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EFFECT OF CAROUSELING ON ANGULAR RATE SENSOR ERROR PROCESSES (POST-PRINT) 1

Effect of Carouselingon Angular Rate Sensor Error Processes

Jussi Collin,Member, IEEE, Martti Kirkko-Jaakkola,Member, IEEE, and Jarmo Takala,Senior Member, IEEE

Abstract—Carouseling is an efficient method to mitigate themeasurement errors of inertial sensors, particularly MEMSgyroscopes. In this article, the effect of carouseling on the mostsignificant stochastic error processes of a MEMS gyroscope,i.e.,additive bias, white noise,1/f noise, and rate random walk, isinvestigated. Variance propagation equations for these processesunder averaging and carouseling are defined. Furthermore, anovel approach to generating1/f noise is presented. The exper-imental results show that carouseling reduces the contributionsof additive bias, 1/f noise, and rate random walk significantlycompared to plain averaging, which can be utilized to improvethe accuracy of dead reckoning systems.

Index Terms—gyroscopes, microelectromechanical systems,noise generators, stochastic processes,1/f noise

I. I NTRODUCTION

M EMS gyroscope (gyro) technology has developedrapidly during the past years, and MEMS gyros have

now been shown to be able to perform even high-precisiontasks such as gyrocompassing (i.e., North seeking by ob-serving the Earth’s rotation rate) [1]–[3]. The small physicalsize, power consumption, and batch manufacturing cost makeMEMS gyros ideal for a variety of applications in, e.g., landvehicles and mobile devices.

The key to high accuracies is sophisticated compensation ofmeasurement errors which exhibit significantly more variationson MEMS gyros than in the case of, e.g., optical sensors.Common strategies are error model calibration when the truerotation is known (usually zero) [4], [5] and deliberatelyaltering the orientation of the sensitive axis of the gyro inorder to separate measurement errors from the input signal.Intentional slewing of the gyro has two main approaches: thesensor can be either rotated continuously, which is referredto as carouseling, or it can be rotated at specific discreteintervals (often180◦), which is known asindexing[6], [7]. Inthe literature, the termsmaytaggingandtwo-point carouselingare also used for indexing. Turntables are commonly usedfor offline calibration of inertial measurement units (IMUs),e.g., [8], but in this article, we study IMU rotations that areapplied during the actual measurement.

J. Collin and J. Takala are with Department of Pervasive Computing,Tampere University of Technology, Finland. E-mail: jussi.collin@tut.fi

M. Kirkko-Jaakkola is with Department of Navigation and Positioning,Finnish Geodetic Institute, Finland.

c© 2014 IEEE. This is a post-print version of a paper publishedin the IEEE Transactions on Instrumentation and Measurement ( DOI10.1109/TIM.2014.2335921 ). Personal use of this materialis permitted.Permission from IEEE must be obtained for all other users, includingreprinting/ republishing this material for advertising orpromotional purposes,creating new collective works for resale or redistributionto servers or lists,or reuse of any copyrighted components of this work in other works.

Rotating IMUs have been studies already since the1960s [9], and much research has been focused on high-quality sensors. However, MEMS gyros whose measurementerrors are less stable than those of, e.g., ring laser gyros,canbenefit even more from IMU slewing because the fluctuatingerror processes are significantly more difficult to estimate. Sig-nificant improvements in pedestrian dead reckoning obtainedusing a foot-mounted rotating IMU have been reported [10],although the test setup was too cumbersome for real-life use.However, a dedicated rotating system is not always necessary,e.g., if the IMU is mounted at the wheel of a land vehicle [11].

The studies of carouseling and indexing errors have beenmostly investigating common inertial navigation error statessuch as biases and scale factors [12]–[15]. In contrast, thisarticle studies the stochastic error components in the output ofa MEMS gyro and analyzes their variance propagation undercarouseling in comparison with non-carouseled averaging.Inaddition to MEMS sensors, the results are relevant for othertypes of rate sensors where non-stationary noise processescause significant errors after integration over time.

Among the possible error processes in the output of aMEMS gyro is1/f (flicker) noise whose name originates fromits power spectral density.1/f noise is a long-memory processand is nontrivial to synthesize [16]–[21]. In this article,weboth use the fractional integral model of1/f noise [22] andpropose a novel approach to generating noise with constantAllan variance at certain averaging times. Synthetic1/f noisecan be used to simulate not only MEMS gyros but also, e.g.,transient circuits [23] or other phenomena where1/f noise isencountered [24], [25].

This article is organized as follows. The stochastic processesused for modeling the most important error processes in theoutput of MEMS gyros are defined in Section II, followedby carouseling analysis in Section III. A novel approach ofsynthesizing noise with constant Allan variance is presentedin Section IV, and experimental validation is carried out inSection V. Finally, Section VI concludes the article.

II. ERROR PROCESSMODELS

In this section, the models used for different error compo-nents in the output of angular rate sensor are described. Theseprocess models are chosen based on [26]. Similar modelshave been employed, e.g., in [27] except for that1/f noisewas not considered in [27]. Errors that are dependent on theinput signal magnitude (scale factor errors) are neglectedinthe following discussion.

2 EFFECT OF CAROUSELING ON ANGULAR RATE SENSOR ERROR PROCESSES (POST-PRINT)

A. Additive Bias

The additive biasb is modeled as a random constant;therefore,

bt = bt−1. (1)

It would be possible to embed the bias in other error processes,but in this article, other error processes are treated as zero-mean.

B. White Noise

White noise, often calledangular random walkin the con-text of gyroscopes, is a random process where the samples aremutually independent. It is assumed that the process has zeromean and a constant varianceσ2

WN. Many factors contributewhite noise to the sensor output. For instance, quantizationnoise is white, and so is thermal noise.

C. Rate Random Walk

Rate random walk (RRW) is the sum of independent andidentically distributed, zero-mean increments, modeled as

rt = rt−1 + qt (2)

whereq1, q2, . . . constitute a white noise process with vari-anceσ2

q . It can be seen that RRW is a Markov process, i.e.,memoryless: the valuert does not depend on other previousor future realizations of the process thanrt−1. An importantsource of RRW are changes in the temperature of the sensor.Other than that, RRW is caused by, e.g., aging of the sensorelement.

D. 1/f Noise

In this article,1/f noise is modeled as a fractional integralof a white noise sequence. Assuming a degree of integra-tion 0 < d < 1, 1/f noise is modeled in discrete time as [28]

ft =

t∑

i=1

Γ (t− i+ d)

Γ (t− i+ 1)Γ(d)wi (3)

where w1, w2, . . . are white noise with varianceσ2w and

Γ denotes the gamma function. As opposed to RRW,1/fnoise is a long-memory process [29]. Although encounteredin various contexts, its physical origin is unknown [25].

III. G YROSCOPECAROUSELING

In this article, the concept of carouseling is defined asfollows; a schematic is shown in Fig. 1. Consider two gyroswith sensitive axesx and y aligned perpendicular to eachother. In carouseling, these sensors are intentionally rotatedon the plane defined by the sensitive axes, and the measure-mentsωx and ωy—deteriorated by biases, noise, and otherimperfections—are used to estimate the true angular rateωabout a fixed “virtual” axisφ = 0 on the plane of the sensitiveaxes. The outputs of the gyros are combinations of the angularrate of interestω and the angular rate about the axisφ = 90◦,denoted here byω⊥:

ωx = −ω sinφ+ ω⊥ cosφ+ ǫx (4a)

ωy = ω cosφ+ ω⊥ sinφ+ ǫy (4b)

Fig. 1. Schematic of gyroscope carouseling

where ǫx and ǫy denote additive sensor measurement errorsas a superposition of the error processes defined in Section II.The instantaneous angular rate at the orientation of interest isthen estimated as

ω = −ωx sinφ+ ωy cosφ = ω + ǫ; (5)

the angular rateω⊥ about the axis perpendicular to the axisof interest could be computed in a similar manner, but, in thisarticle, we focus on the analysis of a single axis of interest.

Since the sensors can only be sampled at discrete intervals,only discrete values of the carouseling angleφ need tobe considered. In the analysis presented in this article, weassume a uniform carouseling rate with periodT secondsand a uniform sensor sampling frequencyN/T hertz for anintegerN . We will focus on the average angular rate duringthe tth carouseling revolution which is estimated as

ωt =1

N

N∑

i=1

− ωx

((t− 1)T + i

T

N

)sin

2πi

N

+ ωy

((t− 1)T + i

T

N

)cos

2πi

N.

(6)

In this section, the effect of carouseling on the measurementerrors present inωx andωy is studied in terms of the stochasticprocesses defined in Section II. Comparison is made to a singlegyroscope that is not carouseled, i.e., whose measurementsaredirectly averaged over intervals ofT seconds.

A. Additive Bias

For simplicity, consider only the behavior of one of the two(physical) gyros during a carouseling revolution. In continuoustime, it is obvious that carouseling cancels the bias because

∫ 2π

0

b sinφ dφ = 0 (7)

and the same applies to the other gyro with cosine coefficients.Fortunately, this is the case in discrete time as well under theassumption that the sampling rate is uniform and an integermultiple of the carouseling frequency. Consider the sum

1

N

N∑

i=1

b sin2πi

N=

b

N

N∑

i=1

sin2πi

N. (8)

Using Euler’s formula, the sum can be interpreted as theimaginary part (or the real part in the case of cosine terms) of

3

the sum ofN roots of unity which is well known to be zerofor all N > 1. In contrast, it is clear that direct averaging hasno influence on the constant additive bias.

B. White Noise

Averaging uncorrelated noise obviously decreases its vari-ance. Computing the variance of the carouseled angular esti-mate yields

varωt =1

N2

N∑

i=1

σ2WN sin2

2πi

N+ σ2

WN cos22πi

N

=σ2WN

N

(9)

which is equal to the variance of the directly averaged whitenoise sequence. Therefore, carouseling does not have advan-tages over direct averaging in terms of white noise.

C. Rate Random Walk

In order to analyze the joint distribution of two consecutivecarouseling revolutions, define theN × N lower triangularcumulative sum matrix

R =

1 0 · · ·

1 1. . .

......

. . .

, (10)

and partition the corresponding2N × 2N cumulative summatrix as

R2 =

[R O

I R

](11)

whereO and I denoteN × N matrices of zeros and ones,respectively. Also define the integrator vector

1 =1

N

[1 1 · · ·

]T∈ R

N , (12)

and the carouseling coefficient vectors

s =1

N

[sin 2π 1

Nsin 2π 2

N· · · sin 2π

]T

c =1

N

[cos 2π 1

Ncos 2π 2

N· · · cos 2π

]T.

(13)

Now, given a white noise vectorq ∈ R2N , two consecutivedirectN -averages of a RRW sequence would be obtained as

[1T 0T

0T 1T

]R2q (14)

where0 is a N × 1 vector of zeros. As the covariance ofqis, by definition,σ2

q I whereI denotes the identity matrix, thecovariance matrix of (14) is computed as

[1T 0T

0T 1T

]R2 σ

2q I R2

T

[1 0

0 1

]

= σ2q

[1TRRT1 1TRIT11T

IRT1 1TRRT1+ 1TII

T1

]

= σ2q

[1N2

∑Ni=1 i

2 1N2

∑Ni=1 iN

1N2

∑N

i=1 iN1N2

∑N

i=1 i2 + N3

N2

]

= σ2q

[2N3+3N2+N

6N2

N+12

N+12

2N3+3N2+N6N2 +N

].

(15)

It can be seen that the two averages are correlated and that thevariance of the second average is approximately proportionalto 4N/3. Every subsequent average will have variance higherby σ2

qN than the previous one, which is intuitively understoodbecause of the process model (2) and can be seen by repeatingthe calculations forR3 etc. A rigorous proof by induction is,however, not given here.

Analogously, the carouseled averages would be computedas

−

[sT 0T

0T sT

]R2qx +

[cT 0T

0T cT

]R2qy (16)

where both gyros have their respective realizations of RRWdriving noise. We will assume that the RRW incrementsqtof the two gyros are statistically independent and identicallydistributed (i.i.d.). In principle, the increments are correlated atleast for the component caused by changes in the ambient tem-perature, but as long as the carouseling periodT is reasonablyshort (e.g., in the order of1 second), temperature fluctuationsduring one carouseling revolution can be neglected in mostapplications. If the sensors are of identical make and modeland originate from the same production batch, the assumptionof identical distributions can be considered reasonable.

Keeping in mind that the elementwise sums of the vectorss

and c equal 0 as discussed in Section III-A, i.e.,1T s =1T c = 0, the covariance of the sine coefficient term in (16)is expressed as

σ2q

[sTRRT s sTRIT ssT IRT s sTRRT s+ sT IIT s

]

= σ2q

[sTRRT s 0

0 sTRRT s

] (17)

which implies that the consecutive carouseled averages areuncorrelated and have equal variances. To compute the valuesof these variances, let us interpretsTRRT s as a numericalintegration according to the rectangle rule:

sTRRT s =N1

N

N∑

i=1

1

N

N∑

j=i

sin2πk

N

2

≈N

∫ 1

0

(∫ 1

x

sin 2πy dy

)2

dx

=N

2π

∫ 1

0

(cos 2πx− 1)2dx =

3N

8π2.

(18)

Similar computations for the cosine term yield an asymptoticproportionality coefficient ofN/(8π2), summing up to a totalvariance ofσ2

qN/(2π2). This is already96 % smaller thanthe asymptotic coefficient4/3 obtained for direct averagingin (15), and by repeating the calculations forR3, . . . onecan see that subsequent carouseled averages have the samevariance as opposed to direct averaging where the variancesincrease linearly. A rigorous proof is again omitted, butthe phenomenon can be intuitively understood based on theMarkov property of RRW and the result obtained in Sec-tion III-A.

Note that the carouseling periodT does not appear explicitlyin (15) and (18). However, as long as the sensor sampling rateis constant, the carouseling periodT is proportional to the

4 EFFECT OF CAROUSELING ON ANGULAR RATE SENSOR ERROR PROCESSES (POST-PRINT)

Fig. 2. Effect of carouseling on1/f noise

number of carouseling pointsN , and on the other hand, thevarianceσ2

q of the RRW driving noise depends on the samplingrate.

D. 1/f Noise

Analogously to the analysis presented for RRW in Sec-tion III-C, define the matrixF ∈ RN×N which produces a1/f noise sequence by multiplying a white noise sequencew.This matrix is unit lower triangular with subdiagonal entriescomputed according to (3); in fact, it is easy to see thatF is also a Toeplitz matrix. Consequently, the productF1effectively computes the cumulative sum of the first columnof F, andFT1 contains the same values in the reverse order.To analyze the convergence of this sum, let us conduct a limitcomparison test and compute the limit of the ratio ofid−1 andthe ith entry of the first row ofFT as i tends to infinity:

limi→∞

Γ (i+ d)

Γ (i+ 1)Γ(d)

/id−1 = lim

i→∞

Γ (i+ d) i1−d

iΓ(i)Γ(d)

= limi→∞

Γ (i+ d)

idΓ(i)Γ(d)

=1

Γ(d)> 0 ∀ d > 0.

(19)

Since the series∑∞

i=0 id−1 is well known to diverge for alld ≥

0, the limit comparison test concludes that the elements in theproductF1 also tend to infinity with increasing timet andpositived.

As opposed to the strictly positive direct averaging vec-tor 1, the carouseling coefficient vectorss andc contain bothpositive and negative entries. In fact, ifN happens to beeven, these vectors containN/2 pairs of opposite numbers.Then, the cumulative carouseled sumFT s or FT c can beinterpreted, as the dimension of the matrixF increases, as thesum ofN/2 different alternating series. The absolute valuesof the terms in these series are decreasing and, therefore,these alternating series converge according to the Leibnizcriterion. Fig. 2 illustrates the behavior of carouseled anddirectly averaged1/f noise withd = 1/2 andN = 300. As

the values inFT1 grow larger than those inFT s andFT c, itcan be expected that the variances computed using the squaredform σ2

w1TFFT1 also increase significantly faster than their

carouseled counterparts.

IV. CONSTANT ALLAN VARIANCE

A popular tool for analyzing gyroscope measurement errorsis the Allan variance. It is especially suitable for the analysis ofindexing because both of them operate based on the differencesof consecutive sample averages. The Allan varianceσ2

A(τ) isa function of averaging timeτ , computed as

σ2A(τ) =

1

2 (M − 1)

M−1∑

j=1

(y(τ)j+1 − y(τ)j)2 (20)

where the values ofy(τ)j and M are obtained by dividingthe datay into disjoint bins of lengthτ , y(τ)j is the averagevalue of thejth bin, andM is the total number of bins [4],[30]. When defined this way,σ2

A(τ) is a statistic, function ofa gyro noise sampley. If y consists of pure1/f noise, itcan be expected thatσ2

A(τ) is independent ofτ [25], [31],[32]. In this section, we introduce a discrete sequence thathas this property; the term ‘sequence’ is used instead of‘stochastic process’ because the data are generated in a non-causal procedure and the variance of the individual randomvariables in the sequence is a function of the length of thesequence.

Algorithm 1 Generating the deterministic sequenceS2n

Input: 1 < n ∈ N

Output: v = S2n

1: v =[− 1

212

]

2: for i = 2, . . . , n do3: v = v ⊗ [1 1] + a1...2i

4: end for

Algorithm 1 describes a procedure to generate a se-quenceS2n of length2n for which

∣∣S2n(τ)j+1 − S2n(τ)j∣∣ = 1, ∀τ = 1, 2, 4, . . . , 2n−1; (21)

the progress of the algorithm is tabulated in Table I. Startingwith the sequenceS2 =

[− 1

2 ,12

], use the Kronecker product⊗

to duplicate the elements ofS2 to obtain[− 1

2 ,−12 ,

12 ,

12

].

Clearly, (21) now holds forτ = 2; in order to make it validfor τ = 1 as well, add the sequencea1...2i to the Kroneckerproduct where

ak =

− 12 if k = 1

+ 12 if k = 2

−ak−2 otherwise.

(22)

It is easy to see that the elements ofa1...2i repeat with aperiod of four. The resulting sequence isS4 = [−1, 0, 1, 0].As an example, the sequenceS2048 is plotted in Fig. 3. Thesequence is quantized at distinct values because it is generatedas a superposition oflog2 2048 = 11 square waves. Accordingto [29], a logarithmic amount of state variables is sufficient tocharacterize a1/f process.

5

Fig. 3. The sequenceS2048 as generated using Algorithm 1

TABLE IGENERATING THE DETERMINISTIC SEQUENCES8

S2 − 1

2

1

2

⊗[1 1] − 1

2− 1

2

1

2

1

2

a1...4 − 1

2+ 1

2+ 1

2− 1

2

S4 −1 0 1 0

⊗[1 1] −1 −1 0 0 1 1 0 0a1...8 − 1

2+ 1

2+ 1

2− 1

2− 1

2+ 1

2+ 1

2− 1

2

S8 − 3

2− 1

2

1

2− 1

2

1

2

3

2

1

2− 1

2

Similarly, to obtain a stochastic sequenceR, replace thedeterministic value1 in (21) by the random variablexτ withzero mean and varianceC2:

R(τ)j+1 − R(τ)j = xτ , ∀τ = 1, 2, 4, . . . , 2n−1. (23)

To obtain such a sequence, drawxi with zero mean and unitvariance and adda1...2ixi instead ofa1...2i to the Kroneckerproduct at line 3 of Algorithm 1. This procedure is tabulatedin Table II. The sequenceR2n can be expressed as a matrix–vector productKx whereK ∈ R2n×n is a constant matrixandx ∈ R2n is an i.i.d. random vector; theith column of thematrix K is computed as the Kronecker product ofa1...2i anda 2n−i × 1 vector of ones. Forn = 2,

K =

−0.5 −0.5−0.5 0.50.5 0.50.5 −0.5

(24)

and then

R4 = K[x1 x2]T . (25)

Under the i.i.d. and unit variance assumptions we have

covx =

[1 00 1

](26)

TABLE IIGENERATING THE STOCHASTIC SEQUENCER4

R2 − 1

2x1

1

2x1

− 1

2x1 − 1

2x1

1

2x1

1

2x1

− 1

2x2 + 1

2x2 + 1

2x2 − 1

2x2

R4 − 1

2x1 − 1

2x2 − 1

2x1 + 1

2x2

1

2x1 + 1

2x2

1

2x1 − 1

2x2

and thus

covR4 = KKT =

0.5 0 −0.5 00 0.5 0 −0.5

−0.5 0 0.5 00 −0.5 0 0.5

. (27)

To obtain the constantC2 = varxτ in (23), express the databin average differences in (20) in the matrix–vector productform Ay as

y(τ)j+1 − y(τ)j =

[−1

τ. . . −

1

τ

1

τ. . .

1

τ

]

y(j−1)τ+1

...yjτ

yjτ+1

...y(j+1)τ

(28)to obtain

A =

−1 1 0 00 −1 1 00 0 −1 1

(29)

and then

covAy = covAKx = AKKTAT =

1 0 −10 1 0

−1 0 1

, (30)

showing that, indeed, the varianceC2 = 1. If this varianceis unknown, it can be shown that (20) without the term12yields an unbiased estimate ofC2, i.e., E

[2σ2

A(τ)]= C2.

However, it is not the minimum-variance unbiased estimator(MVUE), which can be found by using the Moore–Penrosepseudoinverse ofK,

K+ =

[−0.5 −0.5 0.5 0.5−0.5 0.5 0.5 −0.5

]. (31)

Now, K+R4 = [x1 x2]T , and the well known sample variance

of this is the MVUE. Having a theoretical mean value for aprocess with constant Allan variance can help in extending thestatistical models discussed in [27] to1/f -type processes. Theconstant variance property was derived with nonoverlappingAllan variance and does not hold exactly for overlapping Allanvariance estimators [33], [34].

Interestingly,K16×4+0.5 is equal to the standard Gray coderepresentation in matrix form [35, Table 1]. Thus, as

Sm = K[1 1 · · ·

]T, (32)

the sequenceS + n/2 also depicts the number of ones inthe Gray code representation, and bounds for the sequencecan be obtained from number theory [36]. Other interesting

6 EFFECT OF CAROUSELING ON ANGULAR RATE SENSOR ERROR PROCESSES (POST-PRINT)

properties can be found as well: for example, the columns ofthe covariance matrix of the sequence

(KKT

)also follows

the rule defined by (21). Furthermore, all diagonal elementsof the productKKT contain the constant valuen/4, there-fore, the process obtained this way is variance-stationary,unlike the discrete1/f process described in [4]. Provingthe above hypotheses rigorously is left for future work, butcomputer simulations have shown them to hold for at leastR2, . . . , R32768.

The family of noise sequences presented in this section canprovide an alternative view to1/f noise as their Allan varianceis exactly constant at certain averaging times and because thesequences are stationary for a given length. The sequenceshave interesting properties and could be useful in the errorpropagation analysis of MEMS gyro indexing.

V. SIMULATIONS AND EXPERIMENTAL RESULTS

In this section, the validity of the calculations presentedinSection III is shown by computer simulations and confirmedby experimental results. We first analyze RRW and1/f noiseusing simulated data, after which the models are applied toauthentic data measured by a MEMS gyro.

A. Rate Random Walk Simulation

The decrease in RRW caused by carouseling was evaluatedby first generating2× 1000 mutually independent RRW real-izations according to (2) with driving noise varianceσ2

q = 1.Then, both the direct averaging and carouseling operationswere applied withN = 200, resulting in 1000 simulationcases. The variances of the resulting sequences are plottedasfunctions of time (i.e., average block or carouseling revolutionnumber, referred to as data bins) in Fig. 4 along with thevariances predicted by (15) and (17), also including the contri-bution of the cosine term of (16) in the latter. It can be seen thatthe predictions match the simulation realizations quite well.The prediction of the averaged variances was computed byneglecting the lower order terms in (15), but this inaccuracybecomes insignificant quickly when the index of the data binincreases.

B. 1/f Noise Simulation

A simulation of the evolution of the variance of1/f noiseequivalent to that presented in Section III-C for RRW isshown in Fig. 5. The1/f noise sequences were generatedaccording to (3) withσ2

w = 1 and d = 1/2, and theaverages and carouseling were computed usingN = 200.Since they-axis scale is linear in this figure, as opposedto Fig. 4, it can be seen that the variance of averaged1/fnoise increases slowly in comparison with RRW. The rate ofincrease seems to be logarithmic, which would be natural whenconsidering the relation established in (19). In contrast,thecarouseled1/f noise exhibits no visible increasing trend inFig. 5. Furthermore, its variance is significantly smaller thanthe driving noise varianceσ2

w = 1 and a visual inspectionshows that the variance is also smaller than that of averaged1/f noise.

Fig. 4. Effect of carouseling on the variance of rate random walk in thecomputer simulations

Fig. 5. Effect of carouseling on the variance of1/f noise in the computersimulations

C. Real Gyro Data Test

Test data were recorded for one hour at a sampling rateof 100 Hz using a three-axis MEMS gyro [37]; the Allanvariances of thex- andy-gyro data computed according to (20)are plotted in Fig. 6. During the entire test, the true angularrate to be measured by the sensors was zero. It is interestingto notice that thex-gyro exhibited larger variations than they-gyro, but the cause of this discrepancy was not investigatedfurther. The main error parameters for the two gyros areestimated in Table III. Since thex-gyro does not exhibit a clearascending RRW slope, its RRW was estimated at the samevalue ofτ as for they-gyro. Since the variance of1/f noisewas observed to increase very slowly even in the averagedcase in Section V-B,1/f noise is neglected in this analysis.The bias instability values are only mentioned for reference inTable III.

7

Fig. 6. Allan variances of the two gyros used for carouseling

TABLE IIIGYRO ERROR PARAMETERS AS ESTIMATED FROMFIG. 6 [31]

Parameter Unit Value

x-gyro y-gyro

σ2

WN(τ = 1 s) (rad/s)2 3 · 10−7 1 · 10−7

σ2q

(rad/s)2 /s 3 · 10−10 2 · 10−10

Bias instability rad/s 1 · 10−5 6 · 10−5

Fig. 7 shows the directly averaged and carouseled datawith N = 200 samples (consequently,T = 2 s) and therespective predicted confidence intervals which are estimatedas the sum of white noise and RRW variances, i.e., excludingthe contribution of 1/f noise. Note that since (9) reliesheavily on the assumption of equal white noise variances,the average of the white noise variances of the two gyroswas used in the computations. Furthermore, it can be seenthat the gyros exhibit a significant initial bias drift, probablydue to sensor warm-up. However, this phenomenon is notvisible in the carouseled estimate. Errors due to change inambient temperature or due to aging of the sensor [38] aredifficult to model, and carouseling clearly makes these errorsless effective in the output solution. The initial transient periodwas excluded from the computation of Allan variance in Fig. 6and the confidence interval estimation in Fig. 7 except forthe carouseled case.6.5 % of the carouseled angular ratedata points exceed the2σ confidence interval which shouldcorrespond to95 % of the samples. Considering that theconfidence interval was estimated neglecting the contributionof 1/f noise, the result can be regarded as satisfactory.

The effect of carouseling on the resulting angle estimatesobtained by integrating the gyro data is illustrated in Fig.8.The increase in accuracy due to carouseling is dramatic: whilethe directly averaged gyro measurements lead to angle errorsexceeding600◦ after one hour, the carouseled angle errorsaccumulate to only approximately1.5◦ in 60 minutes. Theerrors shown in Fig. 8 include the initial warm-up phase wherethe gyro biases were not yet stabilized.

(a)

(b)

(c)

Fig. 7. Averaged and carouseled gyro data with zero input (solid gray line)and their estimated2σ confidence intervals (dashed black line) withN = 200:(a) averagedx-gyro; (b) averagedy-gyro; (c) carouseled estimate

8 EFFECT OF CAROUSELING ON ANGULAR RATE SENSOR ERROR PROCESSES (POST-PRINT)

(a)

(b)

Fig. 8. Accumulated angle estimation errors when integrating the data shownin Fig. 7 withN = 200, T = 2 s: (a) averagedx- andy-gyros; (b) carouseledestimate

D. Field Test

To show practical application of carouseling we performed atest run with passenger car with an inertial measurement unitsattached to the right-hand side rear (non-steerable) wheel. Thecarouseling axis of interest was chosen to be the vertical axis,i.e., heading gyros were considered. The test vehicle, shownin Fig. 9, included the following measurement units:

• dual-frequency GPS receiver [39] and reference ring-lasergyro unit [40] with sub-degree heading accuracy for thetest period,

• MEMS inertial measurement unit with 3D accelerome-ter [41] and 3D gyroscope [37] (identical to the gyro usedin the static test in Section V-C) fixed to the right-handside rear wheel, and

• identical MEMS inertial measurement unit fixed to thecenter console of the vehicle.

The carouseling was performed by filtering the wheel-basedaccelerometer data to estimateφ [11]. The vehicle was drivenat a slow speed10–12 km/h in a parking lot for12 minutes.The test route is shown in Fig. 10 and samples of rawmeasurements are shown in Fig. 11. Since the test area wasa public parking lot, a constant velocity was impossible tomaintain. Therefore, the number of sampling pointsN wasnot constant during the test but varied from revolution torevolution; the values are plotted in Fig. 12. The stationarysections at the beginning and the end of the test are not seenin Fig. 12, and test vehicle once had to slow down in order

Test vehicle.epsFig. 9. Test vehicle used in field test

Fig. 10. Test route

to give way to another vehicle, causing a peak in the valueof N .

The resulting angular rate estimation errors, as referencedto the ring-laser gyro unit, are plotted in Fig. 13 along withthe predicted confidence intervals. It can be seen that theintervals computed with the error variances correspondingto stationary data do not match the field test errors. Thereare many possible reasons for the discrepancy. For instance,MEMS sensors are sensitive to accelerations and vibration;theinstantaneous carouseling angleφ was not precisely knownand the carouseling rate was not exactly constant during eachrevolution; and scale factor and cross-axis sensitivity errorswere not compensated for. Nevertheless, the results suggestthat the carouseling error equations derived in Section IIIarecorrect but the noise variances were not appropriate for thedata: there is no drifting trend visible in Fig. 13a. Note thatthe data shown in Fig. 13 has a lot of local peaks; these aredue to a bug in the software that writes the gyro measurementsto a memory card.

Errors in the heading estimates obtained by integrating theangular rates are shown in Fig. 14. A constant additive biaswas compensated for in the cabin gyro data; the value of thisbias was determined based on the first30 seconds of dataduring which the test vehicle remained stationary. Withoutsuch a correction, the cabin gyro heading error increasesto 240◦ during the test. Although no bias compensationwas applied to the wheel-mounted gyro data, the carouseledheading estimates are still more accurate than the cabin-fixedestimates because of the mitigation of rate random walk in thecarouseling process. Obviously, the errors encountered inthistest are larger than those seen in Section V-C for the reasonsdiscussed above.

VI. CONCLUSIONS

In this article, the effect of carouseling on various errorprocesses in the output of a MEMS gyro was studied. Itwas shown that in addition to canceling constant biases,carouseling reduces the contributions of rate random walkand 1/f noise but does not mitigate white noise better thanplain averaging. An immense performance improvement was

9

(a)

(b)

Fig. 11. Raw data: (a) measured at wheel; (b) measured insidethe cabin

Fig. 12. Number of carouseling pointsN during the field test

(a)

(b)

Fig. 13. Angular rate estimation error (solid gray line) with 2σ (dashed) and10σ (dotted) intervals predicted based on Table III: (a) carouseled at wheel;(b) averaged inside the cabin

Fig. 14. Heading estimation errors with bias compensation for the cabingyro

observed in the case where the gyro outputs are integrated for,e.g., navigation purposes.

As a side product, an alternative approach of synthesizing1/f noise was proposed. The proposed method generatesvariance-stationary sequences which are not ideal for ana-lyzing the long-time correlation properties of1/f noise, butcould be useful, e.g., in the analysis of gyro indexing systems.Investigating the applicability of the noise produced by themethod is a topic of future studies.

The variance propagation equations for carouseling werederived under the assumptions of negligible scale factor errors,uniform sensor sampling and carouseling rate, and precise

10 EFFECT OF CAROUSELING ON ANGULAR RATE SENSOR ERROR PROCESSES (POST-PRINT)

knowledge of the instantaneous carouseling angle. In real-lifeapplications, particularly the last two of these assumptions donot necessarily hold perfectly, as was seen in the field test.Quantifying the sensitivity of the derived covariance predictionformulas to variable slewing rates and multi-axis carouseling,such as the patterns studied in [1], is left as future work.

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[2] L. I. Iozan, M. Kirkko-Jaakkola, J. Collin, J. Takala, and C. Rusu, “Usinga MEMS gyroscope to measure the Earth’s rotation for gyrocompassingapplications,”Measurement Science and Technology, vol. 23, no. 2, Feb.2012.

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[5] A. Noureldin, T. B. Karamat, M. D. Eberts, and A. El-Shafie, “Perfor-mance enhancement of MEMS-based INS/GPS integration for low-costnavigation applications,”IEEE Transactions on Vehicular Technology,vol. 58, no. 3, pp. 1077–1096, 2009.

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[11] J. Collin, “Vehicle positioning,” PCT Patent application FI2013/050 357,Apr. 2, 2013. [Online]. Available: http://patentscope.wipo.int/search/en/WO2013150183

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Jussi Collin (M’11) received the M.Sc. and Dr.Tech.degrees from the Tampere University of Technology,Tampere, Finland, in 2001 and 2006, respectively,specializing in sensor-aided personal navigation. Heis currently a Senior Research Fellow with theDepartment of Pervasive Computing, Tampere Uni-versity of Technology. His research interests includestatistical signal processing and novel sensor-basednavigation applications. Dr. Collin is a Vice Chairof the IEEE Finland Section Signal Processing andCircuits & Systems Chapter.

11

Martti Kirkko-Jaakkola (S’12-M’14)received his M.Sc. and D.Sc. (Tech.) degrees fromTampere University of Technology (TUT), Finland,in 2008 and 2013, respectively. From 2006 to 2013he worked with the Department of Pervasive Com-puting, TUT. Currently, he is a senior research scien-tist at the Finnish Geodetic Institute, Kirkkonummi,Finland, where his research interests include precisesatellite positioning, low-cost MEMS sensors, andindoor positioning.

Jarmo Takala (S’97M’99SM’02) received theM.Sc. (Hon.) degree in electrical engineering and theDr.Tech. degree in information technology from theTampere University of Technology, Tampere, Fin-land (TUT) in 1987 and 1999, respectively. He wasa Research Scientist with VTTAutomation, Tampere,from 1992 to 1995. From 1995 to 1996, he was a Se-nior Research Engineer with Nokia Research Center,Tampere. From 1996 to 1999, he was a Researcherat TUT. Currently, he is a Professor in computerengineering at TUT and the Dean of the Faculty of

Computing and Electrical Engineering. His current research interests includecircuit techniques, parallel architectures, and design methodologies for digitalsignal processing systems. Prof. Takala is Co-Editor-in-Chief of Journalof Signal Processing Systems and he was Associate Editor of the IEEETransactions on Signal Processing in 2007 - 2011. He was the chair of IEEESignal Processing Society’s Design and Implementation of Signal ProcessingSystems technical Committee in 2011-2013.

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