INTRODUCTION
In the metal cutting industry, form grinding process is still intensively
used for finishing and super finishing operations of threedimensional
profiles surfaces to impart them the desired shape and dimensional accuracy
and the required surface roughness. However, the CNC machine tools used
for this process need to be stopped from time to time to dress the grinding
wheel to expose new and free cutting abrasive grains and to eliminate
the deviation in the grinding wheel profile resulting from the effect
of the wheel nonuniform wear. Usually, the dressing step is achieved
by a diamond tip that is moved according to a secondary CNC programme.
As a result of that, the diameter of the grinding wheel changes and new
calculation of the tool path is required to compensate for the deviation
of the actual wheel size from the initial one.
The accuracy of the form grinding process and different methods of machining
error compensation have been investigated by a number of researchers in earlier
literature. Tso and Yang (1998) developed a mathematical
model of the thermal deformation and cutting forces by utilising the thermal
bending moment and specified grinding energy. They found that the thermal factor
and grinding forces contributed to negative compensation, whilst wheel wear
and deformation contributed to positive compensation. Choi
and Lee (2002) proposed a method for machining error compensation in the
process of cylindrical grinding. The method was based on mechanical modelling
of the work piece supporting system including the work piece itself. Tian
et al. (2008) presented a study dealing with the analysis of dimensional
error variation in CNC grinding. The study also discussed a new intelligent
error precompensation technique that was based on employing a touch probe for
part dimension measurement. Saleh et al. (2007)
developed an ultraprecision intelligent ELID fiveaxis grinding machine that
incorporated error compensation capability. Huang et al.
(2007) followed a compensation method that used the ground profile measured
from a Talysurf profilometer to modify the NC tool path for the next grinding
cycle by offsetting the residual profile error along the normal at the grinding
point. There are also other works that reported various error compensation techniques
and methodologies for improvement of machined surfaces form accuracy using online
approaches for different machining operations which also can be applied successfully
for grinding operations. Park et al. (1999) analysed
the shapegeneration process of a long slender shaft supported by correction
steadies. They obtained a simulation model of form accuracy of the traverse
grinding process from experimental data. Yang et al.
(1996) investigated the accuracy of a horizontal machining centre using
a strategy that was based on realtime error compensation. Quafi
et al. (2000) presented an approach that was established on software
compensation of the geometric, thermal and dynamic errors. This approach made
use of a multisensor monitoring system. Yuan and Ni (1998)
applied a realtime error compensation technique for 10 different machine
tools. They managed to improve the accuracy in ranges between 310 times as
they compensated for geometric and thermal errors. Tian et
al. (2002) was focused on the development of an errorcompensation system
that used the approach of self learning. Dirts and Gutman
(1986) developed a system for active error compensation during machining.
The system provided a support table on which the work piece could be mounted
and controlled in its position through a frictionfree spring type hinge. In
addition, the authors have also published a number of works dealing with grinding
efficiency and control (Adamczyk et al., 2003;
Muller and Wehmeyer, 1990; Wang, 1999).
In contrast with other works, this study elaborates on a methodology, which
is based on mathematical modelling, that can be used to minimise the profile
error resulting from changing the wheel diameter after dressing. It also highlights
the possibility of achieving highly accurate machining tasks that previously
have been performed on fiveaxis machine tools on fouraxis grinding machine
tools, which will result in cost effective production of those timeconsuming
finishing operations. This research project was conducted from Sep. 2006 to
2008.
MATERIALS AND METHODS
Case study and concept of error compensation: In order to compensate
for the effect of the deviation of the actual wheel size from the initial
wheel size on the accuracy of the ground profile, a special grinding procedure
that is based on providing positioning of the grinding wheel along a normal
to any specific point of the ground surface. For example, in a local industry,
the practice followed to perform grinding of the gauge (calibre XΠT90)
is shown in Fig. 1a, which is used for pilger tube inspection,
a fiveaxis CNC grinding machine tool (GG 52) is employed (Fig.
1b). The profile of this gauge is formed by ellipses that are located
on a cylindrical surface. The law according to which the principle axes
of the ellipses are varied is formed based on the optimal reduction approach
followed during the process of plastic deformation of the tube ingot.
For forming complex profiles, the CNC grinding machine is equipped with
a special profileforming kinematical system, which provides the necessary
generating motions required for both, the grinding wheel and work piece.
As shown in Fig. 1, the work piece (1) rotates about
axis C, while the grinding wheel (2) moves along axes X and Z and also
has the ability turn around axis A. This turning of the wheel about axis
A provides the opportunity to position its axis of rotation along any
perpendicular to any point of the machined profile.

Fig. 1: 
Scheme of form grinding of (a) gauge profile XΠT90
on (b) GG 52 fiveaxis CNC machine tool 

Fig. 2: 
Scheme of tool path for form generation 
Therefore, the mentioned above four axes (X, Z, A and C) are adequate
to form the desired profile of the tube gauge. However, the machine tool
possesses another fifth controlled axis (Y) that is used to mount the
grinding wheel perpendicular to the machined profile.
The function of the controlled Y coordinate can be explained using the
scheme of profile generation shown in Fig. 2. The scheme
shows a cross sectional view of the work piece (1) in an instantaneous
contact with the grinding wheel (2). The wheel, having an initial radius
R_{k}, is positioned in such a way that its center is located
at a point O_{kn} of an equidistant and at the same time it touches
the machined profile at point A. If during form generation the axis Y
is not used, the wheel centre will be shifted to a new position located
at point O_{k1}, as a result of wheel dressing. This shift will
takes place along axis X_{1} , which coincides with the centers
line O_{kn}O. Therefore, for the same work piece diameter, the
grinding wheel can be in contact with the machined surface at a different
point À_{1}. Obviously, this will result in an error in
the work piece profile. The controlled Y coordinate may allow correction
and compensation for the error resulting from the change in the wheel
diameter. The correction can be accomplished along the NN (axis X) normal
to the machined profile at the point A. Hence, the dressed wheel (with
the smaller diameter) and the work piece will have the same contact point
fixed before dressing.
This approach will eliminate the error received in the machined profile
and allow us to determine the positioning coordinates of the wheel centre
and at the same time produce a universal file for the cutter location
data (CLDATA) regardless of the actual value of the grinding wheel radius.
However, it is an obvious fact that the wheel radius needs to be always
taken into consideration when programming the tool paths required for
profile generation.
When creating a simulation program for form generation, there is an opportunity
to eliminate the effect of varying the grinding wheel radius I the control
files. Therefore, the given gauge profile (Fig. 1) can
be ground on a fouraxis machine tool with the help of a special model
that correlate the change in the wheel radius with machined profile error.
Mathematical model development for error compensation: In order
to establish a strategy for machining error compensation, we need to determine
the effect of the change in the grinding wheel radius on the profile accuracy.
Based on that, we will develop a program that will be able to implement
this error compensation strategy.
The machined profile error δ(φ), that specifies the deviation
of the actual wheel radius from the initially programmed one, is expressed
as a function of the polar angle φ and is found by using Eq.
1:
where, R_{kn} is nominal (design) radius of the grinding wheel,
R_{k} is actual wheel radius, ρ(φ) is profile curvature
radius and γ(φ) is angle of transmission of motion of the kinematic
couple: grinding wheel and work piece. The angle ε(φ) can be
determined by Eq. 2:
Because the gauge dimension is a standard size (found from tables) and
is given as discrete geometrical values, the calculations of the parameters
in Eq. 1 and 2 should be accomplished
using numerical methods relying on classical relations. In addition, the
calculations should be carried out for the most inconvenient (in terms
of machining feasibility) cross sections of the gauge profile, i.e., along
the profile depths where changes in the two equations parameters can take
on big ranges. Therefore, the profile radius along different depths can
be determined numerically by the following Equation:
The radius vector R_{e}(φ) of the equidistant of the wheel
centre can be determined from the following formula:
where, γ_{o}(φ) is angle of motion transmission of the
profile. Angles γ_{o}(φ) and γ(φ) can be determined
using the geometrical relations from Fig. 2. From the
velocity vector relations:
where, ω is the angular velocity of the work piece, now we have:
In order to determine the angle γ_{o}(φ), the function
R_{e}(φ) will be replaced by R(φ).
RESULTS AND DISCUSSION
Analysis and discussion: The mathematical model developed by the
Eq. 16 was used in an application program to analyse
the profile error resulting from the change in the grinding wheel size
after dressing. Figure 3 shows a screen print of the
program. With the help of the icon Load, the geometrical data of the machined
profile can be imported. The data can be stored in the form of a simple
text file. In addition, the input data of the grinding wheel are also
inserted in another window below that of the profile one. Activating the
icon Profile accuracy, calculation of the profile error can be achieved
as a function of the profile polar angle (curve 1 in Fig.
3). The input characteristics shown herein in Fig. 3
correspond to the specifications of the machined gauge XΠT90 and
a grinding wheel of nominal radius and usable range of 150 and 50 mm,
respectively. The simulation results show that the maximum error, when
an actual wheel radius of 125 mm is used (after dressing), is positive
(+0.11 mm). The maximum error occurs for small values of the profile polar
angle (less than 20°).
To minimise the profile error while maintaining a maximum usable range
of the grinding wheel (from R_{min} to R_{max}), it is
suggested implementing a new control approach for the grinding process.
The suggested approach considers dividing the grinding wheel usable range
(R_{max}R_{min}) into several segments. For each segment,
its mean radius is taken into consideration as the nominal radius of the
grinding wheel.

Fig. 3: 
Interface of application program for profile error
calculation 
For this radius value of each segment, a separate NC tool
path program (using the coordinates X, Z, A and C) is established.
Efficiency of the compensation strategy: The second part of the of the
maximum error calculation program is designed to estimate the efficiency of
the suggested compensation strategy. The program efficiency can be illustrated
through the following example. If, for example, we employ a grinding wheel of
175 mm radius a usable grinding range of 50 mm with five segments (with nominal
radii equal to 130, 140, 150, 160 and 170 mm), the maximum error can be reduced
from ±0.11 to ± 0.027 mm (Fig. 4a). If we further
increase the number of segments to 10, for example, the machined profile error
will further decrease to ±0.013 mm (Fig. 4b).
As it might be earlier noticed from the above discussion, the main drawback
of the suggested control approach and compensation strategy is the increased
number of the required tool path programs (one tool path program for each
segment) depending on the desired profile accuracy. However, as it has
been mentioned earlier, a fouraxis grinding machine can be successfully
employed instead of the fiveaxis one as only four programmed axes (X,
Z, A and C) are sufficient with the proposed compensation strategy.
To arrange the proposed error compensation technique, a fouraxis CNC
grinding machine, equipped with an online control system operated via
a PC, with a special simulation program should be employed (Fig.
5). The simulation program automatically calculates the required controlled
coordinates and transforms them into Gcodes used to control the machine
tool drives. For examples, when a number of five segments are chosen,
the first control program is established for the first mean radius (170
mm) of the first segment. When the grinding wheel radius reaches 150 mm
as a result of wheel dressing, a new control program that will consider
a new nominal radius of 160 mm will be activated. It is worth mentioning
that the simulation time is completely within the machining time and has
no influence on the total performance of the form grinding process.
The developed error compensation strategy and control program can be
also successfully used to overcome different concerns related machining
accuracy of similar profiles on CNC grinding machine tools.

Fig. 4: 
Simulation results of profile error calculation using
grinding wheel of 175 mm radius and 50 mm usable range for the case
of: (a) five segments and (b) ten segments 

Fig. 5: 
Scheme of fouraxis CNC grinding machine 
CONCLUSION
The current study has demonstrated an approach for machining error compensation
that was applied for an industrial case where the profile of a gauge is
currently machined by form grinding on a fiveaxis CNC grinding machine
tool. The approach which is based on a mathematical model considers minimising
the machined profile error resulting from dressing the grinding wheel
and changing its diameter. The simulation results of the gauge profile
accuracy show that the machining error can be dramatically decreased by
dividing the usable range of the grinding wheel into several segments.
For each segment having its own (mean) nominal radius, a separate NC tool
path program needs to be generated. Therefore, the major drawback of this
strategy is the increased number of the secondary tool paths required
for those nominal radii of the wheel usable range segments. However, this
approach can be completely justified when we consider the reduction in
the machining cost achieved as a result of grinding the gauge profile
on a fouraxis CNC machine tool instead of the fiveaxis one. The developed
error compensation strategy and control program can be also successfully
used to overcome different concerns related machining accuracy of similar
profiles on CNC grinding machine tools.
NOMENCLATURE
R_{k} 
= 
Initial radius of grinding wheel 
R_{k1} 
= 
Minimal radius of grinding wheel 
δ(φ) 
= 
Machined profile error 
R_{kn} 
= 
Nominal radius of grinding wheel 
ρ(φ) 
= 
Profile curvature radius 
γ(φ) 
= 
Angle of transmission of motion of the kinematic couple 
γ_{o}(φ) 
= 
Angle of motion transmission of profile 