104 A. Kist et al. In order to drive the total measurable tracking error of the robot etrack =z −z′meas to zero, we implement an Iterative Learning Controller (ILC). In [10, 18] we used the ILC as a feedforward controller within the joint controller of the Stewart platform. In this work, we use the ILC scheme to directly adjust the task-space motion commandz to best compensate etrack. ILC is used when a motion command is executed several times in a row. It stores and processes the tracking error from the current trial j to improve the tracking error in the next trial j+1by learning to generate a control signal ∆ILC to compensate for the plant dynamics. The updating law for this control signal over the trials is: ∆ILC,j+1 =Q· (L· etrack,j +∆ILC,j) (1) Note that etrack,j(t) and ∆ILC,j(t) are the full time signals during the whole motion task of trial j, i.e. t ∈h0, 1 fex i . The learning function Lis used to process the error signal of trial j. To ensure the robustness and convergence of the ILC, the entire signal is processed with a robustness filter Q. In this work, the learning function is chosen as a P-controller, i.e.: L=Lp and the robustness filter is chosen as a zero-phase lowpass filter with a cutoff frequency fQ,cut. Our implementation of the ILC is a purely data-based approach that does not require modeling of the system. However, the parameters Lp and fQ,cut must be tuned to achieve stable convergence of the learning. A further discussion of our ILC implementation and its convergence analysis is omitted here and the reader is referred to our previous publications [10, 18]. The adjusted task-space motion command for the robot is thus zad =z +∆ILC. To enforce the equilibrium condition of the interface forces, the force F resulting from the robot’s motionz′ is measured at the interface of the Experimental Substructure using a force-torque sensor (FTS). The measured force F′, which is generally not equal to the actual interface force due to measurement errors in the FTS, is fed back to the Numerical Substructure. Note that we assume that this signal is not affected by the postprocessing delays in the robot, since we assume to use an additional FTS and do not rely on internal force/torque measurements in the robot. To ensure stable RTHS experiments, a Normalized Passivity Controller (NPC) is implemented1. The NPC monitors the power flow between the substructures and the Transfer System. If the power at the interface to the Experimental Substructure PEXP = ˙z′meas · F′ 2 is greater than the power at the interface to the Numerical Substructure PNUM = ˙z · F′ad, i.e. Perror = PEXP −PNUM ≥0, the experiment is considered active and therefore unstable. In this case, artificial damping force Fd is added to the force feedback to restore the passivity of the experiment. It is calculated as follows: Fd =(Gd · ˜Perror | ˜Ptot| · ˙z if ˜Perror >0 0 if ˜Perror ≤0 (2) Where Ptot = PEXP +PNUM, i.e. the total power, is used to normalize the error power Perror for the calculation of the damping gain. Gd is a tuning parameter. Note also that ˜• indicates filtering with a first order lowpass filter with time constant TLP to smooth the signals. Since the NPC already acts on the outer control loop and not within the robot’s low-level controllers, we have not modified our implementation of [13, 18] and it is referred to that publication for further information on our NPC scheme. The adapted measured interface force F′ad =F′ +Fd is then finally used in the next time step of the numerical simulation. Compared to our previous implementation of the scheme, the adapted control scheme presented here has the advantage of acting completely as an outer-loop controller. No interaction nor integration with internal computations in the hardware components, such as low-level actuator controllers, is required. The only requirements for the used robot/actuator are that its desired task-space motion can be commanded by an external device and that a measurement of the actual task-space motion is available. Apart from these requirements, with this control scheme the hardware components for the robot and the FTS are interchangeable. However, if different hardware components are used, the control parameters of the ILC and NPC may need to be adjusted, but a model of either of these components is not required. In addition, it should be noted that the ILC now aims to drive the total delay induced by the robot/actuator used to zero, thus including processing and communication delays and not just improving low-level actuator tracking performance. 1In our control approach, ILC aims to improve the test fidelity and NPC ensures stability [18]. Even though a high fidelity RTHS experiment is less likely to become unstable, since ILC reduces the tracking error iteratively, in the first iterations the error is usually still large and additional approaches are needed to ensure stability. In our case, this is addressed by the NPC. On the other hand, NPC alone cannot improve the fidelity of the test, as it only prevents instability. Hence the combination of NPC and ILC. 2The actual power at the interface to the Experimental Substructure would require using the actual interface force F. However, in a real experimental setup, only the measured force F′ is available, and we use this force to calculate the power at this interface.
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