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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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Fig. 3 Attitude error
a Heading errorb Levelling errors
Fig. 4 Heading error comparison
IEE Proc.-Radar Sonar Navig., Vol. 151, No. 1, February 200462
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