# MODEL OF THE FINAL SECTION OF NAVIGATION OF A SELF-GUIDED MISSILE STEERED BY A GYROSCOPE

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1 JOURNAL OF THEORETICAL AND APPLIED MECHANICS 50, 2, pp , Warsaw th Anniversary of JTAM MODEL OF THE FINAL SECTION OF NAVIGATION OF A SELF-GUIDED MISSILE STEERED BY A GYROSCOPE Edyta Ładyżyńska-Kozdraś Faculty of Mechatronics, Warsaw University of Technology, Warsaw, Poland Zbigniew Koruba Faculty of Mechatronics and Machine Building, Kielce University of Technology, Kielce, Poland This paper presents the modelling of dynamics of a self-guided missile steeredusingagyroscope.insuchkindsofmissiles,themainelementisa self-guiding head, which is operated by a steered gyroscope. The paper presents the dynamics and the method of steering such a missile. Correctness of the developed mathematical model was confirmed by digital simulation conducted for a Maverick missile equipped with a gyroscope being an executive element of the system scanning the earth s surface and following the detected target. Both the dynamics of the gyroscope and the missile during the process of scanning and following the detected target were subject to digital analysis. The results were presented in a graphic form. Key words: self-guiding, dynamics and steering, steered gyroscope, missile 1. Introduction In the case of automatic steering of self-guided missiles, kinematic equations of spins were related to missile equations of dynamics, using the Boltzmann- Hamel equations(ładyżyńska-kozdraś et al., 2008), which were developed in a relative frame of reference Oxyz, rigidly connected with the missile(żyluk, 2009). Atthemomentofdetectingthetarget,itwasassumehatthemissile automatically passes from the flight on the programmed trajectory to trackingflightofthetargetaccordingtotheassumedalgorithm,inthiscase

2 474 E. Ładyżyńska-Kozdraś, Z. Koruba the method of proportional navigation. Controlling the motion of the missile is carried out by the deflection of control surfaces, i.e. direction steer and heightsteerattheanglesδ V andδ H respectively.thecontrollawsconstitute kinematic relations of deviations of set and realised flight parameters, stabilising the movement of the missile in channels of inclination and deflection. Therealisationofthedesiredflightpathofthemissileiscarriedoutbythe autopilot, which generates control signals based on the compounds derived for the executive system of steering. In the final section of navigation, various types of disturbances may affect the missile, such as wind or shock waves from shells exploding nearby. Therefore, additional stabilisation is necessary, in this case performed by the gyroscope. When searching for a ground target, the gyroscope axis, facing down, strictly outlines defined lines on the earth s surface with its extension. Theopticsystempositionedintheaxisofthegyroscope,withaspecificangle ofview,canthusfinhelightorinfraredsignalemittedbyamovingobject. Therefore, kinematic parameters of reciprocal movement of the missile head andgyroscopeaxisshouldbeselecteodetectthetargetatthehighestprobability possible. After locating the target(receiving the signal by the infrared detector), the gyroscope goes into the tracking mode, i.e. from this point its axis takes a specific position in space, being directed onto the target. 2. The general equations of missile dynamics Dynamic equations of missile motion in flight were derived in quasi-coordinates ϕ,θ,ψandquasi-velocitiesu,v,w,p,q,r(fig.1)usingtheboltzmann- Hamel equations true for mechanical systems in the system associated with the object. The following correlation expresses them in a general form d T T + ω µ π µ k k r=1α=1 γµα r T ω α =Q µ (2.1) ω r where: α,µ,r=1,2,...,k, k numberofdegreesoffreedom, ω µ quasi- -velocities, T kineticenergyexpressedinquasi-velocities, π µ quasi- -coordinates, Q µ generalisedforces, γ r αµ three-indexboltzmannfactors, expressed by the following correlation γ r αµ = k δ=1λ=1 k ( arδ q λ a rλ q δ ) b δµ b λα (2.2)

3 Model of the final section of navigation Fig. 1. Assumed reference system and parameters of the missile in the course of guidance Relations between quasi-velocities and generalised velocities are k k ω δ = a δα q α q δ = b δµ ω µ (2.3) α=1 µ 1 where: q δ generalised velocities, q k generalised coordinates, a δα = =a δα (q 1,q 2,...,q k )and b δα =b δα (q 1,q 2,...,q k ) coordinatesbeingfunctions of generalised coordinates, while the following matrix correlation exists: [a δµ ]=[b δµ ] 1. The Boltzmann-Hamel equations, after calculating the values of Boltzmann factors and indicating kinetic energy in quasi-velocities, a system of ordinary differential equations of the second order was received which describes the behaviour of the missile on the track during guidance. In the frame of reference associated with the moving object Oxyz, they have the following form M V+KMV=Q (2.4) where:mistheinertiamatrix,k kinematicrelationsmatrix,v velocity vector and

4 476 E. Ładyżyńska-Kozdraś, Z. Koruba m S z S y 0 m 0 S z 0 S x 0 0 m S M= y S x 0 0 S z S y I x I xy I xz S z 0 S x I yx I y I yz S y S x 0 I zx I zy I z 0 R Q R 0 P Q P K= 0 W V 0 R Q W 0 U R 0 P V U 0 Q P 0 V=col[U,V,W,P,Q,R] The vector of forces and moments of external forces Q affecting the moving missileisthesumofinteractionsofthecentre,inwhichitismoving.this vectorconsistsofforces:aerodynamicq a,gravitationalq g,steeringq δ and thrustq T.Theflyingmissileissteeredautomatically.Thesteeringisdonein twochannels:inclinationbytiltingtheheightsteerbyδ H anddeflectionby tiltingthedirectionsteerbyδ V where Q=Q g +Q a +Q δ +Q T =col[x,y,z,l,m,n] (2.5) X= mgsinθ+t 1 2 ρsv2 0(C xa cosβcosα+c ya sinβcosα C za sinα) +X Q Q+X δh δ H +X δv δ V Y=mgcosθsinφ 1 2 ρsv2 0 (C xasinβ C ya sinβ)+y P P+Y R R+Y δv δ V Z=mgcosθcosφ 1 2 ρsv2 0 (C xacosβsinα+c ya sinβsinα+c za cosα) +Z Q Q+Z δh δ H L= 1 2 ρsv2 0 l(c mxacosβcosα+c mya sinβsinα C mza sinα)+l P P +L R R+L δv δ V M= mgx c cosθcosφ 1 2 ρsv2 0 l(c mxasinβ+c mza cosβ)+m Q Q +M W W+M δh δ H N=mgx c cosθsinφ 1 2 ρsv2 0l(C mxa cosβsinα+c mya sinβsinα +C mza cosα)+n P P+N R R+N δv δ V

5 Model of the final section of navigation while:m missilemass,t missileenginethrustvector(fig.2),ρ(h) air density at a given flight altitude, l characteristic dimension(total length of the missile body), S area of reference surface(cross-section of rocket body), V 0 = U 2 +V 2 +W 2 velocityofthemissileflight, C xa,c ya,c za,c mxa, C mya,c mza dimensionlesscoefficientsofaerodynamiccomponentforces, respectively:resistancep xa,lateralp ya andbearingp za aswellasthemoment oftiltingm xa,inclinationm ya anddeflectionm za (Fig.2),X Q,Y P,Y R,Z Q, L P,L R,M Q,N P,N R derivativesofaerodynamicforcesandmomentswith respect to components of linear and angular velocities. Fig.2.Forcesandmomentsofforcesactingonthemissileinflight 3. Layout of gyroscopic self-guidance of missiles Figures 3 presents a simplified diagram of the layout of gyroscopic self-guidance ofmissilesontoagrounargetemittinginfraredradiation(e.g.atankora combat vehicle). Figure4showsthegeneralviewofthemissileusedinthescanningand tracking gyroscope, i.e. one which can perform programmed movements while searching for the target and tracking movements after detecting the ground target through an adequate steering mounted on its frames. The equations expressing dynamics of this kind of gyroscope steered by omitting the moments of inertia of its frames, have the following form dω yg2 J gk ( J go cosϑ g +J gk ω gx2 (ω gz2 +ω gy2 sinϑ g )+M k sinϑ g ω gz2 + dφ g ) ω gx2 cosϑ g +η c dψ g =M c

6 478 E. Ładyżyńska-Kozdraś, Z. Koruba Fig.3.Diagramoftheprocessofself-guidingamissileonatarget where dω ( gx2 J gk J gk ω gy2 ω gz2 +J go ω gz2 + dφ ) g dϑ g ω gy2 +η b d( J go ω gz2 + dφ ) g =M k M rk =M b (3.1)

7 Model of the final section of navigation Fig. 4. General view of the gyroscope and assumed systems of coordinates ω gx2 =Pcosψ g Rsinψ g + dϑ g ( dψg ) ω gy2 =(Pcosψ g +Rsinψ g )sinϑ g + +Q cosϑ g ( dψg ) ω gz2 =(Pcosψ g +Rsinψ g )cosϑ g +Q sinϑ g andj go,j gk momentsofinertiaofthegyroscoperotorintermsofitslongitudinalaxisandprecessionaxis,respectively,ϑ g,ψ g anglesofrotationof internalandexternalframesofthegyroscope,respectively,m k,m rk torques drivingtherotorofthegyroscopeandfrictionforcesintherotorbearingin the frame, respectively. ThesteeringmomentsM b,m c affectingthegyroscopeexpressedbyeqs. (3.1),foundonthePRboard,weshallpresentasfollows M b =Π(t o,t w )M p b (t)+π(t s,t k )M s b M c =Π(t o,t w )M p c (t)+π(t s,t k )M s c (3.2) where: Π( )arefunctionsoftherectangularimpulse, t o timemomentof thebeginningofspatialscanning,t w momentofdetectingthetarget,t s momentofthebeginningoftargettracking, t k momentofcompletingthe process of penetration, tracking and laser lighting of the target.

8 480 E. Ładyżyńska-Kozdraś, Z. Koruba Theprogramsteeringmoments M p b (t)and Mp c (t)puttheaxisofthe gyroscopeintherequiredmotionandarefoundbythemethodofsolvingthe inverse problem of dynamics(osiecki and Stefański, 2008) [ M p d 2 b (τ)=π(τ ϑ gz dϑ gz o,τ w ) dτ 2 +b b dτ 1 2 ( dψgz dτ ) 2sin2ϑgz dψ gz dτ cosϑ gz] 1 c b ( d Mc p 2 (τ)=π(τ ψ gz o,τ w ) dτ 2 cos 2 ϑ p gz +b dψ gz c + dψ gzdϑ gz dτ dτ dτ sin2ϑ gz (3.3) where + dϑ gz dτ cosϑ gz) 1 c c τ=tω Ω= J gon g J gk c b =c c = 1 J gk Ω 2 b b =b c = η b J gk Ω ϑ gz =ε ψ gz =σ (3.4) andϑ gz,ψ gz aretheanglesdeterminingthepositionofthegyroscopeaxisin space. For the target tracking status, values of angles determining the given positionofthegyroscopeaxisareequalto where:ε,σaretheanglesdeterminingagivenpositionofthetargetobservation line(tol). The angles ε, σ are defined by the following relationships constituting kinematic equations TOL(Mishin, 1990) where dr e =V pxe V cxe dε r ecosε=v pye V cye (3.5) dσ r e=v pze V cze V pxe =V 0 [cos(ε χ p )cosεcosγ p sinεsinγ p ] V pye = V 0 sin(ε χ p )cosγ p V pze =V 0 [cos(ε χ p )sinεcosγ p cosεsinγ p ] V cxe =V c [cos(ε χ c )cosεcosγ c sinεsinγ c ] V cye = V c sin(ε χ c )cosγ c V cze =V c [cos(ε χ c )sinεcosγ c cosεsinγ c ]

9 Model of the final section of navigation andr e distancebetweenthecentreofgravitymassofpranheground target,v 0,V c velocitiesofpranhegrounarget,γ p =θ α,χ p =ψ β positionanglesoftheprvelocityvector, γ c,χ c positionanglesofthe velocity vector of the ground target. Ifangulardeviationsbetweentherealangles ϑ g and ψ g andrequired anglesϑ gz andψ gz aredenotedasfollows e ψ =ψ g ψ gz e ψ =ψ g ψ gz (3.6) then the tracking steering moments of the gyroscope shall be expressed as follows ( Mb(τ)=Π(τ s de ) ϑ s,τ k ) k b e ϑ k c e ψ +h g dτ ( Mc(τ)=Π(τ s de ) (3.7) ψ s,τ k ) k b e ψ +k c e ϑ +h g dτ where k b = k b J gk Ω 2 k c = k c J gk Ω 2 h g = h g J gk Ω Thus, the steering law for the autopilot, taking into account the dynamics of inclination of steers, shall be expressed as follows d 2 δ m dδ ( m 2 +h mp +k mp δ m =k m (γ p γp )+h dγp ) m dγ p d 2 δ n 2 +h dδ ( n np +k npδ n =k n (χ p χ dχp ) p)+h m dχ p (3.8) where:b m,b n arethecoefficientsofstabilisingsteers,k mp,k np coefficients ofamplificationsofsteerdrives,h mp,h np coefficientsofsuppressionsofsteer drives,k m,k n coefficientsofamplificationsoftheautopilotregulator,h m,h n coefficients of suppressions of the autopilot regulator. Therequiredanglesofpositionγ p,χ p oftheprvelocityvectoraredetermined by the method of proportional navigation(koruba, 2001) dγp =a dϑ g γ dχ p =a χ dψ g (3.9) where:a γ,a χ aretherequiredcoefficientsofproportionalnavigation.

11 Model of the final section of navigation Fig.6.Changeofanglesofattackandglideanglesinfunctionoftime Fig. 7. Angular position of the missile in function of time during guidance Fig.8.Angularpositionofthegyroscopeaxisinfunctionoftime

12 484 E. Ładyżyńska-Kozdraś, Z. Koruba With the parameters selected as in the above, the target was destroyed in 6softheflight. It should be emphasised that the gyroscope scanning and tracking layout proposed in this paper improves the stability of the system of missile selfguidance and increases resistance to vibrations born from the board of the missile itself. References 1.KorubaZ.,2001,DynamicsandControlofaGyroscopeonBoardofanFlying Vehicle, Monographs, Studies, Dissertations No. 25, Kielce University of Technology, Kielce[in Polish] 2. Koruba Z., Ładyżyńska-Kozdraś E., 2010, The dynamic model of combat target homing system of the unmanned aerial vehicle, Journal of Theoretical and Applied Mechanics, 48, 3, Ładyżyńska-Kozdraś E., 2008, Dynamical analysis of a 3D missile motion under automatic control,[in:] Mechanika w Lotnictwie ML-XIII 2008, J. Maryniak(Edit.), PTMTS, Warszawa 4. Ładyżyńska-Kozdraś E., 2009, The control laws having a form of kinematic relations between deviations in the automatic control of a flying object, Journal of Theoretical and Applied Mechanics, 47, 2, Ładyżyńska-Kozdraś E., Maryniak J., Żyluk A., Cichoń M., 2008, Modeling and numeric simulation of guided aircraft bomb with preset surface target, Scientific Papers of the Polish Naval Academy, XLIX, 172B, Mishin V.P., 1990, Missile Dynamics, Mashinostroenie, Moskva, pp Osiecki J., Stefański K., 2008, On a method of target detection and tracking used in air defence, Journal of Theoretical and Applied Mechanics, 46, 4, ŻylukA.,2009,Sensitivityofabombtowinurbulence,JournalofTheoretical and Applied Mechanics, 47, 4, Model końcowego odcinka nawigacji samonaprowadzającego pocisku rakietowego sterowanego giroskopem Streszczenie W pracy zaprezentowano modelowanie dynamiki samonaprowadzającego pocisku rakietowego sterowanego przy użyciu giroskopu. W tego rodzaju pociskach rakietowych atakujących samodzielnie wykryte cele głównym elementem jest samonapro-

13 Model of the final section of navigation wadzająca głowica, której napęd stanowi giroskop sterowany. W pracy przedstawiona została dynamika i sposób sterowania takiego pocisku. Poprawność opracowanego modelu matematycznego potwierdziła symulacja numeryczna przeprowadzona dla pocisku klasy Maverick wyposażonego w giroskop będący elementem wykonawczym skanowania powierzchni ziemi i śledzenia wykrytego na niej celu. Analizie numerycznej poddana została zarówno dynamika giroskopu, jak i pocisku podczas procesu skanowania i śledzenia wykrytego celu. Wyniki przedstawione zostały w postaci graficznej. Manuscript received July 11, 2011; accepted for print September 12, 2011

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