## 2 Influence of External Factors and Gravity on the Movement of the MWR Mechanism
\begin{array}{ll}\frac{\varphi_{Ki}-\varphi_{Pi}}{T_ {1}}t+\varphi_{Pi},&\text{for $t<T_{1}$,}\\ -\frac{\varphi_{Ki}-\varphi_{Pi}}{T_{1}}t+\varphi_{Ki},&\text{for $T_{1}\leq t<T_{1}+T_{2}$,} \end{array} \right. \tag{3}\]  
\[\varphi_{szpi}(t\ =\ \left\{ \begin{array}{ll}(\varphi_{Ki}+\varphi_{Pi})( \frac{t}{T_{1}}-\frac{1}{2\pi}\sin(\frac{2\pi t}{T_{1}}))-\varphi_{Pi},&\text{ for $t<T_{1}$,}\\ -(\varphi_{Ki}+\varphi_{Pi})(\frac{t}{T_{2}}-\frac{1}{2\pi}\sin(\frac{2\pi t}{ T_{2}}))+\varphi_{Pi}\,&\text{for $T_{1}\leq t<T_{1}+T_{2}$,}\end{array} \right. \tag{4}\]  
where \(\varphi_{lzpi}(t)\)--linear function describing the end position and return in \(i\) = 1, 2, 3 axes, \(\varphi_{szpi}(t)\)--sinusoidal function describing the end position and return in the I axis, \(T_{1}\)--time to achieve end position, and \(T_{2}\)--time for a return to the initial position. At the beginning of the research, the mechanism was set in the initial position \(\varphi_{Pi}\) (Figure 4a,c,e), considering the maximum safe ranges of motion for the given movement. They were, respectively, as follows:  
* \(\varphi_{P1}\ =\ -15^{\circ}\ for\ \varphi_{1}(t)\),
* \(\varphi_{P2}\ =\ -35^{\circ}\ for\ \varphi_{2}(t)\),
* \(\varphi_{P3}\ =\ -70^{\circ}\ for\ \varphi_{3}(t)\). Then, the end positions \(\varphi_{Ki}\) were adopted (Figure 4b,d,f). In this way, the range of movements in the experimental studies were determined:  
* \(\varphi_{K1}\ =\ 25^{\circ}\) for \(\varphi_{1}(t)\),
* \(\varphi_{K2}\ =\ 20^{\circ}\) for \(\varphi_{2}(t)\),
* \(\varphi_{K3}\ =\ 60^{\circ}\) for \(\varphi_{3}(t)\). The research started with the linear input signal \(\varphi_{lzi}(t)\) for all axes by the given assumptions. The experiment time was \(T_{1}\ =\ 5\) s. The mechanism handle was not loaded during the tests. The research began with setting the initial position \(\varphi_{Pi}\). Then, the system was stopped, the boundary and safety conditions were checked, and the set signal \(\varphi_{lzi}(t)\), the end position \(\varphi_{Ki}\), and the controller settings were loaded. After checking the correctness of the above input data, the experiment was carried out, which consisted of exciting the movement by the input signal of \(\varphi_{lzi}(t)\). Analyzing the characteristics of the individual axes of the mechanism (Figure 5), it can be observed that only in the case of the \(\varphi_{2}\) axis was there a sudden drop in angular velocity at the beginning of the movement (Figure 5b). It reached the value of \(\omega_{2}\) = \(-13.15\) deg/s for \(t\) = 0.05 s. This was due to the mass of the elements, which the system must move at the start. The control system set the initial position \(\varphi_{2P}\) and then stopped the mechanism. Therefore, a precise active torque M\({}_{2}\) was required so that the mechanism started from this position. The angular velocity stabilized after \(t_{u}\) = 0.25 s. After reaching the set angular velocity value, the maximum difference between the actual value \(\dot{\varphi}_{2}\) and the calculated \(\ddot{\varphi}_{lz2}\) was a maximum of 2.05\({}^{\circ}\)/s. The position error increased until the angular velocity stabilized, then decreased, reaching the set value \(\varphi_{lz2}\) with each step. Figure 4: Range of motion for \(\varphi_{1}\) axis: initial positions \(\varphi_{\mathrm{IP}}\) (**a,c,e**) and end positions \(\varphi_{\mathrm{IK}}\) (**b,d,f**) for \(i\) = 1, 2, 3 for an anatomical range of motion, where the position of the hand is in the anatomical position of the body. The maximum value of the position error \(e_{2}(t)\ =\ 2.95^{\circ}\) was for \(t\) = 0.25 s, while the minimum value \(e_{2}(t)\ =0.05^{\circ}\) occured for \(t\) = 4.90 s (Figure 6). Figure 5: Characteristics of linear input \(\varphi_{12i}(t)\)-input signal, \(\varphi_{1}(t)\)-measurement result, \(\omega_{1}(t)\)-angular velocity: (**a**) adduction abduction movement \(\varphi_{1}(t)\), (**b**) palmar and dorsal flexion \(\varphi_{2}(t)\), (**c**) pronation and supination \(\varphi_{3}(t)\).