International Journal of Machine Tools & Manufacture 111 (2016) 17–30

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International Journal of Machine Tools & Manufacture

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n Corrical Engtechnic)

E-m

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Functional accuracy investigation of work-holding rotary axes in five-axis CNC machine tools

Mehrdad Vahebi Nojehdeh, Behrooz Arezoo n

Mechanical Engineering Department, Amirkabir University of Technology (Tehran Polytechnic), Hafez Ave, Tehran, Iran

a r t i c l e i n f o

Article history:Received 24 June 2016Received in revised form29 August 2016Accepted 5 September 2016Available online 6 September 2016

Keywords:Machine toolFive-axisRotary axesSensitive directionError motionKinematic chain

x.doi.org/10.1016/j.ijmachtools.2016.09.00255/& 2016 Elsevier Ltd. All rights reserved.

espondence to: Postal address: CAD/CAPP/CAineering Department, Amirkabir University, 424 Hafez Ave, Tehran 15875-4413, Iran.ail addresses: arezoo@aut.ac.ir, barezoo@yaho

a b s t r a c t

Rotary axes error motions of five-axis CNC machine tools affect dimensional, geometrical and form ac-curacy of machined features according to structural design and sensitive direction considerations. Workcarrying kinematic chains mostly use one rotary axis in radial Fixed Sensitive Direction (radial FSD)setup. The application keeps workpiece axis coaxial with the axis of rotation, preparing more flexibilityfor machining of complicated features. In spite of conspicuous advantages, in case of poor geometricalquality of machined features, lack of knowledge about error patterns makes the error source isolationprocess more difficult. This paper investigates the consequences of cycling error motions of radial FSDwork-holding rotary axes, in order to prepare intuitive knowledge about possible error patterns. A virtualcutting module equipped with an error-mapping model is generated to simulate the cutting process invicinity of different error motion scenarios. As a novel approach, comparison of measured deviationswith available error patterns facilitates the error source isolation. Experimental results conducted onairfoil cutting process verified the effectiveness of the presented method.

& 2016 Elsevier Ltd. All rights reserved.

1. Introduction

Five-axis CNC machine tools perform precision machining ofcomplex features in automotive, aerospace and power generationindustries to cope with strict quality requirements. Inside of theirkinematic chain, rotary axes are applied for orientation control ofthe cutting tool. Incorrect performance of rotary axes greatly af-fects functional accuracy of machined features and increase com-plexity of root cause analysis. According to single point cuttingprinciple, limited area of cutting tool travels over wide area on theworkpiece. Hence, error motions of work-holding rotary axesgenerate more complicated consequences compared to tool-holding rotary axes. In case of geometrical quality failures of ma-chined features, lack of knowledge about error patterns may leadto costly mistakes in machine tool repair or trouble-shooting tasks.Investigation of error motion consequences in work-holding axes,prepares useful diagnostic knowledge in framework of categorizederror patterns. This approach adopts error modeling techniques toconvert possible error motions to their unique error patterns.

Error modeling techniques have been widely discussed forcompensation of accuracy assessment purposes ([1–8]). Rotaryaxis significantly increased complexity of analysis and variety of

M Research Center, Mechan-of Technology (Tehran Poly-

o.com (B. Arezoo).

multi-axis structural designs [9]. ISO 230-1:2012 [10] recentlyupdated to become a comprehensive source for machine tooltesting procedures including basic concepts for rotary axes.Moreover, innovative measurement methods and data re-presentation schemes were developed to evaluate rotary axes instructural loop of the kinematic chain ([11–13]). Fu [14] applieddifferential motion matrix to model influence of individual errorsof five-axis machine tool, including rotary axes, in order to identifyand compensate geometric errors. Jiang [15] considered positionindependent geometric errors (PIGE) of five-axis machine toolsusing homogeneous transformation matrixes. Huang [16] dis-cussed two different rotary axes modeling concepts and derivedtransformation matrix for each one. Jiang [17] developed specialprobing scheme to measure 12 error components of two rotaryaxes of a five-axis machine tool in single setup. Xiang [18] pro-posed a method to measure, model and compensate 41 positiondependent and independent geometric errors of five-axis CNCmachine tools, using forward and inverse kinematics based onscrew theory.

During recent years, consequences analysis of individual errorshave been developed in parallel to error modeling techniques.Wadhwani [19] investigated faults of rotary bearings in order toclarify two main categories of quality failures including single-point defects versus generalized roughness. Soori [20] considered

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M.V. Nojehdeh, B. Arezoo / International Journal of Machine Tools & Manufacture 111 (2016) 17–3018

estimation of machined workpiece shape deviation through geo-metrical error modeling approach for three axis machining. Zar-garbashi [21] tried to interpret the outcome signals of LVDT1 toolof ball-bar system by FFT2 filtering in order to isolate error sourcesof a rotary axis used in five-axis CNC machine tools. Fesperman[22] developed a data driven virtual machine tool for five-axismachine in order to simulate functional accuracy of the machineincluding tool-holding rotary axes. Chen [23] considered rankingof errors according to their effectiveness on final tool posture errorwithout considering sensitive direction observations. On the otherhand, ISO 230-7:2015 [24] is published by TC393-SC24 to clarifyfundamental definitions and testing methods for rotary axes. Thisdocument has been modified according to Lu [25] and brieflydiscusses error motion consequences.

These studies and other similar works prepared useful materialin error modeling, measuring and compensation of rotary axes andspindles. Nowadays, spindle analyzers are commercially availablethat could prepare a wide range of test results and analysis.Nevertheless, there is lack of knowledge about accuracy char-acteristics of work-holding rotary axes, when they cut the work-piece in vicinity of systematic error motions. The consequence ofsuch error motions may be referred as unique signature to beapplied for diagnostic purposes for continues monitoring of ma-chined features.

The present work is aimed at modeling of cutting process ofradial Fixed Sensitive Direction (FSD) work-holding rotary axes invicinity of systematic errors in order to derive error patterns.Considered errors include dynamic error motions (known as po-sition dependent geometric errors; PDGEs) as well as stationaryposition/orientation errors (known as position independent geo-metric errors; PIGEs) of radial FSD work-holding axes. Other errorsources such as interpolation errors, thermal induced deflections,tool-workpiece non-rigidity are excluded from the scope of thisresearch. The results of this research can be applied for processingof deviations measured on machined features for error sourceisolation purposes.

For this purpose, the paper is organized as follows. Section 2represents definitions and nomenclatures related to axis of rota-tion error motions and their cyclic effects according to lateststandards. This section also establishes radial FSD work-holdingaxes application in kinematic chain. Section 3 discusses kinematicmodeling concepts starting from basic concepts to finalize anadequate model for cyclic error motion handling. Section 4 con-centrates on machining application observations related to radialFSD work-holding axes. This section connects error motions ofaxes of rotation to error patterns of machined workpiece using avirtual cutting module. Section 5 represents categorized results ofsimulations and explains practical data analysis method to elim-inate calculus noises from cyclic number estimation. This sectiondescribes experimental test details carried out on airfoil cuttingprocess to verify the suitability and effectiveness of the method inpractical applications.

2. Characteristics of rotary machining axes

A typical machining rotary axis transforms connected compo-nents around its axis of rotation for machining purposes accordingto commanded target positions and demanded acceleration andspeed. Five-axis CNC machine tools have two rotary machiningaxes in different levels of structural loop. Regardless of application,

1 Linear variable differential transformer.2 Fast fourier transform.3 39th technical committee of ISO.4 Sub committee.

rotary axes have common error motion characteristics. Each errormotion may show random or systematic cyclic behavior thatwould affect functional accuracy of a machining rotary axis. Clause3.3 of ISO 230-7:2015 [24] declares that “Consequences of thismotion on the accuracy of machined workpieces vary dependingon the type of machining application.” Type of machining appli-cation refers to sensitive direction type and orientation, whichincludes fixed, rotating, and varying sensitive direction in frame-work of single or multiple sensitive directions (Fig. 3).

2.1. Error motions of axis of rotation

According to ISO 230-7:2015 [24], for a typical rotary axis, re-gardless of its application, there are six possible error motionsattributed to axis of rotation and six stationary position/orienta-tion deviation attributed to axis average line. The former has dy-namic nature while the latter is due to machine tool axes setupproblem and is independent from axis function. Fig. 1 illustratesthese parameters for [wA'bXYZBt] configuration and Table 1 re-presents their nomenclatures according to ISO 841:2001 [26]. Inresearch papers such as [27], error motions of axis of rotation arecalled Position Dependent Geometric Errors (PDGE). Those errorsrelated to position/orientation deviation of axis average line areknown as Position Independent Geometric Errors (PIGE).

2.2. Cyclic effects of error motions

In spite of linear axes, motion of rotary axes is cyclic in natureover 360-degree repeating travel. Consequently, each error motionof this type of axis may show cyclic behavior synchronized withrotational frequency or in other different frequencies. Integermultiplications of error motions frequencies results in systematicconsequences while non-integer multiplications generate randomconsequences. The first harmony of cyclic error motions is veryimportance and is called fundamental error motion [24] whichresembles the resonance frequency in vibration monitoring ap-plications. Fundamental error motion shows one cycle per re-volution (CPR¼1) and is synchronized with axis revolution fre-quency. Such cyclic error motion may leave systematic con-sequences affecting dimensional or form accuracy of machinedworkpieces. On the other hand, non-repeated error motions orthose repeated in non-integer multiplication of cycles are men-tioned as asynchronous error motions. Consequences of asyn-chronous error motions appear as increased surface roughness inmachined features [24]. The aforementioned cyclic error motionsare represented in Fig. 2.

2.3. Rotary axis applications in cutting process

Based on design and application, multi-axis machine tools mayapply rotary axes to carry either the cutting tool or the workpiece.A most important criterion of axis application is sensitive directionobservations. According to Fig. 3, different types of sensitive di-rections include Fixed Sensitive Direction (FSD), Varying sensitivedirection (VSD), and Rotating Sensitive Direction (RSD). StandardISO230-7:2015 [24] defines FSD as “sensitive direction where thefunctional point in machine coordinate system does not changewith the angular position of the rotating component”. It also de-fines Non-sensitive “direction as direction perpendicular to theworkpiece surface at the functional point”.

Fig. 4 represents two different applications of a typical FSDrotary axis in five-axis machine tool kinematic chain. Fig. 4a showsan illustration of [wA'bXYZB(C)t] configuration in which work-piece mounted on A-axis in radial FSD setup. The secondary rotaryaxis, B-axis, carries the cutting tool in RSD condition of the planeparallel to XZ. Fig. 4b, illustrates another alternative for

Fig. 1. A typical radial FSD work-holding rotary axis (A-Axis) and illustration of error motions according to ISO230-7:2015 [24].

Table 1Error motions nomenclature of a typical rotary axis (A-axis) according to ISO230-7:2015 [24].

Error motions of axis of rotation (PDGE)EXA Axial error motion of A-axis

EYA Radial error motion of A-axis in Y-axis direction

EZA Radial error motion of A-axis in Z-axis direction

EAA Angular positioning error motion of A-axis

EBA Tilt error motion of A-axis around Y-axis

ECA Tilt error motion of A-axis around Z-axisPosition/orientation deviations of axis average line (PIGE)EXOA Error of the position of A-axis in X-axis direction

EYOA Error of the position of A-axis in Y-axis direction

EZOA Error of the position of A-axis in Z-axis direction

EAOA Zero position error of A-axis

( )EB OZ A Error of the orientation of A-axis in B-axis direction; squareness ofA-axis to Z-axis

( )EC OY A Error of the orientation of A-axis in C-axis direction; squareness ofA-axis to Y-axis

M.V. Nojehdeh, B. Arezoo / International Journal of Machine Tools & Manufacture 111 (2016) 17–30 19

aforementioned configuration that is [wA'B'bXYZ(C)t]. In thisconfiguration, both rotary axes are work-carrying axes and theA-axis has radial FSD configuration.

Fig. 2. Categorized representation of cyclic error motion

3. Kinematics of radial FSD work-holding axis

During cutting process, interactions of individual error para-meters determines the final accessible accuracy. An appropriatekinematic modeling approach would simulate such interactionstaking into account structural configurations as well as error mo-tion considerations. This paper adopts Homogenous Transforma-tion Matrix (HTM) calculations for kinematic modeling due to itssuccessful application reports and easiness of use ([4,22,28]).

3.1. Homogenous transformation matrix (HTM)

According to [28], HTM represent N-dimensional positionvector as a (Nþ1) dimensional vector. The actual component ofN-dimensional vector can be found by dividing each componentby the (Nþ1)th component known as scale factor. According to Eq.(1), three O1, O2, and O3 orientation vectors and a position vector(P), can describe spatial transformation of a coordinate frame withrespect to its reference state [14]. Orientation vectors are directionvectors with infinite length and hence their scale factors are zero.On the other hand, position vector represents simple three-di-mensional offset and its scale is set to unity [28].

s of rotary axes, according to ISO 230-7:2015 [24].

Fig. 3. Illustration of different applications for a: Fixed Sensitive Direction (FSD), b: Varying Sensitive Direction (VSD), and c: Rotating Sensitive Direction (RSD) [24].

Fig. 4. Kinematic configuration of a: [wA'B'bXYZ(C)t] and b: [wA'bXYZB(C)t] five-axis machine tool.

M.V. Nojehdeh, B. Arezoo / International Journal of Machine Tools & Manufacture 111 (2016) 17–3020

= =

( )

⎡⎣⎢

⎤⎦⎥

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

T R P

O O O P

O O O P

O O O P0 1

0 0 0 1 1

mn

x x x x

y y y y

z z z z

1 2 3

1 2 3

1 2 3

Fig. 5. Transform of local frame ( ́ ́ ́P:XOY )

3.2. Reference frame (base frame)

During kinematic modeling of a multi-axes system, me-trological reference frame plays key role in designation of errormotions [25]. Here, according to objectives of the present work,the reference frame is connected to base of the machine tool toaddress both cutting tool deviations and rotary axes error motions.

with respect to base frame (Q:XOY ).

M.V. Nojehdeh, B. Arezoo / International Journal of Machine Tools & Manufacture 111 (2016) 17–30 21

3.3. Coordinate frame transformation by HTM

Considering a target point within its local frame, any deviationswith respect to the reference fame can be represented by or-ientation and translation vectors as a 4�4 HTM. As an example,Fig. 5 illustrates the point P connected to its local frame ́ ́ ́XOY anddeviated as much as θ around Z-axis. In the base frame, the point Pis named as Q and geometrical transformation calculations fromlocal frame to base frame are summarized in Eqs. (2) and (3).

θ θ θ θ= ́ − ́ − = ́ + ́ ( )X X Y T Y X Y. cos . sin . sin . cos 2Q P P x Q P P

θ θθ θ=

[ ] − [ ][ ] [ ]

́

́

( )

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

X

Y

T X

Y01

Cos Sin 1

Sin Cos 0 00 0 0 00 0 0 1

.01 3

Q

Q

x P

P

( ) ( )

( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( )( ) ( )θ θ

θ θ θ θ θ θ θ θ

θ θ θ θ θ θ θ θ

θ θ θ θ θ θ θ θ θ= =

( ) − ( ) − ( ) − ( ) ( ) ( )

( ) + ( ) − ( ) + ( ) − ( ) ( )

− + ( ) ( ) + ( ) ( )

( )

θ θ

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

U T R C

E E E E

E E E E

E E E E E. .

cos . sin sin .cos

. cos sin . sin cos

.cos . sin . sin .cos 1

0 0 0 1 8

PQ

C C

CC CC BC XC

CC CC AC YC

BC AC BC AC ZC

Angel θ refers to rotational error and has quite small value suchthat Eq. (3) can be summarized using Tylor series expansion as Eq.(4):

εε=

− ́

́

( )

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

X

Y

T X

Y01

1 1

1 0 0

0 0 0 00 0 0 1

.01 4

Q

Q

Z x

Z

P

P

3.4. Kinematic modeling of rotary axis

In three-dimensional space, deviations of a local coordinatesystem with respect to reference coordinate system can be mod-eled through successive multiplications of transformation ma-trixes. An adequate model of rotary axis should cover all errormotions with realistic assumptions about error references. Ac-cording to Huang [16], there are two distinct possibility to driveerror model for rotary axes. In the present work, “rotary axiscomponent shift” methode is adopted which is accordance withHuang conclusion.

In this case, for a typical radial FSD rotary axis, there are threemain transformation matrixes including translational and rotationerror motion matrixes and the matrix related to axis rotationfunction. According to ISO 230-7:2015 [24], notations related torotary axis error motion, for example C-axis, are as follows:

θθθ

=

( )( )( )

( )

θ( )

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

T

E

E

E

1 0 0

0 1 0

0 0 1

0 0 0 1 5

C

XC

YC

ZC

θ θθ θθ θ

=

− ( ) ( )( ) − ( )

− ( ) ( )( )

θ( )

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

R

E E

E E

E E

1 0

1 0

1 0

0 0 0 1 6

C

CC BC

CC AC

BC AC

( )θθ θθ θ=

( ) − ( )( ) − ( )

( )

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

C

cos sin 0 0sin cos 0 0

0 0 1 00 0 0 1 7

where θ( )TC and θ( )RC are translational and rotational error motion

matrixes respectively and ( )θC describes geometrical transforma-tion due to rotational function of the rotary axis around Z-axis.Real position of a component attached to the axis, after θ degreerotation is equal to successive multiplications of mentioned ma-trixes as shown in Eq. (8):

where ( )θUPQ is the transformation matrix from P (local frame) to Q(base frame) after rotation equal to θ degree around Z-axis.

4. Cutting process modeling in radial FSD work-holding rotaryaxes

Based on kinematic modeling of error motions, an adequateworkpiece machining model should be derived to cover the con-siderations of sensitive direction observations. This model shouldbe applied to simulate real-condition performance of radial FSDrotary axis and consequently to generate error patterns for eachcombination of error motions and their cyclic behaviors. Havingthe error patterns available, makes it feasible to predict positionalas well as geometrical and form deviations of machined features invicinity of the specific error motions. In this section, perfect ro-tation concept versus imperfect one is reviewed. Then a workpiecemachining model is derived for imperfect rotation function.

4.1. Perfect rotation cutting function

Clause 3.1.9 of ISO230-7:2015 [24], defines perfect rotaryspindle/table as “spindle or rotary table having no error motion ofits axis of rotation relative to its axis average line”. In perfectcutting operation of radial FSD work-holding rotary axis, the axismaintains perfect rotational movement around an ideal axis ofrotation i.e. axis average line. At the same time, direction vector ofcutting tool, located in a fixed angular position, coincides with axisaverage line. Rotational movement carries circumferential pointsof workpiece to the cutting zone. During cutting, the points touchfunctional point of the tool and transform radially as much asdepth of cut. Perfect rotation cutting function illustrates differenttransient sections for concentric and eccentric workpieces that isdistinct from similar transient sections seen on imperfect rotaryaxes.

Fig. 6. Cutting a perfect workpiece on a perfect rotary axis, δω is phase difference of observed point versus radial throw vector.

M.V. Nojehdeh, B. Arezoo / International Journal of Machine Tools & Manufacture 111 (2016) 17–3022

4.1.1. Concentric radial FSD machining of perfect workpieceClause 3.1.10 ISO 230-7:2015 [24], defines perfect workpiece as

“rigid body having perfect surface of revolution about a centerline”.Concentric machining of the perfect workpiece on perfect rotaryaxis would result in concentric, round, and form error free ma-chined surface (Fig. 6a).

4.1.2. Eccentric radial FSD machining of perfect workpieceIf perfect rotary axis carry perfect workpiece with radial throw,

the center point of workpiece starts to rotate on circular path withradius equal to radial throw. During cutting process startup, gra-dually increasing the depth of cut generates arc-shaped cir-cumferential cuts of round workpiece. When depth of cut exceedsthe radial throw, the transient arc-shape cuts would change touniform circular cuts. The maximum thickness of transient arc-shaped cuts is dedicated to a circumferential point which has samephase angle (δ =ω 0) as radial throw vector (Fig. 6b). At the middledepth of are-shaped transient thickness, the point with δ π=ω /2phase offset will be removed (Fig. 6c). Consequently, the pointlocated in δ π=ω phase offset would be the closing point of the arc(Fig. 6d).

The above-mentioned discussion results that, if depth of cutexceeds radial throw, eccentric radial FSD machining of a perfectworkpiece on a perfect rotary axis would yield a circular featurethat is free of any form devotions. However, the machined fea-ture's center point will deviate as much as the radial throw. Aftercutting, if a linear displacement sensor is replaced with the cuttingtool, it would show no deviations, neither in tool side nor in op-posite side. In fact, perfect rotary axis guarantees form accuracy ofthe machined features.

4.2. Imperfect rotation cutting function

In real world applications, error motions of rotary axes are aninherent part of axis function. Such imperfections make the ma-chined workpieces prone to form deviations. As a common con-fusion, the general sense of radial error motion is similar to ec-centricity of the workpiece. The former is the result of improperrotating function and the latter is the workpiece setup problem. Inaddition, they may show different effects on machining accuracy.Eccentricity shows transient effect that may lead to an increase ordecrease in size of cut feature while cyclic radial error motioncould leave unacceptable form deviations. Such differences couldbe revealed through cutting analysis of imperfect rotationfunction.

In order to categorize error patterns of any individual errorsources, it is helpful to consider a circular feature as base feature.This strategy prepares new knowledge that facilitates error sourceisolation in case of dimensional inaccuracies of machined features.Similar method has been used by Liu [29] to generate and savespecial deviated pattern of each individual error. During cutting aworkpiece on radial FSD configuration, any undesired distancevariation between cutting tool and rotary axis average line, wouldresult in violence of cutting accuracy. Furthermore, cyclic nature oferror motions provides special patterns for such deviations. Si-mulation of cyclic error motions on circular feature would yieldspecific after-cut pattern that is representative for consideredcyclic error motion and could be referred to as error signature.

As a typical radial FSD milling application, Fig. 7 illustrates acircular sample feature under erroneous cutting condition byA-axis and Table 2 describes its parameters. Each point of raw

Fig. 7. The illustration of a circular feature, while cutting in radial FSD setup.

Table 2Descriptions for parameters used in Fig. 8.

Parameter Description

O Theoretical originOe Deviated origin

θ( )Re Radial error motion in angle θ

PiO Nominal position of ith point

PiT Transient position of ith point

PiC After cut position of ith point

εS Sensitive direction component of radial error motionε _N S Non-sensitive direction component of radial error motionγS Consequence of sensitive direction deviation in functional pointγ _N S Consequence of non-sensitive direction deviation in functional

point

M.V. Nojehdeh, B. Arezoo / International Journal of Machine Tools & Manufacture 111 (2016) 17–30 23

material ( PiO ) have a fixed radial distance (R) from center of the

feature ( O) and an angular distance ( θ ( )ic ) from the functionalpoint. During cutting process, rotary axis transforms the point ( Pi

O )to the cutting zone and shifts it radially inward as much as depthof cut. An ideal cutting tool,5 is aligned parallel to theoretical co-ordinate axes, YOZ .

Radial error motion ( θ( )Re ) demonstrates special 2-D pattern asfunction of rotational angle (θ). Orthogonal components of radialerror motion are ε ( )iZC and ε ( )iYC in sensitive and non-sensitivedirections respectively.

The prepared algorithm first discretizes the base circle to anumber of points ( Pi) and determines angular distance ( θ ( )ic ) of

5 According to [24], ideal cutting tool is capable of cutting in exact accordancewith its position, without deflection, wear, etc.

them with reference to functional point (Eq. (9)). It then appliesperfect rotation ( ( )U iOT ) to transform each point from nominalposition ( Pi

O ) to the functional position ( PiF ), according to Eqs. (10)

and (11) .

δθδθ

δθ π=( )( )

= =

( )

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

PR cos i

R sin ii n n

0. .. .

1

; 1 : ; 2 /

9

iO

= ( ) ( )P U i P. 10iF

OF

iO

( ) θ θθ θ

=( ( )) − ( ( ))( ( )) ( ( ))

( )

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

U ii i

i i

1 0 0 00 cos sin 0

0 sin cos 0

0 0 0 1 11

OF C C

C C

A non-perfect rotational transformation, transfers the points todeviated destination here named as Transient position ( Pi

T ). Thedeviation includes two components in sensitive ( εS) and non-sensitive (ε _N S) directions. For considered base circle (connected toA axis) in YZ plane, these parameters are equal to ε ( )iZA and ( )ε iYArespectively. Each component has its own consequence on relativeposition of workpiece and cutting tool. The component alignedwith sensitive direction (εS) directly changes the tool-work relativedistance while non-sensitive one ( ε _N S), partially takes part indeviations depending on geometry of cutting feature which here isradius of base circle. Eqs. (12) and (13) calculate the effect of errormotion on material deviation in functional position.

= ( ) ( )P U i P. ; 12iT

FT

iF

Center point

Functional point

Cutting toolDepth of Cut

Fig. 8. Illustration of non-sensitive consequence effect on depth of cut.

M.V. Nojehdeh, B. Arezoo / International Journal of Machine Tools & Manufacture 111 (2016) 17–3024

( ) ( )

( ) ( )( ) ( )( )

ε

γ γ

ε ε

γ ε

γ

=( )

− + ( )

=

=

=( )

_

_

_

_

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

⎧⎨⎪⎪

⎩⎪⎪

U ii

i iwhere

i i

i i

i Eq

1 0 0 00 1 0

0 0 1

0 0 0 1

, ;

.1413

FT N S

N S S

N S YA

S ZA

N S

The error component which is parallel to sensitive direction (εS)may result in either positive or negative consequences dependingon its radial direction. For positive consequences, tool penetratesmore in workpiece material than it is required and hence the sizeof machined feature would be smaller and vice versa for negativeone. On the other hand, according to Eq. (14), non-sensitive di-rection deviation cause an increase in relative distance betweenfunctional point of the tool and the workpiece material that alwaysresults in negative consequences, which causes an increase of size.This fact is demonstrated in Fig. 8.

( )( ) ( )γ ε= − = − + ( )_ _i R R R i R 14N S e N S 2 2Eq. (14) implies that Re is equal or larger than R and as a result,

the sign of ( )γ iYC is always negative. Therefore, the non-sensitivedirection deviation increases the size of after cut machinedfeatures.

According to Eqs. (15) and (16), the cutting tool radiallytransforms workpiece material in transient coordinate ( Pi

T ) asmuch as nominal depth of cut, described with a simple Z-directiontransformation matrix. Finally, after-cut position ( Pi

C ) is the re-sultant position of discretized point i and is achievable throughsuccessive multiplications of all aforementioned transformationsmatrixes as Eq. (17).

= ( ) ( )P U i P. ; 15iC

TC

iT

( ) =−∆

∆ = ≤

∆ = ( − ) >( )

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎧⎨⎪

⎩⎪U i

Z

Z ifZ Z

Z Z Z ifZ Z

1 0 0 00 1 0 00 0 10 0 0 1

,

0,

,16

TC

P F

PF

P F

iC

iC

iC

( )= ( ) ( ) ( )P U i U i U i P. . . 17iC TC FT OF iO

According to Eqs. (18) and (19), inverse transformation of PiC

yields after-cut positions,^Pi

O . Difference between radial lengths ofPi

C and nominal radial value is Radial correction value ( ( )R iCor ) forith discretized point and its trace constructs after-cut appearance.

= ( ) ( ) ( )^ − −P U i U i P. . 18iO

OF

FT

iC1 1

( ) = − ( )^

R i P P 19Cor iO

iO

Cyclic effects of error motions can be introduced in calculationsby sinusoidal equations according to Eqs. (20) and (21), where (n)is harmonic number of error motion cycle and is equal to unity forfundamental error motion.

( )ε ε θ= ( ( )) ( )i i. sin 20S ZC C

( )ε ε θ= ( ( )) ( )_ i i. cos 21N S YC CAll algorithms related to cutting process modeling, are in-

tegrated in a virtual cutting module (VCM) in Matlab (2014a) code.The module takes cyclic error motions as input in order to gen-erate after-cut geometrical characteristics of error pattern in out-put. It prepares visual sense of deviated feature supported by arectilinear graph with a matrix background to use in error diag-nostic purposes as error pattern.

5. Simulations and experiments

5.1. Simulations

The prepared model made it possible to generate error patternof machined features in presence of specific cyclic error motion.Such patterns prepare an intuitive illustration of cyclic error mo-tions consequences for radial FSD work-holding axes supported bya characteristic graph. Simulations take a circular feature as basefeature and cut it virtually by VCM in presence of different com-positions of cyclic error motions. Table 3 lists simulation results ofprimary scenarios starting from non-cyclic deviations and ex-tending to ones that are more complicated. The first column re-presents different scenarios manipulating direction of error mo-tions and their CPR number. Second column illustrates error con-sequences on circular feature under specified error motion. Thirdcolumn shows characteristic deviation graph that is suitable fordiagnostic applications. Last column gives an intuitive sense aboutafter-cut appearance of resultant deviations on an airfoil section.

5.1.1. Non-cyclic patternsPractical applications take rotary axes average line as a datum

for machine tool axes setup. In this case, non-cyclic deviations canbe treated as zero position deviation for axes of the machine tool.Obviously, positive or negative non-cyclic deviations, alignedparallel to sensitive direction (Table 3a and b) cause uniform sizechange in machined feature without any form deviations. On theother hand, non-sensitive deviations in both positive and negativedirections (Table 3a), result in size increase. Hence, during externalcutting operations, slight deviation of axis of rotation in non-sensitive direction may affect the size of feature in safe-side whichis diameter increase. Such fact is risky for internal cuttings sinceincreased dimension of internally cut feature cannot be decreasedin principle.

5.1.2. Cyclic patternsCyclic error motions cause planetary movement of axis of ro-

tation around axis average line. They generate more complicatederror patterns in comparison with non-cyclic ones. Table 3c and d

Table 3Error patterns of various possible compositions of error motions.

M.V. Nojehdeh, B. Arezoo / International Journal of Machine Tools & Manufacture 111 (2016) 17–30 25

Table 3 (continued )

M.V. Nojehdeh, B. Arezoo / International Journal of Machine Tools & Manufacture 111 (2016) 17–3026

represent consequences of 1-CPR synchronous cyclic error mo-tions in sensitive and non-sensitive directions, respectively. Theircomparison discovers an interesting difference; Non-sensitiveconsequences show cycles in twice frequencies compared withsensitive ones. Concentrating on form error, 1-CPR cycles in sen-sitive direction (i.e. fundamental error motion) shows no formerror while non-sensitive one shows slightly distorted base circlein orthogonal directions. New pattern looks like an ellipse withmajor axis laying in π/2 phase shift with respect to cycling phase.

Considering deviation of center point of the feature, 1-CPR cyclesin sensitive direction shifts the center of feature as much as half ofcycling domain while non-sensitive one maintains the averagecenter location with no shifts.

ISO 230-7:2015, annex A [24] describes synchronous 1-CPR cyclicerror motion as fundamental error motion and declares that sucherror motion never causes any form deviation. The simulations in thepresent work reveals that, in principle, non-sensitive part of funda-mental error motion generates slight form deviation. However, such

Nominal ProfileMachined Pro�ile

Deviation Vector

Normal Dev. Vector

Pro�ile Normal Vector

Fig. 9. Deviation vector mapping to the normal vector of the surface.

Fig. 11. Structural representation of five-axis [wÁbXZYB(C)t] machine tool used forexperimental verifications.

M.V. Nojehdeh, B. Arezoo / International Journal of Machine Tools & Manufacture 111 (2016) 17–30 27

form deviations are negligible when compared to consequencesraised from error motions aligned with sensitive direction.

The 2-CPR cyclic error motion in sensitive direction generatesmore distorted ellipse pattern with 2π/3 phase shift. Cycles over2-CPR generate polygon-shaped patterns where its number ofsides ate equal to cycle number. For instance, Table 3f and g re-present resultant patterns for 3-CPR and 4-CPR respectively.

5.1.3. Symmetric and asymmetric cycling patterns2-dimensional error motions may show symmetric or asym-

metric deviation of axis of rotation. For all 2-dimensional errormotions, resultant error pattern will follow the predominant cy-cling style which is sensitive direction cycle. For instance, eight-shaped cycling error motion (mentioned in annex A of ISO 230-7[15]) is the result of 1-CPR cycling in one direction and 2-CPR inanother one (Table 3h) and its resultant error pattern looks like2-CPR because of being in sensitive direction.

5.2. Preliminary data preprocessing

Practical applications perform dimensional quality control of freeform or sculptured surfaces via scanning of a section by touch triggerprobes or other solutions. Such measuring methods yield discretizedpositions of machined features and if possible, about normal vectorregarding to each point. According to Fig. 9, in order to process suchpractical data for cyclic number extraction, a preliminary data pro-cessing stage is needed to map deviation of each point in normal di-rection of the feature. This processing is helpful to eliminate the cyclicnoise, during comparison of measured data with nominal values. This

Fig. 10. A five axis machine tool with [wÁbXZYB(C)t] st

algorithm compensates inclination of deviation vectors due to phasedifference between measured and nominal data. After running pre-liminary algorithm, subtraction of nominal deviation vector fromnominal profile yields characteristic graph of error motions.

Characteristic graph of practical data encompass distortionsthat prohibit their exact compatibility with theoretic ones re-presented in Table 3. Meanwhile detection of the cyclic number(CPR) of deviations has a high importance for interpretation ofavailable graphs.

As a comprehensive solution, Discrete Fourier Transform (DFT)algorithm [30] is adopted and customized to estimate cyclenumber of characteristic graph. A number of papers reportedsuccessful application of FFT in spectrum interpretation of rotaryaxes data ([19,21,25]). The main difference in application of FFT inerror motions spectrums rather than its conventional vibrationmonitoring applications is that time domain is changed to degree.

According to Eq. (22), the DFT algorithm maps vibration signal

ructural configuration used for experimental tests.

Fig. 12. Real versus nominal positions of scanned sections.

Fig. 13. Characteristic graph of sections.

6 Coordinate measuring machine.

M.V. Nojehdeh, B. Arezoo / International Journal of Machine Tools & Manufacture 111 (2016) 17–3028

( ( )x n ) as input to frequency domain ( ( )X k ) as output. The algo-rithm is customized to map variations of characteristic graph toCPR number.

( ) ∑= ( ) = = … −( )

π

=

−−X k x n W W e k N. , , 0, 1, , 1

22n

N

Nnk

Nj

N

0

12

where, N is length of signal vector and K is the demanded har-mony number. The mentioned procedure for practical data pre-processing (PDP) is integrated as a compact code in MATLAB(R2014a).

5.3. Experiments

In order to verify practical capabilities of presented method, anexperimental verification is carried out. For this purpose, a[wÁbXZYB(C)t] five-axis CNC machine tool is chosen (see Fig. 10and Fig. 11). A 20 mm flat end mill with 1 mm nose radius is usedfor finishing operation. The machining conditions was such thatthe angle between the tool axis and normal vector to the surface

(inclination angle) is kept fixed at 10 degrees in order to avoidgouging and minimum loss of tool stiffness. The cutting processsuffered from an unknown cause which raised nonconformity dueto exceeded tolerances of machined airfoil. The Á rotary axis of themachine holds the workpiece in radial FSD setup. According toFig. 12, a Trimek-10.07.05-Spark type CMM6 measured the ma-chined airfoils in three profiles located in three different levels ofairfoil height.

The preliminary data processing module (PDP) processed initialCMM results to extract normal deviation vectors as well as char-acteristic graph, and CPR number. Fig. 13 represents magnitude ofnormal deviation vectors as characteristic graphs for each sectionand Fig. 14 illustrates DFT analysis results. DFT analysis revealed1-CPR estimation for cycling number in all three sections. Thisbehavior was compatible with Table 3c, attributing the errorsource to be in the same frequency with the axis of rotation.

The first candidate for such error source is fundamental errormotion of axis of rotation. Hence, as the first approach, a quick testwas conducted to evaluate fundamental error motion of Á axis.Using adequate reference artefacts (in our case a precision cylin-der) and according to clause 5.4.3 of ISO230-7:2015 [24] the

Fig. 14. DFT analysis results for scanned sections and their resultant Harmony numbers.

Fig. 15. Deviation improvement after correction.

M.V. Nojehdeh, B. Arezoo / International Journal of Machine Tools & Manufacture 111 (2016) 17–30 29

fundamental error motion found to be negligible. Also, as com-plementary check, the parallelism of Á rotary axis average linewith respect to X-axis of linear motion was found negligible (ac-cording to DG9 test procedure of ISO 10791-1 [31] and based ongeneral guidelines of clause 10.1.4.1 of ISO230-1:2012 [32]). Therewas only one remained candidate for 1-CPR cycling effect, whichwas datum axis of measurement. During CMM measuring of theairfoil, discrete positions of sections were captured with respect toa datum. This datum was an axis fixed to Á axis via clampingsurfaces of the blade. Misaligned datum caused 1-CPR effect onmeasuring results and light modification on datum surfaces of thework-holding axis fixture lifted undesired deviation. Fig. 15 illus-trates improvement of machined feature deviations after appliedcorrections.

For this experiment, the root cause investigation time de-creased significantly due to presented evidence about cyclic nature

M.V. Nojehdeh, B. Arezoo / International Journal of Machine Tools & Manufacture 111 (2016) 17–3030

of the deviations and similarity of deviation pattern with availableerror signatures depicted in Table 3. Such evidence limited errorsource tracking procedures to the most possible candidates andisolated the source of the problem. It should be mentioned that,capabilities of this methodology is limited to accuracy investiga-tion of work-holding axes and it cannot cope with the other pos-sible sources like interpolation inaccuracies, tool deflection orthermal variations. In the experimental case, such errors wereminimized through thermal stabilization (short finish cutting timeand circulation of chilled coolant) and tool rigidity improvement(carbide cutting tool with minimal overhang).

6. Conclusion

This paper investigates precision cutting performance of radialFSD work-holding rotary axes. Consequences of cyclic error mo-tions are derived as unique error signatures containing a char-acteristic graph. According to type of radial cyclic error motions,resultant error patterns included a composition of dimensionaloffsets, feature center shifts, and form deviations. As cyclic numberincreases, error patterns switch from simply dimensional offset tocomplicated form deviation patterns. This approach can be used asan effective monitoring scheme for five-axis complicated ma-chining processes in which the profiles of machined features aremeasured directly. Conventional quality control idea checks theprofile deviations value for accept/reject decision while presentedmethod observes deviations pattern to match it with error sig-natures and introduce probable error sources. Regardless of fea-ture complexity, the presented method makes it feasible to com-pare the machined feature deviations with available error patternsto detect root cause of deviations. Using this method, costly andcomplicated machining operations can be monitored continuouslywithout need for additional tests for diagnostic purposes.

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Functional accuracy investigation of work-holding rotary axes in five-axis CNC machine toolsIntroductionCharacteristics of rotary machining axesError motions of axis of rotationCyclic effects of error motionsRotary axis applications in cutting process

Kinematics of radial FSD work-holding axisHomogenous transformation matrix (HTM)Reference frame (base frame)Coordinate frame transformation by HTMKinematic modeling of rotary axis

Cutting process modeling in radial FSD work-holding rotary axesPerfect rotation cutting functionConcentric radial FSD machining of perfect workpieceEccentric radial FSD machining of perfect workpiece

Imperfect rotation cutting function

Simulations and experimentsSimulationsNon-cyclic patternsCyclic patternsSymmetric and asymmetric cycling patterns

Preliminary data preprocessingExperiments

ConclusionReferences

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