Third International Symposium on Intelligent Information Technology and Security Informatics
A Cloud Model Inference System Based Alpha-Beta Filter for Tracking of Maneuvering Target
Huang Jianjun, Member, IEEE and Zhong Jiali
Li Pengfei
ATR Key Lab Shenzhen University Shenzhen, 518060, China E-mail:
[email protected],
[email protected]
Air Defense Forces Command Academy Zhengzhou , 450052, China E-mail:
[email protected] membership function, triangle membership function, and so on. Once the membership function is established, the output is unique through fuzzy logic. However, the establishment of membership function is under the influence of subjective consciousness. We often establish the membership function depending on the transcendental knowledge of specialists or statistical method. When the membership function is not well established, the performance of a fuzzy-gain filter will degrade. To deal with the difficulty in establishing appropriate membership functions, we can use the statistical method to describe the membership degree. The cloud model presented by Li Deyi is such a method which can combine fuzziness and randomness together, it mainly reflects the uncertainty of things in the universe or concept in the human knowledge: fuzziness and randomness. Cloud model is the transformation bridge from quantitative concept to qualitative concept and from qualitative concept to quantitative concept. In this paper, an adaptive α-β filter based on cloud model inference is proposed for maneuvering target tracking. The experiment results show that the algorithm performs out existence in mean squared error. The rest of the paper is organized as follows. In Section Ⅱ, the concept of cloud model and relative knowledge are introduced. Section Ⅲ presents the α-β filter based on cloud model. The tracking performance of the proposed method is compared with the methods in [6] and [7] in Section Ⅴ. Section Ⅵ gives the conclusion.
Abstract—An adaptive alpha-beta filter based on cloud model inference is presented for maneuvering target tracking. The proposed tracker incorporates cloud model in a conventional alpha-beta filter by using the rule bank based on cloud model, which utilizes the residue error and the change of residue error in the last prediction to determine the values of alpha and beta, then track the maneuverable target accurately. The experiment results show that the algorithm is satisfactory and effective. Keywords-maneuvering inference; alpha-beta filter
I.
target
tracking;
cloud
mode
INTRODUCT
Significant research efforts have been devoted to the problem of maneuvering target tracking. These methods are by and large model based. They are usually developed under various assumptions about statistical models of the process noise and the measurement noise, and about target dynamics. For example, adaptive algorithms using the “current” statistical model, state or measurement equations of maneuvering target are proposed for maneuvering target tracking in [1-3]. Their tracking performances depend on whether the targets’ states are described accurately. When the target model is consistent with real state, the target can be tracked accurately. However, the movement of target is uncertain, unknown target acceleration appears as an extensive process noise in the target model and the original process noise variance can not cover it, which leads to mismatch between the target models and real patterns. In such case, it may fail to track the targets without divergence. To cope with the problems of modeling, [4] and [5] introduced artificial intelligence in maneuvering target tracking. The algorithm decreases the accuracy requirement of the target models. It does not depend on how accurately the models describe the targets. But it requires large computation. To overcome the problems above, [6] and [7] proposed a fuzzy-gain filter based α-β filter. They use the residue error, and the change of residue error or the change of course angle in the last prediction to determine the values of α and β, and complete the tracking of the maneuvering targets accurately. In fuzzy logic, the membership function represents the certainty degree of a qualitative concept. At present, there are several types of membership functions, such as ladder 978-0-7695-4020-7/10 $26.00 © 2010 IEEE DOI 10.1109/IITSI.2010.119
II.
INTRODUCTION TO THE CLOUD MODEL
A. Cloud and Cloud Drops Assume that U is a quantitative numerical universe of discourse and C is a qualitative concept in U. If x ∈ U is a random realization of concept C, and u ( x) ∈ [0,1] , standing for certainty degree to which x belongs to C, is a random variable with stable tendency:
u : U → [0,1], ∀x ∈ U , x → u ( x) The distribution of x in the universe of discourse U is called a cloud and expressed by C(X). Each x is called a cloud drop. The cloud model has three characters, namely, the Expected value ( Ex ), the Entropy ( En ) and the Hyper-
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Step1: Produce a random value which obeys the normal distribution with a mean En1 and a standard deviation He1
Entropy ( He ), which well integrates the fuzziness and randomness of qualitative concept in a unified way. In the discourse universe, Ex is the position corresponding to the center of the cloud gravity, whose elements are fully compatible with the qualitative linguistic concept; En is a measure of the concept coverage, larger entropy means a more macroscopic concept. En is determined by both randomness and fuzziness of concept; He is a measure of the uncertainty of the entropy En , it is the entropy of entropy. It is determined by randomness and fuzziness of the entropy. At present, several types of cloud models have been developed, such as the ladder cloud, the triangle cloud, and so on. Among all kinds of cloud models, the normal cloud model is the most conventional and is used in this paper.
En1' = NORM ( En1 , He12 )
Then, calculate u1 = exp(− (a1 − Ex1 ) 2 2 En1' 2 ) . Step2: Produce a random value which obeys the normal distribution with a mean En2 and a standard deviation He2 En2' = NORM ( En2 , He22 )
Then, calculate u2 = exp(− (a 2 − Ex2 )2 2 En2' 2 ) . Step3: Calculate certainty degree u through soft “AND”. Step4: Produce a random value which obeys the normal distribution with a mean Eny and a standard deviation Hey En 'y = NORM ( En y , He 2y )
B. Inference based on Cloud Model In cloud model inference, rules are crucial and are expressed as IF-THEN pattern. There are several types of cloud model inference rules [8]. Fig.1 shows a typical rule of type double condition and single conclusion. It is expressed as follow: IF A1, A2 then B where A1 is a qualitative concept ( Ex1 , En1 , He1 ) , A2 is a qualitative concept ( Ex2 , En2 , He2 ) and B is a qualitative concept ( Ex y , En y , He y ) .
Step5: Calculate b = Ex y ± −2 ln(u ) En'y . OUTPUT (b, u ) ; END III.
The conventional α-β filter can be defined by the following equations:
A1
A2
∧
∧
∧
∧
β T
∧
∧
(1)
∧
[ Z ( k ) − X ( k / k − 1)] ∧
X(k / k − 1) = X(k − 1) + T V (k − 1) ∧
V (k / k − 1) = V (k − 1)
(2) (3) (4)
∧
where k represents the number of scan, X ( k ) is the ∧
smoothed target position, X ( k / k − 1) is the target’s ∧
predicted position, Z ( k ) is the target’s observed position, ∧
∧
V ( k ) is the smoothed target velocity. V ( k / k − 1) is the ∧
target’s predicted velocity, V ( k ) is the target’s observed velocity. α, β are two fixed-coefficient filter parameters. T is the radar scan time or the sampling interval. A.
Update the values of α and β From [6], we see that the residue error and the change of residue error in the last prediction are relative to the maneuver of target. The change of residue error can represent the degree of maneuver. So the residue error and the change of residue error are used to determine the values of α and β through cloud inference. The residue error and the change of residue error are used as the inputs. The value of α is the output and the value of β is calculated by β = 2 − α − 2 1 − α . The algorithm of α-β filter based on cloud model inference is shown in Fig.2. The details are discussed in Section B and Section C. Finally, the new α-β filter is used to track targets.
u1 CGx1 CGy
∧
V ( k ) = V ( k / k − 1) +
( Ex y , En y , Hey )
u
∧
X(k ) = X(k / k − 1) + α [ Z (k ) − X(k / k − 1)]
The cloud generator CGx1 and cloud generator CGx 2 are called antecedent cloud generators; the cloud generator CG y is called consequent cloud generator. The inference process of a double condition and single conclusion rule is referred to as rule generator shown in details as follows: Input: a qualitative concept A1 ( Ex1 , En 1 , He1 ) in the universe of discourse U1, a qualitative concept A2 ( Ex2 , En 2 , He2 ) in the universe of discourse U2, a qualitative concept B ( Ex y , En y , He y ) in the universe of discourse U3, a certain value a1 in U1 and a certain value a2 in U2. Output: a cloud drop in U3 drop (b, u ) . BEGIN ( Ex1 , En1 , He1 )
DESIGN OF α-β FILTER BASED ON CLOUD MODEL
drop( y, u)
u2 CGx2
( Ex2 , En2 , He2 )
Figure 1. Double condition and singlec onclusion rule generator
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1 0.9 0.8 0.7
α
0.6
E ΔE
β
0.5 0.4 0.3 0.2 0.1 0
Figure 2. α-β filter based on cloud model inference
0
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Figure 3. The antecedent cloud model
B. Syetem variable Compute the normalized residue error along the x and y coordinate at the kth scan [6]
ΔE (k ) =
⎧ z _ y (k ) − y _ p(k ) ⎪ z _ y (k ) − z _ y (k − 1) , if z _ y (k ) − y _ p( k ) < z _ y ( k ) − z _ y ( k − 1) ⎪ ⎪ z _ y ( k ) − y _ p (k ) Ey (k ) = ⎨ , if z _ y ( k ) − y _ p( k ) > z _ y ( k ) − z _ y ( k − 1) ⎪ z _ y (k ) − z _ y (k − 1) ⎪0, ifz _ y (k ) − y _ p( k ) = z _ y ( k ) − z _ y ( k − 1) = 0 ⎪
The qualitative language values of ”zero” (ZE), ”small” (SP), ”middle” (MP), ”large” (LP) are used to describe the residue error and the change of residue error in the last prediction and transform them into normal clouds shown as follow: C ZE = C(0,0.7,0.01) , x ∈ [0, 0.3]
⎩
(6) where x _ p(k ) and y _ p(k ) are the target’s predicted position along the x and y coordinate at the kth scan respectively; z _ x(k ) and z _ y (k ) are the target’s observed position along the x and y coordinate at kth scan respectively. Based on Ex (k ) and E y (k ) , ΔE x ( k ) and ΔE y (k ) can be calculated as follows:
and
⎧ E y (k ) − E y (k − 1) , if E y (k ) − E y (k − 1) < E y (k − 1) ⎪ E y (k − 1) ⎪ ⎪ E (k ) − E (k − 1) ⎪ y y , if E y (k ) − E y (k − 1) > E y (k − 1) ΔE y (k ) = ⎨ ⎪ E y (k ) − E y (k − 1) ⎪ ⎪0, ifE y (k ) − E y (k − 1) = E y (k − 1) = 0 ⎪⎩
As a result, we calculate E (k ) =
E (k ) and ΔE (k ) as
( Ex2 (k ) + E y2 (k )) 2
(10)
C. Cloud Inference According to cloud inference process, we can update α and β through the steps as follow: 1) Establish antecedent cloud models and consequent cloud models.
(5)
⎧ E x (k ) − Ex (k − 1) , if Ex (k ) − E x (k − 1) < Ex (k − 1) ⎪ Ex (k − 1) ⎪ ⎪ E (k ) − E x (k − 1) ΔE x ( k ) = ⎨ x , if Ex (k ) − Ex (k − 1) > Ex (k − 1) ⎪ Ex (k ) − E x (k − 1) ⎪0, ifE (k ) − E (k − 1) = E (k − 1) = 0 x x x ⎪ ⎪⎩
2
Hence, the range of values that E (k ) and ΔE (k ) each may take is in the interval [0, 1].
⎧ z _ x(k ) − x _ p(k ) ⎪ z _ x(k ) − z _ x(k − 1) , if z _ x(k ) − x _ p(k ) < z _ x(k ) − z _ x(k − 1) ⎪ ⎪ z _ x(k ) − x _ p(k ) Ex (k ) = ⎨ , if z _ x(k ) − x _ p(k ) > z _ x(k ) − z _ x(k − 1) ⎪ z _ x(k ) − z _ x(k − 1) ⎪0, ifz _ x(k ) − x _ p(k ) = z _ x(k ) − z _ x(k − 1) = 0 ⎪ ⎩
and
(ΔEx2 (k ) + ΔE y2 (k ))
C SP = C(0.35,0.7,0.01) C M P = C(0.65,0.7,0.01) C LP = C(1,0.7,0.01), x ∈ [0.7,1] where the first and the last one are half normal clouds. These normal clouds for linguistic quantity are shown in Fig .3. To make sure the value of α is between 0.3 and 0.8 (an empirical range), the qualitative language values of ”zero” (ZE), ”small” (SP), ”middle” (MP), ”large” (LP), ”very large” (VP), ”extremely large” (EP) are used to describe the value of α and transform them into normal cloud shown as follows: C ZE = C(0,0.7,0.01), x ∈ [0, 0.3]
(7)
C SP = C(0.3,0.7,0.01) C MP = C(0.5,0.7,0.01)
(8)
C LP = C(0.6,0.7,0.01) C VP = C(0.8,0.7,0.01) C EP = C(1,0.7,0.01), x ∈ [0.7,1] Here, the first and the last cloud are half normal clouds. These clouds are shown in Fig.4. 2) Establish rule bank based on cloud models
follow: (9)
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TABLEⅠ.
1
RULE BANK BASED ON CLOUD MODEL
0.9 0.8
ΔE
0.7
α
0.6 0.5 0.4 0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 4. The consequent cloud model
Step4: If there have n (n>1) u (i, j ) are positive numbers, the consequent cloud generator will produce n values. We can obtain the value of α through averaging. Step5: Calculate β = 2 − α − 2 1 − α .
From filter function (1), α is used to adjust the proportions of observed position and prediction position in the smoothed position of target. When α=1, the smoothed position is equal to the observed position. When α=0, the smoothed position is equal to the prediction position. Adaptive filter updates the values of filter parameters according to the performance feedback of target tracking. In this paper, the residue error and the change of residue error are used to represent the maneuvering degree, so we use them to establish feedback mechanism to obtain good target tracking performance. The rule bank based on cloud models is established as Tab.Ⅰ, which reflects the following facts: • The value of α is between [0.3, 0.8] with high probability.
3)
•
The value of α increases with the increase of the residue error and change of residue error.
•
IF the residue error =”zero” or ”small” AND the change of residue error =”zero” or ”small” THEN α =”small”.
•
IF the residue error =”middle” or ”large” AND the change of residue error =”middle” or ”large” THEN α=”extremely large”.
IV. EXPERRIMENT RESULTS To show the effectiveness of the proposed algorithm, two scenarios for tracking a maneuvering target are examined. For comparison purposes, we also simulate the methods presented in [6] and [7] separately to track the target. The first scenario uses simulation data shown in Fig.5. The second scenario uses radar measurement data shown in Fig.6. For the first scenario, the results of mean squared estimation error for each algorithm are shown in Fig.7. The tracking results of the radar measured target are shown in Fig.8, and the results of mean squared estimation error for each algorithm are shown in Fig.9. From the Fig.7-9, we can see that the performance of the proposed algorithm is obviously better than the other two methods in mean squared error. Adaptive α-β filter based on cloud model inference system can adjust the values of α and β properly and tracking accurately when the target maneuvers.
Update the values of α and β through double condition single conclusion rule generator shown in Fig.2.
13
The value of α is inferred with the rule bank by the double condition and single conclusion rule generator as follows: Step1: Given a certain residue error and a change of residue error in the last prediction, the two antecedent cloud generators produce two certainty degrees: u1 (i )(i ≤ 4) , u2 ( j )( j ≤ 4) . Step2: Calculate the certainty degree u (i, j )(i ≤ 4, j ≤ 4) (11) u (i, j ) = u1 (i ) AND u2 ( j ) Step3: If only one u (i, j ) is not zero, the consequent cloud generator is enabled by the rule which is located at (i, j ) in TableⅠ, to produce only one consequent value that is the value of α.
12 11 10
y(km)
9 8 7 6 5 4 3
0
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20
30
40
50
x(km)
Figure 5. Simulation target data
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V.
CONCLUSION
1.4
An adaptive α-β filter based on cloud model inference system for maneuvering target tracking has been proposed. To solve the hard problems in target tracking include the uncertain target models and complex of calculation, this paper incorporates cloud model in conventional α-β filter, utilize the residue error and change of residue error to adjust the values of α and β through rule generator based on cloud model. Simulation results show that this algorithm has a good tracking performance.
RMSE of cloud RMSE of fuzzy in [6] RMSE of fuzzy in [7]
1.2
rms errors(km)
1
0.8
0.6
0.4
ACKNOWLEDGMENT This work is supported by Weaponry Equipment Preresearch Foundation of China (Grant No. XXXC80).
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REFERENCES
[3]
[4]
[5]
[6]
[7]
[8]
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Figure 7. Mean squared errors of scenario1
Qing Fang and Chuan Li, “Design of A Modified Algorithm for Maneuvering Targets Tracking,” Interational Conference on Radar, pp.1-5, Oct. 2006. Junliang Chen, Xiaodong Hou, Zheng Qin and Ronghua Guo, “A Novel Adaptive Estimator for Maneuvering Target Tracking,” in Proceedings of the 4th World Congress on Intelligent Control and Automation, vol.4, pp. 2677-2681, June 2002. Fei Chen, Zhongliang Jing and Jianxun Li, “Adaptive Maneuvering Targets Tracking for Multiple Airborne Mobile Platforms and Sensors,“ in International Conference on Mechatronics and Automation, pp. 3756-3760, Aug. 2007. Jianfang Shi, Hongbiao Tang and Qingshuang Zhu, “Maneuvering Target Tracking Based on Wavelet Neural Network,” Fourth International Conference on Natural Computation, vol.2, pp. 599604, Oct. 2008. Sengupta,D. and R.A.Iltis, “Neural solution to the Multi-target tracking data association problem,” IEEE Trans. on Aerospace and Electronic Systems, vol. 25, pp.96-108, Jan. 1989. Keith C C, Chan , Vika Lee and Henry Leung, “Radar tracking for air surveillance in a stressful environment using a fuzzy-gain filter,” IEEE Trans. on Fuzzy Systems , vol.5, pp.80-289, Jan. 1997, LI Pengfei ， YU Jianping and LI Liangqun, “Maneuvering target tracking based on fuzzy adaptiveα- βfilter,” System Engineering and Electronics(Chineese Ed), vol.30, Nov. 2008 Deyi Li 、 Yi Du ， ”Artificial Intelligence with Uncertainty”,National Defense Industry Press(Chineese Ed), 2005.
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Figure 8. Tracking results of scenario2
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Figure 9. Mean squared errors of scenario 2
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Figure 6. Radar measurement data
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