On Mobility Analysis of Linkages Using Group Theory

This paper presents a generalization of Kutzbach-Grubler criterion for mobility analysis of kinematic chains based on group theory. This new mobility criterion applies to exceptional linkages and reduces to a group theoretic representation of Kutzbach-Grubler criterion for trivial linkages. Furthermore, it is shown that using sets and groups of Euclidean displacements, one can distinguish between trivial, exceptional, and paradoxical linkages. Using these concepts, formal definitions of trivial, exceptional, and paradoxical linkages are presented and it is shown that there are two classes of paradoxical linkages. This work builds upon and extends the work of Herve and Fanghella and Galletti in application of group theory to analysis of kinematic chains.

[1]  O. Bottema.  On Grubler's formulae for mechanisms , 1951 .

[2]  F. M. Dimentberg,et al.  A spatial four-link mechanism having two prismatic pairs , 1966 .

[3]  K. H. Hunt.  Prismatic pairs in spatial linkages , 1967 .

[4]  Kenneth J. Waldron,et al.  A study of overconstrained linkage geometry by solution of closure equations — Part II. Four-bar linkages with lower pair joints other than screw joints☆ , 1973 .

[5]  Kenneth J. Waldron.  A study of overconstrained linkage geometry by solution of closure equations — Part 1. Method of study , 1973 .

[6]  Kenneth J. Waldron,et al.  The C?H?C?H- linkage? , 1974 .

[7]  J. Hervé.  Analyse structurelle des mcanismes par groupe des dplacements , 1978 .

[8]  K. H. Hunt,et al.  Kinematic geometry of mechanisms , 1978 .

[9]  J.Eddie Baker.  The EHHH-linkage☆ , 1979 .

[10]  A Compendium of Line-Symmetric Four-Bars , 1979 .

[12]  Jorge Angeles,et al.  Spatial kinematic chains , 1982 .

[13]  J.Eddie Baker.  On completing the determination of existence criteria for overconstrained 4-bars with helical joints , 1982 .

[14]  Pietro Fanghella,et al.  Kinematics of spatial linkages by group algebra: A structure-based approach , 1988 .

[15]  Overconstrained five-bars with prismatic joints and parallel adjacent joint axes☆ , 1989 .

[16]  E. Kuznetsov.  Underconstrained structural systems , 1991 .

[17]  K Wohlhart.  Merging two general goldberg 5R linkages to obtain a new 6R space mechanism , 1991 .

[18]  Cemil Bagci.  Determining General and Overclosing Constraints in Mechanism Mobility Using Structural Finite Element Joint Freedoms , 1992 .

[19]  Karl Wohlhart,et al.  The two types of the orthogonal bricard linkage , 1993 .

[20]  C. Galletti,et al.  Mobility analysis of single-loop kinematic chains: an algorithmic approach based on displacement groups , 1994 .

[21]  C. Galletti,et al.  Metric Relations and Displacement Groups in Mechanism and Robot Kinematics , 1995 .

[23]  J.Eddie Baker.  On 5-revolute kinematic loops with intersecting adjacent joint axes , 1996 .

[24]  B. Ravani,et al.  On Mobillity Analysis of Linkages Using Group Theory , 2002 .