175
1 INTRODUCTION
A free–gyro positioning system is to determine the
positionofavehiclebyusingtwofreegyros.Itisan
active positioning system like an inertial navigation
system(INS)inviewofobtainingapositionwithout
externalsource.However,theFPSistodetermineits
own position by using the nadir angle between the
vertica
l axis of local geodetic frame and the axis of
free gyro, while an INS is to do it by measuring its
acceleration.
In general the INS comprises a set of inertial
measurementunits(IMU’s),bothaccelerometersand
gyros, the platform on which they are mounted,
includingthest
abilizationmechanismifsoprovided,
and the computer that performs the calculations
needed to transf orm sensed accelerations and, in
some mechanizations, angles or angular rates into
navigationally useful information such as position,
velocity and attitude. It is composed of a very
complicatedstructure.
On theother hand,thefree‐gyro positioning and
directionalsystemconsistsof asetoftwosensorsof
gyroaxismotion rate andthreesensors of the body
frame, two free gyros, and the computer tha
t
calculatesnavigationalinformation,position,etc.Itis
comparativelysimplerthantheINS.
Park&Jeong(2004)investigatedhowtodetermine
thegyrovectorsoftwofreegyrosandthepositionof
a vehicle by using the gyros. The errors in the FPS
were invest
igated broadly by Jeong (2005). And the
algorithmic design of free gyroscopic compass and
positioning was suggested by measuring the earth’s
rotationrateontheba
sisofafreegyroscope(Jeong&
Park,2006;Jeong&Park,2011).
Meanwhile, the free‐gyro positioning and
directionalsystemisthoughttohaveitsown errors.
This paper is to analyze such errors theoretically.
A Study on the Errors in the Free-Gyro Positioning
and Directional System
T.G.Jeong
KoreaMaritimeUniversity,Busan,RepublicofKorea
ABSTRACT:Thispaperistodevelopthepositionerrorequationsincludingtheattitudeerrors,theerrorsof
nadirandship’sheading,andtheerrorsofship’spositioninthefree‐gyropositioninganddirectionalsystem.
Indoingso,thedeterminationofship’spositionbytwofreegyrovectorswasdiscussedandthealgorit
hmic
design of the free‐gyro positioning and directional system was introduced briefly. Next, the errors of
transformation matrices of the gyro and body frames, i.e., attitude errors, were examined and the attitude
equationswerealsoderived.Theperturbationsoftheerrorsofthenadirangleincludingship’sheadingwere
invest
igatedineachstagefromthesensorofrate ofmotionofthespinaxistothenadirangleobtained.Finally,
theperturbationerrorequationsofship’spositionusedthenadirangleswerederivedintheformofalinear
errormodelandtheconceptofFDOPwasalsosuggestedbyusingcovarianceofpositionerror.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 8
Number 2
June 2014
DOI:10.12716/1001.08.02.01
176
Firstly, the errors of transformation matrices of the
gyro and body frames, i.e., attitude errors, will be
examinedandtheattitudeequationsbealsoderived.
The perturbations of the errors of the nadir angle
includingship’sheadingwillbeinvestigatedineach
stagefromthesensorofrateofmotionofthespinaxis
tothenadirangleobtained.Finally,theperturba
tion
error equations of ship’s position used the nadir
angles will be derived in the form of a linear error
model and the concept of FDOP will be also
suggestedbyusingcovarianceofpositionerror.
Before the errors involved are discussed, the
overviewofthefree‐gyropositioninganddirectional
systemwillbepresented.
2 OVERVIEWOFFREE‐GYR
OPOSITIONING
ANDDIRECTIONALSYSTEM
2.1 Determinationofship’sposition
Thenadirangle,
,isgivenbyanarbitraryposition
andgyrovectorasshownasequation(1).

cos cos cos
x
e
ut



cos sin sin
yez
utu

 (1)
Here,
e
is the (presumably uniform) rate of
Earthrotation,
isthegeodeticlongitude,
isthe
geodetic latitude and
t denotes time. And
x
u ,
y
u
and
z
u are the components of the gyro vector,
i
g
,
whosesuperscriptindicatestheinertialframe.
Ifweusetwogyrovectorsof
, ,
T
i
aaxayaz
uuu


g
and
, ,
T
i
bbxbbz
uuug in (1), we can determine the
position

,
of a vehicle by using two
correspondingnadirangles
a
and
b
.
Meanwhile the azimuth of the gyro vector from
thenorth,
,isrepresentedbyequation(2):
tan
D
D
E
N
, (2)
where,

sin cos
D
xe
Nu t



sin sin cos
yez
utu

,
 
sin cos
D
xey e
Eu tu t

 .
Oncedeterminingtheposition,wecanalsoobtain
the azimuth of a gyro vector by using Eq. (2). Park
andJeong (2004)already suggestedthe algorithmof
howtodetermineaposition.
Fig.1 shows the measurement quantities in the
localnavigationframe.
Figure 1. Measurement quantities in the local navigation
frame
2.2 Relationbetweenship’sheadingandazimuthofgyro
vector
As Jeong & Park (2006) mentioned, the north
component of the earth’s rotation rate is
cos
e
,
where
depictsthegeodeticlatitudeofanarbitrary
position.Fig.2showsthattheangularvelocitiesofthe
fore‐aftandtheathwartshipcomponentsaregivenby
equation (3) (Titterson, et al., 2004), where
is
shipʹs heading. And it also shows that
is the
azimuthofagyrovectorfromship’shead.
Figure2.Relationbetweenship’sheadingandazimuthofa
gyrovector
cos cos
Nx e


cos sin
Ny e

(3)
By taking the ratio of the two independent
gyroscopic measurement, the heading,
, is
computedby(4).
1
tan
Ny
Nx
 (4)
Meanwhileassumingthatagyrovectoris
away
from ship’s head, its azimuth from North is
representedbyEq.(5).Thereforetheangularvelocity
of the horizontal axis of a gyro,
H
, is given by
equation(6)onthenavigationframeorlocalgeodetic
frame.
177

(5)
cos sin
He


(6)
Equation(6)showsthatifthenorthcomponentof
the earth’s rotation rate can be known on the
navigationframe,thenadirangleofagyrovector,
,
isobtainedby(7),byintegratingEq.(6)incrementally
overatimeinterval.
2
1
t
H
t
dt

(7)
2.3 Algorithmicdesignoffree‐gyropositioningand
directionalsystem
Fig.3andFig.4showthealgorithmicdesignoffree
gyros positioning system mechanization. In this
mechanization two sensors for sensing the motion
rate of the spin axis are mounted in the free gyro.
Three sensors for sensing
the motion rate of the
platform are mounted inorthogonaltriad. From the
sensorsinthegyroframe,thespinmotionrate,
/
g
ig
ω ,
is obtained and from the ones in the body frame,
/
b
ib
ω ,isdetected.Byusingthesum,
/
g
bg
ω
,oftherates
from the free gyro and the ones detected from the
bodysensors,thetransformationmatrix,
b
g
C ,andits
inverse are determined. Therefore the spin motion
rate,
/
g
ig
ω , sensedfromthe freegyro istransformed
into
/
b
ig
ω byusingthematrix,
b
g
C .
Meanwhile the rate of the earthʹs rotation,
/
n
ie
ω ,
and the rate of the vehicle movement,
/
n
en
ω , are
addedtomake
/
n
in
ω .Here
/
n
en
ω isgivenby:
/
cos sin
T
n
en





ω ,
and
/
n
in
ω is also expressed as:
 
/
cos sin
T
n
in e e





ω
.
By using the matrix,
b
n
C , it will be transformed
into
/
b
in
ω
, which is subtracted from the sensed rate
fromthebody,
/
b
ib
ω .Asaresult,
/
b
nb
ω isgenerated.
Byusingthis,thetransformationmatrix,
n
b
C ,andthe
inverseofit,
b
n
C ,areobtained.Andtherateof
/
b
ig
ω
,
istransformedinto
/
n
ig
ω byusingthematrixof
n
b
C
.
Figure 3. Free gyro positioning and directional system
mechanization(1)
Figure 4. Free gyro positioning and directional system
mechanization(2)
For ship’s heading, by using the matrix of
b
n
C ,
/
n
en
ω ischangedinto
/
b
en
ω ,whichissubtractedfrom
/
b
ig
ω , and
D
ω is calculated. Finally, using the
transformation matrix,
n
b
C , we can get the spin
motion rate in the NED frame,
N
ω , where
T
NNxNyNz

ω . As a result, the shipʹs
heading,
,iscalculatedbyusingthecomponentsof
thespinmotionrateaccordingtoequation(4).
Next for ship’s position, let’s look into the nadir
angle(Fig.4).Themotionrateofthespinaxisinthe
navigation frame,
/
,
n
ig
ω is given by
T
LLxLyLz

ω . The azimuth of the gyro
vector from the ship’s head,
, can be obtained by
integrating
Lz
.Thentheazimuthofthegyrovector
fromtheNorth,
,isobtained.
The horizontal angular velocity,
LH
, and the
tiltingrate,
H
,ofthespinaxisarecalculated.And
thenadirangleofthegyrovector,
,canbeobtained
by integrating
H
. Finally we can get the ship’s
positionexpressedby
,
,usingequation(1).
178
Next, using equation (2), we can also obtain the
azimuth of the gyro vector,
, and ship’s heading,
, which are modified by iteration. And ship’s
positionisalsocorrected.Inadditionifonlyweknow
the northward component of ship’s speed, we can
alsoobtaintheheight,
h .
3 PERTURBATIONFORMOFERROR
EQUATIONS
Thispaperderivestheerrorequationsoftheposition
andattitudebyusinglinearerrormodelforms (Jekeli,
2001;Roger,2007).
3.1 Gyroframeandbodyframeerrorequations
Gyro frame error equations can be derived by the
transformation matrix,
b
g
C
. The estimated or
computedmatrixofitisrepresentedby(8):

bb
g
g
CIΞ C , (8)
where
Ξ
is a skew-symmetric matrix, which is
equivalent to the vector,
T
xyz



ξ , and is
given by:
0
0
0
zy
zx
yx







Ξ
.
Thedifferentiation form of
b
g
C
canbe
representedbythefollowing.
/
bbg
g
gbg
CCΩ
(9)
Theerrormatrixof
b
g
C isalsogivenby:
bbb b
g
gg g
CCC ΞC (10)
Takingthederivativeof(10)yields
/
bbbbbg
g
gg ggbg
  C ΞC ΞC ΞC ΞC Ω (11)
Theerrorequationofequation(9)isgivenby:

///
bbg bgbg
g
gbg gbg g bg

CCΩ C Ω C Ω (12)
where the perturbation in angular rate,
/
g
bg
Ω
,
denotestheerror inthe computedvalue,
/
g
bg
Ω , and
isexpressedas:
//
bg g
g
bg bg
ΩΩ Ω. (13)
Substituting (13) into(12) and equating(12)with
(11),wecanget
///
bbg bgbg
g
gbg gbg g bg

 ΞC ΞC Ω C Ω C Ω
. (14)
Substituting (10) into (14) and arranging it
yields:
/
bgg
g
bg b
Ξ C Ω C . (15)
Thisisequivalenttothevectorformgivenby:
/
bg
g
bg
ξ C ω
, (16)
where
/
g
bg
ω is theerrorin the rotation rate of the
gyro frame relative to the body frame. This is
separatedby:
// /
g
ggb
bg ig b ib
ωωC ω (17)
Anerrorofthisequationwillberepresentedby:
// / /
δ
gggbgb
bg ig b ib b ib

 ωωC Ξω C ω . (18)
Substituting(18)into(16)andrearrangingitgives
///
bbgb
ib g ig ib

 ξωξC ωω
. (19)
Equation (19) shows the error dynamics of the
gyroframeattitude.
Similarly, we can get the error equations of the
bodyframeattitudeasthefollowing(20).
///
nnbn
in b ib in


γ
ω
γ
C ωω (20)
Here
γ istheerrorangleofbodyframeattitude
given by
T
xyz

γ . This can be represented
equivalentlybytheskew‐symmetricmatrix,
Γ
.
0
0
0
zy
zx
yx


Γ
(21)
Inaddition,theerror oftherateofmotion ofthe
navigationframe,
/
δ
n
in
ω ,isexpressedas:

/
δcos sin
δsin cos
e
n
in
e








ω
.
179
3.2 Perturbationsoferrorequationsfornadirangleand
ship’sheading
InFig.(3), therateofmotion ofthespin axisin the
navigationframe,
/
n
ig
ω isgivenby;
//
nnbg
ig b g ig
ω CCω .
Wecangettheerrorofthisequationbyusingthe
differentialoperator,
.

///
nnnbgnbg
ig b b g ig b g ig

 ωΓCCΞ C ω CC ω (22)
Here
/
δ
T
n
i g Lx Ly Lz
  


ω
.
Theerrorof theazimuthof thegyrovector from
ship’s head,

, and that of the north,
, are
representedby:
2
1
t
Lz
t
dt
 
,



. (23)
The error of the horizontal rate of the spin axis,
LH
,isexpressedas:
22
Lx Lx Ly Ly
LH
Lx Ly
 


. (24)
And also the error of the tilting rate of the spin
axis,
H
,isobtainedas:
sin cos
HLH LH

 . (25)
Finally, the error of the nadir angle of the gyro
vector,
,isgivenby:
2
1
t
H
t
dt
 
. (26)
Meanwhiletheerrorofship’sheadingisobtained
asthefollowing.
Theerroroftherateofthemotionofthespinaxis
inthebodyframe,
D
ω ,isexpressedas:

// //
δ
bg b g b n n
D
gig g ig n en en

 ωΞC ω C ω C Γω ω (27)
Andtheerroroftherateofthemotionofthespin
axisinthenavigationframe,
N
ω ,isgivenby:

//
nnn
NbDenen

 ωΓC ωΓω ω

//
nbg bg
bgig gig
CCωΞC ω , (28)
where
T
NNxNyNz

ω . Therefore the
errorofship’sheading,
,isrepresentedby:
22
sec
Nx Ny Ny Nx
Nx
 


. (29)
Fig.5showsthe errordynamicsin thenavigation
framewediscussedsofar.
Figure 5. Error dynamics of free gyro positioning and
directionalsystem
3.3 Errorequationsofship’sposition
Fora developmentofthepositionerrorequation,we
differentiallyperturbeqn. (1),assumingthattheearth
rateisaconstant,andthenderivetheperturbationsof
errors resulting in a linear model. The differential
perturbationofeqn.(1)isgivenbyeqn. (30):

cos cos cos sin
ex ey
tu tu
 
 
sin sin
z
u



cos cos sin
yexee
ututt
 





sin cos sin sin cos
xeyez
ututu
 


y
cos sin u cos cos
xe e
ut t
 

(30)
Now, let two gyro‐vectors be
, ,
T
i
aaxayaz
uuu
g and
, ,
T
i
bbxbbz
uuug . And
the corresponding nadir angles of
a
and
b
are
given. The perturbation matrix is arranged by
eqn.(31).
aaa
bbb
Z
Z





(31)
Here,
180
cos cos cos sin
aeaxeay
Z
tu tu




sin sin
az a a
u



cos cos sin
ay e ax e e
ututt
 



,
 
cos cos cos sin
bebxeby
Z
tu tu
 


sin sin
bz b b
u



cos cos sin
by e bx e e
ututt
 



,

sin cos
aax e
ut

 


sin sin cos
ay e az
utu


,

sin cos
bbx e
ut

 


sin sin cos
by e bz
utu

,

cos sin
aax e
M
ut





cos cos
ay e
ut

,

cos sin
bbx e
M
ut





cos cos
by e
ut

.
This matrix can be written in the matrix‐vector
form,
1
 Z Ψ xxΨ Z , (32)
where,
Δ



x
,
1
1
aa
bb





Ψ
,and
Δ
a
b
Z
Z



Z
havebeenintroduced.Thecovariancematrixforeqn.
(32)isgivenby:


11
cov Δ cov Δ
T

x Ψ Z Ψ . (33)
The assumption may be made that the sensor and time
errors show a random behavior resulting in a normal
distribution with expectation value zero and variance,
2
s
.
Therefore, measured sensor and time values are linearly
independent or uncorrelated. The

cov ΔZ is
represented by:
2
cov Δ
s
ZI, (34)
where
I
istheunitmatrix.
Substitutingeqn.(34)intoeqn.(33)yields


12 1 2 1 1
cov Δ
TT
ss


x Ψ I ΨΨΨ
1
22
T
ss


x
ΨΨ Q (35)
where
1
T
x
Q ΨΨ . The cofactor matrix,
x
Q
(Hofmann‐WellenhofB.etal,2001)is a2x2matrix
where two components are contributed by the gyro
vectorsof
i
a
g
and
i
b
g
.Theelementsofthecofactor
matrixaredenotedas:
qq
qq


x
Q
. (36)
In the cofactor matrix the diagonal elements are
used for FDOP which is the geometry of two free
gyros.
22
FDOP q q

. (37)
Therefore the error of a position in the free‐gyro
positioninganddirectionalsystem,
rms
f
d ,isgivenby:
rms s
f
dFDOP
. (38)
Fromeqn.(38)the
rms
f
d canbecomputedeasily.
If in this case two‐dimensional error distribution is
close to being circular, the probability is about
0.63(Kaplan,2006).
4 CONCLUSIONS
Firstofall,thispaperdealtwiththedeterminationof
ship’s position by free gyros and its algorithmic
design briefly. Next, the errors of
transformation
matrices of the gyro and body frames, i.e., attitude
errors, were examined and the attitude equations
werealsoderived.Theperturbationsoftheerrorsof
the nadir angle including ship’s heading were
investigated in turn. Finally, the perturbation error
equations of ship’s position used the nadir angles
werederivedin
theformofalinearerrormodeland
the concept of FDOP was also suggested by using
covarianceofpositionerror.
However,thefree‐gyropositioninganddirectional
systemhasstillmanyproblemstobesolved.Firstthe
error of a position was experimentally verified.
Especially the sensor errors will
have to be
investigated.Inaddition,theadditionaldriftneedsto
be investigated, which occurs when a free gyro is
suppressed by the measures which prevent gimbals
181
lockandtumbling.Thealignmentinthesystemalso
needstobeexamined.
Allthesewillbedealtwithinthenextpapers.
REFERENCES
[1]Hofmann‐Wellenhof B, Lichtenegger H and Collins
J(2001), “GPS, Theory and Practice”, 5
th
edition,
SpringerWienNewYork,p.272.
[2]Jekeli, C.(2001), Inertial Navigation Systems with
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