## 2 Materials and Methods
### Dynamic Error in the Partial System of Motion of Robot Manipulator Links
In Equation (23), the variable is the movement of the link \(q\), the controlling action is the \(U\) voltage, which is applied to the electric drive. Synthesis of the control law. The program control of \(U(t)\) was chosen from the condition of ideal characteristics of the drive motor:  
\[U(t)=\frac{S}{r}\cdot\frac{d\varphi(t)}{dt}=\frac{S}{r}\cdot k_{\mathrm{P}} \cdot\frac{dq(t)}{dt}\text{.} \tag{24}\]  
Equation of dynamic errors. The positioning error was set as  
\[\Psi=q-q_{\mathrm{P}}(t)\text{,} \tag{25}\]  
where \(q_{\mathrm{P}}(t)\) is the the actual movement of the link during time \(t\). The speed error was given as  
\[v=\frac{d\Psi}{dt}=\frac{dq}{dt}-\frac{dq_{\mathrm{P}}(t)}{dt}\text{.} \tag{26}\]  
After substituting the control law of (24) and the relation of \(q=q_{\mathrm{P}}+\Psi\) into the control Equation of (23), we obtain a system of differential equations for the \(\Psi\) error positioning  
\[ \begin{gathered} a\cdot\left(\frac{d^{2}q_{\mathrm{P}}(t)}{dt^{2}} +\frac{d^{2}\Psi}{dt^{2}}\right)+\mathrm{F}(q_{\mathrm{P}}+\Psi)=\eta_{ \mathrm{P}}\cdot k_{\mathrm{P}}\cdot\mathrm{M}\\ \tau\cdot\frac{dM}{dt}+\mathrm{M}=-S\cdot k_{\mathrm{P}}\cdot \frac{dq}{dt}\end{gathered} \text{.} \tag{27}\]  
After reducing the system of differential Equation (27) to one differential equation and simplifying, we obtain:  
\[\tau\tau_{\mathrm{M}}\frac{d^{3}\Psi}{dt^{3}}+\tau_{\mathrm{M}}\frac{d^{2} \Psi}{dt^{2}}+\frac{d\Psi}{dt}=-\tau\tau_{\mathrm{M}}\frac{d^{3}q_{\mathrm{P}}} {dt^{3}}-\tau_{\mathrm{M}}\frac{d^{2}q_{\mathrm{P}}}{dt^{2}}\]\[-\frac{1}{S^{*}}\left[\tau\left(\frac{d\text{F}_{\text{P}}}{dt}+\frac{d\text{F}_{ \text{P}}^{\prime}}{dt}\Psi+\text{F}_{\text{P}}^{\prime}\frac{d\Psi}{dt}\right)+ \text{F}_{\text{P}}+\text{F}_{\text{P}}^{\prime}\Psi\right], \tag{28}\]  
where \(\text{F}_{\text{P}}=\text{F}(q_{\text{P}})\); \(\text{F}_{\text{P}}^{\prime}=d\text{F}/dq_{\text{P}}\); \(S^{*}=\eta_{\text{P}}\cdot k_{\text{P}}^{2}\cdot S\); \(\tau_{\text{M}}\) is the time constant of the system of \(\tau_{\text{M}}=a/S^{*}\). After taking into account the \(1/S^{*}\) parameter as small and \(d\Psi/dt=v\)--the speed error from (28) we obtain the approximate equation  
\[\tau_{\text{M}}\frac{d^{2}v}{dt^{2}}+\tau_{\text{M}}\frac{dv}{dt}+v=-\tau_{ \text{M}}\frac{d^{3}q_{\text{P}}}{dt^{3}}-\tau_{\text{M}}\frac{d^{2}q_{\text{ P}}}{dt^{2}}. \tag{29}\]  
Modeling of dynamic errors. To solve Equation (29), let us assume a sinusoidal law of software acceleration change  
\[\frac{d^{2}q_{\text{P}}(t)}{dt^{2}}=\varepsilon\cdot\sin(\Omega\cdot t),\ \Omega=2\cdot p/t_{\text{P}} \tag{30}\]  
Then, the differential of dependence (30) will be  
\[\frac{d^{3}q_{\text{P}}(t)}{dt^{3}}=\varepsilon\cdot\Omega\cdot\cos(\Omega \cdot t). \tag{31}\]  
After substituting the value of the differentials (30) and (31) into the differential Equation (29), we obtain the differential equation  
\[\tau_{\text{M}}\frac{d^{2}v}{dt^{2}}+\tau_{\text{M}}\frac{dv}{dt}+v=-\tau\cdot \tau_{\text{M}}\cdot\varepsilon\cdot\Omega\cdot\cos(\Omega\cdot t)-\tau_{ \text{M}}\cdot\varepsilon\cdot\sin(\Omega\cdot t). \tag{32}\]  
We will solve the differential Equation (32) in the form of the sum of two decisions:  
\[v(t)=\overset{\sim}{v}(t)+\overset{\sim}{\overset{\sim}{v}}(t), \tag{33}\]  
where  
\[\overset{\sim}{v}(t)\equiv\tau_{\text{M}}\frac{d^{2}v}{dt^{2}}+\tau_{\text{M }}\frac{dv}{dt}+v\ ; \tag{34}\]  
\[\overset{\sim}{\overset{\sim}{v}}(t)\equiv-\tau\cdot\tau_{\text{M}}\cdot \varepsilon\cdot\Omega\cdot\cos(\Omega\cdot t)-\tau_{\text{M}}\cdot \varepsilon\cdot\sin(\Omega\cdot t). \tag{35}\]  
We equate the differential Equation (34) to zero and replace the \(v=\text{e}^{\lambda\cdot t}\) variable.