A Rapid Initial Alignment Alogrithm Based on Strapdown Gyrocompass

LI Si-hai, YAN Gong-mina, YANG Peng-xiangb and QIN Yong-yuan

College of Automation, Northwestern Polytechnical University, Xi’an, 710072, China

ayangongmin@163.com, bpengxiangyang@gmail.com

Keywords: Strapdown Inertial Navigation System(SINS); gyrocompass; fast initial alignment

Abstract. The basic principles for stabilized gyrocompass initial alignment are analyzed in platform

inertial navigation system (PINS), then similar principles and initial alignment algorithms suitable

for programming are proposed for strapdown inertial navigation system (SINS). The scheme of

SINS gyrocompass initial alignment can be divided into four steps, including leveling alignment

with header uncertainty, coarse header alignment, leveling realignment and gyrocompass alignment

for header. By simplifying SINS nonlinear error model under header uncertainty, the formula of

coarse header alignment is deduced. On the assumption of navigation computer having large

memory and powerful computing ability, and basing on the ‘multiformity’ of SINS mathematical

platform and the ability to attitude reverse control, a specific progress for SINS rapid gyrocompass

alignment is introduced and designed in detail. Finally, some tests prove that the proposed

alignment algorithm in this paper is effective.

Introduction

The scheme of gyrocompass initial alignment in platform inertial navigation system (PINS)

usually can be divided into two steps, with leveling alignment firstly and then header alignment.

Header alignment usually adopts the method of gyrocompass alignment after the finishing of

horizontal leveling.SINS alignment is usually divided into two stages: in coarse alignment stage,

coarse calculating navigation system is built through the measurement of inertial devices which use

the earth's rotation angular velocity and gravity acceleration as reference; in precise alignment

stage, misalignment angle is evaluated by optimal estimation method through modern control

theory to get precise attitude matrix[1,2].

The classic SINS analytic coarse alignment method isn’t applicable in vibration environment.

There are many literatures about initial alignment in dynamic environment and also some

application examples, in particular, the development of laser gyroscope and fiber-optic gyroscope

has been putting forward the study of strapdown gyrocompass [3-6]. Essentially, SINS and PINS

are identical in some point of view, the former uses the mathematical tools (attitude matrix,

quaternion or Euler angle) to simulate the latter's entity platform, and the mathematical tools

describe strapdown system's frame with respect to special reference frame. It is well known that

entity platform in PINS has an isolation function from external interference, thus the gyrocompass

can realize initial alignment in a dynamic base environment, similarly, the corresponding

mathematical platform of SINS can also be established according to the characteristics of PINS

gyrocompass alignment to isolate external interference. Classical control theory compared to

modern optimal estimation method, the former has an advantages of not requiring accurate math

model and noise model, moreover, the method to design gyrocompass alignment using classical

control theory application is easy to reach. However, the platform gyrocompass alignment has a

flaw of very long north-seeking time. Whereas, taking OctansIII fiber-gyro compass from iXSea as

an example, it has the ability of accomplishing initial alignment in 3 minutes in dynamic

environment and reaches a precision of 0.2º×sec(L)[5].

Based on the analysis of the platform gyrocompass initial alignment, the principle of strapdown

gyrocompass initial alignment and software programming algorithm are presented. According to the

characteristics of SINS, a specific procedures to shorten strapdown gyrocompass initial alignment

time are designed.

Advanced Materials Research Vols. 532-533 (2012) pp 1563-1567Online available since 2012/Jun/14 at www.scientific.net© (2012) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMR.532-533.1563

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Gyrocompass Initial Alignment Principles in PINS

In this paper, "east-north-up" geographic coordinate, denoted as n, have been selected as

navigation reference frame, and "right-front-up" frame, denoted as b, have been selected as SINS

body frame.

The control principles of PINS gyrocompass initial alignment are shown in Fig.1-Fig.3. The

symbols in figures are define as following:

E∇ and

N∇ - accelerometer bias for east and north direction;

xa and

ya - environment disturbance acceleration;

n

sfxf� and n

sfyf� - accelerometer outputs;

Eε ,

Nε and

Uε - gyro drifts;

Eϕ ,

Nϕ and

Uϕ - platform misalignment angles;

cxω ,

cyω and

czω - control angular rate along east, north and up direction respectively;

ieω ,g, R and L - earth rotation rate, earth gravity, earth radius and local geographic latitude,

assumed as constants.

xE a+∇1/s 1/R 1/s

g

Kx1

Kx2/R

Kx3/s

EVδ cyω yφ

Nε

n

sfxf~

control priniples

yN a+∇1/s 1/R 1/s

g

Ky1

Ky2/R

Ky3/s

NVδ cxωxφ

Eε

n

sfyf~

control principles

Liez cosωφ

Fig 1. East control loop of leveling alignment Fig 2. North control loop of leveling alignment

yN a+∇1/s 1/R 1/s

g

Kz1 Kz2/R

1/s

NVδ cxω xφ

Eε

n

sfyf~

K(s)Uε

Lie cosω

czω zφcontrol principles

Fig 3. Header control loop of compass alignment

Also in these figs, the leveling alignment uses a three-order leveling loop and the header

alignment uses four-order gyrocompass alignment loop on the basis of two-order north control loop

of leveling alignment.

Usually, ( ) /[( ) cos ]z3 z4 ie

K s K s K ω L= + is choose in gyrocompass alignment loop, where

( , , ; 1, 2,3, 4)ijK i x y z j= = are parameters in alignment control principles. According to the compass

alignment performances, the typical parameters of east leveling control loop and header control

loop are shown below: x 1

2 2 2

x 2

3 2

x 3

3

( 2 1 / ) / 1

/ ( )

s

K

K

K g

σ

σ ξ ω

σ ξ

=

= + − =

(1)

z1 z4

2 2

z2

4

z3

2

4 / 1

4 /

s

K K

K

K g

σ

σ ω

σ

= =

= − =

(2)

The performances selection of north control loop is similar to that of east control loop. In (1) and

(2), σ ,ξ and /s

g Rω = are decaying coefficient, damping ratio and schuler frequency respectively.

Under certain alignment accuracy and rapidity request, the decaying coefficient is always adjusted

according to environment interference in practice.

1564 Materials Science and Information Technology II

Strapdown Gyrocompass Initial Alignment Principle and Algorithm

The entity platform in PINS is replaced by mathematical platform in SINS, showed in Fig.4. In

this figure, n

bC� means strapdown attitude matrix, acts as mathematical platform, b

ibω� and b

sff� mean

outputs of gyro and accelerometer, T

c cx cy czω ω ω = ω is the control angular rate vector to

mathematical platform, and [ ]T0 cos sinn

ie ie ieL Lω ω=ω , b

sff� is transformed to

Tn n n n

sf sfx sfy sfzf f f = f � � ��

by matrix n

bC� .

In the platform control principles Fig.1 - Fig.3, part of the signal flow represents the motion law

of the entity platform, another part means alignment control rule. By transplanted the platform

signal flow into strapdown gyrocompass alignment, entity platform is replaced by mathematical

platform, but the control principles don't have to make any change. Such as the east loop, strapdown

gyrocompass leveling alignment is built by the combination of Fig.1 and Fig.4, and it is showed in

Fig.5. In Fig.5, the measurement errors of gyros and accelerometers are implicit in the mathematical

platform solutions. The difference between Fig.1 and Fig.5 is that the former is represented as

platform error angle, which shows the error propagation principle directly, but the later shows that

in mathematical platform. In fact, both entity and mathematical platforms are identical essentially

about error propagation, however, Fig.5 is more convenient for algorithm programming and

comprehension.

b

sf

n

b

n

sf fCf~~~

=

])~~~[(

~~c ×−−= ωCωCωCC

b

n

n

ie

b

n

b

ib

n

b

n

b

�

b

ibω~ b

sff~

n

sff~

cω

b

sf

n

b

n

sf fCf~~~

=

])~~~[(

~~c ×−−= ωCωCωCC

b

n

n

ie

b

n

b

ib

n

b

n

b

�

b

ibω~ b

sff~

n

sfyf~ cxω

1/s 1/R

Kx1

Kx2/R

Kx3/s

EVδ cyωn

sfxf~

n

sfzf~

czω

EPδ

Fig 4. SINS algorithm platform Fig 5. SINS east control loop of leveling alignment

Initial Alignment Method of Strapdown Gyrocompass

The first step of PINS gyrocompass initial alignment is always given a coarse header angle by

external device. In SINS, if the coarse angle is provided first, the course of strapdown gyrocompass

alignment will be the same to that of platform. But in this paper, a gyrocompass initial alignment

method without external coarse header is introduced, and the scheme can be divided into four steps

with detail in the following.

A. Leveling Alignment with Uncertain Header Angle. With uncertain header, it will work by

set the initial value to zero. For the leveling attitude angles, they are generally not very large in the

course of initial alignment for vehicles and ships (such as less than 30º), and they are also assumed

to be zeros in initial value setting. After the initialization of strapdown attitude matrix, Fig.5 and

Fig.6 are used for leveling alignment scheme, while just setting the control angle rate component

czω to zero. Because of the uncertain header angle, header error angle may be very large, the earth

rotation angular rate can be equivalent to gyro drift and seen as a disturbance to leveling alignment.

By analysis of the third-order leveling alignment loop, it is easy to know that the earth rotation

interference doesn't influence the precision of leveling alignment, and the steady precision still lie

on the accelerometers bias. It is assumed that attitude matrix 1

( )n

b htC� is obtained at the leveling

alignment step.

B. Coarse Header Self-Alignment. After the first step, leveling attitude error can reach

requirement of small values at several arc-minute level, while assume header angle error to be large.

In this step, use (3) and (4) to proceed strapdown inertial navigation velocity update despite of large

header error, and the navigation velocities also represent velocity errors in static base.

Advanced Materials Research Vols. 532-533 1565

[( ) ]n n b b n

b b ib n ie= − ×C C ω C ω

�� � �� (3)

n b

b sfδ =V C f� ��

(4)

where [ ]TE N UV V Vδ δ δ δ=V� , and the initial mathematical platform and velocity are set as

1( )n

b htC� and (0) 0δ =V� respectively. Secondly, simplify strapdown inertial navigation nonlinear error

formula under large header error[7] and by ignoring some secondary factors, the relationship

between velocity error and large error angle z

ϕ can be written as

cos (cos 1)E ie zV g Lδ ω ϕ= −��

(5)

cos sinN ie zV g Lδ ω ϕ= −��

(6)

that is

2cos 2 ( ) / ( cos ) 1

z E z z ieV t t g Lϕ ϕϕ δ ω= +

(7)

2sin 2 ( ) / ( cos )

z N z z ieV t t g Lϕ ϕϕ δ ω= −

(8)

At last, set up correcting matrix as

cos sin 0

sin cos 0

0 0 1

z z

z z zϕ

ϕ ϕϕ ϕ

− =

C and make multiplication to the

attitude matrix ( )n

b ztϕC� , then the coarse self-alignment attitude matrix is obtained like this

'( ) ( )n n

b z b z zt tϕ ϕ ϕ=C C C� �

(9)

C. Leveling Re-Alignment. Mathematical platform 1

( )n

b htC� is already obtained in step A, but via

step B of coarse header self-alignment, the leveling error angle in '( )n

b ztϕC� might become rather

large (such as 0.5º). Therefore, a leveling re-alignment under the condition of none-large header

error is needed, where it is different from step A. To this point, all preparation has been done for the

later gyrocompass header alignment.

D. Gyrocompass Header Alignment. After coarse header alignment and precise leveling

alignment, control principles of Fig.5 and gyrocompass control principles will be executed

simultaneously. Using Fig.5 to keep accuracy of east alignment channel, while in gyrocompass

control principles, the main effort is to reach gyrocompass header alignment and also keep north

alignment accuracy.

It is easy to see that step C and D are the same as platform gyrocompass alignment course with

external coarse header angle, and the control parameter setting in these two steps can also refer to

parameter design in platform gyrocompass alignment. But the difference between classic platform

gyrocompass alignment and method in this paper is that a header alignment method is presented

through SINS nonlinear error equation under large header error, and the coarse header error is

obtained from velocity errors.

Alignment Tests and Conclusion

In the tests, gyrocompass initial alignment tests are carried out using laser gyro strapdown

inertial measurement unit (LGSIMU), with gyro drift stability being 0.01º/hr and accelerometer bias

being 0.5×10-4g. Install LGSIMU on vehicle and preheat for preparation. At the beginning, the

vehicle remains static for 300s, then test persons perform motions including opening and closing

door, getting on and off the vehicle, walking on the car, and interference continues for about 300s.

All of the 600s sampling data of gyro and accelerometer are stored to computer for later data

processing. Two data processing algorithms are presented:

1566 Materials Science and Information Technology II

(1) Taking original data from 0s to 300s, traditional Kalman filtering method is used for initial

alignment, then attitude tracking is executed from 300s to 600s and the header tracking is seen as a

reference to gyrocompass alignment result.

(2) Using data from 300s to 600s, initial alignment test is executed according to the procedures

in section 4. A header error of about 3º is achieved in the coarse header-alignment step.

Gyrocompass header alignment step are conducted for 4 times repeatedly and the header errors

between gyrocompass header alignment and attitude tracking above are showed in Fig.6, where the

numbers on curves denote the times of gyrocompass alignment. As can be seen from the figure, the

third curve has been small to less than 0.02º and it can achieve header alignment accuracy

requirement, while the fourth curve no longer reduces error essentially. By comparison, in the

conventional method it takes at least 15min for completing the initial gyrocompass alignment, but

in the improved method it only needs 5min sampling data to finish the alignment, so the improved

gyrocompass initial alignment method presented in this paper can shorten the alignment time

effectively.

Fig 6. Header errors in SINS gyrocompass alignment test

References

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[3] QIN Yongyuan, YAN Gongmin, GU Dongqing,et al. A clever way of SINS coarse alignment

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[4] Sandoval Romero G E. Fiber Optic Gyrocompass Superluminescent Fiber Source. IEEE A&E

Systems Magazine,2005,7:19-20

[5] iXSea Ltd, OctansIII UG Part 1 Introduction MU-OCTIII-002-A.pdf[EB/DK],2004,7

[6] WANG Jin. Study on gyrocompass alignment for SINS. Changsha: National University of

Defense Technology,2005

[7] DING Yangbin, WANG Xinlong, WANG Zhen, et al. Study on unscented Kalman filter

applied in initial alignment of large azimuth misalignment on static base of SINS.Journal of

Astronautics,2006,27(6):1201-1204

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Advanced Materials Research Vols. 532-533 1567

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