## 2 Building a Model of a Robotic System
Although manipulators of the Delta type are well studied, the development of new methods for analyzing the workspace that are more efficient in terms of increasing the accuracy of approximation and versatility of their application for various versions of their architecture is an important task. The task of determining the workspace can be solved with a given accuracy using interval analysis methods [28; 29; 30]. One of the disadvantages of deterministic interval analysis methods is a large number of elements in the covering set. Nevertheless, the article's authors proposed an approach to the transformation of the covering set, which was considered earlier in [31]. It is used to transform the covering set before rendering the stage. The interval analysis method is applied to approximate the set of solutions to a system of nonlinear inequalities that determine the constraints on the parameters of the DeLi robot links (Figure 4). The manipulator has 4 degrees of freedom and includes three RUU kinematic chains. In each of the chains, a rotary drive joint \(A_{i}\) is used to connect to the base, and there are two universal joints--\(C_{i}\) to connect to the working platform and \(B_{i}\) to connect two links together. The base and work platform are equilateral triangles. As an end-effector, we will consider point P--the center of the working platform. We write the boundaries of the workspace in the form of a system of inequalities:  
\[\left\{ \begin{array}{l}\theta_{i}-\theta_{max}\leq 0\\ \theta_{min}-\theta_{i}\leq 0\end{array} \right. \tag{1}\]  
where \(\theta_{i}\) are the angles of rotation in the drive joints, and \(\theta_{min}\) and \(\theta_{max}\) are the minimum and maximum angles, respectively. Based on the solution of the inverse kinematics described in [32], we obtain  
\[\theta_{i}=2\tan^{-1}\left(\frac{-F_{i}\pm\sqrt{E_{i}^{2}+F_{i}^{2}+G_{i}^{2}} }{G_{i}-E_{i}}\right) \tag{2}\]  
where \(E_{i},\ F_{i}\), and \(G_{i}\) are defined as:  
\[E_{1}=2d\left(y_{P}+\frac{a-2c}{2\sqrt{3}}\right)\text{, }F_{1}=2z_{P}d\text{,} \tag{3}\]  
\[G_{1}=x_{P}^{2}+y_{P}^{2}+z_{P}^{2}+\left(\frac{a-2c}{2\sqrt{3}}\right)^{2}+d ^{2}+2y_{P}\bigg{(}\frac{a-2c}{2\sqrt{3}}\bigg{)}-e^{2} \tag{4}\]\[E_{2}=-d\bigg{(}\sqrt{3}\bigg{(}x_{P}+\frac{2c-a}{4}\bigg{)}+y_{P}+\frac{2c-a}{4 \sqrt{3}}\bigg{)},\ F_{2}=2z_{P}d, \tag{5}\]  
\[G_{2}=x_{P}^{2}+y_{P}^{2}+z_{P}^{2}+\bigg{(}\frac{2c-a}{4}\bigg{)}^{2}+\bigg{(} \frac{2c-a}{4\sqrt{3}}\bigg{)}^{2}+d^{2}+2\bigg{(}x_{P}\bigg{(}\frac{2c-a}{4 \sqrt{3}}\bigg{)}+y_{P}\bigg{(}\frac{2c-a}{4\sqrt{3}}\bigg{)}\bigg{)}-e^{2} \tag{6}\]  
\[E_{3}=d\bigg{(}\sqrt{3}\bigg{(}x_{P}-\frac{2c-a}{4}\bigg{)}-y_{P}-\frac{2c-a}{4 \sqrt{3}}\bigg{)},\ F_{3}=2z_{P}d, \tag{7}\]  
\[G_{3}=x_{P}^{2}+y_{P}^{2}+z_{P}^{2}+\bigg{(}\frac{2c-a}{4}\bigg{)}^{2}+\bigg{(} \frac{2c-a}{4\sqrt{3}}\bigg{)}^{2}+d^{2}+2\bigg{(}y_{P}\bigg{(}\frac{2c-a}{4 \sqrt{3}}\bigg{)}-x_{P}\bigg{(}\frac{2c-a}{4}\bigg{)}\bigg{)}-e^{2} \tag{8}\]  
where \(a\) is the side length of a regular triangle of the base, \(c\) is the side length of a regular triangle of the movable platform, \(d\) is the distance between \(A\) and \(B\) joints, \(e\) is the distance between \(B\) and \(C\) joints, and \(x_{P}\), \(y_{P}\), \(z_{P}\) are the coordinates of point P.  
There is an ambiguity in the solution of the inverse kinematics, when for the same coordinates of the point P there are two possible angles of rotation in the drive rotary joint for each of the three kinematic chains. An example of two possible positions of the kinematic chain \(A_{i}B_{i}C_{i}\) is shown in Figure 5. The first position corresponds to \([-]\) in Equation (2), the second position corresponds to \([+]\). Thus, there are eight possible solutions to the inverse kinematics, and, therefore, for each of them, a workspace can be determined. In this case, one should take into account the presence of singularity zones of the mechanism when hit, in which the dynamic loads on the links increase significantly, and the robot loses controllability.