Predictive Control of Redundant Parallel Robots and Trajectory Planning

Transcription

1 Predictive trl Reddat Parallel Rbts ad rajectr Plaig elda K.; öhm J. bstract he paper deals with the desig mdel-based mlti-step discrete predictive ctrl reddatl actated parallel rbts. I the paper there is a itrdcti speciic cmpesati gravitati r vertical parallel rbt cigratis. he the eective desig ctrl actis b predictive algrithms is preseted. Fiall the trajectr plaig a time parameterizati desired rbt paths bth rm iematical ad damical ptimizati pit view is discssed. he paper ccldes b demstrati several eperimets realized real labratr mdel reddat parallel rbt Slidig Star. Itrdcti wadas the rther develpmet idstrial rbts machie tls ad ceters depeds als reslts ad capabilities preset research ctrl desig. I practice qalit lcal decetralized PID-based ctrl is sall available. here eists e qesti here: Is this tpe ctrl siciet ad sae r ewl develped machies sch as parallel rbts? he aswer is t irst view clear. Hwever the lg-r eperimets ca epse besides diiclties with selecti the PID parameters sme prblematic states e.g. atagistic behavir r verlarge eerg csmpti. vercme these desirable states sme mdel descripti eerg decplig is reqired. Herewith reqiremet the mai advatage PID ctrl ctrl witht wledge mdel is disappeared. I case that the mdel is available the the se sme glbal cetralized mdelbased ctrl strateg is mre eective. he sitable selecti sch strateg is geeralized predictive ctrl GP [] csidered i this paper. evertheless the lcal ctrl ca be sell sed as a ast sb-drive-ctrl which desired vales are geerated b mai mdel-based ctrller e.g. b GP.

2 Sbseqet qesti is a trasrmati the demads the rbt mti i.e. t lill sme path give e.g. b techlg t prper rm sable i ctrl desig. he mti is a prcess rig i time therere sme time parameterizati the path i.e. trajectr plaig has t be csidered. I the paper the trajectr plaig will be tlied rm iematical ad damical pit view. he mai dierece is that the iematical wa prvides l time parameterizati give path idepedetl the rbt. O the ctrar the damical wa ses damical mdel depedet real rbt ad perrms simltaesl desig a trajectr shape ad a trajectr time parameterizati its realizati. he eplaati plaig is placed ater sectis dealig with the ctrl desig. del arragemet Parallel rbts represet mlti-bd sstem which damical mdel ca be described i phsical crdiates b agrage s eqatis mied tpe leadig t a sstem dieretial algebraic eqatis [3] s Φ λ g s s where is a mass matri s is a vectr phsical crdiates their mber is sall greater tha mber degrees reedm Φ s is a Jacbia λ is a vectr agrage s mltipliers g is a vectr ther iteral relatis matri cects ipts t apprpriate dieretial eqatis ad s represets gemetrical cstraits. del ca be trasrmed t mre sitable rm i idepedet crdiates labeled as which mber crrespds t mber degrees reedm. Sch trasrmati elimiates agrage s mltipliers ad algebraic cstraits. Herewith the dieretial algebraic mdel chages t mdel rdiar dieretial eqatis R R R R R g R where R is a Jacbia matri represetig ll space Jacbia Φ s. Obtaied mdel has a sitable rm bth r simlati ad r real ctrl.

3 . mpesati gravitati he eects gravitati have t be csidered l i vertical plaar ad geerall spatial rbts. Idividal elemets these rbts chage their psitis ths chage als distribti eects gravitati drives. hriztal plaar rbt Fg z vertical plaar rbt Fg Figre : Distribti eects gravitati drives r dieret rbt cigratis he gravitatial eects themselves are ivlved i the vectr g. simpli ctrl desig the mdel is sitable t trasrm t the llwig rm R R R R R g g R 3 r where a vectr g is a actr g depedig l velcities ad a vectr g r is a actr g ctaiig l elemets cased b eects gravitati. Frther sme terms i the mdel 3 ca be etitled de t their meaig F R gr R Fg Fτ 4 where F g is a vectr rce eects rm gravitati F τ is a vectr geeralized rce eects rm ipts altgether is a vectr F rce resltat cmpted r the crdiates tpts which represet the crdiates mvable platrm. he mdel ca be rewritte t a rm ad simpliied R R R g R R R F g he btaied mdel is sed r ctrl desig. Its rm garatees that term i 5 eqals zers r arbitrar ad zer time derivatives. his prpert is eedl r sed liearizati i et secti. Real ctrl actis are iall cmpted b epressi w ipts rm 4 as llws F 5 R R R F Fg 6

4 . O-lie dierece liearizati r parallel rbts O-lie dierece liearizati is a speciic algrithm which agaist stadard liearizati des t reqire a partial derivatives. It will be eedl r mlti-step predictive algrithms. itrdce it i brie let s arise rm liear ctis mdel 5 which liear term is sppsed t be liearized as llws [] a a 7 et s sppse that crdiates ad derivatives represetig crret rbt state ] [ ] [ are available; zers i are replaced b sitable zer epsil t avid zer divisi. Fiall sppse sme reerece state r lillig the prperties rm secti.: ] is arbitrar [ ] [ r r. he the algrithm the liearizati is give b the llwig epressi.... i i.. ] [ ] [ ] [ ] [ a a 8 he liearized rigial cti ca be epressed as a state matri a a 9 he algrithm is perrmed -lie drig real ctrl with respect t crret state the rbt.

5 3 Predictive ctrl Predictive ctrl is a mlti-step apprach cmbiig eedrward ad eedbac ctrl desig. Feedrward part is realized via predictis based rbt mdel. Feedbac part cected rm tpt measremet cmpesates sme mdel iaccracies ad limited eteral distrbaces. he desig csists i repeated lcal miimizati qadratic criteri. It icldes predictis rm eqatis predictis. 3. Eqatis predictis Eqatis predictis rm the character predictive algrithms [4]: algrithms geeratig ll ctrl actis algrithms geeratig icremets ctrl actis Priciple eqatis predictis is repetitive sel-iserti apprpriate mdel. derive geeral rm eqatis predictis r ll ctrl actis let s sppse discretized state rmlati mdel 5 with liearizati 9 as llws Eqatis predictis epress tre tpts ŷ rm crret state ˆ ˆ ˆ ˆ G G O ] [ ] ˆ ˆ ˆ [ ˆ ˆ i.e.

6 lthgh the parallel rbts have prel astatic character described eqatis cat garatee zer stead-state errr. It is cased e.g. b passive resistaces mechaical baclashes hsteresis. herere let s cs w e eample icremetal algrithm which atrall b itegrative character remves steadstate errr prblem. he lgrithm arises rm simple idea: btai itegrative character let s isert itegratr directl t the mdel 5. Its realizati is represeted b llwig lies g a a g d dt t d ~ 3 d dt d dt a a g a a g ~ ter trasrmati t state-space rmlati ad discretizati the sal discrete state-space mdel is btaied ~ ~ ~ ~ ~ ~ ~ 4 ~ where state csists ~ [ ]. I geeral case it is ecessar t cmpte this eteded state b state bserver. I case sitable selecti state variables i.e. havig phsical iterpretati as i r case the state ca be epressed as llws ~ s 5. Eqatis predictis with mdel 4 are btaied ideticall as ad. I case mdel 4 the real vales ctrl actis are cmpted rm previs vales. Vales ~ represet i view 4 icremets ctrl actis. ~ 6

7 3. iimizati qadratic criteri he secd dametal part predictive desig is a miimizati the criteri. J ˆ j w j Q j Q j j 7 where ad are hrizs; Q ad Q are pealizatis; ad w j are desired vales. Reslt the miimizati is real vales ctrl actis. prvide eective miimizati sqare-rt apprach with matri represetati the criteri 7 is sitable t se J [ ˆ w ] Q Q Q ˆ w Q 8 where Q diag Q ad Q diag Q. Sqare-rt miimizati meas that l sqare-rt criteri represetig a vectr is miimized. he Eclidea rm metied vectr is vale the criteri. he miimizati leads t a sstem algebraic eqatis 9 Q G Q w Q 9 I 9 the predicti is csidered. he sstem 9 itsel ctais mre rws tha clms ad i case reddat ipts represets deiciet ra sstem which is slved b rthgal-triaglar decmpsiti. he decmpsiti redces ecess rws ad ca slve eve liearl depedet clms. 4 rajectr plaig rajectr plaig is e iheret preparative peratis bere startig real ctrl prcess rbt mti. Its bjective is t geerate the reerece ipts w describig desired mti. Real trajectr is sall give b a mber dieret parameters: techlgical e.g. sitable machiig velcities mti rietati ad cstrctial legths radises shapes etc..

8 he plaig ca be csidered either rm iematical pit view where paths the mti are w r rm damical pit view where real paths are w ad l start ad ed pits ad bds are give r sed rbt. 4. Kiematicall ptimal desig rbt trajectries Kiematicall ptimal desig is based elemetar laws iematics. Drig desig the llwig characteristics have t be sccessivel determied legth path ad pssible rage rtati apprimate time r reachig the ed pit path gemetrical parameter r psitis ad rtatis real crdiates segmets ad apprpriate time derivatives et s tlie step b step the determiati metied characteristics. he path legth l ad rtati rage ψ ca be geerall determied as llws l ds ψ ψ ial ψ iital s I cases whe the legth cat be determied aalticall the it ca be determied apprimatel as a sm legths small abscissa segmets sbstittig csidered path. pprimate time ca be determied the basis w path legth ad e.g. w velcities v ω ad acceleratis a α i start ad ed pits via simple iematics laws b epressis a dv dt l t v iital v ial dω ψ α t dt ω ω iital ial I epressis dble itegrati is perrmed ad rm btaied times t ad t the higher labeled as t ial is chse. w it is pssible t determie the gemetrical parameter p t ad ρ t which represets e-dimesial D time parameterizati. Its cmptati arises rm selecti plmials acceleratis e.g. let s csider plmials 3 rd rder ad apprpriate iitial ad ial cditis r s v a ψ ω α a t a a t a t a t 3 3 α t α α t α t α t 3 3

9 dble itegrati the sstem algebraic eqatis r w ceiciets a a a a3 α α α α3 is cmpsed. mpted ceiciets is sed r determiati gemetrical parameter r psitis ad rtatis p s t p l ρ ψ t ρ ψ 3 he parameter ctrls the real plaig i D r 3D space. Fr its cmptati is sitable t select zer iitial ad ial accelerati t redce mber eqatis. he selecti 3 rd rder prvides ctis ad segmetall smth crves p t d derivatives. I the plmial r accelerati is 5 th rder the the crves are ll ctis ad smth p t d derivatives. w it is pssible t prvide real plaig idividal trajectr segmets via parametric crves r via desel sampled geeral crves. t the latter case case geeral crves the gemetrical parameter serves as a selectr crdiates w [ z] rm their apprpriate table give crve. he velcities ad acceleratis ca be determied b epressi a [ v :ed- - v :ed-/s v v [ v v v :ed - :ed- /s /s z /s] z /s v /s] z 4 his cmptati is l apprimati. herere the parameterizati accrac depeds the selecti iitial legth sbstittive abscissa segmets. D abscissa 34 II. 3D arcs 4 arcs I. abscissa arc 56 SE arc 3 s v a eample spatial rbt: Figre : Eample iematicall ptimal desig D ad 3D rbt trajectries

10 4. Damicall ptimal desig rbt trajectries Damicall ptimal desig is ather wa preparig the desired vales r real ctrl. he desig des t se iematical laws bt it is based damical relatis ivlved i the mdel csidered rbt e.g. mdel. he timeparameterized trajectries are geerated b simlati speciic ctrl tas. his secti tlies tw was: rve-based desig l start ad ed pits are give Rage-space desig start ad ed pits ad bds are give 4.. rve-based desig trajectries he damical crve-based desig csidered here is deied this wa: et s have tw pits start ad ed pit ad presmptive time i which the real rbt shld perrm the mti betwee thse pits. path is ree hard cstraits; l ed-pit shld be achieved see Figre 3. z start ed Figre 3: Sitati at damicall crve-based ptimal desig s a sitable wa predictive ctrl ca be sed agai. Predictive desig partl csists i the cmpsiti the eqatis predictis which ivlve the damical rbt mdel ad partl csists i miimizati qadratic criteri 7. he criteri icldes several adjstable parameters: the hrizs ad pealizatis Q ad Q ad als the desired vales w which determie the trasiti rm start t ed pit i the criteri. he vales w csidered here serve l r desig trajectries which represets real desired vales sed r real ctrl. I r case the vales w r desig are deied s that crrespd t ed state respectivel t the ed psiti i.e. w [ w j w ] w j cst. j. distribte the ipt eerg i whle time iterval determied b time i.e. t desig the trajectr with sitabl distribted eerg the described parameters the criteri are sed.

11 I we set ma / s s sitable selected samplig ad where is rder the mdel the the qadratic criteri will csider l last diereces amg predicted ed-pit ad its reerece vale. he hriz represetig iitial isesitivit determies mber ree tpts i.e. tpts witht pealizati which eable the algrithm t shit the reachig the ed pit at later time step at time with apprpriate distribti ipt eerg. hs i the criteri 8 the matri G ad crrespdig diereces w ctai l last rws ad elemets respectivel Q G Q w Q O O Q w w 5 I case rbts liear sstems drig the trajectr desig the mdel parameters have t be chaged accrdig t crret state with simltaes prgressivel shrteed hriz : ma ma - mi mi mi > 6 he shrteig prvides that the time limit is t verr. he algrithm prvides irm distribti the ipt eerg i speciied time... E S Figre 4: Eample damicall plaed trajectr r vertical rbt Slidig Star

12 4.. Rage-space desig trajectries he damical rage-space desig serves r desig trajectries which are deied b start pit ad mber bds e.g. i D r a rb ; see Figre 5 speciig ree crridr r rbt mti. ra crridr ed start rb Figre 5: Sitati at damicall rage-space ptimal desig he rage-space desig prvides plaig smth trajectries r rbt mti withi speciied bds. Fr described desig the predictive ctrl is sed agai. It partl icldes rbt damics ad als prvides ptimal distribti ipt eerg. Used qadratic criteri is mdiied de t ew reqired bds [5] which sbstitte sal desired vales w J ˆ j ra j Qra ˆ j rb j Qrb j Q j j 7 i matri represetati J [ ˆ ra ˆ rb ] Q ra Q rb Qra Q Q rb ˆ ra ˆ rb Q 8 I this desig is als imprtat selecti ew tpt pealizatis Q ra ad Q rb which are selected i sal ctrl prcess as idetit matrices. he pealizatis are t cstat bt drig the desig are damicall chaged accrdig t crret bds.

13 I simple terms chagig bd has higher weight therere the apprpriate pealizati has higher vales ad vice versa. I case that the crret bds have cstat vales the the pealizatis have the same vales..75 [m].5 ra rb.5 S E [m] Figre 6: Eample damicall plaed trajectr via rage-space ctrl. 5 Real eperimets Real eperimets were perrmed the labratr mdel rbt Slidig Star Figre 7. he rbt represets vertical plaar parallel cigrati with reddat actati. Idividal elemets lie dieret levels ptetial eergies ad drig mti chage their w ptetial eerg t therere rce eects gravitati has t be csidered e.g. as it is idicated i secti.. Figre 7: abratr mdel parallel rbt Slidig Star

14 he iematicall ptimal trajectr r eperimets is shw i Figre 8. abscissa 34 II. arcs 4 arcs I. abscissa arc 56 SE arc 3 s v a Figre 8: estig trajectr r rbt Slidig Star he aim the eperimets was t prve ptec algrithms predictive ctrl applied t parallel reddatl actated rbts. Idividal algrithms i cde were implemeted i -Simli evirmet see Figre 8 ad cmpiled t digital sigal prcessr rm dspe cmpa. Figre 9: Simli scheme r predictive ctrl PSD ctrller serves l r cmparis

15 he llwig igre illstrates the reslts eperimets. ime histries ctrl errrs ad ctrl actis represeted b measred crrets drives are shw. errr Figre : ime histries predictive algrithm geeratig icremets ctrl actis he time histries respective the real eperimets prve t sccessl prcess ctrl ad pssibilit t se predictive ctrl mdel based ctrl strateg r imprvemet crret lcal appraches.

16 6 clsi he paper presets brie tlie ther bacgrd predictive ctrl ad als its tilizati i damicall ptimal trajectr plaig. he ther is prved b real eperimets labratr mdel parallel rbt Slidig Star. s a cclsi r tre research it ctrl parallel rbts is a tilizati mdel-based appraches which prvide ptimal distribti ipt eerg spplemeted with ast lcal idepedet ctrllers bild i idividal drives. he trajectr plaig is ather imprtat tas. he damicall ptimal desig trajectries described i the paper establishes basic idea hw t avid bstacles i rbt wrspace. Especiall rage-space predictive ctrl pssibl achieved -lie is iterestig r eedbac-camera ctrl sstems. cwledgemets he athrs tha t pr.. Valáše rm Dept. echaics dies FE eablig them t test algrithms rbt mdel Slidig Star ad t eg. P. Píša rm Dept. trl Egieerig FEE r spprt at implemetati. he are rm zech echical Uiversit i Prage. rever the athrs appreciate id spprt Grat gec the zech Repblic b grats /6/P75 6/8 del-based trl echatric Sstems r Rbtics ad /5/7 5/7 ethds Predictive trl lgrithms ad Implemetati. iteratre [] Ords.; lare D.: State - Space Descripti r GP trllers. I: It. Jral Sstems SI. Vl pp [] elda K.; öhm J.; Valáše.: State-Space Geeralized Predictive trl r Reddat Parallel Rbts. I: echaics ased Desig Strctres ad achies arcel Deer Vl pp [3] Stejsal V.; Valáše. I: Kiematics ad Damics achier arcel Deer ew Yr 996. [4] elda K.; öhm J. et al: I: Kweb 5. [5] Peař J. et al: trl SR sig P based itre Distribti. I: 6 th IF Smpsim liear trl Sstems Stttgart 4.