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UBC Theses and Dissertations

Computer control of a hydraulic press brake Lane, John Alexander 1994

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COMPUTER CONTROL OF A HYDRAULIC PRESS BRAKEBYJohn Alexander LaneB.Sc. (Mechanical Engineering), 1988University of New BrunswickA THESIS SUBMITTED in PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESDepartment of MECHANICAL ENGINEERINGWe accept this thesis as conformingto the required standardUNIVERSITY OF BRITISH COLUMBIAAPRIL, 1994© John Alexander Lane, 1994In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of CicL 9€E-LJThe University of British ColumbiaVancouver, CanadaDate 2S APRL 994DE-6 (2188)AbstractHydraulic press brakes are widely used in industry to form sheet metal intovarious shapes using bending operations. When the press brake is controlled by aComputer Numerical Control (CNC) unit, the desired bend can be achieved without theuse of mechanical gages or manual adjustments, leading to increased flexibility andaccuracy in the manufactured components.In this thesis, a single axis hydraulic press brake is retrofitted for dual axisCNC 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 axiscan be independently controlled by the CNC unit. Each actuator is instrumented withlinear optical encoders which provide feedback for closed-loop servo position control.The hydraulic system is modified to provide constant pressure fluid power to the servovalves. An accumulator is used between the pump and the valves to suppress pressurefluctuations. Pressure transducers are integrated into the ports of the actuator to monitorthe pressure during the operation of the press.The dynamics of the hydraulic servo system, including the valves, actuators andthe ram are modeled. A non-linear model, which includes the influences of pistonposition, is presented. A simplified model is shown to be adequate provided that thepractical ranges of the piston position is considered. The mathematical model is used toexperimentally determine the dynamics using parametric identification techniques. Thewell-damped system is approximated by a first order system with substantial delaybetween the servo-valve amplifier command and the actuator piston motion. A readilyavailable CNC system is retrofitted to the press brake and a delay compensating poleplacement digital control system is developed and implemented for the independent11control of two actuators. The performance of the system is evaluated for a series offorming operations.This thesis provides basic guidelines for the design and analysis of hydraulicallyactuated CNC presses.111Table of ContentsPageAbstract.iiTable of Contents ivList of Tables viiiList of Figures ixAcknowledgments xiiChapter 1 Introduction 1Chapter 2 Literature Survey 32.1 The Brake Forming Process 32.2 Hydraulic Press Brakes 42.3 Hydraulic Supply Systems 72.4 Cylindrical Hydraulic Servo Actuators 9Chapter 3 Modifications Required for Computer Control 143.1 MechanicaL Design Modifications 143.1.1 Ram Gibbing 143.1.2 Positioning System for CNC Design Objectives 173,1.2.2 Mechanical Design 173.2 Hydraulic System 193.2.1 Supply System 203.2.2 Servo-actuator 213.3 Control Computer 22Chapter 4 Modeling of System Dynamics 244.1 Introduction 24iv4.2 Dynamic Model of Servo-actuator .254.2.1 The Load Pressure, Load Flow Model 294.2.2 Improved Frequency Response Model 324.2.3 A Directionally Biased Model for Asymmetrical Actuators 354.2.4 Non-Linear Valve Flow Relation 374.2.5 Non-Linear Actuator Compliance 384.3 Results 424.3.1 The Effect of the LPLF Linearization on the System Response 424.3.2 The Effect ofPiston Area Ratio the System Response 434.3.3 The Effect of Input Signal Amplitude on the SystemResponse 434.3.4 The Effect of Initial Piston Position on the System Response 454.3.5 Effect of Coulomb Friction on the System Response 464.3.6 Validation of the Non-Linear Models 474.4 Conclusions 49Chapter 5 Identification of Servo-actuator Dynamics for Control 505.1 Introduction 505.2 Choice of Identification Signal 505.3 System Response to a Step Input 525.4 Frequency Response Experiments 545.4.1. Experiment Description 545.4.2. Experimental Results 545.5 Parametric Identification 565.5.1 Theory 565.5.1.1 Least Squares Method 57v5.5.1.2 Instrumental Variables Method .595.5.2. Experiment Description 605.5.2.1 Choice of Identification Signal 605.5.2.2 Model Structure 625.5.2.3 Data Analysis 625.5.3. Experimental Results 635.5.3.1 Model Selection 635.5.3.2 Model Validation 635.6 Conclusion 71Chapter 6 Coordinated Motion Control of Press Ram 726.1 Introduction 726.2 Motion Control 736.2.1. Objective 736.2.2. Velocity Profile 736.2.3. The Process to be Controlled 746.2.4 Control Scheme 746.2.4.1 Introduction 746.2.4.2 Design of the Pole-Placement Controller 756.2.4.3 Controller Implementation 786.3 Positioning System Performance 816.3.1 The Dynamic Response of the Ram Positioning System 826.3.2 Determination of Positioning System Dead Band 836.3.3 Determination of Controlled Motion Performance 846.3.3.1 Motion Profile: No Forming Operation 856.3.3.2 Motion Profile: With Forming Operation 88vi6.4 Conclusions .92Chapter 7 Conclusions and Recommendations 93BIBLIOGRAPHY 95APPENDIX A 98Actuator Natural Frequency Calculations 98APPENDIX B 99Derivation of Pole-Placement Control Law Parameters 99APPENDIX C 103Friction Characteristics of Guide System 103APPENDIX D 104Model Parameters 104VIIList of TablesPageTable 5.1. Delay, rise time and steady state gain obtained from step response experiments 53Table 5.2. Comparison of the loss functions computed for a variety of model structures 63Table 5.3. Model Parameters determined from parametric identification 64Table 6.1. Process models and control law parameters for left axis controller 81Table 6.2. Process models and control law parameters for right axis controller 81Table 6.3. Velocity Error Constants for each axis as determined from motion profile tests: no formingloads 86Table 6.4. Results of the Brake forming analysis 89vi”List of FiguresPageFigure 2.1. The brake forming process 3Figure 2.2. A typical press brake 5Figure 2.3. Pressure Compensated Variable delivery hydraulic supply 9Figure 2.4. Components of a hydraulic servo-actuator 10Figure 3.1. Ram orientation mechanism 14Figure 3.2. Degrees of freedom of CNC press brake 15Figure 3.3. Ways box modification 16Figure 3.5. Position measurement system for CNC operation 18Figure 3.6. Schematic Diagram of the original hydraulic system 19Figure 3.7. Schematic diagram of hydraulic system used for CNC operation 20Figure 3.8. Technical illustration of servo-actuator 21Figure 3.9. Block diagram of the Hierarchical Open Architecture Manufacturing CNC system 22Figure 4.1. Typical hydraulic system used for servo positioning 24Figure 4.2. Functional diagram of servo-actuator 25Figure 4.3. Comparison ofvelocity response predicted by LPLF model versus the velocity predicted bythe LC model with piston area ratio R=1 42Figure 4.4. Comparison of the effect of piston area ratio, R, on the velocity response of servoactuator, aspredicted by the LC model 43Figure 4.5. Comparison of the frequency response predicted by the LPLF model to that predicted by theIFR model at various servovalve armature current amplitudes 44Figure 4.6. Comparison of the Velocity response predicted by LC model versus that predicted by theFCC model when the initial position differs from the LC linearization position 45ixFigure 4.7. Comparison of the response predicted by LC model versus that predicted by the FCC modelwhen the initial position corresponds to the LC linearization position 46Figure 4.8. Comparison of the effect of coulomb friction, Fc, on the velocity response of the system aspredicted by the LC model 47Figure 4.9. Comparison of the velocity response predicted by the LC and FCC models to the actualresponse 48Figure 5.1. Velocity response of the left actuator to a step change in valve command voltage 52Figure 5.2. Magnitude response of the right actuator in extension determined for a variety of input signalamplitudes 55Figure 5.3. Phase response of left actuator in extension determined for a variety of input signalamplitudes 56Figure 5.4. Effect of excitation signal amplitude on the steady-state gain predicted by the frequencyresponse experiments (left actuator extending) 61Figure 5.5. A comparison of the experimentally determined frequency response to that predicted by theidentified models: Left actuator extending 65Figure 5.6. A comparison of the experimentally determined frequency response to that predicted by theidentified models: Right actuator extending 65Figure 5.7. A comparison of the experimentally determined frequency response to that predicted by theidentified models: Left actuator retracting 66Figure 5.8. A comparison of the experimentally determined frequency response to that predicted by theidentified models: Right actuator retracting 67Figure 5.9. Comparison of the response of the measured system to that predicted by the parametricallyidentified models: left actuator extending 68Figure 5.10. Comparison of the measured system to that predicted by the parametrically identifiedmodels: right actuator extending 69xFigure 5.11. Comparison of the measured system to that predicted by the parametrically identifiedmodels. Left actuator retracting 70Figure 5.12. Comparison of the measured system to that predicted by the parametrically identifiedmodels: right actuator retracting 70Figure 6.1. Velocity profile for typical bending cycle 73Figure 6.2. Block diagram of a pole-placement controlled system 75Figure 6.3. Press setup for positioning system performance experiments 82Figure 6.4. The response of left and right axis of the positioning system to a series of step changes incommand position. Yl: left axis; Y2: right axis, Yref: reference command 83Figure 6.5. Response of PID controlled positioning system to step changes in reference position 84Figure 6.7. Response of positioning system to motion profile: no forming loads 87Figure 6.8. Plot of the absolute and relative tracking error of each positioning system: no forming loads.87Figure 6.9. Sample work-piece and finished part used for brake-forming tests 88Figure 6.10. Plot of the absolute and relative tracking error of each positioning system: Motion profilewith bending operation 90Figure 6.11. Actuator pressures recorded for motion profile with forming operation 91Figure 6.12. Free-body diagram of forces acting on ram during the dwell operation 91Figure C. 1. Friction force exerted by guide system 103xAcknowledgmentsI would to thank my supervisor Dr. YusefMtintas for his guidance during thiswork. His patience and support is greatly appreciated.I would like to express special thanks to Grant Lindsay of Del Scimieder Hydraulicsfor his helpfhl assistance and technical support throughout the project. I would also liketo thank Adrian Clark, Scott Roberts and Dr. Malcolm Smith for their helpful suggestionsand encouragement.The author wishes to acknowledge the support of the several companies:Accurpress for providing the press brake for this work, Parker Hannifin Corporation forsupplying hydraulic system components, and Atchley Valves for providing the servovalves.xliChapter 1IntroductionA hydraulic press brake is a machine tool used to form bends in metal plate orsheet. In production, a typical press brake is setup to perform a single bend in a batch ofsimilar components. The bend angle is set by trial and error using an adjustablemechanical stop. This setup requires several attempts and is only useful for one bendangle.By comparison, Computer Numerical Control (CNC) press brakes use closed-loopposition control which improves precision and simplifies the setup such that only one trialbend is needed. Since setup information is stored digitally, CNC press brakes can beprogrammed 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 CNCcontrol. 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 torepresent the system dynamics• to investigate ways of improving system performance in the face ofunexpecteddesign shortcomingsIn this work, a hydraulic servo positioning system is designed, modeled andanalyzed. A hydraulic supply for servo positioning system is designed and implemented.High performance servo-valves are mounted to the existing actuators. Mechanicalmodifications are made to allow indcpendent actuator motion. A position feedback1Chapter]: Introduction 2system utilizing linear optical encoders is designed and implemented. The dynamiccharacteristics of the system are experimentally verified using system identificationtechniques. Based on the results of these experiments, a delay-compensating pole-placement control scheme is chosen and implemented. The performance of the CNC pressbrake is evaluated.A survey of both research and industrial literature concerning the brake-formingoperation, press-brakes, hydraulic system modeling and relevant automatic control theoryis presented in Chapter 2.Chapter 3 describes the work required to convert the manually controlled pressbrake to one capable of computer control. Details of the hydraulic and mechanical designsare presented and a description of the electrical hardware used for control.In Chapter 4, models used to represent the dynamics of the hydraulic positioningsystem are presented and applied. An alternate actuator compliance model is developedand investigated. A summary of general recommendations for modeling is given.Chapter 5 describes the system identification experiments conducted on theposition control system. Results of step response, frequency response and parametricidentification tests are presented and discussed. A summary of conclusions for control ispresented.The delay-compensating, pole-placement control scheme is described in Chapter 6.A simplification to the control law is developed. The performance of the positioningsystem 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 2Literature Survey2.1 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 adie. The bend is formed as the punch penetrates the die. Although variations exist, thereare two fundamental types of brake forming operations. When the punch bottoms thework piece in the die the operation is known as coining or bottom bending. When thepunch 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 PIECEFigure 2.1. The brake forming process.In the ideal bottom bending operation, the yield stress of the work piece material isinduced throughout out the area of the bend, causing the work piece to take the shape ofDIE3Chapter 2: Literature Survey 4the punch and die. While bottom bending generally produces the best quality bends, itrequires 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 punchpenetration. Therefore, air bending operations do not require specially mated tooling.Furthermore, since the yield stress of the material is not induced throughout the area ofthe bend, the actuation force required is two to five times lower than that required bybottom bending [2]. Because of this inherent advantage, automation efforts in brakeforming have focused on the air-bending operation [4]. However, the cost of not inducingthe yield stress of the material throughout the area of bend is that a certain amount ofelastic deformation remains at the final punch penetration. This elastic deformation causesthe work piece to ‘spring-back’ when the punch is withdrawn from the die. Compensationmodels, which are capable of approximating spring-back given the desired bend angle andthe material thickness, are frequently used [5].2.2 Hydraulic Press BrakesBrake forming operations are performed on machines known as press brakes. Atechnical illustration of the Accurpress press brake used for this project is shown in figure2.2. This machine is typical of those utilized in the industry. The fundamentalcomponents of the press brake are depicted.The punch is clamped to a large ram which is positioned by two hydraulicactuators. The actuators are driven in parallel by a single or multistage solenoid operatedvalve system. Usually, the hydraulic circuit has provisions for two speeds: one for rapidpositioning and one for the forming operation. The die is fixed to the press bed. Thepositioning system controls the amount the punch penetrates the die. Due to theextremely high forces applied, the ram gibbing is not sufficient to hold the punch parallelChapter 2: Literature Survey 5to the die. As a consequence, press brakes use an auxiliary system to ensure tooling isproperly oriented. The flrndamental differences between manually operated press brakesand automated ones lie in these two systems, which ensure appropriate positioning andorientation of the punch with respect to the die.RAMPUNCHDIEPRESS BED POSITIONINGSYSTEMWith manually operated press brakes, the positioning system and the orientationsystem are distinct mechanisms. The orientation mechanism may consist of a stiffmechanical device or a sensitive hydraulic feedback circuit adjusted to minimize tilt of theram during asymmetrical loading conditions. Typically the positioning system consists ofa calibrated adjustable stop and an electronic switch which triggers a solenoid drivenhydraulic valve. When flow to the actuator is arrested or reversed, the punch penetrationis limited. The accuracy of this system is dependent upon the sensitivity of the switch aswell as the condition of the hydraulic system. All hydraulic systems are subject tovariations in fluid viscosity and bulk modulus. These variations can cause significantdeviations in the final punch penetration over the run of a batch of parts.Figure 2.2. A typical press brake.Chapter 2: Literature Survey 6On a two-axes CNC press brakes the orientation as well as the positioning tasksare handled by an integrated position control system. The flow of hydraulic fluid to eachactuator is controlled by two independent precision single or multistage spool valves. Therelative 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 therelative 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 bychanges in the state of the hydraulic fluid. The benefits of computer-controlledpositioning systems are many.For a given batch of parts requiring the same bend angle, the setup procedure for apress brake involves three steps. First, the punch and die are mounted and adjusted to becoplanar. Next, the clamping mechanism is adjusted to ensure the forming edges areparallel. Finally, the positioning mechanism is set to give the desired bend. Trial and errortesting 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 usuallyless than three [2]. By comparison, the setup procedure for CNC press brakes is muchsimpler. A tooling offset reference is set by bottoming the punch in the die, and apredictive model is used to calculate the desired punch penetration. After a testworkpiece has been bent, the error between the desired work piece and the test piece isentered into the positioning controller and then the press brake is ready to form parts. Inthis way CNC press brakes require much less setup time. Furthermore, once a series ofbends have been tested and calibrated, the CNC press brake can readily switch betweenbends with no extra setup time.Chapter 2: Literature Survey 72.3 Hydraulic Supply SystemsHydraulic servo-actuators used for positioning systems require constant pressuresupply systems. Although constant supply pressure under varying loading conditions isdifficult to achieve, Ahmed and Asok [6] have reported that slight (<10%) fluctuations inthe supply pressure have little effect on the response of the servo-actuator. In order toachieve this constraint, some consideration must be given to the design of the hydraulicsupply system.Systems that supply fluid power to hydraulic systems can be classified into one oftwo categories: constant delivery systems or variable delivery systems. In the simplestform, a constant delivery system designed to provide constant pressure can consist of afixed displacement pump and a pressure reliefvalve (PRy). In order to maintain arelatively constant system pressure the pump would supply a constant flow of fluid to thesupply line and the PRy. Whatever flow is not used by the servoactuator would passthrough the PRV at maximum pressure drop. Although this type of system is inexpensiveto implement, the cost of providing the maximum flow rate at the system pressure is twofold; the overall efficiency of the system is low, and the quantity of heat generated by thislow efficiency can breakdown the hydraulic fluid. For these reasons, constant deliverysystems are rarely used in servoactuator applications.Variable delivery systems have the advantage of being able to provide only enoughflow to satisfy the requirements of the servo-actuator while accommodating some internalleakage. Depending on the type of pump being used, variable delivery flow can beachieved in one of two ways. If a fixed displacement pump is used, the pump wouldprovide flow to a hydraulic accumulator which in turn would supply the servo-actuator.Chapter 2: Literature Survey 8When the system pressure reaches the desired pressure, the pump would be shut off untilthe system pressure falls below a prescribed lower pressure limit. While this type ofsystem is more efficient than a constant delivery system, a compromise must be madebetween the frequency at which the pump is cycled and the maximum pressure fluctuationallowed.Systems employing variable displacement pumps use pressure feedback tocontinuously vary the flow to track the desired system pressure. The fundamentalcomponents of a pressure-compensated variable delivery hydraulic supply system are: avariable displacement pump with pressure feed-back, a hydraulic accumulator, and a checkvalve (figure 2.3). In this system, the pump utilizes an internal spool valve to control theangular displacement of the swash plate. As the swash plate angle increases, so does theflow delivered from the pump. When the pressure in the supply line deviates from thedesired system pressure, the control valve adjusts the output flow-rate to maintian thedesired 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 tosatisfy the flow requirements while the pump is responding to a demand for more flow.Although these systems are generally more expensive than the systems describedpreviously, they are capable of providing the smoothest supply pressure with a minimumof pressure fluctuation. For this reason, they are the most common supply system usedwith precision servoactuators.Chapter 2: Literature Survey 9tnternal Reservoir 1jjJAccumulatorPressure Compensated Variable Displacement PumpFigure 2.3. Pressure Compensated Variable delivery hydraulic supply.2.4 Cylindrical Hydraulic Servo ActuatorsA schematic diagram of a hydraulic servo-actuator used for CNC press brakes ispresented 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 controlledby 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 theload is proportional to the servo-valve command signal. The characteristics of the maincomponents which make up these servo-actuators, the loaded actuator and the servo-valvehave been thoroughly investigated. [6-16].Chapter 2: Literature Survey 10Previous investigations of cylindrical hydraulic servo systems have found that very littleviscous damping is attributable to the actuator[7,17]. When the compliance of thehydraulic fluid within the actuator is considered, a symmetrical actuator at mid-strokebehaves as an under damped second order system [13]. Asymmetrical actuators can bemodeled by a third order system [11]. Depending on the gibbing or guide ways used forthe 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 tothe torque motor driving the primary stage combined with a quadratic lag due to thedynamics of the secondary stage flow controlling spool. Often the dynamics of the flowcontrolling spool are fast enough to render them insignificant. The flow through theservo-valve is proportional to the valve opening and the square root of the pressure dropacross the open port.Stage ValveStage Spool/RodHydraulic CylinderFeedback SpringFigure 2.4. Components of a hydraulic servo-actuator.Chapter 2: Literature Survey 11The ftmndamental difficulties in controlling servo-actuators stem from fivephenomena:i) the nonlinear pressure/flow relationship for the flow through the servo-valveii) the inherent lack of viscous load dampingiii) the presence of significant Coulomb dampingiv) varying actuator compliance andv) variation of characteristics of hydraulic system.Although the non-linear pressure flow relationship can have significant affect onthe large signal response of the servo-actuator, it is generally considered insignificant fortypical operating control signals. Therefore, it is common practice to linearize the flowcharacteristic about the null operating point, where the spool is centered and the portsclosed. This represents the worst-case scenario for closed loop stability because the valvegain is highest and the flow damping is lowest. Alternate methods of analysis have beendeveloped to predict the frequency response of the system which varies with the amplitudeof the input signal [13].In cases where the response of the control valve is much faster than the responseof the loaded actuator, significant research effort has been expended to overcome the lackof viscous damping [9,17, 19]. Some of the earliest schemes to improve damping involvedthe introduction of laminar leakage across the chambers of the actuator. These systemswere simple to implement but reduced the stifibess and efficiency of the actuator.Transient flow networks, unlike the leakage techniques, were non-dissipative but theywere difficult to tune and reduced actuator stiffness. Hydromechanical feedbackmechanisms were developed which added the required damping and increased the actuatorstiffness, but these systems required precision manufacturing operations which werespecific to each valve/actuator/load combination. The cost and flexibility of theseChapter 2: Literature Survey 12methods were improved when electronic feedback of acceleration and pressure signalsreplaced hydromechanical feedback. Modern digital controllers are capable of addressingthis problem in a number ofways, the most common being pole-placement controlschemes [201.The presence of Coulomb damping has the combined effect of reducing theoscillatory nature of the system response while also contributing to the steady state error.To compensate for the absence ofviscous damping, early analyses made attempts tomodel Coulomb damping as a theoretical viscous equivalent [11,21]. Current analysessimply treat this damping as an external force disturbance [22].The compliance of a symmetrical hydraulic cylinder is a function of the bulkmodulus of the hydraulic fluid and the ratio of oil volume residing on each side of thepiston. Since the volume ratio changes with movement of the piston, the compliancevaries as well. To simplif’ modeling, the common practice is to linearize the compliancerelation about the most compliant position. For asymmetrical actuators the change in oilvolume ratio with piston movement is more pronounced. While studies have beenconducted to determine the effect of changing oil volume on the frequency response andstability of simple control systems [11], typical control strategies either assume a constantvalue for actuator compliance, or assume the dynamics of the loaded actuator are muchfaster than the dynamics of the control valve [23-241.The sources of hydraulic system performance degradation are many. In the short term, themain cause of system degradation is the change of state of the hydraulic fluid. Hydraulicfluids used in servo-actuators have two undesirable characteristics: 1) the viscosity of thefluid changes dramatically with temperature and 2) the bulk modulus (or stiffness) of thefluid changes dramatically with the quantity of dissolved gas in the fluid. Over the longterm, regular wear of precision components will cause a change in the characteristics ofChapter 2: Literature Survey 13the system. Moreover, long term wear is accelerated in the presence of contaminates inthe hydraulic fluid.Given the existence of these phenomena, methods to insure precise positioncontrol have recently been investigated using digital adaptive control schemes. Thesecontrollers have been labeled ‘switching’ adaptive controllers because the control law isupdated at a frequency which is an order of magnitude lower than the ioop closingfrequency. Typically the adaptive controller schemes either used a method of recursiveleast squares to identify reduced order models of the open ioop system dynamics (selftuning regulators) or identified the control law parameters directly (model referenceadaptive control) [20,22]. Provisions were made to check the stability of the system andtemporarily halt estimation if the input signal ceases to be ‘persistently exciting’. Severalimplementations used an exponential forgetting factor and/or covariance matrix resettingin the recursive identification to track time varying parameters. The most common controlstrategy chosen was pole-placement, but adaptive optimal controllers have also beeninvestigated [23,25].Chapter 3Modifications Required for Computer ControlThis chapter describes the work done to change the press brake from a manual toCNC control.3.1 Mechanical Design Modifications3.1.1 Ram GibbingThe press brake used in this project was originally designed to allow only onedegree of freedom of ram movement: vertical translation. The gibbing of the press (figureLink Side PlatesTorque TubeGuide-barWaysbox 0RamFigure 3.1. Ram orientation mechanism.14Chapter 3: Instrumentationfor Computer Control 153.1) consisted of a pair of ways boxes which were rigidly connected to the ram, sliding ona pair of parallel guide bars, one bolted to each side plate. To maintain precise orientationof the tooling while undergoing the extreme forces of the forming process, an auxiliarydevice was employed. A torsional link known as a torque tube, was mounted between theside plates of the press. This torque tube was connected to a ways box on each side of theram 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 seconddegree 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 waysbox was modified to provide guide-bar clearance. To ensure the ram remained centeredbetween the side-plates through-out its range of motion a cylindrical sliding surface wasmachined in the lower ways box pad (figure 3.3). The pads were oriented such that theram would remain centred between the guide bars.Figure 3.2. Degrees of freedom of CNC press brake.Chapter 3: Instrumentationfor Computer Control 16Flat padsOriginal Waysbox Modified WaysboxFigure 3.3. Ways box modification3.1.2 Positioning System for CNCThe positioning system originally installed on the press brake is depicted in figure3.4. This system consisted of a parallel guide mechanism which housed limit switchesused to signal the change-of-speed and the ram-stop position. These switches weretripped 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 stopposition. This design was chosen to compensate for deflections which occur in the sideplates of the ram during bending. While the theoretical precision of this system wasbounded by the resolution of the micrometer (0.012 mm) and the repeatability of the stopswitch, the actual precision may have been worst due to the presence of a low frequencydynamic mode of oscillation with a frequency of 21 Hz.forCylindrical PadChapter 3: Instrumentationfor Computer Control 17Micrometer BarrelStop Switch MechanismFigure 3.4. Positioning system used for manually operated press brake3.1.2.1 Design ObjectivesComputer controlled ram motion requires a system to measure the position andorientation 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 frequency3.1.2.2 Mechanical DesignIn order to compensate for side-plate deformations, a design utilizing a measuringslide 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 theram/actuator connections. High-precision ball-joints were used to connect each tie-rod toTie-Rod PostTie-rodChapter 3: Instrumentationfor Computer Control 18its respective tie-rod post and slide mechanism. A lower bound on the length of the tie-rodwas determined by bounding the measurement error due to tie-rod rotation.Tie-rod PostConnection PinTie-rodEncoder Head SliderEncoder Body GuidesFigure 3.5. Position measurement system for CNC operation.In order to achieve the 0.01 (mm) position repeatability, a linear encoder with aresolution of 0.005 (mm) was chosen. This encoder consists of a body which houses afixed scale and a moving read head. The body was mounted to the press bed and the readhead was attached to the translating slide of the linear guide system which was designedand manufactured in-house. The guide system utilizes a pair of cylindrical guides, onefixed, the other adjustable. This built-in adjustment capability eases alignment procedureswhile providing a method of preloading the guide system for greater rigidity. Provisionswere made to allow the attachment of a transducer to measure the velocity of each side ofthe ram with respect to the bed.The tie-rod, tie-rod posts, and the slider were designed to have high stiffness andsmall mass to minimize the effect of their dynamics on the feedback signal.Chapter 3: Instrumentationfor Computer Control 193.2 Hydraulic SystemA schematic diagram of the original hydraulic system is shown in figure 3.6Hydraulic flow was provided by two fixed displacement gear pumps: one for highpressure, the other for high flow. A pair of normally open solenoid valves, weresequenced such that both pumps provided flow during the rapid moves, but only the highpressure pump provided flow for the feed operation. A counter balance valve located inthe rod-side line was used to lock the actuator during idle times.HII3HFor CNC press operation, the original hydraulic system was replaced by a systemrepresented by the schematic shown in figure 3.7. A description of the componentscomprising this system follows.HIGH VOLUME PUMPFigure 3.6. Schematic Diagram of the original hydraulic system.Chapter 3: Instrumentationfor Computer Control 203.2.1 Supply SystemFor high efficiency and good performance, a variable displacement pump withpressure feedback was chosen to deliver the hydraulic power. Since the response time ofthe pump was significantly slower than that of the servo-valves, an accumulator was usedto keep the system pressure within 10% of the desired level. This accumulator also dampsharmonic components of the pump pressure caused by the oscillating pump pistons.A check-valve was placed between the accumulator and the pump to eliminate thepossibility of the accumulator pressure driving the pump in reverse in the event of a poweroutage. A counter-balance valve was used to lock the system whenever the systempressure is lost.Figure 3.7. Schematic diagram of hydraulic system used for CNC operation.Chapter 3: Instrumentationfor Computer Control 21Two ifiters are used to filter the fluid: a coarse low pressure filter located in thereturn line of the original system, and a fine, high-pressure filter in the supply line to keepmetallic pump debris from reaching the sensitive pilot stage of the servo-valves.3.2.2 Servo-actuatorA technical illustration of the servo-actuator used for positioning is shown in figure3.8. 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 ofprototype Atchley 320 servo-valves (20 g.p.m., 85 Hz bandwidth [261) were mounted asclose as possible to the ports of the original actuator.Special in-line transducer fittings were designed and manufactured to house piezoelectric transducers capable of measuring cylinder port pressures.Pressure PortPressure TransducerUpper Transducer FittingPressure TransducerLower Transducer FittingFigure 3.8. Technical illustration of servo-actuator.Chapter 3: Instrumentationfor Computer Control 223.3 Control ComputerThe computer used to provide CNC control was the Hierarchical OpenArchitecture Manufacturing CNC system (HOAM-CNC) developed in-house by the CNCresearch group of the University of British Columbi&s Mechanical Engineeringdepartment. A block diagram of this controller is shown in figure 3.9.r 486PCL SYSTEM MASTER_JMAIN Bus: ISA BusC30 DSPCNC MASTERCNC Bus: DSP LINK80C196KC 1 80C196KC 1Axis ControllerJ Axis ControJFigure 3.9. Block diagram of the Hierarchical Open Architecture Manufacturing CNCsystem.The HOAM-CNC system uses a hierarchical structure to divide control tasks upbetween three computational systems: the system master, the CNC master, and the axiscontrollers. At the axis level several functions are performed: the loop is closed from acommon clock pulse, some position interpolation is done, and controller status variablesare updated. At the mid-level, the CNC master is used to initialize and coordinate eachChapter 3: Instrumentationfor Computer Control 23axis controller. These functions have been implemented by way of a flexible taskscheduling system which is also capable of high level interpolation computations andalternate functions such as data collection or on-line identification. At the top of theHOAM-CNC hierarchy is a PC based host computer which provides a user interface andmass storage. For a detailed description of the HOAM-CNC system see [27].In order to optimize positioning performance each axis of the HOAM-CNC wasmodified to allow the use of a high-bandwidth, scaled control signal. The nine bit outputconmiand voltages were fed to a pair of Parker BD-98 gain amplifiers which were used todrive the servo-valves. Quadrature encoder signals from the positioning system wereconnected directly to inputs on the axis controllers.Chapter 4Modeling of System Dynamics4.1 IntroductionIn order to satisfy given performance constraints, the design of a servo-mechanismrequires detailed information about how a series of individual components will performtogether. Dynamic system models can provide this information if the characteristics ofeach component are known. A typical hydraulic circuit used for servo-positioningsystems (figure 4.1) can be analyzed as two distinct subsystems: a servo-actuator systemand a hydraulic supply system. The hydraulic supply consists of a pressure compensatingvariable displacement pump supplying compliant hydraulic lines. The servo-actuatorsystem consists of a two-stage servovalve connected to a hydraulic actuator which in turnmoves the ram. In the following sections, the physics of the servo-actuator system is/Figure 4.1. Typical hydraulic system used for servo positioning.24Servo-actuatorChapter 4: Modeling ofSystem Dynamics 25described and models used to represent the dynamics are presented.4.2 Dynamic Model of Servo-actuatorThe servo-actuator used to position each joint of the ram can be considered as asystem consisting of three distinct subsystems:a precision flow-control valve, a hydrauliccylinder and a load (figure 4.1).Hydraulic Cylinder LoadFigure 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, usuallyof a flapper or a jet-pipe type (flapper-valve type shown in figure 4.2). When the torquemotor is driven, the primary valve provides a differential pressure across the ends of asecondary closed-centre spool valve. The displacement of the secondary spool is fed backto the torque motor by way of a cantilever spring. This spooi controls the flow to thecylinder. The position of the flow controlling spool as a function of the torque motorcurrent can be represented by a first-order lag due to the torque motor combined with aquadratic lag due to the dynamics of the spool:FextChapter 4: Modeling ofSystem Dynamics 26x(s) K,— 2 2’ia(S) (ts+1)(s +24’ws+w )where:x: spool displacementi: torque motor currentK,: spool valve positioning gainco,,: natural frequency of spool dynamicsC: coefficient of friction for spool valveGenerally, the dynamics of the spool are considerbly faster than the dynamics of the torquemotor 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 gainThe flow through a port of the secondary-stage spool valve has been found to beproportional to the area of the valve opening and the square root of the pressure dropacross the port. Since the area of the valve opening is proportional to the valvedisplacement, the following expressions can be used to represent the flow through theports 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 27q: flow out of port A ofvalveq: flow into port B ofvalveKqa , Kqb: coefficients of flow for ports A and Bp5: supply line pressurept: return line (tank) pressurePa,Pb: pressure acting upon ports A and B of the valveThe hydraulic cylinder, as depicted in figure 4.2 consists of two fluid chambersseparated 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 cylindersexhibit compliance under load. This compliance (or its inverse, stiffness) is a concern toservo-system designers because it limits the maximum bandwidth of the system.To determine a relationship between the chamber pressures, chamber flows andpiston velocity, the continuity equation can be applied to control volumes encompassingeach 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 pistonA: area of the pistonAr: area of the piston minus the area of the rodKb,: coefficient of leakage past the pistonK,: coefficient of leakage past the rod sealsq: rate of change ofvolume due to volume A complianceqC. rate of change ofvolume due to volume B complianceSince fluid compressibility is directly proportional to the volume in which thepressure acts, the compliance of each actuator chamber may be expressed as:Chapter 4: Modeling ofSystem Dynamics 28Ca =—(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 B13e: bulk modulus of the fluidy: piston positionIf the fluid compliance relations, (4.9) and (4.10), are linearized about a particularoperating point (L0), the following expressions for the rate of change of volume withinthe 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 BBy summing the forces acting on the load mass, the following equation for themotion 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 loadBr: effective viscous damping actuatorF: coulomb friction forceF: external load (including gravity)Equations (4.1-4.13) are the fbndamental epressions most models use to studycylindrical hydraulic servo-actuators.Chapter 4: Modeling ofSystem Dynamics 294.2.1 The Load Pressure, Load Flow ModelMost of the research concerning cylindrical hydraulic servo-actuators pertains tothe 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 modelsis worthwhile because they provide the simplest estimation of system performance.Merritt [14] presents a thorough analysis of symmetric hydraulic actuators utilizing asimple 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 bemade. By assuming zero tank pressure, no cavitation and a symmetric actuator, the flowsinto 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 P8where:q1: effective load flow (q—q)Kq: normalized valve flow gainp1: 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 gainSince this linearization yields a zero value for the null flow pressure coefficient, this valueis usually determined empirically from closed-port leakage tests [14].To extend this LPLF analysis to the actuator, the load is assumed to undergorelatively small excursions from a set operating point. Furthermore, Wit is assumed theChapter 4: Modeling ofSystem Dynamics 30fluid in each chamber is of equal compliance, (4.7-4.10) can be linearized and combinedto yield:q1 =_Av+K,p+CL, (4.16)where:K1 = K,1 +CQbV1: total volume of fluid within the cylinder and linesEquation (4.16) can be combined with (4.15) and rearranged to yield, using Laplacenotation:Kx+AvPzj 1’ (4.17)CabS+Kctmwhere:Kctm =K, +K. (4.18)Considering the equation defining the load dynamics (4.13), some assumptionsmust be made to accommodate the coulomb damping term. In the past coulomb dampinghas been modeled with viscous damping 21,23], but modem analysis considers coulombfriction to be an external disturbance. If the damping is assumed to be strictly viscous, theload 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 theservoactuator to the spooi displacement and the external force:v(s) Kx(s) = 2 +2,w,s+co,’ (4.20)Chapter 4. Modeling ofSystem Dynamics 31v,, (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,. 42413AKepq (4.24)X Mr= 4Kj3 (4.25)KF= 1 (4.26)‘t•ab MrIt has been noted that the contribution of viscous damping in this type of drive isvery small [9,17,19]. Hence, it is common practice to assume Br = 0. Using thisassumption, (4.22) and (4.23) reduce to:c = Kctm1]I3eMr , and (4.27)1413 A 2w =I e (4.28)VtMrwhich indicates two phenomena characteristic of all valve-controlled actuators. First, fora cylinder with minimal external leakage, the damping is proportional to the cross portleakage (null position valve leakage and piston leakage). Second, the stiffness of theactuator is proportional to the bulk modulus of the fluid and the piston area while beinginversely 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]. IntuitiveChapter 4. Modeling ofSystem Dynamics 32variations 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 systemperformance. Since the derivation makes use of a control valve linearization about theoperating point with the least damping (the null position), the LPLF model gives a lowerbound for damping within the system. Furthermore, since the actuator complianceexpressions relations (4.9) and (4.10) are linearized about the most compliant pistonposition, this model also provides a lower bound for the bandwidth of the system.4.2.2 Improved Frequency Response ModelWhile the LPLF model can be used to determine the frequency response of aservoactuator system, McCloy and Martin [13] present a more sophisticated model whichconsiders the effect of the spool valve displacement on the damping of the system. Thisimproved frequency response (IFR) model is based upon the LPLF model.Assuming a pure inertia load, (i.e., no damping and no external load), and asymmetric actuator (piston area A,,) the expression describing the load dynamics (4.13)can be simplified to:dvMr_L=Ap(Pb_Pa), (4.29)or,Mdvp,=__4LL. (4.30)Rewriting the non-linear pressure/flow relationship for the LPLF model (4.14) in nonnormalized form yields:q, = Kx.JI —p1. (4.31)Chapter 4: Modeling ofSystem Dynamics 33K,=Substituting (4.30) into (4.31) yields:I Mdvq, =KqXv+_jf_jL. (4.32)dvff4.32 is linearized about the pomt of maximum load velocity, (L 0), the followmgexpression 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 dtwhere:A: flow/pressure linearization constantMcCloy and Martin [13] suggest using a value of A in (4.34) other than 1/2. The criterionused to select A is minimum error between the actual pressure/flow relation and thelinearized 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) toproduce:Mdv VMd2Kx(1+A r ‘)—Av t r Y (4.35)q “ Ap3 dt “ 4J3e’p cit2or, by rearranging:d2 d K’‘ +—2-(4A ‘ vPe )+v (4 Pe , ) = —K x (4 Pe P) (4.36)dt2 dt Vp ‘ VtMr V VtMrChapter 4. Modeling ofSystem Dynamics 34For 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,wwhere:KX 15Mq e r (439)Vand, 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 VtMrIn 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 theprimary stage of the servovalve are included (4.2), an expression for the frequencyresponse 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)PSAPVFrom (4.42) and (4.43) it is evident the IFR model can be used to determine thefrequency response for a particular amplitude of excitation. However, as with the LPLFmodel, the IFR model assumes symmetrical actuators. If the ratio of the cap-side to rodside piston area (R) is not unity, a more appropriate model may be required.Chapter 4. Modeling ofSystem Dynamics 354.2.3 A Directionally Biased Model for Asymmetrical ActuatorsAlthough many investigations of the dynamics characteristics of servosystemsemploying non-symmetrical actuators have been conducted, most either ignore the effectsof fluid compressibility[24] or assume equal fluid chamber compliance. An exception tothis trend is presented by Watton [29]. In this study, the stability and step response of aproportionally-controlled, symmetrical servoactuator are studied for cases where the ratioof the volume of oil on each side of the piston is not unity. An application of the analysisused in this investigation follows.Given the pressure/flow relations (4.3-4.6), linearized relations for the flowthrough 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 0KqaKqb A & B port flow gains when x,, <0KKCb: A & B port pressure coefficients when x 0Kca,Kct,: A & B port pressure coefficients when x <0or, more generally,(4.48)q=dKx+dKp (4.49)where:d : ‘+‘ when x, 0Chapter 4: Modeling ofSystem Dynamics 36d : !l when x <0If the actuator flow expressions, (4.46-4.49), are linearized about a particularpiston position, and external leakage is neglected, the following expressions can bewritten: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,pbPa= 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 bewritten 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+a3v(s) 1 s2+b1+b (4.58)F(s) Mr3+as+awhere:Chapter 4. Modeling ofSystem Dynamics 37B K Ka1 = _r_ + —f- + —-Mr Ca Cba2= (KaKb K12) (‘22CaCb Ca Cb ) Mr Ca Cb ) Mra3= (KaKb _Kip2)Br +Ar2Ka +A2Kb 2ArApKtpCaCbMrb_K2CaCbdK — 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 volumecompliances, Ca and Cb are equal [29].4.2.4 Non-Linear Valve Flow RelationThe models presented so far, which have commonly been used to design hydraulicservo actuator systems, all assume a linearized flow/pressure relationship. Although it hasbeen concluded that this assumption is valid for a significant range of load pressures andvalve openings, the accurate prediction of the large signal response of a hydraulic servorequires the non-linear flow/pressure characteristics of the secondary spool valve to beconsidered. To this end, several non-linear models have been developed. The following isChapter 4. Modeling ofSystem Dynamics 38a description of a particular model used to numerically simulate the response of theservoactuator system.From the manufacturer’s information, the response of the spool of the valve to anarmature 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 fluidcompressibility (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 ComplianceIt has been shown that the change of actuator compliance with piston position hasonly a minor effect on the response of symmetrical actuators [11]. However, forasymmetrical actuators, the relative change of compliance for a change in piston positionis magnified by the ratio of the area on each side of the piston. To investigate thisphenomena, 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 thefollowing assertion: the pressure increase in a control volume is equal to the integral ofthe flow entering the volume divided by the compliance of the fluid in the volume. Forcontrol volume A this assertion is represented by the following expression:Chapter 4. Modeling ofSystem Dynamics 39Pa(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 timefunction 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 yieldsthe 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+ArYfor; <0,J3 KqaxviJpa—ptsign(pa_p,)_Kp(pa_pb)+ApvY (4.70)dt e Vai+Ap(Ly)=1eKqbXViJIPs_PbSIgn(PPb) +K,P(Pb _Pa)jePb+4r’y (4.71)di Vj+AyChapter 4: Modeling ofSystem Dynamics 40The derivation of the pressure causal compliance (PCC) model is based upon asomewhat different assertion: the rate of change of the volume of fluid compressed (thecompliance flow) is proportional to the rate of change of the product of the complianceand 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 functionnotation):dPa (va,+Ap(L_y)) A 474ca — 13e — Pa ( . )For control volume B, this analysis yields:- ‘Pb (Vb, + Ary) Ar 4751e +PbjjVY (. )For the PCC model, substituting (4.74) and (4.75) into (4.62) and (4.63) and solving forand yields the following actuator control volume pressure relations:cit dtforx 0,dKqaXvIPs_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+ArYChapter 4. Modeling ofSystem Dynamics 41for; <0,Kqa XvJIPa PtIS11l(Pa —pj—K1(Pa Pb) + Avy1 + I?)Pa_13 e (4.78)dt e 1+A(L—y)dKqbXvIPs_PbISin(Ps_Pb)+KP(Pb_Pa)+KlePb+ArVYl+)e (479)diSince, for all practical purposes, <<1 and <<1 expressions (4.76-4.79) can besimplified 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 toaugment this expression so as to ensure mathematical validity when the velocity of thepiston is zero. The following expression can be used:F = Mr = ArPb — ApPa BrVy -F +F1 (4.80)where:for: <“cd ArPb ApPa +F1and 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 ofnonlinear state equations which can be solved numerically to determine the system response toa series of inputs.The values of the system model parameters are presented in Appendix D.Chapter 4: Modeling ofSystem Dynamics 42I,,EE>‘>>4-C)0a)>4.3 Results4.3.1 The Effect of the LPLF Linearization on the System ResponseA comparison between the step response predicted by the LPLF model and theresponse predicted by the non-linear LC model for a symmetric actuator with no externalload is shown in figure 4.3. For the valve opening step, (0 <t < 125ms) the responseLPLF Model *LC Model * **200-20-40-60-80-100*- Compliance relation linearizedabout most compliancepiston position, Lc=62.5mm.* *-Area Ratio, R=l- Initial piston position, Lo=Lc.50 100 150 200 2500Time (mS)Figure 4.3. Comparison of velocity response predicted by LPLF model versus the velocitypredicted by the LC model with piston area ratio R=1.predicted by the LPLF model is significantly more oscillatory than that predicted by theLC model. This can be attributed to the fact that the LPLF model is based upon alinearization about the valve null position, the position exhibiting the least damping. Forthe valve closing step (125 <t < 250ms) both models predict an oscillatory response, withthe LC model exhibiting greater damping. Also, from this plot, the capability of the LPLFmodel to predict the steady-state velocity of a symmetrical actuator is exhibited.Chapter 4: Modeling ofSystem Dynamics 43100-20-30F6o-70-804.3.2 The Effect of Piston Area Ratio the System ResponseA comparison of the velocity response of two actuator system: one with asymmetric actuator, the other with an asymmetric actuator is shown in 4.4. While theresponse of the two systems is similar, the steady state velocities are different and thesystems oscillate at slightly different frequencies.4.3.3 The Effect of Input Signal Amplitude on the System ResponseIn order to examine the effect of the amplitude of the servovalve command signalon the response of the system, the frequency response predicted by the LPLF model wascompared to the frequency response predicted by the WR model for various values ofinput armature current. The results are presented in figure 4.5. From this plot, it isevident that the system response exhibits much greater damping as the amplitude of theR=1 *R=1.8**- Compliance relation linearizedabout most compliance pistonposition, Lc=80.4mm.- Initial piston position, Lo=Lc.50 100 150 2000 250Time (mS)Figure 4.4. Comparison of the effect of piston area ratio, R, on the velocity response ofservoactuator, as predicted by the LC model.0—50—100—150—200—250—3000Figure 4.5. Comparison of the frequency response predicted by the LPLF model to thatpredicted by the [FR model at various servovalve armature current amplitudes.Chapter 4: Modeling ofSystem Dynamics 44input signal is increased. Also, as the amplitude of the input signal approaches zero thefrequency response predicted by the IFR model approaches that predicted by the LPLFmodel. This phenomena can be explained by the fact that the LPLF model’s damping termis linearized about the null flow position which is approached as the amplitude of the inputsignal is reduced. The system bandwidth (-3 dB) is approximately 30 Hz.5040302010Fzq.iexcy (Hz)20050 100 150 200Frqi.iezioy (Iz)Chapter 4. Modeling ofSystem Dynamics 45100-20C)-40a)>C0-600-70-804.3.4 The Effect of Irntial Piston Position on the System ResponseIn 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 bythe non-linear model utilizing the flow-causal compliance relation (FCC Model) isexamined. For this comparison, the compliance relation of the LC model is linearizedabout the critical piston position (i.e., the position yielding the maximum actuatorcompliance). In the first comparison, figure 4.6, the response predicted by the LC model iscompared to the response predicted by the FCC model when the initial position is thecritical position. In this case, the models compare quite favorably. In the secondcomparison ,flgure 4.7, the initial position of the actuator is varied from one limit to theother. In this case the FCC model predicts a much better damped response than the LCmodel. These plots indicate two things: i) the systems response tends to be less oscillatoryLC Model*,Lo=80.4mmFCC Model,Lo=1 10mmFCC Model,Lo=1 5mm*- Compliance relation lineanzedabout most compliance pistonposition, Lc=80.4mm.- Initial piston position, Lo=Lc.50 100 150 2000 250Time (mS)Figure 4.6. Comparison of the Velocity response predicted by LC model versus thatpredicted by the FCC model when the initial position differs from the LC linearizationposition.Chapter 4: Modeling ofSystem Dynamics 46away from the most compliant position, ii) the LC model should be linearized about apiston position in which the actuator will operate. However, if the higher order dynamicsof the actuator become significant, the LC model will not provide an accurate descriptionof the system dynamics away from the linearization point.100[10-20-30-600-70-800 50 100 150 200 250Figure 4.7. Comparison of the response predicted by LC model versus that predicted bythe FCC model when the initial position corresponds to the LC linearization position.4.3.5 Effect of Coulomb Friction on the System ResponseThe positioning system has been found to have significant non-linear frictioncharacteristics (see Appendix C). The dominant non-linear characteristics can be modeledby coulomb friction. In order to investigate the effect of this type of friction on the systemresponse characteristics, the system was simulated using the LC model for variousamounts of coulomb friction (figure 4.8). From this plot we can observe that the quantityof coulomb friction in the system has a negligible effect on the steady-state velocity of theLC Model *- - - - - - - -- FCC Model ***- Compliance relation lineazedA about most compliance pistonV1 position, Lc.-**- Initial piston position, Lo=Lc.Time (mS)Chapter 4: Modeling ofSystem Dynamics 47system while the control spool is open, but contributes substantially to the damping of thesystem 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.100io-20-30C). -40a)C0-600-70-80250Figure 4.8. Comparison of the effect of coulomb friction, Fc, on the velocity response ofthe system as predicted by the LC model.4.3.6 Validation of the Non-Linear ModelsA comparison between the measured response of the servoactuator system to amulti-step input and the response predicted by the LC and FCC models is presented infigure 4.9. From this plot we can observe that the LC and FCC models match the trend ofthe actual system reasonably well. However, the actual system response does not exhibitan oscillating response as predicted by the two models. This discrepancy can be attributedNo CoulombFriction*Fc=300 N.*Fc=600N.**- Compliance relation linearizedabout most compliance pistonposition, Lc=80.4mm.- Initial piston position, Lo=Lc.0 50 100 150 200Time (mS)Chapter 4: Modeling ofSystem Dynamics 48to unmodeled leakage across the piston seals which was not readily obtainable using theequipment available.60—. 40E2:.2 -20a)>.40“--60Time (mS)500Figure 4.9. Comparison of the velocity response predicted by the LC and FCC models tothe actual response.Also, the actual system response is generally a bit slower than the responsepredicted by the models, particularly for the case when control valve is required to delivermore flow. While some of this sluggishness can be attributed to the aforementionedunmodeled piston seal leakage, the difference in response time exhibited between the valveopening and valve closing cases can be attributed to unmodeled flow non-linearities withinthe servovalve which cause its response time to be dependent upon the amplitude of theapplied current.Note that the steps in the velocity response of the actual system near the zerovelocity can be attributed to valve deadband.-80** Compliance relation linearizedabout most compliance pistonposition, Lc=80.4mm.‘‘- Initial piston position, Lo=Lc,Fc=280 N.0 100 200 300 400Chapter 4: Modeling ofSystem Dynamics 494.4 ConclusionsA model for the compliance of cylindrical actuators has been developed andapplied. Both the supply system and the servo-actuator have been analyzed. Thesignificant results of this analysis can be summarized as follows:• Neglecting valve dynamics, symmetric actuators respond as second order systemswith 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 canalso be used to represent the dynamics of an asymmetric actuator.• Coulomb friction at the load tends to damp the actuator oscillations when the valve isin the critical position.• A positioning system designed using linear theory should be analyzed for a range ofpiston positions.Chapter 5Identification of Servo-actuator Dynamics for Control5.1 IntroductionWhile dynamic modeling is a useful tool for selecting hydraulic system componentsand 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 thedynamics of hydraulic systems difficult. To accommodate for these effects, a number ofexperiments can be conducted on the system in order to obtain a better representation ofthe system dynamics. These are known as system identification (SI) experiments and thepractical 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 SignalFundamentally, all system identification experiments involve two simple steps: 1)system input excitation and, 2) observation of the system response. Regardless of thetype of SI experiment, the input signal used must be capable of exciting the relevantdynamics of the system. In order to choose an appropriate input excitation signal for ahydraulic system, some consideration must be given to practical limitations. For systemssuch as press brakes which position large masses, care must be taken to avoid potentiallydamaging excitation induced vibrations.For a closed center spool valve, an investigation of the Improved FrequencyResponse (IFR) model reveals two important phenomena:50Chapter 5: Ident/Ication ofServo-actuator Dynamicsfor Control 511. The response of a spool-valve controlled hydraulic cylinder becomes moredamped 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 portsclosed).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 operatingpoint for a position control system, it is reasonable to assume that the smaller the deviationfrom the null-flow position, the more relevant the identified model will be to the task ofcontroller design. Further, the IFR model presented earlier shows that an equivalentlinear frequency response model of the non-linear servo-actuator can be achieved if aperiodic wave form of constant amplitude is used. Given this, a low amplitude periodicwave form with a mean amplitude of zero would seem to be ideal.Unfortunately, valve deadband and hysteresis affect the valve dynamics most at thenull position. Furthermore, the amplitude of the input signal has to be large enough sothat the steady state flow gain identified is not biased by the valve-spool stiction. Parkerand Desjardins [31] have suggested that a Pseudo Random Binary Sequence (PRBS) inputsignal of an amplitude of at least 10% of the maximum amplitude is sufficient to eliminatethe effects of non-linear valve gain and spool friction. In practice, the amplitude ofexcitation signal required will vary with each system.Watton [29] has shown that the steady state flow gain (SSFG) of an asymmetricalactuator 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 excitationsignal chosen causes the load to change direction, the system will undergo an external loadexcitation. Such system excitations are undesirable during identification experiments.Therefore, if accurate estimations of the steady-state velocity response of the actuator areto be obtained, separate identification experiments should be conducted for the extendingand retracting cases.ChapterS: Identfication ofServo-actuator Dynamicsfor Control 525.3 System Response to a Step InputIn order to determine the velocity response of each actuator to a step change invalve command voltage, two sets of experiments were conducted: one in extension andone in retraction. For each experiment, an alternating step input signal of non-zero meanand not passing through zero voltage was applied to the servo-valve amplifier, and thevelocity response of the load was measured. Each experiment was repeated for a numberof differing input amplitudes.The response of the hydraulic system to a typical step change in valve input signalduring extension is shown in figure 5.1. Approximate first-order system parametersobtained from the step response experiments are presented in table 5.1. Note that theextension response differs somewhat from the retraction response. From this table we can!160_______-14k-180 CommandE-16j-200 . > VelocityC.)-18220>-240 .. -20-260 -22-280-240 50 100 150 200 250Time (mS)Figure 5.1. Velocity response of the left actuator to a step change in valve commandvoltage.Chapter 5: Identfication ofServo-actuator Dynamicsfor Control 53see that while the steady state velocity and delay do not change significantly with theamplitude of the excitation signal, the rise time does. Further, the rise time for theextending cases varies somewhat from that of the retracting cases.Left Actuator Extending Valve Opening Valve ClosingStep Amp (V) Gain (mmN.s) Rise (ms) Delay (ms) Rise (ms) Delay (ms)0.100 7.93 12.0 5 9.2 40.150 9.53 15.0 4 7.9 40.250 9.48 12.5 4 7.8 3Left Actuator Retracting Valve_Opening Valve ClosingStep Amp (V) Gain (mmN.s) Rise (ms) Delay (ms) Rise (ms) Delay (ms)0.100 6.75 15.4 4 7.3 30.150 7.74 16.8 4 8.4 30.250 7.53 13.4 4 7.2 3Right Actuator Retracting Valve Opening Valve ClosingStep Amp (V) Gain (mmN.s) Rise (ms) Delay (ms) Rise (ms) Delay (ms)0.100 5.80 6.1 4 5.9 40.150 6.00 5.8 4 5.8 30.250 6.03 6.2 4 6.1 3Right Actuator ExtendingStep Amp (7)0.1000.1500.250Gain (mmN.s)3.954.616.11Valve OpeningRise (ms) (ms)433Valve ClosingRise (ms) (ms)433Table 5.1. Delay, rise time and steady state gain obtained from step response experiments.Chapter 5: Identfication ofServo-actuator Dynamicsfor Control 545.4 Frequency Response Experiments5.4.1. Experiment DescriptionIn order to determine the relevant dynamics of the system, frequencyresponse experiments were conducted on each actuator for the extending and. retractingcases. For each actuator, an input signal was applied to servo-valve input, and thevelocity signal was measured. A PRBS input signal was selected because of the excellentsignal to noise ratio effected in the velocity transducer. To minimize the effects ofhysteresis, deadband, and coulomb friction loading, a DC offset was added to the signalto ensure the valve did not pass through the null position, reversing the direction of theload. Because the WR model indicates that the frequency response of such hydraulicsystems varies with the amplitude of the excitation signal, these experiments wereconducted for a range of excitation signal amplitudes.The data from each experiment was analyzed using a transfer function analysis.5.4.2. Experimental ResultsThe magnitude and phase responses of the left actuator in extension is shown in figures5.2 and 5.3. Note that the peaks at 60Hz for the lower amplitude signals are due toelectrical 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 thesystem decreases as well. This can be attributed to the increased effect of valvecomponent stiction at low valve actuation forces as discussed in [31].2. The band pass frequency of the velocity ioop is approximately 30 Hz. This comparesfavourably with the band pass predicted by LPLF model.Chapter 5: Identfication ofServo-actuator Dynamicsfor Control 558276:. 70wz6458520 20 40 60 80 100 120 140 160 180Frequency (Hz.)—0-—— 50 mV—0-—— 100 mV—&—— 200 mV400mV600 mV800 mVFigure 5.2. Magnitude response of the right actuator in extension determined for a varietyof input signal amplitudesFrequency (Hz.)10 20 30Figure 5.3. Phase response of left actuator in extension determined for a variety of inputsignal amplitudes0040 50-135—0-—— 50 mV—0—-— 100 my—&--— 200 mV400mV600 mV800 myChapter 5: Identfication ofServo-actuator Dynamicsfor Control 563. There exists a dynamic mode due to load dynamics near 140 Hz. As the amplitude ofthe input signal is increased, the frequency of this mode decreases. This frequencydecrease 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 orderdynamics of the valve spool which, like the hydraulic cylinder, is underdamped only forsmall valve openings.5.5 Parametric Identification5.5.1 TheoryIn order to determine a more precise description of the dynamic system model, anumber of parametric identification experiments were conducted. Due to its ability to giveunbiased estimates under less restrictive conditions, the Method of Instrumental Variableswas chosen over the Least Squares Method to determine the model coefficients. A briefdescription 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 isbased on the LS method, it is useful to introduce the LS method first. Least Squares MethodThe discrete time transfer function for the open-loop velocity response of ahydraulics servo actuator has the form:vy(q’) = B(qj (5.1).u(q ) A(qwhereChapter 5: Identfication ofServo-actuator Dynamicsfor Control 57B(qj = b0 +blq+...+bflq(5 2)A(qj =1 +a1q’ +Using the Auto Regressive Moving Average (ARMA) notation, (5.1-5.2) can bereformulated as a time seriesv(k) =b0u(k) +b1u(k — 1)+. . +bflbu(k — flb) —a1v(k—i)—... —ay (k — n) (5.3)orv(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 )Tis the parameter vector ofvariables describing the ‘true’ dynamics of the system. The goalof the parametric identification is to obtain the parameter vector 0 from a data set of Nmeasured regressors(5.7)pT(N)and the measured outputv(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 58e(1) v(1)—(pT( )OE = (5.9)e(N) v(N)—çoT( )Othe least squares estimate of 0 (denoted as 0), is that which minimizes the sum of thesquares 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 minimumoccurs for0= (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 ‘persistentlyexciting’. In simpler terms, the input signal used to construct the data set must be capableof exciting the particular dynamics of the system one wishes to identify. Given that theabove 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 linearlyindependent, 0 has been shown to converge to a biased estimate of 0. Soderstrom andStoica [32]present this phenomena as follows. If the true response of the system is givenbyv(k) = çoT (k)O÷ w(k) (5.12)where w(k) is stochastic disturbance, the difference between the true parameters and theestimated parameters can be written as:N—=o(k)Tco(k)] [41jco(k)Tw(k)] (5,13)Chapter 5: Ident/Ication ofServo-actuator Dynamicsfor Control 59As the number of samples N tends towards infinity, equation (5.13) will not converge tozero unless the expectationEço(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 provideunbiased estimates in the presence of correlated measurement errors. Instrumental Variables MethodThe IV method augments the LS method by the introduction of a vector of signalsor ‘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 andtherefore 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 thansimple LS methods.ChapterS: Ident/1cation ofServo-actuator Dynamicsfor Control 60The approximately optimal IV method was chosen to identif,’ the dynamic systemparameters in this experiment. For a detailed description of this method see Soderstromand Stoica [32].5.5.2. Experiment DescriptionFor 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 velocitytransducer was sampled at 1 millisecond intervals. The collected data was analyzed usingMatlab’s System Identification Toolbox [33]. Experiments were conducted for eachactuator extending and retracting. Choice of Identification SignalOf the two types of signal considered, pseudo random noise and a pseudo-randombinary sequence (PRBS), the PRBS was chosen because of the superior signal to noiseratio effected in the velocity transducer. A DC bias (as in the frequency responseexperiment) was added to the PRBS to prevent reversing the direction of the load. Inorder to determine the lowest amplitude signal capable of minimizing the non-linear valvecharacteristics a number of frequency response experiments were conducted usingexcitation signals of differing amplitudes.The open-loop velocity gain predicted for each amplitude of input signal appliedfor the left actuator in extension is shown in figure 5.4At low input signal amplitudes, the velocity response is dominated by valve nonlinearities. As the amplitude of the excitation signal is increased, more consistentestimates of the open loop gain are obtained. From these experiments, an input signalamplitude of 600mV, or 3% of the maximum valve input, was chosen. This amplitude, theChapter 5: Ident/ication ofServo-actuator Dynamicsfor Control 61lowest input signal amplitude capable of overcoming the valve stiction, was within therange of input signals required during the forming operation.850 0 U84U‘—83 0820 0. 81C).2 80>0.o 0278e.77°76750 100 200 300 400 500 600 700 800PRBS Input Signal Amplitude (my)Figure 5.4. Effect of excitation signal amplitude on the steady-state gain predicted by thefrequency response experiments (left actuator extending). Model StructureIn order to effectively determine the dynamics system parameters using theparametric identification techniques described above, information about the modelstructure must be known. From the frequency response analysis, the third order systemdynamics predicted by the IFR model can be observed: a dominant first order lag due tothe valve dynamics combined with slight oscillatory mode due to the actuator/loaddynamics. From the step response analysis, a first order response with 3-4 ms delay canbe observed.ChapterS: Identfication ofServo-actuator Dynamicsfor Control 625.5.2.3 Data AnalysisFor each situation (left actuator extending, left actuator retracting, right actuatorextending, right actuator retracting) a number of trials were conducted. For each trial theservo-valve command and the voltage across the velocity transducer were sampled at 1millisecond intervals. The 4000 point data sets collected were split into two 2000 pointdata sets: the first was used for estimation of the dynamics system parameters, the secondto validate the estimated model.For each identification data set, the ‘approximately optimal’ IV method was appliedfor a number of model stmctures of first and third orders with two to five milliseconds ofdelay. Using the validation data set, the response for each identified model was comparedto the actual system response. A prediction error function consisting of the sum of thesquares of the error between the actual system response and the simulated response wascomputed to compare the relative goodness of fit for each model.5.5.3. Experimental Results5.5.3.1 Model SelectionA comparison of the loss functions for each structure of model fit to the data ispresented in Table 5.2. The structures corresponding to the minimum value of lossfunction are shown highlighted.The identified model parameters for the candidate structures are presented in Table5.3. Note that model parameters for two alternate candidate structures pertaining to theextension cases have been included.ChapterS: Ident/Ication ofServo-actuator Dynamicsfor Control 63Model B(c[1)=+ Loss FunctionA(q’) ‘ 1+aq+. .+aq’ Left Actuator Right ActuatorExtending Retracting Extending Retracting1 1 2 2065 2376 814 6201 1 3 718 1394 603 4921 1 4 1084 2133 621 10521 1 5 1734 2936 1083 25531 1 6 2036 3021 1876 34083 3 2 1495 3275 260 14823 3 3 234 219 353 1493 3 4 54 80 118 1123 3 5 205 155 548 2983 3 6 467 605 1104 627Table 5.2. Comparison of the loss ftinctions computed for a variety of model structures. Model ValidationIn order to ensure that the identified parameters are reasonably accuratedescriptions of the relevant system dynamics, two sets of validation experiments wereconducted: a frequency response comparison and step response comparison.For the first comparison, the frequency response predicted by each of thecandidate models for each case was compared to the frequency response determineddirectly from sampled data.The measured frequency response of the left actuator extending is compared tothat predicted by the identified candidate models in figure 5.5. Although overestimatingthe steady state gain, the first order model with three sampling delays gives an excellentdescription of the process dynamics for frequencies between 10 and 65 Hz. The firstorder model with four delays accurately predicts the steady state gain, but generally overestimates the response of the system for frequencies between 10 and 110 lIz. The thirdorder model with four delays is capable of representing the dynamics mode due to theChapter 5: Ident/Ication ofServo-actuator Dynamicsfor Control 64load/actuator dynamics, but the amplitude at the resonant mode is significantlyoverestimated.Case ‘1a b d [A] = 1 +aq’+•.•+aq [B] = b0 +Left Extension1 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 Extension1 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 Retraction1 1 3 [1, -0.9022] [699.2, 657.3]3 3 4 [1, -2.0982, 1.9283, -.7785] [448.5, 1204.5, -1676]Right Retraction1 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 identificationThe measured frequency response of the right actuator extending is compared tothat predicted by the identified candidate models in figure 5.6. The first order model withthree sampling delays tends to overestimate the response of the system from steady stateto 15 Hz, while underestimating it everywhere else. The first order model with four delaysaccurately predicts the steady state gain, gives a reasonably accurate description of theresponse of the system for frequencies between 0 and 100 Hz. The third order model withfour delays gives a reasonable description of the system dynamics from steady state to 90Hz, although it incorrectly predicts the dynamic mode.The measured frequency response of the left actuator retracting is compared tothat predicted by the identified candidate models in figure 5.7. While both modelsaccurately predict the steady-state gain of the system, the third order model with fourdelays better represents the dynamics throughout the range of frequencies. Note howeverChapter 5: Identification ofServo-actuator Dynamicsfor Control. 81I::> 720 20 40 60 80 100 120 140 160Frequency (Hz.)65Figure 5.5. A comparison of the experimentally determined frequency response to thatpredicted by the identified models: Left actuator extending.878481>7269660 20 40 60 80 100 120 140 160 180Frequency (Hz.)Figure 5.6. A comparison of the experimentally determined frequency response to thatpredicted by the identified models: Right actuator extending.8784_Experimental— -— First Order Model: d=3—— First Order Model: d=4Third Order Model6966Experimental— -— First Order Model: d=3—— First Order Model d=4Third Order ModelChapterS: Iden4/Ication ofServo-actuator Dynamicsfor Control 66the third order model dramatically overestimates the frequency response of the dynamicmode.\Figure 5.7. A comparison of the experimentally determined frequency response to thatpredicted by the identified models: Left actuator retracting.The measured frequency response of the right actuator extending is compared tothat predicted by the identified candidate models in figure 5.8. The first order model givesa good description of the response of the system from steady state through to 85 Hz. Thethird order model gives a good description of the system dynamics from steady state up to60 Hz, but beyond this gives an poor estimate of the dynamics and incorrectly predicts theof the dynamic mode.For the step response comparison, the response predicted by each of the candidatemodels for each case was compared to a measured response from a data set independentof that used in the identification.8784. 81U,784-’00>726966Experimental— -— First Order Model: d=3Third Order Modelo 20 40 60 80 100 120 140 160 180Frequency (Hz.)Chapter 5: Ident/1cation ofServo-actuator Dynamicsfor Control 678784. 81‘7269660 20 40 60 80 100 120 140 160 180Frequency (Hz.)Figure 5.8. A comparison of the experimentally determined frequency response to thatpredicted by the identified models: Right actuator retracting.The measured response of the left actuator extending is compared to that predictedby the identified candidate models in figure 5.9. While none of the models testedconstitutes an excellent fit, the third order model seems to give the best representation ofthe dynamics and the steady-state gain. Both first order models give reasonable fits withneither being outstanding at all points on the comparison.The measured response of the right actuator extending is compared to thatpredicted by the identified candidate models in figure 5.10. Again, while none of themodels tested constitutes an excellent fit, the third order model seems to give the bestrepresentation of the dynamics and the steady-state gain. The first order model with fourdelays 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=3Third Order ModelChapter 5: Ident/Ication ofServo-actuator Dynamicsfor Control 68.—.. -5-15-20>-25-300 200Figure 5.9. Comparison of the response of the measured system to that predicted by theparametrically identified models: left actuator extending.The measured response of the left actuator retracting is compared to that predictedby the identified candidate models in figure 5.11. In this case, the system is exhibitingthird order or higher dynamics. Although a slight DC bias seems to exist, the third ordermodel gives the best fit. The response predicted by the first order model with three delaysseem to fit the first order dynamics satisfactorily despite the same DC bias.The measured response of the right actuator retracting is compared to thatpredicted by the identified candidate models in figure 5.12. As was the case with the leftactuator retracting, the system response is clearly third order or higher with a higher modefrequency of approximately 135 Hz. While generally underestimating the amplitude of thehigher order oscillations, the response predicted by the third order model does fit theactual response reasonably well. Again, the response predicted by the first order modelwith three delays seems to fit the first order dynamics as well as can be expected.040 80 120 160Time (mS)Chapter 5: Identfication ofServo-actuator Dynamicsfor Control 690I:>-20-25Figure 5.10. Comparison of the measured system to that predicted by the parametricallyidentified models: right actuator extending.15EEioActual Response First Order Model Third Order ModelActual First First ThirdResponse Order Order OrderModel: Model: Modeld=3 d=40 40 80 120 160Time (mS)20025200 40 80 120 1600-5200Figure 5.11. Comparison of the measured system to that predicted by the parametricallyidentified models. Left actuator retracting.Time (mS)Chapter 5: Identfication ofServo-actuator Dynamicsfor Control 7015EE[050Actual Response- — — —— First Order Model Third Order ModelFigure 5.12. Comparison of the measured system to that predicted by the parametricallyidentified models: right actuator retracting.200 40 80 120 160Tirne(mS)200ChapterS: Identfication ofServo-actuator Dynamicsfor Control 715.6 ConclusionFrom the identification experiments a number of conclusions can be drawn:• At very small input signals, the steady state gain of the servo-actuator dropssignificantly and the dynamics of the load are not significant.• As the amplitude of the input signal is increased, the steady-state gain of theservo-actuator increases and an oscillatory mode at 13 0-150 Hz due to theload dynamics is more apparent.• The system response of the servo-actuators in the extending cases can berepresented adequately by a first order model.• The response of the servo-actuators in the retracting cases can be characterizedby a dominant first order lag combined with a low amplitude dynamic mode atapproximately 13 0-150 Hz.Chapter 6Coordinated Motion Control of Press Ram6.1 IntroductionThe objective of the efforts described in this chapter was to design and implementa practical control scheme which accomplished smooth positioning and orientation of thepunch with respect to the die throughout the bend cycle. In order to facilitate coordinatedmotion between the two actuators, a control scheme capable of matching the closed iooppositioning dynamics of the axes was chosen. Model parameters presented in the previouschapter were used exclusively for the controller design.Given the cost sensitivity of manufacturing a machine tool for the industrialmarket, efforts to improve the positioning performance focused on software rather than onhardware. Although velocity and pressure signals were available and could have improvedthe robustness and performance of the system, only position feedback was used so that theincremental cost of CNC implementation would be less. The results indicate that as longas the control law is properly designed, the system behaves satisfactorily without pressureand velocity feedback signals.In this chapter, the term ‘following error’ refers to the difference between thedesired position and the actual position of either axis during movement at a particularvelocity. The term ‘tracking error’ has been used to describe the difference betweenfollowing error of each axis. The term ‘steady-state-positioning error’ refers to thedifference between the desired position and the actual position of either axis when thedesired position is not changing and system dynamics are not prevalent.72Chapter6: CoordinatedMotion Control ofPress Ram 736.2 Motion Control6.2.1. ObjectiveThe motion control task attempted to achieve two fundamental objectives:1. less than 0.25 mm tracking error during the forming process2. a steady-state positioning error of less than 0.0 15 mm under forming load.6.2.2. Velocity ProfileA plot of the desired punch velocity profile for a brake-forming cycle is depicted infigure 6.1. The profile consists of four segments:1. rapid approach to the clearance plane above the work-piece2. brake-forming operation at feed-rate3. dwell in bend4. rapid retract to the start positionTime (Seconds)Figure 6.1. Velocity profile for typical bending cycle.The performance of the positioning system during the bend and dwell operationsdetermines the precision of the press. In these cases, the actuators are either extending or6040‘ 20EE00e> -20-40-60Apph [ 4\\ \\/____Dwell0.0 1.0 2.0 3.0 4.0 5.0Chapter6: CoordinatedMotion Control ofPress Ram 74holding their position against a forming load. For these reasons, the actuator modelscorresponding to the extension cases were used for controller design.6.2.3. The Process to be ControlledThe results of the identification experiments have shown that the response of theactuators can be characterized by a first order lag combined with an under damped modeat approximately 140 Hz. However, for the extension cases, the higher order dynamicsare less prevalent especially for the small input signal amplitudes (100-150 mY) usedduring the forming operation. Moreover, the first order models generally provide gooddescriptions of the dynamic system performance at frequencies up to 100 Hz.Furthermore, considering the computational complexity required to compensate for higherorder dynamics, the first order models were chosen for initial design efforts.6.2.4 Control Scheme6.2.4.1 IntroductionIn order to achieve low tracking error, a control scheme which would facilitatematching the dynamics of each positioning system was desirable. In order to obtain lowsteady-state position error, the closed loop positioning system would require high gain tomaximize load force disturbance rejection. Since the open-loop system dynamicscontained several delays, a delay compensation scheme was deemed necessary to allowhigh 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 capableof 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 Band the poles A. The controller consists of a feed-forward filter T, a feed-back filter S,Chapter6: CoordinatedMotion Control ofPress Ram 75controller poles R and an estimation filter (or observer) A0. These four filters are selectedsuch 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 Amwhere 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 positioncontrol task follows. Design of the Pole-Placement ControllerThe open-loop velocity response of each of the servo-actuators in the extendingcases can be represented by the following discrete time transfer fhnction:YrejProcessyControllerChapter6: CoordinatedMotion Control ofPress Ram 76vy(q_1) — qd (b0’ +b1’cf’) F countsu(q’) — 1+a’q [vsec (6.2)where d is the number of delays, b1’, b2’, and a are the identified system parameters, and1 count/second equals 0.005mm/s of actuator velocity.Adding an integration term to this expression yields:y(q’) — q’d (b1 +b2q’) F countsu(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 ordersystem with a delay equivalent to the open-loop system delay:y(q) — B(q) — (b1q+b2)0 [countsyj(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 positioncontrol loop has a unity steady state gain:b 1+ami+am2 66mO b1+b2Since it is desirable to use only the position transducers for process feedback, anobserver A0 (q) is required to estimate any higher order feedback terms required. In orderto ensure a causal design analysis, Astrom and Wittenmark [34] have shown that thefollowing 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 77degS(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 observerpolynomial 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 werechosen 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 78s. =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 Ia1---—a2ji+a2--?d_II b b— a2 — a0b1= (+i +a1i (6.27)a2r=— nd+1 (6.28)b2s, = 0, 1 = 2,3,.. d + 1 (6.29) Controller ImplementationUsing the forward shift notation, the pole-placement control law can berepresented 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 controllerpolynomials. Given that in the applied case, n, = 0, n8 = I and ii,.=+ 1, the control law:Iu(k) =t0y(k) —s1y(k — i) — ru(k— j) (6.32)1=0 j=IChapter6: CoordinatedMotion Control ofPress Ram 79would require (d + 4) multiplication operations and (d + 4) add operations per ioopclosure. Note that this implementation requires precise representation of parameters t, sand s1 in order to avoid steady-state error due to numerical round-off. For this reason, thefollowing simplification was developed.The steady state gain of a pole-placement controlled system can be represented bythe following relation:bmy(l) = iO = (6.33)‘)ami=O j i k=O joFor systems with an inherent integration such as DC motor positioning systems or servo-hydraulic positioning systems=0Therefore, (6.33) reduces to:bm= ° = ° (6.34)Yrej(’)amjwhich indicates that the steady-state gain of the pole-placement controlled system can bescaled by appropriate choices of S and T. Furthermore, if the desired steady state gain isunity, 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=1Using (6.36), the control law can be further simplified to yield:nd+1u(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 duringscaling computations while guaranteeing appropriately scaled output• it reduces the number of control parameters that need to be represented, lesseningmemory requirements• it reduces the precision in which t0 and s0 need to be represented, further lesseningmemory requirements• it reduces the number and complexity of operations required for implementationThis control algorithm was implemented on two axes of the HOAM-CNC usingIntel 80196 assembly language. A pair of overdamped CLTF poles (4) were chosensuch that the system response was fast enough to satisfy the following-error constraintswhile not deviating significantly from the control valve dynamics. The control lawparameters as well as the process parameters used for design are shown in tables 6.1 and6.2.Chapter6: Coordinated Motion Control ofPress Ram 81Left 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 -0.064Table 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 -0.076Table 6.2. Process models and control law parameters for right axis controller.6.3 Positioning System PerformanceThe equipment setup used for the performance experiments is depicted in figure6.3. The object of these experiments was to evaluate the positioning system’s performancein terms of dynamic response, position deadband, following error, tracking error and theeffects of load force disturbance (compliance).Chapter6: CoordinatedMotion Control ofPress Ram 826.3.1 The Dynamic Response of the Ram Positioning SystemA 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. Thecommand signal and the measured position response are shown in figure 6.4. Negativesteps represent actuator extension.The results of this test indicate that rise time of each axis in extension (for thisamplitude of excitation) is between 11 and 12 ms Furthermore, the extension response ofeach actuator exhibited no overshoot. The left actuator was observed to overshootapproximately 0 -2% occasionally during retraction. This slight overshoot was deemedacceptable given that it does not occur during the forming operation. Moreover, the effectof 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 830.100.00-0.10E -0.20C00.0a--0.40-0.50-0.60Figure 6.4. The response of left and right axis of the positioning system to a series of stepchanges in command position. Yl: left axis; Y2: right axis, Yref: reference command.For comparison purposes, a traditional PD control algorithm was implemented andmanually tuned. The response of the PD controlled system for one of the better tuningsis 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 ms6.3.2 Determination of Positioning System Dead BandThe performance of hydraulic actuators is critically dependent on the performanceof the valve. Valve hysteresis and spool stiction can cause excessive deadband in thepositioning system. In order to determine the effects of these phenomena, each axis of thepositioning system was given an alternating series of small steps in reference position.Both the reference and the actual axis positions were measured at two millisecond0 50 100 150 200 250 300 350 400Time (mS)Chapter6: Coordinated Motion Control ofPress Rain 840.100.00-0.10E -0.20C0-0.3000-0.40-0.50-0.60700 800 900 1000Figure 6.5. Response of PID controlled positioning system to step changes in referenceposition.intervals. The results are presented in figure 6.6. These results indicate that the systemdeadband is smaller that one basic length unit (0.005 mm).6.3.3 Detenmnation of Controlled Motion PerformanceOne of the most important measures of a positioning system’s performance is itsability to follow the desired motion profile. A measure of this following capability, thevelocity error constant (VEC) is defined as the ratio of the following error to the desiredvelocity of the move. While the following error for either axis of the CNC press brake isnot critical, the tracking error is, for this controls the orientation of the tooling. Thetracking error will be proportional to difference between the VEC of each axis.Hampering a press brake’s ability to maintain constant tooling orientation is thetendency for the actuators to be unequally loaded. This occurs frequently because CNCpress brakes are often used to perform a number of bends using a number of sets of0 100 200 300 400 500 600Time (mS)Chapter6: CoordinatedMotion Control ofPress Ram 85Yl-----Y2LjL .... ...0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4Time(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 theaxis dynamics and ensuring both axes are stiff enough not to be affected significantly bydisturbance loads.In order to evaluate the performance of the positioning system with regards tothese measures, two experiments were conducted. For the first experiment, a typicalmachine 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 workpiecesof differing geometry. Motion Profile: No Forming OperationThe object of the motion profile experiment was to determine the ability of thepositioning system to track the desired motion proffle. For this experiment, each actuatorwas given a motion profile representing a typical bending operation: rapid approach, feed- 0.010-0.01-0.02-0.03-0.04Chapter6: CoordinatedMotion Control ofPress Ram 86rate forming operation, dwell, and rapid retract. Sample results of this test are shown infigures 6.7 and 6.8. Note that the rough tracking response during the rapid move is dueto an error in the linear interpolation routine. The velocity error constants for each axisare 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 aslight mismatch of axis response times. Furthermore, for either axis, the VEC forextending case differs from that of the retracting case.• The steady-state position error is not greater than 0.005 mm (one BLU) during thedwell operation.• The relative position difference between each axis (the tracking error) is notgreater than 0.015 mm (3 BLU) during the feed operation.Case: Left Actuator VEC Right Actuator(ms) VEC(ms)Extension 11.8 12.3Retraction 13.4 11.9Table 6.3. Velocity Error Constants for each axis as determined from motion profiletests: no forming loads.Chapter6: CoordinatedMotion Control ofPress Ram 871.251.000.75EE 0.50I00.25C00.000a--0.25-0.50-0.75Figure 6.7. Response of positioning system to motion profile: no forming loads.0.100.08E0. 3.2 3.6 4.0 4.4Time (mS)4.8 5.2 5.6Figure 6.8. Plot of the absolute and relative tracking error of each positioning system: noforming loads.706050E40 EC030 o0a201005.0 6.0 7.0 8.00.0 1.0 2.0 3.0 4.0lime (mS)Chapter6: CoordinatedMotion Control ofPress Ram 886.3.3.2 Motion Profile: With Forming OperationThe object of this experiment was to determine the stiffness of the positioningsystem: that is, the ability to maintain tooling orientation and achieve the desired finalpunch penetration in the presence of typical forming loads. As with the previouslydescribed experiment, each actuator was given a motion profile representing a typicalbending 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 andsecured 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 allowasymmetrical actuator loading. Two pressure transducers, one for each fluid chamber,were installed in the right actuator. A typical work piece and the finished part aredepicted in figure 6.9.A series of tests were conducted forming workpieces of varying thicknesses and1100mmWork piece J1tFinished Part 90)Figure 6.9. Sample work-piece and finished part used for brake-forming tests.Chapter6: CoordinatedMotion Control ofPress Ram 89widths. The total width and geometric centre of the bend, w and c respectively, wererecorded for each test. The hydraulic pressure acting in each chamber of the rightactuator was measured as were the reference and actual positions of each axis during thebend cycle. Sample results of this test are shown in figures 6.10 and 6.11Using the free body diagram shown in figure 6.12 the hydraulic pressure exertedby the right actuator was used to determine a static theoretical applied bending forcedistributionP =- [] (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 6.4W 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 1.812 300 1.5 500 0.65 9 12 13.5 7.3 69 0.64 1.623 600 1.5 500 0.65 13 15 29.9 16.2 77 0.44 0.924 150 3.0 725 0.49 13 38 26.5 27.6 360 0.47 1.385 200 3.0 700 0.51 24 47 41.4 40.2 408 0.59 1.176 200 3.0 700 0.51 21 51 42.7 41.4 421 0.50 1.24Table 6.4. Results of the Brake forming analysis.Note: the computation of the axis compliance values presented in table 6.2 is basedon an assumption that the entire positioning error is due to the disturbance load. Theresults of the previous section indicate that up to 4 to 5 urn of positioning error existsregardless of disturbance loads. Therefore, the axis compliance values determined forrelatively low load forming conditions may be somewhat overestimated. Given this, theChapter6: CoordinatedMotion Control ofPress Ram 900. 0.060)C 0.040i 0.020.00-0.02-0.04compliances of the left and right axis, averaged for the four larger bending loads weredetermined to be:C1, = 0.50 [] (6.42)C2 = 1.18 F1 (6.43)LkNJBased upon the system pressure and the piston area, an upper bound on the force capacityof each actuator was determined to be 50 [kN]. Considering the reduction in valve gaindue 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 4.4lime (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 919 608 Pa50012345678910lime (S)Figure 6.11. Actuator pressures recorded for motion profile with forming operation.ltIIttI11flFbaL:bJ___1Figure 6.12. Free-body diagram of forces acting on ram during the dwell operation.Chapter6: CoordinatedMotion Control ofPress Ram 926.4 ConclusionsIn this chapter, details of the control scheme were presented. A practicalsimplification to the pole-placement control law was developed and implemented. Theperformance of the positioning system can be summarized as follows:• The rise time of the response of the positioning system to a step change inposition 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 thanone 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 7Conclusions and RecommendationsIn this thesis a computer-controlled positioning system for a hydraulic press brakewas designed, modeled and analyzed. Both linear and non-linear system models wereapplied to cylindrical hydraulic actuators. Non-linear state-relations describing the changein actuator compliance with position were developed. An analysis of the actuator systemdynamics was conducted using a variety of models. The results indicate that simple linearmodels are capable of predicting the dynamic mode of an asymmetrical actuator if thevalve flow coefficients are known. However, the piston position and the direction ofpiston movement during critical positioning stages also need to be considered in systemdesign.Hydraulic components for the system supply and servo-actuator were chosen andimplemented.The gibbing of the press-brake was modified to allow rotation in the plane of theram. A position feedback system was designed, implemented and tested. No backlashwas detected.Experiments were conducted to determine the system dynamics of the hydraulicservo-actuators. Step response and frequency response tests indicate a first order modelwith consideration of delay is acceptable. Dynamic system parameters were determinedusing parametric identification techniques.A delay-compensating, pole-placement control law was designed for the modelstructures determined in the identification section. A universal simplification of the poleplacement control law for processes with inherent integration was developed and93Chapter 7: Conclusions and Recommendations 94implemented. This simplification eliminated output scaling errors while also reducingcomputational overhead. Motion profile tests indicated that dynamics of each axis wereadequately matched. Also, the system exhibited less than one basic length unit steadystate position error on a positioning move. Actuator compliances were determined foreach axis through actual bending tests.In order to create a satisfactory position system, this work has effectively outlined aprocedure for designing position control systems for CNC press-brakes. With regard tothis procedure, future work needs to be conducted in identification and control. For thisthesis, all identification experiments were carried out in open-loop. While this wasrelatively safe for this small press, it could be quite dangerous for much larger ones.Closed-loop identification techniques should be investigated.For improved motion control, integrating pole-placement controllers could be usedto enhance the stifihess of the positioning system if numeric sensitivity and valveirregularities were overcome. Also, cross-coupled control strategies should be examinedas a means of reducing axis tracking errors.Finally, since the stiffliess of hydraulic actuators is adversely affected by dissolvedgases in the fluid a system for determining the bulk modulus of a samples of fluid would bea tremendous asset.BifiLIOGRAPHY1. Pourboghrat, F. and Stelson, K.A., “Pressbrake Bending in the Punch-SheetContact Region-Part 1: Modeling Nonuniforinities”, Trans. of ASME, J. Eng.for md., Vol.110, ppl25-130, 1988.2. Anon. “Sheet Metal Bending Methods”, Accurate Manufacturing CompanyNews Release, EM- 105.3. 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Astrom, K.,J. and Wittenmark, B., Computer Confrolled Systems, Prentice-Hall, Inc., 1984.APPENDIX AActuator Natural Frequency CalculationsAn estimate of the natural frequency of the actuator is provided by equation 4.28:14J3A2w=I ep A.1Given: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 volumeat 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.]98Appendices 99APPENDIX BDerivation of Pole-Placement Control Law ParametersFor a system with an open-loop position response characterized by the discrete-timetransfer function:y(q) — B(q) — b1q’’ +b2q’+• •+b1B 1u(q) A(q)— q’”(q”+a1q’’’+...+a)’choose a desired closed-loop system response which has the same order and thesame number of delays:y(q) B(q) (bjq+b2)boYrf(q) 4(q) q1ta(aoq1 +amIq+.+am())where =1. For a unity gain system,1+— i1,nLlJl,flb+lIn order for the controller to be causal, the order of the observer polynomial mustsatisfy the following constraint:degA0(q) 2degA(q)—degAm(q)— 1 B.2deg4(q) 2(fla d)Qa +nd)—ldeg4(q)n+n —1If the control law takes a relatively small amount of time to implement we are free tochoose:Appendices 100degA0(q)=n+n—l B.2For low-noise transducers such as optical encoders, it is desirable to place theobserver polynomials at the origin for the fastest response such that:4 = qfliffldI B.2The 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 — ldAgain, if the controller output signal happens near the beginning of the loop closurecycle, we are free to choose:degR(q) = degS(q)=+ n—such that:R(q) =r0q’+r1q’’+•••+i B.2S(q) =s0q +s1q’+- •+s B.2where:= ‘la + ‘1dThe feed-forward filter is chosen:T(q) = bm04 =b0q’ B.2In determination of the control law coefficients requires the solution of theDiophantine 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 101For a = 2, b = 1, 11d > icollecting terms of B.l0 of like orders yields:q2(fl+nd)_1: a0r =1 B. 11q2(fl+fld)2: a0r1 +a1, =a B.12qfl+fld: aoIr +a1,4+ Zt2 +b1s0 =0 B. 13qfl+fldI: a1r + +b1s + b2s0=0 B. 14qa: a2r +b1s2 + b2s1=0 B. 15qfld•4: b1s3+b2 =0 B.16q’: +b2sfld =0 B. 17q°: b2sfld+l =0 B.18Assuming b1 0 and b2 0, solving B. 17 and B. 18 yields:5÷1=0 B.19S =0 B.20Back substituting into the series of equations inferred between B. 15 and B. 17 wecan determine that:B.21Solving B.13 for 5o=(a+±‘d-1) B.22Solving B.15 for s1:a2rs1=— B.23Solving B.11 -B.13 yields:Appendices 102B.24B.251=ama2—air B.26= —(a_2+a1t_), i = [3:n] B.27Solving B.22 and B.23 into B. 14 and solving for yields:b2 ( b2c1ld+a2Ic---’— B27b ba1 — a2 —- — —-2 b1AppendicesAPPENDIX C103Friction Characteristics of Guide SystemThe contribution of the guide system was measured for a variety of actuator speeds.The results are presented in figure C. 11.51.0z0.510.0___ ___ ____ ___ ___O5-1.0-1.5° Left Actuator0C Right Actuator0—0-— 0-00-0 —0---—00-30 -20 -10 0 10 20 30Piston Velocity, Vy (mmls)Figure C. 1. Friction force exerted by guide system.Appendices 104APPENDIX DModel Parameters VMr=104 [kg]B,.=0 [N.s/m]A=O.00456 [m2JA,. = 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 [mlF=(300,600) [N]p=690 [Mpa]pg=O [MpaJ;=0.005 [ms]g=9.81 [m/s2]


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