A group-theoretic approach to the mobility and kinematic of symmetric over-constrained structures

Some symmetric structures can be finite mechanisms with at least one degree-of-freedom even though they are over-constrained. Exploring group representation theory, this study presents a group-theoretic approach to the mobility of symmetric over-constrained structures, which have both internal mechanism modes and self-stress states. To establish the compatibility equations for the structures, the constraint equations and elementary compatibility matrix for a general link member are given. Then, symmetry subspaces associated with different irreducible representations are established, which are important to derive the symmetry-adapted coordinate system. Using the Great orthogonality theorem, the original compatibility matrix is decomposed into a few independent block matrices along the diagonal. Investigation on the null-space and the left null-space of these blocks enables us to present the symmetry representations for the internal mechanisms and self-stress states. Mobility of some example structures is studied, including 3D C3v symmetric Bricard mechanisms, a 2D C4v symmetric retractable structure, and a 3D Ih symmetric polyhedral structure. The generality and effectiveness of this group-theoretic approach are validated. Kinematic behavior of the numerical models agrees with that of the physical models assembled by LEGO components. It turns out that all the illustrative structures can be transformable while retaining full symmetry.