AN INVESTIGATION INTO THE REDUCTION OF STICK-SLIP FRICTION

Transcription

1 AN INVESTIGATION INTO THE REDUCTION OF STICK-SLIP FRICTION IN HYDRAULIC ACTUATORS William Scott Owen. B.A.Sc, The University of British Columbia, 1990 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MECHANICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 2001 William Scott Owen, 2001

2 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88)

3 Abstract The stick-slip friction phenomenon occurs during the switch from static to dynamic friction. Static friction is the force that opposes the sliding motion of an object at rest. Dynamic friction is the force that opposes the sliding motion of a moving object. Thus, near zero velocity, there is a switch from static to dynamic friction. Generally, static friction is greater than dynamic friction. In order to move an object the applied force must exceed the static friction. Once movement starts the friction force typically decreases as it switches to dynamic friction. However, if the applied force is still at the original magnitude, then the sudden increase in the resultant forces results in an increase in the object's acceleration; namely a jerky motion. In a similar manner, when an object is brought to rest the sudden increase in friction, as the switch from dynamic to static friction occurs, results in an abrupt and premature stopping of the object. Because of the rapidly changing and inconsistent nature of the friction force at low velocities, accurate and repeatable position control is difficult to achieve. In some cases the actuator position controller can reach a limit cycle (hunting effect). Friction compensation at low speeds has traditionally been approached through various control techniques. This work presents an alternative solution, namely, friction avoidance. By rotating the piston and rod, the Stribeck region of the friction - velocity curve is avoided and the axial friction opposing the piston movement is approximately linearized. As a result, simpler, linear control techniques at low speeds may then be utilized. Simulation and experimental results are presented to validate this approach and identify the operating limits for the rotational velocity. The experimental results validate the model. The results show that by rotating the piston, the friction is reduced and the Stribeck curve is eliminated. As the rotational velocity is increased the static friction from the axial motion approaches the static friction of the rotational motion. In order to eliminate the Stribeck curve, the rotating velocity must be located outside the range of the Stribeck area of the rotating friction - rotating velocity curve and into the full fluid lubrication regime. ii

4 Table of Contents Abstract Table of Contents ii Hi List of Tables List of Figures Acknowledgement Dedication vii viii x xi Chapter 1 Introduction Preliminary Remarks Motivation and Objective Thesis Overview 4 Chapter 2 Friction in Hydraulic Actuators Friction in Lubricated Machines Basic and Classical Models of Friction Stribeck Curve Regime I: Static Friction Regime II: Boundary Lubrication Regime III: Partial Fluid Lubrication Regime IV: Full Fluid Lubrication Stick-Slip Friction Static Friction and Rising Static Friction Frictional Memory Friction in Hydraulic Actuators Control of Hydraulic Actuators in the Presence of Friction Model Based Friction Compensation Observer Based Friction Compensation Observer Based Adaptive Friction Compensation 18 iii

5 2.2.2 Friction Avoidance Summary 20 Chapter 3 The Hydraulic Actuator Model Modeling Non-Rotating Model Servo Valve Fluid Flow Pressure Changes Dynamics Friction State Space Model Modeling of the System with Piston Rotation DC Motor Current Dynamics Helical Motion State Space Model Simulation Summary 37 Chapter 4 Friction Identification in Hydraulic Actuators: Experimental Method Introduction Hydraulic Actuator Setup Data Acquisition Position Signal Velocity Determination Pressures Friction Model Determining The Friction Parameters Equation of Motion Friction Parameter Determination Static Parameter Determination Dynamic Parameter Determination 46 iv

6 4.6 Pure Rotational and Linear-Rotating Friction Parameters Power Requirements Summary 48 Chapter 5 Reducing Stick-Slip Friction in Hydraulic Actuators: Experimental Results Pure Rotation Hydraulic Actuator - 0 Rpm Hydraulic Actuator Rotating Friction Parameters Model Validation Power Requirements Position Tracking Summary 70 Chapter 6 Conclusions and Suggestions for Future Work Conclusions Recommendations for Future Work 72 Nomenclature 74 Bibliography 80 Appendix A Hydraulic Actuator Specifications 84 A. 1 Process and Instrumentation Diagram 84 A.2 Physical Parameters 85 A.3 Instrumentation and Hardware 85 A.3.1 Hydraulic Pump 85 A.3.2 Motor 86 A.3.3 Servo Valve 87 A.3.4 Pressure Transducers 88 A.3.5 Potentiometer 88 A.3.6 Data Acquisition 89 Appendix B Simulation and Experimental Values 90 Appendix C Jacobian Linearization 91 v

7 C. 1 Jacobian Linearization and Discretization 91 C.2 Non-Rotating Model State Equations : 92 C.2.1 Position 92 C.2.2 Axial Velocity 92 C.2.3 Pressure 93 C.2.4 Friction 93 C.2.5 Servo Valve Input 94 C.2.6 Jacobians 94 C.3 Rotating Model State Equations 95 C.3.1 Position 95 C.3.2 Axial Velocity 95 C.3.3 Pressure 97 C.3.4 Friction 97 C.3.5 Rotating Velocity 98 C.3.6 Motor Current 99 C.3.7 Motor Voltage Input 100 C.3.8 Servo Valve Input 100 C.3.9 Jacobians 101 vi

8 List of Tables Table 3.1 Viscous Friction Parameter vs. Rotation Speed 34 Table 5.1 LuGre Rotating Friction Parameters 50 Table 5.2 Hydraulic Actuator LuGre Axial Friction Parameters (Positive Velocity, 0 rpm) 51 Table 5.3 Hydraulic Actuator LuGre Axial Friction Parameters (Negative Velocity, 0 rpm) 52 Table 5.4 LuGre Axial Friction Parameters (Positive Axial Velocities, with Rotation) 58 Table 5.5 LuGre Axial Friction Parameters (Negative Axial Velocities, with Rotation) 58 Table A.l Physical parameters 85 Table A.2 Hydraulic Pump 85 Table A.3 Motor and Amplifier 86 Table A.4 Servo Valve 87 Table A.5 Pressure Transducers 88 Table A.6 Potentiometer 88 Table A.7 Data Aquisition 89 Table B.l Simulation and Experimental Values 90

9 List of Figures Figure 1.1 Hydraulic Actuator 1 Figure 1.2 ISE's Hysub Remote Operated Vehicle 3 Figure 1.3 Hydraulic Actuated Manipulator 3 Figure 2.1 Da Vinci's Model of Friction 6 Figure 2.2 Basic Model of Friction 7 Figure 2.3 Classical Model of Friction 8 Figure 2.4 Stribeck Curve for Lubricated Surfaces 9 Figure 2.5 Stribeck Curve Regimes 10 Figure 2.6 Material Contact at Asperities 10 Figure 2.7 Dahl's Spring Model 11 Figure 2.8 Rising Static Friction 13 Figure 2.9 Frictional Memory 14 Figure 2.10 Hysteresis Effect 14 Figure 2.11 Double Acting Hydraulic Actuator 15 Figure 2.12 Sealless Tapered Pistons 19 Figure 3.1 Block Diagram for Hydraulic Actuator System 21 Figure 3.2 Hydraulic Actuator with DC Motor 27 Figure 3.3 Block Diagram for Hydraulic Actuator System with a Rotating Piston 27 Figure 3.4 Helical Motion of an Element on a Rotating Piston 29 Figure 3.5 Vector Components on an Element on the Piston 30 Figure 3.6 Piston Position for the Standard and Rotating Model 33 Figure 3.7 Friction Curves for the Standard and Rotating Model (460 rpm) 34 Figure 3.8 Viscous Friction Parameter vs. Rotation Speed 35 Figure 3.9 Friction versus Velocity for a Slow Angular Velocity (50 rpm avg) 36 Figure 3.10 Step Input - Piston Position 37 Figure 4.1 Experimental Setup 39 Figure 4.2 Hydraulic Actuator, Motor, and Servo-Valve 40 Figure 4.3 Quasi-Static Experiments: Applied Force and Acceleration Force 44 Figure 4.4 Quasi-Static Velocity Curve 45 Figure 5.1 Rotating Friction - Rotating Velocity Curve 49 Figure 5.2 Hydraulic Actuator Axial Friction - Axial Velocity Curve at 0 rpm. 51 viii

10 Figure 5.3 Axial Friction - Axial Velocity Curve at 10 rpm (0.033 m/s) 52 Figure 5.4 Axial Friction - Axial Velocity Curve at 25 rpm (0.083 m/s) 53 Figure 5.5 Axial Friction - Axial Velocity Curve at 50 rpm (0.17 m/s) 54 Figure 5.6 Axial Friction - Axial Velocity Curve at 75 rpm (0.25 m/s) 55 Figure 5.7 Axial Friction - Axial Velocity Curve at 100 rpm (0.33 m/s) 55 Figure 5.8 Axial Friction - Axial Velocity Curve at 125 rpm (0.42 m/s) 56 Figure 5.9 Axial Friction - Axial Velocity, Actuator Slipping at 100 rpm 57 Figure 5.10 Average Percent Axial Static Friction Reduction 59 Figure 5.11 Axial Coulomb Friction Parameter vs. Rotation Speed 60 Figure 5.12 Axial Stribeck Friction Parameter vs. Rotation Speed 60 Figure 5.13 Axial Static Friction Parameter vs. Rotation Speed 61 Figure 5.14 Axial Viscous Friction Parameter vs. Rotation Speed 62 Figure 5.15 Axial Stribeck Velocity Friction Parameter vs. Rotation Speed 63 Figure 5.16 Axial Bristle Spring Constant vs. Rotation Speed 63 Figure 5.17 Axial Bristle Damping Coefficient vs. Rotation Speed 64 Figure 5.18 Hydraulic Actuator Model Comparison at 0 rpm 66 Figure 5.19 Hydraulic Actuator Model Comparison at 10, 25, and 50 rpm 66 Figure 5.20 Hydraulic Power Requirements 67 Figure 5.21 Percent Reduction in Hydraulic Power (at m/s axially) 68 Figure 5.22 Total Power Requirements 69 Figure A.l Hydraulic Actuator PID 84 Figure A.2 Motor amplifier signal conversion 86 Figure A.3 Servo valve signal conversion 87 Figure A.4 Anti-Aliasing Filter 89 ix

11 Acknowledgement There are three people who deserve a very special thank you. I would like to start by thanking my wife, Mary Wells, for her support throughout this degree. Her intellectual stimulation helped to drive my thirst for knowledge. Andrea Zaradic deserves to be thanked as well, not only did she introduce me to my wife, she also gave me the extra push I needed when I was contemplating returning to school for graduate studies. She also told me about a professor at the University of British Columbia who was looking for students. The third person at the top of my list to be thanked is Elizabeth Croft. She took a chance by accepting me as a student and I have not looked back since. Her enthusiasm is never ending and contagious. James McFarlane of International Submarine Engineering must also be thanked. Our first meeting started with a 5:30 am phone call in October 1998 and led to this project. His financial and intellectual contribution is appreciated. The Science Council of British Columbia also contributed to this project through a GREAT Scholarship and their support was invaluable. My family deserves to be thanked. I will start with my brother Bob Owen, those years on 16 th Avenue were full of good times; my sister Marne Owen, who still continues to pursue a higher education while working; and my parents, Scott and June Owen, who always taught us to work hard and go after what we want. Doug Yuen from the machine shop and Glen Jolly from instrumentation must also be recognized. Their technical contribution helped the project along. Fellow students, Damien Clapa, Jason Elliott, David Langlois, and Sonja Macfarlane, with whom I started my degree, should also be recognized. Daniela Constantinescu should also be thanked for her insightfulness that she brought to the lab meetings. x

12 Dedication This thesis is dedicated to my wife, Mary Wells, and our daughter, Patricia Ann Mary Owen, born on July 21, The joy that these two people bring to my life continues to grow.

13 Chapter 1 Introduction 1.1 Preliminary Remarks Hydraulic actuators provide high force, stiffness, and durability suitable for applications in mining, machining equipment, and remote manipulator operations in unstructured environments such as ground, sea, and space applications. Although in some cases, these applications can also utilize pneumatic actuators or electric motors, hydraulic actuators provide a strength and durability that is unparalleled [1]. There is a growing need for such actuators to perform with improved precision and repeatability for manipulation tasks such as remote assembly, repair, and nuclear remediation. A recurring issue with hydraulic actuators is the level of friction present in the system. This friction affects the controllability, accuracy, and repeatability of the actuator. To achieve improved precision and repeatability, especially at low speeds, the problems related to friction in hydraulic actuators must be overcome. In a typical hydraulic actuator, as shown in Figure 1.1, movement of the rod, piston, and hydraulic fluid are subject to friction. The contacts between the rods and the seals, and between the piston o-rings and seals and the cylinder, and the viscous effects of the hydraulic fluid all generate friction. This friction must be overcome before movement can occur. In hydraulic manipulators, friction can reach 30% of the nominal actuator torque [2]. Figure 1.1 Hydraulic Actuator. 1

14 Friction is a complicated phenomenon that is still not fully understood. Friction models that have been used in the literature range from being a simple constant force that opposes motion to a seven parameter model including various behavioral characteristics such as stiction, a negative viscous slope, frictional memory, and hysteresis. Armstrong-Helouvry et al. discuss several of these models in detail [3]. Canudas de Wit et al [4] developed a new model in 1995, the LuGre model, which incorporates many of these effects. Before motion starts, the surfaces are in the static friction regime. A force greater than the static friction is required for movement to commence. As the component starts to move the friction suddenly decreases as it switches to the dynamic friction regime. This sudden change in friction results in a jerky actuator motion, making positional control and repeatability difficult [2, 3, 5, 6, 7, 8]. This effect is commonly referred to as stick-slip friction. Stick-slip friction is a non-linear friction phenomenon and can be found in hydraulic actuators around the zero velocity range. Modeling of this sudden switching is difficult, and precise control of the system usually involves complex system identification and prediction. 1.2 Motivation and Objective The motivation behind this work is best provided through an example. Figure 1.2 is a remote operated vehicle (ROV) from International Submarine Engineering Ltd. (ISE), a robotics and submersible vehicle company located in Port Coquitlam, British Columbia. The ROV can be used for subsurface applications such as maintenance, search and rescue, and underwater research. 2

15 The objective of this research is to investigate a novel approach of avoiding and reducing friction in hydraulic actuators. Stick-slip friction is encountered when the actuator switches directions and must pass through zero velocity. Stick-slip friction is also encountered when the actuator functions at speeds close to zero velocity where the switch from static to dynamic friction occurs. In this case a cycle of sticking and slipping can occur [3]. If near zero velocity operation can be avoided then it is expected that stick-slip friction will not be a problem. By rotating the rod and piston [10] the actuator will be kept in motion, well away from zero velocity, without moving axially. Since there is continuous motion, when the actuator is moved axially, stick-slip friction will be avoided. The end result of reducing stick-slip friction is an expected increase in controllability, accuracy, repeatability, and a reduction of jerk in hydraulic actuators. 1.3 Thesis Overview The research studied the effects of rotating the piston on the friction developed in a hydraulic actuator. A non-rotating hydraulic actuator was modeled using a state space approach in order to obtain a base line of performance. Next, a rotating actuator model was developed to provide some guidance as to what effects the rotation would have on the friction. Finally, experimental measurements validated the non-rotating model, confirmed that rotating the piston would reduce the amount of friction present in a hydraulic actuator, and that the Stribeck curve would be eliminated. The outline of the thesis is as follows: Chapter 2: Friction in Hydraulic Actuators: Friction in lubricated machines is introduced. Particular attention is paid to friction in hydraulic actuators with a discussion on the current methodologies in the industry. The difference between controlling the actuator in the presence of friction, through modeling and observers, and trying to reduce the problem of friction will be highlighted. Chapter 3: The Hydraulic Actuator Model: A state space model for a non-rotating hydraulic actuator will be presented. This model will utilize a friction model that has been adopted from the literature. Then a new model for a hydraulic actuator that incorporates a rotating piston will be presented. The same friction model will be used but it will be a function of multiple velocities instead of just one velocity. The rotating model will provide guidance as to what to expect from the experimental tests. 4

16 Chapter 4: Friction Identification in Hydraulic Actuators: Experimental Method: The method to determine the friction parameters will be presented. Quasi-static experiments were conducted to obtain friction values over a range of velocities that includes the Stribeck curve and extends into the viscous region of the friction - velocity curve. These experiments were used to obtain the axial friction parameters for the hydraulic actuator with no rotation and while rotating at various speeds. These techniques were also applied to identify the rotating friction parameters of the motor, piston, and rod with no axial motion. Chapter 5: Reducing Stick-Slip Friction in Hydraulic Actuators: Experimental Results: The experimental results will be presented. This includes the rotating friction - rotating velocity curve and rotating friction parameters of the system for rotating with no axial motion, and the axial friction - axial velocity curve and the axial friction parameters of the actuator at speeds ranging from 0 rpm to 125 rpm. The axial friction parameters determined through experimentation will be used in the non-rotating model of the hydraulic actuator to validate the model. The power requirements to move axially and to rotate the system will be compared. Chapter 6: Conclusions and Suggestions for Future Work: Conclusions stemming from this research are summarized. Suggestions for further work are provided. 5

17 Chapter 2 Friction in Hydraulic Actuators 2.1 Friction in Lubricated Machines Armstrong-Helouvry, Dupont, and Canudas de Wit [3] provided an extensive survey of friction research, some of which is summarized in the following section. 1 A clear picture of the friction phenomena and the problems found with friction in lubricated machines is important to the development of the friction avoidance technique presented in this thesis Basic and Classical Models of Friction Da Vinci (1519) first postulated that the friction force is proportional to the normal load, as shown in Figure 2.1. Da Vinci believed the coefficient of friction to be dependent on the characteristics of the contact areas and remained constant. In reality, coefficients of friction are dependent on surface characteristics that are dependant on time, temperature, lubrication, and other variables. A friction force opposes the direction of motion and is independent of the contact area between the two surfaces. The net driving force is the difference between the applied force and the frictional force. F, Normal F Net Drive = F Applied - F, Friction F Applied 77777/ ////// F, Friction Figure 2.1 Da Vinci's Model of Friction. 1 The bulk of the information in this section comes from reference [3]. Figures 2.6, 2.7, 2.8, and 2.9 are modeled after those in reference [3]. 6

18 Coulomb (1785) introduced the concept of a dry or Coulomb friction. The friction force opposing motion was constant and independent of the velocity, ^Dynamic M'Dynamic^^Normal> (2*1) where F Dynamic \s the dynamic friction force, F Nomal is the normal force, and Moynamic is the dynamic coefficient of friction. Morin (1833) stated that there was a threshold friction force that had to be overcome before movement occurred: ^Static ~ MSialic ^Normal ' (2-2) where fj-dynamic < fislalic. F Sta tic is the static friction force and u, S tatic is the static coefficient of friction. This is the "Basic Model of Friction" and is shown in Figure 2.2. Friction Force Sialic Coulomb " >- Velocity Figure 2.2 Basic Model of Friction. Reynolds (1866) made a significant contribution to understanding friction with his work in viscous fluid flow. Viscosity is the ability of a fluid to resist shear. More formally, viscosity relates momentum flux to the velocity gradient. It is the property of a fluid that relates applied stress to the resulting strain rate [11]. Applying this property yields: F w. Bx, (2.3) where B is the viscous damping coefficient, F Viscous is the viscous friction, and x is the velocity. The friction model then becomes: F = F \x\ = 0, Sialic Applied F = F + F x > Dynamic Coulomb T 1 Viscous A ^ (2.4) 7

19 This is the "Classical Model of Friction", Figure 2.3. Friction Force Figure 2.3 Classical Model of Friction. Here the model shown is symmetrical. This is not always the case. Friction can be direction dependent as discussed in [5, 12, 13] Stribeck Curve One of the main problems with the Classical model is the discontinuity between static friction and dynamic friction. The Classical model does not provide a sufficient representation of friction, especially for a lubricated application. Under lubrication the discontinuity is softened. However, the nature of the discontinuity is such that it is still difficult to compensate for. Stribeck (1902) developed the Stribeck Curve. The change in static friction to dynamic friction was recognized as being continuous, as shown in Figure 2.4 [3, 14]. The steep negative slope or negative viscous slope shows that there is a continuous change in the friction. 8

20 Friction Force > Velocity Figure 2.4 Stribeck Curve for Lubricated Surfaces. As shown in Figure 2.4, the Coulomb friction parameter is measured at the intersection of the viscous friction curve and the friction axis. The Stribeck friction is the difference between the static friction and the Coulomb friction [3]. The Stribeck curve applies to lubricated surfaces. If the surfaces are dry and unlubricated, then the change from static to dynamic friction can be considered essentially discontinuous, as in the Classical model. The Stribeck Curve can be separated into four different regions, each displaying the different characteristics of friction. These different regimes are: Regime I: Regime II: Regime III: Regime IV: Static Friction (also known as stiction) Boundary Layer Lubrication Partial Fluid Lubrication Full Fluid Lubrication These different regimes are shown in Figure

21 Regime I Regime II Friction Force A ^1 Figure 2.5 Stribeck Curve Regimes Regime I: Static Friction Regime I is the static Friction regime where there is no apparent relative velocity between the contact surfaces and no appreciable sliding or movement occurs. The contact between the two surfaces occurs at asperities (microscopic roughness) as shown in Figure 2.6. The contact area between the two surfaces is relatively small compared to the total area of each surface. This is due to the surface imperfections on each surface and the difficulty in obtaining a purely flat and smooth surface at the microstructure level. Boundary Lubricant Asperities Figure 2.6 Material Contact at Asperities. Figure 2.6 shows a boundary lubricant on the surfaces of the materials. Many lubricants have additives that leave deposits on the surfaces. These deposits help to reduce the coefficient of friction between the surfaces and therefore reduce friction. The choice of lubricant affects both the surface friction and the surface wear. Some lubricants, called way oils, can actually lower 10

22 the level of static friction below the level of Coulomb friction. However, the characteristics of these lubricants change over time and cannot always be relied upon. Through continuous use the lubricant can break down and as wear increases the lubricant becomes dirty, losing the properties that it was chosen for in the first place. Furthermore, in applications such as hydraulics the lubricant is the hydraulic fluid, which will not necessarily have such desirable properties. Before sliding occurs there can be elastic deformation between the asperities. This is known as the Dahl Effect where there is a pre-sliding dislocation. Dahl (1968, 1976, 1977) modeled the asperities as springs, Figure 2.7, where the friction force depends on the displacement and not the velocity: Ffriction = ~k,x. (2.5) When noticeable movement occurs the friction force has reached the breakaway force, that is the static friction level, and the 'springs' have then been broken. Figure 2.7 Dahl's Spring Model. The term static friction is often considered to be somewhat misleading. Friction is considered to be a function of velocity and since there is no velocity or sliding in the static friction regime then friction cannot exist. Polycarpou and Soom (1992) refer to the friction force at zero velocity as a tangential force or a force of constraint Regime II: Boundary Lubrication As previously mentioned, many lubricants have additives that leave a deposit on the surfaces of the materials. This leads to Regime II: Boundary Lubrication. With hydrodynamic lubrication a minimum velocity is required to draw the lubricant in between the surfaces. In Regime II the relative velocity is below that minimum and there is no lubrication except that provided by the boundary lubricant. Movement occurs when the applied force is greater than the static friction 11

23 and the asperities are sheared. This regime of the Stribeck curve experiences solid to solid contact Regime III: Partial Fluid Lubrication In Regime III the velocity has increased to a point where the lubricant begins to be drawn into the area between the surfaces. The fluid lubrication increases and the solid to solid contact decreases. There is a partial support of the surfaces by the fluid and a partial support by the asperities. As the velocity increases the surfaces are supported more and more by the fluid and less by the asperities. As a result the resistance to movement decreases (the friction decreases), and under constant applied force, acceleration of the moving body increases. With increasing acceleration, the velocity increases and more lubricant is drawn in. This positive feedback cycle, results in an unstable system response. The negative slope of the friction curve, referred to as a negative viscous slope, is responsible for this unstable condition, and leads to most of the problems related to friction compensation. When motion is initiated, the applied force increases until it is large enough to overcome the static friction. Then, as motion begins the friction decreases rapidly. The sudden decrease in friction results in the net force being higher than desired and a jerky motion results. A similar phenomenon occurs as the body is brought to rest. As the body decelerates the friction force suddenly increases. The sudden increase in friction results in the net force being lower than desired and the body comes to a sudden halt prior to reaching the desired set point. These starting and stopping effects result in overshooting or undershooting the desired trajectory. Regime III also experiences a frictional memory phenomenon where there is a time lag between a change in velocity or load conditions and the resulting change in friction. This results in a hysteresis effect. Frictional memory will be discussed in Section Regime IV: Full Fluid Lubrication When full fluid lubrication occurs all solid to solid contact has been eliminated and the surfaces are supported entirely by the lubricant. In this area the friction is near linear Stick-Slip Friction Stick-slip friction behaviour, which occurs near zero velocity, covers Regimes I, II, and III. The various factors that contribute to stick-slip friction are: 12

24 Static Friction Rising Static Friction Frictional Memory Negative Viscous Slope Static friction, rising static friction, and frictional memory are considered in the following sections Static Friction and Rising Static Friction Static friction was discussed under Regime I. A sub-topic of static friction is rising static friction. Several researchers considered rising static friction, such as Rabinowicz (1958) and Kato et al. (1972). Static friction is a function of time as shown in Figure 2.8. A F Sialic ^ Time at Zero Velocity Figure 2.8 Rising Static Friction. As shown in Figure 2.8 a short period of rest results in a breakaway force with a value between the Coulomb friction and static friction. A long period of rest results in the breakaway force approaching static friction. It is hypothesized that rising static friction is related to the time it takes for the asperities to effectively weld together, while the surfaces are at rest. Counter to the dwell time theory, several researchers have considered varying the rate of force application. Rabinowicz, Kato, and other's work considered a constant rate of force application, thus dwell time was a function of the force application rate. Johannes et al. (1973) and Richardson and Nolle (1976) investigated independent variation of the force rate and dwell time. They demonstrated that static friction is not a function of dwell time but is a function of the force 13

25 application rate. Recent work by Canudas de Wit et al. [4] showed that static friction is independent of the time at rest but is dependent on the rate of force application Frictional Memory There is a time lag in the change of friction following a change in velocity. It is hypothesized that the physical process is related to the time required to modify the lubricating film after the determining parameters are changed. In other words, it takes time for a system to come to a new steady state when the determining parameters are changed. Empirical models represent frictional memory as a delay in the friction response to velocity: Fpricuon =?velocity (*(' ~ At )), (2.6) where xis the velocity and At is the time lag, Figure 2.9. Frictional memory results in a hysteresis effect as shown in Figure Friction Friction o JO "5 > s=.o %H _o "C fa Velocity o "3 > c o Time Lag Velocity No Frictional Memory Time Frictional Memory Time Figure 2.9 Frictional Memory. CD CD «H O fa c o Acceleration fa Deceleration Velocity Figure 2.10 Hysteresis Effect. 14

26 The same environment can result in two different frictional forces leading to problems in modeling and control. A large hysteresis effect can result in unmodeled disturbances, which, when combined with the general instability of the Stribeck curve, can lead to undesirable limit cycling (hunting) by the controller. 2.2 Friction in Hydraulic Actuators Until recently, not much work had been done on friction in hydraulic actuators. Tafazoli states in his thesis from 1997 "To our best knowledge, there is no published work on friction modeling, estimation, and compensation for hydraulically actuated manipulators" [14]. The literature review showed that there had been very little work on hydraulic actuators and friction prior to Tafazoli determined that friction in hydraulic actuators consumes a large part of the applied actuating force. Tafazoli worked on a work cell for the decapitation of salmon at the Industrial Automation Laboratory at the University of British Columbia. He showed that there is a considerable amount of static and Coulomb friction in the actuator and the guide ways that position the fish [14]. With his work on a mini-excavator Tafazoli showed that the hydraulic actuators again contain a large amount of static friction [14]. Lischinsky et al. [2] showed that in a Schilling Titan II manipulator the joint friction can reach 30% of the nominal actuator torque. They attributed this friction to the tight seals that are required to prevent leaks. Figure 2.11 shows a simplified diagram of a double acting hydraulic actuator. The main contributors to friction will be the lip seals, the piston seals, and the piston o- ring. Hydraulic Ports Lip Seals Cylinder Piston Piston Seal Piston O-Ring 3 Rod 3 Figure 2.11 Double Acting Hydraulic Actuator. 15

27 Lischinsky et al. have also determined that the Schilling Titan II manipulator can have a 25% drop between static and Coulomb friction [2]. The impact of the negative viscous slope in their application is quite significant. Hsu et al. [21] and Kwak et al. [22, 23] considered system identification of friction in hydraulic actuators. Kwak et al. [22] states "The requirements imposed by today's high precision machines motivates the precise simulation of friction between these seals and sliding components..." Recent work by Bonchis et al. [13] showed that friction in asymmetric hydraulic actuators is direction and location dependent. The direction dependency was attributed to the seal exhibiting different characteristics depending on whether the rod was being extended or retracted. The pressure differential between the atmospheric pressure and the chamber also influenced the friction. The chamber will develop a different pressure depending on whether the rod is being extended or retracted. The pressure differential affects the seal and its pressure on the rod. The location dependency was attributed to wear in the actuator. The friction parameters for a hydraulic actuator can change depending upon where in the actuator the piston is located. Typically, for a well placed actuator, the center of the actuator will have more wear, and the ends will not have as much wear. There are two approaches to dealing with friction. The first is to design the control system to compensate for the friction. The alternative is to design the hydraulic actuator such that it avoids friction. Armstrong-Helouvry et al. [3] discuss friction avoidance as a possibility in the design of equipment. This latter method is not used very often as control techniques are the standard approach Control of Hydraulic Actuators in the Presence of Friction Recent approaches to the stick-slip friction problem in hydraulic actuators have been through various control techniques such as: model based friction compensation [14, 15], observer-based friction compensation [14, 18, 19], model/observer-based adaptive friction compensation [2, 4], nonlinear PI control [8], and generalized predictive control [1] where all the nonlinearities in an electrohydraulic system are captured. The above topics that explicitly include friction compensation will be discussed further. 16

28 Model Based Friction Compensation In [14] Tafazoli used both model based and observer based friction compensation in a hydraulic system. The former will be discussed here and the latter will be discussed in the next section. In both applications Tafazoli used the estimated friction to estimate the acceleration: F - F >v Applied Friction,^ _* M where F Applied is the applied force, F Friction is the estimated friction force, M is the system mass, and a. is the estimated acceleration. The estimated acceleration was used in a control law that included position feedback, velocity feedback, and acceleration feedback. In the model based, approach Tafazoli [14, 15] used the modified Tustin model [16] to estimate friction: F Dynamic = («0 ~ «lh" + OC 2 2 \x\) Sign( X ), (2.8) where F Dynamjc is the dynamic friction, x is the velocity, a 0 is the Coulomb friction, a x is the Stribeck friction parameter, and a 2 is the viscous friction parameter. Tafazoli then included a term to represent the static friction near zero velocity: F = F +(F -F exp \ Friction Dynamic \ Applied Dynamic I f ( \ 2\ X V ) (2.9) where D v is a threshold velocity that represents where the switch from static friction to dynamic friction occurs [17]. When the velocity was less than D v equation (2.9) was used; when the velocity was greater than D v, the friction was dynamic and equation (2.8) was used. As the control system did not measure velocity, Tafazoli used a velocity observer. The velocity was estimated in real time using a low pass-filtered differentiator: 7> + l where T v is the filter time constant and s is the Laplace operator. The model based approach proved to be better than a conventional proportional-derivative (PD) controller. However, as Tafazoli realized, his model based approach was not adaptive and 17

29 considered friction to be time invariant. The friction parameters were estimated off-line and were not updated on-line Observer Based Friction Compensation To consider the time variance of friction Tafazoli used a modified Friedland-Park Coulomb friction observer to estimate and compensate for friction in a hydraulic actuator [14, 18, 19]. The Friedland-Park friction observer [24] depends on a measured velocity. As before, Tafazoli incorporated a velocity observer using an on line low pass-filtered differentiator, equation (2.10). This estimated velocity was used in the Friedland-Park observer. The observer provided a more adaptive approach, as the friction estimation relied more on the current estimated velocity and acceleration and not a set friction parameter. The observer also caught the hysteresis found in friction at low velocities whereas the modified Tustin model did not [14]. Tafazoli's observer significantly outperformed a conventional proportional-derivative (PD) controller and was comparable to the model based approach [14] Observer Based Adaptive Friction Compensation Lischinsky et al. [2] applied an adaptive control scheme to a hydraulic system using the LuGre model of friction (reference Section 3.2.5). As the parameter z, representing the average bristle deflection, is not measurable, it has to be estimated using an observer. The friction estimation is then used in a control system with an outer position control loop and an inner torque control loop. The position control is a conventional PD controller. The torque loop considers the dynamics of the system including the estimated friction. Using friction compensation showed a significant improvement over the same system without friction compensation. To consider the time variance of friction Lischinsky et al. [2] included an adaptive control scheme. Equation (3.31) is rewritten as: dz. o* \X\ = x-e-^z, (2.11) dt g(x) where 6 is an adaptive parameter that is estimated during controller operation. The friction parameters, particularly Coulomb friction, may be updated on occasion using the current value for 6 as a multiplier. Lischinsky et al. found the adaptive friction compensation scheme provided the best results. 18

30 2.2.2 Friction Avoidance Friction avoidance is the design of machines such that friction is reduced or avoided. Meikandan et al. [7, 31] looked at sealless, tapered pistons where they found that the friction would be reduced. Meikandan et al. considered three types of pistons: diverging, converging, and a converging-diverging piston, Figure The arrows indicate the direction of fluid flow. ////// ////// ////// ////// ////// ////// Diverging Converging Converging-Diverging Figure 2.12 Sealless Tapered Pistons. Their initial studies were theoretical [31]. The friction in the converging-diverging piston was calculated to be one-fifth of a similar conventional piston with seals. Considering the taper angle, eccentricity of the piston, and the velocity, the converging-diverging piston is preferred. Though it has more leakage than the other two pistons, it always has a positive centering force that keeps the piston in the center of the cylinder. The other two pistons can develop a negative centering force pushing them into the cylinder wall. The diverging piston develops a negative centering force below a critical velocity. The converging piston develops a negative centering force above a critical velocity. Meikandan et al. [7] confirmed their theoretical work with experiments. The diverging piston had high friction values at low velocities, corresponding to the negative centering force causing piston contact with the cylinder wall. At high velocities the friction force was low and viscous in nature. The converging piston had high friction values at high velocities when the negative centering force caused the piston to contact the cylinder wall. At low velocities the friction values were low and viscous in nature. The converging-diverging piston exhibited low friction values that were viscous in nature at all velocities. 19

31 2.3 Summary Friction is problematic in the accurate positioning and repeatability of hydraulic actuators. This is a result of the Stribeck effect; namely the negative viscous slope portion of the friction - velocity curve. The usual approach to compensate for friction is through various control techniques. For example, acceleration feedback using either model based friction estimation or observers to estimate friction has been shown to be successful [14]. Friction avoidance using tapered and sealless pistons has also been employed [7, 31]. This approach reduced the friction significantly but required a high hydraulic fluid flow rate. The approach presented here considers the rotation of the piston and rod to reduce friction. It is expected that once the piston and rod are moving the Stribeck effects will be avoided. 20

32 Chapter 3 The Hydraulic Actuator Model 3.1 Modeling The primary purpose of friction models has been in the use of observers in control systems [2, 3, 4, 14, 15, 24]. The friction model prediction is used by the controller to compensate for frictional disturbances in the system. On the other hand, the purpose of modeling hydraulic actuators in this work is to predict the effects that modifying the design of hydraulic actuators will have on the friction in the actuator. The models will be validated through comparison with empirical results. 3.2 Non-Rotating Model The model for the hydraulic actuator was developed with reference to a number of sources [2, 14, 25, 26, 27, 28]. The friction parameters for the hydraulic actuator are derived from data made available by Tafazoli [14]. Since hydraulic actuators are highly non-linear, Jacobian linearization, was used in the development of the model. The block diagram of the system is shown in Figure 3.1. Controller Servo Hydraulic Actuator Gain = K p Valve 4x4 Model *P» PL ' Z X P Figure 3.1 Block Diagram for Hydraulic Actuator System Servo Valve The flow of the hydraulic fluid to the actuator is controlled with a servo valve. This has been modeled as a first order system [26]. The equation relating the spool position, x iv, to the control voltage, V m, is: 21

33 where K sv is the valve position gain and TSV is the time constant of the valve. The control voltage of the valve is limited to ±10 volts. A high gain in a position feedback loop may produce a voltage greater than the upper and lower limits. To effectively model this limitation, the valve control voltage in the model had upper and lower bounds of ±10 Volts Fluid Flow The position of the servo valve spool controls the flow rate in, Q x, and out, Q 2, of the hydraulic actuator. However, the flow rate also depends on the supply pressure, PS, the tank pressure, PT, and the pressure in the actuator chambers, PT and P2. The following equations determine the flow rate through the servo valve [14, 25, 26, 27]: a = KXSV(S(XJ4P~^ + Si-x^Z-P,), (3.13) Q 2 = Kx (s(x )JP2-PT + S(rx )JPS-P2), (3.14) where K is the servo valve flow gain and S(x sv ) is a switching function that indicates whether the actuator is extending or retracting: S{x ) = \ x >0 = 0 * <0 (3.15) Under ideal conditions there is no leakage and Q x = Q 2 = Q,. Equating Q and Q 2 (equations (3.13) and (3.14)) produces the following relation: PS=PL+P2, (3.16) where the tank pressure is equal to zero. In other words, the sum of the pressures in each of the actuator chambers is equal to the supply pressure. Defining the pressure across the load as [25] PL=P,-P2 (3.17) the following relations can also be established: P{= P 5 + P L and, (3.18) P -P P 2 =^y±. (3.19) Differentiating the above two equations yields the pressure rate of change in each chamber: 22

34 PL and, (3.20) (3.21) Substituting equations (3.18) and (3.19) into either equation (3.13) or (3.14) yields the servo valve flow equation: Q L =Kx s (3.22) Taking fluid compressibility into account, and using the continuity principle, the following equations can be derived [14, 25, 26]: A,x n + V h. Q l =A l x p + 1 h ' and (3.23) A 2 (L-x p ) + V h Qi - Ai x P 7, P- 1 2 ' (3.24) where A x and A 2 are the piston areas, L is the actuator stroke length, V h is the hose volume between the servo valve and the actuator, x p is the piston axial position, x p is the piston axial velocity, and B is the effective bulk modulus of the oil. If the two flows are added, Q t + Q 2 = 2Q L, the result can be used to eliminate the dependency on position: QL = A x + A 2 XP + 2V h +A 2 L PL- (3.25) Pressure Changes Equating the servo valve flow equations (3.13) or (3.14) with the continuity equations (3.23) or (3.24), respectively, the pressure change in each of the hydraulic chambers can be determined: Pi = v f A (S(* )4 p s ~ p. + s <r*»)v^)-4* P l v h + Ax p P2 = V M A /! _, [KX SV{S(X SV )4F 2 + S(-xjJP s -P 2)-A 2 x p V h + A 2\ L X p) (3.26) (3.27) 23

35 There are two reasons that the above two equations cannot be used. First, these equations show the pressure change as being dependent on the piston position, where in fact it only depends on the rate of change in position, ie. velocity. Second, under ideal conditions P X = P 2 ; therefore using equations (3.26) and (3.27) for P x and P 2 produces a dependency in the state space matrices, and the state transition matrix is singular. The model becomes unstable. Under the original assumption of ideal conditions with no leakage, P s = P l + P 2 must be satisfied. Both problems can be solved by considering the pressure to be one state represented by the load pressure, PL, instead of two different states [29]. Instead of working with the individual chamber pressures, the load pressure and load flow can be used to eliminate the dependency on position and to eliminate the singularity. By equating equations (3.22) and (3.25) for QL and solving for P,, the rate of change of the load pressure is: P, = 4/3 2V h +A 2 L Kx. A x + A 2 (3.28) A saturation experienced by the system is pressure [28]. During simulations the pressures in the hydraulic chambers can exceed the supply pressure. As this is not realistic upper and lower bounds, equal to the supply pressure, were placed on the chamber pressures Dynamics Applying Newton's second law to a hydraulic piston yields the equation of motion for an actuator: FActuator ^1 A A ^2 MX p + F p F r t c U o Flood ' (3.29) where F A c t u a t o r is the applied force, F L o a d is the external load, F p F r t c t j o n is the friction opposing axial motion, M is the system mass, and x p is the system acceleration. Setting the applied load to zero and substituting in equations (3.18) and (3.19) the equation of motion may be rewritten as: M A, A-. A + A, P - F L pfriction (3.30) 24

36 During the simulations the external load was set to zero. An external load does not affect the state space equations during linearization. The external load does determine the initial pressures in each chamber during static equilibrium Friction Contact between two surfaces occurs at surface asperities. Relative motion between the two surfaces results in the asperities behaving like a spring-damper system. Movement is resisted until the bond between the asperities breaks, or the asperities are sheared. The force required to break the bond between the two surfaces for movement to start is the static friction [3, 4,6]. Haessig and Friedland [6] developed a model where the asperities were modeled as bristles. Canudas de Wit et al [4] and Lischinsky et al. [2] represented the average deflection of the bristles by a state variable z. This model is known as the LuGre (Lund-Grenoble) model [4]: dz ~ 0 \x\ = i -^-z, (3.31) dt g(x) g(x) = a 0 + a, exp ( \ 2\ X V U J J (3.32) dz F FH«io» = ~oz + o- x + a 2 x, (3.33) dt where F Frictlm is a generic friction force, dz/dt is the rate of bristle deflection, g(x) is a function describing the steady state friction characteristics at a constant velocity [4], v sk is the Stribeck velocity defined as being the most unstable velocity on the Stribeck curve [30], x is a generic velocity, a 0 is the Coulomb friction, a x is the Stribeck friction, a 2 is the viscous friction parameter, cr 0 is the bristle spring constant, and ox is the bristle damping coefficient. The static friction is equal to Cf0 + or,. The friction parameters are divided into four static parameters, a 0, a x, a 2, and v sk, and two dynamic parameters, o~ 0 and CT,. These friction parameters are difficult to estimate since the model is nonlinear in parameters and the average deflection of the bristles cannot be measured. A method of obtaining the friction parameters will be discussed in Chapter 4. 25