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Modeles des Robots Manipulateurs. Application a leur Commande", CEPADUES, Toulouse, France, (1984). *" * Professor, ** Graduate Student, Mechanical Design and Production Department Faculty of Engineering, Cairo University, Giza 12316- Egypt. eg Abstract This paper presents a systematic simulation algorithm for flexible manipulators with revolute joints. This algorithm takes into consideration the flexibility of links and joints, structural damping and joints' internal clearance. Flexible multi-body dynamics formulations are employed to obtain the equations of motion of such manipulators by considering their links as a lumped and as a consistent mass system.

0 m ] O02-- I . 02 11 e 0 0~ o, i o. 5 Fig. 10. E x a m p l e 1 Difference in Orientation from Rigid System at point A Fig. 11. E x a m p l e 2: T w o - L i n k M a n i p u l a t o r Current Advances in Mechanical Design and Production, MDP-7 i!!. I-T~ 35 J -3 (Sec) T'~ne (See) (a) Actuator 1 Torque (b) Actuator 2 Torque Fig. 12. Example 2 Actuator Torques 'Fi ,i /// :L , |~ ~,~ t~,-,,: Ii l++',',=_.. , 1/ o [ "~ os I i s l~,ne (-) 2 2'S ' I . i 3 3s (a) Orientation of Linkl at Point A o os t t l ' * .

Fourth order Runge-Kutta [13] is used for time step integration. 4. Example 1: One-Link Manipulator A single link (Fig. m, (Fig. 8), no damping is considered and the revolute joint is assumed to be perfect. 0 rad/s with no initial strain. Simulation is performed for 4 cases: a rigid body system, 4 and 8 element per link lumped mass, and 4-element per link consistent mass. The expected motion of the link is to be vibrating but its gross motion should be following the motion of the rigid body system.

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