Fuzzy Inference-Based Dynamic Determination of IMM Mode Transition Probability for Multi-Radar Tracking

Transcription

1 Fuzzy Inference-Based Dynamic Determination of IMM Mode Transition Probability for Multi-Radar Tracking Yeonju Eun, Daekeun Jeon CNS/ATM Team, CNS/ATM and Satellite Navigation Research Center Korea Aerospace Research Initute Daejeon, Republic of Korea Abract A new method of determining IMM mode transition probability for multi-radar tracking in air traffic control is presented. In order to improve the accuracy of maneuvering target tracking under a multi-radar environment, dynamic determination of the IMM mode transition probability is udied. The aperiodic measurement inputs from asynchronous multi-radar are also taken into account. In addition to dynamic determination with aperiodic measurement inputs, fuzzy inference is used to adju the mode transition probability from one mode to another. A simulation udy shows that this method for dynamic determination of IMM mode transition probability gives better tracking performance than conventional methods in terms of position accuracy. Keywords IMM; mode transition probability; fuzzy inference; multi-radar tracking; air traffic control I. INTRODUION Target tracking in air traffic control has a number of technical problems to be solved, such as variously maneuvering multitargets, asynchronous sensor input data from multi-sensors, nonlinearity of target maneuvers, limited measurements by radar sensing, and so on. For the various and nonlinear maneuvers of targets, an eimation method is developed based on a hybrid dynamic syem that includes both continuous and discrete syem charactercs. As a representative method of hybrid syem eimation, Interacting Multiple Model (IMM) was suggeed by H. A. P. Blom [] in 984. Since Rong Li and Bar-Shalom [] applied IMM to the aircraft tracking problem, it has been widely used in the surveillance data processing of air traffic control. IMM has improved tracking accuracy performance by providing data on multiple dynamics of various flight modes, such as conant velocity (), conant turn (), and conant acceleration (CA)[3]. The performance requirements of target tracking for air traffic control include small eimation error for conantvelocity maneuvers, a low eimation error peak compared to the measurement, and accurate indication of flight mode derived from the detection of maneuvers. For these required performance indices, the followings are needed to be optimized: the dynamic model of aircraft considering every flight mode; model parameters such as syem process noise and measurement noise; mode transition probabilities [4]. In their research concerning IMM performance improvement through optimization of the sub-model set and model parameters, Rong Li and Bar-Shalom [] udied flight modes and syem process noise, and R. W. Osborne [5] udied measurement noise. For optimization of the mode transition probability of IMM, a dynamic determination using sojourn time is introduced in [6]. However, the average sojourn time for each dynamic model should be determined empirically, and it is known that conant mode transition probabilities are used in mo surveillance data processing syems of realworld air traffic control syem.. In this udy, based on the observation that the measurement inputs from multi-radar is aperiodic and a small mode change probability is acceptable for the small time interval of measurement inputs, dynamic determination of mode transition probability is inveigated. Fuzzy inference is suggeed for the adjument of mode transition probability to utilize the knowledge of eperts about flight mode transitions in air traffic control. II. ALGORITHM DEVELOPMENT A. Determination of mode transition probabilityconsidering the erratic time intervals of input data. The Markov mode transition probability, π ij, is the probability of mode change from i-th mode at time k- to j-th mode at time k, and epressed as in Eq. (). p p p j p p p j π ij = { PM j ( M i ( k ) } = () pi p j p jj Since the effect of mode transition probability on tracking performance does not overshadow design parameters such as

2 flight modes or syem processing noise [4], conant mode transition probability is used in many practical applications. However, in the asynchronous multi-radar environment of air traffic surveillance data processing, the sampling time intervals of measurements change depending on the number of radars, rotation period of each radar, relative positions of a target and the radars, and the detection probability of each radar. Therefore, the mode transition probability is more important in the True Multi-Radar Tracking method, in which each measurement is used at the moment of sensing. Intuitively, the probability of the flight mode changing should be small when the sampling time interval is small, and it would be reasonable to change the mode transition probability in accordance with the time interval of sensing. As a preliminary udy about the relationship between the mode transition probability and the time interval of sensing, a simulation udy was used to check the position accuracy performance of tracking with different values of mode transition probability for various sensing time intervals. Referring to real recorded tracking result data of maneuvering aircraft, an aircraft's trajectory is modeled as a sequence of segments of flight modes, as shown in Table. The transition from one mode to the net is modeled according to fir order lag dynamics. ( = F = ( = F T ( k ) + Γv ( k ) T ( ) + T k T ( k ) + Γv sint = sint T T + T T v T ( k ) ( k ) sint sint v T T ( k ) ( k ) () (3) TABLE I. MODELING OF REFERENCE TRAJEORY FOR SIMULATION STUDY Segment Number Start Time (sec) Duration (sec) Flight Mode Turn Rate (deg/sec) The sensing performance of the radar is modeled as σ R (andard deviation of range error) = 7m and σ ϕ (andard deviation of azimuth error) =.8⁰. The radar is assumed to be located at X = km and y = km in the local frame of the initial target position. IMM with and modes is used for target tracking, and EKF is used for mode. The and modes included in IMM are the Nearly Conant Velocity Model and Nearly Conant Turn Model based on White Noise Acceleration (WNA), and the ate equations are given as Eqs. () and (3). where the ate vector is the position and the velocity in a local D plane, [ ξ ξ η η]. includes the turn rate: [ ξ = ξ η η ]. T is the sampling period. v and v are the white noise processes with zero mean for noise modeling, and they have error covariance of Q = Γσ Γ and Q = Γσ Γ with typical values of σ.m s and v v = σ v =.5m s [4]. The measurement equation is defined as Eq. (4), where w( denotes measurement noise. z( = H( + w( = ( + w( Mode transition probability is given as Eq. (5), where subscript denotes mode and denotes mode. v (4) p p π ij == (5) p p The variables p and p, which measure the probability of maintaining the mode, are given as.95 in a practical air traffic control application. In this udy, the various values are teed to check the relationship between mode transition probability and sampling rates. Fig. 4 shows the simulation results of the Root Mean Square (RMS) tracking position error (for 5 iterations of a Monte

3 Carlo simulation) for various mode transition probabilities and conant sampling rates. The results show that the RMS position error decreases with high-frequency sampling, but the mode transition probability for the minimum RMS error differs depending on the sampling frequency. RMS Position Error (m) T=sec T=sec T=5sec T=4sec T=3sec T=sec T=sec Mode Transition Probability (p ) B. Fuzzy Inference for Adjument of Mode Transition Probability In the dynamic determination of mode transition probability, the probabilities of maintaining modes and are assumed to be the same, i.e. p = p, for the sampling time interval suggeed in the previous section. However, p p in actual mode transition, and the adjument of each p and p is achieved by reflecting the mode transition tendency, which can be inferred from ate variables. In this section, fuzzy inference is inveigated for the adjument of p and p. In order to use epert knowledge in air traffic control to determine the flight mode, turn rate and change of turn rate Δ are chosen as the ate variables for fuzzy inference. Additionally, a sampling time interval is selected to decide whether the adjument is required or not, since smaller sampling time intervals require greater probabilities of mode maintenance. Therefore, the input data for fuzzy inference are sampling time interval T, turn rate, and change in turn rate Δ. The turn rate is derived from in mode, and the change in turn rate Δ is calculated from. Fig.. RMS Position Error for Each Conant Sampling Time and Mode Transition Probability SM BG From the results shown in Fig. 4, the optimal mode transition probability for each conant sampling frequency can be determined by fitting a curve to the data, as shown in Fig. 5 and given as Eq. (6), where it is assumed p = p for simplicity..5 T ~ N Radar T ~ Fig. 3. Membership Functions for Input T T p with min(rms Pos Error) f(t)=-.t -.5T T (sec) Fig.. Be-Fit Curve of the Smalle Mode Transition Probability vs. Conant Sampling Time f ( T ) =.T.5T (6).5 NB NM ZO PM PB c c c3 c4 Fig. 4. Membership Functions for and Δ The membership functions for sampling time interval are SM (Small) and BG (Big), as shown in Fig. 3. T ~ denotes the average sampling time for multiple radars, and N Radar is the number of radars. The membership functions for turn rate and change of turn rate Δ are NB (Negative Big), NM (Negative Medium), ZO (Zero), PM (Positive Medium), and PB (Positive Big). c, c, c 3, and c4 are determined as shown in Eqs. (7) and (8), where and Δ are the andard values of turn rate and change of turn rate. Both variables can be determined based on atcal udies of recorded radar data or by using epert knowledge in air traffic control. or Δ

4 For, c c, c, c ( ) (, -,, ), 3 4 = (7) For Δ, ( c, c, c3, c4 ) = ( Δ, - Δ, Δ, Δ ) (8) Using the fuzzy inference method suggeed here, we can determine whether p is big or small. The fuzzy sets for the output variables, BG (Big), NO (Nominal), and SM (Small), are used as given in Fig SM NO Fig. 5. Fuzzy Sets for Output The fuzzy rules are as follows: ) If T SM, then p NO and p NO ) If T BG and NM and Δ ZO, then p BG and p SM 3) If T BG and PM and Δ ZO, then p BG and p SM 4) If T BG and ZO and Δ ZO, then p SM and p BG 5) If T BG and NB and Δ PM, then p SM and p BG 6) If T BG and PB and Δ NM, then p SM and p BG 7) If T BG and NM and Δ NM, then p BG and p SM 8) If T BG and PM and Δ PM, then p BG and p SM 9) If T BG and Δ NB, then p NO and p NO. ) If T BG and Δ PB, then p NO and p NO. ) If T BG and ZO and Δ NM, then p BG and p SM. ) If T BG and ZO and Δ PM, then p BG and p SM. BG For defuzzification, the weighted average method is used. The output after defuzzification is the normalized amount of required adjument of p. If Δ p indicates this normalized output value from defuzzification, then p can be adjued as given in Eq. (9). p new (.995, p ( Δp.5) ma( Δp )) = min (9) In Eq. (6), ma( Δp ) is the maimum variation of p obtained by fuzzy inference. It is a positive value and a design parameter to be determined by users. III. SIMULATION STUDY The same reference trajectory given in Table is used for the simulation udy. Table II presents the assumed conditions of the radars in this udy. The periods of rotations and the performances of the radars are assumed to be similar, with the required performance of an en-route radar set forth in the Eurocontrol Standard Document for Radar Surveillance in En- Route Airspace and Major Terminal Areas, SUR.ET.S T.-STD-- [7]. The reference trajectory and simulated radar data with error modeling are shown in Fig. 6. Y (km) Position TABLE II. CONDITIONS OF RADARS Radar 3 X (km) Y (km) 5-9 Period of Rotation (Sec) Initial Azimuth of Radar (deg) 4 σ Azimuth (deg) Performa nce σ Range (m) P Detect Reference Trajectory Sensor Data from Radar Sensor Data from Radar Sensor Data from Radar X (km) 4 Fig. 6. Reference Trajectory and Simulated Sensor Data In this simulation udy, and Δ of Eqs. (7) and (8) are assumed as = 3deg/ sec and Δ =.5deg/ sec from epert knowledge, and ma( Δp ) of Eq. (9) is given as.. As a tracking method for asynchronous multi-radar tracking, multiple plot variable update [8] and IMM are used. The simulation results are shown through Figs. 7-, and all of the results in Figs. 7- are obtained from a Monte Carlo

5 simulation of 5 trials. Fig. 7 shows the RMS position error of the tracking results. For comparison with the tracking results associated with dynamic p, two typical values of p are selected:.95 and.98. The change hiories of the mode probabilities are shown in Figs. 8 and 9. Compared with the reference trajectory described in Table, the mode probabilities of Figs. 8 and 9 indicate that the tracker detected the mode transition correctly. RMS Position Error (m) Mode Probability p =.95 p =.98 Dynamic p Time (sec) Fig. 7. RMS Position Error of Tracking Results mode, p =.95 mode, p =.95 mode, Dynamic p mode, Dynamic p Time (sec) Fig. 8. Comparison of Mode Probability Change Hiories Figs. 7-9 indicated that the dynamic p yields a smaller RMS position error than the atic p. For a more readily apparent difference, the comparison of RMS position errors for various atic p and dynamic p values is shown in Fig.. In Fig., the minimum RMS position error of atic p is 78.88m, which is obtained using p =. 99. The RMS position error of dynamic p is 78.74m, which is smaller than the minimum RMS position error of atic p. Mode Probability RMS Position Eimation Errors (meters) mode, p =.98 mode, p =.98 mode, Dynamic p mode, Dynamic p Time (sec) Fig. 9. Comparison of Mode Probability Change Hiory Static p Dynamic p Mode Transition Probability (p) Fig.. RMS Position Error Comparison IV. CONCLUSION A dynamic determination of IMM mode transition probabilities was presented, taking into account erratic intervals of sensor data input from asynchronous multiple radars and tendencies of ate variables. As an empirical method, preliminary udy results about the relationship between atic p and RMS position errors were used to determine p, and the fuzzy inference method was incorporated for the adjument of p according to the tendency of ate variable. Through a simulation udy, it was shown that the suggeed method gives the smalle RMS position error compared with any other atic p. ACKNOWLEDGMENT This work was funded by the Miniry of Land, Transport and Maritime Affairs of Korea.

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