applied sciences
Article A Novel Method for the Measurement of Geometric Errors in the Linear Motion of CNC Machine Tools
1, 1, 1 1 1 2 Xuan Wei †, Zhikun Su † , Xiaohuan Yang , Zekui Lv , Zhiming Yang , Haitao Zhang , Xinghua Li 1,* and Fengzhou Fang 3,*
1 State Key Laboratory of Precision Measuring Technology and Instruments, Tianjin University, Tianjin 300072, China 2 Key Laboratory of Advanced Transducers and Intelligent Control System, Ministry of Education, Taiyuan University of Technology, Taiyuan 030024, China 3 Centre of Micro/Nano Manufacturing Technology (MNMT-Dublin), University College Dublin, Dublin 999015, Ireland * Correspondence: [email protected] (X.L.); [email protected] (F.F.); Tel.: +86-1338-991-6132 (X.L.); +353-01-716-1810 (F.F.) These authors contributed equally to this work. † Received: 31 July 2019; Accepted: 12 August 2019; Published: 15 August 2019
Abstract: In order to improve the accuracy of the linear motion of computer numerical control (CNC) machine tools, a novel method based on a new type of 1-D (1-dimensional) artifact is proposed to measure the geometric errors. Based on the properties of the displacement measurement of a revolutionary paraboloid and the angle measurement of plane mirrors, the 1-D artifact can be applied to identify position errors and angle errors. Meanwhile, the concrete 6 degrees-of-freedom error identification method is described in this paper in sufficient detail. Through measuring the 1-D artifact horizontally and vertically using the machine tool, the geometric errors can be obtained by calculating the deviation between the characteristic parameter of the 1-D artifact measured by the machine tool and that measured by a more precise method, for example, laser interferometry. Experiments were carried out on a coordinate measuring machine, and the validity and accuracy of the method were discussed by comparing the result with the identification error measured by a laser interferometer.
Keywords: CNC machine tools; linear motion; geometric errors; micro displacement measurement; 1-D artifact
1. Introduction With the rapid development of mechanical manufacturing, CNC (computer numerical control) machine tools cover a wide range of applications in the mold manufacturing industry, electronic engineering, and the automotive and aerospace industries [1,2]. Owing to its advantages of being all-purpose with high precision and a large processing scope, it has become a very necessary type of equipment [3]. As is widely known, precision plays a key role in CNC machine tools, and this is a hot topic in the machining automation domain. Generally speaking, there are two approaches to enhance accuracy. One is error avoidance, i.e., improving processing techniques, which is effective but costs plenty of resources. The other is error compensation, which is conducted by obtaining the original errors of CNC machine tools by some measuring method [4]. Clearly, as an economical and efficient method, error compensation has attracted the attention of many researchers. The guide rail of machine tools has 6 degrees-of-freedom errors in linear motion [5]. As the most accurate method, laser interferometers are widely applied to
Appl. Sci. 2019, 9, 3357; doi:10.3390/app9163357 www.mdpi.com/journal/applsci Appl. Sci. 20192019,, 99,, x 3357 FOR PEER REVIEW 22 of of 11 identification of the geometric error of the linear guideway [6,7]. However, laser interferometers haveidentification a high price of the and geometric tedious operation error of the as linear the cost guideway of their [high6,7]. precision. However, laser interferometers have a highThe price R-test, and invented tedious operation by Weikert as theet al. cost [8] of, was their further high precision. used to measure the location errors of a guideThe way R-test, by Ibaraki invented et al. by [ Weikert9]. Yamaji et al. Masashi [8], was improved further used the R to-test measure device theto incorporatelocation errors three of displacementa guide way bysensors Ibaraki designed et al. [ 9to]. simultaneously Yamaji Masashi collect improved far more the R-testinformation device [10 to]. incorporate Another method three isdisplacement to calibrate sensorsthe errors designed with standard to simultaneously parts, such collect as a ball far- morebar and information a ball-plate. [10 ].For Another example, method Cheng is Fangto calibrate realized the the errors determination with standard of the parts, geometri suchc aserror a ball-bar parameters and aof ball-plate. CMMs (coordinate For example, measuring Cheng mFangachines) realized using the a determination 1-D artifact (i.e. of, the a 1 geometric-D ball-bar) error [11 parameters]. Tsutsumi ofproposed CMMs (coordinatea classical measuring method,machines) i.e., using to conduct a 1-D artifactball bar (i.e., tests a in 1-D axial, ball-bar) radial, [11 and]. Tsutsumi tangential proposed directions a classicaland identify measuring PIGEs (method,position- i.e.,independent to conduct geometric ball bar testserrors in) by axial, analyzing radial, the and eccentricities tangential directions of the circular and identifytrajectory PIGEs [12]; this(position-independent method has been accepted geometric by errors)many scholars by analyzing and is the included eccentricities in ISO of10791 the- circular6 [13]. trajectory [12]; this methodIn this article has been, we acceptedpropose bya systematic many scholars method and to is identify included geometric in ISO 10791-6 errors [in13 the]. linear motion of CNCIn thismachine article, tools we based propose on aan systematic innovative method 1-D artifact to identify. The 1 geometric-D artifact errors is composed in the linear of a series motion of surfaceof CNC mirror machine target toolss.based Furthermore, on an innovative each target 1-D includes artifact. Thea revolutionary 1-D artifactis paraboloid composed ofand a seriesa plane of mirrorsurface. mirror The former targets. is Furthermore, designed to each measure target the includes 2-D displacement a revolutionary and paraboloid identify and the a 2 plane-D original mirror. locationThe former error isdesigned of machine to measure tools, and the the2-D latter displacement is to identify and identify the 2-D the original 2-D original angle location error. We error can of obtainmachine 4 tools,degrees and-of the-freedom latter is errors to identify by measuring the 2-D original the 1 angle-D artifact error. usin We cang an obtain angle 4 sensor degrees-of-freedom driven by a machineerrors by tool measuring in one direction. the 1-D artifact Then, usingby rotating an angle the sensor1-D artifact driven 90 by°, the a machine other 2 tooldegrees in one-of- direction.freedom errorsThen, bycan rotating be obtained the 1-D by artifact another 90 ◦measurement., the other 2 degrees-of-freedom As such, the method errors can can realize be obtained the measurement by another ofmeasurement. the geometric As errors such, thein the method linear can motion realize of the machine measurement tools; the of theresult geometric was compared errors in withthe linear that obtainedmotion of by machine laser interferometer tools; the result. was compared with that obtained by laser interferometer.
2. M Measurementeasurement Mechanism Mechanism
2.1. Geometric Geometric Errors Errors of of a a Linear Linear Motion G Guideuide R Railail Any object withoutwithout constraints constraints has has 6 degrees6 degrees of freedom,of freedom, which which means mea itns can it movecan move along along or rotate or rotatearound around any direction. any direction. Due to imperfect Due to manufacture, imperfect the manufacture, guide rail may the have guide micro-movements rail may have in microevery- direction;movements as in such, every each direction axis has; 6as degrees-of-freedom such, each axis has errors 6 degrees [14].-of Take-freedom the X-axis, errors for [14 example:]. Take theFigure X-1axis shows, for the example: 6 degrees-of-freedom Figure 1 shows errors, the namely, 6 degrees position-of-freedom error δ x errors(x), the, namely, horizontal position straightness error error δy(x), the vertical straightness error δz(x), the roll error εx(x), the yaw error εz(x), and the pitch ( ), the horizontal straightness error ( ), the vertical straightness error ( ), the roll error error εy(x). ( ), the yaw error ( ), and the pitch error ( ).
Figure 1. The 6 degrees-of-freedom errors of the X-axis. Figure 1. The 6 degrees-of-freedom errors of the X-axis. 2.2. Design of the 1-D Artifact 2.2. Design of the 1-D Artifact To identify the geometric errors of machine tools, a novel 1-D artifact was designed with measurementTo identify targets the fixed geometric on it as errors Figure of2 a machine illustrates. tools, As Figure a novel2b shows, 1-D artifact the target was is designed composed with of measurementa plane mirror targets with a fixed diameter on it ofas 10Figure mm and2a illustrates. a revolutionary As Figure paraboloid 2b shows, with the a target diameter is composed of 5 mm. ofThe a plane equation mirror of the with revolutionary a diameter of paraboloid 10 mm and satisfies: a revolutionary paraboloid with a diameter of 5 mm. The equation of the revolutionary paraboloid satisfies: 2 2 x y + + == 2z ((unitunit: :m m).) . (1) 0.1547 . 0.1547 . Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 11 Appl. Sci. 2019, 9, 3357 3 of 11 Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 11
(a) (b)
Figure 2. (a()a Sketch) of the 1-D artifact; and (b) sketch of the measurement(b) target.
FigureFigure 2. (2.a) ( Sketcha) Sketch of theof the 1-D 1- artifact;D artifact and; and (b )(b sketch) sketch of theof the measurement measurement target. target. The distance of the two targets is approximately 50 mm, and nine targets were selected for a measuredTheThe distance linear distance range of the of twoofthe 400 targetstwo mm. targets is The approximately angleis approximately sensor 50 can mm, obtain and50 mm, nine 2- Dand targets angle nine wereinformation targets selected were forby selected ameasuring measured for a thelinearmeasured plane range mirror oflinear 400 and mm.range obtain Theof 400 angle2-D mm. micro sensor The displacement angle can obtain sensor 2-D informationcan angle obtain information 2 -byD anglemeasuring information by measuring the revolutionary by the measuring plane paraboloid.mirrorthe plane and obtain mirror 2-D and micro obtain displacement 2-D micro information displacement by measuringinformation the by revolutionary measuring the paraboloid. revolutionary paraboloid.Zekui Lv Lv elaborated elaborated the themechanism mechanism by which by an which angle an sensor angle identifies sensor 2-D identifies micro displacement 2-D micro displacementinformationZekui by informationLv measuring elaborated a byrevolutionary measuring the mechanism paraboloid a revolutionary by in which his paper paraboloid an [15 angle]. Figure in se nsor his3 illustrates paper identifies [15]. the principle Figure2-D micro 3 illustratesof measuringdisplacement the the principle informationoptical path. of Basedby measuring measuring on the theprinciple a opticalrevolutionary of optical path. auto-collimation,paraboloid Based on thein his the principle paper spot displacement [15]. of optical Figure 3 autois linearlyillustrates-collimation, related the the to principle thespot angle displacement of of measuring the measuring is lin theearly surface optical related in path.to the the small Basedangle angle of on the range the measuring principle [16]. On surface the of other optical in thehand,auto small the-collimation, displacement angle range the [16].ofspot the displacement On reflection the other point is hand, lin onearly the the revolutionary related displacement to the paraboloid angle of the of reflectionthe is alsomeasuring linearly point surfacerelated on the in withrevolutionarythe the small tangent angle paraboloid plane range angle is [16]. also at the On linearly reflection the other related point. hand, with Thus, thethe the displacementtangent spot displacementplane ofangle the at reflection is the proportional reflection point point. to on the the Thdisplacementrevolutionaryus, the spot ofdisplacement theparaboloid reflection isis point. alsoproportional linearly related to the displacement with the tangent of the plane reflection angle atpoint. the reflection point. Thus, the spot displacement is proportional to the displacement of the reflection point.
Figure 3. An illustration of the method of measuring the optical path. Figure 3. An illustration of the method of measuring the optical path. 2.3. Identification of GeometricFigure 3. Errors An illustration of the method of measuring the optical path. 2.3. Identification of Geometric Errors We took the X-axis geometric error measurement as an example; measurement for the Y-axis and 2.3. Identification of Geometric Errors Z-axisWe was took similar. the X As-axis Figure geometric4a shows, error the measurement spindle of the machineas an exa toolmple; drives measurement the angle sensor for the in Y the-axisX and Z-axis was similar. As Figure 4a shows, the spindle of the machine tool drives the angle sensor directionWe with took the the 1-D X artifact-axis geometric laid on the errorX-axis measurement horizontally. as The an rollexample; error εmeasurementx(x), the pitch for error theε yY(x-axis), in the X direction with the 1-D artifact laid on the X-axis horizontally. The roll error ( ), the pitch theand position Z-axis error was δsimilar.x(x), and As the Figure horizontal 4a shows, straightness the spindle error ofδ they(x) machinecan be identified tool drives by the measurement angle sensor error ( ), the position error ( ), and the horizontal straightness error ( ) can be identified in the X direction with the 1-D artifact laid on the X-axis horizontally. The roll error ( ), the pitch error ( ), the position error ( ), and the horizontal straightness error ( ) can be identified Appl. Sci. 2019, 9, 3357 4 of 11 of the plane mirrors and revolutionary paraboloids. In the case of good repeatability of the machine tools, the geometric errors remain approximately constant along the same motion path. Therefore, when the 1-D artifact is laid on the X-axis vertically, as shown in the Figure4b, the yaw error εz(x) and the vertical straightness error δz(x) can be identified.
(a) (b)
Figure 4. (a) Horizontal measurement with respect to the X-axis; and (b) vertical measurement with
respect to the X-axis.
Actually, due to the different orientation and position of each target, what the angle sensor obtains is the relative 2-D angle and 2-D position information of targets. Before identifying the geometric errors of the machine tools, the relative angle and position information should be calibrated by a laser interferometer or by machine tools with higher precision. Through comparing the calibration values with the measurement values obtained by machine tools, the geometric errors of the machine tools at the location of the targets can be identified by calculation.
3. Errors Identification Principle The method of angle and position calibration and geometric error calculation was introduced in this section. Before the identification of the errors of the machine tools, calibration of the reference artifact must be implemented; the aim of this was to obtain the relative angle and position information of targets in the workpiece coordinate system via a high accuracy method, such as by laser interferometer and machine tools with higher precision. The result of the calibration, called the calibration values, included the two-dimensional angle and position of each target in the workpiece coordinate system. It is worth mentioning that measurement values were obtained by the same calculation principle using the machine tool to be tested. However, the difference was that they include the two-dimensional angle and position by horizontal measurement and that by vertical measurement. Based on the calibration values and measurement values, the geometric errors could be determined.
3.1. Calibration of the Relative Angle In order to identify the angle errors of machine tools, a high-accuracy angular standard should be provided by calibration. The workpiece coordinate system was established as shown in Figure5, and the relative angles θxi and θyi are the tilt angles of the plane mirror of the ith target around the X-axis and Y-axis, respectively. The first target was selected as the reference target, and the angle of the other targets were determined relative to it. The calibration relative angles could be obtained through measurement compensated by laser interferometer.
1
Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 11
Appl. Sci. 2019, 9, 3357 5 of 11 Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 11
Figure 5. Sketch of the relative angle and position of targets.
The spot displacementFigure on 5. the image sensor was proportional to the angle of the reflective Figure 5. SketchSketch ofof thethe relativerelative angleangle andand positionposition ofof targets.targets. surface, i.e., each pixel corresponds to 2.27 arc seconds. Subpixel spot extraction can improve the The spot displacement on the image sensor was proportional to the angle of the reflective surface, angle The resolution spot displacement by one order on of the magnitude. image sensor By usingwas proportional an angle sensor to the to angle measure of the every reflective plane i.e., each pixel corresponds to 2.27 arc seconds. Subpixel spot extraction can improve the angle resolution mirror,surface, the i.e., displacement each pixel corresponds between the to spot 2.27 reflected arc seconds. by each Subpixel plane mirrorspot extraction and that canrefle improvected by thethe by one order of magnitude. By using an angle sensor to measure every plane mirror, the displacement referenceangle resolution plane mirror by one can order be obtained. of magnitude. Due to Bythe usingsurface an error angle of sensora plane to mirror, measure it is every rational plane to between the spot reflected by each plane mirror and that reflected by the reference plane mirror can computemirror, the the displacement average value between of multiple the spotsamplings. reflected As by such, each the plane calibration mirror andrelative that anglesreflected could by thebe be obtained. Due to the surface error of a plane mirror, it is rational to compute the average value of obtainedreference through plane mirror measurement can be obtained. compensated Due byto athe laser surface interferometer error of a withplane an mirror, angular it lens.is rational to multiple samplings. As such, the calibration relative angles could be obtained through measurement compute the average value of multiple samplings. As such, the calibration relative angles could be 3.2.compensated Calibration by of athe laser Relative interferometer Position with an angular lens. obtained through measurement compensated by a laser interferometer with an angular lens. 3.2. CalibrationThere is a point of the Relativeon every Position revolutionary paraboloid surface at which the tangent plane is parallel to3.2. the C alibrationplane mirror of the around Relative the Position revolutionary paraboloid. This point is defined as the feature point of the revolutionaryThere is a point paraboloid. on every revolutionaryTo quantitatively paraboloid determine surface the relative at which positioning the tangent of plane paraboloids, is parallel it to theThere plane is mirror a point around on every the revolutionary revolutionary paraboloid paraboloid. surface This pointat which is defined the tangent as the plane feature is parallel point canto the be planedefined mirror as the around relative the positioningrevolutionary of paraboloid.the feature Thispoints point of theis defined corresponding as the feature paraboloids. point of Likeof the the revolutionary computation paraboloid. of the relative To quantitativelyangle, the position determine of other the targets relative was positioning also found of relative paraboloids, to the itthe can revolutionary be defined as paraboloid. the relative To positioning quantitatively of the determine feature pointsthe relative of the positioning corresponding of paraboloids, paraboloids. it referencecan be defined target, as shownthe relative in Figure positioning 5. The displacement of the feature values points of andthe corresponding are the relative paraboloids. position ofLike the the feature computation point of of the the th relative targetangle, along thethe positionX-axis and of other Y-axis, targets respectively. was also As found Figure relative 6a shows, to the referenceLike the computation target, as shown of the in Figurerelative5. angle, The displacement the position valuesof otherX targetsand Ywasare also the found relative relative position to the of when the angle sensor moves along the X-axis, the reflection pointFi movesFi from to and the thereference feature target, point ofas theshownith target in Figure along 5. theTheX displacement-axis and Y-axis, values respectively. and As are Figure the 1 6relativea shows, 2 position when spotof the on feature the image point of sensor the th moves target fromalong the 1 toX -axis 2 . and In theY-axis, same respectively. paraboloid, As theFigure translational 6a shows, the angle sensor moves along the X-axis, the reflection point moves from ps1 to ps2 and the spot on displacementwhen the angle of thesensor angle moves sensor, along as mentionedthe X-axis, theabove, reflection is linearly point related moves with from the displacement to and theof the image sensor moves from pc1 to pc2. In the same paraboloid, the translational displacement 1 2 of the the spot on the image sensor. Besides this, the coefficient of linearity can be obtained by the anglespot sensor,on the as image mentioned sensor above, moves is linearly from related1 to with2 . In the the displacement same paraboloid, of thespot the on translational the image measurementsensor.displacement Besides of this, the the the a ngle angle coe ffisensor, cient sensor of as linearity onmentioned the canrevolutionary beabove, obtained is linearly paraboloid. by the related measurement To with conduct the of thedisplacement a anglelarge sensor-scale of measurement,onthe the spot revolutionary on thethe imagedistances paraboloid. sensor. of the BesidesTo feature conduct this, points a large-scale the of coefficient different measurement, revolutionary of linearity the can distancesparaboloids be obtained of theshould feature by thebe determined.pointsmeasurement of diff erent of revolutionary the angle sensor paraboloids on the should revolutionary be determined. paraboloid. To conduct a large-scale measurement, the distances of the feature points of different revolutionary paraboloids should be determined.
(a) (b)
Figure 6. (a)) Relative displacement measurement in one revolutionary paraboloid paraboloid;; and ( b) r relativeelative position measurement (a) of feature points on didifferentfferent revolutionaryrevolutionary paraboloids.paraboloid(b) s. Figure 6. (a) Relative displacement measurement in one revolutionary paraboloid; and (b) relative In order order to to calculate calculate the the relative relative positioning positioning of feature of feature points points in the in workpiece the workpiece coordinate coordinate system, position measurement of feature points on different revolutionary paraboloids. system,the machine the drives machine an angledrives sensor an angle from sensor the measurement from the measurement point M1 to M 2 pointalong the1 toX -axis, 2 along as shown the in Figure6b. The points F and F are defined as the feature points of two revolutionary paraboloids. In order to calculate1 the relative2 positioning of feature points in the workpiece coordinate
system, the machine drives an angle sensor from the measurement point 1 to 2 along the Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 11 Appl. Sci. 2019, 9, 3357 6 of 11
X-axis, as shown in Figure 6b. The points 1 and 2 are defined as the feature points of two Spotrevolutionary images of paraboloids. the featurepoint Spot andimages the of measurement the feature point point and are shownthe measurement in Figure7a,b. point The are position shown ofin Figure the spot 7a,b. center The isposition obtained of the by spot ellipse center fitting, is obtained which can by ellipse realize fitting subpixel, which spot can extraction. realize subpixel It was foundspot extraction. by calibration It was that found each pixel by calibration in the image that corresponds each pixel in to 2.27the image arc seconds corresponds or 1.6 µ m to, 2.27 and the arc resolutionsseconds or of 1.6 the μm angle, and and the position resolutions can reach,of the respectively, angle and position 0.2 arc seconds can reach, and respectively, 0.16 µm by subpixel 0.2 arc spotseconds extraction. and 0.16 Actually, μm by subpixel the spot imagespot extraction. of a feature Actually, point cannot the spot be obtained image of directly, a feature so itpoint is obtained cannot bybe obtained measuring directly, the corresponding so it is obtained plane by mirror. measuring the corresponding plane mirror.
(a) (b)
Figure 7. (a) Spot image of the feature point (by measuring the corresponding plane mirror) mirror);; and ( b) spotspot image of the measurement pointpoint onon aa revolutionaryrevolutionary paraboloid.paraboloid.
As illustrated in Figure6b, the vector relation between two feature points F and F is described As illustrated in Figure 6b, the vector relation between two feature points1 2 1 and 2 is bydescribed the following by the equation:following equation: → → → → F1F2 = F1M1 + M1M2 + M2F2. (2) ⃗ = ⃗ + ⃗ + ⃗. (2) In the X direction, we have: In the X direction, we have:
(X(F X− ) =) =(X( −X ) +) +(X(M −X ))++(X( F −X M )),, (3)(3) 2 − F1 M1 − F1 2 − M 1 2 − 2 where and are the X coordinates of feature points and in the object ordinate system, where XF 22 and X F11 are the X coordinates of feature points F1 and F22 in the object ordinate system, and and are the X coordinates of measuring points and in the object ordinate and XM 22 and X M11 are the X coordinates of measuring points M1 and1 M2 in the2 object ordinate system. Throughsystem. Through measurement measurement by laser by interferometer laser interferometer with a position with a position lens or machine lens or machine tools with tools higher with precision,higher precision,lxM2M1 , defined 2 1 , de asfined the distance as the distance between between measuring measuring points M 1 pointsand M 2,1 can and be calculated.2, can be Incalculated. addition, In the addition, distance the between distance the between measurement the measurement point and the point feature and point the infeature a revolutionary point in a paraboloidrevolutionary can paraboloid be determined can based be determined on the spot based displacement on the spot in an displacement image by the in following an image equation: by the following equation: (XMi XFi) = k(xMi xFi), (4) ( −− ) = ( −− ), (4) where k is is the the conversionconversion coecoefficientfficient from spot displacement displacement to to actual actual displacement displacement calibrated calibrated by by a alaser laser interferometer; interferometer; X Mi andand X Fiareare the the XX coordinatescoordinates ofof thethe measurementmeasurement point point and and feature feature point point of theof theith revolutionary th revolutionary paraboloid paraboloid in the object in the coordinate object coordinate system; andsystem;xMi and andx Fi are and the X coordinates are the X ofcoordinates the measurement of the measurement point and feature point point and offeature the ith point revolutionary of the th paraboloidrevolutionary in the paraboloid image system. in the imageSince system. the aim of the computation of the relative positioning was to obtain the position of each featureSince point the relative aim of tothe the computation feature point of ofthe the relative reference positioning revolutionary was to paraboloid, obtain the thepositionX coordinate of each XfeatureF0 of the point feature relative point to ofthe the feature reference point revolutionary of the reference paraboloid revolutionary was equal paraboloid, to 0. Therefore, the X coordinate in the X direction, of the the feature relative point position of the ofreference the feature revolutionary point of the paraboloidith revolutionary was equal paraboloid to 0. Therefore, can be calculated in the X asdirection, follows: the relative position of the feature point of the th revolutionary paraboloid can be calculated as follows: X = k(xM xF ) + lx + k(x x ). (5) Fi 0 − 0 MiM0 Fi − Mi
= ( − ) + + ( − ). (5) Similarly, in the Y direction, we have: Appl. Sci. 2019, 9, 3357 7 of 11
Similarly, in the Y direction, we have:
Y = k(yM yF ) + ly + k(y y ). (6) Fi 0 − 0 MiM0 Fi − Mi
In these equations, XFi and YFi represent the relative position of the ith feature point in the object system. Besides this, xM0, yM0, xMi, yMi, xF0, yF0, xFi, and yFi are the coordinates of the spot in the image system, and lxMiM0 and lyMiM0 are, respectively, the displacement in the X direction and the Y direction measured by laser interferometer or machine tools with higher precision.
3.3. Calculation of the 6 Degrees-of-Freedom Errors
By the method mentioned in Sections 3.1 and 3.2, the relative angles θxi and θyi and position XFi and YFi of the ith target were calibrated. To ensure repeatability of the measurement system, the angle difference between the first and the last plane mirror should be, where possible, the same as the angle difference of calibration. The alignment can be conducted based on the spot position in the image, which can reduce the cosine error of the measurement effectively. When the 1-D artifact was located on the X-axis of the machine tool horizontally, the measurement value of the relative angles θ0xi and θ0yi and relative position X0Fi and YFi0 of the ith target could be obtained by the machine tool to be tested. In the process of measurement, lxMiM0 and lyMiM0 should be computed by the feedback value of the machine tool. That means that X0Fi, Y0Fi are obtained by the following equations: X0 = k(x0M x0F ) + X0 X0 + k(x0 x0 ), (7) Fi 0 − 0 Mi − M0 Fi − Mi Y0 = k y0 y0 + Y0 Y0 + k y0 y0 , (8) Fi M0 − F0 Mi − M0 Fi − Mi where x0M0, y0M0, x0Mi, y0Mi, x0F0, y0F0, x0Fi, and y0Fi are the coordinates of the spot in the image system when measuring horizontally, and XM0 0, XMi0 , YM0 0, and YMi0 are the measured coordinates of the measurement point in the machine coordinate system. Due to geometric errors in the linear motion of the machine tool, the relative angle and position measured by the machine tool are not the same as the calibration values. The roll error εx(x) and pitch error εy(x) are caused by deviation in the relative angles and the position error δx(x) and horizontal straightness error δy(x) are caused by deviation in the relative position. Thus, by calculating the deviation between the measurement values and calibration values, these errors at the target positions can be identified. The roll error εx(x) and the pitch error εy(x) at the target position can be identified by the following equations: εx(x) = θ0 θ , (9) xi − xi εy(x) = θ0 θ . (10) yi − yi Due to the character of a revolutionary paraboloid, the displacement difference measured by the machine tool includes coupling of angle errors. Hence, the position error δx(x) at the target position can be identified by the following equation:
δx(x) = X0 X kmεy(x ), (11) Fi − Fi − Fi where k and m represent the conversion coefficients from spot displacement in the image to actual displacement and from angle variation to spot displacement in the image, respectively, and εy(xFi) is the pitch error at the target position. Similarly, the horizontal straightness error δy(x) can be identified by the following equation:
δy(x) = Y0 Y kmεx(x ). (12) Fi − Fi − Fi Appl. Sci. 2019, 9, 3357 8 of 11
When the 1-D artifact is located vertically, the measurement values of the relative angles θ00 xi and θ00 xi and relative position X00 Fi and Y00 Fi of the ith target can be obtained by the following equations:
Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 11 X00 = k(x00 M x00 F ) + X00 X00 + k(x00 x00 ), (13) Fi 0 − 0 Mi − M0 Fi − Mi Y 00 ′′ ==k y(00 ′′ −y00 ′′ +) +Z(00 −Z00 +) +k y(00 ′′ −y00 ′′ ,), (14)(13) Fi M0 − F0 Mi − M0 Fi − Mi where x00 M0, y00 M0, x00 Mi, y00 Mi, x00 F0, y00 F0, x00 Fi, and y00 Fi are the coordinates of the spot in the image ′′ = ( ′′ − ′′ ) + ( − ) + − , (14) system when measuring vertically. Unlike horizontal layouts, the Y coordinates of each target are calculatedwhere ′′ based , ′′ on , the′′ displacement, ′′ , ′′ , in′′ the, Z′′ direction, and ′′ of are the the machine. coordinates Thus, of the theX coordinates spot in the image00 system00 when measuring vertically.00 Unlike00 horizontal layouts, the Y coordinates of each target XM0 and XMi and the Z coordinates ZM0 and ZMi of the measurement point in the machine coordinate systemare calculated were used based to computeon the displacement the relative positionsin the Z direction of targets. of the machine. Thus, the X coordinates and and the Z coordinates and of the measurement point in the machine Similarly, the yaw error εz(x) and the vertical straightness error δz(x), instead of the pitch error coordinate system were used to compute the relative positions of targets. εy(x) and horizontal straightness error δy(x), are caused by deviation in the relative angle θyi00 and ( ) ( ) positionSimilarly,YFi00 . Therefore, the yaw error the yaw error andεz ( thex) can vertical be identified straightness by the error equation: , instead of the pitch error ( ) and horizontal straightness error ( ) , are caused by deviation in the relative ′′ εz(x) = θ00 θ , (15) angle and position . Therefore, the yaw erroryi − (yi ) can be identified by the equation: ( ) = ′′ − , and the vertical straightness error δz(x) can be identified by the equation: (15)
and the vertical straightness error ( ) can be identified by the equation: δz(x) = Y00 Y kmεx(x ). (16) Fi − Fi − Fi ( ) = ′′ − − ( ). (16) Thus, the 6 degrees-of-freedom errors of a linear motion guide rail can all be identified. Thus, the 6 degrees-of-freedom errors of a linear motion guide rail can all be identified. 4. Experiment 4. Experiment Experiments in the X direction were carried out in a CMM, and Figure8a shows the measurement setupExperiments when the 1-D in artifact the X was direction located horizontally. were carried Through out in this a CMM, measurement, and Figure two angle 8a shows errors and the twomeasurement position errors setup were when identified. the 1-D artifact was located horizontally. Through this measurement, two angle errors and two position errors were identified.
Artifact laid Artifact laid horizontally vertically
(a) (b)
Figure 8. ((aa)) Measurement Measurement horizontally; and ( b) m measurementeasurement vertically.
( ) ( ) The calculated results of of the the position position error error δ x( x) andand the horizontal straightness error error δ y( x) identifiedidentified throughthrough measurement measurement using using the the 1-D 1-D artifact artifact and and laser laser interferometer interferometer are shownare shown in Figure in Figure9a,b. The9a,b. type The oftype the of laser the interferometerlaser interferometer was an was SJ6000, an SJ6000, produced produced by Chotest. by Chotest. The results of the pitch error εy(x) and the roll error εx(x) are shown in Figure 10a,b. Since the laser interferometer cannot identify the roll error of the machine tool, the measurement result of the roll error was not compared with that measured by laser interferometer. Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 11
Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 11
Appl. Sci. 2019, 9, 3357 9 of 11 Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 11
(a) (b) (a) (b) Figure 9. (a) The position errors of the X-axis; and (b) the horizontal straightness errors of the X-axis. Figure 9. (a) The position errors of the X-axis; and (b) the horizontal straightness errors of the X-axis.
The results of the pitch error ( ) and the roll error ( ) are shown in Figure 10a,b. Since the The results of the pitch error ( ) and the roll error ( ) are shown in Figure 10a,b. Since the laserlaser interferometer interferometer cannot cannot( identifya )identify the the rollroll error ofof the the machine machine tool, tool,(b the) the measurement measurement result result of the of the roll rollerror error was was not not compared compared with with that that measured measured byby laser laser interferometer. interferometer. FigureFigure 9. 9. ((aa)) The The position position errors errors of of the the XX--axis;axis; and and ( (bb)) t thehe horizontal horizontal straightness straightness errors errors of of the the XX--axis.axis.
The results of the pitch error ( ) and the roll error ( ) are shown in Figure 10a,b. Since the laser interferometer cannot identify the roll error of the machine tool, the measurement result of the roll error was not compared with that measured by laser interferometer.
(a) (b)
Figure 10.( a(a)) The pitch errors of the X-axis; and (b) the roll errors(b) of the X-axis. FigureFigure 10. 10.(a) (Thea) The pitch pitch errors errors of of the the XX--axis;axis; and ((bb)) the the roll roll errors errors of of the theX-axis. X-axis. For a vertically oriented 1-D artifact, Figure 11a,b illustrates the calculated results of the vertical straightness error ( ) and the yaw error ( ) measured using the 1-D artifact and a laser For aFor vertically a vertically oriented oriented(a )1 -D 1-D artifact artifact,, Figure Figure 11a,b11a,b illustratesillustrate s the the(b calculated )calculated results results of the of verticalthe vertical interferometer.straightness error δz(x) and the yaw error εz(x) measured using the 1-D artifact and a laser interferometer. straightness errorFigure ( 10.) and(a) The the pitch yaw errors error of the X -(axis; ) measuredand (b) the roll using errors theof the 1 -XD-axis. artifact and a laser interferometer. For a vertically oriented 1-D artifact, Figure 11a,b illustrates the calculated results of the vertical
straightness error ( ) and the yaw error ( ) measured using the 1-D artifact and a laser interferometer.
(a) (b)
FigureFigure 11. 11. (a(a) )The The vertical vertical straightness straightness errors errors of of the the XX-axis;-axis; and and (b (b) )the the yaw yaw errors errors of of the the XX--axis.axis. By comparing the(a geometric) errors of linear motion of the machine tools(b measured) usinga 1-D artifactFigure with 11. those (a) The obtained vertical(a) using straightness a laser interferometer, errors of the itX could-axis; and be seen (b) thatthe yaw the(b) maximum errors of the measurement X-axis. Figure 11. (a) The vertical straightness errors of the X-axis; and (b) the yaw errors of the X-axis. Appl. Sci. 2019, 9, 3357 10 of 11 deviation and mean measurement deviation of the position errors were 1.2 µm and 0.57 µm, while the maximum measurement deviation and mean measurement deviation of the angle errors were 0.8” and 0.32”, which might be caused by processing errors of the revolutionary paraboloids and plane mirrors.
5. Conclusions This article presented an innovative method for the measurement of geometric errors in the linear motion of CNC machine tools. The main innovation was the combination of revolutionary paraboloids and plane mirrors in an array. The former realized the measurement of position and angle, and the latter expanded the measurement range significantly, which made it possible for the method to be applied in error detection for machine tools. In addition, the measurement mechanism and the principle of identifying geometric errors were described in detail. By locating the 1-D artifact horizontally and vertically, a total of six geometric errors of a guide rail moving along the X direction were identified. By comparing with measurement results obtained using a laser interferometer, the feasibility and practicability were confirmed. Incidentally, the measurement system based on the new 1-D artifact could also be applied to identify the squareness errors of machine tools. An L-shaped artifact could be produced by stitching two 1-D artifacts together to include two-dimensional information of different vertical directions in order to measure the squareness. Due to the measurement mechanism, the identification of straightness errors and angle errors did not demand good repeatability of the pose in the workspace. That means the result of these errors would not vary obviously if the pose of the 1-D artifact did not change significantly. However, the positioning error, because of its distribution along the X-axis, was sensitive to the pose of the 1-D artifact. Thus, we obtained the relative position of each target in the workpiece coordinate system to weaken this influence, i.e., to reduce the error caused by lateral tilt of the 1-D artifact from first-order error to second-order error (cosine error). However, the error caused by vertical tilt of the 1-D artifact could not be eliminated. Hence, when the levelness of the machine tool bed is inferior, identification of the positioning error based on this method will not show such good performance. This is also the reason why we selected horizontal measurement to identify the positioning error of machine tools. Besides this, the accuracy of position and angle measurement using this method could also reach a fairly high standard, and the rapidity, scalability, and operability of the method make it easy to automate and integrate. Due to the advantages of a noncontact measurement, the measurement time of an axis based on the method proposed is less than two hours if data processing is automated. Thus, this method has considerable value and potential.
Author Contributions: F.F. and X.L. proposed the method; X.W. designed the measurement experiments and wrote the paper; X.Y. and Z.L. developed system software and processed the data; Z.S. and Z.Y. designed optical structure; H.Z. modified the paper. Funding: This research was financially supported by the National Natural Science Foundation of China (NSFC) (No: 51775378), the National Key R&D Program of China (No.2017YFF0108102) and Natural Science Foundation of Shanxi Province, China (Grant No. 201801D121180). Conflicts of Interest: The authors declare no conflict of interest.
References
1. Liu, H.; Hua, X.; Chen, J.; Rui, Y. Measurement and compensation of machine tool geometry error based on Abbe principle. Int. J. Adv. Manuf. Technol. 2018, 98, 2769–2774. [CrossRef] 2. Wang, X.; Liu, S.; Zhang, G.; Wang, B. Error analysis of the articulated flexible arm CMM based on geometric method. In Proceedings of the Sixth International Symposium on Instrumentation and Control Technology, Beijing, China, 6 November 2006. 3. Xiang, S.; Li, H.; Ming, D.; Yang, J. Geometric error identification and compensation for non-orthogonal five-axis machine tools. Int. J. Adv. Manuf. Technol. 2018, 96, 2915–2929. [CrossRef] 4. Pan, F.; Li, N.; Bai, Y.; Wang, X.; Wu, X. Geometric errors measurement for coordinate measuring machines. Iop Conf. Ser. Earth Environ. Sci. 2017, 81, 012117. [CrossRef] Appl. Sci. 2019, 9, 3357 11 of 11
5. Zhang, G.X. Coordinate Measuring Machines; Tianjin University Press: Tianjin, China, 1999; pp. 405–425. 6. Giacomo, B.D. Calibration of Positional Errors in Coordinate Measuring Machines and Machine Tools Using a Laser Interferometer System. Appl. Mech. Mater. 2015, 798, 303–307. [CrossRef] 7. Anke, G.; Dirk, S.; Gert, G. Self-Calibration Method for a Ball Plate Artefact on a CMM. CIRP Ann. 2016, 65, 503–506. 8. Weikert, S. R-Test, a New Device for Accuracy Measurements on Five Axis Machine Tools. CIRP Ann. Manuf. Technol. 2004, 53, 429–432. [CrossRef] 9. Ibaraki, S.; Oyama, C.; Otsubo, H. Construction of an error map of rotary axes on a five-axis machining center by static R-test. Int. J. Mach. Tools Manuf. 2011, 51, 190–200. [CrossRef] 10. Masashi, Y.; Hamabata, N.; Ihara, Y. Evaluation of Linear Axis Motion Error of Machine Tools Using an R-test Device. Procedia. Cirp. 2014, 14, 311–316. [CrossRef] 11. Cheng, F.; David Lee, B. An innovative method for coordinate measuring machine one-dimensional self-calibration with simplified experimental process. Rev. Sci. Instrum. 2013, 84, 139–144. 12. Tsutsumi, M.; Saito, A. Identification and compensation of systematic deviations particular to 5-axis machining centers. Int. J. Mach. Tools Manuf. 2003, 43, 771–780. [CrossRef] 13. Zhong, G.; Wang, C.; Yang, S.; Zheng, E.; Ge, Y. Position geometric error modeling, identification and compensation for large 5-axis machining center prototype. Int. J. Mach. Tools Manuf. 2015, 89, 142–150. [CrossRef] 14. Wu, B.; Yin, Y.; Zhang, Y.; Luo, M. A new approach to geometric error modeling and compensation for a three-axis machine tool. Int. J. Adv. Manuf. Technol. 2018.[CrossRef] 15. Lv, Z.; Li, X.; Su, Z.; Zhang, D.; Yang, X.; Li, H.; Li, J.; Fang, F. A Novel 2D Micro-Displacement Measurement Method Based on the Elliptical Paraboloid. Appl. Sci. 2019, 9, 2517. [CrossRef] 16. Sun, H.; Geng, A.-H.; Zhang, H.-B.; Wang, X. Precision Angle-Measuring System Based on Linear CCD. MicroComput. Inf. 2009, 25, 61–62.
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).