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Journal of Chongqing University-Eng. Ed. Vol. 1 No. 2 December 2002 Tool selection and collision-free in 5-axis numerical control machining of free-form surfaces* YANG Changqi, QIN Datong, SHI Wankai State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400044, P. R. Chinna Received 29 August 2002; revised 15 October 2002 Abstract: The methodology of 5- axis cutter selection to avert collision for free-form surface machining by flat- end cutters is presented. The combination of different cutters is adopt aiming at short machining time and high precision. The optimal small cutter is determined based on the geometric information of the points where a cutter most probably collide with the machined surface. Several larger cutters are selected to machine the surface in order to find the interference-free area. The difference of machining time for this area between the optimal small cutter and the large cutters is calculated. The functional relationship between the machining time and the radius of a cutter is established, by which the optimal number of cutters is obtained. The combination of cutters, which possesses the minimum overall machining time, is selected as the optimal cutter sizes. A case study has demonstrated the validity of the proposed methodology and algorithms. Keywords: optimal tool selection; collision- free; tool path generation; free- form surfaces; flat- end cutter YANG Changqi (杨长祺): Male; Born 1974; PhD candidate; Research fields: CAD and CAM. * Funded by the Doctorate Degree Program Foundation of the Ministry of Education (No. 2000061120) 1. Introduction It is the most important process to design tool paths for a free-form surface. In order to meet the requirement of machining precision, the scallop height must be less than the permissive error. To achieve this aim, the usual approach is to make tool paths dense enough. However, the denser the tool paths are, the more the machining time is needed. So it is expected to have sparse tool paths and meet the requirement of the machining precision. From a theoretic view point, this can be achieved by using a big cutter. But the larger the cutter is, the more interference between the cutter and the surface occurs, and also the more area of the surface is left inaccessible. Therefore, it is necessary to select the optimum cutter or a combination of different cutters for the aim of less machining time and perfect machining precision. Recent researches concentrate on the issue about how to select the optimal tool in 3-axis numerical control (NC) machining of free- form surface 1-3. Some tool and die makers have found that, by changing from 3-axis to 5-axis milling, efficiency can be improved by 10 to 20 times 4, 5. Furthermore, a ball- end cutter has disadvantages including low cutting efficiency on the neighborhood of the axis, small cubage to contain scraps and abominable cutting condition when the area of little curvature is machined. However there are no such problems for a flat-end cutter. So the flat-end cutter is increasingly paid attention to. This paper presents a new approach of 5- axis cutter selection for free- form surface machining. By this approach, those points on the surface where collision of the cutter with the surface most probably occurs can be found. With the geometric information of these points, the optimum small cutter can be determined. An algorithm of constant scallop height is applied to dispose the surface tool paths using this optimum small cutter. The research work also includes several cutters selected to machine the surface which are larger than the optimal small one. When these larger cutters move along the tool path mentioned above, the interference- free area of the surface can be found. In this area, the difference of machining time between the optimum small cutter and larger ones can be calculated. And there exists a functional relationship between the difference of machining time and cutter radii. the number of optimal cutters can be calculated by this function. If the number is larger than 1, the radii of larger cutter can be calculated and selected. 2. Selection of the optimum small cutter To select the optimum cutter without local gouging and global collision with the surface, the primary task is to find the areas where cutters most probably collide with the surface. It is easy to find these areas for a general regular surface by converting parameter curves into curvature ones. These points having maximal principal curvature are likely to be those where local gouging most probably takes place. But it is difficult to do so for a free-form surface. By scattering the free- form surface into dense points, the points having maximal principal curvature can be found. After these points are found, the optimum small cutter can be selected by comparing the local geometric information of points with the shape of the cutter. 2.1 Shape of cutter When the cutter moves along a tool path, the swept surface of the cutter is formed. Obviously, the swept Journal of Chongqing University-Eng. Ed. Vol. 1 No. 2 21 surface of the cutter is tangential to the surface along the tool paths. So the swept surface satisfies two conditions: 1) it is tangential to the surface; 2) the normal of the swept surface is identical with that of the point on the tool path. The swept surface is defined as ( , )SS p = where p and are parameters of the tool paths (as shown in Fig. 1). The coordinates of the cutter blade are transformed from the global reference frame into the local reference frame by ( , )S p cos cos (cos1)sin sin sincos (cos1)cos sin sin cossin RR RR RR =+ + where R is the radius of cutter; is the tilting angle ; and is the rotating angle. Fig. 1. Relation between cutter and tool-path From the conditions satisfied by the swept surface, an inference can be drawn as follows. Sp(p,0)=rp = r ud d u p + r vd d v p S(p,0)= Rsinx+Rcosy Spp(p,0)= rpp Sp (p,0)=Rsind dp x +Rcos d dp y p(cosx+siny) S (p,0)= R cos cosx R cos sin y+R sinz ns(p,0)= sp(p,0)s (p,0)/sp(p,0)s (p,0)= nr (1) where r=r(u,v) is a point on the surface which is deter- mined by the surface parameters u and v. Thus the first fundamental quantity and second fundamental quantity of the swept surface of the cutter are given by Es= Sp S p= rp r p Fs= Sp S =Rsinrp Gs= S S =R 2 Ls= ns S pp= nrpp Ms=ns S p=Rsin pp p n r r +Rrp(kmaxkmin)cos (sin2)/2 Ns= ns S = Rsin (2) Deriving from the identity n x=0, we have n xp=rp(kmaxkmin) sin cos (3) dS= Sp dp+S d (4) where dS is an arbitrary direction which is xa in the tangent plane of the surface as shown in Fig.2, and can be expressed in terms of x and y of the surface frame. Hence, there exist the parametric increments cos() cos d d sin cos pr p R = (5) Then, the effective curvature of the swept surface of the cutter is obtained as follows. 22 ,eff 22 d2d dd II I d2 d dd sss s s sss L pMpN k EpF pG + = + 2 2 sinsin cosR = 22 maxmin 2 maxmin 2 ()sin2 sin2 sincos ()sin 2cos kk kk 222 maxminmin 2 sinsinsincos ()sin2 coscos kkk (6) Fig. 2. Relation among cutting direction,random direction and principal direction 2.2 Comparison between local geometric information of the surface and cutting shape of the cutter The effective normal curvature of the CC (cutter contact) point in the xz plane can be expressed using Eulers formula for normal curvatures as follows. k=kmincos2 + kmaxsin2 (7) As shown in Fig. 2, there exists = (8) Substituting Eq. (8) into Eq. (7) gives k=kmax(kmaxkmin)(cos2cos2+0.5sin2 sin2+ sin2 sin2) It is necessary to ensure that k of the surface is less than k,eff of the cutter swept surface in order to avoid the interference between the cutter and the surface, i.e. k,eff k= 2 2 max 2 sinsin (cos () cos k R kminsin2()0 When the condition = is satisfied, the minimum of k,eff kis obtained as follows: (k,eff k)min= 2 max 2 sinsin ( cos )k R (9) Since sin2/cos2 0 is always true, the formula (9) can be simplified into (sin)/R kmax 2.3 Avoiding global collision and selecting the optimum small cutter There are two ways to avoid global collision between the cutter and the surface: 1) After the vector of the cutter axis is calculated under the local gouging- free condition, fix the cutter and check if global collision occurs or not. Re-adjust the vector of the cutter axis if global collision does occur. 2) After the range of the vector that the cutter axis can reach is calculated under the global collision-free condition, check local gouging. However, during selecting a cutter, the cutting direction is uncertain. So local gouging is unable to be determined. Thus, the first way is not suitable to the selection of the cutter. YANG Changqi, et al / Tool selection and collision- free in 5- axis numerical control machining Vol. 1 No. 222 A flow chart summarizing a potential procedure for global collision-free machining is shown in Fig. 3, where max is the maximal angle between the vector of the cutter axis and the normal vector of the point p. Fig. 3. The flow chart of searching maximum angle that the cutter axis vector can arrived Using the anti-insertion knots algorithm6, the control points of free- form surface are readjusted. Among the lines connecting grid vertex with CC point, the line that forms a minimum angle with the normal vector of CC point is selected. As a result, the cone used to detect global collision is formed. As for this cone, the line is the generatrix, the CC point is the vertex, and the normal vector of the CC point is the axis. When the vector of the cutter axis locates within cone, there is no global collision. The radius of the optimum small cutter is given by R=(sinmax)/kmax (10) where max is the half conical angle. 3 Selection of the large cutter With the optimum small cutter determined, the tool paths can be marked out using various kinds of programming algorithms. We adopt the algorithm of constant scallop-height which aids in improving 5-axis machined surface quality and automating the non- isoparametric cutter path generation for CAD/CAM systems. The cutting width of a flat-end cutter is given by 7 La=RcosLsinLsinaRcosLsinL+ RcosLcosa Lb=RcosLsinLsinbRcosLsinL+ RcosLcosb L= La +Lb (11) where L is the cutting width; R is the radius of the cutter; L is the inclination angle; L is the tilt angle; a and b are two angles illustrated in Fig. 4. From Eq. (11), it is deduced that the cutting width is proportional to the radius of the cutter when L and L are constant. As the large cutter moves along the tool paths marked out by the optimum small cutter, k,eff k is determined by k,eff k= 2 22 maxmin 2 sinsin (cos ()sin () cos L kk R where RL is the radius of the large cutter.Since 2 2 sin cos 0 is always satisfied, only the following terms need be calculated. Fig. 4. Cutting breadth of the cutter (sin)/RLkmaxcos2() kminsin2() (12) where , and are the calculated angles of the optimum small cutter at the CC point; and kmax and kmin are the principal curvatures at the CC point. There is no local gouging at the CC point when the value of formula (9) is larger than 0. On the other hand, the position of cutter where global collision most probably occurs is the conical generatrix paralleled to the vector of the cutter axis and passed through the CC point. Fig. 5 shows the critical position of the vector of the cutter axis without global collision at the CC point. So there is no interference when the value of formula (9) is larger than 0 and the vector of the cutter axis lies in the cone Fig. 5. The relation between the vector of cutter axis and the cone for checking collision If the values of , and are unequal to those values calculated for the optimum small cutter, it is possible to avoid local gouging for a large cutter. But its cutting breadth has to be decreased in regulating , and to avoid local gouging. The change of the cutting breadth of the adjusted large cutter is very small as compared with that of the small cutter. So the approximation of , and is reasonable, and it also decreases the computational complexity considerably. If there is no global collision and local gouging for a large cutter at certain CC points, the cutting breadth is given by L0+L= L0+ 0R R dL dR = R+ (13) which can be further simplified into La=(La/R0+cossin cosaF(a) cos sinaF(a)(RR0) (14) F() is calculated by Journal of Chongqing University-Eng. Ed. Vol. 1 No. 2 23 2 12 123 3 123 2(1 sin)sin ( ) sin2cos(cos2sin ) 2cos (sin1) sin sin2cos(cos2sin ) DD F DDD D DDD + = + + + where D1, D2, D3 and D4 are respectively given by D1=(R02k/2 cos2 sin2)/2 D2=R0k/2 sin cos sin D3= R02k/2 cos cos sin D4= h R0 sin+ D1 in which k/2 is the surface curvature in the xz plan and R0 is the radius of small cutter. By the same process of reasoning above, Lb can be obtained, and the cutting breadth is given by L=La+Lb (15) After the tool paths of surface are marked out using the optimum small cutter, the values of , , , k/2, kmax and kmin are calculable. These values can be applied to calculating the cutting breadth of different large cutter at every CC point. If there is local gouging at some CC points, the value of L is set at 0. Otherwise, the value of L is determined by Eq. (14). Consequently, the additional cut area can be obtained by 1 ii Ac cL = (16) where ci-1 and ci are the neighboring CC points; and ii cc 1 is the distance between ci-1 and ci. The average cutting rate of the optimum small cutter is determined by V=A/tt (17) where A is the area of the surface; and tt the total machining time using the optimum small cutter For a certain large cutter, the saved time (ts) is calculated by ts=A /V t0 (18) where t0 is the time needed for a milling machine to automatically exchange its cutter. The functional relation between the radius of the cutter and additional cut area can be obtained by using spline fit. After maximizing the spline function, the radius of the cutter can be obtained which is rounded as that of the optimum maximum cutter. 4. Examples Fig. 6 shows the tool paths marked out by the optimum small cutter with a machining error of 0.01 mm. Fig. 7 illustrates the relation between the radius of the cutter and additional cut area. It can be seen in Fig. 7 that when the radius of a selected larger cutter is too large or too small, the additional cut area is 0. The reason is that if the selected cutter is too large, collision occurs at every CC point, so L is set at 0. On the other side, L is less than 0 when a small cutter is smaller than the optimum small cutter. For this surface model, the calculated radius of the optimum small cutter is R1=1 cm and that of the optimum large cutter is R2=2.82 cm which is rounded into 3 cm. Fig. 6. The tool path marked out by the optimum small cutter Fig. 7. The relation between different cutterand additional cutting area 5. Conclusion The algorithm of cutte