Desired Attitude Angles Design Based on Optimization for Side Window Detection of Kinetic Interceptor *

Transcription

1 Poceedings of the 7 th Chinese Contol Confeence July 6-8, 008, Kunming,Yunnan, China Desied Attitude Angles Design Based on Optimization fo Side Window Detection of Kinetic Intecepto * Zhu Bo, Quan Quan, Cai Kaiyuan School of Automation Science and Electical Engineeing, Beijing Unvesity of Aeonautics and Astonautics, Beijing 00083, P.R.China chubo@asee.buaa.edu.cn Abstact: In this pape the chaactes of pitch angle, yaw angle and oll angle of Kinetic Intecepto (KI) with side window duing the teminal guidance pocess ae discussed. In ode to guaantee that KI is able to tack taget missile with high speed continually, a novel method to design the desied attitude angles is pesented. Given an ellipse side window, the constaints fom visibility, angle of attack and the angle of sideslip on attitude angles ae analyzed and illustated in inequalities. The existence of the solution to these inequalities is confimed though analyzing a special solution. An object function is designed and a single-objective nonlinea optimization model is theeby fomulated. The solutions of this optimization poblem, which could be obtained though solving the optimization model, ae egaded as the desied attitude angles. The initial value chosen to solve the poblem by numeical iteative algoithm is also discussed. The validity of the analysis and the effectiveness of the method ae confimed by simulation esults. Key Wods: Kinetic intecepto, Attitude tacking, Desied attitude angles, Nonlinea optimization INTRODUCTION The unexpected aeo-optical effects aise on the head of Kinetic Intecepto due to its high speed in aeosphee []. As a esult, the measuement accuacy of angles of line-of-sight (LOS) is lost badly. In ode to avoid this negative impact, the side window detection technique was adopted [], while a new poblem follows. The vision scope of the imaging infaed seeke is limited by the side window. It is difficult to design the efeence attitude angles (also called the desied attitude angles in this pape) in this situation since the main task of attitude contol is to ensue that the moving taget missile is in the vision scope of seeke duing the whole intecepting pocess [3]. Besides, the angle of attack is often egulated accoding to dynamic foce desied, and the angle of sideslip is always expected to be as small as possible so as to educe the lateal foce duing the intecepting pocess. It is well known that both angle of attack and angle of sideslip ae nonlinea functions with espect to attitude angles and tajectoy angles [4], so the constaints on the two angles can be conveted into the constaints on thee attitude angles. Base on the above analysis, it is necessay to take all the constaints fom visibility, the angle of attack and the angle of sideslip into account when designing desied attitude angles fo KI with side optical widow. Howeve, many eseaches focus on designing new guidance law [5-8]. The desied attitude angles ae obtained though solving the designed guidance law in these methods. But these methods only based on solving guidance law ae no longe applicable because the visibility of taget missile with high speed is not guaanteed without consideing the special chaacteistics of side window. * This wok is suppoted by the 863Pogamme of China (Gant No: 006AA0Z74). 59 In this pape, a novel appoach to design desied attitude angles fo KI with side window is investigated. Given an ellipse side window, the constaints fom visibility, the angle of attack and the angle of sideslip on attitude angles ae analyzed and descibed in inequalities. The existence of the solution to the goup of inequalities is poved though analyzing a special solution when LOS just goes coss the cente of the window. Accoding to the pactical contol equiements, the desied attitude angles ae specified and an object function is designed. By this method, the task is conveted into a single-object nonlinea optimization poblem. This pape is oganized as follows: Section intoduces the poblem in detail. Section 3 designs the object function. And the initial values chosen to solve the poblem by numeical iteative algoithm ae also discussed. Section 4 pesents the simulation tests and elated esults. Section 5 gives some conclusions. PROBLEM STATEMENT The elated notations of the poblem ae given fist and then eleven inequalities used to specify the poblem ae designed. The chaacteistics of the constaints and the solution to the inequalities ae also analyzed in this section.. Notation Two basic chaactes ae supposed fo the KI investigated in this pape. () A stapdown seeke is equipped and the azimuth and elevation of detecting ange of seeke is limited by the side window () The side window is appoximately elliptic and the long axis of the ellipse is in the longitudinal symmetical plane of KI. The detailed definition of inetial coodinate fame, body coodinate fame and othe physical paametes in this

2 pape ae given by [4] and the notations ae given as follows: Axyz The eath coodinate fame Oxy z The body coodinate fame Cxwywz w The side window coodinate fame A The oigin of Axyz O The mass cente of KI and oigin of Oxyz C The cente of window and oigin of Cxwywzw F, F The Focus of ellipse side window H The Seeke B The Head of KI T The taget missile HT Line-of-sight (LOS) M Intesection point of LOS and side widow D Image of point C in the plane Oxz ϑ, ψ, Pitch angle, yaw angle and oll angle of KI ϑ, ψ, Attitude angles of KI at a specified moment ϑ, ψ, The desied attitude angles of KI α, β Angle of attack and angle of sideslip of KI θ, ψ v Flight path angle of KI a The length of long axis of elliptical window d, d The distance between M and F, F The length of line segment xij, yij, zij The coodinates of point i in j coodinate fame σ Miss distance of intecepting mission The side window coodinate fame (see Fig. ) is defined as Cxwywz w, whee x w is along the long axis of side window and the diection pointing to head of KI is defined to be positive. y w is pependicula to the window plane and the diection pointing upwad is defined to be positive. z w is defined accoding to Right-Hand Rule. point O (see Fig. ). The case that the thee points HCTae,, collinea is illustated in Fig.. Fig. Relative position of KI and taget. Fomulation of the Constaints Fo some cetain elative position between KI and taget missile, d and d ae nonlinea functions with espect to ϑ, ψ, and can be ewitten as d ( ϑ, ψ, ) and ( ) d ϑ, ψ,.in that case, the equiement fo visibility of taget is equivalent to the equiement that point M should be inside the window (see Fig. 3). Hence, the constaint is descibed by: g ϑψ,, d ϑψ,, + d ϑψ,, a< 0 ( ) ( ) ( ) ( ) x w z w Fig. the position of taget in body coodinate fame Obviously, the tansfom matix between Oxy z and Cxwywzwcan be easily calculated based on the knowledge of the stuctue paametes of KI and is a constant matix duing the whole pocess of teminal guidance. The pupose of defining the side window coodinate fame is to offe a convenient method to calculate the thee attitude angles accoding to the azimuth and elevation of LOS measued by seeke. In ode to make it easie to analyze the elative position elation between KI and taget, we tanslate point A to 593 Fig. 3 the Position of M in the elliptic window Fo some cetain, v with espect to ϑ, ψ, [4] given as follows: β ( ϑ, ψ, ) = acsin ( cosθ ( cos sin ( ψ ψv) + sinϑ sin cos( ψ ψv) ) sinθcosϑsin ) cosθ( sinϑcos cos( ψ ψv) sin sin ( ψ ψv) ) sinθcosϑcos α( ϑ, ψ, ) acsin θ ψ, α and β ae nonlinea functions The constaints fom angle of attack and angle of sideslip ae given by: g( ϑψ,, ) α( ϑψ,, ) αmax 0 ( ) g3( ϑψ,, ) αmin α( ϑψ,, ) 0 ( 3) g4( ϑψ,, ) β( ϑψ,, ) βmax 0 ( 4) g5( ϑψ,, ) βmin β( ϑψ,, ) 0 ( 5) Whee αmin, α max denote the minimum value and the maximum value of angle of attack espectively and β, β denote the minimum value and the maximum min = cos β max

3 value of angle of sideslip espectively. The fou paametes ae designed accoding to pactical contol equiements. Usually, the angle of sideslip is expected to be as small as possible in pactical flight pocess. Fo this pupose, we can make both β min and β max nea zeos. Sometimes the angle of attack is expected to be a specified value, which is calculated accoding to guidance law, athe than to be designed in a ange. Fo this pupose, we can letαmin = αmax = αexp, whee αexp denotes the specified value of angle of attack. It is not necessay to change the fom of the constaint given by()~(3) in that case. Accoding to the definitions of pitch angle, yaw angle and oll angle [4], the thee angles must satisfy at any time g6 ( ϑψ,, ) ϑ 0 ( 6) g7 ( ϑψ,, ) ϑ0 ( 7) g8 ( ϑψ,, ) ψ 0 ( 8) g9 ( ϑψ,, ) ψ0 ( 9) g0 ( ϑψ,, ) 0 ( 0) g ϑψ,, 0 ( ) ( ).3 Analysis of These Constaints It is well known that the main task of attitude contol fo KI duing the teminal guidance phase is to ensue the taget missile with high speed is always in the vision scope of seeke, the contol equiement descibed by ()must be satisfied all the time. Hence, it is a had constaint. In most cases, the equiements descibed by () ~ (5) ae just the expectations of attitude contol and tajectoy contol and ae not indispensable. Hence, they ae consideed to be soft constaints (except fo the case when the angle of attack is hoped to be a specified value). The equiements descibed by(6) ~ ()ae obtained based on the definitions of the thee attitude angles. They have no impact on the existence of the solution of the inequalities given by () ~ (5)and may have impact on the numbe of the solutions when the poblem is esoluble..4 A Special Solution to the Inequalities In ode to check the solution to the inequalities given by () ~ ()exists, we will analyze the special case when the LOS just goes acoss the cente of the window, i.e., the thee points HCTae,, collinea (see Fig.). Based on geomety of ellipse, d( ϑ, ψ, ) + d( ϑ, ψ, ) eaches the minimum value in that case. The thee attitude angles ae obtained though two otations of body coodinate fame. Fistly, otate Oxyz along axis Oy until T is inside the longitudinal symmetical plane of KI and we have z T 0 ψ = ac tan ( ) xt 0 Then, otate plane Oxy along axis Oz until T is on line HC. We suppose the velocity vecto of the taget missile is in the longitudinal symmetic plane of KI at the moment. So tacking the taget without olling is ealized and we have = 0 ( 3) Based on HCTae,, collinea, we have yt yh yc yh = ( 4) xt xh xc xh whee T [ xt yt zt] = [ xt0 yt0 zt0 ] L ( ϑψ,, ) ( 5) In the above equation ( 5 ), L(, ϑ, ψ ) given in [4] is the tansfom matix between Oxyz and Axyz. Combining()~(5)yields CD OH ϑ = acsin CD ( xt0) + ( zt0) HD y T0 acsin ( 6) whee = ( CD ( x0) + ( z0) HD y0) ( CD y0+ HD ( x0) + ( z0) ) (7) Because the positions of HCTin,, Oxyz is detemined by the stuctue aangement of KI, HD, OH, CD ae constants and can be easily calculated based on the stuctue paametes. xt0, yt0, zt0 can be calculated based on navigation infomation including the elative distance, flight time and elative velocity etc [9]..5 The Specified Poblem In ode to offe a convenience to undestand the poblem, a specified desciption with espect to a cetain moment in the intecepting pocess is given hee. Given the actualϑ, ψ, measued at time t and the,, d ϑ, ψ,, fou nonlinea functions d( ϑ ψ ), ( ) α( ϑ, ψ, ), β ( ϑ, ψ, ) which satisfy the inequalities given by ( ) ~ ( ), the question is how to design the desied ϑ, ψ, fo the next contol time t +Δ t,so that the fou following objects will be satisfied at time t +Δ t : The fou functions d ( ϑ, ψ, ), d ( ϑ, ψ, ), αϑψ (,, ), (,, ) β ϑ ψ will satisfy the constaints given by inequalities()~(). In ode to obtain the maximal visibility magin, we need that the point M is as close to point C as possible. 594

4 In ode to educe oveload, the angle of sideslip β ϑ, ψ, is as small as possible. ( ) In ode to educe oveload (o to satisfy the equiement fo fit aeodynamic foce), the angle of attack α( ϑ, ψ, ) is as small as possible (o is equal to the specified valueα ). Fig. 4 Block diagam of KI attitude contol system Remak : () This pape use visibility magin to illuminates the possibility of visibility. Its meaning is simila to the concept of obustness in contol theoy. We use it hee to expess the possibility of M keeping inside the window in spite of maneuve of taget missile o any small distubance. () Fo intecepting moving taget, the poblem should be solved online and the solution is a sequence of desied attitude angles duing the whole pocess of teminal guidance. (3) The special chaactes of the efeence attitude angles fo KI with slide window and the feasible methods to calculate them ae the two main concens in this pape. Hence, we don t cae othe poblems hee, such as the fom of the attitude contol loop and the implementation of the command (see fig.4). Based on the qualitative desciption of desied attitude angles, this question can be conveted to an optimization poblem with espect to nonlinea constaints given by inequalities()~(). We can get a sequence of the desied attitude angles though solving the optimization poblem online. 3 MAIN RESULTS Geneally speaking, the poblem descibed in section has moe than one feasible solution. Howeve, the optimal one is ou pimay concen. How to define the best solution to the poblem is an open question. A method based on optimization is intoduced in this section. 3. A Single-Object Optimization Model In ode to satisfy the equiements fo visibility, small angle of attack and small angle of sideslip with espect to the inequalities given by()~(), a single-object nonlinea optimization model is given by exp + ( ϑψ,, ) d ( ϑψ,, ) d min f ( ϑψ,, ) = w ϑψ,, a αϑψ (,, ) βϑψ (,, ) + w + w3 αmax βmax st.. gi ( ϑψ,, ) 0, i=, ( 8) whee w, w, w3 ( 0,), satisfying w w w 3, denote the weight coefficients of the thee pats of the object function espectively. d ϑ, ψ, d ϑ, ψ, α ϑ, ψ,, The thee pats ( ) + ( ), ( ) β ( ϑ, ψ, ) ae nomalized by being divided by a, αmax, β max. The expessions of gi ( ϑψ,, ) 0, i=, ae given by the inequalities()~(), in which ( ) is the necessay condition fo continual tacking and can t be modified duing the whole intecepting pocess. Remak : () Based on the analysis fo inequalities ()~(5)in section.3, the values of α max, β max can be designed accoding to the specified equiements in pactical application. () The solution of the poblem descibed in(8)is consideed as the desied attitude angles at time t +Δt athe than the desied attitude angles at time t. 3. The Initial Values of The Poblem Solving the optimization poblem online is dynamic and should be implemented duing the whole pocess of teminal guidance. As fo the initial attitude angles, thee ae two diffeent meanings hee. The attitude angles at the beginning of teminal guidance. They ae detemined by many factos. Howeve, the equiement that the moving taget should be in the vision scope of infaed seeke must be satisfied. The initial values used in iteative algoithms to solve the optimization poblem. 4 SIMULATIONS In ode to contast, two goups of expeiments with the same simulation conditions except fo the method to calculate the efeence attitude angles ae implemented. Method : Calculate the efeence attitude angles though solving typical guidance law. Method : Calculate the efeence attitude angles though solving the single-object optimization poblem pesented in this pape. 595

5 In Method, we choose the actual attitude angles at t as the initial values when iteative algoithms ae used to calculate the efeence angles at t +Δ t. In ode to ensue the taget is in the vision scope of seeke at beginning of teminal guidance, the attitude angles when HCTae,, collinea ae adopted at the moment in the simulations. Both the two methods use Matlab to implement simulations. Unde this condition, local exhaustive algoithm is used to geneate goups of simulated data with popotional spacing in the feasible ange of flight path angle and position coodinates of KI. The othe invaiable simulation paametes ae given in table Table The Invaiable Initial Paametes Relative distance 60km Meeting angle Velocity of KI Velocity of taget 7 400m s 000m s The position of taget ( 09.6, 73.65, 5.000) km The numbe of goups It is well known that intecepting accuacy is one of the most impotant indexes to evaluate the pefomance of KI. Hence, the missing distance is ou pimay concen. The statistical chaacteistics of the goups of data ae descibed in Fig.5-7. Fig. 5 The statistical esults fo Fig. 6 The statistical esults fo σ m σ 0m Fig. 7 The statistical esults fo σ > 0m Fo simulations without any noises, the main eason fo big miss distance is that the seeke can t detect the taget any longe duing the pocess of teminal guidance. Hence, the degee of the intecepting accuacy eflects the stability of LOS in the vision scope of seeke to a geat extent. In addition, thee conclusions ae made based on the simulation esults. () It is necessay to take the constaint fom side window into account when calculating the efeence attitude angles fo attitude contol of KI. () The novel method based on optimization pesented in this pape is an efficient way to ensue the visibility of taget in the intecepting pocess and is able to impove the intecepting accuacy of this KI. (3) In ode to obtain the same accuacy of intecepting mission, the method designed in this pape has geate ange to choose initial condition fo teminal guidance than othes methods without consideing the chaacteistic of side window. 5 CONCLUSION In this pape we pesent a novel method to design the efeence attitude angles fo a KI with side window duing pocess of teminal guidance. The main pupose is to ensue the visibility of taget missile with high speed. The constaints fom both the angle of attack and the angle of sideslip ae also consideed in the method. Simulation esults indicate that it is necessay to take the equiements fo visibility into account when designing. The appoach intoduced in this pape can guaantee that the taget is in the vision scope of seeke effectively. Futhemoe, it is a new way to impove the pefomance of this kind of inteceptos. Fo pactical implementation, it is necessay to conside some key factos, such as the pefomance of actual attitude contol engines and the time consumed to solve the optimization poblem online continuously. These poblems need to be investigated in futhe study. REFERENCES [] THOMPSON R A. Real-time implementation of stuctual and aeo-optical effects fo hypevelocity missiles. Poceedings of SPIE, the Intenational Society fo Optical Engineeing,,996: [] ZHOU Y P, ZHANG R, LI J L. Attitude vaiable stuctue contol fo side window detection. Moden Defense Technology, 006, 34(): 9-3. [3] ZHOU Di. New Guidance Laws fo Homing Missile[M]. National Defense Industial Pess, Beijing, China,

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