Original Article
Conversion method study from
cutter-location points to nonuniform
rational B-spline toolpath NC file
for high-speed machining
Proc IMechE Part C:
J Mechanical Engineering Science
0(0) 1–12
! IMechE 2018
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DOI: 10.1177/0954406218762018
journals.sagepub.com/home/pic
Jian-wei Ma , De-ning Song, Zhen-yuan Jia, Guo-qing Hu and
Wei-wei Su
Abstract
Parts with complex curved surfaces are widely applied and the demands for the machining quality and the machining
efficiency of such complex parts become increasingly higher. In order to realize high-speed and high-quality machining,
the nonuniform rational B-spline interpolator is widely researched and demonstrated to be superior to the conventional
linear or circular interpolators. However, the nonuniform rational B-spline toolpath NC files cannot be directly generated from the computer-aided design (CAD) models by using commercial computer-aided manufacturing (CAM) software. To deal with this problem, a conversion method from the cutter-location points to the nonuniform rational Bspline toolpath numerical control (NC) file is presented. To avoid the bad curve-fitting effect at sharp corners of the
toolpath and to meanwhile reduce the computational burden, the cutter-location pre-processing method, consisting data
segmentation and data simplification, is provided first. Then, the least-square method is employed to fit the cutter
locations to the nonuniform rational B-spline curves, and an iterative fitting approach is proposed for linear/nonuniform
rational B-spline hybrid toolpath generation. Finally, a user interface is designed for displaying the fitting results and
outputting the NC file with nonuniform rational B-spline toolpaths. By using this method, nonuniform rational B-spline
and linear toolpaths hybrid interpolation NC program can be generated for the high-speed machining of complex curved
surface parts with the utilization of the nonuniform rational B-spline interpolator. The experimental results demonstrate
the feasibility and the advantages of the presented method.
Keywords
Nonuniform rational B-spline toolpath, parametric curve interpolation, curve fitting, high-speed machining, complex
curved surface, cutter-location points
Date received: 17 August 2017; accepted: 5 February 2018
Introduction
Parts with complex curved surfaces are widely applied
in the equipment of fields, such as aerospace, automation, and die/molds.1,2 With the development of the
high-end equipment used in these fields, demands for
the machining quality and the machining efficiency of
such complex parts become increasingly higher.
Normally, the curved toolpath of these complex
parts is expressed as continuous short-line or circular
segments, thus using the linear/circular interpolator to
machine these parts.3,4 However, there exists inevitable drawbacks for the linear/circular interpolator.
First, the rapid acceleration/deceleration at the transition points between two adjacent segments must
be executed due to the first-order discontinuity at
these points.5–7 Then, the approximation of the
desired free-form curved toolpath by linear/circular
segments will result in a large file size.7 Both of the
above factors lead to the degradation of machining
quality and efficiency.
To address the shortcomings of the traditional
linear/circular interpolator, the parametric curve
interpolation methods are widely researched and
Key Laboratory for Precision and Non-traditional Machining
Technology of the Ministry of Education, School of Mechanical
Engineering, Dalian University of Technology, Dalian, China
Corresponding author:
Jian-wei Ma, Key Laboratory for Precision and Non-traditional
Machining Technology of the Ministry of Education, School of
Mechanical Engineering, Dalian University of Technology, Dalian
116024, China.
Email: mjw2011@dlut.edu.cn
2
developed to generate the smooth curved toolpath directly without the rapid acceleration/deceleration for
the high-speed and high-precision machining.5–9
Lots of modern computer numerical control (CNC)
systems, such as Siemens (Germany), FANUC
(Japan), and Guangyang numerical control (GNC)
(China) can provide cubic spline or nonuniform
rational B-spline (NURBS) interpolation function.
Taking the GNC numerical control system as an
instance, the G-code of NURBS interpolation is G6.2.
When utilizing these curved interpolation functions,
the parametric curve NC file with the corresponding
requested format must be generated in advance.
However, most existing commercial CAM applications,
such as UG/NX and solid CAM, do not have the capability to generate the curved interpolation toolpath and
can only output the routine linear/circular interpolation
NC files or the cutter location points, which makes it
also difficult to realize the machining with the parametric curve interpolator, although the CNC system provides this function. As a result, research on the
generation method of the parametric curved interpolation toolpath becomes significant for filling up the
vacancy between the CAD model of the complex parts
and the curve interpolator of the CNC system, thus
realizing high-speed and high-quality machining.
To generate the curved toolpath NC file for the
parametric interpolator, Ma and Kruth10 provided
an NURBS curve-fitting method. Similarly, Yu11
also fitted the NURBS curve based on arc-centripetal
parameterization for the parametric toolpath generation. Recently, Yang et al.12 presented a curve-fitting
and optimal interpolation method for CNC machining under the confined error by using the quadratic
B-splines. However, these methods fitted the whole
free-form toolpath as a single parametric curve,
which could hardly keep the sound-fitting precision
at the large-curvature regions. Although Lin et al.13
and Wu et al.14 split the whole toolpath to several
fitting sections, and they utilized all of the cutter locations to fit the NURBS curve at each section, which
might make the algorithms time-consuming or obtain
too many control points of the fitted NURBS curve.
In addition, Lin15 and Deng and Lin16 proposed an
adaptive data-fitting approach by the progressiveiterative approximation for the spline curve fitting.
Garcia-Capulin et al.17 presented a hierarchical genetic algorithm approach for the curve fitting with
B-splines. These methods all considered the whole
path as one curve and fitted the data to splines using
the time-consuming iterations. It can be seen from their
results that the fitting errors are large unless the iteration for extremely much times. Furthermore,
although there are also some other curve-fitting methods,18,19 they are proposed for the mathematical
computations but not for the NURBS-toolpath generation where extra issues, such as the NC-file size and
the toolpath combination of NURBS and linesegment, should be considered. Based on the above
Proc IMechE Part C: J Mechanical Engineering Science 0(0)
discussion, more appropriate NURBS-toolpath generation method is also in urgent need.
As a result, this study takes both fitting precision
and computational burden into consideration and presents a NURBS-toolpath NC-file generation method
from the original cutter-location points for realization
of high-speed machining with the NURBS interpolator.
First, the cutter location points are pre-processed by
data segmentation and simplification, so as to ensure
the sound-fitting precision at sharp corners and the low
computational burden. After that, the simplified data
are utilized to fit the NURBS curves at every sections of
the whole toolpath according to the least squared
method. Then, the fitting precision is judged and the
iterative fitting approach is given. When the fitting precision also cannot be satisfied after the iterative fitting,
the linear interpolation toolpath is exported, otherwise,
the NURBS toolpath is outputted. Finally, the user
interface is designed based on the MATLAB/graphical
user interface (GUI) so as to generate the NURBStoolpath NC file for the high-speed machining of the
complex curved surface parts. Academic contribution
of this approach concentrates on the combination of
data segmentation, data simplification, and leastsquare curve-fitting algorithms for generation of the
piecewise NURBS and line-segment hybrid toolpath,
thus realizing the high-efficiency machining.
Pre-processing of cutter location points
In this section, the cutter-location points obtained
from the CAD model of the complex curved surface
parts are pre-processed in order to ensure a better
performance of the further NURBS curve fitting.
The pre-processing contains two blocks: the data segmentation and the data simplification. During data
segmentation, the integral curved toolpath is split
into several segments at the sharp-corner positions.
Afterward, the cutter-location data at each segment
are simplified into key points by using the data-simplification block. Both of the above two procedures
are conductive to the fitting precision and the computational efficiency of the NURBS curve fitting.
In order to avoid the bad-fitting effect at the largecurvature regions, the cutter-location data are first segmented at the sharp corners. Denote the original desired
cutter-location sequence as {Qri}, i ¼ 1,. . ., nq, where nq
is the total amount of the cutter-location points. Given
the angle condition for the segmentation as q, scan the
cutter-location sequence {Qri} from i ¼ 2 to i ¼ nq 1
and calculate the included angle between the vector
Qri 1Qri and the vector QriQri þ 1 by
!
ðQri Qri1 Þ Qriþ1 Qri
¼ arccos Qri Qri1 Qriþ1 Qri ð1Þ
As shown in Figure 1, compare with q, and if
> q is satisfied, denote the current cutter location
Ma et al.
3
Figure 1. Schematic diagram of data segmentation.
Figure 3. Illustration diagram of data pre-processing.
Figure 2. Schematic diagram of data simplification.
Qri as a segmentation point, thus realizing the data
segmentation. Denote the total number of the data
sections after the toolpath splitting as nc and the
amount of the cutter locations at ith section as ki,
thus the cutter-location sequence of ith section can
be expressed as {Qci,j}, i ¼ 1,. . ., nc, j ¼ 1,. . ., ki.
Considering that the overcrowding cutter locations
is adverse to the fitting precision and the numerical
efficiency, the data simplification is conducted after
the data segmentation. Given the precision condition
for the retrenching data as eq, scan the cutter-location
sequence {Qci,j} from i ¼ 1 to i ¼ nc and j ¼ 2 to
j ¼ ki 1 with the calculation of the distance from
the point Qci,j to the line Qci,j 1Qci,j þ 1 by
)
Qci,jþ1 Qci,j1 Qci,jþ1 Qci,j
Qci,jþ1 Qci,j1
d ¼ ( )
Qci,jþ1 Qci,j1 Qci,jþ1 Qci,j
Qci,jþ1 Qci,j1 Qci,jþ1 Qci,j
ð2Þ
As illustrated in Figure 2, judge the relation
between d and eq. If d > eq is not satisfied, then get
rid of the Qci,j point, otherwise, leave this point and
retain it. Repeat above scanning until the condition
d > eq is satisfied for every reserved cutter locations.
Two notes should be clarified for the data simplification approach as follows. First, the d > eq condition is
judged iteratively point-by-point instead of one-pass
judging after calculating the distances of all points, so
as to retain the key points that can reflect the main
shape of the toolpath. Taking a standard circle as an
instance, the discretized points will have the same d
values computed by equation (2), and in this case,
either no point or all points should be removed if
they are judged in the meanwhile. On the contrary,
when using the proposed iterative judging and removing method by scanning {Qci,j} from i ¼ 1 to i ¼ nc and
j ¼ 2 to j ¼ ki 1, the d value for one point will
be larger once its adjacent point is removed, so that
the key points can be kept to reflect the shape of the
toolpath. Second, the data simplification is started
from j ¼ 2 and end by j ¼ ki 1, and this is because
if the first or the final point is removed, the discontinuous between two adjacent data sections will occur.
As the ending point of ith section is the starting point
of (i þ 1)th section, keep the first and the final points
of each section from being removed can thus ensure
the continuity of the whole toolpath.
Denote the amount of reserved cutter locations
after the data simplification as ti, thus the cutter-location sequence after simplification can be expressed as
{Qi,j}, i ¼ 1,. . ., nc, j ¼ 1,. . ., ti. By conducting above
procedure, the remained cutter locations are all key
points that can reflect the overall shape of the desired
curved toolpath without the redundant points, and
this is conductive to the precise curve fitting with a
lower computational burden.
In order to further explain the cutter-location preprocessing procedure in detail, the data segmentation
and the data simplification are simultaneously plotted
in the same illustration figure (see Figure 3). As shown
in Figure 3 and according to the above definition, the
original cutter-location points are Qr1 to Qr7. After
the data segmentation, at Qr4 where the included
angle between the vector Qr3Qr4 and the vector
Qr4Qr5 is larger than q in this illustration, the
cutter-location points are divided into two segments,
i.e., segment 1 from Qc1,1 to Qc1,4 and segment 2 from
Qc2,1 to Qc2,3. The data simplification is employed
within each segment so as to retain the key points,
instead of all tedious points, to fit the NURBS
curves at every segments. In this illustration, the
point Qc1,3 in the first segment and the point Qc2,2
in the second segment are removed, according to the
distance principle presented above. After the data
4
Proc IMechE Part C: J Mechanical Engineering Science 0(0)
simplification, the final points used for the curve fitting become Q1,1 to Q1,3 in the first segment and Q2,1
to Q2,2 in the second segment.
Conversion from cutter location points
to NURBS-toolpath NC file
In this section, the conversion method from the cutterlocation points to the NURBS-toolpath NC file is presented. First, the curve fitting from the simplified cutter
locations to the NURBS curve is introduced. Then, the
iterative fitting method for the toolpath generation
is given. Finally, the user interface for the NC-file creation is designed.
Fitting of simplified cutter-location data by
NURBS curve
8
m
>
>
>
j
¼
int
i
<
np
>
m
m
>
>
þ
j
u
j
u j
¼
1
i
þ
i
u
: pþi
j1
np
np
ð6Þ
In which, int(x) stands for the maximum integer no
more than x.
Then, the basic functions {Ni,p(u)} are obtained
using the Cox–deBoor recursion formula as
8
>
>
< Ni,1 ðuÞ ¼
In general, the parametric equation of the p-order
NURBS curve C(u) can be expressed as
Pn
Ni,p ðuÞ!i Pi
CðuÞ ¼ Pi¼1
,
n
i¼1 Ni,p ðuÞ!i
Denote the knot vector as U ¼ u1 , upþ1 ,
upþ2 , . . ., un , unþ1 , . . ., um g, where u1 , upþ1 equal to 0
and unþ1 , . . ., um equal to 1. The other knot values are
calculated as Piegl and Tiller20
1, ui 4u 5 uiþ1
0, others
uiþpþ1 u
u
ui
>
>
: Ni,p ðuÞ ¼
Ni,p1 ðuÞ þ
Niþ1,p1 ðuÞ
uiþp ui
uiþpþ1 uiþ1
ð7Þ
u1 4u4unþpþ1
ð3Þ
In which, {Pi} is the set of control points, {!i} is the
set of weights, {Ni,p(u)} is the set of p-order B-spline
basic functions defined on the nonuniform knot
vector U, and U ¼ {u1, u2,. . ., un þ p þ 1}. To fit a
NURBS curve, the parameters in equation (3) in
terms of U, {Ni,p(u)}, {!i}, and {Pi} should all be
determined. Here, the weights {!i} are set as constant
1.0 for every control points so as to avoid the nonlinear calculation, thus saving the computational
time. In addition, the global approximation method
is utilized to fit a curved toolpath as smooth as possible. The determine procedure of knot vector U,
B-spline basic functions {Ni,p(u)}, and control points
{Pi} is presented below.
Given the initial order value pmin and the initial
amount of control points nmin, the spline parameter
corresponding to each simplified cutter location is first
calculated using the endocentric parameterization
method. For the cutter-location sequence {Qi,j},
i ¼ 1,. . ., nc, j ¼ 1,. . ., ti, the spline parameter u j corresponding to each point is calculated as Piegl and
Tiller20
8
< u 1 ¼ 0, u ti ¼ 1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Qi,j Qi,j1 k
: u j ¼ u j1 þ Pti kpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
,
kQi,k Qi,k1 k
k¼2
In which, ‘‘0/0 ¼ 0’’ is stipulated and i ¼ 1, 2,. . ., n.
At last, to obtain the control points {Pi}, i ¼ 1, 2,. . .,
n, let the initial and the final control points equal to the
first and the last cutter locations of the corresponding
data section, respectively, and the other inner control
points are calculated using the least square method. In
detail, they are given as Piegl and Tiller20
8
P1 ¼ Qi,1 , Pn ¼ Qi,ti
>
>
>
>
< 0 P2 1
B .. C T 1
>
@ . A¼ N N R
>
>
>
:
Pn1
ð8Þ
Note that for ith section of the simplified data Qi,j
(j ¼ 1,2,. . .,ti), the number of control points of the
fitted NURBS is n who is adjusted adaptively during
the following iterative fitting procedure, and n is
always smaller than the ti which is the number of
the points of ith section data Qi,j (j ¼ 1, 2,. . ., ti) so
as to keep the advantage of NURBS interpolation in
reducing NC-file size. The control points {Pi}, i ¼ 1,
2,. . ., n, are calculated using equation (8), in which the
matrix N and R are computed as
0
j ¼ 2, . . ., ti 1
1
N2,p ðu 2 Þ
Nn1,p ðu 2 Þ
B
C
..
..
..
N¼@
A
.
.
.
N2,p u ti 1 Nn1,p u ti 1
ð9Þ
ð4Þ
Then, to obtain the knot vector of the fitted curve,
let the spline order p equals to pmin and the amount of
control points n as nmin, thus the amount of the knots
m in the knot vector U can be computed as
m¼nþpþ1
ð5Þ
and
1
N2,p ðu 2 Þr2 þ þ N2,p u ti 1 rti 1
C
B
..
C
R¼B
A
@
.
Nn1,p ðu 2 Þr2 þ þ Nn1,p uti 1 rti 1
0
ð10Þ
Ma et al.
5
In equation (10), rj, j ¼ 2,. . ., ti 1 is obtained as
rj ¼ Qi,j N1,p u j Qi,1 Nn,p u j Qi,ti , j ¼ 2, . . ., ti 1
ð11Þ
By using the above method, the NURBS curve can
be fitted at each data section, then, the fitting precision should be judged before generation of the
NURBS-toolpath NC file. To deal with this problem,
the iterative fitting approach is provided in the following subsection.
Iterative fitting of NURBS curve for
toolpath generation
As the curve fitting method instead of the curve interpolation method in Piegl and Tiller20 is utilized here to
keep the smoothness of the obtained NURBS toolpath, the fitting errors must be constrained. To satisfy
the precision condition during the curve fitting, the
original cutter locations before data simplification
{Qci,j} instead of those after data simplification are
utilized for the calculation of the fitting errors. In
addition to {Qci,j} (j ¼ 1, 2,. . ., ki), the middle points
Qcm,i,j (j ¼ 1, 2,. . ., ki 1) between each two adjacent
cutter locations Qci,j and Qci,j þ 1 (j ¼ 1, 2,. . ., ki 1)
are also utilized to compute the fitting errors, in order
that the chord error can be confined within the given
tolerance thus obtaining enough small fitting error.
For a fitted NURBS curve C(u), the distance from
the cutter location Qci,j or the middle point Qcm,i,j
to the curve is first computed. To reduce the computational burden, the distance is obtained by the adaptive segmentation. Figure 4(a) shows the schematic of
the data points and the fitted curve, while Figure 4(b)
Figure 4. Schematic diagram for calculation of distance from
desired cutter location to fitted curve. (a) Geometry of distances. (b) Calculation procedure of distance.
illustrates the fitting-error rapid computation procedure. As shown in Figure 4(a), although the simplified
points Qi,j are utilized for curve fitting, the original
cutter locations Qci,j, and the middle points Qcm,i,j of
the original cutter locations are used for the fittingerror calculation so as to ensure the fitting precision.
As shown in Figure 4(b), first, the fitted curve C(u) is
divided into nd portions, and the distance between
each equipartition point and the cutter location Qci,j
or the middle point Qcm,i,j is calculated. Then, the
closest point from the curve to the cutter location or
the middle point is recorded, and the curve between
the former and latter adjacent points of this closest
point on the curve is further divided into nd portions.
Similarly, the distance between each equipartition
point and the cutter location Qci,j or the middle
point Qcm,i,j is calculated again. Repeat above procedure until the difference between the former and latter
minimum distances is less than the given precision
condition. Thus, the distance from the desired cutter
location Qci,j to the fitted curve, denoted as di,j, and
the distance from the middle point Qcm,i,j to the fitted
curve, denoted as dm,i,j, can be obtained.
Given the maximum fitting-error tolerance ec, if
equation (12) is satisfied, output the NURBS toolpath; else, add the amount of control points by letting
n ¼ n þ 1 and conduct the curve fitting once again. In
detail, first let n ¼ n þ 1 and then calculate the knot
vector {Ui}, i ¼ 1, 2,. . ., n þ p þ 1, using equation (6),
and after that, compute all of the control points {Pi}
(i ¼ 1, 2,. . ., n) again using equation (8).
di,j 5 ec , j ¼ 1, . . ., ki
dm,i,j 5 ec , j ¼ 1, . . ., ki 1
ð12Þ
Take the maximum amount of the control points
nmax as ti 3 because if the number of control
points is too large, the advantage of the NURBS
interpolation in reducing the file size will disappear.
If equation (12) cannot be satisfied even when
n ¼ nmax, add the spline order value by letting
p ¼ p þ 1 and fit the curve again. Given the maximum
order pmax, if equation (12) also cannot be satisfied
when p ¼ pmax, indicating that the appropriate curve
fitting can hardly be realized, output the linear interpolation toolpath. By using above iterative fitting
method at every cutter-location sections, the integral
toolpath can be generated, and the procedure is
further summarized in Figure 5 for concise, and the
variable parameters illustrated in Figure 5 are summarized in Table 1.
After the above iterative fitting, the NURBS toolpath can be generated, although there exists a few
linear interpolation toolpaths at areas where the fitting precision cannot be satisfied. By using NURBS
toolpath instead of the linear toolpath for machining
of the complex curved surface parts, the more smooth
motion can be obtained, which is beneficial to the
high-speed machining.
6
Proc IMechE Part C: J Mechanical Engineering Science 0(0)
Design of user interface for NURBS-toolpath
NC-file generation
Figure 5. Flow chart of presented toolpath-generation
method for high-speed machining.
To realize the NC machining with NURBS interpolator, the NURBS-toolpath NC file with a required
format of the corresponding CNC system should be
generated in advance. Therefore, a user interface for
NURBS-toolpath NC-file generation is designed here
based on MATLAB/GUI for the convenient utilization. The designed interface is illustrated in Figure 6.
In the ‘‘User Defination’’ panel, the angle condition
for data segmentation (q), the precision conditions
for data simplification (eq) and NURBS fitting (ec),
the minimum and maximum order values (pmin and
pmax), the minimum amount of control points (nmin),
and the feedrate value for machining can be defined
by users. By pushing the ‘‘Load CL File’’ button, a
pop-up dialog for selection of the cutter-location file
will appear and the cutter locations in the selected file
will then be plotted in the figure below this button.
After that, the iterative NURBS-fitting procedure will
be conducted when pressing the ‘‘NURBS Fitting’’
button, and the fitting result along with the original
desired cutter locations will be illustrated in the corresponding figure. In addition, the fitting error, i.e. the
distances from the desired cutter locations to the fitted
curves, will be calculated and shown when pushing the
‘‘Display Fitting Error’’ button, in order that the fitting result can be seen by the users. Finally, the NC
file for high-speed machining with the NURBS interpolator can be generated and displayed in the text box
when clicking the ‘‘Output NC File’’ button. By utilization of this user interface, it becomes extremely
convenient to converse the cutter-location points to
the NURBS-toolpath NC file, so that the gap between
the complex CAD model and the NURBS interpolator can be filled up.
Experimental results
Table 1. Summary of the used parameters.
Parameter
Definition
Qr
Qc
nc
Q
C(u)
d
dm
ec
n
nmin
nmax
p
pmin
pmax
Original cutter location point
Cutter location after data segmentation
Total amount of segmented sections
Cutter location after data simplification
Fitted NURBS curve
Distance from Qc to C(u)
Distance from Qcm to C(u)
Maximum fitting-error limitation
Number of control points during curve fitting
Minimum control points during curve fitting
Maximum control points during curve fitting
Order of the fitted curve
Minimum order of the fitted curve
Maximum order of the fitted curve
NURBS: nonuniform rational B-spline.
To verify the feasibility of the presented methods,
experiments are conducted and the results are illustrated in this section. A VGM 210 machine tool
with the CNC system of GNC61 by GONA
Corporation shown in Figure 7 is utilized here, and
the NURBS-interpolation NC-file format of the
GNC61 system is provided below.
G6.2 P_ (Spline order) K_ (Knot vector) X_ Y_ Z_
(Control points) R_ (Weights) F_ (Feedrate).
KXYZRF
KXYZRF
KXYZRF
KXYZRF
K1:
K1:
...
Ma et al.
7
Figure 6. User interface for NURBS-toolpath NC-file generation.
Figure 8. Butterfly toolpath used for experiments.
Figure 7. Machine tool used for experimental tests.
A butterfly-shape curved toolpath shown in
Figure 8 is employed for the experiments. The CAM
software UG/NX 8.0 is used for generation of the
original cutter-location file. Set the angle condition
for the data segmentation q as 35 , the precision conditions for data simplification and NURBS fitting as
0.05 mm and 0.06 mm, respectively, and the pmin, pmax,
and nmin as 2, 3, and 3, respectively. It should be
noticed that the maximum fitting error condition
can be set arbitrarily according to the machining tolerance, and if the curve fitting cannot satisfy the set
condition, linear interpolation toolpath, instead of
NURBS toolpath, will be generated in order to
ensure the precision. When the error condition is set
relative larger, there will be more NURBS toolpaths
than linear toolpaths, which is beneficial to the efficiency, and on the contrary if the error is set as an
extremely small value, more linear toolpaths will be
generated. By utilizing the cutter-location pre-processing method presented in Pre-processing of cutter
location points section and the iterative fitting
method presented in Conversion from cutter location
points to NURBS-toolpath NC file section, the
obtained linear/NURBS toolpath is illustrated in
Figure 9, where the fitting result is also compared
with the desired cutter locations obtained from the
CAM software. In Figure 9, the fitted toolpaths
with different colors represent different sections after
the data segmentation. As can be seen, a wellfitting effect is got. The real fitting errors between
the desired cutter locations and the fitted toolpaths
are further calculated and illustrated in Figure 10. It
can be seen that in some segments, the fitting error is
large, and this is because the curvature variations of
these segments are relatively large, but even though,
8
Proc IMechE Part C: J Mechanical Engineering Science 0(0)
Figure 9. Comparison of desired cutter locations and fitted toolpaths.
Figure 10. Fitting error between desired cutter locations and
fitted toolpaths when error tolerance is 0.06 mm.
the errors can be effectively confined within the maximum fitting error tolerance set here, i.e., 0.06 mm. In
addition, it should be noted that the fitting error tolerance can be set artificially here. When the tolerance
is changed to 0.03 mm instead of 0.06 mm, the final
fitting errors become those shown in Figure 11, where
it indicates that the presented method can also confine
the fitting error within the set limitation.
The above results are obtained by programing the
proposed method with Cþþ on an i7 core Intel CUP
computer with 3.4 GHz frequency. The total computation time of the two tests are recorded in the program
as about 430 ms, which indicates that the algorithm is
computational efficiency and the computation time can
be neglected. In order to verify the advantages of the
presented method, the existing method proposed in Lin
et al.13 that is utilized for NURBS toolpath fitting is
also employed here as a comparison. Note that
although that existing method contains data segmentation as well, it does not simplify the data points, and all
of the cutter locations are utilized for the curve fitting.
The computation time of the method in Lin et al.13 is
Figure 11. Fitting error between desired cutter locations and
fitted toolpaths when error tolerance is 0.03 mm.
Figure 12. Fitting result of conventional method.
about 480 ms, and the fitting result is shown in
Figure 12. It is seen that the computational time is
longer than the proposed method, and this is because
the conventional method utilizes all of the cutter locations to fit curves, while the proposed method
Ma et al.
9
Table 2. Comparison results of the linear-toolpath and linear/NURBS-toolpath NC files.
NC file
Total number of rows
File size
Pure linear toolpaths
Linear/NURBS hybrid toolpaths
G55 G90
N0090 G01 X0.0 Y0.0 F800
N0100 X.11 Y-.103
N0110 X.18 Y-.231
N0120 X.217 Y-.377
N0130 X.246 Y-.524
N0140 X.304 Y-.815
N0150 X.348 Y-.958
N0160 X.43 Y-1.081
N0170 X.564 Y-1.146
N0180 X.713 Y-1.15
...
N2850 X-.216 Y-.377
N2860 X-.176 Y-.232
N2870 X-.112 Y-.093
N2880 X0.0 Y0.0
N2890 M02
282
6.04 kB
G55 G90 F800
G6.2 P3 K0 X0 Y0
K0 X0.313 Y-0.413
K0 X0.348 Y-0.958
K1
K1
K1
G1 X0.564 Y-1.146
G6.2 P3 K0 X0.564 Y-1.146
K0 X0.646 Y-1.165
K0 X1.044 Y-1.084
K0.00947 X1.633 Y-0.781
...
K1
K1
K1
M02
172
3.07 kB
NURBS: nonuniform rational B-spline.
Figure 13. Comparison of machining times of pure linear toolpath and linear/NURBS toolpath for butterfly toolpath. (a) Linear
toolpath. (b) Linear/NURBS hybrid toolpath.
merely uses the key points after data simplification.
Additionally, it can be seen from Figure 12 that the
fitting errors in zone I and zone II are extremely large,
and the reasons are analyzed below. At zone I, the
conventional method employing all the cutter-location
points to fit a curve, while the proposed method utilizes
simplified points merely. As the curvature variation is
relatively large at zone I, the intensive data can hardly
result in a well fitting result. On the contrary, a better
effect can be got when using the key points to fit a
curve. Additionally at zone II, both of the conventional
and the proposed methods cannot realize a well fitting,
however, the conventional method output the fitted
curve as the toolpath, while the proposed method
output the original linear toolpath to keep a well
machining precision. That is to say, the proposed
method generates linear and NURBS hybrid toolpath,
so as to improve the machining efficiency under the
premise of machining precision being ensured.
To evaluate the machining efficiency of the presented method, the real application is conducted.
The generated hybrid toolpath NC file using the proposed method is compared with the NC file with the
pure linear interpolation toolpaths in Table 2. As can
be seen, the file size of the NC code is reduced from
the original 6.04 kB to the regenerated 3.07 kB when
10
using the NURBS toolpaths instead of the linear toolpaths, and this is beneficial to the file transition and
storage. By using the generated linear/NURBS-toolpath NC file and the traditional linear interpolation
toolpath for machining, the machining times are recorded by the GNC61 system, and they are shown
in Figure 13. When machining with pure lineartoolpath NC file, the machining time is about 10.0 s,
Figure 14. Star toolpath used for experiments.
Proc IMechE Part C: J Mechanical Engineering Science 0(0)
while the time becomes 8.0 s when using the
linear/NURBS-toolpath NC file. Thus, the machining
time is reduced by 20%, and this is because the firstorder discontinuous points are dramatically reduced
when using the NURBS toolpath instead of the shortline segment toolpath, so that the rapid deceleration/
acceleration processes are decreased. Therefore, it is
more appropriate for the NURBS toolpath than the
traditional linear toolpath to realize the high-speed
machining.
Another case with large-scale star toolpath shown
in Figure 14 is then conducted to further verify the
feasibility of the presented method. Also, the conventional method in Lin et al.13 is taken as a comparison.
The maximum fitting error limitation is set as an
extremely small value of 0.005 mm, and other constraints are same as those in the first case. The fitting
result of the two methods are compared in Figure 15,
and the fitting errors are compared in Figure 16. It
can be seen that in zone I, the proposed method generates linear toolpath because the maximum fitting
error is larger than the set limitation in this zone,
while the conventional method generates curved toolpath, and cannot ensure a predefined error constraint.
Furthermore, the computation time of the proposed
algorithm is about 85 ms, and that of the conventional
Figure 15. Comparison of fitting results of different method. (a) Proposed method. (b) Conventional method.
Ma et al.
11
Figure 16. Comparison of fitting errors of different method. (a) Proposed method. (b) Conventional method.
Figure 17. Comparison of machining times of pure linear toolpath and linear/NURBS toolpath for star toolpath. (a) Linear toolpath.
(b) Linear/NURBS hybrid toolpath.
method is about 94 ms, which is similar to the results
of the first case.
In addition, the hybrid NC file generated by the
proposed method consists 143 lines with a file size
of 3.05 kB, while the original linear toolpath NC
file consists 668 lines with a file size of 11.6 kB. As
for the machining time, the linear toolpath takes
about 29 s, while machining with the proposed
hybrid toolpath, the machining time is reduced to
22 s, as shown in Figure 17. Again, the advantages
of the proposed method are demonstrated.
Conclusion
In order to realize high-speed and high-quality
machining, the parametric curve interpolation methods are widely researched and developed to generate
the smooth curved toolpath for the complex curved
surface parts, and the NURBS interpolator possesses
superiorities over conventional linear or circular
interpolators. In this study, a conversion method
from cutter-location points to NURBS-toolpath NC
file is presented in order to fill in the gap between the
CAD model of complex parts and the parametric
curve interpolator provided by the modern CNC systems, thus realizing the high-speed and high-quality
machining of parts with complex curved surfaces. To
ensure a well curve-fitting effect and reduce the computational burden, the cutter-location pre-processing
consisted by data segmentation and data simplification is provided. The least-square method for NURBS
curve fitting is used and an iterative fitting approach is
presented for the generation of the NURBS toolpath.
A user interface for the convenient NURBS-toolpath
NC-file outputting is designed, where the fitting
results and the fitting errors can be seen directly.
Experimental results demonstrate the feasibility of
the presented conversion method. Additionally, the
superiority of NURBS toolpath in realization of
high-speed machining compared with the traditional
12
Proc IMechE Part C: J Mechanical Engineering Science 0(0)
continuous short-line toolpath is also verified. This
research achievements are of vital importance for realizing the high-quality and high-efficient machining
for the complex curved surface parts.
Acknowledgements
The authors wish to thank the anonymous reviewers for
their comments which led to improvements of this paper.
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with
respect to the research, authorship, and/or publication of
this article.
Funding
The author(s) disclosed receipt of the following financial
support for the research, authorship, and/or publication
of this article: This work was supported by National
Natural Science Foundation of China (Grant No.
51575087 and Grant No. 51675081), National Science and
Technology Major Project of China (No. 2016ZX04001002), Innovation Project for Supporting High-level Talent
in Dalian (No. 2016RQ012), Science Fund for Creative
Research Groups (No. 51621064), and the Fundamental
Research Funds for the Central Universities (No.
DUT17LAB13).
ORCID iD
Jian-wei Ma
http://orcid.org/0000-0002-2177-529X
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