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The International Journal of Advanced Manufacturing Technology

Improvement of toolpath quality combining polynomial interpolation with reduction of toolpath points2014 •

The paper presents a new combination of two methods for tool path generation for five-axis machining proposed earlier by the authors. The first method is based on the grid generation technologies whereas the second method exploits the space-filling curve approach. Combination of the two techniques is superior with regard to the conventional methods and with regard to the case when the two methods are applied independently. In particular the algorithm allows us to generate tool paths for workpieces with complex boundaries as well as when the scallop and gouging constraints are changing sharply and irregularly. In this case the conventional methods are inefficient, whereas the proposed algorithms construct the required tool path and reduce the length of the path and the time of the machining. The numerical experiments are complemented by the real machining as well as by the test simulations on the Unigraphics 18.

Numerous works have shown that the linear format of tool path is not well adapted to HSM for it does allow an optimal follow-up of the tool trajectory by the NC unit, nor a good part surface quality. This paper deals with formats of tool trajectory relying on polynomial models. The tool path can be described as polynomial curves as well as polynomial surfaces. Geometrical and dynamical advantages of using such formats are exposed.

Precision Engineering-journal of The International Societies for Precision Engineering and Nanotechnology

A lightweight interpolation algorithm for short-segmented machining tool paths to realize vibration avoidance, high accuracy, and short machining time2019 •

International Journal of Machine Tools and Manufacture

5-axis local corner rounding of linear tool path discontinuities2013 •

2019 •

CNC machine tools incorporate the interpolators to generate reference points due to physical movement. In this study, initially the NURBS curve has been generated and sent to interpolator. By consideration of NURBS curve properties and other dynamic conditions like maximum velocity, the maximum acceleration and the maximum jerk of the reference points have been generated for each sample time. The current interpolators generally work based on 2nd degree Taylor series, but the numerical errors which are produced because of the nature of numerical truncations are not considerate. In this report, the real time interpolation of feed rates has been investigated in order to precisely control position on 3 axis CNC table with Milne-Simpson predictor corrector. In this approach, the numerical errors have been reduced with correction of the main calculating core of the interpolation unit that has been led to the higher accuracy. This claim has been proved by experimental tests on 2 axis CNC t...

This paper presents a smooth spline interpolation technique for five-axis machining of sculptured surfaces. The tool tip and orientation locations generated by the CAM system are first fitted to quintic splines independently to achieve geometric jerk continuity while decoupling the relative changes in position and orientation of the cutter along the curved path. The nonlinear relationship between spline parameters and displacements along the path are approximated with ninth order and seventh order feed correction splines for position and orientation, respectively. The high order feed correction splines allow minimum deviation from the reference axis commands while preserving continuous jerk on three translational and two rotary drives. The proposed method has been experimentally demonstrated to show improvements in reducing the excitation of inertial vibrations while improving tracking accuracy in five-axis machining of curved paths found in dies, molds and aerospace parts.

Computer-Aided Design 36 (2004) 1219–1229
www.elsevier.com/locate/cad
A new format for 5-axis tool path computation, using Bspline curves
Jean Marie Langerona, Emmanuel Ducb,*, Claire Lartiguec,d, Pierre Bourdetd
a
Machining Solutions, Dassault Systèmes, 9 quai Marcel Dassault, BP 310, 92156 Suresnes, Cedex, France
b
LaRAMA, IFMA, Campus de Clermont Ferrand/Les Cézeaux, BP 265, 63175 Aubière, Cedex, France
c
IUT de Cachan, 9 av Division Leclerc, 94234 Cachan, Cedex, France
d
Lurpa, ENS de Cachan, 61 av du président Wilson, 94235 Cachan Cedex, France
Received 22 July 2003; received in revised form 3 December 2003; accepted 10 December 2003
Abstract
This article presents a new format of tool path polynomial interpolation in 5-axis machining. The linear interpolation usually used
produces tangency discontinuities along the tool path, sources of decelerations of the machine tool whereas polynomial interpolation reduces
the appearance of such discontinuities. The new format involves a faster tool path and a better surface quality. However, it imposes a
modification of the process so as to take the interpolation format and the inverse kinematics transformation (necessary to 5-axis machining)
into account. This article deals with the geometrical problem of tool path calculation. Validation tests are detailed. They show that profits
concern the reduction of machining time as well as the quality of the machined surfaces. Indeed, the trajectory continuity avoids the
appearance of marks and facets.
q 2004 Elsevier Ltd. All rights reserved.
Keywords: High speed milling; 5-axis machining; Bspline interpolation; Tool path computation
1. Introduction
The purpose of the machining of free-form surfaces is to
carry out complex shapes according to a given level of
quality, while minimising machining time. This method of
machining relates to mould and dies, aeronautics, or car
industries.
The manufacturing process consists of four main
activities. The first one relates to the design of part
surfaces using a CAD system in order to respect design
requirements.
The second activity consists in calculating the machining
tool trajectory, according to the chosen machining strategy
by means of a CAM system [1]. The machining strategy
defines the geometrical characteristics that are necessary to
carry out the machining. In addition to the geometry of the
part surfaces and that of the tool, one chooses the machine
tool, 3 or 5 axis, the machining type, point milling or
flank milling, the machining direction and the longitudinal
* Corresponding author. Tel.: þ33-4-73-28-80-96; fax: þ 33-4-73-2880-85.
E-mail addresses: Jean-marie_langeron@ds-fr.com (J. M. Langeron);
emmanuel.duc@ifma.fr (E. Duc); claire.lartigue@lurpa.ens-cachan.fr
(C. Lartigue); pierre.bourdet@lurpa.ens-cachan.fr (P. Bourdet).
0010-4485//$ - see front matter q 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.cad.2003.12.002
and side step values. The latter are calculated according to
the geometrical specifications, essentially form deviation
and surface roughness. Then, the calculation algorithm
calculates a set of tool positions defining the tool path by
calculating tangent tool locations at the surface and by
arranging them according to the longitudinal and side steps.
The tool path is calculated considering the given type of tool
path description format, to be read by the Numerical Control
unit (NC unit).
During the third activity, the post-processor ensures the
translation of the APT program file to the ISO program file,
in order to be treated by the NC unit.
The last activity is the realisation of the part on the
machine by the movement of the cutting tool. The machine
tool is a mechanical system, generally using 3 or 5 axis of
displacement. A NC unit controls the machine. In particular,
the NC unit ensures the reading of the program, the
calculation of movement and speed orders of each axis of
the machine tool.
The communication between these four activities goes
through the definition of geometrical languages for the
surface description (IGES, STEP, geometrical model of
surfaces) and for the machining tool path (APT, ISO,
interpolation format of the tool path).
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J.M. Langeron et al. / Computer-Aided Design 36 (2004) 1219–1229
To improve part quality and to decrease machining time,
the process must henceforth be adapted to High Speed
Machining (HSM). In particular, the tool path calculation
and its communication to the NC unit must take the
dynamical behaviour of the machine tool and the real-time
behaviour of the NC unit into account. Heisel and Feinauer
indicate that the form of the calculated tool path and the
interpolation format used generate constraints on the realtime follow-up of the tool path and may cause decelerations
of the machine tool or appearance of chatters during
machining [2]. If the tool path calculation and its
communication to the NC unit are optimised, machining
time is thus minimal while respecting quality requirements.
To solve this problem, we consider that the CAM system
must compute a C2 continuous tool path using polynomial
curves. This improvement should increase the effective
speed of the tool during machining in 3-axis as well as in 5axis machining of free-form surfaces.
To evolve in this direction, NC units now propose to read
polynomial or Bspline curves in 3-axis milling (Heindenhain, Siemens, Fanuc and Num for example). It is thus
necessary to define a dedicated format for the communication of the tool path between the CAM system and the NC
unit, for the APT format only considers linear and circular
interpolation formats. CAM systems must evolve to propose
tool path computation methods well adapted to this new
format. However, in 5-axis the problem is more difficult
since calculation methods must integrate the problem of the
inverse kinematics transformation.
This article presents a new format of tool path
computation and interpolation for 5-axis machining,
based on the Bspline interpolation of the tool path in
the part coordinate system, respecting accuracy requirements. The goal is to define a 5-axis tool path description
format, adapted to the communication between the CAM
software and the NC unit. In this case, the CAM output is
directly expressed through Bspline curves. We propose a
calculation method allowing solving the computation
problem.
Generally the Bspline interpolation is used to build curve
for CAD systems. Some works are related to the curve
interpolation for NC machining, such as papers presented by
Erkorkmaz, Cheng, Xu, Yeh, and Yong [3 –7]. Authors
suggest to use tool path described as polynomial curves, to
enhance the acceleration, speed and chord error control,
during the real-time NC follow-up of the tool path. The
main results consist in a smoother variation of the speed
axis, a reduction of machining time and a diminution of the
chordal error.
Our approach does not consider real-time interpolation
but it is only located at the CAM stage, and concerns the
geometrical calculation of the tool path. To ensure the
follow-up of the tool path, we use an adapted NC unit
Siemens 840D. Many papers study the computation of tool
path with curves in 5-axis milling. Only Fleisig and Spence
presents an off line interpolation of the tool path [8]. In
Section 3, we study the main difference between our both
approaches.
Section 2 exposes problems linked to 5-axis machining.
Section 3 is dedicated to the presentation of the interpolation
format we propose, and the method of calculation adopted.
Section 4 relates comparison tests, which show that
polynomial interpolation leads to important profits in
terms of quality and machining time.
2. 5-Axis machining
Five axis machining is carried out on machine tools
which have two rotation axes (A; B or A; C or B; C)
allowing the tool to be oriented relatively to the part in
addition to the usual axes of translation X; Y; Z: Bohez
presents the various types of machine structures [9]. In
the present work, we are only interested in continuous 5axis machining, the most generally used for aeronautics
parts. Continuous 5-axis machining is used for the
surface milling with ball end or toroı̈dal cutter, and
flank milling with cylindrical or conical cutter.
Five axis machining poses two different types of
problems. The first one can be described as geometrical. It
relates to the tool path calculation in the part coordinate
system (P-system) and also to the inverse kinematics
transformation that is necessary to calculate position
instructions of the 5-axis from the calculated tool path.
The second type is mechanical and relates to both speed and
acceleration calculation and control of the 5-axis using
adapted laws. The most usual interpolation format of the
tool path in 5-axis machining is the linear one. The tool path
calculated using the CAM system is expressed in APT
format and is transformed into ISO format by means of a
post-processor.
2.1. Geometrical problem associated to the 5-axis
tool path computation
As the geometrical problem is concerned, the computing
process of position instructions of the 5-axis can be divided
in four stages.
(1) Initially, the tool path is calculated as a set of tool
positions and tool axis orientations in the given
interpolation format. The tool path is expressed in the
part coordinate system (P-system). To perform this
step, various calculation algorithms are available in
literature. Li, Takeuchi and Choi present such methods
[10 –12].
Let us call Pi ; the ith tool position and Ai the ith tool
axis orientation expressed in the P-system. If we
consider two successive couples ðPi ; Ai Þ and
ðPiþ1 ; Aiþ1 Þ; the tool is supposed to be moving in
J.M. Langeron et al. / Computer-Aided Design 36 (2004) 1219–1229
straight line in the P-system.
PðuÞ ¼ Pi þ uðPiþ1 2 Pi Þ
AðuÞ ¼ Ai þ uðAiþ1 2 Ai Þ
; for 0 # u # 1
ð1Þ
Successive couples are computed in order to respect a
given deviation named machining tolerance. The
distance between the computed tool path and
the theoretical perfect tool path must be closer than
the machining tolerance, along the tool path.
(2) Instructions of the rotation axes are calculated from the
previous set of tool axis orientations [13]. This
calculation leads to solve a trigonometrical system,
which can have zero, one, or several solutions in the
space of authorised situations of the machine.
(3) Next step is the calculation of the coordinates of the
tool TIP in the machine coordinate system (M-system).
The ISO program is generated.
(4) The NC unit interprets the programmed locations of the
5-axis and axis orders are calculated according to
dimensions of the tools and adjustments of the machine
(Fig. 1).
The inverse kinematics transformation relates to stages 2
and 3. Bohez presents a complete analysis of the inverse
kinematics transformation [14]. The post-processor usually
performs the inverse kinematics transformation during the
translation stage of the APT file. The geometrical analysis of
this transformation shows that the transformation is not
linear from the P-system to the M-system. A linear
movement of the tool expressed in the P-system is not
transformed into a linear movement of the tool in the
M-system. The same effect can be observed on the real
speed of the tool relatively to the part. The speed of the
tool relatively to the part is the subtraction of that of the tool
relatively to the M-system minus that of the part relatively
to the M-system (Eq. (2)).
Vtool=part ¼ Vtool=M-system 2 Vpart=M-system
ð2Þ
To solve both problems, post-processors generally add
tool locations on the initially calculated tool path so that
1221
the movement of the tool in the M-system is as close as
possible to the ideal movement. The really treated tool path
is thus computed at two stages: first, in the P-system by the
CAM system, and second, in the M-system by the postprocessor. In the same way, the speed of each axis is
programmed according to time inverse so that the
programmed tool feedrate is respected. In this case, we
consider that the post-processor is of major importance in
the geometrical calculation of the tool path. As a result, the
final error between the real tool path and the ideal one is the
addition of the CAM computation error, the post-processor
error and the NC follow-up error.
Note that, the calculated tool path is adapted to a
dedicated machine tool, with its own kinematics and its
own NC unit. The change of machine tool involves a
new calculation program. Furthermore, when changing
interpolation format, it is necessary to modify the CAM
calculation, the treatment of the NC unit, and the
kinematics transformation carried out by the postprocessor.
Henceforth, all these problems can be avoided with
the use of new NC units for they can perform all tool
path treatments in the P-system. Indeed, NC units carry
out in real-time the tool path interpretation and the
inverse kinematics transformation just after the tool
location calculation (TRAORI function for Siemens
840D) [15]. The tool path calculated by the CAM
system is directly transmitted to the NC unit. The role of
the post-processor is only limited to the translation of the
APT format into the ISO format. In this case, the
geometrical transformation takes place for each calculation of a tool location order. Therefore, any interpolation format is seen the same way as regards the
geometrical transformation. At this point, two complementary activities can be defined:
† the CAM system calculates the tool path according
to the specified machining tolerance in the P-system,
† the NC unit carries out the interpretation and the
follow-up of the tool path in the P-system, and
calculates displacement and speed orders of the motion
axes.
Note that this configuration is well known in 2.5- or 3axis machining where the post-processor just ensures the
necessary translation of languages. The integration of the
inverse kinematics transformation at the level of the NC unit
makes coherent the 5-axis machining with the 3-axis
machining process, and makes also coherent the NC unit
of the machine tools with that of robots.
2.2. Mechanical problems associated to the 5-axis
tool path follow up
Fig. 1. Tool path transformation from P-system to M-system.
At the mechanical level, the problem relates to the
performance of the rotation axes comparatively to
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J.M. Langeron et al. / Computer-Aided Design 36 (2004) 1219–1229
the translation axes. For machines for which rotation axes
are located on the table, axes have to move important
masses and also support significant efforts during machining. As a result, they have low capacities of acceleration.
When the machine has to slow down or to stop the tool
movement, the speed reduction involves significant times of
deceleration and re-acceleration. This effect appreciably
increases machining time, and is amplified in HSM, because
rotation axes must reach greater speeds. Therefore, it can be
economically more interesting to machine in 3-axis at high
speeds and a minimal number of decelerations, than in
5-axis.
Decelerations generally appear when the tool path
presents tangency or curvature discontinuities, or when
the speed of the rotation axes abruptly evolve. To avoid
producing marks on the part or generating a too large
contour lag error, the NC unit decides to slow down, even to
stop. Bloch details some aspects of the NC units well
adapted to HSM [16]. Erkorkmaz shows that discontinuities
are source of high frequency vibrations, which can excite
the natural modes of the machine tool structure [3]. Contrary
to machining with 3-axis, 5-axis machining at high speeds
imposes good clockwork of all the axes, i.e. the most
constant variation of the 5-axis movements between two
programmed successive positions. The number of tool
locations is increased to obtain the same distance between
two successive positions, even if the machining tolerance is
already respected.
To remove tangency discontinuities, CAM systems must
thus compute C2 continuous tool paths, as smooth as
possible, without abrupt variations of the tool orientation.
Basically, the linear interpolation commonly used only
produces a C0 continuous tool path and seems not well
adapted. Previous work showed profits in the part surface
quality and in the management of speeds in 3-axis
machining when using polynomial interpolation [17,18].
Consequently, the use of polynomial interpolation seems a
factor of improvement in 5-axis HSM. Section 3 presents
the interpolation format we have defined, based on the use
of Bspline curves.
3. Polynomial interpolation in 5-axes
be carried out in the part coordinate system (problem A
before problem B) or in the machine coordinate system
(problem B before problem A)?
Concerning the case for which the tool path can be
computed in the M-system, problem B before problem A,
the computation is carried out by the NC unit from the CAM
ISO data. Direct calculation in CAM supposes that the CAM
system carries out the inverse kinematics transformation.
For example, the movements of the axes X; Y; Z; A; C are
defined using five polynomial curves expressed in the Msystem.
As the other case is concerned, problem A before
problem B, the CAM system fits a curve to a sub-set of ideal
tool path positions. In other words, P and A are expressed in
P-System, using Bspline Curves (Eq. (1)). At this stage, tool
path discontinuities may simply be detected. The curve is
directly transmitted to the NC unit, which interprets it
before performing the inverse kinematics transformation.
Therefore, this case seems well adapted to 5-axis machining
using polynomial tool path.
In Section 3.1, we consider that problem A is solved
before problem B. The movement of the tool TIP and the
tool axis orientation are expressed as polynomial curves in
the P-system. The NC unit interprets the tool path, samples
the tool path in the P-system and calculates the tool
movement orders for each sampling period.
3.1. Description of the interpolation format:
the 5xNurbs format
We propose to describe the tool movement as follows.
The tool movement is defined using the two parametric
curves PðuÞ and QðuÞ:
† PðuÞ ¼ ðXtðuÞ; YtðuÞ; ZtðuÞÞ; defines the locus of the tool
TIP locations,
† QðuÞ ¼ ðXhðuÞ; YhðuÞ; ZhðuÞÞ; defines the locus of a
second point belonging to the tool axis.
Note that this description is generic and allows the
expression of the tool path for all types of machining 3- or 5axis, machining in point or flank milling and for all
interpolation formats (Fig. 2).
When using polynomial interpolation in 5-axis machining, two geometrical calculations are necessary, at the CAM
and Post-processor stages:
† problem A: computation of the tool path using a
polynomial curve,
† problem B: calculation of the inverse kinematics
transformation.
Here, the essential difficulty concerns the relative
position of these two problems within the general process.
In other words, does the calculation of the tool path have to
Fig. 2. Definition of the tool path using the 5xNurbs format.
1223
J.M. Langeron et al. / Computer-Aided Design 36 (2004) 1219–1229
The tool path is thus expressed like a polynomial
function C of R towards R6 :
C : R ! R6
0
XtðuÞ
1
0
1
C
C B
B
C
C B
B
B YtðuÞ C B PðuÞ C
C
C B
B
C
C B
B
C
C B
B
C
B ZtðuÞ C B
C
C B
B
u ! CðuÞ ¼ B
C with PðuÞ [ R3
C¼B
C
C B
B
C
B XhðuÞ C B
C
C B
B
C
C B
B
B YhðuÞ C B QðuÞ C
C
C B
B
A
A @
@
i¼0
and QðuÞ [ R
ð3Þ
The calculation of the tool location relatively to the
surface is led by a usual calculation of distance in metric
space. By gathering the coordinates, we construct two
curves PðuÞ and QðuÞ in the P-system.
The two curves are computed so that their relative
distance is equal to a length H:
’H [ R so that ;u; llQðuÞ 2 PðuÞll ¼ H
ð4Þ
The tool axis AðuÞ is thus calculated for each value of the
u parameter by subtraction of the two curves:
AðuÞ ¼ ðQðuÞ 2 PðuÞÞ=llQðuÞ 2 PðuÞll
ð5Þ
H corresponds to the height of the tool in flank milling and the
diameter of the tool in point milling. It defines the size of the
hull containing the tool. The direct computation of PðuÞ and
AðuÞ curves can introduce gouging due to the size of the tool,
whereas the computation of PðuÞ and QðuÞ take that into
account. Considering a point located on the cutting area of the
tool surface, for a given parameter u* ; its distance to the ideal
tool location is less than the distance of the point Qðu* Þ to the
ideal curve Q: Using this format, one can thus quantify the
maximal distance between the side of the tool and the surface
or the distance between the back of the tool and the part, when
the use of PðuÞ and AðuÞ as defined in Eq. (1) does not permit
them. Indeed, the length of AðuÞ is equal to 1. If the
computation of the tool path gives an error equal to 0.01 mm
for Aðu* Þ; then the real error is multiplied by H:
Polynomial interpolation curves are non-uniform cubic
Bsplines. Each one is defined by a set of ðn þ 1Þ control
points and a knot sequence mj representing the variation of
the parameter u along the curve. The degree m is equal to 3.
Both curves P and Q present the same parameter setting in
order to synchronise the two trajectories.
n
X
Ni;m ðuÞPi with u [ ½m0 ; mnþmþ1 and Ni;mþ1;
PðuÞ ¼
i¼0
spline functions
QðuÞ ¼
n
X
ð6Þ
Ni;m ðuÞQi with u [ ½m0 ; mnþmþ1
i¼0
and Ni;mþ1; spline functions
i¼0
with u [ ½m0 ; mnþmþ1
ZhðuÞ
3
For each parameter u; the unit vector of the axis is
obtained using the following relation
n
n
X
X
Ni;m ðuÞðQi 2 Pi Þ= Ni;m ðuÞðQi 2 Pi Þ
AðuÞ ¼
ð7Þ
ð8Þ
Fleisig proposes a different method, for curve
interpolation from a set of tool locations and orientation,
using quintic spline and Bézier curves [8]. The tool path
is then C3 continuous. The goal is to produce a constant
feed and to reduce the angular acceleration. To compare
with our method, PðuÞ is the position curve and ðQðuÞ 2
PðuÞÞ=llQðuÞ 2 PðuÞll is the orientation curve. Fleisig uses
a near arc length parameterised quintic polynomial spline
for the position curve and a near arc-length parameterised quintic spherical bezier curve for the orientation
curve. A third spline is necessary for the re-parameterisation of both curves and the synchronization of the
position and the orientation splines. In our method, both
curves are defined with the same parameterisation and
using the same Bspline format, and are calculated using
the same algorithm at the same level. The use of the
parameterisation curve is thus not necessary. The realtime computation of the tool location and the speed is
ensured by the NC Unit Siemens 840D.
However, the main difference consists in the choice of
the initial tool location set. Fleisig has not concentrated on
this issue, considering that the set of tool locations is given.
With our approach, we select a set of tool locations from the
whole theoretical tool path, defining an initial curve that fits
the theoretical tool path while respecting a specified
machining tolerance. Our algorithm is thus located at the
CAM stage dedicated to the tool path computation. Section
3.2 details the calculation method of tool path as Bspline
curves.
3.2. Method of tool path calculation
The tool path calculation is carried out by interpolation
of a set of particular positions of C2 continuous portions of
the tool path. The tool path calculation is divided into four
steps.
(1) Whatever the tool geometry and the type of machining,
the first step relates to the calculation of a tangent
location of the tool on the surface. Calculation can be
led according to algorithms proposed by Yu, Lee and
Li in Refs. [19 –21].
(2) The second step relates to the organisation of the
successive tool locations so as to respect the
machining tolerance and according to the given
interpolation format [22]. When using the polynomial interpolation, the tool path is calculated from
a set of sampled positions belonging to the
theoretical tool path. The tool path is thus calculated
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J.M. Langeron et al. / Computer-Aided Design 36 (2004) 1219–1229
as polynomial curves so that the distance between
the calculated tool paths to the theoretical one is
lower than the specified machining tolerance while
minimising the geometrical information sent to NC
unit.
(3) The next step concerns the tool path planning in the
transversal direction according to the respect of the
maximum transversal step or the maximum scallop
height allowed. Lee proposes a calculation algorithm
of the maximum step in 5-axis machining [23].
(4) The last step is dedicated to the detection of tool path
discontinuities, which becomes essential in polynomial interpolation. Indeed, polynomial interpolation permits to associate C2 continuous curves to
tool path portions. As a result, those portions must be
at least C2 continuous, and tangency or curvature
discontinuity points must be detected. Choi presents a
method based on the comparison of the relative
position of the tool centre’s points and contact points
in 3-axis machining [24]. Duc proposes to isolate the
portions from tool path having soft variation of
curvature radius [25]. If the discontinuity point is not
precisely detected, the CAM system must add
interpolating points or control points in order to
respect the machining tolerance. In this case, the
calculated curve may present oscillations of great
frequency and curve arcs of low lengths, which
disturbs the follow-up of the tool path by the NC
unit. Consequently, a tool path calculated and
optimised for linear interpolation is not a good
support for an accurate and optimal calculation of a
tool path in polynomial interpolation.
The method we develop relies on the interpolation of tool
path portions using non-uniform cubic Bsplines. Each
portion of the calculated tool path is a curve C of R6 (see
Eq. (3)). C is solution of the problem of interpolation of a set
of tool locations belonging to the theoretical tool path that
exactly machines the surface. We use the method suggested
by Piegl [26]. Note that the tool path interpolation problem
is not the same than the curve interpolation problem, defined
in CAD software. Our problem is not to compute a curve
passing trough a set of points, but to compute a curve in a
given format, which is an approximation of a theoretical
curve. The calculated curve must be located in a pipe
centred on the theoretical curve. The distance between both
curves must be evaluated along the whole curve and be less
than the given machining tolerance, which defines the radius
of the pipe. In addition, the number of passing points must
be minimal, in order to simplify the NC unit treatment.
Let us consider a C2 continuous portion T of the
theoretical tool path. In fact, T corresponds to two portions
of C2 continuous curves, T1 and T2, describing the
movement of the tool TIP and of another point of the tool
axis, distant by H distance of the tool end (see Eq. (4)). Fig. 3
presents an example of such curve T1 or T2 (1). From T; we
Fig. 3. Computation of the tool path.
define a very fine sampling of tool locations, which implies
the calculation of a great number of Mi positions of the tool
along T (2). Each, position Mi is defined by six coordinates
(Xti, Yti, Zti, Xhi, Yhi, Zhi). This sampling less than 0.1 mm
is too fine to be directly sent to the NC unit.
From the initial set, we select a sub-set of n Mi positions
(3). The Mi positions are regularly distributed so as to
minimise the number of interpolating points and in order to
take curvature variations of the tool path into account. The
selected set of Mi positions defines a broken line (4), which
is an approximation of the theoretical tool path, respecting a
given maximal chordal error. The maximal chordal error is
greater than the machining tolerance, so that the number of
chosen points is very small. The number and the locus of the
Mi interpolating points depend on the form of the theoretical
curve and on the given machining tolerance. Several
distributions are possible, but the best one minimises the
number of Mi positions and avoids the appearance of
oscillations on the calculated curve. The calculated tool path
(5) is computed by interpolation of the Mi points. The curve
is accepted, when it is located in the pipe (6) centred on the
theoretical curve (1), and which radius is equal to the
machining tolerance (7).
To ensure at least a C0 continuity at the junction points of
the different curves, we impose that extremities must
coincide. The problem of interpolation is solved according
to the following steps [26]:
† Parameter setting of the curve: the parameter ui of each
Mi position is equal to the arc length of the Mi position
on the curve T1; in order to obtain a near arc length
parameterisation,
† Calculation of the knot sequence: the knot sequence
includes n þ 5 knots, which verify:
m0 ¼ m1 ¼ m2 ¼ m3 ¼ 0; mnþ1 ¼ mnþ2 ¼ mnþ3 ¼ mnþ4
¼ l; mjþ3 ¼
jþ2
1X
u for j ¼ 1; …; n 2 3;
3 j i
where l indicates the length of T1;
† Construction of the problem of interpolation: for each Mi
(Xti, Yti, Zti, Xhi, Yhi, Zhi) position, Mi belongs to the
searched C curve and verifies:
Mi ¼
n
X
j¼0
Nj;m ðui ÞPj ; for i ¼ 0; …; n
ð9Þ
J.M. Langeron et al. / Computer-Aided Design 36 (2004) 1219–1229
Eq. (9) leads to a linear problem of 6·ðn þ 1Þ equations for
6·ðn þ 1Þ unknowns (coordinates of the Pj control points).
† Inversion of the problem and calculation of the control
points of the curve C,
† Checking the respect of the machining tolerance by
calculation of the distance between the calculated curve
and T: If the machining tolerance allowed is not respected,
a refining of the sampling of the Mi positions is carried out.
The verification of the respect of the machining tolerance
is led by analysing the distance between a sampling of C and
T, distance which is calculated according to the norm
1225
defined on the set of the functions of R in R6 : The
computation is based on the calculation of the distance
between a tool location X (X1ðx1; y1; z1Þ; X2ðx2; y2; z2Þ) of
R6 (belonging to C) and the ideal curve.
d6 ðX; TÞ ¼ MinðMaxðllX1; T1ðutÞll; llX2; T2ðutÞllÞ;
ð10Þ
for ut covering the interval of definition of T, ll ll indicates
the Euclidean distance in R3 :
The distance between the curves C and the T is thus
obtained by Eq. (11)
DðC; TÞ ¼ Maxðd6 ðCðucÞ; TÞ; for uc covering the interval
of definition of C:
Fig. 4. 5xNurbs format, example of program.
ð11Þ
1226
J.M. Langeron et al. / Computer-Aided Design 36 (2004) 1219–1229
This method we expose easily computes interpolation
curves. However, the most significant interpolation errors
are located at the extremities. To decrease error, tangents at
the extremity points must be imposed.
In the case of smooth aeronautic parts, this algorithm
describes a 270 mm length tool path, using only one Bspline
with six control points and three arcs, with a machining
tolerance equal to 0.02 mm. This algorithm is integrated to the
Catia CAD/CAM Software. Siemens has developed the
Oricurve function to treat such curves. Fig. 4 presents an
example of a tool path generated using the polynomial format.
XT, YT, ZT and XH, YH, ZH are the coordinates of the
control points, solution of the problem defined in Eqs. (3)
and (9). The parameters DK and PL identify the variation of
the knot sequence for each arc of the curve and W indicates
the weight of the control point, which is equal to 1 for
Bspline curves.
4. Applications
Several test parts were defined and machined in order to
estimate benefits brought by the polynomial interpolation.
They are representative of different type cases in five
continuous axes machining:
† machining of a smooth surface with a toroidal cutter to
maximise the covering rate,
† machining of a cavity with a hemispherical tool to avoid
collisions with the part,
† machining of surfaces having the shape of a turbine blade.
Machining was carried out on the Mikron UCP710
milling centre of Lurpa, equipped with a Siemens 840D NC
unit adapted by Siemens to treat 5xNurbs tool paths. Cutting
tools are carbide tools authorising a spindle speed of
15 000 rpm and feedrates variable between 2000 and
4000 mm/min. Due to the functionality of the Siemens
NC unit, we have compared various modes of interpolation,
the linear interpolation G1, the linear format interpolated in
a polynomial curve in real-time by the NC unit (function
compcad of the Siemens 840D) and the 5xNurbs
interpolation.
Two preliminary analyses were conducted. One concerns
the length of elementary tool paths and the second one
relates to the evolution of interpolation errors along the tool
path. Following, real machining was carried out, and we
have compared machining time and analysed real speed of
the axes. Finally, the quality and the visual aspect of the
machined parts were studied.
Fig. 5. Part PM5X: computed tool path (Bspline/linear).
for the 5xNurbs interpolation. All the tests showed that the
polynomial interpolation makes it possible to decrease
the number of geometrical information necessary to control
the tool. Generally, a tool path portion not having
discontinuity is expressed as only one C2 continuous Bspline
curve from the beginning to the end of the portion. The
elementary arc length is longer than that in linear
interpolation. In Fig. 5, segment ends are represented as
lines on the right for the linear interpolation (A). The ends of
the 51 interior arcs of Bspline (1) are represented on the left
for the Bspline tool path (B). The number of segments is
strongly reduced. Fifty-four control points (2) define the
curve, whereas 129 points are necessary for linear interpolation. The length of each arc is varying between 2 and
8 mm. The length of the arcs is reduced when the curvature
radius of the tool path is reduced. The NC unit can thus read
longer segments. The error between the computed curve and
the theoretical one lies between 2 0.006 and 0.005 mm.
The quality of the calculated tool path depends on the
geometry of the surfaces to be machined and on the accuracy
of the tool path calculation. When the surface presents
tangency discontinuities, this may imply a non-optimised
tool path calculation and may alter the performance of the NC
unit. Initial surface must be C2 continuous. If a loss of
accuracy during the interpolation is authorised at the points
of connection is authorised, this may lead to a smoother
movement and improve the performance of the NC unit.
4.1. Distribution of the arcs of curves
4.2. Kinematic analysis
For this analysis, the length of each elementary tool path
segment is evaluated. The elementary segment is a line
segment for the linear interpolation and an arc of the Bspline
The problem of the axis speed results from the treatment
of the tool path by the NC unit. Indeed, NC must calculate
J.M. Langeron et al. / Computer-Aided Design 36 (2004) 1219–1229
a speed law according to the motorization of each axis,
taking the acceptable variations of contour lag error into
account, and considering the response in vibration of the
machine tool structure. The C1 continuity of the tool path
and the interpolation format play a significant role.
Theoretically, it is impossible to cross a C1 discontinuity
with a non-null speed while ensuring a weak contour lag
error. When the linear interpolation is chosen, either due to
the form to be machined or to the tool path calculation, C1
discontinuities between successive elementary tool paths
exist and the NC unit must use algorithms to detect such
discontinuities and to smooth them. Considering the
adaptation of the algorithm to the tool path, this treatment
is more or less powerful.
For our testing machine tool, rotation axes are less
dynamic than translation axes. The machine is set with weak
gain for the position control loop and a low value of the jerk
to smooth the speed orders and to avoid marks on the parts.
Similarly, each speed variation causes a significant increase
of machining time essentially due to the significant
deceleration time. The polynomial interpolation allows an
important benefit in machining time, by eliminating C1
discontinuities of the tool path present in linear interpolation. As the tool path is C2 continuous, it is not necessary
to slow down. The NC unit better interprets the tool path and
only orders slow-downs when it is really necessary.
Generally, the machine is more dynamic and the tool
presents a more continuous, fast and smooth movement. The
rotation axes can reach the ordered speed more quickly and
maintain it for a longer period of time. The benefit is
particularly significant on complex C2 continuous shapes
that present great variations of curvature. Fig. 6 presents the
evolution of the actual mean feedrate of machining
in function of the interpolation format and the programmed
feedrate. The use of Bspline curve multiplies the average
feedrate by two. The polynomial interpolation, computed in
real-time (function compcad) also allows a gain of
approximately 300 mm/min.
1227
surface and the nominal surface) according to methods
suggested by Jerard and Lartigue [27,28]. A grid of points is
computed on the surface of the part. For each point, the
straight line normal to the surface is computed and oriented
outer of the part. A sampling of the tool path simulates the
machining. The intersection between each line and each tool
location and the associated distance are computed. For each
line, the minimum value is kept.
The variation of geometrical errors is more continuous
when using the polynomial interpolation. Even if maximum
error values are of the same magnitude order, the error is
more homogeneous on great portions of the tool path. In
case of the PM5X test, which presents the machining of a
cavity (see Figs. 7 and 8), errors are concentrated on the
connection between two surfaces. The interpolation of a tool
path by a Bspline curve with a tolerance equal to 0.01 mm
involves significant errors at the ends of the curves.
The surfaces of the machined parts present the same
defects. Marks due to the decelerations in linear interpolation and the A-blows speeds of the rotation axes are
eliminated as for the facets appearing on large curvature
radius (see Fig. 7).
4.3. Quality analysis
To compare the two interpolation formats, we have
simulated the machined surface and then evaluated
geometrical deviations (deviations between the simulated
Fig. 6. Part AI5X: measured average feedrate.
Fig. 7. Part PM5X: simulation of the machining errors G1/5xNurbs.
1228
J.M. Langeron et al. / Computer-Aided Design 36 (2004) 1219–1229
dimensions into account. The model suggested in this
article contains all geometrical information that is necessary
to define algorithms for compensation (in particular the
normal to the tool path for each tool position). In flank
milling, it is sufficient to know if the part is on the left or on
the right of the tool, and to develop algorithms for
compensation of tool diameter variations. Further work
will focus on this point.
Acknowledgements
The work presented in this article relates to the 5xNurbs
project, which gathered Dassault Systèmes, Siemens, Cenit
and Lurpa about the development of a new format of
interpolation in 5-axis machining.
References
Fig. 8. Parts PM5X and AI5X: machined part, elimination of facets and
marks.
5. Conclusion and prospects
HSM requires to guarantee a high speed of the axes of the
machine tool all tool path long. This depends on both the
numerical controller capacities to calculate laws of optimal
accelerations and the CAM system capacity to calculate
accurate tool paths that are adapted to the NC unit treatment.
This article presents a new description format of the tool
path in 5-axis machining. This format uses non-uniform
cubic Bspline curves and expresses the tool movement in
the part coordinate system (P-system). The C2 continuity
introduced by the interpolation format ensures a continuous
evolution of the speed of rotation axes, which have lower
capacities of acceleration.
Test parts are machined, and permit to conclude on
significant benefits in the reduction of machining time as well
as on the quality of the machined surfaces. Indeed, the
smoothness of the tool path avoids the appearance of marks
or facets. Today, the part PM5X was machined on different 5axis machine tools, such as Mikron, Hermlé, Huron and
Forest liné. In all cases, machining time is reduced. The
reader can evaluate directly the performance and the interest
of this method, using the CAM software Catia V5R12 SP2,
from Dassault Systèmes and the NC software 840D 6.3 from
Siemens.
New research is now necessary to improve calculation of
the curve of interpolation by decreasing the number
of control points and by clearly identifying the points of
discontinuity. Finding the best tool path, which respects
the required quality and simplifies the dynamical treatment
of the NC, is still an issue to be addressed.
In addition, it becomes necessary to propose a method of
tool compensation in order to take variations of tool
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Jean Marie Langeron, graduated in 1986 of
the French Aerospace Engeneering school of
ENSMA (Ecole Nationale Supérieure de
Mécanique et d’Aérotechnique) he joins the
(Numerical Control) NC department of DASSAULT SYSTEMES in 1987. In charge of the
Tool Path generator development team of
CATIA NC for a long time, he is now
Consultant Research and Development,
directly attached to the Director Research
and Development of the DELMIA and
CATIA Machining Solutions.
1229
Emmanuel Duc is Assistant Professor at the
French Institute for Advanced Mechanics
(IFMA-Institut Français de Mécanique Avancée) of Clermont Ferrand. His research interests include the tool path computation for High
Speed Milling, particularly in 5-axis machining and pocket machining. Computational
methods are developed using Bspline curve,
and experimental validations are tested on
various machine tool. He has conducted many
studies in aeronautics aera.
Claire Lartigue is Professor at the department
of Mechanical Engineering of IUT CachanUniversity of Paris XI. She is in charge of
research activities in the fields of CAD/CAM
and inspection of sculptured surfaces. These
activities focus in particular on the assessment
and the improvement of the geometrical
quality of machined parts. An experimental
laboratory consisting of a 5-axis HSM center
and digitizing means contributes to the success
of this research.
Pierre Bourdet member of CIRP is a
Professor at Ecole Normale Superieure in
Cachan. This institution is a main place in
France, which educate new teachers. Pierre
Bourdet is doing researches at University
Research Laboratory in Automated Production
(LURPA). The domain of research concerns
the tridimensional geometry for activities of
design, manufacturing and inspection. The
team’s research activities concerning three
topic areas: algorithms for the assessment of
measured points in co-ordinate metrology,
geometrical quality of machined parts and model functional tolerancing
for mechanism design.

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