Computer-Aided Modelling for a Poly Articulated Robotic Arm with Spherical Joints

Transcription

1 Proceedngs of the World Congress on Engneerng 211 Vol III, July 6-8, 211, London, U.K. Computer-ded Modellng for a Poly rtculated Robotc rm wth Sphercal Jonts Voleta Dumtru, Sorn Dumtru bstract: The paper refers to a seres of poly artculated robotc arms, wth multple vertebrae, havng functons of exploraton, nspecton, nterventon and grpng n operatng spaces wth reduced restrctons. t frst, the paper relates about the constructon of the dgtal model of the arm, the analyss technques and the constructon of the prototype from polymer powders usng the 3D technology. We propose an analytc method, wth SMB type hgh performance software, for the assembly of the model, the redundancy, the knematcs and reversed knematcs analyss. Then, t analyses the manoeuvrablty of the arm to obtan an mposed target. Index Terms: poly artculated, robotc arm, modelaton I. INTRODUCTION The conventonal poly artculated mechanc confguratons are dscrete mechansms obtaned by the connecton n cascade of certan rgd bodes artculated by a connecton wth a sngle degree of freedom. The elements of the knematc chan are made of passve components, the actvaton beng performed only at the level of the jont. In contrast wth the classcal structures, the serpentne type models (hyper redundant) are, theoretcally, bult from an nfnte number of jonts, whch allows to obtan a knematc chan descrbed by a contnuous curve that confers t a hypothetcally nfnte moblty. The dscrete mechancal systems can obtan an mproved manoeuvrablty and flexblty by addng several degrees of freedom. Roboson [1] proposed to classfy these types of mechancal systems n three categores: Dscrete, wth conventonal structure; serpentne, snake shape; contnuous. The serpentne systems are composed of numerous jonts connected by very short rgd connectons, beng a very moble structure n the apparent form of a smooth curve, lke a snake body. The contnuous systems are permanently undulatng along or by means of reslent deformaton. We must underlne the fact that there are fundamental dfferences between the three classes of mechancal systems, as we can notce from fgure 1. For the mechancal systems wth dscrete structure, each jont s actuated and rgdly V. Dumtru, Mechancal Engneerng Department, Unversty of Craova, Romana (tel ; fax ; e-mal dumtru.voleta@yahoo.com); S. Dumtru, Electrcal Engneerng Department, Unversty of Craova, Romana;(tel ; fax ;e-mal: sorn_dumtru23@yahoo.com) connected to the next jont, whle the contnuous robot (fg. 1.b) can perform the same operaton by energzng a sngle actuatng element. Ths concept also exsts for the hyperredundant manpulators whch have an addtonal number of degrees of freedom, therefore they need the actuaton of n jonts or t s enough a sngle actuatng element, fgure 1.a. a) b) Fg. 1. The structural dfferences exstent between hyper-redundant and contnuous structure; (a) mechancal system wth hyper redundant structure; (b) Mechancal system wth contnuous structure Poly artculated arms wth dscrete elements are a class of manpulators whch perform exploraton, nspecton, nterventon. Fg. 2. Geometrcal shapes of the peces used to buld up the tentacles In fgure 2 there are shown the geometrcal shapes of the peces that can be used to buld up the tentacles. There can be dentfed smple, symmetrcal or asymmetrcal geometrcal shapes. There can be varants wth a sngle type of pece, wth two types or three types of peces and rarely wth several constructve types of peces. sngle pece or a group of peces forms a geometrcal module [2], fgure 3. The smple structure of these manpulators permts the performance of some lght mechansms wth a wde operaton range, capable to operate n restrcted areas and wth a dffcult access. lso, the hyper-redundant mechansms can be used to poston some small loads that do not need a precse postonng, lke mnmum nvasve nspectons and exploratons. The analyss and the ISSN: (Prnt); ISSN: (Onlne)

2 Proceedngs of the World Congress on Engneerng 211 Vol III, July 6-8, 211, London, U.K. modulaton of these systems are really complex and a great number of researchers have tred to offer solutons. The snake lke robots have been nvestgated for a long tme for numerous applcatons that beneft of an ncreased dexterty and hgh capacty to avod obstacles [3], [4]. Due to the subsstent hgh number of freedom degrees, the effcent resoluton for the nverse knematcs was one of the man theoretcal problems that have delayed ther use. Chrkjan and Burdck [5] solved ths problem by usng the varaton calculaton methods. Prevous artculated constructons of shake robots were cumbersome for the constructors, as they needed a lot of mechancal jonts, so they had mechancal problems, namely clearances. In order to prevent these problems, a few papers by Walker and Gravagne [6] were concentrated on the knematcs and the control of the spne constructons. Dfferent concepts are also presented n [7], [8], [9], whch nspred ths paper. The authors have presented tools n the shape of a snake usng as materal super elastc NT that has a hgher rgdty than the SMs, but the heatng Fg. 3 Geometrcal modules made up of one or two peces problem and the complexty of the multple degrees of freedom (DOF) s stll relevant, therefore creatng DOF lmted mechansms and artculated systems actuated by flexble cables. In the nverse knematc analyss, Han and Rudolph [1] had obtaned remarkable results that represented the bass of the nverse knematc theory for the hyper redundant robots wth II. THEORETICL ND TECHNOLOGICL CONSIDERTION Techncal poly artculated hyper redundant models are complex structures that operate n a trdmensonal space. Constructvely, the poly artculated robotc arm s composed of a hgh number of jonts connected by very short rgd jonts (vertebrae), consttutng a moble structure whch has an apparent form of a smooth curve smlar to a snake body. Each vertebra s desgned for a smple ntegraton and to facltate the performance of the requests. These requrements nclude: more sectons for bendng along the hyper redundant arm; aspect (range 1:6 wth a vdeo capacty at the end). In table I and II are shown the functonal requrements and the desgn specfcatons for a poly artculated mechancal system. Constructvely, the poly artculated robotc arm conssts n two concentrc tubes (fgure 4). I wll refer to these tubes as beng two mechansms, an exteror one and the other an nsde one. ISSN: (Prnt); ISSN: (Onlne) Table I FUNCTIONL REQUIREMENTS Requrements Specfcatons 3D complex confguraton Task drected desgn Multple bendng sectons Spnnng of the lnks: ± 15 Forward/ backward Forward/ Backward: ± 22 mm movement daptablty/ Reconfguraton of the components Manoeuvrablty Smple, compact Rapd and stable fxaton Safety durng operaton Mnaturzng (reduced szes/ small loads) Increase the feld of vson spect report Range 1:6 Each mechansm enclosed several adjacent elements smlar to some rgd vertebra, nterconnected by cables (wres), dsposed n seres. Each lnk s made wth nversed segments (opposte) sngle or double curved (sphercal surfaces) and foreseen wth three sem-crcular longtudnal channels. When the nsde mechansm s nserted nto the exteror mechansm, the three operatonal ports (channels) can be algned, formng a three dmensonal space used to pass the nstruments from the frst end (proxmal) to the second one (dstal) of the robotc arm. The vertebrae are mantaned n contact by means of cables (wres); 3 cables for the exteror mechansm, dsposed at 12 degree between them and a central cable for the nsde mechansm, fgure 1.4. By actuatng two cables of the exteror mechansm t s possble to rotate the fnal lnk wth an angle of ± 1 n any drecton. By combnng the unque confguratons of the exteror/ nteror lnks (proxmal, ntermedary and dstal) of the exteror/ nteror mechansms we offer the tentacular robotc arm the ablty to make any route defned n the 3D space. Table II DESIGN SPECIFICTIONS Desgnng concept Movements generated n 3D Multple vertebrae ddtonal lnks nterconnected by cables (wres) Internal/ external mechansm Gudng devce for the mechansm ctuators for the actuaton ctuators for control Modular constructon Smple attachng/ removng devce Super elastc alloys Plastc materals Exteror endoscope mechancally attached External vdeo system dequate desgn The shape t can take based on the route made s mantaned by couplng the nteror mechansm on the rgd mode; Fg. 4. Structural vew of the poly artculated robotc arm The materals for the vertebrae and cables: thermal plastc polymers polyamde (duramd), hgh densty polyethylene, garolte (consoldated fbre). The features of the thermal plastc polymers are shown n table III.

3 Proceedngs of the World Congress on Engneerng 211 Vol III, July 6-8, 211, London, U.K. Table III FETURES OF THE THERML PLSTIC POLYMERS Polymer Densty (g/cm 3 ) T v C T t C Isotropc module (GPa) Crystallzaton module (GPa) P-6 1, ,8 16 HDPE ,4-2 3 III. THE DIGITL MODEL Ths paragraph descrbes the CD modelaton technques n order to establsh the optmal for of the exteror/ nteror vertebrae (proxmal, ntermedary, dstal) startng from a basc structure represented by the IGES surface and fnally reachng to the vrtual model of an ntermedary external vertebra. Ths model can serve to the performance of vrtual assembles that wll serve to numerc smulatons, respectvely to ts practcal performance usng the nonconventonal technology of Rapd prototypng. The CD modelaton of the vertebrae was performed usng conventonal technques avalable n CD Sold Work 21 software. We started by sketchng out a shape that would satsfy from the functonal pont of vew the needs of such vertebrae. These needs consst n the possblty of the vertebrae to be fxed by cables (wres). Thus, n the frst phase, t was modulated an ntermedary exteror vertebra whch fulfls to a certan extend the functonal requrements mposed to such a constructon. For the performance of ths vertebra are needed n modelaton operatons lke: jet mouldng, the creaton of plans, etc. In order to make the assembly, t was consdered the optmzaton of the shape for the ntermedary exteror vertebra, so that t should have two major functonaltes: the capacty for a precse postonng; the possblty to keep the full contact of the two adjacent enttes durng the requests. In fgure 5 s shown the basc sketch after whch we desgned the ntermedary exteror vertebrae. In order to force the free desgn of the sketch, between the consttuent geometrc elements were establshed, dependng on the case, relatons of co axalty, symmetry, parallelsm, vertcalty, etc. From the constructve pont of vew, the prototype of the ntermedary exteror vertebra s characterzed by the followng elements (fgure 5): transmsson outlet (1), wth a 7.5 mm dameter allows the ntermedary nteror vertebra, wth a 6.5 +/-,1 mm dameter, to pass; the frst cone (2); the second cone (3); the hemsphere (4), wth a 14.5 mm dameter, receves the second end of the prevous adjacent vertebra; the hemsphere (5), wth a 14.5 mm dameter, receves the frst end of the posteror adjacent vertebra; transmsson outlets (6), three of them wth a 1.25 mm dameter provdng the gudng of the operatng cables (wres); holes (7), wth a 1.25 and 2.5 mm provde the releasng of the cables operatng (wres); sem-crcular channels (8), three of them, dentcal, equdstant, together wth the exteror sem-crcular channels provde the gudance of the operatng cables (wres). In fgure 6 s shown the fnal model of the ntermedary exteror vertebra. The proxmal exteror vertebra, the dstal exteror vertebra, as well as the nteror vertebrae (proxmal, ntermedary, and dstal) had been shaped usng the same classcal technque of modellng lke for the ntermedary exteror vertebra. Fg. 5. Basc sketch after whch the ntermedary exteror vertebra was In order to verfy the wanted functonalty of the desgned vertebrae (exteror, nteror), t was performed ts assembly wth 2 exteror vertebrae, respectvely 33 nteror. Fg. 6. The fnal model of the ntermedary exteror vertebra Ths assembly was obtaned usng the Sold Works 21 software, fgure 7. In order to make the vertebrae we used one of the rapd prototypng technques (rapd manufacturng of the prototypes 3D models), namely the prototypng technque by 3D prntng. The 3D prototypng system Z prnter 31 produced by Z-Corporaton and owned by the company CD Work Craova, wth the followng features: the szes of the prototype 2 x 25 x 33; resoluton (1 mcrons); speed (24 mm/hour); materal (polyamde, polystyrene). Fg. 7. The poly artculated robotc arm assembly desgned wth Sold Work 21 software IV. THE MVEBRBILITY OF THE ROBOTIC RM ISSN: (Prnt); ISSN: (Onlne)

4 Proceedngs of the World Congress on Engneerng 211 Vol III, July 6-8, 211, London, U.K. The movement of a robotc arm, even a poly artculated hyper-redundant one or any other type, can be seen from two angles: from the knematc pont of vew; from the dynamc pont of vew. The knematcs studes the movement of the robotc arm. In a knematc analyss s calculated, at a gven moment, the poston, the speed and the acceleraton of each segment of the manpulator and all the dervates of the poston varables wthout takng nto consderaton the forces that cause ths movement. In order to analyse a mechanc system, the MBS type performant software allow several operatons and namely: the analyss of the feasblty (the assembly of the model), redundancy, statc balance, the knematc analyss, the dynamc analyss and the nverse dynamc. These can be performed separately or together n a certan sequence dependng on the type of analyss and the degree of freedom (DOF) of the analysed system. The degree of freedom of a mechancal system s gven by the number of movements that are not knematcally determned: DOF = DOM ND where, DOM (Degrees of Moblty) represents the degree of moblty, namely the number of the generalzed coordnates that defne all the elements n the mechansm and the DN (Drven Number) s the number of drven/ generated movements (rhenome restrctons, tme dependent). The movement of a rgd body s descrbed by sx generalzed coordnates: the coordnates that are orgns for the marker attached to the body and the drectons of the axs of ths marker relatvely to the axs of the global reference system (nertal). ccordngly, the degree of lberty s equvalent wth the total number of ndependent coordnates n the system: DOF = 6 x n r where n s the number of moble bodes n the system and r s the number of restrctons mposed n the lmts of certan admtted tolerances. The knematcs deals wth aspects of redundancy, avodng collsons and avodng sngulartes. The redundancy conssts n the dentfcaton and elmnaton of the redundant connectons (addtonal) from the over constraned mechancal system. In the shapng phase, the elmnaton of the redundances mples the elmnaton of the duplcates and the replacement of the jonts from the ntal module wth other type of jonts of nferor classes, n the condtons n whch the ntal movement of the mechancal system must be observed. The redundant constrans correspond to the movements that are not produced n the mechancal system, the redundant space (R) beng actually the reverse of the space assocated by movement (S), namely R = 6 S. Thus, the degree of lberty of the mechancal systems that contan redundances can be expresses by the relaton: DOF = 6 x n (r R). For drect knematc we consdered a seral knematc structure wth 2 rgd vertebrae, wth lengths l >, = 1, 2, 2, connected wth sphercal jonts for a poly artculated robotc arm. It was consdered the lengths of the elements and the relatve movements from the rotatonal spheres: φ 1, φ 21,... φ 192, l It s calculated the poston and the drecton of the fnal effort, namely the poston of the characterstc pont M O 2 and the drecton of the coordnate system attached to the effectors (O 2 X 2 Y 2 Z 2 ), wth respect to the reference coordnate system O X Y Z, through the method of homogenous operators [11]. In ths respect, to the basc element s assocated the Cartesan coordnate system O X Y Z (reference system) and the elements characterstc to the rght Cartesan coordnate systems O 1 X 1 Y 1 Z 1 (system 1), O 2 X 2 Y 2 Z 2 (system 2), O 2 X 2 Y 2 Z 2 (system 2). The coordnate systems 1, 2, result after the modulaton of relatve movements of knematc jonts through homogenous operators. The relatve movement (rotaton around the z axs) from the rotaton jont, formed between the basc element O and element 1, s modelled by the rotaton operator: z 1 R1 1 (4.1) Ths characterzes the poston and the orentaton of the coordnate system 1 wth respect to reference system OXYZ. The nverse operator characterzes the poston and the drecton of the reference coordnate system wth respect to the coordnate system 1. The operator s matrxes and are: 1 Cnm Snm Snm Cnm 1 1 Cnm Snm mn Snm Cnm 1 a whch, Cnm cos, nm S sn a nm nm The relatve movement from the rotaton jont formed between the 1 and 2 element s modelled by the rotaton operator:, whch characterzes the poston and the drecton of the coordnate system 2 wth respect to the coordnate system 1. The poston and the drecton of coordnate system 2, wth respect to the reference coordnate system, are gven by the product operator:,2,1 1, ,2 (4.2) The product operators k17 (k=, 1, 2, ) characterzes the poston and the drecton of the coordnate system k. These operators matrxes are: k f f... f.... f f... f k2 k2 k k2 k2 k (4.3) k 2 where, f j ; = 2, 3, 4,, 18; j = 1, 2, 3,, 18; k =, 1, 2,, 18; From the analyss of the operator result the followng conclusons the orgn of the coordnate system 2 s postoned wth respect to the reference system through the coordnates: 2 x 1, 21, d32 f ; y1, 21, d32 f (4.4) z 1, 21, d32 f181 In the nverse knematc analyss of the poly artculated spatal robotc arm wth 2 elements s pursued the determnaton of the relatve movements from the knematc ISSN: (Prnt); ISSN: (Onlne)

5 Proceedngs of the World Congress on Engneerng 211 Vol III, July 6-8, 211, London, U.K. jonts, necessary to realse a certan poston and drecton of the effectors soldary wth the coordnate system 19,2. Knowng the numerc values of the elements of the operator s matrx 1617 (noted 17 ), gven under the form: b b b... b.... b b... b (4.5) re determned the values of the parameters 1, 21,... so that the poly artculated system to 219 realze the poston and the drecton mposed. In order to solve the problem of nverse knematc we start from the relaton: (4.6) where, the operator s matrx s gven under the form (4.5). Forwards, we follow the successve separaton of the unknown values φ 1, φ 21,... φ 192 n the rght member of the relaton (4.41) by successve multplcaton of ths relaton on the left wth nverse operators , 12, Thus, we obtan the followng relatons: k2 = k2, k = 1, 2, 3,..., 19 (4.7) Then, the operator s matrxes k2 are: k k f... f.... k k... k k2 k2 k k2 k2 k (4.8) k 6 where, the elements: k j ( = 2, 3,..., 18); (j = 1, 2,..., 18); (k = 1, 2,..., 19) are expressed dependng on the relatve movements wthn the jonts and elements b j ( = 2, 3,..., 15); (j = 1, 2,..., 15) of the matrx 2. From the matrx equatons (4.7) result the followng equatons: k2 k2 kj h (4.9) j Forwards, the expressons of the matrxes are systematzed: k17 s k17, (k = 1, 2,..., 2) and are wrtten the equatons out of whch are determned the 2 th relatve movements from the knematc jonts, are characterstc to the analysed knematc chan, (φ 1, φ 21 φ 182 ): (4.1) The dynamc model of a mechancal structure s analytcally represented by a system of dfferental equatons that defne the connectons occurrng between the generalzed coordnates or ther dervates and the forces that actuate upon each segment of the manpulator. The models used to determne these dfferental equatons are numerous, Lagrange Euler, Newton Euler, D lambert s generalzed prncple, beng only a few of the classcal procedures used to calculate the dynamc model for the poly artculated mechansms [ 12]. Poly artculated movement mechansm s made n accordance wth the governng equatons:, H q q c q q G q Q (4.11) where: q the poston vector of the jont; q - - the speed vector; q - the acceleraton vector; H the nerta matrx; C descrbes the corols and centrfugal effects; G gravtatonal loadng; Q the generalzed force/ torque vector assocated wth q. The Lagrange form can be wrtten usng the Denavt-Hartenberg parameters and s very effcent from the conventonal pont of vew. The Lagrange equatons result from the equlbrum of the energy equaton and t has the followng form: d L d t q (4.12) where: L=K-P, K the knetc energy of the system; P the potental energy of the system; the torque that operates on the jont; q the poston of the jont. The knetc energy K of a mechansm wth n bodes s the sum of the knetc energes K of ndvdual bodes: n K K (4.13) 1 The knetc energy K of an ndvdual connecton s obtaned by ntegratng the knetc energy of the dfferental mass along the arm: K d (4.14) where: s the knetc energy of the dfferental mass of the element fxed n the jont n the r pont. Ths knetc energy s dependent of ts mass and speed, d m, respectvely v. The v speed of the r pont s obtaned through the dfferentaton of the global poston of the vector r : d d v r r r (4.15) dt dt Replacng the tme dervates wth the varables of the q j jont, the speed of the r pont can be descrbed as follows: k v q r ugqj r j1 q (4.16) j j1 ISSN: (Prnt); ISSN: (Onlne)

6 Proceedngs of the World Congress on Engneerng 211 Vol III, July 6-8, 211, London, U.K. where: the matrx u g defnes the speed of a fxed q j pont wthn a jont n the global coordnate system as a functon of the varables for the q j jont. The knetc energy of an jont can be expressed as follows: 1 T K dk Trace upjurq pq (4.17) r 2 p1 r1 where J s the pseudo-nertal matrx for the arm, and s calculated usng the relaton (4.18): 2 x dm xyd m xzd m xd m 2 xyd m y dm yzd m yd m T J r r d m 2 xzd m yzd m z dm zd m xd m yd m zd m d m V. THE NLYSIS OF THE OPERTING SPCE (MOVEMENT) The operatng space s the space where we can fnd the fnal effector n operaton, for the performance of the functons, accordng to the destnaton and the establshed features. Generally, t s a volume, partcularly s a surface. In the consdered case, the movement range of the dstal exteror vertebrae (nstruments) s a volume and namely a cone wth the vertex n the orgn (fgure 8). The desgned jonts can be tlted up to ± 1 degrees. The angle was establshed by the major dmensons of the exteror vertebrae, respectvely the nteror ones. To determne the shape and the sze of the operatng space, we wll create a software applcaton that permts to ntroduce the dmensons of the elements and the length of the fxed element, and to smulate the movement and the determnaton of the knematc parameters of the smplfed model of a robotc arm, there wll be developed a model usng the Smulnk-Sm Mechancs software from the Matlab programme nspecton, nterventon and grppng n operatng spaces wth reduced restrctons. It s descrbed the constructve soluton for the vrtual model. The vrtual prototype s conceved as a parameterzed model, easy adaptable for varous nterventons, dependng on the szes of the dscrete elements. The mathematc modelaton of the poly artculated robotc arm takes nto consderaton both the knematc approach and the dynamc one. The drect knematcs s used to test the models of nversed knematcs. The nversed knematcs s used for the complete smulaton for the movement of the robotc arm. In ths respect, we consder for each segment the rotaton angles around the arses of the vertebrae. The resulted elements usng the technology of 3D Rapd prototypng are wthn the standards mposed by the shape of the sze, the structure and the performance tme (from the CD models to physcal models made of plastc materal). REFERENCES [1] Robnson, G., J. B. C. Daves, Contnuum Robots- State of the rt, Proc IEEE Internatonal Confernce on Robotcs and utomaton, Detrot, Mchgan, pp ; [2] I. Staretu, Ssteme de prehensune (calcul s constructe), Edtura LUX LIBRIS, Brasov, 21; [3] N. Smaan, Snake-lke Unts Usng Flexble Backbones and ctuaton Redundancy for Enhanced Mnaturzaton, Proceedngs of the 25 IEEE Internatonal Conference on Robotcs and utomatton,pg , Barcelona, Span, prl 25; [4] Ka Xu, N. Smaan, ctuaton Compensaton for Flexble Surgcal Snake Robots wth Redundant Remote ctuaton, Proceedngs of the 26 IEEE Internatonal Conference on Robotcs and utomatton, pp , Orlando, Florda, May 26; [5] G. S. Chrkjan, J. Burdck, Knematcs of Hyper-redundant Robot Locomoton wth pplcatons to graspng, Proceedngs of the IEEE Internatonal Conference on Robotcs and utomatton, pp , tlanta,1993; [6] I. Walker, I, Gravagne, Manpularty, Force ann Complance for Planar Contnuum Manpulators, IEEE Transactons on Roboycs and utomaton, vol 18, No 3, June 23; [7]. Degan, H Choset, l. Wolf, T. Ota, M. Zenat, Percutaneous Intrapercardal Intervantons Usng a Hghly rtculated Robotc Probe, Internatonal Conference on Bomedcal Robotcs and Bomechatroncs, Psa, Italy, February, 26; [8] J. Pers, D. Reynaerls, H. Van Brussel, G. De Gersem, H-W Tang, Desgn of an dvanced Tool Gudng System for Robotc Surgery, Internatonal Conference on Robotcs and utomatton,tape, Tawan, 23; [9] N. Smaan, R. Taylor, P. Flnt, Dextereous System for Laryngeal surgery Internatonal Conference on Robotcs and utomatton, New Orleans, L.. Unted Stated, 24; [1] L Han, Lee Rudolph, Unfed pproach for Inverse Knematcs of a Spatal Chan wth Sphercal Jonts; [11] Fl. Dudta, D. Daconescu, Gr. Gogu, Mecansme rtculate, Inventca Cnematca, Ed. Tehnca, Bucurest, 1989; [12] Knematcs of rtculated Mult Body Mechansms, IH-27 Hydrauljarjestelmen Mallnnus Ja Smulont Jat Cocurss, Salvador Esque 25-6; [13] N. Cross, Engneerng desgn methods: strateges for product desgn, Wley, Fg. 8. The operatng space (movement) of the poly artculated robotc arm VI. CONCLUSION The poly artculated techncal models are complex structures, wth multple vertebrae, that operate n the three dmensonal space. They perform functons of exploraton, ISSN: (Prnt); ISSN: (Onlne)