PROBLEMS AND PROCEDURES FOR KINEMATIC DESIGN OF MANIPULATORS

Transcription

1 第八屆全國機構與機器設計學術研討會 台北科技大學 2005 年 12 月 2 日 PROBLEMS AND PROCEDURES FOR KINEMATIC DESIGN OF MANIPULATORS Marco Ceccarell LARM: Laboratory of Robotcs and Mechatroncs DMSAT - Unversty of Cassno Va D Baso 43, Cassno (Fr), Italy ceccarell@uncas.t ABSTRACT- Ths survey overvews the currently used procedures for a knematc desgn of manpulators that can be grouped n three man approaches: extenson of Synthess of Mechansms, applcaton of Screw Theory, applcaton of 3D Knematcs/Geometry. The survey s presented by llustratng basc concepts and man formulatons n order to emphasze on problem formulaton, computatonal efforts, and future challenges. KEYWORDS: Robotcs, Robot Knematcs, Desgn of Manpulators. 1. INTRODUCTION Knematc desgn of manpulators refers to the determnaton of the dmensonal parameters of a knematc chan,.e. lnk lengths and lnk angles. Once the knematc archtecture of a manpulator s szed by means of a knematc desgn, a manpulator can be completely defned by means of a mechancal desgn that specfes all the szes and detals for a physcal constructon. Indeed, knematc desgn s a fundamental step n a desgn procedure of any mechancal system and ts accuracy wll affect strongly the basc propertes of a mechancal systems. In the case of manpulators the knematc desgn s of a partcular mportance snce the manpulator tasks can be performed when the knematc archtecture has been properly conceved or chosen and specfcally syntheszed (.e. knematc desgn). Several approaches have been formulated for the knematc desgn (.e. synthess) of mechansms and many of them have been specalzed for robotc manpulators. General procedures and specfc algorthms both for general knematc archtectures and specfc desgns of manpulators have been proposed n a very rch lterature. A lmted lst of references s reported wth the am to gve to the readers basc sources and suggestons for further readng on the topc. In ths paper a tentatve survey of current ssues s presented by usng basc concepts and formulatons n order to emphasze on problem formulaton and computatonal efforts. Indeed, a great attenton s stll addressed to knematc desgn of manpulators by robot desgners and researchers manly wth the am to mprove the computatonal effcency, generalty and optmalty of the algorthms, even wth respect to new and new requrements for robotc manpulatons. Ths survey has overvewed the currently avalable procedures for knematc desgn of manpulators that can be grouped n three man approaches, namely extenson of Synthess of Mechansms (for example: Precson Pont Technques, Workspace Desgn, Inverson algorthms, Optmzaton formulaton), applcaton of Screw Theory, applcaton of 3D Knematcs/Geometry (for example: Le Group Theory, Dual Numbers, Quaternons, Grassmann Geometry). A knematc desgn procedure s amed to obtan closedform formulaton and/or numercal algorthms, whch can be used not only for desgn purposes but even to nvestgate effects of desgn parameters on desgn characterstcs and operaton performance of manpulators. Usually, there s a dstncton between open-chan seral manpulators and closed-chan parallel manpulators. Ths dstncton s also consdered as a constrant for the knematc desgn of manpulators and n fact dfferent procedures and formulaton have been proposed to take nto account the pecular dfferences n ther knematc desgn. Nevertheless, recently, attempts have been made to formulate a unque vew for knematc desgn both of seral and parallel manpulators, manly wth an approach usng optmzaton problems. Future challenges n the feld of robot desgn can be recognzed manly n the aspects for computatonal effcency and n concevng new manpulator archtectures wth a fully nsght of desgn degeneracy both of the knematc possbltes and proposed numercal algorthms. 1

2 2. THE PROBLEM OF KINEMATIC DESIGN OF MANIPULATORS The manpulator archtecture of a robot s composed of an arm mostly for translaton movements, a wrst for orentaton movement, and an end-effector for nteracton wth the envronment and/or external objects, as shown n Fg. 1. Generally, the term manpulator refers specfcally to the arm desgn, but t can also nclude the wrst when attenton s addressed to the overall manpulaton characterstcs of a robot. A knematc study of robots deals wth the determnaton of confguraton and moton of manpulators by lookng at the geometry durng the moton, but wthout consderng the actons that generate or lmt the manpulator moton. Therefore, a knematc study makes possble to determne and desgn the moton characterstcs of a manpulator but ndependently from the mechancal desgn detals and actuator s capablty. A knematc chan can be of open archtecture, when referrng to seral connected manpulators, or closed archtecture, when referrng to parallel manpulators, as n the examples shown n Fg. 2. The knematc model of a manpulator can be obtaned n the form of a knematc chan or mechansm by usng the above-mentoned schemes for jonts and rgd lnks through essental dmensonal szes for connectons between two jonts. The moblty of a manpulator s due to the degrees of freedom (d.o.f.s) of the jonts n the knematc chan of the manpulator, when the lnks are assumed to be rgd bodes. In order to determne the geometrcal szes and knematc parameters, one can usually refer to a scheme lke that n Fg. 3 by usng a H D notaton, n agreement wth a procedure proposed by Hartenberg and Denavt n Fg. 1: A scheme for the manpulator archtecture of a robot wth the arm, wrst, and end-effector. a) b) Fg.2: Planar examples of knematc chans of manpulators: a) seral chan as open type; b) parallel chan as closed type. Ths scheme gves the mnmum number of parameters that are needed to descrbe the geometry of a lnk between two jonts, but also ndcates the jont varables. The jonts n Fg. 3 are ndcated as bg black ponts n order to stress attenton to the lnk geometry and H D parameters. Fg. 3: A knematc scheme for manpulator lnk parameters accordng to the H- D notaton. In partcular, referrng to Fg. 3 for j-lnk, the j-frame X j Y j Z j s assumed as fxed to j-lnk, wth the Z j axs concdng wth the jont axs, wth the X j axs lyng on the common normal between Z j and Z j+1 pontng to Z j+1. The orgn O j and the Y j axs are determned to obtan a Cartesan frame as shown n Fg. 3. The knematc parameters of a manpulator can be defned accordng to the H D notaton n Fg. 3 as: - a j, whch s named as the lnk length that s measured as the dstance between the Z j and Z j+1 axes along X j ; - j, whch s named as the twst angle that s measured as the angle between the Z j and Z j+1 axes about X j ; - d j+1, whch s named as the lnk offset that s measured as the dstance between the X j and X j+1 axes along Z j+1; - j+1, whch s named as the jont angle that s measured as the angle between the X j and X j+1 axes about Zj j+1 When a jont can be modeled as a rotaton par, the angle j+1 s the correspondng knematc varable. When a jont can be modeled as a prsmatc par, the dstance d j+1 s the correspondng knematc varable. The other H D parameters can be consdered as dmensonal parameters of the lnks. The H D notaton s very useful for the formulaton of the poston problems of manpulators through the socalled transformaton matrx by usng matrx algebra. The poston problem of manpulators, both wth seral and parallel archtectures, conssts of determnng the poston and orentaton of the end-effector as a functon of the manpulator confguraton that s gven by the lnk poston that s defned by the jont varables. In general, the poston problem can be consdered from 2

3 dfferent vewponts dependng on the unknowns that one can solve n the followng formulatons: - Knematc Drect Problem n whch the dmensons of a manpulator are gven through the dmensonal H D parameters of the lnks but the poston and orentaton of the end-effector are determned as a functon of the values of the jont varables; - Knematc Inverse Problem n whch the poston and orentaton of the end-effector of a gven manpulator are gven, and the confguraton of the manpulator chan s determned by computng the values of the jont varables. A thrd knematc problem can be formulated as: - Knematc Indrect Problem (properly Knematc Desgn Problem ) n whch a certan number of postons and orentatons of the end-effector are gven but the type of manpulator chan and ts dmensons are the unknowns of the problem. Although general concepts are common both for seral and parallel manpulators, peculartes must be consdered for parallel archtectures chans. One can consder as generalzed coordnates the poston coordnates of the center pont P of the movng platform wth respect to a fxed frame (Xo Yo Zo), Fg. 4, and the drecton s descrbed by Euler angles defnng the orentaton of the movng platform wth respect to a fxed frame. A matrx R defnes the orthogonal 3 3 rotaton matrx defned by the Euler angles, whch descrbes the orentaton of the frame attached to the movng platform wth respect to the fxed frame, Fg. 4. Let A and B be the attachment ponts at the base and movng platform, respectvely, and d the leg lengths. Let a and b be the poston vectors of ponts A and B n the fxed and movng coordnate frames, respectvely. Thus, the Inverse Knematcs Problem can be solved by usng A B p Rb a (1) to extract the jont varables from leg lengths. The length of the -th leg can be obtaned by takng the dot product of vector A B wth tself, for :1,,6 n the form Fg. 4: Parameters for knematc problems of parallel manpulators. The Drect Knematcs Problem descrbes the mappng from the jont coordnates to the generalzed coordnates. The problem for parallel manpulators s qute dffcult snce t nvolves the soluton of a system of nonlnear coupled algebrac equatons (1), and has many solutons that refer to assembly modes. For a general case wth planar base and platform, the Drect Knematcs Problem may have up to 40 solutons. A 20- th degree polynomal can be derved whose solutons lead to 40 mutually symmetrc assembly modes. The poston and orentaton of the movable plate can be computed by usng the geometry of the mechancal desgn of plates and legs, separately. In partcular, the plate desgns provde the possblty to deduce specfc formulaton as functon of the typcal solutons. Generally, the plate desgns are characterzed by havng the jont connectons n the plane plate and typcal desgn confguratons are llustrated n Fg. 5. In partcular, the general confguraton 6S of Fg. 5 a) of the Gough Stewart Platform refers to the case of sx sphercal jonts S that are nstalled on both plates on the vertces of a hexagon. The case of Fg. 5 b), that s named 3S plate desgn, s characterzed by the geometry of an equlateral trangle n whch vertces are located n the sphercal jont centers. Ths desgn plate and ts modfcatons s usually used n parallel manpulators wth a reduced number of d.o.f.s. Smlar s the plate desgn of Fg. 5 c), whch s an llustratve example of the many combnatons that can be obtaned from the desgn plate of Fg. 5 b) by substtutng the sphercal jonts wth revolute jonts or by a combnaton of these. Indeed, even the general plate desgn of Fg. 5 a) can gve a great varety of manpulator archtectures when sphercal jonts are substtuted by revolute and prsmatc jonts or a combnaton of these. Unversal jonts or Hooke jonts can be used to obtan the equvalent moblty of S jonts. d 2 t p Rb a p Rb a (2) Fg. 5: Plate desgns for attachments n the plates of parallel manpulators: a)general 6S desgn n the Gough Stewart Platform; b) 3S trangular desgn; c) S-2R-R plate desgn. 3. PROCEDURES FOR KINEMATIC DESIGN OF MANIPULATORS Synthess deals wth reverse problems of Analyss. Analyss s the Drect Problem formulaton of Knematcs of mechansms and manpulators n whch the pose,.e. poston and orentaton, s formulated as functon of the knematc varables of nput actuaton, when the knematc chan s gven. Thus, Synthess of mechansms and manpulators deals wth the calculaton 3

4 of the knematc chan as functon of manpulatve tasks. Characterstc manpulatve tasks of manpulators concern wth manpulaton of objects as movement and orentaton of grasped objects or end-effector tself durng a sutably programmed moton of a manpulator. But manpulaton nclude also other aspects of functonal and operaton characterstcs, and nowadays mechatronc approaches are also used to consder those other aspects. Calculaton of knematc chan of mechansms and manpulators s usually attached through three problems, namely type synthess, number synthess, and dmensonal synthess. Number synthess concerns wth the determnaton of the number of lnks and jonts n the chan, whch are useful or necessary to obtan a desred moblty and manpulaton capablty of a manpulator. Smlarly, the type synthess concerns wth the determnaton of the structure of the knematc chan,.e. the type of jonts and knematc archtecture, that are useful or necessary to obtan a desred moblty and manpulaton capablty of a manpulator. Fnally, dmensonal synthess (.e. knematc desgn) concerns wth the calculaton of the lnk szes and range moblty of jonts that are useful or necessary to obtan a desred moblty and manpulaton capablty of a manpulator. Type synthess and number synthess are related to morphologes of manpulator archtectures and today they are approached wth desgner s own experence or through complex desgn procedures that most of the tme can be understood as data bases n nformatcs expert systems. The tradtonal desgn actvty on manpulators s ndeed stll recognzed n the problem of the dmensonal desgn of a manpulator when ts knematc archtecture s gven. Ths s the problem that s surveyed n ths paper. The manpulatve tasks that are used as desgn data and constrants, are related manly to knematc features such as workspace, path plannng, Statc accuracy; but other aspects can be also consdered wthn the Mechancs of robots, such as sngulartes, stffness behavor, dynamc response. One aspect of relevant sgnfcance for manpulator desgn s the workspace analyss that s often used as desgn means yet, besde as a crtera for evaluatng the qualty of desgned solutons. Postonng and orentaton capablty can be evaluated by computng poston and orentaton workspaces that gve the reachable regons by the manpulator extremty or endeffector as functon of the moblty range of the manpulator jonts. Poston workspace refers to reachable ponts by a reference pont on manpulator extremty, and orentaton workspace descrbes the angles that can be swept by reference axes on manpulator extremty. Once the workspace ponts (both n poston and orentaton) are determned, one can use them to perform an evaluaton of workspace characterstcs and feasblty evaluaton of knematc desgn solutons. In partcular, a cross-secton area can be determned by selectng from the computed workspace ponts those that lay on a cross-secton plane under examnaton. Thus, the shape can be llustrated by the result n a plot form. The computaton of the value of a cross-secton area can be obtaned by usng a grd evaluaton or an algebrac formula. By referrng to the scheme of Fg. 6 for a grd evaluaton, one can calculate the area measure A as a sum of the scannng resoluton rectangles over the scanned area as I J A = 1 j=1 A P j (3) j by usng the A P j entres of the bnary matrx that are related to the cross-secton plane for A. Alternatvely, one can use the workspace ponts of the boundary contour of a cross-secton area that can be determned from an algebrac formulaton or usng the entres of the bnary matrx. Thus, referrng to the scheme of Fg. 7 and by assumng as computed the coordnates of the cross-secton contour ponts as an ordnate set (r j, z j ) of the contour ponts H j wth j=1,, N, the area measure A can be computed as N z z r A (4) j =1 1,j 1 1,j 1,j r1,j 1 By extendng the above-mentoned procedures, the workspace volume V can be computed by usng the grd scannng procedure n a general form A Pjk j k I J K V (5) =1 j1 k=1 n whch P jk s the entry of a bnary representaton n a 3D grd. Fg. 6: A general scheme for bnary representaton and evaluaton of manpulator workspace. 4

5 consderng the manpulator extremty/end-effector as a coupler lnk whose knematcs s the purpose of the formulaton. Thus, Drect and Inverse Knematcs can be formulated and Synthess problems can be attached by usng Precson Ponts as those ponts (.e. poses n general) at whch the pose and/or other performances are prescrbed as to be reached and/or fulflled exactly. Ths can be expressed as F F (8) X Fg. 7: A scheme for computng workspace volume of seral open-chan manpulators wth revolute jonts. By usng the boundary contour ponts through the Pappus-Guldnus Theorem the workspace volume V can be computed wthn the bnary mappng procedure as n whch F s the performance evaluaton at -th precson pont whose coordnates X are functon of the mechansm confguraton that can be obtanng by solvng closure equatons through any tradtonal methods for mechansm analyss. I 2 V 2 P j + (6) =1 J j=1 or wthn the algebrac formulaton n the form j 2 2 z z r N V 1,j 1 1,j 1,j r1,j 1 (7) 2 j =1 when the workspace volume s a sold of revoluton. Therefore, t s evdent that the formula of Eq. (6) has a general applcaton, whle Eq. (7) s restrcted to seral open-chan manpulators wth revolute jonts. But smlar approaches and formulaton can be proposed and used for a numercal evaluaton of workspace characterstcs of parallel manpulators. Smlarly, hole and vod regons, as unreachable regons, can be numercally evaluated by usng the formulas of Eqs (4) to (7) to obtan the value of ther cross-sectons and volumes, once they have been prelmnarly determned. Orentaton workspace can be smlarly evaluated by consderng the angles n a Cartesan frame representaton. Summarzng, the desgn problem for manpulators can be formulated as a set of equatons, whch gve the poston and orentaton of a manpulator n term of ts extremty together wth addtonal expressons for requred performance n term of sutable crtera evaluatons. 3.1 PROCEDURES FROM SYNTHESIS OF MECHANISMS Snce manpulators can be treated as spatal mechansms, the tradtonal technques for mechansm desgn can be used once sutable adaptatons are formulated to consder the pecularty of the open chan archtecture. Two ways can be approached as referrng to general model for closure equatons: elaboraton of closure equatons for the open polygon ether by addng a fcttous lnk wth ts jonts ether by usng the coordnates of the manpulator extremty, as shown n the llustratve example of Fg.8. In any case, the tradtonal technques for mechansms are used by Fg. 8 Closng knematc chan of a3r manpulator by addng a fctous lnk and sphercal jonts. Thus, desgn requrements and desgn equatons can be formulated for the Precson Ponts, whose maxmum number for a mathematcal defned problem can be determned by the number of varables. But, the pose accuracy and plannng as well as the performance value away from the Precson Ponts wll determne errors n the behavor of the manpulator moton, whose evaluaton can be stll computed by usng the desgn equatons n proper procedures. Precson Ponts technques for mechansms have been developed for path postons, but for manpulator desgn the concept has been extended to performance crtera such as workspace boundary ponts and sngulartes. Thus, new specfc algorthms have been developed for manpulator desgn by usng approaches from Mechansm Desgn but wth specfc formulaton of the pecular manpulatve tasks. Approaches such as Newton-Raphson numercal technques, dyad elmnaton, Graph Theory modelng, moblty analyss, Instantaneous knematc nvarants have been developed for manpulator archtectures as extenson of basc propertes of planar mechansms that have been nvestgated and used for desgn purposes snce the second half of 19-th century. Of course, the complexty of 3D archtectures have been requested the development of new more effcent calculaton means, such as a sutable use of Matrx Algebra, 3D Geometry consderatons, and Screw Theory formulaton. 3.2 APPLICATION OF 3D GEOMETRY 5

6 AND SCREW THEORY Three dmensonal Geometry of spatal manpulators has requred and requres specfc consderaton and nvestgaton on the 3D characterstcs of moton. Thus, dfferent mathematzatons can be used by takng nto account of generalty of 3D moton. Dual numbers and quaternons have been ntroduced n the last decades to study Mechansm Desgn and they are specfcally appled to study 3D propertes of rgd body moton n manpulator archtectures. The structure of mathematcal propertes of rgd body moton has been also addressed for developng new Algebra theores to practcal applcaton for desgn purposes of spatal mechansms and manpulators. Recently Le Group Theory and Grassman Geometry have been adapted and successfully appled to develop new calculaton means for desgnng new solutons and characterzng manpulator desgn n general frames. A group G s a non-empty set endowed wth a closed product operaton n the set satsfyng some defnton condtons. A subset H of elements of a group G s called a subgroup of G f the subset H consttutes a group whch has common group operaton wth the group G. Furthermore, a group G s called a Le group f G s an analytc manfold and the mappng GxG to G s analytc. The set {D} of rgd body motons or dsplacements s a 6-dmensonal Le group of transformatons, whch acts on the ponts of the 3- dmensonal Eucldean affne space. The Le subgroups of {D} play a key role n the moblty analyss and synthess of mechansms. Fg. 9: 4 Hybrd sphercal-sphercal dscontnuously movable 7R mechansm, (Lee and Hervè 2004) For example n Fg.9, (Lee and Hervè 2004), n the hybrd sphercal-sphercal spatal 7R mechansm has a mechancal bond {L(4,7)} between lnks 4 and 7 as the ntersecton set of two subsets {G1} and {G2}: {G1}={R(O1,uZ1)}{R(O2,uZ2)}{R(O1,uZ3)}{R(O2,uZ4)} {G2}={R(O2,uZ7)}{R(O2,uZ6)}{R(O1,uZ5)}. (9) Hence, the relatve moton between lnks 4 and 7 s depcted by {L(4,7)}= {G1} {G2} = {R(O1,uZ1)}{R(O2,uZ2)} {R(O1,uZ3)}{R(O2,uZ4)} {R(O2,uZ7)}{R(O2,uZ6)} {R(O1,uZ5)}. In general, {R(O1,uZ1)}{R(O2,uZ2)}{R(O1,uZ3)} {R(O2,uZ4)} {R(O1,uZ5)} {R(O2,uZ6)} s a 6-dmensonal knematc bond and generates the dsplacement group {D}.Therefore,{R(O1,uZ1)}{R(O2,uZ2)}{R(O1,uZ3)}{R(O 2,uZ4)}{R(O1,uZ5)}{R(O2,uZ6)} {R(O2,uZ7)} = {D} {R(O2,uZ7)} = {R(O2,uZ7)}. Ths yelds that the A-G-B- F-C-E-D 7R chan has one dof when all knematc pars move and consequently {L(4,7)} ncludes a 1- dmensonal manfold denoted by {L(1/D)(4,7)}. If all the pars move and jont axes do not ntersect agan, any possble moblty characterzed by ths geometrc condton stops occurrng and we have {L(4,7)} {L(1/D)(4,7)}. Grassman Geometry and further developments have been used to descrbe the Lne Geometry that can be assocated wth spatal moton. Plucker coordnates and sutable algebra of vectors are used n Grassman Geometry to generalze propertes of moton of a lne that can be fxed on any lnk of a manpulator, but manly on ts extremty. Screw Theory was developed to nvestgate the general moton of rgd bodes n ts form of helcodal (screw) moton n 3D space. A screw entty was defned to descrbe the moton and to perform computaton stll through vector approaches. A unt screw s a quantty assocated wth a lne n the three-dmensonal space and a scalar called ptch, whch can be represented by a 6 x 1 vector as $ = [ s, r x s + s ] T (10) where s s a unt vector pontng along the drecton of the screw axs, r s the poston vector of any pont on the screw axs wth respect to a reference frame and s the ptch of the screw. A screw of ntensty s represented by $= $. When a screw s used to descrbe the moton state of a rgd body, t s often called a twst, represented by a 6 x 1 vector as $ = [, v ] T, where represents the nstant angular velocty and v represents the lnear velocty of a pont O whch belongs to the body and s concdent wth the orgn of the coordnate system. Screw Theory has been appled to manpulator desgn by usng sutable models of manpulator chans, both wth seral and parallel archtectures, n whch the jont moblty s represented by correspondng screws. Thus, screw systems descrbe the moton capablty of manpulator chans and therefore they can be used stll wth a Precson Pont approach to formulate desgn equatons and characterstcs of the archtectures. In Fg.10 an llustratve example s reported as based on the fundamental so-called Screw Trangle model for effcent computatonal purposes, even to deduce closed-form desgn expressons. 6

7 Fg. 10: A scheme of screw trangle for 3R manpulator desgn. 3.3 OPIMIZATION PROBLEM DESIGN The dualty between seral and parallel manpulators s not anymore understood as a competton between the two knematc archtectures. The ntrnsc characterstcs of each archtecture make each archtecture as devoted to some manpulatve tasks more than an alternatve to the counterpart. The complementartes of operaton performance of seral and parallel manpulators make them as a complete soluton for manpulatve operatons. The dfferences but complementartes n ther performance have gven the possblty n the past to treat them separately, manly for desgn purposes. Several analyss results and desgn procedures have been proposed n the last two decades. Manpulators are sad useful to substtute/help human bengs n manpulatve operatons and therefore ther basc characterstcs are usually referred to human manpulaton performance aspects. A well-traned person s usually characterzed for manpulaton purpose manly n terms of postonng skll, arm moblty, arm power, movement velocty, and fatgue lmts. Smlarly, robotc manpulators are desgned and selected for manpulatve tasks by lookng manly to workspace volume, payload capacty, velocty performance, and stffness. Therefore, t s qute reasonable to consder those aspects as fundamental crtera for manpulator desgn. But generally snce they can gve contradctory results n desgn algorthms, a formulaton as multobjectve optmzaton problem can be convenent n order to consder them smultaneously. Thus, the optmum desgn of manpulators can be formulated as where T s the transpose operator; X s the vector of desgn varables; F(X) s the vector of objectve functons that express the optmalty crtera, G(X) s the vector of constrant functons that descrbes lmtng condtons, and H(X) s the vector of constrant functons that descrbes the desgn prescrptons. There are a number of alternatve methods to solve numercally a mult-objectve optmzaton problem. In partcular, n the example of Fg. 11 the proposed multobjectve optmzaton desgn problem has been solved by consderng the mn-max technque of the Matlab Optmzaton Toolbox that makes use of a scalar functon of the vector functon F (X) to mnmze the worst case values among the objectve functon components. One can search the mnmum value by usng Sequental Quadratc Programmng that s successfully used for solvng optmzaton problems wth non-lnear objectve functons and constrants and several varables. Ths numercal procedure works n such a way that at each step k a soluton s found along a search drecton wth varable update k.the teraton contnues untl the vector of varables converges. A numercal procedure has been developed so that the formulaton for the manpulator desgn has been easly ncluded wthn the solvng procedure for the optmzaton problem by usng the facltes of the Optmzaton Toolbox of Matlab, whch permts an easy arrangement for an optmum desgn wth analytcal expressons. The heren used numercal procedure from commercal software package for the soluton of a mult-objectve optmzaton desgn problem s outlned n the flowchart of Fg.11. mn T 1 ( X), f 2 ( X),..., f N ( X) F( X) mn f (11) subjected to G ( X) 0 (12) H ( X) 0 (13) Fg.11: A general scheme for optmum desgn procedure by usng mult-objectve optmzaton problem solve by commercal software 7

8 The problem for achevng optmal results from the formulated mult-objectve optmzaton problem conssts manly n two aspects, namely to choose a proper numercal solvng technque and to formulate the optmalty crtera wth computatonal effcency. Indeed, the solvng technque can be selected among the many avalable ones, even n commercal software packages, by lookng at a proper ft and/or possble adjustments to the formulated problem n terms of number of unknowns, non-lnearty type, and nvolved computatons for the optmalty crtera and constrants. On the other hand, the formulaton and computatons for the optmalty crtera and desgn constrants can be deduced and performed by lookng also at the pecularty of the numercal solvng technque. Those two aspects can be very helpful n achevng an optmal desgn procedure that can gve solutons wth no great computatonal efforts and wth possblty of engneerng nterpretaton and gude. Snce the formulated desgn problem s ntrnscally hgh nolnear, the soluton wll be obtaned when the numercal evoluton of the tentatve solutons due to the teratve process converges to a soluton that can be consdered optmal wthn the explored range and therefore a soluton can be consdered an optmal desgn but as a local optmum n general terms. Ths last remark makes clear once more the nfluence of sutable formulaton wth computatonal effcency for the nvolved crtera and constrants n order to have a desgn procedure, whch s sgnfcant from engneerng vewpont and numercally effcent. 4.EXPERIENCES AT LARM IN CASSINO At LARM: Laboratory of Robotcs and Mechatroncs n Cassno, snce the begnnng of 90 s a research lne has been dedcated to the development of analyss formulaton of manpulator performances, that could be used n proper desgn problems by takng advantage of the pecularty of the solvng technques n commercal software. Results are reported n papers whose a selecton s reported n the references as regardng wth seral and parallel manpulators; more detals and further references can be found n the LARM webpage Snce the manpulatve characterstcs are fundamental for the operaton and desgn of both manpulator archtectures, the author and hs co-workers have attempted analyss procedures that could be used wth few adjustments and no great computatonal efforts n the performance analyss for both seral and parallel archtectures. The several experences have been summarzed n a recent textbook publshed by the author and t s reported n the references. Workspace has been analyzed to characterzed ts manfold and to formulate effcent evaluaton algorthms. Scannng procedure and algebrac formulaton for workspace boundary have been proposed as outlned n the Fgs. 6 and 7. Results can be obtaned lke the llustratve examples n Fg. 12. a) b) Fg.12: Illustratve examples of results of workspace determnaton through : a) scannng procedure of Fg.6; b) algebrac formulaton of workspace boundary. Attempts have been proposed for an nverson of the algebrac formulaton whose desgn algorthm can gve all possble soluton lke for the reported case of 3R manpulator n Fg.13. Further study has been carred out to characterze the geometry of rng (nternal) vods as outlned n Fg.14. a) b) Fg.1.3: Desgn solutons for 3R manpulators by nvertng algebrac formulaton for workspace boundary: a) all possble solutons; b) feasble workspace desgns. 8

9 Fg.16: Determnaton of sngularty confguraton of a wre parallel manpulator by lookng at the descrptve geometry of the manpulator archtecture Fg.14: Manfolds for rng vod of 3R manpulators. A workspace characterzaton has been completed by lookng at desgn constrants for solvable workspace n the form of the so-called Feasble Workspace Regons. The case of 2R manpulators has been formulated and general topology has been determned for desgn purposes, as reported n Fg. 15. Stffness of manpulators has been studed through models wth lumped parameters to deduce both for numercal and expermental dentfcaton of the entres of stffness matrx. The effcency and expermental feasblty s based on a parttonng of the stffness nto matrces for separate effect evaluaton of stffness coeffcents, wrench statcs transmsson, and knematc confguraton. Recently, a fnal goal has been focused on the development of a desgn procedure that has been based both on mult-crtera optmzaton problem and sutable numercal evaluaton of performances. An attempt has been proposed to obtan a unque desgn procedure both for seral and parallel manpulators through the objectve formulaton f1( X ) 1 V pos V pos ' Vor ' J f 2 ( X ) 1 Vor mndet f3( X ) (14) det J f 4 ( X) 1 5 ( X) 1 f U d Y d 0 U Y g g where V pos and V or values correspond to computed pèosotn and orentaton volume V and prme values descrbe prescrbed data; J s the manpulator Jacoban; U d and U g are complant dsplacements along X, Y, and Z-axes, Y d and Y g are complant rotatons about, and ; d and g stand for desgn and gven values, respectvely. Illustratve example results are reported n Fg.15: General geometry of Feasble Workspace Regons for 2R manpulators depcted as grew area. Sngularty analyss and stffness evaluaton have been approached to obtan formulaton and procedure that are useful also for expermental dentfcaton, operaton valdaton, and performance testng. Sngularty analyss has been approached by usng arguments of Descrptve Geometry to represent sngularty condtons for parallel manpulators through sutable formulaton of Jacobans va Cayley-Grassman determnates or doman analyss. Fgure 16 shoe examples how usng tetrahedron geometry n parallel manpulators has determned straghtforward the shown confguratons. Fg. 17: Evoluton of the functon F and ts components 9

10 versus number of teratons for an example of PUMAlke robot optmal desgn: poston workspace volume as f 1 ; orentaton workspace volume as f 2 ; sngularty condton as f 3 ; complant translatons and rotatons as f 4 and f 5. Fg. 18: Evoluton of desgn H-D parameters versus number of teratons for the example of PUMA-lke robot n Fg F 12 Objectve functons f 3 f2 f 1 0 f 5 f teraton Fg. 19: Evoluton of the functon F and ts components versus number of teratons for an example of CaPaMan optmal desgn. 300 D esgn parameters [mm] bk hk ck teraton Fg. 20: Evoluton of desgn parameters versus number of teratons for the example of CaPaMan optmal desgn n Fg.19. ak sk r p bk sk hk rp ak ck The above-mentoned example survey from the LARM experences s just to show the potentalty and feasblty of possble results n researchng for better desgn procedure of manpulators through many open problems both on conceptual and methodologcal aspects. 5. FUTURE CHALLENGES The topc of knematc desgn of manpulators, both for robots and mult-body systems, addresses and stll wll address attenton for research and practcal purposes n order to acheve better desgn solutons but even more effcent computatonal desgn algorthms. An addtonal aspect that cannot be consdered of secondary mportance, can be advsed n the necessty of updatng desgn procedures and algorthms for mplementaton n modern current means for the Informatcs Technology (hardware and software) that s stll evolvng very fast. Thus, future challenges for the development of the fled of knematc desgn of manpulators and mult-body systems at large can be recognzed, besde the nvestgaton for new desgn solutons, n: - more exhaustve desgn procedures, even ncludng mechatronc approaches; - updated mplementaton of tradtonal and new theores of Knematcs nto new Informatcs frames. Research actvty s often drected to new solutons but because the reached hghs n the fled manly form theoretcal vewponts, manpulator desgn stll needs a wde applcaton n practcal engneerng. Ths requres better understandng of the theores at level of practcng engneers and user-orented formulaton of theores. Thus, the above-mentoned challenges can be ncluded n a unque frame, whch s orented to a transfer of research results to practcal applcatons of desgn solutons and procedures. Mechatronc approaches are needed to acheve better desgn solutons by takng nto account the constructon complexty( ntegraton of current solutons and more of future systems n whch the mechancal counterpart s overwhelmed by many sub-systems of dfferent natures. Although the mechancal aspects of manpulaton wll be always fundamental because of the mechancal nature of manpulatve tasks, the desgn and operaton of manpulators and mult-body systems at large wll be more and more nfluenced by the desgn and operaton of the other sub-systems for sensors, control, artfcal ntellgence, and programmng through a multdscplnary approach/ntegraton. Ths aspect s completed by the fact that the Informatcs Technology provdes day by day new potentaltes both n software and hardware for computatonal purposes but even for techncal supports of other technologes. Ths pushes to re-elaborate desgn procedures and algorthms n sutable formulaton and logcs that can be used/adapted for mplementaton n the evolvng Informatcs. Addtonal efforts are requested by system users and practtoner engneers to operate wth calculaton means (codes and procedures n commercal software packages) that are more and more effcent n term of computaton tme and computatonal results (numercal 10

11 accuracy and generalty of solutons) as well as more and more user-orented desgn formulaton n term of understandablty of desgn process and ts theory.. ths s a great challenge: snce whle more exhaustve algorthms and new procedures (wth mechatronc approaches) are requested, nevertheless the success of future developments of the fled strongly depends on the capablty of the researchers of expressng the research result that wll be more and more specalst (and sophstcated) products, n a language (both for calculaton and explanatory purposes) that should not need a very sophstcate expertse. 6. CONCLUSIONS Snce the begnnng of Robotcs the complexty of the knematc desgn of manpulators has been solved wth a varety of approaches that are based on Theory of Mechansms, Screw Theory, or Knematcs Geometry. Algorthms and desgn procedures have evolved and stll address research attenton wth the am to mprove the computatonal effcency and generalty of formulaton n order to obtan all possble solutons for a gven manpulaton problem, even by takng nto account other features n a mechatronc approach. REFERENCES The reference lst s lmted to man works for further readng and to author s man experences. Ctaton of references has not ncluded n the text snce the subjects refer to a very reach lterature that has not ncluded for space lmts. Angeles J., Fundamentals of Robotc Mechancal Systems, Sprnger-Verlag, NewYork, Angeles J., The Robust Desgn of Parallel Manpulators, 1 st Int. Colloquum Coll. Research Center 562, Braunschweg, 2002, pp Bhattacharya S., Hatwal H., Ghosh A., On the Optmum Desgn of Stewart Platform Type Parallel Manpulators, Journal of Robotca, Vol, 13, 1995, pp Burdck, J. W., A Classfcaton of 3R Regonal Manpulator Sngulartes and Geometres, Mechansm and Machne Theory, 1995, Vol. 30, No.1, pp Carretero J. A., Podhorodesk R. P., Nahon M. A., Gosseln C. M., Knematc Analyss and Optmzaton of a New Three Degree-of Freedom Spatal Parallel Manpulator, ASME Journal of Mechancal Desgn, Vol.122, No.1, 2000, pp Ceccarell M., A Synthess Algorthm for Three- Revolute Manpulators by Usng an Algebrac Formulaton of Workspace Boundary, ASME Journal of Mechancal Desgn, Vol. 117, 1995, pp Ceccarell M., Optmal Desgn and Locaton of Manpulators, n: Computatonal Dynamcs n Multbody Systems, Kluwer, Dordrecht, 1995, pp Ceccarell M., A Formulaton for the Workspace Boundary of General N-Revolute Manpulators, Mechansm and Machne Theory, Vol. 31, 1996, pp Ceccarell M., Desgnng Two-Revolute Manpulators for Prescrbed Feasble Workspace Regons, ASME Journal of Mechancal Desgn, 2002, Vol. 124, pp Ceccarell M., An Optmum Desgn of Parallel Manpulators: Formulaton and Expermental Valdaton, 1 st Int. Coll. Collaboratve Research Center 562, Braunschweg, 2002, pp Ceccarell M., Fundamentals of Mechancs of Robotc Manpulaton, Kluwer, Dordrecht, Ceccarell M., Carbone G, A Stffness Analyss for CaPaMan (Cassno Parallel Manpulator), Mechansm and Machne Theory, Vol. 37, No. 5, 2002, pp Ceccarell M., Carbone G, Numercal and expermental analyss of the stffness performance of parallel manpulators, 2nd Internatonal Colloquum Collaboratve Research Centre 562, Braunschweg, pp.21-35, Ceccarell M., Lann C., A Mult-Objectve Optmum Desgn of General 3R Manpulators for Prescrbed Workspace Lmts, Mechansm and Machne Theory, 2004, Vol. 39, No. 2, pp Ceccarell M., Ottavano E., An Analytcal Desgn for CaPaMan wth Prescrbed Poston and Orentaton, 2000 ASME Bennal Mechansms and Robotcs Conf, Baltmore, 2000, paper DETC2000/MECH Ceccarell M., Ottavano E., Desgn Problems for Parallel Manpulators n Assemblng Operatons, IFAC Workshop n Intellgent Assembly and Dsassembly, Bucharest, 2003, pp Ceccarell M., Carbone G., Ottavano E., An Optmzaton Problem Approach for Desgnng both Seral and Parallel Manpulators, Proc. of MUSME 2005, Int. Symposum on Multbody Systems and Mechatroncs, Uberlanda, 2005, Paper No. 3- MUSME05. Duffy J., Analyss of Mechansms and Robot Manpulators, Arnold, London, Duffy J., Statcs and Knematcs wth Applcatons to Robotcs, Cambrdge Unversty Press, Dukkpat R. V., Spatal Mechansms, CRC Press, Erdman A. G. (Edtor), Modern Knematcs, Wsley, Freudensten F., Prmrose E.J.F., On the Analyss and Synthess of the Workspace of a Three-Lnk, Turnng- Par Connected Robot Arm, ASME Journal of Mechansms, Transmssons and Automaton n Desgn, 1984, Vol.106, pp Gosseln C. M., The Optmum Desgn of Robotc Manpulators Usng Dexterty Indces, Robotcs and Autonomous Systems, 1992, Vol. 9, pp Gosseln C., Angeles J., The Optmum Knematc Desgn of a Planar Three-Degree-of-Freedom Parallel Manpulator, ASME Journal of Mechansms, Transmssons, and Automaton n Desgn, Vol. 110, 1988, pp Grace A., Optmzaton Toolbox User s Gude, The Matlab Works Inc., Gupta, K. C., Roth, B., Desgn Consderatons for Manpulator Workspace, ASME Journal of 11

12 Mechancal Desgn, 1982, Vol. 104, pp Hao F., Merlet J.-P., Mult-Crtera Optmal Desgn of Parallel Manpulators Based on Interval Analyss, Mechansm and Machne Theory, Vol. 40, 2005, pp Lann C, Saramago S.F.P., Ceccarell M., Optmal Desgn of 3R Manpulators by Usng Classcal Technques and Smulated Annealng, Journal of the Brazlan Socety for Mechancal Scences, 2002, Vol. XXIV, pp Lee C, Hervè J.M., Synthess of Two Knds of Dscontnuously Movable Spatal 7R Mechansms through the Group Algebrac Structure of Dsplacement Set, Proceedngs of the 11th World Congress n Mechansm and Machne Scence, 2004, Tanjn, paper CK49. Manoochehr S., Sereg A.A:, A Computer-Based Methodology for the Form Synthess and Optmal Desgn of Robot Manpulators, Journal of Mechancal Desgn, 1990, Vol. 112, pp Merlet J.-P., Desgnng a Parallel Manpulator for a Specfc Workspace, Int. Journal of Robotcs Research, Vol. 16, No. 4, 1997, pp Merlet J-P.: Parallel Robots, Kluwer, Dordrecht, Neto J., Sntess de mecansmos, Edtoral AC, Ottavano E., Carbone G., A Procedure for the Multobjectve Desgn of Parallel Manpulators, Int. Journal of Mechancs and Control, vol.04, n.02, 2003, pp Ottavano E., Ceccarell M., Lann C., A Characterzaton of Rng Vod n Workspace of Three- Revolute Manpulators", Xth IFToMM World Congress, Oulu, 1999, Vol.3, pp Ottavano E., Ceccarell M., Optmal Desgn of CAPAMAN (Cassno Parallel Manpulator) wth Prescrbed Orentaton Workspace, Internatonal Journal Robotca, 2002, Vol.20, pp Paden B., Sastry S., Optmal Knematc Desgn of 6R Manpulators, The Int. Journal of Robotcs Research, 1988, Vol.7, No. 2, pp Park F.C., Optmal Robot Desgn and Dfferental Geometry, Transacton ASME, Specal 50 th Annversary Desgn Issue, 1995, Vol.117, pp Roth B., On the Screw Axes and Other Specal Lnes Assocated wth Spatal Dsplacements of a Rgd Body, ASME Jnl of Engneerng for Industry, Feb 1967, pp Roth B., Analytcal Desgn of Open Chans, 3rd Int. Symposum on Robotcs Research, MIT Press, 1986, pp Schonherr J., Evaluaton and Optmum Desgn of Parallel Manpulators Havng Defned Workspace, Proc. of ASME Bennal Robotcs Conference, Baltmore, 2000, Paper DETC2000/MECH Schraft R.D., Wanner M.C., Determnaton of Important Desgn Parameters for Industral Robots from the Applcaton Pont f Vew: Survey Paper, Proceedngs of ROMANSY 84, Kogan Page, London, 1985, pp Selg J.M.: Geometrcal Foundatons of Robotcs, Worls Scentfc, London, Seerng W.P., Schenman V., Mechancal Desgn of an Industral Robot, Ch.4, Handbook of Industral Robotcs, Wley, New York, 1985, pp Shmano B.E., Roth B., Dmensonal Synthess of Manpulators, 3rd RoManSy, 1980, pp Sobh T.M., Toundykov D.Y., Knematc synthess of robotc manpulators from task descrptons optmzng the tasks at hand, IEEE Robotcs & Automaton Magazne, Vol.11, n.2, 2004, pp Stoughton R., Ara T., A Modfed Stewart Platform Manpulator wth Improved Dexterty, IEEE Transactons on Robotcs and Automaton, Vol. 9, No. 2, 1993, pp Takeda Y., Funabash H., Knematc Synthess of In- Parallel Actuated Mechansms Based on the Global Isotropy Index, Journal of Robotcs and Mechatroncs, Vol. 11, No., 5, 1999, pp Tsa L.W.: Robot Analyss: The Mechancs of Seral and Parallel Manpulators, Wley, New York, Tsa M.J., Lee H.W., On the Specal Bases of Two- and Three-Screw Systems, ASME Jnl Of Mechancal Desgn, Sept. 1993, pp Ucker J.J., Pennock G.R., Shgley J.E, Theory of Machnes and Mechansms, Oxford Unversty Press, Vanderplaats G., Numercal Optmzaton Technques for Engneers Desgn, McGraw-Hll, New York, Vjaykumar R., Waldron K.J., Tsa M.J., Geometrc Optmzaton of Seral Chan Manpulator Structures for Workng Volume and Dexterty, The Int. Journal of Robotcs Research, 1986, Vol.5, No. 2, pp