Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 48, 1, pp. 5-26, Warsaw 2010 DYNAMICS AND CONTROL OF A GYROSCOPE-STABILIZED PLATFORM IN A SELF-PROPELLED ANTI-AIRCRAFT SYSTEM Zbigniew Koruba Zbigniew Dziopa Izabela Krzysztofik Kielce University of Technology, Faculty of Mechatronics and Machine Building, Kielce, Poland e-mail: ksmzko@tu.kielce.pl; zdziopa@tu.kielce.pl; pssik@tu.kielce.pl The paper presents a mathematical model of a triaxial gyroscopic plat- form on amoving platform base (motor vehicle). Control software plat- forms are designated with the inverse dynamics of the duties, while the control correction–using theLQRmethod.The consideredplatformcan beused as an independent observationbase for systems, cameras,parcels or gun machines. In the present study, it is shown in its application to stabilization of anti-aircraftmissile launchers. Key words: antiaircraft system, gyroscope, dynamics and control, gyroscope-stabilized platform 1. A physical model of a launcher in a self-propelled anti-aircraft missile system Military vehicles require versatile equipment to perform numerous tasks, for instance, observationdevices including television and infrared cameras andwe- apon such as guns, missile launchers, etc. It is essential that reliable reference systems be used to maintain the equipment and weapon constant orientation and effective operation, irrespective of external disturbances such as the ve- hicle motion. In this study, we consider a three-axis gyroscope platform employed to angularly stabilize a launcher in a self-propelled missile system. The launcher mounted on a vehicle consists of two main parts (Dziopa, 2004-2008). One is a pedestal placed directly on the vehicle. The basic mo- tion of the pedestal is very much dependent on that of the carrier. The other 6 Z. Koruba et al. element is a turret mounted on the pedestal. Therefore, the turret basic mo- tion is a combination of the basic motion of the vehicle and the motion of the turret resulting from target detection and tracking processes. There is a thermovision camera fixed on the turret which sends images to the operator’s control desk. Sitting in the vehicle in front of amonitor screen, he determines themotion of the turret. The turret consists of twomain elements: a platform and a system of four guide rails to launch four missiles. The guide rails are fixed on the platform, symmetrically in relation to a vertical plane passing through the centre of the turret mass. On each side of this plane, there are two guide rails, one above the other. Theplatform can rotate in relation to the pedestal in accordance with the angle of azimuth ψpv, where ψpv is an angle of the platform deviation. The guide rail system is mounted on the platform and together they form a kinematic rather than rotary pair. The guide rail system, therefore, can rotate in relation to the base in accordance with the elevation angle ϑpv. This angle, ϑpv, is the angle of pitch of the guide rail system. After the platform and the guide rail system move to the position of target interception, the launcher does not change its configuration. The ana- lysis of the system performance commences on target interception, therefore, in the assumed model the basic motion of the launcher is reduced to the ba- sic motion of the carrier. This means that the basic motion of the launcher is closely related to the basic motion of the vehicle. The turret is an object with inertial characteristic dependent on the target position with respect to the anti-aircraft system. The turret mass remains stable, yet the moments of inertia and themoment of deviation change. Once the target is locked on, the turret characteristic remains unchanged. The launcher was modelled as two basic masses and eight deformable ele- ments (Fig.1). To improve the legibility of the diagram in Fig.1, the launcher does not include the guide rail system. Figure 2, then, is a supplement of Fig.1. Thepedestal is a perfectly stiffbodywithmass mw andmoments of inertia Iwx and Iwz. The pedestal is mounted to the vehicle body by means of four passive elastic-damping elements with linear parameters kw11 and cw11, kw12 and cw12, kw13 and cw13, and kw14 and cw14, respectively. The turret is a perfectly stiffbodywithmass mv,moments of inertia Ivx and Ivz andmoment ofdeviation Ivxz. It ismounted to thepedestal bymeansof fourpassive elastic- damping elementswith linear parameters kw21 and cw21, kw22 and cw22, kw23 and cw23, and kw24 and cw24, respectively. The inertial characteristic of the turret is dependent on the actual position of its component objects, i.e. the platform and the guide rail system. The platform is a perfectly stiff body Dynamics and control of a gyroscopic-stabilized platform... 7 Fig. 1. A physical model of the launcher Fig. 2. A physical model of the guide rails with mass mpl and main central moments of inertia Iplξ′v, Iplη′v, Iplζ′v. The four-guide-rail system is also a perfectly stiff body with mass mpr and main central moments of inertia Iprξpv, Iprηpv, Iprζpv. The positions of the body of the pedestal with mass mw and moments of inertia Iwx and Iwz and those of the body of the turret with mass mv, moments of inertia Ivx and Ivz andmoment of deviation Ivxz, at anymoment are determined in right-handedCartesian orthogonal coordinate systems. The reference systems are as follows: a) Coordinate systems determiningmotion of the pedestal: 0wxwywzw – the coordinate system moving in the basic motion with respect to the ground-fixed coordinate system 0xyz. The condition that the cor- 8 Z. Koruba et al. responding axes 0wxw‖0x, 0wyw‖0y and 0wzw‖0z are parallel is always satisfied. If the basic motion of the pedestal is not disturbed, then the point 0w coincides with the centre of pedestal mass at any moment. Swxwywzw – the coordinate system moving, in a general case, in translatory motion with respect to the 0wxwywzw coordinate system. The origin of the coordinate system Sw coincides with the centre of pedestal mass at any moment. The condition that the corresponding axes Swxw‖0wxw, Swyw‖0wyw and Swzw‖0wzw are parallel is always satified.Disturbances to the basic motion cause that the centre of pedestal mass Sw moves along the 0wyw axis, which means that the translatory motion in the assumedmodel is reduced to a straight-line motion. Swξwηwζw – the coordinate systemmoving, in a general case, in rotarymotion about a fixed point with respect to the Swxwywzw coordinate system. The axes Swξw, Swηw and Swζw are rigidly connectedwith the pedestal body as they are its main central axes of inertia. Disturbances to the basic motion cause that the pedestal body rotates about the Swzw axis in accordance with a change in the pitch angle ϑw and about the Swxw axis in accordance with a change in the tilt angle ϕw, whichmeans that the rotary motion about a fixed point in the assumed model is reduced to two rotary motions. If there are no disturbances to the basic motion of the pedestal, then the coordinate systems 0wxwywzw,Swxwywzw and Swξwηwζw coincide at anymo- ment. In themodel, the pedestal is an element of a 3Dvibrating system,which perfoms complexmotion in relation to the 0wxwywzw coordinate system.This motion is a combination of a straight-line motion of the centre of mass Sw in accordnace with a change in the yw coordinate, rotary motion about the Swzw axis in accordance with a change in the pitch angle ϑw and rotary mo- tion about the Swxw axis in accordance with a change in the tilt angle ϕw. b) Coordinate systems determiningmotion of the turret: 0vxvyvzv – the coordinate system performing the basic motion in relation to the ground-fixed coordinate system 0xyz. The condition that the corresponding axes 0vxv‖0x, 0vyv‖0y and 0vzv‖0z are parallel is always satisfied. If there are nodisturbances to the basicmotion of the pedestal, then the point 0v coincides with the centre of pedestal mass at any moment. Svxvyvzv – the coordinate system performing, in a general case, translatory motion in relation to the 0vxvyvzv coordinate system. The origin of Dynamics and control of a gyroscopic-stabilized platform... 9 the coordinate system Sv coincides with the centre of the turret mass at any moment. The condition that the corresponding axes Svxv‖0vxv, Svyv‖0vyv and Svzv‖0vzv are parallel is always satified. Disturbances to the basicmotion cause that themass centre of the turret Sv moves along the 0vyv axis, which means that in the assumed model, the translatory motion is reduced to a straight line motion. Svξvηvζv – the coordinate systemmoving, in a general case, in rotarymotion about a fixed point in relation to the Svxvyvzv coordinate system. The Svξv, Svηv and Svζv axes are rigidly connected with the turret body as they are the main central axes of inertia if the following conditions are met: ψpv =0 and ϑpv =0. Disturbances to the basic motion cause that the turret body rotates about the Svzv axis in accordancewith a change in the pitch angle ϑv and about the Svxv axis in accordance with a change in the tilt angle ϕv, which means that, in the assumed model, the rotary motion about a fixed point is reduced to two rotarymotions. If therearenodisturbances to thebasicmotionof the turret, the coordinate systems 0vxvyvzv, Svxvyvzv and Svξvηvζv coincide at any moment. In the model, the turret is an element of a 3D vibrating system, which performs complex motion in relation to the 0vxvyvzv reference system consisting of a straightline motion of the mass centre Sv in accordance with a change in the yv coordinate, rotarymotion about the Svzv axis in accordancewith a change in the pitch angle ϑv and rotary motion about the Svxv axis in accordance with a change in the tilt angle vpv. In the general case, the position of the Swξwηwζw coordinate system in re- lation to the Swxwywzw coordinate system is determinedby theBryant angles ϑw and ϕw. The application of these angles leads to an isometric sequential transformation Rϑwϕw, which is a combination of two consecutive revolutions ϑw and ϕw. The transformation Rϑwϕw has the following form Rϑwϕw =    cosϑw sinϑw 0 −sinϑw cosϕw cosϑw cosϕw sinϕw sinϑw sinϕw −cosϑw sinϕw cosϕw    (1.1) We consider low values of angular vibrations of the launcher pedestal, thus, if there is such a degree of approximation, we can assume that sinϑw = ϑw cosϑw =1 sinϕw = ϕw cosϕw =1 and neglect the ratios of these angles. 10 Z. Koruba et al. The transformation Rϑwϕw as amatrix has the following form Rϑwϕw =    1 ϑw 0 −ϑw 1 ϕw 0 −ϕw 1    (1.2) Generally, theposition of the Svξvηvζv coordinate system in relation to the Svxvyvzv coordinate system is determined by the Bryant angles ϑv and ϕv. The application of these angles leads to an isometric sequential transforma- tion Rϑvϕv, which is a combination of two consecutive revolutions ϑv and ϕv. The transformation Rϑvϕv has the following form Rϑvϕv =    cosϑv sinϑv 0 −sinϑv cosϕv cosϑv cosϕv sinϕv sinϑv sinϕv −cosϑv sinϕv cosϕv    (1.3) We consider low values of angular vibrations of the launcher turret, thus, if there is such a degree of approximation, we can assume that sinϑv = ϑv cosϑv =1 sinϕv = ϕv cosϕv =1 and neglect the ratios of these angles. The transformation Rϑvϕv has the following matrix form Rϑvϕv =    1 ϑv 0 −ϑv 1 ϕv 0 −ϕv 1    (1.4) The turret inertia characteristic is dependent on the actual position of its component objects at the moment the target is intercepted. The turret configuration is determined basing on positions of the platform and the guide rail system.The position of the platformbodywithmass mpl andmoments of inertia Iplξ′v, Iplη′v, Iplζ′v and the position of the body of the guide rail system withmass mpr andmoments of inertia Iprξpv, Iprηpv, Iprζpv are determined in right-handed Cartesian orthogonal coordinate systems. The reference systems are the following coordinate systems: a) Coordinate systems defining position of the platform: Svξ ′ vη ′ vζ ′ v – the coordinate system rotated about the angle ψpv in relation to the Svξvηvζv coordinate system. The Svξ ′ v, Svη ′ v and Svζ ′ v axes are rigidly connected with the platform body so that they are the main central axes of inertia. The operator rotates the platform by the tilt angle ψpv in relation to the target position. Dynamics and control of a gyroscopic-stabilized platform... 11 b) Coordinate systems defining position of the guide rail system: Svξpvηpvζpv – the coordinate systemrotatedby ϑpv in relation to the Svξ ′ vη ′ vζ ′ v coordinate system. The Svξpv, Svηpv and Svζpv axes are rigidly connec- ted with the body of the system of guide rails so that they are themain central axes of inertia. The operator rotates the platform by the pitch angle ϑpv in relation to the target position. Themutual position of the coordinate systems discussed above is determi- ned by the Bryant angles ψpv and ϑpv. The application of these angles leads to a transformation in form of a transformation matrix. The transformation Rψpv from the Svξvηvζv coordinate system to the Svξ ′ vη ′ vζ ′ v coordinate system has the following form (Fig.3) Rψpv =    cosψpv 0 −sinψpv 0 1 0 sinψpv 0 cosψpv    (1.5) Fig. 3. Transformation of the Svξ ′ v η′ v ζ′ v coordinate system in relation to the Svξvηvζv coordinate system The transformation Rϑpv from the Svξ ′ vη ′ vζ ′ v coordinate system to the Svξpvηpvζpv coordinate system has the following form (Fig.4) Rϑpv =    cosϑpv sinϑpv 0 −sinϑpv cosϑpv 0 0 0 1    (1.6) The position of the Svξpvηpvζpv coordinate system in relation to the Svξvηvζv coordinate system is determined by the Bryant angles ψpv and ϑpv, 12 Z. Koruba et al. Fig. 4. Transformation of the Svξpvηpvζpv coordinate system in relation to the Svξ ′ v η′ v ζ′ v coordinate system as shown in Fig.5. The application of these angles lead to an isometric se- quential transformation Rψpvϑpv, which is a combination of two consecutive revolutions ψpv and ϑpv. The transformation Rψpvϑpv has the following form Rψpvϑpv =    cosψpv cosϑpv sinϑpv −sinψpv cosϑpv −cosψpv sinϑpv cosϑpv sinψpv sinϑpv sinψpv 0 cosψpv    (1.7) Fig. 5. Transformation of the Svξpvηpvζpv coordinate system in relation to the Svξvηvζv coordinate system 2. A mathematical model of the launcher of the self-propelled anti-aircraft missile system There are six degrees of freedom resulting from the structure of the model describing disturbances to the launcher basic motion in space. Dynamics and control of a gyroscopic-stabilized platform... 13 Three independent generalized coordinates were assumed to determine po- sitions of the pedestal withmass mw andmoments of inertia Iwx, Iwz at any moment: yw – vertical displacement of the centre of the launcher pedestal mass Sw, ϕw – angle of rotation of the launcher pedestal about the Swxw axis, ϑw – angle of rotation of the launcher pedestal about the Swzw axis. Three independent generalized coordinates were assumed to determine po- sitions of the turret with mass mv, moments of inertia Ivx, Ivz and moment of deviation Ivxz at anymoment: yv – vertical displacement of the centre of the launcher mass Sv, ϕv – angle of rotation of the launcher about the Svxv axis, ϑv – angle of rotation of the launcher pedestal about the Svzv axis. For the launcher, the equations of motion are mwÿw +fyw =0 Iwzϑ̈w +fϑw =0 Iwxϕ̈w +fϕw =0 (2.1) and (mv +mp1+mp2+mp3+mp4)ÿv +fyv =0 (2.2) Iϑvϑ̈v +fϑv =0 Iϕvϕ̈v +fϕv =0 where fyw – function of coordinates yw,ϑw,ϕw,yv,ϑv,ϕv,yn,ϑn,ϕn and their deri- vativeswith respect to time, specifying the analytical formof restitution, dissipative and gravity forces, including the generalized static displace- ment, fϑw,fϕw – function of coordinates yw,ϑw,ϕw,yv,ϑv,ϕv,yn,ϑn,ϕn and their derivatives with respect to time, specifying the analytical form of resti- tution and dissipative forces moments acting in the direction of ϑw and vpw coordinate, respectively, with the static generalized displacement, fyv – function of coordinates yw,ϑw,ϕw,yv,ϑv,ϕv,ξp1,ξp2,ξp3,ξp4 and their derivatives with respect to time, specifying the analytical form of re- stitution, dissipative, gravity, inertia and gyroscopic forces, taking into account the static generalized displacement, fϑv – function of coordinates yw,ϑw,ϕw,yv,ϑv,ϕv,ξp1,ξp2,ξp3,ξp4 and their derivatives with respect to time, specifying the analytical form of mo- ments of forces restitution, dissipative, gravity, inertia and gyroscopic forcesmoments, taking intoaccount the static generalizeddisplacements, 14 Z. Koruba et al. fϕv – function of coordinates yw,ϑw,ϕw,yv,ϑv,ϕv,ξp1,ξp2,ξp3,ξp4 and their derivatives with respect to time, specifying the analytical form of mo- ments of forces restitution, dissipative, gravity, inertia and gyroscopic, including the static generalized displacement, Iϑv,Iϕv – reduced moment of inertia resulting from the movement pursuant to the coordinate ϑv and ϕv, respectively. Functions fyw, fϑw, fϕw, fyv, fϑv, fϕv and reducedmoments of inertia Iϑv, Iϕv need to be written in long mathematical expressions, the analytical form of which is presented in the monograph by Dziopa (2008). Some of the physical quantities are included in Fig.1 and Fig.2. 3. Numerical simulation of the launcher motion The launcher isdirectly subjected todisturbances generatedduringthe launch. Excitations caused by the launch of each of the four missiles are passed on- to the pedestal through the turret. The launcher vibrations result also from the excitation generated by the vehicle moving across a battlefield. The pe- destal mounted on the vehicle passes the disturbances to the turret and the missiles being launched. Examples of the angular acceleration variations for the pedestal and the turret in the pitch and tilt motions are shown in Figs. 6 and 7. Fig. 6. Angular acceleration of the pedestal: (a) in pitch motion, (b) in tilt motion The standard deviation of the pedestal angular acceleration in the pitch motion ϑ̈w is σϑ̈w = 11.5348rad/s 2. The standard deviation of the pedestal angular acceleration ϕ̈w in the tilt motion is σϕ̈w =7.9355rad/s 2. The standard deviation of the turret angular acceleration ϑ̈v in the pitch motion is σ ϑ̈v =28.0760rad/s2. Dynamics and control of a gyroscopic-stabilized platform... 15 Fig. 7. Angular acceleration of the turret: (a) in tilt motion, (b) in pitch motion The standard deviation of the turret angular acceleration ϕ̈v in the tilt motion is: σϕ̈v =53.9298rad/s 2. The paper presents the concept of application of a three-axis gyroscope platformmounted on the pedestal of a self-propelled anti-aircraft systemwith the aim of stabilizing the launcher, i.e. eliminating the undesired angular mo- tions of the vehicle and themissiles being launched.The principle of operation of the system is presented in a schematic diagram in Fig.8. It is predicted that, except for the launcher, there may be another gyroscope-stabilized system fixed on the platform. This system, responsible for the space scanning and target tracking, is able to detect the target au- tomatically while the vehicle moves. The target is then tracked until it is destroyed by one of the missiles in the anti-aircraft missile system. 4. A simplified model of motion of the three-degree gyroscope platform (TGP) Figure 9 showsa schematic diagramof a three-axis platformequippedwith two three-degree gyroscopes (Pavlov, 1954; Pavlovskǐı, 1986). It is required that thereareat least two framesof theplatform: innerandouter.Theplatformand the frames are equippedwith angular displacement sensors and transmitters of controlmoments (Pavlov, 1954;Pavlovskǐı, 1986).Thegyroscopes aremounted inside the platform in such away that themeasurement axes of the gyroscope are parallel to the corresponding axes of the platform frames. One gyroscope has themain axis parallel to the Oxp axis of the platform, therefore, it is able to measure the platform rotations about the other two axes, Oyp and Ozp. The main axis of the other gyroscope, however, is parallel to the Oyp axis of the platform and, therefore, it is able tomeasure the platform rotations about 16 Z. Koruba et al. Fig. 8. Schematic diagram of the principle of operation of the self-propelled anti-aircraftmissile systemwith a three-axis gyroscope-stabilized platform Dynamics and control of a gyroscopic-stabilized platform... 17 axes Oxp and Ozp. In the three-axis platform, the motions about the three axes of suspension interact. The two stabilization systems affect each other, whichmeans that if there are any disturbances to one axis, they are passed to the other two axes. Fig. 9. General view of the three-axis gyroscope-stabilized platformmounted on a wheeled vehicle Inaddition, since theplatform is subjected tovibrations andother external disturbances, it is necessary that the control parameters be optimally selected both at the design stage and under operational conditions. The model presented below describes the gyroscope platform control in a closed-loop system, where the control parameters are optimized using the LQRmethod (Koruba, 2001). Due to limited space, the model is a linearized model. Let us consider a case when the angular displacements of the gyroscope axes and the platform elements from the initial positions are small. If we neglect the ratios of velo- 18 Z. Koruba et al. cities as low order quantities and assume that the gyroscopes are astatic and the inertia of their frames is negligible, we have: — equations describingmotion of the gyroscopes: Jgk(ϑ̈g1− ψ̈p − ṙ ∗)+Jgong1(ψ̇g1+ ϑ̇p +q ∗)= Mkg12 −Mr2g1 Jgk(ϑ̈g2+ φ̈p + ṗ ∗)+Jgong2(ψ̇g2− ψ̇p −r ∗)= Mkg22 −Mr2g2 (4.1) Jgk(ψ̈g1+ ϑ̈p + q̇ ∗)+Jgong1(ψ̇p − ϑ̇g1+r ∗)= Mkg11 −Mr1g1 Jgk(ψ̈g2− ψ̈p − ṙ ∗)+Jgong2(φ̇p − ϑ̇g2+p ∗)= Mkg21 −Mr1g2 —equations describingmotion of the platform elements (platform, inner fra- me, outer frame) [Jxp +Jgk +mpl 2 p + l 2 g1p(m1g1 +m2g1 +m3g1)+ l 2 g2p(m1g2 +m2g2 + +m3g2)](φ̈p + ṗ ∗)+Jgkϑ̈g2−Jgong2(ψ̇p + ψ̇g2+r ∗)+Vpmplp(ψ̇p +r ∗)+ +Vp(m1g1 +m2g1 +m3g1)lg1p(ψ̇g1+ ϑ̇p +q ∗)+ −Vp(m1g2 +m2g2 +m3g2)lg2p(ϑ̇p +q ∗)= Mkp3−Mrp (Jyrw +Jyp +Jgk +mpl 2 p)(ϑ̈p + q̇ ∗)+Jgkψ̈g1+Jgong1(ψ̇p − ϑ̇g1+r ∗)+ +mplpV̇p +Vp[−2(m1g1 +m2g1 +m3g1)lg1p(φ̇p +p ∗)+ (4.2) +(m1g2 +m2g2 +m3g2)lg2pφ̇p] = Mkp2−Mrrw [Jzrz +Jzrw +Jzp +2Jgk + l 2 g1p(m1g1 +m2g1 +m3g1)+ l 2 g2p(m1g2 +m2g2 + +m3g2)](ψ̈p + ṙ ∗)−Jgkϑ̈g1−Jgkψ̈g2−Jgong1(ϑ̇p + ψ̇g1+ q ∗)+ +Jgong2(φ̇p + ϑ̇g2+p ∗)+ [lg1p(m1g1 +m2g1 +m3g1)+ −lg2p(m1g2 +m2g2 +m3g2)]V̇p +Vpmplp(ϑ̇p − φ̇p)= Mkp1−Mrrz where Jgo,Jgk are moments of inertia of the gyroscope rotors; Jxp,Jyp,Jzp,Jyrw,Jzrz – moments of inertia of the platform elements; m1gi,m2gi,m3gi, i =1,2 –masses of the rotor and the inner and outer frames of gyroscopes 1 and 2, respectively; lp, lg2p, lg2p – distances between the centres of gravity of the platform, gyroscope 1, gyroscope 2 and the geometric center of platform rotation, respectively; ϑg1,ψg1,ϑg2,ψg2,φp,ϑp,ψp – angles determining the position of particular axes of rotation of the gyroscope and platform elements; ng1,ng2 – angular velocities of the rotors of gyroscopes 1 and 2, respectively; Vp – linear velocity of the vehicle; p ∗,q∗,r∗ – angular velocities of the vehicle; Mri –moments of friction forces in the bearings of the axis of rotation of particular gyroscopes and platform elements; Mkgi,Mkpi – stabilization moments generated by the correction motors of the gyroscope and platform elements; respectively. Dynamics and control of a gyroscopic-stabilized platform... 19 5. Optimal selection of the control parameters for the three-axis gyroscope platform on a movable base Let us write the equations of motion of the controlled platform in the vector- matrix form ẋ=Ax+Bup (5.1) The vector up shows a pre-programmed open-loop control (Dziopa, 2006a), the schematic diagram of which is presented in Fig.10. Fig. 10. Schematic diagram of control of the gyroscope platform in the open-loop system To assure the platform stability, it is necessary to apply an additional corrective control uk to the closed-loop system.Then, the equationsdescribing motion of the controlled platform will become ẋ ∗ =Ax∗+Buk (5.2) where: x∗ =x−xp is the deviation between the real and desiredmotions; xp is the desired vector of state of the analyzed gyroscope platform. The lawof stabilization control uk is determinedbyusing the linear-square optimization LQRmethod (Koruba, 2001) with a functional in the form J = ∞ ∫ 0 [(x∗)⊤Qx∗+u⊤kRuk] dt (5.3) The law is presented in the following form uk =−Kx ∗ (5.4) 20 Z. Koruba et al. Fig. 11. Schematic diagram of control of the gyroscope platform in the closed-loop system where u= [Mk1,Mki+1,Mkk] ⊤ x= [ψg1, ψ̇g1,ϑg1, ϑ̇g1,ψg2, ψ̇g2,ϑg2, ϑ̇g2, Φ̇p, ϑ̇p, ψ̇p] ⊤ xp = [ψg1z, ψ̇g1z,ϑg1z, ϑ̇g1z,ψg2z, ψ̇g2z,ϑg2z, ϑ̇g2z, Φ̇pz, ϑ̇pz, ψ̇pz] ⊤ The coupling matrix K found in Eq. (5.4) is derived from the following relationship K=R−1B⊤P (5.5) Thematrix P is a solution to the algebraic Riccati equation A ⊤ P+PA−2PBR−1B⊤P+Q=0 (5.6) In Eqs. (5.5) and (5.6), the matrices of weights R and Q reduced to the diagonal form are matched experimentally; the search begins at equal values qii = 1 2ximax rii = 1 2uimax i =1,2, . . . ,n (5.7) where ximax is the maximum range of changes in the i-th value of the state variable, uimax – maximum range of changes in the i-th value of the control variable. Dynamics and control of a gyroscopic-stabilized platform... 21 Figure 12 presents a simplified schematic diagram of the control and cor- rection of the three-axis gyroscope platform. Fig. 12. Schematic diagram of control of a TGP in the open-loop system 6. Results Figures 13-20 show the performance of the stabilization platform. There is a clear difference in the systemoperation resulting fromtheparameter selection. Figures 17 and 18 show the performance of the platform affected by ki- nematic excitations of the pedestal. The dynamics is illustrated in Figs. 6 and 7. Corrective controls clearly protect the platform from the influence of the pedestal. When a disturbance occurs, the platform remains in the transitional pro- cess for a relatively long period of time, if the selection of the regulator pa- rameters is not optimal. However, if the regulator parameters are optimized with the LQRmethod, the platform returns to the initial position immediate- 22 Z. Koruba et al. Fig. 13. Displacements of the platform for the initially selected parameters of regulators, (a) time-dependent angular displacements, (b) time-dependent changes in angular velocities Fig. 14. Angular displacements of the platform for the optimized parameters of regulators, (a) time-dependent angular displacements, (b) time-dependent changes in angular velocities Fig. 15. Angular displacements of gyroscope 1 (a) for the initially selected parameters of regulators, (b) for the optimized parameters of regulators Dynamics and control of a gyroscopic-stabilized platform... 23 Fig. 16. Optimized correctionmoments of (a) the platform, (b) gyroscope 1 Fig. 17. Time-dependent angular displacements due to kinematic motion of the pedestal, (a) without corrective controls, (b) with corrective controls Fig. 18. Time-dependent changes in angular velocities resulting from kinematic motion of the pedestal (a) without corrective controls, (b) with corrective controls 24 Z. Koruba et al. Fig. 19. Pre-programmedmotion of the platform around a circular cone (a) for the initially selected parameters of regulators, (b) for the optimized parameters of regulators Fig. 20. Optimized correctionmoments in pre-programmedmotion of (a) the platform, (b) gyroscope 1 ly (Fig.14). Similar variations of the angular quantities and their derivatives in function of time can be observed for the platform gyroscopes (Fig.15). It should be emphasized that the values of the optimized correction moments of the platform and one of the gyroscopes are relatively small (Fig.16). As canbe seen inFigs. 19 and20, the platformmoves in apre-programmed motion around a circular cone. Figure 19 shows the pre-determined and real trajectories in the coordinates ψp and ϑp. If the parameters of regulators are optimized, only the initial phase of the platform operation does not coincide with the pre-determined one, which is due to external disturbances. After a short period of time, the platform performs a pre-programmedmotion. Figu- re 20 shows a diagram of correction moments generated by the stabilization motors of the platform and gyroscope 1, so that the platform can perform the pre-programmedmotion. Dynamics and control of a gyroscopic-stabilized platform... 25 Rererences 1. Dziopa Z., 2004, The dynamics of a rocket launcher placed on a self-propelled vehicle,Mechanical Engineering, 81, 3, 23-30, ISSN 1729-959 2. Dziopa Z., 2005, An analysis of physical phenomena generated during the launch of a missile from an anti-aircraft system, The Prospects and Develop- ment of Rescue, Safety and Defense Systems in the 21st Century, PolishNaval Academy, Gdynia, ISBN, 83-87280-78-X, 296-303 3. Dziopa Z., 2006a, An anti-aircraft self-propelled system as a system determi- ning the initial parameters of themissile flight,Mechanics in Aviation ML-XII 2006, PTMTS, ISBN 83-902194-6-8, 223-241 4. Dziopa Z., 2006b, Modelling an anti-aircraft missile launcher mounted on a roadvehicle,Theory ofMachines andMechanisms, Vol.1,University of Zielona Góra and PKTMiM, ISBN 83-7481-043-2, 205-210 5. Dziopa Z., 2006c, The missile coordinator system as one of the objects of an anti-aircraft system, 6th International Conference on Armament Technology: Scientific Aspects of Armament Technology, MilitaryUniversity of Technology, ISBN 83-89399-27-X, 221-229 6. DziopaZ., 2008,TheModelling and Investigation of theDynamicProperties of the Sel-Propelled Anti-Aircraft System, Kielce University of Technology,Kielce 7. 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] 8. Koruba Z., Osiecki J., 1999, Construction, Dynamics and Navigation of Close-Range Missiles, Part 1, University Course Book No. 348, Kielce Univer- sity of Technology Publishing House, PL ISSN 0239-6386,Kielce [in Polish] 9. Mishin V.P. (edit.), 1990,Dinamika raket, Mashinostroenie,Moskva 10. MitschkeM., 1977,Dynamics of aMotorVehicle,WKŁ,Warszawa [inPolish] 11. Pavlov V.A., 1954, Abiacionnye giroskopicheskie pribory, Gos. Izdat. Obo- ronnǒı Primyshlennosti, Moskva 12. Pavlovskǐı M.A., 1986,Teoriya giroskopov, Byshcha Shkola, Kiev 13. Svetlickǐı V.A., 1963, Dinamika starta letatel’nykh apparatob, Nauka, Mo- skva 26 Z. Koruba et al. Dynamika i sterowanie platformy giroskopowej s samobieżnym zestawie przeciwlotniczym Streszczenie W pracy przedstawiony jest model matematyczny trzyosiowej platformy girosko- powej na ruchomej podstawie (pojeździe samochodowym). Sterowania programowe platformy wyznaczone są z zadnia odwrotnego dynamiki, natomiast sterowania ko- rekcyjne – za pomocą metody LQR. Rozpatrywana platforma może znaleźć zastoso- wanie jako niezależna pdstawa dla układów obserwacyjnych, kamer, działek czy też karabiniemaszynowych.Wniniejszymopracowaniupokazane jest jej zastosowaniedo stabilizacji wyrzutni przeciwlotniczych pocisków rakietowych. Manuscript received March 14, 2009; accepted for print April 3, 2009