Dynamic Load Carrying Capacity of Flexible Manipulators Using Finite Element Method and Pontryagin s Minimum Principle

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1 Journal o Optmzaton n Industral Engneerng 1 (13) 17-4 Dynamc Load Carryng Capacty o Flexble Manpulators Usng Fnte Element Method and Pontryagn s Mnmum Prncple Moharam H. Korayem a,*, Mostaa Nazemzadeh b, Hamed Rahm nahooj c a Proessor, Robotc Research Laboratory, College o Mechancal Engneerng, Iran Unversty o Scence and echnology, ehran, Iran b Phd canddate, Robotc Research Laboratory, College o Mechancal Engneerng, Amrkabr Unversty, ehran, Iran c MSc, Department o Mechancal Engneerng, Damavand Branch, Islamc Azad Unversty, Damavand, Iran Receved 5 September, 11; Revsed 8 May, 1; Accepted 7June, 1 Abstract In ths paper, ndng Dynamc Load Carryng Capacty (DLCC) o lexble lnk manpulators n pont to-pont moton was ormulated as an optmal control problem. he nte element method was employed or modellng and dervng the dynamc equatons o the system. he study employed ndrect soluton o optmal control or system moton plannng. Due to olne nature o the method, many dcultes such system nonlneartes and all types o constrants can be catered or and mplemented easly. he applcaton o Pontryagn s mnmum prncple to ths problem was resulted n a standard two-pont boundary value problem (PBVP), solved numercally. hen, the ormulaton was developed to nd the maxmum payload and correspondng optmal path. he man advantage o the proposed method s that the varous optmal trajectores can be obtaned wth derent characterstcs and derent maxmum payloads. hereore, the desgner can select a sutable path among the numerous optmal paths. In order to very the eectveness o the method, a smulaton study consderng a two-lnk lexble manpulator was presented n detals. Keywords: Flexble Manpulator; Fnte Element Method; Pontryagn Mnmum Prncple. 1. Introducton Most ndustral robots n use today are composed o heavy and st lnks to satsy the requred repeatablty and accuracy. hese lnks, thereore, have nherently a large nerta, requrng n turn a long tme to complete the moton and more power consumpton n the actuators. o ncrease the productvty by ast moton and to complete a moton wth small energy consumpton, ndustral robot manpulators are requred to have lght weght and lexble structures. On the other hand, the manpulators are typcally used to repeat a prescrbed task a large number o tmes, so even small mprovements n ther perormance may result n large monetary savng. Senstvty analyss o the geometrc parameters such as length, thckness and wdth on the maxmum delecton o the end eector and vbraton energy o a sngle lnk lexble manpulator was nvestgated by Korayem et al. (1). Fndng the ull load moton or a pont-to-pont task can maxmze the productvty and economc usage o the manpulators. homas et al. (1985) used the load * Correspondng author E-mal: hkorayem@ust.ac.r capacty as a crteron or szng the actuators o robotc manpulators at the desgn stage. In ther study, they consdered the maxmum load n the neghbourhood o a robot conguraton. he rst ormulaton to obtan the maxmum payload o a manpulator n pont to pont moton was presented by Wang and Ravan (1988). hey used the teratve lnear programmng (ILP) method to solve the problem. Wang et al. (1) solved the optmal control problem wth the drect method n order to determne the maxmum payload o a rgd manpulator. he basc dea o ths work s to parameterze the jont trajectores by usng B-splne unctons and tunng the parameters n a nonlnear optmzaton untl a local mnmum that satses the constrants acheved. hs method leaks rom lmtng the soluton to a xed-order polynomal as well as complexty ssues arose n derentatng torques wth respect to jont parameters and payload due to ther constrants and dscontnuty. Korayem and Gharblu (4) were presented a computatonal algorthm or maxmum load determnaton 17

2 Moharam H. Korayem et al./ Dynamc Load Carryng Capacty o... va lnearzng the dynamc equaton and constrants on the bass o Iteratve Lnear Programmng (ILP) approach or lexble moble manpulators. But because o some ILP approach s dcultes, n ther work the lnk lexblty has not been consdered ether n the dynamc equaton or smulaton procedure. In contrast, the optmal control method s known as an approprate method n the cases where the system has a large number o degree o reedom, especally when nonlnear terms are large and luctuatng, e.g., n problems wth consderaton o lexblty n jonts or lnks, gravty acceleraton or havng hgh speed moton. Furthermore, optmzaton o the varous objectves s targeted by means o ths approach. On the other hand, because o the nature o the optmal control problem, many dcultes lke system nonlneartes and all types o constrants may be catered or and mplemented easly. hus, ths method was wdely used as a powerul and ecent tool n analysng the nonlnear system, such as path plannng o the derent types o rgd and moble manpulators (Bertolazz et al. (5), Bessonnet and Chesse (5), Calles and Rentrop (8) and Korayem et al. (11)). On the other hand, n case o dynamc modellng lexble manpulators, the nte element method (FEM) has been used to solve very complex structural engneerng problems durng the past years. One o the man advantages o FEM over the most o other approxmate soluton methods s the act that FEM can handle rregular geometres routnely. Another sgncant advantage o FEM, especally over analytcal soluton technques s the ease wth whch nonlnear condtons can be handled. A comprehensve comparson o FEM wth other avalable methods or dynamc robot analysng s addressed by Korayem and Rahm (11). In nte element modellng o dynamcal manpulators, the elastc deormatons are analysed by assumng a known rgd body moton and later superposng the elastc deormaton wth the rgd body moton (Usoro et al. (1986)). Dogan and Isteanopulos (7) developed nte element models to descrbe the delecton o a planar two-lnk lexble robot manpulator. Zhang et al. (4) proposed dynamc equaton o planar cooperatve manpulators wth lnk lexblty n the absolute coordnates wth the moshenko beam theory and the nte element method. Rashdar et al. (1) used nte element method to model a sngle lnk lexble manpulator by dvdng the system nto 1 elements. hen, they presented optmzaton o nput shapng technque or vbraton control o the system usng genetc algorthms. Zebn (1) presented theoretcal nvestgaton nto the dynamc modellng and characterzaton o a constraned two-lnk lexble manpulator usng nte element method. hen, the nal derved model o the system was smulated to nvestgate the behavour o the system. Mohamed and okh (4) derved the dynamc model o a sngle-lnk lexble manpulator usng FEM and then studed the eed-orward control strateges or controllng the vbraton. Yue et al. (1) used the nte element method or descrbng the dynamcs o the system and computed the maxmum payload o knematcally redundant lexble manpulators. Fnally, they numercally smulated a planar lexble robot manpulator to valdate ther research work. Nowadays the advantages o optmal control theory are well establshed and a host o ssues related to ths technque have been studyng specally n the eld o optmal moton plannng o robots (Brot et al. (1), Korayem (13) and Bjorkenstam et al. (13)). Rahm et al. (9) proposed ndrect soluton o open-loop optmal control method to trajectory optmzaton o lexble lnk/jont manpulator n the pont-to-pont moton. In the mentoned work, despte ILP based studes the complete orm o the obtaned nonlnear equaton was used. hus, unlke the prevous ILP based works to solve the problem lnearzng equatons was not requred. However, the paper employed assumed modes method to derve the robot dynamc moreover; nng the ull load was not consdered n ths research study. he man objectve o the presentng paper s to provde a nonlnear dynamc modellng and optmal control o lexble manpulators n order to determne the dynamc load carryng capacty o such robots. he paper rstly deals wth the nonlnear modellng o the general lexble lnks robot manpulators. hen, the optmal control problem that wth employng o Pontryagn's mnmum prncple supports the executon o the optmzaton soluton o model s expressed as a bre revew; subsequently, an applcaton example wth the two-lke lexble manpulator s presented and dscussed to demonstrate the eectve perormance o the proposed approach. Lastly, the paper s concluded wth hghlghtng the eature propertes o the proposed model.. Dynamc Modelng he nte element method s broadly used to derve dynamc equatons o elastc robotc arms. Researcher usually used the Euler Bernoull beam element wth multple nodes and Lagrange shape uncton to acheve the reasonable nte element model. he node number can be selected accordng to requrement on precson. But, ncreasng the node number may enlarge the stness matrx and t cause to long and complex equatons. Hence, choosng the proper node number s very mportant n the nte element analysng. he overall nte element approach nvolves treatng each lnk o the manpulator as an assemblage o n elements o length l. Consder lnk to be dvded nto elements '1', '',..., 'j',... 'n ' o equal length, l, where n s the number o elements o the th lnk. Let us dene the ollowng notaton, where subscrpt j reer to the j th element o lnk. OXY s the nerta system o coordnates, O X Y s the body-xed system o coordnates attached 18

3 Journal o Optmzaton n Industral Engneerng 1 (13) 17-4 to lnk. u,j-1 s the lexural dsplacement at the common juncton o elements '(j - 1)' and 'j' o lnk. u,j s the lexural slope at the tp o the common juncton o elements '(j - 1)' and 'j' o lnk. hs slope s measured wth respect to axs O X. For each element the knetc energy, j, and potental energy, V j, are computed n terms o a selected system o generalzed coordnate, q, and ther rate o change wth respect to tme, q. It s convenent to dene r as the poston vector o lnk n the nerta reerence rame n terms o the poston o each pont n the body-xed coordnate system: 1( j 1) l1 x1 j r1, y1 j 1 L1 1 1 L r u n u n (1) 1 ( j 1) l x j 1 1 y j or,3, NL (Number o Lnks), where 1 s the transormaton matrx rom O XY to ts prevous body-xed coordnate system. It s obvous that O X Y OXY s the nerta system o coordnates. 1 cos( 1) sn( 1) sn( 1) cos( 1) 1 cos( un sn( un or, 3, NL, 1 1 ) ) sn( u cos( u n1 n1 ) ) where x j s the dstance along O X n a body-xed coordnate system rom node (j-1), l s the length o the elements n the th lnk and s the jont angle between lnk and -1. Fnally, yj s dened as the element dsplacement and expresses the deormaton o each lnk due to ts orgnal shape: 4 j k ( xj ) u, jk ( t), k 1 y (3) where u s lexural dsplacement at the common juncton o elements '(j-1)' and 'j' o lnk. k s the shape unctons (Hermtan unctons) o a beam element and obtan as: () 1 x 3 x x x 1 3 l x x x1 l l x 3 l x, l 3 x, l 3, x x 4 x x. l l Consequently, knetc energy, j, and potental energy, V j, or the j th element o lnk can be computed by the ollowng equaton: 1 j And V V j l m g l r r m dx j t t gj V ej 1 j 1l xj 1 l y j 1 dxj EI y j xj In above equaton, the potental energy s conssted o two parts. One part s due to gravty ( V gj ) and another s related to elastcty o lnks ( V ej ). m and dx j (4) (6) (5) EI are the th mass and the lexural rgdty o element, respectvely. Ater that, or each o these elements the knetc energy and potental energy V are computed n terms o a j j selected system o n generalzed varables q q q..., q ) and ther rate o change q. hese ( 1,, n energes are then combned to obtan the total knetc energy,, and potental energy, V, or the entre system. Fnally, usng Lagrange equatons the equatons can be wrtten n compact orm as: M ( q) q C( q, q ) G( q) U, (7) where M s the nerta matrx, C s the vector o Corols and centrugal orces, G descrbes the gravty eects and U s the generalzed orce nserted nto the actuator. 3. Formulaton o the Optmal Control Problem 3.1. Statement o the optmal control problem By denng X X X q, q 1, rewrte n state space orm as:, Eq.(7) can be X ( t) ( X ( t), U ( t)) (8) 19

4 Moharam H. Korayem et al./ Dynamc Load Carryng Capacty o... For the maxmum payload determnaton problem the state departng rom the ntal condtons x( t ) x must reach the nal condtons x( t ) x durng the overall tme t n such a way that the maxmum payload can be carred. Generatng optmal movements can be acheved by mnmzng a varety o quanttes nvolvng drectly or not some dynamc capactes o the mechancal system as t J ( u) L( X ( t), U( t)) dt (9) t where the Lagrangan L may be speced n qute vared manners. In the presentng paper, the attenton s restrctng to dene the perormance measure as: t J ( ).5 ( R X W X ) dt t (1) In the presented study, the objectve uncton s dened as a uncton o the actuators veloctes and torques to mnmze energy consumpton o the system Lu and Dong (1). he objectve uncton expressed by (1) s mnmzed over the entre duraton o the moton. he rst term n (1) s presented to mnmze the total torque consumpton o the system. he second and thrd terms are mnmzed the overall state varables durng the moton. In the above equaton, X W X s the generalzed squared norm o the state vector wth respect to a symmetrc, sem-dente weghtng matrx W, R s the generalzed squared norm o the nput vector regardng to a symmetrc, dente weghtng matrx R. hs can combne, or nstance, energy consumpton, actuatng torques, travellng tme or boundng the velocty magntude or maxmum payload. By denng U as a set o admssble control torque over the tme nterval the mposed bound o torque or each motor can be expressed as: U { U U U } (11) I U be a set o admssble control torque over the tme nterval t [ t, t ], or a speced payload, the optmal control problem s to obtan the U ( t) U n such a manner that the objectve crteron n Eq. (9) s mnmzed subject to the moton equatons, boundary values and torque constrants. 3.. Necessary condton or optmalty Now as the ormulaton o the optmal control problem has been completed, the soluton o optmal problem should be ormulated. In the presentng paper, an ndrect soluton o the optmal control s employed to solve the path plannng problem. hs technque provdes an excellent tool to calculate optmal trajectory wth hgh accuracy or robots that nclude, n ths case, lexble arms. hs method can overcome the hgh nonlnearty * nature o the optmzaton problem n spte o usng complete nonlnear states. Accordngly, the method s a good canddate or the cases where the system has a large number o degrees o reedoms or hgh nonlneartes such as the lexble manpulators. By mplementng Pontryagn's mnmum prncple or solvng optmzaton problems the necessary condtons or optmalty are obtaned as stated on the bass o varatonal calculus. Denng the Hamltonan uncton as: * H ( X, U, Y, mp, t) Y ( X, U, t) L( X, U, mp, t) (1) n addton to costate tme vector-uncton Y(t) that veryng the costate vector-equaton (or adjont system) Y * H X (13) and the mnmalty condton or the Hamltonan as: * H U, (14) X * H Y leads to transorm the problem o optmal control nto a non-lnear mult-pont boundary value problem, that there exst some numercal technques or solvng such problems. Hence, the mportant task s to be achevng the explct ormulaton o condtons (8), (13) and (14). Notceably, these calculatons need to compute the Jacoban matrces that requre handlng huge amounts o arthmetc operatons when copng wth complex dynamcal systems. he ullment o such requrements wth remanng all nonlnear state and control constrants s the man advantage o the presentng research study. here exst some numercal technques or solvng such problems, a number o whch have been reported n assocated lterature such as those by Krck (9). 4. Smulaton or a wo Lnk Flexble Manpulator In ths secton, a lexble two lnk manpulator wth the concentrated payload o mass m p connected to the second lnk as depcted n Fg. 1 s consdered to smulaton. Fg. 1. A two-lnk manpulator wth lexble lnks.

5 Journal o Optmzaton n Industral Engneerng 1 (13) 17-4 he physcal parameters o the model used n these smulaton studes were 6 E1 I1 EI 1 kg. m, 1 m. L1 L 1 m and m 5 kg By denng the state vectors as ollows: X X 1 Q Q x1 x3 x5 x7 x9 x11, x x x x x x. 4 6 he sate space equaton o the system can be wrtten as: (15) x x, x F ( ) ; 1...6, (16) 1 where F () can be obtaned rom Eq. (7). And the boundary condton can be expressed as: x 1 () = / rad, x 3 ( ) = / 3 rad x 1( t ) = /6 rad, x 3( t ) = /3 rad (17) x ( ) x ( t ), x ( ) x ( t 5 ) x ( ) x ( t 7 7 ) In order to derve the equatons assocated wth optmalty condtons, penalty matrces can be selected as: W dagw1, w, w3, w4, w5, w6 ; (18) R dagr1, r. So the objectve uncton s obtaned by substtutng Eq. (18) Into Eq. (1) as below 1 6 L = r 1u1 + ru w x (19) 1 hen, by consderng the costate vector as Y y1 y... y1, the Hamltonan uncton can be expressed rom as: H 1 r r wx y x, () 1 1 where x, 1,..., 1 can be substtuted rom Eq. (16). Usng Eq. (13) derentatng the Hamltonan uncton wth respect to the states, result n costate equatons as ollows: H y, 1,,1 (1) x he control uncton n the admssble nterval can be computed usng Eq. (14), by derentatng the Hamltonan uncton wth respect to the torques and settng the dervatve equal to zero. hen, by applyng motors torque lmtaton, the optmal control becomes: U U U PU U otherwse ; 1, () U U U he actuators whch are used or medum and small sze manpulators are the permanent magnet D.C. motor. he torque speed characterstc o such D.C. motors may be represented by the ollowng lnear equaton: u1 u 1 S1 x11 ; u1 1 S1 x11 S x1 ; u S x1 where S / m, (3) and are the stall torque and th maxmum no-load speed o motor, respectvely. In the presentng study, these parameters are consdered as w m 3.5 rad/s and s 3 N.m. Fnally, 4 nonlnear ordnary derental equatons are obtaned by substtutng Eq.(3) nto Eqs. (1) and (16), whch wth 4 boundary condtons gven n Eq. (17) construct a two pont boundary value problem (PBVP). Nowadays there are numerous nluental and ecent commands or solvng such problems that are avalable n derent sotware such as MALAB, MAEMAHICA or FORRAN. hese commands by employng competent methods such as shootng, collocaton, and nte derence solve the problem. In ths study, BVP4C command n MALAB whch s based on the collocaton method s used to solve the obtaned problem. he detals o ths numercal technque are gven n Shampne et al. (). 5. Results and Dscusson In ths smulaton, ndng the maxmum payload value carred between the ntal and nal pont, durng the overall tme t 1. 5 second s presented. Usng the obtaned equatons at secton and on the bass o the presented control method n secton 3, the robot path plannng problem s nvestgated by ncreasng the payload mass untl the maxmum allowable load s determned. he penalty matrces are consdered to be W= (,, 1, 1, 1, 1) and R =dag(.1). he maxmum payload or these values o penalty matrces s ound to be 1 kg. he obtaned angular veloctes and torque curves graphs or a range o m are shown n Fg.s and 3. It p can be ound that, ncreasng the m p results n enlargng the velocty values. Also, as shown n the gures, ncreasng the payload ncreases the requred torque untl the maxmum payload, so that or the last case the torque curves lay on ther lmts. Hence, t s the most possble values o the torques and ncreasng the payload that can lead to volatng the boundary condtons. m 1

6 Moharam H. Korayem et al./ Dynamc Load Carryng Capacty o mp= kg orque o Motor (N.m ) Lower bound orque o Motor (N.m) 1-1 Lower bound mp=.5 kg me (s) me (s) 4 3 mp= kg mp=1 kg mp=.5 kg orque o Motor (N.m ) Angular Velocty (rad/s) mp=1 kg -3 Lower bound me (s) Fg. - Angular veloctes and torques o motors Frst jont 4 3 mp= kg 3 orque o Motor (N.m ) Lower bound orque o Motor (N.m) 1-1 Lower bound mp=.5 kg me (s) me (s)

7 Journal o Optmzaton n Industral Engneerng 1 (13) mp= kg 3 mp=1 kg mp=.5 kg orque o Motor (N.m) 1-1 Angular Velocty (rad/s) mp=1 kg - -3 Lower bound me (s) Fg. 3. Angular veloctes and torques o motors Second jont 6. Concluson Full load moton plannng o lexble manpulators or a gven two-end-pont task n pont-to-pont moton, based on ndrect soluton o optmal control problem has been addressed n ths paper. We employed the nte element method to model and derve the nonlnear dynamc equatons o lexble manpulator. It was ound that n the presence o nonlnear and hghly luctuated terms n dynamc equatons, open loop optmal control approach s a superor canddate or generatng the ull load moton path. he Hamltonan uncton has been ormed and the necessary condtons or optmalty have been derved based on Pontryagn's mnmum prncple. he obtaned equatons establshed a two pont boundary value problem whch was solved by numercal technques. Fnally, smulatons or a two-lnk planar manpulator wth lexble lnks were carred out and the ecency o the presented method was llustrated. he obtaned results demonstrate the power and ecency o the method to overcome the hgh nonlnearty nature o the optmzaton problem whch wth other methods may be very dcult or mpossble. he optmal trajectory and correspondng nput control obtaned usng ths method can be used as a reerence sgnal and eed orward command n control structure o lexble manpulators. 7. Reerences [1] Bertolazz, E., Bral, F. and Da Lo, M. (5). Symbolc numerc ndrect method or solvng optmal control problems or large multbody systems. Multbody System Dynamcs. 13(), [] Bessonnet, G. and Chessé, S. (5). Optmal dynamcs o actuated knematc chans, Part : Problem statements and computatonal aspects. European Journal o Mechancs A/Solds, 4: [3] Bjorkenstam, S., Gleeson, D., Bohln, R., Carlson, J.S. and Lennartson, B. (13). Energy ecent and collson ree moton o ndustral robots usng optmal control. Proceedngs o the Internatonal Conerence on Automaton Scence and Engneerng, IEEE Press, [4] Brot, S., Arakelan, V. and Le Baron, J.P. (1). Shakng Force Mnmzaton o Hgh-Speed Robots va Optmal rajectory Plannng. Advances n Mechansms Desgn: Mechansms and Machne Scence, 8, [5] Calles, R. and Rentrop, P. (8). Optmal control o rgd-lnk manpulators by ndrect methods. GAMM-Mtt. 31(1), [6] Dogan, M., and Isteanopulos, Y. (7). Optmal Nonlnear Controller Desgn or Flexble Robot Manpulators wth Adaptve Internal Model. IE Control heory and Applcatons, 1(3), [7] Krk, D.E. (197). Optmal Control heory, an Introducton. Prentce-Hall Inc., Upper Saddle Rver, New Jersey. [8] Korayem, M.H. and Gharblu, H. (4). Analyss o wheeled moble lexble manpulator dynamc motons wth maxmum load carryng capactes. Robotcs and Autonomous Systems, 48(-3), [9] Korayem, M.H., Nazemzadeh, M. and Azmrad, V. (11). Optmal rajectory Plannng o Wheeled Moble Manpulators n Cluttered Envronments usng Potental Functons. Scenta Iranca, ransacton B, Mechancal Engneerng, 18(5), [1] Korayem, M.H., and Rahm, H.N. (11). Nonlnear Dynamc Analyss or Elastc Robotc Arms. Fronters o Mechancal Engneerng, 6(), [11] Korayem, M.H., Shae, A.M. and Azm, A. (1). Parametrc study o a sngle lexble lnk based on moshenko and Euler-Bernoull beam theores. Internatonal Research Journal o Appled and Basc Scences, 3(7), [1] Korayem, M.H., Nazemzadeh, M. and Rahm, H.N. (13). rajectory optmzaton o nonholonomc moble manpulators departng to a movng target amdst movng obstacles. Acta Mechanca, 4(5),

8 Moharam H. Korayem et al./ Dynamc Load Carryng Capacty o... [13] Lu, S. and Dong, S. (1). Modelng and expermental study or mnmzaton o energy consumpton o a moble robot. Proceedngs o the Internatonal Conerence on Advanced Intellgent Mechatroncs, [14] Mohamed, Z., and okh, M.O. (4). Command shapng technques or vbraton control o a lexble robot manpulator. Mechatroncs, 14, [15] Rahm, H.N., Korayem M.H. and Nkoobn, A. (9). Optmal Moton Plannng o Manpulators wth Elastc Lnks and Jonts n Generalzed Pont-to-Pont ask. ASME-IDEC/CIE 33rd Mechansms and Robotcs Conerence, 7(B), [16] Rashdar, M., Ahmad, D., and Rashdar, A. (1). Optmal Input Shapng Vbraton Control o a Flexble Manpulator Usng Genetc Algorthm. Advances n Mechancal Engneerng and ts Applcatons, (), [17] Shampne, L.F., Rechelt, M.W. and Kerzenka, J. (). Solvng boundary value problems or ordnary derental equatons n MALAB wth bvp4c. Avalable at tutoral, Accessed: /1/6. [18] homas, M., Yuan-Chou, H.C. and esar, D. (1985). Optmal actuator szng or robotc manpulators based on local dynamc crtera. ASME Journal o Mechansms, ransmssons and Automaton n Desgn, 17, [19] Usoro, P.B., Nadra, R. and Mahl, S.S. (1986). A nte element/lagrange approach to modelng lghtweght lexble manpulators. Journal o Dynamc Systems, Measurement, and Control, 18, [] Wang, L.. and Ravan, B. (1988). Dynamc load carryng capacty o mechancal manpulators. Part : computatonal procedure and applcatons, Journal o Dynamc Systems, Measurement, and Control, 11, [1] Wang, C.Y.E., moszyk, W.K. and Bobrow, J.E. (1). Payload maxmzaton or open chaned manpulator: ndng motons or a Puma 76 robot. IEEE ransacton o Robotc and Automaton, 17(), [] Yue, S. so, S.K., and Xu, W.L. (1). Maxmum dynamc payload trajectory or lexble robot manpulators wth knematc redundancy. Mechansm and Machne heory, 36, [3] Zebn,. (1) Modelng and Control o a wo-lnk Flexble Manpulator usng Fuzzy Logc and Genetc Optmzaton echnques. Journal o Computers, 7(3), [4] Zhang, C.X., and Yu, Y.Q. (4). Dynamc analyss o planar cooperatve manpulators wth lnk lexblty. Journal o Dynamc Systems, Measurement, and Control, 16,