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mühendislikdergisi

Cilt: 3-9

Dicle Üniversitesi Mühendislik Fakültesi

78

Data Association for Simultaneous

Localization and Mapping in

Clutter Environment

Extended abstract

Robot or vehicle has been tried to build up a map and simultaneously localize its position within an unknown environment or to update its position and map which was called problem in the early of 1990. Smith et al. (1990) called his problem simultaneous localization and mapping (SLAM). They shows SLAM is general problem that while mapping the environments measurement noises statistically dependent on before their values and monotonically growing with map building, so robot/vehicle incorrectly localize their position and obtain environment mapping. There are several theoretical and applicable studies available in the literature. There are some specific problems available with this method. For example, minimization of observation and measurement noises, increasing the number of object in the building environment and robot/vehicle remembering the occurrence of its position which is known as data association(correspondence) in the OLWHUDWXUHHWF«

Recent studies considering these problems and have tried to propose a solution with using different statistical methods based on Bayes theorem. While some of these studies trying to minimize the effects of observation noises on the mapping and position errors, some of them have developed algorithms to reduce the processing times cause of increasing the number of objects related environments causing problems in real-time applications (Montemerlo, M. et al 2001, Kim C. et al 2008). However, some studies have focused on the data association problem, which is known as remembering the occurrence of the robot/vehicle, in other words trying to solve the problem of uncertainty of the object location (Neira J. and 7DUGyV-'5H[ H. Wong et al 2010). These studies can be listed as Kalman- based estimators, sequential monte carlo approaches, known as particle filters and their derivatives, and expectation maximization based estimators.

When observation noise statistically increases over time, measurement data can be obtained in complex and leads to formation of measurement uncertainty. SLAM method takes advantage of the hallmarks of an autonomous robot/vehicle location information while the surrounding the objects. If the landmarks

are obtained the correct information, position of autonomous robot/vehicle is used to obtain the correct measurement.

In some cases, the number of landmarks and to be close each other leads to the interference which landmark is arrival of the measurement landmarks. In this situations, declining the performance of estimators, leading to increase mean square error of mapping and positions of autonomous robot/vehicle, so SLAM problem causing the results in a number of uncertainties with improper obtaining building of WKH PDS DQG WKH YHKLFOH¶V SRVLWLRQ DQG KHDGLQJ angle. In this case, there is a need for data association.

Successful data association is provided by observed measurement results from itself correctly association. In the SLAM problem, the estimator is able to forecast the new landmarks, recognize the false alarms (incorrect measurements) and follow the measurements correctly. The most basic algorithm for data association is nearest neighbor method. This method uses Mahalanobis distance during processing. Mahalanobis distance calculates the distance between measured and predicted observation. Algorithm accepts predicted target position closest to the measured position as valid measurement. According to observation measurement creating the acceptance region for next renewal of landmarks, acceptance region is referred to as gate. However, the measurement may not be associated with the nearest landmark, in NN filter not interested in this situation. Therefore, updated state vector may lead to divergence. It is also observed in dynamic environments, are not performing well (Rex H. Wong et al 2010).

If number of landmarks is very high and landmarks to be close to each other in noisy observations, NN algorithms do not give better results addressed in the previous studies. Because of this, predictive value of actual measurement from the nearest measurement is known as a reference to valid measurement in the gate. NN algorithm shortens the processing time and not used all the measurements to reach conclusions quickly, so the situation away from the optimal structure. Probabilistic data association (PDA) algorithm calculates the probability of being the target measurement all measurement in the gate. PDA calculates a combined value of innovation, as it tries to solve stochastic problem of uncertainty. Using the relational likelihood of hypothesis provides a unified innovation, and with it creates a unified valid gate.

During the operation, algorithm accepts the independent of target and not interfering neighboring landmarks. However, in SLAM problem landmarks are correlated with each other, make mutual interference cannot be ignored. Therefore, it is not suitable multiple object or target tracking and dynamic situations, or algorithm should be run for each target. This is a retreat for PDA and emphasizes joint PDA (JPDA) for multiple target tracking. JPDA has been developed for following all targets in a loop.

There are some studies available using JPDA for solving data association problem of SLAM in the literature. (Rex H. Wong et al 2010)¶ VWXG\ XVHG JPDA for solving of wireless sensor networks in SLAM problem and 3-scan JPDA algorithm was used. They said that on the basis of noisy sensor information and possible false repercussions that the signal has white noise and this leads to uncertainty in the position and angle of incoming signals. In other JPDA based approach proposed by Zhou et al. Their proposed method is measurement-oriented, using Depth-First-Search (DFS) algorithm for generation of hypothesis.

These studies have been generally used to solve the problem of data association in SLAM problem. However, performance of the algorithms use the filters is not covered. A number of comparisons were only made during the uncertainties. Proposed study taking into account previous studies have focused on

two new approaches on a more appropriate for SLAM problem. First, there is an alternative to be presented the problem of data association problem of SLAM in high noisy and uncertainty in feature-based environment. Developed algorithm for the problem of SLAM application is proposed for the first time used related environment and scenario. Another improvement is the used filter. Extended Kalman filter (EKF) is generally preferred in previous studies (Rex H. Wong et al 2010, Montemerlo, M. et al 2002). In this study, unscented Kalman filter (UKF) is considered to be more successful in minimizing observation noise problem in SLAM. UKF tries to estimate posteriori probability distribution of state by selecting a certain number of sigma points on probability distribution with a non-linear function. This method allows filter less time to make accuracy and processing speed than FastSLAM based particle filters. JPDA based UKF is for the first time used for SLAM problem in this study. The correct filter used with solution of the uncertainty of situation is thought to achieve the desired result is more favorable.

Keywords: Simultaneous Localization and Mapping,

unscented Kalman Filter, data association, joint probabilistic data association.

80

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gözlemleQPLúWLU 'L÷HU ELU L\LOHúWLUPH LVH NXOODQÕODQ V]JHoWLU 'DKD |QFHNL oDOÕúPDODUGD JHQHORODUDNJHQLúOHWLOPLú.DOPDQV]JHFLWHUFLK HGLOPLúWLU (Rex H. Wong ve ark. 2010, Montemerlo, M. ve ark 2001 Bailey T. 2001). %X oDOÕúPDGD NRNXVX] .DOPDQ V]JHFL (Unscented Kalman filter, UKF) SLAM SUREOHPLQLQ J|]OHP JUOWVQQ PLQLPL]H edilmesinde daha EDúDUÕOÕ RODFD÷Õ GúQOHUHN

82

WHUFLKHGLOPLúWLU. UKF süzgeci deterministik bir \DNODúÕPOD model durumunun VRQFXO RODVÕOÕ÷ÕQÕ GD÷ÕOÕP ]HULQGH EHOLUOL VD\ÕGD VLJPD QRNWDVÕ VHoHUHN GR÷UXVDO ROPD\DQ ELU IRQNVL\RQOD WDKPLQ HWPH\H oDOÕúPDNWDGÕU %X \|QWHP V]JHFLQ NHVWLULP GR÷UXOX÷XQX YH LúOHP KÕ]ÕQÕ )DVW6/$0WDEDQOÕSDUoDFÕNV]JHoOHULQGHQGDKD NÕVD VUHGH \DSPDVÕQD RODQDN VD÷ODPDNWDGÕU SLAM probOHPLQLQ o|]PQGH $QNÕúKDQ + (2012), Kim C. ve ark. (2008), ve Bailey T. (2002) \DSPÕú ROGXNODUÕ oDOÕúPDlarda bu süzgecin daha X\JXQ VRQXoODU YHUGL÷Lni J|]OHPOHPLúOHUGLU8.)LOHELUOLNWHJHOLúWLULOPLú JPDA DOJRULWPDVÕ EWQOHúLN RODUDN SLAM SUREOHPL LoLQ EX oDOÕúPDGD |QHULOPLúWLU. ÇDOÕúPDGD GXUXP EHOLUVL]OL÷LQLQ o|]P ve GR÷UX V]JHo NXOODQÕPÕ LOH LVWHQLOHQ X\JXQ VRQXFDXODúÕODFD÷ÕGúQOPúWU.

dDOÕúPDQÕQLOHUOH\HQE|OPOHULúXúHNLOGHGLU,,, E|OPGH SUREOHP WDQÕPÕ ,,,. b|OPGH veri LOLúNLOHQGLUPHhipotezleri, IV. b|OPGHNRNXsuz kalman süzgeci, V. b|OPGH deneysel sonuçlar YH WDUWÕúPD son olarak VI. b|OPGH VRQXo YH GH÷HUOHQGLUPH\HUDOPDNWDGÕU.

3UREOHP7DQÕPÕ

%XE|OPGHSDUD]LW\DQNÕOÕRUWDPGDSLAM için YHUL LOLúNLOHQGLUPH DOJRULWPDVÕ DQODWÕODFDNWÕU %D\HV \DSÕVÕ\OD ELUOLNWe stokastik modelin VRQFXORODVÕOÕ÷Õ

^

`

^

` ^

`

^

`

0: 0: 0 0: 1 0: 0 0: 1 0: , | , , | , , | , , | , k k k k k k k k k k k P x m Z U x P z x m P x m Z U x P z Z U   (1)

J|VWHULOLU %urada xk k DQÕQGD URERWXQ GXUXPX m LVH JOREDO KDULWDGDNL J|]OHPOHQHQ QHVQH

LúDUHWOHULQL VÕQÕU WDúODUÕ, Z_0:k ve U_0:k bütün |OoPOHU YH NRQWURO YHNW|UQ J|VWHUPHNWHGLU

0

x LVH URERWXQ EDúODQJÕo GXUXPXQX

vermektedir. 6RQFXO RODVÕOÕNOD J|]OHP PRGHOL

^

_k| _k,

`

P z x m RODUDN J|VWHULOLU YH z , _k

^

z z1 2, ,...,zm k

`

k DQÕQGDNL |OoPOHULQ WRSODPÕQÕ

verir ve |OoP

( , )

k k k

z h x m _ G (2)

olarak modellenir. Burada G_k NN(0(0,(0,(0,(0,Q_kk)¶ GÕU YH|OoPWDKPLQL

/ 1 Ö

Ö ( k k ) k

z h x   G (3)

olarak verilir. gOoP WDKPLQL YH|OoPQIDUNÕ iQRYDV\RQYHNW|UüQYHULU, / 1 Ö Ö ( k k ) v z z z h x    (4) YHLQRYDV\RQNRYDU\DQVÕ / 1 / 1 Ö Ö ( _{k k} ) _v[ ( _{k k} )]T _k S h x 

¦

h x  Q (5)

olarak verilir. h x(Ö_{k k/ 1}) |OoPIRQNVL\RQX6 v

GXUXP NRYDU\DQVÕ YH G_k VÕIÕU RUWDODPDOÕ

|OoPQ EH\D] JUOWVGU Qk NRYDU\DQVÕQD

sahiptir. øVWDWLVWLNVHOJHoHUOLOLNNDSÕVÕJ|]OHPYH HQ \DNÕQ |OoP DUDVÕQGD HúOHúWLUPH\L VD÷ODU (Zhou B. and Bose N, 1993) %X WDQÕPGDQ \ROD

oÕNDUDN Mahalanobis mesafesi (ya da

QRUPDOOHúWLULOPLúLQRYDV\RQNDresi- NIS), 1 T x n M v S v J d (6)

RODUDN WDQÕPODQÕU (Rex H. Wong ve ark. 2010).

NIS VHUEHVWOL÷LQ N boyutu ile Chi-Kare (F2)

GD÷ÕOÕPÕQDVDKLSWLUJn NDSÕ HúLN GH÷HULGLUNN

V]JHFL |OoPQ JHOHQ VLQ\DOLQLQ JHoHUOLOL÷LQL test etmek için EXHúLNGH÷HULQLNXOODQÕU Bütün RODVÕ|OoPOHUDUDVÕQGDKDQJLVLQHVQHQLQWDKPLQ HGLOHQ GH÷HULQH HQ \DNÕQ LVH R JHoHUOL |OoP RODUDN DWDQÕU (÷HU oRNOX KHGHIOHUGH NN V]JHFLQGH NDSÕODU VW VWH oDNÕúÕUVD VHQV|U |OoPOHUL ELUELULQGHQ ED÷ÕPVÕ] ROVD ELOH EHOLUVL]OLN GXUXPX RUWD\D oÕNDU V]JHo EX

GXUXPGD GR÷UX YHUL LOLúNLOHQGLUPHVL

\DSDPD\DELOLU g]QLWHOLN WDEDQOÕ SLAM

SUREOHPLQGHEWQVHQV|U|OoPOHULELUELULQGHQ ED÷ÕPVÕ] ROVDODU ELOH WDKPLQ HGLOHQ |OoPOHU robotun EHQ]HU SR]LV\RQ KDWDVÕ WDUDIÕQGDQ LVWDWLVWLNVHO RODUDN ELUELUL\OH LOLúNLOLGLU %X durumdan, PDA oRNOXKHGHIWDKPLQLGXUXPXQX

o|]HPH]\DGDWHNUDUOÕRODUDNKHUELUKHGHILoLQ V]JHo DOJRULWPDVÕ WHNUDU NRúWXUXOXU dRNOX

KHGHI SUREOHPLQL o|]PHN LoLQ JPDA

algoULWPDODUÕ|QHULOPLúWLU.

+HGHIOHULQNPHVLk DQÕQGDX_k { , ,..., }x x_{1 2} x_n

RODUDN GúQOUVH LOJLOL ]DPDQGD EX NPH LOH LOLúNLOHQGLULOHQ JHoHUOL |OoPOHULQ WRSODPÕ m olarak kabul edilir ve k DQÕQGDNL |OoPOHU

NPHVLZ_k { , ,...,z z_{1 2} z_M} olarak \D]ÕOÕU

%XUDGDKHUELU|OoP\DKHGHIWHQ\DGDSDUD]LW

\DQNÕGDQ JHOPHNWHGLU 3DUD]LW \DQNÕ x olarak ₀

\D]ÕOÕU7DKPLQHGLOHQKHGHI|OoP x_{zmzm , ölçüm} x m¶LQLQRYDV\RQYHNW|ULVH Ö x m m x zx z  Öz m m z zzz z (7)

oODUDN\D]ÕOÕU (Y.Bar-Shalom ve ark. 2009). Her

bir x KHGHIL LoLQ ELUOHúWLULOPLú

D÷ÕUOÕNODQGÕUÕOPÕúLQRYDV\RQ 1 M x x x m m m zx

¦

Ex xz m m z

¦

Exz (8)

burada E_mx hedef x¶ WHQ JHOHQ LOLúNLOHQGLULOPLú

RODVÕOÕ÷Õ J|VWHUPHNWHGLU YH E₀x t DQÕQGDNL

|OoPOHULQ KLoELULVLQLQ KHGHIWHQ JHOPHGL÷L RODVÕOÕ÷ÕJ|VWHULU { ( ) | } ( ) x k x m P k Z am I E

¦

) ) (9) 0 1 1 M x x m m E 

¦

E (10) burada m=1,2,…M; x=0,1,…,N 1 ( ) 0 x m ölçüm hedeften gelmektedir a ölçüm hedeften gelmemektedir ­ ) ® ¯ (11)

burada )( )k LOJLOL ]DPDQGD ELUOHúLN LOLúNLVHO ROD\ODUÕJ|VWHULUYH 1 ( ) M x( ) m m k I k ) x( ) m( I (12)

\D]ÕOÕU, I_mx( )k ELUH\VHO LOLúNLVHO ROD\Õ J|VWHULU

ˆ ROD\ODUÕQ NHVLúLPLQL J|VWHULU YH amx( ))

JHoHUOLOLN PDWULVLQGHNL |OoP z ve hedef x DUDVÕQGDNLLOLúNLVHOKLSRWH]LJ|VWHULU [ ( )]x m a : ₎ (13) 0 1 2 1 2 1 1 1 1 1 2 2 2 2 2 1 2 . ... 1 1 1 N N N N M M M M x x x x z a a a z a a a z a a a § · ¨ ¸ : _{¨ ¸} ¨ ¸ ¨ ¸ ¨ ¸ © ¹ z · 1 N a ¸ 1 ·· 1 z ¸ N ¸ 2 N aN¸¸ ¨ ¸ ¨ ¨¨ ¸¸¸ ¸ ¸ ¸¸ z ¸ N¸¸ N ¹ M a_MN¸¸ a (14)

(úLWOLN ¶ GDQ, k ]DPDQÕQGDki tüm ölçümlerin ELUOHúLN ROD\ODUÕQÕQ RODVÕOÕ÷Õ HúLWOLN ¶ We YHULOPLúWLU 1 1 { ( ) | } { ( ) | ( ), } 1 _{( | ( ), ) { ( )}} k k k k P k Z P k Z k Z p Z k Z P k c   ) ) ) ) (15)

Buradaki normalizasyon sabiti c |OoPOHULQ ELUOHúLN |QFO \R÷XQOX÷X YH ( ))k ’ GDNL EWQ GH÷HUOHULQ WRSODPÕQÕ J|VWHUPHNWHGLU. BLUOHúLN RODVÕOÕN\R÷XQOXNIRQNVL\RQX 1 1 0 ( | ( ), ) [ ( ) | ( ), ] k k M x k m m m p Z k Z p z k I k Z   )

–

(16)

Hedef x LOH LOLúNLOHQGLULOHQ m |OoPQ, Gauss \R÷XQOX÷XQDED÷OÕROGX÷XQGDQ 1 1 [ ( ) | ( ), ] Ö ( ; , ) ( ) 1 ( ) 0 x k m m x x x m m m m x m p z k k Z N z z S e÷HU D A e÷HU D I   ­ ) ® ₎ ¯ (17) 1RUPDO\R÷XQOXNGD÷ÕOÕPIRQNVL\RQX 1 1 ( ( ) ( ) ) 1 1/ 2 2 Ö ( ; , ) (2 ) e x x x T m m m x x m m m z S z x G m N z z S P SS   x(((((((((( x) () () () () () () (111 x T)) ) (18)

gösterilir. Burada PG GR÷UX|OoPOHULQRODVÕOÕ÷Õ m heGHI[¶LQNDSÕVÕQÕQLoHULVLQGHNL|OoPzz _mmxx

ve S LQRYDV\RQ YHNW|U YH NRYDU\DQVÕGÕUmx

gOoPOHU H÷HU KHU KDQJL ELU KHGHIOH LOLúNLOHQGLULOPH]VH R ]DPDQ |OoPOHU LOJLOL

NDSÕQÕQ GÕúÕQGD RODFDNWÕU YH KHVDED

NDWÕOPD\DFDNWÕU  HúLWOL÷LQGHNL LNLQFL IDNW|U ELUOHúLN ROD\ODUÕQ |QFO RODVÕOÕ÷ÕQÕ YHUPHNWHGLU

<DQOÕú DODUPODUÕQ WRSODP VD\ÕVÕ m0 olarak

WDQÕPODQÕU 'R÷UX |OoPOHULQ VD\ÕVÕ mc=M-m0

olarak verilir, M EXUDGD WRSODP JHoHUOLOLN DODQÕ LoHULVLQGHNL |OoPOHULQ VD\ÕVÕQÕ YHUPHNWHGLU %XUDGDQ|QFORODVÕOÕN { ( )} { ( ) | ( ), ( )} { ( ), ( )} P k P k G M PG M ) ) ) ) ) ) (19)

ifade edilir. Burada ( )G ) LNLOL G]HQGH KHGHI

DOJÕODPDJ|VWHULFLVLGLU\DGD 1 ( ) M x( ) 1 1,..., x m m a x N G )

¦

) d (20)

ve ( )M ) olayGDNL \DQOÕú |OoPOHULQ VD\ÕVÕQÕ

YHUPHNWHGLU %WQ KHGHIOHU LNLOL J|VWHULFL LOLúNLVLQHED÷OÕRODUDNWDQÕPODQÕUVDW_m( )) ilgili olayda m |OoP LOH EWQ KHGHIOHULQ LOLúNLVLQL J|VWHUPHNWHGLU 1 ( ) N x( ), 1,..., m m x a m M W )

¦

) (21) 1 ( ) M[1 m( )] m M )

¦

W ) (22) ¶XQLNLQFLWHULPL 1 0 1 { ( ), ( )} N ( ) (1_{D t D t}) _F( ) t P_{G ) M )}

–

P G _P GP m ₍₂₃₎

ifade edilir. Burada PF \DQOÕú DODUPODU LoLQ

3RLVVRQ RODVÕOÕN \R÷XQOXN IRQNVL\RQXQX

J|VWHUPHNWHGLU (Y.Bar-Shalom ve ark. 2009):

0 0 0 ( ) ( ) ! m A F A P m e m O O  ₍₂₄₎

burada O \DQOÕú |OoPOHULQ X]D\VDO \R÷XQOX÷X

ve A JHoHUOLOLN E|OJHVLQLQ DODQÕQÕ YHUPHNWHGLU

(úLN GH÷HULQLQ GH LúOHPH NDWÕOPDVÕ\OD

ROXúWXUXODQNDSÕE|OJHVLQLQDODQÕ 1/2 ( ) A S JS k (25) oODUDNYHULOLU%XUDGDQHúLWOLN9) 1 0 1 0 ! { ( )} ( ) (1 ) ! ! N A D t D t t m A P k e P P M m O O G G   )

–

 (26)

öOoP H÷HU KHU KDQJL ELU KHGHIOH

LOLúNLOHQGLULOPH]VHLNLER\XWOXX]D\GDJ|]OHPA DODQÕQGD G]JQ GD÷ÕOÕPD VDKLSWLU GHQLOLU <DQOÕú DODUPODU LoLQ G]JQ \R÷XQOXN

fonksiyonu m0¶ ÕQ VW RODUDN A-m0 gibi

WDQÕPODQÕU Rex H. Wong ve ark. 2010). (17) ve

ELUOHúWLULOHUHNWHNUDUHOGHHGLOLUVH 0 1 1 Ö ( k| ( ), k ) m M ( ; ,x x) m m m m p Z ₎k Z  A

–

N z z S ₍₂₇₎ YHLOHWHNUDU\D]ÕOÕUVD 0 1 1 1 { ( ) | } Ö ( ; , ) ( ) (1 ) k m M x x m m m m m N D t D t t P k Z N z z S c P P W G G O  ) u 

–

–

(28) RODUDNWDQÕPODQÕU SLAM LoLQøOLúNLVHO+LSRWH]2OXúWXUXOPDVÕ

PDA DOJRULWPDVÕ SDUD]LW \DQNÕOÕ RUWDPGD ELU

KHGHILQ WDNLEL LoLQ X\JXQ VRQXoODU

YHUHELOPHNWHGLU dRNOX KHGHI WDNLEL LoLQ OLWHUDWUGH JPDA DOJRULWPDODUÕ |QHULOPLúWLU Bu \|QWHPGH RUWDPGDNL EWQ KHGHIOHULQ PDA PDQWÕ÷Õ LOH ELU G|QJGH WDNLS HGLOPHVL DPDoODQPÕúWÕr. JPDA DOJRULWPDVÕ ELUGHQ oRN KHGHILQ WDNLELQLQ \DQÕ VÕUD ELUELULQH \DNÕQ VH\UHGHQ YH\D NHVLúHQ KHGHIOHUGH GH X\JXQ

sonuçlar verebilmektedir (Pakfiliz, 2004, Y.Bar-Shalom ve ark. 2009). JPDA DOJRULWPDVÕQGD ELOLQHQ VD\ÕGDNL KHGHI L]LQLQ ROXúWXUXOPDVÕ LoLQ |OoPQ KHGHIOH LOLúNLOHQGLULOPH LKWLPDOOHUL HQ VRQYHULVHWLED]DOÕQDUDN\DSÕOÕU3DNILOL]. %LUGHQ ID]OD KHGHILQ ROGX÷X GXUXPODUGD NDUúÕODúÕODQ HQ |QHPOL SUREOHP LNL YH\D GDKD

fazla KHGHI LoLQ ROXúWXUXODQ JHoHUOLOLN

NDSÕODUÕQÕQNHVLúLPE|OJHOHULQLn üst üste binerek NHVLúPHVL SUREOHPLGLU %X JLEL GXUXPODUGD V]JHo SHUIRUPDQVÕ D]DOPDNWDGÕU ùLPGL LNL KHGHI YH DOÕQDQ DOWÕ |OoPOH ROXúWXUXODQ JHoHUOLOLN PDWULVLQGHQ EDKVHGLOLUVH úHNLO  ROXúWXUXODQ NDSÕ |OoPOHU YH KHGHIOHUL J|VWHUPHNWHGLU ùHNLO *HoHUOLOLNPDWULVLYHNPHOHQPHVL (Pakfiliz 2004). %XVHQDU\R\DED÷OÕRODUDNROXúWXUXODQJHoHUOLOLN PDWULVL¶GDNLJLELRODFDNWÕU 0 1 2 1 2 3 4 5 6 1 1 1 1 1 1 1 0 1 { } 1 0 1 1 0 0 1 1 0 x z x x x z z z a z z z : (29)

Burada x0; x1 ve x2 hedefleri bilinirken, parazit

\DQNÕ \D GD \HQL KHGHI RODUDN J|VWerilir. Bu matrise dayanarak JPDA DOJRULWPDVÕQGD ED]Õ NDEXOOHU\DSÕODELOLU

- %LU|OoPVDGHFHELUKHGHIWHQJHOHELOLU - +HU KDQJL ELU KHGHI ELUGHQ ID]OD |OoPOH

LOLúNLOHQGLULOHPH]

- 3DUD]LW\DQNÕGXUXPXEXNXUDOOD

NÕVÕWODQDPD]

Temel olarak bu kabullerde x0 süWXQX KDULo

WXWXOXU YH GL÷HU VWXQODU LoLQ ELU VÕUDGDNL elemandan sadece bir eleman sorumlu olur. 9HUL LOLúNLOHQGLUPH KLSRWH]LQLQ VD\ÕVÕ QHVQH LúDUHWOHUL VD\ÕVÕ YH |OoPOHUOH H[SRQDQVL\HO RODUDN DUWÕú J|VWHUGL÷LQGHQ DQD QRNWD ROD\ODU

VHWLQLQ SHUIRUPDQVÕ HWNLOHPHGHQ QDVÕO

UHWLOHFH÷LGLU

Geleneksel olarak geçerlilik matrisi içerisinde VÕIÕU ROPD\DQ HOHPDQODUGDQ JHoHUOL ELUOHúLN

KLSRWH]OHU UHWLOHUHN \RUXFX DUDúWÕUPD

algoULWPDODUÕNXOODQÕOPDNWDGÕU Rex H. Wong ve

ark. 2010) %X PHWRW KÕ] YH KHVDSODPD karmaúÕNOÕ÷Õ ROPDGÕ÷Õ GXUXPODUGD \DQL SLAM SUREOHPLQL J|] |QQH DOÕQGÕ÷ÕQGD nesne LúDUHWOHULQLQ VD\ÕVÕQÕQ D] ROGX÷XQGD uygun VRQXoODU YHUHELOPHNWHGLU gWH \DQGDQ H÷HU JHoHUOLOLN PDWULVLQGH ELUOHULQ VD\ÕVÕ oRN ID]OD ROXUVD EX GXUXP JHoHUOLOL÷LQL \LWLUPHNWHGLU *HUoHN ]DPDQOÕ X\JXODPDODUGD KÕ] YH KDIÕ]D problemi RODUDNRUWD\DoÕNPDNWDGÕU.

'DKD |QFHNL oDOÕúPDODUGa DFS (Y.Bar-Shalom ve T. Fortman 1988) \|QWHPL PDQWÕNOÕ LOLúNLVHO KLSRWH]OHU NXUXOPDVÕ LoLQ WHUFLK HGLOPLúWLU Bu oDOÕúPDGDGD')6\|QWHPLQGHQ\DUDUODQÕOPÕúWÕU ùHNLO  WHNUDU LQFHOHQHFHN ROXUVD DOWÕ |OoP YH LNL KHGHI ROGX÷X ELOLQPHNWHGLU +HU ELU |OoP \D KHGHIOHUGHQ \D GD SDUD]LW \DQNÕGDQ JHOPHNWHGLU %X \]GHQ KHU ELU |OoP LoLQ o RODVÕOÕN YDUGÕU YH  KLSRWH] KHVDSODQPDVÕ gerekmektedir. Bu hipotezleULQ KHSVL PDQWÕNOÕ ROPD\DELOLU PDQWÕNOÕ KLSRWH]OHU RUWDN RODUDN KDULoWXWXODELOLUYH')6\|QWHPLEXLúLQo|]P için \DUGÕPFÕ ROPDNWDGÕU 'HWD\OÕ ELOJL LoLQ B. Zhou, and N. Bose 1993)¶e EDNÕODELOLU

JPDA - UKF Güncelleme ve

7DKPLQ$GÕPÕ

j¶ LQFL KHGHILQ GXUXP JQFHOOHPHVL x_{k k}j_| (Y. Bar-Shalom ve ark. 2009)¶WHYHULOGL÷LJLEL

( ) | 0 | 1 | 1 ( ) j m k j j j k k j k k ij k k i x E x  

¦

E x i (30)

86

Burada m k j’inci hedef için geçerli _j( )

|OoPOHULQVD\ÕVÕx_{k k}j_{| 1} durum tahmini, x_{k k}j_| ( )i i’nci geçeUOL |OoP NXOODQDUDN \DSÕODQ UKF

güncellemesi ve ijE LOJLOL LOLúNLVHO ROD\ODUÕ

göstermektedir (Y. Bar-Shalom ve ark. 1995, 2009). KoYDU\DQVLoLQGXUXPJQFHOOHPHVL | 0 | 1 ( ) | | | | | 1 [ ( ) ( ( ) )( ( ) ) ] j j j k k j k k m k j j j j j T k k k k k k k k k k ij i P P P i x i x x i x E E     

¦

(31)

RODUDN KHVDSODQÕU 'XUXP WDKPLQL x_{k k}j_{| 1} ,

NRYDU\DQVÕP_{k k}j_{| 1}  WDKPLQ HGLOHQ KHGHI |OoP

| 1 j k k

z _ ve onun inovasyon NRYDU\DQVÕS UKF _kj

WDKPLQ DGÕPÕQGD GHWD\OÕ RODUDN ek A’ da DQODWÕOPDNWDGÕU

'HQH\VHOdDOÕúPDODUYH7DUWÕúPD

Bu bölümde, daha önceden Tim Bailey (2002) WDUDIÕQGDQJHOLúWLULOHQ\D]ÕOÕPGDQ\DUDUODQÕODUDN

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