This paper is a study of the duality between the statics of a variety of structures and the kinematics of mechanisms. To provide insight into this duality, two new graph representations are introduced; namely, the flow line graph representation and the potential line graph representation. The paper also discusses the duality that exists between these two representations. Then the duality between a static pillar system and a planar linkage is investigated by using the flow line graph representation for the pillar system and the potential line graph representation for the linkage. A compound planetary gear train is shown to be dual to the special case of a statically determinate beam and the duality between a serial robot and a platform-type robot, such as the Stewart platform, is explained. To show that the approach presented here can also be applied to more general robotic manipulators, the paper includes a two-platform robot and the dual spatial linkage. The dual transformation is then used to check the stability of a static system and the stationary, or locked, positions of a linkage. The paper shows that two novel platform systems, comprised of concentric spherical platforms inter-connected by rigid rods, are dual to a spherical six-bar linkage. The dual transformation, as presented in this paper, does not require the formulation and solution of the governing equations of the system under investigation. This is an original contribution to the literature and provides an alternative technique to the synthesis of structures and mechanisms. To simplify the design process, the synthesis problem can be transformed from the given system to the dual system in a straightforward manner.

1.
Girvin
,
S. M.
, 1996, “
Duality in Perspective
,”
Science
0036-8075, New Series, Genome Issue, October,
274
(
5287
), pp.
524
525
.
2.
Maxwell
,
J. C.
, 1864, “
On Reciprocal Figures and Diagrams of Forces
,”
Philos. Mag.
0031-8086, Series 4,
27
, pp.
250
261
.
3.
Davies
,
T.
, 1983, “
Mechanical Networks—III, Wrenches on Circuit Screws
,”
Mechanism and Machine Theory
,
Pergamon
, New York, Vol.
18
(2), pp.
107
112
.
4.
McGuire
,
W.
,
Gallagher
,
R. H.
, and
Ziemian
,
R. D.
, 2000,
Matrix Structural Analysis
, 2nd ed.,
Wiley
, New York.
5.
Phillips
,
J.
, 1984,
Freedom in Machinery, Vol. 1
,
Cambridge University Press
, Cambridge, England.
6.
Gosselin
,
F.
, and
Lallemand
,
J.-P.
, 2001, “
A New Insight into the Duality Between Serial and Parallel Non-Redundant and Redundant Manipulators
,”
Robotica
,
Cambridge University Press
, Cambridge, Vol.
19
(4), pp.
365
370
.
7.
Ball
,
R. S.
, 1998,
The Theory of Screws
,
Cambridge University Press
, Cambridge, England. (Originally published in 1876 and revised by the author in 1900, now reprinted with an introduction by H. Lipkin and J. Duffy).
8.
Collins
,
C. L.
, and
Long
,
G. L.
, 1995, “
On the Duality of Twist/Wrench Distributions in Serial and Parallel Chain Robot Manipulators
,”
IEEE International Conference on Robotics and Automation
, May, Vol.
1
, pp.
526
531
.
9.
Mohamed
,
M. G.
, and
Duffy
,
J.
, 1985, “
A Direct Determination of the Instantaneous Kinematics of Fully Parallel Robot Manipulators
,”
ASME J. Mech., Transm., Autom. Des.
0738-0666, Vol.
107
(
2
), pp.
226
229
.
10.
Duffy
,
J.
, 1996,
Statics and Kinematics with Applications to Robotics
,
Cambridge University Press
, Cambridge, England.
11.
Waldron
,
K. J.
, and
Hunt
,
K. H.
, 1991, “
Series-Parallel Dualities in Actively Coordinated Mechanisms
,”
The International Journal of Robotics Research
,
The MIT Press
, October, Vol.
10
(5), pp.
473
480
.
12.
Davidson
,
J. K.
, and
Hunt
,
K. H.
, 2004,
Robots and Screw Theory: Applications of Kinematics and Statics to Robotics
,
Oxford University Press
, New York.
13.
Swamy
,
M. N. S.
, and
Thulasiraman
,
K.
, 1981,
Graphs, Networks, and Algorithms
,
Wiley
, New York.
14.
Shai
,
O.
, 2001, “
The Duality Relation Between Mechanisms and Trusses
,”
Mechanism and Machine Theory
,
Pergamon
, New York,
36
(3), pp.
343
369
.
15.
Shai
,
O.
, and
Mohr
,
Y.
, 2004, “
Towards Transferring Engineering Knowledge Through Graph Representations: Transferring Willis Method to Mechanisms and Trusses
,”
Eng. Comput.
0264-4401,
20
(
1
), pp.
2
10
.
16.
Shai
,
O.
, 2001, “
The Multidisciplinary Combinatorial Approach and its Applications in Engineering
,”
AIEDAM—AI for Engineering Design, Analysis and Manufacturing
, Vol.
15
(
2
), pp.
109
144
.
17.
Shai
,
O.
, 2002, “
Utilization of the Dualism Between Determinate Trusses and Mechanisms
,”
Mechanism and Machine Theory
,
Pergamon
, New York, Vol.
37
(11), pp.
1307
1323
.
18.
Shai
,
O.
, and
Rubin
,
D.
, 2003, “
Representing and Analyzing Integrated Engineering Systems Through Combinatorial Representations
,”
Eng. Comput.
0264-4401,
19
(
4
), pp.
221
232
.
19.
Shai
,
O.
, and
Pennock
,
G. R.
, 2005, “
A Study of the Duality Between Planar Kinematics and Statics
,”
ASME J. Mech. Des.
(in press).
20.
Shai
,
O.
, and
Polansky
,
I.
, 2004, “
Finding Dead Point Positions of Linkages Through Graph Theoretical Duality Principle
,”
ASME J. Mech. Des.
(in press).
21.
Uicker
,
J. J.
, Jr.
,
Pennock
,
G. R.
, and
Shigley
,
J. E.
, 2003,
Theory of Machines and Mechanisms
, 3rd ed.,
Oxford University Press, Inc.
, New York.
22.
Dobrjanskyj
,
L.
, and
Freudenstein
,
F.
, 1967, “
Some Applications of Graph Theory to the Structural Analysis of Mechanisms
,”
ASME J. Eng. Ind.
0022-0817,
89
, pp.
153
158
.
23.
Stewart
,
D.
, 1965, “
A Platform with Six Degrees of Freedom
,”
The Institution of Mechanical Engineers
, Great Britain, Vol.
180
(
15
), pp.
371
384
.
24.
Yan
,
H.-S.
, and
Wu
,
L.-L.
, 1989, “
On The Dead-Center Positions of Planar Linkage Mechanisms
,”
ASME J. Mech., Transm., Autom. Des.
0738-0666,
111
(
1
), pp.
40
46
.
25.
Merlet
,
J. P.
, 1989, “
Singular Configurations of Parallel Manipulators and Grassmann Geometry
,”
The International Journal of Robotics Research
,
The MIT Press
, Vol.
8
(5), pp.
45
56
.
26.
Waldron
,
K. J.
, and
Kinzel
,
G. L.
, 2004,
Kinematics, Dynamics, and Design of Machinery
, 2nd ed.,
Wiley
, New York.
You do not currently have access to this content.