the drift-rate-ramp coefficient.
By forming and operating on the clusters of data containing
an input given by (22), we obtain
σ2(T) =R2T2
2
.
(23)
This indicates that the drift-rate-ramp noise has a slope of +1
in the log–log plot of σ(T) versus T, as shown in Fig. 5. The
amplitude of drift rate ramp R can be obtained from the slope
line at T = 21/2.
The rate PSD associated with this noise is
SΩ(f) =
R2
(2πf)3.
(24)
It should be noted that there might be a flicker acceleration
noise with 1/f3PSD that leads to the same Allan-variance T
dependence [1].
D. Estimation Quality of Allan Variance
With real data, gradual transitions would exist between the
different Allan standard-deviation slopes. A certain amount of
noiseorhashwouldexistintheplotcurveduetotheuncertainty
of the measured Allan variance [11]. In practice, the estimation
of the Allan variance is based on a finite number of independent
clusters that can be formed from any finite length of data.
The confidence of the estimation improves as the number of
independent clusters is increased.
Definingtheparameterδ asthepercentageerrorinestimating
the Allan standard deviation of the cluster due to the finiteness
of the number of clusters
δ =σ(T,M) − σ(T)
σ(T)
(25)
where σ(T,M) denotes the estimate of the Allan standard
deviation obtained from M independent clusters; σ(T,M) ap-
proaches its theoretical value σ(T) in the limit of M approach-
ing infinity. A lengthy but straightforward [16] calculation
shows that the percentage error is equal to
σ(δ) =
1
?
2?N
n− 1?
(26)
where N is the total number of data points in the entire run, and
n is the number of data points contained in the cluster.
Equation (26) shows that the estimation errors in the region
of short cluster length T are small as the number of independent
clusters in these regions is large (small). On the contrary, the
estimation errors in the region of long cluster length T are
large as the number of independent clusters in these regions is
small. For example, if there are 2000 data points and the cluster
sizes of 500 points are used, the percentage error in estimating
σ(T) is approximately 40%. On the other hand, for the cluster
containing only ten points, the percentage error is about 5%.
III. TEST ENVIRONMENT
Three different-grade IMUs were involved in evaluating the
use of Allan variance in modeling inertial-sensor noise. The
IMUs include the Honeywell CIMU navigation-grade IMU,
the Honeywell HG1700 tactical-grade IMU, and the Systron
Donner MotionPak II-3g MEMS-grade IMU. The test was held
at room temperature for seven days at the Inertial Laboratory,
Mobile Multi-Sensor System Group, Department of Geomatics
Engineering, University of Calgary. Some 4-h static data were
collected in each test, and the most stable 2-h data were
extracted for analysis. The test layout and the equipment used
in this test are shown in Figs. 6 and 7. The following sections
detail the characteristics of the tested IMUs and the data acqui-
sition systems.
A. CIMU—Navigation-Grade IMU
The Commercial Inertial Measurement System (CIMU)
[Fig. 6(b)] is a relevant small (i.e., a cube of 13.4 cm in height
with 19.3-cm length and 16.9-cm width) navigation-grade IMU
manufactured by Honeywell International, Inc. The gyro in the
run bias is about 0.0022◦/h, and the random walk is about
0.0022◦/h1/2. The accelerometer in the run bias is about 25 µg,
and the noise is about 0.00076 m/s/h1/2(0.0025 FPS/h1/2).
B. HG1700—Tactical-Grade IMU
The Honeywell HG1700 [Fig. 6(b)] is a cylinder that is
7.6 cm in height and 9.4 cm in diameter and is a light-
weight and low-cost tactical-grade IMU that utilizes three
GG1308 miniature ring laser gyros along with three Honeywell
RBA-500 resonant beam digital accelerometers to measure the
angular rate and linear acceleration. This IMU has a gyro bias
repeatability of better than 3◦/h, a gyro scale-factor accuracy of
better than 150 ppm, and a gyro random-walk PSD level of less
than 0.15◦/h1/2. The accelerometer residual bias is less than
1000 µg, the scale-factor stability is 300 ppm, and the linearity
is 500 ppm.
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