Tracking a 3D maneuvering target with passive sensors

I. INTRODUCTION

Tracking a 3D Maneuvering Target With Passive Sensors

FRANCOIS DUFOUR Ecole Normale SuNrieure France

MICHEL MARITON MATRA France

The fusion of wasuremenls from passive infrared sensors for the tracking of a target maneuvering in a 3D space in the presence of clutter is studied. Based on a hybrid system description of the maneuver scenarios, the interacting multiple models (IMM) approach is used a* lhe false alarms generated by clutter are acco-dated through probabilistic data association (PDA). The importance of accurate models is emphasized and we exhibit the corresponding behavior of the IMM dynamics.

Manuscript received October 7, 1990; revised December 10, 1990.

IEEE Log No. 9100913.

This work was supported by DRET (DGA, Paris) under Grant 891357 and by NATO under Grant 890885.

Authors’ addresses: E Dufour, Dept. de Physique AppliquCe, Ecole Normale Sufirieure, 61 AV. du Pdt Wilson, 94230 Cachan, France; M. Mariton, Labratoire de Baitement des Images et du Signal, MATRA BP 235-Les Miroirs, 38 boulevard Paul Cezanne, 78052 St Quentin en Yvelines Cedex, France, (address correspondence to this author).

0018-9251/91/0700-0725 $1.00 @ 1991 IEEE

The case of maneuvering targets is a well-established topic of the tracking literature. Many solutions have been proposed using variable dimension filters [3], adaptive process noise [14], and multiple model techniques [9, 111. The recent monograph [4] provides an in-depth review of these and other related techniques (see also [5] for a radar oriented perspective) .

A promising approach is the interacting multiple models (IMM) algorithm originally proposed by Blom [6]. This algorithm is based on a hybrid system description of the maneuver scenarios, where the Occurrence of target maneuvers is explicitly included in the kinematic equations through regime jumps. To the usual state variable xf E R” (position, speed,. . .) a discrete regime variable rf E {1,2, ..., M } is appended with rf = 1 corresponding to a no-maneuver situation and r, = 2 to M corresponding to different maneuvering hypothesis. The transitions of the regime variable are modeled with a Markov chain. There exists a large body of results from hybrid systems research to support the design of tracking applications, both in continuous-time [18, 161 and discrete-time [S, 10, 171, and, if needed, non-Markovian transitions could be taken into account [S, 151. Compared with other multiple model approaches the main advantage of hybrid systems based solutions is that they are not subject to the so-called “oblivious detection” difficulty. The a priori jump probabilities captured in the regime dynamics make hybrid algorithms alert to detect regime changes whereas classical solutions are biased toward the no-maneuver hypothesis after long quiescent periods.

The potential of the IMM solution has now been thoroughly investigated and confirmed through simulations [7] and it is available as part of the MULTIDAT software [4]. In the presence of clutter, the IMM has to be complemented to take into account the uncertainty of measurements origin. It was shown by Houles and Bar-Shalom [13] that the probabilistic data association (PDA) logic is an efficient solution for that aspect.

Existing applications of the IMM algorithm have emphasized the case of radars (active sensors), as in [6 or 21 for civil aircraft. The case of a passive sensor was first considered in [13] where the azimuth and elevation measured by an infrared search and track (IRST) sensor was fused with the range and azimuth measured by a radar.

The contribution of this work is twofold. First a novel application of the IMM algorithm is studied where passive-only sensors (two IR sensors) are fused for tracking a target maneuvering in three dimensions. Second more accurate models of target motion are proposed to improve performance. When general models are used to describe the maneuveringperiods, it is

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 27, NO. 4 JULY 1991 725

shown here that the IMM behavior is not satiqactory, in that the innovations associated with the dilgperent models do not discriminate between the corresponding target maneuvering regimes. The tuning of the Markov chain transition matrix, i.e., a priori information, is then crucial to obtaining the correct ordering of the a posteriori regime probabilities. On the contrary, a more satisfactory behavior of the IMM algorithm is obtained by careful4 selecting the target motion models in the different regimes.

The paper is organized as follows. Section I1 describes the application and discusses passive fusion concepts for 3D tracking. Section 111 recalls the IMM and PDA equations and Sections IV focuses on the modeling issue. Simulation results are reported in Section V together with recommendations on the selection of models.

11. APPLICATION: PASSIVE SENSORS FUSION

The emergence of passive sensors like IRST and FLIR (forward-looking infrared) has a significant impact on the design of weapons systems that traditionally depended on radars. The fusion of IR and EM (electromagnetic) sensors make the system less susceptible to target counter-measures and to destruction of one sensor by a preemptive strike. Also the physical measurements of IR and EM sensors nicely complement each other. A surveillance radar delivers range and azimuth but elevation is not usually available because of ground clutter; on the other hand the azimuth and elevation measurements of a passive sensor do not allow an instantaneous range determination. Also the quality of the azimuth and elevation delivered by an IR sensor can improve the angular tracks provided by the radar.

Passive sensors will clearly have a role to play in applications where radars have been traditionally been employed as stand-alone sensors, but it is also clear that an efficient integration of these new sensors requires dedicated data fusion algorithms. This work reports on parts of recent efforts in this direction and considers the a priori more difficult problem of passive-only fusion.

When the passive sensor platform is allowed to move freely, it is possible to recover range observability by selecting an appropriate path for the platform (this solution has been investigated in the sonar case in [12]). In other applications (e.g., a Command and Control (C2) system for air defense) the sensor platforms have very slow mobilities compared with the target dynamics and this solution is not feasible. A solution is then to use several passive sensors and to fuse their information in some way to estimate the range.

To be more specific, consider the situation depicted in Fig. 1. Each IR sensor measures two angles, azimuth and elevation, and two main architectures and

Target t R3 i'

Sensor 1 Sensor 2

1

Fig. 1. Geometry of the scenario.

Y Y Y Y \ I

~ __...__ 3D estimates ...____. Solution 1 Solution 2

Fig. 2. Tho fusion systems.

algorithmic solutions are possible to perform fusion. First, some tracking algorithm can be implemented at the sensor level (locally) to produce tracks of concatanated angles. The azimuth and elevation track files are then passed to a central fusion node where the range is estimated to generate 3D target estimates. This is called trackfusion. Second, the raw angle measurements can be passed to the fusion node where they are directly fed into a 3D tracking filter. This is called detection fusion. Fig. 2 summarizes the two architectures.

The choice between the two solutions has many system implications and depends on the specifics of the application, it will thus not be discussed in detail here. However it is possible to compare the two solutions on a general basis. It is true that solution 1 has lower bandwidth requirements for the communication from sensory nodes to the fusion node. The 2D filtering of raw data eliminates most false alarms before transmission while the complete set of measurements (including false alarms) is transmitted with the second solution. On the other hand, the prefiltering of solution 1 degrades performance even if the common process noise correction is taken into account. This degradation is most significant when hard target maneuvers are present. The tracking of multiple targets with solution 2, though possible, is made cumbersome by the many combinations involved at the data association step. A possible

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recommendation is thus to use the two solutions in a complementary fashion, with solution 1 associated with a surveillance mode, covering a large area, tracking multiple targets with slow dynamics and measurements rates and solution 2 associated with a track-and-fiie mode, the sensors being focused on a narrow field-of-view centered on a single target with high dynamics and measurement rates. In the sequel we present a detection fusion design (solution 2). The target is maneuvering and uses counter-measures which generate a localized clutter (false alarms).

111. HYBRID-MODEL-BASED ALGORITHM: IMMPDA FILTER

In this section we introduce the algorithm coupling the IMM tracker, to account for maneuvers, and the PDA processing of multiple detections.

A. Hybrid Systems

The target encounter is modeled by a hybrid system

x k t l = A ( C k ) x k + vk (1) where x k E R" is the usual state (e.g., position and velocity) and c k E { 1,2,. . . , M} is the target regime (e.g., ck = 1 when the target moves at constant speed and ck = 2 when the target accelerates). As is the case below, n may also depend on ck. The process v k is a zero-mean white Gaussian noise with known covariance R" (ck) . We also need nonlinear state models as

x k t l = g ( c k , x k ) + vk-

r i j = P { c k t l = j I ck = i}

(2)

(3)

The jumps of ck are described by a Markov chain

or, in vector form,

P k t l = n p k (4)

with p k = [P{Q = I}, ..., P{ck = M}]' and n =

The measurements yk E R"' are a nonlinear ( X i j ) i , j = I , M *

function of the state

yk = h(xk) + w k (5 ) where wk is a zero-mean white Gaussian noise with known covariance R". It is possible to include regime-dependent measurements in a hybrid filter [16] but this is not needed here. For our application, the nonlinearity h converts Cartesian target coordinates to azimuth and elevation angles as shown below.

B. Hybrid Filters

The contribution of this work is not the IMMPDA filter itself but rather the discussion of the role of

new and more accurate target models and the introduction of a new application to passive sensor fusion. The reader is therefore referred to [4, 7, 131 for a comprehensive presentation of this filter and the algorithm is recalled here without proofs.

Hybrid filters are essentially infinite dimensional filters and, consequently, approximations are needed to arrive at implementable nearly optimal solutions. The IMM concept is based on a single approximation (a sum of Gaussian variables is replaced by a single Gaussian variable) and it belongs to the class of merging suboptimal hybrid filters. In this class it was shown, both through simulations [7] and analysis [2], that the IMM compares favorably with its closest competitor, the generalized pseudo-Bayes algorithm (GPBA) of order 2 [l]. Roughly speaking, the IMM provides the performance of the GPBA-2 with much less computational requirements. The other class of suboptimal hybrid filters is based on pruning unlikely regime sequences [19], but it was shown in [2] that pruning may produce loss of track by eliminating valid sequences when a maneuver occurs.

The PDA concept is an efficient way to handle data origin uncertainty where measurements are first validated through gating and then weighted according to the likelihood of their origin. It has been favorably compared with its competitors, including nearest neighbors and multiple hypothesis associations [4].

In the following paragraphs we detail the IMMPDA Filter equations.

Notations: We consider the kth time step. At the end of step k - 1, & - l / k - l ( j ) , the associated error covariance R$- l ,k - l ( j ) and the a posteriori probabilities P ( c k - 1 = j 1 Z k - l ( j ) } for each regime ck = j , j = 1 , . . . ,M have been computed.

Markovian transitions to compute M estimates and the associated covariances (2;- l , k - ( j ) and R&k- (1)). These variables serve as inputs to M Kalman filters running in parallel to perform state extrapolations and updates ( 2 k / k ( j ) and R t l k ( j ) ) . The innovations ( & ( j ) ) are finally fed to the regime probabilities update function. The outputs of step k are thus 2 k / k ( j ) , R t I k ( j ) and P{ck = j I Z k ( j ) } for j = 1, ..., M.

The first function is to expand the regime

We use the following notations.

(iil,ii2,. . .,a,) is the canonical basis of R". mk is the number of measurements validated

at time k for model i (cf, the validation principle of $6).

k for model i, 2, = {zi, ..., z;*} where z: is the j th validated measurement at instant k.

2 k is the set of measurements validated at time

z, = { 2 1 , ..., 2 k } .

i k - l / k - l ( i ) = E { X k - 1 I Z k - 1 , C k - 1 = i). 2,&+l(~) = E ( x k - 1 I Z k - m = j } . & / k - l ( i ) = E { X k I Z k - 1 , C k = i}.

DUFOUR & MARITON: TRACKING A 3D MANEUVERING 'JARGET WITH PASSIVE SENSORS 727

Rkx-l/k-l(i) = E((xk-1- &-l/k-l(i))(xk-l- fk-l/k-l(i))' I Zk-1,Ck-1 = i}.

a,tl/k-l(i))' I Zk-1,Ck = i}. Rkx+l/k-l(i) = E((xk-1- q-l/k-l(i))(Xk-l-

Rkx/k-l(i) = E{(& - fk/k-l(i))(Xk - 4c /k - l ( . j ) ) ' I

Pk-l/k-l(i) = P{Ck-1 = i I Zk-1). pklk-l(i) = P{Q = i I Zk-1). h p - l ( i ) = E{yk I Z k - 1 , ~ = i} .

Zk-1,Ck = i}.

YkJk-1 = E{yk I z k - 1 ) .

RkIk-#) = E{CYk - h l k - l ( i ) ) C Y k - jklk-l(i))' I

R&(i) = - &,k-l( i ) )CYk - Y k l k - l ( i ) ) f I z,-,,c, = i}.

Z k - 1 , ~ = i}.

measurement corresponds to the target for model i.

measurement at instant k for model i.

probabilities of the above events.

The event $(i) is def ied as no validated

The event eL(i) is defied as zL is the correct target

p;(i) = P{8;(i) I Zk}, 0 5 15 mk are the

Pd is the probability to detect the target. Pg is the probability (at g sigma) to validate a

Vo,(i) is the volume of the validation region at time

.?&(i) = E{xk I B;(i),Zk,ck = i} .

target measurement within the gate.

k for model i.

R:lk(i) = E { ( X k - i:lk(i))(Xk - iL1k(iNf I @(i),zk,Ck = i}.

~ k ( i ) = z: - jjklk-l(i).

2) Regime Probabilities Extrapolation: M

3) State and Covariance Expansion:

+ ( & - l l k - l ( f ) - .c-lp-l(i))

x ( & - l l k - l ( l ) - &Ik-l(i))').

4) State and Covariance Extrapolation: When the state model is linear, we get the usual Kalman equations

iklk-l(i) = A ( C k = i)Pk+_llk-1 (8)

R&-l(i) = A(ck = i)R;+llk-lA'(ck = i ) + R". (9)

However we also use nonlinear state models and the nonlinearity xk = g(xk-l) + V k must then be linearized near ik+_llk-l(i). With (a2g'/ax2) the

Hessian of the fth component of g and ( d g l d x ) the Jacobian of g (computed at 31-llk-l(i)), we obtain

4clk-l(i) = g ( K l p - l ( i ) )

and

5 ) Measurement Linearization: With (d2h1/ax2) the Hessian of the fth component of h and (ahlax) the Jacobian of h (computed at i?+-l(i)), we obtain

j k l k - l ( i ) = h(%clk-l(i))

6 ) Measurement Vdidation: The measurement z; is validated if and only if:

(4 - Yk,k-l)'(R{lk-l)-'(2: - j k l k - 1 ) < g (12)

where R{lk-l stands for the largest among the model conditioned covariances (R{lk-l = R{lk-l(io) such that det(RLlk-l(iO)) 2 det(Rilk-l(i)) for any i = 1, . . . ,M) and

pg = 1 - (1 + g2)e -gZ /2 (13) (the volume of the validation region is Vok(i) = r2/2 det(R{lk-l)g2 for model i at instant k ) .

7) Association Probabilities: We assume Poissonian false alarms with a known rate X (there are m F = m false alarms with probability P{mF = m } = e- x VOk(i ) (A Vok (i))" / m !) so that

7

728 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 27, NO. 4 JULY 1991

for 1 5 j 5 mk, and, where N(x ,Y ,R) is a Gaussian law with mean T and variance R, and

bk (i) = bk(i) + a i ( i )

(16) (a is the normalization constant).

and

bk(i) = J-7- 1 - PdPg (17) IV. TARGET MOTION MODELS

As any model-based algorithm, the IMM algorithm 8) for hfeasurement 1: For an observation cannot perform better than the models on which it

relies. In this paragraph we introduce three different sets of models to describe target maneuver scenarios. To each of these sets corresponds a different dynamic behavior and performance of the algorithm.

z;(i) we have

i ~ l k ( i ) = i k l k - l ( i ) + Rib-i(i)(R;lk-i(i))-lzL(i) (18)

and

Rlkp(i) = Rkxlk-i(i) - R~rk-l(i)(R;~k-l(i))-'R;~k-l(i). A. First Set of Models

(19) (R:lk(i) does not depend on 1 and it is noted R;lk(i) in the sequel). In the absence of validated measurements,

f & k ( i ) = J k l k - l ( i )

We start with the set of models most often used in

Thrget motion is described with three models

Model 1.1: For flight segments with constant

the multiple models tracking literature.

the update step is skipped

and x = (x l , x* ,x3 ,X l ,x2 ,X3) t (25)

(M = 3):

(20) speed, a state vector of order 6 (n = 6 ) is used

R&k(i) = Rkxlk-l(i). (21) and the discrete-time dynamics matrix

where &(i) = pL(i)ZL(i). 10) Regime Probabilities Update: 0 0 i T 3 0 0 T 2

0 i T 3 0 0 T 2

Note that this model describes flight segments with constant averaged speed (as opposed to a sample pklk(i) = - Cvok(i) ' -mkP;'$'(mk)

p = l pathwise constant speed). Model 1.2: For flight segments where the target

maneuvers, a state vector of order 8 (n = 8) is considered

( m k

x N Z k p ( i ) , O , R ; l k - l ( i ) )

+ vok(i)-"'y,o(mk) x = ( X l , x2, x3, Xl , x2,*3, %, X2If (28)

729 DUFOUR & MARITON TRACKING A 3D MANEUVERING TARGET WITH PASSIVE SENSORS

and the discrete-time dynamics matrix

1 0 0 T O O i T 2

O l O O T O 0 O O l O O T 0 0 0 0 1 0 0 T 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0

k = e F =

0

iT2 0

0 T 0 0 1

0

0 0 1 0

0 0 1 0 0 1 0 0 0 cos(w) -sin(w) 0 0 0 0 sin(w) cos(w) 0

W W

1 - cos(w) sin(w) W

A =

RV =

0 9 - 4 0 0 9 - 3 0 0 $2

0 0 3 - 4 0 0 9 3 0 0

i T 3 0 O T 2 0 0 T 0

O i T 3 0 0 T 2 0 0 T

0 O $ T 3 0 O T 2 0 0

$ 2 - 2 0 0 T 0 0 1 0

(29)

0 0 aT4 0 0 4T3 R“= I

$T3 0 0 T 2 0 0

With this model we consider maneuvers in the XlX2 plane for a flight at nearly constant altitude (in the simulations below the target flies at constant X3). Of course we could also have considered vertical maneuvers by increasing the state vector with a vertical acceleration term.

Model 1.3: variables as the second model but a larger 0,. This was first suggested in [13] to account for the transitions between constant speed segments (described by model 1.1) and turns (described by model 1.2).

A third model is used, with the same

0,”. (37)

B. Second Set of Models

Again three models are used. Model 2.1:

Model 2.2:

For flight segments with constant

For turns, the exact kinematics of a average speed model 1.1 is again selected.

mobile turning with a constant angular rate w are used. This leads to a state vector in R6 with

1 = (x~,x2,X3,xl,x2,x3ij)’ . (31) We have in continuous-time

x, = -x,w

x 2 = x1w (32) { x 3 = 0

and a continuous-time dynamics matrix

(33)

where 0, is the null matrix and I, the identity matrix of dimension n * n, with

(34)

It can be shown by recurrence that

F” = (G) (35)

(36) For the covariance of the process noise we take

0 $T4 0 0 i T 3 0

$T4 0 0 i T 3 0

0 0 i T 3 0 0 T 2

0 $T3 0 0 T 2

Model 2.3: For U > 0 model 2.2 describes a counterclockwise turn, and model 2.3 is its natural counterpart for a clockwise turn.

Obviously models 2.2 and 2.3 assume that the rotation rate of the target is known. This is generally the case for a civilian aircraft [2] because its maneuvers are constrained by flight rules, especially when approaching an airport. For a military aircraft this assumption is less natural, but it has often been reported that pilots in combat or attack situations tend to fly on the limit of their flight envelope, so that taking as a known w the maximum g turning rate is not unrealistic.

C . Third Set of Models

Finally we consider the generalization of model 2.2 to the case where w is not known. Because we then have to estimate U , only two models are needed and Lj will take positive (respectively negative) values for the turn described by model 2.2 (respectively model 2.3) in the previous solution.

730 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 27, NO. 4 JULY 1991

Model 3.1: For flight segments with constant speed, a model similar to model 1.1 is again selected, but the noise input is modified. Whereas in model 1.1 we described speed as constant on the average here we consider a consrant speed. This leads to a covariance matrix as

0 c: 0 0 0 0 0 0 0 a ; o o o 0

R " = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \ 1 0 0 0 0 0

0 1 0 0 0 0

. (45)

0 0 1 0 0 0 R"= I

The use of this model leads to less volatile estimates of speed when the target is not maneuvering.

Model 3.2 We include the rotation rate w into the state vector

The continuous-time kinematics of the turn can then be written exactly

The sampled version of this equation is

with Id the identity of R7 and Lf the Lie derivative with respect to f(Lf(Id)x = f ( x ) and Lf o Lf(Id), =

at x ) . Up to second order in T we get o f ( x ) where ( J f ) x is the Jacobian o f f computed

for

X3 + TX3 T2

Xi - T X ~ W - -X1w2 2 T 2 .

X 2 + T X I W - -X2w2 2 X3

\ W

V. SIMULATION RESULTS

The two IR sensors are located along the x1 axis with sensor 1 at x1 = 0 and sensor 2 at x1 = 1 = loo00 m. The sampling rate is T = 1 s. Using the detection fusion architecture, the azimuth and elevation angles, ai and ei, measured by sensor i, are transmitted to the fusion node where the measurement vector (a1,e1,a2,e2)' is formed at each time step. The observation equation is then of the form (5) with

arctan (2) \ I

We assume that the raw angles errors are independent and the measurement noise wf has a diagonal covariance R" = 14~:. The false alarms are generated from a Gaussian distribution with (T = 3 mrd centered on the true target position. The number of false alarms is uniformly distributed between 0 and 2 and the average delay between false alarms is also a random variable with uniform distribution over [0 s, 15 SI.

trajectory. As shown in Fig. 3 the target makes three circular turns with rectilinear segments connecting them. The speed modulus is kept constant throughout (= 300 m/s). The target starts at t = 0 s at X I = loo00 m, X2 = 15000 m, X, = 200 m with X I = -300 m/s, X2 = 0, X3 = 0. The segments are defined as follows.

The different algorithms were tested using the same

1st segment.

2nd segment.

Rectilinear flight until the plane Xi = 3000 m (from t = 0 s to t = 23 s).

Circular turn for -3 rd with acceleration 70 m/s2 (from r = 23 s to r = 38 s) .

731 DUFOUR & MARITON TRACKING A 3D MANEUVERING IARGET WITH PASSIVE SENSORS

3rd segment. Rectilinear flight until the plane

4th segment. Circular turn for +3 rd with

5th segment. Rectilinear flight until the plane

6th segment. Circular turn for +3 rd with

7th segment. Rectilinear flight until final time

X I = 90oO m (from t = 38 s to t = 60 s).

acceleration 70 m/s2 (from t = 60 s to t = 75 s).

XI = 3OOO m (from t = 75 s to t = 90 s).

acceleration 70 m/s2 (from t = 90 s to t = 105 s).

t = 147 s (from t = 105 s to t = 147 s).

B. Single Model Reference

For comparison purposes, we also designed a single model Kalman tracker. The model was chosen as model 1.1 with U, = &% m/s2 and U, = 0.003 rd and

X = 0.00035 (rd)-2

g = 4

A. Parameter Values

The following values have been chosen for the

First Set of Models: model parameters

Model 1.1: U, = 5 m/s2 and ow = 0.004 rd Model 1.2 U, = 7.5 m/s2 and U, = 0.004 rd Model 1.3: U, = 40 m/s2 and U, = 0.004 rd

with

0.2 0.25 0.55

X = 0.00035 (rd)-2

g = 4

Pd = 1.

Second Set of Models: Model 2.1: U, = &% m/s2 and U, = 0.003 rd Model 2.2: U, = &@ m/s2 and U, = 0.003 rd Model 2.3: U, = &% m/s2 and U, = 0.003 rd

with

0.98 0.01 0.01

0.2 0.05 0.795

X = 0.00035 (rd)-2

g = 4

P d = 1

w = 0.3 rd/s.

Third Set of Models: Model 3.1: U, = a m and U, = 0.003 rd Model 3.2: U, = a m, U, = 0.05 rd/s and

U, = 0.003 rd with

X = 0.00035 (rd)-2

g = 4

P d = 1.

C. Performance Analysis

The three IMMPDA algorithms associated with the three sets of models presented above were simulated. Their performance and behaviors are compared here.

1) First Algorithm: The probabilities of the three regimes are plotted on Fig. 4. The correct regime has the largest probability during each segment and the turns are quickly detected. The expected role of model 1.3 is apparent on Fig. 5 where the regime probabilities have been zoomed around the onset of the 6th segment maneuver. It is seen that the probability of regime 3 rises at the start of the turn and then gives way to a dominant probability of being in regime 2. Once the turn is completed regime 1 takes over again.

The corresponding error on the location of the target is plotted on Fig. 6. Natural transients are observed at the onset of maneuvers and the error remains acceptable. However, as can be seen from Fig. 7, the error is not significantly reduced compared with that obtainable using the single model filter. Both curves show peak errors around 160 m. To understand this phenomenon, the errors associated with the three regimes are plotted on Fig. 8. The finding is that these errors are comparable for the three regimes both on quiescent and maneuvering segments. This is contrary to the natural behavior of the IMM dynamics where it would be expected to have the smallest errors for regime 1 (respectively regime 2) during rectilinear segments (respectively maneuvering segments). The explanation is that the models 1.1, 1.2 and 1.3 are not discriminating enough; although model 1.1 works better during rectilinear flight it does so only marginally and models 1.2 and 1.3 also do quite a good job. Symmetrically, it appears that model 1.1 is quite able to track the target when it maneuvers. Also the state expansion step mixes the regime-conditioned estimates in a way that helps the filters based on the “wrong” models to come back on track. The tuning of the regime transition matrix (i.e., a priori information) is then important to recover the correct ordering of the regime probabilities, as shown on Fig. 4, because the a posteriori information conveyed by the innovations conditioned on regime

732 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 27, NO. 4 JULY 1991

Trajectory in the x-y plan: - tuget. - - esrimatc xi04 1.8

1.2 - I -

0.8 -

0.6 -

0.4 -

-0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

x position (m) xi04

Fig. 3. ' h e and estimated trajectories (estimate is that of second IMM).

5

733

"Q 20 40 60 80 LOO I20 140 160

1 (sec)

Fig. 4. Regime pmbabdities for first IMM.

I

n e

(Sec)

FQ. 5. Zoom at start of turn.

DUFOUR & MARITON: TRACKING A 3D MANEWERING TARGET WITH PASSIVE SENSORS

I

734

200

150- b k 5 3 100- - L

50-

Position error for [MM

180

-

I

160

140

I20

100

80

60

40

20

0 0 20 40 60 80 100 I20 140

I fgec)

Fig. 6. Position error with fust IMM filter.

Posttim error for Mmcmodel

'0 20 40 60 80 I00 I20 140 160

t (sec)

Fig. 7. Position e m with single model filter.

Position cmr fa: - MI. - - M2. ... M3

250:

I (xc)

Fig. 8. Model errors for first IMM.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 27, NO. 4 JULY 1991

t (Sec)

Fig. 9. Regime probabilities for second IMM.

Poricion mor for: - MI. - - M2. ... M3 160

I ' 0 20 40 60 80 100 120 140

1 (sec)

Fig. 10. Model ermm for second IMM.

140

I

"0 20 40 60 80 loo 120 140

I (SCC)

Fig. 11. Positim ecror with sccond IMM filter.

a

DUFOUR & MARITON TRACKING A 3D MANEUVERING TARGET WITH PASSIVE SENSORS

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0 e

loo I 2 0 140 160

t (W

Fig. 12 Regime probabilities for third IMM.

Position amr fa IMM

I so

160

140

I10

100

80

60

40

20

OO 20 40 60 so loo I 2 0 I40 160

I (=)

Fig. 13. Position emr with third IMM tilter.

"0 20 40 60 80 loo 110 I 4 0 160

t (=)

Fig. 14. Regime probabilities for modified third IMM.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 27, NO. 4 JULY 1991

Ill /

0 20 40 (d) SO IOU 120 IJO IN)

f 3x1

Fig. 15. Position e m with modified third IMM filter and single model filter.

hypothesis are not contrasted enough. Also it must be noted that the primary objective of the tracker is to reduce unartaiuty on the kinematics of the target and that a correct ordering of the multiple models probabilities is not an objective per se but only as a means to improve tracking performance. In this respect the comparison between Figs. 6 and 7 is disappointing.

2) Second Algorithm: The idea of the second set of models is to improve the above results by more carefully modeling the maneuvers. The three probabilities displayed on Fig. 9 show that the regimes are correctly ordered and it is seen on Fig. 10 that the dynamics of these probabilities are now significantly driven by the a posteriori information contained in the innovations; the errors of model 21 are smallest during rectilinear flight and larger than that of model 2.2 (resp. 23) during counterclockwise (resp. clockwise) turns. 'Ib this more satisfactory behavior of the IMM internal dynamics corresponds a tracking performance improvement as seen on Fig. 11 where the peak errors at the onset of the first maneuver are reduced to 100 m (compared with 160 m with the first version of the algorithm).

provides a satisfactory performance and a natural behavior of the IMM dynamics. However it depends on the hypothesis that the turning rate is known. For situations where this assumption is too unrealistic we can use the third set of models where the turning rate is estimated through nonlinear state dynamics. The probabilities are as shown on Fig. 12 and again a correct ordering is obtained. Because of the delay in estimating w at the onset of a maneuver, this solution produces rather large peaks as seen on Fig. 13. However once the w estimate has converged a very good tracking performance is obtained during turns.

3) n i r d Algorithm: The second algorithm

We finally modified this third algorithm by using a different transition rates matrix

0.7 0.3 ' = (0.3 0.7)

With larger offdiagonal entries this matrix favors regime transitions and results in much more volatile sample paths of the probabilities (see Fig. 14). This shows the possibility of a different behavior of the IMM algorithm where no single regime is allowed to rise significantly above the others. This could be called "regime mixing." It provides the best performance as shown on Fig. 15 where the occurrence of maneuvers is now hardly uoticeable (the peak error is around 60 m (compared with 160 m with the single model filter) and only slightly above the errors caused by noise and false alarms during quiescent segments)). It is believed that this kind of mixing should play an important role in future applications of the IMM idea, in particular because it opens the way to treat the maneuvers of military aircraft by weighting models associated to maximum g turns and rectilinear flights. A 3g turn would then be tracked by probabilistically mixing a tracker with a 7g model and a tracker with a constant speed model.

VI. SUMMARY AND CONCLUSION

We studied a passive sensor fusion system where a single target maneuvering in clutter was tracked by fusing the angles-only measurements in a 3D filter. Good tracking performance has been achieved without active sensors even in the presence of extreme maneuvers.

Regarding the multiple model concept, we have analyzed the hybrid-system-based IMM algorithm and in particular the necessity of accurate models.

DUFOUR & MARITON. TRACKING A 3D MANEWERING 7XRGE.T WITH PASSIVE SENSORS 737

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There appears to be a kind of equivalence between the complexity of a single model in the IMM bank and the number of models to be included in that bank. For example, our third modeling includes a nonlinear model (model 3.2) and thus introduces additional complexity, however it saves a filter with respect to our second modeling where two linear models (models 2.2 and 2.3) were needed for positive and negative turns. More detailed complexity and performance analysis will be required to understand this alternative.

ACKNOW L EDGM ENTS

The authors carefully acknowledge the useful comments of Professor D. D. Sworder of the University of California at San Diego.

REFERENCES

Ackerson, G. A., and Fu, K. S. (1970) On state estimation in switching environments. IEEE Transactions on Automatic Control, AC-15 (Feb. 1970), 10-17.

Poursuite de cibles manoeuvrantes par des algorithmes markoviens hybrides. In Actes du &e Colloque GRETSI, Juan-les-Pins, France, June 1989.

Variable dimension filter for manewering target tracking. IEEE Transactions on Aerospace and Electronics Systems,

Barret, I., and Vacher, P. (1989)

Bar-Shalom, Y., and Birmiwal, K. (1982)

AES-18 (Sept. 1982), 621-629. Bar-Shalom, Y., and Fortmann, 'I E. (1988)

Tracking and Data Association. New York Academic Press, 1988.

Multiple Target Tracking with Radar Applications. Dedham, MA: Artech House, 1986.

A sophisticated tracking algorithm for ATC surveillance data. In Proceedings of the International Radar Conference, Paris, France, May 1984.

The interacting multiple model algorithm for systems with markovian switching coefficients. IEEE Transactions on Automatic Control, 33 (Aug. 1988), 780-783.

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Campo, L., Mookejee, P., and Bar-Shalom, Y. (1990) State estimation for systems with sojourn time dependent markov model switching. IEEE Transactions on Automatic Control, 36 (Dec. 1990), 238-243.

State estimation for discrete systems with switching parameters. IEEE Transactions on Aerospace and Electronics Systems, AES-14 (May 1978), 418-425.

Chizeck, H. J., Willsky, A. S., and Castanon, D. A. (1986) Discrete time Markovian jump linear quadratic optimal control. International Journal of Control, 43 (1986), 213-231.

Gauvrit, M. (1984) Bayesian adaptive filter for tracking with measurements of uncertain origin. Automatica, 20 (1984), 217-224.

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Optimal observer motion for localization with bearing measurements. Computers and Mathematics with Applications, 18 (1989), 171-180.

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Jaminski, A. (1%9)

Mariton, M. (1989)

Mariton, M. (1990)

Millnert, M. (1982)

Sworder, D. D. (1976)

'hgnait, J. K. (1982)

738 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 27, NO. 4 JULY 1991

Francois Dufour was born on September 9, 1%5. He received the Agregation de Sciences Physiques in 1989 from E o l e Normale Superieure de Cachan, France, where he is currently pursuing doctoral studies in stochastic control and signal processing.

Michel Mariton was born on September 27, 1959. He received the Agregation de Sciences Physiques in 1982 from Ecole Normale Superieure de Cachan, France. He received the D.Sc. degree in 1986 from Centre National de Recherche Scientifique (CNRS), Paris.

During his doctoral studies, he was a research associate with CNRS Signal and Systems Laboratory, and his doctoral work was awarded the Best Thesis Prize in Control Theory from AFCET. He received a CNRS Young Researcher Medal in 1989. After a stay at the University of California, San Diego, as a Visiting Scholar in 1987, Dr. Mariton joined MATRA in 1988 where he is presently Deputy Head of the Signal and Image Processing Laboratory with MATRA MS2i. His research interests are in target tracking, image processing and systems science.

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