A general mobility analysis framework to reveal nonintuitive kinematic characteristics of rigid origami

Rigid origami exhibits intriguing reconfigurability governed by geometric constraints imposed by its creases. This intrinsic relationship gives rise to nonintuitive kinematic characteristics, including the number of degrees of freedom (DOFs) and their corresponding motion paths in complex configurations. Here, we develop a general mobility analysis framework to reveal the indeterminate kinematic characteristics of rigid origami. By first analytically deriving kinematic compatibility conditions up to the fourth order, we clarify the role of high-order compatibility analysis in determining rigid foldability. The results show that origami structures with infinitesimal mechanisms can be classified into three categories: regular origami, bifurcated origami, and shaky origami. These kinematic characteristics can be revealed by second-order analysis in most cases, with only a few cases requiring third-order analysis. Based on the analytical results, we then develop an angle-dominated simulation algorithm for tracing the finite motions of rigid origami, including single- and multi-DOF regular origami as well as bifurcated origami. The proposed framework is validated on 20 previously reported origami structures and 31 newly designed structures, including planar origami, polyhedral origami, and thick origami, confirming the generality of our approach. This work elucidates the nonintuitive kinematic behaviors of rigid origami by integrating analytical derivation, numerical simulation, and prototype validation, advancing the mobility analysis of rigid origami and other spatial linkages.