Performance improvement of in-flight alignment forautonomous vehicle under large initial heading error

H.S. Hong, J.G. Lee and C.G. Park

Abstract: The paper presents a new method of in-flight alignment (IFA) for a strapdown inertialnavigation system (SDINS) aided by a super short baseline (SSBL) system under a large initialheading error. To handle the large heading error, a new attitude error model is introduced in whichattitude errors are divided into heading and levelling errors using a newly defined horizontal frame.Some navigation error dynamic models are derived from the attitude error model for indirectfeedback filtering of the IFA system. A Kalman filter with position measurements is designed toestimate navigation errors as an indirect feedback filter. Simulation results show that the newlydeveloped model works effectively for coarse alignment of a large heading error. The proposed IFAmethod very quickly decreases the heading error from more than 408 to about 58.

1 Introduction

A strapdown inertial navigation system (SDINS) is used toobtain position, velocity, and attitude information forvehicles, such as aeroplanes, ships, cars, and guidedweapons [1, 2]. For SDINS, the accuracy of attitudeinformation governs the accuracy of velocity and positionestimation. In the process of calculating navigationvariables, initial attitude errors propagate other navigationvariable errors because attitude angle terms are integratedand applied for velocity and position calculations.

Frequently, in SDINS applications, vehicles are forced toinitialise their attitude while moving [3, 4]. For example, anautonomous underwater vehicle (AUV) with SDINS usuallyestimates its initial attitude using only an in-flight alignment(IFA) method. Keeping an AUV stationary is time- andspace-consuming. In many practical situations, an AUV hasto be periodically realigned to prevent divergence due tounexpected bumps. Presently available IFA methods aregenerally adequate for the small initial attitude error caseand require only coarse calculations of initial attitude.However, the accuracy of such a coarse calculation of initialheading angle is very poor when the vehicle is moving. Amethod is required that estimates heading angle for the largeerror case. Some research has been conducted regarding thelarge heading error case [4–7]. Some INS error formu-lations for a large heading error are presented in [4–6], anda nonlinear filtering application is presented in [7].Rogers [4] considers the heading error as an initial wander

azimuth error to separate the large heading error F smalllevelling errors. The initial wander azimuth error can bedirectly estimated by position measurement. The estimationefficiency of this method is proportional to the rate andmagnitude of position change. However, if the vehicle ismoving at only several metres per second, the change invehicle position over a few minutes is small, and the wanderangle error cannot be estimated quickly.

This paper considers an SDINS error formulation andextends Rogers’ work. A new navigation error model ispresented to accommodate a large heading error alignmentfor a vehicle that travels much less quickly than an airplane.With AUVs, the heading error is generally much larger thanlevelling errors. If levelling errors can be held to smallvalues, separation of heading error and levelling errors ispossible via a new horizontal frame introduced in the model.In addition, an IFA method using the new linear error modelis proposed to estimate heading error without initial headinginformation. A Kalman filter is employed for alignment.Computer simulation shows the model’s effectiveness inIFA under large initial heading error, and also demonstratessatisfactory performance of the filter. Another simulationcompares the proposed error model with the conventionalsmall angle model in the worst case.

2 Navigation error dynamic model

2.1 Coordinate systems and the SDINSmodel

The coordinate systems used in this paper are inertialframe (i-frame), Earth-fixed frame (e-frame), navigationalframe (n-frame), horizontal frame (h-frame), and body frame(b-frame).

The i-frame is fixed with the centre of the Earth as theorigin. The e-frame coincides with the i-frame at the originbut rotates with the Earth rate. The n-frame is a local levelframe in which the vertical axis is parallel with the gravityvector. Axes of the n-frame is point in the directions north,east, down in that order. In the n-frame, the rate vectors arerepresented as [1, 2]

q IEE, 2003

IEE Proceedings online no. 20030698

doi: 10.1049/ip-rsn:20030698

H.S. Hong is with Samsung Electronics Co., Korea

J.G. Lee is with the School of Electrical Engineering & Automatic ControlResearch Center, Seoul National University, Kwanak-ku, Seoul 151-744,Korea

C.G. Park is with the School of Mechanical and Aerospace Engineering &The Institute of Advanced Aerospace Technology, Seoul NationalUniversity, Kwanak-ku, Seoul 151-744, Korea

Paper first received 28th May 2002 and in revised form 29th May 2003

IEE Proc.-Radar Sonar Navig., Vol. 151, No. 1, February 2004 57

vnin ¼ vn

ie þvnen

vnie ¼ VN 0 VD½ �T ¼ V cos L 0 �V sin L½ �T

vnen ¼ rN rE rD½ �T ¼ _ll cos L � _LL �_ll sin L

� �T

where vnin is the rate of the n-frame relative to the i-frame

expressed in the n-frame, vnie is the Earth rate, vn

en is the rateof the n-frame relative to the e-frame, L represents latitude,l represents longitude, and V is the magnitude of the Earthrate.

The h-frame is a local level frame that rotates withheading angle c from the n-frame. Figure 1 shows therelationship between the n-frame and the h-frame.

The dynamic equations of the position, velocity, andattitude of the SDINS are summarised as follows:

_LL ¼ VN

Rm þ h; _ll ¼ VE

ðRt þ hÞ cos Lð1Þ

_vvn ¼ Cnb f b � ð2vn

ie þvnenÞ � vn þ gn ð2Þ

_CCnb ¼ Cn

bVbnb ð3Þ

where Rm ¼ Roð1 � 2e þ 3e sin2 LÞ; Rt ¼ Roð1 þ e sin2 LÞ;Ro is the radius of the Earth at the equator, e representsflattening, h is height, gn is the gravitational forcerepresented in the n-frame, f bð¼ ½ fx fy fz�TÞ representsspecific force measured at the accelerometer,vnð¼ ½VN VE VD�TÞ is the velocity of the vehiclerepresented in the n-frame, vb

nbð¼ ½ox oy oz�TÞ is therate of the b-frame relative to the n-frame, and Vz

xy is theskew-symmetric matrix for the rate vz

xy:

2.2 Attitude error dynamic model

Attitudes are defined in this paper by roll ð�Þ; pitch ð�Þ; andyaw ðcÞ angles, which constitute a direction cosine matrix(DCM) [1, 2]. The yaw angle is equivalent to the headingangle. The roll and pitch angles represent the levellingangles. Considering the heading and the levelling anglesseparately, the DCM between the n-frame and the b-framecan be divided into two parts:

Cnb ¼ Cn

hChb; Cn

h ¼cosc � sinc 0

sinc cosc 0

0 0 1

264

375

Chb ¼

cos � sin� sin � cos� sin �

0 cos� � sin�

� sin � sin� cos � cos� cos �

264

375

ð4Þ

The estimation error of the DCM is defined as

dCnb ¼ CC

nb � Cn

b ð5Þ

The attitude estimation errors are equivalent to (5). In theconventional small angle error model, the estimate CCn

b isexpressed as

CCnb ¼ ðI � hdqn�iÞCn

b;

hdqn�i ¼0 �d#D d#E

d#D 0 �d#N

�d#E d#N 0

24

35 ð6Þ

where dqnð¼ ½d#N d#E d#D�TÞ is a small angle errorvector represented in the n-frame.

The estimation error of the DCM is

dCnb ¼ �hdqn�iCn

b ð7Þ

If the heading error is large, (7) has a large modelling errorbecause attitude angle errors are all assumed to be small.Therefore, we need an alternative attitude error model forthe large heading error case.

By using the h-frame, the estimate CChb can be expressed as

(8) with the small levelling error model and the estimate CCnh

can be expressed as (9) with the large heading error model:

CChb ¼ I � hdwh�i

Ch

b;

hdwh�i ¼0 �d’z d’y

d’z 0 �d’x

�d’y d’x 0

24

35 ð8Þ

CCnh ¼ Cn

h þ dCnh ð9Þ

where dwhð¼ ½d’x d’y d’z�TÞ is a small angle errorvector represented in the h-frame and dCn

h is defined as

dCnh �

d cosc �d sinc 0

d sinc d cosc 0

0 0 0

24

35 ð10Þ

Using (8) and (9), (5) is rewritten as

dCnb ¼ CCn

hCChb � Cn

b

¼ Cnh þ dCn

hð Þ I � hdwh�i

Chb � Cn

b ð11Þ

The second-order term of the errors in (11) is small enoughto be ignored, so the DCM error dCn

b can be approximated as

dCnb � dCn

hChb � Cn

hhdwh�iChb

¼ dCnh � Cn

hhdwh�i

Chb ð12Þ

Note that dCnb is divided into two parts: the heading error

term and the leveling error term. The DCM error dCnb is

expressed as

dCnb ¼ ECh

b ð13Þ

where

Fig. 1 Relationship between n-frame and h-frame

IEE Proc.-Radar Sonar Navig., Vol. 151, No. 1, February 200458

E � dCnh � Cn

hhdwh�i ¼a2 �a1 �d’E

a1 a2 d’N

d’y �d’x 0

264

375

¼d coscþ d’z sinc �d sincþ d’z cosc

d sinc� d’z cosc d coscþ d’z sinc

d’y �d’x

264

�d’x sinc� d’y cosc

d’x cosc� d’y sinc

0

375 ð14Þ

There are no second-order error terms in the error matrix E,so the DCM error is represented by a linear sum of theheading and the levelling errors. By differentiating (13), adifferential equation of the attitude errors is obtained as

d _CCnb ¼ _EECh

b þ E _CChb ¼ _EECh

b þ EChbV

bhb

¼ _EEChb þ ECh

bVbhbCb

hChb ¼ ð _EE þ EVh

hbÞChb ð15Þ

On the other hand, we can derive another differentialequation, (16), by differentiating (5):

d _CCnb ¼ _CCCC

nb � _CCn

b ¼ CCnbVV

bnb � Cn

bVbnb

¼ CCnhCC

hbVV

bnb � Cn

bVbnb ð16Þ

Inserting (8) and (9) into (16) and ignoring the second-ordererror terms results in

d _CCnb � ðEVh

nb þ CnhdV

hnbÞCh

b ð17Þ

where dVhnb is VV

h

nb �Vhnb:

From (15) and (17), the differential equation of the errormatrix E becomes

_EE þ EVhhb ¼ EVh

nb þ CnhdV

hnb ð18Þ

The linear differential equation of attitude errors isrearranged as

_EE ¼ EVhnb � EVh

hb þ CnhdV

hnb ¼ EVh

nh þ CnhdV

hnb ð19Þ

The calculations of Vhnh and Cn

hdVhnb are

Vhnh ¼

0 � _cc 0_cc 0 0

0 0 0

24

35 ð20Þ

CnhdV

hnb ¼ dVn

nbCnh

¼0 �doD doE

doD 0 �doN

�doE doN 0

24

35Cn

h ð21Þ

where

_cc ¼ oy sin� sec �þ oz cos� sec �;

� is the roll angle; � is the pitch angle

dvnnb ¼ doN doE doD½ �T¼dvn

ib�dvnin

¼dvnib�ðdvn

ieþdvnenÞ¼dvn

ib

�

VD�rNRttRt þh

� �dL� rN

Rt þhdhþ 1

Rt þhdVE

�rERmmRmþh

dL� rERmþh

dh� 1Rmþh

dVN

�ðVN þrN sec2 Lþ rDRtt

Rt þhÞdLþrN tanL

Rt þhdh� tanL

Rt þhdVE

2666664

3777775

Rmm ¼ @ðRmÞ@L

¼6R0esinLcosL; and

Rtt ¼@ðRtÞ@L

¼2R0esinLcosL

The linear differential equations are represented for eacherror variable as follows:

_aa1 ¼ _cca2þdoD cosc ð22aÞ

_aa2 ¼� _cca1�doD sinc ð22bÞ

d _’’x ¼ _ccd’y�doE sinc�doN cosc ð23aÞ

d _’’y ¼� _ccd’x�doE coscþdoN sinc ð23bÞ

2.3 Velocity and position error dynamicmodel

The velocity error vector dv is defined as

dvn ¼ vvn � vn ð24ÞWe can obtain a differential equation of velocity errors bydifferentiating (2):

d_vvn ¼ CCnb ff b � ð2vvn

ie þ vvnenÞ � vvn � Cn

b f b

þ ð2vnie þvn

enÞ � vn ð25ÞInserting (13) into (25) and ignoring the second-order errorterms results in the following linear differential equation:

d_vvn ¼ �ð2vnie þvn

enÞ � dvn þ EChb f b þ Cn

bdf b

� ð2dvnie þ dvn

enÞ � vn ð26ÞThe position error vector dp ¼ ½dL dl dh�T is defined as

dL ¼ LL � L; dl ¼ ll � l; dh ¼ hh � h ð27ÞThe linear differential equations of the position errors arederived from (1).

d _LL ¼ rERmm

Rm þ hdL þ rE

Rm þ hdh þ 1

Rm þ hdVN ð28Þ

d_ll ¼ rN sec L tan L � Rtt

Rt þ h

�dL � rN sec L

Rt þ hdh

þ sec L

Rt þ hdVE ð29Þ

d_hh ¼ �dVD ð30Þ

3 Design of in-flight alignment system

3.1 State-space dynamic model

Navigation error dynamic models composed of attitudeerror differential equations, velocity error differential

IEE Proc.-Radar Sonar Navig., Vol. 151, No. 1, February 2004 59

equations and position error differential equations werederived in Section 2. The state-space system model can beformed from the differential equations of the navigationerrors. The error state variable x(t) consists of navigationerrors and sensor bias errors. In the model, w1ðtÞ is whiteprocess noise with zero mean and covariance Q(t). Thelinear time-varying stochastic system model is

_xxðtÞ ¼ FðtÞxðtÞ þ w1ðtÞ ð31Þ

xðtÞ ¼ dpT ðdvnÞT ðdwhÞT aT HT eT bTa

� �T

w1ðtÞ ¼ 01�3 wTa wT

g wTg 01�3 01�3 01�3

� �T

where H is the accelerometer bias (random constant) error, erepresents the gyroscope bias (random constant) error, ba isthe aided sensor bias (random constant) error, wa is theaccelerometer white noise, and wg is the gyroscope whitenoise.

The time-varying system matrix F(t) follows:

F ¼

F11 F12 03�2 03�2 03�3 03�3 03�3

F21 F22 F23 F24 F25 03�3 03�3

F31 F32 F33 02�2 02�3 F36 02�3

F41 F42 02�2 F44 02�3 F46 02�3

09�19

266664

377775

F11 ¼

rERmmRm þ h

0rE

Rm þ h

rN sec L tan L � RttRt þ h

� �0 � rN sec L

Rt þ h

0 0 0

26664

37775;

F12 ¼

1

Rm þ h0 0

0 sec LRt þ h

0

0 0 �1

2664

3775

F21 ¼

� 2VN þrN sec2 Lþ rDRtt

Rt þh

� �VE þ

rERmmRmþh

VD

2VN þrN sec2 Lþ rDRtt

Rt þh

� �VN þ 2VD�

rNRttRt þh

� �VD

�rERmmRmþh

VN � 2VD�rNRttRt þh

� �VE

2666664

0 �rDVERt þh

þ rEVDRmþh

0rDVNRt þh

�rNVDRt þh

0 � rEVNRmþh

þ rNVERt þh

3777775

F22 ¼

VDRm þ h

2VD þ 2rD �rE

�2VD � rDVN tan L þ VD

Rt þ h2VN þ rN

2rE �2VN � 2rN 0

26664

37775;

F23 ¼�f h

z sinc �f hz cosc

f hz cosc �f h

z sinc

�f hy f h

x

264

375

F24 ¼�f h

y f hx

f hx f h

y

0 0

24

35; F25 ¼ Cn

b

F31¼

ðVD�rNRttRtþh

Þcosc�rERmmRmþh

sinc 0 �rN coscRtþh

�rE sincRmþh

�ðVD�rNRttRtþh

Þsinc�rERmmRmþh

cosc 0rN sincRtþh

�rEcoscRmþh

264

375

F32 ¼� sinc

Rm þ hcoscRt þ h

0

� coscRm þ h

� sincRt þ h

0

264

375; F33 ¼ 0 _cc

� _cc 0

" #

F36 ¼ � Chbð1; 1Þ Ch

bð1; 2Þ Chbð1; 3Þ

Chbð2; 1Þ Ch

bð2; 2Þ Chbð2; 3Þ

" #

F41¼VN þrN sec

2 LþrDRttRtþh

� �cosc 0 �rN tanL

Rtþhcosc

� VN þrN sec2 LþrDRtt

Rtþh

� �sinc 0

rN tanLRtþh

sinc

264

375

F42 ¼0 tan L

Rt þ hcosc 0

0 � tan LRt þ h

sinc 0

24

35; F44 ¼ 0 _cc

� _cc 0

� �

F46 ¼Ch

bð3;1Þcosc Chbð3;2Þ cosc Ch

bð3;3Þ cosc�Ch

bð3;1Þ sinc �Chbð3;2Þ sinc �Ch

bð3;3Þ sinc

" #

where Cbaði; jÞ is the ith row and jth column element of Cb

a:

3.2 Measurement model

The position information adequately compensates for thelarge initial heading error. Subtracting the positionmeasurement of the aided sensor from the SDINS positionestimates induces a measurement model:

yðtkÞ ¼ ppINS � pAID ¼ p þ dp � p � dpAID

¼ dp � dpAID ð32Þ

where INS represents SDINS position estimates and AIDrepresents position data from an aided sensor.

A measurement model is formulated by adding whitenoise, w2; with a zero mean and covariance RAID:

yðtkÞ ¼ HðtkÞxðtkÞ þ w2ðtkÞ ð33Þ

where

HðtkÞ ¼1 0 0 01�13 �1 0 0

0 1 0 01�13 0 �1 0

0 0 1 01�13 0 0 �1

24

35

Remark 1: From the state-space dynamic model, one can seethat the attitude error state variables are directly related tothe velocity error variables, which can be estimated byposition measurement. Therefore, F23 and F24 of the systemmatrix F(t) are crucial to the estimation performance.

Remark 2: If the heading error is very large, the systemmatrix F(t) experiences a large modelling error becausesome elements of F(t) are composed of the attitudeestimates. In this case, F23 is contaminated by the largeheading error, and consequently, the purity of F24 becomesimportant. Therefore, the accuracy of CCh

b significantlyaffects the performance of the heading estimation. For

IEE Proc.-Radar Sonar Navig., Vol. 151, No. 1, February 200460

this, the pitch angle ð�Þ and the roll angle ð�Þ can becalculated by a levelling method to constitute elements thatare composed of Ch

b in the system matrix F(t) [3]. If thelevelling errors can remain small, the proposed error modelcan be most effective.

3.3 Navigation error compensation

In the indirect feedback method, the estimated error statevector xxðtkÞ ¼ ½dppT ðdvvnÞT ðdwwhÞT aaT HHT eeT �Tfrom the Kalman filter is used to compensate for navigationerrors [8]. The compensation for the DCM error isaccomplished using

CCnbFIN ¼ CCn

bINS � dCCnb ¼ CCn

bINS � EECChb ð34Þ

where CCnbINS is the estimate of the SDINS before

compensating for the error and CCnbFIN is the final estimate

after compensating for the error.EE is composed of the estimates of aa1; aa2; d’’x; and d’’y

according to (14), i.e.

EE ¼aa2 �aa1 �d’’E

aa1 aa2 d’’N

d’’y �d’’x 0

24

35

where d’’N ; d’’E can be calculated from d’’x; d’’y; and cc:Compensation for the velocity and position errors must be

made to obtain the final navigation solution to the followingequations:

vvnFIN ¼ vvn

INS � dvvn ð35Þ

LLFIN ¼ LLINS � dLL; llFIN ¼ llINS � dll;

hhFIN ¼ hhINS � dhhð36Þ

where vvnINS; LLINS; llINS; and hhINS are the estimates of the

SDINS and vvnFIN ; LLFIN ; llFIN ; and hhFIN are the final estimates

after compensating for errors.

4 Simulation results

Monte-Carlo simulations were carried out to demonstratethe effectiveness of the proposed IFA filter with the newnavigation error dynamic model. Two Monte-Carlo simu-lations show that the proposed filter is adequate for caseswith large initial heading errors. The target system was amoving AUV required to align its attitude. Some additionalsensors offer position measurement relative to the referenceship or relative to the e-frame. For example, the super shortbaseline (SSBL) acoustic positioning system is a goodadditional sensor that offers position data relative to thereference ship having a GPS receiver. For the IFAsimulation of an AUV, SSBL is selected as an additionalsensor.

4.1 Simulation 1: various heading error case

Conditions of simulation 1 are listed here. The specifica-tions ð1�Þ of the sensors are shown in Table 1.

. Total trial number of the Monte-Carlo simulation: 100

. Total running time of a trial: 280 s

. Speed of the vehicle: 2 m/s (average speed)

. Random initial error distribution

Heading error distribution: Nð0; ð45 degÞ2ÞLevelling error distribution: Nð0; ð1 degÞ2ÞHorizontal position error distribution: Nð0; ð5mÞ2ÞVertical position error distribution: Nð0; ð5mÞ2Þ

To confirm the performance of estimating the heading anglefor various cases, the initial estimate of the heading angle israndomly generated with a normal distribution. A 19-orderfilter is designed to estimate the navigation errors asexplained in Section 3. The initial error covariance is setto the random initial error distribution, and the noisecovariance of the filter is set to the sensor error specifica-tions in Table 1.

The accuracy of an SSBL system is inversely pro-portional to the distance between the reference ship and theAUV. During IFA, the AUV maintains (within severalhundreds of metres) its distance from the reference ship.Therefore, we assume the accuracy ð1�Þ of the SSBLsystem is 1–2 m in this situation [9, 10]. The positionaccuracy ð1�Þ of the reference ship is assumed to be 3 mwith the help of a Differential Global Positioning System(DGPS).

Figure 2 shows the trajectory of an AUV. The AUVmanoeuvres under the sea, changing its heading direction toimprove observability of the filter [11, 12] and keeping itsdepth constant.

In this simulation, the pitch angle ð�Þ and roll angle ð�Þ;which are the elements of Ch

b; can be calculated by thefollowing equations to form the system matrix F(t):

� ¼ sin�1ð fx=gÞ� ¼ sin�1ð�fy þ ozVx=g cos �Þ

Figure 3 shows the mean and standard deviation of headingand levelling errors. In Fig. 3a, the standard deviation ð1�Þof a heading angle error rapidly decreases from a largeinitial value of more than 408 to near 58. When the headingerror approaches a steady state value, the mean of theheading error is almost 08 and the standard deviation isabout 2.58. Figure 3b shows the behaviour of the levelling

Fig. 2 Trajectory of AUV during in-flight alignment

Table 1: Error specifications ð1sÞ of gyroscope,accelerometer, and SSBL

Error Random constant White noise

Sensor

Gyroscope 1�=h 0:3�=ffiffi

hp

Accelerometer 1000mg 50mg

SSBL (relative position) 1 m 2 m

IEE Proc.-Radar Sonar Navig., Vol. 151, No. 1, February 2004 61

errors, which converge to small values. These results showthat the proposed algorithm is very effective for a largeinitial heading error.

4.2 Simulation 2: worst case

The second Monte-Carlo simulation shows that theproposed filter with the new error model is better than afilter with the conventional small angle error model whenthe initial heading error is very large. Simulation conditions

for simulation 2 are the same as those of simulation 1 withone exception: the initial heading error is fixed at a verylarge value, i.e. 908. Figure 4 shows that with an alignmentfilter using the conventional small angle error model, theheading error converges to a large value. However, theproposed model enables an AUV to carry out IFA even inthis worst case scenario.

5 Conclusions

A new IFA method is proposed for an SDINS with a largeinitial heading error. A notable feature of the proposedmethod is the separation of large heading errors from smalllevelling errors using the h-frame. In the attitude errordynamic model introduced here, the small angle assumptionis employed only for the levelling errors. The large headingerror is handled by employing a perturbed sinusoidal term.Linear dynamic equations of navigation errors are obtainedfrom the small angle assumption and the perturbedsinusoidal term. They are applied to a Kalman filter todesign an IFA filter.

The filter and model proposed was applied to a low gradeSDINS aided by an SSBL system. Its performance isdemonstrated via a Monte-Carlo simulation. The simulationresults show that the proposed method is effective incoarsely estimating a large heading error. For a headingerror of 908 the proposed method reduces the error to about208, which is almost 75% less than that of a method usingthe conventional small angle error model.

The proposed method is useful for navigating a vehiclesuch as an AUV, which initially has a large heading error.

6 Acknowledgment

The authors gratefully acknowledge the financial support ofthe Agency for Defense Development and the AutomaticControl Research Center of Seoul National University.

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12 Goshen-Meskin, D., and Bar-Itzhack, I.Y.: ‘Observability analysis ofpiece-wise constant systems, Part II: Application to inertial navigationin-flight alignment’, IEEE Trans. Aerosp. Electron. Syst., 1992, 28, (4),pp. 1068–1075

Fig. 3 Attitude error

a Heading errorb Levelling errors

Fig. 4 Heading error comparison

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