# Design and Analysis of Yoshimura Tubular Origami Mechanisms
## 3 The Topological Matrix Representation Method for Yoshimura Origami Configurations  
### Description of the Crease Matrix  
In the Yoshimura origami planar creases, there are essentially three types of crease shape elements, as shown in Figure 10. The base length \(l\) and width \(w\) are used to describe the characteristic information of each crease shape element. In addition to the standard crease shape element, an inclination angle \(\theta\) is introduced to describe the degree of deviation for the left- and right-biased crease shape elements. As analyzed in Section 3.1, the specific left and right declination angle values are expressed as the magnitude of the angle between the line connecting the vertex of the mountain line to the midpoint of the base and the midsagittal line of the base. It should be noted that, as shown in Figure 10, the base line can correspond to either a boundary line or a valley line.  
Thus, the symbol \(\mu_{l}^{w}\) is used to represent the standard crease shape element, \(\sideset{}{4}{v}{l}{w}{l}\), for the left-biased crease shape element, and \(\sideset{}{4}{v}{l}{w}{l}\) for the right-biased crease shape element. In these symbols, the subscripts \(l\), \(w\), and \(\theta\) are used to describe the basic characteristic information, including the base length, width, and the left or right inclination angle of the crease shape element.  
Figure 8: Right-biased homogeneous tubular origami mechanism.  
Figure 9: Sibling–heterotrophic integrated tubular origami mechanism.  
With the symbol representation method for the crease shape elements mentioned above, we can now establish a crease matrix representation method for Yoshimura origami planar creases. The crease matrix is denoted by \(\mathbf{M}\), as shown in Equation (1). The following are some explanations regarding the crease matrix \(\mathbf{M}\):  
\[\mathbf{M}=\begin{bmatrix}M_{1,1}&M_{1,2}&\cdots&M_{1,j}\\ M_{2,1}&M_{2,2}&\cdots&M_{2,j}\\ \vdots&\vdots&\ddots&\vdots\\ M_{i,1}&M_{i,2}&\cdots&M_{i,j}\end{bmatrix} \tag{1}\]  
As illustrated by the specific meaning of the crease matrix in Figure 11, each element within the crease matrix \(\mathbf{M}\) represents the following:  
(1) The number of rows \(i\) in the crease matrix \(\mathbf{M}\) represents the number of layers in the Yoshimura origami planar creases, while the number of columns \(j\) represents the number of complete common edges, i.e., boundary lines or valley lines, in each layer of planar creases.  
(2) Each element in the rows of the crease matrix \(\mathbf{M}\) represents a complete crease shape element within each layer of the planar creases.  
(3) Each element in the crease matrix \(\mathbf{M}\) can represent different types of crease shape elements, such as the standard crease shape element \(u_{l}^{w}\), the left-biased crease shape element \(\prescript{\wedge,d}v_{l}^{w}\), and the right-biased crease shape element \(\prescript{\wedge d}o_{l}^{w}\).  
Figure 11: Specific meaning of the crease matrix.  
Figure 10: Description of crease shape elements.  
It is important to note that the connection between the layers of planar creases will remain intact only if the length of the boundary lines or valley lines corresponding to each crease shape element is equal. Therefore, it is assumed that the complete common edge length within each layer of planar creases must be equal.  
In this section, we introduce the concept of a topological matrix for origami, which enhances our ability to represent and design the spatial form of the Yoshimura origami mechanism. Once the relationship between the plane creases and spatial forms is established, the origami configuration's topological matrix serves as a more intuitive mathematical tool for representing the designed origami mechanism.