Investigation of the Use of Commercial Robotic Arms for Real-Time Hybrid Substructuring 105 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 F0ad = F0 +Fd is then finally used in the next time step of the numerical simulation. z, ˙z z0, ˙z0 NPC Fd Transfer System Delayτ Delayτ Controlled Actuator Robot FTS ILC F0 F if Perror >0 PEXP PNUM Experimental Substructure g z0 F zex z Numerical Substructure g F0ad F0ad z0meas ∆ILC zad ˙z0meas Fig. 3Signal flow of the whole RTHS loop used in this work. It includes the control scheme using a combination of ILC and NPC to ensure robust and high-fidelity RTHS experiments, which we adapted from our previous work outlined in [18] to act as a pure outer-loop controller. Figure adapted from [21]. 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 taskspace 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. SIMULATIVE INVESTIGATION To validate the performance of the adapted control scheme presented in the previous section, we performed simulations, namely purely virtual RTHS experiments. Thereby, the Transfer System and the Experimental Substructure are also simulated numeri1In 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 F0 is available, and we use this force to calculate the power at this interface. Fig. 3 Signal flow of the whole RTHS loop used in this work. It includes the control scheme using a combination of ILC and NPC to ensure robust and high-fidelity RTHS experiments, which we adapted from our previous work outlined in [18] to act as a pure outer-loop controller. Figure adapted from [21]. Simulative Investigation To validate the performance of the adapted control scheme presented in the previous section, we performed simulations, namely purely virtual RTHS experiments. Thereby, the Transfer System and the Experimental Substructure are also simulated numerically. The virtual RTHS experiment is implemented in MATLAB®/Simulink® (version R2024a, MathWorks®). We use three different dynamic models for the actuator. To model the dynamics of industrial robotic arms, we use theRobotic System Toolbox inMATLAB®. In addition to the available functions for calculating the robot kinematics, we use the Joint Space Motion Model block inIndependent Joint Motionmode with its default parameters to simulate the closed-loop motion of the robot. The algorithms of this toolbox are based on the definition of a rigidBodyTree object in MATLAB®. To simulate theKUKA® KR16 robotic arm, we custom-built the rigidBodyTree object from CAD data. We also import a rigidBodyTree object for the ABB® IRB 120 robot arm, which is already importable from the Robotic System Toolbox. For our in-house Stewart platform, a dynamic and kinematic model is available from previous work. The dynamics of its legs have been identified up to 100Hz and a standard cascaded individual joint controller with velocity feedforward is used for joint control. See [17, 21] for more information on the Stewart platform, its system identification, and the implemented joint controller. To also model the effects of delays, two Transport Delay blocks are implemented within the signal flow as shown in fig. 3. These blocks delay a signal by the time delayτ. In our simulation we assume that the delay for the pre- and postprocessing is the same. We also assume ideal transfer behavior of the FTS, i.e. F′ =F. The control scheme is implemented as described in the previous section. In the virtual RTHS experiment, the Experimental Substructure is also simulated. The simplicity of the system design also allows for easy modeling of the Experimental Substructure as it is a simple one-dimensional oscillator. For a more stable numerical simulation, we added a small damper with damping ratio dEXP in parallel to the spring kEXP. To model ground contact, we activate a ground reaction force on mEXP as soon as it touches the ground. This reaction force is based on a nonlinear spring-damper model as described in [22] (note that we have increased the vertical ground stiffness of this model to106 kgm−2 to minimize ground penetration). In table 1 all relevant parameters for the simulation can be found, including the chosen Simulink® solver and the sample time ∆TRT. Figure 4 shows the result of a virtual RTHS experiment with a delay value3 of τ = 10ms and the KUKA® KR16 as actuator without ILC (Trial 1) and after ILC convergence (Trial 20) compared to the reference solution. The reference solution is the actual behavior of the system in fig. 2 and what the RTHS experiment should ideally predict. In fig. 4a the interface displacement z′meas of the virtual RTHS experiment is shown compared to the reference solution zref. The result 3Note that we implement two delays, one for receiving and preprocessing and one for postprocessing and sending, with the same value τ, cf. fig. 3. Thus, the total artificial delay in addition to the robot’s trajectory tracking errors is always 2τ.
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