Keywords:

ierre nd wrie gurie in ris @wgAF his thesis is undertken in the wiroEnno rootique group @wsgyfA direted y rofF téphne égnierF wy thesis work is (nnilly supported y the prnheEgomté egion nd the gentre xtionl de l eherhe ienti(que @gxAF pirst of llD s would like to thnk my supervisors téphne égnierD xiols endre'D nd qilgueng rwng for their enourgementD helpful guidneD gret reD vlule dviesD nd ontinuous nd indispensle ssistneF téphne égnierD professor t wgD guided me through ll the yers in the good diretion of my reserhF st is relly my pride to perform my hFhF reserh under his supervision nd it is ft tht this thesis would not hve een possile unless his supervisionF qilgueng rwngD reserher in vortoire de hotonique et de xnostrutures @vxAD worked on the helil miroswimming eforeF re helped me lot t the eginning of my thesis to e e fmilir with the sujetF xiols endre'D professor t pemtoE in fesnçonD helped me lot in the visul servoing ontrol prtF ris help is very importnt for me to (nish my thesisF s would like to thnk ll of them for their vilility for numerous disussions nd their motivting investments in the reserh through my thesisF s would lso like to thnk inn rliyoD ssistnt professor in our temD for his gret support on on(gurtion of the system in vinuxY ylvin ledelD the mehnil engineerD for the mehnil prts tht he mde for meY ghristophe qrnd for the rpid prototyping prtsY homs eonD reserher in )uid mehnisD for the vlule disussion out visous liquids nd his help of the mesurement of the visosity of the liquidsF s would lso like to thnk ll the urrent nd former memers of ssD speilly the memers of our tem wsgyfD for their friendship nd supportF hey hve mde the entire grdute experiene edutionlD entertiningD nd enjoyleF e hd lot of interesting disussionsF s9m hppy to list their nmes hereX ten yhin erhmiE nsD oukeyn fouheoutD eude folopionD wokrne foudoudD vur gohenD ellister pourtetD inming vuD edeni wohnd yusidD qillume willetD henjing xiD géile oretD entoine eillEhu)osF et lstD s wnt to give ll my loving thnks to my prentsF hey hve lost lot time to hve me round due to my reserh rodF ithout their enourgement nd understnding it would hve een impossile for me to (nish this workF s would lso like to thnk the fmily festD who is my host fmily when s rrived in prne t villeF he whole fmily is very kind nd dorleF s9m very lose to the fmily during my eight yers in prneF hey helped nd supported my lotF s would lso like to thnk my der oyfriend for shring the plesing nd di0ult moments during ll my grdute yersF ris enourgement nd supports helps me lot to (nish my thesisF QHGHIGPHIR in ris Abstract: relil miroswimmers ple of propulsion t low eynolds numers hve een proposed for numerous pplitionsD rnging from in vitro tsks on lE onEEhip @eFgF trnsporting nd sorting miro ojetsY mehnil omponents miro ssemlyFFFA to in vivo pplitions for minimlly invsive mediine @eFgF trgeted drug deliveryY rhytherpyY hyperthermiFFFAD due to their miro sizes nd essiility to tiny nd lustered environmentsF everl kinds of mgnetilly tuted helil swimmers with di'erent geometry prmetersD hed shpesD nd mgneti positioning hve een proposed in prior worksF roweverD the in)uene of the geometry prmetersD the hed shpe nd the mgneti positioning @hedD oted tilFFFA hs not een lerly studiedF es fr s we knowD the existing helil miroswimmers re primrily openEloop ontrolledD due to the omplexE ity of the ontrol of the mgneti (eld tuting the helil propulsionD nd the limited numer of feedk options proessing the required preisionF his thesis ims toX

• ompre the swimming performnes of helil swimmers with di'erent designs to further improve their design nd to hrterize their swimming propertiesF

• relize visul servo ontrol of helil swimmersF ledEup helil miroswimmers t the millimeter sle re designed nd swim t low eynolds numersF he design of these sledEup helil miroswimmers n e guideline for the miroEfrition of helil miroswimmersF e visul servo ontrol of the sledEup helil miroswimmer orienttion in the Qh speD nd pth following on the horizontl plne hve een relizedF he ontrol method will e pplied on helil miroswimmers in future worksF Résumé : v9utilistion de mirongeurs hélioidux ples de se mouvoir dns des liquides à file nomre de eynolds trouve son intérêt dns euoup d9pplitionsX de tâhes inEvitro dns des lortoires sur pue @trnsport et tri de miroEojetsY sE semlge de miroEomposntsFFFAD à des pplitions inEvivo en médeine miniEinvsive @livrison interne et ilée de médimentsD uriethérpieD thermothérpieFFFAY gre à leur dimensions mirosopiques et gilité permettnt l9ès à des endroits normlement très restreintsF lusieurs types de ngeurs hélioidux tionnés mgnétiquement possédnt divers prmètres géométriquesD formes de tête et positions de l prtie mgnétique ont été proposés dns de préédents trvuxF gependntD l9in)uene de tous es prmètres n9 ps lirement été étudiéeF À notre onnissneD les mirongeurs hélioidux dns l9étt de l9rt sont priniplement ontrôlés en oule ouverteD en rison de l omplexité de l ommnde du hmp mgnétique tionnnt l propulsionD et du nomre limité de retours ynt des ritères stisfisntsF gette thèse vise àX

• omprer les performnes de déplement de ngeurs hélioidux ve des onE eptions di'érentes (n d9méliorer leur design et de les rtériserF

• réliser un sservissement visul de ngeur hélioidlF our se fireD des ngeurs hélioidux de tilles millimétriques ont été onçus et sont mis en onditions à file nomre de eynoldsF v oneption de es 4millingeurs4 servir de se à l oneption de mirongeursF ne ommnde oule fermée pr retour visuel de l9orienttion d9un mirongeur hélioidl dns un espe QhD et un suivi de trjetoires sur pln horizontl ont été e'etuésF gette méthode de ommnde ser pr l suite ppliquée à des mirongeurs hélioiduxF General Introduction 1

1 Introduction 3

IFI wotivtions F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F R IFIFI sn vitro pplitions F F F F F F F F F F F F F F F F F F F F F F F F F R IFIFP sn vivo pplitions F F F F F F F F F F F F F F F F F F F F F F F F F S IFP wimming t low eynolds numers F F F F F F F F F F F F F F F F F F F F T IFPFI eynolds numer F F F F F F F F F F F F F F F F F F F F F F F F F F F T IFPFP xvierEtokes equtions F F F F F F F F F F F F F F F F F F F F F F F U IFPFQ eiprol motion nd nonEreiprol motion F F F F F F F F F F F V IFPFR voomotion of miroEorgnisms F F F F F F F F F F F F F F F F F F F W IFPFS fioEinspired miroswimmers F F F F F F F F F F F F F F F F F F F F F IH IFQ relil miroswimmers F F F F F F F F F F F F F F F F F F F F F F F F F F F F II IFQFI relil miroswimmer frition proess F F F F F F F F F F F F F II IFQFP tte of rt helil miroswimmers F F F F F F F F F F F F F F F F F IR IFQFQ relil propulsion model F F F F F F F F F F F F F F F F F F F F F F F PI IFR wiroroots tution F F F F F F F F F F F F F F F F F F F F F F F F F F F F PR
IFRFI etution method of miroroots F F F F F F F F F F F F F F F F F F PR IFRFP tte of the rt of mgneti tution systems F F F F F F F F F F F PU IFS wiroroots loliztion F F F F F F F F F F F F F F F F F F F F F F F F F F F PU IFSFI iletromgneti nd mgneti trking F F F F F F F F F F F F F F F PV IFSFP wgneti esonne smging @wsA F F F F F F F F F F F F F F F F PV IFSFQ gomputed omogrphy @gA nd )uorosopy

F F F F F F F F F F F PW IFSFR ltrsound F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F PW IFSFS ision F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F QH IFSFT ummry F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F QH IFT isul servoing for miroroots F F F F F F F F F F F F F F F F F F F F F F F QI IFTFI sntrodution of visul servoing F F F F F F F F F F F F F F F F F F F QI IFTFP fsi omponents of visul servoing F F F F F F F F F F F F F F F F QP IFTFQ smgeEfsed isul ervoing @sfA F F F F F F F F F F F F F F F F QQ IFTFR ositionEfsed isul ervoing @fA F F F F F F F F F F F F F F QR IFTFS isul servoing for miroroots F F F F F F F F F F F F F F F F F F F QR IFU yverview of thesis F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F QS
2 Overview of the system 39 PFI ledEup relil wiroswimmers @rwsA F F F F F F F F F F F F F F F F F RH PFIFI hy the sledEup helil miroswimmersc F F F F F F F F F F F F RH PFIFP hesign nd frition of sledEup helil miroswimmers F F F F RH PFIFQ wimming t low eynolds numers F F F F F F F F F F F F F F F F RQ PFP Qh relmholtz oil pirs system F F F F F F F F F F F F F F F F F F F F F F F RS viii Contents PFPFI relmholtz oils priniple F F F F F F F F F F F F F F F F F F F F F F F RS PFPFP Qh relmholtz oil pirs system design nd relistion F F F F F F RS PFPFQ ision system implementtion F F F F F F F F F F F F F F F F F F F F RU PFPFR wgneti (eld lirtion nd genertion F F F F F F F F F F F F F RV PFPFS ummry F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F SR PFQ otting ermnent wgnet wnipultor @wwA F F F F F F F F F F F F SR PFQFI ww priniple

F F F F F F F F F F F F F F F F F F F F F F F F F F F SS PFQFP ww design nd relistion F F F F F F F F F F F F F F F F F F F F SS PFQFQ ottion frequeny ontrol F F F F F F F F F F F F F F F F F F F F F F SU PFQFR ummry F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F SV PFR isionEsed trking F F F F F F F F F F F F F F F F F F F F F F F F F F F F F TH PFRFI rking of rw F F F F F F F F F F F F F F F F F F F F F F F F F F TH PFRFP fryentre of the helil swimmer F F F F F F F F F F F F F F F F F TI PFRFQ yrienttion of the helil swimmer F F F F F F F F F F F F F F F F F TI PFS wimming hrteristis nlysis F F F F F F F F F F F F F F F F F F F F F F TR PFSFI ottion frequeny nlysis F F F F F F F F F F F F F F F F F F F F F TR PFSFP rnsltion veloity nlysis F F F F F F F F F F F F F F F F F F F F F TS PFSFQ ropulsion mtrix identi(tion F F F F F F F F F F F F F F F F F F F TS PFT ummry F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F TU
3 Swimming characteristics of helical swimmers 69 QFI qeometry prmeters of helil tils F F F F F F F F F F F F F F F F F F F F UH QFIFI e rief introdution to the design of experiments F F F F F F F F F UH QFIFP ghoie of the prmeters F F F F F F F F F F F F F F F F F F F F F F UH QFIFQ wimming performne of the rws with di'erent geometry pE rmeters F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F UI QFIFR smpts of the geometry prmeters F F F F F F F F F F F F F F F F UP QFP ottionl propulsion hrteristis F F F F F F F F F F F F F F F F F F F F UQ QFPFI ottionl propulsion of rws with di'erent mgneti positionings UQ QFPFP ottionl propulsion of rws with di'erent hed shpes F F F F UU QFPFQ ptors in)uening rottionl propulsion F F F F F F F F F F F F F F UW QFPFR ghoie of the helil swimmer design F F F F F F F F F F F F F F F F VH QFQ ropulsion veloityD dvne veloity nd torque nlysis F F F F F F F F F VI QFQFI ropulsion mtrix identi(tion cs M tp rnsformtion mtrix from the test pttern to the side mer oordinte system cs R tp ottion mtrix from the test pttern to the side mer oordinte system cs t tp rnsltion mtrix from the test pttern to the side mer oordinte system ct M cs rnsformtion mtrix from the top mer to the side mer oordinte sysE tem ct M tp rnsformtion mtrix from the test pttern to the top mer oordinte sysE tem ct R cs ottion mtrix from the top mer to the side mer oordinte system ct R tp ottion mtrix from the test pttern to the top mer oordinte system ct t cs rnsltion mtrix from the top mer to the side mer oordinte system ct t tp rnsltion mtrix from the test pttern to the top mer oordinte system ims g fryenter of the rw on the side mer imge plne imt g fryenter of the rw on the top mer imge plne w M cs rnsformtion mtrix from the side mer to the world oordinte system w M ct rnsformtion mtrix from the top mer to the world oordinte system w Q = ( w X, w Y, w Z) T goordintes of point Q in the world system List of abbreviations xv w R cs ottion mtrix from the side mer to the world oordinte system w R ct ottion mtrix from the top mer to the world oordinte system w t cs rnsltion mtrix from the side mer to the world oordinte system w t ct rnsltion mtrix from the top mer to the world oordinte system w Q = ( w X, w Y, w Z, 1) T romogeneous oordintes of point Q in the world system

F F F F F F F F F F F F F F F F F F F VI QFQFP ropulsion veloity nd dvne veloity F F F F F F F F F F F F F F VR QFQFQ epplied torque nlysis F F F F F F F F F F F F F F F F F F F F F F F VU QFR ypenEloop Qh steering ontrol F F F F F F F F F F F F F F F F F F F F F F F F VU QFRFI ypenEloop steering priniple F F F F F F F F F F F F F F F F F F F F F VV QFRFP hiretion ngle steering hrteriztion F F F F F F F F F F F F F F VW QFRFQ snlintion ngle steering hrteriztion F F F F F F F F F F F F F WR QFRFR hiretion ngle steering on stiky sustrte F F F F F F F F F F F WS QFS ummry F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F WS

General introduction

wiroroots hve the potentil for numerous pplitions rnging from in vitro tsks on lEonEEhip @eFgF trnsporting nd sorting miro ojetsY miroEssemly of mehniE l omponentsFFFA to in vivo pplitions for minimlly invsive mediine @eFgF trgeted drug deliveryY rhytherpyY hyperthermiFFFAD due to their miro sizes nd essiilE ity to tiny nd lustered environmentsF ome ontextsD suh s miro)uidi hnnels for in vitro tsks nd lood vessels or urinry systemD often imply tht the miroroots hve to swim in )uidF roweverD t the mirosleD the )uid eomes extremely visous nd the eynolds numer of the miroswimmers drmtilly lowersF he orksrew type rotting propulsion proposed y urell is one of the swimming tehnique suitle for swimming t low eynolds numersF wgneti relil miroswimmersD inspired y iF oli teriD re driven y rotting mgneti (eldD nd onvert the rotry motion to liner motion t low eynolds numersF por more thn ten yersD reserhers hve developed severl kinds of mgnetilly tuted helil swimmers with di'erent geometry prmetersD hed shpesD nd mgE neti positioningF roweverD the in)uene of the geometry prmetersD the hed shpes nd the mgneti positioning hs not een lerly demonstrtedF hereforeD the deE sign of the helil swimmers is still not (xedF es fr s we knowD the existing helil swimmers re primrily openEloop ontrolledD due to the omplexity of the ontrol of the mgneti (eld tuting the helil propulsionD nd the limited numer of feedk options with the required preisionF hereforeD the two mjor ojetives of this thesis reX

• to ompre the swimming performnes of helil swimmers with di'erent designs to further improve their design nd to hrterize their swimming propertiesF

• to relize visul servo ontrol of helil swimmersF sn order to hieve the ojetivesD sledEup helil miroswimmersD whih re de(ned s the helil swimmers with overll lengths higher thn 1 mmD ut still swim t the low eynolds numersD re proposed in this thesisF he )uid motion round the heliE l swimmer is only hrterized y the eynolds numerF hereforeD the swimming properties of helil swimmers should e the sme if they swim t the sme eynolds numersD whih mke dimensionless study possileF he sledEup helil miroswimE mers filitte the frition with di'erent designs nd the oservtion y visionF he sledEup helil miroswimmers used in the experiments re presented in ghpter PF he experimentl setupD inluding mgneti tution system nd vision system re lso presentedF he mgneti tution system is omposed of three orthogonlly rrnged relmholtz oil pirsF he generted rotting mgneti (eld is ontrolled y the supplied urrentsF he relmholtz oil pirs re di0ult to e sled upF hereforeD rotting permnent mgnet mnipultor is uilt for the igger version of the sledEup helil miroswimmersF he vision system is omposed of mer with n optil lens nd mer with n endosope onneted y oupler lens in order to solve the line of sight prolemF he ryenter nd the orienttion of helil swimmers re relEtime trkedF ome methods of the estimtion of the swimming hrE teristisD suh s rottion frequenyD trnsltion veloityD nd the propulsion mtrix of the helil swimmers re lso introdued in this hpterF he in)uene of geometry prmeters of the helil swimmers on their swimming performnes re studied in ghpter QF he design of the geometry prmeters of the helil swimmers is (xed fter the swimming performnes omprisonF henD the rottionl propulsion hrteristis of the helil swimmers with di'erent hed shpes nd mgneti positioning re studiedF he in)uene ftor on the rottionl propulsion hrteristis is disussedF he design of the hed shpe nd the mgneti positioning re then determinedF he design of the sledEup helil miroswimmers n e guideline for the frition of helil miroswimmersF he propulsion mtrix of the helil swimmer is identi(edF he propulsion liner veloity is predited using the estimted propulsion mtrixF et lst in this hpterD n openEloop steering of the helil swimmer is presentedF fsed on the steering hrteriztion of the helil swimmersD Qh steering strtegy is proposed to improve the e0ienyF he visul servo ontrol of the sledEup helil miroswimmers is disussed in ghpter RF he vision system with two mers is (rstly lirtedF e Qh pose estimtionD inluding the orienttion nd the position of the helil swimmers in the Qh spe is then estlishedF e ontrol lw is designed for the mgneti (eldF he visul servo ontrol for the orienttion of the helil swimmers is relized in the Qh speF es helil swimmers re nonholonomi systemsD pth following tsk is plnnedF he Ph pth following tsk on the horizontl plne is proposed sed on the estlished stte model of the helil swimmers nd the ontrol lw of the mgneti (eldF he visul servo ontrol of the Ph pth following is vlidted y following stright line with the initil position oth on nd o' tht lineF Qh pth following tsk in the spe is s well disussed for the perspetivesF wore omplited pths should e tested in future worksF yne vlidted y the sledEup helil miroswimmersD the visul servo ontrol n e pplied to helil miroswimmers y using the optil lenses suitle to the resolutionF ome perspetives for future works is s well disussed t lstF relil miroswimmers ple of propulsion t low eynolds numer hve the poE tentil to pply on miro)uidD lEonEEhipD nd minimlly invsive mediineF hue to the sling of physil e'etsD loomotion of miroswimmers is di'erentF rditionl frition nd powerEsupply methods eome unfesile t miroslesF sn this hpE terD stte of the rt out the helil miroswimmers is introduedD inluding the stte of the rt of their loomotionD fritionD tutionD loliztion nd visul servo ontrolF 1.1 Motivations wiroroots inlude the roots from I mirometer to I millimeter ghillet PHIQF wiE roroots re shown on sle rD with some intuitive smll ojetsD suh s n iF oli teriD red lood ellD nd n nt in pigure IFIF ntetheredD ontrollleD moile miroroots hve een proposed for numerous pplitions rnging from in vitro tsks on lEonEEhip to in vivo pplitions for minimlly invsive mediine eott PHHUF elthough the spei( pplitions of miroroots re still in n erly onept stgeD the diretion in whih the (eld is heding is rpidly solidifyingF pigure IFIX he miro nd nno sleF vet us summrize the pplition of miroroots from in vitro to in vivo ppliE tionsF 1.1.1 In vitro applications hue to their miro sizes nd essiility to tiny nd lustered environmentsD miroE roots n e used for vrious in vitro tsksD suh s mnipultion of miroEojetsF e helil miroswimmer of 38 µm in length ple of pushing nd rotting mirosphere of 6 µm in dimeter in )uid is demonstrted in hng PHHWF he mnipultion of miroEojets enles miroEssemlyF es for potentil pplitionsD dvned mnuE fturing relies on the miroEssemly of smll mehnil omponentsD using miroE gents mnipulting miroEojets in order to uild ssemled systemsF por exmpleD wiroEyptoEiletroEwehnil ystems @wyiwA sense or mnipulte optil sigE nls on very smll sle using integrted mehnilD optilD nd eletril systemsD therefore their frition requires preise miroEssemly systemF enother exmple is the fritionD pkging nd interonnetion of di'erent miroEomponents in the wiro iletroni wehnil ystem @wiwA where miroEssemly is vitlF wiroroots re lso suitle (elds for industril iotehnology nd mediine useF hey re promising tools for the tretment of iologil ells euse of their high throughput nd high repetilityF sndeedD miroroots operte t the sme sle s orgni ellsD llowing e0ient in vitro intertions in order to move nd sort ellsF sntegrtion of miro)uidi hip nd rootis sed on miroEeletromehnil systems tehnology is innovtion for iomediineF eurte motion ontrolD high propulsion powerD nd the pumping mehnism of motion permit the miroroot to lod multiple ojets nd trnsport them to desired lotions in the miro)uidi hip nhez PHIID rwng PHIQF hey perform dptle tsks on orpusles suh s lood ells sized from 6 -10 µmF e system onsists of severl miroroots is deE veloped in ruper PHHRD tht n utomtilly sort di'erent ells y trnsporting them to di'erent repositoriesF righEspeed mirorooti tution driven y permE nent mgnets in miro)uidi hip n output fore of the order of millinewtons rgiwr PHIPD rgiwr PHIQF hese mgnetilly tuted miroroots re pE le of delivering miroeds in mirohnnels peng PHIIF 1.1.2 In vivo applications sn vivo pplitions of miroroots is espeilly in minimlly invsive surgeryF winiE mlly invsive proedures re linked with vriety of ptientEoriented ene(ts rnging from redution of reovery timeD medil omplitionD infetion risksD nd postoperE tive pin to inrese qulity of reD inluding preventtive reF et the smll sles worldD miroroots hve the potentil to perform tsks tht re urrently di0ult or impossile endik IWWVD wk PHHID grrozz PHHQF ome lotions in the humn ody would eome ville for wireless interventionD inluding the irultory sysE temD the urinry systemD nd the entrl nervous systemD if we were le to rete miroroots with mximum dimension of only few millimetres or lessF wiroroots will undoutedly led to the development of therpiesF xelson lssi(ed the medil miroroot tsks in xelson PHIHF wiroroots n e used for trgeted therpy inluding trgeted drug deliveryD rhytherpyD nd hyperE thermiF rgeted drug delivery n inrese simultneously the drug onentrtion in region of interest nd redue the risk of side e'et in the rest of the odyF frhytherE py is the plement of rdiotive soure ner unwnted ells suh s tumorF he unwnted ells lose to the rdiotive soure n e killed y the rdited energyF ryperthermi refer to the lol delivery of het energy to destroy unwnted ellsF wiE roroots n lso e used to remove mteril y mehnil mensF wiroroots use rotry motion to perform ltion tskD for exmpleD in the removl of ftty deposits from the internl wlls of lood vesselsF wiroroots n t s simple stti strutures whose positions re ontrollleD suh s stents to keep vsulr pssgewys openD oE lusions to log lood vessel tht nourishes tumorD nd implnts like eletrodes for rin stimultionF wiroroots n lso e used to trnsmit informtions tht would otherwise e di0ult or impossile to otinF et the sme timeD their positions n e used to lolize the unknown internl phenomen suh s internl hemorrhgeF ome tsks require very simple struturesD mking them fesile nerEterm golsF pigure IFP summrizes the ove medil tsksF sn lots of ontextsD suh s in miro)uid hnnels for in vitro tsks nd in the lood pigure IFPX xumerous in vivo medil pplitions of miroroots xelson PHIHF vessels or urinry system for medil pplitionsD the miroroots hve to swim in )uidF he miroroots working in )uidE(lled lumensD tuesD nd vities re lled miroswimmersF sn this thesisD the reserh fous on the miroswimmersF 1.2 Swimming at low Reynolds numbers es the swimmers re sled down to the mirosleD the fundmentl physis governing the devies remins the smeD ut the reltive importne of physil e'ets hnges urell IWUUF sn this setionD the physis of the swimmers t the mirosle is introE duedF henD the loomotion suitle for swimming t the mirosleD inspired y the miroEorgnisms is presentedF 1.2.1 Reynolds number he eynolds numer is dimensionless quntityF ysorne eynolds de(ned the eynolds numer s the rtio of the inertil fores to the visous fores hundred yers goF e typil inertil stress is given y fernoulliElike dynmi pressureD whih n e expressed s σ inertial ∼ ρv 2 D nd leding to n inertil fore f inertial ∼ ρv 2 D where v is the veloityD ρ is the density of the )uidF e typil visous stress on rigid ody is given y σ viscous ∼ ηv/LD leding to typil visous fore on the ody of the form f viscous ∼ ηv/LD where L is hrteristi liner dimensionD η is the dynmi visosity of the )uid vug PHHWF hus the eynolds numers n e expressed s followsX

Re = inertil fores visous fores = ρvL η = vL ν @IFIA
where ν is de(ned s the kinemti visosity of the )uidF ine the hrteristi liner dimension is smllD swimming in mirosles is de(ned s swimming t low eynolds numerF he eynolds numers for some swimming miroorgnisms re listed in le IFIF sn wter @ρ ≈ 10 3 kg/m 3 D η ≈ 10 -3 Pa • sAD swimming teriumD suh s iF oliD moves with eynolds numer of 10 -5 -10 -4 F e humn spermtozoon trvelling round 200 µm/s moves with eynolds numer of 10 -2 F vrger ilites suh s rmeium with length of 100 µm trvelling t out 1 mm/s hs eynolds numer of 0.1F

-∇p + ∇ • T = ρ ∂ v ∂t + ρ( v • ∇) v @IFPA
where T is the stress tensor nd ρ is the )uid densityF he onservtion of mss n e expressed sX ∂ρ ∂t + ∇ • (ρv) = 0 @IFQA sn the se of the inompressile )ow of xewtonin )uidsD the inompressile )ow ssumes onstnt visosity nd densityF e simpli(tion of the xvierEtokes equtions is otined elowX

   -∇p + η∇ 2 v = ρ ∂ v ∂t + ρ( v • ∇) v ∇ • v = 0 @IFRA
where η is the onstnt dynmi visosityF et low eynolds numer regimesD the visous fores dominte efore the inerti foresF he inerti terms of the xvierEtokes equtions re thus negligileF he xvierEtokes eqution n e simpli(ed s followsX

η∇ 2 v = ∇p ∇ • v = 0 @IFSA
his eqution is only ext for Re = 0D ut it is onsidered s vlid pproximtion for Re 1F his eqution is liner nd independent of timeF hese properties led to some importnt onsequenes for loomotionD s we disuss elowF 1.2.3 Reciprocal motion and non-reciprocal motion es pointed out oveD the linerity nd timeEindependene of tokes iqution @IFSA leds to rteEindependene propertyF sf ody undergoes surfe deformtionD the distne trvelled y the swimmer etween two on(gurtions does not depend on the rte t whih the surfe deformtion oursD ut only on its geometryD whih mens the sequene of shpes the swimmer psses through etween these two on(gurtionsF en importnt onsequene of rteEindependent swimming t low eynolds numers is the soElled 4sllop theorem4X if the sequene of shpes displyed y swimmer deforming in timeEperiodi fshion is identil when viewed fter timeEreversl trnsformtionD then the swimmer nnot move on vergeF xote tht the ondition is not tht the motion e stritly timeEreversl invrintD with the sme forwrd nd kwrds rteD ut only tht the sequene of shpes is the sme when viewed forwrd or kwrd in timeF his kind of motion is termed s reiprol motionF eiprol motion nnot e used for loomotion t low eynolds numersF urell illustrted the futility of reiprol motion with the exmple of sllopF e sllop opens its shell slowly nd loses it fstD squirting out wterD to move hedF et low eynolds numersD the sllop moves hed y losing its shellD ut it9s k to where it strted y opening its shellF xo net displement n e retedF hereforeD nonEreiprol motions re required for net displement t low eynolds numersF pigure IFQX he PEhinged theoretil swimmer proposed y urell urell IWUUX @A he reiprol series of on(gurtions efgfe leds to k nd forth motion onlyD known s the 4sllop theorem4F @A he nonEreiprol series of ngle on(gurtions efghe retes net displement fter whole yleF sn his ppersD urell proposed simple exmple of nonEreiprol motion y theoretil twoEhinged swimmer omposed of three rigid links rotting out of phse with eh otherD whih is depited in pigure IFQF he two hinges o'er two hegrees of preedom @hopA nd the swimmer n go through series of ngle on(gurtionsF e reiprol motion is illustrted y pigure IFQD whih is series of on(gurtions efgfeF sf this sequene is reserved in timeD the new sequene is efgfeD whih is identil with the origin sequeneF he nonEreiprol series of on(gurtions efgheD s shown in pigure IFQD whih n e reversed in time into the series of on(gurtion ehgfeD results in net displement fter one yleF roweverD this is only hypothetil nonEreiprol motionF e hve not yet (gured out whether it exists in the ntureF sn miroEniml kingdomD there re t lest two other solutions to the prolem of swimming t low eynolds numersD whih is introdued in the next setionF 1.2.4 Locomotion of micro-organisms xture hs gret del to teh us out miroswimmer designF wiroEorgnisms re le to swim t low eynolds numer y generting nonEreiprol motionsF urell desried two tehniques tht miroorgnisms dopt to generte nonEreiprol motion in his highly ited pper 4vife t vow eynolds xumers4 urell IWUUX )exile or osilltion @pigure IFRA nd orksrewEtype rotting propulsion @pigure IFRAF pigure IFSX @A e spermtozo hs )exile tilF st stoks its til to generte )exile or osilltion to dvneF @A en imge of n iF oli yISUXrU terium with undle of )gellF he undle of )gell is driven into helil shpe to generte orksrewElike propulsionF he )exile or osilltion is used y some eukryotes suh s spermtozo @see pigure IFSA to propel themselvesF iukryoti )gell re tive orgnelles tht deform to rete pddling motionD suh s trveling wves or irulr trnslting movementsF he orksrewEtype rotting propulsion is used y some prokryotes suh s iF oli teri @see pigure IFSA ferg IWUQD ferg PHHRF iFoli ells onsist of rodEshped ody nd undle of pssive )gellF he undle of )gell is driven y moleulr rotry motor into helil shpe to generte orksrewElike motionF 1.2.5 Bio-inspired microswimmers e numer of rooti swimming methods hve een shown to work t reltively smll slesD ut will hve redued e'etiveness s size dereses to the mirosle euse they mke use of reiprol motion eott PHHWF rious miroswimmers designsD s shown in pigure IFTD inspired y the nture miroEorgnismsD use nonEreiprol motions n e dpted to mke e'etive motions t low eynolds numersF pigure IFTX @A plexile em tthed to mgneti hedF @A plexile mgneti miroEed tilF @A igid helil til (xed to mgneti hedF e )exile )gellum is the esiest mehnism inspired from the miroEretureF st is one dimensionl strutureF he hllenge is to frite simultneously )exile nd stle struture t mirosleF e )exile em n e tuted y uniform osillting mgneti (eld while tthed to mgneti hed quo PHHVF enother pproh to form )exile )gellum is to use selfEssemled mgneti edsF he mgneti fore etween mgneti eds is limited nd fst rottion of these ed hins n led to disssemly due to the )uidi drg fores welle PHHQF reneD hreyfus onneted the mgneti prtiles with hxe strnds hreyfus PHHSF he overll length of is pproximtely 24 µmF sn ftD hin of eds tuted with n osillting (eld does not result in nonEreiprol motionF roweverD tthing pylod to one end of the hin reks the motion symmetry of the trvelling wve long the ed hinF he trnsport of single red lood ell is demonstrtedF e kink in the hin lso reks the symmetry nd llows propulsion t mirosle oper PHHTF wore reentlyD )exile )gellum with ultured ontrtile rdiomyoytes in presented in illims PHIRD whih is ple of selfEpropulsion without externl mgneti (eldF he min hllenge of the )exile )gell is the motion ontrol with preisionF otting rigid helil )gell whih onvert the rotry motion to liner moE tionD is nother pproh of propulsion t low eynolds numers inspired y E teril propulsionF e entimeter sled helil til tthed to rotry motor is tested in the visous liquid to simulte the propulsion t low eynolds numers fehkm PHHSD fehkm PHHTF roweverD the motor is di0ult to e sled down to mirometerF hereforeD reserhers use mgnetilly tuted helil swimmersF ertiE (il rigid helil )gell tthed to mgneti heds with severl tens of mirometer re presented in fell PHHTD hng PHHWD rwng PHIHD whoney PHIID rigid heliE l til with mgneti oting is presented in qhosh PHHWF wore detils of these helil miroswimmers re presented in etion IFQFPF he struture of rigid helil miE roswimmers re more solid thn selfEssemled )exile )gellF he motion ontrol of the mgneti helil swimmers n e relized y the ontrol of the rotting mgneti (eldF e helil swimmer n e propelled forwrds or kwrds simply y reversing the diretion of rottionF hereforeD the helil )gell tuted y rotting (eld re more suitle for ontrol nd future pplitions in miroslesF yne of the min hllenges for helil miroswimmers is the frition of the Qh omplex strutures t the mirosleD whih is introdued s followsF 1.3 Helical microswimmers sn reent yersD helil miroswimmers hve eome growing re of interest for miroroot reserhersF sn this setionD the stte of the rt of the miroEfrition proess of the helil miroswimmers is introldued t (rstF henD the stte of the rt of the helil miroswimmers is presentedF et lstD the helil propulsion model is presentedF 1.3.1 Helical microswimmer fabrication process sn reent yersD thnks to the development of miroEfritionD numer of frition methods hve een found tht re ple of smllEsle helix fritionF he min hllenge for friting miroroots re the repetility of the proessD nd the onE trol over design prmetersF hree miroEfrition methods of helil miroswimmers is presentedF 1.3.1.1 Self-scrolling method he (rst funtionl helil miroswimmers is frited in PHHTD using the selfEsrolling tehnique of snqesGqes ilyer fell PHHTF vots of helil miroswimmers re fE rited sed on this selfEsrolling method rwng PHHWD hng PHHWD hng PHIHD rwng PHIHD rwng PHIIF he frition proess is illustrted shemtilly in pigE ure IFUF he initil lyers re grown on semiEinsulting qllium ersenide @qesA using woleulr fem ipitxy @wfiA system equipped with vlved rker for es nd solid soures for q nd sn @pigure IFUAF efter thin qes u'er lyerD 500 nm thik sri(il elqes lyer is depositedF e 15.6 nm thik lyer onsisting of sn 15 q 85 es is deposited on the top of the sri(il lyerF sn 15 q 85 es is qes with IS7 of q toms repled y sn tomsF e 15.6 nm thik lyer of qes is deposited on the top of the snqes lyerF snqes hs the sme struture s qesD ut its rystl mesh size is slightly superior thn tht of qesD euse sn tomi ry is superior to tht of qF et the intersetion of the ilyerD dpttion of eh mesh over the other genertes internl strinF pigure IFUX fsi frition proess sequene @EgAX initil plnr ilyer ptterned through onventionl mirofrition tehniquesD ssemles itself into QEh strutures during wet eth relese euse of the internl strinF wetl pds nd support rs re reted through negtive photolithogrphy nd liftEo' rwng PHHWF he initil pttern n e reted through photolithogrphy @pigure IFUAF efter the development of photoresistD etive son ithing @siA is used to trnsfer the pttern to rion @pigure IFUAF he rs to whih the struture (xed t the end re reted through liftEo' proess with negtive photoresist @pigure IFUdD eD fAF he 10 nm thik gr lyer serves s n dhesive lyerF he 100 nm thik eu lyer is used for eletril onttsF pinllyD P7 hydro)uori id @rpA queous solution is used to seletively eth the elqes sri(il lyer under the snqesGqes ilyer @pigure IFUgAF huring this wet ethingD the ptterned ilyer urled up releses the internl strin nd forms QEh struturesF efter the wet eth releseD the hips re rinsed in deionized wter nd susequently in isopropyl loholF hey re dried with superritil gy 2 dryer so tht the strutures would not e dmged from surfe tensionF he diretion of the srolling is determined y the nisotropy in sti'ness of the snqesGqes ilyerF pigure IFV illustrtes tht the srolling diretion of the rion is `IHHb on @HHIA qes wferF he rdius of the helix is djusted y tuning the thikness of the rionF e squre plte nikel hed is mounted on the helil (lment for mgneti tutionF he squre width nd thikness of the nikel hed n e djusted with the lithogrphy step nd metl deposition stepD respetivelyF 1.3.1.2 GLancing Angle deposition (GLAD) method sn PHHWD even smller helil miroswimmers re frited y qveh methodD whih is presented in qhosh PHHWF his method uses spheril seeds on whih helil pillrs re grown through the rottion of tilted stge during evportionF he proess is shown in pigure IFWF pigure IFVX @A snqesGqes ilyer is ptterned y photolithogrphy in rionElike shpe for the helil tilF @A he rionElike struture selfEforms helix tethered t one end y relesing the internl stress fter wet ething of the sri(il lyerF he red rrow indites the srolling diretion of the rionD iFeFD `IHHb on @HHIA qes wfer hng PHIHF he helil strutures re freed from the wfer y sonitionF hey re mgnetized y depositing olt lyer long one hlf of the helix nd permnently mgnetizing them perpendiulr to their helil xes in susequent stepF pigure IFWX @A qveh frited helil swimmersF illrs re deposited t n ngle nd under onstnt rottion of the stgeD resulting in helies on the spheril seedsF @A Qh lithogrphy with diret lser writing toolF @IA olymeriztion of the photoresist t the fol point of the lserF @PA development nd susequent rinsing nd @QA mgneti metl otingF eyer PHIQF en externl rotting mgneti (eld tutes the mgnet to rotteF he strength of the pplied mgneti (eld vries from 50 mT to 70 mTF he mehnism swimD propelled y wves trvelling long the spirlF he swimming veloity inresed linerly with the exittion frequeny nd depended on the shpe of spirlF eording to the swimming performnes tested t low eynolds numer @Re < 1AD reserhers predited tht the spirlEtype mehnism is suitle for swimming in the miroworldF 1.3.2.2 A magnetic screw capable of travelling through meat ome yers lterD the sme group developed mgneti srew ple of swimming through gel nd orgnisml orgn sshiym PHHHD sshiym PHHID sshiym PHHIF his mehnism onsists of xdpef mgnet @2 mm in dimeter nd 7.5 mm in lengthA nd rss tip with spirl shpe @4 mm in lengthAD s shown in pigure IFIIF he intensity of the pplied (eld vries from 4 mT to 10 mTF he mgneti srew rottes synhronously with the externl rotting (eld until ertin frequeny lled 4stepEout frequeny4F et this frequenyD the swimming veloity shows mximumF sn pigure IFIIX @A e mgneti srew swimming mhine designed y sshiym in PHHHF @A he mgneti srew is ple of trvelling through the ovine metF higher frequeny thn stepEout frequenyD the rottion of the mehnism n not synE hronize to the externl frequenyD the swimming veloity deresesF wimming perE formnes re tested in wide rnge of eynolds numers from 6 × 10 -7 to 430F he veloity of the mehnism depends on the environmentl mediF he mehnism n swim t mximum veloity of 0.022 mm/s t Re = 6 × 10 -7 F hey predit tht this mgneti srew n still e sled down to mirometerEsized mhineF he mehnism is lso exmined in the ovine met @see pigure IFIIAF he strength of the pplied mgneti (eld is 15 mTD nd the frequeny is 10 HzF hough the tion is not very smoothD the mehnism runs through the met t out 1 mm/sF he result suggests tht the helil mgneti mehnism hs gret potentil to run in the humn orgnsF 1.3.2.3 Articial bacterial agella of a size comparable to E. coli sn PHHUD the (rst helil swimmers of size omprle to iF oliD whih re ple of swimming in oth wter nd pr0n oilD lled 4erti(il fteril plgell @efpA4 re mde y group t ir urih fell PHHUF hey demonstrte some hrteristis of the efp lter hng PHHWD hng PHHWD hng PHIHD hng PHIHF efp onsists of helil til mde y selfEsrolled qesGsnqes ilyer nd thinEsqureEplte softEmgneti mde y grGxiGeu trilyer hed on one endD s shown in pigure IFIPF he length of efp is pproximtely 38 µmD the dimeter nd the thikness of helil til is pproximtely 2.8 µm nd 30 nmF he dimension of the squreEplte is 4.5 µm × 4.5 µm × 200 nmF he thikness of xi lyer is 180 nmF he eynolds numer of efp is in the rnge of 10 -4 F pigure IFIPX erti(il teril )gell mde y ir urih hng PHHWF he swimming loomotion of efp is provided y low strength mgneti (eld genE erted y three orthogonl eletromgneti oil pirsF he )ux density of the pplied (eld in the experiment vries from 1 mT to 2 mTF efp veloity inreses linerly regrding the frequeny until rehing mximum vlue nd then the veloity redues nd eomes less deterministi with the inresing (eld frequenyF his phenomen is onsistent with the previous lowEeynoldsEnumer experiments of the helil swimE mers t the millimeter sle sshiym PHHHF he mximum veloity is 18 µm/s with 2 mT (eldD whih is equivlent to HFS odyElength per seondF his mximum relE tive veloity of efp is not yet s high s iF oliD whih n swim IH odyElength per seondF efp n s well e steered in wter y pplied mgneti torqueF efp n hieve some simple tsksD suh s trnsltion nd rottion of mirospheres with 6 µm of dimeterF 1.3.2.4 Helical NanoBelt actuated by an electro-osmotic force e group t ss nd vx demonstrted tht eletroEosmoti propulsion of relil xnofelt @rxfA shows muh higher swimming performnes in terms of swimming veloity nd mnipultion fore rwng PHIHD rwng PHIIF iletroEosmosis propulE sion is presented in detil in etion IFRFIF rxf hs similr geometry s efpD helil til tthed to hed in tue formD s shown in pigure IFIQF he dimeters of the tue hed nd the til re 2.1 µmF he length of the hed is 12 µm nd the totl length of rxf is 74 µmF woreoverD 10 nm thik ferromgneti metl lyer is dded to the surfe of helil tilF heir entire surfe eomes funtionl to propulsion in omprison to the efp9s soft mgneti hed s the only funtionl prt nd the til s pssive prtF pigure IFIQX relil nnoelt @rxfA mde y ss in PHIH rwng PHIHF hese eletroEosmoti propelled rxf swimmers hieves speeds of PR ody lengths per seond under n externl eletri (eld of 239 V/mmF iF oli teri moves t pproximtely IH ody lengths per seond ferg PHHRF he heethD s the fstest lnd nimlD hs speed of 110 km/hD pproximtely PS ody lengths per seond y omprison rildernd IWSWF rxfs9 swimming diretions n e esily ontrolled y di'erentiting the (eld grdientF nusul nd disontinuous pumping propulsion is oserved from highEspeed mE er nlysesF he swimming performne ppers to depend highly on the surfe onditionsF 1.3.2.5 Articial magnetic glass nano-structured propellers sn PHHWD group t rrvrd niversity presented even smller helil swimmers with dimeter of 200 nm nd length of 2 µm using qveh method qhosh PHHWF he helil miroswimmer is shown in pigure IFIRF e permnent mgneti (lm with thikness of 30 nm is evported onto one side of the swimmerF hey demonstrte tht the mgneti mteril is not lwys restrited to the hedD ut my e diretly pplied to the surfe of the helil til s wellF pigure IFIRX iw imge of n individul glss srew with nnoEstrutured heliity qhosh PHHWF he helil miroswimmers n e propelled t speeds of out 40 µmD whih is equivlent to PH ody lengths per seondD with rotting mgneti (eld of 6 mT t out 150 HzF hese rti(il swimmers re not only similr in size to teril ellD ut they lso moves t the similr speedF he helil miroswimmers n drw urved line with preprogrmmed mgneti (eldF hey re lso le to push miroeds with dimeter of 5 µmF 1.3.2.6 Millimeter-scaled helical swimmers at the low Reynolds number e millimeterEsled helil miroswimmer using miro wire eletril dishrge mhinE ing is frited in whoney PHIID s shown in pigure IFISF he helil swimmer is mde from xitinolD with ylindril mgneti hedF he mgneti hed is permnent mgnet @nikelEplted xdpefAF he 5 mm long helix hs 3.5 turnsF he outer dimeter is 1 mmD the pith is 1 mmD the helix rion width is 0.2 mmF he helil swimmer is tuted y three orthogonlly rrnged relmholtz oil pirsF he helil swimmer swim in visous liquids to simulte the low eynold numE er regimeF st rottes t 2.23 Hz to lne its own weight of 8.9 mgF he uthor shows openEloop ontrolled swimming of the helil swimmerD whih enles 4E turn4 trjetoryF en imge sequene of the swimming is shown in pigure IFISF he pigure IFISX @A nning eletron mirogrph of the frited helixF @A smge seE quene of 4Eturn4 trjetory whoney PHIIF swimming veloity in the experiment is up to 0.35 mm/sD whih is equivlent to HFHU ody length per seondF 1.3.2.7 Helical microswimmers with object holder wore reentlyD ottori presents lw shped heded helil miroswimmers to trnsE port ojets in ottori PHIPF he helil miroswimmers re mde of polymerD nd frited y Qh diret lser writing rpid prototyping method presented in etion IFQFIF he helil miroswimmers with ojet holder is shown in pigure IFITF he dimeters of the helil miroswimmers vry from 1 µm to 8 µmF he lengths vry from 4 µm to 64.5 µmF he xiGi thin ilyers re deposited on the surfe of the polymer helil miroswimmers y eletron em evportion for mgneti tution nd improvement of surfe ioomptiilityF he nikel lyer is ferromgnetiD whih ontriutes to the propulsionF rnsporttion of 6 µmEdimeter polystyrene prtiles in puri(ed wter is demonE strted using individul helil miroswimmers with miroEholder frited onE jointly with the helil odyF he proedure of ojet trnsport n e seprted into four stgesX pprohingD lodingD trnsporting in Ph nd QhD relesingD s shown in pigure IFITF he miro prtile n e trnsported over one surfe nd steered to nother lower surfeF henD it n e trnsported k to the initil higher surfeF 1.3.2.8 Summary he spei(tions of the helil swimmers introdued presented ove re summrized in le IFPF hey re lso shown on sle r with the iF oli teri in pigure IFIUF he helil swimmers with n overll length eyond 1 mm re de(ned s sledEup miroswimmersD euse they re still in the low eynolds numer regimeF hey hve di'erent geometry prmetersD hed shpesD nd mgneti positioningF heir swimming performnes re di'erent s wellF roweverD no one hs demonstrted the in)uene of the geometry prmetersD hes shpesD nd mgneti positioning on the swimming performneF pigure IFITX @A olymer helil swimmer with lw shped holder hedF @A rnsE port proedureX @iA pprohingD @iiA lodingD @iiiAE@ivA trnsporting Ph nd QhD nd @vA relesing ottori PHIPF pigure IFIUX he helil swimmers in the literture shown on sle rF prom hereD one of the min ojetives of this thesis is to ompre the swimming performne of the helil swimmers with di'erent geometry prmetersD hed shpesD nd mgneti positioningD to (nd the in)uene ftors on the swimming performneD in order to further optimize the design of the helil miroswimmersF roweverD mking di'erent geometry prmeters nd hed shpes requires severl weeks of works nd high ostF sn this thesisD sledEup helil miroswimmers t the millimeter sle re used in visous liquidsD in order to filitte the frition nd oservtionF he reson nd the fesiility of the sledEup helil miroswimmers e disussed in detil in etion PFIFIF es the miroEfrition proess of the helil miroswimmers is msteredD the helil swimmers t the millimeter sle will e esily sled down in future worksF qry nd rnok qry IWSS developed in IWSS simpli(ed esistive pore heory @pA to derive n expression for the propulsive veloity of seEurhin spermtozoF p is very intuitive pproh for modelling slender ody dynmis t low eynolds numersF he underlying ssumption of p is tht the hydrodynmi fores re proportionl to the lol ody veloityD with the onstnt proportionlity eing de(ned s the oe0ient of resistneF p is used to determine the fores used y veloity on n in(nitesimlly smll length of helix y vighthill in vighthill IWUTF p tkes the veloity v of one of these smll segments of the helixD deomposes the veloity into omponents prllel nd perpendiulr to the segmentD nd reltes them to prllel nd perpendiulr drg fores ting on the segment with the di'erentil foresX

df ⊥ = ε ⊥ v ⊥ ds @IFTA df = ε v ⊥ ds @IFUA
where ε ⊥ nd ε re the visous drg oe0ientsD whih hve numer of empiril pproximtionsF o get the totl drg fores on the helil swimmerD the lol (lment fores df ⊥ nd df n e trnsformed into the ody oordinte system nd integrted long the whole length of the helixF por stedyEstte motionD the externl pplied fores nd torques on the helil swimmer hve to lne the propulsive fore nd torque reted y its rottionF hereforeD the reltionship etween the externl pplied fores nd torque with the liner veloity nd rottion speed re then estlishedF eordE ing to the integrtion long one diretion or three diretionsD two di'erent propulsion models re otined nd introdued in the following setionsF

a = 2πnσ ε cos 2 θ + ε ⊥ sin 2 θ sin θ @IFIHA b = 2πnσ 2 (ε -ε ⊥ ) cos θ @IFIIA c = 2πnσ 3 ε ⊥ cos 2 θ + ε sin 2 θ sin θ @IFIPA
where n is the numer of the turns of the helixD θ is the ngle etween the xis of the helix nd the diretion of srollingD σ is the ryon of the helixD nd the onstnt ε nd ε ⊥ re the visous drg oe0ientsF sn se of mgneti (eld tuted helil swimmerD the fundmentl ontrol input is the rottion frequeny of the mgneti (eldD the non)uidi pplied fore n e onsidered s the seond inputF he helil swimmer rottes in syn with the pplied (eldD nerly instntneously rhing n equilirium phse shift suh tht the mgneti torque ounterlnes the )uidi drg torqueF st is more instrutive in this se to rerrnge the liner equtions with the non)uidi pplied fore f nd the rottion veloity ω of the helil swimmer s the input vrilesF pigure IFIW shows the ehvior oserved with this type of swimmingF his PEhop xil model hs een used with suess to study the fesiility of helil swimming eott PHHW nd to hrterize si swimming properties hng PHHWD u PHIPF 1.3.3.2 3D helical propulsion model wgneti tuted helil miroroot re sed on the ssumption tht the miroswimE mers ehve similrly to teriD whih re pproximtely neutrlly uoyntF hereE foreD the reserhers pointed the helil swimmer in diretion of desired veloity rond IWWTD sshiym PHHHF roweverD mnEmde helil swimmers re typilly hevE ier thn their )uid mediumF hey sink down during the horizontl propulsion due to their own weightD resulting in veloity tht is not ligned with the thrust diretionF hereforeD reserhers pointed the helil swimmers in the diretion hving n ngle with the horizontl plne to overome its grvity hng PHHWF whoney proved in pigure IFIWX ulittive ehvior of helilEpropeller swimmingD with rottion freE queny s the ontrol vrile eott PHHWF his pper whoney PHII tht there is unique omintion of ngle nd rottion speed to hieve the desired veloityF sn order to (nd this omintion of ngle nd rottion speedD whoney etlished TEhop helil propulsion modelF vet the helix oordinte frme lie with the x h xis prllel to the entrl xis of the helixD nd the y h nd z h xis e rrnged s in pigure IFPHF he origin of this oordinte system is loted t the enter of the helixD denoted y O h F pigure IFPHX relix oordinte frmeF fy integrting the p long the three diretionsD the reltionship etween the fore f h nd torque τ h to trnsltionl veloity v nd ngulr veloity ω round O h is otined s followsX

f h τ h = A h B h B T h C h v ω @IFIQA whereX A h =   a h11 0 0 0 a h22 0 0 0 a h33   , B h =   b h11 0 b h13 0 b h22 0 0 0 b h33   , C h =   c h11 0 c h13 0 c h22 0 c h13 0 c h33   @IFIRA
A h D B h D nd C h onstitute the propulsion mtrixF he prmeters of the propulsion mtrix depend on the shpe nd the size of the helil propeller nd the visosity of the )uidD whih is omputed in whoney PHIIF sn this thesisD only the Ih helil propulsion model is usedF he prmeters of the Ih propulsion mtries of the helil swimmers used in the experiments re identi(ed in ghpter QF 1.4 Microrobots actuation sn this setionD the tution of the miroswimmers is disussedD speilly the mgneti tution methodF henD stte of the rt of the mgneti tution systems is presentedF 1.4.1 Actuation method of microrobots he storgeD hrvest nd trnsmission of power is ruil point in wireless miroroot tutionF ithout ddressing power from the eginningD it is di0ult to trnsition from prototypes to fully miniturized wireless deviesF here re two min tution method to the miroswimmersX the onord power nd the trnsmitted powerF 1.4.1.1 Onboard power ftteries n o'er inexpensive power soureD ut the totl deliverle energy sles down with volumeF hereforeD they re not promising for mirorootsF wiwEsed power genertors provide higher energy densities thn trditionl genE ertors nd tteriesF everl reserhers onvert vrious types of energy into eletril energy toson PHHQF ome use onord hemil fuels s well s gluose fuelD nd others hrvest mehnil energy from the environment in the form of virtionsF he operting environmentD suh s temperture nd pr is onstrinedF e temperture grdient retes voltge di'erentilD whih mkes the use therml energy nother potentil option hilvo IWWWF snsted of generting the loomotion for mirorootD wrtel uses diretly miroE orgnisms tht re le to swimD suh s swrm of wgnetotti teri @wfA wrtel PHHWF hey n e lolized nd ontrolled y wgneti esonne smging @wsA systemF rious strtegies re proposed to funtionlize the wfD inluding ttking nnoprtiles to the wfD pushing or pulling n tthed miroEojet with one or more wfD or using swrm of wf to move n untthed ojetF ghllenging issues suh s ontrol of the orienttion nd genertion of su0ient fores must still e ddressedF 1.4.1.2 Transmitted power en lterntive pproh is to design no power storge or genertion systemD ut wireless power trnsmission system to tute the miro devieF wo solutions re introdued elowX the eletroEosmoti fore nd the mgneti foreF Electro-osmotic forceX rwng used eletroEosmosis fore to move the helil swimmers rwng PHIIF iletroEosmosis relies on the priniple thtD on the surfe of ny ody plunged in liquidD lol hemil retions hppen etween solid nd liquidF hey drive to displement of eletri hrgesD whih results in opposed hrges ppering on eh side of the thinElyer surfe of the solidF en eletri potentil is reted owing to the presene of this thin interfil lyer nmed tern lyerD known s the zet potentil ζF es soon s the potentil estlishedD pplying n eletri hg (eld triggers reltive motion of opposed eletri hrgesF he result is )uid movement @pump e'etA if the solid wlls re tthedF ytherwiseD movement of smll ojets while )oting over the liquid is oservedF pigure IFPI presents the eletroEosmosis propulsion prinipleF he eletroEosmoti propulsion is very e0ient rwng PHIIF roweverD the use of eletroEosmosis fore is limitedF st n only e pplied for moving miroEsize ojetsF pigure IFPIX iletroEosmosis propulsion for helil swimmer rwng PHIIF Magnetic force and torqueX usiEstti nd lowEfrequeny mgneti (eld is nother pproh to pply fores nd torques diretly to mgneti mterils t disE tne without the need of ny tethersD or diret onttF woreoverD the humn ody is trnsprent to mgneti (eldD whih mens the mgneti permeility of the humn ody is pproximtely the sme s tht of n ir vuumF here re no signi(nt intertion of tissue with lowEfrequeny mgneti (eldsF ell mgnetized ojets re exerted oth fore nd torque within n externlly imposed mgneti (eldF he mgneti fore nd torque developed on mgnetized ojet re expressed s follows tiles IWWVX

-→ f m = Vm ( - → m • ) - → B dV m @IFISA -→ τ m = Vm - → m × - → B dV m @IFITA
where V m is volume of the mgnetized ojetD -→ B is the )ux density of the pplied (eld @T )D -→ m is the mgnetiztion of the ojet @A/m )F st is lso possile to desrie the pplied mgneti (eld y the mgneti (eld intensity

- → H @A/m )D where - → B = µ 0 - → H nd µ 0 = 4π × 10 -7 T • m/A
is the permeility of the free speF he mgnetiztion of the ojetD whih generlly vries ross the odyD n e modelled s onstnt throughout the ody with vlue equl to the verge mgnetiE ztionF e often onsider the totl dipole moment -→ M of mgneti odyD whih is the produt of the mgneti ody volume nd the verge mgnetiztionF he mgneti fore nd torque s funtion of the dipole moment n e expressed sX

-→ f m = ( -→ M • ) - → B @IFIUA -→ τ m = -→ M × - → B @IFIVA
prom the reltionshipD we get tht the mgneti fore is proportionl to the grdient of the mgneti (eldD whih t to move the ojet in the (eld to lol mximumD nd the mgneti torque is proportionl to the mgneti (eldD whih t to lign the mgnetiztion of n ojet with the (eldF he tution methods for miroswimmers re summrized in le IFQ with their dvntges nd disdvntgesF sn this thesisD we need n tution method pplile for swimmers t the millimeter sleD whih is le to e sled down to the mirosleF he tution of the swimmers should e esily ontrolledF hereforeD the mgneti tution method is hosen s the tution method for the helil swimmers in this thesisF xextD the mgneti tution systems re presentedF le IFQX ummry of the tution methods for the miroswimmersF is the use of eletromgnetsF wxeil designed mgneti stereotxis systemD whih onsisted of three orthogonl pirs of oils in 4helmet4 on(gurtionD to ontrol mgE neti (elds throughout humn hed wxeil IWWSF rowever this design llowed only prtil ontrol euse full torque nd fore ontrol is not possile owing to the singuE lrities in the workspeF esin used omintion of uniform mgneti (eld generted y relmholtz oil pir nd mgneti grdient generted y wxwell oil pir to ontrol miroroot with elliptil shpe esin PHHTF he oil pirs re rigidly onnetedD nd n roE tte round the workspe to ontrol the orienttion nd diretion of the mgneti mirorootF he system llows P hopD one in trnsltion nd one in rottionF uummer of ir group designed eletromgneti system nmed ytowgD whih ontins eight eletromgnets @see pigure IFPPAF ytowg system enles SEhop @QE hop position nd PEhop pointing orienttionA wireless mgneti ontrol of fully unE tethered miroroot uummer PHIHF pigure IFPPX @A rel imgeD nd @A onept imge of ytowg eletromgneti system for the ontrol of introulr miroroots uummer PHIHF wny reserhers use threeEpirErelmholtz oils orthogonlly rrnged @see pigE ure IFPQA to generte uniform rotting mgneti (eld t the workspe hng PHIHD whoney PHIID willer PHIPF hree orthogonl relmholtz oil pirs llows QEhop onE trol in rottionF enother devie tht mny reserhers use is position ontrolled roE tting permnent mgnet @see pigure IFPQA pountin PHIHD whoney PHIID u PHIPF he otting permnent mgnet n e mounted on TEhop rooti mnipultor or simple trnsltorF hree orthogonl relmholtz oil pirs nd rotting permnent mgnet re disussed in detil in ghpter PF 1.5 Microrobots localization por the losedEloop motion ontrol of helil miroswimmersD the min hllenge is to lolize nd trk the helil swimmers positionF rious methods hve een proposed to lolize nd trk mirorootF ptil resolution is n importnt ftor in the pigure IFPQX @A hree relmholtz oil pirs orthogonl rrnged whoney PHIIF @A e rotting permnent mgnet mounted on skw wotomn wrS TEhyp rooti mnipultor whoney PHIPF determintion of miroroots positionsF he mximum rte t whih the position estimtion n e updted is lso n importnt ftor for relEtime ontrolF inergy e'etiveness should e s well onsideredF 1.5.1 Electromagnetic and magnetic tracking he priniple in eletromgneti trking is the use of pir of deviesX one ts s (eld genertorD the other ts s sensorF he genertor emits lowEfrequeny eletromgneti (eld tht indues voltgeD whih is s funtion of distne nd orienttionD on the pikEup oils of the sensorF he ury of these systems depends on the presene of other mgneti ojets in the environmentF wny ftors need to e onsideredD suh s mteril propertiesD shpeD size nd position these ojets reltive to the (eld genertor or sensorD s well s the frequeny used for loliztionF wgneti trking is n lterntive methodology for wireless medil mirorootsF e mgnet is enpsulted in the devie to e trkedD nd n rry of externl mgE netoresistive sensors re used to mesure the mgnet9s (eld ru PHHTF en ssumption is tht the mgnet ehves s mgneti point dipoleD whose (eld in the Qh spe is disussed in etion PFQFIF he ssumption is vlid when the mesurement is performed t lrge distne ompred with the ody length of the mgnetF wgneti trking is promising when there re lineEofEsight prolemsD euse humn ody is trnsprent to stti nd lowEfrequeny mgneti (eldF glirtion is usully required efore the tul implementtion of the trking lgorithm to minimize the e'et of kground noiseF 1.5.2 Magnetic Resonance Imaging (MRI) e linil wgneti esonne smging @wsA system is powerful non invsive imgE ing tehnique the hs plyed nd will ply n importnt role in the medil ommunityF e linil ws hs the ility to generte lrge mgneti (eldsF perromgneti ojetsD or mgnetotti teri re trked nd nvigted under relEtime wsF he typil (eld strength is 1.5 T or stronger xelson PHIHF he ws systems n lso e used to provide miroroots loomotion pilitiesF ulmur present in his pper ulmur PHHS the loliztion of groups of nnoprtiules in n ws systemF wrms of mgnetotti teri re lolized s single ojets in wrtel PHHWF he sptil resolution of the ws system is simiE lr to tht of the erly multislie gomputed omogrphy @gA system nkin PHHSD rehing 0.25 mm × 0.25 mm × 1.5 mm in 3.0 T ws mhine hik PHHSF sn the sme workD n isotropi resolution of 0.57 mm is hievedD ut the sn time is 25 sD whih is too long for mirorooti pplitionsF he men error of 3.0 T ws mE hine is less thn 1.0 mm ezmi PHHQF fesides the limittion of the resolution of ws mhineD nother drwk of ws loliztion is the limited hoie of mteril for frition of miroroots ghinzei IWWWF perromgneti ojets use imge rtifts tht re sometimes lrger thn the ojet to e lolized pelfoul PHHRF sn the se of mgnetilly tuted mirorootsD n dditionl mgneti (eld grE dient for propulsion or rottionEdedited rotting mgneti (eld would e requiredF roweverD mgneti (eld grdientEsed propulsion or rottion due to rotting mgneti (eld would e lrgely limited when ws hs to e used with n oservtion devieD euse of the on)it etween the imging nd propelling mgneti (eld rwng PHIHF his prolem n e voided y using nonEmgneti oservtion devieF 1.5.3 Computed Tomography (CT) and uoroscopy gomputed tomogrphy is medil imging proedure tht uses omputerEproessed E rys to produe tomogrphi imges or 9slies9 of spei( res of the odyF he ptient is pled etween n Ery soure nd detetor rryF e set of projeted Ery imges gthered from vrious view points is used to reonstrut the interior of the (nl imge in QhF ixel resolution n reh 0.8 mm for 512×512 pixel mtrix pitton PHHUD nd n isotropi pixel size of 0.195 mm is hieved reently hurye PHHVF g sns re not well suited for imging regions of soft tissue surrounded y lrge volumes of highEdensity mteril suh s oneD euse of the underlying proess of imge reonstrution sed on rdition ttenutionF pluorosopy lso uses n Ery soure nd )uoresent reeiverF righEresolution Ph rdiogrphi imges n reh out 100 µm pixel resolution entonuk PHHHF foth g nd )uorosopy re suitle for in vivo pplitionsD nevertheless the rdition dosge tht potentil ptient reeives should e tken into onsidertionF iven more ruil is the dosge reeived y the medil st'D whih n e redued @ut not nelledA thnks to teleEmnipultion tehniquesF 1.5.4 Ultrasound por loliztion in soft tissueD only ultrsound omines good resolutionD miniml dE verse helth e'etsD high speedD sfetyD dequte frme rtes @up to IHH frmes per seondA u PHHSD nd low ost ilkomerson IWWUF he mjor drwks of ultrE sound re relted to low signlEtoEnoise rtio nd the presene of strong wve re)etors suh s ones nd ir poketsF hese my produe rtifts in ultrsound imges or shield n ultrsound signlF he disdvntge of ultrsoni imging my e prtilly overome through pssive loliztion tehniquesD sed on ultrsound trnsmissionF sn pplitions of the loliztionD frequenies up to 3 MHz re resonleD resulting in resolution of out 500 µm nd penetrtion depth of 15 -20 cm ilkomerson IWWUF he loliztion of wireless medil miroroot n e omplished with miE roroot tht ts s n emitter y rrying n ultrsound trnsduer onordF e ntilever or ny other mehnil struture n t s suhD virting t its resonnt frequenyF he emitted signl is mesured y set of reeivers pled on the surfe of the ptient9s odyF hrough the use of ws dt gthered o1ineD loliztion ury n e gretly improved plukiger PHHUF 1.5.5 Vision ision is useful rooti sensor sine it mimis the humn sense of vision nd llows for nonontt mesurement of the environmentF he imges re trnsmitted optilly y mers in omintion with optil lensesF yviouslyD vision n not e used for trkE ing miroroots in ptient9s ody euse of lineEofEsight prolemsF he illuminting onditions hve to e onsidered s wellF roweverD vision hs severl dvntges ompred with other imging methodF pirstlyD high sptil resolution n e hieved y mirosopi lensesF he resolution is limited y the light di'rtionF hereforeD resolution of 200 nm n e rehedF he limittion n e overome y using mer in omintion with iwD s shown in ievers PHHSF ome miroroots re tuted y mgneti (eldF enother dvntge of vision is to void the on)it etween the imging nd the tuting mgneti (eldF hirdlyD gret energies re required to generte mgneti (eld or to emit rdition for wsD g nd ultrsoundF st is not the se for visionD whih is energyEsving methodF 1.5.6 Summary he loliztion methods of the miroroots re summrized in le IFR with their dvntges nd disdvntgesF sn this thesisD the experiments re ll in vitro experiE mentsF he non invsive imging tehniques or mini invsive imging tehniques re not neessry to e tken into onsidertionF he min riteri re the simpliity of use nd the resolution of the systemF es resultD vision is good ndidte for the loliztion of the helil swimmersD due to the good resolution of the vision system with suitle optil lensesD nd the ese of useF he vision system n e implnted y mer onneted to omputer with suitle optil lensF xotie tht the vision method is not promising for in vivo pplitionD whih requires non invsive imging tehniquesF he losedEloop motion ontrol of the miroroots with vision feedk refers to visul servo ontrolD whih is lso lled visul servoingF he stte of the rt of visul servoing is introdued in the next setionF le IFRX ummry of the loliztion methods for the mirorootsF isul servoingD lso lled visul servo ontrolD is the fusion of results of mny elemenE tl res inluding highEspeed imge proessingD omputer vision nd ontrol theoryF et the erly stgeD typilly visul sensing nd mnipultion re omined in n openEloop systemD 4looking4 then 4moving4F he ury of the resulting opertion depends diE retly on the ury of the visul sensor nd the root endEe'etor ruthinson IWWTF en lterntive to inresing the ury of these susystems is to use visulEfeedk ontrol loopF e losedEloop position ontrol for root endEe'etor y mhine viE sion referred to s visul servoingF he term 4visul servoing4 ppers to hve een (rst introdued y rill nd rk rill IWUW in IWUW to distinguish their pproh from erlier experiments where the system lternted etween piture tking nd movingF sn IWVHD nderson nd eiss nderson IWVH introdued txonomy of visul servo systemX

• hynmi lookEndEmove systemsX he vision system provides input to root ontroller tht then uses joint feedk to internlly stilize the rootF

• hiret visul servo systemsX he visul ontroller diretly omputes the input to the root joints nd root ontroller is elimintedF es pointed out y ruthinson ruthinson IWWTD nerly ll of the reported systems used the (rst pprohD dynmi lookEndEmove systemF here re two merEroot on(gurtions for visul servoingX eyeEinEhnd on(guE rtion nd eyeEtoEhnd on(gurtionF he eyeEinEhnd on(gurtion onsists in mer mounted on the end e'etor of the rootD s shown in pigure IFPRF sn the eyeEtoEhnd on(gurtionD the mer is (xed in the workspeD s shown in pigure IFPRF es reE grds s the pplition for mirorootsD it is impossile to mount visul sensor on miroroot onsidering the sizeF ynly the eyeEtoEhnd on(gurtion is onsidered for mirorootsF pigure IFPRX opX @A iyeEinEhnd on(gurtionF @A iyeEtoEhnd on(gurtionF fotE tomX ypposite imge motion produed y the sme root motionF ghumette PHHU 1.6.2 Basic components of visual servoing he im of visionEsed ontrol shemes is to minimize n error e(t)D whih is typilly

de(ned y ghumette PHHTX e(t) = s( c p(t)) -s * @IFIWA
where s * is the desired vlues of the feturesD s is the urrent vlues of the feturesD nd c p(t) is the reltive pose etween the mer nd the ojet t instne tF hen the ontrol vriles re de(ned in the mer frmeD the vrition of the visul feture s relted to the reltive movements etween the mer nd the sene is given yX

ṡ = ∂s ∂ c p ∂ c p ∂t + ∂s ∂t = L s • c v o + ∂s ∂t @IFPHA
where L s ∈ R k×6 is the visul fetures intertion mtrix tht represents the di'erentil reltionship etween the fetures s nd the mer frme ruthinson IWWTF c v o = v c -v o presents the reltive instntneous veloity etween the mer frme nd the ojet frme expressed in the mer frmeF v c is the instntneous mer veloity nd v o is the instntneous ojet veloityF ∂s ∂t presents the vrition of s due to the ojet own motionF sn the se tht the ojet is rigid @no deformtionsAD ∂s ∂t disppersD thenX ṡ = L s v c @IFPIA sing @IFIWA nd @IFPIAD we otin the reltionship etween mer veloity nd the time vrition of the errorX

ė = L e v c @IFPPA
where L e = L s F sing lssil ontrol lw suh tht the visul error e is expeted to onverge to zero exponentilly @ ė = -λeAD the reltionship eomes s followX

v c = -λL e + e @IFPQA
where L e + ∈ R 6×k is hosen s the wooreEenrose pseudoEinverse of L e D tht is

L e + = (L e T L e ) -1 L e T @IFPRA
when L e is of full rnk TF his hoie llow ė = -λL e L + e e nd v c to e minimlF hen k = 6D if det L e = 0 it is possile to invert L e D giving the ontrolX v c = -λL -1 e e @IFPSA sn rel visul servo systemD it is impossile to know perfetly in prtie either L e or L + e F hereforeD n pproximtion or n estimtion of one of those two mtries must e relizedF foth the pseudoinverse of the pproximtion of the intertion mtrix nd the pproximtion of the pseudoinverse of the intertion mtrix re noted s L + e F he ontrol lw n e expressed s followsX v c = -λ L + e e @IFPTA his is the si design implemented y most visul servo ontrollersF isul servoing shemes minly di'er in the wy tht s is designedF sn the following setionsD we introdue two di'erent pprohesX @IA n smgeEfsed isul servoing @sfA in whih s onsists of set of fetures tht re immeditely ville in the imge dtY @PA oseEfsed isul servoing @fA in whih s onsists of set of Qh prmetersD whih must e estimted from imge mesurementsF 1.6.3 Image-Based Visual Servoing (IBVS) sn sfD the error is de(ned diretly in terms of Ph imge feture prmetersF isul fetures n e seleted in Ph imge spe s point oordintesD prmeters representing stright lines or ellipsesD region of interestD ontoursF sn ll sesD the prmeters de(ning the internl mer lirtion re requiredF pirst dvntge of sf is tht the positioning ury is roust to errors in mer lirtions nd imge noiseD sine the fetures re diretly expressed in the imge spe ruthinson IWWTD ghumette PHHTF enother dvntge of sf is tht the trget n e onstined to remin visile during the exeution of tsk if oth the initil nd desired imge feture lotions re within the mer (eld of view ghumette IWWVF roweverD sf is not without its shortomingsF pirstlyD it is hrd to predit the trjetory of the end e'etor nd root my reh its joint limits wrey PHIHF purE thermoreD the trjetories of the end e'etor re often suoptiml ghumette PHHTF pinllyD usul sf is only lol symptotilly stle nd my fil in the presene of lrge displement to relize ghumette IWWVD gerver IWWWD in whih se the mer my ross singulrity of the intertion mtrix ghumette PHHTF 1.6.4 Position-Based Visual Servoing (PBVS) sn fD fetures re extrted from the imge nd used to ompute or estimte the reltive pose etween the ojet nd the mer frmes in grtesin spe or to reonstrut the Qh oordintesF gomputing tht pose from set of mesurements in one imge requires the mers intrinsi prmeters nd the Qh model of the ojet oservedF gompre to sfD f hs glol symptoti stility of ontroller to stilize the pose of the mer from ny initil ondition if Qh estimtion is perfetF f llows the mer to follow theoretilly n optiml trjetory in the grtesin speD ut generlly not in the imge speF irrors in lirtion nd imge mesurement propgte errors in the Qh worldF iven smll errors in the imge mesurements n led to errors in the pose tht n impt signi(ntly the ury of the system ghumette PHHTF purthermoreD there is no ontrol in the imge whih implies tht the trget ojet my leve the mer (eld of view wlis IWWWF 1.6.5 Visual servoing for microrobots wiroroots re inresingly lled upon to perform more mnipultion tsksF roweverD these opertions re primrily openEloopD due to the limited numer of feedk options with the required preision urtohvil PHHWF wnipultions re often performed in onjuntion with devies suh s optil mirosopesD or iwD whih ll provide sensory feedk in the form of monoulr imgeF yne mjor hllenge of performing visul servoing tsks for miroroots is the imge qulityF elEtime visul feedk from mirosopes is typilly noisy nd lurred when oserving miroroots in highEspeed motionD s shown in pigure IFPSF st retes signi(nt hllenges for n imge proessing systemF he mjor hllenge of performing visul servoing tsks inside iw is lning the needs of imge qulity with relEtime imgingF he sequentil snning used to rete iw imge neessittes lower frmertes thn those ville with optil mersF ome visul servoing tsks for miroots with simple geometries hve een relisedF por exmpleD wikw desried oservtion system for monitoring miro mss xis lignment systemD nd ontroller sed on the visul servoing method for ontrolling miro mhine wikw IWWTF he system n e ontrolled sed on visul inforE mtion on imge plnes rther thn in Qh speF he mislignment error of optil (res re deresed due to the visul servo ontrollerF pigure IFPSX smge sequene of helil miroswimmer in motion hng PHHWF geh model sed methods hve een pplied to help more preise motion feedk for miroroots with more omplited geometriesF urtohvil used to trk miroE gripperD whih is shown in pigure IFPTD in relEtime inside iw urtohvil PHHWF mdzte proposed utomted mirossemly system using Qh poseEse visul servo ontrolD supplied y geh modelEsed trking lgorithm mdzte PHHWF he ojets for ssemly re shown in pigure IFPTF pigure IFPTX @A wirogripper trked inside iw urtohvil PHHWF @A wiroEojet for ssemly mdzte PHHWF roweverD s fr s we knowD the relEtime visul servoing tsks hs not een pE plied for helil miroswimmersD due to omplexity of the ontrol of the mgneti (eld tuting the helil propulsionD nd the low imge qulity reted y the omplexity of the geometry nd the high speed rottion @see pigure IFPSAF hereforeD the other min ojetive of this thesis is to relize visul servo ontrol of the helil swimmersF 1.7 Overview of thesis es presented in the hpterD helil miroswimmers ple of helil propulsion inE spired y iFoli teril swimming ehviors re promising for swimming t low eynolds numersD euse the helil propulsion is nonEreiprol motionF he helil miroswimmers hve gret potentil for numerous pplitions rnging from in vitro tsksD suh s miroEmnipultion nd miroEssemlyD to in vivo tsksD suh s trgeted drug deliveryD if the motion of the helil miroswimmers is ontrollleF he mjor hllenge is to relize losedEloop motion ontrol of the helil miroswimE mersF sn order to relize the ontrolD the hrteriztion of the swimming properties is ruilF everl mgnetilly tuted rti(il helil swimmers hve een proposed nd presented in this hpterF hey hve di'erent geometry prmetersD hed shpesD nd mgneti positioningF end s fr s we knowD none of the proposed helil swimmers hs een ontrolled y loseEloop ontrollerF hereforeD the two mjor ojetives of this thesis reX

• to ompre the swimming performnes of helil swimmers with di'erent designs to further improve their design nd to hrterize their swimming propertiesF

• to relize visul servo ontrol of helil swimmersF ledEup helil miroswimmersD whih re de(ned s the helil swimmers with overE ll lengths higher thn 1 mmD ut still swim t the low eynolds numersD re proposed in this thesisF he sledEup helil miroswimmers filitte the frition nd lolE iztion y visionF he investigtion of the swimming properties of the di'erent helil swimmers re presented in ghpter QF he reliztion of visul servo ontrol of the helil swimmers is demonstrted in ghpter RF he detiled pln of this thesis is desried s followsX ghpter P is n overview of the systemD inluding the helil swimmersD the exE periment setupD nd some methods of the swimming hrteristis estimtionF he spei(tions nd the frition of ll sledEup helil miroswimmers used in this thesis re presentedF he experimentl setup inludes mgneti tution system nd vision systemF he mgneti tution system is desried from the design to the genertion of the ontrolled mgneti (eldF e relEtime trking method with the ryenter nd the orienttion is presented s wellF he methods of the estimtion of the swimming hrteristisD suh s rottion frequenyD trnsltion veloityD nd the propulsion mtrix of the helil swimmer is introdued t lstF sn ghpter QD the in)uene of the geometry prmeters of the helil swimmers on their swimming performnes is studiedD in order to further improve the design of the helil swimmersF henD the rottionl propulsion hrteristis of the helil swimmers with di'erent hed shpes nd mgneti positionings re investigtedF he design of the helil swimmers is disussed oth t the mirosle nd the millimeter sleF he propulsion mtries of the helil swimmer in di'erent visous liquids re identi(edF he liner propulsion veloity is simulted sed on the estimted propulsion mtrixF et lstD n openEloop steering ontrol of the helil swimmers is presentedF he steering of the helil swimmers hve not een extensively studied in the litertureF roweverD the steering is very importnt to relize Qh motion of the helil swimmersF e strtegy of steering is proposed to improve the steering e0ienyF sn ghpter RD the lirtion of the mer systems is t (rst presentedF henD pose estimtionD inluding the orienttion nd the position of the helil swimmers in the Qh spe is presentedF he ontrol lw of the tuting mgeti (eld is estlishedF he visul servo ontrol for the orienttion of the helil swimmers is relized in the Qh speF th following tsks re plnned for helil swimmersF he Ph pth following tsk on the horizontl plne is relized y following stright line with the initil position oth on nd o' tht lineF Qh pth following tsk in the spe is s well disussed for future pplitionsF ome perspetives re disussed t lstF Chapter 2 Overview of the system sn this hpterD the helil swimmers used in the experiments is presentedD s well s the visous liquidsF he mgneti tution systems used re the three orthogonlly rrnged relmholtz oil pirs nd rotting permnent mgnet mnipultorF he onE trol of the strength nd the rottion frequeny of the rottion mgneti (eld generted y the mgneti tution systems re presented respetivelyF he visul trking of the helil swimmers with its ryenter nd xis on the Ph imge is then introduedF et lstD the methods to nlyse the rottion frequeny nd the swimming veloity of the helil swimmers is presented in detilD whih is useful to the hrteriztion of the swimming property of the helil swimmersF 2.1 Scaled-up Helical Microswimmers (SHMs) es presented eforeD in this thesisD the helil swimmers used in the experiments re the sledEup helil miroswimmers t the millimeter sleF sn this setionD the reson nd the fesiility of using the rws is presented in detilF he design nd frition of the di'erent rws is lso desriedF 2.1.1 Why the scaled-up helical microswimmers? he ontrol of helil miroswimmers hs not een lerly relized minly due to the limited oservtion tools to hrterize the swimming hrteristis nd to identify the propulsion mtries in the mirosleF p to nowD di'erent helil miroswimmers hve een proposed s introdued in etion IFQFPF hese helil miroswimmers hve di'erent geometry prmeters @dimeterD pithD widthD numer of turnsAD di'erent hed shpes inluding squreEplteD ylinder nd sphereD nd mgneti positioningF nderstnding how these geometry prmetersD hed shpes nd mgneti positionE ing in)uene the swimming performne llows improvements in helil miroswimmers designF wking di'erent geometry prmeters nd hed shpes requires severl weeks of work nd high ostF st is then not suitle to modify the geometry prmetersF e then hve the ide to sle the helil miroswimmers up to the millimeter sleF he rws re fesileD euse the )uid motion round the helil swimmer is only hrterized y dimensionless prmeterX the eynolds numer @see etion IFPFIAF hereforeD the propulsion ehviors of helil swimmers should e the sme if they swim t the sme eynolds numersD whih mke dimensionless study possileF o emulte t n upperEsle the environment of mirosle swimmers swimming t low eynolds numersD one n use more visous liquidD suh s glyerolF yne the optiml design for the helil swimmers t the millimeter sle is determinedD it n e pplied s guideline to the frition of the helil swimmers t the mirosleF he heE lil swimmers n e sled down to the mirosle without di0ultiesD euse the miroEfrition proess of helil miroswimmers is mstered s presented in etion IFQFIF sn dditionD the rws n e esily viewed with regulr optil deviesF he qulity of imges should e good enough for trkingF hereforeD they re lso good ndidte to test the visul servo ontrol t the (rst stepF yne the visul servo ontrol is vlidted on the rwsD it n e pplied to helil miroswimmers y using suitle optil devies with high resolution in future worksF he designs of the rws with di'erent geometry prmetersD hed shpesD nd mgneti positioning re presented s followsF 2.1.2 Design and fabrication of scaled-up helical microswimmers he rws re lssi(ed nd nmed y their inner dimetersF wo tegories of rws re mdeX rwIs re those with inner dimeters of 1 mmY nd rwSs re the rws with inner dimeters of 5 mmF

SHM1s

wo tegories of rwIs re mdeF rwIs in the (rst tegory re mde of titniumD with mgneti lyers oted on the surfe of helil tilsF rwIs in the seond tegory re mde of resineD hving non mgneti tils nd mgneti hedsF hey re ll nmed rwIEmh @mh mens mgneti hedD nd is the numerAF pigure PFIX rwIs with inner dimeters of 1 mmD re mde of titniumF hey re oted y ferromgneti lyers @nikelAF SHM1s with magnetic tails: he titnium rwIs re mde y i.materialise @http://i.materialise.com/ AF hey re presented in pigure PFIF heir geometry prmE eters re sled from rxf s wellD whih re presented in le PFIF eording to their hed shpesD the rwI without hed is nmed rwIEnoneD nd the rw with ylindril hed is nmed rwIEylinderF hey re oted y ferromgneti mterilX nikelF por the thik lyer otingD eletroplting is usedF itnium oxides very esilyF he titnium is thus deoxidized y hydrohlori id right efore the eletropltingF he titnium models re pled onto the negtive eletrodes nd the positive eletrode is pled ner the modelsF hese re immersed in nikel th with the pplition of eletri potentil etween the two eletrodesF he deposition rte is lirted onto the dummy silion surfe oted with titnium lyerF he end of the helil til is used to mke n eletri ontt during the eletrolyti deposition proessD thus the lst pith is not otedF he thikness of nikel lyers re estimted s 20 -50 µmF SHM1s with magnetic heads: he resine rwIEmhs re mde y 3d prod @http://www.3dprod.com/ AF hey re presented in pigure ??F he dvntge of the resine rwIEmhs is the low ost nd the quik fritionD whih llows frition of lrge numers of rwIEmhs with di'erent geometry prmeters @pithD widthD numers of turnsAF he di'erent geometry prmeters re de(ned y 2 3 ftoril designD whih re depited in le PFPD using design of experiments methodF he design of experiments is presented in more detil in etion QFIF he miniml thikness imposed y 3d prod is 0.3 mmF ih rwIEmh hs ylindril tue hedD with 0.5 mm thik slit t the middleF e irleEplte permnent mgnet @upermgnete EIFSE HFSExA mgnetized in the dimeter with 1.5 mm of dimeter nd 0.5 mm of thikness is seted in the slitF xote tht rwIEmhV is the sledEup model of rxfF he fmily of rwSs with their nmes re depited in pigure PFQF ell rwSs geometry prmeters re sled from rxfD whih ppered in rwng PHIHD exept for the thiknessF st mens tht they re lso sledEup from rwIEmhVF he geometry prmeters spei(tions of rxf nd rwSs re desried in le PFIF he si strutures of rwS re mde of erylonitrile futdiene tyrene @efA @eflus RHHA y rpid prototyping mhine @trtsys himension UTVA using pused heposition wodeling @phwA methodF he thermoplstis re lique(ed nd deposited y n extrusion hedD whih follows toolEpth de(ned y v (leF he mterils re deposited in lyers s (ne s 0.25 mmF hereforeD the thikness of rwSs is designed s 1 mm thik for hving enough lyers to enfore the strutureF pigure PFQX rwSs with inner dimeters of 5 mmF @AE@dA he (rst turn of helil til is over y mgnetsF @eA he whole helil til is overed y mgnetsF he rwSs hve mgneti helil tils nd nonEmgneti hedsF heir (rst pithes @see pigure PFQ @AE@dAA or entire surfe @see pigure PFQ @eAA of their helil tils re overed with xdpef permnent mgnets from upermgnete @5 mm×1.5 mm×1 mmAF he mgneti lyer oted on the surfe of the helil tils re not uniformF he heds of the rwSs re in di'erent shpesX without hed @pigure PFQ @AAD with ylindril hed @pigure PFQ @AAD spheril hed @pigure PFQ @AA nd squre hed @pigure PFQ @dAAF hose heds re not mgnetiF 2.1.3 Swimming at low Reynolds numbers es explined in etion PFIFID to simulte the low eynolds numer environmentsD visous liquids must e usedF qlyerol solutions re good ndidtesD esuse glyerol is wterEsoluleD olorlessD odourlessD visous liquidF qlyerol solutions with di'erent onentrtions show di'erent visositiesF he onentrtions of the glyerol solutions used in the experiments re QH7D SH7D TH7D UH7D US7D VH7D VS7D WH7D WS7D nd IHH7F he eynolds numer n e lulted s in eqution @IFIAD using the hrterE isti liner dimension nd the veloity of the rwD the density nd visosity of the liquidF he visosity of the liquid is mesured y flling ll visometer @frook(eld upIHAF he visosity of glyerol solutions is sensitive to the environmentl temperE tureF he mesured experimentl temperture is out 23 • CF he density of the liquid is mesured y portle density meter @hwe QSAF por the rottionl propulE sion hrteristis studiesD the trnsltionl veloities re onsidered s zeroF sn order to lulte the eynolds numers for rottionl propulsion hrteristisD we ssume tht the liner veloities re I pith per seondF he hrteristi liner dimensions re tken s the hydruli dimeters of the rwsF he hydruli dimeter of tue is de(ned s the di'erene etween its outer dimeter nd inner dimeterD whih is its thiknessF le PFQ summrizes the mesured visosities of glyerol solutions used nd the lulted eynolds numers t whih the two sizes of rws swim in the rottionl propulsion experimentsF sn wterD swimming terium suh s iFoli hs Re ≈ 10 -5 -10 -4 F e humn spermtozoon moves with Re ≈ 0.01F he lrger ilites suh s prmeium swims with Re ≈ 0.1 vug PHHWF he rxfs swim with Re ≈ 3.4 × 10 -4 -3.6 × 10 -2 rwng PHIIF sn the experiments for hrteriztion of rottionl propulsion properE tiesD the eynolds numers t whih the rws swim re Re ≈ 0.001-3F st is tehnilly di0ult to further derese the eynolds numers t whih the rws swimF xeverE thelessD with these experimentl onditionsD the rottionl propulsion ehviors of the rws re studied t low eynolds numersF he rws re tuted y rotting mgneti (eldF sn the following two setionsD two mgneti tution systems used in the experiments re presentedX Qh relmholtz oil pirs system nd otting ermnent wgnet wnipultor @wwAF 2.2 3D Helmholtz coil pairs system e Qh relmholtz oil pirs system onsists of three orthogonlly rrnged relmholtz oil pirs ple of generting rotting mgneti (eld round ny xis in the Qh spe is used in the experiments to tute the helil swimmersF sn this setionD the priniple of the relmholtz oil pir is t (rst introduedF he design nd the frition of the Qh relmholtz oil pirs system re then desriedF henD the vision system is presented fterwrdsF he ontrol of the rottion mgneti (eld genertion oth in the Ph nd Qh spe is presented t lstF 2.2.1 Helmholtz coils principle e relmholtz oil pir is mde up of twoD identilD irulr oils ligned on the sme xis nd seprted y distne h equl to the rdius R of the oilF ih oil rries n equl eletril urrent )owing in the sme diretionF his rrngement of oils is invented y qermn physiistD rermnn von relmholtzD over entury goF relmholtz oils hve een used in pplitions in whih highly uniform mgneti (elds re requiredF he mgneti (eld produed y relmholtz oils is homogeneous t the enter of the oilsF he mgneti )ux density norm B t the enter of the oils is desried yX

B = 4 5 3/2 µ 0 nI R @PFIA
where n is the numer of turns per oilD I is the urrent )ux density running through the oilsD R is the rdius of oilD nd µ 0 = 4π × 10 -7 T • m/A = 1.26 × 10 -6 T • m/A is the permeility onstntF he simultion of the mgneti (eld generted y relmholtz oil pir y gomsol is depited y pigure PFRF he mgneti )ux density norm long the xis diretion of the oil pir nd long the diretion perpendiulr to the xis nd t the middle of the two oils is shown in pigure PFRF he mgneti )ux density norm long these two diretion shows the homogeneity of the (eld t the enter of the oil pirF yne relmholtz oil pir n generte uniform mgneti (eld t the region of interest in one diretionF hereforeD three orthogonl rrnged relmholtz oil pirs n generte uniform mgneti (eld t the region of interest in ny diretion in the speF purthermoreD it n generte rotting uniform mgneti (eld round ny xis in the speF 2.2.2 3D Helmholtz coil pairs system design and realisation st is deided tht the smll oils hve n inner open spe lrger thn 8 cm for the ese of useF his would llow enough room for smples essF he relmholtz oils pirs re designed for mximum 15 mT t the enter of the oils speF he mximl urrent in one oil is designed t out 3 A due to the the heting prolem of the resistnesF he dimeter of the smll oils is deided to e 50 mmF ith the dimeter of the oil nd the urrentD the numer of turns is estimted s SHHF he dimeter of the opper wire is hosen s 0.8 mmF his dimeter of the opper wire is reltively high in order to void overheting of the wireF he width of the oils is designed to hve good ompromise etween the width nd the thikness of the oilsF sn the he geh design of the three relmholtz oil pirs with mehnil support for the oils @lue prtA is depited y pigure PFSF e plteform is designed t the enter of the relmholtz oils to hold the ontiner with the rw in the liquidF he plteform should e semiEtrnsprent nd e le to disperse the lightD so tht vih n e pled t the ottom of the support to klight the rw t the workspe of the relmholtz oils through the plteformF he mehnil support for the oils nd the supports of the smll nd medium oils re mde of ef y rpid prototype mhine @himension UTV eriesAF he supports of the ig oils is mde of luminium for two resonsX (rstlyD the dimension surpsses the prototype mhine pity @203 mm × 203 mm × 305 mmAY seondlyD luminium is good ondutor of hetF he resistnes of the ig oils re higherD so the ig oils rete more hetF he supports mde of luminium re good het rditorsF he opper wires re mounted mnullyF he mesured resistnes of eh oils re shown ih set of relmholtz oils is driven y n eh SHGS REEhg servompli(er of wxon motor ontrol mpli(erD ple of 5 A ontinuous urrent nd 10 A pek urE rentF he mpli(ers re limented y huEvmd QHHERV hg power supplyD ple of 6.7 A urrent nd 48 V voltgeF he mpli(ers re used on urrent ontrol modeD the output urrent re thus onstnt for (xed input voltgeF enlog ommuniE tion etween the g nd the mpli(ers is omplished with ensory TPT enlog nd higitl sGy rd @TPTAF 2.2.3 Vision system implementation e stereo vision system with two mersD the top mer nd the side merD re integrted in the systemF es shown in pigure PFTD the top mer is on the topD nd the side mer is eside the relmholtz oilsF he two mers re oth (rewire mersF he top mer is ike pHQPfF he mximl frme rte is PHV fps t full resolution @640 × 480AF e lens with 25 mm of fol length is mounted on the top merF he side mer is quppy ro pHQPF he mximl frme rte is VP fps t full resolution @656 × 492AF hue to the limited side openingD n endosope @fipolA with dimeter of 2.7 mm is onneted on the side mer y oupler lensF he fol length of the oupler lens is 30 mmF pigure PFTX Qh relmholtz oil pirs experiment setup with two mersF 2.2.4 Magnetic eld calibration and generation por the tution of the mgneti rws with Qh relmholtz oil pirs setupD the rotting mgneti (eld should e generted with desired mgneti (eld strength nd rottion frequenyF wo methods to generte rotting mgneti (eld with relmholtz oil pirs re introdued elowF he (rst method is the genertion of rotting mgE neti (eld round un xis xD yD or zF es the mgneti (eld is in the plnes yOzD xOzD or xOy nytimeD it n e de(ned s the Ph rotting mgneti (eldF he seond method is the genertion of rotting mgneti (eld round ny xis in the speF herefore it n e de(ned s the Qh rotting mgneti (eldF sn order to generte the rotting mgneti (eld with desired mgneti (eld strengthD the relmholtz oil pirs should e lirtedF 2.2.4.1 Magnetic eld calibration o lirte the relmholtz oil pirs mens to (nd the reltionship etween the strength of the mgneti (eld generted t the enter of the oil pirs nd the urrents pssing through the oilsF prom eqution @PFIAD s the numer of turn @nA nd the resistne for one oil pir @RA re (xedD the generted mgneti (eld )ux density @B 0 A depends only on the urrent @IA through the oilsF he urrents pss through oil pir re onstnt for (xed output voltge of the gs rdF hus the generted mgneti (eld depends only on the output voltge of the gs rdF he reltionship etween the generted mgneti (eld nd the ommnded output voltge of the gs rd is lirted in the enter of the workspe using rirst qwHV gussmeterF pigure PFU depits the mesured mgneti )ux density t the enter of the relmholtz oil pirs generted individully y the three relmholtz oil pirsF he pity of this relmholtz oil pirs is limited y the output power of the power supply pplied to the ig oilsF he prtil mximl mgneti (eld tht the Qh relmholtz oil pirs n generte is out 12 mTF pigure PFUX wgneti )ux density of the mgneti (eld generted individully y the three relmholtz oil pirs in funtion of the output voltge of the gs rdF sn the following experimentsD rotting mgneti (elds with the mgneti )ux density of B 0 = 5 mT nd B 0 = 10 mT t the enter of the workspe re usedF le PFT desries the output voltge of the gs rd orresponding to eh oils to generte 5 mT or 10 mT mgneti (eldF le PFTX pei(tions of the gs rd output voltges to generte mgneti (eld with mgneti )ux densities of B 0 = 5 mT nd B 0 = 10 mTF wgneti )ux density @B 0 A U x @A U y @A U z @A 5 mT I I IFP 10 mT P P PFR 2.2.4.2 World coordinate system rereD the world oordinte system R w = (O, x, y, z) is de(ned y the diretions of the mgneti (eld generted y the three relmholtz oil pirsF he enter O is de(ned y the enter of the three oil pirsF he x xis is de(ned s the mgneti (eld diretion when we pply urrent to the smll oil pirF husD the x xis is in the diretion of the xis of the smll oil pirD whih n e shown in pigure PFVF imilrlyD the y xis is de(ned s the mgneti (eld diretion when we pply urrent to the medium oil pirD nd the z xis is de(ned s the mgneti (eld diretion when we pply urrent to the ig oil pirF wke sure tht the orienttion of the vetor spe (x, y, z) is rightEhndedD or the urrent through the ig oil pir should e hngedF pigure PFVX orld oordinte system R w = (O, x, y, z) is de(ned y the diretions of the mgneti (eld generted y the three relmholtz oil pirsF

B ⊥x =   B x B y B z   =   0 B 0 • cos(2πf t) B 0 • cos(2πf t + d phase )   @PFPA where d phase = - π 2
presents the phse di'ereneD note tht d phase = π 2 if the mgneti (eld rottes round -xD f is the rottion frequenyD nd B 0 is the mgneti )ux density generted t the enter of the workspeF e simultion of iqution @PFPA with gomsol multiphysis of the genertion of rotting mgneti (eld round the xis x with the relmholtz oil pirs is illustrted in pigure PFWF he frequeny of the rotting (eld is 1 HzF he red rrows show the diretion of the mgneti (eld t the instnt tF his demonstrted the rottion of the mgneti (eldF

U x @A U y @A U z @A d phase @CGE π 2 A x H P PFR B z : - -x H P PFR B z : + y P H PFR B x : - -y P H PFR B x : + z P P H B y : - -z P P H B y : +
e grphi user interfe of the genertion of the rotting mgneti (eld is illustrted in pigure PFIHF he output voltges of the gs rd nd the phse di'erene should e referred to le PFUF sn the progrmD the smple time is 40 msF he genertion of the rotting mgneti (eld is vlidted y the movement of mgneti pole in the generted (eldF his progrm of the genertion of the Ph rotting mgneti (eld is very useful for the rw9s swimming hrteristion experimentsF

=   u x u y u z   =   n y -n x 0   @PFQA with n x = 0 or n y = 0F xote tht v is vetor orthogonl oth to u nd nF vet v e v = n × uF v =   v x v y v z   =   n x n z n y n z -n 2 x -n 2 y   @PFRA
xote tht ũ nd ṽ re the unit vetors of u nd vD so tht ũ nd ṽ re the sis vetors of the plne orthogonl to the xis nF husD

ũ =   ũx ũy ũz   = 1 u   u x u y u z   = 1 u 2 x + u 2 y + u 2 z   n y -n x 0   @PFSA nd ṽ =   ṽx ṽy ṽz   = 1 v   v x v y v z   = 1 v 2 x + v 2 y + v 2 z   n x n z n y n z -n 2 x -n 2 y   @PFTA emrk tht if n x = 0 nd n y = 0D
we n hose ũ = (1, 0, 0) T nd ṽ = (0, 1, 0) T F henD the prmetri eqution of the vetoril irle with the rdius r nd norml vetor n n e written sX

       x = r • ( ũx cos(2πf t) + ṽx sin(2πf t)) y = r • ( ũy cos(2πf t) + ṽy sin(2πf t)) z = r • ( ũz cos(2πf t) + ṽz sin(2πf t))
@PFUA his prmetri eqution n e written in mtrix form y developing ũ nd ṽX

  x y z   = r cos(2πf t) u   n y -n x 0   + r sin(2πf t) v   n x n z n y n z -n 2 x -n 2 y   @PFVA
husD the rotting mgneti (eld B ⊥n = (B x , B y , B z ) T round the vetor n n e expressed sX

B ⊥n =   B x B y B z   = B 0 cos(2πf t) u   n y -n x 0   + B 0 sin(2πf t) v   n x n z n y n z -n 2 x -n 2 y   @PFWA with u = n 2 y + n 2 x nd v = (n x n z ) 2 + (n y n z ) 2 + (-n 2
x -n 2 y ) 2 D nd where f is the rottion frequenyD t is the timeD B 0 is the mgneti )ux density t the enter of the workspeF he mgneti )ux density B 0 depends only on the output voltge of the gs rdF henD the ontrol vriles of the systemD whih re the output voltges for the three oils @U s , U m , U b AD n e expressed sX

  U s U m U b   = cos(2πf t) u   U x n y -U y n x 0   + sin(2πf t) v   U x n x n z U y n y n z -U z (n 2 x + n 2 y )   @PFIHA
where U x D U y D nd U z re respetively the output voltge from the gs rd to the smllD mediumD nd ig oils to generte mgneti (eld of B 0 F he vlues n e referred to le PFTF his genertion method of the Qh rotting mgneti (eld round the xis n is vlidted y the oinidene of the xis n of the rotting (eld nd the xis of the rwIEmhV whih is rotting round its own xis in the rotting (eldF he lultion of the norml vetor of the xis of rw y vision is introdued in etion RFPFIF 2.2.5 Summary he ompositionD ontrol vrilesD nd the output of the Qh relmholtz oil pirs sysE tem integrted with vision system is summrized in le PFVF pigure PFIP desries shem of the Qh relmholtz oils systemF he urrents pssing through the relmholtz oils re ontrolled y the g through gs rd nd three mpli(ersF he mpli(ers re limented y power supplyF he Qh relmholtz oil pirs system is designed for the visul servo ontrol of the rwF he rw is ontrolled y the mgneti (eld generE ted y the Qh relmholtz oilsF he position @gA nd orienttion @θA of the rw in the imge n e ommunited with the g through visionEsed trking lgorithmF he trking nd the estimtion methods re introdued lter in etion PFRF le PFVX ummry of the Qh relmholtz oil pirs system with vision systemF Item Description epplition rotting mgneti (eld genertion gomposition Q relmholtz oil pirs 8 P mers 8 P vihs 8 gs rd 8 Q mpli(ers 8 power supply gontrol vriles

U s D U m D nd U b ystem output B(n, f )
pigure PFIPX hem of the relmholtz oils systemF he ontrol vriles of the system re the output voltges of the gs rd U s D U m D nd U b F he frequeny nd the xis of the rotting mgneti (eld re ontrolledF he vision system returns the position @gA nd orienttion @θA of the rwF

Rotating Permanent Magnet Manipulator (RPMM)

es the relmholtz oils system re di0ult to sle up to the size required for medil useD euse of the energy e0ieny nd ostD tuting the mgneti roots with ww is onsideredF sn this setionD the design of the frition of ww system is introduedF he frequeny of the rottion mgneti is ontrolled y sh ontrollerD whih is presented fterwrdsF 2.3.1 RPMM principle vots of reserhers tute the mgneti roots with ww vee PHHWD pountin PHIHD whoney PHIPD whoney PHIPF he dipole moment of the mgnet is noted s MD whih is lwys perpendiulr to its rottion xis ΩF he mgneti (eld generted y ww n e urtely modelled with the pointEdipole model purlni PHHIF he mgneti (eld H t position p reltive to the enter of the ww n e expressed s followsX

H = 1 4π|p| 3 3pp T |p| 2 -I M @PFIIA where I is the identity mtrixF xote tht H = 3pp T |p| 2 -ID then the mgneti (eld H is expressed sX H = 1 4π|p| 3 HM @PFIPA
whoney proved in his pper whoney PHIP tht the mgneti (eld vetor t every point in speD generted y rotting mgneti dipole suh s permnent mgnetD rottes round (xed xis ωF his desired rottion xis of the lol mgneti (eld t the position p is norml to the plne ontining HD suh tht H T ω = 0F he lol (eld rottion xis ω t the position p is given s followsX pigure PFIRX @A he Qh gti model of the ww systemD nd @A the ww system used in the experiments X e permnent mgnet is rigidly onneted with motorD whih is mounted on liner stgeF en eletri hg motorD wxon motor @12 VD 15 WAD is used in the following experE iments to rotte the permnent mgnetF he mximl output of the motor is PVTH turnsGmin @47 HzAF he motor is driven y REEhg servompli(er of wxon motor ontrol mpli(erD ple of 2 A ontinuous urrentF enlog ommunition etween the g nd the mpli(ers is omplished with ensory TPT enlog nd higitl sGy gs rd @TPTAF e wxon inrementl enoder @Q hnnelsD SHH ounts per turnA is used to omunite the ngulr position of the motor to the gs rdD in order to relize frequeny ontrolF he motor is (xed on liner stgeD in order to omine the rottion with trnsltionF he liner stge used in the following experiments is xewmrk ivEPSH liner motion stge with trvel rnge of 250 mm nd mximl speed of 220 mm/sF pigure PFIR presents respetively Qh ti model of the ww systemD nd the ww system used in the experimentsF 2.3.3 Rotation frequency control he rottion frequeny of the rotting mgneti (eld of the ww is ontrolled y the rottion frequeny of the motorD on whih the mgnet is mountedF sn this setionD the rottion frequeny ontrol of the motor y roportionlEsntegrlEherivtive @shA ontroller is introduedF 2.3.3.1 The principle of PID controller e sh ontroller lultes n error vlueD whih is de(ned s the di'erene etween mesured vrile nd desired setpointF he ontroller ttempts to minimize the error y djusting the proess ontrol inputsF he sh ontroller lultion involves three seprte onstnt prmetersX the proportionlD the integrl nd derivtive vluesD denoted D sD nd hF reuristillyD these vlues n e interpreted in terms of timeX depends on the present errorD s on the umultion of pst errorsD nd h is predition of future errorsD sed on urrent rte of hnge erki IWWPF he output of the sh ontrollerD u(t)D n e expressed in funtion of the error e(t) s followsX

ω = H -1 Ω |H -1 Ω| @PFIQA with H -1 = 1 2 (H -I)F
u(t) = K p e(t) + K i t 0 e(t) dt + K d d dt e(t) @PFIRA
where K p is proportionl ginD K i is integrl ginD nd K d is derivtive ginF he three gins re tuning prmetersF 2.3.3.2 PID controller algorithm he input prmeter of our system is the voltge sent to the mpli(er from the gs rdF he mpli(er is on urrent modeD whih mens tht the output urrent of mpliE (er is (xed for (xed input voltgeF he output of the system to e ontrolled is the rottion speed of the motorF he inrementl enoder returns the ngulr position Φ of the motor t n instnt tF he urrent rottion speed n e lulted s dΦ/dtF he error should e the di'erene etween the mesured rottion speed nd the desired rottion speedF elgorithm I is implemented in the progrm to lulte the output voltge of the gs rd in funtion of the rottion speed errorF he vriles of the lgorithm re the desired rottion speed @setpeedAD the mesured rottion speed @mesurepeedAD nd the smple time @dtAF he smple time is the time of every lultion of voltge outputF he output voltge of the gs rd is lulted every 50 msF ine the smple time is very shortD the integrl nd derivtive of the error n e expressed s error×dt nd @errorprevious errorAGdtF he D sD h gins K p D K i D nd K d determintion is presented s followsF Algorithme 1 sh ontroller lgorithm Result: glulte the output voltge out ommunited to the motor Input: setpeedD mesurepeedD dt Initialization

previousirror ← 0 integrl ← 0 Start e ← setpeed E mesurepeed integrl ← integrl C error×dt derivtive ← @error E previousirrorAGdt out ← K p × errorC K i × integrl CK d × derivtive
previousirror ←error wit@dtA goto start 2.3.3.3 P, I, D gains determination iegierExihols tuning method is used to determine the D sD h gins K p D K i D nd K d F his method is developed y iegler nd xihols iegler IWRPF st egins with setting the integrl gin K i nd the derivtive gin K d to zeroF he proportionl gin K p is then inresed from zero until it rehes the ultimte gin K u D t whih the output of the ontrol loop osilltes with onstnt mplitudeF he osilltion period is noted s P u F he ultimte gin K u nd osilltion period P u re used to set the D sD h ginsF le PFWX D sD h gins determintion y iegierExihols methodF Control type

K p K i K d HFSK u s HFRSK u IFPK p /P u sh HFTK u PK p /P u K p P u /8
prom some experimentsD the ultimte gin of the system is otined s K u = 0.5F he osilltion period is P u = 0.6 sF husD the D sD h gins of the system re lulted s K p = 0.3D K i = 1D K d = 0.02F e test for desired rottion frequeny t 5 Hz is relised to vlidte the D sD h ginsF pigure PFIS illustrtes the rottion frequeny of motor revolution in funtion of time with the D sD h gins determined eforeF he time of sttionry regime estlishment is less thn 1 sF he inury of the sttionry regime is less then 10%F he D sD h gins re eptleF he frequeny of the rotting mgneti (eld is then ontrolledF 2.3.4 Summary he ompositionD ontrol vrilesD nd the output of the ww system integrted is summrized in le PFVF e permnent mgnet is mounted on motorF pigure PFIP desries shem of the ww systemF he urrent pssing through the motor is ontrolled y the g through gs rd nd n mpli(erF he mpli(er is limented pigure PFITX hem of the ww systemF he ontrol vriles of the system re the output voltges of the gs rd U out F he frequeny of the rotting mgneti (eld generted y the system is ontrolledF

Vision-based tracking

o quntify the displement nd the orienttion of the rwD no solution other thn visul trking is in the mentime roustD urte nd t relEtimeF e roust nd relEtime visul trking is one of the keys to suessful visul servoing tskF sn this setionD relEtime trking method of i nd method sed on the imge moments for the estimtion of the ryenter nd the xis of the trked ojet re introduedF 2.4.1 Tracking of a SHM he trking of rw in the experiments is relised y isul ervoing ltform @iA wrhnd PHHSF i is omplete rossEpltform lirry developed nd mintined y the sxse vgdi temD tht llows prototyping nd developing pE plitions in visul trking nd visul servoingF fklight system is used in the experimentsF fy using klightD the imge of rw is set of onneted pixels with the sme gry levelF pigure PFIU shows n imge of rwIEmhV in the liquid with klightF sn order to hve etter ontrst with klightD the rws re pinted into lk olorF hereforeD rw is onsidered s lo @vphot lss in iA in imgesD de(ned y the set of onneted pixels with the sme gry levelF he initil dot of the rw is de(end y the funtion dot.initTrackingD whih llows to de(ne the dot y looking the pixels onneted with the initil pixel with the gry levelF he initil pixel is de(ned y lik on one pixel in the imge of the rwF he trking funtion dot.Track llows to trk the dot with the sme grey level of the initil dotF en exmple of the trked dot of rwIEmhV is shown in pigure PFIU with green pixelsF pigure PFIUX @A en imge of rwIEmhVF @A efter imge proessing with iD rwIEmhV is trked @green pixelsAF 2.4.2 Barycentre of the helical swimmer he ryentre of rw is lulted y i using imge momentsF he generl Ph (p + q) th order moments of greyElevel imge f (x, y) re de(ned s rorn IWVTX m pq = I

x p y q f (x, y) dxdy @PFISA with p, q = 0, 1, 2... he integrtion is over the whole imge IF imilrlyD one n ompute the moments of region in the imgeD of n ojet y integrting only over the region RF he zerothD (rst nd seond imge moments n e otined from iF he re of the ojet n e found when p = q = 0 @A = m 00 AF he entroid of the ojet is de(ned s the enter of mss of (gure of the sme shpe with the onstnt mss per unit reF he enter of the mssD in turnD is the point where ll the mss of the ojet ould e onentrted without hnging the (rst moment of the ojet out ny xisF sn Ph seD the (rst moment out the xExis is por exmpleD the ryentre of rwIEmhV is lulted nd mrked with red ross using iD s shown in pigure PFIUF 2.4.3 Orientation of the helical swimmer he orienttion of n ojet in the imge n e de(ned y the diretion of its xis of lest inertiF o do soD we (nd the line for whih the integrl of the squre of the distne to points in the ojet is minimumF ht integrl isX

E = R r 2 f (x, y) dxdy @PFIVA
where r is the perpendiulr distne from the point (x, y) to the lineF qiven point (x, y) on the ojetD suppose tht the point (x 0 , y 0 ) is the losest point on the lineF he distne r etween the point nd the xis is written yX

r 2 = (x -x 0 ) 2 + (y -y 0 ) 2 @PFIWA
es the point (x 0 , y 0 ) is on the xisD x 0 nd y 0 n e written y prmetri equtions with the prmeter s s followsX

x 0 = -ρ sin θ + s cos θ @PFPHA y 0 = ρ cos θ + s sin θ @PFPIA
where ρ is the distne from the origin to the losest point on the xisD nd θ is the ngle etween the x xis nd the xis of the ojetD with θ ∈ -π 2 , π 2 F ustituting the prmetri equtions for x 0 nd y 0 into the expression of rD we otinX r 2 = (x 2 + y 2 ) + ρ 2 + 2ρ(x sin θ -y cos θ) -2s(x cos θ + y sin θ) + s 2 @PFPPA pigure PFIVX yrienttion of the rw in imgeF sn order to (nd the losest point (x 0 , y 0 )D @PFPPA should e di'erentited with respet to s nd set to zeroD whih leds toX s = x cos θ + y sin θ @PFPQA his result n now e sustituted k into the prmetri equtions for x 0 nd y 0 F prom whereD the distne r n e thus written s followsX r 2 = (x sin θ -y cos θ + ρ) 2 @PFPRA e now ddress the minimiztion of

E = R (x sin θ -y cos θ + ρ) 2 f (x, y) dxdy @PFPSA
y di'erentiting with respet to ρ nd setting the result to zeroD whih leds toX

x sin θ -y cos θ + ρ = 0 @PFPTA with (x, y) the enter of the ojet reF es the xis of lest seond moment psses through the enter of reD the oordintes re hnged to x = x -x nd y = y -yF e hveX

x sin θ -y cos θ

+ ρ = x sin θ -y cos θ @PFPUA hus E = a 0 sin 2 θ -b 0 sin θ cos θ + c 0 cos 2 θ @PFPVA
where aD b nd c re the seond moments given yX

a 0 = R (x ) 2 f (x, y) dx dy @PFPWA b 0 = 2 R (x y )f (x, y) dx dy @PFQHA c 0 = R (y ) 2 f (x, y) dx dy @PFQIA nd aD bD c n e omputed yX a 0 = m 21 m 00 -x 2 @PFQPA b 0 = 2 m 11 m 00 -xy @PFQQA c 0 = m 02 m 00 -y 2 @PFQRA
henD i n e expressed in the form s followsX his orienttion lultion method is t (rst veri(ed o'Eline y four pitures of rw with di'erent orienttions orresponding to the four di'erent ses @see pigure PFIWAF he rw is well trked with the presenttion of its ryenter nd orienttionF sn hpter RD rel time visul trking of the rw using this detetion of its ryenter nd orienttion is presentedF

E = 1 2 (a 0 + c 0 ) - 1 2 (a 0 -c 0 ) cos 2θ - 1 2 b 0 sin 2θ @PFQSA
(a) (b) (c) (d) (e) θ = -64.5 • (f) θ = -39.4 • (g) θ = 27.5 • (h) θ = 62.6 •
pigure PFIWX @AE@dA he rws with di'erent orienttions orresponding to the four di'erent sesF @eAE@hA he trked rws with their ryenters @red rossesAD their orienttions @red linesAD nd the lulted θ with respet to the lol xis iF

Swimming characteristics analysis

he swimming hrteristis of helil swimmer n e desried y the rottion frequenyD the trnsltion veloityD nd its propulsion mtrixF sn this setionD the methods to estimte the rottion frequeny nd trnsltion veloity of the rw re introduedF e method to identify the propulsion mtrix of the rw is presented s wellF 2.5.1 Rotation frequency analysis he rottion frequeny of rw is lulted o'Eline y n imge sequene reorded during the experimentF he numer of turns is ounted etween two imge frmesF he time n e lulted y the frme numers nd the frme rteF he method is detiled y n exmple of rwIEmhV tuted y 10 mT rotting mgneti (eld t 12 Hz in WH7 glyerol solutionF en imge sequene is reorded t 60 Hz frme rte @during one seondD TH imges re tkenAF e rndom frme is tken s the initil positionD for exmple the IPth frme s shown in pigure PFPHF yne turn is ounted when the end of the til repsses its initil positionF et IUth frme in the exmpleD s shown in pigure PFPHfD the rwIEmhV rotted one turn from IPth frmeF e ount the numer of turn during t lest one seond @TH frmesAF rwIEmhV (nished its IPth turn t the end of one seond @the UPnd frmeAF ht mens tht rwIEmhV rotted IP turns in I seondF hereforeD the ngulr speed of rwIEmhV rotted in WH7 glyerol solution with 10 mT rotting mgneti (eld t 12 Hz is 12 HzF enother exmple is rwIEmhV tuted y 5 mT rotting mgneti (eld t 12 Hz in WH7 glyerol solutionF prom the IPth frmeD rwIEmhV (nished its Tth turn t UUth frmeF TS frmes took 1.083 sF hereforeD rwIEmhV used 1.083 s to rotte T turnsF

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l)
pigure PFPHX en imge sequene of rwIEmhV tuted y 10 mT rotting mgneti (eld t 12 Hz in WH7 glyerol solutionF he frme rte of the imge sequene is 60 HzF he numer on the imge desries the numer of the frmeF he ngulr speed of rwIEmhV tuted y 5 mT rotting mgneti (eld t 12 Hz in WH7 glyerol solution is thus 6 ÷ 1.083 = 5.54 HzF

Translation velocity analysis

he trnsltion veloity of rw is lulted y the pixels it trvelled per seondD nd then onverted to millimetres per seondF he method is detiled with the exmple of rwIEmhS trvelled in pure glyerol solution with 10 mT rotting mgneti (eld t 3 HzF he xis of the rotting (eld is in the horizontl plneF etuted y the rotting mgneti (eldD rwIEmhS propelled forwrdD nd t the sme time it snk down euse of its own grvityF ynly the dvne veloity is lulted in order to ompre the swimming performneF pigure PFPI depits the position of rwIEmhS respetively t 2 sD 4 sD 6 sD 8 sF hnks to pixel rulerD the horizontl displement is IH pixels etween 2 s nd 4 sD W pixels 4 s nd 6 sD nd W pixels 6 s nd 8 sF he dvne veloity of rwIEmhS n e onsidered s onstntD euse the horizontl displement is uniformF fetween 2 s nd 8 sD the dvne veloity of rwIEmhS is lulted s RFVQ pixels per seondF he irle mgneti hed of 1.5 mm dimeter took PT pixels in the imgeF herefterD the dvne veloity is onverted to 4.83 ÷ 26 × 1.5 = 0.28 mm/sF he veloity of rw n e lulted in reltime s wellF he position of rw n e de(ned y the trked ryenterF he veloity of the rw n e then determined y its moved ryenterF 2.5.3 Propulsion matrix identication sn etion IFQFQD the reltionship etween the fore nd torque exerted on helil swimmer nd its trnsltion veloity nd ngulr speed is given in @IFWAF rereD method to estimte the propulsion prmeters through two experiments is introduedF he rw is pled vertilly in the liquidF e de(ne the inverse diretion of pigure PFPIX y1ine trnsltion veloity mesureX fusion imge of rwIEmhS tuE ted y 10 mT rotting mgneti (eld t 3 Hz in IHH7 glyerol solution t 2 sD 4 sD 6 sD 8 sF the grvity fore s the positive diretion for trnsltionF he positive diretion of rottion is de(ned y the right hnd ruleD s shown in pigure PFPPF he frited rws respet the rightEhnded hirlityF st mens tht positive pplied torque n generte positive propulsive veloityF fy djusting the rottion frequeny of the mgneti (eldD rw n swim upwrd nd downwrdD depending on the lne etween propulsionD grvity nd uoyny fores ting on the rwF pigure PFPPX pwrd diretion is de(ned s the positive diretion for f nd vD the rottion diretion in whih rwIEmhV rottes to generte positive fore is de(ned s the positive diretion for τ nd ωF he (rst experiment is tht (nd the rottion frequeny ω 1 with tht rottion freE queny the height of rw is motionlessF st mens tht the trnsltion veloity of the rw v 1 is zeroF prom @IFWAD sine the trnsltion veloity is nullD the pplied fore f app n e expressed s followsX

f app = av 1 + bω 1 = bω 1 @PFQUA
where the pplied fore f app is the pprent weight @the resultnt of the grvity nd uoyny foresA of the rwF henD the prmeter b n e lulted sX b = f app ω 1 @PFQVA he seond experiment is dropping the rw in vertil diretion in the liquid without ny externl mgneti fore nd torqueF he dropping veloity is mesured s v 2 nd ngulr speed is mesured s ω 2 F he pplied fore nd torque n e written sX

f app = av 2 + bω 2 @PFQWA τ app = bv 2 + cω 2 @PFRHA
with the pplied torque eing zeroF e n getX

a = f app -bω 2 v 2 @PFRIA c = -bv 2 ω 2 @PFRPA
he propulsion prmeters aD bD nd c re then identi(edF he presented estimtion methods of the rottion frequenyD the trnsltion veE loityD nd the propulsion mtrix of the helil swimmers in this setion re used to nlyse the swimming properties of the helil swimmers in the next hpterF 2.6 Summary sn this hpterD ll of the types of the ledEup relil wiroswimmers @rwsA used in the experiments re presentedF he rws re designed with di'erent geometry prmetersD mgneti positioningD nd hed shpesF he rws swim in visous liqE uids to simulte the low eynolds numer regimeF wo mgneti tution systemsD Qh orthogonlly rrnged relmholtz oil pirs system nd rotting permnent mgE net mnipultorD for the genertion of the rotting mgneti (eld re fritedF he ontrol of the strength nd the rottion frequeny of the rottion mgneti (eld genE erted y the mgneti tution systems is presentedF e relEtime trking method of the rw with the estimtion of its ryenter nd xis in the Ph imge is s well introdued in this hpterF he method of the estimtion of the rottion frequenyD the trnsltion veloityD nd propulsion mtrix of the rwD whih desries the swimE ming hrteristis of the rwD is presented in this hpterF sn the next hpterD the swimming hrteristis of the presented rws is studied nd ompredF sn this hpterD the swimming performne of the rws with di'erent geometry prmeters is ompredF he rottionl propulsion hrteristis of helil swimmers with di'erent mgneti positionings nd hed shpes re investigtedF he design of the mgneti positionings of helil swimmers is disussed oth t the mirosle nd t the millimeter sleF he liner veloity generted y the propulsion is studied sed on the estimted propulsion mtriesD s well s the required mgneti torqueF en openEloop Qh steering of helil swimmers is lso studiedF e Qh steering strtegy is proposed to improve the steering e0ienyF 3.1 Geometry parameters of helical tails sn this setionD the geometry prmeters in)uene on the swimming performne is studiedF e hesign of ixperiments @hoiA is used to relize the geometry prmeters design nd to determine the impts of the geometry prmetersF 3.1.1 A brief introduction to the design of experiments e design of experiments is series of test runs in whih purposeful hnges re mde to the input vriles of system of proess nd the e'ets on the response vriles re mesuredF ptoril design is invented y onld pisher in the IWPHs nd IWQHs for n griulturl reserh pisher IWQSF sn ftoril designsD multiple ftors re investigted simultneously during the testF ine the IWRHsD this method hs een suessfully pplied in the militry nd in the industryF hoi llows to redue the overll numer of experimentsD s well s to explore the intertions etween ftors qoupy PHHTF ome terms re ommonly used in design of experiments wson PHHQF e 4ftor4 is de(ned s ontrollle experimentl vrileD whih in)uenes the response vrileF he 4response4 is the outome of n experimentF ih ftor must ssume some vluesD de(ned s levelsF he hnges tht our on the men of the vlues of the response vrile orrespond to the 4e'ets4F fesides the e'ets used y the ftorsD the e'ets reted y the intertion of the ftors n e determinedF hese intertions orrespond to omined e'etsD where the e'et of eh ftor depends on the levels of the other ftorsF 3.1.2 Choice of the parameters sn our seD the ojetive of the experiment is to identify the geometry prmeters of helil swimmer tht hve signi(nt e'ets on its swimming performneF his leds to n improvement of the geometry designF pirstlyD quntity vlue whih desries the swimming performne of helil swimmer should e hosen s the response of the systemF his response is the linE er dvne veloity of the helil swimmer in 3 Hz rotting mgneti (eld with mgneti )ux density of 10 mT t the enterF eondlyD the ftors should e determinedF es the study is dimensionlessD wht we re interested in is only the shpe of the rwsD not the sizeF hereforeD the inner dimeters of the rws re (xed t 1 mmF yther geometry prmeters re hngedF hree input ftors re thus identi(edX the pithD the widthD nd the numer of turnsD denoted y AD B nd CF e simple PElevel full ftoril design is hosen ording to the ojetive of the experimentD whih requires 2 3 = 8 di'erent prmeters design on(gurtionsF henD the high level @+A nd the low level @-A should e de(ned for eh ftorX the pith is ompred etween 1.5 mm nd 4 mmY the width is ompred etween 0.5 mm nd 1.2 mmY nd the numer of turns is ompred etween IFS turns nd QFS turnsF le QFI depits the test mtrixF his tle of test mtrix will guide us in designing the geometry prmeters of the rwsF he designed eight rws with di'erent geometry prmeters re desried in le QFPF hey re presented in pigure PFPD nd nmed from rwIEmhI to rwIEmhVF he frition of the eight rwIEmhs is presented in etion PFIFPFIF le QFIX est mtrix of the full ftoril design for Q ftors nd P levels pisher IWQSF hesign pith @AA width @BA n of turns @CA elthough rwIEmhT shows the est swimming performneD its struture is too frgile to e used due to the width of the helil tilF hereforeD the est geometry prmeters design of the rws is rwIEmhVF xote tht the geometry prmeters of rwIEmhV re sled up from rxf rwng PHIIF sn the following experimentsD only rwIEmhV nd the rws with the sme shpe of rwIEmhV @the geometry prmeters re lso sled up from rxfD suh s rwSsA is usedF

A × B B × C A × C I - - - + + + P + - - - + - Q - + - - - + R + + - + - - S - - + + - - T + - + - - + U - + + - + - V + + + + + + vevel - 1.

Rotational propulsion characteristics

he rottionl propulsion hrteristi of rw is de(ned s the urve of the reE sponse rottion frequeny of the rw s funtion of the frequeny of the externl rottion mgneti (eldF sn this setionD the rottionl propulsion hrteristis of the rws with di'erent mgneti positionings nd hed shpes re studied nd ompredF efterwrdsD the hed shpe nd mgneti positioning design of the helil swimmer is disussed oth t the millimeter sle nd t the mirosleF

Rotational propulsion of SHMs with dierent magnetic positionings

sn the (rst pleD the rottionl propulsion hrteristis of rws with di'erent mgE neti positionings re studiedF he presented rti(il helil swimmers hve di'erent mgneti positioningsX with mgneti hed or with oted mgneti tilF rowE everD the in)uene of mgneti positioning on propulsion hrteristis hs not een shown yetF rereD the rottionl propulsion hrteristis of the rws with the sme geometry prmeters nd hed shpeD ut with di'erent mgneti positioningsD re investigted nd ompredF 3.2.1.1 SHM1-mh8 with a magnetic head he rottionl propulsion hrteristi of rwIEmhV is investigted in di'erent visE ous liquids tht ensure the low eynolds numer onditionsD with rotting mgneti (elds of di'erent strengthsF xote tht rwIEmhV is rw with mgneti hedD s shown in pigure PFPF he visous liquids re WH7 nd WS7 glyerol solutionsD s well s pure glyerolF he orresponding eynolds numers re HFHHVD HFHHRD nd HFHHI whih n e found in le PFQF he rotting mgneti (eld is generted y Qh relmholtz oil pirs setupF he mgneti )ux density t the enter of the workspe is 5 mT or 10 mTF pigure QFP depits the rottionl propulsion hrteristi of rwIEmhV in di'erent visous liquids respetively with 5 mT nd 10 mT rotting mgneti (eldsF he method of rottion frequeny estimtion of the rw is presented in etion PFSFIF rwIEmhV rottes t the eginning in syn with the rotting mgneti (eld until it rehes mximl rottion frequenyF feyond this mximl rottion frequenyD the rottion frequeny of rwIEmhV dereses gently with the externl rottion frequeny of the mgneti (eldF

(a) (b)
pigure QFPX ottion frequeny of rwIEmhV in rotting mgneti (eld generted y Qh relmholtz oil pirs with mgneti (elds of @A 5 mT nd @A 10 mTF rwIEmhV shows utEo' frequeny with gentle delineF he mximl rottion frequeny is de(ned s the utEo' frequenyF he vlue of the utEo' frequeny strongly depends on the visosity of the liquidD euse rw hs to overome the rottionl )uidi drg torque whih depends on the visosity of the liquidF he )uidi drg torque inreses with the visosity of the liquidF husD the vlue of the utEo' frequeny dereses with the visosity of the liquidF he vlue of the utEo' frequeny lso depends on the strength of the mgneti (eldF he mgneti (eld pplies mgneti torque on the rw whih overomes the )uidi drg torqueF hereforeD the vlue of the utEo' frequeny inreses with the strength of the rotting mgneti (eldF feyond the utEo' frequenyD the ville mgneti torque is not su0ient to keep the rw rotting synhronously with the pplied rotting mgneti (eldF ithin 10 mT rotting mgneti (eld of 40 HzD the rottion frequeny of the rwIEmhV is 0 Hz in three di'erent glyerol liquidsF Discussion he utEo' frequeny n explin the presene of mximl trnsltion speed of the helil miroswimmers with mgneti hed in prior works sshiym PHHID hng PHHWF he utEo' frequeny should e tken in onE sidertion for future pplitionsD euse the swimming performne of helil miroswimmer n e strongly limited due to the high rottion frequeny of the pplied mgneti (eldF 3.2.1.2 SHM1-cylinder with an uniformly coated magnetic tail he rottionl propulsion hrteristi of rws with n uniformly oted mgneti tils is investigtedF he rw used in the following experiments is rwIEylinder with 20 -50 µm thik oted mgneti lyer presented in etion PFIFPFID whih hs the sme geometry prmeters s rwIEmhVF rwIEylinder is uniformly oted with 20 -50 µm thik ferromgneti lyerF he response rottion frequeny of rwIEylinder in pure glyerol liquid with 10 mT rotting mgneti (eld n not inrese with the frequeny of the rotting mgneti (eld higher thn 1 HzF he frequeny of the rwIEylinder in pure glyerol is lwys 0.08 HzF e suppose tht the rottion frequeny of rwIEylinder is in sturtionF heres the totl dipole moment of rwIEylinder is muh lower thn tht of rwIEmhVD euse the ferromgneti oting lyer is thinD so tht the mgneti torque exerted on the rw is very wekF his sturtion is hieved t frequeny lower thn 1 HzF sn order to vlidte our ssumptionD stronger mgneti (elds nd less visous liquids re usedF sn the following experimentsD the wwD whih is introdued in etion PFQD is usedF he mgneti )ux density t the zone of interest is out 30 mTF he visous liquids used re QH7D TH7D VH7D nd WH7 glyerol solutionsF he orresponding eynolds numers re respetively HFSRD HFIID HFHQ nd HFHHVD whih n e found in le PFQF he rottionl propulsion hrteristi of rwIEylinder in di'erent visous liqE uids is shown in pigure QFQD whih is di'erent from tht of the rwIEmhV with mgneti hedF et the (rst stepD the rottion frequeny of rwIEylinder synhroE nizes with the rotting mgneti (eldF henD the rottion frequeny of rwIEylinder loses the synhroniztion with the externl rotting (eldD ut it keeps inresingF piE nllyD the rottion frequeny stilizes t (xed vlueD whih is slightly smller thn its mximumF his vlue is lled the sturtion frequeny of rwIEylinderF he sturtion frequeny of rwIEylinder depends strongly on the eynolds numerF et reltively high eynolds numer @Re = 0.54AD the sturtion frequeny is not rehed efore 25 HzF et reltively low eynolds numer @Re = 0.008AD the sturtion frequeny is rehed efore 1 HzF he sturtion frequeny is 0.4 HzF he rwIEylinder rottes t the sturtion frequeny in highEfrequeny rotting mgneti (eldsD suh s 80 HzF Discussion pigure QFQX rwIEylinder with 20 -50 µm thik oted mgneti lyer shows sturtion of frequeny with mgneti (eld of out 30 mT generted y wwF sn the se of millimeterEsled helil swimmersD the rw with n uniformly oted mgneti til is not interestingX euse its totl dipole moment is not strong enoughD due to the lyer thikness limited y the oting proessD ompred to the mgneti heds whih re mde of permnent mgnetsF hereforeD the sturtion frequeny of the rw with oted mgneti til is low ompred to the utEo' frequeny of the rw with mgneti hedF roweverD the helil swimmer with n uniformly oted mgneti til will e very interesting t the mirosleF et the mirosleD the totl dipole moment of the mgneti hed nd the oted mgneti til hve the sme order of mgnitudeF es resultD the utEo' frequeny of helil miroswimmer with mgneti hed nd the sturtion of helil miroswimmer with mgneti oted til should e similr in the sme environmentsF he di'erene is tht eyond the utEo' frequenyD the swimming performne of helil miroswimmer with mgneti hed will strongly dereseD while the performne of helil miroswimmer with oted mgneti til will notF woreoverD in highEfrequeny rotting mgneti (eldD the helil miroswimmer with mgneti hed will stop rottingD nd the helil miroswimmer with oted mgneti til will rotte t the sturtion frequenyF e predit tht the performne of the helil miroswimmer with oted mgneti til is more outstnding thn tht of the helil miroswimmer with mgneti hedD whih should e veri(ed in future worksF 3.2.1.3 SHM5-cylinder with a non-uniformly coated magnetic tail ometimesD the mgneti oting is not perfetly uniform due to the proess of the miroEfritionsF he in)uene of the nonEuniformity of the mgneti oting lyer on the rottionl propulsion hrteristi of rw is studied s followsF he rwSEylinder with nonEuniformly oted mgneti lyerD whih is introE dued in etion PFIFPFPD is used in the experimentF es the rwSEylinder is too long to e pled in the workspe of the Qh relmholtz oil pirsD the ww is used to generte rotting mgneti (eldF ell tht the mgneti )ux density t the zone of interest is out 30 mTF pigure QFR depits the rottionl propulsion hrteristi of rwSEylinderF rwSEylinder rottes t (rst synhronously with the pplied rotting (eld until mximl rottion frequenyD whih is the sme s in the two previous sesF he di'erene is thtD eyond this mximl rottion frequenyD rwSEylinder suddenly stops rottingF ome osilltions of rwSEylinder re oservedF his mximl rottion frequeny is de(ned s utEo' frequenyF feyond this utEo' frequenyD the ville mgneti torque is no longer su0ient to keep the helil swimmers rottingF he vlue of the utEo' frequeny depends on the eynolds numer s wellD whih dereses with the eynolds numerF pigure QFRX foth @A rwSEylinder nd @A rwSEnone with nonEuniformly oted mgneti lyer exhiit utEo' frequeny with rutl stop with mgneti (eld out 30 mT generted y wwF Discussion roweverD the rottion performne of the rw with nonEuniformly oted mgneti (eld dereses rutlly eyond the utEo' frequenyD whih is di'erent from the rw with mgneti hedF st mens tht the nonEuniformly oted mgneti lyer of the helil swimmer ould strongly limit its swimming performneF sf the helil miroswimmers re designed with oted mgneti tilsD the oting proess should e treted refully to ensure the uniformity of the mgneti lyersD to keep their swimming performneF

Rotational propulsion of SHMs with dierent head shapes

sn the stte of the rt of rti(il helil swimmersD severl di'erent hed shpes for helil swimmers hve pperedF roweverD the in)uene of hed shpe on rottionl propulsion hs not een demonstrted yetF he rottionl propulsion hrteristis of the rws with the sme geometry prmeters nd the sme mgneti positioningD ut with di'erent hed shpes re investigted nd ompredF rwSs re used in the following experimentsF ell the rwSs hve the sme geomeE try prmeters nd mgneti positioningD s shown in pigure PFQF he only di'erene of the rwSs is the hed shpeF vet us strt with rwSEnoneD the rw without hedF he rottionl propulsion hrteristi of rwSEnone in di'erent visous liquids is shown in pigure QFRF rwSEnone shows utEo' frequeny with rutl stopF he rottionl propulsion hrteristi of rwSEnone is similr to tht of rwSEylinder shown in pigure QFRF henD the rottionl propulsion hrteristi of rwSEsqure nd rwSEsphereD rws with squre nd sphere hedsD re studiedF hey ll exhiit utEo' frequeny with rutl stopF fefore the utEo' frequenyD they ll rotte synhronously with the externl rotting (eldF pigure QFS summrizes the vlues of the utEo' frequeny of the rwSs with mgneti tils nd di'erent heds s funtion of the eynolds numersF rwSEnone hs the highest vlues of the utEo' frequenyD euse it does not hve hed to rete dditionl )uidi drg torqueF he squre hed retes more drg torque thn the spheril nd ylindril hedsD therefore rwSEsqure hs the lowest vlues of the utEo' frequenyF pigure QFSX lues of the utEo' frequeny of rwSs with the sme geometry prmE eters nd mgneti positioningD ut with di'erent hed shpesF Discussion he hed shpe of helil swimmer does not in)uene the rottionl propulsion hrteristiF he helil swimmers with the sme geometry prmeters nd mgneti positioningD ut with di'erent hed shpes exhiit the sme rottionl propulsion hrteristiX the utEo' frequenyF elthough the hed shpe does in)uene the vlues of the utEo' frequenyD this in)uene is not signi(ntF

Factors inuencing rotational propulsion

he rws with di'erent mgneti positioningsX mgneti hedsD uniformly or nonE uniformly oted mgneti tilsD exhiit di'erent rottionl propulsion hrteristisF rereD the ftor in)uening rottionl propulsion hrteristi is disussedF ih possile mgnetiztion diretion of the rw whih ontriutes to propulsion is de(ned s mgnetiztion phseF e (nd tht the rottionl propulsion hrteristi depends on the mgnetiztion phse of the helil swimmersX

• rwIEmhV with mgneti hed only hs one mgnetiztion phseD whih is in the plne of the mgneti plteF st rottes synhronously with the externl rotting mgneti (eld until utEo' frequenyF feyond this utEo' frequenyD the rottion frequeny of rwIEmhV exhiits gentle deline towrds zeroF

• rwSs with nonEuniformly oted mgneti helil tils hve severl mgnetiE ztion phsesD whih depend on the numer of the smll mgnets overed on the helil tilsF hey rotte in syn with the externl rotting (eld until utEo' frequenyF efter they lose the synhroniztion with the externl rotting (eldD the rottion frequeny exhiits rutl stop to zeroF eording to the rottionl propulsion hrteristisD we predit tht the helil miroswimmers with uniformly oted mgneti tils will e the most promising rE ti(il helil miroswimmersD euse they exhiit sturtion of frequenyF heir propulsion performne will not sustntilly derese with the externl rotting mgE neti (eld t high rottion frequenyF heres the rottion frequeny of the helil miroswimmers with plte mgneti hed will derese eyond the utEo' frequenyF roweverD it is di0ult to predit the vlues of the utEo' frequeny nd the vlues of the sturtion for helil miroswimmers from the results of the rwsD euse those vlues lso depend on the strength of the externl mgneti (eld nd the totl dipole moment of the helil miroswimmersF he oting proess during miroEfrition should e refully mngedD euse if the oted mgneti lyer is not uniformD the propulsion performne will e lrgely limitedF he predition regrding the rottionl propulsion hrteristi of helil miroswimmers t the mirosle will e vlidted in future worksF 3.2.4 Choice of the helical swimmer design ine the hed shpe hs no signi(nt in)uene on the rottion performne of the helil swimmerD it n e designed ording to the requirementsF por exmpleD ylindril hed n e good design to hold ojetsF he hoie of the mgneti positioning design of the helil swimmer is disussed in two sesX t the millimeter sle nd t the mirosleF 3.2.4.1 At the millimeter scale es disussed in etion QFPFIFPD t the millimeter sleD the totl dipole moment of the rw with n uniformly oted mgneti til is not strong enough due to the limited thikness of the oted mgneti lyerF he vlue of the sturtion frequeny of the rw with oted mgneti til is low ompred to the vlue of the utEo' frequeny of the rw with mgneti hedD whih is mde of permnent mgnetF hereforeD rw with permnent mgnet hed is more promising of the swimming t the milE limeter sleF reneforthD only rwIEmhV is used in the visul servoing experimentsD euse it shows the est performne on the utEo' frequeny nd swimming veloityF 3.2.4.2 At the microscale roweverD t the mirosleD the helil swimmer with uniformly oted mgneti til will e more promisingD euse the totl dipole moment of the mgneti hed nd the oting mgneti til hve the sme order of mgnitudeF he mgneti torques pplied on the helil miroswimmers with the sme totl dipole moment re the sme with the sme rotting mgneti (eldF es resultD the vlue of the utEo' frequeny of helil miroswimmer with mgneti hed nd the vlue of the sturtion frequeny of helil miroswimmer with n uniformly oted mgneti til should e similrF fut eyond this vlue of the rottion frequeny of the externl mgneti (eldD the rotE tion performne of the helil miroswimmer with mgneti hed deresesD nd the rottion performne of the helil miroswimmer with n uniformly oted mgneti til stilizes t this rottion frequenyF woreoverD in highEfrequeny rotting mgneti (eldD the helil miroswimmer with mgneti hed stops rottingD nd the helil miroswimmer with oted mgneti til rottes t the sturtion frequenyF sn onE lusionD we predit tht the optiml mgneti positioning of the helil miroswimmer is the helil miroswimmer with n uniformly oted mgneti tilD nd the oting proess should e refully mnged to ensure the uniformity of the oted lyerF 3.3 Propulsion velocity, advance velocity and torque analysis sn this setionD the propulsion veloity generted y the helil propulsionD nd the required mgneti torque re studiedF he liner nd ngulr veloities nd the pplied fores nd torques of rw re relted y the propulsion mtrixF hereforeD the propulsion mtries of rwIEmhV re t (rst identi(ed in di'erent visous liquidsF henD the liner veloity of rwIEmhV is studied sed on the estimted propulsion mtrixF he mgneti torque required to mintin the propulsion is lso nlysed sed on the propulsion mtrixF 3.3.1 Propulsion matrix identication he urrent propulsion mtrix identi(tion method hs shown in etion PFSFQD thtX

a = f app -bω 2 v 2 @QFUA b = f app ω 1 @QFVA c = -bv 2 ω 2 @QFWA
where f app is pplied non)uidi fore on the rwIEmhVF es the rw swims vertillyD the pplied fore is thus the pprent weight f a @the resultnt of the grvity nd uoyny foresA of the rwF ω 1 is the rottion veloity t whih the height of rwIE mhV sty motionlessD ω 2 nd v 2 is the rottion nd trnsltion veloity t whih rwIE mhV drops down freely without pplied fores nd torquesF he propulsion mtrix of rwIEmhV is t (rst estimted in pure glyerol liquidF he eynolds numer is 0.001F he pplied mgneti (eld is 10 mTF sn order to estimte the pplied fore f app D the grvittion nd uoyny fores of rwIEmhV should e estimtedF he mss of rwIEmhV is mesured t 0.02 g with its mgneti hedF he volume of rwIEmhV without its mgneti hed V helix is estimted through gti softwre model s 9 × 10 -9 m 3 F he hed is ylinder with 1.5 mm dimeter @dA nd 0.5 mm thikness @kAF he volume of the hed is lulted s follows

V head = π × d 2 2 × k = 1 × 10 -9 m 3 @QFIHA
he uoyny fore exerted y rwIEmhV in glyerol is then estimted sX

f buoyancy = ρg(V helix + V head ) = 1 × 10 -4 N @QFIIA
he pplied non)uidi fore f app D whih is the pprent weight in this seD is then lulted s

f app = f a = G -f buoyancy = -1 × 10 -4 N @QFIPA
xote tht the diretion of the grvity is de(ned s the negtive diretion @f pigure PFPP in etion PFSFQAF he rottion frequeny orresponding to ω 1 is mesured s 1.9 HzF prom @QFVAD the prmeter b is then estimted sX

b = f app ω = -1 × 10 -4 1.9 × 2π = -8.38 × 10 -6 N • s @QFIQA
he free dropping veloity is mesured s 0.59 mm/sF roweverD the rottion veloity of the free dropping ould not e oserved y eyesF enother method of prmeters estimtion should e usedF es presented in etion IFQFQFID in the se of the mgneti (eld tuted helil swimmersD the trnsltion veloity n e expressed s funE tion of the rottion veloity with the non)uidi pplied fore s seond inputF he reltionship etween the trnsltion veloity nd the rottion veloity is liner if the pplied fore is (xedD nd n e expressed sX

v = f app a - b a ω @QFIRA
he mesured trnsltion veloities of rwIEmhV s funtion of the rottion veloities is depited in pigure QFT with the non)uidi pplied fore is the pprent weight of rwIEmhVF he liner reltionship is vlidted experimentllyF he red line is the liner regression lulted y the si (tting tool of wtlF he eqution is given sX v = 4.9 × 10 -5 ω -5.6 × 10 4 @QFISA he prmeter a n e estimted y the slope of the line sX a = -b 4.9 × 10 -5 = 0.17 N • s/m @QFITA xote tht the vlue of a n e lso lulted y the vEintereptF pigure QFTX rnsltion veloity of rwIEmhV s funtion of rottion frequeny in glyerol with 10 mT mgneti (eldF he red line is the liner regression y wtlF hen with the expression @QFUAD the rottion veloity of the free dropping ω 2 is otined sX

ω 2 = f app -av 2 b = -0.106 rad/s @QFIUA
he orresponding rottion frequeny is 0.017 HzF st mens tht rwIEmhV tkes 58 s to rotte one turn in free droppingF his explins why no rottion is oserved during the free dropping experimentsF et lstD from @QFWA the prmeter c n e estimted sX c = 4.66 × 10 -8 N • m • s @QFIVA henD the propulsion mtries of rwIEmhV in WS7 nd WH7 glyerol solutions re estimted in the sme wyF he rottion frequeny rwIEmhV used to lne its pprent weight re respetively 4 Hz nd 11.3 Hz in WS7 nd WH7 glyerol solutionsF he orresponding ngulr veloities re lulted in le QFSF he liner reltionE ships etween the liner veloity nd the rottion veloity of rwIEmhV in the two glyerol solutions re shown in pigure QFU nd QFUF he free dropping veloities re mesured nd ngulr veloities during the free dropping re estimtedF hey re ll summrized in le QFSF le QFSX engulr nd liner veloities used to estimte the propulsion mtries of rwIEmhV in di'erent visous liquidsD where

ω 1 D v 1 D v 2 re mesuredD nd ω 2 is estiE mtedF 7 glyerol ω 1 @rdGsA v 1 @mmGsA ω 2 @rdGsA v 2 @mmGsA IHH IIFWQ H EHFII EHFSW WS PSFIP H EHFPS EHFWI WH UHFWT H EHFRI EPFIQ (a) (b)
pigure QFUX rnsltion veloity of rwIEmhV s funtion of rottion frequeny in @A WS7 glyerol nd @A WH7 glyerol solutions with 10 mT mgneti (eldF he red line is the liner regression y wtlF he estimted prmeters of the propulsion mtries of rwIEmhV in three glyerol solutions with di'erent visosities re summrized in le QFTF le QFTX ummry of the estimted aD bD c prmeters of the propulsion mtries of rwIEmhV in di'erent visous liquidsF he prmeters a nd c re positiveD nd they inrese with the visosity of the liquidF he prmeter b is negtiveD nd it dereses with the visosity of the liqE uidF hese prmeters re used to estimte the propulsion veloity generted y the propulsion nd the required mgneti torqueF

7 glyerol @ N • s/mA @ N • sA @ N • m • sA IHH 0.

Propulsion velocity and advance velocity

es presented in etion IFQFQFPD in the helil miroswimmers pplitionsD we point them in diretion t n ngle with the horizontl plne to overome its grvityF rereD we de(ne the ngle etween the xis of the rw nd the horizontl plne yOz s the inlintion ngleD whih is noted s θ i D s shown in pigure QFVF he inlintion ngle vries from 0 • to 90 • F pigure QFVX snlintion ngle θ i is de(ned s the ngle etween the xis of the rw nd the horizontl plne yOzF sn the se of helil swimmer tuted y pure rotting mgneti (eldD the pplied non)uidi fore only depends on the pprent weight nd the inlintion ngleD whih n e expressed sX f app = f a sin θ i @QFIWA where f a is the pprent weight of the rwF sn the se of mgneti (eld tuted rwD the fundmentl input is the rottion frequeny of the mgneti (eldD nd the non)uidi pplied fore n e onsidered s the seond inputF es the non)uidi pplied fore depends on the inlintion ngleD thus the inlintion ngle n e the seondry inputF he propulsion veloity of rwIEmhV sed on its propulsion mtrix s funtion of the rottion frequeny of the mgneti (eld nd its inlintion ngle n e expressed sX

v = f a sin θ i a - b a ω @QFPHA
with ω the ngulr veloity of rwIEmhVF es proved erlierD the rw rottes in syn with the rotting mgneti (eld until the utEo' frequenyF hereforeD the propulsion veloity of rwIEmhV n e expressed s X

v = f a sin θ i a - b a 2πf @QFPIA
efore the utEo' frequenyD with f the rottion frequeny of the pplied mgneti (eldF ell tht the utEo' frequeny of rwIEmhV in pure glyerol with 10 mT mgneti (eld is 5 HzF he propulsion veloity of rwIEmhV with di'erent inlintion ngles in pure glyerol with 10 mT mgneti (eld etween -5 Hz nd 5 Hz is then simulted nd shown in pigure QFWF pigure QFWX istimted propulsion veloity of rwIEmhV in the propulsive diretion sed on the propulsion mtrix in pure glyerolF he dvne veloity is de(ned s the liner veloity on the horizontl plneD whih depends on the propulsion veloity nd the inlintion ngleF he dvne veloity of the rwD noted s v h D n e esily lulted sX

v h = v cos θ i @QFPPA
he propulsion veloity is expressed in iqution @QFPIAD then we otinX

v h = f a sin 2θ i 2a - b a 2πf cos θ i @QFPQA
with f the rottion frequeny of the mgneti (eld nd θ i the inlintion ngle of rwIE mhVF he dvne veloities of rwIEmhV in pure glyerol with di'erent inlintion ngles re estimted nd shown in pigure QFIHF he dvne veloity of rwIEmhV in pure glyerol with 3 Hz rotting mgneti (eld nd with n inlintion ngle of 0

• is (a) (b)
pigure QFIHX istimted liner dvne veloity of )oting rwIEmhV on the horizontl plne s funtion of @A the rottion frequeny of the mgneti (eld nd @A the inlintion ngle sed on the propulsion mtrix in pure glyerolF lulted s 0.93 mm/s sed on the estimted propulsion mtrixD whih is mesured s 1.09 mm/s in etion QFIFQ @f le QFQAF he error is estimted s IS7F his liner dvne veloity on the horizontl plne n lso e expressed s funtion of the inlintion ngleD s shown in pigure QFIHF e notie tht the dvne veloity of rwIEmhV on the horizontl n e negtiveD for exmple rwIEmhV with inlintion ngles from 32 • with 1 Hz rotting mgneti (eldD whih should e voided in the future for rw swimming ontrolF prom with 2 Hz rotting mgneti (eldD the dvne veloity of )oting rwIEmhV should e positive with ny inlintion ngleF pigure QFIIX viner dvne veloity of rwIEmhV sinking down to the ottom on the horizontl plne s funtion of @A the rottion frequeny of the mgneti (eld nd @A the inlintion ngle sed on the propulsion mtrix in pure glyerolF hen the rw sinks down to the ottomD the pplied non)uidi fore is 0 N if the frition is negletedF he frition n e negleted if the inlintion ngle of the rw is not 0 • D euse the ontt etween the rw nd ottom is just point onttF sn the se of the rw sinking down to the ottomD the liner dvne veloity on the horizontl plne is lwys positiveD whih is shown in pigure QFII nd QFIIF he liner veloity on the horizontl plne dereses with the inlintion ngleD nd the derese rte inresesF he dvne veloity of the rw is lrgely limited if the inlintion ngle is higher thn 60 • F

Applied torque analysis

he mgneti torque required to mintin propulsion is estimted sed on the propulE sion mtrixF prom iqution @IFVA sed on the propulsion modelD the required pplied torque should eX τ = bv + cω @QFPRA epling the expression of the propulsion veloity y iqution @QFPIAD the required torque n e expressed sX

τ = b a f a sin θ i - b a 2πf + cω @QFPSA
he required torques re estimted t the vertil position of rwIEmhVD whih mens the inlintion ngle θ i = 90 • F he required torque t the vertil position is higher thn t ny other position for the sme propulsionF he estimted required mgneti torques for the propulsion of rwIEmhV efore the utEo' frequenies in di'erent visous liquids sed on the estimted propulsion mtries re depited in pigure QFIPF wore torque is required for rw to rotte in more visous liquids @t lower eynolds numerA t the sme frequenyF heoretillyD the required mgneti torques for the mximl rottion frequeny of the rw t di'erE ent eynolds numers should e the smeD s shown in pigure QFIPF he di'erene of the required torques for the rottion of rwIEmhV t the mximl rottion frquenyD whih is the utEo' frequeny presented in etion QFPFIFID in di'erent visous liquids is due to the is of the propulsion mtrix estimtionF he mgneti torque provided y mgneti tution system to one mgneti devie is limitedF sf the required torque to mintin helil swimmer rotting t frequeny surpsses the mximum mgneti torque tht the tution system n provide to itD the helil swimmer n not rotte t tht frequenyF ht is why there is mximl rottion frequeny @utEo' frequenies nd sturtions of frequenyA for mgnetilly tuted helil swimmerF st lso explins how the vlue of the utEo' frequeny or the vlue of the sturtion frequeny inreses with the eynolds numerD euse less torque is required to mintin the rottion of the helil swimmer t higher eynolds numerF 3.4 Open-loop 3D steering control he steering of the rw is very importnt to relize Qh motionD ut it hs not een extensively studied in the stte of the rtF sn this setionD n openEloop Qh steering ontrol of the rw is investigtedF he steering e0ieny is hrterized y pigure QFIPX @A imulted mgneti torque required for the propulsion of rwIEmhV in di'erent visous liquids sed on the estimted propulsion mtriesF @A heoretil mgneti torque required for the propulsion of helil swimmer in di'erent visous liquidsF the response time to step funtionF e strtegy to improve the steering e0ieny in openEloop steering ontrol is proposed t the endF 3.4.1 Open-loop steering principle he openEloop steering of the rw n e pplied oth on the inlintion ngle nd the diretion ngleF he inlintion ngle θ i hs een de(ned in the lst setion s the ngle etween the xis of the rw nd the horizontl plne yOzF rere we de(ne the diretion ngle θ d s the ngle etween the z xis nd the projetion of the helil swimmer xis on the yOz plneD whih is shown in pigure QFIQF fy ontrolling the diretion ngle of the helil swimmerD its dvne diretion projeted on the horizontl plne n e ontrolledF et the equilirium regimeD the helil swimmer rottes synhronously with the externl rotting mgneti (eld round -→ u t F he xis diretion of the rw should e in the diretion of -→ u t F he mgnetiztion of the rw is noted s

-→ M t D nd the mgneti (eld is noted s -→ B t F foth -→ M t nd -→ B t re
-→ τ t+ = --→ M t+ × --→ B t+ @QFPUA
steering is openEloopF his step ommnd is relized y step funtion of the mgneti (eldD whih n e expressed sX B = B(n i , f, t),

0 s t < 2 s B(n, f, t), t 2 s @QFQHA
where n i is the initil xis of rwIEmhVD nd n is the ommnded xis of rwIEmhVF n = (n x , n y , n z ) T n e expressed y the diretion ngle θ d nd the inlintion ngle

θ i sX n =   n x n y n z   =   sin θ i cos θ i sin θ d cos θ i cos θ d  
@QFQIA me lultion is pplied for n i D nd the initil diretion ngle is (xed t 0 • F he funtion B(n, f, t) is given y iqution @PFWAF he rottion frequeny f of rwIEmhV is kept t 3 Hz in the following experimentsF rwIEmhV is )oting in pure glyerolF he positive diretion of the diretion ngle steering is de(ned s the lokwise diretionF he diretion ngle steering of rwIEmhV is hrterized with di'erent inlintion nglesD suh s 0

• D 45 • D nd 65 • F 3.4.2.1 Inclination angle of 0 •
st is wellEknown thtD with n inlintion ngle of 0 • D if the diretion ngle hnges y 180 • D the helil swimmer will just reverse its rottion diretion to reverse the dvne diretionF roweverD the position of the hed reltive to its ody does not hngeF sn other wordsD the hed n not e lwys kept in the front of or t the k of the helil swimmerF oD we n sy tht helil swimmer n not turn y 180 • with n inlintion ngle of 0 • F he step response to smll diretion ngle steeringD suh s 30 • ngleD is shown in pigure QFIRF he response shows (rst order system hrteristisF he response time is 4.6 sF en osilltion of the diretion ngle ppers t 3 HzF es the frequeny of the osilltion orresponds extly to the rottion frequeny of the rwD we dedue tht this osilltion is used y the dissymmetry of the helil morphology reltive to its xisF he osilltion will llow the nlysis of the rw9s selfErottion phse in future worksF he diretion of rwIEmhV n not e steered towrds 90 • with n inlintion ngle of 0 • F e oserve tht rwIEmhV stopped its selfErottion fter the pplition of the step ommnd to the diretion ngleD euse its xis is no longer long the xis of the rotting mgneti (eldF he mgneti (eld prllel to the xis of rwIEmhVD whih ontriutes to the steeringD is not strong enough to steer rwIEmhV towrds 90 • F he diretion ngle of rwIEmhV stys t 0

• 90 • 180 • -90 • 0 • RFT ∞ ∞ ∞ 45 • PFT SFV SFT RFP 65 • PFQ RFR RFV QFS
he responses of the diretion ngle of rwIEmhV to the step ommnd of 30 • D -90 • D nd 180 • show (rst order system hrteristisF roweverD the response to 90 • ngle diretion steering shows seond order system hrteristisF pigure QFIT nd pigure QFIU depit the imge sequenes of rwIEmhV during the diretion ngle steering t 90 • nd -90 • respetively from the top nd the side mersF prom the pprent length of the rwIEmhV viewed on the imges from the top merD we dedue tht the inlintion ngle of rwIEmhV inreses during the diretion ngle steering towrds 90 • F his inrese of the inlintion ngle is not ovious for the diretion ngle steering towrds -90 • from the pprent lengthF sn the next hpterD we introdue method to estimte the xis of the rw in the Qh spe from the two imgesD in order to lerly he hnge of the inlintion ngle of the rw is de(ned s the inlintion ngleF sn the following experiments of inlintion ngle steering of rwIEmhVD the diretion ngle is kept t 0 • F hereforeD the inlintion ngle of rwIEmhV n e onsidered s the ngle etween the xis z nd the xis of rwIEmhV projeted on the imge tken from the side mer trked y iF he inlintion ngle steering is hrterized y the inlintion ngle response to step ommndF his step ommnd is relized y step funtion of the mgneti (eldD whih is expressed in iqution @QFQHAF he rottion frequeny of rwIEmhV is 3 HzF he initil inlintion ngle is 10 • F he rw is )oting in pure glyerolF he inlintion ngle steering of rwIEmhV requires to steer its inlintion ngle towrds respetively 30 • D 60 • D nd 90 • F he inlintion steering is hieved for ll the ommnd nglesF pigure QFIW shows the step responses to the di'erent inlintion ngle steeringsF ell the responses exhiit (rst order system hrteristisF pigure QFIWX tep responses of the inlintion ngle steering t 30 • D 60 • D nd 90 • with the initil inlintion ngle t 10 • F husD we develop Qh steering strtegy y omining inlintion nd diretion steering for the openEloop steering of the rwF sn the se where the rw n not respond to the diretion ngle steeringD we n inrese the inlintion ngle of the rw to relize the diretion ngle steeringF por exmpleD the diretion ngle steering of rwIEmhV from 0 • to 90 • with 0 • inlintion ngle n not e hievedD s presented in etion QFRFPFIF rere we use the Qh steering strtegy to (rstly inline rwIEmhV t 45 • F he diretion ngle steering t 90 • n e hievedD nd then we ly rwIEmhV down to 0 • F hereforeD the diretion ngle steering of rwIEmhV from 0 • to 90 • with 0 • inlintion ngle n e hieved y using this Qh steering strtegyF 3.4.4 Direction angle steering on a sticky substrate e helil swimmer is not lwys )oting in liquidsX sometimes it sinks down to the ottomF e stiky sustrteD suh s olydimethylsiloxne @hwAD is used to simulte high surfe frition t the mirosleF he diretion ngle steering of rwIEmhV is studied with di'erent inlintion ngles on hw sustrteF le QFV summrizes the response times to diretion ngle diretion steering of 30 • D 90 • D 180 • D nd -90 • on hw sustrteF he response time to the direE tion ngle steering of 30 • with 0 • inlintion ngle on hw is muh greter thn tht in )oting onditionsD whih revels tht the steering performne is limited y high surfe fritionF he diretion ngle steering performne is gretly improved y inresing the inlintion ngle of rwIEmhVF le QFVX esponse times of diretion ngle steering of rwIEmhV with di'erent inlintion ngles on hw @unitX sAF gommnd ngle of diretion steering snlintion ngle

30 • 90 • 180 • -90 • 0 • UFV ∞ ∞ ∞ 45 • RFH TFW VFS SFT 65 • PFU SFT RFT QFW
he diretion ngle steering of 90 • of rwIEmhV with 0 • inlintion ngle n not e hievedF fy using the proposed Qh steering strtegyD we (rst inrese the inlintion ngle of rwIEmhV to 45 • F henD the diretion ngle steering of 90 • is hievedF e predit tht this Qh steering strtegy n improve steering performne t the mirosleF 3.5 Summary sn this hpterD the swimming performne of the rws with di'erent geometry pE rmeters hs een ompredF e found tht the pith of the helil swimmer is the most in)uentil ftor on the swimming performneF he swimming performne of the rw is improved y using longer pithF he rottionl propulsion hrteristis of the rw with di'erent mgneti posiE tionings re lso ompredF he rw with mgneti hed shows utEo' frequenyD nd the rw with oted mgneti lyer shows sturtion of frequenyF he hed shpe does not in)uene the rottionl propulsion hrteristisF et the millimeter sleD the rw with mgneti hed shows etter rottion performne thn the rw with oted mgneti tilD euse the totl dipole moment of the oted thin mgE neti lyer is not s strong s the permennt mgneti hedF es resultD the utEo' frequeny of the rw with mgneti hed is muh higher thn the sturtion freE queny of the rw with oted mgneti tilF roweverD t the mirosleD s the totl dipole moment of the mgneti hed nd the oted mgneti lyer re omprleD we predit tht the rw with oted mgneti lyer hs etter rottion performneD euse the rottion frequeny of the rw with mgneti hed dereses eyond the utEo' frequeny down to zeroD nd the rottion frequeny of the rw with oted mgneti lyer stilizes t nonEzero vlue lled the sturtion frequenyF xote tht the mgneti lyer should e uniformD otherwise the rw with nonEuniformly oted mgneti lyer risks to hve utEo' frequeny s wellF he predition regrding the rottionl propulsion of helil swimmers t the mirosle will e vlidted in future worksF he design of rwIEmhV is hosen for the visul servo ontrol experimentD euse of its performne t the millimeter sleF he propulsion mtries of rwIEmhV is estimted in di'erent visous liquidsF he liner veloity nd dvne veloity generted y the helil propulsion is simulted sed on the estimted propulsion mtriesD s well s the required mgneti torque for the propulsionF en openEloop steering ontrol oth on the diretion ngle nd the inlintion ngle of rwIEmhV is studiedF he steering e0ieny of the diretion ngle steering of rwIEmhV n e inresed y inresing its inlintion ngleF hereforeD we proposed Qh steering strtegy tht inreses the rw9s inlintion ngle if its diretion ngle steering n not e hievedF he losedEloop steering of the rw is studied in the next hpterF Chapter 4

Visual servo control of helical swimmers he openEloop ontrol of the orienttion of helil swimmers hs een studied in the previous hpterF sn this hpterD losedEloop ontrol of the orienttion of helil swimmers in the Qh spe y visul servoing is demonstrtedF henD pth following tsk on the Ph horizontl plne of helil swimmers is proposed sed on the ontrol lw of the mgneti (eldF he visul servo ontrol of the Ph pth following is tested with stright lineF e Qh pth following tsk in the spe is lso disussed for the perspetivesF elEtime orienttion nd position of the helil swimmer re required for the visul servoing tsksF he methods for the reonstrution of the Qh position nd orienttion of helil swimmers from trking in the top nd side views re presentedF sn order to relize the Qh reonstrution of helil swimmers from multiple imgesD we strt the hpter with the mer lirtionF 4.1 Camera calibration 4.1.1 Introduction e mer is mpping etween the Qh world nd Ph imge rrtley PHHRF he mer lirtion is neessry step in Qh omputer vision in order to extrt metE ri informtion from Ph imgesF fsed on the dimension of lirtion ojetsD most lirtion tehniques n e lssi(ed into four tegoriesX E Qh referene ojetEsed lirtion egrwl PHHQX hese methods employ lE irtion ojet tht hs preise Qh geometri feturesF glirtion n e done very e0ientlyF he lirtion ojet usully onsists of two or three plnes tht re orE thogonl to eh otherF ometimesD plne undergoing preisely known trnsltion is lso usedD whih equivlently provides Qh referene pointsF his pproh requires n expensive lirtion pprtus nd n elorte setupD euse its ury diretly depends on the qulity of the lirtion ojetF E Ph plneEsed lirtion hng PHHHD turm IWWWX his kind of methods uses speillyEdesigned plnr pttern with ornersD dots or ny fetures tht n e esily extrted from the imge t few di'erent orienttionsF rere ginD the ury of the lirtion depends on the qulity of the lirtion plnr ptternF etD Ph pttern is fr simpler to mke urtely thn Qh ptternX lser printer does the jo perfetlyF E Ih lineEsed lirtion hng PHHRD rmmrstedt PHHSX his method uses Ih stik with three or more olliner pointsF e mer n e lirted y oserving moving line round (xed pointF he min dvntge of this method is tht no speil pttern is neededF e n for exmple use string of lls hnging from the eilingF his tehnique is reltively newF pew softwres re villeF E selfElirtion rrtley IWWUX his kind of lirtion tehniques uses only point orrespondenes in multiple viewsD without needing to know either the Qh lotion of the points or the mer lotionF fut this requires stronger kground in omputer vision nd more omplex lgorithmsF sn the lst ouple of yersD lirtion with Ph pprtus turned to e the est hoie in most situtions euse of its ese of use nd good ury wedioni PHHRF woreoverD there exists severl open softwres tht re oth esy to use nd e0ient @mer lirtion toolox for wtlD ypengAF es fr s we re onernedD i lso implements onvenient mer lirtion lgorithmF 4.1.2 Notation and perspective projective model he projetion mtrix relting point of the Qh ojet to its Ph imge should e de(nedF pigure RFI shows mer with enter of projetion C nd the optil xis z c F he imge plne is t fous tht is t the fol length f wy from CF e use R w = (O, x, y, z) to denote the Qh world oordinte systemD R c = (C, x c , y c , z c ) to denote the mer oordinte systemD R ip = (o ip , u, v) to denote the Ph imge pixel oordinte systemD nd R im = (o im , x, y) to denote the Ph imge metri oordinte systemF vet us onsider Qh point Q whih oordintes in the world oordinte system re given y w Q = ( w X, w Y, w Z) T F his point written in the mer oordinte system pigure RFIX erspetive projetive modelF

is c Q = ( c X, c Y, c Z)
T F he trnsformtion from the point Q written in the world oordinte system to the mer oordinte system is omposed of rottion c R w nd trnsltion c t w F he reltionship etween the homogeneous oordintes

c Q = ( c X, c Y, c Z, 1) T nd w Q = ( w X, w Y, w Z, 1) T n e expressed sX c Q =     c R w c t w 0 0 0 1     w Q @RFIA
where c R w nd c t w re lled the extrinsi prmeters of the merF iquivlentlyD the extrinsi prmeters de(ne the position of the mer enter nd the mer9s heding in world oordinte systemF he point q is the imge of the point Q on the mer9s imge plne with metri oordintes im q = ( im x, im y) T F he min property of the mer model is tht the reltionship etween the Qh point Q nd its imge projetion q is liner projetiveF he reltionship etween the point Q nd its imge q n e written sX

s im q = P c Q @RFPA
where s is n ritrry sle ftorD P is 3×4 mtrix lled the perspetive projetion mtrix without loss of generlityF he fol length of the mer n e ritrrily tken to 1D then P n e simply written sX P = I 33 0 13 @RFQA sn generlD the origin of the Ph imge pixel oordinte system does not oinide with tht of the Ph imge metri oordinte system where the z c xis intersets the imge plneF vet (u 0 , v 0 ) e the trnsltionF hereforeD the reltionship etween the homogeneous oordintes of the imge point q in Ph imge pixel oordinte system ip q = ( ip x, ip y, 1) T nd Ph metri oordinte system im q = ( im x, im y, 1) T without distortion n e written sX

ip q =   p x x + u 0 p y y + v 0 1   =   p x 0 u 0 0 p y v 0 0 0 1   K im q @RFRA
where p x nd p y orrespond respetively to the sle ftor @in pixelsA long the x nd y xisF he mtrix K is lled the mer lirtion mtrixF xote tht K only depends on the intrinsi prmeters of the merF sn onlusionD mer lirtion is neessry to determine the extrinsi nd inE trinsi prmeters of the merF he trnsformtions etween the oordinte systems re presented in pigure RFPF he reltionship etween the Qh point Q in world oordiE nte system w Q = ( w X, w Y, w Z, 1) T nd its imge projetion q in Ph pixel oordinte system ip q = ( ip x, ip y, 1) T n e written sX

ip q ≡ K [I 33 0 13 ] c R w c t w 0 31 1 w Q @RFSA
where ≡ is the projetive equlityD whih mens the equlity up to ny nonEzero sle ftorF pigure RFPX rmeters to e determined in mer lirtionF sn dditionD one must tke into ount the lens distortionF he trnsformtion of the imge point m from meters to pixels n e expressed sX

u = u 0 + p x x(1 + k ud r 2 ) @RFTA v = v 0 + p y y(1 + k ud r 2 ) @RFUA
where r is the rdil distneD with r 2 = x 2 + y 2 F he trnsformtion of the imge point m from pixels to meters n e expressed sX

x = (u -u 0 )(1 + k du r 2 )/p x @RFVA y = (v -v 0 )(1 + k du r 2 )/p y @RFWA
where r 2 = ((u -u 0 )/p x ) 2 + ((v -v 0 )/p y ) 2 D nd k du is de(ned s the distorted to undistorted distortion prmeterD nd k ud is de(ned s the undistorted to distorted distortion prmeterF 4.1.3 Calibration of the vision system es presented in etion PFPFQD the vision system is omposed of two mersX one is esides the Qh relmholtz oil pirsD nd the other one is on the topD s shown in pigure RFQF en endosope is onneted to the side mer insted of optil lens in order to solve the line of sight prolemF he mer lirtion mtries omposed of the intrinsi prmeters of the top nd the side mers @K t nd K s A need to e lirtedF he trnsformtion mtries etween the two mer oordinte systems nd the world oordinte systemD s shown in pigure RFQ need to e determinedF he lirtion of the vision system is studied s follows with the determintion of the lirtion mtriesD nd the trnsformtion mtries etween the world oordinte system nd the two mer oordinte systemsF pigure RFQX @A he vision system omposed of the top nd the side mers with the Qh relmholtz oil pirsF @A rnsformtion etween the world nd the mer oordinte systemsF 4.1.3.1 Identication of intrinsic parameters e test pttern with 6 × 6 dots mtrixD s shown in pigure RFRD is used to lirte the intrinsi nd extrinsi prmeters of the merF he metri distne etween two dots is 4 mmF he origin of the test pttern oordinte system R tp = (O tp , x tp , y tp , z tp ) is de(ned s the enter of the dot in the (rst row nd the (rst olumnF he x tp xis is in the diretion of the olumnD the y tp xis is in the diretion of the rowD nd the z tp xis is de(ned s z tp = x tp × y tp D s shown in pigure RFRF he positions of the dots in the test pttern oordinte system re knownF he positions of the dots in the Ph pixel imge oordinte system n e estimted y the trking funtion in iF welve pitures with di'erent positions of the test pttern re tken y onvention to estimte the mer prmetersF pigure RFR shows the di'erent positions of the test ptternD nd for eh positionD the test pttern oordinte system is lulted y the funtion of iF he intrinsi prmeters for the top mer re p x = 3722.94D p y = 3723.41D u 0 = 546.99D nd v 0 = 452.54F he distortion is estimted s k ud = -0.261 nd k du = 0.265F hereforeD the lirtion mtrix without distortion of the top mer K

t n (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l)
pigure RFRX welve pitures with di'erent positions of the test pttern re tken to estimte the mer prmetersF he test pttern oordinte system for eh piture is omputed nd mrked y the funtion of iF xote tht R ct = (C t , x ct , y ct , z ct ) denotes the top mer oordinte systemD nd R cs = (C s , x cs , y cs , z cs ) denotes the side mer oordinte systemF he test pttern is put t the position where it ould e viewed y oth mersD s shown in pigure RFSF he imges viewed y the top nd the side mer re depited respetively in pigure RFS nd RFSF pigure RFSX @A glirtion for the determintion of the trnsformtion mtrix etween the two mer oordinte systemsF @A smge viewed y the top merF @A smge viewed y the side merF he trnsformtion mtries ct M tp nd cs M tp from test pttern oordinte sysE tem R tp respetively to the top nd the side mer oordinte systems R ct nd R cs depend on the rottion nd the trnsltion mtries etween themD with ct M tp = ct R tp ct t tp 0 31

1

nd cs M tp = cs R tp cs t tp 0 31
1 F he rottion nd trnsltion mtries re shown in pigure RFTF pigure RFTX rnsformtion etween the two mer oordinte systemsF he reltionship etween point Q in mer oordinte systems @ ct Q in R ct nd cs Q in R cs A nd in test pttern oordinte system @ tp QA n e expressed sX

ct Q = ct M tp tp Q @RFIPA cs Q = cs M tp tp Q @RFIQA hereforeD the reltionship etween the point Q in two mer oordinte systems n e expressed sX ct Q = ct M tp cs M -1 tp ct Mcs cs Q @RFIRA
ith the imges tken y the top nd the side mer of the test ptternD the two trnsformtion mtries from the test pttern oordinte system to the two mer oordinte systems re otined sX

    @RFITA
emrk tht the enter of the top mer oordinte system with lens is estimted on the sensor plne of the top merF fut the enter of the side mer oordinte system is not on the sensor plne of the side merF he enter of the side mer oordinte system is shown in pigure RFTF henD the trnsformtion mtrix ct M cs from the side mer to the top mer oordinte system n e estimted sX

ct M cs = ct M tp cs M -1 tp =    
-0.0220 0.0152 -0.9996 0.0448 -0.9995 -0.0222 0.0217 -0.0084 -0.0219 0.9996 0.0157 0.2518 0 0 0 1

    @RFIUA
his trnsformtion mtrix ct M cs from the side mer to the top mer oordinte

system n e pproximted sX

ct M cs = ct M tp cs M -1 tp =     0 0 -1 0.04 -1 0 0 -0.01 0 1 0 0.25 0 0 0 1     @RFIVA
he (rst three olumns of this mtrix presents the xes of the the side mer oordiE ntes expressed in the top mer oordinte systemF he lst olumn of this mtrix presents the enter of the side mer oordinte system expressed in the top mer oordinte systemF 4.1.3.3 Transformation between the world and the camera coordinate systems he world oordinte system R w = (O, x, y, z) is de(ned y the threeEorthogonl rE rnged relmholtz oil pirsD s presented in etion PFPFRFPF e remind here tht the origin O of the world oordinte system is de(ned t the enter of the relmholtz oils setupF he x xis is de(ned s the mgneti (eld diretion when is pplied positive urrent to the smll oil pirF husD the x xis is in the diretion of the xis of the smll oil pirD whih n e shown in pigure RFUF imilrlyD the y xis is de(ned s the mgneti (eld diretion when is pplied positive urrent to the medium oil pirD nd the z xis is de(ned s the mgneti (eld diretion when is pplied positive urrent to the ig oil pirF he rottion mtrix w R ct from the top mer oordinte system R ct to the world oordinte system R w n e written y expressing the three xes of the top mer system in the world oordintesD whih is pproximtelyX

w R ct =   0 0 -1 1 0 0 0 -1 0   @RFIWA
he relmholtz oil pirs system is pled where the optil enter of the top mer projets on the origin of the world oordinte systemF he trnsltion mtrix w t ct from R ct to R w n e expressed y the oordintes of the enter C t in world oordinte systemD whih is pproximtelyX w t ct = (0.25, 0, 0) T F pigure RFUX rnsformtion etween the world nd the mer oordinte systemsF he two mtries n e veri(ed y the lirtion of i using test pttern with the origin of the test pttern oinident with the enter of the relmholtz oil pirs setupF sn onlusionD the trnsformtion mtrix w M ct from the top mer oordinte system R ct to the world oordinte system R w n e expressed sX

w M ct =     0 0 -1 0.25 1 0 0 0 0 -1 0 0 0 0 0 1     @RFPHA
he trnsformtion mtrix w M cs from the side mer oordinte system R cs to the world oordinte system R w n e expressed sX

w M cs = w M ct ct M cs =     0 -1 0 0 0 0 -1 0.04 1 0 0 0.01 0 0 0 1     @RFPIA
ell the trnsformtion mtries re identi(edF he vision system with two mers is therefore lirtedF he lirtion mtries of the two mers nd the trnsforE mtion mtries etween the world nd the mer oordinte systems re used for the pose estimtion of the helil swimmerF his si lgorithm llows to estimte the prmeters of the mersF his estiE mtion method is not perfetF roweverD the highly preise mer lirtion is very expensive in timeD nd it is not the ojetive of this thesisF 4.2 Visual servo control for the 3D orientation en openEloop steering ontrol of the rw is presented in etion QFRF sn this setionD visul servo ontrol of the mgneti (eld to hieve the steering of the rw to ny trget orienttion in the Qh spe is studiedF sn order to relize the ontrolD the relEtime orienttion of the rw in the Qh spe should e t (rst estimtedF 4.2.1 Real-time orientation estimation in the 3D space he orienttion of the rw is onsidered s the xis of the rw in the Qh speF e method to estimte the norml vetor of the rw xis -→ n in world oordinte system n is introdued hereD using the xes tht re trked in the two imges tken with the top nd the side mers with known lirtion nd posesF eE ll tht R ct = (C t , x ct , y ct , z ct ) depits the oordinte system of the top merD R imt = (O imt , x imt , y imt ) depits the undistorted metri imge oordinte system of the top merD nd where ct h t = (h tx , h ty , h tz ) T is the unit norml vetor of the plne P 1 F es P 1 psses through the origin C t D then we hve d = 0F

R ipt = (O ipt , u ipt , v ipt )
he point P D with ct P = ( ct X, ct Y, ct Z) T in the top mer oordinte systemD is point on the xis of the rwD for exmple the ryenter of the rwF husD P is in the plne P t F hereforeD we hveX

h tx ct X + h ty ct Y + h tz ct Z = 0 @RFPQA henD h tx ct X ct Z + h ty ct Y ct Z + h tz = 0 @RFPRA fy de(nitionD the point p t with im p t = ct X ct Z , ct Y ct Z
T is the imge of the point P in the metri imge plne of the top merF

im pt = ct X ct Z , ct Y ct Z , 1
T is its homogeneous oordintesF e n write the reltionship under the mtrix formX ct h T t imt pt = 0 @RFPSA hen we multiply y K -1 t K t D whih is the identity mtrix ID with K t the intrinsi prmeters mtrix of the top merF

ct h T t K -1 t (K t imt pt ) = 0 @RFPTA
prom the reltionship etween the metri oordintes nd the pixel oordintes in the imge plneD we hveX

ct h T t K -1 t ipt pt = 0 @RFPUA e de(ne K -T t = (K -1 t ) T = (K T t ) -1 D then (K -T t ct h t ) T ipt pt = 0 @RFPVA vet us introdue ipt h t = K -T t ct h t K -T t ct h t @RFPWA henD we hve ipt h T t ipt pt = 0 @RFQHA
he point p t D whih is the imge of the point P @ryenter of the rwAD n e trked in the imge plne y iD s presented in etion PFRFPF he oordintes of p t in the imge pixel plne n e expressed s ipt pt = (u p , v p , 1) T F enother point q t D whih is the imge of point Q on the xis of the rwD n e otined from the ryenter nd the xis orienttion of the rwF his point in the top mer imge plne n e expressed s ipt qt = (u q , v q , 1) T F hen ipt h T t = (h ipx , h ipy , h ipz ) T n e lulted from these two points p t nd q t D withX

       h ipx u p + h ipy v p + h ipz = 0 h ipx u q + h ipy v q + h ipz = 0 h 2 ipx + h 2 ipy + h 2 ipz = 1 @RFQIA
he unit norml vetor of the plne P t written in the top mer oordinte system R ct n e otined from the inverse of @RFPWAX

ct h t = K T t ipt h t K T t ipt h t @RFQPA
fy the sme methodD the unit norml vetor -→ h s of the plne P s de(ned y the projetion enter C s of the side mer nd the xis of the rw -→ n written in the side mer oordinte system R cs n e otined yX

cs h s = K T s ips h s K T s ips h s @RFQQA
where K t is the intrinsi prmeters mtrix of the top merD nd ips h s is lulted from two points on the xis of the rw in the imge of the side merF he xis of the rw is the intersetion of the two plnes P t nd P s F st then n e lulted y the vetor produt of the two norml vetors -→ h t nd -→ h s of the two plnesF hereforeD the xis of the rw -→ n in the top mer oordinte system ct n n e expressed sX

ct n = ct h t × ct h s = ct h t × ( ct R cs cs h s ) @RFQRA
where ct R cs is the rottion mtrix from the side mer oordinte system R cs to the top mer oordinte system R ct D whih is given in setion RFIFQFPF henD the norml vetor of the rw9s xis in Qh world oordinte system n e written sX n = w R ct ct n @RFQSA where w R ct is the rottion mtrix from the top mer oordinte system R ct to the

u   n y -n x 0   + B 0 sin(2πf t) v   n x n z n y n z -n 2 x -n 2 y   B ⊥n -sign@B ⊥n • n * A • λ n × n *   n x n y n z   B n @RFRRA
where B 0 is the mgneti )ux density generted in the enter of the workspeD f is the frequeny of the rottionD u =

n 2 y + n 2 x D v = (n x n z ) 2 + (n y n z ) 2 + (-n 2
x -n 2 y ) 2 D nd λ is the ontrol prmeterF sn our systemD the mgneti (eld is ontrolled y the output voltges of gs rd to the three relmholtz oil pirsF eording to @PFIHAD the output voltges @U s , U m , U b A pigure RFIHX flokEdigrm of the visul servo ontrol of the rw orienttion in the Qh speF n e expressed sX

  U s U m U b   = cos(2πf t) u   U x n y -U y n x 0   + sin(2πf t) v   U x n x n z U y n y n z -U z (n 2 x + n 2 y )   -sign(B ⊥n • n * ) • λ U n × n *   n x U y /U x n y U z /U x n z   @RFRSA
where λ U is the ontrol prmeter for the output voltgeF U x D U y D nd U z re respeE tively the output voltge from the gs rd to the smllD mediumD nd ig oils to generte mgneti (eld of mplitude B 0 F he vlues re given in le PFTF eording to our knowledgeD the ontrol lw of the mgneti (eld for the losedEloop ontrol of the orienttion steering of helil swimmers is estlished for the (rst timeF 4.2.3 Results he results of the visul servo ontrol of the rw orienttion in the Qh spe re presented in this setionF pirstD the estimtion method of the orienttion of the rw is vlidted in the Qh speF henD the ontrol prmeter λ U is determinedF et lstD the visul servo ontrol of rw orienttion steering is studied with di'erent ommnd nglesF 4.2.3.1 Real-time axis estimation sn order to verify the method of the reonstrution of the xis of the rw in the Qh speD the rw is trked in n openEloop orienttion ontrolF rereD one exmple of input ngles for the openEloop ontrol is diretion ngle of θ d = 60 • nd n inlintion ngle of θ i = 60 • F henD the Qh orienttion should e lulted ording to the ryenter trked on the two imgesD to dedue the relEtime diretion nd inlintion nglesD nd ompre with the input diretion nd inlintion nglesF pigure RFII presents rwIEmhV trked y the top nd the side mers during the openEloop orienttion ontrolF he rw is trked in oth imgesF he xes of the rw re estimted nd mrked with red linesF ith the intrinsi prmeters of the vetorsF hereforeD ording to @RFRSAD the ontrol prmeter should not e higher thn QF gonsequentlyD we ompre only the visul servo ontrol of the orienttion of the rw with the ontrol prmeter under threeF he ontrol prmeter is hosen y the performne of the visul servo ontrol of the diretion ngle of rwIEmhVF he diretion ngle is ontrolled from 0 • to 60 • y step funtionD with the inlintion ngle kept t 40 • F he rottion frequeny is 2 HzF pigure RFIP depits the responses of the diretion ngle lulted y the relEtime estimtion to the sme step funtion with di'erent ontrol prmeters λ U = 1, 2, 3F he ontrol prmeter is hosen s PD euse of the response time nd resonle overshootingF he inlintion ngle of rwIEmhV is lso monitoredF pigure RFIP depits the relEtime inlintion ngle of rwIEmhVF e oserve tht there is n impulse of the pigure RFIPX esponses of the relEtime @A diretion ngle nd @A inlintion ngle of rwIEmhV to the sme step funtion of diretion ngle ontrol with di'erent ontrol prmeters λ U = 1, 2, 3F inlintion ngleD lthough the ontrol is only on diretion ngleF he inlintion ngle inreses nd then dereses to the initil vlue he inlintion ngle inreses more with higher ontrol prmeterD euse the mgneti torque exerted on rwIEmhV is higherF pigure RFIQX rjetories of rwIEmhV for the visul servo ontrol of the diretion ngle with di'erent ontrol prmeters on n unit sphereF he trjetory of the visul servo ontrol of the diretion ngle of the rw with di'erent ontrol prmeters is tred in pigure RFIQF he movements re desried in the referene frme reltive to the enter of the hed of rwIEmhVF he trjetory of the end point of rwIEmhV9s til is thus tred on n unit sphereD with the enter de(ned s the enter of rwIEmhV9s hedF here re stti errors etween the trget orienttion nd the (nl orienttion of the rwIEmhVF hese errors re the sme for the di'erent prmetersD nd re out -2 • F ht is out Q7F hese errors re due to the lirtion of the mersF yne the ontrol prmeter is hosenD the diretion ngle ontrol with more omE mnd ngles n e tested in the following experimentsF 4.2.3.3 Visual servo control of dierent direction angles he visul servo ontrol of the diretion ngle with hosen ontrol prmeter λ U = 2 is (rstly vlidted with zero ngle ommndF st mens tht visul servo ontrol of the diretion ngle without initil error is testedF he diretion ngle of rwIEmhV does not hngeF rwIEmhV keeps going in the sme diretionF he visul servo ontrol of diretion ngle with smll ngle ommnd is tested with 5 • F he ngulr error etween the trget diretion ngle nd the (nl diretion ngle of rwIEmhV fter the visul servo ontrol of the diretion ngle with 5 • is shown in pigure RFIRF e n see tht the visul servo ontrol is sensitive to smll ngle ommnd suh s 5 • F pigure RFIRX engulr error of the visul servo ontrol of the diretion ngle of rwIE mhVF vterD the visul servo ontrol is tested with trget ngles tht re every 60 • in turnF hey re 60 • D 120 • D 180 • D -120 • D nd -60 • F ell these visul servo ontrol of the diretion ngle re with the sme inlintion ngle of 40 • F st mens tht rwIE mhV n steer t ny diretion in the horizontl plne with 40 • of inlintion ngleF he ngulr errors etween the trget diretion ngle nd the (nl diretion ngle of rwIEmhV re shown in pigure RFIRF hese errors vry from -6 • to 3 • F he desired n * is given in solute vlueF his error n e removed y omputing n * from pir of desired orienttion of the rw tken y the mersF he trjetories of the end of rwIEmhV for the visul servo ontrol of di'erent diretion ngles with initil diretion ngle of 0 • nd inlintion ngle of 40 • on n unit sphere re shown in pigure RFISF emrk tht for the visul servo ontrol of diretion ngles with negtive ommnd nglesD suh s -60 • nd -120 • D the inlintion dereses nd then inreses during the ontrolF everselyD for the visul servo ontrol of diretion ngles with positive ommnd nglesD the inlintion inreses nd then dereses during the ontrolD whih is shown in pigure RFIPF p to disturneD the pigure RFISX rjetories of rwIEmhV for the visul servo ontrol of di'erent diretion ngles with inlintion ngle of 40 • on n unit sphereF ontrol tkes the shortest pth on the unit sphere s expeted y using the geodesi error n × n * F sf one would like to follow prllel pth @inlintion ngle onstntA on the unit sphereD then one should tke desired orienttion s funtion of timeX n * (t)F 4.2.3.4 Visual servo control for both direction and inclination angles ine rwIEmhV n steer t ny diretion in the horizontl plne with 40 • of inE lintion ngleD we omine the visul servo ontrol of the diretion ngle with the inlintion ngleF he initil diretion ngle nd inlintion ngle of rwIEmhV re respetively 0 • nd 40 • F rereD the diretion ngle ommnd is (xed t 60 • F he inE lintion ommnd ngle of the inlintion ngles re tested with 30 • nd 60 • F he trjetories of the end of rwIEmhV for the visul servo ontrol re tred on n unit sphereD s shown in pigure RFITF he visul servo ontrol of the orienttion of the rw in the Qh spe is vlidtedF eording to our knowledgeD the losedEloop ontrol of the orienttion of the rw is relized for the (rst timeF he losedEloop steering ontrol of helil swimmers is pigure RFITX rjetories of the rw for the visul servoing ontrol for oth diretion nd inlintion ngles on n unit sphereF ruil to relize ontrolled Qh motionF 4.3 Visual servo control for the path following es the visul servo ontrol of the orienttion of the rw hs een sueededD we re mitious to relize more omplited tsksD for exmple pth following tsk for the rwF xotie tht the rw is nonholonomi systemF xonholonomi systems reD roughly spekingD mehnil systems with onstrints on their veloity tht re not derivle from position onstrints floh PHHSF he on(gurtion of the rw is repE resented y the position nd orienttion of its min odyF he rw must reh nd follow geometri pth in the grtesin spe strting from given initil on(guE rtion @on or o' the pthAF por this tskD time dependene is not relevnt euse one is onerned only with the geometri displement etween the rw nd the pth vumond PHHIF sn this ontextD the ommnd inputs n e ritrrily sled with respet to time without hnging the resulting pth of the rwF 4.3.1 Real-time position estimation in the 3D space he orienttion of the rw in the Qh spe is presented in etion RFPFIF sn order to know the distne etween the rw nd the pthD the position of the rw should lso e knownF he relEtime position of the rw is desried y its ryenter in the Qh speF e method to estimte the relEtime position of the rw in the Qh spe from its positions in the two imges tken with the top nd the side mers with known lirtions nd poses is introdued s followsF his proess requires the intersetion of two known rys in speD nd is ommonly known s tringultionF hen noise is presentD the two rys will not generlly meetD in whih se it is neessry to estimte the est solution of the point in Qh speF he ryenter of the rw in Qh spe is mesured s g t nd g s in the two imges tken with the top nd side mersF he two mer perspetive projetion mtries P t nd P s orresponding to the two imges re knownF ine there re errors in the mesured points g t nd g s D the rys kEprojeted from the points re skewedF st mens tht there will not e point G in Qh spe whih extly stis(es g t = P t G nd g s = P s GD nd the imges points do not stisfy the epipolr onstrint g T s Fg t = 0F

e desirle feture of the method of tringultion used is tht it should e invrint under trnsformtion of the pproprite lssF henote y τ tringultion method to ompute the Qh spe point G from the mesured points nd pir of mer mtriesF G = τ (g t , g s , P t , P s ) @RFRUA he tringultion is sid to e invrint under trnsformtion H ifX τ (g t , g s , P t , P s ) = H -1 τ (g t , g s , P t H -1 , P s H -1 ) @RFRVA his mens tht tringultion using the trnsformed mers results in the trnsE formed pointF sf the mer mtries re known only up to n 0ne @or projetiveA trnsformtionD then it is lerly desirle to used n 0ne @or projetiveA invrint tringultion method to ompute the Qh spe pointF hereforeD it is inpproprite to minimize errors in the Qh speF ynly imge distnes re minimized with simple liner tringultion methodsF he liner tringultion method is the diret nlogue of the hiret viner rnsE formtion @hvA method rrtley PHHRF sn eh imgeD sed on the projetive model of the merD we hve the reltionship etween the point on the metri imge nd the point in the Qh mer oordinte systemX imt g ≡ (I 0) ct G ims g ≡ (I 0) cs G @RFRWA he points in the metri imge oordinte system n e otined from the points mesured in the pixel imge oordinte system sX

imt g = K -1 ct ipt g ims g = K -1 cs ips g @RFSHA
he points in the metri imges oordinte system nd the Qh points in mer oorE dinte systems re linerly dependentF hereforeD the ross produts re then zeroF

imt g × ct G = 0 ims g × cs G = 0 @RFSIA
ith the trnsformtion etween the two mer oordinte systemsD we n writeX

ct G = ct R cs cs G + ct t cs @RFSPA
with ct R cs the rottion mtrix nd ct t cs the trnsltion mtrix from the side mer to the top merD whih re known from the lirtionF he ross produts re thenX

imt g × ( ct R cs cs G) + imt g × ct t cs = 0 ims g × cs G = 0
@RFSQA e n omine the two equtions in mtrix form suh tht sX

imt g × ct R cs ims g × cs G = -imt g × ct t cs 0 @RFSRA e note [m] × the skewEsymetri mtrix of the vetor m = (m x , m y , m z ) T suh thtX [m] × =     0 -m z m y m z 0 -m x -m y m x 0     @RFSSA
he ross produt of the vetors m nd x n e expressed in mtrix formX [m] × x = m × xF hen we solve the system @RFSRA with ingulr lue heomposition @hA method to get the point q in the side mer oordinte system cs GF he position of the ryenter G in the Qh world oordinte system n e expressed with the trnsformtion from R cs to R w sX

w G = w M cs cs G @RFSTA
he Qh position of the ryenter of the helil swimmer in the world oordinte system n e then determined from the ryenter on the two imgesF he position of the ryenter is used to determine the distne etween the rw nd the referene pthF xote tht this is lso the si lgorithm to estimte point in the Qh speF e etter performne n e hieved y using dvned methods for omputer visionD whih is ostly in time nd is not the ojetive of this thesisF 4.3.2 2D Path following on the horizontal plane por lots of inEvitro pplitions of helil swimmersD suh s the pplitions for lEonE EhipD the ltitude of the helil swimmer is onstnt nd usully t the enter of the hnnelsF he plnned pth to e followed is thus only on the horizontl plneF he helil swimmer is rElike rootF st mens tht helil swimmer n only dvne in the diretion of its xisD ut n not instntneously move sidewysF he ontrol of the pth following of the rw on the horizontl plne n e inspired y n uniyle modelF vet us strt with the nottionsF

        ṡ = v h cos θ e 1 + d/R ḋ = v h sin θ e θe = Ω x - v h cos θ e R + d @RFTSA
4.3.2.3 Control law designed with chained form he ontrol ojetive is to ensure the onvergene of the rw to the referene pth CF hereforeD the stte vriles d nd θ e re expeted to e rought nd kept t zeroF d = 0 mens tht the ryenter of the rw G elongs to pth CF θ e = 0 ensures the diretion ngle of the rw is in the sme diretion s the tngent of the pth CF he stte spe model of the rw @RFTSA is lerly nonlinerF fell proposed to linerize the model round the equilirium d = θ e = 0 using liner systems theoryF sn tht seD ontrol design does not rely on the tul model of the rwD ut on n pproximted oneF mson hve estlished in ontrol theory tht modelds of moile roots n e onverted into lmost liner modelsD nmely hined formsD in n ext wy mson IWWSF uh n pproh is ttrtive nd used y lots of reserhers for rElike moile roots vumond PHHID huilot PHHPD sine it llows us to use liner systems theory while still relying upon the tul nonliner model of the rwF his pproh is followed herefterF he model of the rw @RFTSA n e onverted into hined form dedited to systems with two inputs nd limited to dimension threeF he generl hined system of dimension three is written sX

   ẋ1 = m 1 ẋ2 = x 3 m 1 ẋ3 = m 2 @RFTTA where X = (x 1 , x 2 , x 3 ) T nd M = (m 1 , m 2 )
T re respetively the stte nd ontrol vetorsF sn order to verify tht hined form system is lmost linerD just reple the time derivtive y derivtion with respet to the stte vrile x 1 F sing the nottionsX d dx 1

x i = x i nd m 3 = m 2 m 1 @RFTUA the hined form @RFTTA n e rewritten sX    x 1 = 1 x 2 = x 3 x 3 = m 3 @RFTVA
he lst two equtions of system @RFTVA onstitute liner systemF vet us onvert the model of the rw into hined form y using the hnge of the oordintesF x 1 = s @RFTWA onsidering @RFTIAD whih imposesX

m 1 = ẋ1 = v h cos θ e 1 + dc @RFUHA
he virtul ommnd vrile m 1 is ssoited with the evolution of the urviliner oordinteF woreoverD for the simpliity of the trnsformtion into hined formD we hooseX x 2 = d @RFUIA st follows thenX ẋ2 = v h sin θ e = x 3 m 1 @RFUPA hereforeD from @RFUHA nd @RFUPAD the lst stte vrile x 3 is given yX

x 3 = (1 + dc) tan θ e @RFUQA
pinllyD from @RFTTA the lst ontrol vrile m 2 is given yX

m 2 = ẋ3 = d dt ((1 + dc) tan θ e ) @RFURA = tan θ e d dt (1 + dc) + (1 + dc) d dt tan θ e @RFUSA = c ḋ tan θ e +
d = - 1 c D v = 0 nd tan θ e = π 2
[π]F prom prtil point of viewD the rw respets these onditions one properly initilizedF xow tht the model of the rw is onverted to liner modelD the ontrol tehniques for liner system n e ppliedF e use simple expression for the virtul ontrol lwX

m 3 = -k t x 3 -k d x 2 (k t , k d ) ∈ R +2 @RFUVA
where k t nd k d re the ontrol prmeters nd re stritly positiveF epling @RFUVA into @RFTVA givesX x 2 + k t x 2 + k d x 2 = 0 @RFUWA whih implies tht oth x 2 nd x 3 onverge to 0F sn view of @RFUIA nd @RFUQAD d nd θ e onverge to 0 s wellF he referene pth following is therefore hievedF woreoverD sine the evolution of the error dynmis @RFUWA is driven y x 1 = sD the gins (k t , k d ) impose settling distne insted of settling timeF gonsequentlyD the helil swimmer trjetory will e identil if the initil error is the smeF he guidne performnes will e veloity independentF he tul ontrol lw expression on the ngulr veloity of the steering of the rw is s followsD y reporting @RFUHA nd @RFUUA into @RFUVAX

Ω x = - cos 3 θ e 1 + dc k d d + k t (1 + dc) tan θ e + dc ds d tan θ e + c tan 2 θ e (1 + dc) + v h cos θ e R + d @RFVHA
where v h is the dvne veloity on the horizontl plneD whih is usully onstntD nd lulted from @QFPQAF por the se tht the referene pth is stright lineD the ontrol of the ngulr speed of the steering on the horizontl plne n e simpli(ed sX

Ω x = -cos 3 θ e (k d d + k t tan θ e ) @RFVIA
he trget diretion ngle n e otined y the relEtime diretion ngle nd the ngulr veloity of the steering of the rwD whih n e expressed sX

θ * d (t) = θ d (t) + Ω x dt @RFVPA
with dt the smple timeF he trget orienttion of the rw @n * A n e lulted with the trget diretion ngle @θ * d A nd trget inlintion ngle @θ * i A from @QFQIAF he inlintion ngle n e t 0 • for some tsksD suh s the trnsport of miro ojets in miro hnnelsF he inlintion ngle n lso e used to ompenste the grvity in the se of the rwsF he ontrol of the mgneti (eld @RFRRA depends on the relEtime orienttion @nA nd the trget Qh orienttion @n * A of the rwD nd n e expressed y @RFRRAF henD the ontrol of the output voltge of the gs rd n e expressed s @RFRSA with the relEtime orienttion @nAD the trget orienttion n * F pigure RFIWX flokEdigrm of the visul servo ontrol of pth following of the rw on the Ph horizontl plneF sn onlusionD the lokEdigrm of the visul servo ontrol of pth following of the rw on the Ph horizontl plne is shown in pigure RFIWF he Qh orienttion nd position of the rw n e otined from the trked rw in the imges tken y the two mers using the proposed estimtion methodsF he distne nd diretion ngle error etween the rw to the referene pth n e then lultedF sing the proposed ontrol lwD the relEtime trget orienttion of the rw n e otinedF he ontrol of the mgneti (eld n e estlished y the relEtime nd trget orienttion of the rwF he output voltges of the gs rd n e onverted from the mgneti (eldF eording to our knowledgeD the ontrol lw of the pth following of helil swimmers is proposed for the (rst timeF e visul servo ontrol of the pth following of the rw with referene pth of stright line on the horizontl plne is tested nd presented s followsF

Results

pirstD the estimtion method of the ryenter of the rw is vlidted in the Qh speF he visul servo ontrol of the pth following is tested with stright line in the Ph horizontl plneF he trjetories of the rw di'er depending on the ontrol prmetersD whih is shown s followsF 4.3.3.1 Real-time position estimation sn order to verify the method of the reonstrution of the ryenter of the rw in the Qh speD rwIEmhV is trked in n openEloop ontrol from the top nd side imgesF he diretion ngle nd the inlintion ngle of rwIEmhV in the openEloop ontrol re respetively θ d = 0 • nd θ i = 40 • F he rottion frequeny of the mgneti (eld is 2 HzF ith this inlintion ngle nd rottion frequenyD rwIEmhV moves on the sustrteF he dvne veloity is estimted s v h = 0.47 mm/s from @QFPQA nd pigure QFIIF he relEtime Qh position of the ryenter of rwIEmhV @G = (x, y, z) T A is estimted with the method presented in etion RFQFIF pigure RFPH depits the evolution of the oordintes xD yD nd zF he trjetory of the ryenter of rwIEmhV tred on the horizontl plned is presented in pigure RFPHF pigure RFPHX @A elEtime position of the ryenter of the rw in the Qh speF @A rjetories of the ryenter of the rw re tred on the horizontl plne yOzF ith this openEloop ontrolD rwIEmhV should move forwrd in the z diretionF hereforeD the oordinte z inresesF he urrent dvne veloity of the rw is lulted s v h = 0.3 mm/sD whih is smller thn the estimted dvne veloityF e ssume tht it is due to the frition etween the rw nd the sustrteF he ltitude of the rw x is kept t onstntF roweverD the oordinte y of the rw is not onstntD whih should eF he Qh relmholtz oil pirs re not perfetF hereforeD the mgneti (eld generted t the enter is not perfetly uniformF wgneti grdient my e retedF he ft tht rwIEmhV does not dvne extly in the z diretion is due to the presene of the mgneti grdientF e expet tht this error n e eliminted with losedEloop ontrolD whih is now disussed s followsF 4.3.3.2 Trajectories for the visual servo control of the path following henD the visul servo ontrol of the pth following of rwIEmhV is tested with refE erene pth of stright line on the Ph horizontl plneF he funtion of the stright line is z = 0F he rw is initilly o' the pthD with n initil distne error of out -5 mmD nd without initil diretion ngle errorF rjetories of the ryenter of the rw for the visul servo ontrol of the Ph pth following with di'erent ontrol prmeters re tred in pigure RFPIF he hoie of the ontrol prmeters depends on the requirement of the systemX rehing the referene quikly or trvelling lessF por the three sesD with k t = 0.1 nd k d = 0.1D the trvelled pth of the rw is the shortestY with k t = 0.1 nd k d = 0.8D the rw rehes the referene pth the fstestF pigure RFPIX isul servo ontrol of Ph pth following on the horizontl plne with di'erent prmetersF en imge sequene of the rw tken y the top mer during the visul servo ontrol of the Ph pth following of stright line is shown in pigure RFPPF he referene pth is drwn with lue line on the imgeF he rw rehes the referene pthD then dvnes long this referene pthF e oserve tht the diretion ngle of the rw is negtive for the ryenter of the rw dvning long the z xisF he generted propulsive fore hs omposition is the -y diretionD in order to ounterlne the mgneti fore reted y the mgneti grdientF efterwrdsD the trjetories of the openEloop ontrol nd visul servo ontrol of the rw to dvne in the z diretion re ompredF he visul servo ontrol of

= ṅ × n c + n × ṅc @RFVWA = (Ω × n) × n c -n × ∂n c ∂s ṡ @RFWHA = n c × (n × Ω) -n × ∂n c ∂s ṡ @RFWIA
with Ω the steering ngulr veloity of the rwF ell tht [m] × is the skewEsymetri mtrix of the vetor m = (m x , m y , m z ) T D whose expression is given in @RFSSAF hen the derivtion of the orienttion error n e expressed sX

ėθ = [n * ] × [n] × Ω -n × ∂n * ∂s ṡ @RFWPA with [n c ] × [n] × =   -n yc n y -n zc n z n yc n x n zc n x n xc n y -n zc n z -n xc n x n zc n y n xc n z n yc n z -n xc n x -n yc n y   @RFWQA
hereforeD with the expression of the ṡD ėθ n e expressed sX ėθ =

[n c ] × [n] × Ω -n × ∂n c ∂s n T c 1 + d/R v @RFWRA
sn onlusionD the stte spe model of the rw in the world oordinte system n e expressed sX

ḋ ėθ 4×1 =        -→ SG T -→ SG 0 1×3 -n × ∂n c ∂s n T c 1 + d/R [n c ] × [n] ×        4×6 v Ω 6 ×1 @RFWSA
he liner veloity n e expressed s v = vn v D where n v is totl veloity diretion of the rwF emrk thtD in the mirosleD euse the grvity of the rw is negletedD this diretion is in generl oinides with the orienttion of the rw n v = nF he stte spe model of the rw n e expressed sX

ḋ ėθ 4×1 =        -→ SG T -→ SG n v 0 1×3 -n × ∂n c ∂s n T c n 1 + d/R [n c ] × [n] ×        4×4 L T d,θ v Ω 4×1 @RFWTA
where L T d,θ is nmed the Qh intertion mtrix of the rwD y referene to the tsk funtion pproh mson IWWIF emrk tht with the projetion on the horizontl plne yOzD we (nd the sme kinemti equtions with @RFTSAF he ontrol lw design for the nonliner system @RFWTA should e studied in future worksF e hope tht helil swimmers n follow more omplited Qh referene pths in the futureF 4.4 Summary sn ghpter QD n openEloop ontrol of the orienttion of the rw hs een studiedF sn this hpterD losedEloop ontrol of the orienttion of the rw in the Qh spe y visul servoing is presentedF he visul servo ontrol of the orienttion of the rw is oseEfsed isul ervoing @fAF hereforeD method to estimte the orienttion of the rw in the Qh spe from its xes trked in the two imges tken with the top nd the side mers is proposed in the (rst pleF he intrinsi prmeters of the two mers nd the trnsformtion mtries eE tween the mer oordinte systems nd the world oordinte system need to e preliminrily determinedF he ontrol of the mgneti (eld is sed on the orientE tion error etween the relEtime Qh orienttion of the rw nd the trget orienttionF he mgneti (eld n e deomposed in the diretions perpendiulr to the xis of the rwD whih ontriutes to the rottion of the rwD nd in the diretion prllel to the xis of the rwD whih ontriutes to steer the rwF he visul servo ontrol of the orienttion of the rw in the Qh spe is tested with di'erent ommnd ngles oth for the diretion nd inlintion nglesF eording to our knowledgeD the losedEloop ontrol of the orienttion of helil swimmers is relized for the (rst timeF henD pth following tsk is proposed for the rwF he rw is desried y the its position nd orienttionF he im of the pth following tsk is to minimize the distne nd orienttion errors to the referene pthF e method to estimte the relEtime Qh position of the rw from its ryenter trked in the two imges tken with the top nd the side mers is proposedF e ontrol lw design with hined form is proposed for the Ph pth following tskF he visul servo ontrol for the pth following of the rw is tested with referene pth of stright line on the plneD nd with the rw initilly oth on nd o' the pthF eording to our knowledgeD the losedEloop ontrol for the pth following tsk of helil swimmers is relized for the (rst timeF e Qh stte model of the rw is proposedF he ontrol design for the Qh pth following of the rw in the spe will e studied in future worksD s well s more referene pthsD suh s irle or helixF Conclusion and perspectives relil miroswimmers ple of propulsion t low eynolds numers hve the poE tentil pplitions on miro)uidD lEonEEhipD nd minimlly invsive mediineD due to their miro sizes nd ontrollilityF everl mgnetilly tuted rti(il helil swimmers hve pperedF hey hve di'erent geometry prmetersD hed shpesD nd mgneti positioningsD ut their in)uenes on swimming performne hve not een demonstrtedF es fr s we knowD none of them hve een ontrolled y loseEloop ontrollerF hereforeD this thesis ims to improve the design of helil swimmers nd to relize their losedEloop ontrolF sn order to improve the design of helil miroswimmersD we hve ompred the swimming performne of sledEup helil miroswimmers @rwsA t the millimeter sle with di'erent geometry prmeters in visous liquidsF e hve found tht the most in)uentil ftor is the pithF he rws with longer pithes show etter swimE ming performneF he rottionl propulsion performne of the rws with di'erent hed shpes nd mgneti positioning re ompredF he rw with mgneti hed shows utEo' frequenyD ut the rw with n uniformly oted mgneti til shows sturtion of frequenyF he hed shpes did not in)uene the rottionl propulsion hrteristis @utEo' frequeny or sturtion of frequenyAD ut they in)uene on the vlues of the utEo' frequenyF e predit tht t the mirosleD s the totl dipole moment of the mgneti hed nd the oted mgneti lyer re similrD the rw with n uniformly oted mgneti til is etterD euse its rottion frequeny stilizes t the sturtion frequenyD nd will not derese down to zero with the externl pplied (eld eyond the utEo' frequenyF he mgneti lyer should e oted uniformlyD otherwise the rw with nonEuniformly oted mgneti lyer risks hving utEo' frequeny s wellF he predition of the rottionl propulsion hrteristis of the helil swimmers t the mirosle will e vlidted in future worksF roweverD t the millimeter sleD the rw with mgneti hed shows etter rottionl propulsion performneD euse the totl dipole moment of the oted thin mgneti lyer is not s strong s the mgneti hed9sF es resultD the vlue of the utEo' frequeny is muh higher thn the vlue of the sturtion of frequeny for rw t the millimeter sleF he liner propulsion veloity of the rw hs een studied sed on the estimted propulsion mtrixF en openEloop ontrol of the orienttion of the rw inluding the diretion ngle nd the inlintion ngle of the rw hs een studiedF he diretion ngle steering e0ieny of the rw n e improved y inresing its inlintion ngleF hereforeD Qh steering strtegy of the rw hs een proposedF e losedEloop ontrol of the orienttion of the rw is proposed using poseEsed visul servoingF e method of relEtime orienttion estimtion of the rw in the Qh spe is introduedF he intrinsi prmeters of the two mers nd the trnsformE tion mtries etween them nd the world oordinte system need to e preliminrily determinedF he ontrol lw of the mgneti (eld is designed proportionl to the geodesi error of the Qh orienttion of the rw to the trget orienttionF he mgE neti (eld n e deomposed in the diretions perpendiulr to the xis of the rwD whih ontriutes to its selfErottionD nd in the diretion prllel to the xis of the rwD whih ontriutes to its steeringF eording to our knowledgeD the visul servo ontrol of the orienttion of the rw in the Qh spe is relized for the (rst timeF henD pth following tsk hs een proposed for the rwD s rw is nonE holonomi systemF he im of the pth following tsk is to minimize the distne nd orienttion errors to the referene pthF e method to estimte the relEtime Qh posiE tion of the rw hs een lso proposed in order to estimte the distne to the pthF he ontrol lw for the pth following on the Ph horizontl plne hs een designed with lineristion of the system thnks to the hined formsF he visul servo ontrol of the pth following of the rw hs een tested with referene pth of stright lineD nd with the rw initilly oth on nd o' the pthF eording to our knowledgeD the losedEloop ontrol of the pth following of the helil swimmers is relized for the (rst timeF he pth following tsks in the Qh spe will e studied in future worksF he results of the entire work in this thesis will e pplied on helil miroswimE mers t the mirosleD s the miroEfrition of the helil miro swimmers hs een msteredF e will t (rst vlidte the sturtion frequeny of the helil miroswimE mer with uniformly oted mgneti tilD nd show tht its swimming performne will not derese with the high rottion frequeny of the externl mgneti (eldF eE ondlyD the losedEloop ontrol of the orienttion of the rw will e pplied on helil miroswimmers y using suitle optil devies with high resolutionF he visul servo ontrol of the Ph pth following tsk on the horizontl plne will e lso pplied on helil miroswimmersF purthermoreD the ontrol lw of the Qh pth following tsk should e studiedF wore omplited Qh referene pthsD suh s Qh lineD irle or helixD will e tested t (rst on the rwsD then on helil miroswimmersF e hope tht the Qh motion of helil miroswimmers n e ontrolled y loseEloop ontroller in order to hieve more tsks for in vitro nd in vivo pplitions in the ner futureF

  mgneti (eld ontrol R c = (C, x c , y c , z c ) gmer oordinte system R cs = (C s , x cs , y cs , z cs ) goordinte system of the side mer R ct = (C t , x ct , y ct , z ct ) goordinte system of the top mer R ims = (O ims , x ims , y ims ) Ph metri imge oordinte system of the side mer R imt = (O imt , x imt , y imt ) Ph metri imge oordinte system of the top merR im = (o im , x, y) Ph imge metri oordinte system R ips = (O ips , u ips , v ips ) Ph pixel imge oordinte system of the side mer R ipt = (O ipt , u ipt , v ipt ) Ph pixel imge oordinte system of the top mer R ip = (o ip , u, v) Ph imge pixel oordinte systemR w (O, x, y, z) orld oordinte system µ 0 ermeility onstnt ω engulr veloity of the selfErottion of the rw Ω x engulr veloity of the steering of the rw on the horizontl plne eld generted y the Qh relmholtz oil pirs B n wgneti (eld prllel to the xis n B ⊥n wgneti (eld perpendiulr to the xis n h s xorml vetor of the plne de(ned y the enter of the side mer nd the xis of the rw in the world oordinte system h t xorml vetor of the plne de(ned y the enter of the top mer nd the xis of the rw in the world oordinte system xiv List of abbreviations K gmer lirtion mtrix K s glirtion mtrix of the side mer K t glirtion mtrix of the top mer n = (n x , n y , n z ) T exis of the rw in the world oordinte system di'erent of the urrents θ d hiretion ngle of the rw θ * d engle etween the urrent tngent of the referene pth nd the z xis θ * d rget diretion ngle of the rw θ e hiretion ngle error etween the rw nd the referene pth θ i snlintion ngle of the rw

  density generted t the enter of the oil pirs B x wgneti (eld generted y smll relmholtz oil pirs B y wgneti (eld generted y wedium relmholtz oil pirs B z wgneti (eld generted y ig relmholtz oil pirs c ropulsion prmeter c(s) gurvture of the referene pth d histne etween the rw nd the referene pth f ottion frequeny G Qh position of the ryenter of the rw g fryenter of of n ojet in the imge k d gontrol prmeter for Ph pth following tsk k t gontrol prmeter for Ph pth following tsk m pq Ph (p + q) th order moments of greyElevel imge n xumer of turns R dius of urvture of the referene pth Re eynolds numer s gurviliner oordinte of the referene pth U b yutput voltge of ig relmholtz oil pirs U m yutput voltge of medium relmholtz oil pirs U s yutput voltge of smll relmholtz oil pirs U x wximum output voltge of smll relmholtz oil pirs for B 0 U y wximum output voltge of medium relmholtz oil pirs for B 0 U z wximum output voltge of ig relmholtz oil pirs for B 0 v viner propulsion veloity of the rw v h viner dvne veloity of the rw on the horizontl plne

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.1 In vitro applications . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.2 In vivo applications . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Swimming at low Reynolds numbers . . . . . . . . . . . . . . . . 6 1.2.1 Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.2 Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.3 Reciprocal motion and non-reciprocal motion . . . . . . . . . . . 8 1.2.4 Locomotion of micro-organisms . . . . . . . . . . . . . . . . . . . 9 1.2.5 Bio-inspired microswimmers . . . . . . . . . . . . . . . . . . . . . 10 1.3 Helical microswimmers . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.1 Helical microswimmer fabrication process . . . . . . . . . . . . . 11 1.3.2 State of art helical microswimmers . . . . . . . . . . . . . . . . . 14 1.3.3 Helical propulsion model . . . . . . . . . . . . . . . . . . . . . . . 21 1.4 Microrobots actuation . . . . . . . . . . . . . . . . . . . . . . . . 24 1.4.1 Actuation method of microrobots . . . . . . . . . . . . . . . . . . 24 1.4.2 State of the art of magnetic actuation systems . . . . . . . . . . 27 1.5 Microrobots localization . . . . . . . . . . . . . . . . . . . . . . . 27 1.5.1 Electromagnetic and magnetic tracking . . . . . . . . . . . . . . 28 1.5.2 Magnetic Resonance Imaging (MRI) . . . . . . . . . . . . . . . . 28 1.5.3 Computed Tomography (CT) and uoroscopy . . . . . . . . . . . 29 1.5.4 Ultrasound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.5.5 Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.6 Visual servoing for microrobots . . . . . . . . . . . . . . . . . . . 31 1.6.1 Introduction of visual servoing . . . . . . . . . . . . . . . . . . . 31 1.6.2 Basic components of visual servoing . . . . . . . . . . . . . . . . 32 1.6.3 Image-Based Visual Servoing (IBVS) . . . . . . . . . . . . . . . . 33 1.6.4 Position-Based Visual Servoing (PBVS) . . . . . . . . . . . . . . 34 1.6.5 Visual servoing for microrobots . . . . . . . . . . . . . . . . . . . 34

  tehniques to generte nonEreiprol motionX @A he )exile or osilltionX )gell rete ptterns suh s trveling wvesF @A he orksrewEtype rotting propulsionX moleulr motor spins pssive teril )gell eott PHHWF

  1.3.3.1 1D helical propulsion modelhe Ih helil propulsion model is proposed y urell in urell IWUUF ynly two degrees of freedom re onsideredX rottion t ngulr speed ω round its xis nd trnsltion t veloity v long tht xis @see pigure IFIVAF fy integrting the p long only one diretion of the helil xisD the result is 2 × 2 propulsion mtrix relting the veloity nd rottion speed to the externl fore nd torqueF pigure IFIVX PEhop propulsion of helil swimmerX rottion round its xis nd trnsltion long tht xis eott PHHWF 2 × 2 mtrix is lled the propulsion mtrix P of the helil propellerF he hydrodynmiists ll this mtrix the resistne mtrixF he onstnts aD bD c nd d re proportionl to the )uid visosity η nd depend otherwise only on the shpe nd the size of the helil propellerF sf every dimension of the propeller is inresed y the ftor kD the new propulsion mtrix P hs elements a = kaD b = k 2 bD c = k 2 c nd d = k 3 dF urell proved in his pper urell IWWU tht the propulsion mtrix must e symE metrilF he motion is tully desried y only three onstntsX trnsltionl motion re oupled y the o'Edigonl prmeter bF reneD n externl pplied fore n indue rotry motionD nd vie versD torque n indue trnsltionl motionF urell pointed out tht the propulsive e0ieny is more or less proportionl to the squre of the o'Edigonl element of the mtrixF hese propulsion prmeters re omputed s eott PHHWX

  -up Helical Microswimmers (SHMs) . . . . . . . . . . . . 40 2.1.1 Why the scaled-up helical microswimmers? . . . . . . . . . . . . 40 2.1.2 Design and fabrication of scaled-up helical microswimmers . . . . 40 2.1.3 Swimming at low Reynolds numbers . . . . . . . . . . . . . . . . 43 2.2 3D Helmholtz coil pairs system . . . . . . . . . . . . . . . . . . . 45 2.2.1 Helmholtz coils principle . . . . . . . . . . . . . . . . . . . . . . . 45 2.2.2 3D Helmholtz coil pairs system design and realisation . . . . . . 45 2.2.3 Vision system implementation . . . . . . . . . . . . . . . . . . . . 47 2.2.4 Magnetic eld calibration and generation . . . . . . . . . . . . . 48 2.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.3 Rotating Permanent Magnet Manipulator (RPMM) . . . . . . 542.3.1 RPMM principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.3.2 RPMM design and realisation . . . . . . . . . . . . . . . . . . . . 55 2.3.3 Rotation frequency control . . . . . . . . . . . . . . . . . . . . . 57 2.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.4 Vision-based tracking . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.4.1 Tracking of a SHM . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.4.2 Barycentre of the helical swimmer . . . . . . . . . . . . . . . . . 61 2.4.3 Orientation of the helical swimmer . . . . . . . . . . . . . . . . . 61 2.5 Swimming characteristics analysis . . . . . . . . . . . . . . . . . 64 2.5.1 Rotation frequency analysis . . . . . . . . . . . . . . . . . . . . . 64 2.5.2 Translation velocity analysis . . . . . . . . . . . . . . . . . . . . . 65 2.5.3 Propulsion matrix identication . . . . . . . . . . . . . . . . . . . 65 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

  pigure PFRX @A he simultion of the wgneti (eld generted y relmholtz oil pir y gomsolF @A wgneti )ux density long the xes of the two oilsD whih is homogeneous t the enter of the two oilsF est rolling onditionsD the thikness of the SHH turns opper wire of 0.8 mm dimeter is estimted s 13.3 mm with width of 24 mmF efter the estimtion of the spe tht SHH turns of opper wire would tkeD the dimeters of the medium nd the ig oils n e determinedF pinllyD the prmeters of the three relmholtz oil pirs re summrized in le PFRF le PFRX pei(tions of threeEpirErelmholtz oils design for fa15 mTF

  pigure PFSX @A hree orthogonl rrnged relmholtz oil pirs geh designF @A hree orthogonl rrnged relmholtz oil pirs experimentl setupF in le PFSF he resistnes of two oils in the sme pir re nerly identilD whih ensure the urrents pssing through the two oils in the sme oil pir re nerly identilF ht in)uenes the homogeneity of the mgneti (eld generted y the oil pirF nexpeted mgneti grdient will e retedD if the urrents re not identilF le PFSX wesured resistnes @ Ω A of eh oils to vlidte tht the two oils of eh pir

  (a) t=0 s (b) t=0.125 s (c) t=0.25 s (d) t=0.375 s (e) t=0.5 s (f) t=0.625 s (g) t=0.75 s (h) t=0.875 s pigure PFWX imultion of Ph rotting mgneti (eld with frequeny of 1 HzF le PFU summrizes the gs rd output voltges nd the phse di'erene to generte rotting mgneti (eld with mgneti )ux density of B 0 = 10 mT t the enter nd round the xes xD yD or zF le PFUX pei(tions of the gs rd output voltges nd the phse di'erene to generte rotting mgneti (eld with mgneti )ux density of B 0 = 10 mT round the xes xD yD or zF exis of rottion

  pigure PFIHX qrphi user interfe to generte Ph rotting mgneti (eldF 2.2.4.4 3D rotating magnetic eld control he genertion of rotting mgneti (eld round n xis in Qh spe is more omE plitedF e method to generte rotting mgneti (eld round n xis -→ n D expressed in the world oordinte system with n = (n x , n y , n z ) T D is introdued s followsF qiving rotting mgneti (eld round the vetor n is equivlent to write the prmetri eqution of vetoril irle with the norml vetor n nd with the rdius physilly presenting the mgneti (eld strengthF n u v O pigure PFIIX otting mgneti (eld round the xis nF xote tht u nd v re the se vetors in the plne orthogonl to nF gonsider u vetor in the plne whih is perpendiulr with nD therefore u nd n re orthogonlF e n let u e u

  ine H is onstruted using the unit vetor of pD H hnges only with the diretion of pF hereforeD the lol (eld rottion xes re invrint to the distne in the sme diretionF he xil nd rdil positionsD whih re de(ned respetively s the positions on the xis ω nd the positions spnned y the rotting MD hve een used in prior works for tution due to their simpliityF he lol (eld rottion xes for those positions re prllel to the ww rottion xisD s shown in pigure PFIQF he mgneti roots re exlusively operted in xil nd rdil positions of wwD the pplied (eld rottes round n xis prllel to the rottion xis Ω of the ww whoney PHIPF 2.3.2 RPMM design and realisation he permnent mgnet used is set of ten permnent mgnet rings from supermagneteF he inner nd outer dimeter of these rings re respetively 6 mm nd 15 mmF he totl length of the mgnet is 60 mmF he mgneti (eld mgnitudes round the mgnet re mesured using rirst qwHV gussmeterF he mgneti (eld strength t the opertion positionD whih is de(ned s 2 cm from the enter of the wwD is out 30 mTF he hole of the mgnet is used to (x itself with housingF he Qh design of the housing is shown in pigure PFIRF he housing is mde of ef nd relized y pigure PFIQX hen dipole mgnet with moment M rottes round n xis Ω with M perpendiulr to ΩD the (eld vetor t ny given postion rottes roundD nd is perpendiulr toD onstnt xisF he xis t vrious positions re illustrted with lrge lue rrows whoney PHIPF rpid prototyping methodF he housing is (xed on the xis of motorF he permnent mgnet is driven y the motor with the onnetion of the housingF (a) (b)

  time (ms) measured rotation speed(hz) pigure PFISX ottion frequeny revolution for desired rottion frequeny t 5 Hz with the D sD h gins K p = 0.3D K i = 1D K d = 0.01F y power supplyF he ww system is designed only for the swimming properties hrteriztion of the rwsF xo visul feedk is returned to the gF le PFIHX ummry of rotting permnent mgnet mnipultor

  g = (x, y) is the position of the ojet in the imgeF hereforeD the position of the ojet n e lulted y g = (x, y)

  sn order to minimize iD i should e di'erentited with respet to θ nd set the result to zeroF e getX tan 2θ = b 0 (a 0 -c 0 ) @PFQTA unless b 0 = 0 nd a 0 = c 0 F sf b 0 = 0 nd a 0 = c 0 D the ojet is too symmetri to llow us to (nd n xis in this wyD sphere for exmpleF E is independent of θF roweverD the vlue of θ n e otined with only the vlue of tan 2θD euse there re three possiilitiesX θ = arctan(tan 2θ)D θ = arctan(tan 2θ) + π 2 or θ = arctan(tan 2θ)to de(ne the θD the width nd the height of the trked dot should e ompredF pour ses should e distinguishedX IF if width b heightD nd tan 2θ > 0D then θ = arctan(tan 2θ) -π 2 PF if width `heightD nd tan 2θ < 0D then θ = arctan(tan 2θ) QF if width `heightD nd tan 2θ > 0D then θ = arctan(tan 2θ) RF if width b heightD nd tan 2θ < 0D then θ = arctan(tan 2θ) + π 2

  parameters of helical tails . . . . . . . . . . . . . . . . 70 3.1.1 A brief introduction to the design of experiments . . . . . . . . . 70 3.1.2 Choice of the parameters . . . . . . . . . . . . . . . . . . . . . . 70 3.1.3 Swimming performance of the SHMs with dierent geometry parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.1.4 Impacts of the geometry parameters . . . . . . . . . . . . . . . . 72 3.2 Rotational propulsion characteristics . . . . . . . . . . . . . . . . 73 3.2.1 Rotational propulsion of SHMs with dierent magnetic positionings 73 3.2.2 Rotational propulsion of SHMs with dierent head shapes . . . . 77 3.2.3 Factors inuencing rotational propulsion . . . . . . . . . . . . . . 79 3.2.4 Choice of the helical swimmer design . . . . . . . . . . . . . . . . 80 3.3 Propulsion velocity, advance velocity and torque analysis . . . 81 3.3.1 Propulsion matrix identication . . . . . . . . . . . . . . . . . . . 81 3.3.2 Propulsion velocity and advance velocity . . . . . . . . . . . . . . 84 3.3.3 Applied torque analysis . . . . . . . . . . . . . . . . . . . . . . . 87 3.4 Open-loop 3D steering control . . . . . . . . . . . . . . . . . . . 87 3.4.1 Open-loop steering principle . . . . . . . . . . . . . . . . . . . . . 88 3.4.2 Direction angle steering characterization . . . . . . . . . . . . . . 89 3.4.3 Inclination angle steering characterization . . . . . . . . . . . . . 94 3.4.4 Direction angle steering on a sticky substrate . . . . . . . . . . . 95 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

  angle of 45 • henD rwIEmhV is inlined up with n inlintion ngle of 45 • F he diretion steerings t 30 • D 90 • D 180 • D nd -90 • ngles re respetively studiedF he step responses to the di'erent diretion ngle steering re shown in pigure QFISF he diretion steerings of rwIEmhV t 180 • n e hieved with 45 • inlintion ngleF xot only the dvne pigure QFIRX tep response to the 30 • diretion ngle steering with 0 • inlintion ngleF diretionD ut lso the hed position hngesF he hed stys t the k of the helil swimmer no mtter wht the dvne diretion isF he 90 • ngle diretion steering n e esily hieved with 45 • inlintion ngleF he response times to the diretion ngle steering re summrized in le QFUF st shows tht the steering performneD whih is hrterized y the response time to step ommndD is improved y inlining rwIEmhV up t 45 • F le QFUX esponse times of diretion ngle steering of rwIEmhV with di'erent inlintion ngles @unitX sAF gommnd ngle of diretion steering snlintion ngle 30

  pigure QFISX tep responses to the 30 • D 90 • D 180 • D nd -90 • ngle diretion steering with 45 • inlintion ngleF oserve the relEtime inlintion ngle of the rwF he inrese of the inlintion ngle is used y the inerti of the selfErottion of rwIEmhVF his lso explins why rwIEmhV turned towrds the ounter lokwise diretion for the diretion ngle steering of 180 • F o vlidte this ssumption in future worksD we hve to frite the rw with the sme geometry prmeters ut with the leftEhnd hirlityD nd see wether it turns towrds the lokwise diretion for the diretion ngle steering of 180 • F pigure QFITX smge sequene of 90 • ngle diretion steering with 45 • inlintion ngleF @AE@fA show the top view of rwIEmhVD nd @gAE@lA show the side viewF (sequene of -90 • ngle diretion steering with 45 • inlintion ngleF @AE@fA show the top view of rwIEmhVD nd @gAE@lA show the side viewF 3.4.2.3 Inclination angle of 65 • he diretion ngle steering of rwIEmhV is tested with n even higher inlintion ngle suh s 65 • F he step responses to the 30 • D 90 • D 180 • D nd -90 • diretion ngle steerings with 65 • inlintion ngle re depited in pigure QFIVF he hrteristis of diretion steering responses with 65 • inlintion ngle re similr to those with 45 • inlintion ngleF heir response times re summrized in le QFUF st shows tht the diretion ngle steering with 65 • inlintion ngle is more e0ient thn tht with 45 • inlintion ngleF pigure QFIVX tep responses to the 30 • D 90 • D 180 • D nd -90 • diretion ngle steering with 65 • inlintion ngleF sn onlusionD in openEloop diretion ngle steeringD the steering performne n e improved y inresing the inlintion ngle of the rwF 3.4.3 Inclination angle steering characterization

Contents 4 . 1

 41 Camera calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.1.2 Notation and perspective projective model . . . . . . . . . . . . . 98 4.1.3 Calibration of the vision system . . . . . . . . . . . . . . . . . . . 100 4.2 Visual servo control for the 3D orientation . . . . . . . . . . . . 106 4.2.1 Real-time orientation estimation in the 3D space . . . . . . . . . 107 4.2.2 Control law design . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.3 Visual servo control for the path following . . . . . . . . . . . . 117 4.3.1 Real-time position estimation in the 3D space . . . . . . . . . . . 117 4.3.2 2D Path following on the horizontal plane . . . . . . . . . . . . . 119 4.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.3.4 Perspectives for 3D path following . . . . . . . . . . . . . . . . . 128 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

  for the side mer re p x = 751.71D p y = 749.72D u 0 = 298.62D nd v 0 = 208.29F he distortion is estimted s k ud = -0.372 nd k du = 0.414F hereforeD the lirtion mtrix without distortion of the side mer K s n of the top nd the side mers re then identi(edF hey re used to onvert for exmple the imge point of the ryenter of the helil swimmer from the metri imge oordinte system to the pixel imge oordinte systemD nd vie versF 4.1.3.2 Transformation between the camera coordinate systems yne the reltive position of the top nd the side mer is (xedD the trnsforE mtion mtrix etween the two mer oordinte systems ct M cs should e foundF

  880 -0.129 -0.457 0.006 0.224 0.736 -0.639 -0.011 0.419 -0.665 -0.618 0

  depits the undistorted pixel imge oordinte system of the top merF imilrlyD R cs = (C s , x cs , y cs , z cs )D R ims = (O ims , x ims , y ims ) nd R ips = (O ips , u ips , v ips ) depit respetively the oordinte systemD the undistorted metri imge oordinte systemD nd the undistorted pixel imge oordinte system of the side merF R w = (O, x, y, z) depits the world oordinte systemD whih is de(ned in setion RFIFQFQF pigure RFVX hetermintion of the xis of the rw in the world oordinte systemF he projetion enter of the top mer C t nd the xis of the rw de(nes plne (C t , -→ n ) nmed s P t D s shown in pigure RFVF he eqution of the plne P t in the top mer oordinte system R ct n e written sX h tx x + h ty y + h tz z + d = 0 @RFPPA

  world oordinte system R w D whih is given in setion RFIFQFQF he orienttion of the rw is onsidered n s wellF yne the orienttion of the rw in the Qh spe n = (n x , n y , n z ) T is knownD the diretion ngle θ d nd the inlintion ngle θ i of the rw previously de(ned in etion QFRFI n e esily otined yX θ inlintion ngles re used to nlyse the orienttion steering of the helil swimmerD nd the diretion ngle is used s n intermedite ontrol vrile for the Ph pth following tskF 4.2.2 Control law design sn generlD the ontrol vrile of the helil swimmer system is the mgneti (eld BF sn our systemD the ontrol vriles re the output voltges of the gs rdD whih depends on the mgneti (eld BF e ontrol lw of the mgneti (eld B to hieve the Qh orienttion steering of the rw is designed s followsF he norml vetor of the rw9s relEtime orienttion is noted s -→ n the mgneti torque exerted on the rw n then e written sX (a) B ⊥n • n * > 0 (b) B ⊥n • n * < 0 pigure RFWX wodelling of the mgneti (eld for rw steering with the projetion of B ⊥n on the trget diretion of the rw n * @A long n * D nd @A long -n * F positive --→ B ⊥n • -→ n * > 0D the mgneti (eld --→ B n whih ontriutes to the steering should e in the diretion of --→ n F sf this projetion is negtiveD --→ B n should e in the sme diretion s + -→ n F hereforeD --→ B n in the world oordinte system B n n e expressed sX B n = -sign@B ⊥n • n * A • λ n × n * n @RFRQA with λ the ontrol prmeterF sn onlusionD the lokEdigrm of the visul servo ontrol of the rw orienttion in the Qh spe is depited in pigure RFIHF he input of the system is the trget orientE tion n * of the rwF he Qh xis of the rw n = (n x , n y , n z ) T in the world oordinte system n e onstruted from the relEtime trking on the imges otined with the top nd the side mersF he ontrol lw of the mgneti (eld to steer rw t the trget orienttion n * in the Qh spe n e written sX B = B 0 cos(2πf t)

  pigure RFIIX rwIEmhV trked from @A the top nd @A the side mers for openEloop orienttion ontrol t the diretion ngle θ d = 60 • nd the inlintion ngle θ i = 60 • F two mers nd the trnsformtion mtries etween the two mers nd the world oordinte systemD the orienttion of the rw t instnt t in Qh world oordinte system is estimted sX n = (0.8758, 0.4150, 0.2466) T @RFRTA eording to @RFQTA nd @RFQUAD the diretion ngle θ d nd the inlintion ngle θ i of the rw re dedued s θ d = 59.3 • nd θ i = 61.1 • F he errors for the diretion ngle nd inlintion ngle re estimted respetively s IFP7 nd IFV7F hese errors re eptle for the orienttion steering studyF he method of the estimtion of the orienttion of the rw in the Qh spe from the xes trked in the two imges tken with the top nd the side mers with known lirtions is then vlidtedF 4.2.3.2 Determination of the control parameter he ontrol prmeter λ U should e determinedF he output voltge for the ig oils is limited to 3 V @f etion PFPFRFIAF he mximl error etween the relEtime xis of the rw nd the trget xis is n × n * = 1D euse oth n nd n * re unit

  the Ph pth following of the rw on the horizontl plne is depited in pigure RFIUX • C(s) is the prede(ned referene pth with respet to the urviliner oordiE nte s of the urveF he urve should e twie di'erentile in R 2 for poseE sed shemD nd one di'erentile for imgeEsed shem gheruini PHHVD gheruini PHHVF he ontrol is poseEsed in our seD so the urve should e twie di'erentileF• θ d is the diretion ngle of the rwF pigure RFIUX relil swimmer modelling of the Ph pth following on the horizontl plneF• -→ v h is the dvne veloity of the rw on the horizontl plneF• G is the ryentre of the rwF• S is the urviliner oordinte of the nerest point on the pth to the rwF• d = SG is the distne etween the rw nd the pthF • -→ i s is the tngent of pth C t SF -→ j s is perpendiulr to -→ i s F • cdenotes the urvture of the pth CD nd R denotes the rdius of urvture t SD with R = 1/cF • θ c is the ngle etween the urrent tngent of the pth nd the z xisD whih n e onsidered s the diretion ngle of the pthF • θ e = θ d -θ c de(nes the diretion ngle error of the rw to the pthF • θ * d is the lulted trget diretion ngle of the rw for the ommnd to hieve the pthF 4.3.2.2 Kinematic equations he im of the pth following ontrol is to minimize the lterl nd ngulr errorsF e propose here to express the stte oordintes with respet to the pth CF wore preiselyD the position G nd the diretion ngle θ d of the rw re desried y the urviliner oordinte sD the distne error of the rw to the pth de(ned s d = SG D nd the diretion ngle error of the rw to the pth de(ned s θ e = θ d -θ c F he stte vetor is vriles of the system re the liner dvne veloity nd the ngulr veloity of the steering of the rw on the horizontl plneX u = v h Ω x @RFSVA he rel ontrol vriles re the output voltge of the gs rd (U s , U m , U b )D whih ontrol the mgneti (eld B generted y the Qh relmholtz oil pirsD n e dedued from the intermedite ontrol vrilesF he derivtion of s is relied on pigure RFIVD in whih O denotes the urvture enter of the pth C t the urviliner oordinte SF he interept theorem llows us to writeX pigure RFIVX herivtion of s equtionF the lterl error d is expressed s followsX ḋ = v h sin θ e @RFTPA he derivtion of the diretion ngle error θ e n e expressed yX θe = θdθc @RFTQA where θd = Ω x is the ngulr veloity of the steering of the rw on the horizontl plneD nd θc = dθ c ds ds dt = c ṡF hereforeD the derivtion of the diretion ngle error is thenX θe = Ω x -v h cos θ e R + d @RFTRA sn onlusionD the stte spe model of the rw moving on the Ph horizontl plne is expressed sX 

  sequene of the visul servo ontrol of the Ph pth following for stright line of the rw with the ontrol prmeters k t = 0.1 nd k d = 0.8 tken y the top merF the rw for stright line pth following with initil position on the pth is testedF he trjetory of the ryenter of the rw for the visul servo ontrol to follow the stright line z = 0 on the horizontl plne yOz is tred in pigure RFPQF his trjetory is ompred with the trjetory of the open loop ontrol with the sme initil positionF yserved on pigure RFPQD the movement of the rw is pertured y the unexpeted mgneti grdientF por the losedEloop ontrol y visul servoingD the rw returns onto the referene line fter the perturtionF hereforeD the visul servo ontrol of the pth following of the rw is dvntgeousD euse the rw n follow the referene pth with the presene of the unexpeted mgneti grdientF pigure RFPQX gomprison of the trjetories of the openEloop ontrol nd visul servo ontrol of rw to dvne in the z diretionF eording to our knowledgeD the losedEloop ontrol of the pth following of helil swimmers is relized for the (rst timeF wore omplited referene pthsD suh s irle or squre will e tested in future worksF 4.3.4 Perspectives for 3D path following por the pplitions in n open speD for exmple for the inEvivo pplitions in eyeE llsD or in the urinry systemD the plnned pth to e followed is in generl in the Qh speF hi'erent from the pth in the horizontl plne for the pplitions for lEonE EhipD the ltitude of the pth vries s wellF por the Qh pth following in the speD the kinemti equtions is written diretly in the Qh speF he ontrol lw design will e studied in futures worksF e now present the nottionsF 4.3.4.1 Notations he nottions of the Qh pth following of the rw in the spe is depited in pigE ure RFPRX • C(s) is the prede(ned referene pth with respet to the urviliner oordinte s of the urveF he urve should e twie di'erentile in R 3 F • -→ n is the relEtime xis of the rwF • -→ v p is the propulsive veloity of the rwD whih is in the sme diretion s -→ n F pigure RFPRX relil swimmer model of the Qh pth followingF • -→ v is the totl veloity of the rwD onsidering the grvity of the rwD whih is in the diretion of -→ n v F • θ d is the diretion ngle of the rwF • θ i is the inlintion ngle of the rwF • G is the ryentre of the rwF • S is the urviliner oordinte of the nerest point on the pth to the rwF • d = SG is the distne etween the rw nd the pthF • -→ n c is the tngent of pth C t SF • θ c is the ngle etween the urrent tngent of the pth nd the z xisD whih is the diretion ngle of the pthF • c(S) denotes the urvture of the pth CF • -→ n * is the lulted trget orienttion of the rw to hieve the pthF 4.3.4.2 Kinematic equations imilr to the Ph pth followingD the im of the ontrol is to minimize the distne nd orienttion errors to the referene pthF he position P nd the orienttion -→ n of the rw re desried respetively y the urviliner oordinte sD the distne error of the rw to the pth de(ned s d = |SG|D nd the orienttion error of the rw to the pthF he orienttion error of Qh pth following is un vetorD whih is de(ned s e θ = n × n c F he stte vetor n e then written sX q vriles of the system re the totl veloity nd the totl ngulr veloity of the rw in the Qh speX u = v Ω @RFVRA he rel ontrol vriles re the output voltge of the gs rd (U s , U m , U b )D whih ontrol the mgneti (eld B generted y the Qh relmholtz oil pirsD n e dedued from the intermedite ontrol vrilesF imilr to @RFTIAD ṡ in the Qh spe n e expressed sX vetor is de(ned s e θ = n × n c @RFVVA henD the derivtion of the orienttion error e θ n e expressed in R w sX ėθ

  le PFIX pei(tions of the rxf nd the rws rwng PHIHD u PHIPF

	2.1.2.2 SHM5s
	Parameters		HNB	SHM5	SHM1-none/-cylinder
	snner dimeter	2.1 µm	5 mm	1 mm
	hikness	27.2 nm	1 mm	0.3 mm
	ith	14 µm	20 mm	4 mm
	idth	2.5 µm	7.2 mm	1.44 mm
	vength	62 µm	90 mm	18 mm
	x of turns		RFS	RFS		RFS
	eight		!	@A 1.75 g	@A 0.04 g
				@A 2.08 g	@A 0.05 g
				@A 2.63 g
				@dA 1.85 g
				@eA 4.98 g
	wgneti mteril	xikel	xdpef	xikel
	wgneti lyer		30 nm	1 mm	20 -50 µm
	1mm	(a) SHM1-mh1		(e) SHM1-mh5
		(b) SHM1-mh2		(f) SHM1-mh6
		(c) SHM1-mh3		(g) SHM1-mh7
		(d) SHM1-mh4		(h) SHM1-mh8
	pigure PFPX rwIs with inner dimeters of 1 mmD mde of resineF heir geometry
	prmeters re de(ned y 2 3 ftoril designF wgneti heds re seted in the slits
	of the hedsF
	le PFPX qeometry prmeters spei(tions of resine rwIEmhsF
	rwIEmh	dimeter	pith	width	n of pith	length
		@ mmA	@ mmA	@ mmA		@ mmA
	I	I		IFS	HFS	IFS	PFPS
	P	I		R	HFS	IFS	T
	Q	I		IFS	IFP	IFS	PFPS
	R	I		R	IFP	IFS	T
	S	I		IFS	HFS	QFS	SFPS
	T	I		R	HFS	QFS	IR
	U	I		IFS	IFP	QFS	SFPS
	V	I		R	IFP	QFS	IR

  le PFQX wesured visosity of di'erent glyerol solution nd lulted eynolds numers t whih rwSs nd rwIs swim in the rottionl propulsion experimentsF

	% of Glycerol viscosity ( mPa • s)	density ( g/m 3 )	Re(SHM5) Re(SHM1)
	QH7	QFP	IFHW	!	HFSR
	SH7	VFS	IFIR	PFTV	!
	TH7	ITFI	IFIU	IFRS	HFII
	UH7	QPFR	IFIW	HFUQ	HFHT
	US7	RQFU	IFPH	HFSR	!
	VH7	TTFS	IFPI	HFQT	HFHQ
	VS7	IPRFT	IFPP	HFPH	!
	WH7	PTHFH	IFPQ	HFHW	HFHHV
	WS7	SSPFH	IFPR	!	HFHHR
	IHH7	ISPRFH	IFPT	!	HFHHI

  is lulted o1ine from the reorded videosF he method of the lultion is presented in etion PFSFPF por eh rwD (ve tests re relizedF he verge veloity of the S tests is used for the e'et estimtionsF rwIEmhTs re very frgileD nd roke fter three testsF husD only three tests re used for the veloity estimtion of rwIEmhTF le QFQ shows the swimming veloities for eh test nd the verge veloities of the rwIEmhs with di'erent geometriesF

	5 mm 4 mm le QFPX pei(tions of the eight rws for geometry prmeters studyF 0.5 mm IFS vevel + 1.2 mm QFS dimeter @mmA pith @mmA width @mmA n of pith length @mmA rwIEmhI IFS IFS HFS IFS PFPS rwIEmhP IFS R HFS IFS T rwIEmhQ IFS IFS IFP IFS PFPS rwIEmhR IFS R IFP IFS T rwIEmhS IFS IFS HFS QFS SFPS rwIEmhT IFS R HFS QFS IR rwIEmhU IFS IFS IFP QFS SFPS rwIEmhV IFS R IFP QFS IR 3.1.3 Swimming performance of the SHMs with dierent geometry parameters he rwIEmhs re tuted y rotting mgneti (eld generted y the Qh relmholtz oil pirsD whih is presented in etion PFPF he mgneti )ux density t the enter is 10 mTD nd the frequeny of the rotting (eld is 3 HzF he xis of the rotting (eld is in the horizontl plneF hen tuted y the rotting mgneti (eldD the rws propelled forwrd in IHH7 glyerol in syn with the externl (eldD nd t the sme time they snk down euse of their own grvityF ynly the dvne veloity is useful to ompre the swimming performneF mgneti (eldD expressed in mm/sF wimming veloity unI unP unQ unR unS Average velocity rwIEmhI HFPR HFQR HFRT HFQV HFQR 0.35 rwIEmhP HFVR HFVS HFVT HFUR IFHP 0.86 rwIEmhQ HFPS HFHW HFIP HFIQ HFIR 0.15 rwIEmhR HFVU HFUU HFVU HFVT HFVT 0.85 rwIEmhS HFRW HFQR HFRI HFPV HFQT 0.37 rwIEmhT IFRP IFTP IFPU xe xe 1.44 rwIEmhU HFPQ HFPH HFHV HFIS HFIR 0.16 rwIEmhV IFHU IFPR HFWS HFWW IFPQ 1.09 3.1.4 Impacts of the geometry parameters Y A = 0.86 + 0.85 + 1.44 + 1.09 4 -0.35 + 0.15 + 0.37 + 0.16 4 = 0.80 @QFIA Y B = 0.15 + 0.85 + 0.16 + 1.09 4 -0.35 + 0.86 + 0.37 + 1.44 4 = -0.19 @QFPA Y C = 0.37 + 1.44 + 0.16 + 1.09 4 -0.35 + 0.86 + 0.15 + 0.37 4 = 0.21 @QFQA Y A×B = 0.35 + 0.85 + 0.37 + 1.09 4 -0.86 + 0.15 + 1.44 + 0.16 4 = -0.04 @QFRA Y B×C = 0.35 + 0.86 + 0.16 + 1.09 4 -0.15 + 0.85 + 0.37 + 1.44 4 = 0.07 @QFSA Y A×C = 0.35 + 0.15 + 1.44 + 1.09 4 -0.86 + 0.85 + 0.37 + 0.16 4 = -0.22 @QFTA hese e'ets of the ftors nd their intertions re depited in pigure QFIF he min ftor whih in)uenes helil swimmer9s performne the most is the pithD euse the e'et of the pith Y A is the highestF he rws with 4 mm of pith swim fster thn those with 1.5 mm of pith with the sme rotting mgneti (eldF fesidesD the numer of the pith hs non ignorle in)ueneF ine the e'et of the numer of he swimming veloity le QFQX wimming veloities of di'erent rwIEmhs in IHH7 glyerol with 10 mT the pith Y

he e'et Y of the ftors nd the possile intertions re evluted y the di'erene etween the verge response vlues for eh ftor or intertion t its highest level @CAD nd tht t its lowest level @EAD whih n e expressed sX C is positiveD the rws with QFS turns show etter swimming performne thn those with IFS turnsF he width of the helil til shows negtive e'et on the swimming performneD euse Y B is negtiveF he rws with smller widths of the helil til show etter swimming performneF xone of the intertion etween the ftors show signi(nt e'etF pigure QFIX i'et of ftors nd the intertions on swimming veloityF

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ITUD PHHTF @gited on pge PVFA ruthinson IWWT F ruthinsonD qFhF rger et FsF gorkeF A tutorial on visual servo controlF ootis nd eutomtionD siii rnstions onD volF IPD noF SD pges TSI !TUHD ot IWWTF @gited on pges QID QP nd QQFA rwng PHHW qilgueng rwngD rideki rshimotoD hominik tF fellD vixin F hongD frdley tF xelson et F honF Piezoresistive InGaAs/GaAs Nanosprings with Metal ConnectorsF xno vettersD volF WD noF PD pges ppF SSR!STID perury PHHWF @gited on pges II nd IPFA rwng PHIH qF rwngD F rliyo et F égnierF Remotely Powered Propulsion of Helical NanobeltsF sn ootis iene nd ystemsD PHIHF @gited on pges IID ITD PWD RP nd RQFA rwng PHII qF rwngD F friveD vF gourudD eF gvnnD yF edelkrimD sF oertEhilipD eF fevertosD sF gnesD F rliyo et F égnierF Electroosmotic propulsion of helical nanobelt swimmersF he snterntionl tournl of ootis eserhD volF QHD noF UD pges VHT!VIWD PHIIF @gited on pges IID ITD PSD RR nd UQFA rwng PHIQ qilgueng rwngD son elexndru svnD toel egnusD rugo lmonD éstien elvoD xiols ghilletD téphne égnier et enneEwrie rghiriE qosnetF Mobile microrobotic manipulator in microuidicsF ensors nd eE tutors eX hysilD PHIQF @gited on pge SFA sshiym PHHH uF sshiymD uFsF eriD wF endoh et eF mzkiF Spiral-type micromachine for medical applicationsF sn wiromehtronis nd rumn ieneD PHHHF wr PHHHF roeedings of PHHH snterntionl ymposium onD pges TS !TWD PHHHF @gited on pges IRD IT nd PPFA sshiym PHHI uF sshiymD wF endohD eF mzki et uFsF eriF Swimming micromachine driven by magnetic torqueF ensors nd etutors eX hysilD volF WID noF IEPD pges IRI ! 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seleted pplitionsF futterworthErenemnnD fostonD tuin IWWIF @gited on pge UFA urtohvil PHHW fFiF urtohvilD vFF hong et fFtF xelsonF Real-time Rigid-Body Visual Tracking in a Scanning Electron MicroscopeF snterntionl tournl of ootis eserhD volF PVD noF RD pges RWV!SIID wrh PHHWF @gited on pges QR nd QSFA uummer PHIH wFF uummerD tFtF eottD fFiF urtohvilD F forerD eF engul et fFtF xelsonF OctoMag: An Electromagnetic System for 5-DOF Wireless Mi-cromanipulationF ootisD siii rnstions onD volF PTD noF TD pges IHHT !IHIUD deF PHIHF @gited on pge PUFA vug PHHW iF vug et FF owersF The hydrodynamics of swimming microorgan-ismsF eports on rogress in hysisD volF UPD noF WD pge HWTTHID PHHWF @gited on pges U nd RRFA vumond PHHI tenEul vumondD editeurF v rootique moileF rité sgPF rerE més sieneD risD PHHIF @gited on pges IIU nd IPPFA vee PHHW tuEung veeD fyungkyu uim et ehEun rongF A exible chain-based screw propeller for capsule endoscopesF snterntionl tournl of reision ingineering nd wnufturingD volF IHD noF RD pges PU!QRD PHHWF @gited on pge SSFA vighthill IWUT tmes vighthillF Flagellar HydrodynamicsF sew eviewD volF IVD noF PD pges ITI!PQHD IWUTF @gited on pge PIFA wk PHHI wt wkF MInimally invasive and robotic surgeryF teweD volF PVSD noF SD pges STV!SUPD PHHIF @gited on pge SFA whoney PHII erthur F whoneyD tohn gF rrzinD ierhr fmerg et tke tF eottF Velocity Control with Gravity Compensation for Magnetic Helical Microswimmers. edvned ootisD volF PSD noF VD pges IHHU!IHPVD PHIIF @gited on pges IID IUD IVD PQD PRD PU nd PVFA whoney PHIP eFF whoney et tFtF eottF Control of untethered magnetically actuated tools with localization uncertainty using a rotating permanent magnetF sn fiomedil ootis nd fiomehtronis @fiooAD PHIP Rth siii e iwf snterntionl gonferene onD pges ITQP !ITQUD june PHIPF @gited on pge SSFA whoney PHIP eFF whoneyD hFvF gownD uFwF willer et tFtF eottF Control of untethered magnetically actuated tools using a rotating permanent magnet in any position. pges QQUS!QQVHF siiiD PHIPF @gited on pges PVD SS nd STFA wlis IWWW izio wlisD prnçois ghumette et ylvie foudetF 2 1/2 D Visual Servo-ingF ootis nd utomtionD siii rnstions onD volF ISD pges PQV!PSHD IWWWF @gited on pge QRFA wrhnd PHHS iF wrhndD pF pindler et pF ghumetteF ViSP for visual servoing: a generic software platform with a wide class of robot control skillsF ootis eutomtion wgzineD siiiD volF IPD noF RD pges RH !SPD deF PHHSF @gited on pge THFA wrey PHIH wohmmed wreyF Contributions to control modeling in visual servoing, task redundancy, and joint limits avoidance. heseD niversité ennes ID héemre PHIHF @gited on pge QRFA wrtel PHHW ylvin wrtelD whmood wohmmdiD yujdi pelfoulD ho vu et ierre ouponneuF Flagellated Magnetotactic Bacteria as Controlled MRItrackable Propulsion and Steering Systems for Medical Nanorobots Operating in the Human Microvasculature. he snterntionl tournl of ootis eserhD volF PVD noF RD pges SUI!SVPD PHHWF @gited on pges PR nd PWFA wson PHHQ oert vF wsonD ihrd pF qunst et tmes vF ressF ttistil deE sign nd nlysis of experimentsF svi isi sx yfefsvs exh essgF svi 8 yxD Pnd éditionD PHHQF @gited on pge UHFA wxeil IWWS FqF wxeilD ogers gF itterD fF ngD wFeF vwsonD qFF qilliesD uFqF ikD iFqF uteD sss rowrd wFeF et wFF qrdyF Characteristics of an improved magnetic-implant guidance systemF fiomedil ingineeringD siii rnstions onD volF RPD noF VD pges VHP!VHVD IWWSF @gited on pge PUFA wedioni PHHR qerrd wedioni et ing fing ungF imerging topis in omputer viE sionF rentie rll D pper ddle iverD xtD eD PHHRF @gited on pge WVFA welle PHHQ F welleD yFqF glder¤ R nD wFeF uio et qFqF pullerF Microstructure evolution in magnetorheological suspensions governed by Mason number. hys ev i tt xonlin oft wtter hysD volF TVD noF R t ID pge HRISHQD PHHQF @gited on pge IHFA wikw IWWT wF wikwD uF oshidD wF uuot et F worimitsuF Visual servoing for micro mass axis alignment deviceF sn sntelligent oots nd ystems 9WTD sy WTD roeedings of the IWWT siiiGt snterntionl gonferene onD volume QD pges IHWI!IHWT volFQD IWWTF @gited on pge QRFA willer PHIP uFwF willerD eFF whoneyD F hmid et tFtF eottF Proprioceptive magnetic-eld sensing for closed-loop control of magnetic capsule endoscopesF sn sntelligent oots nd ystems @syAD PHIP siiiGt snterntionl gonferE ene onD pges IWWR!IWWWD PHIPF @gited on pge PUFA xelson PHIH fFtF xelsonD sFuF uliktsos et tFtF eottF Microrobots for Minimally Invasive MedicineF ennul eview of fiomedil ingineeringD volF IPD noF ID pges SS!VSD PHIHF @gited on pges SD T nd PWFA eyer PHIQ uthrin iF eyerD vi hng et frdley tF xelsonF Bio-inspired magnetic swimming microrobots for biomedical applicationsF xnosleD volF SD noF RD pges ppF IPSW!IPUPD PHIQF @gited on pge IQFA urell IWUU iFwF urellF Life at low Reynolds number. emerin tournl of hysisD volF RSD noF ID pges Q!IID IWUUF @gited on pges TD UD VD W nd PIFA urell IWWU iFwF urellF The eciency of propulsion by a rotating agellum. ro xtl ed i eD volF WRD noF PID pges IIQHU!IID IWWUF @gited on pge PPFA nkin PHHS F nkinF CT and MRIF urgeryD volF PQD noF SD pges ppF ITP!ITSD PHHSF @gited on pge PWFA oper PHHT wrus operD émi hreyfusD ten fudryD wr permigierD térôme fiE etteD rowrd e toneet al. On the dynamics of magnetically driven elastic lamentsF tournl of pluid wehnisD volF SSRD pges ITU!IWHD PHHTF @gited on pge IHFA mson IWWI gF msonD wFvF forgne et fF ispiuF oot ontrolX the tsk funtion pprohF yxford engineering siene seriesF glrendon ressD IWWIF @gited on pge IQIFA mson IWWS glude msonF Control of chained systems application to path following and time-varying point-stabilization of mobile robotsF eutomti gontrolD siii rnstions onD volF RHD noF ID pges TR!UUD IWWSF @gited on pge IPPFA nhez PHII nhezD ee olovevD w rrzim et yq hmidtF Microbots Swimming in the Flowing Streams of Microuidic ChannelsF tournl of the emerin ghemil oietyD volF IQQD noF RD pges UHI!UHQD PHIIF @gited on pge SFA nderson IWVH eF gF nderson et vF eissF Image-based visual servo control using relational graph error signalsF sn IWVH snterltionl onferene on yernetis nd soietyD IWVHF @gited on pge QIFA hik PHHS pritz hikF Whole-body MRI at high eld: technical limits and clinical potentialF iurF diolFD volF ISD noF SD pges ppF WRT!WSWD PHHSF @gited on pge PWFA ievers PHHS F ievers et F ptikowF Visual servoing of a mobile microrobot inside a scanning electron microscopeF sn sntelligent oots nd ystemsD PHHSF @sy PHHSAF PHHS siiiGt snterntionl gonferene onD pges IQSH!IQSRD PHHSF @gited on pge QHFA leigh IWVV wF eF leighD tF F flke et xF vironF The propulsion of mucus by cilia. he emerin review of respirtory diseseD volF IQUD noF QD pges UPT!URID wrs IWVVF @gited on pge UFA turm IWWW FpF turm et FtF wynkF On plane-based camera calibration: A general algorithm, singularities, applicationsF sn gomputer ision nd ttern eognitionD IWWWF siii gomputer oiety gonferene onFD volume ID pges P volF @xxiiiCTQUCTTQAD IWWWF @gited on pge WVFA mdzte PHHW fF mdzteD F ernouldD F hemeleD xF ve portEit et iF wrE hndF Real-time vision-based microassembly of 3D MEMSF sn edvned sntelliE gent wehtronisD PHHWF esw PHHWF siiiGewi snterntionl gonferene onD pges VV!WQD PHHWF @gited on pge QSFA endik IWWV pF endikD FF stryD FF pering et wF gohnF Applications of micromechatronics in minimally invasive surgeryF wehtronisD siiiGewi rnstions onD volF QD noF ID pges QR!RPD IWWVF @gited on pge SFA huilot PHHP fF huilotD gF griouD F wrtinet et wF ferdutF Automatic Guidance of a Farm Tractor Relying on a Single CP-DGPSF eutonF ootsD volF IQD noF ID pges SQ!UID tuillet PHHPF @gited on pge IPPFA ottori PHIP F ottoriD vF hngD pF iuD uF urwzykD eF prnoEyreg¤ R n et fFtF xelsonF Magnetic Helical Micromachines: Fabrication, Controlled Swimming, and Cargo TransportF edvned wterilsD volF PRD noF TD pges ppF VII!VITD perury PHIPF highlighted s the front overF @gited on pges IQD IV nd IWFA ruper PHHR F ruperD eF uortshkD wF thnishD rF rulsen et F ptikowF Transporting cells with mobile microrobotsF xnoiotehnologyD sii roeedings ED volF ISID noF RD pges IRS!ISHD PHHRF @gited on pge SFA ilkomerson IWWU hF ilkomerson et hvid vyonsF A system for ultrasonic beaconguidance of catheters and other minimally-invasive medical devicesF ltrsonisD perroeletris nd prequeny gontrolD siii rnstions onD volF RRD noF PD pges RWT!SHRD IWWUF @gited on pges PW nd QHFA illims PHIR frin t illimsD ndeep enndD tgnnthn jgopln et w her e ifF A self-propelled biohybrid swimmer at low Reynolds number. xt gommunD volF SD pge QHVID PHIRF @gited on pge IHFA u PHHS ioEghen uD ghngErong ruD vei unD tF en et uFuF hungF Highfrequency high frame rate ultrasound imaging system for small animal imaging with linear arraysF sn ltrsonis ymposiumD PHHS siiiD volume QD pges IRQI!IRQRD PHHSF @gited on pge PWFA u PHIP F uD qF rwngD xF endre' et F égnierF Scaled-Up Helical Nanobelt Modeling and Simulation at Low Reynolds NumbersF sn sge9IP siii snterntionl gonferene on ootis nd eutomtionD pges RHRS ! RHSID PHIPF @gited on pges PPD PU nd RPFA esin PHHT ueml ferk esinD url ollmers et frdley tF xelsonF Modeling and control of Untethered Biomicrorobots in a Fluidic environment using Electromagnetic FieldsF snterntionl tournl of ootis eserhD volF PSD noF SETD pges SPU!SQTD PHHTF @gited on pge PUFA hng PHHH F hngF A exible new technique for camera calibrationF ttern enlE ysis nd whine sntelligeneD siii rnstions onD volF PPD noF IID pges IQQH ! IQQRD nov PHHHF @gited on pge WVFA hng PHHR hengyou hngF Camera calibration with one-dimensional objectsF tE tern enlysis nd whine sntelligeneD siii rnstions onD volF PTD noF UD pges VWP !VWWD july PHHRF @gited on pge WVFA hng PHHW vF hngD tFtF eottD vF hongD fFiF urtohvilD hF fell et fFtF xelsonF Articial bacterial agella: Fabrication and magnetic controlF epplF hysF vettFD volF WRD noF TD PHHWF @gited on pges RD ISD PP nd QSFA hng PHHW vF hngD tFtF eottD vF hongD uFiF eyerD fFiF urtohvilD rF hngD gF fergeles et fF tF xelsonF Characterizing the Swimming Properties of Articial Bacterial FlagellaF xno vettersD volF WD pges QTTQ!QTTUD ytore PHHWF @gited on pges IID ISD PP nd USFA hng PHIH vF hngD uFiF eyer et fFtF xelsonF Articial bacterial agella for mi-cromanipulationF v ghipD volF IHD pges PPHQ!PPISD PHIHF @gited on pges IID IQD IS nd PUFA hng PHIH vF hngD uFiF eyerD F etitD fFiF urtohvil et fFtF xelsonF Motion control of articial bacterial agellaF sn xnotehnology @siiiExexyAD PHIH IHth siii gonferene onD pges VWQ !VWTD ugF PHIHF @gited on pge ISFA iegler IWRP tF qF iegler et xF fF xiholsF Optimum Settings for Automatic Con-trollersF rnstions of ewiD volF TRD pges USW!UTVD IWRPF @gited on pge SVFA Publications IF T. XuD qF rwngD xF endre'D nd F égnier wodeling nd wimming roperty ghrteriztions of ledEp relil wiE roswimmersF Mechatronics, IEEE/ASME Transactions onF volFD noFWWD ppFIDIID HF PF T. XuD qF rwngD xF endre'D nd F égnier he rottionl propulsion hrteristis of sledEup helil miroswimmers with di'erent heds nd mgneti positioningF Advanced Intelligent Mechatronics (AIM), 2013 IEEE/ASME International Conference onD ppFIIIREIIPHD WEIP tuly PHIQD Best Paper Finalist QF T. XuD qF rwngD xF endre'D nd F égnier ledEup helil nnoelt modeling nd simultion t low reynolds numersF Robotics and Automation (ICRA), 2012 IEEE International Conference onD ppFRHRSERHSID IREIV wy PHIP RF T. XuD qF rwngD xF endre'D nd F égnier ghrteriztion of hreeEdimensionl teering for relil wimmersF Robotics and Automation (ICRA), 2014 IEEE International Conference onD wy QI E tune UD PHIR

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1.7 Overview of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Acknowledgements

he reserh desried in this thesis ws rried out from ytoer PHIH to perury PHIR t the snstitut des système intelligent et de ootique @ssA in the niversity

pigure QFIQX wgneti helil swimmer steering modelF he inlintion ngle nd diretion ngle of steering re respetively θ i nd θ d F where the mgneti (eld n e deomposed in the diretions perpendiulr nd prllel to the diretion of -→ u t F henD the mgneti (eld

hereforeD the mgneti torque exerted on the rw fter the xis of the rotting mgneti (eld hnges n then e written sX

es the mgnetiztion --→ M t+ is perpendiulr to the -→ u t diretionD this mgneti torque n e deomposed in the diretions perpendiulr nd prllel to -→ u t s wellF he torque --→ τ ut is the mgneti torque in the diretion of -→ u t D whih mkes the rw rotte round its tul xis -→ u t F he mgneti torque --→ τ ⊥ut is the mgneti torque perpendiulr to -→ u t D whih mkes the rw rotte round the diretion of -→ u t × -→ M t F ht mens the diretion of the xis of the rw hngesF he xis diretion hnge of the rw is lled the steering of the rwF 3.4.2 Direction angle steering characterization he hnge of the diretion ngle of the rw is de(ned s the diretion ngle steeringF he diretion ngle of rwIEmhV in this setion is onsidered s the ngle etween the xis z nd the xis of the rw projeted on the imge tken from the top mer trked y iF st is not extly the diretion ngle in the Qh speF he ext diretion ngle of the rw will e lulted from the xis of the rw in the Qh spe for the visul servo ontrolD whih is introdued in the next hpterF he diretion ngle steering is hrterized y the response to step ommnd to the diretion ngle with onstnt inlintion ngleF hereforeD this diretion ngle

es the mgnetiztion of the rw is lwys perpendiulr to its own xis -→ n D the mgneti torque n lso e deomposed in the diretions perpendiulr nd prllel to -→ n F he torque -→ τ n is the mgneti torque oliner to - → n D whih mkes the rw rotte round its urrent xis -→ n F he torque --→ τ ⊥n is the mgneti torque perpendiulr to -→ n D whih mkes the rw rotte round -→ M × -→ n F ht mens the steering of the rw n e mnged y ontrolling these two torquesF 4.2.2.1 Self-rotation control he mgneti (eld --→ B ⊥n whih mkes the rw rotte round its relEtime xis nD with n = (n x , n y , n z ) T in the world oordinte systemD depends on nD the frequenyD nd the timeF

where B ⊥n is --→ B ⊥n expressed in the world oordinte systemF st n e expressed in mtrix form s @f etion PFPFRFRAX

x -n 2 y ) 2 F B 0 is the mgE neti )ux density t the enter of the relmholtz oil pirsD whih depends on the output voltge of the gs rd @f etion PFPFRFIAF

Steering control