ZHANG Kai,XIONG Jiajun,and FU Tingting

1.Department of Graduation,Air Force Early Warning Academy,Wuhan 430019,China;

2.Early Warning Intelligence Department,Air Force Early Warning Academy,Wuhan 430019,China

Abstract:Aiming at handling complicated maneuvers or other unpredicted emergencies for hypersonic glide vehicle tracking,three coupled dynamic models of state estimation based on the priori information between guidance variables and aerodynamics are presented.Firstly,the aerodynamic acceleration acting on the target is analyzed to reveal the essence of the target’s motion.Then three coupled structures for modeling aerodynamic parameters are developed by different ideas:the spiral model with a harmonic oscillator,the bank model with trigonometric functions of the bank angle and the guide model with the changing rule of guidance variables.Meanwhile,the comparison discussion is concluded to show the novelty and advantage of these models.Finally,a performance assessment in different simulation cases is presented and detailed analysis is revealed.The results show that the proposed models perform excellent properties.Moreover,the guide model produces the best tracking performance and the bank model shows the second;however,the spiral model does not outperform the maneuvering reentry vehicle(MaRV)model markedly.Keywords:hypersonic glide vehicle,state estimation,dynamic model,aerodynamic parameter,guidance variable.

1.Introduction

State estimation of hypersonic glide vehicle(HGV)based on the radar measurements has been known to be a key problem for successful early-warning and interception[1].As the HGV reenters the atmosphere,its violent maneuver behaviors induced by control system may greatly reduce the tracking accuracy[2].Compared with employment of sophisticated filter algorithms or sensor models,good motion models largely determine the bound of estimation error and are often more beneficial to improving the estimation performance.It is greatly necessary to study the modeling problem for HGV to handle the complicated maneuvers and improve the estimation accuracy.

The so-called wide-band kinematic models such as the white-noise acceleration,Wiener-process acceleration,Singer and Jerk model are usually applied to tracking maneuvering targets[3–6].These models can cover many motion patterns,but they may not be sufficient to accurately estimate the states of HGV which has violent maneuver behaviors and special motion characteristics.As a matter of fact,if some prior information such as motion type oraerodynamics can be obtained beforehand,the flexibility of motion models is expected to be properly promoted.To match the skip or turn motions of HGV,Wang et al.[7 – 10]modeled the target’s acceleration as a stochastic process with sine wave(SW)autocorrelation.The SW model can perform well when the target is skipping in vertical direction,but may show mismatch if the target turns right or left.Liang and Qin et al.[11–14]took advantage of the interacting multiple model(IMM)to handle the skip and turn maneuvers.The IMM is confirmed with excellent adaptability for maneuvering targets,whereas the problem of computation complexity has not yet been completely solved.

For the reentry vehicle flying in the atmosphere,modeling aerodynamics can effectively improve the state estimation accuracy[15,16],which uses aerodynamic parameters to present the unknown maneuvers and builds the dynamic model by state augmentation[17–23]or input estimation[24–26].For ballistic reentry vehicle(BRV)tracking,Li and Jilkov[15]listed a summary of dynamic models which obtain many scholars’approvals.The BRV models mainly focuson the identification of the drag coefficient(αd)[15],ballistic coefficient(β=1/αd)[17]or its function(γ=ρ/β)[24,25],on the assumption of a particular model structure.Regarding maneuvering reentry vehicle(MaRV)tracking,a prevailing dynamic model is jointly estimated with the aerodynamic parameters(αd,αtandαc)[15].The classical dynamic models mentioned above have clear physical meanings,but their tracking accuracy drops sharply when the model is mismatched or the aerodynamic force changes dramatically.Moreover,they generally model aerodynamic parameters by using independent Gauss or Markov processes.However,the uncoupled structures cannot entirely present the coupled aerodynamics of HGV.

As we know,the maneuvers of HGV are induced by variation of the guidance variables(i.e.,bank angle and angle of attack).Based on this idea,Hough[27]used the changing law of guidance variables to deduce the aerodynamic acceleration model.However,some parameters in the model depend on off-line statistics of the flight data,which means the method is difficult to apply in practice.In order to identify the maneuver behavior on-line in the terminal phase,Karelahti et al.[28,29]tried to estimate the guidance coefficient by assuming a certain type of the proportional guidance law.However,the terminal guidance identification usually has special structure and strong constraints,which is difficult to adapt to the HGV tracking problem in the midcourse phase.

According to the analysis mentioned above,it can be concluded that taking the priori information of guidance variables into consideration can make for modeling,but how to model the coupled aerodynamics is an urgent problem to be solved.For this purpose,three coupled dynamic models for state estimation are presented by deducing the relationship between the aerodynamics and the guidance variables.Firstly,the effect of each component of aerodynamic acceleration acting on the target is analyzed to reveal the necessity of modeling aerodynamics.Then three coupled structures of aerodynamics are designed by different particular ideas.Meanwhile,discussion of model comparison is concluded to show the novelty of the models.The performance simulation compared with the classical model in different simulation cases is presented,and detailed performance analysis is revealed later.

2.Aerodynamics analysis

2.1 Aerodynamic acceleration analysis

The target’s vector motion can be described by the following equation:

whereaAis the aerodynamic acceleration;aGis the gravity acceleration.TheaGcan be estimated via target’s location indirectly,butaAwhich directly determines the maneuvers is unknown.The key to the state estimation problem of HGV is how to modelaA.For facilitation of analysis,aAis generally decomposed in the velocity-turn-climb(VTC)coordinate system(CS)[15]:

whereuv,utanducare triplet of unit vectors which define the orthogonal body frame of VTC-CS;φandαare the bank angle and angle of attack,respectively;DandLare the drag acceleration and lift acceleration,respectively;Sandmare the target’s reference area and mass,respectively;vis the speed;ρis the atmospheric density;CD(α)andCL(α)are the drag coefficient and lift coefficient,respectively,which are functions ofα[30].

In order to reveal the relationship betweenaAand the guidance variables,the time derivatives ofaAcan be expressed as

whereωindicates the turn rate of the target.DefiningΓas the maneuver plane,it is clear thatωis orthogonal toΓ.The sketch of the derivative vector is shown in Fig.1,and deeper information of each term in(4)is given below:

Fig.1 Sketch of derivative vector

I denotes the motion in the planeΓ.This term is perpendicular toωandaA.With the hypothesis of constant lift and drag(i.e.,no banking),the motion is confined to the so-called coordinated turn model;

II denotes the motion out of the planeΓ,which is attributed to the rotation of the lift vector.With the hypothesis of constant,the motion is confined to the so-called spiral model;

III is attributed to variation of the drag.This term contributes to the motion inside the planeΓ;

IV is attributed to variation of the lift.This term contributes to out-of-plane motion.

The analysis shows that the complicated maneuvers of HGV are composed of some simple motion models.Besides,guidance variables conduct these terms via lift and drag indirectly.It means that consideration of relationship betweenaAand guidance variables can contribute to handling complicated maneuvers of HGV.

2.2 Aerodynamic parameter definition

Unlike accounting for the drag in the BRV model,both the drag and lift should be considerate in the HGV model.A popular approach to describeaAis to use a kinematic state-dependent factorρv2/2 to transform these uncertainties(CD,CL,S,andm,etc)into the format of so-called aerodynamic parameters,andaAcan be described in the east-north-up(ENU)-CS as follows[15]:

wherev=andvzare the components ofvin the ENU-CS,respectively;Ax,AyandAzare the components ofaAin the ENUCS,respectively;αv,αtandαcare the drag parameter,turn force parameter and climb force parameter along the corresponding direction of three axes in the VTC-CS,respectively.

Aerodynamic parameters can present the target’s maneuvers in each direction directly:(i)αt＞0(orαt＜0)represents turning left(or right);(ii)αc＞0(orαc＜0)represents climbing(or diving).Referring to(2),(3)and(5),the relationship between aerodynamic parameters and guidance variables can be expressed as follows:

It can be found that the variation of aerodynamic parameters is determined by guidance variablesαandφ.Meanwhile,the guidance variables lead to the coupling between aerodynamic parameters.The classical MaRV model(MM)in[17–26]is a decoupled way in essence.Hence,in the next section we try different ways to take the coupled relationship into account for aerodynamic modeling.

3.Aerodynamic modeling

In this section,three original dynamic models with different coupled structures,which have potential to express the aerodynamics of HGV,are described in the following subsections.

3.1 Spiral model

The key idea of spiral model(SM)is to assume the HGV undergoing the spiral motion based on the moment equilibrium[20,21].The SM accounts for the correlation betweenαtandαcto improve estimation accuracy.According to(6)and assumingαis nearly constant,the derivatives ofαtandαccan be expressed as follows:

According to(7),a steady harmonic oscillator existing betweenαtandαccan be concluded as follows:

whereis the spiral rate.

In(8),the harmonic oscillator uses the spiral rateϖto reflect the severity of maneuvers.ϖcan be used as the augmented state variable to represent the coupled relationship betweenαtandαc.As a consequence,modeling the statistical property ofϖby a first-order Markov process,the dynamic of augmented parameters in the SM can be expressed as

wherewv,wt,wcandwϖare zero-mean Gaussian white noise sequences for the corresponding augmented parameters;λϖis the maneuver frequency forϖ.

3.2 Bank model

Like the SM,the bank model(BM)also takes the correlation betweenαtandαcinto consideration.The difference is that the latter uses the bank angleφto present trigonometric functions betweenαtandαc.

The BM assumes that the HGV yaws left or right induced by the bank to turn(BTT)control.It means that the bank between lateral motion and lift motion is indicated by bank angleφ.According to this idea,theαtandαcare modified by(6)as follows:

whereαl=CL(α)S/mis defined as the lift parameter.

According to(10),the modifications aligned withαlandφshow a concise structure forαtandαc.We can useαlandφas the modified parameters to represent the negative correlation betweenαtandαc.Therefore the modified parameters in the BM are presented as follows:

wherek=sinφis used to substitute for trigonometric functions(i.e.,sinφor cosφ).

Referring to(5),(6)and(11),the components ofresulting from the modified parameters are represented by

The statistical properties of aerodynamic parameters caused by maneuvers are represented by the first-order Markov processes.The dynamic of augmented parameters in the BM is expressed as follows:

wherewlandwkare zero-mean Gaussian white noise sequences for the corresponding augmented parameters;λlandλkare the maneuver frequency forandk,respectively.

3.3 Guide model

The guide model(GM)no longer simply treats aerodynamic parameters as the random processes,but models them by changing rule of guidance variablesαandφ.For lacking information of the target’s control system,we can assume that variation ofαandφobeys the first-order time delay processes as follows[27]:

whereταandτφare the time constants which represent the changing speed of corresponding guidance variables;αgandφgare the guidance commands which represent the expected values of corresponding guidance variables.

Since the HGV glides at high speed in near space,sharp adjustment ofαgmay make aerodynamic characteristics complex and increase burden on the target’s control and thermal protection system.Hence,αis usually used as an auxiliary guidance variable to perform small-scale adjustment based on the nominal angle of attack[31].It means thatαg-αis very small,soCD(αg)andCL(αg)can be approximated as the first-order Taylor expansion equations:

Referring to(15)and(16),the derivatives ofαvandαlcan be expressed as

whereαvgandαlgare the guidance commands ofαvandαl,respectively.

According to(4),the derivatives ofαtandαccan be expressed as

In this way,the dynamic of augmented parameters in the GM can be built by(17)and(19):

The guidance commandsαvg,αlgandφgare time-varying parameters which result in the target maneuvers.To ensure availability and self-adaptability of the GM,it is necessary to identify the guidance commands online.For this reason,the guidance commands are assumed to be constant within a short time.Combining(15)and(17),the guidance commands are approximated on account of the estimated aerodynamic parameters:

Note that the guidance command in(22)can not be identified in the filter startup phase(k＜n);therefore it can be assumed that the target does not maneuver(i.e.,αg=α&φg=φ),the GM is simplified to Gaussian-Wiener model consequently.

4.Discussion of model comparison

In summary,the dynamic models mentioned above rely on the following idea:quantify the unknown aerodynamic accelerationsaAthrough aerodynamic parameters or their variants scaled by a kinematic state-dependent factorρv2/2,then model these parameters as the well designed process and augment the state vector by them.In order to reveal the intrinsic difference from other similar models,four dynamic models(i.e.,the MM[15]and the proposed models)shown in Table 1 are discussed below.

The MM gives rise to independence of the aerodynamic parameters(αd,αtandαc)in three channels.Its uncoupled structure is deemed to accommodate wide-band variation,but loses some prior information as well.

Table 1 Dynamic models for the HGV

Based onαd,αtandαc,the SM uses the additional augmented stateϖto cover the spiral motion between turn and climb.It assumes thatωsvaries when HGV is maneuvering.This consideration may contribute to the improvement of estimation accuracy.However,it also increases complexity because of a thirteen-state vector.

The BMfocuseson the feature of bank motion with coupled kinematic equationand have clear physical meanings:αvandαlrepresent the drag and lift,respectively;φindicates the direction of lift.Through this form of parametric modeling,the BM considers the coupled relationship betweenαtandαc,but it does not increase the computational complexity.

The GM models aerodynamic parameters by the first order time-delay process of guidance variables indrectly.Analyzing(20)under different flight conditions,the following conclusions can be drawn:i)When the HGV is not maneuvering,guidance variables keep constant(i.e.,αg=αandφg=φ),(20)can be simplified as the Gauss-Wiener model;(ii)When the HGV is climbing(or diving),αkeeps constant andφchanges(i.e.,αg=αandφg=φ),(20)can be simplified as a model related withαvgandαlg;(iii)When the HGV is turning left(or right),αchanges andφkeeps constant(i.e.,αg=αandφg/=φ),(20)can be simplified as a model related withφg.In summary,it is clear that the GM can match the maneuver adaptively and may help to improve the estimation accuracy.

In a word,although these models are augmented by parameters similarly,the MM has the loosest structure.By consideration of prior information,the coupled models mentioned in this paper should have better performance than the MM.

5.Filtering framework

The state equationf(X)ofXis discretized for filtering by the following equation:

wheretis the sampling period;F(Xk|k)is the Jacobian matrix off(Xk|k,tk)with respect toXk|k.

The measurement vector obtained in the spherical CS is defined asZ=(R,A,E)T,whereRis the radial distance,Ais the pitch angle,andEis the azimuth.According to the relationship between the ENU-CS and the spherical CS,the measurement equation is expressed as follows:

wherenR,nAandnEare zero mean Gaussian white noise sequences for the observations.

6.Simulation and performance comparison

This section presents the overall perspective of comparative performance assessment for the proposed dynamic models.Discussion in the following attempts to reveal their validity and utility for state estimation.

6.1 Simulation facilities and criteria

Simulation environment is designed as follows:(i)Target parameters:a typical vehicle CAV-H[32]is applied to the simulation,the predetermined glide duration is set to 180 s,strong maneuver happens att≈35 s and weak maneuver happens att≈90 s;(ii)Sensor parameters:the measurements are generated from the simulation trajectory with zero-mean white noise,and the standard deviationRdenotes 500 m in range,0.01 rad in pitch angle and 0.01 rad in azimuth,the sampling periodT=0.1 s.

Four dynamic models mentioned in Section 4 are compared.The Monte Carlo simulation is performed 100 times for the simulation example.In order to compare the performance explicitly,the root mean square errors(RMSEs)of each case are calculated.Note thatαtandαcof the BM are calculated indirectly from the estimated statesαlandkby(11).

6.2 Performance comparison

Fig.2 and Fig.3 show the position and velocity RMSE results.It can be seen that:(i)When the target is maneuvering strongly,the estimation errors of the four models become somewhat large,and then gradually decrease.

The estimation errors of the proposed models are approximately 95%,80%and 75%of that in the MM.Clearly,the convergence speed of GM is fastest;(ii)When the target is maneuvering weakly,the estimation errors of MM and SM increase significantly,but the position estimation errors of BM and GM fluctuate softly,and their velocity estimation errors are approximately 75%and 60%of that in the MM;(iii)When the target does not maneuver,the difference in performance is not obvious.

Fig.2 RMSE of position estimation for different filters

Fig.3 RMSE of velocity estimation for different filters

Fig.4 and Fig.5 show the aerodynamic parameter estimation and the corresponding RMSE results.In order to represent the self-adaption of GM,the guidance commandsαtgandαcg(i.e.,the expected values ofαtandαc)are calculated fromαlgandφgby(11).It can be seen that:(i)When the aerodynamic parameters are changing drastically,the guidance commands decrease or increase rapidly in the direction of the corresponding parameters’change trend;consequently the GM converges faster than the other three models.The estimation errors of MM and SM are nearly of the same order,the performance of BM is better than those of MM and SM;(ii)When the aerodynamic parameters are changing slightly,the performance differences of the four models are small,and the guidance commands are nearly equal to the estimated aerodynamic parameters.

Fig.4 Aerodynamic parameters estimation of different dynamic models

Fig.5 RMSE in aerodynamic parameter estimation of different dynamic models

Analysis of the above results shows that:(i)The reason why the SM does not show outstanding performance may be that the assumption of harmonic motion is over constrained for state estimation;(ii)The guidance commands reflect the change trend of maneuver quickly;therefore the GM can adaptively match the flight state online and show better performance.

In order to study the influence of the estimator on these models,a comparative performance assessment with various conditions for different estimators as well as measurement parameters is presented.The cases with various conditions are shown in Table 2.

Table 2 Cases with various conditions for comparison of performance

Table 3 shows the average RMSEs of estimated states.It can be seen that:(i)Estimation accuracy of UKF is higher than that of EKF,but the performance of the filtering algorithm is not obvious compared with the model;(ii)When the noise variance is doubled,the average RMSEs of position and velocity increase by approximately 100%and 35%respectively,and the average RMSEs of aerodynamic parameters increase by approximately 15%–40%;(iii)When the sampling period is doubled,the average RMSEs of position and velocity increase by approximately 70%and 25%,respectively,and the average RMSEs of aerodynamic parameters increase by approximately 20%–200%.

Analysis of the above simulation shows that:(i)The filter has less influence on the performance of the models,and using EKF as the estimator can improve the computational efficiency;(ii)Reducing the noise variance helps to improve the position and velocity estimation;(iii)Increasing the sampling frequency helps to improve aerodynamic parameter estimation;(iv)Similar to previous results,the SM shows slightly better performance than the MM,and the BM provides better tracking performance than the SM and the MM.Meanwhile,the GM is the best model among these models,further confirming the conclusion mentioned above.

Table 3 Performance comparison of state estimation errors

7.Conclusions

Aiming at handling complicated maneuvers or other unpredicted emergencies,three coupled dynamic models are developed for state estimation of HGV.The novelty of the proposed models lies on some coupled information between guidance variables and aerodynamics:the SM with a harmonic oscillator,the BM with trigonometric functions of bank angle and the GM with changing rule of guidance variables.Compared with the works of the predecessors,the proposed models no longer simply treat the aerodynamic parameters as independent random processes,but model them in consideration with the prior knowledge mentioned above.The new structures can match the aerodynamics better and be closer to the real motion of HGV.

A comparative assessment based on the Monte Carlo method is carried out with different cases.The simulation reveals that the GM produces the best estimation accuracy and the BM shows the second.Notwithstanding consideration of the harmonic motion,the SM does not outperform the MM markedly.This research provides a new idea for how to improve the estimation accuracy of the dynamic model by utilizing the coupled information between guidance variables and aerodynamics.