duc emmanuel revised

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calculation methods of tool-paths. For each step, we characterize the nature of the

errors and we define associated algorithms for the error evaluation in terms of devia-

tions. Among others, this leads us to define algorithms for evaluating the location

error resulting from the initial step. The value of this error is significant, and not

influence-free for the following calculations. Other algorithms concern the evaluation

of the chordal errorand the scallop error, linked to the discretization steps. There-

fore, algorithms we suggest allow the assessment of the calculation methods of the

NC tool-path respecting the specified tolerances and criteria.

2. Methods of tool-path calculation

2.1. The issue

The issue of tool-path calculation is to transform the surface geometry of the part into

tool-path geometry, in a format understandable by the Numerical Controller (NC).

Nowadays, the most usual format remains the linear one : the tool-path consists of a

succession of line segments. The tool-path generation consists in the calculation of a

set of successive points. Next, the linear interpolation between two successive points

is performed by the NC. The calculation of the successive points defining the tool-

path generally relies on the surface geometry, the tool geometry and the datum of a

machining strategy. The machining strategy defines the driving direction of the tool

and two discretization steps. Common calculation methods consist of three main

steps :

- calculation of the tool location

- calculation of a single path in the driving direction

- calculation of adjacent single paths in the perpendicular direction

In addition, a fourth step can be considered, corresponding to the detection of interfe-

rences and collisions between the tool and the surface. This last step is obviously

necessary to determine an interference free tool-path.

Each calculation step is based on geometric approximations and on calculation

models so as to respect the authorized tolerances, tolerances generally linked to geo-metric specifications associated to the surface to be machined. Unfortunately, both

calculation hypothesis and approximations are too strong and lead to the non respect

of the tolerances. Let us analyze the loss in precision all tool-path calculation long.


3/20

2.2. Tool location

The first step is the calculation of the tool location, when the tool is tangent to the

surface. It is based on the geometric model of the part surface and is function of the

tool geometry. To define the various relations, let us consider a tool tangent to a sur-

face ( figure 1) and the following definitions :

Figure 1.Location of the tool tangent to the surface

- CC (cutter contact), is the contact point between the tool and the surface,

- CE is the center point of the tool,

- CL (cutter location), is the extremity point of the tool,

- , is the normal to the surface at the CC point, and , is the direction of the tool

axis,

- r is the radius of the ball endmill tool and the small radius of the filleted (or toric)

endmill tool, and, R the large one.Therefore, we can consider the location of the characteristic points of the tool in

function of the geometry of the tool endmill.

In the case of the ball endmill, the locations are given by equations (1) :

[1]

In the case of the filleted endmill, the locations are given by equations (2) :

[2]

Several algorithms can be used to perform this calculation. Among these, we can quote

the most usual ones that can be classified into four families ([JEN 96],[DRA 97] ) :

CC

CE CE

CCCL

CL r r

R

u u

n

n u

OCE

OCC

rn+= OCL

OCE

ru OCC

rn ru+= =

ku nu n

-----------------= OCE

OCC

rn Rk u

k u-----------------+ +=

OCL

OCE

ru OCC

rn Rk u

k u----------------- ru+ += =


4/20

a) The direct resolution of equation (1) or (2)

The direct resolution imposes a preliminary sampling of the surface into Cc points.

This sampling is in most cases performed without taking into account possible inter-

ferences between the tool and the surface.

b) The calculation of the offset surface

The sampling of the offset surface leads to the set of tool locations. However, the cal-

culation of the offset surface is rarely exact and requires approximations. Therefore,

the offset surface, on which tool locations are defined, constitutes an approximation

of the real one [KIM 95].

c) The method of the inverse offset surface

This method generally leads to interference free tool-paths. The precision of the cal-culated tool-path is linked to the discretization step of the grid ([SAI 91],[SUZ 91]).

d) The meshing of the surface

The locations of the tool tangent to the surface are calculated from the meshing of the

surface. Thus, the tool is tangent to the facets representing the surface [KUR 92].

Each method relies on approximation and discretization phases. Moreover, due to the

complexity of the calculation, risks of interferences between the tool and the surface,

may appear. Errors occurring in this calculation step are essential. Indeed, deviations

on the tool locations obviously affect the next steps.

2.3. Calculation of the single path

The single path is an ordered succession of points (corresponding to tool locations)

built according to the machining strategy. Indeed, the single path corresponds to one

trajectory of the tool in the driving direction. The averaging of the points along the

single path is carried out so as to respect the machining tolerance, and both the mini-

mal and maximal distances between successive points.

The calculation of the single path poses several problems. The main one concerns the

respect of the machining tolerance. The single path can be represented by a set of

linear segments, if we consider the linear interpolation between points. This set of

linear segments constitutes a profile that must be distant from the initial surface of a

value less than the machining tolerance.In most cases, the evaluation of the respect of the machining tolerance is carried out

using a 2D model, i.e. a plane representation of the tool locations. This respect is eva-

luated through the calculation of the chordal deviation . To calculate the chordaldeviation, Kuragano and George break this error up into two errors :

- error due to the dividing of the curve into chords (distance between the line lin-

king the contact points and the surface),


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- error due to the tool movement (distance between the lines linking the contact

points and the machined points) ([KUR 92],[GEO 95]).

Considering two successive tool locations C1 and C2, P1 and P2 are the corresponding

machined points. Let R be the radius of the circle approximating the surface locally,

and r the tool radius, the total deviation is then given by (figure 2) :

[3]

Figure 2.Evaluation of the chordal deviation [KUR 92] and the scallop height hThe objective is then to calculate successive tool locations so that the chordal devia-

tion is less than the machining tolerance. However, the calculation of the chordaldeviation relies on strong hypothesis, not always verified. In particular, it supposes:

- the tangency of the tools on the surface (i.e a unique contact point),

- the continuity of the surface between two successive tool locations,

- the 2D modeling of the surface.

Therefore, this generally implies the whole calculated tool-path to not respect the

machining tolerance, in particular for the portions located between two successive

tool locations

2.4. Calculation of the whole tool path

The next step concerns the calculation of adjacent single paths by fixing an accepta-

ble maximum distance. This calculation method is comparable with the previous one.

From a single path, the adjacent path is calculated considering the tool geometry, the

machining strategy and discretization parameters. This planning is performed in the

perpendicular direction of the driving direction according to the maximum scallop

height allowed.

When using a ball endmill tool, Lin and Koren present the most usual model for the

evaluation of the scallop height (figure 2) [LIN 96]. This model is bidimensional,

R 1 cos( ) r 1 cos( )+=

surface SR

r

C1

C2

P2

P1

Ball endmill

p

Rhr

h


6/20

considering the location of the tool for two adjacent single paths. Let r be the tool

radius, p the distance between two successive adjacent paths, R the local curvature

radius of the surface, evaluated in the plane in which p is defined, and h the scallop

height, then h is given by equation (4) for a convex surface and (5) for a concave one.

[4]

[5]

The approximation is obtained by supposing a very large curvature radius R relati-

vely to the tool radius r. Recent methods tend to guarantee the most constant scallop

height all tool-path long ([SUR 94], [SAR 97a], [SAR 97b]).

Here again, the calculation of the scallop height relies on strong hypothesis. For

example, the calculation model supposes that the tool locations for two successive

paths are tangent to the surface. This hypothesis is almost never verified considering

the effect of the machining tolerance. This point will be discussed next.

2.5.Interferences and discontinuities

The identification of discontinuities makes it possible to avoid collisions between the

tool and the surface, and the interferences where the tool is no longer locally tangent

to the surface. This also involves a better respect of the machining parameters since

the models used are assumed continuous.

One distinguishes two types of discontinuities (figure 3) :

- discontinuities on the surface, resulting from the design of the surface or due to

modeling errors of the CAD system,

- discontinuities on the tool-path, near points of multi-tangency or near undercuts.

Figure 3.Discontinuities on the surface and on the tool-path

h R r+( ) 1 p2R-------

2 r2 R r+( )p2R

--------------------2

R= R r p8hrR

r R+-------------

h R R r+( ) 1 p2R-------

2 r2 R r+( )p2R

--------------------2

= R r p8hrR

R r-------------


7/20

The detection of such discontinuities can be performed during the first step of calcu-

lation of the tool locations ; the tool locations are calculated avoiding possible inter-

ferences. The discontinuity detection can also be carried out during the last step by

elimination of all the singularities of the calculated tool-path : self-intersections, non-

machined portions, ... ([CHO 89],[LAI 94]).

3. Tool-path assessment

As previously exposed, the tool-path calculation is a complex problem which suppo-

ses a complete modeling of the surface to be machined in a format interpretable by

the numerical controller (NC). The four described steps use a simple modeling of the

surface so as to rapidly calculate reliable tool-paths. Each model used has a field of

validity, but the complexity of the surfaces made that the limits are often broken. The-

refore, we can plan to check the precision of the calculated tool-paths a posteriori, in

order to avoid obtaining wrong parts.

3.1.Introduction to tool-path assessment

The assessment of the calculated tool-path can be performed following two stand-

points :

- checking that the calculated tool-path respects the specified machining parameters,

- checking that the machined part meets the geometrical specifications.

This last point relies on machining simulations. Therefore, the following checking

can be thought of :

- visual checking of the free-form obtained,

- analysis and correction of the collisions,

- identification of non machined portions,

- checking of the respect of the machining tolerances.

3.2. Usual methods for tool-path assessment

The machining simulation consists in the construction of the envelope surface gene-

rated by the tool movement. This surface corresponds to the union of all the envelope

surfaces associated to elementary tool-path. Generally, the envelope surface is

approximated using methods adapted to each checking : methods of realistic rende-

ring for visual evaluation, or evaluation methods of the distance between the surface

and the tool-path for error evaluation. All these techniques are based on a discretiza-


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tion of both the surface and the tool-path. Three principal methods exist, the "point-

vector" technique , the Z-buffer method, and the solid modeling technique.

The "point-vector" technique consists in the calculation of the distance between the

machined surface and the nominal surface (to evaluate the form deviation, for exam-

ple). From a point on the surface, a line is built following a given direction. Then,

intersection points of this line with all the elementary tool-paths are calculated, and

the distance between each intersection point and the surface is evaluated. The smal-

lest distance, which leaves less material, is the error at the considered point. The set

up of this technique supposes the resolution of the following three problems :

- discretization of the surface,

- orientation of the direction vectors,- modeling of the tool-path.

The technique used by Chappel takes into account the movements of a ball endmill

tool only, but oriented in an unspecified way in space [CHA 83]. A set of "point-vec-

tors" is built on the surface, so that the points belong to the surface and the end of the

vectors belongs to the rough surface of the part. Calculation is thus reduced to calcu-

lation of intersections between segments and a cylinder. The length of the vector indi-

cates the thickness left on the part relatively to the nominal one, after the passage of

the tool. Jerard applied this concept to the machining of free-form surfaces ([JER

89a], [JER 89b]).

Figure 4. The technique "point-vector" [JER 89b]

An initial sampling of the surface is performed. Then, a planar grid, which is perpen-

dicular to the tool axis is projected onto the surface so as to only retain a set of points.

To each point, one associates a vector, the direction of which is normal to the surface,

or is parallel to the direction of the tool axis. In this last case, calculation is simplified

but less precise.

The technique of the Z-buffer is a general method which can also be applied to chec-

king methods based on the solid modeling, or on the tool-path visualization [KIM

95b]. This method relies on the construction of a grid in an xy plane. To each point of

xyz

Tool movementenvelope

Intersection points

Direction vectors

a) orientation in b) oriention in the Z-axisthe normal direction


9/20

the grid, one associates a vertical segment of an initial altitude Z. During machining,

the intersection points between segments and the tool-path are calculated. For each

segment, the point for which the altitude is the lowest is retained. Therefore, we

obtain a Z-buffer representation of the tool-path. The errors of simulation come from

the step of the grid and the modeling of the tool-path.

Figure 5.Z-buffer method [KIM 95b], application to the part figure 10 (3-axis mil.)

To evaluate errors on the calculated tool-path, the Z-buffer representation of the final

surface, of the rough surface and of the machined one must be carried out. Therefore,

machining errors, non-machined portions, volumes of removed material, and colli-

sions are obtained when considering subtractions of the different Z-buffer representa-

tions.

The solid modeling technique of simulation is based on Hanadas work [HAN 94].

This method relies on a solid modeling of the envelope of the tool-path which is onlypossible in 3-axis milling. Each elementary movement of the tool corresponds to a

primitive solid which is substracted to the solid modeling of the initial surface. We

then obtain the contribution of the elementary movement to the machined surface.

This solution is precise but very expensive in calculation time..

Figure 6. Solid modeling of the tool-path [HAN 94]

Previously exposed methods present advantages and drawbacks. The information

given is generally limited. Taking advantages of these methods, we developed assess-

ment methods for calculated tool-path that allow both the checking of the geometrical

Tool axisdirection

x

yz

Z-value

Z-map plane

rotational sweep linear sweep boolean operation


10/20

specifications and the checking of the machining parameters. Methods are general

and can be used for various types of tool geometry, and whatever the tool-path geo-

metry (linear or polynomial).

4. Our methods for the assessment of calculated tool-path

The methods suggested rely on the geometry of the tool and the geometry of the sur-

face. The first objective is to propose methods that allow checking that the tool-path

calculation algorithm provides the respect of the specified machining parameters.

These methods coupled to evaluation methods of geometrical deviations on the

machined surface, and to inspection methods allow the performance assessment of

the machining process. Moreover, the comparison between errors due to the tool-path

calculation to those resulting from the milling on the machine tool can be done [DUC

98a].

Among the methods developed, we next detail three of them. The first one concerns

the intrinsic precision of the CAM system, i.e. its capability to calculate tangent loca-

tions of the tool on the surface. The others are used so as to check that the machining

parameters are respected for the whole tool-path.

4.1. Preliminary analysis : calculation of the relative distance between a surface

and a surface of revolution

Whatever the type of tool geometry, it can be modeled by a surface of revolution:

sphere, tore, cone or cylinder. Several types of distances can be calculated between

this type of surface and the surface to be machined. If we consider couples of points

which normal on their respective surface are colinear, we obtain two couples repre-

sentative of the smallest and the largest distances between both surfaces. None of

these distances corresponds to the distance characteristic of a location error.

To evaluate the location error, it is necessary to restrict the machined surface to a

curve, defines as the most probable contact curve between the tool and the surface.

When the tool is tangent to the surface, the contact exists. In the case of interference

between the tool and the surface, the zone of contact is a surface and the contact

curve is defined considering the points most in interference. If there is no contact, the

curve is defined as the closest curve of to the surface of revolution (figure 7).

The location error is calculated from the position of the point on the contact curve

furthest away from the tool axis. The error corresponds to the distance of the point to

the tool axis minus the tool radius. To calculate this distance a sampling of the axis is

carried out. Then, each point is projected onto the surface to be machined, so as to


11/20

define the most probable contact curve. Note that the contact curve is defined by a set

of points. Each projected point obtained is thus projected orthogonally onto the sur-

face of the tool so as to calculate its corresponding distance. Instead of evaluating the

distance to the surface, the distance to the axis of the surface is calculated. The local

value of the tool radius is substracted to the calculated distance. Therefore, the obtai-

ned value can be positive or negative when an interference exists.

Figure 7.Distance between a surface and a surface of revolution

4.2.Evaluation of the location errors

The purpose is to evaluate the relative distance between the tool, in a given position,

and surface to be machined. The objective of tool-path calculation is to define pas-

sage points which are tangent to the surface, i.e. tool positions which distance to the

surface is null. If the tool is in interference with the surface, the evaluated deviation is

negative, whereas if the tool does not touch the surface, the deviation is positive.

Indeed, the precision of the calculation of the tool locations is strongly linked to the

tool-path calculation method and to the modeling of the surface. For instance, the Z-

buffer method relying on a sampling of the surface does not provide the same loca-

tion errors than a direct method of calculation. The complexity of the calculation of

tangent locations of the tool makes that none of the existing methods lead to an error

free tool-path. As a result, the location error reflects the quality of the algorithms cho-

sen.

To calculate the relative distance between the tool and the surface, the modeling of

the surface geometry and of the active part of the tool is necessary. Both of them can

be carried out by exact surface modeling or by meshing. In fact, the meshing of the

tool surface is not necessary, only the sampling of the characteristic element associa-

ted to the tool is required. These geometric elements are very simple : point for a ball

surface

tool axis

contact curve

tool


12/20

endmill tool, line segment for the conical tool and circle for the filleted one (toric).

On the other hand, according to the type of tool and associated movements, the sur-

face must be meshed. The precision of the evaluation of the error location is directly

linked to the precision of the surface meshing.

Hereafter we study the error location analysis for the 3 types of usual tool geometries.

The ball endmill tool

The characteristic element is a point : the tool center P0 (figure 8). The error is calcu-

lated as the distance between the tool center and the surface minus the tool radius.

Figure 8.Location error for the ball endmill tool and filleted endmill tool

Note that, as the calculation is carried out by projecting the tool center onto the sur-face (cf 4.1), several projected points may exist. Therefore, the location error is eva-

luated from equation (6), considering the smallest distance between the tool center

and the surface

[6]

The filleted endmill tool

In this case, the characteristic element is a circle. The previous method is extended

for all the points of the circle, and the location error corresponds to the smallest

deviation when all the points of the circle are considered. Note that the deviation can

be positive or negative according to possible interferences. Therefore, the smallest

value may correspond to the largest interference. The sampling of the circle is carried

out by a set of points determined so that the distance between each chord, defined by

two successive points, and the circle is less than 1 micrometer. Then, the problem can

be solved by dichotomy.

The cylindrical or the conical tool

P2

P0

P1

tool

tool axis

surface

projected curve

P0

P1

location_error min d P0

Pi

( )( ) r_tool= i 1 .. n, ,=


13/20

The location error is directly induced by the relative distance between a surface of

revolution and a surface, as previously exposed (cf 4.1). The sampling of the tool

axis is projected onto the surface. For each point of the obtained curve, the calcula-

tion of the deviation to the tool axis minus the value of the tool radius at the conside-

red point gives the location error.

Applications of the methods for calculation of error locations have been done in pre-

vious works [DUC 98b]. As expected, results have shown that usual calculation

methods of the tool-path used by CAM system create location errors in the resulting

tool-path.

4.3.Method to evaluate the respect of the machining tolerance

During the elementary movement of the tool between two tangent locations, the tool

follows a trajectory depending on the interpolation format : linear, circular or polyno-

mial. Except for some particular cases, the tool cannot be tangent to the surface

throughout the whole movement. Then, there exists a deviation between the expected

surface and the machined one. This deviation must remain less than a threshold: the

machining tolerance. Therefore, if the deviation is greater than the machining tole-

rance, an error exist, called the chordal error. Thus, the evaluation of the chordal

error consists in the evaluation of the deviation from the envelope surface of the ele-

mentary movement of the tool to the nominal surface, and to compare this deviationto the specified value.

In the case of 3-axis milling with a ball endmill tool, the envelope of the tool move-

ment is a pipe surface which spine is given by the elementary tool-path. Taking the

example of the linear interpolation, the elementary tool-path is a linear segment, whe-

reas it is a free-form curve for the polynomial interpolation. The generatrix of the

pipe surface is given by the projection of the tool in a plane perpendicular to the ele-

mentary trajectory (figure 9).

The calculation of the distance from this envelope surface to the surface to be machi-

ned is carried out as in paragraph 4.1.

[7]

Generally, the projection of the elementary tool-path onto the surface is performed

using a sampling (figure 9). Therefore the chordal error is given by the following

equation (8) :

[8]

chord_ error max d M C,( ) r_tool( )= M Proj(C Surface )

chord_error max d Mi

C,( ) r_tool( )= i 1 .. n, ,=


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For a sampling with 10 points on a linear segment, the error of evaluation is lower

than 1 %, when applying the usual models. Thus, the discretization permits to quan-

tify precisely if the machining tolerance is respected, when the value of chordal_error

is less than the machining tolerance specified. However, it is not possible to evaluate

if low-size details are forgotten during the tool-path calculation. Nevertheless, this

calculation method is largely used in [VAL 00] for a tool-path in a B-spline format.

Figure 9.Evaluation of the chordal error

In 5-axis milling, the movement is more complex due to the evolution of the tool axis

direction considering the two associated rotation movements. Nevertheless, it can be

possible to evaluate the relative position between the machined surface and the nomi-

nal one by building the envelope surface of the tool movements. This calculation is

carried out considering the speed vector of each point of the tool during its move-

ments. Therefore, a point M of the tool surface belongs to the envelope surface of the

tool-path if . where is the speed vector of the point M, and is

the normal to the surface at the M point. The case of the toric tool with the linear

interpolation leads to an implicit equation (9) of the envelope surface, function of the

toric surface parameters (,) and the kinematics parameters of the tool-path

:

[9]

We have to point out that the construction of the envelope surface is based on a preli-

minary step of discretization. The envelope surface is then defined considering a set

of points. The calculation of the distance between the envelope surface and the nomi-

nal allows the verification of the respect of the machining parameters. Note that, the

previous calculation gives more than the chordal error, for in 5-axis, errors are also

envelope of the elementary tool path

tool path C

evaluation points

tool path projected onto the surface Mi

machined part

VM nM 0= VM nM

VM1 C( , )

( )tan( )cos VM1 u1 ( )sin VM1 u2+( )

R ( )sin u1

c

R ( )cos u2

c

u VM1+( )------------------------------------------------------------------------------------------------------------------=


15/20

linked to the evolution of the tool axis direction. As a result, the previous calculation

leads to the evaluation of the geometric deviation, representative of the deviation

between the machined surface and the expected one.

Same remarks can be done in 5-axis milling by the tool flank. The figure 10 is an

illustration of the calculation of the envelope surface in flank milling. The curves are

representative of the envelope surface, and are obtained from a sampling throughout

the tool movement of equation (9). The distance calculation between the envelope

surface and the nominal one is then carried out through the envelope curves [DUC

98b], and leads to the set of geometrical deviations (between the nominal surface and

the machined one) which must be compared to the authorized value. Such evaluation

method is of great interest to assess the methods of tool-path calculation in flank mil-ling. The most usual application concerns aeronautics parts.

Figure 10. Calcul of the envelope surface in flank milling

4.4.Method to evaluate the respect of the maximum scallop height

The left peak on the part is due to the sweeping of surface by ball or toric endmill

tools. As in previous sections, the objective of the calculation method is to check that

the scallop height is less than the specified value.

Most of the checking methods recommend the search of the intersection curve

between two successive single paths and the calculation of the distance of this curve

to the surface (hc0) (figure 11).

We estimate that this result does not give a value faithful to reality. Indeed, in func-

tion of the location of the point, it may happen that the tool is not tangent to the sur-

face, but rather located at a distance equal to the machining tolerance value

(figure 11). To solve this problem, we propose to consider the intersection curves

between the three closer single paths (hc1) (figure 11). Therefore, we directly obtain

surface

envelope curves

tool-path curves n VM


16/20

the form left by the passage of the tool whatever the free-form surface to be machined

and the format of the tool-path. We calculate the track left by three successive paths,

in a plane perpendicular to the speed vector at the considered point. The scallop hei-

ght is equal to the distance between the chord connecting the intersection points to

the track of the tool.

Figure 11. Usual calculation of the scallop height - our model

This calculation method relies on the calculation of the intersection of a circle and a

pipe surface. In the plane perpendicular to the displacement, the track of the tool

movement is a circle. This one cuts the envelope surface of the closest displacement.

Considering the linear interpolation, the envelope surface is a cylinder (figure 12).

Figure 12. Calculation of a point of the scallop curve

Let us consider a circle C, which center is O, the radius of which is R, and for which

constitutes a vector basis. Let Cy be the cylinder of radius R, which axis is

defined by a point C1 and y the unit vector .

hc1

plane

elementary tool-path

hc0

R

M

O

C

Cy

C1

a1

u1 u2,( )

a1


17/20

The point M, belonging to the scallop curve is defined so that its distance to the cylin-

der axis is equal to the value of the radius R :

[10]

[11]

The result is a polynomial equation of degree two in sine and cosine, that can be sol-

ved numerically.

The application concerns the machining of a portion corresponding to the linking of

two surfaces (figure 13). The surface is machined with a ball endmill tool with a

machining tolerance equal to 0,01 mm, and with a maximal scallop height allowed

equal to 0,003 mm. Two machining directions are tested : in the direction of the lin-

king and in the normal one. The tool-path is calculated using a CAM system.

Figure 13. Tool-path and evaluation of hc1

The respect of the maximal allowed value is assessed using the previous calculation

method of hc1 and considering a set of points distant of 1mm onto the tool-path. The

comparison with the usual evaluation of hc0 is also carried out. Results are presented

table 1. Results obtained with hc1 and hc0 are different. The calculation of hc1 indu-

ces a larger number of points in error, for which the value is greater than 0.003 mm.

M C OM R u1cos R u2sin+=

M Cy C1

M a1 R=

R2

C1M a1( )

2=

R2

C1

O R u1

cos R u2

sin+ +( ) a1

( )2

=

0 C1O a1( )

2R u1cos a1( )

2R u2sin a1( )

2+ +=

2R cos u1 a1( ) C1O a1( ) 2R2

cos sin u1 a1( ) u2 a1( )+ +

2R sin u2 a1( ) C1O a1( ) R2

+

test A evaluation of hc1 test B evaluation of hc1


18/20

In fact, the calculation of hc0 is faithful to the calculation method of the tool-path.

Only a few number of points are in error with hc0 that corresponds to the required

precision for the tool-path calculation. However, the calculation of hc1 seems a better

representation of the relative location of the single paths. Indeed, points that do not

belong to a scallop curve due to the influence of the machining tolerance appear. In

the same way, we can consider that values of hc1 between 0.003 mm and 0.01 mm are

directly linked to the combination of the scallop error and the chordal error. As a

result, the calculation is impossible for a larger number of points, considering that the

calculation requires the correct definition of three adjacent paths.

In figure 13, points for which hc1 is greater than 0.003 mm are represented. It can be

seen that errors obtained with test A are concentrated near the linking zone, whereas

they are better averaged with the other machining direction. Moreover, values aregreater with test A than with test B. This can be due to the limits of the CAM system

and its tool-path calculation method. Indeed, the linking zone constitutes a geometric

singularity difficult to take into account.

To conclude, the evaluation of the scallop error with the proposed method is an indi-

cator that shows in particular limits of the tool-path calculation methods.

Table 1 : Evaluation of hc1 and hc0 for tests A and B

Test A Test B

Number of calculated points 13759 6462

hc1 < 0,003 mm 11442 4601

hc1 < 0,004 mm 12259 5954

0,003 < hc1 < 0,01 mm 1221 1675

0,01 < hc1 < 0,1 mm 100 0

hc1 > 0,1 mm 0 0

no scallop 740 52

impossible calculation 156 134hc0 < 0,003 mm 12919 4687

hc0 < 0,004 mm 13579 6428

0,003 < hc0 < 0,01 mm 662 1741

0,01 < hc0 < 0,1 mm 2 0

hc0 > 0,1 mm 0 0

impossible calculation 76 34


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5. Conclusion

The evaluation of the precision of calculated tool-path is an important issue due to the

increase in the expected quality of the machined surfaces. In this paper, we suggest

methods to evaluate errors linked to tool-path calculation methods. In particular, the

methods allow the evaluation of the location error, the chordal error, and the scallop

error. Each error is associated to a step of tool-path calculation. Previous works have

shown the interest of the location error and its influence on the following calculation

steps.

The evaluation of the chordal error is presented. It is based on the calculation of the

envelope surface of the tool movements. Only elementary tool movements are consi-

dered in 3-axis milling, whereas the whole envelope surface corresponding to the

whole tool movement is necessary in 5-axis. Indeed, in 5-axis, the evolutions of the

tool axis direction must also be envisaged. Therefore, the calculation of the distance

between the envelope surface and the nominal surface leads to the evaluation of the

chordal error in 3-axis and to the a global error in 5-axis (the geometrical deviation).

This last calculation is particularly interesting for the assessment of tool-path calcula-

tion methods in flank milling.

Concerning the last point, the suggested method for the evaluation of the scallop error

is different for usual methods exposed in literature. Indeed, it is based on the evalua-

tion of scallop curves for three successive single paths. The resulting indicator of the

scallop error may bring out limits of the tool-path calculation method.

6. References

[CHA 83] I.T Chappel, The use of vectors to simulate material removed by numerically

controlled milling, Computer Aided Design, vol. 15, no. 3, p. 156-158, 1983

[CHO 89] B.K Choi, C.S Jun, Ball-end cutter interference avoidance in NC machining of

sculptured surfaces, Computer Aided Design, vol. 21, no. 6, p. 371-378, 1989

[CHO 94] B.K Choi, Y.C Chung, J.W Park, D.H Kim, Unified CAM-system architecture

for die and mould manufacturing, Computer Aided Design, vol. 26, no. 3, pp. 235-243,

Mars 1994.

[DRA 97] D. Dragomatz, S. Mann, A classified bibliography of litterature on NC milling

path generation, Computer Aided Design, vol. 29, No. 3, pp. 239-247, 1997[DUC 98a] E Duc, C Lartigue, F Thibaut, A test part for the machining of free form surfa-

ces, Improving Machine Tool Performance 98, vol. 1, pp. 423-435, 1998

[DUC 98b] E Duc, Usinage de formes gauches, contribution lamlioration de la qualit

des trajectoires dusinages, doctorat de lEcole Normale Suprieure de Cachan, 1998

[GEO 95] K.K George, N Ramesh Babu On the effective tool path planning algorithms for

sculptured surface manufacture, Computers Industrial Engineering, vol. 28, no. 4, p. 823-

838,1995


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duc emmanuel revised

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