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 • Transformation equation: end effector (P) l • Limitations: link upper right (ur) link upper left (ul) elbow 2 (e 2) • All lengths = l link lower left (ll) elbow 1 (e 1) link lower right (lr) y origin (0) x

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

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

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 Open. GL (for graphics) • The user can modify control parameters (kv 1 = kv 2, kp 1 = kp 2) 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, 1989. • Woo, M. , Neider, J. , Davis, T. , and Shreiner, D. Open. GL Programming Guide. 3 rd ed. Addison-Wesley Publishing Company, 1999.