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Computer control of a hydraulic press brake 1994
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Title | Computer control of a hydraulic press brake |
Creator |
Lane, John Alexander |
Date Created | 2009-02-25 |
Date Issued | 2009-02-25 |
Date | 1994 |
Description | Hydraulic press brakes are widely used in industry to form sheet metal into various shapes using bending operations. When the press brake is controlled by a Computer Numerical Control (CNC) unit, the desired bend can be achieved without the use of mechanical gages or manual adjustments, leading to increased flexibility and accuracy in the manufactured components. In this thesis, a single axis hydraulic press brake is retrofitted for dual axis CNC control. The ram is positioned using two parallel hydraulic cylinder type actuators. One servo-valve and amplifier is dedicated to each actuator so that the motion of each axis can be independently controlled by the CNC unit. Each actuator is instrumented with linear optical encoders which provide feedback for closed-loop servo position control. The hydraulic system is modified to provide constant pressure fluid power to the servo valves. An accumulator is used between the pump and the valves to suppress pressure fluctuations. Pressure transducers are integrated into the ports of the actuator to monitor the pressure during the operation of the press. The dynamics of the hydraulic servo system, including the valves, actuators and the ram are modeled. A non-linear model, which includes the influences of piston position, is presented. A simplified model is shown to be adequate provided that the practical ranges of the piston position is considered. The mathematical model is used to experimentally determine the dynamics using parametric identification techniques. The well-damped system is approximated by a first order system with substantial delay between the servo-valve amplifier command and the actuator piston motion. A readily available CNC system is retrofitted to the press brake and a delay compensating pole placement digital control system is developed and implemented for the independent control of two actuators. The performance of the system is evaluated for a series of forming operations. This thesis provides basic guidelines for the design and analysis of hydraulically actuated CNC presses. |
Extent | 2593272 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2009-02-25 |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080925 |
Degree |
Master of Applied Science - MASc |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of |
Degree Grantor | University of British Columbia |
Graduation Date | 1994-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/5064 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0080925/source |
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COMPUTER CONTROL OF A HYDRAULIC PRESS BRAKE BY John Alexander Lane B.Sc. (Mechanical Engineering), 1988 University of New Brunswick A THESIS SUBMITTED in PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of MECHANICAL ENGINEERING We accept this thesis as conforming to the required standard UNIVERSITY OF BRITISH COLUMBIA APRIL, 1994 © John Alexander Lane, 1994 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature) Department of CicL 9€ E-LJ The University of British Columbia Vancouver, Canada Date 2S APRL 994 DE-6 (2188) Abstract Hydraulic press brakes are widely used in industry to form sheet metal into various shapes using bending operations. When the press brake is controlled by a Computer Numerical Control (CNC) unit, the desired bend can be achieved without the use of mechanical gages or manual adjustments, leading to increased flexibility and accuracy in the manufactured components. In this thesis, a single axis hydraulic press brake is retrofitted for dual axis CNC control. The ram is positioned using two parallel hydraulic cylinder type actuators. One servo-valve and amplifier is dedicated to each actuator so that the motion of each axis can be independently controlled by the CNC unit. Each actuator is instrumented with linear optical encoders which provide feedback for closed-loop servo position control. The hydraulic system is modified to provide constant pressure fluid power to the servo valves. An accumulator is used between the pump and the valves to suppress pressure fluctuations. Pressure transducers are integrated into the ports of the actuator to monitor the pressure during the operation of the press. The dynamics of the hydraulic servo system, including the valves, actuators and the ram are modeled. A non-linear model, which includes the influences of piston position, is presented. A simplified model is shown to be adequate provided that the practical ranges of the piston position is considered. The mathematical model is used to experimentally determine the dynamics using parametric identification techniques. The well-damped system is approximated by a first order system with substantial delay between the servo-valve amplifier command and the actuator piston motion. A readily available CNC system is retrofitted to the press brake and a delay compensating pole placement digital control system is developed and implemented for the independent 11 control of two actuators. The performance of the system is evaluated for a series of forming operations. This thesis provides basic guidelines for the design and analysis of hydraulically actuated CNC presses. 111 Table of Contents Page Abstract.ii Table of Contents iv List of Tables viii List of Figures ix Acknowledgments xii Chapter 1 Introduction 1 Chapter 2 Literature Survey 3 2.1 The Brake Forming Process 3 2.2 Hydraulic Press Brakes 4 2.3 Hydraulic Supply Systems 7 2.4 Cylindrical Hydraulic Servo Actuators 9 Chapter 3 Modifications Required for Computer Control 14 3.1 MechanicaL Design Modifications 14 3.1.1 Ram Gibbing 14 3.1.2 Positioning System for CNC 16 3.1.2.1 Design Objectives 17 3,1.2.2 Mechanical Design 17 3.2 Hydraulic System 19 3.2.1 Supply System 20 3.2.2 Servo-actuator 21 3.3 Control Computer 22 Chapter 4 Modeling of System Dynamics 24 4.1 Introduction 24 iv 4.2 Dynamic Model of Servo-actuator .25 4.2.1 The Load Pressure, Load Flow Model 29 4.2.2 Improved Frequency Response Model 32 4.2.3 A Directionally Biased Model for Asymmetrical Actuators 35 4.2.4 Non-Linear Valve Flow Relation 37 4.2.5 Non-Linear Actuator Compliance 38 4.3 Results 42 4.3.1 The Effect of the LPLF Linearization on the System Response 42 4.3.2 The Effect ofPiston Area Ratio the System Response 43 4.3.3 The Effect of Input Signal Amplitude on the System Response 43 4.3.4 The Effect of Initial Piston Position on the System Response 45 4.3.5 Effect of Coulomb Friction on the System Response 46 4.3.6 Validation of the Non-Linear Models 47 4.4 Conclusions 49 Chapter 5 Identification of Servo-actuator Dynamics for Control 50 5.1 Introduction 50 5.2 Choice of Identification Signal 50 5.3 System Response to a Step Input 52 5.4 Frequency Response Experiments 54 5.4.1. Experiment Description 54 5.4.2. Experimental Results 54 5.5 Parametric Identification 56 5.5.1 Theory 56 5.5.1.1 Least Squares Method 57 v 5.5.1.2 Instrumental Variables Method .59 5.5.2. Experiment Description 60 5.5.2.1 Choice of Identification Signal 60 5.5.2.2 Model Structure 62 5.5.2.3 Data Analysis 62 5.5.3. Experimental Results 63 5.5.3.1 Model Selection 63 5.5.3.2 Model Validation 63 5.6 Conclusion 71 Chapter 6 Coordinated Motion Control of Press Ram 72 6.1 Introduction 72 6.2 Motion Control 73 6.2.1. Objective 73 6.2.2. Velocity Profile 73 6.2.3. The Process to be Controlled 74 6.2.4 Control Scheme 74 6.2.4.1 Introduction 74 6.2.4.2 Design of the Pole-Placement Controller 75 6.2.4.3 Controller Implementation 78 6.3 Positioning System Performance 81 6.3.1 The Dynamic Response of the Ram Positioning System 82 6.3.2 Determination of Positioning System Dead Band 83 6.3.3 Determination of Controlled Motion Performance 84 6.3.3.1 Motion Profile: No Forming Operation 85 6.3.3.2 Motion Profile: With Forming Operation 88 vi 6.4 Conclusions .92 Chapter 7 Conclusions and Recommendations 93 BIBLIOGRAPHY 95 APPENDIX A 98 Actuator Natural Frequency Calculations 98 APPENDIX B 99 Derivation of Pole-Placement Control Law Parameters 99 APPENDIX C 103 Friction Characteristics of Guide System 103 APPENDIX D 104 Model Parameters 104 VII List of Tables Page Table 5.1. Delay, rise time and steady state gain obtained from step response experiments 53 Table 5.2. Comparison of the loss functions computed for a variety of model structures 63 Table 5.3. Model Parameters determined from parametric identification 64 Table 6.1. Process models and control law parameters for left axis controller 81 Table 6.2. Process models and control law parameters for right axis controller 81 Table 6.3. Velocity Error Constants for each axis as determined from motion profile tests: no forming loads 86 Table 6.4. Results of the Brake forming analysis 89 vi” List of Figures Page Figure 2.1. The brake forming process 3 Figure 2.2. A typical press brake 5 Figure 2.3. Pressure Compensated Variable delivery hydraulic supply 9 Figure 2.4. Components of a hydraulic servo-actuator 10 Figure 3.1. Ram orientation mechanism 14 Figure 3.2. Degrees of freedom of CNC press brake 15 Figure 3.3. Ways box modification 16 Figure 3.5. Position measurement system for CNC operation 18 Figure 3.6. Schematic Diagram of the original hydraulic system 19 Figure 3.7. Schematic diagram of hydraulic system used for CNC operation 20 Figure 3.8. Technical illustration of servo-actuator 21 Figure 3.9. Block diagram of the Hierarchical Open Architecture Manufacturing CNC system 22 Figure 4.1. Typical hydraulic system used for servo positioning 24 Figure 4.2. Functional diagram of servo-actuator 25 Figure 4.3. Comparison ofvelocity response predicted by LPLF model versus the velocity predicted by the LC model with piston area ratio R=1 42 Figure 4.4. Comparison of the effect of piston area ratio, R, on the velocity response of servoactuator, as predicted by the LC model 43 Figure 4.5. Comparison of the frequency response predicted by the LPLF model to that predicted by the IFR model at various servovalve armature current amplitudes 44 Figure 4.6. Comparison of the Velocity response predicted by LC model versus that predicted by the FCC model when the initial position differs from the LC linearization position 45 ix Figure 4.7. Comparison of the response predicted by LC model versus that predicted by the FCC model when the initial position corresponds to the LC linearization position 46 Figure 4.8. Comparison of the effect of coulomb friction, Fc, on the velocity response of the system as predicted by the LC model 47 Figure 4.9. Comparison of the velocity response predicted by the LC and FCC models to the actual response 48 Figure 5.1. Velocity response of the left actuator to a step change in valve command voltage 52 Figure 5.2. Magnitude response of the right actuator in extension determined for a variety of input signal amplitudes 55 Figure 5.3. Phase response of left actuator in extension determined for a variety of input signal amplitudes 56 Figure 5.4. Effect of excitation signal amplitude on the steady-state gain predicted by the frequency response experiments (left actuator extending) 61 Figure 5.5. A comparison of the experimentally determined frequency response to that predicted by the identified models: Left actuator extending 65 Figure 5.6. A comparison of the experimentally determined frequency response to that predicted by the identified models: Right actuator extending 65 Figure 5.7. A comparison of the experimentally determined frequency response to that predicted by the identified models: Left actuator retracting 66 Figure 5.8. A comparison of the experimentally determined frequency response to that predicted by the identified models: Right actuator retracting 67 Figure 5.9. Comparison of the response of the measured system to that predicted by the parametrically identified models: left actuator extending 68 Figure 5.10. Comparison of the measured system to that predicted by the parametrically identified models: right actuator extending 69 x Figure 5.11. Comparison of the measured system to that predicted by the parametrically identified models. Left actuator retracting 70 Figure 5.12. Comparison of the measured system to that predicted by the parametrically identified models: right actuator retracting 70 Figure 6.1. Velocity profile for typical bending cycle 73 Figure 6.2. Block diagram of a pole-placement controlled system 75 Figure 6.3. Press setup for positioning system performance experiments 82 Figure 6.4. The response of left and right axis of the positioning system to a series of step changes in command position. Yl: left axis; Y2: right axis, Yref: reference command 83 Figure 6.5. Response of PID controlled positioning system to step changes in reference position 84 Figure 6.7. Response of positioning system to motion profile: no forming loads 87 Figure 6.8. Plot of the absolute and relative tracking error of each positioning system: no forming loads.87 Figure 6.9. Sample work-piece and finished part used for brake-forming tests 88 Figure 6.10. Plot of the absolute and relative tracking error of each positioning system: Motion profile with bending operation 90 Figure 6.11. Actuator pressures recorded for motion profile with forming operation 91 Figure 6.12. Free-body diagram of forces acting on ram during the dwell operation 91 Figure C. 1. Friction force exerted by guide system 103 x Acknowledgments I would to thank my supervisor Dr. YusefMtintas for his guidance during this work. His patience and support is greatly appreciated. I would like to express special thanks to Grant Lindsay of Del Scimieder Hydraulics for his helpfhl assistance and technical support throughout the project. I would also like to thank Adrian Clark, Scott Roberts and Dr. Malcolm Smith for their helpful suggestions and encouragement. The author wishes to acknowledge the support of the several companies: Accurpress for providing the press brake for this work, Parker Hannifin Corporation for supplying hydraulic system components, and Atchley Valves for providing the servo valves. xli Chapter 1 Introduction A hydraulic press brake is a machine tool used to form bends in metal plate or sheet. In production, a typical press brake is setup to perform a single bend in a batch of similar components. The bend angle is set by trial and error using an adjustable mechanical stop. This setup requires several attempts and is only useful for one bend angle. By comparison, Computer Numerical Control (CNC) press brakes use closed-loop position control which improves precision and simplifies the setup such that only one trial bend is needed. Since setup information is stored digitally, CNC press brakes can be programmed to perform a number of different bends at any particular instant in time. In this thesis, a single axis hydraulic press brake is retrofitted for dual axis CNC control. The focus of this thesis is the servo system used to position the forming tools. The objective of this work is two fold: • to reduce uncertainty during the design stage by investigating models used to represent the system dynamics • to investigate ways of improving system performance in the face ofunexpected design shortcomings In this work, a hydraulic servo positioning system is designed, modeled and analyzed. A hydraulic supply for servo positioning system is designed and implemented. High performance servo-valves are mounted to the existing actuators. Mechanical modifications are made to allow indcpendent actuator motion. A position feedback 1 Chapter]: Introduction 2 system utilizing linear optical encoders is designed and implemented. The dynamic characteristics of the system are experimentally verified using system identification techniques. Based on the results of these experiments, a delay-compensating pole- placement control scheme is chosen and implemented. The performance of the CNC press brake is evaluated. A survey of both research and industrial literature concerning the brake-forming operation, press-brakes, hydraulic system modeling and relevant automatic control theory is presented in Chapter 2. Chapter 3 describes the work required to convert the manually controlled press brake to one capable of computer control. Details of the hydraulic and mechanical designs are presented and a description of the electrical hardware used for control. In Chapter 4, models used to represent the dynamics of the hydraulic positioning system are presented and applied. An alternate actuator compliance model is developed and investigated. A summary of general recommendations for modeling is given. Chapter 5 describes the system identification experiments conducted on the position control system. Results of step response, frequency response and parametric identification tests are presented and discussed. A summary of conclusions for control is presented. The delay-compensating, pole-placement control scheme is described in Chapter 6. A simplification to the control law is developed. The performance of the positioning system is evaluated in terms of system response, dead-band, following error, and stiffness. The results are presented and discussed. Finally, Chapter 7 is a summary of conclusions arrived at through this work. Recommendations for future work are suggested. Chapter 2 Literature Survey 2.1 The Brake Forming Process “Brake forming is a method of forming straight-line bends in sheets and plates” [1]. In a brake forming operation a sheet metal workpiece is positioned between a punch and a die. The bend is formed as the punch penetrates the die. Although variations exist, there are two fundamental types of brake forming operations. When the punch bottoms the work piece in the die the operation is known as coining or bottom bending. When the punch does not bottom the work-piece in the die the operation is known as air-bending [2]. The brake forming process is illustrated in figure 2.1. WORK PIECE Figure 2.1. The brake forming process. In the ideal bottom bending operation, the yield stress of the work piece material is induced throughout out the area of the bend, causing the work piece to take the shape of DIE 3 Chapter 2: Literature Survey 4 the punch and die. While bottom bending generally produces the best quality bends, it requires very high actuation forces and specially mated tooling for each angle of bend [3]. In an air bending operation, the angle ofbend is determined by the amount of punch penetration. Therefore, air bending operations do not require specially mated tooling. Furthermore, since the yield stress of the material is not induced throughout the area of the bend, the actuation force required is two to five times lower than that required by bottom bending [2]. Because of this inherent advantage, automation efforts in brake forming have focused on the air-bending operation [4]. However, the cost of not inducing the yield stress of the material throughout the area of bend is that a certain amount of elastic deformation remains at the final punch penetration. This elastic deformation causes the work piece to ‘spring-back’ when the punch is withdrawn from the die. Compensation models, which are capable of approximating spring-back given the desired bend angle and the material thickness, are frequently used [5]. 2.2 Hydraulic Press Brakes Brake forming operations are performed on machines known as press brakes. A technical illustration of the Accurpress press brake used for this project is shown in figure 2.2. This machine is typical of those utilized in the industry. The fundamental components of the press brake are depicted. The punch is clamped to a large ram which is positioned by two hydraulic actuators. The actuators are driven in parallel by a single or multistage solenoid operated valve system. Usually, the hydraulic circuit has provisions for two speeds: one for rapid positioning and one for the forming operation. The die is fixed to the press bed. The positioning system controls the amount the punch penetrates the die. Due to the extremely high forces applied, the ram gibbing is not sufficient to hold the punch parallel Chapter 2: Literature Survey 5 to the die. As a consequence, press brakes use an auxiliary system to ensure tooling is properly oriented. The flrndamental differences between manually operated press brakes and automated ones lie in these two systems, which ensure appropriate positioning and orientation of the punch with respect to the die. RAM PUNCH DIE PRESS BED POSITIONING SYSTEM With manually operated press brakes, the positioning system and the orientation system are distinct mechanisms. The orientation mechanism may consist of a stiff mechanical device or a sensitive hydraulic feedback circuit adjusted to minimize tilt of the ram during asymmetrical loading conditions. Typically the positioning system consists of a calibrated adjustable stop and an electronic switch which triggers a solenoid driven hydraulic valve. When flow to the actuator is arrested or reversed, the punch penetration is limited. The accuracy of this system is dependent upon the sensitivity of the switch as well as the condition of the hydraulic system. All hydraulic systems are subject to variations in fluid viscosity and bulk modulus. These variations can cause significant deviations in the final punch penetration over the run of a batch of parts. Figure 2.2. A typical press brake. Chapter 2: Literature Survey 6 On a two-axes CNC press brakes the orientation as well as the positioning tasks are handled by an integrated position control system. The flow of hydraulic fluid to each actuator is controlled by two independent precision single or multistage spool valves. The relative position of each actuator is measured by a high resolution position transducer. The movement of each actuator or axis is controlled by a digital feedback control scheme. Typically, CNC press brakes use gibbing which allows some tilting of the ram but the relative orientation of the tooling is ensured by linking each axis at the control level. Because these systems use feedback control the positioning is much less affected by changes in the state of the hydraulic fluid. The benefits of computer-controlled positioning systems are many. For a given batch of parts requiring the same bend angle, the setup procedure for a press brake involves three steps. First, the punch and die are mounted and adjusted to be coplanar. Next, the clamping mechanism is adjusted to ensure the forming edges are parallel. Finally, the positioning mechanism is set to give the desired bend. Trial and error testing of the forming operation is performed until the finished bends are within tolerance. With manually operated press brakes, this procedure is repeated many time, not usually less than three [2]. By comparison, the setup procedure for CNC press brakes is much simpler. A tooling offset reference is set by bottoming the punch in the die, and a predictive model is used to calculate the desired punch penetration. After a test workpiece has been bent, the error between the desired work piece and the test piece is entered into the positioning controller and then the press brake is ready to form parts. In this way CNC press brakes require much less setup time. Furthermore, once a series of bends have been tested and calibrated, the CNC press brake can readily switch between bends with no extra setup time. Chapter 2: Literature Survey 7 2.3 Hydraulic Supply Systems Hydraulic servo-actuators used for positioning systems require constant pressure supply systems. Although constant supply pressure under varying loading conditions is difficult to achieve, Ahmed and Asok [6] have reported that slight (<10%) fluctuations in the supply pressure have little effect on the response of the servo-actuator. In order to achieve this constraint, some consideration must be given to the design of the hydraulic supply system. Systems that supply fluid power to hydraulic systems can be classified into one of two categories: constant delivery systems or variable delivery systems. In the simplest form, a constant delivery system designed to provide constant pressure can consist of a fixed displacement pump and a pressure reliefvalve (PRy). In order to maintain a relatively constant system pressure the pump would supply a constant flow of fluid to the supply line and the PRy. Whatever flow is not used by the servoactuator would pass through the PRV at maximum pressure drop. Although this type of system is inexpensive to implement, the cost of providing the maximum flow rate at the system pressure is two fold; the overall efficiency of the system is low, and the quantity of heat generated by this low efficiency can breakdown the hydraulic fluid. For these reasons, constant delivery systems are rarely used in servoactuator applications. Variable delivery systems have the advantage of being able to provide only enough flow to satisfy the requirements of the servo-actuator while accommodating some internal leakage. Depending on the type of pump being used, variable delivery flow can be achieved in one of two ways. If a fixed displacement pump is used, the pump would provide flow to a hydraulic accumulator which in turn would supply the servo-actuator. Chapter 2: Literature Survey 8 When the system pressure reaches the desired pressure, the pump would be shut off until the system pressure falls below a prescribed lower pressure limit. While this type of system is more efficient than a constant delivery system, a compromise must be made between the frequency at which the pump is cycled and the maximum pressure fluctuation allowed. Systems employing variable displacement pumps use pressure feedback to continuously vary the flow to track the desired system pressure. The fundamental components of a pressure-compensated variable delivery hydraulic supply system are: a variable displacement pump with pressure feed-back, a hydraulic accumulator, and a check valve (figure 2.3). In this system, the pump utilizes an internal spool valve to control the angular displacement of the swash plate. As the swash plate angle increases, so does the flow delivered from the pump. When the pressure in the supply line deviates from the desired system pressure, the control valve adjusts the output flow-rate to maintian the desired system pressure. Since the response time of this type of pump is generally slower (50-l2Oms) than response time of a typical servovalve (4-2Oms) an accumulator is used to satisfy the flow requirements while the pump is responding to a demand for more flow. Although these systems are generally more expensive than the systems described previously, they are capable of providing the smoothest supply pressure with a minimum of pressure fluctuation. For this reason, they are the most common supply system used with precision servoactuators. Chapter 2: Literature Survey 9 tnternal Reservoir 1jjJ AccumulatorPressure Compensated Variable Displacement Pump Figure 2.3. Pressure Compensated Variable delivery hydraulic supply. 2.4 Cylindrical Hydraulic Servo Actuators A schematic diagram of a hydraulic servo-actuator used for CNC press brakes is presented in figure 2.4. The load is connected to the output shaft of a hydraulic cylinder. The cylinder consists of two chambers: the piston-side chamber and the rod-side chamber. For high performance systems, flow to each chamber of the cylinder is typically controlled by a high-precision multistage critically-centered spool valve often called a servo-valve. An electronic amplifier supplies a command signal to the valve. The output velocity of the load is proportional to the servo-valve command signal. The characteristics of the main components which make up these servo-actuators, the loaded actuator and the servo-valve have been thoroughly investigated. [6-16]. Chapter 2: Literature Survey 10 Previous investigations of cylindrical hydraulic servo systems have found that very little viscous damping is attributable to the actuator[7,17]. When the compliance of the hydraulic fluid within the actuator is considered, a symmetrical actuator at mid-stroke behaves as an under damped second order system [13]. Asymmetrical actuators can be modeled by a third order system [11]. Depending on the gibbing or guide ways used for the system, significant amounts of Coulomb damping may also be present [18]. The dynamics of a two-stage servo-valve can be represented by a simple lag due to the torque motor driving the primary stage combined with a quadratic lag due to the dynamics of the secondary stage flow controlling spool. Often the dynamics of the flow controlling spool are fast enough to render them insignificant. The flow through the servo-valve is proportional to the valve opening and the square root of the pressure drop across the open port. Stage Valve Stage Spool /Rod Hydraulic Cylinder Feedback Spring Figure 2.4. Components of a hydraulic servo-actuator. Chapter 2: Literature Survey 11 The ftmndamental difficulties in controlling servo-actuators stem from five phenomena: i) the nonlinear pressure/flow relationship for the flow through the servo-valve ii) the inherent lack of viscous load damping iii) the presence of significant Coulomb damping iv) varying actuator compliance and v) variation of characteristics of hydraulic system. Although the non-linear pressure flow relationship can have significant affect on the large signal response of the servo-actuator, it is generally considered insignificant for typical operating control signals. Therefore, it is common practice to linearize the flow characteristic about the null operating point, where the spool is centered and the ports closed. This represents the worst-case scenario for closed loop stability because the valve gain is highest and the flow damping is lowest. Alternate methods of analysis have been developed to predict the frequency response of the system which varies with the amplitude of the input signal [13]. In cases where the response of the control valve is much faster than the response of the loaded actuator, significant research effort has been expended to overcome the lack of viscous damping [9,17, 19]. Some of the earliest schemes to improve damping involved the introduction of laminar leakage across the chambers of the actuator. These systems were simple to implement but reduced the stifibess and efficiency of the actuator. Transient flow networks, unlike the leakage techniques, were non-dissipative but they were difficult to tune and reduced actuator stiffness. Hydromechanical feedback mechanisms were developed which added the required damping and increased the actuator stiffness, but these systems required precision manufacturing operations which were specific to each valve/actuator/load combination. The cost and flexibility of these Chapter 2: Literature Survey 12 methods were improved when electronic feedback of acceleration and pressure signals replaced hydromechanical feedback. Modern digital controllers are capable of addressing this problem in a number ofways, the most common being pole-placement control schemes [201. The presence of Coulomb damping has the combined effect of reducing the oscillatory nature of the system response while also contributing to the steady state error. To compensate for the absence ofviscous damping, early analyses made attempts to model Coulomb damping as a theoretical viscous equivalent [11,21]. Current analyses simply treat this damping as an external force disturbance [22]. The compliance of a symmetrical hydraulic cylinder is a function of the bulk modulus of the hydraulic fluid and the ratio of oil volume residing on each side of the piston. Since the volume ratio changes with movement of the piston, the compliance varies as well. To simplif’ modeling, the common practice is to linearize the compliance relation about the most compliant position. For asymmetrical actuators the change in oil volume ratio with piston movement is more pronounced. While studies have been conducted to determine the effect of changing oil volume on the frequency response and stability of simple control systems [11], typical control strategies either assume a constant value for actuator compliance, or assume the dynamics of the loaded actuator are much faster than the dynamics of the control valve [23-241. The sources of hydraulic system performance degradation are many. In the short term, the main cause of system degradation is the change of state of the hydraulic fluid. Hydraulic fluids used in servo-actuators have two undesirable characteristics: 1) the viscosity of the fluid changes dramatically with temperature and 2) the bulk modulus (or stiffness) of the fluid changes dramatically with the quantity of dissolved gas in the fluid. Over the long term, regular wear of precision components will cause a change in the characteristics of Chapter 2: Literature Survey 13 the system. Moreover, long term wear is accelerated in the presence of contaminates in the hydraulic fluid. Given the existence of these phenomena, methods to insure precise position control have recently been investigated using digital adaptive control schemes. These controllers have been labeled ‘switching’ adaptive controllers because the control law is updated at a frequency which is an order of magnitude lower than the ioop closing frequency. Typically the adaptive controller schemes either used a method of recursive least squares to identify reduced order models of the open ioop system dynamics (self tuning regulators) or identified the control law parameters directly (model reference adaptive control) [20,22]. Provisions were made to check the stability of the system and temporarily halt estimation if the input signal ceases to be ‘persistently exciting’. Several implementations used an exponential forgetting factor and/or covariance matrix resetting in the recursive identification to track time varying parameters. The most common control strategy chosen was pole-placement, but adaptive optimal controllers have also been investigated [23,25]. Chapter 3 Modifications Required for Computer Control This chapter describes the work done to change the press brake from a manual to CNC control. 3.1 Mechanical Design Modifications 3.1.1 Ram Gibbing The press brake used in this project was originally designed to allow only one degree of freedom of ram movement: vertical translation. The gibbing of the press (figure Link Side Plates Torque Tube Guide-bar Waysbox 0 Ram Figure 3.1. Ram orientation mechanism. 14 Chapter 3: Instrumentationfor Computer Control 15 3.1) consisted of a pair of ways boxes which were rigidly connected to the ram, sliding on a pair of parallel guide bars, one bolted to each side plate. To maintain precise orientation of the tooling while undergoing the extreme forces of the forming process, an auxiliary device was employed. A torsional link known as a torque tube, was mounted between the side plates of the press. This torque tube was connected to a ways box on each side of the ram by mechanical links. One goal of the retrofit was to add the capability to create bends with up to three (3) degrees of trim without adjusting the tooling. This required the addition of a second degree of freedom of ram motion: rotation within the plane of the ram (figure 3.2). In order to allow this rotation, the torque-tube was disconnected and each ways box was modified to provide guide-bar clearance. To ensure the ram remained centered between the side-plates through-out its range of motion a cylindrical sliding surface was machined in the lower ways box pad (figure 3.3). The pads were oriented such that the ram would remain centred between the guide bars. Figure 3.2. Degrees of freedom of CNC press brake. Chapter 3: Instrumentationfor Computer Control 16 Flat pads Original Waysbox Modified Waysbox Figure 3.3. Ways box modification 3.1.2 Positioning System for CNC The positioning system originally installed on the press brake is depicted in figure 3.4. This system consisted of a parallel guide mechanism which housed limit switches used to signal the change-of-speed and the ram-stop position. These switches were tripped by a moving slide which was connected to the ram by way of a tie-rod mechanism. A micrometer barrel, attached to the moving slide, was used to fine-tune the final stop position. This design was chosen to compensate for deflections which occur in the side plates of the ram during bending. While the theoretical precision of this system was bounded by the resolution of the micrometer (0.012 mm) and the repeatability of the stop switch, the actual precision may have been worst due to the presence of a low frequency dynamic mode of oscillation with a frequency of 21 Hz. for Cylindrical Pad Chapter 3: Instrumentationfor Computer Control 17 Micrometer Barrel Stop Switch Mechanism Figure 3.4. Positioning system used for manually operated press brake 3.1.2.1 Design Objectives Computer controlled ram motion requires a system to measure the position and orientation of the ram. The goals for the design of this system were as follows: • side-plate deformation compensation • 0.01 mm position repeatability • provisions for both position and velocity transducers • high resonant frequency 3.1.2.2 Mechanical Design In order to compensate for side-plate deformations, a design utilizing a measuring slide tie-rod connected to the ram (similar to the original system) was chosen (figure 3.5). To simplif’ the ram orientation task, the tie-rod posts were placed at the axes of the ram/actuator connections. High-precision ball-joints were used to connect each tie-rod to Tie-Rod Post Tie-rod Chapter 3: Instrumentationfor Computer Control 18 its respective tie-rod post and slide mechanism. A lower bound on the length of the tie-rod was determined by bounding the measurement error due to tie-rod rotation. Tie-rod Post Connection Pin Tie-rod Encoder Head Slider Encoder Body Guides Figure 3.5. Position measurement system for CNC operation. In order to achieve the 0.01 (mm) position repeatability, a linear encoder with a resolution of 0.005 (mm) was chosen. This encoder consists of a body which houses a fixed scale and a moving read head. The body was mounted to the press bed and the read head was attached to the translating slide of the linear guide system which was designed and manufactured in-house. The guide system utilizes a pair of cylindrical guides, one fixed, the other adjustable. This built-in adjustment capability eases alignment procedures while providing a method of preloading the guide system for greater rigidity. Provisions were made to allow the attachment of a transducer to measure the velocity of each side of the ram with respect to the bed. The tie-rod, tie-rod posts, and the slider were designed to have high stiffness and small mass to minimize the effect of their dynamics on the feedback signal. Chapter 3: Instrumentationfor Computer Control 19 3.2 Hydraulic System A schematic diagram of the original hydraulic system is shown in figure 3.6 Hydraulic flow was provided by two fixed displacement gear pumps: one for high pressure, the other for high flow. A pair of normally open solenoid valves, were sequenced such that both pumps provided flow during the rapid moves, but only the high pressure pump provided flow for the feed operation. A counter balance valve located in the rod-side line was used to lock the actuator during idle times. HII3H For CNC press operation, the original hydraulic system was replaced by a system represented by the schematic shown in figure 3.7. A description of the components comprising this system follows. HIGH VOLUME PUMP Figure 3.6. Schematic Diagram of the original hydraulic system. Chapter 3: Instrumentationfor Computer Control 20 3.2.1 Supply System For high efficiency and good performance, a variable displacement pump with pressure feedback was chosen to deliver the hydraulic power. Since the response time of the pump was significantly slower than that of the servo-valves, an accumulator was used to keep the system pressure within 10% of the desired level. This accumulator also damps harmonic components of the pump pressure caused by the oscillating pump pistons. A check-valve was placed between the accumulator and the pump to eliminate the possibility of the accumulator pressure driving the pump in reverse in the event of a power outage. A counter-balance valve was used to lock the system whenever the system pressure is lost. Figure 3.7. Schematic diagram of hydraulic system used for CNC operation. Chapter 3: Instrumentationfor Computer Control 21 Two ifiters are used to filter the fluid: a coarse low pressure filter located in the return line of the original system, and a fine, high-pressure filter in the supply line to keep metallic pump debris from reaching the sensitive pilot stage of the servo-valves. 3.2.2 Servo-actuator A technical illustration of the servo-actuator used for positioning is shown in figure 3.8. The compact design of the press precluded the use of an off the shelf servo-actuator, so the original cylinders were used with only minor plumbing changes. A pair of prototype Atchley 320 servo-valves (20 g.p.m., 85 Hz bandwidth [261) were mounted as close as possible to the ports of the original actuator. Special in-line transducer fittings were designed and manufactured to house piezo electric transducers capable of measuring cylinder port pressures. Pressure Port Pressure Transducer Upper Transducer Fitting Pressure Transducer Lower Transducer Fitting Figure 3.8. Technical illustration of servo-actuator. Chapter 3: Instrumentationfor Computer Control 22 3.3 Control Computer The computer used to provide CNC control was the Hierarchical Open Architecture Manufacturing CNC system (HOAM-CNC) developed in-house by the CNC research group of the University of British Columbi&s Mechanical Engineering department. A block diagram of this controller is shown in figure 3.9. r 486PC L SYSTEM MASTER_J MAIN Bus: ISA Bus C30 DSP CNC MASTER CNC Bus: DSP LINK 80C196KC 1 80C196KC 1 Axis ControllerJ Axis ControJ Figure 3.9. Block diagram of the Hierarchical Open Architecture Manufacturing CNC system. The HOAM-CNC system uses a hierarchical structure to divide control tasks up between three computational systems: the system master, the CNC master, and the axis controllers. At the axis level several functions are performed: the loop is closed from a common clock pulse, some position interpolation is done, and controller status variables are updated. At the mid-level, the CNC master is used to initialize and coordinate each Chapter 3: Instrumentationfor Computer Control 23 axis controller. These functions have been implemented by way of a flexible task scheduling system which is also capable of high level interpolation computations and alternate functions such as data collection or on-line identification. At the top of the HOAM-CNC hierarchy is a PC based host computer which provides a user interface and mass storage. For a detailed description of the HOAM-CNC system see [27]. In order to optimize positioning performance each axis of the HOAM-CNC was modified to allow the use of a high-bandwidth, scaled control signal. The nine bit output conmiand voltages were fed to a pair of Parker BD-98 gain amplifiers which were used to drive the servo-valves. Quadrature encoder signals from the positioning system were connected directly to inputs on the axis controllers. Chapter 4 Modeling of System Dynamics 4.1 Introduction In order to satisfy given performance constraints, the design of a servo-mechanism requires detailed information about how a series of individual components will perform together. Dynamic system models can provide this information if the characteristics of each component are known. A typical hydraulic circuit used for servo-positioning systems (figure 4.1) can be analyzed as two distinct subsystems: a servo-actuator system and a hydraulic supply system. The hydraulic supply consists of a pressure compensating variable displacement pump supplying compliant hydraulic lines. The servo-actuator system consists of a two-stage servovalve connected to a hydraulic actuator which in turn moves the ram. In the following sections, the physics of the servo-actuator system is / Figure 4.1. Typical hydraulic system used for servo positioning. 24 Servo-actuator Chapter 4: Modeling ofSystem Dynamics 25 described and models used to represent the dynamics are presented. 4.2 Dynamic Model of Servo-actuator The servo-actuator used to position each joint of the ram can be considered as a system consisting of three distinct subsystems:a precision flow-control valve, a hydraulic cylinder and a load (figure 4.1). Hydraulic Cylinder Load Figure 4.2. Functional diagram of servo-actuator. The flow control valve used for the positioning system is a two-stage servovalve. The primary stage consists of an electronic torque motor driving a primary valve, usually of a flapper or a jet-pipe type (flapper-valve type shown in figure 4.2). When the torque motor is driven, the primary valve provides a differential pressure across the ends of a secondary closed-centre spool valve. The displacement of the secondary spool is fed back to the torque motor by way of a cantilever spring. This spooi controls the flow to the cylinder. The position of the flow controlling spool as a function of the torque motor current can be represented by a first-order lag due to the torque motor combined with a quadratic lag due to the dynamics of the spool: Fext Chapter 4: Modeling ofSystem Dynamics 26 x(s) K, — 2 2’ia(S) (ts+1)(s +24’ws+w ) where: x: spool displacement i: torque motor current K,: spool valve positioning gain co,,: natural frequency of spool dynamics C: coefficient of friction for spool valve Generally, the dynamics of the spool are considerbly faster than the dynamics of the torque motor so that (4.1) can be approximated by the following expression: x(s) K, (42) ia(S) (r1s+1)’ where: K: effective spool valve positioning gain The flow through a port of the secondary-stage spool valve has been found to be proportional to the area of the valve opening and the square root of the pressure drop across the port. Since the area of the valve opening is proportional to the valve displacement, the following expressions can be used to represent the flow through the ports of a spool valve: forx O, q,, =KqaXviJPs —psign(F pa) (4.3) b =KX/IPb —pts1gn(pb —J) (4.4) for x <0, q, = Kqa X,q1IPa — Pt IS1(Pa — (4.5) Ib = Kqb X Jp — Pb sign(p — Pb) (4.6) where: Chapter 4. Modeling ofSystem Dynamics 27 q: flow out of port A ofvalve q: flow into port B ofvalve Kqa , Kqb: coefficients of flow for ports A and B p5: supply line pressure pt: return line (tank) pressure Pa,Pb: pressure acting upon ports A and B of the valve The hydraulic cylinder, as depicted in figure 4.2 consists of two fluid chambers separated by a piston. The rod connecting the piston to the rod is assumed to be rigid. Because the fluid on either side of the piston is mildly compressible, hydraulic cylinders exhibit compliance under load. This compliance (or its inverse, stiffness) is a concern to servo-system designers because it limits the maximum bandwidth of the system. To determine a relationship between the chamber pressures, chamber flows and piston velocity, the continuity equation can be applied to control volumes encompassing each chamber to obtain the following expressions: qa_qc Klp(PaPb)=ApVy (4.7) qb qQ, +K,P(Pb Pa)+1<lePb = rVY (4.8) where: v: velocity of the piston A: area of the piston Ar: area of the piston minus the area of the rod Kb,: coefficient of leakage past the piston K,: coefficient of leakage past the rod seals q: rate of change ofvolume due to volume A compliance qC. rate of change ofvolume due to volume B compliance Since fluid compressibility is directly proportional to the volume in which the pressure acts, the compliance of each actuator chamber may be expressed as: Chapter 4: Modeling ofSystem Dynamics 28 Ca =—(Vaj+Ap(L—y)), (4.9) Cb =_(Vbi+ArY), (4.10) where: 1’ , V: volume of fluid in the line leading to chambers A and B 13e: bulk modulus of the fluid y: piston position If the fluid compliance relations, (4.9) and (4.10), are linearized about a particular operating point (L0), the following expressions for the rate of change of volume within the actuator may be written: q_4i!. (4.11) qC,4P (4.12) where: : linearized compliance of fluid in chamber A : linearized compliance of fluid in chamber B By summing the forces acting on the load mass, the following equation for the motion of the load can be written: F = Mr = ArPb ApPa BrVy —F,sign(v)+F (4.13) where: Mr: combined effective mass of piston and load Br: effective viscous damping actuator F: coulomb friction force F: external load (including gravity) Equations (4.1-4.13) are the fbndamental epressions most models use to study cylindrical hydraulic servo-actuators. Chapter 4: Modeling ofSystem Dynamics 29 4.2.1 The Load Pressure, Load Flow Model Most of the research concerning cylindrical hydraulic servo-actuators pertains to the case where there is equal area on each side of the piston: i.e. symmetric actuators. Although most hydraulic presses do not employ such actuators, an analysis of such models is worthwhile because they provide the simplest estimation of system performance. Merritt [14] presents a thorough analysis of symmetric hydraulic actuators utilizing a simple linear model based on a load pressure, load flow (LPLF) simplification. In order to reduce (4.1-4.13) to a linear model numerous assumptions need to be made. By assuming zero tank pressure, no cavitation and a symmetric actuator, the flows into and out of the spool valve can be equated so that the following expression, normalized with respect to the supply pressure, holds: q, =KqXvjIl_PL, (4.14) V P8 where: q1: effective load flow (q — q) Kq: normalized valve flow gain p1: effective load pressure (Pa — Pb) If the spool only undergoes small disturbances about the null flow position, (4.14) can be linearized to yield: q1 — Kq; — Kp,, (4.15) where: ICE: null flow pressure gain Since this linearization yields a zero value for the null flow pressure coefficient, this value is usually determined empirically from closed-port leakage tests [14]. To extend this LPLF analysis to the actuator, the load is assumed to undergo relatively small excursions from a set operating point. Furthermore, Wit is assumed the Chapter 4: Modeling ofSystem Dynamics 30 fluid in each chamber is of equal compliance, (4.7-4.10) can be linearized and combined to yield: q1 =_Av+K,p+CL, (4.16) where: K1 = K,1 + CQb V1: total volume of fluid within the cylinder and lines Equation (4.16) can be combined with (4.15) and rearranged to yield, using Laplace notation: Kx+Av Pz j 1’ (4.17) CabS+Kctm where: Kctm =K, +K. (4.18) Considering the equation defining the load dynamics (4.13), some assumptions must be made to accommodate the coulomb damping term. In the past coulomb damping has been modeled with viscous damping 21,23], but modem analysis considers coulomb friction to be an external disturbance. If the damping is assumed to be strictly viscous, the load equation (4.13) can be rewritten, using Laplace notation: MrSVy +BrVy = —App, +F, (4.19) Combining (4.17) and (4.19) yields the following transfer functions for the response of the servoactuator to the spooi displacement and the external force: v(s) K x(s) = 2 +2,w,s+co,’ (4.20) Chapter 4. Modeling ofSystem Dynamics 31 v,, (s) = KFt (rabs +1) (4 21) F(s) s2+2Cco,s+w, where: J +--- I (4.22)1 A ‘ v, 4Ap\lI3eMr ‘ ‘ = I’e (1+”m’) (4.23) V:M,. 42 413AKepq (4.24)X Mr = 4Kj3 (4.25) KF = 1 (4.26) ‘t•ab Mr It has been noted that the contribution of viscous damping in this type of drive is very small [9,17,19]. Hence, it is common practice to assume Br = 0. Using this assumption, (4.22) and (4.23) reduce to: c = Kctm1]I3eMr , and (4.27) 1413 A 2 w =I e (4.28) VtMr which indicates two phenomena characteristic of all valve-controlled actuators. First, for a cylinder with minimal external leakage, the damping is proportional to the cross port leakage (null position valve leakage and piston leakage). Second, the stiffness of the actuator is proportional to the bulk modulus of the fluid and the piston area while being inversely proportional to the stroke. Because of their inherent simplicity, (4.27) and (4.28) are often used during the preliminary stages of servoactuator design [28]. Intuitive Chapter 4. Modeling ofSystem Dynamics 32 variations of (4.27) and (4.28) exist for the design of servoactuators utilizing non- symmetrical hydraulic cylinders. The chief virtue of the LPLF model is its conservative estimate of system performance. Since the derivation makes use of a control valve linearization about the operating point with the least damping (the null position), the LPLF model gives a lower bound for damping within the system. Furthermore, since the actuator compliance expressions relations (4.9) and (4.10) are linearized about the most compliant piston position, this model also provides a lower bound for the bandwidth of the system. 4.2.2 Improved Frequency Response Model While the LPLF model can be used to determine the frequency response of a servoactuator system, McCloy and Martin [13] present a more sophisticated model which considers the effect of the spool valve displacement on the damping of the system. This improved frequency response (IFR) model is based upon the LPLF model. Assuming a pure inertia load, (i.e., no damping and no external load), and a symmetric actuator (piston area A,,) the expression describing the load dynamics (4.13) can be simplified to: dv Mr_L=Ap(Pb_Pa), (4.29) or, Mdv p,=__4LL. (4.30) Rewriting the non-linear pressure/flow relationship for the LPLF model (4.14) in non normalized form yields: q, = Kx.JI —p1. (4.31) Chapter 4: Modeling ofSystem Dynamics 33 K,= Substituting (4.30) into (4.31) yields: I Mdv q, =KqXv+_jf_jL. (4.32) dvff4.32 is linearized about the pomt of maximum load velocity, (L 0), the followmg expression for the load flow is obtained: q, = Kxf(1+.j ;‘;; (4.33) or the more general form: q,=Kqxv(1+A Mr L), (4.34) Ap dt where: A: flow/pressure linearization constant McCloy and Martin [13] suggest using a value of A in (4.34) other than 1/2. The criterion used to select A is minimum error between the actual pressure/flow relation and the linearized relation (4.34). This criterion yields a value of 2/3. Assuming no actuator leakage (K = 0), (4.34) can be combined with (4.16) to produce: Mdv VMd2Kx(1+A r ‘)—Av t r Y (4.35)q “ Ap3 dt “ 4J3e’p cit2 or, by rearranging: d2 d K’ ‘ +—2-(4A ‘ vPe )+v (4 Pe , ) = —K x (4 Pe P) (4.36)dt2 dt Vp ‘ VtMr V VtMr Chapter 4. Modeling ofSystem Dynamics 34 For a harmonic input with a fixed amplitude of the form: x(t) = X sin(a#), (4.37) the frequency response can be expressed by the relation: v(joi) = K (4.38) x(ja)) 2 _(02 +2j(X)w,w where: KX 15Mq e r (439)V and, as in the LPLF model (4.27 and 4.28): 14$A2 4/3AK = j e and K = e p q (4.40) VtMr X VtMr In order to produce a harmonic spool motion, a harmonic current signal of the form: 1a(t) = ‘a sin(ot), (4.41) has to be applied to the torque motor of the servo valve. When the dynamics of the primary stage of the servovalve are included (4.2), an expression for the frequency response of the load velocity with respect to the servovalve input signal can be written: v (flu) = I<KX (442) a(J) CO,2 — 2 —2’rco,o(I)+j(tj io, —t7w3 +2WiWf(Ia)) where: KKI 15Mq I al e r (4.43) PSAPV From (4.42) and (4.43) it is evident the IFR model can be used to determine the frequency response for a particular amplitude of excitation. However, as with the LPLF model, the IFR model assumes symmetrical actuators. If the ratio of the cap-side to rod side piston area (R) is not unity, a more appropriate model may be required. Chapter 4. Modeling ofSystem Dynamics 35 4.2.3 A Directionally Biased Model for Asymmetrical Actuators Although many investigations of the dynamics characteristics of servosystems employing non-symmetrical actuators have been conducted, most either ignore the effects of fluid compressibility[24] or assume equal fluid chamber compliance. An exception to this trend is presented by Watton [29]. In this study, the stability and step response of a proportionally-controlled, symmetrical servoactuator are studied for cases where the ratio of the volume of oil on each side of the piston is not unity. An application of the analysis used in this investigation follows. Given the pressure/flow relations (4.3-4.6), linearized relations for the flow through each port of the spool can be written: for x O, qa=Kqxv-K0pa (4.44) qb=Kqbxv+Kcbpb (4.45) for x, <0, qa=KqaxvKcp (4.46) qb=Kqbxv+Kbp1, (4.47) where: Kqa,Kqb: A & B port flow gains when x 0 KqaKqb A & B port flow gains when x,, <0 KKCb: A & B port pressure coefficients when x 0 Kca,Kct,: A & B port pressure coefficients when x <0 or, more generally, (4.48) q=dKx+dKp (4.49) where: d : ‘+‘ when x, 0 Chapter 4: Modeling ofSystem Dynamics 36 d : !l when x <0 If the actuator flow expressions, (4.46-4.49), are linearized about a particular piston position, and external leakage is neglected, the following expressions can be written: q, = —Av + K, (Pa — Pb) + Ca (4.50) qb (4.51) Combining (4.48-4.49) with (4.50-4.5 1) yields: dKqaXv_dKcaPa = —Av + K, (Pa — Pb ) + Ca (4.52) dKqbXv+dKcbpb = ArVy + K1 (Pa — Pb) — Cb (4.53) These expressions can be rewritten using Laplace notation: dK x +A v +K,pb Pa = Ca5+(dKca+Kip) (4.54) — dKqbXv + Av — ‘ipPa 4 55Pb — — CbS +(dKCb + Ic) ( . ) Considering (4.13), the following expression describing the dynamics of the load can be written using Laplace notation: MrVyS+BrVy = ArPb — ApPa +F (4.56) Combining (4.54-4.56) yields the following transfer fhnctions: vt(s) = — dK(S+ 1) (45) x1,(s) s +a1s +a2s+a3 v(s) 1 s2+b1+b (4.58)F(s) Mr3+as+a where: Chapter 4. Modeling ofSystem Dynamics 37 B K K a1 = _r_ + —f- + —- Mr Ca Cb a2 = (KaKb K12) (‘22 CaCb Ca Cb ) Mr Ca Cb ) Mr a3 = (KaKb _Kip2)Br +Ar2Ka +A2Kb 2ArApKtp CaCbMr b_K 2 CaCb dK — Ar ( KaKqb — KipKqa) + A ( KbKqa — KlpKqb) xv - CaCbMr — ArKqbCa+ApKqaCb — Ar ( KaKqb — KjpKqa) + A ( KbKqa — KjpKqb) Ka=dKci + K1 Kb=°’Kth + K, It has been shown that (4.57) and (4.58) reduce to the LPLF expressions (4.20) and (4.21) respectively, when the piston area ratio is unity and the actuator volume compliances, Ca and Cb are equal [29]. 4.2.4 Non-Linear Valve Flow Relation The models presented so far, which have commonly been used to design hydraulic servo actuator systems, all assume a linearized flow/pressure relationship. Although it has been concluded that this assumption is valid for a significant range of load pressures and valve openings, the accurate prediction of the large signal response of a hydraulic servo requires the non-linear flow/pressure characteristics of the secondary spool valve to be considered. To this end, several non-linear models have been developed. The following is Chapter 4. Modeling ofSystem Dynamics 38 a description of a particular model used to numerically simulate the response of the servoactuator system. From the manufacturer’s information, the response of the spool of the valve to an armature current can be represented by a first order lag, such that: x(s) = K1 (459) la(S) ‘rs+1’ Combining expressions (4.3-4.8) and solving for the rate of change of volume due to fluid compressibility (or the compliance flows): for; 0, ca = Kqa xJIF — psign(p — Pa) A,, Vy — Kip(Pa Pb)’ (4.60) = ‘qb XV.,)[Pb —p(sign(p —ps) — Ar Vy - K,P(pb Pa) — KlePb (4.61) for; <0, = Kq ; — p Sign(p — + A,, vy —K1,,(p3 — Pb), (4.62) = Kqb XjlP —pbsign(p3 — Pb) A,. Vy - KJP(pb —pa) — KIePb (4.63) 4.2.5 Non-Linear Actuator Compliance It has been shown that the change of actuator compliance with piston position has only a minor effect on the response of symmetrical actuators [11]. However, for asymmetrical actuators, the relative change of compliance for a change in piston position is magnified by the ratio of the area on each side of the piston. To investigate this phenomena, two alternative models were developed: one based on a flow causal relation, the other base on a pressure causal relation. The derivation of the flow causal compliance (FCC) model is based upon the following assertion: the pressure increase in a control volume is equal to the integral of the flow entering the volume divided by the compliance of the fluid in the volume. For control volume A this assertion is represented by the following expression: Chapter 4. Modeling ofSystem Dynamics 39 Pa(t) = f1(.)ca(t)dt (4.64) Since the compliance of the control volume (4.9) is a function of the piston position, (4.64) can be rewritten: Pa(t) = f3ej ( + (_))dt (4.65) Differentiating (4.65) and solving for the compliance flow yields (dropping the time function notation): q,,—_4(v,1+A(L_y)). (4.66) A similar expression can be derived for control volume B: qd, =5_Vbi+ArY. (4.67) Substituting (4.66) and (4.67) into (4.62) and (4.63) and solving for and yields the following control volume pressure relations for the FCC model: for; O, dPap Kqaxv]ps_paIsign(ps_pa)_Klp(pa_pb)+ApvY (4.68)dt e Va,+Ap(L—y) KqbXvgPb —pslgn(Pb Pt)lp(PbPa)ePbr1’y (4.69)di Vbi+ArY for; <0, J3 KqaxviJpa—ptsign(pa_p,)_Kp(pa_pb)+ApvY (4.70)dt e Vai+Ap(Ly) =1e KqbXViJIPs_PbSIgn(PPb) +K,P(Pb _Pa)jePb+4r’y (4.71)di Vj+Ay Chapter 4: Modeling ofSystem Dynamics 40 The derivation of the pressure causal compliance (PCC) model is based upon a somewhat different assertion: the rate of change of the volume of fluid compressed (the compliance flow) is proportional to the rate of change of the product of the compliance and the pressure. For control volume A this can be written as: q(t) = _(C(t)p(t)) (4.72) Substituting (4.9) for Ca (t), (4.72) can be rewritten: ca (t) = -—(v1+ A (L — y(t)))pa (t)) (4.73) Differentiating the expression in parenthesis of (4.73) yields (dropping the time function notation): dPa (va,+Ap(L_y)) A 474ca — 13e — Pa ( . ) For control volume B, this analysis yields: - ‘Pb (Vb, + Ary) Ar 475 1e +PbjjVY (. ) For the PCC model, substituting (4.74) and (4.75) into (4.62) and (4.63) and solving for and yields the following actuator control volume pressure relations: cit dt forx 0, d KqaXvIPs_PaI5ifl(Ps_Pa)_KP(Pa_Pb)+APVY1+PP) I3e e (4.76)dt V,1+A(L—y) KgbxvgIPbPtIsign(Pb_Pt)+Klp(Pb_Pa)+KIePb+Arv,1+) Pbp e (4.77)dt e Vbj+ArY Chapter 4. Modeling ofSystem Dynamics 41 for; <0, Kqa XvJIPa PtIS11l(Pa —pj—K1(Pa Pb) + Avy1 + I?) Pa_13 e (4.78) dt e 1+A(L—y) d KqbXvIPs_PbISin(Ps_Pb)+KP(Pb_Pa)+KlePb+ArVYl+) e (479) di Since, for all practical purposes, <<1 and <<1 expressions (4.76-4.79) can be simplified to the expressions derived using the FCC model (4.68-4.7 1). While the load relation (4.13) is valid while the piston is moving, it is desirable to augment this expression so as to ensure mathematical validity when the velocity of the piston is zero. The following expression can be used: F = Mr = ArPb — ApPa BrVy -F +F1 (4.80) where: for: < “cd ArPb ApPa +F1 and for: ArPb — ApPa + F1 I ‘‘cd =Fsign(v) (4.81) These equations (4.59-4.63, 4.68-4.71, 4.80-4.81) can be arranged as a series ofnon linear state equations which can be solved numerically to determine the system response to a series of inputs. The values of the system model parameters are presented in Appendix D. Chapter 4: Modeling ofSystem Dynamics 42 I,, E E >‘ > > 4- C) 0 a) > 4.3 Results 4.3.1 The Effect of the LPLF Linearization on the System Response A comparison between the step response predicted by the LPLF model and the response predicted by the non-linear LC model for a symmetric actuator with no external load is shown in figure 4.3. For the valve opening step, (0 <t < 125ms) the response LPLF Model * LC Model * ** 20 0 -20 -40 -60 -80 -100 * - Compliance relation linearized about most compliance piston position, Lc=62.5mm. * * -Area Ratio, R=l - Initial piston position, Lo=Lc. 50 100 150 200 2500 Time (mS) Figure 4.3. Comparison of velocity response predicted by LPLF model versus the velocity predicted by the LC model with piston area ratio R=1. predicted by the LPLF model is significantly more oscillatory than that predicted by the LC model. This can be attributed to the fact that the LPLF model is based upon a linearization about the valve null position, the position exhibiting the least damping. For the valve closing step (125 <t < 250ms) both models predict an oscillatory response, with the LC model exhibiting greater damping. Also, from this plot, the capability of the LPLF model to predict the steady-state velocity of a symmetrical actuator is exhibited. Chapter 4: Modeling ofSystem Dynamics 43 10 0 -20 -30 F6o -70 -80 4.3.2 The Effect of Piston Area Ratio the System Response A comparison of the velocity response of two actuator system: one with a symmetric actuator, the other with an asymmetric actuator is shown in 4.4. While the response of the two systems is similar, the steady state velocities are different and the systems oscillate at slightly different frequencies. 4.3.3 The Effect of Input Signal Amplitude on the System Response In order to examine the effect of the amplitude of the servovalve command signal on the response of the system, the frequency response predicted by the LPLF model was compared to the frequency response predicted by the WR model for various values of input armature current. The results are presented in figure 4.5. From this plot, it is evident that the system response exhibits much greater damping as the amplitude of the R=1 * R=1.8* * - Compliance relation linearized about most compliance piston position, Lc=80.4mm. - Initial piston position, Lo=Lc. 50 100 150 2000 250 Time (mS) Figure 4.4. Comparison of the effect of piston area ratio, R, on the velocity response of servoactuator, as predicted by the LC model. 0—50 —100 —150 —200 —250 —300 0 Figure 4.5. Comparison of the frequency response predicted by the LPLF model to that predicted by the [FR model at various servovalve armature current amplitudes. Chapter 4: Modeling ofSystem Dynamics 44 input signal is increased. Also, as the amplitude of the input signal approaches zero the frequency response predicted by the IFR model approaches that predicted by the LPLF model. This phenomena can be explained by the fact that the LPLF model’s damping term is linearized about the null flow position which is approached as the amplitude of the input signal is reduced. The system bandwidth (-3 dB) is approximately 30 Hz. 50 40 30 20 10 Fzq.iexcy (Hz) 200 50 100 150 200 Frqi.iezioy (Iz) Chapter 4. Modeling ofSystem Dynamics 45 10 0 -20 C) -40 a) > C 0 -60 0 -70 -80 4.3.4 The Effect of Irntial Piston Position on the System Response In this section, a comparison of step response predicted by the non-linear servo- actuator model utilizing a linearized compliance relation (LC Model) and that predicted by the non-linear model utilizing the flow-causal compliance relation (FCC Model) is examined. For this comparison, the compliance relation of the LC model is linearized about the critical piston position (i.e., the position yielding the maximum actuator compliance). In the first comparison, figure 4.6, the response predicted by the LC model is compared to the response predicted by the FCC model when the initial position is the critical position. In this case, the models compare quite favorably. In the second comparison ,flgure 4.7, the initial position of the actuator is varied from one limit to the other. In this case the FCC model predicts a much better damped response than the LC model. These plots indicate two things: i) the systems response tends to be less oscillatory LC Model*, Lo=80.4mm FCC Model, Lo=1 10mm FCC Model, Lo=1 5mm * - Compliance relation lineanzed about most compliance piston position, Lc=80.4mm. - Initial piston position, Lo=Lc. 50 100 150 2000 250 Time (mS) Figure 4.6. Comparison of the Velocity response predicted by LC model versus that predicted by the FCC model when the initial position differs from the LC linearization position. Chapter 4: Modeling ofSystem Dynamics 46 away from the most compliant position, ii) the LC model should be linearized about a piston position in which the actuator will operate. However, if the higher order dynamics of the actuator become significant, the LC model will not provide an accurate description of the system dynamics away from the linearization point. 10 0 [10 -20 -30 -60 0 -70 -80 0 50 100 150 200 250 Figure 4.7. Comparison of the response predicted by LC model versus that predicted by the FCC model when the initial position corresponds to the LC linearization position. 4.3.5 Effect of Coulomb Friction on the System Response The positioning system has been found to have significant non-linear friction characteristics (see Appendix C). The dominant non-linear characteristics can be modeled by coulomb friction. In order to investigate the effect of this type of friction on the system response characteristics, the system was simulated using the LC model for various amounts of coulomb friction (figure 4.8). From this plot we can observe that the quantity of coulomb friction in the system has a negligible effect on the steady-state velocity of the LC Model * - - - - - - - - - FCC Model ** * - Compliance relation lineazed A about most compliance piston V1 position, Lc. - ** - Initial piston position, Lo=Lc. Time (mS) Chapter 4: Modeling ofSystem Dynamics 47 system while the control spool is open, but contributes substantially to the damping of the system when the control spool is closed. Since the valve-closing condition is the most critical in terms of position control, this friction may lend a stabilizing effect to the overall system dynamics. 10 0 io -20 -30 C) . -40 a) C 0 -60 0 -70 -80 250 Figure 4.8. Comparison of the effect of coulomb friction, Fc, on the velocity response of the system as predicted by the LC model. 4.3.6 Validation of the Non-Linear Models A comparison between the measured response of the servoactuator system to a multi-step input and the response predicted by the LC and FCC models is presented in figure 4.9. From this plot we can observe that the LC and FCC models match the trend of the actual system reasonably well. However, the actual system response does not exhibit an oscillating response as predicted by the two models. This discrepancy can be attributed No Coulomb Friction* Fc=300 N.* Fc=600N.* * - Compliance relation linearized about most compliance piston position, Lc=80.4mm. - Initial piston position, Lo=Lc. 0 50 100 150 200 Time (mS) Chapter 4: Modeling ofSystem Dynamics 48 to unmodeled leakage across the piston seals which was not readily obtainable using the equipment available. 60 —. 40 E2: .2 -20 a) > .40 “--60 Time (mS) 500 Figure 4.9. Comparison of the velocity response predicted by the LC and FCC models to the actual response. Also, the actual system response is generally a bit slower than the response predicted by the models, particularly for the case when control valve is required to deliver more flow. While some of this sluggishness can be attributed to the aforementioned unmodeled piston seal leakage, the difference in response time exhibited between the valve opening and valve closing cases can be attributed to unmodeled flow non-linearities within the servovalve which cause its response time to be dependent upon the amplitude of the applied current. Note that the steps in the velocity response of the actual system near the zero velocity can be attributed to valve deadband. -80 ** Compliance relation linearized about most compliance piston position, Lc=80.4mm. ‘‘- Initial piston position, Lo=Lc, Fc=280 N. 0 100 200 300 400 Chapter 4: Modeling ofSystem Dynamics 49 4.4 Conclusions A model for the compliance of cylindrical actuators has been developed and applied. Both the supply system and the servo-actuator have been analyzed. The significant results of this analysis can be summarized as follows: • Neglecting valve dynamics, symmetric actuators respond as second order systems with damping which is proportional to the amplitude of the valve opening. • If the oil volume on each side of the piston is nearly equal, a second order model can also be used to represent the dynamics of an asymmetric actuator. • Coulomb friction at the load tends to damp the actuator oscillations when the valve is in the critical position. • A positioning system designed using linear theory should be analyzed for a range of piston positions. Chapter 5 Identification of Servo-actuator Dynamics for Control 5.1 Introduction While dynamic modeling is a useful tool for selecting hydraulic system components and determining the general order of the assembled system, the effects of valve deadband, hysteresis, stiction, spool leakage, and transport delay make the precise modeling of the dynamics of hydraulic systems difficult. To accommodate for these effects, a number of experiments can be conducted on the system in order to obtain a better representation of the system dynamics. These are known as system identification (SI) experiments and the practical result is an approximate linear model which can be used for the controller design. In this chapter, the three methods chosen to identif,’ the system dynamics are discussed. 5.2 Choice of Identification Signal Fundamentally, all system identification experiments involve two simple steps: 1) system input excitation and, 2) observation of the system response. Regardless of the type of SI experiment, the input signal used must be capable of exciting the relevant dynamics of the system. In order to choose an appropriate input excitation signal for a hydraulic system, some consideration must be given to practical limitations. For systems such as press brakes which position large masses, care must be taken to avoid potentially damaging excitation induced vibrations. For a closed center spool valve, an investigation of the Improved Frequency Response (IFR) model reveals two important phenomena: 50 Chapter 5: Ident/Ication ofServo-actuator Dynamicsfor Control 51 1. The response of a spool-valve controlled hydraulic cylinder becomes more damped as the amplitude of the input signal is increased. 2. The overall gain of the spool-valve is largest at the null flow position (all ports closed). These two phenomena have led Merritt [14] to suggest that the null flow position, should be used for controller design. Since the null-flow position is the critical operating point for a position control system, it is reasonable to assume that the smaller the deviation from the null-flow position, the more relevant the identified model will be to the task of controller design. Further, the IFR model presented earlier shows that an equivalent linear frequency response model of the non-linear servo-actuator can be achieved if a periodic wave form of constant amplitude is used. Given this, a low amplitude periodic wave form with a mean amplitude of zero would seem to be ideal. Unfortunately, valve deadband and hysteresis affect the valve dynamics most at the null position. Furthermore, the amplitude of the input signal has to be large enough so that the steady state flow gain identified is not biased by the valve-spool stiction. Parker and Desjardins [31] have suggested that a Pseudo Random Binary Sequence (PRBS) input signal of an amplitude of at least 10% of the maximum amplitude is sufficient to eliminate the effects of non-linear valve gain and spool friction. In practice, the amplitude of excitation signal required will vary with each system. Watton [29] has shown that the steady state flow gain (SSFG) of an asymmetrical actuator extending will not be equal to the SSFG of the same actuator retracting. Furthermore, if significant Coulomb friction is present at the load, and the input excitation signal chosen causes the load to change direction, the system will undergo an external load excitation. Such system excitations are undesirable during identification experiments. Therefore, if accurate estimations of the steady-state velocity response of the actuator are to be obtained, separate identification experiments should be conducted for the extending and retracting cases. ChapterS: Identfication ofServo-actuator Dynamicsfor Control 52 5.3 System Response to a Step Input In order to determine the velocity response of each actuator to a step change in valve command voltage, two sets of experiments were conducted: one in extension and one in retraction. For each experiment, an alternating step input signal of non-zero mean and not passing through zero voltage was applied to the servo-valve amplifier, and the velocity response of the load was measured. Each experiment was repeated for a number of differing input amplitudes. The response of the hydraulic system to a typical step change in valve input signal during extension is shown in figure 5.1. Approximate first-order system parameters obtained from the step response experiments are presented in table 5.1. Note that the extension response differs somewhat from the retraction response. From this table we can !160 _______ -14k -180 CommandE -16j -200 . > Velocity C.) -18220 > -240 .. -20 -260 -22 -280 -24 0 50 100 150 200 250 Time (mS) Figure 5.1. Velocity response of the left actuator to a step change in valve command voltage. Chapter 5: Identfication ofServo-actuator Dynamicsfor Control 53 see that while the steady state velocity and delay do not change significantly with the amplitude of the excitation signal, the rise time does. Further, the rise time for the extending cases varies somewhat from that of the retracting cases. Left Actuator Extending Valve Opening Valve Closing Step Amp (V) Gain (mmN.s) Rise (ms) Delay (ms) Rise (ms) Delay (ms) 0.100 7.93 12.0 5 9.2 4 0.150 9.53 15.0 4 7.9 4 0.250 9.48 12.5 4 7.8 3 Left Actuator Retracting Valve_Opening Valve Closing Step Amp (V) Gain (mmN.s) Rise (ms) Delay (ms) Rise (ms) Delay (ms) 0.100 6.75 15.4 4 7.3 3 0.150 7.74 16.8 4 8.4 3 0.250 7.53 13.4 4 7.2 3 Right Actuator Retracting Valve Opening Valve Closing Step Amp (V) Gain (mmN.s) Rise (ms) Delay (ms) Rise (ms) Delay (ms) 0.100 5.80 6.1 4 5.9 4 0.150 6.00 5.8 4 5.8 3 0.250 6.03 6.2 4 6.1 3 Right Actuator Extending Step Amp (7) 0.100 0.150 0.250 Gain (mmN.s) 3.95 4.61 6.11 Valve Opening Rise (ms) 11.2 7.5 7.6 Delay (ms) 4 3 3 Valve Closing Rise (ms) 8.9 6.4 6.1 Delay (ms) 4 3 3 Table 5.1. Delay, rise time and steady state gain obtained from step response experiments. Chapter 5: Identfication ofServo-actuator Dynamicsfor Control 54 5.4 Frequency Response Experiments 5.4.1. Experiment Description In order to determine the relevant dynamics of the system, frequency response experiments were conducted on each actuator for the extending and. retracting cases. For each actuator, an input signal was applied to servo-valve input, and the velocity signal was measured. A PRBS input signal was selected because of the excellent signal to noise ratio effected in the velocity transducer. To minimize the effects of hysteresis, deadband, and coulomb friction loading, a DC offset was added to the signal to ensure the valve did not pass through the null position, reversing the direction of the load. Because the WR model indicates that the frequency response of such hydraulic systems varies with the amplitude of the excitation signal, these experiments were conducted for a range of excitation signal amplitudes. The data from each experiment was analyzed using a transfer function analysis. 5.4.2. Experimental Results The magnitude and phase responses of the left actuator in extension is shown in figures 5.2 and 5.3. Note that the peaks at 60Hz for the lower amplitude signals are due to electrical noise observed in both the command signal as well as the transducer signal. From these results, a few observations can be made: 1. As the amplitude of the excitation signal is decreased, the steady-state gain of the system decreases as well. This can be attributed to the increased effect of valve component stiction at low valve actuation forces as discussed in [31]. 2. The band pass frequency of the velocity ioop is approximately 30 Hz. This compares favourably with the band pass predicted by LPLF model. Chapter 5: Identfication ofServo-actuator Dynamicsfor Control 55 82 76 :. 70 w z 64 58 52 0 20 40 60 80 100 120 140 160 180 Frequency (Hz.) —0-—— 50 mV —0-—— 100 mV —&—— 200 mV 400mV 600 mV 800 mV Figure 5.2. Magnitude response of the right actuator in extension determined for a variety of input signal amplitudes Frequency (Hz.) 10 20 30 Figure 5.3. Phase response of left actuator in extension determined for a variety of input signal amplitudes 0 0 40 50 -135 —0-—— 50 mV —0—-— 100 my —&--— 200 mV 400mV 600 mV 800 my Chapter 5: Identfication ofServo-actuator Dynamicsfor Control 56 3. There exists a dynamic mode due to load dynamics near 140 Hz. As the amplitude of the input signal is increased, the frequency of this mode decreases. This frequency decrease can be attributed to an increase in valve damping as predicted by the IFR model. 4. At low amplitudes of input signal, the system exhibits a dynamic mode near 100 Hz. Since this mode disappears at higher amplitudes, it is likely caused by the higher order dynamics of the valve spool which, like the hydraulic cylinder, is underdamped only for small valve openings. 5.5 Parametric Identification 5.5.1 Theory In order to determine a more precise description of the dynamic system model, a number of parametric identification experiments were conducted. Due to its ability to give unbiased estimates under less restrictive conditions, the Method of Instrumental Variables was chosen over the Least Squares Method to determine the model coefficients. A brief description of theses identification schemes follows. Soderstrom and Stoica [32] present excellent descriptions of the Least Squares (LS) method as well as the Instrumental Variables (IV) method. Since the IV method is based on the LS method, it is useful to introduce the LS method first. 5.5.1.1 Least Squares Method The discrete time transfer function for the open-loop velocity response of a hydraulics servo actuator has the form: vy(q’) = B(qj (5.1). u(q ) A(q where Chapter 5: Identfication ofServo-actuator Dynamicsfor Control 57 B(qj = b0 +blq+...+bflq (5 2) A(qj =1 +a1q’ + Using the Auto Regressive Moving Average (ARMA) notation, (5.1-5.2) can be reformulated as a time series v(k) =b0u(k) +b1u(k — 1)+. . +bflbu(k — flb) —a1v(k—i)—... —ay (k — n) (5.3) or v(k) = çoT(k)O (5.4) where: q(k) =(u(k)...u(k—nb) -v(k— 1)... Vy(kfla)) (5.5) and = (be.. .b a1.. .a )T is the parameter vector ofvariables describing the ‘true’ dynamics of the system. The goal of the parametric identification is to obtain the parameter vector 0 from a data set of N measured regressors (5.7) pT(N) and the measured output v(1) (5.8) v(N) If a set of independent measurement errors with zero mean and variance ?L,2 (white noise) exist such that: ChapterS: Identfication ofServo-actuator Dynamicsfor Control 58 e(1) v(1)—(pT( )O E = (5.9) e(N) v(N)—çoT( )O the least squares estimate of 0 (denoted as 0), is that which minimizes the sum of the squares of the measurement errors: V(O)=.te2(k) (5.10) It has been shown that if the loss function, (5.10) has a unique minimum, this minimum occurs for 0= (5.11) Note that in order for this minimum to exist, co(k)pT (k) must be non-singular. For this reason, care must be taken to choose an excitation signal which is ‘persistently exciting’. In simpler terms, the input signal used to construct the data set must be capable of exciting the particular dynamics of the system one wishes to identify. Given that the above conditions hold, 0 has been shown to converge to 0 for large N. However, if the elements of the measurement error vector s are not linearly independent, 0 has been shown to converge to a biased estimate of 0. Soderstrom and Stoica [32]present this phenomena as follows. If the true response of the system is given by v(k) = çoT (k)O÷ w(k) (5.12) where w(k) is stochastic disturbance, the difference between the true parameters and the estimated parameters can be written as: N — = o(k)Tco(k)] [41jco(k)Tw(k)] (5,13) Chapter 5: Ident/Ication ofServo-actuator Dynamicsfor Control 59 As the number of samples N tends towards infinity, equation (5.13) will not converge to zero unless the expectation Eço(k)Tw(k = 0 (5.14) If a correlation exists between the measurement error and the regression vector, 5.14 will fail. While this bias may be small for systems with a high signal to noise ratio, other methods, such as the Instrumental Variable method have been shown to provide unbiased estimates in the presence of correlated measurement errors. 5.5.1.2 Instrumental Variables Method The IV method augments the LS method by the introduction of a vector of signals or ‘instruments’ Z(k) = (u(k). . .u(k — b) (k-).. . (k - ha ))T (5.15) which are uncorrelated to the disturbance w(k). Note that the instruments: {h1y(1_)J1yQt_uia)] (5.15) are determined from the measured output. Thus the function to be minimized becomes: v(e) = .t(z(k)T e(k))2 (5.16) and the IV estimate of the parameter vector is: N N T = [z(k)T(k)] [Z(k)v(k)] (5.17) The determination of the instruments involves filtering of the measured output and therefore requires an apriori estimation of the stochastic disturbance term w(k). Typically this estimation is performed on an independent data set using the LS method. For this reason, IV methods require a significant amount more computation time than simple LS methods. ChapterS: Ident/1cation ofServo-actuator Dynamicsfor Control 60 The approximately optimal IV method was chosen to identif,’ the dynamic system parameters in this experiment. For a detailed description of this method see Soderstrom and Stoica [32]. 5.5.2. Experiment Description For this experiment, a small amplitude excitation signal was applied to the servo- valve amplifier. This excitation signal as well as an output voltage from the velocity transducer was sampled at 1 millisecond intervals. The collected data was analyzed using Matlab’s System Identification Toolbox [33]. Experiments were conducted for each actuator extending and retracting. 5.5.2.1 Choice of Identification Signal Of the two types of signal considered, pseudo random noise and a pseudo-random binary sequence (PRBS), the PRBS was chosen because of the superior signal to noise ratio effected in the velocity transducer. A DC bias (as in the frequency response experiment) was added to the PRBS to prevent reversing the direction of the load. In order to determine the lowest amplitude signal capable of minimizing the non-linear valve characteristics a number of frequency response experiments were conducted using excitation signals of differing amplitudes. The open-loop velocity gain predicted for each amplitude of input signal applied for the left actuator in extension is shown in figure 5.4 At low input signal amplitudes, the velocity response is dominated by valve non linearities. As the amplitude of the excitation signal is increased, more consistent estimates of the open loop gain are obtained. From these experiments, an input signal amplitude of 600mV, or 3% of the maximum valve input, was chosen. This amplitude, the Chapter 5: Ident/ication ofServo-actuator Dynamicsfor Control 61 lowest input signal amplitude capable of overcoming the valve stiction, was within the range of input signals required during the forming operation. 85 0 0 U84 U ‘—83 0 820 0 . 81 C) .2 80 > 0. o 0278 e .77 °76 75 0 100 200 300 400 500 600 700 800 PRBS Input Signal Amplitude (my) Figure 5.4. Effect of excitation signal amplitude on the steady-state gain predicted by the frequency response experiments (left actuator extending). 5.5.2.2 Model Structure In order to effectively determine the dynamics system parameters using the parametric identification techniques described above, information about the model structure must be known. From the frequency response analysis, the third order system dynamics predicted by the IFR model can be observed: a dominant first order lag due to the valve dynamics combined with slight oscillatory mode due to the actuator/load dynamics. From the step response analysis, a first order response with 3-4 ms delay can be observed. ChapterS: Identfication ofServo-actuator Dynamicsfor Control 62 5.5.2.3 Data Analysis For each situation (left actuator extending, left actuator retracting, right actuator extending, right actuator retracting) a number of trials were conducted. For each trial the servo-valve command and the voltage across the velocity transducer were sampled at 1 millisecond intervals. The 4000 point data sets collected were split into two 2000 point data sets: the first was used for estimation of the dynamics system parameters, the second to validate the estimated model. For each identification data set, the ‘approximately optimal’ IV method was applied for a number of model stmctures of first and third orders with two to five milliseconds of delay. Using the validation data set, the response for each identified model was compared to the actual system response. A prediction error function consisting of the sum of the squares of the error between the actual system response and the simulated response was computed to compare the relative goodness of fit for each model. 5.5.3. Experimental Results 5.5.3.1 Model Selection A comparison of the loss functions for each structure of model fit to the data is presented in Table 5.2. The structures corresponding to the minimum value of loss function are shown highlighted. The identified model parameters for the candidate structures are presented in Table 5.3. Note that model parameters for two alternate candidate structures pertaining to the extension cases have been included. ChapterS: Ident/Ication ofServo-actuator Dynamicsfor Control 63 Model B(c[1) = + Loss Function A(q’) ‘ 1+aq+. .+aq’ Left Actuator Right Actuator Extending Retracting Extending Retracting 1 1 2 2065 2376 814 620 1 1 3 718 1394 603 492 1 1 4 1084 2133 621 1052 1 1 5 1734 2936 1083 2553 1 1 6 2036 3021 1876 3408 3 3 2 1495 3275 260 1482 3 3 3 234 219 353 149 3 3 4 54 80 118 112 3 3 5 205 155 548 298 3 3 6 467 605 1104 627 Table 5.2. Comparison of the loss ftinctions computed for a variety of model structures. 5.5.3.2 Model Validation In order to ensure that the identified parameters are reasonably accurate descriptions of the relevant system dynamics, two sets of validation experiments were conducted: a frequency response comparison and step response comparison. For the first comparison, the frequency response predicted by each of the candidate models for each case was compared to the frequency response determined directly from sampled data. The measured frequency response of the left actuator extending is compared to that predicted by the identified candidate models in figure 5.5. Although overestimating the steady state gain, the first order model with three sampling delays gives an excellent description of the process dynamics for frequencies between 10 and 65 Hz. The first order model with four delays accurately predicts the steady state gain, but generally over estimates the response of the system for frequencies between 10 and 110 lIz. The third order model with four delays is capable of representing the dynamics mode due to the Chapter 5: Ident/Ication ofServo-actuator Dynamicsfor Control 64 load/actuator dynamics, but the amplitude at the resonant mode is significantly overestimated. Case ‘1a b d [A] = 1 +aq’+•.•+aq [B] = b0 + Left Extension 1 1 3 [1, -0.9197] [704.9, 920.6] 1 1 4 [1, -0.8712] [1572.7, 702.44] 3 3 4 [1, -2.057, 1.885, -0.76611 [1890.6, -1813.5, 1120.4, -206] Right Extension 1 1 3 [1 ,-0.91 70] [738.0, 417.1] 1 1 4 [1, -0.8609] [982.3, 505.3] 3 3 4 [1, -1.9342, 2.4757, -1.3156] [1335.8, -361.65, -308.06, 1760.7] Left Retraction 1 1 3 [1, -0.9022] [699.2, 657.3] 3 3 4 [1, -2.0982, 1.9283, -.7785] [448.5, 1204.5, -1676] Right Retraction 1 1 3 [1, -0.8558] [1111, 558.3] 3 3 4 [1, -2.1225, 2.0106, -0.8145] [2131, -2808 2361 -818.4] Table 5.3. Model Parameters determined from parametric identification The measured frequency response of the right actuator extending is compared to that predicted by the identified candidate models in figure 5.6. The first order model with three sampling delays tends to overestimate the response of the system from steady state to 15 Hz, while underestimating it everywhere else. The first order model with four delays accurately predicts the steady state gain, gives a reasonably accurate description of the response of the system for frequencies between 0 and 100 Hz. The third order model with four delays gives a reasonable description of the system dynamics from steady state to 90 Hz, although it incorrectly predicts the dynamic mode. The measured frequency response of the left actuator retracting is compared to that predicted by the identified candidate models in figure 5.7. While both models accurately predict the steady-state gain of the system, the third order model with four delays better represents the dynamics throughout the range of frequencies. Note however Chapter 5: Identification ofServo-actuator Dynamicsfor Control . 81 I:: > 72 0 20 40 60 80 100 120 140 160 Frequency (Hz.) 65 Figure 5.5. A comparison of the experimentally determined frequency response to that predicted by the identified models: Left actuator extending. 87 84 81 >72 69 66 0 20 40 60 80 100 120 140 160 180 Frequency (Hz.) Figure 5.6. A comparison of the experimentally determined frequency response to that predicted by the identified models: Right actuator extending. 87 84 _ Experimental — - — First Order Model: d=3 — — First Order Model: d=4 Third Order Model 69 66 Experimental — - — First Order Model: d=3 — — First Order Model d=4 Third Order Model ChapterS: Iden4/Ication ofServo-actuator Dynamicsfor Control 66 the third order model dramatically overestimates the frequency response of the dynamic mode. \ Figure 5.7. A comparison of the experimentally determined frequency response to that predicted by the identified models: Left actuator retracting. The measured frequency response of the right actuator extending is compared to that predicted by the identified candidate models in figure 5.8. The first order model gives a good description of the response of the system from steady state through to 85 Hz. The third order model gives a good description of the system dynamics from steady state up to 60 Hz, but beyond this gives an poor estimate of the dynamics and incorrectly predicts the of the dynamic mode. For the step response comparison, the response predicted by each of the candidate models for each case was compared to a measured response from a data set independent of that used in the identification. 87 84 . 81 U, 78 4-’ 0 0 >72 69 66 Experimental — - — First Order Model: d=3 Third Order Model o 20 40 60 80 100 120 140 160 180 Frequency (Hz.) Chapter 5: Ident/1cation ofServo-actuator Dynamicsfor Control 67 87 84 . 81 ‘72 69 66 0 20 40 60 80 100 120 140 160 180 Frequency (Hz.) Figure 5.8. A comparison of the experimentally determined frequency response to that predicted by the identified models: Right actuator retracting. The measured response of the left actuator extending is compared to that predicted by the identified candidate models in figure 5.9. While none of the models tested constitutes an excellent fit, the third order model seems to give the best representation of the dynamics and the steady-state gain. Both first order models give reasonable fits with neither being outstanding at all points on the comparison. The measured response of the right actuator extending is compared to that predicted by the identified candidate models in figure 5.10. Again, while none of the models tested constitutes an excellent fit, the third order model seems to give the best representation of the dynamics and the steady-state gain. The first order model with four delays predicts a response very similar to the third order model, but without the higher- order dynamics. The first order model with three delays tends to overestimate the steady- state gain while underestimating the rise time of the system. Experimental — - — First Order Model: d=3 Third Order Model Chapter 5: Ident/Ication ofServo-actuator Dynamicsfor Control 68 .—.. -5 -15 -20 >-25 -30 0 200 Figure 5.9. Comparison of the response of the measured system to that predicted by the parametrically identified models: left actuator extending. The measured response of the left actuator retracting is compared to that predicted by the identified candidate models in figure 5.11. In this case, the system is exhibiting third order or higher dynamics. Although a slight DC bias seems to exist, the third order model gives the best fit. The response predicted by the first order model with three delays seem to fit the first order dynamics satisfactorily despite the same DC bias. The measured response of the right actuator retracting is compared to that predicted by the identified candidate models in figure 5.12. As was the case with the left actuator retracting, the system response is clearly third order or higher with a higher mode frequency of approximately 135 Hz. While generally underestimating the amplitude of the higher order oscillations, the response predicted by the third order model does fit the actual response reasonably well. Again, the response predicted by the first order model with three delays seems to fit the first order dynamics as well as can be expected. 0 40 80 120 160 Time (mS) Chapter 5: Identfication ofServo-actuator Dynamicsfor Control 69 0 I: >-20 -25 Figure 5.10. Comparison of the measured system to that predicted by the parametrically identified models: right actuator extending. 15 E Eio Actual Response First Order Model Third Order Model Actual First First Third Response Order Order Order Model: Model: Model d=3 d=4 0 40 80 120 160 Time (mS) 200 25 20 0 40 80 120 160 0 -5 200 Figure 5.11. Comparison of the measured system to that predicted by the parametrically identified models. Left actuator retracting. Time (mS) Chapter 5: Identfication ofServo-actuator Dynamicsfor Control 70 15 E E [0 5 0 Actual Response - — — — — First Order Model Third Order Model Figure 5.12. Comparison of the measured system to that predicted by the parametrically identified models: right actuator retracting. 20 0 40 80 120 160 Tirne(mS) 200 ChapterS: Identfication ofServo-actuator Dynamicsfor Control 71 5.6 Conclusion From the identification experiments a number of conclusions can be drawn: • At very small input signals, the steady state gain of the servo-actuator drops significantly and the dynamics of the load are not significant. • As the amplitude of the input signal is increased, the steady-state gain of the servo-actuator increases and an oscillatory mode at 13 0-150 Hz due to the load dynamics is more apparent. • The system response of the servo-actuators in the extending cases can be represented adequately by a first order model. • The response of the servo-actuators in the retracting cases can be characterized by a dominant first order lag combined with a low amplitude dynamic mode at approximately 13 0-150 Hz. Chapter 6 Coordinated Motion Control of Press Ram 6.1 Introduction The objective of the efforts described in this chapter was to design and implement a practical control scheme which accomplished smooth positioning and orientation of the punch with respect to the die throughout the bend cycle. In order to facilitate coordinated motion between the two actuators, a control scheme capable of matching the closed ioop positioning dynamics of the axes was chosen. Model parameters presented in the previous chapter were used exclusively for the controller design. Given the cost sensitivity of manufacturing a machine tool for the industrial market, efforts to improve the positioning performance focused on software rather than on hardware. Although velocity and pressure signals were available and could have improved the robustness and performance of the system, only position feedback was used so that the incremental cost of CNC implementation would be less. The results indicate that as long as the control law is properly designed, the system behaves satisfactorily without pressure and velocity feedback signals. In this chapter, the term ‘following error’ refers to the difference between the desired position and the actual position of either axis during movement at a particular velocity. The term ‘tracking error’ has been used to describe the difference between following error of each axis. The term ‘steady-state-positioning error’ refers to the difference between the desired position and the actual position of either axis when the desired position is not changing and system dynamics are not prevalent. 72 Chapter6: CoordinatedMotion Control ofPress Ram 73 6.2 Motion Control 6.2.1. Objective The motion control task attempted to achieve two fundamental objectives: 1. less than 0.25 mm tracking error during the forming process 2. a steady-state positioning error of less than 0.0 15 mm under forming load. 6.2.2. Velocity Profile A plot of the desired punch velocity profile for a brake-forming cycle is depicted in figure 6.1. The profile consists of four segments: 1. rapid approach to the clearance plane above the work-piece 2. brake-forming operation at feed-rate 3. dwell in bend 4. rapid retract to the start position Time (Seconds) Figure 6.1. Velocity profile for typical bending cycle. The performance of the positioning system during the bend and dwell operations determines the precision of the press. In these cases, the actuators are either extending or 60 40 ‘ 20 E E 0 0 e > -20 -40 -60 Apph [ 4\\ \ \/____Dwell 0.0 1.0 2.0 3.0 4.0 5.0 Chapter6: CoordinatedMotion Control ofPress Ram 74 holding their position against a forming load. For these reasons, the actuator models corresponding to the extension cases were used for controller design. 6.2.3. The Process to be Controlled The results of the identification experiments have shown that the response of the actuators can be characterized by a first order lag combined with an under damped mode at approximately 140 Hz. However, for the extension cases, the higher order dynamics are less prevalent especially for the small input signal amplitudes (100-150 mY) used during the forming operation. Moreover, the first order models generally provide good descriptions of the dynamic system performance at frequencies up to 100 Hz. Furthermore, considering the computational complexity required to compensate for higher order dynamics, the first order models were chosen for initial design efforts. 6.2.4 Control Scheme 6.2.4.1 Introduction In order to achieve low tracking error, a control scheme which would facilitate matching the dynamics of each positioning system was desirable. In order to obtain low steady-state position error, the closed loop positioning system would require high gain to maximize load force disturbance rejection. Since the open-loop system dynamics contained several delays, a delay compensation scheme was deemed necessary to allow high controller gain. A pole-placement control scheme, similar to that used by Watton [22] was chosen because it utilizes a performance-based design procedure and is capable of compensating for process delays. A block diagram of a pole-placement controlled system is shown in figure 6.2. The process to be controlled is represented by polynomials of open loop system zeros B and the poles A. The controller consists of a feed-forward filter T, a feed-back filter S, Chapter6: CoordinatedMotion Control ofPress Ram 75 controller poles R and an estimation filter (or observer) A0. These four filters are selected such that the controlled system responds according to a chosen response: Figure 6.2. Block diagram of a pole-placement controlled system. y BT Bm (6.1)AR+BS Am where 4 contains the poles, and Bm the zeros of the desired closed loop transfer thnction (CLTF). A detailed description of the pole-placement design process can be found in [34]. A description of how the pole-placement control scheme was adapted to the position control task follows. 6.2.4.2 Design of the Pole-Placement Controller The open-loop velocity response of each of the servo-actuators in the extending cases can be represented by the following discrete time transfer fhnction: Yrej Process y Controller Chapter6: CoordinatedMotion Control ofPress Ram 76 vy(q_1) — qd (b0’ +b1’cf’) F counts u(q’) — 1+a’q [vsec (6.2) where d is the number of delays, b1’, b2’, and a are the identified system parameters, and 1 count/second equals 0.005mm/s of actuator velocity. Adding an integration term to this expression yields: y(q’) — q’d (b1 +b2q’) F counts u(q1) — (1+aq1+a2q2) [ (6.3) or, for convenience, using the forward shift operator: y(q) (b1q+b2) Ecounts 6qna(q2±q1±) L v (.4) The desired system response was chosen to be that of a damped second order system with a delay equivalent to the open-loop system delay: y(q) — B(q) — (b1q+b2)0 [counts yj(q) — 4(q) — qfld (q2 +a1q’ + am2) [count (6.5) where y(q) is the reference position, and b is gain chosen such that the position control loop has a unity steady state gain: b 1+ami+am2 66mO b1+b2 Since it is desirable to use only the position transducers for process feedback, an observer A0 (q) is required to estimate any higher order feedback terms required. In order to ensure a causal design analysis, Astrom and Wittenmark [34] have shown that the following conditions must be met: deg4(q) 2degA(q) — degA (q) —1 (6.7) degR(q) degS(q) (6.8) degR(q) degT(q) (6.9) Chapter6: CoordinatedMotion Control ofPress Ram 77 degS(q)< degA(q) (6.10) degR(q) deg4(q)+degA(q)—degA(q) (6.11) Therefore, given (6.7) the order of the observer poiynomial was chosen: deg4(q)=n+1 (6.12) Since the linear encoders provide low-noise position measurements, the observer polynomial poles were placed at the origin for the fastest estimation convergence: (6.13) In order to meet causality constraints (6.7-6.11) the controller polynomial orders were chosen such that: degR(q)=degS(q)=degT(q)=n +1 (6.14) Therefore, the controller polynomials were chosen to be: R(q) = + qfld +• + r, (6.15) S(q) =s0q’’1+s1q+• •+sq’ + s,.1 (6.16) T(q) =t0q”’ +t1q” +•. +tflq + tnd+I b0A(q) = (6.17) Solving the Diophantine equation symbolically for the controller parameters (Appendix B) yields: f(nd =i), ri=amiai (6.18) ( “2 ‘ “2 = —a1-)r± (6.19) a1 — a2 — a0 — 2 b1 (6.20) (6.21) Chapter6: CoordinatedMotion Control ofPress Ram 78 s. =0, i = 2,3,.. •nj +1 (6.22) f(nd >i), r=amiai (6.23) r2=am (6.24) for] > 2,j d’ = —(a1i+a2,) (6.25) (( b I a1---—a2ji+a2--?d_I I b b — a2 — a0 b1 = (+i +a1i (6.27) a2r = — nd+1 (6.28)b2 s, = 0, 1 = 2,3,.. d + 1 (6.29) 6.2.4.3 Controller Implementation Using the forward shift notation, the pole-placement control law can be represented by: R(q)u(q) = T(q)y,.j(q) — S(q)y(q) (6.30) Rewriting 6.30 using discrete time series notation yields: (6.31) where n, n and r are the respective orders of the feed-forward, feed-back and controller polynomials. Given that in the applied case, n, = 0, n8 = I and ii,. = + 1, the control law: I u(k) =t0y(k) —s1y(k — i) — ru(k — j) (6.32) 1=0 j=I Chapter6: CoordinatedMotion Control ofPress Ram 79 would require (d + 4) multiplication operations and (d + 4) add operations per ioop closure. Note that this implementation requires precise representation of parameters t, s and s1 in order to avoid steady-state error due to numerical round-off. For this reason, the following simplification was developed. The steady state gain of a pole-placement controlled system can be represented by the following relation: bmy(l) = iO = (6.33) ‘) am i=O j i k=O jo For systems with an inherent integration such as DC motor positioning systems or servo- hydraulic positioning systems =0 Therefore, (6.33) reduces to: bm = ° = ° (6.34) Yrej(’) amj which indicates that the steady-state gain of the pole-placement controlled system can be scaled by appropriate choices of S and T. Furthermore, if the desired steady state gain is unity, then: (6.35) For ,z =0 and ,z3 = 1, 6.35 can be rewritten: Chapter6: CoordinatedMotion Control ofPress Ram 80 = to, (6.36) Substituting the relation: Ay(k) = y(k) —y(k— i) (6.37) into the control law (6.32) and rearranging yields: (1 \ u(k) =t0yrej(k) — —1)—s0Ay(k) — ru(k — j) (6.38) 1=0 j=1 Using (6.36), the control law can be further simplified to yield: nd+1 u(k) = tO(Yref(k) — y(k —1)) —s0iXy(k) — r1u(k — j) (6.39) which, in comparison with (6.32), requires only (d + 3) multiplication operations and (d + 3) add operations per ioop closure. This control law implementation (6.39) accomplishes the following: • it eliminates the possibility of the steady-state error due to numerical round-off during scaling computations while guaranteeing appropriately scaled output • it reduces the number of control parameters that need to be represented, lessening memory requirements • it reduces the precision in which t0 and s0 need to be represented, further lessening memory requirements • it reduces the number and complexity of operations required for implementation This control algorithm was implemented on two axes of the HOAM-CNC using Intel 80196 assembly language. A pair of overdamped CLTF poles (4) were chosen such that the system response was fast enough to satisfy the following-error constraints while not deviating significantly from the control valve dynamics. The control law parameters as well as the process parameters used for design are shown in tables 6.1 and 6.2. Chapter6: Coordinated Motion Control ofPress Ram 81 Left Axis (Yl) Pole-Placement Design Parameters [B] b0 +b1q’ 1573 702 [A]= 1+aq+a2q 1 -1.871 0.871 [Am ]=1+a1q +a2q 1 -1.429 0.4724 [T]=t0 2.36 [S] = S0 +s1q 21.62 -19.260 [R]= i+,q1.. 1 0.443 0.429 0.418 0.408 -0.064 Table 6.1. Process models and control law parameters for left axis controller. Right Axis (Y2) Pole-Placement Design Parameters [B] = b0 +b1q 982 505 [A]=1+a1q’+a2 1 -1.861 0.861 {A]=1+aq’+a 1 -1.429 0.4724 [T]=t0 2.85 [SJ=s+s1q 21.07 -18.225 [R] = t +rq’+.. +tq’ 1 0.398 0.373 0.352 0.335 -0.076 Table 6.2. Process models and control law parameters for right axis controller. 6.3 Positioning System Performance The equipment setup used for the performance experiments is depicted in figure 6.3. The object of these experiments was to evaluate the positioning system’s performance in terms of dynamic response, position deadband, following error, tracking error and the effects of load force disturbance (compliance). Chapter6: CoordinatedMotion Control ofPress Ram 82 6.3.1 The Dynamic Response of the Ram Positioning System A simple step response test was conducted to verify the dynamics response of each axis. For this test, each axis was given a reference command signal consisting a series of steps. The step amplitude was set as large as possible while avoiding controller saturation. The command signal and the measured position response are shown in figure 6.4. Negative steps represent actuator extension. The results of this test indicate that rise time of each axis in extension (for this amplitude of excitation) is between 11 and 12 ms Furthermore, the extension response of each actuator exhibited no overshoot. The left actuator was observed to overshoot approximately 0 -2% occasionally during retraction. This slight overshoot was deemed acceptable given that it does not occur during the forming operation. Moreover, the effect of any response overshoot would be mitigated somewhat by velocity profiling. Figure 6.3. Press setup for positioning system performance experiments. Chapter6: CoordinatedMotion Control ofPress Ram 83 0.10 0.00 -0.10 E -0.20 C 0 0. 0 a- -0.40 -0.50 -0.60 Figure 6.4. The response of left and right axis of the positioning system to a series of step changes in command position. Yl: left axis; Y2: right axis, Yref: reference command. For comparison purposes, a traditional PD control algorithm was implemented and manually tuned. The response of the PD controlled system for one of the better tunings is shown in figure 6.5. From the following observations can be made: • The rise time is between 20 and 30 ms • Both axis exhibit between 6 and 9% overshoot during the extension and retraction. • The settling time for each axis is greater than 300 ms 6.3.2 Determination of Positioning System Dead Band The performance of hydraulic actuators is critically dependent on the performance of the valve. Valve hysteresis and spool stiction can cause excessive deadband in the positioning system. In order to determine the effects of these phenomena, each axis of the positioning system was given an alternating series of small steps in reference position. Both the reference and the actual axis positions were measured at two millisecond 0 50 100 150 200 250 300 350 400 Time (mS) Chapter6: Coordinated Motion Control ofPress Rain 84 0.10 0.00 -0.10 E -0.20 C 0 -0.30 0 0 -0.40 -0.50 -0.60 700 800 900 1000 Figure 6.5. Response of PID controlled positioning system to step changes in reference position. intervals. The results are presented in figure 6.6. These results indicate that the system deadband is smaller that one basic length unit (0.005 mm). 6.3.3 Detenmnation of Controlled Motion Performance One of the most important measures of a positioning system’s performance is its ability to follow the desired motion profile. A measure of this following capability, the velocity error constant (VEC) is defined as the ratio of the following error to the desired velocity of the move. While the following error for either axis of the CNC press brake is not critical, the tracking error is, for this controls the orientation of the tooling. The tracking error will be proportional to difference between the VEC of each axis. Hampering a press brake’s ability to maintain constant tooling orientation is the tendency for the actuators to be unequally loaded. This occurs frequently because CNC press brakes are often used to perform a number of bends using a number of sets of 0 100 200 300 400 500 600 Time (mS) Chapter6: CoordinatedMotion Control ofPress Ram 85 Yl -----Y2 Lj L .... ... 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Time(S) Figure 6.6. Response of positioning system to small changes in reference position. tooling setup along the press bed. In the absence of coupling (be it mechanical, hydraulic, or controller coupling) between the axes, close axis tracking is achieved by matching the axis dynamics and ensuring both axes are stiff enough not to be affected significantly by disturbance loads. In order to evaluate the performance of the positioning system with regards to these measures, two experiments were conducted. For the first experiment, a typical machine cycle motion profile was used to determine the VEC. For the second experiment, the same motion profile was used to perform a forming operations on several workpieces of differing geometry. 6.3.3.1 Motion Profile: No Forming Operation The object of the motion profile experiment was to determine the ability of the positioning system to track the desired motion proffle. For this experiment, each actuator was given a motion profile representing a typical bending operation: rapid approach, feed- 0.05 0.04 0.03 0.02 E .- 0.01 0 -0.01 -0.02 -0.03 -0.04 Chapter6: CoordinatedMotion Control ofPress Ram 86 rate forming operation, dwell, and rapid retract. Sample results of this test are shown in figures 6.7 and 6.8. Note that the rough tracking response during the rapid move is due to an error in the linear interpolation routine. The velocity error constants for each axis are presented in table 6.3. From these results the following observations can be made: • The following error during the feed-rate move is less than 0.1 mm • The velocity error constants for each of the axes are not equal. This is due to a slight mismatch of axis response times. Furthermore, for either axis, the VEC for extending case differs from that of the retracting case. • The steady-state position error is not greater than 0.005 mm (one BLU) during the dwell operation. • The relative position difference between each axis (the tracking error) is not greater than 0.015 mm (3 BLU) during the feed operation. Case: Left Actuator VEC Right Actuator (ms) VEC(ms) Extension 11.8 12.3 Retraction 13.4 11.9 Table 6.3. Velocity Error Constants for each axis as determined from motion profile tests: no forming loads. Chapter6: CoordinatedMotion Control ofPress Ram 87 1.25 1.00 0.75 E E 0.50 I 0 0.25 C 0 0.00 0 a- -0.25 -0.50 -0.75 Figure 6.7. Response of positioning system to motion profile: no forming loads. 0.10 0.08 E0.06 0.04 0.02 0.00 -0.02 2.8 3.2 3.6 4.0 4.4 Time (mS) 4.8 5.2 5.6 Figure 6.8. Plot of the absolute and relative tracking error of each positioning system: no forming loads. 70 60 50 E 40 E C 0 30 o 0 a 20 10 0 5.0 6.0 7.0 8.00.0 1.0 2.0 3.0 4.0 lime (mS) Chapter6: CoordinatedMotion Control ofPress Ram 88 6.3.3.2 Motion Profile: With Forming Operation The object of this experiment was to determine the stiffness of the positioning system: that is, the ability to maintain tooling orientation and achieve the desired final punch penetration in the presence of typical forming loads. As with the previously described experiment, each actuator was given a motion profile representing a typical bending operation: rapid approach, feed-rate forming operation, dwell, and rapid retract. The tooling consisted of a 30 degree punch and a 90 degree die which were aligned and secured to the press. The respective centres of the punch and die were located 500 (mm) from the left actuator, (approximately 1/3 of the distance between the actuators) to allow asymmetrical actuator loading. Two pressure transducers, one for each fluid chamber, were installed in the right actuator. A typical work piece and the finished part are depicted in figure 6.9. A series of tests were conducted forming workpieces of varying thicknesses and 1 100mm Work piece J 1t Finished Part 90) Figure 6.9. Sample work-piece and finished part used for brake-forming tests. Chapter6: CoordinatedMotion Control ofPress Ram 89 widths. The total width and geometric centre of the bend, w and c respectively, were recorded for each test. The hydraulic pressure acting in each chamber of the right actuator was measured as were the reference and actual positions of each axis during the bend cycle. Sample results of this test are shown in figures 6.10 and 6.11 Using the free body diagram shown in figure 6.12 the hydraulic pressure exerted by the right actuator was used to determine a static theoretical applied bending force distribution P =- [] (6.40) as well as the force exerted by the left actuator for the dwell operation: R1 = R)— [kN] (6.41) The results of these test are shown in table 6.4 W C — c ‘i — Yref ‘2 — Y,.j R1 R2 F C1 C2Case [mm] [mm] [mm] 1 [pm] [pm] [kN] [kN] E/mi {J {) 1 150 1.5 425 0.70 4 5 7.1 3.0 67 0.50 1.81 2 300 1.5 500 0.65 9 12 13.5 7.3 69 0.64 1.62 3 600 1.5 500 0.65 13 15 29.9 16.2 77 0.44 0.92 4 150 3.0 725 0.49 13 38 26.5 27.6 360 0.47 1.38 5 200 3.0 700 0.51 24 47 41.4 40.2 408 0.59 1.17 6 200 3.0 700 0.51 21 51 42.7 41.4 421 0.50 1.24 Table 6.4. Results of the Brake forming analysis. Note: the computation of the axis compliance values presented in table 6.2 is based on an assumption that the entire positioning error is due to the disturbance load. The results of the previous section indicate that up to 4 to 5 urn of positioning error exists regardless of disturbance loads. Therefore, the axis compliance values determined for relatively low load forming conditions may be somewhat overestimated. Given this, the Chapter6: CoordinatedMotion Control ofPress Ram 90 0.16 0.14 0.12 0.10 10.08 i 0.06 0) C 0.04 0 i 0.02 0.00 -0.02 -0.04 compliances of the left and right axis, averaged for the four larger bending loads were determined to be: C1, = 0.50 [] (6.42) C2 = 1.18 F1 (6.43) LkNJ Based upon the system pressure and the piston area, an upper bound on the force capacity of each actuator was determined to be 50 [kN]. Considering the reduction in valve gain due to pressure drop, a practical bound on the actuator capacity should be much lower. 4.8 5.2 5.6 6.02.8 3.2 3.6 4.0 4.4 lime (mS) Figure 6.10. Plot of the absolute and relative tracking error of each positioning system: Motion profile with bending operation. Chapter6: CoordinatedMotion Control ofPress Ram 91 9 60 8 Pa 50 012345678910 lime (S) Figure 6.11. Actuator pressures recorded for motion profile with forming operation. ltIIttI11flFb a L:bJ _ _ _ 1 Figure 6.12. Free-body diagram of forces acting on ram during the dwell operation. Chapter6: CoordinatedMotion Control ofPress Ram 92 6.4 Conclusions In this chapter, details of the control scheme were presented. A practical simplification to the pole-placement control law was developed and implemented. The performance of the positioning system can be summarized as follows: • The rise time of the response of the positioning system to a step change in position was 11-12 ms for each axis. • The dead-band was found to be less than one basic length unit (BLU=O.OO5mm). • The steady-state positioning error (under no load) was found to be less than one BLU. • The compliance of each axis of the positioning system was determined to be: Left axis (Yl): 0.50 [] Right axis (Y2): 1.18 [‘k] Chapter 7 Conclusions and Recommendations In this thesis a computer-controlled positioning system for a hydraulic press brake was designed, modeled and analyzed. Both linear and non-linear system models were applied to cylindrical hydraulic actuators. Non-linear state-relations describing the change in actuator compliance with position were developed. An analysis of the actuator system dynamics was conducted using a variety of models. The results indicate that simple linear models are capable of predicting the dynamic mode of an asymmetrical actuator if the valve flow coefficients are known. However, the piston position and the direction of piston movement during critical positioning stages also need to be considered in system design. Hydraulic components for the system supply and servo-actuator were chosen and implemented. The gibbing of the press-brake was modified to allow rotation in the plane of the ram. A position feedback system was designed, implemented and tested. No backlash was detected. Experiments were conducted to determine the system dynamics of the hydraulic servo-actuators. Step response and frequency response tests indicate a first order model with consideration of delay is acceptable. Dynamic system parameters were determined using parametric identification techniques. A delay-compensating, pole-placement control law was designed for the model structures determined in the identification section. A universal simplification of the pole placement control law for processes with inherent integration was developed and 93 Chapter 7: Conclusions and Recommendations 94 implemented. This simplification eliminated output scaling errors while also reducing computational overhead. Motion profile tests indicated that dynamics of each axis were adequately matched. Also, the system exhibited less than one basic length unit steady state position error on a positioning move. Actuator compliances were determined for each axis through actual bending tests. In order to create a satisfactory position system, this work has effectively outlined a procedure for designing position control systems for CNC press-brakes. With regard to this procedure, future work needs to be conducted in identification and control. For this thesis, all identification experiments were carried out in open-loop. While this was relatively safe for this small press, it could be quite dangerous for much larger ones. 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Astrom, K.,J. and Wittenmark, B., Computer Confrolled Systems, Prentice- Hall, Inc., 1984. APPENDIX A Actuator Natural Frequency Calculations An estimate of the natural frequency of the actuator is provided by equation 4.28: 14J3A2 w=I ep A.1 Given: Half mass of ram: Mr = 104 [kg] Cap side piston area: A = 0.00456 [m2] Rod side piston area: A,. = 0.00253 [m2] Total actuator fluid volume at the most compliant position: = 455e [m3] Bulk modulus of fluid: = 689.5e [NI m2] Using the average piston areas to accommodate for actuator asymmetry: — 14(689.5e6 )( 0.00456 + 0.00253)2 — (455ej(104) = 860 [radls] or, f,=137 [Hz.] 98 Appendices 99 APPENDIX B Derivation of Pole-Placement Control Law Parameters For a system with an open-loop position response characterized by the discrete-time transfer function: y(q) — B(q) — b1q’’ +b2q’+• •+b1 B 1 u(q) A(q) — q’”(q”+a1q’’’+...+a)’ choose a desired closed-loop system response which has the same order and the same number of delays: y(q) B(q) (bjq+b2)bo Yrf(q) 4(q) q1ta(aoq1 +amIq+.+am()) where =1. For a unity gain system, 1+ — i1,n Ll Jl,flb+l In order for the controller to be causal, the order of the observer polynomial must satisfy the following constraint: degA0(q) 2degA(q)—degAm(q)— 1 B.2 deg4(q) 2(fla d)Qa +nd)—l deg4(q)n+n —1 If the control law takes a relatively small amount of time to implement we are free to choose: Appendices 100 degA0(q)=n+n—l B.2 For low-noise transducers such as optical encoders, it is desirable to place the observer polynomials at the origin for the fastest response such that: 4 = qfliffldI B.2 The order of the controller poles must be such that: degR(q) deg4(q) + deg4 (q) — degA(q) degR(q)(n ±d)(fla±fld) degR(q) ‘a + n — ld Again, if the controller output signal happens near the beginning of the loop closure cycle, we are free to choose: degR(q) = degS(q) = + n — such that: R(q) =r0q’+r1q’’+•••+i B.2 S(q) =s0q +s1q’+- •+s B.2 where: = ‘la + ‘1d The feed-forward filter is chosen: T(q) = bm04 =b0q’ B.2 In determination of the control law coefficients requires the solution of the Diophantine equation: A(q)R(q) + B(q)S(q) = A,(q)A (q) Expanding B.9: q(a0q +a1q’+ •+a,, )(r0q + • •+, ) + B.1O (b1q’ +b2q’+.. .+bflI)(sOq’- +s1q’’+• .+s,) = qq(q + aq”’+• Appendices 101 For a = 2, b = 1, 11d > icollecting terms of B.l0 of like orders yields: q2(fl+nd)_1: a0r =1 B. 11 q2(fl+fld)2: a0r1 +a1, =a B.12 qfl+fld: aoIr +a1,4+ Zt2 +b1s0 =0 B. 13 qfl+fldI: a1r + +b1s + b2s0 =0 B. 14 qa: a2r +b1s2 + b2s1 =0 B. 15 qfld•4: b1s3+b2 =0 B.16 q’: +b2sfld =0 B. 17 q°: b2sfld+l =0 B.18 Assuming b1 0 and b2 0, solving B. 17 and B. 18 yields: 5÷1=0 B.19 S =0 B.20 Back substituting into the series of equations inferred between B. 15 and B. 17 we can determine that: B.21 Solving B.13 for 5o = (a+ ±‘d-1) B.22 Solving B.15 for s1: a2r s1=— B.23 Solving B.11 -B.13 yields: Appendices 102 B.24 B.25 1=ama2—air B.26 = —(a_2+a1t_), i = [3:n] B.27 Solving B.22 and B.23 into B. 14 and solving for yields: b2 ( b2 c1ld+a2Ic---’ — B27b b a1 — a2 —- — —- 2 b1 Appendices APPENDIX C 103 Friction Characteristics of Guide System The contribution of the guide system was measured for a variety of actuator speeds. The results are presented in figure C. 1 1.5 1.0 z 0.5 10.0 ___ ___ ____ ___ ___ O5 -1.0 -1.5 ° Left Actuator 0 C Right Actuator 0 —0-— 0- 0 0 -0 —0---— 0 0 -30 -20 -10 0 10 20 30 Piston Velocity, Vy (mmls) Figure C. 1. Friction force exerted by guide system. Appendices 104 APPENDIX D Model Parameters V Mr=104 [kg] B,.=0 [N.s/m] A=O.00456 [m2J A,. = 0.00253 [m2] V=455e [m3] Pe689.5e [N/rn2] V1=20.56e-6 [m3]; ,=35.05e-6 [m3J; K = 6.04e-6 [rn5/N. s] K1=0 [m5/N.s] Kie = 0 [m5/N. s] Kq = 144.6e-6 [m3/s] L=O.125 [ml F=(300,600) [N] p=690 [Mpa] pg=O [MpaJ ;=0.005 [ms] g=9.81 [m/s2]
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