106 A. Kist et al. Table 1 Parameters used in the simulation. For an exact formulation of the initial conditions of the system shown in fig. 2 and the trajectoryzex(t) see [18, 21]. We have also added a small damping to the Experimental Substructure to allow a more stable numerical simulation of it. variable value variable value fex 0.25Hz g 9.81ms−2 mNUM 9.6187kg mEXP 0.3813kg kNUM 10000kgs−2 dNUM 500kgs−1 kEXP 8650kgs−2 dEXP 10kgs−1 Lp 0.5 fQ,cut 4Hz Gd 800kgs−1 TLP 0.01s Simulink® Solver ode3 ∆TRT 0.001s ·10−2 7.4 7.2 7 6.8 1 1.5 2 t / s z / m 2.5 3 Reference RTHS: Trial 1 RTHS: Trial 20 (a) The interface displacement z0meas of the virtual RTHS experiment before and after convergence of the ILC compared to the reference solutionzref. 1 1.5 2 t / s 2.5 3 0 −5 −10 −15 −20 5 F / N (b) The interface forceF0ad of the virtual RTHS experiment before and after convergence of the ILC compared to the reference solution Fref. Fig. 4Result of a virtual RTHS experiment with a delay value of τ =10ms and the KUKA® KR16 as actuator without ILC (Trail 1) and after ILC convergence (Trial 20) compared to the reference solution. In fig. 5 the robustness of our controller to different amounts of delayτ in the virtual RTHS experiments is investigated. Again we use the KUKA® KR16 as actuator. We show the tracking error etrack over time for the different delay values without ILC (Trial 1) and after its convergence (Trial 20) in fig. 5a, while in fig. 5b we visualize the convergence of the ILC over the trials by calculating the relative root-mean-square (RMS) tracking error e RMS,rel track = RMS(etrack) MAX(|z|) per trial. It becomes apparent that our control scheme can cope with varying amounts of delay without retuning its parameters, since it significantly reduces the total tracking error and thus compensates for the delays. Note, however, that with a robustness filter Qless than 1 for frequencies above fQ,cut, the residual error after convergence is non-zero. Furthermore, this residual error depends on the transfer behavior of the reference system itself as well as the transfer behavior of the Transfer System. See [18] for the convergence analysis of the ILC in RTHS. This explains why the tracking error is not completely zero after convergence, and its final value also depends on the chosen delay value. In [18] we also show that a convergence criterion must be satisfied for ILC in RTHS. Thus, there is a limit to how much the delay τ can be increased without violating the convergence criterion. This can be seen in fig. 5 for the results for τ =60ms as the error diverges over the iterations. However, after retuning the ILC by adjusting the robustness filter Q, e.g. by choosingfQ,cut =2Hz, stable convergence could be achieved also for τ =60ms as shown by the light green values in fig. 5b. In fig. 6 we show the robustness of our controller to different actuators in the virtual RTHS experiments. The delay is chosen as τ =10ms for the simulations with all three actuators. Again, the applicability and robustness of our approach is demonstrated as the tracking error converges similarly for all three actuator types and a significantly lower residual tracking error is achieved after convergence. EXPERIMENTAL RESULTS The experimental realization of the RTHS test including the proposed control scheme is visualized in fig. 7. A dSpace® MicroLabBox dS1202 [23] is used to run the numerical time-integration of the Numerical Substructure and the control scheme in real-time. Both are developed in MATLAB®/Simulink® (version R2016b, MathWorks®) on a Host PC and compiled and exe- (a) The interface displacement z′meas of the virtual RTHS experiment before and after convergence of the ILC compared to the reference solutionzref. ·10−2 7.4 7.2 7 6.8 1 1.5 2 t / s z / m 2.5 3 Reference RTHS: Trial 1 RTHS: Trial 20 (a) The interface displacement z0meas of the virtual RTHS experiment before and after convergence of the ILC compared to the reference solutionzref. 1 1.5 2 t / s 2.5 3 0 −5 −10 −15 −20 5 F / N (b) The interface forceF0ad of the virtual RTHS experiment before and after convergence of the ILC compared to the reference solution Fref. Fig. 4Result of a virtual RTHS experiment with a delay value of τ =10ms and the KUKA® KR16 as actuator without ILC (Trail 1) and after ILC convergence (Trial 20) compared to the reference solution. In fig. 5 the robustness of our controller to different amounts of delayτ in the virtual RTHS experiments is investigated. Again we use the KUKA® KR16 as actuator. We show the tracking error etrack over time for the different delay values without ILC (Trial 1) and after its convergence (Trial 20) in fig. 5a, while in fig. 5b we visualize the convergence of the ILC over the trials by calculating the relative root-mean-square (RMS) tracking error e RMS,rel track = RMS(etrack) MAX(|z|) per trial. It becomes apparent that our control scheme can cope with varying amounts of delay without retuning its parameters, since it significantly reduces the total tracking error and thus compensates for the delays. Note, however, that with a robustness filter Qless than 1 for frequencies above fQ,cut, the residual error after convergence is non-zero. Furthermore, this residual error depends on the transfer behavior of the reference system itself as well as the transfer behavior of the Transfer System. See [18] for the convergence analysis of the ILC in RTHS. This explains why the tracking error is not completely zero after convergence, and its final value also depends on the chosen delay value. In [18] we also show that a convergence criterion must be satisfied for ILC in RTHS. Thus, there is a limit to how much the delay τ can be increased without violating the convergence criterion. This can be seen in fig. 5 for the results for τ =60ms as the error diverges over the iterations. However, after retuning the ILC by adjusting the robustness filter Q, e.g. by choosingfQ,cut =2Hz, stable convergence could be achieved also for τ =60ms as shown by the light green values in fig. 5b. In fig. 6 we show the robustness of our controller to different actuators in the virtual RTHS experiments. The delay is chosen as τ =10ms for the simulations with all three actuators. Again, the applicability and robustness of our approach is demonstrated as the tracking error converges similarly for all three actuator types and a significantly lower residual tracking error is achieved after convergence. EXPERIMENTAL RESULTS The experimental realization of the RTHS test including the proposed control scheme is visualized in fig. 7. A dSpace® MicroLabBox dS1202 [23] is used to run the numerical time-integration of the Numerical Substructure and the control scheme in (b) The interface force F′ad of the virtual RTHS experiment before and after convergence of the ILC compared to the reference solutionFref. Fig. 4 Result of a virtual RTHS experiment with a delay value of τ = 10ms and the KUKA® KR16 as actuator without ILC (Trail 1) and after ILC convergence (Trial 20) compared to the reference solution. clearly shows that our adapted control scheme achieves the desired behavior and proves its applicability, as the RTHS solution after convergence of the ILC is almost identical to the reference solution. Thus, the ILC is able to almost completely mitigate the total error induced by the transfer system on the interface displacement. In fig. 4b the interface force F′ad of the virtual RTHS experiment is shown compared to the reference solution Fref. The results show that the RTHS experiment predicts a slightly earlier contact after convergence of the ILC compared to the reference solution. The reason is that the ILC aims at shifting the motion command for the robot to compensate the total measured error on the interface displacement etrack =z −z′meas, which includes an additional delay for measuring, postprocessing, and transmitting the actual motion of the robot z′. The Experimental Substructure and the force measurement by the FTS, which is assumed not to be part of the robot, are not affected by this additional delay. Since the ILC takes this delay into account, it shifts the motion command too much from the perspective of the Experimental Substructure, resulting in the early contact compared to the reference solution. This aspect must be taken into account when interpreting the results when the RTHS experiment and the control scheme are implemented as presented. In addition, we mention that even if the NPC was in use, it barely intervened, since the RTHS loop was hardly active4 during the whole (virtual) experiment (The additional damping force was below0.2N during the whole experiment). 4In terms of passivity analysis, ’active’ refers to the case where the Transfer System induces additional energy into the RTHS loop, which was hardly the case here.
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