CONTROL AND CORRECTION OF A GYROSCOPIC PLATFORM MOUNTED IN A FLYING OBJECT


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1 JOURNAL OF THEORETICAL AND APPLIED MECHANICS 45,1,pp.4151,Warsaw2007 CONTROL AND CORRECTION OF A GYROSCOPIC PLATFORM MOUNTED IN A FLYING OBJECT Zbigniew Koruba Faculty of Mechatronics and Machine Buiding, Kielce University of Technology The work is concerned with the optimal control and correction of a threeaxis gyroscopic platform fixed on board of a flying object. The deviations from the predetermined motion are minimized by means of a method of programmed control, an algorithm of the optimal correction control, and selection of optimal parameters for the gyroscopic platform. Key words: gyroscope, optimisation, control, navigation Notation A,B stateandcontrolmatrices,respectively; x,u stateandcontrol vectors,respectively; x p,u p set(programmed)stateandcontrolvectors, respectively;u k correctioncontrolvector;k amplificationmatrix;q,r weight matrices; OGP oneaxis gyroscopic platform; TGP twoaxis gyroscopic platform; FO flying object; TTGP threeaxis gyroscopic platform; M p g1,mp g2,mp g3,mp g4 programmedcontrolmomentsappliedtotheinnerand outerframesoftherespectiveplatformgyroscopes; M p k1,mp k2,mp k3 programmed correction moments applied to the respective stabilization axes of theplatform; µ g,µ p dampingcoefficientsrelativetothestabilizationand precessionaxesoftheogp;µ c coefficientofdryfrictioninthegyroscope bearings;d c diameterofthegyroscopebearingjournal;n c normalreaction inthegyroscopebearings;ϑ gz,ψ pz anglesdeterminingthesetpositionofthe gyroscopeandtheogpaxesinspace,respectively;k k,k g,h g amplification coefficientsoftheogpclosedloopcontrol;j go,j gk longitudinalandlateral momentsofinertiaofthegyroscoperotor,respectively;n g gyroscopespin velocity;φ p,ϑ p,ψ p anglesdeterminingspatialpositionoftheplatform;φ pz, ϑ pz,ψ pz anglesdeterminingthesetpositionoftheplatforminspace;p,q, r angularvelocitiesofthefomotion(kinematicinteractionwiththegyroscopicplatform);p o,q o,r o amplitudesoftheangularvelocitiesp,q,r,
2 42 Z. Koruba respectively;ν z frequencyofthefoboardvibrations;ν p setfrequencyof 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 platforms need to be characterized by high operational reliability and high accuracy in maintaining the predetermined motion. 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 of weapons, such as unmanned aerial 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 threeaxis 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 platform will be written in the vectormatrix 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 a mechanical system, system parameters and its constraintsatwhichthesetmotionistheonlypossiblemotionofthesystem. In practice, the problems are frequently associated with special cases in which 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. Thenextstepistoformulatethedesiredsignals(Koruba,2001).Letu p stand for the desired(set, programmed) control vector. The control problem
3 Control and correction of a gyroscopic platform involvesdeterminationofthevariationsofcomponentsofthevector u p in function of time, which are actually the control moments about the gyroscope axis and the motion being determined by the desired angles(ones defining the set position in space of the platform). Now, transform Eq.(2.1) into the form Bu p =ẋ p Ax p (2.2) Thequantityx p inexpression(2.2)isthedesiredstatevectoroftheanalyzed gyroscopicplatform.theestimatedcontrolmomentsu p,areknownfunctions of time. Here, we are considering openloop control of a gyroscopic platfrom, thediagramofwhichisshowninfig.1. Fig. 1. Layout of the gyroscopic platform openloop control Now,letuscheckwhatmotionofthegyroscopicplatformisexcitedbythe controls. By substituting them into the righthand sides of Eqs.(2.2), we get ẋ =Ax (2.3) wherex =x x p isadeviationfromthedesiredmotion. The inverse problem is explicit for the derivatives of the state variables in relationtotime ẋ,butforafixedmotion,i.e.withinlimits,whent. The question is: will the solutions to the above equations describe also the set motionofthegyroscopicplatform?iftheinitialaxisangleissettobethesame astherequiredone:x(0)=x p (0),wewillobtaindesiredangulardisplacements of the gyroscopic platform. However, if the initial platform position is different from the desired position, then, even though the velocities of the state variables coincide with the desired ones, the platform will not realize the desired motion in the desired location in space.
4 44 Z. Koruba Apart from the above mentioned inconsistency of the programmed control, the platform performance is affected by external disturbances, mainly kinematic interaction with the base(flying object board), nonlinear platform motion (large gyroscope axis angles), friction in the suspension bearings, manufacturing inaccuracy, errors of the measuring instruments, etc. Fig. 2. Layout of the gyroscopic platform closedloop control Additionalclosedloopcorrectioncontrolu k (Fig.2),mustbeappliedto make the gyroscopic platform move along a set path. Then, the equations describing motion of the controlled platform will have the following form ẋ =Ax +Bu k (2.4) Thecorrectioncontrollaw u k,willbedeterminedbymeansofalinearquadratic optimisation method with a functional in the form J= Thelawwillbepresentedas 0 [(x ) Qx +u kru k ]dt (2.5) u k = Kx (2.6) The coupling matrix K in Eq.(6) is estimated from the following relationship A P+PA 2PBR 1 B P+Q=0 (2.7)
5 Control and correction of a gyroscopic platform The matrix P is a solution to Riccati s algebra equation K=R 1 B P (2.8) Theweightmatrices, Rand Q,inEqs.(2.7)and(2.8),rearrangedinto the diagonal form are selected at random(koruba, 2001), yet the search starts fromthevaluesequalto q ii = 1 2x imax r ii = 1 2u imax i=1,2,...,n (2.9) where:x imax maximumvariationrangeoftheithstatevariablevalue;u imax maximum variation range of the ith control variable value. Bysubstituting(2.6)into(2.4),weobtaintheequationsofstateinanew form again ẋ =(A BK)x =A x (2.10) where A =A BK (2.11) Recallthatitisimportanttomaketheplatformasstableaspossible.This means the transition processes resulting from the switching on the control system or a sudden disturbance must be reduced to a minimum. 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 threeaxis gyroscopic platform Foratwoaxisplatform,thestateandcontrolvectorsxanduaswellasthe stateandcontrolmatricesaandb,areasfollows [ x= ϑ p, dϑ p dτ,ψ p, dψ p dτ,ϑ g, dϑ g dτ,ψ g, dψ ] g dτ u=[m k1,m k2,m gw,m gz ] h py η gw h A= pz η gz h py 0 n 0 (η pw +η gw ) 0 n n n 0 h pz 0 n n 0 (η pz +η gz )
6 46 Z. Koruba while τ=ω g t h py = h py J p Ω g 0 c p c p c B= p c p c gw c gz Ω g = J gon g Jgw J gz h pz = h pz J p Ω g η gw = η gw J gw Ω 2 g η pw = η pw J pw Ω 2 g η gz = ηgz J gzω 2 g η pz = η pz J pz Ω 2 g c p1 = 1 J py Ω g c p2 = 1 J pz Ω g c gw = 1 J gw Ω g c gz = 1 J gz Ω g A block diagram of the control and the correction for a twoaxis platform isshowninfig.3. Fig. 3. Diagram of the TGP closedloop control However,inthecaseofathreeaxisplatform,wehave x= [ ψ p, dψ p dτ,ϑ p, dϑ p dτ,φ p, dφ p dτ,ψ g1, dψ g1 dτ,ϑ g1, dϑ g1 dτ,ψ g2, dψ g2 dτ,ϑ g2, dϑ ] g2 dτ u=[m k1,m k2,m k3,m g1,m g2,m g3,m g3 ]
7 Control and correction of a gyroscopic platform h pz η p η p h py η p h px η p3 0 0 A= h pz 0 A η p1 0 A η p A A 2 0 η p A h px η p h pz A 2 0 η p A 2 0 η p4 A 1 = JΣ gw A 2 = JΣ gz J Σ gz J Σ gw 0 c c c c c c c B= c c c c 7 while τ=ω g t Ω g = J gon g JgwJ Σ gz Σ η p1 =η p1 +η gz η p2 =η p2 +η gw η p3 =η p3 +η gz η p4 =η p1 +η gw η p1 = η p J pz Ω g η gw = η g JgwΩ Σ g h py = h py J py Ω g c 2 = 1 J py Ω 2 g c 5 =c 7 = 1 J Σ gw Ω2 g η p2 = η p J py Ω g η gz = η g JgzΩ Σ g η p3 = ηp J pxω g h px = h px J px Ω g h pz = h pz J pz Ω g c 1 = 1 J pz Ωg 2 c 3 = 1 J px Ωg 2 c 4 =c 6 = 1 Jgz ΣΩ2 g Figure4showsalayoutofthecontrolandthecorrectionofathreeaxis gyroscopic platform.
8 48 Z. Koruba Fig. 4. Outline of the TTGP closedloop control 4. Obtained results and conclusions AssumethatthesetmotionoftheTGPisdescribedbythefollowingfunctions of time(the programmed motion is set around the circumference) ψ pz =ψ o pzsinν p t ϑ pz =ϑ o pzcosν p t and that there is an external disturbance(kinematic interaction with the base alongtheox p,oy p,oz p axesintheformofthefoboardharmonicvibrations)inthetimeinterval = t 1,t 2 occurringwithangularvelocityequal to ψpz o =0.25rad ϑo pz =0.25rad ν p=1.5 rad s r o=5 rad s ν z =50 rad s and the disturbing moments act only in relation to the precession axis and have the form M z =M v z+m t z=µ g r s +m t csgn(r s )
9 Control and correction of a gyroscopic platform where:m t c =0.5d cµ c N c,withthetgpparametersequalto J go = kgm 2 J gk = kgm 2 J p = kgm 2 n g =600 rad s µ g =0.01Nms m t c= Nms 2 Figures 5 through 8 concern the investigations of the twoaxis gyroscopic platform case. We confirm the operational efficiency of the closedloop 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 the TGP and TTGP cases, no data concerning the latter have been included. Fig.5.VariationsoftheTGPangularpositioninfunctionoftimecausedbyinitial conditions:(a) without correction control,(b) with correction control The work discusses the results of some preliminary investigations of dynamics and control of a two and threeaxis gyroscopic platform fixed on board ofaflyingobject.torealizetheprogrammedmotionoftheplatform,weneed 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 Golubiencew modified method be used to determine the optimal parameters of that system. The optimized parameters of the controlled one, two, and threeaxis gyroscopic 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. Itshouldbenotedthatthedatawereobtainedfortwoandthreeaxis platforms with threerate gyroscopes. Such gyroscopes are used as control
10 50 Z. Koruba Fig. 6. Timedependent variations of the TGP correction moments Fig.7.ThesetanddesiredtrajectoriesofundisturbedTGPmotion:(a)intheopen system,(b) with feedback Fig.8.ThesetandrealundisturbedTGPmotion:(a)intheopensystem,(b)with feedback
11 Control and correction of a gyroscopic platform sensors. Thus, the operational accuracy of the platforms depends mainly on the operational accuracy of the applied gyroscopes. Since threerate 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, Leningrad 3.KorubaZ.,2001,DynamicsandControlofGyroscopeontheBoardofAerial Vehicle[in Polish], Monographs, Studies, Dissertations No. 25, Kielce University of Technology, p. 285, Kielce 4. Koruba Z., 1999, Selection of the optimum parameters of the gyroscope system on elastic suspension in the homing missile system, Journal of Technical Physics, 40, 3, Sterowanie i korekcja platformy giroskopowej umieszczonej na pokładzie obiektu latającego Streszczenie W pracy przedstawione jest sterowanie optymalne i korekcja trzyosiowej platformy giroskopowej, umieszczonej na pokładzie obiektu latającego. Odchylenia od ruchu zadanego są minimalizowane za pomocą sterowania programowego, algorytmu optymalnego sterowania korekcyjnego oraz wyboru optymalnych parametrów platformy giroskopowej. Manuscript received August 3, 2006; accepted for print October 4, 2006
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