SUBTITLE A Multiple Model Filter Without Markov Switching

When tracking with multi-sensor systems, the set of sensors used may be asynchronous and there may be communications delays between sensor platforms and the fusion center. Despite these conditions, it is desirable that each sensor maintains an accurate track. It has been recognized for some time that the use of a multiple model filter is superior to the use of a single model filter for tracking maneuvering targets. However, existing multiple model tracking algorithms use Markov switching, assuming that the likelihoods of the target state switching between kinematic models are known. The objectives of this paper are twofold. First, it will present a Multiple Model (MM) tracking algorithm, called the ARMM algorithm, that does not assume a priori knowledge of the target transition probability matrix. This work attempts to relax some of the assumptions found in the most widely used MM tracking algorithm. Second, it will be shown that the ARMM algorithm can also be used as the second, and final, stage in a logical process for fusing asynchronous tracks from multiple sensors that use different kinematic models in their individual track filters. 1.0 INTRODUCTION Assume the existence of an algorithm that can optimally fuse asynchronous track data from multiple sensors using filters with different target motion models. An optimal, Multiple Model (MM) filter would be a special case of this algorithm and could be obtained by: (1) requiring the measurement provided to each filter to be the same, and (2) assuming that the measurements are taken at the same time (synchronously). From this realization, the authors were led to consider an analytic approach used to derive two, asynchronous, track fusion algorithms for multiple sensors, whose filters use the same model, as the approach for deriving a MM filter. This resulting algorithm, which will be called the ARMM filter, uses only the filter residuals to provide an on-line capability for weighting the output of the two local tracks. This algorithm does not require a priori knowledge of the target state transition likelihoods. Additionally, it will be demonstrated how tracks of the same target from different sensors and generated by filters using different kinematic models (it is possible that the individual tracks can each be from a different sensor or, in other instances, that some of the sensors are producing more than one track from filters with different target models) can be combined to produce a fused system track using the ARMM. This is accomplished by applying either of the track fusion algorithms multiple times and then combining the results of the individual applications with the ARMM. This paper provides building blocks to be used REPORT DOCUMENTATION PAGE Form Approved OMB No. 0704-0188 Public reporting burder for this collection of information is estibated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing this collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burder to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 1. REPORT DATE (DD-MM-YYYY) 01-03-1998 2. REPORT TYPE Conference Proceedings 3. DATES COVERED (FROM TO) xx-xx-1998 to xx-xx-1998 4. TITLE AND SUBTITLE A Multiple Model Filter Without Markov Switching Unclassified 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) Rice, T. R. ; Alouani, A. T. ; 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME AND ADDRESS Systems Research and Technology Department Naval Surface Warfare Center Dahlgren Division Dahlgren, VA22448-5100 8. PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME AND ADDRESS Director, CECOM RDEC Night Vision and Electronic Sensors Directorate, Security Team 10221 Burbeck Road Ft. Belvoir, VA22060-5806 10. SPONSOR/MONITOR'S ACRONYM(S) 11. SPONSOR/MONITOR'S REPORT NUMBER(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT APUBLIC RELEASE , 13. SUPPLEMENTARY NOTES See Also ADM201041, 1998 IRIS Proceedings on CD-ROM. 14. ABSTRACT When tracking with multi-sensor systems, the set of sensors used may be asynchronous and there may be communications delays between sensor platforms and the fusion center. Despite these conditions, it is desirable that each sensor maintains an accurate track. It has been recognized for some time that the use of a multiple model filter is superior to the use of a single model filter for tracking maneuvering targets. However, existing multiple model tracking algorithms use Markov switching, assuming that the likelihoods of the target state switching between kinematic models are known. The objectives of this paper are twofold. First, it will present a Multiple Model (MM) tracking algorithm, called the ARMM algorithm, that does not assume a priori knowledge of the target transition probability matrix. This work attempts to relax some of the assumptions found in the most widely used MM tracking algorithm. Second, it will be shown that the ARMM algorithm can also be used as the second, and final, stage in a logical process for fusing asynchronous tracks from multiple sensors that use different kinematic models in their individual track filters. 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT Public Release 18. NUMBER OF PAGES 20 19. NAME OF RESPONSIBLE PERSON Fenster, Lynn [email protected] a. REPORT Unclassified b. ABSTRACT Unclassified c. THIS PAGE Unclassified 19b. TELEPHONE NUMBER International Area Code Area Code Telephone Number 703767-9007 DSN 427-9007 Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39.18 in the design of track fusion systems, where the tracks are provided by different sensors. These tracks may be asynchronous, have different communications delays, and be generated using different models of the same target. This paper is organized as follows. In section 2, background on target tracking and track fusion, as it relates to MM algorithms, is presented. Next, the analytical approach is presented in section 3. Two track fusion algorithms derived using the analytical approach are presented in section 4. In section 5, the derivation of the ARMM algorithm is presented, while in section 6, simulation results obtained using the ARMM to track a simulated target are given. A summary and conclusions are given in section 7. 2.0 BACKGROUND 2.1 TARGET TRACKING For many target tracking applications, it is not possible to predict the beginning and end of a target maneuver. Hence, when tracking a target with unknown and variable dynamics, it is difficult to decide on the dynamics model to be used by the tracking filter during a given time segment of its trajectory. Existing approaches can be divided into two classes: the Single Model (SM) approach and the Multiple Model (MM) approach. In the SM approach only a single model filter is used at a given time (see [1]-[4], among others). The tracking starts with a filter that uses a Constant Velocity (CV) model and a low process noise until a maneuver is detected. Upon maneuver detection, the state process noise of the filter is increased in [1], while [2] estimates the target acceleration and treats it as an input to the filter. In [3], the filtered state is augmented during a maneuver, while in [4] the target acceleration is treated as a bias whose estimate is used to correct the output of the CV filter. The success of the single model approach is heavily dependent on the quality of the maneuver detector used. The MM approach employs multiple filters at any given time, each with a single different model. The outputs of these filters are then weighted to obtain the final target track. Probability techniques are used to determine the weights to be assigned to these tracks. Since a number of models (filters) are used in parallel, contrary to the SM approach, there is no hard switching between models. This approach eliminates the need for maneuver detection and filter initialization. It is worth noting that, conceptually, this MM approach can lead to an optimal solution to the tracking problem. However, the optimal solution will become intractable due to the exponential growth with time of the number of track histories. For this reason sub-optimal solutions are considered (see [5] and [6]), where a limited number of filters are used. For example, the filter used in [6], known as the Interacting Multiple Model (IMM) filter, uses only two models: a CV and a CA (Constant Acceleration) model. One of the shortcomings associated with existing MM approaches is that the knowledge of the probability that governs the likelihood of transitions between dynamic models (called switching coefficients) is assumed known. This assumption requires a priori knowledge about the target’s trajectory and is clearly not reasonable when it comes to tracking unknown targets in real time. As a result, it is necessary to use offline techniques, such as running multiple Monte Carlo experiments using different trajectories and a number of “candidate” switching coefficients, to determine the “best” switching coefficients. Another, perhaps mild, deficiency of the IMM is it’s use of the approximation that a sum of Gaussian probability functions (pdfs) is itself a Gaussian pdf 2.2 TRACK FUSION In the past, track fusion algorithms have not been considered as the primary means of fusing multisensor data for several reasons. First, fusion algorithms that could be shown to optimally combine tracks from sensors that reported their tracks asynchronously (either because a target’s position was measured at different times or because of communications delays or both) did not exist. This reason has recently been negated by [7] for the case where all of the sensors use track filters having the same dynamical model. In order to obtain a globally optimal solution for this case, it is necessary to feed the fused track back to each of the sensors to be used as the input state [7]. Of course, this still leaves to be solved the case where there are multiple subsets of track filters with each subset using a different dynamical model in it’s filters. Second, single model filters, such as the Kalman filter, do not realistically reflect the true covariance of the track except when their dynamical model matches the dynamics of the target. Otherwise, the covariance provided by the filter is smaller than the true covariance. Since most known track fusion algorithms utilize the inverse of the covariance of the track to weight the contribution of its state vector to the fused state vector, filters that are not doing well because of a model mismatch are contributing more than they should to the fused track. If one decides to use the SM approach to alleviate this problem, there are still grave shortcomings. SM approaches can quickly discern when a target starts maneuvering, however, they cannot accurately determine when the target has stopped maneuvering and returned to straight and level motion and will continue to track with the maneuver model filter. When this occurs, the SM model will not provide a good estimate of the covariance. Also, when the SM model transitions from one model to another, one must wait for the new filter to settle before the covariance can be considered meaningful. On the other hand, the MM approach provides a soft switching mechanism between models that precludes the need for filter initialization and settling. Also, the ability of MM algorithms to detect a maneuver and respond to that maneuver by increasing the weight given to the filter with the more accurate target motion model is good. Therefore, MM algorithms seem to be the best candidate for filtering data from multiple sensors when the tracks from these sensors are to be fused. However, a logical approach to fusing MM algorithms is needed. 3.0 ANALYTICAL APPROACH Begin by considering the following generic system. There exist a target being observed by one or two sensors whose dynamics of motion are assumed to be governed by ) ( ) ( ) ( t W G t AX t X + = & , (1) where ) (t X is the target state and ) (t W is a zero mean Gaussian process with covariance ) ( ) ( ] ) ( ) ( [ τ δ τ − = ′ t t q W t W E . (2) There are two different paths that can be taken from this point. 3.1 INITIAL ASSUMPTIONS FOR THE FIRST PATH The first path, which will be denoted as the MSSM (Multi-Sensor, Single Model) process, assumes that there are two sensors, each with a single Kalman filter to produce local tracks and each using the same dynamical model of the target. Also, it is assumed that, in general, the time at which each sensor takes a measurement of the target’s position is different, in other words the sensors are asynchronous, 2 , 1 , 1 = < < − i t t t k i k (see Figure 1). The measurement model for the i th filter, i=1, 2, is given by ) ( ) ( ) ( i i i i i i t V t X H t z + = , (3) where ) ( k i t V is a zero mean, white, Gaussian, measurement noise process. Here ) ( ) ( ) , ( ) ( 1 1 k k k k k t W t X t t t X + Φ = + + , (4) ) ( 1 1 ) , ( k k t t A k k e t t − + + = Φ , (5) ∫ + + Φ = 1 ) ( ) , ( ) ( 1 k k t t k k d W G t t W τ τ τ , (6) kj k j k t Q t W t W E δ ) ( )] ( ) ( [ = ′ , (7) kj k i j i k i t R t V t V E δ ) ( )] ( ) ( [ = ′ , (8) ∫ + + Φ′ ′ + Φ = 1 ) , 1 ( ) ( ) , 1 ( ) ( k k t t d k t G Gq k t k t Q τ τ τ τ . (9) and the prime is used to denote matrix transpose (See [7] or [8]). 3.2 INITIAL ASSUMPTIONS FOR THE SECOND PATH The second path, denoted as the SSMM (Single Sensor, Multi-Model) process, assumes that there is only one sensor and two Kalman filters are processing it’s measurements. Therefore the tracks produced by these sensors are synchronous. However, it is assumed that one filter has a CV model (call this LP1) while the other has a CA model (call this LP2), which leads to two target motion models given by: LP1: ) ( ) ( ) , ( ) ( 1 1 1 1 1 1 1 k k k k k t W G t X t t t X + Φ = + + , (10) LP2: ) ( ) ( ) , ( ) ( 2 2 2 1 2 1 2 k k k k k t W G t X t t t X + Φ = + + , (11) where ) , 0 ( ~ ) ( 1 1 q N t W k , (12) ) , 0 ( ~ ) ( 2 2 q N t W k , (13) ' 1 ] [ ) ( z z y y x x t X k & & & = , (14) ' 2 ] [ ) ( z z z y y y x x x t X k & & & & & & & & & = . (15) It is worth noting here that ) ( ) ( 2 1 k k t MX t X = , (16) where