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Estimation of Maximum Allowable Step Length for Five-Axis Cylindrical Machining
Journal of Manufacturing Processes Vol. 2mo. 1 2000
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Estimation of Maximum Allowable Step Length for Five-Axis Cylindrical Machining Yean-Ren Hwang and Min-Tse Ho, Dept. of Mechanical Engineering, National Central University, Taiwan
Abstract Five-axis cylindrical machining is the only way to machine some complicated sculptured surfaces, such as the wheel surfaces of turbo compressors. Current computer-aided manufacturing (CAM) algorithms for five-axis cylindrical machining only consider the surfaces' geometric information when generating cutter contact points (CC points). Hence, the step length, which is the distance between two CC points, is determined only by the surface parameters for these CAM algorithms. Because of tool-axis rotation, the actual cut trajectory within each step is no longer a straight line passing through two consecutive CC points for five-axis machining. Thus the cut error and the maximum allowable step length should depend on the structural parameters of NC machines as well as the surface's geometry. This paper develops a new algorithm to estimate the maximum allowable step length based on the cut error of five-axis cylindrical machining, and shows that this algorithm provides better estimation than traditional algorithms. Keywords: Step Length, Five-Axis NC, Tool Path Generation, Cylindrical Machining
Introduction Five-axis NC machining has several advantages over fixed-axis machining, including better surface finish, higher removal rates, and a shorter setup time.'*2Furthermore, some sculptured surfaces such as turbine blades, marine propellers, and tire grooves can only be machined by five-axis NC machine^.^ Due to these reasons, more and more research has been devoted to five-axis NC machining in past few decades.'-9 End milling (face milling) and cylindrical milling (side milling) are two major methods for five-axis NC machining. Most recent research and development has been focusing on the end milling m e t h ~ d ' $mainly ~ * ~ - ~because of its variety of applications. However, the end milling method cannot be used to machine some complicated surfaces, such as the wheel surface of turbo compressor^.^ This is because the space between two blades is too small,
requiring the tool-axis direction be kept tangential to the blades' surface^.^ Five-axis cylindrical milling is more effective. Instead of tips, the side of the cutting tool is used to machine the part for cylindrical milling. Therefore, the cutter contact point (CC point) is on the side of the cutting tool. Figure 1 shows the geometric relation between the cylindrical cutter and the machined surface, and Figure 2 shows a flat end cutter. In past decades, few paper^^?^ have addressed the problems of five-axis cylindrical milling, probably due to difficulties in implementation. The structure of five-axis NC machines is basically categorized into three types: spindle-tilting, table-tilting, and spindle-table-tilting, according to the machines' rotational The tool axis of a spindle-tilting type has two rotation degrees of freedom. Therefore, the tool axis is rotated in two directions to achieve the desired orientation. On the other hand, the table with the machined part is rotated along two axes for table-tilting type machines. For spindle-table-tilting type machines, both the table and spindle have one rotational degree of freedom. This paper focuses on cylindrical milling for spindle-tilting type five-axis NC machines, as shown in Figure 3. As shown in Figure 4, there are three major stages to generate NC codes for most current CAM algorithms. In the first stage, the CC points are generated from the geometry of the sculptured surfaces with two frequently used algorithms. The first algorithm generates the CC points along constant parameter lines of the machined surfaces: whereas the second algorithm generates the CC points along a curve or line in Cartesian space4 For both algorithms, the step length, which is equal to the distance between two consecutive CC points, is chosen such that the cut error tolerance requirement is satisfied. Hence, the step length is also determined at this stage for both algorithms.
Journal of Manufacturing Processes Vol. 2 N o . 1 2000
Figure I Machining with a Cylindrical Cutter Figure 3 Spindle-TiltingType Rive-Axis NC Machine
Generate CC polnts
Stage 1
v Generate CL points and CO angles
Figure 2 Machining with a Flat End Cutter
In the second stage, the cutter location points (CL points) and the tool's cutter orientation angles (CO angles) are generated from the CC points. Because the tool axis of five-axis machines can be rotated in two directions, the CL point and CO angles are not unique for each CC point. The main considerations at this stage are generating CL points and CO angles to minimize the cusp height and avoid gouging and interference with the machined park5-' A different approach proposed by Li and Jerard8 generates CL points and CO angles by dividing the surface into triangle meshes and placing the tool onto these meshes. The last stage is post-processing, in which the structure of five-axis NC machines is considered and the CL points and CO angles are converted to
Stage 2
+ Post-process~ng
Stage 3
Figure 4 Three Major Stages for Generating NC Machining Codes
machine codes (G and M codes). Because five-axis NC machines have different rotational axes, the machine codes will vary. Research literature about post-processing techniques is minimal because the work of current post-processors mainly focuses on coordinate transformation. Most current CAM algorithms consider only geometric information of sculptured surfaces and not the structures of five-axis NC machines when generating CC points, CL points, and CO angles at
Journal of Manufacturing Processes Vol. 2mo. 1 2000
the first and second stages. Although five-axis NC machines may have different structures and sizes, current CAM algorithms will generate the same CC, CL, and CO data because the surface geometry is the same. Only the machine codes are different because the structure of five-axis NC machines is considered at the post-processing stage. Because the CC points are the same, the step length of each step is the same for different five-axis NC structures. As will be shown in a later section, the cut error for different structures of five-axis machines (with different rotational axes and dimensions) differs for the same step length. This is due to the effect of the rotation axes. Therefore, to fulfill the requirement of the same error tolerance, the step length and CC points should be selected according to the structures of five-axis NC machines as well as the surface geometry. The analysis of cut error for different structures of five-axis NC machines is crucial to generation of CC points and determination of the step length. For a given cut tolerance, the main concern of NC tool path generation is to have as large a step length as possible to reduce the amount of data and machining time." The machine codes generated from the CL points are sequences of segment commands. Each command represents a cut step or a fast movement step of the tool. Linear interpolation (that is, the GO1 command for cutting and GOO command for fast moving) is the most commonly used method to move the tool. The cut trajectory is a straight line between two CC points. Detailed analysis of the allowable step length and scallop height for three-axis machining has been addressed by Sarma and Dutta." The tool axis for five-axis machining is no longer fixed in one direction. In each step, the toolholder is moved linearly from the start position to the end position, and the tool axis is rotated linearly from the start angle to the end angle. The cut trajectory, which is the actual cutting curve, is affected by the tool-axis rotation as well as the toolholder translation. In general, the cut trajectory is not a straight line between two CC points for five-axis NC machining. Hence, previous step length analysis in Sarma and Duttal' is not suitable for fiveaxis NC machining. By considering the effect of tool-axis rotation, a new estimation method for the maximum allowable step length for five-axis cylindrical machining is proposed below.
Maximum Allowable Step Length for Three-Axis Cylindrical Milling The cut error estimation for three-axis machining has been discussed in Sarma and Duttal1 and Zeid!2 The curve along the cut path is approximated by an arc, as shown in Figure 5. The cut trajectory is a straight line passing through the two consecutive CC points. The cut error is the maximum distance between this straight line and the arc.
where R and 8 represent the radius and angle of the approximated arc, respectively, and ef and s represent the cut error the step length of this step, respectively. Equation (1) can be rearranged as follows:
Because ef is much smaller than R,
For a specified cut tolerance, em,,, s has to satisfy the following condition:
The upper bound of s, denoted by sf, is the maximum allowable step length for fixed-axis cylindrical milling given in Eq. (4). When the step length is chosen less than sf,the cut error is guaranteed to be less than the specified cut tolerance em,,. Based on this relation, an appropriate step length can be chosen and the next CC point then determined. Equation (4) is true only for fixed-axis machining because the tool is always held in one direction and the cut trajectory is a straight line. For five-axis (that is, four and five-axis) NC machines, the tool axis may rotate along one or two axes in each cut step. The cut trajectory is not a straight line, although the toolholder movement is a straight line. Figure 6 shows the tool movement of spindle-tilting cylindrical NC machines. When tool rotation exists, the cut trajectory is a curve instead of a line passing through two consecutive CC points. In this case, the cut error is actually larger than the value estimated by Eq. (3), and the upper bound of the step length chosen
I
I
Figure 5 Fixed-Axis Cylindrical Machining Figure 7 Tool Motion of One Single Step
denoted by L. In Figure I, the tool axis is along the X axis. Define the vector c from C to A , and equals V =(0,-~,-b)~
(5)
If the tool is first rotated about the X axis with an then about the Z axis with an angle y, as angle shown in Figure 7, the vector from the holder's rotation center to the tool corner arc's center is equal to T($,y ) ~where , the transformation matrix
+,
Figure 6 Variable-Axis Cylindrical Machining
should be smaller than sf in Eq. (4) to fblfill the same cut error tolerance.
Maximum Allowable Step Length for Five-Axis Cylindrical Milling Rotation of Cylindrical Cutters Two major types of tools used for cylindrical milling are flat head and drum head cutters. A drum head cutter, as shown in Figure 1, is mounted to a toolholder whose rotational center is denoted as C. The center of the tool's edge arc is denoted as A. The radius of the tool edge is denoted by p, the distance between A and the central line of the tool is denoted by b, and the horizontal distance between A and C is
Note that the first matrix of T(+,y) is a rotational matrix about the Z axis, and the second matrix is the rotational matrix about the X axis. A typical flat head cutter (shown in Figure 2) has a toolholder center denoted as C and a tip point denoted as A . The distance between C to the bottom of the tool is denoted by L, and the radius of the cutter is denoted by r. The vector G becomes
In general, the tip of a flat head tool is used for five-axis cylindrical milling. Because a point can be considered as a circle with zero radius, the derivation for the drum head cutter can be applied to the flat head cutter by setting p = 0 and b = r. Therefore, only the case for the drum head cutter is discussed in the following sections.
Journal of Manufacturing Processes Vol. 2mo. 1 2000
Maximum Allowable Step Length Figure 8 shows the tool movement of one cut step for a spindle-tilting NC machine, where and in Figure 8 represent the position vectors of the first and second CC points, respectively. Because the step length is small, the curve between and pi is approximated by an arc with its center at 0. Denote R as the radius of the approximated arc and 0 as the angle between and Let ? and ii be the unit tangential vector and unit normal vector of this arc at 6 , respectively. A local coordinate system is formed with its origin at 0, its Zl axis parallel to E , its 6 axis parallel to t,and its XI axis parallel to f = E . The tool rotation angles with respect to the X,Y,Z, coordinate system at the CC points are closely related to the lead and tilt angles at these CC p-oints. Denote the tool rotation angles at 4 as and yl and the tool rotation angles at P, as and y2. The positions of the toolholder's rotation center at and pi are denoted by El and c2, respectively. The coordinate information about 6 , pi, C, , and is listed as follows:
8 6
8
+,
q.
Figure 8 Tool Motion of Step
8
c2
Hence, the cut trajectory ~ ( h for ) 05A
where the subscript I indicating that these coordinates are with respect to the X,YI& coordinate system. In this cut step, the toolholder's - rotation center is moved along the line segment C,C, .The holder position, denoted by C(h), can be written in terms of and as follows:
c, e,,
The distance between the tool cut point and the arc, d(A), is as follows:
where A is a parameter varying from 0 to 1. At the same time, the tool's rotational angles change linearly from and yl to +z and 7,.
When id(h)l reaches its minimum, d(A) reaches its maximum. Therefore, the cut error of this step is
5
1 is
e = max {d(h)}= R - minx{lP(h)l} he[O,l]
where A+ = 4, -4, and Ay = y1 - ?,.The trajectory of the tool corner arc's center A(h) can be obtained from C(h), +(A), and ?(A).
hs[O,l]
Because I ~ ( h )=l ~A(h) A(h), minimum /A(h)l happens when
Journal of Manufacturing Processes Vol. 2mo. 1 2000
and j(h) = ( R + ~ ) ( ohsin0, , 1- h + hcose):
The solution to Eq. (22), AO, guarantees min{l@)l} and $~j~(~)l}. Th'is means that the cut error hs[O !I of this step can be obtained by substituting this h0 into Eq. (19). Due to the fact that the step length of each cut step is small, the angle variation of the tool axis should be small during the machining step. This means that hA+ and AAy are small for A E [0,1]. Hence, T(+(A), y(X)) can be approximated by using Taylor expansion with respect to A+ and Ay as follows.
-V +-(h2-A) d~(@(h)J(h))
dl?
2
dh
= (R+,)(o, sine,
By substituting ~ ( hand )
.(h) ,Eq. (22) becomes d3L
Define
"
=(4 4 R+P):
ii, = (R + p)(~,sine, 1 + cose):
Equation (23) implies that Equation (29) can be written as follows:
T(C$~,~,)can be derived from Eq. (23) by assigning A equals 1. where
Therefore,
Journal of Manufacturing Processes Vol. 2No. 1 2000
+,,
Lambda for different tool angle Increment 0.545 054-
L=60 b = 2
-
R=50 p = 8
0535
0.53 -
2 0.525 -
//
$
0.2
n
0.52
-
0.515
-
0.51
-
0 505
-
?/Deatl
0.5 0
gamma = 0.0
/
01 Delta p h (rad) ~
0.2
Figure 9 Lambda Values for Different Delta Phi
Cut error for d~fferenttool angle Increment 04 L=60 b = 2
038
R=50 p = 8
-E
E
0.36 -
Constant Angles w.r.t. the Machined Surface
In Eq. (34), A0 is not written as an analytical form in terms of 0. This increases the difficulties of deter/ ' / mining the step length of each step during machin. /--ing. "Normal to part" and "constant lead and tilt ,/angles" machining are two commonly used methods for five-axis cylindrical machining. In fact, the "normal to part" machining is just a special case of "con// I stant lead and tilt angles" machining when the lead and tilt angles are kept at zero. Hence, the discussion Delta gamma = 0 0 is focused on constant angle-machining. When the lead angle a and tilt angle f3 are kept 01 0.2 constant along the cut path, 4, = a and yl = P, A+ Delta phi (rad) = -0, and Ay = 0. Therefore
-_-C
0.2
0.34 -
-/;/;
L
6
m
0.15
03203
Note that X0 depends on L, b, p, R, 0, yl, A+, and Ay. The machine parameters L, b, and p can be considered fixed values as long as there is no tool change during the machining process. The radius of the approximated arc R depends on the curvature along the cut trajectory. When the position of the first CC point and the cut path trajectory are determined, R can be found. For many five-axis machining strategies, the tool orientation angles are based on the geometry of the machined surface only. Typical examples are "normal to part" and "constant lead angle" machining. For these cases, the start tool angles and yl are determined from the start CC point, and 0, A+, and Ay depend on the step length of this cut step.
-
' / -
0 28
0
Figure 10 Cut Errors for Different Delta Phi
Equation (34) is a cubic equation and its solution, AO,can be solved.13The cut error of this step can be obtained by substituting A0 into Eq. (19). Figure 9 shows A0 for a simulation case when A+ and Ay vary from zero to 0.2 radians for specified parameters. When both A+ and Ay are zero, A0 is equal to 0.5. This means that the maximum cut error occurs at the middle of the cut step when the tool is fixed in one orientation. This fact coincides with the fixedaxis NC machining. When either A+ and Ay increases, A0 also increases. At the same time, the maximum cut error does not occur at the middle of the cut step (A0 = 0.5) when the tool angle varies within this step. As shown in Figure 10, the cut error also increases when A+ and/or Ay increase.
[;:]
- ~ c o s a s+ i binasin] Lcosacosp- bsinacosp = 8' 4, Lsina + bcosa
where
q3 = Lsina
+ bcosa
Journal of Manufacturing Processes Vol. 2mo. 1 2000
Because the step length is assumed small, 8' sin0=0, c 0 s 8 = 1 - ~ and ,
i(hu)i
-
jR+
-
L
4 = ( R + p)(0, 0, -8' /2)T
(43)
Hence, the cut error of this step is equal to
Therefore, Eq. (33) becomes where s = R0 is the step length of this step. Therefore, for a given e,,,,
The step length for this step to fulfill the cut tolerance e,,, can be found as follows:
The above equation can be written as follows:
where 0(03) and 0(B4).represent the terms with 0, and 04, respectively. Because 0 is assumed to be small, the terms with 0(03) and 0(04) can be neglected. This implies that h0 = 112 for constant lead and tilt angle machining because R + p and R + p + q3 are not zero. Therefore,
and
Because 0'14 is very small, (i(P)/can be further approximated by applying binomial expansion as follows:
This means that the maximum allowable step length for five-axis cylindrical milling, denoted by s,, is equal to
where sf is the maximum allowable step length for three-axis cylindrical machining. Note that s, depends on q3, which is related to L and b. This again shows that the structural dimensions of fiveaxis NC machines affect the maximum allowable step length for the same specified em,,. Figure I 1 shows that value of s, is smaller than sf for the same em,, for different tool angles. The flat head cutter is the most commonly used tool for cylindric_al machining because of its simplicity. Because ~ ( h )coincides with p ( h ) , p can be set to zero and b equals the radius of the cutter for a flat head cutter. This implies that q, = b for "normal to part" machining because a is always equal to zero. Therefore, the estimation equation [Eq. (54)] can be further simplified as follows:
The above equation shows that s, is closely related to the radius of the flat head cutter. When the radius of the flat head cutter is larger, the maximum allow step length becomes smaller. Therefore, a smaller step length should be chosen when a larger
Journal of Manufacturing Processes Vol. 21No. 1 2000
Step length vs. &,,for
different tool angle (NURB)
9 -
..-.
-E" E
8
-
7
-
6
-
Computer 2nd
Radius Approximate
5 -
4 7
Step Length Estimation 1
C 0
0.05
0.1 . ,e (mm)
0.15
0.2
Determine Next CC, CL, CO Data Along Tool Path
Figure 11 Step Length vs. em., for Different q3
diameter flat head cutter is used for five-axis cylindrical milling. As shown in Figure 12, the step length estimation algorithm is integrated with the CAD system. Both the CAD geometry and the tool parameters are considered for generating five-axis toolpaths. Following a cut path, the curvature at a CC point is obtained by taking the second-order derivative along that direction. The approximate radius of the curve is equal to the inverse of the curvature at each CC point. Based on the tool parameters and the approximate radius at each CC point, the algorithm estimates the maximum allowable step length and determines the next CC point, CL point, and CO angle. These CL points and CO angles are exported to an NC programming file, which will be used for controlling CNC machines. In the verification example, a flat head cutter is used because of its simplicity and wide usage in industry. Because A(h) is actually the cut trajectory for a flat head cutter, p can be set to zero and b equals the radius of the cutter r. This implies that q, = r for "normal to part" machining because a is always equal to zero. Therefore, the estimation equation [Eq. (54)] can be further simplified as follows:
The above equation shows that s, is closely related to the radius of the flat head cutter. When the radius of the flat head cutter is larger, the maximum allowable step length becomes smaller. Therefore, a smaller step length should be chosen when a larger
C Output to NC Machine
Figure 12 Function Block Chart
diameter flat head cutter is used for five-axis cylindrical milling. The ruled surface in Figure 13 is used for verification of the estimation algorithm proposed in this section. The curve of this ruled surface on the constant Y-plane is shown in Figure 14.
where R, = 10, q, = q + 7~12and q varies from 0.37 to 2.37 radians. When the machining direction is along the X direction, the radius of the approximated arc equals R,q. The maximum allowable step length can be obtained from Eq. (54),
The corresponding CC points are shown as the '+' sign in Figures 13 and 14 when em, is specified as 0.01. Both figures show that the step length is smaller where the curve's curvature is larger.
Conclusion Traditional tool path algorithms only consider the machined surface geometry when generating CC points data. Using these algorithms, the step lengths
Journal of Manufacturing Processes Vol. 2/No 1 2000
Figure 13 C C Points for Ruled Surface with Spiral Curve Cross Section
will be the same for different types of NC machines. However, the actual cut trajectory for five-axis cylindrical milling machines is not a straight line passing through two consecutive CC points due to tool-axis rotation. This means that the maximum allowable step length should depend on the structural parameters of NC machines. The algorithm proposed in this paper estimates to be the cut error based on the surface's geometry as well as the NC structural parameters. For the same e,,,, the maximum allowable step length for fiveaxis cylindrical milling is estimated smaller than that of fixed-axis cylindrical milling.
Figure 14 C C Points for Single Tool Path
AXISMachining," Int'l Journal of Productzon Research (v34, n l , 1996), pplll-135. 8. Susan X. Li and Robert G. Jerard, "5-Axis Machining of Sculptured Surfaces with a Flat-End Cutter," Computer Azded Design (v26, n3, 1994), pp165-178. 9. Y. Takeuch~, M. Nagasaka, and K. Morishige, "5-AXIS Control Machining with Top and Slde Cutting Edges of Ball End Mill," Journal of the Japan Soczety of Preciszon Engg. (v61, n4, Apnl 1999, pp561-565 10. Y. Koren and R.S. Lin, "Five-Axis Surface Interpolators:' Annals of the CIRP (v44, 1995), pp379-382. 11. R. Sarma and D. Dutta, "The Geometry and Generation of NC Tool Paths," Journal of Mechanical Design (June 1997), pp253-258. 12. I. Zeid, CAD/CAM Theov andPractice (New York. McGraw-Hill, 1991). 13. S.M. Selby, Standard Mathematzcal Tables (CRC Press, 1964).
References 1. G.W. Vickers and K.W. Quan, "Ball-Mills versus End-Mills for Curved Surface Mach~ning,"ASME Journal of Engg for Industuy (vl11, Feb. 1989), pp22-26. 2. H.D. Cho et a]., "Five-Axis CNC Milling for Effecbve Machining of Sculptured Surfaces:' Int 'I Journal of Production Research (v3 1, n l 1, Nov. 1993), pp2559-2573 3. Xiong-Wei Liu, "Five-Axis NC Cylindrical Milling of Sculptured Surfaces," Computer Azded Design (~27,1112,1995), pp887-894. 4. G.C. Lmey and T.M. Ozsoy, "NC Machining of Free-Form Surfaces:' Computer Aided Destgn (v19, n2, 1987), pp85-90. 5. B.K. Choi, C S. Lee, J.S. Hwang, and C.S. Jun, "Compound Surface Modeling and Machimmg," ComputerAidedDesign(v20, n3,1988), pp127-136. 6. B.K. Choi, J.W. Park, and C.S. Jun, "Cutter Location Data Optimization in 5-Axis Surface Machining:' ComputerAided Design (v25, n6, June 1993), pp377-386. 7. Y.S Lee and T.C. Chang, "Machined Surface Error Analysis for 5-
Authors' Biographies Yean-Ren Hwang is an associate professor m the Dept of Mechanical Engineering at the National Central Un~versity(NCU), Taiwan He rece~ved a BS degree from the Nat~onalTaiwan University in 1983 and an MS degree from the Georgia Inst~tuteof Technology in 1986. He then worked In automobile industry before gaining a PhD degree from the Univers~tyof California at Berkeley m 1993. He worked for Unigraphics Div of EDS before he joined NCU in 1994. His research interests include surface modeling, multiple-axis NC machining, and computer vision. s degree in the Min-Tse Ho is a graduate student and completed h ~ BS Dept. of Mechanical Engineertng at the National Central University, Taiwan, in 1998. His research includes five-axis NC mach~ningand surface modeling
0
Estimation of Maximum Allowable Step Length for Five-Axis Cylindrical Machining Yean-Ren Hwang and Min-Tse Ho, Dept. of Mechanical Engineering, National Central University, Taiwan
Abstract Five-axis cylindrical machining is the only way to machine some complicated sculptured surfaces, such as the wheel surfaces of turbo compressors. Current computer-aided manufacturing (CAM) algorithms for five-axis cylindrical machining only consider the surfaces' geometric information when generating cutter contact points (CC points). Hence, the step length, which is the distance between two CC points, is determined only by the surface parameters for these CAM algorithms. Because of tool-axis rotation, the actual cut trajectory within each step is no longer a straight line passing through two consecutive CC points for five-axis machining. Thus the cut error and the maximum allowable step length should depend on the structural parameters of NC machines as well as the surface's geometry. This paper develops a new algorithm to estimate the maximum allowable step length based on the cut error of five-axis cylindrical machining, and shows that this algorithm provides better estimation than traditional algorithms. Keywords: Step Length, Five-Axis NC, Tool Path Generation, Cylindrical Machining
Introduction Five-axis NC machining has several advantages over fixed-axis machining, including better surface finish, higher removal rates, and a shorter setup time.'*2Furthermore, some sculptured surfaces such as turbine blades, marine propellers, and tire grooves can only be machined by five-axis NC machine^.^ Due to these reasons, more and more research has been devoted to five-axis NC machining in past few decades.'-9 End milling (face milling) and cylindrical milling (side milling) are two major methods for five-axis NC machining. Most recent research and development has been focusing on the end milling m e t h ~ d ' $mainly ~ * ~ - ~because of its variety of applications. However, the end milling method cannot be used to machine some complicated surfaces, such as the wheel surface of turbo compressor^.^ This is because the space between two blades is too small,
requiring the tool-axis direction be kept tangential to the blades' surface^.^ Five-axis cylindrical milling is more effective. Instead of tips, the side of the cutting tool is used to machine the part for cylindrical milling. Therefore, the cutter contact point (CC point) is on the side of the cutting tool. Figure 1 shows the geometric relation between the cylindrical cutter and the machined surface, and Figure 2 shows a flat end cutter. In past decades, few paper^^?^ have addressed the problems of five-axis cylindrical milling, probably due to difficulties in implementation. The structure of five-axis NC machines is basically categorized into three types: spindle-tilting, table-tilting, and spindle-table-tilting, according to the machines' rotational The tool axis of a spindle-tilting type has two rotation degrees of freedom. Therefore, the tool axis is rotated in two directions to achieve the desired orientation. On the other hand, the table with the machined part is rotated along two axes for table-tilting type machines. For spindle-table-tilting type machines, both the table and spindle have one rotational degree of freedom. This paper focuses on cylindrical milling for spindle-tilting type five-axis NC machines, as shown in Figure 3. As shown in Figure 4, there are three major stages to generate NC codes for most current CAM algorithms. In the first stage, the CC points are generated from the geometry of the sculptured surfaces with two frequently used algorithms. The first algorithm generates the CC points along constant parameter lines of the machined surfaces: whereas the second algorithm generates the CC points along a curve or line in Cartesian space4 For both algorithms, the step length, which is equal to the distance between two consecutive CC points, is chosen such that the cut error tolerance requirement is satisfied. Hence, the step length is also determined at this stage for both algorithms.
Journal of Manufacturing Processes Vol. 2 N o . 1 2000
Figure I Machining with a Cylindrical Cutter Figure 3 Spindle-TiltingType Rive-Axis NC Machine
Generate CC polnts
Stage 1
v Generate CL points and CO angles
Figure 2 Machining with a Flat End Cutter
In the second stage, the cutter location points (CL points) and the tool's cutter orientation angles (CO angles) are generated from the CC points. Because the tool axis of five-axis machines can be rotated in two directions, the CL point and CO angles are not unique for each CC point. The main considerations at this stage are generating CL points and CO angles to minimize the cusp height and avoid gouging and interference with the machined park5-' A different approach proposed by Li and Jerard8 generates CL points and CO angles by dividing the surface into triangle meshes and placing the tool onto these meshes. The last stage is post-processing, in which the structure of five-axis NC machines is considered and the CL points and CO angles are converted to
Stage 2
+ Post-process~ng
Stage 3
Figure 4 Three Major Stages for Generating NC Machining Codes
machine codes (G and M codes). Because five-axis NC machines have different rotational axes, the machine codes will vary. Research literature about post-processing techniques is minimal because the work of current post-processors mainly focuses on coordinate transformation. Most current CAM algorithms consider only geometric information of sculptured surfaces and not the structures of five-axis NC machines when generating CC points, CL points, and CO angles at
Journal of Manufacturing Processes Vol. 2mo. 1 2000
the first and second stages. Although five-axis NC machines may have different structures and sizes, current CAM algorithms will generate the same CC, CL, and CO data because the surface geometry is the same. Only the machine codes are different because the structure of five-axis NC machines is considered at the post-processing stage. Because the CC points are the same, the step length of each step is the same for different five-axis NC structures. As will be shown in a later section, the cut error for different structures of five-axis machines (with different rotational axes and dimensions) differs for the same step length. This is due to the effect of the rotation axes. Therefore, to fulfill the requirement of the same error tolerance, the step length and CC points should be selected according to the structures of five-axis NC machines as well as the surface geometry. The analysis of cut error for different structures of five-axis NC machines is crucial to generation of CC points and determination of the step length. For a given cut tolerance, the main concern of NC tool path generation is to have as large a step length as possible to reduce the amount of data and machining time." The machine codes generated from the CL points are sequences of segment commands. Each command represents a cut step or a fast movement step of the tool. Linear interpolation (that is, the GO1 command for cutting and GOO command for fast moving) is the most commonly used method to move the tool. The cut trajectory is a straight line between two CC points. Detailed analysis of the allowable step length and scallop height for three-axis machining has been addressed by Sarma and Dutta." The tool axis for five-axis machining is no longer fixed in one direction. In each step, the toolholder is moved linearly from the start position to the end position, and the tool axis is rotated linearly from the start angle to the end angle. The cut trajectory, which is the actual cutting curve, is affected by the tool-axis rotation as well as the toolholder translation. In general, the cut trajectory is not a straight line between two CC points for five-axis NC machining. Hence, previous step length analysis in Sarma and Duttal' is not suitable for fiveaxis NC machining. By considering the effect of tool-axis rotation, a new estimation method for the maximum allowable step length for five-axis cylindrical machining is proposed below.
Maximum Allowable Step Length for Three-Axis Cylindrical Milling The cut error estimation for three-axis machining has been discussed in Sarma and Duttal1 and Zeid!2 The curve along the cut path is approximated by an arc, as shown in Figure 5. The cut trajectory is a straight line passing through the two consecutive CC points. The cut error is the maximum distance between this straight line and the arc.
where R and 8 represent the radius and angle of the approximated arc, respectively, and ef and s represent the cut error the step length of this step, respectively. Equation (1) can be rearranged as follows:
Because ef is much smaller than R,
For a specified cut tolerance, em,,, s has to satisfy the following condition:
The upper bound of s, denoted by sf, is the maximum allowable step length for fixed-axis cylindrical milling given in Eq. (4). When the step length is chosen less than sf,the cut error is guaranteed to be less than the specified cut tolerance em,,. Based on this relation, an appropriate step length can be chosen and the next CC point then determined. Equation (4) is true only for fixed-axis machining because the tool is always held in one direction and the cut trajectory is a straight line. For five-axis (that is, four and five-axis) NC machines, the tool axis may rotate along one or two axes in each cut step. The cut trajectory is not a straight line, although the toolholder movement is a straight line. Figure 6 shows the tool movement of spindle-tilting cylindrical NC machines. When tool rotation exists, the cut trajectory is a curve instead of a line passing through two consecutive CC points. In this case, the cut error is actually larger than the value estimated by Eq. (3), and the upper bound of the step length chosen
I
I
Figure 5 Fixed-Axis Cylindrical Machining Figure 7 Tool Motion of One Single Step
denoted by L. In Figure I, the tool axis is along the X axis. Define the vector c from C to A , and equals V =(0,-~,-b)~
(5)
If the tool is first rotated about the X axis with an then about the Z axis with an angle y, as angle shown in Figure 7, the vector from the holder's rotation center to the tool corner arc's center is equal to T($,y ) ~where , the transformation matrix
+,
Figure 6 Variable-Axis Cylindrical Machining
should be smaller than sf in Eq. (4) to fblfill the same cut error tolerance.
Maximum Allowable Step Length for Five-Axis Cylindrical Milling Rotation of Cylindrical Cutters Two major types of tools used for cylindrical milling are flat head and drum head cutters. A drum head cutter, as shown in Figure 1, is mounted to a toolholder whose rotational center is denoted as C. The center of the tool's edge arc is denoted as A. The radius of the tool edge is denoted by p, the distance between A and the central line of the tool is denoted by b, and the horizontal distance between A and C is
Note that the first matrix of T(+,y) is a rotational matrix about the Z axis, and the second matrix is the rotational matrix about the X axis. A typical flat head cutter (shown in Figure 2) has a toolholder center denoted as C and a tip point denoted as A . The distance between C to the bottom of the tool is denoted by L, and the radius of the cutter is denoted by r. The vector G becomes
In general, the tip of a flat head tool is used for five-axis cylindrical milling. Because a point can be considered as a circle with zero radius, the derivation for the drum head cutter can be applied to the flat head cutter by setting p = 0 and b = r. Therefore, only the case for the drum head cutter is discussed in the following sections.
Journal of Manufacturing Processes Vol. 2mo. 1 2000
Maximum Allowable Step Length Figure 8 shows the tool movement of one cut step for a spindle-tilting NC machine, where and in Figure 8 represent the position vectors of the first and second CC points, respectively. Because the step length is small, the curve between and pi is approximated by an arc with its center at 0. Denote R as the radius of the approximated arc and 0 as the angle between and Let ? and ii be the unit tangential vector and unit normal vector of this arc at 6 , respectively. A local coordinate system is formed with its origin at 0, its Zl axis parallel to E , its 6 axis parallel to t,and its XI axis parallel to f = E . The tool rotation angles with respect to the X,Y,Z, coordinate system at the CC points are closely related to the lead and tilt angles at these CC p-oints. Denote the tool rotation angles at 4 as and yl and the tool rotation angles at P, as and y2. The positions of the toolholder's rotation center at and pi are denoted by El and c2, respectively. The coordinate information about 6 , pi, C, , and is listed as follows:
8 6
8
+,
q.
Figure 8 Tool Motion of Step
8
c2
Hence, the cut trajectory ~ ( h for ) 05A
where the subscript I indicating that these coordinates are with respect to the X,YI& coordinate system. In this cut step, the toolholder's - rotation center is moved along the line segment C,C, .The holder position, denoted by C(h), can be written in terms of and as follows:
c, e,,
The distance between the tool cut point and the arc, d(A), is as follows:
where A is a parameter varying from 0 to 1. At the same time, the tool's rotational angles change linearly from and yl to +z and 7,.
When id(h)l reaches its minimum, d(A) reaches its maximum. Therefore, the cut error of this step is
5
1 is
e = max {d(h)}= R - minx{lP(h)l} he[O,l]
where A+ = 4, -4, and Ay = y1 - ?,.The trajectory of the tool corner arc's center A(h) can be obtained from C(h), +(A), and ?(A).
hs[O,l]
Because I ~ ( h )=l ~A(h) A(h), minimum /A(h)l happens when
Journal of Manufacturing Processes Vol. 2mo. 1 2000
and j(h) = ( R + ~ ) ( ohsin0, , 1- h + hcose):
The solution to Eq. (22), AO, guarantees min{l@)l} and $~j~(~)l}. Th'is means that the cut error hs[O !I of this step can be obtained by substituting this h0 into Eq. (19). Due to the fact that the step length of each cut step is small, the angle variation of the tool axis should be small during the machining step. This means that hA+ and AAy are small for A E [0,1]. Hence, T(+(A), y(X)) can be approximated by using Taylor expansion with respect to A+ and Ay as follows.
-V +-(h2-A) d~(@(h)J(h))
dl?
2
dh
= (R+,)(o, sine,
By substituting ~ ( hand )
.(h) ,Eq. (22) becomes d3L
Define
"
=(4 4 R+P):
ii, = (R + p)(~,sine, 1 + cose):
Equation (23) implies that Equation (29) can be written as follows:
T(C$~,~,)can be derived from Eq. (23) by assigning A equals 1. where
Therefore,
Journal of Manufacturing Processes Vol. 2No. 1 2000
+,,
Lambda for different tool angle Increment 0.545 054-
L=60 b = 2
-
R=50 p = 8
0535
0.53 -
2 0.525 -
//
$
0.2
n
0.52
-
0.515
-
0.51
-
0 505
-
?/Deatl
0.5 0
gamma = 0.0
/
01 Delta p h (rad) ~
0.2
Figure 9 Lambda Values for Different Delta Phi
Cut error for d~fferenttool angle Increment 04 L=60 b = 2
038
R=50 p = 8
-E
E
0.36 -
Constant Angles w.r.t. the Machined Surface
In Eq. (34), A0 is not written as an analytical form in terms of 0. This increases the difficulties of deter/ ' / mining the step length of each step during machin. /--ing. "Normal to part" and "constant lead and tilt ,/angles" machining are two commonly used methods for five-axis cylindrical machining. In fact, the "normal to part" machining is just a special case of "con// I stant lead and tilt angles" machining when the lead and tilt angles are kept at zero. Hence, the discussion Delta gamma = 0 0 is focused on constant angle-machining. When the lead angle a and tilt angle f3 are kept 01 0.2 constant along the cut path, 4, = a and yl = P, A+ Delta phi (rad) = -0, and Ay = 0. Therefore
-_-C
0.2
0.34 -
-/;/;
L
6
m
0.15
03203
Note that X0 depends on L, b, p, R, 0, yl, A+, and Ay. The machine parameters L, b, and p can be considered fixed values as long as there is no tool change during the machining process. The radius of the approximated arc R depends on the curvature along the cut trajectory. When the position of the first CC point and the cut path trajectory are determined, R can be found. For many five-axis machining strategies, the tool orientation angles are based on the geometry of the machined surface only. Typical examples are "normal to part" and "constant lead angle" machining. For these cases, the start tool angles and yl are determined from the start CC point, and 0, A+, and Ay depend on the step length of this cut step.
-
' / -
0 28
0
Figure 10 Cut Errors for Different Delta Phi
Equation (34) is a cubic equation and its solution, AO,can be solved.13The cut error of this step can be obtained by substituting A0 into Eq. (19). Figure 9 shows A0 for a simulation case when A+ and Ay vary from zero to 0.2 radians for specified parameters. When both A+ and Ay are zero, A0 is equal to 0.5. This means that the maximum cut error occurs at the middle of the cut step when the tool is fixed in one orientation. This fact coincides with the fixedaxis NC machining. When either A+ and Ay increases, A0 also increases. At the same time, the maximum cut error does not occur at the middle of the cut step (A0 = 0.5) when the tool angle varies within this step. As shown in Figure 10, the cut error also increases when A+ and/or Ay increase.
[;:]
- ~ c o s a s+ i binasin] Lcosacosp- bsinacosp = 8' 4, Lsina + bcosa
where
q3 = Lsina
+ bcosa
Journal of Manufacturing Processes Vol. 2mo. 1 2000
Because the step length is assumed small, 8' sin0=0, c 0 s 8 = 1 - ~ and ,
i(hu)i
-
jR+
-
L
4 = ( R + p)(0, 0, -8' /2)T
(43)
Hence, the cut error of this step is equal to
Therefore, Eq. (33) becomes where s = R0 is the step length of this step. Therefore, for a given e,,,,
The step length for this step to fulfill the cut tolerance e,,, can be found as follows:
The above equation can be written as follows:
where 0(03) and 0(B4).represent the terms with 0, and 04, respectively. Because 0 is assumed to be small, the terms with 0(03) and 0(04) can be neglected. This implies that h0 = 112 for constant lead and tilt angle machining because R + p and R + p + q3 are not zero. Therefore,
and
Because 0'14 is very small, (i(P)/can be further approximated by applying binomial expansion as follows:
This means that the maximum allowable step length for five-axis cylindrical milling, denoted by s,, is equal to
where sf is the maximum allowable step length for three-axis cylindrical machining. Note that s, depends on q3, which is related to L and b. This again shows that the structural dimensions of fiveaxis NC machines affect the maximum allowable step length for the same specified em,,. Figure I 1 shows that value of s, is smaller than sf for the same em,, for different tool angles. The flat head cutter is the most commonly used tool for cylindric_al machining because of its simplicity. Because ~ ( h )coincides with p ( h ) , p can be set to zero and b equals the radius of the cutter for a flat head cutter. This implies that q, = b for "normal to part" machining because a is always equal to zero. Therefore, the estimation equation [Eq. (54)] can be further simplified as follows:
The above equation shows that s, is closely related to the radius of the flat head cutter. When the radius of the flat head cutter is larger, the maximum allow step length becomes smaller. Therefore, a smaller step length should be chosen when a larger
Journal of Manufacturing Processes Vol. 21No. 1 2000
Step length vs. &,,for
different tool angle (NURB)
9 -
..-.
-E" E
8
-
7
-
6
-
Computer 2nd
Radius Approximate
5 -
4 7
Step Length Estimation 1
C 0
0.05
0.1 . ,e (mm)
0.15
0.2
Determine Next CC, CL, CO Data Along Tool Path
Figure 11 Step Length vs. em., for Different q3
diameter flat head cutter is used for five-axis cylindrical milling. As shown in Figure 12, the step length estimation algorithm is integrated with the CAD system. Both the CAD geometry and the tool parameters are considered for generating five-axis toolpaths. Following a cut path, the curvature at a CC point is obtained by taking the second-order derivative along that direction. The approximate radius of the curve is equal to the inverse of the curvature at each CC point. Based on the tool parameters and the approximate radius at each CC point, the algorithm estimates the maximum allowable step length and determines the next CC point, CL point, and CO angle. These CL points and CO angles are exported to an NC programming file, which will be used for controlling CNC machines. In the verification example, a flat head cutter is used because of its simplicity and wide usage in industry. Because A(h) is actually the cut trajectory for a flat head cutter, p can be set to zero and b equals the radius of the cutter r. This implies that q, = r for "normal to part" machining because a is always equal to zero. Therefore, the estimation equation [Eq. (54)] can be further simplified as follows:
The above equation shows that s, is closely related to the radius of the flat head cutter. When the radius of the flat head cutter is larger, the maximum allowable step length becomes smaller. Therefore, a smaller step length should be chosen when a larger
C Output to NC Machine
Figure 12 Function Block Chart
diameter flat head cutter is used for five-axis cylindrical milling. The ruled surface in Figure 13 is used for verification of the estimation algorithm proposed in this section. The curve of this ruled surface on the constant Y-plane is shown in Figure 14.
where R, = 10, q, = q + 7~12and q varies from 0.37 to 2.37 radians. When the machining direction is along the X direction, the radius of the approximated arc equals R,q. The maximum allowable step length can be obtained from Eq. (54),
The corresponding CC points are shown as the '+' sign in Figures 13 and 14 when em, is specified as 0.01. Both figures show that the step length is smaller where the curve's curvature is larger.
Conclusion Traditional tool path algorithms only consider the machined surface geometry when generating CC points data. Using these algorithms, the step lengths
Journal of Manufacturing Processes Vol. 2/No 1 2000
Figure 13 C C Points for Ruled Surface with Spiral Curve Cross Section
will be the same for different types of NC machines. However, the actual cut trajectory for five-axis cylindrical milling machines is not a straight line passing through two consecutive CC points due to tool-axis rotation. This means that the maximum allowable step length should depend on the structural parameters of NC machines. The algorithm proposed in this paper estimates to be the cut error based on the surface's geometry as well as the NC structural parameters. For the same e,,,, the maximum allowable step length for fiveaxis cylindrical milling is estimated smaller than that of fixed-axis cylindrical milling.
Figure 14 C C Points for Single Tool Path
AXISMachining," Int'l Journal of Productzon Research (v34, n l , 1996), pplll-135. 8. Susan X. Li and Robert G. Jerard, "5-Axis Machining of Sculptured Surfaces with a Flat-End Cutter," Computer Azded Design (v26, n3, 1994), pp165-178. 9. Y. Takeuch~, M. Nagasaka, and K. Morishige, "5-AXIS Control Machining with Top and Slde Cutting Edges of Ball End Mill," Journal of the Japan Soczety of Preciszon Engg. (v61, n4, Apnl 1999, pp561-565 10. Y. Koren and R.S. Lin, "Five-Axis Surface Interpolators:' Annals of the CIRP (v44, 1995), pp379-382. 11. R. Sarma and D. Dutta, "The Geometry and Generation of NC Tool Paths," Journal of Mechanical Design (June 1997), pp253-258. 12. I. Zeid, CAD/CAM Theov andPractice (New York. McGraw-Hill, 1991). 13. S.M. Selby, Standard Mathematzcal Tables (CRC Press, 1964).
References 1. G.W. Vickers and K.W. Quan, "Ball-Mills versus End-Mills for Curved Surface Mach~ning,"ASME Journal of Engg for Industuy (vl11, Feb. 1989), pp22-26. 2. H.D. Cho et a]., "Five-Axis CNC Milling for Effecbve Machining of Sculptured Surfaces:' Int 'I Journal of Production Research (v3 1, n l 1, Nov. 1993), pp2559-2573 3. Xiong-Wei Liu, "Five-Axis NC Cylindrical Milling of Sculptured Surfaces," Computer Azded Design (~27,1112,1995), pp887-894. 4. G.C. Lmey and T.M. Ozsoy, "NC Machining of Free-Form Surfaces:' Computer Aided Destgn (v19, n2, 1987), pp85-90. 5. B.K. Choi, C S. Lee, J.S. Hwang, and C.S. Jun, "Compound Surface Modeling and Machimmg," ComputerAidedDesign(v20, n3,1988), pp127-136. 6. B.K. Choi, J.W. Park, and C.S. Jun, "Cutter Location Data Optimization in 5-Axis Surface Machining:' ComputerAided Design (v25, n6, June 1993), pp377-386. 7. Y.S Lee and T.C. Chang, "Machined Surface Error Analysis for 5-
Authors' Biographies Yean-Ren Hwang is an associate professor m the Dept of Mechanical Engineering at the National Central Un~versity(NCU), Taiwan He rece~ved a BS degree from the Nat~onalTaiwan University in 1983 and an MS degree from the Georgia Inst~tuteof Technology in 1986. He then worked In automobile industry before gaining a PhD degree from the Univers~tyof California at Berkeley m 1993. He worked for Unigraphics Div of EDS before he joined NCU in 1994. His research interests include surface modeling, multiple-axis NC machining, and computer vision. s degree in the Min-Tse Ho is a graduate student and completed h ~ BS Dept. of Mechanical Engineertng at the National Central University, Taiwan, in 1998. His research includes five-axis NC mach~ningand surface modeling