55
1 INTRODUCTION
Thetrackingradarsystemhasa wide application in
both the military and civilian fields. In the military,
tracking is essential for fire control and missile
guidance whereas in civilian application it is useful
for controlling traffic of manned maneuverable
vehicles such as ships, submarines and aircrafts
whichrequireaccuratetra
cking.Trackingfiltersplay
thekeyroleoftargetstateestimationfromwhichthe
trackingsystemisupdated.Oneofthetrackingfilters
inusetodayinmanytrackingapplicationsistheα‐β‐
γfilterwhichisadevelopmentoftheα‐βfilteraimed
attrackinganacceleratingtargetsincetheα‐βfilt
eris
only effective when the target model input is a
constantvelocitymodel.
Duetotheessentialrolethattrackingfiltersplay
in a tracking system, many researchers have taken
quite an interest in understanding the theory and
application leading to valuable insights into design
developmentsandimprovement.Intheearlyworkof
BenedictandBordner(1962),theaut
horsbasedtheir
analysisoftheα‐βfilteronthefrequencydomain(Z‐
transform).Theyproposedarelationshipbetweenthe
α andβfiltering coefficients derived from a pole
matchingtechniqueinordertooptimizethetracker’s
ability to reduce noise and achieve a good transient
performa
nce.Thisledtowhatisknowntodayasthe
Benedict‐Bordnerrelationship.Simpson(1963)further
extended this study to the α‐β‐γ filter by including
theaccelerationtermthusarrivingattheoptimization
conditionbetweenthefilterweightcoefficients.
Kalata(1984)proposedtheuseofatrackingindex
thatrelatesthefilt
ercoefficientsandisa functionof
position uncertainty due to target maneuverability,
radar measurement uncertainty and update time
interval.Heutilizedthe trackingindexparameterto
derive implicit closed form equations of the
A Study on the Performance Comparison of Three
Optimal Alpha-Beta-Gamma Filters and Alpha-Beta-
Gamma-Eta Filter for a High Dynamic Target
T.G.Jeong
KoreaMaritimeandOceanUniversity,Busan,Korea
A.W.Njonjo
J
omoKenyattaUniversityofAgricultureandTechnology,Juja,Kenya
B.F.Pan
KoreaMaritimeandOceanUniversity,Busan,Korea
ABSTRACT:Theα‐β‐
γ
tracking filterisusefulfortrackingaconstantaccelerationtargetwithzerolagerrorin
thesteadystate.It,however,depictsaconstantlagerrorforamaneuveringtarget.Variousalgorithmsoftheα‐
β‐γtrackingfilterexistinliteratureandeachoneofthempresentsitsownuniquechallengesandadvantages
dependingonthedesignrequirement.
This study invest
igates the operation of three α‐β‐γ tracking filter design methods which include Benedict‐
Bordner also known as the Simpson filter, Gray‐Murray filter and the fading memory constant acceleration
filter.Thesefiltersarethencomparedbasedontheabilitytoreducenoiseandfollowamaneuveringta
rgetwith
minimumlagerror, againstthejerkymodelα‐β‐γ‐η.Resultsobtainedfromsimulationsoftheinputmodelof
the target dynamics under consideration indicate an improvement in performance of the jerky model in
comparisonwiththeconstantaccelerationmodels.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 11
Number 1
March 2017
DOI:10.12716/1001.11.01.05
56
smoothing coefficients which resulted in optimal
performance. A more convenient way to determine
the optimal filtering weights was investigated by
Gray and Murray (1993) whereby a damping
parameter that computes the smoothing coefficients
directlywasderivedanalytically.
Njonjoetal(2016)investigatedtheperformanceof
the fading memory α‐β‐γ filter on
a high dynamic
targetwarship.Theresearchconcludedthatthefilter
was capable of tracking the highly maneuvering
vessel with a relatively good accuracy in terms of
noise reduction. This research was further extended
byPanetal(2016)wherethefilterwasoptimizedin
ordertoimproveitstracking
abilitybyreducingthe
noise further. The optimization procedure involved
varying the value of the discounting factor, ξ, with
the residual error and determining the ξ that
corresponds to the minimum error. The study
concludedthattheoptimalfilteruniquelyvarieswith
the initial speed and average speed of the
target
underconsideration.
In this study, different algorithms of the steady
state Kalman filter are investigated and compared
based on capability to reduce noise of a highly
maneuvering target and hence provide quality
estimates. The study focusses on performance
comparisonoftheBenedict‐Bordnerfilteralsoknown
as the Simpson filter,
Gray‐Murray filter and the
fading memory α‐β‐γ filter, also known as the
critically damped filter. These filters are then
comparedwiththeoptimalα‐β‐γ‐ηfilter.
2 THEORYOFTHEΑ‐Β‐ΓFILTER
Theα‐β‐γfilterisasteadystateKalmanfilterwhich
assumesthattheinputmodelofthetarget’sdynamics
is a constant
acceleration model.The model has a
low computational load since only two steps are
involved that is, estimation and updating the
prediction estimates of position, velocity and
acceleration as shown in Equations 1‐6. In addition,
thesmoothingcoefficientsofthefilterareconstantfor
agivensensorwhichfurthercontributes
toitsdesign
simplicity.Theselectionoftheweightingcoefficientis
animportantdesignconsiderationasitdirectlyaffects
the error reduction capability. The optimal filter of
three different designs of the α‐β‐γ filter are
investigated and compared based on their ability to
reduce tracking error and improve the tracking
response.
Thethreedesignsdifferintheirselectionof
thesmoothingcoefficientsα,βandγ.
Prediction;
),1(
2
2
)1()1()( n
s
A
t
n
s
tn
s
n
p
VPP
(1)
),1()1()( n
s
tAn
s
n
p
VV (2)
).1()( n
s
An
p
A (3)(3)
Smoothing;
)),()(()()( n
p
n
o
n
p
n
s
PPPP
(4)
)),()(()()( n
p
n
o
t
n
p
Vn
s
PPV
(5)
)).()((
2
2
)()( n
p
n
o
t
n
p
An
s
PPA
(6)
whereo,pandsdenotetheobserved,predictedand
smoothed state parameters respectively; P, V and A
are the target’s position, velocity and acceleration
respectively;tisthesimulationtimeinterval;andnis
thesamplenumber.
2.1 Benedict‐Bordnermodel
The optimal filter is obtained when the
condition in
Equation7issatisfied.
.
2
2
(7)
The design of this filter does not specify the
optimalpositionsmoothingcoefficient,α,henceitis
chosenbasedonthesystemapplication.Itisproposed
to vary α with observed high frequency power
fluctuations of the tracking error residual or the
innovation,
))()(( n
p
n
o
PP .
The Benedict‐ Bordner filter coefficient
relationshipbecomesanoptimalthirdordertracking
filterwhentheconditioninEquation8issatisfied;
.0)
2
(2
(8)
2.2 Gray‐MurrayModel
This filter is an extension of the Kalata filter
coefficients relationship which employs the tracking
index to compute a damping parameter which is
consequentlyusedtocalculatethepositionsmoothing
coefficient,α.Thetrackingindexisdeterminedfrom
therelationshipgiveninEquation9.
.
2
v
w
t
(9)
where t = target tracking period;
=
maneuverabilitynoise;and
=measurementnoise.
Thedampingparameter,r,iscomputedasshown
inEquation10.Theposition,velocityandacceleration
gainparameters,α,β,andγ,arecomputedexplicitly
asshowninEquations11‐13.
57
,
4
)
2
8()4(
r (10)
,
2
1 r
(11)
,14)2(2
(12)
.
2
2
(13)
2.3 Thefadingmemoryfiltermodel
The fading memory filter has three real roots and
represents the filter minimizing the discounted least
squares error for a constantly accelerating target as
discussedBrookner(1998).Theposition,velocityand
acceleration gain coefficients are computed from the
dampingparameter,ξ,whichis
thediscountingfactor
and whose value was investigated through an
optimization process and found to depend on the
initial and average speed of the target under
considerationaswasdeterminedbyPanetal(2016).
Thesmoothingcoefficientsareobtainedasshownin
Equations14‐16.
,
3
1
(14)
),1()1(5.1
2
(15)
.)1(
3
(16)
3 THEORYOFα‐β‐γ‐ηFILTER
The α‐β‐γ‐η filter is a constant gain, four‐state
tracking filter.The four state vector includes
position, velocity, acceleration and jerk, a time
derivative of acceleration. The jerk is modelled as a
constant and includes zero mean white Gaussian
noise. Equations 17‐ 20 are
the prediction equations
for position, velocity, acceleration and jerk
respectively where they are updated from the
estimated state thereby lowering the tracking error.
Equations 21‐24 are the smoothing equations which
are computed by adding a weighted difference
between the observed and the predicted position to
theforecaststate.
Prediction;
),1(
6
3
)1(
2
2
)1()1()( n
s
j
t
n
s
A
t
n
s
tn
s
n
p
VPP
(17)
),1(
2
2
)1()1()( n
s
j
t
n
s
tAn
s
n
p
VV (18)
),1()1()( n
s
tjn
s
An
p
A
(19)
).1()( n
s
jn
p
j
(20)
Smoothing;
)),()(()()( n
p
n
o
n
p
n
s
PPPP
(21)
)),()(()()( n
p
n
o
t
n
p
Vn
s
PPV
(22)
)),()((
2
2
)()( n
p
n
o
t
n
p
An
s
PPA
(23)
)).()((
3
6
)()( n
p
n
o
t
n
p
jn
s
PPj
(24)
wherethesubscriptso,pand sdenotetheobserved,
predicted and smoothed state parameters
respectively; P, V and A are the target’s position,
velocity and acceleration respectively; t is the
simulationtimeinterval;andnisthesamplenumber.
The filter weight constants, α, β, γ and
η, are
computed using the fading memory filter model as
shown in Equations 25‐28 (Brookner, 1998). ξ is the
discounting factor that minimizes the least squares
errorforaconstantjerkmodelinput.Thesmoothing
constants are determined from the value of the
discountingfactorhencetheoptimizationof
thefilter
isappliedontheξasillustratedbyPanetal(2016).
,
4
1
(25)
),
2
111411()1(
6
1
2
(26)
),1()1(2
3
(27)
.)1(
6
1
4
(28)
58
4 SIMULATION
4.1 Inputmodeloftarget’sdynamics
The simulation tests were carried out on a high
dynamictargetmovingattheinitialspeedof50m/s
as observed from a stationary own ship. A sample
signal of n=1000 data samples was investigated at
samplingintervaltimeoft=3s
whichcorresponds to
the time of one aerial rotation of the radar antenna.
The target’s initial position as observed from the
radar range measurements was (573, 1038.4) after
scan‐conversion to produce Cartesian coordinates.
The input model employed to generate the target
dynamicsisasshownbelowinEquations29‐
30.
,10)]3cos(5)3sin(9
)2cos(6)7.0sin(8)99.0cos(7)2.1sin(10[
iwiwi
wiwiwiwia
i
X
(29)
)].2sin(22)3.0cos(20[ wiwib
i
Y
(30)
The resulting data was then sampled at intervals
of three seconds to obtain the true tra jectory of the
targetasshowninFigure1.
Figure1.Target’struetrajectory.
4.2 Noiseaddition
The observation measurement obtained from the
radarsensorcontains an error which was accounted
for by corrupting the true positions with zero mean
random white Gaussian noise with a standard
deviation,σ,of10m.Figures2a&2bshowtheerror
distributionintheobservation.
Figure2a.East‐westerrordistributionintheobservation.
Figure2b.North‐southerrordistributionintheobservation.
4.3 Filtergaincoefficientselectionandcomputation
4.3.1 FiltergaincoefficientsselectionusingtheBenedict‐
Bordnermodel
Since this design method does not provide an
analytical solution for determining the position
smoothing coefficient α, in this study, the position
smoothingcoefficientwasdeterminedexperimentally
throughatrialanderrormethod
byplottingitagainst
the corresponding innovation which is the total
residual obtained from the difference between the
observedposition and predicted position trajectories
as shown in Figure 3. The interval evaluated was
selected based on the stability constraints provided
for by Jury (1964) for the α‐β‐γ tracking filter. The
value
ofαthatbestreducedtheinnovationwasfound
to be α=0.86. Equations 7 & 8 were then used to
compute the values of the velocity and acceleration
smoothingcoefficientsasshowninTable3.1.
Figure3. Total residual between observed and predicted
positions against corresponding value of position
smoothingcoefficient,α.
59
Table1. Smoothing coefficients obtained from Benedict‐
Bordnermodel.
_______________________________________________
αβγ
_______________________________________________
0.860.64884.4409x10
‐16
_______________________________________________
4.3.2 FiltergaincoefficientsselectionusingtheGray‐
Murraymodel
The maneuverability and measurement noise
variances were determined experimentally by an
iterative trial and error method by changing the
values of maneuverability and measurement error
variances while simultaneously feeding the
measurementdatatothefilterforeacherrorvariance.
The
output was then used to compute cumulative
positional error which was then plotted against
corresponding error va riances. The purpose of this
procedure was to determine the error variance
coefficientcorrespondingtotheleasterror.Fromthe
Figures 4‐7, the values of the maneuverability and
measurement error variance coefficients
corresponding
totheminimumresidualerrorare10
‐3
and 1 respectively. Consequently, the respective
standard deviations are estimated to be σ
w=0.03162
andσ
v=1.
The tracking index was, therefore, computed as
Λ=0.2846 and, consequentlythe damping parameter,
r=0.6873. The smoothing coefficients are then
computedusingEquations11‐13andareobtainedas
displayedinTable2.
Table2.SmoothingcoefficientsobtainedfromGray‐Murray
model.
_______________________________________________
αβγ
_______________________________________________
0.52770.19560.0101
_______________________________________________
Figure4. Cumulative error difference between observed
and predicted positions against maneuverability error
variance.
Figure5. Cumulative error difference between true and
smoothedpositionsagainstmaneuverabilityerrorvariance.
Figure6. Cumulative error difference between observed
and predicted positions against measurement error
variance.
Figure7. Cumulative error difference between true and
smoothedpositionsagainstmeasurementerrorvariance.
4.3.3 Filtergaincoefficientsselectionusingthefading
memorymodel
The optimal value of the damping parameter ξ
was experimentally found to be 0.62 for a
maneuveringtargetwithaninitialspeedof 50.4 m/s
(Pan et al. 2016). The Equations 14‐16 were then
employedtocomputetheoptimalfiltering
coefficients
whichwereobtainedasshowninTable3.
60
Table3. Smoothing coefficients obtained from fading
memoryfiltermodel.
_______________________________________________
αβγ
_______________________________________________
0.73790.31880.0467
_______________________________________________
4.3.4 Filtergaincoefficientsselectionusingthejerky
model
The optimal value of the damping parameter ξ
was determined through an iterative trial and error
methodandfoundtobe0.74foramaneuveringtarget
with an initial speed of 50.4 m/s and sampled at
intervalsof3seconds(Pan
etal.2016).Equations25‐28
werethenemployedtocomputetheoptimalfiltering
coefficientsasshownonTable4.
Table4. Smoothing coefficients obtained from jerky filter
model.
_______________________________________________
α βγη
_______________________________________________
0.70010.30850.0612 7.6163
‐04
_______________________________________________
5 FILTERPERFORMANCECOMPARISON
Inthisstudy,thecomparisonofthefilterswasbased
onthefollowingperformanceindicesi.e.trackingand
estimation error reduction, sensitivity of the filter to
targetmaneuversandoutputdatastability.
5.1 α‐β‐γfilterresultsandperformancecomparison
Figures8‐10 show the true, observed,
predicted and
smoothed positions trajectories obtained from the
trackingproblemusingthevariousα‐β‐γfiltermodels
under consideration in this study. The figures
represent the positional trajectories for the Benedict‐
Bordner filter, Gray‐Murray model, the Fading
memoryfiltermodelrespectively.Ofthethreemodels
underinvestigation,theGray‐Murraymodel
appears
tofollowthetargetquitewellwithhighsensitivityto
changes in target maneuvers as indicated by the
stabilityandsteadinessofthetrajectoriesasthetarget
transitionsfromonepointtothenext.Inaddition,the
output trajectories which include the predicted and
smoothed position trajectories can be
observed to
transition very smoothly and closely to the true
trajectory for the entire duration of the tracking
period.Thefadingmemorymodelperformsnearlyas
well as the Gray‐Murray model except for a few
fluctuations of data samples at several points along
thetarget’scurvesindicatingareducedsensitivity
at
these points on the targets’ trajectories as it
maneuvers. As for the Benedict‐Bordner model,
showninFigure8,thefilterperformsworst,basedon
sensitivity to target maneuvers and data stability,
comparedto the other two α‐β‐γfilters as indicated
bythevisiblyclearjerky motion at the beginning
of
thetrackingprocess.However,astrackingcontinues
thetrajectoriesstabilizeandthetrackingaccuracycan
beseentoalsoimprove.
Figure8.Target’sTrue,Observed,PredictedandSmoothed
Position,Benedict‐Bordnermodel.
Figure9.Target’sTrue,Observed,PredictedandSmoothed
Position,Gray‐Murraymodel.
Figure10. Target’s True, Observed, Predicted and
SmoothedPosition,fadingmemorymodel.
Table5showsthetotalpredictionandestimation
errors obtained using the different α‐β‐γ filters.
Estimation error is obtained by computing the
deviationoftheestimateddatafromthetrueposition
foreachsample.Similarly,predictionerror indicates
howfarthepredictedpositiondeviatesfromthetrue
position hence it is the
tracking error. These results
showthatthefadingmemorymodelhasthehighest
accuracy in both tracking and estimation of the
positionofthehighdynamictargetamongtheα‐β‐γ
filters as can be seen from the small error values
obtained, followed by the Gray‐Murray model. The
Benedict‐Bordnermodel
performstheworstinterms
of tracking and estimation noise reduction for both
prediction and estimation as indicated by the
resultinglargeerrorsvalues.Thiscanbeexplainedby
thefactthatthedesignofthisfilterisbasedprimarily
on the requirement for satisfying a good transient
response. And since
performance of a filter is a
tradeoffbetweenagoodtransientresponseandnoise
reduction, the filter then performs poorly when
applied to meet the requirement for tracking error
reduction.
61
Table5. Summary of the total tracking and estimation
accuracyobtainedfromtheα‐β‐γfilters.
______________________________________________
FiltertypeTrackingerror Estimationerror
____________________________
mm
______________________________________________
Benedict‐Bordner 26,32611,677
Gray‐Murray2107111,693
Fadingmemory 19,62210,653
______________________________________________
5.2 Comparisonofα‐β‐γfilterresultswiththejerky
model
Figure 11 shows the true, observed, predicted and
smoothedtrajectoriesobtained usingthejerk model.
Thecurvescanbeobservedtoeasilyfollowthehighly
maneuvering target with greater sensitivity as
indicated by the steadiness in the predicted and
smoothedtrajectories
andareductionoffluctuations
thatwereobservedinthetrajectoriesobtainedusing
the fading memory α‐β‐γ filter. However, the Gray‐
Murraymodelstill maintains agreatersensitivityto
target maneuvers and has a higher stability in its
outputdataleadingtosteadiertrajectories.
Figure11. Target’s True, Observed, Predicted and
SmoothedPosition,jerkymodel.
Thetotaltrackingerrorandtotalestimationerror
areobtainedasshowninTable6.Theresultsindicate
an improvement in both tracking and estimation
accuraciesonapplyingthejerkmodelincomparison
with the fading memory α‐β‐γ filter model. The
accuracyintrackingisthereforeimprovedby1,733.27
m equivalent to
approximately 9%. Similarly, the
estimation accuracy is increased by 419.49 m on
employingthejerkmodelfilter.
Table6. Summary of the total tracking and estimation
accuracyobtainedfromtheα‐β‐γfilters.
_______________________________________________
FiltertypeξTrackingerror Estimationerror
___________________________
mm
_______________________________________________
α‐β‐γfilter 0.62 1962210,653
α‐β‐γ‐ηfilter 0.74 17,85910227
_______________________________________________
6 CONCLUSION
This study investigated the performance of three
conventional α‐β‐γ filter models under the same
initial conditions to track a high dynamic target
undergoing random velocity changes. The
performance of the filters was evaluated based on
abilitytofollowthemaneuveringtargetsteadilyand
closelywithminimumjerkymotionsand
withoutloss
oftarget.Itwasalsoafunctionofnoisereductionin
theestimationandpredictionresults.
Of the three filters, the Benedict‐Bordner filter
performedtheworstastheresultingtrajectorieswere
characterizedbyovershootingatvariouspointsofthe
target’scurves.
The critically damped filter, on the
other hand,
performed efficiently in terms of noise reduction in
bothpredictionandestimationwhichisvisiblyclear
from the high accuracy obtained compared to the
Gray‐Murray filter. In addition to demonstrating a
goodcapabilityof following themaneuvering target
witheaseandsteadiness,thecriticallydampedfilter
wasalso
easytoimplementduetoitssimplicityand
lowcomputational load. However, the Gray‐Murray
filterdepictedabettersensitivitytotargetmaneuvers
whichwasvisiblefromtheobtainedsmoothcurvesof
thepositiontrajectoriesindicatingahigherefficiency
infollowingthehighlymaneuveringtarget.
Onapplyingthejerkmodel,
animprovementwas
realisedinbothnoisereductionandabilitytofollow
themaneuveringtargetwithlessfluctuationson the
trajectories.Tracking accuracy was improved by
approximately 9% compared to the constant
acceleration filter. The jerky model was therefore a
furtherenhancementoftheconstantaccelerationfilter
in terms of increasing
data stability through a
reduction of fluctuations especially at points of
suddenspeedandcoursechanges.
Future studies will investigate the tracking
performance of the filter while both the observing
shipandthehighdynamictargetareonmotion.
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