ZHANG Junfeng（張軍峰），WU Xiaoguang（武曉光），WANG Fei（王 菲）

College of Civil Aviation，Nanjing University of Aeronautics and Astronautics，Nanjing 211106，China

Introduction

Aircraft trajectory prediction is one of the research focuses in air traffic management automation field［1］.Fast and accurate trajectory prediction is the fundamental basis for conflict detection and resolution［2］，arrival and departure management［3］and trajectory-based operation［4］，which can in turn ensure flight safety，improve operational efficiency，and alleviate controller workload.

Trajectory prediction is the process of predicting the future progress of individual aircraft on the basis of the current aircraft state，estimates of pilot and controller intent，expected environmental conditions and computer models of aircraft performance and procedures.There are two kinds of methods for trajectory prediction：one is based on aircraft performance model and the other optimal estimation theory.The former utilizes various types of aircraft performance parameters and meteorological information to realize aircraft trajectory prediction through the total energy model（TEM）［5］.Such method is applied a wide range due to the base of aircraft data（BADA）［6］，released by EUROCONTROL Experimental Center（EEC）.However，if unable to accurately determine the flight intention，it is difficult to guarantee the accuracy of trajectory prediction［7］.Furthermore，wind and other weather factors should be considered，which increase the difficulty to solve the trajectory prediction problem.The latter is also applied far and wide such as Kalman filter，α-β filter［8］，which has achieved a certain effect.But a single Kalman filter cannot provide good estimates for a hybrid system with various modes，which is the case in aircraft trajectory prediction.So the aircraft trajectory prediction can be treated as a stochastic linear hybrid systems （SLHS）estimation problem solved by multiple model approach［9］.

Multiple model approach，as an important way to solve the SLHS estimation problems，is known to have a computational cost that grows exponentially with time，and thus suboptimal algorithms such as generalized Psuedo-Bayesian （GPB），interacting multiple model （IMM）algorithms have been proposed.In particular，the IMM algorithm，which has excellent performance with low computational cost，computes the state estimate using a weighted sum of the estimates from a bank of Kalman filters that are matched to different modes of the system，and successfully applied to the aircraft trajectory prediction［10］.

Mode transition detection is the key of SLHS estimation problem，so the hybrid estimation based trajectory prediction method depends greatly on timely and accurate detection of flight mode transitions.The standard IMM algorithm assumes that the residual is zero mean，the mode probability can be calculated through the likelihood function.Usually such assumption is not justified owing to the incompleteness of the mode sets in IMM algorithm［11］.Some authors have also considered complete mode sets［12］or a variable structure IMM（VS-IMM）algorithm［13］，in which the set of modes is chosen from a finite class of mode sets based on the continuous state estimate.However，when all modes operated non-optimally，the likelihood probabilities of each mode are similar，which leads to the inaccurate mode estimates and results in poor state estimation.Residual-mean IMM （RMIMM）［14］algorithm is proposed by Hwang et al.，which designs a novel likelihood function that uses the mean of the residual produced by each Kalman filter.The designed likelihood function gives clearer and sharper differences between the correct mode and the other modes，so that the number of false mode estimates decreases relative to the standard IMM.But the RMIMM algorithm still follows the assumption of residuals being zero mean［15］，and only considers the characteristics of single Kalman filter's（KF's）residual［16］and ignores the mixing process in the standard IMM algorithm.

In this paper，a modified IMM （M-IMM）algorithm is proposed through the performance study of IMM algorithm and the influence analysis of input interaction for mean deviation of the residuals.Then such proposed M-IMM algorithm is also applied to the aircraft trajectory prediction.And the simulation results indicate that the proposed algorithm is able to realize not only the reduction of false mode estimates but also the improvement of prediction accuracy.

1 M-IMM Algorithm

The dynamics of an aircraft as a discrete-time stochastic linear hybrid system is described by：

where x（k）is the state variable at time k；Am（k）is the state transition matrix；wm（k）（k）is the process noise in flight mode m（k），which is zero-mean，Gaussian sequences with covariance Qm（k）：

z（k）is the measurement variables at time k；Cm（k）is the measurement matrix；v（k）is the measurement noise，which is zero-mean，Gaussian sequences with covariance R：

m（k）∈ ｛1，2，…，r ｝is the flight mode，and a Markov transition of the flight mode is given by：

whereμ（k）∈Rris the mode probability at time k，andΠ ＝｛πij｝∈Rr×ris the mode transition matrix.

1.1 Mixing

The input to KFj at time kis adjusted by weighting the output of each KF with the mixing probability as the weight：

Then the covariance corresponding to Eq.（4）is：

1.2 KF

The state estimation and its corresponding covariance are computed based on r KFs running in parallel.The KF is mainly composed of two steps：predicting and updating.The predicting step is described as follows：

In the updating step，firstly based on predicting state and covariance，the residual is defined as：

and the KF-computed residual covariance matrix is

Furthermore，the KF gain is：

Lastly，the KF state estimate and its corresponding covariance matrix are updated using：

1.3 Mode probability update

The performance of the IMM algorithm （as well as other hybrid estimation algorithms in general）depends on the operating scenario.Mathematically，an operating scenario is described in discrete time by the following dynamic system（which can be called as the true system）：

Note that the system is general in the sense that ATmay or may not belong to the mode setof the IMM algorithm.In other words，the true system could be more general than the hybrid system model（Eqs.（1）and（2）），assumed by the IMM algorithm.After all，in the application to aircraft prediction，t he mode setmay not cover the whole dynamic modes of aircraft.Therefore，it should not be arbitrarily assumed that likelihood functionΛj（k）is Gaussian pdf with zero mean.

Based on the above definition，the state estimation error for KF j can be defined as：

and the error for the mixed initial condition：

Substituting Eqs.（6）and （12）into Eq.（8），the residual can be rewritten as

Substituting Eq.（11）into Eq.（15），

Based on the definitionsΔAj＝AT-AjandΔCj＝CT-Cj，

Expand the above equation：

Using Eq.（14），then：

Similarly，substituting Eq.（9）into Eq.（13），the state estimation error for KFj can be written as：

Using Eqs.（8）and（12）：

i.e.，

Namely：

Based on the definitionsΔAj＝AT-AjandΔCj＝CT-Cj，and using Eq.（14），then：

As the measurement matrix of any flight mode remains unchanged for trajectory prediction，i.e.，ΔCj＝0，Cj＝CT＝C.Then the mean value of residual and state estimation error are computed by taking the conditional expectation on Eqs.（16）and（17）respectively：

Since in the IMM framework there is no true model for sure，i.e.，with probability 1，a new definition of the mean of the residual is proposed shown as Eq.（20）：a weighted sum of the mean of the residual computed by each KF with the mode probability estimate as the weight.Similarly，a new definition of the mean of the state estimation error shown as Eq.（21）is proposed as a weighted sum of the mean of the state estimation error corresponding to KF j with the same weight.

where，ΔAi＝Aj-Ai，i.e.，suppose mode j corresponding to the true mode，and

The M-IMM algorithm is proposed based on the above analysis，whose general structure is similar to the standard IMM algorithm except for the“Mode Probability Update”step.In the proposed M-IMM algorithm，likelihood function shown as Eq.（22）is used to update mode probability：

1.4 Combination output

The state estimate is a weighted sum of the estimates fromr KFs and the mode estimate is the mode which has the highest mode probability：

2 Simulation Results

2.1 Simulation preparation

The numerical case and radar trajectory case of aircraft horizontal movement are chosen to conduct the simulation validation.The behavior of an aircraft in a local navigation frame（ξ-η）is considered withξ-axis pointing east andη-axis pointing north.Let the continuous state vector be：

And the aircraft dynamics is described by Ref.［17］：

where Tsis the sampling time，wξand wηare the process noises，φ ＝0denotes the aircraft in constant velocity（CV）mode，andφ ＝1denotes the aircraft in coordinate turn（CT）mode.

The mode transition matrix is chosen as：Π ＝

2.2 Numerical case

A numerical case for flight trajectory is designed with initial position （x0，y0）＝ （10，15）km，heading 270°，aircraft speed v ＝480kn ＝890km／h，composed of six segments，which is shown as Fig.1.

Fig.1 Aircraft trajectory diagram

The parameters chosen in this simulation are as follows：sampling time Ts＝1s，the process noise with mean zero and covariance matrices：

Suppose the root mean squared error （RMSE）of position measurements is 50m，i.e.，the root mean squared（RMS）position error of each direction in a local navigation frame（ξ-η）i s 25m，then the measurement noise with mean zero and covariance matrix：

The trajectory prediction comparison is shown in Fig.2 about approaches from single KF （CV），IMM algorithm，RMIMM algorithm to M-IMM algorithm.

The conclusion drawn from Fig.2is that the trajectory prediction performance of KF is poor as a single KF cannot provide good estimates for a hybrid system with various mode.However，IMM algorithm， RMIMM algorithm and the proposed M-IMM algorithm are able to realize the aircraft trajectory prediction.

In addition，simulation results in Figs.3and 4show how the M-IMM reduces the number of false mode estimates compared with the IMM and RMIMM algorithms.

Fig.2 Aircraft trajectory prediction diagram

Fig.3 Mode probabilities of IMM （a），RM-IMM（b），and M-IMM （c）algorithms

Fig.4 Estimated mode results of IMM （a），RMIMM（b），and M-IMM（c）algorithms

The quantitative comparison of the above hybrid estimation algorithm is shown in Table 1，in which RMSE is chosen as the criteria for prediction accuracy：

Table 1 Trajectory prediction results for numerical case

It can be concluded from Table 1that the proposed MIMM algorithm is able to not noly reduce the number of false mode estimates but also increase the prediciton pricision.

2.3 Radar trajectory case

Figure 5is radar trajectories of arrival，departure and overflight at Beijing Terminal Airspace from 14∶00to 18∶00in Feb.19，2012.

The overflight ETD871is chosen as the simulation case，whose trajectory is shown in Fig.6，where the Beijing Terminal Airspace border is discribed by dotted line.

Fig.5 Radar trajectories at Beijing Terminal Airspace

Fig.6 Radar trajectory of flight ETD871

The parameters chosen are as follows：sampling time Ts＝4s（radar update cycle），the covariance matrices of process noise，

And the measurement noise is mean zero with covariance matrix：

The quantitative comparison of the above hybrid estimation algorithm is shown in Table 2，in which RMSE is chosen as the criteria for prediction accuracy.

Table 2 Trajectory prediction results for radar trajectory case

It can be concluded from Table 2that compared with the standard IMM and RMIMM algorithms，the prediction accuracy is improved and the flight mode estimated error rate is reduced through the proposed M-IMM algorithm.

3 Conclusions

Timely and accurate detection of flight mode transitions is the key of aircraft trajectory prediction.On one hand，the proposed M-IMM algorithm has inherited the architecture of standard IMM algorithm，which effectively guarantees the timely detection of the flight mode transitions.On the other hand，the proposed M-IMM algorithm has improved the accuracy of flight mode detection by abandoning the false assumption that likelihood function with zero mean Gaussian distribution.However，flight mode detection by the proposed algorithm still has a certain lag.Therefore，an important direction for the future research lies on trying to reduce the detection lag by concerning the state-dependent mode transition hybrid estimation algorithm.

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