Movement control of nonrigid mechanical systems with a changing vector of parameters and number of freedom degrees

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1 Movee corol of org echacal syses wh a chagg vecor of paraeers a ber of freeo egrees V.Y. Rovsy, V.M. Shaov, V.M. Glov Absrac. The paper scsses he peclares of he corolle ovee yacs of fleble echacal syses wh e-varyg ber of freeo egrees. The sraeges seqece of sch osaoary obecs corol s sae. These sraeges garaee he hgh accracy of he corol a apg of he elasc oscllaos. Bloc schees of he corol syse are sggese ha realze hese corol sraeges a ffere sages of he obec. I. INTRODUCTION AND STATEMENT OF A TASK May ffere ypes of ovg echacal obecs ha clearly ehb he properes of fleble l-freqecy oscllag syses wh screely e-varyg ber of freeo egrees are well ow. Typcal eaples of sch echacal syses (MS) are he orb-asseble large space srcres (LSS) []. Space a erwaer roboc oles ha chage her srcre rg he operao a have log fleble aplaor ls or fleble payloas ca be cosere as sch obecs. Slar yacs ehb eoc lsory blgs ha are cosrce o ovg basee wh acve sably syses []. These obecs are creae earhqae-proe zoes. A prcpal feare of sch MS s a rg carrer boy (a boy) a aache o hrogho he assebly soe fleble elees (carre boes). Sch cosrco aes possble o solve he corol probles of screely evolvg srcre (DES) wh he se of eqaos ha are shape as a seqece of oel-physcal oels (MPM) [3] M : = (); + = (), =,, (, N); () = +, = ; ( ) = M( ) I, = where ϑ q s he corolle coorae of he carrer boy; s he coorae of he rasfer (rg) oo; s he aoal oo of he carrer boy e o he flece of he fleble elees;, are he faeal freqeces a he ecably coeffces of he elasc oes; s he ber of he fleble carre elees a he -h sage of he assebly; N s he oal ber of aache elees; M () s he corol aco; s he corol Sppore by Rssa Foao for Basc Research (Proec ). V.Y. Rovsy, V.M. Shaov, V.M. Glov, Ise of Corol Sceces, Rssa Acaey of Sceces, Profsoyzaya 65, 7997, Moscow, Rssa Tel: ; e-al : rov@p.rss.r law (he p sgal of he oreao syse acaor evce); I = Ic( ) s he era oe of he cosrco a he -h sage of he assebly, M, ( =,,,..., N) efes MPM of he obec a he -h sage of s assebly he orb. Ie = efes he MPM of he carrer boy: M : = ( ), ( ) = M( ) I. () A hs sage, carrer boy s se p, oree a sablze wh he accracy, whch s ee for he e assebly sages. Wh creasg he vale of, he oel () becoes ore coplcae sce he ber of freeo egrees a he era oe also crease. Accorg o he geeral Raylegh s heore, a cree of he era oe leas o ecrease freqeces. They close wh he freqecy of he "rg" corolle oo. I s well ow ha, as a resl, a qaly corol becoes probleac a oo sably ay arse. Ths sably ca be case by he "capre" of he reglaor by elasc oscllaos. A = N eqaos () efes copleely asseble cosrco. The above scsso sggess ha, whe esgg he corol syse, he followg hree qalavely ffere ypes of he corolle obec coo shol be sgshe.. The al ype ( = ) volves he carrer boy oreao wh respec o he reqre reco a s sablzao wh accracy ha s ecessary for he frher assebly.. Oce he frs cosrco fleble elee ( = ) a soe oher fleble elees ( * ) are aache o he asseble obec ha begs ehb he properes of a fleble MS, whch s characerze by he presece of oe or several coparavely hgh-freqecy ( Hz) vbrao oes. 3. As he ber of he fleble elees creases ( * < N), he asseble cosrco rs o a har-ocorol syse. Sch syse s sgshe by a bg era oe of he aache boes a low elasc oes freqeces (<, Hz). These freqeces close wh he faeal freqecy of he "rg" oo of he obec. I he paper he followg ass are solve: he rasforao of yacal properes of a screely evolvg srcre ha s beg chage accorace wh prescrbe cosrco assebly seqece; eerao of he rasforao boares Bewee he boares he asseble cosrco reas he properes ha correspo o oe of hree aforeeoe ypes of he syse coo: a) rg boy, b) fleble obec wh sgfcaly affece syse yacs of he cosrco elasc oscllaos, c) fleble l-freqecy cosrco ha reqres a eeso of he observao

2 vecor, so ha he esre corolle yacs ca be acheve; he corol syse esg of screely evolvg fleble obec wh he se of he seqece of algorhs ha correspo o he obec coo a plee a sable corol of he a boy wh regars o elasc oscllaos a prove a hgh accracy o all sages of he assebly. II. TRANSFORMATION OF THE DES DYNAMICAL PROPERTIES For brevy as he corol obec wll be cosere he cosrco of "brella" ype [3] ha s show Fg.. Ths cosrco s sable for escrbg sch obecs as с y о ϑ α о y o o ϕ y y b of he aforeeoe sprgs. K s = N = s he oal ber of he cosrco elees. Trasforao of he yacal properes of he DES ereae srcres ca be reflece by he MPM () coeffces,, I ( ), =, ; =, N. Calclao of c hese coeffces a bg vales of he ber reqres ch e. For solvg of he as ha s efe he rasforao of he DES yacal properes hrogho he assebly s covee o se he pacage of progras [3] for coper ervao of he DES aheacal graphoel. As he op proc of hs pacage, oreover of coper vsalzao he graph-oel, we have wo raglar arces =, = a row ar ( I ) =..., =, N. I I I I I Fg. s show he eaple of hs pacage se for he "spral" ype of he obec assebly (Fg. 3) wh he paraeers: K=, =, s, ( s = 5), =, K, = g = 5 g, I = 5 g, r =,5 ;, l =, =, 6 (all elees are he sae). Fg..Crre srcre of he DES. he bg space rao-elescopes a space solar-reflecors [,]. As he MS s he oaly of rg boes oe of he s he carrer boy (, I ). Ohers (carre boes,, I ) are he elees ha are aache o he carrer boy oe or aoher orer. A he aachg pos of he elees here are he sprgs ha ae he flebly of he carre boes a resrc her splacees. Frher plaeparallel oo of all boes s cosere. For efeess he reglar srcre s cosere (for eaple bg copo reflecor []). Raally (Fg. ) K chas are aache o he carrer boy a he pos o ( α, r = oo, =, K ). Each he -h cha has s coece ae rg elees of a pvo ype. The paraeers of he elees are:, I, l, r (ass, oe of era, legh a he sace fro he po o o s ceer of ass, =,,...,,..., s, s he ber of he elee he -h cha, s s he ber of he las elee whe he cha s o copleely asseble ye). Cha eforaos s efe by he elascy coeffces a) b) c) ( ) a ( ) Fg.. Chagg of he MPM coeffces hrogho he obec assebly. Fg. a shows ha he faeal freqeces rage of he elasc oscllaos we as he ber creases. A hs he lowes freqecy ( ) ecreases ha leas µ I ( ) с

3 ε o cog ogeher he freqecy wh he freqecy ( ) of rg oo. he syse). Ths leas o sep-wse cog ogeher freqeces a (, 4 ). Ths cog ogeher of aforeeoe freqeces occrs also for ore geeral case of PD corol of lfreqecy obecs. I Fg. 5 s show he eaple of coper cosrce of he raecory hoograph of he characersc eqao roos he space of hree esos ( α,, ). Alog he hr as ha coplees he plae of cople varable o orhogoal rhero he ber of aache elee s p ase. Sch approach o he aalyss λ Fg. 3. The "spral" ype of he obec asse = Sary coeffce of elasc oscllaos ecably () = [( Iс() I) ] a he egree of ecably µ ( ) = wh a he cree of he ber crease also. Ths fac caes ha srbg flece of he elasc oscllaos o he corol qaly grows, he cosrco rs o he har-o-corol syse a s reqre o have ore perfec corol algorh. Le s coser he process of cog ogeher of he lowes freqecy a he freqecy of he rg oo (rg obec wh he oe of era Iс( )) a PD corol algorh. I hs case ( ) = M( ) Ic ( ) = ( + ). The freqecy s efe by characersc eqao p + I ()( ) с + p =, p=. Obae wh he help of coper he graph ( ) = ( ) ( ) of cog ogeher he freqeces a s show Fg. 4. I s obvos ha a he assebly of he frs row elees ( =,, = ) he freqeces a coe ogeher slowly ( ( ),). Ths s eplae by sall cree of he sary oe of era. A he aachg he frs elee of he seco row ( = 3 ) he oe of era I creases sgfcaly (proporoally o sqare of he 3 sace of he aache ass fro he ceer of era of,4,,,8,6,4,, = ( ) Fg. 4. The graph of freqeces a cog ogeher. α ε λ Fg. 5. The roos raecory hrogho DES assebly. of he syse yacal properes aes possble o coec he cofgrao of he roos srbo wh he crre vale of he ber. Fg. 5 llsraes he geeral eaple of he raecory behavor of he syse () characersc eqao roos λ ( ) = α ± a roos λ ( ) = α ± wh lear PD corol algorh a =, 6. The bg sace bewee of he raecory al pos of he roos λ ( ) a λ( ) alog he as shows esseal flece of he oa oe ~ ( ) o he rg boy ovee (). A creasg of he ber aforeeoe propery s reae val oly a low. A passage fro oe row of he assebly o he e row s occrre p-le characer of he cog ogeher he freqeces a. The sace ecreases a fro a vale of he ber * ( or eaple * = 36 ) he sace bewee he roos becoes oo lle ( ε ) orer o garaee esre yacs of he corolle DES a s ecessary o se ore coplcae corol algorh. III. CONTROL STRATEGY TRANSFORMATION THROUGHOUT THE DES ASSEMBLY Above was eere he presece of hree ypes of he corolle obec coo hrogho s assebly: rg boy elasc MS esseally fleble MS. For each ype of he obec coo s reqre parclar approach o he corol algorh esg. λ

4 A. The sraegy of he DES corol he al sage of s coo For realzao of he reqre qaly of he obec corol frs of all he base corol algorh (,, ) s syhesze. A leas hs algorh s garaee esre yacs rg he frs phase of he DES esece as he rg boy. Maheacal oel of he rg obec correspos o eqao (). For he cocreeess of he vesgao a ag o acco ha alos all corol syses se he o-boar coper scree aalog of PD algorh s chose as base oe ( ) = [ ˆ( ) + ˆ( )], =,,,...,. (3) The se of hs scree algorh leas very ofe o he ecao of he cosrco elasc oscllaos. I (3) ˆ( ) he esao of he easre coorae. For he process of esao s se s vales of he coorae (), =, s, rg he screeess pero T. The vale ˆ( ) s calclae as he frs fferece of he coorae ˆ( ). As he syse s scree he corol aco () s scoos a s cosa rg he screeess pero T. Throgho rege of he obec sablzao ha s he a oe ea zoes a hyseress he acaor evce characerscs lea o he ao-oscllaos. Toay here are ay well-ow algorhs ha garaee esre yacs of he sablzao processes (, χ=τ τ χ *, τ s he par of he sable l cycle pero τ whe a (), τ s he oe whe Γ ( ) =, χ * s he assble vale of he coeffce of he l cycle qaly). The ovee ha correspos o hs l cycle ca be cosere as he referece oe. Sce s reqre ha he flece of he elasc oscllaos o he syse yacs wll be eglgble = ε, (4) all rase processes hrogho he obec assebly s e o he referece ovee. B. The sraegy of he DES corol he sage of he elasc MS Oce he frs fleble elee ( = ) a soe oher oes ( ) are aache o he carrer boy he obec rs o elasc MS. Usally a hs case scree base corol eces elasc oscllaos of he cosrco ha efor he sable l cycle a, as he resl, coo (4) s broe. The corol accracy ecreases. Who rasforao of he corol sraegy he hgh aple of he elasc oscllaos ca be as a corse of he syse sably [5]. The oher proble for he elasc MS corol s crease of he DES era oe (Fg., c) ha leas o ecrease of he corol aco effecveess a o crease of he yac errors. A las he a proble he corol algorh syhess for hs sage of he obec coo s creasg hrogho he assebly eso of oel (), p-le chagg all s coeffces a ecrease of he. a a Γ Iaccracy of he DES echacal paraeers calclao he syse esgg (a coseqely of oel () coeffces) a he sae for he al vales of he ew elasc oes ha occr a each e sage of he obec assebly reqre o oly rasforao of he corol sraegy b he se of aapve corol. A he sall vales of he ber he faeal freqeces of he elasc oscllaos are coparavely hgh a far away fro he freqecy of he "rg" oo (Fg. 5). I hs case s covee o apply as a corol sraegy he approach wh he se of ellge agosc [6]. Ths approach es he g of he base algorh ha garaees boh he "rg" oo sablzao a he elasc oscllaos apg. The essece of hs approach s followg. I s wellow [5] ha he ecao of he elasc oscllaos occrs a each swchg of he scree corol aco. Iesy of he oscllaos ρ () = (), whe ecees soe crcal level ρ, leas o sably of he corol syse cr (aly a he epese of he oa oe creasg). Coseqely, he vale ρ () a he eqaly ρ() < ρ ca be se boh for agoscs of he syse cr coo a as he corol sgal a = f( ρcr ρ( )) he loop of he base algorh (,,, λ) paraeer λ g (aapao). A hs he base algorh flece o he oscllag copoe = + of he oel () ca be esae by qasevelope ρ(, λ) = Ev[((, λ),)] a erval ha s eqal o several peros of he l cycle. Ths qasevelope afer wo-sage approao ca be presee by epoeal crve ρ(, λ) = Ev[((, λ),)]. The vale of epoeal crve e ν ( λ ) efes he rae of he oa oe aple chagg. The sg ν ( λ ) efes he characer of hs chagg. A sgν = he oa oe coverges, a sgν = + verges. Ths, for ay fe vale λ [ λ, λa ], ([ λ, λ a ] s assble rage of he paraeer λ g) he reglaor flece o he elasc copoe () ca be efe by he sgle ber ν = ν( λ ). Chagg he vale λ a calclag he e ν ( λ ) we oba "oel" fco ν ( ) = ν λ ca be obae. Ths fco esaes he base algorh flece o he obec of he elasc oscllaos. Toaly of he "oel" fcos ϒ = { ν ( λ)}, ( =, ; * ), each of ha has soe local eres (clg global ), s se as a foraoal sofware for he ellge agoscs sbsyse of he oscllao copoe () crre coo a for g of he base algorh paraeer λ. I [6] was show ha for esgg of he DES corol syse a each sage of he obec assebly s ecessary o solve he e wo ass: ) o efe he ber of he oa oe sg he efe vale of s freqecy ; ) o choose fro he oaly ϒ = { ν ( λ)}, ha s ep he coper as he owl-

5 ege base, he correspog o ber "oel" fco ν ( λ ) a o choose ew vale of he paraeer λ [ λ, λa] ha garaees he flflle of wo coos sg[ ν ( λ )] = a ν( λ) = ν. A hs case he ew corol aco [ (, λ)] reas goo qaly of he corol of he obec rg ovee a a he sae e realzes aal rae of he oa oe apg. Bloc schee of he sablzao syse of he DES as he fleble MS ha has aoal loop of he oscllag copoe agoscs a aapve correco of he base algorh s show Fg. 6. Base loop DES Corol evce K Dgal sesor z = [ ] Base algorh ([],, λ ) λ ν Iforao ole of he sbsyse of he base algorh g z () Ic Daa base ϒ = { ν ( λ)} a sbsyse of he paraeer λ g Fg. 6. Bloc schee of he aapve sablzao syse of he DES as he ce l he sa * = + τ whe he aforeeoe phase wll be as opal. Ths e- fleble MS. Here he aplfcao coeffce K ( ) of he corol elay ca be roce oly a par of swchg pos. evce s g accorg o he vale of he era oe Opal phase s he phase a whch he oa I ( ) c ha s calclae avace. Ths g realzes oe's aple afer he swchg wll be he salles cosacy of he corol aco effecveess fro all possble oes. I epes o he reco of he corol aco swchg. The opal phase s efe as ( ) = cos, N hrogho all sages of he obec's assebly. Aforeeoe procere of he qase- follows [7]: ( ) velope ρ() ae ν λ esao s realze he forao ole of he sbsyse of he base algorh g. π sg =+, = (5) π ( + ) sg =, =,,,... Ths ole has efcao evce of he oa oe freqecy a he evce of he e ν (, λ ) calclao. As For eaple he vale of he e-elay τ he p sgal s se he ae base Z ha cosss of he swchg po ha s characerze by he coo he base aa z[] l (aples of he recfe sgal (( )) = ε, sg ( ) = (ε s he ea zoe of he relay fco) wll be as follows: zl, ) a he ae base T = { [ l]}. I he rege of he oa oe he ffereces [] l = { [] l [ l ]},5 [ π ( )] π, τ τ = (6) of he aace elees of he base aa T = { [ l]} coce wh he sepero,5 τ [3 π ( )] π < π, of he oscllao copoe ha has aal aple. Afer he average operasa whe ( where = ( ) s he oa oe phase a he - ) = ε. L o τ = [] l, ( L = T ) he oa oe I [7] was show ha for he syse ovee sably L opal phase of swchg s be a leas a he oe-half of = he swchg pos ha occr a each pero of he l freqecy = πτ ca be calclae. For he eerao of he oa oe ber he ffereces Fg. 7. The a loop of he corol syse s epce by a = cycle. Bloc schee of sch ype corol syse s show are aalyze a s asse = o le. Ths loop cles a aoal l wh wo g paraeers K ( =, ) for = =., τ. The frs paraeer K s he g aplfcao coeffce ha s eee for he aeace Correce corol aco [ (, λ)] aps oa of he cosa level of he corol aco = MKIc ( ) oe by opal way. A he sae e oher elasc oes wh he varable ass-era properes of he asseble ca be crease a oe of he wll be as ew oa obec. oe. The he process of he paraeer λ g s repeae. Descrbe processes of seqeal apg of he oa oes ae place hrogho he obec assebly a o o reqre aoal cospo eergy for corol. C. The sraegy of he DES corol he sage of he esseally fleble MS The a efcecy of he prevos sraegy of corol hs sage of he obec's presece s possbly o esae he qesevelope ρ() T rg he observao erval sr ha s accepable for corol. Ths s he resl of cog ogeher he lowes freqecy of he elasc oscllaos wh he freqecy of he "rg" ovee (Fg. 5). A hs case resoace processes ca occr a he syse becoes sable. As he base of he sraegy of corol for he esseally fleble MS ca be se he oe ha was sggese [7]. I hs ype of corol s se he esaos of he oa oe phases he sas of he corol aco swchg. The e-elay τ for corol aco swchg s ro-

6 The seco g coeffce τ plees he corol by he e-elay of he relay corol aco, whch swches wh respec o he base algorh reqrees. The esao of crre phase of he oa oe s obae wh he help of Kala fler [8]. The eaple of coper slao of he sggese syse for oe sage of he obec assebly s show Fg. 8. As he obec correspog o eqaos () a = 6 was chose he large space srcre wh he era oe 4 I ( = 6) = g. Oher paraeers are gve he able. c As he oa oe a he al sage of he corol was. Ths oe s sbece o he corol aco flece he os srogly becase s egree of ecably () µ = =,88 he os hgh. A al erval of he slao ( = c) he loop of e-elay of he corol aco swchg was o operae. I hs case he corol aco ( ) cases he crease of he elasc oe aple o he vale 3 A, ra ha s close o he crcal oe. осц. осц. осц.3 DES Base loop Corol evce Dgal sesor K, τ [] os Kala fler z= [] ẑ Base algorh = (,) z - z Iforao ole of he corol syse by e-elay swchg Fg. 7. Bloc-schee of corol syse for DES as he ( ), =,6 (, ) τ Ic () =, f = π,7,,5,5,8 5,,7,3,5,,4, (),5,5,,,5,3 Fg. 8. Processes a rege of sablzao DES a phase corol for a case =. I orer o preve he capre of he reglaor by elasc oscllaos a sably of he syse ovee a = c he algorh of phase corol [ (,, )] was apple. The ervals of he e-elay of he corol aco swchg are shae (see oscllogra ). As he resl he oa oe aple was ecrease very qcly. IV. CONCLUSION Sggese approaches ha realze aapve correco of he base algorh wh sg he elees of ellge agoscs a he eho of phase corol garaee apg of elasc oscllaos who creasg cospo of he eergy for corol. Refereces.. Beey I. A ereely large ye lra lghwegh space elescope a array. (Feasbly assesse of a ew cocep). Beey Desgs, Ic.4645 Qarer Charge Dr. Aaale, VA 3, Saleh A., Ael H. Opal corol of aapve/sar lsory blg srcres. Joral of Coper- Ae Cvl a Ifrasrcre Egeerg, 3, 998, P Rovsy V.Y., Krova I.N., Shaov V.M., Glov V.M. Graph-oels of orbal assebly a yacs of a large space srcre. // Proceegs of 6-h IFAC Sypos o Aoac Corol Aerospace. Preprs, v. (E. A. Nebylov). 4. pp Byaas V.I. Mlrror corollable srcres // Space researches, v.8, 5, 99, P ( Rssa). 5. Rovsy V.Y., Shaov V.M., Specfc relay corol of fleble saelles Proceegs of he 5-r IFAC Sypos o Aoac Corol Space, Geoa Rovsy V.Y., Zelyaov S.D., Shaov V.M., Glov V.M. The esg eho of robs corol by fleble spacecraf. 6-h Cogress IFAC, Prage, Czech, Jly 4-8, 5. Fll paper (Mo-E-TO/6) o CD-ROM. 7. Rovsy V.Y., Shaov V.M. Ae corol algorhs fleble saelles sg forao o he phase of elasc oscllaos. Proceegs of he 6-r IFAC Sypos o Aoac Corol Space Yerlova T.V., V.M. Shaov, A.S. Yerlov, V.G. Borsov Recrre esao of he agle oo cooraes of fleble obecs of aerospace echqes // Joral Aerospace eqpe, 6, 4, P ( Rssa).