Radar Tracking with an Interacting Multiple Model and Probabilistic Data Association Filter for Civil Aviation Applications

Transcription

1 Sensors 03, 3, ; doi:0.3390/s Article OPEN ACCESS sensors ISSN Radar racing with an Interacting Multiple Model and Probabilistic Data Association Filter for Civil Aviation Applications Shau-Shiun Jan * and Yu-Chun Kao Institute of Civil Aviation, National Cheng Kung University, ainan 700, aiwan; aocnsl@gmail.com * Author to whom correspondence should be addressed; ssan@mail.ncu.edu.tw; el.: (ext. 6369); Fax: Received: March 03; in revised form: 4 May 03 / Accepted: 5 May 03 / Published: 7 May 03 Abstract: he current trend of the civil aviation technology is to modernize the legacy air traffic control (AC) system that is mainly supported by many ground based navigation aids to be the new air traffic management (AM) system that is enabled by global positioning system (GPS) technology. Due to the low receiving power of GPS signal, it is a maor concern to aviation authorities that the operation of the AM system might experience service interruption when the GPS signal is ammed by either intentional or unintentional radio-frequency interference. o maintain the normal operation of the AM system during the period of GPS outage, the use of the current radar system is proposed in this paper. However, the tracing performance of the current radar system could not meet the required performance of the AM system, and an enhanced tracing algorithm, the interacting multiple model and probabilistic data association filter (IMMPDAF), is therefore developed to support the navigation and surveillance services of the AM system. he conventional radar tracing algorithm, the nearest neighbor Kalman filter (NNKF), is used as the baseline to evaluate the proposed radar tracing algorithm, and the real flight data is used to validate the IMMPDAF algorithm. As shown in the results, the proposed IMMPDAF algorithm could enhance the tracing performance of the current aviation radar system and meets the required performance of the new AM system. hus, the current radar system with the IMMPDAF algorithm could be used as an alternative system to continue aviation navigation and surveillance services of the AM system during GPS outage periods.

2 Sensors 03, Keywords: radar; Kalman filter; interacting multiple model; probabilistic data association filter; air traffic management. Introduction According to the International Civil Aviation Organization s plan, the current technology trend is to develop and implement Communications, Navigation, Surveillance, and Air raffic Management (CNS/AM) systems based on the Global Positioning System (GPS) to replace legacy Air raffic Control (AC) systems based on ground-based radar []. he Next Generation Air ransportation System (NextGen) will modernize the air traffic control system in the United States, and many of the foundational elements required to meet the predicted capacity and efficiency improvements rely on widespread use of precision Positioning, Navigation, and iming (PN) services provided by the GPS []. he Federal Aviation Administration (FAA) needs to ensure a sufficient bacup PN capability to reduce the riss of aviation users if GPS becomes unavailable. As described in the FAA Flight Plan [], the air traffic system of the future will be much more dependent on GPS. his strategic goal will not be achievable without a NextGen Alternate PN (APN), especially in the event of a GPS loss of service. he APN program ensures alternate PN services provided by the AC system, minimizes the economic impact of GPS outages, and supports air transportation s timing needs. he existing legacy navigation and surveillance infrastructure, i.e., VHF Omnidirectional radio Range (VOR), Distance Measuring Equipment (DME), Non-Directional Beacon (NDB), and secondary radar, may not achieve the desired level of performance for NextGen operation [3]. o improve AC tracing performance, one solution is to upgrade the AC radar beacon system. However, this solution is expensive. A more cost-effective solution is to improve the AC tracing algorithms, which does not require the implementation of new radar facilities. he present study thus proposes a radar tracing algorithm that meets the requirements of APN systems. he maor challenge and difficulty for AC tracing is tracing target maneuvering in a cluttered environment [4 7]. Bar-Shalom and Blom [4] introduce a tracing algorithm called the Interacting Multiple Model (IMM) estimator, which provides tracing estimates with significant noise reduction and fast response to sequences of aircraft maneuver modes [4,5]. However, the tracing of an aircraft in a cluttered environment might be a challenge due to the several observations for a single aircraft under such environment [6,7]. hat is, some tracing measurements do not originate from the target aircraft. herefore, the present study utilizes the Probabilistic Data Association (PDA) filter [6,7] to assign weights to the validated measurements. he PDA filter can extend the tracing capability to a highly cluttered environment. In order to gain possible improvement on the tracing performance, this study combines the IMM estimator and PDA filters to create an IMMPDA filter (IMMPDAF). Most studies on radar tracing algorithms use simulated data for analysis. o further evaluate the IMMPDAF, this study uses real flight radar data collected by the Civil Aeronautics Administration (CAA) of aiwan. A filter that uses the nearest neighbor method and the standard Kalman filter (NNKF) is commonly used for radar tracing systems. herefore, this paper compares the tracing performance of the IMMPDAF and the NNKF in terms of positioning accuracy. Finally, the validation of the IMMPDAF under APN

3 Sensors 03, requirements is discussed. Due to the complexity of the IMMPDAF, this paper also conducts a computational load study on the IMMPDAF and the NNKF. he rest of this paper is organized as follows: Section introduces the algorithms of the IMM estimator and the PDA filter. hen, the IMM estimator and the PDA filter are integrated. he dynamic model setting of the IMMPDAF for the AC scenario is also discussed in Section. he implementation of the IMMPDAF to a radar system is discussed in Section 3. he tracing performance of the IMMPDAF is compared with that of the NNKF. In addition, the computation loads of these filters are evaluated. Section 4 concludes this wor.. IMMPDAF.. IMM Estimator he IMM estimator is a good candidate for a radar tracing system [4]. he IMM estimator provides significant noise reduction and a fast response. Previous investigations have found that the IMM estimator is a cost-effective technique for tracing a maneuvering target [4]. In the IMM estimator, several Kalman filters are used in parallel. Each Kalman filter uses a different dynamic model. he IMM estimator can be separated into four steps. In the first step, the state estimate and covariance are mixed. hese mixed estimates and covariances are the inputs of the Kalman filter. he equations are: U i ( ) = r l= pu i l i pu ( ) ( ) r 0 i i i= l + X ( ) = X ( ) U ( ) () { } r i i i i i= P ( ) = U ( ) P ( ) + X ( ) X ( ) X ( ) X ( ) (3) In Equations ( 3), the subscripts i,, and l indicate different Kalman filters. here are up to r different dynamic models of Kalman filter, the total number for u i, X 0, X + i, P 0 and P + i is r, and the total number for U i and p i is r. u i () represents the model probabilities in the previous time (). Since the initial model probabilities have little impacts on the results over time, they can be conveniently chosen. In this study, the initial model probabilities are set to be rectangular functions. p i is an element of the Marov transition matrix; there is probability p i that the target will transit from model i to model at each time step. U i represents the conditional probability of the target in the model state, which transited from the i model state. According to [5], the IMM algorithm performance is robust to the choice of transition matrix. X + i () and P + i () are the updated state and covariance, respectively, from the Kalman filter in the previous time step (). X 0 () and P 0 () are the mixed state and the mixed covariance, respectively, for each Kalman filter as inputs at time. hen, the second step implements the multiple models of the Kalman filter: ()

4 Sensors 03, X ( ) F X ( ) 0 ( ) ( ) = P = F P F + Q ( ) ( ) ( ) = ( ) ( ) K = P ( ) H H P ( ) H + R + X X K Z H X + P I K H P ( ) = ( ) + ( ) ( ) (4) (5) he details of Kalman filter can be found in [8]. In Equations (4,5) the subscript indicates it is the th Kalman filter, and Equations (4,5) are applied for each Kalman filter. In Equation (4), F is the dynamic model and Q is the covariance of the process noise. X () and P () are the uncorrected predicted state and covariance, respectively. In Equation (5), H is the observation matrix, R is the covariance of the measurement noise, and K is the Kalman gain. Z() is the measurement at time and I is the identity matrix. X + () and P + () are the corrected predicted state and covariance, respectively. In the third step, the updated model probabilities are calculated as: S( ) = H P ( ) H + R y ( ) = Z( ) H X ( ) d ( ) = y ( ) S( ) y ( ) d ( ) (6) exp Λ ( ) = π S ( ) In Equation (6), the subscript indicates it is the th Kalman filter; exp[] means the exponential operation and represents calculating the determinant of the matrix. he lielihood function Λ is the probability density function of the estimated measurement H X (), which is a normal distribution with mean Z() and covariance S (). In Equation (7), the subscripts i and indicate different Kalman filters. u () represents the latest model probabilities at time. In the last step, the corrected predicted state X + () and P + () covariance from the Kalman filters are combined. he weighting factor is the latest model probabilities u (): r ( ) = + ( ) ( ) (7) r (8) = X X u { } ( ) = ( ) ( ) + ( ) ( ) ( ) ( ) P u P X X X X = u ( ) = r i= ( ) Λ ( ) ( ) Λ ( ) u u X() and P() are the combinations of the system state and covariance from to rth Kalman filters, respectively. Using Equations ( 9), the IMM estimator calculates the system state and covariance from time to. i (9)

5 Sensors 03, PDA Filter he PDA filter is designed for tracing a target in a cluttered environment based on the Kalman filter [6,7]. he PDA filter obtains an estimator which incorporates all the returning measurements that might originate from the target of interest rather than select only one of them. he resulting estimator sets a gate to determine whether the measurements are valid. he association probabilities of the target being traced for each validated measurement are then calculated. hese probabilities are assigned to all of the validated measurements with different weights. Next, the validated measurements are combined with different weights based on location, and then the combined measurement is used to update the state estimate of the target. he PDA filter can extend the tracing capability to a highly cluttered environment. he probabilities are used for weighting the correction and covariance in the PDA filter. he basic assumptions and theories for the PDA filter can be found in [6,7]. Here, a brief introduction of its functions and equations is given. Because the PDA filter is based on the Kalman filter, the first equations are the same as those for the Kalman filter: + X ( ) = F X ( ) zˆ ( ) = H X ( ) + P ( ) = F P ( ) F + Q Similarly, the first step is to predict the system state and covariance. In Equation (0), the subscript indicates it is the th PDA filter, X + () is the calculated state at time, F is the dynamic model, and X z is the predicted () is the predicted state at time. H is the observation matrix and ( ) measurement at time. P + () is the calculated covariance at time and Q is the process noise covariance. he second step is to determine which measurements can be used to update the state and covariance: Z ( ) ( ) ˆ i zi z( ) ( ) = ( ) + = S H P H R ( ) ( ) ( ) i i ˆ (0) () Z S Z γ () he main purpose of Equation () and () is to validate the measurements by Chi-Square test, and it includes all the measurements returned by all the radars. In Equation (), the total number of measurements is unnown; the subscript i indicates it is the ith measurement. Z i is the innovation of ith measurement for th PDA filter, z i () is the ith measurement at time and R is the covariance of the measurement noise. Equation () is the validation equation. γ is the threshold corresponding to the gate probability. γ can be obtained from Chi-Square tables for a chosen gate probability. Once the ith measurement passes the Chi-Square test in Equation (), it can be utilized in the rest of the PDA filter. hen, the association probabilities of each validated measurement are calculated as:

6 Sensors 03, V( ) ( 4π 3) γ S( ) = N zi( ) ; zˆ ( ) ; S( ) P D Li ( ) = mv( ) ( ) i m PD PG + L n n = β i ( ) = PD PG m D G + n= n L ( ), i =, m, i = 0 P P L ( ) (3) (4) In Equation (3), the subscript indicates it is the th PDA filter, V() is the volume of the validation region, m is number of total validation measurements, and N zi ( ) ; zˆ ( ) ; S( ) represents the normal distribution, centered at ẑ( ) with variance S(). L i is the lielihood of the ith measurement for th PDA filter, β i is the associated probability of the ith measurement for th PDA filter. In Equations (3) and (4), P D and P G are the target detection probability and the gate probability, respectively. β i () is the association probability of the validated measurements. Note that β 0 () is the probability that all measurements are not valid [7]. he suggested values of P D and P G can be found in [5]. P D is required to be larger than 90% for primary surveillance radar sensor and 98% for secondary surveillance radar sensor. P G is the probability that the true measurement from the target is detected and validated; and P G is usually chosen to be at least 95%. If P G is set too low, the gate γ will be too small that no measurement can pass the validation and it could be end up losing trac. If P G is set too high on the other hand, the gate γ will be too large that causes too much false alarms (i.e., not true measurement from the target) and reduction on the accuracy [5]. he last step is to update the system state and covariance. In this step, only the validated measurements and their association probabilities are used: m v( ) = βi Zi ( ) i= W( ) = P ( ) H S ( ) P ( ) P ( ) W ( ) S ( ) W ( ) P = W i Zi Zi v v W i= = m ( ) ( ) β ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + X = X + W v + P 0 P 0 P P ( ) = β ( ) + ( β ) ( ) + ( ) (5) (6) In Equations (5,6), the subscript indicates it is the th PDA filter, v() is combined innovation; W() is Kalman gain; P () and P () are the covariances of state updated. In Equation (6), X + () and P + () are the calculated system state and covariance, respectively at time. he propagation of the PDA filter is shown in Equations (0 6).

7 Sensors 03, IMMPDA Filter he IMM estimator and PDA filter can be combined as shown in Figure. he Kalman filters in the IMM estimator are replaced by the PDA filters. At the top, Equations ( 3) are implemented to deal with the data from the previous epoch. hen, the PDA filters [Equations (0 6)] use the data from Equations ( 3) as the initial values and generate the calculated states and covariances. he next step is to use Equations (6,7) to obtain the latest model probabilities. he last step is to utilize Equations (8,9) to combine the states and covariances with different weights, which are related to the latest model probabilities. he combinations of states and covariances, X() and P() are the outputs of the IMMPDAF. In Figure, the IMMPDAF has three different models of PDA filter. he number of models depends on the number of dynamic motions of the target. In this paper, the tracing target is a commercial airline aircraft. he selection of suitable motion models for an AC system is discussed in the next section. Figure. Flow chart of the IMMPDAF with three models.

8 Sensors 03, uning of IMMPDAF As described above, the IMMPDAF can be considered as a modification of the Kalman filter. he Kalman filter is tuned by changing the process noise covariance Q matrices and the measurement noise covariance R matrices. he matrices Q and R adust the Kalman gain. he R matrices represent the accuracy of the measurement device; larger R matrices mean that the measurements can be trusted less. In most practical situations, matrices Q and R are unnowns. In this paper, a run-time estimate method [9] is used to obtain the measurement noise covariance R. he adaptive measurement noise covariance is estimated in Equation (7). v(n)is the innovation in time epoch n, obtained from Equation (5); and v avg () is average innovation value from time to time : v = avg ( ) n= v ( n) avg avg v( n) v ( ) v( n) v ( ) n= R ( ) = (7) From the conclusions of [9], the run-time estimate method can handle changes of measurement noise covariance, and converge faster than that of a standard Kalman filter..5. Dynamic Models of Commercial Airline Aircraft he ey to successful target tracing is selecting suitable dynamic models that fit the target motion [0,]. Ideally, each motion should have a correct dynamic model. In an AC scenario, the motions of commercial airline aircraft can be summarized as ascent, descent, en-route, turning, and change of speed. hese motions can be described as three basic dynamics: constant velocity motion, accelerated motion, and turning motion [,]. Following [ 5] and our previous wor [6], this study uses the constant velocity model, the acceleration model, and the turning rate estimator model, respectively given in Equations (8 0), respectively, for the IMMPDAF. For AC radar tracing, the measurement is the target 3-dimensional position (x, y, z) of the tracing target in WGS-84 coordinates. In Equations (8 0), X is the system state, x, y, z are the positions, x, y, z are the velocities, and ω is the turning rate. F represents the dynamic model and t is the time interval. Generally speaing, the time epoch is close to 5 seconds. Q is the covariance of process noise and σ is zero-mean Gaussian white noise. he σ value is tuned by experience and testing, and the value is related to the measurement noise covariance: x y z I ti X = = x 0 I 3 y z 3 3, F, t I3 0 Q = σ 0 ti3 (8)

9 Sensors 03, x y z I ti X = F = x y z 3 t t I3 I3 3 Q = σ t I3 ti3 3 3,, 0 I3 ωt 0 0 t 0 0 x ωt 0 0 t 0 0 y z t 0 X = x, F ω t = ωt 0 0, y z ω t ωt 0 0 ω t t t t t t t Q = 0 0 t σ 3 t t t t 0 t σ ω σ (9) (0) 3. Experimental Results and Discussion he experiment data are from the CAA of aiwan. he data formats include radar data (ASERIX AC048) and ADS-B (ASERIX AC0). he radar data were recorded from local time :00:00

10 Sensors 03, midnight on 8 March 0. Hundreds of commercial airline aircraft were traced and recorded. wo cases are used here to investigate whether the performance of the IMMPDAF significantly varies in the area surrounding aiwan. he first case target was from east head to aiwan and the second case target flight away from aiwan to north. he radar data was processed using the IMMPDAF and the NNKF. he ADS-B data have the target Callsign and GPS calculated positions; these were set as the initial values of the two methods and regarded as the true positions. As described in the previous section IMMPDAF includes three filters of different dynamic models shown in Equations (8 0). he parameters of initial model probabilities and transition matrix are shown in Equation (). he initial values of model probabilities have minor effect on the performance, and the IMMPDAF is very robust to the choice of transition matrix. Since the values set for P D is 0.9 and P G is 0.999, and the degree of freedom is 3, the gate threshold obtained from Chi-square table is approximately 6. [ 3 3 3] u initial = p = he performance requirements for APN can be found in [7]. he minimum performance requirements for APN are shown in able. he FAA categorized the airspace into several zones [7]. En-route is the airspace at flight level (FL) 80 to FL 600 (8,000 to 60,000 feet). erminal is the airspace starting at 500 feet above the ground and extending out to 5 statute miles from the airport; it then goes up at a -degree angle to FL 80. Under 500 feet, the APN system needs to support the LNAV/non-precision approach. Accuracy (95%) is defined as the errors in the horizontal 95% under the accuracy requirement. hat is, according to a normal distribution, 95% of error (error mean + standard deviations) has to be under the accuracy requirement. 3.. Case able. Required performance for APN. Navigation Surveillance Accuracy (95%) Separation NAC p En-route 0 nmi 5 nmi 0. nmi 4 nmi nmi erminal nmi 3 nmi 0.05 nmi LNAV 0.3 nmi In Figure, the target heads to aiwan aoyuan International Airport (IAA: PE, ICAO: RCP) from east to west. he interval between epochs is 5 s. Note that a lot of surveillance radar systems have a scan time of close to 5 s. For example, ASR- surveillance radar has a scan time of 4 to 6 seconds. he tracing performance is shown in Figure 3. In case, the target starting at en-route airspace, so the accuracy requirement begins from nmi. As shown in Figure 3, the NNKF experiences the instability at the beginning and the IMMPDAF performs more stable. he reason of the difference is the strategy to deal with the low quality measurements. While the NNKF ust chooses to trust one of the ()

11 Sensors 03, measurements and updates, the IMMPDAF gives all the validated measurements weightings and updates with all of them. From Figure 3, the tracing performance of both the IMMPDAF and the NNKF meets the accuracy requirement. he IMMPDAF outperforms the NNKF, as shown in able. he legends for Figures 3 to 5 are illustrated below. Figure. Flight path for case (target heading toward aiwan). 5.5 Case Latitude aiwan arget course Longitude Figure 3. Performance of IMMPDAF and NNKF for Case..5 Case Error in Horizontal (nmi) IMMPDAF error mean + standard deviations IMMPDAF error NNKF error mean + standard deviations Epoch (5 s) 0.5 NNKF error Accuracy requirement 3.. Case In Case, the target taes off from aiwan aoyuan International Airport and flies northeast as depicted in Figure 4. he tracing performance is shown in the upper plot of Figure 5, and the bottom plot of Figure 5 is the zoom-in plot of the upper one which shows the tracing performance for the LNAV/non-precision approach region. Note that the NNKF results initially exceed the accuracy requirement of APN whereas the IMMPDAF results remain under the accuracy requirement. hat is,

12 Sensors 03, the NNKF method does not meet the accuracy requirement (95%) in this case. As a result, the IMMPDAF outperforms the NNKF. Figure 4. Flight path for Case (target flying away from aiwan). 8 Case 7 6 Latitude aiwan arget Course Longitude Figure 5. Performance of IMMPDAF and NNKF for Case..5 Case 0.5 Case Error in Horizontal (nmi).5 Error in Horizontal (nmi) Epoch (5 s) Epoch (5 s) he mean accuracies of the horizontal positioning are summarized in able. he IMMPDAF outperforms the NNKF for all tested cases. In comparison to the use of the NNKF, the average improvement on the positioning (tracing) performance gained from the proposed IMMPDAF is about 50%. he overall mean accuracies of the horizontal positioning are also listed in able.

13 Sensors 03, able. Overall mean accuracies of the horizontal positioning for IMMPDAF and NNKF Computation Load Accuracy (95%) IMMPDAF NNKF CASES 0.3 nmi 0.9 nmi CASES nmi 0.58 nmi AVERAGE nmi 0.75 nmi For all the experiments, the computations were performed in MALAB (R008a). A PC with an Intel Core Duo E GHz CPU and GB of RAM was used. he computation performance was measured using a MALAB function. he computation performance results are listed in able 3. he values are total seconds that running a case needed. Each test runs 0 times and to obtain the average result. he average result is shown that both algorithms in the single target AC radar tracing scenario are below the radar scan time an epoch (5 seconds).without coding optimization, the IMMPDAF requires about nine times than that of the NNKF. able 3. Computation performance of IMMPDAF and NNKF for Cases to. Lower values are better. 4. Conclusions Cost ime (s) IMMPDAF NNKF CASE s.49 s CASE 5.0 s.734 s AVERAGE.43 s.49 s his paper presented a filter based on the Interacting Multiple Model (IMM) estimator and the Probabilistic Data Association (PDA) filter, and this filter is thus called IMMPDAF, which was applied to the radar tracing system. he real flight radar data was used to evaluate the tracing performance of the IMMPDAF, and the conventional Nearest Neighbor Kalman Filter (NNKF) was used as the baseline to show the performance gained from the proposed filter. Experimental results show that the tracing performance of the IMMPDAF is better than that of the NNKF. In one test case (i.e., Case ), the IMMPDAF meets the accuracy requirement of the Alternate Positioning, Navigation, and iming (APN), but the NNKF results exceeded the APN accuracy requirement for some periods of time. Although the computation load of the IMMPDAF is nine times that of the NNKF, the IMMPDAF is still suitable for AC radar since processing time of each epoch is below the radar scan time (5 s). he proposed IMMPDAF gained about 50% improvement on tacing performance than that of the NNKF. Importantly, the radar tracing with the proposed IMMPDAF met the performance requirements of the APN. hat is, the current radar with the proposed IMMPDAF is capable to provide the navigation and surveillance services for the Air raffic Management (AM) system during periods of GPS outages.

14 Sensors 03, Acnowledgments his wor was supported by the National Science Council in aiwan under research grant NSC 0-68-E MY3, and the real flight radar data is provided by aiwan Civil Aeronautics Administration. Authors gratefully acnowledge the support. his paper is an extension of wor presented at the Institute of Navigation GNSS 0 Conference, Nashville, N, USA, 7 September 0 [8]. Conflict of Interest he authors declare no conflict of interest. References. Global Air Navigation Plan for CNS/AM Systems, nd ed.; International Civil Aviation Organization: Montreal, QC, Canada, 00.. NextGen Implementation Plan; Federal Aviation Administration: Washington, DC, USA, Concept of Operations for NextGen Alternative Positioning, Navigation, and iming; Federal Aviation Administration: Washington, DC, USA, Blom, H.A.P.; Bar-Shalom, Y. he interacting multiple model algorithm for systems with Marovian switching coefficients. IEEE rans. Automat. Contr. 988, 33, Blacman, S.S.; Popoli, R. Design and Analysis of Modern racing Systems; Artech House: Boston, MA, USA, Bar-Shalom, Y.; se, E. racing in a cluttered environment with probabilistic data association. Automatica 975,, Bar-Shalom, Y.; Daum, F.; Huang, J. he Probabilistic Data Association Filter-Estimation In the Presence of Measurement Origin Uncertainty. IEEE Contr. Syst. Mag. 009, 9, Welch, G.; Bishop, G. An Introduction to Kalman Filter; Department of Computer Science University of North Carolina: Chapel Hill, NC, USA, Kang, J.G.; You, B.J. A Run-time Estimate Method of Measurement Error Variance for Kalman Estimator. In Proceedings of the 6th IEEE International Symposium on Robot and Human interactive Communication, RO-MAN 007, Jeu, Korea, 6 9 August 007; pp Li, X.R.; Jilov, V.P. A survey of maneuvering target tracing: Dynamic models. IEEE rans. Aerosp. Electron. Syst. 000, 39, Li, X.R.; Bar-Shalom, Y. Design of an interacting multiple model algorithm for tracing in air traffic control systems. In Proceedings of the 3nd IEEE Conference on Decision and Control, San Antonio, X, USA, 5 7 December Bar-Shalom, Y.; Li, X.R.; Kirubaraan,. Estimation with Applications to racing and Navigation; John Wiley & Sons: New Yor, NY, USA, 00; pp Singer, R.A. Estimating optimal tracing filter performance for manned maneuvering targets. IEEE rans. Aerosp. Electron. Syst. 970, AES-6, Fitzgerald, R.F. Simple tracing Filters: steady state filtering and smoothing performance. IEEE rans. Aerosp. Electron. Syst. 980, AES-6,

15 Sensors 03, Dufour, F.; Mariton, M. racing a 3D maneuvering target with passive sensors. IEEE rans. Aerosp. Electron. Syst. 99, 7, Kao, Y.C.; Jan, S.S. Validation of Interacting Multiple Model Estimator Implementation for Radar racing System. In Proceedings of IGNSS Symposium 0, Sydney, Australia, 5 7 November Narins, M.; Eldredge, L.; Enge, P.; Harrison, M.; Kenagy, R.; Lo, S. Alternative Position, Navigation, and iming he Need for Robust Radionavigation. In Proceedings of the Royal Institute of Navigation NAV0 Conference, London, UK, 30 November December Kao, Y.C.; Jan, S.S. Interacting Multiple Model and Probabilistic Data Association Filter on Radar racing for AM System. In Proceedings of ION GNSS 0, Nashville, N, USA, 7 September by the authors; licensee MDPI, Basel, Switzerland. his article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (