Stray light from the sun is one of the most significant factors affecting image quality for the optical system of a spacecraft. This paper proposes a method to design a deployable supporting mechanism for the sunshield based on origami. Firstly, a new type of space mechanism with single-closed loop was proposed according to thick-panel origami, and its mobility was analysed by using the screw theory. In order to design a deployable structure with high controllability, the tetrahedral constraint was introduced to reduce the degree of freedom (DOF), and a corresponding deployable unit named tetrahedral deployable unit (TDU) was obtained. Secondly, the process to constructing a large space deployable mechanism with infinite number of units was explained based on the characteristics of motion and planar mosaic array, and kinematics analysis and folding ratio of supporting mechanism were conducted. A physical prototype was constructed to demonstrate the mobility and deployment of the supporting mechanism. Finally, based on the Lagrange method, a dynamic model of supporting mechanism was established, and the influence of the torsion spring parameters on the deployment process was analysed.