Algorithms for the evaluation of errors in the calculation of NC tool-paths Emmanuel DUC, Claire LARTIGUE Laboratoire Universitaire de Recherche en Production Automatisée Ecole Normale Supérieure de Cachan 61 avenue du Président Wilson 94235 CACHAN cedex – France E-mail : [email protected] ABSTRACT : The assessment of tool-path calculation methods 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 these tool-path calculation methods. In particular, the methods allow the evaluation of the location error, the chordal error, and the scallop error, each corresponding to a step of the tool-path calculation. KEY WORDS : CAM, Free Form machining, Tool-path assessment

1. Introduction Nowadays, the quality of free-form surfaces takes advantage of High Speed Milling (HSM). The use of HSM implies new requirements in the process. As the cutting process is improved, the final shape of the part results from the tool motions. Thus, an error in the calculation of the tool-path may lead to a low-quality part surface. Therefore, evaluating the performance of the tool-path calculation method is now a real need. The paper deals with algorithms for evaluating errors linked to the calculation of the tool-path. A first analysis shows that, whatever the method used, calculation of the NC tool-path relies on approximations, often source of errors. More generally, calculation methods are sequential, and consist of three main steps: calculation of toolpath locations, calculation of a single path in the driving tool direction, calculation of adjacent paths in the perpendicular direction. The most significant errors due to approximation techniques concern the use of a 2D modeling of the geometry whereas the problem is 3D. Furthermore, both last steps use discretization criteria not necessarily well defined. Nevertheless, errors cannot be avoided. However, they need to be in accordance with the specified tolerances and criteria, and this for the complete calculated tool-path. In this paper, we present the algorithms we defined for evaluating the errors in the

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 deviations. 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 error and the scallop error, linked to the discretization steps. Therefore, 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 toolpath 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 interferences 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 geometric 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.

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 surface ( figure 1) and the following definitions : u

u R

CE CC

CE CL

r

n

r CL

CC

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, - n , is the normal to the surface at the CC point, and u , 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) : OC E = OC C + rn

OC L = OC E – ru = OC C + rn – ru

[1]

In the case of the filleted endmill, the locations are given by equations (2) : u∧n k = ----------------u∧n

k∧u OC E = OC C + rn + R ----------------k∧u

[2]

k∧u OC L = OC E – ru = OC C + rn + R ----------------- – ru k∧u 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] ) :

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 interferences 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 calculation 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 calculated 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 minimal 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 evaluated through the calculation of the chordal deviation δ. To calculate the chordal deviation, Kuragano and George break this error up into two errors : - error due to the dividing of the curve into chords (distance between the line linking the contact points and the surface),

- 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 δ is then given by (figure 2) : δ = R ( 1 – cos θ ) + r ( 1 – cos θ )

[3]

Ball endmill C2

C1 r

p θ

θ

δ

P1 surface S

θ

P2 θ

r R

h’

h R

Figure 2. Evaluation of the chordal deviation δ [KUR 92] and the scallop height h The objective is then to calculate successive tool locations so that the chordal deviation δ is less than the machining tolerance. However, the calculation of the chordal deviation 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 acceptable 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,

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. 2 p 2 ( R + r )p 2 h = ( R + r ) 1 –  ------- – r – -------------------- – R  2R 2R

8hrR R » r ⇒ p ≈ ------------r+R

[4]

2 p 2 ( R + r )p 2 h = R – ( R + r ) 1 –  ------- – r – ------------------- 2R 2R

8hrR R » r ⇒ p ≈ ------------R–r

[5]

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

The detection of such discontinuities can be performed during the first step of calculation of the tool locations ; the tool locations are calculated avoiding possible interferences. 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, nonmachined portions, ... ([CHO 89],[LAI 94]).

3. Tool-path assessment As previously exposed, the tool-path calculation is a complex problem which supposes 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. Therefore, 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 standpoints : - 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 generated 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 rendering 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-

tion of both the surface and the tool-path. Three principal methods exist, the "pointvector" 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 example). 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 smallest 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-vectors" 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 calculation of intersections between segments and a cylinder. The length of the vector indicates 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]). Tool movement envelope

Intersection points x Direction vectors y b) oriention in the Z-axis a) orientation in the normal direction Figure 4. The technique "point-vector" [JER 89b] z

An initial sampling of the surface is performed. Then, a planar grid, which is perpendicular 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 checking 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

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. Z-value z

y x

Tool axis direction

Z-map plane

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 collisions are obtained when considering subtractions of the different Z-buffer representations. The solid modeling technique of simulation is based on Hanada’s work [HAN 94]. This method relies on a solid modeling of the envelope of the tool-path which is only possible 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.. rotational sweep

linear sweep

boolean operation

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

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 geometry (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 surface. 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 locations 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 representative 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

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 surface 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 obtained value can be positive or negative when an interference exists.

surface tool axis contact curve tool

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 passage 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 Zbuffer method relying on a sampling of the surface does not provide the same location 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 chosen. 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 associated to the tool is required. These geometric elements are very simple : point for a ball

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 surface 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 calculated as the distance between the tool center and the surface minus the tool radius. tool axis tool

surface P0

P0 P1 P2

projected curve P1

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 surface (cf § 4.1), several projected points may exist. Therefore, the location error is evaluated from equation (6), considering the smallest distance between the tool center and the surface location_error = min ( d ( P 0 P i ) ) – r_tool

i = 1, .., n

[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

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 calculation of the deviation to the tool axis minus the value of the tool radius at the considered point gives the location error. Applications of the methods for calculation of error locations have been done in previous 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 polynomial. 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 tolerance, 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 elementary movement of the tool to the nominal surface, and to compare this deviation to the specified value. In the case of 3-axis milling with a ball endmill tool, the envelope of the tool movement 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, whereas 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 elementary trajectory (figure 9). The calculation of the distance from this envelope surface to the surface to be machined is carried out as in paragraph 4.1. chord_ error = max ( d ( M, C ) – r_tool )

M ∈ Proj(C → Surface )

[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) : chord_error = max ( d ( M i, C ) – r_tool )

i = 1, .., n

[8]

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 quantify 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. envelope of the elementary tool path tool path C evaluation points tool path projected onto the surface Mi machined part 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 nominal 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 movements. Therefore, a point M of the tool surface belongs to the envelope surface of the tool-path if V M ⋅ n M = 0 . where V M is the speed vector of the point M, and n M 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 (β,θ) and the kinematics parameters of the tool-path (V M1,Ω C) : – ( cos ( θ ) V M1 ⋅ u 1 + sin ( θ ) V M1 ⋅ u 2 ) tan ( β ) = -----------------------------------------------------------------------------------------------------------------( R sin ( θ ) u 1 ⋅ Ω c – R cos ( θ ) u 2 ⋅ Ω c + u ⋅ V M1 )

[9]

We have to point out that the construction of the envelope surface is based on a preliminary 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 nominal 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

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 milling. The most usual application concerns aeronautics parts.

surface envelope curves tool-path curves

n

VM

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 function of the location of the point, it may happen that the tool is not tangent to the surface, 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

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 height is equal to the distance between the chord connecting the intersection points to the track of the tool.

plane

θ

hc0

hc1 elementary tool-path

R

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).

Cy a1 C1

O C M

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 ( u 1, u 2 ) 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 a 1 .

The point M, belonging to the scallop curve is defined so that its distance to the cylinder axis is equal to the value of the radius R : M ∈ C ⇒ ∃θ M ∈ Cy ⇒ 2

R = ( C1 M ∧ a1 )

OM = R cos θu 1 + R sin θu 2

[10]

C1 M ∧ a1 = R

2

2

R = ( ( C 1 O + R cos θu 1 + R sin θu 2 ) ∧ a 1 ) 2

2

2

0 = ( C 1 O ∧ a 1 ) + ( R cos θu 1 ∧ a 1 ) + ( R sin θu 2 ∧ a 1 )

2

[11]

2

+ 2R cos θ ( u 1 ∧ a 1 ) ( C 1 O ∧ a 1 ) + 2R cos θ sin θ ( u 1 ∧ a 1 ) ( u 2 ∧ a 1 ) + 2R sin θ ( u 2 ∧ a1 ) ( C 1 O ∧ a 1 ) – R

2

The result is a polynomial equation of degree two in sine and cosine, that can be solved 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 linking and in the normal one. The tool-path is calculated using a CAM system.

test A

evaluation of hc1

test B

evaluation of hc1

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

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

Test B

Number of calculated points hc1 < 0,003 mm

13759 11442

6462 4601

hc1 < 0,004 mm

12259

5954

0,003 < hc1 < 0,01 mm

1221

1675

0,01 < hc1 < 0,1 mm

100

0

0

0

no scallop impossible calculation hc0 < 0,003 mm

740 156 12919

52 134 4687

hc0 < 0,004 mm

13579

6428

662

1741

0,01 < hc0 < 0,1 mm

2

0

hc0 > 0,1 mm

0

0

impossible calculation

76

34

hc1 > 0,1 mm

0,003 < hc0 < 0,01 mm

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 are greater 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 indicator that shows in particular limits of the tool-path calculation methods.

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 considered 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 calculation 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 evaluation 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 Thiébaut, A test part for the machining of free form surfaces, Improving Machine Tool Performance ’98, vol. 1, pp. 423-435, 1998 [DUC 98b] E Duc, Usinage de formes gauches, contribution à l’amélioration de la qualité des trajectoires d’usinages, doctorat de l’Ecole Normale Supérieure 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. 823838,1995

[HAN 94] T. Hanada, S. Natsume, T. Hoshi, Development of automated toolpath interference check system by using solids models, International Japan Society of Precision Engineering, vol .28, no. 4, p. 379-380, 1994 [HWA 92] J.S Hwang, Interference-free tool-path generation in the NC machining of parametric compound surfaces, Computer Aided Design, vol. 24, no. 12, pp. 667-676, 1992 [HOC 97] S Hock, high speed cutting (HSC) in die and mould manufacture, 1st Internatio nal German and French Conference on HIGH SPEED MACHINING, 1997 [JEN 96] C.G Jensen, D.G Anderson, A review of numerically controlled methods for finish-sculptured-surface machining, IIE Transactions, Vol. 28, pp. 30-39, 1996 [JER 89a] R.B Jerard, R.L Drysdale, J. Magewick, Methods for Detecting Errors in Numerically Controlled Machining of Sculptured Surfaces, IEEE computer Graphics & Applications, p. 26-39, 1989 [JER 89b] R.B Jerard, S.Z Hussaini, R.L Drysdale, B. Schaudt,Approximate methods for simulation and verification of numerically controlled machining programs,The Visual computer, no. 5, p. 329-348, 1989 [KIM 95] K.I Kim, K. Kim, A new machine strategy for sculptured surfaces using offset surface, International Journal of Production Research, vol .33, no. 6, pp. 1683-1697, 1995 [KIM 95b] C.B Kim, S. Park, M.Y Yang, Verification of NC tool path and manual and automatic editing of NC code, International Journal of Production Research, vol. 33, no. 3, p. 659-673, 1995 [KUR 92] T. Kuragano, FRESDAM system for design of aesthetically pleasing free-form objects and generation of collision-free tool paths, Computer Aided Design, vol. 24, no. 11, p. 573-581, 1992 [LAI 94] J.Y Lai, D.J Wang, A strategy for finish cutting path generation of compound surfaces, Computers in Industry, vol. 25, p. 189-209, 1994 [LIN 96] R.S Lin, Y. Koren, Efficient tool-path planning for machining free-form surfaces, Transaction of ASME, Journal of Engineering for Industry, vol. 118, p. 20-28, 1996 [LON 87] G.C Loney, T.M Oszoy, NC machining of free form surfaces, Computer Aided Design, vol. 19, no. 2, p. 85-90, Mars 1987 [SAR 97a] R. Sarma, D. Dutta, The Geometry and Generation of NC Tools Paths - Journal of Mechanical Design, vol 119, p 253-258, 1997 [SAR 97b] R. Sarma, D. Dutta, An integrated system for NC machining of multi-patch surfaces, Computer Aided Design, vol 29, no 11, p 741-749, 1997 [SAI 91] T. Saito, T. Takahashi - NC machining with G-buffer Method - Computer Graphics, vol. 25, no. 4, p. 207-216, Juillet 1991 [SCH 95] H Schulz, High-speed milling of dies and moulds, cutting conditions and techno logy, Annals of the CIRP, Vol. 44, No. 1, pp. 35-38, 1995 [SUR 94] K. Suresh, D.C.H Yang, Constant Scallop-height Machining of Free-form Surfaces, Journal of Engineering for Industry, vol 116, p 252-259, 1994 [SUZ 91] H Suzuki, Y Kuroda, M Sakamoto, S Haramaki, H van Brussel, Development of the CAD/CAM System based on parallel processing and inverse offset method, Transputing’91 Proceeding of the world Transputer User Group (WOTUG) conference, 1991 [TUS 93] J. Tustly, High-Speed Machining, Annals of the CIRP, Vol. 42, No. 2, pp. 733738, 1993 [VAL 00] L. Valette, E. Duc, C. Lartigue, A method for computation and Assessment of NC Tool-paths in free form curve format, Improving Machine Tool Performance ’00, 2000