XY-Theta Positioning Table with Parallel Kinematics and Unlimited Theta Rotation

Size: px

Start display at page:

Download "XY-Theta Positioning Table with Parallel Kinematics and Unlimited Theta Rotation"

Alice Price
6 years ago
Views:

1 XY-heta Postonng able wth Parallel Knematcs and Unlmted heta Rotaton Ilan A. Bonev 1, Aleander Yu 2, Paul Zsombor-Murra 2 1 Department of Automated Manufacturng Engneerng, École de echnologe Supéreure, Montreal, QC, Canada lan.bonev@etsmtl.ca 2 Department of Mechancal Engneerng, McGll Unverst, Montreal, QC, Canada Abstract Parallel robots are ncreasngl beng used n ndustr for postonng and algnment. he have the advantage of beng rgd, quck, and accurate. However, most parallel robots have ver lmted rotatonal capabltes and n cases unlmted rotaton s necessar, the onl opton used to be addng a rotar table. here however ests one overlooked XY-heta parallel robot desgn that can reach an pont wthn a crcular workspace at an orentaton. hs unque desgn conssts of three pvots movng ndependentl along a common crcular path. hese pvots are connected ndvduall to three lnear gudes attached to a moble platform n a smmetrc Y shape. B studng the knematcs of ths parallel robot, ths paper shows that not onl s the detrous workspace large but the knematc accurac s hgh and unform. I. INRODUCION Workspace capabltes were never a strong pont for an parallel robot structure [1]. Partcularl, orentatons acheved b these robots are tpcall ver lmted and depend on the poston of the moble platform. o ncrease the rotaton capabltes of some parallel robots, addtonal motors are usuall added n seres. hs however, ncreases the nerta of the moble part, thus reducng the speed of the robot. Probabl the onl 6-degree-of-freedom (DOF) full-parallel robot that has unlmted rotatonal capabltes about an as s the Rotopod [2], commercalzed b the Parallel Robotc Sstems Corporaton (Fg. 1). In ths parallel robot, s legs of fed length connect a moble platform to a base, through sphercal jonts. he centers of the base sphercal jonts move ndependentl along a common crcular path under the acton of s motors. here ests a ver smlar planar parallel robot, whch s however stll unknown to the publc [3]. hs parallel robot has a rare archtecture (jont sequence) that, although well known and prevousl studed n the lterature [4], has never been gven an partcular nterest. It s a pecular desgn that features a unque crcular path, on whch all ntermedate revolute jonts move. hese jonts slde along three lnear gudes attached to the moble platform n a smmetrc Y shape. he resultng mechansm has a sngulart-free crcular workspace (under the natural assumpton that the lnear gudes have lmted lengths) n whch an orentaton can be acheved. Furthermore, the detert of ths robot s farl hgh and unform. Fg. 1. he Rotopod R-seres heapod (courtes of PRSCo). hs paper wll analze the knematcs of ths unknown parallel robot n the followng sectons. It wll be shown that the drect and the nverse knematcs for ths robot are smple, that ts sngulartes are outsde the feasble workspace that s nsde the crcular track. Fnall, the local detert analss of ths robot wll reveal that the robot s farl accurate wthn ts range of moton. II. BRIEF DESCRIPION OF HE PARALLEL ROBO In ths paper, the term lnear bearng wll be used to refer to the lnear segment on the moble platform along whch a pvot, denoted b O ( = 1, 2, 3), ma translate (Fg. 2). Naturall, t s assumed that these lnear bearngs (or lnear gudes) have fnte lengths and form a smmetrc Y shape. he pvots wthn those lnear bearngs travel along a crcular path, whch allows the robot to rotate wthout lmts about the center of the crcular path, when all motors move at the same rate. A base reference frame O s fed at the center of the crcular path, and a moble reference frame C s fed at the center of the moble platform n such a wa that the as concdes wth lnear bearng 2.

2 he followng vector equaton can now be wrtten: uuuur uuuur uuur CO = OO OC. (3) It elds two algebrac equatons: ρ cos θ = cos φ, (4) ρsn θ = sn φ, (5) Fg. 2. Schematc of the parallel robot under stud. Let the angle between the base as and the moble as be denoted b θ (the orentaton of the moble platform) and the coordnates of pont C (the ntersecton of the three lnear bearngs) wth respect to the base frame (the poston of the moble platform) be denoted b and. he poston (, ) and orentaton (θ) of the moble platform wll be referred to as the pose of the moble platform. On the other hand, the locaton of ponts O on the crcular path s defned b the angle φ, whch s the angle between the base as and the vector OO. he three angles φ wll be referred to as the nput varables. Fnall, for smplct, t s assumed that the radus of the crcular path s of unt length. III. INVERSE KINEMAICS he nverse knematc problem s to fnd the nput varables that correspond to a gven pose of the moble platform. Geometrcall, solvng the nverse knematc problem of ths mechansm conssts of fndng the ntersecton ponts of lnes CO wth the crcular path. hus, for a gven pose of the moble platform, ths mechansm allows two possble solutons for each nput varable (Fg. 2), assumng that pont C s nsde the crcular path. However, for onl one of each of the two solutons, the correspondng pont O s nsde the lnear bearngs. hus, there s onl one practcall realsable (.e., feasble) soluton for the nput varable φ. Let us denote then b m the unt vector along each lnear bearng, drected outwards as shown n Fg. 2. B defnton, we have: cos θ m = sn θ, (1) π 2 π θ =θ + ( 1). (2) 6 3 ρ s the dstance (postve or negatve) from C to O along m. Combnng the above two equatons to elmnate ρ and rearrangng leads to the followng smple equaton that defnes the nverse knematc problem: sn φ θ = cosθ sn θ. (6) ( ) In order to have a soluton for the above equaton, ts rghthand sde should be between 1 and +1. It can be easl shown that ths rght-hand sde s actuall the drected dstance from O to lnear bearng, and ths dstance should be less than 1 n order to have a soluton. hus, assumng that pont C s nsde the crcular path, the two solutons for leg are: φ =θ + arcsn cos θ sn θ (7) and ( ) ( ) φ =π+θ arcsn cos θ sn θ. (8) It can be shown that onl the frst soluton s feasble,.e., pont O s nsde the lnear bearng. IV. DIREC KINEMAICS he drect knematc problem s to fnd the pose of the moble platform that corresponds to a gven set of nput varables. For a gven set of nput varables (.e., gven ponts O ), ths mechansm allows theoretcall two poses for the moble platform [6]. Both poses have the same poston but a 180º dfference n the orentaton. However, onl one of these poses s practcall realsable (.e., ponts O are nsde the lnear bearngs). Furthermore, t can be easl shown, that f an of the angles of trangle O 3 s greater than 120º, then the center C would have to be outsde the crcular path, whch means that both poses wll not be practcall realsable. Assumng that none of the angles of the trangle O 3 s greater than 120º, the correspondng practcall realsable pose can be easl obtaned through the followng geometrc constructon, based on the noton of frst Fermat pont. In an trangle O 3 n whch none of the angles s greater than 120º (Fg. 3), f equlateral trangles are constructed outsde trangle O 3 and ther etreme vertces (ponts P ) are connected wth the opposte vertces of trangle O 3, the resultng lnes (P O ) wll ntersect at a sngle pont. hs pont, F, s called frst Fermat pont (also known as orrcell pont, or frst sogonc center) and has the propert that all nscrbed angles are equal,.e., O 1 FO 3 = O 3 FO 2 = O 2 FO 1 = 120º. (hs pont also has the nterestng propert that t mnmzes the sum of the dstances to the vertces of the trangle O 3.) herefore, ths frst Fermat pont s actuall the center pont C.

3 V. VELOCIY ANALYSIS Net, the relatonshp between the rates of the nput varables (.e., the nput rates) and the lnear and angular veloct of the moble platform at ts center C (.e., the output rates) need to be found. hese wll be necessar nor onl for calculatng the speed of dsplacements but also for evaluatng the local detert of the parallel robot under stud. hs s done b dfferentatng (6) wth respect to tme, and after rearrangng and wrtng n matr form, the followng s obtaned: & sn cos θ θ ρ = cos( φ θ) φ & &. (14) θ & hs equaton can also be wrtten n matr form as: [ ] Em ρ q& = m o φ &. (15) Fnall, wrtng the above equaton for all values of and placng the resultng equatons n matr form, elds: Fg. 3. Solvng the drect knematcs of the parallel robot under stud. o fnd the coordnates of pont C and the orentaton, θ, of the moble platform, the followng smple calculatons wll be performed. Let p denotes the vector connectng pont O to pont P. herefore, t can be easl shown that vector p can be wrtten as: p 1 3 p = ( o j + ok) + E( o j o k), (9) p E = 0 1, (10) and (, j, k) = (1, 2, 3) or (2, 3, 1) or (3, 1, 2). Now, takng lnes O 1 P 1 and O 2, for eample, ther ntersecton pont s C and has the followng coordnates: = = ( O )( ) ( )( ) 1 P 1 P 1 P1 ( O P )( O P ) ( O P )( O P ) ( O )( ) ( )( ) 1 P 1 P 1 P1 ( O P )( O P ) ( O P )( O P ) , (11) (12) Fnall, the orentaton of the moble platform remans to be found. hs can be done b measurng the angle between vector CO 2, and the base as: π θ= atan2 ( O, ). (13) 2 2. Em1 ρ1 m1o1 0 0 Zq& Em2 ρ 2 q& = 0 m2o2 0 φ & Λ φ. & (16) Em 3 ρ3 0 0 m3o3 hus, the Jacoban matr of ths parallel robot s 1 J = Λ Z. (17) VI. SINGULARIIES Sngulartes are specal confguratons at whch the veloct equaton (16) degenerates,.e., one or both matrces Z and Λ become sngular. We see that Λ s sngular when a lnear bearng s tangent to the crcular path. hs mples that the center C wll be outsde or on the crcular path, whch means that some ρ wll be negatve, whch s an unfeasble confguraton. herefore, we ma state that, as long as the pvots reman nsde the lnear bearngs, there wll be no such sngulart. he geometrc nterpretaton of matr Z s more dffcult to see wthout some background n lne geometr. he rows of ths matr are n fact the lne coordnates of the lnes normal to the lnear bearngs and passng through the pvots O, epressed n the moble frame. hs means that matr Z s sngular f and onl f these three lnes are lnearl dependant (all parallel to a sngle lne or all ntersectng at a sngle pont). Due to the geometr of the moble platform, these lnes can never be parallel to a sngle lne. Also, f we constran ponts O to be nsde the lnear bearngs, then the onl possblt for these lnes to ntersect at one pont s for the three pvots to concde wth pont C, whch s mpossble under the assumed feasblt condtons. herefore, we ma state that as long as the pvots are constraned to move nsde the lnear bearngs (or equvalentl, as long as pont C s constraned to move wthn the crcular path), ths mechansm does not have an sngulartes.

4 VII. WORKSPACE ANALYSIS he most advantageous feature of ths mechansm s that ts local propertes over are dentcal for an orentaton of the moble platform, but rotated at the orentaton angle. hus, the set of locatons that are attanable b pont C for a gven orentaton (referred to as constant-orentaton workspace [7]), under some gven feasblt condtons, are needed to be found onl for the orentaton θ = 0. In practce, the lnear bearngs wll not reach pont C (Fg. 4). Let ε be the offset dstance from pont C to the begnnng of the feasble range for pont O. Once agan, a geometrc method ma be used to determne the workspace of ths robot. For a constant orentaton, the ntersecton of three regons (see the green hatched regon) needs to be computed. hs would be the constant-orentaton workspace. hen, t s obvous that for another orentaton, the workspace would be the same but rotated about O. Now, what s certanl of man nterest s the area that s accessble b pont C for an orentaton (the so-called detrous workspace). What s great about ths robot s that ths area s a dsk (the one n red). For brevt, the eact geometrc algorthm wll not be presented here. hs algorthm ma be used to optmze the detrous workspace of ths robot and set the optmum length of the feasble range for pont O (.e., the length of the lnear bearngs). It s eas to show that ths optmum length s equal to 1 ε, n whch case the detrous workspace of ths mechansm s a dsk of radus 1 ε. Note that ths detrous workspace s easl achevable n practce. Indeed, wth a reasonabl small value for ε, there would be no other mechancal lmts restranng the workspace. Furthermore, the resultng detrous workspace wll have no sngulartes and, as t wll be shown n the net secton, the robot wll have a farl hgh and unform accurac nsde ths crcular detrous workspace. VIII. LOCAL DEXERIY ANALYSIS A detert analss can be appled to the sstem now that the Jacoban matr s known. Detert s defned as the capablt of the robot to make accurate movements and s a measure of ts knematc accurac [5]. o characterze the accurac of a robot s detert, an epresson was establshed usng ts Jacoban matr. he Jacoban matr s used because t s the lnear transformaton between Cartesan and actuator veloctes. he local detert s the qualt of that transformaton. he local detert nde wll be defned as [5]: 1 ξ =, (18) J J 1 1 J = tr JJ 3. (19) he local detert s lmted between 0 and 1 the hgher ts value, the more accurate the robot. When the local detert s equal to 1, t s sad that the robot s sotropc at the gven pose. Inversel, when the local detert s equal to zero, t s sad that the robot s at a sngular confguraton. As alread mentoned, an knematc performance nde for ths parallel robot wll lead to the same plot over the poston workspace, but the plot wll be rotated b the orentaton angle. herefore, the local detert plot wll be shown onl for θ = 0. Fgure 5 shows the local detert contour plot for the parallel robot under stud, over a detrous workspace of radus. As can be seen, the closer the center of the moble platform to the center of the crcular path, the hgher the local detert. he mamum value s nearl 5. he local detert drops eponentall when nearng the crcular path, but, nsde the detrous workspace of radus, the local detert drops onl to about 0.79, whch s a relatvel small decrease. herefore, the parallel robot has a farl hgh and unform detert. 1 Detert plot for θ = Fg. 4. Detrous workspace of the parallel robot under stud Fg. 5. Local detert contour plot for the parallel robot under stud.

5 IX. CONCLUSION A bref but complete stud of the knematcs of an unknown planar parallel robot wth crcular detrous workspace was presented n ths paper. It was shown that smple elegant solutons est to all problems nvolved n the knematc desgn and control of ths robot. Fnall, t became obvous that the parallel robot has a farl hgh and unform knematc accurac nsde ts crcular detrous workspace. hus, a good mechancal desgn n whch the moble platform s supported on the three pvots wll ensure a ver rgd structure that s sutable for precson postonng. X. REFERENCES [1] J.-P. Merlet, Parallel Robots, 2 nd ed., Sprnger, he Netherlands. [2] I.A. Bonev and C.M. Gosseln, Geometrc Algorthms for the Computaton of the Constant-Orentaton Workspace and Sngulart Surfaces of a Specal 6-RUS Parallel Manpulator, ASME Desgn Engneerng echncal Conferences, Montréal, Canada, September 29 October 2, [3] A. Schedegger and R. Lecht, Postonng Devce, US Patent No. 6,622,586, fled on December 21, 2001, ssued on September 23, [4] I.A. Bonev, D. Zlatanov, and C.M. Gosseln, Sngulart Analss of 3-DOF Planar Parallel Mechansms va Screw heor, ASME Journal of Mechancal Desgn, Vol. 125, No. 3, pp , [5] C. Gosseln, he optmum desgn of robotc robots usng detert ndces, Robotcs and Autonomous sstems, Vol. 9, pp , [6] J.-P. Merlet, Drect Knematcs of Planar Parallel Manpulators, Proceedngs of the 1996 IEEE Internatonal Conference on Robotcs and Automaton, Mnneapols, Mnnesota, Aprl, [7] J.-P. Merlet, C.M. Gosseln, and N. Moul, Workspaces of Planar Parallel Manpulators, Mechansm and Machne heor, Vol. 33, No. 1/2, pp. 7-20, 1998.

Pose, Posture, Formation and Contortion in Kinematic Systems

Pose, Posture, Formation and Contortion in Kinematic Systems 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,