# 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
 \right]_{p\times N}=\left[ \begin{array}{cccc}s_{11}&s_{12}&\cdots&s_{1N}\\ s_{21}&s_{22}&\cdots&s_{2N}\\ \vdots&\vdots&\ddots&\vdots\\ s_{p1}&s_{p2}&\cdots&s_{PN}\end{array} \right] \tag{3}\]  
where the parameters \(i\) and \(j\) indicate the column and row, respectively (\(j\) = 1, 2,..., P; I = 1, 2,..., N):  
According to Table 1 and Equation (2), whether there is a carrier (the black solid points in Table 1) at the slot position can be determined, and the specific arrangement rules (expressed by the matrix \(S_{c}\)) of the carrier in the track can be obtained, as follows:  
\[ \begin{array}{ll}S_{c}=\left[S_{j}^{C}\right]_{p\times N}(i\text{ and }j\text{ are the positive integer})\\ \text{where }S_{ji}^{C}=\left[s_{m,i}^{1}(j,i)\ s_{m,i}^{2}(j,i)\ s_{m,i}^{3}(j,i)\right] \end{array}  \tag{4}\]  
where the matrix \(S_{ji}^{C}\) represents the carrier state in each type of track unit. The intersecting relation between any two tracks can form a different intersection point; that is, 18 collision relationships can be obtained between the carriers, as shown in Figure 6. No. 3 is the interior carrier collision relationship passing through MtP, and No. 4 is that through ScP in track-2; thus, Nos. 3 and 4 are notably different. No. 6 passes through the lower surface ScP in track-2, No. 7 passes through the lower surface ScP in track-1, and No. 13 passes through the lower surface ScP in track-1. Therefore, Nos. 6, 7, and 13 are also different from each other. No. 9 passes through the ScP on the upper surface in track-1, and No. 10 passes through the ScP on the upper surface in track 2. Therefore, No. 9 is also different from No. 10. So far, all the carrier collision relationships are listed in Figure 6.  
According to Equations (1)-(4), the intersection relationship between two track units can be expressed by \(s(x_{k},i_{k},j_{k},\phi_{k})s(x_{t},i_{t},j_{t},\phi_{t})\). Thus, the equations for the intersecting relationship of 18 pairs of tracks in Figure 6 were obtained (\(q_{ij}^{1}-q_{ij}^{18}\)), which are as follows:  
\[s(x,i,j,\phi^{+})s(z,i,j,\phi^{-})=s(x,i,j,\phi^{-})s(z,i,j,\phi^{+})=q_{ij}^ {1} \tag{5}\]  
\[s(y,i+1,j-1,\phi^{-})s(y,i+2,j-1,\phi^{+})=s(y,i+1,j-1,\phi^{+})s(y,i+2,j-1, \phi^{-})=q_{ij}^{2} \tag{6}\]  
\[s(x,i+2,j,\phi^{+})s(z,i+2,j,\phi^{-})=s(x,i+2,j,\phi^{-})s(z,i+2,j,\phi^{+})= q_{ij}^{3} \tag{7}\]  
\[s(x,i+4,j,\phi^{+})s(z,i+4,j,\phi^{-})=s(x,i+4,j,\phi^{-})s(z,i+4,j,\phi^{+})= q_{ij}^{4} \tag{8}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Rotation** & \multicolumn{4}{c}{**Type**} \\ \hline
**Rotation** & **x** & **y** & **z** & **u** & **v** & **w** \\ \hline
**\(\phi^{-}\)** & **y** & **z** & **u** & **v** & **w** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cont. where  
\[q_{ij}^{14}=\underset{g=1,2,3}{\cap}\Big{\{}(s_{u}^{g}(i+3,j+2)-1)(s_{y}^{4-g}(i+ 5,j+2)-1)\
eq 0\Big{\}};\] \[q_{ij}^{15}=\underset{g=1,2,3}{\cap}\Big{\{}(s_{x}^{g}(i,j)-1)(s_ {z}^{4-g}(i,j+3)-1)\
eq 0\Big{\}};\] \[q_{ij}^{16}=\underset{g=1,2,3}{\cap}\Big{\{}(s_{x}^{g}(i+2,j+3)-1 )(s_{z}^{4-g}(i+2,j)-1)\
eq 0\Big{\}};\] \[q_{ij}^{17}=\underset{g=1,2,3}{\cap}\Big{\{}(s_{x}^{g}(i+3,j+1)- 1)(s_{z}^{4-g}(i+1,j+1)-1)\
eq 0\Big{\}};\]  
Finally, to avoid collision between carriers at any time, the logical relationship (5-22) of two intersected tracks is used, and the different carrier positions must satisfy Equation (23). \[T_{ji}=\prod_{k=1}^{P}\prod_{l=1}^{N}\left(\tau_{ji}\tau_{kl}\right)\text{, }(i\text{ and }j\text{ are the positive integer}) \tag{23}\]  
By substituting Equation (23) into Equation (4), the initial positions of carrier arrangement are obtained, and the carrier arrangement of each track can be represented by the parameter \(S_{c}\). Figure 6: Intersecting relationship between two tracks in which carriers may collide.