Simulations and Trajectory Tracking of Two Manipulators Manipulating a Flexible Payload

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1 Smuatons and rajectory racng of wo Manpuators Manpuatng a Fexbe Payoad Peng Zhang Coege of Communcaton Engneerng Jn Unversty Changchun, Chna Yuan-chuan L Coege of Communcaton Engneerng Jn Unversty Changchun, Chna bstract Manpuatng fexbe payoads are beng extensvey studed because of the potenta appcatons. So far, the modeng method sn t perfect and there are few studes about the trajectory tracng of them. In ths paper the modeng and trajectory tracng of manpuatng fexbe payoads by robot manpuators are studed. he fnte eement method (FEM s used to approxmate the vbraton of fexbe payoad. In order to restran the dsturbance of vbraton to trajectory tracng the adaptve sdng mode contro aw s desgned. he stabty of the system s proven by stabty theory. Smuaton resuts vadate the modeng method and contro aw. Keywords manpuators, fexbe payoad, adaptve, trajectory I. INRODUCION Because of the potenta appcatons of manpuatng fexbe payoad n ndustry, such as n automobe ndustry and shpbudng, t strs a great dea of nterest. fexbe beam whch one end s fxed manpuated by robot manpuator was studed by. Yuawa, J.Q. Wu[],[]. In ndustry some assemby tass, such as nsertng a fexbe part nto a hoe, requre to consder the fexbty of objects and these can found n [3],[4],[5]. Durng the producton of prnted wrng boards(pwb, a arge number of eastc sheets need to be assembed to form a boo[6],[7]and the trajectory pannng was studed n them. J.K. Ms[8],[9]frsty proposed the rgd contro method and studed the robotc fxtureess assemby of sheet meta by ths method. Sun and Lu[],[],[] who used the rgd contro method have done a great dea of wor n manpuatng fexbe payoad by mutpe robot manpuators whch advanced the studes of manpuatng fexbe payoad. t present, most of wor about manpuatng fexbe payoad ony consdered the poston contro whch moves a fexbe payoad from an orgna poston/orentaton to prearranged poston/orentaton. In[3] the trajectory tracng of two manpuators manpuatng fexbe payoad was studed usng sdng mode contro and a fxed estmated tem repaced a the eements except the rgd eements. In ths paper the trajectory tracng w be studed and the adaptve sdng mode contro aw w be used. Compared to the assumed mode method, the fnte eement method (FEM has some advantages that every coordnate n FEM has ts physca meanng and the numbers of coordnate can be chosen accordng to the precson so the FEM s used to approxmate the vbraton dynamcs here and the confguraton of pnned-pnned s used. he mode of the system s got n tas space because compared to the jont space and end-effector space, t doesn t requre to pan the trajectory of snge robot manpuator n the tas space so t can reduce the cacuaton. Because of the choce of system coordnates the defnton of vbraton coordnates w have some dffcuty and t s soved propery n ths paper. In the trajectory tracng of manpuatng fexbe payoad the dsturbance of vbraton can t be negected. In order to restran the effect of vbraton the adaptve sdng contro aw s desgned. Compared to [4] the regressor matrx doesn't requre to be cacuated and the chaterng when the contro aw traverses the sdng surface w reduce. he stabty of the system s proven by stabty theory and smuaton resuts vadate the modeng method and contro aw. he organzaton of the paper s as foows. Secton the mode of the system s derved and the decomposton of t s done. Secton 3 the adaptve sdng contro aw s desgned and the stabty of the system s proven. he smuaton resuts are shown n secton 4. he concusons are gven at ast.. Mode of the system II. MODEL In [6], Sun ndcated that besdes the camped-free mode, the other mode can be aso used f the rgd body of the fexbe payoad can be determned. In ths paper the pnnedpnned mode w be used. he vrtua n s defned as the dot ne as n Fg. and t s aso the rgd body of the fexbe payoad. In the process of moton the manpuators and vrtua n form a cosed-oop as cooperatng rgd payoad. wo assumptons are made. here s no deformaton n the process of moton so the vrtua n and the ast n of manpuators w eep parae; he payoad s grasped rgdy. he poston and veocty of any pont n fexbe payoad are as foows R r R r y x y ( cos( θ x ( sn( θ y ( sn( θ x θ ( cos( θ y y ( /8 /$5. 8 IEEE RM 8

2 R R y x y cos( θ x sn( θ y sn( θ x cos( θ y y cos( θ sn( θ he fgure s as foowng θ L θ L θ L 3 sn( θ cos( θ x ( x, y, y x, ( ( y R R r ( x, y θ R 3 Fg. Confguraton of cooperatve manpuatng θ R θ R It can be seen that the expressons of are dfferent and t w be expaned ater. he expressons of s s as foows 4 j= ( j ( y = ϕ ( x q ( x x x x ϕ = ϕ 3 = x 3 3 3x x x x ϕ 3 = 3 4 = ϕ (4 R s the dspacement of some pont of the payoad correspondng to Cartesan coordnates, s the poston of the payoad s orgn n Cartesan coordnate, x, y s the ange of payoad correspondng to Cartesan coordnates, the subscrpt, r express eft and rght of the payoad s orgn, s the ength of snge eement, s th eement from the orgn, s the Jacoban matrx between object coordnate and Cartesan coordnate, x, y are oca coordnates. he netc energy of the fexbe payoad s = E Er E = R R dx E he potenta of t s ρ (5 y p = E pr E p E p = EI ( dx x E he fexbe payoad s dvded nto two eements. We defne that q, q are q, q, p 3, p4 are equa to qr, qr are q 3, q 4, qr3, qr4 are q 5, q 6 respectvey because of the defnton of R. he coordnates of vbraton are q ~ q 6. (6 Ignorng the gravty, the Lagrange equaton s as foows d dt L L ( ( = p p L = E E p he mode of the fexbe payoad s f (7 p Bp G p p = f (8 p = [ x, y, θ, q] f ony acts to x, y, θ. Because the two ends of fexbe payoad are fxed, the q, q 5 are. Dvdng the mode of fexbe payoad nto two parts: rgd moton and vbraton, the foowng equatons are got he equatons of rgd moton he equatons of vbraton q B B q = f (9 q B B q K q ( F = he mode of snge manpuator s foowng M ( θ θ C ( θ, θ θ G = τ J ( θ f ( he reatons between are = J ( θ θ = J ( x θ = J ( θ J ( x = J θ = J x J ( In ( the θ s expressed by x and each sde of ( s eft-mutped by J. he dynamc mode of manpuator n tas space s as foows J M ( θ J ( J C ( θ, θ J J M ( θ J J G = J τ J J (θ f M C G = u J ( x f (3 he reatons between the f are as foows J ( x f = f (4

3 he f can be dvded nto two parts movng force f m and nterna force f I. he sum of nterna forces s, so we can get f = f f J ( x f = (5 m I Form (9, (3, (4and(5the mode of system s as foows u M p Cp G (6 It can be shown that the dynamcs of the system derved usng Euer-Lagrange formuaton has the foowng propertes. M s symmetrca and postve defnte and the sub-tems made up of M are aso symmetrca and postve defnte; p ( M C p = f we choose C propery; 3 K F s symmetrca and postve defnte. B. he decomposton of mode From the prevous dervatons the rgd coordnates and vbraton are n coupng. In ths secton mode of rgd moton w be decomposed. From the equaton ( we can get ( Fq q = x B K (7 q KF q = x B (8 From (6 and (8 the foowng equaton w be got ~ ~ M C G Mx C f ( q, q = u (9 f ( q, q = B q In (9 the frst three tems are made up of rgd coordnates and the atter three tems s made up of rgd and vbraton coordnates. he reatons x ( M C x = s st satsfed from the dervatons. heorem. From (8 and gven the x,, x beng bounded, the states of vbraton w be bounded. Proof. We can see the x,, x are bounded because the nputs are bounded. he x can be expressed by trgonometrc functon from the transton of Fourer. ssumng the x has the smpe expresson as foows K F x = η sn( ωt ( So the rght sde of (8 can be expressed as η sn( ωt σ and t s bounded. From [5] the foowng equatons w be got q = η sn( ωt σ η sn( ωt ς ( I q he dmensons of vbraton are more than one so the ( s anaytc expresson. η sn( ωt σ s reated to orgna vaue and physca characterstc of the matera. η sn( ωt ς s compusve vbraton because of x. Because, K F are symmetrca and postve defnton and x,, x are bounded the q are bounded and t can be got that q, q are bounded smary. From the proof the f ( q, q w be bounded. From [4] there are b, b, b that are satsfed wth the foowng reaton. ~ ~ ( M C f ( q, q = Ψ b x b x Ψ < b ( III. RJECORY RCKINGY In order to trac a pannng trajectory and avod the dsturbance of vbraton the adaptve sdng mode contro aw s desgned. hs contro aw needn t compute the regressor matrx and the chaterng w reduce. Let us defne the state error and sdng surface as foows x d r = d Λe s = r e = x and Λ s postve dagona matrx. Let us assume b ˆ ˆ, bˆ, b are the estmatons of b, b, b. ~ ˆ ~ b = b b, ˆ ~ b = b b, b ˆ = b b. Because b, b, b are ~ fxed we get ˆ ~ ˆ ~ b = b, b = b, ˆ b = b. heorem. he system s stabe f the contro aw s foowng u 3 u = u u u3 u = Mr Cr G u = Ks ( bˆ bˆ x bˆ = ( bˆ bˆ x bˆ s x s s x ε ˆ s s b =, b ˆ = x, b ˆ s > ε s ε s = (3 K, Λ s postve dagona matrx.,, are postve. ε s the accurate range. Proof. We start by defnng the nonnegatve functon V = Ms From (9and (3 we get s ~ b ~ b ~ b

4 ~ ~ ~ ~ V = s Ms s Ms bb bb = s ~~ ( u u Ψ = s Ks s u s Ψ When equaton s satsfed s > ε and from ( and (3 the foowng V s Ks s u s ( b b x b x = s Ks s ( bˆ bˆ x bˆ x x s ( b b x b It used s s = s here. ~ ~ ~ V s Ks s ( b b x b x From (3 we get ( ( M M M u ( C C C [ u C q G G ( C F q C q K ] (6 It has been proven a the states on the rght sde of (6 are bounded[7] so the nterna s bounded. IV. SIMULION o verfy the modeng method and contro aw the smuaton was done on two dentca three DOF panar manpuators (Fg.. he parameters of the robot manpuators and the payoad are that the masses of three ns are m = m = 8g and m3 = 4g ; the ength of three ns are L =. 7m, L =. 6m and L 3 =. m ; the centers of ns are L g =.4m, Lg =. 35m, L3 g =. m. he ength of payoad s L =. 5m and the densty s ρ = g. he m parameters of contro ow are EI =, ε =., C =., K =, Λ = 5, = = =. he trajectory s as foows x = [. sn(π t,.75. sn(πt,. sn(πt ] d x = [.4π cos(πt,.4π cos(πt,.4π cos(πt ] d p V s Ks < (4 he proof s over and so the system s stabe. In the paper the amount of manpuators s two so =,. It shoud be seen from the prevous dervaton that n the process of controer desgnng the u s provded by two manpuators so the force/ torque provded by snge manpuator s as foowng u = ( M u u3 J ( x f I r C r G (5 Here we assume the force/ torque offerng to the fexbe payoad s provded by two manpuators averagey. It s defned J ( x fi = FI. From (3, (6 and (5 the nterna force can be descrbed as foowng F I = u ( C Cq K C C ( C q G F q

5 Fg. rajectory of payoad Fg.3 Vbraton Fg.4 Error of trajectory

6 Fg.5 Interna force From the smuaton resuts t can be seen that the trajectory tracs the pannng trajectory qucy. Frsty, the man nfuence to vbraton s the physca characterstc of the fexbe payoad. t about t=.5s the orgna vbraton because of the physca characterstc dsappears and then the vbraton s compeed by the rgd moton. From the smuaton resuts of nterna force t can be seen the effect of vbraton to nterna s more dstnct than to rgd poston and ths can be expaned by the physca characterstc. he frequency of vbraton s smar to the frequency of rgd moton. he smuaton resuts prove the heorem.. V. CONCLUSIONS he modeng and trajectory tracng of manpuatng fexbe payoads by robot manpuators are studed n ths paper. he fnte eement method (FEM s used to approxmate the vbraton of fexbe payoad. Comparng the three spaces the system mode s got n tas space. In order to restran the dsturbance of vbraton to trajectory tracng the adaptve sdng mode contro aw s desgned. It s proven that the modeng method s accurate and the contro aw s vad through the smuaton resuts. We ddn t propose a drect contro method n ths paper that can guarantee the vbraton convergng zero and we w do ths wor next. CKNOWLEDGMEN hs wor s supported by Natona Natura Scence Foundaton of Chna, grant number (63753, REFERENCES []. Yuawa M. Uchyama and H. Inooa. Handng of a Constraned Fexbe Object by a Robot. Proceedngs - IEEE Internatona Conference on Robotcs and utomaton [] J.Q. Wu, Z.W. Luo M. Yamata K. Ito. Gan Schedued Contro of Robot Manpuators for Contact ass on Uncertan Fexbe Objects. Proceedngs of the IEEE Internatona Conference on Systems, Man and Cybernetcs [3] Y.F. Zhang, R. Pe and C.Y. Chen. Strateges for utomatc ssemby of Deformabe Objects. Proceedngs - IEEE Internatona Conference on Robotcs and utomaton [4] H. Naaga K. Ktaga and H. suune. Study of Inserton as of a Fexbe Beam nto a Hoe. Proceedngs - IEEE Internatona Conference on Robotcs and utomaton [5] Jr. Werner Kraus and B.J. McCarragher. Force Feds n the Manpuaton of Fexbe Materas. Proceedngs - IEEE Internatona Conference on Robotcs and utomaton [6] J. K. Ms and Ing, Jerry G.-L. Robotc Fxtureess ssemby of Sheet Meta Parts Usng Dynamc Fnte Eement Modes: Modeng and Smuaton. Proceedngs - IEEE Internatona Conference on Robotcs and utomaton [7] W. Nguyen and J. K. Ms. Mut-Robot Contro For Fexbe Fxtureess ssemby of Fexbe Sheet Meta uto Body Parts. Proceedngs - IEEE Internatona Conference on Robotcs and utomaton [8] Y.F. Zheng and M. Z. Chen. rajectory pannng for two manpuators to deform fexbe beams. Proceedngs - IEEE Internatona Conference on Robotcs and utomaton [9] M.Z. Chen and Y.F. Zheng. Vbraton-free movement of deformabe beams by robot manpuator. Proceedngs - IEEE Internatona Conference on Robotcs and utomaton [] Y.H. Lu and D. Sun. Stabzng a Fexbe Beam Handed by wo Manpuators Va PD Feedbac. IEEE ransactons on utomatc Contro.,VOL. 45(: 59-64, [] D Sun and Y.H. Lu. Poston and Force racng of a wo-manpuator System Manpuatng a Fexbe Beam. Journa of Robotc Systems, Vo.8(4: 97-. [] D. Sun and Y.H. Lu. Poston and Force racng of a wo- Manpuator System Manpuatng a Fexbe Beam Payoad. Proceedngs - IEEE Internatona Conference on Robotcs and utomaton [3] -Yahmad, mer S. bdo, Jam. etc. Modeng and contro of two manpuators handng a fexbe object [J]. Journa of the Frann Insttute. Vo.344(5, pp [4] Sze San Chong and Xnghuo Yu. Robust daptve Sdng Mode Controer for Robotc Manpuators. IEEE Worshop on Varabe Structure Systems.996. pp3-35 [5] Y.Z. Lu, W.L.Chen, L.Q. Chen. Mechancs of Vbraton. Hgher Educaton Press. Chna. 998 [6] D. Sun. Cooperatve Contro of wo-manpuator Systems Handng Fexbe Objects. Ph.D. dssertaton. Dept. Sys. Eng. & Eng. Management. Chnese Unv. Hong Kong, 997 [7] P. Zhang and Y.C. L, Poston/Force Contro of wo Manpuators Handng a Fexbe Payoad Based on Fnte-eement Mode 7 Internatona Conference on Robotcs and Bommetcs