The Pantograph - Sushi Suzuki

doutfanaticalMechanics

Nov 14, 2013 (3 years and 5 months ago)

55 views

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
























)
sin(
)
sin(
)
cos(
)
cos(
2
1
2
1
0




l
y
x
P
x

y


Transformation equation:





Limitations:




All lengths =
l

l

1

2

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

2

80
10
2



80
10
1



Forward Kinematics


The Jacobian and singularities


Jacobian Matrix





The Jacobian is not invertible when its determinant equals 0






Singularities occur when











)
cos(
)
cos(
)
sin(
)
sin(
2
1
2
1
0




l
J
P
)
tan(
)
tan(
)
sin(
)
cos(
)
cos(
)
sin(
0
)
sin(
)
cos(
)
cos(
)
sin(
)
det(
2
1
2
1
2
1
2
1
2
1
0


















p
J



n



2
1
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:


)
(
2
1
)
(
2
1
2
1
2
2
2
2
2
1
2
lr
ll
l
e
e
e
p
p
v
v
m
v
v
m
v
m
L





)
)(
(
2
1
)
(
2
1
2
2
2
2
lr
ll
ZZs
ZZu
ZZl
ur
ul
u
I
I
I
v
v
m








Dynamics


Joint and link velocities

]
)
)
cos(
)
(cos(
)
)
sin(
)
sin(
[(
2
2
2
1
1
2
2
2
1
1
2
2
0

















l
v
p
]
)
2
/
)
cos(
)
(cos(
)
2
/
)
sin(
)
sin(
[(
2
2
2
1
1
2
2
2
1
1
2
2
0

















l
v
urc
]
)
)
cos(
2
/
)
(cos(
)
)
sin(
2
/
)
sin(
[(
2
2
2
1
1
2
2
2
1
1
2
2
0

















l
v
ulc
2
1
2
2
1
0


l
v
e

2
2
2
2
2
0


l
v
e

2
1
2
2
0
4


l
v
lrc

2
2
2
2
0
4


l
v
llc

1
0
0






ul
lr
2
0
0







ur
ll
Dynamics



Lagrangian in terms of
θ
1

and
θ
2

)
(
)
(
8
1
)
3
5
3
1
(
)
(
2
1
)
cos(
2
2
2
2
2
2
2
1
2
1
2
1
e
p
mrotor
i
o
s
u
l
e
p
m
m
l
Q
I
d
d
m
m
m
m
m
l
R
R
Q
L























Dynamics


Equations of Motion
































































2
1
2
1
2
2
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
2
1
2
2
2
1
1
)
sin(
)
cos(
)
cos(
)
)
cos(
)
sin(
(
)
)
cos(
)
sin(
(
















































Q
R
Q
Q
R
Q
R
Q
R
L
L
dt
d
i
i
i

Equations of motion:







Control Law:

Control


Partitioned Controller I








)
,
(
)
(



B
M

























2
1
2
2
2
1
2
1
2
1
)
sin(
)
,
(
)
cos(
)
cos(
)
(











Q
B
R
Q
Q
R
M






'
)
,
(
),
(






B
M


E
K
E
K
p
v
d








'




d
E
Control


Partitioned Controller II


System simplifies to:








The controller will act in a critically damped when:






0
)
,
(
)
(
)
,
(
)
(
















E
K
E
K
E
B
E
K
E
K
M
B
M
p
v
p
v
d



















2
1
0
0
v
v
v
k
k
K









2
1
0
0
p
p
p
k
k
K
2
,
1
;
2


i
k
k
pi
vi
Control


Block Diagram

)
(

M


System

p
K
v
K
d




'



d


d

)
,
(



B



E

E
+

+

+

+

+

+

+

-

-

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:

cm
l
gm
.
, Q
gm
.
R
23

,
53
1
37
5
2
2



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.
OpenGL
Programming Guide.

3
rd

ed. Addison
-
Wesley Publishing
Company, 1999.