Effects of System Parameters and Controlled Torque on the Dynamics of Rigid-Flexible Robotic Manipulator

Transcription

1 Journal of Robotcs, Networkng and Artfcal Lfe, Vol. 3, No. (September 016), Effects of System Parameters and Controlled orque on the Dynamcs of Rgd-Flexble Robotc Manpulator Sachndra Mahto Mechancal Engneerng Department, NERIS, Arunachal Pradesh , Inda E-mal: Abstract hs work llustrates the effects of varous system parameters on the dynamcs of flexble lnk of revolute-jonted rgd-flexble manpulator. Flexble lnk s consdered as a Euler-Bernoull beam and fnte element based on Langrange approach s employed for dynamc analyss. A comparatve study s carred out for comparatve dynamc response for the varaton of system parameters and controlled torque exctaton. Keywords: Rgd-flexble revolute manpulator, Euler-Bernoull beam, Shape optmzaton, Fnte element method, Parametrc study 1. Introducton In the last few decades, dynamcs and control of the flexble manpulators have receved a consderable research attenton. Flexble robotcs systems have several advantages over the conventon systems. However, due to lghtweght of flexble systems, stffness s low and there s a serous problem of vbraton. Sometmes, to retan the advantages of the flexble system, some of the lnks are made flexble and some are rgd. Stll the dynamc behavor mprovement, optmal desgn and proper control strategy s the research nterest for flexble manpulator. Most of the researchers contrbuted ther works n dfferent ways of dynamc modelng and control aspects of flexble robotc manpulators. Sunada and Dubowsky [1] developed a lumped parameter FEM model for analyzng the complete behavour of ndustral robotc manpulator wth complex-shape flexble lnks. Fakuda and Arakawa [] studed the modelng and dynamc characterstcs of two-lnk flexble robotc arms and controlled the vbraton by takng nto account the gravty, payload, and the coupled vbraton between the frst and second arm. Usoro et al. [3] developed fnte element models to descrbe the deflecton of a planar mult-lnk model. Ower and Vegte [4] used a Lagrangan approach to model the planar moton of a manpulator consstng of two flexble lnks and two rotary jonts. Benat and Morro [5] developed a Lagrangan approach for the dynamcs of chan wth flexble lnks.. Bayo [6] used FEM to deal wth mult-lnk flexble manpulator consderng moshenko beam theory and ncludng nonlnear corols and centrfugal effects for the elastc behavor. Jonker [7] presented a nonlnear fnte element based formulaton for analyzng the dynamc behavor of flexble manpulators. De Luca and Sclano [8] presented closed-form equatons of moton for planar flexble mult-lnk robot arm. Morrs and Madan [9] studed the accurate modelng based on Lagrange-Euler formulaton of a two lnk flexble manpulators. 116

2 Sachndra Mahto Dogan and Iftar [10] carred out the modelng and control of two-lnk robot manpulator whose frst lnk s rgd and second lnk s flexble. Everett et al. [11] showed that t s possble to desgn a two-lnk flexble manpulator that has nearly poston nvarant frst natural frequency wth wde separaton between the frst two natural frequences, to have ts behavour lke rgd manpulator. Chen [1] developed a lnearzed dynamc model for mult-lnk planar flexble manpulator. Yang et al. [13] studed the tp trajectory trackng control for flexble mult-lnk manpulator usng Lagrangan assumed mode method. Zhang et al. [14] derved a partal dfferental equaton model for a flexble two-lnk manpulator usng Hamlton's prncple. Some researchers presented ther works on shape optmzaton of statc/rotatng beams. Karhaloo and Nordson [15] determned the optmum taperng of a cantlever beam carryng an end mass to maxmze fundamental frequency. Wang [16] addressed optmum desgn of a sngle lnk manpulator to maxmze ts fundamental frequency. Yoo et al. [17] used the assumed mode method for dynamc modellng of rotatng flexble manpulator for modal analyss and shape optmzaton to ncrease the fundamental frequency of the beam. Dxt et al. [18] presented a fnte element model of sngle lnk robotc manpulator for revolute as well as prsmatc jont. hey used SQP for optmzng beam shapes under dfferent optmzaton condtons. From the above survey, t s observed that there s no much research contrbuton for the effects of system parameters n the dynamcs of rgd-flexble robotc manpulator. In ths paper, the dynamc behavor of robotc manpulator s presented for the varaton of system parameters and mprovement through control strategy.. Modelng and Soluton echnque he fnte element formulaton has been adopted here as descrbed by Usoro et al. [3]. Fg. 1(a) shows rgd-flexble manpulator comprsed of two lnks, two jonts and tp load. he lnks are clamped on the jonts. Shoulder jont (jont 1) s located at the orgn of XOY represents the statonary co-ordnate frame and elbow jont (jont ) s located at the orgn of X 1 O 1 Y 1 and X O 1 Y whch represent the movng co-ordnate frames. manpulator s consdered slender. Flexble lnk s treated as a Euler-Bernoull beam and gravty force s neglected. Consder a pont P n the th element on the manpulator at a dstance ' x ' from the elbow hub. he pont P attans the poston P wth respect to nertal frame of reference (XOY) after havng rgd body moton θ 1 (t) and θ (t) of shoulder and elbow jont respectvely and flexural deflecton w( xt, ). Flexural deflecton w(,) xt of pont P s approxmated n fnte element technque usng Hermtan shape functons as w xt, Nw Nw Nw Nw N W ( ) = = { } , (1) W = w w w + w. where N = N1 N N3 N4 and { } 1 1 Fg. 1. (a) Confguraton of rgd-flexble manpulator, th (b) ypcal element wth sx dof In FEM formulaton, the manpulator s dvded nto fnte number of elements wth each element havng sx degrees of freedom. Detal of th element of the lnk s shown n Fg. 1(b), where θ 1 and θ are the hub rotaton of shoulder and elbow jonts respectvely, w -1, w and w +1, w + are the transverse deflecton and slope at the frst and second node of the element respectvely. he poston vector of P wth respect to nertal system XOY s gven by Eq.. For the predcton of approxmate dynamc behavour of the optmzed beams, smaller exctaton torque s consdered for lnearzaton of system modellng. Global poston vector of the pont P under smaller angular and flexural dsplacement s gven by 117

3 Effects of System Parameters X a+ L1 + b+ ( 1) h+ x r = op = Y = ( a+ L ) θ + (( 1) h+ x + b)( θ + θ ) + N { W}, () In fnte element method, varables are converted nto nodal varables. ( ) r = op = f θ, θ, w, w, w, w. (3) Let Z = θ θ w w w w, then absolute velocty of the pont P of the flexble lnk s obtaned as r r = Z. (4) t Z.1 Knetc Energy Computaton of the Lnk Element he Knetc energy of the th element of the lnk s gven by e 1 h r r K = m d. x (5) 0 t t We have r r = Z r r Z. (6) t t Z Z Substtutng Eq. 6 n Eq. 5, we have 1 e h K = Z r r m d xz, (7) 0 Z Z and the elemental mass matrx s gven by h e r r M = m dx 0 Z Z M11 M1 M13 M14 M15 M16 M1 M M3 M4 M5 M 6 e M31 M3 M =. (8) M41 M4 P ( 4X4) M51 M 5 M61 M6 All the constants of the above matrx n Eq. 8 are obtaned by proper ntegraton.. Elastc Potental Energy of the Lnk Element he potental energy of the th element of lnk due to elastc deformaton s gven by 1 h 1 h e w '' '' V = EI d x = { W} EI N N d x{ W}. 0 x 0 (9) hus, elemental stffness matrx s gven by h e '' '' K = EI N N dx e EI h 1 6h K = 3. (10) h 0 0 6h 4h 6h h h 1 6h 0 0 6h h 6h 4h.3 Lagrange s Equaton of Moton n Dscretzed Form Frst lnk s rgd and posses knetc energy only. However, beng second lnk flexble t posses both knetc and potental energy. he knetc energy and the potental energy of the system are obtaned by computng energy of rgd lnk and the energy of each element of the flexble lnk and then summng over all the elements. he global mass matrx and global stffness matrx can be obtaned as 1 1 = [ q ] [ M][ q ] and V = [ q] [ K][ q] (11) respectvely. Here [ q] = q1 q w1 w.. wn+ 1 wn+ s the global nodal vector. he Lagrangan of the system s gven by L= -V and then Lagrange s equatons of moton of ths dynamc system may be wrtten as L L = q, t F (1) q q where F q s the generalzed force vector. Due to modellng lnearzaton, global mass and stffness matrces are constant and equaton of moton of undamped system s expressed as M + K = F (13) [ ]{ q} [ ]{ q} { }. Global load vector {F} and global nodal dsplacement vector {q} for ' n ' number of fnte elements are gven by { F} = τ τ

4 Sachndra Mahto 3. Optmzaton Procedure o retan and optmze the advantage, upper lmt of the optmzed mass (M * ) s constrant to the prescrbed mass (M). In the rgd-flexble robotc system, flexble lnk s consdered for the shape optmzaton. X = [d 1 d... d n ] s the desgn vector wth d ndcatng dameter of the th fnte element of lnk. lnk lengths n ts natural frequences, statc tp deflecton, hub angles, and dynamc response are plotted n Fg. to Fg. 5. Natural frequences of the flexble lnks decrease wth ncrease of ts length and vce versa. Statc lnk deflecton ncreases wth ncrease of ts length. Hub angles ncrease wth decrease of ts length and vce-versa. Resdual vbraton ncreases wth ncrease of length of the flexble lnk. able 1. Dfferent Optmzaton Problems Optmzaton Problem/Objectve Constrant Maxmzaton of fundamental beam M * -M 0 frequency Permssble Bound : X UB < X<X LB Mnmum and maxmum dameter of the beam elements (X) are denoted by X LB, X UB respectvely. he MALAB functon fmncon employng sequental quadratc programmng (SQP) technque s used for constraned optmzaton of nonlnear functon. 4. Results and Dscusson Modelng of ths system s hghly complex and nonlnear n nature. However, lnearzed model s preferred and consdered n ths work to predct ts approxmate dynamc behavor. Structural dmensons of ths revolute-jonted rgd-flexble robotc manpulator for numercal experments are taken as lengths 0.75m and 0.75m, dameters 0.03m and 0.01m, mass denstes 710 kg/m 3 and 710 kg/m 3, hub rad 0.0m and 0.01m, hub nertas 0.03 kgm and 0.03 kgm for frst and second lnk respectvely. Young's modulus 7.11x10 10 N/m and mass of the motor 0.1 kg s consdered. Exctaton torque τ = τ m snπt Nm s consdered for both the jonts for seconds acton. orque ampltude 3.0 Nm and 1.0 Nm are consdered for shoulder and elbow jonts respectvely. Statc load of 1N s consdered at the tp of the manpulator for comparatve statc beam deflectons. he rato of payload (M p ) to the mass of second lnk (M ) s denoted by µ ; rato of motor mass (M m ) to mass of the frst lnk (L 1 ) s denoted by µ 1 and rato of the second lnk length (L ) to frst lnk length (L 1 ) s denoted by (L * ). Fg.. Beam frequences of flexble lnk for dfferent lnk lengths Fg. 3. Statc deflecton of flexble lnk due to 1N force at tp for dfferent lnk lengths 4.1 Effects of Dfferent Lnk Length As the second lnk s flexble, ts length s vared wth respect to the frst lnk length. Effects of varaton of Fg. 4. Hub angles due to exctaton torques for dfferent lnk lengths 119

5 Effects of System Parameters Fg. 5. Vbraton resdual of the tp of flexble lnk for dfferent lnk lengths Fg. 7. Dynamc flexble tp deflecton for dfferent payloads 4. Effects of Dfferent Payload, Motor Mass and Hub Inerta Robotc system means to take load and due to flexble lnks, there s always flexural vbraton. Changes n the dynamcs of the manpulator due to the change of payload, motor mass and hub nerta are plotted n Fg. 6 to Fg. 11. Hub angle decrease and resdual vbraton of flexble lnk tp ncreases wth the ncrease of payload and vce-versa. By ncreasng the motor mass whch s actng as a payload for lnk 1 has sgnfcant effect on shoulder jont angle for a gven set of torques but less effect on elbow jont, however resdual vbraton ncreases. By ncreasng the hub-nerta of the shoulder jont, hub angles of shoulder jont decreases and elbow jont ncrease for a gven set of appled torque and very less effect on the resdual vbraton of the flexble lnk tp. Fg. 8. Hub angles for dfferent motor mass for set of appled orques Fg. 9. Dynamc flexble tp deflecton for dfferent motor Mass Fg. 6. Hub angles for dfferent payloads for set of appled torques Fg. 10. Hub angles for dfferent hub nerta for set of appled torques 10

6 Sachndra Mahto Fg. 11. Dynamc flexble tp deflecton for dfferent hub nerta 4.3 Effects of Mass Dstrbuton of Lnk Optmal desgn has a great mportance n engneerng applcatons. Shape optmzaton s done as per the optmzaton problem (able 1) to ncrease the fundamental frequency and the optmzed shape of the flexble lnk s shown n Fg. 1, where the mass of the lnk s re-dstrbuted. It s observed that mass s concentrated more towards the root sde n shape optmzaton. Statc lnk deflecton due to 1N force at tp, natural frequences of the optmzed flexble lnk are plotted n Fg. 13 and Fg. 14 and ts dynamcs (hub angles and resdual vbraton) are plotted n Fg. 15 and Fg. 16 respectvely for set of exctaton torque. It s observed that parameters vz., statc tp deflecton, natural frequences, hub angle and resdual vbraton are mproved. Fg. 14. Comparson of natural frequences Fg. 15. Comparson of jont angles for set of appled torques Fg. 16. Comparson of tp resduals for set of appled torques 4.4 Effects of Controlled orque Fg. 1. Optmzed shape of flexble lnk Fg. 13. Statc beam deflecton due to 1 N force at the tp of optmzed lnk Ygt [19] presented the poston and dervatve (PD) control torque for sngle lnk revolute-jonted flexble manpulator as gven below τ = K ( θ θ ) K ( θ). (14) PD p f v he feedback gans K p and K v depend upon the equvalent rgd system parameters and are expressed as and 1 3 p h 3 p n K = ( J + ml + M L ) f, (15) 1 3 v h 3 p n K = ( J + ml + M L ) f. (16) 11

7 Effects of System Parameters where 'm' s the mass per unt length, 'L' s the lnk length, 'J h ', 'θ f ' s the fnal angular poston, and 'f n ' s the fundamental frequency. PD controller s able to stablze the system but vbraton of the flexble beam can not be controlled. Ge et al. [0] extended the work of Yagt [19] and ntroduced energy-based robust (EBR) control law and added the nonlnear deflecton feedback to mprove the performance of the PD controller by addng the nonlnear control term as gven by t t = t K ylt (, ) θσ ( ) yl (, σ)d σ, (17) EBR PD f 0 where K f s the gan constant of robust control, σ s the dummy varable, θ s the hub angular acceleraton and y s the deflecton at the tps. Above control law s also applcable for mult-lnk robotc revolute jonted systems. Elbow angle due to the controlled torque s plotted n Fg. 17 and resdual vbraton s shown n Fg. 18. It s observed that hub angle s acheved faster by controlled torque and tp vbraton s lesser than the snusodal exctaton. Fg. 17. Comparson of elbow angles for dfferent exctaton Fg. 18. Comparson of dynamc tp response for dfferent exctaton 5. Concluson In ths work, fnte element analyss of revolute-jonted rgd-flexble manpulator has been performed through lnear modelng. Classcal nonlnear optmzaton s used to solve the constraned shape optmzaton. From the numercal experments, t s observed that there s a great role of system parameters (lnk lengths, payloads, hub-nerta, lnk shape) n ts system dynamcs and should be analyzed properly durng the desgn. It s also observed that controlled torque mproves the system dynamcs further. References 1. W. Sunada and S. Dubowsky, On the dynamc analyss and behavour of ndustral robotc manpulators wth elastc members, ASME Journal of Mechansm, ransmssons and Automaton n Desgn, 105(1) (1983), pp Fukuda and A. Arakawa, Modelng and control characterstcs for a two-degrees-of-freedom couplng system of flexble robotc arms, JSME, Seres C, 30 (1987), pp P.B. Usoro, R. Nadra and S.S. Mahl, S.S., A fnte element/lagrangan approach to modelng lght weght flexble manpulators, ASME Journal of Dynamc System, Measurement and Control, 108 (1986), pp J.C. Ower and J.V. Vegte, Classcal control desgn for a flexble manpulator: modelng and control system desgn, IEEE Journal of Robotcs and Automaton, 3(5) (1987), pp M. Benat and A. Morro, Dynamcs of chan of flexble lnks, ASME J. of Dyn. Sys., Meas., and Control, 110 (1988), pp E. Bayo, moshenko versus Bernoull-Euler beam theores for nverse dynamcs of flexble robots, Internatonal Journal of Robotcs and Automaton, 4(1) (1989), pp B. Jonker, A fnte element dynamc analyss of flexble manpulators, he Internatonal Journal of Robotcs Research, 9(4) (1990), pp A. DeLuca and B. Sclano, Closed form dynamc model of planar multlnk lghtweght robots, IEEE ransactons on Systems, Man and Cybernetcs SMC-1, 4 (1991), pp

8 Sachndra Mahto 9. A.S. Morrs and A. Madan, Statc and dynamc modelng of a two-flexble-lnk robot manpulator, Robotca, 14(3) (1995), pp A. Dogan and A. Iftar, Modelng and control of a two-lnk flexble robot manpulator, In Proceedngs of the IEEE Internatonal Conference on Control Applcatons, (1998), pp L.J. Everett,. Jennchen and M. Compere, Desgnng flexble manpulators wth the lowest natural frequency nearly ndependent of poston, IEEE rans. on Robotcs and Automaton, 15(4) (1999), pp W. Chen, Dynamc modelng of mult-lnk flexble robotc manpulators, Computers and Structures, 79() (001), pp W. Yang, W.L. Xu and S.K. so, Dynamc modelng based on real-tme deflecton measurement and compensaton control for flexble mult-lnk manpulators, Dynamcs and Control, 11(1) (001), pp X. Zhang, W. Xu, S.S. Nar, and V.S. Chellabona, PDE modelng and control of a flexble two-lnk manpulator, IEEE ransactons on Control Systems echnology, 13() (005), pp B.L. Karhaloo and F.I. Nordson, Optmum desgn of vbratng cantlever. Journal of Optmzaton, heory and Applcatons. 11(6) (1973), pp F.Y. Wang, On the External Fundamental Frequences of one Lnk Flexble Manpulators. he Internatonal Journal of Robotcs Research, 13 (1994), pp H.H. Yoo, J.E. Cho, and J. Chung, Modal analyss and shape optmzaton of rotatng cantlever beams, Journal of Sound and Vbraton, 90 (006), pp U.S. Dxt, R. Kumar, and S.K. Dwvedy, Shape optmzaton of flexble robotc manpulators, ASME Journal of Mechancal Desgn, 18 (006), pp A.S. Ygt, On the stablty of PD control for a two-lnk rgd-flexble manpulator, J. of Dyn. Sys., Meas., and Control, 116 (1994), pp S.S. Ge,.H. Lee, G. Zhu, Energy-based robust controller desgn for mult-lnk flexble robots, Mechatroncs, 6(7) (1996), pp