Pose, Posture, Formation and Contortion in Kinematic Systems
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1 Pose, Posture, Formaton and Contorton n Knematc Systems J. Rooney and T. K. Tanev Department of Desgn and Innovaton, Faculty of Technology, The Open Unversty, Unted Kngdom Abstract. The concepts of pose, posture, formaton and contorton are defned for seral, parallel and hybrd knematc systems. Workspace and jontspace structure s examned n terms of these concepts. The nter-relatonshps of pose, posture, formaton and contorton are explored for a range of robot workspace and jontspace types. Introducton Robot manpulators (seral, parallel and hybrd) have two spaces assocated wth them. Usually the workspace s consdered to be the physcal space n whch the robot s embedded and through whch t executes movement. At each nstant each lnk of the robot occupes a poston and orentaton n the workspace. Smlarly the jontspace s usually consdered to be the more abstract space defned by the (drven) jont varables. At each nstant the values of the (drven) jont varables determne the state of the robot n the jontspace. Earler work n ths feld used concepts, such as closure and assembly confguraton, derved from the dsplacement analyss of sngle loop knematc lnkages and ther assocated seral systems [Rooney and Duffy, 97]. More recently there has been consderable nterest n parallel systems [Hunt and Prmrose, 993; Gosseln and Merlet, 994; Duffy, 996; Merlet,996; Tanev, 998], but most work to date has focussed on ether the workspace or the jontspace but not usually on both. Sometmes defntons and concepts assocated wth both spaces have been somewhat mprecse, or confused, or even occasonally contradctory. Ths stuaton s unsatsfactory especally when the mappng between the spaces s to be nvestgated. One am of ths paper s to establsh some termnology and defntons that are generc and consstent, so that they may be appled wth common justfcaton to seral, parallel and hybrd knematc systems. Pose, Formaton and Workspace In a knematc system, one way to locate a rgd lnk n space requres the specfcaton of a poston and an orentaton wth respect to some (global) coordnate frame that s consdered to be fxed. The pose of the lnk s defned to be the combnaton of ts poston and orentaton and as the lnk moves t traces out a sequence of poses. A typcal knematc system conssts of several nterconnected lnks wth each lnk havng ts own pose at each nstant. The confguraton of the complete system s therefore specfed by a set of poses (one for each lnk) that change wth tme as the system moves. However, attenton s usually focussed on the end-effector (for seral systems) or the movng platform (for parallel systems), and hence on the pose of just one lnk.
2 The workspace of a robot s often defned n terms of the 'reach' of the system, specfed by ts partcular geometrcal parameters. Usually the prmary defnton mplctly nvolves the orentaton of the end-effector, snce the orentaton, sze and shape of the latter determne the maxmum and mnmum reaches of the robot. The workspace of a D (3D) system s then seen as a twodmensonal (three-dmensonal) Eucldean space. However, n a D knematc system the pose of a lnk nvolves three varables (two for poston and one for orentaton of the lnk), and so t should be represented by a pont n a threedmensonal workspace. Moreover, ths s not a Eucldean space snce the orentaton varable s cyclc wth perod π and dstances between pars of ponts n ths space are not Pythagorean. Smlarly, n a 3D knematc system the pose of a lnk nvolves sx varables (three for poston and three for orentaton of the lnk), and so t should be represented by a pont n a sxdmensonal workspace. Agan, ths s not a Eucldean space snce the orentaton varables are cyclc wth perod π or π and dstances between pars of ponts n ths space are not Pythagorean. These consderatons lead to the followng generc defntons: The workspace of a robot knematc system s the space defned by all those varables requred to specfy the poston and orentaton of the end-effector or movng platform wth respect to a global fxed frame. Ths general defnton of workspace refers to the type of robot knematc system (such as seral 3R manpulator arm, or parallel 3-RPR platform) rather than to a partcular system wth partcular geometry. The workspace of any specfc robot geometry s then a subset of the general workspace wth varous geometrcal lmts mposed by the szes and shapes of the lnks etc.; The pose of a lnk s the combnaton of ts poston and orentaton specfed wth respect to a global fxed frame. It s just a pont n the workspace; A formaton s defned to be a collecton of related poses for one of the lnks, such as the end-effector or movng platform. It s a subset of the workspace and the workspace s thereby parttoned nto separate formatons. 3 Posture, Contorton and Jontspace An alternatve way to locate a rgd lnk n space s to specfy some of ts relatve spatal relatons wth respect to (local) coordnate frames attached to other lnks. For a knematc system the jont varables determne the relatve spatal relatons between adjacent connected lnks. The posture of any lnk s defned to be the combnaton of (drven) jont varables that locate the lnk n space [Rooney and Earl, 983], and as the lnk moves t passes through a sequence of postures. Thus both the pose and the posture of a lnk locate t n space. A typcal knematc system conssts of several nterconnected lnks, wth each lnk havng ts own posture at each nstant. The confguraton of the complete system s therefore specfed by a set of postures (one for each lnk) that change wth tme as the system moves. However, attenton s usually focussed on the end-effector (for seral systems) or the movng platform (for parallel systems), and hence on the posture of just one lnk. These consderatons lead to the followng generc defntons: The jontspace of a robot knematc system s the space defned by all those (drven) jont varables requred to specfy the poston and orentaton of the end-effector or
3 movng platform. Ths general defnton of jontspace refers to the type of robot knematc system (such as seral 3R manpulator arm, or parallel 3-RPR platform) rather than to a partcular system wth partcular geometry. The jontspace of any specfc robot geometry s then a subset of the general jontspace wth varous lmts mposed by the ranges of moton of the jonts etc.; The posture of a lnk s the combnaton of jont varable values specfyng ts poston and orentaton wth respect to local frames at the drven jonts. It s just a pont n the jontspace; A contorton s defned to be a collecton of related postures for one of the lnks, such as the end-effector or movng platform. It s a subset of the jontspace and the jontspace s thereby parttoned nto separate contortons. 4 Seral, Parallel and Hybrd Knematc Systems The number of (drven jont) varables requred for the posture of an end-effector or movng platform lnk s the same as the number of (poston and orentaton) varables requred for the pose of the lnk. As any lnk moves through a sequence of poses t also moves through a correspondng sequence of postures. However, n general t s possble at each nstant for each pose to correspond to more than one posture and conversely for each posture to correspond to more than one pose (Fgure ). Jontspace Workspace Posture Pose Contorton Formaton Fgure. Mappng between the jont space and the workspace. Typcally, seral knematc systems have several postures assocated wth a gven pose, but they always have just one pose assocated wth any gven posture. For example n 3D a general geometry 6R robot manpulator arm may have up to sxteen possble postures for a gven pose [Duffy and Crane, 980; Raghavan and Roth, 990], and n D a general geometry 3-dof robot manpulator arm (wth only revolute and/or prsmatc jonts) may have up to two postures for a
4 gven pose. To see how ths s possble, consder a planar RRR seral system wth the end-effector fxed n some pose. The mddle revolute jont can then adopt two confguratons wthout changng the pose of the end-effector, thereby leadng to two postures for that one pose (Fgure ). Fgure. Postve and negatve angle for a revolute jont, and a schematc representaton. We can show ths algebracally. Thus wth three revolute jonts, R, R, and R 3 n seres, there are two closure equatons: x = a cos( θ ) + a cos( θ + θ ) + a cos( θ + θ + θ ) () y = a sn( θ ) + a sn( θ + θ ) + a sn( θ + θ + θ ) () where a j are the lnk lengths, θ ( =,,3) are the jont angles, and ( x, y ) are the coordnates of a pont on the end-effector wth respect to a global coordnate frame. From eqn. () and eqn. () we obtan two solutons for θ, ( cos( φ) ) ( sn( φ) ) x a + y a a a θ =± arccos aa (3) where φ = θ+ θ + θ3 s the orentaton of the end-effector wth respect to a global coordnate frame. The two solutons for θ gven by eqn. (3) determne exactly two postures of the RRR manpulator. Smlarly, consder a planar RPR seral system wth the end-effector fxed n some pose. The prsmatc jont can then adopt two confguratons wthout changng the pose of the end-effector, agan leadng to two postures for that one pose (Fgure 3). In ths case the prsmatc jont must be constructed so that t allows sldng n the negatve drecton and hence can have a negatve value for the jont varable. Agan we can show ths algebracally. Thus wth one revolute jont R, one prsmatc jont P, and a second revolute jont R 3, n seres, there are two closure equatons. In ths case, the coordnates of a pont on the end-effector are: x = a sn( θ ) + S cos( θ ) + a cos( θ + θ ) (4) 34 3 y = a cos( θ ) + S sn( θ ) + a sn( θ + θ ) (5) 34 3
5 where a j are the lnk lengths, θ ( =,3) and S are the jont varables, and ( x, y ) are the coordnates of a pont on the end-effector wth respect to a global coordnate frame. From equatons (4) and (5) we obtan two values for S, [ φ ] [ φ ] 34cos( ) 34sn( ) ( 3) S =± x a + y a a a (6) where φ = θ+ θ3 s the orentaton of the end-effector wth respect to a global coordnate frame. The two solutons for S gven by eqn. (6) determne exactly two postures of the RPR manpulator. Fgure 3. Postve and negatve dstance for a prsmatc jont, and a schematc representaton Table shows the number of postures obtaned n ths way for the RRR and RPR, and addtonally for sx other planar three-jont systems consstng of just revolute and prsmatc jonts. Each of the four systems contanng at least two revolute jonts has two postures, whereas each of the remanng four systems contanng at least two prsmatc jonts has just one posture. In Table a pose or posture s denoted by the symbol L j, where L represents the end-effector lnk, the subscrpt j ndexes the pose and the superscrpt ndexes the posture. Ths notaton s a general one and t s also used later n Tables and 3 for parallel and hybrd systems, respectvely. In contrast to the stuaton wth seral systems, parallel knematc systems are usually consdered (correctly) to have several poses assocated wth a gven posture, but are also consdered (ncorrectly) to always have just one posture assocated wth any gven pose. For example the Stewart platform may have up to possble poses for a gven posture [Lazard and Merlet, 994], or the general Stewart type platform up to 40 possble poses [Husty, 996]. In general a parallel manpulator system may have more than one pose for each posture and smultaneously more than one posture for each pose. Ths s shown schematcally n Fgure where several postures (one from each contorton) map to each pose and where each posture maps to several poses (one n each formaton). To see how ths s possble, consder one of the legs of a three-legged 3-RRR parallel system wth the platform fxed n some pose.
6 Table. Examples of dfferent postures of planar seral manpulators. SERIAL MANIPULATORS Jontspace / POSTURES RRR RPP RRP PRP PRR PPR RPR PPP
7 Table. Examples of dfferent poses and postures of planar parallel manpulators. PARALLEL MANIPULATORS Workspace/ POSES Jontspace/ POSTURES L 3-RRR L L 6 L 3 L L L RPR 5 6 L L3 L L
8 Table 3. Examples of dfferent poses and postures of a planar hybrd manpulator. HYBRID MANIPULATOR Workspace/ POSES Jontspace/ POSTURES (RPRRR)R L L 3 L 4 As wth the seral system consdered earler (Fgure ), the mddle revolute jont n each leg can adopt two confguratons wthout changng the pose of the platform, thereby leadng to eght postures n total for that one pose. Wth general geometry ths 3-RRR system also has up to sx poses for each posture, as s well known [Hunt, 983]. The upper half of Table shows a partcular example of a 3-RRR parallel manpulator system that has specal geometry. Ths system has only four poses for each posture but a full complement of eght postures for each pose. (Agan the Lj notaton s used here to label and dstngush the dfferent confguratons.) The planar three-legged 3-RPR parallel system may be treated n a smlar way to the 3-RRR system f the prsmatc jonts n the former are physcally desgned to allow negatve-valued jont dsplacements. As wth the seral system consdered earler (Fgure 3), the prsmatc jont n each leg can adopt two confguratons wthout changng the pose of the platform, thereby leadng to eght postures for that one pose. Wth general geometry ths 3-RPR system (lke the 3-RRR) also has up to sx poses for each posture, as s well known [Hunt, 983; Gosseln and Merlet, 994]. The lower half of Table shows a partcular example of a 3-RPR parallel manpulator system that has specal geometry [Duffy, 996]. Ths system has only four poses for each posture but a full complement of eght postures for each pose. (Agan the Lj notaton s used here to label and dstngush the dfferent confguratons.) Fnally, f we consder hybrd (combnatons of seral and parallel) knematc systems we fnd that (as wth the parallel case) a hybrd system has n general more than one posture for each pose and more than one pose for each posture. Table 3 shows the (RPRRR)R hybrd system, where agan the Lj notaton s used to label and dstngush the dfferent confguratons.
9 5 Workspace and Jontspace Structure Snce n general the mappng between jontspace and workspace s many-many (Fgure ) the geometrcal and the topologcal structures of workspaces and jontspaces are often complcated. However, they may be explored n terms of pose, posture, formaton and contorton. As a frst step we consder brefly the RRR and RPR seral systems. Fgure 4. Jontspace (contorton ) for the RRR planar seral manpulator. Fgure 5. Workspace of the RRR planar seral manpulator. Fgure 4 shows one of the two contortons (of postures) n the jontspace of the RRR planar seral manpulator. Fgure 5 shows the sngle formaton (of poses) n ts workspace, correspondng to both contortons n the jontspace. Smlarly, Fgure 6 shows one of the two contortons (of postures) n the jontspace of the RPR planar seral manpulator, and Fgure 7 shows the sngle formaton (of poses) n ts workspace, correspondng to both contortons n the jontspace.. A further paper on the structure of workspaces and jontspaces s planned. Fgure 6. Jontspace (contorton ) for the RPR planar seral manpulator. Fgure 7. Workspace of the RPR planar seral manpulator.
10 6 Dscusson and Conclusons The paper has examned some aspects of the relatonshp between jontspace and workspace for robot manpulators. It has concentrated on planar knematc systems for clarty but the approach apples equally to spatal systems. The concepts of pose and formaton for workspace, and posture and contorton for jontspace have been ntroduced and explored for a range of seral, parallel and hybrd systems. Workspace and jontspace structures have been examned brefly, and a further paper s planned to explore ths n more detal and to apply the concepts n the wder context of forward and reverse knematcs. References Duffy, J. (996). Statcs and Knematcs wth Applcatons to Robotcs. Cambrdge Unversty Press. Duffy, J., Crane, C (980). A dsplacement analyss of the general spatal 7-lnk 7-R mechansm,. Mechansm and Machne Theory 5: Gosseln, C.M., and Merlet, J-P. (994). The drect knematcs of planar parallel manpulators: specal archtectures and number of solutons. Mechansm and Machne Theory 9(8): Hunt, K.H., (983). Structural knematcs of n-parallel-actuated robot-arms. Journal of Mechansms, Transmssons, and Automaton n Desgn 05: Hunt, K.H., and Prmrose, E.J.F. (993). Assembly confguratons of some on-parallel-actuated manpulators. Mechansm and Machne Theory 8():3 4. Husty, M., (996). An algorthm for solvng the drect knematcs of general Stewart-Gouth platforms. Mechansm and Machne Theory 3(4): Lazard, D. and Merlet, J-P. (994). The (true) Stewart platform has confguratons. Proceedngs of IEEE Internatonal Conference on Robotcs and Automaton. San Dego: Merlet, J-P. (996). Drect knematcs of planar parallel manpulators. Proceedngs of the IEEE Internatonal Conference on Robotcs and Automaton, Mnneapols, Mnnesota: Raghavan, M. and Roth, B. (989) Knematc analyss of the 6R manpulator of general geometry. In Mura, H. and Armoto, S. (eds), The Ffth Internatonal Symposum on Robotcs Research, MA:MIT Press, Rooney, J., and Duffy, J. (97). On the closure of spatal mechansms. Proceedngs of the th ASME Mechansms Conference & Internatonal Symposum on Gearng and Transmssons. San Francsco, CA, USA: paper no. 7-Mech-77. Rooney, J., and Earl, C.F. (983). Manpulator postures and knematc assembly confguratons. Proceedngs of the Sxth World Congress on Theory of Machnes and Mechansms, New Delh, Inda: Tanev, T.K. (998). Forward dsplacement analyss of a three-legged four-degree-of-freedom parallel manpulator. In Lenarcc, J. and Husty, M.L. (eds), Advances n Robot Knematcs: Analyss and Control, Kluwer Academc Publshers, Dordrecht:
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