# Designing of the Tracks and Modeling of the Carrier Arrangement in Square Rotary Braiding Machine
## 2 Track-Splicing Process  
### General Steps of Track Splicing  
In order to save space, the main steps of the track-splicing method using the CAC method are as follows:  
The carrier arrangement characteristics (CAC) method has been presented in [24]. It comprises the existence and arrangement characteristics of carriers in the horn gear slot in the tracks. It is a mathematical description of the carrier state in the slot of the track. The CAC is different from the \"number of carrier arrangement characteristics\", and the CAC needs to be calculated first. Then, the \"number for the carrier arrangement characteristics\" is calculated second. Figure 2 shows the splicing mechanism from the first track unit (1) to the final one (\(j+1\)).  
1. Determine the characteristics of the carrier arrangement in the 4 \(\times\) 4 track (\(M_{b_{j}}\)). This step can determine the existence of the carriers in the 4 \(\times\) 4 track units, indicated by 1, 2, 3,..., (\(j+1\)) in Figure 2, in the arrangement period. The characteristics of the carrier arrangement \(M_{b_{j}}\) represent the CAC set of carriers that do not collide with each other in the tracks, while the elements in \(M_{b_{j}}\) only represent the existence of the carriers in the slots in an arrangement period of the carrier.
2. Determine the characteristic number of carrier arrangement for track units 1, 2. This step can determine the number of carriers at the slots in the track composed of the first two track units (1 and 2 in Figure 2), called dim. [M] and the dim. [M] is the number of elements in the CAC, that is, the dimension of the CAC. There is no collision constraint between the carriers, and it is independent of the track unit number. \[\xi_{L_{1}}=\text{dim.}\big{\{}M_{b_{1}}\cup M_{b_{2}}\big{\}}\] 3. Determine the carrier arrangement characteristics of the first sub-track, composed of the first and second track units. This step can determine the existence of the carriers in the first sub-track composed of the first track unit \"1\" and the second track unit \"2\", as shown in Figure 2. \[M_{\text{pa}(1-2)}=M_{b_{1}}\cap M_{b_{2}}\] 4. Determine the characteristic carrier arrangement number of the first \(j\)th sub-track, composed of \(j\) track units ranging from the first track unit \"1\" to the \(j\)th track unit (\(j\)): \[\xi_{L_{j}}=\text{dim.}\big{\{}M_{\text{Pa}(1-j)}\cup M_{b(j+1)}\big{\}}\] 5. Determine the carrier arrangement characteristics of the track spliced by the first (\(j+1\)) track units, where the existence of the carriers in the track units--from the first unit \"1\" to the final one (\(j+1\)) in Figure 2--in the arrangement period is obtained: \[M_{\text{pa}(1-(j+1))}=M_{\text{Pa}(1-j)}\cap M_{b(j+1)}\] 6. Determine the arrangement period (T) of the carriers in the spliced track unit. \(f_{i}\) is the summation of the \(i\)th track unit, m is the number of track units, and \(v\) is the number of track loops. \(\xi_{L_{j}}\) indicates the number of carriers and the arrangement characteristics of the \(j\)th track unit (formed by the equivalent track units that are formed by the first \(j\) track units and the (\(j+1\))th track unit shown in Figure 2). \[T=\sum_{i=1}^{m}f_{i}-\sum_{j}^{v}\xi_{L_{j}}\]