Synthesis, Mobility, and Multifurcation of Deployable Polyhedral Mechanisms With Radially Reciprocating Motion

Abstract

Extending the method coined virtualcenterbased (VCB) for synthesizing a group of deployable platonic mechanisms with radially reciprocating motion by implanting dualplanesymmetric 8bar linkages into the platonic polyhedron bases, this paper proposes for the first time a more general singleplanesymmetric 8bar linkage and applies it together with the dualplanesymmetric 8bar linkage to the synthesis of a family of onedegree of freedom (DOF) highly overconstrained deployable polyhedral mechanisms (DPMs) with radially reciprocating motion. The two 8bar linkages are compared, and geometry and kinematics of the singleplanesymmetric 8bar linkage are investigated providing geometric constraints for synthesizing the DPMs. Based on synthesis of the regular DPMs, synthesis of semiregular and Johnson DPMs is implemented, which is illustrated by the synthesis and construction of a deployable rectangular prismatic mechanism and a truncated icosahedral (C60) mechanism. Geometric parameters and number synthesis of typical semiregular and Johnson DPMs based on the Archimedean polyhedrons, prisms and Johnson polyhedrons are presented. Further, movability of the mechanisms is evaluated using symmetryextended rule, and mobility of the mechanisms is verified with screwloop equation method; in addition, degree of overconstraint of the mechanisms is investigated by combining the Euler's formula for polyhedrons and the Grأ¼bler–Kutzbach formula for mobility analysis of linkages. Ultimately, singular configurations of the mechanisms are revealed and multifurcation of the DPMs is identified. The paper hence presents an intuitive and efficient approach for synthesizing PDMs that have great potential applications in the fields of architecture, manufacturing, robotics, space exploration, and molecule research.