Structural Design and Kinematic Analysis of Thick-Panel Origami Mechanism with Third-Order Magic Square Topology and Flat Deployed Surface

Abstract

This paper proposes a connection-based design method for thick-panel origami mechanisms with third-order magic square topology. A two-vertex seven-crease mechanism is introduced as the basic module, and crease assignment matrices and module-splicing conditions are established to describe the mountain–valley assignments and boundary-connection relationships. By selecting the configuration with the minimum row-major code as the canonical representative among equivalent configurations under rotation, turnover, and mirror reflection, 23 nonredundant configurations are obtained. Based on the module-splicing relationships, kinematic compatibility criteria are established, identifying 19 rigidly foldable configurations. Surface-flattening methods are developed for 3M1V and 3V1M vertices by constructing an equivalent spatial 6R mechanism and employing double-parallelogram mechanism to eliminate the valley-fold constraints on the working surface, respectively. Mechanism No. 8 is selected for modular-expansion design because all valley-fold constraints on the working surface can be eliminated. Its geometric constraints and deployed/folded ratio are analyzed, and modular-expansion methods in the row and column directions are proposed. Simulations and prototype experiments verify the kinematic model and show that the expanded mechanism exhibits deterministic motion and deploys smoothly into a flat configuration without evident interference. The proposed method provides a systematic framework for topological reconstruction, rigid-foldability evaluation, surface flattening, and modular expansion of thick-panel origami deployable structures.