## 3 Results and Discussion  
The following restrictions are adopted for modeling the positioning error of the manipulator. There is no flexibility in the connections and links and, accordingly, there are no high-frequency oscillations of the links. The mass of the links of the robot manipulator in relation to the mass of the object being moved is on average 100 or more times higher than the mass of the object being moved. Modeling of the Position and Orientation Error of the Robot Gripper Due to the Temperature Deformation of the Manipulator Links  
We will simulate the position and orientation error of the robot gripper due to temperature deformations of the third link for the manipulator scheme shown in Figure 3.  
Figure 3: Kinematic scheme of the robot in the initial position. To take into account the temperature deformations of the third link, we introduce an additional fourth local system of axes of O\({}_{4}\)x\({}_{4}\)y\({}_{4}\)z\({}_{4}\), combining the origin of O\({}_{4}\) with the pole M of the gripper. Let us set the following parameters of the manipulator: \(l_{2}=1\) m; \(l_{3y}=l_{3z}=0.5\) m; \(q_{10}=q_{30}=0\); \(q_{20}=0.1\) m; \(\Delta q_{1}=\Delta q_{3}=0.001\) rad; \(\Delta q_{2}=10^{-4}\) m; \(\Delta t=+20^{\circ}\)C.  
For the initial position of the manipulator, in which the angular coordinates \(q_{1}=q_{3}=0\), all rotation matrices are unitary, i.e.,  
\[\mathrm{A}_{0,1}=\mathrm{A}_{1,2}=\mathrm{A}_{2,3}=\mathrm{A}_{0,2}=\mathrm{A }_{0,3}=\mathrm{I}=\mathrm{diag}[1,\,1,\,1], \tag{57}\]  
and we also have  
\[r_{\mathrm{M}}^{(3)}= \begin{bmatrix}0\\ l_{3y}\\ l_{3z}\end{bmatrix} = \begin{bmatrix}0\\ 0.5\\ 0.5\end{bmatrix} ,\ r_{\mathrm{M}}^{(2)}= \begin{bmatrix}0\\ l_{3y}+l_{2}\\ l_{3z}\end{bmatrix} = \begin{bmatrix}0\\ 1.5\\ 0.5\end{bmatrix} ,\ r_{\mathrm{M}}^{(1)}= \begin{bmatrix}0\\ l_{3y}+l_{2}\\ l_{3z}+q_{20}\end{bmatrix} = \begin{bmatrix}0\\ 1.5\\ 0.6\end{bmatrix} ,\ \text{m}.\]  
From the kinematic scheme (Figure 3), we determine the vectors of a small linear displacement of the \(O_{i}\) point at the fixed point of \(O_{i-1}\) (\(i=4,\,3,\,2,\,1\)):  
\[\xi_{4}^{(4)}= \begin{bmatrix}0;\ \alpha\cdot l_{3y}\cdot\Delta t;\ \alpha\cdot l_{3z}\cdot\Delta t \end{bmatrix} ^{\mathrm{T}}= \begin{bmatrix}0;\ 1.2;\ 1.2\end{bmatrix} ^{\mathrm{T}}\cdot 10^{-4}\ \text{m};\]  
\[\xi_{3}^{(3)}= \begin{bmatrix}0;\ \alpha\cdot l_{2}\cdot\Delta t;\ 0 \end{bmatrix} ^{\mathrm{T}}= \begin{bmatrix}0;\ 2.4;\ 0\end{bmatrix} ^{\mathrm{T}}\cdot 10^{-4}\ \text{m}; \tag{58}\]  
\[\xi_{2}^{(2)}= \begin{bmatrix}0;\ 0;\ \alpha\cdot q_{20}\cdot\Delta t+\Delta q_{2 }\end{bmatrix} ^{\mathrm{T}}= \begin{bmatrix}0;\ 0;\ 1.24\end{bmatrix} ^{\mathrm{T}}\cdot 10^{-4}\ \text{m};\]  
\[\xi_{1}^{(1)}= \begin{bmatrix}0;\ 0;\ 0\end{bmatrix} ^{\mathrm{T}}.\]  
From the same scheme, we determine the vectors of small rotations of \(\phi_{i}^{(i)}\). After taking into account that the rotation axis of \(\Delta q_{1}\) is \(x_{1}\), and the rotation of \(\Delta q_{3}\) is the \(x_{3}\) axis, we obtain  
\[\phi_{1}^{(1)}=[\Delta q_{1};\ 0;\ 0]^{\mathrm{T}},\ \phi_{3}^{(3)}= \begin{bmatrix}0;\ 0;\ \Delta q_{3} \end{bmatrix} ^{\mathrm{T}},\ \phi_{2}^{(2)}=\phi_{4}^{(4)}= \begin{bmatrix}0;\ 0;\ 0 \end{bmatrix} ^{\mathrm{T}}. \tag{59}\]  
We form a skew symmetric matrix of small turns by (3):  
\[\Theta_{1}= \begin{bmatrix}0&0&0\\ 0&0&-\Delta q_{1}\\ 0&\Delta q_{1}&0\end{bmatrix} ,\ \Theta_{3}= \begin{bmatrix}0&-\Delta q_{3}&0\\ \Delta q_{3}&0&0\\ 0&0&0\end{bmatrix} ,\ \Theta_{2}=\Theta_{4}=0\,.\]  
We determine the linear errors of the position of the gripper pole in the reference system of the axes according to (1):  
\[\Delta r_{\mathrm{MI}}^{(0)}=\mathrm{A}_{0,1}\Big{(}\Theta_{1}r_{\mathrm{M}}^{ (1)}+\xi_{1}^{(1)}\Big{)}=\Theta_{1}r_{\mathrm{M}}^{(1)}= \begin{bmatrix}0&0&0 \\ 0&0&-0.001\\ 0&0.001&0\end{bmatrix}\begin{bmatrix}0\\ 1.5\\ 0.6\end{bmatrix} = \begin{bmatrix}0\\ -0.6\\ 1.5\end{bmatrix}