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Manipulator Dynamics

Introduction to ROBOTICS. Manipulator Dynamics. Dr. Jizhong Xiao Department of Electrical Engineering City College of New York jxiao@ccny.cuny.edu. Manipulator Dynamics. Mathematical equations describing the dynamic behavior of the manipulator For computer simulation

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Manipulator Dynamics

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  1. Introduction to ROBOTICS Manipulator Dynamics Dr. Jizhong Xiao Department of Electrical Engineering City College of New York jxiao@ccny.cuny.edu

  2. Manipulator Dynamics • Mathematical equations describing the dynamic behavior of the manipulator • For computer simulation • Design of suitable controller • Evaluation of robot structure • Joint torques Robot motion, i.e. acceleration, velocity, position

  3. Manipulator Dynamics • Lagrange-Euler Formulation • Lagrange function is defined • K: Total kinetic energy of robot • P: Total potential energy of robot • : Joint variable of i-th joint • : first time derivative of • : Generalized force (torque) at i-th joint

  4. Manipulator Dynamics • Kinetic energy • Single particle: • Rigid body in 3-D space with linear velocity (V) , and angular velocity ( ) about the center of mass • I: Inertia Tensor: • Diagonal terms: moments of inertia • Off-diagonal terms: products of inertia

  5. Velocity of a link A point fixed in link i and expressed w.r.t. the i-th frame Same point w.r.t the base frame

  6. Velocity of a link Velocity of point expressed w.r.t. i-th frame is zero Velocity of point expressed w.r.t. base frame is:

  7. Velocity of a link • Rotary joints,

  8. Velocity of a link • Prismatic joint,

  9. Velocity of a link The effect of the motion of joint j on all the points on link i

  10. Kinetic energy of link i • Kinetic energy of a particle with differential mass dm in link i

  11. Kinetic energy of link i Center of mass Pseudo-inertia matrix of link i

  12. Manipulator Dynamics • Total kinetic energy of a robot arm Scalar quantity, function of and , : Pseudo-inertia matrix of link i, dependent on the mass distribution of link i and are expressed w.r.t. the i-th frame, Need to be computed once for evaluating the kinetic energy

  13. Manipulator Dynamics • Potential energy of link i : Center of mass w.r.t. base frame : Center of mass w.r.t. i-th frame : gravity row vector expressed in base frame • Potential energy of a robot arm Function of

  14. Manipulator Dynamics • Lagrangian function

  15. Manipulator Dynamics The effect of the motion of joint j on all the points on link i The interaction effects of the motion of joint j and joint k on all the points on link i

  16. Manipulator Dynamics • Dynamics Model

  17. Manipulator Dynamics • Dynamics Model of n-link Arm The Acceleration-related Inertia matrix term, Symmetric The Coriolis and Centrifugal terms Driving torque applied on each link The Gravity terms

  18. Example Example: One joint arm with point mass (m) concentrated at the end of the arm, link length is l , find the dynamic model of the robot using L-E method. Set up coordinate frame as in the figure

  19. Example

  20. Example Kinetic energy

  21. Example • Potential energy • Lagrange function • Equation of Motion

  22. Example: Puma 560 • Derive dynamic equations for the first 4 links of PUMA 560 robot

  23. Example: Puma 560 • Set up D-H Coordinate frame • Get robot link parameters • Get transformation matrices • Get D, H, C terms

  24. Example: Puma 560 • Get D, H, C terms

  25. Example: Puma 560 • Get D, H, C terms ……

  26. Example: Puma 560 • Get D, H, C terms

  27. Example: Puma 560 • Get D, H, C terms

  28. Thank you!

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