Research of Strapdown Inertial Navigation System Monitor Technique Based on Dual-axis Consequential

Rotation

CHENG Jianhua, LI Mingyue, CHEN Daidai, CHEN Li SHI Junyu

College of Automation Beijing Aerospace Times Laser Inertial Technology Company Ltd

Harbin, Heilongjiang Province, China Beijing, China dream-flying-a@163.com Shijunyu173@sina.com

Abstract - According to the accumulation characteristic of SINS positioning error caused by inertial sensor error, this paper brought the rotation monitor technique to improve the SINS accuracy. Dual-axis consequential rotation scheme is designed by deeply analyze toward the existing single-axis and two-axis scheme. This scheme is selected as the final scheme by analyzing modulation results of constant gyro drift, the gyro scale error and the deviation of the gyro installation error. Computer simulation shows that this scheme has better modulation than single-axis and two-axis continuous scheme and can improve position accuracy of SINS effectively. Index Terms - Strapdown Inertial Navigation System, Error Analyze, Rotation Modulation, sensor error.

I. INTRODUCTION

Strapdown Inertial Navigation System (SINS) based on the optical gyro has various advantages, such as minor size, light weight, high dependability, low cost. And SINS has been progressively superseding the platform inertial navigation system, and being widely used aviation, aerospace, voyage and land navigations, it represents the future trend of inertial navigation system.

As the inertial components of SINS directly fixed on the carrier, the inertial components were impacted by the working conditions of oscillation, temperature, a wide range of angular motion and linear motion and other factors, leading to quite a large system dynamic errors and components errors, which cause the accumulation errors of navigation inevitably, and restrict SINS from application of high-precision navigation.

As an effective measure to improve the precision of navigation and positioning on a long-time continuously working condition for SINS, this topic has also arouse lots of interests of many scholars. By introducing single-axis, dual-axis or triple-axis rotation structures, this technique could modulate the errors of the inertial components effectually so as to greatly improve the accuracy of the inertial navigation system, without introducing any external navigation infor-mation. For example, the widely used MK39 and MK49 Ship’s Inertial Navigation System adopted the single-axis and dual-axis rotation monitor technique respectively, which improve the accuracy of SINS from 1 mile/8h to 1mile/24h[1]. Therefore, researching and contriving an appropriate scheme of rotation monitor technique have important practical significance for improving the long-term accuracy of SINS.

II. ANALYSIS OF THE ROTATION MONITOR SYSTEM

A. Analysis the errors of SINS The error of the inertial components (gyroscopes and

accelerometers) is the most important source of the system error, including its main errors the gyro constant drift and the acceleration bias ε , ∇ , respectively , the scale errors ( gKδ , aKδ ) and the installation errors ( gKδ , aKδ ).

The constant drift will inspire not only three kinds of periodic oscillating errors, but the longitude error accumulated over time, as (1) shown:

tt tztytx

ies ϕεϕεε

ωϕδλ sincostan −−= (1)

Where, ieω is the rotary angular rate of the earth, ϕ is the

latitude, txε , tyε , tzε are the gyros constant drifts coordinating to the directions of east, north and up of the geographical coordinate system respectively, t is the time,

sδλ is the longitude steady state error. By (1) can be seen that the longitude error of SINS will

increase with the time. Therefore, the measures to compensate or modulate for the inertial component errors are required for improving the long-term accuracy of SINS.

Assumed the equivalent east, north and up gyros constant drifts with periodic oscillation characteristics, namely:

��

��

�

===

tttttt

tztz

tyty

txtx

ωεεωεεωεε

sin)(sin)(sin)(

and ies ωωω >>>> (2)

Where, sω is the angular frequency of Schuler oscillation. In the gyro drift as in (2), the steady state error of latitude

is:

yyy

yiesxx

xs tB

tA

t εωω

ϕωωωεωωωϕ sin

sincos)(

22

+=Δ

zzz

zies tC

εωω

ϕωωω sincos2

− (3)

Proceeding of the IEEE International Conference on Information and Automation Shenzhen, China June 2011

978-1-61284-4577-0270-9/11/$26.00 ©2011 IEEE 203

tystxies tA

tA

t εω

ϕωϕωω

εωϕωω

δλ ���+

�−= coscos

secsin

tan)(

22

tzies tA

εω

ϕωω

ϕωω��

�

�+− sincos

sin22 (4)

Where, ))(( 2222 ωωωω −−= iesA . By (4) can be seen, the longitude error contains only a

tiny constant component, not including the error accumulated over time. The problem of SINS positioning error accum- ulated over time will be solved, if the SINS gyro drift with the form (2) or can transform into this form through certain systematic technologies.

In order to explain the error characteristics inspired by the gyro constant drift and periodic drift, two simulated environments are provided for the comparison of the errors:

(1) hyx /0.01�== εε ;

(2) h�tyx /600)/sin(20.01 �⋅== εε . The simulation position is 45.7°N and 126.6°E, the

longitude errors corresponding to the above two conditions are shown in Fig.1. And the error causing by the periodic oscillation of the gyro drift greatly attenuates than the constant drift. So the modulation change the constant drift into the periodic oscillation drift can greatly improve the positioning precision of SINS.

The SINS rotation monitor technique employs the rotary structures can modulate the inertial components constant error into the periodic oscillation error, so as to the improvement of SINS accuracy.

Figure.1 Comparison of the accuracy errors simulations

B. Single-axis rotation monitor system The single-axis rotation monitoring system introduces a

single degree of freedom rotation structure to modulate the inertial components error. And the rotation can be one-way continuous rotation, intermittent rotation or reciprocating rotation. For example, when SINS rotates around the axis zr with the angular rate of ω , the rotation of the IMU framework coordinate system (coordinate system r ) relative to the carrier coordinate system makes the gyro drift along the axes of r coordinate system been modulated. The modulation effects of the gyro drift and accelerometer bias are:

���

∇⋅=∇⋅=

rbr

b

rbr

b

TT εε

(5)

Take gyroscope as an example, after the rotating of the framework coordinate system, the modulation of the gyro drift is:

��

��

�

=+=−=

tztz

tytxty

tytxtx

tttt

εεωεωεεωεωεε

1

1

1

cossinsincos

(6)

Where, 1txε , 1tyε , 1tzε are the modulated gyros constant drifts in the geographical coordinate systems in the directions of east, north and up.

Since the existence of the scale error and the installation error, the introduction of rotation structure will inspire a new source of error. When the rotation structure rotates around the axis zr , the asymmetry errors of the scale factor will make the accumulation of equivalent gyro drift in one rotation cycle as follows:

��

�

��

�

�

+=

+=

=

���

−+

++

ωπ

ωπ

ωπ

δδπδω

δδω

ϕπωδω

δω

/2

0

/2

0

/2

0

)(2

)(cos0

gzgzbibz

gygxieb

iby

bibx

KKdt

KKdt

dt

(7)

When taking the scheme with clockwise and counter clockwise around the axis zr , the equivalent gyro drift is:

���

���

�

+=

=

=

���

−+ωπ

ωπ

ωπ

δδπδω

δω

δω

/4

0

/4

0

/4

0

)(4

0

0

gzgzbibz

biby

bibx

KKdt

dt

dt

(8)

Equation (6) is the gyro drift along x, y-axis change into periodic drift after the rotation. Meanwhile, the accelerometer bias along x, y-axis is modulated to be periodic[3]. However, the scheme of single-axis rotation can’t completely modulate all of the gyro drifts along three axes, so it has not radically solved the problem that the SINS errors accumulated over time.

In this case, the rotation axis’s stabilities of the drift and scale factor will have a huge impact to the entire SINS. Compared with the scale factor stability of laser gyroscope better than 0.5ppm, the fiber optic gyro (FOG) is difficult to achieve this accuracy presently. So, SINS based on FOG has to adopt more degree of rotation scheme to improve long-term accuracy.

C. Dual-axis rotation monitor system Since the scheme of single-axis rotation can not modulate

the gyro drifts along three axes concurrently, the solution by increasing the rotation degree of freedom, such as dual-axis or triple-axis rotation structure, must be adopted to improve the modulation effects.

Dual-axis rotation monitor system modulates the inertial components error by introducing the rotation structure with two degree of freedom. And the manner of its rotations can be

204

continuous rotation, consequential rotation or reciprocating rotation. As shown in Fig.2, the system can rotate around the east axis and the up axis.

Figure.2 Dual-axis rotation monitor system

Once the scheme of dual-axis rotation monitor system was adopted, the design and selection of the rotation scheme is essential for the modulation results. For example, when SINS was rotating around the Z, X-axis simultaneously with the angular rate ω , the equivalent gyro drift is:

rXZ

b CC εε ⋅⋅= (9) Where, ZC , XC are coordinate transformation matrix rotation

around the axes of roz , rox :

���

�

�

�−=

1000cossin0sincos

11

11

tttt

CZ ωωωω

(10)

���

�

�

�

−=

ttttCX

22

22

cossin0sincos0

001

ωωωω (11)

The equivalent gyro drift after the modulation goes to:

���

�

�

�

+−+

+−=

���

�

�

�

tttttttttt

tzty

tztytx

tztytx

tz

ty

tx

ωεωεωωεωεωεωεωωεωε

εεε

cossincossincossin

sinsincoscos2

2

1

1

1

(12) We can learn from (12), even though the degree of

freedom of the rotation structure has been increased, the modulation results can not be improved. Therefore, for dual-axis rotation monitor system, a reasonable design of the rotation scheme is demanded to modulate the components error to the max.

III. ANALYSIS OF THE MODULATION EFFECT OF DUAI-AXIS CONSEQUENTIAL ROTATION

The inertial component constant error, the scale error and the installation error are the three important error sources of SINS. So the design of the scheme of rotation monitor system mainly works over the modulation of these three errors.

By (6), when SINS rotating around one axis, the gyro drifts along another two axes are modulated at the same time,

which attracts many academicians to research the modulation manner of dual-axis consequential rotation, by designing a reasonable rotation manner and rotation sequence, to realize the effective modulation of the components error[5][6][7].

The designed dual-axis consequential rotation orders are shown in Fig.3.

(a) rotation orders 1-4 (b) rotation orders 5-8

Figure.3 dual-axis consequential rotation manner

According to Fig. 3, the 8 rotation sequences are: Order 1: clockwise rotation around the X-axis for 180°

from A to B; Order 2: clockwise rotation around the Z-axis for 180°

from B to C; Order 3: counter clockwise around the X-axis for 180°

from C to D; Order 4: counter clockwise the X-axis for 180° from D to

A; Order 5: clockwise rotation around the Z-axis for 180°

from E to F; Order 6: counter clockwise the X-axis for 180° from F to

G; Order 7: clockwise rotation around the Z-axis for 180°

from G to H; Order 8: counter clockwise the east axis for 180° from H

to E;

A. Analysis of the modulation effect of dual-axis consequential rotation to the constant error

To simplify the analysis, assuming the frame coordinate system r r rox y z , the carrier coordinate system b b box y z and

the geographic coordinate system t t tox y z coincide at the initial time, from which we can calculate the equivalent gyro drifts during the 8 rotation progresses, as follows:

riri

br

bib T δωδωε )(== (13)

Where, brT is the transformation matrix from the coordinate

system r to b , rirδω is the gyro drift along the axes lines of

coordinate system r , 1,.....,8i = . The X-axis, for example, its equivalent gyro drift in the

geographical coordinate system is:

205

�����

�

�����

�

�

+−==

−=−=

+−=−=

−==

tt

tt

tt

tt

tytxtx

txtx

tytxtx

txtx

tytxtx

txtx

tytxtx

txtx

ωεωεεεε

ωεωεεεε

ωεωεεεε

ωεωεεεε

sincos

sincos

sincos

sincos

8

7

6

5

4

3

2

1

(14)

In a rotation cycle, according to the integral calculation of (14), the equivalent gyro drift becomes zero, as well as the Y, Z-axis:

���

���

�

=++

=++

=++

���

ωπ

ωπ

ωπ

εεε

εεε

εεε

/

0

821

/

0

821

/

0

821

0)(

0)(

0)(

dt

dt

dt

tztztz

tytyty

txtxtx

�

�

�

(15)

As shown in (15), the scheme of dual-axis consequential rotation monitor can simultaneously modulate the gyro drift along three axes lines effectively.

B. Analysis of the modulation effect of dual-axis consequential rotation to the scale error

When the gyro scale error exists, its matrix is:

0 00 00 0

gx

g gy

gz

KK K

K

δδ δ

δ

� � �= � � �

(16)

Where, gxKδ , gyKδ , gzKδ are the scale errors of the gyro x , y , z , respectively.

By analysing, the equivalent gyro drift error modulated by the scale error during the progress of rotating is:

bir

rbg

br TKT ωδε = (17)

According to (17), the equivalent drift of rotation sequences from 1 to 4 can be obtained[8].

Order 1:

���

�

�

�

���

�

�

�

=ϕωϕω

ω

δδδδ

δε

sincos

00

00

12

211

ie

ie

gg

gg

gx

KKKK

K (18)

Where,

tKtKK gzgyg ωδωδδ 221 sincos +=

ttKttKK gzgyg ωωδωωδδ cossincossin2 +−= .

Order 2:

���

�

�

�

+���

�

�

�

=ωϕω

ϕωδ

δδδδ

εsin

cos0

0000

34

43

2

ie

ie

gz

gg

gg

KKKKK

(19)

Where, tKtKK gygxg ωδωδδ 223 sincos += ;

ttKttKK gygxg ωωδωωδδ cossincossin4 +−= .

Order 3:

���

�

�

� −

���

�

�

�

=ϕωϕω

ω

δδδδ

δε

sincos

00

00

12

213

ie

ie

gg

gg

gx

KKKK

K (20)

Order 4:

���

�

�

�

−���

�

�

�

=ωϕω

ϕωδ

δδδδ

εsin

cos0

0000

34

43

4

ie

ie

gz

gg

gg

KKKKK

(21)

The angular rate of the framework rotation meets ω >> ieω , so the integral calculations of the equivalent drifts in the sequences from 1 to 4 approach to zero. Which means the manner of dual-axis in turn rotation will not introduce the navigation and positioning error inspired by the scale error during the rotary motion of rotation structure.

C. Analysis of the modulation effect of dual-axis consequential rotation to the installation error

As the equivalent gyro drift caused by the installation error can be expressed as:

bir

rb

gzxgzy

gyxgyz

gxygxzb

rbir

rb

br T

EEEEEE

TEgTT ωδδ

δδδδ

ωδε���

�

�

�

==0

00

(22)

According to (22), the equivalent drifts in the orders from 1 to 4 can be gained[8].

Order 1:

���

�

�

�

+−+=

ωωδωδωωδωδε)cossin(

)sincos(0

1

tEtEtEtE

gzygyz

gzygyz

Order 2:

���

�

�

�++−

=0

)cossin()sincos(

2 ωωδωδωωδωδ

ε tEtEtEtE

gyxgxy

gyxgxy

Order 3:

���

�

�

�

−+−−−−=

))(cossin())(sincos(

0

3

ωωδωδωωδωδε

tEtEtEtE

gzygyz

gzygyz

206

Order 4:

���

�

�

�−−−−+−

=0

))(cossin())(sincos(

4 ωωδωδωωδωδ

ε tEtEtEtE

gyxgxy

gyxgxy

An equation:

dtdt )( 432

/

0 1 εεεεεωπ

+++=� � (23)

According to (23), the integral calculations of the equivalent drifts in the orders from 1 to 4 are as follows:

���

���

�

=+=

+=

=+=

�� ��

000)(

sin2)()(

000)(/

0

dt

tdtEEdt

dt

z

gxygzyy

x

εωωε

εωπ

(24)

And the orders from 5 to 8 become:

���

���

�

=+=

+−=

=+=

�� ��

000)(

sin2)()(

000)(/

0

dt

tdtEEdt

dt

z

gxygzyy

x

εωωε

εωπ

(25)

From the sum of (24) and (25), the manner of dual-axis consequential rotation will not introduce the navigation and positioning error inspired by the installation error during the rotary motion of rotation structure.

Through the above analysis, the scheme of dual-axis in turn rotation monitor system can simultaneously achieve the modulations of the constant error, the scale factor error and the installation error, without introducing any new navigation and positioning error inspired by the installation error because of the angular rate ω of the framework rotation.

IV. SIMULATION

Assume the error sources of simulation as follows: The gyro drifts are 0.003 /x y z hε ε ε= = = � . The acceleration

biases are 53 10x y z g−∇ = ∇ = ∇ = × . The scale errors are 66 10x y zK K Kδ δ δ −= = = × . The installation errors are

68 10gyx gzy gxzE E Eδ δ δ −= = = × , 51 10gxyEδ −=− × , 66 10gzx gyxE Eδ δ −= − = × .

The angular rate of the framework rotation is 45 / minω = �.

Fig. 4 to Fig. 8 denote the simulation results.

Figure.4 The position error inspired by the constant error of single-axis continuous rotation

Figure.5 The position error inspired by the constant error of dual-axis

continuous rotation

Figure.6 The position error inspired by the constant error of dual-axis

consequential rotation

Figure.7 The position error inspired by the scale error of dual-axis

consequential rotation

207

Figure.8 The position error inspired by the installation error of dual-axis

consequential rotation

Comparing Fig. 4, Fig. 5 and Fig. 6, the continuous rota- tion of single-axis and dual-axis can not modulate the constant error completely so that can not suppress the divergence of SINS positioning error. And the scheme of dual-axis in turn rotation can simultaneously achieve the effective modulation of the constant error along three axes lines, causing the positioning error attenuated. From the Fig.7 and Fig. 8, the dual-axis in turn rotation can effectively modulate the equivalent inertial component error introducing by the scale error and the installation error.

V. CONCLUSION

In order to improve strapdown inertial navigation system performance, the scheme of dual-axis rotation scheme is adopted, which is based on the analysis of single-axis and dual-axis rotation characteristics. Through the theoretically analysis and simulation, this scheme can modulate the constant error, the scale error and the installation error effectively. Therefore it is an effectively method to improve long-time navigation and positioning precision of SINS.

ACKNOWLEDGEMENT

This paper is supported by the Fundamental Research Finds for the Central Universities(HEUCF11401).

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