## 2 Influence of External Factors and Gravity on the Movement of the MWR Mechanism
Figure 6: Characteristics of the position error \(e_{2}(t)\) for linear input signal \(\varphi_{122}(t)\). The following experiment consisted of reaching the given angular position and returning to the initial position defined by the function \(\varphi_{lzpi}(t)\). To examine the system's response at higher angular velocities, the total motion duration was determined as \(T_{1}+T_{2}~{}=~{}7\) s. Consequently, it was required that the mechanism reach the predetermined position \(\varphi_{\text{K2}}\) in \(T_{1}~{}=~{}3.5\) s and then return to the original position \(\varphi_{P2}\) within \(T_{2}~{}=~{}3.5\) s. The experimental data of the \(\varphi_{2}\) axis, under the application of linear actuation for the complete cycle of \(\varphi_{lzp2}\), are shown in Figure 7a. Using a linear input signal, a sharp decrease in the angular velocity \(\omega_{2}\) right at the start of the test, dropping to \(-8.55^{\circ}\) per second at 0.25 s, can be noticed. This caused a slight positioning error in the mechanism right after it started \((-0.4^{\circ})\). Improvement can be seen when comparing this to the results from using the linear input signal \(\varphi_{lz2}\). However, angular velocity stabilized after 0.54 s, which was a disadvantage because a higher angular velocity was set in the experiment. After reaching the maximum position (\(\varphi_{2}~{}=~{}20^{\circ}\)), a sudden change in the angular velocity \(\omega_{2}\) can be noticed. This also resulted in angular position oscillations, which decreased to t = 4.25 s (\(t_{o}\) = 0.75 s). This was caused by the beginning of the second phase of the movement, in which there was a step change in the set speed \(\varphi_{lzp2}~{}=~{}-15.71^{\circ}_{\text{s}}\). The position error \(e_{2}~{}(t)\) increased until the angular velocity stabilized, then decreased until the system reached the highest position (\(\varphi_{2}~{}=~{}20^{\circ}\)). The maximum value of the position error \(e_{2}~{}(t)\) was 2.55\({}^{\circ}\) for t = 3.65 s, while the minimum value was 0.56\({}^{\circ}\) for t = 3.50 s (Figure 7b). Figure 7: Characteristics of linear repetition input: (**a**) \(\varphi_{lz2}(t)\)-input signal, \(\varphi_{2}(t)\)-measurement result, \(\omega_{2}(t)\)-angular velocity; (**b**) position error: \(e_{2}(t)\). The next stage consisted of enforcing the sinusoidal input signal \(\varphi_{sz2}\) for the \(\varphi_{2}(t)\) axis. Such an input reflects the real motion much better because it can distinguish between two phases of motion (acceleration and deceleration). The tests were conducted for the same parameters \(T_{1},\varphi_{P2},\) and \(\varphi_{K2}\) as the linear reference inputs. The results are shown below. For the sinusoidal input signal, one can observe a decrease in the angular velocity \(\omega_{2}\) at the beginning of the movement. It is similar to the linear input. Time \(t_{u}\) is also almost identical to a linear signal and is \(t_{u}\) = 0.25 s (Figure 8a). The position error \(e_{2}(t)\) increases significantly at the beginning of the movement until t = 0.65 s, where the regulation system gradually compensates. For the first half of the movement, it remains at about \(2^{\circ}\) and then decreases (Figure 8b). The subsequent tests were performed for one complete cycle, according to the \(\varphi_{szp2}\) function. The range of motion \(\varphi_{P2},\)\(\varphi_{K2}\) and its parameters \(T_{1}\ =\ T_{2}\ =\ 3.5\) s were the same as for the case of the linear input signal (Figure 9). With a sinusoidal input signal \(\varphi_{szp2}\), the angular velocity \(\omega_{2}\) decrease can be observed at the beginning of the movement. It reached the value of \(-23.85^{\circ}/\)s for t = 0.025 s and was much larger than in the case of the previous study. This resulted in an error in the angular position \(\varphi_{2}\) of the mechanism immediately after the start (\(-2.02^{\circ}\)). The reason for this was the higher input value of the angular velocity (Figure 9a).