Analysis of the multisensor multitarget tracking resource allocation problem Pierre Dodin
Julien Verliac
Vincent Nimier
ONERA Chatillon
ONERA Chatillon
ONERA Chatillon
[email protected]
[email protected]
[email protected]
This paper deals with a study of the multisensor management problem. The main tool is the classical assignment formulation, using KullbackLeibler entropy as costs. In order to use the bene t brought by the data fusion, coalitions or pseudosensors must be created at each step of time, creating an exponential calculus of all the possible sensor partitions. We compare this method and a prede nite strategy using di erent scenarios. Abstract -
Keywords:
Tracking, resource allocation, combinatorial
optimization, entropy
1
the eÆciency of the method than the mean precision, we have used on the second scenario validation gates. These gates, computed with the same noise generation, allow us to evaluate the number of lost targets for each method. It is not, contrarily to the mean precision, a criterion computed by a mean, therefore it better illustrates the \adaptive" behavior of the method. In part 2 we give a short description of the IMMKF algorithm equations, required to compute the Kullback-Leibler's criterion applied to tracking (section 3). In part 4 we discuss di erent formulations for the assignment problem (section 4), and we nish with a numerical study (section 5).
Introduction
The multisensor multitarget tracking resource allocation problem consists of the sensor's optimal assignment at each step of time, to minimize errors on all the target's positions. This problem is considerably complexi ed if we take into account the possibility of many sensors observing one target, to use the bene t brought by the data fusion. One can nd in Nash's article [12] the rst interpretation of the problem in term of assignment, but it's just about \pure" assignment (one to one). It's within the framework of detection and classi cation [8, 16] that the allocation problem has been completely set. The criterion used in these two papers is the Kullback Leibler's entropy [8, 10], and the assignment can be computed with a linear program [16]. We use a centralized Kalman lter's predicted mesurements at each time step to compute the criterion. We will see in the section 4 that the linear program formulation proposed is problematic because of the integral constraints. One of the most interesting aspects of the criterion is that it's working as well in tracking as in detection. The paper [15] suggests to use this criterion in tracking, with the help of an IMMKF [15]. However this paper does not give any indication on the optimization method used. In this article we use the optimization principle proposed by Schmaedeke [16], with some nuances (Section 4). Therefore, our optimization uses the Kullback Liebler criterion computed by a IMMKF. We compare this method with a prede nite strategy to estimate the gain brought on manuvring targets. Two scenarios are studied: One with manuvering targets during a short time, the other with constancy manuvring targets. In order to obtain another criterion to evaluate
2
The IMMKF
To manage ltering during manuvres, a multiple model Kalman lter, or IMMKF is used. One can write linear di erential equations satis ed by the state vector Xk and the measure vector Yk . Xk+1 Yk
= =
Fk (k )Xk + Gk (k )wk Hk (k )Xk + vk
These equations are function of a parameter k which is a Markov chain with values in 1; ; N according to transition probability ji . In our simulation, we have considered state vectors of dimension four, containing information about position and speed in the horizontal plane. We are working with cartesian coordinates, therefore to take into account measurement errors in (r; ) , we use a special covariance matrix on measurement noisevk (see [18]). The state vector Xk , the measurement vector Yk , and the covariance matrix on measurement noise Rk can be written as follows: 2
3 xk 6 yk 7 7 Xk = 6 4 x_k 5 Yk y_k
Rk
=
=
r2 sin2 k + rk2 2 cos2 k (r2 rk2 2 )sink cosk
rk sink rk cosk
(r2
rk2 2 )sink cosk 2 2 r cos k + rk2 2 sin2 k
we have used a \strong noise-weak noise" IMMKF in our simulation, it means that the IMMKF has two
models :
2
6 6 Gk (i) = i 2 6 4
dt3
0
dt4 4
2
dt4
03
0
0
dt3
3
3
The
criterion:
Kullback-
Leibler's entropy
7 7 7 5
For the random gaussians vectors q0 (X ) and q1 (X ), with means X0 and X1 , and variance 0 and 0 , the 0 2 0 dt discrimination q1 with respect to q0 is given by : Z q1 (X ) 1 IMMKF algorithm D(q1 ; q0 ) = q1 (X ) log( ) dX = ftr[0 1 (1 q0 (X ) 2 Fusion of estimates for the model j 1 j )g 0 + (X1 X0 )(X1 X0 )T )] log( jj The ltering process starts with an a priori state 0j bj (k 1jk 1), a state covariance matrix Pj (k 1jk 1), X and associated probabilities j (k 1) for each model. The computation of the discrimination is linked with Starting with these datas, one can compute the pre- the IMMKF ltering. The discrimination gives the \distance" between the state probability density of a dicted probability of the model : X target if one makes a measurement and the state probj = ji i (k 1) ability on the opposite case. Let consider the case i without measurement: One predicts the target state The fused probability of the model : at step k with the IMMKF equations; therefore we have Xbj (kjk 1) and Pj (kjk 1) for each dynamic i (k 1) ijj = ji model j . One computes : 4
03
dt
2
dt2
dt
2
0
2
j
The fused state estimation for the model j : X b0j (k 1jk 1) = bi (k 1jk 1)ijj X X i
Pj (k jk
1) =
bj (k jk X
1) =
X j
X
j Pj (k jk
1)
bj (k jk j X
1)
and the variance estimation of the fused state for the model j : X Hence one obtains the state probability density of the bi (k 1jk 1) target: P0j = ijj fPi (k 1jk 1) + [X i 1 1 b (k jk 1))T b bi (k 1jk 1) X b0j (k 1jk 1)]T g po (X jZ ) = j2P (k jk 1)j 2 expf (X X X0j (k 1jk 1)][X 2 b (k jk 1))g P (k jk 1) 1 (X X Filtering for each model j It is classical Kalman ltering equations, see for exam- In the case where we decide to measure the target at step tk , the history of the mesurements becomes Z = ple [18] z [ Z . The probability of the target to be in the state X and to follow the dynamic model j knowing the Actualization of models probabilities measurement history Z is noted p(X; j jZ ). On one The models likehood is computed with ltering inno- hand: vations j (k), the covariance of the ltering innovation po (X; j jZ ) = 0j p0 (X jZ ) Sj (k ), and a Gaussian assumption for the likehood. on the other hand Therefore one can write for the model j p(X; j jZ ) = j p(X jZ ) 1 j = (2jSj (k)j) 2 exp( 12 j (k)T Sj (k) 1 j (k)) allowing us to compute model j probability updating : We want to compute the discrimination of p(X; j jZ ) with respect to po (X; j jZ ): j j j = P XZ p(X; j jZ ) D= p(X; j jZ ) log( )dX i i i po (X; j jZ ) j j
0
0
0
0
0
0
0
0
Combination of estimates
The state global estimate Xb (kjk) and its covariance matrix P (kjk) obtained with the output of the IMMKF lter are given by : X b (k jk ) = bj (k jk )j X X j
P (k jk ) =
X i
j fPj (k
1jk
Extending this argument to each possible mesurement: D=
0
Z
@ (z )
1 0 p(X; j jZ ) )dX A p(z jZ )dz p(X; j jZ ) log( po (X; j jZ ) (X )
XZ j
0
This expression can be simpli ed under some hypothesis (see [15]): X j 1 T D= f j log( )g + trfP (k jk 1) 1 (P (k jk ) P (k jk 1) b b [X (kjk) Xj (kjk)] g 0 2 j
1) + [Xb (kjk)
bj (k jk )] X
+(Xb (kjk)
1)) )g 1 log( jP (kjk)j ) 2 jP (kjk 1)j Further, we will note Gij the gain D calculated for each couple (pseudo-sensor i, target j ). 4
b (k jk X
1))(Xb (kjk)
b (k jk X
T
matrice ALP 0 2
Assignment problem formula-
4
tion
At each time step, an assigment problem is solved, using Kullback-Liebler's criterion as costs. In the paper of Wayne Schmaedeke [16], it is suggested to solve the allocation problem with the following linear program : max
2s
t X1 X
Gij xij
(1)
submit to
(2)
i=1 j =1 2s
X1
2
i J (k) j =1
xij
8
0
5
10
15 nz = 76
20
j
= 1; ; t
(3)
k
k
= 1; ; s
(4) Said di erently, one must compute :
0
max v (P ) P
8ij
(5) The basic sensors are numbered from 1 to s. The pseudo sensors can be numbered from s + 1 up to 2s 1. For each basic sensor k, let J (k) be the set of integers consisting of k and the integer numbers of the pseudo sensors which contain sensor k in their combination. The integral condition is not necessary if the constraints polyhedra is integral, or if the constraints matrix is totally unimodular. Note that this condition is automatically veri ed for assignment problems. Unfortunately, the following matrix is not totally unimodular, because one cannot check that each square submatrix has a determinant value equal to 0,1 or -1: In fact let us consider the scenario with 3 sensors versus 4 targets, the constraints matrix is shown gure 1. One extracts the submatrix containing the columns 1,3,6,9,12,15,and 18. It's the matrix: 0 1 1 0 0 1 0 0 0 B 0 0 1 0 0 0 1 C B C B 0 1 0 0 0 1 0 C B C C A=B B 0 0 0 0 1 0 0 C B 1 1 0 1 1 0 1 C B C @ 0 0 1 1 1 0 0 A 0 0 0 0 0 1 1 The determinant calculus gives 2. It means that the corresponding linear program solutions will be in 12 . E ectively if one takes as a basis the corresponding variables, and if one calls b the second member of the linear program 1, formally the basis solution will be : 1 com(A)T xb = A 1 b = det(A) Where com(A) is the cofactor matrix of A. Therefore the only possibility is the systematic calculus of every sensors partitions P , grouped in sP pseudosensors. xij
25
Figure 1: Constraints matrix. Each dot is a 1.
1
xij
i=1 t X X
6
Where v(P ) is the value of the program : max
sP X t X
Gij xij
i=1 j =1
submit to xij 1
sP X i=1 t X
j
= 1; ; t
xij
i
i = 1; ; sP
xij
0
8ij
j =1
Because of the exponential nature of the calculus, this limits the maximum number of sensor in a \coalition". However, pratically one observes that the maximal coalition in the three sensors case is very rarely employed. This has a simple explanation: The more the sensors number in a coalition is high, the less remains ressources for other objectives. Then the gain brought by the fusion must be consequent. If we limit the maximum size of the coalitions, the problem stays arti cially polynomial with respect to the problem size. 5
Numerical study
We propose in this section a numerical study on two di erent scenarios. We present the di erent parameters of the simulation: we have taken as parameter and transition matrix : 0:99 0:01
= [0:1; 150] = 0:01 0:99
The three sensors will be placed in ( 500; 1000); ( 300; 1000) and (0; 1000). The mesures with Gaussian noise are computed using one meter for the distance covariance, and one milliradian for the azimuth covariance. The four planes have as initial state vector: coordinates plane1 plane2 plane3 plane4
x
-600 -600 -600 -600
y
4000 3900 3800 3700
x_ 100 100 100 100
y_
0 0 0 0
Trajectoire relle 4000
3500
3000
2500
2000
1500
1000 −600
−500
−400
−300
−200
−100
0
100
200
−100
0
100
200
Trajectoire relle 4000
The initial models vector probability are set to [ 21 ; 12 ]. In the following array we describe the manuvres in the scenarios. The accelerations are in m:s 2 . For the rst scenario: time 11 t 31 32 t 34 plane 3 in x 0 0 plane 3 in y -60 0 plane 4 in x 0 0 plane 4 in y -90 0 time 35 t 55 56 t 75 plane 3 in x 0 0 plane 3 in y 60 0 plane 4 in x 0 0 plane 4 in y 90 0 For the second scenario : time 11 t 31 32 t 34 plane 3 in x 0 0 plane 3 in y 0 0 plane 4 in x -62 30 plane 4 in y -150 150 time 35 t 55 56 t 75 plane 3 in x -49 0 plane 3 in y -150 -150 plane 4 in x 0 0 plane 4 in y -150 -150
3 sensors versus 4 targets, rst scenario The scenario studied here is a duel between 3 sensors and 4 targets. The rst two targets are non manuvring targets, whereas the two last manuvre at the same time. The third starts a 6g turn and after recovers its initial trajectory. The fourth do the same manuvre, but at 9g. The scenario has a total duration of 75 periods. We have performed 100 tests to compute the quadratic error : 1 75 i=1 kXi 75 X
3500
3000
2500
2000
1500
1000
500 −600
−500
−400
−300
−200
Figure 2: First and second scenario. strategy DG prede nite error target 1 5.9057 5.1177 error target 2 4.9369 3.8727 error target 3 9.5014 10.2358 error target 4 8.4723 10.0100 total error 28.8163 29.2364 We can observe that the DG (Gain Discrimination) strategy decreases the error on the more manuvring targets and tries to avoid to loose the non-manuvring targets. We can also count during the 75 periods how many time the di erent grouping strategies have been used (See the array below). We can remark that in this scenario, the gain brought by the fusion does not compensate the lost of available ressource for other targets, it means that we are close to pure assignment (one to one). Nevertheless we will see for the second scenario, that the robustness brought by the data fusion makes a di erence if we take into account validation windows. strategy alone 1 ^ 2 3 2 ^ 3 1 1 ^ 3 2 in % of time 91.24 0.81 0.58 7.32 strategy 1^2^3 in % of time 0.04
3 sensors versus 3 targets
It's the same scenario, but we have deleted the fourth target. The prede nite strategy 1 consist of assigning the three sensors to the three targets by circular peron the 100 generations of measures with noise. The mutation. The prede nite strategy 2, consist of assignprede nite strategy consists of assigning the four tar- ing a target to the group 1 ^ 2, an other to the sensor gets to the three sensors by circular permutation. The 3, and then perform circular permutations between the results obtained are: two coalitions and all the targets. bi k X
2
As an example, we can show the gain discrimination graphic describing the fused gain discrimination (computed with the fused information of all the sensors) for all the targets in fonction of time: On the gures 4 and 5 we show the di erent sensortargets allocations in fonction of time. The gure 4 show the allocations of sensors 1 and 2. he gure 5 show the allocations of the sensor 3 and the allocation intensity for each target, a line of lenght 2 means that there is a two sensors fusion on the corresponding target.
3
2.5
2
1.5
1
0.5
0
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10
V (k + 1; ) = fz
: [z zb(k + 1jk)] 1 S (k + 1) [z zb(k + 1jk )] g
40
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Validation gates
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Figure 3: Fused gain discrimination for all the targets in fonction of time.
0 10
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activite
1
activite
1
0
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1 activite
To validate the optimization method, an other criterion than the quadratic eror can help us to understand the adaptive behavior of this one. In fact, to compute the number of lost targets during the simulation gives an information which is not calculated with a mean. The procedure is described as follows: at each t, one veri es that the measure on the target T performed by the sensor C is inside its validation gate. Said di erently, one must veri es that the measure is in the volume:
20
activite
strategy alone 1 ^ 2 3 2 ^ 3 1 1 ^ 3 2 in % of time 84.8 7.26 1.21 6.64 strategy 1^2^3 in % of time 0.12
3.5
activite
In this scenario, the two manuvring targets perform 12g manuvres in the direction of the sensors. strategy DG prede nite error target 1 5.4157 4.9731 error target 2 5.6899 3.9962 error target 3 11.4807 11.8081 error target 4 7.0184 9.5314 total error 29.6048 30.3088
cible 1 cible2 cible3 cible4
activite
3 sensors versus four targets, second scenario
4
activite
strategy GD prede nite 1 prede nite 2 error target 1 4.1629 4.1049 3.6866 error target 2 3.6401 3.4598 3.5584 error target 3 6.2239 6.6999 8.7529 total error 14.0269 14.2645 15.9979 We notice that coalitions strategies are more often used. strategy alone 1 ^ 2 3 2 ^ 3 1 1 ^ 3 2 in % of time 75 2.13 1.76 20.7 strategy 1^2^3 in % of time 2.5
0
0 10
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Figure 4: Allocations of sensors 1 ( rst column) and 2 If it is not, one takes only the predicted measure. The (second column) results obtained on the 100 test are in the following array:
[3] H.Durrant-Whyte et J.Manyika, Data Fusion
3 activite
activite
1
0
and Senasor Management, a decentralized information-theoretic approach, Ellis Horwood,
2 1 0
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10 3
1
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1994. [4] H.Durrant-Whyte et J.Manyika, An Information-
theoretic Approach to Management in Decentralized Data Fusion, SPIE vol 1828, snesor fusion V,
activite activite
activite
activite activite
activite
1992. 1 [5] K.J.Hintz et E.S.McVey, Multi-Process Con0 strained Estimation, IEEE Transaction on Sys0 10 20 30 40 50 60 70 10 20 30 40 50 60 70 tems, Man and Cybernetics vol 21 no 1, jan3 vier/fevrier 91. 1 2 [6] K.J.Hintz et G.A.McIntyre, An Information The1 oretic Approach to Sensor Scheduling, SPIE vol 2755, 1996. 0 0 10 20 30 40 50 60 70 10 20 30 40 50 60 70 [7] J.Kapur et K.Kesavan, Entropy Optimisation 3 1 Principles with applications, Academic Press, 2 1992. 1 [8] K.Kastella, Discrimination Gain to Optimize De0 tection and Classi cation, IEEE Transaction on 0 10 20 30 40 50 60 70 10 20 30 40 50 60 70 Systems,Man and Cybernetics, part A :Systems and Humans, vol 27 no1, janvier 1997. Figure 5: Allocations of sensor 3 ( rst column) and [9] K.Kastella, Joint Multitarget Probabilities for Deallocation intensity (second column) tection and Tracking, Procedings of SPIE, Acquisition, Tracking and Pointing XI. vol 3086 p122128, avril 1997. strategy DG prede nite [10] S.Musick et K.Kastella, The Search for Optimal lost target1 0 0 Sensor Management, SPIE vol 2759, mars 1996. lost target2 0 0 lost target3 30 40 [11] S.Musick et K.Kastella, Comparison of sensor lost target4 20 25 Management Strategies for Detection and Classi cation, 9th National Symposium on Sensor Fusion, 1996. 6 Conclusion [12] J.M.Nash Optimal Allocation of Tracking ReWe have seen in the di erent simulations that the gain sources, Proceedings IEEE Conference on Decibrought by the optimization which takes account the sion and Control, p1177, 1977. fusion aspect, must be seen as a robustness improvment of the tracking. The optimization aspect is com- [13] A Papoulis, Probability, Random Variables and Stochastic Processes, MacGrawHill Book Complexi ed by the fusion ascpect, therefore exponentiality pany, Second edition Part 3, 1994. of the algorithm can be problematic if one increases the number of sensors. Nevertheless, three elements can [14] P.L.Rothman et S.G.Bier, Evaluation of Sengive a hope for a real time application. First the limsor Management Systems, IEEE CH 2759 p1747, ited interest of big coalitions which help us to reduce 1989. the complexity. Second the possibility of suboptimal optimization by \neighnourhood" change, or more pre- [15] W.Schmaedeke et K.Kastella, Information Based Sensor Management and Immkf, Spie Conference cisely a transfert of one or more sensors from a coalition on Signal and Data Processing of Small Targets, towards an other. Third, a di erent point of view than 1998. the centralized one may help us to investigate how this heavy calculus could be distributed. [16] W.Schmaedeke, Information based Sensor Management, SPIE vol 1955, 1993. References [17] G.A. Watson, W.D.Blair et T.R.Rice, Enhanced Electronicaly Scanned Aray Rsource Management [1] Y Bar-Shalom et Xiao-Rong Li, MultitargetTrough multisensor Integration, SPIE vol 3163, Multisensor Tracking :principles and techniques, 1997. YBS, 1995. [18] Brian D. O. Anderson et John B. Moore, Optimal [2] D.M.Buede et E.L.Wlatz, Issues in Sensor ManFiltering, Prentice-Hall, 1979. agement, IEEE TH 0333, p839-842, 1990. 2
