The Pantograph
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Transcript of The Pantograph
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.