Thermal Calibration for the Accelerometer Triad Based on the Sequential Multiposition Observation

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 62, NO. 2, FEBRUARY 2013 467

Thermal Calibration for the Accelerometer TriadBased on the Sequential Multiposition Observation

Jie Yang, Wenqi Wu, Yuanxin Wu, and Junxiang Lian

Abstract—This paper presents a thermal calibration approachfor the accelerometer triad without any temperature-controlledincubator. This approach is implemented by the sequential mul-tiposition observation of the gravity, the raw output, and thetemperature values of accelerometers. Based on the fact that thenorm of measurement outputs of the accelerometer triad ideallyequals to the gravity value in constant thermal conditions, a multi-position least squares estimation procedure is utilized to solve theindividual reference-frame thermal parameters, which relaxes theorientation accuracy requirement of the turntable. According tothe multiposition observation of a gravity vector in two referenceframes, the constrained quaternion optimization is used to solvethe relative attitude among different reference frames. Assumingthe superposition relation of the gravity vector and the rotationaxis direction, the multiposition calibration of the inner triadrelationship is implemented for the navigation computation pur-pose. Comparisons of specific force measurement and navigationposition in two experiments illustrate that the thermal drift errorscan be greatly reduced in the after-power-on process. It shows thatthe sensitivity axis of the accelerometer may change as large as fivearcseconds in direction.

Index Terms—Accelerometer, constraint optimization, multipo-sition, temperature compensation, thermal calibration.

I. INTRODUCTION

INERTIAL navigation is a completely autonomous naviga-tion method that can provide the position, velocity, and

attitude information. The inertial measurement unit (IMU)generally comprises a gyroscope triad and an accelerometertriad. Calibration of the IMU must be implemented to esti-mate a set of parameters, which transforms the raw outputof the gyroscope triad to angular velocity and the raw outputof the accelerometer triad to linear acceleration [1], [2]. Theparameters of the accelerometer triad include scale factors,misalignments, biases, and lever-arm parameters, which can beefficiently estimated by ad hoc calibration using an expensive

Manuscript received March 7, 2012; revised July 10, 2012; accepted July 11,2012. Date of publication August 27, 2012; date of current versionDecember 29, 2012. This work was supported in part by the National NaturalScience Foundation of China under Grant 61174002, by the Foundation forthe Author of National Excellent Doctoral Dissertation of PR China underGrant FANEDD 200897, by the Program for New Century Excellent Talents inUniversity under Grant NCET-10-0900, and by the Fok Ying Tung EducationFundation under Grant 131061. The Associate Editor coordinating the reviewprocess for this paper was Dr. Salvatore Baglio.

The authors are with the Department of Automatic Control, Collegeof Mechatronics and Automation, National University of Defense Technol-ogy, Changsha 410073, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]).

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.2012.2212608

precision turntable in laboratory environment. In recent years,a promising IMU multiposition calibration approach has beenproposed to relax the orientation requirement of the turntable[3]–[12]. The multiposition calibration of the accelerometertriad is usually performed in constant thermal conditions after along warm-up stage and assumes that the calibration parametersare almost constant.

Estimation methods need to be settled for the accelerometertriad in the thermal calibration. Two kinds of estimation meth-ods can be applied to calculate scale factors, misalignments,and biases of the accelerometer triad, namely, by minimizingcost functions [3]–[6] or by solving least squares functions[7]–[10]. In [3], Newton’s method was used to minimize thecost function, which is the error between the squared mag-nitude of the input parameters and the squared magnitude ofthe output. In [4], the minimization of the cost function wasnumerically performed using the damped Newton’s methodwhere the damping factor at each iteration is corrected by aline search procedure. In [5], the downhill simplex optimizationmethod was used to minimize the cost function. In [7]–[9],the authors utilized the combined case least squares method toimplementing the minimization of the cost function. In [10],a Kalman filter was used to estimate the calibration parame-ters. Thus, iterative methods are mostly utilized to solve thecalibration parameters and to achieve high estimation accuracy,but the need of an initial rough estimate makes them incon-venient. Zhang et al. [11] proposed an improved multipositioncalibration for solving the unknown parameters without anyinitial guess. However, the accelerometer temperature drift isstill left as an unresolved problem. On the other hand, manytrials have been made to establish thermal models of calibrationparameters to attack this problem [13]–[21]. Aggarwal et al.[15] explored the effects of thermal variations on biases andscale factors at different temperature values through the thermalchamber and then proposed three-order polynomial thermalmodels for ADI microelectromechanical system sensors. Toinvestigate the thermal property in varying temperature con-ditions, Aggarwal et al. [16] also concerned the thermal rampexperiment from which a simple polynomial temperature modelis developed for the inertial sensor biases and scale factors.After compensating the thermal errors, the inertial navigationsolution was significantly improved. These two trials have toutilize the turntable with a temperature-controlled incubatorthat requires the precise orientation information. To improve therobustness to the turntable error, in [21], an indirect calibrationtechnique to estimate the body-frame drift induced by the vari-ation of the accelerometer sensitivity axes due to temperaturechanges is proposed. However, this estimation method depends

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468 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 62, NO. 2, FEBRUARY 2013

on and is limited to the accuracy of horizontal accelerometermeasurements. Another important issue is the calibration of therelationship between the gyroscope triad and the accelerometertriad in constant thermal conditions. In [12], it is utilized witha nonvertical rotation axis observation method to attack thisproblem, in which the specific force measurement errors of theaccelerometer triad caused by the inner lever arm during theIMU rotation were not considered.

This paper proposes a thermal calibration method for the ac-celerometer triad that does not need any temperature-controlledincubator. This paper is organized as follows. Section IIpresents the thermal models of the accelerometer triad, wherethe scale factors, biases, and misalignments are considered asthermal calibration parameters. Section III describes the ac-celerometer thermal calibration method based on the sequentialmultiposition observation. The calibration of intertriad relation-ship is reported in Section IV. Section V gives the calibrationresults of thermal models and the inner triad relationship ofa navigation-grade IMU. Two experiments have validated theimprovement of specific force measurement and navigationposition of the IMU according to thermal models. Conclusionsand discussions are drawn in Section VI.

II. THERMAL ANALYSIS AND MODELS OF THE

ACCELEROMETER TRIAD

Consider an accelerometer triad consisting of three almostorthogonally mounted accelerometers. Because the specificforce measured by accelerometers has an obvious drift alongwith the temperature change, the novel measurement model ofthe accelerometer triad in thermal conditions is represented as

f̃a(T )=ka(T )Sa

p (T )Rps(T ,T0)R

sb(T0)f

b(T0)+ba(T )+va

(1)

where the 3-D vector of T denotes the temperature valuesof accelerometers; the 3-D vector of T 0 denotes the constanttemperature values after the warm-up stage; the 3-D vector off b(T 0) denotes the input specific force expressed in the bodyframe, i.e., b-frame (see below for the body frame definition)at temperature T 0; the 3 × 3 matrix of Rs

b(T 0) denotes thedirection cosine matrix from b-frame to s-frame (see belowfor the s-frame definition); the 3 × 3 matrix of Rp

s(T ,T 0)denotes the direction cosine matrix from s-frame to p-frame(see below for the p-frame definition); the 3 × 3 matrix ofSa

p(T ) denotes the nonorthogonal transformation from p-frameto a-frame (see below for the a-frame definition); the 3 × 3matrix of ka(T ) denotes the scale factor matrix; the 3-D vectorof ba(T ) denotes the bias vector; the 3-D vector of f̃

a(T )

denotes the raw output of accelerometers; and the 3-D vector ofva is assumed to zero-mean Gaussian white noise. For highlyaccurate accelerometers in this paper, the noise va is less than10 μg/

√Hz. The accelerometer scale factors and biases in (1)

are, respectively, defined as functions of temperature values

ka,x ∼ ka,x(T x),ka,y ∼ ka,y(T y),ka,z ∼ ka,z(T z) (2)ba,x ∼ ba,x(T x), ba,y ∼ ba,y(T y), ba,z ∼ ba,z(T z) (3)

where ka,i(T i) denotes the thermal scale factor of the ithaccelerometer, and ba,i(T i) denotes the thermal bias of the

Fig. 1. Graphical representation of the relationship between p-frame anda-frame.

ith accelerometer. Define the frame of sensitivity axes of ac-celerometers, i.e., a-frame, so that xa denotes the x sensitivityaxis of the accelerometer, ya denotes the y sensitivity axis ofthe accelerometer, and za denotes the z sensitivity axis of theaccelerometer. The sensitivity axis of the ith accelerometer canalso be influenced by the corresponding temperature changes;therefore, the nonorthogonal transformation matrix Sa

p(T ),transforming the specific force from the reference frame tothe sensitivity axes of accelerometers, also depends on thermalconditions. Define the thermal reference frame, i.e., p-frame,so that xp coincides with the accelerometer sensitivity axisxa, yp lies in the xaya plane, and zp constitutes a right-handed orthogonal frame with xp and yp. Note that p-frame isdefined by x and y sensitivity axes of accelerometers, whichmeans that the thermal effect on the orientation of p-frameis only determined by T x and T y . After the warm-up stage,T x reaches T x0, T y reaches T y0, and T z reaches T z0. Then,s-frame is exactly p-frame at temperature T 0. According to theabove definitions, the graphical representation of the relation-ship between p-frame and a-frame can be given as follows.

In the given representation, zayz denotes the projection of za

in the ypzp plane, and zaxz denotes the projection of za in thexpzp plane. In Fig. 1, two thermal assumptions can be givenproperly as follows.

1) The orientation of the ith sensitivity axis of the ac-celerometer depends only on its own temperature.

2) Because of the specific definition of p-frame, the orien-tation of xp depends only on T x, the orientation of yp

depends on T x and T y , and the orientation of zp alsodepends on T x and T y .

Thus, the orientation of each axis as a function of thecorresponding temperature can be described as follows:

Ori(xa)∼f1(T x),Ori(ya)∼f2(T y),Ori(za)∼f3(T z)Ori(xp)∼f1(T x),Ori(yp)∼f4(T x,T y),

Ori(zp)∼f5(T x,T y) (4)

YANG et al.: THERMAL CALIBRATION FOR ACCELEROMETER TRIAD 469

where Ori(·) denotes the orientation of the corresponding axis,and fi(·), (i = 1, 2, . . . , 5) denotes the function of the temper-ature values. From (4), we can draw the conclusion that za

and zp have different orientations and effecting temperatures,and in fact, they are independent of each other due to thespecific definition of p-frame. On one hand, (for a-frame)the temperature of z-axis accelerometer does not depend onthe direction of x and y sensitivity axes of accelerometers.On the other hand, (for p-frame) the axis zp indeed dependson the direction of x and y sensitivity axes of accelerometers,and its effecting temperature is also related to those of x andy temperature axes of accelerometers. As the separation angleformed by the two axes, which can be denoted as the dotproduct of these two unit vectors, depends on the orientationof each axis, the nonorthogonal transformation matrix Sa

p(T )can be expressed as (5), which is shown in the bottom ofthe page, where (ya · xp)(T x,T y) denotes the dot productof the y sensitivity axis of the accelerometer unit and thex-axis in p-frame relying on T x and T y , (za · xp)(T x,T z)denotes the dot product of the z sensitivity axis of the ac-celerometer unit and the x-axis in p-frame relying on T x andT z , and (za · yp)(T x,T y,T z) denotes the dot product of thez sensitivity axis of the accelerometer unit and the y-axis inp-frame relying on T x, T y , and T z . These three dot productscan be approximated by three small thermal misalignments,i.e., τ yx(T x,T y), τ zx(T x,T z), and τ zy(T x,T y,T z). Thesethree thermal misalignments are caused by the nonorthogonalmounting error of the accelerometer triad. In addition, (ya ·yp)(T x,T y) denotes the dot product of the y sensitivity axis ofthe accelerometer unit and the y-axis in p-frame relying on T x

and T y , and (za · zp)(T x,T y,T z) denotes the dot product ofthe z sensitivity axis of the accelerometer unit and the z-axis inp-frame relying on T x, T y , and T z . These two dot products areroughly equal to 1. From (5), the projection of the ith sensitivityaxis of the accelerometer in p-frame takes the following form:

ax = [1 0 0]T , ay(T x,T y)

= [τ yx(T x,T y) τ yy(T x,T y) 0]T , az(T x,T y,T z)

= [τ zx(T x,T z) τ zy(T x,T y,T z) τ zz(T x,T y,T z)]T .

(6)

From (6), the constraint equations need to be satisfied as follows:

τ 2yx(T x,T y)+τ 2

yy(T x,T y)=1 (7)τ 2zx(T x,T z)+τ 2

zy(T x,T y,T z)+τ 2zz(T x,T y,T z)=1. (8)

The orthogonal transformation matrix Rps(T ,T 0)

in (1) transforms the specific force in s-frame at thetemperature T 0 to p-frame at temperature T . Because thethermal reference frame is defined by the x and y sensitivityaxes of accelerometers, Rp

s(T,T 0), transforming a vectorfrom s-frame to p-frame, is also only influenced by x and ytemperature axes of accelerometers. Define the Euler rotationsequence from s-frame to p-frame as follows: first, aroundthe y-axis with ψ; second, around the z-axis with θ; andfinally, around x-axis with φ, or equivalently, as shown atthe bottom of the page. According to (5) and (9), we can getthe nonorthogonal transformation matrix Sa

s(T ,T 0) froms-frame to the sensitivity axes of accelerometers. Ignoringsmall quantities over the first order gives (10), as shown at thebottom of the page. From (10), the orientation trajectory of the

Sap(T ) =

⎡⎣ xa · xp 0 0(ya · xp)(T x,T y) (ya · yp)(T x,T y) 0(za · xp)(T x,T z) (za · yp)(T x,T y,T z) (za · zp)(T x,T y,T z)

⎤⎦

=

⎡⎣ 1 0 0τ yx(T x,T y) τ yy(T x,T y) 0τ zx(T x,T z) τ zy(T x,T y,T z) τ zz(T x,T y,T z)

⎤⎦

≈

⎡⎣ 1 0 0τ yx(T x,T y) 1 0τ zx(T x,T z) τ zy(T x,T y,T z) 1

⎤⎦ (5)

Rps(T ,T 0) =

⎡⎣ cosψ cos θ sin θ −sinψ cos θ−cosψ sin θ cosφ+ sinψ sinφ cos θ cosφ sinψ sin θ cosφ+ cosψ sinφcosψ sin θ sinφ+ sinψ cosφ −cos θ sinφ −sinψ sin θ sinφ+ cosψ cosφ

⎤⎦ψ,θ,φ(T ,T 0)

(9)

Sas(T ,T 0)=Sa

p(T )Rps(T ,T 0)

≈

⎡⎣ 1 θ(T x,T x0,T y,T y0) −ψ(T x,T x0,T y,T y0)τ yx(T x,T y)−θ(T x,T x0,T y,T y0) 1 φ(T x,T x0,T y,T y0)τ zx(T x,T z)+ψ(T x,T x0,T y,T y0) τ zy(T x,T y,T z)−φ(T x,T x0,T y,T y0) 1

⎤⎦ (10)


TABLE I36-POSITION OBSERVATION SCHEME

sensitivity axis of the accelerometer in a stable reference frameassociated with the temperature changes can be analyticallydescribed. The direction cosine matrix Rs

b(T 0) in (1) rotates avector from the stable b-frame defined by the gyroscope triadto the stable s-frame defined by the accelerometer triad attemperature T 0. The b-frame is similarly defined by the x andy sensitivity axes of gyroscopes. The xb axis coincides with thegyroscope sensitivity axis xg , yb lies in the xgyg plane, andzb constitutes a right-handed orthogonal frame with xb and yb.The calibration of the gyroscope triad is sufficiently exploredin [11] and [12], which is not repeated due to space limit.

The thermal calibration for the accelerometer triad is to es-timate parameters such as scale factors, misalignments, biasesof accelerometers in the individual reference frame, the rela-tionship among different reference frames in varying thermalconditions, and the relationship between the gyroscope andaccelerometer triads in constant thermal conditions.

III. THERMAL CALIBRATION FOR THE

ACCELEROMETER TRIAD

In this section, the accelerometer thermal calibration is de-scribed based on the sequential multiposition observation ofthe gravity, the raw output, and the temperature values. Thiscalibration process is to solve the individual reference-frameparameters and the relationship between different referenceframes.

A. Thermal Calibration for the IndividualReference-Frame Parameters

The utilization of the multiposition observation informationdepends on the fact that the norm of measurement suppliedby the accelerometer triad ideally equals to the magnitude ofspecific force. Particularly in static cases, the gravity value is areliable observation quantity [3]–[12] given by

(fp(T ))Tfp(T )=gTg (11)

fp(T )=(ka(T )Sa

p (T ))−1

(f̃a(T )−ba(T )−va

)=(ka(T )Sa

p (T ))−1

f̃a(T )−f0(T )−δfp

(12)

f0(T )=(ka(T )Sa

p (T ))−1

ba(T )≈(ka(T ))−1ba(T )(13)

δfp=(ka(T )Sa

p (T ))−1

va≈(ka(T ))−1va (14)

where g denotes the gravity vector. The given equation foundsthe accelerometer thermal calibration based on the sequentialmultiposition observation. In [12], the least measurement set forthe multiposition observation is proven to need nine positions,and extra positions will result in more accurate calibrationresults. In this paper, a 36-position observation scheme, whichis a full observation set designed by the maximum sensitivityprinciple founded in [12], has been utilized to enhance theperformance of the accelerometer triad, as shown in Table I.In addition, some other observation schemes as subsets ofthe proposed 36-position observation scheme are described inTable II. Accuracy comparison by the condition numbers of theleast squares estimation and the standard deviation of residualgravity errors will be implemented in Section V.

The thermal calibration relies on the three-axis turntablewithout a temperature-controlled incubator, and the tempera-ture of accelerometers needs to be measured at any time. Theaccelerometer triad is consecutively arranged as the 36-positionobservation scheme for several cycles after the IMU is poweredon. Let f̃

a

i,j(T i) denote the ith raw output of the accelerometerin the jth position at temperature T i. After l(l ≥ 3) cyclesof the 36-position observation scheme, we will get l groupsof raw output of the ith accelerometer in the jth position atdifferent temperature, denoted as f̃

a

i,j,k(T i,j,k). To investigatethe relationship between the raw outputs of accelerometersand temperatures, model the raw outputs by the least squaresregression [16] to fit a second-order curve as

f̃a

i,j(T i,j) = f̃a0

i,j + f̃a1

i,jT i,j + f̃a2

i,jT2i,j (15)

where f̃a0

i,j , f̃a1

i,j , and f̃a2

i,j are the corresponding scalar coef-ficients. Meanwhile, the other-order regression methods suchas one-order regression and three-order regression will alsobe used in Section V. The residual between the actual rawoutput and the fitted raw output should be less than the standarddeviation of measurement noise. This can be considered as ameasure of different-order regression methods.


TABLE IIOTHER OBSERVATION SCHEMES

Using the fitted model (15), the raw outputs at 36 positionscan be computed at some temperature. Then, a multipositioncalibration method in constant thermal conditions, e.g., in [11]and [12], will be utilized to estimate the accelerometer param-eters. Consequently, the temperature-dependent accelerometerparameters, i.e., ka(T ), Sa

p(T ), and ba(T ), can be determinedby solving an unknown symmetry matrix in [11] and [12].In addition, the analysis of the measurement accuracy of theaccelerometer triad as measurement noise is provided in Ap-pendix A. The calibrated parameters can be fitted as functionsof their relevant temperatures by the least squares regressionmethod to establish the corresponding thermal models. For thescale factor of the ith accelerometer, we have

ka,i(T i) = ka,i,0 + ka,i,1T i + ka,i,2T2i (16)

for the bias of the ith accelerometer, we have

ba,i(T i) = ba,i,0 + ba,i,1T i + ba,i,2T2i (17)

and for the misalignment angles, we have

τ yx(T x,T y) = τ yx,0+τ yxx,1T x+τ yxy,1T y

+τ yxx,2T2x+τ yxy,2T

2y (18)

τ zx(T x,T z) = τ zx,0+τ zxx,1T x+τ zxz,1T z

+τ zxx,2T2x+τ zxz,2T

2z (19)

τ zy(T x,T y,T z) = τ zy,0+τ zyx,1T x+τ zyy,1T y+τ zyz,1T z

+τ zyx,2T2x+τ zyy,2T

2y+τ zyz,2T

2z (20)

τ yy(T x,T y) = τ yy,0+τ yyx,1T x+τ yyy,1T y

+τ yyx,2T2x+τ yyy,2T

2y (21)

τ zz(T x,T y,T z) = τ zz,0+τ zzx,1T x+τ zzy,1T y+τ zzz,1T z

+τ zzx,2T2x+τ zzy,2T

2y+τ zzz,2T

2z. (22)

Then, the least squares regression method can be uti-lized to estimate these thermal coefficients, respectively, inAppendix B. Equations (16)–(18) depict the thermal modelsof the accelerometer triad. Based on these thermal models, thethermal drift errors induced by temperature variations can beeffectively compensated.

B. Thermal Calibration for the Relationship BetweenDifferent Reference Frames

For navigation computation purposes, we also need to ex-press the specific force in some fixed orthogonal frame. Weknow that the accelerometer temperature will approach a stablestate after the warm-up stage and after the p-frame reaches thes-frame, as defined in Section II.

As described in (9), the orthogonal transformation betweens-frame and p-frame can also be calculated according to thesequential 36-position observation scheme. First, we can getthe corresponding calibration parameters in constant thermalconditions by substituting the temperature values T i0 into thethermal models described by (16)–(18). Then, the raw output ofaccelerometers in the jth position can also be derived throughthe thermal models in (15). Consequently, the specific forcemeasurement in constant thermal conditions, ignoring the mea-surement noise, can be derived by (1) as

fsj (T 0)=fp

j (T 0)=(ka(T 0)S

ap(T 0)

)−1(f̃a

j (T 0)−ba(T 0)).

(23)

Then, the standardized specific force vector lpj (T ) in the jthposition can be transformed into the lsj(T 0) case through theRs

p(T ,T 0) relation. Thus, we can get the following:

lpj (T ) = Rps(T ,T 0)l

sj(T 0). (24)

In addition, the normalized vectors take the following forms:

lpj (T ) =fp

j (T )∣∣fpj (T )

∣∣ lsj (T 0) =fs

j (T 0)

|fsj (T 0)|

. (25)

Substituting (9) into (24), the corresponding component equa-tions are given as follows:⎡⎣l

px

j (T )

lpy

j (T )lpz

j (T )

⎤⎦=

⎡⎣Rp

s,1,1(T ,T 0) Rps,1,2(T ,T 0) Rp

s,1,3(T ,T 0)Rp

s,2,1(T ,T 0) Rps,2,2(T ,T 0) Rp

s,2,3(T ,T 0)Rp

s,3,1(T ,T 0) Rps,3,2(T ,T 0) Rp

s,3,3(T ,T 0)

⎤⎦

×

⎡⎣l

sxj (T )

lsyj (T )lszj (T )

⎤⎦. (26)


Then, Rps(T ,T 0) can be solved by the constrained opti-

mization [22], [23] in contrast to the least squares methodthat is more apt to error disturbances and makes Rp

s(T ,T 0)obtained usually deviate from being an orthogonal matrix.The constrained optimization is used to calculate the atti-tude quaternion qps (T ,T 0) considering the constrained equa-tion of ‖qps ((T ,T 0)‖2 = 1. Then, the direction cosine matrixRp

s(T ,T 0) can be transformed from qps (T ,T 0). However,the least squares estimation of Rp

s(T ,T 0) directly from (26),which is an unconstrained method, can always result in thedeviation from the property of an orthogonal matrix, such as theproperty of special orthogonal (3). Thus, the constrained opti-mization is more preferable to calculating the attitude matrix.

According to the obtained Rps(T,T 0), the three Euler angles

can be solved as follows:

ψ(T ,T 0) = tan−1

(−Rp

s,1,3(T ,T 0)

Rps,1,1(T ,T 0)

)

φ(T ,T 0) = tan−1

(−Rp

s,3,2(T ,T 0)

Rps,2,2(T ,T 0)

)

θ(T ,T 0) = sin−1(Rp

s,1,2(T ,T 0)). (27)

As stated in Section II, the relative orientation between p-frameand s-frame is affected only by ΔT x and ΔT y . When thesetwo temperature difference approach to zero, the p-frame isexactly the s-frame with the thermal Euler angles equating tozero. Based on different Euler angles and different temperaturedifferences, the thermal models of these three Euler angles aremodeled as follows:

ψ(T ,T 0) =ψx,1ΔT x + ψy,1ΔT y

+ ψx,2(ΔT x)2 + ψy,2(ΔT y)

2 (28)θ(T ,T 0) = θx,1ΔT x + θy,1ΔT y

+ θx,2(ΔT x)2 + θy,2(ΔT y)

2 (29)φ(T ,T 0) =φx,1ΔT x + φy,1ΔT y

+ φx,2(ΔT x)2 + φy,2(ΔT y)

2. (30)

In addition, ΔT i = T i − T i0 in (28)–(30) denotes the temper-ature difference between the varying temperature values and thebalancing temperature values of accelerometers.

First, we choose the state vector shown at the bottom of thepage. These thermal parameters can be solved by a least squaresmethod applied to the following:

Hx = zΔ=

⎡⎣ · · ·Hk

· · ·

⎤⎦x =

⎡⎣ · · ·zk

· · ·

⎤⎦ (32)

where the system matrix element Hk of 3 × 12 dimension takesthe form in (33), which is shown at the bottom of the page. Themeasurement values are given as follows:

zk = [ψ(T ,T 0) θ(T ,T 0) φ(T ,T 0) ]T . (34)

Combining (31)–(34), the thermal coefficients of Euler anglescan be effectively estimated as follows:

x =(HTH

)−1HT z. (35)

IV. CALIBRATION OF THE RELATIONSHIP BETWEEN

THE GYROSCOPE AND ACCELEROMETER TRIADS

In this section, the calibration method of the relationshipbetween the gyroscope and accelerometer triads in constantthermal conditions is designed. This method is implementedby assuming the parallel property of the gravity vector androtation axis direction. Relying on this property, the specificforce measurement of accelerometers in static cases can also begiven by the rotation observation of gyroscopes and the innertriad relationship. This method avoids the compensation ofinner lever-arm parameters that was omitted in [12]. By rotatingthe IMU into different positions, three different expressions ofa vector in two frames can be found to solve the relationshipbetween these two frames. This indicates that the expressionsboth in b-frame and s-frame satisfy the constraints as follows:

lsi (T 0) = Rsb(T 0)l

bi (T 0), (i = 1, 2, 3). (36)

Then, the orthogonal transformation matrix Rsb(T 0) can be

determined using the constrained optimization [22], [23]. Withthe same Euler rotation sequence in Section II, Rs

b(T 0) canbe expressed as (37), which is shown at the bottom of thepage, where αx denotes the rotation angle around the y-axis,

x = [ψx,1 ψy,1 ψx,2 ψy,2 θx,1 θy,1 θx,2 θy,2 φx,1 φy,1 φx,2 φy,2 ]T (31)

Hk =[lpj (T )×

] ⎡⎣ [ ΔT x ΔT y ΔT 2x ΔT 2

y ] 01×4 01×4

01×4 [ ΔT x ΔT y ΔT 2x ΔT 2

y ] 01×4

01×4 01×4 [ ΔT x ΔT y ΔT 2x ΔT 2

y ]

⎤⎦ (33)

Rsb(T 0) =

⎡⎣ cosαx cosαy sinαy −sinαx cosαy

−cosαx sinαy cosαz + sinαx sinαz cosαy cosαz sinαx sinαy cosαz + cosαx sinαz

cosαx sinαy sinαz + sinαx cosαz −cosαy sinαz −sinαx sinαy sinαz + cosαx cosαz

⎤⎦ (37)


Fig. 2. Relationship of the IMU, the gravity, and the rotation axis.

αy denotes the rotation angle around the z-axis, and αz denotesthe rotation angle around the x-axis.

As shown in Fig. 2, the collimation of the gravity vector andthe rotation axis of the turntable can be utilized to measurethe vector in two frames. The observation scheme of No. 25,No. 29, and No. 33 in Table I can be used for computing theinner triad relationship effectively.

First, we can measure the unit rotation axis lbi by rotatingthe IMU around the vertical rotation axis in a clockwise andcounterclockwise manner, where the earth rotation componentis canceled out. Then, the unit vector expression of the rotationaxis is given as follows:

lbi (T 0) =

t∫0

ωb+ib (τ)dτ −

t∫0

ωb−ib (τ)dτ∣∣∣∣ t∫

0

ωb+ib (τ)dτ −

t∫0

ωb−ib (τ)dτ

∣∣∣∣(38)

where t denotes the rotation time, ωbib denotes the angular

velocity measured by the gyroscope triad, the superscript “+”denotes clockwise rotation, and the superscript “−” denotescounterclockwise rotation.

Meanwhile, the specific force measurements hold the follow-ing relation during the IMU rotation:

fsoR

(t)=−gs=fso(t)−fs

R(t)=fsa(t)−fs

r(t)−fsR(t) (39)

where fsoR

(t) denotes the specific force measurement at thepoint OR; fs

o(t) denotes the specific force measurement atthe point O; fs

r(t) and fsR(t) denote the specific force mea-

surement errors caused by the inner lever arm and the outerlever arm, respectively; and fs

a(t) denotes the specific forcemeasurement of accelerometers. In addition, the specific forceerrors take the following forms, respectively [24], [25]:

fsR(t) = −ωR×(ωR×R), (fs

R(t))T ωR = 0 (40)

fsr(t) =Ss

a(T 0)far(t)

=Ssa(T 0)

⎡⎣aT

x (ωR × (ωR×rssx))aTy

(ωR×

(ωR×rssy

))aTz (ωR×(ωR×rssz))

⎤⎦ (fs

r(t))T ωR �= 0

(41)

where R denotes the vector distance from the center ofs-frame to the rotation axis, named as the outer lever arm,and rssi denotes the vector from the center of s-frame to theith accelerometer in s-frame, named as the inner lever arm.Therefore, we have to estimate these inner lever-arm parametersand compensate fs

r(t) to get the specific force on the IMU cen-ter. However, the estimation of these lever-arm parameters canalways be intractable. Fortunately, when the IMU is stationary,the lever-arm problems can be neglected because of very smallmagnitude of the Earth’s rotation rate, and the measurement ofspecific force input on the accelerometer triad holds

fsoR

(t) = −gs = fso(t) = fs

a(t). (42)

Therefore, the unit specific force vector measured by the ac-celerometer triad in the ith position can be derived as

lsi (T 0) =(ka(T 0)S

as(T 0))

−1(f̃a(T 0)− ba(T 0)

)∣∣∣(ka(T 0)S

as(T 0))

−1(f̃a(T 0)− ba(T 0)

)∣∣∣ . (43)

Through selecting a set of three different positions by adjustingthe turntable, Rs

b(T 0) can be readily determined, and morepositions will result in better accuracy. The physical variationof the rotation axis would degrade the computation accuracyof the rotation axis vector; therefore, a slow rotation rate isrecommended to depress the axis disturbance during rotations.In addition, only the gravity, raw output, and temperature valuesare utilized in the sequential multiposition observation. It re-laxes the requirement of the turntable precision and effectivelyprevents the influence of the imprecise turntable referenceinformation on the specific force measurement. As a result, themeasurement of the gravity vector in s-frame can be improved.After the intertriad relationship calibration, the varying-framemeasurement and navigation in different thermal conditionsare achieved.

V. CALIBRATION RESULTS AND

EXPERIMENT VALIDATION

A. Calibration Results

A navigation-grade IMU was calibrated consisting ofthree laser gyroscopes and three quartz accelerometers (seeTable III), using a three-axis turntable of two-arcsecond ac-curacy (1σ) (Fig. 3). A high-precision temperature probe wasstuck onto the inner surface of each accelerometer. The tem-perature signal was collected with sample time of 1 min andthe specific force signal with the frequency of 100 Hz. Theaccelerometer triad was calibrated by the 36-position obser-vation scheme for five times after power-on. The IMU stayedfor 1 min in each stationary position, at which the averageaccelerometer output was used in the computation for thepurpose of depressing the measurement noise. A warm-up stageof at least 3 h was needed before the accelerometer temper-atures reach the stable states. Each 36-position observationscheme takes about 36 min. The fifth 36-position observation


TABLE IIIPERFORMANCE OF GYROSCOPES AND ACCELEROMETERS IN THE IMU

Fig. 3. Three-axis turntable without a temperature-controlled incubator.

Fig. 4. Output fitting of the x accelerometer.

scheme was performed after the temperature is stable, tocalculate the scale factors, misalignments, and biases of theaccelerometer triad in constant thermal conditions. The calibra-tion parameters of the gyroscope triad are known as the priorinformation.

The fitted errors of the raw output of accelerometers forNo.1 posture of 36-position observation scheme are depictedin Figs. 4–6. For each accelerometer, the raw output is fitted,respectively, by the one-order, two-order, and three-order re-gression methods. The residual errors fitted by three different


Fig. 5. Output fitting of the y accelerometer.

Fig. 6. Output fitting of the z accelerometer.

regression methods are less than 10 μg for each accelerometer,which is the standard deviation of measurement noise. Herein,the second-order regression method is more preferable forsmaller fitted errors than the one-order regression case andfor comparable errors but fewer regression coefficients thanthe three-order regression case. The condition numbers ofleast squares estimation for calibration parameters in constantthermal conditions by different observation schemes are, re-spectively, described in Fig. 7. The residual gravity errors areshown in Fig. 8 and the standard deviation in Fig. 9. Theconclusion can be drawn that more positions will result inlower condition numbers and lower residual gravity errors aswell as better estimation accuracy for calibration parameters.Table IV gives the calibration results of the accelerometer triadwith and without compensating thermal drift errors. In fact,the second-order regression model can be used to get enoughaccuracy obtained by the least squares curve fit method of theseparameters.

The quantities of second-order coefficients in Table IV are re-spectively small compared with the zero-order coefficients, andthe higher-order regression model is not necessary to fit theseparameters. The thermal effects on the calibration parameters

Fig. 7. Condition numbers by different observation schemes.

Fig. 8. Residual gravity errors by different observation schemes.

Fig. 9. Standard deviation of residual gravity errors by different observationschemes.


TABLE IVCALIBRATION RESULTS OF THE NONTHERMAL AND THERMAL MODELS OF THE ACCELEROMETER TRIAD

are illustrated for the scale factors in Fig. 10, biases in Fig. 11,misalignments between the accelerometer input axes and itsassociated reference frame in Figs. 12 and 13, and thermal Eulerangles from s-frame to p-frame in Figs. 14–16. The maximumchanges of scale factors are, respectively, 355.32, 393.09, and319.72 ppm. The maximum changes of biases are, respectively,67.41, 5.33, and 1.17 μg. The maximum change of misalign-ments as described by nondiagonal elements of Sa

p(T ) is 3.43

arcsecond, whereas the other two misalignments described bythe diagonal elements are almost constant with changes lessthan 0.001 arcsecond. The orientation trajectories of threesensitivity axes of accelerometers in s-frame are depicted inFigs. 17–19. The maximum separation angle among the varyingsensitivity axes is 3.13 arcseconds for the x accelerometeraxes, 6.58 arcseconds for the y accelerometer axes, and 0.78arcseconds for the z accelerometer axes.


Fig. 10. Thermal changes of scale factors.

Fig. 11. Thermal changes of biases.

Fig. 12. First set of thermal changes of misalignments.

Fig. 13. Second set of thermal changes of misalignments.

Fig. 14. Thermal changes of ψ.

Fig. 15. Thermal changes of θ.


Fig. 16. Thermal changes of φ.

Fig. 17. Thermal changes of the x sensitivity axis of the accelerometer ins-frame. (Start position: red line. End position: green line.)

Fig. 18. Thermal changes of the y sensitivity axis of the accelerometer ins-frame. (Start position: red line. End position: green line.)

Fig. 19. Thermal changes of the z sensitivity axis of the accelerometer ins-frame. (Start position: red line. End position: green line.)

Fig. 20. Raw output of gyroscopes in test I.

B. Experiment Validation

Two validation experiments were performed in the labora-tory: a static test (Test I) and a turntable-tumbling test (Test II).The measurement histories were, respectively, shown inFigs. 20 and 21 for gyroscopes and in Figs. 22 and 23 foraccelerometers. The static test was implemented in the after-power-on process where the temperature drift of the raw outputof the y accelerometer was much obvious, as shown in Fig. 22.The turntable-tumbling test was performed in a similar temper-ature condition as the static case, and the 36-position calibrationscheme was performed for five times again in Figs. 21 and 23.Norm errors of specific force measurements were, respectively,given in Figs. 24 and 25. For Test I, the standard deviationis 91.07 μg, without any thermal consideration, whereas thestandard deviation is reduced to 3.26 μg after the thermal com-pensation. As for Test II, the standard deviation is 102.98 and6.63 μg before and after the thermal compensation, respec-tively. The calibration parameters of the gyroscope triad arethe same for these two cases whether the thermal drift is takeninto account or not; therefore, the thermal performance ofaccelerometers becomes vital in the navigation computation. As


Fig. 21. Raw output of gyroscopes in test II.

Fig. 22. Raw output of accelerometers in test I

Fig. 23. Raw output of accelerometers in test II

shown in Fig. 26 and 27, the inertial navigation performance isgreatly improved by more than 50% after the thermal compen-sation, for both tests I and II.

Fig. 24. Norm errors of specific force measurements before and after thethermal compensation in test I.

Fig. 25. Norm errors of specific force measurement before and after thethermal compensation in test II.

Fig. 26. Navigation position error before and after the thermal compensationin test I.


Fig. 27. Navigation position error before and after the thermal compensationin test II.

VI. CONCLUSION AND DISCUSSIONS

Thermal calibration of the accelerometer triad is a necessarystage for high-accuracy inertial navigation. This paper hasproposes a thermal calibration method of the accelerometertriad based on the sequential multiposition observation of thegravity, the raw output, and temperature values of accelerome-ters. Ideally, the norm of the IMU specific force measurementshould be equal to the magnitude of the gravity in static cases,whether the thermal conditions are considered or not. The maincontribution of this paper can be summarized as follows.

1) By the newly devised thermal calibration models, theaccelerometer parameters as a function of temperature areinvestigated, such as the scale factors, the biases, and themisalignments between the accelerometers.

2) In addition, the relative direction changes of the sensitiv-ity axes of accelerometers are successfully calibrated andcarefully examined.

3) The relationship between the gyroscope and accelerom-eter triads is successfully calibrated without consideringthe size effect error of the specific force measurement ofaccelerometers.

After the calibration procedure, the experiments reveal thatthe sensitivity axis of the accelerometer may change as large asfive arcseconds in direction, which is firstly introduced in this

paper. Validation tests prove the effectiveness of the proposedthermal calibration method.

Some improvements can be made on the model of accelerom-eters. The nonlinear errors, such as squared errors, cross-coupling errors, scale-factor nonlinearities, and scale-factorasymmetries, are not considered in the thermal model. Thenonlinear errors are not calibrated because they are relativelysmaller quantities as compared with the systemic thermal driftsof scale factors, biases, and misalignments. In this regard,the calibration of the accelerometer triad in constant thermalconditions is not implemented in the full-scale range. In ad-dition, the proposed thermal calibration method relies on theambient temperature, and it will certainly show a degradedperformance in other ambient temperature. Therefore, differentthermal models calibrated in different ambient temperatureshould be investigated, and a combined thermal model may wellbe used for harsh applications.

APPENDIX A

From (11)–(14), we can get the observation equation (A1),as shown at the bottom of the page, where KA is the symmetrymatrix and takes the following form:

KA=

[KT

aKa KTa f0

fT0 Ka fT

0 f0

],Ka=

(ka(T )Sa

p(T ))−1

. (A2)

Supposing that the measurement noise of each accelerometerhas the Gaussian distribution with the variance of V i, (i=x,y, z), respectively, and is independent, the measurement noiseafter some measurement period has the following property:

E (δfpi ) = 0, E

((δfp

i )Tδfp

i

)= Vi (A3)

where E(·) is the operator of mathematical expectation.Then, (A1) is reduced to[(E(f̃a(T )

))T

−1

]KA

[E(f̃a(T )

)−1

]−

∑i=x,y,z

V i= |g|2. (A4)

From (A4), the norm of the IMU specific force measurementis affected by the noise variance. In the validation for theinput–output model, the calibration parameters are relativelyreliable if the residual squared gravity error is less than the sumof the noise variance.

(fp(T ))T fp(T ) =((

ka(T )Sap(T )

)−1f̃a(T )− f0(T )− δfp

)T ((ka(T )Sa

p(T ))−1

f̃a(T )− f0(T )− δfp

)=

(Kaf̃

a(T )− f0(T )

)T (Kaf̃

a(T )− f0(T )

)− 2

(Kaf̃

a(T )− f0(T )

)T

δfp + (δfp)T δfp

=

([Ka f0 ]

[f̃a(T )−1

])T ([Ka f0 ]

[f̃a(T )−1

])− 2

(Kaf̃

a(T )− f0(T )

)T

δfp + (δfp)T δfp

=[ (

f̃a(T )

)T

−1]KA

[f̃a(T )−1

]− 2

(Kaf̃

a(T )− f0(T )

)T

δfp + (δfp)T δfp

=[ (

f̃a(T )

)T

−1]KA

[f̃a(T )−1

]− 2 (fp(T ))T δfp − (δfp)T δfp

= |g|2 (A1)


APPENDIX B

The detailed implementation of least squares estimation forthermal coefficients of scale factors, biases, and misalignmentsis respectively described as follows.

For the scale factor of the ith accelerometer, the followingleast squares equation is given as

F sf,iXsf,i =Y sf,iΔ=

⎡⎣ · · ·1 T i,k T 2

i,k

· · ·

⎤⎦⎡⎣ka,i,0

ka,i,1

ka,i,2

⎤⎦

=

⎡⎣ · · ·ka,i(T i,k)

· · ·

⎤⎦ . (B1)

Thus, the least squares solution can be derived as

Xsf,i =(F T

sf,iFTsf,i

)−1F T

sf,iY sf,i. (B2)

For the bias of the ith accelerometer, the least squares equationis given as

F bias,iXbias,i =Y bias,iΔ=

⎡⎣ · · ·1 T i,k T 2

i,k

· · ·

⎤⎦⎡⎣ ba,i,0ba,i,1ba,i,2

⎤⎦

=

⎡⎣ · · ·ba,i(T i,k)

· · ·

⎤⎦ (B3)

with the least squares solution of

Xbias,i =(F T

bias,iFTbias,i

)−1F T

bias,iY bias,i. (B4)

For the misalignment angles of the accelerometer triad, we havethe observation equation (B5), shown at the bottom of the page,where the corresponding elements in (B5) take the forms in(B6)–(B15), also shown at the bottom of the page. Therefore,the least squares solution is derived as

Xmis =(F T

misFTmis

)−1F T

misY mis. (B16)

FmisXmis =Y misΔ=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

· · ·Fk,mis,τyx

01×5 01×7 01×5 01×7

01×5 Fk,mis,τzx01×7 01×5 01×7

01×5 01×5 Fk,mis,τzy01×5 01×7

01×5 01×5 01×7 Fk,mis,τyy01×7

01×5 01×5 01×7 01×5 Fk,mis,τzz

· · ·

⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎣Xmis,τyx

Xmis,τzx

Xmis,τzy

Xmis,τyy

Xmis,τzz

⎤⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

· · ·τ yx(T x,k,T y,k)τ zx(T x,k,T z,k)

τ zy(T x,k,T y,k,T z,k)τ yy(T x,k,T y,k)

τ zz(T x,k,T y,k,T z,k)· · ·

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(B5)

F k,mis,τyx= [ 1 T x,k T y,k T 2

x,k T 2y,k ] (B6)

F k,mis,τzx= [ 1 T x,k T z,k T 2

x,k T 2z,k ] (B7)

F k,mis,τzy= [ 1 T x,k T y,k T z,k T 2

x,k T 2y,k T 2

z,k ] (B8)

F k,mis,τyy= [ 1 T x,k T y,k T 2

x,k T 2y,k ] (B9)

F k,mis,τzz= [ 1 T x,k T y,k T z,k T 2

x,k T 2y,k T 2

z,k ] (B10)

Xmis,τyx= [ τ yx,0 τ yxx,1 τ yxy,1 τ yxx,2 τ yxy,2 ]

T (B11)

Xmis,τzx= [ τ zx,0 τ zxx,1 τ zxz,1 τ zxx,2 τ zxz,2 ]

T (B12)

Xmis,τzy= [ τ zy,0 τ zyx,1 τ zyy,1 τ zyz,1 τ zyx,2 τ zyy,2 τ zyz,2 ]

T (B13)

Xmis,τyy= [ τ yy,0 τ yyx,1 τ yyy,1 τ yyx,2 τ yyy,2 ]

T (B14)

Xmis,τzz= [ τ zz,0 τ zzx,1 τ zzy,1 τ zzz,1 τ zzx,2 τ zzy,2 τ zzy,2 ]

T (B15)


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Jie Yang received the B.S. degree in navigation andcontrol from National University of Defense Tech-nology, Changsha, China, in 2007. He is currentlyworking toward the Ph.D. degree at National Univer-sity of Defense Technology.

His research interests include the developmentof parameter identification and state estimationmethodologies for inertial navigation systems andinertial-based integrated navigation systems.

Wenqi Wu received the B.S. degree from TianjinUniversity, Tianjin, China, in 1988 and the M.S. andPh.D. degrees from National University of DefenseTechnology, Changsha, China, in 1991 and 2002,respectively.

He is currently a Professor with the Departmentof Automatic Control, College of Mechatronics andAutomation, National University of Defense Tech-nology. His research interests include global nav-igation satellite system, inertial, and multisensorintegrated navigation theory and application.

Yuanxin Wu received the B.S. and Ph.D. degreesin navigation from National University of DefenceTechnology, Changsha, China, in 1998 and 2005,respectively.

From 2005 to 2007, he was with the NationalUniversity of Defense Technology as a Lecturer.From February 2009 to February 2010, he was aVisiting Postdoctoral Fellow with the Departmentof Geomatrics Engineering, University of Calgary,Calgary, AB, Canada. He is currently an AssociateProfessor with the National University of Defense

Technology. His current research interests include inertial navigation systems,inertial-based integrated navigation systems, and state estimation theory.

Dr. Wu was the recipient of 2008 Top 100 National Excellent DoctoralDissertations in China, 2010 New Century Excellent Talents in University inChina, and 2011 the Fok Ying Tung Education Fund.

Junxiang Lian received the B.E. and Ph.D. degreesfrom National University of Defense Technology,Changsha, China, in 2000 and 2007, respectively.

From March to August 2012, he was a Visit-ing Scholar with the Division of Control, Schoolof Electrical and Electronic Engineering, NanyangTechnological University, Singapore. He is cur-rently a Lecturer with NUDT. His research inter-ests include motion measurement of small vehicles,vision/inertial measuring unit fusion, and integratednavigation.

Documents

Thermal Calibration for the Accelerometer Triad Based on the Sequential Multiposition Observation