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This is to certify that the

thesis entitled

TOOL SWEEP GENERATION
FOR GENERIC CUTTERS

presented by
Hari Krishna Kosaraju

has been accepted towards fulfillment
of the requirements for

MASTERS degreein MECHANICAL ENGINEERING

52%

Major professor

Date 2/ 3

0-7639 MS U is an Affirmative Action/Equal Opportunity Institution

 

 

 

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DATE DUE DATE DUE DATE DUE ll

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MSU Is An Affirmative Action/

TOOL SWEEP GENERATION
FOR GENERIC CUTTERS

By

Hari Krishna Kosaraju

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

Master of Science

Department of Mechanical Engineering

1993

ABSTRACT

TOOL SWEEP GENERATION
FOR GENERIC CUTTERS

By

Hari Krishna Kosaraju

Software for verification of NC toolpaths must in turn, be verified to be performing
correctly and accurately. Testing of this software is facilitated by having available test
surfaces representing the envelope of the cutter during a single tool motion. However, for
five-axis motions with sufficiently general cutter shapes, this surface is not practical to
define analytically. This work represents a method developed to generate relatively
accurate surfaces approximating the desired shape. In particular, APT cutters and ball-
ended cutters were considered as examples for analysis. Surfaces accurate to within less
than 0.0001 in. were generated for a three-axis move; while surfaces accurate to
approximately one-thousandth of an inch were achieved for a five-axis move, for
moderate angles of rotation. For higher angles of rotation, the errors were higher due to
the discrete nature of the cutter model that was utilized. Trade-offs were identified

between desired accuracy and required levels of discretization of several variables.

iii

ACKNOWLEDGEMENTS

I thank Dr. Erik Goodman who guided me from the outset of this study, and steered me to
successful completion. I also thank Ms. Leslie Hoppensteadt for the valuable guidance
that she provided, and corrected my course when required. Further, I would also like to

thank Dr. Clark Radcliffe, and Dr. Brian Feeny for their direction.

1.0
1.1

2.0
2.1
2.2
2.3

3.0
3.1
3.2

4.0
4.1
4.2
4.3
4.4

iv

TA LE F

List of Tables

List of Figures

Introduction
Description of the Problem

Surface Representations
Background
Bezier Surfaces

B-Spline Surfaces

Analysis
Three-axis move

Five-axis move

Results
Three-axis move
Four-axis move
Five-axis move

Conclusions

vi

vii

12

15
16
19

31
31
34
35

 

 

LiSt 0f References

Appendix A
Definition of the APT Cutter

Appendix B
Program

Appendix C

Example of a program File

Appendix D
Example of an IGES File

Appendix E
I-DEAS

Appendix F
Coons Patch
Ferguson Surfaces

Cutter paths for Numerical Control

Appendix G

List of References

 

66

68

70

106

111

116

119
120
123

130

 

‘hfifi.»o.fi_~\&.—A—n~_ .. J-*Ar ~

vi

W

3.1) Parameters used for cutter of Fig. 3.1 for three-axis move.

3.2) Parameters used for the five-axis move.

4.1) Parameters used for the ball-ended cutter for the three-axis move.

4.2) Parameters used for the APT cutter for a three-axis move.

A5.1) Interface Capabilities of IDEAS.

16

19

31

32

117

 

 

1.1)

1.2)

2.1)

3.1)

3.2)

3.3)

3.4)

4.1)

4.2)

4.3)

vii

LI [1?

Cartesian Coordinate System.

Active cutting profile changes for four-axis and

five-axis moves of a flat-end cutter for example.

Characteristic Polyhedron of a cubic Bezier patch.

APT Cutter used for analysis.

Section of Cutter that does not have a circular section

in its profile.

Cell Structure used for finding the cut values.

Magnified subcell being used to show cause of certain errors.

Ball-ended Cutter used for analysis in a three-axis move.

An example of a surface cut by a ball-ended cutter in a

three-axis move. This surface was generated in I-DEAS.

Verified Surface of the ball-ended cutter for the three-axis

14

28

29

3O

30

41

42

:

 

4.4)

4.5)

4.6)

4.7)

4.8)

4.9)

4.10)

viii

move by using the NC Verification Program. Tolerances

of the order of ten-thousandth of an inch were achieved.

APT Cutter used for analysis for both the three-axis move

and the five-axis move.

Surface generated by I-DEAS by the APT Cutter for a

three-axis move.

Verified Surface of the APT Cutter for the three-axis move.
Tolerances of the order of ten-thousandth of an inch were

achieved.

Verified Surface of the APT Cutter for the four-axis move.
Tolerances of the order of one-thousandth of an inch were

achieved.

Outlines of the five individual surfaces generated in I-DEAS
obtained from an APT cutter for a five-axis move with a 5-

degree rotation about the X and Z axes.

Shaded image of the five individual surfaces generated in
I-DEAS for the five-axis move with a 5-degree tip about the

X and Z axes.

Outline of the surface generated in I-DEAS obtained by

joining the five individual surfaces for a five-axis move with

43

45

46

47

48

49

 

4.11)

4.12)

4.13)

4.14)

4.15)

4.16)

4.17)

a S-degree tip about the X and Z axes.

Shaded image of the surface obtained by joining the five
individual surfaces for a five-axis move with a S—degree tip

about the X and Z axes. Surface generated in I-DEAS.

Verified surface of the APT cutter for a five-axis move with a
S-degree tip about the X and Z axes. Tolerances of the order

of a thousandth of an inch were achieved.

Verified Surface of the APT Cutter for the five-axis move
with a 5-degree rotation about the X and Z axes. Figure shows

interesting patterns due to various errors.

Outline of the surface obtained for a five-axis move with a
lO-degree tip about the X and Z axes. Surface generated in
I-DEAS.

Shaded image of the surface obtained for a five-axis move
with a 10-degree tip about the X and Z axes. Surface
generated in I-DEAS.

Verified surface for the five-axis move showing the error
patterns for the tolerance limits specified for a 10-degree

rotation about the X and Z axes.

Verified surface for the five-axis move showing the maximum

50

51

52

53

54

55

56

4.18)

4.19)

4.20)

4.21)

4.22)

4.23)

error for a 10-degree tip about the X and Z axes. Tolerances

of 0.004 inch were achieved.

Verified surface for the five-axis move for a 15-degree tip
about the X and Z axes showing the error patterns when the
cutter surface considered is only 10 degrees about the central

profile. Tolerance levels very high.

Outline of the surface generated by IDEAS for a 15-degree
tip about the X and Z axes and considering 20 degrees of the

cutter surface about the central profile.

Shaded image of the surface generated by I-DEAS for a 15-
degree tip about the X and Z axes and considering 20 degrees

of the cutter surface about the cenual profile.

Verified surface for a five-axis move with a 15-degree tip about
the X and Z axes and considering 20 degrees of the cutter
surface about the central profile. Errors down to order of

thousandths of an inch.

Verified surface for a five-axis move with a 30-degree tip
about the X and Z axes and considering 30 degrees about
the central profile. Tolerance levels very high

indicating the cutter surface considered Still not enough.

Surface showing end surface of revolution for the three-axis

57

58

59

61

62

A.1)

F.l)

F2)

E3)

E4)

xi

move.

A generic APT Cutter.

Parametric Surface Patch.

Machining a surface using a ball mill cutter.

Tolerances for part and drive surfaces.

Common normals on the surfaces.

63

69

126

127

128

129

 

Lil WIS

This document focuses on the problem of transforming cutter toolpaths to three-
dimensional surfaces. This problem is compounded by the fact that cutters may have
vastly different shapes and parameters. This study was aimed at modeling very generic
cutters. Then it is easy to examine special case cutters like the ball mill or the end mill, by

simply changing certain parameters and setting other parameters to zero.

In current practice the geometry of a component to be manufactured on a
numerically controlled machine tool is often coded from a drawing in a part-programming
language such as APT. Recent developments in computer processing of two-and-a—half
and three-dimensional geometries are paving the way for design systems that create a
component description as a computer stored model rather than a drawing. To manufacture
components designed in this way, the part programming approach is inappropriate, since

the geometry of the component is already in computer form.

In either case, the toolpath finally generated must usually be checked for geometric
errors -- removing too much or too little material. NC Toolpath Verification Software has
been developed at Michigan State University and other places for this purpose. However,
robust testing of this verification software, and graphical illustration of its correct
operation, requires that software be available to generate examples of the envelopes of
tools being swept through tool motions in three, four, and five axes, for quite generic

cutter shapes. This study is aimed at developing a tool to satisfy this need.

 

 

—__L‘ has...» .77 .

L1 H P BLEM

The description of a generic cutter is shown in Appendix A. The problem is to find
a surface cut by a generic cutter for a three-axis move and a general five-axis move. It has
to be noted that this problem is quite distinct from the usual problem of generating the
toolpaths given part surfaces and cutters, and the toolpath verification problem. A three-
axis move is defined by the translation of the cutter along a straight line path in three-
dimensional space; while a five-axis move is defined by the translation and rotation about
this straight line. For the problem considered, a three—axis move is defined by the
translation of the cutter along the Z-axis. A five-axis move for a cutter is defined by the
translation along the Z-axis and also by rotations about the X and Z axes for the coordinate
system shown in Fig 1.1. This does not cause a lack of generality, since any surface
obtained in this fashion can then be transformed in 3-D space to obtain surfaces positioned

and oriented anywhere in 3—D space.

These developed surfaces are to be used as test surfaces for the NC Milling
Toolpath Verification Software being developed by the MSU Case Center. The need arises
for independent test surfaces to verify the algorithms of the NC Verification software. To
test the accuracy of the operation of this software for any new cutter, independent test
surfaces have to be created. This is achieved by calculating the tool envelopes for various
three-axis, four-axis and five-axis moves. The initial and final positions and the initial and
final orientations of the cutter are specified and the cutter is assumed to translate and
change its orientation in a linear fashion in between the two specified positions. These
surfaces were developed for different cutters and were used to test the NC Verification

Software.

 

 

 

3

For a three-axis move, the problem is very simple. The tool envelope is calculated
by performing a sweep operation. Since the active cutting edge does not change in a linear
three—axis move, a sweep operation can be performed on the active cutting edge, which is
defined by the profile in a plane normal to the direction of the tool motion. This is the
method used for a three-axis move. The active cutting edge is calculated based on the
description of the cutter, and then a sweep operation is performed on this active cutting
edge. This task is achieved in l-DEAS (see Appendix E), a solid modeling package, which
is run automatically from a program file created by the FORTRAN 77 software developed
in this work. The surface is created from I-DEAS and'is written as an IGES file, which can

then be used as the test surface for the NC Verification program.

For a five-axis move, however, a sweep operation cannot be performed due to the
fact that the active cutting edge changes continuously. For example, for a Z-axis move, the
cutter not only translates along the coordinate axis but also rotates about the X and Z axes
(see Fig. 1.1). Due to this, the active cutting edge changes continuously. This is illustrated
in Fig. 1.2. Therefore, the approach taken is to discretize the cutter surface, and then
perform the necessary transformations on these discrete points. These transformed points
are then mapped into a cell structure, and the points having the maximum X value are
taken as the points on the cutter performing the cutting operation. The reason a cell
structure is used is because the transformed points cannot be used directly. The obvious
choice would be to use the maximum X values at each tool position on the profile.
However this cannot be used since a part of the tool might be performing the cutting
operation for regions before or after it, which will not be taken into account if the
maximum X values are taken directly for each tool position. The instantaneous cutting

profile is a non-planar curve, not a planar cross-section as in the three-axis move.

This cell structure has a length of 0.5 inches along the Y-axis and a length of 1.2

 

   

 

inches along the Z-axis. The cell structure has 12 cells in the Y-direction and 24 cells in
the Z-direction. I-DEAS can create a surface from a maximum of 36 intersecting mesh of
curves. Hence, the reason for dividing into a 12X24 cell structure (Fig 3.3). The maximum
X value in a particular cell is taken as the value corresponding to the center of the cell.
Hence, we have a 12X24 rectangular array of points, from which 36 splines can be
created. This is similar to sweeping a cutter across a vertical bed of nails and the end
points of the nails indicate the extent to which the cutting operation is being done. The
values of 0.5 in. and 1.2 in. for the Y and Z distances respectively, can be changed per the
discretion of the user depending on the tolerance requirements. A Z—depth buffer scheme
is used to find the correct X-coordinates of the transformed points. Each cell is further
subdivided into a 25x25 subcell structure and initially all the cells are assigned a depth
value (X) of zero. Then each of the transformed points are mapped onto a particular cell
by using a mapping function from cartesian coordinate space to an index, that assigns it to
a particular subcell. If the depth (X) of this new transformed point is greater than the depth
of the cell, then this depth is assigned as the new depth value; else the old value is
retained. All the points on the surface of the cutter are mapped in this fashion onto the cell
structure and the depth value of the corresponding cell is upgraded continuously, finally
storing only the maximum X-value corresponding to that cell. To account for the cases
where there might be “misses” in the innermost cells, i.e., the innermost cell in each of the
12X24 cells might not get any points mapped onto it, and to improve the X estimate in
other cases, a linear interpolation scheme was used in which the X value corresponding to
the centre of the central subcell is calculated by linear interpolation of the Y values for the
two adjacent subcells on the top and the bottom, the Z values of the two adjacent subcells
on the side and using Y values for one pair of diagonal subcells, and Z values for another
pair of diagonal subcells. The maximum of these five values is stored as the maximum X
value for the central subcell. Due to this, there are some errors introduced because the

points are being taken to be at the center of the cell when the points are actually

somewhere else in the subcell. (see Fig. 3.4) The reason for taking the values to be at the
center of the cell is to make the splines very smooth. Otherwise, the points could be

anywhere in the subcell due to which the splining is badly affected, leading to large errors.

(Note: The surface desired can always be created using a Z-axis move, then trans-
formed to represent any arbitrary tool move. Only a partial “cutaway” envelope is desired,

as viewing the entire envelope would visually occlude most regions of interest.)

 

 

Fig 1.1 - Cartesian Coordinate System.

 

 

 

 

Fig. 1.2 - Active cutting profile changes for four-axis and five-axis
moves of a flat-end cutter for example.

2.1 W

When a surface is machined using a cutter such as a ball ended cutter, the tool’s
center moves on another surface parallel to the cut surface, which is offset from the

original surface by an amount equal to the cutter radius, R.

There are a number of exact analytical methods to model surfaces. Some of these
methods are based on a framework built up of two intersecting families of curves. A
simple mathematical element used to model a surface is a parametric patch. The network
of intersecting curves divides a surface into an assembly of topologically rectangular
patches, each of which has as its own boundaries, two curves (namely u and v curves in
parametric space). It is assumed that u and v run from O to 1 along the relevant boundaries.
Hence r(u,v), 0<u, v<1 represents the interior of the surface patch, while r(u,0), r(u,1),
r(0,v) and r(1,v) represent the four known boundary curves. The problem of defining a
surface patch is, hence, a problem of finding a suitably well defined function r(u,v) that
reduces to the correct boundary curve when u=0, u=1, v=0 or v=I.(fig. 2.1). Associated
with every patch is a set of boundary conditions. These may include the four corner points,
the four corner curves, the tangent vectors, and the twist vectors. The surfaces are usually
described using what are usually called blending functions. These are so called because
their effect is to blend together four separate boundary curves to give a well defined

surface.

There are a number of patches that are defined in parametric space in difi‘erent
ways (Appendix F). The following sections discuss the various analytical methods for
modeling surfaces in parametric space. The various methods available to model surfaces

are discussed, and then the method that was used for our problem is described.

NW

The cubic Bezier [3] curve is simply a Ferguson curve (see Appendix F)
reformulated in a way that avoids any need for tangents to be specified. There is a similar
close relationship between the Ferguson and the Bezier surface patches. A Bezier patch is
defined in terms of a ‘characteristic polyhedron’, which is specified in terms of the
position vectors of its 16 vertices (Fig 2.1). The patch is in a sense an approximation to the
polyhedron; though in general the only points they have in common are the corner points
r00, r03, r30 and r33. The configuration of the characteristic polyhedron gives a good
indication of the general shape of the corresponding patch [4], and modification of one or
more vectors rij alters the patch in a predictable way.'Ihe advantage in defining such a kind

of patch is that no gradient or twist vectors need be specified.

The equation defining each patch is given by,

3 3
r(u,v) = Z zrijgi(u)gj(v),05u,v51 (2.1)
i=w=o
where
_ 3! 3-]; _
ggn—Zfiijfi$n_n ,k-QL23 02)

The gk are the Bernstein basis functions [1,2]. It is easy to check that r(0,v) is a
cubic Bezier curve whose characteristic polygon, is defined by r00, r01, rm and r03. Fig.

2.2 shows that this curve is one of the boundaries of the patch. Correspondingly, the other

10

three patch boundaries r(1,v), r(u,0) and r(u,1) are curves of the same type.
The Bezier patch equation (2.2) can be defined in matrix form as
r(u, V) = G (u) BGTW) (2.3)

where B = ['ij] is the 4X4 matrix of polyhedron vertices and

0(a) = [good g, (u) g2(u) g3(u)]

=|:(1—u)3 3u(1—u)23u2(1—u)u3:|
1 o 00 (2.4)
-3 3 O 0
= 2
[luu (13] 3 —630
-13 -31
=UM

Similarly C(v) = MTV, with M as defined by (2.4) and v = [1 v v2 v3]. Using these

results, the patch equation can be expressed as

r(u, v) = UMBMTV (2.5)
The Ferguson patch derived in the last section was shown to be
r(u, V) = UCQCTV (2.6)
where Q is defined by (2.16) and C by (2.14). For the two to be equivalent,
CQCT = MBMT (2.7)
or Q = (C'IM)B(C'1M)T (2.8)

On inverting C and multiplying out the right hand side we get,

11

—

No.0) r(o.1) r,(o,0) r,(o,1)
r(1.0) r(1,1) r,(1,0) r,(1.1)

 

 

Q :-
ru (0, 0) ru(0,1) ruv (0, O) ruv(0, 1)
_ru(1,0)ru(1,1) r,,(1,0) ru,(1.1)_
r
’00 '03 3 (’01 ‘ ’00) 3 ('03 ’ '02)
_ ’30 ’33 3 ('31 " '30) 3 (’33 " '32)

3('10"’oo) 3("13"03) 9("oo"1o"01 “11) 9(’02"’12"03+'13)

 

_3 ('30 ‘ ’20) 3 (’33 " ’23) 9 (’20 " ’30 " '21+ '31) 9 ('22 " '32 ' '23 + '33)_

(2.9)

From the last equation, the gradient and twist vectors at the patch comers,
expressed solely in terms of the vertices of the characteristic polyhedron, are obtained.
The gradient vectors depend only upon points lying on the boundary of the polyhedron,
while the four interior points r11, r12, rm and r22 influence the values of the cross
derivatives. The advantage of using this kind of a patch is that there is no necessity for
either the gradients or the twist vectors. 'Ihus Ferguson’s simplifying assumption, namely
that the cross derivatives are zero, can be avoided. Only the position vectors of the 16

polyhedron vertices need be supplied.

The foregoing approach has been implemented in the UNISURF surface design

system developed by Bezier [5,6] for the Renault car company.

 

 

 

 

1‘61

that '

curt-a

ofcha

region

defined 1'
having a
mathemar
finfimnon_
130thch
COlilpler pc
single B-Spl

the use Of citl

 

 

12

mm

The B-spline surfaces are related to B-spline curves [7,8] in exactly the same way
as Bezier curves are related to Bezier surfaces (section 2.2). Like the Bezier surface patch
the B-spline surface is defined in terms of a characteristic polyhedron. This mimics fairly
closely the gross geometrical properties of a surface, which is defined for a topologically

rectangular set of polyhedron vertices rij, i = 0, 1, 2, ....... , p, j = O, 1, 2, ........ ,q by

r(u,v) = 2P: irij~4,i+1(u)N4,j+l(v) (2.10)
i = Oj = 0
Here we are using cubic B-splines. The properties of B-splines [9,10,11] ensure
that the surface defined by equation (2.10) will have overall continuity of gradient and
curvature.
The advantage of using B-splines is that they are only locally non-zero. The effect
of changing a single vector coefficient is to change the B-spline surface only locally in the

region for which u and v lie in these ranges; the rest of the surface is unaltered.

A cubic Bezier patch, as that defined in section 2.2, is based on a polyhedron
defined by just 16 vertices. A single patch of this kind can represent a surface element
having a fairly simple topography. Moreover, the Ferguson and cubic spline patches are
mathematically equivalent to Bezier surfaces, and they consequently have the same
limitation. However, there is no limit on the number of vertices defining the characteristic
polyhedron of a cubic B-spline surface. It is therefore possible, by choosing a sufficiently
complex polyhedron, to represent a highly convoluted surface element by means of a
single B-spline equation such as equation (2.10). Hence, an individual cubic B-spline
surface, based on cubic polynomials, has capabilities that in other systems would require

the use of either a composite surface or of a single patch based on polynomials of degree

""5"“??—

13

higher than cubic. This advantage is reinforced by the local modification property and the
automatic second order continuity. Furthermore, another advantage of the B—spline
formulation is its ability to preserve arbitrarily high degrees of continuity over complex
surfaces. Also as mentioned earlier any change in the local shape of a B-spline surface is
not propagated throughout the entire surface. These characteristics make the B-spline
surface suitable for use in an interactive environment. IGES files can be written for Non-
uniforrn rational B-spline (NURBS) surfaces, and the MSU Verification software can read

IGES NURBS written by IDEAS.

l4

r22

 

 

 

r12
32
1'02 21
r11
31
r01
13 /"20
1'10
V
u 1'30
l'03
r00

Fig 2.1 - Characteristic Polyhedron of a cubic Bezier patch.

1.9. W

The surfaces generated by generic cutters for three-axis moves are easy to
generate. The move involves only the translation of the profile of the cutter, which is
obtained by the projection of the cutter onto the vertical plane normal to the toolpath,
namely, the X-Y plane for the cartesian coordinate system shown in Fig 1.1. The surface
obtained by translating this projected profile of the cutter along the Z-axis is the envelope
of the cutter. The surfaces generated by this method can be arbitrarily accurate and satisfy
almost zero tolerance limits, since this is an analytical solution and there are no
approximations being made in the generation of this surface. The end surfaces of
revolution are also obtained by sweeping the profile through 90 degrees, therefore, these

surfaces are also very accurate.

However, for a five-axis move, the same technique cannot be used since the cutting
profile is changing continuously as the cutter orientation changes along the move. The
cutting profile is not constant due to the continuous change in the orientation of the cutter.
This can be seen by the illustration in Fig. 1.2. As a result, a single cutting profile cannot
be utilized for calculating the surface generated by a five-axis move. Hence, there is no
easy way to generate this surface. The surfaces generated for five-axis moves are,
therefore, not completely accurate, using the algorithm developed here to generate the
surface. These surfaces involve numerous approximations in the generation of the surface,
due to which the surfaces are designed only to meet specified tolerance limits and error
conditions. The approximations made in the generation of the surface for the five-axis
moves and the inaccuracies introduced due to these surfaces are discussed later in this

chapter.

15

16

am

In this section, the surface generation technique for a three-axis move is discussed.
The surfaces generated are highly accurate as described earlier. For the sake of analysis,
the APT cutter shown in Fig. 3.1 is considered. This is a projection of the cutter onto the
X-Y plane. The cutter is obtained by rotating the profile about the Y-axis. The parameters

used to describe the cutter are shown below in Table 3.1. (see Appendix A)

Table 3.1: Parameters used for cutter of Fig. 3.1 for three-axis move.

 

 

,“patameter j

d

 

 

 

 

 

 

l——_———-
N H: 0

 

 

Only one side of the cutter is considered for analysis since the two surfaces
produced are mirror images. The cutter edge is divided into three sections: a straight line,
a circular are followed by another straight line. The four end points for the three sections
are the four end points corresponding to the straight line section, circular section and

finally followed by the straight line section as seen in Fig. 3.1. The four points are defined

as follows: x1 = 0 (3.1)
Y1 = 0 (3.2)
21 = 0 (3.3)

x2 = e+reos(1t-a) (3.4)

17
y2 = f— rsinm—a) (3.5)
22 = 21 (3.6)
x3 = e+rcos (b) (3.7)
)3 =f+ rsintb) (3.8)
23 = 21 (3.9)
x4 = x3+ (y3-L) tan (b) (3.10)
M = L (3.11)
Z4 = 21 (3.12)

The end points, namely 1,2,3, and 4 (see Fig. 3.1) are calculated and then a profile
is created by drawing a straight line between points 1 and 2, a circular arc between points

2 and 3, and finally another straight line between points 3 and 4 as shown in Fig. 3.1.

These points, after being calculated, are written in a file called a program file (see
Appendix C for an example of a program file) by the software developed. This program
file is used as the input file for I-DEAS. The FORTRAN program also writes another file
calculating the initial and final vectors of the axis, and the cutter tip location points. In I—
DEAS the Solid Modeling family is first opened; and then using the Construction
Geometry task, the profile is created as specified earlier. This profile is then translated
through the required distances as specified by the user along the three cartesian coordinate

8X88.

A skin group is created from the profiles at the start and end locations. This skin
group is saved, and then the task is changed to the Object Modeling task in I-DEAS. In
this task an object is now created using the skin function. Further the profile is also rotated
to obtain the end surface of revolution. The two surfaces are joined to obtain a single

surface. Non-Uniform Rational B-Splines (see section 2.3) are used while creating the

 

 

 

 

-m-‘H .—

 

18

“race. This object is then stored on the work bench, and an IGES file is written out for

this surface.

The IGES file created from I-DEAS is then used as the input test file for the
Michigan State NC Verification Software. The cutter parameters and the cutter locations
are also specified by the developed software and are stored in a file. This file also serves as
an input file for the NC Verification software along with the IGES file obtained for the
surface from I-DEAS. This surface is then verified by moving the tool along the surface.
As seen in the results section if the surface is all green, then the surface produced meets
the required tolerance limits. If the surface is red or is any of the colors below red (as
shown in the legend), then the surface is being gouged by the cutter. However, if the
surface is blue or any of the colors above blue; then the cutter is not touching the surface at
all, i.e, the cutter is not cutting the required surface. If the test surface created does not
produce an all green picture, there are two possible reasons: 1) The test surface produced
is not sufficiently accurate to meet the specified tolerances, or 2) There is an error of some
sort in the verification software. For the first case, the profiles may be recalculated by
discretizing the profiles further, to try to obtain more ideal surfaces. For the second case
the surfaces are achieving the design goal - namely, to help in debugging and testing of the
NC Verification software. The various cutters used and the corresponding surfaces
obtained are discussed in the results section, and examples of outcomes 1) and 2) will be

described.

 

 

19

1.2 Elie-axis Move

In this section the surface generation technique for a general five-axis move is
described. A five-axis move consists of translations along the three coordinate axes and
rotations about the X and Z axes for the coordinate system shown in Fig. 3.1. The surfaces
produced by this method are not very accurate as there are certain approximations
involved in the design of these surfaces. Hence, these surfaces are designed to meet pre-
specified tolerance levels. The cutter that was used for the analysis of a three-axis move is
again considered in this section. A profile of this cutter is shown in Fig. 3.1. The initial
tool surface of the cutter is obtained by rotating this profile about the Y-axis. The

parameters used for this cutter are shown below in Table 3.2. (Appendix A).

Table 3.2: Parameters used for the five-axis move.

 

 

 

 

 

 

 

 

 

 

 

Only one half of the cutter is considered for analysis and the surface for this is
generated. The cutter edge is again divided into three sections: a straight line, a circular
are, followed by another straight line (Fig. 3.1). Since some cutters like the one shown in
Fig 3.2 do not have a circular are as one of the sections, the user is provided with an option

to either include or exclude the circular portion for analysis. The same objective can also

 

 

20

be achieved by setting the radius to be equal to zero.

The four end points for the three sections of the cutter shown in Fig. 3.1 are the
same as described earlier. The coordinates of the four points are calculated as described

for the three-axis move. (Equations (3.1) to (3.9))

Once the four end points are calculated, the profile can be created by drawing a
straight line between points 1 and 2, a circular are between points 2 and 3 and finally a
straight line between points 3 and 4. This profile is discretized with the user having the
option to choose as many points (denoted by mm and specified by the user) in each section
as he wants. The coordinates for the discretized points in section 1 (namely the first

straight line section), denoted by (xp(i),yp(i),zp(i)) are calculated by using the following

 

 

 

equations:
. (i- 1) (yz-yl)
yptn = +y1 (3.13)
mm
077(1) -)’1) (Xz-Xl)
mm) = x1+ (3.14)
(YZ’yl)
(ypti) -y1) (22-21)
zp(i) = 21+ (3.15)

(yZ-yl)

The second part of the profile is next discretized with the user again having the
option to specify the number of points in the second section, i.e., the circular arc section,

using the following equations:

mm) = f— rsin(angle) (3.16)
xp(i) = e+rcos (angle) (3.17)
Zp(i) = Z1 (3.18)

The angle specified in the above equations is the angle between the line joining the

point on the circular arc and the center of the circular arc, and the horizontal. This angle

21

changes Continuously as the position on the are changes and is dependent on a if the point
on the arc is below the horizontal and is dependent on b if the point on the arc is above the
horizontal. Furthermore, the counter i is also incremented from (mm + 2) to (mm + nnnl *
2 + 1) as each point is being calculated, nnnl being the number of points in half the

circular section.

Finally the points on the third section are calculated using the straight line

equations as specified below

 

 

. jty4-y3)
YPU) = —————+)’3 (3.19)
nnn2

(xpa)-y)(x -x)

xp(i) = x3+ 3 4 3 (3.20)
0’4’Y3)

(yp(D-)’)(z —2 )

zp(i) = 23+ 3 4 3 (3.21)

(y4 -y3)
Here j is just a counter that keeps getting incremented as each new point on the

profile is being calculated, and nnnZ is the number of points in the last section. The
counter i being incremented from (mm + mml * 2 + 2) to (mm + nnnl * 2 + nnn2 + l).
The user has the choice to make the discretization as fine as necessary. Hence, the profile

is now discretized per the user’s specifications.

After one profile has been discretized, more profiles are calculated on the cutter by
transforming the above discretized profile. The profile calculated above is rotated about
the Y-axis, by certain increments specified by the user, up to a certain angle also specified
by the user. The transformation matrix for a rotation [3 about the Y-axis is given by

cos(B) 0 —sin(B)
RB = 0 1 0 (3.22)

sin([3) 0 cos(B)

 

 

'7" .r-tr

22

Hence, the profiles on the cutter are calculated by premultiplying RB by p = (XPU),

WU), Zp(j)) to obtain the following transformation equations:

xp (if) = xp 0') cos (B) + 2p (i) sin ([3)

mad) = W ti)
zp(i,j) = —xp (j)sin(B) +zp (j)eos([3)

(3.23)
(3.24)
(3.25)

In the above equations, [3 is the angle of rotation about the Y-axis; i is the counter

corresponding to each profile; and j is the counter corresponding to the points on each

profile. Hence, for the given profile, all the points are transformed to obtain a new profile

on the cutter for a certain angle. Then the angle is incremented, and the transformations

for all the points on the profile are repeated for this new angle. In this way all the profiles

on the cutter up to the specified angle are calculated.

The user next has to specify the initial and final positions and orientations. Based

on the number of increments in the angle along the traverse, specified by the user, the

increments along the three coordinate axes and the increments in the angle can be

calculated using the following equations:

(xe—xa)
delx = ——
(prOf—l)

dely - (ye‘ya’

 

_ (prof— 1)
(262- 2(1)
delz = ——
(prof— 1)
deltheta = (thetaZ-thetal)
(pmf— 1)
delgamma = (gammaZ — gammal)

 

(prOf- 1)

(3.26)

(3.27)

(3.28)

(3.29)

(3.30)

0“!

 

 

 

23

where (xe, ye, ze) are the coordinates of the end position, (xa, ya, za) are the
coordinates of the initial position, theta] and thetaZ are the initial and final rotations about
the X-axis, and gamma] and gammaZ are the initial and final rotations about the Z-axis.
All these are to be specified by the user, and the program calculates the increments along
each axis and the increments in the angles using the above formulae, prof being the

number of profiles to be calculated along the traverse in the Z-direction.

The transformations about the X-axis and the Z-axis are next applied to each point
on every profile of the cutter. The transformation matrix for the rotations about the X-axis
by an angle 9, is given by,

1 0 0

R9 = 0 005(9) sin(9) (3.31)
0 -sin(9) cos (9)

and the transformation matrix for the rotation about the Z—axis, by an angle Y is
given by
cos (Y) sin (Y) 0

R7: —sin (Y) cos (Y) 0 (3'32)
0 0 1

The overall transformation matrix for the rotation about both the X and Z—axes is

given by

cos (y) sin (7) cos (9) sin (y) sin (9)
R = —sin (Y) cos (Y) cos (9) cos (Y) sin (9) (333)
0 -sin (9) cos (9)

The transformation vector is obtained by postrnultiplying p, the position vector by

the rotation matrix R.

The components of the transformed vector are hence given by,

   

we" zany—,— .._. .—

 

 

24

xtran5(i,j. k) = xp(i,j) cos (y) -yp(i,j) sin (y) +xinitial (2.34)

xp (i,j) sin (7) cos (9) +yp (i.j) cos (7) cos (9)

t .’ .’k = e e e e e
yransuj ) —Zp(l,j)sin(9)+ylnltlal (235)

and finally

xp(i.j) sin <6) sin (7) +yp<i.j) cos (7) Me) (3.36)

t 1 2k : . . ' ' '
2 fans“ I ) +ZP(I.J) cos (6) +zmztlal

These transformations are performed for all the points. As each point is
transfonned it is mapped onto a cell structure. The reason a cell structure is used is
because the transformed points cannot be used directly. The seemingly obvious approach
would be to use the maximum X values at each tool position on the profile. However this
cannot be used since a part of the tool might be perfonning the cutting operation for
regions before or after it, which will not be taken into account if the maximum X values
are taken directly for each tool position. Another way of understanding it is that the active
cutting profile in a five-axis move is not represented by any planar profile of the cutter, but
rather is a complex, continuously changing curve on the cutter surface. The cell structure
used for the examples has a length of 0.5 inches along the Y-axis and a length of 1.2 inches
along the Z-axis. The cell structure has 12 cells in the Y—direction and 24 cells in the Z-
direction. I-DEAS can create a surface from an intersecting mesh of at most 36 curves.
This is the reason for dividing into a 12X24 cell structure (Fig 3.3). The maximum X
value in a particular cell is taken as the value corresponding to the center of the cell.
Hence, we have a 12X24 rectangular array of points, from which 36 splines can be
created. This is similar to sweeping a cutter across a vertical bed of nails and the end
points of the nails indicate the extent to which the cutting operation is being done. The
values of 0.5 in. and 1.2 in. for the Y and Z distances respectively, can be changed as per
the discretion of the user depending on the tolerance requirements. A Z-depth buffer

scheme is used to find the correct X-coordinates of the transformed points. Each cell is

 

 

25

further subdivided into a 25x25 subcell structure and initially all the cells are assigned a
depth value (X) of zero. Then each of the transformed points are mapped onto a particular
cell by using a mapping function from cartesian coordinate space to an index, that assigns
it to a particular subcell. If the depth (X) of this new transformed point is greater than the
depth of the cell, then this depth is assigned as the new depth value; else the old value is
retained. All the points on the surface of the cutter are mapped in this fashion onto the cell
structure and the depth value of the corresponding cell is upgraded continuously, finally
storing only the maximum X-value corresponding to that cell. To account for the cases
where there might be “misses” in the innermost cells, i.e., the innermost cell in each of the
12X24 cells might not get any points mapped onto it, and to improve the X estimate in
other cases, a linear interpolation scheme was used in which the X value corresponding to
the centre of the central subcell is calculated by linear interpolation for X using (X,Y)
values for the two adjacent subcells on the top and the bottom, interpolating the (X,Z)
values of the two adjacent subcells on the side and using (X,Y) values for one pair of
diagonal subcells, and (X,Z) values for the other pair of diagonal subcells. The maximum
of these five values is stored as the maximum X value for the central subcell. The reason
for using the maximum X-value instead of an average of the five different values is
because some of these might be zero due to the cells not getting any hits. Hence the
maximum of the five values is stored as the maximum X value for the central subcell. Due
to this, there are some errors introduced, in part because the (Y,Z) position of the X value
is being taken to be at the center of the cell, when the points are actually somewhere else
in the subcell, (see Fig. 3.4). The reason for taking the values to be at the center of the cell
is to make the splines very smooth. Otherwise, the points could be anywhere in the
subcell, due to which the splining is badly affected, leading to large errors. When the
program finishes all the calculations, the values stored in the cells correspond to the points

like the vertical bed of nails after sweeping a cutter across it.

 

 

 

26

The cutter is then moved to a new tool position, and the points on the cutter that
perform the cutting operation are again calculated after incrementing the angles and
positions by the proper increments as illustrated above. These transformed points are
again mapped onto the cell structure as illustrated above; changing the X-value of the
particular cell that it maps into, if the new value is greater than the value of the stored one,

corresponding to that cell.

These resulting 12X24 arrays of points are all stored in a file called a program file
(see Appendix C for an example of a program file), which serves as an input for I-DEAS.
Five such rectangular arrays were created to account for the full surface of the cutter. A
single surface cannot be taken, since the points would then be too far apart, leading to
large errors. Initially all the points are plotted in three-dimensional space in I-DEAS, and
then a mesh of intersecting curves is created by fitting splines between these points. The
splines are created from points both on the surface of the cutter (Y-direction) and along the
traverse (Z-direction). Since there are 24 points along the Z direction and 12 points in the
Y-direction, we have 24 splines running in the Y-direction and 12 splines in the Z-
direction. I-DEAS creates cubic B-splines through these points. Even though the order of
the spline can be changed, a cubic spline was found to introduce smaller errors than a
higher-order spline. Hence, a cubic spline was chosen as the curve to be fit through these
points. These are all done in the Construction Geometry task of the Solid Modeling

family, under the control of the program file generated by the FORTRAN77 program.

I-DEAS can create a surface from an intersecting mesh of curves when the number
of curves is less than or equal to 36. The splines can be unevenly distributed in the two
directions. Hence 24 splines were chosen in the Y—direction and 12 splines in the Z-
direction. Hence we have 24 points over a 1.2 inch tool move while there are 12 points for

a 0.5 inch height. To account for the full cutter, five such surfaces, each of 0.5—inch height,

 

 

 

27

were taken and joined later on.

A skin group is next created from the intersecting mesh of curves. This skin group
serves as the wireframe for creating a surface. It is saved and then the Object Modeling
task is opened up in I-DEAS. In this task an object is created from the skin group by using
the skin function. This object is stored on the work bench and an IGES file is written for

this surface.

The FORTRAN77 software also writes another file specifying the start and end
positions and orientations. This file, the IGES file written out in I-DEAS, and the cutter
parameters are then used as the input test files for the Michigan State NC Verification
Software. The cutter parameters and the cutter locations are also specified. The surface is
then verified as if the software moved the tool along the surface. As seen in the results
section, if the surface is all green, then the surface produced meets the specified tolerance
limits. If the surface is red or is any of the colors below red as shown in the legend, then
the surface is being gouged by the cutter. However, if the surface is blue or any of the
colors above blue; then the cutter is not within the tolerance limit from the surface, i.e, the
cutter is not cutting down to the required surface. If the surface does not meet the specified
tolerance limits, the surfaces may need to be recalculated by discretizing the profiles
further. Alternatively, the output might be illustrating a bug in the verification software,
which would then achieve the required goal. The various cutters used and the

corresponding surfaces obtained are discussed in the results section.

 

 

x_..__...

 

 

W

 

 

 

28

Figttrv 3.1 , APT t-uttt-r- list-(l for iuritly

 

 

 

 

 

 

 

 

 

 

 

 

 

 

30
A 24,12
12 cells in 0.5 in.
this direction.
.__..\
1,1 \
Y

 

 

4 cells in this direction. __>

Each of these 12X24 cells further
subdivided into a 25x25 subcell structure.

Fig 3.3 - Cell Structure used for finding the cut values.

 

Actual value with maximum X
value might be at this point but
it is being taken to be at the
center of the cell. This leads to
errors in calculation.

 

 

 

Fig. 3.4 - Magnified subcell being used to show
cause of certain errors.

 

 

4,1 REsuL'rs

In the last section, the different algorithms used for the three-axis moves and the
five-axis moves were discussed. As explained earlier, the three-axis move involved the
sweep of a projected profile, and hence tolerances of the order of a ten-thousandth of an
inch were achieved. However, due to the discretization of the profile for the five-axis
move and errors introduced due to the splining process while creating the surface,
tolerances of the order of a thousandth of an inch were achieved for moderate angles of
rotation about the X and Z axes, and as the angle of rotation increased, the errors also

increased.

MW

For the three-axis move, two different types of cutters were used for analysis. The
first cutter used was a ball ended cutter. The parameters used for the ball ended cutter are
shown below.

Table 4.1: Parameters used for the ball-ended cutter for the three-axis move.

 

   

 

Parameter

 

 

 

 

 

 

 

 

 

 

 

 

 

The cutter used is shown in Fig. 4.1. This cutter was the cutter obtained by using

31

32

the above parameters and was generated in I-DEAS. The surface is obtained by sweeping
this cutter in a three-axis move, i.e., with translations along the three coordinate axes (Fig.
4.2). This surface was obtained by translating the cutter by a distance of 10in. along the
X-axis, a distance of 5 in. along the Y-axis, and a distance of 10in. along the Z-axis. The
projected profile was calculated by calculating the three end points of the two sections,
namely the circular section for the ball part and the straight section for the cylindrical part
of the cutter (see Fig 4.1). The surface was then generated by sweeping the profile as
described in the earlier section (Section 3). This surface was then tested by using the
Michigan State University NC Verification program. The resulting verified surface with
the tolerance limits is shown in Fig. 4.3. As seen from this figure, the surface generated
was accurate up to 0.0001 in. This is shown by the fact that the surface is all green, and the
error bounds for this were 0.0001 for both the inward and outward tolerance. Hence, no

gouging or missing the surface occurs for these specified tolerance values.

Next, an APT cutter was looked at for analysis. The parameters used for the APT
cutter are shown below.

Table 4.2: Parameters used for the APT cutter for a three-axis move.

 

 

l—

 

 

 

I—

 

 

 

 

 

This cutter is shown in Fig. 4.4. The cutter was generated in I-DEAS by

calculating the control points of the projected profile and then revolving it about the Y-axis

 

33

by 360 degrees. The surface was generated by performing a three-axis move on the cutter.
The cutter was translated by a distance of 5 in. along the X axis, 7 in. along the Y-axis, and
a distance of 7 in. along the Z-axis. The surface generated is shown in Fig. 4.5. The
surface was generated by sweeping a continuous profile. The profile was calculated from
the four end control points of the two straight sections and the circular section (see Fig.
4.4). The surface was then verified by the verification program. The surface was generated
with errors less than 0.0001 in. This is shown by the fully green portion of the verified
surface shown in Fig. 4.6. As seen from the legend, the errors are less than ten thousandths
of an inch. Hence, highly accurate surfaces can be generated by the algorithm described in
Section 3.1 for a three-axis move. The end surface of revolution was also created by
revolving the profile and this was then joined to the surface obtained for the three-axis
move. When the IGES file for this surface was written and verified (see Fig. 4.23) it was
seen that tolerances of the order of ten-thousandth of an inch were achieved. A bug in the
verifier which caused the apparent gouging at the top of the cutter was found from the

picture, and subsequently was easily fixed.

 

   

34

31W

Next, a four-axis move was performed. The APT cutter described earlier in Section
4.1 was used for analysis. The same parameters as that used for the three-axis move and
listed in Table 4.2 were used for this cutter. The cutter was generated by taking a projected
profile and then revolving it about the Y-axis by 360 degrees. The cutter was made to
perform a four-axis move, namely, translations about the three cartesian coordinate axes
and rotation about the Z-axis. For the example considered for analysis, the cutter was
actually tipped by 10 degrees over a 3 in. move along the Z-axis. The cutter started off in
the vertical position and was made to r0tate gradually about the Z-axis reaching a final
orientation of 10 degree rotation about the Z-axis after a translation of 3 in. along the Z-
axis. The surface obtained was then verified using the NC Verification Software. It was
found that for the four-axis move the errors were still less than a thousandth of an inch.
The surface can be seen in Fig. 4.7. The legend shows that the errors are less the 0.0008 in.
This includes both inward tolerances and outward tolerances. This implies that the
maximum errors introduced in approximating the surface were less than 0.0008 in. for a
four-axis move when the cutter is tipped by 10 degrees about the axis of movement. This
was sufficiently accurate to meet the testing and illustration needs of the NC verification
software. Note that there is a blue line of what appears to be missed tolerances at the
bottom of the surface in Fig. 4.7, as well as the red band at the top. The blue band was
caused by another bug in the four-axis and five-axis verification algorithm, and was again
easily fixed when made evident by this picture, (Naturally, this sharp-pointed cutter had
not been among those against which the verification software code was initially tested, and

the bug is not exposed by more usual cutter shapes).

35

43W

A five-axis move was next considered for analysis. The cutter used was an APT
cutter, the description of which is shown in Table 4.2. A five-axis move consists of
translation about the three cartesian coordinate axes and also rotations about the Z-axis
and the X-axis. In this case, the cutter profile was discretized. Due to the discretized nature
of the cutter, the surfaces generated were specified to meet certain tolerance values. First a
five-axis move for small angles was considered with the cutter translating by 1.2 inches
along the Z—axis and tipping the cutter by 5 degrees about the X-axis and 5 degrees about
the Z-axis respectively. The surfaces generated by IDEAS from a mesh of curves can be
generated only from a maximum of 36 intersecting curves. However, the 36 curves can be
distributed in any fashion. For a 1.2 inch move along the Z-axis, 24 points were
considered along the traverse; and 12 points were taken in the vertical direction. The 12
points taken were for a height of 0.5 inch. Five such surfaces were created, and the
resulting smaller surfaces were joined in I-DEAS by using the sewing operation in I-
DEAS. Each of the 12X24 cells was further subdivided into a 25x25 subcell structure.
The nine innermost subcells were considered for obtaining the value corresponding to the
central subcell, which was the only value used in the 12X24 cell structure for spline
fitting. The maximum of the values corresponding to the center subcell and those obtained
from linear interpolation between the 8 surrounding subcells (total of five values) were
used to obtain the value corresponding to the cell. This leads to errors in the calculation of
the value corresponding to the cell since the value at any point in the subcell is being
translated to the center of the cell. Since the height is 0.5 inches, and this is being
subdivided into 12 cells, and each of these cells is further subdivided into 25 subcells; the
maximum error in Y due to this could be 0.5/(12X25) = 0.001666 when the point
performing the cutting operation (having the maximum X value) is at the very corner of

the cell. Furthermore when all the points are calculated, a mesh of curves is being created

 

 

36

in l—DEAS by fitting cubic splines through these points resulting in a 12X24 intersecting
mesh of curves. This splinin g process could introduce errors since a cubic B-spline was
used to interpolate between the points. Also the cutter was simulated by discretizing it
very finely. Finally we could also have round-off errors introduced even though double

precision values were used.

For the five-axis move, the number of points considered was 2500 points on a
profile, and 41 profiles were considered on the surface of the cutter. These profiles were
considered at 0.5 degree increments up to an angle of 10 degrees on either side of the
central profile (namely, the profile obtained by looking at the cutter from the direction of
traverse, i.e., Z-axis and projected onto the X-Y plane). This discretized nature of the
profile also leads to some errors in calculation since the actual point performing the
deepest cutting Operation could be a different one, other than the one obtained from the
2500 points on each profile considered for analysis. The outlines of the five different
surfaces created are shown in Fig. 4.8. These surfaces are created in I-DEAS from the
program file (see Appendix C for an example of a program file) that was generated as an
output from the software (Appendix B). Fig 4.9 shows the shaded image of these difi‘erent
surfaces, while Fig 4.10 shows the single surface obtained by joining the five individual
surfaces. Fig. 4.11 shows the shaded image of the whole surface. This surface is written as
an IGES file and is then loaded into the verification software to test the software. The
resulting verified surface is seen in Fig. 4.12. It can be seen that the inward tolerances are
less than a thousandth of an inch while the outward tolerances are also of the same order.
Hence, for a very small angle (5 degrees about the two axes), the tolerances obtained are
approximately at the desired limits of a thousandth of an inch. Some very interesting
patterns can be seen when the tolerance limits are changed to vary from 0.0001 in. instead
of 0.001 in. as seen earlier. The patterns seen are due to errors involving all the factors

listed above, the primary error being from the splining process. It was found that when the

 

 

 

37

program was run with the same parameters but with 1 degree increments on either side of

the central cutter profile instead of 0.5 as used earlier, the inward tolerances increased to
0.005, an increase in error by five times. This can be explained by the fact that now there
are more misses obtained due to the discretization. While the actual cutting is done by
some other point, the large increments in angle limit the number of points that are being

considered and, hence, cause the increase in error.

Another test case was considered next where the angles of rotation about the X and
Z axes were increased to 10 degrees for a movement of 1.2 in. along the Z-axis. The cutter
was made to start off in the vertical direction and was then tipped by the angles specified,
by the time the cutter reached the final position. In this case, the number of points
considered on each profile was 3000. The number of profiles considered was 41 again as
earlier, with the increment in the angle for each profile being 0.5 degrees. Hence, the cutter
surface was again simulated for an angle of 10 degrees on either side of the central cutter
profile. The same cell structure as described earlier was used. Each of the 12X24 cells was
further subdivided into a 25x25 subcell structure. The cutter transformations were
calculated, and the surface created is shown in Fig 4.14. Fig 4.15 shows the same surface
but shaded to show the five different surfaces. These five different surfaces were then
joined in I-DEAS. The IGES file was written for this surface and then loaded into the
Verification software. Fig 4.16 shows the error patterns for the surface when the inward
and outward tolerance limits are specified to be 0.001 inch. The errors were found to
progressively increase towards the top right hand comer of the surface. This indicates that
the amount of cutter surface considered was insufficient and the actual cutting operation is
being done by a part of the cutter that is beyond the $10 degree section of the cutter
surface used in developing the test surface. The maximum error was found to be less than
0.004 in. at the upper right hand comer. This is illustrated in Fig 4.17. The completely

green surface for the error bounds shown illustrates the fact that the errors are less than

 

38

0.004 in. The surface was found to be gouged by a maximum of less than 0.004 in. at the
upper right hand comer. This can be explained by the fact that due to misses in these cells,
the linear interpolation pulls the surface inwards, causing the surface to appear gouged.
Another test case was run by increasing the increment size between each profile to 1
degree and increasing the cutter surface considered to 20 degrees on either side of the
initial active cutter profile. This reduced the error at the upper right hand corner but
increased the error in the middle of the surface to about 0.005 in. These errors appeared as
random red patches on the surface. These can be explained by the fact that we now get
errors in the surface due to misses on the surface of the cutter, rather than due to the angle

of the cutter. Also there might not be enough points at any particular section of the cutter.

The case where the cutter tips by 15 degrees was next considered for analysis. The
cutter is again translated by 1.2 inches and is made to tip by 15 degrees about both the X
and Z axes for this traverse. The cutter starts off in the vertical position initially and then
rotates gradually, making a final angle of 15 degrees about the two specified axes at the
end of traverse. The number of points considered was again 3000 on each profile. Initially,
the case was considered where the number of profiles was unchanged. The cutter surface
was considered up to an angle of 10 degrees on either side in 0.5 degree increments. This
surface was created in I-DEAS, and the IGES file was written out for this surface. When
this surface was verified, the surface was found to have enors increasing upward and
toward the right; showing maximum errors at the upper right hand corner. This surface is
shown in Fig. 4.18. The surface has progressively increasing errors toward the right hand
top corner, with the maximum error being at the top and being of the order of 0.03 in. The
cutter surface was changed to include 20 degrees on either side of the cutter, in 0.5 degree
increments. The outline of the surface generated in IDEAS is seen in Fig. 4.19. The
shaded image is shown in Fig. 4.20. The IGES file for this particular surface was created

and loaded into the Verification software. It was found that when the cutter surface angle

 

 

39

was Changed to include 20 degrees on either side with 0.5 degree increments, the errors
were found to be less than 0.005 in. This surface is shown in Fig. 4.21. The small red
patches seen are due to “misses” in the corresponding cells. When all the nine subcells
considered for analysis of a particular cell have misses, then due to the linear
interpolation, the surface is pulled inward and the surface appears as if it is gouged. A
similar phenomenon was noticed for the 5—degree move initially, when only 2000 points
were used on the profile. However, by increasing the number of points on the profile of the
cutter to 2500 for the S-degree move, the problem of “misses” was taken care of and all
the cells were able to get hits. However, in this case for the 15-degree move, the number of
points could not be increased further since very large arrays were being used and memory
problems resulted from an increase in the number of points on the profile. The arrays used

were very large and the memory storage required was not available from the computer.

As a case of special interest, the case where the cutter is rotated by 30 degrees
about the X and Z axes at the end of the tool move was considered. The number of points
on each profile was again 3000. Initially, the cutter surface up to 10 degrees on either side
of the projected cutter profile was used with 0.5 degree increments. The resulting surface
was created in l-DEAS and the IGES file written for this particular surface. The IGES file
was loaded into the Verification software and checked for errors. This resulted in very
large errors of the order of a tenth of an inch. The errors were small at the bottom of the
surface and increased progressively towards the upper right hand comer. This suggests
that the cuter surface segment used was not enough and that a larger section should have
been used. The cutter surface was gradually increased to 55 degrees on either side of the
initial active cutter profile, with 0.5 degree increments between the profiles. It was found
that the errors were reduced to the order of a hundredth of an inch. The errors still
increased progressively towards the upper right hand corner, indicating that the cutter

surface considered might not have been enough. However, when the cutter surface

 

40

considered was increased, the computer ran out of memory due to the large array storage
involved in this. Further, due to the large memory storage, the computers take an
enormous amount of time paging for information; and consequently, the program runs
very slowly. Hence there is a trade off between accuracy and speed. If faster processes are
required, then smaller arrays would have to be used, resulting in larger errors and vice
versa. Also there is a trade off between memory availability and accuracy. If more virtual
memory was available for use, then the surface of the cutter could be discretized even
more finely resulting in very accurate computations, since as the number of points on the
cutter surface increases, it approaches the limiting condition of a continuous cutter.
However, since there is a limitation on the amount of memory available to us, the price has

to be paid in the form of accuracy, resulting in errors in the calculations.

       

tllaeqooot-eo-Ie-egllcbrt 'I“?~:O‘.t

  

 

41

 

i
-'. . E;
.‘ 7:; Y
3 > _
'l‘ 1 _,:3 J
.22 1
312:3 E
3.: _. .l
.k '.
n I “f:

 

 

 

 

Figllrv 4.1 - Ball-vmlml Cltttvr' trsml for analysis in it throw-axis rmwv.

42

 

 

 

 

Plum-«- 4.‘

Au vxurrrplv ()f it ~~Ilr'l'iu't- cut by :t ltilll-t‘llllt'll ruttt-r in :t tlrrt-tunxis Illtn‘t'.
This hlll'filt't' was gl‘ltt‘l'flfl‘tl in I-DEAS.

 

tn) ‘vwfil'

Lin' 3.4 5w

Figurt- 4.3 ,

 

Yttrifiml Surface of the hall-t-rrtlt-d t-uttvr for the three-axis move by using
the NC Verification Program. Ttrlt‘l'itlltft's of the order of ttIu-tlruusaudtlr
nl' an inch Wt‘l’t‘ m-lrit-Vt-(l.

 

 

 

 

 

44

 

Figure—1.4 - APT (‘Irttt-r' llM‘tl for analysis for lmtlr tlrt- tlii'vv-uxis ulth' :tml tlu- fivv-
:rxis umvv.

 

 

45

 

 

Fiuurt- 4.3 ,

Snr'fm-t- gt-ru-r-ittt-tl lry I-DEAS by the APT (‘uttt-r fur- :1 lhl't't‘n‘txi.‘ umvv.

 

 

46

Pun-t

 

taut, an] 3:):-

feco'r hat )1

”my. D-t all"

Add.

 

Figurt- 4.0 Vt‘r'ifit'd Surface of the APT Cuttor fur Illt‘ throw-axis mnvv. Tolerances
of the order- r)f ton-thousandth of an inch wot-v aclrit-VHI.

 

 

 

47

die-t3»-

nryts‘ ll” 11l'

$1.1

 

Figurt- 4.7 , \'t-1'ifio(l Surface of the APT Cutter for tho bur-axis move. Tolerance s
of tho order of thousandth of an int-h were achieved.

 

48

LA

 

 

Figlu‘v 4.8 — ()llt-lillt‘s of tho fivr- iutlivitluitl Slll'fitt't'h gt‘ut'l'tttt‘ll in I-DEAS olrtttirrt-tl
from an APT vuttvr for it livv-uxis mow with it 5 (It'gl't't' rotation about

thv X and Z HXt‘h.

 

 

 

49

 

Ls

 

Figurt- «1.9 » Sharlt-tl image of thi- fin- iutlivitlual surfaces gt-uvrutt-tl in I-DEAS for tlu-

livv—axis urovt- with :r 5 (It-grow tip about tht- X and Z ttXt's.

 

 

 

 

 

 

 

50

Y
J 4”
Figur-t- 4.10 - Outline of the stir-fart- gt-ru-r'ittt-tl in I-DEAS olttttint-tl by joining tlrt- fiw-

individual srtr'fiu't-s for it fivv-zrxis urovr- with it 5 (it‘gl't't' tip about the» .\
and Z :txvs.

 

 

l
l
l

 

51

 

is

Fiuur-t- -1.11 , Shudml inmgt- of ”iv stir-fum- olrtnint-tl Iry joining, tht- fivr- individual sur-
faces for u fivv-uxis “1on with it .3 tlt'gn-t- tip about the X turd Z :ths.
Surf‘urv gt‘llt'l'ittt'tl in I-DEAS.

 

 

 

     

 

J‘s.

Figurt- 1.12 Verified surface of the APT ruttr-i- for u fivwatxis move with a 5 degree
tip about the X and Z axvs. Tolt-ranrt-n ol' the «ruler of thousandth of an
inch were achieved

Pout; an: i; .-

)nd 4 .AL Value

Fm" 439'. A!
ng- J-v it)”

‘Am

 

 

F1 rurv 4.13 ~ Vr-r'ified Surface of the. APT C‘uttt-r for tho firm-axis move with a 5 degree
rotation about the X and Z ilxvs. Figure shows intt-rcsting puttcrns due

to various errors.

54

 

 

’<
N
X

Figur-c «1.1-I - ()utlinv of tho Mll'ft'll‘t' olrtuint-tl for it two :txis tnovv with it ill «ll-grt-t- tip
:llHnlt thv X and Z more. Surfiu-t- gt-rn-r'ittt-tl in I-DEAS.

55

 

 

 

Figtu't- 4.15 ,

Shitdt-(I inmgt- of tho sln'fm-t- Uhtilillt‘d for u fivc—itxis mow- with :1 ll)
ult-grt-t- tip ulrout tln- X and Z uxr-s. Sur'fut-t- gt-nt-rutvtl in I—DEAS.

 

56

9mm vs 2::-

'w. ..t 111w

Can. .2 131.93
1‘ n» '. ult’r
a.” 3.); “:11:
Penn v .2:th

Wm;- a-Hil"
our

 

Figurt- 4.16 Vorificd surface for the five-axis mow- showing thc error patterns for the
tolcr'ance limits specified for a ll) «lt-gr-t-t- rotation about. the X and Z

t'lXt'S.

 

 

 

 

 

 

Figurr- 4.17 ~ Vorifiml surfiu-c for tln- h'wuaxis mow- showing tht- maximum t'r'ror for
u [U <lt-gt't-t- tip about thv X and Z :rch. Tolt-t'iun't-s of (LOU—l inch wort-
itt'lllt'Yt'tl.

 

 

5.7.“, .

58

 

Figuri- 4.18 - Verified surface for the fivc-axis mow for a 15 dt-grr-c tip about the X and
Z axcs showing tlrc err-or [)attt‘l'lh wln-n the ruttr-r- surfact- considcred is

only 10 degrees about tho central [)l'lrfilt‘. Tult'l'illlt't' lvvr-ls vor'y high.

_ a”
'35

 

 

 

 

-.—____H..___

59

 

A #mw, /\__ H -_ _ _______ w “W”

\\/

f£\

4* ;
Figlu‘t- 4.19 ’ ()utlilu' of thv .s‘lu‘fnm- gt'm‘ratml by I-DEAS for n 15 clvgn-t- tip about

rlu- X and Z nxvs mul «'onsitluring 20 (It-grows of tlu- vuttvr hlll'filc't' about
tlu- rvun'ul profilv.

 

 

 

6O

 

 

 

 

Figun- 4.20 Slliult'tl iluagt- of tho sul'fm-o gvllt'l'éltvd by I-DEAS for :1 15 «It-gnu- tip
about tho X and Z uxvs mul t‘Ullhitlt'l'illL’, 20 clt-gn-u-s of tlu- vutu-r surfact-
about tlu- m-utrnl profilv.

 

Oauu rd 2:0
0.. 4 .al «I»:
2.! .nl.u

’ u C‘lut‘.

 

Figurv 4.21 , Vurifiml surface for a fivv-axis mow- with a 15 «It-grep tip about the X
and Z axos and considering 2U Ilt‘gl't‘t'h ol' tlu- ruth-r surfarv about tho

u-utral profile. Errors down to Hl‘llt'l‘ ol' thousandth of an inch.

 

 

 

 

 

Figun- 4.22 ~ Vt-rih'ml surfarv fo - a livv-uxis mow with n 30 tlt-gn-o tip about tlu- X and
Z mum and rousitlvriug 3t) (lt'gi'vvs about tlu- l‘('lltl‘ill profilv. Tolt-rum'r
low-l5 wry high imliratiup; tlu- Fultt'l' surfarv rousitlvrml still not vuough.

 

__ ,,

rotate rd L:-

l

JAIA
L’s—v ‘1'
,.-u
tom! '3'»
F).r;- D‘vnl"
w“

 

Figurt- -l."3 - Surfaro showing mu] surfarv of rvvolution for the thrt-v-axis move.

   

 

9.9. W

1) For the algorithm used, tolerances of the order of one ten-thousandth of an inch
were achieved for three-axis moves. It can be seen that for this particular algorithm highly
accurate surfaces were generated for the three—axis moves for both the APT cutter and the

ball-ended cutter.

 

2) For a five-axis move, tolerances of the order of a thousandth of an inch were
achieved for moderate angles of rotation. For angles up to 15 degree rotations about the
two axes for a move of 1.2 inches along the Z-axis, it can be seen that the tolerances

obtained were of the order of a thousandth of an inch.

3) For larger angles, the errors were found to increase. This is due to a restriction
on the availability of memory. It can be seen from the 10 degree and the 15 degree moves
that the errors increase towards the top right hand comer. Ideally we would like to take the
whole of the cutter surface into account. However due to the restriction on the availability
of memory, only a part of the cutter surface has to be taken into account. Further even for
this part of the cutter, the cutter surface can be discretized only up to a certain level,

though it is desired that the cutter surface be discretized very finely.

4) There is a trade off between memory and accuracy. The higher the accuracy, the
higher the memory requirements for very accurate surface generation. It has to be noted
that if the surface could be discretized very finely and enough of the cutter surface could

be considered, then the surfaces generated would be very accurate. However we have to

65

trade off the available memory with accuracy.

5) There is a trade off between speed and accuracy. It was found that when the
program was being run, to obtain very accurate surfaces very large arrays had to be used.
This resulted in very slow processing of the program due to the large amount of time taken
for paging the arrays. Typically each run for the surfaces generated took about a day. If
less accurate surfaces are desired then smaller arrays can be used and this speeds up the

process.

 

6) The algorithm to date has been able to detect two errors in the NC Verification
Software. The bands of pink and blue seen at the top and bottom of the surfaces for the
three-axis and four-axis moves were found to be errors in the NC Verification software

that were later corrected.

 

 

 

 

 

W

1) Ralston, A. (1965), A First Course in Numerical Analysis, McGraw-Hill.

2) Goult, R. J., Hoskins, R. F., Milner, J. A. and Pratt, M. J. (1973), Applicable

Mathematics, MacMillan.

3) Bezier, P. (1968), ‘How Renault uses Numerical Control for Car Body Design
and Tooling’, SAE Paper 680010

4) Bezier, P. (1972), Numerical Control: Mathematics and Applications, Mley.

5) Bezier, P. (1974), ‘Mathematical and Practical Possibilities of UNISURF”. In
Computer Aided Geometric Design, (R. E. Bamhill and R. F. Reisenfeld, eds.), Academic
Press.

6) Bezier, P. (1974), ‘UNIS URF System: Principles, Program, Language’, Proc.
1973 PROLAMAT Conference, Budapest (J. Hatvany ed.), North Holland Publishing
Company, Amsterdam.

7) Gordon, W. J. and Reisenfeld, R. F. (1974), ‘B-Spline Curves and Surfaces’. In
Computer Aided Geometric Design (R. E. Bamhill and R. F. Reisenfeld, eds.), Academic
Press.

67

8) Faux, I. D. and Pratt, M. J. (1985), ‘Computational Geometry for Design and
Manufacture’, John Wiley & Sons.

9) Cox, M. G. (1972), ‘The Numerical Evaluation of B-Splines’, .1. Institute of
Mathematical Applications. 10, 134-149.

10) De Boor, C. (1972), ‘On Calculating with B-Splines’, J. Approx. Th. 6, 50-62.

11) De Boor, C. (1978), A Practical Guide to Splines, Springer-Verlag, New York.

 

 

 

 

 

 

 

APPENDIX A

FTH

For three-dimensional machining, the profile of the part surface can be any combination of
the surface allowed in APT. A contouring motion is defined on the basis of the part profile. The
path of the cutter profile is continuously controlled to generate the desired geometry of the part
surface. The statement defining a cutter in the APT language usually has seven parameters and it

is described as follows (Fig A.l):

CU'I'I'ER/d,r,e,f,a,b,L
where

d = the diameter of the circle generated by the intersection of the end and the side surfaces
of the cutter.

r = the radius of the corner circle.

e = the distance from the comer circle center to the cutter center line.

f = the distance from the corner circle center to the plane passing through the cutter end
center and perpendicular to the cutter center line.

a = the angle(degrees) between the cutter end surface and the plane perpendicular to the
cutter axis (it lies in the range of 0 S a < 90).

b = the angle(degrees) between the cutter side surface and the cutter axis (it lies in the
range of -90 < b < 90).

L = the cutter height measured from the control point of the cutter.

It has to be noted that usually, the corner circle should be tangent to the end and side

surface.

68

69

 

 

 

 

 

 

 

 

 

 

Fig. A.l: A generic APT Cutter

 

 

 

 

5 w.-3___

 

 

 

APPENDIX B

 

7-? *r l.

 

00000000000

 

ERQSERAM

LAST MODIFIED : JULY 15, 1993

***************************************************************
*

* PROGRAM TO CALCULATE THE TOOL ENVELOPE

*

: PREPARED BY

* KOSARAJU HARI KRISHNA

I

*

*********

***************************************************************

PROGRAM MAIN

DOUBLE PRECISION ANG,ANG_TEMP,AI

DOUBLE PRECISION XP_BETA(45,3050),YP_BETA(45.3050)
DOUBLE PRECISION ZP_BETA(45,3050)

DOUBLE PRECISION A1,BI,R1 ,L1,El,Fl

DOUBLE PRECISION XA11,YA11,ZA11

DOUBLE PRECISION XE11,YE11,ZE11

DOUBLE PRECISION XA1.XA2,XA3,XA4

DOUBLE PRECISION YA1,YA2.YA3,YA4

DOUBLE PRECISION 2A1 ,ZA2,ZA3,ZA4

DOUBLE PRECISION PROF__INCREMENT

INTEGER PROPS

DOUBLE PRECISION HEIGHT

DOUBLE PRECISION P12,PII,ANGLE

DOUBLE PRECISION XP(3OSO),YP(3050),ZP(3050)

DOUBLE PRECISION XTRANS(4S,3050),YTRANS(45,3050)
DOUBLE PRECISION ZTRANS(45,3OSO)

DOUBLE PRECISION XF(l85,185),YF(185,18S),ZF(185,185)
DOUBLE PRECISION XF3(225,225),YF3(225,225),ZF3(225,225)
REAL Z_INDEX,Y_INDEX

70

 

 

 

71

INTEGER Z__INDE1,Y_INDE1
INTEGER ARC
INTEGER NNN,NNN 1 ,NNN 2

P11=3.l41592654
XA1=0.0
YA1=0.0
ZAI=0.0
READ IN THE COORDINATES OF THE CUTTER
PRINT *, "INPUT THE ANGLE OF LOWER FACE OF CUTTER, A”
READ(*,"') A12
PRINT *, "INPUT THE ANGLE OF UPPER FACE OF CUTTER, B"
READ(*,*) Bl2
Bl: -(B12/180*PIl)
PRINT ‘, ”INPUT THE RADIUS OF THE CIRCULAR PART OF CUTTER,R"
READ(*,*) R1
PRINT "', "INPUT THE LENGTH OF THE CUTTER, L"
READ("',*) Ll
PRINT *, "INPUT THE HEIGHT OF THE CENTER, F"
READ(*,*) F1
PRINT ", "INPUT THE OFFSET OF THE CENTER, E"

READ(*,*) El

CALCULATING THE COORDINATES OF THE CUTTER
P12 = 3.141592654/2
ZA2 = ZAI
2A3 = 2A1
2A4 = 2A1
SL1 = SQRT( F1**2 + E1**2)
SL2 = SQRT ( SL1**2 - Rl**2)

 

 

 

 

 

72

XA2 = XA1+ E1 + R1 * COS(PI2 -A1)
YA2 = YA1+ Fl - R1 * SIN(P12 - A1)
YA3 = YA1+ F1+ Rl * SIN(Bl)
YA4 = YAI + L1
XA3 = XA1+ E1+ RI * COS(Bl)
XA4 = XA3 -(YA1+ L1 - YA3) ' TAN(B1)
PRINT *,"DO YOU WISH TO PERFORM A 3-AXIS MOVE OR A GENERAL
IFIVE AXIS MOVE"
PRINT *, "SELECT ONE OF THE FOLLOWING"
PRINT *, "l - THREE AXIS MOVE"
PRINT *, "2 - FIVE AXIS MOVE"
READ(*,*) N1
IF(Nl.EQ. 1)THEN
GO TO 1001
ELSEIF(N1.EQ.2)THEN
GO TO 2001
ENDIF

2001 PRINT *, " "

PRINT *, ”INPUT THE ANGLE ON THE CUTTER TILL WHERE YOU WOULD
ILIKE To TAKE THE PROFILES INTO ACCOUNT"
READ(*,*) ANGLE
PRINT *, "INPUT THE ANGULAR INCREMENT BETWEEN EACH PROFILE"
READ(*,*) PROF_INCREMENT
PROFS=ANGLEIPROF_II~ICREMENT*2 + 1
THE PROFILE IS BEING DISCRETIZED
SPLI'I'I'ING THE FIRST PART(LINE) OF THE CUTTER
PRINT ',"INPUT THE NUMBER OF POINTS IN THE FIRST SECTION "
RBAD("',*) NNN
Do 11 I = l,NNN+1
YP(I)= (I-l)*(YA2-YAl)/NNN + YAl

 

11

41

21

 

 

 

73

IF(YA2.EQ.YA1)THEN
XPa) a XAI + (I-l)*(XA2-XAl)/NNN
ZP(I) = 2A1 + (I-1)*(ZA2-ZAl)/NNN
ELSE
XP(I)=XAI +( (YP(I)-YA1)/(YA2-YA1))*(XA2-XA 1)
ZPCI)=ZA1 +( (YP(I)-YAl)/(YA2-YA1))*(ZA2-ZA1)
ENDIF
CONTINUE
SPLITI'ING THE CIRCULAR PART OF THE CUTTER
PRINT *,"DO YOU HAVE A CIRCULAR ARC AS PART OF THE CUTTER"
PRINT *," INPUT 1 FOR YES AND 2 FOR NO"
PRINT *," "
READ(*,") ARC
ARC = 1
IF(ARC.EQ.1)THEN
GO TO 21
ELSEIF(ARC.EQ.2)THEN
GO TO 31
ELSE
PRINT *,"PLEASE INPUT AGAIN"
GO TO 41
ENDIF
PRINT", " "
ANG=P12-Al
ANG_TEMP=ANG
PRINT *,"INPUT THE NUMBER OF POINTS IN THIS SECTION "
READ(*,*) NNNl
DO 22 I = NNN+2, NNN+NNN1+1
ANG = ANG - ANG_TEMP/nnn1

YP(I)= Fl-Rl*SIN(ANG)

 

bl

 

 

 

74

XP0)= E1 + R1*COS(ANG)
ZP(1)=ZA1
22 CONTINUE
ANG=O
ANG_TEMP=BI
DO 23 I = NNN+NNN1+2NNN+NNN1*2+1
ANG = ANG + ANG_TEMP/NNN1
ZP(I)=ZA1
YP(I)=F1+R1*SIN(ANG)
XP(I)=E1 + R1*COS(ANG)
23 CONTINUE
31 PRINT *, " "
C SPLITTLNG THE LAST PART OF THE CUTTER
J=l
PRINT", "INPUT THE NUMBER OF POINTS IN THE LAST SECTION”
READ(*,*) NNN2
DO 24 I = NNN+NNN1*2+2,NNN+NNN1*2+NNN2+1
YP(I)= (I)*(YA4-YA3)/NNN2 + YA3
XP(I)=XA3 +( (YP(I)-YA3)/(YA4-YA3))*(XA4-XA3)
ZP(I)=ZA3 +( (YP(I)-YA3)/(YA4-YA3))*(ZA4-ZA3)
J=J+1
24 CONTINUE
C THE PROFILE HAS BEEN DISCRETIZED
C FINDING THE DIFFERENT PROFILES OF CUTTERS
II = l
J = l
K = l
ANGLEl = ANGLE
ANGLE = -ANGLE
DO 111 AI = -ANGLE1,ANGLE1,PROF_INCREMENT

 

 

75

1F(AN0LELE.0)THEN
ANGLE = PROF_INCREMENT * (J -l) - ANGLEl
J=J+l
ELSEIF(ANGLE.GT.O)THEN
ANGLE = PROF_INCREMENT * (K+l )
K=K+l
ENDIF
PROFILE_POINTS=NNN+NNN1*2+NNN2+1
C CALCULATING THE DIFFERENT PROFILES
BETA = ANGLE*P11/ 180
D0123 J] = I,PROFILE_POINTS
XP_BETA(II,JI)= XP(Jl) * COS(BETA) + ZP(JI) * SIN(BETA)
YP_BETA(Il,JI)= YP(J1)
ZP_BETA(11,J1)= - XP(Jl) * SIN(BETA) + ZP(J1)* COS(BETA)
123 CONTINUE
I 1 = II + 1
111 CONTINUE
C THE DIFFERENT PROFILES HAVE BEEN CALCULATED
C CALCULATING THE TRANSFORMATIONS AND FINDING THE MAXIMUM X
PRINT *,"INPUT THE INITIAL ROTATIONS ABOUT THE X-AXIS,THETA1"
READ(*,*) THETA12
THETAI = THETA12 *PIl/180
PRINT *,”INPUT THE FINAL ROTATION ABOUT THE X-AXIS,THETA2"
READ(*,*) THETA22
THETA2 = THETA22 "‘ PIl/180
PRINT *,"INPUT THE INITIAL ROTATION ABOUT THE Z-AXIS,GAMMA1"
READ(*,*) GAMMA12
GAMMA] = GAMMA12 *PIl/l80
PRINT *,"INPUT THE FINAL ROTATION ABOUT THE Z-AXIS,GAMMA2"
READ(*,*) GAMMA22

3“... an.

 

76

GAMMAz = GAMMA22 * PI 1/180
PRINT *,"INPUT THE NUMBER OF PROFILES To BE CALCULATED"
READ(*,*) 11
PRINT *, "INPUT THE X-COORDINATE OF THE INITIAL LOCATION"
READ(*,*) XAll
PRINT *, "INPUT THE Y-COORDINATE OF THE INITIAL LOCATION"
READ(*,"') YAll
PRINT *, "INPUT THE Z-COORDINATE OF THE INITIAL LOCATION"
READ(*,*) 2A11
PRINT *, "INPUT THE X-COORDINATE OF THE FINAL LOCATION"
READ(*,*) XEll
PRINT *,"INPUT THE Y-COORDINATE OF THE FINAL LOCATION"
READ(*,*) YEll
PRINT *, "INPUT THE Z-COORDINATE OF THE FINAL LOCATION"
READ(*,*) ZEll
XAHI=XAH
YAlll = YAll
ZAl 11 = ZAll
xell = 0.0
zell = 1.2
yell=0.0
XElll = XEll
YElll = YEll
2E] 11 = ZEll
DELX a (XEll - XAll)/(I_Z-l)
DELY = (YEll . YAll)/(I_Z-1)
DM = (ZEll . ZAll)/(I_Z-l)
DEL_THETA = (THETA2 - THETA1)/(I_Z-l)
DEL_GAMMA = (GAMMA2 - GAMMAl)/(I_Z-l)
HEIGHT = 0.5

 

 

 

 

flJH-u-ur ‘—

 

 

77

D0 127111 = 124
D0 128 III = 1,60
YF3(111,J11)=YA1+(JI l-l)*(HEIGHT-YAl)/ll
ZF3(Ill,Jll)=ZAl+(Ill-l)*(ZEll-ZAll)/23
128 CONTINUE
127 CONTINUE
DO4S6lll=1,180
DO 456111 = 1,180
XF(Il,Jl) = 0.0
456 CONTINUE
Y_SCALE = 0.5/275
Y_SCAL = 550
z_SCALE = 1.2/575
z_SCA = 575/12
DO 2111(1 =1,I_Z
DO 221 II = 1,PROFILE_POINTS
DO 23111: 1,PROFS
GAMMA = GAMMAI + (Kl - 1)*DEL_GAMMA
THETA = THETAI + (Kl - l)*DEL_THETA
x_INIT = XA11+ DELx *(Kl - 1)
Y_INIT = YAll + DELY *(Kl - l)
z_INIT = ZAll + DELZ *(K1 -1)
XTRANS(IIJ1)=XP_,BETA(IIJ1)*COS(GAMMA) - YP_BETA(II,J1)"'
ISIN(GAMMA) + x_INIT
YTRANS(Il ,Jl)=XP_BETA(IlJ1)*SIN(GAMMA)*COS(THETA) +Y_INIT +
1YP_BETA(II,J1)*COS(THETA)*COS(GAMMA) - 2P_BETA(II,JI)*SII~I(THETA)
ZTRANSGIJ1)=XP_BETA(IIJ1)*SIN(THETA)*SIN(GAMMA) +Z_INIT +
1YP_BETA(II,I l)*COS(GAMMA)*SIN(THETA) + ZP_BETA(Il,J1)*COS(TI-IETA)
Y_INDEX = INT((YTRANS(Il J1)+12*Y_SCALE)*Y_SCAL +0.5)+1
z_INDEX = INT((ZTRANS(IlJl)+l2*Z_SCALE)*Z_SCA + O.5)+l

78

IFiYJNDEXLT. 1.0R.Y_INDEX.GT. 1500.0R.Z_INDEX.LT. I .OR.Z_INDEX.GT.

1600)THEN
GO TO 231
ENDIF
IF(MOD((Y_INDEX-12),25).EQ.0.0)THEN
Y_INDE1=INT(Y_INDEX/25)*3 + 1
ELSEIF(MOD((Y_INDEX-l3),25).EQ.0.0)TH EN
Y_INDE1=INT(Y_INDEX/25)*3 + 2
ELSEIF(MOD((Y_INDEX-14),25).EQ.0.0)THEN
Y_INDE1=INT(Y_INDEX/25)*3 + 3
ELSE
Y_INDE1 = -l
ENDIF
IFCMOD((Z_INDEX-12),25).EQ.O)THEN
Z_INDE1=INT(Z__INDEX/25)*3 + l
ELSEIF(MOD((Z_INDEX- l 3),25).EQ.0)THEN
Z_INDEI=INT(Z_INDEX/25)*3 + 2
ELSEIF(MOD((Z_INDEX-l4),25).EQ.0)THEN
Z_INDE1=INT(Z_INDEX/25)*3 + 3
ELSE
Z_INDE1 = -1
ENDIF
IF(Y_INDE1.EQ.- l .OR.Z_INDEI .EQ.- l)TH EN
GO TO 231
ENDIF
IF(XTRANS(IIJ1).GE.XF(Z_INDE1,Y_INDE1))THEN
XF(Z_INDE1,Y_INDE1) = XTRANSU l ,J l)
YF(Z_INDE1,Y_.INDEI)= YTRANS(II,J1)
ZF(Z_INDE1,Y_INDE1) = ZTRANSGIJI)
ENDIF

 

 

79

231 CONTINUE
221 CONTINUE
211 CONTINUE
DO 567111: 2.71.3
DO 568 111 = 2,167.3
ZE_MID=ZF(Ill.J11)
YE_MID=YF(Ill,Jll)
XE_E=XF(111,J11)
IF(ZF(Ill+1J11+1)EQZF(Ill-lJll-1))THEN
GOTOIOI
ENDIF
XE_A=XF(Ill-l,J11-1)+(ZE_MID-ZF(IIl-lJll-1))*(XF(111+1,
1]11+l)—XF(Ill-1,Jll-l))/(ZF(lll+l,Jll+l)-ZF(Ill-l,Jll-1))
101 IF(ZF(III+1,J11).EQZF(IIl-IJll))THEN
print",ill,jll
print*,ZF(Ill+1,J11),ZF(lll-1,Jll)
GOT0102
ENDIF
XE_B =XF(Ill-I,Jll)+(ZE_MID-ZF(111-1,Jl1))*(XF(Ill+l,
lJll)-XF(Ill-1,Jl1))/(ZF(I11+1Jll)-ZF(Ill-lJll))
102 IF(YF(Ill-l,Jll+l).EQ.YF(lll+lJll-l))THEN
GOT0103
ENDIF
XE_C=XF(Ill-I,Jll+l)+(YE_MID-YF(11l-l,J11+l))*(XF(Ill-l,
lJll+1)—XF(111+IJll-1))/(YF(Ill-1,Jl1+1)-YF(Ill+1,Jll-1))
103 IF(YF(Ill,J1I+1).EQ.YF(111,J11-l))THEN
GOT0104
ENDIF
XE_D=XF(IllJll-l)+(YE_MlD-YF(11lJll-l))*(XF(Ill,

lJll+l)-XF(IIlJll-l))/(YF(III,JIl+l)-YF(Ill,Il1-1))

 

 

80

104 XE_MID = MAX (XE_E.XE_A,XE_B,XE_C,XE_D)

IIQ.F1 = INT(II l/3)+l
JKLFI = INT(Jll/3)+l
XF3(IKLF1JKLF1)=XE_MID

S68 CONTINUE

567 CONTINUE
DO 555111: 1.24
J 11 =1
XF3(11 1.111) = 0

555 CONTINUE

C FORMATTING THE OUTPUT TO CREATE A PROGRAM FILE FOR I-DEAS
jk2 = 24
jkl = 12
OPEN(UNIT=9 .FILE='meshl .prg',TYPE=’UNKNOWN')
OPEN(UNIT =10,FILE='mesh2.prg',TYPE='UNKNOWN')
OPEN(UNIT=1 1 .FILE='mesh3 .prg'.TYPE='UN KN OWN')
OPENCUNIT=l2,FILE='meSh4.prg'.TYPE='UNKNOWN')
OPEN(UNIT=1 3 .FILE='m esh5 .prg',TYPE='UNKNOWN')
WRITE(9.*) "K : SO"
WRITE(9.*) "K : CG"
WRITE(9,*) "K : SE"
WRITE(9,") "K : M3"
WRITE(9,"') "K : !!CR"
WRITE(10.*) "K : !!CR"
WRITE(9,*) "K : P"
WRITE(10.“) "K : P"
WRITE(9,*) "K : K"
WRITE(10.*) "K : K"
WRITE(11.*) ”K: !!CR"
WRITE(11.*) "K : P"

 

El

 

81

WRITE(11,*) "K : K"
WRITE(12."') "K : !!CR"
WRITE(12,*) "K : P"
WRITE(12.*) "K : K"
WRITE(13,*) "K : !!CR"
WRITE(13,*) "K : P"
WRITE(13,*) "K : K"
DO 351 J] = 1sz
DO 251 Kl = l,jk1
WRITE(9.33) "K : ".XF3(Jl,K1).YF3(Jl.K1),ZF3(Jl.K1)
33 FORMAT(A4.T6,E14.7.1X,El4.7.lX.El4.7)
251 CONTTNUE
DO 252 Kl =jkl.((2*jkl)-1)
WRITE(IO,33) "K : ".XF3(J1,K1),YF3(JI.Kl),ZF3(J1.Kl)
252 CONTINUE
DO 253 Kl = ((2*jkl)-l).((3*jkl)-2)
WRI'I'E(11,33) "K : ".XF3(Jl.K1).YF3(Jl,K1 ),ZF3(Jl.K1)
253 CONTINUE
Do 254 Kl = ((3*jkl)-2),((4*jkl)-3)
WRITE(12.33) "K : ".XF3(J1,K1),YF3(J1,K1 ),ZF3(JI.K1)
254 CONTINUE
DO 255 Kl = ((4*jkl)-3),((5*jkl)-4)
WRITE(13.33) "K : ".XF3(J1,K1),YF3(J1,Kl),ZF3(Jl.I(l)
255 CONTINUE
351 CONTTNUE
WRITE(9,*) "K : D"
WRITE(10.*) "K : D"
WRITE(10,*) "K : !!CR"
WRITE(10.*) "K : SP"
WRI'I'E(10,*) "K : L"

WRITE(II.*) "K: D"

WRITE(11.*) "K: !!CR"
WRITE(11.*) "K : SP"
WRITE(11.*)"K: L"

WRITE(12,*) "K : D"

WRITE(12.*) "K: !!CR"
WRITE(12.*) "K : SP"
WRITE(12,*) "K : L"

WRITE(13.*) "K : D"

WRITE(13."') "K : !!CR"
WRITE(13,*) "K : SP"
WRITE(13,*) "K : L"

WRITE(9,*) "K: !!CR"
WRITE(9.*) "K : SP"

WRITE(9.*) "K : L"

NUMBER_P=1

DO 451 II = 131a

DO 551 K1 = 1.11:1
IF(NUMBER_P.LT.10)THEN
WRITE(9.75) "K : p". NUMBER_P
WRITE(10,75) "K : p", NUMBER_P
WRITE(II,75) "K : p", NUMBER_P
WRITE(12.75) "K : p", NUMBER_P
WRITE(13,75) "K : p". NUMBER_P
ELSEIF(NUMBER_P.LT.100)THEN
WRITE(9.76) ”K : p", NUMBER_P
WRITE(10,76) "K : p". NUMBER_P
WRITE-201.76) "K : p". NUMBER_P
WRITE(12,76) "K : p", NUMBER_P
WRITE(I3,76) "K : p", NUMBER_P

82

 

83

ELSEIF(NUMBER_RLT.1000)THEN
WRITE(9.77) "K : p", NUMBER_P
WRITE(10,77) "K : p", NUMBER_P
WRITE(11,77) "K : p", NUMBER_P
WRITE(12.77) "K : p", NUMBER_P
WRITE(I3,77) "K : p", NUMBER_P
ELSEIF(NUMBER_P.LT. 10000)THEN
WRITE(9.78) "K : p". NUMBER_P
WRITE(10,78) "K : p". NUMBER_P
WRITE(11,78) "K : p", NUMBER_P
WRITE(12.78) "K : p". NUMBER_P
WRITE(13.78) "K : p", NUMBER_P
ELSE
WRITE(9.79) "K : p". NUMBER_P
WRITE(10,79) "K : p”, NUMBER_P
WRITE(11,79) "K : p", NUMBER_P
WRITE(12.79) "K : p", NUMBER_P
WRITE(13.79) "K : p". NUMBER_P
75 FORMAT(A5.T6.II)
76 FORMAT(A5,T6.12)
77 FORMAT(A5,T6.I3)
78 FORMAT(A5.T6.I4)
79 FORMAT(AS.T6,IS)
ENDIF
NUMBER_P = NUMBER_P + l
551 CONTINUE
WRITE(9,*) "K : "
WRITE(10,*) "K : "

WRITE(10.*) "K : "

 

 

WRITE(11,*) "K : "
WRITE(11.*) "K : "
WRITE(12,*) "K : "
WRITE(12.*) "K : "
WRITE(13."') "K : "
WRITE(13,*) "K : "
WRITE(9.*) "K : "

451 CONTINUE
INNN = 1
NUMBER_P=1
DO 53 J1 = leI
DO 63 K1 = 1Jk2
IF(NUMBER_P.LT. 10)THEN
WRITE(9.75) "K : p", NUMBER_P
WRITE(10,75) "K : p". NUMBER_P
WRITE(11,75) "K : p". NUMBER_P
WRITE(12.75) "K : p", NUMBER_P
WRITE(13.75) "K : p", NUMBER_P
ELSEIF(NUMBER_RLT.100)THEN
WRITE(9.76) “K : p", NUMBER_P
WRITE(10,76) "K : p", NUMBER_P
WRITE(11,76) "K : p", NUMBER_P
WRITE(12.76) "K : p". NUMBER_P
WRITE(13.76) "K : p", NUMBER_P
ELSEIF(NUMBER_RLT.1000)THEN
WRITE(9.77) "K : p". NUMBER_P
WRITE(10,77) "K : p". NUMBER_P
WRITE(11,77) "K : p". NUMBER_P
WRITE(12.77) "K : p", NUMBER_P
WRITE(13.77) "K : p", NUMBER_P

 

 

85

ELSEIF(NUMBER_P.LT. 10000)THEN
WRITE(9.78) "K : p", NUMBER_P
WRITE(10,78) "K : p", NUMBER_P
WRITE(11,78) "K : p", NUMBER_P
WRITE(12.78) "K : p", NUMBER_P
WRITE(13.78) "K : p", NUMBER_P
ELSE
WRITE(9.79) "K : p". NUMBER_P
WRITE(10,79) "K : p". NUMBER_P
WRITE(11,79) "K : p", NUMBER_P
WRITE(12.79) "K : p", NUMBER_P
WRITE(13.79) "K : p", NUMBER_P
ENDIF
NUMBER_P = NUMBER_P + jkl
63 CONTINUE
WRITE(9.*) "K : "
WRITE(9,*) "K : "
WRITE(10.*) "K : "
WRITE(10,*) "K : "
WRITE(11,*) "K : "
WRITE(11,"') "K : "
WRITE(12.*) "K : "
WRITE(12,*) ”K : "
WRITE(13,*) "K : "
WRITE(13.*) "K : "
INNN = INNN + 1
NUMBER_P = INNN
53 CONTINUE
WRITE(9.*) "K : D"
WRITE(9,*) "K : llCR"

WRITE(9,*) "K : SK"
WRI'I'E(9.*) "K: MC"
WRITE(9.*) "K : L"

WRITE(IL") "K : D"
WRITE(11,*) ”K: !!CR"
WRITE(11,*) "K : SK"
WRITE(11,*) "K : MC"
WRWE(11.*)"I( : L"
WRITE(10,*) "K : D"
WRITE(10.*) "K : !!CR"
WRITE(10.*) "K : SK"
WRITE(10,*) "K : MC"
WRITE(10.*) "K : L"
WRITE(12.*) "K : D"
WRITE(12,*) "K : !!CR"
WRITE(12."') "K : SK"
WRITE(12.*) "K : MC"
WRITE(12,*) "K : L"
WRITE(13,*) "K : D"
WRITE(13,*) "K : !!CR"
WRITE(13,*) "K : SK"
WRITE(13,*) "K : MC"
WRITE(13,*) "K : L"
NUMBER_P=1

Do 533K1=1.jk2
IF(NUMBER_P.LT.10)THEN
WRITE(9.75) "K : c", NUMBER_P
WRITE(10,75) "K : c". NUMBER_P
WRI'I'E(11.75) "K : c". NUMBER_P

WRITE(12.75) "K : c". NUMBER_P

 

86

 

WRITE(13.75) "K : c", NUMBER_P
ELSEIF(NUMBER_RLT.100)THEN
WRITE(9.76) "K : c", NUMBER_P
WRITE(10,76) "K : c", NUMBER_P
WRITE(11,76) "K : c". NUMBER_P
WRITE(12.76) "K : c", NUMBER_P
WRITE(13.76) "K : c", NUMBER_P
ELSEIF(NUMBER_PLT.1000)THEN
WRITE(9.77) "K : c". NUMBER_P
WRITE(10,77) "K : c". NUMBER_P
WRITE(11,77) "K : c", NUMBER_P
WRITE(12.77) "K : c". NUMBER_P
WRITE(13.77) "K : c". NUMBER_P
ELSEIF(NUMBER_PLT.10000)THEN
WRITE(9.78) "K : p", NUMBER_P
WRITE(10,78) "K : p", NUMBER_P
WRITE(11,78) "K: p". NUMBER_P
WRITE(12.78) "K : p", NUMBER_P
WRITE(13.78) "K : p". NUMBER_P
ELSE

WRITE(9.79) "K : p", NUMBER_P
WRITE(10,79) "K : p", NUMBER_P
WRITE(11,79) "K: p", NUMBER_P
WRITE(12.79) "K : p", NUMBER_P
WRITE(13.79) "K : p". NUMBER_P
ENDIF

NUMBER_P = NUMBER_P + 1

533 CONTINUE

WRITE(9,"') "K : "

WRITE(10,*) "K : "

87

 

 

WRITE(11,*) "K: "
WRITE(12.") "K : "

WRITE(13.*) "K : "

NUMBER_P = jk2 +1

DO 633 II = jk2+1,jk2+jkl
IF(NUMBER_P.LT. 10)THEN
WRITE(9.75) "K : c", NUMBER_P
WRITE(10,75) "K : c". NUMBER_P
WRITE(11,75) "K : c", NUMBER_P
WRITE(12.75) "K : c". NUMBER_P
WRITE(13.75) "K : c", NUMBER_P
ELSEIF(NUMBER_P.LT.]00)THEN
WRITE(9.76) "K : c", NUMBER_P
WRITE(10,76) "K : c". NUMBER_P
WRITE(11,76) "K : c", NUMBER_P
WRITE(12.76) "K : c". NUMBER_P
WRITE(13.76) "K : c". NUMBER_P
ELSEIF(NUMBER_RLT.IOOO)THEN
WRITE(9.77) "K : c", NU M BER_P
WRITE(10,77) "K : c", NUMBER_P
WRITE(11,77) "K : c", NUMBER_P
WRITE(12.77) "K : c", NUMBER_P
WRITE(13.77) "K : c". NUMBER_P
ELSEIF(NUMBER_P.LT. 10000)THEN
WRITE(9.78) "K : p", NUMBER_P
WRITE(10,78) "K : p", NUMBER_P
WRITE(11,78) "K : p". NUMBER_P
WRITE(12.78) "K : p". NUMBER_P
WRITE(13.78) "K : p", NUMBER}
ELSE

 

 

 

 

WRITE(9.79) "K

89

: p", NUMBER_P

WRITE(10,79) "K : p". NUMBER_P

WRITE(11,79) "K: p", NUMBER_P

WRITE(12.79) "K I p", NUMBER_P

WRITE(13.79) "K : p". NUMBER_P

ENDIF

number _p = NUMBER_P + l

633 CONTINUE
WRITE(9,*) "K :
WRITE(9,*) "K :
WRITE(9.*) "K:
WRITE(9.") "K:
WRITE(9,*) "K :
WRITE(9,"') "K :

WRITE(10,*) "K :
WRITE(10.*) "K :
WRITE(10,*) "K :
WRITE(10,*) "K :
WRITE(10.*) "K :
WRITE(10.*) "K :
WRITE(11,"‘) "K :
WRITE(11,‘) "K:
WRITE(11,*) "K:
WRITE(11.*) "K :
WRITE(11,*) "K :
WRITE(11,*) "K :
WRITE(12,*) "K :
WRITE(12,"‘) "K :
WRITE(12,*) "K :
WRITE(12,*) "K : .

lMA"
S"
STO"

SAVE"

!MA"
5"
STO"

SAVE"

lMA"

WRITE(12."') "K

WRITE(12.") "K :
WRITE(13,"') "K :
WRITE(13.*) "K :
WRITE(13,*) "K :
WRITE(13,‘) "K :
WRITE(13,*) "K :
WRITE(13,*) "K :

WRITE(9,*) "K :
WRITE(10,*) "K
WRITE(11,*) "K
WRITE(12,*) "K
WRITE(13.*) "K
WRITE(9.*) "K :
WRITE(9,*) "K:
WRITE(9.*) "K:
WRITE(9,*) "K :
WRITE(9,*) "K :
WRITE(9.*) "K :
WRITE(9,*) "K:
WRITE(9,*) "K :
WRITE(9,*) "K :
WRITE(10,*) "K

WRITE(10,*) "K :
WRITE(10,*) "K :
WRITE(10,*) "K :
WRITE(10.‘) "K :
WRITE(10.*) ”K :
WRITE(10,"') "K :
WRITE(10,*) "K :

: S"
STO"

SAVE"

!MA"
S"

ST "
skinl"

: skin2"
: skin3"
: skin4"
: skinS"
!!TA"
0..

CR"
SK”
SKIN] "
!MA"

STO"

: !!TA"

0"
CR"
SK"

!MA"
STO"

SKIN2"

 

90

 

 

 

 

V-A@

WRITE(10,*) "K :
WRITE(11,*) "K:
WRITE(11,*) "K:
WRITE(11,*) "K:
WRITE(ll.*) "K :
WRITE(11,*) "K:
WRITE(11,*) "K:
WRITE(11,*) "K:
WRITE(11,*) "K :
WRITE(11,*) "K :
WRITE(12.*) "K :
WRITE(12.*) "K :

WRITE(12,*) "K

WRITE(12,*) "K :
WRITE(12,*) "K :
WRITE(12,*) "K :
WRITE(12,*) "K :
WRITE(12,*) "K :
WRITE(12,*) "K :
WRITE(13.*) "K :
WRITE(13.*) "K :

WRITE(13,*) "K

WRITE(13.*) "K :
WRITE(13.*) "K :
WRITE(13,*) "K :
WRITE(13.*) "K :
WRITE(13.*) "K :
WRITE(13.*) "K :

!!TA"

CR"

SK"

SKIN3"

!MA"

ST II

!!TA"

: CR"

SK"

SKIN4"

!MA"

STO"

!!TA"

0"

: CR"

SK"
SKINS"
!MA"

STO"

WRITE(9.*) "K : !MA"

WRITE(9,*) "K : WR"

 

91

 

 

 

WRITE(9.*) "K :
WRITE(9,*) "K :
WRITE(9,*) "K:
WRITE(9.*) "K:
WRITE(9.*) "K:
WRITE(9.*) "K :
WRI'I'E(9.*) "K:
WRITE(9.*) "K :
WRITE(9.") "K:
WRITE(9.*) "K:
WRITE(9,*) "K:
WRITE(9.*) "K:

WRITE(10,*) "K :
WRITE(10.*) "K :

WRI'I'E(IO,"') "K

WRITE(10,*) "K : "
WRITE(10,*) "K :
WRITE(10,*) "K :
WRITE(10,*) "K :
WRITE(10,‘) "K :

WRITE(10,*) "K

WRITE(10,*) "K :
WRITE(10,*) "K :
WRITE(10,’) "K :
WRITE(10,*) "K :
WRITE(10,*) ”K :
WRITE(11,"') "K :
WRITE(11,*) "K :
WRITE(11,’) "K :
WRITE(11,") "K :

92

CO"

1"

!MA"
WR"

: CO"

P"
0..
3..
: G"
R"

I"

D"

!MA"
WR"
CO"
In

 

 

WRITE(11,*) "K :
WRITE(11,‘)"K :
WRITE(11,*) "K
WRITE(llfi") "K:
WRITE(11,*) "K:
WRITE(11,*) "K :
WRITE(11,*) "K :
WRITE(11,*) "K :
WRITE(11,*) "K :

WRITE(11,*) "K :

WRITE(12.*) "K :
WRITE(12,*) "K :
WRITE(12.*) "K :
WRITE(12.*) "K :

WRITE(12.*) "K :
WRITE(12.*) "K :

WRITE(12,*) "K :

WRITE(12,*) "K :
WRITE(12,*) "K :
WRITE(12,*) "K :
WRITE(12,*) "K :

WRITE(12,*) "K : '

WRITE(12.*) "K :
WRITE(12.*) "K :
WRITE(13.*) "K :
WRITE(13,*) "K :
WRITE(13,*) "K :
WRITE(13,*) "K :
WRITE(13,*) "K :
WRITE(13.*) "K :

93

C"
POI

: O"

S"
G"

R"

U"

1..

D"

E"
!MA"
WR"
CO"
1..

C"

P"

!MA"
WR"
CO"
1..

C"

 

wq-‘r

 

 

 

94

WRITE(13,*) "K : o"
WRITE(13.*) "K : S"
WRITE(13,*) "K : G"
WRITE(13.") "K : R"
WRITE(13.*) "K : U"
WRITE(13,*) "K : I"
WRITE(13.") "K : D"
WRITE(13,*) "K : E"

WRITE(9.*) "K : a=",al

 

WRITE(9,*) "K : b=",b1
WRITE(9,*) "K : r=",rl
WRITE(9,*) "K : l=".ll l
WRITE(9,*) "K : e=".el
WRITE(9,*) "K: f=".fl
WRITE(9,*) "K : "
WRITE(9.*) "FS: "
WRITE(9,"') "K : testl.igs;~"
WRITE(9,*) "K : YES"
WRITE(9.*) "FD: "
WRITE(9,*) "K: !!!!DE"
WRITE(9.*) "K : Y"
WRITE(9,*) "K : T "
WRITE(9.*) "K : CG"
WRITE(9.*) "K: DE"
WRITE(9,*) "K : L"
WRITE(9.*) "K : p1.p288"
WRITE(9,*) "K : cl.c36"
WRITE(9,*) "K: "
WRITE(9,*) "K : "
WRITE(9,*) "K : REDI"

95

WRITE(9.*) "K:
WRITE(Q’t) "K

DE"
: G"
WRITE(9.*) "K : Y"
WRI'T'E(9."‘) "K : REDI"
WRITE(9.*) "K : SAVE"
WRITE(9.*) "K : "
WRITE-(103) "K : "
WRFI'E(10.*) "FS: "

WRITE(10,*) "K : test2.igs;~"

 

WRITE(10,*) "K : YES"
WRITE(10,*) "FD: " I
WRITE(10,*) "K : !!1!DE"

WRITE(10.*) "K : Y"

WRIT1300.") "K : TA"

WRITE(IOf) "K

WRITE(IOf) "K :
WRH'EUQ") "K :

WRITE(10,*) "K
WRITE(10,*) ”K

WRITE(IOf) "K :
WRITE(IW") "K :
WRI11500:") "K :
WRITE(IOI) "K :
WRITE(ION "K :
WRITE(10,*) "K :
WRITE(IOJ') "K :
WRITE(10.*) "K ;
WRITE(10,*) "K :
“”5010 "K:

: CG"

DE N
L"

: pl ,p288"

: cl ,c36"

"

REDI"

DE '1

WRITE-(113) "FS: "

96

WRITE(IIf) "K : testB.igs;~"
WRrrE(11.*) "K : YES"
WRITE(IL'") "FD: "

WRITE(11.*) "K: IIIIDE"

WRI'1'1'3(11."')"K:
“”501.” "K:
WRITE-(113) "K:
“”501.” "K:
WRITE(Uf) "K:
WRITE(IL") "K:
WRUE(11.*) "K:
WRITE-(113) "K:
WRITE(III") "K:
WRITE(11."')"K:
WRITE(IL'") "K:
WRITE(11.*) "K:
WRITE(11,*) "K ;
WRm(11.*)"K:
WRITE(11,*) "K :

Y"
TA"
CG"
DE"

L"

p1 ,p288"
c1 .c36"

REDI"
DE"
G.

y.
REDI"

SAVE"

WRITE(11.*) "K: "
WRITE(12,*)"K : "
WRITE(IZJ') "FS: "
WRITE(IZ") "K : test4.igs;~"
WRITE(12."')"K : YES"
WRITE-(12,") "FD: "
WRITE(12.*) "K : IIIIDE"
““1502.” "K : Y"
WRITE(12.*) "K : TA"
“(Rm—302.") "K : CG"
WRU'E(12.*) "K : DE"

WRITE(12,*) "K
WRITE(12,*) "K

WRITE(12.*) "K

WRITE(12,"') "K :
WRITE(12,*) "K :
WRITE(12,*) "K :
WRITE(12,*) "K :
WRITE(12,*) "K :

WRITE(12,*) "K :

WRITE(12,*) "K

WRITE(12,*) "K :
WRITE(12,*) "K :
WRITE(13.*) "K :
WRITE(13,*) "FS:
WRITE(13.*) "K :

WRITE(13,"') "K :

 

97

: L"
: p1.p288"

: cl ,c36"

REDI"

DE II

Y"

: REDI"

SAVE"

test5.igs;~"

YES"

WRITE(13,*) "FD: "

WRITE(I3,*) "K :
WRITE(13,‘) "K :
WRITE(13,*) "K :
WRITE(13.*) "K
WRITE(13,*) "K :
WRITE(13.*) "K :
WRITE(13.*) "K :
WRITE(13,*) "K
WRITE(13,*) "K :
WRITE(13,"') "K :
WRITE(13.*) "K
WRITE(13,*) "K :
WRITE(13,*) "K :

!!llDE"
Y"

T N

: CG"

DE"
L"

p1.p288"

: cl .036"

: REDI"

DE H
G"

 

 

 

 

WRITE(13,*) "K
WRITE(13.*) "K

WRITE(13,*) "K :
WRITE(13,*) "K :
WRITE(13,*) "K :

WRITE(13,*) "K

WRITE(13.*) "K :

WRITE(I3,*) "K

WRITE(13,*) "K :

WRITE(13,*) "K :

98

: Y"

: REDI"

SAVE"

!!TA"

: 0"

MA"

: G"

WRITE(13,*) "K : 2

WRITE(13.*) "K

WRITE(13,"') "K :
WRITE(13.*) "K :
WRITE(13,*) "K :
WRITE(13,"‘) "K :
WRITE(13,*) "K :
WRITE(13,*) "K :
WRITE(13,*) "K :
WRITE(13.*) "K :

WRITE(13.*) "K

WRITE(13,*) "K :

WRITE(13."') "K

WRITE(13.*) "K :
WRITE(13,*) "K :
WRITE(13.*) "K :
WRITE(13.*) "K :
WRITE(13,*) "K :
WRITE(13,*) "K :

WRITE(13."') "K :

: G"

3"

4"

G"

AU"
!TA"
SUI.

: CR"

SW"

: L"

l"

WRITE(13.*) "K :
WRITE(133') "K :

WRITE(13.*) "K

WRITE(13.*) "K :
WRITE(13,*) "K :
WRITE(13.*) "K :
WRITE(13.*) "K :
WRITE(13,*) "K :
WRITE(13,*) "K :
WRITE(13,*) "K:
WRITE(13.*) "K :
WRITE(13.*) "K :
WRITE(13.*) "K :
WRITE(13,*) "K :
WRITE(13,*) "K :
WRITE(13,*) "K :
WRITE(13.*) "K :

 

99

!MA"

: CO"

I"

C"

0..
S"

G"

R"
CU"
0.001"
U"

D"

E"

WRITE(13,"') "FS: "

WRITE(13,"‘) "K :
WRITE(13,*) "K :

surface.igs;~

YES"

WRITE(13."') "FD: "

WRITE(13.*) "K :
WRITE(13.*) "K :
WRITE(13.*) "K :
WRITE(13,*) "K :
WRITE(13.*) "K :
WRITE(13.*) "K :
WRITE(13,*) "K :

WRITE(13.*) "K

WRITE(13,*) "K :

”!!TA"
0"
MA"
STO"
SAVE"

Y"

: EXIT"

Y"

 

100

XA11_TEMp=0

YA11_TEMP=L1

ZA11_TEMP=0
XEll=0
YE11=L1
2E1 l=0

CALCULATING THE AXIS COORDINATES

x_AXISl = XA11_TEMP *COS(GAMMAl)-YA11_TEMP * SIN(GAMMAI)
Y_AXISI = XA11_TEMP *SIN(GAMMA1)*COS(THETA1)+YA1l_TEMP *
ICOSCI'HETAI)*COS(GAMMA1) - ZA11_TEMP * SIN(THETAI)
z_AXISI = XA11__TEMP *SIN(THETA1)*SIN(GAMMA1) + YA11__TEMP "'
ICOS(GAMMAI)*SIN(THETA1) + ZA11_TEMP * COS(THETAI)
OPEN(UNIT=8 ,F ILE='fi VC.C1',TYPE='UNKNOWN')
X_AXIS2 = XE] l*COS(GAMMA2)-YE11 * SIN(GAMMA2)
Y_AXIS2 = XEll *SIN(GAMMA2)*COS(THETA2)+YEl1 '-
1COS(IHETA2)*COS(GAMMA2) - ZEl 1* SIN(THETAZ)
z_AXISZ = XEll *SIN(THETA2)*SIN(GAMMA2) + YEll "'
lCOS(GAMMA2)*SIN(THETA2) + ZEII * COS(THETAZ)
WRITE(8,*)XA111,YA111,ZA111,X_AXISI,Y_AXISI,Z_AXISI
WRITE(8,*) XEl 1 1,YE1 ll,ZEl I 1,X_AXISZ.Y_AXISZ,Z_AXISZ

1001 PRINT *, " "

C CALCULATING THE COORDINATES OF THE CUTTER

C

PRINT *, "INPUT THE X-COORDINATE OF THE INITIAL LOCATION"
READ(*,*) XAI

PRINT *, "INPUT THE Y-COORDINATE OF THE INITIAL LOCATION"
READ(*,*) YAl

YA9 = YA]

PRINT *, "INPUT THE Z-COORDINATE OF THE INITIAL LOCATION"

READ("‘,*) ZAI

 

101

c PRM "'. "INPUT THE X-COORDINATE OF THE FINAL LOCATION"

C

C

READ(*,*) XEI

PRINT *,"INPUT THE Y-COORDINATE OF THE FINAL LOCATTON"
READ(*.*) YE]

PRINT *, "INPUT THE Z-COORDINATE OF THE FINAL LOCATION"
READ(*.*)ZE1

PRINT *, "INPUT THE DEPTH TO WHICH THE CUTTER GOES INITIALL "
READ(*.*) DEPTH

ZAll = 2A]

YA2 = YA]

YA3 = YA]

YA4 = YA]

SL1= SQRT(F1**2 + E1**2)

SL2 = SQRT ( SL1**2 - Rl**2)

XA2 = XA1+ E1 + R1 * COS(P -Al)

ZA2=ZA1+ F1 - R1 * SIN(P- A1)

2A3 = ZAl + F1 + Rl * SIN(BI)

2A4 = 2A] + Ll

XA3 = XA1+ El + RI * COS(Bl)

XA4 = XA3 -(ZA1+ L1 - 2A3) * TAN(Bl)
OPEN(UNIT=4,FILE='seq.prg'.TYPE='UNKNOWN')

WRITE(4,*) "K : SO"

WRITE(4,*) "K : CG"

WRITE(4.*) "K : CR"

WRITE(4.*) "K: L"

WRITE(4.*) "K : PP"

WRITE(4,*) "K : K"

WRITE(4,*) "K

WRITE(4,*) "K

WRITE(4,*) "K :
WRITE(4,*) "K :
WRITE(4,*) "K :
WRITE(4,*) "K :
WRITE(4.*) "K :
WRITE(4.*) "K :
WRITE(4,*) "K :
WRITE(4,*) "K :
WRITE(4.‘) "K :
WRITE(4,*) "K :
WRITE(4,*) "K :
WRITE(4.*) "K :
WRITE(4.*) "K :
WRITE(4,‘) "K :
WRITE(4,*) "K :
WRITE(4.*) "K :
WRITE(4,*) "K :
WRITE(4,*) "K :

WRITE(4,*) "K :

WRITE(4.*) "K

WRITE(4,*) "K :

WRITE(4,‘) "K

WRITE(4.*) "K :
WRITE(4,*) "K :

WRITE(4,‘) "K :

:cl

: ",XA1,ZA1

: ",XA2. 2A2

D"
!!CR"

A"

S"

K"

". E1,F1

". XA2,ZA2
", XA3ZA3
D"

!!CR"

Ln

PP"

Kn

", XA3.ZA3
". XA4.ZA4
D"

!!CR"

PC"

Lu

C2"

: 03"

D"

!MA"

PR II

 

102

 

 

   

WRITE(4,*) "K
WRITE(4,*) "K

WRITE(4,*) "K :

WRITE(4,*) "K :

WRITE(4,‘) "K

WRITE(4.*) "K :
WRITE(4,*) "K :
WRITE(4,*) "K :
WRITE(4,*) "K :
WRITE(4,*) "K:
WRITE(4,*) "K :
WRITE(4,*) "K:
WRITE(4,*) "K :
WRITE(4,*) "K:
WRITE(4,*) "K:
WRITE(4.*) "K :
WRITE(4.*) "K :
WRI'I‘E(4,*) "K :
WRITE(4,*) "K :
WRITE(4,*) "K :
WRITE(4,*) "K :

WRITE(4,‘) "K :

: STO"

: PROFILE_1"

!!DE"

L"

:c1"

c2"

c3"

D"

DE '0

P"

!!CR"

SK"

Z"

WRITE(4,*) "K : "

WRITE(4.*) "K :
WRITE(4,"') "K :
WRITE(4.*) "K :

WRITE(4,‘) "K :

 

103

 

 

 

WRITE(4,*) "K:
WRITE(4.*) "K:
WRITE(4.*) "K:
WRITE(4.*) "K:
WRITE(4.*) "K:
WRITE(4.*) "K:
WRITE(4.") "K:
WRITE(4,*) "K:
WRITE(4,*) "K:
WRITE(4,*) "K:
WRITE(4.*) "K :
WRITE(4.*) "K:
WRITE(4,*) "K:
WRITE(4,*) "K: '
WRITE(4,"') "K:
WRITE(4.*) "K:
WRITE(4,*) "K:
WRITE(4.*) "K:
WRITE(4,*) "K:
WRITE(4,*) "K:
WRITE(4,*) "K :
WRITE(4,*) "K :
WRITE(4.*) "K:
WRITE(4.*) "K:
WRITE(4,*) "K:
WRITE(4,*) "K:

WRITE(4,*) "K :

SKINI"

!!TA"

0"

CR ll

SK"

SKINl"

!MA"

STO"

OBJECT 1"

!MA"

1..
D"
E"
a=".al
b=".bl

7!

tr]

104

105

WRITE(4,*) "K : l=",ll
WRITE(4,*) "K : e=".el
WRITE(4.*) "K : f=",f1
WRITE(4,*) "K : "
WRITE(4,*) "FS: "
WRITE(4.*) "K : test.igs;~"
WRITE(4,*) "K : YES"
WRITE(4,*) "FD: "
WRITE(4,*) "K : SAVE"
WRITE(4,*) "K : Y"

2000 END

*********P********/‘*********/*********/*********/*********/*********/**

   

_ - ..l't
0 a”. I.|‘I.‘I'...H.' ""'“'3.'.'"".. "'"t'.".’
u‘al' l ‘

 

APPENDIX C

 

The following shows an example of a program file that was produced to create the surface
for a five-axis move with a rotation of 15 degrees about the X-axis and a rotation of 15

degrees about the Z-axis at the end position, with the cutter being vertical in the initial

EXAMPLE QF A PRQQRAM FILE

position.

NKWWWWWKWNWNKWKNNNWWWWWWNWWWKWKWWW7‘1

:80

:00

:SE

:M3

: !!CR

:P

:K

: 0.0000000E+00 0.0000000E+00 0.0000000E+00
: 0.2624361E-01 0.4545455E-01 0.0000000E+00
: 0.5293053E-01 0.9090909E-01 0.0000000E+00
: 0.7898218E—01 0.1363636E+00 0.0000000E+OO
: 0.1051450E+00 0.1818182E+00 0.0000000E+00
: 0.1320346E+OO 0.2272727E+00 0.0000000E+00
: 0.1581724E+00 0.2727273E+00 0.0000000E+00
: 0.1843440E+00 0.3181818E+00 0.0000000E+00
: 0.2114576E+00 0.3636364E+00 0.0000000E+00
: O.2376567E+00 0.4090909E+00 0.0000000E+00
: 0.2647924E+00 0.4545455E+00 0.0000000E+00
: 0.2911977E+00 0.5000000E+00 0.0000000E+00
: 0.0000000E+00 0.0000000E+00 0.5217391E-01
: 0.2758704E-01 0.4545455E-01 0.5217391E-01

: 0.5397850E-01 0.9090909E-01 0.521739lE-01

: 0.8082776E—01 0.1363636E+00 0.5217391E-01

: 0.1079499E+00 0.1818182E+00 0.5217391E-Ol
: 0.1347947E+00 0.2272727E+00 0.5217391E-01
: 0.1620175E+00 0.2727273E+00 0.5217391E-01
: 0.1891462E+00 0.3181818E+00 0.5217391E-01
: 0.2161873E+00 0.3636364E+00 0.5217391E-01
: 0.2436805E+00 0.4090909E+00 0.5217391E-01
: 0.2705285E+00 0.4545455E+00 0.5217391E-01
: 0.2982397E+00 0.5000000E+00 0.5217391E-01
: 0.0000000E+00 0.0000000E+00 0.1043478E+00
: 0.2800061E-01 0.4545455E-01 0.1043478E+OO

: 0.5527150E-01 0.9090909E-01 0.1043478E+00

: 0.8293294E-01 0.1363636E+00 0.1043478E+00

106

 

 

 

fififif

NWWKWNKNKNWWNKWKWWWWWKKKNRNWWWWKKNWWKKWK'

107

0. 1 108879E+00 0.1818182E+00 0.1043478E+00

'. 0.1381219E+00 0.2272727E+00 0.1043478E+00
: 0.1658659E+00 0.2727273E-I-00 0.1043478E+00
: 0.1938777E+00 0.3181818E+00 0.1043478E+00

: 0.4039252E+00 0.4090909E+00 0.1200000E+01
: 0.4463525E+00 0.4545455E+00 0.1200000E+01
: 0.4885214E+OO 0.5000000E+00 0.1200000E+01
:D

: !!CR

:SP

:p13
:p14
:p15
:p16
:p17
:p18
:p19
:p20
:p21
:p22
:p23
:p24

: p25
: p26
: p27
: p28
: p29

 

ytttfifl)
g;:p31
sz32

717<FN1317=73

:p72
:p84

:p96

:p108
:p120
:pl32
:p144
:p156
:p168
:p180
:p192
:p204
:p216
:p228
:p240
:p252
:p264
:p276
:p288

7=7<$N 7=7<5N 7=7<FN 737<=N 737<5N 7=7§5N 7:755“ 7:7:5‘ FIVSFN 717<FN 7:7SFN 73?§'

:p33
:p34
:p35
:p36

:p12
:p24
:p36
:p48

:MCHK
:SKL
:th

:c1
:c2
:c3
:c4

 

108

gzcs
ch6

:c8

:c9

:c10
:cll
:c12
:c13
:cl4
:c15
:c16
:c17
:c18
:c19
:c20
:c21
:c22
:c23
:c24

: c25
: c26
: 027
: c28
: 029
: 030
: c3]
: c32
: C33
: C34
: c35
: C36

: SAVE
: !MA
:STO
:skinl
: !!TA
:CR

: SK
: SKIN]

6N7:WZSNNWNK§W§NNKWNWNWWKNWWKKNKKNWNWNWNWWNWW

 

109

110

1.0471975825423
K: b= 0.52359879127113

0.50000000000000
1.8660000000000

. r= 0.50000000000000

K
K:1=

KKKKKKKKKKKKKKKKK

:a=
:f=

Kze
K
K:

FS:

K : test.igs;~

KzYES

$13511
::1,p288
l,c36

1%
: DLPc mDGYEA m
mKKKKKKKKKKKKKKKKKK

 

 

ream

APPENDIX D

 

EXAMPLE 52F AN IQES FILE

GEOMETRY CREATED BY SDRC I-DEAS VI SOLID MODELING
.,4Hback,8HteSt.igS.
51HSDRC I~DEAS VI: Solid_Modeling.IBM ,TYPE-IGES30..32.8,24,8.56,8HtG

111

G 1

est.igs,l.0,1,2HIN,,,13H9307l1.002109,3.9E-05,2.0E+00,,,4,; G 3
128 1 1 1 0 0 0 000000000D l
128 0 4 1105 0 0D 2

128, 22, 46, 3, 3, 0, 0, 1, 0, 0. IP 1
.0000000E+00, .0000000E+00, .0000000E+00, .0000000E+00, 1P
.1013165E+00, .1516875E+00, .2011921E+00, .2500170E+00, 1P
.2974528E+00. .3441730E+00, .3896845E+00, .4347022E+00, 1P
.4789001E+00, .5227740E+00, .5661415E+00, .6093253E+00, 1P
.6522664E+00, .6951437E+00, .7380213E+00, .7809516E+00, 1P
.8241245E+00, .8674732E+00, .9113160E+00, .1000000E+01, 1P
.1000000E+01. .1000000E+01. .1000000E+01, 1P 8
.0000000E+00, .0000000E+00, .0000000E+00, .0000000E+00, 1P
.4347487E-01, .6521247E-01, .8695064E-01, .1086886E+00, 1P
.1304270E+00, .1521646E+00, .1739025E+00, .1956396E+00, 1P
.2173778E+00, .2391155E+00, .2608552E+00, .2825951E-t-00, 1P
.3043355E+00, .3260743E+00, .3478134E+00, .3695526E+00, 1P
.3912935E+00, .4130344E+00. .4347753E+00, .4565148E+00, 1P
.4782536E+00, .4999922E+00, .5217311E+00. .5434695E+00, 1P
.5652083E+00, .5869463E+00, .6086854E+00, .6304245E+00, 1P
.6521655E+00, .6739061E+00, .6956468E+00. .7173856E-l-00, 1P
.7391241E+00, .7608619E+00, .7826015E+00, .8043414E+00, 1P
.8260834E+00, .8478249E+00, .8695663E+00, .8913066E+00, 1P
.913046OE+00, .9347851E+00, .9565238E+00, .1000000E+01, 1P
.1000000E+01, .1000000E+01, .1000000E+01, IP 21
.1000000E+01, .1000000E+01. .1000000E+01, .1000000E+01, 1P
.1000000E+01, .1000000E+01. .1000000E+01, .1000000E+01, 1P
.1000000E+01, .1000000E+01, .1000000E+01, .1000000E+01, 1P
.1000000E+01. .1000000E+01, .1000000E+01, .1000000E+01, 1P
.1000000E+01, .1000000E+01, .1000000E+01, .1000000E+01, 1P
.1000000E+01, .1000000E+01, .1000000E+01, .1000000E+01, 1P
.1000000E+01, .1000000E+01. .1000000E+01, .1000000E+01, 1P
.1000000E+01, .1000000E+01. .1000000E+01, .1000000E+01, 1P
.1000000E+01, .1000000E+01, .1000000E+01, .1000000E+01, 1P
.1000000E+01. .1000000E+01, .1000000E+01, .1000000E+01, 1P
.1000000E+01, .1000000E+01, .1000000E+01, .1000000E+01, 1P
.1000000E+01, .1000000E+01, .1000000E+01, .1000OOOE+01, 1P
.1000000E+01, .1000000E+01, .1000000E+01, .1000000E+01. 1P

“QM-AWN

\O

10
11
12
13
14
15
16
17
18
19
20

22
23
24
25
26
27
28
29
30
3 1
32
33
34

S

 

1

2

 

112

 

.1000000E+01, .1000000E+01, .IOOOOOOE+01, .1000000E+01,

:4956503E+00, .1062340E+01, .1909242E+01. .4956485E+00,

.1058905E+01,
.1962534E+01,
.4956476E+00.
.9426219E-100.
.1515192E+01.
.5217355E+00,
.9996656E+00,
.1613054E+01,
.5217422E+00.
.1041944E+01,
.1704185E+01,
.5217399E+00.
.1064105E+01,
.1795341E+01,
.5217373E+00,
.1068986E+01,
.1886494E+01.
.5217373E+00,
. 105567 8E+01 .
.1985099E+01,
.5217382E+00.
.9557156E+00.
.1537715E+01,
.5478207E+00.
.1015832E+01,
.1635814E+01.
.5478284E+00,
.1052567E+01,
.1727011E+01,
.5478255E+00,
.1070167E+01,
.1818132E+01,
.5478237E+00,
.1070670E+01,
.1909244E+01,
.5478255E+00,
.1052349E+01 .
.2000000E+01,

.1932025E+01,
.4956519E+00,
.1041992E+01,
.1500000E+01,
.5217385E+OO.
.9857751E+00,
.1590402E+01,
.5217342E+OO.
.1033334E+01.
.1681432E+01,
.5217359E+OO,
.1060224E+01,
.1772578E+Ol ,
.5217372E+00,
.1069413E+01,
.1863674E+01,
.5217368E+OO.
.1061842E+Ol,
.1962521E+01,
.5217391E+OO.
.9461338E+OO,
.1515208E+01,
.5478153E+00,
.100336OE+01,
.1613033E+01,
.5478373E+OO,
.1045317E+01,
.1704191E+01,
.5478283E+OO,
.1067342E+01,
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.1886472E+01.
.5478282E+00,
.1058594E+01,
.1985106E+01,
.5478252E+00.

.4956500E+00,
.1046771E+01,
.2000000E+01.
.5217370E+00,
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.1023437E+01,
.1658588E+01,
.5217357E+00,
.1055278E+01.
.1749777E+01,
.5217378E+00,
. 1068 803E+01 ,
.1840933E+01,
.5217396E+00.
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.1500000E+01,
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.1063535E-I-01,
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.1072534E+01,
.1863685E+01,
.5478253E+00,
.1064733E+01,
.1962527E+01.
.5478260E+00,
.9496481E+00,

.1052835E+01,
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.5217381E+00,
.1012167E+01,
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.1049201E+01,
.1727013E+01,
.5217361E+00,
. 1066990E+01,
.1818128E+01,
.5217369E+00.
. 1067665E+01,
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.5217378E+00,
.1049556E+01,
.2000000E+01.
.5478262E+00,
.9699207E+00.
.1567864E+01,
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.1027031E+01,
.1658590E-l-01,
.5478295E+00,
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.5478252E+00,
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.1840914E+01,
.5478245E+00,
. 1068222E+01,
.1932029E+01.
.5478215E+00,
.1047460E+01,
.1500000E+01,

1P

 

35

IP 634

1P
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672

 

.5739135E+OO,
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113

.1515195E+01,
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.6260915E+00,
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.1840902E+01,

.57 39 1 35E+00.
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. 1515204E-l-01,
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.5999961E+00.
.1052197E+01,
. 1704198E+01,
.6000003E-000.
.1073763E+01,
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.5999996E+00,
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.6260828E+00,
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.6260878E+00.
. 1062742E+01,
. 1727007E+01,
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. 1079660E+01,
.1818133E+01,
.6260875E+00.

1P
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.1081204E+01,
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.65217 19E+00.
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.1011396E+01,
.1590436E+01,
.7043469E+00,
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1863691E+01.
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1073460E+01,
1962516E+01,
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9603470E+00,
1515220E+01.
6521885E+00,
1018179E+01.
1612996E+01,
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6782586E+00,
1034090E+01,
1635821E+01,
6782596E+00,
1069445E-l-01,
1727006E+01.
6782623E+00.
1086126E+01,
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6782617E+00.
1085613E+01,
1909243E+01.
6782612E+00,
1066202E+01,
2000000E+01,
7043530E+00.
9914083E+00.
1567785E+01,
7043487E+00.
1048380E+01,
1658612E+01,

114

.6260883E+00,
.1077089E+01,
.1932035E-I-01,
.6260878E+00,
.1055675E+01,
.1500000E+01,
.6521704E+00,
.1004022E+01,
.1590425E+01,
.6521729E+00,
.1050830E+01,
.1681421E+01,

.6521736E+00,

.1076681E+01,
.1772574E+01,
.6521730E+00,
.1084830E+01,
.1863689E+01.
.6521723E+00.
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.1795350E+01,
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.9771258E+00,
.1537768E+01,
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.1635819E+01,
.7043459E+00,

. 1079663E-l-01,
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.6260881E+00,
. 1060702E+01 .
.2000000E+01,
.6521701E+00,
984159813400.
.1567818E-l-01,
.6521777E+00.
.1041325E+01,
.1658605E+01,
.6521815E+00.

.1071975E+01,

. 17497 84E+01 .
.6521766E+00,
. 1084421E+01.
.1840905E+01,
.6521748E+00,
.1080056E-I-01,
.1932038E+01,
.6521753E+00.
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.67 82596E+00,
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. 1772574E+01,
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. 1087 879E+01 ,
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.67 82614E+00.
.1079204E+01,
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.67 82598E+00,
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.1515197E+01.
.7043763E+00.
.1025614E+01,
.1612986E+01,
.7043476E+00,
.1065993E+01.

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764

 

 

 

 

.1681418E+01,
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.1044860E+0],
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.7565219E+OO,
.1079569E+01,
.1727000E+01,
.7565217E+00.
.1095583E+01,

.7043498E+00.
.1078551E+01,
.1749791E+01,
.7043435E+00.
.1090680E+01 .
.18409108+01,
.7043488E+00,
.1085857E+01,
.1932042E+01,
.7043446E+00.
.1063923E+01,
.1500000E+01,
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.7304389E+OO,
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.1515106E+Ol,
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.1072803E+01,
.1704215E+01.
.7565269E+00,
.1093186E+01,
.1795357E+01,

115

.1072853E+01,
.1727003E+01,
.7043453E+00,
.1089281E+01,
.1818131E+Ol.
.7043469E+00,
.1088560E+01.
.1909242E+01.
.7043476E+OO.
. 1069002E+01,
.2000000E+01,
.7304335E+00,
.9952091E+00,
.1567771E+01,
.7304456E+00,
.1051897E+01,
.1658605E+01,
.7304350E+OO,
.1081787E+01,
.1749802E+01,
.7304412E+00,
.1093803E+01,
.1840913E+01,
.7304380E+OO.
.1088734E+01,
.1932041E+01,
.7304360E+00.
.1066672E+01,
.1500000E+01.
.7565326E+00,
.1018861E+01,
.1590446E+01,
.7565236E+OO,
.1064782E+Ol ,
.1681415E+01,
.7565231E+00,
.1089720E+01,
.1772569E+01.
.7565204E+00,

.1704209E+01,
.7043518E+00.
.1086814E+01,
.1795352E+01,
.7043442E+00,
.1090226E+01,
.1886467E+01,
.7043507E+00.
.1075537E+01,
.1985126E+01.
.7043447E+00,
.9807271E+00,
.1537764E+01,
.7304310E+00,
.1041187E+01.
.1635821E-I—01,
.7304374E+00,
.1076244E+01,
.1727001E+01,
.7304361E+00.
.1092433E+01,
.1818130E+01,
.7304389E+00.
.1091506E+01.
.1909242E+01,
.7304386E+00,
.1071810E+01,
.2000000E+01,
.7565339E+00,
.9990057E+00,
.1567743E+01,
.7565313E+00,
. 1055490E+01,
.1658619E+01,
.7565325E+00,
.1085170E+01,
.1749796E+01.
.7565210E+00,
.1096909E+01,

1P
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765
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77 1
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APPENDIX E

     

 

I-DEAS

I-DEAS offers a comprehensive package for mechanical design engineers.
This package provides capabilities for

* solid modelling,

* system assembly,

* kinematics,

* finite element pre/post processing,

* finite element solution.

* system dynamics,

* drafting,

* test data analysis, and

* project relational database.

These capabilities are packaged as families of software. I-DEAS consists of the following
families:

1. The solid modelling family which provides modelling of objects, assemblies, and
mechanics.

2. The engineering analysis family which provides mesh generation and output display for
FEM applications. analysis and system dynamics.

3. The system dynamics family which is an interactive system modelling and analysis
program used to design complex mechanical systems.

4. The drafting family for dimensioning objects and for graphical output.

5. The test data analysis family which provides different methods for testing and statistics.

116

 

117

The solid model family is an interactive computing tool that allows to:
* create three dimensional objects.
* assemble three dimensional objects into systems.

* analyze and evaluate objects in three-dimensional Space.

Solid modelling offers four methods to create and manipulate solid geometry:

1. Boolean operations with primitives.

2. Generating three dimensional objects by translating or revolving two dimensional
outlines(profiles).

3. Generating three dimensional objects by skinning operations using a set of cross
sections.

4. Assembling an object from points, facets. surfaces. and bodies.
For communication with other systems I-DEAS provides a set of interfaces, that fulfil
different requirements of applications. I-DEAS interfaces with other systems via the

following files:

Table 5.1: Interface Capabilities of I-DEAS.

 

 

 

 

 

Output from I-DEAS Input to I-DEAS
Layout_To_Draftin g Universal Files
Universal Files IGES files
Programme files Programme files
IGES files

 

Calma_IGES files
Hewlett_Packard_MI

 

 

Computer_Vision

IIF

 

 

 

 

 

 

118

It was these capabilities of IDEAS that were used. A Fortran program was written, which
when executed creates a program file (see Appendix C) that serves as an input for I-
DEAS. Further this program file when executed in I-DEAS outputs an IGES file that

serves as the input test surface for the Verification program.

It has to be noted that when I-DEAS creates a surface from an intersecting mesh of curves,

the maximum number of intersecting splines that it can use is only 36.

APPENDIX F

 

119

EICQQBSBEISII

The most elementary of a class of surfaces, originally studied by Coons[3] is
known as the Coons patch and is constructed solely in terms of information given on its
boundary and certain auxiliary scalar functions of u and V. This surface is obtained by
linear interpolation. A surface patch can be created from only two of its boundaries, say
r(0,v) and r(I,v) by using linear interpolation and is given as

r1(u,v) = (1—u)r(0,v)+ur(l,v) (1)

Similarly. the linear interpolation gives a surface in the v-direction as

r2(u,v) = (l—v)r(u,0)+vr(u,1) (2)

Adding these two surface patches and removing the unwanted linear interpolants

in the u and V direction, namely

r3(u,v)=(l—u)(l—v)r(0,0) +u(l—v)r(l,0) +(1—u)vr(0, 1) +uvr(1, 1)

(3)
The surface r = r1 + r2 - r3 is conveniently expressed in the matrix form as
r (0: V) (1" V)
r u, v = 1 -— + , 0 , l
( ) [( u) "lanai [r(u )r(u )][ v ]
(4)

0, 0 0, 1 —
_[(l_u) u]r( )r( ) (I V)
r(1,0) r(1,1) v
Successive substitution of u = O, u = 1. v = 0, v = 1 Show that the patch defined by

equation (4) has the four original curves as its boundaries. The auxiliary functions a, (l-u).

v. (l-v) are the blending functions.

 

 

120

MW

An important generalization of equation (4) can be made. The linear blending
functions in the patch result from the use of uniform linear interpolation. In fact we may
replace (l-u), u by 0(0(u), 0L1(u) where the blending functions are now any functions.
Similarly, we can replace (l-v), v by a0(v), 0t](v) in equation (4). The new blending
functions are customarily chosen to be continuous and monotonic over the entire interval.
In practice, polynomial blending functions are generally used for analytical and
computational convenience. Hence, given a network of curves, we can construct a
composite surface. This surface only has positional continuity across patch boundaries.
The gradient continuity essential for most practical applications is achieved by using a less
elementary kind of patch defined n0t only in terms of its boundary curves but also in terms

of its cross boundary slopes.

The simplest set of polynomial blending functions satisfying the conditions given

by equations (5), (6), (7) and (8)

a0(0).—.1

a (1)=0
O (5)

a1(l)=1
50(0) = 50(1) = [11(0)= 51(1) = O (6)
[30(0) = [30(1) = 131(0) = [31(1) = 0 (7)

andfinally
B—<0)=1.B‘<1>=0

0 0 (8)

B—1(0)=0.B_1(1)=1

 

”

; I new ' ‘"

J“-1‘t‘7-=-:'

 

 

121

comprises of the following cubic equations:

a0(u) = 1- (3142—2143) (9)
:11 (u) = 3:42—2:43 (10)
[3004) = u— (2112—113) (11)
[31 (u) = —u2+u3 (12)

Using these cubics, we obtain the Ferguson cubic curve segment. When using this
set of blending functions, we are therefore constructing a composite surface on a curve

network made up of Ferguson curves[4]. The blending function vectors can then be

 

writtenas
F(u) -—- [c2000 a1(u) 13004) 15.00]
1 0 0 0' (13)
=[luu2u3]_03g_12_01=UC
2 —2 1 1-
and

r
a”) 10-3 2 1

T a(v) 00 3 -2 " 7"
F = - —

(v) [30») 01-21 v2 CV (14)
13(1’) 00—11 3

 

 

 

 

Then the surface patch becomes

 

 

 

~ fiw., ._

 

122

mm = UCQCTV (15)

where the matrix Q is defined as

"40.0) r(o,1) r.,(0.0) r.,(0, 1)
MO) r(1,1) r.,(1.0) r.(1.1)
Q: (16)

ru(0. 0) ruto. 1) ru,(o, 0) ru,(0. 1)

_r,,(1.0)r,,(1,1)ru,(1.0)r,,.(1,1)

 

 

_l

The patch boundaries for the Ferguson patch are blended in such a way that the
cross derivatives at patch corners are implicitly assumed to be zero, so that the lower right
hand matrix 2 X 2 submatrix of Q is a null matrix. Because of these attributes, Ferguson

curves are widely used in aircraft design.

Forrest[5.7,8] states that the assumption of cross derivatives being equal to zero at
all mesh intersections can lead to local flattening of the surface near patch corners.
However, this effect can be minimized by arranging for the mesh lines to lie roughly along
the lines of curvature of the surface. Since there is no twist in such a surface along all such

lines.

If we permit non-zero twist vectors at patch corners, then we are faced with the
problem of how to determine them. One way of doing this is to extend Ferguson’s

originally proposed method for calculating ru and rv [1,4,6]

 

123

MW

When a surface is machined using a cutter such as a ball-ended cutter, the tool
center moves on another parallel surface, offset from the original by an amount equal to
the cutter radius.

When designing such a surface, a sequence of points and normals for the surface
are calculated. The offset is added along the surface normal at each point, together with a
displacement along the tool axis to determine the position of the cutter tip (for tool setting
purposes it is convenient to use the cutter tip rather than the tool center as the reference
point). The new points determine the linear segments of the cutter path, so that the
machining accuracy is a function of the different cutter parameters such as those described

in Appendix A and the spacing of the curves specified by the user in the data[10].

Offset surfaces are also required to make allowance for the skin thickness in sheet
metal surfaces. Consider a parametric surface r = r1(u.v) and the corresponding offset
surface at a distance d along the normal n1(u,v) given by

r2 = r1 (u. V) +dn1 (u. V) (17)

The normals of correspondin g points on r and r2 have the same direction but have
opposite sense if d exceeds one of the principal radii of curvature [2,10]. In machining a
surface, the cutter radius must not exceed the minimum radius of curvature in the surface,
otherwise gouging of the surface will occur. If sections of the offset surface are plotted, the
danger points will appear as cusps or even loops [1]]. When the curvature of the parallel
surface is infinite. the surface has ridges at certain points. This occurs when the offset is

made on the concave side of the surface.

The comparatively simple cutter path calculations described above are inadequate

for general NC programming for three reasons[9]. First, the user has to choose suitably

 

124

spaced curves on the offset surface to ensure that the machined surfaces meet the Specified
tolerances. Second, the use of more complex cutter Shapes for practical reasons makes
analytical determination of the offset surfaces impractical. Finally, the surfaces are not
unbounded, a common problem being to machine a surface that is bounded on one or more
sides by surfaces that must also be machined accurately.

Referring to Fig F.2 we have surface S], which is being currently machined, and
another surface 82. which is to be machined later[ 10]. To cut along the boundary curve C,
the cutter must be moved in such a way as to maintain the correct machining tolerances
with respect to 81 (known as the part surface) and 82 (known as the drive surface). These
tolerances define in effect a pair of offset surfaces for S] and 82. so that the accuracy of
the boundary curve is defined only indirectly (Fig F.3). It has to be noted that the terms

drive and part surface are reversed when 82 is machined with S] as the boundary.

In order to machine S] completely, a number of cutter paths have to be defined
whose spacing represents a compromise between excessive cusp height and long

machining times.

The cutter passes are usually defined by means of a series of intermediate drive
surfaces. terminated by a pass along the boundary. If the surface S] is also bounded by a
third surface 83, each cutter pass must terminate with the cutter also within the tolerance
of surface S3. This surface is referred to as a check surface. The cutter path for each pass
is approximated by a series of linear segments. The end points of these segments are each
defined by the position of the cutter tip when the tool is within the tolerance of the part
surface. the current drive surface, and one of a sequence of pseudo check surfaces whose

spacing determines the lengths of the steps taken along the path.

The role of the pseudo check surface is to help maintain the required tolerances of

 

__"\.."'1. .- .,. 1...: 0‘ lili 4 _-e
-.1.: nl-"Ql.'C‘ '0 ‘ “H"

 

125

the part, drive, and possibly check surfaces. If the accuracy of the intersection curve is
important, the tolerances may have to be carefully chosen when the angle between the

surface normals is small.

In order to define the cutter path tangent, unit normals have to be defined on the
surfaces S1 and 82. However, the exact intersection curve is not followed, and the unit
normals are taken in the direction of the common normal PIN] and P2N2 as shown in Fig
F.4. Thus for example, PIN] is normal to the part surface S] and the tool surface, and
PIN] represents the shortest distance between the tool and the part surface for any given

position T of the cutter tip.

The points P1. N1, P2, N2 are calculated iteratively. Starting at N where the tool
normal has direction n, the intersection P of the tool normal PN and the surface normal is
calculated. If the angle is less than a certain specified value, the points P and N are
accepted. If not, N is modified using estimates of the curvature of the surface at P. The
common normals I11 and n2 are used at every stage to define the tangent direction of the
cutter path for any given tool position T. If T is not Within tolerance. this is only an
approximation to the cutter path tangent. which is itself ill-defined due to the tolerances

allowed for S] and 82.

Starting with the tool within tolerance of the part and drive surfaces. the
subsequent tool movements are calculated by the computer in order to obtain the next tool

point.

 

 

126

 

 

Fig. F.1 - Parametric Surface Patch.

 

127

 

 

 

 

 

 

 

(FF ‘\
S3
\ /________.._

 

 

 

 

 

 

 

 

 

 

82

Fig F.2 - Machining a surface using a ball mill cutter.

 

128

 

drive surface part surface

yrance tolerance
82 \

.3. \

Sl

 

 

boundary curve

anywhere
inside this box.

Fig F.3 - Tolerances for part and drive surfaces.

 

129

81

 

 

 

 

Fig F.4 - Common Normals of the surfaces.

   

 

APPENDIX G

 

 

LIST OF REFERENCES

l) Mortenson, Michael J. (1985), ‘Geometric Modeling’, John Wiley & Sons, Inc.
2) Willmore, T. J. (1958), ‘Diflerential Geometry’, Oxford University Press.

3) Coons, S. A. (1967), ‘Surfaces for Computer Aided Design of Space Forms’,
Report MAC-TR—4l, Project MAC, M.I.T.

4) Ferguson, 1. C. (1963). ‘Multivariable Curve Interpolation’, Report NO. D2-

22504, The Boeing CO., Seattle. Washington.

5) Forrest, A. R. (1972), ‘Interactive Interpolation and Approximation by Bezier

Polynomials’, Computer J. 15. 1, 71-79.

6) Ferguson, J. C. (1964), ‘Multivariate Curve Interpolation’, Journal ACM. 11, 2,
221-228

7) Forrest, A. R. (1972), ‘Mathematical Principles for Curve and Swface
Representation’. In Curved Surfaces in Engineering (proc. Conference at Churchill

College. Cambridge, 1972), IPC Science and Technology Press Ltd.
8) Forrest, A. R. (1980), ‘The Twisted Cubic Curve: a Computer-Aided Geometric
Design Approach’, Computer Aided Design, 12, 4, 165-172.

9) Bezier. P. (1968), ‘How Renault uses Numerical Control for Car Body Design

130

 

 

 

131

and Tooling’, SAE Paper 680010

10) Faux. I. D. and Pratt, M. J. (1985), ‘Computational Geometryfor Design and

Manufacture’, John Wiley & Sons.
11) Flutter A. G. (1976), ‘The POLYSURF System’, Proceedings 1973
PROLAMAT Conference, Budapest (J. Hatvany. ed.), North Holland Publishing Company.

Amsterdam.

12) I-DEAS Reference Manual 5