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The Pantograph

The Pantograph. by Kevin Bowen and Sushi Suzuki. Introduction About the Pantograph. The Pantograph is a 2 DOF parallel mechanism manipulator The device will be used for haptic, biomechanic, and teleoperation research in the MAHI lab

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The Pantograph

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  1. The Pantograph by Kevin Bowen and Sushi Suzuki

  2. Introduction About the Pantograph • The Pantograph is a 2 DOF parallel mechanism manipulator • The device will be used for haptic, biomechanic, and teleoperation research in the MAHI lab • We will derive the forward kinematics and dynamics, devise a state-space controller, and program a simulation to test our theoretical model • Ultimately, this will help us control the real pantograph upon its completion

  3. y x Forward Kinematics Geometry and Coordinate Setup • Transformation equation: • Limitations: • All lengths = l end effector (P) l link upper right (ur) link upper left (ul) elbow 2 (e2) elbow 1 (e1) link lower left (ll) link lower right (lr) origin (0)

  4. Forward Kinematics The Jacobian and singularities • Jacobian Matrix • The Jacobian is not invertible when its determinant equals 0 • Singularities occur when

  5. Dynamics Lagrangian Dynamics • Assumptions: Elbows and pointer are point masses, links are homogeneous with length l, shoulder is just cylinder part with mass of whole shoulder • The Energy Equation:

  6. Dynamics Joint and link velocities

  7. Dynamics Lagrangian in terms of θ1 and θ2

  8. Dynamics Equations of Motion

  9. Control Partitioned Controller I • Equations of motion: • Control Law:

  10. Control Partitioned Controller II • System simplifies to: • The controller will act in a critically damped when:

  11. + + System + + + + - - + Control Block Diagram

  12. Simulation Description • Programmed using C++ and OpenGL (for graphics) • The user can modify control parameters (kv1 = kv2, kp1 = kp2) and the destination location (only position control) of the pantograph. • The user also can “poke” at the circular end effector using the IE 2000 joystick (with force feedback) and act as a disturbance force to the system. • The destination locations are bounded by physical constraints (10 < θ1 < 80, 10 < θ2 < 80) but the simulation itself is not. Therefore, unrealistic configuration of the pantograph can be reached. • Approximations:

  13. Simulation Screen Capture

  14. Conclusion Where to go from here • We were able to derive the forward kinematics and dynamic characteristics of the pantograph using its geometric properties • The simulation of our theoretical model shows that a partitioned controller should be appropriate for position control of the pantograph • Upon completion of the pantograph we will be able to apply our theoretical model and determine its accuracy • Future goals: study of human arm dynamics, teleoperation, high fidelity haptic feedback, and hopefully virtual air hockey.

  15. References The books and people that helped us • Craig, J.J. Introduction to Robotics: mechanics and control. 2nd ed. Addison-Wesley Publishing Company, 1989. • Woo, M., Neider, J., Davis, T., and Shreiner, D. OpenGL Programming Guide. 3rd ed. Addison-Wesley Publishing Company, 1999.

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