REDUNDANCY RESOLUTION OF CABLE-DRIVEN PARALLEL MANIPULATORS

Transcription

1 REDUNDANCY RESOLUION OF CABLE-DRIVEN PARALLEL MANIPULAORS by Maryam Agah A thess submtted to the Department of Mechancal and Materals Engneerng In conformty wth the requrements for the degree of Doctor of Phlosophy Queen s Unversty Kngston, Ontaro, Canada (September 1) Copyrght Maryam Agah, 1

2 Abstract In ths thess, the redundancy resoluton and falure analyss of Cable-Drven Parallel Manpulators (CDPM) are nvestgated. A CDPM conssts manly of a Moble Platform (MP) actuated by cables. Cables can only apply force n the form of tenson. So, to desgn a fully controllable CDPM, the manpulator has to be redundantly actuated (e.g., by usng redundant cables, external force/moment or gravty). In ths research, the redundancy resoluton of planar CDPMs s nvestgated at the knematc and dynamc levels n order to mprove the manpulator safety, relablty and performance, e.g., by avodng large tenson n the cables that may result n hgh mpact forces, and avodng large MP veloctes that may cause nstablty n the manpulator, or on the contrary, by ncreasng the cable tensons and the stffness for hgh-precson applcatons. he proposed approaches are utlzed n trajectory plannng, desgn of controllers, and safe dynamc workspace analyss where collson s mmnent and the safety of humans, objects and the manpulator tself are at rsk. he knematc and dynamc models of the manpulator requred n the desgn and control of manpulators are examned and smulated under varous operatng condtons and manufacturng automaton tasks to predct the behavour of the CDPM. In the presented research, some of the challenges assocated wth the redundancy resoluton are resolved ncludng postve tenson requrement n each cable, nfnte nverse dynamc solutons, slow-computaton abltes when usng optmzaton technques, falure of the manpulator, and elastcty of cables that has a sgnfcant role n the dynamcs of a heavy loaded manpulator wth a large workspace. Optmzaton-based and non-optmzaton-based technques are employed to resolve the redundancy of CDPM. Dependng on the advantages and dsadvantages of each method, task requrements, the used redundancy resoluton technque, and the objectve functon sutable optmzaton-based and non-optmzaton-based routnes are employed. Methodologes that could combne redundancy resoluton technques at varous levels (e.g., poston, velocty, acceleraton, and torque levels) are proposed.

3 Acknowledgements Frst and foremost, my utmost apprecaton to my thess supervsor, Professor Ron Anderson, whose nvaluable advce, gudance, patence, sncerty, encouragement and motvatng teachng methods I wll never forget. My grattude for hs knd and constant support s beyond words. Many thanks to all general staff who helped me complete my thess. Specal thanks to Ms. Gayle Laporte, who untl her day of retrement had knd concern and consderaton regardng students lves and ther academc requrements, Ms. Gabrelle Whan wth her postve energy, and Ms. Jane Daves for her kndness and nspratonal smle n assstng students. I gratefully acknowledge Dr. Arunma Khanna for helpng me overcome my fears and teachng me how to lve lfe one day at a tme. I would also lke to thank my dear frend and colleague, Amr Morad, wth whom I had the pleasure of workng n the same laboratory and sharng deas. Hs valuable comments on the stffness of cable-drven manpulators are greatly apprecated. he encouragement from my frends, especally Ada Rezae (wth her genune analytcal nsght), Sevda Bahtyar (for understandng me and welcomng me to her home), Ehsan Ghazvnan (for hs generous frendshp), Shaghayegh Oghbae (wth her open mnd and beautful heart), Sasan aghzadeh (for hs helpful comments on control of robot manpulators), Shohreh Keshavarz and Bahman Kash (for ther hosptalty and goodwll), Amrreza Rouh (wth hs unque way of cheerng me up), Paweena (Sue Sue) U-hanual (wth her endless energy and sense of humor), Mchelle Savoe (whose home s always sweet home), Arash Kanan (for hs steady companonshp), and Adn Jabbar (for helpng me stay actve) throughout my study s warmly apprecated. Specal thanks to my dear parents and ssters, who always gave me the confdence and strength to focus on my goals. her uncondtonal support and voces flled wth love always gave me energy and motvaton. Last but not least, I would lke to thank Mehrdad, my dear husband, for hs love and never-endng support, for always beng there for me, whether physcally or emotonally, and for havng fath n me. I want to thank hm for hs encouragement when I was desperate or unfocused, and, most of all, for always supportng my decsons despte the hardshps they put hm through.

4 able of Contents Abstract... Acknowledgements... Lst of Fgures... v Lst of ables... x Notaton... x Chapter 1 Introducton Background and Motvaton Mechancs and Control of Robotc Manpulators Knematcs Jacoban Matrx and Sngulartes Statcs Dynamcs Poston Control Lterature Revew Resolvng Redundancy at Varous Levels Cable-Drven Parallel Manpulators and Redundancy Resoluton heory Generalzed Inverse of the Jacoban Matrx Weghted Generalzed Inverse of the Jacoban Matrx Knematc Analyss of Robot Manpulators Generalzed Inverse Method for Resolvng Knematc Redundancy Resolvng Redundancy at Dynamc Level Pont-wse Optmzaton Crtera and Functonal Constrants for Resolvng Knematc Redundancy Augmented Jacoban echnque for Resolvng Knematc Redundancy Extended Jacoban echnque for Resoluton of Knematc Redundancy Objectves of Research Proposed Work hess Outlne and Contrbutons Chapter Redundancy Resoluton of Cable-Drven Parallel Manpulators Introducton Modelng v

5 ..1 Knematc Analyss Dynamc Analyss Optmzaton-Based and Non-Optmzaton-Based Methods Optmzaton-Based Method Mnmzng Cable ensons for a Gven rajectory Resolvng Redundancy When Mnmzng Velocty Non-Optmzaton-Based Method Smulaton Results Smulaton Results When Mnmzng Cable ensons Smulaton Results When Mnmzng Velocty Smulaton Results When Usng Non-Optmzaton-Based Methods Conclusons Chapter 3 Impact Reducton n Cable-Drven Parallel Manpulators Introducton Impact Modelng Cable Elastcty Expermental Determnaton of Elastcty Modulus of a Steel Wre Rope Cable Dynamcs Modelng Collson Detecton rajectory Plannng and Impact Reducton Smulaton Results Conclusons... 9 Chapter 4 Force Capablty and Cable Falure Introducton Force/Moment Capabltes Maxmum Capablty Mnmum Capablty Falure Analyss Controller Desgn Smulaton Results Dynamc Statc Cable Falure Poston Response under PID Control v

6 4.5 Conclusons... 1 Chapter 5 Contact Force and Vbraton Analyss n Machnng Applcatons Introducton Modelng Knematcs of the Cutter and rajectory Plannng n Machnng Applcatons Mllng Parameters Dynamc Modelng for Mllng Operaton Vbraton Analyss Smulaton Conclusons Chapter 6 Conclusons Summary Future Work References v

7 Lst of Fgures Fgure 1.1. (a) An example of a seral manpulator wth sx DOF [4], (b) abstract render of a Hexapod platform (Stewart Platform) [3] Fgure 1.. (a) NIS RoboCrane, developed by the Natonal Insttute of Standards and echnology, (photo by N.E. Wasson Jr., US echnologes), (b) FALCON-7 ultrahgh speed manpulator, adapted from [58] Fgure.1. Example planar 3-DOF cable-drven parallel manpulators Fgure.. Coordnates and varables Fgure.3. (a) Free-body dagram of moble platform, (b) free-body dagram of pulley of cable Fgure.4. (a) Moble platform trajectory, (b) confguraton of moble platform Fgure.5. (a) Cable length rates, (b) mnmum to mantan postve tenson n cables Fgure.6. (a) Cable tenson wthout null space contrbuton, (b) cable tensons wth null space contrbuton Fgure.7. Actuator torques Fgure.8. (a) Moble platform trajectory, (b) confguraton of moble platform Fgure.9. Components of moble platform velocty: (a) actual velocty of the moble platform, x, and the predefned component of moble platform velocty, x o, (b) varable porton of the moble platform velocty,.e., x var x xo Fgure.1. (a) Solutons of p 1, p, p 3 and v to mantan postve tensons n the cables, (b) cable length rates Fgure.11. (a) Partcular soluton of cable tensons, (b) cable tensons Fgure.1. Actuator torques Fgure 3.1. Coordnates and varables for planar parallel manpulators durng collson Fgure 3.. (a) Free-body dagram of the moble platform durng collson, (b) dagram of cable wth elastc and dampng effects Fgure 3.3. (a) Wre rope components, adapted from [38], (b) cross-secton of a 77 wre rope, and (c) correct way of measurng wre rope dameter Fgure 3.4., lb testng machne Fgure 3.5. Spral trajectory of the moble platform for t c t t oc Fgure 3.6. Orgnal and modfed trajectory of moble platform Fgure 3.7. Varable porton of the moble platform trajectory v

8 Fgure 3.8. Change n confguraton of moble platform Fgure 3.9. Acceleraton of moble platform Fgure 3.1. Orgnal and modfed acceleraton of moble platform Fgure (a) Cable length rates, (b) mnmum to mantan postve tenson n cables Fgure 3.1. enson n cables Fgure Actuator torques Fgure Deflecton of cables Fgure Deflecton dfference for three dampng coeffcents Fgure 4.1. Block scheme of operatonal space feedback lnearzaton/computed torque control for planar cable-drven manpulator Fgure 4.. (a) Desred trajectory of moble platform, (b) change n confguraton of moble platform Fgure 4.3. (a) Force and moment result for maxmum force capablty, (b) magntude and drecton of maxmum external force correspondng to force results n the adjacent plot Fgure 4.4. (a) Cable tensons correspondng to maxmum force capablty, (b) Mnmum norm cable tensons correspondng to zero external forces and moments (.e., mnmum force capablty) Fgure 4.5. Maxmum statc force capablty for = 5 deg: (a) Fgure 4.6. v F ext, (b) x ext y F M ext correspondng to maxmum statc force capablty for = 5 deg z Fgure 4.7. Cable tensons correspondng to maxmum statc force capablty, (a) cable 1, (b) cable, (c) cable 3, (d) cable Fgure 4.8. Mnmum statc force capablty for = 5 deg: (a) Fgure 4.9. F ext, (b) x ext y F M ext correspondng to mnmum statc force capablty for = 5 deg z Fgure 4.1. Cable tensons correspondng to mnmum statc force capablty, (a) cable 1, (b) cable, (c) cable 3, (d) cable Fgure Cable tensons correspondng to maxmum force capablty when cable 1 fals, frst falure approach Fgure 4.1. Maxmum force capablty correspondng to cable tensons n Fgure 4.11, frst falure approach: (a) forces/moments, (b) magntude and drecton of the maxmum external force Fgure Optmzaton of cable tensons maxmzng external forces when cable 1 fals, second falure approach

9 Fgure Maxmum force capablty correspondng to cable tensons n Fgure 4.13, second falure approach: (a) forces/moments, (b) magntude and drecton of the maxmum external force Fgure Cable tensons correspondng to mnmum force capablty when cable 1 fals, frst falure approach Fgure Mnmum force capablty correspondng to cable tensons n Fgure 4.15, frst falure approach: (a) forces/moments, (b) magntude and drecton of the mnmum external force Fgure Optmzaton of cable tensons mnmzng external forces when cable 1 fals, second falure approach Fgure Mnmum force capablty correspondng to cable tensons n Fgure 4.17, second falure approach Fgure (a) Desred and actual trajectory of moble platform, (b) trackng error Fgure 4.. Commanded and actual, (a) cable tensons, (b) actuator torques Fgure 5.1. (a) Example applcaton of cable-drven parallel manpulator n mllng operaton, (b) schematc representaton of cutter travel dstance l c to reach full depth of cut Fgure 5.. Conventonal mllng operaton, showng orentaton and angular velocty of the mllng cutter, c and c respectvely, tooth orentaton, x th, depth of cut, d ct, feed per tooth, f, chp depth of cut, t ch, workpece speed, v c, radus of the cutter, D, and cuttng forces F t and F r : (a) before mllng, (b) durng mllng, and (c) durng mllng representng cuttng forces Fgure 5.3. Schematc llustraton of cutter, workpece and desred path of cutter Fgure 5.4. Schematc llustraton of cuttng angle range per tooth, c, to reach full depth of cut, d ct Fgure 5.5. (a) Free-body dagram of moble platform and cutter, (b) free-body dagram of cutter, (c) free-body dagram of moble platform, (d) alternate free-body dagram of moble platform replacng reacton forces wth equvalent forces Fgure 5.6. (a) Moble platform trajectory, (b) confguraton of moble platform Fgure 5.7. enson n cables Fgure 5.8. Actuator torques Fgure 5.9. Natural frequences of planar cable manpulator Fgure 5.1. Fundamental natural frequency and correspondng mode shape Fgure Mode shape correspondng to fundamental frequency n = 55 rad/s at nstant 1 t = 5.6 s... 15

10 Fgure 5.1. Mode shape correspondng to second natural frequency n = 53 rad/s at nstant t = 5.6 s Fgure Mode shape correspondng to thrd natural frequency n = rad/s at nstant 3 t = 5.6 s x

11 Lst of ables able 5.1. Approxmate energy requrements n cuttng operatons (at drve motor, corrected for 8% effcency; multply by 1.5 for dull tools) [56] x

12 Notaton Subscrpts: ntal aug ex ext ext f f h j k o obj p P R t t j v var x y at torque level augmented extended external external force fnal homogenous soluton, related to null term ndex, teraton number ndex, teraton number teraton number, ndex predefned/gven/desred objectve functon partcular (mnmum norm) soluton centre of mass pont reference/desred tme nstant tme nstant j at knematc level Varable porton x coordnate y coordnate

13 Superscrpts: lower bound # generalzed (Moore-Penrose) nverse 1 nverse + upper bound transposed Overscrpts: weghted generalzed nverse lne segment. (dot) frst-tme dervatve.. (double-dot) second-tme dervatve Varables: c ext nc s cuttng angle range per tooth orentaton of external force on moble platform drecton of cable orentaton of normal contact force orentaton of poston vector of the centre of the obstacle wth respect to mass centre of moble platform t x y pulley angle tme step poston dsplacement n x poston dsplacement n y rotatonal dsplacement orentaton of moble platform x

14 c w v k c orentaton of mllng cutter angle of the desred lnear path arbtrary scalar arbtrary scalar at velocty level coeffcent of knetc frcton orentaton of poston vector of contact pont wth respect to mass centre of moble platform nc orentaton of r B/P n X, Y) collson angle n movng frame th orentaton of tooth c max m mn p tenson n faled cable tenson n cable maxmum allowable cable tenson desred tenson n cable mnmum allowable cable tenson partcular (mnmum norm) tenson n cable n th A a a x a y a s natural frequency tooth frequency total metallc area of the wre/cable constant coeffcent x coordnate of poston vector of anchor y coordnate of poston vector of anchor known constant coeffcent xv

15 b b s constant coeffcent known constant coeffcent c C fw constant coeffcent, cos th constant coeffcent Closs loss coeffcent of actuator c m c w D d D c d ct E f F f F nc F r F t F extx F exty g I p I z k k vscous dampng coeffcent at motor shaft dampng coeffcent of cable cutter radus mnmum dstance between moble platform and obstacle cable dameter depth of cut elastcty (Young s) modulus feed per tooth frcton force normal contact force radal cuttng force tangental cuttng force external force n x drecton external force n y drecton gravty moment of nerta of spool moment of nerta of moble platform number of faled cables, number of cables wth dfferent tensons stffness of cable xv

16 l l c l l o l w m m p undeflected length of cable cutter travel dstance length of cable undeflected length of cable length of cut, length of workpece dmenson of the task space, number of poston/orentaton coordnates mass of moble platform M ext z external moment n z drecton M RR n materal removal rate number of degrees of freedom, number of cables, number of confguraton varables n th P c p PC PQ number of teeth on the cutter perphery cuttng power varable coeffcent lne segment between the centre of the cutter and the contact pont dstance between the centre of the obstacle and the mass centre of moble platform PW 5 lne segment between mass centre of moble platform and ntal poston of the contact pont r r B/P r C/P r C/Px r C/Py r f degree of redundancy dstance between the B and pont P dstance between contact pont and mass centre of moble platform x poston of contact pont wth respect to mass centre of moble platform y poston of contact pont wth respect to mass centre of moble platform rato of radal to tangental cuttng forces xv

17 r ob r p r Q/O r Q/Oy r Q/Px radus of obstacle radus of spool x poston of the centre of the obstacle n fxed frame y poston of the centre of the obstacle n fxed frame x poston of the centre of the obstacle wth respect to mass centre of moble platform r Q/Py y poston of the centre of the obstacle wth respect to mass centre of moble platform rw 5 /W 3 dstance parallel to workpece surface before whch cutter reaches the workpece rw 6 /W 4 dstance cutter travels parallel to workpece surface after materal removal s over s( + ) sn( + ) s t t c c t ch t cut t f t m t oc U ext u t v c sn tme collson nstant torque on the cutter spndle chp depth of cut (undeformed chp thckness) cuttng tme fnal tme nstant torque of motor tme nstant at whch the modfed trajectory merges nto the desred trajectory tme nstant to clear the obstacle n the orgnal trajectory work done by the external forces and moments total specfc energy for cuttng workpece velocty, feed rate xv

18 vc t tangental velocty of contact pont v N w c W3W 5 perpheral speed of the cutter wdth of the cut lne segment parallel to workpece surface before whch cutter reaches the workpece W4W 6 lne segment cutter travels parallel to workpece surface after materal removal s over x y poston/orentaton coordnate of end effector (or moble platform) confguraton varable Ponts: A B C O P Q W 1 W W 3 W 4 W 5 W 6 anchor pont of cable attachment pont of cable on the moble platform contact pont fxed reference orgn centre of mass pont centre of obstacle edge corner of the cuttng surface of the workpece edge corner of the cuttng surface of the workpece frst contact pont on the workpece the cuttng path passes through last contact pont on the workpece the cuttng path passes through ntal poston of the contact pont fnal poston of the contact pont Vectors: vector of cable orentatons xv

19 v (.) corr l W τˆ a c e f h max mn p a c e F F c F e F ext vector of spool rotatons arbtrary vector arbtrary vector at velocty level vector of knemtc functons (.) correctonal tenson healthy cables provde vector of change n cable lengths wrench change vector of actuaton forces/torques, vector of cable tensons force/torque vector actual cable tensons force/moment vector correspondng to addtonal task force/moment vector correspondng to basc task cable tensons correspondng to falure homogeneous soluton of nverse dynamcs vector of maxmum allowable cable tensons vector of mnmum allowable cable tensons mnmum norm (partcular) soluton of nverse dynamcs poston vector of anchor constant vector, coeffcent vector trackng error vector of forces and moments on moble platform force and moment vector correspondng to addtonal task force and moment vector correspondng to basc task vector of force and moment exerted by the end effector on the envronment xx

20 F ext external force on moble platform f F f F nc f obj g h h a h v J frcton force normal contact force vector of objectve functon(s) vector of gravtatonal terms, free/arbtrary vector n the homogeneous soluton at acceleraton level free/arbtrary vector n the homogeneous soluton at knematc level free/arbtrary vector n the homogeneous soluton at knematc level the th column of the Jacoban transposed matrx J k l n J q r r C/P r Q r Q/P free/arbtrary vector vector of magntude and drecton of cable free vector spannng the null space of Jacoban matrx jont dsplacement vector resdual vector poston vector of contact pont wth respect to mass centre of moble platform poston vector of the centre of the obstacle n fxed frame poston vector of the centre of the obstacle wth respect to mass centre of moble platform R x reacton force at the shaft bearngs appled by the moble platform on the cutter n x drecton R y reacton force at the shaft bearngs appled by the moble platform on the cutter n y drecton S sparse vector n whch only element correspondng to the cable wth dfferent tenson has a value of 1 xx

21 sgn(.) vector whose components are gven by sgn functons of the vector wthn parentheses S N a c u v C W w c W f x coeffcent row vector vector of actuator torques actual torque commanded actuator torques vector of mode shape velocty of contact pont vector of external forces and moments (wrench) gravtatonal force on cutter external wrench as a result of falure dsplacement vector of moble platform, vector of poston and orentaton coordnates of moble platform, arbtrary vector x c x C x o x P x R x var y cuttng trajectory poston vector of contact pont vector of predefned trajectory of moble platform poston vector of pont P n fxed frame reference/desred trajectory vector of varable porton of moble platform trajectory arbtrary vector, confguraton vector Matrces: A C C loss C m arbtrary matrx dagonal matrx correspondng to the centrfugal and Corols terms loss coeffcent matrx postve defnte dagonal matrx of motors dampng coeffcent and xx

22 F s F v I I p J J J c J e J f dagonal matrx of Coulomb frcton dagonal matrx of vscous frcton coeffcent dentty matrx postve defnte dagonal matrx of spools moment of nerta Jacoban matrx matrx relatng moble platform velocty to rate of change of cable orentatons Jacoban matrx related to knematc functons Jacoban matrx related to the end effector Jacoban transposed matrx whose column correspondng to faled cable s replaced by zero K J K c K d K g K K p K w M N R p S k S N k porton of generalzed stffness matrx Cartesan stffness matrx postve defnte dagonal dervatve gan matrx generalzed stffness matrx postve defnte dagonal ntegral gan matrx postve defnte dagonal proportonal gan matrx postve defnte dagonal matrx of cables stffness nerta matrx matrx whose columns correspond to the orthonormal bass for the null space of J postve defnte dagonal matrx of spool rad sparse matrx coeffcent matrx W W l postve defnte weghng matrx postve defnte weghng matrx correspondng to resdual vector r xx

23 W m postve defnte weghng matrx correspondng to vector x Symbols: f(.) functon of varable nsde parentheses, functonal relatonshp between jont space and operatonal space Eucldean norm (.) (X", Y") sgn(.) X Y (X, Y) (X, Y) nfntesmal change change n varable partal dervatve knematc/constrant functon n terms of varable wthn parentheses cutter coordnate system sgnum functon coordnate axs coordnate axs movng coordnate system fxed coordnate system xx

24 Chapter 1 Introducton 1.1 Background and Motvaton Snce 198s, employng robot manpulators has drawn a consderable attenton to repettve operatons, mass producton, hgh precson, and n hazardous envronments to assure the accuracy and reduce the cost nvolved [39, 11, 116]. Snce robot manpulators can be programmed off the lne, they can be used for varous tasks and applcatons, so, they can be mmune to obsolescence. Robots have been prmarly used for applcatons that have well-arranged (rather artfcal) envronments free of obstacles, and wth no unexpected dsturbance. Settng up well-arranged and easy to reach envronments costs more than buldng the robots themselves. o avod or ease the lmtatons robot applcatons have, a robot has to have functons that allow t to understand ts stuaton, and to plan and control ts moton consstent wth the stuaton [91]. Such a robot s sad to be dexterous and autonomous. A dexterous robot can carry out varous knds of tasks n varous stuatons. If a robot s requred to have dexterty, t should have two or more arms, rather than one; three or more fngers, rather than a smple jaw-type grpper; and seven or more jonts for each arm, rather than sx or less [91]. hs type of mechancal redundancy s dvded nto knematc and actuaton redundancy. Knematc redundancy s found n open-loop and closedloop manpulators, and actuaton redundancy s found n only closed-loop manpulators. Redundancy can also be n the form of sensng,.e., havng more sensors than theoretcally necessary (e.g., sensng the exstng passve unsensed jonts of a manpulator [96]). Usually when 1

25 hgh relablty s requred, sensng n redundancy s used. he addtonal sensors and actuators provde the manpulator wth sensng and mechancal redundances. A robot manpulator s made up of several bodes/lnks connected by jonts. he Degrees of Freedom (DOF) of a manpulator are the number of ndependent parameters requred to fully specfy the confguraton of the manpulator. DOF of a manpulator defnes the number of actuators (.e., the number of actuated jonts) that the manpulator requres to be fully controllable. Industral robots that must be able to poston and orent end effectors (or output lnks that perform the requred task) arbtrarly n 3-dmensonal workspace have sx actuators. Weldng robots usually have fve or less actuators because they may not need rotaton about the weldng torch. Robot manpulators can be classfed accordng to ther archtecture. Seral manpulators consst of an open-loop chan of lnks connected by jonts from the base to the end effector. An example of a seral manpulator s shown n Fgure 1.1(a). A manpulator s sad to be parallel f ts knematc structure takes the form of a closed-loop chan of lnks connected by jonts. A parallel manpulator conssts of a moble platform (end effector, output lnk) connected to a fxed base by several branches/legs/lmbs. Fgure 1.1(b) llustrates an example of a parallel manpulator known as Stewart platform or Gough-Stewart platform wth 6 prsmatc jonts. A manpulator s called a hybrd manpulator f t conssts of both open-loop and closed-loop chans. If all lnks of a manpulator move n a plane or n parallel planes, then the manpulator s called planar. Seral manpulators have larger workspace and better manoeuvrablty than the parallel manpulators; however, these manpulators generally have lower accuracy, lower payload capacty, and hgher vbraton, n comparson wth the parallel manpulators because of ther cantlever structure. On the other hand, parallel manpulators, n general, have a hgher stffness,

26 hgher accuracy and hgher payload capacty than the seral manpulators snce multple chans carry the load and handle the end effector. However, they have more complex mechansm and smaller workspace because of the constrants created by the closed-loop knematc chan(s). (a) (b) Fgure 1.1. (a) An example of a seral manpulator wth sx DOF [4], (b) abstract render of a Hexapod platform (Stewart Platform) [3]. Knematcally redundant manpulators have more DOF than are necessary to perform a gven task, whch means that for a gven end effector velocty nfnte number of jont veloctes exsts. In closed-loop manpulators, redundancy can also be n the form of actuaton f actuated jonts are more than necessary, whch means that for a gven end effector trajectory and external forces/moments nfnte number of actuator torques/forces exsts. Redundant manpulators can use ther degree of redundancy to satsfy addtonal desrable tasks. Redundancy gves the manpulator great versatlty and broad applcablty to avod obstacles, reduce the mpact forces durng collson, avod structural lmtatons (e.g., due to jont lmts), mnmze jont forces/torques, reach behnd an object, avod sngulartes (e.g., confguratons at whch moblty of the manpulator s reduced and t would not be possble to mpose an arbtrary moton to the end effector), and to mprove safety, relablty and performance of the manpulator. In severe 3

27 envronments (e.g. space and underwater) where the manpulator s prone to falure of jonts a redundant manpulator s more relable. So, redundancy s ncluded n varous systems that need hgh relablty. It should be noted that redundant manpulators have dsadvantages too. hey have more jonts and actuators. her structure s more complex, bulker and heaver and they requre more complcated control algorthms whch results n more necessary computatons. So, there s a trade-off between the advantages and dsadvantages of a redundant manpulator. Cable-drven parallel manpulators are a specal knd of parallel manpulator wth multple cables attached to a moble platform and wth the advantage of havng larger workspace the moble platform can reach, beng able to be dsassembled and reconfgured, ncreased manoeuvrablty, and beng lghtweght, transportable and safe (due to the lack of the heavy rgd body branches). he lght weght and the long range of cables allow hgh speed moton, as well. he potental applcatons of cable-drven manpulators nclude both terrestral applcatons, e.g., manufacturng, medcal and entertanment; and the space applcatons. Cables can only apply force n the form of tenson (.e., pullng the moble platform but not pushng t). herefore, to desgn a fully controllable cable-drven parallel manpulator, the manpulator has to be redundantly actuated. hus, at least n + 1 cables are requred for an n-dof manpulator to keep postve tenson n all cables [36, 58, 59, 19]. For cable-drven parallel manpulators where t s necessary to move the end effector of the manpulator from pont to pont rapdly, dynamcs of the manpulator plays an mportant role. In most redundancy resoluton schemes there are dynamc nteractons between the end effector moton and the self-moton (or null-moton) of the manpulator. Self-moton refers to those jont veloctes that result n zero moton of the end effector. At torque level the null term of actuator 4

28 forces/torques s nterpreted as portons of actuator forces/torques that result n zero forces/moments the end effector could apply/resst. he proper use of null term s of great mportance n redundancy resoluton. In the proposed research, the redundancy resoluton of cable-drven parallel manpulators s studed at varous knematc and dynamc levels n order to perform desrable tasks whle mantanng postve tensons n all cables. Some termnologes used n the study of mechancs, modelng and control of robot manpulators are explaned n Secton 1.. A lterature revew of work relevant to ths research s gven n Secton 1.3. he theory supportng redundancy resoluton technques are gven n Secton 1.4. he theory of generalzed nverses and weghted generalzed nverses are brefly revewed n Secton and Secton 1.4., respectvely. Local and global methods to resolve knematc redundancy of manpulators are dscussed n Secton Generalzed nverse methods to resolve redundancy are descrbed n Secton Redundancy resoluton technques at dynamc level are revewed n Secton Resoluton of redundancy usng varous performance crtera s dscussed n Secton he objectves and the scope of the proposed research are gven n Secton 1.5. Secton 1.6 gves an ntroducton to the proposed research. Fnally, the thess outlne and ts contrbutons are presented n Secton Mechancs and Control of Robotc Manpulators In the followng subsectons (Secton 1..1 to Secton 1..5) some termnologes used n the study of mechancs, modelng and control of robot manpulators are explaned. An understandng of the mechancs of robot manpulators s necessary n desgn, path plannng and control of manpulators. 5

29 1..1 Knematcs Knematc analyss deals wth the dervaton of relatve motons among varous lnks of a gven manpulator [116]. Knematc analyss ncludes forward knematcs and nverse knematcs, whch could also be dvded nto poston, velocty, and acceleraton analyses. Forward knematc analyss s to calculate poston, velocty and acceleraton of the end effector for a gven jont dsplacement, velocty and acceleraton. Inverse knematc analyss s to calculate jont dsplacement, velocty and acceleraton for a gven poston, velocty and acceleraton of the end effector. Forward and nverse knematc analyses can be solved by varous methods of analyss ncludng geometrc vector analyss, matrx algebra and screw algebra [116]. A better understandng of the knematcs s the frst concern n the desgn and control of robot manpulators [116]. 1.. Jacoban Matrx and Sngulartes Generally, n performng velocty analyss of a manpulator t s convenent to defne a matrx quantty called the Jacoban matrx of the manpulator. Jacoban matrx n a seral manpulator specfes a mappng from veloctes n jont space to veloctes n task space (operatonal space, Cartesan space). At certan confguratons, called sngulartes, ths mappng s not nvertble. In contrast to seral manpulators, the Jacoban matrx n a parallel manpulator specfes a mappng from veloctes n task space to veloctes n jont space. A parallel manpulator s sad to be at a sngular confguraton when ether the mappng does not exst or, f the mappng exsts, t s not nvertble. It should be noted that the Jacoban matrx of a manpulator s not necessarly confguraton dependant. 6

30 1..3 Statcs Statcs deals wth the relatons of forces that produce equlbrum among varous members of a robot manpulator [116]. he forces exerted on a manpulator arse from varous sources ncludng gravtatonal forces, external forces/moments, frctonal forces, forces/torques appled by the actuators, and nerta forces. Consderng these forces are essental when desgnng a robot manpulator n order to sze the parts of the manpulator properly. Of course, nerta forces are excluded from the statc force analyss Dynamcs Dynamcs deals wth the forces/torques requred to cause the moton of a mechansm. So, the nerta forces are ncluded n the dynamc analyss. Dynamc analyss deals wth dervaton of the equaton of moton of a gven manpulator. Dynamc analyss problems are dvded nto drect and nverse analyses. In the drect analyss, the moton of manpulator s calculated as a functon of tme for a gven jont forces/torques. In the nverse dynamcs, the requred jont forces/torques are calculated for a gven jont trajectory as a functon of tme. Dynamc analyss s the bass of desgn specfcatons, advanced control of constraned robotc systems and trajectory plannng. Inverse dynamc model s mportant for real-tme, model-based control of a robot manpulator [116]. Dynamc analyss problems can be solved by varous methods of analyss ncludng the Newton-Euler equatons, Lagrangan equatons of moton, and the prncple of vrtual work [116] Poston Control Most robot manpulators are drven by actuators whch supply a force or a torque to cause moton of the lnks. In ths case, the actuator torques are computed usng an algorthm (lnear or nonlnear) to produce the desred moton. o desgn such algorthms, the problem of dynamcs s essental. he man objectves of a poston control system are to compensate the errors n 7

31 knowledge of the parameters of a system automatcally, and to adjust to dsturbances whch tend to perturb the system from the desred trajectory [39]. o acheve ths, the control algorthm computes the requred actuator torque nputs by usng the nformaton obtaned from the poston and velocty sensors. 1.3 Lterature Revew For knematcally redundant manpulators, tasks, n general, nvolve the end effector moton and achevng addtonal performance usng redundancy, through the self-moton of the manpulator [34]. In the followng subsectons, a lterature revew of dfferent redundancy resoluton schemes at poston, velocty, acceleraton and torque levels s presented Resolvng Redundancy at Varous Levels Let the dmenson of the task space of an n -DOF manpulator be m, where m 6. If m n the manpulator s sad to be knematcally redundant. Consderng the functonal relatonshp between the n 1 vector of jont dsplacement q and the m 1 vector of end effector dsplacement x as x f(q) ( 1.1) the relaton between the n 1 vector of jont velocty q and the m 1 vector of end effector velocty x s defned as where J s the x Jq ( 1.) m n Jacoban matrx. It should be noted that the Jacoban matrx for a parallel manpulator corresponds to the nverse Jacoban of a seral manpulator [116]. In other words, the Jacoban matrx for a parallel manpulator maps the end effector velocty from the task space onto the jont space,.e., q Jx. 8

32 For redundant manpulators (.e., n m) the general soluton to equaton ( 1.) s typcally presented n the form where # # q J x ( I J J) h ( 1.3) v # 1 J J ( JJ ) s the generalzed (Moore-Penrose) nverse of the Jacoban for a redundant manpulator [75, 91, 16], and I s the dentty matrx. he frst term on the rght-hand sde of equaton ( 1.3) s the mnmum norm (partcular) soluton and the second term s the homogenous soluton that maps the free vector h v to the null space of J. he addton of homogenous soluton results n no end effector velocty. Local and global methods are two possble approaches to resolve knematc redundancy [1]. Whtney [118] was one of the frst researchers to resolve redundancy at the velocty level by mnmzng the knetc energy of the manpulator. He used weghted generalzed nverse solutons (explaned n Secton 1.4.) whch locally mnmze q Wq for some postve defnte weghng matrx W such as the nerta matrx. he homogenous soluton of equaton ( 1.3) can be used to optmze addtonal performance crteron [35, 53, 81, 91, 15]. Legeos [75] used h v n equaton ( 1.3) to optmze a crteron functon subject to makng the end effector follow a prescrbed trajectory. He proposed that the gradent vector of a scalar functon be used as the arbtrary vector h v. He used jont range avalablty as an llustratve example of a crteron functon. Balleul [], Chang [33], Seraj [17] and Ma et al. [8] presented alternatve formulatons for locally optmzng a secondary crteron by augmentng the Jacoban matrx. Some of the appled secondary crtera nclude jont range avalablty [68, 69, 75], sngularty avodance [16, 19], varous measures of dexterty (such as manpulablty measure,.e., a factor referrng to the ease of arbtrarly changng the poston and orentaton of the end effecter) [17, 18], obstacle avodance [67, 81, 14], and local jont torque mnmzaton [37, 53, 61]. 9

33 Balleul [1] showed that the generalzed nverse approach,.e., usng only the mnmum norm soluton of equaton ( 1.3), does not produce jont space trajectores that avod knematc sngulartes. Another practcal problem wth the generalzed nverse approach shown by Klen and Huang [69] s that the closed trajectores n task space do not generally correspond to closed jont space trajectores. hs repeatablty problem was solved by Balleul [] usng the extended Jacoban method. he extended Jacoban method s based on augmentaton of the Jacoban matrx usng secondary performance crtera, n order to form an nvertble square Jacoban matrx. he method wll be explaned n Secton Roberts and Macejewsk [14] also presented a technque to generate a repeatable generalzed nverse that possessed desrable propertes (such as the mnmzaton of the norm of jont veloctes, knetc energy, etc) of a partcular generalzed nverse. Redundancy can also be resolved based on augmentaton of the forward knematcs of the manpulator. he augmentaton method wll be explaned n Secton and Secton An early approach to augment the Jacoban matrx was proposed by Macejewsk and Klen [81] for obstacle avodance. Seraj [17] presented a repeatable approach to control the manpulator confguraton over the entre moton based on the augmentaton of the Jacoban matrx. He resolved redundancy at poston level. Chang et al. [34] and Park et al. [11] also used another form of augmented Jacoban matrx to resolve redundancy at knematc and dynamc levels. hey augmented the Jacoban matrx wth a mnmum number of ts null space vectors. Khatb [63] was one of the frst researchers who formulated the dynamcs of a manpulator n the task space and gave a method for redundancy resoluton usng the manpulator dynamcs n the task space. He resolved redundancy at torque level and mnmzed the knetc energy usng 1

34 weghted generalzed nverse of the Jacoban matrx (as explaned n Secton 1.4.5). A study of the dynamc propertes of redundant manpulators was also presented by Khatb [6]. o consder the dynamc effects of redundancy resoluton, the nverse knematcs of redundant manpulators must be resolved at acceleraton level [8]. Hollerbach and Suh [53] resolved redundancy at acceleraton level and presented an expresson to mnmze the norm of jont forces/torques. hey examned methods for resolvng knematc redundances consderng the effect on jont force/torque. o ncorporate dynamcs, they were one of the frst researchers who resolved redundancy at acceleraton level rather than at velocty level. hey used the formulated generalzed nverse n terms of acceleraton and ncorporated t nto dynamcs and showed the effect of redundancy resoluton on jont force/torque. hey used the jont acceleraton null space vector, weghted by torque range (as explaned n Secton 1.4.5), to mnmze jont force/torque. Hollerbach and Suh [53] showed that usng knematc redundancy n order to mnmze the jont forces/torques n the dynamc control of redundant manpulators may result n nstablty problem (due to the hgh jont veloctes). he lack of a knematc crteron n the local optmzaton process leads hgh jont veloctes. In order to prevent hgh jont veloctes and hgh jont forces/torques, Ma et al. [78-8] proposed a balancng technque takng nto account the norm of jont veloctes and jont forces/torques. hey also resolved redundancy at the jont acceleraton level to consder the manpulator dynamcs and the effect of knematc crteron on the manpulator jont forces/torques. Alba-Gomez et al. [13], Lahouar et al. [73], and Zomaya et al. [133] studed the problem of path/trajectory plannng. Alba-Gomez et al. [13] nvestgated the optmal confguratons of parallel manpulators consderng the sngular values of the Jacoban matrx. hey employed all the feasble orentatons of the end effector that could be used to accomplsh the desred task for trajectory 11

35 plannng. Zomaya et al. [133] and Lahouar et al. [73] proposed algorthms sutable for on-lne trajectory plannng n dynamc envronments to avod obstacles. Hrose and Ma [5] also resolved redundancy at acceleraton level. hey decomposed the redundant DOF of a gven manpulator nto nonredundant subsets, where only the selected jonts were accelerated/decelerated. hey performed nverse knematc and dynamc analyses consderng the jont forces/torques, power consumpton of the actuator or the rate of obstacle avodance. hey carred out the procedure n parallel for all the DOF combnatons and selected the optmal combnaton set based on the obtaned evaluaton. Kazerounan and Nedungad [61], Hsu et al. [54], Adl et al. [7], Cheng et al. [35], Nokelby et al. [94], Zbl et al. [13], and Garg et al. [47], dscussed redundancy resoluton usng manpulator dynamcs. Kazerounan and Nedungad [61] resolved redundancy at acceleraton level and compared the results usng weghted and unweghted generalzed nverses. Hsu et al. [54] presented a dynamc control law to track a gven end effector trajectory and to resolve redundancy to accomplsh desrable subtasks. Adl et al. [7] studed the effect of nternal forces on the structural stffness and dynamc behavour of parallel manpulators. Cheng et al. [35] proposed a scheme for computng the nverse dynamcs and formulated basc control algorthms usng ther proposed approach. Garg et al. [47] extended a numercal, optmzaton-based scalng factor method (presented n [94]) and an analytcal method (presented n [13]) to determne the wrench capabltes of redundant spatal parallel manpulators. Seraj [17] developed a control strategy for redundant manpulators. he redundancy was resolved by mposng a set of knematc constrants, n task space or jont space, on the manpulator to perform an addtonal task. hese knematc functons and the pose (poston and orentaton) of the end effector coordnates were augmented to form the generalzed coordnates 1

36 for the manpulator. Seraj s [17] augmented Jacoban guaranteed a repeatable soluton (.e., the closed trajectores n the work space are mapped to closed trajectores n the jont space so that for cyclc tasks the manpulators wll return to ts startng confguraton [14]). Km et al. [64] developed unfed statc models for control of a redundant manpulator. hey developed orthogonalty propertes between the partcular and homogenous solutons and derved the decoupled moton of these two solutons and addressed the optmal control problem based on the decoupled motons. Km et al. [65] presented mpact control and plannng strateges for reducng the mpact force resultng from the collson of a knematcally redundant manpulator wth ts envronment assumng the mpact event had some fnte duraton. Park et al. [1] developed a method (called the knematcally decoupled jont space decomposton) as a coordnate transformaton between the jont space and the task space. hey used the method to analyse the moton of redundant manpulators n a unfed way from knematcs to dynamcs. Nakamura and Hanafusa [93] proposed a method to solve the globally optmal redundancy control problem. hey presented an algorthm for computng globally optmal jont space trajectores for redundant manpulators Cable-Drven Parallel Manpulators and Redundancy Resoluton RoboCrane, shown n Fgure 1.(a), s an early cable-drven parallel manpulator developed at the Natonal Insttute of Standards and echnology (NIS) [14] n early 198s for use n shppng ports. In RoboCrane, the gravtatonal force on the end effector s consdered to provde the cables wth the postve tensons. Many other cable-drven manpulators were also developed such as the CHARLOE [3] for use on Internatonal Space Staton and ultra-hgh 13

37 speed cable-drven parallel manpulator FALCON desgned by Kawamura et al. [58], shown n Fgure 1.(b). Redundancy resoluton technques can be extended to cable-drven manpulators. General knematc, dynamc and stffness analyses, as well as, the control and desgn of cable-drven manpulators are nvestgated n [15, 16, 18, 19, -4, 6-9, 31, 36, 4, 4-46, 48-51, 55, 57-6, 66, 7-7, 74, 76, 77, 8, 83, 85-9, 97-99, 1, 13, 19, , 119-1, 131]. Shen et al. [19] derved the manpulablty of cable-drven manpulators and examned a 3-cable planar cable robot wth pont-mass moble platform and formed a set of manpulatng forces as the set of forces that the 3 cables could exert on the moble platform. Wre Actuator Unt Rod Hand Frame (a) (b) Fgure 1.. (a) NIS RoboCrane, developed by the Natonal Insttute of Standards and echnology, (photo by N.E. Wasson Jr., US echnologes), (b) FALCON-7 ultrahgh speed manpulator, adapted from [58]. he lower overall stffness of cable-drven manpulators compared wth rgd parallel manpulators s another challenge n cable manpulators. he lower overall stffness may cause less rgdty and postonng accuracy. he stffness of a cable-drven manpulator depends both 14

38 on the cables stffness (elastcty) and the null porton of the cables tensons the latter of whch can be used to mprove the overall stffness [4]. he stffness of cable-drven manpulators s studed n [18, 19, 4, 36, 4, 43, 58, 59, 7, 85, 88-9]. Choe et al. [36] nvestgated stffness analyss of a cable-drven manpulator and proposed a desgn to reduce vbraton caused by elastcty of cables. Kawamura et al. [58, 59] derved knematcs and dynamcs of a hgh speed cable-drven parallel manpulator. hey analysed the moton stablty and nvestgated the nonlnear elastcty of the cable-drven manpulator. hey used non-lnear elastcty of cables and the nternal forces arsng from the redundant actuaton to reduce the undesrable vbratons of cabledrven parallel manpulators. Dao and Ma [4] studed the vbraton of cable manpulators due to the transversal vbraton of cables. Du et al. [4] presented a dynamc model for cable-drven parallel manpulators wth cables of slowly tme- varyng length and showed that t s necessary to take nto consderaton the cable dynamcs for manpulators of long-span cables. Wllams [119] presented a cable-suspended haptc nterface concept to provde 6-DOF wrench (force and moment) feedback to a human operator n vrtual realty or remote applcatons for lghter, safer, more dexterous, and more economcal operaton. Jeong et al. [55] presented a parallel cable mechansm for measurng a robot pose of 6-DOF and estmated the workspace of the manpulator. adokoro et al. [11] proposed a moton smulator for vrtual sensaton of acceleraton by applyng a parallel cable-drven manpulator and demonstrated the effectveness of ther desgn expermentally. Wllams and Gallna [1, 11] ntroduced translatonal planar cable-drven manpulators and presented knematc, statc and dynamc modelng as well as the control archtecture. Some researchers [16, 43, 57, 97, 98, , 1, 11] nvestgated how to desgn postve tenson controllers for cable-drven manpulators wth redundant cables to follow prescrbed trajectores. Kamshma et.al. [57] proposed a control algorthm to provde postve 15

39 cable tensons by addng a small dsplacement to the desred length of cables to produce nternal forces. Lahouar et al. [7] studed a collson free path plannng method and valdated the smulatons expermentally. Pham et al. [1] proposed an approach to check the exstence of postve torques and modeled the torque optmzaton problem n a standard lnear programmng form to obtan the optmal soluton. Gosseln and Grener [48] addressed the determnaton of the cables tensons and proposed to use a p-norm (e.g. a 4-norm) to optmze the dstrbuton of the forces n a cable-drven parallel manpulator. Dependng on the applcaton of the manpulator, knowng the range of applcable loads on the moble platform s of great nterest. Moreover, n dynamc envronments where collson s mmnent, for safety reasons, t s desred to reduce the force/moment that the manpulator apples and to know the maxmum forces and moments that the moble platform apples/ressts wthout damagng the envronment/manpulator. It s also benefcal to know whether the moble platform can mantan ts poston and orentaton (pose) wthout any external forces and moments, and f not, what the mnmum external forces and moments are n order to keep the desred pose/trajectory of the moble platform. he force capablty of redundantly-actuated parallel manpulators [47, 94, 13] and cable-drven parallel manpulators [8-3, 49, 5, 71, 13] s dscussed and analytcal and optmzaton-based methods are presented. Determnng the workspace, whch s that volume of space the end effector of the manpulator can reach, s crtcal n any robotc desgn, moton plannng and control. o avod movng the manpulator nto mpossble postons and orentatons that could damage the manpulator and ts envronment, the workspace analyss s carred out pror to manufacturng. he workspace analyss of cable-drven parallel manpulators s nvestgated n [16,, 7-9, 43-45, 49, 5, 55, 71, 8, 83, 13, 1, 11]. 16

40 he problem of trajectory control s dscussed n [16, 43, 57-59, 74, 97-99, 114, 115, 1-1, 131] where control strateges are presented. Falure analyss s a crtcal element when desgnng a manpulator and ts components. Consderng the cable-drven parallel manpulators, a falure can be any malfunctonng n the cable actuatng mechansm that results n dfferent cable length rate and cable tenson for nstance [95]. Any falure n the actuatng mechansm can affect the performance of the manpulator. he mechancal falures of parallel and seral manpulators were dscussed n [96] to study the falure modes of parallel manpulators wth ther effects on the DOF, actuaton and constrant. Falure of cable-drven parallel manpulators was nvestgated n [8, 88, 89, 95], where methodologes for nvestgatng the effects of cable falure were presented and crtera for recovery from these falures were establshed. 1.4 heory he confguraton of a knematcally redundant manpulator can be changed wthout changng the poston and orentaton of the end effector or the object grasped by the end effector. A human arm s a typcal example, whch has 7 DOF. A hand requres a total of sx DOF (three to hold an object at a certan poston n space, and three more to orent t), so, the human arm has one redundant DOF to poston and orent the grasped object. Because of ths degree of redundancy, t s possble to move the elbow freely whle graspng a fxed object. Knematc redundancy can be used to avod jont lmts, obstacles, and sngularty and to mnmze energy consumpton. he decson at tme nstant t to change the confguraton of the manpulator (wthout changng the poston and orentaton of the end effector) affects the confguraton of the manpulator at tme nstants greater than t, so, knematc redundancy has causalty and, therefore, forms a global problem that can be optmzed by vewng the entre moton [91]. 17

41 Actuaton redundancy,.e., another knd of redundancy, s found n only closed-loop mechansms (e.g., cable-drven parallel manpulators). For example, f an object s grasped by three fngers wth three DOF each, the mechansm has freedom to adjust how hard to squeeze the object, because the total closed-loop mechansm has nne actuators, whle the object moton s specfed by only sx ndependent varables. he actuaton redundancy of mult-branch mechansms s represented n the relatonshp between the forces and moments appled to the end effector or object and the forces and moments that the ndvdual branches exert at the contact ponts. A general closed-loop mechansm usually has actuated jonts and unactuated jonts. he dmenson of the workspace commonly defnes the number of actuated jonts. Actuatng some of the unactuated jonts of a nonmult-branch results n actuaton redundancy, whch s dstnct from that n mult-branch closed-loop mechansms n a sense that the redundancy s not represented n terms of the forces/moments at the ends of the ndvdual branches [91]. So, the parametrc representaton of actuaton redundancy s dfferent for mult-branch and nonmult-branch closedloop manpulators [91]. Actuaton redundancy concerns only the determnaton of forces and moment. It does not concern the determnaton of the confguraton of the manpulator. Determnng the forces and moments at tme nstant t does not affect the determnaton at tme nstants greater than t. So, actuaton redundancy does not have causalty and, therefore, forms a local problem that can be optmzed by vewng the relatonshp at tme nstant t [91]. In the followng subsectons the background theory requred to resolve redundancy and the general redundancy resoluton schemes are revewed. 18

42 1.4.1 Generalzed Inverse of the Jacoban Matrx he tasks of a manpulator can be formulated usng the redundancy and the task-decomposton approach where the generalzed nverses are wdely used, especally to express the soluton to the nverse velocty and force analyss of seral manpulators and forward velocty and force analyss of parallel manpulators. he generalzed nverse formulatons n ths secton are adapted from [5, 41, 64, 16]. he generalzed nverses are used to express the solutons to the system of lnear equatons havng the form not be nvertble [41]. Ax y, where the matrx A has elements wth dfferent physcal unts and may For example, the lnear system of equaton ( 1.),.e., # x Jq, has a soluton gven by q J x. If the matrx # J satsfes the followng four condtons: # JJ J J ( 1.4) # # # J JJ J ( 1.5) # # JJ JJ # # J J J J then, J # s called the generalzed (Moore-Penrose) nverse of J. For every fnte mn real matrx J, there s a unque nm real matrx J # satsfyng equatons ( 1.4), ( 1.5), ( 1.6) and ( 1.7) [41]. ( 1.6) ( 1.7) 1.4. Weghted Generalzed Inverse of the Jacoban Matrx In many physcal problems, the components of x or y n Ax = y may have dfferent physcal dmensons. Even f the components are physcally consstent, the sgnfcance of magntude can be dfferent. For example, f y s the actuator forces/torques vector of a robot manpulator, a moderate value of torque of a large motor would have crtcal meanng to a small motor. In these cases, t would be necessary to evaluate the magntude of resdual vector r = y Ax and the 19

43 magntude of soluton x based on a sutable weghng of the components. Dependng on the mnmzaton of the norm of the weghted vector x,.e., x W x (where W m s a weght matrx), m or mnmzaton of the norm of the weghted resdual vector r,.e., r W r (where W l s a weght l matrx), or both, three weghted generalzed nverse matrces are derved to select a soluton among a set of mnmum norm and/or least square solutons. he theory of generalzed nverses of matrces that has been wdely appled to the area of robotcs s revewed n [5, 41, 64] Knematc Analyss of Robot Manpulators he nverse knematc model of the manpulator s used to map the trajectory of the end effector from the task space (operatonal space, Cartesan space) onto the jont space. Snce the nverse dynamc equatons requre the end effector trajectory n the jont space, the nverse knematc model s needed to calculate the trajectory n the jont space. As also explaned n Secton 1.3.1, local and global methods are two possble approaches to resolve knematc redundancy [1]. Local methods solve equaton ( 1.),.e., x Jq, for q n terms of q and x at each tme nstant and then generate an entre jont space path by ntegratng q [1]. Global methods try to defne the jont trajectory n terms of the entre moton of the end effector. When usng local and global approaches there s a trade-off between the computaton cost and the global optmalty. Global methods are essentally noncausal, snce they use the entre moton of the end effector, and as a result are not sutable n real tme algorthms. he globally optmal control approach s sutable for off-lne trajectory plannng for tasks requrng strct optmalty, e.g., obstacle avodance n a complcated workspace and energy mnmzaton [91]. Local methods, n contrast, are more sutable for real tme algorthms such as sensor-based obstacle

44 avodance because they use the current nformaton for the next step determnaton n jont moton plannng. Local methods are less computatonally demandng than global methods [1]. he man dsadvantage of local methods s the unavodable occurrence of algorthmc sngulartes [1] Generalzed Inverse Method for Resolvng Knematc Redundancy An early approach was proposed by Whtney [118] to solve equaton ( 1.) for the jont velocty q wth a prescrbed end effector velocty x (t) to mnmze the norm of the jont velocty q. Whtney named ths method resolved moton rate control. he mnmum norm soluton s gven as # q J x ( 1.8) where # 1 J J ( JJ ) s the generalzed nverse of the Jacoban for a redundant manpulator. he general soluton to the nverse velocty analyss of a knematcally redundant manpulator,.e., the general soluton to equaton ( 1.), usng the resolved moton method s gven as equaton ( 1.3),.e., # # q J x ( I J J) hv Resolvng Redundancy at Dynamc Level he equatons of moton of an n-dof seral manpulator can be wrtten n the compact matrx form whch represents the jont space dynamc model [16] q C( q, q ) q g( q) Fv q Fs sgn( q ) J ( q F ( 1.9) ext τ M( q) ) where τ represents the n 1 vector of actuaton forces/torques, M (q) s an n n nerta matrx, q s the n 1 vector of jont acceleratons, C( q, q) q s an n 1 vector correspondng to the centrfugal and Corols terms, g (q) represents the n 1 vector of gravtatonal terms, F q denotes vscous frcton forces/torques, Fs sgn(q ) represents Coulomb frcton forces/torques, sgn(q) corresponds to the n 1 vector whose components are gven by the sgn of the jont v 1

45 veloctes, and F ext denotes the m 1 vector of force and moment exerted by the end effector on the envronment. J ( q) F denotes the forces/torques nduced at the jonts due to the contact ext forces/moments exerted by the envronment on the end effector. he n n nerta matrx M (q) s a symmetrc, postve defnte and, n general, confguratondependent matrx. In the case of redundant manpulators, the jont forces/moments that can be used to produce a gven end effector force/moment vector are not unque. he forces/moments actng on the end effector, F, could n fact be appled by the actuators at the jonts usng the relatonshp τ J ( q) F ( 1.1) where τ represents the jont forces/torques. For a redundant manpulator, gven end effector trajectory and external forces/moments, equaton ( 1.1) would be one of the solutons of jont forces/torques that provdes the requred forces/moments at the end effector [63]. In fact, there exst an nfnte number of actuator torques/forces that result n the same forces/moments at the end effector. Khatb [63] expressed the dynamcs of a manpulator n task space and showed that for a redundant manpulator where J s the weghted generalzed nverse of the F J ( q) τ ( 1.11) n m Jacoban matrx J correspondng to the soluton that mnmzes the manpulator s nstantaneous knetc energy (.e., mnmzng 1 q Mq ). J s calculated as he generalzed jont force vector wll then be [63] J M J ( JM J ) ( 1.1)

46 where τ J ( q) F [ I J ( q) J ( q)] ( 1.13) h s an arbtrary jont force/torque vector. he second term on the rght-hand sde of h equaton ( 1.13) s nterpreted as portons of actuator forces/torques that result n zero forces/moments the end effector could apply/resst. o resolve the redundancy at acceleraton level the velocty equaton ( 1.) s dfferentated. So, x Jq Jq ( 1.14) he jont acceleraton n equaton ( 1.9) can be substtuted by # # q J ( x Jq ) ( I J J) ( 1.15) h a where # 1 J J ( JJ ) corresponds to the soluton that mnmzes the norm of jont acceleraton q, and # (I J J) h s a vector n the null space of J. Hollerbach and Suh [53] were the frst a researchers to resolve redundancy at acceleraton level usng the null space vector # (I J J) h. a Substtutng equaton ( 1.15) nto equaton ( 1.9) results n τ MJ ( # # x Jq ) M( I J J) ha C( q, q ) q g( q) Fv q Fs sgn( q ) J ( q) F ( 1.16) ext Hollerbach and Suh [53] presented an expresson for h a to mnmze the jont forces/torques by placng the actuator forces/torques closest to the mdpont of the jont force/torque lmts,.e., mnmzng τ τ τ ( 1.17) where + and are the upper and lower bounds of jont forces/torques, respectvely. Hollerbach and Suh [53] calculated h a as where τˆ and τˆ are # # τˆ τˆ h a M( I J J) ( 1.18) 3

47 τ ˆ τ MJ # ( x Jq ) C( q, q ) q g( q) F q F sgn( q ) J ( q) F τ ˆ τ MJ # ( x Jq ) C( q, q ) q g( q) F q F sgn( q ) J ( q) F v v s s ext ext ( 1.19) ( 1.) If the avalable jont torque range s dfferent, a weghng matrx can be employed n equaton ( 1.17) Pont-wse Optmzaton Crtera and Functonal Constrants for Resolvng Knematc Redundancy Pont-wse optmzaton crtera are used n local methods where the entre moton of the manpulator s not utlzed. In other word, n local methods, the redundancy s resolved nstantaneously wthout consderng the entre task space path. In contrast to global methods, local methods only use nformaton at each tme nstant to determne the next step n plannng the jont trajectores. Knematc optmalty crtera may nclude mnmzng jont velocty, obstacle avodance consderng varous dstance measures (dstance from an obstacle to a pont on the manpulator that s closest to the obstacle) [67, 81, 16], jont lmt avodance [69, 75], knematc sngularty avodance, and maxmzng the manpulablty measure [18]. In these approaches, based on local optmzaton of knematc crtera the redundancy s resolved at the jont velocty level Augmented Jacoban echnque for Resolvng Knematc Redundancy he tasks to be done by a manpulator can be dvded nto several subtasks wth a prorty order. Each subtask s performed usng the remanng DOF after mplementng subtasks wth hgher prorty [15], e.g., a weldng task can be broken nto the poston and orentaton control of the end effector, where the poston control s gven frst prorty. In addton to a weldng task, a task of trackng a trajectory through a workspace wth obstacles can be broken nto poston and 4

48 orentaton control of the end effector and avodng obstacles where the former s pror to the latter [15]. It s assumed that each subtask has the form of ether a desred trajectory of sutable varables or a certan crteron [15]. he form of desred trajectory s sutable when the operator can gve a trajectory that acheves the gven subtask. If the operator does not know the actual desred trajectory but knows how to evaluate the trajectory, t would be of the from of a certan crteron, e.g., the subtask of keepng the manpulator away from the sngulartes or keepng the jont angles wthn ther lmts, the confguraton of the manpulator s evaluated usng a crteron [15]. Local methods that produce the jont space trajectores q correspondng to a prescrbed end effector trajectory x, are easy to mplement and can adapt to unpredcted changes n x [1]. A set of knematc functons s defned n task space or jont space reflectng the desrable confguraton that wll be acheved n addton to the specfed end effector moton. he new resulted generalzed coordnates contan the knematc functons and the pose of the end effector. he method can also be used for optmzaton of knematc objectve functons or satsfyng a set of knematc nequalty constrants, e.g. augmented Jacoban technque s a repeatable method (.e., the closed trajectores n the work space are mapped to closed trajectores n the jont space so that for cyclc tasks the manpulators wll return to ts startng confguraton [14]) n obstacle avodance problems [17]. A set of r knematc functons ( q), ( q),, ( )} are defned n task space or jont space where { 1 r q r n m s the degree of redundancy. hese knematc functons represent the addtonal desrable tasks. Knematc functons parameterze the nternal moton of the manpulator wthout changng the velocty of the end effector [17]. Once the knematc functons are defned they can 5

49 6 be combned wth the set of m ( 6 m ) poston and orentaton coordnates of the end effector x m x x,,, 1 x to obtan a set of n confguraton varables },,, { 1 n y y y as },,, { )} (, ), ( ), (,,, { }, { n r m y y y x x x q q q φ x y ( 1.1) where y s an 1 n confguraton vector. he new augmented forward knematc model relatng the confguraton vector y to the jont angle vector q s ) ( ) ( ) ( q y q φ q x y ( 1.) Dfferentatng equaton ( 1.) results n ) ( ) ( ) ( ) ( ) ( aug c e t t t t q q J q q J q J q q φ q x y ( 1.3) where aug J s the n n augmented Jacoban matrx, ) ( e q J s an n m matrx related to the end effector and ) ( c q J s an n r matrx assocated wth the knematc functons. he knematc functons have to be defned such that the augmented Jacoban matrx would not be rank defcent. So, t can be sad that the augmented Jacoban technque may have algorthmc sngulartes. he augmented Jacoban can also be used to relate the end effector forces/moments to the jonts forces/torques as ) ( ) ( ) ( aug t t F q J τ ( 1.4) Equaton ( 1.4) can be rewrtten as ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( c c e e c e c e t t t t t F q J F q J F F q J q J τ ( 1.5)

50 where F e s an m 1 force/moment vector correspondng to the basc task and F c s an r 1 force/moment vector correspondng to the addtonal task. So the total jont forces/torques can be rewrtten as the contrbutons of the basc and the addtonal tasks,.e., e and c respectvely, to the overall jont forces/torques,.e., e ( t) τc( t) e e c τ( t) τ J F J F ( 1.6) c Extended Jacoban echnque for Resoluton of Knematc Redundancy he augmented Jacoban technque can also be used to optmze any desred knematc objectve functon whch leads to the extended Jacoban method proposed by Balleul [, 1]. he extended Jacoban technque s a repeatable local method (because formng a square augmented Jacoban matrx results n a unque nverse velocty soluton) for jont space path generaton whch makes use of knematc optmalty crtera (representng the addtonal desrable tasks). Consderng an objectve functon f obj whch s to be optmzed n performng the moton, Balleul [] proves that there wll be r constrants ( q), j 1,, r, where r s the number of f degrees of redundancy and j( q) s equal to obj nj. Vector nj (q) spans the null space of the q Jacoban matrx. If the end effector traces a trajectory j x (t) correspondng to the jont trajectory q (t) where an objectve functon f obj s extremzed at each tme nterval, then f ( q) 1( q) r ( q) x (t) ( 1.7) where f(q) s a functonal relatonshp between jont space and operatonal space. Dfferentatng both sdes of equaton ( 1.7) results n 7

51 f q x (t) q q ( 1.8) he coeffcent matrx f Jex s square and s referred to by Balleul [] as q q the extended Jacoban. If the extended Jacoban s non-sngular along a trajectory of nterest the jont velocty would be obtaned as 1 x (t) q (t) J ex ( 1.9) Of course at any pont where f J s sngular the extended Jacoban wll also be sngular. q Note that any method whch resolves knematc redundancy through knematc constrants on jont space moton nvolves algorthmc sngulartes [1]. Ponts n the jont space where equaton ( 1.8) cannot be solved for some value of x are called algorthmc sngulartes assocated wth the constrant(s) ( q), j 1,, r [1]. j he addtonal task requrements can be expressed as a set of knematc nequalty constrants φ q) c (, where c s a constant vector. Usng redundancy n avodng knematc sngulartes, collson and jont lmts are examples where nequalty constrants can be used. 1.5 Objectves of Research Knematc analyss s requred n the desgn and control of manpulators and dynamc modelng s requred for mproved force and moton control when the manpulator has to provde hgh 8

52 veloctes and acceleratons whch s a common property n cable-drven parallel manpulators. A dynamc model can be used for computer smulaton of a robotc system. Usng dynamc modelng, the nput actuator forces/torques to follow a desred trajectory are determned. he choce of force/torque nput from an nfnte number of possble force/torque nputs a cabledrven parallel manpulator can have depends on the crteron used to resolve redundancy. he goal of the research s to fnd optmal redundancy resoluton technques applcable to cabledrven parallel manpulators and fnd effcent control strateges consderng dynamc algorthm to calculate optmal actuator forces/torques. he use of a varety of performance crtera and ntroducng alternatve methods, such that several desrable tasks can be acheved, les wthn the scopes of the proposed research. Resolvng redundancy nvolves employng optmzaton technques to a great extent. So, avalable optmzaton schemes wll be revewed and dependng on the advantages/dsadvantages of each method, task requrements, the selected redundancy resoluton technque, and the objectve functons sutable optmzaton routnes wll be employed. o obtan an optmum result when resolvng redundancy several ssues ncludng the performance crtera of nterest, postve tensons n cables and so on have to be taken nto consderaton. In other words, the nverse dynamc solutons of cable-drven parallel manpulators are lmted compared to that of general types of redundant manpulators snce postve tensons n cables have to be guaranteed, as well. Snce a dynamc model can also be used to develop sutable control strateges, approaches to desgn postve tenson controllers for cable-drven parallel manpulators wth redundant cables wll be studed. 9

53 Dependng on the applcaton requrements and choce of controller, redundancy wll be resolved at dfferent levels, e.g. velocty, acceleraton, or torque level and relevant velocty, acceleraton, or torque control schemes wll be employed whle resolvng redundancy. In order to formulate a secondary task (as a performance crteron) nvolvng force control, t s necessary to resolve redundancy at the acceleraton level. However, ths leads to the problem that undesrable or unstable moton of the manpulator can arse due to self-moton of the manpulator when the cable length rates are not ncluded n redundancy resoluton. So, nvestgatng methodologes that could combne redundancy resoluton at velocty and acceleraton levels would be of great nterest. Pont-wse optmzaton of redundancy (.e., satsfyng an objectve functon locally, or usng a local performance crteron) may result n dscontnuous actuator forces/torques solutons. Examples of such approaches are gven n [5, 53, 61, 78-8]. It should be noted that dscontnuty n actuator forces/torques may also occur when usng the generalzed nverse methods, the augmented Jacoban, or the null space contrbuton, detals of whch have been explaned n Secton to Secton Oh and Agrawal [97] proposed a method to fnd the globally contnuous tensons by trajectory plannng. Whether the avalable redundancy resoluton schemes may be modfed to avod the nstabltes and dscontnutes s a subject of the proposed research. One possblty (besdes Oh and Agrawal s [97] approach) to avod dscontnuty s to select a contnuous free vector when usng the null space contrbuton (explaned n Secton.3.1). Hgh jont veloctes are another ssue when resolvng redundancy at dynamc level. o avod hgh jont veloctes, Hollerbach and Suh [53] suggested to weght the local optmzaton crteron wth a knematc term. So, f a means can be found to balance between the torque mnmzaton crteron and the cable length rates or ther dervatves (.e., acceleraton 3

54 components), the performance of the technques could be ncreased n the sense of global stablty. Another possblty to avod hgh jont veloctes s to employ and modfy balancng technques proposed by Ma et al. [78-8] that take nto account the norm of jont veloctes and jont forces/torques n order to mprove local torque optmzaton technques. he broader queston s whether any local algorthm can be completely successful, or whether only a global resoluton of redundancy can ultmately be problem-free. Varous sutable performance crtera such as mnmum norm cable length rates, obstacle avodance and manpulablty measure wll also be addressed and examned for cable-drven parallel manpulators. If applcable, the redundancy resoluton technques wll be extended to cable-drven parallel manpulators wth dfferent layouts, or cable-drven parallel manpulators wth more degrees of actuaton redundancy, or cable-drven parallel manpulators wth knematc and actuaton redundancy (.e., a closed-loop manpulator wth knematc redundancy that s redundantly actuated [9]) consderng knematcs, dynamcs and control analyses. 1.6 Proposed Work Redundant manpulators are a consderable research subject n the feld of robotcs. Varous optmzaton technques have been appled to resolve redundancy of redundant manpulators. he am of the proposed research s to resolve some of the challenges assocated wth the redundancy resoluton of redundant cable-drven manpulators, and to mprove the manpulator safety, relablty, performance, etc. by redundancy. One of the man challenges s that cables act only n tenson;.e., the actuated cable tensons should be obtaned postve n the nverse dynamc results. he other challenge s the exstence of nfnte nverse dynamc solutons when redundancy n actuaton s used (when there are more actuators than degrees of freedom). In other 31

55 words, obtanng a unque soluton among the nfnte solutons s complcated and needs several consderatons such as avodng negatve tensons n cables, reducng the actuator forces/torques, reducng cable length rates, etc. herefore, cable actuaton and redundancy add more complexty to the nverse dynamcs and redundancy resoluton n the cable-drven parallel manpulators. hese ssues wll be addressed n the proposed research and the smulaton results wll be gven. As dscussed n the lterature revew (Secton 1.3), there are many publcatons concernng redundancy resoluton of seral and parallel manpulators ncludng planar cable-drven parallel manpulators. In most of the publshed research, the redundancy of cable-drven parallel manpulators s resolved at torque level. he objectve of ths research s to beneft from potental advantages of cable-drven parallel manpulators and extend and modfy exstng redundancy resoluton technques used n general redundant seral and parallel manpulators to cable-drven parallel manpulators. o ncorporate redundancy at varous knematc and dynamc levels, n the proposed research redundancy wll be resolved at poston, velocty, acceleraton, and torque levels. he exstng approaches (revewed n Secton 1.4) ncludng the generalzed nverse methods, usng optmzaton crtera such as mnmzng cable length rates, mnmzng actuator forces/torques, and modfyng moble platform velocty are extended to the research n progress. he reason to choose such performance crtera s to avod large tenson n cables and large cable length rates that may cause nstablty n the manpulator. Moreover, functonal constrants to avod sngulartes, avod obstacles, avod structural lmtatons (e.g., jont lmts), ncrease manpulablty, and low-energy-consumng wll also be used n the proposed research. It could also be nvestgated how to utlze redundancy n order to reduce the potentally damagng effects caused by nteracton of robotc manpulators wth ther envronment. 3

56 When the closed trajectores n task space do not have to correspond to closed jont space trajectores the generalzed nverse methods (.e., wthout null space contrbuton) can be appled. However, to optmze addtonal crteron functons (e.g., mnmzng actuator forces/torques, mnmzng cable length rates, and modfyng velocty of the moble platform) the null space projecton s used. In other words, among nfnte nverse dynamc solutons (.e., nfnte number of actuator torques/forces), or among nfnte nverse knematc solutons (.e., nfnte number of jont veloctes), the one s chosen that results n the optmzed crteron functon. It should be noted that the mentoned crteron functons can be employed n generalzed nverse methods, and also when the null space contrbuton s used. Examples of usng such crteron functons wll be gven n the followng chapters. he null space projecton represents the redundancy left after achevng the desred forces/moments at the moble platform or after achevng the desred trajectory. In one of the redundancy resoluton schemes used n ths research, the degree of redundancy of cable-drven manpulators s utlzed to acheve the mnmum actuator forces/torques requred to mantan postve cable tensons, snce the cables cannot push. Smlar works are addressed n [97, 113, 115, 1, 11] where the redundancy was resolved at the torque level. In order to have a comparson reference, the research work s started by duplcatng the exstng works as gven n Secton hen, the redundancy resoluton technques are modfed and expanded usng smlar redundancy resoluton approaches appled to redundant seral manpulators. Snce n cable-drven manpulators negatve tenson n cables cannot be acheved, redundancy can be resolved at torque level to mnmze actuator torques not less than a postve threshold [97, 113, 115, 1, 11]. So, n the proposed research, t wll also be nvestgated f the redundancy can be resolved at poston, velocty and acceleraton levels and then extend the problem such that 33

57 postve cable tensons are preserved. he challenge nvolved s how to use actuaton redundancy n order to optmze a knematc objectve functon as well as mantanng postve tenson n all cables. It wll be nvestgated how to ncorporate knematc redundancy and actuaton redundancy, e.g., usng a knematcally redundant cable-drven parallel manpulator n addton to ts actuaton redundancy. An example s usng a 3-DOF ( translatons and 1 rotaton) planar cable-drven parallel manpulator that s used for postonng tasks. o desgn a fully controllable 3-DOF planar cable-drven parallel manpulator, at least four cables are needed (wthout usng gravty). So, for a -DOF task, the manpulator has degrees of redundancy: the actuaton redundancy (due to one extra cable) and the knematc redundancy (snce the DOF of the manpulator s greater than the dmenson of the task space). In the research presented, a comprehensve lterature revew was undertaken to categorze the avalable redundancy resoluton technques appled to robot manpulators and to study advantages and dsadvantages of each method. he problem of nfnte nverse dynamc soluton due to actuaton redundancy was dscussed for an example planar cable-drven parallel manpulator. he knematc and dynamc modelng of planar cable-drven parallel manpulators are developed usng Newton-Euler approach and the obtaned results are smulated to verfy the effectveness of the redundancy resoluton technques. o valdate the resultng jont veloctes obtaned from a gven end effector trajectory, the knematc analyss s carred out usng two geometrcal approaches: (1) knematc analyss of each branch as a sngle cable, () knematc analyss of the manpulator as a whole by elmnatng the rows (n the velocty analyss) that correspond to the change n the drecton of each cable axs, and then augmentng the cable length rates. 34

58 From the knematc analyss, the confguraton of the manpulator s calculated nstantaneously. he obtaned results are then used n the force and moment balance equatons. Usng the actuaton redundancy, the actuator forces/torques are then calculated n order to mnmze the norm of actuator forces/torques whle mantanng postve cable tensons. he feasble regon of the free vector n the homogeneous soluton at whch postve tensons n cables are guaranteed s also calculated usng the nequalty constrants. Note that the postve tensons n cables act as the constrants n the optmzaton problem. Mnmum norm actuator forces/torques are used as a performance crteron to resolve actuaton redundancy. Dependng on the optmzaton problem and the requred executon effcency, a sutable optmzaton routne s then selected n MALAB. Dependng on the degrees of redundancy, several performance crtera can be combned n order to accomplsh several subtasks. Each subtask s performed usng the degrees of redundancy that reman after all the subtasks wth hgher prorty have been mplemented. Examples of multobjectve functons would be mnmzng the actuator forces/torques as well as mnmzng the cable length rates, or mnmzng the actuator forces/torques as well as avodng obstacles, and mprovng the manpulator safety, relablty, performance, etc usng varous measures (e.g., dexterty measures). So, strateges for dealng wth competng objectves wll be dscussed. In the meantme, the generalzed nverse methods wll reman the reference method for the sake of comparson whle applyng other methods. Many researchers [, 1, 33, 34, 81, 11, 14, 17] have used augmented forms of Jacoban matrx to utlze redundancy at varous levels. o apply the augmented Jacoban method, the possblty of formng a square Jacoban matrx at knematc level (relatng the end effector veloctes to cable length rates) or at torque level (relatng the moble platform forces/moments to 35

59 actuator forces/torques) wll be nvestgated usng sutable performance crtera ncludng object avodance, manpulablty measure and mpact reducton. he nvestgaton may start from force analyss leadng to velocty analyss or vce versa. Snce the am s to form a square matrx, the procedure can begn by addng extra columns (dependng on the degrees of redundancy) to the nverse Jacoban matrx n velocty analyss or addng extra rows (dependng on the degrees of redundancy) to the Jacoban matrx n force analyss. he formulaton results wll be smulated usng MALAB and MAPLE n order to study the effectveness of each approach on reducng the norm of actuator forces/torques, cable tensons, cable length rates, and modfyng the velocty of the moble platform, and so on. he obtaned results wll be compared wth the generalzed nverse methods as the reference technque. Usng generalzed nverse methods (wthout null space contrbuton) mples that postve tenson constrants are not consdered n the formulaton. So, usng the null space contrbuton, t wll be shown how effectvely postve tenson n the cables s mantaned. he correct executon of a specfc moton prescrbed to the moble platform of a manpulator s assgned to the control system whch shall provde the jont actuators of the manpulator wth the commands consstent wth the desred moton trajectory [16]. Control of moble platform moton requres an accurate analyss of the characterstcs of the manpulator, actuators, and sensors [16]. Modelng a robot manpulator s a necessary bass to fndng moton control strateges [16]. he feedback control of cable-drven manpulators s, n general, more challengng compared to ther counterpart parallel manpulators [97]. In the proposed research, knematc and dynamc models wll also be used to develop sutable control strateges n order to acheve postve cable tensons. he safety and relablty of the manpulator should be consdered whle resolvng redundancy and desgnng controllers. 36

60 1.7 hess Outlne and Contrbutons he goal of the proposed research s to resolve redundancy of cable-drven parallel manpulators consderng the complcatons assocated wth the nverse dynamcs of such manpulators. One of the man concerns n the nverse dynamcs of these manpulators s that the cables act n tenson, and therefore, the calculated tensons for the actuated cables should be postve. he other concern s because of the exstence of nfnte solutons for actuator torques/forces (due to actuaton redundancy). So, obtanng a unque soluton among the nfnte solutons s complcated and needs several consderatons such as avodng negatve tensons n cables, reducng the actuator torques/forces, and reducng cable length rates. Although there are publcatons to resolve actuaton redundancy of cable-drven parallel manpulators by mnmzng actuator forces/torques, no paper has looked at the problem by optmzng performance crtera other than mnmzng/maxmzng actuator forces/torques as well as stffness and workspace ndces. So, the proposed work ncorporates the redundancy n generatng postve cable tensons as well as optmzng desrable performance crtera (e.g., mnmzng actuator forces/torques, mnmzng cable length rates, and ncreasng relablty of the manpulator). In ths research, the exstng redundancy resoluton technques appled to general seral and parallel manpulators are also employed and modfed for cable-drven manpulators. Several technques (e.g., trajectory trackng control, velocty control and force control schemes) can be employed for controllng a manpulator. he technque and ts mplementaton may have a sgnfcant nfluence on the manpulator performance and then on the possble range of applcatons [16]. So, developng sutable control strateges (e.g., n trajectory trackng control, velocty control and/or force control, subject to postve cable tensons) usng knematc and 37

61 dynamc models of cable-drven parallel manpulators wll also be nvestgated n the proposed research. he control schemes wll be examned wthn the scope of redundancy resoluton consderng safety, relablty and performance of the manpulator. A cable-drven manpulator equpped wth sutable and practcal control strateges that s capable of resolvng redundancy by optmzng desrable performance crtera, has a wder range of applcaton n the ndustry. he study s contnued n the followng manner. In Chapter, the redundancy resoluton of cabledrven parallel manpulators s nvestgated. he knematc and dynamc modelng of an example planar cable-drven parallel manpulator s developed n Secton..1 and Secton.., respectvely. Secton.3 covers the redundancy resoluton of cable manpulators usng optmzaton-based (Secton.3.1) and non-optmzaton-based (Secton.3.) procedures. he redundancy resoluton schemes at the torque and velocty levels are gven n Secton and Secton.3.1., respectvely. Smulaton results at the torque and velocty levels are then developed n Secton.4 n order to verfy the effectveness of the redundancy resoluton technques at the torque and velocty levels. he optmzaton-based results are dscussed n Secton at the torque level and n Secton.3.1. at the velocty level. he nonoptmzaton-based results are gven n Secton.3.. he concluson of ths chapter s n Secton.5. In Chapter 3, redundancy resoluton of cable-drven parallel manpulators durng collson s nvestgated. he dynamc modelng of an example planar cable-drven parallel manpulator durng collson s presented n Secton 3.. he cable elastcty and dampng are dscussed n Secton 3.3. he collson detecton method s explaned n Secton 3.4. Mult-objectve redundancy resoluton scheme s gven n Secton 3.5. Smulaton results are dscussed n Secton 38

62 3.6 n order to verfy the effectveness of the redundancy resoluton technque and to compare the elastcty and dampng effects of cables. he chapter concludes wth a dscusson n Secton 3.7. In Chapter 4, the maxmum and mnmum forces and moments the moble platform can apply (or resst) are determned. In Secton 4., the force and moment capabltes are nvestgated at the dynamc level for a gven trajectory, as well as at the statc level for a gven pose of the moble platform. he falure analyss s gven n Secton Secton 4.3 presents a control scheme based on feedback lnearzaton. he smulaton results are reported n Secton 4.4 wth the dynamc and statc force capabltes respectvely n Secton and Secton 4.4., the effect of cable falure n Secton 4.4.3, and the Cartesan trajectory control employng a Cartesan PID controller n Secton he chapter concludes wth Secton 4.5. Chapter 5 covers the applcaton of a cable-drven manpulator n a mllng operaton. he knematc and dynamc modelng of an example planar cable-drven parallel manpulator s presented n Secton 5., where Secton 5..1 covers the trajectory plannng of the moble platform and the knematcs of the mllng operaton, Secton 5.. provdes the force relatonshps and power requrements n cuttng operatons, and Secton 5..3 ncludes the dynamc modelng for the proposed mllng operaton. he vbraton analyss of the manpulator s carred out n Secton 5.3, where the effect of elastc stffness on the stffness of the manpulator s nvestgated. he smulaton results are reported n Secton 5.4. he chapter concludes wth Secton 5.5. he conclusons of the work developed n ths thess are gven and dscussed n Chapter 6. he advantages and possble applcatons of cable-drven parallel manpulators are ponted out as well as the lmtatons of the current work. Suggestons for future work studes are also presented n Chapter 6. 39

63 Chapter Redundancy Resoluton of Cable-Drven Parallel Manpulators 1.1 Introducton Varous technques have been appled for redundancy resoluton of redundant manpulators. When resolvng redundancy at the torque level, gven the external forces and moments on the moble platform, because of the exstence of nfnte solutons for the actuator forces/torques, some actuator forces/torques (referred to as the homogenous soluton) result n zero forces/moments at the moble platform. Usng the homogenous soluton, desrable performance crtera can be acheved n order to mprove the performance and have a fal-safe manpulator. Moreover, to lower the overall cost, the average sze of actuators can be reduced by optmzng the dstrbuton of the forces n the cables. When redundancy n actuaton s used n parallel cable-drven manpulators, obtanng a unque soluton among the nfnte nverse dynamc solutons s complcated and needs several consderatons such as avodng negatve tensons n cables, reducng the actuator forces/torques, and reducng cable length rates. herefore, cable actuaton and redundancy add more complexty to the nverse dynamcs and redundancy resoluton n the cable-drven parallel manpulators. In ths chapter, t s attempted to resolve some of the challenges assocated wth the redundancy resoluton of cable-drven parallel manpulators, beneft from potental advantages of cable manpulators, and extend and modfy exstng redundancy resoluton technques proposed by other researchers. 1 Parts of the works presented n ths chapter have been publshed n [8, 1]. 4

64 he redundancy resoluton of planar cable-drven parallel manpulators s nvestgated at the torque level usng optmzaton-based and non-optmzaton-based methods. Some of the challenges assocated wth the redundancy resoluton of cable-drven parallel manpulators, ncludng postve tenson requrement n each cable, nfnte nverse dynamc solutons, and slow-computaton abltes when usng optmzaton technques are addressed. he knematc and dynamc modelng of an example planar cable-drven parallel manpulator s developed n Secton..1 and Secton.., respectvely. Secton.3 covers the redundancy resoluton of cable manpulators usng optmzaton-based (Secton.3.1) and non-optmzatonbased (Secton.3.) procedures. he redundancy resoluton schemes at the torque and velocty levels are gven n Secton and Secton.3.1., respectvely. Smulaton results at the torque and velocty levels are then developed n Secton.4 n order to verfy the effectveness of the redundancy resoluton technques at the torque and velocty levels. he optmzaton-based results are dscussed n Secton at the torque level and n Secton.3.1. at the velocty level. he non-optmzaton-based results are gven n Secton.3.. he concluson of ths chapter s n Secton.5.. Modelng In ths secton, the redundancy resoluton of planar cable-drven parallel manpulator s nvestgated. Fgure.1 shows varous manpulator layouts and Fgure. shows the coordnates and parameters used for the analyss of a planar cable-drven parallel manpulator. he fxed coordnate system (X, Y), located at O, s attached to the base, whle the movng coordnate system, (X, Y), s attached to the moble platform at ts centre of mass P wth poston vector of x P = [x, y] n (X, Y). All vector expressons wll be n the base reference frame unless otherwse stated. he poston of the base attachment pont of each cable (anchor) A s 41

65 a = [a x, a y ], r B/P s the dstance between the attachment pont of cable on the moble platform B and pont P, s the orentaton of the poston vector r B/P n X, Y) and s constant, and l = [l cos, l sn ] s the vector of the magntude and drecton of each cable. Cable 4 Cable 1 Moble platform Cable Cable 3 Cable 3 Cable 4 Moble platform Cable 1 Cable Cable 1 Cable 4 Cable 3 Cable Moble platform (a) (b) (c) Fgure.1. Example planar 3-DOF cable-drven parallel manpulators. Y O a 1 X l 1 x P Y B 4 θ 1 P B 3 B 1 B X A 1 α 1 Fgure.. Coordnates and varables. Followng the knematc analyss n Secton..1, the dynamc problem s frst formulated n Secton.. n the task space and then formulated n terms of the cable lengths and dervatves of cable lengths. Usng the knematc and dynamc analyses, the redundancy of the manpulator s resolved at varous levels (as wll be explaned n Secton.3). 4

66 43..1 Knematc Analyss he pose (poston and orentaton) of the moble platform can be wrtten as ) sn( ) cos( / / P B P B y y x x r l a r l a y x, n,, 1 (.1) where l l x cos, l l y sn, s the orentaton of the moble platform, and n (n 4) s the number of cables. he nverse velocty soluton that relates the moble platform velocty = [,, ] to the cable length rates = [,, ] s derved usng the dervatve of equaton (.1) as y x y x s r s c s r s c l l n n P B n n P B n n J ) ( ) ( / 1 1 / (.) where c, s and s( + ), = 1,, n, stand for cos, sn and sn( + ), respectvely, and the Jacoban matrx J s the coeffcent matrx of the moble platform velocty. It should be noted that wthn the context of seral manpulators the Jacoban matrx s referred to as the coeffcent matrx of jont veloctes. he alternate forward velocty soluton of equaton (.) s n l l y x 1 # J (.3) where J # s the generalzed (Moore-Penrose) nverse of J,.e., J # = (J J) -1 J. Usng technques based on the generalzed nverse of matrces may, n general, lead to non-nvarant and nconsstent results (.e., results that are not nvarant wth respect to changes n the reference frame and/or changes n the dmensonal unts used to express the problem [41]). In such cases, x x y l 1 l n l

67 the generalzed nverse could be weghted usng a sutable weghng metrc to solve lnear physcal systems wthout producng nconsstences and errors resultng from mxed physcal unts n the problem formulaton. Accordng to Doty et al. [41], when the left-hand sde of equaton (.) s unt consstent and J has full column-rank, then J # s nvarant to the choce of any weghng metrc. herefore, usng a weghng metrc wll not be requred... Dynamc Analyss For the dynamc modelng, equaton (.3) s dfferentated as d n (.4) dt # x y l l # J J l l 1 n 1 O Y X α Y P C F ext M ext X Y O β X r p c m B mg (a) (b) Fgure.3. (a) Free-body dagram of moble platform, (b) free-body dagram of pulley of cable. Consderng the free-body dagram of the system of the moble platform shown n Fgure.3(a), the dynamc force and moment balances can be wrtten as J 1 m n p m p Mx g W x y m I z 44 p F g F M extx ext y extz (.5)

68 where m p and I z are the mass and moment of nerta of the moble platform respectvely, = [ 1,, n ] s the vector of cable tensons, and are the components of the F extx F exty external force F, s the external moment actng on the moble platform, M s the ext f M extz nerta matrx, x = [ x, y, ] s the vector of lnear and angular acceleratons of the moble platform, g s the vector of gravtatonal force wth g = 9.81 m/s, and W represents the external forces and moments (wrench). Consderng the free-body dagram of the pulleys shown n Fgure.3(b), the dynamcs of actuators can be expressed as 1 rp 1 n cm τ r pn 1 I p1 β c mn β I pn where s the torque motor exerts, r p s the radus of spool, c m s the vscous dampng coeffcent at motor shaft, I p s the moment of nerta of spool, β, and β are respectvely the vectors of spool rotatons, angular velocty and angular acceleraton. For forward knematcs, pulley angles are related to pose of the moble platform x. All are defned to be zero when the moble platform s located at the orgn of the fxed frame (X, Y). For the consdered conventon shown n Fgure.3(b), a postve change n angle wll cause a negatve change n the length of cable. hus β (.6) R p β l (.7) where R p s a postve defnte dagonal matrx of spool rad and the vector of change n cable length s l l l, where l s the vector of ntal cable lengths. Usng the nverse poston soluton n equaton (.1), the general cable length l s gven by 45

69 46 / / 1 / 1 1 / 1 )) sn( ( )) cos( ( )) sn( ( )) cos( ( 1 1 n P B n n P B n P B P B n y n x y x r a y r a x r a y r a x l (.8) where l s calculated usng equaton (.8) gven the ntal pose of the moble platform. he soluton of equaton (.7) s l R β 1 p (.9) he angular veloctes of spools, β, are related to the cable length rates, l, as l β p n p r r (.1) By dfferentatng equaton (.1), the angular acceleraton of spool,, s related to l as l β p n p r r (.11) o relate the spools angular acceleraton, β, to x and x by replacng l n equaton (.11), equaton (.) s dfferentated as x J Jx l t (.1) Due to the nature of parallel manpulators, the drect knematcs of these manpulators s complcated. hus, for the sake of parametrc formulatons and to expand the dfferental form of

70 the Jacoban matrx, the transposed of the Jacoban matrx s used n the followng formulatons nstead of the Jacoban matrx tself. he second term on the rght-hand sde of equaton (.1) s expanded as J J J J n J, 1,, n (.13) 1 where J s the th column of the Jacoban transposed matrx J. Consderng the Jacoban matrx used n equaton (.), for the planar manpulator, the transpose of the Jacoban matrx s n terms of the cable orentaton and moble platform orentaton. herefore, J can be wrtten as J J J (.14) he frst term on the rght-hand sde of equaton (.14) can be expanded as J r sn cos B / P cos( ) and the second term on the rght-hand sde of equaton (.14) can be expanded as 31 (.15) J r B / cos( ) P 31 (.16) where can be wrtten as 1 x (.17) hus, usng equaton (.15) to equaton (.17), equaton (.14) can be rewrtten n terms of = [ 1,, n ] and x as 47

71 48 x α J 3 3 / 3 / ) cos( ) cos( cos sn P B n P B r r (.18) he frst term on the rght-hand sde of equaton (.18) could be expanded n terms of x. Dfferentatng equaton (.1) results n P B P B l c r c l s s r s l c 3 3 / / 1 ) ( ) ( x, n,, 1 (.19) he soluton of l ],, [ s gven by x 3 3 / / 1 ) ( ) ( P B P B c l r l c l s s r s c l, n,, 1 (.) o relate the drectons for the axes of cables to the velocty of the moble platform, the second row of equaton (.) s used to construct the overall soluton of l ],, [ as x J α (.1) where J s 3 / / ) ( ) ( 1 n n n n P B n n n n P B c l r l c l s c l r l c l s n J (.) Havng the tenson n the cables and substtutng β from equaton (.1), and β gven by equaton (.11) nto equaton (.6), the actuator torques,, wll be expressed n Cartesan form as Jx R I x J R I J R C τ R p p p p p m p t (.3)

72 where C m and I p are postve defnte dagonal matrces of motors dampng coeffcent and spools moment of nerta, respectvely. Alternatvely, the backward dfference method could be used to approxmate n (.6) as ( tk ) ( tk 1) ( tk ) (.4) t where t = t k t k 1 s the tme step, the tme ndex k corresponds to the teraton number for the tme step, the rotaton of spool, s and l s gven by equaton (.). 1 l, 1,, n (.5) rp Havng the tenson n the cables and substtutng from equaton (.5), and gven by equaton (.4) nto equaton (.6) the relaton between the actuator torques,, and the velocty of the moble platform wll be ( tk ) R pτ( tk ) CmR p J( tk ) x ( tk ) I pr p J( tk ) x ( tk ) J( tk1 ) x ( tk1 ) (.6) t.3 Optmzaton-Based and Non-Optmzaton-Based Methods Consderng the mnmum and maxmum allowable tenson n the cables, mn and max, respectvely, the soluton to equaton (.5) s gven by 1 n τ p τh, 1,, n mn max (.7) he frst term on the rght-hand sde of equaton (.7) s denoted by p whch s the mnmum norm (partcular) soluton of equaton (.5). Gven the external forces/moments actng on the moble platform, p can be wrtten as 49

73 τ p J # M x y g W (.8) he partcular soluton p s derved from the generalzed nverse of matrx J and J # s the generalzed nverse of J, gven by J # = J (J J) -1 (.9) If J n equaton (.5) has full row-rank, then J # s nvarant to the choce of any weghng metrc, e.g., refer to Doty et al. [41]. herefore, usng a weghng metrc wll not be requred. When the lne segments PB are all collnear wth ther correspondng cables, J does not have full row-rank. he second term on the rght-hand sde of equaton (.7) s denoted by h whch s the homogeneous soluton that maps the free n1 vector k to the null space of J. # τ ( I J J ) k (.3) h he homogeneous soluton h s nterpreted as a porton of cable tensons that result n zero forces/moments at the moble platform. he homogeneous soluton h could also be wrtten as τ Nλ h (.31) where N s an nr matrx (r beng the degree of actuaton redundancy) whose columns correspond to the orthonormal bass for the null space of J and may be determned usng the sngular value decomposton, and s an r1 arbtrary vector. Consderng the lower and upper tenson lmts n the cables, the feasble regon of s descrbed by the common nterval bounded by n lnear nequaltes. For example, for a manpulator wth one degree of redundancy,.e., havng one extra cable, reduces to a scalar and the feasble regon of becomes one-dmensonal. Furthermore, two degrees of redundancy (.e., two extra cables) or three degrees of redundancy (.e., three extra cables) respectvely result n twodmensonal (-D) or three-dmensonal (3-D) vector. In other words, for one, two, or three 5

74 degrees of redundancy, the feasble doman s typcally a lne n 1-D space, plane n -D space, or volume n 3-D space, respectvely..3.1 Optmzaton-Based Method Consderng equaton (.31), the determnaton of depends on the optmzaton of a crteron functon, e.g., mnmzng the norm of the actuator forces/torques. It should be noted that throughout ths thess, the term norm stands for the -norm, and all mnmzatons correspond to the -norm of the relevant vector. he constrant tenson functon s τ τ τ p Nλ (.3) mn τ max where τ mn and τ max are the vectors of mnmum and maxmum allowable cable tensons respectvely. Substtutng equaton (.4) nto equaton (.7) results n d 1 n 1 n 1 n (.33) dt # # # # # # J M J l l J MJ l l J g J W Nλ Equaton (.33) represents the nverse dynamc equaton of the cable-drven parallel manpulator n terms of cable lengths and ther dervatves. In the followng subsectons, the resoluton of redundancy consderng the mnmzaton of cable tensons wll be referred to as the redundancy resoluton at the torque level, and the resoluton of redundancy consderng the mnmzaton of velocty wll be referred to as the redundancy resoluton at the velocty level. Moreover, the term postve tenson refers to tensons that are greater than the mnmum allowable cable tenson Mnmzng Cable ensons for a Gven rajectory o resolve the redundancy at the torque level, for gven trajectores of the moble platform, a s dentfed (f t exsts) at each nstant such that mnmum norm actuator forces/torques s acheved 51

75 avodng negatve tenson n the cables. Gven the trajectory of the moble platform, t s requred to know the Jacoban matrx at each tme nstant to construct the constrant functon of equaton (.7) as a functon of the decson varable. hus the optmzaton problem s formulated as follows: mnmze τ τ subject to τ mn τ p 1 n Nλ τ max (.34) Consderng equaton (.34), for each pose of the moble platform, the value of s calculated (f t exsts) such that mnmum postve cable tensons are mantaned. Gven the trajectory of the moble platform, x(t) = [x(t) y(t) (t)], and usng the nverse velocty analyss of equaton (.), the Jacoban matrx at each tme nstant s calculated recursvely. Usng equaton (.) and substtutng the ntal Jacoban matrx, J t, and the moble platform trajectory at t 1, x y t, the vector of cable length rates at t 1, l 1 1 l n t 1, s calculated as t x y t l 1 l n t J (.35) Followng that, the vector of cable lengths l at t 1, l l 1 n t s obtaned as l l t l (.36) t1 t t 1 where l s the vector of cable length rates, and t s the tme ncrement. By substtutng l nto t 1 equaton (.1), the orentaton of cable, α, at t j s derved for = 1,, n. herefore, the Jacoban matrx at t j s obtaned. By repeatng the procedure, the Jacoban matrx at each tme nstant wll be derved and substtuted nto equaton (.3) n order to dentfy such that mnmum postve cable tensons are mantaned. As explaned n equaton (.34), the objectve functon to be used n the 5

76 optmzaton problem s to mnmze the norm of tensons n the cables subject to postve cable tensons Resolvng Redundancy When Mnmzng Velocty In order to resolve redundancy at the velocty level consderng the mnmzaton of the moble platform velocty or the norm of cable length rates, equaton (.3) and the moble platform velocty should be related. In fact, the trajectory of the moble platform should be modfed nstantaneously such that ether the mnmum norm of the cable length rates or the mnmum norm of the moble platform velocty s acheved subject to postve cable tensons. A smlar approach was proposed by Oh and Agrawal [97] to mnmze the sum of the norms for x(t) = [x(t) y(t) (t)] and ts dervatves at specfc tme nstants. hey used a fnte collocaton grd n tme to form a fnte number of nequalty constrants of equaton (.3). In the approach proposed n ths secton, ether the norm of the cable length rates or the norm of the moble platform velocty s mnmzed at each tme nstant. In ths approach, for a gven moble platform trajectory, x o (t) = [x o (t) y o (t) o (t)], the trajectory s modfed at each tme nstant such that the objectve functon and the constrant functon are satsfed. For ths purpose, the moble platform trajectory s chosen to have the followng form x a a t a t a t a t ( t t) a t f 5 f x ( t) y b b1t bt b3t b4t ( t f t) b5t ( t f t) p t ( t f t) (.37) c c1t ct c3t c4t ( t f t) c5t ( t f t) p 3 ( t t) p where t f s the fnal tme nstant, a, b and c are the known constant coeffcents, and p 1, p and p 3 are the unknown varable coeffcents (the values of p 1, p and p 3 wll be calculated at each tme nstant). he frst term on the rght-hand sde of equaton (.37) s denoted by vector 53

77 x o (t) = [x o (t) y o (t) o (t)], and the second term by vector x var (t, p 1, p, p 3 ) = [x var (t, p 1 ) y var (t, p ) var (t, p 3 )]. Dfferentatng equaton (.37) wth respect to tme results n where x, x o and x t, p, p, p ) x o ( t) x ( t, p, p, ) (.38) ( 1 3 var 1 p3 x var are the tme dervatves of x, x o and x var, respectvely. x var s referred to as the varable porton of the moble platform velocty. he arguments of x, ther dervatve wll be omtted n the followng paragraphs. x o and x var and he ffth order x o, y o and o trajectores are chosen to satsfy eghteen ntal and fnal boundary x and condtons of the moble platform trajectory, x, and ts dervatves,.e.,, x, x, y, y, y,,, f x, x, x, y, y, y,,,, respectvely. he eghteen coeffcents a, b and c are defned from these eghteen boundary condtons. It should be noted that the addton of the second term on the rghthand sde of equaton (.37), x var, does not affect the boundary condtons of the moble platform trajectores and ther dervatves because of the chosen form of t 3 (t f t) 3. Usng ths addtonal term on the rght-hand sde, the constrant functon represented by equaton (.3) wll become a functon of p 1, p and p 3, and n the optmzaton problem p 1, p and p 3 are changed such that the objectve functon s satsfed as well as achevng postve cable tensons. hree objectve functons are defned and ether of them could be used n order to resolve redundancy at velocty level. he objectve functons are lsted as mnmzng x Mx mp x mp y I z (.39) mnmzng l l l l (.4) 1 n mnmzng var var var var var x Mx mx my I (.41) z where x, var y and var are the components of the varable porton of the moble platform var velocty. o avod unt nconsstency as a result of mxed physcal unts n the velocty vector x and x var, the weghted norm of the vectors s used as the objectve functons. In other words, the 54

78 objectve functons of equaton (.39) and equaton (.41) respectvely mnmze the weghted norm of the moble platform velocty x and the varable porton of the velocty x var. Snce the frst term on the rght-hand sde of equaton (.37) s constant, the frst dervatve of the second term (.e., x var ) s used n the optmzaton problem of equaton (.41). he am of usng the frst dervatve of the varable term on the rght-hand sde of equaton (.37) (.e., thrd objectve functon, s to mnmze the weghted norm of x var ) to defne the x var such that x n equaton (.38) traces the gven moble platform velocty (.e., x o ) as close as possble. All three objectve functons are subject to τ mn τ, p1, p, p3) τ p( t, p1, p, p3) N( t, p1, p, p3) v ( t λ τ (.4) where v s an r1 arbtrary vector at the velocty level s an arbtrary vector that s smlar to at torque level. So, after calculatng p 1, p and p 3, at each tme nstant, the mnmum v s calculated (f t exsts) that mantans postve tenson n all cables. Resolvng redundancy at the velocty level s only useful for applcatons n whch the specfed ntal and fnal poses of the manpulator are of nterest, such as pck and place, and spot weldng. Snce the trajectory s modfed nstantly, the proposed redundancy resoluton scheme may cause jerky moton n addton to dscontnuty n cable tensons as the manpulator moves along the trajectory. Gven the trajectory of the moble platform as a functon of p 1, p and p 3, and usng the nverse max velocty analyss, the Jacoban matrx at each tme nstant s derved as a functon of p 1, p and p 3 usng a smlar recursve procedure explaned n Secton In other words, equaton (.35) can be rearranged as J( t j, p1, p, p3) x ( t j1, p1, p, p3) l ( t j1, p1, p, p3) (.43) 55

79 and equaton (.36) as l, p, p, p ) l( t, p, p, p ) t l( t, p, p, ) (.44) ( t j1 1 3 j1 1 3 j 1 p3 By substtutng l (t j+1, p 1, p, p 3 ) nto equaton (.1), cos α and sn α at t j+1 are obtaned as functons of p 1, p and p 3 and as a result the Jacoban matrx at t j1 s derved as a functon of p 1, p and p 3. J (t j+1, p 1, p, p 3 ) s then substtuted nto equaton (.4), τ mn τ j1, p1, p, p3) τ p ( t j1, p1, p, p3) ( t Nλ τ (.45) v max n order to dentfy p 1, p, p 3 and v such that the objectve functon s satsfed mantanng postve tenson. In fact, p 1, p and p 3 appearng n the objectve functon are changed such that the norm of l l or x M x or x varmx var s mnmzed subject to postve tenson n the cables..3. Non-Optmzaton-Based Method o reduce the computatonal tme that s desrable n real tme applcatons, nstead of performng an optmzaton procedure, t s nvestgated f a vald soluton for the cable tensons exsts gven the external forces and moments on the moble platform. Consderng the partcular soluton p of equaton (.8), when p s negatve or f the maxmum tenson lmt s volated, usng the actuaton redundancy, the tenson n the cables s modfed satsfyng the tenson constrants. herefore, consderng p,.e., the mnmum norm soluton of n equaton (.5), f the desred tenson n cable s, s replaced wth as m m m S [ 1 n] (.46) where S s a sparse n1 vector n whch only th element, correspondng to the cable wth dfferent tenson, has a value of 1. Substtutng the soluton of cable tensons gven by equaton (.7) nto equaton (.46) results n 56

80 S τ S k (.47) m p # # where S S ( I J J ) and t corresponds to the th row of ( I J J ). he mnmum norm N arbtrary vector k s then obtaned as # # # N m N N k S S S J Mx g W (.48) # 1 # where S S ( S S ) s the generalzed nverse of the th row of ( I J J ). Gven the pose of N N N N the moble platform, a feasble soluton for k s characterzed by a convex regon bounded by n lnear nequaltes on the elements of k consderng the nequalty constrants of equaton (.7). On condton that the feasble regon of each element of k s not empty, there exsts a soluton for cable tensons such that [,, ] = W. Otherwse, the tenson constrants cannot be F extx F exty M extz satsfed. As an example, f =, then k S N p. If a soluton exsts for k, t can be concluded that the gven pose s wthn the workspace. he procedure can be generalzed when k cables have dfferent tensons, where the mnmum norm arbtrary vector k s calculated as k S # Nk # # S S J Mx g W m1 m mk N k k (.49) where S k s a sparse matrx wth n rows and k r columns n whch only elements s,k = 1, the degree of actuaton redundancy s r = n m, m s the dmenson of the task space, ndex # # 1 corresponds to the cable wth dfferent tenson, S S ( I J J ), and S S ( S S ). It should be noted that for r degrees of redundancy, r extra constrants (e.g., desred tensons n r cables) result n a closed form soluton for vector k (only f k has a feasble regon). herefore, f there are more than r desred constrants on specfc cable tensons, there may not be a soluton for vector k. N k # k Nk Nk Nk Nk 57

81 .4 Smulaton Results In the followng subsectons, the smulaton results of redundancy resoluton of the planar cabledrven manpulator, shown n Fgure.1(a), are presented. Comparng the layouts of Fgure.1, the layout shown n Fgure.1(a) offers a larger statc workspace and a larger range of orentatons wthout nterference problems between the cables and the moble platform [84]. So, the layout of Fgure.1(a) s selected for the smulatons. For the smulaton, the anchor postons are {a 1, a, a 3, a 4 } = {[ 1,.75], [1,.75], [1,.75], [ 1,.75] } (unts n meters), and the angular postons of cable attachment ponts on the moble platform wth respect to the movng frame are defned by { 1,, 3, 4 } = {18,,, 18} (unts n degrees). he mass m p, moment of nerta I z, and radus r B/P of the moble platform are respectvely kg,.144 kg.m, and.15 m. he moment of nerta of each spool I p, the radus of each spool r p, and the vscous dampng coeffcent at each motor shaft c m are respectvely.8 kg.m,.5 m, and.1 Nms. Consderng the conventon shown n Fgure.3(b), all spool angles are defned to be zero ntally. he upper lmt on the actuator torques s calculated based on the maxmum allowable cable tenson of 315 N for a steel cable wth a breakng strength of 16 N [] and a safety factor of 4. It s assumed that no external forces/moments act on the moble platform,.e., [ F ext, F x ext, M y ext ] z = [,, ]. he optmzaton problems are carred out n MALAB usng the fmncon functon to verfy the optmzaton procedures dscussed n Secton Smulaton Results When Mnmzng Cable ensons As explaned n.3.1.1, the optmzaton problem s formulated as follows: mnmze τ τ subject to τ mn τ τ p Nλ τ max (.5) 58

82 [deg] y [m] x [m] and for each pose of the moble platform, a value s calculated (f t exsts) such that mnmum postve cable tensons are mantaned. Wth four cables, the constrant functon of equaton (.5) s reduced to four lnear nequaltes n terms of, where s reduced to a scalar. he termnaton tolerances placed on constrant volatons were chosen as 1 3 N, on the objectve functon as 1 3 N, and on the estmated parameter values (.e., ) as 1 4 N. he ntal and fnal boundary condtons ( x, x, x, y, y, y,,, ),f are assumed to be (,,,,,,,, ) and (.5 m,,,.5 m,,, 1 deg,, ) f, respectvely, wth t[, 1] s and tme step of t =.1 s. So, a ffth order polynomal s used for the moble platform trajectory satsfyng eghteen boundary condtons of x(t), and ts frst and second dervatves. Fgure.4(a) and Fgure.4(b) respectvely show the desred moton and confguratons of the moble platform, regardless of whether postve tenson n the cables s acheved or not tme [sec] (a).5 Fgure.4. (a) Moble platform trajectory, (b) confguraton of moble platform (b) Fgure.5(a) represents the cable length rates calculated usng equaton (.35). It should be noted that the determnaton of cable length rates s ndependent of the optmzaton of the tenson n the cables. Fgure.5(b) shows the mnmum that guarantees postve tenson n the cables. Snce 59

83 mnmum [N] the change n the orentaton of the moble platform s small (.e., 1 deg) the optmzaton was termnated successfully. If the change n the orentaton of the moble platform s not small enough, the pose of the moble platform may not le wthn the avalable workspace of the manpulator and the tenson constrants may not be met. (a) Fgure.5. (a) Cable length rates, (b) mnmum to mantan postve tenson n cables. he plots of the partcular solutons p (.e., wthout null space contrbuton) are gven n Fgure.6(a). As t can be seen, tensons n the thrd and fourth cables are negatve. o mantan postve tensons n the thrd and fourth cables, null space contrbuton s used. Fgure.6(b) llustrates the tenson hstores resultng from the substtuton of mnmum, shown n Fgure.5(b), nto equaton (.31) and then pluggng h nto equaton (.7). he mnmum allowable tenson of N n the fourth cable shows that the frst three cables are crtcal for mantanng the confguratons shown n Fgure.4(b). Substtutng the cable tensons and the velocty of moble platform nto equaton (.6) results n the actuator torques shown n Fgure.7. In addton to the -norm of the cable tensons used n equaton (.5), hgher norms were used, but for the gven example, no sgnfcant change n the results was observed tme [sec] (b)

84 4 [N.m] 3 [N.m] [N.m] 1 [N.m] p4 [N] 4 [N] p3 [N] 3 [N] p [N] [N] p1 [N] 1 [N] tme [sec] (a) tme [sec] (b) Fgure.6. (a) Cable tenson wthout null space contrbuton, (b) cable tensons wth null space contrbuton tme [sec] Fgure.7. Actuator torques..4. Smulaton Results When Mnmzng Velocty As explaned n.3.1., for a gven moble platform trajectory,.e., x o (t) = [x o (t) y o (t) o (t)] of equaton (.37), the trajectory s modfed at each tme nstant such that: 61

85 [deg] y [m] x [m] mnmze x var subject to Mx τ var mn m x p τ τ var p m Nλ p v y var τ I max z var (.51) and a value s calculated for p 1, p, p 3 and v (f t exsts), at each tme nstant, that mantans postve tenson n the cables. Wth four cables, the constrant functon of equaton (.4) s reduced to four lnear nequaltes n terms of v, where v s reduced to a scalar. he smulaton parameters n ths secton are the same as n Secton.4.1. For comparson, the gven trajectory of the moble platform, x o, n ths secton s the same as the desred trajectory of the moble platform at the torque level (dscussed n Secton.4.1). Fgure.8(a) and Fgure.8(b) show the moton of the moble platform whle mantanng postve tenson n the cables. From Fgure.8, t can be seen that the moble platform moves towards the up-rght corner of the base (.e., towards the thrd anchor) untl t reaches ts predefned fnal poston and orentaton. he termnaton tolerances placed on the constrant volatons were chosen as 1 3 N, on the objectve functon as 1 3 J, and on the estmated parameter values as 1-4 (m/s 6 for p 1 and p, rad/s 6 for p 3, and N for v ) tme [sec] (a) Fgure.8. (a) Moble platform trajectory, (b) confguraton of moble platform (b) 6

86 (a) Fgure.9. Components of moble platform velocty: (a) actual velocty of the moble platform, x, and the predefned component of moble platform velocty, x o, (b) varable porton of the (b) moble platform velocty,.e., x var x xo. Fgure.9 shows the actual velocty of the moble platform x and the predefned components of moble platform velocty x o and the varable porton of the moble platform velocty 63 x var x xo. he coeffcents a, b and c are determned from the eghteen predefned boundary condtons ( x, x, x, y, y, y,,, ),f wth t[t, t f ]. From the optmzaton problem represented by equaton (.51), t s desred that the actual velocty of the moble platform, x, be as close to x o as possble. he varable porton of the moble platform velocty llustrated n Fgure.9(b) shows how close x s to x o. As t can be seen from Fgure.9(a), for tme nstants before t =.45 s, the sold curves of x, y and le under the dash-dotted curves of x o, y o and o, respectvely. So, for the frst half of the moton of the moble platform the soluton of equaton (.51) has resulted n velocty components (.e., x, y and ) whch have lower magntudes compared to the correspondng components of gven velocty of the moble platform x o (.e., x o, y o and Nevertheless, for the second half of the moton of the moble platform, ths s not the case for x o ).

87 and when compared wth x o and o, respectvely. It should be noted that for comparson, x o was chosen the same as the desred velocty of the moble platform, x, when resolvng redundancy at the torque level n Secton Snce at each tme nstant a new set of p 1, p, p 3 and v that satsfes the objectve and constrant functons s calculated, there s no guarantee that the solutons of p 1, p, p 3 and v are contnuous. he optmzaton tolerances and the ntal guess for the optmzaton parameters can also affect the solutons of p 1, p, p 3 and v. An approach to avod dscontnutes s the globally contnuous soluton proposed by Oh and Agrawal [97] that determnes a specfc set of p 1, p, p 3 and v, for the entre moton of the moble platform usng dscrete poses of the moble platform along the path. hey used trajectory parameterzaton n conjuncton wth a fnte collocaton grd n tme to ensure smooth tensons durng the path. However, t should be noted that ther proposed soluton mnmzes the moble platform velocty at specfc tme nstants (not at each tme nstant),.e., x n equaton (.38) traces x o as close as possble at specfc tme nstants. In fact, ther proposed scheme determnes one constant set of p 1, p, p 3 and v over [, t f ] to steer the moble platform between gven boundary condtons satsfyng postve tenson n the cables. Fgure.1(a) shows the solutons of p 1, p, p 3 and v that guarantee postve tenson n the cables. Fgure.1(b) represents the cable length rates resultng from the substtuton of p 1, p, p 3 and v, shown n Fgure.1(a), nto equaton (.43). Consderng Fgure.1(b) and the trajectory of the moble platform that results n the confguratons shown n Fgure.8(b), as the moble platform moves the frst and fourth cables are extended whereas the second and thrd cables are shortened, whch s smlar to Fgure.5(a). he change n the length of the frst, thrd and fourth cables are more notceable compared to the second cable. 64

88 p4 [N] 4 [N] p3 [N] p [N] p1 [N] p 1 [m/s 6 ] 3 [N] [N] 1 [N] p [m/s 6 ] p 3 [rad/s 6 ] v [N] tme [sec] (a) (b) Fgure.1. (a) Solutons of p 1, p, p 3 and v to mantan postve tensons n the cables, (b) cable length rates tme [sec] (a) tme [sec] (b) Fgure.11. (a) Partcular soluton of cable tensons, (b) cable tensons. Fgure.11(a) shows the plots of the partcular solutons p wth negatve tensons n the thrd and fourth cables. Null space contrbuton s used to mantan postve tensons n the thrd and fourth cables. Fgure.11(b) llustrates the tenson hstores resultng from the substtuton of p 1, p, p 3 65

89 4 [N.m] 3 [N.m] [N.m] 1 [N.m] and v, shown n Fgure.1(a), nto equaton (.45). As t can be seen from Fgure.11(b), the tenson n the fourth cable s approxmately N. In the optmzaton problem of equaton (.51) used n ths work, the mnmum tenson n the cables was consdered to be N. Smlar to Fgure.6(b), Fgure.11(b) shows that the fourth cable s not crtcal for mantanng the confguratons shown n Fgure.8(b) tme [sec] Fgure.1. Actuator torques. Fgure.1 shows the actuator torques obtaned from the substtuton of cable tensons and the moble platform velocty nto equaton (.6)..4.3 Smulaton Results When Usng Non-Optmzaton-Based Methods For a gven trajectory, the method dscussed n Secton.3. was employed n order to fnd the mnmum norm postve cable tensons. he executon tme of the developed MALAB code usng non-optmzaton-based method, explaned n.3., resulted n the same results as n optmzaton-based method, explaned n Secton.3.1.1, but wth shorter computatonal tme of.99 seconds. he executon tme of the optmzaton-based method, explaned n Secton.3.1.1, 66

90 was about 1.99 seconds. Durng ths tme, the cable tensons were determned for 11 poses n both methods. It can be concluded that for a gven trajectory, usng the optmzaton-based methods to calculate the mnmum norm tenson n the cables can ncrease the computatonal tme sgnfcantly. hus, t s recommended to avod usng optmzaton-based methods f there exst any other nonoptmzaton-based methods..5 Conclusons In ths chapter, two approaches to resolve actuaton redundancy of planar cable-drven parallel manpulators were nvestgated usng optmzaton-based and non-optmzaton-based methods. In the frst approach, a prescrbed trajectory of the moble platform was followed and the norm of cable tensons was mnmzed whle mantanng postve tenson n the cables. In the second approach, the desred moble platform trajectory was modfed at each tme nstant such that the mnmum norm velocty of the moble platform or mnmum norm cable length rates was acheved, subject to postve cable tensons. o reduce the computatonal tme that s desrable n real tme applcatons, a non-optmzaton procedure was proposed to calculate a vald soluton for the cable tensons gven the desred trajectory of the moble platform. Smulatons of a 3-DOF planar cable-drven parallel manpulator was developed mnmzng ether the norm of cable tensons or the norm of the varable porton of the moble platform velocty. Based on the optmzaton results, t was observed that usng the null space contrbuton the cable tensons can be kept postve successfully. However, t should be noted that ths s not always the 67

91 case, e.g., when the manpulator has a large orentaton value or when the change n the orentaton of the moble platform s not small enough. Based on the optmzaton crtera, the tolerances used n the optmzaton routne, and the optmzaton scheme, the torque level and velocty level approaches used contnuous moble platform trajectores and produced contnuous cable tensons. At the velocty level, the modfed trajectory of the moble platform and the cable length rates were contnuous as well. he choce of a proper optmzaton scheme, whether to mnmze the tenson n the cables or to mnmze the velocty components, depends on the applcaton of the manpulator. For example, resolvng redundancy at the velocty level s, n general, recommended for trajectory plannng [97] and/or for applcatons when the specfed ntal and fnal poses of the manpulator and/or ntal and fnal veloctes and acceleratons of the moble platform are of nterest. Comparng the two descrbed redundancy resoluton technques, the computatonal procedure at the torque level s faster and less complcated than that at the velocty level. Usng the non-optmzaton-based method, the executon tme was reduced to less than one-thrd of that of the optmzaton-based procedure. hus, t s recommended to employ non-optmzaton methods nstead of optmzaton methods, f any exst. 68

92 Chapter 3 Impact Reducton n Cable-Drven Parallel Manpulators 3.1 Introducton In ths chapter, redundancy resoluton of cable-drven parallel manpulators durng collson s nvestgated. Elastc cable modelng s neglected n some researches snce they have less concern about elastc and dampng effects on the dynamcs of the manpulator when used n small workspace wth lght-weght moble platforms and low acceleratons. However, n the work presented, the effect of cable dynamcs on the dynamcs of whole system s nvestgated as well. A smple model for cable dynamcs contanng elastc and dampng behavours s used. In dynamc envronments where collson s mmnent, for safety reasons, t s desred to reduce the tenson n the cables such that the force/moment that the manpulator apples, e.g., the contact forces, are mnmzed durng collson wthout damagng the envronment/manpulator. For mpact reducton and trajectory plannng purposes, the collson s frst detected consderng the mnmum dstance between the moble platform and a specfed obstacle. he effect of mpact durng collson s reduced and the error n the trajectory of the moble platform after collson s mnmzed consderng the effects of dampng and stffness of cables. he dynamc modelng of an example planar cable-drven parallel manpulator durng collson s presented n Secton 3.. he cable elastcty and dampng are dscussed n Secton 3.3. he collson detecton method s explaned n Secton 3.4. A mult-objectve redundancy resoluton scheme s gven n Secton 3.5. Smulaton results are dscussed n Secton 3.6 n order to verfy Parts of the works presented n ths chapter have been publshed n [1, 11]. 69

93 the effectveness of the redundancy resoluton technque and to compare the elastcty and dampng effects of cables. he chapter concludes wth a dscusson n Secton Impact Modelng In ths secton, the dynamcs of a planar cable-drven parallel manpulator shown n Fgure 3.1 s nvestgated for redundancy resoluton n the presence of a statonary obstacle. Fgure 3.1 shows the coordnates and parameters used for the analyss of a planar cable-drven parallel manpulator that are smlar to Fgure.. When collson occurs, the moble platform becomes n contact wth the obstacle at pont C, as shown n Fgure 3.1. he tangental and normal drectons to the surface of the moble platform at pont C are denoted by t and n, respectvely. he poston vector of contact pont C s r C/P = [r C/P cos nc, r C/P sn nc ] n X, Y. As shown n Fgure 3.1, the poston of the centre of the obstacle Q wth respect to O and P are r Q = [r Q/Ox, r Q/Oy ] and r Q/P = [r Q/Px, r Q/Py ], respectvely. Y O X x P Y B 4 t B 3 C Q n θ nc X a 1 l 1 θ 1 P B 1 B A 1 α 1 Fgure 3.1. Coordnates and varables for planar parallel manpulators durng collson. 7

94 o nvestgate the effect of collson n the redundancy resoluton of cable-drven parallel manpulators, the mpact s ncluded n the modelng. he knematc and dynamc modelng explaned n Secton. s employed for the purpose of mpact modelng. Y O X Y t C P F nc α nc θ nc n X Y O X B α F f A B l (a) (b) Fgure 3.. (a) Free-body dagram of the moble platform durng collson, (b) dagram of cable wth elastc and dampng effects. Fgure 3.(a) shows the free-body dagram of the moble platform when collson occurs. F nc s the normal contact force and F f s the frcton force actng on the moble platform n the opposte drecton of the moton of the platform. Consderng Fgure 3.(a), n the absence of any other external force/moment, the external forces and moments on the moble platform, F and M ext, z ext f can be wrtten as and F ext f F F ( 3.1) nc f M F r sn( ) F r cos ( )sgn( v ) ( 3.) extz nc C/P nc nc f C/P nc nc C t where nc defnes the orentaton of the normal contact force F nc, the moble platform n t-drecton, and sgn( v Ct v c s the velocty of pont C on t ) s a sgnum functon. When a force F s appled to 71

95 the centre of the moble platform because of the task requrements, then the total external force would be Fext f Fnc Ff F and the moment equaton wll be the same as equaton ( 3.). For a statonary obstacle, velocty of pont C n X, Y, v C, s gven by By projectng v C onto tangental drecton t, vc t x rc / P sn( nc) v C ( 3.3) y rc / P cos( nc) v C t s obtaned as x sn( ) y cos ( ) r ( 3.4) nc For the crcular moble platform, the normal contact force F nc wll always pass through the geometrcal centre of the moble platform,.e., nc C/P nc nc ( 3.5) herefore, for a statonary obstacle, equaton ( 3.) s smplfed to M F r sgn( v ) ( 3.6) extz f C/P C t For a non-crcular moble platform, equaton ( 3.5) s not vald. Consderng F f = k F nc, where k s the coeffcent of knetc frcton, equaton (.5) s wrtten as where c s J 1 Mx g Fncc ( 3.7) n cosnc k cos( nc sgn( vc )) t c snnc k sn( nc sgn( v )) ( 3.8) Ct k rc/p sgn( vc ) t he soluton of equaton ( 3.7) s gven by equaton (.7),.e., 7

96 1 n τ p τh, 1,, n mn max where the mnmum norm (partcular) soluton, p, s ( 3.9) τ p J # Mx g F nc c ( 3.1) and the homogeneous soluton, h, s the same as equaton (.3) and equaton (.31). 3.3 Cable Elastcty Wre rope, or cable, typcally conssts of several strands of wre wrapped around a metallc of fbrous core. Wre ropes have the ablty to resst relatvely large axal loads n comparson to bendng and torsonal loads. Elastcty of wre has a sgnfcant role n the dynamcs of a heavy loaded manpulator wth a large workspace. he stran of the wre along ts longtude n the elastc model of the wres generates poston errors n the desred moton of the moble platform. hus, elastc modelng s requred for accurate poston control of the system. D c (a) (b) (c) Fgure 3.3. (a) Wre rope components, adapted from [38], (b) cross-secton of a 77 wre rope, and (c) correct way of measurng wre rope dameter. 73

97 3.3.1 Expermental Determnaton of Elastcty Modulus of a Steel Wre Rope he varous components of a wre rope (or cable) are shown n Fgure 3.3(a). he strands of a wre rope carry the major porton of the load actng on a wre rope [38]. Fgure 3.3(b) and (c) show the cross secton of a 77 rope and correct way to measure the dameter of the rope, respectvely. Fgure 3.4., lb testng machne. In the work presented, the tensle testng of a wre rope was carred out and the elastcty modulus of a 77 steel wre rope was determned. Several tests were run on 1/16 n dameter, 77 steel wre ropes wth varous lengths on a, lb tenson-compresson testng machne. he expermental setup and the testng machne are shown n Fgure 3.4. A clp gauge wth a nomnal gauge length of n was used to determne the stran n an ndvdual wre rope. he wre was 74

98 wound around the pulleys to prevent slppng on the specmen. he specmen was loaded up to 1 lb, and the axal stran per n of wre was measured. hen, a load deformaton curve was obtaned, where the expermental data were ftted lnearly. he slope of the ftted lne s EA/l, where E s the elastcty modulus, A s the total metallc area of the wre, and l s the undeflected length of wre, that s, n. For ths experment, the undeflected length of wre s the clp gauge length. he metallc area of the wre rope was calculated usng the dameter of the wre whch was measured wth a mcrometer callper, as shown n Fgure 3.3(c). Havng the total metallc area, the obtaned expermental elastcty modulus was about 79 GPa Cable Dynamcs Modelng o nclude the effects of elastcty and dampng, the cables are modeled as a sprng and damper, shown n Fgure 3.(b). Neglectng the mass of each cable, the tenson n each cable that acts along the cable, can be wrtten as l lo E A cw ( l ) l o ( 3.11) l o where E s the effectve Young s modulus of cable, the cross-secton area of cable s A = πd c / 4, where D c s the dameter of the cable, as shown n Fgure 3.3(c), l o s the undeflected length of cable, c w s the dampng coeffcent of cable, and l l s the deflecton o rate of the cable,.e., the stran rate. he frst term on the rght-hand sde of equaton ( 3.11) represents the elastc behavour of the cable that s a lnear functon of the stran, l = l l o, and the axal stffness of the cable, E A / l o. he second term s the dampng force that represents the effect of dampng as a result of the frcton between the brads of the cable. It s assumed that the dampng force has a lnear relatonshp wth the stran rate. 75

99 76 Usng the backward dfference method, undeflected cable length rate at nstant t k, ) ( k o t l s approxmated as t t l t l t l k o k o k o ) ( ) ( ) ( 1 ( 3.1) Substtutng l,, and o l, gven by equaton (.), equaton ( 3.9) and equaton ( 3.1), respectvely, nto equaton ( 3.11) results n a second order polynomal as below: ) ( ) ( ) ( ) ( ) ( ) ( 1 k k o k w k o w k k o w t A l E t l t l c t l t c A E t t l t c ( 3.13) where the unknown of the second order polynomal s the undeflected length of cable, l o. Intally at k = 1, the moble platform s at rest and the second term on the rght-hand sde of equaton ( 3.11) s zero. herefore, the undeflected length of cable, l o s calculated as o A E t t A l E t l ) ( ) ( ) ( ( 3.14) Havng the trajectory of the moble platform, x = [x, y, ], the cable lengths are calculated as / / ) sn( ) cos( P B P B y x r y a r x a l ( 3.15) and wll then be substtuted nto equaton ( 3.13) to solve for the undeflected length of cable, l o. o satsfy the postve tenson constrant n each cable, not only l should be postve but the stran rate, l o l, should also be such that mn. 3.4 Collson Detecton By montorng the speed of the motors usng encoders nstalled at each actuator, dynamc collson can be detected. In ths secton, for the smulaton purposes, the collson s detected knowng the locaton of the obstacle and assumng that the obstacle could not be avoded.

100 he dstance between the moble platform and the obstacle s montored at each tme nstant. When collson occurs, the gven trajectory of the moble platform s modfed for the subsequent nstants after collson such that the mnmum dstance between the moble platform and the obstacle s greater than a threshold. When the moble platform and the obstacle are crcular, the mnmum dstance d s formulated as d = PQ (r C/P + r ob ) ( 3.16) where PQ s the magntude of the poston vector r Q/P and r ob s the radus of the obstacle. Gven the poston of the centre of the obstacle r Q, and pose (poston and orentaton) of the moble platform, the poston of Q wth respect to P s r Q/P = [ r Q / O x / O y x, rq y ] ( 3.17) Usng the orentaton of r Q/P, nc, gven by r 1 Q / P y nc tan ( 3.18) rq / Px the orentaton of the normal contact force F nc s determned by nc = π + (θ nc + φ) ( 3.19) hen, nc s substtuted nto equaton ( 3.8) to construct the soluton of cable tensons as gven by equaton (.7). For general non-crcular shapes, the calculaton of the mnmum dstance d and the orentaton of the normal contact force, nc have to be modfed accordngly. 3.5 rajectory Plannng and Impact Reducton When collson occurs, the gven trajectory of the moble platform has to be modfed whle keepng postve tenson n each cable and satsfyng addtonal desrable crtera,.e., mnmzng the norm of tenson n the cables for a gven normal contact force F nc and mnmzng the trajectory devaton from the desred trajectory of the moble platform. 77

101 In the work presented, before collson occurs, redundancy s resolved at the torque level whle tracng the gven trajectory of the moble platform. herefore, the optmzaton problem before collson s formulated as follows: mnmze τ τ 1 n subject to τ τ τ Nλ τ mn p max ( 3.) For each pose of the moble platform, the value of s calculated such that mnmum cable tensons are mantaned subject to postve tenson n each cable. When collson occurs, the mnmum postve tenson n the cables s determned for each tme nstant such that the magntude of the normal contact force F nc does not exceed a certan value. For the subsequent nstants, the trajectory of the moble platform s modfed whle collson s avoded and the modfed trajectory traces the gven moble platform trajectory as close as possble. hen, usng the redundancy of the manpulator, the mnmum tenson n the cables s calculated such that the tenson of each cable s postve. If the postve tenson n each cable s not acheved, the trajectory s modfed agan and the tenson mnmzaton procedure s repeated. For ths purpose, the trajectory of the moble platform s gven as xo ( t) t tc xo ( t) x var ( t) tc t t 1 oc x( t) x y ( 3.1) xo ( t) x var ( t) toc t t m xo ( t) tm t t f where t c s the nstant at whch the mnmum dstance d becomes zero and collson occurs, t oc s the tme nstant to clear the obstacle n the orgnal trajectory x o, t m s the nstant at whch the modfed trajectory merges nto the desred trajectory, and t f s the fnal tme nstant. he gven trajectory of the moble platform s x o = [x o (t), y o (t), o (t)] and vector x var ( t) [ x ( t), y ( t), ( t)] j varj var j var s the j varable porton of the trajectory. he gven trajectores for x o, y o and o are chosen to be ffth order 78

102 polynomals satsfyng eghteen ntal and fnal boundary condtons of the trajectory, x, and ts dervatves. he varable porton of the moble platform trajectory, nstant, t c < t < t m. x varj, s calculated at each tme In order to have contnuous xvar and y 1 var for t > t 1 c, after collson occurs, the orgnal trajectory of the moble platform s changed n x and y drectons n a way that the moble platform follows a spral path, as depcted n Fgure 3.5. On the condton that the postve tenson constrant s satsfed, s kept unchanged,.e., var =. In other words, for t 1 c < t t oc, f postve tenson n each cable cannot be acheved, the orentaton of the moble platform wll then be changed. he trajectory s modfed at t = t c + t such that for t c t t oc PQ( t) as bs s ( t) ( 3.) where a s and b s are known constant coeffcents, and s corresponds to the orentaton of r P/Q,.e., r P/Q = [ PQ cos s, PQsn s the boundary condtons of PQ and s at t c and t oc as ], as shown n Fgure 3.5. Coeffcents a s and b s are defned usng b s PQ( toc) PQ( tc ) ( t ) ( t ) s oc s c ( 3.3) and a s PQ( toc) bs s ( toc) PQ( tc ) bs s ( tc ) ( 3.4) where PQ, the magntude of the poston vector r Q/P gven by equaton ( 3.17), s obtaned at t c and t oc usng the orgnal trajectory x o as PQ t r x ( t ) r y ( t ( ) Q / O o Q/O o ) x y ( 3.5) where t could be t c or t oc. Smlar to nc n equaton ( 3.19), the angle s s determned at t c and t oc as 79

103 ( t ) tan s 1 r r Q / Oy Q / Ox y x o o ( t ) ( t ) ( 3.6) he angle s at each tme nstant t, t c < t < t oc, s gven by t t s ( t) oc t t c s ( tc ) s ( tc ) s ( t ) ( 3.7) oc c By substtutng s from equaton ( 3.7) nto equaton ( 3.), PQ s calculated for t c < t < t oc. Followng that, the poston of the moble platform s obtaned as x t), y( t) r PQ( t) cos ( t), r PQ( t) sn ( t) ( Q / Ox s Q / O y s ( 3.8) Y O X P at t oc s at t oc s at t c Q P at t c Fgure 3.5. Spral trajectory of the moble platform for t c t t oc. When the moble platform clears the obstacle, the trajectory of the moble platform merges nto the desred trajectory x o wthn a specfed tme span, t m t oc, whch ndcates how fast x merges nto x o. Smlar to the desred trajectory, the modfed trajectores x and y, for t oc < t t m, are chosen to be ffth order polynomals satsfyng eghteen boundary condtons of x and y, and ther dervatves at t oc and t m. In other words, the ffth order polynomals x and y, for t oc < t t m, are determned such that 8

104 ( x, x, x) ( x, x, x) toc tm ( x, x, x ) o o o o o toc ( x, x, x ) o tm ( x, x, x var1 var1 var1 toc ) ( 3.9) and ( y, y, y) ( y, y, y) toc tm ( y, y, y ) o o o o o toc ( y, y, y ) o tm ( y, y, y var1 var1 var1 toc ) ( 3.3) Smlar to the tme nterval t c t t oc, provded that the postve tenson constrant s mantaned, s kept unchanged,.e., var =. Usng of x var, j x var, j = 1,, the constrant functon represented by equaton ( 3.9) wll become a functon j y var and j varj. he goal after collson, t c < t t f, s to trace the gven trajectory as close and smooth as possble and mnmze the tenson n each cable. herefore, the objectve functon s to mnmze x x x y and τ τ, one at a tme, such that the var j var j varj obstacle s avoded and mn (t, varj varj x var, y j var, j var ) j max. 1 n In the smulaton, the cables are allowed to cross over the obstacle whch could occur when the obstacle and the cables are on dfferent planes. After calculatng x var, j y var and j varj, at each tme nstant (t > t c ), the mnmum s calculated that mantans postve tenson n all cables. he velocty and acceleraton components of the moble platform are then approxmated usng the backward dfference method as and x tk tk tk ( ) x ( 1 x ( ) ) ( 3.31) t x tk tk tk ( ) x ( ) ( 1 x ) ( 3.3) t he results are subsequently substtuted nto equaton ( 3.7) for tenson optmzaton purposes. 81

105 3.6 Smulaton Results In ths secton, the smulaton results of redundancy resoluton of the planar cable-drven manpulator, shown n Fgure 3.1, are presented. Smlar to Secton.4, the anchor postons are {a 1, a, a 3, a 4 } = {[ 1,.75], [1,.75], [1,.75], [ 1,.75] } (unts n meters). he angular postons of cable attachment ponts on the moble platform wth respect to the movng frame are defned by { 1,, 3, 4 } = {5, 315, 45, 135} (unts n degrees). he mass m p, moment of nerta I z, and radus r B/P of the moble platform are respectvely kg,.144 kg.m, and.15 m. he moment of nerta of each pulley I p, the radus of each pulley r p, and the vscous dampng coeffcent at each motor shaft c m are respectvely.8 kg.m,.5 m, and.1 Nms. Consderng the conventon shown n Fgure.3(b), all pulley angles are defned to be zero ntally. A typcal value of E = 57.3 GPa has been chosen, for a 77 steel strand provded n [5]. he dameter of the cable s D c = 1.6 mm and the cross-secton area of each cable s A =.16 mm. Further nformaton on 77 stanless steel wre ropes and avalable optons can be found n [1]. he followng range of dampng coeffcents per unt length of cable, c w l Ns, s nvestgated for each cable. he radus and poston of the obstacle are r ob =.5 m and [r Q/Ox, r Q/Oy ] = [.5 m,.1 m]. he ntal and fnal boundary condtons ( x, x, x, y, y, y,,, ) are (,,,,,,,, ) and (.35 m,,,.35 m,,, deg,, ) f, respectvely, wth t[, 1] s and tme step of t =.1 s. he mnmum allowable tenson of each cable s mn = N. Smlar to Secton.4, the upper lmt on the actuator torques s 315 N for a steel cable wth a breakng strength of 16 N [] and a safety factor of 4. Wth four cables, the constrant functon n equaton ( 3.) s reduced to four lnear nequaltes n terms of, where s reduced to a scalar. At the nstant of collson, the maxmum allowable 8

106 y, y o [m] x, x o [m] normal contact force F nc s 5 N. he termnaton tolerances placed on constrant volatons are chosen as.1 N, on the objectve functon as.1 N, and on the estmated parameter as.1 N. he smulaton results are obtaned usng the fmncon functon n MALAB. Fgure 3.6 shows the gven and modfed trajectores of the moble platform,.e., x and x o, respectvely. he vertcal dash-dotted lnes correspond to nstants t c =.19 s, t oc = 5.77 s and t m = 8.77 s. For t c < t t oc, the mnmzaton of x var has resulted n the x component of the trajectory wth lower magntude compared to x o. Due to the locaton of the obstacle, tracng the x component of the trajectory has been delayed. On the other hand, due to the spral path, as defned by equaton ( 3.), the mnmzaton of y var, for t c < t t oc, has resulted n y components wth larger magntudes compared to y o..4. x x o t c t oc t m y y o Fgure 3.6. Orgnal and modfed trajectory of moble platform. As explaned n the prevous secton, for t oc < t t m, x and y are ffth order polynomals satsfyng eghteen boundary condtons at t = t oc and t = t m. Although the obstacle s cleared at t = t oc, to avod dscontnuous trajectory at t = t oc, the trajectory s modfed untl t merges nto the desred trajectory smoothly., o [deg] o tme [sec] 83

107 var [deg] y var [m] x var [m].5 t c t oc t m tme [sec] Fgure 3.7. Varable porton of the moble platform trajectory. Fgure 3.7 shows the varable porton of the moble platform trajectory, x var = x x o. From Fgure 3.7, t can be seen how close x s to x o. For t c < t t oc, x var corresponds to x var and for t 1 oc < t t m, x var corresponds to x var Fgure 3.8. Change n confguraton of moble platform. 84

108 Fgure 3.8 shows how the moble platform moves around the obstacle untl t clears the obstacle and then contnues the orgnal trajectory. Fgure 3.9 depcts the actual acceleraton of the moble platform. Collson results n jumps n the velocty and acceleraton of the moble platform at t c + t =. s. For t. s, the trajectory of the moble platform s modfed such that the magntude of the acceleraton should not exceed 1g. At nstant t c + t =. s, the maxmum deceleraton n x drecton and the maxmum acceleraton n y drecton, are 4.3 m/s and 5.93 m/s, respectvely. Due to the trajectory and redundancy of the manpulator, the orentaton of the moble platform remans unchanged and so do ts dervatves. Fgure 3.9. Acceleraton of moble platform. o compare the results wth the orgnal acceleraton of the moble platform, the results shown n Fgure 3.9 are magnfed along wth the orgnal acceleraton of the moble platform, as shown n Fgure 3.1. For t oc t t m, x and y correspond to ffth order polynomal trajectores satsfyng the boundary condtons at t oc and t m, as explaned n the prevous secton. 85

109 Fgure 3.1. Orgnal and modfed acceleraton of moble platform t c t m 4 t oc [N] tme [sec] (a) (b) Fgure (a) Cable length rates, (b) mnmum to mantan postve tenson n cables. 86

110 Fgure 3.11(a) represents the cable length rates resultng from the substtuton of x nto equaton (.). Consderng Fgure 3.11(a) and the trajectory of the moble platform that results n the confguratons shown n Fgure 3.8, as the moble platform moves the frst cable s released whereas the thrd cable s wound around the thrd pulley. Fgure 3.11(b) shows the mnmum that guarantees postve tenson n each cable. Fgure 3.1 llustrates the tenson hstores. It can be seen that the second cable s not crtcal (except for the collson nstant) for mantanng the confguratons shown n Fgure 3.8. Substtutng the cable tensons and the angular velocty and acceleraton of the pulleys nto equaton (.6) results n the actuator torques shown n Fgure he peaks n the acceleraton components of Fgure 3.9 have resulted n jumps n the cable tensons and actuator torques at the correspondng tme nstant t c + t =. s [N] [N] 3 [N] 4 [N] tme [sec] 5 1 tme [sec] Fgure 3.1. enson n cables. 87

111 3 [N.m] 4 [N.m] 1 [N.m] [N.m] tme [sec] 5 1 tme [sec] Fgure Actuator torques. l 1 [m] 4 x l [m] 4 x 1-4 No dampng c w l =. Ns c w l =. Ns x x 1-4 l 3 [m] 5 1 tme [sec] l 4 [m] 5 1 tme [sec] Fgure Deflecton of cables. 88

112 l 1ND - l 1c [m] [m] w3 l 3 w4 l 4 [m] [m] l 3ND - l 3c w1 l 1 w l 5 x x tme [sec] l ND - l c l 4ND - l 4c 5 x Fgure Deflecton dfference for three dampng coeffcents. x tme [sec] c w l = Ns c w l = Ns Fgure 3.14 shows the amount the cables are stretched. As gven n equaton ( 3.13), the soluton of l o depends on the dampng coeffcent of cables. hree values were selected for the dampng coeffcent of the cables: c w l = Ns, c w l = Ns and c w l =. he deflecton l = l l o was calculated solvng the second order equaton ( 3.13) for l o, and then subtractng l o from the deflected length of the correspondng cable, l. For the geometrc and materal propertes of the cables, the dampng forces n cables, c ( l l ) n equaton ( 3.11), are much smaller than the w o elastc forces, l l E A l o o n equaton ( 3.11). herefore, no sgnfcant dfference between the results s notced, except for the nstant of collson. When the dampng s neglected, hgher peaks are notceable after collson. For the calculated cable tensons, t was verfed that the cables are stretched,.e., l >, durng the moton of the moble platform. Due to the dependency of l o on the dampng coeffcent of cables gven n equaton ( 3.13), the maxmum cable deflecton, whch 89

113 happens at t c + t =. s, l that belongs to the fourth cable, s approxmately.34 mm for c w l =,.33 mm for c w l = Ns, and.6 mm for c w l = Ns, as shown n Fgure o nvestgate the effect of dampng coeffcents, the dfference between l when there s no dampng (.e., l ND when c w l = ) and when c w l = Ns s plotted n Fgure he deflecton dfference s also plotted for c w l = Ns. As shown n Fgure 3.15, the effect of havng dampng s only notceable at the nstant of collson. 3.7 Conclusons In the work presented n ths chapter, the dynamcs of cable-drven parallel manpulators was nvestgated consderng a smple model for cables that ncluded ther elastcty and dampng effects. An approach to resolve the actuaton redundancy of planar cable-drven parallel manpulators was examned to mnmze the cable tensons and devaton n the trajectory whle clearng an obstacle wth postve tenson n each cable. Before collson, a prescrbed trajectory of the moble platform was followed and the norm of vector of cable tensons was mnmzed. he nfnte solutons for cable tensons were utlzed to dentfy a vector of cable tensons such that the magntude of the contact force was kept below an assgned value whle the cable tensons were mnmzed. After collson, the desred moble platform trajectory was modfed at each tme nstant such that the moble platform traced ts predefned trajectory as close and smooth as possble. After collson, a spral path was proposed untl the obstacle was cleared. Once the obstacle was cleared, the trajectory was modfed such that the trajectory merged nto the gven trajectory wthn a specfc tme span satsfyng the boundary condtons on poston, velocty and acceleraton. 9

114 Once the trajectory was modfed, the mnmum norm tenson n the cables and the actuator torques were determned. he smulaton of a 3-DOF planar cable-drven parallel manpulator was developed satsfyng objectve functons and constrants. hree dampng coeffcents were used n the smulaton. No sgnfcant varatons, except for the nstant of collson, were notced regardng the effect of dampng on the deflecton of cables. Accordng to the results, the dampng effects could be neglected n a small workspace wth short cables. Based on the optmzaton crtera, the tolerances used n the optmzaton routne, the optmzaton scheme, and the desred contnuous moble platform trajectores, contnuous cable tensons were produced before and after collson. 91

115 Chapter 4 Force Capablty and Cable Falure Introducton Dependng on the applcaton of the manpulator, e.g., assembly and pck-and-place applcatons, knowng the range of applcable loads on the moble platform s of great nterest. Moreover, n dynamc envronments where collson s mmnent, for safety reasons, t s desred to reduce the force/moment that the manpulator apples and to know the maxmum forces and moments that the moble platform apples/ressts wthout damagng the envronment/manpulator. It s also benefcal to know whether the moble platform can mantan ts poston and orentaton (pose) wthout any external forces and moments, and f not, what the mnmum external forces and moments are n order to keep the desred pose/trajectory of the moble platform. Falure analyss s a crtcal element when desgnng a manpulator and ts components. Consderng the cable-drven parallel manpulators, a falure can be any malfunctonng n the cable actuatng mechansm that results n dfferent cable length rate and cable tenson for nstance [95]. Any falure n the actuatng mechansm can affect the performance of the manpulator. o provde relablty and safety n the operaton of autonomous and ntellgent robotc systems, t s desrable that a robot controller can montor the actuator performance on-lne and detect actuator faults and slack cables (n cable-drven parallel manpulators). Dependng on the task and envronment of the cable-drven manpulator, the actuator wth faled or slack cable may be kept operatng, or be put nto a safe mode to avod unnecessary damages. 3 Parts of the works presented n ths chapter have been publshed n [8]. 9

116 In the presented work, the maxmum and mnmum forces and moments the moble platform can apply (or resst) are determned at the dynamc level for a gven trajectory, as well as at the statc level for a gven pose of the moble platform. Optmzaton and non-optmzaton-based methods are utlzed to fnd the mnmum norm tenson n the cables correspondng to the mnmum and maxmum force and moment capabltes. Some of the challenges assocated wth the redundancy resoluton of cable-drven parallel manpulators, ncludng postve tenson requrement n each cable, nfnte nverse dynamc solutons, slow-computaton abltes when usng optmzaton technques, falure of the manpulator, and also desgnng an effectve control structure consderng the former concerns to follow a prescrbed trajectory wth small trackng error are addressed. he dynamc and statc force and moment capabltes are nvestgated n Secton 4.. he falure analyss s gven n Secton Secton 4.3 presents a control scheme based on feedback lnearzaton. he smulaton results are reported n Secton 4.4 wth the dynamc and statc force capabltes respectvely n Secton and Secton 4.4., the effect of cable falure n Secton 4.4.3, and the Cartesan trajectory control employng a Cartesan PID controller n Secton he chapter concludes wth Secton Force/Moment Capabltes o nvestgate the maxmum and mnmum force capabltes of the moble platform n any drecton, a constraned optmzaton (for the maxmum force) and a non-optmzaton-based method (for the mnmum force) are carred out to fnd the mnmum norm of cable tensons vector consderng the lmts of cable tensons. he tenson lmts are the mnmum and maxmum allowable tensons the cables can provde n order to avod havng slack cables, and breakng the cables or saturatng the actuators. Havng the components of the maxmum/mnmum force that 93

117 the moble platform can apply/resst, the drecton of the appled/ressted force can be calculated. Due to the unt nconsstency n the components of wrench W n equaton (.5), the maxmzaton and mnmzaton are performed for the force and moment separately. For the dynamc case, gven the trajectory of the moble platform, the mnmum norm tenson that results n maxmum/mnmum external force/moment s calculated subject to postve cable tenson. For the statc case, the acceleraton vector x n equaton (.5), as well as the rotatonal velocty and acceleraton of the spool n equaton (.6) are zero, and gven the pose of the moble platform, the mnmum norm tenson s calculated to acheve the maxmum/ mnmum force capabltes such that postve tenson n the cables s mantaned Maxmum Capablty o determne the maxmum external force/moment the moble platform can apply/resst, optmzaton problem s formulated as follows: maxmze F F or ext x ext y M extz subject to F F M extx exty extz x M y g J 1 n ( 4.1) and, = 1,, n mn max o avod unt nconsstency, nstead of optmzng force and moment separately, the work done by the external forces and moments, U ext, durng each tme nterval t = t k t k 1, can be optmzed. In other words, havng the lnear and rotatonal dsplacements of the moble platform,.e., x, y and, durng t, the tenson n the cables are vared wthn ther lmts such that the work done by the external forces and moments are maxmzed. he work U ext s gven by 94

118 U ext F ext Fext M ext x y x y z ( 4.) herefore, the objectve functon can be wrtten as maxmze U ext F x F y M ( 4.3) extx exty extz 4.. Mnmum Capablty For the mnmzaton problem,.e., the smallest force capablty, the goal s to nvestgate whether the moble platform can mantan ts desred pose wthout any external force/moment, and f not, what the mnmum requred external force/moment s n order to mantan the desred pose/trajectory of the moble platform. o reduce the computatonal tme that s desrable n real tme applcatons, nstead of performng an optmzaton procedure, t s nvestgated f a vald soluton for the cable tensons exsts assumng the external force/moment s [ F ext, F x ext, M y ext ] = [,, ] (.e., the mnmum external wrench, f possble). hus, usng z the null space contrbuton, the soluton of cable tensons n equaton (.5) s calculated smlar to equaton (.7) as where τ and τ are p h τ 1 n τ p τh mn max ( 4.4) τ p J # Mx g ( 4.5) τ ( I J J ) k h # where the generalzed nverse of the transposed Jacoban, J # s gven by equaton (.9). he ( 4.6) homogeneous soluton h could also be wrtten as τh Nλ, gven by equaton (.31). he soluton of cable tensons usng the non-optmzaton-based method s gven by equaton (.47) n Secton

119 4..3 Falure Analyss In Secton 4., the maxmum and mnmum external force and moment the moble platform can apply/resst, consderng the lmts of cable tensons, were formulated. Havng the results for the maxmum and mnmum external force and moment, the cable tensons are calculated when one of the cables fals. he cable falure formulatons presented n ths secton are adapted from [95]. he relatonshp between the cable forces, the nerta terms, and the external forces and moments on the moble platform s gven n equaton (.5). When cable fals, t cannot mantan the desred tenson. If ths level of cable tenson affects the requred external forces and moments then the faled cable may result n the falure of the manpulator. When the tenson of cable has a value dfferent from the requred tenson, the external forces and moments on the moble platform, W f, wll be W f Mx g J Mx g 1 n j1 J j J J where s column of J representng the axs of cable, τ [ 1, and c s the J tenson n faled cable. he change n the force/moment capablty of moble platform wll be j J W = W W f = J ( f ) = J ( c) ( 4.8) If the remanng cables could provde the lost tenson due to the faled cable, there wll be no change n the force/moment capablty of the manpulator. o fully compensate for the lost moble platform force/moment,.e., for W =, the correctonal tenson the remanng cables provde should be τ corr J # f J ( ) J c ) ( 4.9) where n, the th column of J s replaced by zero,.e., J [ J 1 J ], and J f n τ f ( 4.7) ( ) c # f J ( τ τ f f c n ] f n J # f J f f ( J J f ) 1 that maps the lost force/moment capablty, 96 J ( c), to the orthogonal

120 complement of the null space of the reduced Jacoban matrx J f. After applyng the correctonal tenson, the devaton n force/moment capablty wll be zero usng # f f the lost force/moment could be fully recovered. f W ( I J J )J ( τ τ f ) ( 4.1) he formulaton can be extended to the case when k cables have dfferent tenson. he correctonal tenson the remanng cables should provde, for W =, wll be where n J f τ corr k # J f J 1 ( ) J c # f J ( τ τ, k columns of J correspondng to the faled cables are replaced by zero. f ) ( 4.11) o calculate the maxmum/mnmum external force/moment when one or more cables fal, two approaches are proposed. In the frst approach, for the manpulator wth faled cable(s), the external wrench W s substtuted wth the result of maxmum/mnmum external force/moment calculated usng the optmzaton procedure presented n Secton 4., and then, tenson n each cable s calculated. If the tenson constrant s not satsfed, the manpulator cannot mantan ts force/moment capablty. In that case, f the calculated s less than the mnmum allowable mn, t s replaced by mn, and f t s greater than the maxmum allowable max, t s replaced by max. hen, the vector of cable tensons n equaton (.5) s updated and W s calculated based on the updated cable tensons as F F M extx exty extz x M y g J 1 97 n ( 4.1) he second approach to calculate the force capablty n the presence of a falure s to perform an optmzaton method when one of the cables fals. In other words, for the manpulator wth a faled cable, tenson n the cables are changed usng the smlar procedure n Secton 4.,

121 satsfyng tenson constrants, such that maxmum/mnmum external force/moment s acheved, where the optmzaton varables are cable tensons. 4.3 Controller Desgn hs secton presents the control scheme and the dynamc algorthm for calculatng the actuator torques and nstantaneous pose of the moble platform consderng the uncertanty n the actuaton torque coeffcents. A PID controller s used to mantan the desred performance n the presence of actuaton degradaton and cable falure, to montor the degradaton on-lne, and to detect slack cables. In the presented approach, the torque loss coeffcents are ncorporated n the dynamc model formulaton wth the commanded actuator torques as plant nputs nstead of the actual jont torques. he manpulator dynamc model s often not known accurately due to the presence of Coulomb frcton, backlash, payload varaton, large flexblty, unknown dsturbances, large dynamc couplng between dfferent lnks, and tme-varyng parameters such as frcton and those parameters relatng to the robot age (wear and tear) [13]. When the manpulator dynamcs model s not perfect, the analyss of the resultng closed loop system becomes more dffcult. When decouplng and lnearzng a system are not performed by the controller, or are naccurate or ncomplete, the overall closed loop system remans nonlnear [39]. Stablty and performance analyss s much more dffcult for nonlnear systems. One method of stablty analyss s called Lyapunov stablty analyss n whch one s requred to propose a generalzed energy functon that: (1) has contnuous frst partal dervatve and s strctly postve for all state vectors except at zero where t would be zero, and () has nonpostve dervatve, where here, the dervatve means the change n the generalzed energy functon along all system trajectores [39]. 98

122 hese propertes may not necessarly be global, wth correspondngly weaker or stronger stablty results [39]. he dea s to show a postve defnte energy-lke functon of state to always decrease or reman constant. hus, the system would be stable n the sense that the sze of the state vector s bounded [39]. When the dervatve of the energy functon s strctly less than zero, asymptotc convergence of the state to the zero vector can be concluded. In ths secton, the controllers are consdered based on feedback lnearzaton theory to asymptotcally stablze the system to the commanded (reference) pose x R = [x R, y R, R ], whle keepng postve tenson n the cables. A model-based control scheme wth a PID term s mplemented to reduce the trackng error e = x R x. he control archtecture, shown n the block dagram of Fgure 4.1, conssts of four man parts: the PID controller, the nverse dynamcs soluton, vrtual to real torque calculaton wth mnmum torque estmaton satsfyng postve cable tenson condton, and drect dynamcs soluton. o compute the torque control law, equaton (.5) s rewrtten as J τ M( x K e K e K edt) g W ( 4.13) R p d where matrx gans K p, K d and K are postve defnte dagonal matrces accomplshed ndependently for the x, y and motons even though the dynamcs model s coupled. As explaned n Secton.3, the soluton to equaton ( 4.13) s gven by wth p as τ p # mn τ τ p τ h, 1,, n max ( 4.14) J M( x K e K e K edt) g W ( 4.15) R p d and h as gven by equaton (.3) and equaton (.31). 99

123 x R g W C m β I β p Dsturbance x R + x e PID K e K p d e K + edt + + x Inverse dynamcs M + R p + + c + + calculaton a Plant Drect dynamcs x Double ntegrator x Fgure 4.1. Block scheme of operatonal space feedback lnearzaton/computed torque control for planar cable-drven manpulator. 1

124 Gven the actual Cartesan values, x, Jacoban and coeffcent matrces on the rght-hand sde of equaton (.3), and substtutng the actual velocty of the moble platform and the cable tensons of equaton ( 4.14) nto equaton (.3), the torque control law s obtaned n a standard Cartesan form as c R p τ C m R J x I t p J I pr p pr p where c s the vector of commanded actuator torques. Jx ( 4.16) Note that the actual feedback values, x, and forward pose knematcs are used n the formulaton. Alternatvely, n practce, due to uncertanty, plus sensor nose problems and the problem of dgtally twce dfferentatng the sensor feedback x, the reference Cartesan values, x R, could be chosen nstead of the actual feedback values from actuator encoder sensors [1]. he actual torque wll be less than the commanded torque because of power loss due to frcton, slp, heat generaton. For smulaton purposes, a loss coeffcent matrx, C loss, s defned that relates the commanded torque, c, to the actual torque, a, as C ( 4.17) a loss where C loss dag[ Closs,, C ] 1 loss and C n loss, = 1,, n, s the loss coeffcent of actuator. he coeffcent matrx, C loss, s an nn postve defnte dagonal matrx. Usng the actual feedback values, x, and the actual torque, a, gven by equaton ( 4.17), the vector of actual cable tensons, a, s determned by rearrangng equaton (.3) as τ a c J R 1 p a CmJ I p x I pjx ( 4.18) t Note the actual tenson a, can no longer guarantee the postve tenson condton. o have postve cable tensons the actual tenson a s requred to be equal to or greater than mn that s 11

125 C loss hus, on the condton that R 1 p c C m J I p J x I pjx τ t mn ( 4.19) C c 1 loss R p τ mn C m R 1 p J I R p 1 p J x I t 1 pr p Jx ( 4.) the actual cable tensons, a, wll satsfy the postve tenson constrant. Otherwse, any cables wth negatve tenson are consdered slack and ther actual tensons wll be replaced by zeros. Alternatvely, f there s no cable/actuator falure, the cable tensons could ntally be calculated consderng actuators loss coeffcents such that the tenson n all cables be postve. In that case, by replacng c n equaton ( 4.) wth c n equaton ( 4.16), the lower bound tenson constrant n equaton ( 4.14) s updated as 1 τ mn τ C loss ( 4.1) herefore, gven the actuators loss coeffcent matrx, C loss, the soluton to equaton ( 4.13) s recalculated as 1 loss mn τ τ p τ h C, 1,, n max ( 4.) and as a result, the actual tenson, a, gven by equaton ( 4.18) s guaranteed to be equal to or greater than mn. Once the actual cable tensons are obtaned, the acceleraton of the moble platform s updated by rearrangng equaton (.5) as 1 1 x M J τ g W ( 4.3) a Usng the backward dfference method or double ntegratng the acceleraton of the moble platform, x, pose of the moble platform, x, wll be updated and used as poston feedback shown n Fgure 4.1.

126 13 Note there s not generally drect access to Cartesan poston x feedback va sensors. Instead, the feedback s calculated usng the encoder feedback for each cable pulley angle to determne the cable lengths l usng equaton (.7) as β R l l p. hese lengths are then used as the nputs to the forward poston knematcs soluton of equaton (.1). Snce the pose of the moble platform s the same for all cables, equaton (.1) s used for any two cables and j for the calculaton of moble platform orentaton as ) sn( ) cos( ) sn( ) cos( / / / / j P B j j j P B j j P B P B j y y j x x y y x x r l a r l a r l a r l a ( 4.4) o solve equaton ( 4.4) for the orentaton of the moble platform,, equaton ( 4.4) could be expanded and smplfed n terms of cos or sn. he results would be two second order polynomal equatons where the unknown varable s ether cos or sn. One of the polynomals, n terms of cos, s gven by cos cos 1 fw fw fw C C C ( 4.5) where / / / / / / / / sn sn cos cos cos 1 x x x x j j x x x x j j j j j P B P B fw j P B P B j j fw j P B P B P B P B fw l a l a r r C r r l a l a C r r r r C ( 4.6) A smlar polynomal equaton could be wrtten n terms of cos or sn usng the second row of equaton ( 4.4). hus, m equatons, m beng the dmenson of the task space, could be derved n the form of equaton ( 4.5) where each equaton has two roots and each root gves two possble

127 orentatons,. he actual orentaton would be the common soluton among these 8m possble solutons. Once the moble platform orentaton s determned, the Cartesan poston x s calculated usng equaton (.1) for feedback n the control archtecture shown n Fgure Smulaton Results In ths secton, the smulaton results of redundancy resoluton of the planar cable-drven manpulator, shown n Fgure., are presented. For the smulaton, the anchor postons are {a 1, a, a 3, a 4 } = {[ 1,.75], [1,.75], [1,.75], [ 1,.75] } (unts n meters), and the angular postons of cable attachment ponts on the moble platform wth respect to the movng frame are defned by { 1,, 3, 4 } = {5, 315, 45, 135} (unts n degrees). he mass m p, moment of nerta I z, and radus r B/P of the moble platform are respectvely kg,.144 kg.m, and.15 m. he moment of nerta of each spool I p, the radus of each spool r p, and the vscous dampng coeffcent at each motor shaft c m are respectvely.8 kg.m,.5 m, and.1 Nms. Consderng the conventon shown n Fgure.3(b), all spool angles are defned to be zero ntally. he desred trajectores for x, y and are chosen to be ffth order polynomals satsfyng eghteen ntal and fnal boundary condtons of the trajectory, x, and ts dervatves. he ntal and fnal boundary condtons ( x, x, x, y, y, y,,, ),f are (,,,,,,,, ) and (.35 m,,,.35 m,,, deg,, ) f, respectvely, wth t[, 1] s and tme step of t =.1 s. he mnmum allowable tenson of each cable s mn = N. he upper lmt on the actuator torques, max, s calculated based on the maxmum allowable cable tenson of 315 N for a steel cable wth a breakng strength of 16 N [] and a safety factor of 4. 14

128 y [m] x [m] Wth four cables, the constrant functon n equaton ( 4.4) s reduced to four lnear nequaltes n terms of gven n equaton (.31), where s reduced to a scalar. he termnaton tolerances placed on constrant volatons are chosen as.1 N, on the objectve functon as.1 N for force optmzaton and.1 (Nm) for moment optmzaton, and on the estmated parameter (.e., cable tensons) as.1 N. he smulaton results are obtaned usng the fmncon functon n MALAB. For the smulaton, a computer wth a GHz Dual-Core CPU was used Dynamc Fgure 4. shows the desred trajectory and the confguraton of the moble platform along the gven trajectory. [deg] tme [sec] -1 1 (a) Fgure 4.. (a) Desred trajectory of moble platform, (b) change n confguraton of moble platform. he ntal guess for the optmzaton varables (tenson n the cables) at each tme nstant was chosen consderng the results of the optmzed varables n the prevous tme nstant. For the ntal tme step, the maxmum/mnmum allowable tenson n the cables was chosen as the ntal.5 guess f maxmzaton/mnmzaton was gong to be performed. (b) 15

129 m extz [N.m] F exty [N] F ext [N] F extx [N] For the gven trajectory, shown n Fgure 4.(a), the external forces and moment that could be ressted by the moble platform correspondng to the maxmum force capablty are plotted n Fgure 4.3(a). he maxmum magntude of the force, as shown n Fgure 4.3(b), s approxmately 5 N at the end of the trajectory. As t can be seen from Fgure 4.3(a) and (b), for the gven trajectory, as the moble platform moves away from the orgn of the base frame, the external forces and moment ncrease. Gven the external force and moment, there are nfnte tenson solutons for equaton (.5), and Fgure 4.4(a) depcts the mnmum norm postve tenson soluton that can provde maxmum external force the moble platform can apply/resst tme [sec] ext [deg] tme [sec] (a) (b) Fgure 4.3. (a) Force and moment result for maxmum force capablty, (b) magntude and drecton of maxmum external force correspondng to force results n the adjacent plot. For the mnmum force capablty, t s desred that the external force/moment on the moble platform be zero satsfyng tenson constrants. o nvestgate that, the nverse dynamc problem s solved for zero external force/moment, and the mnmum norm tenson s calculated. In the smulaton, f p, calculated usng equaton ( 4.5), volates tenson lmts, s replaced wth m = mn f < mn, or wth = max f > max. hen, usng equaton (.49) wth external wrench p m p 16

130 W =, the arbtrary vector k s obtaned and substtuted nto equaton ( 4.6). Once p and calculated, the cable tensons are obtaned usng equaton ( 4.4). τ h are For the same trajectory, shown n Fgure 4.(a), the mnmum norm tenson n the cables that can provde zero external force and moment s shown n Fgure 4.4(b). From the second plot of Fgure 4.4(b), t can be seen that durng the frst.45 s perod of the trajectory, cable, 3 and 4 have tensons more than N n order to follow the desred trajectory. For the subsequent nstants, cable 1, 3 and 4 are capable of mantanng the desred trajectory wthout the help of cable. Fmax1 [N] Fmax [N] Fmax4 [N] Fmax3 [N] tme [sec] (a) Fgure 4.4. (a) Cable tensons correspondng to maxmum force capablty, (b) Mnmum norm cable tensons correspondng to zero external forces and moments (.e., mnmum force capablty) he executon tme of the developed MALAB code usng both optmzaton-based (for maxmum force capablty) and non-optmzaton-based (for mnmum force capablty) methods was about 36 seconds. However, usng the optmzaton-based method for both maxmum and mnmum force capabltes ncreases the computatonal tme to about 56 seconds. Fmn [N] Fmn4 [N] Fmn1 [N] Fmn3 [N] tme [sec] 1 (b) 4.4. Statc 17

131 In ths secton, the maxmum and mnmum force capabltes are determned for the statc case for moble platform orentaton of = 5 deg. In the smulaton, the optmzaton s carred out for a range of platform postons wthn x and y lmts consderng the anchor postons, where 1 x 1 (meter) and.75 y.75 (meter) wth a step of.1 m n each drecton. Fgure 4.5 shows the maxmum statc force capablty. he color bars of Fgure 4.5(a) and (b) respectvely correspond to F ext and F x ext y, where the unts are n Newtons. As the horzontal and vertcal dstances from the centre of the base frame ncrease, the magntude of the maxmum that the moble platform can apply/resst, respectvely ncrease to a maxmum of 9 N approxmately, at the boundary of the statc workspace close to anchor postons. Fgure 4.6 shows the moment correspondng to the maxmum statc force capablty. he color bar represents the ntensty of moment M extz, where the unt s Nm. Fext x and F ext y (a) Fgure 4.5. Maxmum statc force capablty for = 5 deg: (a) (b) F ext, (b) x ext y F. 18

132 Fgure 4.6. M ext correspondng to maxmum statc force capablty for = 5 deg. z (a) (b) (c) (d) Fgure 4.7. Cable tensons correspondng to maxmum statc force capablty, (a) cable 1, (b) cable, (c) cable 3, (d) cable 4. 19

133 Fgure 4.7 shows the cable tensons correspondng to the maxmum statc force capablty. he color bars correspond to the tenson ntensty, where the unts are n Newtons. he presented workspace, where 1 x 1 (meter) and.75 y.75 (meter), s dvded nto four quadrants n a counter clockwse order such that the anchor poston of cable 1 s n quadrant 1 where x and y are both negatve, anchor poston of cable s n quadrant where x s postve and y negatve, and so on for quadrants 3 and 4. Accordng to the results, the cable correspondng to each quadrant can provde the mnmum tenson n that quadrant. he results of external forces and moment, and the cable tensons for the mnmum force capablty are depcted n Fgure 4.8 to Fgure 4.1, respectvely. Accordng to the results (excludng the areas near the boundares at the anchor postons), t can be concluded that for the gven orentaton of the moble platform, no sgnfcant external forces/moments are requred n order to mantan the gven pose of the moble platform. For the statc case, usng the optmzaton-based method presented n Secton 4., the executon tme of the developed MALAB code was about 14 mnutes. Durng ths tme, the force capablty was determned for 3351 poses. (a) (b) Fgure 4.8. Mnmum statc force capablty for = 5 deg: (a) F ext, (b) F x ext y 11

134 Fgure 4.9. M ext correspondng to mnmum statc force capablty for = 5 deg. z (a) (b) (c) (d) Fgure 4.1. Cable tensons correspondng to mnmum statc force capablty, (a) cable 1, (b) cable, (c) cable 3, (d) cable