- Email: [email protected]
Bearings-Only Tracking for Maneuvering Targets in Clutter
3a-093
Copyright © 1996IFAC 13th Triennial Wurld Congress, San Fmncisco, USA
BEARINGS-ONLY TRACKING FOR MANEUVERING TARGETS IN CLUTTER Don Lerro· and Yaakov Bar-Shalom·· ·Engtl1"'~rtng
"Th e
Tcchnl!logy
UIl;Uer~ l t'}
C ~ nt.er,.rl.1ystIC,
eT
(}j Con,-uchcut, St01T&, CT
Abstract.. In recursive henrings-cnly trn.ckinp;, the Cartesian coordinate Extended Kalman Filter or the modified polar =ordina(.;, filter provide poor convergence and erratk hehaVlOr nllfl to th"., I=k of imtial targt':t rang... informatlon rllqllirM for proper filter Imtialb:atlOll, VVhile the advantage of a recut'8ive estimator IS that it allows for models which account for ac('clerrl.tions of maneuvermg targeh, it call1lot be relied upon because of mitiah-.;ation problems. The maximum likelihood estimatcr combined with the probabalistic data alisocia~iolL, .ML/PDA, has ueell ~hown to be ;:!'Il dfectlve e~tunator for tracking non_mant>llvt>ring targ ... ts in clutter. This batch method IS shown to provide t he accurate mitlaimatJOn required for rE"<'ursive filter~ once the ra.nl/;e observab~lity IS sufficient rollow.lnl/; an own~hlp manue\·er. Furthermore" t~e cility of handling man<.euvers and clutter is obtamed by uSing a recursIve Interactmg Multiple l'Iilodel Probaballstlc Oat.a i\s~0ciation with Amplitude Informa.tion(IMMPDAFAI) following t.hf' bat..::h ML/PDA initl,alb:ation. The resulting filter, called the Batch/R':"cu["sive IMl\.lPDAFAI (BRIMM) is shown ';0 provide reliable track mitialization as well as accurate state estimation through target maneuvers in the presence of dutter. Subsequently, it '$ demonstrated how the !:;"!\IPD;\FAI can be u~-=-d for .:Id.:lph!.l ~ d~CI$lon., ab(}ut ownslHp mafleul' t f"$ loWlt'J upon the ttlrget ijtate e8timatlOtl to enhance The tnrg ... t ,")h~ ... rvahility after a target maneuver has occurred Key Words. BE'B.ring9-on~y tracking, clutter, maneuverinl/; targets, probabalistic data a.5soci
1. i'lTROD1)CTTO'l
A nonlinear estimatioll problem of interest is the passive Lracking of moving targets u~ing only hearing or line of sight (LOS) angle measuremen ts. The problem is to determine the relative target position and velocity with respect to an ownship sensor platform. The solution for this problem requiret> an owns hip acceleration for observability of the range to the target. Since t.he problem i~ nonlinear, the usual approach for reemsive estimation is to employ an Extended l{aJrnan Filter (EKFL Beca.use the LOS is an incomplete pot>ition observation it cannot be convcrtcd into Cartesian coordinates to allow for linear filtering as ill (LB93J. In recun,ive bearings-only [racking the use of the Cartesian coordinate Extended Kalman Filter has been t>hown to provide erratic estimation results and unstable behavior (Aid79), even without the detrim('ntal effects caused by the pres€'nce of false detectiont> or clutter. The inaccuracy of the estimates is c:ansed by the lack of initial target range information prior to thl! first ownship manu ever resulting in poor Cartesian position and velocity estimates. Similar problems are encountered hy the vlodi:fied Polar filter, even though this filter has bettN convergenCf! properties than the Cartesian El{F for initial ranges f'stirna.tf's whkh a,re larger than the true target range(Aid83). Furthermore, theRe. algorithms have not been examined in practical scenarios, i.e .. with target maneuvers and fahe alarm~. The performance of the Kalrnan filter (rep;a.rdle::;s of tht' coordinates) in tbe presence of false alarmt> hal; been shown to be poor (BL95).
4196
The Maximum Likelihood (,\.fL) solution which provideR estimates based on a. batch of measurements, obtained over a period of time which encompasses an ownship maneuver, has been t>hown to provide estimates which achieve the eramerRao Lowt'r Bound (CRLB) ,i.e" a. covariall<.:e consistent with the inverse of the FD,1. However, robust performance in a cluttered environment with maneuvering targets is desirable. A method that iududes Probabalistic Data Association (PDA) in the ML framework, ).1L-PDA (JB90), has been shown to be robust in a cluttered environment (or a constant velocity target. This method has been extended to improve estimation for low observable targets by using the target amplitude feature (KB95). The limitation with the ::\1L approach is, however, that it cannot be applied to the situation where tar~ett> und('r,l!;o maneuvers. Recursive methods which incorporate models for the acceleration of tbe target ill maneuvers can provide accurate estimatiun ami reliable track mallltellaru':e [or ~he problem of tracking a maneuvering target. In particular, the Interacting ,,\1 uItiple :\lodel Probabalistic Data Association Filter with Amplitude Informat.ion (I.\1.\1PDAFAI), which has been shown to be robust when tracking low SNR maneuvcring targets in cluttcr for the active sonar problem (81.95) is a good candidate for the pasRive bearing~-only problem. In this paper a Hew approach for the bearings-only tracking problem is df>veloped where the ML-PDA batch method is used :first to provide consisknt initialization for a recursive estimator. Once the accepted initial estimate and covariance
are obtained from the :\1L-PDA algorithm, then a recursive approach consisting of the I\tr.\fPDAFAI is used to provide reliable track maintenance for the bearings only problem in clutter. The resulting metllod, called the Batch Recursive nn1PDAFAI (BRnB1), improve8 position and velof.:ity e;;stimation over current recursive methods when tracking both non-maneuvering ;~nd maneuvering targets. The advantage of this approach is that key tracking decisions can be made automatically by a.ssessing the likelihood of target models (pre~ent/absent, maneuvering/non-maneuvering) to provide rapid and accurate decisiollf for both track acceptance and false track dismi.ss.d in track formation. Furthermore, by assessing model probabilities, the nrMPDAFAI provides the ability to continue tracking through target maneuvers, determine the on~et and termination of these manuevers) and quickly dismissco; lost tracks It will also be shown how this method can be used for adapt~v( drocisions about ownship maneuvers based upon the available target state estimation to enhance the target observability.
The Probabalistic Data Ass.ociation Filter equations which incorporate the target amplitude feature (LeB93) are i(klk - 1 I ~ Fi(k
-1Ik - I) - "o(k - I)
(6)
which is a linear prediction ilnd has associated covariance P(klk - I) ~ FP(k -
JI! -I)F' + Q
t.he predicted measurement
i~
(7)
nonlinearly related to the state
i(klk - I) ~ h [i(klk - 11]
(8)
with a covariance
S(k) ~ h, (k)P(klk - I)h~(k)
+R
which relies on the evaluatioll of the Jacobian filter gain is
(9) h~(k)
and the
(10 ) A validation region centered around the predicted measurement is set up to select the ~et of measurements to be associakd probabilistically t.o the target. The validation region
2. TilE RECURSIVE BEARINGS-ONLY TRACKING PROBLE.ll
lS
[z - ;(klk -1)j'S-'(k)[(, - z(klk -1)] 0;, In Cartesian coordinat.es thf' discrete dynamic equation for the relative target state is x(k
+ IJ ~
Fx(k)
+ J'dk) -
",,(k)
(I)
The stat.e vector consists of t.he relative position and velocities
(11)
where I determines the siz~ of the validation region. The set of validat.ed measuremcats which exceed the detection t.hreshold at time k is (12)
where mk is the number of validated measurements received at time k. The updated state is
(2)
i(klk) ~ i(klk - I)
+ W(k)_(k)
(13)
where where ~o, 'fJ-;, are the own ship position coordinates and ~r .11t are the target position coordinates. The process noise.tJ(k), is used to model unpredictable target accelerations in each coordinate assumed to be zero mean. white and Gaussian with equal variance q in each coordinate. The vector ua(k) models the;; own platform mot.iOll. The measurement equation is
z(k)
~
O(k)
+ w(k)
m(k)
v(kl ~
(14)
flJvj(k)
J=l
is the combined innovat.ion formed from all the validat(-o. mcasurcments. The innovation associated wit.h the,;th measurement is.
(3)
where 9(k) is the line of sight angle to the target
L
vJ(k) ~ zJ(k) - z(klk-I}
(15)
The association probabilitiei< are
(4) The measurement. noise, -w(k). is assumed to he white.Gaussian. zero mean with variance R. The relative target range is (5)
4197
i
~
I, ... , m(k)
(16)
and ~o
=
b
.
b + ),!l1(k) to L, ........ 1=1
J
"
(17)
due to the muJtimodaJ na t ure of t he log.likelihood {unction . T he test. statistic to be ll sed fo r track a cceptance is
wl)(.' r ~
(18)
6= ~[Z",i ,]- NI'
(24)
-/Nu (19) The probability tha.t the ta.rget measurement falls in the va.lidation region is PG and N[l-'j 0, 5] is the normal pdf with argument tI , mean zero, and covariance S. The amplitude likelihood rat.io
(20) is used to modify the slillldard PDAF associa.tion probabilitieto, where pQ" (a) is the pdf of the amplit.ude d ue (·0 noise only which exceeds t.he detection threshold and p;(a) is the pdf of the amplitude which exceeds t he t hreshold a n<1 originated from the target . T he proba.hility of detection is PD. .>. is the spat iaJ density of faJ se m~a.s llrem ~ nts . This density is number of fal se alarms per unit area. hence it. is direc tly related to the probability of faJse alarm, Pj Q , and the resolution cell size of the processed. measure ment s.
where J.' and tT : which d epen d on the values of PD and >., are also derived in (KB95 ). The observa.bilit.y of the solution is re fl ected. in the FIM. An accept.ed ML-PDA solution ma.y exist with such poor observability due to the targetjowlI ship scenario tha.t t.he estimates are too poor to initiaJize the rt.!(;ursive est.imator. The recursive estimators have been sh(lwn to require an initia1 estimate t.hat i~ reasonably dose 10 I.he tru e target. location to perform reliably and ensure convergenc.e(Aid79). Therefore, an additional t.C!ll based on th e invf!r!;e of the Fnf is pe rformed. Since the observability in thf. bearings only problem is dominated by errors in the range component of the solu tion, it is requi n .'tI t hat (t.he firs t o rde r appro:timation ofthe) standa.rd de viatio n o f I.he range erro r of the solu tion satisfy
(25) where do is a problem depend ent th res hold . This test ensu.res sufficiently accurate rec ursive initi alization a nd recursive estimation .
3. ML/PDA 11ETHOD OF TRACK l'lITIALlZATlON T he )..t L- PDA presented in (JB90) presents a t echnique for estimat ing the target sta.te for a constant velocity target in cilal.e r using bearing-only measurements acquired over time . T his tech nique combined with tht, target amplitud e f('!atllr ~ was s hown to improve t.he convergence of the est imat io n fo r low SNR ta.rge ts (KBD5). The CRLR which acconnt!'! for th e a.mplitude information and a method of validating
(2 1) The '\i L-P DA estimated t.arget sl.ate at th e batch,it{ l) . hoils covariance
th~
initial t im e of
(22) wh ere N
J = "(PL' , .\", . ,)
L
~ [v,e( k))[V.,e(k)],
(23)
k=l
where Q2(PD. AV!I.!1) is the information reduc tion factor due tht' preSefl(."C of false m ca.... nrcnu: uts which de pends o n the "aJU(:s of Pr> a.nel ). . This has bf.'C n derived with ;md with ~ out t he a.m plit ll d~ l~at llre ill {K Il9.1 ,. A valid ation test for the t rack is necessary s ince the maximization procedure may (;ou\'erge (.0 a local minimum res ult ing in an unreaJi st ic {rack to
4198
4.
ADAPTIVE OWNSHIP MANEUVER FOR TARGET OBSERVABILITY E)l HANCEME)lT
Adaptive decisions about ownship m ane uve rs can be made based on the estimated ta.rgl ~ t s~ate pro\'id ed t he estimalion of t he target st.ate i.s accu rate S UcJl ~ haL the o nset and termina.tion of the target man euver can be dete rmined. The on- linr. a.bility to estima.te t he ta.rget (;ourse accu rately will provide t.he mea.ns for an o ..... ns hip mane uver decision to enhance the observahility of the target o nce t he targe t mane uvcr has terminated . This ca.n he accomplis hed by performing a test. a fter the man ell\'er has been sensed. T he mos t accura.t.e IIU!aIlS of sensin g t he manf:!11Ver ill th ro ug h the course estimate . \A/hen a target. undergoes a coordinated turn the course r.,stimate becomes ver.\· poor. A test can be devised to sense when the course ac.cllr:a.ey of the filter stabilizes, which indica.tes the manuever has tl!rminated. Based UpOll the new target course an own ship mam'!llver can be made to improve the target obsen'abi1ity and , hence , the estimation accu.racy. An est.imate of the course varian ce can be obtained from a first order a.pproximation as _2
iJ I'll
'.2
+
\.
P33 .S
-
2E, f7 P1:J
(26)
where .i is t he T.uget s peed estima.te . Com parison of the course 5tandaJd deviation to all acceptance threshold (e.g.: 20° ) will be used to deleUlliue the te rmina.{l()n of the T.arget manc u vcr which indicates t ilat o ne s hould perform a platform maueu\'cr iu t.he diren io n of the ta rge t to close the range .
5. SI'JULATJON RJo:SULTS A:'lD PERFOR'.IANCE CO}lI'ARIC;ON The performance of the Batch-Recursive n"n'JPDAFAI (BRnL\1) is a..'lscssed [or both lIonmaneuvering and maneuveTing targets. The performance i~ examined for fOIlI target and oWlIl>hip platform scenarios lIsing two nn,1PDAFAI COllfigurations of target. motion models. The surveillance region is 180 0 comprised of 6(J hf!aring cells of 1.1° each and the standard deviation of t,ht~ bea.rilll,!; measurements has been taken as 0.9°, The target amplitude has a Rayleigh pdf with an average S~R of HdB. The t.hreshold is set at 2.0 hence PD = ,9, P Ju = .135 and A = .OitSjde,q. Therefore the expected number of false alarms observed in the surveillance region is 8.1 per measurement period.
5.1. Targd awl Ownship $cf'flarios The first two scena.rios illusl rat.ed in Figures I and 2 depict the tarp;et a.nd oWlIship trajectory for a constant velocity target over 65 measurements where the time between bearing measurements is 20s. In both scenarios the ownship executes a maneuver (90'" (~ourse <.hange) starting at k=10. The ta.rget maintains a speed of 10 at/s and the ownship maintains a speed of 14 m/::; In Scenario 2 the oWllship platform executes an additional mallcuver (90" course change) toward the target at k=45. Scenario:3 shown in Figure 3 represents thc same oWIIship platform motion as Scenario 1 b1lt now the target undergo!";; (I 1ROo coordina.ted tnrn starting at k=:30 whkh laq,R for If) ~amples ending at k=40{.9"/s). Scenario 4, shown in Figure 4, is similar to Scenario 3 except the ownship platform executes another maneuver (90" COllrse change) toward the targct at k=4·':i. One additional scenario is examined which is similar to Scenario 4 but the last ownship maneuver does not occur at the fixed time k=45. The last ownship leg oc'Curs aJapLively based upon the estimation as described section 5. This scenario is used to demoTI!'!trate the capabilit,v of dderlllinin,L', oWtl~hip manuevers based upon the on-line estimation.
5.2. Tnu:k Formation: AfL-PDA The batch 11 L- PDA was used over the first 20 measurements to perform track formation and the acceptance test in Section 5 was performed. The t.est. was designed to accept 95% of the valid target t.rac.ks for S~R=13 dB. The observed results were consist.ent: 98% pa...:;sed the test for each target scenario from 200 run:'l. The validated tracks from the batch are used as initial estimateR for t.he no.1PDAFAI.
5.3. Tilt: IAfMPD.4FAI COllfi,qm'atious Two configurations of the nL\fPDAFAI, based upon different Rets of maneuver moneh-, were used to track the target through the manuever. Both configurations utilize model Ml fo.r the constant course and speed portions of the trajectory
4199
while the second configuration employs left and right coordinated turn models. Configuration 1. The mode f'et consists of three second order kinematic. models with the following design parameters
MO: "no target" model with the same proce.'3S noise as
,u1. A! 1: white noi~e acceleration with variam:oe q1 = 0.Of2. ,\,12: white noise acceleration with variance q2 = 0.1 2 .
Confi,quHltion 2. The mode set consists of models MO and
M1 and A1 J : coordinated turn model for a left turn AJ4: coordinated turn model for a right turn. The htter models include it white noise acceleration with low va.riance q3 = q4. = 0.0]2 in each coordinate. The turn rate should be choosen to handle a range of typical target manel1vers. The assumed tum rate is n = .5°/8 for these models. Note this is half of the turn rate used by the target in these simulat.ions (u~inp; the actual turn rate provides similar results).
.5.4. The
Re8ult.~
The estimation performance is provided in terms of the RMS position errors, R~1S course errors: and R~1S speed errors over 65 time int.ervals (of 211s each) using 200 :\'Ionte Carlo trials. It. must be noted tha,l the 200 trials are "post :\·iLPDA acceptance" runs where the target has passed acceptance a.t measurement interval 20. A lost track is declared if the ta.rget measurement falls outside the largest track gate for all models or the probability of the "no target" model exceed" .95. The HRL\1M filter RMS position error, RMS course error, and R ?l'fS !>peed error for Sc(~nario 1 are "hown for both configurations described above in Figures 5, 6, and 7: respectively. The adva.ntage of poerforming the additional owns hip maneuver to dose the range to the target in Scenario 2 is obvious from the improvement of the estimation errors for both 1.M.MPDAFAI configurations. The R.MS position: course and speed for ScelLMio 2 are shown in Figurcs 8, 9. and 10. Configuration 1 and Configuratipeed ~st.imation error i ... crlmparable for the two configurations. The accurate course estimation using coordina.ted turn models allows for an on-line adaptive maneuver based on th.~ estimated target course according to the tcst in Sil'Ction 5. Comparing ConfigUlation 2 and Configuration 2A it.
Copyright © 1996IFAC 13th Triennial Wurld Congress, San Fmncisco, USA
BEARINGS-ONLY TRACKING FOR MANEUVERING TARGETS IN CLUTTER Don Lerro· and Yaakov Bar-Shalom·· ·Engtl1"'~rtng
"Th e
Tcchnl!logy
UIl;Uer~ l t'}
C ~ nt.er,.rl.1ystIC,
eT
(}j Con,-uchcut, St01T&, CT
Abstract.. In recursive henrings-cnly trn.ckinp;, the Cartesian coordinate Extended Kalman Filter or the modified polar =ordina(.;, filter provide poor convergence and erratk hehaVlOr nllfl to th"., I=k of imtial targt':t rang... informatlon rllqllirM for proper filter Imtialb:atlOll, VVhile the advantage of a recut'8ive estimator IS that it allows for models which account for ac('clerrl.tions of maneuvermg targeh, it call1lot be relied upon because of mitiah-.;ation problems. The maximum likelihood estimatcr combined with the probabalistic data alisocia~iolL, .ML/PDA, has ueell ~hown to be ;:!'Il dfectlve e~tunator for tracking non_mant>llvt>ring targ ... ts in clutter. This batch method IS shown to provide t he accurate mitlaimatJOn required for rE"<'ursive filter~ once the ra.nl/;e observab~lity IS sufficient rollow.lnl/; an own~hlp manue\·er. Furthermore" t~e c
1. i'lTROD1)CTTO'l
A nonlinear estimatioll problem of interest is the passive Lracking of moving targets u~ing only hearing or line of sight (LOS) angle measuremen ts. The problem is to determine the relative target position and velocity with respect to an ownship sensor platform. The solution for this problem requiret> an owns hip acceleration for observability of the range to the target. Since t.he problem i~ nonlinear, the usual approach for reemsive estimation is to employ an Extended l{aJrnan Filter (EKFL Beca.use the LOS is an incomplete pot>ition observation it cannot be convcrtcd into Cartesian coordinates to allow for linear filtering as ill (LB93J. In recun,ive bearings-only [racking the use of the Cartesian coordinate Extended Kalman Filter has been t>hown to provide erratic estimation results and unstable behavior (Aid79), even without the detrim('ntal effects caused by the pres€'nce of false detectiont> or clutter. The inaccuracy of the estimates is c:ansed by the lack of initial target range information prior to thl! first ownship manu ever resulting in poor Cartesian position and velocity estimates. Similar problems are encountered hy the vlodi:fied Polar filter, even though this filter has bettN convergenCf! properties than the Cartesian El{F for initial ranges f'stirna.tf's whkh a,re larger than the true target range(Aid83). Furthermore, theRe. algorithms have not been examined in practical scenarios, i.e .. with target maneuvers and fahe alarm~. The performance of the Kalrnan filter (rep;a.rdle::;s of tht' coordinates) in tbe presence of false alarmt> hal; been shown to be poor (BL95).
4196
The Maximum Likelihood (,\.fL) solution which provideR estimates based on a. batch of measurements, obtained over a period of time which encompasses an ownship maneuver, has been t>hown to provide estimates which achieve the eramerRao Lowt'r Bound (CRLB) ,i.e" a. covariall<.:e consistent with the inverse of the FD,1. However, robust performance in a cluttered environment with maneuvering targets is desirable. A method that iududes Probabalistic Data Association (PDA) in the ML framework, ).1L-PDA (JB90), has been shown to be robust in a cluttered environment (or a constant velocity target. This method has been extended to improve estimation for low observable targets by using the target amplitude feature (KB95). The limitation with the ::\1L approach is, however, that it cannot be applied to the situation where tar~ett> und('r,l!;o maneuvers. Recursive methods which incorporate models for the acceleration of tbe target ill maneuvers can provide accurate estimatiun ami reliable track mallltellaru':e [or ~he problem of tracking a maneuvering target. In particular, the Interacting ,,\1 uItiple :\lodel Probabalistic Data Association Filter with Amplitude Informat.ion (I.\1.\1PDAFAI), which has been shown to be robust when tracking low SNR maneuvcring targets in cluttcr for the active sonar problem (81.95) is a good candidate for the pasRive bearing~-only problem. In this paper a Hew approach for the bearings-only tracking problem is df>veloped where the ML-PDA batch method is used :first to provide consisknt initialization for a recursive estimator. Once the accepted initial estimate and covariance
are obtained from the :\1L-PDA algorithm, then a recursive approach consisting of the I\tr.\fPDAFAI is used to provide reliable track maintenance for the bearings only problem in clutter. The resulting metllod, called the Batch Recursive nn1PDAFAI (BRnB1), improve8 position and velof.:ity e;;stimation over current recursive methods when tracking both non-maneuvering ;~nd maneuvering targets. The advantage of this approach is that key tracking decisions can be made automatically by a.ssessing the likelihood of target models (pre~ent/absent, maneuvering/non-maneuvering) to provide rapid and accurate decisiollf for both track acceptance and false track dismi.ss.d in track formation. Furthermore, by assessing model probabilities, the nrMPDAFAI provides the ability to continue tracking through target maneuvers, determine the on~et and termination of these manuevers) and quickly dismissco; lost tracks It will also be shown how this method can be used for adapt~v( drocisions about ownship maneuvers based upon the available target state estimation to enhance the target observability.
The Probabalistic Data Ass.ociation Filter equations which incorporate the target amplitude feature (LeB93) are i(klk - 1 I ~ Fi(k
-1Ik - I) - "o(k - I)
(6)
which is a linear prediction ilnd has associated covariance P(klk - I) ~ FP(k -
JI! -I)F' + Q
t.he predicted measurement
i~
(7)
nonlinearly related to the state
i(klk - I) ~ h [i(klk - 11]
(8)
with a covariance
S(k) ~ h, (k)P(klk - I)h~(k)
+R
which relies on the evaluatioll of the Jacobian filter gain is
(9) h~(k)
and the
(10 ) A validation region centered around the predicted measurement is set up to select the ~et of measurements to be associakd probabilistically t.o the target. The validation region
2. TilE RECURSIVE BEARINGS-ONLY TRACKING PROBLE.ll
lS
[z - ;(klk -1)j'S-'(k)[(, - z(klk -1)] 0;, In Cartesian coordinat.es thf' discrete dynamic equation for the relative target state is x(k
+ IJ ~
Fx(k)
+ J'dk) -
",,(k)
(I)
The stat.e vector consists of t.he relative position and velocities
(11)
where I determines the siz~ of the validation region. The set of validat.ed measuremcats which exceed the detection t.hreshold at time k is (12)
where mk is the number of validated measurements received at time k. The updated state is
(2)
i(klk) ~ i(klk - I)
+ W(k)_(k)
(13)
where where ~o, 'fJ-;, are the own ship position coordinates and ~r .11t are the target position coordinates. The process noise.tJ(k), is used to model unpredictable target accelerations in each coordinate assumed to be zero mean. white and Gaussian with equal variance q in each coordinate. The vector ua(k) models the;; own platform mot.iOll. The measurement equation is
z(k)
~
O(k)
+ w(k)
m(k)
v(kl ~
(14)
flJvj(k)
J=l
is the combined innovat.ion formed from all the validat(-o. mcasurcments. The innovation associated wit.h the,;th measurement is.
(3)
where 9(k) is the line of sight angle to the target
L
vJ(k) ~ zJ(k) - z(klk-I}
(15)
The association probabilitiei< are
(4) The measurement. noise, -w(k). is assumed to he white.Gaussian. zero mean with variance R. The relative target range is (5)
4197
i
~
I, ... , m(k)
(16)
and ~o
=
b
.
b + ),!l1(k) to L, ........ 1=1
J
"
(17)
due to the muJtimodaJ na t ure of t he log.likelihood {unction . T he test. statistic to be ll sed fo r track a cceptance is
wl)(.' r ~
(18)
6= ~[Z",i ,]- NI'
(24)
-/Nu (19) The probability tha.t the ta.rget measurement falls in the va.lidation region is PG and N[l-'j 0, 5] is the normal pdf with argument tI , mean zero, and covariance S. The amplitude likelihood rat.io
(20) is used to modify the slillldard PDAF associa.tion probabilitieto, where pQ" (a) is the pdf of the amplit.ude d ue (·0 noise only which exceeds t.he detection threshold and p;(a) is the pdf of the amplitude which exceeds t he t hreshold a n<1 originated from the target . T he proba.hility of detection is PD. .>. is the spat iaJ density of faJ se m~a.s llrem ~ nts . This density is number of fal se alarms per unit area. hence it. is direc tly related to the probability of faJse alarm, Pj Q , and the resolution cell size of the processed. measure ment s.
where J.' and tT : which d epen d on the values of PD and >., are also derived in (KB95 ). The observa.bilit.y of the solution is re fl ected. in the FIM. An accept.ed ML-PDA solution ma.y exist with such poor observability due to the targetjowlI ship scenario tha.t t.he estimates are too poor to initiaJize the rt.!(;ursive est.imator. The recursive estimators have been sh(lwn to require an initia1 estimate t.hat i~ reasonably dose 10 I.he tru e target. location to perform reliably and ensure convergenc.e(Aid79). Therefore, an additional t.C!ll based on th e invf!r!;e of the Fnf is pe rformed. Since the observability in thf. bearings only problem is dominated by errors in the range component of the solu tion, it is requi n .'tI t hat (t.he firs t o rde r appro:timation ofthe) standa.rd de viatio n o f I.he range erro r of the solu tion satisfy
(25) where do is a problem depend ent th res hold . This test ensu.res sufficiently accurate rec ursive initi alization a nd recursive estimation .
3. ML/PDA 11ETHOD OF TRACK l'lITIALlZATlON T he )..t L- PDA presented in (JB90) presents a t echnique for estimat ing the target sta.te for a constant velocity target in cilal.e r using bearing-only measurements acquired over time . T his tech nique combined with tht, target amplitud e f('!atllr ~ was s hown to improve t.he convergence of the est imat io n fo r low SNR ta.rge ts (KBD5). The CRLR which acconnt!'! for th e a.mplitude information and a method of validating
(2 1) The '\i L-P DA estimated t.arget sl.ate at th e batch,it{ l) . hoils covariance
th~
initial t im e of
(22) wh ere N
J = "(PL' , .\", . ,)
L
~ [v,e( k))[V.,e(k)],
(23)
k=l
where Q2(PD. AV!I.!1) is the information reduc tion factor due tht' preSefl(."C of false m ca.... nrcnu: uts which de pends o n the "aJU(:s of Pr> a.nel ). . This has bf.'C n derived with ;md with ~ out t he a.m plit ll d~ l~at llre ill {K Il9.1 ,. A valid ation test for the t rack is necessary s ince the maximization procedure may (;ou\'erge (.0 a local minimum res ult ing in an unreaJi st ic {rack to
4198
4.
ADAPTIVE OWNSHIP MANEUVER FOR TARGET OBSERVABILITY E)l HANCEME)lT
Adaptive decisions about ownship m ane uve rs can be made based on the estimated ta.rgl ~ t s~ate pro\'id ed t he estimalion of t he target st.ate i.s accu rate S UcJl ~ haL the o nset and termina.tion of the target man euver can be dete rmined. The on- linr. a.bility to estima.te t he ta.rget (;ourse accu rately will provide t.he mea.ns for an o ..... ns hip mane uver decision to enhance the observahility of the target o nce t he targe t mane uvcr has terminated . This ca.n he accomplis hed by performing a test. a fter the man ell\'er has been sensed. T he mos t accura.t.e IIU!aIlS of sensin g t he manf:!11Ver ill th ro ug h the course estimate . \A/hen a target. undergoes a coordinated turn the course r.,stimate becomes ver.\· poor. A test can be devised to sense when the course ac.cllr:a.ey of the filter stabilizes, which indica.tes the manuever has tl!rminated. Based UpOll the new target course an own ship mam'!llver can be made to improve the target obsen'abi1ity and , hence , the estimation accu.racy. An est.imate of the course varian ce can be obtained from a first order a.pproximation as _2
iJ I'll
'.2
+
\.
P33 .S
-
2E, f7 P1:J
(26)
where .i is t he T.uget s peed estima.te . Com parison of the course 5tandaJd deviation to all acceptance threshold (e.g.: 20° ) will be used to deleUlliue the te rmina.{l()n of the T.arget manc u vcr which indicates t ilat o ne s hould perform a platform maueu\'cr iu t.he diren io n of the ta rge t to close the range .
5. SI'JULATJON RJo:SULTS A:'lD PERFOR'.IANCE CO}lI'ARIC;ON The performance of the Batch-Recursive n"n'JPDAFAI (BRnL\1) is a..'lscssed [or both lIonmaneuvering and maneuveTing targets. The performance i~ examined for fOIlI target and oWlIl>hip platform scenarios lIsing two nn,1PDAFAI COllfigurations of target. motion models. The surveillance region is 180 0 comprised of 6(J hf!aring cells of 1.1° each and the standard deviation of t,ht~ bea.rilll,!; measurements has been taken as 0.9°, The target amplitude has a Rayleigh pdf with an average S~R of HdB. The t.hreshold is set at 2.0 hence PD = ,9, P Ju = .135 and A = .OitSjde,q. Therefore the expected number of false alarms observed in the surveillance region is 8.1 per measurement period.
5.1. Targd awl Ownship $cf'flarios The first two scena.rios illusl rat.ed in Figures I and 2 depict the tarp;et a.nd oWlIship trajectory for a constant velocity target over 65 measurements where the time between bearing measurements is 20s. In both scenarios the ownship executes a maneuver (90'" (~ourse <.hange) starting at k=10. The ta.rget maintains a speed of 10 at/s and the ownship maintains a speed of 14 m/::; In Scenario 2 the oWllship platform executes an additional mallcuver (90" course change) toward the target at k=45. Scenario:3 shown in Figure 3 represents thc same oWIIship platform motion as Scenario 1 b1lt now the target undergo!";; (I 1ROo coordina.ted tnrn starting at k=:30 whkh laq,R for If) ~amples ending at k=40{.9"/s). Scenario 4, shown in Figure 4, is similar to Scenario 3 except the ownship platform executes another maneuver (90" COllrse change) toward the targct at k=4·':i. One additional scenario is examined which is similar to Scenario 4 but the last ownship maneuver does not occur at the fixed time k=45. The last ownship leg oc'Curs aJapLively based upon the estimation as described section 5. This scenario is used to demoTI!'!trate the capabilit,v of dderlllinin,L', oWtl~hip manuevers based upon the on-line estimation.
5.2. Tnu:k Formation: AfL-PDA The batch 11 L- PDA was used over the first 20 measurements to perform track formation and the acceptance test in Section 5 was performed. The t.est. was designed to accept 95% of the valid target t.rac.ks for S~R=13 dB. The observed results were consist.ent: 98% pa...:;sed the test for each target scenario from 200 run:'l. The validated tracks from the batch are used as initial estimateR for t.he no.1PDAFAI.
5.3. Tilt: IAfMPD.4FAI COllfi,qm'atious Two configurations of the nL\fPDAFAI, based upon different Rets of maneuver moneh-, were used to track the target through the manuever. Both configurations utilize model Ml fo.r the constant course and speed portions of the trajectory
4199
while the second configuration employs left and right coordinated turn models. Configuration 1. The mode f'et consists of three second order kinematic. models with the following design parameters
MO: "no target" model with the same proce.'3S noise as
,u1. A! 1: white noi~e acceleration with variam:oe q1 = 0.Of2. ,\,12: white noise acceleration with variance q2 = 0.1 2 .
Confi,quHltion 2. The mode set consists of models MO and
M1 and A1 J : coordinated turn model for a left turn AJ4: coordinated turn model for a right turn. The htter models include it white noise acceleration with low va.riance q3 = q4. = 0.0]2 in each coordinate. The turn rate should be choosen to handle a range of typical target manel1vers. The assumed tum rate is n = .5°/8 for these models. Note this is half of the turn rate used by the target in these simulat.ions (u~inp; the actual turn rate provides similar results).
.5.4. The
Re8ult.~
The estimation performance is provided in terms of the RMS position errors, R~1S course errors: and R~1S speed errors over 65 time int.ervals (of 211s each) using 200 :\'Ionte Carlo trials. It. must be noted tha,l the 200 trials are "post :\·iLPDA acceptance" runs where the target has passed acceptance a.t measurement interval 20. A lost track is declared if the ta.rget measurement falls outside the largest track gate for all models or the probability of the "no target" model exceed" .95. The HRL\1M filter RMS position error, RMS course error, and R ?l'fS !>peed error for Sc(~nario 1 are "hown for both configurations described above in Figures 5, 6, and 7: respectively. The adva.ntage of poerforming the additional owns hip maneuver to dose the range to the target in Scenario 2 is obvious from the improvement of the estimation errors for both 1.M.MPDAFAI configurations. The R.MS position: course and speed for ScelLMio 2 are shown in Figurcs 8, 9. and 10. Configuration 1 and Configurati