he Knematc Analyss of a Symmetrcal hree-degree-of-freedom lanar arallel Manpulator Damen Chablat and hlppe Wenger Insttut de Recherche en Communcatons et Cybernétque de Nantes, rue de la Noë, 442 Nantes, France Damen.Chablat@rccyn.ec-nantes.fr Abstract resented n ths paper s the knematc analyss of a symmetrcal three-degree-of-freedom planar parallel manpulator. In opposte to seral manpulators, parallel manpulators can admt not only multple nverse knematc solutons, but also multple drect knematc solutons. hs property produces more complcated knematc models but allows more flexblty n trajectory plannng. o take nto account ths property, the noton of aspects,.e. the maxmal sngularty-free domans, was ntroduced, based on the noton of workng modes, whch makes t possble to separate the nverse knematc solutons. he am of ths paper s to show that a non-sngular assembly-mode changng trajectory exst for a symmetrcal planar parallel manpulator, wth equlateral base and platform trangle. Index erms arallel manpulator, Sngularty, Aspect, Assembly modes, Workng modes. F I. INRODUCION OR two decades, parallel manpulators have attracted the attenton of more and more researchers who consder them as valuable alternatve for robotc mechansms [-]. As stated by a number of authors [4], conventonal seral knematc machnes have already reached ther dynamc performance lmts, whch are bounded by hgh stffness of the machne components requred to support sequental jonts, lnks and actuators. hus, whle havng good operatng characterstcs (large workspace, hgh flexblty and manoeuvrablty), seral manpulators have dsadvantages of low precson, low stffness and low power. Also, they are generally operated at low speed to avod excessve vbraton and deflecton. Conversely, parallel knematc machnes offer essental advantages over ther seral counterparts (lower movng masses, hgher rgdty and payload-to-weght rato, hgher natural frequences, better accuracy, smpler modular mechancal constructon, possblty to locate actuators on the fxed base) that should lead to hgher dynamc capabltes. However, most exstng parallel manpulators have lmted and complcated workspace wth sngulartes, and hghly nonsotropc nput/output relatons [5]. Hence, the performances may sgnfcantly vary over the workspace and depend on the drecton of the moton. A well-known feature of parallel manpulators s the exstence of multple drect knematc solutons (or assembly modes). hat s, the moble platform can admt several postons and orentatons (or confguratons) n the workspace for one gven set of nput jont values [6]. he dual problem arses n seral manpulators, where several nput jont values correspond to one gven confguraton of the end-effector. o cope wth the exstence of multple nverse knematc solutons n seral manpulators, the noton of aspects was ntroduced [7]. he aspects were defned as the maxmal sngularty-free domans n the jont space. For usual ndustral seral manpulators, the aspects were found to be the maxmal sets n the jont space where there s only one nverse knematc soluton. Many other seral manpulators, referred to as cuspdal manpulators, were shown to be able to change soluton wthout passng through a sngularty, thus meanng that there s more than one nverse knematc soluton n one aspect. New unqueness domans have been characterzed for cuspdal manpulators [8], [9]. A defnton of the noton of aspect was gven by [0] for parallel manpulators wth only one nverse knematc soluton. hese aspects were defned as the maxmal sngularty-free domans n the workspace. A second defnton was gven by [] for parallel manpulators wth several nverse knematc solutons. hese aspects were defned as the maxmal sngularty-free domans n the Cartesan product of the workspace wth the jont space. However, t was shown n [2] that t s possble to lnk several drect knematc solutons wthout meetng a sngularty, thus meanng that there exsts cuspdal parallel manpulators. hs property was found for partcular lnks lengths. However, [] conjectured that such propertes cannot exst for symmetrcal parallel manpulator. he am of ths paper s to show that a symmetrcal -DOF planar parallel manpulator can change assembly mode wthout meetng a sngularty. We mean by symmetrcal, a manpulator wth equlateral base and platform trangles. hs paper s organzed as follows. Secton II descrbes the planar -RRR parallel manpulator studed, whch s used all along ths paper. Secton III recalls the noton of aspect for parallel manpulators. A non-sngular assembly-mode changng trajectory s shown for the symmetrcal planar parallel manpulator. he workspace and the generalzed IRCCyN: UMR CNRS 6596, Ecole Centrale de Nantes, Unversté de Nantes, Ecole des Mnes de Nantes
2 aspects are calculated usng octree models. II. RELIMINARIES A. arallel manpulator studed he manpulator under study s a planar three-dof manpulator comprsng three parallel RRR chans shown n Fg.. hs manpulator s used to llustrate the example n ths paper. hs manpulator has frequently studed, n partcular n [6-5]. l m O m 2 2 2 nverse-knematcs matrces of the manpulator, defned as ( c b) ( c b) E( p c) A ( c2 b2) ( c2 b2) E( p c2) (4a) ( ) ( ) ( ) c b c b E p c ( c b) E( b a) 0 0 0 ( c2 b2) E( b2 a2) 0 0 0 ( ) ( ) c b E b a (4b) C. Sngulartes For the planar manpulator studed, such confguratons are reached whenever the axes C, 2 and ntersect (possbly at nfnty), as depcted n Fg. 2. In the presence of such confguratons, the manpulator cannot resst any torque appled at the ntersecton pont I. y C l 2 2 A x m l Fgure : he -RRR parallel manpulator studed he actuated jont varables are the rotaton of the three revolute jonts located on the base (, 2, ). he Cartesan varables are the coordnate ( xy, ) of the operaton pont and the orentaton of the platform. he passve and actuated jonts wll always be assumed unlmted n ths study. onts A, and, (respectvely C, and ) le at the corners of an equlateral trangle, whose geometrc center s 0 (respectvely ). Moreover, l l l2 l 6, wth l denotng the length of A, m m m2 m 6, wth m denotng the length of C, r r r2 r 0, wth r denotng the length of A O and s s s2 s 5, wth s denotng the length of C, n unts of length that need not be specfed n the paper.. Knematc Relatons he velocty p of pont can be obtaned n three dfferent forms, dependng on whch leg s traversed, namely, p = E( b a ) + E( c b ) E( p c ), [,] () wth matrx E, 0 E 0 We would lke to elmnate the three dle jont rates, 2 and from eq. (), whch we do by dot-multplyng eq. () by ( c ) b, thus obtanng ( c b ) p =( c b ) E( b a )+( c b ) E( p c ), (2) [,] Equaton (2) can now be cast n vector form, namely, At ρ wth t p and ρ 2 () wth ρ thus beng the vector of actuated jont rates. Moreover, A and are, respectvely, the drect-knematcs and the I C A Fgure 2: Example of parallel sngularty For the manpulator under study, the seral sngulartes occur whenever ( b a ) ( c b ) lm for at least one value of, as depcted n Fg. for =,.e. whenever the ponts A,, and C are algned. D. Workng modes he noton of workng modes was ntroduced n [] for parallel manpulators wth several solutons to the nverse knematc problem and whose matrx s dagonal. A workng mode, denoted Mf, s the set of mechansm confguratons for whch the sgn of ( j,, n for a parallel manpulator wth n degrees of freedom) does not change and does not vansh. A mechansm confguraton s represented by the vector (X, q), whch permts us to locate the moble platform as well as all the lnks.. ( X, q) W Q such that sgn( )=cst for j,, n Mf and det( ) 0 herefore, the set of workng modes ( Mf, j I ) s obtaned whle usng all permutatons of sgn of each term. he manpulator under study has eght workng modes, as depcted n Fg. 4, that we call now (a), (b),..., (h). Each workng mode s defned accordng to the sgn of as s gven s n table. 2
ablty of a parallel manpulator to change ts nverse knematc soluton depends on the bounds n the passve and actuated jonts. hs problem s not taken nto account n our study snce unlmted jonts are assumed. C A 2 Fgure : Example of seral sngularty when A, and C are algned 2 A C (a) (b) (c) (d) 2 A C (e) (g) Fgure 4: he eght workng modes (f) (h) Fgure 4 (a) (b) (c) (d) (e) (f) (g) (h) N N N N N N N N N N N N able : he eght workng modes of the manpulators studed wth N (resp. ) denoted negatve values of (resp. postve values) Accordng to each workng mode, the parallel sngularty locus changes n the workspace, as shown n Fg. 5. In generally, the Fgure 5: he same platform confguraton wth two jont confguratons (sngular on the left and none sngular on the rght) E. Octree Models Octree models are herarchcal data structures based on recursve subdvson of the space, respectvely [6]. hey are useful for representng complex dmensonal shapes lke workspaces [0]. A close method s used n [7] that dvdes the workspace nto boxes. hs method does not use recursve subdvson but nterval analyss methods [8] to buld the dextrous workspace. However, t does not make t possble to perform oolean operatons or to make path-connectvty analyss easly. he frst method permts us to calculate easly all knd of space and the computng tme s lmted as a functon of the accuracy. he second one s more exact but requres more ablty to be mplemented. In both cases, we can characterze spaces whose dmensons are ether lengths or angles. Snce the structure of the octree model has an mplct adjacency graph, path-connectvty analyses and trajectory
4 plannng can be carred out naturally. he optmal constructon method of a 2 k -tree s derved from the shape, whch recalls the tree. he most nterestng approach conssts n testng successvely all the nodes present n the maxmal depth, followng an order of numberng whch quckly allows nodes to be grouped and thus, smplfes the 2 k -tree. he order of numberng for ths algorthm s based on Morton's sweepng [9]. he nverse or drect knematc model s used to calculate 2 k -tree. he fgure 6 represents the octree model of ts Cartesan workspace where the frst and the second axs represent the poston and the thrd axs the orentaton of the moble platform. QA Q ; QA s connected. he seral aspects are the maxmal sngularty-free domans n the jont space for one gven workng mode. For each workng mode, there exsts, at least, one aspect where det( A ) s postve and another one where det( A ) s negatve. However, such regons can be dsjont. In table 2, we assocated the aspects wth a workng mode for whch det( A ) s postve. For the workng mode (a), there exst four aspects and for the other ones, there s only one aspect. Due to the symmetrcal propertes of the mechansm, there exst also aspects where det( A ) s negatve. Workng modes (a) (b) (c) (d) (e) (f) (g) (h) N fgure 7 8 9 0 2 4 able 2: he projecton of the generalzed aspects on the workspace when det( A ) 0 for each workng mode x y Fgure 6: he Cartesan workspace III. WORKSACE ANALYSIS A. Aspect defntons he noton of aspect was ntroduced by [7] to cope wth the exstence of multple nverse knematc solutons n seral manpulators. Recently, the noton of aspect was defned for parallel manpulators wth only one nverse knematc soluton to cope wth the exstence of multple drect knematc solutons [0] and for parallel manpulators wth multple nverse and drect knematc solutons (the generalzed aspects []). For the manpulator studed, we use the second defnton. he generalzed aspects A are defned as the maxmal sets n W Q so that A W Q ; A s connected; A ( X, q) Mf such that det( A ) 0 In other words, the generalzed aspects A are the maxmal sngularty-free domans of the Cartesan product of the reachable workspace (called W) wth the reachable jont space (called Q). he projecton of the generalzed aspects onto the workspace yelds the parallel aspects WA so that, WA W ; WA s connected. he parallel aspects are the maxmal sngularty-free domans n the workspace for one gven workng mode. he projecton of the generalzed aspects onto the jont space yelds the seral aspects QA so that, Fgure 7: he four parallel aspects for the workng mode (a) and det( A ) 0 Fgure 8: he parallel aspect for the workng mode (b) and det( A ) 0 Fgure 0: he parallel aspect for the workng mode (d) and det( A ) 0 Fgure 9: he parallel aspect for the workng mode (c) and det( A ) 0 Fgure : he parallel aspect for the workng mode (e) and det( A ) 0
5 Fgure 2: he parallel aspect for the workng mode (f) and det( A ) 0 Fgure : he parallel aspect for the workng mode (g) and det( A ) 0 osture () osture (2) Fgure 4: he parallel aspect for the workng mode (h) and det( A ) 0 he calculaton of the generalzed aspects can be performed by 6 usng a 2 octree model or two octree models. We use the second method. he frst one s the projecton of the generalzed aspect onto the reachable workspace and the second one the projecton onto the reachable jont space for a gven workng mode and a gven sgn of det( A) constant. o obtan these results wth an accuracy of 0.09 for the poston and,4 degrees for the orentaton, the computng tmes s 90 seconds wth an AMD Athlon X processor 2500 + and the maxmum memory used s 80 Mb. he connectvty analyss of each doman requres 20 seconds.. Non-sngular posture changng trajectores In [2], a non-sngular posture changng trajectory was found for a -RR planar manpulator. However, t appears that ths trajectory passes close to a sngular confguraton. hs property was confrmed n [0] for the same manpulator and for a -RRR planar manpulator wth non-symmetrcal geometry [20]. Accordng to the assumpton n [], we can thnk that such propertes may not exst for the mechansm studed. However, we shown that a non-sngular confguraton changng trajectores exsts. For the followng nput jont values: =5.86260, 2 =.277470, = 5.2885 Four drect knematc solutons are found (Fgure 5 and able ). We notce that solutons and 4 are n the same generalzed aspect (he parallel aspect assocated s depcted n the fgure 8). osture () osture (4) Fgure 5: he four drect knematc solutons for =5.86260, 2=.277470, = 5.2885 osture N x y n degrees ().02.956 57.50 (2) 0.705 2.75 46.85 () 4.68-5.4 2.5 (4) -0.57 2.720 26.5 able : Four drect knematc solutons for the same jont values A frst method to confrm ths property s to evaluate the determnant of A and (able 4) and to fnd out a trajectory between these two postures. det(a) osture () 07.990-4.2-4.008.827 osture (4) 5.868 -.02-5.997 2.720 able 4: Evaluaton of det( A ) and generalzed aspect for the two pose n the same We fnd a non-sngular contnuous trajectory between postures () and (4) by passng through an ntermedate posture (5) (Fgure 6) whose poston s (-0.987;.90) and orentaton s 2.5 degrees. etween these three postures, a lnear nterpolaton s defned to stay n the same generalzed aspect. he values of det(a),, and are evaluated and each value of these ndces s normalzed by ts maxmum value as t s shown n fgure 7.
6 Fgure 6: he ntermedate posture (5) for the non-sngular changng trajectory Wth ths result, we have proofed that a non-sngular assembly mode trajectory s possble for a symmetrcal planar -RRR parallel manpulator. Insde such trajectory, not any knematc ndex, derved from the Jacoban matrces, permts us to recognze such property..2 0.8 0.6 0.4 0.2 det( ) A 2 0 t 0 0. 0.2 0. 0.4 0.5 0.6 0.7 0.8 0.9 osture () osture (5) osture (4) Fgure 7: Varatons of the normalzed values of det(a),, and along the trajectory (t) between postures () and (4) C. Characterstc surfaces o separate the drect and nverse knematc solutons, the unqueness domans are determned for the parallel manpulator wth one nverse knematc soluton n [0] and for parallel manpulator wth several nverse knematc solutons n [20]. he boundares of the unqueness domans are defned by the characterstc surfaces [20]. For the generalzed aspect (b), we can compute the characterstc surface that permts us to solate the assembly modes where t s possble to realze non-sngular assembly mode changng trajectores (Fgure 8). Fgure 8: he parallel sngulartes and the characterstc surfaces assocated wth the generalzed aspect (b) IV. SUMMARY AND CONCLUSIONS A knematc analyss of a planar -RRR parallel manpulator wth symmetrcal propertes was presented n ths paper. he eght workng modes have been characterzed and generalzed aspects have been found out. Insde such domans, any contnuous trajectores are possble. In such domans, there are non-sngular changng trajectores but not any knematc ndex can recognze such property. An example of non-sngular changng trajectory s gven and the characterstc surface are computed whch permt, n a future works, to defne closely the unqueness domans of the manpulator studed. REFERENCES [] H. Asada and J.J. Slotne, Robot Analyss and Control, John Wley & Sons, (986). [2] K.S. Fu, R. Gonzalez and C.S.G. Lee, Robotcs: Control, Sensng, Vson, and Intellgence, McGraw-Hll, (987). [] J.J. Crag, Introducton to Robotcs: Mechancs and Control, Addson Wesley, (989). [4] L.W. sa, Robot Analyss, he Mechancs of Seral and arallel Manpulators, John Wley & Sons, (999). [5] D. Stewart, A latform wth Sx Degrees of freedom, roceedngs of the Insttuton of Mechancal Engnners Vol. 80, art, No. 5, 7-86, (965). [6] J-. Merlet, arallel robots, Kluwer Academc ubl., Dordrecht, he Netherland, (2000). [7]. orrel, A study of manpulator nverse knematc solutons wth applcaton to trajectory plannng and workspace determnaton, roc. IEEE Int. Conf on Rob. And Aut., pp 80-85, (986). [8] h. Wenger, A new general formalsm for the knematc analyss of all nonredundant manpulators, IEEE Robotcs and Automaton, pp. 442-447, (992). [9] J. El Omr, Analyse Géométrque et Cnématque des Mécansmes de ype Manpulateur, hèse, Nantes, (996). [0] h. Wenger and D. Chablat, Unqueness Domans n the Workspace of arallel Manpulators, IFAC-SYROCO, Vol. 2, pp. 4-46, -5 Sept., Nantes, (997).
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