## 3 Results and Discussion
### Modeling of Static Deformations and Manipulator Errors under Elastic Pliability of Robot Links  
We will perform simulation of static deformations and errors of the manipulator. For example, consider the kinematic scheme of the elastic robot manipulator in the initial position (Figure 4). The elastic pliability of drives of kinematic pairs is modeled by elastic elements that have stiffness of \(c_{1}\), \(c_{2}\), and \(c_{3}\). The gripper is loaded with technological loads of \(\text{F}_{\text{x}}\), \(\text{F}_{\text{y}}\), \(\text{F}_{\text{z}}\), and moments of \(\text{M}_{\text{x}}\), \(\text{M}_{\text{y}}\), \(\text{M}_{\text{z}}\). We set arbitrary parameters of the manipulator elements for modeling deformations and errors. We accept the following: \(l_{2}=1\) m; \(l_{3\text{y}}=l_{3\text{z}}=0.5\) m; \(q_{10}=q_{30}=0\); \(q_{20}=0.1\) m; \(\text{e}_{1}=\text{e}_{3}=\text{e}_{\text{ang}}=2\cdot 10^{-6}\) rad/N\(\cdot\)m; \(\text{C}_{1}=\text{C}_{2}=1/\text{e}_{\text{ang}}=0.5\cdot 10^{6}\) N\(\cdot\)m/rad; \(\text{e}_{2}=\text{e}_{\text{lin}}=4\cdot 10^{-6}\) m/N, \(\text{C}_{2}=1/\text{e}_{\text{lin}}=0.25\cdot 10^{6}\) N\(\cdot\)m/rad; G = 200 N; \(b=0.5\) m.  
Other data necessary for modeling were obtained from the results of modeling the position and orientation error of the robot gripper due to the temperature deformation of the manipulator links:  
\[\text{A}_{0,1}=\text{A}_{1,2}=\text{A}_{2,3}=\text{I}=\text{diag}[1,\,1,\,1],\]  
\[r_{\text{M}}^{(3)}= \begin{bmatrix}0\\ 0.5\end{bmatrix} ,\,r_{\text{M}}^{(2)}= \begin{bmatrix}0\\ 1.5\\ 0.5\end{bmatrix} ,\,r_{\text{M}}^{(1)}= \begin{bmatrix}0\\ 1.5\\ 0.6\end{bmatrix} ,\,\text{m};\]  
\[\xi_{1}^{(1)}=\xi_{3}^{(3)}=[0;\,0;\,0]^{\text{T}},\,\xi_{2}^{(2)}=[0;\,0;\, \Delta q_{2}]^{\text{T}}\,\text{m}; \tag{60}\]  
\[\vartheta_{1}^{(1)}=[\Delta q_{1};\,0;\,0]^{\text{T}},\,\,\vartheta_{3}^{(3)} =[0;\,0;\,\Delta q_{3}]^{\text{T}},\,\,\vartheta_{2}^{(2)}=\vartheta_{4}^{(4) }=[0;\,0;\,0]^{\text{T}};\]  
Figure 4: Kinematic scheme of the elastic manipulator in the initial position of the gripper with a technological load. \[\Theta_{1}= \begin{bmatrix}0&0&0\\ 0&0&-\Delta q_{1}\\ 0&\Delta q_{1}&0\end{bmatrix} ,\Theta_{2}=0,\ \Theta_{3}= \begin{bmatrix}0&-\Delta q_{3}&0\\ \Delta q_{3}&0&0\\ 0&0&0\end{bmatrix} .\]  
Modeling of deformations of elastic elements \(\Delta q_{i}\). We load the gripper with a unit force \(\mathrm{F_{x}}=1\) in the absence of other loads. Deformation \(\Delta q_{1}=0\), because the \(\mathrm{F_{x}}\) force is parallel to the axis of rotation \(\mathrm{x_{1}}\) and its moment relative to this axis is zero. Deformation \(\Delta q_{2}=0\), because the \(\mathrm{F_{x}}\) force acts perpendicular to the direction of \(q_{2}\) movement. The deformation will be as:  
\[\Delta q_{2}=-\mathrm{F_{x}}l_{3\mathrm{y}}/\mathrm{C_{3}}=-1\cdot 10^{-6}\ \mathrm{ rad},\]  
(the minus sign means that the \(\mathrm{F_{x}}\) force tries to turn the third link clockwise relative to the \(z_{3}\) axis of rotation). We load the gripper with the \(\mathrm{F_{y}}=1\) force in the absence of other loads. The deformation will be:  
\[\Delta q_{1}=-\mathrm{F_{y}}\cdot\left(l_{3\mathrm{y}}+q_{20}\right)/\mathrm{ c_{1}}=-1.2\cdot 10^{-6}\ \mathrm{rad}.\]  
Deformations \(\Delta q_{2}=0\) and \(\Delta q_{3}=0\). After loading the gripper with a unit force \(\mathrm{F_{z}}=1\), we will have:  
\[\Delta q_{1}=\mathrm{F_{z}}\cdot\left(l_{3\mathrm{y}}+l_{2}\right)/\mathrm{c_{ 1}}=3\cdot 10^{-6}\ \mathrm{rad};\ \Delta q_{2}=\mathrm{F_{z}}/\mathrm{c_{2}}=4\cdot 10 ^{-6}\ \mathrm{rad};\ \Delta q_{3}=0.\]  
When loading the gripper with single moments, it is necessary to take into account that arbitrary moment loading does not cause linear deformations, i.e., \(\Delta q_{2}=0\).