# Designing of the Tracks and Modeling of the Carrier Arrangement in Square Rotary Braiding Machine
## 3 Analysis of the Carrier Arrangement  
### Modeling of the Carrier Position  
In order to achieve full carrier arrangement in the slots of the track, as shown in Figure 3, the carriers are arranged in accordance with \"10101010\", and the arrangement period is eight, which leads to the collision problem. To ensure that the carriers function smoothly--that is, without collision--under the guidance of the designed 484 cross-break track, it is necessary to study the specific numerical result of the carrier arrangement. This will solve the collision problem of carriers in the braiding process, after obtaining the repeat arrangement of the carriers. In the 484 cross-breaks track braiding system, the possible collisions are positioned at the self-intersection points (ScP) and the multi-track intersection points (MtP) between two different tracks (see track-1 and track-2 in Figure 2(b)). In order to establish a mathematical model of the carriers' positions around the ScP and MtP, respectively, six basic track units were deconstructed from the 484 cross-break tracks (Figure 4), as shown in Figure 4(b). The deconstruction process is shown in Figure 4(a). Both Type \(y\) and Type \(v\) pass through MtP, and are located above and below the surface of the 484 cross-break track unit, respectively. Although they appeared to be mirrored, more than two MtP points must be analyzed to obtain a more accurate arrangement of the carriers that they do not collide at all the MtPs' position. Types \(x\) and \(z\) pass through ScP, which is located at the end of each 484 cross-break track unit. Type \(u\) and \(w\) pass through MtP, located inside of the 484 cross-break track unit. Therefore, the six basic tracks are of different types. In this paper, the position of each type of track unit in the 484 cross-break track, and the direction of the carriers in the track, are defined to represent the corresponding azimuth feature sets (AFS) of the basic track units, which are represented by the symbol \(s(m,r,c,\phi)\). The value of \(m\) represents the type of basic track unit (type-\(x\), type-\(y\), type-\(z\), type-\(u\), type-\(v\), or type-\(w\)), \(r\) represents the position of the track in the row, \(c\) represents the position of the track in the column, and \(\phi\) represents the direction of the carrier's movement in the track driven by the horn gear. The value of the parameter \(\phi\) has  
Figure 3: Schematic of 8 \(\times\) 8 484 cross-break track. two rotations, \(\phi^{+}\) and \(\phi^{-}\), as shown in Table 1. For example, the track in the \(i\)th column and the \(j\)th row is denoted by \(s_{ji}=s(f(i),j,i,\phi)\). \[f(i)=\left\{ \begin{array}{cc}x,z\,&i=2a\\ y,u,w,&v,&i=2a-1\end{array} \right. \tag{1}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Rotation**} & \multicolumn{5}{c}{**Type**} \\ \cline{2-7}  & x & y & z & u & v & w \\ \hline \hline \multirow{2}{*}{\(\phi^{+}\)} & \multirow{2}{*}{2} & \multirow{2}{*}{2} & \multirow{2}{*}{2} & \multirow{2}{*}{2} & \multirow{2}{*}{2} & \multirow{2}{*}{2} & \multirow{2}{*}{2} \\  & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Position and direction of the carriers in each basic track type. Figure 4: Minimum unit of 484 cross-break track. Figure 5: Units of basic track type, derived from Figure 4.  
The corresponding position and movement direction of the carriers in each track are shown in Table 1, and whether there is a carrier at the position of the slot of the \(i\)th column and \(j\)th row is indicated by the value of parameter \(s_{m}^{n}\), where the symbol m = \(x\), \(y\), \(z\), \(u\), \(v\), \(w\); n = 1, 2, 3. The specific expression is as follows:  
\[s_{m}^{n}=\left\{ \begin{array}{ll}0,&\text{there is no carrier in the slot}\\ 1,&\text{there is a carrier in the slot}\end{array} \right. \tag{2}\]  
The state of the carriers in the slots of each horn gear is represented by the matrix S,  
\[S=\left[ \begin{array}{c}S_{1}\\ S_{1}\\ \vdots\\ S_{p}\end{array} \right]=\left[ \begin{array}{cccc}\tau_{ji}\end{array}