Design Of One-DOF Rigid And Flat Foldable Cubes

Rigid origami is a branch of origami with great potentials in engineering applications to deal with rigid-panel folding.One of the challenges is to compactly fold the polyhedrons made from rigid facets with a single degree of freedom.Yet,there is little work on how to fold polyhedral structure compactly into a flat configuration.Mathematicians proved that it is impossible to fold a sealed polyhedron rigidly.Hence,most of the existing researches focus on rigid folding schemes for the cubical structure without top facet.There has been no solution that makes origami cubes with all six facets satisfy rigid foldability,flat foldability and one degree of freedom(DOF)at the same time.Therefore,this thesis will explore the possibility to solve this challenge by introducing one diagonal cutting on the top facet of the cube.Firstly,a modified rigid and flat foldable crease pattern has been proposed by adding two extra diagonal creases and one diagonal incision on the basis of the well-known non-rigid but flat foldable crease pattern.Based on the equivalence between the rigid origami and the spherical linkages,the origami cube with this crease pattern forms the 4R-5R-4R-5R spherical linkage loop at four vertices on the bottom facet.Actually,the spherical linkage loop composed of the bottom and four side facets that drives the origami cube to realise the folding performance with one degree of freedom.It should be noticed that the cube’s top facet with a cutting works as the follow-up structure,whose detailed kinematic analysis is not carried out.Moreover,in order to explore all the possible schemes to realise the rigid and flat folding performance with one DOF,there are sixteen(=2~4)possible distributions of the diagonal creases on the bottom and four side facets.Considering the rotational and flipping symmetry in effective distributions,three other effective crease patterns are proposed,which can be equivalent to 4R-4R-5R-5R spherical linkage loop,4R-4R-4R-6R spherical linkage loop and 5R-4R-5R-4R spherical linkage loop,respectively.Due to the different assemblies of spherical linkages,each cube has its own folding performance and symmetric properties.Finally,three-dimensional software models and card physical models have been fabricated to demonstrate the folding process and validate the study.In addition to folding cubes,the new method can be readily extended to the prism structures with symmetric quadrilateral bottom.For each rigid origami structure in this thesis,its foldability derives from the motion of the spherical linkage loop at the bottom,so the height of origami structure in each case can be optional without considering the top surface.In this thesis,through the innovative design and theoretical analysis of the crease patterns,the cubical or prismatic origami structures not only have the rigid and flat foldability,but also can realize the single degree of freedom folding and unfolding movement,simplifying the control needed for the configuration transformation.The results are suitable for variable rigid materials,which can be readily utilized to design deployable structures for various engineering applications including building,small satellites,cube-shaped cartons,containers,etc.