8 C. Insam et al. optimum scaling factor. However, if the scaling factor is too high, the algorithm diverges. In the tuning process for MFIIC, experience on different trajectories showed that a scaling value of αMFIIC =50 yields good results. The MFIIC algorithm was trained for 26 iterations. The identified transfer functions stated in Sect. 1.3.2 were used for MBDC and the state controller ˜C was set up using the LQ control law. The state controller contains only two states x for each leg, namely the leg length and the leg velocities coming from the modeled plant ˜GS,i. The weighting in the cost function of the LQ control law was chosen such that the control effort was almost neglected: The costs for slow and oscillating behavior were weighted 1012 (106) times higher than the cost for the voltage output for the leg length (leg velocity). For MBDC we use a low-pass filter with cutoff frequency at 100 Hz in the implementation. This dampens unwanted highly dynamic effects. This is necessary because system identification was carried out up to 100 Hz and therefore the dynamics above this frequency cannot be predicted correctly. 1.4 Results and Discussion This section presents the results of the experiments described in Sect. 1.3. The desired position trajectory that was carried out by the Stewart Platform varied in amplitude Aand frequency f. These were applied with each of the three feedforward control schemes (see Sect. 1.2) to investigate their tracking performance. Experiments were conducted with amplitudes of A ∈ {1, 3, 5}mm and frequencies of f ∈ {0.25,0.5,1, 2}Hz. We also performed experiments with higher frequencies, but all methods could only achieve maximum amplitudes of 0.5mm, therefore we do not show the results. This implies that mechanical limits like maximum velocity and motor voltages were achieved. 1.4.1 Convergence of MFIIC First, we needed to check the convergence of our implemented MFIIC algorithm. The results are shown in Fig. 1.5. In Fig. 1.5a, the desired trajectory and the achieved trajectory over the time interval 1 f are shown for the first, the 12th and the 26th (which was the last) iteration. As the iterations increase, the achieved displacement comes closer to the desired displacement, especially during the dynamic scenarios. However, when the direction of motion changes and the velocity approaches zero, it takes some time for the Stewart Platform to overcome the friction, thus the error is large at these points. The plot on the right, Fig. 1.5b, shows the root-mean-square (RMS) error between the desired reference signal td and the true 0 0.2 0.4 0.6 0.8 1 –1 –0.5 0 0.5 1 Time in s Achieved displacement inmm Desired 1st iteration 12th iteration 26th iteration (a) 0 10 20 30 10–3 10–2 10–1 Number of Iterations RMS error in mm2 MFIIC (b) Fig. 1.5 The error between the desired signal td,Z and the achieved displacement t Z inZ-direction using MFIIC as the feedforward control scheme is shown for f =1Hz and A=1mm. (a) Desired trajectory (black, dashed line) and achieved displacement over time in the 1st (blue), the 12th (green) and the 26th (orange) iteration. (b) RMS error over the number of iterations
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