# Design and Analysis of Yoshimura Tubular Origami Mechanisms
## 3 The Topological Matrix Representation Method for Yoshimura Origami Configurations
### Description of the Morphological Matrix  
In the Yoshimura origami spatial structure, through analysis, there are three types of cells that exist through analysis, standard cells, left-biased cells, and right-biased cells, each composed of corresponding cell elements. The internal constraints of the crease pattern elements corresponding to the cells can result in either a fully overlapped or non-overlapped spatial configuration, which leads to different characteristic information being conveyed even by the same type of cell. In this context, \"fully overlapped\" refers to the phenomenon where the planes at both axial ends of the cylindrical state can approximately coincide, which is referred to as the fully overlapped spatial configuration of the Yoshimura origami structure. Conversely, the non-overlapped configuration is the opposite of this. Additionally, there may be various cell elements within the cells. Describing the spatial configuration information of such mixed-type cells can be more complex. Given these situations mentioned above, further decomposition for each type of cell is necessary so that resultant cell elements can describe their own spatial configuration information more comprehensively. By decomposing the cells, the characteristic information of each cell element can be described using height \(h\). In addition to the standard cell elements, an additional torsion angle \(n\) is required to describe the axial torsion of the left-biased and right-biased cell elements. As shown in Figure 12, the standard cell element can be denoted by the symbol \(U\). The fully overlapped standard cell element is denoted by \(U_{1}\), while the non-overlapped standard cell element, which has a certain spatial height, as previously analyzed, is denoted by \(U_{2}^{h}\).Unlike the standard cell element, the left-biased and right-biased cell elements may not only have spatial height but also a certain axial torsion angle. For the left-biased and right-biased cell elements, the skew angle value of the crease pattern element corresponds to the axial torsion angle in the origami spatial configuration. The determination of the torsion angle and its direction is further explained as follows:  
As shown in Figure 13, by taking a cross-sectional view of a cell element from the origami spatial configuration, the left-biased and right-biased cell elements are derived from the standard cell element. Therefore, a hypothetical cross-section corresponding to the  
Figure 12: Standard cell element. standard cell element can be established, represented by the black dashed triangular line. The blue dashed triangular line represents the actual position of the cell element formed after derivation from the standard cell element. The angle formed during the derivation process is defined as the axial torsion angle \(n\) of the left-biased and right-biased cell elements. The direction is determined as follows: the left-biased cell element corresponds to a counterclockwise axial torsion angle, while the right-biased cell element corresponds to a clockwise axial torsion angle. As shown in Figure 14, the left-biased cell element and the right-biased cell element are represented by the symbols \(V\) and \(O\), respectively. For the left-biased cell element \(V\), \({}^{\wedge,n}V_{1}\) represents the fully overlapped left-biased cell element with a counterclockwise axial torsion angle, and \({}^{\wedge,n}V_{2}^{h}\) represents the non-overlapped left-biased cell element with a counterclockwise axial torsion angle. For the right-biased cell element \(O\), \({}^{\wedge n}O_{1}\) represents the fully overlapped right-biased cell element with a clockwise axial torsion angle, and \({}^{\wedge n}O_{2}^{h}\) represents the non-overlapped right-biased cell element with a clockwise axial torsion angle. With the symbolic representation of the cell elements established above, we can now develop the method for describing the morphological matrix of the Yoshimura origami spatial structure.