  Rigid foldability allows an origami pattern to fold about crease lines
without twisting or stretching component panels. It enables folding of rigid
materials, facilitating the design of foldable structures. Recent study shows
that rigid foldability is affected by the mountain-valley crease (M-V)
assignment of an origami pattern. In this paper, we investigate the rigid
foldability of the square-twist origami pattern with diverse M-V assignments by
a kinematic method based on the motion transmission path. Four types of
square-twist origami patterns are analyzed, among which two are found rigidly
foldable, while the other two are not. The explicit kinematic equations of the
rigid cases are derived based on the kinematic equivalence between the rigid
origami pattern and the closed-loop network of spherical 4R linkages. We also
propose a crease-addition method to convert the rigid foldability of the
non-rigid patterns. The motion compatibility conditions of the modified
patterns are checked, which verify the rigid foldability of the modified
patterns. The kinematic analysis reveals the bifurcation behaviour of the
modified patterns. This work not only helps to deepen our understanding on the
rigid foldability of origami patterns and its relationship with the M-V
assignments, but also provides us an effective way to create more rigidly
foldable origami patterns from non rigid ones.
