## 3 Parallel Manipulator Mathematical Model  
Let us consider the scheme of a parallel manipulator based on a tripod (Figure 2). The tripod includes three rods of variable length \(l_{1},\ l_{2},\ l_{3}\), which are connected by rotary joints \(D_{1},\ D_{2},\ D_{3}\) to the base and spherical joints \(E_{1},\ E_{2},\ E_{3}\) to the working platform. The base and work platform are equilateral triangles. When changing the lengths of the rods, the working platform moves along the \(Z_{1}\) axis at a distance \(z_{1}\), and rotates around the \(X_{1}\) axis at an angle \(\psi\) and around \(Y_{1}\) at an angle \(\theta\). In addition, there are additional degrees of freedom--displacement along the \(X_{1}\) axes at a distance \(x_{1}\) and along \(Y_{1}\) at a distance \(y_{1}\) and rotation relative to \(Z_{1}\) by an angle \(\alpha\), which are determined by the formulas [20]. Figure 1: Diagram of a multi-robotic system—(**a**) serial manipulator, (**b**) parallel manipulator. Figure 2: Parallel manipulator—(**a**) without a movable base, (**b**) on a movable base. \[\alpha=Tan^{-1}\bigg{(}\frac{\sin\psi\sin\theta}{\cos\psi+\cos\theta}\bigg{)} \tag{1}\]  
\[x_{1}=\frac{r}{2}(\cos\theta\cos\alpha+\sin\psi\sin\theta\sin\alpha-\cos\psi\cos\alpha) \tag{2}\]  
\[y_{1}=-r\cos\psi\sin\alpha \tag{3}\]  
The input coordinates are the lengths of the drive links \(l_{1}\), \(l_{2}\), \(l_{3}\), the output coordinates are the coordinates of the point \(O^{\prime}\) of the end effector: \(x_{O^{\prime}},~{}y_{O^{\prime}}\), \(z_{O^{\prime}}\). Point \(O^{\prime}\) is located at a distance \(h\) from the center of the movable platform. To determine the workspace, it is necessary to first determine the set of permissible values of the linear and angular coordinates of the center \(O\) of the movable platform, and then, for these values, determine the set of coordinates \(O^{\prime}\) of the end effector. Coordinates \(O^{\  
\[E_{2}^{(0)}=M_{1\_0}E_{2}^{(1)}=\left[ \begin{array}{c}\cos\varphi\Big{(}x_{1}-0.5 r\Big{(}M_{11}-\sqrt{3}M_{12}\Big{)}+x_{0}\Big{)}-\sin\varphi\Big{(}z_{1}-0.5r \Big{(}M_{31}-\sqrt{3}M_{32}\Big{)}\Big{)}\\ y_{1}-0.5r\Big{(}M_{21}-\sqrt{3}M_{22}\Big{)}\\ \sin\varphi\Big{(}x_{1}-0.5r\Big{(}M_{11}-\sqrt{3}M_{12}\Big{)}+x_{0}\Big{)}+ \cos\varphi\Big{(}z_{1}-0.5r\Big{(}M_{31}-\sqrt{3}M_{32}\Big{)}\Big{)}\\ 1\end{array} \right], \tag{24}\]  
\[E_{3}^{(0)}=M_{1\_0}E_{3}^{(1)}=\left[ \begin{array}{c}\cos\varphi\Big{(}x_{1} -0.5r\Big{(}M_{11}+\sqrt{3}M_{12}\Big{)}+x_{0}\Big{)}-\sin\varphi\Big{(}z_{1}- 0.5r\Big{(}M_{31}+\sqrt{3}M_{32}\Big{)}\Big{)}\\ y_{1}-0.5r\Big{(}M_{21}+\sqrt{3}M_{22}\Big{)}\\ \sin\varphi\Big{(}x_{1}-0.5r\Big{(}M_{11}+\sqrt{3}M_{12}\Big{)}+x_{0}\Big{)}+ \cos\varphi\Big{(}z_{1}-0.5r\Big{(}M_{31}+\sqrt{3}M_{32}\Big{)}\Big{)}\\ 1\end{array} \right]. \tag{25}\]  
We calculate the integers corresponding to the coordinates of the point \(O^{\prime}\) to describe the form of a partially ordered set of integers,  
\[\left\{ \begin{array}{c}x_{O^{\prime}}^{(Z)}=\Big{[}\frac{x_{O^{\prime}}}{ \delta}\Big{]},x_{O^{\prime}}\in\mathbb{R},\;x_{O^{\prime}}^{(Z)}\in\mathbb{Z }\\ y_{O^{\prime}}^{(Z)}=\Big{[}\frac{y_{O^{\prime}}}{\delta}\Big{]},y_{O^{\prime}} \in\mathbb{R},\;y_{O^{\prime}}^{(Z)}\in\mathbb{Z}\\ z_{O^{\prime}}^{(Z)}=\Big{[}\frac{z_{O^{\prime}}}{\delta}\Big{]},z_{O^{\prime}} \in\mathbb{R},\;z_{O^{\prime}}^{(Z)}\in\mathbb{Z}\end{array} \right., \tag{26}\]  
where \(\delta\) is the approximation accuracy. In addition to restrictions on the permissible ranges of drive lengths, it is necessary to exclude the mechanism from falling into singularity zones in which the control of the mechanism is lost. We use the method proposed by [PERSON] [21], which is based on the analysis of the Jacobi matrix of the mechanism obtained by differentiating the constraint equations and describing the transition from generalized velocities in driving kinematic pairs to angular or linear velocities of the end effector.