## 3 Results and Discussion
### Modeling of Dynamic Errors in the Partial System of Motion of Robot Links
Dynamic errors of the position of \(\Psi=q_{1}-q_{\text{p1}}\) i and velocity of \(\mathbf{v}=\frac{d\Psi}{dt}=\left(\frac{dq_{1}}{dt}\right)-\left(\frac{dq_{1 \text{p1}}}{dt}\right)\) are determined by integrating the differential Equation (29), which for the first link will have the form:  
\[\tau_{1}\tau_{\text{M}}\frac{d^{2}v_{1}}{dt^{2}}+\tau_{\text{M}}\frac{dv_{1}} {dt}+v_{1}=-\tau_{1}\tau_{\text{M}}\frac{d^{3}q_{\text{p1}}}{dt^{3}}-\tau_{ \text{M}}\frac{d^{2}q_{\text{p1}}}{dt^{2}}. \tag{68}\]  
where \(\tau_{1}=0.05\ \text{s}\) is the electromagnetic engine time constant; \(\tau_{\text{M}}\) is the time constant of the system for the first link is determined by the equation  
\[\tau_{\text{M}}=\frac{a_{11}}{\eta_{\text{p1}}.5_{1}.k_{\text{p1}}^{2}},\]  
\[\tau_{\text{M}}=\frac{78.8}{0.9\text{-}0.135.51^{2}}=0.25\ \text{s}.\]  
The software acceleration will change according to the sinusoidal law  
\[\frac{d^{2}q_{\text{p1}}(t)}{dt^{2}}=\varepsilon\text{-}\sin(\Omega\cdot t), \ \varepsilon=3.85\ \text{s}^{-2},\ \Omega=3.93\ \text{s}^{-1}. \tag{69}\]  
The velocity error is defined by Equation (50)  
\[v(t)=3.52.10^{3}.\text{e}^{-\frac{t\cdot(\tau_{\text{M}}+\sqrt{\frac{\tau_{ \text{M}}^{2}-4\tau_{\text{M}}}{2\tau_{\text{M}}}})}{2\tau_{\text{M}}}}-3.52.10 ^{3}.\text{e}^{-\frac{t\cdot(\tau_{\text{M}}-\sqrt{\frac{\tau_{\text{M}}^{2} -4\tau_{\text{M}}}{2\tau_{\text{M}}}})}{2\tau_{\text{M}}}}\]  
\[+\frac{\left(\tau_{\text{M}}{}^{2}\varepsilon\Omega+(\tau_{\text{M}})^{2} \varepsilon\Omega^{3}-\tau\tau_{\text{M}}\varepsilon\Omega\right)\text{cos}( \Omega\cdot t)}{(\tau_{\text{M}})^{2}\Omega^{4}-2\tau_{\text{M}}\Omega^{2}+ \tau_{\text{M}}{}^{2}\Omega^{2}+1}-\frac{\tau_{\text{M}}\varepsilon\text{-} \sin(\Omega\cdot t)}{(\tau_{\text{M}})^{2}\Omega^{4}-2\tau_{\text{M}}\Omega^{2} +\tau_{\text{M}}{}^{2}\Omega^{2}+1}. \tag{70}\]  
Using Equation of (56), we determine the positioning error  
\[\Psi(t)=-3.52.10^{3}.\frac{2\tau_{\text{M}}}{\tau_{\text{M}}+\sqrt{\tau_{\text{ M}}^{2}-4\tau_{\text{M}}}}\left(1-\text{e}^{\frac{t\cdot(-\tau_{\text{M}}-\sqrt{\frac{ \tau_{\text{M}}^{2}-4\tau_{\text{M}}}{2\tau_{\text{M}}}})}{2\tau_{\text{M}}}} \right)^{-1}\right)\]\[+3.52\cdot 10^{3}.\frac{2\tau_{\rm M}}{\tau_{\rm M}-\sqrt{\tau_{\rm M}^{2} -4\tau_{\rm M}}}\left(1-\left({\rm e}^{\frac{\tau(-\mathsf{M}+\sqrt{\zeta_{\rm M }^{2}-4\tau_{\rm M}})}{2\cdot\tau_{\rm M}}}\right)^{-1}\right)\] \[+\frac{\left(\tau_{\rm M}{}^{2}\epsilon+(\tau\tau_{\rm M})^{2} \epsilon\Omega^{2}-\tau\tau_{\rm M}\epsilon\right)\!\sin(\Omega\!\cdot\!t)}{( \tau\tau_{\rm M})^{2}\Omega^{4}-2\tau_{\rm M}\Omega^{2}+\tau_{\rm M}{}^{2} \Omega^{2}+1}+\frac{\tau_{\rm M}\epsilon\!\cdot\!(\cos\left(\Omega\!\cdot\!t \right)-1)}{\Omega\!\left((\tau\tau_{\rm M})^{2}\Omega^{4}-2\tau\tau_{\rm M} \Omega^{2}+\tau_{\rm M}{}^{2}\Omega^{2}+1\right)}\;. \tag{71}\]  
The time error simulation graphs for the first link are shown in \[\tau_{1}\tau_{\text{M}}\frac{d^{2}v_{1}}{dt^{2}}+\tau_{\text{M}}\frac{dv_{1}} {dt}+v_{1}=-\tau_{1}\tau_{\text{M}}\frac{d^{3}q_{\text{p1}}}{dt^{3}}-\tau_{ \text{M}}\frac{d^{2}q_{\text{p1}}}{dt^{2}}. \tag{68}\]0  
The analysis of the simulation results shows that the exact analytical solution of the system of differential equations for the dynamic positioning error enables the maximum accuracy of robot links positioning. For our case, the positioning error for the partial motion system is \(1.008\cdot 10^{-4}\) degrees, or \(6.048\cdot 10^{-3}\) min. More precise movement can be obtained by changing the drive characteristics of the robot links. For example, a change in the power supply voltage of the electric drive. The available elastic pliable elements cause elastic deformations. The calculation of elastic forces of deformation makes it possible to determine the vector of errors, which is the first column of the compliance matrix of the gripper.