Model-based control of a one-dimensional pendulum actuated with Pleated Pneumatic Artificial Muscles with adaptable stiffness

Transcription

1 Facultet Ingeneurswetenschaen Vakgroe Werktugkunde Model-based control of a one-dmensonal endulum actuated wth Pleated Pneumatc Artfcal Muscles wth adatable stffness Bruno Mera y Duran Proefschrft ngedend tot het behalen van de academsche graad van Burgerljk Werktugkundg-Elektrotechnsch Ingeneur Academejaar 4-5 Promoter: Prof. dr. r. Drk Lefeber Co-Promotor: Prof. dr. r. Patrck Kool

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3 Abstract Model-based control of a one-dmensonal endulum actuated wth Pleated Pneumatc Artfcal Muscles wth adatable stffness Due to the comressblty of ar and the drong force to contracton characterstc of the muscle, a jont actuated by two of these muscles set n antagonstcally oston, shows a comlant behavour and can be adated whle controllng oston. Ths comlance adataton enhances the ossbltes of exlotaton of natural dynamcs so that energy consumton and control actvty s decreased. The man urose of the roject s to nvestgate the exlotaton of natural dynamcs n combnaton wth the jont trajectory control. The jont trackng controller tracks the desred trajectory whle adatng the jont comlance, as such that the natural regme corresonds as much as ossble to the reference trajectory. Currently the endulum s assembled and ts hardware comonents have been tested. Ths thess reorts on the desgn and constructon of the endulum. The effect on valve control actvty and energy consumton by changng the comlance characterstcs are shown by means of a smulaton and exerments on the hyscal endulum. A technque to change the stffness onlne has been evaluated. The next stes n the study are a correct estmaton of the system s arameters and the extenson of the system to a three dmensonal endulum, n that way that t can be used for mlementaton on the walkng bed Lucy, from the Vrje Unverstet Brussel.

4 amenvattng tude van een modelgebaseerde controle van een slnger aangedreven door balgactuatoren met regelbare stjfhed. Een scharnergewrcht, aangedreven door een stel antagonstsch ogestelde seren, vertoont een soeel en elastsch gedrag te wjten aan de samendrukbaarhed van lucht en de dalende kracht - contracte karakterstek van de ser. Bovenden kan deze soeelhed ngesteld worden, onafhankeljk van de oste van het gewrcht. De varabele stjfhed, het omgekeerde van de soeelhed, beïnvloedt de egenfrequente van het systeem en bedt dus de mogeljkhed om de natuurljke dynamca van de slnger ut te buten om zo het energeverbruk en de controleactvtet te vermnderen. Het hoofddoel van dt roject s te komen tot een controle van de slnger waarbj de utbutng van de natuurljke dynamca van het systeem gecombneerd wordt met een trajectsturng. De trajectsturng volgt een ogelegd traject, terwjl de stjfhed van de structuur zodang wordt ngesteld dat de slngerbewegng volledg bnnen de natuurljke dynamca van het systeem valt. De slnger s gebouwd en de bjhorende hardware comonenten zjn getest. Deze thess geeft een beschrjvng van het ontwer en de bouw van de slnger. De nvloed van de varabele stjfhed o het energeverbruk en de kleenactes wordt geïllustreerd aan de hand van smulates en exermenten o de werkeljke slnger. Een technek om de otmale stjfhed langsheen een ogelegd traject te bealen en te varëren werd onderzocht. De volgende staen bnnen dt onderzoeksgebed zjn de correcte bealng van de arameters van het systeem en een utbredng naar een dredmensonale slnger, om de ogedane kenns te kunnen gebruken voor verdere ontwkkelng van de tweeotge staende robot Lucy.

5 Résumé Contrôle basé sur un modèle, d une endule, actonnée ar des muscles artfcels neumatques lés avec une radeur adatable. Grâce à la comressblté de l ar et de la caractérstque force-contracton décrossante du muscle, une jonture actonnée ar deux muscles antagonstes, résente un comortement comlant, dont la soulesse eut être controlée ndéendamment de la oston. Cette soulesse adatve ermet de bénéfcer du régme naturel du système our ans dmnuer la consommaton d énerge et l effort de contrôle. L objectf de cette thèse est de combner l exlotaton du mouvement naturel de la endule avec un contrôle en oston de tye suv de trajectore. Une trajectore sécfée est suve ar un contrôle en oston endant que la soulesse de la jonture est adatée de telle façon que le régme naturel corresond le lus ossble à la trajectore de référence. La endule est actuellement assemblée et les comosant testés. Cette thèse donne une déscrton de la constructon de la endule. L nfluence de la soulesse de la jonture sur l effort de contrôle et la consommaton d énerge est llustrée ar des moyens de smulatons et d exérmentatons sur la endule réelle. Une technque qu ermet de détermner la soulesse otmale tout au long d une trajectore secfée est étudée. La rochane étae dans le domane est d estmer correctement les aramètres du système et d étendre ce sytème vers une endule trdmensonelle. De cette façon, l ncororaton de la technque ourra se fare sur le robot bède Lucy.

6 Acknowledgements Na een jaar van ure fun n Barcelona, heeft het mj veel moete gekost om me weer n het leven van een ngeneursstudent te storten. Tjdens de erode waar k vorg jaar o één van de vele terrasjes aan het geneten was van het mooe weer, heb k nu de meeste tjd doorgebracht n het labo, vloekend o de zoveelste kle de maar net wlde werken. Na hard werk te hebben geleverd, ben k dan toch gekomen tot een (zelf)bevredgend resultaat.dt afstudeerwerk s uteraard net zonder de hul en steun van anderen tot stand gekomen en daarom dt dankwoord. Ten eerste wl k mjn romotor Prof. Lefeber en co-romotor Prof. Kool bedanken om mj de mogeljkhed te geven tot deze endverhandelng. Bram Vanderborght en Björn Verrelst zou k wllen bedanken voor de begeledng, de goede raad, de hul bj raktsche roblemen en vooral voor het nalezen van de endverhandelng. Een groot deel van deze thess werd verwezenljkt n samenwerkng met mjn klasgenoot Peter Beyl. We hebben dan ook vele uren samen doorgebracht n het labo en zou hem wllen bedanken voor de vlotte samenwerkng. Voor de technsche hul kon k steeds terecht bj Ronald Van Ham, Jean-Paul cheens, André Plasschaert, Therry Lenor en Gabrël Van den Nest en dank hen hervoor. Jors Naudet en Mchael Vandamme wl k bedanken voor hun hul bj het rogrammeren. Verder zou k nog mjn vrenden wllen bedanken voor hun aanwezghed en het lezer dat we doorheen de jaren aan de VUB hebben beleefd. Thomas, k ben je enorm dankbaar voor je hul bj het maken van de User Interface. Nathale, jou wl k secaal bedanken, net alleen voor het nalezen van mjn roefschrft, maar ook voor je steun en de vele koffeauzes de mj er steeds weer boveno helen.

7 Pa y Mam, os quero agradecer or vuestro nterés, aoyo y amor durante toda m carrera unverstara. n vosotros, no hubera sdo lo msmo. Os quero mucho! Tenslotte zou k mjn ouders wllen bedanken, voor het vertrouwen, de steun en de vrjhed de ze me door alle jaren heen gegeven hebben. Aan hen heb k dan ook alles te danken. Brussel Jun 5 Bruno

8 Contents Introducton.... Motvaton and goal.... Background nformaton Aroach Outlne...7 Desgn...8. Introducton...8. Mechancal Desgn Pleated Pneumatc Artfcal Muscle The one-dmensonal jont set-u Introducton Knematcs of the one-dmensonal endulum Adatable assve behavour of a revolute jont Desgn of the one-dmensonal endulum Constructng the exermental set-u The frame The valve system The neumatc crcut The exhaust slencer The suort Electronc desgn Valve system seed-u crcutry Pressure ensor Incremental encoder Data acquston card...33

9 .3.5. afety, suly and transformaton board oftware Concluson...39 Control Introducton Jont trackng controller Mathematcal formulaton for comlance adataton The smulaton model of the endulum Descrton of the endulum set-u The mechancal equatons of the model The thermodynamcal equatons of the model Comlete smulaton model Energy consderatons Concluson...55 Results and dscusson Introducton The smulaton model Evaluaton of the jont trajectory trackng controller General consderatons Results and dscusson Exlotaton of the natural dynamcs Exerments Results and dscussons...63 mulaton : the mortance of settng a sutable stffness...63 mulaton : evaluaton of the mathematcal formulaton...67 mulaton 3: changng the stffness onlne The hyscal endulum Evaluaton of the jont trajectory trackng controller Exlotaton of the natural dynamcs General consderatons...8

10 Results and dscusson Concluson...86 General conclusons and future ersectves...88 Aendces A Dynamc model of the endulum...9 B afety Board...94 Bblograhy Robotcs Programmng Data acquston card...97

11 Lst of Fgures Fgure.: Industral robot wth heavy electrcal drves and the soft actuator AMAC... Fgure.: The actve walker Asmo and the assve walker Dense... Fgure.3: The bedal walkng robot Lucy...3 Fgure.4: CAD drawng and hotograh of the hyscal endulum...4 Fgure.: The Pleated Pneumatc Artfcal Muscle...9 Fgure.: Theoretcal forces at ressure levels, and 3 bar as a functon of the contracton... Fgure.3: Theoretcal maxmum muscle dameter as functon of contracton Fgure.4: Forces at ressure levels, and 3 bar and volume resultng from the olynomal fttng as a functon of the contracton... Fgure.5: The antagonstc muscle setu and the ossblty to adat both oston and comlance ndeendently...3 Fgure.6: Basc confguraton of the ull rod and leverage system...4 Fgure.7: Influence on muscle contracton by changng d and α... Fgure.8: tatc and closed muscle torque characterstcs... Fgure.9: tatc and closed muscle torque characterstcs at dfferent ressure levels, wth the desgn arameters set to the one used n the exermental set-u... Fgure.: Exloded and assembled vew of the basc frame...3 Fgure.: Vew of the connecton late and the oston lmter...4 Fgure.: Vew of the valve sland...5 Fgure.3: chematc overvew of the neumatc crcut...6 Fgure.4: Vew of the exhaust slencers...7 Fgure.5: Vew of the suortng structure of the endulum...8 Fgure.6: chematc overvew of the control hardware...9 Fgure.7: Vew of the seed-u crcutry...3

12 Fgure.8: Vew of the ressure sensor...3 Fgure.: Vew of the safety, suly and transformaton board...35 Fgure.: The Control tab of the GUI...37 Fgure.: The Model tab of the GUI...37 Fgure.3: The Valves Test tab of the GUI...38 Fgure 3.: chematc overvew of the control structure...4 Fgure 3.: Bang-bang ressure control scheme...44 Fgure 3.3: The endulum model...48 Fgure 3.4: tructure of the comlete smulaton model...5 Fgure 4.: The jont angle θ for erfect and erroneous arameter estmatons of the model...59 Fgure 4.: Torques of the jont for erfect and erroneous arameter estmatons of the model...6 Fgure 4.3: Muscle ressure wth valve acton for erfect model....6 Fgure 4.4: Actual and desred jont angle wth closed muscles for = 6 Nm and = 3 Nm...63 Fgure 4.5: Actual and desred jont angle and angular velocty for = 6 Nm and = 3 Nm...64 Fgure 4.6: Actual and requred ressure n muscle and for Fgure 4.7: Actual and requred ressure n muscle and for = 6 Nm..65 = 3 Nm..65 Fgure 4.8: Energy consumton and valve acton n functon of, for dfferent trajectores...68 Fgure 4.9: Jont angle, ressure and frequency for a trajectory wth lnear varyng frequency...69 Fgure 4.: tffness K ~ and nose due to cotg( ) term for the secfc trajectory...7 Fgure 4.: Actual and requred ressure and valve acton for a trajectory wth lnear varyng frequency and onlne changed...7 Fgure 4.: Actual and requred ressure and valve acton for a trajectory wth lnear varyng frequency and a constant...7 Fgure 4.3: n functon of the frequency of the mosed trajectory....73

13 Fgure 4.4: Effect on the valve actons due to devaton on the onlne changng arameter...74 Fgure 4.5: Angular velocty, comuted torque and absolute ressure for frst trackng exerment...75 Fgure 4.6: Angular velocty, comuted torque and absolute ressure for a trackng exerment by usng a flter on the angular velocty Fgure 4.7: Measured and desred jont angle θ and ressure course for muscle...79 Fgure 4.8: Intal trajectory and mosed trajectory durng exerments...8 Fgure 4.9: Energy consumton and valve acton for one erod of dfferent trajectores n functon of the stffness arameter Fgure 4.: Influence of the stffness arameter...8 on the valve acton and the absolute ressure n the muscle...83 Fgure 4.: The uncontrolled oscllaton of the endulum for three dfferent stffness arameters after a controlled erod of seconds...84 Fgure A.: chematc overvew of the studed model...9 Fgure B.: Electronc scheme of the safety, suly and transformaton board 94

14 Lst of Tables Table.: Coeffcents of the olynomal force functon aroxmaton... Table.: Coeffcents of the olynomal volume fttng aroxmaton... Table 3.: Inertal arameters of the endulum model...49 Table 4.: Pressure reacton levels of the bang-bang ressure controller, used n smulaton...58 Table 4.: Ar mass flow, exergy and valve acton durng one erod of a sne wave trajectory wth frequency of Hz for = 6 Nm and = 3 Nm...66 Table 4.3: Calculated and smulated otmal values of for dfferent sne waves trajectores...67 Table 4.4: Pressure levels of the bang-bang controller for the hyscal endulum...8 Table 4.5: Exermental, calculated and smulated otmal values of for dfferent sne-wave trajectores...85

15 Nomenclature Acronyms AICH DACOUT DIO PPAM UB Analog Inut Channel Analog Channel Outut Dgtal Inut/Outut Pleated Pneumatc Artfcal Muscle Unversal eral Bus controller ymbols α coeffcent denotng centre of gravty of the endulum lnk α angular desgn arameters of the jont muscle setu ε muscle contracton c ε c contracton of muscle at chosen central oston θ ρ ar densty at standard condtons kg/m³ ε contracton of muscle n a jont setu θ angular jont oston rad( ) c θ chosen central oston n the jont setu rad( ) Δ ~ control varable of delta- control unt bar τ actuator torque Nm a reacton level of bang-bang ressure controller bar b reacton level of bang-bang ressure controller bar b dstance between orgn O and onts B m B fxed base onts of the jont muscle setu c reacton level of bang-bang ressure controller bar C neumatc valve flow constant td.l/mn/bar C ( q, q& ) centrfugal/corols matrx d dstance between R and onts D m D movng onts of a jont muscle setu D ( q) nerta matrx e reacton level of bang-bang ressure controller bar f reacton level of bang-bang ressure controller bar f ( θ ) dmensonless force functon of muscle n the jont setu l f ε, R dmensonless muscle force functon f coeffcents of a olynomal force fttng F t muscle tracton force N g acceleraton of gravty m/s²

16 G centre of gravty of art of the endulum G ( q) gravtatonal torque/force vector I moment of nerta of art kgm² ( θ ) k jont stffness functon for a jont setu Nm/bar K jont stffness Nm knetc energy J l ntal muscle length m l base susenson bar length of the muscle jont setu m b l length from vot to art m l actual length of muscle n the jont setu m m m& ar ar mass flow through valves of muscle kg/s m mass of art kg n number of fbres olytroc exonent gauge ressure bar stffness arameter of delta- control unt Nm ntal gauge ressure when muscle was closed bar P ntal absolute ressure when muscle was closed bar generalzed coordnate q Q θ generalzed torque assocated to q r ( θ ) leverage arm of muscle n the jont setu m R rotaton ont of the jont muscle setu t tme s t ( θ ) torque functon of muscle n a jont setu Nm/bar T jont torque Nm temerature K T torque generated by muscle n the jont setu Nm U otental energy J l v ε, R dmensonless volume functon v coeffcents of olynomal muscle volume fttng V enclosed volume of muscle m³ V ntal volume when muscle was closed m³ W work J ubscrt ar atm d of comressed ar atmosherc dervatve muscle number

17 sotherm s art number at sothermal condtons samle uerscrt a analog d dgtal ex of exhaust valve n of nlet valve suly of comressed ar suly ~ requred value ^ calculated wth estmated arameter values dervatve wth resect to tme

18 Chater Introducton. Motvaton and goal Nowadays, robots are wdely used n all knds of roducton and assembly chans to assst n very hard, reettve and labour ntensve tasks wth otmum erformance. Most of the robots bult, use heavy electrcal drves, makng these machnes heavy and ower demandng and not safe for humans. Robots n assembly halls are comletely solated and can not be accessed when workng. As the feld of robotc solutons s stll exandng and more robots wll be used n the mmedate surroundngs of eole, there s a need for safety aganst all ossble accdents. Therefore, soft actuators are beng develoed. uch soft actuators offer a natural comlance, makng them human frendly. At the Unversty of Psa, an electromechancal Varable tffness Actuaton [] motor has been develoed, and secfcally desgned for machnes and robots hyscally nteractng wth humans. Another examle of such an actuator s the AMAC (Actuator wth Mechancally Adjustable eres Comlance)[4], whch uses two motors, one for controllng the oston and the other for controllng the stffness, the nverse of comlance. Fgure.: Industral robot wth heavy electrcal drves and the soft actuator AMAC

19 For the last ten years the Robotcs and Multbody Mechancs Research Grou of the Vrje Unverstet Brussel s conductng research n the doman of legged bedal robots. Beds can be dvded nto man categores: on the one hand the full actuated robots, whch don't use natural or assve dynamcs at all, lke Asmo [7] and QRIO [8], on the other hand the "Passve walkers" who don't requre actuaton at all to walk down a sloed surface or only use a lttle actuaton just enough to overcome energy losses, due to frcton and mact effects, when walkng over level ground. An examle of such a robot s the Dutch robot Dense. Fgure.: The actve walker Asmo and the assve walker Dense The man advantage of "assve walkers" s that they are hghly energy effcent but unfortunately they are of lttle ractcal use. They have dffcultes to start, can't change ther seed and cannot sto n the same way a comletely actuated robot can. But on the other hand, a comletely actuated robot consumes a lot of energy. As energy consumton s an mortant ssue for beds, t remans mortant to exlot the natural dynamcs by tryng to ncororate the unforced moton of a system nstead of gnorng or avodng t. The goal of the Robotcs and Multbody Mechancs Research Grou s to develo a robot whch combnes these two categores. Ths robot should be able to adat the natural dynamcs as a functon of the mosed walkng moton. For Chater : Introducton

20 ths research a jont wth controllable comlance s needed n order to nfluence the natural dynamcs. But the varable stffness actuators mentoned revously are too comlex and too heavy for an alcaton such as an autonomous bedal robot. Therefore, an nterestng alternatve s the leated neumatc artfcal muscle (PPAM), develoed at the Robotcs and Multbody Mechancs Research Grou. It s beleved that these neumatc actuators have nterestng characterstcs, whch can be used n the feld of legged robotcs. In ths context, the develoment of a lanar walkng robot actuated wth leated neumatc artfcal muscles (PPAM) was started. The robot has been gven the name Lucy (fgure.3). It weghs 3 kg and s 5 cm tall. It uses muscles to actuate 6 n jonts, as such Lucy s only able to walk n the sagttal lane. A sldng mechansm revents the robot from fallng sde-wards. The goal of ths roject s to create a lghtweght bed, whch s able to walk n a dynamcal stable way whle exlotng the adatable assve behavour of the leated neumatc artfcal muscles n order to reduce energy consumton and control efforts. The robot s currently able to move n a quas-statc way, but stffness n the jonts s stll set constant [],[4]. Fgure.3: The bedal walkng robot Lucy The man urose of ths master thess s to nvestgate the adatable assve behavor of the Pleated Pneumatc Artfcal Muscle, whch allows the stffness Chater : Introducton 3

21 of a jont, actuated by two antagonstcally couled muscles, to be vared onlne. Because the research on ths toc wll be used for further mlementaton on Lucy, the nvestgaton s done on one modular art of Lucy. Wth such a modular art a sngle endulum set-u, shown n fgure.4, was created n order to nvestgate energy consumton reducton wth mosed swngng moton of the endulum whle the stffness of the jont s changed. The dea n ths work s to change the natural frequency n combnaton wth ure trajectory control n such a way that control efforts are mnmzed. Ths strategy s tested on both a hyscal and mathematcal comuter smulaton model. Fgure.4: CAD drawng and hotograh of the hyscal endulum. Background nformaton The frst offcal recordngs of assvely walkng toys goes back tll 888 (e.g. Falls atent). Insred by the observaton of human data, n whch the muscles of the swng leg are actvated only at the begnnng and the end of the swng hase, Morawsk & Wojceszak were the frst to gve a mathematcal formulaton Chater : Introducton 4

22 for these toys. They concluded that n human locomoton the moton of the swng leg s merely a result of gravty actng on an unactuated double endulum. Insred by ths calculatons, t was McGeer [5] who ntroduced the concet of Passve Dynamc Walkng. McGeer showed that a smle lanar mechansm wth two legs could be made to walk n a stable way down a slght sloe wth no other energy nut or control. Ths system acts lke two couled endula. The stance leg acts lke an nverted endulum, and the swng leg acts lke a free endulum attached to the stance leg at the h. Gven suffcent mass at the h, the system wll have a stable trajectory that reeats tself and wll return to ths trajectory even f erturbed slghtly. McGeer also constructed a hyscal examle. Hs research was contnued by the grou of Andy Runa [3]. They studed the models of McGeer n more detal and extended the two dmensonal model to three dmensons whle buldng several Passve Walkers. Usng natural frequences of the robot s a useful concet for energy effcent walkng. However, the natural frequency of a system s defned by nherent system roertes lke: dmensons, mass dstrbuton, collson behavor and jont comlance. As a result, the egenfrequency of these models s set durng constructon, whch lmts the dfferent assve walkng atterns and walkng seeds. In order to change the natural frequency of the system, assve elements wth varable comlance were mlemented. In ths context the grou of Takansh develoed the two-legged walker WL-4, where a comlex non-lnear srng mechansm makes changes n stffness ossble. A more elegant way to mlement varable comlance s to use neumatc artfcal muscles. tffness can easly be changed by changng the aled ressures. Research on ths toc was done by van der Lnde [] and Wsse [5] through mlementaton of McKbben s tye of neumatc artfcal muscles..3 Aroach Pneumatc artfcal muscles have the ossblty to adat the stffness whle controllng oston. Exlotaton of the natural dynamcs of a walkng robot by varyng the comlance characterstcs can be aroached n two ways: a more bologcal nsred aroach and an analytcal aroach. The frst oton s to Chater : Introducton 5

23 desgn the robot, takng nto account all the nertal and comlance arameters, n such a way that the system erforms a moton close to walkng wthout actuaton, by makng the system oscllate at ts natural frequency. By controllng the jont comlance, the moton characterstcs are then adated and as such adat the walkng attern. Concernng dynamc robot stablty, ths s however a comlex task and wll robably result n a small range of feasble moton atterns. A second aroach s to desgn jont trajectores for a secfc robot confguraton, so that dynamc stablty s ensured f these trajectores are tracked by a trajectory trackng controller. In order to reduce valve control actvty, the jont trajectory trackng unt then adats the comlance of the dfferent jonts, so that the natural moton best fts the gven trajectores. As a result the global stablty s ensured due to the calculated trajectores, whle energy consumton s lowered by adatng the jont comlances. But of course, settng the ranges n whch the natural moton corresonds to the calculated trajectores requred for dynamc stablty, asks for a roer desgn of all nertal and jont desgn arameters. o that n the end, a combnaton of these two dfferent aroaches wll gve nterestng results. In ths context, two master theses were roosed by the Robotcs and Multbody Mechancs Research Grou of the Vrje Unverstet Brussel. In ths reort the model based aroach wll be dscussed. My colleague, Peter Beyl, wll carry out a more bologcal aroach, by usng Fuzzy Logc controllers, artfcal neural networks (ANN), central attern generators (CPG), Look U Tables or combnatons of all strateges mentoned above. As we both needed an exermental set-u to check our smulaton results, the frst art of our thess conssted of desgnng and constructng the endulum. Therefore, the followng stes were executed: Determnaton of the dmensons of the leverage mechansm and consequently the jont torque characterstcs. Buldng, testng and debuggng the electronc comonents. Buldng a rogram lbrary, used to send and receve data by use of a data acquston board. Chater : Introducton 6

24 Assemblng the endulum Imlementaton of the jont trajectory controller Wrtng a user nterface to communcate wth the endulum and extract the data nformaton. Imlementaton of a controller for settng the jont stffness n order to ncororate natural dynamcs. Measurng and analyzng the results. The frst fve tasks were done together wth my colleague Peter Beyl. The remanng tasks were carred out by myself, snce a comletely dfferent aroach was chosen..4 Outlne Ths thess roceeds as follow: Chater (Desgn) gves an extensve descrton of the mechancal desgn, wth the jont torque characterstcs as an mortant dscusson. The electronc desgn, needed to control the valves whch set the ressures n the muscles s also gven. Those ressures defne the jont stffness and oston. A manual gude of the user nterface s gven, n order that further research on ths toc can be done by other eole. Chater 3 (Control) reorts on the controller used for trackng a desred trajectory n combnaton wth exlotaton of the natural dynamcs. The trackng control strategy conssts of a multlevel constructon of several essental blocks, tryng to coe wth the nonlnear structure by usng model-based feedforward technques. The method used to take nto account the natural dynamcs wll be formulated. In order to evaluate ths control archtecture a comuter smulaton model was bult. A descrton of the modellng of the robot dynamcs and the thermodynamcs of the muscle/valve system s gven, followed by an overvew of the comlete smulator. ome energy consderaton wll be gven, so that the method roosed for ncororatng the natural dynamcs can be evaluated. Chater : Introducton 7

25 Results and dscussons on both the smulaton model and the hyscal endulum are gven n chater 4. For both cases, the erformances of the jont trajectory trackng controller wll be dscussed. In the context of the exlotaton of the natural dynamcs, smulaton and exermental results wll be gven. Fnally, a comarson between smulaton and exerments on the real endulum wll be dscussed, n order to conclude whether or not an onlne changng stffness can be used for further mlementaton on Lucy. General conclusons are drawn n chater 7, followed by the lannng of future research n the doman of exlotaton of natural dynamcs. Chater : Introducton 8

26 Chater Desgn. Introducton The man goal of the endulum s to nvestgate whether the PPAM can be an nterestng actuator for exlotaton of natural dynamcs n combnaton wth trajectory trackng. The endulum set-u has been desgned for that urose and should be sutable for exerment on the followng man tems: Evaluaton of a trajectory trackng control strategy wth the secfc neumatc system. Evaluaton of the adatable assve behavour of the PPAM and the nfluence on the natural dynamcs of the system. Ths chater starts wth a descrton of the mechancal desgn of the endulum. The PPAM and ts characterstcs wll be dscussed. The knematcs of a onedmensonal jont setu controlled by two antagonstcally ostoned muscles, are reorted. ecal attenton s devoted to the desgn of the actuator connecton. The electroncs needed to control the ressures n the muscles wll be exlaned n the second aragrah, electronc desgn. In the thrd aragrah, software, a user manual of the guded user nterface wll be gven, so that the exermental set-u can be easly used by other ersons. Constructng the endulum by just one erson would have taken too much tme n the context of a master thess. Therefore, my colleague Peter Beyl and I worked as a team. As a consequence, all the work, excet from the GUI, referred to n ths chater has been carred out together.

27 . Mechancal Desgn... Pleated Pneumatc Artfcal Muscle The PPAM consst of a membrane made of an aromatc olyamde such as Kevlar to whch a thn lner of olyroylene s attached n order to make the membrane artght. The membrane of ths muscle s arranged nto radally lad out folds that can unfurl free of radal stress when nflated. Renforcng hgh tensle Kevlar fbers are ostoned n each crease. The hgh tensle longtudnal fbres of the membrane transfer tenson, whle the folded structure allows the muscle to exand radally, whch avod energy losses and hysteress. The folded membrane s ostoned nto two end fttngs, whch close the muscle and rovde tubng to nflate and deflate the enclosed volume. An eoxy resn fxes the membrane and the Kevlar fbres to the end fttngs. Fgure.: The Pleated Pneumatc Artfcal Muscle Daerden [] extensvely dscusses several characterstcs concernng the PPAM, for nelastc as well as elastc membranes. In ths work manly two characterstcs are mortant: generated tracton and enclosed volume for each contracton. The frst s used for jont torque dmensonng and control uroses, whle the latter s used to redct jont comlance wth closed muscles. Furthermore, the maxmum Chater : Desgn 9

28 dameter when the muscle s fully bulged should be taken nto account when desgnng the dfferent jonts of the robot n order to rovde enough sace for the muscle to bulge. When nflated wth ressursed ar, the muscle generates a undrectonal ullng force along the longtudnal axs. Generally, f the number of fbres s large enough, the generated muscle force deends on the aled gauge ressure ( ), the contracton (ε ) and the two arameters, ntal muscle length ( l ) and l slenderness ( ) [4] R The tracton characterstc s gven by : F t l = l f ( ε, ) (.) R l wth f ( ε, ) the dmensonless force functon as defned by Daerden []. R Fgure.: Theoretcal forces at ressure levels, and 3 bar as a functon of the contracton Usng l = mm, R =.5 mm and n = 4, results n the theoretcal force characterstcs dected n fgure., [4]. The tracton as a functon of contracton s drawn for dfferent aled gauge ressures:, and 3 bar. The grah shows the non-lnear character of the generated muscle force. For small Chater : Desgn

29 contractons, the forces are extremely hgh, whle for large contractons, the forces dro to zero. For the endulum, contractons wll be bounded somewhere between 5 and 35 %. The frst lmt s set to bound the stresses on the fbres and consequently extends the lfetme of the muscle. And beyond 35 % contracton, forces dro too low to be of ractcal use. Fgure.3 dects the theoretcal maxmum dameter to contracton for the consdered muscle []. Ths aroxmated maxmum dameter should be taken nto account durng the desgn of the jonts of the endulum n order to rovde enough sace for the muscle to bulge. Fgure.3: Theoretcal maxmum muscle dameter as functon of contracton To comare the theoretcal characterstcs wth those of the hyscal PPAM, statc load tests were carred out by Verrelst. A bref dscusson on the results wll be gven. For more nformaton we refer to [4]. A devaton between measured data and theoretcal model of the force was observed. For dmensonng urose, the theoretcal model can be used. However, for the trajectory trackng control, an accurate estmaton of the real force functon s requred. nce the force functons of dfferent muscles are very smlar, a 4 th order olynomal functon ft on the ressure scaled measured data was erformed n order to acheve a better force estmaton. The force functon can be exressed as: 3 F t = l f ( ε ) = l ( f ε + f ε + f ε + f + f ε ) (.) 4 3 Chater : Desgn

30 Fnally, for reason of rogrammng convenence, a olynomal fttng was erformed on the theoretcal data for the enclosed muscle volume: V ε ) = l v( ε ) = l ( v ε + v ε + v ε + v ε + v ε + ) (.3) ( v Those equatons are much easer to handle than the numercal soluton derved from the mathematcal model as derved by Daerden and Verrelst. In table. and table., the coeffcents of resectvely the force and volume fttng are gven. The values are vald when the generated force F t s exressed n N, the ntal muscle length l n m, the ressure exressed n bar, the volume gven n ml and the contracton ε exressed n %. Both olynomal fttngs are shown n Fgure.4 f f f f 3 f Table.: Coeffcents of the olynomal force functon aroxmaton v v v v 3 v 4 v Table.: Coeffcents of the olynomal volume fttng aroxmaton Force (N) 5 bar bar 3 bar Volume (ml) Contracton (%) Contracton (%) Fgure.4: Forces at ressure levels, and 3 bar and volume resultng from the olynomal fttng as a functon of the contracton Chater : Desgn

31 ... The one-dmensonal jont set-u... Introducton Pneumatc artfcal muscles can only exert a ullng force. In order to have a bdrectonal workng revolute jont, two muscles are couled antagonstcally. The antagonstc coulng of the two muscles s acheved wth a ull rod and leverage mechansm, as s dected n fgure.5. The lever arm can be vared n such a way that the hghly non-lnear force-length characterstc of the PPAM s transformed to a more flattened torque-angle relaton. Whle one muscle contracts and rotates the jont n ts drecton, the other muscle wll elongate. Jont oston Δ τ θ Jont tffnes Σ K Fgure.5: The antagonstc muscle setu and the ossblty to adat both oston and comlance ndeendently. The gauge ressure dfference between the two muscles determnes the generated torque (T ), and consequently also angular oston (θ ). Furthermore, both ressures can be ncreased (decreased) n such a way that jont stffness ( K ) rases (lowers) wthout affectng angular oston. Thus both oston and comlance can be controlled ndeendently.... Knematcs of the one-dmensonal endulum The knematcs of the endulum have the same structure as one modular art of Lucy. For the sake of convenence, the same dscusson wll be gven as n [4] Chater : Desgn 3

32 The basc confguraton of the ull rod and leverage mechansm s dected n fgure.6 Y θ C α α D d D d l m r r R l b l m B B O X b b Fgure.6: Basc confguraton of the ull rod and leverage system The two muscles are connected at one sde of the system to a fxed base n the onts B and B resectvely. The other ends of the muscles are attached to a Chater : Desgn 4

33 votng art at the onts D and D, of whch the rotaton axs asses through a ont R. The rods are assumed to be rgd. An orthogonal X, Y - coordnate system s defned. The X -axs s algned wth the base onts B and B, whle the vertcal Y -axs ntersects the hyscal votng ont R and les along the base susenson bar of the ull rod mechansm. The essental arameters to be determned durng the desgn rocess of the jont are the followng: b s the dstance between the orgn O and the ont B. d s the dstance between the votng ont R and the ont α s the angle between the vector RD and art. ( α s not orented and always ostve) l m s the actual length of muscle D. RC ont on the rotatng l b s the length of the base susenson bar, measured between the orgn O and the vot ont R. θ reresents the rotaton angle, measured between RC and the Y -axs. (θ s orented, counter-clockwse s ostve) The generated torque can be found by combnng the tracton force generated by the muscles and the orthogonal leverage arms r and r : T ( θ ) = T ( θ ) T ( θ ) = = = F ( θ ) r ( θ ) F t l t ( θ ) t t ( θ ) ( θ ) r f ( θ ) r ( θ ) l ( θ ) f ( θ ) r ( θ ) (.4) (.5) wth t ( ) and t ( ) the torque functons deendng on the angular oston of θ θ the jont. To calculate the force functons, we wll use the olynomal functon ftted on the measured force data as descrbed n the frst aragrah. Chater : Desgn 5

34 The vectors B D and RD are beng calculated wth followng equatons: B D B RD RD D = = = [ b d sn( α θ ), lb + d cos( α θ )] [ d sn( α + θ ) b, lb + d cos( α + θ )] [ d sn( α θ ), d cos( α θ )] [ d sn( α + θ ), d cos( α + θ )] = (.6) (.7) (.8) (.9) The exresson for r (θ ) can then be found as: B D RD r (θ ) = (.) B D The muscle contracton ε as functon of the rotaton angle θ s gven by: l l l B D + B D ε ( θ ) + (.) l c m c m m c = = ε + = ε l l c The contracton ε (θ ) s defned wth resect to c ε, whch s the contracton of muscle at a chosen central oston θ c. In our case θ c =. The arameters are fxed durng the jont desgn rocess Adatable assve behavour of a revolute jont A PPAM has two sources of comlance, beng gas comressblty, and the drong force to contracton characterstc []. The latter effect s tycal for neumatc artfcal muscles whle the frst s smlar to standard neumatc cylnders. Jont stffness, the nverse of comlance, for the consdered revolute jont, can be obtaned by the angular dervatve of the torque characterstc: Chater : Desgn 6

35 K dt dt dt = = dθ dθ d dt d = t + t dθ dθ dθ dt dθ (.) d The terms reresent the share n stffness of changng ressure wth dθ contracton, whch s determned by the acton of the valves controllng the jont and by the thermodynamc rocesses takng lace. If the valves are closed and f we assume olytroc comresson/exanson, the ressure changes nsde a muscle are a functon of volume changes [9]: P V = P V (.3) n o n o wth: P = P + (.4) atm leadng to: n d Vo dv = n( Patm + o ) n+ (.5) dθ V dθ Wth P, V the absolute ressure and volume of muscle, P o the absolute ntal ressure, Vo the ntal volume when the valves of muscle were closed, and o the gauge ressure and ntal gauge ressure resectvely, P atm the atmosherc ressure and n a olytroc exonent. The olytroc exonent s ntroduced to descrbe devatons from the sentroc exanson/comresson. An sentroc rocess assumes reversble adabatc thermodynamc condtons and the exonent becomes n ths case n γ = c c =. 4 for dry ar [9]. = v Durng the jont desgn rocess, t s ensured that the torque to angle and the volume to angle characterstcs of a jont are monotonous functons. Meanng that the dervatves dt dθ and dv dθ kee the same sgn wthn the range of Chater : Desgn 7

36 moton for whch the jont was desgned. Thus, referrng to fgure.6, ncreasng the jont angle θ wll ncrease the torque functon t ( ), whle the volume of the resectve muscle wll decrease. Indeed the larger θ, the lesser muscle s contracted. Consequently, the generated force s bgger. On the contrary, less contracton means that the muscle gets thnner and that volume decreases. Thus dt dθ > and dv dθ < θ. For the other muscle n the antagonstc setu, the actons are ooste: dt dθ and dv dθ. Combnng (.), (.3) < and (.5) wth ths nformaton gves: > K = k θ + + ) P (.6) ( ) o k ( θ ) o katm ( θ atm wth k ( θ ) k ( θ ) k atm ( θ ) V = t( θ ) n n V n o + V = t ( θ ) n n V = k ( θ ) + k n o + dv V + dθ V dv V + dθ V ( θ ) n o n n o n dt dθ dt > dθ dt > dθ dt dθ The coeffcents k ( ), k ( ), k ) are determned by the geometry of the θ jont and ts muscles. θ atm (θ From equaton (.6), we can conclude that when muscles n an antagonstc jont setu are closed, a assve srng element wth an adatable stffness s created. The stffness s controlled by a weghted sum of both ntal gauge ressures. nce stffness deends on a sum of gauge ressures whle oston s determned by dfferences n gauge ressure, the angular oston can be controlled whle settng stffness. How to use ths nterestng roerty and how to ncororate t nto the control of the endulum n order to reduce energy consumton and valve control, wll be dscussed n detal n the next chater. Chater : Desgn 8

37 ...4. Desgn of the one-dmensonal endulum All the equatons descrbed above, n combnaton wth the muscle force functon, are used to desgn the characterstcs. A descrton on how the desgn arameters of the former aragrah are chosen s now gven. As mentoned before, the endulum conssts of one modular art of the bed Lucy. Therefore, some of the arameters are already fxed, namely: b l l m b = 4mm = mm = 35mm The central oston of the endulum was set to θ = For desgnng the remanng arameters, some requrements should be met: c We want an angle range between - and, or larger. Wthn the angle range, the contracton has to be lmted between 5% and 35%. A smaller contracton leads to stresses that are too hgh for the muscle fbres. At hgher contracton, the forces dro too low for ractcal use. Enough torque should be rovded by the antagonstc setu to hold a mass of kg at the maxmum of the angle range. Therefore statc torque characterstcs of the jont should be comared wth the torque needed to hold a mass of kg. tablty has to be nsured. The torque characterstc of the jont wth closed muscles s mortant here. As the stffness of the jont s gven by dt dt K =, ths characterstc has to be ncreasng ( K = > ). In that dθ dθ case, the jont stffness s ostve. If that s not the case, the swng moton wll be amlfed and nstablty wll occur. Besdes ths, wthn the angle range, we want ths torque characterstc to be as lnear as ossble. The jont stffness wll then be almost constant wthn our workng doman. Chater : Desgn 9

38 The dameter of the muscles should be taken nto account durng the desgn, n order to rovde enough sace for the muscle to bulge. The jont range s determned va the mnmum and maxmum angles at the mnmum and maxmum contracton of the muscles. Angle range and torque characterstcs are determned as a functon of the jont alcaton. The desgn of all these functonaltes s comlex and s lnked to the secfc moton of the endulum. A small Matlab rogram has been wrtten to nvestgate the nfluence on the jont characterstcs by changng the desgn arameters. The characterstcs are shown n Fgure.7 and.8. Fgure.7 shows the contracton of one muscle n functon of the jont angle. Maxmum and mnmum lmts of the contracton are also lotted. In the left lot d s changed. In the rght lot, the contractons for dfferent values of α are gven α =6 en ε =9% Contracton ε (%) Mnmal contracton (%) Maxmal contracton (%) 4 35 d =3 mm en ε =9% Contracton ε (%) Mnmal contracton (%) Maxmal contracton (%) d =mm d =5mm d =3mm d =35mm 5 α =45 α =6 α = Θ (degrees) α = Θ (degrees) Fgure.7: Influence on muscle contracton by changng d and α Fgure.8 shows two dfferent torque characterstcs. On the left sde, the statc torque of the jont s lotted for a maxmum ressure of 3 bar n both muscle and. The nfluence on the statc torque characterstc by changng d s also Chater : Desgn

39 shown. On the rght sde, the torque characterstc of the jont wth closed muscles s lotted to analyze the jont stffness when muscles are closed. Ths characterstc s found by settng 3 bar nto the muscle and closng the muscle at zero degrees. By movng the endulum out of s equlbrum, a reellng torque s generated. Ths torque characterstc s lotted on the rght. 5 α =6, ε =9% and, =3 bar tatc Torque (τ - τ ) (Nm) Mnmal torque needed (Nm) α =6, ε =9% and, =3 bar 5 d =35mm Closed muscles torque (Nm) d =35mm d =3mm 5 d =3mm 5 d =5mm d =mm d =5mm d =mm Θ (degrees) Θ (degrees) Fgure.8: tatc and closed muscle torque characterstcs Based on these lots the desgn arameters were set to the followng values: d = 3mm α ε = 6 = 9% As we can see on fgure.7, for these values the contracton stays between the lmts of 5% and 35% for an angle range between -35 degrees and 35 degrees. In fgure.9, the statc torque characterstcs and torque characterstcs wth closed muscle were redrawn wth the chosen desgn arameter values. Ths was done for dfferent ressure levels Chater : Desgn

40 Wthn the angle range, Fgure.9 shows that enough torque can be rovded to lft the mass. If we now look at the torque characterstcs of the endulum wth closed muscles, we see that there s an almost lnear evoluton of the torque between - degrees and degrees. The angle range s now fxed and all requrements are met. α =6, d =3mm, ε =9 α =6, d =3mm, ε =9 5 bar bar 3 bar 4 bar Mnmal Torque Needed (Nm) 5 bar bar 3 bar 4 bar tatc torque (τ - τ ) (Nm) 5-5 Closed muscles torque (Nm) Θ (degrees) Θ (degrees) Fgure.9: tatc and closed muscle torque characterstcs at dfferent ressure levels, wth the desgn arameters set to the one used n the exermental set-u...3. Constructng the exermental set-u..3.. The frame As was argued revously, a ull rod and leverage mechansm was selected to oston two muscles n an antagonstc setu. The basc frame n whch ths system s ncororated, s dected n fgure.. The modular unt s made of two slats at the sde, whch are connected arallel to each other by two lnkng bars. A jont rotary art, rovded wth roller bearngs, s foreseen for the Chater : Desgn

41 connecton wth an other modular unt. The fxed base for the ull rods mechansm ncludes two rotary axes at whch the muscles are attached. The small rotatons of these axes are guded by sldng bearngs ostoned n the frame. All the arts of the basc frame are made of a hgh-grade alumnum alloy, AlMg, aart from the bolts and nuts, requred to assemble the frame. ldng bearngs Rotary axes Roller bearng Jont rotaton onts Fxed base of muscle ull rod mechansm Fgure.: Exloded and assembled vew of the basc frame Two more lnkng bars are connected to the jont rotary art of the basc frame. Ths s the swngng art of the endulum. The mass s attached to the screw thread connectng the two lnkng bars. The mass conssts of smle dscs, recuerated from dumbbells. In order to attach the dscs and because the dameters of the dscs centre holes are dfferent to the dameter of the screw thread, centerng rngs were fabrcated. The dscs are colour marked, so that the mass of each dsc s known. They can be easly changed to alter the load. The muscles are ostoned crosswse to allow comlete bulgng. At one sde they are attached to the frame va the fxed rotary base and at the other sde the nterface to the next modular unt s rovded va the leverage mechansm. Two connecton lates showed n fgure., are fxed to the next modular unt and ncororate the leverage mechansm. Agan sldng bearngs are used to gude the rotatons of both rotary axes. The oston of the rotaton onts determnes the dmensons of the leverage mechansm and consequently jont torque characterstcs. The mathematcal formulaton of the torque as a functon of force relaton was gven n secton... The connecton lates ncororate the arameters α, α, d and d of the leverage mechansm for both muscles. nce Chater : Desgn 3

42 these arameters have a large nfluence, the connecton late system s the one whch can be changed easly, besdes muscle dmensons, to alter jont torque characterstcs. Therefore the two lates have to be relaced wth only dfferent ostoned holes for the sldng bearngs. Poston lmter Connecton late Fgure.: Vew of the connecton late and the oston lmter At the jont rotaton sde, an angular oston lmter s rovded. Ths devce s equed wth two screws, whch can be regulated searately n order to set the jont rotaton range. The lmter s used for the two followng reasons: Avod sngular jont confguratons n the ull rod and leverage mechansm. Ths haens when the axs of the muscle s algned wth the jont rotaton ont and the muscle attachment ont n the leverage mechansm. In ths stuaton the muscle can serously damage the leverage mechansm when ncreasng ressures would by requred by the controller. Lmt the contracton between 5 and 35 %. Chater : Desgn 4

43 ..3.. The valve system The force develoed by the muscles s roortonal to the ressure nto the muscle. Therefore, a rad and accurate ressure control s needed to set stffness and oston. Fast swtchng on/off valves are used. The neumatc solenod valve 8 / NC made by Matrx, weghs only 5 g. They have a reorted swtchng tme of about ms and flow rate of 8 td.l/ mn. The ermtted ressure dfference over the valve ranges, for each tye, between... 6 bar and... 8 bar resectvely. To ressurse and deressurse the muscle, whch has a varyng volume u to 4 ml, t s best to lace a number of these small on/off valves n arallel. Obvously the more valves used, the hgher the electrc ower consumton, rce and weght wll be. mulatons of the ressure control on a constant volume were done by Van Ham et al. n the context of the develoment of Lucy and ths led to the comromse of nlet and 4 outlet valves. The dfferent number between nlet and outlet comes from the asymmetrc ressure condtons between nlet and outlet and the am to create equal muscle s nflaton and deflaton flows. For detaled nformaton on the smulatons s referred to [3]. Fgure.: Vew of the valve sland Chater : Desgn 5

44 The 6 valves are brought together n a valve sland wth secal desgned nlet and outlet collectors after removng arts of the orgnal housng materal. A hotograh of the valve sland s gven n fgure.. The total weght of ths devce s less than 5 g. [4] Partcles of eoxy resent n the muscles and other drtness can affect the good workng of the valves. To rotect the valves, a flter gauze s laced at the suly connecton of the valve sland and between the muscle and the valve sland. Leaks were detected there where the electrc wrng s assng through the valve sland. To revent the ar to escae through these holes, a sulementary rubber jont s laced between valve and valve sland The neumatc crcut Fgure.3 gves an overvew of the neumatc crcut, whch s used to regulate the suly ressure of the dfferent muscles of the endulum. The neumatc scheme shows the dentcal neumatc crcuts of whch each drves one antagonstc muscle. The valve sland s dected wth searately nlet and exhaust, whch each of them s reresented by two / electrcally actuated valve symbols. These two symbols reresent the reacton levels of the valve system. The number of actual valves, whch are ncluded n each confguraton are dected as well. The rght and left muscle are connected to the suly ressure regulatng unt by searate tubes. The uly ressure regulatng unt Ar suly Inlet valves # # Pressure sensor Exhaust valves # lencer Deressurze valve Pendulum Muscle Muscle #3 3 Hgh ressure regulator Inlet valves # # Exhaust valves # lencer Hgh ressure valve Pressure sensor #3 Fgure.3: chematc overvew of the neumatc crcut Chater : Desgn 6

45 ressure regulatng unt conssts of a mechancal unt that determnes the ressure n the crcut. The crcut s nterruted wth an electrcally actuated 3/ valve. The exhaust of ths valve s connected to an electrcally actuated / deressurzng valve n order to deflate the comlete endulum The exhaust slencer Addtonally, a slencer s added at the exhaust of each valve sland. Wthout a Exhaust slencers slencer, the mmedate exanson to atmosherc condtons of the comressed ar at the exhaust creates a lot of nose. A slencer conssts of a closed ermeable tube whch makes the ressursed ar leave the volume slowly, resultng n a strongly reduced nose generaton. But generally, a slencer also obstructs the dynamc erformance of muscle deflaton, snce a ressure rse n the slencer lowers the exhaust arflow. It s therefore mortant to use large slencers wth good ermeable Fgure.4: Vew of the exhaust materal [4]. As the basc frame of the slencers endulum s not movng and the valves very large exhaust slencers. For mlementaton on Lucy t should be taken nto account, that the valves sland are connected to the movng robot, so smaller exhaust slencers are used The suort As neumatc muscles can easly exert forces u to 5 N and the endulum s swngng at frequences where dynamc forces are beng mortant, a strong suortng structure has to be rovded to hang u the endulum. Besdes ths, Chater : Desgn 7

46 the suortng structure has to be as such that the endulum can be easly dsmounted and nverted. everal confguratons can be used for ths urose. Fgure.5 shows a cture of the used suortng structure. It has been decded to use a rectangular ole that s clamed to the wall. One sde of the slats of the basc frame s ressed aganst the ole wth screw thread and rectangular tubes used as washers. Holes were made nto the ole n order to fx both nverted and non- nverted endulum. To avod oscllatons of the basc frame due to dynamcal forces generated by the swng moton, the other slat s attached to a rectangular frame. The endulum s now restng n a knd of symmetrcal cage, and can easly be dsassembled and nverted. In the future a second rectangular frame should be used, because oscllatons were stll observed at hgher frequences of the swng moton. Pole Washer Plate Rectangular frame Fgure.5: Vew of the suortng structure of the endulum Each valve sland, together wth ts electronc crcut, s mounted on a late. Each late s clamed at one sde of to the ole. It can be easly ostoned where desred. Chater : Desgn 8

47 .3 Electronc desgn In fgure.6 an overvew of the control hardware used to control jont oston and stffness, s gven. A multfuncton I/O data acquston card from NI-DAQ s used to exchange data wth a central PC, used to control the whole endulum. User Interface Central PC IA databus Emergency to AT-MIO-6E- Data acquston card Pressure suly unt eed-u crcutry eed-u crcutry Valve sland afety/uly Transformaton Board Valve sland Pressure extensor muscle Pressure flexor muscle Jont Angle Jont tffness Alarm sgnal Fgure.6: chematc overvew of the control hardware Pressures are measured wth absolute ressure sensors and the angular oston s catured wth an ncremental encoder. The valves of the two slands are Chater : Desgn 9

48 controlled by dgtal mcro-controller sgnals after beng transformed by the seed-u crcutry n order to enhance swtchng seed of the valves. In the next sectons detaled nformaton s gven about the dfferent elements of the control hardware..3.. Valve system seed-u crcutry To have a rad and accurate ressure control, swtchng tme of the valves should be as lttle as ossble. In order to enhance the oenng tme of the Matrx valves, the manufacturer rooses a seed-u n tenson crcutry, shown n hotograh.7. everal ractcal tests, for whch s referred to [3], have resulted n an oenng and closng tmes of about ms. An oenng voltage of 3 V s beng aled durng ms. Fgure.7: Vew of the seed-u crcutry The data acquston card commands the valves va dscrete 5 V on/off sgnals. Two crcuts control searately the two nlet valves and another two control the Chater : Desgn 3

49 exhaust valves. Hereby three valves are controlled smultaneously by one crcut. For the electronc scheme of the seed-u crcutry we refer to [4, D.].3.. Pressure ensor To have a good dynamc ressure measurement, the ressure sensor, shown n fgure.8, s ostoned nsde the muscle. The ressure sensor and ts electroncs are the same as used for Lucy, excet from the AD-converter. nce ths sensor s nsde the muscle volume, an absolute ressure sensor s rovded. In order to ass through the entrance of a muscle, the sze of the sensor and ts electroncs has to be small ( mm). An absolute ressure sensor, CPCAFC, from Honeywell has been selected for ths urose. The sensor measures absolute ressure values u to s (6.9 bar) and has an accuracy of about mbar. The outut of the ressure sensor s amlfed by a dfferental amlfer. As the ressure sensor oututs an analogue sgnal, secal attenton should be gven to ossble nose nterferences. Hgh recson and accuracy s requred, so a cable consstng of three-wre sgnal leads s used. The thrd sgnal lead, or sheld, s necessary and s grounded at the sgnal source to reduce common-mode nose. In secton..4., more nformaton on how ths nose was reduced, s gven. AD-Converter (not used) Dfferental Amlfer Pressure ensor Fgure.8: Vew of the ressure sensor Because the electroncs of the ressure sensors are handmade, they dffer from each other. That s why a calbraton has to be done. By dong ths, offset and senstvty of each ressure sensor are determned, so that at the same ressure Chater : Desgn 3

50 level, the same voltage s outut for both sensors. For several ressure levels, the outut voltage of the left and rght ressure sensors has been acqured. Ths outut voltage s roortonal to the aled ressure n the muscle, so by means of smle lnear fttng of the measured data, voltage offset and senstvty can be found Incremental encoder The angular oston s measured wth an ncremental encoder (HEDM654, Aglent). uch a devce converts rotary dslacements nto dgtal ulse sgnals. Ths tye s an otcal encoder, whch consst of a rotatng dsk, a lght source and a hotodetector (lght sensor). The dsk, whch s mounted on the rotatng shaft, has attern of oaque and transarent sectors coded nto the dsk. As the dsk rotates, these atterns nterrut the lght emtted onto the hotodetector, generatng a dgtal or ulse sgnal outut. The encoder s ncremental, whch means that the encoder does not outut absolute oston. To determne drecton and reference oston, 3 channels are used. Usng two code tracks (A and B) wth sectors ostoned 9, f A leads B, for examle, the dsk s rotatng n a clockwse drecton. If B leads A, then the dsk s rotatng n a counter-clockwse drecton. The thrd outut channel s used as a zero or reference sgnal, whch sules a sngle ulse er revoluton. In our case, only channel A and B are used. We assume that the oston where no muscle s nflated, corresonds wth zero degrees and s therefore taken as reference oston. When usng a quadrature encoder wth the data acquston card, there are two otons. For smle alcatons, you can connect the encoder drectly to the counter/tmer of the data acquston card, wthout any extra logc or sgnal condtonng. In ths confguraton, the counter wll ncrement on state transtons on channel A. Deendng on the Fgure.9: Vew of the state of B at those transtons, the counter wll ncremental encoder count u or down. Whle ths method s very Chater : Desgn 3

51 smle to mlement, t has a coule of otentally serous drawbacks. If the encoder dsk s not rotatng, but s vbratng enough back and forth to cause actve transtons on channel A, wthout changng the state of channel B, then each movement wll be ncorrectly counted. Another roblem results when encoder oututs nclude nose or jtter that s large enough to be erroneously counted as a vald state transton. Therefore a clock converter devce (L784, LI Comuter systems) s used. Ths devce converts the A and B sgnals nto a clock and u/down sgnal that can be connected drectly to the data acquston card. The UP/DN outut ndcates the drecton of rotaton. When n X4 mode, resoluton s ncreased four tmes as the CLK outut wll now ulse once on every transton of ether the A or B sgnals. The L784 ncludes low-ass flters to revent mscounts due to nose and jtter. In addton, the L784 uses dual one-shots to revent the mscountng roduced by vbraton, or dther, as descrbed revously Data acquston card The AT-MIO-6E- data acquston card has 6 sngle-ended or 8 dfferental analog nuts wth a samlng rate of kamles/s, analog outut channels and 8 dgtal I/O channels bult n. A Data Acquston ystem Tmng Controller (DAQ-TC) s avalable. The DAQ-TC ncludes counter/tmer devces, eght of whch are desgned to control the tmng of analog nut and analog outut oeratons. The remanng two counter/tmers are 4-bt u/down counter/tmers avalable for a wde varety of tmng and countng alcatons and wll be used for retrevng nformaton on the jont angle and also used as an nternal clock. Each of the two 4-bt counter/tmers of the DAQ-TC ncludes three nut sgnals (OURCE, GATE, and UP_DOWN) and two outut sgnals (OUT and INTERRUPT). The DAQ card s nstalled n a Pentum 5 Mhz comuter. When rogrammng the DAQ hardware, any alcaton develoment envronment (ADE) can be used but n ether case a NI-DAQ drver must be nstalled. Ths drver has an extensve lbrary of functons that can be called Chater : Desgn 33

52 from the ADE. Those functons can be found n []. The drver also carres out many of the comlex nteractons, such as rogrammng nterruts, between DAQ and comuter. In the begnnng Matlab was used to rogram the DAQ card, but comlcated rogrammng and low samlng rate made us change our mnd and we fnally decded to use Vsual C++. The mnmum control erod durng whch all control calculatons are done s ms. Therefore a controller samlng tme of ms s used. A cable connector s used to connect the data acquston card wth the sensors and valves slands. The ressure sensor oututs a non-referenced sngle-ended (NRE) analog sgnal. The left muscle ressure sensor s connected to analog nut 6 (PIN 5), the rght muscle ressure sensor to analog nut 7 (PIN 7). As we mentoned before, analog sgnals can be dsturbed by nose. A roer wrng system and correct groundng of the sgnal crcut can strongly reduce ths roblem. To measure a grounded sgnal source wth a sngle-ended confguraton, the DAQ card was confgured n the NRE nut confguraton. The sgnal s then connected to the desred analog nut (ACH6 for left ressure sensor and ACH7 for rght ressure sensor) of the DAQ card, and the sgnal local ground reference s connected to AIGND. The ground ont of the sgnal should now be connected to the AIENE n. Any otental dfference between the DAQ card ground and the sgnal ground aears as a common-mode sgnal at both the analog nut (ACH6/ACH7) and AIGND and ths dfference s rejected by the nternal electroncs of the DAQ card. If ths method s not used, a dfference n ground otentals would aear as an error n the measured voltage. As mentoned before, the ncremental encoder has 3 outut channels, but only wll be used. The clock converter transforms these two sgnals nto CLK and UP/DOWN. The ulse sgnal, CLK, s connected to Programmable Functon Inut (PFI) GPCTR_OURCE and the drecton sgnal, UP/DOWN, s connected to the UP_DOWN (DIO6) nut. The DAQ-card s confgured for a smle event count wth hardware u/down control. In ths confguraton, the DAQ card wll control the counter, n such a way that t wll count u/down on every transton of ether A or B. Each valve sland has 6 valves to be controlled, so 8 dgtal sgnals are needed. Because one of the dgtal channels s already occued by one of the sgnals of Chater : Desgn 34

53 the encoder, an analog sgnal s used. The valves are connected through DIO DIO8, excet from DIO6. The remanng valve s connected to DACOUT DACOUT s used to determne the control tme of the whole set-u, by means of a scoe. The remanng 4-bt counter/tmer s confgured n such a way that an nternal base clock of khz s used as tmer. The tme, the encoder counter and sensors values are read at the begnnng of every control loo and valve states are changed f necessary afety, suly and transformaton board Clock converter Pressure sensors connecton Emergency sto connecton 5 V eed-u crcutry suly / valve suly Encoder connecton 3/ valve suly 4 V eed-u crcutry suly 5 V board suly UP/DWN CLK AIense,, Alarm gnal V ressure sensors suly Fgure.: Vew of the safety, suly and transformaton board Chater : Desgn 35

54 The maxmum absolute ressure n the muscles, wthout damagng them, s about 4,5 bar. The suly ressure, however, s 7 bar. If workng roerly, the ressure s lmted to 4 bar by the controller. If a roblem occurs wth the controller, the ressure can exceed ths lmt, and the muscles wll be damaged. To avod such a stuaton and to rotect the muscles aganst hgh ressure, we mlemented a safety crcut, whch handles all the alarm sgnals, orgnatng from the ressure sensor nsde the muscle and an emergency sto. Whenever an alarm sgnal s actvated, the 3/ suly valve of the two ressure regulatng neumatc crcuts s closed, whle the deressursng valve s oened n order to deflate all the muscles. The safety, suly and transformaton board showed n fgure., conssts of the safety crcut, the connecton to all voltage suly sources and to the sgnals from the ressure sensors and encoder. On ths board, the encoder sgnals are also transformed by the clock converter..4 oftware As mentoned before, Vsual C++ s used to rogram the DAQ card. To make exerments and testng more user frendly, a grahcal user nterface (GUI) was develoed. A bref exlanaton on how to use the GUI wll now be gven. The GUI conssts of dfferent tabs. Each tab has some arameters, to be flled n by the user. The frst tab to aear s the Control tab and s shown n fgure.. In ths tab, arameters lke samlng rate, acquston tme and PID gans used by trajectory trackng controller are set. These values can be vared onlne. A secal attenton s gven to the comlance calculaton. More nformaton on the defnton of s gven n chater 3 In the second tab shown n fgure., the Model arameters are set. As exlaned n the mechancal desgn, we can alter the load of the endulum by removng or addng dscs. By clckng on the check boxes, the user can enter the dscs used durng the exerments. Other masses, free oscllaton erod and Chater : Desgn 36

55 atmosherc ressure are also to be flled n, so that the endulum can be modelated. Fgure.: The Control tab of the GUI Fgure.: The Model tab of the GUI Chater : Desgn 37

56 The Trajectory tab, s used to secfy the ntal angle, the offset and the frequency of the desred trajectory we want to be tracked by the trackng controller. In the Plottng tab, the user can secfy the fle where data has to be wrtten to. Because lottng data n Vsual C++ s too comlcated, we rather use Matlab. Therefore, data acqured n Vsual C++ has to be sent to Matlab. The Matlab engne s loaded n C++ to access Matlab s rebult grahcs functons. By usng ths engne, we are able to send the data to the Matlab worksace and lot them drectly from Vsual C++. After each exerment the data s automatcally sent to Matlab and lotted. The oton to lot dfferent tye of data durng the exerment or test, can also be enabled. Plottng the data durng the exerment takes a lot of tme, so t should only be done when no control acton s needed. Once the data s lotted, Matlab worksace wll oen, so that the data can be manulated f necessary. To calculate the energy consumton durng the exerment, a Matlab M-fle s made. Tyng CalcEnergy nto the worksace wll gve the valve acton and energy consumton. More nformaton on how energy s beng calculated wll be gven n the next chater. Fgure.3: The Valves Test tab of the GUI Chater : Desgn 38

57 As we exerenced a lot of roblems makng the valves sland to work roerly, a last tab s used to test the valves. Inut and outut valves can be tested, by choosng the desred valve and to start the test. An overvew of the valves test tab s gven n fgure.3..5 Concluson In ths chater, the hard- and software needed to construct the endulum was descrbed. By analyzng the jont characterstcs, the muscles and actuators connecton arameters were determned. To acqure the ressures n the muscles and the jont angle, and to control the valves, a data acquston card s used. The control of the endulum s done by a PC. To make the exermental setu user frendly a GUI was mlemented. Chater : Desgn 39

58 Chater 3 Control 3.. Introducton The man urose of ths work s to ncororate the exlotaton of the natural dynamcs by adatng jont stffness n combnaton wth trajectory trackng. In secton 3., the man focus s on the develoment of a trackng controller, whch ncororates the actuator characterstcs and dynamc model of the robot. The control strategy conssts of a multlevel constructon of several essental blocks, tryng to coe wth the non-lnear structure usng model-based feedforward technques. Ths trackng controller wll be ncororated n a smulaton model and n the hyscal set-u. ecton 3.3 gves a mathematcal formulaton for comlance adataton, n order to reduce energy consumton and control actvty durng the jont trajectory trackng. A smulaton model of a one-dmensonal endulum wth one antagonstc muscle ar actuatng the jont, s resented n secton 3.4. Ths model s used to show the mortance of arorate stffness selecton n order to reduce control actvty. A jont trackng controller s ncororated n a smulaton model. Fnally, some energy consderatons concernng the roosed control strategy are gven n secton Jont trackng controller The jont trackng controller has to command the valves of the two muscles n order to track an mosed desred trajectory. The comlete system ncororates several nonlneartes such as the non-lnear behavour of the endulum confguraton and the nonlneartes ntroduced by the antagonstc muscle set-u. The trackng controller s desgned n a modular way, whch means that the

59 controller can be easly adated for an alcaton wth another mechancal confguraton, but wth an analogue antagonstc muscle actuator set-u. Therefore, the controller s multstage, of whch each stage deals wth the dfferent nonlneartes searately. A schematc overvew of the roosed control structure s gven n fgure 3.. The controller conssts of three arts: a feedback lnearzaton module, a delta- unt and a bang-bang ressure controller. The feedback lnearzaton or comuted torque module s a standard nonlnear control technque, whch deals wth the nonlnear behavour of the mechancal endulum confguraton []. The delta- unt translates calculated torques nto desred muscle ressure levels, cong wth the nonlneartes ntroduced by the muscle actuaton system. Fnally, the bang-bang ressure controller commands the valves n order to set the requred ressures n the muscles. Ths control structure wll be mlemented n a comuter smulaton of the endulum, as well as n the control of the hyscal set-u. As s sad before, the endulum conssts of one modular art of Lucy, whch means that the used trackng controller wll have the same structure as descrbed n [4]. For the sake of convenence, the same dscusson on the control strategy wll be gven. ome of the modules were slghtly changed: an ntegrator s added to the comuted torque method, and of course the model dynamcs and arameters are not the same. ecal attenton s also gven to the estmaton of these arameters. ~ ~ & && ~ &&& ~ θ, θ, θ, θ tffness ( ) Control Desred Trajectory Comuted Torque &~ θ & Dˆ τ ~ Delta_P Control ~ ~ Valves Valves Bang-Bang Actons Pressure Control Actons Model, θ &~ K D K P K I θ & C ˆ, G ˆ θ θ, & θ θ ~ θ Fgure 3.: chematc overvew of the control structure Chater 3: Control 4

60 In the frst module, the comuted torque ~ τ s calculated. The comuted torque method conssts n decomosng the controller nto two arts. One art s modelbased n that t makes use of the modelled dynamcs of the artcular system under control. The second art of the control s error-drven n that t forms error sgnals by dfferencng desred and actual varables and multles ths by gans. It s also called the servo orton[]. To fnd the model-based art, the rgd body dynamcs can be wrtten down: τ = D ˆ ( θ ) & θ + Cˆ( θ, & θ ) & θ + Gˆ ( θ ) (3.) where D ˆ ( θ ) s the nerta matrx, C ˆ( θ, & θ ) s the centrfugal and corols term, G ˆ ( θ ) s the gravty term and θ the jonts ostons of the real system. In our case, the system s one-dmensonal. Consequently, those matrces wll consst of only one term. The symbol ^ denotes that the resectve exressons are calculated wth estmated arameter values. The man roblem s to fnd an otmal model of the endulum. We can lnearze ths equaton by choosng τ = α τ + β, wth α = Dˆ ( θ ) and β = C ˆ( θ, & θ ) & θ + Gˆ ( θ ) (3.) Fllng ths nto the rgd body mechancs (3.) gves: τ = & θ (3.3) As we see, the resultng equaton s that of unt-mass system. The desgn of the servo oston s now very smle: gans are chosen as f we were controllng systems comosed of sngle unt masses. ~ ~ ~ ~ τ && & = θ + K D ( θ & θ ) + K P ( θ θ ) + K I ( θ θ ) dt (3.4) Chater 3: Control 4

61 The symbol reresents requred values. The gans K I, K P and K D are manually tuned n order to nfluence controller erformance. The followng exresson for the comuted torque s thus obtaned: ~ ~ ~ ~ ~ τ = Cˆ( θ, & θ ) & + Gˆ + Dˆ && & θ ( θ ) ( θ ) θ + K + K + K D ( θ & θ ) P ( θ θ ) I ( θ θ ) dt (3.5) The requred ressures values to be set n the muscles are then calculated by the delta- control unt. These two gauge ressures of both muscles are generated as follows: ~ ~ = + Δ~ tˆ ( θ ) = Δ~ tˆ ( θ ) (3.6a) (3.6b) wth a arameter that s used to nfluence the sum of ressures and consequently the jont stffness, Δ ~ nfluences the dfference n ressure of the two muscles and consequently the generated torque. The functons t ˆ ( θ ) and t ˆ ( θ ) are calculated wth estmated values of the muscle force functons and geometrcal arameters of the ull rod and leverage mechansm of the actuaton system. Exresson (.5) allows lnkng the requred torque to the requred ressure values n the muscles: ~ τ = ~ tˆ ( θ ) ~ tˆ ( θ ) = (ˆ t ( θ ) + tˆ ( θ )) ~ (3.7) Δ If the calculated ressure values ~ and ~ of equatons (3.6) are set n the muscles, the generated torque deends only on Δ ~ and s ndeendent of the jont stffness arameter, n case the modellng would be erfect. Ths means that jont stffness s changed wthout affectng the jont angular oston. Chater 3: Control 43

62 Feedng back the knee angle θ and ntroducng the torque τ ~, exresson (3.7) can be used to determne the requred ~ Δ : ~ ~ τ Δ = tˆ ( θ ) + tˆ (3.8) ( ) θ The delta- unt s actually a feedforward calculaton from torque level to ressure level, usng the cnematc model of the muscle actuaton system. The calculated Δ ~ affects the torque requred to track the desred trajectory, whle s ntroduced to determne the sum of the ressures, whch nfluences the stffness of the jont as was dscussed n secton...3. Increasng the comlance of the jont. lowers In the last control block the desred gauge ressures calculated by the delta- unt are comared wth the measured gauge ressure values after whch arorate valve actons are taken by a bang-bang ressure controller. We defne the ressure error as ( = ~ ), wth ~ the desred ressure calculated by the error delta- unt and the ressure measured n the muscle. The bang-bang ressure control scheme s gven n fgure 3.. Acton a b c d e f error Fgure 3.: Bang-bang ressure control scheme If the ressure dfference stays between c and d, no valve acton takes lace (dead zone) and the muscle stays closed. In ths stuaton the muscle s actng as a comlant assve element. If the ressure error s ncreasng and reaches level e, one nlet valve s oened n order to make the ressure rse to the desred Chater 3: Control 44

63 level. If one oened nlet valve s not enough and error s stll ncreasng and becomes larger than f, a second nlet valve s oened. The nlet valves are closed agan when the dfference dros below level d. The same aroach s used for negatve values of error, but now exhaust valves wll be oened. If error s beyond level a, 4 exhaust valves wll be oened nstead of Mathematcal formulaton for comlance adataton In secton...3 a formulaton of the comlance was gven for closed muscles. As shown, a weghted sum of both ressures n the antagonstc muscle set-u determnes the jont comlance, whle ressure dfferences determne the generated torque and consequently also the jont oston. Ths means that comlance can be set whle controllng the oston. Ths s nterestng, because by settng the arorate jont stffness control actvty can be reduced whle trackng a desred trajectory. In the revous aragrah, the arameter, used to nfluence the sum of ressures and consequently the jont stffness, was ntroduced. The queston s now: what should be the value of? In other words, what should be the jont stffness, so that the natural moton best fts the desred trajectores? In ths secton a mathematcal formulaton develoed by Verrelst [4], s gven for estmatng an arorate value of. The startng-ont for the estmaton rocedure s to ft the natural ressure sloes wth the requred ones. The desred ressures deend on the desred trajectory. The delta- unt n combnaton wth the comuted torque module determnates the requred ressures as exlaned n the revous secton. Pressure changes wth closed muscles are nfluenced by. Chater 3: Control 45

64 Chater 3: Control 46 The requred ressure sloes are calculated by dervng equatons (3.6) wth resect to the trajectory θ ~ : θ θ θ θ θ θ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ d d d dt t d d d d d dt t d d Δ = Δ + = (3.9a) (3.9b) Takng nto account equaton (3.8), the dervatves can be exanded by: ( ) ( ) = Δ τ θ θ θ ~ ~ ~ ~ ~ d dt d dt K t t t t d d (3.) wth K ~ (equaton (.)) reresentng the stffness assocated wth the desred trajectory and τ ~ the torque calculated by the comuted torque module. On the other hand, combnng equaton (.5), vald for closed muscles, wth equatons (3.6) yelds: ( ) ( ) Δ = + Δ = θ θ θ θ θ θ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ d dv V V n P d dv V V t n d d d dv V V n P d dv V V t n d d n n atm n n n n atm n n (3.a) (3.b) The dea s to match for each muscle the requred ressure sloe wth the sloe assocated wth the natural dynamcs, by selectng an arorate value. Once the desred trajectory s known, exressons (3.9) and (3.) can be evaluated n every ont θ. ubsequently a value s searched n order to match as much as ossble both ressure sloes. For each θ, equatons (3.9a) and (3.a) and equatons (3.9b) and (3.b) are thus resectvely combned and each solved for a value j.

65 dt n dv n dv d Δ~ = ( Patm ) t dθ t V dθ + Δ~ + V dθ ~ ~ ~ ~ dθ (3.a) = t dt ~ dθ t n V dv ~ dθ ( P Δ~ ) atm n V θ dv dδ~ ~ dθ ~ dθ θ (3.b) Note that the ntal volume V j volume V j., when closng a muscle s set equal to the actual From here on, there are two otons. We can use a constant, as exlaned by Verrelst. The j are calculated n each ont θ, searated by equal tme ntervals along the desred trajectory, and a mean s then taken to select one : = z z [ + ] = (3.3) wth z the number of onts chosen to evaluate equatons (3.) Ths means that a constant s used along the whole trajectory. Probably, a constant wll only be sutable for some smle trajectores. As we want a method for any trajectory, a varable s needed. One soluton s to use the same dea of comarng the ressure sloes for the comlete trajectory at once, but for the calculaton a dfferental formulaton on has to be solved. To avod comlcated calculatons, t was chosen for evaluatng the equatons (3.) at every samle tme durng the control of the endulum. At every control samle tme, the angle θ s acqured and both for both muscles s then calculated wth tme a dfferent s set. and are calculated. The mean + =, so that at every samle As we see n the equatons (3.a) and (3.b) a correct estmaton of the volume s needed. o the volumes of the end fttngs and tubng s should be added. If ths s not done, the equatons (3.a) and (3.b) wll not gve the most sutable Chater 3: Control 47

66 and an erroneous stffness wll be set. The nfluence of the end fttngs and tubng volumes wll be exlaned n more detal n the next chater. As mentoned n the revous chater, secal attenton was gven to the comlance calculaton durng the desgn of the GUI. On the Control tab the user can choose between three dfferent otons: he can set a constant by gvng t a certan value, or by calculatng t wth the method descrbed above, or use a varable, also calculated by the latter method The smulaton model of the endulum Descrton of the endulum set-u G m, I,, l X τ O Y G, m, l θ Fgure 3.3: The endulum model The endulum model conssts of arts: the lnk, and the mass. The mass s modelled as a ont mass and s ostoned at a dstance l of the rotaton ont. The length of the lnk s l, ts mass m and the moment of nerta about ts centre of mass G s I. The locaton of the centre of mass G of the lnk s gven by OG = α l, wth α =.769. The arameter values are gven n table 3. Chater 3: Control 48

67 l (m) m (kg) ( kgm ) I Table 3.: Inertal arameters of the endulum model An accurate estmaton of those arameters s crucal, as well for the comuter smulaton as for the jont trackng controller. In general the model of the endulum wll not be erfect. Thnk for examle of the frcton model, ths s not taken nto account n our model. To llustrate the nfluence of the arameter errors on the jont trackng controller we can frst look at the real endulum dynamcs: τ = D ( θ ) & θ + C( θ, & θ ) & θ + G( θ ) (3.4) Although, for our control law, we use equaton (3.) : τ = α τ + β, wth α = Dˆ ( θ ) and β = C ˆ( θ, & θ ) & θ + Gˆ ( θ ) where n ths case D ˆ ( θ ), Gˆ ( θ ) and C ˆ( θ, & θ ) are the estmated nerta, gravty and corols/centrfugal term. Decoulng and lnearzng wll not therefore be erfectly accomlshed when arameters are not known exactly. tartng from equaton (3.5), the closed-loo equatons for the system are gven by: [( D Dˆ )&& θ + ( C Cˆ )& θ + ( G Gˆ )] E& + K E& + K E + K E dt = D (3.5) D P I ˆ We clearly see that f the arameters are exactly estmated, the rght hand dsaears and no control error wll be made. As mentoned before, ths wll not be the case n our hyscal setu. Therefore, some devatons on these model arameters were ncororated n the smulaton. Chater 3: Control 49

68 3.4.. The mechancal equatons of the model The equaton of moton s gven by (3.4): τ = D ( θ ) && θ + C( θ, & θ ) & θ + G( θ ) Assumng the model as beng frctonless, we can now calculate the elements of the nerta, corols and gravty terms (Aendx A): D( θ ) = ml C( θ, & θ ) = G( θ ) = g + m α l ( m l + m α l ) sn( θ ) + I (3.6a) (3.6b) (3.6c) The thermodynamcal equatons of the model The equatons descrbng the thermodynamc rocesses n the neumatc system were already dscussed by [4]. As we need these equatons to understand how the comuter smulaton, exlaned n the next secton, works, those equatons wll be reeated here. The thermodynamc rocesses n the two muscles/valve systems are descrbed by four frst order dfferental equatons. Two equatons determne the ressure changes n both muscles of the endulum and the remanng two descrbe conservaton of mass n the resectve muscle volume. In assumton that the ressursed ar behaves lke a erfect gas, the erfect gas law comletes the set of equatons requred to run the smulaton. The frst law of thermodynamcs, whle neglectng the flud s knetc and otental energy and assumng a olytroc rocess, can be wrtten for each muscle as [4, Aendx B]: su n ex ( rt m& rt m& ( P ) V& ) n & = ar ar ar ar atm + V (3.7) Chater 3: Control 5

69 wth r the dry ar gas constant. su T ar s the temerature of the suly ar and T ar the temerature n muscle. The total orfce flow through the oened nlet valves and exhaust valves of muscle are gven by n m& ar and ex m& ar resectvely. The latter two can be calculated wth the followng equatons, whch reresents a normalzed aroxmaton of a valve orfce flow defned by the Internatonal tandard IO6358 [989]: m& ar = CP ρ u 93 u T ar Pd Pu b b f Pd P u b (3.8a) m& ar = CP ρ u 93 u T ar f P P d u b (3.8b) wth ρ the ar densty at standard condtons. C and b are two flow constants charactersng the valve system. The constant C s assocated wth the amount of ar flowng through the valve orfce, whle b reresents the crtcal ressure rato at whch orfce ar flows become maxmal. Pu and P d are the ustream and downstream absolute ressures, whle u T ar ar s the ustream temerature. When chokng occurs, equaton (3.8b) s vald, otherwse equaton (3.8a) s used. The muscles are controlled by a number of fast swtchng on/off valves. Once the actons (oenng or closng) of the valves are known, all the ar flows can be calculated n order to be substtuted n (3.7). The temerature n the muscle s calculated wth the erfect gas law: PV Tar = (3.9) m r ar wth P ( + P ) = atm the absolute ressure n muscle. The total ar mass ar gven by ntegraton of the net mass flow enterng muscle : m s m & ar = m& m& (3.) n ar ex ar Chater 3: Control 5

70 Comlete smulaton model The structure of the comlete smulaton model s gven n fgure 3.4 and s based on the smulaton model of Lucy. The kernel of these smulatons s based on three equaton blocks, whch ntegrates frst order dfferental equatons only. The dfferental equatons are numercally ntegrated usng a 4 th order Runge-Kutta method wth ntegraton tme ste of 5 μs. P T su ly su ly Valve system model m & / m& n ar ex ar Runge-Kutta numercal ntegraton Integraton tmeste 5 μs Equatons of moton Thermodynamc dfferental equatons τ V V & θ, ω Antagonstc muscle model Valve acton T ar Thermodynamc state equatons t θ,ω,,t Delay observer Valve control Control unt Jont trajectory trackng Exlotng Natural Dynamcs amlng tme 5 μs Desred trajectory Fgure 3.4: tructure of the comlete smulaton model The frst block contans the equatons of moton gven by (3.4). nce the equatons of moton are second order, these equatons have to be transformed nto a set of frst order equatons. In order to do ths, the angular velocty ω s ntroduced: ω = & θ (3.) Chater 3: Control 5

71 In our case, the equaton of moton can now be rewrtten as two frst order equatons: ( θ ) [ τ C( θ ω) G( θ )] & ω = D, & θ = ω (3.) The thermodynamcs of the jont s characterzed by four frst order dfferental equatons on ressure (3.7) and ar mass (3.) ( frst order equatons for each muscle). Fnally the two thermodynamc state equatons (3.9) comlete the set. The lnk between the equatons of moton and the thermodynamc dfferental equatons s gven by the antagonstc muscle model. In ths model the generated torque τ (3.7) of the jont s calculated wth the ressure nformaton of the thermodynamc block. To calculate ths torque the olynomal forces (.) must be known. These are functons of the contracton (.), whch can be calculated by knowng the angle nformaton from the equatons of moton. Addtonally, to determne the ressure changes n the thermodynamc dfferental equaton block, muscle volume and volume changes are needed. They are calculated wth the olynomal volume functons (.3). Consequently, angle and angular velocty are requred from the equatons of moton. tartng from the desred trajectory and by takng the natural dynamcs nto account, the control unt determnes the arorate valve actons and sends valve control sgnals to the delay observer. The valves have an observed oenng and closng tme delay of about ms [3] Ths s ncororated n the delay observer. In the smulaton a delay tme of oenng or closng the valve of ms s used. The valve system model wll then calculate the ar mass flow rates for each muscle. Therefore, temerature and ressure n the muscles are requred from the thermodynamc dfferental equatons block. Addtonally, devatons can be ntroduced on the mass, centre of gravty and nerta arameters and on the force functons, n order to check the robustness of the controller, as was exlaned n the revous secton. Chater 3: Control 53

72 3.5. Energy consderatons The man urose of ths thess s to nvestgate the exlotaton of the natural dynamcs of the system n order to reduce energy consumton and control acton. Therefore, t wll be nterestng to have an dea of the energy consumton and control acton that takes lace durng the smulaton or exerment. As the valves are controlled by the control unt, the valve actons wll gve us an dea of the control acton. The control sgnals to acton the valves have two ossble values: for an oen valve and for a closed valve. To have an dea of the amount of control acton, the control sgnals for the nut as well as for the outut valves are accumulated durng the exerment or smulaton. The tme er erod, durng whch the nut/outut valves are oen s then calculated wth: sum( Valve gnals) Valve Acton ( ms / erod) = * amletme * (3.3) f T ne Wave acquston Wth amle Tme, the samle tme at whch data s acqured and not the controller samle tme. In the case of the hyscal endulum, the samle tme at whch data s acqured and the controller samle tme are the same. The thermodynamc condtons of the ressurzed ar also determne energy consumton. o aart from valve actons, t s nterestng to consder actual ar mass enterng and leavng the total system. The energy consumton deends not only on the ar mass flows but s related to the thermodynamc condtons of the comressed ar suly source. It s not straghtforward to calculate the actual energy needed to ower the endulum, snce ths deends on how the ressurzed ar of the neumatc suly source has been created. One way to gve an dea of energy consumton s to calculate the exergy assocated wth the artcular neumatc ar mass flow. Exergy s n fact the max-mum amount of energy, wth resect to the surroundng envronment, whch can be transformed nto useful work. For a comressor, the mnmal work Chater 3: Control 54

73 needed to comress ar from ressure level to s done at sothermal condtons and can be calculated as follows [9]: W& sotherm = m& rt (3.4) ln hereby assumng the ar to behave as a erfect gas. The symbol m& reresents the total ar mass flowng through the comressor at ressure level, r s the dry ar gas constant and T s the temerature the ar at ressure level exressed n Kelvn. Usng ths equaton we can calculate the exergy of the suly source as well as the exergy of the ar leavng the muscles. The exergy of the suly source for muscle s gven by: W n sotherm n suly mar rtatm dt = &, ln (3.5a) atm, whle the outut exergy can be calculated wth: W out sotherm out mar rtar dt = &, ln (3.5b) atm Durng the smulaton, the absolute suly level s set at 7 bar, the atmosherc absolute ressure at bar and the atmosherc temerature s 93 K Concluson In ths chater a control archtecture structure for the endulum was dscussed. The structure covers the exlotaton of the natural dynamcs n combnaton wth jont trajectory trackng. The jont trajectory trackng controller uses a comuted torque method to coe wth the nonlnear behavor of the endulum Chater 3: Control 55

74 confguraton. The calculated torques are then transformed nto desred muscle ressure levels by a delta- unt, cong wth the nonlneartes ntroduced by the muscle actuaton system. Fnally, a bang-bang ressure controller commands the valves n order to set the requred ressures n the muscles. The exlotaton of the natural dynamcs s mortant, as t wll decrease energy consumton and control acton. In ths context, two slghtly dfferent mathematcal formulatons were gven to redct an adequate stffness settng. Frst, a sutable constant stffness was dscussed. In the second formulaton, the stffness s changed onlne, whch makes t more sutable for comlcated trajectores. A smulaton model, whch ncororates ths control archtecture as well as the modellng of the endulum dynamcs and the thermodynamc rocesses, whch take lace n the muscles, were descrbed. Ths smulaton model wll be used to nvestgate exlotaton of the natural dynamcs and artcularly to test the mathematcal formulaton descrbed for that urose. Fnally, some energy consderatons were made, n order to have an dea about the amount of energy consumton and valves acton durng the smulaton or exerment. Chater 3: Control 56

75 Chater 4 Results and dscusson 4.. Introducton In the revous chaters, the exermental setu and smulaton model as well as a control archtecture, whch combnes the exlotaton of the natural dynamcs wth a trajectory trackng controller, were descrbed. To evaluate whether the roosed controller can meet trackng requrements and deal wth the natural dynamcs of the system, t was ncororated nto the comuter smulaton and the hyscal endulum. Ths chater wll gve results on ths theme for both smulaton and ractcal set-u. Results on the smulaton model wll be gven n secton 4.. Frst of all the trajectory trackng control wll be evaluated. The nfluence of erroneous model arameter estmatons wll be shown. econdly, the nfluence of the jont comlance on the energy consumton and valve actons wll be dscussed. In ths context, secal attenton s gven to the evaluaton of the mathematcal formulatons gven n the revous chater for estmatng the arorate jont stffness. In secton 4.3, trackng exerments on the hyscal set-u wll be dscussed. uccessvely, the ossblty to decrease control actvty and energy consumton by settng a sutable stffness, wll be shown on the hyscal endulum.

76 4.. The smulaton model 4... Evaluaton of the jont trajectory trackng controller 4... General consderatons As mentoned n chater 3, the control archtecture conssts of 3 man blocks: a feedback lnearzaton module, a delta- unt and a bang-bang ressure controller. The comuted torque ncororates a model-based module and a servo control. The model-based art requres a good estmaton of the model arameters, as was exlaned before. In order to nvestgate the nfluence of a wrong estmaton of the model arameters, devatons on the mass, nerta, centre of gravtes and force functon were ntroduced nto the smulaton. A smulaton wth % devaton, whch means that the arameter estmatons are exact, wll be comared wth a smulaton wth % devaton on the nerta, 7.5 % on the centre of gravty and 5% on the mass and the force functon. The servo orton of the comuter torque module requres a correct tunng of the gans. Ths was done ntutvely, but n such a way that stablty s stll ensured and control actons are mnmzed wthout affectng the trackng accuracy. error (mbar) Valve acton a - Oen all exhaust valves b -6 Oen one exhaust valve c -5 Close all exhaust valves d 5 Close all nlet valves e 6 Oen one nlet valve f Oen all nlet valves Table 4.: Pressure reacton levels of the bang-bang ressure controller, used n smulaton Chater 4: Results and dscusson 58

77 The bang-bang controller comares the requred ressure to be set nto the muscle wth the measured gauge ressures and takes the arorate valve actons deendng on the varous reacton levels. These levels were also manually tuned n order to mnmze energy consumton by takng nto account the trajectory trackng accuracy. The values of the ressure reacton levels are gven n table 4. The controller samlng tme s set to ms and a valve delay tme of ms s ntroduced. The valve delay tme corresonds wth the real data, recorded by Verrelst. As was mentoned n chater, a controller samlng tme of ms s used for the control of the hyscal endulum. Next, the trajectory trackng control shall be evaluated by mosng a desred sne-wave trajectory wth a frequency of Hz and an amltude of 5. The stffness arameter s set to Nm. Ths s chosen as an examle to valdate the trajectory trackng, wthout takng nto account the natural dynamcs of the system 4... Results and dscusson 6 4 Perfect model ( Mean error on Θ =.948, Maxmum error on Θ = ) q (degrees) qdes (degrees) Tme (s) Erroneous model ( Mean error on Θ =.75, Maxmum error on Θ = ) q (degrees) qdes (degrees) Tme (s) Fgure 4.: The jont angle θ for erfect and erroneous arameter estmatons of the model Chater 4: Results and dscusson 59

78 - Perfect model Torque (Nm) Desred torque (Nm) frcton CT(model art) (Nm) P acton (Nm) I acton (Nm) D acton (Nm) Tme (s) - Erroneous model Torque (Nm) Desred torque (Nm) frcton CT(model art) (Nm) P acton (Nm) I acton (Nm) D acton (Nm) Tme (s) Fgure 4.: Torques of the jont for erfect and erroneous arameter estmatons of the model Fgure 4. dects the grahs of the angle θ for the erfect and erroneous model. The frst erod was omtted because some settlng tme s needed and s not nterestng for the urose of ths secton. As we see, the trackng controller unt can stll coe wth the ntroduced errors. In fact the trackng errors are very small. The dfference between both smulatons, and thus the nfluence of erroneous arameter estmaton, s better seen on fgure 4.., were the requred torques, calculated by the nverse dynamc control block, and the actual aled torques are dected. The model-based art and the servo ortons are also lotted. In the erfect model, a lttle dfference s observed between requred and aled torque, due to a mnmum ressure error requred for the bang-bang controller to actvate. The desred torque s almost equal to the model art of the comuted torque. Ths s what we exected, because the model art was calculated wth exact arameters. The servo orton deals wth the dfference n requred and measured angle oston and velocty. The erroneous model shows the nfluence of the erroneous estmaton of the model arameters. Frst of all, there s a bgger dfference between the aled and requred torque because of the nfluence of Chater 4: Results and dscusson 6

79 the force functon errors. The model-based art of the comuted torque dffers substantally of the calculated torque, due to the ntroducton of devatons on the mass and nerta. The servo orton has to take more acton, as the model-based art s usng wrong arameters Perfect model (muscle ) valve acton ressure (bar) requred ressure (bar) Tme (s) Detal erfect model (muscle ) valve acton ressure (bar) requred ressure (bar) Tme (s) Fgure 4.3: Muscle ressure wth valve acton for erfect model. Fgure 4.3. dects requred and actual gauge ressures and valve actons taken by the bang-bang controller for muscle. A closed valve s reresented by a horzontal lne at the bar ressure level. An oen nlet valve s reresented by a small eak uwards, whle an oen outlet valve s reresented by a downwards eak. Larger eaks means that nlets or 4 outlets are oened. From fgure 4.3. we can conclude that the bang-bang controller s able to track the desred ressure qute accurately. No acton taken by the valves, means that the ressure error error s stuated n the dead zone of the bang-bang controller. Ths s the case between.78 s and.86 s. Durng ths erod muscle s closed, and the natural dynamcs of the system s exloted. It s now clear that the dead zone of the bang-bang controller wll be very mortant to decrease the energy consumton and should be exloted as much as ossble. Ths wll be dscussed n the followng secton. Chater 4: Results and dscusson 6

80 4... Exlotaton of the natural dynamcs 4... Exerments If we set some ressure values n both muscles of the antagonstc setu, close them and then release the endulum from a oston dfferent of the equlbrum, the endulum starts oscllatng wth a certan frequency. If ths frequency corresonds wth the desred trajectory, no valve acton needs to be taken. Ths means that f stffness s set n such a way that the natural dynamcs of the system suts the desred trajectory, energy consumton wll be mnmzed durng trackng. The frst smulatons were done n ths context. A constant, the arameter to nfluence the sum of the ressures and consequently the jont stffness, was set. Ths wll be done for two dfferent values of, n order to show the mortance of settng a correct stffness for mnmzng the energy consumton. The desred trajectory agan s a sne-wave wth a frequency of Hz and an amltude of 5. In a second seres of smulatons, the mathematcal formulaton roosed by Verrelst and descrbed n chater 3 wll be evaluated for dfferent snusodal trajectores. tll, a constant wll be used. Fnally, the ossblty to track a random trajectory wth exlotaton of the natural dynamcs, wll be nvestgated n the thrd seres of smulatons. The stffness s changed onlne n order to set the most sutable stffness at every control samle tme. No arameter devatons are ntroduced n the smulatons. The controller samlng tme and the valve delay tme are stll ms, resectvely ms. Chater 4: Results and dscusson 6

81 4... Results and dscussons mulaton : the mortance of settng a sutable stffness We can llustrate the nfluence of the stffness by consderng the unactuated oscllaton wth closed muscles of the endulum as was exlaned n the revous secton and comare t wth the desred trajectory. In fgure 4.4, one erod of the unactuated oscllaton s shown for dfferent values of Hz sne wave that has to be tracked., as well as the 6 4 s = 6 Nm q (degrees) qdes (degrees) Jont angle ( ) Tme (s) 6 4 s = 3 Nm q (degrees) qdes (degrees) Jont angle ( ) Tme (s) Fgure 4.4: Actual and desred jont angle wth closed muscles for = 6 Nm and = 3 Nm For both stffness, the oscllaton does not ft the desred trajectory, but for = 3 Nm, we can see that the base frequency s near to Hz. We can exect that for = 3 Nm, the valve actons and energy consumton wll be less than n the case of = 6 Nm, as the natural dynamcs fts best the mosed trajectory. To nvestgate ths assumton, a smulaton of an actuated oscllaton was done for = 6 Nm and = 3 Nm. Fgure 4.5. dects the jont oston Chater 4: Results and dscusson 63

82 and angle velocty of the two smulatons for one erod. There s no sgnfcant dfference between the two smulatons concernng the jont angle and angle velocty. The trackng s slghtly better for the case where = 3 Nm. Jont angle ( ) Jont angle ( ) s = 6 Nm angle desred angle -3.6 angle velocty -45 desred ang vel Tme (s) s = 3 Nm angle desred angle -3.6 angle velocty -45 desred ang vel Tme (s) Angular velocty ( /s) Angular velocty ( /s) Fgure 4.5: Actual and desred jont angle and angular velocty for = 6 Nm and = 3 Nm In fgure 4.6 and 4.7 the gauge ressures and the valve actons are dected. To get a clearer vew on the valve actons, erods are shown on the rght of each fgure. For the same reason as before, the frst erod s not shown. In fgure 4.7 the mean ressure s twce as hgh as n fgure 4.6, ths because of the dfference n value of. A very clear dfference can be observed between the two smulatons. Lookng at the valve acton, we can mmedately conclude that the case where = 3 Nm, s the most arorate one to track a sne wave wth a frequency of Hz. That s what we already suggested before. For examle, between 4.9 and 4.9 seconds no acton Chater 4: Results and dscusson 64

83 .5 s =6Nm.5 s =6Nm Pressure muscle (bar).5 valve acton ressure requred ressure Tme (s) Pressure muscle (bar) valve acton ressure requred ressure Tme (s).5.5 Pressure muscle (bar).5 valve acton ressure requred ressure Tme (s) Pressure muscle (bar) valve acton ressure requred ressure Tme (s) Fgure 4.6: Actual and requred ressure n muscle and for = 6 Nm.5 s =3Nm.5 s =3Nm Pressure muscle (bar).5 valve acton ressure (bar) requred ressure (bar) Tme (s) Pressure muscle (bar) valve acton ressure (bar) requred ressure (bar) Tme (s).5.5 Pressure muscle (bar).5 valve acton ressure (bar) requred ressure (bar) Tme (s) Pressure muscle (bar) valve acton ressure (bar) requred ressure (bar) Tme (s) Fgure 4.7: Actual and requred ressure n muscle and for = 3 Nm s taken by the valves of muscle (or between.45 and 3.5s). The reason for ths s that the ressure sloe nduced by the natural dynamcs of the system, fts the ressure sloe requred to track the desred trajectory. We can also Chater 4: Results and dscusson 65

84 understand why the ressure levels of the bang-bang controller are so mortant durng the exlotaton of the natural dynamcs. The ressure error of muscle near to 4.75 seconds has a maxmum value of 59 mbar, whch means that error s stll stuated n the dead zone. A smaller dead zone would have requred the bang-bang controller to take acton. A larger dead zone wll ncrease the exlotaton of the natural dynamcs, but a larger devaton on the trajectory trackng wll be observed. A comromse has to be made between control acton and trackng error. Table 4. gves a summary of the ar mass flow, the exergy and the valve acton for both smulatons over one erod. The valve actons are defned as n (3.3) = 6 Nm = 3 Nm Muscle Muscle Total Muscle Muscle Total Arflow nut. mg 7.9 mg 8. mg.7 mg 3.9 mg 35.6 mg Arflow outut.4 mg 6.6 mg 8. mg 4. mg. mg 36. mg Exergy nlet 9.7J 7.7 J 37.4 J 3.6 J.3 J 5.9 J Exergy outlet 6.8 J 7. J 3.8 J.4 J. J 3.5 J Inlet valve acton 39.8 ms 36. ms 75.8 ms 7.7 ms 4.9 ms.6 ms Outlet valve acton 34.4 ms 4.8 ms 76. ms.5 ms 6.7 ms 7. ms Table 4.: Ar mass flow, exergy and valve acton durng one erod of a sne wave trajectory wth frequency of Hz for = 6 Nm and = 3 Nm The most nterestng value s the exergy nlet, because ths wll gve an ndcaton of the energy consumton of the system. As we can see the total nut exergy n the case of = 3 Nm equals 5.9 J, whch s much lower than 37.4 J n the case of = 6 Nm. Therefore, from ths exerment we can conclude that a sutable stffness affects n a ostve way the energy consumton and control acton durng the trajectory trackng. Chater 4: Results and dscusson 66

85 mulaton : evaluaton of the mathematcal formulaton The man goal s to see whether the mathematcal formulaton gven by Verrelst s usable. Therefore, n ths smulaton, jont sne wave trajectores wth dfferent frequences are mosed. Table 4.3 gves an overvew of the dfferent sne wave trajectores for whch the mathematcal formulaton was evaluated. In the second column the calculated values of, are gven. The values n the thrd column gve the otmal, for the corresondng trajectory. For each trajectory, the latter values were obtaned by varyng mnmzed. and evaluate where the energy consumton and valve actons are ne wave frequency (Hz) Calculated (Nm) smulaton (Nm) Table 4.3: Calculated and smulated otmal values of waves trajectores for dfferent sne Fgure 4.8 dects the energy consumton and valve actons er erod for the dfferent sne waves n functon of. For each trajectory, there s a where energy consumton and valve acton s mnmum. Ths s the most otmal. By comarng the calculated and the otmal, we can conclude that the mathematcal formulaton s ndeed sutable for settng the stffness. The otmal s ncreasng as the frequency s ncreasng. Ths was exected because s nfluencng the stffness and a hgher oscllaton frequency requres a hgher stffness. Chater 4: Results and dscusson 67

86 Total nlet exergy (J) Frequency =.5 Hz Exergy Valve acton Inlet valves acton (ms) Total nlet exergy (J) Frequency =.75 Hz Exergy Valve acton Inlet valves acton (ms) Total nlet exergy (J) s (Nm) Frequency = Hz Exergy Valve acton s (Nm) Inlet valves acton (ms) Total nlet exergy (J) s (Nm) Frequency =.5 Hz Exergy Valve acton s (Nm) Inlet valves acton (ms) Fgure 4.8: Energy consumton and valve acton n functon of dfferent trajectores, for In the revous dscusson only sne wave trajectores were mosed. For these trajectores the assve behavour fts the mosed trajectory f ths case a constant value s justfed. s set rght. In We have to remember that the goal of ths thess s to nvestgate the ossblty to exlot natural dynamcs for mlementng t later nto the bed Lucy. In general, the trajectores for such a bed are not just sne waves wth only one frequency comonent. For such a stuaton, a constant wll not be sutable as shown n fgure 4.9. In ths fgure, a sne wave wth a lnear ncreasng frequency from.5 Hz to Hz s mosed. Only the ressure course for one muscle s shown here. The trajectory can stll be tracked but we can clearly see that the natural dynamcs are only exloted n the tme nterval [,s 3s ], or n other words, for a certan frequency range of the mosed trajectory. At a frequency hgher than.9 Hz even 4 exhaust valves are oened, n order to ncrease the flow rate and consequently track the requred ressure course. Note also that the mean ressure nsde the muscle remans constant durng the whole smulaton, due to a constant = Nm. Chater 4: Results and dscusson 68

87 Jont angle ( ) Pressure muscle (bar) Frequency q (degrees) qdes (degrees) Tme (s) Frequency valve acton ressure requred ressure Tme (s) Fgure 4.9: Jont angle, ressure and frequency for a trajectory wth lnear varyng frequency The concluson of ths smulaton s that a varyng wll be more sutable for trajectores dfferent from a ure sne wave. Ths wll be the dscusson of the next smulaton. mulaton 3: changng the stffness onlne The urose of ths smulaton s to evaluate the method of usng a varyng or stffness, as descrbed n chater 3. Ths was done by mosng the same trajectory as n fgure 4.9., Frst, some comments have to be gven on the equatons that were used durng ths smulaton. In chater 3 a comlete descrton of the mathematcal formulaton was gven. Usng these equatons led to unsatsfyng results: the stffness K ~ assocated wth the desred trajectory, became nfntve n some onts. Chater 4: Results and dscusson 69

88 The exlanaton for ths henomenon can be found by startng from the dynamcally requred stffness, derved from the equaton of moton: ~ ~ ~ dt d d θ ~ K = ~ = ~ d g sn(θ ) d d + θ θ dt 3 ~ 3 ~ d θ dt ~ d θ = d ~ + g cos(θ ) d 3 = 3 dt dθ dt ~ dθ + g dt ~ cos(θ ) (4.) wth: ~ θ = A cos(ω t) ω = ω + ωt ωmax ωmn ω = tsmulaton (4.a) (4.b) (4.c) The dervatves of the desred trajectory θ ~ are gven by: ~ dθ = A sn dt ~ d θ = Acos dt 3 ~ d θ = A sn 3 dt ωacos ( ω. t + ω. t )( ω + ω. t) ( ω. t + ω. t )( ω + ω. t ) ωasn( ω. t + ω. t ) 3 ( ω. t + ω. t )( ω + ω. t) 4ωAcos( ω. t + ω. t )( ω + ω. t) ( ω. t + ω. t )( ω + ω. t) (4.3a) (4.3b) (4.3c) Thus we have for K ~ : ~ ( +. t) 6ω cotg( ω.t + ω.t ) g cos(θ) ~ + K = d ω ω (4.4) We see that a cotg() functons arses for ths secfc trajectory. The course of K ~ s dected n the frst lot n fgure 4.. In the second lot, the cotg( ) term s gven. The latter exlans why K ~ s beng nfntve. The cotg( ) term can be Chater 4: Results and dscusson 7

89 seen as a nose sgnal dsturbng K ~ and s therefore neglected. The resultng course of K ~ s gven n the thrd lot. K(Nm/rad) wth nose Nose (Nm) K(Nm/rad) wthout nose Tme (s) Tme (s) Tme (s) Fgure 4.: tffness K ~ and nose due to cotg( ) term for the secfc trajectory K ~ s now gven by: K ~ ~ = d( ω + ω. t) gcos(θ ) (4.5) + In the grah at the bottom of fgure 4., we can observe that the requred stffness s ncreasng as the frequency ncreases. Ths s what we exected, because f the frequency of the mosed trajectory ncreases, a hgher stffness of the antagonstc setu wth closed muscles wll be needed n order to exlot natural dynamcs. Usng ths equaton, runnng the smulaton for an mosed trajectory wth frequency varyng between.5 Hz and Hz and usng a varyng, gves us the ressure course as dected n fgure 4.. To comare ths to the stuaton where a constant was used, the ressure course from fgure 4.9 was redrawn n fgure 4.. Comarng fgure 4. and 4., we can conclude that valve Chater 4: Results and dscusson 7

90 acton s extremely decreased. The natural dynamcs are now beng exloted durng the whole smulaton, ths by makng varable. 4 Frequency Varable s Pressure muscle (bar) valve acton ressure requred ressure Tme (s) Fgure 4.: Actual and requred ressure and valve acton for a trajectory wth lnear varyng frequency and onlne changed. 4 Frequency Constant s = Nm Pressure muscle (bar) valve acton ressure requred ressure Tme (s) Fgure 4.: Actual and requred ressure and valve acton for a trajectory wth lnear varyng frequency and a constant. Chater 4: Results and dscusson 7

91 Because the requred oscllaton frequency s ncreasng, energy has to be njected nto the system. Ths exlans why n fgure 4. more nlet valve actons are taken durng the whole trajectory. Once, the frequency remans constant, the nlet valve actons wll decrease, snce no addtonal energy has to be njected. Note that the ressure s ncreasng as the frequency of the mosed trajectory ncreases. As mentoned before, the requred stffness rases as the frequency rases, whch means that the sum of ressures n the muscles has to ncrease. Ths also means that ncreases, as t reresents a arameter to nfluence the sum of the ressures. The evoluton of 4.3. along the trajectory s shown n fgure s calculated(nm) s (Nm) s (Nm) s calculated + 5% devaton(nm) s calculated - 5% devaton(nm) 6 s (Nm) Frequency (Hz) Fgure 4.3: n functon of the frequency of the mosed trajectory. Devatons on were ntroduced to nvestgate the effect on the valve control when the onlne calculaton of s not gvng the correct value. Ths can be nterestng to know, snce we can already exect that the onlne calculaton of the otmal for the smulaton wll not necessarly gve the otmal for the hyscal endulum. The nfluence on the valve actons s gven n fgure 4.4. On the left we can see the valve actons durng the whole smulaton and on the Chater 4: Results and dscusson 73

92 rght a detaled vew s gven. We can draw the same concluson as before: a wrong varable value of wll ncrease the valve actons. Therefore, t wll be mortant to have a good model of the endulum setu, such that the method wth varable stffness can be used durng exerments. Valve actons.5.5 Frequency Valve actons Frequency Valve actons( s +5%) Valve actons( s +5%) Valve actons( s -5%) Tme (s) Valve actons( s -5%) Tme (s) Fgure 4.4: Effect on the valve actons due to devaton on the onlne changng arameter Wth ths smulaton, the mortance of usng a varable for trackng trajectores wth not only one frequency comonent, has been showed. Usng a varable, wll result n a varable mean of the ressure n each muscle searately. Therefore, the antagonstc setu should be desgned n such a way that stays between a certan range, n order to lmt the gauge ressure n the muscles between bar and 3.5 bar. Or, n the case of the bed Lucy, the generated trajectores should take nto account those lmts f natural dynamcs should be exloted. Chater 4: Results and dscusson 74

93 4.3. The hyscal endulum Evaluaton of the jont trajectory trackng controller In the revous secton the jont trajectory trackng controller was evaluated by means of comuter smulatons. The hyscal endulum was bult n order to erform some trackng exerments, and to adat the smulaton model accordng to the exermental data. In ths secton the erformance wll be dscussed and an overvew wll be gven of whch arameters dffers from the smulaton model. A correct estmaton of these arameters was not done, but the mortance should not be neglected. Angular velocty ( /s) Torque(Nm) Pressure (bar) Frst trackng exerment Measured Desred tme (s) Total torque CT(model art) P acton I acton D acton tme (s) 3.5 Pressure muscle 3 Desred ressure Valve acton tme (s) Fgure 4.5: Angular velocty, comuted torque and absolute ressure for frst trackng exerment In order to nvestgate f exlotaton of the natural dynamcs s ossble, a good workng trajectory trackng controller has to be mlemented. In a ractcal setu, some addtonal roblems arse, e.g. nose on the ressure sensor sgnal or small errors n the jont angle nformaton. An even more mortant roblem s the measurng of the angular velocty. The angular velocty s requred by the comuted torque module, and n order to have a stable controller, ths arameter Chater 4: Results and dscusson 75

94 has to be well known and as much as ossble free of nose. As we cannot drectly measure the angular velocty, t should be calculated numercally, startng from the jont angle acqured wth the ncremental encoder. Therefore, a frst order numercal method was used. Fgure 4.5 shows the angular velocty, the comuted torque erformance and the absolute ressure n muscle for the frst trackng exerment. A sne wave wth a frequency of.75 Hz was mosed and a constant = 4 Nm was used. As we can see, a lot of nose aears on the angular velocty sgnal. Ths wll have an mact on the trackng controller as the servo orton wll have to coe wth ths nose. We clearly see that the D- acton s tryng to elmnate ths nose. The delta- unt transforms the nosy comuted torque to ressure levels, whch means that a lot of valve acton wll be taken by the bang-bang controller as shown at the bottom of fgure 4.5. The valves are swtchng a lot, and most of the tme the 4 outlet or nlet valves are oened n order to track the ressure course. Wth regard to energy consumton, ths s unaccetable. Angular velocty ( /s) Torque(Nm) Pressure (bar) Frst trackng exerment Measured Desred tme (s) Total torque CT(model art) P acton I acton D acton tme (s) 3.5 Pressure muscle 3 Desred ressure Valve acton tme (s) Fgure 4.6: Angular velocty, comuted torque and absolute ressure for a trackng exerment by usng a flter on the angular velocty. To avod such roblems, a frst order dgtal flter was used on the angular velocty. A blnear transformaton was done n order to convert an analog to a Chater 4: Results and dscusson 76

95 dgtal flter. The blnear transformaton s a mathematcal mang of varables. In dgtal flterng, t s a standard method of mang the s or analog lane nto the z or dgtal lane. It transforms analog flters, desgned usng classcal flter desgn technques, nto ther dscrete equvalents [9]. The frst order analog flter used on the hyscal endulum s gven by: H a ( s) = s + ω a (4.6) Pre-warng the flter desgn gves: cut off ω d T ω = tan a (4.7) T wth T s, the samle tme and flter, gven by: ω the cut-off ulsaton of the desred dgtal cut off d cut off ω d = π f f cut off = Hz T = ms cut off (4.7a) (4.7b) (4.7c) The blnear transformaton s gven by: z H d (4.8) ( z) = H a T z + whch results n the dgtal flter: b z + b H d ( z) a z + a = (4.9) Chater 4: Results and dscusson 77

96 Wth a = +, a = (4.9a) Tωa Tωa b = b = + Tω (4.9b) a The ntroducton of a flter greatly mroved the erformance of the jont trajectory controller. Fgure 4.6 shows the angular velocty, comuted torque and the absolute ressure when a flter s used. Very remarkable s the effect of the flter on the valve actons and s a frst ste n the correct drecton concernng energy consumton. The gans used by the servo orton of the comuted torque, were manually tuned n such a way that stablty s stll ensured and control actons are mnmzed wthout affectng the trackng accuracy. As was exlaned, n chater 3 t s mortant to have a good estmaton of the model arameters, n order to reduce the servo orton to a mnmum. No frcton model was used durng the exerment. Also the exstng of small ar leaks n the muscles and tube connectons were not taken nto account. In the future, a fnetunng of the model should be made. The jont angle s gven n fgure 4.7. We can see that the control unt s sutable to track ths trajectory. Durng the frst erod, trackng s not so accurate. An exlanaton can be found n the ressure course also dected n fgure 4.5. In the begnnng, the endulum s at rest at an angle of 5. The muscles are nflated and a certan ressure s set nto the muscle. Once the exerment starts, a ressure much lower than the actual ressure s needed. All the outlet valves are oened n order to track the ressure course, but ths s not enough. The ressure gradent over the valves s nsuffcent, so that the tme constant of deflaton s too large n comarson to the requrements assocated wth the mosed ressure course. Ths can cause the endulum to become unstable. A reason for ths unwanted henomena, s the use of an exhaust slencer, as t obstructs the dynamc erformance of the muscle deflaton. The Chater 4: Results and dscusson 78

97 ressure rse n the slencer lowers the exhaust arflow. Durng the constructon, ths slencer was made as large as ossble to avod such stuatons. 6 4 Measured Requred Jont angle ( ) tme (s) Pressure (bar) Pressure muscle Desred ressure Valve acton tme (s) Fgure 4.7: Measured and desred jont angle θ and ressure course for muscle Other solutons for ths roblem are ncreasng the outlet valves, or settng the ntal muscle ressure at a lower level. The frst oton s mortant f hgh oscllaton frequences are needed, snce hgh ressure gradents over the valves wll then be requred. The reason why the ntal ressure s set at such a hgh level requres an exlanaton. In the begnnng, the dea was to start at an ntal angle of zero degrees and to use a sne functon for the mosed trajectory. Because of a dscontnuty n the angular velocty ( θ &(t ) =, θ(t & = ) = Aω ), nstablty = + occurred. Therefore, we oted to start at an ntal angle equal to the amltude of the mosed trajectory. The trajectory used to brng the endulum n ts ntal oston as dected n fgure 4.8. By usng ths trajectory, the dscontnuty of the angular velocty s elmnated. When the ntal angle s reached, a certan settlng tme s needed for the endulum to stay at rest. The settlng tme can serously be decreased by ncreasng the stffness of the jont. Ths means that Chater 4: Results and dscusson 79

98 has to be ncreased durng the trackng of the ntal trajectory, and as a consequence the ntal ressure wll be at a hgher level. 6 4 Intal trajectory Requred trajecory exerment Jont angle ( ) tme (s) Fgure 4.8: Intal trajectory and mosed trajectory durng exerments Durng the exerments an ncreasng devaton on the jont angle was observed as the tme ncreased due to some mscounts of the encoder sgnal. Ths was not a roblem n the context of ths thess, because the acquston tme durng the exerments was relatvely short. But n the future, the thrd sgnal of the ncremental encoder should also be ncororated, n order to reset the encoder counter when the endulum asses trough hs reference angle. Based on ths secton, we can conclude that the roosed control archtecture s able to track sne wave trajectores. In order to reduce energy consumton, the exlotaton of the natural dynamcs s requred. Ths s shown n the next secton Exlotaton of the natural dynamcs General consderatons We want to nvestgate the exlotaton of the natural dynamcs of the system n order to reduce energy consumton. A method to estmate the energy consumton was gven n chater 3. Of course, ths wll not gve us the correct value, but t s nterestng to have an dea of the energy consumton and control acton takng lace durng the exerment. If a correct estmaton s needed, Chater 4: Results and dscusson 8

99 measurement of the arflow, ar temerature and muscle volume wll be needed. In our case, we only want to nvestgate whether the PPAM can be used to exlot the natural dynamcs of the system and therefore the formulaton from chater 3 wll do. The calculaton of the energy s only started after two seconds of acquston, when the endulum s n regme. As mentoned before, the dead zone of the bang-bang controller has an mortant nfluence on energy consumton. Changng the ressure levels of the bang-bang controller wll ncrease or decrease valve actons. In order to nvestgate only the nfluence of the stffness on the energy consumton and valve actons, these ressure levels were set to constant values gven by table 4.4. These values were manually tuned, so that control effort s mnmzed by takng nto account the trajectory trackng accuracy P <. 5bar P. 5bar error (mbar) error (mbar) Valve acton a - - Oen all exhaust valves b -6-3 Oen one exhaust valve c -4-5 Close all exhaust valves d 4 5 Close all nlet valves e 6 3 Oen one nlet valve f Oen all nlet valves Table 4.4: Pressure levels of the bang-bang controller for the hyscal endulum Results and dscusson In the context of the exlotaton of the natural dynamcs, one exerment was done on the hyscal endulum. The man goal s to fnd out f energy consumton and control acton can be mnmzed by settng the arorate stffness and to determne the value of the otmal stffness. Therefore, the stffness arameter was vared, and for dfferent trajectores, energy consumton and valve acton were calculated. The same exerment was done Chater 4: Results and dscusson 8

100 n the comuter smulaton model, of whch results were already shown n fgure 4.8. Remember that n the smulaton model, the otmal and the found wth the mathematcal formulaton gven by Verrelst were comared. We concluded that the method of the ressure sloes was sutable for sne trajectores wth one frequency comonent, as the two values were almost the same. In the case of the hyscal endulum, energy consumton and valve acton, defned by (3.3), for one erod of dfferent trajectores are gven n fgure 4.9. Total nut exergy vs s Total outut exergy vs s Exergy (J) Hz.75 Hz Hz Exergy (J) Hz.75 Hz Hz 5 Valve actons (ms) 3 4 s (Nm) Valve actons (frequency =.75 Hz) 5 5 In valves muscle In valves muscle Out valves muscle Out valves muscle 3 4 s (Nm) Exergy (J) 3 4 s (Nm) Influence amltude on total nut exergy degrees degrees 3 4 s (Nm) Fgure 4.9: Energy consumton and valve acton for one erod of dfferent trajectores n functon of the stffness arameter We can clearly dentfy otmal values for, where energy consumton and valve actons are mnmzed. Ths s also llustrated by fgure 4.,where the valve actons, requred to track a trajectory wth a frequency of.75 Hz and an amltude of 5 degrees s dected. In ths fgure two dfferent values of were set, a wrong and the otmal value. If we look between.3s and.7s, no valve acton s taken for = 9Nm, because the natural ressure course fts the requred one. Comarng wth = 3Nm, we can conclude that t s mortant Chater 4: Results and dscusson 8

101 to set the rght stffness n order to exlot the natural dynamcs of the endulum. Ths was already concluded n smulaton, but now t s confrmed n ractce. s = 3 Nm Pressure (bar) Pressure muscle Desred ressure Valve acton Pressure (bar) tme (s) s = 9 Nm (Otmal) Pressure muscle Desred ressure Valve acton tme (s) Fgure 4.: Influence of the stffness arameter the absolute ressure n the muscle on the valve acton and Another way to llustrate how nfluences the natural dynamcs, s the unactuated oscllaton of the endulum for the two dfferent stffness arameters. Ths s gven n fgure 4.. After seconds of control, the muscles are closed. If the stffness s sutable, when = 9Nm, the base frequency of the uncontrolled oscllaton aroxmates.75 Hz durng the two frst erods. Whle when = 3Nm, the base frequency s stuated around Hz, snce the stffness of the jont s hgher. When = Nm the base frequency s about.4 Hz, snce the stffness of the jont s lower. We only consder the two frst erods, because due to frcton and leakage of the neumatc system, the uncontrolled oscllaton wll dam and frequency wll change. Chater 4: Results and dscusson 83

102 Jont angle ( ) Jont angle ( ) Jont angle ( ) 5 Measured Angle ( s = 3Nm) Desred angle tme (s) 5 Measured Angle ( s = ot = 9Nm) Desred angle tme (s) 5 Measured Angle ( s = Nm) Desred angle tme (s) Fgure 4.: The uncontrolled oscllaton of the endulum for three dfferent stffness arameters after a controlled erod of seconds As was mentoned before and shown n fgure 4., settng a dfferent stffness changes the mean ressure. Ths means that the stffness s bounded between two values, snce the absolute ressure n the muscle s lmted between and 4.5 bar. Varyng the stffness tll those lmts were reached, showed that the range of for ths endulum s bounded between 6 and 4 Nm In fgure 4.9, the nfluence of the amltude on the nut exergy s shown. The amltude has lttle nfluence on the otmal stffness, lke was exected, but energy consumton seems to be hgher. Due to the nonlnear torque to angle relaton, the shae of the assve trajectory devates from a ure sne-wave. The devaton from the sne-wave ncreases for larger amltudes. Consequently, more valve swtchng s needed for larger movements. Ths clearly shows the mortance of desgnng a jont wth a lnear torque characterstc when muscles are closed, when sne-wave trajectores wth one frequency comonent are mosed. Chater 4: Results and dscusson 84

103 In order to see whether the method of the ressure sloes can be used on the ractcal setu, a resume of the otmal values of the exermental setu s gven n table 4.5 and comared wth the calculated and smulaton values. Exermental ne wave frequency (Hz), otmal ( Nm) Calculated (Nm) smulaton, otmal ( Nm) Table 4.5: Exermental, calculated and smulated otmal values of dfferent sne-wave trajectores for The exermental values are near to the calculated and smulated values. From ths we can conclude that the method of the ressure sloe could be used, but only as an aroxmaton of the otmal stffness. The stffness wll not be set at hs otmal value, but stll energy consumton and valve acton wll be decreased and natural dynamcs wll be exloted. The error that s made by usng ths method, can be decreased by a better arameter estmaton, e.g. leaks n the tubng and muscle and frcton. The most mortant arameters to be estmated, are the volume of the muscle and the olytroc exonent. The volume s calculated usng the estmated olynomal fttng descrbed n secton..., whch does not take nto account the volumes of the end fttngs and tubng s. Besdes ths, the estmated olynomal fttng was done on a theoretcal model of the muscle. For the mathematcal descrton of the enclosed volume, the leated olyester membrane s aroxmated by consderng at each arallel secton a crcular membrane attern nstead of the leated structure. No exermental data on the volume has ever been acqured. The olytroc exonent s ntroduced to descrbe devatons from the sentroc exanson/comresson takng lace when muscles are closed and the endulum s oscllatng. The value of the exonent should be exermentally estmated and may deend on the secfc rocess. The estmaton of these arameters was not done durng ths thess, but s very mortance f a varyng stffness has to be Chater 4: Results and dscusson 85

104 used n combnaton wth trajectory trackng. But stll, romsng results were roduced Concluson In ths chater the trajectory trackng control structure as descrbed n chater 3 was evaluated on a comuter smulaton model and a hyscal endulum. The smulaton model showed that the roosed control archtecture coes wth the nonlneartes ntroduced by the actuator characterstcs and dynamc model of the endulum. The mortance of a correct estmaton of the model arameters was demonstrated by ntroducng devatons on the nerta, the mass, the centre of gravty and on the force functon. The mlementaton of the control unt nto the hyscal endulum requred the ntroducton of a flter to reduce the nose on the calculated angular velocty. A trackng exerment llustrates that the control unt s able to follow a sne-wave. The man dfferences wth the smulaton model were also dscussed. In the context of the exlotaton of the natural dynamcs, 3 exerments were erformed on the comuter smulaton model and one on the hyscal endulum. In the frst exerment, the smulaton results showed the mortance of settng the rght jont stffness n order to reduce control effort and energy consumton. In the second exerment, the mathematcal formulaton, descrbed n secton 3.3, was evaluated. Therefore, the constant stffness obtaned by ths method was comared wth the otmal stffness found by tunng the stffness manually. We concluded that the mathematcal formulaton can ndeed be used to estmate the otmal stffness, when a sne-wave wth one frequency comonent s mosed. In order to exlot the natural dynamcs for trajectores wth more than one frequency comonent, the stffness was changed onlne n exerment 3. A snewave trajectory wth lnear varyng frequency was mosed. By usng the method of varyng stffness, the control unt was able to track the desred trajectory n combnaton wth the exlotaton of the natural dynamcs. The exerment on the hyscal endulum was erformed to fnd out f energy consumton and control acton can ractcally be mnmzed by settng the arorate stffness and to determne the value of the otmal stffness. For Chater 4: Results and dscusson 86

105 dfferent trajectores, the otmal stffness was found. By comarng the exermental values of the stffness arameter wth those found n the smulaton, we were able to conclude that the method of the ressure sloes can be used to aroxmate the otmal stffness. A better arameter estmaton of the endulum s model and the bang-bang controller s ressure level wll decrease the error made on the estmaton of the stffness. Once these arameters are correctly estmated, we should be able to vary the stffness along the trajectory. Therefore, adatng the endulum model to all exermental data and tunng the bang-bang controller s the next ste to take n order to comletely exlot the natural dynamcs. Chater 4: Results and dscusson 87

106 Chater 5 General conclusons and future ersectves Ths thess reorts on the desgn, constructon and control of a one-dmensonal endulum actuated by a ar of leated neumatc artfcal muscles (PPAM). Pneumatc artfcal muscles have very nterestng characterstcs towards exlotaton of natural dynamcs of the system. Due to the comressblty of ar and the drong force to contracton characterstc, a jont actuated by two antagonstcally ostoned muscles has a comlant behavour, whch can be adated whle controllng oston. The man urose s to use ths adatable jont comlance to combne exlotaton of natural dynamcs combned wth jont trajectory control, n order to reduce control effort and energy consumton sgnfcantly. The jont trajectory controller tracks the mosed trajectory, whle changng the jont stffness, as such that the natural trajectory corresonds as much as ossble to the reference trajectores. The nvestgaton on ths theme s done on both a comuter smulaton model and a hyscal endulum. The latter, together wth ts soft- and hardware, had to be desgned and constructed. A lot of tme was sent on testng and debuggng the dfferent comonents. Chater dscusses both mechancal and electronc desgn requred for a well workng hyscal setu. The mechancal desgn focuses on the actuator connecton, as t defnes the jont characterstcs of the antagonstc muscle setu and consequently the comlance. Therefore a formal descrton of the jont knematcs s gven and t s shown that the jont oston s nfluenced by dfferences n both muscle ressures; whle the comlance of the jont, wth closed muscles, s set by a weghted sum of ressures. The basc frame, suortng structure and neumatc crcut are assembled to form one modular art of Lucy. In order to control the ressure n a muscle, fast swtchng on/off valves are used. The electroncs requred to acton these valves together wth the ressure sensors and the ncremental encoder are tested and descrbed. A data

107 acquston card serves as a data transfer agent between the exermental setu and a comuter, on whch the comlete control of the endulum s mlemented. The control archtecture used to combne jont trajectory trackng wth exlotaton of the natural dynamcs, s exlaned n chater 3. The jont trajectory trackng controller conssts of a multlevel constructon of several essental blocks, whch try to coe wth the system s nonlneartes at searate levels. A model-based feedforward controller calculates the requred torques to track the desred jont trajectores. These calculated torques are then transformed nto desred muscle ressure levels by a delta- unt, cong wth the nonlneartes ntroduced by the muscle actuaton system. Fnally, a bang-bang ressure controller wth dead-zone commands the valves n order to set the requred ressures n the muscles. A mathematcal formulaton s gven n order to ncororate the natural dynamcs n the control unt. The startng ont for fndng an arorate constant stffness, s to ft the natural ressure sloes, ths s when both muscles are closed, wth the requred ones. The better these two ft, the less acton wll be taken by the bang-bang controller. Consequently, energy consumton and control effort are decreased. Based on ths formulaton, an addtonal method s gven to change the stffness onlne. In order to evaluate the trackng controller and the ressure sloe method, a smulator, based on the one used for Lucy, s develoed. Chater 4 gves an overvew of the results that are obtaned wth the comuter smulaton model and the ractcal setu. For both, trackng exerments are erformed n order to evaluate the robustness of the trajectory trackng controller. From these exerments, the concluson can be drawn that the roosed control structure s ndeed sutable to track mosed trajectores. Great mortance s attached to the ncororaton of the natural dynamcs of the system. Frst of all, the smulaton model s used to show the nfluence of settng a constant sutable stffness on the energy consumton and control effort when a sne-wave wth only one frequency comonent s mosed. The otmal stffness values are found for dfferent frequences of the mosed sne-wave. These values are then comared to the constant values obtaned wth the method of ressure sloes. Based on these results, we can consder that the method can be used for settng the sutable stffness. We can draw the same concluson for the method of changng the stffness onlne. Ths s nterestng for further research on Lucy, Chater 5: General conclusons and future ersectves 89

108 snce the requred trajectores are not ure sne-waves and thus an onlne varyng stffness wll be needed n order to exlot natural dynamcs of the comlete robot. Exerments on the hyscal endulum are erformed, n order to fnd the otmal stffness for mosed sne-wave trajectores wth one frequency comonent. These otmal values are comared wth the smulaton values. The results are very romsng, as both values are stuated near to each other. Ths means that a method s found to control the stffness of the hyscal endulum n order to exlot the natural dynamcs n combnaton wth trajectory trackng. Two man stes should be taken to mrove the erformance the stffness control: A correct estmaton of the endulum s arameters. The muscle volumes, the olytroc exonent and the frcton are the most mortant arameters to be estmated. Fne tunng of the ressure levels of the bang-bang controller. The dead zone should be exloted as much as ossble, wthout losng trackng accuracy. Before usng ths technque on Lucy, a study should be done on a double endulum, n order to nvestgate the coulng effects, whch can lead to chaotc behavour of the system. Fnally, the same nvestgaton should be done on an nverted endulum. To conclude, n ths master thess the frst stes n the drecton of exlotaton of natural dynamcs by usng PPAM s were made. Chater 5: General conclusons and future ersectves 9

109 Aendx A Dynamc model of the endulum G m, I,, l X τ O Y G, m, l θ Fgure A.: chematc overvew of the studed model In ths secton the equaton of moton for the model dected n fgure A. s derved. A Newton-Lagrange formulaton of the equaton of moton s used for ths urose. The model has only one DOF θ : d dt K K U + = Q & θ θ θ θ (A.) K s the total knetc energy and U the otental energy. Q θ s the generalzed force assocated wth the endulum. The knematc exressons for the ostons and veloctes of the centers of mass for the dfferent lnks are gven by: OG OG v v G G = = l = α T l( snθ,cosθ ) T α l ( snθ,cosθ ) T ( cosθ, snθ ) & θ T l ( cosθ, snθ ) & θ = (A.) (A.3) (A.4) (A.5)

110 The total otental energy of the endulum s equal to: U = m gy + m gy (A.6) G G Whch results n: ( m l α m ) cosθ U = g + (A.7) l The artal dervatve becomes: U = g θ ( m l + α m ) snθ l (A.8) The knetc energy K of art s gven by: K = mvg + I ω (A.9) wth ω the angular velocty for art. For the two arts ths becomes: K K = = m l & θ ( m α l + I )& θ (A.) The total knetc energy s the sum of these two terms: ( m l + m α l I )& θ K = + (A.) Now the dfferent dervatves of the knetc energy can be calculated: d K = dt & θ K = θ ( m l + m α l + I ) & θ (A.) (A.3) Chater A: Dynamc model of the endulum 9

111 Chater A: Dynamc model of the endulum 93 The torque τ aled n the vot ont reresents the generalzed force. o the equaton of moton for ths model can be summarzed as followed: ) ( ), ( ) ( θ θ θ θ θ θ τ G C D + + = & & & & (A.4) Wth: ( ) ( ) θ α θ θ θ α θ sn ) ( ), ( ) ( l m m l g G C I l m m l D + = = + + = & (A.5)

112 Aendx B afety Board Fgure B.: Electronc scheme of the safety, suly and transformaton board Chater B: afety board 94

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