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An investigation into the reduction of stick-slip friction in hydraulic actuators Owen, William Scott 2001

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A N INVESTIGATION INTO THE REDUCTION OF STICK-SLIP FRICTION IN HYDRAULIC  ACTUATORS  W i l l i a m Scott  Owen.  B . A . S c , The University of British Columbia, 1990  A THESIS SUBMITTED IN P A R T I A L F U L F I L L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF  M A S T E R OF A P P L I E D S C I E N C E  in  T H E F A C U L T Y OF G R A D U A T E S T U D I E S D E P A R T M E N T OF M E C H A N I C A L  ENGINEERING  We accept this thesis as conforming to the required standard  T H E U N I V E R S I T Y OF B R I T I S H C O L U M B I A August 2001 © William Scott Owen, 2001  In  presenting this  degree at the  thesis  in  partial  fulfilment  of  the  requirements  University  of  British  Columbia,  I agree that the  freely available for reference and study. I further agree that copying  of  department  this thesis for scholarly or  by  his  or  her  purposes may be  representatives.  It  is  for  an  Library shall make it  permission for extensive  granted by the understood  head of my  that  publication of this thesis for financial gain shall not be allowed without permission.  Department The University of British Columbia Vancouver, Canada  DE-6 (2/88)  advanced  copying  or  my written  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  Table of Contents Abstract  ii  Table of Contents  Hi  List of Tables  vii  List of Figures  viii  Acknowledgement  x  Dedication Chapter 1  xi Introduction  1  1.1  Preliminary Remarks  1  1.2  Motivation and Objective  2  1.3  Thesis Overview  4  Chapter 2 2.1  6  Friction in Lubricated Machines  6  2.1.1 Basic and Classical Models of Friction  6  2.1.2  8  2.1.3  2.2  Friction in Hydraulic Actuators  Stribeck Curve 2.1.2.1  Regime I: Static Friction  10  2.1.2.2  Regime II: Boundary Lubrication  11  2.1.2.3  Regime III: Partial Fluid Lubrication  12  2.1.2.4  Regime IV: Full Fluid Lubrication  12  Stick-Slip Friction  12  2.1.3.1  Static Friction and Rising Static Friction  13  2.1.3.2  Frictional Memory  14  Friction in Hydraulic Actuators  15  2.2.1 Control of Hydraulic Actuators in the Presence of Friction  16  2.2.1.1  Model Based Friction Compensation  17  2.2.1.2  Observer Based Friction Compensation  18  2.2.1.3  Observer Based Adaptive Friction Compensation  18 iii  2.2.2 2.3  Friction Avoidance  Summary  Chapter 3  19 20  The Hydraulic Actuator Model  21  3.1  Modeling  21  3.2  Non-Rotating Model  21  3.2.1  Servo Valve  21  3.2.2  Fluid Flow  22  3.2.3  Pressure Changes  23  3.2.4  Dynamics  24  3.2.5  Friction  25  3.2.6  State Space Model  26  3.3  Modeling of the System with Piston Rotation  27  3.3.1  DC Motor  28  3.3.1.1  Current  28  3.3.1.2  Dynamics  28  3.3.2  Helical Motion  29  3.3.3  State Space Model  31  3.4  Simulation  32  3.5  Summary  37  Chapter 4  Friction Identification in Hydraulic Actuators: Experimental Method  38  4.1  Introduction  38  4.2  Hydraulic Actuator Setup  39  4.3  Data Acquisition  40  4.3.1  Position Signal  41  4.3.2  Velocity Determination  42  4.3.3  Pressures  42  4.4  Friction Model  42  4.5  Determining The Friction Parameters  43  4.5.1  Equation of Motion  43  4.5.2  Friction Parameter Determination  44  4.5.2.1  Static Parameter Determination  45  4.5.2.2  Dynamic Parameter Determination  46 iv  4.6  Pure Rotational and Linear-Rotating Friction Parameters  47  4.7  Power Requirements  48  4.8  Summary  48  Chapter 5  Reducing Stick-Slip Friction in Hydraulic Actuators: Experimental Results.49  5.1  Pure Rotation  49  5.2  Hydraulic Actuator - 0 Rpm  50  5.3  Hydraulic Actuator Rotating  52  5.4  Friction Parameters  59  5.5  Model Validation  64  5.6  Power Requirements  67  5.7  Position Tracking  69  5.8  Summary  70  Chapter 6  Conclusions and Suggestions for Future Work  72  6.1  Conclusions  72  6.2  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  C. 1 Jacobian Linearization and Discretization C.2  Non-Rotating Model State Equations C.2.1  C.3  91 :  92  Position  92  C.2.2 Axial Velocity  92  C.2.3 Pressure  93  C.2.4 Friction  93  C.2.5  94  Servo Valve Input  C.2.6 Jacobians  94  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  98  Rotating Velocity  C.3.6 Motor Current  99  C.3.7 Motor Voltage Input  100  C.3.8  100  Servo Valve Input  C.3.9 Jacobians  101  vi  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  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  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 0.008 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  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  Dedication This thesis is dedicated to my wife, Mary Wells, and our daughter, Patricia Ann Mary Owen, born on July 21, 2001. The joy that these two people bring to my life continues to grow.  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  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  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  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  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. A clear picture of the friction 1  phenomena and the problems found with friction in lubricated machines is important to the development of the friction avoidance technique presented in this thesis. 2.1.1  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 = FApplied - F,Friction  F Applied  77777/  F,Friction  •  //////  Figure 2.1 Da Vinci's Model of Friction. 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]. 1  6  Coulomb (1785) introduced the concept of a dry or Coulomb friction.  The friction force  opposing motion was constant and independent of the velocity,  M'Dynamic^^Normal>  ^Dynamic where F  \s  Dynamic  the dynamic friction force, F  (2*1)  is the normal force, and Moynamic is the  Nomal  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 ' where  fJ-  Dynamic  < fi  Slalic  (2-2)  . F tic is the static friction force and u,tatic is the static coefficient of Sta  S  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.  Bx,  w  where B is the viscous damping coefficient, F  Viscous  (2.3)  is the viscous friction, and x is the velocity.  The friction model then becomes: F =F Sialic Applied F =F +F Dynamic Coulomb 1  1  \x\ = 0, T  1  x> 0 Viscous  (2.4) A  ^ 7  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]. 2.1.2  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  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:  Static Friction (also known as stiction)  • Regime II:  Boundary Layer Lubrication  • Regime III:  Partial Fluid Lubrication  • Regime IV:  Full Fluid Lubrication  These different regimes are shown in Figure 2.5.  9  Friction Force  A Regime I  ^1  Regime II  Figure 2.5 Stribeck Curve Regimes.  2.1.2.1 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  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. 2.1.2.2 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  and the asperities are sheared. This regime of the Stribeck curve experiences solid to solid contact. 2.1.2.3 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 2.1.3.2. 2.1.2.4 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. 2.1.3  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  • Static Friction • Rising Static Friction • Frictional Memory • Negative Viscous Slope Static friction, rising static friction, and frictional memory are considered in the following sections. 2.1.3.1 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  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. 2.1.3.2 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 2.10. Friction  Friction  o  o  "3 >  JO  "5 >  s= .o %H _o "C fa  Time Lag  co Velocity  Velocity  Time No Frictional Memory  Time Frictional Memory  Figure 2.9 Frictional Memory.  CD CD «H  O  fa  Acceleration  co  fa  Deceleration Velocity Figure 2.10 Hysteresis Effect. 14  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 1996.  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 oring. Hydraulic Ports  Cylinder  Piston Lip Seals  3 3  Rod  Piston Seal  Piston O-Ring  Figure 2.11 Double Acting Hydraulic Actuator. 15  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. 2.2.1  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  2.2.1.1 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  >v  Applied  - F  Friction  ,^ _*  M where F  Applied  is the applied force, F  is the estimated friction force, M is the system mass,  Friction  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: Dynamic  F  where F  Dynamjc  ~ «lH" + OC \x\) Sign( ) ,  (2.8)  2  = ( « 0  2  X  is the dynamic friction, x is the velocity, a is the Coulomb friction, a is the 0  Stribeck friction parameter, and a  2  x  is the viscous friction parameter. Tafazoli then included a  term to represent the static friction near zero velocity:  F  Friction  =F  Dynamic  +(F \  Applied  -F  \|exp  Dynamic I  f V  (  • \ 2\  X  (2.9) )  where D is a threshold velocity that represents where the switch from static friction to dynamic v  friction occurs [17]. When the velocity was less than D equation (2.9) was used; when the v  velocity was greater than D , the friction was dynamic and equation (2.8) was used. As the v  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 is the filter time constant and s is the Laplace operator. v  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  considered friction to be time invariant. The friction parameters were estimated off-line and were not updated on-line. 2.2.1.2  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]. 2.2.1.3  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 , dt g(x)  (2.11)  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  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 2.12. 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  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  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. *P Controller Gain = K  p  Servo Valve  »  PL '  Z  Hydraulic Actuator 4 x 4 Model X  P  Figure 3.1 Block Diagram for Hydraulic Actuator System.  3.2.1  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 , to the control iV  voltage, V , is: m  21  where K is the valve position gain and T is the time constant of the valve. The control SV  sv  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. 3.2.2  Fluid Flow  The position of the servo valve spool controls the flow rate in, Q , and out, Q , of the hydraulic x  2  actuator. However, the flow rate also depends on the supply pressure, P , the tank pressure, P , S  T  and the pressure in the actuator chambers, P and P . The following equations determine the T  2  flow rate through the servo valve [14, 25, 26, 27]: a = KX (S(XJ4P~^ SV  + Si-x^Z-P,),  (3.13)  + S(rx„)JP -P ),  (3.14)  Q = Kx„(s(x„)JP -P 2  2  T  S  2  where K is the servo valve flow gain and S(x ) is a switching function that indicates whether sv  the actuator is extending or retracting: S{x„)  =\  x„>0  = 0  *„<0  (3.15) Under ideal conditions there is no leakage and Q = Q = Q,. Equating Q and Q (equations x  2  2  (3.13) and (3.14)) produces the following relation:  P =P +P , S  L  (3.16)  2  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]  P =P,-P L  (3.17)  2  the following relations can also be established:  P +P P=  5  L  {  P  and,  (3.18)  -P  P =^y±. (3.19) Differentiating the above two equations yields the pressure rate of change in each chamber: 22 2  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 =Kx L  (3.22)  s  Taking fluid compressibility into account, and using the continuity principle, the following equations can be derived [14, 25, 26]:  Q =A x + l  l  Qi - Ai  p  A,x + V . ' and n  1  h  h  (3.23)  A (L-x ) + V 7, P-2 ' 2  x P  p  h  (3.24)  1  where A and A are the piston areas, L is the actuator stroke length, V is the hose volume x  2  h  between the servo valve and the actuator, x is the piston axial position, x is the piston axial p  p  velocity, and B is the effective bulk modulus of the oil.  If the two flows are added, Q + Q = 2Q , the result can be used to eliminate the dependency t  2  L  on position: QL 3.2.3  A +A x  =  2  +  P  X  2V +A L h  2  (3.25)  PL-  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  v  h  P = 2  V V  M  h  +  (S(*„)4 s ~ . p  A  A  + Ax  +  (3.26)  <r*»)V^)-4* l  s  P  p  /! _  2\  A  p  L  , [KX {S(X )4F + SV  SV  2  (-xjJP -P )-A x  S  s  2  2  p  (3.27)  p)  X  23  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 = — P ; therefore X  2  using equations (3.26) and (3.27) for P and P produces a dependency in the state space x  2  matrices, and the state transition matrix is singular. The model becomes unstable. Under the original assumption of ideal conditions with no leakage, P = P + P must be satisfied. s  l  2  Both problems can be solved by considering the pressure to be one state represented by the load pressure, P , instead of two different states [29]. L  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 Q and L  solving for P , , the rate of change of the load pressure is: 4/3 P,  =•  2V +A L h  A +A  Kx.  x  2  (3.28)  2  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. 3.2.4  Dynamics  Applying Newton's second law to a hydraulic piston yields the equation of motion for an actuator: FActuator  where  F  A c t u a t o r  ^1 A  is the applied force,  F  ^2  A  L o a d  —  MX  p  + F  p F r t c U o  „  is the external load,  Flood  F  p F r t c t j o n  '  (3.29) is the friction opposing  axial motion, M is the system mass, and x is the system acceleration. Setting the applied load p  to zero and substituting in equations (3.18) and (3.19) the equation of motion may be rewritten as: A, A-. M  A + A,  P  L  - F pFriction  (3.30)  24  During the simulations the external load was set to zero. A n 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. 3.2.5  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 — =i dt  °~ \x\ -^-z, g(x) 0  (  g(x) = a + a, exp 0  V  FH«io»  F  where F  (3.31)  • \ 2\ X  U J  (3.32)  J  dz =°~oz + o- — + a x, dt x  (3.33)  2  is a generic friction force, dz/dt is the rate of bristle deflection, g(x) is a function  Frictlm  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  is the Coulomb friction, a is the Stribeck friction, a  0  x  is the viscous friction  2  parameter, cr is the bristle spring constant, and o is the bristle damping coefficient. The static x  0  friction is equal to Cf + or,. 0  The friction parameters are divided into four static parameters, a , a , a , and v , and two 0  x  2  sk  dynamic parameters, o~ and CT, . These friction parameters are difficult to estimate since the 0  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  The LuGre model is suitable for a state space approach and, unlike other models, does not require a separate component for static friction [3, .14]. Instead, the deflections of the bristles are used to calculate the pre-displacements and thus determine the pre-displacement forces (static friction).  The above equations (3.31), (3.32), and (3.33) can be combined to give one equation for friction that can be used in the actuator equations of motion: O~ O~AX\Z  FFriction — O~ Z + <7 X • 0  0  X  a + a exp 0  (  • \2 \ X  x  (3.34)  + a x. 2  )  V  The velocity along a specific direction of motion replaces the generic velocity (i) to obtain a specific directional friction. For example, for FpFrjctjon „ x = xp. 3.2.6  State Space Model  Including equations (3.28), (3.30), and (3.31), a 4 x 4 state space model of the actuator can be formed. The state variables are: [x, x2 x3 x ] = \xp xp PL z\ , 4  and the state space equations are:  A  x  A  (3.35)  —x,  Xj  2  A +A  2  x  CT CT,X  2  0  (  M  ' a +a 0  4B 2V +A L h  Kx. S(xv^^^  x  S(-x )J^l  +  v  4  X  2  ,  ,2\  22  a  X  ,(3.36)  exp \  A  V  skJ  A +A  r  l  2  and (3.37)  2  a + a exp 0  x  V  'X  2\  (3.38)  N  J  26  It can be seen from the selected equations that the state space model is highly non-linear. The equations are linearized and discretized for a computer simulation. Jacobian linearization for non-linear systems is discussed in Appendix C. 3.3  Modeling of the System with Piston Rotation  Rotating the piston and rod generates a relative velocity between the rod and seals and between the piston seals and the cylinder wall. Once the velocity is sufficiently high, and the rotation is maintained, the switch from static friction to dynamic friction will no longer occur, as the actuator only operates in the dynamic regime. When the piston moves axially, the axial friction axial velocity curve is linear and the Stribeck region is avoided. However, without a sufficient rotating velocity the axial friction versus axial velocity relation will remain nonlinear.  The hydraulic actuator model was modified to include rotating the piston, resulting in a 6 x 6 state space model. Rotation of the piston was modeled with a permanent magnet brushless DC motor coupled to the rod, Figure 3.2. I— Hydraulic Ports DC Motor  /  EE  <5  Lip Seals  Coupling  Cylinder  Ll Piston Piston Seal Piston O-Ring  Figure 3.2 Hydraulic Actuator with DC Motor. The block diagram for the system is shown in Figure 3.3. v. Controller Gain = K p  p'  x  Servo Valve  Motor and Hydraulic Actuator 6 x 6 Model  PL ' > e»I z  x  >•  Figure 3.3 Block Diagram for Hydraulic Actuator System with a Rotating Piston.  27  3.3.1  DC Motor  Torques and angular velocities are usually used to model motors. To remain consistent with the hydraulic forces and force units, the motor torques and angular velocities were changed into forces and 'linear' velocities acting on the piston surface. 3.3.1.1  Current  Applying Kirchoffs voltage law and taking into account the back electromotive force the following equation represents the DC motor [26]:  L where K  bemf  L  m  (3.39)  L R,  m  m  cyl  is the back electromotive force constant, L is the motor inductance, R m  cyl  is the  radius of the piston, R is the motor resistance, V is the voltage applied to the motor, i is the m  m  m  motor current, i is the rate of change of the motor current, and x is the rotating velocity of the m  e  system. The angular velocity, co , of the system is: g  cyl  K  3.3.1.2 Dynamics The equation of motion for rotation is:  J_  jy  M where F  eFriclion  (3.41)  OFriction  cyl  K  is the friction opposing the rotating motion, K is the torque constant, and x is t  e  the rotating acceleration. The torque of the motor, r ,\s: m  r =K,i . m  (3.42)  m  Dividing by the radius of the piston produces the equivalent force,  lied  , acting on the piston  surface:  Ki ^Supplied  '  (3.43)  cyl  K  F  gFricljon  is equation (3.34) where the velocity is the angular velocity, x : e  28  p0Friction — 0~ Z Q  Xg  + (TjXg  f  2  x  U*  V  3.3.2  + ax  g  (3.44)  ( Xg • }  a + a exp 0  Z  J  J  Helical Motion  Combining the rotating and axial motion yields a helical motion for each element on the piston, as shown in Figure 3.4. The direction along the helical motion, as shown, is considered to be positive. By maintaining the same direction of rotation the helical motion will always have the same sign regardless of the axial motion. helical, x  h  rotating, x  e  Figure 3.4 Helical Motion of an Element on a Rotating Piston. During operation, the helical friction force opposing the helical motion is modeled as: CT CT, \X \Z h  0  FhFriction ~ °~0  Z  G\ h X  • + oi x^, 2  (3-45)  g( h) X  where F  hFriclion  is the friction opposing the helical motion and x is the velocity in the helical h  direction.  The friction forces opposing the axial and rotating motions are approximated by the vector components of the helical friction force, Figure 3.5.  29  pFrictior,  F  v  N  hFriction  ^  Figure 3.5 Vector Components on an Element on the Piston. The rotating friction and axial friction components may be rewritten as: F,  ^hFriction  a  F  3  n  <  (3.46)  ^  =— F  pFriction  .  (3.47)  hFriction '  x  h  x x since -^- = sin(0) and -7^-= cos(0). x  An assumption was made that the axial and rotating  h  components of F  hFriclion  are approximately equal to F  pFriclion  and F  , respectively.  eFriclion  To eliminate the dependency between x , x , and x in the state space equations, setting p  h  x  + e'  =  h  x  x  l s  eliminated 2  p  h  in equations  = sign(x ) - sign{^x + x )-1 h  g  e  (3.46)  and (3.47).  Noting  that,  and ^jx + x\ = ^jx + x , the two system equations 2  2  2  p  9  (3.30) and (3.41) are then rewritten as:  x  pr  1 M  A, A~,  A. + A,  Q~o Z X  P-  (3.48)  P  •\j  x  + P  e  g(^ l + l )  x  x  x  and 1 m  M  GQXQZ  - <j x + + xl g x  cyl  R  e  0*0\ 8 zx  CCjXg  (3.49)  The motor torque and the hydraulic pressure combine into a resultant force that produces the helical motion. If there is no rotary motion then the helical velocity defaults to equal the axial 30  velocity and, likewise, if there is no axial motion then the helical velocity defaults to equal the rotating motion.  During operation it is the friction force opposing the helical force that is relevant. It is expected that the friction opposing the motion can have a range of parameters depending upon the influence of both the rotating and axial motion. 3.3.3  State Space Model  Including equations (3.28), (3.31), (3.39), (3.48), and (3.49) a 6 x 6 state space model can be formed that represents a hydraulic actuator complete with a rotating piston. The state variables are: [x, x x x x x ] = \x x P z x i \ , 2  3  4  5  6  p  p  L  g  m  and the resulting state space equations are:  X^ "™ X 2  Ml  I  i  2  i  J  s +  p  (A + AA  2  J  (TQX2 X^ I —" "~" ^TtX-y  XI  2  f  0  AB  2V +A L H  I  X2  ...+.  a  (3.50)  5  + a, exp  2  ' x l  V V  v  +  *  (3.51)  x ^ ~ ^ J)  Kx,  (3.52)  2  r~  2  X ^ —— ^ X 2  X  a + ax exp 0  '  ^x +x  v v  2  2  k  y2  2 5  ^  (3.53)  J)  31  J_ M  0~QO~\ 4 5 X  -<7 X l  5  +•  X  , and 2  a + a expj 0  x  V  V  *k  v  K. x. ^bemf^S f  R 6 X  M  Rcy!  (3.54)  J J  (3.55)  The model of the motor and thus the current, equation (3.55) was included in the state space model of the actuator. The time constant of the motor is much great than the bandwidth of the hydraulic system. The model of the motor could have been eliminated from the state space model but was left in for completeness.  The motor could have been modeled separately but that would have introduced a time lag between the back electromotive force and the resulting current.  Such a model would have  required the speed of rotation from the actuator model to be fed back to the motor model in a negative feedback loop. Simulation results showed the difference between the two models was less than one percent so the combined model was chosen. 3.4  Simulation  The baseline axial (ie. non-rotating) simulation results are consistent with the published behaviour of hydraulic actuators in the literature [1, 14, 23, 32]. The parameter values used in the simulation are found in Appendix B. The position control loop for the piston was closed using a simple proportional controller to facilitate evaluation of the model behaviour without the controller influencing the outcome.  Simulations were then run with the piston rotating over a range of speeds from 0 to 800 rpm. A sinusoidal position reference input was used to evaluate the tracking ability of both the rotating and non-rotating models while the piston velocity passed back and forth through the Stribeck region. Figure 3.6 shows the piston position for the standard model and the rotating model for a simulation run time of 1.25 seconds. The lag between the models and the desired piston position is apparent near time zero and can be attributed to stiction. When the piston is rotated the phase  32  shift decreases. As the angular velocity of the piston is increased there is a corresponding increase in the improvement of the phase shift. 0.061  1  1 „  ™  1 T  ,  ™  „  Piston Rotation = 460 RPM 0.05  K  P  ,  = 5 0 0  standard Rotating  1  —  1  Phase Shift 8.55 degrees  r  7.65 degrees  / /  0.04 £ 0.03 a  o  0.01 X -0.01  d  Input  0.6 0.8 1 Time (s) Figure 3.6 Piston Position for the Standard and Rotating Model.  0  0.2  0.4  1.2  The friction curves for the two systems are shown in Figure 3.7 for a simulation run time of four seconds. The top curve is the total (helical) friction versus the helical velocity. As the piston starts to rotate the system friction passes through the Stribeck curve and then proceeds along the viscous portion of the curve to an operating region where it remains.  The bottom curve shows the axial friction versus the axial velocity with the coefficient of determination (R ) given for each model. The typical friction - velocity curve is apparent for the 2  non-rotating model where the velocity accounts for only a portion of the changes in friction. When the piston is rotated the axial friction - axial velocity curve is a linearized, predictable, and dependent function of velocity. The magnitude of the axial friction has decreased as well. This translates into a smaller hydraulic power requirement.  33  Helical Friction vs Helical Velocity 3000 |  1  1  2000 o  i  ^ ^ ^ ^ ^  Start Up  PH  i  Operating Region  | 10i 13 X i  /  i  i  0.2  0  i  0.4 0.6 Helical Velocity (m/s) Axial Friction vs Axial Velocity  1000  i  0.8  Standard Model R^ = 0.92612 500  e  0  o  Rotating Model R = 1 F = 2700 Vel + 0.004 2  P  P  CCJ  •S -500 0.1 0.15 -0.05 0 0.05 0.2 Axial Velocity (m/s) Figure 3.7 Friction Curves for the Standard and Rotating Model (460 rpm).  -0.2 -1000  -0.15  -0.1  The value of the viscous friction coefficient that was used during the simulation was 2318 N/m/s. The slope of the linearized friction curve is given as 2700 N/m/s. In the simulations the slopes of the linearized friction curves are near the value of the viscous friction coefficient. This suggests that all the factors that influence friction, except the viscous friction, are reduced to some degree. The variation of the slope or viscous friction parameter is evident in Table 3.1 and Figure 3.8. Table 3.1 Viscous Friction Parameter vs. Rotation Speed. RPM  Viscous Friction Parameter (N/m/s)  0  2320  120  3600  290  2940  460  2700  630  2580  800  2520  34  Viscous Friction Parameter Variation with Speed 4000  1  1500  s-  PH  § o o  1000  co  200  400  600  800  1000  Rotation Speed (rpm) Figure 3.8 Viscous Friction Parameter vs. Rotation Speed. As the angular velocity is decreased the operating region lengthens and approaches the Stribeck region. As this happens, the relation between the axial friction and axial velocity becomes nonlinear, as shown in Figure 3.9. In this case the friction dependence on velocity is no longer first order. However, the non-linear phenomena such as hysteresis are still eliminated. If the angular velocity slows down below the Stribeck limit, non-linear phenomenon begin to reappear.  During simulation, it was found that a minimum rotational velocity of more than twice the Stribeck velocity (0.07 m/s, R = 0.99) was required for hysteresis to be avoided. This speed 2  indicates the outer limit of the Stribeck region. For the axial friction to be completely linearized the required velocity was 0.5 m/s (R = 1.0). 2  35  Helical Friction vs Helical Velocity 1000 a o  IC 500 13  X  1000  a *HJ  o  F  0.1 0.15 Helical Velocity (m/s) Axial Friction vs Axial Velocity = -39358 V e l - 47 V e l + 5318 Vel +1.25 3  0.25  2  500 0  Rotating Model R = 0.99 Standard Model R = 0.93  -500 -1000 -0.2  -0.15  -0.1  -0.05 0 0.05 0.1 0.15 0.2 Axial Velocity (m/s) Figure 3.9 Friction versus Velocity for a Slow Angular Velocity (50 rpm avg).  To simulate a pick and place motion the simulation was run with several consecutive step inputs. Figure 3.10 provides the results. When the piston is rotated the rise times are faster and the steady state errors are improved. Simulations showed that rotating the piston is equivalent to removing damping from the system.  36  Piston Position  0.06 0.05  SS Error Standard 0.2378 mm Rotating 0.0450 mm SS Error Improvement = 50%  Piston Rotation = 460 RPM K =400 P  SS Error \ Standard 0.0096 mm V Rotating 0.0014 mm i\SS Error Improvement = 86%  0  Standard Rotating • Input (  -0.01 0  0.2\  0.4  0.6 0.8 Time (s) Figure 3.10 Step Input - Piston Position.  1.2  The steady state error improvement is small in magnitude. However, as discussed in Section 1.2 a small error in the joint actuators can result in a large error in the end effector's position. The percent improvement does show significant improvement. 3.5  Summary  The simulations show that the axial friction is linearized when the piston and rod are rotated. Linearizing axial friction and eliminating the Stribeck effect will improve the controllability of hydraulic actuators.  The simulations showed that the steady state error and phase lag are  reduced. The results motivate the further investigation of rotating the piston and rod to reduce friction in hydraulic actuators.  37  Chapter 4 Friction Identification in Hydraulic Actuators: Experimental Method 4.1 Introduction The purposes of these experiments are two fold. The first purpose is to evaluate the benefit, and characterize the effect of rotating the piston in a hydraulic actuator on the reduction and linearization of friction. Linearizing the friction eliminates the non-linear effects found in the Stribeck area of the friction - velocity curve. When linearized, the friction then becomes more predictable. The second purpose is to validate the hydraulic actuator model, which can be used to predict friction for both design (friction elimination) and control (friction compensation) purposes. A fourth order state space model was developed to simulate a hydraulic actuator under normal operating conditions. A sixth order model was also developed to simulate the effect of rotating the piston. The model showed that by rotating the piston at a sufficient velocity, hysteresis is avoided, and the friction can be linearized.  One major difference expected between the model results and the experimental results is the required rotational velocity. The friction parameters used in the model for rotating the motor and piston are the same as the friction parameters used for axial motion of the piston, 588 N for static friction. This results in a high static friction that the motor must overcome in order to rotate. It is expected that the rotating friction parameters are considerably less. For example, Lischinsky et al. [2] determined that the first joint, a hydraulic actuator, of their Titan Schilling II manipulator had a static friction level of about 200 Nm. Canudas de Wit et al. [33] showed that their DC motor had a static friction level of 0.29 Nm. These two friction levels are quite different. It is expected that there will be a significant difference in the rotating friction and axial friction levels when the correct values are determined.  This will result in a lower rotating  velocity.  38  Hz. This was deemed acceptable [34] as the fastest component in the system was the valve with a maximum frequency response of 13 Hz.  The motor speed was set to a constant level and was not changed during each test run so its response time, given its much faster time constant, did not affect the results. The control of the motor speed was accomplished through a dedicated amplifier for the motor.  The signal to the servo valve was run in open loop for friction parameter identification experiments.  A position feedback controller was used for tracking experiments.  A simple  proportional controller was used so that the controller did not influence the response of the system. A low gain of 100 V/m was used so that the friction phenomena near zero velocity were more clearly observed.  To eliminate noise, the position signal was averaged off line with a window size of 10 before differentiating. This averaging behaved as a 100 Hz filter. The resulting velocity was again averaged producing a signal that was effectively filtered at 10 Hz. As the open loop bandwidth of the system was measured at 1 Hz, this was considered acceptable.  The data filtered to 10 Hz was used to determine the static parameters.  This produced, on  average, over 250 data points per test run.  In order to acquire a sufficient number of data points for the dynamic parameters the data filtered to 100 Hz was used. This again provided over 250 data points per test run. Data points where the applied force was greater than the static friction (a + Cf,) were removed in order to focus on 0  the static friction regime. 4.3.1  Position Signal  The position was obtained by using a linear potentiometer. The potentiometer was calibrated by measuring the displacement of the motor and actuator in 1 mm increments. One hundred data points were taken. The resulting graph was linear. The curve fit of the data produced an average error of zero with a standard deviation of 0.07 mm. The 95% confidence interval for 95% of the error population [35] was ±0.16 mm.  41  Given the above position signal accuracy, at a sampling frequency of 1000 Hz, the linear speed of the actuator was limited to 160 mm/s in order to obtain accurate measurements within the 95% confidence interval. Velocity curves were compared at the sampling rate of 1000 Hz and the filtered rates of 100 Hz and 10 Hz. There was no loss of data. In other words, the velocity did not have high frequency components, and the slower effective sampling rates were sufficient to accurately reflect the velocity data. 4.3.2  Velocity Determination  Using the same filter as Tafazoli [14] the velocity was obtained from the position signal off-line using a low pass-filtered differentiator.  The filter was an order 16 digital Finite-Impulse  Response filter. It is a non-causal, high order, delay free differentiator. 4.3.3  Pressures  Pressure transducers were used to measure the supply pressure, return pressure, and the pressure in each cylinder chamber. The supply pressure was held at 712 psi while the return pressure was zero. 4.4  Friction Model  The model used for the friction term,  F  is the LuGre model as discussed earlier, repeated  ,  pFrictjon  here for convenience: dz . — -x dt  cr  0  x  g(x)  g(x) = a + a exp 0  (4.1)  , and  x  _,  z,  (4.2)  dz  Fpncior, =0-02 + 0-, — + a x.  dt  The model has static and dynamic parameters.  (4.3)  2  The static parameters are a , a cr ,and 0  v  2  while the dynamic parameters are o~ and cr,. The static parameters characterize the steady 0  state static map between velocity and friction [2, 4]. This includes static, Coulomb, Stribeck, and viscous friction.  The dynamic parameters characterize the dynamic response of friction  42  including stiction. The parameters are difficult to determine as they appear nonlinearly in the equations [36]. 4.5  Determining The Friction Parameters  In order to confirm the accuracy of the friction model and the hydraulic actuator model, the friction parameters were determined.  Model simulations were then run with the correct  parameters, duplicating the experimental test runs for comparison. In [37] Wassink discusses the repeatability of friction.  Friction is history dependant and  experimental tests may often exhibit different friction values. Tests must be performed close together to have any confidence in the comparison of values.  With this in mind, up to ten runs were conducted consecutively over a short period of time for each rotation speed. The friction parameters were calculated for each run and the average of these values are used in the following discussion and in the model simulations. 4.5.1  Equation of Motion  With the applied load set to zero, the equation of motion for an actuator (equation (3.29)) is rewritten to solve for friction: F  „=P,A-P A -Mx  pFriclio  2  2  p  This equation requires knowledge of the acceleration.  (4.4) During quasi-static experiments the  acceleration is low enough to consider the velocity to be constant at any instant in time. Figure 4.3 shows a typical curve for the applied force and acceleration force plotted against time. It can be seen that omitting the acceleration is justified. This approach was also used by Tafazoli [14].  43  Applied Force 600 g400 o  £200  Acceleration Force  10  15 Time (s)  20  25  30  Figure 4.3 Quasi-Static Experiments: Applied Force and Acceleration Force. The applied force can then be considered to be the friction force: P A P A. pFriction x  4.5.2  x  2  2  (4.5)  Friction Parameter Determination  Quasi-static experiments were conducted by using a sinusoidal signal with a period of 200 seconds to control the servo-valve in open loop. The control voltage input to the servo valve was: V„=\.0-l  .0 sin(400;r t-nlT).  (4.6)  This input allowed the control voltage to increase from zero in a very slow manner, maximizing the amount of data collected near zero velocity while minimizing the amount of data in the viscous region. The experiments were run with a duration of 35 seconds.  This produced a slowly varying sine wave resulting in velocities varying from zero in the static regime to velocities in the viscous regime. Figure 4.4 shows a typical velocity-time curve.  44  15 20 Time (s) Figure 4.4 Quasi-Static Velocity Curve.  35  The resulting friction - velocity curve is split into a static regime and a dynamic regime. The split is determined by observing how the LuGre friction model behaves. The switch from static to dynamic was chosen as 0.6 mm/s. The dynamic regime and static regime are used to determine the static parameters and the dynamic parameters, respectively. Using a set velocity to represent the transition from static to dynamic motion was considered by Tafazoli [14] and Karnopp [17]. 4.5.2.1 Static Parameter Determination With quasi-static tests with near zero acceleration the velocity can be considered to be constant. dz At constant velocity the friction enters steady state and — = 0 . Solving equation (4.1) for z dt and substituting into equation (4.5) produces: (  P\ A  ^2 A  -a0  +a  x  x  ^  f  p  sign(x ) + a x  x  exp  p  2  p  (4.7)  J)  A nonlinear least squares optimization function from Matlab {Isqnonlin) [38] was utilized to determine the four static parameters. With F - P A - P A ss  X  X  2  2  taken as the measured steady state 45  friction and F the estimated friction (from the right-hand side of equation (4.7)), the cost ss  function to be minimized can be written as [33]: # data point s  .min.  [*".(**)-tH*)]  Z  (4-8)  2  The original parameter estimates for the non-linear least squares were obtained through trial and error. 4.5.2.2 Dynamic Parameter Determination The dynamic parameters are more difficult to determine because the internal state z is not dz measurable. At velocities between zero and 0.6 mm/s — is assumed to be constant. Then dt (  equation (4.2) is approximated by g(x ) = a + a since exp 0  {  V  • \2 \  X  \y ) sk  1 since x —> 0. J  Assuming the rate of deflection of the bristles to be negligible at near zero velocities, equation (4.3) can be written as [2]: FFriction = ^  (4.9)  This provides a simpler equation for z :  z=^ ^ ,  (4.10)  dz dz which can be used in equation (4.1) to eliminate z and <r . A n estimate, — , for — for each dt dt data point is obtained: 0  - = x-^" ''°"W. c  a +a  dt  0  (4.11)  {  Then z is estimated using Euler integration [36] to obtain z . This allows the estimation, <J , of 0  <7 from equation (4.9), rewritten as: 0  a =  f  Fpri  0  m  .  (4.12)  An estimate, d , for the other dynamic parameter, cr,, is found through a linearized description x  of the LuGre friction model and the equation of motion [2]:  46  \ A -P A =Mx +  p  2  2  p  (cr, + a )x 2  p  +  (4.13)  ax, 0  p  where the bristle deflection is estimated with the micro-displacements of the actuator and the rate of bristle deflection is estimated with the near zero velocities. The parameter <7, is chosen such that equation (4.13) has near critical damping, (4.14) In this case the damping coefficient was chosen to be 0.9. It should also be noted that the viscous friction was neglected from equation (4.13).  At near zero velocities the friction -  velocity curve is in the static regime and approaching the boundary lubrication regime. Viscous friction is not yet developed and is therefore neglected.  Curve fitting is applied to equation (4.3) to solve for a and CJ, using & and a, as the original 0  0  estimates. Here, again, viscous friction is neglected.  The friction parameters for the hydraulic actuator are presented in Section 5.2. 4.6  Pure Rotational and Linear-Rotating Friction Parameters  The approach to determine the rotating friction parameters for rotation with no axial motion was the same as that used for the axial motion with no rotation. The torque and rotating velocity of the motor and piston was obtained from the motor amplifier. The torque was converted into a force acting on the piston surface. This force was used as the applied force in the equations from Section 4.5 in place of the hydraulic pressures. The control voltage input to the motor was V =1.0-1.0sin(400^r^-^/2) m  (4.15)  The rotating friction parameters for rotation with no axial motion are presented in Section 5.1.  The axial friction parameters were determined for the hydraulic actuator while under a constant speed of rotation. The approach is identical to that outlined in the preceding sections. Each rpm has a corresponding set of axial friction parameters. These parameters are presented in Section 5.3.  47  4.7  Power Requirements  The mechanical power was calculated to determine the cost of rotating the piston. The power for the non-rotating piston can be calculated from [39]:  (P,A.  -P A )Ax 2  2  n  Power = y - = {P A ~ A )x The power for rotating the motor, piston, and rod is calculated from: p  p  hyd  }  t  2  2  p  Power =r co . e  m  (4.16) (4.17)  e  Substituting in equations (3.40) and (3.43) the rotating power becomes: e= *p n«i e-  Power  F  (- )  i  4  P  18  The total power is then equal to: Power , ={P A -P A )x Tola  X  X  2  2  p  +F ^ x , .  (4.19)  The hydraulic and total power requirements are presented in Section 5.6. 4.8  Summary  The determination of the friction parameters was split into two regimes, static parameters and dynamic parameters. The static parameters were determined from the dynamic friction region of the friction - velocity curve. The dynamic parameters were determined from the static friction region of the friction - velocity curve.  Rotating friction parameters were determined for rotation of the hydraulic system with no axial motion. Axial friction parameters were also determined for axial motion of the hydraulic system with no rotation. Axial friction parameters were also determined for the hydraulic actuator for a range of piston rotation speeds. The results are presented in Sections 5.1, 5.2, and 5.3.  The hydraulic power and total power (rotating and hydraulic) was calculated to determine if rotating the piston was economically beneficial. The results are presented in Section 5.6.  48  Chapter 5 Reducing Stick-Slip Friction in Hydraulic Actuators: Experimental Results 5.1 Pure Rotation The rotating friction - rotating velocity curve for rotation with no axial motion (i.e., no translation of the hydraulic actuator) produced a typical Stribeck curve, Figure 5.1. The LuGre rotating friction parameters are provided in Table 5.1. The rotating friction - rotating velocity curve from each test run was consistent, as suggested by the standard deviations. The negative viscous slope has a drop of 30%. 60  r  50  40  | 30 o  20  10  0.1  0.2  0.3 0.4 0.5 0.6 Velocity (m/s) Figure 5.1 Rotating Friction - Rotating Velocity Curve  0.7  49  Table 5.1 LuGre Rotating Friction Parameters. LuGre Friction Parameter  Average Value  Standard Deviation  Coulomb Friction  34.7 N  0.5  Stribeck Friction  14.6 N  3.9  Static Friction  49.3 N  4.4  8.3 N/m/s  1.4  0.042 m/s  0.012  Viscous Friction Stribeck Velocity  OC  W  ske  v  Bristle Spring Constant  °oe  300 N/m  70  Bristle Damping Coefficient  °~\e  120 N/m/s  13  5.2  Hydraulic Actuator - 0 Rpm  The axial friction - axial velocity curve for the hydraulic actuator with zero rotation produced the curve shown in Figure 5.2. The LuGre axial friction parameters for the hydraulic actuator for positive velocities are provided in Table 5.2.  The axial friction parameters for negative  velocities are provided in Table 5.3. The subscripts p and n refer to positive and negative velocities, respectively.  The results show that friction is direction dependent in hydraulic actuators, which is in agreement with work done by Bonchis et al. [13]. The standard deviations indicate a wide range of possible friction values, which is typical for friction and indicative of the problems found with controlling hydraulic actuators. The standard deviations for the hydraulic actuator are much higher than those for the motor.  The influence of the lubricant and its viscosity can be seen. During pure rotation, lubricant shears against the moving surfaces, but is not under bulk displacement. While moving axially, lubricant is being displaced, and the viscosity of the lubricant will have a greater influence on the friction curve.  The negative viscous slope has a drop of 30 %. The drop occurs over a much shorter range of velocities when compared to pure rotation. Thus, the negative viscous slope is steeper for the 50  hydraulic actuator, which suggests greater control problems would be encountered in the control of the hydraulic actuator. 600 h  1  -0.01  1  I  1  L _  -0.005  0 0.005 0.01 Axial Velocity (m/s) Figure 5.2 Hydraulic Actuator Axial Friction - Axial Velocity Curve at 0 rpm.  Table 5.2 Hydraulic Actuator LuGre Axial Friction Parameters (Positive Velocity, 0 rpm). LuGre Friction Parameter  Average Value  Standard Deviation  Coulomb Friction  330 N  30  Stribeck Friction  150N  15  Static Friction  480 N  45  22 800 N/m/s  2 500  0.0024 m/s  0.0003  2.6 x 10 N/m  0.6 x 10  b  1.1 x 10 N/m/s  0.1 x 10  4  Viscous Friction Stribeck Velocity Bristle Spring Constant  «2„  skp  V  b  Co  P  Bristle Damping Coefficient  4  51  Table 5.3 Hydraulic Actuator LuGre Axial Friction Parameters (Negative Velocity, 0 rpm) LuGre Friction Parameter  Average Value  Standard Deviation  Coulomb Friction  «o„  290 N  25.0  Stribeck Friction  «.„  HON  15  400 N  40  Viscous Friction  28 200 N/m/s  1 800  Stribeck Velocity  0.0031 m/s  0.0003  Bristle Spring Constant  2.6 x 10 N/m  0.8 x 10  Bristle Damping Coefficient  33.9 x 10 N/m/s  14.2 x 10  Static Friction  5.3  <*0«  + « . „  b  4  b  4  Hydraulic Actuator Rotating  The axial friction - axial velocity curves for the hydraulic actuator with a piston rotation of 10 rpm (0.033 m/s) produced the curve shown in Figure 5.3.  6001-  c o  Figure 5.3 Axial Friction - Axial Velocity Curve at 10 rpm (0.033 m/s) 52  The Stribeck curve has been reduced in size.  However, it still exists and can affect the  performance of the actuator. Comparing with the rotating friction - rotating velocity curve, Figure 5.1, 10 rpm (0.033 m/s) is still within the negative viscous slope of the Stribeck curve for the rotating piston. In this case only partial lubrication has occurred and as the actuator moves axially it must finish lubricating the surface to surface contacts.  As the speed of rotation is increased the system moves towards full fluid lubrication and the Stribeck curve is eliminated during axial movement. This can be seen in Figures 5.4 and 5.5 where the rotational speed has been increased to 25 and 50 rpm. Referring again to Figure 5.1, at rotational speeds of 25 rpm (0.083 m/s) and 50 rpm (0.17 m/s) the rotating friction response has moved away from the Stribeck curve into the full fluid regime. As a result, the effective axial friction - axial velocity curve becomes near linear. 6001-  400 h  200  co  •S o o  • l-H  Is  '3  M00  -400  -600 -0.01  -0.005  . . , , 0 .„ , . , 0.005 0.01 Axial Velocity (m/s) Figure 5.4 Axial Friction - Axial Velocity Curve at 25 rpm (0.083 m/s).  53  600  400 h  Figure 5.5 Axial Friction - Axial Velocity Curve at 50 rpm (0.17 m/s) Rotating the piston and rod breaks the bonds between the asperities. Then, when axial motion commences the axial friction opposing axial motion is reduced. The negative viscous slope is eliminated if the rotating velocity is in the full fluid lubrication area of the rotating friction rotating velocity curve. However, even at higher rotational speeds there appears to be a 'static' friction component in the axial direction. But, the switch from the static regime to the dynamic regime is continuous.  As the speed of rotation is increased further there is no significant improvement in the axial friction reduction as shown in Figure 5.6 through to Figure 5.8. It is apparent in these results that, when the piston and rod are rotated, the classical friction - velocity model (Figure 2.3), with the static friction equal to the Coulomb friction, more closely represents the axial friction - axial velocity curve. A more simplified model can then be used to estimate axial friction.  54  600  400 h  Figure 5.7 Axial Friction - Axial Velocity Curve at 100 rpm (0.33 m/s)  600  400  ,200  co  •g o UH  <200  -400  -600 -0.01  -0.005  0 0.005 0.01 Axial Velocity (m/s) Figure 5.8 Axial Friction - Axial Velocity Curve at 125 rpm (0.42 m/s). At 100 rpm, Figure 5.7, some system related problems were noticed. The actuator was observed to slip during movement in some test runs and this is reflected in Figure 5.9. A discontinuity can be found in several places. In this particular system, 100 rpm is close to a resonant mode and a disturbance in the performance can be seen. The instability originates near 75 rpm and peaks at 100 rpm.  56  600 h  Figure 5.9 Axial Friction - Axial Velocity, Actuator Slipping at 100 rpm. Table 5.4 shows the LuGre axial friction parameters corresponding to positive axial velocities for a range of rotating speeds.  Table 5.5 shows the LuGre axial friction parameters  corresponding to negative axial velocities for the same range of rotating speeds. As the rotation speed is increased the axial friction values become more stable and thus more predictable. For example, the standard deviation for axial static friction at 0 rpm is 44.3 N . It decreases to 13.1 N at 50 rpm and to 6.9 N at 125 rpm. The instability of the parameters at 100 rpm is evident with a corresponding increase in the standard deviation.  57  Table 5.4 LuGre Axial Friction Parameters (Positive Axial Velocities, with Rotation). Friction Parameters  a  (N)  0p  0 (0) 330  Std Dev  a (N)  125 (0.42)  50  0  0  lp  2p  skp  6  0p  4  Xp  0  30 40  Std Dev 15 « o + oc (N) 480 Std Dev 45 a (N/m/s) 22800 Std Dev 2500 v (m/s) 0.0024 Std Dev 0.0003 er (10 N/m) 2.6 Std Dev 0.6 cr (10 N/m/s) 1.1 Std Dev 0.1 P  Speed of dotation (rpm (m/s)) 25 50 75 100 (0.083) (0.17) (0.25) (0.33)  30 150  Xp  10 (0.033)  0 50  25 50  15  0 40  15 50  10  1 50  15 40  70  0 50  10 50  0  0 40  40 90  0  10 40  15  25  10  37200 34800 31500 33400 31400 1100 1700 14 1300 4500 0.0017 0.0030 0.0029 0.0030 0.0035 0.0004 0.0004 0.0007 0.0005 0.0015 0.4 0.4 0.3 1.3 0.7 0.06 1.1 0.09 0.1 0.6 8.4 2.6 4.1 2.1 12.3 8.5 3.2 3.5 2.0 14.1  30300 1200 0.0032 0.0006 0.2 0.08 4.7 5.7  Table 5.5 LuGre Axial Friction Parameters (Negative Axial Velocities, with Rotation). Friction Parameters  0 (0)  10 (0.033)  Speed of Rotation (rpm (m/s)) 25 50 75 100 (0.083) (0.25) (0.33) (0-17)  a  290  10  0  0p  (N) Std Dev  25 110  «,„ (N) «o + a (N) P  lp  15 400  (N/m/s)  2p  Std Dev V S  kp  ( ) m/s  Std Dev cr (10 N/m) 6  Qp  0.0031 0.0003 2.6  lp  0.0024 0.0004  14.2  1600 0.0026 0.0005 0.3  0.8 1.4  0.0025 0.0008  0.08  0.0023 0.0007  0.1  31300 0.0020 0.0009  0.03  30600 1700 0.0026 0.0007 0.2  0.2 0.4  0.02  15  5100  0.3  0.3 0.06  20  2900  0.2  0.4 0.09  33200  15 30  10  1400  0.3  0.4 1.8  32500  20 40  10  1 30  10 30  0 1  40  10 30  35600  0 30  10.0  1600  0.8  (10 N/m/s) 33.9 Std Dev  44500  0.9  4  30  30  0  1  10.0 70  1800  Std Dev a  60  40 28200  0  0 30  135  Std Dev a  10 125  Std Dev  0  125 (0.42)  0.05 0.3  0.1  0.04 58  O $  600  Positive Velocity (solid) Negative Velocity (dash) Rotating Only  500 400 h 1 300 200 100 0  (D  ®  (D  100 40 . 60 80 120 Rotation Speed (rpm) Figure 5.11 Axial Coulomb Friction Parameter vs. Rotation Speed  o  20  O ^  600  Positive Velocity (solid) Negative Velocity (dash) Rotating Only  500 400 300 200 100  0 0  20  40  60 80 100 120 Rotation Speed (rpm) Figure 5.12 Axial Stribeck Friction Parameter vs. Rotation Speed.  O •  600  Positive Velocity (solid) Negative Velocity (dash) Rotating Only  500  C) g,  c o  400  300  r  o *+-» CO  t/3  200  100  f  0 0  20  §  ?  ?  100 60 80 Rotation Speed (rpm) Figure 5.13 Axial Static Friction Parameter vs. Rotation Speed. 40  # 120  The effective axial Coulomb friction parameter converges to zero when the piston and rod are rotated at speeds above 20 rpm. The Stribeck friction parameter is reduced and the static friction approaches that of the rotating static friction with no axial motion. The axial friction - axial velocity curves for axial motion have an inflection point, which, with the curve fitting of the LuGre model, requires a non-zero value for the Stribeck velocity and thus, the Stribeck friction.  If the Classical model of friction were to be utilized instead of the LuGre model, the effective axial Coulomb friction becomes equal to the axial static friction with the Stribeck friction equal to zero.  61  5.5 5  •  4.5 4 o  3.5 3  CO  & PH  C  o 1/3  3 O o  2.5 V  2 O ^  1.5 1  Positive Velocity (solid) Negative Velocity (dash) Rotating Only  0.5 0 0  20  40  60  80  100  120  Rotation Speed (rpm) Figure 5.14 Axial Viscous Friction Parameter vs. Rotation Speed. The viscous friction parameter can be seen to increase and then decrease, much like the Stribeck curve. The viscous friction parameter can be seen to change in a similar manner to that predicted by the simulations, reference Figure 3.8.  62  0.07 0.06 -  2  Positive Velocity (solid) Negative Velocity (dash) Rotating Only  O •  0.05 0.04 0.03  13  0.02  h  0.01  $  oh 0  20  i  40  9 i  i  *  60 80 100 120 Rotation Speed (rpm) Figure 5.15 Axial Stribeck Velocity Friction Parameter vs. Rotation Speed.  O $  i 0  20  40  i  Positive Velocity (solid) Negative Velocity (dash) Rotating Only  I  60 80 100 Rotation Speed (rpm) Figure 5.16 Axial Bristle Spring Constant vs. Rotation Speed.  A 120  10 9  O $  i  Positive Velocity (solid) Negative Velocity (dash) Rotating Only  8 O  7 6  CD  5h  i  O  O  bO  c "a E  4  CO  Q  PQ  4 o  20  100 120 60 80 Rotation Speed (rpm) Figure 5.17 Axial Bristle Damping Coefficient vs. Rotation Speed. 40  The bristle spring and damping constants appear to approach the values of the motor. The magnitudes have been reduced and are more consistent, but they are still higher than those of the motor. 5.5  Model Validation  The rotating model developed in Chapter 3 provided valuable insight as to what the result would be when the piston and rod are rotated. However, where the model showed that friction would be linearized, the experiments showed that the friction is not completely linearized. The most notable difference in the axial friction, as predicted by the model, is the elimination of the Stribeck curve.  A significant difference between the rotating model and the experiments is the friction values for pure rotation and pure axial motion. Using static friction as a base of comparison, the rotating model used the same values for both forms of motion with the static friction estimated at 588 N . When rotation occurred and axial motion commenced the transfer between the same friction values was seamless and the model behaved well. However, the experiments showed that the 64  actual values for rotating and axial static friction are quite different. Static friction for rotating with no axial motion was 50 N while static friction for axial motion with no rotation was 480 N . When these two widely different values are used in the rotating model the discontinuity resulted in the model becoming unstable.  Based on the above observation, the non-rotating model, developed in Chapter 3 was implemented with the experimentally computed axial friction values. This model was compared to the experimental curves. The results are plotted in Figure 5.18 for zero rotation and Figure 5.19 for rotation speeds of 10, 25 and 50 rpm. The results show that the LuGre friction model and the non-rotating hydraulic actuator model agree well with the experiments. The non-rotating model closely predicts the axial friction - axial velocity behaviour. The simulations for 75, 100, and 125 rpm also agreed well with the experimental results.  A few modifications were required. One can consider the axial friction - axial velocity curve for zero rotation, Figure 5.18. The value for <7,„, the bristle damping coefficient for negative velocities, resulted in an unstable model. The value for <r,„ was reduced by two thirds to 11.3 x 10 N/m/s. This new value is within the 95% confidence interval for 95% of the population but 4  outside the 95% confidence interval for the mean. This new value produced stable results that agreed with the friction values that were measured experimentally.  One can also consider the axial friction - axial velocity curves for various rotation speeds, Figure 5.19. The values of o , the bristle damping coefficient for positive velocities, are considerably Xp  higher than o~ and again produced unstable results. Thus, the values for cr,„ were substituted ln  for a . lp  The simulations were stable and corresponded with the experimental results. The  experimental values for the bristle spring constant for positive velocities and negative velocities,( <j  0p  and  <7 „), 0  are similar to each other suggesting that the bristle damping  coefficient for positive and negative velocities, (ox  and <7,„), should also be similar.  65  were over whelmed by the sluggish response of the pressure change. It is expected that for a smaller actuator and a higher supply pressure, both outside the limits of this experimental system, that improvements would be seen.  Canudas de Wit et al. [5] state that typical errors caused by friction are steady state errors and tracking lags. Steady state error is related to static friction and the tracking lag is related to the viscous friction. If static friction can be eliminated or reduced then the steady state error will be reduced. If the friction in a system can be reduced or eliminated then the tracking lag will be improved. However, since the viscous friction cannot be removed there will be an upper limit on the possible improvement.  Since rotating the rod and piston reduces the static friction and system friction it is expected that the tracking performance of a hydraulic system with rod and piston rotation would be improved. However, as friction is reduced damping is removed from the system. This results in an increase in transient oscillations. A slight increase in oscillation was observed in response to an axial step input when the piston was rotated. 5.8  Summary  Rotating the piston and rod eliminates the Stribeck effect and reduces axial friction in a hydraulic actuator. A minimum critical rotating velocity is required for the near linearization of the axial friction. This critical velocity is located in the full fluid regime of the rotating friction rotating velocity curve.  Comparison with experimental results showed that the non-rotating hydraulic actuator model accurately predicted the axial friction behaviour of the actuator with no rotation. The nonrotating model, using the appropriate friction parameters can also be used to predict the axial friction behaviour at various speeds of rotation.  If a simple friction model is desired, with the piston rotating at sufficient speeds, the axial friction - axial velocity curve approaches the Classical friction - velocity model with static friction equal to Coulomb friction.  70  Rotating the piston reduces the hydraulic power requirements. However, the power required to rotate the piston exceeds the power saved by reducing axial friction.  71  Chapter 6 Conclusions and Suggestions for Future Work 6.1 Conclusions Hydraulic actuators are highly non-linear. One of the main contributors to the non-linearities is friction, particularly near zero velocity. Simulations and experimental results of this work have shown that rotating the piston at a sufficient velocity linearizes the axial friction opposing the axial motion of the piston. Both the Stribeck effect and the hysteresis effect were shown to be eliminated above a critical rotating piston speed. Rotating the piston also resulted in a reduction of the axial friction and thus a reduction in the damping of the system.  The non-rotating hydraulic actuator model, incorporating the LuGre friction model, accurately predicted the friction behaviour of the hydraulic actuator. This model effectively predicts the axial friction behaviour when the piston and rod are rotated when the appropriate axial friction parameters are utilized.  Reducing the friction in the hydraulic system reduced the hydraulic power requirements. This equates to smaller hydraulic power packs and therefore reduced costs.  However, in this  particular application, rotating the piston required more overall power as the power required for rotating was greater than the power saved from reducing the friction.  Thus, the benefits of  reduced friction and increased accuracy must be evaluated against an increase in power costs.  Another expected benefit of reducing or eliminating stick-slip friction is a reduction in fatigue of the system. With a decrease in jerk, the amount of stress and fatigue will be reduced.  6.2 Recommendations for Future Work Future work includes evaluating the tracking ability of a rotating hydraulic actuator. The effect that rotating the piston has on hunting (limit cycling) in a system requires evaluation.  72  A further step will be to consider various control systems such as a feed forward control loop to provide a canceling signal for friction, adaptive control, and acceleration feed back to capture the dynamics of the system.  Control strategies for turning the rotation on and off to maintain a minimum helical velocity can also be considered.  The combined axial and rotating velocities can be minimized during  operation.  The practicality of implementing a rotating piston and rod will also require evaluation. Instead of rotating the rod and piston, one may consider rotating the cylinder and holding the piston and rod secure.  It is expected that rotating will introduce extra wear in the system. The wear of the lip seals, orings, piston seals, and the cylinder wall will need evaluation.  73  Nomenclature Parameter  Physical Property  Units  A  Piston surface areas  m  A  Piston surface areas  m  B  Viscous damping coefficient  C  Capacitance  juF  D  Velocity level for switch from static to dynamic friction  m/s  2  v  F  2  2  Ns/m  Actuation force applied to piston  N  F  Force applied to an object  N  F  Equivalent force applied by the motor  N  Coulomb friction force  N  Resultant driving force  N  Dynamic friction force  N  F  Generic friction force  N  F  Estimate for friction  N  F  Helical friction force  N  F  Axial friction force  N  F  Rotary friction force  N  External load on the actuator  N  Normal force  N  Steady state friction  N  Estimate of steady state friction  N  F  Static friction force  N  F  Friction as a function of velocity  N  F  Viscous friction force  N  H{s)  Velocity filter, low pass-filtered differentiator  Actuator  Applied  ±  dApplied  F  Coulomb  F  Drive  1  F  Dynamic  [•riction  Friction  hb riction  pFriction  OFriction  F  Load  1  F  Normal  t Static  velocity  Viscous  74  Parameter  Physical Property  /  Identity matrix  J  Manipulator Jacobian  J,  Jacobian state transition matrix  J  Jacobian input matrix  K  Servo valve flow gain  U  Back electromotive force  K f oem  P  K  Units  m /s/JPa 2  V/rad/s  Proportional Gain  V/m  Servo valve position gain  m/V  K,  Motor torque constant  Nm/A  L  Actuator stroke length  mm  L  Motor inductance  M  System mass  kg  Px  Pressure in chamber one  Pa  Pi  Pressure in chamber two  Pa  PL  Load pressure  Pa  p  Supply pressure, constant  m  s  P  T  P°™ hyd er  Power  Tolal  Power  g  Henries  MPa  Tank pressure  Pa  Hydraulic power  W  Total power  W  Rotating power  W  Qx  Flow rate into chamber one  m /s  Qi  Flow rate into chamber two  m /s  QL  Load flow  m /s  R  Resistance  ohms  R  Coefficient of determination  cyl  Hydraulic piston radius  K  Motor resistance  2  R  3  3  3  mm ohms  75  Parameter S(x„)  Physical Property  Units  Switching function Velocity filter time constant  s  Daqbook analog output to motor  V  Daqbook analog output to servo valve  V  v  Hose volume between the servo valve and actuator (Typical x2)  m  v  Unfiltered signal  V  v  Applied voltage to the motor  V  V  Filtered signal  V  v  Servo valve control voltage  V  X  Vector of end-effector coordinates in Cartesian space  m  AX  End effector's position error in Cartesian space  m  v daqm  v daqsv  h  in  m  out w  a dz dt dz dt  3  Estimate for acceleration  m/s  Rate of change of bristle deflection  m/s  Estimate for the rate of change of bristle deflection  m/s  Describes part of the steady state friction characteristics at a constant velocity [4]  N  h  Discrete time step  s  L  Motor current  A  L  Rate of change of motor current  A  k  Discrete time  s  K  Dahl's spring constant  q  Vector of joint positions in joint space  m  Aq  Joint position errors in joint space  m  q>  Joint position of the i joint in joint space  m  s  Laplace operator  t  Time variable  s  Time lag  s  At  th  2  N/m  76  Parameter  Physical Property  «/  Input variables  u,  Small deviations about the input variable's operating point  u  Input variables operating point  "k  Discrete input variable vector  i0  Units  Stribeck velocity  m/s  Estimate for Stribeck velocity  m/s  Axial Stribeck velocity for positive velocities  m/s  *kn  Axial Stribeck velocity for negative velocities  m/s  ,k6  Rotating Stribeck velocity  m/s  v  skp  V  V  X  Generic position  m  X  Generic velocity  m/s  X  Generic acceleration  m/s  2  x,,  Reference position  m  x„  End effector's x-coordinate in Cartesian space  m  x  Piston helical position  m  x  h  Piston helical velocity  m/s  x  h  Piston helical acceleration  m/s  Xi  State variables  Xi  State variable derivatives  h  * i  Small deviations about the operating point  i  Small deviations about the operating points derivatives  iO  Operating points of the state variables  iO  Operating points of the state variable derivatives  xi  Current state variable deviation vector  X  X  X  X  * k+l  X  2  Future state variable deviation vector  P  Piston axial position  m  P  Piston axial velocity  m/s  X  X  77  Parameter 'P X  Physical Property  Units  Piston axial acceleration  m/s  2  Servo valve spool displacement  m  Xg  Piston angular position  m  Xg  Piston angular velocity  m/s  Xg  Piston angular acceleration  m/s  y.  End effector's y-coordinate in Cartesian space  m  z  Average bristle deflection  m  z  Estimate for the average bristle deflection  m  End effector's z-coordinate in Cartesian space  m  *  z  A(h)  Forced response of a system Torque produced by the motor  0(h)  2  Nm  Free response of a system Damping coefficient Coulomb friction  N  Estimate for Coulomb friction  N  Axial Coulomb friction for positive velocities  N  Axial Coulomb friction for negative velocities  N  09  Rotating Coulomb friction  N  «.  Stribeck friction  N  Estimate for Stribeck friction  N  <*lp  Axial Stribeck friction for positive velocities  N  «.„  Axial Stribeck friction for negative velocities  N  Rotating Stribeck friction  N  « 0  <x  0p  A  « 1  a  Viscous friction coefficient  N/m/s  a  Estimate for viscous friction coefficient  N/m/s  Axial viscous friction coefficient for positive velocities  N/m/s  Axial viscous friction coefficient for negative velocities  N/m/s  2  2  « 2 „  78  Parameter oc  2e  Physical Property  Units  Rotating viscous friction coefficient  N/m/s  Static friction  N  «Op+«l„  Axial static friction for positive velocities  N  « 0 „  Axial static friction for negative velocities  N  Rotating static friction  N  +  « ! „  p  Effective bulk modulus (oil)  MPa  Bristles stiffness coefficient  N/m  Estimate for bristles stiffness coefficient  N/m  Axial bristles stiffness coefficient for positive velocities  N/m  Axial bristles stiffness coefficient for negative velocities  N/m  Rotating bristles stiffness coefficient  N/m  Bristles damping coefficient  N/m/s  Axial bristles damping coefficient for positive velocities  N/m/s  Axial bristles damping coefficient for negative velocities  N/m/s  °\8  Rotating bristles damping coefficient  N/m/s  O".  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A Esposito, Fluid Power with Applications, 4 Ed., Prentice Hall, Upper Saddle River, th  NJ, 1997.  83  Appendix A Hydraulic Actuator Specifications A.l  Process and Instrumentation Diagram  The process and instrumentation diagram (PID) is provided in Figure A . l .  1£0 V A C L P h  208 V A C 3 P h  Figure A . l Hydraulic Actuator PID. 84  A.2 Physical Parameters The physical parameters of the system are given in Table A. 1: Table A.1 Physical parameters. Parameter  Measured / Calculated Value  Unit  System Mass  15.0  Stroke Length  0.172  kg m  Hose Volume (x 2)  1.961 x 10"  m  Piston Diameter  0.0635  m  Rod Diameter  0.0254  m  Piston Areas (x2)  0.00266  m  5  3  2  A.3 Instrumentation and Hardware A.3.1 Hydraulic Pump A gear type hydraulic pump was used to supply a constant pressure to the hydraulic system. Table A.2 Hydraulic Pump. Manufacturer  Hyseco  Motor  3 hp  Voltage  208 V 3 Ph 60 Hz  Maximum Pressure  9.0 Mpa (1300 psi)  Operating Pressure  4.9 (712 psi)  85  A.3.2  Motor  A permanent magnet DC motor was coupled to the piston to provide rotation. Table A.3 Motor and Amplifier. Manufacturer  Indramat  Motor Model  M K D 071B-061-GP1  Amplifier Model  D K C 1.1-030-3  Motor Voltage  208 V 3 Ph 60 Hz  Amplifier Controller Voltage  24 VDC  Command Voltage  [-10 ... +10] V D C  Speed Output  0-10 V D C  Torque Output  0-10 VDC  Speed Range  1 - 30 000 rpm  Controller Speed Limit for Experiments  1 - 500 rpm  Torque Constant  0.77 Nm/A  Windings Inductance  0.0104 H  Windings Resistance  1.65 ohms  Back Electromotive Force  0.6685 V/rad/s  Mass  8.8 kg  Operating Ambient Temperature  0° to 45° C  Nominal Torque  8Nm  The following circuit was used to convert the 0 to 5 V D C analog signal from the controller into a 0 to +10 V D C signal for the motor amplifier, Figure A.2: R = \0k& {Typxl) I—V\A/  1  Figure A.2 Motor amplifier signal conversion. 86  A.3.3 Servo Valve A direct drive servo proportional control valve regulated the flow to the hydraulic actuator. Table A.4 Servo Valve. Manufacturer  Moog  Model  D633 (R-16-K-01-M-0-N-S-M-2)  Time Constant  12 ms  Supply Voltage  24 V D C  Command Voltage  [-10 ... +10] V D C  Output  4 - 20 raA  Rated Flow  40 1/min at 1000 psi pressure drop  Max Flow  75 1/min  Fluid Temperature Range  -20° to 80° C  The following circuit was used to convert the 0 to 5 VDC analog signal from the controller into a -10 to +10 V D C signal for the servo valve, Figure A.3:  i—wv  R = \QkQ(TypxA) i  AA/V  'I WV  V„, = -10 —»+\ov  5V V  daqsv  = 0^5V  Figure A.3 Servo valve signal conversion.  87  A.3.4 Pressure Transducers Pressure transducers were used to measure the supply pressure, tank pressure, and the pressure in each hydraulic chamber. Table A.5 Pressure Transducers. Manufacturer  M S L Measurement Specialties Ltd.  Model  MSP-300-2500-P-Z  Pressure Rating  2500 psi  Accuracy  ± 1% full scale output (includes nonlinearity, hysteresis, and repeatability)  Excitation Input  12 VDC  Output  1-5 VDC  Operating Temperature Range  -40° to 85° C  Compensated Temperature Range  0° to 55° C  A.3.5 Potentiometer The displacement of the hydraulic actuator and the motor was measured with a linear potentiometer. Table A.6 Potentiometer. Supplier  Duncan Electronics, a BEI Electronics Company  Model  6300-100  Electrical Travel  0.200 m  Resistance  16 kilo-ohms  Manufacturer's Accuracy Standard  ±0.38 mm  Best  ±0.075 mm  Calculated Accuracy  ±0.16 mm  Repeatability  Within 0.013 mm  Actuation Force  0.56 N  Input  5 VDC  Output  1-5VDC  88  A.3.6  Data Acquisition  An IOTech Daqbook 120 was used for data acquisition. Labview software was used to control the data acquisition. Table A.7 Data Acquisition. Supplier  IOTech  Model  Daqbook 120  Resolution  12 bit  Supply Voltage  15 V D C  Analog Output (x 2)  0-5 VDC  Analog Input  8 Differential 6 : 0 - 5 VDC 2: 0 - 10 V D C  Before entering the daqbook the signals were passed through an anti-aliasing filter, Figure A.4, with a cut off frequency of 194 Hz. This is appropriate with a sampling frequency of 1000 Hz and a resulting Nyquist frequency of 500 Hz. R = S20n  Figure A.4 Anti-Aliasing Filter.  89  Appendix B Simulation and Experimental Values The values for the physical parameters used in the model simulations and experiments are found in Table B . l . The simulation values are relevant for Chapter 3. The values for experimental and model validation pertain to Chapter 4 and Chapter 5. Table B.l Simulation and Experimental Values. Parameter  Physical Property  Simulation Values  Experimental & Model Validation  A  Piston surface areas  6.33 x l O m  2.66xlO m  A  Piston surface areas  6.33xlCT m  F  External load on the actuator  2  Load  1  Servo valve flow gain  K  -4  2  4  2  ON 5.9xlO  -3  2  2.66xlO m -3  2  ON m /s/^  -4  2  5.9X10" m lsl-JPa 4  2  Back electromotive force  0.6685 V/rad/s  0.6685 V/rad/s  K„  Servo valve position gain  0.005 m/V  0.005 m/V  K,  Motor torque constant  0.77 Nm/A  0.77 Nm/A  L  Actuator stroke length  60 mm  172 mm  0.0104 Henries 34 kg  0.0104 Henries 15 kg  2.94 MPa  4.91 MPa  OPa  ~0Pa  19 mm  31.75 mm  1.65 ohms  1.65 ohms  L  Motor inductance  M  System mass  P  Supply pressure, constant  P  Tank pressure  m  s  T  cyl  R  Hydraulic piston radius Motor resistance  K  v  Hose volume between the servo valve and actuator (Typical x2) Servo valve control voltage  8.9x 10~ m  1.961X10 m  ±10 V  ± 10 V  Stribeck velocity  0.032 m/s  m/s  Servo valve spool displacement  ±0.6 mm  ± 0.6 mm  Servo valve time constant  12 ms  12 ms  Coulomb Friction  370 N  N  «.  Stribeck friction  217N  N  a  Viscous friction coefficient  2318 N/m/s  N/m/s  588 N  N  Effective bulk modulus (oil)  1724 MPa  1724 MPa  Bristles stiffness coefficient  5.77xlO N/m  N/m  Bristles damping coefficient  2.28x10" N/m/s  N/m/s  h  y„  sv  X  t »  2  Static Friction P  5  3  6  -5  3  90  Appendix C Jacobian Linearization C.l  Jacobian Linearization and Discretization  The state space equations were linearized as follows [34]: x* = J x'+ J u*,  (C.l)  ' = *, ~ x ,  (C.2)  x'=x,-x ,  (C.3)  u*=u,-u ,  (C.4)  J =^-,md dx,  (C.5)  x  u  x  i0  l0  l0  x  J  u = { ^ >  (C6)  aw,  where x, are the state variables, x, are the state variable derivatives, u are the input variables, j  x , x, , and u, are operating points, x*, x*, and u* are small deviations about the operating i0  0  0  points, J is the Jacobian state transition matrix calculated about the operating points, and J is x  u  the Jacobian input matrix calculated about the input operating points, x is a vector representation of x,.  The state equations were then discretized for computer simulation [34].  x =0{h)xl+A{h)ul  {C.l)  M  where h is the time step, k is the current time, and k +1 is one time step into the future. 0(h) is the free response of the system: 0{h) = exp{J h),  (C.8)  x  and A(h) is the forced response of the system:  A{h) = j; [0{h)-I\j . x  u  (C.9)  Then the state variables of interest are: =**+*<>•  (CIO) 91  Details of the individual linearizations of the non-rotating state space equations ((3.35), (3.36) (3.37), and (3.38)) and the rotating state space equations ((3.50), (3.51), (3.52), (3.53), (3.54), and (3.55))are given in the following sections. C.l  Non-Rotating Model State Equations  C.2.1 Position Equation (3.35) describes the position of the actuator: (C.ll) The state equations are: dx,  = 0,  3x,  3x  (C.12)  (C.13) 2  (C.14)  dx. dx,  = 0.  dx  (C.l 5)  4  C.2.2 Axial Velocity Equation (3.36) describes the axial velocity of the actuator:  ( A - A )  p  ,  (A+A)  ^ 0 ^ 1 "^4  f  M  a  0  + «, exp V  P^2 2\  .  (C.16)  fx ^  )  The state equations are: 3x  2  dx,  = 0,  (C.17)  92  f ( \2\ 2a o~ o~ x x x exp_ 2 X  x  dx _ 1 2  dx  2  M  - cTj  O~ O~ X sign(x ) x  0  -a +  4  2  0  x  0  3x  J ,(C18)  x  exp  v * v  j  J) (C.19)  2M  3  (7 CT \X \ 0  •<j +•  M  4  4  +4)  2  2  2  a +a  dx =  3x _ 1  2  V  exp  3x  x  +•  2  a +a  0  X  2  (C.20)  0  'x  1  a +a 0  x  exp  V  A  C.2.3 Pressure Equation (3.37) describes the load pressure of the actuator: f  46  f  m—:  A  + S{-x P)j+x s  3  sv  2V +A L V h  n;  iA+A).  (C.21)  v  2  The state equations are: dx  3  dx,  2B{A +A )  3x,  2V + A L '  X  h  dx _42BKxJ  2  (C.23)  2  S{x ) , S{-x )^  3  sv  2V +A L h  (C.22)  dx  3  dx,  = 0,  2  sv  +•  V  3  3*3  dx.  (C.24)  J  = 0.  (C.25)  C.2.4 Friction Equation (3.38) describes the load average bristle deflection (friction) of the actuator: <7 |x |x 0  a  0  2  4  'x ^ + ax exp _ V  2\  (C.26)  J 93  The state equations are: 3*4  = 0,  (C.27)  dx,  f  di  4  dx.  2a o~ x x x exp x  <J sign(x )x 0  = 1-  2  2  2  4  V  4  a + a exp 0  0  v"sk a + a exp 2  x  0  J  V  (C.28)  <xx  2  ^  x  \ skJ V  V  v  dx = 0, dx. 4  dx dx  J)  (C.29)  <7 x  4  0  2  4  a + a exp V  fx ^  2\  (C.30)  2  x  0  C.2.5  'x ^  1  y  Servo Valve Input  The state equations with respect to the input (servo valve opening) are:  dx, ' =0, dx  (C.31)  dx2 - = o, dx  (C.32)  V  dx,  4/3K  dx  2V +A L  sv  H  S{xJ  S  P  ~  3  X  + S(-x„l  (C.33)  2  2  dx  A  ox. = 0.  (C.34)  C.2.6 Jacobians The Jacobian state transition matrix is:  94  dx, 3x,  J  R  dx, 3x dx, 3x,  dx, dx,  dx, 3x  3x 3x  dx 3x  3x, 3i 3x,  3x, 3x 3x  dx.  4  3x, 3x  2  3x,  3x.  3  4  3x, 3x  2  3  2  3  3  3  4  dx, 3x  4  3x 3x  4  (C.35)  3  3x  4  and the Jacobian input matrix is: 3x, 3x  2  Bx„ 3x  J.. =  (C.36)  3  3x  4  3x„, C.3 Rotating Model State Equations C.3.1 Position Equation (3.50) describes the axial position of the actuator: (C.37) The state equations are: 3x, dx, 3x, 3x  = 0,  (C.38)  = 1,  (C.39)  = 0,  (C.40)  = 0.  (C.41)  2  3x, 3x  3  3x, 3x, C.3.2 Axial Velocity  Equation (3.51) describes the axial velocity of the actuator:  95  1 M  A* A^  I  2  J  +  I  0*0*2*4  J 2  1  2  V2  - C T , X  2  + *5  X  0*1*4*2  •••+•  ' a + «, exp  f2 Y  +  2\\  (C.42)  012X2  r  0  V  J J  sk  V  The state equations are: dx  = 0.  2  dx,  dx _ 1  0*0*4  2  3x7" M |  0*0*2 * 4  + * 5  V*2  + X  (x  2 2  5  3  (C.43)  0 * , - « 2  0'o  +  i*4  '*2 +*ni 2  )  2  C r  a + a exp 0  x  V  V  sk  v  J J  2a (7 cr x x exp 2  x  0  x  4  V  •••+•  sk  v  x  J J  sk  v  (C.44)  '* * ^ 2 2 +  a + a exp 0  V  2 5  JJ)  V  dx _ (A + A ) 2  x  dx  2  3x7 M|  0" *2  V*2  +  a ax  +•  O  _  5  X  (C.45)  2M  3  3x _ 1  2  Q  x  *2  a + a exp 0  2  dx  *5  x  V  dx  (C.46)  2  0,  V  v  *  JJ)  (C.47)  5  dx  2  dx  (C.48)  A  96  C.3.3 Pressure Equation (3.52) describes the load pressure of the actuator: 4/?  2V„ + A L  A +A  Kx.  x  2  (C.49)  2  The state equations are: dx, dx, dx _  2  dx,  dx,  (C.51)  2V„ +A L 2  V2/? Kx  3  (C.50)  2/7(/I +A )  3  dx  = 0,  S{x„)  s}  2V +A,L  TJPS  k  S(-xJ  +  X j " ^P + X3 S  (C.52) J  (C.53)  dx  4  dx  3  dx  5  dx  3  0,  (C.54)  (C.55)  dx  fi  C.3.4 Friction Equation (3.53) describes the average bristle deflection (friction) of the actuator: x  4  C T Q X -\JX ~t~ x^  ^ x "i" X g  2  4  2  a  0  < + a, exp  (C.56)  x\+x ^'  (  2  The state equations are: dx  4  dx,  = 0,  dx. dx  (C.57)  CTQ 2  2 2  +x  2  -r A  4l X  5  +  5  X  a  0  X  2  X  4  ( + or, exp  \ V  (x\+x ^ 2  J  J)  97  2a o~ x x ^x + xj exp l  0  2  A  2  V V  v  *  J)  (C.58)  a + a, exp| 0  dx = 0. dx.  (C.59)  4  3x  (C.60)  4  dx  A  a + a, exp 0  dx.  <7 x x 0  3*5  4  5  f  <Jxl + X  x x^  f  r  •yjxl + Xa + «, exp V V  2  2  2+  0  '  2a (7 x x -yjxl + x] exp x  Q  4  J  *k  v  J)  'xl+xl^  5  '  f~2  a + a, exp| 0  V V  ,  (C.61)  i\W  2  v„. ">k  J))  dx =0.  (C.62)  4  dx. C.3.5 Rotating Velocity  Equation (3.54) describes the transverse velocity of the actuator:  J_ M  ^l 6  °~0 4 S  X  Rcyl  X  ^x\  +  X  x]  0~Q(T, X  •c x +[  5  a + cf, exp Q  4  (C.63)  X$  X2  I  v i  X^  v  J)  The state equations are: dx  5  dx,  = 0,  (C.64)  98  ox _ 1 5  dx  M  2  2a a a x x x x  ^0 ^ 2 ^4 ^5 (x  2 2  +X  Q  2  A  exp  5  V V  +•  )2  2 5  x  <  a +a 0  ox  (C.65)  x\+x ^  f  exp  x  JJ  sk  v  V  2  J  sk  V  J)  5  (C.66)  dx.  3x _ 1 5  ox  4  I  M  ax 0  5  2  2  <T CT X  5  5  + — V * 2 + * 5  2  2 5  'x  a, exp  1  2 2  V  +  x  (C.67)  ^  2 5  J J J  sk  V  cr (j x ' 'x or + a, exp  a +  - cr, -  3  (x +X  5  f  0  ox _ 1 ox M  l  0  +  2  )  0  1  4  2  x ^ 2  +  0  V V  2a <7 <7,x x l  0  4  2 5  2  X  exp  a +a 0  x  dx  5  exp  _  dx  6  JJ  sk  v  (C.68)  'x +x ^ 2  sk  J J  *5  V V v  "**  V  2  sk  y  K,  J  J)  (C.69)  MR  cyl  C.3.6 Motor Current Equation (3.55) describes the motor current: m  mo  Rcyl  (C.70)  The state equations are: ^  = 0,  (C.71)  OX,  f i = o.  (C.72)  OX,  99  3*6  _Q  3-*6  _Q  (C.73)  (C.74)  3x, K bemf  dx. 3*5  L  R m  (C.75)  cyl  3*6  (C.76)  dx„ C.3.7 Motor Voltage Input The state equations with respect to the first input (motor voltage) are: dx,  dv  = 0,  (C.77)  = 0,  (C.78)  = 0,  (C.79)  = 0,  (C.80)  = 0,  (C.81)  m  2  dx-. 3 3F  m  3*4  3F  m  5  3F  m  3*  1  6  (C.82)  3^ C.3.8 Servo Valve Input The state equations with respect to the second input (servo valve opening) are: dx, 3*  0.  (C.83)  = 0,  (C.84)  JV  dx  2  dx.. dx. dx  sv  4/3K 2V +A L h  2  +  S(- x.  4  P +x s  3  (C.85)  100  3x 4  =0,  (C.86)  dx =0, dxsv  (C.87)  dx 5  dx,  (C.88)  C.3.9 Jacobians The Jacobian state transition matrix is:  J, =  dx,  dx.  dx,  dx  3x  3x  2  dx,  3x,  2  3x  3  3x  2  3x  2  3x  dx,  dx  3x  3*3  3  5  3x  2  3* 3x  4  3x  5  3*6  3  3x  3  3*3  3x  5  6  2  2  3x  3  3x  4  3*  4  3*4  3*4  3x  2  3x  3  3x  4  3x  5  3*6  5  3x  5  3x  5  3x  5  3x  4  3x  5  3*6  3x  dx, dx  dx  5  3x  dx,  dx  3*  3* 3x  3x  2  3  dx  6  4  3x  dx,  dx,  dx,  3x 3x  3*  4  dx  3x,  2  3  4  3*6  5  2  3*3  3x  6  3*6  3*6  3*6  3*6  2  3x  3*4  3x  3*6  3  5  (C.89)  4  and the Jacobian input matrix is: 3x,  W 3x  m  2  w  m  J  T —  u  ~  3x  3  3^ 3x  4  m  dv  dx, 3* 3x  iV 2  3*, 3x  v  3  3*, 3x  v  3xm  3*, 3x  3*6  3*. .„ 3*  5  (C.90)  4  v  5  s  6  3* . iV  101  

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