## 2 Materials and Methods
### Geometric and Kinematic Errors of Robots  
The error of the manipulator is the deviation of the actual values of the kinematic characteristics from their program values. It can be an error of coordinates, speed, or acceleration. Geometric error is caused by geometric deviations: manufacturing and assembly inaccuracy, temperature and force deformations, gaps in kinematic pairs, and a number of others. Kinematic errors are caused by inaccurate operation of the actuators of the robot elements. Consider the effect of temperature deformation on positioning accuracy.  
To take into account the temperature deformations of the third link, an additional fourth local system of axes of \(O_{4}x_{4}y_{4}z_{4}\) is introduced, parallel to the \(O_{3}x_{3}y_{3}z_{3}\) axes and having the \(O_{4}\) origin, aligned with the M pole of the gripper. Then, the linear errors of the position of the \(M\) pole of the gripper in the reference system of the axes are determined by the equation:  
\[\Delta r_{\text{M}}^{(0)}=\sum_{s=1}^{4}\Delta r_{\text{Ms}}^{(0)}\,\ \Delta r_{\text{Ms}}^{(0)}=A_{0,s}\left( \cdot\Theta_{s}\cdot r_{\text{M}}^{(s)}+\tilde{\sigma}_{s}^{(s)}\right)\,, \tag{1}\]  
where \(\Delta r_{\text{M}}^{(0)}=\left[\Delta x_{\text{M}}^{(0)},\Delta y_{\text{M}} ^{(0)},\Delta z_{\text{M}}^{(0)}\right]^{\text{T}}\) is the sought vector of linear errors; \(r_{\text{M}}^{(\text{s})}=\left[x_{\text{M}}^{(\text{s})},y_{\text{M}}^{( \text{s})},z_{\text{M}}^{(\text{s})}\right]^{\text{T}}\) is the vector of coordinates of the M pole in the \(i\)-th local system of axes; \(\tilde{\epsilon}_{\text{s}}^{(\text{s})}=\left[\tilde{\epsilon}_{\text{sx}}^ {(\text{s})},\tilde{\epsilon}_{\text{sy}}^{(\text{s})},\tilde{\epsilon}_{ \text{sz}}^{(\text{s})}\right]^{\text{T}}\) is the vector of a small linear displacement of the O\({}_{\text{i}}\) point for the O\({}_{\text{i}-1}\) fixed point due to the combined influence of the inaccuracy of processing the \(\Delta q_{\text{i}}\) linear coordinate and the temperature deformation of the link.  
\[\Delta l_{t}=\alpha\cdot l\cdot\Delta t, \tag{2}\]  
where \(\alpha\) is the thermal expansion coefficient (for steel \(\alpha=1\), \(2\cdot 10^{-5}\)\(1/^{\circ}\)C); \(\Delta t\) is the temperature increase; \(l\) is the link length; \(\Theta_{\text{i}}\) is the skew-symmetric matrix of small rotations of the \(i\)-th system of axes due to the inaccuracy of angular coordinates processing of \(\Delta q_{i}\):  
\[\Theta_{\text{i}}=\begin{bmatrix}0&-\theta_{\text{iz}}&\theta_{\text{iy}}\\ \theta_{\text{iz}}&0&-\theta_{\text{ix}}\\ -\theta_{\text{iy}}&\theta_{\text{ix}}&0\end{bmatrix}. \tag{3}\]  
For example, for a rotating kinematic couple with the \(\mathbf{z_{s}}\) axis of rotation we have:  
\[\theta_{\text{iz}}=\Delta q_{\text{i}}\ ;\ \theta_{\text{ix}}=\theta_{\text{ iy}}=0.\]  
The orientation error of the gripper in the reference system of the axes is determined by the ratio:  
\[\vartheta^{(0)}=\sum_{\text{i=1}}^{3}A_{0,\text{i}}\cdot\theta_{\text{i}}^{( \text{i})}\,, \tag{4}\]  
where \(\theta_{\text{i}}^{(\text{i})}=\left[\theta_{\text{ix}}^{(\text{i})},\theta_{ \text{iy}}^{(\text{i})},\theta_{\text{iz}}^{(\text{i})}\right]^{\text{T}}\) is the vector of small angular deviations.  
For example, if the axis of rotation of the pair is the \(z_{i}\) axis, then \(\theta_{\text{i}}^{(i)}=\left[0;\ 0;\ \Delta q_{\text{i}}\right]^{\text{T}}\).