DYNAMIC MODELING AND PATH PLANNING OF FLEXIBLE-LINK MANIPULATORS

Transcription

1 Int. J. Mech. Eng. & Rob. Res. M Nazemzadeh et al., Research Paper ISSN ol., No., Aprl IJMERR. All Rghts Reserved DYNAMIC MODELING AND PAH PLANNING OF FLEXIBLE-LINK MANIPULAORS M Nazemzadeh *, M aher and Sh Nazer *Correspondng Author: M Nazemzadeh, mn.nazemzadeh@gmal.com In ths paper, dynamc modelng and path plannng of flexble-lnk manpulators are presented. Each lnk of flexble manpulator s modeled by fnte number of elements, and the dsplacement of element s formulated based on nodal coordnates and shape functons of beam element. hen, the knetc and potental energy of the system s developed usng the dsplacement n the reference coordnate systems. hen, by employng the Lagrange prncple, the nonlnear dynamc model of the system s derved and valdated. Keywords: Flexble-lnk, Manpulator, Dynamc modelng, Fnte element method INRODUCION Flexble-lnk manpulators exhbt many advantages over ther tradtonal rgd ones: they have lght weght, ther motors are smaller, and ther producton s frugal. Because of these novel features, the applcaton of flexble manpulators are exceedngly developed durng last decades, and they have been acheved an mportant role n many felds of scence such as surgcal operaton (Kumar et al., ; and Lao et al., 8), nuclear applcaton (Jansen et al., 99; and Perrot et al., ), and aero space structures (Satoko and Kazuya, ; and Zhong, ). hus, the dynamc modelng and analyss of such system s mportant and treated by some authors: Book (999) analyzed the dynamc behavor of flexble manpulators based on recursve lagrangan method. Moreover, a Newton-Euler approach s presented n Rakhsha and Goldenberg (985) to model the dynamc of a flexble robot. Meghdar and Fahm () used an analytcal method to decouple the dynamc equatons of elastc manpulators. Furthermore, a lumped model of a planer flexble manpulator s presented n Megahed and Hamza (). Sngh (Megahed and Hamza, ) used an extended Hamlton s prncple to derve the equaton of moton of the flexble manpulator. Korayem and Rahm () and Korayem et al. ( and ) presented the dynamc Department of Mechancs, Damavand Branch, Islamc Azad Unversty, Damavand, Iran.

2 Int. J. Mech. Eng. & Rob. Res. M Nazemzadeh et al., modelng of flexble manpulator systems, based on assumed mode method. In ther method, the flexble behavor of the system s modeled va egen functons multpled by modal coordnate of the system. In ths paper, the mathematcal analyss and dynamc modelng of flexble manpulator s presented based on fnte element method. Each lnk of the system s modeled by fnte number of elements, and the dsplacement vector of each pont of the robot s formulated n the reference coordnate by means of fnte formulaton of beam element. hen, the knetc and potental energes of the system are presented, and the dynamc model of the system s derved usng Lagrange prncple. Fnally, smulaton results are presented. FINIE ELEMEN FORMULAION FOR MAHEMAICAL MODEL OF HE SYSEM o present the mathematcal and dynamc model of the flexble manpulators, the system wth m number of lnks, each lnk s dvded to n elements wth length of l. As the total dsplacement of each pont of the flexble manpulator can be presented as r. Accordng to Fgure, the reference coordnate system s shown by OXY, and the local coordnate system attached to th lnk s assumed as O XY. he parameters of the flexble manpulator are shown as follows: s j th element of th lnk, r s dsplacement vector of element, r o s dsplacement vector of th jont, represents angular dsplacement of th jont, n s number of elements of th lnk, L s length of th lnk, m s mass per length of th lnk, g s gravtatonal constant of earth, l s length of j th element of th lnk, E s elastcty modulus of th lnk, I s moment of nerta of th lnk, represents rotaton matrx between local and global coordnate system. Fgure shows a flexble-lnk manpulator. o present the total dsplacement vector of th element of the system n the global coordnate system, ths vector s assumed as a summaton of dsplacement of O, and the deflecton of the lnk n the local coordnate O X Y : r Fgure : he Flexble Manpulator r, r r, f r O ( j ) l x y...() Where y s the deflecton of element due to flexblty of system n the local coordnates. By mplementaton of fnte element method, ths dsplacement s presumed a summaton of Hermtan shape functon multpled to nodal coordnate of the element (Zenkewcz et al., 5): x,t y k( x ) uk...() k Where k shows the shape functon and u k are the nodal coordnate of the systems, and are gven as (Zenkewcz et al., 5):

3 Int. J. Mech. Eng. & Rob. Res. M Nazemzadeh et al., x x...() l ( x ) l x x ( x ) x...() l l ( x ) l x x...(5) l x x ( x )...(6) l l As the dsplacement vector of element s formulated, the knetc energy of the element s stated as follows: l r m t r. dx t x l...(7) If, the vectors z u u u u u u u u Equaton (7) can be rewrtten as: z M ( l, p) M z l r m zl r. z p and are defned, the dx...(8) Besde, the potental energy of the element s shown as, and s a summaton of gravtatonal potental energy, and elastc potental energy g e : e g...(8) he gravtatonal potental energy s gven as: g l m g...(9) r dx And the elastc energy of the system s: y l e EI ( )dx K...() x Where the stffness matrx K s presented as: K EI l 6l 6l 6l l 6l l 6l 6l 6l l 6l l...() Now, the generalzed coordnate vector s defned as q, and the total knetc and potental energy of the system can be wrtten as: as ( q,q ) m n j ( q ) m n j...() hen, the Lagrange functon s ntroduced L( q, q ), and the Lagrange prncple s developed: he prncple of Lagrange for dynamc systems s expressed as: d dt L L Q j q j q...() j Where q j represents the generalzed coordnates, Q j s the generalzed external force. hus, by mplementaton of Lagrange prncple, the nonlnear dynamc equatons of the system are summarzed as follows: q q B M q f,...() As n Equaton () s presented, the nonlnear dynamc model of the system s developed, and no lnearzaton s done. hus, the nonlnear terms affect the dynamc of the system. It must be notced that for each lnk of the flexble manpulator, the frst node s concded 5

4 Int. J. Mech. Eng. & Rob. Res. M Nazemzadeh et al., on the jont of the lnk. hus, these nodal coordnates are zero: u ( t ), u( t )...() DYNAMIC MODEL OF A SINGLE LINK MANIPULAOR For a sngle-lnk flexble manpulator, as the lnk modeled by one element, the generalzed coordnate vector of the system s q u u, where s angular dsplacement of the robot jont, u and u are the elastc deflecton and slope of the end pont of the flexble manpulator. Moreover, the rotaton matrx of the system s: hus, the dynamc equaton of the system can be derved, usng Lagrange prncple. o smulate the dynamc behavor of the system, the parameters are gven as: m = 5 kg, L = m, I = 5e-9, E = e9 pa, g = 9.8. he smulaton results are as follows: Fgure : Angular Dsplacement of Flexble Manpulator cos( ) sn( ) sn( ) cos( )...(5) And the total dsplacement of any pont of the robot s: x xcos( ) sn( ) ( L r x x sn( ) cos( ) ( L x x ) u ( L L x x ) u ( L L x ) u L x ) u L Fgure : Dsplacement Mode Shape of the Flexble Manpulator...(6) So, the knetc and potental energes of the sngle-lnk flexble manpulator are stated as: L u 7L u L u ml 78 u u Lu L uu 78 Lu 7L u...(7) g mgl 6u cos( ) Lu cos( ) 6L sn( ) EI...(8) e u Lu u Lu...(9) L CONCLUSION In ths paper, the nonlnear dynamc analyss of the flexble manpulators has been studed usng fnte element method. he total 6

5 Int. J. Mech. Eng. & Rob. Res. M Nazemzadeh et al., dsplacement vector of the system has been formulated usng Hermtan shape functon. Hence, the total dsplacement of the elastc arm n reference coordnate system has been presented, the Lagrange prncple has been used to derve the nonlnear dynamc moton of the elastc manpulator. Fnally, the proposed method has been employed to derve the dynamc equatons of a sngle-lnk manpulator, and some smulatons are done. REFERENCES. Book W J (999), Recursve Lagrangan Dynamcs of Flexble Manpulator Arms, Int. J. of Robotcs Research, ol., pp Jansen J F, Burks B L, Babcock S M, Kress R L and Hamel W R (99), Long- Reach Manpulaton for Waste Storage ank Remedaton, n ASME Dyn. Syst. Control. Dv., DSC, ol., pp Korayem M H and Rahm H N (), Nonlnear Dynamc Analyss for Elastc Robotc Arms, Fronters of Mechancal Engneerng, ol. 6, No., pp Korayem M H, Rahm H N and Nkoobn A (), Mathematcal Modelng and rajectory Plannng of Moble Manpulators wth Flexble Lnks and Jonts, Appled Mathematcal Modellng, ol. 6, No. 7, pp Korayem M H, Rahm H N, Nkoobn A and Nazemzadeh M (), Maxmum Allowable Dynamc Payload for Flexble Moble Robotc Manpulators, Latn Amercan Appled Research, ol., No., pp Kumar R, Berkelman P, Gupta P, Barnes A, Jensen P S, Whtcomb L L and aylor R (), Prelmnary Experments n Cooperatve Human/Robot Force Control for Robot Asssted Mcro-Surgcal Manpulaton, Proc. of IEEE Int. Conf. on Robotcs and Automaton, ol. 6, pp Lao H, Suzuk H, Matsumya K, Masamune K, Doh and Chba (8), Fetus-Supportng Flexble Manpulator wth Balloon-ype Stablzer for Endoscopc Intrauterne Surgery, Int. J. of Medcal Robotcs and Computer Asssted Surgery, ol., pp Megahed S M and Hamza K (), Modelng and Smulaton of Planar Flexble Lnk Manpulators wth Rgd p Connectons to Revolute Jonts, Robotca, ol., pp Meghdar A and Fahm F (), On the Frst-Order Decouplng of Dynamcal Equatons of Moton for Elastc Multbody Systems as Appled to a wo-lnk Flexble Manpulator, Multbody Systems Dynamcs, ol. 5, pp. -.. Perrot Y, Corder J J, Frconneau J P, Masonner D, Martn E and Palmer J D (), Development of a Long Reach Artculated Manpulator for IER n essel Inspecton Under acuum and emperature, Fuson Eng. Desgn, ol. 69, pp Rakhsha F and Goldenberg A A (985), Dynamc Modelng of a Sngle Lnk Flexble Robot, Proc. of the IEEE Int. Conf. on Robotcs and Automaton, pp

6 Int. J. Mech. Eng. & Rob. Res. M Nazemzadeh et al.,. Satoko A and Kazuya Y (), Adaptve Reacton Control for Space Robotc Applcatons wth Dynamc Model Uncertanty, Advanced Robotcs, ol., pp Zhong J L (), Dynamc Modelng and braton Control of a Space Flexble Manpulator Usng Pezoelectrc Actuators and Sensors, Advanced Materals Research, ol. 5, pp Zenkewcz O C, aylor R L and Zhu J Z (5), he Fnte Element Method, ts Bass and Fundamentals, Elsever Butterworth Henemann. 8