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January 2014 published online 9Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture

Guofu Ding, Shaowei Zhu, Elssawi Yahya, Lei Jiang, Shuwen Ma and Kaiyin Yanprocess

Prediction of machining accuracy based on a geometric error model in five-axis peripheral milling

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Original Article

Proc IMechE Part B:J Engineering Manufacture0(0) 1–11� IMechE 2014Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/0954405413516611pib.sagepub.com

Prediction of machining accuracybased on a geometric error model infive-axis peripheral milling process

Guofu Ding, Shaowei Zhu, Elssawi Yahya, Lei Jiang,Shuwen Ma and Kaiyin Yan

AbstractMachining accuracy is the most critical indicator to evaluate the machining quality of parts in metal cutting industry.However, it is difficult to be identified before real cutting, because of a variety of error sources presented in a machiningprocess system, such as assembly inaccuracy of machine tool, deformation caused by temperature variation and dynamiccutting force, tool wear, servo lag and so on. Consequently, it is difficult to determine whether a new machining processcan satisfy accuracy requirements beforehand. Traditionally, a machining process is validated through the ‘‘trial and error’’approach, which is time consuming and costly. If machining accuracy can be predicted to a large extent, a rational pro-cess can be planned to ensure the precision of parts and even to maximize resource utilization without trial cuts. Forthis purpose, this work focuses on machining accuracy prediction for five-axis peripheral milling based on the geometricerrors. An error synthesis modeling method is proposed to integrate the geometric errors of the process system,including machine tool geometric error, workpiece locating error, cutting tool dimension error and setup error. From amulti-body system point of view, all these errors are synthesized to generate position error of the cutting contact pointin the workpiece coordinate system. Then the machining error is obtained by projecting the position error to the work-piece normal vector, which can be measured by a coordinate measuring machine. The prediction model has been evalu-ated by a cutting test with our in-house-developed prototype software. The result shows that the proposed method isfeasible and effective.

KeywordsMachining accuracy, accuracy prediction, error modeling, error synthesis model, five-axis peripheral milling

Date received: 1 April 2013; accepted: 18 November 2013

Introduction

Owing to the presence of different error sources inmilling, a new process should be proved to satisfy theaccuracy requirement before formal production.Traditionally, it is mainly validated through the ‘‘trialand error’’ method. Obviously, this method is time con-suming and costly. If the machining accuracy can bepredicted before cutting, there are twofold benefits: (1)the process can be validated to satisfy the accuracyrequirement of final parts and further to maximizeresource utilization rate, and (2) compensations can beused to improve the machining accuracy. This is mean-ingful to reduce the production cost.

In the past several decades, many researches havefocused on this topic. Typically, in 1977, Wu1 first pro-posed forecasting compensatory control (FCC), whichwas widely adopted to improve machining accuracy.

Then, in 1995, the International Academy forProduction Engineering (CIRP) started a workinggroup ‘‘Modelling of Machining Operations’’ to developpredictive models for machining performance in orderto facilitate effective planning of machining operationsto achieve optimum productivity, quality and cost.2

Generally, previous studies focus on different errorsources, mainly including geometric error and thermalerror of machine tool, cutting force caused deformation

Institute of Advanced Design and Manufacturing, School of Mechanical

Engineering, Southwest Jiaotong University, Chengdu, China

Corresponding author:

Shaowei Zhu, Institute of Advanced Design and Manufacturing, School of

Mechanical Engineering, Southwest Jiaotong University, No.111, 1st

North Segment, 2nd Ring Road, Chengdu 610031, China.

Email: [email protected]

at Southwest Jiaotong University on April 27, 2014pib.sagepub.comDownloaded from


and workpiece locating error. The following relatedworks are classified according to the error sources.

Machine tool geometric error

To predict the machining error caused by geometricerror of machine tool structures, a variety of modelingmethods have been developed. As early as 1960s, Leete3

established a geometric error model of a three-axismachine tool by triangular relationship. Then, Fouriertransform,4 variational approach,5 homogeneous coor-dinate transformation,6 rigid body kinematics7 andmulti-body system (MBS)8 were sequentially applied tomachine tool geometric error modeling. And a homoge-neous coordinate transformation (HTM)-based model-ing method was widely adapted.9,10

Particularly, given the flexibility of the machine toolstructures, Wang and Ehmann11 proposed a method tomeasure the total position error of a range of nodes inmachining space by telescoping ball-bars directly, andthen get the position error of an arbitrary tool tipthrough interpolation algorithm.12

Machine tool thermal error

Since thermal deformation was first found to be a mainfactor affecting the precision of machine tools in 1933,many studies have focused on machine tool thermalerror prediction and compensation. In the past fewdecades, finite element method (FEM),13 HTM,14

regression analysis (RA),15 neural network (NN),16

gray system theory17 and fuzzy inference18 wereadopted to establish thermal error prediction model.

FEM provides a solution for the temperature andthermal deformation of a machine tool, but its predic-tion accuracy is low owing to insufficient understand-ing of the boundary conditions.16 Additionally, thetemperature field and thermal deformation of machinetools change over time; FEM cannot describe this time-varying phenomenon in complex machiningconditions.17

In theory, the HTM-based model is feasible. But inpractice, it is difficult to implement because definederror parameters are difficult to detect in real time.

RA-based model is suitable for online real-time com-pensation because of its simple computing process andfast computing speed. However, its prediction precisionis not good, and it cannot describe the complex non-linear relationship between the temperature field distri-bution and the thermal deformation of machine tools.19

For the NN modeling, the biggest challenge is therobustness of the model. Consequently, some improvedand hybrid models were developed.19,20

Cutting force caused tool-workpiece deformation

To predict the deformation caused by cutting force, thecutting force should be predicted at first. In the lastdecades, the analytical method based on cutting

geometry calculation and cutting force coefficientsidentification was widely adopted to establish a cuttingforce model.21–23 At present, three types of cuttingforce models are widely used:22 (1) characterizing theeffects of the shearing on the rake face and the rubbingat the cutting edge by a single coefficient,24,25 (2) separ-ating the shearing and rubbing effects with two inde-pendent coefficients26 and (3) using the normal forcecoefficient, the friction force coefficient and the chipflow angle to characterize the cutting force model.27

For the deformation caused by cutting force, tooldeflection is mainly calculated according to cantileverbeam theory;28 workpiece deformation is mainly calcu-lated through FEM,21 but it is time consuming and theprediction precision is low.

Workpiece locating error

According to the error source, previous researches onworkpiece locating error could be divided into twoaspects: (1) geometric error (workpiece offsets owing togeometric error of fixture and workpiece) and (2) defor-mation error (locators and workpiece deformation dueto clamping force and cutting force).

For the first aspect, a Jacobian matrix was widelyadopted to establish the relationship between the sourceerrors and the workpiece position/orientation errors.29

Giving the source errors, the workpiece locating errorcan be calculated by this relationship model. In addi-tion, other methods such as the method of moments30

and Monte Carlo simulation31 were also adopted toestablish the workpiece positioning error caused by geo-metric error.

For the other aspects, besides the traditional elasticmechanic which was adopted to establish the deforma-tion model,29 finite element analysis (FEA) was popu-larly adopted too.32

All above-mentioned error modeling mainly focusedon one kind of error source. However, the machiningprocess is affected by many error sources, and the pro-portion is varied in different processes. So these modelscannot predict the machining accuracy of final partsaccurately in all cases. Consequently, several modelswere proposed to integrate more error sources. Forexample, Yuan and Ni33 established an error synthesismodel including machine tool geometric error and ther-mal error based on rigid body and small error assump-tions. Suneel et al.34 proposed an integrated productquality model using artificial neural networks (ANN).Li et al.35 proposed a machining accuracy predictionmodel integrated GM (1, 1) model, Markov chainmodel and Taylor approximation method. The ANNand Gray theory–based models have high predictionaccuracy because of taking all error factors intoaccount, but a large number of experiment data areneeded. In addition, these models are only suitable fora specific process.

The previous researches have confirmed that MBSand HTM are correct and feasible for machine tool

2 Proc IMechE Part B: J Engineering Manufacture 0(0)



geometric error modeling. And the synthesis modelsconsidering more error sources can improve predictionprecision. Furthermore, this article establishes an ana-lytical synthesis model integrating machine tool geo-metric error, workpiece locating error, cutting tooldimension error and setup error based on MBS to pre-dict the machining error of any cutting position.According to the machining errors of some pickedpoints, the dimension and form accuracy of final partscan be further predicted, and error compensation andprocess optimization can be used. Prototype softwareand cutting test are used to show that the proposedmodel is feasible and effective.

Error source analysis

The error sources exist in the whole machining processand affect final machining error in different ways.According to the process sequence, this study dividedthe error sources into three groups: (1) before cutting,(2) in cutting and (3) after cutting, as shown in Figure 1.

Errors generated before cutting

Usually, there are three steps before cutting in a com-puter numerical control (CNC) machining process:computer-aided design (CAD), computer-aided processplanning (CAPP) and computer-aided manufacturing(CAM).

In CAD, if a machining surface is described by apoint cloud, there is always a deviation called fittingerror between the fitted surface and the originalsurface.

In CAPP, machining method, machine tool, cuttingtool, cutting parameters, cutting scheme and clampingscheme are drafted. Once a machining process isplanned, the geometric error of the machine tool andthe workpiece locating error are determined subse-quently. The cutting parameters will further affect thecutting force, surface roughness and servo trackingerror in cutting.

In CAM, a planned process is converted to numeri-cal control (NC) codes. Usually, the machine tools can-not control the tool feeding along freeform curves, butcan control straight lines and arcs. Therefore, in pro-gramming, a series of discrete points are selected andconnected by straight line segments to replace a curve.Consequently, the deviation called interpolation errorbetween the curve and the straight segments is resulted.In addition, scallop height is left over between two adja-cent cutting trajectories, and rounding error is gener-ated because of rounding.

Errors generated in cutting

In metal cutting, the process system (machine tool, cut-ting tool, workpiece and fixture) is affected by cuttingforce and cutting heat.

For machine tools, the cutting force brings theelastic deformation of the machine tool structures,though it is usually small because of good stiffness ofmachine tools. And local temperature increase bringsthermal deformation of machine tool structures, espe-cially the spindle. Moreover, there is always trackingerror between the actual position and the orderposition.

For cutting tool, its geometry is different from thenominal one because of manufacturing error and wear.And the tool axis may not coincide with the spindle axisduring setup. Moreover, the cutting force brings elasticdeformation, especially for slender tools, and the cut-ting heat brings thermal deformation, though it is slightbecause of the presence of cutting fluid.

For fixture, there are geometric error and setuperror as same as cutting tool. And elastic deformationand thermal expansion of locators may emerge due toclamping force, cutting force and cutting heat.

For workpiece, especially thin wall parts, elasticdeformation still happens owing to cutting force andclamping force. And the workpiece probably expandsbecause of cutting heat and ambient temperatureincreasing. In addition, its position and orientationerrors in the machining space also affect final machin-ing accuracy.

Errors generated after cutting

After cutting, the machining accuracy is usuallydetected by a coordinate measuring machine (CMM).But the measured results are always different from theactual positions because of the existence of

Figure 1. The existence of error sources in a machiningprocess.

Ding et al. 3



measurement errors. Moreover, the part may bedeformed due to stress relief after it is dismounted fromthe fixture.

Error synthesis model

To predict the machining error more precisely, thisstudy establishes an error synthesis model integratingthe geometric errors of five-axis peripheral milling pro-cess system, including machine tool geometric error,workpiece locating error, cutting tool dimension errorand setup error. Corresponding prediction process isproposed as briefly shown in Figure 2. First, all the errorparameters are measured and stored in a database. Thenthe error parameter values are loaded into the errorsynthesis model to predict the position error and themachining errors of some picked machining positions.Moreover, the dimension and form errors of the partscan be further predicted based on the acquired machin-ing errors. Incidentally, the prediction results can beused for error compensation and process optimization.

Error parameter definition

Machine tool geometric error. It is well known that anunconstrained object has six degrees of freedom

(DOFs). Accordingly, its position error can bedescribed by six parameters along the DOFs respec-tively. Similarly, for machine tool, there are sixgeometric error parameters of each component.Taking X-slideway for example, its six geometric errorparameters (DXx, Dyx, Dzx, Dax, Dbx and Dgx) areshown in Figure 3. For a three-axis machine tool, thereare 21 geometric error parameters, including position-ing error, straightness error, pitch error, yaw error, rollerror of each axis and perpendicularity errors betweenevery two axes (Ddij, i = x, y, z; j = x, y, z).

For multi-axis machine tool, each rotation axisbrings its six geometric error parameters too. Figure 4shows the six geometric error parameters (DxC, DyC,DzC, DaC, DbC and DgC) of a C-rotary table.

Workpiece locating error. The purpose of locating is toensure that the workpiece is in a correct position in themachining space during the whole cutting process. Butit always deviated from the ideal position owing to thegeometric error and setup error of locators. The devia-tion can be described by the error parameters along sixDOFs (Dxw, Dyw, Dzw, Daw, Dbw and Dgw), respec-tively, as shown in Figure 5.

Cutting tool dimension error and setup error. Cutting tooldimension error mainly includes radius error (DRT)and length error (DLT). In addition, the clearancebetween the tool holder and the tool post leads to setuperror when tool installation is done. As the tool rotatesaround the axis of the spindle, the setup error can bedescribed by the error parameters along five DOFs(DxT, DyT, DzT, DaT and DbT), where DxT, DyT andDzT are consistent with tool run-out errors.

Error synthesis modeling

According to the theory of MBS, a machining processsystem can be viewed as a kind of MBS composed ofworkpiece, fixture, cutting tool and machine tool. Andthe machine tool can be viewed as a sub-MBS

Figure 2. The proposed prediction process.

Figure 3. Geometric error parameters of X-slideway.




composed of workbench, column, spindle, slidewaysand so on. Taking a XFYZBA five-axis machine toolfor example, its structure is shown in Figure 6 and thetopological construction of the process system is shownin Figure 7.

Set up a coordinate system for each ‘‘body’’ and usea 434 HTM to represent the transformation relation-ship between a pair of adjacent bodies; the ideal cuttingcontact point (CCP) Pideal in the workpiece coordinatesystem (WCS) can be described as equation (1), and theideal tool orientation Videal can be described as equation(2) (see Appendix 2 for all equations).

However, in an actual machining process, owing tothe presence of errors, the actual transformationmatrices are different from the ideal ones.

For cutting tool, owing to the dimension error andsetup error, the actual CCP and tool orientation willdeviate from the ideal position. From the NC code, theideal tool tip (TP) coordinate Xi, Yi, Zi (i = 1,2,.,n)and rotation angle Ai, Bi (i = 1,2,., n) can be obtaineddirectly, and the ideal tool orientation in WCS can befurther obtained by equation (3) according to the topo-logical construction of the process system.

In five-axis peripheral milling, the interpolation stepis usually small. Therefore, the tangent vector at thisposition can be looked as the vector from currentTP i (i = 1,2, ., n) to next one i+1, as shown inequation (4).

Obviously, the normal vector at this position is thecross product of the tool axis vector vt,i and the tangentvector ti. Consequently, the unit normal vector can beobtained by equation (5).

Ideally, the position of CCP in a tool coordinate sys-tem (TCS) is shown in equation (6). However, owing tothe radius error DRT, the actual position of CCP inTCS should be shown as equation (7). Moreover, owingto the length error DLT and setup errors, the actualtransformation matrix of the tool with respect to thespindle is shown as equation (8).

For spindle, owing to the presence of geometricerror, its actual transformation matrix with respect toA-axis is shown as equation (9).

For A-axis, owing to the presence of geometric error,its actual transformation and rotation matrixes withrespect to B-axis are shown as equations (10) and (11).

Similarly, the actual transformation and rotationmatrixes of B-axis with respect to Z-axis are shown asequations (12) and (13).

Figure 4. Geometric error parameters of a C-rotary table.

Figure 5. Workpiece locating error parameters.

Figure 6. Structure of the machine tool.

Ding et al. 5



For Z-axis, owing to the presence of geometric error,the actual transformation matrix of Z-axis with respectto Y-axis is shown in equation (14).

Similarly, the actual transformation matrixes ofY-axis and X-axis with respect to the column are shownin equations (15) and (16).

For the workpiece and fixture, the geometric errorand setup error lead to workpiece position and orienta-tion errors, and then result in deviations of the CCPs.Therefore, the actual transformation matrix of work-piece and fixture with respect to X-axis is shown asequation (17).

Especially, the column is looked as the referencecoordinate system without errors, because it is fixedwith ground.

Consequently, in conclusion, the actual CCP Pactual

and tool orientation Vactual in WCS can be described asequations (18) and (19). And the position deviationbetween the actual and the ideal CCPs in WCS is shownas equation (20).

Finally, the machining error e can be obtained byprojecting the position error E to the unit normal vec-tor n, as equation (21) shows.

Development of prototype software

In order to implement the proposed error synthesismodel, prototype software was developed based on

Visual Studio 2010. As shown in Figure 8, the proto-type software can be divided into three modules: themain program module, the database management mod-ule and the display module.

The main program module coordinates all modulesto complete machining accuracy prediction, displayand output. The process information is set through theinterface artificially. The position error and machiningerror of each prediction point are computed by theerror synthesis model. The dimension error and formerror of the features are computed by the accuracy pre-diction model. And the proportions of individual errorfactors are also computed. The computation result isexported to the graphic display module for display.

In the database management module, Oracle isadopted to store and manage the process information,error parameters and prediction positions, includingthe functions of data addition, modification, deletionand inquiry. Especially, the error parameters are associ-ated with the process information. For example, oncethe machine tool is selected, its geometric error para-meters are linked and involved in the error calculationautomatically.

The display module is based on the OpenGL graphicinterface. The three-dimensional (3D) part model canbe shown, translated, rotated and zoomed in this mod-ule. By receiving the error prediction results from themain program module, the magnitude and direction ofthe machining errors are displayed on the 3D partmodel. Moreover, curves of the machining errors and ahistogram of the proportions of individual error factorsare drawn to show the prediction result intuitively.

Figure 9 shows the flowchart of this prototype soft-ware. Previously, the process information, error para-meters and prediction positions are added in the

Figure 7. Topological construction of the process system.

Figure 8. Module structure of the prototype software.

Figure 9. Flowchart of the prototype software.




database. The error parameters are associated with themachine tools, cutting tools and fixtures, and the pre-diction points are associated with the machining fea-tures. Then, load the 3D model of the prediction part,which will be shown in a view window. Next, select theprocess system; the error parameters and predictionpositions are selected at the same time. After loadingthe NC code, the error calculation begins, the programmatches the NC code and the prediction positions auto-matically, and the machining errors are computed bythe error synthesis model. The computation result issent to display on the part model and plot error curve.Furthermore, the dimension error and the form error ofthe selected feature are computed by the accuracy pre-diction model. Afterward, the proportions of individualerror factors are calculated and shown by a histogram.Finally, the process information and prediction resultsare exported to a text report.

Experiment and discussions

To verify the feasibility and effectiveness of the pro-posed model, an experiment was conducted on aXFYZBA structure five-axis machine tool (Figure 6).The finishing process of the part like a distorted letter‘‘S’’ was selected. Figure 10 shows its tool path.

The experiment contains four steps: (1) error para-meters measurement and identification, (2) machiningerror prediction, (3) real cutting and (4) machiningerror measurement and result analysis. First, the laserinterferometer–based 12-line method36 was adopted toidentify the machine tool geometric error parameters.The positioning errors of the 12 lines were measured bya Renishaw XL-30 laser interferometer system, andthen the 21 geometric error parameters at each mea-surement node were identified according to the 12-linealgorithm. Figure 11 shows the positioning error identi-fication results of X-axis, Y-axis and Z-axis as an exam-ple. In the measurement, thermal error has little effect,because of small change of ambient temperature andshort continuous running time of machine tool.

After the workpiece was clamped on the workbench,the coordinates of some picked points on the workpiecesurface are detected by a Renishaw probe. Then theworkpiece locating errors were identified by the methodproposed in literature37 and the results are listed inTable 1. In this case, only the angular errors aroundX-axis (Daw) and Y-axis (Dbw) affect final machiningaccuracy.

Before finishing operation, the actual radius andlength of the tool were measured by a tool detectingdevice on the machine tool while the tool is rotating.Table 2 shows the nominal values and the measuredvalues, and the dimension errors can be obtainedaccordingly. In this measurement, the dimension errorscontain the setup errors of the tool because it isrotating.

Then, on one hand, the error parameters, processinformation and prediction points were added into thedatabase and the machining errors of the picked predic-tion points were predicted and displayed by the soft-ware. On the other hand, the finishing cutting wascompleted and the machining errors of the picked pre-diction points on the machined part (Figure 12) weremeasured by a CMM. Figure 13 shows the predictionresult and the measurement result contrastively. In thefigure, the prediction error is the difference between themeasurement result and the prediction result (measuredvalue minus predicted value).

Figure 10. Tool path of the experiment part.

Figure 11. Identified positioning errors of X-axis, Y-axis andZ-axis.

Figure 12. The machined test part.

Ding et al. 7



In the prediction, machine tool geometric error,workpiece locating error, tool dimension error andsetup error were measured. But other error sourcessuch as deformation, thermal error, tracking error andso on were not included. To reduce the effect by theseerrors, the cutting parameters were selected safely andthe machine tool was preheated in the cutting process.

From Figure 13, it can be seen clearly that the pre-dicted machining errors are close to the measured ones.The mean error (ME) is 0.0035 mm and the root meansquare error (RMSE) is 0.0090 mm. Obviously, theerror curves in Figure 13 show that the proposed modelcan predict the machining error to a large degree. Andit is shown that the proposed method is correct andfeasible.

Conclusion

The feasibility of MBS and HTM used for machinetool geometric error modeling has been proven in theprevious researches.6–10 In this study, an error synthesismodel is established for five-axis peripheral millingbased on MBS and HTM, which integrates machinetool geometric error, workpiece locating error, cuttingtool dimension error and setup error. By the synthesismodel, the machining error of any cutting position canbe predicted as measured by CMM, which can be

further used for dimension error and form error predic-tion, even error compensation and processoptimization.

The proposed model has been implemented into aprototype software system. The error parameters areidentified and stored in a database. A cutting test wascarried out to test the practicability and effectiveness ofthe proposed method. To reduce the effect by othererror sources, the cutting parameters were selectedsafely and the machine tool was preheated. The resultshows consistent trend of the predicted and measuredmachining errors (Figure 13). It can be seen that theproposed model can predict the machining error to alarge degree.

Further researches containing more error sourceswill be conducted in the future.

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This work is supported by (1) the Special Fund ofHigh-end CNC Machine Tools and BasicManufacturing Equipment (2010ZX04015-011), (2)Sichuan Applied Basic Research Plan (2012JY0092)and (3) the Fund of New Century Excellent Talents(09-0665) in China.

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Appendix 1

Notation

Ai, Bi, Ci

(i = 1,2,.,n)ideal rotation angle around A-axis,B-axis and C-axis

A A-axisB B-axisC column/C-axisE position error of CCP in WCSF fixtureLT ideal tool lengthni (i = 1,2,.,n) unit normal vector at CCP iPactual actual CCP in WCSPideal ideal CCP in WCSPt ideal CCP in tool coordinate system

(TCS)Pt0 actual CCP in TCS

RT ideal tool radiusmRn (m = Z, Band k = B, A)

ideal rotation matrixes of body nwith respect to body m

mRn0 (m = Z, B

and k = B, A)actual rotation matrixes of body nwith respect to body m

S spindle

Ding et al. 9



ti (i = 1,2,.,n) tangent vector at CCP iT tooljTk (j = C, X, F,Y, Z, B, A, Sand k = X, F,W, Y, Z, B, A,S, T)

ideal transformation matrixes ofbody k with respect to body j

jTk0 (j = C, X, F,

Y, Z, B, A, Sand k=X, F, W,Y, Z, B, A, S, T)

actual transformation matrixes ofbody k with respect to body j

Vactual actual tool orientation in WCSVideal ideal tool orientation in WCSvt,i (i = 1,2,.,n) ideal tool orientation in WCSVt tool orientation in TCSW workpiecex current location of X-axisy current location of Y-axisz current location of Z-axisXi, Yi, Zi

(i = 1,2,.,n)ideal tool tip (TP) in WCS

X X-axisY Y-axisZ Z-axise machining error/normal erroru rotation angle of the spindleDdxz(z) perpendicularity error of Z-axis with

respect to X-axisDdyz(z) perpendicularity error of Z-axis with

respect to Y-axisDdxy(y) perpendicularity error of Y-axis with

respect to X-axisDLT tool length errorDRT tool radius errorDxi (i = x, y, z,w, A, B, C, S, T)

displacement error of component ialong X-axis

Dyi (i = x, y, z,w, A, B, C, S, T)

displacement error of component ialong Y-axis

Dzi (i = x, y, z,w, A, B, C, S, T)

displacement error of component ialong Z-axis

Dai (i = x, y, z,w, A, B, C, S, T)

angular error of component iaround X-axis

Dbi (i = x, y, z,w, A, B, C, S, T)

angular error of component iaround Y-axis

Dgi (i = x, y, z,w, A, B, C, S, T)

angular error of component iaround Z-axis

Appendix 2

Equations

Pideal =(CTX � XTF � FTW)�1 � CTY�YTZ � ZTB � BTA � ATS � STT � Pt ð1Þ

Videal=ZRB � BRA � Vt ð2Þ

where F, W, C, X, Y, Z, B, A, S, T represent the fix-ture, workpiece, column, X-axis, Y-axis, Z-axis, B-axis,A-axis, spindle and tool, respectively; Pt = [px py pz 1]

T

and Vt = [0 0 1 0]T represent the position of CCP and

the tool orientation in the tool coordinate system(TCS); jTk (j = C, X, F, Y, Z, B, A, S and k = X, F,W, Y, Z, B, A, S, T) are the ideal transformationmatrixes of body k with respect to body j; mRn (m = Z,B and k = B, A) are the ideal rotation matrixes ofbody n with respect to body m

vt, i =vx, ivy, ivz, i

24

35=

cos (Bi) 0 sin (Bi)0 1 0

� sin (Bi) 0 cos (Bi)

24

35�

1 0 00 cos (Ai) � sin (Ai)0 sin (Ai) cos (Ai)

24

35 �

001

2435

=sin (Bi) cos (Ai)� sin (Ai)

cos (Bi) cos (Ai)

24

35 ð3Þ

ti =tx, ity, itz, i

24

35=

Xi+1 � Xi

Yi+1 � Yi

Zi+1 � Zi

24

35 ð4Þ

ni =nx, iny, inz, i

24

35=

vt, i3tivt, i3tij j ð5Þ

Pt, i =px, ipy, ipz, i

24

35=RT �

nx, iny, inz, i

24

35 ð6Þ

P0t, i =

p0

x, i

p0y, i

p0z, i

264

375=(RT +DRT) �

nx, iny, inz, i

24

35 ð7Þ

STT=

1 0 DbT DxT0 1 �DaT DyT

�DbT DaT 1 LT +DLT +DzT0 0 0 1

2664

3775

ð8Þ

zT0s =

cos (u+Dgs(u)) � sin (u+Dgs(u)) Dbs(u) Dxs(u)sin (u+Dgs(u)) cos (u+Dgs(u)) �Das(u) Dys(u)�Dbs(u) Das(u) 1 Dzs(u)

0 0 0 1

2664

3775

ð9Þ

where f is the rotation angle of the spindle

BT0

A =

1 �DgA(A) DbA(A) DxA(A)DgA(A) cos (A+DaA(A)) � sin (A+DaA(A)) DyA(A)�DbA(A) sin (A+DaA(A)) cos (A+DaA(A)) DzA(A)

0 0 0 1

2664

3775

ð10Þ

BR0

A =

1 �DgA(A) DbA(A) 0DgA(A) cos (A+DaA(A)) � sin (A+DaA(A)) 0�DbA(A) sin (A+DaA(A)) cos (A+DaA(A)) 0

0 0 0 1

2664

3775

ð11Þ

ZT0

B =

cos (B+DbB(B)) �DgB(B) sin (B+DbB(B)) DxB(B)DgB(B) 1 �DaB(B) DyB(B)

� sin (B+DbB(B)) DaB(B) cos (B+DbB(B)) DzB(B)0 0 0 1

2664

3775

ð12Þ




ZR0

B =

cos (B+DbB(B)) �DgB(B) sin (B+DbB(B)) 0DgB(B) 1 �DaB(B) 0

� sin (B+DbB(B)) DaB(B) cos (B+DbB(B)) 00 0 0 1

2664

3775

ð13Þ

YT0

Z =

1 �Dgz(z) Dbz(z) Dxz(z)+Ddxz(z)Dgz(z) 1 �Daz(z) Dyz(z)+Ddyz(z)�Dbz(z) Daz(z) 1 z+Dzz(z)

0 0 0 1

2664

3775

ð14Þ

where Ddxz(z) and Ddyz(z) are the perpendicularityerrors of Z-axis with respect to X-axis and Y-axis,respectively; z is the current location of Z-axis

CT0

Y =

1 �Dgy(y) Dby(y) Dxy(y)+Ddxy(y)Dgy(y) 1 �Day(y) y+Dyy(y)�Dby(y) Day(y) 1 Dzy(y)

0 0 0 1

2664

3775

ð15Þ

where Ddxy(y) is the perpendicularity error of Y-axiswith respect to X-axis; y is the current location ofY-axis

CT0

X =

1 �Dgx(x) Dbx(x) x+Dxx(x)Dgx(x) 1 �Dax(x) Dyx(x)�Dbx(x) Dax(x) 1 Dzx(x)

0 0 0 1

2664

3775

ð16Þ

where x is the current location of X-axis

XT0

F � FT0

W =

1 �Dgw Dbw DxwDgw 1 �Daw Dyw�Dbw Daw 1 Dzw

0 0 0 1

2664

3775 ð17Þ

Pactual= (CT0

X � XT0

F � FT0

W)�1 � CT0Y � YT0

Z �ZT

0

B � BT0

A � AT0

S � ST0

T � P0

t ð18Þ

Vactual =ZR

0

B � BR0

A � Vt ð19Þ

E=Pactual � Pideal ð20Þ

e=ET � n= Pactual,x � Pideal, x Pactual, y � Pideal, y½

Pactual, z � Pideal, z� �nxnynz

24

35 ð21Þ

Ding et al. 11



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