Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 50, 2, pp. 473-485, Warsaw 2012 50th 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 e-mail: lady@mchtr.pw.edu.pl Zbigniew Koruba Faculty of Mechatronics and Machine Building, Kielce University of Technology, Kielce, Poland e-mail ksmzko@tu.kielce.pl This paper presents themodelling of dynamics of a self-guidedmissile ste- ered using a gyroscope. In such kinds of missiles, the main element is a self-guiding head, which is operated by a steered gyroscope. The paper presents the dynamics and the method of steering such a missile. Cor- rectness 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 themissile (Żyluk, 2009). At the moment of detecting the target, it was assumed that the missile automatically passes from the flight on the programmed trajectory to trac- king flight of the target according to the assumed algorithm, in this case – 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 he- ight steer at the angles δV and δH respectively. The control laws constitute kinematic relations of deviations of set and realised flight parameters, stabi- lising the movement of the missile in channels of inclination and deflection. The realisation of the desired flight path of the missile is carried out by the 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 af- fect 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. The optic system positioned in the axis of the gyroscope, with a specific angle of view, can thus find the light or infrared signal emitted by amoving object. Therefore, kinematic parameters of reciprocal movement of the missile head and gyroscope axis should be selected to detect the target at the highest pro- bability 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 ofmissilemotion inflightwerederived inquasi-coordinates ϕ, θ, ψ and quasi-velocities U, V , W , P, Q, R (Fig.1) using the Boltzmann- Hamel equations true for mechanical systems in the system associated with the object. The following correlation expresses them in a general form d dt ∂T∗ ∂ωµ − ∂T∗ ∂πµ + k ∑ r=1 k ∑ α=1 γrµα ∂T∗ ∂ωr ωα = Q ∗ µ (2.1) where: α,µ,r = 1,2, . . . ,k, k – number of degrees of freedom, ωµ – quasi- -velocities, T∗ – kinetic energy expressed in quasi-velocities, πµ – quasi- -coordinates, Q∗µ – generalised forces, γ r αµ – three-index Boltzmann factors, expressed by the following correlation γrαµ = k ∑ δ=1 k ∑ λ=1 (∂arδ ∂qλ − ∂arλ ∂qδ ) bδµbλα (2.2) Model of the final section of navigation... 475 Fig. 1. Assumed reference system and parameters of the missile in the course of guidance Relations between quasi-velocities and generalised velocities are ωδ = k ∑ α=1 aδαq̇αq̇δ = k ∑ µ−1 bδµωµ (2.3) where: q̇δ – generalised velocities, qk – generalised coordinates, aδα = = aδα(q1,q2, . . . ,qk) and bδα = bδα(q1,q2, . . . ,qk) – coordinates being func- tions of generalised coordinates, while the following matrix correlation exists: [aδµ] = [bδµ] −1. TheBoltzmann-Hamel equations, after calculating thevalues ofBoltzmann 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 MV̇ +KMV =Q (2.4) where: M is the inertia matrix, K – kinematic relations matrix, V – velocity vector and 476 E. Ładyżyńska-Kozdraś, Z. Koruba M=          m 0 0 0 Sz −Sy 0 m 0 −Sz 0 Sx 0 0 m Sy −Sx 0 0 −Sz Sy Ix −Ixy −Ixz Sz 0 −Sx −Iyx Iy −Iyz −Sy Sx 0 −Izx −Izy Iz          K=          0 −R Q 0 0 0 R 0 −P 0 0 0 −Q P 0 0 0 0 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] Thevector of forces andmoments of external forces Qaffecting themoving missile is the sum of interactions of the centre, in which it is moving. This vector consists of forces: aerodynamic Qa, gravitational Qg, steering Qδ and thrust QT . The flyingmissile is steered automatically. The steering is done in two channels: inclination by tilting the height steer by δH and deflection by tilting the direction steer by δV Q=Qg +Qa +Qδ +QT = col[X,Y,Z,L,M,N] (2.5) where X =−mgsinθ+T − 1 2 ρSV 20 (Cxacosβcosα+Cya sinβ cosα−Cza sinα) +XQQ+XδHδH +XδV δV Y = mgcosθsinφ− 1 2 ρSV 20 (Cxa sinβ −Cya sinβ)+YPP +YRR+YδV δV Z = mgcosθcosφ− 1 2 ρSV 20 (Cxacosβ sinα+Cya sinβ sinα+Czacosα) +ZQQ+ZδHδH L =− 1 2 ρSV 20 l(Cmxacosβcosα+Cmya sinβ sinα−Cmza sinα)+LPP +LRR+LδV δV M =−mgxccosθcosφ− 1 2 ρSV 20 l(Cmxa sinβ +Cmzacosβ)+MQQ +MWW +MδHδH N = mgxccosθ sinφ− 1 2 ρSV 20 l(Cmxacosβ sinα+Cmya sinβ sinα +Cmzacosα)+NPP +NRR+NδV δV Model of the final section of navigation... 477 while: m –missile mass, T –missile engine thrust vector (Fig.2), ρ(H) – air density at a given flight altitude, l – characteristic dimension (total length of themissile body), S – area of reference surface (cross-section of rocket body), V0 = √ U2+V 2+W2 – velocity of the missile flight, Cxa, Cya, Cza, Cmxa, Cmya, Cmza – dimensionless coefficients of aerodynamic component forces, respectively: resistance Pxa, lateral Pya andbearing Pza aswell as themoment of tilting Mxa, inclination Mya and deflection Mza (Fig.2), XQ, YP , YR, ZQ, LP , LR, MQ, NP , NR – derivatives of aerodynamic forces andmoments with respect to components of linear and angular velocities. Fig. 2. Forces andmoments of forces acting on the missile in flight 3. Layout of gyroscopic self-guidance of missiles Figures 3presentsa simplifieddiagramof the layoutof gyroscopic self-guidance of missiles onto a ground target emitting infrared radiation (e.g. a tank or a combat vehicle). Figure 4 shows the general view of the missile used in the scanning and tracking gyroscope, i.e. one which can perform programmedmovements 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 Jgk dωyg2 dt cosϑg +Jgkωgx2(ωgz2 +ωgy2 sinϑg)+Mk sinϑg −Jgo ( ωgz2 + dΦg dt ) ωgx2 cosϑg +ηc dψg dt = Mc 478 E. Ładyżyńska-Kozdraś, Z. Koruba Fig. 3. Diagram of the process of self-guiding a missile on a target Jgk dωgx2 dt −Jgkωgy2ωgz2 +Jgo ( ωgz2 + dΦg dt ) ωgy2 +ηb dϑg dt = Mb (3.1) Jgo d dt ( ωgz2 + dΦg dt ) = Mk −Mrk where Model of the final section of navigation... 479 Fig. 4. General view of the gyroscope and assumed systems of coordinates ωgx2 = P cosψg −Rsinψg + dϑg dt ωgy2 =(P cosψg +Rsinψg)sinϑg + (dψg dt +Q ) cosϑg ωgz2 =(P cosψg +Rsinψg)cosϑg − (dψg dt +Q ) sinϑg and Jgo, Jgk – moments of inertia of the gyroscope rotor in terms of its lon- gitudinal axis and precession axis, respectively, ϑg, ψg – angles of rotation of internal and external frames of the gyroscope, respectively, Mk, Mrk – torques driving the rotor of the gyroscope and friction forces in the rotor bearing in the frame, respectively. The steering moments Mb, Mc affecting the gyroscope expressed by Eqs. (3.1), found on the PR board, we shall present as follows Mb =Π(to, tw)M p b (t)+Π(ts, tk)M s b Mc =Π(to, tw)M p c (t)+Π(ts, tk)M s c (3.2) where: Π(·) are functions of the rectangular impulse, to – time moment of the beginning of spatial scanning, tw – moment of detecting the target, ts – moment of the beginning of target tracking, tk – moment of completing the process of penetration, tracking and laser lighting of the target. 480 E. Ładyżyńska-Kozdraś, Z. Koruba The program steering moments Mp b (t) and Mpc (t) put the axis of the gyroscope in the requiredmotion and are found by themethod of solving the inverse problem of dynamics (Osiecki and Stefański, 2008) M p b (τ)=Π(τo,τw) [d2ϑgz dτ2 + bb dϑgz dτ − 1 2 (dψgz dτ )2 sin2ϑgz − dψgz dτ cosϑgz ] 1 cb Mpc (τ)=Π(τo,τw) (d2ψgz dτ2 cos2ϑpgz + bc dψgz dτ + dψgz dτ dϑgz dτ sin2ϑgz (3.3) + dϑgz dτ cosϑgz ) 1 cc where τ = tΩ Ω = Jgong Jgk cb = cc = 1 JgkΩ 2 bb = bc = ηb JgkΩ and ϑgz, ψgz are the angles determining the position of the gyroscope axis in space. For the target tracking status, values of angles determining the given po- sition of the gyroscope axis are equal to ϑgz = ε ψgz = σ (3.4) where: ε, σ are the angles determining a given position of the target observa- tion line (TOL). The angles ε, σ are defined by the following relationships constituting kinematic equations TOL (Mishin, 1990) dre dt = Vpxe −Vcxe − dε dt recosε = Vpye −Vcye dσ dt re = Vpze −Vcze (3.5) where Vpxe =V0[cos(ε−χp)cosεcosγp − sinεsinγp] Vpye =−V0 sin(ε−χp)cosγp Vpze = V0[cos(ε−χp)sinεcosγp − cosεsinγp] Vcxe = Vc[cos(ε−χc)cosεcosγc − sinεsinγc] Vcye =−Vc sin(ε−χc)cosγc Vcze = Vc[cos(ε−χc)sinεcosγc − cosεsinγc] Model of the final section of navigation... 481 and re – distance between the centre of gravity mass of PR and the ground target, V0, Vc – velocities of PRand the ground target, γp = θ−α, χp = ψ−β – position angles of the PR velocity vector, γc, χc – position angles of the velocity vector of the ground target. If angular deviations between the real angles ϑg and ψg and required angles ϑgz and ψgz are denoted as follows eψ = ψg −ψgz eψ = ψg −ψgz (3.6) then the tracking steering moments of the gyroscope shall be expressed as follows Msb(τ)=Π(τs,τk) ( kbeϑ −kceψ +hg deϑ dτ ) Msc(τ)=Π(τs,τk) ( kbeψ +kceϑ +hg deψ dτ ) (3.7) where kb = kb JgkΩ 2 kc = kc JgkΩ 2 hg = hg JgkΩ Thus, the steering law for the autopilot, taking into account the dynamics of inclination of steers, shall be expressed as follows d2δm dt2 +hmp dδm dt +kmpδm = km(γp −γ ∗ p)+hm (dγp dt − dγ∗p dt ) d2δn dt2 +hnp dδn dt +knpδn = kn(χp −χ ∗ p)+hm (dχp dt − dχ∗p dt ) (3.8) where: bm, bn are the coefficients of stabilising steers, kmp, knp – coefficients of amplifications of steer drives, hmp, hnp – coefficients of suppressions of steer drives, km,kn – coefficients of amplifications of theautopilot regulator, hm,hn – coefficients of suppressions of the autopilot regulator. The required angles of position γ∗p, χ ∗ p of the PR velocity vector are deter- mined by themethod of proportional navigation (Koruba, 2001) dγ∗p dt = aγ dϑg dt dχ∗p dt = aχ dψg dt (3.9) where: aγ, aχ are the required coefficients of proportional navigation. 482 E. Ładyżyńska-Kozdraś, Z. Koruba 4. Obtained results and final conclusions The tested model of navigation and the steering of the self-guiding missile describes the fully autonomous motion of the Maverick combat vessel, which is to directly attack and destroy a ground target after being detected and identified. Figures 5-8 show selected results of digital simulation of the dynamics of the missile during self-guidance on a detected ground target. It was assumed that the missile was launched from an aircraft-carrier moving at a speed of 200m/s at a height of 400m.The target wasmoving along an arc of a circle at the speed of 10m/s. The parameters of the steered gyroscope were as follows Jgk =2.5 ·10 −4kgm2 Jgo =5.0 ·10−4kgm 2 ng =600 rad s ηb = ηc =0.01 Nms rad while the coefficients of its regulator had the values kb =31.480 Nm rad kc =2.986 Nm rad hg =31.525 Nms rad The coefficients of proportional navigation and parameters of the regulator of PR autopilot were as follows aγ =3.5 aχ =3.5 km =2.703 Nm rad kn =11.439 Nm rad hm =9.887 Nms rad Fig. 5. Spatial trajectory of a self-guiding missile Model of the final section of navigation... 483 Fig. 6. Change of angles of attack and glide angles in function of time Fig. 7. Angular position of the missile in function of time during guidance Fig. 8. Angular position of the gyroscope axis in function of time 484 E. Ładyżyńska-Kozdraś, Z. Koruba With the parameters selected as in the above, the target was destroyed in 6s of the flight. It should be emphasised that the gyroscope scanning and tracking layout proposed in this paper improves the stability of the system of missile self- guidance and increases resistance to vibrations born from the board of the missile itself. References 1. Koruba Z., 2001,Dynamics and Control of a Gyroscope on Board of an Fly- ing Vehicle, Monographs, Studies, Dissertations No. 25, Kielce University of Technology, Kielce [in Polish] 2. Koruba Z., Ładyżyńska-KozdraśE., 2010, The dynamicmodel of combat target homing system of the unmanned aerial vehicle, Journal of Theoretical and Applied Mechanics, 48, 3, 551-566 3. Ładyżyńska-Kozdraś E., 2008, Dynamical analysis of a 3D missile motion under automatic control, [In:] Mechanika w Lotnictwie ML-XIII 2008, J. Ma- ryniak (Edit.), PTMTS,Warszawa 4. Ładyżyńska-Kozdraś E., 2009, The control laws having a form of kinema- tic relations between deviations in the automatic control of a flying object, Journal of Theoretical and Applied Mechanics, 47, 2, 363-381 5. 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, 101-113 6. Mishin V.P., 1990,Missile Dynamics, Mashinostroenie,Moskva, pp. 463 7. Osiecki J., StefańskiK., 2008,Onamethodof targetdetectionand tracking used in air defence, Journal of Theoretical and Applied Mechanics, 46, 4, 909-916 8. Żyluk A., 2009, Sensitivity of a bomb to wind turbulence, Journal of The- oretical and Applied Mechanics, 47, 4, 815-828 Model końcowego odcinka nawigacji samonaprowadzającego pocisku rakietowego sterowanego giroskopem Streszczenie Wpracy zaprezentowanomodelowanie dynamiki samonaprowadzającegopocisku rakietowego sterowanego przy użyciu giroskopu. W tego rodzaju pociskach rakieto- wych atakujących samodzielnie wykryte cele głównym elementem jest samonapro- Model of the final section of navigation... 485 wadzająca głowica, której napęd stanowi giroskop sterowany.W pracy przedstawio- na została dynamika i sposób sterowania takiego pocisku. Poprawność opracowanego modelumatematycznegopotwierdziła symulacja numerycznaprzeprowadzonadla po- cisku klasy „Maverick” wyposażonego w giroskop będący elementem wykonawczym skanowaniapowierzchni ziemi i śledzeniawykrytegonaniej celu.Analizienumerycznej poddana została zarówno dynamika giroskopu, jak i pocisku podczas procesu skano- wania i śledzenia wykrytego celu.Wyniki przedstawione zostaływ postaci graficznej. Manuscript received July 11, 2011; accepted for print September 12, 2011