Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 45, 1, pp. 41-51, Warsaw 2007 CONTROL AND CORRECTION OF A GYROSCOPIC PLATFORM MOUNTED IN A FLYING OBJECT Zbigniew Koruba Faculty of Mechatronics and Machine Buiding, Kielce University of Technology e-mail: ksmzko@tu.kielce.pl Thework is concernedwith theoptimal control and correctionof a three- axis gyroscopicplatformfixedonboardof a flyingobject.Thedeviations from the predeterminedmotion areminimized bymeans of amethod of programmedcontrol, an algorithmof the optimal correction control, and selection of optimal parameters for the gyroscopic platform. Key words: gyroscope, optimisation, control, navigation Notation A, B – state and control matrices, respectively; x, u – state and control vectors, respectively; xp, up – set (programmed) state and control vectors, respectively; uk – correction control vector; K – amplificationmatrix; Q,R – weight matrices; OGP – one-axis gyroscopic platform; TGP – two-axis gyro- scopic platform; FO – flying object; TTGP – three-axis gyroscopic platform; M p g1, M p g2, M p g3, M p g4 – programmed control moments applied to the inner and outer frames of the respective platform gyroscopes; M p k1 , M p k2 , M p k3 – pro- grammed correction moments applied to the respective stabilization axes of the platform; µg, µp – damping coefficients relative to the stabilization and precession axes of the OGP; µc – coefficient of dry friction in the gyroscope bearings; dc –diameter of thegyroscopebearing journal; Nc –normal reaction in the gyroscope bearings; ϑgz,ψpz –angles determining the set position of the gyroscope and theOGP axes in space, respectively; kk, kg, hg – amplification coefficients of theOGP closed-loop control; Jgo, Jgk – longitudinal and lateral moments of inertia of the gyroscope rotor, respectively; ng – gyroscope spin velocity; Φp, ϑp, ψp – angles determining spatial position of the platform; Φpz, ϑpz, ψpz – angles determining the set position of the platform in space; p ∗, q∗, r∗ – angular velocities of the FO motion (kinematic interaction with the gy- roscopic platform); p∗o, q ∗ o, r ∗ o – amplitudes of the angular velocities p ∗, q∗, r∗, 42 Z. Koruba respectively; νz – frequency of the FO board vibrations; νp – set frequency of the platform vibrations. 1. Introduction Gyroscopic platforms mounted on board of flying objects, especially homing rockets and unmanned aerial vehicles, are used as a reference for navigation instruments. They also provide the input for sighting and tracing systems, target coordinators, and television cameras (Kargu, 1988;Koruba,1999, 2001). Consequently, gyroscopic platformsneed tobecharacterizedbyhighopera- tional reliability and high accuracy inmaintaining the predeterminedmotion. Since the operating conditions include vibrations and external disturbances, it is essential that the platform parameters be properly selected both at the design and operation stages. This study examines applications of the three–axis gyroscopic platform to navigation, control and guidance ofweapons, such as unmannedaerial vehicles and guided bombs, when pinpoint accuracy in locating a target is required. The presented algorithm of selection of optimal control, correction and damping moments for the three-axis gyroscopic platform ensures stable and accurate platform operation even under conditions of external disturbances and kinematic interactions with the FO board. 2. Control of the gyroscopic platform position The equations of a controlled gyroscopic platformwill bewritten in thevector- matrix form ẋ=Ax+Bu (2.1) To determine the programmed controls u, use a general definition, which says that the inverse problems of dynamics (Dubiel, 1973) involve estimation of the external forces acting on amechanical system, system parameters and its constraints at which the set motion is the only possiblemotion of the system. In practice, the problems are frequently associated with special cases inwhich it is necessary to formulate algorithms determining the control forces that assure the desired motion of the dynamic system irrespective of the initial conditions of the problem, though such a procedure is not always possible. The next step is to formulate the desired signals (Koruba, 2001). Let up stand for the desired (set, programmed) control vector. The control problem Control and correction of a gyroscopic platform... 43 involves determination of the variations of components of the vector up in function of time, which are actually the control moments about the gyroscope axis and themotion being determined by the desired angles (ones defining the set position in space of the platform). Now, transform Eq. (2.1) into the form Bup = ẋp−Axp (2.2) The quantity xp in expression (2.2) is the desired state vector of the analyzed gyroscopic platform.The estimated control moments up, are known functions of time. Here, we are considering open-loop control of a gyroscopic platfrom, the diagram of which is shown in Fig.1. Fig. 1. Layout of the gyroscopic platform open-loop control Now, let us checkwhatmotion of the gyroscopic platform is excited by the controls. By substituting them into the right-hand sides of Eqs. (2.2), we get ẋ ∗ =Ax∗ (2.3) where x∗ =x−xp is a deviation from the desiredmotion. The inverse problem is explicit for the derivatives of the state variables in relation to time ẋ, but for a fixed motion, i.e. within limits, when t → ∞. The question is: will the solutions to the above equations describe also the set motion of the gyroscopic platform? If the initial axis angle is set to be the same as the requiredone: x(0)= xp(0),wewill obtaindesiredangular displacements of the gyroscopic platform.However, if the initial platformposition is different fromthedesiredposition, then, even though thevelocities of the state variables coincidewith the desired ones, the platformwill not realize the desiredmotion in the desired location in space. 44 Z. Koruba Apart from the abovementioned inconsistency of the programmed control, the platformperformance is affected by external disturbances,mainly kinema- tic interaction with the base (flying object board), non-linear platformmotion (large gyroscope axis angles), friction in the suspension bearings, manufactu- ring inaccuracy, errors of the measuring instruments, etc. Fig. 2. Layout of the gyroscopic platform closed-loop control Additional closed-loop correction control uk (Fig.2), must be applied to make the gyroscopic platform move along a set path. Then, the equations describingmotion of the controlled platform will have the following form ẋ ∗ =Ax∗+Buk (2.4) The correction control law uk, will be determined by means of a linear- quadratic optimisation method with a functional in the form J = ∞∫ 0 [(x∗)⊤Qx∗+u⊤kRuk] dt (2.5) The law will be presented as uk =−Kx∗ (2.6) The couplingmatrix K in Eq. (6) is estimated from the following relation- ship A ⊤ P+PA−2PBR−1B⊤P+Q=0 (2.7) Control and correction of a gyroscopic platform... 45 Thematrix P is a solution to Riccati’s algebra equation K=R−1B⊤P (2.8) The weight matrices, R and Q, in Eqs. (2.7) and (2.8), rearranged into the diagonal formare selected at random (Koruba, 2001), yet the search starts from the values equal to qii = 1 2ximax rii = 1 2uimax i =1,2, . . . ,n (2.9) where: ximax –maximumvariation rangeof the ith statevariable value; uimax – maximum variation range of the ith control variable value. By substituting (2.6) into (2.4), we obtain the equations of state in a new form again ẋ ∗ =(A−BK)x∗ =A∗x∗ (2.10) where A ∗ =A−BK (2.11) Recall that it is important tomake the platform as stable as possible. This means the transition processes resulting from the switching on the control systemora suddendisturbancemustbereduced toaminimum.Thus, selection of the optimal parameters of the system described by Eq. (2.10), and then application of the Golubiencew modified optimization method (Dubiel, 1973) are carried out. 3. Control and correction of the two- and three-axis gyroscopic platform For a two-axis platform, the state and control vectors x and u as well as the state and control matrices A and B, are as follows x= [ ϑp, dϑp dτ ,ψp, dψp dτ ,ϑg, dϑg dτ ,ψg, dψg dτ ]⊤ u= [Mk1,Mk2,Mgw,Mgz] ⊤ A=    0 1 0 0 0 0 0 0 0 −hpy 0 0 0 ηgw 0 0 0 0 0 1 0 0 0 0 0 0 0 −hpz 0 0 0 ηgz 0 0 0 0 0 1 0 0 0 hpy 0 − √ n 0 −(ηpw+ηgw) 0 − √ n 0 0 0 0 0 0 0 1 0 √ n n 0 hpz 0 √ n n 0 −(ηpz +ηgz)    46 Z. Koruba B=    0 cp1 0 0 0 −cp1 0 0 0 0 0 cp1 0 0 0 −cp2 0 0 0 0 0 cgw 0 0 0 0 0 0 0 0 0 cgz    while τ = Ωgt Ωg = Jgong√ JgwJgz ηgw = ηgw JgwΩ 2 g ηgz = ηgz JgzΩ2g hpy = hpy JpΩg hpz = hpz JpΩg ηpw = ηpw JpwΩ 2 g ηpz = ηpz JpzΩ 2 g cp1 = 1 JpyΩg cp2 = 1 JpzΩg cgw = 1 JgwΩg cgz = 1 JgzΩg A block diagram of the control and the correction for a two-axis platform is shown in Fig.3. Fig. 3. Diagram of the TGP closed-loop control However, in the case of a three-axis platform, we have x= [ ψp, dψp dτ ,ϑp, dϑp dτ ,Φp, dΦp dτ ,ψg1, dψg1 dτ ,ϑg1, dϑg1 dτ ,ψg2, dψg2 dτ ,ϑg2, dϑg2 dτ ]⊤ u= [Mk1,Mk2,Mk3,Mg1,Mg2,Mg3,Mg3] ⊤ Control and correction of a gyroscopic platform... 47 A=    0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 −hpz 0 0 0 0 0 ηp1 0 0 0 0 0 ηp1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 −hpy 0 0 0 0 0 ηp2 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −hpx 0 0 0 0 0 ηp3 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 hpz 0 A1 0 0 0 −η̂p1 0 A1 0 0 0 −ηp1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 A2 0 0 0 A2 0 −η̂p2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 A1 0 0 0 hpx 0 0 0 0 0 −η̂p3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 hpz 0 0 0 A2 0 −ηp1 0 0 0 A2 0 −η̂p4    A1 = √√√√J Σ gw JΣgz A2 =− √√√√J Σ gz JΣgw B=    0 c1 0 0 0 0 0 −c1 0 0 0 0 0 −c1 0 0 0 c2 0 0 0 0 0 −c2 0 0 0 0 0 0 0 0 0 c3 0 0 0 0 0 −c3 0 0 0 0 0 0 0 0 0 c4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c7    while τ = Ωgt Ωg = Jgong√ JΣgwJ Σ gz η̂p1 = ηp1+ηgz η̂p2 = ηp2+ηgw η̂p3 = ηp3+ηgz η̂p4 = ηp1+ηgw ηp1 = ηp JpzΩg ηp2 = ηp JpyΩg ηp3 = ηp JpxΩg ηgw = ηg JΣgwΩg ηgz = ηg JΣgzΩg hpx = hpx JpxΩg hpy = hpy JpyΩg hpz = hpz JpzΩg c1 = 1 JpzΩ 2 g c2 = 1 JpyΩ 2 g c3 = 1 JpxΩ 2 g c4 = c6 = 1 JΣgzΩ 2 g c5 = c7 = 1 JΣgwΩ 2 g Figure 4 shows a layout of the control and the correction of a three-axis gyroscopic platform. 48 Z. Koruba Fig. 4. Outline of the TTGP closed-loop control 4. Obtained results and conclusions Assume that the setmotion of theTGP is describedby the following functions of time (the programmedmotion is set around the circumference) ψpz = ψ o pz sinνpt ϑpz =ϑ o pz cosνpt and that there is an external disturbance (kinematic interaction with the base along the Oxp, Oyp, Ozp axes in the form of the FO board harmonic vibra- tions) in the time interval ∆ = 〈t1, t2〉 occurring with angular velocity equal to ψopz =0.25rad ϑ o pz =0.25rad νp =1.5 rad s r∗o =5 rad s νz =50 rad s and the disturbing moments act only in relation to the precession axis and have the form Mz = M v z +M t z = µgrs+m t c sgn(rs) Control and correction of a gyroscopic platform... 49 where: mtc =0.5dcµcNc, with the TGP parameters equal to Jgo =5 ·10−3kgm2 Jgk =2.5 ·10−4kgm2 Jp =2.5 ·10−2kgm2 ng =600 rad s µg =0.01Nms m t c =2 ·10−4Nms2 Figures 5 through 8 concern the investigations of the two-axis gyroscopic platform case.We confirmthe operational efficiency of the closed-loop optimal control which minimizes the deviations from the set motions to admissible values and kinematic interaction with the base (the flying object board). As there is great similarity between theTGPandTTGPcases, nodata concerning the latter have been included. Fig. 5. Variations of the TGP angular position in function of time caused by initial conditions: (a) without correction control, (b) with correction control Thework discusses the results of some preliminary investigations of dyna- mics and control of a two- and three-axis gyroscopic platform fixed on board of a flying object. To realize the programmedmotion of the platform, we need to apply controls determined from the inverse problem of dynamics. Then to correct and stabilize this motion, we have to introduce control with feedback. Furthermore, it is required that the Golubiencewmodifiedmethod be used to determine the optimal parameters of that system. The optimized parameters of the controlled one-, two-, and three-axis gy- roscopic platform allows us: a) to reduce the transition process to minimum; b) to minimize the overload acting on the platform; c) to minimize values of the gyroscope control moments, which may affect technical realizability of the control. It should be noted that the data were obtained for two- and three-axis platforms with three-rate gyroscopes. Such gyroscopes are used as control 50 Z. Koruba Fig. 6. Time-dependent variations of the TGP correctionmoments Fig. 7. The set and desired trajectories of undisturbed TGPmotion: (a) in the open system, (b) with feedback Fig. 8. The set and real undisturbed TGPmotion: (a) in the open system, (b) with feedback Control and correction of a gyroscopic platform... 51 sensors. Thus, the operational accuracy of the platforms depends mainly on theoperational accuracyof theappliedgyroscopes. Since three-rate gyroscopes are easier to control, relatively little energy needs to be supplied to alter the spatial position of the platform (Koruba, 2001). References 1. Dubiel S., 1973, Generalized constraints and their application to the study of controllability of flying objects [in Polish], Supplement Bulletin of Military University of Technology, Warsaw 2. Kargu L.I., 1988, Gyroscopic Equipments and Systems, Sudostroyenye, Le- ningrad 3. Koruba Z., 2001,Dynamics and Control of Gyroscope on the Board of Aerial Vehicle [inPolish],Monographs,Studies,DissertationsNo.25,KielceUniversity of Technology, p.285, Kielce 4. Koruba Z., 1999, Selection of the optimum parameters of the gyroscope sys- tem on elastic suspension in the homing missile system, Journal of Technical Physics, 40, 3, 341-354 Sterowanie i korekcja platformy giroskopowej umieszczonej na pokładzie obiektu latającego Streszczenie W pracy przedstawione jest sterowanie optymalne i korekcja trzyosiowej platfor- my giroskopowej, umieszczonej na pokładzie obiektu latającego.Odchylenia od ruchu zadanego są minimalizowane za pomocą sterowania programowego, algorytmu opty- malnego sterowania korekcyjnego oraz wyboru optymalnych parametrów platformy giroskopowej. Manuscript received August 3, 2006; accepted for print October 4, 2006