ZS175柴油機機體三面鉆削組合機床總體及右主軸箱設(shè)計【含12張CAD圖紙】

資源目錄里展示的全都有，所見即所得。下載后全都有,請放心下載。原稿可自行編輯修改=【QQ：401339828 或11970985 有疑問可加】 Uncertainty propagation in calibration of parallel kinematic machinesBernhard Jokiel Jr.a, John C. Ziegertb, Lothar BiegcaSandia National Laboratories, Albuquerque, NM 87185-0958, USAbUniversity of Florida Machine Tool Research Center, 237 MEB, Gainesville, FL 32611, USAcSandia National Laboratories,1Albuquerque, NM 87185-0958, USAAbstractThis paper outlines in detail a method for determining the uncertainty present in the kinematic parameters (joint locations, initial strutlengths, and spindle location and orientation) for parallel kinematic devices after calibration. The uncertainty estimation method using MonteCarlo simulations was applied to a sequential method for determining the kinematic parameters of fully assembled Hexel Tornado 2000 (a63 Stewart platform) milling machine. Results for the uncertainty present in the kinematic parameters of a Hexel Tornado 2000 millingKeywords: Stewart platform; Calibration; Kinematic parameters; Uncertainty.1. IntroductionOver the last decade, multi-axis machine tools based onparallel kinematic mechanisms (PKMs) have been devel-oped and marketed worldwide as alternatives to traditionalserial stacked-slide, orthogonal machine architectures. Thegeneral difference between PKM and orthogonal serialmechanisms is the arrangement of the actuators. In orthog-onal serial mechanisms, individual actuators responsible formotion in individual Cartesian degrees of freedom (DOF)are joined end to end in a serial chain connecting thestationary ground frame to the moving frame. In PKMarchitectures, individual actuators are not typically arrangedto independently control a single Cartesian DOF. Insteadthe actuators are arranged so that each actuator is connectedbetween the stationary ground frame and the moving frame,so that a combination of actuator motions controls the mo-tion of the moving frame in Cartesian space. A popular typeof PKM is the Stewart platform (Figure 1). The Stewartplatform connects a moveable platform to a stationaryground frame by six extensible links or struts, allowing forcontrolled motion of the platform in all six DOF.Generally for PKM machines, the Cartesian position andorientation of the tool point carried on the platform isobtained from a kinematic model of the particular machine.Accurate positioning of these machines relies on the accu-rate knowledge of the parameters of the kinematic modelunique to the particular machine. The parameters in thekinematic model include the spatial locations of the jointcenters on the machine base and moving platform, the initialstrut lengths, and the strut displacements. The strut displace-ments are readily obtained from sensors on the machine.However, the remaining kinematic parameters (joint centerlocations, and initial strut lengths) are difficult to determinewhen these machines are in their fully assembled state. Thesize and complexity of these machine generally makes itdifficult and somewhat undesirable to determine the remain-ing kinematic parameters by direct inspection such as in acoordinate measuring machine. In order for PKMs to beuseful for precision positioning applications, techniquesmust be developed to quickly calibrate the machine bydetermining the kinematic parameters without disassemblyof the machine. A number of authors have reported tech-niques for calibration of PKMs. Soons 1,2, Masory, 3,Zhuang et.al. 4,5, Ropponen 6 In another paper 7, theauthors have reported on work recently completed by theUniversity of Florida and Sandia National Laboratories oncalibration of PKMs, which describes a new technique tosequentially determine the kinematic parameters of an as-sembled parallel kinematic device.Calibration of PKMs, or any other type of machine,begins by collecting sets of measurements. The collectedmeasurement data is supplied to an algorithm, which com-putes the model parameters. As with any metrology task,1Sandia is a multiprogram laboratory operated by Sandia Corporation,a Lockheed Martin Company, for the United States Department of Energyunder Contract DE-AC0494AL85000.Precision EngineeringJournal of the International Societies for Precision Engineering and Nanotechnology25 (2001) 4855machine after calibration using a SMX 4,000 laser tracker are shown.there is uncertainty present in the collected measurementdata. The uncertainty present in the measurement data willpropagate through the parameter identification algorithmand result in errors in the computed kinematic parameters.The errors in the recovered kinematic parameters will thencreate positioning errors when the kinematic model is usedfor machine control. The purpose of this paper is to explorethe issue of propagation of uncertainty in the calibration ofPKM devices. We will describe a methodology for evalu-ating the uncertainty in the recovered parameters, and theresulting error bounds in machine positioning. We willdemonstrate this methodology for the new calibrationmethod we have developed and reported in another paper7.This paper is organized in the following manner. Section2 describes the methodology for evaluating uncertaintypropagation in PKM calibration. Section 3 contains a briefsynopsis of the calibration methodology used in this work.In Section 4, the error budget for the calibration measure-ments is developed. Section 5 gives the results of the un-certainty propagation simulation. Section 6 compares thesimulation results to experimental results performed on themachine.2. Methodology for evaluating uncertainty propagationIn any machine calibration task, uncertainty is introducedin two ways. First, the calibration method involves perform-ing measurements with a specified set of instruments. Thesemeasurement instruments will have an associated uncer-tainty in their output, which is a function of the physicalprincipals and construction of the instruments themselves,as well as the environment in which the measurements takeplace.In conjunction with the uncertainty associated with theexternal calibration instruments, there may be additionaluncertainty introduced by the machine itself. The feedbackdevices on the struts of PKM machines typically cannotdirectly measure the absolute distance between correspond-ing pairs of joint centers, only the change in length of thestrut from some “home” position at which the absolute jointcenter distance is assumed to be known. Changes in thethermal state of the machine may cause this assumed valueto be incorrect and/or to fluctuate over time. The strutdisplacement feedback devices also have a finite resolutionand some uncertainty associated with their output. Geomet-rical effects not included in the kinematic model, such asnon-spherical joint motion, also create an effective strutlength error which is not sensed by the strut feedbackdevices. Therefore, the machines repeatability and absolutepositioning accuracy during the calibration measurementscontributes to the overall uncertainty of the basic data usedby the calibration algorithm.The uncertainties present in the external measurementinstrument combine with the uncertainties in the machinemotions to create errors in the data used by the calibrationalgorithm to compute the parameters of the machines ki-nematic model. Since error is present in the collected data,there will be error present in the computed model parame-ters as well. Since these computed model parameters do notidentically match the actual physical parameters of the realmachine, positioning errors will result when the recoveredparameters are used during positional control of the ma-chine. These positioning errors due to incorrect parameter-ization of the model will be compounded with the uncer-tainties inherent to the machine itself, resulting in the finalvolumetric accuracy of the machine.This is an important point that should not be overlooked.The uncertainties arising from the machine itself (i.e., re-peatability, sensor resolution, unmodeled thermal and geo-metric effects) affect the overall positioning performance ofthe machine twice. First, they add uncertainty to the basicmeasurement data that is used by the calibration algorithm.Second, after the calibration is complete, and the newlyparameterized kinematic model is used for machine control,these same machine uncertainties still exist and contributeto positioning errors.We propose the following methodology for evaluation ofuncertainty propagation through the PKM calibration pro-cess:1. Construct an error budget for the measurement de-vices used during calibration to predict their uncer-tainty contributions.2. Construct an error budget for the machine itself toestimate the magnitude of strut length error which canoccur due to thermal effects, sensor resolution anduncertainty, non-sphericity of the joint motions, andother effects inherent to the particular machine whichmay cause non-repeatability of positioning.3. Combine these two uncertainty sources to estimatethe overall uncertainty present in the collected mea-surement data used by the calibration algorithm toobtain model parameters.Fig. 1. A 12-joint (six-six) Stewart platform device.49B. Jokiel et al. / Precision Engineering 25 (2001) 48554. Construct a “perfect” dataset from the machinemodel, which assumes no uncertainties exist in themeasuring instruments or machine.5. Perform a Monte Carlo simulation of the calibrationprocess by running the calibration algorithm multipletimes, each time corrupting the “perfect” dataset withrandomly generated errors assuming a uniform distri-bution over the uncertainty bounds predicted by thecombined error budget analyses.6. Analyze the distribution of errors in the model pa-rameters obtained from the Monte Carlo simulation.7. Using the machine kinematic model with each set ofmodel parameters obtained from the calibration sim-ulation, simulate the machine motions during a par-ticular machine accuracy performance test (i.e., - cir-cular ball bar tests, laser interferometric displacementmeasurements, etc.), being sure to include the uncer-tainty associated with the machine itself in the simu-lation.8. Analyze the simulated performance data.The results of this simulation will show the range ofresults one is likely to achieve on the final machine posi-tioning performance tests from a particular calibration pro-cedure. This procedure is outlined in the flowchart in Figure2. It is also possible to modify these simulations to examinethe effect of various contributors to the overall positioningperformance of the machine. For example, one might as-sume that the machine itself is perfect to determine howuncertainties in the external calibration measurement instru-ments contribute to machine positioning errors. Conversely,one can assume that the external measurements are perfectand examine the propagation of various sources of machineerror, such as thermal effects or non-sphericity of jointmotions, to the final positioning accuracy performance.3. Calibration methodologysequential determinationtechniqueThe approach outlined in Section 2 was used to evaluateuncertainty propagation in calibration of PKM devices for anew method for sequential determination of kinematic pa-rameters in PKMs developed by the authors and reportedelsewhere 10. As a convenience to the reader, this methodis briefly described here. The parameter identificationmethod uses a spatial coordinate measuring device such asa laser tracker or a laser ball bar. The calibration is per-formed in four steps:1. Location of a central reference frame (R) and themachine frame (M).2. Identification of the spatial locations of the jointscenters of rotation.3. Determination of the spindle orientation and noselocation.4. Determination of the initial strut lengths.3.1. Locating the central reference and machine referenceframesIn general, several locations of the spatial coordinate-measuring device may be required to complete all of thenecessary measurements. Therefore, a single, stable refer-Fig. 2. Uncertainty propagation algorithm flowchart.Fig. 3. Locations of coordinate reference frames used during calibration.50B. Jokiel et al. / Precision Engineering 25 (2001) 4855ence frame, R, is necessary to which all of the coordinatedata collected in various frames may be transformed (Figure3). The R frame consists of three gauge points secured to theworktable. Utilizing the spatial coordinate measuring de-vice, the locations of the R gauge points are measuredrelative to the measuring devices coordinate system, andthe homogeneous coordinate transformation (HTM) be-tween the measurement devices coordinate system and theR frame (MeasTR) is computed using the following equa-tions.The HTM between the R system and machines referenceframe (M) may also be determined at this time. Using thespatial measurement device, the plane of the worktable,the desired location of the X-axis and the machine originare measured. Using the unit normal vector of the best-fitplane, the unit vector of the X-axis direction vectorprojected into the best-fit plane, and the coordinates ofthe desired machine origin projected into the best-fitplane the HTM of the M frame relative to the measure-ment device frame, (MeasTM), is constructed. The HTMrelating the R and M frames (MTR) is then calculated bymatrix multiplication.3.2. Joint center location identificationStewart platform machines require the use of sphericalor Hooke joints to connect the struts to the machine baseand to the moveable platform. These joints allow the strutends to rotate about fixed points on the machine base andplatform as the machine geometry changes. If a strut isheld at a fixed length and rotated about one of its joints,all of the points on the link move with spherical motionabout a common center of rotation. This fact can beleveraged to determine the location of the joints centerof rotation.Two gauge points are affixed to either side of one strut sothat the gauge-point centers and the strut centerline lie in thesame plane. Three gauge points are affixed to the platform,which define a platform reference coordinate system PR.The platform is then commanded to move along an arbi-trary, predetermined path designed to hold the strut in ques-tion at an arbitrary fixed length. As the platform moves, thisfixed-length strut rotates about its joint centers, and pointson the strut trace a spherical path in space (Figure 4). Atseveral locations along the path, the machine motion ispaused, and the spatial coordinates of the two link and threeplatform gauge points are measured. This “move-pause-measure” sequence is repeated until the desired number ofplatform poses has been reached.The gauge point coordinates may be expressed relative totwo different coordinate systems, the M system, or the PRsystem. Gauge point coordinates expressed relative to the Msystem lie on the surface of a sphere whose center is thecenter of rotation of the base joint of the examined strut. Thesame gauge point coordinates expressed relative to the PRsystem lie on the surface of a sphere whose center is thecenter of rotation of the platform joint for the examinedstrut. The locations of the base and platform joint centersmay be determined by fitting the appropriately transformedstrut gauge point coordinates to the equation of a sphere.Assuming the joints produce spherical motion, the calcu-lated coordinates of the center of the best-fit sphere is thecenter of rotation of the joint in question. This method isrepeated sequentially for each of the six struts to recover all12 of the base and platform joint center locations.MeasTR5Measx RMeasy RMeasz RMeasr10001#Measx R5Measr22Measr1uMeasr22Measr1uMeasz R5Measx R3Measr32Measr1uMeasr32Measr1uMeasy R5Measz R3Measx R(1)T 5Measx MMeasy MMeasn planeMeaspO0001#Measx M5Measu x2 Measu xzMeasn plane!Measn planeuMeasu x2Measu xzMeasn plane!Measn planeuMeasy M5Measn plane3Measx MMeaspO5Measp 2 Measp zMeasn plane!Measn planeMTR5 MeasTM!21 MeasTR(2)51B. Jokiel et al. / Precision Engineering 25 (2001) 48553.3. Locating the spindle noseIn order for the machine controller to be able to controlthe path of the tool point, it is necessary to express thelocations of the platform joints relative to the spindle noseand spindle centerline orientation. This is accomplishedthrough the following measurement procedure. A fixtureholding a gauge point is attached to a tool holder mountedin the spindle. The toolholder is rotated slowly by hand. Atseveral different locations, the motion is stopped and thecoordinates of the gauge point are measured. These coordi-nates are then transformed into the PR system. Ideally, thesepoints should all lie on a perfect circular path. After the datacollection sequence is complete, the measured coordinatesare fit to a plane, and then are fit to a circle in the best-fitplane. The unit normal vector of the best-fit plane is theorientation vector of the spindle relative to the PR system.The coordinates of the center of the circle projected alongthe positive direction of the normal vector by the offsetdistance of the fixture and tool holder are the coordinates ofthe center of the spindle nose (Figure 5). Since the platformjoint locations and the spindle location and orientation areboth known relative to the PR system, the locations of theplatform joints may be expressed relative to the spindlenose.3.4. Determining the initial strut lengthsOnce all of the joint centers of rotation have been lo-cated, the initial strut lengths may be determined. The plat-form is retracted to its home position and the coordinates ofthe PR gauge points are measured. The locations of the basejoints and the platform joints are all expressed relative to thePR system, and the straight-line distances between the eachof the six strut joint pairs are calculated. This “home-measure-calculate” sequence is repeated several times. Theinitial strut length for each strut is taken to be the mean ofthe calculated strut lengths for each strut over all of therepetitions.4. Error budgetThe degree of success of the sequential calibrationmethod relies heavily upon the stability of the length of thefixed-length strut during machine motions, and the qualityof measurements of the gauge point coordinates. If themachine exhibited perfect repeatability, and the externalmeasurement device gave perfect measurements, then therecovered model parameters would be exact. However,there are thermal, geometric, and controller effects in themachine, which may cause the actual strut length to varyduring the measurement cycle. In addition, the externalmeasurement instrument is incapable of producing perfectmeasurements. Therefore, in order to estimate the amount ofuncertainty present in the recovered kinematic parametersand the propagation of the kinematic parameter uncertain-ties to the tool tip, an error budget for the constant strutlength and measurement device must be constructed.Fig. 4. Planar view of sweeping motion of constant length strut.Fig. 5. Locating the spindle nose location and centerline orientation.Table 1Summary of base joint half strut uncertaintyUncertainty ComponentValue (mm)Base Joint Sphericity0.013Strut Flexibility0.003Thermal Effects0.007Command Mismatch0.005Least Count Motion0.001RSS0.016Table 2Summary of platform joint-half strut uncertaintyUncertainty ComponentValue (mm)Platform Joint Sphericity0.013Strut Flexibility0.003Thermal Effects0.007RSS0.01552B. Jokiel et al. / Precision Engineering 25 (2001) 48554.1. Strut length error budgetFive factors were identified to influence the nominallyfixed 400 mm length of the strut between the measurementtarget fixture and the base joint. They are 1 base jointmotion sphericity, 2 strut axial flexibility, 3 strut elon-gation due to thermal effects, 4 strut length commandmismatch, and 5 least count servo motion (Table 1).The base joints on the machine tested were a capturedsphere design. Strut axial motion due to the non-sphericityof the base joint is roughly equal to half the sphericity of theform of the trapped sphere. From CMM measurements thesphericity of the spher