ME5286CourseNotes09-09 - Manipulator Dynamics: Two...

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Unformatted text preview: Manipulator Dynamics: Two Degrees-of-freedom Manipulator Dynamics Objective: Calculate the torques necessary to overcome dynamic effects Consider 2 dimensional example Based on Lagrangian approach The Lagrangian Generalized coordinate Torque or force For Mass M 1 For mass M 2 , the Cartesian coordinates are The velocities are The velocity squared Thus: Note: And the kinetic energy The Lagrangian is then Differentiating the Lagrangian for T 1 Thus on joint 1, the torque is Differentiating the Lagrangian for T 2 The torque on joint 2 is then Inertial or sensitivity matrix Centripetal matrix Effective inertias Coupling inertias D 12 = D 21 Centripetal forces acting on joint i due to a velocity at joint j D 111 = D 222 = 0 & D 122 = D 211 Coriolis forces acting on joint i due to velocities at joints j & k D 112 = D 121 & D 212 = D 211 Manipulator Dynamics: Multiple Degree-of-freedom Systems In general, the dynamic equation for a manipulator in terms of a six dimensional displacement vector q (6 joints, 7 links and a gripper) is = 6 x 6 inertia matrix = 6 x 6 viscous friction matrix = 6 x 1 vector deFning Coriolis and centrifugal forces = 6 x 1 vector deFning the gravity terms = 6 x 1 vector of input generalized forces Bottleneck Computation of joint torques in order to maintain joint positions, velocities, and accelerations Variety of methods used, based on: Lagrangian techniques Newton-Euler Formulation Table look-up approaches Computation of Joint Torques Fastest Approach: Table look-up approach, but large memory requirement Most reasonable approach to date: Iterative form of Newton Euler with dynamics referred to links’ internal coordinate system (Luh, Walker & Paul, 1980) In 1980, solution for 6 DOF manipulator on PDP11/45 with ¡oating point: Assembly language: 4.5 msec FORTRAN: 33.5 msec (too slow) Inertial effects of drive and transmissions on dynamics Effect of Activator/Drive on Inertial Effects: (Ignoring compliance and dissipative losses) J L Equiv = Equivalent inertia of load & motor/transmission on load shaft Typically Thus may contribute a large constant inertia to the overall effective inertia …...
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This note was uploaded on 11/25/2009 for the course MECH 371 taught by Professor Craig during the Spring '09 term at Hong Kong Institute of Vocational Education.

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ME5286CourseNotes09-09 - Manipulator Dynamics: Two...

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