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International Journal of Machine Tools & Manufacture 101 (2016) 65–78
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International Journal of Machine Tools & Manufacture
http://d0890-69
n CorrE-m
altintas@
journal homepage: www.elsevier.com/locate/ijmactool
Modeling and compensation of volumetric errors for five-axis machinetools
Sitong Xiang a, Yusuf Altintas b,n
a School of Mechanical Engineering, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai 200240, PR Chinab Manufacturing Automation Laboratory, Department of Mechanical Engineering, The University of British Columbia, Vancouver, BC, Canada, V6T 1Z4
a r t i c l e i n f o
Article history:Received 10 August 2015Received in revised form4 November 2015Accepted 11 November 2015Available online 1 December 2015
Keywords:Five-axis machine toolVolumetric errorError measuringError compensationScrew theory
x.doi.org/10.1016/j.ijmachtools.2015.11.00655/& 2015 Elsevier Ltd. All rights reserved.
esponding author.ail addresses: [email protected], xiangmech.ubc.ca (Y. Altintas).
a b s t r a c t
This article proposes a method to measure, model and compensate both geometrically dependent andindependent volumetric errors of five-axis, serial CNC machine tools. The forward and inverse kinematicsof the machine tool are modeled using the screw theory, and the 41 errors of all 5 axes are representedby error motion twists. The component errors of translational drives have been measured with a laserinterferometer, and the errors of two rotary drives have been identified with ballbar measurements. Thecomplete volumetric error model of a five-axis machine has been modeled in the machine's coordinatesystem and proven experimentally. The volumetric errors are mapped to the part coordinates along thetool path, and compensated using the kinematic model of the machine. The compensation strategy hasbeen demonstrated on a five-axis machine tool controlled by an industrial CNC with a limited freedom,as well as by a Virtual CNC which allows the incorporation of compensating all 41 errors.
& 2015 Elsevier Ltd. All rights reserved.
1. Introduction
There are 21 known geometric errors in three-axis machinetools [1], and 41 errors exist for five-axis serial machine tools. Theerrors have integrated effects in determining the orientation andposition errors of the tool tip relative to the workpiece in five axismachine tools. The modeling and compensation of these volu-metric errors are needed to improve the accuracy of the machinein the five-axis machining of parts [2].
The volumetric error compensation of multi-axis machine toolshas 3 engineering steps: the kinematic modeling, measurementand modeling of axis errors, and their compensation during thepositioning of the machine along the tool path. The kinematics ofthe machine have been modeled by applying the homogeneoustransformation matrix (HTM) [1], by using the screw theory [3,4],by the product of exponential model [5], or by the differentiablemanifold-based method [6]. This paper adopts the screw theory-based, generalized modular kinematic model of the five-axis ma-chines reported previously [3].
The geometric errors of machine axes are measured by directand indirect methods as reviewed by Schwenke [7] and Ibaraki [8].The laser interferometer is mostly used in measuring the geo-metric errors of translational axes directly, and the geometric
[email protected] (S. Xiang),
errors of rotary axes are identified indirectly by measurementswith a ballbar [9], R-test [10,11], touch trigger probe [12,13], ma-chining tests [14] and tracking interferometer [15,16]. Once thegeometric errors of the axes are measured and modeled as afunction of position in the Machine Coordinate System (MCS), theyare translated to the tool orientation and tool tip position using aforward kinematic model of the machine in the Part CoordinateSystem (PCS). The errors are compensated by transforming thetool position and orientation errors to drive components via theinverse kinematic model of the machine in PCS. Lei and Hsu [17]presented a compensation algorithm for five-axis machine toolsand analyzed the singularity problems. They [18] compensated thetool axis orientation errors first, followed by the compensation ofthe translational errors. Zhu [19] presented an identification ap-proach for recognizing 6 error parameters of rotary axes via theballbar test and verified the compensation of a five-axis machinetool on a “S” shape tool path. Huang [20] merged an iterativecompensation method into the post-processor and generated anew, error-compensated NC program.
Commercial CNC systems have look-up tables which can befilled with axis errors at each position of the machine within itsoperating volume. However, they allow for the compensation ofonly translational errors, but not the deviations of tool orienta-tions needed in five-axis machining applications [7].
This paper presents a detailed modeling, measurement andcompensation method for the volumetric errors of five-axis ma-chine tools as outlined in the flow chart given in Fig. 1. Yang et al.[21] used the screw theory to identify and compensate the 11
Fig. 2. Configuration of the five-axis machine tool.
Fig. 1. Flowchart of compensation strategy.
S. Xiang, Y. Altintas / International Journal of Machine Tools & Manufacture 101 (2016) 65–7866
position-independent geometric errors (PIGEs), i.e., the squarenessof 3 translational axes and 4 linear offsets and 4 angular tilts oftwo rotary axes. This article extends Yang's method [21] by in-cluding both position-dependent and position-independent 41geometric errors with a proposed, unified compensation methodwith the introduction of error twists.
Henceforth, the paper is organized to present the measure-ment, modeling and compensation algorithms used in Fig. 1. Thekinematic model of a sample five-axis machine tool is modeledwith the screw theory in Section 2. The concept of error twists isintroduced to build a 3D volumetric error map of the machine inSection 3. Section 4 presents a methodology to identify all theerrors of rotary axes, and Section 5 demonstrates the compensa-tion strategy of volumetric errors. The paper is concluded in Sec-tion 6 by summarizing the effectiveness of the method and itspractical application in CNC systems.
2. Kinematic model of five-axis machine tools
Although any serial, five-axis kinematic configuration can bemodeled with the generalized kinematic model developed in ourVirtual CNC [3], a machine tool with 3 translational and a trunnionwith 2 rotary drives is used to illustrate the proposed modeling ofvolumetric errors and their compensation (Fig. 2). The MCS is
defined at the intersection of the centerlines of the two rotaryaxes. The objective is to predict the relative error between the tooltip and workpiece clamped on the table as the drives move withinthe workspace of the machine. The kinematics of the machine ismodeled using screw theory as presented in [3] and summarizedfor the particular five-axis machine used here.
The motion commands to the five drives (x, y, z, θA, θC) are usedto predict the position and orientation of the tool tip relative to the
Fig. 3. Error twist.
S. Xiang, Y. Altintas / International Journal of Machine Tools & Manufacture 101 (2016) 65–78 67
workpiece (X, Y, Z, I, J, K) using the forward kinematics of themachine (see CAM section in Fig. 1).
The five-axis machine tool is represented by the workpiece andthe tool chains. If the individual drives of the machine move (x, y,z, θA, θC) amount, the new coordinates of the workpiece and the
tool tip in MCS can be evaluated using screw motions ( θξ̂⋅e ) withtwists (ξ) as:
⎫⎬⎪⎭⎪
( )( )
( ) ( )= ⋅ ⋅ ⋅
= ⋅ ⋅⋯
( )
θ θξ ξ ξ
ξ ξ
^ ⋅( − ) ^ ⋅ − ^ ⋅ −
^ ⋅ ^ ⋅
g e e e g
g e e g
0
0 1
bwx
bw
bty z
bt
X A A C C
Y Z
where, gbw(0) and gbt(0) are the initial 4�4 motion matrices of theworkpiece and the tool tip relative to the MCS. The transformationmatrix from the tool tip to the workpiece can be obtained from Eq.(1) as:
( )
( ) ( )
= ⋅ = ⋅
= ⋅ ⋅ ⋅ ⋅ ⋅ ( )θ θξ ξ ξ ξ ξ
−
− ^ ⋅ ^ ⋅ ^ ⋅ ^ ⋅ ^ ⋅
g g g g g
g e e e e e g0 0 2
wt wb bt bw bt
bwx y z
bt
1
1 C C A A X Y Z
The twist for a rotary axis is expressed as:
⎡⎣ ⎤⎦ξ ω ω= = × ( )v v q, 3
where, ω is the unit vector in the positive direction of the rotaryaxis-line and q is any point on the axis-line expressed in MCS, i.e.,for A-axis, ω¼[1 0 0]T, q¼[x 0 0]T. The screw motion of the rotaryaxes has the form [22]:
⎡
⎣⎢⎢
⎤
⎦⎥⎥
( ) θω ωω=
− ( × ) +
( )
θθ θ
ξω ω
^⋅^
×^
×
ee eI v v
0 1 4
T3 3
1 3
where θ is the angular rotation of the axis. For the A-axis, theθω̂e can be expanded as:
⎡
⎣⎢⎢
⎤
⎦⎥⎥
⎡
⎣⎢⎢
⎤
⎦⎥⎥
( )
( )
θ θ
θ θ
ω ω= + ^ ⋅ + ^ ⋅ −
= + − ⋅ + − ⋅ −( )
θω̂×
×
e I
I
sin 1 cos
0 0 00 0 10 1 0
sin0 0 00 0 10 1 0
1 cos5
A A
3 32
3 3
2
The^operator converts vector to matrix as^ = ×ab a b. The screwmotion for translational axes is expressed as:
⎡⎣⎢
⎤⎦⎥
⎡⎣⎢⎢
⎤⎦⎥⎥
θξ = =
⋅
( )θξ̂⋅ ×
×e
v I v0
,0 1 6
3 3
1 3
where, v is the unit vector of the positive direction, i.e., for X-axis,v¼[1 0 0]T. The detailed derivation of the screw motion can befound in References [3,22].
Compared with the traditional HTM method, the screw theory-based modeling has 2 advantages: the motions are defined in theMCS, and therefore no local coordinate system is needed; it pro-vides explicit mathematical solutions to the inverse kinematics.
3. Modeling of volumetric errors
The volumetric errors of the machine are modeled by in-troducing 1 error twist for each component error of the drive. Theerror twists are then mapped to the machine's working space anddrives with the aid of the kinematic model of the five-axis ma-chine as follows.
3.1. Error twists (ξe)
Error twists are used to model the geometric errors betweenthe tool tip and the workpiece. Assume that the ideal position ofthe machine is at Pd, and the ideal direction of the linear axis orrotation axis of the rotary drive is ωd (Fig. 3). However, the realmachine position, perhaps at P and the axis may have an angularerror of θe. The ideal twist ξd and the actual twist ξ with geometricerrors can be expressed as:
⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦ξ ω ω ξ ω ω= × = × ( )q q, 7d d d dT T
The change from the ideal twist ξd to the actual twist ξ can be
regarded as the result of an error motion θξ̂ ⋅e e e. The error twist ξeconsists of an angular geometric error θe around the commonperpendicular line of the ideal and the actual axis lines, and thelinear position error d of the axis. The error twist ξ ω= [ ]v
e e eTcan
be expressed as:
⎫
⎬⎪⎪
⎭⎪⎪
θ θ θ
θ θ θ
ω ω ω
ω ω ω ω
=×
= =−
=× ( × )
+×
=×
+−
( )
hd
hd
q q
vq q q q q
sin,
sin sin 8
ed
ee
e
d
e
ed
ee
d
e
d d
e
When the angular geometric error is zero (θe¼0), the actualaxial-line is parallel to the ideal axis-line, and thus the twist vectorcontains only translational errors ( ω = = ∞h0,e e ) and the errortwist changes to:
⎡
⎣⎢⎢
⎤
⎦⎥⎥ξ =
−
( )d
q q
0 9e
d
When d¼0, i.e. = =h q q0,e d, the actual and ideal axes coincidewithout linear errors, and hence the error twist indicates onlyangular errors:
⎡⎣⎢
⎤⎦⎥
⎡⎣⎢
⎤⎦⎥θ θ
ξ ωω ω ω ω
= =× ( × ) ×
( )
v qsin sin 10e
e
e
d
e
d
e
T
As an example, let's assume that the rotary drive C has a tilterror (εyc) around the Y-axis. The coordinate system of the C driveis defined in MCS as = [ ]O O Oq x y z
T (see Fig. 3 and Fig. 4).Based on Eq. (8), ωe and ve can be expressed as:
Fig. 4. Example of error twist.
Table 141 geometric errors of the AC table tilting five-axis machine tool.
X-axis Y-axis Z-axis Squareness error A-axis C-axis PIGEs of Rotaryaxes
δxx δyy δzz Sxy δxa δxc δxoc Sboaδyx δxy δxz Syz δya δyc δyoc Scoaδzx δzy δyz Sxz δza δzc δyoa Saocεxx εxy εxz εxa εxc δzoa Sbocεyx εyy εyz εya εyc (ISO 230-7)εzx εzy εzz εza εzc
S. Xiang, Y. Altintas / International Journal of Machine Tools & Manufacture 101 (2016) 65–7868
⎧
⎨⎪⎪⎪
⎩⎪⎪⎪
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
⎡
⎣⎢⎢
⎤
⎦⎥⎥
ω
ω
= [ ]
= × = ×
( )
O
O
O
v q
0 1 0
010 11
e
e e
x
y
z
T
which leads to the error twist ξ ω= [ ]ve e eT. The geometric errors of
all axes can be defined in a similar fashion.
3.2. Modeling of volumetric errors with error twists
Table 1 lists 41 geometric errors of the five-axis machine tool.Each axis has 6 position-dependent geometric errors (PDGEs). Alinear axis has 1 positioning, 2 straightness and 3 angular (roll,pitch and yaw) errors as shown in Fig. 5. For example, the errorsfor the linear X-axis are δxx, (δyx, δzx), (εxx, εyx, εzx). A rotary axishas 1 axial error indicating the linear offset of the axis of rotation,
Fig. 5. Definition
2 radial errors, 1 angular positioning error and 2 tilt errors, i.e., theerrors of rotary drive C are δzc, (δxc, δyc), εzc, and (εxc, εyc). Inaddition, 3 PIGEs, the squareness errors (Sxy, Syz, Sxz), lie betweenthe 3 linear axes (X, Y, Z), and 8 PIGEs (δxoc, δyoc, δyoa, δzoa, Saoc,Sboc, Sboa, Scoa) lie between the 2 rotary axes [23] (see Fig. 6).
Parameters θξ̂ ⋅e eiI
eiIand θξ̂ ⋅e ei
D
eiDrepresent the PIGE and PDGE matrices
of the axis i (i¼X, Y, Z, A, C), respectively.The PDGE matrix of the axis i can be evaluated by the products
of its 6 error motions as
( )( ) ( )δ δ δ ε ε ε= ( ) ( ) ( )
= ⋅ ⋅ ⋅ ⋅ ⋅ ( )
θ
δ δ δ ε ε ε
ξ
ξ ξ ξ ξ ξ ξ
^
^ ^ ^ ^ ⋅ ^ ⋅ ^ ⋅δ δ δ ε ε ε
e T T T R R R
e e e e e e 12
xi yi zi xi yi zieiD
eiD
xi xi yi yi zi zi xi xi yi yi zi zi
The modeling of PIGEs is however detailed as follows:
3.3. Squareness errors of the translational axes
For the five-axis machine tool with a configuration shown inFigs. 2, 3 squareness errors of linear axes are illustrated in Fig. 6.When measuring the squareness errors, the X-axis is set as thereference, and the squareness error is positive when the anglebetween 2 axes is greater than 90 degrees.
The PIGE models of translational axes are obtained from theerror twists as follows:
⎧⎨⎪⎩⎪ ( )
=
= ⋅ ( )
θ
θ ξ
ξ ξ
ξ ξ
^ ⋅ ^ ⋅
^ ⋅ ^ ⋅ ^ ⋅ −
e e
e e e 13
S
S S
eYI
eYI
eSxy xy
eZI
eZI
eSyz yz eSxz xz
3.4. Modeling of 8 PIGEs of rotary axes
ISO 230-7 defines 8 PIGEs of 2 rotary axes, which are displayedin Fig. 6. Similar to the modeling of squareness errors, the PIGEmodels of the A- and C-axis are expressed as:
⎧⎨⎪⎩⎪
( )
( )
= ⋅ ⋅ ⋅
= ⋅ ⋅ ⋅ ( )
θ δ δ
θ δ δ
ξ ξ ξ ξ ξ
ξ ξ ξ ξ ξ
^ ⋅ ^ ⋅ ^ ⋅ ^ ⋅ ^ ⋅ −
^ ⋅ ^ ⋅ ^ ⋅ ^ ⋅ ^ ⋅ −
δ δ
δ δ
e e e e e
e e e e e 14
S S
S S
eCI
eCI
xoc xoc yoc yoc Saoc aoc Sboc boc
eAI
eAI
yoa yoa zoa zoa Sboa boa Scoa coa
After adding all the error motion matrices into Eq. (1), the finalforward kinematics model of the five-axis machine tool can beobtained as:
s of PDGEs.
Fig. 6. Definitions of PIGEs.
Fig. 7. Measurements set-ups of laser interferometer. (a) Positioning error, (b) Angular error, (c) Straightness error, (d) Squareness error.
Fig. 8. Definition of PIGEs of AC rotary axes (relative notation).
S. Xiang, Y. Altintas / International Journal of Machine Tools & Manufacture 101 (2016) 65–78 69
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎫
⎬
⎪⎪⎪⎪
⎭
⎪⎪⎪⎪
( )
( )
( ) ( )
( )
= ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅
= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅( )
θ θ θ θ
θ θ θ
θ θ θ θ
ξ ξ ξ ξ ξ
ξ ξ ξ
ξ ξ ξ ξ ξ ξ
^ ⋅ − ^ ⋅ ^ ⋅ ^ ⋅ ^ ⋅ −
^ ⋅ ^ ⋅ ^ ⋅ −
^ ⋅ ^ ⋅ ^ ⋅ ^ ⋅ ^ ⋅ ^ ⋅
g e e e e e
e e e g
g e e e e e e g
0
015
bwe x
bw
bte y z
bt
X eXD
eX eAI
eA eAD
eA A A
eCI
eC eCD
eC C C
eYI
eY YD
eY eY eZI
eZ Z eZD
eZ
( )= ⋅ = ⋅ ( )−
g g g g g 16wte
wbe
bte
bwe
bte1
4. Geometric errors identification and measurement
All the 21 geometric errors of linear axes except for 3 roll errorshave been measured by a laser interferometer. The measurementset-ups for the positioning, angular, straightness and squarenesserrors of linear axes are shown in Fig. 7. A ballbar is used tomeasure 1 axial, 2 radial, 1 angular positioning and 2 tilt errors, aswell 4 PIGEs per rotary axis.
4.1. Measuring patterns for PIGEs and PDGEs of rotary axes
It is noted that the PIGEs of the rotary axes shown in Fig. 6 aredefined in the MCS of the five-axis machine tool shown in Fig. 2.
For five-axis machines with table or spindle tilting [24] config-urations, these error parameters (Eq.(14)) are not suitable for ki-nematic modeling because the PIGEs of the “higher” rotary axis are
S. Xiang, Y. Altintas / International Journal of Machine Tools & Manufacture 101 (2016) 65–7870
influenced by the motion of the “lower” rotary axis. For example,assume the C-axis has a tilting error around the Y-axis of the MCSat A¼0. However, this error tilts the C-axis around the Z-axis ofthe MCS at A¼-90. Therefore, for the five-axis machine tool de-picted in Fig. 2, the relative notation suggested by Ibaraki[11,12,14] is to be used here instead of the absolute notationproposed in ISO 230-7 standards. These two notations can convertto each other [8]. Eight PIGEs of relative notations are illustrated inFigs. 8, and 12 PDGEs are presented in Fig. 5.
The patterns shown in Fig. 9 are used for ballbar tests as sug-gested by Tsutsumi [25,26]. H indicates the vertical distance be-tween the table-side ball and the center-line of the A-axis (Fig. 9a–c), while L represents the horizontal distance between the table-side ball and the center-line of the C-axis (Fig. 9d–f). Before run-ning the ballbar tests, the backlash of the linear axes is measuredand compensated by the CNC controller.
The kinematic model of the table-side ball can be expressedbased on error twists as:
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
( ) ( )
δδ
δ
γ β
γ
β
ε ε δ
ε ε δ
ε ε δ
α αα α
δβ
β
ε ε δ
ε ε δ
ε ε δ
= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
= ⋅
−
−⋅
−
−
−⋅
( − + ) − ( − + )( − + ) ( − + )
⋅ ⋅−
⋅
−
−
−⋅
( − ) − ( − )( − ) ( − ) ⋅
( )
θ θ θ θ θ θξ ξ ξ ξ ξ ξ− −
^ ⋅ ^ ⋅ ^ ⋅ − ^ ⋅ ^ ⋅ ^ ⋅ −
−
−
xyz
PIGE PDGE T PIGE PDGE T
L
He e e e e e
L
H
a a
a a
c c
c cL
H
1
0
1
0
1
1 0 0
0 1 0
0 0 1
0 0 0 1
1 0
1 0 0
0 1 0
0 0 0 1
1
1
1
0 0 0 1
1 0 0 00 cos sin 0
0 sin cos 0
0 0 0 1
1 0 0 00 1 0
0 0 1 00 0 0 1
1 0 0
0 1 0 00 1 0
0 0 0 1
1
1
1
0 0 0 1
cos sin 0 0sin cos 0 00 0 1 00 0 0 1
0
1
17
T
T
TA A A ideal C C C ideal
xAX
yAX
zAX
AY AY
AY
AY
PIGE
za ya xa
za xa ya
ya xa za
PDGE
AY AY
AY AY
T
yCA
CA
CA
PIGE
zc yc xc
zc xc yc
yc xc zc
PDGE T
eAI
eAI
eAD
eAD
A A eCI
eCI
eCD
eCD
C C
A A A ideal
c C C ideal
When the A-axis is active and the C-axis is stationary (Fig. 9.a-c), the matrices PDGEC and TC-ideal become unit matrices. Similarly,when the C-axis is active and the A-axis is stationary (Fig. 9d–f),the matrices PDGEA and TA-ideal become unit matrices. However,the PIGEs of A- and C-axis will always influence the table-side balleven when the angle of the A- or C-axis is constant.
4.2. Decoupling method of PIGEs and PDGEs
Here the A-axis is given as an example to demonstrate thedecoupling method of PIGEs and PDGEs. From Eq.(17), the devia-tions in the X, Y and Z directions can be evaluated as:
⎧⎨⎪⎪
⎩⎪⎪
Δ δ ε ε
Δ δ ε ε
Δ δ ε ε
= + ⋅ ⋅ ( − ) + ⋅ ⋅ ( − ) + _= + ⋅ − ⋅ ⋅ ( − ) + _= − ⋅ − ⋅ ⋅ ( − ) + _ ( )
H a H a f
L H a f
L H a f
cos sin
cos
sin 18
x xa ya za PIGE x
y ya za xa PIGE y
z za ya xa PIGE z
where, the deviations caused by PIGEs (fPIGE) are expressed as:
⎧
⎨
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
( ) ( )( )
( ) ( )( )
δ β β γ
δ γ δ
α β
δ β δ
β α
_ = + ⋅ + ⋅ ( − )⋅ + ⋅ ( − )⋅
_ = + ⋅ + ⋅ − − ⋅ −
⋅ + ⋅ − ⋅
_ = − ⋅ + ⋅ − − ⋅ −
⋅ − ⋅ − ⋅ ( )
f H H a H a
f L a H a
L a
f L a L a
H a
cos sin
cos cos
sin
sin cos
sin 19
PIGE x xAX CA AX AX
PIGE y yAX AX yCA
AX CA
PIGE z zAX AX yCA
CA AX
Fig. 10 shows the sensitive directions of the ballbar, and giveserror decompositions in radial, tangential and axial patterns of theA-axis.
Step 1 Calibration of PIGEs: First assume the PDGEs do notexist and the 8 PIGEs can be obtained by reading the eccentricitiesmeasured by the ballbar as [9]:
Radial direction of the A axis:
⎧⎨⎩δδ
= −= − ( )
ey y
ez z 20RA AX
RA AX
Axial direction of the A axis:
⎪
⎪⎧⎨⎩
γ
β
= ⋅
= − ⋅ ( )
ey H
ez H 21
AA AX
AA AX
Radial direction of the C axis:
⎪
⎪⎧⎨⎩
δ β β
δ δ α
= − − ⋅ − ⋅
= − − + ⋅ ( )
ex x H H
ey y y H 22
RC AX CA AX
RC AX CA AX
Axial direction of the C axis:
⎪
⎪⎧⎨⎩
β β
α
= ⋅ + ⋅
= − ⋅ ( )
ex L L
ey L 23
AC CA AX
AC AX
Step 2 Calibration of PDGEs: However, the PIGEs always existwith PDGEs, and they affect the ballbar readings together. Theballbar readings in axial (ρa), radial (ρr) and tangential (ρt) direc-tions are shown as follows.
Axial direction of the A axis:
ρ Δ= ( )x 24a
Fig. 10. Sensitive directions of the ballbar and error decompositions.
Table 2Eight PIGEs evaluated by the proposed simultaneous and Tsutsumi's [9] methods.
Type αAX βAX γAX βCA δxAX δyAX δzAX δyCA
Separately calculated [9] �10.3″ �57.5″ �46.0″ 78.1″ 53 μm 25 μm �20 μm 12.6 μmSimultaneously Calculated 6.3″ �37.5″ �31.5″ 48.7″ 46 μm 21 μm �12 μm 20.5 μm
Fig. 9. Measuring patterns for A- and C-axis [26].
S. Xiang, Y. Altintas / International Journal of Machine Tools & Manufacture 101 (2016) 65–78 71
Radial direction of the A axis:
ρ Δ Δ= ⋅ ( − ) − ⋅ ( − ) ( )z a y acos sin 25r
Tangential direction of the A-axis:
ρ Δ Δ= ⋅ ( − ) + ⋅ ( − ) ( )z a y asin cos 26t
The influence of PIGEs can be extracted by substituting thePIGEs obtained in Step 1 as:
⎧
⎨⎪⎪
⎩⎪⎪
⎡⎣ ⎤⎦⎡⎣ ⎤⎦
ρ ρ
ρ ρ
ρ ρ
′ = − _
′ = − _ ⋅ ( − ) − _ ⋅ ( − )
′ = − _ ⋅ ( − ) + _ ⋅ ( − ) ( )
f
f a f a
f a f a
cos sin
sin cos 27
a a PIGE x
r r PIGE z PIGE y
t t PIGE z PIGE y
By substituting Eq. (19) into Eq. (27), the axial, radial and tan-gential errors can be listed as:
Fig. 11. PDGEs of A-axis defined by two different methods.
S. Xiang, Y. Altintas / International Journal of Machine Tools & Manufacture 101 (2016) 65–7872
⎧
⎨
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪( )
( ) ( )
ρ δ ε ε
ρ δ δ ε ε
ρ δ δ ε ε
ε
′ = + ⋅ ⋅ ( − ) + ⋅ ⋅ ( − )
′ = ⋅ ( − ) − ⋅ ( − ) − ⋅ ⋅ ( − ) − ⋅
⋅ ( − )
′ = ⋅ − + ⋅ ( − ) − ⋅ − ⋅ ⋅
− + ⋅ ⋅ − ( )
H a H a
a a L a L
a
a a H L
a L a
cos sin
cos sin cos
sin
sin cos
sin cos 28
a xa ya za
r za ya ya za
t za ya xa ya
za
Three tests with different values of H and L ((H, 0), (H1, 0), (H,L)) are conducted to calibrate the 6 PDGEs of the A-axis [19]. The3 groups of ballbar readings are recorded: (ρa1, ρr1, ρt1), (ρa2, ρr2,ρt2) and (ρa3, ρr3, ρt3). Substituting these ballbar readings into Eqs.(27), (28), the final results of 6 PDGEs are obtained as:
⎧
⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪
⎡⎣ ⎤⎦⎡⎣ ⎤⎦
( ) ( )( ) ( ) ( )
( ) ( )( )
( )
( )
δ
ρ ρ
ρ ρ
ερ ρ ρ ρ
δδ ρ
ερ ε ρ δ
ερ δ δ ε
δ ρ ε ε
=
′ ⋅ − + ′ ⋅ −
+ − ′ ⋅ − + ′ ⋅ −
−
=′ − ′ ⋅ ( − ) + ′ − ′ ⋅ ( − )
− ⋅ ( − )
=⋅ ( − ) − ′
( − )
=( − )⋅ ′ + ⋅ − ( − )⋅ ′ −
= −′ − ⋅ ( − ) + ⋅ ( − ) + ⋅ ⋅ ( − )
⋅ ( − )= ′ − ( − ) − ( − ) 29
H a a
H H a a
H H
a a
H H a
aa
a H a
La a L a
L a
H a H a
cos sin
2 cos sin
cos sin
sin
cossin
cos sin
cos sin sin
cos
cos sin
za
r t
r t
xar r t t
yaza r
zat xa r ya
yar za ya za
xa a ya za
2 2
1 1 1
1
2 1 2 1
1
1
3 3
3
1
Step 3 Iteration: Substituting the PDGEs back into Eq. (18), therevised PIGEs can be obtained, which are then used to recalculatePDGEs. After iteration, the values of PIGEs and PDGEs are obtained.
Similarly, the PDGEs and PIGEs of the C-axis can be obtainedwith the same 3 steps with the patterns described in Fig. 9. Inaddition, this decoupling method can also be applied to the or-thogonal measuring patterns proposed in Ref. [26].
4.3. Experimental results
After compensating the positioning errors of 3 translationalaxes with the CNC, ballbar tests are conducted using the measur-ing patterns shown in Fig. 9. PIGEs and PDGEs are decoupled si-multaneously. PIGEs results calculated separately by the Tsutsu-mi's method [9] are compared against the proposed decouplingmethod in Table 2. The set-up errors of the ballbar can be analyzedusing the method presented in Refs. [27, 28].
PDGEs of the A- and C-axis defined by the proposed simulta-neous method are compared with Zhu's [19] separate measure-ment method in Fig. 11 and Fig. 12, respectively. Zhu models PDGEswithout considering PIGEs. The differences between the results ofthese 2 methods mainly come from the mutual influences be-tween the PIGEs and PDGEs. As shown in Eq. (27), when decou-pling the PDGEs, the influence of the PIGEs should be removed andvice versa. Usually the PDGEs are relatively smaller than PIGEs,and therefore, the mutual influence between them may have littleinfluence on decoupling the PIGEs but a large influence on PDGEs.Simultaneously decoupling these 2 groups of errors can obtainmore accurate results than decoupling them separately.
5. Compensation of volumetric errors
The proposed flowchart of the volumetric compensation algo-rithm for five-axis machine tools is shown in Fig. 1.
NC part programs are generated by a Computer Aided Manu-facturing (CAM) system, and the tool tip position and tool axisorientation vector ( = [ ]X Y Z I J KP T), i.e., the Cutter Location(CL) file, are represented in PCS. A postprocessor transforms themotions to drive commands (G-codes) in PCS using the inversekinematics model of the machine (Step 1). The CNC considers thejerk, acceleration and velocity profiles and generates drive com-mands ( θ θx y z, , , ,A C) at discrete time intervals (i.e., T¼1 ms) [29].However, the real tool position becomes ′ = [ ′ ′ ′ ′ ′ ′]X Y Z I J KP T
due to geometric errors of the machine which can be predicted by
Fig. 12. PDGEs of C-axis defined by two different methods.
S. Xiang, Y. Altintas / International Journal of Machine Tools & Manufacture 101 (2016) 65–78 73
the forward kinematics of the machine with the volumetric errormodel (Eq. (16) (Steps 2, 4, and 5 as presented in Section 3). Thepredicted tool position and orientation with geometric errors (P′)are compared against the ideal desired trajectory (P) to predict theerror components (Step 6) as:
⎡⎣ ⎤⎦Δ = − ′ − ′ − ′ − ′ − ′ − ′ ( )X X Y Y Z Z I I J J K K 30T
If the translational ( Δ Δ Δx y z, , ) and orientation (Δ Δ ΔI J K, , ) er-rors are larger than the set tolerances, they are added to the de-sired trajectory (P) (Step 7) as:
⎡⎣ ⎤⎦Δ= + ( )P P 31c
where Pc is the new tool position command vector with geometricerror compensation. Pcis passed through the inverse kinematics ofthe machine to generate drive commands with geometric errorcompensation components ( θ θx y z, , , ,c c c A Cc c
) (Step 8).However, this linear method does not compensate all the errors
because of the nonlinear kinematics of five-axis machine tools. Ifthe volumetric errors are large, the residual errors could still beout of tolerance, and a second iteration may be needed when themachine errors are large.
Fig. 13. Set up to reduce Abbe e
The compensation of geometric errors is carried out by map-ping the measured errors from MCS to PCS where the actualcompensation is made (Step 5), and the evaluation of inverse ki-nematics is done (Step 8).
5.1. Mapping of geometric errors from MCS to PCS
The MCS is set in the CNC by the manufacturer where the axiserrors are measured, and the PCS is set at a convenient location onthe part by the NC programmer. Since the error measurements areconducted in MCS while the actual compensation application is inPCS, the geometric error models should be mapped from MCS toPCS and then substituted into the volumetric error model.
As shown in Fig. 13, the laser interferometer is aligned near thecenterline of the ball screw to reduce the Abbe errors. First the axisis moved to the end of the stroke, then the laser interferometer isfixed on the slide or table close to the stroke endpoint and mea-surements are collected.
As shown in Fig. 14, the error measurements are conducted inMCS. The axis is moved from one end (Point A) to the other (PointB), and the errors are measured along the axis relative to the PointA. The errors captured by the laser interferometer are fitted to a
rror during measurements.
Fig. 14. Error mapping from Machine Coordinate System (MCS) to Part Coordinate System (PCS).
S. Xiang, Y. Altintas / International Journal of Machine Tools & Manufacture 101 (2016) 65–7874
third order polynomial as a function of axis position xm in MCS as:
= ( ) ( )error f x 32m
When the part is placed in the workspace of the machine, thepart error at its coordinate center OP is zero, although machine hasnon-zero error ( = ( )error f xOP MCS OP, ) relative to its coordinate sys-tem (MCS). The objective is to compensate the machine errorsalong the tool path on the part by considering the location of thetool. If the tool is at positionxpcsrelative to PCS whose center isxOP ,the error at the tool center point in MCS becomes:
= ( ) ← = + ( )e f x x x x 33x MCS m m OP pcs,
However, the actual error on the part at its coordinate system(PCS) is evaluated as:
( )= + − ( ) ( )e f x x f x 34x PCS OP pcs MCS OP MCS,
This mapping of error from MCS to PCS is carried out for alltranslational and angular motions along the tool path beforecompensating them correctly within the CNC.
After mapping all the geometric error models from MCS to PCS,they are fitted to polynomials and substituted into the volumetricerror model (Step 5), which leads to the translational (Δ Δ Δx y z, , )and orientation (Δ Δ ΔI J K, , ) errors (Eq. (30)), and the compensatedtrajectory Pc (Eq. (31)). Pcis passed through the inverse kinematicsof the machine to generate drive commands with geometric errorcompensation components ( θ θx y z, , , ,c c c A Cc c
) (Step 8).
5.2. Inverse kinematics for rotary and translational axes
The tool orientation ( = [ ]I J KO T) is achieved by command-ing angular positions ( θ θ,A C) to 2 rotary drives (A,C), which areevaluated from the following inverse kinematics model. The or-ientation of the tool tip relative to the workpiece = [ ]I J KO T is
Fig. 15. Experimental results of erro
expressed as a function of drive positions in PCS as:
⎡⎣⎢
⎤⎦⎥
⎡⎣⎢
⎤⎦⎥= ( )
( )g x y z a cO r
0, , , ,
0 35wt PCSot
where, = [ ]r 0 0 1otT. The orientation of the tool is only de-
termined by the rotary axes, and it is independent of translationalmovements. Hence Eq. (35) can be simplified as:
⎡⎣⎢
⎤⎦⎥
⎡⎣⎢
⎤⎦⎥= ⋅
( )θ θξ ξ^ ⋅ ^ ⋅e eO r
0 0 36otC C A A
Since the twists ( ξ ξ,A C) are defined in Eq. (3), the corre-sponding rotary drive positions (θ θ,A C) can be extracted using thePaden-Kahan sub-problems of the screw theory as explained in[3,22]. The derivation process is briefly demonstrated here. Eq.(36) can be transformed as:
⎡⎣⎢
⎤⎦⎥
⎡⎣⎢
⎤⎦⎥
( )= =( )
θ θξ ξ^ ⋅ ^ ⋅ −e er
c O0 0 37otA A C C
The vector c has the following relationship with 3 coefficientsα, β and γ, which can be calculated with = =u r v O,ot .
α β γω ω ω ω= + + ( × ) ( )c 38C A C A
⎫
⎬
⎪⎪⎪
⎭
⎪⎪⎪
( )( )
( )( )
α β
γα β αβ
ω ω ω ω
ω ω
ω ω ω ω
ω ω
ω ωω ω
=−
−=
−
−
= ±‖ ‖ − − −
‖ × ‖ ( )
u v v u
u
1,
1
2
39
CT
A AT
CT
CT
A
CT
A CT
AT
CT
A
CT
A
C A
2 2
2 2 2
2
Equations ⋅ =θξ̂ ⋅e u cA A and ⋅ =θξ̂ ⋅(− )e v cC C can be solved by usingPaden-Kahan sub-problem 1 shown as:
r compensation for linear axes.
Table 3Experimental parameters of axial ballbar measurements for rotary axes.
Axis name L (mm) H (mm) Axis travel Initial position of table-side ball in MCS (mm) Initial position of spindle-side ball in MCS (mm)
A-axis 0 49.770 30°-�60° (0,24.885,�456.237) (100,24.885,�456.237)C-axis 100 49.490 0°-360° (100,0,�449.849) (100,0,�349.849)
S. Xiang, Y. Altintas / International Journal of Machine Tools & Manufacture 101 (2016) 65–78 75
⎫⎬⎪⎭⎪( )( )θ
ω ω ω ω
ω
′ = − ⋅ ⋅ ′ = − ⋅ ⋅
= ′ × ′ ′ ′ ( )a
u u u c c c
u c u c
,
tan 2 , 40
A A A A
A A
T T
T T
⎫⎬⎪⎭⎪( )( )θ
ω ω ω ω
ω
′ = − ⋅ ⋅ ′ = − ⋅ ⋅
= − ′ × ′ ′ ′ ( )a
v v v c c c
v c v c
,
tan 2 , 41
C C C C
C C
T T
T T
If the tool tip position is defined by vector = [ ]X Y ZP T , itcorresponds to the coordinates of 3 translational and 2 rotarydrives as:
⎡⎣⎢
⎤⎦⎥
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
⎛
⎝
⎜⎜⎜⎜
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
⎞
⎠
⎟⎟⎟⎟
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
( ) ( ) ( )
( ) ( )( )
θ θ= = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
= ⋅ ⋅ ⋅ + ⋅ ⋅
θ θ
θ θ
ξ ξ ξ ξ ξ
ξ ξ
− ^ ⋅ ^ ⋅ ^ ⋅ ^ ⋅ ^ ⋅
− ^ ⋅ ^ ⋅
42
g x y z g e e e e e g
g e e
xy
zg
P1
, , , ,
0001
0 0
0
1 0 0 00 1 0 00 0 1 00 0 0 1
0 0 00 0 00 0 00 0 0 0
0
0001
wt A C bwC C A A X x Y y Z z
bt
bwC C A A
bt
1
1
Substituting the evaluated positions of the rotary drives (θ θ,A C)into Eq. (42), the positions ( x y z, , ) of 3 translational drives aresolved from the inverse kinematics model as:
⎡
⎣
⎢⎢⎢
⎤
⎦
⎥⎥⎥
⎡⎣⎢
⎤⎦⎥
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
( ) ( )( ) ( )= ⋅ ⋅ −
( )
θ θξ ξ^ ⋅ − ^ ⋅ −
xyz e e g gP
0
01
0
0001 43
bw btA A C C
In summary, the tool tip position and tool orientation( [ ]X Y Z I J K T) are transformed to drive positions
θ θ[ ]x y z A CTwith the inverse kinematic model of the machine.
6. Simulation and experimental results
Compensation experiments have been conducted on a five-axismachine (Quaser UX600) controlled by a Heidenhein iTNC530CNC. The 3 translational and 2 rotary axes have been compensatedby entering look-up tables into the CNC. As shown in Fig. 15, thepositioning errors of the X, Y and Z drives are reduced from( μm50, 25, 55 ) to ( μm5, 1, 2 ), respectively.
After completely compensating the positioning errors of3 translational axes (Fig. 15), 20 errors of 2 rotary axes have beenmeasured with ballbar tests (Fig. 9) using the parameters listed inTable 3. Here the axial measurements are analyzed as an example.
Substituting the 20 identified error parameters of the rotaryaxes (Fig. 11 and Fig. 12) into Eq.(17), the position of the table-sideball O1 in MCS can be obtained. When conducting ballbar tests, theRotation Tool Center Point function is turned on. Translational axesfollow the rotation automatically, and they follow the circularmotion so that the spindle-side ball O2 is kept stationary to thetable-side ball O1. For the axial measurement of the C-axis, theposition of the spindle-side ball O2 in MCS is decided by the X- andY-axis as:
⎛⎝⎜
⎞⎠⎟
⎛⎝⎜
⎞⎠⎟
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
= ⋅ ⋅ ⋅ ⋅ ⋅−
( )
θ θ θξ ξ ξ ξ ξ^ ⋅ ^ ⋅ ^ ⋅ ^ ⋅( − ) ^ ⋅e e e e eO
1000
349.8491 44
y x2
eYI
eY YD
eY eY X eXD
eX
The distance between the balls O1 and O2 is the ballbar lengthand the measured error can be evaluated as:
ρ = ‖ − ‖ − ( )lO O 451 2
where l is the reference length of the ballbar. However, the com-mercial CNC compensates only 3 linear (δxx, δyy, δzz) and 2 angularpositioning (εxa, εzc) errors but not all 41 errors, hence there is notmuch improvement in the accuracy of rotary drives (Fig. 16a–b). Toverify the proposed volumetric error model, the simulated ballbarresults (uncompensated) are compared with the real measure-ments (uncompensated) in Fig. 16. The simulated ballbar resultscan be calculated by Eqs. (17), (44) and (45). Fig. 17 shows thedifferences between the ballbar measurements and the simulatedresults for uncompensated rotary drives. The sources of errors inthe prediction are due to the identification inaccuracy of 20 geo-metric errors of the rotary axes, servo mis-match and the influenceof gravity on the ballbar measurements.
If we assume that the geometric errors are modeled andmeasured accurately, all 41 errors can be compensated to improvethe accuracy of the machine which is demonstrated in simulationsfor 2 rotary drives (Fig. 16c–d).
The full compensation has been demonstrated on a virtual five-axis CNC platform [30] by simultaneously considering all 41 errors.The compensated and uncompensated tool tip positions and toolaxis orientations along a helical tool path are shown in Fig. 18. Toclearly show the volumetric errors and the compensation results,some relatively large geometric errors were deliberately sub-stituted into the forward kinematics model. The total position er-ror is reduced from 5 mm to 10 μm after 2 iterations. The referencetool path shows the ideal tool path without considering any geo-metric error, while the simulated tool path presents the one withgeometric errors.
For real cases, all the component errors of the five axes weremeasured by the laser interferometer and the ballbar. Amongthese errors, the position dependent errors were mapped fromMCS to PCS, fitted by third order polynomials and then substitutedinto the volumetric error model (Section 5.1) embedded in theVirtual CNC system. The final contouring error of the tool pathmainly comes from the servo dynamics and the volumetric errors.A partial enlarged view is shown in Fig. 19, and it can be found thatbefore compensation, the total contouring error of the final toolpath is as large as 60 μm, but is reduced to 7 μm after volumetricerror compensation. The orientation errors are also fullycompensated.
7. Conclusion
This paper presents a systematic modeling, measurement andcompensation of both position-independent and position-depen-dent geometric errors of five-axis CNC machines. When the mea-sured errors of 2 rotary drives (i.e. with a ballbar) are decoupledinto 12 position-dependent and 8 position-independent geometric
Fig. 17. The error between the measured and predicted ballbar results for two rotary drives.
Fig. 16. Error compensation of rotary axes. (a)(b) ballbar measurements before and after compensating angular positioning errors with CNC, (c)(d) Simulated results beforeand after compensating all 41 errors.
S. Xiang, Y. Altintas / International Journal of Machine Tools & Manufacture 101 (2016) 65–7876
errors, they can be combined with the 21 geometric errors oftranslational axes directly measured with a laser interferometer.The 41 decoupled geometric errors are mapped into error twists,
which facilitates the calculation of not only the volumetric error ofthe machine at any position, but their compensation in the CNC byemploying the kinematic model of the five-axis machine. The
Fig. 19. Partial enlarged view before and after compensation.
Fig. 18. Helix tool path before and after compensation with large errors.
S. Xiang, Y. Altintas / International Journal of Machine Tools & Manufacture 101 (2016) 65–78 77
current commercial CNCs, which only allow for the compensationof error at each discrete position through look–up tables, can in-tegrate the proposed strategy to predict the geometric errors ofthe machine at any position along the tool path and compensatethem in real time. The proposed concept has been demonstratedin an open five-axis virtual CNC system built in-house.
The proposed research can be extended to include measure-ment uncertainties and the inclusion of thermal deformations ofthe machine.
Acknowledgments
This research was sponsored by the NSERC (NSERC IRC J260683)-Pratt & Whitney Canada Research Chair, the ChinaScholarship Council (No. 201406230151), and National NaturalScience Foundation projects of the People's Republic of China51275305, the Shanghai Municipality Military and Civilian Projectsfor the Construction of Industrial Development System [No.JMJH2013002], the Liaoning Province Scientific and Technical
S. Xiang, Y. Altintas / International Journal of Machine Tools & Manufacture 101 (2016) 65–7878
Innovation Key Special Projects [No. 201301001] and the ShanghaiMunicipality Minhang District Industry-University-Research Co-operation Projects [No. 2013MH111].
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