A Technique for Deriving Equational Conditions on the Denavit-Hartenberg Parameters of 6R Linkages that are Necessary for Movability
This work addresses a foundational problem in robotics and mechanism design by providing necessary conditions for linkage mobility, though it is incremental as it builds on existing bond theory.
The paper tackles the problem of identifying necessary conditions for mobility in closed 6R linkages, which are typically rigid, by deriving equational conditions on Denavit-Hartenberg parameters for the first time, resulting in the discovery of a new mobile linkage.
A closed 6R linkage is generically rigid. Special cases may be mobile. Many families of mobile 6R linkages have been characterised in terms of the invariant Denavit-Hartenberg parameters of the linkage. In other words, many sufficient conditions for mobility are known. In this paper we give, for the first time, equational conditions on the invariant Denavit-Hartenberg parameters that are necessary for mobility. The method is based on the theory of bonds. We illustrate the method by deriving the equational conditions for various well-known linkages (Bricard's line symmetric linkage, Hooke's linkage, Dietmaier's linkage, and recent a generalization of Bricard's orthogonal linkage), starting from their bond diagrams; and by deriving the equations for another bond diagram, thereby discovering a new mobile 6R linkage.