# Designing of the Tracks and Modeling of the Carrier Arrangement in Square Rotary Braiding Machine
## 2 Track-Splicing Process
### The Splicing Process of 8 \(\times\) 8 484 Cross-Break Track  
The splicing schematic of the 8 \(\times\) 8 484 cross-break track is shown in Figure 3. It can be seen that the 8 \(\times\) 8 484 cross-break track is composed of four 4 \(\times\) 4484 cross-break track units: 4 \(\times\) 4484 cross-break (1), 4 \(\times\) 4484 cross-break (2), 4 \(\times\) 4484 cross-break (3), and 4 \(\times\) 4484 cross-break (4). As shown in Figure 3a, the numbers in brackets represent the order of track unit splicing, and 4 \(\times\) 4 indicates that the track consists of four rows and four columns, with the horn gear configuration, for a total of 16 (16 = 4 \(\times\) 4) horn gears. The number 484 represents that each track first includes a 4-slot horn gear, then connects a total of 8n-slots (8n = 4 \(\times\) 2n, where n is a positive integer) horn gear system composed of a 2n 4-slot horn gears and, finally, connects a 4-slot horn gear. \"Cross-break\" means that the trajectory of the track first follows the horizontal direction (cross), then the +45\({}^{\circ}\) or \(-45^{\circ}\) direction (break), as shown in Figure 3b. The meanings of ScP and MtP in Figure 3b are explained in Section 3.1. 1. Determine the CAC of the first to fourth 4 \(\times\) 4 track units: \[M_{\text{b}_{i}}=\left[ \begin{array}{c}f^{4}\\ e^{4}\end{array} \right]\cup\left[ \begin{array}{c}f^{4}\\ e^{4}\end{array} \right]\cup\left[ \begin{array}{c}f^{4}\\ e^{4}\end{array} \right]\cup\left[ \begin{array}{c}f^{4}\\ e^{4}\end{array} \right]=\left[ \begin{array}{c}f^{4}\\ e^{4}\end{array} \right],\;\text{i}=1,2,3,4\]
2. Determine the CAC carrier arrangement number in the first sub-track, composed of track unit (1) and track unit (2): \[\xi_{L_{1}}=\text{dim. }\bigg{\{}\left[ \begin{array}{c}f^{4}\\ e^{4}\end{array} \right]\cup\left[ \begin{array}{c}f^{1}_{1}\\ e^{1}\end{array} \right]\cup\left[ \begin{array}{c}f^{4}\\ e^{4}\end{array} \right]\bigg{\}}=3\]
3. Determine the CAC of track unit (1) and track unit (2): \[M_{TR_{1-2}}=M_{\text{b}_{1}}\cap M_{\text{b}_{2}}=\left[ \begin{array}{c}f ^{4}\\ e^{4}\end{array} \right]\cap\left[ \begin{array}{c}f^{4}\\ e^{4}\end{array} \right]=\left[ \begin{array}{c}f^{4}\\ e^{4}\end{array} \right]\]
4. Determine the CAC carrier arrangement number in the second sub-track, consisting of track units (1), (2), and (3): \[\xi_{L_{2}}=\text{dim. }\bigg{\{}\left[ \begin{array}{c}f^{4}\\ e^{4}\end{array} \right]\cup\left[ \begin{array}{c}f^{1}_{1}\\ e^{1}\end{array} \right]\cup\left[ \begin{array}{c}f^{4}\\ e^{4}\end{array} \right]\cup\left[ \begin{array}{c}f^{4}\\ e^{4}\end{array} \right]\bigg{\}}=4\]
5. Determine the CAC of track unit (1), track unit (2), and track unit (3): \[M_{TR_{1-3}}=M_{TR_{1-2}}\cap M_{\text{b}_{3}}=\left[ \begin{array}{c}f^{4} \\ e^{4}\end{array} \right]\cap\left[ \begin{array}{c}f^{4}\\ e^{4}\end{array} \right]=\left[ \begin{array}{c}f^{4}\\ e^{4}\end{array} \right]\]
6. Determine the CAC number in the third sub-track, which consists of track units (1), (2), (3), and (4): \[\xi_{L_{3}}=\text{dim. }\bigg{\{}\left[ \begin{array}{c}f^{4}\\ e^{4}\end{array} \right]\cup\left[ \begin{array}{c}f^{1}\\ e^{1}\end{array} \right]\cup\left[ \begin{array}{c}f^{4}\\ e^{4}\end{array} \right]\cup\left[ \begin{array}{c}f^{4}\\ e^{4}\end{array} \right]\cup\left[ \begin{array}{c}f^{1}_{1}\\ e^{1}\end{array} \right]\cup\left[ \begin{array}{c}f^{4}\\ e^{4}\end{array} \right]\bigg{\}}=6\]  
Figure 2: Diagram of the mechanism of track splicing. (7) Determine the arrangement period of the carrier in the 8 \(\times\) 8 484 cross-break track:  
\[T^{*}_{{}_{TR}}=\sum_{i=1}^{4}f_{i}-\sum_{j=1}^{3}\xi_{L_{j}}=21-(3+4+6)=8\]  
Therefore, the arrangement period of the carrier in the 8 \(\times\) 8 484 cross-break track is 8, and its CAC set is \(\left[ \begin{array}{cc}f^{4}&e^{4}\end{array} \right]^{T}\). However, in each 484 cross-break track units obtained by applying the method in [24], it is not possible to determine which of the eight adjacent slots has a carrier and which do not.