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Design and Workspace Analysis of a 6-6 Cable-Suspended Parallel Robot Design and Workspace Analysis of a 6-6 Cable-Suspended Parallel Robot

Jason Pusey, Abbas Fattah and Sunil Agrawal
Elena Messina and Adam Jacoff
Department of Mechanical Engineering
Intelligent Systems Division
University of Delaware
National Institute of Standards and Technology
Newark, DE, 19716
Gaithersburg, MD 20899-8230
USA
pusey, fattah, agrawal@me.udel.edu
USA

Abstract

In this paper, we study the design and workspace
of a 6-6 cable-suspended parallel robot. The workspace volume is
characterized as the set of points where the centroid of the MP (MP)
can reach with tensions in all suspension cables at a constant
orientation. This paper attempts to tackle some aspects of optimal
design of a 6DOF cable robot by addressing the variations of the
workspace volume and the accuracy of the robot using different
geometric configurations, different sizes and orientations of the MP.
The global condition index is used as a performance index of a robot
with respect to the force and velocity transmission over the whole
workspace. The results are used for design analysis of the cable-robot
for a specific motion of the MP.


1. INTRODUCTION

Parallel robots generally have larger load capacities,
faster and more accurate motions and larger stiffness
throughout their workspace as compared to the serial ones
[6]. These attributes associated with the parallel robots
make them quite attractive in real world applications. The
increased popularity of parallel robots has opened many
questions.
One particular question that this paper attempts to
address is what is an optimal design? An optimal design
of a robot generally refers to one which has the largest
workspace, fewest singularities, large force output, high
stiffness throughout its workspace, high accuracy, etc.
Maximizing the workspace volume is one of the goals of an
optimal design. Performance indices are defined to measure
the different aspects that encompass an optimal design.
This paper focuses on a particular type of parallel robot,
namely the cable suspended parallel robot. Traditional
parallel robots are driven by prismatic actuators. Some
common prismatic actuators include hydraulic and
pneumatic pistons and motor driven lead screws. These
systems tend to be large, heavy, and cumbersome. Cable
suspended parallel robots are slightly different from the
traditional parallel robots. In a cable suspended parallel
robot, the moving platform (MP) is suspended and
manipulated by the attached cables that are connected to the
base.
Performance indices are defined to measure the different
aspects that encompass an optimal design. The
performance index is a term used when referring to
methods that compare the quality of one parallel robot
design to another. One common performance index used is
the condition number of the Jacobian matrix of the robot.
The condition number, further explained in section IV, is
used to determine how close the robot is, in its particular
position and orientation, to a singularity configuration.
Another commonly used performance index is the global
condition index (GCI), which is a measure of kinematic
dexterity of the robot over the whole workspace [5].
Some reported research on cable-suspended robots are
NIST Robocrane [1], ultra-high speed robot, tendon-driven
Stewart platforms, parallel wire mechanism for measuring a
robot pose, controller designs for cable-suspended robot
[2], and design and workspace analysis of planar cable-
suspended robots ([3], [4]). A three dimensional cable
suspended robot, the ROBOCRANE, was created by the
National Institute of Standard Technology (NIST). A
systematic analysis for the particular geometry selected in
the design of the cable robots by NIST, is still lacking.
This paper attempts to tackle some aspects of optimal
design of a 6DOF cable robot by addressing the variations
of the workspace volume and the accuracy of the robot
using different geometric configurations, different sizes and
orientations of MP. The workspace volume for the cable
robots is characterized by the set of points where the center
of mass of moving platform can be positioned while all
cables are in tensions.
The organization of the paper is as follows: Section II
explains the detailed kinematic and geometric modeling of
the 6-6 cable robot. Section III uses the modeling to obtain
the forces in the cables and defines the workspace for the
cable robot. Section IV deals with global condition index
(GCI), which are used to measure the quality of
performance of the robot over the whole workspace. Based
on the underlying modeling and workspace analysis,
workspace volume and GCI are simulated in Section V for
different geometry configurations, sizes of moving platform
and orientations. Section VI uses the results of Section V
to attain the best design of the cable-robot for a desired
motion of the moving platform.

II. KINEMATIC AND GEOMETRIC MODELING

The model used is that of a 6-6 cable suspended parallel
robot. The base points of the manipulator b
1
, , b
6
are all
contained within the same plane (z
O
= 0) as shown in Fig. 1.
The points are placed at some radial distance
base
r
from the
base coordinate system F
O
that is located at O the center of
the base platform (BP). The MP similarly has a set of
connection points a
1
, , a
6
located at a distance
end
r
from
the moving coordinate frame F
E
attached to O
E
the center
of mass of the MP. These points are located on the (z
E
= 0)
plane relative to frame F
E
.
Proceedings of the 2003 IEEE/RSJ
Intl. Conference on Intelligent Robots and Systems
Las Vegas, Nevada · October 2003
0-7803-7860-1/03/$17.00 © 2003 IEEE
2090 The position vector of point b
i
on the base is defined by

( )
( )
cos
sin
0
base
i
O
i
base
i
r
r
= b
(1)

where
base
r
is the radial distance from the base coordinate
frame F
O
. The variable
i denotes the angular location of
point b
i
on BP with respect to axis x
0
.
The position vectors of the connection points on the MP are
as follows:

( )
( )
cos
sin
0
end
i
E
i
end
i
r
r
= a
(2)
The variable
i denotes the angular location of each
connection point on the MP with respect to x
E
. The position
vector from point b
i
to point a
i
, i.e., the cable vector shown
in Fig. 2, can be expressed with respect to frame F
O
as

b
5

z
E


a
2

a
1

a
6

a
5

a
4

a
3

b
1

b
2

b
6

b
3

b
4

x
o


y
o


z
o


y
E


x
E


F
O


F
E


mg

O

O
E



Fig. 1 A schematic of a 6-6 cable robot under study



O
O
O
E
O
i
E
E
i
i
=
+ l
p
R
a
b
(3)

where
E
O
p
represents the position vector of point O
E
with
respect to O.
O
E
R
is the rotation matrix of MP with
respect to BP using a fixed axis rotation sequence of , , and about x
0
, y
0
, and z
0
axes, respectively. The
magnitude of each
O
i
l
vector is

2
2
2
2
O
i
i
ix
iy
iz
l
l
l
l
=
=
+
+
l

(4)
Using the time derivative of the kinematic constraint
equations, the relation between the cable velocity vector
q&

and the twist of MP, i.e., t can be related by the Jacobian
matrix as follows:

=
q Jt
&

(5)

where

1
2
3
4
5
6
T
l
l
l
l
l
l
= q
& & & & & &
&

(6)
T
O
T
O
T
E
E
= t
p &
(7)
Here,
T
E is the angular velocity vector of the MP with
respect to frame F
O
. The Jacobian matrix of the cable robot
is written such that its i
th
row is

T
T
O
O
O
i
i
i
i
i
i
l
l
=
×
l
l
j
a
(8)


z
E


E
a
i

O
b
i

x
o


y
o


z
o


y
E


x
E


F
O


F
E


O
l
i


O
p
E


b
i


a
i

O

O
E



Fig. 2 A sketch of cable i and position vectors of BP and
MP

III. WORKSPACE ANALYSIS

The static equilibrium of the cable suspended parallel
robot is used to find the force of each cable. Force and
moment balance on the MP is

(
)
6
1
6
1
E
i
i
O
O
i
i
i
m
=
=
= =
=
×
=
F
g
T
0
M
a T
0

(9)

where it can be seen that there are no moments applied to
MP and the only external force is the gravitational force.
All other external forces and moments are ignored. To
relate the external forces represented by (9) to the forces in
2091 the cables, the