FGAs-Based Data Association Algorithm for Multi-sensor Multi-target Tracking

V ol. 16　 N o . 3 　　　　　　　　　　　　 CHIN ESE JO U RN A L OF A ERO N A U TICS　　　　　　　　　　　　　 A ugust 2003

FGAs-Based Data Association Al gorithm for Mul ti-sensor Multi-target Tracking ZHU L i-li, ZHAN Huan-chun, JING Ya-zhi ( Faculty 302, N anj ing University of A eronautics and A str onautics, N anj ing　210016, China) Abstract: 　 A nov el data association algor ithm is dev eloped based on fuzzy g enetic alg orithms ( F GA s ) . T he static par t of data association uses one FG A to determine both t he lists o f compo site measurement s and the solutions o f m-best S-D assig nment. In the dy namic part of data association, the results o f the m-best S -D assignment ar e then used in turn, w ith a K alman filter state estimator , in a multi-populat ion F GA -based dynamic 2D assig nm ent algo rithm to estimate the stat es of the mov ing tar gets ov er time. Such an assig nment -based data association algo rithm is demonstr ated on a simulat ed passive sensor tr ack for mation and maint enance pr oblem. T he simulation results show its feasibility in multi-sensor multi-targ et tracking . M oreover , algo rithm development and r eal-time pr oblems are briefly discussed . Key words : 　 multi -targ et tracking; data associatio n ; FG A ; assig nment problem ; K alman filter 一种基于 模糊遗传算法的多传 感器多目标跟踪数 据关联算法. 朱力立, 张焕春, 经亚枝. 中国航 空学报( 英文版) , 2003, 16( 3) : 177- 181. 摘　要: 基于模 糊遗传算法 发展了一 种新的数据 关联算法。数据关 联的静态 部分靠一个 模糊遗传 算法来得出量测组合序列和 S -D 分配的 m 个最优 解。 在数据关 联的动态部分, 将得到的 S -D 分配 的 m 个最优解在一个基于多种群模 糊遗传算法的动态2D 分配算法中依靠一个卡 尔曼滤波估计器 估计出移动目标各个时刻的状 态。 这一基于 分配的数据关联算 法的仿真试验内容 为被动式传感器 的航迹形 成和维持 的问题。仿真试验 的结果表明 该算法在 多传感器多 目标跟踪中 应用的可 行性。 另外, 对算法发展和实时性问题进行了简单讨论。 关键词: 多目标跟踪; 数据关联; 模糊遗传算法; 分配问题; 卡尔曼滤波器 文章编号: 1000-9361( 2003) 03-0177-05　　　中图分类号: V 243　　　文献标识码: A

　　T he mult i-t arg et t racking problem can be di-

t he data associat ion is formulated as the con-

vided int o tw o interrelat ed tasks of st at e estimat ion

strained combinat orial opt im izat ion problem. How -

and data associat ion. Association is t he decision

ever, the S-D assignment problem ( either a st atic

process linking observation of a common orig in in the presence of f alse alarm s and missed det ect ions.

one or a dynam ic one ) is know n to be NP-

For centralized f usion, stat ic ( quasi-state) association ( measurements -t o -measurements ) is used for

at ion t echnique [ 2,

track f ormat ion and g enerat ion of t he composit e measurements for t rack m aint enance, and dynamic

it y. In addition, m-best assignment algorit hm s[ 6,

[ 1]

association is used f or t rack m aint enance . In recent years, multidimensional ( i. e. S -D ) assignment algorit hm s[ 1-4] w ere show n t o be ef fective in dat a associat ion for m ult isensor m ultitarget tracking in t he presence of clutt er . In assignment , Received dat e: 2003-01-13; R evision received dat e: 2003-04-25 A rt icle U RL: ht t p: / / ww w . hkx b. n et . cn/ cja/ 2003/ 03/ 0177/

hard [ 2, 3] . T hen t he successive L agrangian relax5]

w as developed to construct sub-

opt imal solutions with pseudo-polynomial complex7]

obt ain top m best assignment solut ions by repeatedly using t he st andard S -D assignment algorit hm . A st andard genetic algorithm ( GA ) based st atic data associat ion ( f or st andard S-D assignment ) w as proposed in t he presence of missed det ections [ 8] only .

ZHU Li-li, ZHA N Huan-chun, JIN G Y a-zhi

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T he f ocus of this paper is to present an F GAsbased mult idimensional assignment algorithm that the aut hors developed. One m odif ied f uzzy GA

S

ci 1 …is =

( M HT ) . M oreover, t his algorit hm is essentially a combinat ion max imum likelihood ( M L) approach .

∑ s= 1

u( is) ln

( F GA ) can obt ain m-best S-D assignm ent solutions in one run . One modif ied m ulti-populat ion FGA has the pot ential of eff icient implement ations of a suboptimal M ultiple Hypot hesis T racking

CJA

( u( is ) - 1) ln( 1 - P D s) P Ds

s

2 Rs T

1 2

+

u( is) 2

- 1

{ [ Zsi s - X p ] Rs [ Z sis - X p ] } w here

( 2)

is t he volume of t he f ield of view of sensor s and u( is ) is a binary indicator funct ion . Xp is t he M L stat e estimation of t rue target p ( here, t he s

[ 1, 2, 6, 7]

conversion of coordinat es is omitt ed for sim plif icat ion ) . Zsi s is one measurement originated

1　Problem F ormulation T he problem formulation follow s t he w ork

[ 1-7]

of Bar-Shalom and his co-w orkers. Now briefly describe t he problem f orm ulat ion in this sect ion ,

from p , and is m odeled as Xp plus addit iv e w hit e Gaussian noise N ( 0, Rs ) . Moreover , P D s is t he non -unit y det ect ion probabilit y of sensor s.

and om it detailed ex posit ion. F urt hermore, t he

T o determine t he m-best assig nment s, one

problem f ormation discussed here is applicable to

only need t o rank the S-D assignm ent solutions in order of increasing cost ( dif ferent f rom the w ay in t he lit erat ure[ 1, 3, 6, 7] ) . Define t he m -best assig n-

tracking problems w it h sy nchronized sensors and low speed t arget s. 1. 1　Static assignment probl em T he st at ic assig nm ent problem considered in [ 1-5]

this w ork is a modified version of the one in t he lit erature . T he major dif ference is t hat t he lat t er must be divided int o t w o part s of the S -D assig nment and t he m-best S -D assignment , w hile t he former is a w hole . In a multisensor-multit arget scenario [ 1,

2, 6, 7]

,

there are S list s of m easurem ent s from S sensors w hich are synchronized and provide lists at discret e time samples t = 1, …, T . F or t he stat ic assig nment problem , t he goal is to associat e t he S list s of

ments in the feasible solution space w it h t he m least cost s as: a1 , …, am , w ith their cost s of t he assignment ( or hy pothesis) c( a1) , …, c( am ) , respectively. 1. 2　Dynamic 2D assignment probl em [ 1, 6, 7]

T he dy namic problem is solved aft er each scan to updat e t he t racks, st art ing w ith t he second scan . T he goal is to associat e t he t-th list of composite measurement s w it h the list of t racks form ed at time inst ant t - 1. According to a sec[ 1, 7]

ond-order kinem at ic model, t he t arget st at e X( t) = ! ( t, t - 1) X( t - 1) +

is

ns measurement s obtained at time inst ant t , s= 1,

G( t - 1) W( t - 1)

…, S, w hich is S dimensions. During t he course of

w here ! ( ) is t he st at e t ransit ion matrix , and G ( ) is t he dist urbance m at rix . T he process noise vector W ( ) is modeled as a w hit e, zero -m ean

im plement at ion, the m-best assignm ents are det ermined and ranked in order of increasing cost . T he [ 2, 6, 7] generalized S -D assignment problem is n

n

1

s

min∑… ∑ci 1 …is i = 0 1

( 1)

i= 0 s

w here ci 1 …i S is t he cost of associating t he S-t uple of measurements ( i1 , …, is ) , is = 1, …, ns .

is a bina-

ry indicat or variable indicating t he association of this S -tuple. Not e t hat , in Eq. ( 1) , is = 0 is a dummy measurement f rom a list w it h the considerat ion of missed det ect ion. T he cost ( negat ive log a[ 2, 6, 7] rit hm of t he generalized likelihood ratio ) is

( 3)

Gaussian random variable wit h know n covariance matrix Q( ) . T he composite measurement s are assumed to be a linear funct ion of t he target stat e corrupted by m easurem ent noise [ 1, 7] , Z( t) = H( t) X( t) + V( t)

( 4) w here H ( ) is t he measurement m at rix , and V( ) is zero-m ean w hit e measurement noise w it h known covariance m at rix R( ) . Denote t he number of composite m easurements corresponding t o a1 , …, am by ∀1, …, ∀m , re-

FGA s-Based D at a A ssociat ion A lgorit hm f or M ult i-sensor M ulti-t arget Tracking

A ugust 2003

spectively. L et k= 1, …, m. Def ine t he true measurem ent probability

[ 7]

m

P{ z } =

∑ ∑

k= 1 m

by

d zk exp( c( a1 ) - c( ak ) )

k= 1

exp( c( a1 ) - c( ak ) )

tion of the 2D assig nment problem can be divided int o tw o phases : Phase 1: m m inimization costs and t heir mea1

m in ∑∑cy i z j # y i z j i= 0 j = 0

! 　　! 　　!

{ cy i zj } =

( 6)

∀

N

m

m in ∑∑cy i z j # y i z j i= 0 j = 0

w here # y i z j is a binary assignment variable. Phase 2: minimizat ion costs for each t rack from all t he corresponding m easurement -t rack pairs, follow ing the result s of st ep 1. arg min { cy i z j } y

( 7)

i

　　T he cost of assigning m easurement z i t o t rack [ 7]

y i is

cy i zj =

0　　　　　 　　　 ∃ ( y i , z j ) P{ z j } - ln ∃ ( y 0, z j )

if i = 0 or j = 0 if - ln( ) < 0

∞　　　 　　　　　 ot her ( 8) w here the likelihood f unction calculat ion ∃ ( y i , z j ) from a Kalman filter st at e estimator is t

∃ ( y i, z j ) =

∏

gorit hm. 2. 1　Description of the FGAs [ 9, 10] , fuzzy t ools or Fuzzy In t he lit erat ure L og ic-based t echniques are used f or modeling different GA component s or adapting GA control param eters, respectively , w it h the goal of im proving performance. Generally , GAs result ing f rom such a w ay are called F uzzy GAs ( F GAs) . Moreover, [ 9]

∀

N

fundament al association alg orit hms. Since t he GA is a w ell-known algorithm , just brief ly describe t he FGAs here. Aft er that , discuss t he assignment al-

( 5)

w here d zk is 1, if z ∈ak . Ot herw ise , d z k is zero . L et y i , i = 0, …, N , denote one track f rom the t rack list wit h the time st am p t - 1. T he solu-

surem ent-t rack pairs,

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2 S( k)

-

many research results ex hibited the bet ter performance of FGAs, than the st andard GAs . An FGA is m ore eff icient than a standard GA in solving the t raveling salesman and ot her combinatorial [ 10] opt imization problem s . In preliminary studies, t he authors developed an F GA, which adopts 6 fuzzy log ic cont rollers for adapt ing control parameters ( i. e. selective pressure, crossover probabilit y and mut ation probability ) of a modified GA . 2. 2　FGA-based static assignment sol ution Based on t he prelim inary st udies, t he component s of t he F GA w ere modified according to t he st at ic assignment problem . T he main cont ents w ill be provided in this subsect ion. T he chromosome represent at ion is decoded as a sy mbol string . T he alleles are the serial number of measurem ents corresponding to the target s detect ed by a sensor. Each chromosome is one list of measurement f rom one sensor . S chrom osom es make an individual. T hus, the g enot ypes are uniquely mapped ont o t he S list s of m easurem ent s

1 2

k= 1

t

1 exp d T ( k) S- 1( k) d( k) 2∑ k= 1

( 9)

w here d( k ) is the m easurement residual and S( k) is the residual covariance. In addit ion , the likelihood of false alarm s ∃ ( y 0, z j ) is assumed uniformly [ 2, 7] , i. e . probable over each sensor' s field of view S

∃ ( y 0, z j ) =

∏ s= 1

u( i )

1

s

s

( 10)

2　Algorithm Description T he solut ion approaches adopt FGAs as t he

from S sensors. T he virt ue of such a represent ation is that it is fit for any measurem ent dat a t ype . T he leng th of each chromosom e is equal t o t he maximum leng th of the measurem ent lists. For a short list, one can use dumm y measurements t o fill it . Generally , t he S lists of raw measurements include many eff icient associat ion modes. Hence, t he initial populat ion is achieved by copying t he individual, w hich denot es the raw measurem ents ( S list s) . M oreover, a generation gap is adopt ed as t he Elit ist M odel f or t he same reason. T he goal of t he assignm ent algorit hm is to

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ZHU Li-li, ZHA N Huan-chun, JIN G Y a-zhi

CJA

globally m inimize the cost ( see Eq. ( 1) ) . T o ensure that the result ing f it ness values are non -neg a-

leles of a part icular chromosome k are t he serial ′ number of ∀k composit e m easurem ent s in ak . T he

tive, individuals are assig ned fitness according to

alleles of a part icular chromosome k are t he serial number of tracks to be associated ( i. e. N tracks) .

their rank in the populat ion rat her t han their raw perf ormance . T hus, a selective pressure is used to limit t he reproduct ive range. T he select ion method

′

′

T he leng th of chromosomes k and k is t he big one ′ betw een ∀k and N . Dummy measurem ents or dum-

spread and zero bias. T he crossover operation is Partly M apping Crossover for symbolic code series.

′ my tracks are used t o keep ∀k = N . Chromosome k ′ and k make an indiv idual. T hus, the genotypes are uniquely mapped onto m easurem ent -t rack

T he mut ation operation is Random ly T wo-Point

pairs. T he init ial population is achieved by copy ing

Int erchange M ut at ion for transposition represent ation of com binatorial optimizat ion problems . T he

t he individual.

is St ochast ic U niversal Sam pling w it h m inimum

stop crit erion of t he FGA is a maximum g enerat ion

Once t he m subpopulat ions hav e been built , one can t hen be ready t o t rack maintenance. Each

number.

subpopulat ion assigns ∀k composite m easurem ent s

T he F GA -based static assignment algorit hm ident if ies t he t argets and est imat es t heir states by

( from t he latest scan) t o the N most likely previ-

ML est imation. It searches a population in parallel by probabilist ic t ransition rules. Only the cost function and corresponding fitness levels directly influence t he direct ions of search. It is im portant to note that such an algorithm provides a number of potent ial solut ions to t he given problem . Hence , one can choose the f inal solution by ranking t he ob-

′

ous tracks using it s global cost minim izat ion func′ tion in Eq. ( 6) . Specially, ∀′ 1 , …, ∀ m should replace ∀1, …, ∀m . Aft er t he preset g enerat ion num ber, all subpopulat ions y ield t heir measurem ent-t rack pairs w it h t he best fitness level. A decision rule Eq. ( 7) is t hen used t o yield the updat ed t racks. Specially, in this approach, the t rack init ia-

ject ive associat ion cost . F urtherm ore, one can se-

tion rule and t he track maintenance rule are defined as f ollows . A new track can be born aft er

lect m best association solutions simultaneously for the m-best assignm ent wit hout repeat ing run.

t wo successive scans w it h measurement s assigned t o t he track. An old track can be eliminated aft er

2. 3　FGA-based dynamic assignment sol ution

t hree successive scans w it h no measurement assig nment t o t he t rack .

T he FGA in t he dynamic assignm ent algorit hm is a symbolic coded mult i-populat ion FGA. T he genet ic operat ion of t his FGA is sim ilar to t he one discussed in the previous subsect ion . M oreover, t he st op criterion is the same. T he dif ference is t he populat ion num ber and t he individual m eaning. Before implementing the dynam ic 2D assig nment algorithm, Eq. ( 5) is used t o select the com-

3　Present ation of Simulation In this sect ion , a simulat ed passive mult i-sensor mult i-t arget t racking problem is solved with t his F GAs-based dat a associat ion alg orit hm . T he goal is to est imate t he f easibilit y of this algorithm. T he problem is a 7 sensors 5 t arg et s scenario . T he simulated measurement data set includes 10

posit e measurements and calculat e t heir probabilities. T he selection discards t hose com posit e mea-

t ime samples of measurements f rom 7 sensors. For more details on this scenario , e. g . targets sim ula-

surem ents w ith t heir probabilities less than a cer-

tion and passive sensor specification, see Ref . [ 7, 9] . T his scenario results in very complex candidat e

tain threshold. Denote the number of selected composite measurem ents corresponding t o a1 , …, ′ ′ a by ∀ , …, ∀m , respect ively . L et k = k = 1, …, m . m

′ 1

T he sam e technique is applied t o f orm all m subpopulat ions corresponding t o a1, …, am . T he al-

associations, in t he presence of false alarms and [ 3 , 7 , 9] . missed detections T he preset parameters of t he alg orit hm are t he f ollow ing .

FGA s-Based D at a A ssociat ion A lgorit hm f or M ult i-sensor M ulti-t arget Tracking

A ugust 2003

For t he F GA-based st at ic assignment phase, the populat ion size is 80, t he g enerat ion gap is 0. 6, and t he st op generation is 150. 20 best solutions are selected f rom the S -D assig nment result s. T he select ive pressure and the probability of m ut ation and crossover are adapt ive cont rolled by t he FGA. For t he FGA-based dynamic 2-D assig nment phase , t he number of subpopulations is 20. For each subpopulat ion, t he populat ion size is 40, t he generat ion gap is 0. 9, and the stop generation is 50. Each subpopulat ion has its select ive pressure , the probabilit y of mutation and crossover . Fig. 1 show s the ex periment results. T hese result s are similar to t hose results in Ref . [ 7] . T his denot es t he feasibility of this alg orithm in such a sim ulat ed scenario, although t his scenario is simple for st ate estimation ( for t he target mot ion models [ 7]

w ere constant velocit y ) .

Fig . 1　T racking r esults

4　Conclusions An FGAs-based dat a association algorit hm for mult i-sensor multi-t arget t racking is developed in this paper, and eit her the FGAs or the m-best S-D assignment t echnique is modified. T he f easibilit y of t he alg orit hm w as demonst rated using a passive mult i-sensor mult i-t arget tracking problem . Due to the limited t est ing in t he present w ork , t he algorit hm requires f urt her analy sis and testing , using bot h simulat ed and real m ulti-sensor m ulti-target data. Alt hough the algorit hm has show n it s feasibilit y, it s practicality seems t o be hampered because of the real-t im e problem . Generally , there are tw o

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w ay s to solve t he real-t im e problem, i. e. parallel algorithm and hardw are-ty pe algorit hm. T he latter w ay is adopt ed. Desig n of hardw are-type F GA is presently underw ay , and design of t he hardw are -t ype st at e est im at or is planned for future w orks. References [ 1] 　 K irbarajan T , Wan g H, Bar-Sh alom Y , et al. E ff icient mult isensor fusion using multidimensional dat a association [ J] . IEEE Transact ions on A erospace and E lect ron ic Systems, 2001, 37 ( 2) : 386- 398. [ 2] 　 Deb S , Y eddan apudi M , Pat t ipat i K R . A generalized S-D assignment algorit hm f or mult isensor -mult it arget stat e est imat ion [ J] . IEEE T ransactions on A erospace and Elect ronic Syst ems, 1997, 33 ( 2) : 523- 536. [ 3] 　 Chunmun M R, K irubarajan T , Pat tipati K R , et al. Fast dat a associat ion using multidimensional assignment w it h clust ering [ J ] . IEEE T ran sact ion s on A erospace and E lectronic Syst ems, 2001, 37 ( 3) : 898- 911. [ 4] 　 K irubarajan T , Bar-Shalom Y , Patt ipat i K R. M ult iassignment f or t racking a large number of overlapping object s[ J] . IEEE Transact ions on A erospace and Elect ronic Syst ems, 2001, 37 ( 1) : 2- 19. [ 5] 　 Pat tipati K R , D eb S, Bar -Sh alom Y , et al. A new relaxat ion algorithm and passive sensor dat a associat ion[ J] . IEEE Transact ions on A ut omat ic Con trol, 1992, 37 ( 2 ) : 198213. [ 6] 　 Popp R L, Pat t ipat i K R, Bar-Sh alom Y . Dynamically adapt able m-best 2-D assignment algorit hm and mult ilevel parallelizat ion [ J ] . IEEE Transact ions on A erospace and Elect ronic Syst ems, 1999, 35 ( 4) : 1145- 1159. [ 7] 　 Popp R L, Pat t ipat i K R, Bar-Sholm Y . m-best S -D assign ment algorit hm w ith applicat ion t o multit arget tracking [ J] . IEEE Transact ions on A erospace and E lect ron ic Systems, 2001, 37 ( 1) : 22- 37. [ 8] 　 王宁, 郭立, 金大胜, 等. 遗传 算法在多传感 器多目标静 态 数 据 关联 中 的应 用[ J] . 数 据 采集 与 处 理, 1999, 14 ( 1) : 18- 21. Wang N , Guo L, J in D S , et al. A pplication of genet ic algorithm in mult isensor mult it arget st at ic data associat ion[ J] . Journal of D at a Acquisit ion & Processing, 1999, 14 ( 1) : 18 - 21. ( in Ch in ese) [ 9] 　 Herrera F, Loz ano M . Fuzzy genetic algorith ms: issues and models[ Z] . U R L: cit eseer. nj. nec. com / 16799. ht ml. [ 10] 　Herrera F, Loz ano M . A daptation of genet ic algorith m paramet ers based on fuzzy logic cont rollers [ A ] . In: Herrera F , V erdegay J L ed. Genet ic A lgorit hms and Soft Comput ing [ C ] . Heidelberg : Physica -V erlag , 1996. 95- 125.

Biographies: ZHU Li-l i　 Born in 1972, he is cur rently a gr aduate student for P h. D. degree in A utom ation Eng ineer ing at the N anjing U niv ersity o f Aer onautics and A stro nautics. His current resear ch inter ests are applications invo lving data fusion, multitarg et tr acking , fuzzy control and g enetic alg orit hm . E-mail: ali zhu@ 163. com