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A split target detection and tracking algorithm for ballistic missile tracking during the re-entry phase
Defence Technology xxx (xxxx) xxx
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A split target detection and tracking algorithm for ballistic missile tracking during the re-entry phase Muhammad Asad a, Sumair Khan b, Ihsanullah a, Zahid Mehmood a, Yifang Shi d, Sufyan Ali Memon c, Uzair Khan a, * a
Department of Electrical and Computer Engineering, COMSATS University Islamabad, Abbottabad Campus, Pakistan Department of Computer Science, COMSATS University Islamabad, Abbottabad Campus, Pakistan Electrical Engineering Department, Indus University, Karachi, Pakistan d Department of Automation, Hangzhou Dianzi University, Xiasha Higher Education Zone, 2nd Street, Hangzhou, 310018, China b c
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 June 2019 Received in revised form 5 December 2019 Accepted 26 December 2019 Available online xxx
In the re-entry phase of a ballistic missile, decoys can be deployed as a mean to overburden enemy defenses. This results in a single track being split into multiple track-lets. Tracking of these track-lets is a critical task as any miss in the tracking procedure can become a cause of a major threat. The tracking process becomes more complicated in the presence of clutter. The low detection rate is one of the factors that may contribute to increasing the difficulty level in terms of tracking in the cluttered environment. This work introduces a new algorithm for the split event detection and target tracking under the framework of the joint integrated probabilistic data association (JIPDA) algorithm. The proposed algorithm is termed as split event-JIPDA (SE-JIPDA). This work establishes the mathematical foundation for the split target detection and tracking mechanism. The performance analysis is made under different simulation conditions to provide a clear insight into the merits of the proposed algorithm. The performance parameters in these simulations are the root mean square error (RMSE), confirmed true track rate (CTTR) and confirmed split true track rate (CSTTR). © 2020 Production and hosting by Elsevier B.V. on behalf of China Ordnance Society. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Keywords: Split event probability JIPDA Data association Ballistic missile Estimation
1. Introduction Split target tracking is a sub-field of multi-object tracking where one object or track splits into more than one child tracks. Birth of a new track transforms the single object tracking problem to a multiobject tracking problem. Defense system has many applications related to the split target tracking such as ballistic missile tracking, which is discussed in this work as one of the applications of the split target tracking problem. Short-range and medium-range ballistic missiles can be easily intercepted in the re-entry phase. However, in the re-entry phase, decoys may be deployed, which in turn may be detected as multiple objects for the radar. The tracker has to be intelligent so as to track each object (both decoy and the real target) in the presence of clutter as any object can carry a potential threat. It is a challenging
* Corresponding author. E-mail addresses: [email protected] (M. Asad), [email protected] (S. Khan), [email protected] (Ihsanullah), [email protected] (Z. Mehmood), [email protected] (Y. Shi), [email protected] (S.A. Memon), [email protected] (U. Khan). Peer review under responsibility of China Ordnance Society
task for the tracking system to track a ballistic missile in the reentry phase because of limited time for the object tracking, sensor noise, and less a priori information about the target. The state of the ballistic missile such as position, velocity, and ballistic coefficients are estimated through different filters [1,2]. Performance parameters for these filters are accurate in terms of estimation and processing time. The probabilistic data association (PDA) [3e5] is the Bayesian solution for the problems related to the single-scan data association. The PDA algorithm works well for the cases when there is only a single target to be tracked but PDA algorithm does not provide any framework for false track discrimination. The false track discrimination problem is solved in the integrated probabilistic data association algorithm (IPDA) [6e8] along with the target trajectory state update. Two-step multiple hypothesis trackers (MHT) [9e11] which is itself a multi-scan target tracking algorithm is used to achieve the accuracy in tracking of the target [12]. [13] proposed a target tracking algorithm in a densely cluttered environment using multiscan data association approach by considering the target existence as an event. The high clutter density performance of the MHT algorithm is improved by combining it with the modified logic-based
https://doi.org/10.1016/j.dt.2019.12.008 2214-9147/© 2020 Production and hosting by Elsevier B.V. on behalf of China Ordnance Society. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article as: Asad M et al., A split target detection and tracking algorithm for ballistic missile tracking during the re-entry phase, Defence Technology, https://doi.org/10.1016/j.dt.2019.12.008
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track initiation method. It is used in high clutter density for the split and merge issues along with the measurement noise and missdetection cases [14]. In a multi-target tracking system, the problem of measurement to track association is rectified by in-cooperating the joint association events into target tracking [15]. Multi-target tracking algorithms are compared for the single-scan and multi-scan data association problem [16]. The JIPDA algorithm is used to track the multi-target scenario in Ref. [17], but the track split scenario is not considered as an event and the target is tracked by conditioning on only birth and death events. The multiple target tracking problem is also addressed in Ref. [18], where authors have assumed the extended target scenario. In Ref. [19], authors have reduced the computational complexity of multi-target tracking algorithm by utilizing the Markov chain for random data associations. The target tracking problem is divided into different events such as a birth event, death event, merge event and split event for the target. To improve the target tracking performance and completeness, its state should be updated by conditioning over all the events. The earlier tracking algorithms assumed only birth, death and merge events, but the split is only considered as a birth of a new track. In Ref. [20], split target tracking is accomplished by using the IMM-JPDA algorithm. This algorithm provides no mathematical formula for split detection and confirmation. No track quality measure is provided in the same algorithm [21]. also utilizes the IMMJPDA algorithm to track the missiles (ballistic and intercontinental) in maneuvering and to detect the decoys [22]. uses the IMM-like algorithm for split target tracking in a clutter-less environment. Thus no data association and track quality measures are provided, at the same time this algorithm has not presented any framework for split event detection and confirmation. In the proposed work a new split event detection and tracking mechanism is integrated with the JIPDA algorithm and is termed as split event JIPDA (SE-JIPDA). In the best knowledge of authors, no significant mathematical contribution is made in the literature to track the split targets by considering the split as an event. The major contributions of the proposed work are: C A novel mathematical framework is proposed for split event detection and confirmation. C Tracking and management of track-lets. C Integration of split event detection/tracking algorithm with the existing JIPDA algorithm. C Performance validation of the proposed algorithm in the cluttered environment.
Fig. 1. Ballistic missile trajectory with three phases.
_’ xk ¼ ½x y x_ y
(1)
where x and y are position coordinates, and x_ and y__ are velocities in both x and y coordinates, respectively. Target trajectory state propagates as
xk ¼ Fxk1 þ wk
(2)
where F is state transition matrix, and xk is zero mean white Gaussian noise with covariance ℚ .
F¼
I2 02
ℚ¼
TI2 I2
0:25T 4 I2 0:5T 3 I2
(3) 0:5T 3 I2 T 2 I2
(4)
02 is second order zero matrix and I2 is second order identity matrix. T is the sample time. The object measurement model is
yk ¼ Hxk þ vk
(5)
where H is measurement matrix, and vk is white Gaussian measurement noise.
H ¼ ½I2
02
(6)
In Section 2 mathematical modeling of SE-JIPDA is presented, Section 3 provides simulation and analysis of results, and in Section 4 the proposed work is concluded. 2. Mathematical model Ballistic missile has three phases; boost phase, mid-course phase and the re-entry phase as shown in Fig. 1. Anti-ballistic missile defense system mostly works in the re-entry phase. Ballistic missile deploys the decoys in the re-entry phase, this employs that a single target splits into multiple targets. In this work, a split event joint integrated probabilistic data association (SE-JIPDA) is developed for proper detection of split event and tracking of these split tracks in the cluttered environment.
2.2. Tracking process 2.2.1. Track initialization Tracks in each scan are initialized using two point differencing method.
Vres ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðYÞ ðYÞ 2 ðXÞ ðXÞ 2 yk yk1 þ xk xk1 T
Vthreshold
(7)
where k is index for scan and Vthreshold is equal to 80 m=s . xXð:Þ and
2.1. Target measurement and motion model
yYð:Þ are the XeY coordinates of ith measurement yk ðiÞ2yk , where yk
In this study, the model under consideration is constant velocity model. The trajectory state xk is equal to
is the measurement set received at scan k. The ð:Þ operator in subscript symbolizes the scan index for the purpose of generalization.
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The criteria in (8) is applied to every ith measurement from the measurement set yk to identify the selected measurements in the validation gate.
. i . 1 Pdt Pgt P ctk Y k1 P qtk ð0Þ Y k . P ctk ; qk ð0Þ Y k ¼ 1 Pdt Pgt P ctk Y k1
and
2.2.2. Measurement selection
t yk ðiÞ H b x kjk1
’
t ðS t Þ1 yk ðiÞ H b x kjk1 g
(8)
S t is the measurement innovation covariance matrix calculated by t standard Kalman filter. H b x kjk1 is predicted position of the target. g
is the gating threshold, and its value is selected as 13.3 due to 3s limit on the standard deviation.
ith
measurement yk ðiÞ in The measurement likelihood for the the measurement vector yk given target existence [8] is calculated as
p yk ðiÞ ctk ;Y k1 ¼pr;k mF ðmk Þ @
t
o n P qtk ð0Þ Y k ¼
X
o n P εjY k
(15)
ε2Xðt;0Þ
n o P ctk Y k is posteriori probability of object existence. qtk ðiÞ is the detection event for i > 0, and non-detection event for i ¼ 0.
2.3.1. Target existence state update The target existence state is updated as [16].
n o 2 o n o 1 Pdt Pgt P ctk Y k1 X k o n ¼4 P ctk Y P ε Y k k1 t t t 1 Pd Pg P ck Y ε2Xðt;0Þ n
1 mk X X X 1Pdt Pgt 1 o o n AðεÞþ n AðεÞA 1Pdt Pgt P ctk Y k1 ε2Xðt;0Þ P ctk Y k1 i¼1 ε2Xðt;1Þ (9)
2 þ4
mk X
X
i¼1 ε2Xðt;1Þ
3 5
3 n o k 5 : P ε Y (16)
where Pdt is probability of detection for each track t. Pgt is gating probability. Pfctk Y k1 g is prior probability of target existence. ε2
Xðt; 0Þ is the set of feasible events allocated zero measurement to track. ε2Xðt; 1Þ is the set of events allocated one measurement to track, pr;k is clutter PDF [8], mF ðmk Þ is probability of clutter and follows the Poisson distribution, where AðεÞ ¼
Y h2T0 ðεÞ
h h
1 Pd Pg
Y
o ph ðiðh; εÞ Þ n h h h Pd Pg P ck Y k1 k
rk
h2T1 ðεÞ
(10) h
and rk is clutter density. pk ðiðh; εÞÞ is the measurement likelihood in event ε with respect to the target h under normal distribution. ith
The data association probability btk ðiÞ for each ith measurement is computed by using the formulation provided in Refs. [8,16]. btk ðiÞ
is used for each ith measurement as its weight associated with the track. btk ðiÞ is defined as
i h P ctk ; qtk ðiÞ Y k n o ;i 0 bk ðiÞ ¼ P ctk Y k t
(11)
For i > 0,
i t P ctk ; qk ðiÞ Y k ¼
X
o n P εjY k
(12)
ε2Xðt;1Þ
where
n o AðεÞ : P ε Y k ¼ P ε AðεÞ For i ¼ 0,
2.3.2. Target trajectory state update Each measurement is used to update the target trajectory state using prior trajectory state and its predicted error covariance matrix.
t t t b x kjk1 þ Kk xtk ðiÞ ℍ b x kjk1 x kjk ðiÞ ¼ b
(17)
Ptkjk ðiÞ ¼ ðI Kk HÞPtkjk1 ;
(18)
t where b x kjk ðiÞ is the target trajectory state estimate. Ptkjk ðiÞ is the
trajectory state error covariance matrix conditioned on the ith t measurement. b x kjk1 is the predicted target trajectory state, and Ptkjk1 is the predicted state error covariance matrix.
2.2.4. Data association
h
(14)
2.3. Tracking update process
2.2.3. Measurement likelihood
0
h
(13)
2.4. Split tracking algorithm The split tracking algorithm contains the following stages. 2.4.1. Split track initialization At scan k, the validation gate is formed using a predicted state of parent track tparent . There should be more than one measurements inside the validation gate to start the split track initialization procedure as shown in Fig. 2 (for a special case of a single split). A cluster of measurements is formed at any scan k. Cluster formation mechanism and conditions are summarized below. a) Number of clusters depend on the choice on number of possible splits. Its a good exercise to consider every measurement in the validation gate to be a candidate of a possible split track. Here, in this study, two splits are considered. b) If their are more number of measurements than the number of possible split tracks (as depicted in Fig. 2), then the data
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validation gate is equal to
sI k1 sI sI psI k ðiÞ ¼ p yk ðiÞ qk ðiÞ; ck ; Y ¼N
t
t b ysI k ðiÞ; H x kjk ; S
(22)
(23)
LsI k is the measurement likelihood ratio calculated as t t t t LsI k ¼ 1 Pd Pg þ Pd Pg
mk X psI ðiÞ k
(24)
rk
i¼1
and
i h k sI ¼ P csI k ; qk ðiÞ Y
Fig. 2. Initialization pattern of tentative split tracks.
association is calculated for each ith measurement in the validation gate as
btk ðiÞ ¼
.
Pdt Pgt ptk ðiÞ rk
(19)
where ptk ðiÞ ¼ N
2 o sI k ¼4 P ck Y
i¼1
rk
n o 3 k1 n o 1 Pdt Pgt P csI X k Y k sI o n P ε Y 5 1 Pdt Pgt P ctk Y k1 εsI 2XðsI;0Þ
þ4
trajectory state, rk is clutter density, Pdt is detection probability and Pgt is gating probability, as already defined in the earlier sections. Lk is defined as mk X ptk ðiÞ
(25)
εsI 2XðsI;1Þ
2
t t x kjk is the parent track estimated ytk ðiÞ; H b x kjk ; S t , b
Lk ¼ 1 Pdt Pgt þ Pdt Pgt
n o P εsI Y k ;
where
n
Lk
X
mk X
X
3 o P ε Y 5 n
sI k
i¼1 εsI 2XðsI;1Þ
(26) and the joint event probability is updated using
;
(20)
where mk defines the number of measurements in the validation t gate. The measurements with highest probability bk ðiÞ associated with the predicted position of parent track tparent are used to define the center of each cluster. Each cluster becomes a source of tentative split track. In Fig. 2, the measurements [y1 ;y2 ] have higher data association probabilities (19) as compared to [y3 ; y4 ]. Thus the former set of measurements are selected as a center of each cluster. c). If their are more number of measurement than the number of possible split tracks, then these measurements are associated with either cluster based upon their normalized distance square measure with respective clusters.
n o AðεÞ P εsI Yk ¼ P ε AðεÞ
(27)
Each cluster becomes a tentative split track. The measurement in each cluster is used to update the tentative split track trajectory sI b sI state x kjk and its error covariance matrix Pkjk .
sI sI I sI b b x kjk ðiÞ ¼ b y x kjk1 þ K sI ðiÞ ℍ x kjk1 k k
(28)
sI sI ℙsI kjk ðiÞ ¼ I K k ℍ ℙkjk1
(29)
sI th tentative split track. b K sI x kjk1 k is standard Kalman gain for the sI
and PsI kjk1 are the predicted trajectory state and its associated error By following the identical procedure more clusters can also be formed. In this work for the sake of simplicity, only two clusters are assumed to be formed. Whenever, tentative split tracks share measurements, joint integrated probabilistic data association (JIPDA) algorithm is used for measurement to track association. In this particular scenario, it becomes a multi-target tracking problem. 2.5. Data association probability bIk ðiÞ of each selected measurement is calculated as
bsIk ðiÞ ¼
P
n
csIk ; qsIk ðiÞ Yk
o
n o k P csI k Y
covariance matrix for the sIth split. Initialization of these parameters is carried out using the parent track tp . Each split track update its respective state in each scan. The prior information at this stage is only provided by the parent track tp . Using the prior information each cluster is updated with the measurements associated with it. Thus each cluster provides a state for each tentative split and each cluster is propagated to the next scan as sI sI b x kjk x kþ1jk ¼ Fb
(30)
sI ℙsI kþ1jk ¼ Fℙkjk F’ þ ℚ
(31)
(21)
sI is index of split track and i is index of measurement. Measurement likelihood of the ith measurement in the
2.5.1. Split event probability P sI s The split event probability is conditioned on the target existence event such that the target exists and it splits into more (in this case
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two) targets. A situation may arise such that, target exists and it does not split with probability 1 P sI s . The event that target exists and splits and the event that target exists and does not split are mutually exclusive and exhaustive events. sI
PksI þ P k ¼ 1
(32)
5
sI k1 k1 sI sI csI ; Yk1 ¼ p y þ p y p yk csI ; Y ; S c ; Y ; S k k k k k k k o n o n sI k1 k1 sI sI LsI þ 1 P SsI ¼ pr;k mf msI k k P Sk ck ; Y k ck ; Y n o o n sI k1 k1 sI sI 1 P SsI þ LsI ¼ pr;k mf msI k k ck ; Y k P Sk ck ; Y (39)
and sI
P k ¼ 1 PksI
(33)
th track after tentative The split event probability P sI k of the sI
Thus (34) becomes
o n sI sI k P sI k ¼ P Sk ck ; Y
split event SsI k is computed as,
k1 sI o p yk SsI o n n k ; ck ; Y sI k1 k sI sI sI P SsI ¼ Pk ¼ P Sk ck ; Y k c k ; Y sI k1 p yk ck ; Y
¼
! LsI k n o o n sI k1 sI sI k1 1 P SsI þ LsI k ck ; Y k P S k c k ; Y
(40)
o n sI k1 : P SsI k c k ; Y (34)
where
h i k1 k1 sI csI ; Y k1 sI sI ¼ p y þ p y p yk csI ; Y ; S c ; Y ; S k k k k k k k
2.6.1. Confirmation of split track All the tentative split tracks are checked for the status of split track confirmation. There are two conditions for the split track confirmation.
n o sI k1 k1 sI P SsI ¼ p yk SsI k ; ck ; Y k c k ; Y sI sI sI k1 c ; Y k1 P ; Y S þp yk Sk ; csI k k k (35)
i) Condition on separation distance. ii) Condition on split confirmation probability threshold. Both conditions (41) and (42) should be satisfied at once by the tentative split track to get the status of confirmed split track.
As the non split track is equivalent to its non existence. Thus
X mk mk X n o k1 k1 k1 k1 sI sI sI sI ¼ P qk ðiÞjSsI ¼ p yk SsI p yk ; qk ðiÞjSsI p yk qk ðiÞ; SsI k ; ck ; Y k ; ck ; Y k ; ck ; Y k ; ck ; Y i¼0
As, sIth split track event is directly related to its respective event sI of existence ðSsI k ⇔ck Þ, thus (36) becomes
mk X psI ðiÞ k1 sI k ¼ pr;k mf ðmk Þ 1 Pdt Pgt þ Pdt Pgt p yk SsI k ; ck ; Y r ðiÞ i¼1 k
!
¼ LsI k (37) The non-split event collapse into the non-existence event, thus
sI k1 ¼ pr;k mf ðmk Þ p yk Sk ; csI ; Y k
2.6. The measurement likelihood in (35) becomes
(36)
i¼0
(38)
sI sI b xk 1 b xk 2
T 1 sI1 sI2 sI2 1 b b ℙsI x > Ds þ ℙ x k k k k
(41)
and
c P sI s tS ;
(42)
where Ds is separation distance in meters between the two tentative split tracks. For this particular scenario, it is selected as 20 m tcS is the split confirmation threshold and is selected as 0.95. sI1 and sI2 are the indexes of any two tentative split tracks under considerh i sI ation. b x kjk ; ℙsI is the split track trajectory estimate and its error kjk covariance matrix calculated using standard JIPDA algorithm (whenever tracks share some measurements). The confirm split tracks propagation scenario is presented in Fig. 3. In the initial scans, track-lets are identified as tentative split tracks. If both the conditions in Eqs. (41) and (42) are satisfied, then tentative split tracks attains the status of confirmed split track in scan k þ Ns . Where Ns is the number of additional scans used for the split track confirmation.
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M. Asad et al. / Defence Technology xxx (xxxx) xxx Table 1 Algorithm for SE-JIPDA 1. Split event (SE) Initialization i At time k, measurements Yk from sensor. ii Split event initialize when two or more measurements in the validation gate. iii Clusters are formed based on association probability. iv When tentative split tracks share measurements Joint Integrated Probabilistic h i sI Data Association (JIPDA) is used for calculating b x kjk ; P sI kjk v Data association probability is calculated. vi Measurement likelihood for each measurement is calculated. pk ðiÞ ¼ p yk ðiÞ qk ðiÞ; ck ; Y k1 2.Computation of Split probability P skI 1 0 n o LsI sI k k sI sI AP sI @ Pk ¼ P Sk ck ; Y ¼ kjk1 sI sI þ LsI 1 Pkjk1 k Pkjk1 where
n o sI k1 sI ¼ P SsI c ;Y Pkjk1 k k And
0 1 msIk X psI sI sI k1 k ðiÞ A @1 P t P t þ P t P t LsI ¼ pr;k mf msI d g d g k ¼ p y k Sk ; ck ; Y k j¼1
Fig. 3. Progress of confirmed split tracks.
2.6.2. Merger of tentative split tracks If the tentative split tracks persist to follow the same path in parallel for “Nm ” scans, then these tracks will be merged as they will only increase the computational complexity. The process is depicted in Fig. 4. In this particular case if the tentative split tracks satisfy the check (43) for “Nm ” consecutive scans, then the tentative split tracks are merged to create a single track.
rk
3. Confirmation of Split event Two conditions for split confirmation 1 sI T sI sI sI 1 2 b i b xk 1 b xk 2 þ P sI x k 2 > Ds P sI xk 1 b k k c i ii P sI s tS 4. Merging of split tracks follow same path for number of scans 1 sI T sI sI sI 1 2 b b xk 2 þ P sI x k 2 < Dm P sI xk 1 b xk 1 b k k
sI sI b xk 1 b xk 2
T
1 2 P sI þ P sI k k
1
sI sI b xk 1 b xk 2
< Dm
(43)
where Dm is the merging threshold between the tentative split tracks. In this study, }Nm } is selected as 7 scans, and Ds is selected as 5 m. Pseudo code for split event JIPDA (SE-JIPDA) is given in Table 1, which shows the overall flow of the split event detection and tracking mechanism as discussed in the previous sections. 3. Simulation study Ballistic missile trajectory is shown in Fig. 5. Results are presented for two different cases. In Case-1, target and its decoy are far from each other as compared to Case-2, where the target and its decoy is close to each other. Both cases are presented in Fig. 5. Initial position of the
Fig. 4. Parallel progress of two tentative split tracks.
Fig. 5. Ballistic missile trajectory with decoys.
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parent target is [100 m;1290 m]. At scan 29, the parent target splits into two track-lets creating a multi-target scenario in the surveillance area. Each track-let is considered as a target to be tracked in the presence of clutter. In Case-1, the velocities are [2500;-500] m= s and [2400;-700] m=s for parent target and split target, respectively. In Case-2, the velocities are [2500;-500] m=s and [2400;600] m=s for parent target and split target, respectively. The surveillance area under observation is [3000 m 800 m]. In each case, the comparison is made for different values of detection probability Pd (for clarity in the figures, Pdt is replaced by Pd ), while keeping the same clutter density. Simulations are carried out in MATLAB with a total of 100 Monte Carlo runs with each Monte Carlo run having 50 scans.
3.1. Case-1 analysis Simulation results in Case-1 are compared for different values of Pd (1; 0:85;0:7) by keeping the same clutter density (r ¼ 106 m2 ). With the low probability of detection, the RMSE increases as shown in Fig. 6. The RMSE plot is cumulative for both parent and split track. In the splitting phase, the RMSE increases due to the fact that both parent track and split track RMSE’s are added up. Fig. 7 shows the confirmed true track rate for the parent track. A small variation can be observed in the scan interval 27 to 30, when parent target
Fig. 8. Case-1: Confirmed split true track rate (CSTTR) of split track r ¼ 106 m2 ;Pd ¼ 1; 0:85; 0:7.
Fig. 9. Case-2: Cumulative RMSE for both parent and split tracks r ¼ 106 m2 ; Pd ¼ 1; 0:85; 0:7.
Fig. 6. Case-1:Cumulative RMSE for both parent and split tracks r ¼ 106 m2 ; Pd ¼ 1; 0:85; 0:7.
shares the measurements with the split target. Confirmed split true track rate (CSTTR) characteristic is analyzed for the Case-1 for different values of Pd as shown in Fig. 8. It is a unique performance parameter of the proposed algorithm, as it helps in the analysis of split detection and its tracking. In Fig. 8, the CSTTR starts to increase showing the presence of split and attains
Fig. 7. Case-1:Confirmed true track rate (CTTR) of parent track r ¼ 106 m2 ; Pd ¼ 1; 0:85; 0:7.
Fig. 10. Case-2: Confirmed true track rate (CTTR) of parent track r ¼ 106 m2 ; Pd ¼ 1; 0:85; 0:7.
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3.4. Comparison with IPDA and ITS algorithms
Fig. 11. Case-2: Confirmed split true track rate (CSTTR) of split track r ¼ 106 m2 ;Pd ¼ 1; 0:85; 0:7.
the maximum value till the end of the complete run.
3.2. Case-2 analysis In Case-2, both target and decoy are close to each other. In this particular case, multiple tracks share measurements for a greater number of scans as compared to Case-1. In Figs. 9e10, there is comparison of RMSE and CTTR for r ¼ 106 m2. It is visible that RMSE increases and CTTR slow down with a decrease in the value of Pd . Fig. 11 shows the CSTTR for the Case-2.
Integrated track splitting (ITS) is a multi scan single target tracking algorithm. The ITS algorithm relates more number of scans to determine the target trajectory state estimate in the presence of clutter. Integrated probabilistic data association (IPDA) is a single scan single target tracking algorithm. IPDA assumes point target assumption and merges all PDF’s as one Gaussian PDF in each scan. The single PDF propagates in the next scan for measurement selection and update process. The proposed SE-JIPDA algorithm is compared with the IPDA and ITS algorithms. Results are presented in Figs. 12e13 for cumulative RMSE and CTTR for the Case-1, respectively. Similarly, Figs. 14e15 presents the results for Case-2 for both cumulative RMSE and CTTR, respectively. It is evident that SE-JIPDA performance is much better as compared to the IPDA and ITS algorithms in terms of RMSE and CTTR. Split event true track rate (CSTTR) cannot be compared with both algorithms (ITS and IPDA) as they do not provide the procedure for split event detection and its state update.
4. Conclusion In this work, split event detection and tracking algorithm is
3.3. Comparison of Case-1 and Case-2 The CTTR performance for both cases is identical as the same clutter density and target motion model is considered. Due to the large separation in Case-1, targets share measurement for the less number of scans as compared to Case-2 (small separation). Thus the CTTR performance in Case-1 (Fig. 7) is better in the measurement sharing region as compared to Case-2 (Fig. 10). The CSTTR in the Case-1 (Fig. 8) is also fast as compared to Case-2 (Fig. 11). Case-1 confirms the split track at faster rate as compared to the Case-2. The RMSE is also affected in both the cases due to the different separation distances in both cases.
Fig. 13. Case-1: Cumulative CTTR of both parent and child tracks r ¼ 106 m2 and Pd ¼ 0:7.
Fig. 12. Case-1: Cumulative RMSE of both parent and child track r ¼ 106 m2 and Pd ¼ 0:7.
Fig. 14. Case-2: Cumulative RMSE of both parent and child tracks r ¼ 106 m2 and Pd ¼ 0:7.
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Fig. 15. Case-2: Cumulative CTTR of both parent and child tracks r ¼ 106 m2 and Pd ¼ 0:7.
proposed in the presence of clutter. The proposed algorithm is termed as split event joint integrated probabilistic data association filter (SE-JIPDA). This algorithm provides a mechanism for split event detection and it tracks each split in the presence of clutter. The split event detection mechanism helps to monitor the performance of both the parent and split targets. The performance of the proposed SE-JIPDA algorithm is compared with different algorithms under different simulation conditions. The results show a significant improvement in terms of true track rate and root mean square error performance compared to the existing algorithms. In the future, this work could be extended to multi-scan tracking algorithms and multiple model systems. References [1] Singh NK, Bhaumik S, Bhattacharya S. A comparison of several nonlinear filters for ballistic missile tracking on re-entry. In: IEEE first international conference on control, measurement and instrumentation (CMI); 2016. p. 459e63. 2016. [2] Julier S, Uhlmann J. Unscented filtering and nonlinear estimation’. Proc IEEE Mar 2004;92(3):401e22. [3] Bar-Shalom Y, Fred D, Huang J. The probabilistic data association filter’. IEEE Control Syst 2009;29(6):82e100.
9
[4] Mahemuti B, Inoue D, Kakugo A, Konagaya A. Investigation of the microtubule dynamics with probabilistic data association filter. In: 2016 IEEE 11th annual international conference on nano/micro engineered and molecular systems (NEMS). Sendai, Japan; 2016. p. 101e6. [5] Agamennoni G, Fontana S, Siegwart RY, Sorrenti DG. Point clouds registration with probabilistic data association. In: 2016 IEEE/RSJ international conference on intelligent robots and systems (IROS). Daejeon: South Korea; 2016. p. 4092e8. [6] Sajjad S, Khan U. A performance analysis of single target tracking algorithm for the imagery data’. Islamabad: 2nd International Conference on Space; 2016. p. 20e2 (ICS 2016) Sep. 2016. [7] Musicki D, Evans R, Stankovic S. Integrated probabilistic data association’, Automatic Control. IEEE Trans. 1994;39(6):1237e41. [8] Challa S, Moreland M, Musicki D, Evans R. Fundamentals of object tracking’. Cambridge University Press; 2011. [9] Makris A, Prieur C. Bayesian multiple-hypothesis tracking of merging and splitting targets. IEEE Trans Geosci Remote Sens Dec. 2014;52(12):7684e94. [10] Zhao K, Meuter M, Muller-Schneiders S, Pauli J. A novel multi-hypothesis tracking framework for lane recognition,” 17th International Conference on Information Fusion (FUSION). 2014. p. 1e8. Salamanca. [11] Blackman SS. Multiple hypothesis tracking for multiple target tracking’. IEEE Aerosp Electron Syst Mag Jan 2004;19(1):5e18. [12] Musicki D, La Scala BF, Evans RJ. Integrated track splitting filter - efficient multi-scan single target tracking in clutter’. IEEE Trans Aerosp Electron Syst Jan 2007;43:1409e25. [13] Khan U, Song TL. Target tracking with a two-scan data association algorithm extended for the hybrid target state’. IET Radar, Sonar Navig 2015;9:1330e7. [14] Li D, Zhang Y, Lin Y, Liu J. A novel track initiation method for track splitting and merging’. OCEANS 2016 - Shanghai; April 2016. p. 1e7. [15] Genovesio A, Olivo-Marin JC. Split and merge data association filter for dense multi-target tracking’. In: Proceedings of the 17th international conference on pattern recognition, vol. 4. ICPR); 2004. p. 677e80. Aug. 2004. [16] Musicki D, Evans R. Joint integrated probabilistic data association: JIPDA’. IEEE Trans Aerosp Electron Syst 2004;40(3):1093e9. [17] Asad M, Khan S, Arif M, Mehmood Z, Durrani S, Khan U. Ballistic missile tracking in the presence of clutter using multi-target tracking algorithm’, vol. 2017. IBCAST; 2017. [18] Huang Y, Song TL, Lee WJ, Kirubarajan T. Multiple detection joint integrated track splitting for multiple extended target tracking’. Signal Process 2019;162: 126e40. [19] Huang Y, Song TL, Cheagal DH. Markov chain realization of multiple detection joint integrated probabilistic data association’. Sensors 2018;19(1):112. [20] Bar-Shalom Y, Chang KC, Blom HAP. Tracking of splitting targets in clutter using an interacting multiple model joint probabilistic data association filter. In: Proceedings of the 30th IEEE conference on decision and control, brighton, UK, vol. 2; 1991. p. 2043e8. [21] Dunton TM. Ballistic missile tracking using the interacting multiple model joint probabilistic data association filter. Monterey, California: Master’s Thesis, Naval Postgraduate School; 2013. [22] Sigalov D, Michaeli T, Oshman Y. Tracking a splitting target in clutter using the IMM methodology. In: 2012 IEEE 27th convention of electrical and electronics engineers in Israel. Eilat; 2012. p. 1e5.
Please cite this article as: Asad M et al., A split target detection and tracking algorithm for ballistic missile tracking during the re-entry phase, Defence Technology, https://doi.org/10.1016/j.dt.2019.12.008
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Defence Technology journal homepage: www.elsevier.com/locate/dt
A split target detection and tracking algorithm for ballistic missile tracking during the re-entry phase Muhammad Asad a, Sumair Khan b, Ihsanullah a, Zahid Mehmood a, Yifang Shi d, Sufyan Ali Memon c, Uzair Khan a, * a
Department of Electrical and Computer Engineering, COMSATS University Islamabad, Abbottabad Campus, Pakistan Department of Computer Science, COMSATS University Islamabad, Abbottabad Campus, Pakistan Electrical Engineering Department, Indus University, Karachi, Pakistan d Department of Automation, Hangzhou Dianzi University, Xiasha Higher Education Zone, 2nd Street, Hangzhou, 310018, China b c
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 June 2019 Received in revised form 5 December 2019 Accepted 26 December 2019 Available online xxx
In the re-entry phase of a ballistic missile, decoys can be deployed as a mean to overburden enemy defenses. This results in a single track being split into multiple track-lets. Tracking of these track-lets is a critical task as any miss in the tracking procedure can become a cause of a major threat. The tracking process becomes more complicated in the presence of clutter. The low detection rate is one of the factors that may contribute to increasing the difficulty level in terms of tracking in the cluttered environment. This work introduces a new algorithm for the split event detection and target tracking under the framework of the joint integrated probabilistic data association (JIPDA) algorithm. The proposed algorithm is termed as split event-JIPDA (SE-JIPDA). This work establishes the mathematical foundation for the split target detection and tracking mechanism. The performance analysis is made under different simulation conditions to provide a clear insight into the merits of the proposed algorithm. The performance parameters in these simulations are the root mean square error (RMSE), confirmed true track rate (CTTR) and confirmed split true track rate (CSTTR). © 2020 Production and hosting by Elsevier B.V. on behalf of China Ordnance Society. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Keywords: Split event probability JIPDA Data association Ballistic missile Estimation
1. Introduction Split target tracking is a sub-field of multi-object tracking where one object or track splits into more than one child tracks. Birth of a new track transforms the single object tracking problem to a multiobject tracking problem. Defense system has many applications related to the split target tracking such as ballistic missile tracking, which is discussed in this work as one of the applications of the split target tracking problem. Short-range and medium-range ballistic missiles can be easily intercepted in the re-entry phase. However, in the re-entry phase, decoys may be deployed, which in turn may be detected as multiple objects for the radar. The tracker has to be intelligent so as to track each object (both decoy and the real target) in the presence of clutter as any object can carry a potential threat. It is a challenging
* Corresponding author. E-mail addresses: [email protected] (M. Asad), [email protected] (S. Khan), [email protected] (Ihsanullah), [email protected] (Z. Mehmood), [email protected] (Y. Shi), [email protected] (S.A. Memon), [email protected] (U. Khan). Peer review under responsibility of China Ordnance Society
task for the tracking system to track a ballistic missile in the reentry phase because of limited time for the object tracking, sensor noise, and less a priori information about the target. The state of the ballistic missile such as position, velocity, and ballistic coefficients are estimated through different filters [1,2]. Performance parameters for these filters are accurate in terms of estimation and processing time. The probabilistic data association (PDA) [3e5] is the Bayesian solution for the problems related to the single-scan data association. The PDA algorithm works well for the cases when there is only a single target to be tracked but PDA algorithm does not provide any framework for false track discrimination. The false track discrimination problem is solved in the integrated probabilistic data association algorithm (IPDA) [6e8] along with the target trajectory state update. Two-step multiple hypothesis trackers (MHT) [9e11] which is itself a multi-scan target tracking algorithm is used to achieve the accuracy in tracking of the target [12]. [13] proposed a target tracking algorithm in a densely cluttered environment using multiscan data association approach by considering the target existence as an event. The high clutter density performance of the MHT algorithm is improved by combining it with the modified logic-based
https://doi.org/10.1016/j.dt.2019.12.008 2214-9147/© 2020 Production and hosting by Elsevier B.V. on behalf of China Ordnance Society. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).
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track initiation method. It is used in high clutter density for the split and merge issues along with the measurement noise and missdetection cases [14]. In a multi-target tracking system, the problem of measurement to track association is rectified by in-cooperating the joint association events into target tracking [15]. Multi-target tracking algorithms are compared for the single-scan and multi-scan data association problem [16]. The JIPDA algorithm is used to track the multi-target scenario in Ref. [17], but the track split scenario is not considered as an event and the target is tracked by conditioning on only birth and death events. The multiple target tracking problem is also addressed in Ref. [18], where authors have assumed the extended target scenario. In Ref. [19], authors have reduced the computational complexity of multi-target tracking algorithm by utilizing the Markov chain for random data associations. The target tracking problem is divided into different events such as a birth event, death event, merge event and split event for the target. To improve the target tracking performance and completeness, its state should be updated by conditioning over all the events. The earlier tracking algorithms assumed only birth, death and merge events, but the split is only considered as a birth of a new track. In Ref. [20], split target tracking is accomplished by using the IMM-JPDA algorithm. This algorithm provides no mathematical formula for split detection and confirmation. No track quality measure is provided in the same algorithm [21]. also utilizes the IMMJPDA algorithm to track the missiles (ballistic and intercontinental) in maneuvering and to detect the decoys [22]. uses the IMM-like algorithm for split target tracking in a clutter-less environment. Thus no data association and track quality measures are provided, at the same time this algorithm has not presented any framework for split event detection and confirmation. In the proposed work a new split event detection and tracking mechanism is integrated with the JIPDA algorithm and is termed as split event JIPDA (SE-JIPDA). In the best knowledge of authors, no significant mathematical contribution is made in the literature to track the split targets by considering the split as an event. The major contributions of the proposed work are: C A novel mathematical framework is proposed for split event detection and confirmation. C Tracking and management of track-lets. C Integration of split event detection/tracking algorithm with the existing JIPDA algorithm. C Performance validation of the proposed algorithm in the cluttered environment.
Fig. 1. Ballistic missile trajectory with three phases.
_’ xk ¼ ½x y x_ y
(1)
where x and y are position coordinates, and x_ and y__ are velocities in both x and y coordinates, respectively. Target trajectory state propagates as
xk ¼ Fxk1 þ wk
(2)
where F is state transition matrix, and xk is zero mean white Gaussian noise with covariance ℚ .
F¼
I2 02
ℚ¼
TI2 I2
0:25T 4 I2 0:5T 3 I2
(3) 0:5T 3 I2 T 2 I2
(4)
02 is second order zero matrix and I2 is second order identity matrix. T is the sample time. The object measurement model is
yk ¼ Hxk þ vk
(5)
where H is measurement matrix, and vk is white Gaussian measurement noise.
H ¼ ½I2
02
(6)
In Section 2 mathematical modeling of SE-JIPDA is presented, Section 3 provides simulation and analysis of results, and in Section 4 the proposed work is concluded. 2. Mathematical model Ballistic missile has three phases; boost phase, mid-course phase and the re-entry phase as shown in Fig. 1. Anti-ballistic missile defense system mostly works in the re-entry phase. Ballistic missile deploys the decoys in the re-entry phase, this employs that a single target splits into multiple targets. In this work, a split event joint integrated probabilistic data association (SE-JIPDA) is developed for proper detection of split event and tracking of these split tracks in the cluttered environment.
2.2. Tracking process 2.2.1. Track initialization Tracks in each scan are initialized using two point differencing method.
Vres ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðYÞ ðYÞ 2 ðXÞ ðXÞ 2 yk yk1 þ xk xk1 T
Vthreshold
(7)
where k is index for scan and Vthreshold is equal to 80 m=s . xXð:Þ and
2.1. Target measurement and motion model
yYð:Þ are the XeY coordinates of ith measurement yk ðiÞ2yk , where yk
In this study, the model under consideration is constant velocity model. The trajectory state xk is equal to
is the measurement set received at scan k. The ð:Þ operator in subscript symbolizes the scan index for the purpose of generalization.
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3
The criteria in (8) is applied to every ith measurement from the measurement set yk to identify the selected measurements in the validation gate.
. i . 1 Pdt Pgt P ctk Y k1 P qtk ð0Þ Y k . P ctk ; qk ð0Þ Y k ¼ 1 Pdt Pgt P ctk Y k1
and
2.2.2. Measurement selection
t yk ðiÞ H b x kjk1
’
t ðS t Þ1 yk ðiÞ H b x kjk1 g
(8)
S t is the measurement innovation covariance matrix calculated by t standard Kalman filter. H b x kjk1 is predicted position of the target. g
is the gating threshold, and its value is selected as 13.3 due to 3s limit on the standard deviation.
ith
measurement yk ðiÞ in The measurement likelihood for the the measurement vector yk given target existence [8] is calculated as
p yk ðiÞ ctk ;Y k1 ¼pr;k mF ðmk Þ @
t
o n P qtk ð0Þ Y k ¼
X
o n P εjY k
(15)
ε2Xðt;0Þ
n o P ctk Y k is posteriori probability of object existence. qtk ðiÞ is the detection event for i > 0, and non-detection event for i ¼ 0.
2.3.1. Target existence state update The target existence state is updated as [16].
n o 2 o n o 1 Pdt Pgt P ctk Y k1 X k o n ¼4 P ctk Y P ε Y k k1 t t t 1 Pd Pg P ck Y ε2Xðt;0Þ n
1 mk X X X 1Pdt Pgt 1 o o n AðεÞþ n AðεÞA 1Pdt Pgt P ctk Y k1 ε2Xðt;0Þ P ctk Y k1 i¼1 ε2Xðt;1Þ (9)
2 þ4
mk X
X
i¼1 ε2Xðt;1Þ
3 5
3 n o k 5 : P ε Y (16)
where Pdt is probability of detection for each track t. Pgt is gating probability. Pfctk Y k1 g is prior probability of target existence. ε2
Xðt; 0Þ is the set of feasible events allocated zero measurement to track. ε2Xðt; 1Þ is the set of events allocated one measurement to track, pr;k is clutter PDF [8], mF ðmk Þ is probability of clutter and follows the Poisson distribution, where AðεÞ ¼
Y h2T0 ðεÞ
h h
1 Pd Pg
Y
o ph ðiðh; εÞ Þ n h h h Pd Pg P ck Y k1 k
rk
h2T1 ðεÞ
(10) h
and rk is clutter density. pk ðiðh; εÞÞ is the measurement likelihood in event ε with respect to the target h under normal distribution. ith
The data association probability btk ðiÞ for each ith measurement is computed by using the formulation provided in Refs. [8,16]. btk ðiÞ
is used for each ith measurement as its weight associated with the track. btk ðiÞ is defined as
i h P ctk ; qtk ðiÞ Y k n o ;i 0 bk ðiÞ ¼ P ctk Y k t
(11)
For i > 0,
i t P ctk ; qk ðiÞ Y k ¼
X
o n P εjY k
(12)
ε2Xðt;1Þ
where
n o AðεÞ : P ε Y k ¼ P ε AðεÞ For i ¼ 0,
2.3.2. Target trajectory state update Each measurement is used to update the target trajectory state using prior trajectory state and its predicted error covariance matrix.
t t t b x kjk1 þ Kk xtk ðiÞ ℍ b x kjk1 x kjk ðiÞ ¼ b
(17)
Ptkjk ðiÞ ¼ ðI Kk HÞPtkjk1 ;
(18)
t where b x kjk ðiÞ is the target trajectory state estimate. Ptkjk ðiÞ is the
trajectory state error covariance matrix conditioned on the ith t measurement. b x kjk1 is the predicted target trajectory state, and Ptkjk1 is the predicted state error covariance matrix.
2.2.4. Data association
h
(14)
2.3. Tracking update process
2.2.3. Measurement likelihood
0
h
(13)
2.4. Split tracking algorithm The split tracking algorithm contains the following stages. 2.4.1. Split track initialization At scan k, the validation gate is formed using a predicted state of parent track tparent . There should be more than one measurements inside the validation gate to start the split track initialization procedure as shown in Fig. 2 (for a special case of a single split). A cluster of measurements is formed at any scan k. Cluster formation mechanism and conditions are summarized below. a) Number of clusters depend on the choice on number of possible splits. Its a good exercise to consider every measurement in the validation gate to be a candidate of a possible split track. Here, in this study, two splits are considered. b) If their are more number of measurements than the number of possible split tracks (as depicted in Fig. 2), then the data
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validation gate is equal to
sI k1 sI sI psI k ðiÞ ¼ p yk ðiÞ qk ðiÞ; ck ; Y ¼N
t
t b ysI k ðiÞ; H x kjk ; S
(22)
(23)
LsI k is the measurement likelihood ratio calculated as t t t t LsI k ¼ 1 Pd Pg þ Pd Pg
mk X psI ðiÞ k
(24)
rk
i¼1
and
i h k sI ¼ P csI k ; qk ðiÞ Y
Fig. 2. Initialization pattern of tentative split tracks.
association is calculated for each ith measurement in the validation gate as
btk ðiÞ ¼
.
Pdt Pgt ptk ðiÞ rk
(19)
where ptk ðiÞ ¼ N
2 o sI k ¼4 P ck Y
i¼1
rk
n o 3 k1 n o 1 Pdt Pgt P csI X k Y k sI o n P ε Y 5 1 Pdt Pgt P ctk Y k1 εsI 2XðsI;0Þ
þ4
trajectory state, rk is clutter density, Pdt is detection probability and Pgt is gating probability, as already defined in the earlier sections. Lk is defined as mk X ptk ðiÞ
(25)
εsI 2XðsI;1Þ
2
t t x kjk is the parent track estimated ytk ðiÞ; H b x kjk ; S t , b
Lk ¼ 1 Pdt Pgt þ Pdt Pgt
n o P εsI Y k ;
where
n
Lk
X
mk X
X
3 o P ε Y 5 n
sI k
i¼1 εsI 2XðsI;1Þ
(26) and the joint event probability is updated using
;
(20)
where mk defines the number of measurements in the validation t gate. The measurements with highest probability bk ðiÞ associated with the predicted position of parent track tparent are used to define the center of each cluster. Each cluster becomes a source of tentative split track. In Fig. 2, the measurements [y1 ;y2 ] have higher data association probabilities (19) as compared to [y3 ; y4 ]. Thus the former set of measurements are selected as a center of each cluster. c). If their are more number of measurement than the number of possible split tracks, then these measurements are associated with either cluster based upon their normalized distance square measure with respective clusters.
n o AðεÞ P εsI Yk ¼ P ε AðεÞ
(27)
Each cluster becomes a tentative split track. The measurement in each cluster is used to update the tentative split track trajectory sI b sI state x kjk and its error covariance matrix Pkjk .
sI sI I sI b b x kjk ðiÞ ¼ b y x kjk1 þ K sI ðiÞ ℍ x kjk1 k k
(28)
sI sI ℙsI kjk ðiÞ ¼ I K k ℍ ℙkjk1
(29)
sI th tentative split track. b K sI x kjk1 k is standard Kalman gain for the sI
and PsI kjk1 are the predicted trajectory state and its associated error By following the identical procedure more clusters can also be formed. In this work for the sake of simplicity, only two clusters are assumed to be formed. Whenever, tentative split tracks share measurements, joint integrated probabilistic data association (JIPDA) algorithm is used for measurement to track association. In this particular scenario, it becomes a multi-target tracking problem. 2.5. Data association probability bIk ðiÞ of each selected measurement is calculated as
bsIk ðiÞ ¼
P
n
csIk ; qsIk ðiÞ Yk
o
n o k P csI k Y
covariance matrix for the sIth split. Initialization of these parameters is carried out using the parent track tp . Each split track update its respective state in each scan. The prior information at this stage is only provided by the parent track tp . Using the prior information each cluster is updated with the measurements associated with it. Thus each cluster provides a state for each tentative split and each cluster is propagated to the next scan as sI sI b x kjk x kþ1jk ¼ Fb
(30)
sI ℙsI kþ1jk ¼ Fℙkjk F’ þ ℚ
(31)
(21)
sI is index of split track and i is index of measurement. Measurement likelihood of the ith measurement in the
2.5.1. Split event probability P sI s The split event probability is conditioned on the target existence event such that the target exists and it splits into more (in this case
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two) targets. A situation may arise such that, target exists and it does not split with probability 1 P sI s . The event that target exists and splits and the event that target exists and does not split are mutually exclusive and exhaustive events. sI
PksI þ P k ¼ 1
(32)
5
sI k1 k1 sI sI csI ; Yk1 ¼ p y þ p y p yk csI ; Y ; S c ; Y ; S k k k k k k k o n o n sI k1 k1 sI sI LsI þ 1 P SsI ¼ pr;k mf msI k k P Sk ck ; Y k ck ; Y n o o n sI k1 k1 sI sI 1 P SsI þ LsI ¼ pr;k mf msI k k ck ; Y k P Sk ck ; Y (39)
and sI
P k ¼ 1 PksI
(33)
th track after tentative The split event probability P sI k of the sI
Thus (34) becomes
o n sI sI k P sI k ¼ P Sk ck ; Y
split event SsI k is computed as,
k1 sI o p yk SsI o n n k ; ck ; Y sI k1 k sI sI sI P SsI ¼ Pk ¼ P Sk ck ; Y k c k ; Y sI k1 p yk ck ; Y
¼
! LsI k n o o n sI k1 sI sI k1 1 P SsI þ LsI k ck ; Y k P S k c k ; Y
(40)
o n sI k1 : P SsI k c k ; Y (34)
where
h i k1 k1 sI csI ; Y k1 sI sI ¼ p y þ p y p yk csI ; Y ; S c ; Y ; S k k k k k k k
2.6.1. Confirmation of split track All the tentative split tracks are checked for the status of split track confirmation. There are two conditions for the split track confirmation.
n o sI k1 k1 sI P SsI ¼ p yk SsI k ; ck ; Y k c k ; Y sI sI sI k1 c ; Y k1 P ; Y S þp yk Sk ; csI k k k (35)
i) Condition on separation distance. ii) Condition on split confirmation probability threshold. Both conditions (41) and (42) should be satisfied at once by the tentative split track to get the status of confirmed split track.
As the non split track is equivalent to its non existence. Thus
X mk mk X n o k1 k1 k1 k1 sI sI sI sI ¼ P qk ðiÞjSsI ¼ p yk SsI p yk ; qk ðiÞjSsI p yk qk ðiÞ; SsI k ; ck ; Y k ; ck ; Y k ; ck ; Y k ; ck ; Y i¼0
As, sIth split track event is directly related to its respective event sI of existence ðSsI k ⇔ck Þ, thus (36) becomes
mk X psI ðiÞ k1 sI k ¼ pr;k mf ðmk Þ 1 Pdt Pgt þ Pdt Pgt p yk SsI k ; ck ; Y r ðiÞ i¼1 k
!
¼ LsI k (37) The non-split event collapse into the non-existence event, thus
sI k1 ¼ pr;k mf ðmk Þ p yk Sk ; csI ; Y k
2.6. The measurement likelihood in (35) becomes
(36)
i¼0
(38)
sI sI b xk 1 b xk 2
T 1 sI1 sI2 sI2 1 b b ℙsI x > Ds þ ℙ x k k k k
(41)
and
c P sI s tS ;
(42)
where Ds is separation distance in meters between the two tentative split tracks. For this particular scenario, it is selected as 20 m tcS is the split confirmation threshold and is selected as 0.95. sI1 and sI2 are the indexes of any two tentative split tracks under considerh i sI ation. b x kjk ; ℙsI is the split track trajectory estimate and its error kjk covariance matrix calculated using standard JIPDA algorithm (whenever tracks share some measurements). The confirm split tracks propagation scenario is presented in Fig. 3. In the initial scans, track-lets are identified as tentative split tracks. If both the conditions in Eqs. (41) and (42) are satisfied, then tentative split tracks attains the status of confirmed split track in scan k þ Ns . Where Ns is the number of additional scans used for the split track confirmation.
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M. Asad et al. / Defence Technology xxx (xxxx) xxx Table 1 Algorithm for SE-JIPDA 1. Split event (SE) Initialization i At time k, measurements Yk from sensor. ii Split event initialize when two or more measurements in the validation gate. iii Clusters are formed based on association probability. iv When tentative split tracks share measurements Joint Integrated Probabilistic h i sI Data Association (JIPDA) is used for calculating b x kjk ; P sI kjk v Data association probability is calculated. vi Measurement likelihood for each measurement is calculated. pk ðiÞ ¼ p yk ðiÞ qk ðiÞ; ck ; Y k1 2.Computation of Split probability P skI 1 0 n o LsI sI k k sI sI AP sI @ Pk ¼ P Sk ck ; Y ¼ kjk1 sI sI þ LsI 1 Pkjk1 k Pkjk1 where
n o sI k1 sI ¼ P SsI c ;Y Pkjk1 k k And
0 1 msIk X psI sI sI k1 k ðiÞ A @1 P t P t þ P t P t LsI ¼ pr;k mf msI d g d g k ¼ p y k Sk ; ck ; Y k j¼1
Fig. 3. Progress of confirmed split tracks.
2.6.2. Merger of tentative split tracks If the tentative split tracks persist to follow the same path in parallel for “Nm ” scans, then these tracks will be merged as they will only increase the computational complexity. The process is depicted in Fig. 4. In this particular case if the tentative split tracks satisfy the check (43) for “Nm ” consecutive scans, then the tentative split tracks are merged to create a single track.
rk
3. Confirmation of Split event Two conditions for split confirmation 1 sI T sI sI sI 1 2 b i b xk 1 b xk 2 þ P sI x k 2 > Ds P sI xk 1 b k k c i ii P sI s tS 4. Merging of split tracks follow same path for number of scans 1 sI T sI sI sI 1 2 b b xk 2 þ P sI x k 2 < Dm P sI xk 1 b xk 1 b k k
sI sI b xk 1 b xk 2
T
1 2 P sI þ P sI k k
1
sI sI b xk 1 b xk 2
< Dm
(43)
where Dm is the merging threshold between the tentative split tracks. In this study, }Nm } is selected as 7 scans, and Ds is selected as 5 m. Pseudo code for split event JIPDA (SE-JIPDA) is given in Table 1, which shows the overall flow of the split event detection and tracking mechanism as discussed in the previous sections. 3. Simulation study Ballistic missile trajectory is shown in Fig. 5. Results are presented for two different cases. In Case-1, target and its decoy are far from each other as compared to Case-2, where the target and its decoy is close to each other. Both cases are presented in Fig. 5. Initial position of the
Fig. 4. Parallel progress of two tentative split tracks.
Fig. 5. Ballistic missile trajectory with decoys.
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parent target is [100 m;1290 m]. At scan 29, the parent target splits into two track-lets creating a multi-target scenario in the surveillance area. Each track-let is considered as a target to be tracked in the presence of clutter. In Case-1, the velocities are [2500;-500] m= s and [2400;-700] m=s for parent target and split target, respectively. In Case-2, the velocities are [2500;-500] m=s and [2400;600] m=s for parent target and split target, respectively. The surveillance area under observation is [3000 m 800 m]. In each case, the comparison is made for different values of detection probability Pd (for clarity in the figures, Pdt is replaced by Pd ), while keeping the same clutter density. Simulations are carried out in MATLAB with a total of 100 Monte Carlo runs with each Monte Carlo run having 50 scans.
3.1. Case-1 analysis Simulation results in Case-1 are compared for different values of Pd (1; 0:85;0:7) by keeping the same clutter density (r ¼ 106 m2 ). With the low probability of detection, the RMSE increases as shown in Fig. 6. The RMSE plot is cumulative for both parent and split track. In the splitting phase, the RMSE increases due to the fact that both parent track and split track RMSE’s are added up. Fig. 7 shows the confirmed true track rate for the parent track. A small variation can be observed in the scan interval 27 to 30, when parent target
Fig. 8. Case-1: Confirmed split true track rate (CSTTR) of split track r ¼ 106 m2 ;Pd ¼ 1; 0:85; 0:7.
Fig. 9. Case-2: Cumulative RMSE for both parent and split tracks r ¼ 106 m2 ; Pd ¼ 1; 0:85; 0:7.
Fig. 6. Case-1:Cumulative RMSE for both parent and split tracks r ¼ 106 m2 ; Pd ¼ 1; 0:85; 0:7.
shares the measurements with the split target. Confirmed split true track rate (CSTTR) characteristic is analyzed for the Case-1 for different values of Pd as shown in Fig. 8. It is a unique performance parameter of the proposed algorithm, as it helps in the analysis of split detection and its tracking. In Fig. 8, the CSTTR starts to increase showing the presence of split and attains
Fig. 7. Case-1:Confirmed true track rate (CTTR) of parent track r ¼ 106 m2 ; Pd ¼ 1; 0:85; 0:7.
Fig. 10. Case-2: Confirmed true track rate (CTTR) of parent track r ¼ 106 m2 ; Pd ¼ 1; 0:85; 0:7.
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3.4. Comparison with IPDA and ITS algorithms
Fig. 11. Case-2: Confirmed split true track rate (CSTTR) of split track r ¼ 106 m2 ;Pd ¼ 1; 0:85; 0:7.
the maximum value till the end of the complete run.
3.2. Case-2 analysis In Case-2, both target and decoy are close to each other. In this particular case, multiple tracks share measurements for a greater number of scans as compared to Case-1. In Figs. 9e10, there is comparison of RMSE and CTTR for r ¼ 106 m2. It is visible that RMSE increases and CTTR slow down with a decrease in the value of Pd . Fig. 11 shows the CSTTR for the Case-2.
Integrated track splitting (ITS) is a multi scan single target tracking algorithm. The ITS algorithm relates more number of scans to determine the target trajectory state estimate in the presence of clutter. Integrated probabilistic data association (IPDA) is a single scan single target tracking algorithm. IPDA assumes point target assumption and merges all PDF’s as one Gaussian PDF in each scan. The single PDF propagates in the next scan for measurement selection and update process. The proposed SE-JIPDA algorithm is compared with the IPDA and ITS algorithms. Results are presented in Figs. 12e13 for cumulative RMSE and CTTR for the Case-1, respectively. Similarly, Figs. 14e15 presents the results for Case-2 for both cumulative RMSE and CTTR, respectively. It is evident that SE-JIPDA performance is much better as compared to the IPDA and ITS algorithms in terms of RMSE and CTTR. Split event true track rate (CSTTR) cannot be compared with both algorithms (ITS and IPDA) as they do not provide the procedure for split event detection and its state update.
4. Conclusion In this work, split event detection and tracking algorithm is
3.3. Comparison of Case-1 and Case-2 The CTTR performance for both cases is identical as the same clutter density and target motion model is considered. Due to the large separation in Case-1, targets share measurement for the less number of scans as compared to Case-2 (small separation). Thus the CTTR performance in Case-1 (Fig. 7) is better in the measurement sharing region as compared to Case-2 (Fig. 10). The CSTTR in the Case-1 (Fig. 8) is also fast as compared to Case-2 (Fig. 11). Case-1 confirms the split track at faster rate as compared to the Case-2. The RMSE is also affected in both the cases due to the different separation distances in both cases.
Fig. 13. Case-1: Cumulative CTTR of both parent and child tracks r ¼ 106 m2 and Pd ¼ 0:7.
Fig. 12. Case-1: Cumulative RMSE of both parent and child track r ¼ 106 m2 and Pd ¼ 0:7.
Fig. 14. Case-2: Cumulative RMSE of both parent and child tracks r ¼ 106 m2 and Pd ¼ 0:7.
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Fig. 15. Case-2: Cumulative CTTR of both parent and child tracks r ¼ 106 m2 and Pd ¼ 0:7.
proposed in the presence of clutter. The proposed algorithm is termed as split event joint integrated probabilistic data association filter (SE-JIPDA). This algorithm provides a mechanism for split event detection and it tracks each split in the presence of clutter. The split event detection mechanism helps to monitor the performance of both the parent and split targets. The performance of the proposed SE-JIPDA algorithm is compared with different algorithms under different simulation conditions. The results show a significant improvement in terms of true track rate and root mean square error performance compared to the existing algorithms. In the future, this work could be extended to multi-scan tracking algorithms and multiple model systems. References [1] Singh NK, Bhaumik S, Bhattacharya S. A comparison of several nonlinear filters for ballistic missile tracking on re-entry. In: IEEE first international conference on control, measurement and instrumentation (CMI); 2016. p. 459e63. 2016. [2] Julier S, Uhlmann J. Unscented filtering and nonlinear estimation’. Proc IEEE Mar 2004;92(3):401e22. [3] Bar-Shalom Y, Fred D, Huang J. The probabilistic data association filter’. IEEE Control Syst 2009;29(6):82e100.
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Please cite this article as: Asad M et al., A split target detection and tracking algorithm for ballistic missile tracking during the re-entry phase, Defence Technology, https://doi.org/10.1016/j.dt.2019.12.008