Geometric constraints and motion branch variations for reconfiguration of single-loop linkages with mobility one

This paper explores essence of geometric constraints induced reconfiguration of single-loop kinematic chains with mobility one. Reconfigurable kinematic chains are firstly classified into four categories in light of variations of parameters in the Chebychev-Grübler-Kutzbach mobility criterion. Within these four categories, single-loop kinematic chains with mobility one but distinct motion branches are further classified in accordance with degeneration of degree-of-freedom of certain revolute joints. With the essence of reconfiguration, the interrelationship of motion-branch changes through constraint singularity induced transitory positions of reconfigurable single-loop linkages is revealed in the context of reciprocity of screws. Four basic geometric constraints leading to transitory positions are explored by analysing the Bennett plano-spherical linkage, a kinematic embodiment of Grassmann varieties. Geometric constraints induced screw-system variation and motion branch changes of a novel asymmetric 7R linkage and the line- and plane-symmetric Bricard 6R linkage with capability of reconfiguring their motion branches are subsequently analysed for interpreting the fundamentals explored.