Thesis Modelling Simulation And Control Of A Hydraulic
Institutionen för teknik och design, TDFahrzeugmechatronik, FK03Modelling, simulation and control of a hydraulic craneModellierung, Simulation und Steuerung eines hydraulischen KransModellera, simulera och styra av en hydraulisk kranSubmitted for the Degree of Master of Science in Automotive Mechatronicsat Munich University of Applied SciencesVäxjö, September 2007Examensarbete nr: TD 002/2008Författare: Alexander Heinze (780228-P191)Avdelningen för Maskinteknik
Organisation/ Organization/ InstitutionVÄXJÖ UNIVERSITETInstitutionen för teknik och designSchool of Technology and DesignFörfattare/ Author/ AutorAlexander HeinzeDokumenttyp/ Type of document/ DokumententypMS WordHandledare/ Supervisor/AufsichtspersonTorbjörn Ekevid(Växjö University)Examinator/ Examiner/PrüferProf. Dr. PeterWolfsteiner(Munich University ofApplied Sciences)Titel/ Title/ TitelModelling, Simulation And Control Of A Hydraulic CraneNyckelord/ Keywords/ Schlüsselwörtercrane dynamics, friction maps, evaluation of dynamic cylinder friction, LuGre model, crane-tip controlUtgivningsår/ Year of issue/ Jahrder Veröffentlichung2007InternetSpråk/ Language/ SpracheEnglishhttp://www.vxu.se/tdAntal sidor/ Number of pages/Seitenanzahl137
AbstractThe objective of this thesis is to develop a model that represents the dynamics of ahydraulically operated forestry crane. The model was derived with the traditional EulerLagrange formalism and considered the crane mechanics, three double-acting hydrauliccylinders and the valve control unit. On the basis of the derived model we reproduced theentire crane model in MATLAB in order to run simulations herewith. This gave us thepossibility to do parameter changes for further studies of the crane in motion.Another major goal within the thesis work was to estimate cylinder friction of the hydraulicactuators. We build up a test rig and used a double-acting cylinder for determing its frictionalbehaviour. For this, we ran open-loop experiments in order to create velocity-friction mapsthat represented the static friction force of the cylinder. In this concern, we varied systempressure and cylinder load to study their influence on the friction force. By means of thederived static friction maps we approached the cylinder’s dynamic friction behaviour andapplied both step and ramp control inputs to examine the spring-damping characteristics ofthe microspoic bristles in the contacting area. The dynamic friction experiments have beenexerted in the fashion of the LuGre model. As a result we acquired different nominal frictionparameters that we necessarily used to develope adequate friction models.A third objective of this thesis was to establish a crane-tip control. Instead of a traditionalcontrol, providing a direct relationship between joystick input and cylinder extension, the focuswas to build up a control for the end-effector’s trajectory in a two-dimensional frame. This wasachieved by using inverse kinematics in order to determine the required joint angles thatcorresponded to the desired position of the crane-tip.The work also contains a CD including all developed MATLAB models that have been writtenwithin this project.
SummaryHydraulic cranes are very popular for carrying out hauling operations of forestry machines. Inchapter 1 we will give an overview of such vehicles followed by the problem definition and adescription of the tools that have been used during the project work.A small hydraulic crane was provided by Rottne Industri AB to do experimental work in thelaboratory hall of Växjö University. In chapter 2, the laboratory crane and its constituent partsare described in detail. This involves the mechanical structure, the connected hydraulicsystem, sensors and peripheral equipment. Sketches of the crane elements are alsopresented in this chapter.The mechanical model will be derived in chapter 3, beginning with the definition of rotationmatrices followed by a set-up of necessary kinematic chains. Finally the derivation of theEuler-Lagrange equations allows us for examinig the dynamical behaviour of the crane.In chapter 4 the hydraulic model will be described. It is divided into two major parts, namelythe mathematical model of the spool valve and the mathematical model of the hydrauliccylinders. In both cases we will foremost present a static model which will be followed by thedynamic model in order to derive the corresponding equations of motion.Friction modeling is done in chapter 5. At the beginning of this chapter we will describe themain friction phenomenon including the traditional static friction model and the dynamic bristleinterpretation in the fashion of the LuGre model. Open-loop experiments at the laboratorycrane will be carried out to examine the frictional behaviour of the used hydraulic cylinders.The results provide us the cylinder’s nominal friction parameters which, in turn allows us tobuild up adequate friction models.The main control principle is described in chapter 6. The traditional control task of a hydrauliccrane is accomplished via two joysticks that control each cylinder separately. We will presenta more convenient method using a sophisticated crane-tip control. In this concern, we willfocus on inverse joint determination combined with traditional PID control.We used MATLAB for all kind of modeling and simulation during the thesis work. The mainprogram files are represented in chapter 7 but can also be found on the attached CD in theappendix.
AcknowledgementsThis thesis work was carried out in the department of technology and design (TD) at VäxjöUniversity, Sweden. It was initiated within the research group Heavy Vehicles in colaborationwith Växjö University and Rottne Industri AB.The author would like to express his gratitude to all those who confirmed the permission andthus made it possible to complete the thesis work at Växjö University.Most notably, I would like to convey my thanks to the project supervisor Torbjörn Ekevid atVäxjö University for providing advisory support and professional advice of any kind. I alsowant to thank Matz Lennels at Växjö University for giving me advisable help regarding controlprinciples of Euler-Lagrange systems. Furthermore I am deeply indepted to Anders Hultgrenat Kalmar University for his stimulating suggestions and outstanding encourangement fromthe very beginning of this thesis work, especially for the intense and valuable discussionsabout cylinder friction on many nights in the laboratory hall. I would also like to thank RottneIndustri AB whose courtesy of the hydraulic crane enabled us to experience the real-cranebehaviour along friction experiments that have been directly carried out at the labcrane.Finally I would like to give my special thanks to my professor at Munich University of AppliedSciences for encouraging me working abroad but also for his help and feedback on numerousissues of this thesis workAlexander HeinzeVäxjö, 29th September 2007
SymbolsGeneral notation:vryvX rYZXAYvr (x)XPosition vector of length y referred to the X-framePosition vector of distance between part Y and part Z referred to the X-frameRotation matrix from frame Y to frame XValue of x-direction of vector rFrames:IeInertial frame1e/ 2 e/ 3 e/ 4 eJoint frames (origin in the center of 1st, 2nd, 3rd and 4th joint)c1e/ c 2 e/ c3 eCyldiner frames (origin in the center of the opening joint of 1st, 2nd and 3rdcylinder)Indices:x1 , x2 , x3Piston extension from cylinder center of 1st, 2nd and 3rd cylinderc1s , c2 s , c3 sCenter of opening joint of 1st, 2nd and 3rd cylinderc1e , c2e , c3eCenter of closing joint of 1st, 2nd and 3rd cylinderj1 , j2 , j3Center of 1st, 2nd and 3rd jointcg1 , cg 2 , cg 3Center of gravity of 1st, 2nd and 3rd linkcg t1 , cg t 2Center of gravity of 1st and 2nd torque linkt1Joint center of torque link 1 (opening)t2Joint center of torque link 2 (opening)t3Joint center of torque link 3 (closing)ctct d ,α d , β dCrane-tipDesired crane-tip postion and corresponding joint anglesSymbols:qαVector of generalized coordinatesβ2nd joint variablex33rd joint variableTTtransKinetic energy1st joint variableTranslational kinetic energy
TrotRotational kinetic energyVQPotential energyωAngular velocitymIJgMassInertia tensorJacobian matrixfForce vectorMhyMass matrixVector of gyroscopic and active forceslcLength of cylinder framelosOffset start (opening joint – start of cylinder frame)loeOffset end (end of cylinder frame – closing joint)lpLength of cylinder pistonlrLength of cylinder rodlcOffset end (end of cylinder frame – closing joint)ϕ1Slope angle of 1st cylinder referred to 1 e -frameϕ2Slope angle of 2nd cylinder referred to 2 e -frameε1 , ε 2Slope angles of 1st and 2nd torque link referred to 2 e -frameψiAuxillary angles in 1st cylinder frameθiAuxillary angles in 2nd cylinder framer2Swivel radius of 3rd joint about 2nd jointr3Swivel radius of crane-tip about 3rd jointrdiagDistance between 2nd joint and crane-tipxsSpool strokeuControl inputp0Tank pressurepSSupply pressurepNNominal pressureC0Constant discharge coefficientQNNominal flowReReynolds numberνKinematic fluid viscosityGeneralized forcesGravity acceleration vectorState vectorptrTransition pressurep A , pBCylinder chamber pressures
ωnNatural frequencyωdRinging frequencyζDamping ratiokcSpring stiffnessτTime constantA1 , A2Cross-sectional area of upper and lower cylinder chamber ( A1 A2 )F frFriction forceFgGravitational forceV0,1 , V0, 2Initial cylinder volumesV1 , V2Dynamic cylinder volumesdcDiameter of cylinder framedrDiameter of cylinder rodρDensity of hydraulic fluidqint / qextInternal / External leakage flowEoilBulk modulus of oil elasticityC1 , C 2Oil constantsk int / kextLeakage coefficientsμzStatic friction coefficientAverage bristle deflectionFCCouloumb friction forceFNNominal forceFvViscous friction forcekvViscous friction coefficientDamping factorδvViscous friction gradientFSStatic frictionF StStribeck frictionvσStribeck velocityδσStribeck gradientα 0 ,α 1,α 2Static friction parametersσ 0 ,σ 1Dynamic friction parametersk p , ki, kdPID control gainseControl error
Table of contents1BACKGROUND. 111.11.21.31.41.51.61.72A TYPICAL FORWARDER IN USE . 11VEHICLE CONSTRUCTION OF A FORWARDER . 12THE MAIN WORKING PRINCIPLE . 13PROBLEM DEFINITION . 15GOAL . 15RECENT APPROACHES AND PREVIOUS WORK . 16TOOLS . 17THE LABORATORY CRANE. 182.1SYSTEM DESCRIPTION. 182.2DESCRIPTION OF THE CRANE COMPONENTS AND ASSEMBLIES [7] . 192.2.1 Rigid Beams. 192.2.2 Hydraulic system. 192.2.2.12.2.2.22.2.2.32.2.3Sensors . 222.2.3.12.2.3.22.2.3.32.2.3.42.2.4Hydraulic cylinders. 20Valve package . 20Hydraulic power unit . 21Pressure sensors. 22Load sensors . 23Angular sensors. 23Position sensor . 24Peripheral equipment. 252.2.4.12.2.4.22.2.4.3dSpace system . 25Amplifier box . 25Interface box . 262.3SKETCHES OF THE CRANE ELEMENTS . 262.3.1 Link sketches . 272.3.1.12.3.1.22.3.1.32.3.1.42.3.1.52.3.23Sketch of the 1st link . 27Sketch of the 2nd link. 27Sketch of the torque link . 28Sketch of the 3rd link . 28Sketch of the 4th link . 29Cylinder sketches. 29MECHANICAL MODEL. 313.1KINEMATICS OF THE CRANE . 313.1.1 Rotation matrices . 333.1.1.13.1.1.23.1.2Rotation matrices of link frames . 33Rotation matrices of cylinder frames. 34Kinematic chains. 403.1.2.1Determination of forward kinematics. 403.1.2.2Determination of inverse kinematics . 413.1.2.2.1 Joint variable α . 443.1.2.2.2Joint variableβ. 45
3.1.2.2.3Joint variablex3. 463.1.3 Workspace . 463.2DYNAMICS OF THE CRANE . 473.2.1 Euler-Lagrange formulation . 483.2.2 Kinetic energy . 483.2.3 Potential energy . 503.2.4 Generalized forces. 513.2.5 Equations of motion . 524HYDRAULIC MODEL. 544.1MATHEMATICAL MODEL OF THE SPOOL VALVE . 544.1.1 Static model of the spool valve . 544.1.1.14.1.1.24.1.1.3Valve configuration . 54Working principle . 56Flow equations . 584.1.2 Dynamics of the spool valve . 614.2MATHEMATICAL MODEL OF THE CYLINDER . 644.2.1 Static model of the cylinder. 644.2.2 Dynamics of the cylinder. 665MODELLING OF HYDRAULIC FRICTION . 705.1FRICTION PHENOMENA . 705.1.1 Static friction model. 715.1.1.15.1.1.25.1.1.35.1.1.45.1.2Coulomb friction. 71Viscous friction . 72Static friction (Stiction) . 73Stribeck friction . 74Dynamic friction model. 765.1.2.15.1.2.2Dynamic properties of friction . 76The LuGre Model. 775.2ESTIMATION OF NOMINAL FRICTION PARAMETER . 805.2.1 Experimental set-up . 805.2.2 Static parameter estimation . 835.2.3 Dynamic parameter estimation . 895.2.3.1Estimation of dynamic parameterσ05.2.3.2Estimation of dynamic parameterσ 1 . 925.2.46. 89Influence on friction due to variations in load and pressure . 94CONTROL OF THE CRANE . 966.1CONTROL TASK . 966.2INVERSE JOINT CONTROL . 976.2.1 PID control . 986.2.2 Control algorithm. 996.2.3 Optimization . 1006.3PERFORMANCE OF CRANE MODEL WITH PID-CONTROL . 1017CRANE MODELLING IN MATLAB. 1067.1COMPLETE SET OF STATE SPACE EQUATIONS . 106
7.27.37.4MATLAB-CODE ‘CRANE MAIN’ . 107MATLAB-CODE ‘DIMENSION’. 117MATLAB-CODE ‘PLOT CHARTS’. 1208CONCLUSIONS AND OUTLOOK. 1279APPENDIX. 1289.1DATA SHEETS . 1289.2CD-ROM . 1339.2.1 Content. 1339.2.2 CD-Rom attachment . 13310BIBLIOGRAPHY . 134
Background1 Background1.1 A typical forwarder in useSwedish economy is characterized by a market heavily orientated towards foreign trade inwhich the agricultural sector is of great importance. Within this branch, timber productsaccount for an essential part of the export market. This emanates from a highly forested landutilization in Sweden (see Figure 1), recently not least because of improved transportationand distribution systems. Consequently agricultural technology plays an important role in theforest-based industry. In the middle part of southern Sweden there exist numerousmanufacturers for forestry machines, all of them offer products to facilitate and speed-up themachining in wood proceedings. Some of the companies are joining, together with theUniversities of Växjö and Kalmar, a research group named Heavy Vehicles (swed. TungaFordon), whose ambition is to develop know-how and advancement in technology in order tomake the forestry vehicles become more efficient.One of the members within this research group is Rottne Industri AB, a leading manufacturerin logging machinery that has his headquarters located in Rottne, nearby the town of Växjö. InSweden, Rottne Industri covers approximately 15% of the market for forestry machines with aproduction capacity of up to 200 machines per year. The company develops andmanufactures a wide range of forwarders and harvester machineries covering everything fromfirst thinning to clear felling. All of the machines are equipped with a hydraulic crane forhandling the timber logs. It is a major objective within the sector to continually improve theefficiency and reliability of the forestry machines, predominantly by introducing moreautomated steering control. For research purposes Rottne Industri has provided one of theirforwarders that is now located in the laboratory of the technology department at VäxjöUniversity. On the basis of this prototype of a forwarder, practicable research activities likesensor feedback and programmable control could be done in the laboratory hall of VäxjöUniversity.Figure 1: Land utilization in Sweden11
BackgroundForestry work often consists of monotonous work steps that could be performed automaticallyfacilitating the human work and thus save time and energy. A main limitation but also anessential way to enhance the efficiency of forwarders is the control interface of the crane.Nowadays the crane is manoeuvred manually by well-trained and experienced operators.They undertake the task of controlling each cylinder separately in order to move the crane-tipto its desired position. Consequently the operator has to use several levers and buttonssimultaneously which complicates a precise and effective realization of the handlingoperation. In respect of this, the forestry industry is particularly interested in a more direct andeasier way of controlling the crane in order to relief the driver from parallel operations. Theanswer to this problem is a control mechanism which allows controlling the crane-tip in aplane Cartesian frame, namely the up-down motion with one lever and the forward-backwardmotion using a second lever.In comparison to harvester machines that are additionally utilized in operations for felling,delimbing and cutting trees, a typical forwarder is used to haul the log from the stump to aroadside landing. Beside stability and flexibility, good tracking is one of the major conditions inorder to carry out good log-hauling in a densely wooded working environment. Primarilyforwarders are equipped with a powerful hydraulic system that has made them extremelypopular within the class of heavy articulated vehicles. Since most of the forwarders consist ofmany moving parts, they are regarded as high maintenance wheelers. In the followingsections a short description of the constructive aspects and the control principle are given.1.2 Vehicle construction of a forwarderDepending on performance and load carrying capabilities, forwarders nowadays exist inseveral versions that not only vary in dimension and size but also in power transmission,hydraulic equipment and steering. Whereas light weight built machines are employed inlogging operations with load capacities of eight metric tonnes, Rottne’s Solid Rapid forwarderfor example can carry payloads of up to 16 tonnes.Figure 2: Rottne forwarder SMV Rapid [15]12
BackgroundThe machines are characterized by a robust frame design with powerful articulated joints,wide wheels and a load area with a low center of gravity for providing a good groundclearance. Hence forwarders have an excellent stability which makes them use the full reachand lifting capacity of the loader. Commonly they have a high ground pressure which allowsclimbing over rocks and stumps as well as travelling through deep snow or wet lands.Forwarders are basically equipped with a 24 volt electrical system for control and monitoringoperations. The motion of each crane link is quick and easy actuated by hydraulic cylindersthat are connected to a powerful hydraulic system. The system together with its workingprinciple is specified in chapter 1.3. Generally the manipulator is a forward linkage of threebeams with an additional telescopic extension arm. With this configuration the forwarder hasa broad reach and a flexible operating mode which makes it a particularly suitable vehicleunder exceedingly difficult conditions in the forest.1.3 The main working principleThere exist several classifications of robot manipulators with variations in power source(hydraulic, electric, pneumatic), control method (servo / non-servo, closed loop / open loop)and application area (assembly / non-assembly). Depending on its field of application, arobotic system is also defined by its geometry. Several ways of linked motions are possible,e.g.:- articulated manipulators (three revolute joints)- spherical or SCARA manipulators (two revolute and one prismatic joint)- cylindrical manipulator (one revolute and two prismatic joints)- cartesian manipulator (3 prismatic joints)The laboratory forwarder from Rottne Industri is of the type of an articulated manipulator. It isalso termed anthropomorphic manipulator because its joints are placed in a human specificlinkage. The system design is shown in Figure 3 but it should be annotated that the Rottneforwarder is additionally equipped with an extractable telescopic forearm on which the gripperis eFigure 3: Anthropomorphic manipulator13
BackgroundThe configuration in Figure 3 allows the gripper for reaching a wide working area, still beingmost flexible and accessible. Often robotic systems are also valued in accuracy (attribute ofhow close the end-effector can come to a given point) and repeatability (attribute of how closethe end-effector can return to a previously taught point), that are both highly dependent on theresolution of the signals, the backlash in the joints, the control algorithm and several outerinfluences.The control of a robotic system involves several tasks, beginning from signal perception anddata processing to the actuating operations of the cylinder. A typical robotic paradigm of theanthropomorphic manipulator is given in Figure 4:sensorspower supplycomputercontrollermechanical armprogram storage/networkend-of-arm toolinginput device/teach pendantFigure 4: Robotic paradigmBy using an input device, e.g. a teach pendant, the desired operation is commanded to theCPU. Sensors provide the necessary feedback information about the environment, the jointangles or the position of the end-effector. The adaptive control algorithm is implemented inthe computer network and is responsible for calculating and generating a suitable outputsignal to the actuators that will move the mechanical structure.Forwarders are heavy machines that use a powerful hydraulic system to do work. The controlprinciple is based upon articulated steering joint with double acting cylinders. A craneoperator easily controls the valves from within a cabin and consequently distributes the fluiddirectly through the circulated load-sensing hydraulic system. Hydraulic fluid is transmittedunder high pressure throughout the hydraulic hoses into the cylinder chambers. Thus, thecylinder force acts on the linkage and manoeuvres the coupled arms of the c
cylinder) Indices: x1, x2, x3 Piston extension from cylinder center of 1 st, 2nd and 3rd cylinder c1s ,c2s,c3s Center of opening joint of 1 st, 2nd and 3rd cylinder c1e ,c2e,c3e Center of closing joint of 1 st, 2nd and 3rd cylinder j1, j2, j3 Center of 1 st, 2nd and 3rd joint cg1,cg2,cg3 Center of gravity of 1 st, 2nd and 3rd link cgt1,cgt2 .
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