Rigid origami is an important subset of origami with broad engineering applications from space structures to metamaterials. The rigid foldability of an origami pattern is determined by both the geometric parameters and the mountain-valley crease assignment. In this paper, by using the equivalent relationships between origami vertices and spherical linkages, a systematic method was proposed to analyze the motion of the triangle-twist pattern with varying distribution of mountain and valley creases, and its rigid folding types were identified. Moreover, kirigami technology was applied to the rigid folding pattern without changing its degree of freedom, from which a new kind of overconstrained 6R linkage was developed. The theoretical method proposed in this paper can be readily extended to study other types of origami patterns, which will in turn help to design structures with large deployable ratio as well as some new spatial linkages.

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