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An accurate calibration method for accelerometer nonlinear scale factor on a low-cost three-axis turntable

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2014 Meas. Sci. Technol. 25 025102

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Meas.Sci.T echnol.25(2014)025102(7pp)doi:10.1088/0957-0233/25/2/025102 An accurate calibration method for accelerometer nonlinear scale factor on a low-cost three-axis turntable

Jianye Pan,Chunxi Zhang and Qingzhong Cai

School of Instrumentation Science and Optoelectronics Engineering,Beijing University of Aeronautics

and Astronautics,Beijing100191,People’s Republic of China

E-mail:panjianye1987@https://www.360docs.net/doc/301088772.html,

Received18September2013,revised2November2013

Accepted for publication21November2013

Published9January2014

Abstract

Strapdown inertial navigation system(SINS)requirements are very demanding on gyroscopes

and accelerometers as well as on calibration.To improve the accuracy of SINS,high-accuracy

calibration is needed.Adding the accelerometer nonlinear scale factor into the model and

reducing estimation errors is essential for improving calibration methods.In this paper,the

inertial navigation error model is simpli?ed,including only velocity and tilt errors.Based on

the simpli?ed error model,the relationship between the navigation errors(the rates of change

of velocity errors)and the inertial measurement unit(IMU)calibration parameters is

presented.A tracking model is designed to estimate the rates of change of velocity errors.With

a special calibration procedure consisting of six rotation sequences,the accelerometer

nonlinear scale factor errors can be computed by the estimates of the rates of change of

velocity errors.Simulation and laboratory test results show that the accelerometer nonlinear

scale factor can be calibrated with satisfactory accuracy on a low-cost three-axis turntable in

several minutes.The comparison with the traditional calibration method highlights the

superior performance of the proposed calibration method without precise orientation control.

In addition,the proposed calibration method saves a lot of time in comparison with the

multi-position calibration method.

Keywords:inertial measurement unit,accelerometer nonlinear scale factor,error model,

calibration

(Some?gures may appear in colour only in the online journal)

1.Introduction

Inertial navigation is entirely self-contained and can provide information including position,velocity and attitude at a high rate.It is now widely used as the main navigation means in missiles,aircraft,robots and other autonomous vehicles[1].An inertial measurement unit(IMU)consists of three orthogonal accelerometers and three orthogonal gyroscopes.Their measurements,speci?c forces and angular rates are used to compute attitude,velocity and position.IMU calibration is a process of determining the coef?cients that transform the raw outputs of inertial sensors to meaningful quantities of interest.Traditional calibration methods are usually performed by comparing the IMU outputs with known reference information obtained by precise orientation control, which requires specialized high-accuracy equipment[2].

The IMU calibration methods that can provide accuracy on a low-cost two-axis or three-axis turntable have been widely discussed for many years.Some of them are based on the Kalman?lter.Three sets of rotation sequences were designed to estimate the IMU calibration parameters by tracking the rates of change of velocity errors[3].In addition,Lee presented several rotation sequences to estimate the IMU calibration parameters by tracking the rates of change of velocity errors [4].Mark designed two paths to calibrate gyroscope errors and accelerometer errors,respectively.First of all,gyroscope errors

Table1.Frame de?nition.

Frames Description

i frame The inertial frame

e frame The Earth frame

b frame The body frame

t frame The true geographic frame

c frame The compute

d geographic frame

p frame The platform frame

were estimated independently of the accelerometer errors by a special path,then accelerometer errors were estimated by another special path following gyroscope error compensation [5].Two sets of rotation sequences were designed by Savage. Comparing the speci?c force before rotation with the speci?c force after rotation,the gyroscope and accelerometer errors can be obtained[6].Camberlein described a calibration method developed by the SAGEM company,and it provided an overview of the Kalman?lter,along with a typical calibration procedure to estimate the IMU calibration parameters[7].Like the methods based on the Kalman?lter,the multi-position calibration methods also do not require a high-precision turntable,and these have been widely discussed in recent years[1].

In some applications,such as marine navigation and gravity measurement,the nonlinear scale factor should be introduced into the accelerometer error model[8],but so far it has been largely neglected or treated less seriously.In traditional calibration methods,although the accelerometer nonlinear scale factor can be estimated using the least squares algorithm,the accuracy is limited by the accuracy of the turntable.An improved multi-position calibration method using the particle swarm optimization algorithm was proposed to estimate the accelerometer nonlinear scale factor,and this could achieve the needed accuracy on a low-cost three-axis turntable[2].However,the multi-position calibration method needs24rotation sequences for calibration and it takes a long time for tens of minutes.

In this current work,an accurate calibration method for the accelerometer nonlinear scale factor on a low-cost three-axis turntable is presented.A special calibration procedure is designed,forcing individual sensor errors to contribute to different components of velocity error and its rate of change. After accelerometer bias is estimated by the rate of change of the north velocity error,the accelerometer nonlinear scale factor can be computed by the rate of change of the upward velocity error.The calibration procedure,consisting of six rotation sequences,takes only a few minutes.

The paper is organized as follows.In section2,the inertial navigation error model is simpli?ed,including only velocity and tilt errors.In section3,based on the simpli?ed error model,the relationship between the navigation errors(the rates of change of velocity errors)and the IMU calibration parameters is presented.A tracking model is designed to estimate the rates of velocity error components.Data collection rotation sequences are designed,forcing the accelerometer nonlinear scale factor to contribute to the rates of velocity error components.Observation equations for each rotation are built to compute the accelerometer nonlinear scale factor.In section4,the proposed calibration method is compared with

the traditional calibration method by simulation and laboratory testing.The results prove that the accelerometer nonlinear scale factor can be accurately calibrated by the proposed calibration method on a low-cost three-axis turntable.In section5,the conclusion is discussed.

2.Simpli?ed error model

The frames used in this paper are de?ned in table1.

The classical inertial navigation algorithm can be described as[9]

˙C t

b

=C t b

ωb ib?C b t

ωt ie+ωt et

×

˙v t=C t b f b?(2ωt ie+ωt et)×v t+g t

˙L=v t

N

(R+h)

˙λ=v t

E

sec L

(R+h)

˙h=v t

U

(1) where the IMU positions L,λand h are the latitude,longitude and height,respectively.Ground velocity in the geographic frame v t=

v t E v t N v t U

T

,where the subscripts stand for east, north and upward velocity components,respectively.C b t is the body attitude matrix with respect to the geographic frame

(rotating from the geographic frame to the body frame).ωb

ib is the body angular rate measured by gyroscopes.ωt

ie

= [0ωie cos Lωie sin L]T is the Earth rotation rate vector in the geographic frame.ωt et=[?v t N v t E v t E tan L]T

(R+h) is the angular rate of the geographic frame with respect to the Earth frame,expressed in the geographic frame.

f b=[f b x f b y f b z]T is the speci?c force measured by accelerometers.

g t=[00?g]T is the gravity vector in the geographic frame and R is the radius of sphere of the Earth. The3×3matrix(·×)is de?ned so that the cross product satis?es a×b=(a×)b for two arbitrary vectors.

In the classical inertial navigation algorithm,the relationship of the attitudes,velocities and positions are complex.When the IMU is static,the classical inertial navigation algorithm can be simpli?ed by the known zero velocity and the known position[9]

˙C t

b

=C t b

ωb ib?C b tωt ie

×

˙v t=C t b f b+g t.(2) Assuming the azimuth and attitude errors are small angles, the classical inertial navigation error model corresponding to the classical inertial navigation algorithm is expressed as[9]˙φ=φ×ωt

it

+δωt it+εt

δ˙v t=?φ×f t+δv t×

2ωt ie+ωt et

+v t×

2δωt ie+δωt et

+δf t

δ˙L=δv t N

(R+h)?δh·v t N

(R+h)2

δ˙λ=δL·v t E sec L tan L

(R+h)+sec L·δv t E

(R+h)?δh·v t E sec L

(R+h)2

δ˙h=δv t U(3)

whereωt

it

is the angular rate of the geographic frame with respect to the inertial frame,expressed in the geographic frame,εt=[εEεNεU]T is the gyroscope error in the geographic

frame and δf t =[δf E δf N δf U ]T is the accelerometer error in the geographic frame.

Similar to the classical inertial navigation algorithm,the classical inertial navigation error model can also be simpli?ed when the IMU is static.The derivations are described as follows.

The small rotation angle vector of the computed geographic frame around the platform frame is de?ned as ψ.

The small rotation angle vector of the true geographic frame around the computed geographic frame is de?ned as θ.

The small rotation angle vector of the true geographic frame around the platform frame is de?ned as φ.

The relationship of the vectors ψ,θand φis described as [10]

φ=ψ+θ.

(4)

The relationship of the vector ψand the gyroscope error can be expressed in the computed geographic frame as [10]

d ψd t

c

+ωc ic ×ψ=εc .(5)When the position is known to be a constant and the velocity is known to be zero,the computed geographic frame is coincident with the true geographic frame.In other words,the vector θcan be seen as a zero vector.So equation (5)can be written as d φd t

t

+ωt it ×φ=εt .(6)When the velocity is known to be zero,the vector ωt et is a zero vector.Considering the equation ωt it =ωt ie +ωt et ,the vector ωt it is equal to the vector ωt ie .Equation (6)can be expressed as

˙φ+ωt ie ×φ=εt .(7)The differential equation of velocity in the computed

geographic frame is [10]

˙v

c =(I ?φ×)C t b (f b +δf b )+g t .(8)

With the high order item neglected,equation (8)can be

derived as

˙v

c =f t ×φ+f t +g t +δf t .(9)

When the IMU is static,f t is equal to ?g t ,and equation (9)

can be written as

˙v

c =?g t ×φ+δf t .(10)

The differential equation of velocity in the true geographic

frame is

˙v

t =0.(11)

Associating equation (10)with equation (11),we can get

the velocity error equation in the true geographic frame

δ˙v

t =?g t ×φ+δf t .(12)

Equations (7)and (12)are the simpli?ed error model and

can be used for alignment and calibration when the IMU is static.

Based on equations (7)and (12),the attitude error mechanization and the velocity error mechanization are shown in ?gures 1and 2,respectively.From these ?gures,it can be seen that the velocity error will not affect the attitude error when the velocity is known to be

zero.

Figure 1.Attitude error

mechanization.

Figure 2.Velocity error mechanization.

3.Calibration method on a low-cost three-axis turntable

The process of the proposed calibration can be described as follows.First,the IMU is aligned using its own gyrocom-pass alignment.After alignment,the IMU is switched to navigate and a Kalman ?lter is started to estimate the rate of change of the velocity error using a tracking model.Finally,the accelerometer nonlinear scale factor can be computed by the rates of change of the velocity error components,rotating from an initial orientation at t =0to a ?nal orientation at t =T .

3.1.Relationship between the navigation errors and the IMU calibration parameters

To improve the accelerometer accuracy by compensation,the nonlinear scale factor should be introduced into the sensor model because some accelerometer calibration coef?cients change as a function of the products of speci?c force components [11].An accelerometer error model can be expressed as

???δf b x δf b y δf b z ???=???aB x aB y aB z ???+???aSF x aMA xy aMA xz aMA yx aSF y aMA yz aMA zx aMA zy aSF z ??????f b x f b y f b

z ???

+

k 2x k 2y k 2z ???f b x f b x f b y f b y f b z f b

z ???

(13)

Figure3.Calibration procedure.

whereδf b=[δf b xδf b yδf b z]T is the accelerometer error in the body frame,and aSF,aMA and aB are accelerometer scale factor errors,misalignments and bias,respectively.k2= [k2x k2y k2z]T is the accelerometer nonlinear scale factor.

A gyroscope error model is generally expressed as

???εb x

εb y

εb z

?

??=

?

?

gB x

gB y

gB z

?

?+

?

?

gSF x00

gMA yx gSF y0

gMA zx gMA zy gSF z

?

?

?

??

ωb x

ωb y

ωb z

?

??(14)

whereεb=[εb xεb yεb z]is the gyroscope error in the body frame,and gSF,gMA and gB are gyroscope scale factor errors, misalignments and drifts,respectively.

The gyroscope and accelerometer errors in the geographic frame can be expressed by equations(15)and(16)[10]

εt=C t bεb(15)

δf t=C t bδf b.(16) Associating equations(13)–(16)with equations(7)and (12),the relationship between the navigation errors and the IMU calibration parameters is obtained.Furthermore,at the beginning and end of the rotation sequences,the rates of change in velocity error are used to estimate the calibration parameters in equations(13)and(14).

The difference between the rates of change of the two horizontal velocity error components,rotating from an initial orientation at t=0to a?nal orientation at t=T,are given as δ˙v E(T)?δ˙v E(0)= δf E?g φN(17)

δ˙v N(T)?δ˙v N(0)= δf N+g φE.(18)

The sum of the upward velocity error component’s rate of change,rotating from an initial orientation at t=0to a?nal orientation at t=T,is given as

δ˙v U(T)+δ˙v U(0)= δf U(T)+ δf U(0).(19) The difference between the upward velocity error component’s rates of change,rotating from an initial orientation at t=0to a?nal orientation at t=T,is given as

δ˙v U(T)?δ˙v U(0)= δf U(T)? δf U(0).(20)

3.2.Rates estimate for velocity error components

The calibration process assumes the availability of the rate of change of velocity error.This availability is satis?ed by using a tracking model,implemented in a Kalman?lter algorithm,to produce estimates of the rate of change of velocity error.The tracking?lter dynamics model for the east velocity component is[3]

?

??

δv n E(T)

δ˙v n E(T)

δ¨v n E(T)

?

??=

?

?

1 t0

01 t

001

?

?

?

??

δv n E(T? t)

δ˙v n E(T? t)

δ¨v n E(T? t)

?

??+Q

E

(21)

where Q E is the process noise vector.

The measurement equation for the east velocity component is[3]

Z E=

100

δv n E(T)δ˙v n E(T)δ¨v n E(T)

T

+R E(22) where R E is the measurement error.

The tracking?lter dynamics models and the measurement equations for the north and upward velocity components are similar to those for the east velocity component,and so will not be rewritten here.

3.3.Data collection rotation sequences and observation equations

Rotation sequences force individual sensor errors to contribute to different components of velocity error and its rate of change. In this paper,six rotation sequences are designed and the calibration procedure is shown in?gure3.For each of these rotations,the initial orientation of the sensor case is along a cardinal axis;that is,east.Data observed at the beginning and end of theπrotations are used to extract component errors. As little as10s at each observation point is suf?cient.

Associating with rotation1,equations(20)and(18)can be respectively expressed as

δ˙v U(T)?δ˙v U(0)=?2aB z?2g2·k2z(23)

δ˙v N(T)?δ˙v N(0)=?2aB y?g·π·gSF x.(24) Continuing with rotation2,similar to the equation(24) with rotation1,equation(18)with rotation2can be written as δ˙v N(T)?δ˙v N(0)=2aB y?g·π·gSF x.(25) Similarly,the following equations are obtained with rotations3–6

δ˙v U(T)?δ˙v U(0)=?2aB x?2g2·k2x(26)

Recorded error

Recorded error Calibration parameters

Simulated error (traditional method)(proposed method)Accelerometer nonlinear scale factors 20/100/50021.5/97.4/503.120.4/201.1/501.0for X ,Y and Z axes,respectively (μg g ?2)20/100/50017.6/98.3/501.919.6/200.7/501.220/100/50018.7/102.6/497.620.8/199.2/498.7Average estimate errors (μg g ?2)

–

1.7/

2.3/2.5

0.5/0.9/1.2

Time (s)

T h e r a t e o f c h a n g e o f v e l o c i t y e r r o r (m /s

2)Figure 4.The rate of change of velocity error estimates for rotations

1–2.

δ˙v N (T )?δ˙v N (0)=?2aB z ?g ·π·gSF y (27)δ˙v N (T )?δ˙v

N (0)=2aB z ?g ·π·gSF y (28)δ˙v U (T )?δ˙v

U (0)=?2aB y ?2g 2·k 2y (29)δ˙v N (T )?δ˙v N (0)=?2aB x ?g ·π·gSF z (30)δ˙v N (T )?δ˙v

N (0)=2aB x ?g ·π·gSF z .(31)

From equations (23),(26)and (29),both accelerometer

bias and the nonlinear scale factor contribute to the rate of change of the upward velocity error.Associating equation (24)with equation (25),aB x can be estimated by the difference between the rate of change of the north velocity error,rotating from an initial orientation at t =0to a ?nal orientation at t =T .Similarly,aB y and aB z can be estimated by equations (27),(28),(30)and (31).After the accelerometer bias is estimated,the accelerometer nonlinear scale factors k 2x ,k 2y ,k 2z can be computed by equations (23),(26)and (29).

The traditional methods compare raw sensor data measured on the rate table with projections of reference rate and gravity for calibration,so results depend on the accuracy of the turntables.The proposed calibration method uses the velocity and attitude indications of IMU to estimate calibration coef?cients,including the accelerometer nonlinear scale factors.This method is relatively robust with regard to instrumentation errors because it has the advantage of using a navigation algorithm that can determine the initial attitude and the amount of change in the angular position.The turntable is used only to rotate the IMU orthogonally from one ?xed position to another,without precise orientation control.

4.Simulation and laboratory testing

In traditional calibration methods,the accelerometer nonlinear scale factor can be estimated based on the least squares algorithm,comparing the IMU outputs with the known references provided by the turntable.The traditional calibration methods are compared with the proposed method by simulation and laboratory testing in this section.

4.1.Simulation

To compare the proposed calibration method with the traditional calibration method,six simulation tests are performed using MATLAB.The simulation conditions are assumed as follows.The accelerometer nonlinear scale factor is set as 20,100and 500μg g ?2,respectively.The rotation angle error of the turntable is set as 3 (1σ)and 3 (1σ),respectively.

In the proposed calibration process,the rate of change of the velocity error is estimated using a tracking model.For example,the rate of change of velocity error estimates for rotations 1–2are shown in ?gure 4.From the ?gure,it can be seen that the accelerometer nonlinear scale factor only affects the rate of change of the upward velocity error,consistent with the equations (23)–(31)in section 3.In other words,the accelerometer nonlinear scale factor can be estimated by the rate of change of upward velocity error.The calibration results with two different rotation angle errors of the turntable are shown in tables 2and 3,respectively.When the rotation angle error of the turntable is 3 (1σ),the average estimate errors are 2.2μg g ?2for the traditional method and 0.9μg g ?2for the proposed method,respectively.When the rotation angle error of the turntable is 3 (1σ),the average estimate errors are 25.8μg g ?2for the traditional method and 1.0μg g ?2for the proposed method,respectively.With the improvement in the accuracy of the turntable from 3 (1σ)to 3 (1σ),the calibration accuracy of the traditional method will be improved at 23.6μg g ?2.In order to meet the calibration accuracy requirement for a high-accuracy accelerometer,a high-accuracy turntable is needed for the traditional method.However,the average estimate errors of the proposed method performed on the two different turntables are close,with a difference of only 0.1μg g ?2.That is,the accuracy of the proposed method does not depend on the accuracy of the turntable.

https://www.360docs.net/doc/301088772.html,boratory testing

In the laboratory test,an IMU is calibrated using the proposed calibration method as well as the traditional method for

Recorded error

Recorded error Calibration parameters

Simulated error (traditional method)(proposed method)Accelerometer nonlinear scale factors 20/100/500 5.4/136.9/536.220.8/201.3/500.8for X ,Y and Z axes,respectively (μg g ?2)20/100/50040.9/72.3/488.320.7/200.5/501.520/100/50056.5/88.5/475.719.3/198.9/498.4Average estimate errors (μg g ?2)

–

24.0/25.4/27.9

0.7/1.0/1.3

Table 4.Calibration results of the accelerometer nonlinear scale factor.

Traditional calibration Proposed calibration Calibration parameters

method

method

Accelerometer nonlinear scale factors ?112.5(for X -axis)?88.9(for X -axis)for X ,Y and Z axes,respectively (μg g ?2)

?38.1(for Y -axis)?68.2(for Y -axis)96.8(for Z -axis)

129.4(for Z -axis)

-5

Position n u m b er

N o r m e r r o r s o f s p e c i f i c f o r c e (g )

Figure 5.Norm errors of speci?c force measurement for static IMU.

comparison.The IMU consists of three gyroscopes with a bias stability of 0.01?h ?1and three accelerometers with a bias stability of 10μg.A high-accuracy three-axis turntable with an accuracy of 5 (1σ)is used in this experiment.First of all,the gyroscopes and accelerometers are compensated with temperature errors.This compensation prevents the gyroscope drifts and accelerometer bias from being affected by temperature errors.Then the IMU is compensated with gyroscope scale factor errors,misalignments and drifts,and accelerometer scale factor errors,misalignments and bias.After compensation,the IMU is mounted on the center of the turntable and the accelerometer nonlinear scale factor is calibrated using the traditional method and the proposed method,respectively.In the calibration process,in order to simulate the low-cost turntable,the accuracy of the high-accuracy turntable is reduced to 3 (1σ)to control the orientation by human intervention.The calibration results of the accelerometer nonlinear scale factor are shown in table 4.The calibration results of the accelerometer nonlinear scale

Table https://www.360docs.net/doc/301088772.html,pensation modes.

Compensation number Compensation description Compensation 1Not compensated

Compensation 2Compensated by the traditional calibration results

Compensation 3

Compensated by the proposed calibration results

factor using the traditional method and the proposed method are inconsistent,with a difference of about 28.7μg g ?2for each axis.The accuracy of the traditional method depends on the accuracy of the turntable,so the calibration result of one axis may be much larger than the true value,while the other axes may be much smaller.The results of which methods are closer to the true values of the accelerometer nonlinear scale factors are validated in the following test.In addition,the proposed method takes not more than 5min for calibration,while the traditional method takes about 40min.

To compare the accuracy of the calibration results,a high-accuracy turntable is required.The high-accuracy three-axis turntable with an accuracy of 5 (1σ)described above can meet the requirements.A set of accelerometer data is collected at 14static positions.The accelerometers are compensated with the traditional calibration results and the proposed calibration results for comparison.The modulus of the speci?c force measured by accelerometers is compared with the gravity theoretically.Norm errors of the speci?c force measurements with different compensation modes (in table 5)are shown in ?gure 5,with standard deviations of 20.7,10.3and 4.6μg for compensations 1–3,respectively.

From the laboratory test results shown above,it can be seen that the traditional calibration method,limited by the accuracy of the turntable,cannot meet the accelerometer nonlinear scale factor calibration requirements,while the calibration method described in this paper can achieve the accuracy on a low-cost three-axis turntable.Furthermore,the proposed calibration method using only six rotation sequences takes just a few minutes,saving a lot of time compared with the multi-position calibration

methods[2]using24rotation sequences,which takes tens of minutes.

5.Conclusion

In this paper,an accurate calibration method for the accelerometer nonlinear scale factor is proposed.First of all, a simpli?ed error model is built,including only velocity and tilt errors.Based on the error model,the relationship between the navigation errors(the rates of change of velocity errors) and the IMU calibration parameters is presented.The rate of change of velocity error is estimated using a tracking model.A special calibration procedure is designed,forcing individual sensor errors to contribute to different components of velocity error and its rate of change.In this calibration procedure,both accelerometer bias and nonlinear scale factor contribute to the rate of change of the upward velocity error. After accelerometer bias is estimated by the rate of change of the north velocity error,the accelerometer nonlinear scale factor can be obtained.The accurate calibration method for the accelerometer nonlinear scale factor,described in this paper, features several de?nite advantages.These are:

?high accuracy,

?use of a low-cost three-axis turntable,

?fast,taking only several minutes.References

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