140 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 57, NO. 1, JANUARY 2008

Analysis and Modeling of InertialSensors Using Allan Variance

Naser El-Sheimy, Haiying Hou, and Xiaoji Niu

AbstractIt is well known that inertial navigation systems canprovide high-accuracy position, velocity, and attitude informationover short time periods. However, their accuracy rapidly degradeswith time. The requirements for an accurate estimation of navi-gation information necessitate the modeling of the sensors errorcomponents. Several variance techniques have been devised forstochastic modeling of the error of inertial sensors. They arebasically very similar and primarily differ in that various signalprocessings, by way of weighting functions, window functions, etc.,are incorporated into the analysis algorithms in order to achievea particular desired result for improving the model characteriza-tions. The simplest is the Allan variance. The Allan variance isa method of representing the root means square (RMS) random-drift error as a function of averaging time. It is simple to computeand relatively simple to interpret and understand. The Allan vari-ance method can be used to determine the characteristics of theunderlying random processes that give rise to the data noise. Thistechnique can be used to characterize various types of error termsin the inertial-sensor data by performing certain operations on theentire length of data. In this paper, the Allan variance techniquewill be used in analyzing and modeling the error of the inertialsensors 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 asystematic characterization of various random errors containedin the inertial-sensor output data. Being a directly measurablequantity, the Allan variance can provide information on the typesand magnitude of the various error terms. This paper coversboth the theoretical basis for the Allan variance for modeling theinertial sensors error terms and its implementation in modelingdifferent grades of inertial sensors.

Index TermsAllan variance, error analysis, gyroscopes, iner-tial navigation, inertial sensors.

I. INTRODUCTION

AN INERTIAL measurement unit (IMU) typically outputsthe vehicles (e.g., aircraft) acceleration and angular rate,which are then integrated to obtain the vehicles position,velocity, and attitude. The IMU measurements are usuallycorrupted by different types of error sources, such as sensor

Manuscript received June 30, 2005; revised September 3, 2007. This workwas supported in part by the Geomatics for Informed Decision NetworkCentre of Excellence (GEOIDE NCE) and in part by the Natural Sciences andEngineering Research Council (NSERC) of Canada.

N. El-Sheimy is with the Mobile Multisensor Research Group, Departmentof 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 T2C4R7, Canada (e-mail: haiying407@gmail.com).

X. Niu is with Shanghai SiRF Technology Co., Ltd., Shanghai 200000, China(e-mail: xjniu@sirf.com).

Color versions of one or more of the figures in this paper are available onlineat 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 IMUmeasurements in the navigation algorithm, these errors willbe accumulated, leading to a significant drift in the positionand velocity outputs. A standalone IMU by itself is seldomuseful since the inertial-sensor biases and the fixed-step in-tegration errors will cause the navigation solution to quicklydiverge. Inertial systems design and performance predictiondepends on the accurate knowledge of the sensors noisemodel. The requirements for an accurate estimation of naviga-tion information necessitate the modeling of the sensors noisecomponents.

The frequency-domain approach for modeling noise by us-ing the power spectral density (PSD) to estimate the transferfunctions is straightforward but difficult for nonsystem analyststo understand. Several time-domain methods have been devisedfor stochastic modeling. The correlation-function approach isthe dual of the PSD approach, which is being related as theFourier 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 tothe coefficients of a difference equation, which is expressed asan 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, orwide dynamic range. They work best with a priori knowledgebased on a model of few terms [1]. Yet, several time-domainmethods have been devised. They are basically very similar andprimarily differ in that various signal processings, by way ofweighting functions, window functions, etc., are incorporatedinto the analysis algorithms in order to achieve a particulardesired result of improving the model characterizations. Thesimplest 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 measurablequantity, it can provide information on the types and magnitudeof various noise terms. Because of the close analogies toinertial sensors, this method has been adapted to random-driftcharacterization of a variety of devices [1], [8][12].

Put simply, the Allan variance is a method of representingthe root mean square (RMS) random-drift errors as a functionof averaging times. It is simple to compute and relatively simpleto interpret and understand. The Allan variance method can beused to determine the characteristics of the underlying randomprocesses that give rise to the data noise. In this paper, thistechnique is used to characterize various types of noise termsin different inertial-sensor data.

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EL-SHEIMY et al.: ANALYSIS AND MODELING OF INERTIAL SENSORS USING ALLAN VARIANCE 141

Although the Allan variance statistic remains useful forrevealing broad spectral trends, the Allan variance does notalways determine a unique noise spectrum because the mappingfrom the spectrum to the Allan variance is not one-to-one. Thisputs a fundamental limitation on what can be learned about anoise process from the examination of its Allan variance.

In the following, the mathematical definition of the Allanvariance 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 aninput. A model is hypothesized which, although excited bywhite noise, has the same output characteristics as the unitunder test. The phase information is uniquely determined fromthe magnitude response. Thus, for a linear time-invariant sys-tem, by having a knowledge of the output only, and assuminga white-noise input, it is possible to characterize the unknownmodel [14]. Such models are not generally unique; thus, certaincanonical 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 foranalyzing or characterizing data and for stochastic modeling.The PSD, or spectrum analysis, is also better suited to analyzingperiodic or nonperiodic signals than the other methods [1].

The basic relationship for stationary processes between thetwo-sided PSD S() and the covariance K()the Fouriertransform pairis expressed by

S() =

ejK()d. (1)

The transfer-function form of the stochastic model may bedirectly estimated from the PSD of the output data (on theassumption of an equivalent white-noise driving function).

For linear systems, the output PSD is the product of theinput PSD and the magnitude square of the system transferfunction. If state-space methods are used, the PSD matrices ofthe input and output are related to the system-transfer-functionmatrix by

Soutput() = H(j)Sinput()HT(j) (2)

whereH system-transfer-function matrix;HT complex conjugate transpose of H;Soutput output PSD;Sinput input PSD.Thus, for the special case of the white-noise input, (Sinput is

equal to some constant value, i.e., N2i ), the output PSD directlygives the system transfer function.

B. Allan Variance

For the Allan variance, the idea is that one or more white-noise sources of strength N2i drive the canonical transfer func-tion(s), resulting in the same statistical and spectral propertiesas the actual device.

In this paper, Allans definition and results are related to fivebasic noise terms and are expressed in a notation appropriatefor inertial-sensor data reduction. The five basic noise termsare quantization noise, angle random walk, bias instability, raterandom walk, and rate ramp.

Assume that there areN consecutive data points, each havinga sample time of t0. Forming a group of n consecutive datapoints (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 (t), thecluster average is defined as

k(T ) =1T

tk+TtK

(t)dt (3)

where k(t) represents the cluster average of the output ratefor a cluster which starts from the kth data point and containsthe n data points. The definition of the subsequent clusteraverage is

next(T ) =1T

tk+1+Ttk+1

(t)dt (4)

where tk+1 = tk + T .Performing the average operation for each of the two adja-

cent clusters can form the difference

k+1,k = next(T ) k(T ). (5)For each cluster time T , the ensemble of s defined by (5)

forms a set of random variables. The quantity of interest is thevariance of s over all the clusters of the same size that can beformed from the entire data.

Thus, the Allan variance of length T is defined as [8]

2(T ) =1

2(N 2n)N2nk=1

[next(T ) k(T )

]2. (6)

Obviously, for any finite number of data points (N), a finitenumber of clusters of a fixed length (T ) can be formed. Hence,(6) represents an estimation of the quantity 2(T ) whose qual-ity of estimate depends on the number of independent clustersof a fixed length that can be formed. The Allan variance canalso be defined in terms of the output angle or velocity as

(t) =

t(t)dt. (7)

The lower integration limit is not specified since only theangle or velocity differences are employed in the definitions.The angle or velocity measurements are made at discrete times

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142 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 57, NO. 1, JANUARY 2008

given by t = kt0, k = 1, 2, 3, . . . , N . Accordingly, the notationis simplified by writing k = (kt0).

Equations (3) and (4) can then be redefined by

k(T ) =k+n k

T(8)

and

next(T ) =k+2n k+n

T. (9)

According to (6), the Allan variance is estimated as follows:

2(T )=1

2T 2(N2n)N2nk=1

(k+2n2k+n+k)2. (10)

There is a unique relationship that exists between 2(T ) andthe PSD of the intrinsic random processes. This relationship is

2(T ) = 4

0

df S(f) sin4(fT )

(fT )2(11)

where S(f) is the PSD of the random process (T ).In the derivation of (11), it is assumed that the random

process (T ) is stationary in time. This assures that the auto-correlation function of (T ) is not dependent on time, and theautocorrelation function is even, which is a necessary conditionin the derivation of (11). The detailed derivations can be foundin [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 whenpassed through a filter with the transfer function of the formsin4(x)/(x)2. This particular transfer function is the result ofthe method used to create and operate on the clusters.

Equation (11) is the focal point of the Allan-variance method.This equation will be used to calculate the Allan variance fromthe rate-noise PSD. The PSD of any physically meaningfulrandom process can be substituted in the integral, and anexpression for the Allan variance 2(T ) as a function of clusterlength can be obtained. Conversely, since 2(T ) is a measurablequantity, a loglog plot of (T ) versus T provides a directindication of the types of random processes, which exist inthe inertial-sensor data. The corresponding Allan variance ofa stochastic process may be uniquely derived from its PSD;however, there is no general inversion formula because thereis no one-to-one relation [8].

It is evident from (11) and the previous interpretation thatthe filter bandwidth depends on T . This suggests that differenttypes of random processes can be examined by adjusting thefilter bandwidth, namely, by varying T . Thus, the Allan vari-ance provides a means of identifying and quantifying variousnoise terms that exist in the data. It is normally plotted as thesquare root of the Allan variance (T ) versus T on a loglogplot. To estimate the amplitude of different noise components,it is convenient to let n = 2j , j = 0, 1, 2, . . . [5].

Fig. 1. (T ) plot for quantization noise.

C. Representation of Noise Terms in Allan VarianceThe following subsections will show the integral solution

for a number of specific noise terms, which are either knownto exist in the inertial sensor or are suspected to influence thedata. The definition is defined in [1] and [11], and the detailedderivations are given in [8].1) Quantization Noise: The quantization noise is one of the

errors introduced into an analog signal by encoding it in digitalform. That noise is caused by the small differences betweenthe actual amplitudes of the points being sampled and the bitresolution of the analog-to-digital converter [13].

For a gyro output, for example, the angle PSD for such aprocess, as given in [14], is

S(f) = TsQ2z

(sin2(fT )(fT )2

) TsQ2z , f