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 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

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

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

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:

Dynamics Joint and link velocities

Dynamics Lagrangian in terms of θ 1 and θ 2

Dynamics Equations of Motion

Equations of motion: Control Law: Control Partitioned Controller I

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

Control Block Diagram System

Simulation Description Programmed using C++ and OpenGL (for graphics) The user can modify control parameters (k v1 = k v2, k p1 = k p2 ) 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 < θ 1 < 80, 10 < θ 2 < 80) but the simulation itself is not. Therefore, unrealistic configuration of the pantograph can be reached. Approximations:

Simulation Screen Capture

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.

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