Intelligent Tracking Algorithm Using Fuzzy-Based Adaptive βα − Filter for Maneuvering Target

Peng-fei Li1, 2 Jing-xiong Huang1, 2 Wei-xin Xie1 Liang-qun Li1

School of Information Engineering, Shenzhen University Shenzhen 518060, P. R. China

Air Defense Forces Command Academy Zhengzhou 450052, P.R. ChinaE-mail: lpf_cool@sina.com.cn

Abstract

In this paper, a new adaptive βα − filter is proposed for maneuvering target tracking based on fuzzy logic. Firstly, we analyzed that the change of course angle and the measurement residual could reflect the maneuverability of the target real time, then the two parameters are used as the input variables of the fuzzy logic, and the fuzzy logic rules of the maneuverability are designed to calculate the parameters of the βα − filter. The experiment results show a higher performance in maneuvering targets tracking problems and the method could be easily applied to real project.

1. Introduction

Maneuvering target tracking has always been the highlight in the target tracking field. A lot of scholars have been studied on the maneuvering target tracking in various environments, and many effective tracking algorithms for maneuvering target have been proposed, such as the interacting multiple model(IMM) method[1,2], the generalized pseudo-Bayesian(GPB)method[3], the particle filtering method[4]. However, their techniques require the computational efforts and are not fit for engineering application. In fact, the βα − trackers are still quite well received in the tracking community [5]. The βα − filter with the proper

βα − parameters can provide an optimal solution of the Kalman filtering process for the stationary target tracking problem. However, when the target maneuvers, the quality of the position and velocity estimates provided by the βα − filter can be degraded significantly. In order to track the maneuver target accurately, the magnitude of αand β must be adjusted adaptively.

Recently, fuzzy logic is extensively applied to maneuvering target tracking with intelligent adaptation capabilities [6,7]. Chan [6] presents a method using βα − filter and fuzzy logic in order to track a maneuvering target, the inputs of its fuzzy

system are the prediction error and the change of error in the last prediction, but the value of α and β which are gotten by this way could not change with the maneuver of the target real time. Bahari [7] presents a method which uses the heading change and the measurement residual as the inputs of the fuzzy system, but the heading change defined in the paper could not reflect the dynamics of the target, because it uses the measurements which are corrupted by Gaussian noises. To cope with these problems, this paper proposed a fuzzy adaptive

βα − filter to track the maneuvering target. Based on the change of course angle and the measurement residual, we define the fuzzy logic rules to determine the values of α and β for the next prediction. Any change in target dynamics will, therefore, result in changes in α and β .

2. The adaptive βα − filter based on fuzzy logic rules

2.1. The conventional βα − filter

The conventional βα − tracker is defined by the following equations:

( ) ( ) )1(11| −+−=− kTVkxkkx ssp (1)

( ) ( )11| −=− kVkkV sp(2)

( ) ( ) ( ) ( )[ ]1|1| −−+−= kkxkxkkxkx pmps α (3)

( ) ( ) ( ) ( )[ ]1|1| −−+−= kkxkxT

kkVkV pmpsβ (4)

where ( )1| −kkxp denotes the target’s predicted position at the kth scan, ( )1| −kkVp

the

target’s predicted velocity, ( )kxs the smoothed target position, ( )kV s the smoothed target velocity, ( )kxm the target’s observed position T is the sampling interval.

2.2. The analysis of target maneuver

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Radar output signal has no exact mathematical relation with target maneuver, but with no doubt there exist a complex nonlinear mapping between them. To map the input vector to target acceleration vector it is important to find the effective input elements. The measurement residual is the very important parameter to detect the target maneuver. And the change of course angle could also reflect the maneuver of the target. In order to track the maneuvering target accurately, we proposed a new method to detect the maneuver target and set βα −parameters with both the measurement residual and the change of course angle. 1. The change of course angle ( θΔ ):The change of course angle ( θΔ ) is defined as follows:

))2()2|1((1 −−−−= kxkkxangle spψ (5)

))1()2|((2 −−−= kxkkxangle spψ (6)

21 ψψθ −=Δ (7)

1ψ is the angle of the smoothed target position at the thk )2( − scan and the target’s predicted position at the thk )1( − scan, 2ψ is the angle of the smoothed target position at the thk )1( − scan and the target’s predicted position at the kth scan, these are called the course angle. It is shown in Figure 1.

X

Yprediction

point at (k-1)th scan

sm oothed point at (k-2 )th

scan

prediction poin t at kth

scan

1 2

The Track

sm oothedpoint at (k-1)th

scan

Figure 1. The definition of Figure 2. The change of course angle course angle

Figure 2 shows the phenomenon which reflects the change of θΔ with the target maneuver in tracking a maneuver target scene. In the scene, the target maneuvers variously at the th30 to th50 scans and th75 to th95 scans, and in the rest time, the target moves with non-maneuver. From Figure 2 we could see that when the target moves with non-maneuver, the course angle hold the line and the change of course angle is close to zero, and when the target maneuvers, θΔ increases rapidly. That is because when the target begins to maneuver, the measurement )(kxm will change greatly. From the equation (3) and (4) we know that the )(kxs and

)(kVs will also change greatly because of the change of )(kxm , that will make

)|1( kkx p + departure from the real target trajectory and at last lead to 2ψ and θΔ increase rapidly. As a result, we could use the characteristic of θΔ to

detect the maneuverability of the target, and improve the precision of estimation.2. The measurement residual ( ν ): The measurement residual is the very important parameter for maneuver detection. As the residual is the form of vector, it could not be used as the input variable of the fuzzy system, in this paper, the inner-product of the residual is used as the input variable of the fuzzy system:

1))-k|(kx(k)(x1))-k|(kx(k)(x pmT

pm −−=ν

2.3. The set of α and β based on fuzzy logic rules

We use the change of course angle and the measurement residual to obtain the dynamic information of the maneuvering target, and then use the fuzzy system to adjust the magnitude of α and β dynamically to track the maneuver target. The proposed method is illustrated in Figure 3.

Fuzzification Fuzzy rules Defuzzification

FilterMeasurement

Target tracking

ν

α

βα −

Figure 3. Configuration of the proposed algorithm

The target tracking based on fuzzy logic contains four steps: 1) Since different targets have different

maneuverability, the magnitude of θΔ and νmay vary from target to target. To design a general fuzzy tracker which can be used with different types of targets, we normalize the θΔand ν in form of θ ′Δ and ν ′ .

2) The crisp input data should be fuzzified. 3) The fuzzy logic rules should be designed to

generate the output fuzzy variables. 4) Defuzzify the output fuzzy variables into

non-fuzzy variables. Because the second and third step are the most

important and complex step, it should be discussed as follows: 1. Fuzzification of the input variables

The values are mapped into some fuzzy sets defined in the universe of discourse of θ ′Δ and ν ′ .These fuzzy sets are labeled in the linguistic terms of zero (ZE), small positive (SP), medium positive (MP), and large positive (LP). The specific membership function for each fuzzy set is defined by the trapezoidal functions shown in Figure 4. The intervals between the cores of the fuzzy sets (which are all of equal width) are made boundaries of them. Such choice of membership functions ensures that

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the chance of each input data point falling into each fuzzy set be roughly equal, and that the probability of each fuzzy rule being fired is more or less the same.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.2

0.4

0.6

0.8

1.0

ZE SP MP LP

Figure 4. Membership function for the fuzzy sets of θ′Δ and ν ′

The fuzzy sets for output variable α are labeled in the linguistic terms of ZE, SP, MP, LP, very large positive (VP), and extremely large positive (EP). The specific membership function for each fuzzy set is defined by the triangular functions shown in Figure5. Unlike that of θ ′Δ and ν ′ , the universe of discourse is divided up into six instead of four fuzzy sets, and the core of these fuzzy sets are not equally spaced. The central region is more finely divided into four different fuzzy sets, SP, MP, LP, and VP. This is because the region for optimum αis found more often in [ ]8.0,3.0 [6].

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.0

0.2

0.4

0.6

0.8

1.0 ZE SP MP VP

0.9

LP EP

Figure 5. Membership function for the fuzzy sets of α

2. Fuzzy logic rulesWhen designing the βα − tracker, the value of

α should be determined firstly according to the stability law of filter, and then the value of β will be determined with α . If the coefficients are too small, the filters have good noise reduction ratio but can be too unresponsive to dynamic targets. If the coefficients are too large, the filters have good tracking capabilities but can be too sensitive to noise. In order to track the maneuvering target intelligently, the algorithm should be a feedback process. The tracker will adaptively adjust its track parameters according to the track result, and improve the precision of estimation. In this paper, when the absolute of the change of course angle and measurement residual increase rapidly, it means the smoothed target position deviate from the true trajectory and precision of the estimation becomes worse. In order to track the target better, the

algorithm needs to augment the magnitude of α ,and decrease the influence of the prediction value to the result. We use the fuzzy logic method to adjust the magnitude of α , the finalized set of fuzzy associations (i.e. the entries of the Table 1) exhibit the following characteristics.

The magnitude of α should be increased along with 'θΔ and 'ν .Make sure the universe of α are taken to be [ ]8.0,3.0 with a high probability. When the 'θΔ and 'ν belong to ZE or SP, it means the target maneuvers smoothly. At this time, the tracker needs to make the magnitude of α small so as to reduce the influence of the noisy observation to the prediction. So the entry from the top left-hand corner and another two entries next to it are set to SP. When the 'θΔ and 'ν belong to MP or LP, it means the target do a high maneuver. In such cases, setting α to EP could greatly speed up the learning of the new target dynamics. When the 'θΔ and 'ν belong to the other fuzzy sets, it means the target do a gentle maneuver. In such cases, α should be set neither too large nor too small.

Table 1. Fuzzy associations for α

α θ ′Δ

'ν

With the input and output variables defined above, the fuzzy rules can be expressed as

C THEN B' ANDA IF ===Δ ανθThe output of α is evaluated through the

Max-Min compositional rule of inference technique and center of gravity (COG) defuzzification.

The magnitude of β can be evaluated by the equation (9)

ααβ −×−−= 122 (9)

3. Simulation results and analysis

In order to evaluate the performance of the proposed βα − tracker for the maneuver target, two target scenarios with different dynamics are performed. For comparison purpose, we also simulate Keith C.C. Chan’s method and the conventional one.

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3.1. The two scenarios

First scenario: In this simulation, the sampling time is sT 1= . The initial position of the target is given by )2500,4330( mm with an initial speed of )/5.7,/13( smsm and on a constant speed of 30 and speed until st 75= , then it starts to maneuver with acceleration value 2/1 smux −=

2/1 smu y −= until st 99= , and targets moves at a constant speed given by ( )smsm /13,/9 −− up to the end of this simulation at st 135= . In this scenario, the measurements are corrupted by zero-mean Gaussian noises, with the standard deviation m5=σ .The target trajectory is shown in Figure 6.

Figure 6. The trajectory of Figure 7. The trajectory of scenario 1 scenarios 2

Second scenario: In this scenario, we use the real data collected by low altitude detection radar. The details of the radar are given below:

scan rate )/( srev : 3, scan range )(km : 21, measuring range accuracy )(m : 20, measuring angle accuracy (deg) : 5.0 .

The radar is located at the position )0,0( kmkm and the initial position of the target is measured at

)819.2,008.0( kmkm . The radar collected 40 points, which are shown in Figure 7.

3.2 Tracking results analysis

Figure 8 shows the values of the parameter α in the first experiment scenario.

Figure 8. The magnitude of α

In Figure 8, it shows that the values of the parameter α could satisfy the tracking rule of the maneuvering target.

The root mean squared (RMS) position errors are shown in Figure 9 and Figure 10. It shows that the performance of the proposed algorithm is superior to

Chan’s algorithm and the conventional one.

Figure9. RMS position error Figure10. RMS position error of scenario 1 of scenario 2

4. Conclusions

In this paper, a new adaptive βα − filter algorithm based on fuzzy logic is proposed for maneuvering target. We analyze the relationship of the change of course angle and the measurement residual with the maneuver of the target. Then the two parameters are used as the input variables of the fuzzy logic, and designed the fuzzy logic rules by the maneuverability to calculate the parameters of the

βα − filter. The experiment results show a higher performance in maneuvering targets tracking problems than Chan’s method proposed in paper [6] and the conventional one. And it could be easily applied to project.

References

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[3] J. K. Tugnait, “Detection and estimation for abruptly changing systems”, Automatica, 1982, 18, (5), pp.607-615.

[4] M. F. Bugallo, Shanshan Xu, J. Miguez,, P. M. Djuric, “Maneuvering target tracking using cost reference particle filtering”, Acoustics, Speech, and Signal Processing, 2004. Proceedings. (ICASSP '04). IEEE International Conference on. Volume 3,17-21 May 2004 Page(s):iii - 968-71 vol.3

[5] Y, Bar_shalom and X. LI, Multitarget_Multisensor tracking: Principles and Techniques, Storrs, CT: YBS Publishing, 1995.

[6] Keith C.C. Chan, Vika Lee, and Henry Leung, “Radar tracking for air surveillance in a stressful environment using a Fuzzy-Gain filter”, IEEEtransaction on fuzzy systems, vol. 5, NO. 1, February 1997

[7] M. H. Bahari, A. Karsaz, H. Khaloozadeh, “High maneuver target tracking based on combined Kalman filter and fuzzy logic”, IEEE Information, Decision and Control, 2007. IDC '07 12-14 Feb. 2007,pp:59 – 64

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