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UBC Theses and Dissertations

Computer control of a hydraulic press brake Lane, John Alexander 1994-12-31

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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 thesisin partial fulfilment of therequirements for an advanceddegree at the University ofBritish Columbia, I agree thatthe Library shall make itfreely available for reference andstudy. I further agreethat permission for extensivecopying of this thesisfor scholarly purposes may begranted by the head ofmydepartment or byhis or her representatives.It is understood thatcopying orpublication of this thesis for financialgain shall not be allowedwithout my writtenpermission.(Signature)Department ofCicL 9€E-LJThe University of British ColumbiaVancouver, CanadaDate2SAPRL 994DE-6 (2188)AbstractHydraulic press brakes are widely used inindustry to form sheet metal intovarious shapes using bending operations. When thepress brake is controlled by aComputer Numerical Control (CNC) unit,the desired bend can be achieved withouttheuse of mechanical gages or manual adjustments, leadingto increased flexibility andaccuracy in the manufactured components.In this thesis, a single axis hydraulic pressbrake is retrofitted for dual axisCNC control. The ram is positioned usingtwo parallel hydraulic cylinder typeactuators.One servo-valve and amplifier is dedicatedto each actuator so that the motion ofeach axiscan be independently controlled by the CNCunit. Each actuator is instrumentedwithlinear optical encoders which provide feedbackfor closed-loop servo positioncontrol.The hydraulic system is modifiedto provide constant pressure fluidpower to the servovalves. An accumulator is used betweenthe pump and the valvesto suppress pressurefluctuations. Pressure transducersare integrated into the portsof the actuator to monitorthe pressure during the operationofthe press.The dynamics of the hydraulic servosystem, including the valves, actuatorsandthe ram are modeled. A non-linear model,which includes the influences ofpistonposition, is presented. A simplified modelis shown to be adequateprovided that thepractical ranges of the piston position isconsidered. The mathematicalmodel is used toexperimentally determine the dynamics usingparametric identificationtechniques. Thewell-damped system is approximatedby a first order system with substantialdelaybetween the servo-valve amplifiercommand and the actuatorpiston motion. A readilyavailable CNC system is retrofittedto the press brake and a delay compensatingpoleplacement digital control system isdeveloped and implemented for theindependent11control of two actuators. The performance of the system isevaluated for a series offorming operations.This thesis provides basic guidelines for thedesign and analysis of hydraulicallyactuated CNC presses.111Table of ContentsPageAbstract.iiTable of ContentsivList of TablesviiiList of FiguresixAcknowledgmentsxiiChapter 1 Introduction1Chapter 2 Literature Survey32.1 The Brake Forming Process32.2 Hydraulic Press Brakes42.3 Hydraulic Supply Systems72.4 Cylindrical Hydraulic Servo Actuators9Chapter 3 Modifications Required forComputer Control143.1 MechanicaL Design Modifications143.1.1 Ram Gibbing143.1.2 Positioning System for CNC163.1.2.1 Design Objectives173,1.2.2 Mechanical Design173.2 Hydraulic System193.2.1 Supply System203.2.2 Servo-actuator213.3 Control Computer22Chapter 4 Modeling of SystemDynamics244.1 Introduction24iv4.2 Dynamic Model of Servo-actuator.254.2.1 The Load Pressure, LoadFlow Model294.2.2 Improved Frequency ResponseModel324.2.3 A Directionally BiasedModel for Asymmetrical Actuators354.2.4 Non-Linear Valve FlowRelation374.2.5 Non-Linear ActuatorCompliance384.3 Results424.3.1 The Effect of the LPLFLinearization on the SystemResponse 424.3.2 The Effect ofPiston AreaRatio the System Response434.3.3 The Effect of InputSignal Amplitude on theSystemResponse434.3.4 The Effect of InitialPiston Position on the SystemResponse 454.3.5 Effect of CoulombFriction on the SystemResponse 464.3.6 Validation of the Non-LinearModels474.4 Conclusions49Chapter 5 Identification ofServo-actuator Dynamicsfor Control505.1 Introduction505.2 Choice of IdentificationSignal505.3 System Responseto a Step Input525.4 Frequency ResponseExperiments545.4.1. Experiment Description545.4.2. Experimental Results545.5 Parametric Identification565.5.1 Theory565.5.1.1 Least Squares Method57v5.5.1.2 Instrumental VariablesMethod.595.5.2. Experiment Description605.5.2.1 Choice of IdentificationSignal605.5.2.2 Model Structure625.5.2.3 Data Analysis625.5.3. Experimental Results635.5.3.1 Model Selection635.5.3.2 Model Validation635.6 Conclusion71Chapter 6 Coordinated MotionControl of Press Ram726.1 Introduction726.2 Motion Control736.2.1. Objective736.2.2. Velocity Profile736.2.3. The Process tobe Controlled746.2.4 Control Scheme746.2.4.1 Introduction746.2.4.2 Design of the Pole-PlacementController756.2.4.3 Controller Implementation786.3 Positioning System Performance816.3.1 The Dynamic Responseofthe Ram PositioningSystem 826.3.2 Determination of PositioningSystem Dead Band836.3.3 Determination of ControlledMotion Performance846.3.3.1 Motion Profile:No Forming Operation856.3.3.2 Motion Profile:With Forming Operation88vi6.4 Conclusions.92Chapter 7 Conclusions and Recommendations93BIBLIOGRAPHY95APPENDIX A98Actuator Natural Frequency Calculations98APPENDIX B99Derivation of Pole-Placement Control LawParameters 99APPENDIX C103Friction Characteristics of Guide System103APPENDIX D104Model Parameters104VIIList of TablesPageTable 5.1. Delay, rise time and steadystate gain obtained from step response experiments53Table 5.2. Comparison of the loss functions computed fora variety of model structures 63Table 5.3. Model Parameters determined fromparametric identification64Table 6.1. Process models and control law parameters forleft axis controller81Table 6.2. Process models and control law parameters forright axis controller81Table 6.3. Velocity Error Constants for each axisas determined from motion profile tests: no formingloads86Table 6.4. Results of the Brake forming analysis89vi”List of FiguresPageFigure 2.1. The brake forming process3Figure 2.2. A typical press brake5Figure 2.3. Pressure Compensated Variable deliveryhydraulic supply9Figure 2.4. Components of a hydraulicservo-actuator10Figure 3.1. Ram orientation mechanism14Figure 3.2. Degrees of freedom of CNCpress brake15Figure 3.3. Ways box modification16Figure 3.5. Position measurement system forCNC operation18Figure 3.6. Schematic Diagram ofthe original hydraulic system19Figure 3.7. Schematic diagram of hydraulicsystem used for CNC operation20Figure 3.8. Technical illustration of servo-actuator21Figure 3.9. Block diagram ofthe HierarchicalOpen Architecture ManufacturingCNC system 22Figure 4.1. Typical hydraulic system usedfor servo positioning24Figure 4.2. Functional diagram ofservo-actuator25Figure 4.3. Comparison ofvelocity responsepredicted by LPLF model versusthe velocity predictedbythe LC model with piston area ratio R=142Figure 4.4. Comparison ofthe effect of piston area ratio, R, on thevelocity response of servoactuator,aspredicted by the LC model43Figure 4.5. Comparison of the frequencyresponse predicted by the LPLFmodel to that predictedby theIFR model at various servovalvearmature current amplitudes44Figure 4.6. Comparison ofthe Velocityresponse predicted by LC modelversus that predicted by theFCC model when the initial position differsfrom the LC linearizationposition 45ixFigure 4.7. Comparison ofthe response predictedby LC model versus that predictedby the FCC modelwhen the initial position corresponds tothe LC linearization position46Figure 4.8. Comparison of the effect of coulombfriction, Fc, on the velocity responseof the system aspredicted by the LC model47Figure 4.9. Comparison of the velocity responsepredicted by the LC and FCC modelsto the actualresponse48Figure 5.1. Velocity response of the left actuatorto a step change in valve commandvoltage 52Figure 5.2. Magnitude response ofthe right actuator in extension determinedfor a variety of input signalamplitudes55Figure 5.3. Phase response of left actuatorin extension determined for a varietyof input signalamplitudes56Figure 5.4. Effect of excitationsignal amplitude on the steady-state gainpredicted by the frequencyresponse experiments (left actuatorextending)61Figure 5.5. A comparison of theexperimentally determined frequencyresponse to that predictedby theidentified models: Left actuator extending65Figure 5.6. A comparison oftheexperimentally determined frequencyresponse to that predictedby theidentified models: Right actuator extending65Figure 5.7. A comparison of the experimentallydetermined frequency responseto that predicted by theidentified models: Left actuator retracting66Figure 5.8. A comparison ofthe experimentallydetermined frequency responseto that predicted by theidentified models: Right actuatorretracting67Figure 5.9. Comparison ofthe responseof the measured system to that predictedby the parametricallyidentified models: left actuator extending68Figure 5.10. Comparison of the measuredsystem to that predicted by theparametrically identifiedmodels: right actuator extending69xFigure 5.11. Comparison of the measured systemto that predicted by the parametricallyidentifiedmodels. Left actuator retracting70Figure 5.12. Comparison of the measured systemto that predicted by the parametricallyidentifiedmodels: right actuator retracting70Figure 6.1. Velocity profile for typical bendingcycle73Figure 6.2. Block diagram of a pole-placementcontrolled system75Figure 6.3. Press setup for positioning systemperformance experiments82Figure 6.4. The response of left and rightaxis of the positioning system to aseries of step changes incommand position. Yl: left axis; Y2: rightaxis, Yref: reference command83Figure 6.5. Response of PID controlled positioningsystem to step changes in referenceposition 84Figure 6.7. Response of positioningsystem to motion profile: no formingloads 87Figure 6.8. Plot ofthe absolute andrelative tracking error of each positioningsystem: no forming loads.87Figure 6.9. Sample work-piece andfinished part used for brake-formingtests 88Figure 6.10. Plot of the absoluteand relative tracking error of each positioningsystem: Motion profilewith bending operation90Figure 6.11. Actuator pressures recordedfor motion profile with formingoperation 91Figure 6.12. Free-body diagramof forces acting on ram during the dwelloperation 91Figure C. 1. Friction force exertedby guide system103xAcknowledgmentsI would to thank my supervisor Dr. YusefMtintasfor his guidance during thiswork. His patience and support is greatly appreciated.I would like to express special thanksto Grant Lindsay of Del ScimiederHydraulicsfor his helpfhl assistance and technical supportthroughout the project. I would also liketo thank Adrian Clark, Scott Roberts and Dr. MalcolmSmith for their helpful suggestionsand encouragement.The author wishes to acknowledge the supportof the several companies:Accurpress for providing the press brake forthis work, Parker Hannifin Corporationforsupplying hydraulic system components,and Atchley Valves for providingthe servovalves.xliChapter 1IntroductionA hydraulic press brake is a machine toolused to form bends in metal plate orsheet. In production, a typical press brake is setupto perform a single bend in a batch ofsimilar components. The bend angle is setby trial and error using an adjustablemechanical stop. This setup requires several attemptsand is only useful for one bendangle.By comparison, Computer NumericalControl (CNC) press brakes use closed-loopposition control which improves precision and simplifiesthe setup such that only one trialbend is needed. Since setup informationis stored digitally, CNC press brakescan beprogrammed to perform a number of differentbends at any particular instantin time.In this thesis, a single axis hydraulic pressbrake is retrofitted fordual axis CNCcontrol. The focus ofthis thesis is the servosystem used to position the formingtools.The objective ofthis work is twofold:• to reduce uncertainty during the designstage by investigating models usedtorepresent the system dynamics• to investigate ways of improving systemperformance in the face ofunexpecteddesign shortcomingsIn this work, a hydraulicservo positioning system is designed, modeledandanalyzed. A hydraulic supply for servo positioningsystem is designed and implemented.High performance servo-valves are mountedto the existing actuators. Mechanicalmodifications are made to allowindcpendent actuator motion. A positionfeedback1Chapter]: Introduction2system utilizing linear optical encoders is designed andimplemented. The dynamiccharacteristics of the system are experimentally verifiedusing system identificationtechniques. Based on the results of theseexperiments, a delay-compensatingpole-placement control scheme is chosen and implemented.The performance of the CNC pressbrake is evaluated.A survey of both research and industrialliterature concerning the brake-formingoperation, press-brakes, hydraulic system modelingand relevant automatic control theoryis presented in Chapter 2.Chapter 3 describes the work requiredto convert the manually controlledpressbrake to one capable of computercontrol. Details of the hydraulicand mechanical designsare presented and a description of the electricalhardware used for control.In Chapter 4, models used to representthe dynamics of the hydraulic positioningsystem are presented and applied. An alternateactuator compliance model isdevelopedand investigated. A summary ofgeneral recommendations for modelingis given.Chapter 5 describes the systemidentification experiments conductedon theposition control system. Results ofstep response, frequency responseand parametricidentification tests are presentedand discussed. A summary of conclusionsfor control ispresented.The delay-compensating, pole-placementcontrol scheme is described inChapter 6.A simplification to the control lawis developed. The performanceofthe positioningsystem is evaluated in terms ofsystem response, dead-band, followingerror, and stiffness.The results are presented and discussed.Finally, Chapter 7 is a summary ofconclusions arrived at throughthis work.Recommendations for future work aresuggested.Chapter 2Literature Survey2.1 The Brake Forming Process“Brake forming is a method of forming straight-linebends in sheets and plates” [1].In a brake forming operationa sheet metal workpiece is positioned betweena punch and adie. The bend is formed as the punch penetratesthe die. Although variationsexist, thereare two fundamental types of brake formingoperations. When the punch bottomsthework piece in the die the operation is knownas coining or bottom bending. Whenthepunch does not bottom the work-piece inthe die the operation is knownas air-bending[2]. The brake forming process is illustratedin figure 2.1.WORK PIECEFigure 2.1. The brake forming process.In the ideal bottom bending operation,the yield stress of the work piecematerial isinduced throughout out the area ofthe bend, causing the work pieceto take the shape ofDIE3Chapter 2: Literature Survey4the punch and die. While bottom bending generally produces the bestquality bends, itrequires very high actuation forces and specially mated tooling foreach angle of bend [3].In an air bending operation, the angle ofbend is determinedby the amount of punchpenetration. Therefore, air bending operations do not require speciallymated tooling.Furthermore, since the yield stress ofthe material is notinduced throughout the area ofthe bend, the actuation force required is two to five timeslower than that required bybottom bending [2]. Because ofthis 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 ofbend is that a certain amount ofelastic deformation remains at the final punch penetration. Thiselastic deformation causesthe work piece to ‘spring-back’ when the punch is withdrawnfrom the die. Compensationmodels, which are capable of approximating spring-backgiven the desired bend angle andthe material thickness, are frequently used [5].2.2 Hydraulic Press BrakesBrake forming operations are performed onmachines known as press brakes. Atechnical illustration of the Accurpress press brakeused for this project is shown in figure2.2. This machine is typical of those utilizedin the industry. The fundamentalcomponents of the press brake are depicted.The punch is clamped to a large ram which ispositioned by two hydraulicactuators. The actuators are driven in parallel by asingle or multistage solenoid operatedvalve system. Usually, the hydraulic circuit hasprovisions for two speeds: one for rapidpositioning and one for the forming operation. Thedie is fixed to the press bed. Thepositioning system controls the amount the punch penetratesthe die. Due to theextremely high forces applied, the ram gibbing is not sufficientto hold the punch parallelChapter 2: Literature Survey5to the die. As a consequence, press brakes use an auxiliary systemto ensure tooling isproperly oriented. The flrndamental differences between manuallyoperated press brakesand automated ones lie in these two systems, which ensure appropriatepositioning andorientation of the punch with respect to the die.RAMPUNCHDIEPRESS BED POSITIONINGSYSTEMWith manually operated press brakes, the positioningsystem and the orientationsystem are distinct mechanisms. The orientation mechanismmay consist of a stiffmechanical device or a sensitive hydraulic feedback circuitadjusted to minimize tilt of theram during asymmetrical loading conditions. Typicallythe positioning system consists ofa calibrated adjustable stop and an electronic switch whichtriggers a solenoid drivenhydraulic valve. When flow to the actuator isarrested or reversed, the punch penetrationis limited. The accuracy of this system is dependentupon the sensitivity of the switch aswell as the condition of the hydraulic system. Allhydraulic systems are subjecttovariations in fluid viscosity and bulk modulus. Thesevariations can cause significantdeviations in the final punch penetration over the runof a batch of parts.Figure 2.2. A typical press brake.Chapter 2: Literature Survey6On a two-axes CNC press brakes the orientationas well as the positioning tasksare handled by an integrated position control system. The flow ofhydraulic fluid to eachactuator is controlled by two independent precision singleor multistage spool valves. Therelative position of each actuator is measured by a high resolutionposition transducer.The movement of each actuator or axis is controlledby a digital feedback control scheme.Typically, CNC press brakes use gibbing which allowssome tilting of the ram but therelative orientation of the tooling is ensuredby linking each axis at the control level.Because these systems use feedback control the positioningis much less affected bychanges in the state of the hydraulic fluid. The benefitsof computer-controlledpositioning systems are many.For a given batch of parts requiring thesame bend angle, the setup procedure forapress brake involves three steps. First, the punch anddie are mounted and adjustedto becoplanar. Next, the clamping mechanism is adjustedto ensure the forming edges areparallel. Finally, the positioning mechanism is set togive the desired bend. Trial and errortesting ofthe forming operation is performed until thefinished bends are within tolerance.With manually operated press brakes, this procedureis repeated many time, not usuallyless than three [2]. By comparison, thesetup procedure for CNC press brakes ismuchsimpler. A tooling offset reference is setby bottoming the punch in the die, and apredictive model is used to calculate the desiredpunch penetration. After a testworkpiece has been bent, the error between thedesired work piece and the test piece isentered into the positioning controller and then thepress 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 brakecan readily switch betweenbends with no extra setup time.Chapter 2: Literature Survey72.3 Hydraulic Supply SystemsHydraulic servo-actuators used for positioning systemsrequire constant pressuresupply systems. Although constant supply pressureunder varying loading conditions isdifficult to achieve, Ahmed and Asok [6] have reportedthat slight (<10%) fluctuations inthe supply pressure have little effect on the responseof the servo-actuator. In order toachieve this constraint, some consideration must begiven to the design of the hydraulicsupply system.Systems that supply fluid power to hydraulicsystems can be classified into one oftwo categories: constant delivery systems or variabledelivery systems. In the simplestform, a constant delivery system designedto provide constant pressure can consist ofafixed displacement pump and a pressure reliefvalve(PRy). In order to maintain arelatively constant system pressure the pump wouldsupply a constant flow of fluid to thesupply line and the PRy. Whatever flow is notused by the servoactuator wouldpassthrough the PRV at maximum pressure drop. Althoughthis type of system is inexpensiveto implement, the cost of providing the maximum flowrate at the system pressure is twofold; the overall efficiency of the system is low, and thequantity of heat generated by thislow efficiency can breakdown the hydraulic fluid. Forthese reasons, constant deliverysystems are rarely used in servoactuator applications.Variable delivery systems have the advantage ofbeing able to provide only enoughflow to satisfy the requirements of the servo-actuator whileaccommodating some internalleakage. Depending on the type of pump beingused, variable delivery flow can beachieved in one of two ways. If a fixed displacementpump is used, the pump wouldprovide flow to a hydraulic accumulator which in turn would supplythe 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 thistype ofsystem is more efficient than a constant delivery system, a compromise mustbe madebetween the frequency at which the pump is cycled and the maximum pressure fluctuationallowed.Systems employing variable displacement pumps use pressure feedbacktocontinuously vary the flow to track the desired system pressure. Thefundamentalcomponents of a pressure-compensated variable delivery hydraulicsupply 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 spoolvalve to control theangular displacement of the swash plate. As the swash plate angleincreases, so does theflow delivered from the pump. When the pressure in the supply linedeviates from thedesired system pressure, the control valve adjusts the output flow-rateto maintian thedesired system pressure. Since the response time of thistype of pump is generally slower(50-l2Oms) than response time of a typical servovalve (4-2Oms) anaccumulator is used tosatisfy the flow requirements while the pump is responding to a demand formore flow.Although these systems are generally more expensive than the systemsdescribedpreviously, they are capable of providing the smoothest supply pressurewith a minimumof pressure fluctuation. For this reason, they are the most common supply systemusedwith precision servoactuators.Chapter 2: Literature Survey9tnternal Reservoir1jjJAccumulatorPressure Compensated Variable DisplacementPumpFigure 2.3. Pressure Compensated Variabledelivery hydraulic supply.2.4 Cylindrical Hydraulic ServoActuatorsA schematic diagram of a hydraulicservo-actuator used for CNC press brakesispresented in figure 2.4. The load is connectedto the output shaft of a hydraulic cylinder.The cylinder consists of two chambers: the piston-sidechamber and the rod-side chamber.For high performance systems, flowto each chamber of the cylinder is typically controlledby a high-precision multistage critically-centered spoolvalve often called a servo-valve.An electronic amplifier supplies a command signalto the valve. The output velocity of theload is proportional to the servo-valve commandsignal. 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 Survey10Previous investigations of cylindrical hydraulic servo systems havefound that very littleviscous damping is attributable to the actuator[7,17]. When thecompliance ofthehydraulic fluid within the actuator is considered, a symmetrical actuatorat mid-strokebehaves as an under damped second order system [13].Asymmetrical actuators can bemodeled by a third order system [11]. Depending on thegibbing or guide ways used forthe system, significant amounts of Coulomb damping mayalso be present [18].The dynamics of a two-stage servo-valve canbe represented by a simple lag due tothe torque motor driving the primarystage 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 andthe 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 Survey11The ftmndamental difficulties in controlling servo-actuators stemfrom fivephenomena:i) the nonlinear pressure/flow relationship for theflow 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 relationshipcan have significant affect onthe large signal response of the servo-actuator,it is generally considered insignificantfortypical operating control signals. Therefore, it iscommon practice to linearize the flowcharacteristic about the null operating point, where thespool is centered and the portsclosed. This represents the worst-case scenario forclosed loop stability because thevalvegain is highest and the flow damping is lowest. Alternatemethods of analysis have beendeveloped to predict the frequency responseof the system which varies with theamplitudeof the input signal [13].In cases where the response of the control valve is muchfaster than the responseofthe loaded actuator, significant research effort hasbeen expended to overcome the lackof viscous damping [9,17, 19]. Some of the earliest schemesto improve damping involvedthe introduction of laminar leakage across the chambersof the actuator. These systemswere simple to implement but reduced the stifibess andefficiency 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 requireddamping and increased the actuatorstiffness, but these systems required precision manufacturingoperations which werespecific to each valve/actuator/load combination. Thecost and flexibility oftheseChapter 2: Literature Survey12methods were improved when electronic feedback of accelerationand pressure signalsreplaced hydromechanical feedback. Modern digital controllersare capable of addressingthis problem in a number ofways, the most common being pole-placementcontrolschemes[201.The presence of Coulomb damping has the combined effect of reducingtheoscillatory nature of the system response while also contributingto the steady state error.To compensate for the absence ofviscous damping, early analyses madeattempts 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 isa function of the bulkmodulus of the hydraulic fluid and the ratio of oil volumeresiding on each side of thepiston. Since the volume ratio changes with movementofthe piston, the compliancevaries as well. To simplif’ modeling, the commonpractice is to linearize the compliancerelation about the most compliant position. For asymmetricalactuators the change in oilvolume ratio with piston movement is more pronounced. Whilestudies have beenconducted to determine the effect of changing oil volumeon the frequency response andstability of simple control systems [11], typical controlstrategies either assumea constantvalue for actuator compliance, or assume the dynamicsof 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 ofstate of the hydraulic fluid. Hydraulicfluids used in servo-actuators have two undesirablecharacteristics: 1) the viscosity of thefluid changes dramatically with temperature and 2) the bulk modulus(or stiffness) ofthefluid changes dramatically with the quantity of dissolvedgas in the fluid. Over the longterm, regular wear of precision components will causea change in the characteristics ofChapter 2: Literature Survey13the system. Moreover, long term wear is accelerated inthe presence of contaminates inthe hydraulic fluid.Given the existence of these phenomena, methods toinsure precise positioncontrol have recently been investigated using digital adaptivecontrol schemes. Thesecontrollers have been labeled ‘switching’ adaptive controllersbecause the control law isupdated at a frequency which is an order of magnitudelower than the ioop closingfrequency. Typically the adaptive controller schemeseither used a method of recursiveleast squares to identify reduced order models of theopen ioop system dynamics (selftuning regulators) or identified the control law parametersdirectly (model referenceadaptive control) [20,22]. Provisions were madeto check the stability of the system andtemporarily halt estimation ifthe input signal ceasesto be ‘persistently exciting’. Severalimplementations used an exponentialforgetting factor and/or covariance matrixresettingin the recursive identification to track time varyingparameters. The most commoncontrolstrategy chosen was pole-placement, but adaptive optimalcontrollers have also beeninvestigated [23,25].Chapter 3Modifications Required for Computer ControlThis chapter describes the work done to change thepress brake from a manual toCNC control.3.1 Mechanical Design Modifications3.1.1 Ram GibbingThe press brake used in this project was originally designedto allow only onedegree of freedom of ram movement: vertical translation. The gibbingof the press (figureLink Side PlatesTorque TubeGuide-barWaysbox0RamFigure 3.1. Ram orientation mechanism.14Chapter 3: Instrumentationfor Computer Control153.1) consisted of a pair of ways boxes which were rigidly connectedto the ram, sliding ona pair of parallel guide bars, one bolted to each side plate. To maintainprecise orientationof the tooling while undergoing the extreme forces of the formingprocess, an auxiliarydevice was employed. A torsional link known as a torque tube,was mounted between theside plates of the press. This torque tube was connectedto a ways box on each side of theram by mechanical links.One goal of the retrofit was to add the capabilityto create bends with up to three(3) degrees of trim without adjusting the tooling. Thisrequired the addition of a seconddegree of freedom of ram motion: rotation within the planeof the ram (figure 3.2).In order to allow this rotation, the torque-tubewas 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 motiona cylindrical sliding surface wasmachined in the lower ways box pad (figure 3.3). Thepads 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 Control16Flat padsOriginal WaysboxModified WaysboxFigure 3.3. Ways box modification3.1.2 Positioning System for CNCThe positioning system originally installedon the press brake is depicted in figure3.4. This system consisted of a parallelguide mechanism which housed limit switchesused to signal the change-of-speed and the ram-stopposition. These switches weretripped by a moving slide which was connectedto the ram by way of a tie-rod mechanism.A micrometer barrel, attached to the moving slide,was used to fine-tune the finalstopposition. This design was chosen to compensate fordeflections which occur in the sideplates of the ram during bending. While the theoreticalprecision ofthis system wasbounded by the resolution ofthe micrometer (0.012 mm)and the repeatability ofthe stopswitch, the actual precision may have been worst dueto the presence of a low frequencydynamic mode of oscillation with a frequency of21 Hz.forCylindrical PadChapter 3: Instrumentationfor Computer Control17Micrometer BarrelStop SwitchMechanismFigure 3.4. Positioning system used for manuallyoperated press brake3.1.2.1 Design ObjectivesComputer controlled ram motion requiresa system to measure the position andorientation of the ram. The goals for thedesign 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 (similarto the original system) was chosen (figure 3.5).To simplif’ the ram orientation task, the tie-rodposts were placed at the axes of theram/actuator connections. High-precision ball-joints wereused to connect each tie-rod toTie-Rod PostTie-rodChapter 3: Instrumentationfor Computer Control18its respective tie-rod post and slide mechanism. A lower bound on thelength of the tie-rodwas determined by bounding the measurement error due to tie-rod rotation.Tie-rod PostConnection PinTie-rodEncoder Head SliderEncoder BodyGuidesFigure 3.5. Position measurement systemfor CNC operation.In order to achieve the 0.01 (mm) positionrepeatability, a linear encoder with aresolution of 0.005 (mm) was chosen. This encoder consistsof a body which houses afixed scale and a moving read head. The body was mountedto the press bed and the readhead was attached to the translating slide of thelinear guide system which was designedand manufactured in-house. The guide system utilizesa pair of cylindrical guides, onefixed, the other adjustable. This built-in adjustmentcapability eases alignment procedureswhile providing a method of preloading the guide systemfor greater rigidity. Provisionswere made to allow the attachment of a transducerto measure the velocity of each side ofthe ram with respect to the bed.The tie-rod, tie-rod posts, and the slider were designedto have high stiffness andsmall mass to minimize the effect of their dynamics on the feedbacksignal.Chapter 3: Instrumentationfor Computer Control193.2 Hydraulic SystemA schematic diagram ofthe original hydraulicsystem is shown in figure 3.6Hydraulic flow was provided by two fixed displacementgear pumps: one for highpressure, the other for high flow. A pair of normallyopen solenoid valves, weresequenced such that both pumps provided flow duringthe rapid moves, but only the highpressure pump provided flow for the feed operation.A counter balance valve locatedinthe rod-side line was used to lock the actuatorduring idle times.HII3HFor CNC press operation, the original hydraulicsystem was replacedby 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 originalhydraulic system.Chapter 3: Instrumentationfor Computer Control203.2.1 Supply SystemFor high efficiency and good performance,a variable displacement pump withpressure feedback was chosen to deliver thehydraulic power. Since the response time ofthe pump was significantly slower than thatof the servo-valves, an accumulator wasusedto keep the system pressure within 10% of the desiredlevel. This accumulator also dampsharmonic components ofthe pump pressurecaused by the oscillating pump pistons.A check-valve was placed between theaccumulator and the pump to eliminatethepossibility of the accumulator pressure driving the pumpin reverse in the event of a poweroutage. A counter-balance valve was usedto lock the system whenever the systempressure is lost.Figure 3.7. Schematic diagram of hydraulicsystem used for CNC operation.Chapter 3: Instrumentationfor Computer Control21Two ifiters are used to filter the fluid: a coarse low pressurefilter located in thereturn line of the original system, and a fine, high-pressure filter in thesupply line to keepmetallic pump debris from reaching the sensitive pilotstage of the servo-valves.3.2.2 Servo-actuatorA technical illustration of the servo-actuator used for positioningis shown in figure3.8. The compact design of the press precluded theuse of an offthe shelf servo-actuator,so the original cylinders were used with only minor plumbingchanges. A pair ofprototype Atchley 320 servo-valves (20 g.p.m., 85Hz bandwidth[261)were mounted asclose as possible to the ports of the original actuator.Special in-line transducer fittings were designed andmanufactured to house piezoelectric transducers capable of measuring cylinderport 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 theHierarchical OpenArchitecture Manufacturing CNC system (HOAM-CNC) developedin-house by the CNCresearch group of the University of British Columbi&sMechanical Engineeringdepartment. A block diagram ofthis controller is shown in figure 3.9.r486PCLSYSTEM MASTER_JMAIN Bus: ISA BusC30 DSPCNC MASTERCNC Bus: DSP LINK80C196KC180C196KC1Axis ControllerJ Axis ControJFigure 3.9. Block diagram of the Hierarchical Open ArchitectureManufacturing CNCsystem.The HOAM-CNC system uses a hierarchical structureto divide control tasks upbetween three computational systems: the system master, theCNC 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, andcontroller status variablesare updated. At the mid-level, the CNC master is used to initialize and coordinateeachChapter 3: Instrumentationfor Computer Control23axis 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 thetop of theHOAM-CNC hierarchy is a PC based host computer which providesa user interface andmass storage. For a detailed description of the HOAM-CNC systemsee [27].In order to optimize positioning performance each axis of the HOAM-CNCwasmodified to allow the use of a high-bandwidth, scaled control signal. Thenine bit outputconmiand voltages were fed to a pair of Parker BD-98 gain amplifiers whichwere used todrive the servo-valves. Quadrature encoder signals from the positioningsystem wereconnected directly to inputs on the axis controllers.Chapter 4Modeling of System Dynamics4.1 IntroductionIn order to satisfy given performance constraints, the design ofa servo-mechanismrequires detailed information about how a series of individualcomponents will performtogether. Dynamic system models can provide this information if thecharacteristics ofeach component are known. A typical hydraulic circuitused for servo-positioningsystems (figure 4.1) can be analyzed as two distinct subsystems:a servo-actuator systemand a hydraulic supply system. The hydraulic supply consistsof a pressure compensatingvariable displacement pump supplying compliant hydrauliclines. The servo-actuatorsystem consists of a two-stage servovalve connectedto a hydraulic actuator which in turnmoves the ram. In the following sections, the physics ofthe servo-actuator system is/Figure 4.1. Typical hydraulic system used for servopositioning.24Servo-actuatorChapter 4: Modeling ofSystem Dynamics25described and models used to represent the dynamics are presented.4.2 Dynamic Model of Servo-actuatorThe servo-actuator used to position each joint of theram can be considered as asystem consisting of three distinct subsystems:a precision flow-controlvalve, 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 positioningsystem is a two-stage servovalve.The primary stage consists of an electronictorque motor driving a primary valve, usuallyof a flapper or a jet-pipe type (flapper-valvetype shown in figure 4.2). When the torquemotor is driven, the primary valve providesa differential pressure across the ends ofasecondary closed-centre spool valve. Thedisplacement ofthe secondary spool is fedbackto the torque motor by way of a cantilever spring. Thisspooi controls the flow to thecylinder. The position of the flow controllingspool as a function ofthe torque motorcurrent can be represented by a first-order lagdue to the torque motor combined withaquadratic lag due to the dynamics of the spool:FextChapter 4: Modeling ofSystem Dynamics26x(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 fasterthan the dynamics of the torquemotor so that (4.1) can be approximated by the followingexpression:x(s) K,(42)ia(S)(r1s+1)’where:K: effective spool valve positioning gainThe flow through a port of the secondary-stage spoolvalve has been found to beproportional to the area ofthe valve opening and the square root of thepressure dropacross the port. Since the area ofthe valve opening isproportional to the valvedisplacement, the following expressions can be usedto represent the flow through theports of a spool valve:forxO,q,, =KqaXviJPs —psign(Fpa)(4.3)b=KX/IPb—pts1gn(pb—J)(4.4)for x <0,q, = Kqa X,q1IPa — Pt IS1(Pa —(4.5)Ib= Kqb XJp — Pbsign(p — Pb)(4.6)where:Chapter 4. Modeling ofSystem Dynamics27q: 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 consistsof two fluid chambersseparated by a piston. The rod connecting the pistonto the rod is assumed to be rigid.Because the fluid on either side of the piston is mildlycompressible, hydraulic cylindersexhibit compliance under load. This compliance (or itsinverse, stiffness) is a concern toservo-system designers because it limits the maximumbandwidth of the system.To determine a relationship between the chamberpressures, chamber flows andpiston velocity, the continuity equation canbe applied to control volumes encompassingeach chamber to obtain the following expressions:qa_qcKlp(PaPb)=ApVy (4.7)qb qQ, +K,P(PbPa)+1<lePb=rVY(4.8)where:v: velocity ofthe pistonA: area of the pistonAr:area of the piston minus the area of therodKb,: 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 BcomplianceSince fluid compressibility is directly proportionalto the volume in which thepressure acts, the compliance of each actuator chamber maybe 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 linearizedabout a particularoperating point (L0), the following expressions for the rateof 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, thefollowing 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 epressionsmost models use to studycylindrical hydraulic servo-actuators.Chapter 4: Modeling ofSystem Dynamics294.2.1 The Load Pressure, Load FlowModelMost of the research concerning cylindricalhydraulic servo-actuatorspertains tothe case where there is equal area on eachside of the piston: i.e. symmetricactuators.Although most hydraulic pressesdo not employ such actuators, an analysisof such modelsis worthwhile because they providethe simplest estimation of systemperformance.Merritt [14] presents a thorough analysisof symmetric hydraulic actuatorsutilizing asimple linear model based on a loadpressure, load flow (LPLF)simplification.In order to reduce (4.1-4.13)to a linear model numerous assumptionsneed to bemade. By assuming zero tank pressure,no cavitation and a symmetricactuator, the flowsinto and out of the spool valve canbe equated so that the followingexpression,normalized with respectto the supply pressure, holds:q,=KqXvjIl_PL,(4.14)VP8where:q1: effective load flow(q—q)Kq: normalized valve flowgainp1: effective load pressure(Pa — Pb)Ifthe spool only undergoes smalldisturbances about the nullflow position, (4.14)can be linearized to yield:q1 — Kq; — Kp,,(4.15)where:ICE:null flow pressure gainSince this linearization yieldsa zero value for the null flow pressurecoefficient, this valueis usually determined empiricallyfrom closed-port leakage tests [14].To extend this LPLF analysisto the actuator, the load is assumedto undergorelatively small excursions from aset operating point. Furthermore,Wit is assumed theChapter 4: Modeling ofSystem Dynamics30fluid in each chamber is of equal compliance,(4.7-4.10) can be linearizedand combinedto yield:q1=_Av+K,p+CL,(4.16)where:K1 = K,1 +CQbV1: total volume of fluid within the cylinderand linesEquation (4.16) can be combinedwith (4.15) and rearrangedto yield, using Laplacenotation:Kx+AvPzj 1’(4.17)CabS+Kctmwhere:Kctm=K,+K. (4.18)Considering the equation definingthe load dynamics (4.13),some assumptionsmust be made to accommodatethe coulomb damping term.In the past coulomb dampinghas been modeled with viscousdamping 21,23], but modemanalysis considers coulombfriction to be an external disturbance.Ifthe damping is assumedto 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 transferfunctions for the response oftheservoactuator to the spooidisplacement and the external force:v(s)Kx(s)= 2+2,w,s+co,’(4.20)Chapter 4. Modeling ofSystem Dynamics31v,, (s)= KFt (rabs+1)(4 21)F(s) s2+2Cco,s+w,where:J +---I (4.22)1A ‘ v,4Ap\lI3eMr‘ ‘= I’e (1+”m’)(4.23)V:M,.42413AKepq(4.24)XMr=4Kj3(4.25)KF= 1(4.26)‘t•abMrIt has been noted that the contributionof viscous damping in thistype of drive isvery small [9,17,19]. Hence,it is common practice to assumeBr= 0. Using thisassumption, (4.22) and (4.23) reduceto:c= Kctm1]I3eMr, and(4.27)1413A2w=Ie(4.28)VtMrwhich indicates two phenomenacharacteristic of all valve-controlled actuators.First, fora cylinder with minimal externalleakage, the damping is proportionalto the cross portleakage (null position valveleakage and piston leakage). Second, thestiffness oftheactuator is proportional to the bulkmodulus of the fluid and the piston areawhile beinginversely proportional to the stroke.Because of their inherent simplicity, (4.27)and(4.28) are often used during the preliminarystages of servoactuator design [28]. IntuitiveChapter 4. Modeling ofSystem Dynamics32variations of (4.27) and (4.28) exist for thedesign of servoactuators utilizingnon-symmetrical hydraulic cylinders.The chief virtue of the LPLF modelis its conservative estimate ofsystemperformance. Since the derivationmakes use of a control valve linearizationabout theoperating point with the least damping (thenull position), the LPLF modelgives a lowerbound for damping within the system. Furthermore,since the actuator complianceexpressions relations (4.9) and (4.10) arelinearized about the most compliantpistonposition, this model also provides a lower boundfor the bandwidth ofthesystem.4.2.2 Improved FrequencyResponse ModelWhile the LPLF model canbe used to determine the frequencyresponse of aservoactuator system, McCloy andMartin [13] present a moresophisticated model whichconsiders the effect of the spoolvalve displacement on thedamping of the system. Thisimproved frequency response (IFR)model is based upon the LPLFmodel.Assuming a pure inertia load, (i.e.,no damping and no externalload), and asymmetric actuator (piston area A,,)the expression describing theload dynamics (4.13)can be simplified to:dvMr_L=Ap(Pb_Pa),(4.29)or,Mdvp,=__4LL.(4.30)Rewriting the non-linear pressure/flowrelationship for the LPLF model (4.14)in nonnormalized form yields:q, = Kx.JI —p1. (4.31)Chapter 4: Modeling ofSystem Dynamics33K,=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,(L0), the followmgexpression for the load flow is obtained:q,= Kxf(1+.j;‘;;(4.33)or the more general form:q,=Kqxv(1+AMrL),(4.34)Ap dtwhere:A: flow/pressure linearization constantMcCloy and Martin [13] suggest using a valueof A in (4.34) other than 1/2.The criterionused to select A is minimum error between the actualpressure/flow relation and thelinearized relation (4.34). This criterion yieldsa value of 2/3.Assuming no actuator leakage (K= 0), (4.34) can be combined with (4.16)toproduce:Mdv VMd2VKx(1+Ar‘)—Avt r Y(4.35)q “Ap3 dt“4J3e’pcit2or, by rearranging:d2 d K’‘+—2-(4A‘ vPe)+v (4Pe,)= —K x (4PeP)(4.36)dt2 dtVp‘VtMrVVtMrChapter 4. Modeling ofSystem Dynamics34For a harmonic input with a fixed amplitudeof 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 15Mqe r(439)Vand, as in the LPLF model (4.27 and 4.28):14$A24/3AK=jeand K= e p q(4.40)VtMrXVtMrIn order to produce a harmonic spool motion,a harmonic current signal ofthe form:1a(t)= ‘asin(ot),(4.41)has to be applied to the torque motorofthe servo valve. When thedynamics of theprimary stage of the servovalve areincluded (4.2), an expressionfor the frequencyresponse of the load velocity withrespect to the servovalve inputsignal can be written:v (flu)= I<KX(442)a(J)CO,2— 2—2’rco,o(I)+j(tjoio,—t7w3+2WiWf(Ia))where:KKI15Mq Iale r(4.43)PSAPVFrom (4.42) and (4.43) it is evidentthe IFR model can be used to determinethefrequency response for a particularamplitude of excitation. However,as with the LPLFmodel, the IFR model assumes symmetricalactuators. If the ratio of the cap-sideto rodside piston area (R) is not unity, amore appropriate model may be required.Chapter 4. Modeling ofSystem Dynamics354.2.3 A Directionally Biased Model forAsymmetrical ActuatorsAlthough many investigations of the dynamicscharacteristics of servosystemsemploying non-symmetrical actuators havebeen conducted, most either ignore theeffectsof fluid compressibility[24] or assume equalfluid chamber compliance.An exception tothis trend is presented by Watton [29].In this study, the stability andstep response of aproportionally-controlled, symmetrical servoactuatorare studied for cases where theratioof the volume of oil on each side of the pistonis not unity. An application of theanalysisused in this investigation follows.Given the pressure/flow relations (4.3-4.6),linearized relations for the flowthrough each port ofthe spoolcan be written:for x O,qa=Kqxv-K0apa(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 whenx 0KqaKqbA & B port flow gains whenx,, <0KKCb:A & B port pressurecoefficients when x 0Kca,Kct,:A & B port pressure coefficientswhen x <0or, more generally,(4.48)q=dKx+dKp(4.49)where:d :‘+‘when x, 0Chapter 4: Modeling ofSystem Dynamics36d :!lwhen x <0If the actuator flow expressions, (4.46-4.49), arelinearized about a particularpiston position, and external leakage is neglected,the following expressions canbewritten: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 rewrittenusing Laplace notation:dKx +A v+K,pbPa= Ca5+(dKca+Kip)(4.54)— dKqbXv+ Av— ‘ipPa4 55Pb —— CbS+(dKCb+ Ic)( . )Considering (4.13), the following expressiondescribing the dynamics of theload can bewritten using Laplace notation:MrVyS+BrVy = ArPb — ApPa +F (4.56)Combining (4.54-4.56) yields the followingtransfer fhnctions:vt(s) =—dK(S+1)(45)x1,(s) s +a1s +a2s+a3v(s)1 s2+b1s+b(4.58)F(s)Mr3+as+awhere:Chapter 4. Modeling ofSystem Dynamics37B K Ka1= _r_+ —f- + —-Mr Ca Cba2= (KaKbK12) (‘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+ K1Kb=°’Kth+ K,It has been shown that (4.57) and (4.58) reduceto the LPLF expressions (4.20)and (4.21) respectively, when the piston area ratio isunity and the actuator volumecompliances,CaandCbare equal [29].4.2.4 Non-Linear Valve Flow RelationThe models presented so far, which have commonly been usedto design hydraulicservo actuator systems, all assume a linearized flow/pressurerelationship. Although it hasbeen concluded that this assumption is valid for a significantrange of load pressures andvalve openings, the accurate prediction ofthe large signal responseof a hydraulic servorequires the non-linear flow/pressure characteristics of thesecondary spool valve to beconsidered. To this end, several non-linear models have been developed.The following isChapter 4. Modeling ofSystem Dynamics38a description of a particular model used to numerically simulate the responseof theservoactuator system.From the manufacturer’s information, the response of the spool ofthe valveto 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 rateof change of volume due to fluidcompressibility (or the compliance flows):for; 0,ca= Kqa xJIF— psign(p— Pa)A,, Vy— Kip(PaPb)’(4.60)= ‘qbXV.,)[Pb—p(sign(p —ps)— ArVy- K,P(pbPa)— KlePb(4.61)for; <0,= Kq; —pSign(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 actuatorcompliance with piston position hasonly a minor effect on the response of symmetrical actuators[11]. However, forasymmetrical actuators, the relative changeof compliance for a change in piston positionis magnified by the ratio of the area on eachside of the piston. To investigate thisphenomena, two alternative models were developed: onebased 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 controlvolume is equal to the integral ofthe flow entering the volume divided by the compliance ofthe fluid in the volume. Forcontrol volume A this assertion is represented by the following expression:Chapter 4. Modeling ofSystem Dynamics39Pa(t)= f1(.)ca(t)dt(4.64)Since the compliance of the control volume (4.9) is a function ofthe piston position,(4.64) can be rewritten:Pa(t)= f3ej(+ (_))dt (4.65)Differentiating (4.65) and solving for the complianceflow yields (dropping the timefunction notation):q,,—_4—(v,1+A(L_y)). (4.66)A similar expression can be derived for controlvolume 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 relationsfor the FCC model:for; O,dPapKqaxv]ps_paIsign(ps_pa)_Klp(pa_pb)+ApvY(4.68)dteVa,+Ap(L—y)KqbXvgPb —pslgn(PbPt)lp(PbPa)ePbr1’y(4.69)diVbi+ArYfor; <0,J3KqaxviJpa—ptsign(pa_p,)_Kp(pa_pb)+ApvY(4.70)dteVai+Ap(Ly)=1eKqbXViJIPs_PbSIgn(PPb)+K,P(Pb_Pa)jePb+4r’y(4.71)di Vj+AyChapter 4: Modeling ofSystem Dynamics40The derivation of the pressure causal compliance (PCC)model is based upon asomewhat different assertion: the rate of change ofthe volume of fluid compressed (thecompliance flow) is proportional to the rate of changeof the product of the complianceand the pressure. For control volume A this canbe written as:q(t)=_(C(t)p(t))(4.72)Substituting (4.9) forCa(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)) A474ca —13e— Pa ( . )For control volume B, this analysis yields:- ‘Pb(Vb,+ Ary)Ar4751e+PbjjVY(. )For the PCC model, substituting (4.74)and (4.75) into (4.62) and (4.63) andsolving forand yields the following actuator controlvolume pressure relations:cit dtforx0,dKqaXvIPs_PaI5ifl(Ps_Pa)_KP(Pa_Pb)+APVY1+PP)I3ee(4.76)dt V,1+A(L—y)KgbxvgIPbPtIsign(Pb_Pt)+Klp(Pb_Pa)+KIePb+Arv,1+)Pbp e(4.77)dteVbj+ArYChapter 4. Modeling ofSystem Dynamics 41for; <0,Kqa XvJIPa PtIS11l(Pa —pj—K1(Pa Pb)+ Avy1+ I?)Pa_13 e(4.78)dte1+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) canbesimplified to the expressions derived using the FCC model (4.68-4.71).While the load relation (4.13) is valid while the piston is moving, it isdesirable 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: <“cdArPb 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 arrangedas a series ofnonlinear state equations which can be solved numerically to determinethe system response toa series of inputs.The values of the system model parameters are presented in Appendix D.Chapter 4: Modeling ofSystem Dynamics42I,,EE>‘>>4-C)0a)>4.3 Results4.3.1 The Effect of the LPLF Linearization on theSystem ResponseA comparison between the step response predicted by the LPLF model andtheresponse predicted by the non-linear LC model for a symmetric actuatorwith 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 2002500Time (mS)Figure 4.3. Comparison of velocity response predictedby LPLF model versus the velocitypredicted by the LC model with piston area ratio R=1.predicted by the LPLF model is significantly more oscillatorythan that predicted by theLC model. This can be attributed to the fact that theLPLF model is based uponalinearization about the valve null position, the position exhibiting theleast damping. Forthe valve closing step (125 <t < 250ms) both modelspredict an oscillatory response, withthe LC model exhibiting greater damping. Also, fromthis plot, the capability ofthe LPLFmodel to predict the steady-state velocity ofa symmetrical actuator is exhibited.Chapter 4: Modeling ofSystem Dynamics43100-20-30F6o-70-804.3.2 The Effect of Piston Area Ratio theSystem ResponseA comparison of the velocity response of two actuatorsystem: one with asymmetric actuator, the other with an asymmetric actuatoris shown in 4.4. While theresponse of the two systems is similar, the steady statevelocities are different and thesystems oscillate at slightly different frequencies.4.3.3 The Effect of Input Signal Amplitude onthe System ResponseIn order to examine the effect of the amplitude ofthe servovalvecommand signalon the response of the system, the frequency responsepredicted by the LPLF model wascompared to the frequency response predictedby the WR model for various values ofinput armature current. The results are presentedin figure 4.5. From this plot, itisevident that the system response exhibitsmuch greater damping as the amplitudeoftheR=1*R=1.8**- Compliance relation linearizedabout most compliance pistonposition, Lc=80.4mm.- Initial piston position, Lo=Lc.50 100 150 2000250Time (mS)Figure 4.4. Comparison ofthe effect of piston arearatio, R, on the velocity response ofservoactuator, as predicted by the LC model.0—50—100—150—200—250—3000Figure 4.5. Comparison of the frequencyresponse predicted by the LPLF modelto thatpredicted by the [FR model at various servovalvearmature current amplitudes.Chapter 4: Modeling ofSystem Dynamics44input signal is increased. Also, as the amplitude of the input signalapproaches zero thefrequency response predicted by the IFR model approaches thatpredicted by the LPLFmodel. This phenomena can be explained by the fact thatthe LPLF model’s damping termis linearized about the null flow position which is approached as theamplitude of the inputsignal is reduced. The system bandwidth (-3 dB) isapproximately 30 Hz.5040302010Fzq.iexcy (Hz)20050 100 150200Frqi.iezioy (Iz)Chapter 4. Modeling ofSystem Dynamics45100-20C)-40a)>C0-600-70-804.3.4 The Effect of Irntial Piston Position on theSystem ResponseIn this section, a comparison of step response predicted by the non-linearservo-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 ofthe LCmodel is linearizedabout the critical piston position (i.e., the position yielding the maximumactuatorcompliance). In the first comparison, figure 4.6, theresponse predicted by the LC model iscompared to the response predicted by the FCC modelwhen the initial position is thecritical position. In this case, the models compare quitefavorably. In the secondcomparison ,flgure 4.7, the initial position ofthe actuatoris varied from one limit to theother. In this case the FCC model predictsa much better damped response than theLCmodel. These plots indicate two things: i) the systemsresponse tends to be less oscillatoryLCModel*,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 2000250Time (mS)Figure 4.6. Comparison ofthe Velocity response predictedby LC model versus thatpredicted by the FCC model when the initial position differsfrom the LC linearizationposition.Chapter 4: Modeling ofSystem Dynamics46away from the most compliant position, ii) the LC model should be linearizedabout apiston position in which the actuator will operate. However, ifthe higher order dynamicsofthe actuator become significant, the LC model will not provide anaccurate descriptionofthe system dynamics away from the linearization point.100[10-20-30-600-70-800 50 100 150200 250Figure 4.7. Comparison of the response predictedby LC model versus that predictedbythe FCC model when the initial position correspondsto the LC linearization position.4.3.5 Effect of Coulomb Friction onthe System ResponseThe positioning system has been found to have significantnon-linear frictioncharacteristics (see Appendix C). The dominant non-linearcharacteristics can be modeledby coulomb friction. In order to investigate the effect of thistype of friction on the systemresponse characteristics, the system was simulated using theLC model for variousamounts of coulomb friction (figure 4.8). From this plotwe can observe that the quantityof coulomb friction in the system has a negligible effect on thesteady-state velocity of theLC Model*- - - - - - - -- FCC Model***- Compliance relation lineazedAabout most compliance pistonV1position, Lc.-**- Initial piston position, Lo=Lc.Time (mS)Chapter 4: Modeling ofSystem Dynamics47system while the control spool is open, but contributes substantiallyto the damping of thesystem when the control spool is closed.Since the valve-closing condition is the most critical interms 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 ofcoulomb friction, Fc, on the velocity responseofthe system as predicted by the LC model.4.3.6 Validation of the Non-Linear ModelsA comparison between the measured response of theservoactuator system to amulti-step input and the response predictedby the LC and FCC models is presented infigure 4.9. From this plot we can observe that theLC and FCC models match the trend ofthe actual system reasonably well. However, the actualsystem response does not exhibitan oscillating response as predicted by the two models. This discrepancycan be attributedNo CoulombFriction*Fc=300N.*Fc=600N.**- Compliance relation linearizedabout most compliance pistonposition, Lc=80.4mm.- Initial piston position, Lo=Lc.0 50 100 150200Time (mS)Chapter 4: Modeling ofSystem Dynamics48to unmodeled leakage across the piston seals which was not readily obtainableusing theequipment available.60—. 40E2:.2-20a)>.40“--60Time (mS)500Figure 4.9. Comparison of the velocityresponse predicted by the LC and FCCmodels tothe actual response.Also, the actual system response is generally a bitslower than the responsepredicted by the models, particularly for the case whencontrol valve is required to delivermore flow. While some of this sluggishness canbe attributed to the aforementionedunmodeled piston seal leakage, the difference in responsetime exhibited between the valveopening and valve closing cases canbe attributed to unmodeled flow non-linearities withinthe servovalve which cause its response timeto be dependent upon the amplitude of theapplied current.Note that the steps in the velocity response of theactual 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 300400Chapter 4: Modeling ofSystem Dynamics494.4 ConclusionsA model for the compliance of cylindrical actuators has been developed andapplied. Both the supply system and the servo-actuator have beenanalyzed. Thesignificant results of this analysis can be summarizedas follows:• Neglecting valve dynamics, symmetric actuatorsrespond as second order systemswith damping which is proportional to the amplitude of the valveopening.• Ifthe oil volume on each side of the piston is nearlyequal, a second order model canalso be used to represent the dynamics of an asymmetricactuator.• Coulomb friction at the load tends to damp the actuatoroscillations when the valve isin the critical position.• A positioning system designed using linear theoryshould be analyzed for a range ofpiston positions.Chapter 5Identification of Servo-actuator Dynamics forControl5.1 IntroductionWhile dynamic modeling is a useful tool for selectinghydraulic system componentsand determining the general order of the assembled system, theeffects of valve deadband,hysteresis, stiction, spool leakage, and transport delay make theprecise modeling of thedynamics of hydraulic systems difficult. To accommodate forthese effects, a number ofexperiments can be conducted on the system in order toobtain a better representation ofthe system dynamics. These are known as systemidentification (SI) experiments and thepractical result is an approximate linear model whichcan be used for the controller design.In this chapter, the three methods chosen to identif,’ thesystem dynamics are discussed.5.2 Choice of Identification SignalFundamentally, all system identification experiments involvetwo simple steps: 1)system input excitation and, 2) observation of thesystem response. Regardless ofthetype of SI experiment, the input signal used must becapable of exciting the relevantdynamics of the system. In order to choose an appropriateinput excitation signal forahydraulic system, some consideration mustbe given to practical limitations. For systemssuch as press brakes which position largemasses, care must be takento avoid potentiallydamaging excitation induced vibrations.For a closed center spool valve, aninvestigation of the Improved FrequencyResponse (IFR) model reveals two important phenomena:50Chapter 5: Ident/Ication ofServo-actuator Dynamicsfor Control511. The response of a spool-valve controlled hydrauliccylinder becomes moredamped as the amplitude of the input signalis increased.2. The overall gain of the spool-valve is largestat 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. Sincethe null-flow position is the criticaloperatingpoint for a position control system, it is reasonableto assume that the smaller the deviationfrom the null-flow position, the more relevant the identifiedmodel will be to the task ofcontroller design. Further, the IFR modelpresented earlier shows that an equivalentlinear frequency response model of the non-linearservo-actuator can be achievedif aperiodic wave form of constant amplitudeis used. Given this, a lowamplitude periodicwave form with a mean amplitude of zerowould seem to be ideal.Unfortunately, valve deadband and hysteresisaffect the valve dynamics mostat thenull position. Furthermore, the amplitudeof the input signal hasto be large enough sothat the steady state flow gain identified isnot biased by the valve-spool stiction.Parkerand Desjardins [31] have suggested thata Pseudo Random BinarySequence (PRBS) inputsignal of an amplitude of at least 10% of themaximum amplitude is sufficientto eliminatethe effects of non-linear valve gain and spoolfriction. In practice, theamplitude ofexcitation signal required will vary with eachsystem.Watton [29] has shown that the steady stateflow gain (SSFG) of anasymmetricalactuator extending will not beequal to the SSFG of the same actuatorretracting.Furthermore, if significant Coulomb frictionis present at the load,and the input excitationsignal chosen causes the loadto change direction, the system will undergoan external loadexcitation. Such system excitations are undesirableduring identification experiments.Therefore, if accurate estimations of thesteady-state velocity response ofthe actuator areto be obtained, separate identification experimentsshould be conducted for the extendingand retracting cases.ChapterS: Identfication ofServo-actuator Dynamicsfor Control525.3 System Response to a Step InputIn order to determine the velocity response of each actuatorto a step change invalve command voltage, two sets of experiments were conducted: one in extensionandone in retraction. For each experiment, an alternatingstep input signal of non-zero meanand not passing through zero voltage was applied to the servo-valveamplifier, and thevelocity response of the load was measured. Each experiment wasrepeated for a numberof differing input amplitudes.The response of the hydraulic system toa typical step change in valve input signalduring extension is shown in figure 5.1. Approximatefirst-order system parametersobtained from the step response experiments are presented in table5.1. Note that theextension response differs somewhat from the retractionresponse. From this table we can!160_______-14k-180CommandE-16j-200. >VelocityC.)-18220>-240 ..-20-260-22-280-240 50 100 150 200 250Time (mS)Figure 5.1. Velocity response of theleft actuator to a step change in valve commandvoltage.Chapter 5: Identfication ofServo-actuator Dynamicsfor Control53see that while the steady state velocity and delay do not change significantlywith theamplitude of the excitation signal, the rise time does. Further,the rise time for theextending cases varies somewhat from that of the retractingcases.Left Actuator ExtendingValve Opening ValveClosingStep Amp (V) Gain (mmN.s) Rise (ms) Delay(ms) Rise (ms) Delay (ms)0.100 7.93 12.05 9.2 40.150 9.53 15.04 7.9 40.250 9.48 12.54 7.8 3Left Actuator Retracting Valve_OpeningValve ClosingStep Amp (V) Gain (mmN.s) Rise(ms) Delay (ms) Rise (ms) Delay(ms)0.100 6.75 15.44 7.330.150 7.7416.8 4 8.430.250 7.53 13.44 7.2 3Right Actuator Retracting ValveOpening Valve ClosingStep Amp (V) Gain (mmN.s) Rise (ms)Delay (ms) Rise (ms) Delay(ms)0.100 5.806.1 4 5.940.150 6.00 5.84 5.8 30.250 6.03 6.24 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 steadystate gain obtained from step responseexperiments.Chapter 5: Identfication ofServo-actuator Dynamicsfor Control545.4 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 extendingand. retractingcases. For each actuator, an input signal was applied to servo-valveinput, and thevelocity signal was measured. A PRBS input signalwas 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, reversingthe direction oftheload. Because the WR model indicates that the frequencyresponse of such hydraulicsystems varies with the amplitude of the excitationsignal, these experiments wereconducted for a range of excitation signal amplitudes.The data from each experiment was analyzed usinga transfer function analysis.5.4.2. Experimental ResultsThe magnitude and phase responses of the left actuatorin extension is shown in figures5.2 and 5.3. Note that the peaks at 60Hzfor the lower amplitude signals are duetoelectrical noise observed in both the command signalas well as the transducer signal.From these results, a few observations canbe made:1. As the amplitude ofthe excitation signal is decreased,the steady-state gain of thesystem decreases as well. This can be attributed tothe increased effect of valvecomponent stiction at low valve actuationforces as discussed in [31].2. The band pass frequency of the velocityioop is approximately 30 Hz. This comparesfavourably with the band pass predictedby LPLF model.Chapter 5: Identfication ofServo-actuator Dynamicsfor Control558276:.70wz6458520 20 40 60 80 100 120140 160 180Frequency (Hz.)—0-——50 mV—0-—— 100 mV—&—— 200 mV400mV600 mV800 mVFigure 5.2. Magnitude response of the right actuator inextension determined for a varietyof input signal amplitudesFrequency (Hz.)10 20 30Figure 5.3. Phase response of left actuator in extensiondetermined for a variety of inputsignal amplitudes0040 50-135—0-——50 mV—0—-—100 my—&--—200 mV400mV600 mV800 myChapter 5: Identfication ofServo-actuator Dynamicsfor Control563. There exists a dynamic mode due to load dynamics near 140 Hz. As theamplitude ofthe input signal is increased, the frequency of this mode decreases.This frequencydecrease can be attributed to an increase in valve damping as predictedby the IFR model.4. At low amplitudes of input signal, the system exhibitsa dynamic mode near 100 Hz.Since this mode disappears at higher amplitudes, it is likely causedby the higher orderdynamics ofthe valve spool which, like the hydraulic cylinder, isunderdamped only forsmall valve openings.5.5 Parametric Identification5.5.1 TheoryIn order to determine a more precise descriptionofthe dynamic system model, anumber of parametric identification experimentswere conducted. Due to its abilityto giveunbiased estimates under less restrictive conditions,the Method of InstrumentalVariableswas chosen over the Least Squares Methodto determine the model coefficients. A briefdescription of theses identification schemesfollows.Soderstrom and Stoica [32] present excellentdescriptions 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 LSmethod first. Least Squares MethodThe discrete time transfer functionfor the open-loop velocity response ofahydraulics servo actuator has the form:vy(q’) = B(qj(5.1).u(q)A(qwhereChapter 5: Identfication ofServo-actuator Dynamicsfor Control57B(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 parametervector 0 from a data set ofNmeasured regressors(5.7)pT(N)and the measured outputv(1)(5.8)v(N)If a set of independent measurement errorswith zero mean and variance ?L,2 (white noise)exist such that:ChapterS: Identfication ofServo-actuator Dynamicsfor Control58e(1) v(1)—(pT(1)OE =(5.9)e(N) v(N)—çoT(N)Othe least squares estimate of 0 (denoted as 0), is that which minimizesthe sum of thesquares of the measurement errors:V(O)=.te2(k)(5.10)It has been shown that if the loss function, (5.10) hasa 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 excitationsignal which is ‘persistentlyexciting’. In simpler terms, the input signal used to constructthe data set must be capableof exciting the particular dynamics of the system one wishesto identify. Given that theabove conditions hold, 0 has been shown to convergeto 0 for large N.However, if the elements of the measurement errorvector s are not linearlyindependent, 0 has been shown to converge to a biasedestimate of 0. Soderstrom andStoica [32]present this phenomena as follows. If thetrue response ofthe system is givenbyv(k) = çoT(k)O÷ w(k)(5.12)where w(k) is stochastic disturbance, the differencebetween the true parameters and theestimated parameters can be written as:N—=o(k)Tco(k)][411jco(k)Tw(k)](5,13)Chapter 5: Ident/Ication ofServo-actuator Dynamicsfor Control59As the number of samples N tends towards infinity, equation (5.13) will not convergetozero unless the expectationEço(k)Tw(k) = 0 (5.14)If a correlation exists between the measurement error and theregression vector,5.14 will fail. While this bias may be small for systems with a high signalto noise ratio,other methods, such as the Instrumental Variable method have beenshown to provideunbiased estimates in the presence of correlated measurement errors. Instrumental Variables MethodThe IV method augments the LS method by the introductionof 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 theinstruments:{h1y(1_)J1yQt_uia)](5.15)are determined from the measured output. Thus the functionto be minimized becomes:v(e) = .t(z(k)T e(k))2 (5.16)and the IV estimate of the parameter vector is:N NT= [z(k)T(k)][Z(k)v(k)](5.17)The determination of the instruments involves filtering ofthe measured output andtherefore requires an apriori estimation of the stochastic disturbanceterm w(k).Typically this estimation is performed on an independentdata set using the LS method.For this reason, IV methods require a significant amount morecomputation time thansimple LS methods.ChapterS: Ident/1cation ofServo-actuator Dynamicsfor Control60The approximately optimal IV method was chosen to identif,’the dynamic systemparameters in this experiment. For a detailed description of this methodsee Soderstromand Stoica [32].5.5.2. Experiment DescriptionFor this experiment, a small amplitude excitation signalwas applied to the servo-valve amplifier. This excitation signal as well as an output voltagefrom the velocitytransducer was sampled at 1 millisecond intervals. The collecteddata was analyzed usingMatlab’s System Identification Toolbox [33]. Experiments wereconducted for eachactuator extending and retracting. Choice of Identification SignalOf the two types of signal considered, pseudo randomnoise and a pseudo-randombinary sequence (PRBS), the PRBS was chosen becauseofthe superior signal to noiseratio effected in the velocity transducer. ADC bias (as in the frequency responseexperiment) was added to the PRBSto prevent reversing the direction ofthe load.Inorder to determine the lowest amplitudesignal capable of minimizing the non-linearvalvecharacteristics a number of frequency responseexperiments were conducted usingexcitation signals of differing amplitudes.The open-loop velocity gain predicted foreach amplitude of input signal appliedfor the left actuator in extension is shown in figure5.4At low input signal amplitudes, the velocityresponse is dominated by valve nonlinearities. As the amplitude of the excitation signal isincreased, more consistentestimates of the open loop gain are obtained. From theseexperiments, an input signalamplitude of 600mV, or 3% of the maximum valve input,was chosen. This amplitude, theChapter 5: Ident/ication ofServo-actuator Dynamicsfor Control61lowest input signal amplitude capable of overcomingthe valve stiction, was within therange of input signals required during the forming operation.850 0 U84U‘—83 08200. 81C).280>0.o0278e.77°76750 100 200 300 400500 600 700800PRBS Input Signal Amplitude (my)Figure 5.4. Effect of excitation signalamplitude on the steady-state gainpredicted by thefrequency response experiments (left actuatorextending). Model StructureIn order to effectively determinethe dynamics system parameters usingtheparametric identification techniques describedabove, information about the modelstructure must be known. From the frequencyresponse analysis, the third ordersystemdynamics predicted by the IFR model canbe observed: a dominant first order lagdue tothe valve dynamics combined with slightoscillatory mode due to the actuator/loaddynamics. From the step response analysis,a first order response with 3-4 ms delaycanbe observed.ChapterS: Identfication ofServo-actuator Dynamicsfor Control625.5.2.3 Data AnalysisFor each situation (left actuator extending, left actuatorretracting, right actuatorextending, right actuator retracting) a number oftrials wereconducted. For each trial theservo-valve command and the voltage across the velocity transducerwere sampled at 1millisecond intervals. The 4000 point data sets collected were splitinto 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 ‘approximatelyoptimal’ IV method was appliedfor a number of model stmctures of first and thirdorders with two to five milliseconds ofdelay. Using the validation data set, the response foreach identified model was comparedto the actual system response. A prediction error function consistingof the sum of thesquares of the error between the actual system responseand the simulated response wascomputed to compare the relative goodnessof fit for each model.5.5.3. Experimental Results5.5.3.1 Model SelectionA comparison of the loss functions for eachstructure of model fit to thedata ispresented in Table 5.2. The structures correspondingto the minimum value of lossfunction are shown highlighted.The identified model parameters forthe candidate structures are presented inTable5.3. Note that model parameters for two alternatecandidate structures pertainingto theextension cases have been included.ChapterS: Ident/Ication ofServo-actuator Dynamicsfor Control63Model B(c[1)=+Loss FunctionA(q’)‘1+a1q+. .+aq’Left Actuator Right ActuatorExtending Retracting Extending Retracting1 1 2 2065 2376 8146201 1 3 718 1394 6034921 1 4 1084 2133621 10521 1 5 1734 2936 108325531 1 6 2036 3021 187634083 3 2 1495 3275260 14823 3 3 234 219 3531493 3 4 54 80118 1123 3 5 205155 548 2983 3 6 467 6051104 627Table 5.2. Comparison of the loss ftinctions computed fora variety of model structures. Model ValidationIn order to ensure that the identifiedparameters are reasonably accuratedescriptions of the relevant system dynamics, twosets of validation experiments wereconducted: a frequency response comparison and stepresponse comparison.For the first comparison, the frequencyresponse predicted by each of thecandidate models for each case was comparedto the frequency response determineddirectly from sampled data.The measured frequency responseofthe left actuator extending iscompared tothat predicted by the identified candidate models infigure 5.5. Although overestimatingthe steady state gain, the first order model with threesampling delays gives an excellentdescription ofthe process dynamics forfrequencies between 10 and 65 Hz. Thefirstorder model with four delays accurately predicts thesteady state gain, but generally overestimates the response of the system for frequenciesbetween 10 and 110 lIz. Thethirdorder model with four delays is capable ofrepresenting the dynamics modedue to theChapter 5: Ident/Ication ofServo-actuator Dynamicsfor Control64load/actuator dynamics, but the amplitude at the resonant mode is significantlyoverestimated.Case ‘1a bd[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, -28082361 -818.4]Table 5.3. Model Parameters determined fromparametric identificationThe measured frequency response ofthe right actuator extending is compared tothat predicted by the identified candidate models in figure5.6. The first order model withthree sampling delays tends to overestimate the responseofthe system from steady stateto 15 Hz, while underestimating it everywhere else.The first order model with four delaysaccurately predicts the steady state gain, gives areasonably accurate description of theresponse of the system for frequenciesbetween 0 and 100 Hz. The third ordermodel withfour delays gives a reasonable description of the systemdynamics from steady stateto 90Hz, although it incorrectly predicts the dynamic mode.The measured frequency response of the left actuatorretracting is compared tothat predicted by the identified candidate models in figure5.7. While both modelsaccurately predict the steady-state gain ofthe system, the third order model with fourdelays better represents the dynamics throughout therange 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 determinedfrequency responseto thatpredicted by the identified models: Left actuatorextending.878481>7269660 20 40 60 80 100 120 140160 180Frequency (Hz.)Figure 5.6. A comparison of the experimentallydetermined frequency responseto thatpredicted by the identified models: Rightactuator 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 overestimatesthe frequency response of the dynamicmode.\Figure 5.7. A comparison of the experimentallydetermined frequency responseto thatpredicted by the identified models:Left actuator retracting.The measured frequency responseof the right actuator extendingis compared tothat predicted by the identified candidate modelsin figure 5.8. The first order modelgivesa good description of the response of thesystem from steady state throughto 85 Hz. Thethird order model givesa good description of the system dynamicsfrom steady state upto60 Hz, but beyond this gives an poorestimate of the dynamics andincorrectly predicts theof the dynamic mode.For the step response comparison,the response predicted by eachof the candidatemodels for each case was comparedto a measured response froma 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 160180Frequency (Hz.)Chapter 5: Ident/1cation ofServo-actuator Dynamicsfor Control678784. 81‘7269660 20 40 60 80 100 120 140 160180Frequency (Hz.)Figure 5.8. A comparison oftheexperimentally determined frequencyresponse to thatpredicted by the identified models:Right actuator retracting.The measured response of the leftactuator extending is comparedto that predictedby the identified candidate models in figure 5.9. Whilenone of the models testedconstitutes an excellent fit, the third order modelseems to give the best representationofthe dynamics and the steady-stategain. Both first order modelsgive reasonable fits withneither being outstanding at all points onthe comparison.The measured response of theright actuator extending is comparedto thatpredicted by the identified candidate modelsin figure 5.10. Again, while noneof themodels tested constitutes an excellent fit,the third order model seemsto give the bestrepresentation of the dynamics andthe steady-state gain. The first ordermodel with fourdelays predicts a response very similarto the third order model, but withoutthe higher-order dynamics. The first order model withthree delays tends to overestimatethe steady-state gain while underestimating the rise timeofthe system.Experimental— -— First Order Model: d=3Third Order ModelChapter 5: Ident/Ication ofServo-actuator Dynamicsfor Control68.—.. -5-15-20>-25-300200Figure 5.9. Comparison of the response ofthe measured system to that predictedby theparametrically identified models: left actuatorextending.The measured response of the left actuatorretracting is compared to that predictedby the identified candidate models in figure 5.11. Inthis case, the system is exhibitingthird order or higher dynamics. Althougha slight DC bias seems to exist, the third ordermodel gives the best fit. The response predictedby the first order model with three delaysseem to fit the first order dynamics satisfactorily despitethe same DC bias.The measured response ofthe right actuatorretracting is compared to thatpredicted by the identified candidate modelsin figure 5.12. As was thecase with the leftactuator retracting, the system response is clearlythird order or higher witha higher modefrequency of approximately 135 Hz. Whilegenerally underestimating the amplitudeof thehigher order oscillations, the response predictedby 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 dynamicsas well as can be expected.040 80 120160Time (mS)Chapter 5: Identfication ofServo-actuator Dynamicsfor Control690I:>-20-25Figure 5.10. Comparison of the measured systemto that predicted by the parametricallyidentified models: right actuator extending.15EEioActual Response First Order ModelThird Order ModelActual First FirstThirdResponse Order OrderOrderModel: Model: Modeld=3 d=40 40 80120 160Time (mS)20025200 4080 120 1600-5200Figure 5.11. Comparison of the measured systemto that predicted by the parametricallyidentified models. Left actuator retracting.Time (mS)Chapter 5: Identfication ofServo-actuatorDynamicsfor Control7015EE[050Actual Response - — — — — First OrderModel Third Order ModelFigure 5.12. Comparison of the measuredsystem to that predictedby the parametricallyidentified models: right actuator retracting.200 40 80120 160Tirne(mS)200ChapterS: Identfication ofServo-actuator Dynamicsfor Control715.6 ConclusionFrom the identification experiments anumber of conclusions can bedrawn:• At very small input signals, the steadystate gain of the servo-actuatordropssignificantly and the dynamics of theload are not significant.• As the amplitude ofthe input signal is increased,the steady-state gain of theservo-actuator increases and an oscillatorymode at 13 0-150 Hz dueto theload dynamics is more apparent.• The system response of the servo-actuatorsin the extending cases canberepresented adequatelyby a first order model.• The response of the servo-actuatorsin the retracting cases canbe characterizedby a dominant first order lag combined witha low amplitude dynamic mode atapproximately 13 0-150 Hz.Chapter 6Coordinated Motion Control of Press Ram6.1 IntroductionThe objective of the efforts described in thischapter was to design and implementa practical control scheme which accomplished smooth positioningand orientation ofthepunch with respect to the die throughout the bend cycle. In orderto facilitate coordinatedmotion between the two actuators, a control schemecapable of matching the closed iooppositioning dynamics of the axes was chosen. Modelparameters presented in the previouschapter were used exclusively for the controller design.Given the cost sensitivity of manufacturinga machine tool for the industrialmarket, efforts to improve the positioningperformance focused on software rather thanonhardware. Although velocity and pressuresignals were available and could haveimprovedthe robustness and performance of the system, only positionfeedback was used so that theincremental cost of CNC implementation would be less. Theresults indicate that as longas the control law is properly designed, the system behavessatisfactorily without pressureand velocity feedback signals.In this chapter, the term ‘following error’ refersto the difference between thedesired position and the actual position ofeither axis during movement at a particularvelocity. The term ‘tracking error’ has been used todescribe the difference betweenfollowing error of each axis. The term ‘steady-state-positioningerror’ refers to thedifference between the desired position and the actualposition of either axis when thedesired position is not changing and system dynamicsare not prevalent.72Chapter6: CoordinatedMotion ControlofPress Ram736.2 Motion Control6.2.1. ObjectiveThe motion control task attemptedto achieve two fundamentalobjectives:1. less than 0.25 mm tracking error duringthe forming process2. a steady-state positioning error of less than0.0 15 mm under formingload.6.2.2. Velocity ProfileA plot ofthe desired punch velocity profilefor a brake-forming cycleis depicted infigure 6.1. The profile consists of foursegments:1. rapid approachto the clearance plane above the work-piece2. brake-forming operationat feed-rate3. dwell in bend4. rapid retract to the start positionTime (Seconds)Figure 6.1. Velocity profile for typicalbending cycle.The performance of the positioningsystem during the bend anddwell operationsdetermines the precision of the press.In these cases, the actuators are eitherextending or6040‘ 20EE00e> -20-40-60Apph[4\\\\/____Dwell0.0 1.0 2.03.0 4.0 5.0Chapter6: CoordinatedMotion Control ofPress Ram74holding their position against a forming load. For thesereasons, the actuator modelscorresponding to the extension cases wereused for controller design.6.2.3. The Process to be ControlledThe results of the identification experimentshave shown that the response oftheactuators can be characterized by a first order lagcombined with an under damped modeat approximately 140 Hz. However, for the extensioncases, the higher order dynamicsare less prevalent especially for the small inputsignal amplitudes (100-150 mY)usedduring the forming operation. Moreover, thefirst order models generally providegooddescriptions of the dynamic system performanceat frequencies upto 100 Hz.Furthermore, considering the computationalcomplexity required to compensatefor higherorder dynamics, the first order models werechosen for initial design efforts.6.2.4 Control Scheme6.2.4.1 IntroductionIn order to achieve low trackingerror, a control scheme whichwould facilitatematching the dynamics of each positioningsystem was desirable. In order toobtain lowsteady-state position error, the closed looppositioning system would requirehigh gain tomaximize load force disturbance rejection.Since the open-loop system dynamicscontained several delays, a delay compensationscheme was deemed necessaryto allowhigh controller gain. A pole-placement controlscheme, similar to that usedby Watton[22] was chosen because it utilizesa performance-based design procedureand is capableof compensating for process delays.A block diagram of a pole-placementcontrolled system is shown in figure6.2.The process to be controlled is representedby polynomials of open loop system zeros Band the poles A. The controller consists ofa feed-forward filter T, a feed-back filterS,Chapter6: CoordinatedMotion Control ofPress Ram75controller poles R and an estimation filter (or observer)A0. These four filters are selectedsuch that the controlled system responds accordingto a chosen response:Figure 6.2. Block diagram of a pole-placementcontrolled system.yBTBm(6.1)AR+BSAmwhere4contains the poles, andBmthe zeros of the desired closedloop transfer thnction(CLTF).A detailed description of the pole-placementdesign process can be found in[34].A description of how the pole-placementcontrol scheme was adaptedto the positioncontrol task follows. Design of the Pole-PlacementControllerThe open-loop velocity responseof each of the servo-actuatorsin the extendingcases can be represented by the following discretetime transfer fhnction:YrejProcessyControllerChapter6: CoordinatedMotion ControlofPress Ram76vy(q_1)— qd(b0’+b1’cf’)Fcountsu(q’)— 1+a1’q [vsec(6.2)wheredis the number of delays, b1’,b2’, and a are the identified system parameters, and1 count/second equals 0.005mm/s of actuatorvelocity.Adding an integration term to this expressionyields:y(q’)— q’d(b1 +b2q’)Fcountsu(q1) — (1+a1q1+a2q2)[(6.3)or, for convenience, using the forwardshift operator:y(q) (b1q+b2)Ecounts6qna(q2±q1±)Lv(.4)The desired system response was chosento be that of a damped secondordersystem with a delay equivalent to the open-loopsystem delay:y(q) — B(q) — (b1q+b2)b0[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 thepositioncontrol loop has a unity steady state gain:b1+ami+am266mOb1+b2Since it is desirable to use only theposition transducers for processfeedback, anobserver A0(q) is required to estimateany higher order feedback termsrequired. In orderto ensure a causal design analysis, Astrom andWittenmark [34] have shown thatthefollowing conditions mustbe 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, theobserverpolynomial 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 polynomialorders 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+Ib0A(q) = (6.17)Solving the Diophantine equation symbolically for the controllerparameters (Appendix B)yields:f(nd=i),ri=amiai (6.18)(“2‘“2=—a1--—a2)r±a-—(6.19)a1 — a2 — a0 —2b1(6.20)(6.21)Chapter6: CoordinatedMotion Control ofPress Ram78s. =0, i = 2,3,.. •nj +1 (6.22)f(nd>i),r=amiai (6.23)r2=ama (6.24)for] > 2,jd’= —(a1i+a2,) (6.25)(( b Ia1---—a2ji+a2--?d_IIb 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-placementcontrol 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 seriesnotation yields:(6.31)where n, n andrare the respective orders ofthe 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 Ram79would require(d+ 4) multiplication operations and(d+ 4) add operations per ioopclosure. Note that this implementation requires precise representationof 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 controlledsystem can be represented bythe following relation:bmy(l)= iO=(6.33)‘)ami=Oji k=OjoFor systems with an inherent integration such as DCmotor 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-placementcontrolled system can bescaled by appropriate choices of S and T. Furthermore,if the desired steady stategain isunity, then:(6.35)For ,z =0 and ,z3 = 1, 6.35 can be rewritten:Chapter6: CoordinatedMotion Control ofPress Ram80= to, (6.36)Substituting the relation:Ay(k) = y(k) —y(k— i)(6.37)into the control law (6.32) and rearrangingyields:(1 \u(k)=t0yrej(k)— —1)—s0Ay(k) — ru(k— j)(6.38)1=0j=1Using (6.36), the control law can be furthersimplified 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 operationsand(d+ 3) add operations per ioop closure.This control law implementation(6.39)accomplishes the following:• it eliminates the possibility of thesteady-state error due to numericalround-off duringscaling computations while guaranteeingappropriately scaled output• it reduces the number of controlparameters that need to be represented,lesseningmemory requirements• it reduces the precision in whicht0 and s0 need to be represented,further lesseningmemory requirements• it reduces the number and complexityof operations required for implementationThis control algorithmwas implemented on two axes ofthe HOAM-CNC usingIntel 80196 assembly language.A pair of overdamped CLTF poles(4) were chosensuch that the system responsewas fast enough to satisfy the following-errorconstraintswhile not deviating significantly fromthe control valve dynamics. The controllawparameters as well as the process parametersused for design are shown in tables6.1 and6.2.Chapter6: Coordinated Motion Control ofPress Ram81Left Axis (Yl) Pole-Placement DesignParameters[B] b0 +b1q’1573 702[A]= 1+a1q+a2q1 -1.871 0.871[Am]=1+a1q +a2q1 -1.429 0.4724[T]=t0 2.36[S] = S0 +s1q21.62 -19.260[R]= i+,q1+.. 1 0.4430.429 0.418 0.408-0.064Table 6.1. Process models and controllaw parameters for left axis controller.Right Axis (Y2) Pole-PlacementDesign Parameters[B] = b0 +b1q982 505[A]=1+a1q’+a2q1 -1.861 0.861{A]=1+aq’+a1 -1.429 0.4724[T]=t02.85[SJ=s+s1q21.07 -18.225[R] = t+rq’+..+tq’1 0.398 0.373 0.3520.335 -0.076Table 6.2. Process models andcontrol law parameters for rightaxis controller.6.3 Positioning System PerformanceThe equipment setupused for the performance experiments isdepicted in figure6.3. The object of these experimentswas to evaluate the positioning system’sperformancein terms of dynamic response, positiondeadband, following error, trackingerror and theeffects of load force disturbance (compliance).Chapter6: CoordinatedMotion Control ofPress Ram826.3.1 The Dynamic Response of the Ram PositioningSystemA simple step response test wasconducted to verify the dynamics response ofeach axis.For this test, each axis was given a reference commandsignal consisting a series ofsteps.The step amplitude was set as large aspossible while avoiding controller saturation.Thecommand signal and the measured position responseare shown in figure 6.4. Negativesteps represent actuator extension.The results of thistest 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 actuatorwas observed to overshootapproximately 0 -2% occasionally during retraction.This slight overshoot was deemedacceptable given that it does not occur during the formingoperation. Moreover, the effectof any response overshoot would be mitigated somewhatby velocity profiling.Figure 6.3. Press setup for positioning systemperformance experiments.Chapter6: CoordinatedMotion Control ofPress Ram830.100.00-0.10E -0.20C00.0a--0.40-0.50-0.60Figure 6.4. The response of left and right axis ofthepositioning system to a series ofstepchanges in command position. Yl: left axis; Y2: rightaxis, Yref: reference command.For comparison purposes, a traditional PDcontrol algorithm was implemented andmanually tuned. The response ofthe PD controlledsystem for one of the better tuningsis shown in figure 6.5. From the following observationscan be made:• The rise time is between 20and 30 ms• Both axis exhibit between6 and 9% overshoot during the extensionand retraction.• The settling time for each axis is greaterthan 300 ms6.3.2 Determination of Positioning System DeadBandThe performance of hydraulic actuatorsis critically dependent on theperformanceof the valve. Valve hysteresis and spool stiction cancause excessive deadband in thepositioning system. In order to determine theeffects ofthese phenomena, each axis ofthepositioning system was given an alternatingseries of small steps in referenceposition.Both the reference and the actual axis positionswere measured at two millisecond0 50 100 150 200250 300 350 400Time (mS)Chapter6: Coordinated Motion Control ofPress Rain840.100.00-0.10E-0.20C0-0.3000-0.40-0.50-0.60700 800 900 1000Figure 6.5. Response of PID controlled positioningsystem to step changes in referenceposition.intervals. The results are presented in figure 6.6.These results indicate that thesystemdeadband is smaller that one basic length unit (0.005mm).6.3.3 Detenmnation of ControlledMotion PerformanceOne of the most important measures ofa positioning system’s performanceis itsability to follow the desired motion profile. A measureof this following capability, thevelocity error constant (VEC) is definedas the ratio ofthe following error to thedesiredvelocity of the move. While the followingerror for either axis of the CNC press brakeisnot critical, the tracking error is, for this controls theorientation ofthe tooling. Thetracking error will be proportional todifference between the VEC of each axis.Hampering a press brake’s ability to maintainconstant tooling orientation is thetendency for the actuators to be unequallyloaded. This occurs frequently becauseCNCpress brakes are often used to performa number of bends using a number of setsof0 100 200 300 400 500 600Time (mS)Chapter6: CoordinatedMotion Control ofPress Ram85Yl-----Y2LjL .... ...0.0 0.2 0.4 0.6 0.8 1.01.2 1.4Time(S)Figure 6.6. Response of positioning system to small changesin reference position.tooling setup along the press bed. In the absence ofcoupling (be it mechanical, hydraulic,or controller coupling) between the axes, close axis trackingis achieved by matching theaxis dynamics and ensuring both axes are stiff enoughnot to be affected significantlybydisturbance loads.In order to evaluate the performance of thepositioning system with regardstothese measures, two experiments were conducted. Forthe first experiment, a typicalmachine cycle motion profile was used to determine theVEC. For the second experiment,the same motion profile was used toperform a forming operations on severalworkpiecesof differing geometry. Motion Profile: No Forming OperationThe object of the motion profile experiment wasto 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 Ram86rate forming operation, dwell, and rapid retract. Sample resultsofthis test are shown infigures 6.7 and 6.8. Note that the rough tracking response duringthe rapid move is dueto an error in the linear interpolation routine. The velocity errorconstants for each axisare presented in table 6.3.From these results the following observationscan be made:• The following error during the feed-rate move isless than 0.1 mm• The velocity error constants for each of the axesare 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 retractingcase.• The steady-state position error is not greaterthan 0.005 mm (one BLU) duringthedwell operation.• The relative position difference betweeneach axis (the tracking error) is notgreater than 0.015 mm (3 BLU) during the feed operation.Case: Left Actuator VECRight Actuator(ms) VEC(ms)Extension 11.812.3Retraction 13.411.9Table 6.3. Velocity Error Constants for each axisas determined from motion profiletests: no forming loads.Chapter6: CoordinatedMotion Control ofPress Ram871.251.000.75EE 0.50I00.25C00.000a--0.25-0.50-0.75Figure 6.7. Response of positioning systemto 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 trackingerror 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 Ram886.3.3.2 Motion Profile: With Forming OperationThe object of this experiment was to determine the stiffnessof the positioningsystem: that is, the ability to maintain tooling orientationand achieve the desired finalpunch penetration in the presence oftypical formingloads. As with the previouslydescribed experiment, each actuator was given a motionprofile representing a typicalbending operation: rapid approach, feed-rate formingoperation, dwell, and rapid retract.The tooling consisted of a 30 degree punch anda 90 degree die which were aligned andsecured to the press. The respective centres of the punchand die were located 500 (mm)from the left actuator, (approximately 1/3 ofthe distance between the actuators)to allowasymmetrical actuator loading. Two pressuretransducers, one for each fluid chamber,were installed in the right actuator. A typicalwork piece and the finished partaredepicted in figure 6.9.A series of tests were conducted formingworkpieces of varying thicknesses and1100mmWork pieceJ1tFinished Part 90)Figure 6.9. Sample work-piece andfinished part used for brake-formingtests.Chapter6: CoordinatedMotion Control ofPress Ram89widths. The total width and geometric centre of thebend, w and c respectively, wererecorded for each test. The hydraulic pressure actingin each chamber of the rightactuator was measured as were the reference and actualpositions of each axis during thebend cycle. Sample results of this test are shownin figures 6.10 and 6.11Using the free body diagram shown in figure6.12 the hydraulic pressure exertedby the right actuator was used to determinea static theoretical applied bending forcedistributionP=-[](6.40)as well as the force exerted by the left actuator for thedwell operation:R1 = R)— [kN](6.41)The results ofthese test are shown intable 6.4WC— c ‘i — Yref ‘2— Y,.j R1 R2 FC1 C2Case[mm][mm] [mm]1 [pm] [pm] [kN][kN]E/mi{J {)1 150 1.5 425 0.704 5 7.1 3.0 67 0.501.812 300 1.5 500 0.65 912 13.5 7.3 690.64 1.623 600 1.5 500 0.65 1315 29.9 16.2 77 0.44 0.924 150 3.0 725 0.49 1338 26.5 27.6 360 0.471.385 200 3.0 700 0.51 2447 41.4 40.2 408 0.591.176 200 3.0 700 0.51 2151 42.7 41.4 421 0.501.24Table 6.4. Results ofthe Brake forminganalysis.Note: the computation ofthe axis compliancevalues presented in table 6.2 isbasedon an assumption that the entire positioningerror is due to the disturbanceload. Theresults of the previous section indicatethat up to 4 to 5urnof positioning error existsregardless of disturbance loads.Therefore, the axis compliance valuesdetermined forrelatively low load forming conditionsmay be somewhat overestimated. Giventhis, theChapter6: CoordinatedMotion Control ofPress Ram900. 0.060)C0.040i 0.020.00-0.02-0.04compliances of the left and right axis, averaged for the four larger bendingloads 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 theforce capacityof each actuator was determined to be 50 [kN]. Considering the reductionin valve gaindue to pressure drop, a practical bound on the actuator capacity shouldbe 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 errorof each positioning system:Motion profile with bending operation.Chapter6: CoordinatedMotion Control ofPress Ram919608 Pa50012345678910lime (S)Figure 6.11. Actuator pressures recordedfor motion profile with forming operation.ltIIttI11flFbaL:bJ___1Figure 6.12. Free-body diagramof forces acting on ram during the dwell operation.Chapter6: CoordinatedMotion Control ofPress Ram926.4 ConclusionsIn this chapter, details ofthe control scheme were presented. Apracticalsimplification to the pole-placement control law was developed andimplemented. Theperformance of the positioning system can be summarizedas follows:• The rise time of the response of the positioningsystem to a step change inposition was 11-12 ms for each axis.• The dead-band was found to be less than one basiclength unit(BLU=O.OO5mm).• The steady-state positioning error (under noload) was found to be less thanone BLU.• The compliance of each axis ofthe positioningsystem 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 hydraulicpress brakewas designed, modeled and analyzed. Both linear and non-linear system modelswereapplied to cylindrical hydraulic actuators. Non-linear state-relations describingthe changein actuator compliance with position were developed. An analysis ofthe actuator systemdynamics was conducted using a variety of models. The results indicate that simplelinearmodels are capable of predicting the dynamic mode of an asymmetricalactuator ifthevalve flow coefficients are known. However, the piston positionand the direction ofpiston movement during critical positioning stages also need tobe considered in systemdesign.Hydraulic components for the system supply and servo-actuator werechosen andimplemented.The gibbing of the press-brake was modified to allow rotation inthe plane of theram. A position feedback system was designed, implementedand tested. No backlashwas detected.Experiments were conducted to determine thesystem dynamics of the hydraulicservo-actuators. Step response and frequency response testsindicate a first order modelwith consideration of delay is acceptable. Dynamicsystem parameters were determinedusing parametric identification techniques.A delay-compensating, pole-placement control law was designed forthe modelstructures determined in the identification section. A universalsimplification of the poleplacement control law for processes with inherentintegration was developed and93Chapter 7: Conclusions and Recommendations94implemented. This simplification eliminated output scaling errorswhile also reducingcomputational overhead. Motion profile tests indicatedthat dynamics of each axis wereadequately matched. Also, the system exhibited lessthan one basic length unit steadystate position error on a positioning move. Actuator complianceswere determined foreach axis through actual bending tests.In order to create a satisfactory position system, thiswork has effectively outlined aprocedure for designing position control systems forCNC press-brakes. With regard tothis procedure, future work needsto be conducted in identification and control. For thisthesis, all identification experiments were carriedout in open-loop. While this wasrelatively safe for this small press, it couldbe quite dangerous for much larger ones.Closed-loop identification techniquesshould be investigated.For improved motion control, integratingpole-placement controllers couldbe usedto enhance the stifihess of the positioning system ifnumeric sensitivity and valveirregularities were overcome. Also, cross-coupledcontrol strategies should be examinedas a means of reducing axis tracking errors.Finally, since the stiffliess of hydraulic actuatorsis adversely affected by dissolvedgases in the fluid a system for determining thebulk modulus of a samples of fluid wouldbea tremendous asset.BifiLIOGRAPHY1. Pourboghrat, F. and Stelson, K.A., “PressbrakeBending in the Punch-SheetContact Region-Part 1: Modeling Nonuniforinities”, Trans. ofASME, J. Eng.for md., Vol.110, ppl25-130, 1988.2. Anon. “Sheet Metal Bending Methods”, AccurateManufacturing CompanyNews Release, EM- 105.3. 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A., “Implementation ofa Hierarchical Open ArchitectureMultiprocessor Computer Numerical Controller”,Graduate Thesis, TheUniversity of British Columbia, 1993.28. Neal, T. P., “Performance Estimation forElectrohydraulic Control Systems”,Moog Technical bulletin, No. 126, 1974.29. Watton, J., “Further contributions to the Response andStability ofElectrohydraulic Servo Actuators with UnequalAreas - Part 1, SystemModeffing.”, Proc. ASME Winter Annual Meeting, 198530. Anon., “Parker Hydraulic Products and Total SystemsEngineering:Catalogue 0108”, Parker Hannifin Corporation.31. Parker, G. A. and Desjardins, Y. C., “A Comparison ofTransfer FunctionIdentification Methods for an Electro-hydraulicSpeed Control System”, ThirdInternational Fluid Power Symposium, Paper E2,pp 1-27, 1973.32. Soderstrom, T. and Stoica, P., System Identification, PrenticeHall Inc., 1989.33. Anon., Matlab Reference Guide, The MathworksInc., 1992.34. Astrom, K.,J. and Wittenmark, B., Computer ConfrolledSystems, Prentice-Hall, Inc., 1984.APPENDIX AActuator Natural Frequency CalculationsAn estimate of the natural frequency of the actuator is providedby equation 4.28:14J3A2w=IepA.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:= 689.5e6 [NI m2]Using the average piston areas to accommodate foractuator asymmetry:—114(689.5e6 )(0.00456+ 0.00253)2—(455ej(104)= 860 [radls]or,f,=137 [Hz.]98Appendices99APPENDIX BDerivation of Pole-Placement Control LawParametersFor a system with an open-loop position response characterizedby 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 whichhas 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, theorder ofthe observer polynomialmustsatisfy the following constraint:degA0(q) 2degA(q)—degAm(q)— 1B.2deg4(q)2(flad)Qa+nd)—ldeg4(q)n+n —1If the control law takes a relatively small amountof time to implement we arefree tochoose:Appendices100degA0(q)=n+n—lB.2For low-noise transducers such as optical encoders, it is desirableto place theobserver polynomials at the origin for the fastest response such that:4=qfliffldIB.2The order ofthe 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 happensnear the beginning ofthe loop closurecycle, we are free to choose:degR(q) = degS(q)=+ n—such that:R(q) =r0q’+r1q’’+•••+iB.2S(q) =s0q +s1q’+- •+sB.2where:=‘la+‘1dThe feed-forward filter is chosen:T(q)= bm04=b0q’ B.2In determination of the control lawcoefficients 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”’+•Appendices101Fora= 2,b= 1,11d> icollecting terms of B.l0 of likeorders yields:q2(fl+nd)_1:a0r=1 B. 11q2(fl+fld)2:a0r1 +a1, =aB.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 =0B. 17q°: b2sfld+l=0 B.18Assuming b1 0 and b2 0, solving B. 17and B. 18 yields:5÷1=0 B.19S =0B.20Back substituting into the series ofequations inferred between B. 15 and B. 17wecan determine that:B.21Solving B.13 for5o=(a+±‘d-1)B.22Solving B.15 for s1:a2rs1=—B.23Solving B.11 -B.13 yields:Appendices102B.24B.251=ama2—air B.26= —(a_2+a1t_), i=[3:n] B.27Solving B.22 and B.23 into B. 14 and solving foryields:b2 ( b2c1ld+a2Ic---’—B27b ba1 — a2 —- — —-2b1AppendicesAPPENDIX C103Friction Characteristics of Guide SystemThe contribution of the guide system was measured fora variety of actuator speeds.The results are presented in figure C. 11.51.0z0.510.0___ ___ ____ ___ ___O5-1.0-1.5°Left Actuator0CRight Actuator0—0-— 0-00-0 —0---—00-30 -20 -10 0 1020 30Piston Velocity, Vy (mmls)Figure C. 1. Friction force exerted by guide system.Appendices104APPENDIX DModel Parameters VMr=104[kg]B,.=0 [N.s/m]A=O.00456 [m2JA,. = 0.00253 [m2]V=455e [m3]Pe689.5e6[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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