Extension of Graph Theory to the Duality Between Static Systems and Mechanisms

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 behveen 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]  Jean-Pierre Merlet.  Singular Configurations of Parallel Manipulators and Grassmann Geometry , 1989, Int. J. Robotics Res..

[2]  Offer Shai,et al.  Representing and analysing integrated engineering systems through combinatorial representations , 2004, Engineering with Computers.

[3]  H. Saunders,et al.  Matrix Structural Analysis , 1979 .

[4]  Hong-Sen Yan,et al.  On the Dead-Center Positions of Planar Linkage Mechanisms , 1988 .

[5]  Gordon R. Pennock,et al.  Theory of Machines and Mechanisms , 1965 .

[6]  Kenneth J. Waldron,et al.  Series-Parallel Dualities in Actively Coordinated Mechanisms , 1988, Int. J. Robotics Res..

[7]  Offer Shai,et al.  A Study of the Duality Between Planar Kinematics and Statics , 2006 .

[8]  J. Maxwell,et al.  XLV. On reciprocal figures and diagrams of forces , 1864 .

[9]  T. H Davies.  Mechanical networks—III Wrenches on circuit screws , 1983 .

[10]  Offer Shai,et al.  The Duality Between Planar Kinematics and Statics , 2005 .

[11]  Offer Shai,et al.  Utilization of the dualism between determinate trusses and mechanisms , 2002 .

[12]  Offer Shai,et al.  The duality relation between mechanisms and trusses , 2001 .

[13]  Kenneth J. Waldron,et al.  Kinematics, dynamics, and design of machinery , 1998 .

[14]  Joseph Duffy,et al.  A Direct Determination of the Instantaneous Kinematics of Fully Parallel Robot Manipulators , 1985 .

[15]  Seth Earl Elliott,et al.  The theory of screws , 2022 .

[16]  Steven M. Girvin,et al.  Duality in Perspective , 1996, Science.

[17]  Jean-Paul Lallemand,et al.  A new insight into the duality between serial and parallel non-redundant and redundant manipulators , 2001, Robotica.

[18]  Joseph Duffy,et al.  Statics and Kinematics with Applications to Robotics , 1996 .

[19]  Ferdinand Freudenstein,et al.  Some Applications of Graph Theory to the Structural Analysis of Mechanisms , 1967 .

[20]  Offer Shai,et al.  Transforming engineering knowledge through graph representations: transferring the Willis method to linkages and trusses , 2004, Engineering with Computers.

[21]  D. Stewart.  A Platform with Six Degrees of Freedom , 1965 .

[22]  Joseph K. Davidson,et al.  Robots and Screw Theory: Applications of Kinematics and Statics to Robotics , 2004 .

[23]  Curtis L. Collins,et al.  On the duality of twist/wrench distributions in serial and parallel chain robot manipulators , 1995, Proceedings of 1995 IEEE International Conference on Robotics and Automation.

[24]  Offer Shai,et al.  Finding Dead-Point Positions of Planar Pin-Connected Linkages Through Graph Theoretical Duality Principle , 2006 .

[25]  Offer Shai,et al.  The multidisciplinary combinatorial approach (MCA) and its applications in engineering , 2001, Artificial Intelligence for Engineering Design, Analysis and Manufacturing.

[26]  Dieter Jungnickel,et al.  Graphs, Networks, and Algorithms , 1980 .