Electric Motio Platfor for Use i Silatio Tecology Desig a Optial Cotrol of a Liear Electroecaical Actator Evže Töel Abstract Tis paper looks at te esig a optial cotrol of a electroecaical liear actator to be se i a six-egrees-of-freeo otio platfor applicatio itee for se i silatio tecology. Te paper reacts to recet calls i te silatio istry to replace yralic cyliers wit electroecaical actators wile keepig te kieatic a yaic paraeters affecte. Te paper provies a copariso of bot syste types wit a escriptio of te esig of optial cotrol for electroecaical actators. Keywors electroecaical liear actator, Stewart platfor, silatio tecology, optial cotrol A. History I. INTRODUCTION Motio platfors wit six egrees of freeo, also kow as exapos, are possibly te ost poplar robotic aiplators se i silatio tecology. Tis parallel ecais was first escribe by V. E. Gog [], wo costrcte a octaeral exapo to test te beavior of tyres sbjecte to forces create rig airplae laigs. Te first ocet proviig a etaile escriptio of tis strctre se as a airplae cockpit silator was pblise i 965 by D. Stewart [] (ece te ae of te strctre). Te Stewart platfor is a close kieatic syste wit six egrees of freeo a six ajstable legt ars (see Fig. ). Copa to oter siilar strctres, its ai avatage is ig rigiity a a ig iptpower-to-evice-weigt ratio. Mascript receive e 3,. Evže Töel is wit te Departet of Electric Drives a Tractio, Czec Tecical Uiversity i Prage, Czec Repblic (eail: toee@fel.cvt.cz) a e is wit te copay Pragolet, s. r. o. (www.pragolet.cz, e-ail: toeel.jr@pragolet.cz). Fig. : Crret state otio platfor wit yralic cyliers. Util te ieties of te last cetry, te ai obstacle ierig ore itesive applicatio evelopet was te isfficiet coptig power of te available arware. Deteriig te positio of a exapo is sigificatly ore ifficlt ta tat of covetioal serial strctres. Wit regar to te evice beig cotrolle i real tie, te ai calleges are te trasforatio of cooriates a spee of resolvig ateatical proceres [3]. Despite te coptig power isse avig bee ore or less reove i recet years, irect kieatic trasforatio (i.e. trasforig te legt of ars to te positio of te frae) still reais a callege. B. Use i Silatio Tecology Te Stewart platfor is freqetly se i silatio tecology to silate otio effects i veicle or airplae silators. By sig tis eqipet, it is possible to silate te forces actig po te pilot (river) rig te fligt (jorey), ts brigig te silator eve closer to reality. Te cocept a role of otio effect silatio i traiig is iscsse, for istace, i [4]; apart fro escribig te strctre of silators, te book expressly erlies te role of tis aspect rig eergecy evet traiig. Wit otio effects beig geerally perceive before oter kis of perceptios [5], tey provie te first possibility of etectig esi a ageros beavior of te airplae or veicle. C. Crret Stats De to te sigificat weigt of te silator cockpit fitte wit te reqi aiovisal eqipet a
cotrols (te weigt of te cockpit escribe i [6], for istace, aots to.5 toes) a te ee to acieve ig yaics levels, te platfor s liear actators are sally ipleete as yralic cyliers i silatio tecology applicatios. I spite of tis soltio eetig te above reqireets for sfficiet power a yaics levels, it sffers fro te faetal rawbacks of all yralic systes. Tese rawbacks icle, i particlar, iger spatial, teporal a fiacial istallatio reqireets resltig fro te ee to eploy a sfficietly powerfl yralic aggregate a pipig. I aitio, yralic soltios sffer fro relatively ig oise levels, wic caot be avoie by separatig te aggregate fro te platfor, as tis wol ievitably reslt i losses i te yralic pipig, traslatig ito ecreases i yaics. Moreover, te aggregate caot be eclose i a so-proof box as tis soltio oes ot esre sfficiet issipatio of excess eat; istallig te aggregate i a close roo reqires a aircoitioig it to be se, frter icreasig te costs. Te rawbacks of yralic systes icle eviroetal aspects too. I particlar, yralic oil as to be replace after a certai ber of operatio ors a ca, i cases of syste alfctios a breakows, case local polltio. Terefore, ay alfctios of yralic systes ave to be resolve wit tost care a attetio. Hece, i istaces were it is evisage tat te evice will be trasporte o a reglar basis (sc as te ligt sports aircraft silator escribe i [7]) it is ofte preferable to se a electric otio syste, wic, beig ore afforable a reqirig sigificatly sorter istallatio ties, akes te evice ore attractive to cstoers. Copa to yralic soltios, electric systes beefit fro ay avatages i particlar c less oise, iger eergy efficacy a ore sopisticate cotrol etos. However, besies teir iisptable avatages, electric systes also ave several isavatages. Te electroecaical trasissio is sbject to iger frictio, icreasig te wear a tear of te actator. Terefore, te lifetie of electric systes is typically soewat sorter ta tat of yralic soltios. I aitio, electric systes reqire a procere to brig te evice to a safe alt after expecte power cts. I te case of yralic systes, te safety laig procere is cate for sig oil fro a appropriately size yralic acclator. Wit regar to UPS its sigificatly icreasig te price of te syste, power failre eergecies are typically ale sig ecaical locks wic, i case of a expecte power ct, fix te platfor i its crret positio. Noeteless, eergecy escets of platfors wit a electric rive still reai probleatic. De to teir ay pros, tere ave bee growig calls to replace, i certai applicatios, yralic cyliers wit electric liear actators wile keepig teir static a yaic properties. Te first certifie aircraft silator i te worl sig a electric otio syste was fiise i 6 a, accorig to [8], tis tre will prevail i ftre as well. Te followig sectio provies a etaile escriptio of a electric liear actator itee for te applicatio etioe above. Te ai epasis as bee pt o te preparatio of a ateatical oel wic will be sbseqetly se to esig te optial cotrol. II. ELECTROMECHANICAL LINEAR ACTUATOR Wit regar to te forces reqi, te electroecaical actator is base o covertig rotary otio to liear otio via a ball screw. Beefittig fro ig efficiecy, rigiity a accracy ball screws are ofte se i acie tool costrctio. Te esig of te wole electroecaical actator is sow i etail i Fig.. Fig. : Electroecaical actator. A. Mateatical Moel of te Electroecaical Actator I orer to frter aalyse te electroecaical syste a, i particlar, wit respect to te ee to esig te optial cotrol, tis sectio eals wit te ateatical oel to be se. However, it is ot ecessary at tis poit to fie-te all oel paraeters, te ai ai beig to esre tat te oel reflects all sigificat yaic properties of te electroecaical actator a tat it works wit qatities wic ca be easily erive or irectly eas i te syste. Te followig assptios were ae before esigig te ateatical oel: for ecooical reasos, positio or velocity is eas oly at oe locatio (otor saft or ball screw). Terefore, te electroecaical syste will be oelle as a syste wit oe egree of freeo; wit respect to te above, te oel will igore te torsioal rigiity effects of te ball screw a te ecaical copliace of te coectio betwee te saft a te ball screw. As sow later by te cotrol reslts, tese factors ave oly a ior ipact o te
qality of cotrol wit respect to te criteria se to assess reglatio qality. Te ifferetial eqatio of otio ca be efie ieiately after replacig te syste wit a sigle virtal boy wit te geeralize ass sbject to all forces a oets F. Calle te Geeralize Forces Meto, te basic eqatio ca be writte as: x x x F. () I aitio, te followig properties apply (see, for istace, [9]): te kietic eergy of te geeralize ass is eqal to te s of all kietic eergies of te eleets of te syste, te virtal work of te geeralize force is eqal to te s of te virtal works of all forces a oets i te syste. Te kietic eergy of te ecaical syste ca be expresse as: E k x I x, () were is te ass of te loa, x te isplaceet of te ball screw, I te oet of iertia of te rotary parts of te syste a φ te rotatio agle. Te relatiosip betwee te traslatioal positio x a te rotatio agle is efie by te coversio costat k, wile: k x. (3) By sbstittig ito te previos expressio, we obtai te followig eqatio: I k. (4) Accorig to te virtal work priciple, te followig relatiosip ols tre: F were x N j W j M x g x F x F F sg is te frictio force. t t t (5) Te above expressio ca be se to eterie te geeralize force F : F x M k g F sg, (6) were M is te otor torqe, te loa ass a g te gravitatioal acceleratio. By sbstittig ito te basic Geeralize Forces Meto eqatio, we obtai te eqatio of otio for te ecaical part of te actator: t I k x M k g F sg x. t (7) Te rivig torqe M is geerate by a AC servootor cotrolle by a servo river. Wit regar to tese evices beig sippe by te afactrer wit te opti crret / oet reglator settigs, te evice sall be oelle as a first-egree yaic syste wit te tie costat τ. Wit respect to sall tie costats, te reaiig yaic properties of te servo rive ca be left ot of cosieratio, playig oly a egligible role i te overall beavior of te syste a beig irrelevat for te cotrol esig. Servo rive yaics ca be expresse wit te trasfer fctio G el s kel M s s s, (8) were is te variable correspoig to te reqi oet a, by te sae toke, te ipt sigal of te syste a k el te electric costat of te otor. B. Desig of te Optial Cotrol For te sake of coveiece, te ateatical oel will be writte sig atrices of state a trasfore ito te iscrete for. No-liearity case by frictio forces ca be copesate for by aig to or sbtractig fro te ipt sigal te vale Ft correspoig to te frictio force F t, a terefore will be left ot of cosieratio. Below are te eqatios of state of te syste: were A C k x ˆ Axˆ Bˆ y Cxˆ Dˆ, D, B kel, (9) x, xˆ x M g,, ˆ () Te correspoig iscrete for of te syste for te saplig perio T ca be writte as: were xˆ y T A xˆ T B ˆ T T Cxˆ T Dˆ T, ()
A B e e AT T AT e At t B. () I esigig te optial cotrol, te followig qality criteria will be cosie: Te reglate variable will be te isplaceet of te ball screw x. I silatio tecology (were te evice i qestio is se to reproce forces), syste respose tie (i.e. trasissio bawit) is te ost iportat a essetial qality paraeter. Wit regar to sall steay-state eviatios fro te esi positio ot beig perceptible to persos sittig i te silator cockpit, reglatio accracy is ot critical i tis applicatio. I spite of tis reqireet ot beig as critical as, for istace, i acie tools cotrol, were siilar ecaiss ivolvig ball screws are ofte se, it step respose sol reslt i sall oversoots oly. Cotrol as to be sfficietly robst to react flexibly to cages i te loa, wic ca be a vale fro te preset iterval,. ax I ters of cotrol teory, it is avisable to se as c iforatio abot te cotrolle syste as possible. Ieally, we sol be able to eiter irectly easre or soeow erive all state variables. I tis case, te cotrol law eqals to (see, for istace, []): K xˆ x Kr ref, (3) were K x is te row vector, K r te scalar, ref te g referece / esi positio a te k el k costat copesatig te effects of te loa. It ca be prove tat sig state space cotrol (state feeback loops) it is teoretically possible to cotrol te beavior of te syste as reqi []. I practice, owever, oe is liite by te ipt sigal, wic aots to fiite vales fro te rage, i ax. Te followig sectios focs o eteriig te optial state space cotrol to reglate te positio of te electroecaical liear actator. Te optial cotrol is oe wic iiises te optiality criterio; let s ow efie a qaratic criterio base o te cotrol qality reqireets set fort ereiabove: i i N i N y oversoot (4) y q y ref, (5) wit pealisig te ipt sigal if it excees te allowe liits: q ax ax q i i, (6) i ax a oversoot pealisig te cotrol i case of oversoots rig positive it step resposes (i.e. oootoos resposes): oversot q3x x x, (7) Te qatities q, q a q 3 are te asses a, at te sae tie, oralisatio coefficiets of iivial eleets of te criterio. C. Miiisig te Criterio via a Geetic Algorit Followig o fro te above, te cotrol strategy i eqatio (3) is copletely escribe by te vector K K x K r, wit cotrol qality beig eterie by te criterio. Hece, i eteriig te optial cotrol, te goal is to fi te vector K wit te ii vale of (te criterio). Tis isse ca be approace sig a geetic algorit eployig te priciples of evoltio biology (crossbreeig a tatio) to solve coplex probles. Eac iivial i te poplatio is escribe by teir croosoe (i or case te vector K), wit te probability of tis geetic iforatio beig passe to te ext geeratio beig irectly proportioal to its qality (lower vales of te criterio ). I aitio, tere is a sall cace tat tis geetic iforatio will tate (rao cages of gees witi te croosoe). Te iitial poplatio of several raoly cose iivials will be left to evolve er te siple evoltio rles efie above. After several geeratios, we select te best iivial wose croosoe cotais te ieal soltio for or proble (essetially, we are breeig te soltio for te opti cotrol proble). Wit regar to te atre of tis paper, it is ot possible to provie ere copreesive iforatio o te properties of geetic algorits a covergece coitios. However, a etaile escriptio ca be fo, for istace, i []. Te followig cart (Fig. ) sows te algorit covergece we searcig for te optial cotrol vale after 4 geeratios. were, at te optiisatio orizo N, y pealises te eviatio fro te esi positio:
Fctio vale 7.5 7 6.5 6 5.5 5 Best Fctio Vale: 4.695 4.5 3 4 Iteratio Fig. : Deteriig te opti cotrol. Te criterio a te wole algorit se to eterie its ii vale ca be easily ipleete for exaple i te MATLAB-Silik eviroet, as sow i Fig. 4. D. Te Robstess Coitio Te esige cotrol as to esre tat te syste reais stable after a loa cage a for all loa vales fro te relevat iterval. It ca be prove by silatio tat te syste will reai stable if te force exerte by te loa oes ot excee te axi force wic ca be geerate by te rive. Terefore: k k g ax el ax. (8) If te above coitio is et te syste sall reai stable a loa cages ot copesate for by te ipt sigal (see Eqatio (3)) will ave a ipact o te size of te steay-state eviatio oly. E. Ipleetig te Cotrol o a Real Syste I orer for state space cotrol to be possible, all state variables ave to be kow i eac cotrol step. I practice, owever, te electroecaical syste is K Ts z- K Ts z- q -K- q _y (error) _oversot _li.5 ref -K- Kr _li K* Kx ref y()=cx()+d() x(+)=ax()+b() Discrete State-Space x x M x,ref Scope q3 -K- To Workspace _oversot < K Ts z- Fig. : Ipleetatio of te oel of te electroecaical actator. Fig. sows te silatio reslts (respose to step cages of te positio) of te optial state reglator cotrollig te syste er te paraeters efie i Tab.. Tese paraeters correspo to te syste sow i Fig.. Tab. : Paraeters of te electroecaical actator. Nae Vale Uit k 8π ra/ k el. N/V I 5.549. -4 kg. <...5> kg g 9.8 /sec τ. s T. s Ft.4 V F t 6π N Silatio reslts sow tat step resposes o ot reslt i oversoots a oly sall eviatios fro te esi steay-state positio ca be observe. eqippe wit oe sesor oly, aely te positio sesor fitte o te otor saft. Te reaiig state variables (velocity a oet) ave to be erive accorigly. Velocity ca be eterie by ifferetiatig te crret positio, a te crret oet at te saft of te otor ca be ascertaie base o te ipt sigal via relatiosip (8). F. Coparig Dyaic Properties of Electroecaical a Hyralic Systes I tis case, yaic properties cover, i particlar, te trasissio bawit, i oter wors te ipt sigal freqecy rage wic ca be trasitte by te cage / apee syste. I literatre, tis axi freqecy is efie as te freqecy we te aplite of te otpt sigal rops to -3 B. Fig. sows a copares te reslts of freqecy caracteristics easreets cocte for yralic a electroecaical actators. Te positio of te yralic actator is cotrolle by a typical PID reglator, wereas te electroecaical actator ses
te reslts of te optial state space reglator escribe ere. As give i te Figre, te trasissio freqecy bawit of te electroecaical actator eqals to twice tat of te yralic syste. [V] III.CONCLUSION Tis paper escribes te process of esigig a electroecaical actator wic ca be se as a sitable replaceet for yralic cyliers i silatio tecology applicatios sig six-egree-offreeo otio platfors. Proviig a escriptio of te optial state space cotrol, te paper copares te operatioal a yaic properties of yralic a electroecaical systes, provig tat te latter ca acieve better yaics reslts. However, oe as to take ito cosieratio a ecrease i te lifetie of te evice as a coseqece of icrease wear a tear resltig fro ecaical frictio. I aitio, esrig a safe st-ow procere i case of expecte power cts ca be qite costly. I spite of tese rawbacks, crret tres a cstoer ea sow tat electroecaical otio platfors will graally replace crret yralic systes, te ai reasos, apart fro better yaic properties, beig sigificatly lower oise, icrease ease of istallatio a better eergy efficiecy. Magite [B] Pase [eg] -3-4 -6-8 -..4.8.3.6 Freqecy [Hz] Electric actator Hyralic actator - -4-6 -8..4.8.3.6 Freqecy [Hz] Fig. : Meas freqecy caracteristics of te real syste (sigal %). x/t [/sec] x [] M [N] -..4.6.8. -...4.6.8..5..4.6.8 -..4.6.8 tie [s] Fig. : Silatio reslts. REFERENCES Positio Ref [] E. Gog, Cotribtio to Discssio of Papers o Researc i Atoobile Stability, Cotrol a Tyre perforace, Proc. Ato Div. Ist. Mec. Eg., pages 39-394, 956-957. [] D. Stewart, A Platfor wit Six Degrees of Freeo, UK Istittio of Mecaical Egieers Proceeigs 965-66, Vol 8. [3] K. Li,. M. Fitzgeral, F. L. Lewis, Kieatic Aalysis of a Stewart Platfor. IEEE Trasactios o Robotics, vol. 4, No, 8 93. [4] A. T. Lee, Fligt Silatio. Asgate Pblisig Liite, 5. [5] E. Töel, Silace poybovýc vjeů a pilotí treažér. Tesis, 98, p. 6. [6] E. Töel, Silatig Motio Effects Usig a Hyralic Platfor wit Six Degrees of Freeo. I Sborík eziároí koferece IASTED Moellig a Silatio 8, 8 Sept., 8 Gaboroe, Botswaa: ISBN 978--88986-763-5. [7] E. Töel, Silátor lekéo a ltralekéo sportovío letala. [olie] ttp://www.pragolet.cz/claky/fligt_silator_of_ligt_a_l traligt_aircraft.tl [8] S. Mrty, Electric Actators Replace Hyralics i Fligt Silators. [olie] <ttp://acieesig.co/article/electricactators-replace-yralics-i-fligt-silators->. [9]. Nožička, Mecaika a teroyaika. Prage: ČVUT, 99, ISBN 8--47-, 37 p. []. A. Rossiter, Moel Base Pictive Cotrol. CRC Press, 3, ISBN -8493-9-4, 38 p. [] Havlea, V.; Šteca,.: Teorie yaickýc systéů. Prage: ČVUT,, ISBN 8--97-3, 48 p. [] V. Mařík, O. Štěpáková,. Lažaský, Uělá iteligece 3. Acaeia, 3, ISBN 8--47-6, 475 p.