The Pantographby Kevin Bowen and Sushi Suzuki

IntroductionAbout 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 KinematicsGeometry and Coordinate Setup

)sin()sin(

)cos()cos(

21

21

0

ly

x

P

x

y

• Transformation equation:

• Limitations:

• All lengths = l

l

12

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)

1 28010 2 8010 1

Forward KinematicsThe Jacobian and singularities

• Jacobian Matrix

• The Jacobian is not invertible when its determinant equals 0

• Singularities occur when

)cos()cos(

)sin()sin(

21

210

lJ P

)tan()tan(

)sin()cos()cos()sin(

0)sin()cos()cos()sin()det(

21

2121

21210

pJ

n 21

DynamicsLagrangian 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:

)(2

1)(

2

1

2

1 2222

21

2lrllleeepp vvmvvmvmL

))((2

1)(

2

1 2222lrllZZsZZuZZlurulu IIIvvm

DynamicsJoint and link velocities

]))cos()(cos())sin()sin([( 22211

22211

220 lvp

])2/)cos()(cos()2/)sin()sin([( 22211

22211

220 lvurc

]))cos(2/)(cos())sin(2/)sin([( 22211

22211

220 lvulc

21

221

0 lve 22

222

0 lve 21

220

4l

vlrc 22

220

4l

vllc

100 ullr 2

00 urll

Dynamics Lagrangian in terms of θ1 and θ2

)(

)(8

1)

3

5

3

1(

)(2

1)cos(

2

222

22

212121

ep

mrotoriosulep

mmlQ

IddmmmmmlR

RQL

DynamicsEquations of Motion

2

1

21

22

21

2

1

21

21

21212

1212

122122211

)sin()cos(

)cos(

))cos()sin((

))cos()sin((

QRQ

QR

QR

QR

LL

dt

di

ii

• Equations of motion:

• Control Law:

ControlPartitioned Controller I

),()( BM

21

22

21

21

21

)sin(),(

)cos(

)cos()(

QB

RQ

QRM

' ),(),( BM

EKEK pvd ' dE

ControlPartitioned Controller II

• System simplifies to:

• The controller will act in a critically damped when:

0

),()(),()(

EKEKE

BEKEKMBM

pv

pvd

2

1

0

0

v

vv k

kK

2

1

0

0

p

pp k

kK

2,1;2 ikk pivi

ControlBlock Diagram

)(M System

pK vK

d '

d

d

),( B

E E

+

+

+ +

++ +

-

-

SimulationDescription

• 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: cml gm., Q gm.R 23 ,531375 22

SimulationScreen Capture

ConclusionWhere 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.

ReferencesThe 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.