Interacting multiple model filter with error monitoring and recovery technique of perception net

J.W. Choi, T.H. Fang and T.L. Song

Abstract: The paper is concerned with the problem of tracking a manoeuvring target. The error monitoring and recovery technique of the perception net is utilised to improve the tracking performance for a highly manoeuvring target. The interacting multiple model (IMM) filter is well known as being the most cost-effective hybrid-state estimation scheme. Both the detection of a manoeuvring target and the compensation of the estimated state can be achieved by employing a properly implemented error monitoring and recovery technique. IMM and full IMMPDA filters which employ an error monitoring and recovery technique show a better tracking performance compared with the conventional IMM and IMMPDA filters for a highly manoeuvring target, even in a clutter environment. They also reduce the peak values of the estimation errors when manoeuvring starts and finishes. The effectiveness of the proposed method is validated through simulations by comparing it with a conventional IMM filter. A similar evaluation is performed for a conventional IMMPDA filter.

1 Introduction

The problem of tracking a manoeuvring target has drawn a great deal of attention. Much research has concentrated on tracking a manoeuvring target since Singer [ I ] proposed a target model including manoeuvring motion in 1970. The techniques, representative of two general approaches to manoeuvre adaptive filtering, can be outlined. The first approach is to detect a manoeuvre and to cope with the manoeuvre. This approach can use an input estimation filter (Bogler [2] and Whang et al. [3]), a variable dimen- sion filter (Bar-Shalom et al. [4]) and a variable dimension filter with an input estimation filter (Park et al. [Sj). The other approach is to use a multiple dynamic model and for this, there is the interacting multiple model (IMM) filter (Blom and Bar-Shalom [6]).

The IMM filter [6, 71 is well known as a suboptimal hybrid filter and is known to be one of the most cost- effective hybrid state estimation schemes. The interacting multiple model with probabilistic data association (IMMPDA) filter [8-101, which combines the probabilistic data association (PDA) filter [ 11, 121 with the IMM algorithm, can be considered as a very economical filter for tracking a manoeuvring target in a clutter environment.

0 IEE, 2003 IEE Proceedings online no. 20030079 DOI: 10.1049/ip-rsn:20030079 Paper first received 13th August 2001 and in final revised form 27th November 2002 J.W. Choi and T.H. Fang are with the School of Mechanical Engineering and Research Institute of Mechanical Technology, Pusan National University, Pusan 609-735, Korea T.L. Song is with the School of Electrical and Computer Engineering, Hanyang University, Kyunggi-do 425-79 1, Korea

5 8

The IMM algorithm, however, tends to adapt slowly from a non-manoeuvre mode to a manoeuvre mode or vice versa, although a sudden manoeuvre can occur in a true system. The reason is that the model probability is dependent on the past model probabilities and therefore all the subfilters cannot properly represent the change of motion of a target.

The perception net [13, 141, as a structural representa- tion of the sensing capabilities of a system, is formed by the interconnection of logical sensors with three types of modules: a feature-transformation module (FTM), a data- fusion module (DFM) and a constraint-satisfaction module (CSM). By observing the output and input of the DFM an input data error can be detected. The error monitoring and recovery based on this function make it possible to detect and identify errors, and to calibrate possible biases involved in the sensed data and the extracted features. A technique which combines the conventional IMM algo- rithm with the error monitoring and recovery of the perception net is proposed in this paper, in order to track a highly manoeuvring target with sudden movements. Additionally, the method, which is proposed in a clear environment, is extended to the IMMPDA for tracking a manoeuvring target in a clutter environment. The results presented in this paper are a generalisation of the results proposed in [15, 161.

The Kalman filter is widely used as a tracking filter to estimate the position, velocity and acceleration of a target in real-time, because of its recursive structure and sequen- tial data processing capability. In the Kalman filter the current predicted state, which is calculated from the state at the previous time, is combined with the measurement that is acquired by the physical sensor at the current time [ 171. Perception-net-based geometric data fusion is the same as Kalman filtering, and in order to introduce the perception net in target tracking, we also prove it through an algebraic approach.

IEE Proc.-Rudur Sonur Nuvig.. Vol. 150, No. 2, April 2003

2 IMMPDA algorithm

The target dynamics are modelled in Cartesian coordinates as

x'(k + 1) = F'x'(k) + vi@) (1) (2) z(k) = Hx(k) + w(k)

where x'(k) is an n-dimensional state vector at time k for model t, z(k) is an m-dimensional measurement vector at time k, and v(k) and w(k) are zero-mean, mutually inde- pendent, white Gaussian noise vectors with covariance matrices Q(k) and R(k), respectively.

The basic assumption of the IMMPDA filter is that the system obeys one of a finite number of models with known parameters, and model switches occur as a Markov chain with known transition probabilities. In this approach, the state estimate is computed under each possible model hypothesis, with as many filters as the number of assumed models, and with each filter using a different combination of the previous model conditioned estimates.

Mt(k) is the event that model Mt with t = 1, 2, . . . , N is effective during the kth sampling period, Q,(k) is the event that zi(k) is the correct measurement from the target, and zk is the cumulative set of measurements. Then the basic equations of the IMMPDA filter are as follows:

The estimate of the state and its covariance are obtained by using the total probability theorem with respect to the measurement-origin events and the model events as follows:

(7)

P(klk) = 5 pt(k){Pi(klk) + [") - i ( k l k ) ] i= 1

x [2'(klk) - i(klk)IT} (8)

The reader is referred to [S-101 for further details.

3 Perception net

The perception net, as a structural representation of the sensing capabilities of the system, is based on a geometric data-fusion method. The perception net is formed by the interconnection of logical sensors with three types of modules: a feature, transformation module (FTM), a data, fusion module (DFM) and a constraint-satisfaction module (CSM). In this Section, we briefly describe the FTM, the DFM and the error monitoring and recovery technique.

It is assumed that a noise is bounded by an uncertainty hyper-volume, or ellipsoid, and that the size of the uncer- tainty ellipsoid is small enough for a good linear approx- imation around the nominal point in the feature transformation. Formally, the uncertainty, de, of a sensor value, <, is represented as an ellipsoid of the following form

where R, represents a symmetric weight matrix determin- ing the size and shape of the ellipsoid.

The uncertainties propagate through the input-output relationships between the input vectors and the output vectors of the modules. Let us define the mapping relation- ship between the input vector, <, and the output vector, y, of the FTM or the DFM. Then the uncertainty propagation can be approximated as a first-order Jacobian relationship with the assumption that f is smooth and d< is small, as follows:

dy x af d< = J(< , p)d< (12) a< where p represents a parameter vector associated with the module and J(<, p ) represents the Jacobian relationship between dy and de. The uncertainty of <, represented as an ellipsoid of (9), can now propagate to the uncertainty of d<, represented as an ellipsoid in terms of dy, through (12). Substituting d < = p ( < , p)dy, obtained from (12), in (9) gives

dyr(J+)Tf2,J+dy 5 1 (13)

ay ( J + ) ~ R ? J + (14)

where represents the pseudo-inverse of J. The symmetric weight matrix, a-,, is defined as

Consider the two measurements, and C2, defined in the two measurement spaces, where their uncertainties are bounded by the weight matrices, and at2, respectively. The geometric data-fusion method starts by defining the augmented space, E= [Cy <;IT and Q,=Diag[Q,,Q,2]; such measurement data is represented in the augmented space as (E, a=). Then the problem of fusing (Cl , a,,) and (e2, at2) is equivalent to obtaining y by minimising

Then y , by minimising (1 5), and the uncertainty bound, associated withy are given by

Y = (Q,, + ~ t 2 > - ' ( f & $ , + f2,,<,) sz, = ((AAT + BBT)-I}T(ARt,AT + BR,*BT)

(16)

(AAT + B B y (17)

where A and B are given by A =(atl +9,2)-'RtI and B = (atl + L?,2)-'Qc2, respectively. The propagation through the FTM is from (9) to (14). The propagation through the DFM is from (1 5 ) to (1 7).

The DFM of the perception net makes it possible to monitor errors, i.e. biases and faults in sensing. In general, the error monitoring and recovery technique consists of three steps. The first step is to detect an error in the input data by examining the output of the DFM. If an error is detected, then the second step is applied. That is, the second step is to identify the source (sensor) which caused the error. Finally, the third step is to replan the faulty sensor.

The inconsistency of the input data of the DFM can be evaluated based on the ellipsoidal representation of the uncertainty bound: the input data of the DFM are said to be inconsistent if the ellipsoidal uncertainty bounds of the input data have no common intersection.

59 IEE Proc-Radar Sonar " J i g . , Vol. 150, No. 2, April 2003

The existence of a common intersection among the input data can be evaluated by: if

2 Ily - tilln, F 1, for i = 1 , 2

with the output, y, of the DFM, then there exists a common intersection among the ellipsoidal uncertainty bound of ti.

The isolation of the error sources can be done through the net hierarchy. By applying the above error detection method to the DFMs, the logical sensors associated with the DFMs can be classified as likely-in-error, unlikely-in- error or possibly-in-error. These classifications are propa- gated through the net to extend the classification to other logical sensors connected through the hierarchy. The cross- checking of these classifications propagated through the net hierarchy provides further isolation of the errors.

Once the source is isolated, then the system takes action to repair the errors and to recover from the error. The system needs to replan the task based on the isolated errors. However, the identification and replanning of the error sources cannot be implemented by the general method, but should be implemented by a suitable method depending on the specified systems. The reader is referred to [13, 141 for details.

DFMl

4 IMM filter with error monitoring and recovery technique

DFM2 DFM3

Consider that the state of the target is a two-dimensional vector consisting of position and velocity (in one Cartesian coordinate) for a constant velocity model (t = t,), xfV = [x k]: or a three-dimensional vector consisting of position, velocity and acceleration for a constant acceleration model (t = la), x'" = [x k XI? When the tracking sensor measures the target position along the dimension being analysed and provides an output equation such as (2 ) , the measurement matrix associated with XI" is H = [ l 01 and the measure- ment matrix associated with x'. is H = [ l 0 01. Hence, HHT= 1 is clear in both cases. Moreover, in three Cartesian coordinates, it is trivial that HHT=Z3 3 . This property will be used in theorem 1.

The error monitoring and recovery technique of the perception net is effective for the geometric data fusion method. Therefore, in order to apply the error monitoring and recovery technique to a tracking filter design, it is necessary to know how the geometric data-fusion method is connected to the Kalman filter, on which the tracking filter is based.

Theorem 1: Let tl (k) be the predicted state of the Kalman filter, i(klk - l), and t 2 (k ) be

C2(k) = HTz(k) ( 1 8 )

If the covariance matrices of the state vector and the measurement vector are non-singular and HH' = Z,,, x m ,

then the output, y, of the DFM which has inputs <,(k) and &(k) is the same as the measurement update estimation of the Kalman filter using the predicted state, i (k lk - l), and the measurement, ~ ( k ) .

Proof It is assumed that the weight matrices, Q5,(k) and Qz(k,, are the inverse covariance matrices associated with the state vector and measurement vector, P(klk - I)-' and R(k)-', respectively. Then, the weight matrix of C2(k) follows ( 1 2)-( 14)

From (16), the output of the DFM with inputs ('(4 and t 2 (k ) is given by

y = { ~ ( k l k - I)-' + H ~ R ( ~ ) - ' H J - '

x IP(klk - 1)-'tl ( k ) + HTR(k)- 'Ht2(k) ) (20)

From the matrix inversion lemma [ 181, it can be shown that

y = {P(klk - 1)-' + HTR(k)-'H)-'P(klk - 1)-'tl(k)

+ {P(klk - 1)-' + HTR(k)-'H)-'HTR(k)-'Ht2(k) = t1(k)-P(klk- l)H*{HP(kIk- 1)HT + R ( k ) } - ' H t , ( k )

= { I - G'(k)HJ<I(k) + G'(k)HS,(k) (21)

G'(k) = P(klk - l)HT{HP(k(k - 1 ) H T + R(k))-' (22)

+ ~ ( k l k - I ) H ~ { H P ( R I ~ - + ~ ( k ) } - I ~ 5 ~ ( k )

where

Substituting i(1clk - 1) and HTz(k) in t , ( k ) and t2(k) , respectively, gives

(23) Equation (23) is the same as the measurement update equation of the Kalman filter and G'(k) is the same as the Kalman gain matrix. Therefore, it has been proved that two

y(k) = ;(klk - 1) + G'(k){z(k) - H;(klk - 1))

are identical. 0

Using theorem 1, the error monitoring and recovery technique can be applied to the detection of an error in the input data of a Kalman filter. Therefore, the incon- sistency between the predicted state and the measurement can be detected by checking whether there is a common intersection between the uncertainty ellipsoids of the predicted state and the measurement.

The detection of an inconsistency between the predicted state and measurement in the IMM algorithm can be performed through the test

If the inconsistency is detected in the DFM of a subfilter, it means that the predicted state or the measurement has an error.

One cycle of the IMM filter with three models is repre- sented in Fig. 1 as the net hierarchy of the perception net. DFMs 1-3 correspond to Kalman measurement updates at the subfilters and DFM 4 corresponds to the combination of the models' conditional estimates. When DFMs 1-3 detect

I I I

Fig. 1 models

Perception net representation of IMMJilter with three

IEE Proc-Radar Sonar Navig., Vol. 150, No. 2. April 2003 60

inconsistencies between their inputs, DFMs 1-3 are marked as 'A' to indicate that they are 'possibly-in-error'.

For the isolation of the error source, it is necessary to know which input data have an error. When the incon- sistencies are detected in every DFM, it can be considered that the predicted states in all models have errors or that the measurement has an error. Each predicted state is depen- dent on the previous model probabilities, and is calculated by mixing the estimate. Therefore, although the target manoeuvre occurs, the predicted state of the constant acceleration model is not immediately reflected in the estimation of the track. For this reason, inconsistencies are detected in the fusion module for some interval after the target manoeuvre occurs. Finally, if there is not a persistent bias in the measurement, the inconsistencies of the DFMs arise from the error in the predicted states. In addition, it can be considered that the error included in the predicted states comes from the manoeuvre of the target. All the predicted states are classified as 'likely-in-error' and are marked as ' x '. Also, the measurement is classified as 'unlikely-in-error' and is marked as '0'.

The classifications of the input data are propagated through the net. Once the input data are identified through checking the inconsistency of the DFM, the sensor, which supplies the identified input data, can also decide whether it has some faults or not. In other words, if the sensor provides the input data which are classified as likely-in- error, then the sensor is determined as a defective sensor. Therefore, if the inconsistencies are simultaneously detected in DFMs 1-3 of Fig. 1, the logical sensors, which have the target model and generate the predicted states, are classified as defective sensors. The physical sensor which acquires the measurement is classified as an indefectible sensor. When the following equation is satis- fied for the predicted states, we can consider the logical sensors as defective sensors:

where U ( . } is a unit step function and g is the length of a fixed-size window.

If the logical sensors turn out to be defective sensors, it is desirable to put a weight on the measurement data supplied by the indefectible sensor, i.e. the physical sensor, for more favourable filtering. However, it is well known that the conventional IMM filter is superior in terms of the estimation error and computational burden. In order to keep the advantages of the IMM filter, and to restrain the additive computational burden if possible, the fundamental structure of the IMM filter is not changed and the estimate, I(klk), is renewed in the modified structure. In addition, based on the geometric data fusion concept, combining the estimate, I(klk), with the measurement, ~ ( k ) , means that the estimate approaches the measurement. Therefore, in order to put a weight on the measurement when the logical sensors turn out to be defective, the estimate is fused with the measurement, and the associated weight matrices are the inverse matrices of the covariance matrices. Since the data fusion is the same as the Kalman measurement update from theorem 1, it is carried out as follows:

?(klk) = i ( k l k ) + G"(k){z(k) - Hi(kIk)} (27) P ( k l k ) = {I - GO(k)H)P(klk) (28)

GU(k) = P(k(k)HT{HP(klk)HT + R(k))-' (29)

where the superscript a denotes the estimation value of the time at which the logical sensors are identified to be defective.

Note that the correlation between the estimate and the measurement is ignored in (27)-(29) for practical reasons. In particular, the correlation between the two sets of data can be obtained only by excessive calculations and the resulting fusion strategy would be very much complicated if the measurement was weighted. The simulations shown here appear to indicate that omitting the correlation does not adversely affect the quality of the fused result.

5 IMMPDA filter with error monitoring and recovery technique

There are two problems in employing the error monitoring and recovery technique in the IMMPDA filter. The first problem is that it is difficult to detect the inconsistencies between the measurements with the clutter and the predicted states in the error monitoring method. The second is that it is difficult to recover the error caused by the possibility that the measurement may not be unique in a clutter environment. In order to settle these problems, it is necessary to extend the method of error monitoring in a clear environment and reprocess the measurement for recovery.

The IMMPDA filter is driven by the raw data of mk + N, which consists of mk validated measurements and N predicted states. By the measurement update, m k . N states are estimated in the process of filtering. The estimate 2; (klk) is obtained by fusing the predicted state 12-'(klk - 1) with the ith measurement zi(k). Then the weighted distance for detecting the inconsistency between the predicted state and the measurement is defined as

The inconsistency between the predicted $ate and measurement can then be investigated through d, , (k) 5 1 in the same way as (24). Since validated measurements are not constant, the extension of (26) can be represented by the following relation:

If inconsistencies are simultaneously detected in every DFM of the subfilters and are continuously detected in the sampling interval g, (3 1) holds.

Once (31) is satisfied, it means that all the predicted states include the error and that there is not a model in the IMMPDA filter that can effectively describe the target motion. Therefore it is desirable to include a weight on the measurement for more favourable filtering. However, since the measurement is not unique in a clutter environ- ment, the measurement cannot be fused with the estimate of the IMMPDA filter in the same way as (27)-(29). Thus, it is necessary to make a unique measurement to put a weight on the measurement, and this can be calculated by employing the minimum mean-square estimate.

The minimum mean-square estimate of the measurement is given by

IEE Proc.-Radar Sonar Nuvig., Vol. 150, No. 2, April 2003 61

where p:(k) is the probability that the measurement zi(k) is from the target for the model event M,(k). The covariance matrix associated with the measurement estimation is represented in the following:

Y(klk) E E{[z(k) - i(klk)][z(k) - i (k lk )]Tlzk} N

t=l = c P,(WY'(klk) + [i'(klk)

- i(klk)J[i'(klk) - i(klk)lT) (33)

Y'(klk) in (33) can be rewritten as * l h

Y'(klk) = C E { [ z ( k ) - i ' (klk)][z(k) - i ' (klk)JT i=O

I ~ i ( k > , ~ t ( k ) , zk Ip{Qi(k> IM/(~), zk I

+ i'(kl k)i'(klky I Qk), Mt(k), zk IB@)

= 2 E(z(k)z(k)T - i'(klk)~(k)~ - ~(k)i'(klk)~ i=O

= Y : + Y ; + ( Y y + Y: (34)

The first term of the right-hand side in (34) is given by

Y'1 = 2 E{z(k)z(k)Tle;(k), M/(k) , Z"p:(k) i=O nik

i=O = C { z i ( k k ( k ) T + R:(k))Pj(k) (35)

where

~ : ( k ) = EI[z(~) - z,(k)i[z(k) - ~ ~ ( k ) i ~ i ~ , ( k ) , M,(k), zk) = R(k) for i = 1 , 2 , . . . , ink (36)

Rh(k) is the covariance matrix associated with the predicted measurement. Also,

k

P%k) = 1 - CBKk) (37) i= 1

and substituting (36) and (37) in (35) yields

r: = 2 B:(k)z;(k)zi(k)T + ( 1 - Bb(k)JR(k) i=O

+ Bb(k)S'(klk - 1) (3 8)

where S'(klk - 1) is the predicted measurement covariance matrix conditioned on the M'(k). The second term of the right-hand side in (34) is given by

U: = - ~ E l i ( k l k ) ~ ( k ) T I O i ( k ) , M,(k), ZklB:(k> i=O

= - i (k I k)i'(k I k ) (39)

The third term of the right-hand side in (34) is the transpose matrix of f i ( k ) , and the last term is given by

m k

Y: = E{i(klk)i(klk)Tloi(k), M/(k ) , zk}p;(k) i=O

= i(klk)i(klk)T (40)

Substituting (38), (39) and (40) in (34) yields

Y'(klk) = 3 B:(k)zi(k)zi(k)T + Bh(k)S'(klk - 1) i=O

+ { 1 - Pb(k)}R(k) - i(klk)i'(klk)T

- i'(klk)i(klk)T + i(klk)i(klk)T (41)

Y(klk) can be obtained by substituting (41) in (33).

62

Since the unique measurement is extracted, the measure- ment can be combined with the estimation of the IMMPDA filter. Finally, substituting i ( k ) in z(k) of (27), and Y(k) in R(k) of (29), yields the final estimate.

If an inconsistency between the predicted state and the measurement is detected, it is decided provisionally that the input data has an error. The continuously detected error in a fixed-size window is caused by the predicted state. Although fundamentally the error comes from the manoeuvring of the target, the detected error is not judged to come from the manoeuvring of the target. It is considered that the error is caused by the unmodelled dynamics for the motion change. This approach is more general for the actual system.

In spite of a mistake in manoeuvre decision, the above- mentioned method hardly has an effect on the tracking performance. The proposed method, however, could improve the tracking performance, especially when a manoeuvre starts or stops. In conclusion, the IMMPDA filter employing the proposed method takes advantage of the IMMPDA filter and it also reduces the maximum value of the estimation error.

6 Simulations and results

The performance of the proposed method is compared with the standard IMM and IMMPDA estimator using Monte Carlo simulations. A target that turns in two-dimensional space is considered [ 5 ] .

For the simulations, the sampling time T= 1 s, and the measurement noise covariance R I 1 = R22 = 144 m2 and R I 2 = R Z 1 = 5 m2 are used. The conventional IMM and IMMPDA filters considered have three models [8]. The first model, M I , is a second-order kinematic model with white noise acceleration. Both the second model, M2, and the third model, M 3 , are third-order kinematic models with white noise acceleration increments with different variances. M I , M2 and M3 have process noises with a standard deviation of 2 m/s2, 10 m/s2, and 4 m/s2, respec- tively. The assumed model switching probabilities are shown in Table 1.

The value of the effective window length is taken to be 2. The detection probability is assumed as 0.98. The size of the validation gate is 6.

The target is on a constant course and speed until 20 s, it starts to manoeuvre at 20 s and stops manoeuvring at 40 s. Four manoeuvre-input scenarios are considered for the performance analysis. When U , and uV indicate the manoeuvre input the x direction, and the y direction in two-dimensional space, respectively, each scenario of manoeuvre input acceleration is made up of a manoeuvre level as follows:

Simulation 1 : U , = -40 d s 2 , uY = 40 d s 2 (without clutter)

Simulation 2 : U , = -60 d s 2 , u ~ , = 60 d s 2 (without clutter)

Simulation 3 : U, = -25 d s 2 , uY = 25 d s 2 (with clutter)

Simulation 4 : U , = -40 d s 2 , uY = 40 d s 2 (with clutter)

Table 1: Model switching probabilities

MI M2 M3

MI 0.85 0.15 0 M2 0.33 0.34 0.33 M3 0 0.15 0.85

IEE Proc.-Radar Sonar Navig., Vol. 150. No. 2, April 2003

2 -

6- 1 - 0 7- a - E

0 -

-1

Fig. 2 Trajectories for simulations I , 2, 3 and 4

I

20

I8l

16

- proposed filter , I

I , I , I , I . , I

35 - ---.. IMM filter - proposed filter

30 - - E 3

where simulations 1 and 2 are performed in the clear environment, and simulations 3 and 4 are performed in the clutter environment. The number of clutter points is given by a Poisson random variable with a clutter density of 15 per km2 for simulations 3 and 4. Fig. 2 shows the target trajectories for simulations 1, 2, 3 and 4. We assume that the sensor is located at the origin of coordinate and the target starts at the point marked as ' x '.

One hundred Monte Carlo simulations are performed. Figs. 3-6 give the position RMS (root mean square) errors for both the conventional IMM and IMMPDA filters and the proposed filter. Solid lines show the proposed filter behaviour and dots show the conventional filter behaviour.

In simulation 1, shown in Fig. 3, the RMS error in the proposed filter is nearly equal to those in the conventional IMM filter before the manoeuvre starts. Since the target manoeuvres, the proposed filter detects a target manoeuvre and the RMS error decreases more quickly than those of the conventional IMM filter. The RMS errors when stop- ping manoeuvring are similar to those at the beginning of the manoeuvring. There is only a slight difference between the two filters in the case of RMS errors of velocity.

In simulation 2, a target that turns rapidly is considered. In this case, the results show that the peak errors of the proposed filter are remarkably reduced while the target manoeuvres. Comparing Fig. 4 with Fig. 3, it is found that the peak error of the proposed filter is bounded in the neighbourhood of the standard deviation of the measure- ment noise.

l 8

E E @ 25- e &

e ._ s 20- ._

._ ._ U1

a 2 15- 10-

I

0 10 20 30 40 50 60

time, s

Fig. 4 tional IMMjilter (simulation 2)

Position RMS errors of the proposed filter and conven-

In simulations 3 and 4, the manoeuvring target is considered in a clutter environment. Figs. 5 and 6 show the RMS errors of the position in the IMMPDA filter and the proposed filter. It is clear from the simulation results that the IMMPDA with the error monitoring and recovery technique provides superior tracking performance, compared with the conventional IMMPDA filter. It is confirmed that the derived method in a clear environment can be efficiently extended to a clutter environment.

From the results of simulations 1, 2, 3 and 4, it is found that the higher the manoeuvre input, the higher the peak error reduction. In addition, if the manoeuvre input is decreased or the covariance of the process noise is increased, the effectiveness of the proposed filter is reduced. Finally, we conclude that the proposed filter can be more adaptable to a highly manoeuvring input level and that the proposed filter is quite simple without the degrada- tion of the conventional IMMPDA filter. However, the renew procedures of the proposed filer inevitably increase the computational burden. The increased FLOPS (floating- point operations per second) for IMM and IMMPDA filters are 5.16% and 7.17% during a simulation time of 60 s, respectively.

-.--- IMM filter I - proposed filter

I I

0 10 20 30 40 50 60 10 20 30 40 50 60

time, s time, s

6'

Fig. 3 tional IMMjilter (simulation I )

IEE Proc.-Radar Sonar Navig.. Vol. 150, No. 2, April 2003

Position RMS errors of the proposed filter and conven- Fig. 5 tional IMMPDA jilter (simulation 3)

Position RMS errors of the proposed jilter and conven-

63

40

351 E 30- Yf e b 25-

0

a

c c 8 20-

cn 15-

10

- - - - - IMM filter

-

Automatic Control Research Center), Seoul National University.

5; 10 20 30 40 50 60

time, s

Fig. 6 tional IMMPDA filter (simulation 4)

Position RMS errors of the proposed filter and conven-

7 Conclusions

In this paper, the modified IMM and IMMPDA schemes are derived by adding the error monitoring and recovery technique of the perception net to conventional IMM and IMMPDA filters, respectively. Since the error is caused by the mode change of the target, the improved performance of the proposed filter can be obtained by putting a weight on the measurement which is not disturbed by the assumed model of the conventional filter. Its effectiveness is vali- dated through simulations. The proposed method shows a better performance for a rapidly manoeuvring target as well as reducing the maximum values of the estimation errors, especially when a manoeuvre starts and finishes.

8 Acknowledgments

The authors gratefully acknowledge financial support from the Agency for Defense Development and from ACRC (the

9 References

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