approach to machining error compensation

8/4/2019 Approach to Machining Error Compensation

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Integrated Inspection and Machining for Maximum Conformance to Design TolerancesH. A. EIMaraghy (I ), A. Barari , G. K. Knopf

Industrial and Ma nufacturing Systems Engineering, University of Windsor, Windsor, Ontario, CanadaMechanical and M aterials Engineering, University of Western Ontario, London, Ontario, Canad a

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AbstractDesigners intent for the form, fit and function of p roducts is expressed by design tolerances the conformanceto which is the main objective of manufacturing processes. A methodology for maximizing the adherence tothe specified tolerances using an integrated machining and inspection system is presented. Considering thedesired tolerance envelope of the part, an error decomposition technique is developed to model machiningerrors caused by the systematic and non-systematic errors in the machine tool. The model is used toadaptively plan the final m achining cuts, based on inspection feedback, to enhance the g eometric accuracy ofthe final product and is illustrated by an example. This approach reduces scrap and rework and theirassociated costs.Keywords:Optimization, Error Compensation, Design Tolerance

1 INTRODUCTIONThe accuracy of machine tools is a critical factor thataffects the accuracy of manufactured products. Any toolmotion error causes a one-to-one corresponding error inthe work piece geometry. Errors in machine tool motionsare essentially produced by geometric imperfections of thestructural elements a nd by thermal and loa d induced errorscaused by the machining process itself. It is impossible toeliminate all these errors by modifications in design and/ormanufacture of the hardware. As a result, developing anerror models for the machine tools and software errorcompensation are important research issues. In late1970s, researchers focused mostly on the characteristics,classification an d beha viours of machine errors in the threedimensional workspace [I]. Later in 1980s,implementations and applications of theoretical errormodels to compensate the actual machine tool error weredeveloped [2], [3], [4]. Machine tool errors are mostlymeasured during off line calibration to obtain the modelscoefficients. The resulting error model is applied throughthe machine controller to modify the actual cuttingcomman ds. The m odel is an analytically expression for thegeometric errors of a machine. However, obtaining itscoefficients, which vary acc ording to the working loads a ndthermal cond itions, is expensive an d time con suming. Thiswould cause the temporal validity of the error model to bein serious doubt. Hence, research shifted to findingalternative approaches recognizing that the geometry ofthe machined pa rt reflects the contribution of all errors andits accuracy is the ultimate objective. Research related torequired sensors, logistics and system architecture for on-line measurement of the workpiece in recent years havebeen reported [5][6][7]. Some research tried to findanalytical relationship between the actual workpieceaccuracy and the coefficients of the machine tool errormodel [8].This paper belongs to a new generation of errorcompensation techniques. It focuses on modelling themachining errors instead of machine tool errors. Thisapproach has some obvious advantages since themachining errors represent the sum of interactions of allindependent machine tool errors and their variations. Inaddition, the generated error model is developed in -process and he nce can readily be applied for on-line error

compensation of the current work-piece. The Developedmachining error model lends itself to directly compensatefor errors though machining commands instead of thecontroller pulses that has been the trend to date. Someapproaches used least square line or curve fitting,[9][1 O][ ll], or neural network learning system [I2 1 toestimate or model machining errors from the measuredpoints, for 2D contours or turning processes.The proposed method maximizes conformance to thedesign tolerances and is applicable for 3D sculpturedsurfaces. The obtained machining errors model ensuresthat the wo rkpiece features fit within the desired toleranceenvelope while minimizing further machining corrections.The proposed method is suitable for use in integratedinspection and machining systems with on-line in-processinspection or intermittent measurement and geometricfeedback.2 BACKGROUND2.1 Machining ErrorsFigure 1 shows the relationship between any point definedby vector p,, in the desired geometry or machiningcomman d and the corresponding point in the actual cuttingresult defined by vector p, , where index i is the point index.The correspo nding machining error vector is defined by E ~ .The relationship can be described as:p* = p + & T (11Machine tool errors are either systematic or non-systematic. Systematic errors are quasi-static; they varyvery slowly with time and are related to the structure of themachine itself [13]. This type of errors includes geometricimperfections, thermal deformations and alignment errors.Non-systematic errors are the minor part of the machinetool total error and are due to som e uncontrollable sourcessuch as machine tool vibration, spindle vibration or toolchatter [2]. Systematic errors are estimated to account for%70 of machining total error and have been observed tobe as high as 70-120 pm for production class machines[14]. A rigid body kinematic model, based on thehomogeneous coordinate transformation, is usuallyemployed to m odel machine tool quasi-static errors.


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2.2 Rigid body KinematicsMulti-axis machines are composed of a sequence ofelements or links connected by joints to provide motions.With rigid body kinematics, each element and joint can bemodelled using homogeneous coordinate transformationsto describe the position and orientation of one object withrespect to several different coordinate systems. In a typicalmachine tool, with prismatic joints for each translation axis,there exist errors in six degrees of freedom in addition ofthe intended motion. By assigning a coordinate frame to aslide and using a homogen eous transformation matrix, it ispossible to describe the motion of the slide in a referencecoordinate system [2]. Therefore, a machine tool errormodel can be derived for specific machine types. Theactual position and the orientation of the slide coordinateframe are different from that of the ideal slide due to thesesix unwanted motions. Using the small angleapproximation, the desired X slide motion, H x, with all theunwanted motions, can be represented by the followingmatrix [14]:

1xRz+ Sxy 1IXRY -SZX XRX 1 xTzHX =l o 0 0Motions of the other slides, Hy and H z , are representedsimilarly [2]. In the no tations used to depict param etric

Figure 1: Desired and actual geometries.errors, R means rotation, and T means translation. The lefthand lowercase letter means the moving slide and the righthand lowercase letter means the error direction. In ageneric three-axis machine tool, three squareness errorsare defined as the constant parametric errors. They can bedefined as Sxy, Syz and Szx where in these notations Smeans squareness, and the two following letters indicatethat the error is between these two reference axe s. Since amachine tool can be considered as a chain of linkages, thespatial relationship between the cutting tool and the work-piece can be easily determined. The transformation matrixof total system, H , for a generic three-dimensional m achinetool is as follow:H = H, - Hu - H, (3 )The forth column of matrix is the actual position of thecutter relative to the reference coordinate system.Therefore, for any desired point in the machiningcommands, with coordinates of p = [OX DY DZ IT,

the actual point due to quasi-static machine tool error is pwhich is calculated as follows where ( j = [0 0 0 IT :p =H . j (4 )3 QUASISTATIC ERROR AS A LINEAR OPERATORIn common practices, equation (4) is used to compensatefor machine tool errors by the controller. The derivation ofmatrix H requires heavy symbolic manipulation andrequires many simplifications. Ho wever, since equation (4)represent a point-to-point relationship between p an d p , itwould be much more efficient to find a u nified operator thatdirectly relates the whole nominal geometry to the actualmachined surface. Such op erator can be app lied directly tothe machining commands, instead of the controller pulses,with obvious advantages. Therefore, the relationshipbetween the nominal and actual geometry can beexpressed as follow:s~ =OQ.D, (5)Where, DG is the desired geometry and SG is the actualmachined surface effected by all systematic machine toolerrors. By inspection of the second derivatives of p in (4),it appears that each component of p is a linear function ofcomponents of the desired point p . Therefore, a Jacobianof p with respect to p extracts the coefficients of therequired operator as follows:

The quasi-static errors operator, OQ, s obtained by:0, = J . V - ( J . p s ) . j T ( 7 )Where, matrix V is an identity matrix with all zero in its lastcolumn and homogeneous vector p s is the symmetry of pwith respect to the reference point. Using ( 7 ) , he quasi-static errors operator has a form of:

1 -X RZ -S X Y W~(X RX ,.. ) W ~ ( X R X ,.. )OQ=[o 1 W ~ ( X R X ,.. ) W ~ ( X R X ,.. )0 XRX W ~ ( X R X ,. . )+ 1 WE(XRX,.. )

0 0 0 1In this model, W, to W3 an d W 4 to WE are second and thirdorder polynomial functions of independent machine toolerrors. Therefore, the quasi-static errors of machine toolscan be represented by a linear transformation matrix, OQ,which can be applied to the total work-piece geometry tocompensate for m achine tool errors.It can be proven mathematically that any linear operatorcan be decomposed into two multiplied linear operators.The most suitable linear operator for machining errorscompensation is a rigid body transformation because itlends itself to the tolerance zon es definition utilized in thestandards [15], an d since it is orthogonal; it would be easyto correct machining errors by simply translating androtating the coordinates of machining commands withoutaffecting the geometry. A method for extracting therequired rigid body transformation from the total machiningerror matrix, OQ, s presented next.


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4 MACHINING ERROR CORRECTION AND MAXIMUMCONFOORMANCE TO TOLERANCE

The d esign intents for form, fit an d function of products arerepresented by geometrical and dimensioning tolerances.They de fine criteria for acc epting or rejectin g the final partduring inspection and quality control. It is essential forsome products such as mould and dies used to produceauto-parts, to ensure that these machined parts meetspecifications. Any effort to ensure acceptance isworthwhile considering their high cost. An integratedmachining and inspection system together with theproposed error correction method are very useful in thesecases to evaluate the actual geometric deviations ofworkpiece in time to make the necessary errors correction.The tolerance zones definition and required condition forconformance to tolerance are prescribed in the standardsof mathem atical definition of dimensioning and tolerancingprinciples, ASME Y14.5 [15]. The constraints defined bytolerances describe a tolerance envelope for the partgeometry as shown in Figure 1 for a profile feature.Mathematically a point p,', conforms to the tolerance whenthere exists a point p , on the nominal surface with asurface normal of n,and some u, or which:p j = pi +n,uAccording to this definition, m achined part conform,s to thedesired profile tolerance envelope if all points p , of thesurface lie within the upper or lower tolerance, tu or tl, forsome corresponding point p , on the nominal surface. Thedetermination of this corresponding point, p , , is achallenging task [16]. In the current approach, allcorresponding points, p,s, are defined such that themaximum conformance with the design tolerances isachieved.Considering equation (9), the tolerance envelope for thepart features can be defined by two rigid bodytransformations when it is topologically constrained by thetolerance envelopes of the related datum features.Therefore, a rigid body transformation that achievesmaximum conformance to the tolerance envelope is therequired limit for correcting machining errors detected byinspection.In Figure 2(a), the tolerance envelope for a single two -dimensional feature is il lustrated. Portion (b) shows theresult of intermittent measurem ent represented by discretedata points.Any measured point that lies outside the upper limit of thetolerance envelope represents under cut material that canbe corrected. Points under the lower tolerance envelopelimit represent an over cut zone that is not possible tocorrect. Con sidering these facts an d using equation (9), anerror function, named Fonformance residual, is defined forany measured point, p , as follow:

-tl s u s tu (9)

I 0 if dui +dli = t u+ t l{ {dui+dlj > tu + l.zi = duj if duj < dljif { d q + d l j > t u + t l

duj > dljWhere, du, an d dl,, are the Euclidean distances of themeasured point, p , , rom the lower and upper limits oftolerance envelope of a substitute geometry SG.Tolerance envelope of a substitute geometry SG is theproduct of rigid body transformation of the desiredgeometry, DG

Figure 2: Maximum conformance to tolerance m ethod.

SG= T(f). DG (13)Where, n, s the corresponding surface normal vector. T(f)is the rigid body transformation matrix defined by vectorvariable f, which consists of three rotation and threetransformation parameters. It can be seen that the errorvalue defined by (10) for any measured point is a functionof vector variable f. The maximum conformance to thetolerance envelope is achieved when the maximum of theerror function, given by equation (14) is minimized.obj = ~ ; n~ y x) (14)

t

The resulting transformation matrix, T', is the majordecomposed part of the total error operator. It maximizesconformance to the specified tolerances. The Inverse ofthis orthogonal matrix is the linear operator used directlyfor correcting detected machining errors. There is one-to-one mapping between points on the actual part surfaceand their corresponding points on the nominal surface.This m akes local correction of residual errors possible.Figure 2-(c) shows the result of optimisation. The totalmachining error is decomposed into two components: aCompensable error, .zC,,and a residual error E ~ , . he first is


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corrected globally for the whole machining surface.Residual errors are corrected locally and iteratively.5 OPTIMIZATIONThe optimisation problem defined in (14) has the followingcharacteristics:0 Objective function is highly nonlinear with 6 variables.0 In order to avoid any over cut in the work-piece; adiscontinuity in the objective function is created whichmakes its solution difficult.0 A fast optimisation algorithm is required for on-lineapplications.0 A direct search method, which does not need thegradient of the objective function, is employed; hencethe solution is independent of the part geometry.

The Nelder-Mead Simplex method is used. Itsconvergence de pends on the initial simplex (starting point)and the function continuity [17]. Although the least squarebest fit is the most likelihood estimation to fit a substitutefunction into a set of discrete d ata, it tends to overestimatethe geometric deviation and doesn't conform to thetolerance zone definitions in the standards [16], [15].However, its result is useful as a quick and reasonableestimation of the initial simplex of the main prob lem. Thesum of square errors to be m inimized is:

The objective function switches to (1 5) automaticallywhenever the simplex search gets trapped in the infinitypart of the error function. The use of these techniquesresulted in robust convergence properties for the search.6 SIMULATION AND VALIDATION EXPRIMENTS6.1 Simulation of machining errorsIn order to validate of the proposed method, we need toinvestigate how close the optimum rigid bodytransformation matrix obtained by (14), T', is to thetransformation matrix presented in (8), OQ.A simulation ofthe machining errors is presented in this section. Twoactual calibration results of two typical vertical millingmachine tools are reported in [I3 1 (data set # 1) and [8](data set #2) have been use d to rep resent the quasi-staticpart of the ma chining error.Using the m odel presented in (8), the quasi-static operatorcaused by error set # I is:

[IOOOOO -.00005 0.00038 0.0590010.00000 1.00000 -.00006 0.126000.00000 0.00002 0,99999 0.03049Ql = ~0 .00 000 0.00000 0.00000 1 o o o o o ~

The second data set produces another quasi-static errorsoperator as follow:

o w = : .00000 0.00004 0,99999 0.030990.00000 0.00000 0.00000 1 ooooo

It can be observed that the determinant of both matricesdeviates slightly from unity and that they are very close tobeing orthogonal transformation matrices.In order to simulate the non-systematic part of themachining error, a normally distribution error is added tothe quasi-static errors in all three X, Y an d 2 directions.

1 OOOOO -.00007 0.00027 0.028000.00000 1.00000 -.0000 9 0 .05999] (17)

The distribution of the applied noises is chosen such thatthey represent almost 30% of the total machining errors.The distribution of non-systematic errors has a me an equalto zero an d standard deviation of 10 pm , which m eans thatthe non-systematic errors statistically vary between -30 to30 pm in any spatial direction. Simulation and numericalexperiments for variety of geometric primitives andsculptured surface were cond ucted.Figure (3) shows the ma chining error simulation for a cubicB-spline surface with overall dimensions of270mmx200mmx60mm created using data set # I andnon-systematic errors normally distributed with m ean of 0.0and standard deviation of 0.O lmm . In order to evaluate thegenerated surface, an Equiparametric sampling method isused [I8 1 where a sample of points representative of thesurface are equally distributed in the parametric space ofthe B-Spline surface (u-v space). For measurementpurposes, 100 sample points were picked from theresulting surface and analysed using a tolerance zoneevaluation algorithm to evaluate the actual tolerance of themachined surface. The results show that surface errorsrange between -0.06631 mm and 0.13039 mm. Since theupper and lower limits of tolerances equal to the 0.05mm,the simulated machined geometry doses not conform tothe tolerance envelope. In this figure, the m easured pointsindicated by a circle are those that lie above of the desiredgeometry and represent a positive error value or under cut.

Figure (3) Machining and Inspection using data set # I .Figure (4) shows the machining simulation of the samegeometry, at the same machine location, with the samenon-systematic error distribution using the se cond machinetool (data set #2). Upon inspecting 100 points of theresulting surface, it can be seen that the surface errorsrange between -0.00431mm to 0,11843 mm. The createdsurface is within the lower tolerance limit but violates theupper tolerance limit.6.2 Maximizing Conformance to Design ToleranceThe application of the proposed methodology to the worstsurface (surface created by data set # I ) is presented.The rigid body transformation, between the nominalsurface or, in fact, machining commands coordinatesystem and the machine coordinate system, to ensure thatthe machined surface lies completely within the tolerancezone is obtained by optimising using (14) as follows:Rotation about the X axis=0.00035 rad, about the Y axis=-0.00002 rad and about the 2 axis=0.00007 radTranslation in the X direction= -0.00011 mm, in the Ydirection= 0.00011 m m 8, in the 2 direction= -0.01082 mm


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points within the machining comma nds that correspondto those points mach ined surface.In order to obtain the desired e nd result, these correctionsare performed iteratively, with intermittent inspection, untilcomplete conformance is achieved.First, an Equiparametric sampling me thod is used to definethe measurement points. Then, based on the location ofthese p oints, the m achined surface is divided to patches. Acompensation value, which is the difference betweenmaximum and minimum errors of each patch, is calculated.Let the surface parameters associated with a patchvertices be na med u7 hrough u4 an d v7 hrough v4 and leta be the associated error compensa tion value.

Figure (4) Inspection and machining using data set #2.

Figure (6) Residuals of conformance to +/- 0.02 tolerance.Then the compensation command is generated by:

Figure (5) Resu lts of conformance to tolerance of+/- 0.05.With this transformation, the surface completely conformsto the specified tolerance envelope and as the result theconformance residual is equal to 0.00000. Hence, furtherlocal corrections are not required. This result is illustratedin Figure (5).The conformance of the same surface with tightertolerance spec ifications (i.e. smaller u pper and lowertolerance limits equal to 0.02) is analysed. Figure (6)shows the dark areas where conformance residual errors(Eq. 10) remain. The max!mum conformance is achievedby transformation matrix, T , with the following parameters:Rotation about the X axis=-0.00029 rad, about the Y axis=-0.00025 rad and about the 2 axis=0.00009 radTranslation in the X direction= -0.00002 mm, in the Ydirection= 0.00083 mm 8, in the 2 direction= -0.04733 m mThe minimum error is -0.01998 mm and it conforms to thelower limit of the tolerance envelope. The maximum errorof 0.0699 mm is much higher than the desired tolerance.The maximum residual of conformance is 0.0499 mm. Itshould be corrected by extra machining.The optimisation process presents two important pieces ofinformation for use in the com pensation process:

the transformation matrix, T', helps obtain themaximum global conformance between the nominaland machined surfaces. The inverse of this matrix isused to make the necessary changes in the machiningcommand s to correct these errors.the correspondence map which defines regions on thenominal surface where local corrections are neededand the amount of correction to be applied to various

Seven corrective steps were required to completelyremove all residual errors and achieve total surfaceconformance to the tolerance envelope. Figure (7) showsthe residual error in some of these steps. The number ofsurface patches that ne eded corrections de creased quicklyafter the first two compensation steps. The amount ofmaterial to b e mac hined in the later steps of compe nsationis much less than the earlier steps.

Figure (7) Iterative Mach ining Errors CompensationThe maximum Residual surface error in the variouscompensation steps is presented in the Figure (8). It canbe seen that, first, due to the better alignment between thepart and machine coordinate frames which maximizes theconformance to the specified tolerance through rigid bodytransformation, the residual error is reduced by %60. Then,in 7 consecutive compensation steps, completeconformance to the tolerance envelope was achieved.


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Maximum Error in Compensation Steps(mm)

Figure (8) Compensation Steps7 SUMMARY AND DISCUSSIONA methodology for compensating for machining errorsinstead of machine tool errors has been presented. Itemphasizes the importance of considering the ultimateoutcome of machining in which machining errors aremanifest and devising adaptive corrective action base d onclose coordination b etween inspection and machining.The proposed approach aims at maximizing conformanceto tolerance specification before the final cuts are made.This would ensure 100% acceptance of expensive partssuch as complex dies and moulds and reduces waste. Itidentifies the minimum amount of corrective machiningrequired and where it should be applied. These correctionsare applied directly to machining instructions rather thanthe less user -oriented controller signa ls.This unique approach is applicable to several variations ofintegrated inspection and machining systems such as on-line and intermittent inspection or repetitive manufacture.This method is mathematically applicable to parts withmultiple features where consideration of all its datumreferences is necessary. The features' datum can bemodelled as constraints in formulating the optimisationproblem. This reduces the solution space but correspondscloser to the requirements of more complex parts. Also inorder to avoid possibility of over cutting for the constrainedfeatures, the determination of the machining erroroperator can be determined at a safe distance before thefinishing cut by constructing an intermediate surface withinthe CAD model with which to compare measurements.The accuracy and required time for iterative compensationis also related to the density of sampling points (Eq.18). Anadaptive sampling algorithm can potentially increase itsspeed and efficiency.8 ACKNOWLEDGMENTSThe authors acknowledge the valuable contributions thatProfessor Waguih EIMaraghy, of the IMS Center at theUniversity of Windsor, m ade to this research.Funding by the Natural Science 8, Engineering ResearchCouncil of Canada (NSERC) 8, Auto21 Network of Centersof Research Excellence and collaboration with Canada'sNational Research Council 8, its Integrated ManufacturingTechnologies Institute (IMTI) are acknowledged .9 REFERENCES[ I ] Hocken, R. J., Simpson, A,, Brochardt, B., Lazar, J.,Reeve, C., Stein, P., 1977, Three DimensionalMetrology, Annals of CIRP, 26/2:403-408.

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46/1:471-474.

PC A06/MF A01; PB2002-101144/XAB, 1-77.

approach to machining error compensation

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