An Approach for Type Synthesis of Overconstrained 1T2R Parallel Mechanisms

Transcription

1 A Approach for Type Sythesis of Overcostraied 1T2R Parallel Mechaisms C. Dog 1, H. Liu 1, Q. Liu 1, T. Su 1, T. Huag 1, 2 ad D. G. Chetwyd 2 1 Key Laboratory of Mechaism Theory ad Equipmet Desig of State Miistry of Educatio, Tiaji Uiversity, Tiaji , Chia, liuht@tju.edu.c 2 School of Egieerig, The Uiversity of Warwick, Covetry CV4 7AL, UK Abstract. This paper presets a simple ad illustrative approach for type sythesis of a family of overcostraied parallel mechaisms havig oe traslatioal ad two rotatioal movemet capability. This family features a spatial limb plus a member of a class of plaar symmetrical likages, coected by a revolute joit either with the machie frame at the base lik or with the platform at the output lik. Criteria for selectig suitable structures amog umerous cadidates are proposed by cosiderig the realistic ad practical requiremets of recofigurability, movemet capability, compoet desig ratioality, etc. Exploitig the structures obtaied ad examied by the criteria proposed leads to a ovel 5-DOF hybrid module amed TriMule, which offers a lightweight, cost effective, ad flexible desig particularly suitable for cofigurig various robotized maufacturig cells. Key words: Parallel robots, coceptual desig, type sythesis. 1 Itroductio Overcostraied 1T2R (T-traslatio; R-rotatio) parallel mechaisms ca have several advatages over their couterparts that lack overcostraits: higher stiffess, greater cost-effectiveess, ad easier-to-obtai explicit iverse/forward kiematics, for example, because the commo costraits allow fewer joits to be used. However, type sythesis for parallel mechaisms of this kid is by o meas a easy task because the Grübler Kutzbach formula is uable to idetify the virtual costraits produced by the special geometric arragemets of joit axes, ad is thereby o loger valid to determie the umber of degrees of freedom (DOF) i the structure. The last few decades have see tremedous efforts towards type sythesis of lower mobility overcostraied parallel mechaisms i geeral [1, 2] ad those havig 1T2R movemet capability i particular [3, 4]. Methods curretly available iclude those based upo screw theory [1, 5], group theory [3, 4, 6], liear trasformatio [7], ad may others [8]. These methods are geeral by first Author's versio 1

2 2 C. Dog, H. Liu, Q. Liu, T. Su, T. Huag ad D. G. Chetwyd creatig idividual limbs ad the settig the assembly coditios at the fial step. However, the procedures are ot easily uderstadable due to their abstractio ad/or complicated mathematics. Therefore, it is highly beeficial to develop a simple, illustrative ad effective approach that is easily uderstood ot oly by academia but also ad eve more importatly by desig egieers. Aother challegig issue ecoutered i type sythesis is that although a large umber of structures have bee geerated, there is a lack of criteria to evaluate which are the better oes eve if they have the same 1T2R movemet capability [3]. Therefore, the developmet of criteria to select suitable structures for specific applicatios is also a ope issue to be ivestigated. This paper presets a simple, easily visualized yet effective approach for type sythesis of a family of overcostraied 1T2R parallel mechaisms. By fully exploitig the commo costraits provided by a plae, Sectio 2 presets a methodology to sythesize 1T2R overcostraied parallel mechaisms featurig a spatial limb plus a member of a class of plaar symmetrical likages, resultig i two subfamilies with ad without a properly costraied o-actuated limb. Focused o realistic mechaism desig, Sectio 3 ivestigates the criteria for selectig suitable structures amog umerous cadidates. Exploitig the structures obtaied ad equipped with the criteria proposed, Sectio 4 presets a ovel 5- DOF hybrid PKM module before the coclusios are draw i Sectio 5. 2 Methodology Viewed i terms of kiematic iversio [5], Fig. 1 demostrates a geeral structure of the proposed family of overcostraied 1T2R parallel mechaisms. Members belogig to this family feature a spatial limb plus a plaar likage lyig i a plae deoted by. The two ed liks of the spatial limb are represeted by Body I ad Body III, whilst those of the plaar likage are deoted by Body II ad Body III, respectively. Body I ad Body II are coected by a revolute joit, deoted by R, with its joit axis - parallel to. I additio, there are actuated Body Ⅰ Body Ⅱ Spatial limb Plaar parallel likage Author's versio Body Ⅲ Fig. 1 Geeral structure of a family of overcostraied 1T2R parallel mechaisms

3 A Approach for Type Sythesis of Overcostraied 1T2R Parallel Mechaisms 3 prismatic joits for achievig high rigidity, each deoted by P, i the spatial limb ad i each of two idetical limbs of the plaar likage. The plaar likage may also have oe properly costraied o-actuated limb. The term properly costraied here meas that the umber ad type of degrees of freedom of the limb is exactly the same as those of the output lik. Usig these descriptios, two subfamilies of overcostraied 1T2R parallel mechaisms ca be sythesized by givig Body I (or Body III) either of two roles i the system. I the first subfamily, Body I is take as the machie frame, ad Body III thereby as the output lik of the plaar likage to which the platform is attached, leadig to Body II beig the base lik of the likage. Kiematic iversio produces the secod subfamily where Body I is take as the platform, Body II as the output lik, ad Body III as the machie frame to which the base lik is attached. Hece, 1T2R platform motio arises from 1T1R iteral motio of the output lik ad 1R motio of the platform about the axis - with respect to either the output lik or the machie frame. As show i Fig. 2(a), assume that the plaar likage has a basic form represeted by a six-bar likage with a sigle-loop closure. I each actuated limb, the joit coectig the limb to the output lik is a R joit, while the ier joit ad the joit coectig the limb with the base lik are either a P or a R joit. The, the axis - is cofied to be parallel to the commo ormal to the two R joits at either the base lik or the output lik as show i Fig. 1. Note that the output lik i the basic form has 2T1R iteral mobility. Hece, a 1T iteral motio must be restricted by imposig oe additioal costrait wrech (a pure force) with its axis parallel to. The, 1T2R movemet capability of the platform ca be achieved by simultaeously addig a 1R motio about the axis -. The required costrait wrech ca be geerated by either of two ways: (i) the use of a properly costraied o-actuated limb embedded betwee two actuated limbs for achievig structural symmetry as show i Fig. 2(b), or (ii) the use of a 5-DOF Base lik Properly costraied o-actuated limb Output lik wc (a) (b) wc wc (c) Author's versio Fig. 2 (a) A basic form of plaar likage, (b) A properly costraied o-actuated limb, (c) A 5-DOF spatial limb

4 4 C. Dog, H. Liu, Q. Liu, T. Su, T. Huag ad D. G. Chetwyd Output lik Output lik wc wc (a) RP limb (b) PR limb Fig. Fig. 33. Costrait Costrait wrech wrech imposed im- by a properly costraied Output lik o-actuated limb spatial limb as show i Fig. 2(c). Cosequetly, two classes of overcostraied 1T2R parallel mechaisms i each subfamily ca be geerated with ease usig the procedures itroduced as follows. Class A: mechaism usig a properly costraied o-actuated limb Members belogig to this class feature a 6-DOF spatial limb plus a stad-aloe 1T1R plaar parallel mechaism cotaiig a properly costraied o-actuated limb, coected by a R joit to the machie frame at the base lik i the first subfamily, or to the platform at the output lik i the secod. Two types of such costraied limbs, deoted by RP ad PR (P stads for a o-actuated prismatic joit), are available for achievig structural symmetry. Both provide the output lik with a costrait wrech (a force) lyig i, passig through the R joit wc axis, ad beig ormal to the P joit directio as depicted i Fig. 3. As a result, the 1T iteral motio of the output lik parallel to the wrech axis is restricted. This wc,2 wc,3 wc,1 (a) R(2(RPR)-RP)-UPS wc,1 / Output lik wc,2 wc,3 Output lik (b) (2(RPR)-PR)R-UPS Author's versio wc,2 wc,3 wc,2 wc,1 wc,1 wc,3 Output lik Output lik (c) (2(PRR)-RP)R-PUS (d) (2(PRR)-PR)R-PUS Fig. 4 Typical overcostraied 1T2R parallel mechaisms of class A Pure force Pure couple

5 A Approach for Type Sythesis of Overcostraied 1T2R Parallel Mechaisms 5 class uses a 6-DOF spatial limb to provide the platform with a actuatio to geerate 1R motio about the axis -. Cosiderig oly those structures simultaeously actuated by three iteral or exteral P joits, Fig. 4 shows four overcostraied 1T2R parallel mechaisms, where S ad U deote a spherical ad uiversal joit. Class B: mechaism without properly costraied o-actuated limb Members belogig to this class geerally feature a 5-DOF spatial limb plus the 2T1R six-bar uder-costraied plaar likage as show i Fig. 2(a), coected by a R joit as discussed i Class A. Numerous 5-DOF spatial limbs are available to do this [5], but we here cosider oly those havig four R joits ad oe P joit for practical use. Withi this category, the locatio ad directio of the costrait wrech ca uiquely be determied by the coditios: (i) the axis of oe R joit, deoted by R 1, is parallel to the wrech axis, (ii) the axes of the other three R joits, each deoted by R 2, itersect the wrech axis at a commo poit, ad (iii) the axis of the P joit is ormal to the wrech axis, amely ormal to the R 1 joit axis. By sequetially orderig all joits from the machie frame to the platform ad utilizig joit substitutios, these coditios reveal four possible limb structures, deoted respectively by SPR 1, UPR 2R 1, R 1PS ad PR 1S. To assemble the spatial limb with the plaar likage, the directio of the R 1 joit axis must be placed correctly with respect to its locatio ad the directio of the permitted 1T iteral motio of the output lik. For example, if the R 1 joit coects the spatial limb with the platform, the joit axis should be parallel to the Base lik 1T permitted motio wc Base lik wc Output lik R 1 / Output lik R 1 Base lik 1T permitted motio R 1 Base lik 1T permitted motio R 1 / Output lik Output lik Author's versio 1T permitted motio wc wc (a) (b) Fig. 5 Locatio ad directio of the costrait wrech ad the R1 joit axis

6 6 C. Dog, H. Liu, Q. Liu, T. Su, T. Huag ad D. G. Chetwyd wc, 2 (a) R(2-RPR)-SPR 1 wc, 2 wc, 3 wc, 1 wc, 3 / Output lik (c) R(2-RPR)-R 1PS / Output lik Fig. 6 Typical overcostraied 1T2R parallel mechaisms of class B Pure force Pure couple commo ormal of the axes of the two R joits coectig the output lik as show i Fig. 5(a). But if the R 1 joit coects the spatial limb with the machie frame, the joit axis should be parallel to ad ormal to the directio of the 1T permitted motio as show i Fig. 5(b). Similarly, if the P joit coects the spatial limb with the machie frame, the P joit axis should be parallel to the directio of the 1T permitted motio ad the R 1 joit axis parallel to as show by the dashed lie i Fig. 5(b). Fig. 6 shows some typical overcostraied 1T2R parallel mechaisms, all itegratig three R 2 joits ito a S joit to achieve a two-lik limb desig ad oly cosiderig those simultaeously actuated by three iteral P joits though may other possibilities exist. 3 Criteria for Selectig Suitable Structures wc, 1 (b) (2-RPR)R-SPR 1 Output lik Selectig suitable structures for 1T2R parallel mechaisms from amog the huge umber of cadidates is a challegig issue i developig 5-DOF hybrid robotized modules where high rigidity, high dyamic resposes are the essetial requiremets. By cosiderig the realistic ad practical requiremets of recofigurability, movemet capability, compoet desig ratioality, etc., four importat criteria are proposed as follows. Criterio 1: A suitable structure should have a relatively large ratio of workspace volume agaist footprit so that the relevat PKM ca be itegrated wc,1 wc, 2 3 wc, 2 wc, 1 (d) (2-RPR)R-R 1PS wc, Output lik wc, 3 Author's versio

7 A Approach for Type Sythesis of Overcostraied 1T2R Parallel Mechaisms 7 ito a rigid yet compact module, as exactly exhibited by the Sprit Z3 head, the Tricept as well as the Execho, for cofigurig various maufacturig cells. Criterio 2: A suitable structure should have idetical or early idetical actuated limbs (due to the overcostraits). This ot oly leads to cost effective desigs but also is valuable for achievig a appropriate movemet capability. Criterio 3: A suitable structure must allow the mai body of all lower mobility limbs to be give shapes havig high bedig ad/or torsioal stiffess/ mass ratios. Criterio 4: A suitable structure should have a explicit solutio to its iverse kiematics, a importat issue for CNC cotrol. A explicit solutio to the forward kiematics is ot essetial, but is very useful for rapid olie moitorig of the platform poses. 4 A Example Exploitig the structures obtaied i Sectio 2 ad the criteria preseted i Sectio 3, we propose a ovel 5-DOF hybrid module, amed TriMule [9], show i Fig. 7(a). Its essetial structure is the overcostraied 1T2R parallel mechaism show i Fig. 4(a). A critical feature is that this base lik is elaborately desiged ito a three-i-oe part that locates the rear R joits of the two actuated RPR limbs, ad the R ad P joits of the RP limb. Compared with the o-costraied 1T2R parallel mechaism used i the Tricept (see Fig. 7(b)), the ew desig ca theoretically save up to six R joits. This ca be doe by replacig the frot S joits of two UPS limbs by R joits, ad itegratig the rear U joits of the two UPS limbs ad the U joit of UP limb ito the three-i-oe compoet. However, by the criteria proposed i Sectio 3, it is preferable i practice to use two RPS limbs istead of two RPR limbs i the plaar likage. This is because costrait bedig ad torsioal momets imposed o these limbs ca the be avoided completely though these loads will ayway be carried to a great extet by the properly costraied o-actuated RP limb. A sigificat potetial advatage of the TriMule over the Tricept arises from its itegratio of the joits coectig the base lik ad the machie frame ito a three-i-oe part supported at each side by Three-i-oe part Author's versio (a) (b) Fig. 7 3D view of (a) TriMule ad (b) Tricept

8 8 C. Dog, H. Liu, Q. Liu, T. Su, T. Huag ad D. G. Chetwyd a roller bearig requirig oly a small outer frame. It is this ovelty that ideed offers a lightweight, cost effective ad flexible desig of a 5-DOF hybrid module particularly suitable for cofigurig various robotized maufacturig cells. 5 Coclusios A approach is proposed for type sythesis of overcostraied 1T2R parallel mechaisms, resultig i a family comprisig a spatial limb plus a member of a class of plaar likages, coected by a revolute joit. By cosiderig the realistic ad practical requiremets of recofigurability, movemet capability, compoet desig ratioality, etc., four importat criteria are proposed for selectig suitable structures from umerous cadidates, leadig to a lightweight, cost effective ad flexible desig of a ovel 5-DOF hybrid module amed TriMule. Ackowledgmets The research is partially supported by Natioal Key Basic Research Program (2014CB046603) ad Natioal Natural Sciece Foudatio of Chia ( ). Refereces 1 Huag, Z. ad Li, Q.: Geeral methodology for type sythesis of symmetrical lower-mobility parallel maipulators ad several ovel maipulators, It. J. Robot. Res., 21(2), , (2002) 2 Fag, Y. ad Tsai, L.-W.: Structure sythesis of a class of 4-DOF ad 5-DOF parallel maipulators with idetical limb structures, It. J. Robot. Res., 21(9), , (2002) 3 Li, Q. ad Hervé, J.-M.: 1T2R parallel mechaisms without parasitic motio, IEEE Tras. Robot., 26(3), , (2010) 4 Li, Q. ad Hervé, J.-M.: Type sythesis of 3-DOF RPR-equivalet parallel mechaisms, IEEE Tras. Robot., 30(6), , (2014) 5 Fag, Y. ad Tsai, L.-W.: Structure sythesis of a class of 3-DOF rotatioal parallel maipulators, IEEE Tras. Robot. Autom., 20(1), , (2004) 6 Hervé, J.-M.: The Lie group of rigid body displacemets, a fudametal tool for mechaism desig, Mech. Mach. Theory, 34(5), , (1999) 7 Gogu, G.: Structural sythesis of fully-isotropic parallel robots with Schöflies motios via theory of liear trasformatios ad evolutioary morphology, Eur. J. Mech. A/Solids, 26(2), , (2007) 8 Yag, T. ad Liu, A. ad Ji, Q. ad Luo Y. ad She, H. ad Hag, L.: Positio ad orietatio characteristic equatio for topological desig of robot mechaisms, J. Mech. Des., 131(2), , (2009) 9 Huag, T. et al..: A ovel 5-DOF hybrid robot with multi-axes gimbal holder, Chia Patet, No. ZL , (2015) Author's versio