River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Dynamic Substructures, Volume 4 Andreas Linderholt Matt Allen Walter D’Ambrogio Proceedings of the 38th IMAC, A Conference and Exposition on Structural Dynamics 2020 River Publishers
Conference Proceedings of the Society for Experimental Mechanics Series Series Editor Kristin B. Zimmerman, Ph.D. Society for Experimental Mechanics, Inc., Bethel, CT, USA
The Conference Proceedings of the Society for Experimental Mechanics Series presents early findings and case studies from a wide range of fundamental and applied work across the broad range of fields that comprise Experimental Mechanics. Series volumes follow the principle tracks or focus topics featured in each of the Society’s two annual conferences: IMAC, A Conference and Exposition on Structural Dynamics, and the Society’s Annual Conference & Exposition and will address critical areas of interest to researchers and design engineers working in all areas of Structural Dynamics, Solid Mechanics and Materials Research.
River Publishers Dynamic Substructures, Volume 4 Proceedings of the 38th IMAC, A Conference and Exposition on Structural Dynamics 2020 Andreas Linderholt • Matt Allen • Walter D’Ambrogio Editors
Published, sold and distributed by: River Publishers Broagervej 10 9260 Gistrup Denmark www.riverpublishers.com ISBN 978-87-7004-999-3 (eBook) Conference Proceedings of the Society for Experimental Mechanics An imprint of River Publishers © The Society for Experimental Mechanics, Inc. 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, or reproduction in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Preface Dynamic Substructures represents one of eight volumes of technical papers presented at the 38th IMAC, A Conference and Exposition on Structural Dynamics, organized by the Society for Experimental Mechanics, and held in Houston, Texas, February 10–13, 2020. The full proceedings also include volumes on Nonlinear Structures and Systems; Dynamics of Civil Structures; Model Validation and Uncertainty Quantification; Special Topics in Structural Dynamics & Experimental Techniques; Rotating Machinery, Optical Methods & Scanning LDV Methods; Sensors and Instrumentation, Aircraft/Aerospace, Energy Harvesting & Dynamic Environments Testing; and Topics in Modal Analysis & Testing. Each collection presents early findings from experimental and computational investigations on an important area within Structural Dynamics. Coupled structures, or substructuring, is one of these areas. Substructuring is a general paradigm in engineering dynamics where a complicated system is analyzed by considering the dynamic interactions between subcomponents. In numerical simulations, substructuring allows one to reduce the complexity of parts of the system in order to construct a computationally efficient model of the assembled system. A subcomponent model can also be derived experimentally, allowing one to predict the dynamic behavior of an assembly by combining experimentally and/or analytically derived models. This can be advantageous for subcomponents that are expensive or difficult to model analytically. Substructuring can also be used to couple numerical simulation with real-time testing of components. Such approaches are known as hardware-in-the-loop or hybrid testing. Whether experimental or numerical, all substructuring approaches have a common basis, namely the equilibrium of the substructures under the action of the applied and interface forces and the compatibility of displacements at the interfaces of the subcomponents. Experimental substructuring requires special care in the way the measurements are obtained and processed in order to assure that measurement inaccuracies and noise do not invalidate the results. In numerical approaches, the fundamental quest is the efficient computation of reduced order models describing the substructure’s dynamic motion. For hardware-in-the-loop applications, difficulties include the fast computation of the numerical components and the proper sensing and actuation of the hardware component. Recent advances in experimental techniques, sensor/actuator technologies, novel numerical methods, and parallel computing have rekindled interest in substructuring in recent years leading to new insights and improved experimental and analytical techniques. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Växjö, Sweden Andreas Linderholt Madison, WI, USA Matt Allen L’Aquila, Italy Walter D’Ambrogio v
Contents 1 Comparison of Feedforward Control Schemes for Real-Time Hybrid Substructuring (RTHS) ............... 1 Christina Insam, Mert Göldeli, Tobias Klotz, and Daniel J. Rixen 2 Proposed 12-DOF Shaker Control of BARC Structure ............................................................. 15 Kevin Napolitano and Melissa Schmidt-Landin 3 Mechanical Environment Test Specifications Derived from Equivalent Energy in Fixed Base Modes ........ 27 Troy J. Skousen and Randall L. Mayes 4 Implementing Experimental Substructuring in Abaqus ............................................................ 43 Benjamin Moldenhauer, Matt Allen, Daniel Roettgen, and Brian Owens 5 Vibration Test Design with Integrated Shaker Electro-Mechanical Models ...................................... 63 Ryan Schultz 6 Reproducing a Component Field Environment on a Six Degree-of-Freedom Shaker ........................... 73 Jelena Paripovic and Randall L. Mayes 7 In-Situ Source Characterization for NVH Analysis of the Engine-Transmission Unit .......................... 79 Ahmed El Mahmoudi, Francesco Trainotti, Keychun Park, and Daniel J. Rixen 8 Using Modal Projection Error to Predict Success of a Six Degree of Freedom Shaker Test .................... 93 Tyler F. Schoenherr, Janelle K. Lee, and Justin Porter 9 On Dynamic Substructuring of Systems with Localised Nonlinearities........................................... 105 Thomas Simpson, Dimitrios Giagopoulos, Vasilis Dertimanis, and Eleni Chatzi 10 Source Characterization for Automotive Applications Using Innovative Techniques ........................... 117 J. Harvie and D. de Klerk 11 Impact of Junction Properties on the Modal Behavior of Assembled Structures ................................ 127 Jean-Baptiste Chassang, Adrien Pelat, Frédéric Ablitzer, Laurent Polac, and Charles Pezerat 12 Quantifying Joint Uncertainties for Hybrid System Vibration Testing............................................ 131 Nadim A. Bari, Manuel Serrano, Safwat M. Shenouda, Stuart Taylor, John Schultze, and Garrison Flynn 13 Damping Identification and Model Updating of Boundary Conditions for a Cantilever Beam................ 139 Nimai Domenico Bibbo and Vikas Arora 14 An Experimental Substructure Test Object: Components Cut Out From a Steel Structure ................... 149 Andreas Linderholt 15 Frequency Based Model Mixing for Machine Condition Monitoring ............................................. 157 Samuel Krügel and Daniel J. Rixen 16 Using a Machine Learning Approach for Computational Substructure in Real-Time Hybrid Simulation ... 163 Elif Ecem Bas, Mohamed A. Moustafa, David Feil-Seifer, and Janelle Blankenburg vii
viii Contents 17 On the Stability of a Discrete Convolution with Measured Impulse Response Functions of Mechanical Components in Numerical Time Integration......................................................................... 173 Wolfgang Witteveen, Lukas Koller, and Florian Pichler 18 Development of an Electrodynamic Actuator for an Automatic Modal Impulse Hammer ..................... 189 Johannes Maierhofer and Daniel J. Rixen
Chapter 1 Comparison of Feedforward Control Schemes for Real-Time Hybrid Substructuring (RTHS) Christina Insam, Mert Göldeli, Tobias Klotz, and Daniel J. Rixen Abstract In order to meet the high demands in testing, actuators must be able to follow their desired displacement with high precision. Feedforward control enables high tracking performance of actuators. In combination with feedback controllers, an actuator can follow a prescribed trajectory quickly, stably and robustly under varying conditions. In RealTime Hybrid Substructuring (RTHS), a method where parts can be tested under realistic boundary conditions, high tracking performance of the actuator is vital. It not only increases fidelity of the RTHS test outcome—meaning that the test replicates the environment and boundary conditions of the test specimen well—but it also prevents the RTHS loop from becoming unstable. Hence, research is carried out in the field of control schemes being applied to RTHS systems. In this work, the existing cascaded feedback control of the position controlled Stewart Platform is expanded by three different feedforward control schemes: model-based dynamic feedforward, modeling-free iterative learning control and velocity feedforward. The tracking performances are compared and discussed using a commanded sine trajectory. Results reveal that modeling-free iterative learning control and velocity feedforward outperform model-based dynamic feedforward and follow the desired trajectory with high amplitude and phase accuracy. Velocity feedforward is simple and requires almost no implementation effort. Thus it is recommended for applications with stiff actuators. In contrast, modeling-free iterative learning control is recommended for tasks where the actuator is not stiff compared to the test specimen. As all these feedforward control schemes improve the tracking performance compared to feedback control, the fidelity of the RTHS test will improve using them. Keywords Feedforward control for RTHS · Parallel manipulators · Model-based dynamic feedforward · Modeling-free iterative learning control · Velocity feedforward 1.1 Introduction In Real-Time Hybrid Substructuring (RTHS), mechanical components can be tested with realistic boundary conditions. By realistic boundary conditions we mean that the mechanical component is excited by the same forces and displacements that it will be subject to in future applications. Using RTHS, we investigate whether the test specimen will withstand the loads in operation, dynamically influence the movement of the whole structure as intended and functions correctly. This is achieved by running a co-simulation of the structure surrounding the mechanical component (test specimen) in the future application while testing the test specimen on a test bench. The test specimen is referred to as the experimental part (EXP) and the co-simulated surrounding structure is referred to as the numerical part (NUM). 1 The co-simulation and the test specimen are coupled at their interface points in real-time by a so-called transfer system (TS). It consists of an actuator, a force-torque sensor and a digital signal processor [1]. Current applications of RTHS can be found in civil and mechanical engineering. Applications in civil engineering include testing of buildings under earthquake loads, testing of tall buildings under wind loads and testing of road/rail bridges under wind and wave loads. In mechanical engineering, the method has been applied to common problems in the aerospace and 1Note, that instead of naming it parts, one can also find the naming components or substructures in literature. C. Insam( ) · M. Göldeli · T. Klotz · D. J. Rixen Chair of Applied Mechanics, Faculty of Mechanical Engineering, Technical University of Munich, Garching, Germany e-mail: christina.insam@tum.de; mert.goeldeli@tum.de; tobias.klotz@tum.de; rixen@tum.de © The Society for Experimental Mechanics, Inc. 2021 A. Linderholt et al. (eds.), Dynamic Substructures, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47630-4_1 1
2 C. Insam et al. GN G −1 E Fext Fint − z GTS z NUM EXP Fint TS Fig. 1.1 Coupling in RTHS: The numerical part (NUM, in blue) outputs displacements z that are realized by the transfer system (TS, in orange). The experimental part (NUM, in green) reacts to these displacements by forces Fint automotive/transportation sector as well as in robotics and manufacturing systems. Olma et al. [2] In the automotive sector, e.g., semiactive Macpherson suspension systems [3] or vehicle axles [2] have been tested using RTHS. Biomechanical analysis of endoprosthesis have been performed by [4] and preliminary studies for testing of prosthetic feet have been carried out by [5]. A commonly used way to couple the RTHS loop is visualized in Fig. 1.1. The numerical part (NUM, transfer function GN, in blue) is simulated for one numerical time integration step. The solution of this time integration step yields the displacement at the interface points z. The displacements are applied to the experimental part (EXP, transfer function GE, in green)2 by the position controlled actuator (the upper part of TS, in orange). The experimental part reacts to the real displacements z 3 by restoring forces Fint that are measurable by the force-torque sensor (lower part of TS, in orange) at the interface. These forces F int are transferred to the numerical part by the negative sign (action and reaction) and act together with external forces Fext on the numerical part. The next numerical time integration step is performed. The dynamics of the digital signal processor are neglected here. In the case of ideal coupling, the actuator outputs z =z and the force-torque sensor outputs F int =Fint, i.e., GTS =1. If both conditions are satisfied, both parts—the numerical and the experimental—are in equilibrium and compatibility is fulfilled. In case compatibility and equilibrium are fulfilled, the RTHS test emulates the dynamic behavior of the overall structure perfectly. By overall structure we denote the virtual and experimental part combined in one mechanical system. If the true dynamic behavior is replicated, it is stated that the RTHS test has high fidelity. While the dynamics of the force-torque sensor are often negligible andF int =Fint can be assumed, the actuator introduces its own dynamics that are unwanted in RTHS. This is an important aspect that deteriorates the fidelity of the RTHS test. The test can also become unstable if the introduced dynamics and time lag are too large. Over the last two centuries, control schemes that aim at achieving better tracking, i.e., trying to achieve z =z, have been developed. One of the first ideas was called polynomial extrapolation and was developed by [6]. This control scheme alters the input z that is sent to the actuator but does not improve the position control of the actuator itself. Improved control schemes include adaptive controllers [7], sliding mode controllers with adaption layers [8] or model-based controllers [9]. In this contribution, we investigate three different types of feedforward controllers that extend an existing feedback control scheme. In Sect. 1.2, we introduce three different feedforward control schemes. The test bench and the experimental setup are described in Sect. 1.3. In Sect. 1.4, results are displayed and discussed. Lastly, in Sect. 1.5 we summarize the results and give recommendations. 1.2 Feedforward Control Schemes Control can be divided into feedforward and feedback control. Feedforward control calculates the input to the actuator based on the desired trajectory z. The goal is that the actuator follows the desired trajectory as fast and as accurately as possible, while maintaining stability and robustness. If the dynamic behavior of the actuator is perfectly known, one can find a feedforward controller that uses the inverted dynamics and perfect reference tracking can be achieved. Thus, the feedforward controller determines the command input response, i.e., the dynamics with which the actuator follows the desired trajectory. In reality however, there are modeling errors and disturbances. Hence an additional controller, called feedback controller, 2The transfer function is written as G−1 E in the figure, as in our definition of the transfer function it receives forces and outputs displacements. 3Real achieved values are denoted with a • throughout the whole paper.
1 Comparison of Feedforward Control Schemes for Real-Time Hybrid Substructuring (RTHS) 3 z z i Fint GS Pz PIv PIi Fint FF Memory z ud GTS id zFF ez utot u itot z z Fig. 1.2 The transfer system for RTHS experiments with actuator control and force feedback. The current P-PI-PI cascade to control the Stewart platform with transfer function GS is extended by feedforward control (FF). The signals that are needed for feedforward control are shown for model-based dynamic feedforward control in blue, model-free inversion-based iterative learning control in green and velocity feedforward in orange needs to be added to deal with such uncertainties. The parameters of the feedback controller determine the dynamic response to disturbances [10]. In this section, we explain how an existing cascaded control scheme (feedback control) can be expanded by three different types of feedforward controllers. Because it is intended to use the actuator within RTHS, Fig. 1.2 visualizes the entire transfer system with transfer function GTS. It comprises the actuator with the existing control loop and the signals that are used for feedforward control as well as the force feedback signal. The dashed line pointing fromFint to the robot representation indicates the actuator’s sensitivity with respect to external forces. This means that the actuator is not infinitely stiff, and depending on external forces, its dynamics change. The phenomenon is called control-structure interaction (CSI) and is further described in [11] for hydraulic actuators. The existing control loop of the Stewart Platform, which is our actuator and is explained in Sect. 1.3, can be seen in black in Fig. 1.2. This Stewart Platform is driven by electric motors. To control electric motors, cascaded control has proved to be successful [12]. The cascaded control consists of three loops, where the inner loop is the current control loop, the middle loop is the velocity control loop and the outer loop is the position control loop. The dynamics of the inner loop are the fastest, while the dynamics of the outer loop are the slowest. The position loop is controlled by a proportional controller Pz and outputs the desired velocity ˙zd. The velocity loop with velocity error ˙zd − ˙z is controlled by a proportional-integral controller PIv and outputs the desired current. The current loop with current error id −i is again controlled by a proportional-integral controller PIi and sends a voltage to the electric motors. 1.2.1 Model-Based Dynamic Feedforward In model-based dynamic feedforward control (MBDC), a model of the plant dynamics is used to generate an appropriate feedforward signal. The block diagram for MBDC is shown in Fig. 1.3. This feedforward controller is an independent control loop, where the controller ˜C in state space controls the model of the plant ˜GS. Note that the modeled transfer function for theplant GS is denoted by ˜GS. If the model of the actuator is perfect, i.e., ˜GS =GS and linear behavior can be assumed, the real actuator response is the same as the actuator response of the plant model in the feedforward controller [10, 13, 14]. In order to set up a state controller that controls the states x, the feedforward control matrices ˜Mx and ˜Mu are necessary, where the matrix ˜Mx outputs the desired states xd. This control loop is like a simulated control loop, where no disturbances occur. The control voltage ˜uthat is calculated using the state controller ˜Cis used as feedforward command signal and added to the output voltage ud of the P-PI-PI cascade (see blue arrows in Fig. 1.2). This scheme assumes linear behavior of the plant, which is not true for many actuators. It has the advantage that the state controller ˜Cin the feedforward controller can be
4 C. Insam et al. C Mu ∼ ∼ ∼ ∼ ∼ Mx GS z x xd u Fig. 1.3 In model-based dynamic feedforward control, a state controller ˜C is designed such that the command input response of the actuator is highly dynamic chosen with speed and agility because there are no disturbances, such that the command input response is quick [10, 13, 14]. The choice of the state controller ˜Cis arbitrary and possible methods are pole placement or a linear quadratic (LQ) control law. For our experiments, we chose the LQ control law, which is also known as the Riccati controller and is referred to as optimal control. It considers the time and oscillation that arises to go from one state to another state and also the effort (output voltage) that it takes to perform the state transition. A cost function containing both properties is set up, which is optimally fulfilled by the LQ control law. The two properties can be weighted depending on the desired control characteristics [15]. Note that the model of the actuator transfer function ˜GS needs to be identified in the experimental setup where it will be used. The reason is that its dynamics change depending on the external forces exerted on the actuator (CSI), which are the forces Fint coming from the experimental part in the case of RTHS. Because the goal is to cancel the dynamics of the actuator during operation, the dynamics need to be identified under operating conditions. A related approach, where the inverse of ˜GS is used to generate the feedforward signal, has been successfully applied to RTHS by [16–19]. Nevertheless, when inverting the actuator transfer function, additional dynamics need to be added in order to get a realizable inverse transfer function ˜GS−1 . MBDC circumvents this problem due to the independent state space control loop. 1.2.2 Model-Free Inversion-Based Iterative Feedforward Control Iterative learning control can be used for systems that perform a task repeatedly, such as scanning mechanisms of micromirrors, rotating discs or printers. From one iteration to the next, this control scheme learns from the error made and tries to reduce it by the iterative learning control law. In RTHS, applications also exist where the experiment can be done several times in a row and iterative learning control can be applicable. Hochrainer and Puhwein [20] applied iterative learning control to RTHS and showed its applicability and usefulness. The idea of iterative learning control was originally proposed by [21] and developed by [22–24] as early as in the 1980s. Over the past three decades, different methods have developed, some using model knowledge about the actuator from system identification, others trying to learn the transfer function through the cycles as well. In this work, we propose an implementation that is based on a model-free method called model-free inversion-based iterative feedforward control (MFIIC). This method was proposed by [25] and further improved by [26]. In MFIIC, the following data are recorded for the jth iteration: the position error signal ez,j = z −z j, the achieved displacement z j and the desired velocity ˙ zj, which is the output of the position controller Pz (see the green arrows in Fig. 1.2). As the reference trajectory z is the same in each iteration, we do not need to add the index j. For each time step of the entire cycle j +1, the feedforward signal is calculated by: ˙zFF,j+1 = ˙zFF,j + ˙ zj z j · ez,j, (1.1) where the feedforward command from the previous iteration j is denoted by ˙zFF,j. Thus, the feedforward signal that is added in iteration j +1 is based on signals from cycle j. The feedforward signal reduces the error, i.e., ez,j+1 < ez,j if
1 Comparison of Feedforward Control Schemes for Real-Time Hybrid Substructuring (RTHS) 5 the algorithm converges. The main idea is to approximate the inverse transfer function of the plant for each time step of the cycle by a static factor, which is ˙zj z j in our case. Here, the transfer function of the plant includes the PI-controllers for velocity and current (PIi and PIv) and the Stewart Platform (GS). This is because the feedforward signal is introduced on velocity level, which means that the transfer path that the feedforward signal is being put through includes the PI-controllers and the Stewart Platform. The original idea proposed by [25, 26] is slightly different: they use the feedforward signal ˙zFF to approximate the transfer function. Furthermore, they perform the calculation (1.1) in the frequency domain, meaning that all signals are transferred to the frequency domain by the Fourier transform, the calculation is performed and then the feedforward signal is transferred back to the time domain by the inverse Fourier transform. However, in our application the approach described in (1.1), which is in the time domain, performs better. Note that we could also inject the feedforward signal on the position, current or voltage level. In these respective cases, we would need to adapt (1.1) and use other signals to calculate the static factor that approximates the transfer function of the plant. In MFIIC no model knowledge of the actuator and external forces is necessary. Hence, it could be of great benefit in RTHS. The transfer function that is learned over the cycles represents the transfer function under operating conditions and possible CSI does not need to be considered. 1.2.3 Velocity Feedforward A very simple approach is velocity feedforward (VFF). Instead of only prescribing a position trajectory, its derivative, the velocity, is also prescribed [12]. Therefore, the feedforward block only contains a derivative block. The calculated feedforward signal must be injected on the velocity level, see orange errors in Fig. 1.2. This approach is only applicable if the existing control scheme contains a velocity controller, such as is the case in a P-PI-PI cascaded controller. The other two approaches can also be applied to standard PID-controllers or black box systems, as is the case in industrial robots. The approach of VFF is simple to implement. However, it does not consider CSI and thus is only successfully applicable if the actuator is stiff compared to the environment it is interacting with. 1.3 Experimental Setup As previously mentioned, we use a Stewart Platform as an actuator for our RTHS experiments. In order to make RTHS tests with high fidelity, good tracking performance of the actuator is vital. Hence, we want to improve the tracking performance of this Stewart Platform and extend the existing P-PI-PI cascade for position control by feedforward controllers. 1.3.1 Stewart Platform Stewart Platforms, as visualized in Fig. 1.4, consist of six legs that are variable in length. They belong to the group of parallel manipulators, which possess the property that they are stiff actuators with high dynamics. Our Stewart Platform is driven by six electric motors that are controlled by a cascaded controller. The upper platform has six degrees of freedom, i.e., translations in X, Y and Z and rotations about the respective axes, namely φ, θ, ψ. The existing P-PI-PI cascade for position control is implemented decentralized, i.e., each leg is controlled individually. Coupling between legs is assumed to be negligible and controlled by the feedback controller. 1.3.2 System Identification In order to implement MBDC, which was introduced in Sect. 1.2, system identification of the Stewart Platform needs to be performed, meaning that ˜GS needs to be found. As in the existing Stewart Platform, each leg is controlled independently and system identification of each leg is carried out individually as well. In literature, different ways for system identification are
6 C. Insam et al. Fig. 1.4 Stewart Platform used in the experiments stated, see e.g., [27]. The main idea is that all frequencies of interest are excited by the input variable and the output response is measured. In our Stewart Platform, the current control loop is implemented on the servo controllers (hardware) and the current signals cannot be accessed. We assume that the dynamics of the current control loop can be neglected as this loop is supposed to be very fast. Hence, the input signal for system identification of the Stewart Platform is itot and the output is z . The transfer function GS,i for each leg i =1. . . 6writes GS,i =GS,i(s) = Z i (s) Utot,i(s) ≈ Z i (s) Itot,i(s) . (1.2) The variables are written in capital letters to clarify that they are in the frequency domain with s = iω. The Laplace variable is denoted by s, but omitted throughout the rest of this paper. Because the Stewart Platform can only operate safely if it is controlled by a feedback controller, we decided to use closed-loop system identification. In closed-loop system identification it is possible to identify the controlled plant without the input of the controller, even though the feedback controller is active. For this, the requirement is that the correlation between the excitation signal and the plant output (z i) is high, i.e., coherence must be guaranteed (as a rule of thumb larger than 75%). There are different types of excitation signals that can be used, e.g., white noise, pseudo-random binary signal or multi-sine signal. Based on experience, we chose the multi-sine signal for excitation, which is a sum of multiple sine signals. The phases of these sine signals during summation need to be chosen according to the so-called Schroeder phases. The Schroeder phases distribute the sine signals in the time domain and thus spread the input power [27]. The system identification was performed in the frequency range from 0.5 to 100 Hz for each leg. We selected a frequency resolution of 0.5 Hz. In order to inject enough energy for each individual frequency during the experiments, we split the whole frequency range into 10 smaller frequency ranges, namely[0.5,10], [10.5, 20], . . . , [90.5,100] Hz. We performed 10 measurements for each leg, where the multi-sine signal in each of the experiments comprised 20 different frequencies. Each measurement was performed for a time span of 200 s. In the post-processing of the measured data, a phase correction needs to be performed. The sampling frequency during measurement was 1000 Hz, i.e., the sampling time was 1 ms. The discrete measuring introduces frequency-dependent phase delay, as during 1 ms the oscillation goes on. For example, for the measurement at 100 Hz, a sampling frequency of 1000 Hz means that a phase of −36◦ is artificially introduced. Hence, the respective phase needs to be subtracted from the measured phase values of GS,i to obtain the real phase values of GS,i. Based on considerations in [28], the dynamic behavior of the Stewart Platform can be represented with PT1-dynamics and an integrator in this frequency range. In the considered frequency range, only dynamics coming from the mechanical system are relevant. The dynamics from the electric system and the controllers are relevant in a higher frequency regime and do not
1 Comparison of Feedforward Control Schemes for Real-Time Hybrid Substructuring (RTHS) 7 Table 1.1 Summary of the transfer functions of the Stewart Platform’s legs i =1. . . 6 i =1 i =1 i =3 i =4 i =5 i =6 ˜GS,i 1011 s·(s+122.9) 1053.1 s·(s+55.5) 1123.5 s·(s+54.7) 780.9 s·(s+55.3) 988.1 s·(s+66.2) 1070.9 s·(s+59) have an influence up to 100 Hz. Therefore, we fitted a transfer function ˜GS of the following form to the measured data of the real plant GS: ˜GS,i(s) = k s · (Ts +1) = k · ωc s · (s +ωc) , with ωc = 1 T . (1.3) This dynamic behavior can also be interpreted as a mass with damping and no stiffness. This is quite illustrative, as the torques coming from the electric motors need to push the mass of the Stewart Platform’s legs and upper platform, which possess damping (friction, compliance, . . . ) but almost no stiffness. We identified the transfer functions listed in Table 1.1 for each leg. The cutoff frequencies, i.e., the frequency above which the amplitudes fall with a slope of −40 dB/decade is at frequencies of approximately 8–10 Hz. The amplitude falls below zero above 2–3 Hz, which implies that the input signal is mainly attenuated above and the dynamic limits of the Stewart Platform are in this range. The task of the feedforward controllers is to optimally use and even increase the dynamic range of the actuator to its mechanical limits such as velocity limit and maximum motor voltage. 1.3.3 Benchmark Problem To compare the tracking performance of the three different feedforward control schemes in combination with the existing P-PI-PI cascade, the following benchmark problem is used: A sine trajectory with frequency f and amplitude Aaround the initial positiont0 should be followed. Hence, the desired position trajectory td is of the form td(t) = ⎛ ⎜⎜ ⎜⎜ ⎜⎜ ⎜⎝ td,X td,Z td,Z td,φ td,θ td,ψ ⎞ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎠ =t0 + ⎛ ⎜⎜ ⎜⎜ ⎜⎜ ⎜⎝ 0 0 A· sin(2πft) 0 0 0 ⎞ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎠ , with t ∈ 0, 1 f (1.4) and the Stewart Platform is at its initial position t0 for t < 0 and t > 1 f . This trajectory command only prescribes the trajectory in Stewart Platform’s Z-direction, which is the vertical direction, and the other translations (XandY) and rotations (φ, θ, ψ) are set to 0. This trajectory commandtd is transformed by inverse kinematics, which gives the kinematic relationship between the upper platform and the leg lengths, to the desired leg lengths zi (compare to Fig. 1.2), for each leg i =1. . . 6. Between two successive sine trajectories there was a pause of 1 s at positiont0. 1.3.4 Parameters Setting for the Experiments To keep the results comparable, all experiments have been conducted several times with the same conditions, such as the same maximum actuator velocity of 40 mm s and the same constants in the P-PI-PI cascaded feedback controller. For MBDC, a scaling factor αMBDC was introduced to scale the feedforward signal. This is necessary because of the nonlinear transfer behavior of the Stewart Platform, i.e., the amplification from input voltage to leg displacement depends on the amplitude of the input voltage. In the tuning process, a continuous sine command with f = 2 Hz was used and the factor αMBDC was tuned for each leg to its optimum. The obtained values lie in the range of (1.8,3.8). Likewise, a scaling factor αMFIIC was used for MFIIC. Here, the scaling factor determines the convergence rate and the achievable minimum error with this scheme. If the scaling factor is chosen too small, the convergence is slow and the remaining error is large compared to the
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
1 Comparison of Feedforward Control Schemes for Real-Time Hybrid Substructuring (RTHS) 9 achieved displacement t inZ-direction (td,Z and t Z). The RMS error was calculated by rms = f 1000 · 1000 f i=1 td,Z −t Z 2 , (1.5) because the sampling frequency was 1000 Hz and thus the number of measured samples over the time period 1 f was 1000 f . The error in the first iteration corresponds to the error of the P-PI-PI cascade, as learning begins in the second iteration based on the error signal from the first iteration. We see that the algorithm converges and the error decreases over time. Our maximum number of iterations was 26 and convergence was in the order of ≈1 j2 , where j counts the iterations. This means that doubling the number of iterations reduces the error to 1 4 in the considered range. However, there will be a convergence limit due to mechanical limits and maximum motor voltages. Nevertheless, the potential of MFIIC and its convergence looks promising. Using MFIIC, the learned feedforward signal can be saved and used in future experiments. Hence, the results shown in the following sections take the feedforward signal of MFIIC in its converged state, i.e., from the 26th iteration. 1.4.2 Comparison of the Feedforward Control Schemes We first show the representative course over time for A = 1mm and f = 0.5 Hz for the existing P-PI-PI cascade and the three different feedforward control schemes in Fig. 1.6. It can be seen that the achieved displacement t Z comes closer to the desired displacement td,Z for all feedforward control schemes. In Fig. 1.6b, the error between desired and achieved displacement, i.e., tdZ −t Z, is visualized. Here, it is also obvious that all feedforward control schemes reduce the error. The MFIIC and VFF especially reduce the magnitude of the error to approximately 10–20%. The largest magnitude of the error occurs at the turning points of the trajectory, where the velocity is zero and static friction has large influence. In order to make general statements about the tracking performance of all methods for different trajectories, we investigate the results for all amplitudes A∈ {1, 3, 5}mm and frequencies f ∈ {0.25,0.5,1, 2}Hz in a synchronization subspace plot (SSP). The SSP plot can be used to analyze the tracking performance for sine trajectories. The desired displacement td,Z is plotted over the achieved displacement t Z. In the case of ideal tracking, a slope with an incline of 45◦ results. If the incline is lower it means that amplitude overshoot has occurred, if the incline is higher, it means that the Stewart Platform undershot. An ellipse forms in the case of phase errors: In the case of a phase lag, the ellipse is passed through clockwise, and in the case of phase lead the ellipse is passed through counterclockwise. Figure 1.7 visualizes the SSP plots for the measured amplitudes and frequencies. All amplitudes A∈ {1, 3, 5}mmwere only investigated for frequencies f ∈ {0.25,0.5}Hz. For frequency f =1 Hz we investigated amplitudes A ∈ {1, 3}mm 0 0.5 1 1.5 2 –1 –0.5 0 0.5 1 Time in s Displacement inmm Desired P-PI-PI MBDC MFIIC VFF (a) 0 0.5 1 1.5 2 –0.2 –0.1 0 0.1 0.2 Time in s Error inmm P-PI-PI MBDC MFIIC VFF (b) Fig. 1.6 The plots show the results for the P-PI-PI cascade (in black, dashed line), the MBDC (in blue, solid line), the MFIIC (in green, solid line) and the VFF (in orange, solid line) for the desired trajectory withf =0.5Hz andA=1mm. (a) The desired (td,Z) and achieved displacement t Z in Z-direction. (b) The error signal between desired and achieved displacement over time
10 C. Insam et al. f = 0. 25Hz ¨ –1 –0.5 0 0.5 1 –1 –0.5 0 0.5 1 td,Z inmm A= 1 mm ¨ –2 0 2 –2 0 2 A= 3 mm –4 –2 0 2 4 –4 –2 0 2 4 A= 5 mm f = 0. 5Hz ¨ –1 –0.5 0 0.5 1 –1 –0.5 0 0.5 1 td,Z inmm –2 0 2 –2 0 2 –4 –2 0 2 4 –4 –2 0 2 4 f = 1 Hz –1 –0.5 0 0.5 1 –1 –0.5 0 0.5 1 td,Z inmm –2 0 2 –2 0 2 f = 2 Hz −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 tZ inmm td,Z inmm Fig. 1.7 Synchronization subspace plots are shown for amplitudes A∈ {1, 3, 5}mm and frequencies f ∈ {0.25, 0.5, 1, 2}Hz. The results of the P-PI-PI cascade (in black, dashed), the MBDC (in blue, solid), the MFIIC (in green, solid) and the VFF (in orange, solid) are shown
1 Comparison of Feedforward Control Schemes for Real-Time Hybrid Substructuring (RTHS) 11 and for f = 2 Hz we only investigated A = 1 mm. The reason is that our velocity limit was set to 60 mm s throughout the experiments to prevent the Stewart Platform from damage and execution of these experiments included such high velocities. From the SSP plots, it is clear that all feedforward schemes (MBDC, MFIIC, VFF) significantly improve the phase error compared to the P-PI-PI cascade alone. The P-PI-PI cascade suffers from phase lag (the ellipse in the SSP plot are passed through clockwise) and for higher frequencies (f ∈ {1, 2}Hz) from amplitude errors (undershoot) as well. For f =0.25Hz, the MBDC improves the tracking performance only slightly. For f =1 Hz, the amplitude error is larger than for the P-PI-PI cascade, though the phase error decreases. For f = 2 Hz, MBDC outperforms the P-PI-PI cascade, because the optimum value αMBDC for MBDC was tuned for each leg at the excitation frequency of f =2 Hz. MBDC suffers from amplitude undershoot, especially for higher frequencies, which can be observed from its incline >45◦ in the SSP plot. MFIIC and VFF perform comparably. They reduce the phase and amplitude error the most for all frequencies and amplitudes. The tracking performance achieved is almost ideal (straight line, 45◦ incline) in the investigated frequency range. 1.4.3 Coupling Between Directions The Stewart Platform consists of six legs and each of the legs is actuated and controlled independently and coupling effects are neglected. Thus, we need to investigate how large the effect of coupling is and how much the other directions are influenced by a desired movement in Stewart Platform’s Z-direction. We representatively show the movement during one sine trajectory with A = 1mm and f = 0.5 Hz in Fig. 1.8 for the X-direction, i.e., t X. The results are similar for the Ydirection as well as the rotations φ, θ and ψ. We can see that for all methods, the movement induced in X-direction is of the same order of magnitude and the maximum value is 0.04 mm. As the desired amplitude is A = 1 mm, i.e., the entire movement inZ-direction is 2 mm, this induced movement is <5%. We consider this to be acceptable for our ongoing RTHS experiments. 1.4.4 Discussion In this study, we aim to find an appropriate feedforward control scheme that improves the tracking performance of the Stewart Platform. The Stewart Platform is used for RTHS experiments, where the tracking performance of the actuator substantially influences the quality of the RTHS test outcome. The better the tracking performance, the higher the fidelity of the RTHS test. There are also other aspects than just the tracking performance that determine the applicability of the schemes to RTHS, which are investigated here. Table 1.2 gives advantages and disadvantages for MBDC. Model-based schemes require a significant amount of effort in the system identification procedure. The quality of system identification determines the quality of the model-based 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 –4 –2 0 2 4 ·10–2 Time in s Displacement in mm P-PI-PI MBDC MFIIC VFF Fig. 1.8 Induced movement inX-direction during a desired trajectory inZ-direction
12 C. Insam et al. Table 1.2 Pros and cons of the dynamic feedforward control scheme (MBDC) Pros Cons Fine-tuning possible (αMBDC) System identification necessary Dynamics of command response can be chosen as desired Scaling of signal needed (αMBDC) No inversion of Gneeded Experimental setup needs to be known during system identification CSI can be considered in system identification Discontinuities in demanded signal cause jerk in feedforward Table 1.3 Pros and cons of model-free inversion based iterative feedforward control (MFIIC) Pros Cons Fine-tuning possible Determination of αMFIIC by trial No preparation effort (modeling/system identification) Conducting of experiments time-consuming CSI implicitly considered and thus flexible No convergence if RTHS unstable in first iteration Good convergence and great tracking performance achievable Not applicable if time-varying material behavior or investigation of bifurcations (rupturing, crack propagation) are investigated with RTHS Table 1.4 Pros and cons of velocity feedforward (VFF) Pros Cons Least time consuming: no preparation and during experiment no effort In small frequencies worse than MFIIC (small velocity) Error feedforward to further improve Only applicable if actuator stiff compared to experimental part (in our case moving freely without payload, hence fulfilled) Minimal effort, significant achievement / feedforward control. Nevertheless, system identification helps us understand the dynamic behavior of the actuator and reveals the dynamic potential of the actuator. In RTHS, the actuator interferes with dynamically unknown experimental parts EXP. In order to have perfect dynamic compensation of the actuator, the system identification needs to be performed with the experimental part already mounted on the actuator. This makes system identification necessary for each experimental specimen and each experimental setup. If the test specimen’s resonance frequency is being met during system identification, the test specimen could be damaged during this process and cannot be used for the RTHS experiment. Another difficulty arises with this feedforward control scheme if the dynamics of the test specimen change over time, and hence, change during system identification. This leads to inaccurate RTHS experiments. A great advantage, however, is that it can be used for black-box systems, such as industrial robots. System identification can be performed without having access to inner-loop signals and also the feedforward signal can be injected on position level. In Table 1.3, the advantages and disadvantages of MFIIC are listed. One great advantage is that no preparatory work prior to the RTHS test is necessary. The dynamic behavior of the interaction between the actuator and the experimental part are learned throughout the iterations of the RTHS test. Because the MFIIC needs to be trained over several iterations, the execution of the experiments is time consuming. Similar to MBDC, MFIIC cannot be applied, if the test specimen exhibits dynamic behavior that changes over time. There are also RTHS experiments where the test specimen is intended to break throughout the experiment. These kinds of experiments cannot be conducted using MFIIC as the feedforward control scheme. Another difficulty arises if the RTHS experiment is unstable during the first iteration. This happens if the time delay and dynamics of the actuator are too large. In this situation, the actuator cannot learn the correct dynamic behavior. One possibility would be to combine passivity-based approaches with MFIIC to circumvent this problem. The advantages and disadvantages of VFF are listed in Table 1.4. It is superior to the other two methods regarding preparatory time and time to execute the test, as it is straightforward to implement and no convergence problems arise. This scheme works with high accuracy if the stiffness of the actuator can be regarded high compared to the test specimen, meaning that CSI is low. However, if this is not the case, VFF does not further improve the tracking performance as e.g., MBDC and MFIIC. Neither of the feedforward control schemes observes passivity of the coupled RTHS experiments and whether the RTHS experiment is stable or not. An additional observer should be implemented to guarantee stability and thus safety for the users and test specimen.
1 Comparison of Feedforward Control Schemes for Real-Time Hybrid Substructuring (RTHS) 13 1.5 Conclusion In this work, we investigate the applicability of three different feedforward control schemes to improve the tracking performance of a Stewart Platform. The Stewart Platform is used for Real-Time Hybrid Substructuring (RTHS) tests and the existing position controller is a P-PI-PI cascaded feedback controller. The methods that we investigated are modelbased dynamic feedforward (MBDC), model-free inversion-based iterative feedforward (MFIIC) and velocity feedforward (VFF). All methods improve the tracking performance of the Stewart Platform. MBDC is a feedforward control scheme that requires significant effort for system identification, but enables the testing of experimental parts that dynamically influence the movement of the actuator. MFIIC learns the dynamic interaction between the actuator and the experimental part. It can be applied to improve tracking performance in RTHS if the RTHS test results are inaccurate but stable. VFF is a very simple, yet powerful approach, as it achieves high accuracy. Its successful application is limited to actuators that are stiff compared to the experimental specimen, because it cannot improve tracking performance if the actuator’s dynamic behavior is influenced by the experimental part. Future work will include investigations of the applicability of feedforward schemes for RTHS tests. References 1. Saouma, V., Sivaselvan, M.: Hybrid Simulation: Theory, Implementation and Applications. Taylor & Francis, London (2008) 2. Olma, S., Kohlstedt, A., Traphöner, P., Jäer, K.-P., Trächtler, A.: Observer-based nonlinear control strategies for Hardware-in-the-Loop simulations of multiaxial suspension test rigs. Mechatronics 50, 212–224 (2018) 3. Fallah, M.P.M., Bhat, R., Xie, W.F.: Optimized control of semiactive suspension systems using H-infinity robust control theory and current signal estimation. IEEE/ASME Trans. Mechatron 17, 767–778 (2012) 4. Herrmann, S.W.: Dynamic testing of total hip and knee replacements under physiological conditions by. Dissertation (2014) 5. Insam, C., Bartl, A., Rixen, D.J.: A step towards testing of foot prosthesis using real-time substructuring (RTS). In: IMAC – Conference & Exposition on Structural Dynamics (2019) 6. Horiuchi, T., Inoue, M., Konno, T., Namita, Y.: Real-time hybrid experimental system with actuator delay com- pensation and its application to a piping system with energy absorber. Earthq. Eng. Struct. Dyn. 28(10), 1121–1141 (1999). https://doi.org/10.1002/(SICI)1096-- 9845(199910)28:10$<$1121::AID-EQE858$>$3.0.CO;2-O 7. Bartl, A., Mayet, J., Rixen, D.J.: An adaptive approach to coupling vibration tests and simulation models with harmonic excitation. In: IEEE/ASME International Conference on Advanced Intelligent Mechatronics(AIM), pp. 262–267 (2019) 8. Maghareh, A.: Nonlinear robust framework for real-time hybrid simulation of structural systems: design, imple- mentation, and validation. Dissertation, Purdue University (2017) 9. Carrion, J., Spencer, B.F.: Real-time hybrid testing using model-based delay compensation. Smart Struct. Syst. 4 (2006). https://doi.org/10. 12989/sss.2008.4.6.809 10. Föllinger, O.: Regelungstechnik. ISBN:978-3-8007-3231-9. VDE Verlag (2013) 11. Dyke, S.J., Spencer, B.F., Quast, P., Sain, M.K.: Role of control-structure interaction in protective system design. J. Eng. Mech. 121(2), 322–338 (1995). https://doi.org/10.1061/(ASCE)0733-9399(1995)121:2(322) 12. Sygulla, F., Wittmann, R., Seiwald, P., Berninger, T., Hildebrandt, A.-C., Wahrmann, D., Rixen, D.: An EtherCAT- based real-time control system architecture for humanoid robots, pp. 483–490 (2018). https://doi.org/10.1109/COASE.2018.8560532 13. Pellegrini, E., Spirk, S., Pletschen, N., Lohmann, B.: Experimental validation of a new model-based control strategy for a semi-active suspension system. In: 2012 American Control Conference (ACC), pp. 515–520 (2012) 14. Pellegrini, E., Pletschen, N., Spirk, S., Rainer, M., Lohmann, B.: Application of a model-based two-DOF control structure for enhanced force tracking in a semi-active vehicle suspension. In: 2012 IEEE International Conference on Control Applications, pp. 118–123 (2012) 15. Anderson, B.D.O., Moore, J.B.: Optimal Control: Linear Quadratic Methods. Prentice Hall PTR, Englewood Cliffs (1990) 16. Carrion, J., Spencer, B.F.: Model-based strategies for real-time hybrid testing. In: NSEL Report Series (2007) 17. Phillips, B.M., Spencer, B.F.: Model-based multiactuator control for real-time hybrid simulation. J. Eng. Mech. 139(2), 219–228 (2013). https:// doi.org/10.1061/(ASCE)EM.1943-7889.0000493 18. Phillips, B.M., Spencer, B.F.: Model-based feedforward-feedback actuator control for real-time hybrid simulation. J. Struct. Eng. 139(7), 1205–1214 (2013). https://doi.org/10.1061/(ASCE)ST.1943-541X.0000606 19. Fermandois, G.A., Spencer, B.F.: Model-based framework for multi-axial real-time hybrid simulation testing. Earthq. Eng. Eng. Vib. 16(4), 671–691 (2017). https://doi.org/10.1007/s11803-017-0407-8 20. Hochrainer, M.J., Puhwein, A.M.: Investigation of nonlinear dynamic phenomena applying real-time hybrid simulation. In: Kerschen, G. (ed.) Nonlinear Structures and Systems, vol. 1, pp. 125–131. Springer International Publishing, Cham (2020) 21. Uchiyama, M.: Formation of high-speed motion pattern of a mechanical arm by trial. Trans. Soc. Instrum. Control Eng. 14(6), 706–712 (1978). https://doi.org/10.9746/sicetr1965.14.706 22. Arimoto, S., Kawamura, S., Miyazaki, F.: Bettering operation of robots by learning. J. Robot. Syst. 1(2), 123–140 (1984) 23. Casalino, G., Bartolini, G.: A learning procedure for the control of movements of robotic manipulators. In: IASTED Symposium on Robotics and Automation, pp. 108–111 (1984) 24. Craig, J.: Adaptive control of manipulators through repeated trials. In: Proceedings of ACC, San Diego (1984) 25. Kim, K., Zou, Q.: A modeling-free inversion-based iterative feedforward control for precision output tracking of linear time-invariant systems. IEEE/ASME Trans. Mechatron. 18(6), 1767–1777 (2013). https://doi.org/10.1109/TMECH.2012.2212912
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