## 5 Determination of the Interference of the Links of the Mechanism  
The interferences of the links of the mechanism can be divided into three groups:  
- Interference at small angles between links connected by joints. - The interference of links with platforms. - The interference of links that are not connected to each other. The first group can be determined, taking into account the restrictions on the angles of rotation in the joints \(B_{i}\) and \(C_{i}\) (Figure 10):  
\[\left\{ \begin{array}{c}\alpha_{i}\in[\alpha_{\min};\alpha_{\max}]\\ \beta_{i}\in[\beta_{\min};\beta_{\max}]\\ \gamma_{i}^{(i)}\in[\gamma_{\min};\gamma_{\max}]\end{array} \right\} \tag{14}\]  
The angles can be determined using the formula for the cosines between vectors:  
\[\alpha_{i}=\frac{(x_{Ai}-x_{Bi})(x_{Ci}-x_{Bi})+(y_{Ai}-y_{Bi})(y_{Ci}-y_{Bi}) +(z_{Ai}-z_{Bi})(z_{Ci}-z_{Bi})}{de} \tag{15}\]  
\[\beta_{i}=\frac{\sqrt{3}((x_{Bi}-x_{Ci})(x_{p}-x_{Ci})+(y_{Bi}-y_{Ci})(y_{p}- y_{Ci})+(z_{Bi}-z_{Ci})(z_{p}-z_{Ci}))}{ce} \tag{16}\]  
\[\gamma_{i}^{(\rm BI)}=\frac{(x_{Ci1}-x_{Bi1})(x_{Bi2}-x_{Bi1})+(y_{Ci1}-y_{ Bi1})(y_{Bi2}-y_{Bi1})+(z_{Ci1}-z_{Bi1})(z_{Bi2}-z_{Bi1})}{he} \tag{17}\]  
\[\gamma_{i}^{(\rm BI2)}=\gamma_{i}^{(\rm CI1)}=-\gamma_{i}^{(\rm CI2)}=- \gamma_{i}^{(\rm BI1)} \tag{18}\]  
where \(x_{A1}=z_{A1}=z_{A2}=z_{A3}=x_{B1}=0\), \(x_{A2}=\frac{a}{4}\), \(x_{A3}=-\frac{a}{4}\), \(y_{A1}=-\frac{a}{2\sqrt{3}}\), \(y_{A2}=y_{A3}=-\frac{a}{4\sqrt{3}}\), \(y_{B1}=y_{A1}-d\cos(\theta_{1})\), \(z_{Bi}=-d\sin(\theta_{i})\), \(x_{B2}=x_{A2}+\frac{\sqrt{3}d\cos(\theta_{2})}{2}\), \(y_{B2}=y_{A2}+\frac{d\cos(\theta_{2})}{2}\), \(x_{B3}=x_{A3}-\frac{\sqrt{3}d\cos(\theta_{3})}{2}\), \(y_{B3}=y_{A3}+\frac{d\cos(\theta_{3})}{2}\), \(z_{Ci}=z_{p}\), \(x_{C1}=x_{p}\), \(y_{C1}=y_{p}-\frac{c}{\sqrt{3}}\), \(x_{C2}=x_{p}+\frac{c}{2}\), \(y_{C2}=y_{C3}=y_{P}+\frac{c}{2\sqrt{3}}\), \(x_{C3}=x_{P}-\frac{c}{2}\), \(x_{B1}=y_{B31},z_{B31}=z_{B32}=z_{B3}\). The second group of interference includes possible interference of links with platforms \(A_{1},\ A_{2},\ A_{3}\) and \(C_{1},\ C_{2},\ C_{3}\). In this case, the interference with the platform \(C_{1},\ C_{2},\ C_{3}\) is excluded by condition (14). The interference of links \(A_{i},\ B_{i}\) and \(B_{1},B_{i2}\) with platform \(A_{1},\ A_{2},\ A_{3}\) is also excluded by condition (1). Let us add the conditions for the appearance of the remaining interference, namely links \(B_{i1},\ C_{i1}\) and \(B_  
The platform is an equilateral triangle with points \(A_{1},\ A_{2},\ A_{3}\) being the midpoints of the sides. The coordinates of the vertices \(A_{1}^{\prime},\ A_{2}^{\prime},\ A_{3}^{\prime}\) are defined as:  
\[z_{A_{1}^{\prime}}=z_{A_{2}^{\prime}}=z_{A_{3}^{\prime}}=x_{A_{3}^{\prime}}=0,\ x_{A_{1}^{\prime}}=-\frac{a}{2},\ x_{A_{2}^{\prime}}=\frac{a}{2},\ y_{A_{1}^{\prime}}=y_{A_{2}^{ \prime}}=-\frac{a}{2\sqrt{3}}, \tag{23}\]  
\[y_{A_{3}^{\prime}}=\frac{a}{\sqrt{3}} \tag{24}\]  
As a result, it is required to check in the two-dimensional plane XOY whether the points \(D_{l1}\) and \(D_{l2}\) belong to the triangle \(A_{1}^{\prime}A_{2}^{\prime}A_{3}^{\prime}\). If at least one of the points is included in the triangle, then there is interference with the platform. We define the third group of interference using an approach based on determining the minimum distance between the segments drawn between the centers of the joints of each of the links. In [22; 23], a similar condition is used, but the approach has drawbacks. In particular, the authors propose to determine the interference of the segments on the auxiliary plane, and not the distance between the nearest points. This does not allow identifying such an interference of links in which there is no interference of the axes. The approach proposed in the current work is as follows. We construct an auxiliary plane to determine the interference of the links of the mechanism. This plane is parallel to the axis of one of the links and to which the axis of the other link belongs.