Friction Effects in Mechanical System Dynamics and Control

Transcription

1 Budapest University of Technology and Economics Faculty of Mechanical Engineering Department of Mechatronics, Optics and Mechanical Engineering Informatics Friction Effects in Mechanical System Dynamics and Control Csaba Budai Supervisors: Dr. László Kovács Budapest University of Technology and Economics, Budapest, Hungary McGill University, Montréal, Québec, Canada Dr. Péter Korondi Budapest University of Technology and Economics, Budapest, Hungary A thesis submitted to the Géza Pattantyús-Ábrahám Doctoral School of Mechanical Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy March 2017, Budapest

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3 In Memory of my Grandmother, Mária Nagy ( )

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5 Nyilatkozat Alulírott, Budai Csaba kijelentem, hogy ezt a doktori értekezést magam készítettem, valamint abban csak a megadott forrásokat használtam fel. Minden olyan részt, amelyet szó szerint, vagy azonos tartalomban, de átfogalmazva más forrásból átvettem, egyértelműen, a forrás megadásával megjelöltem. Budapest, március Budai Csaba A dolgozat bírálatai és a védésről készült jegyzőkönyv a későbbiekben, a Budapesti Műszaki és Gazdaságtudományi Egyetem, Gépészmérnöki Karának Dékáni Hivatalában érhetőek el.

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7 ACKNOWLEDGEMENTS Acknowledgment First of all, I would like to thank to my PhD supervisor László Kovács for everything I have learned from him for the last few years. I am very grateful for his advices and for the common brainstormings as well as for the constant help, support and useful comments during our work together in the process of doing my dissertation. I would especially thank to him that even being in Canada thus far from here; he has always supported my work. I would like to thank the help of my current and also my previous supervisors, György Lipovszki and Péter Korondi, and also that I could do my PhD in the Deparment of Mechatronics, Optics and Mechanical Engineering Informatics. Also many thanks to Rita Kiss, who continuously supported me and helped my work with her valuable advices and to Ibolya Vő for her assistance in each and every matter when I turned for help. I would like to thank to Antal Huba and Ambrus Zelei the opponents of my internal defense for their thorough, deep and precise criticism of my thesis and for their useful advices. I thank to Antal Huba for supporting me and helping me in my profession for nearly a decade. I am very grateful for him for inciting me and that he somehow always had some time for me when I needed an advice. I will always be thankful to Gábor Stépán for his help and his invaluable advices during our joint work as well as in drawing up the main statements of my thesis. I am also very grateful that he affected me in starting my career in higher education. I also thank the Department of Applied Mechanics to give me the opportunity to use their well-equipped lab which I had needed for constructing and executing my experiments. I am thankful to Csaba Hős, that I could learn the basic theory of the piecewise-smooth dynamical systems. I thank my co-authors, Brigitta Szilágyi, József Kövecses, Gábor Csernák and Gergely Gyebrószki the common work and all I learned from them. I also thank Gábor Sziebig and Zsuzsanna Lévai for their friendly support during my stay in Norway, thank you for helping me spending a wonderful time in Norway. Additional thank to Lubomir Dechevski vii

8 for the valuable conversations in this period. I would like to thank my friends for their continuous, never ending support and encouragement in these years. Special thanks to my friends and colleagues Máté Antali, Árpád Takács and Gábor Manhertz, they always stand by me, no matter what happened. Additional thank to Zoltán Juhász and Zsombor Hajdu for their support in understanding the program Mathematica. I am thankful to János Józsa for his support and valuable advices which helped moving forward in my career. And at last but not least, I would like to thank my parents and my family for their love, patience, encouragement and support in achieving my goals and dreams, without them I could not have reached this far. If you want to find the secrets of the universe, think in terms of energy, frequency and vibration. Nikola Tesla

9 CONTENTS Contents 1 Introduction Friction phenomena and models Friction problems in mechatronics State of the art Thesis outline Basic task in mechatronics: positioning Modeling of the experimental setup Model of dry friction Discrete-time control realization Validating experimental results Simplest representative model Piecewise analytical simulation Piecewise analytical solution Evolution of the discrete states New Results Dynamics of the sampled-data sliding oscillator Stability analysis by neglecting friction Energetic stability analysis Effective viscous damping Passivity condition with effective viscous damping Vibration decay and stability analysis Critical initial conditions Special subset of system parameters ix

10 CONTENTS 3.4 New Results Modeling of the damped-oscillator with dry friction Dynamic analysis of the continuous-time damped- oscillator with dry friction Motion induced by initial condition Piecewise smooth analysis Limit cycles and vibrations Relationship between the initial conditions Application to discrete-time position control New Results Sampled-data oscillator dynamics with combined viscous damping and friction Stability analysis without friction Combined effect of dry friction and viscous damping Verification of results New Results Appendix 71 List of Symbols 75 List of Figures 77 List of Tables 79 Bibliography 81 List of Related Publications 89 x

11 Chapter 1 Introduction Modeling of friction is one of the most difficult tasks in mechanics. This dissipation effect can have strong influence on the system dynamics and can cause complex dynamic behavior. The effect of friction is especially important in positioning automation problems. The applications subjected to position control often demand high accuracy and fast operation at the same time. The necessary performance fulfilling the above mentioned requirements is limited by not only the effect of friction but also by the digital nature of the applied motion controllers. This research was motivated by the recognition of the interplay of the effect of these phenomena on the system dynamics. The later presented experimental results show that the response of these systems are often non-intuitive and unexpected. The presented research focuses on the understanding of the special characteristics of the dynamics induced by friction and sampling. 1.1 Friction phenomena and models In the literature, the definitions related to the modeling of friction phenomena, are often mixed with each other. During this thesis, the usage of these definitions are based on the following terminology. Generally, friction is the resistance to motion which occurs when the surfaces in dry and/or lubricated contact are moving relative to each other, and the arising resistive force which is acting in the opposite direction of the relative motion called friction force [1]. During friction, the total mechanical energy is decreasing and it is mostly converted to thermal energy. However, the underlying mechanisms between the contacting surfaces are still not perfectly known because it depends on many factors, e.g., the contact geometry, the 1

12 CHAPTER 1. INTRODUCTION materials of the contacting bodies, the presence of lubrication, temperature, etc. The reason of the damped motion is that the energy is dissipated. However, it is also possible to damp the motion without contacts, let us think about the effect of eddy current. Friction phenomena appears in fact most, if not all, mechanical systems and it has strong influence on the system dynamics. Due to these facts, the friction phenomena is analyzed by researchers over the centuries and several friction models were developed [2, 3]. Historically, the first observations were carried out by da Vinci in 1493, however, his results become hidden for centuries. Later, in 1699, the same results were rediscovered by Amontons [4], and these were further developed by Coulomb [5] in It was concluded that the friction force is proportional to the normal load, but it is independent of the apparent contact surface area, and acting in the opposite direction to the motion. These observations led to the first model of friction, the so-called Coulomb friction model. The major limitation of this model is, that the friction force is not determined at rest resulting discontinuity for the friction force at zero relative velocity. However, later it was shown that the friction force depends on the true contact area [6]. The apparent contact surface area means the projected surface of the geometric object to the contact surface, while the true contact surface area defines the surface in contact between the geometric object and the surface [7]. Based on references [8, 9], the friction force is greater than or equal to the kinetic (Coulomb) friction force at rest, which is called static friction force. The static friction force counteracts the external forces in a certain extent and keeps the bodies at rest. The external force which is required to overcome the static friction force and initiate the motion is called the break-away force. With these, a modified Coulomb model is considered, called as the stiction model where the friction force is defined at rest using the static friction force. However, that case when the static friction force is equal to the kinetic friction force is also often called as the Coulomb model in the literature. The difference between the magnitude of the static and kinetic friction forces leads to the so-called stick-slip phenomenon [10, 11]. It means that during the motion, when the relative velocity between the contacting surfaces become zero, the friction force becomes enough high to stop the motion resulting sticking and if it is overcome by the external forces, the motion is initiated called slipping [3]. Based on the stiction model, the transition between the static and the kinetic friction 2

13 1.1. FRICTION PHENOMENA AND MODELS force has a sudden jump. This transition was experimentally investigated by Stribeck [12], and based on his observations, the friction force is decreasing with increased sliding velocities. This phenomena is the so-called Stribeck effect, which was later described by different mathematical expressions [13 17]. These formulas are summarized in [18]. For reducing the amount of wear and friction, and also for decreasing the arising heat, machine elements in contact are typically lubricated with some kind of oil or grease. These lubricants are often modeled as Newtonian fluids, and as a result, the corresponding friction force is proportional to the sliding velocity [19]. The corresponding friction model is the viscous friction model, while the underlying dissipation mechanism is also often called viscous damping. On the other hand, due to the presence of the lubricants, the friction force is depending on the evolution of the fluid layer thickness of the lubricant which also depends on many factors, e.g., the viscosity of lubricant, the sliding velocity and the properties of the contacting surfaces, temperature. At motions with high velocity, the fluid layer becomes thick enough to separate the contacting surfaces from each other, and then the friction force is described by the viscous friction model. At low sliding velocity regions, there is no fluid layer built up. In this region the simple Coulomb model is sufficient by considering also the static friction force. The transition between the low and high velocity regions shows, that the friction force is decreasing with increased sliding velocities described by the generalized Stribeck curve. The evolution of the fluid layers are presented detail in [1, 17]. The friction models mentioned above, are often called as the classical models of friction [17, 20]. In other papers, these are mentioned as the static friction models [18, 21, 22], due to the fact that these only describe the steady-state friction characteristics as a function of relative velocity between the contacting surfaces. Using these models, difficulty comes from the zero velocity detection in simulations. To solve this problem switching and/or smoothed models were introduced, e.g., the Karnopp model [23], and the smooth Coulomb model which uses the hyperbolic tangent function [2]. Later, it was shown by the experimental investigation of friction force [9,24], that it is not only the function of velocity, but it also depends on the displacement due to the the elasticity of the contacting surfaces. This phenomenon, when small motions occur at microscopic level without sliding, is called pre-sliding displacement. The first model which is capable to capture this phenomenon was the Dahl friction model [25]. The starting point of the development of 3

14 CHAPTER 1. INTRODUCTION this model was the stress-strain curve from classical solid mechanics which results a springlike behavior of this kind of microscopic displacement. It was the first dynamic friction model where an additional degree-of-freedom was introduced in order to describe the small scale elastic behavior by a first-order differential equation. On the other hand, Dahl model does not capture the Stribeck effect and cannot predict the stick-slip phenomena. To overcome this problem, an extension of the Dahl model was constructed, in the form of the so-called LuGre friction model [26]. The advantage of this friction model is that it contains only few parameters and therefore it is relatively easy to find its parameters experimentally [7]. Further dynamic models can be found in the literature, such as the Leuven friction model [27] which is an improved LuGre model including the hysteretic effect of friction [18]. This model was also further developed resulting the generalized Maxwell slip model, where several single mass-spring oscillators are connected in parallel in order to evaluate the cases of sticking or sliding separately [28]. Further friction models can be found in references [29 36]. 1.2 Friction problems in mechatronics In mechatronics, the dominant sources of dissipation are the actuators, bearings and the motion transmission elements in the drive-train. The dissipative effects of the above mentioned various type machine elements in contact have strong influence on the whole system dynamics and can cause complex dynamic behavior. Therefore, it is important to properly select one friction model that suits the best for the investigated problem. For example, in [37], micro stick-slip motion systems were experimentally analyzed and the results were compared by simulations applying different friction models. It was concluded that the application of LuGre model suits the best for matching the simulation and the experiment. In [38], a single DC motor was used in order to identify and compare the different friction models. It was concluded that the generalized Maxwell slip model suits the best for the experimental results. On the other hand, sometimes there is no need for sophisticated friction models and the simplest Coulomb model is sufficient. For example, in systems where the relative velocity remains constant, the simplest Coulomb friction model will suffice [18], or as it was presented in [39], the limit cycle arising in positioning experiment with DC motor can be explained 4

15 1.3. STATE OF THE ART by the Coulomb friction. Even in these simple cases, the controlled mechanical systems can show intricate dynamics, and chaotic or even stochastic motion might take place in certain parameter ranges [40 44]. In some applications, especially in robotics, the presence of friction is not desirable. In industrial robots, the accurate motion control is an important objective, but the friction can cause large positioning error [45]. Therefore, based on the friction models presented above, friction compensation algorithms [46] are used in order to provide better positioning accuracy. However, the sophisticated control algorithms may require considerable computational time, which result in a slower effective sampling rate. Despite of these facts, many times continuoustime techniques are used to design the controllers of the robotic systems [47, 48]. From engineering point of view, these simplifications can be a good approximation in a lot of cases, but sometimes, the digital effects, like sampling and quantization, cannot be neglected. For example, the non-smooth effects, like quantization, may lead to chaotic motions even at high sampling rates [49, 50]. On the other hand, even in case of the simple proportional-derivative controller, the positioning accuracy can be improved by increasing the proportional control gain, but at the same time, the system becomes less robust to parameter variations and might get unstable for large gain values, because these gains are limited by the digital realization of the controller [51]. The effect of sampling on the system dynamics was analyzed in [51 55] for example. Reference [51] introduces tools and design methods for discrete-time controlled robots, and [52 54] focus on the analytical investigation of the dynamics of digital force control. Reference [55] presents the stability and bifurcation analysis of a discrete-time controlled robots. But, in these studies the effect of the dry friction is considered as a stabilizing effect and it is neglected. This provides sufficient conservatism in many cases in the control design, but cannot be applied when dry friction is the dominant, or the only source of physical dissipation. 1.3 State of the art In many cases, dry friction can be a dominant source of dissipation, and can also greatly affect the nature of dynamic behavior due to its non-smooth properties. On the other hand, very little attention has been paid to the analysis of the effects of dry friction on the stability of systems subjected to discrete-time controllers. Only few papers include some considerations 5

16 CHAPTER 1. INTRODUCTION for Coulomb friction and time discretization [56 58]. In these papers, the common goal was to give a passivity-based approximation for system stability. On the other hand, in [57, 58] can also be observed the above mentioned special vibration shape with concave vibration envelope, but in this paper, there is no physical explanation or analysis was provided for this type of vibrations. It was only concluded that these typically arise in the passive domain of operation. It can be concluded, that still how the friction effects influence the stability of systems subjected to discrete-time control is not well understood and left without discussion in the literature. Therefore, the main purpose of the research summarized in this dissertation is the investigation of friction effects on the dynamics of mechanical systems with discrete-time control. 1.4 Thesis outline The present thesis discusses the effects of friction on the dynamics of controlled mechanical systems. The corresponding results are presented in four chapters. In order to analyze the interplay between the friction effects and the controlled dynamics of mechanical systems, an experimental setup was designed and assembled. The modeling of this experimental setup is presented in Chapter 2 which includes detailed information on the mechanical, electric and control assumptions. During the measurements, a special concave shape vibration envelope was observed in the time history. Chapter 2 also focuses on the understanding of this special characteristic of the damped motion, and it aims to identify the simplest representative dynamic model. Later, this model is called as the sampled-data sliding-oscillator model. The system of equations describing the dynamics of the controlled mechanical system is solved in a series of analytical steps. Chapter 3 deals with the examination of the dynamics of the sampled-data slidingoscillator described above. In order to understand the effect of dry friction, first, a reference model is considered and analyzed without friction. It is followed by the analysis of the effect of dry friction on the dynamic behavior of the sampled-data sliding-oscillator model. First, some results of the corresponding literature are reproduced using passivity analysis. Then, energetic considerations are combined with classical describing function analysis to obtain an improved stability condition. Later, the stability properties of the sampled-data sliding 6

17 1.4. THESIS OUTLINE oscillator model are analyzed by determining the consecutive vibration peaks. At the end of this chapter, the stability properties of the sampled-data sliding-oscillator model are further analyzed. The key element of this analysis is the determination of the dominant vibration frequency of the controlled motion. This is used to show how the dry friction can stabilize the otherwise unstable, digitally controlled motion. In Chapter 4, the analysis of an effective continuous-time system model is presented. This continuous-time system model is also used for the analysis of the stability properties of the sampled-data sliding-oscillator model. In this analysis the important element is the introduction of an unstable system model with negative viscous damping. Finally, in Chapter 5, the sampled-data sliding-oscillator model is extended by considering viscous damping and dry friction. This makes it possible to analyze the combined stabilization effect of both of the two main physical dissipation mechanisms. 7

18 CHAPTER 1. INTRODUCTION 8

19 Chapter 2 Basic task in mechatronics: positioning Positioning is a basic task in automation, where the control system aims to drive the device into a desired position. In order to analyze the interplay of digital nature of the applied controller and the dissipation due to friction, a simple experimental setup was designed and built. It is noted that the digital effects are originated from the temporal and from the spatial discretization, called sampling and quantization, respectively, in data acquisition and in the determination of control inputs. Based on the analogous relation of different systems [59], the positioning problem is analyzed by rotating motion. Thus, the choice of the commercially available sensors and actuators are much better then in case of translation. Therefore, the core component of the experiment is a single link, pendulum-like, direct drive robot operating around its stable equilibrium position presented in Fig The single link robot consists of a rod directly attached to a brushed DC motor (Maxon A-max, ) shaft avoiding the additional effects of the gearboxes and clutches. The DC motor was also equipped with a 500 cpr (counts per revolution) magneto-resistive, incremental encoder (HEDS-5540). In order to achieve an easy-to-use measurement equipment, the controller was operated in a Matlab Simulink Real-Time Workshop (RTW) [60] environment to provide a flexible user interface for setting the desired and the initial positions of the robot. The single link was driven using a self-developed control board which consists of a PIC (24FJ128GA010) micro-controller which communicates with a Simulink RTW based controller via RS-232 9

20 CHAPTER 2. BASIC TASK IN MECHATRONICS: POSITIONING 3 y 1 2 x ϕ 4 Figure 2.1: Experimental setup: 1 - micro-controller based signal processing and communication unit, 2 - H-bridge inverter, 3 - brushed DC motor with encoder, 4 - link communication protocol. The joint angle is measured by the encoder and based on the processed encoder signals, the joint angle is transmitted to a high-level control PC where the necessary control torque is calculated, and the corresponding pulse-width modulation (PWM) duty-cycle is sent back to the micro-controller and the commanded torque is amplified by a unipolar PWM H-bridge inverter. During experiments, in order to get the best choice to understand the coupled effect of sampling and friction on system dynamics the simplest proportional controller was used. The schematic diagram of the position control loop can be seen in Fig It is noted that this experimental setup is capable of testing complicated control architectures. Furthermore, a constant sampling frequency was selected, and the initial angular position ϕ 0 of the link was varied. The experimental result, presented in Fig. 2.3 shows that the response of systems with discrete-time controller and dry friction can be non-intuitive and unexpected. The measured vibration signal in Fig. 2.3 has a special concave shape envelope and it leads to a stop in finite time. This special concave shape envelope in the time history of vibration is very different from the vibration shape of the basic continuous-time systems damped by viscous and/or dry friction. The schematic diagram of the different kind of envelope shapes are presented in Fig On the left panel of Fig. 2.4, a typical vibration is presented of a damped motion with viscous damping resulting convex vibration envelope. In the middle of Fig. 2.4, when the motion is damped by Coulomb friction, the amplitude decay is shown 10

21 2.1. MODELING OF THE EXPERIMENTAL SETUP Power Supply Load 12 V DC τ(t) H-Bridge Inverter u 0 (t) DC Motor ϕ d (t j ) MATLAB Simulink RTW Controller Encoder Count PWM signal Microcontroller Measured position ϕ(t) Encoder Desired position δ u PWM duty-cycle Figure 2.2: Schematic diagram of the position control loop to be linear. Whereas, the right panel of Fig. 2.4 shows the special characteristic vibration decay with concave envelope of the digitally controlled systems with dry friction. This chapter focuses on the understanding of this special characteristic of the damped motion and it aims to identify the simplest possible dynamic model where a discrete-time controller is also considered that can explain the experimentally observed non-intuitive vibration decay. 2.1 Modeling of the experimental setup The encoder signals are interfaced to the controlling PC through the micro-controller board using external interrupt inputs, and the encoder resolution is improved by the 4 times quadrature technique. It results, that the effective encoder resolution is 2000 cpr. Often, the angular position signal generated by the encoder is modeled as a uniform mid-riser quantizer [58]. But, as it can be seen later, the effect of spatial discretization is not necessary for the understanding of the concave envelope vibration; therefore it was neglected from the model of the experimental setup. The core component of the experimental setup is a permanent magnet, DC motor (20 W Maxon A-max 32, ) with graphite brushes. The stator of the motor is an AlNiCo magnet, and the armature has core-less winding without therefore the iron losses. The 11

22 CHAPTER 2. BASIC TASK IN MECHATRONICS: POSITIONING 0.4 concave envelope 0.2 Position [rad] Time [s] Figure 2.3: Measured time history with friction and sampling commutator consists of fixed graphite brushes and copper contacts, separated into 13 commutator segments. Based on this segmentation, the torque ripple [61] can be determined approximately as 0.75 %. Even despite the nominal torque is quite small, i.e., 44.4 mnm, the exerted motor torque is considered to be constant and the effect of torque ripple is neglected. Neglecting also the temperature dependency of the armature resistance, the model of the used DC motor can be derived as a coupled electro-mechanical system. Based on the equivalent electric circuit of the DC motor, the electric equation is u 0 (t) = Ri(t) + L di(t) dt + e(t), (2.1) where u 0 is the input voltage, i is the motor current and e denotes the induced voltage, often called as back electro-motive force (EMF), as a function of time t. In addition, R and L convex linear concave Figure 2.4: Schematic diagram of the different kind of envelope shapes: convex (left), linear (middle), concave (right) 12

23 2.1. MODELING OF THE EXPERIMENTAL SETUP denote the armature resistance and inductance, respectively. The mechanical equation is J r dω(t) dt = τ m (t) τ fr (t) τ L (t), (2.2) where ω is the angular velocity of the motor shaft, J r denotes the second moment of inertia of the rotor, τ m is the electric torque exerted by the motor, τ fr and τ L are the torques due to the dissipation and the external load. In addition, the coupling equations of the electrical and mechanical equations are τ m (t) = K m i(t) and e(t) = K e ω(t), (2.3) where K m and K e denote the motor torque and electric constants, respectively. Substituting back the coupling equations in Eq. (2.3) into Eqs. (2.1)-(2.2), the combined electro-mechanical equation of motion of a DC motor can finally be derived as LJ r ω(t) + RJ r ω(t) + K m K e ω(t) = K m u 0 (t) Rτ fr (t) L τ fr (t) Rτ L (t) L τ L (t), (2.4) where the single and double dots refer to the first and second time derivatives. Symbols ω and ω denote the angular acceleration and the angular jerk, respectively. The numerical values of DC motor parameters are collected in Tab Examining the combined electro-mechanical equation of motion of a DC motor in Eq. (2.4) without any load, τ L = 0, and introducing the electric and the mechanical time constant as t e = L/R and t m = RJ r /(K m K e ), the characteristic equation of Eq. (2.4) can be determined as LJλ 2 + RJ r λ + K m K e = 0 t e t m λ 2 + t m λ + 1 = 0. (2.5) Using the parameters in Tab. 2.1, the time constants are t e = 3.91 ms and t m = 0.51 ms. Table 2.1: The given and the derived parameters of the used DC motor u n 30 V J r 44.0 gcm 2 R 7.17 Ω K m 46.1 mnm/a L mh K e mv/rpm J L kgcm 2 k t mnm/rad k L 2.23 Nm/rad b R mnm s/rad 13

24 CHAPTER 2. BASIC TASK IN MECHATRONICS: POSITIONING Based on the characteristic equation in Eq. (2.5), the two characteristic roots λ 1,2 are λ 1,2 = 1 ( 1 ± 1 4 t ) e 2t e t m (2.6) which are real when t m t e > 4 R 2 J r K m K e L > 4. (2.7) Using the time constants given above, the ratio of the mechanical and electric time constants is t m /t e = 0.13 and λ 1,2 are complex. Therefore, without any load, the effect of the armature inductance can cause vibrations. In the experimental setup, the link is directly attached to the DC motor shaft equipped. During the experiments the link was commanded to maintain its vertical, hanging-down equilibrium position, ϕ d = 0. Considering a direct drive actuation, the equation of motion of the load is τ L (t) = J L ϕ(t) + k t sin(ϕ(t)) J L ϕ(t) + k t ϕ(t), (2.8) where ϕ is the angular position measured from the equilibrium position and J L is the combined second moment of inertia of the link and the encoder with respect to the rotor shaft and k t is a torsional stiffness term due to gravity. The numerical values of the load parameters are shown in Tab Substituting back the equation of the load in Eq. (2.8) into Eq. (2.4) and also using ω(t) = ϕ(t), the combined electro-mechanical equation of motion of a DC motor modifies as LJ... ϕ(t) + RJ ϕ(t) + (K m K e + Lk t ) ϕ(t) + Rk t ϕ(t) = K m u 0 (t) Rτ fr (t) L τ fr (t), (2.9) where J = J r + J L J L. Introducing the effective torsional stiffness k L = K m K e /L due to the armature inductance and the effective viscous damping due to the armature resistance b R = K m K e /R, the the combined electro-mechanical equation of motion of a DC motor in Eq. (2.9) can be given in the form b R J... ( ϕ(t) + J ϕ(t) + b R + b ) R k t k L k L ϕ(t) + k t ϕ(t) = b R K e u 0 (t) τ fr (t) b R k L τ fr (t). (2.10) The numerical values of the parameters k L and b R are given in Tab Since the ratio of b R /k L 0, the corresponding terms in Eq. (2.10) can be neglected resulting the following 14

25 2.1. MODELING OF THE EXPERIMENTAL SETUP equation of motion J ϕ(t) + b R ϕ(t) + k t ϕ(t) = τ u (t) τ fr (t), (2.11) where the control input torque τ u = b R u 0 /K e = K m u 0 /R Model of dry friction The friction torque can be determined assuming Coulomb friction. It models the effective friction effects due to the brushes and also to the bearings. According to the Coulomb friction model assuming that the magnitudes of the static and the kinetic friction torques are equal, the friction torque is given in the form τ C sgn ( ϕ(t)), if ϕ(t) 0 τ fr (t) =, (2.12) [ τ C, τ C ], if ϕ(t) = 0 where τ C denotes the magnitude of the kinetic or Coulomb friction torque. As it can be seen in Eq. (2.12), the friction force f fr is a multivalued function at zero velocity. It results if the velocity is zero, the motion will stop in a so-called sticking region. It will be presented later in Sec In this section, it will be presented when the velocity is zero outside the sticking region, the motion can (still) be analyzed by using the definition of Coulomb friction τ fr (t) = τ C sgn ( ϕ(t)). (2.13) On the other hand, in order to achieve the good results during the simulations the presence of the uncertainty region is also taken into account. Furthermore, the systems with Coulomb friction are a type of piecewise smooth dynamical system, called Filippov system, that is when the uncertainty region is not defined, the so-called chattering phenomenon occurs [62]. The magnitude of the Coulomb friction torque τ C can initially be approximated as τ C = K m i 0 = mnm, where i 0 is the no load current [61]. Note, that the friction torque can depend on the contact force in general, therefore it will slightly vary as the link rotates with different speeds. In the experiments the link typically stopped in the range of 1 2 degrees which corresponds to the friction torque limits τ C = ( ) mnm. Using the Coulomb friction law in Eq. (2.13) and neglecting the small viscous damping 15

26 CHAPTER 2. BASIC TASK IN MECHATRONICS: POSITIONING term b R, the simplified equation of motion of the DC motor can be given in the form J ϕ(t) + k t ϕ(t) + τ C sgn ( ϕ(t)) = τ u (t). (2.14) This assumption is also supported by the experimental observation that the uncontrolled system clearly has a linear vibration envelope which corresponds to Coulomb friction. It is presented as red dashed line in Fig. 2.5, even when the amplifier was connected, but τ u was set to zero. For the further calculations, the friction torque was identified to be τ C = mnm which value gives the best matching result with the experiments linear envelope measurement Position [rad] Time [s] Figure 2.5: Experimental result for parameter identification Discrete-time control realization The necessary control torque τ u is computed by a discrete-time proportional controller where the reference position ϕ d is set to zero and it is realized by a zero-order-hold (ZOH) considering a unit delay; thus τ u (t) = k p ϕ(t j 1 ), t [t j, t j+1 ), t j = jt s, j = 0, 1, 2..., (2.15) where k p denotes the proportional control gain and ϕ(t j 1 ) is the sampled position and delayed control input. In addition, t j denotes the jth sampling instant and t s is the sampling 16

27 2.1. MODELING OF THE EXPERIMENTAL SETUP time. The considered unit delay models the fact that the Simulink model transmits the output torque first, and processes the received input later in the same control cycle. The schematic diagram of the operation of the continuous-time driving force reconstruction by a zero-order-hold is presented in Fig The left panel of this figure shows the case when the driving torque is reconstructed as follows. At the beginning of the time interval [t j 1, t j ), at time instant t j 1, the sampled position data is ϕ(t j 1 ). It is represented by green line in the figure. Using this sampled position data, the controller determines the necessary control torque. Until the next sampling instant t j, the determined control torque is kept constant. It is the simplest case of the discrete-time control, where it was assumed that the necessary time for the computation of the driving torque is negligible. On the other hand, when the computation time is non-negligible, between the actuating and the sampling instant there is a unit delay. In this case, using the sampled position data ϕ(t j 1 ) the driving torque is determined, but the actuation with this control torque happens one sampling time later, at the time instant t j. It results that the driving torque is a piecewise linear function of time which depends on the sampled position data. ϕ ϕ(t j ) ϕ ϕ(t j 1 ) t s ϕ(t) t j 1 t j t j+1 t j+2 t t j 1 t j t j+1 t j+2 t τ u τ u t j 1 t j t j+1 t j+2 t t j 1 t j t j+1 t j+2 t Figure 2.6: Schematic diagram of the operation of the Zero-Order-Hold driving torque reconstruction: without delay (left), considering a unit delay (right) When the control law in Eq. (2.15) is also substituted back into Eq. (2.14), the following effective single-dof mechanical model can be obtained J ϕ(t) + k t ϕ(t) + τ C sgn ( ϕ(t)) = k p ϕ(t j 1 ), t [t j, t j+1 ). (2.16) 17

28 CHAPTER 2. BASIC TASK IN MECHATRONICS: POSITIONING The angular velocity is proportional to the supply voltage of the motor. Thus, a simple way to control the speed of a DC motor is using a pulse-width modulated (PWM) signal to set the average supply voltage. It can be written as u 0 (t) = δ u δ max u n, (2.17) where δ u is the calculated duty-cycle varying during the control and δ max is the maximum of the achievable PWM duty-cycle. The value of δ max is limited by the settings (e.g., selected PWM frequency) of the microcontroller and the efficiency of the H-bridge inverter. The achievable maximum value of δ max is 800 due to the selected 20 khz PWM frequency in the applied micro-controller and due to the efficiency of the H-bridge inverter it is limited to δ max = 794. Note, that during experiments, the nominal voltage u n = 12 V was used. Based on these, the control torque τ u = k p ϕ(t j 1 ) can be also expressed as τ u (t) = K m R u 0(t) = K m δ u (t) u n k p ϕ(t j 1 ) = K m δ u (t) u n. (2.18) R δ max R δ max It results that the necessary value of the duty-cycle is δ u (t) = R K m δ max u n k p ϕ(t j 1 ) = κϕ(t j 1 ) (2.19) where κ is the control gain that can directly be selected and changed during the experiments. Its relationship to proportional position control gain k p is k p = K m R Validating experimental results u n δ max κ κ. (2.20) In the experiment shown in Fig. 2.7, the parameter κ was selected to be κ = 2.8 Nm/rad, which results that k p = mnm/rad. A measured time history of vibrations is shown in black solid line in Fig In the same figure, the blue line shows the result of the numerical simulation implemented in Matlab by using the 4-th order Runge-Kutta scheme for the integration with fixed time-step t s /100 considering the initial conditions ϕ 0 = rad and ω 0 = 0 rad/s (see Tab. 2.2). 18

29 2.1. MODELING OF THE EXPERIMENTAL SETUP Position [rad] 0.2 measurement simulation Time [s] Figure 2.7: Simulation versus experimental results It is noted that during the simulation, Eq. (2.16) was transformed to the form of a firstorder differential equation system, and in the first integration step, the motion was initiated with x 0 > 0 and with negative velocity. Furthermore, the friction torque was considered as τ C = mnm. The simulation result is in good agreement with the measurement and clearly shows the concave nature of the decaying vibrations. The larger errors in the amplitudes in the negative direction are due to the asymmetric friction properties of the experimental device which is not included in the model. The possible asymmetric nature of Coulomb friction was also reported for DC motors in references [21, 63], and it might partially be caused also by the small, but inherent, offset of the amplifier. Comparing the simulation results and the experimental data, it can also be observed that there is a slight difference in the frequencies of the vibrations close to the finite stop time. It can be concluded, that the system presented in Eq. (2.16) can describe qualitatively the characteristics of that kind of damped motion which has the special concave vibration decay. Table 2.2: Parameters for calculations κ 2.8 Nm/rad ϕ rad t s 5 ms ω 0 0 rad/s 19

30 CHAPTER 2. BASIC TASK IN MECHATRONICS: POSITIONING 2.2 Simplest representative model In the previous section a pendulum-like position controlled system was investigated. Here, the pendulum is changed to a rotating disk which helps to eliminate the nonlinear effects due to large angular displacements and the speed and position dependent normal force loading the motor axis. This setup can be shown in Fig. 2.8 where the disk ensures that the only (mechanical) nonlinear effect is due to dry friction. Otherwise, the control realization and the electrical parameters are the same as those of the previously described model. Disk Encoder DC motor Figure 2.8: Brushed DC motor with encoder and with the disk It results, without considering the effect of torsional stiffness, the following equation of motion J ϕ(t) + τ C sgn ( ϕ(t)) = k p ϕ(t j 1 ), t [t j, t j+1 ). (2.21) Using the same experimental setup with another parameter settings collected in Tab The parameter κ was selected to be κ = 0.5 Nm/rad, which results that k p = 15.5 mnm/rad. The experimental data presented in Fig. 2.9 with the solid black line, and the presented time history of vibration shows further as well the concave vibration decay. In the same figure, the solid red line corresponds to the numerical simulation of the sampled-data system. This sim- Table 2.3: Parameters for calculations κ 0.5 Nm/rad ϕ rad t s 10 ms ω 0 0 rad/s J 431 gcm 2 τ C mnm 20

31 2.2. SIMPLEST REPRESENTATIVE MODEL Rotation [rad] experiment simulation Time [s] Figure 2.9: Simulation versus experimental results Rotation [rad] set of ϕ 0 ϕ 0 =1.301 ϕ 0 =1.288 ϕ 0 =1.382 ϕ 0 =1.150 ϕ 0 =1.090 t s =0.01 s k p = Nm/rad stable motion unstable motion Time [s] Figure 2.10: Experimentally predicted stability limit (unstable limit cycle) ulation was implemented in Mathematica by using the command NDSolveValue utilizing event handlers for the sampling and for detecting velocity reversals. During simulation, the combined second moment of inertia of disk and the rotor is J, and the friction torque was selected to be τ C in order to achieve a good (quantitative) agreement with the experimental result. During the experiments it was realized that the concave shape becomes more pronounced at larger initial positions and it shows a certain dependence on the initial conditions. Examples for this are shown in Fig where green and red lines correspond to the cases of stable and unstable motions, respectively. In all stable cases the vibration envelope is concave. Furthermore, it can be concluded that there has to be an unstable limit cycle around the desired zero reference position, because of the motion is sensitive to the initial position. If the effective model in Eq. (2.21) is further simplified without considering the unit delay in the model of the continuous-time driving torque reconstruction, the following effective 21

32 CHAPTER 2. BASIC TASK IN MECHATRONICS: POSITIONING Rotation [rad] experiment simulation with delay simulation without delay Time [s] Figure 2.11: Simulation versus experimental results model can be considered J ϕ(t) + τ C sgn ( ϕ(t)) = k p ϕ(t j ), t [t j, t j+1 ), (2.22) The simulation result is shown as dashed green line in Fig and it is compared with the previous simulation and with the new experimental data. Based on this, it can be concluded that the simulation results, which does not include the effect of the additional unit delay, and the experimental results are initially in a very good agreement. However, a slight difference can be recognized between the two time histories in their frequencies towards the end of the motion. When simulations with and without a unit delay are compared, it can be seen that the effect of delay is negligible in case of the current parameter setting. Based on these, the corresponded unit delay is not negligible from the model of the experimental setup, but, from the viewpoint of the identification of the concave vibration shape, the unit delay is not necessary. Although, the neglected unit delay can have an effect on the system dynamics in case of different parameters, the results clearly show that the simplest model that qualitatively reproduce the observed concave envelope vibrations is the one in Eq. (2.22). Further simplification in the control model will results in a qualitatively different behavior. If in Eq. (2.22) the effect of sampling is neglected, then the control law reduces to a continuous-time proportional controller with zero reference position. This simplified equation of motion forms the model of a sliding-oscillator described by the following equation of motion J ϕ(t) + τ C sgn ( ϕ(t)) = k p ϕ(t) ϕ(t) + ω 2 nϕ(t) + f 0 ω 2 n sgn ( ϕ(t)) = 0, (2.23) 22

33 2.2. SIMPLEST REPRESENTATIVE MODEL where the undamped natural angular frequency ω n = k p /J and f 0 = τ C /k p. Considering the initial conditions as follows ϕ(0) = ϕ 0 and ϕ(0) = 0, the consecutive piecewise segments of the solution can be combined in a special closed form ϕ(t) = (ϕ 0 f 0 (2n + 1) cos (ω n t)) + ( 1) n f 0, with n = ωn t, (2.24) π where denotes the floor function. This solution with specific system parameters is presented as the solid blue curve in Fig ϕ(t) ±φ(t) ±Φ(t) Position [rad] ϕ 0 = 1 rad f 0 = 0.05 rad ω n = 2π rad/s Time [s] Figure 2.12: Time history of vibration of the sliding-oscillator In Eq. (2.24), the term multiplying the cosine function gives the amplitude decay, and considering a positive offset with f 0 the upper linear vibration envelope is approximated by the function ωn t φ(t) = ϕ 0 2f 0. (2.25) π It is presented as the green sawtooth shape function in Eq. (2.24) which shows that every second local minimum of this function sits exactly on the vibration peaks. These local minima can be connected by a continuous function by removing the floor function. Then the approximated vibration envelope and the exact amplitude decay function is described by ±Φ(t) with ω n Φ(t) = ϕ 0 2f 0 t, (2.26) π and the corresponding lower and upper amplitude decay function are shown in red in Fig

34 CHAPTER 2. BASIC TASK IN MECHATRONICS: POSITIONING If the effect of Coulomb friction is neglected in Eq. (2.23), the model further simplifies to the model of a mass-spring oscillator. It follows, that the vibration has no amplitude decay as the total mechanical energy is conserved. Finally, if in Eq. (2.22) the effect of Coulomb friction is neglected, the equation of motion becomes J ϕ(t) = k p ϕ(t j ), t [t j, t j+1 ) (2.27) and the desired zero position is always unstable (see later in Sec. 3.1). Based on the above analysis, if the motion is stabilized by Coulomb friction, the unstable nature of the sampled-data systems due to the temporal discretization results in concave envelope vibrations. The concave envelope vibrations is identified as the following sampleddata generalized mechanical system model where the continuous-time generalized driving force f u is realized by a zero-order-hold resulting that f u is constant over a sampling period mẍ(t) + f C sgn (ẋ(t)) = f u (t j ), t [t j, t j + t s ), (2.28) where x(t) represents the generalized coordinate as a function of time t corresponding to the modeled degree-of-freedom, m is the generalized mass that takes its meaning based on the definition of x and f C denotes the magnitude of the generalized dry friction force. In addition, t j denotes the jth sampling instant and t s is the sampling time. 2.3 Piecewise analytical simulation The equation of motion presented in Eq. (2.28) forms a non-homogeneous ordinary differential equation between two consecutive sampling instants assuming that the direction of motion does not change. Thus, the piecewise linear system of Eq. (2.28) can be solved for the consecutive sampling instants for the discrete state variables collected in z j = x j, (2.29) where x j = x(t j ) and v j = ẋ(t j ) denotes the jth sampled position and velocity data, respectively. v j 24

35 2.3. PIECEWISE ANALYTICAL SIMULATION Piecewise analytical solution The general solution of the non-homogeneous differential equation in Eq. (2.28) for the sampling period t [t j, t j+1 ) assuming ẋ(t) > 0 is x gen (t) = f u f C 2m t2 + c 1 t + c 2, (2.30) where the constants c 1 and c 2 can be determined by using the initial conditions x(t j ) = x j and ẋ(t j ) = v j. As a results c 1 = v j t j m (f u f C ) and c 2 = x j v j t j + t2 j 2m (f u f C ). (2.31) Substituting these into Eq. (2.30), the specific solution is x spec (t) = f u f C 2m (t t j) 2 + v j (t t j ) + x j. (2.32) The state-variables at the end of the jth sampling period are determined by using the following equations x(t j+1 ) = x spec (t j+1 ) and ẋ(t j+1 ) = ẋ spec (t j+1 ) which result in the matrixarray form z j+1 = 1 t s z j + t2 s/(2m) f u t2 s/(2m) f C. (2.33) 0 1 t s /m t s /m By means of these pieces of solutions and using the discrete-time proportional controller, f u = k p x j, the following non-homogeneous discrete map can be created z j+1 = Wz j w sgn (ẋ(t)), t [t j, t j+1 ) (2.34) with W = 1 p/2 t s, w = ψ/2, where p = k pt 2 s p/t s 1 ψ/t m, s ψ = f Ct 2 s m. (2.35) Evolution of the discrete states The following method relies on the piecewise analytical solution of the equation of motion presented in Eq. (2.28) by taking into consideration the switched dynamic behavior due 25