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I n te llig e n t R o u g h M a c h in in g o f S cu lp tu red P a rts
byHui Li
B.A., Chang-Sha Institute of Technology, China, 1982 M .A.Sc., Chang-Sha Institute of Technology, China, 1987
A D issertation Subm itted in P artial Fullfillment of the Requirements for th e Degree of
DOCTOR OF PH ILO SO PH Y
in the Department of M echanical Engineering We accept this dissertation as conforming
to the required stan d ard
Dr. jB. W. Vickers (Dept, of Mechanical Engineering), Co-supervisor
Dr. ZXDçjng (D ept, of Mechanical Engineering), Co-supervisor
Dr. N. ^ jilp rij(D e p t. of Mech. Eng.), D epartm ental Member
Dr. R. P. Podhorodeski (Dept, of Mech. Eng.), Departm ental Member
j. Kirtm (D ept.
Dr. R. L. Kirtm (D ept, df Electrical Engineering), Outside M ember
Dr^ Y. A ltintas (Mech. Eng.. U. of British Columbia), External Examiner v '
© Hui Li, 1996 University of V ictoria
All rights reserved. This dissertation may not be reproduced in whole or in part, by m im eograph or other means, w ithout th e permission of the author.
11
A b str a c t
Sculptured parts, characterized by interconnected and bounded param etric surface patches, are widely used in aerospace, automobile, shipbuilding and plastic mold industries due to th e ir functional and aesthetic properties. However, adoption of these sculptured surfaces on mechanical products increases th e com plexity of m anufacturing and p u ts forward a challenge to achieve high m achining quality and productivity, as well as low machining cost.
Machining of sculptured parts is mostly carried out on a m illing m achine. The milling process can be divided into: rough cut (roughing) and fine cut (finishing) operations. Rough m achining is used to remove excess stock m aterial, while finish machining is aim ed a t generating adequate tool paths for producing the final shape of th e p art. W hen a sculptured part is m achined from prism atic stock, a large am ount of rough cut, up to 90 percent of the to ta l m achining, is required. C utting tim e reduction in rough machining can considerably im prove the efficiency of sculptured p a rt m achining, lower production cost.
This research focuses on th e productivity im provem ent of sculptured part rough milling m achining th a t is affected essentially by CNC tool p a th ajid m achin ing param eters. Two m ajo r strategies, machining p ath strateg y and m achining param eter strategy are investigated. A number of new m ethods are introduced to generate highly productive CNC tool path and machining p aram eters.
Study on m achining p a th strategy involves approaches of generating 2^D CNC tool path trajectory, creating new tool path patterns, and autom atically identify ing optim al tool p a th p a tte rn . W hile reseeirch on m achining p a ram eter strategy fo cuses on the m inim ization of cutting time, based upon th e changing pairt geometry during machining an d m anufacturing constraints. A m ethod th a t incorporates an existing milling process model into the cutting param eter optim ization to predict instantaneous c u ttin g force and identify the most effective c u ttin g param eters is introduced. An im proved model cofficient determ ination schem e using numerical optim ization and artificial neural network techniques is developed, and extensive cutting tests are carried to allow the milling process model to fit into the cutting param eter optim ization. A m ethod for the autom ated form ulation and solution of the cutting tim e m inim ization problem is also introduced to allow im portant machining param eters, including the number of cutting layers, d e p th of cut, feed ra te and cross-cutting d ep th , to be determined w ithout hum an intervention.
The research d irectly contributes to autom ated sculptured p a rt machining, and has a great p o ten tial to produce significant economical benefits to manufac turing industry. T h e stu d y also establishes a platform for fu rth er research and development on intelligent sculptured part machining.
Ill
Examiners:
Dr. G. W. Vickers (D ept, of Mechanical Engineering), Co-supervisor
Dr. Z. Dtmg' (Dept, of M echanical Engineering), Co-supervisor
Dr. N. DjilalZ(Dept, of M ech. Eng.), Depaxtmentai Member
Dr. R. P. Podhorodeski (D ept, of Mech. Eng.). Departm ental Member
Dr. R. L. Kirlin (Dept, of E lectrical Engineering), Outside Member ______________________________________________ Dr./Y. Altintas (Mech. Eng.. U. of British Columbia), External Exam iner
IV
C o n te n ts
T itlep age i A b stract ii C ontents iv List o f F igures ix List o f Tables XA cknow ledgem ents x i
Foreword x ii
D ed ication xiii
C hapter
1
Introduction
1
1.1 Machining P a th S tr a te g y ... 2
1.2 Machining Paxam eter S tra te g y ... 6
1.3 L iterature Review of Related W o r k ... 10
1.3.1 Sculptured Surface Description M e t h o d s ... 10
1.3.2 Com puter-A ided Process Planning ... 12
1.3.3 A daptive Control of Machining P r o c e s s ... 12
1.3.4 C u tter S e lec tio n ... 13
1.3.5 W orkpiece O r ie n ta tio n ... 14
1.4 O utline of the T h e s i s ... 14
C hapter 2 O verview o f In tellig en t R ough M achining o f S cu lp tu red P arts 18 2.1 S tructure and D a ta Flow of the S y s te m ... 18
2.2 Approaches of Selecting Tool Path P a tte rn s ... 21
C hapter 3 Tool P ath G en eration and O ptim al Tool P a th P a tte r n Iden tification 23 3.1 Representation of A C utting Layer for Tool P a th G eneration . . 23
Contents________________________________________________________________ v
3.2.1 Basic Tool P ath P a tte rn s for C utting Layers with A Single
I s l a n d ... 30 Stock-offset p attern ... 30 Component-offset p a t t e r n ... 32 Stock/component-offset p a tte rn ... 34 Parallel-offset p a t t e r n ... 36 Proportional-blending-offset p a tte rn ... 39 Max-min-offset p a t t e r n ... 42
3.2.2 Tool P a th Patterns for C u ttin g Layer with No Islands . . . 43
3.2.3 Tool P a th Patterns for C u ttin g Layer with M ultiple Islands 43 3.2.4 Convex-hull Based Tool P a th P a t t e r n s ... 44
3.3 P roductivity Assessment of Tool P a th P attern s ... 44
3.4 An Exam ple of Productivity A s s e s s m e n t ... 45
3.5 A utom ated O ptim al Tool P a th P a tte rn Id e n tific a tio n ... 46
3.5.1 O ptim al Tool P a th P a tte rn Identification Using Fuzzy P a t tern A n a l y s i s ... 48
3.5.2 Fuzzy Set and Fuzzy P a tte rn C l u s t e r i n g ... 48
3.5.3 Shape Based C u ttin g Layer C l u s t e r i n g ... 49
3.5.4 P rim itive Layer Shape A n a l y s i s ... 52
3.5.5 O ptim al Tool P ath P a tte rn Id e n tifica tio n ... 54
3.5.6 An Example for A u to m ated O ptim al Tool P ath P atte rn I d e n tif ic a tio n ... 54
3.6 Tool P a th P a tte rn Identification by Shape I n d e x ... 56
C h a p t e r 4 I m p r o v e m e n t o f M illin g P r o c e s s M o d e l f o r O p tim a l R o u g h M a c h in in g o f S c u l p t u r e d P a r t s 58 4.1 Imm ersion C utting Geom etry based Milling Process Model . . . . 58
4.2 Load Cell for M easurement of C u ttin g Force and T o r q u e ... 62
4.2.1 Design and C alibration of Load Cell ... 62
4.2.2 Physical Characteristics of th e Load C e l l ... 64
4.3 C u ttin g E x p e rim e n ts ... 65
4.3.1 Experim ent L a y o u t... 65
4.3.2 Calculation of C u ttin g M o m e n t ... 66
4.3.3 Experim ental Results ... 68
4.4 Im provem ent of Milling Process M o d e l ... 69
4.4.1 Average Cutting Force M e t h o d ... 69
4.4.2 O ptim ization M e th o d ... 71
4.4.3 A lternative for D eterm ing Model Param eters by Artificial Neural Network ... 72
4.4.4 Modeling R e s u l t s ... 73
Contents vi
4.6 Simplified Force Model for Feed Rate S ch ed u lin g ... 75
C h a p t e r 5 M a c h in in g P a r a m e t e r O p tim iz a tio n 8 6 5.1 C utting Tim e C a l c u l a t i o n ... 86
5.2 Imposed C o n s t r a i n t s ... 88
5.2.1 Geom etric C o n s t r a i n t s ... 88
5.2.2 Physical Constraints of The Machine Tool and C u ttin g System 88 5.3 Solution M ethod ... 89 5.3.1 Approach 1 ... 91 5.3.2 Approach 2 ... 91 C h a p te r 6 A T e st E x a m p le 93 C h a p te r 7 C o n c lu s io n 98 B ib lio g ra p h y 99 A p p e n d ix A N o m e n c la tu r e 113 A p p e n d ix B F u zzy S e t a n d F u z z y P a t t e r n C lu s te rin g 115 B .l Fuzzy S e t ... 115
B.2 H ard Clustering and Fuzzy Clustering ... 115
B.3 Fuzzy c-Means P attern Clustering A lg o rith m ... 116
A p p e n d ix C S c u lp tu r e d S u rfa c e D e s ig n b y B e ta 2 S p lin e 118 C.l p r e l im i n a r y ... 118
C.2 Beta2-Spline Curve and S u r f a c e ... 119
C.3 Beta2-spline Curve Interpolation P r o b l e m ... 120
C.4 Beta2-spline Surface Interpolation P ro b le m ... 121
C.5 Interpolation A l g o r i t h m ... 121
C.6 Calculation of Interpolating E r r o r ... 122
C.7 Iterative Error-adjusting A lg o rith m ... 125
C.8 Convergence Analysis of the Error-Adjusting A lg o rith m ... 126
C.8.1 For th e Case of Beta2-Spline Curve I n t e r p o l a t i o n ... 126
C.8.2 For th e case of Surface Interpolation ... 127
vil
A p pend ix D
C alculation o f B e n d in g M om ent of E n d-M illin g 132
A p pendix E
Artificial N eural N etw ork 135
E .l The Back-propagation A lg o r ith m ... 135
E.2 Cascade-Correlation A rc h ite c tu re ... 137
E.3 Design and C alibration of a Load Cell for E n d -M illin g ... 138
A p pend ix F 2D Curve O ffsettin g in Tool Path G eneration 140 F .l I n tro d u c tio n ... 140
F.2 T e rm in o lo g y ... 143
F.3 Process D e g e n e ra c ie s ... 144
F.4 A lg o r ith m ... 148
vm
L ist o f F ig u res
1.1 Tool Path Layout S t r a t e g i e s ... 16
1.2 Tool Path P atterns of C ontour-m ap Machining ... 17
1.3 Machining Paréimeters in Contour-m ap Machining ... 17
2.1 System of Intelligent Rough M achining of Sculptured P art . . . . 19
3.1 Representation Tree of A C u ttin g Layer ... 24
3.2 An Example of S culptured P a rt and Its C o n t o u r s ... 27
3.3 T hree Cases of Inserting N ode in Representation Tree ... 29
3.4 Tool P ath G eneration Using Parallel P a t t e r n ... 38
3.5 Feasible Tool P ath P a t t e r n s ... 40
3.6 Layer Shape w ith No I s la n d s ... 44
3.7 Layer Shapes w ith M ultiple I s l a n d s ... 45
3.8 An Example of P ro d u ctiv ity A s s e s s m e n t... 47
3.9 C utting Layer Shape D e s c r i p t i o n ... 50
3.10 O ptim al Tool P a th P a tte rn Id e n tific a tio n ... 53
3.11 A Cutting L a y e r... 55
3.12 Clustering R e s u l t ... 56
3.13 P art Size Based Approach for P a tte rn S e le c tio n ... 57
4.1 End Milling O p e r a t i o n ... 59
4.2 Param eters of th e A ltintas Milling Process M o d e l ... 78
4.3 Various Cases of C utting E n g a g e m e n t... 78
4.4 Layout of Load Cell and E x p e r i m e n t ... 79
4.5 Calibration of Load C e l l ... 80
IX
4.7 C utting Force Signal P ro c e s s in g ... 82
4.8 Modeling of Kt and Kr ... S3 4.9 M aximum C u ttin g Force P r e d ic tio n ... 84
4.10 Influence of Helical Angle ... 85
6.1 Simulation Results of Sculptured P art M a c h i n in g ... 95
6 .2 Simulation results by using constant feed r a t e ... 96
6.3 Simulation results of applying different cutiing force models . . . . 97
C .l B eta2-spline curve,its control vertices and jo in t p o i n t s ... 120
C.2 B eta2-spline surface, its control hull and border points ... 122
C.3 Interpolating a set of 2D p o i n t s ... 130
C.4 Interpolating a m esh of 3D P o i n t s ... 131
E .l Feed-forward Neural N e t w o r k ... 135
E.2 Cascade-correlation Neurzil Network ... 139
F .l M ultiple Normal Directions in Offset Curve G e n e r a tio n ... 142
F.2 Degeneracy of Case 1: Reverse Edge ... 145
F.3 Degeneracy of Case 2: S e lf-in te rse c tio n ... 146
F.4 Degeneracy of Case 3: L o o p s ... 147
F.5 Degeneracy of Offset Curve G e n e r a tio n ... 147
F.6 Degeneracy tre a tm e n t in Step 6 ... 150
F.7 Degeneracy tre atm e n t in Step 7 ... 151
L ist o f T ab les
3.1 O ptim al Tool P ath P attern vs. C u ttin g L a y e r ... 46
4.1 C u ttin g Param eter Range Used in the E x p e r im e n ts ... 65
4.2 Overall Modeling Error of Kt «uid Kr ... 74
4.3 Constajits My, Mr, Pt and Pr ... 74
X I
A c k n o w le d g e m e n ts
First of all, 1 am very thankful to m y supervisors, Drs. G. W. Vickers and Z. Dong, for th eir encourage, guidance ajid support of my work, and for th eir extraordineiry patience in revising, correcting my thesis.
I acknowledge the valuable technical discussions an d inputs from m any individ uals, including Professor Y. A ltintas of D epartm ent of M echanical Engineering, U niversity of British Columbia, and several previous graduate students in Ad vanced M anufacturing Labortory a t th e University of V ictoria, Drs. H. Yao and M. M ilroy and Mr. J. Wu. In particular, I am grateful to th e Mr. A. Makosinski and M r. M. Ly, research scientists of the departm ent, for th eir assistances during the course of the research. The assistance from faculty and staff in the D epart m ent of M echanical Engineering, University of V ictoria and the finatial support from U niversity of Victoria and NSERC axe also highly appreciated.
Xll
Xlll
D e d ic a tio n
C h a p te r 1
I n tr o d u c tio n
Sculptured paxts are characterized by complex free-form surfaces and non-éinalytical contours. Increasing application of sculptured surfaces to th e profile of industrial products has been found due to their functional an d aesthetic properties.
Mzinufacturing of sculptured parts is generally carried out through two stages: rough machining and finish m achining. Rough m achining is used to remove ex cess stock m aterial, with productivity as the m ajor concern. Finish machining is aim ed at generating adequate tool paths for producing the final shape of the part w ith geometric accuracy as th e m ajor concern. W hen a sculptured part is ma chined from prism atic stock a large amount of rough machining, up to 90 percent of the total machining, is required [Puttre 92]. This is due to th e significant shape difference between the sculptured part and the stock. M achining tim e reduction in rough machining can considerably improve the productivity of sculptured part m achining and subsequently lower production costs.
T he objective of this research is to develop strategies and approaches to m in imize the CNC (Com puter Numerical Control) rough machining tim e for sculp tu red parts, given part geometry, workpiece m aterial and m achining constraints. M achining tim e is determ ined by the tool path (cu tte r trajectory) and machining param eters including feed rate, depth of cut and cutting speed. Two types of strategies, machining path strategy and machining parameter strategy, are intro duced in this work to achieve the above objective. Machining path strategy deals w ith autom ated CNC tool p a th generation. Machining parameter strategy deals
1.1 Machining Path Strategy
w ith identification of th e m ost productive m achining param eters.
1.1 M achining P a th S trategy
Basically, two general approaches are widely used for generating CNC tool path for sculptured p art rough machining; offset approach and contour-map (or 2 |D ) approach.
In the offset approach [Vickers 92], tool paths are iteratively generated on off set surfaces obtained by expanding or (offsetting) the sculptured surface along 3-D surface normals. Each layer is offset a distance equal to a prescribed cutting depth. A bail end-m ill is usually needed for th e offset machining. This strategy is illustrated in Fig. 1.1 (a).
In the contour-map approach [Vickers 92], tool paths are iteratively generated on parallel cross sectional layers through the sculptured surface. A t each cutting layer, tool paths are laid out following a specific 2D tool p ath p a tte rn in order to remove the excess m aterial between th e com ponent and stock shapes. A n end mill is used. In addition, a final finishing cu ttin g pass along the surface is usually needed to clear the terracing steps created in th e rough cutting. A ball-nose end- mill can be used to carry out this tcisk as illustrated in Fig. 1.1 (b).
A hybrid approach [Vickers 92], is the com bination of the two stated ap proaches. This approach follows the contour-map strategy to remove excess m a terial when th e cu tte r is away from th e sculptured surface, and switches to the 3D offset strategy in th e near neighbourhood of th e sculptured surface, in order to avoid creating terracing steps on the finished surface. A ball end-m ill or ball nose end-mill is usually used in this case. This strateg y is illustrated in Fig. 1.1 (c).
1.1 Machining Path Strategy
Different tool p ath generation strategies lead to totally different tool paths, and heavily influence th e pro d u ctiv ity of rough machining. Generally speaking, for a large vaxiety of sculptured p arts, the offset approach is less efflcient due to th e fact th a t a high feed ra te can seldom be used. This is due to the need for plunge cu ttin g and the poor cu ttin g condition associated with ball end-mills. The contour-m ap approach is m ore efficient due to the fact th a t th e more productive flat-end m ill can be used an d th a t three dim ensional machining is transferred into m achining a series of two dim ensional slices. In this case m ore efficient tool path p atte rn s can be used. Moreover, it has been noted th a t a lot of mechanical parts are of an inherent 2 |D n atu re, and com plicated parts usually produced in two phases: 2 |D roughing and 3D-5D finishing [Held 89]. Based on several statistics, nearly 80 percent of all m echanical parts, m ade by means of chip removing, can be cut by using 2 |D m illing [Held 89].
In contour-map m achining the c u tte r moves in certain predefined styles or tool p ath p attern s at each cu ttin g layer. Frequently used 2D tool p ath p at terns include stock-offset (or window fram e) and parallel-offset (or sta ir case)
[Wang 87, Vickers 92] as shown in Fig. 1.2 (a) and (b). Depending on th e shape of th e c u ttin g layer and the size of the c u tte r used, the productivity of the tool paths for m achining each cu ttin g layer is influenced by those tool p ath patterns. In most current CNC surface m achining system s, only the stock-offset and parallel-offset tool p a th p attern s are used in contour-map machining. T he pattern selection is based purely on experience, and once a p a tte rn is selected it is usually applied to all cu ttin g layers.
For tool p ath generation, previous research has been focused on machining accuracy (dim ensional and geom etric accuracy) and surface finish. P roductivity of th e tool p ath has recently been brought into consideration [Wang 87, Bala 90,
1.1 Machining Path Strategy
G uyder 90, Held 91]. Wang et ai. conducted productivity analysis on tool path orientation and tool p ath p attern s {parallel-offset and stock-offset) for face milling of convex polygons [Wang 87]. T he vjiriation of m achining tim e of these two p at terns can reach a factor of two, depending on c u ttin g orientation. On the average, th e parallel-offset p attern is more eflBcient than stock-offset pattern [Wang 87].
Bala and Chang developed an approach for planning 2 jD tool paths to m ini m ize cutting tim e [Bala 90]. T he approach deals w ith prism atic paxts. T he cutting area enclosed by the stock and island contours is decomposed into a num ber of groups. The connectivity of these groups forms a graph and can be easily con structed. The sequence of c u tte r moves is based on th e connecting features among groups which can be found by searching the graph. C u tting tim e can be reduced by shortening non-cutting c u tte r retractions. G uyder proposed another scheme for 2 |D milling [Guyder 90]. In this scheme, the tool p ath is formed by a series of offsets generated from stock and p art contours. Having generated the offsets, an optim al cutting sequence for machining these offsets is determ ined in aecordeince w ith a num ber of milling guidelines or criteria, such as gouge free milling, pre venting over milling and m achining being started from th e inner most offset, etc. Held presented an approach for reducing com putation cost in tool path generation [Held 91]. A special technique of com putational geometry, Voronoi Diagram , was used to efficiently generate offset curves used as tool paths in pocketing.
In summary, these previous researches focus on c u tte r trajectory com putation. Tool path p attern and p a tte rn selection are seldom addressed. Wang studied two existing tool path p atterns for special simple cuttin g layer formed by convex poly gon with no islands.
1.1 Machining Path Strategy
w ith four new p attern s for contour-map machining axe introduced and investi gated. The performance, in term s of productivity, of these tool path p attern s is estim ated by the machining tim e or length of tool path. The optim al or most productive tool p ath p attern for a given cutting layer is the one with minimum machining tim e (or shortest tool p ath length).
To identify the optim al tool p ath p attern for a given cutting layer, the sim plest and most direct approach is to evaluate and compare the machining tim e of tool paths derived from the six tool p a th patterns. This exhaustive comparison requires generation of tool paths of all th e tool path patterns, which involves ex tensive numerical com putation.
Intuitively, cutting layers w ith sim ilar geometric shapes may be expected to have the same optim al tool p ath p attern s. Motivated by this heuristic, a layer shape clcissification based approach was developed to identify the optim al tool path pattern. In this approach, the cu ttin g layers of th e sculptured part are clas sified into a few groups based on th e ir geometric similarities. In each group, a cutting layer is selected to function as a prim itive layer shape for th a t group. The tool path patterns are then applied to each prim itive cutting layer to generate tool paths. The optim al tool path p attern s axe identified by the exhaustive com par ison of machining tim e or tool p ath length for each prim itive layer shapes. The most productive tool p ath pattern for any given cutting layer is identified in two steps: firstly, m atching this cutting layer to a layer group; secondly, assigning the optim al tool path p a tte rn for the prim itive cutting layer in this group to a given cutting layer.
Two problems are encountered in using the above approach. One is th e dif ficulty to have an accurate representation of cutting layers and the other is
non-1.2 Machining Parameter Strategy 6
existence of crisp difference am ong th e cutting layer shapes. To haxidle this un certainty and am biguity in cutting layer clustering, fuzzy set m athem atics is used.
Extensive study in this work has been carried out on th e productivity of tool p a th patterns for various cutting layer shapes. This stu d y shows th a t th e size of the islands in the cutting layer and th e size of cutter have a m ajor influence on machining tim e. Some patterns axe suitable for cu ttin g layers with relatively sm all islands, others may work well for cutting layers with big islands. A paxt size based approach is thus developed for identifying th e optim al tool p a th p attern . In this approach, a shape index 7 defined as the ratio of c u tte r’s cross sectional axea over the cutting area in th e following equation, is used to estim ate th e relative size of cutting area with respect to cu tte r size.
where, Ad is the cross section area of cutter; is stock a rea and Ap is th e axea of all the islands. A num ber of thresholds axe set for this shape index for testing th e applicability of various tool p ath patterns. These thresholds axe obtédned from previous m achining experience or productivity analysis of tool p a th patterns. T h e tool path p attern s th at pass the above threshold test axe used for generating tool paths and hence calculating machine time. T he optimad tool p ath pattern identification is then made from th e comparison of m achining tim e of selected tool p ath patterns.
1.2 M achining P aram eter S tra teg y
Machining param eters, especially feed rate ( /) and depth of cut ( aocial depth of cut da and radial depth of cut dc), have a significant influence on m achining time of contour-map machining. Figure 1.3 illustrates these m achining p aram eters used in contour-map machining. Using larger depth of cut and feed ra te can reduce
ma-1.2 Machining Parameter Strategy
chining tim e. However, incautiously selected large m achining param eters m ight cause serious problems in m achining, such as c h a tte r, tool breakage, poor surface finish, etc., which would dcimage the cu tte r, workpiece or even NC m achine.
T h e traditional approach for selecting m achining param eters is based on ac cum ulated machining experience, th a t is either held by m achinists or stored in th e form of machining handbook. T he m achining param eters determ ined by th is experience based approach axe usually safe to use, b u t conservative. To fully u ti lize th e machine power and achieve m axim um productivity without violating th e m achining constraiints, a m achining process m odel needs to be established. T h e m achining process model constructs inherent relations among m achining p aram eters, machining constraints and m achining tim e. Based on this model, safe a n d productive machining param eters can be determ ined.
M any machining process models have been proposed [McGoldrick 83, Sm ith 91, A ltinas 91, Armarego 92]. These models can be classified into two categories: M a terial Removal R ate (MRR) based model and Im m ersion C utting G eom etry (IC G ) based model.
T h e typical M RR based model as given in th e following equation can be found in a machining handbook [Handbook 1]
P = P u{M R R ) = P u { d M ) (1-2) where, P is average cutting power th a t is proportional to the m aterial rem oval rate; is a constant called unit power whose value is experim entally d eterm in ed for various materials. Another M RR based model proposed in [McGoldrick 83] is described as follows:
1.2 M achining Parameter Strategy 8
T = (1.4)
where, P is average cu ttin g power; T is average cutting torque; N is spindle speed; Til - 7Î4 and m i - m^ are constants determ ined by cutting experim ents.
The advantage of these MRR baised models is their sim ple m athem atical for mulations. However, they do not consider the cyclic nature of m illing, and only estim ate the average cutting force, torque or power. In addition, the machining constants in th e m odel depend on all elements of the machining process, workpiece m aterial, cu tter, m achine, etc. Change in any element requires recalibration of all the constants.
ICG based m odels deal with the instantaneous cutting force acting on the cutter. This kind of model considers immersion cutting geom etry configuration which is formed by instantaneous rotational position of cu tter, helical angle of end-mill, startin g and exiting cutting angle and depth of cut. A typical ICG based model proposed in [Altinas 91] is given as:
dFT,i{<t>,z) = KThi{4>,z)dz\
dFR^i{4>,z) = KfidFT,i{<f>,z) (1.5) where, dFx,i and dFn^i are elemental tangential and radial cu ttin g forces on the ith tooth; (f> is th e instantaneous angular location of reference flute; Kt and Kr
are cutting param eters called specific cutting pressure; hi(0, z) an d dz are instan taneous chip thickness and chip width; chip thickness A,(<^, z) is approxim ated by St sin 4>i{z) with St being the feed per tooth.
The resultant cu ttin g force on the cu tter is calculated by integrating all the elemental cu ttin g forces along each tooth, which is shown as follows:
1.2 Machining Param eter Strategy
where, F is the resultant cu ttin g force; Foto{<i>) is such a paxt in the model th at is only dependent on th e cutting immersion geometry.
This ICG based m odel has three m ajor advantages. T he first is the capability to model the instantaneous cutting force. The second is th e separation of cutting immersion geometry from cutting calibration param eters Kt and Kr which allows quick re-calibration for a new machining environment when using a new cutter or new m aterial. T he th ird is th a t the feed rate of an end mill can be readily determ ined from the m odel shown in Equation 1.6.
Many other approaches have been developed to im prove the accuracy of the machining process models. Vauious additional influences, such ais cutter runout [Kline 83], eccentricity an d deflection [Armairego 92], as well as the dynamic prop erties of machine tool [Smith 91] have been considered in these approaches. A general overview and classification of these models is given by Sm ith and Tlusty in [Smith 91].
In this work, the ICG based model is used for m achining param eter determ i nation. In order to ob tain this kind of model for the m illing process, a dedicated load cell was built to m easure the cutting force and to construct the cutting pa ram eters Kt and Kr from th e experim ental cutting force d a ta . Optim ization and Artificial Intelligence (AI) techniques were used to obtain m ore accurate cutting param eters and hence im prove the model. A simplified m odel, removing the in fluence of end m ill’s helical angle from the model, is proposed for contour-map machining. This model dram atically simplifys the procedure for identifying the optim al machining p aram eter (feed rate).
1.3 Literature Review o f Related W ork 10
1.3 L iterature R eview o f R elated W ork
1.3.1 Sculptured Surface D escrip tion M eth od s
A sculptured surface is usually represented by a collection or sum of intercon nected and bounded param etric surface patches together with blending and in terpolation formulas. Numerous m ethods have been developed for describing param etric surfaces. Herm it bicubic surface, Bezier surface, Coons patch and B-spline surface are the commonly used m ethods in correct CA D /CA M system [Coons 64, Bezier 72, Zeid 91]. B-spline surfaces have gained increasing popular ity due to their flexibility for interactive curved surface specification and ability for designing both standard analytic shapes (conics, quadrics, etc.) and free-form curves and surfaces [Riesenfeld 73, Piegl 91]. A B-spline surface patch defined by an (n 1) X (m - f l ) array of control points is given by the following equation:
n m
P ( u , u) = ^ ^ PijNi^k{u)Nj, l{y), 0 < U < Umax, 0 < U < U^ax (1 .7 )
i=o i=o
where A,-,jt(u) (and N j j { v ) ) is the B-spline function defied as:
= ( u - Ui) + (Uitt - ") (1.8)
Ui+k-1 — Ux+k — n,+i
and
=
( O:
"JtherwfsT*')
(19)
A nother sculptured surface description m ethod is Beta-spline, introduced by B arsky [Barsky 83]. Beta-spline surface, is a generalization of uniform cubic B- spline surface. A bicubic Beta-spline surface patch defined by 4 x 4 array of control points Pij is given as:
3 3
= ^ X ^ 7^.j6.(A ,/?2,« )6j(^ i,)02, u), 0 < u < 1, 0 < u < 1 (1.1 0)
1=0 j= 0
where,
3
brWl,02xU) = ^ Cgr{0i,02)u^,O < 1,T = 0, 1,2,3 (1-11)
1.3 Literature Review o f Related Work 11 and, C = {Cgr} = 1 8 C o o C O l Co2 CQ3 ClO Cii Ci2 Ci3 C20 C21 C22 C23 . C 3 0 C 31 C 3 2 C 3 3 2/^1 4^1 + 4/?i + ^2 —6^1 6 ^1(^1 ~ 1) —{6^1 + 601 + 302) 2 Q0i 6)9^ + 3;92 0 0 0
L -
20^
20^ +
201+ 2 /? i +
202 201+
201+ 202+
22 J
6 = 201 + A0i + A0i + 02 + 2The unique characteristics of Beta-spline are the shape p aram eters 0i and 02 in its basis functions. T hese shape param eters add much flexibility for curve or surface design [Barsky 8 8]. For instance, in the case of curve design, increasing 01 will make th e curve closer to the tangent direction more on one curve segment than on its previous one; increasing 02 will maice the curve m ore polygonal. At the extrem e case, when 02 approaches to infinity, or 0\ = 0 , 02 = 0, th e Beta-spline curve will coincide w ith its control polygon. When /3i = I, /?2 = 0, a Beta-spline curve will become a B-spline curve.
Beta2-spline curves or surfaces are a special case of the B eta-spline curves and surfaces [Barsky 85], w ith 0 \ being set to be 1. The first derivatives and curvatures of Beta2-spline curves or surfaces are continuous at the jo in ts in th e case of curves or at the borders in the case of surfaces. W ith only one varying shape p aram eter, Beta2-spline is a sim ple and com putationally more efficient rep resen tatio n th an the general caise, which m akes it very useful in application. A d etailed discussion of Beta2-spline surface and an algorithm of surface interpolation using Beta2-splines are given in A ppendix E.
1.3 Literature Review o f Related Work 12
1.3.2 C o m p u te r - A id e d P r o c e s s P la n n in g
Process planning is a com m on tcisk in discrete part m anufacturing. It is th e m an ufacturing activity responsible for the conversion of design d a ta to m anufacturing instructions. More specifically, process planning is defined as a function w ithin a m anufacturing facility th a t establishes which processes and param eters are to be used to convert a p a rt from its initial form to a final form predefined in an engineering drawing [Chang 91].
Process plan can be generated eith er manually or by com puter-aided system s. In the m anual approach, a process plan is created by a m anufacturing process p lan ner who examines a new paxt (engineering drawing) and determ ines appropriate m anufacturing procedures. T he accum ulated machining experience is critical to the success of the plan. T his m anual approach becomes inefiicient and u n m an ageable when th e num ber of process plans and revisions to those plans increases. Com puterized approaches have been developed to autom ate process planning. W ith the com puter speed and consistency, better and faster process plans can be generated. Two approaches for com puter-aided process planning (C A PP) are cur rently being pursued: variant and generative. The variant approach uses a library retrieval procedure to find stan d ard plans for similar components. The stan d a rd plans are created m anually by process planners. T he generative approach is con sidered more advanced as well as more difficult to develop. In a generative process planning system , process plans are generated autom atically for new com ponents without referring to existing plans. Typical examples of variant and generative process planning system can be found in [Tulkoff 81, Nan 85].
1.3 .3 A d a p tiv e C o n tr o l o f M a c h in in g P ro c e s s
A daptive control of the m achining process, which seeks the optim al perform ance of the machining process, has been an interesting research topic in m anufacturing
en-1.3 Literature Review o f Related Work 13
gineering. Extensive work has been done in this axea [Centner 64, McGoIdrick 83, Sm ith 91]. Centner ajid Idelsohn developed axi adaptive control system for the milling process. The system consists of three m ajo r parts: num erical control unit, sensors and adaptive controller. T he program m ed input data, including cu tte r location, feed rate and spindle speed, are fed into th e num erical control unit. The sensors pick up the feedback signals such as torque, vibration, and tem perature during machining. These signals axe then sent to the adaptive controller, which in turn estim ates the cutting performance in term s of m etal removal rate or sur face finish. Based upon performance assessm ent, the controller adjusts the feed rate and spindle speed on-line to m aintain high cutting performance. A sim ple trial-and-error m ethod to modify the m achining param eters, feed rate and cu ttin g speed in the adaptive controller has been used in this approach [Centner 64].
1.3 .4 C u t t e r S e le c tio n
Bala and Chang presented an approach for au to m atic c u tte r selection [Bala 90]. T he basic idea of this approach is to select th e largest cu tter which satisfies the given constraint th a t m aterial left behind after roughing m ust be capable of being removed in one finishing pass. Based on this approach, different cutting layers may require different cu tte r sizes. Lee et al. proposed an optim ization process for reducing unnecessary tool changes [Lee 91]. T his optim ization process “m erges” cutting layers in the case where: (1) two subsequent cutting layers i and z + 1 require different c u tte r sizes r,- and r,+i; (2) th e machining tim e after merging cutting layers is less th a n or equal to th e m achining tim e of th e current set of cutting layers. After merging, a small cu tte r between r,- and r,+i is chosen for cutting the merged layer.
1.4 Outline o f the Thesis 14
1 .3 .5 W o rk p ie c e O r ie n ta tio n
T he problem of optim al workpiece orientation for a single setu p waa studied by Haghpassand [Haghpassand 91]. In his approach, a G aussian m ap (G m ap) was used to m easure th e tool access capability and a nonlinear optim izatio n technique was used to determ ine workpiece orientation.
A G m ap for a sculptured surface is formed by the intersections of surface nor m als, which axe originated at th e center of a sphere w ith u n it radius, and the surface of the sphere. Since th e G m ap is formed on a sph erical surface 5^(a,/9) (or 5^(x, y, ± \ / l — — y^)), it is difficult to process. A central projection is used to convert the Gm ap in space into an equivalent planax space E ^ { X , Y ) [Haghpassand 91]. Existing 2D algorithm s from com pu tatio n al geom etry can thus be applied on the plane, «ind th e solution can be m apped back onto th e sphere by inverse central projection. W orkpiece orientation for a single setup was formu lated as am optim ization problem in [Haghpassand 91] for surface finish machining. T h e orientation was optim ized to m inim ize th e overall angle betw een th e surface norm als and the milling tool axis.
1.4 O u tlin e o f t h e T h e s is
T he objective of this research is to develop strategies and approaches to minimize the CNC (C om puter Numerical C ontrol) rough m achining tim e for sculptured parts. Research is focused on optim al tool path p attern identification, machining param eter optim ization and milling process modeling. A n overview of the pro posed approach w ith its system stru c tu re and d a ta flow is given in C hapter 2.
In chapter 3, feasible tool p ath p a tte rn s used in contour-map m achining which is a dom inant approach for roughing sculptured parts a re discussed. A fuzzy p attern clustering based and a geom etry shape based m e th o d are introduced to identify th e optim al tool path p atte rn . A scheme for representing com plex cutting
1.4 Outline o f the Thesis 1 5
layers with nested contours is also presented in this chapter.
In chapter 4, two types of m illing process models , m aterial removal rate based model and cutting im m ersion geom etry bcised model, are investigated. Improve m ent on the existing ICG based milling process model, by using optim ization or artificial neural network technique to obtain more accurate mechanics constants, is presented. In addition, a simplification scheme is proposed for modeling cutting force of regular helicad end-m ills by using a process m odel for straight end-mills. This chapter also includes a novel design of a load cell th a t is used for obtaining cutting force signals an d establishing the machining process model.
In chapter 5, some strategies for obtaining the optim al m achining param eters to reduce rough m achining tim e are presented. This optim ization procedure is initially form ulated as m ixed nonlinear programming problem w ith 2 N + 1 design variables { N is the n u m b er of cutting layers). By assum ing evenly distributed depth of cut and using a milling process model to obtain feed rate, this optim iza tion is simplified as a m inim ization of machining tim e w ith respect to a single variable, W, the num ber of cutting layer.
Fincilly, examples of sculptured part rough machining using the proposed ap proach are presented in C h ap ter 6. The conclusion and contribution of this re search are addressed in C h ap ter 7.
1.4 Outline o f the Thesis 1 6
/ I
(a) Offset Macliining
(b) Contour-map Macfiining
z
j i ' i 3 4 t
(c) Hybrid Machining
1.4 Outline o f the Thesis 17
Stock
Tool path Island
(a) Parallel-offset pattern (b) Stock-offset pattern Figure 1.2: Tool P a th P a tte rn s of Contour-m ap M achining
Cutter
1 8
C h a p te r 2
O v e r v ie w o f I n te llig e n t R o u gh M a c h in in g o f S cu lp tu red
P a rts
An intelligent rough m achining system for sculptured surface parts is developed in this work. T h e objective of this system is to m inim ize rough machining tim e by using optim al tool p a th patterns, cutter paths an d m achining parameters. New tool p a th strategies and machining param eter strateg ies are applied to achieve this objective.
2 .1 S tru ctu re and D a ta Flow of th e S y ste m
T he stru ctu re and d a ta flow of this system is illu strated in Figure 2.1. The input to this system is e ith er scattered surface points or a m athem atical surface model of stock and p art. T h e original surface d ata can be modified and interpolated by a beta-spline surface. T h e cutting volume inform ation, th e dimension of the volume to be cut and th e average orientation of the norm al vectors of the p art surface, are evaluated from th e stock and part models. Tool p a th layout strategy, one of offset m achining, contour-m ap machining and hy b rid machining, is determ ined based on this evaluation.
W hen th e contour-map approach is selected, th e next step in tool path gen eration is to determ in e the num ber of cutting layers, or depth of each cutting layer, used for m achining the part. Traditionally, th e num ber of cutting layers is selected by th e C A D /C A M system operator based on th e previous experience.
2.1 Structure and Data Flow o f the System 19
Optimal Roughing Procedure
I
B a B B B B B B H E B B B B B H B B B B H W
[
Stock / Part Surface Mesh DataI
CAD Systems
|_^ u ^ T O _^ te^ j^ n _^ te^ S p n n eJ-a-
k Cutting Volume Evaluation H
t
~jTool Path Layout Strategy Determination
-'Z.V— Z
,
I Contour - Map Machining |
I Offset Machining
I
Estimation of Optimal Number of Cutting Layers Direct Comparison! h Hybrid Machining I Fuzzy Shape Clustering
Tool Path Pattern I Cutting Layei Selection by Shape Index
Load Cell Design
1
Clustering Pattem Productivity Analysis^ . . . “ 1 I Optimal Tool Path Pattem Pattem Productivity Assessmentg
\
selection by Tool Path LengthLoad Cell
Calibration Tool Path Generation
I
Machinery
Experiment
H
Process Milling ModelingK
Machining Parameter (feed rate) Optimization
CNC Code Generation
I
Optimization of the Number of Cutting Layers
Optimization of the Depth of Cut
I^^^^CNC^^iningJ |^M ^in^S im ula^
Machining Parameter Database
2.1 Structure and Data Flow o f the System 20
In the proposed approach, am optimad number of cutting layers is determ ined to minimize rough machining tim e. In order to bypass the extensive com putation of determ ining tool path for all six potential tool path patterns at each cutting layer, a geom etry m apping approach for each cutting layer is used to find an equivalent cutting volume which is a sim ple paxailelepiped with height equal to depth of cut amd width equal to cross feed. T he feed rate for machining these parallelepipeds cam be easily calculated from Equation 1.6, and machining tim e for each of them cam be simply determ ined by th e length of the pairallelepiped and the feed rate. T he to tal machining tim e for th e whole p art cam be approxim ated by sum m ation of the cutting tim e of each parallelepiped. A prelim inary stu d y indicated th at the function of to tal machining tim e with respect to the num ber of cutting layers shows a concave shape on two dimensional space. T he op tim al solution of this function cam be obtained using quadratic search technique [Reklaitis 83].
For each cutting layer, cross section contours of stock amd islands are first generated from the intersection between a hunt plane parsing though current cutting layer and surface models of stock and part. These cross section contours may contain m ultiple nested loops (islands). Depending on th e appearance and the num ber of islands in the cutting layer, the cutting layers cam be grouped into three categories: no island, single island and multiple islands. In order to find the correct cutting region in th e cutting layer, a special d a ta structure, called representation tree is constructed to represent the relation or topology among these contours. Each cutting region is formed by a node in one level amd its child nodes in the next level of the tree. A number of tool p a th patterns, two for no islands, eight for single islands and four for m ultiple islands, are used for machining each cutting region. T hree parallel approaches are available to identify the correct p attern.
2.2 Approaches o f Selecting Tool Path Patterns 21
2 . 2 A p p ro a c h e s o f S e le c tin g T o o l P a t h P a t t e r n s
A p p ro a c h 1 In this approach, all th e feasible tool path patterns are applied to the cutting region to generate th e tool paths and calculate m achining time. T he p a tte m with m inim um m achining tim e is selected as th e o p tim al pattern for tool path generation.
A p p ro a c h 2 As discussed in C h ap ter 1, the size of part and the size of c u tte r have a m ajor influence on th e productivity of tool path p attern s. In this approach, a shape index defined as th e ratio of c u tte r’s cross section area over the cutting area, is used to estim ate th e relative size of cu ttin g are a w ith respect to cu tter size. A n um ber of thresholds are set for this shape index for testing the applicability of certain tool path patterns. T he tool p ath s and the machining tim e of th e tool p ath patterns th at passed th e threshold test are calculated. The o p tim al p a tte m is selected simply by com parison of the machining tim e.
A p p ro a c h 3 This is a p attern clustering and p attern matching based approach. The procedure to find the op tim al tool p ath p attern by this approach has been discussed in C hapter 1.
Having identified the optim al tool p a th p a tte m , cutter retractions, plunges, rapid traverse and ramps are added to generate a full tool path. In ad d itio n , optim al feed rates are determ ined using th e milling process model for each c u tte r move. An ICG based milling process model is used in this approach. In order to build this process model, a load cell was m ade and calibrated to m easure th e cutting forces and torques. A series of cu ttin g experim ents is carried out for var ious workpiece m aterials and m illing cu tters, and the mechanics constants of th e milling process model are determ ined and stored in a database.
2.2 Approaches o f Selecting Tool Path P atterns 22
Finally, a post-processor is called to generate th e NC m achine code for cutting th e whole paxt, and the code can be sent to a CN C m achine for cutting or sent to a m achining simulation module for verification before actual cutting. A num ber of exam ples of this approach for sculptured p a rt rough machining are discussed in C h a p te r 6.
23
C h a p ter 3
T ool P a th G e n e r a tio n and O p tim a l T o o l P a th P a tte r n
Id e n tific a tio n
3.1 R ep resen tation o f A C u ttin g Layer for T ool P ath G en eration
A cutting layer is formed by contours of stock and islands which are generated from the intersection between a hunt plane passing though the cu ttin g layer and surface models of stock and part as shown in Fig. 3.1 (a). These contours are sorted and arranged in such a way to form a tree (called representation tree) rep resenting th e topological stru ctu re of the cutting area in th e cuttin g layer. In the tree, each parent node in a level and its child nodes in the next level form a sub-cutting layer w ith the restriction that each node can be tre a te d only once, i.e., either a parent node or a child node, as illustrated in Fig. 3.1 (b). Using this representation scheme, the cutting area can be precisely defined and quickly searched in tool p ath generation.
The stock model, defined by function F s { x ,y ,z ), can be either a closed com posite surface or a cylindrical surface generated by a sweeping m ethod. T h e p art model, defined by function F^,(z,y,z), can be either a closed com posite surface, or an open com posite surface extracted from the designed p art. T h e contour of the stock on a cu ttin g layer is éJways a closed curve. However, th e contour of the part can contain a num ber of open curves. An exam ple of sculptured p a rt, stock and a hunting plane is illustrated in Figure 3.2 (a). A cutting layer formed by the contours of the p art and stock in the hunting plane is shown in Figure 3.2 (b).
3.1 Representation o f A Cutting Layer fo r Tool Path Generation 24
(a) A example of cutting layer (b) Presentation tree of the cutting layer Figure 3.1: Representation Tree of A C utting Layer
To construct the representation tree, th e stock contour (a closed loop) is put on the root, each loop in part contours is tre a te d as a child node somewhere below th e root. Different cases need different treatm en ts. For pocketing, only contours of the paxt need to be processed. For m achining a protrusion, b o th stock contour and part contour have to be considered. If p art contours contain open curves, these open curves axe connected to form closed regions (islands) each of which represents a set of points on the part surface projected on the plaxie of the cutting layer. The islands, for example, formed for the cutting layer shown in Fig. 3.2 (a) is illustrated in Fig. 3.2 (b). T he islands (or loops) in the cu ttin g layer axe added to the tree one by one until ail of th em axe processed. Tw o operations axe needed in the tree construction: © is designed for adding a node or sub-tree to a tree and © is designed for deleting a node or sub-tree from the tree. The algorithm for constructing the representation tree is presented below. Com m ents on the algorithm are p u t into a pair of parentheses (* •••♦ ).
3.1 Representation o f A Cutting Layer fo r Tool Path Generation 25 A lgorith m o f building rep resen tation tree
Synopsis: R J r e e = b u ild J r e e (F s { x ,y ,z ),F p { x ,y ,z ) ,h ) Input: F a{x,y,z) — surface model of stock
F p {x ,y ,z) — surface model of part
h — cu rren t level of cutting layer
R eturn: R J r e e — R epresentation tree of the cu ttin g layer a t level h
Procedure:
b egin
Stock = Contour generated from stock surface F s{x^y^z) a t level h Loops = Contours generated from part surface F p { x ,y ,z ) a t level h Co = {c \ c E Loops and c is an open curve };
Cc = A closed curve in Loops] I f a = $ ,
Islands = Form islands from open curves Co', R j r e e = S to c k 0 Islands; else If Co = $ , R J r e e = S to ck 0 Cc, Loops = Loops — Cc, else
Isla n d s = Form islands from open curves Co', R J r e e = S to ck 0 Islands;
en d if
do w hile Loops ^ 0 ,
Loop = A closed curve in Loops; L o o p .in jC = F A L S E ;
C = R j r e e ; do w h ile C ^ 0 ,
3.1 Representation o f A Cutting Layer fo r Tool Path Generation 26
S’p = {5 1 S 6 C a n d S C P}', I f P D Loop,
Naj, = in s e r t jiode{Sp — P, Loop)-,
R J r e e = { R jtree — Sp) © (P © Nap); LoopJ.n-0 = T R U E;
break ; (* jump out of the current do loop en d if C = C - S p ; enddo If LoopjinJC = F A L S E , G = $ ; C = RJree; do w hile C ^
P = A node or loop on top level in Rj.ree; Sp = { S \ S £ C a n d P c P } ; If P C Loop, 0 = 0 ® Sp-, endif 0 = 0 - Sp-, endd o
R-tree = {R jtree — O) ® {Loop © G); en d if
enddo endif
end
where, the function of subroutine in s e r t jnode is to add a node to a tree . It can be called recursively. T h e detail algorithm of in se r t jnode is given as follows:
3.1 Representation o f A Cutting Layer fo r Tool Path Generation 2 7
stock
hunting plane
(a) Open surface
part contours
(b) Cutting layer
3.1 Representation o f A Cutting Layer fo r Tool Path Generation 28
A lg o r ith m o f in s e r t a n o d e o r lo o p in t h e r e p r e s e n ta tio n t r e e
Synopsis: N tr e e = insert jnode{Tree^ Loop) Input: Tree — original tree stru ctu re
Loop — loop to be inserted into the tree Return: N tree — new tree stru ctu re
Procedure:
begin
n = be the num ber of nodes or loops existed in Tree; case = 0;
0 = Tree; do w hile G 7^
Isla n d = A node on top level in tree G; = {5 I 5 6 G and S C Island}', I f Isla n d D Loop,
Si = insert jT.ode{Sg © Isla n d , Loop)', N tr e e = (G © Sg) © {Isla n d © Si);
case = I;
break ; (* jum p out of th e current do loop *)
elseif Isla n d C Loop,
Bg = Bg © Sg', case = 2; en d if G = G © Sg\ enddo I f case = 0, N tr e e = T ree © Loop; elseif case = 2,
3.2 Tool Path Patterns f o r Contour-map Machining 29
N tr e e = {Tree © Bg) © {Loop © Eg);
e n d e n d
The three cases of inserting a node in a representation tree in the above al gorithm are illustrated in Fig. 3.3 (a)-(c). T he representation tre e of th e cutting layer given in Fig. 3.1 (a) is built as shown in Fig. 3.1 (b). All th e sub-cutting layers are indicated by enclosing dashed boxes. T he root node 0 represents the stock contour, and other nodes represent paxt contours.
C 5 (
^
(a) CaseO
o
0
(b) Case 1 (c) C ase 2
Figure 3.3: Three Cases of Inserting Node in R epresentation Tree
3 .2 T o o l P a t h P a t t e r n s f o r Contour-map M a c h in in g
Six basic tool path p attern s are investigated for tool p ath generation. Among them , stock-offset and parallel-offset are existing pattern s used in current sur face NC machining packages. Component-offset, stock/component,
proportional-3.2 Tool Path Patterns fo r Contour-map M achining 30
blending-offset and max-min-offset axe new p attern s introduced in this work. These p attern s can be use to generate a tool p ath for cu ttin g layer with single island. Stock-offset and parallel-offset patterns can be also used for cutting layer w ith m ultiple islands or w ithout any islaxid. In addition, two more convex-hull based p attern s are presented for cutting layer with single or multiple islands.
3 .2 .1 B a s ic T o o l P a t h P a t t e r n s fo r C u t t i n g L a y e rs w ith A S in g le I s la n d Stock-offset p a t t e r n
T he stock-offset tool path p a tte m (or window fram e p a tte rn [Wang 87]), illus tra te d in Figure 3.5 (a), is generated by moving th e cu ttin g tool along a series of curves offset inwards from the stock contour. Interm ediate tool p ath loops are created in the vicinity of th e component as the offset curves and the com po nent shape intersect. C utting starts from the stock contour and goes inward to the island contours. The cu tter trajectories S jp a th an d Ijpath for clearing th e boundaries of stock and islands are calculated first. If S -path and Ijpa th have no intersection, add S-path to th e tool p ath and let a curve which is the offset of th e stock contour with an predefined offset distance to be the new stock contour. O therw ise, the intersection of S-path and Ijp a th divides th e cutting layer into a num ber of loops. The offsets of the loops within th e c u ttin g area are taken as th e new stock contours. The tool paths of the sub-cutting layers formed by th e new stock contours and islands contained in them are th en generated recursively. T h e algorithm of stock-offset tool p ath generation is presented below. W here, sym bol ® indicates a operation on two loops and returns a logical tru e value if they have intersection, otherwise return logical false value.
Synopsis: P a th = stock-of fs e t{ S to c k , Islands, d, u)
Input: Stock — stock contour in the given cu ttin g layer Isla n d s — part contour in the given cu ttin g layer d — C u tter’s diam eter
3.2 Tool Path Patterns fo r Contour-map Machining 31
V — C utting overlap ratio
Return: P ath — Tool P ath for machining th e give cutting layer
Procedure: begin Ostock = o f fse t{ S to c k , —d/2); Isle = o f f s e t { I s l a n d s ,d / 2 ) ; P a th = d o w h ile Ostock ^ $ , O = A loop in Ostack; I f (O n Isle) = Oi = o f f s e t { 0 , d * v — d / 2); P a th = P a th W O W s t o c k - o f f s e t { O i , ^ ,d , v ) ; else I f (O (g) Isle) =
Sisle — {Loop I Loop € I s le and Loop C 0 } I f o f f s e t { 0 , —d/2) C o f f s e t { S i s l e ,d / 2 ) , P a th = P a th W Sisle; e lse Oi = o f f s e t { 0 , d * v — d /2); P a th = P a th Ü O W sto c k jo f fset{O x, S is le , d^v); e n d if
Isle = Isle - Sisle;
else
T = [ S I S € I s le and S ® O ^ 0}; Loops — close loops formed by O and T ; Isle = I s le — T
d o w h ile Loops ^
3.2 Tool Path Patterns fo r Contour-map Machining 32 5c = {5 I 5 G Isle and 5 C C}; 0 \ = o f f s e t { C , d * v — d / 2); P a t h — P a t h ^ C W stockjof f s e t { O i , Ss,d^v); I s l e = I s le — Sc Loops = Loops — C e n d d o e n d if e n d if Ostock = O stock — O; e n d d o e n d Component-offset p a t t e r n
T he component-offset tool p a th p a tte rn , illustrated in Figure 3.5 (b), is g en erated (dong a series of curves offset outw ards from the component contour. C u ttin g tool motion starts at the outside and is lim ited to motion w ithin th e stock b o u n d ary. Any offset curve paths th a t go outside the confines of th e stock profile are replaced either by non-cutting fast feed rate motions (for short distances) or by rapid traverse (and associated retra ct and plunge) motions along the boundary. This p a tte m is frequently used in practice due to its simplicity [Vickers 90]. T h e disadvantage is th at it contains numerous non-cutting fast feed rate and rap id traverse motions as well as num erous plunge and retract m otions. T he alg o rith m of component-offset tool p ath p a tte rn generation is given below;
Synopsis: P a th = com ponent{Stock, Isla n d , d,v)
Input: Stock — stock contour in the given cutting layer isla nd — p art contour in the given cutting layer
