The current research of the construction of reconfigurable deployable polyhedral mechanisms (RDPMs) has predominantly focused on regular and Archimedean polyhedra. However, the construction methods for RDPMs based on non-regular polyhedra remain largely unexplored. This paper presents a novel method to construct RDPMs with the circumscribed sphere. First, the mathematical constraints for the polyhedra based on circumscribed spheres are proposed, including single, double, and multiple circumscribed circles. By utilizing the reconfigurable properties of angulated elements, a series of reconfigurable angulated elements (RAEs) with different included angles are designed. A general method for constructing RDPMs is proposed by inserting RAEs into the vertex axes and the face axes of non-regular polyhedra with circumscribed spheres. Two RDPMs are constructed, which can achieve Hoberman sphere motion (HSM) and radially reciprocating motion (RRM). Next, the degree of freedom in HSM configuration is determined by using screw theory. Finally, RDPMs with a large ratio are constructed by inserting straight elements into RAEs. A prototype is manufactured to verify the feasibility of the proposed method. This research proposes a generalized method for constructing RDPMs with the circumscribed sphere, thus significantly expanding the range of polyhedral bases applicable to RDPMs.
Zhang et al. (Thu,) studied this question.