## 4 Experiments and Results
### Influence of the Spring Constant  
In the last experiment, an influence of a spring constant \(k\) was taken into consideration. According to the Hook's law, the spring constant determines the size of deformation by force loading. The higher the force, the higher the deformation of the elastic body (e.g., spring). Thus, it is expected that the higher spring constants would deteriorate the performance of both used controllers. In line with this, the responses of the controllers by controlling the non-linear robotic mechanism loaded with springs of three various constants were applied in our experiments: \(k=420\) N/m, \(k=500\) N/m, and \(k=570\) N/m. However, in the previous experiment, where the step responses at reference velocity \(\omega_{r}=10\) rad/s and \(\omega_{r}=30\) rad/s were observed, the spring with constant \(k=420\) N/m was used. As a result, in this test, these results need to be supplemented with the results obtained by both controllers when the non-linear robotic mechanism was loaded with springs of constants \(k=500\) N/m and \(k=570\) N/m.  
As can be shown in Figure 12, it becomes more difficult for a DC-motor to overcome the load by tightening springs, or increasing the spring constant. Hence, reference current \(i_{r}\) significantly increases and occasionally hits the maximum reference current (\(\pm 1.5\) A). This means, transient peaks increase for linear PI-controller, while the same decrease for OAC-DES. The same finding applies also for disturbance peaks: these increase for the linear PI-controller and the same ones roughly decrease, or stay unchanged for the OAC-DES. The quality of the proposed controller and its benefit thus come into play at higher loads. However, a significant deterioration is observed for the OAC-DES in case of reference velocity \(\omega_{r}=0\). Compared to Figure 10, the OAC-DES restlessness undoubtedly increases. Restlessness most clearly becomes observable at current reference \(i_{r}\) step response and it is expected, that it will even worsen by further tightening the load. Linear PI-controller on the other hand constitutes smooth current reference \(i_{r}\), which treats the DC-motor more generously--does not cause instantaneous changes, and thus lower the restlessness. Despite this fact, linear PI-controller velocity step responses become unacceptable for a decent use in practice.  
Figure 13 depicts controllers step responses, when the spring constant is set at \(k=570\) N/m, which makes the DC-motor difficult to overcome the load. Therefore, the reference current \(i_{r}\) becomes  
Figure 12: Step response at reference velocity \(\omega_{r}=10\) rad/s, spring constant \(k=500\) N/m.  
regularly saturated at maximum actual current (\(\pm 1.5\) A). This fact causes the following phenomenon: when the spring is loaded at its peak, actual velocity \(\omega_{a}\) automatically drops and cannot attain the reference velocity \(\omega_{r}\). Phenomenon is not caused by controller itself and thus cannot be eliminated easily; the limited actual current should be increased to avoid it, but this might harm existing DC-motor, which should be replaced with a larger one in that case.  
Figure 13: Step response at reference velocity \(\omega_{r}=10\) rad/s, spring constant \(k=570\) N/m.  
As expected, by introducing the heaviest load, disturbance peaks for linear PI-controller become even worse, while transient peaks become partially eliminated. For the OAC-DES, velocity step response becomes occasionally biased due to the reference current saturation, but it exhibits a reliable performance otherwise. Again, reference current \(i_{r}\) restlessness becomes noticeable for the OAC-DES.