# The Pantograph - Sushi Suzuki

Mechanics

Nov 14, 2013 (4 years and 6 months ago)

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The Pantograph

by Kevin Bowen and Sushi Suzuki

Introduction

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)

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

]
)
)
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

-
Wesley Publishing Company, 1989.

Woo, M., Neider, J., Davis, T., and Shreiner, D.
OpenGL
Programming Guide.

3
rd