Distributed Multirate Interacting Multiple Model (DMRIMM) Filtering with Out-of-Sequence GMTI Data 1

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1 Distributed Multirate Interacting Multiple Model (DMRIMM) Filtering with Out-of-Sequence GMTI Data Lang Hong Shan Cong Devert Wicker Department of Electril Engineering Target Recognition Branch Wright State University Air Force Research Laboratory Dayton, OH Wright-Patterson AFB, OH Abstract - This paper develops a distributed approach for fusing GMTI data with out-of-sequence (OOS) measurements. A multirate interacting multiple model (MRIMM) fusion algorithm is developed for e ectively fusing multirate information. To incorporate an OOS measurement, two major steps are needed: retrospection from current time to OOS time updating the current estimate with the OOS measurement, which imposes a high computation memory burden on implementing OOS ltering. The multirate approach provides an excellent framework for e±cient information retrospection forward update. The combination of global MRIMM fusion lol MRIMM tracking proves to be powerful for tracking fusing maneuvering nonmaneuvering targets in an environment of OOS measurement reporting. Keywords: Tracking, multirate ltering, interacting multiple models, out-of-sequence measurements. Introduction Ground moving target indictor (GMTI) radar has demonstrated its powerful surveillance/ reconnaissance pability in many military law enforcement operations. Its ability to timely provide detailed information throughout the theater is critil to real-time comm control in a battle eld. In the near future, a surveillance operation may include a network of GMTI platforms. With each GMTI radar operating at several modes including wide area search sector search, this GMTI radar network shall provide an unprecedented amount of data regarding numerous targets in a surveillance region, from small units to major enemy forces. As there are thouss of objects in a surveillance region, an enormous tracking e ort is needed to correlate radar measurements into target trajectories. Meanwhile, under a combat condition, the measurements from multiple platforms won't be synchronized, beuse each individual platform has its own sn rate the communition network nnot guarantee to deliver measurements on time. Asynchronized measurements inevitably make the order of measurements uncertain, which creates an out-of-sequence (OOS) reporting phenomenon. Tracking with OOS measurements poses a challenge to GMTI tracker design. Although the delay of each OOS measurement may be only a fraction of a sn to a few sns, this challenge nnot simply be ignored, beuse this phenomenon is due to the nature of networking is expected to occur frequently. Processing of networked GMTI data with OOS measurements n be performed in two approaches: centralized distributed. In the centralized approach, reports (including OOS measurements) from all GMTI radars are sent to a central processing station (most likely, on the ground) a global picture of the combat scenario is formed, which is then sent to all GMTI platforms if needed. While the centralized fusion is straightforward, time delay of the availability of the global picture to lol users could be of concern, when lol combat decisions need to be made in real time. The time delay issue could be much alleviated if GMTI reports are processed in groups according to their geographil lotions /or their sampling rates. We have successfully developed an e ective e±cient centralized technique for fusing GMTI data with OOS measurements using the MRIMM approach [7]. This paper addresses a distributed fusion approach for processing GMTI data with OOS measurements. A functional diagram of the proposed approach is shown in Figure, sensors A B are grouped together in processor sensors C D are fused in processor. Due to the constraints on computational resources, both processors employ the MRIMM approach. The introduction of the MRIMM for ltering with OOS measurements aims at resolving three key issues: Supported in part by the Target Recognition Branch, Sensor ATR Technology Division, Air Force Research Laboratory by the Air Force Electronic System Center. 54

2 () an e±cient processing structure for information retrospection, () an e±cient memory structure for storing historil information, (3) an e±cient computational structure for tracking nonmaneuvering maneuvering targets. The development of the MRIMM is based on our recently developed multirate techniques [7, 8, 9,,,, 3, 4, 5, 6]. Other related work on OOS measurements processing n be found in [6, 8, 9, ]. k - τ A τ B most recent OOS measurement k - k - κ k Processor : MRIMM-OOS Filtering Processor : MRIMM-OOS Filtering τ D τ C MRIMM-OOS Track Fusion current measurement Sampling Time at Sensor A Arrival Time at Processor Sampling Time at Sensor B Timing Stamp Sequence at Processor Fused Track with OOS Measurements Timing Stamp Sequence at Processor k - κ k most recent OOS measurement current measurement Sampling Time at Sensor D Arrival Time at Processor Sampling Time at Sensor C Figure. Distributed fusion of GMTI data with OOS measurements using the MRIMM. Review of Multirate Filtering Multirate processing for stochastic systems has been developed by the rst author over the last dede for computational e±ciency, performance robustness, memory/communition e±ciency. A variety of multirate processing algorithms have been developed for di erent applitions. A partial list of the development n be found in references [7, 8, 9,,,, 3, 4, 5, 6]. In general, multirate ltering is referred toaprocessingschemetheupdaterateisslower than the measurement sampling rate. However, if the measurements are acquired at multiple rates (multirate measurements), multirate ltering n be directly applied. Figure demonstrates a multirate ltering scheme measurements could either be full-rate or multirate. For full-rate measurements, a measurement decomposition module is used to decompose a string of full-rate measurements into a set of lower rate measurements. The commonly used decomposition approach is based on the wavelets theory which has a nice property of orthogonality in mappings. A simple example of half-rate decomposition is given here. Given a set of full-rate measurements, z k, the half-rate measurements n be derived using two-tap lters z kl = h()z k + h()z k () z kh = g()z k + g()z k () z kl z kh are lowpass highpass quantities available only at every two sampling points, thus the name, half-rate. After the decomposition, multirate measurements are sent to an optimal multirate ltering module which runs on multirate models. Different kinds of multirate models have been developed [9, 3, 4] which are a group of interval models different from conventional point models. In some applitions, interval models signi ntly outperform point models. The optimal multirate ltering module generates multirate estimates which n be directly used in many applitions. However, if full-rate estimates are needed, a reverse mapping (multirate mapping) n be used. The following is obtained by using a multirate lter. () Higher computational e±ciency. Since most of the computational complexity is in the ltering process, a lower update rate will result in a lower computational burden; the lower the rate, the more the savings. () Lower memory requirements. A lower rate of estimates is equivalent to lower memory storage requirements. The compression property of the multirate lter reduces the bwidth requirements when communiting multirate estimates over a network. (3) Higher performance robustness. The multirate interval models exploit better multisn information, which creates better performance in some applitions. (4) E±cient information retrospection. Due to the nature of multisn processing, the reverse mapping from multirate estimates to full-rate estimates (retrospection) is simple straightforward. full-rate measurements measurement decomposition multirate measurements full-rate estimate output multirate models optimal multirate filtering multirate mappings multirate estimate output Figure. A functional diagram of multirate ltering. 55

3 3 Distributed Multirate Interacting Multiple Model Fusion A multisensor dynamic system with switching models is given by M ch k ;k =[M ch k ;M ch k ] (8) M k 3;k =[M k 3;M k ;M k ;M k ]: (9) x k+ = F(M k )x k + v k (M k ) (3) z j k = H j (M k )x k + w j k (M k );j=;:::;n (4) M k fm i g r i= r is the number of models a same set of models is used by all sensors the central system. At time t k, the lth model history is denoted by x^ k-4 k-4 ch x^ k-4 k-4 x^ k-4 k-4 Mixing _ x k-4 _ ch x k-4 M k ch x^ M ch k- k- ch K- x^ M ch k- k- k ch x^ k- k- M x^ k-3 k-3 x^ k-3 M k- k- x^ k- k- _ x x^ k- M k- M k-4 k-3 k-3 x^ k- k- x^ k- k- k x^ k k x^ k k x^ k k ch x^ k k ch x^ k k x^ k k ch x^ k k x^ k k Mixing _ x k Level _ ch x k _ x k Level Level M k;l = fm i;l ; :::; M i k;l g; l =;:::;r k x^ k-3 k-3 x^ k- k- x^ k- k- x^ k k i k is a model sequence index. As time progresses, the number of possible apriorihistory sequences increases exponentially, as indited by the range of l. The a posteriori state estimate n be obtained by evaluating the conditional mean ^x k = Efx k j k g (5) k = fz l g k l= z l = fz i lg N i=: (6) The conditional pdf n be generated using the total probability theorem p[x k j k ]= r k X l= p[x k jm k;l ; k ]P fm k;l j k g: (7) Again, the number of terms necessary to lculate the pdf, p[x k j k ], increases exponentially with time. A good approximation approach for a single-sensor system is the interacting multiple model (IMM). The multirate approach to the single-sensor IMM algorithm has been developed by Hong [9] each model operates at an update rate proportional to the model's assumed dynamics computational e±ciency is achieved. In the following, an algorithm for multisensor multirate interacting multiple model tracking will be developed. In the following, a three-model (one maneuvering constant acceleration () model two other nonmaneuvering constant highpass (ch) models) multirate (quarter-rate/half-rate/full-rate) distributed IMM algorithm will be developed. A signal owchart of DM- RIMM for two platforms is shown in Figure 3. Due to the speci cs of multirate modeling, each multirate universal model set involves four time instants fmk ;Mk ;k;m ch k 3;kg x^ k-3 k-3 x^ k- k- x^ k- k- Combined Fusion Outputs Figure 3. A signal owchart of DMRIMM. Using the total probability theorem, the pdf (p[x k j k ]) is expressed as p[x k j k ] = p[x k j k ;M k 3;k]¹ k 3;k + z ch p[x k jz ch ;z ch ; ;M ch k ;k]¹ ch k ;k +p[x k jz ; ;Mk ]¹ x^ k k k () ¹ k 3;k = P fm k 3;kj k g; () ¹ ch k ;k = P fm ch k ;kjz ch ;z ch ; g () ¹ k z ch = P fmk jz ; g: (3) are hlaf-rate decomposed measurements z is a quarter-rate decomposed measurement. Level estimate updates n be rried out as p[x k j k ;Mk 3;k ] = p[z k jx k ;M k ] C p[x k jx k ;Mk ]p[z k jx k ;M k ] p[x k jx k ;M k ]p[z k jx k ;M k ] p[x k jx k 3 ;M k ]p[z k 3 jx k 3 ;M k 3 ] p[x k 3 jx ;M k 3 ]p[x j ;M k 3;k ]dx :::dx k (4) which is a full-rate Kalman ltering process from (k-4) to k, generating ^x k 3jk 3, ^x k jk, ^x k jk ^x kjk 56

4 (Figure 3) their associated estimate error covariances. Since z k = fz j k gn j= (Eq. (6)), the estimate at level, e.g., ^x k ijk i, is actually the fused estimate of multisensor measurements, with an assumption that the target kinematics n be characterized by a model. Since we are interested in a distributed ltering scheme, i.e., lol measurements are processed lolly only the su±cient statistics are sent to the central unit, we need to exp the central ltering process as a fusion process, which will be detailed as follows. p[x k i j k i ;Mk 3;k]; i =; ; ; 3 = p[z k i jx k i ;M k i] p[x k i jx k i ;Mk i] C p[x k i j k i ;M k 3;k]dx k i (5) p[z k i jx k i ;Mk i ] NY = p[z j k i jk i ;j ;Mk i ] p[x k i j k i;j ;Mk i ] p[x k i j k i ;j ;M j= k i ] (6) the lol measurement history, k i;j = fz j l gk i l=, is used. Eqs. (5) (6) together describe a model fusion process: ² Previously fused estimate ^x R xk i p[x k i j k i ;M propagated to k i (Eq. (5)). k i jk i = k 3;k ]dx k i ² Each lol processor generates its own lol estimate at time k i transmits its lol estimate via p[x k i j k i;j ;Mk i ]tothecentralunit(eq. (6). ² Since the lol propagated quantity p[x k i j k i ;j ] is contained in the central propagated quantity, it needs to be removed before fusion, forming an information increment from lol processor j: p[x k i j k i;j ;M k i ] p[x k i j k i ;j ;M k i is ] (Eq. (6). ² Lol information incremetns are combined with the weight being a likelihood function: p[z j k i jk i ;j ;Mk i ] (Eq. 6). ² The combined information increment from all lol processors is then fused with the central propagated quantity to generate the current fused result (Eq. (5)). For the linear system speci ed in Eqs. (3) (4), the fusion (without feedback) equations become (P k ijk i ) =(P k ijk i ) + NX (P j ) (7) j= ^x k ijk i = P k ijk i [(P k ijk i ) ^x k ijk i + NX ^x j k ijk i ]; i =; ; ; 3 (8) j= the information increments are de ned as (P j ) =(P j k ijk i) (P j k ijk i ) (9) ^x j k ijk i = (P j j k ijk i ) ^x k ijk i (P j j k ijk i ) ^x k ijk i : () Similar ltering fusion processes exist for half-rate quarter-rate ch models. For the half-rate process at level, we have the following p[x k jz ch ;z ch ; ;Mk ;k ch ] = p[zch LH jx k k ;Mk ch ] C p[x k jx k ;Mk ch ]]p[z ch jx k ;Mk ch ] p[x k jx ; [M ch k ]p[x j ;M ch k ;k ]dx dx k () which is a half-rate ltering fusion process, yielding ^x ch k jk ^x ch kjk, their covariances. The halfrate fusion process n be further described as (P ch k ijk i ) =(P ch k ijk i ) + NX (P chj ) j= () ^x ch k ijk i = P ch k ijk i [(Pch ch k ijk i ) ^x k ijk i + NX ^x chj k ijk i ] (3) j= the half-rate information increments are de ned as (P chj ) =(P chj k ijk i ) (P chj k ijk i ) (4) ^x chj k ijk i = (P chj chj k ijk i ) ^x k ijk i (P chj chj k ijk i ) ^x k ijk i : (5) The quarter-rate ltering fusion process n be derived in a similar manner. Now we will derive the model probability mixing fusion formulas for the fusion unit. For the model, we have ¹ k 3;k = C = P fmk 3;k jk g 3Y p[z k i j k i ;Mk i]p fmk 3j g (6) i= 57

5 p[z k i j k i ;M k i] = NY j= p[z j k i jk i ;M k i] (7) is a fusion of lol likelihood functions. P fmk 3 j g in Eq. (6) n be further exped into a mixing process P fm k 3 j g = p ¹ + pch ¹ch + p ¹ = ¹ : (8) p, pch,p are model transitional probabilities at t. Substituting Eqs. (7) (8) into Eq. (6) results in a model probability mixing updating process. Similar model probability mixing updating processes n be obtained for the half-rate ch quarter-rate ch models. Finally, we will derive a mixing process for the multiple model fused estimates. p[x j ;Mk 3;k ]in Eq. (4) n be further exped over another universal multirate model set resulting in p[x j ;M k 3;k ] fm ;M ch k 6;;M k 7;g = ¹ p[x jmk 3;k ; f^x j ; P j g]p ¹ + ¹ p[x jmk 3;k ; f^xch j ; Pch j g]pch ¹ch + ¹ p[x jmk 3;k ; f^x j ; P j g]p ¹ : (9) Under the assumption that the pdf's in Eq. (9) are Gaussian, one n see that Eq. (9) describes a mixture of Gaussian pdf's, n be approximated by a single Gaussian pdf using moment matching. The approximation is detailed as follows p[x j ;M k 3;k ]»= N [x ; Efx jmk 3;k ; [ ¹ ^x j p ¹ + ¹ ^x ch j pch ¹ch + ¹ ^x j p ¹ ]g;cov[ ]] (3) which is a mixing operation for the fused estimate at t. The mixed estimate for the model is denoted as x in (Figure 3). Similar mixing operations for fused ch models n be derived to generate x ch x as in (Figure 3). The overall fusion output as de ned by Eq. () is also characterized by a Gaussian mixture n be simpli ed by ^x kjk = ¹ k 3;k ^x kjk + ¹ ch k ;k^x ch kjk + ¹ k ^x kjk (3) P kjk = ¹ k 3;k[P kjk +(^x kjk ^x kjk )( ) > ]+ ¹ ch k ;k[p ch kjk +(^xch kjk ^x kjk )( )> ]+ ¹ k ;k[p kjk +(^x kjk ^x kjk )( ) > ]: (3) 4 MRIMM for Out-of-Sequence GMTI Measurements When each platform observes out-of-sequence measurements, the issue of distributed fusion with OOS measurements need to be addressed. Assume each lol processor implements a multirate interacting multiple model (MRIMM) algorithm with OOS processing pability sends lol OOS update to the central fusion unit to update the fusion results. An e±cient e ective optimal MRIMM algorithm for OOS processing has been developed by the authors n be found in [7]. A signal ow chart of MRIMM with OOS processing pability is shown in Figure 4 This paper is focused on OOS fusion updates. With the assumption that OOS update is performed at quarterrate (Figure 4), we n fuse the information increments due to the incorporation of an OOS measurement. Initialization Mixing full-rate measurement To multirate central fusion unit Combined lol outputs OOS measurement Mixing Quarter-rate CH model Half-rate CH model Full-rate CA model time (Level ) Combined lol output with OOS measurement (Level ) (Level ) OOS information increments to central fusion unit Figure 4. A signal owchart of MRIMM with OOS measurements. 5 Simulations In this section, a set of scenarios of two-platform single target with di erent maneuvering magnitudes is studied. A sample scenario of two platform tracking is shown in Figures 5 6 the true target trajectory is in solid curve, the on-time measurements are 58

6 in `*' additional OOS measurements are in `o'. The delay time is any between to 3 sns the maneuver in x y coordinates during time [4, 76] is [g -g] during [77, ] is [-g g]. For the rest of the time, the target makes no maneuver. Three models are used: one constant acceleration (CA) model for maneuvering targets, one half-rate constant highpass (CH) one quarter-rate highpass (CH) for nonmaneuvering targets. The measurement noise variance is, the sampling period is seconds. The model transitional probability matrix is given by 3 :98 : : H ij = 4 : :98 : 5 : : : :98 The stard deviation of measurement errors in both x y coordinates is m. Table shows the Monte Carlo simulation results of the distributed MRIMM- OOS algorithm the maneuvers range from to 4g. Both lol estimate RMS error fusion RMS error are given in Table. For comparison, distributed MRIMM results for no OOS measurements are shown in Table. For a particular scenario, (g =),Figure7 show the traces of the combined lol estimate covariance matrices with OOS measurements the trace of the fused estimate covariance. The traces distributed MRIMMwithnoOOSmeasurementsareshowninFigure 8. The reduced traces in notches in Figure 8 are due to the incorporation of OOS measurements. Table : RMS errors of distributed MRIMM with OOS measurements (5 Monte Carlo runs) Platform Platform Fusion Acceleration combined x combined y combined x combined y combined x combined y a x =;a y = a x =;a y = a x =;a y = a x =3;a y = a x =4;a y = Table : RMS errors of distributed MRIMM with no OOS measurements (5 Monte Carlo runs) Platform Platform Fusion Acceleration combined x combined y combined x combined y combined x combined y a x =;a y = a x =;a y = a x =;a y = a x =3;a y = a x =4;a y = By comparing Tables, Figures 6 7, it becomes obvious: () correct incorporation of additional OOS measurements reduces the estimate errors; () fusion of combined lol estimates generates superior results. 6 Conclusions A novel approach for distributed ltering with OOS measurements has been developed. Multirate interacting multiple model fusion (MRIMMF) e ectively resolves the issue of lacking single global model in a multiple model setup. On each lol platform, multirate interacting multiple model ltering proves to be e±cient e ective for information retrospection OOS measurement updates. Additional advantages of data storing computational processing when using the multirate structure are also obvious. The combination of global MRIMM fusion lol MRIMM ltering is powerful for tracking fusing maneuvering nonmaneuvering targets in an environment of OOS measurement reporting. 59

7 4 x 4 Platform : true, on-time OOS measurements.5 x 4 Combined traces: Blue-Platform, Green-Platform, red-fusion 3.5 y Traces x x 5 Figure 5. A sample scenario: platform samples Figure 7. Traces of combined estimate covariances for platforms, fusion with OOS measurements..5 x 4 Combined traces: Blue-Platform, Green-Platform, red-fusion 4 x 4 Platform : true, on-time OOS measurements 3.5 Traces y x x 5 Figure 6. A sample scenario: platform samples Figure 8. Traces of combined estimate covariances for platforms, fusion without OOS measurements. 6

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