49
1 INTRODUCTION
Trackingfiltersareanessentialpartoftargettracking
astheyplaythekeyroleof trackingerrorreduction
andmakingaccurateestimations.Theα‐βandα‐β‐γ
filters, which are steady state Kalman filters for
tracking constant velocity and constant acceleration
targets respectively, are limited in their capacity to
followahighmaneuveringtarget,definedbyajerky
motion, with good accuracy. The jerky motion is
reduced considerably when the ta
rget tracking
equations are modelled to make provision for a
constant‐jerk.Thesecondorderα‐β‐γfilter,therefore,
becomes a four‐state filter when the design is
extendedtoincludetheconstantjerkparameter.
Several a
pproaches have been introduced in the
recentpastinanattempttodesignfilteringequations
that model for the constant jerk. They have been
foundtoout‐performtheconstantaccelerationα‐β‐γ
filterintermsoferrorreductionandabilitytofollow
a maneuvering target with jerky movements.
Mehrotra (1997) suggests a jerk model for tra
cking
highly maneuvering targets marred by unexpected
changesinthespeed.Thesimulationresultsindicate
betterperformanceofthejerkmodelthanthatofthe
lower order filters. The study undertaken by Chen
(2008) also showed an improvement in the target
trackingaccuracywhenthe
α‐β‐γ‐σfilterwasutilized
in tracking a constant jerk model compared to the
conventional α‐β‐γ filter. Similarly, Seyyed (2009)
designed the steady state Kalman filter α‐β‐γ‐η, an
extension of the α‐β‐γ filter, for tracking high
A Study of Optimization of α-β-γ-η Filter for Tracking a
High Dynamic Target
T.G.Jeong&B.F.Pan
KoreaMaritimeandOceanUniversity,Busan,Korea
A.WNjonjo
J
omoKenyattaUniversityofAgricultureandTechnology,Nairobi,Kenya
ABSTRACT:Thetrackingfilterplaysakeyroleinaccurateestimationandpredictionofmaneuveringvessel’s
positionandvelocity.Differentmethodsareusedfortracking.However,themostcommonlyusedmethodis
theKalmanfilteranditsmodifications.Theα‐β‐γfilterisoneofthespecialcasesofthegeneralsolutionpro‐
videdbytheKalmanfilt
er.Itisathirdorderfilterthatcomputesthesmoothedestimatesofposition,velocity
andaccelerationforthenthobservation,andalsopredictsthenextpositionandvelocity.Althoughfoundto
trackamaneuveringtargetwithagoodaccuracytha
ntheconstantvelocity,α‐βfilter,theα‐β‐γfilterdoesnot
performimpressively under highmaneuvers suchaswhen thetarget is undergoingchanging accelerations.
Thisstudy,therefore,aimstotrackahighlymaneuvering targetexperiencingjerkymotionsduetochanging
accelerations.Theα‐β‐γfilterisextendedtoincludethefourthstatetha
tis,constantjerktocorrectthesudden
changeofaccelerationinordertoimprovethefilter’sperformance.Resultsobtainedfromsimulationsofthe
inputmodelofthetargetdynamicsunderconsiderationindicateanimprovementinperformanceofthejerky
model,α‐β‐γ‐η,algorithmascomparedtotheconstantaccelerationmodel,
α‐β‐γintermsoferrorreduction
andstabilityofthefilterduringtargetmaneuver.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 11
Number 1
March 2017
DOI:10.12716/1001.11.01.04
50
maneuveringtargetandconcludedthatcomparedto
the standard α‐β‐γ filter, the new design was more
superior in terms of ability to follow a jerky model
withagoodaccuracy.
In their research, Wu (2011) went further and
proposedanevolutionaryprogrammingbasedα‐β‐γ‐
σfilterthatprovidedanoptimalsimulationtechnique
for a maneuvering target with jerky movement. In
additiontobeinghighlyaccurateandefficientinthe
predictionofthetargettrajectory,thisnewfour‐state
filter was associated with a reduced computational
time.
Njonjo(2016)investigatedtheperformanceofthe
fadingmemoryα‐β‐γfilteronahighdynamictarget
warship. The
research concluded that the filter was
capable of tracking the highly maneuvering vessel
with a relatively good accuracy in terms of noise
reduction.ThisresearchwasfurtherextendedbyPan
(2016) where the filter was optimized in order to
improveitstrackingabilitybyreducingthenoisetoa
minimum.
The optimization procedure involved
varying the value of the discounting factor,ξ, with
the residual error and determining theξthat
corresponds to the minimum error. The study
demonstrated that the optimal filter uniquely varies
withtheinitialspeedandaveragespeedofthetarget
underconsideration.
Theinvestigationpresentedinthis
studyisaimed
at improving the accuracy of the jerky motion of a
high maneuvering target. The design examined is a
fourstate filter, which will be referredto as α‐β‐γ‐η
filter,usingthe fadingmemory filter algorithm.The
filter is first optimized, then its performance is
compared with that of
the optimal memory α‐β‐γ
filterasfurtherdiscussedbyPan(2016)ba sedonits
abilitytofollowthejerkymotionwithahigherorder
ofaccuracy.
2 α‐β‐γ‐ηFILTERMODEL
The α‐β‐γ‐η filter is a constant gain, four‐ state
trackingfilter.Thefourstatevectorincludesposition,
velocity, acceleration and jerk, a time
derivative of
acceleration. The jerk is modelled as a constant and
includeszeromeanwhiteGaussiannoise.
The algorithm involves two major stages of
computations,thatis,predictionandcorrectionsteps.
Equations 1‐4 are the prediction equations for
position, velocity, acceleration and jerk respectively
where they are updated from the
estimated state
therebyloweringthetrackingerrorEquations5‐8are
the smoothing equations which are computed by
adding a weighted difference between the observed
andthepredictedpositiontotheforecaststate.
Prediction;
)1(
6
3
)1(
2
2
)1()1()( n
s
j
t
n
s
A
t
n
s
tn
s
n
p
VPP
(1)
)1(
2
2
)1()1()( n
s
j
t
n
s
tAn
s
n
p
VV
(2)
)1()1()( n
s
tjn
s
An
p
A (3)
)1()( n
s
jn
p
j
(4)
Smoothing;
))()(()()( n
p
n
o
n
p
n
s
PPPP
(5)
))()(()()( n
p
n
o
t
n
p
Vn
s
PPV
(6)
))()((
2
2
)()( n
p
n
o
t
n
p
An
s
PPA
(7)
))()((
3
6
)()( n
p
n
o
t
n
p
jn
s
PPj
(8)
where;
the subscripts o, p and s denote the observed,
predicted and smoothed state parameters
respectively.
P,VandAare thetarget’sposition, velocity and
accelerationrespectively.
t is the simulation time interval and; n is the
samplenumber.
The filter weight constants,α,β,γ
andη, are
computed using the fading memory filter algorithm
as shown in Equations 9‐ 12 and areextracted from
Brookner (1998). Theξrepresents the discounting
factor that minimizes the least squares error for a
constant jerk input model of target dynamics. The
smoothingconstantsaredeterminedfromthevalueof
the discounting factor hence the optimization of the
filterisappliedontheξasillustratedbyPan(2016).
4
1
(9)
)111411(
22
)1(
6
1
(10)
)1(
3
)1(2
(11)
4
)1(
6
1
(12)
3 SIMULATIONOFα‐β‐γ‐ηFILTER
A target model vessel on motion with the initial
relativevelocityof50.4m/sandattheinitialposition
(573,1038.4)ontheCartesiancoordinatesasobserved
from the radar range measurements from own ship
was considered for simulation. The sample
measurements were collected at
the time interval of
three seconds which corresponds to the radar scan
rate of 20 rev per minute. Table 1 below shows a
summaryoftheinitialinputingeneratingtheoriginal
targetmotion.
Table1.TheInitialInputoftargetmotion
_______________________________________________
Position RelativeSpeed TimeInterval SampleSize
(x.y)m/ss(n)
_______________________________________________
(573,1038.4) 50.431000
_______________________________________________
51
3.1 Inputmotionmodelofthetargetdynamics
The model equations used in generation of the
originaltargetmotionareasshowninEquations13&
14.
;10)]3cos(5)3sin(9
)2cos(6)7.0sin(8)99.0cos(7)2.1sin(10[
iwiwi
wiwiwiwia
i
X
(13)
)];2sin(22)3.0cos(20[ wiwib
i
Y
(14)
Figure1.Target’strueposition
NoiseAddition
Since the measurements from the radar sensor
containerrors,thiswastakenintoaccountbyadding
a zero mean random white Gaussian noise with a
standard deviation,σ, of 10 m to the true position
sample.Thiserrordistributionintheobservationisas
showninFig.2and
Fig.3.
Figure2.East‐Westerrorintheobservation
Figure3.North‐Southerrorintheobservation
3.2 Determinationofoptimalksi,ξ
Thefilterweightconstantsα,β,γandηvaluesdepend
onthevalueofthediscountingfactor,ξasshownin
Equations 9 ‐12. Therefore, in order to obtain the
optimal smoothing coefficients, optimization of the
critically damped filter focusses on adjusting theξ
valueexperimentallythroughtrialand
errormethod.
The process of optimization begins by
computationof thetotal transient errorwhich is the
sumofthesquaresofthedifferencebetweenthetrue
trajectoryandthepredictedtrajectory.Thepurposeof
theoptimizationistofindthediscountingfactorthat
minimizes this error then use this
information to
compute the optimal smoothing constants. This is,
therefore, achieved by plotting a range of the
discountingfactor,whichliesintheinterval[0,1]and
withastepincreaseof0.01,againstthecorresponding
transient error. Thirty simulation tests are then
carriedoutforeachξandtheaverageobtained.
The
resultingfigureisasshowninFig.4below.According
tothisgraph,thevalueofξcorrespondingtotheleast
residualerroris0.74.Fig.5and6servetoshowthe
consistency regarding the optimalξand therefore
upholdthevalidityoftheresultsastheybothindicate
asimilar
valueoftheoptimaldiscountingfactor.They
represent plots of the total error difference between
the true and smoothed trajectories and between the
observed and predicted trajectories respectively
againstcorrespondingξvalues.
Figure4. Average Summation of the error difference
betweentrueandpredictedpositionsagainstcorresponding
valueofthediscountingfactor,ξfor30simulations
Figure5. Average Summation of the error difference
between true and smoothed positions against
corresponding value of the discounting factor,ξfor 30
simulations
52
Figure6. Average Summation of the error difference
between observed and predicted positions against
corresponding value of the discounting factor,ξfor 30
simulations
4 PERFORMANCEOFTHEFILTER
The performance of the optimal α‐β‐γ‐η filter is
evaluated based on its ability to follow the
maneuveringtargetwitheaseandwithanincreased
reduction of overshooting along the prediction
trajectory due to jerky movements. In addition, the
filter’s performance is further investigated based on
its capability
of reduction of tracking error. The
performance was then compared with that of the
optimalα‐β‐γfilterundersimilarconditionsoftarget
motion.
Using the obtained optimal discounting factor
(ξ=0.74)theoptimalfilterweightswerecomputedand
consequently the predicted and smoothed trajectory
were obtained. Fig.7 shows the true, observed,
predicted and smoothed trajectories. The curve
enclosed in the rectangle is enlarged for clearer
viewingasshowninFig.8.Thepositionaltrajectories
obtained from simulation tests carried out under
similarconditionsusingtheoptimalα‐β‐γfilterareas
shownbelowinFig.9.Fromtheobtainedtrajectories,
theα‐β‐γ‐ηfiltercanbeobserved
toeasilyfollowthe
highlymaneuveringtargetwithgreatersensitivityas
indicated by the steadiness in the predicted and
smoothed trajectories. In contrast, the predicted and
smoothed trajectories resulting from the α‐β‐γ filter
areseentohaveerraticchangesandovershootingat
variouspointsonthetrajectoriesindicatingthefilter’s
inability
torespond welltorandomchangesinspeed
and quick maneuvers particularly for this type of
trajectory.
As mentioned above, it is visibly clear that after
optimization the α‐β‐γ‐η filter has a higher
performancethantheα‐β‐γfilter.Quantitatively,the
totaltrackingerrorandtotalestimationerrorareused
toevaluate theperformance
oftwofilters.Asshown
in the tablebelow, the α‐β‐γ filter demonstrates the
bestperformancewhenξ=0.64withthetotaltracking
error amounting to 19,592.20 m. On the contrary,
whenξ=0.74 the α‐β‐γ‐η filter depicts its optimal
performance with total tracking error resulting to
17,858.93 m. The accuracy in tracking is
therefore
improved by 1,733.27 m equivalent to 8.9 % higher.
Similarly, the estimation accuracy is increased by
419.49monemployingtheα‐β‐γ‐ηfilter.
Figure7.Target’sTrue,Observed,PredictedandSmoothed
Position,α‐β‐γ‐ηfilter
(optimalξ=0.74)
Figure8. Enlarged View of target’s True, Observed,
PredictedandSmoothedPositioncorrespondingtoFig.7
Figure9.Target’sTrue,Observed,PredictedandSmoothed
Position(ξ=0.64),α‐β‐γfilter
Table2. Summary of the total tracking and estimation
accuracyobtainedfromtheoptimalfilters
_______________________________________________
FiltertypeξTrackingerrorEstimationerror
mm
_______________________________________________
α‐β‐γfilter 0.64 19592.2010646.02
α‐β‐γ‐ηfilter 0.74 17858.9310226.53
_______________________________________________
5 CONCLUSION
Target motions defined by high maneuvering and
jerky movements such as high dynamic vessels
requirepromptandaccuratepredictionsofthetarget
dynamics.Standardfilterssuchastheα‐β‐γfilterare
incapable of tracki ng the randomly changing
accelerations with good accuracy since they are
designedtotrackconstantacceleration
motionsonly.
53
Therefore,ahigherordersteadystatefilterisrequired
totakecareofthejerkymaneuvers.
The results obtained from simulation tests of the
high dynamic target model indicate a better
performanceofthejerkmodelfilterintermsofability
to follow the maneuvering target with a greater
sensitivity
compared to the α‐β‐γ filter under the
same operating conditions of target motion. This
attributeisduetotheinclusionofthejerksmoothing
constant that is responsible for taking care of the
suddenchangesinaccelerationandtargetmaneuvers.
Inaddition,thetracking accuracy increased by8.9%
forthejerk
filter model compared to theα‐β‐γ filter
model. On the downside, the computational load of
theα‐β‐γ‐ηfilterishigherthanthatoftheα‐β‐γfilter
asindicatedbythe increased numberofexpressions
showninsection2.
Inthisstudy,atheoreticalmodelwasadoptedto
generate the target’s dynamics. However, in
a real
environmentthedynamicsofthetargetarecalculated
usingradarmeasurementsofrangeandbearingand
are presented on the ARPA system. The authors
thereforeintendtoapplytheresultsofthisstudytoa
real situation in the near future whereby the range
andangularbearingswillbe
obtainedinordertotest
theeffectivenessoftheproposedalgorithm.
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