## 3 Results and Discussion
### Modeling of Static Deformations and Manipulator Errors under Elastic Pliability of Robot Links
The unit moment \(\mathrm{M_{x}}=1\) causes deformation only in the first kinematic pair, because it has an \(\mathbf{x_{1}}\) axis of rotation:  
\[\Delta q_{1}=\mathrm{M_{x}}/\mathrm{c_{1}}=1/0.5\cdot 10^{6}=2\cdot 10^{-6}\ \mathrm{ rad}\ ;\ \Delta q_{2}=0\ ;\ \Delta q_{3}=0.\]  
All three deformations from the moment \(\mathbf{M_{y}}=1\) are equal to zero, therefore \(\Delta q_{1}=\Delta q_{2}=\Delta q_{3}=0\). The unitary moment \(\mathbf{M_{z}}=1\) creates deformation only in the third kinematic pair with the \(\mathbf{z_{3}}\) axis of rotation:  
\[\Delta q_{1}=\Delta q_{2}=0;\ \Delta q_{3}=\mathrm{M_{z}}/\mathrm{c_{3}}=1/0.5 \cdot 10^{6}=2\cdot 10^{-6}\ \mathrm{rad}.\]  
The results of simulation of deformations are summarized in the Table 1.  
Calculation of elements of the E matrix of elastic deformations. For the calculation, we will use expressions (18) and (19), data from the table, the column number of which corresponds to the column number of the matrix E, the elements of which we are looking for. We define in the following sequence:  
Elements of the first column:  
\[\Delta r_{\mathrm{M}}^{(0)}(\mathrm{F_{x}})=\mathrm{A_{0,3}}\cdot\left(\Theta _{3}\cdot r_{\mathrm{M}}^{(3)}+\xi_{3}^{(3)}\right)=\Theta_{3}\cdot r_{ \mathrm{M}}^{(3)}= \begin{bmatrix}0&-\Delta q_{3}&0\\ \Delta q_{3}&0&0\\ 0&0&0\end{bmatrix} \cdot \begin{bmatrix}0\\ 0.5\\ 0.5\end{bmatrix} = \begin{bmatrix}0.5\\ 0\\ 0\end{bmatrix} \cdot 10^{-6}\ \mathrm{m};\]  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \multicolumn{2}{c}{**Deformations**} & \(\mathbf{F_{x}}\) = \(\mathbf{1}\) & \(\mathbf{F_{y}}\) = \(\mathbf{1}\) & \(\mathbf{F_{z}}\) = \(\mathbf{1}\) & \(\mathbf{M_{x}}\) = \(\mathbf{1}\) & \(\mathbf{M_{y}}\) = \(\mathbf{1}\) & \(\mathbf{M_{z}}\) = \(\mathbf{1}\) \\ \hline \(\Delta q_{1}\) & \(10^{-6}\ \mathrm{rad}\) & \(0\[\vartheta^{(0)}(\mathrm{F_{x}})=\mathrm{A_{0,3}}\cdot\vartheta_{3}^{(3)}= \begin{bmatrix} 0\\ 0\\ \Delta q_{3}\end{bmatrix} = \begin{bmatrix}0\\ 0\\ -1\end{bmatrix} \cdot 10^{-6}\;\mathrm{rad};\]  
Elements of the second column:  
\[\Delta r_{\mathrm{M}}^{(0)}(\mathrm{F_{y}})=\Theta_{1}\cdot r_{\mathrm{M}}^{(1 )}= \begin{bmatrix}0&0&0\\ 0&0&-\Delta q_{1}\\ 0&\Delta q_{1}&0\end{bmatrix} \cdot \begin{bmatrix}0\\ 1.5\\ 0.6\end{bmatrix} = \begin{bmatrix}0\\ 0.72\\ -1.8\end{bmatrix} \cdot 10^{-6}\;\mathrm{m};\]  
\[\vartheta^{(0)}(\mathrm{F_{y}})=\vartheta_{1}^{(1)}= \begin{bmatrix}\Delta q_{ 1}\\ 0\\ 0\end{bmatrix} = \begin{bmatrix}-1.2\\ 0\\ 0\end{bmatrix} \cdot 10^{-6}\;\mathrm{rad};\]  
Elements of the third column:  
\[\Delta r_{\mathrm{M}}^{(0)}(\mathrm{F_{z}})=\Theta_{1}\cdot r_{\mathrm{M}}^{(1 )}+\xi_{2}^{(2)}= \begin{bmatrix}0&0&0\\ 0&0&-\Delta q_{1}\\ 0&\Delta q_{1}&0\end{bmatrix} \cdot \begin{bmatrix}0\\ 1.5\\ 0.6\end{bmatrix} + \begin{bmatrix}0\\ 0\\ 0\end{bmatrix} = \begin{bmatrix}0\\ -1.8\\ 8.5\end{bmatrix} \cdot 10^{-6}\;\mathrm{m};\]  
\[\vartheta^{(0)}(\mathrm{F_{z}})=\vartheta_{1}^{(1)}= \begin{bmatrix}3\\ 0\\ 0\end{bmatrix} \cdot 10^{-6}\]  
Elements of the fourth column:  
\[\Delta r_{\mathrm{M}}^{(0)}(\mathrm{M_{x}})=\Theta_{1}\cdot r_{\mathrm{M}}^{(1 )}= \begin{bmatrix}0&0&0\\ 0&0&-2\\ 0&2&0\end{bmatrix} \cdot \begin{bmatrix}0\\ 1.5\\ 0.6\end{bmatrix} = \begin{bmatrix}0\\ -1.2\\ 3\end{bmatrix} \cdot 10^{-6}\;\mathrm{m};\]  
\[\vartheta^{(0)}(\mathrm{M_{x}})=\vartheta_{1}^{(1)}= \begin{bmatrix}2\\ 0\\ 0\end{bmatrix} \cdot 10^{-6}\;\mathrm{rad};\]  
All elements of the fifth column are zero; elements of the sixth column;  
\[\Delta r_{\mathrm{M}}^{(0)}(\mathrm{M_{z}})=\Theta_{3}\cdot r_{\mathrm{M}}^{(3 )}= \begin{bmatrix}0&-2&0\\ 2&0&0\\ 0&2&0\end{bmatrix} \cdot \begin{bmatrix}0\\ 0.5\\ 0.5\end{bmatrix} \cdot 10^{-6}= \begin{bmatrix}-1\\ 0\\ 3\end{bmatrix} \cdot 10^{-6}\;\mathrm{m};\]  
\[\vartheta^{(0)}(\mathrm{M_{z}})=\vartheta_{3}^{(3)}= \begin{bmatrix}0\\ 0\\ 2\end{bmatrix} \cdot 10^{-6}\;\mathrm{rad}.\]  
Based on the results of the calculations, we form a matrix of elastic pliability of the gripper in the initial position of the robot:  
\[\mathrm{E}=