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This ~tate · "f·the·an rererenwlrxl illustrutcs recently developed computerized fhaun: design and verification technology. kcying on their centr~1 mic in manufacturing processes--cmployinglhe latest compulrr lechnology to minimize costs. incrca.~c: productivity. and as.u~ product quality. Renccling the authors' extensive experience in manufacturing and lixturing fur industry. Comput~r-Aid~d Fixture /Hsign discU$ses the fundamentals of cumput~r· aided thtu~ design (CAFD) techniques ... covers the uses of a fixture component database and n "group technology" (GT) fixtu~·dcsign retrieval ~ystem .. .illuslrates modular fixtu~.' in cllmplcx configurations :tmcnllble to frequent adjustment and improvement. .. introduces the newly developed automated modular tixture configurution design h!chnique . .. integrutcs computer·aided design. process plan· ning. tooling. and manufacturing ... rcview~ three gencrutillns uf CAFD systems. demonstr.lling their progressive increases in efficiency ond their growing need for more sophisticated computer analysis . .. and mon:o Containing ncarly I ()(Kl references. drawings. photographs. and equations.
Computu-Aidtd Fixture Design is a versatile rererenee for mechanical. manufacturing. industrial. and software engineers. and an excellent lexl for IIdvanced undel}!raduutc and graduate studc:nt' in these disciplines.
about the authors . .. YtMtS" (KF.vts) Ro;o.;G is an Associate Pmfessor of Mechanical EnginL'Cring at the Worcester P\.lytcchnic Institute:. Worcester. Massachusctts. The author or coauthor of over lOO journal aniclcs. conference presentations. and book chapters. he is a mc:mber of tht American SocielY of Mechanical Engineers. the Society of Manufacluring Engineers. and the Chinese Mechanical Engineering Society. Dr. Rong receiwd the B.S. degree (1~81) in mechanical engineering from Humin University of Science :md Technology. Harbin. China. the M.S. degree (1984) in manufacturing engineering from Tsinghua University. Bcijing. China. the M.S. degree (1987) in industrial engineering from the University ofWisconsin- Mudison. and the Ph.D. degree (1989) in mechanical engineering front the University of Kenlucky. Lexington . Y"OXtASIl (STEPfl ENS) ZIW is the: Director of Research Administration ut the Beijing Inslitute of Machinery Industry. Beijing. China.. A professor of mechanical engineering ut TsinghulI UniversilY. Beijing. for uver 30 years. he is Ihe author of more than 50 technical books. joumalllniclc.s. book chapters. and translations. He is a member of the Society of Manufllcturing Engineers and u senior member of the Chinese Mechanical Engineering Association. Professor Zhu graduated from Tsinghua University. Bcijing. China. in 1953.
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CIMPITEI-AIIEI FIXTlIE IESIGN Villi. (KEVII) .11. Worcester Polytechnic Institute Worcester, Massachusetts
VlIXIII. (SIIPlEIS) ll. Belling Insmute of Machinery Industry Belling, China
n
MARC EL DE KKER, iN C.
NEW YOItK • BASH
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Preface
Fixtures are imponant in both traditional manufacturing and modem flexible manufHCturing systems (FMS). which directly affect manufacturing quality,
productivity. and cost of products. The time spent on designing and fabri· eating fixtures significantly contributes 10 the production cycle in improving current products and developing Dew ones. Therefore. much attention has been paid 10 me study of fixturing in manufacturing. In machining processe,
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14.6
:s 1.6 1.6-3.2 3.3-4.8 4.9-6.4 6.5-8.0 8.1-9.6 9.7-11 11-13.4 13.5-14.5 >14.6
:s 1.0 1-2 2.1-3 3.1-4 4.1-5 5.1-6 6.1-7 7.1-8 8.1-9 >9
0.1
0.1
1-3 4-10 11-30 31-100 101-500 501-1,000 (1-3) X 103 (3-5) X 103 5,001- 70,000 >7000
0-60 61-120 121-240 240-600 601-1,000 (1-3) X 103 (3.01-6) X 103 (6.1-1.6) X 104 16,000-32,000 >32,000
2 3 4 5 6 7 8 9
Table 9 Index
0 1 2 3 4 5 6 7
8 9
Coding Scheme of the Linear Code H Material 1
Blank 2
Heat treatment 3
Superalloy Alloy steel Carbon steel Cast iron Composite Non-Fe Copper alloy Alum alloy Plastic-wood Others
Bar Cold-roll bar Tube Casting Sharpened bar Sheet Forging Welding Inject modeling Others
Color harden Annealing Normalizing Aging Hardening Tempering Segmentation Nitride Electroplating Others
Operation type 4 VM -center VM -center HM -center HM-center Grind Shape Broach Weld Others
top mill side mill top mill side mill
GT-Based Computer-Aided Fixture Design
147
in Table 7. For example, D 25 = 3 means that the second locatable surface is perpendicular to the fifth locatable surface (or the norm of the locatable hole). Once the linear and matrix codes are developed, the fixturing feature information can be extracted from the product design model and process planning. The information is represented in a quantitative format which can be used in fixture planning and design.
5.6
FIXTURE-DESIGN SIMILARITY ANALYSIS
Fixture design traditionally depends on experienced people who usually formulate similar fixturing methods in their mind when they deal with a workpiece to be fixtured. According to statistics, in the manufacturing industry more than 70% of fixture designs are generated by modifying existing designs that are similar. In order to effectively make use of expert knowledge in existing fixture designs, the similarity between fixture designs needs to be identified. The fixturing feature information provided in the linear and matrix codes is tested and compared with the information stored in a fixturedesign database. Therefore, the fixture-design similarity can be examined between an incoming fixture design requirements and an existing fixture design. A modified similarity coefficient method is applied when a doubleweighed average similarity coefficient is defined, which is dominated by a critical factor. The most critical factor in the fixture design planning is the locating method. If the locating methods of two fixture designs are the same, there is a basis for comparing their similarity. If the locating methods are different, it means the two fixture designs may be totally different. The critical factor coefficient, Kij , can be defined as K = {I, IJ 0,
if the locating methods of two fixture designs are the same otherwise
(3)
The similarity coefficient between the two fixture designs is given by N
Kij Sjj
=
2: {WjjnWFn}
- - " - : I -- - - N
(4)
2: W Fn n:1
where
and j are indexes of two fixture designs to be compared, n is a
Chapter 5
148
fixturing feature index, W ijn is a weight-average similarity coefficient between fixture designs i and j at the feature n, and W Fn is a weight factor assigned for the feature n. The weight-average similarity coefficient can be defined as ~
L {[ I
Will!
- i (A ikn -
= K~I
AJ~I,) i/Rkn)W I'~n} (5)
k
LW
Hn
K~I
where A ikn is the code value of item k in fixture design i at the feature n, AJkn is the code value of item k in fixture design j at the feature n, R kn is the range of item k in the feature n, and W Fkn is a weight factor assigned for item k of feature n. Because the matrix code structure is used, the expression of the similarity coefficient becomes more complex. The factor I(A ikn - A jkn ) I accounts for the fact that, for a given item, the distance between attributes implies the similarity between fixture designs in the specific feature. The factor [I - I(A ikn - A jkn ) I/Rkn ] expresses a similarity score between fixture designs i and j in the item k at the feature n. The product of W Fkn and [I - I(Aikn A jkn ) IlRkn ] is a weighted score of the similarity of fixture designs i and j in item k at feature n. The product of W ijn at all the features and Kij with a summation gives the similar coefficient between two fixture designs. Therefore, the similarity of two fixture designs can be calculated by Eq. (5). When A ikn and A jkn are not comparable (e.g., work material and heat treatment), the weight-average similarity coefficient becomes k
L
{F iikn }
K=I
(6)
where Fjkn is a coefficient to represent the similarity of feature k and
F _{I, IJkn -
Kn
5.7
0,
if A.kn - Al~n otherwise
=
°
is the number of items in fixture designs at the feature n.
IMPLEMENTATION
Based on the fixture structure analysis, fixturing feature analysis, and fixturedesign similarity description, a GT-based modular-fixture-design system is
GT-Based Computer-Aided Fixture Design
149
developed, which includes the fixturing feature extraction and similar fixture-design retrieval functions. There are three major modules in the system: 1. 2. 3.
Fixturing requirement identification (input) Fixture design modification Fixture design information management and documentation
The first is an information input module with a fixturing feature extraction scheme, in which the information can be input into the system under the on-screen menu promotion. The second is a fixture-design retrieval module. With a similar design analysis scheme, the most similar design can be identified for modifications. The third one is a documentation module which provides interface functions for the other modules (i.e., input and output file management). Potentially, the fixturing feature information can be used in the fixturing surface selection where a rule base needs to be established.
5.7.1
Fixturing Feature Extraction
When the fixturing feature extraction scheme is implemented, the part design and manufacturing plan information is interactively input into the system through an interface with a CAD system. The locatable surfaces are chosen, based on user's justification and preference on primary/secondary locating surfaces. The output information can be used for designing or/and retrieving a fixture design. Figure 8 is a block diagram showing the procedure for implementing the fixturing feature extraction scheme. Figure 9 illustrates a menu tree to show the organization of the information input module. Figure 10 shows pop-up menus on computer screen for the input of surface interrelationships. Figure 11 shows pop-up menus specifying the operational information.
5.7.2
Fixture-Design Comparison
The fixture-design similarity analysis can be conducted when the similar feature coefficient and corresponding detail information are specified. According to the design input information, five similar features can be categorized as (1) comparable linear codes, (2) incomparable linear codes, (3) primary locating surface features, (4) secondary locating surface features, and (5) tertiary locating surface features. Different weight factors are assigned to these similar features because some features may be relatively more important than others in the fixture design. The weight factors just reflect the difference. The values of these factors can be determined based on a fixture-design analysis and on human experience.
Chapter 5
ISO
locatable surfaces --- planes --- holes
Figure 8
Fixturing feature extraction in a computer-aided fixture-design system.
The weight factors are considered in two levels. In the first level, they are used to calculate the similarity of each similar feature between any two tixture designs, and in the second level, they are used to calculate the tixturedesign similar coefficient.
151
GT-Based Computer-Aided Fixture Design
~I
Infonnation input
11 Operational into.
I
Operation
Geometrical Info. Operational Into. Fixturing features Interrelationships
Geometrical Info. r---
Material
Length
-
I leat-treatment
Width
Blank
Height
Quit
J2uit r---
Plane fiX1uring teatures Area
'--
Operational Info. Surface finish Dimension To!.
Dimension ToL
Form Tol.
Form Tol.
Batch size
Clampability
Annual demand Other oper info.
Auxiliary
In-surface tixturing features
L-
r-
Fixturing features
Radius No. of critical dimensions No. oflocatable planes
Surface fmish f---
Dimension T 01.
No. of locatable in-surface f - - . - Form ToL No. oflocatable ex-profiles f - - . - Depth Auxiliary
-
Interrelationships Relation of critical dimensions
'--
Ex-profile tix1uring features
Relation of locatable planes Relation of locatable in-surfaces Relation of locatable ex-proftles
Size Surface tinish Dimension Tol. Form Tol. Length Clampability Auxiliary
Figure 9
The pop-up menu of the information input module.
Chapter 5
152
Interrelationship
Relation of critical dimensions Relation of locatable planes Relation of locatable in-surfaces Relation of locatable ex-profiles
11
Relation beween loc able ex-pr ~fIles
Relation between locatable !od locatabl« surfaces
Relation
betw~en
locatable plane and ocatable surfaces
Relation between the CD and locatable surfaces pI
p2
p~
hI
h2
h3
e1
c2
0 0 ~ 0 0 ~ 0 CD200@O®OOO
CDI
~
cm O~000000
• ••
perpendicular
Figure 10
parallel
others
The pop-up menu for interrelationship information input.
GT-Based Computer-Aided Fixture Design
153
Input the other operational infor.
Figure 11
f
J. HOMe End
.----1
Esc
Menu for operational information input.
In the fixture-design similarity analysis, seven factors are directly comparable, including the size of workpiece (length, width, and height), dimensional and form tolerance, batch size, and annual demand. Therefore, Eq. (5) is simplified as
W jjl
=
2: {[I
- I (Ak - A jk ) IlRk]WFk }
_K_ - I_ _ _ _~-------
(7)
The values of the weight factors are assigned according to its importance in the fixture design. Among these factors, the accuracy requirements of the operation are the most important, followed by the dimensional factors. The batch size and annual demand are relatively less important. W Fn is defined as W F1
= W = W = 0.7, F2
F3
W F4 = W F5
= 1.0,
and
W F6
= W = 0.5 F7
The other operational similarity feature contains four factors: work materials, blank, heat treatment, and operation type. They are not numerically comparable. Therefore, Eq. (6) is applied: 4
W jj2 =
~ 2: K=I
{F jj2k }
(8)
154
Chapter 5
Based on the fixturing feature information, the possible locating method and the corresponding locating surfaces, as well as the surface features can be identified. Usually, each locating method specifies three locating surfaces where the similarity refers to the fixturing features of the three locating surfaces, including the spatial relationships between the locating surfaces. The similarity becomes a comparison of selected locating surfaces as well as surface features and surface relationships, which can be evaluated by applying Eq. (5). If the input information of these features is not directly comparable, Eq. (6) is used. The value of K may vary when a different locating method is applied: K
L {FUnk}
W'Jn
= _k~_I_K--
n
= 3, 4,
5, 6
(9)
Because there are five similar features considered, Eq. (4) can be written as
Kij S,]
{t. Wu.W,.}
= --~----
LW
( 10)
Fn
n~1
where W Fn is the weight factor of each similar feature, which are assigned in our study as W F1 = 1, W F2 = 0.8, W F3 = 1, W r4 = W r5 = 0.7, W F6 = 0.5, and 2:~=1 W Fn = 4.7. The assignment of the weight factors reflects the relative importance of each similarity feature in the fixture-design comparison.
5.7.3
Documentation and File Management
The file management menu is provided to perform the documentation and file management functions. These functions include the save and open workpiece input information files, the setup fixture design code library, which includes all the setup and interface procedures, and the open fixture design code files. Figure 12 shows the pop-up menu for file management.
5.8
CASE STUDY
By using the GT-based fixture design system, the fixturing features of a given workpiece can be extracted, the similar fixture design can be identified, and the existing fixture design can be retrieved.
GT-Based Computer-Aided Fixture Design ---------- ----
Give d file
Figure 12
---~------~
nd~e'
155 - -
t 1 HOl'1e End .. -1 Esc
Menu for file management.
Figure 13 shows a workpiece of pump cylinders. The big hole and one side surface are to be machined. There are three critical dimensions. By following the menu instruction provided in the system, fixturing features are extracted as follows:
************************************** Input Information
************************************** No. of locatable planes, p = 3 No. of locatable internal surfaces, q = 3 No. of locatable external profiles, m = 1 No. of critical dimensions, n = 3 Part length = 5.8 in. Part width = 3.44 in. Part height = 3.5 in. Part tightest dimensional tolerance = 0.0074 mm Part tightest form tolerance = 0.0025 Batch size = 100 Annual demand = 10000 Operational type: horizontal milling operation Material: Carbon-steel Blank: Casting Heat treatment: Normalizing
Chapter 5
156
Figure 13
Sample part # I.
Therefore, the linear and matrix codes are obtained as follows: Part geometric information: G = [3. 2. 3, 5,4, 3,7]
Operational information: H = [2, 3, 2, 3]
Locatable plane surfaces:
4
3
2
2
2
']
U= 4 2 2 2 2 2 [2 2 2 2 3 I
GT-Based Computer-Aided Fixture Design
157
Locatable internal surface:
Locatable external profile: W = [3, 1, 1, 1, 3, 1]
Locating-machining surface relationship:
C=
1 1 [ 2
1 2 1
Locating surface relationships
Figure 14 shows another workpiece in which the large hole is to be bored on a horizontal milling machine. There are two critical dimensions in the X and Z directions. By following the menu instruction, the information input file can be obtained similarly.
************************************** Input Information
************************************** No. of locatable planes, p = 3 No. of locatable internal surfaces, q = 2 No. of locatable external profiles, m = 1 No. of critical dimensions, n = 2 Part length = 5 in. Part width = 3.25 in. Part height = 1.625 in. Part highest dimensional tolerance = 0.005 mm Part highest form tolerance = 0.0025 mm Batch size = 100 Annual demand = 10000
Chapter 5
158
Figure 14
Sample part #2.
Operational type: vertical milling operation Material: cast-iron Blank: casting Heat treatment: normalizing The linear and matrix codes are obtained as: Part geometric information: G
= [3,
2, 1, 4, 4, 3, 7J
Operational information: H
= [3, 3, 2, OJ
/59
GT-Based Computer-Aided Fixture Design
Locatable plane surfaces:
U
=
[~
3 2 2
2 2 2
2 2 2
2 3 3
lJ
Locatable internal surface:
V-_[22
;]
2 2
Locatable external profile: W = [3, 1, 1, 1, 1, 2]
Locating - machining surface relationship: C
= [:
4 1
I
4
4 4
4 4
;]
Locating surface relationship:
The similarity coefficients can be calculated between the workpieces in the two examples. Based on these codes and fixture-design rules, all possible locating methods can be identified (Zhu, 1994). When a common locating method is applied to the both cases, they can be compared because Ki) = 1. First, the linear codes are compared. When Eq. (7) is applied, Aik and Ajk are determined by the G code. By considering the weight factors of W fn> W ijl can be obtained:
L W ijl =
K=I
{[I - I(Ak - Ajk)llRdWFd 7
LWFl< K=I
0.7
+ 0.7 + 0.56 + 0.9 + I + 0.5 + 0.5 5.1
= 0.757
160
Chapter 5
With Eq. (8) and the linear code H, W jj2 can be found:
As mentioned, when the matrix codes are obtained, possible locating methods can be determined based on fixture-design rules. If the same locating method is identified for both workpieces, the similarity of fixturing features can be compared. For example, in this case study, plane surfaces can be selected as locating surfaces for both sample workpieces. Therefore. the similarity identification becomes comparisons of primary locating surfaces (first row of matrix U), secondary locating surfaces (second row of matrix U), third locating surfaces (third row of matrix U), and surface relationships between these locating surfaces (in matrix D). The results are obtained as
Finally, the similarity coefficient between the two fixture designs for the two workpieces is determined as (,
2: 2:
WijnWf'n
Si)
=
= 0.70
n=l (,
Wf'n
n=l
Figure 15 illustrates the fixture designs for the two sample workpieces. The fixture designs are not unique and may not be the optimal designs. It should be noted that there can be more than one similarity coefficient to be calculated when different locating methods are applied. The technique presented in this chapter provides a means of identifying the most similar fixture designs based on all possible locating methods. When the similarity coefficient between the two workpieces is high, the locating method and fixture structure can be shared in the design; that is, one fixture design can be obtained from recalling and retrieving the other one. An interactive operation on the fixture-design modification is still necessary in the current stage. A more advanced case-based reasoning method for automated variational fixture design technique is under development; this will be applicable for dedicated fixture designs also.
161
GT-Bas
ed
computer-Aided Fixture Design
(a)
figure 15 Fixture design for (a) sample part #1 and (b) sample part #2.
162
Chapter 5
REFERENCES Asada, H., and A. By (1985), Kinematics Analysis of Work part Fixturing for Flexible Assembly with Automatically Reconfigurable Fixtures, in Proc. of IEEE lilt. COlli on Robotics and Automation, Vol. RA-I, No. 2, pp. 86-93. Bai, y, and Y Rong (1996), Machining Accuracy Analysis for Computer-aided Fixture Design, ASME Transaction: journal of Manufacturing Science and Engineering, Vol. 118, pp. 289-300; partially presented at ASME, WAM, 1993, PED Vol. 64, pp. 507-512. Bausch, J. J., and K. Youcef-Toumi (1990), Kinematics Methods for Automated Fixture Reconfiguration Planning, in Proc. of IEEE Int. COlli Oil Robotics mu/ Automation, pp. 1396-1401. Berry, D. C. (1982), Application of CAD/CAM to Fixture Design, in Proc. 1st Biennial Iflt. Machine Tool Techflolog)' Coni, Chicago, pp. 43 -66. Chen, M. ( 1(89), Computer-aided T-slot Modular Fixture Design in Milling Machine Center, M.S. Thesis, Beijing Mechanical Industry Institute. Chou, Y c., V. Chandru, and M. M. Barash (1989), A Mathematical Approach to Automatic Configuration of Machining Fixtures: Analysis and Synthesis, jourflal of Engineering for Industry, Vol. 111, pp. 299-306. Ferreira, P. M., B. Kochar. C. R. Liu, and V. Chandru (1985), AIFIX: An Expert System Approach to Fixture Design, in Winter Annual Meetillg, ASME, 1985, PED Vol. 56, pp. 73-81. Grippo, P. M., M. V. Grandhi, and B. S. Thompson (1987), The Computer-Aided Design of Modular Fixturing Systems, International journal of AdwlIlced Manufacturing Technology, Vol. 2, No. 2, pp. 75-88. Hoffman, E. G. (1991), jig and Fixture Design, 3rd Ed., Delmar, New York. Mani, M., and W. R. D. Wilson (1988), Automated Design of Workholding Fixtures Using Kinematic Constraint Synthesis, in Proc. 16th North American Manufacturing Conj, pp. 437 -444. Markus, A. (1998), Strategies for the Automated Generation of Modular Fixtures, in Proceedings Manufacturing International, pp. 97 - 103. Menassa, R. J., and W. R. DeVories (1991), Optimization Methods Applied to Selecting Support Positions in Fixture Design, journal of Engineering for Industry, Vol. 113, pp. 412-418. Miller, A. S., and R. G. Hannam (1985), Computer-aided Design using a Knowledgebase Approach and its Application to the Design of Jigs and Fixtures, Proceedings (~f' the Institute of Mechanical Engineers, Vol. 199, No. B4. Nee, A. Y c., and A. Senthil Kumar (J 991), A Framework for an Object/Rule-Based Automated Fixture Design System, Annals of the CIRP, Vol. 40, No. I, pp. 147-151. Nee, A. y, S. Prombanpong, and A. Senthil Kumar (1991), A State-of-Art-Review of Flexible and Computerized Fixture Design Systems, in 5th Ita. Mamif. Conj, China, Vol. 2, pp. B206- B212. Nnaji, B.O., S. Alladin, and P. Lyu (1990), Rules for an Expert Fixturing System on a CAD Screen Using Flexible Fixtures, journal of Intelligent Manufacturing, Vol. I, pp. 31-48.
GT-Based Computer-Aided Fixture Design
/63
Pham, D. T., and A de Sam Lazaro (1990), AUTOFIX-An Expert CAD System for Jigs and Fixtures, International journal of Machine Tools and Malll(faclure, Vol. 30, No. 3, pp. 403-411. Rong, Y, and Y Zhu (1992), Application of Group Technology in Computer-aided Fixture Design, International journal of S.vstems Automatioll: Research & Applicatiom', Vol. 2, pp. 395 -405. Rong, Y, T. Chu, and S. Palaniappan (1992), Fixturing Feature Recognition For Computer-Aided Fixture Design, in Intelligent Manufacturing Systems (R. P. Judd and N. A. Kheir, eds.), Pergamon Press, Elmsford, NY, pp. 97-100. Rong, Y, X. Liu, and A. Wen (1996), Feature Reasoning based Manufacturing Planning, Manufacturing Systems, Vol. 25, No. 3, pp. 271-276; presented at the 27th CIRP Seminar on Manufacturing Systems, Ann Arbor, MI, 1995, pp. 320328. Rong, Y, J. Zhu, and S. Li (1993), Fixturing Feature Analysis for Computer-aided Fixture Design, ASME, WAM, Ped Vol. 64, pp. 267 -271. Trappey, J. C., and C. R. Liu (1990), A Literature Survey of Fixture Design Automation, International journal of Advanced Manufacturing Technology. Vol. 5, pp. 240-255. Zhu, J. (1994), Fixturing Feature Analysis and Fixture Design Planning in Computeraided Fixture Design Systems, M.S. Thesis, Southern Illinois University at Carbon dale.
6 Autontated Fixture Configuration Design
Flexible fixturing is a necessary aspect of flexible manufacturing systems (FMS) and computer-integrated manufacturing systems (CIMS). Modular fixtures are most widely used in industry for job and batch production. Computer-aided fixture design (CAFD) has become a research focus in implementing FMS and CIMS. Fixture configuration design is an important issue in the domain of CAFD. A review of the current research in CAFD indicates that a major problem impeding the automated generation of fixture configurations is the lack of studies on fixture structures. This chapter presents an investigation of fundamental structures of dowel-pin-based modular fixtures and fixturing characteristics of commonly used modular-fixture elements. A modular-fixture element assembly relationship graph (MFEARG) is designed to represent combination relationships between fixture elements. Based on MFEARG, algorithms are developed to search all suitable fixturing unit candidates and mount them into appropriate positions on a baseplate with interference checking. A prototype system for automated design of dowelpin modular-fixture configurations is introduced in this chapter. Examples of fixture design are given at the end of the chapter.
6.1
INTRODUCTION
Reducing production-cycle time and responding to the rapid change of product designs is a means of surviving and thriving in the competitive market for most manufacturing companies. Manufacturing planning, including tooling, makes a major contribution in the production cycle. With the devel164
Automated Fixture Configuration Design
165
opment of CNC technology, which makes machining time much shorter than ever, the attempt to reduce manufacturing time is focused on decreasing the time involved in workpiece setup. Flexible fixturing has become an important issue in FMS and CIMS (Thompson and Gandhi, ] 986; Nee and Senthil Kumar, 1991). There are several categories of flexible fixture such as phasechange materials and modular, adjustable, and programmable fixtures, of which modular fixtures are widely used in industry (Trappey and Liu, 1990; Zhu and Zhang, 1990). Modular fixtures were originally developed for small-batch production to reduce the fixturing cost, where the dedicated fixtures were not economically feasible. The flexibility of the modular fixture is derived from the large number of fixture configurations from different combinations of the fixture element which may be bolted to a baseplate (Thompson and Gandhi, 1986). Modular-fixture elements can be disassembled after processing a batch of parts and reused for new parts. Modularfixture configuration design is a complex and highly experience-dependent task. This impedes further applications of modular fixtures. Lack of skillful fixture designers is a common problem in industry. The development of CAFD systems is necessary to make manufacturing systems truly flexible. Figure 1 shows an outline of fixture-design activities in manufacturing systems, including three steps: setup planning, fixture planning, and fixture configuration design. The objective of setup planning is to determine the number of setups needed, the orientation of workpiece in each setup, and the machining surfaces in each setup. The setup planning could be a subset of process planning. Fixture planning is used to determine the locating, supporting, and clamping points on workpiece surfaces. The task of fixture configuration design is to select fixture elements and place them into a final configuration to locate and clamp the workpiece. As more and more CNC machines and machining centers are employed, many operations can be carried out within a single setup, which needs to be ensured by a well-designed fixture configuration. This chapter focuses on automated fixture configuration design (AFCD). Some previous research on setup planning can be found in the computeraided process planning (CAPP) area (Joneja and Chang, 1989; Chang, 1992; Ferreira and Liu, 1988). Most of the research in the CAFD area was on fixture planning, including a method for automated determination of fixture location and clamping derived from a mathematical model (Chou et aI., 1989); an algorithm for selection of locating and clamping positions which provided the maximum mechanical leverage (De Meter, 1993); kinematicanalysis-based fixture planning (Menassa and De Vries, 1990; Mani and Wilson, ] 988), and rule-based systems developed by European researchers to design modular fixtures for prismatic workpieces (Markus et aI., 1984; Pham and de Sam Lazaro, 1990).
166
Chapter 6
Product Design
(CAD)
Geometric Representation
1
Setup Planning --- Operation Sequence --- Workpiece Orientation Fixture Planning --- Locating SurfacesIPoints --- Clamping SurfacesIPoints --- Supporting Surtac1esllJOllltl
Process Planning (CAPP)
Fixture Configuration ---Fixture Element Selection ---Position and Orientation Determination
Production System NC programming CAM MRP
Figure 1
Fixture Assembly Drawing Element List Robotic Assembly
Fixture design in manufacturing systems.
In the area of AFCD, relatively less literature can be found. Given locating and clamping points on workpiece surfaces, fixture elements can be selected to hold the workpiece based on computer-aided design (CAD) graphic functions (Sakal and Chou, 1991). A two-dimensional (2-0) modular-fixture synthesis algorithm was developed for polygonal parts (Goldberg and Brost, 1994). Whybrew and Ngoi (1990) presented a method to automatically design the configuration of T-slot-based modular-fixturing elements. The key feature of the system was the development of a matrix spatial representation technique which permitted the program to search and identify both objects and object intersections. It was also able to determine the position of objects during the design process. However, the limitation of the method was that only the blocks whose edges were parallel or perpendicular to each other could be represented. Therefore, the design system could only layout the fixture elements in such a way that all the edges of fixture elements were parallel or perpendicular to each other. Trappey et al (1993), presented a methodology for determining the location and orientation of dowel-pin based modular fixture in a 2-D projection basis. It only presented
Automated Fixture Configuration Design
167
detailed research on selecting the fixed point between baseplate and bottom modular-fixture elements and did not describe the rule to select the suitable modular-fixture elements and the method of combining them. The fixturedesign methodology in the case of the 3-2-1 fixture layout method was applied in the study. The major problems involved in AFCD include selection of locators and clamps which make contact with the workpiece, determination of the heights of these units to hold the workpiece, placement of locating and clamping units around the workpiece and on the baseplate, determination of connections between fixture elements, and interference checking among fixture units and with the workpiece and machining envelope. In this chapter, the fundamental structure of dowel-pin-based modular-fixture and fixturing characteristics of commonly used modular fixture elements are first investigated. An MFEARG is introduced to represent basic combination relationships between modular-fixture elements. Based on MFEARG, algorithms are implemented to choose all suitable fixturing unit candidates. Algorithms of mounting fixture units on baseplates are also discussed in this chapter. The input of the system is workpiece representation, workpiece orientation, fixture planning, and machining envelope. This information is extracted from a CAD model of workpieces with process planning information. Its output is a fixture assembly drawing displayed on the computer screen or plotted as hardcopy, and a list of modular-fixture elements as well as their position coordinates and orientations.
6.2
ANALYSIS OF MODULAR FIXTURE STRUCTURES
Figure 2 sketches a dowel-pin-type modular-fixturing system which includes a library of a large number of standard fixture elements (Hoffman, 1987). With combinations of the fixture elements, an experienced fixture designer can build fixtures for a variety of workpieces. In order to automatically generate a fixture configuration design, the issues for the following problems are presented in the remaining sections: 1. 2.
The selection of suitable fixture elements and combinations of these elements into desired functional units The methodology to mount (position) the fixture units (or elements) in appropriate positions and orientations on a baseplate without interference with the space already occupied by the workpiece, machining envelope, or other fixture units mounted in advance
It should be noted that kinematic constraints, locating accuracy, fixturing stability, and fixturing deformation are also important in fixture planning and
/68
Chapter 6
Figure 2
A sketch of BJuco Technik modular fixturing system.
fixture configuration design. Once a fixture configuration design is finished, the design performances need to be verified, which are not presented in this chapter (Rong and Bai, 1996~ Rong et aI., 1994, 1995~ Zhu et aI., 1993). Verification results are the feedback information to the fixture configuration design module for alternative designs, if necessary. Fixturing features of a workpiece have been analyzed, including geometric, operational, and fixturing surface information (Rong et aI., 1993). Once a fixture structure is decomposed into functional units, fixture elements, and functional surfaces, the fixture-design process becomes a search for a match between the fixturing features and fixture structure (Rong and Zhu, 1993). In the application of modular fixtures, a fixture-element assembly relationship database is built up based on the analysis of the fixture structure.
6.2.1
Decomposition of Modular Fixture Structure
The advantage of modular fixtures is its adaptability for various workpieces by changing the configuration combinations of fixture elements. Modularfixture structures can be decomposed into functional units, elements, and functional surfaces. By applying set theory, a fixture body can be defined as a set or an assembly of fixture elements. Let F denote a fixture and e j (i = I, 2, ... , ne) a fixture element in F, where ne is the number of fixture elements in F, i.e., (1)
This is a representation of a fixture at the level of fixture elements.
Automated Fixture Configuration Design
169
A fixture consists of several subassemblies. Each subassembly performs one or more fixturing functions (usually one). These kinds of subassembly in a fixture are considered fixture functional units. In a fixture unit, all elements are connected one with one another directly where only one element is connected directly with the baseplate and one or more elements in the subset are contacted directly with the workpiece serving as the locator, clamp, or support. Let U i denote a fixture unit in a fixture. From the above description, we have (2)
where nei is the number of elements in unit Ui. Therefore, a representation of a fixture at the level of fixture units can be written in the following way: F = {Uili E nul F
= {{eiiU
E neilli E nu}
(3) (4)
where nu is the number of units in fixture F. Dividing a fixture structure into functional units and gIVIng detailed analyses on the functional units plays a key role in automated modularfixture designs. A fixture element consists of several surfaces which can either serve as a locating, clamping, or supporting surface in contact directly with the workpiece (which is named a fixturing-functional surface) or serve as supporting or supported surfaces in contact with other fixture elements (which are called assembly-functional surfaces). Therefore, an element can be represented by (5)
where Sik denotes the functional surface k on fixture element i and nsi is the number of functional surfaces the element i contains. By combining formulas (4) and (5), a fixture can be represented at the level of fixture surfaces in the form (6)
In this way, a fixture structure is decomposed into three levels, i.e., unit, element, and functional surface levels. A conceptual sketch of the fixture structure decomposition is shown in Fig. 3.
170
Chapter 6 Fixture Structure
Function Unit
Fixture Element
Function Surface
Figure 3
Fixture structure tree.
Based on the investigation of various application examples of dowelpin modular fixtures and also for the purpose of automated fixture configuration design, a fixture structure can be classified into seven types of unit (substructure): Vertical Locating Unit (VLU), Horizontal Locating Unit (HLU), Vertical-Horizontal Combination Locating Unit (VHCLU), Vertical Clamping Unit (VCU), Horizontal Clamping Unit (HCU), Vertical Supporting Unit (VSU), Horizontal Supporting Unit (HSU). Fixture units are composed of modular-fixture elements. The functional surfaces of a fixture element perform the tasks of locating, supporting, and clamping. All of the above units are mounted on a baseplate. Figure 4 shows the fixture structure decomposition for dowel-pin modular-fixture systems.
6.2.2
Fixture Units and Elements
In general, a fixture unit consists of several fixture elements where usually only one element is in contact with the workpiece by its fixturing-functional surface to serve as a locator, supporter, or clamp. All fixture elements in a fixture unit are connected through their assembly-functional surfaces. This fixturing-functional surface in a fixture unit is defined as an acting surface of the fixture unit. Each unit must have at least one acting surface which performs the fixturing function. Usually, the acting surface is a plane or a cylindrical surface. The acting plane of a fixture unit can be described by a point on the plane and the external normal vector of the plane. The center of the fixturing plane is chosen as the point to describe the plane. The acting cylindrical surface of a fixture unit can be described by a point on the axis of the cylinder and the vector of axis. The center point of the acting surface is defined as an acting point of the unit and the distance between the surface of baseplate and the acting point is defined as an acting height of the fixture unit. The acting direction of a fixture unit can also be defined by the direction of the external normal vector of the acting surface.
171
Automated Fixture Configuration Design
-f
Vertical Locating Unit (VLU)
-f fI
Surface and Edge Bar
~
Adjustable LocatingBar ..... .
V-Pad ..... .
Ver11,'cal and Horizontal Combination Umt (VHCLU)
Surface and Edge Bar D I S ...r: d Edg BI k ua Ullace an e oc ... ...
Vertical Clamping Unit (VCU)
Horizontal
--C
Clampin~
I
Unit (HCU)
Vertical Supporting Unit (VSU)
-f
~
Side Surface ..... .
Adjusbnent Stop
Horizontal Locating Unit (HLU)
Fixture Structure
Top Surface
-f
Top Surface Side Surface ......
Clamping Support Clamping Bar Speed Clamp with Adjustable Block Serrated Edge Clamp
Adjustable Bar V-Pad
Adjustable Stop
Horizontal Suppo . Unit (HSU)
Unit Level
Figure 4
Element Level
Decomposition of modular-fixture structures.
Surface Level
Chapter 6
172
For fixture units, the most important parameter in fixture design is the acting height. Figure 5 shows the acting heights of different fixture units in a fixture design. In general cases, several fixture elements need to be assembled together to achieve the acting height. The acting heights of fixture units are the parameters which must be known before suitable fixture elements can be selected. The fixture element selection to form a fixture unit is based on a fixture element assembly relationship analysis as shown in the next section. Fixture configuration design is a process of selecting fixture elements from a fixture element library and allocating them together in space according to a certain sequence. In AFCD, a fixture element database needs to be built up, in which the geometry information such as geometric profile, the edges, and surfaces of a fixture element is represented in its own (local) coordinate system. To represent the position and orientation of a fixture element in the fixture system, global and local coordinate systems need to be defined. If the global coordinate system which is associated with the fixture baseplate is defined by O(X, Y, Z), the local coordinate system of fixture element i can be defined by three orthogonal unit vectors (u" Vi, w;) with a local origin Pi(X, y, z), as seen in Figure 6. Once a fixture configuration is built up, the position and orientation of each fixture element needs to be determined. Parameters (Pi' a", ay, a" b", by, bJ are used to represent the position and orientation of the fixture element i in the global coordinate
Workpiece
. h2).
z
z
x Block ( I, w, h )
Cylinder ( r, h)
hi
Bracket ( h ' 12,
Figure 11
y
x
y
W,
hi, h2 )
Three categories of modular-fixture element.
180
Chapter 6
To understand the assembly relationship between fixture elements, assembly features together with the geometric information need to be defined and used to represent modular-fixture elements. The following functional surfaces are defined as assembly features of fixture elements: (1) supporting faces, (2) supported faces, (3) locating holes, (4) counterbore holes, (5) screw holes, (6) fixing slots, (7) pins, and (8) screw bolts. Figure 12 shows the fixture assembly features. A supporting face is the surface that can be used to support other fixture elements or the workpiece. A supported face is the surface that is supported by other fixture elements in a fixture design. A locating hole is the hole machined to a certain accuracy level and can be used as a locating datum with locating pins. Counterbore holes and fixing slots are used to fasten two elements with screw bolts. In a modular-fixture system, assembly features of elements such as locating hole, counterbore hole, screw hole, pin, and screw are designed with standard dimensions. Other parameters of an assembly feature are the position and orientation of the feature in the element's local coordinate system. The homogeneous transformation is used in this research to describe the position and orientation of features. Let F denote the feature position and orientation of an element, which can be represented by F
= (V,
p)l
(9)
where V = ( v, Vy VI 0 ) is the homogeneous representation of the orientation vector V of feature F and v,,, vY' and VI. are the directional cosines of V. P = (x Y z l) is the homogeneous coordinate of origin of feature F. If F is a face-type feature, its origin P is a point on the face, and the orientation vector V is normal to the face and points out from it (Fig. 12a). If F is a hole-type feature, its origin P is the center of the hole end circle and V points outward along the axis of the hole (Fig. I 2b). If F is a pintype feature, its origin P is the center on the tip of the shaft and V points outward along the axis of the shaft (Fig. 12c). In the case of fixing slots, the origin P and vector V are defined as shown in Fig. 12d. In modular-fixture systems, locating holes, counterbore holes, screw holes, and fixing slots are designed perpendicular to the supporting or supported face of an element. The locating holes, counterbore holes, and fixing slots of a supported element are used to locate and fix the supported element to a supporting element. They are defined as associate assembly features with the supported face. Because of the standard design, their relative positions and orientations are known in the local coordinate system of the fixture element and can be extracted from the vector of the supported face.
181
Automated Fixture Configuration Design
r
v
v
~ I
I
(b)
v
(c)
(d) Figure 12 slot.
Assembly features: (a) face type; (b) hole type; (c) pin type; (d) fixing
Similarly, locating holes and screw holes of a supporting element are used to locate and fix a supported element on the supporting element. They are also defined as associate assembly features with the supporting face. Their positions and orientations can be extracted from the vector of the supporting face in the database. It should be noted that a fixture element may serve as
Chapter 6
J82
a supporting element to a supported element in a fixture and may serve as a supported element to another supporting element. Because the number of assembly features on a face may vary, a linked list structure is used in MFEARDB to represent the fixture elements (Fig. 13). In the MFEARDB, fixture element information is organized into four levels-an element list, element records, functional surfaces, and associate assembly features. In an element record, a fixture element identification code and shape type is first defined. The geometric dimensions are retrieved from element parameters. Associate assembly features are represented in terms of their assembly features on a functional surface, which provides a convenient way to find all associate assembly feature information for a specific surface. This will help in understanding assembly relationship, which is mainly carried out according to supporting-supported face pairs. In the data structure, if there are no more assembly features associated with a functional face, the pointer just points to a symbol NIL, which represents the end of list. Therefore, this approach has the advantage of saving memory space.
Element List
Element Record ID
Supported Face 1 Record IndexlID Vx Vy Vz Associate Locating Hole Pointer Associate Counterbore Pointer Associate fixing Slot Pointer
Associate
x x
Supported Face M Pointer # of Supporting Face Supporting Face I Pointer Supporting Face P Pointer
Figure 13
z
-
Associate Screw Hole Pointer
y
z
Supporting Face I Record ID
z
A linked list data structure representing fixture.
/83
Automated Fixture Configuration Design Element Record
310020 Surface and Edge Bar Block 3 90
Supported Face 1 Record 1
15
Next 2 75 15
20
20
~----+I
o o
30 20
Nil
Nil Nil
SPDF I Ptr 1 SPGF 1 Ptr
Supporting Face 1 Record 1
Nil
o o -1
LH tr SHptr
Next 2
30
60
15
15
Nil
o Figure 14
Data structure representing an edge-bar element.
Figure 14 shows an example of the data structure for an edge-bar element, where two functional surfaces (supporting and supported faces) and three types of associate assembly features (two locating holes, two screw holes, and one counterbore hole) can be identified with position and orientation information. The dimensions of the assembly features are standardized with a specific series of modular-fixture systems.
6.3.2
Mathematical Reasoning of Assembly Relationship
When a data structure is designed to represent fixture element and mating relationships are defined between fixture elements, the assembly relationships between fixture elements can be obtained through a reasoning or in-
184
Chapter 6
ference procedure. Actually, the fixture configuration design is similar to an assembly process. Some previous work in assembly area provides valuable information for analyzing assembly relationships between modular fixture elements (Ambler, 1975; Lee, 1985a; 1985b).
(a)
Mating Relationship Between Assembly Features
Mating relationships have been used to define assembly relationships between part components. Researchers defined their own mating assembly relationship according to the application area. In this research, five types of relationship are defined between assembly features for the purpose of understanding the assembly relationship between modular-fixture elements (Fig. 15): 1.
Against. Face 1 is against face 2 when they are coplanar and with opposite normals. This is the assembly relationship between a supporting face of an element and a supported face of another. Let F, = (V" PI)T and F2 = (V2' P2)T denote the positions and orientations of face 1 and face 2, respectively. The against condition can be represented by (10)
2.
3.
4.
where M is a mirror transformation matrix. Align. A hole aligns another hole when their vectors lie along the same line but in opposition. This is the assembly relationship between two holes. Similarly, let FI = (V" p))T and F2 = (V 2 , P 2 )T denote the positions and orientations of hole 1 and hole 2, respectively. The align condition can be represented by
where K is a constant. Fit. A pin fits a hole when their vectors lie along the same line but in opposition. This is an assembly relationship between a pin and a hole. In the same way, let F, = (V), PI)T and F2 = (V z, pzf denote the positions and orientations of the pin and the hole, respectively. The fit condition can be represented by
Screw fit. A screw blot fits a screw hole when their vectors lie along the same line but in opposition. Let FI
= (V"
p)T and FI
= (V
l ,
185
Automated Fixture Configuration Design
Ft
(b) Align
(a) Against
(c) Fit
(d) Screw fit
vector V2 points to reader
(e) Across
Figure 15
Five basic relationships between fixtures.
P2f denote the positions and orientations of the screw blot and the hole, respectively. The screw fit condition can be represented by
186
Chapter 6
5.
Across. A fixing slot crosses a screw hole when the vector of the fixing slot and the vector of the screw hole are coplanar and perf pendicular. Let FI = (VI' PIY and F2 = (V2' P 2)T denote the positions and orientations of the fixing slot and the screw hole, respectively. The across condition can be represented by ( 14)
These five types of mating relationship cover the assembly relationships between assembly features of fixture elements in most fixture designs. (b)
Assembly Criteria Between Fixture Elements
In order to establish the MFEARDB, the possible assembly relationships between fixture elements need to be evaluated. By examining typical fixture assembly structures, the following criteria in four cases for assembling two fixture elements are employed in modular-fixture configuration design (Fig. 16). Let El donate a supporting fixture element and E2 a supported element.
Case 1. E2 can be assembled into a position on El if the following conditions are satisfied: (I) A supporting face of El is against a supported face of E 2 • The face on El covers most of the face on E 2 • (2) At least two locating holes of El align with locating holes of E 2 • (3) One or more counterbore holes of E2 align with the screw holes of El' (4) The body of El does not intersect the body of El' The second half of condition I is a fuzzy condition. It ensures a firm connection between elements. Condition 2 ensures a high locating accuracy between two elements because locating pins can be inserted into locating holes accurately. Condition 3 ensures that two elements can be fixed together by using screws. Condition 4 is obviously an important criterion for interference free. Once these conditions are satisfied, an assembly relationship between fixture elements El and E2 is identified and can be added to the MFEARDB. Case 2. E2 can be assembled into a position on El if the following conditions are satisfied: (I) the same as condition I in Case 1; (2) the same as condition 3 in Case l, and (3) the same as condition 4 in Case 1. The case is the same as the last one except the requirement of the locating hole alignment. In this case, locating accuracy can be only ensured in the direction of the vector of the supporting or supported face. Case 3. E2 can be assembled into a position on El if the following conditions are satisfied: (I) the same as condition I in Case I. (2)
187
Automated Fixture Configuration Design locating holes counterbore hole
case 2: Locating tower on edge block
o case I: Edge block on console
screw holes
~~~~~~~rt fixing slot
o
o
o case 4: Adjustable locating stop
o case 3: Surface bar on console
Figure 16
Four cases of assembling two fixture elements.
188
Chapter 6
a fixing slot of Ez is across a screw hole of El, and (3) the same as condition 4 in Case 1. This case is similar to Case 2. Again, in this case, the locating accuracy can be only ensured in the direction of the vector of the supporting or supported face. Case 4. Ez can be assembled into a position on El if the following conditions are satisfied: (I) a screw of Ez fits the screw hole of El when E2 is a screw bolt and (2) the same as condition 4 in Case I. This kind of assembly case is usually used in an adjustable locating fixture unit. The relative position between two elements is fixed by a nut.
(c)
Inference Assemb(v Relationship Between Element Pairs
Suppose two fixture elements El and E2 are an assembly pair. The assembly features and geometry of the two fixture elements are retrieved from MFEDDB. Let FI = (VI' PI)T denote a supporting face of El and F2 = (V z, pzfr a supported face of E z. Assume PI! and P IZ are any two locating holes on the supporting face and P ZI and Pn are any two locating holes on the supported face. Note that VI' P" P", P 12 and V z, P z, P ZI ' Pn are represented in the fixture element local coordinate systems. If we can find a position and orientation that satisfies the conditions (1) FI against F2 and (2) P" and P I2 align with P ZI and P 22 , respectively, the assembly position and orientation of Ez on El can be obtained from solving assembly mating equations. Our purpose is to find the position and orientation of Ez on El in El's local coordinate system. The local coordinate systems of El and Ez are first made coincidence. Then, after a series of transformations, E2 is translated and rotated to a position and orientation that the relationship between E z and El satisfies above conditions. Based on the mating conditions, we have
(15)
where T is a transformation matrix calculated from
including rotation transformation matrices about the x, y, and z axes and a translation transformation matrix. T is further represented as
189
Automated Fixture Configuration Design
T
=
cos ~ cos "f -sin ex sin ~ cos 'Y - cos ex sin 'Y ( -cos 0: sin [3 cos "f + sin 0: sin "f x
cos ~ sin "f -sin ex sin ~ sin 'Y + cos ex cos 'Y -cos ex sin ~ sin "f - sin ex cos "f
~
sin sin ex cos cos ex cos
~
o~O)
~
z
y
The solution of above equations implies a potential assembly relationship between El and E 2. Solution (x, y, z) is the position coordinate of E2 on El in El's local coordinate system, and solution (ex, ~, "I) is the orientation coordinate of E2 on El in El's local coordinate system. Furthermore, we should check whether conditions 3 and 4 in Case 1 are satisfied for El and E2 in the above position and orientation (x, y, z, ex, ~, "I). If the checking is completed (x, y, z, ex, ~, "I) will store in the MFEARDB as an assembly relationship between El and E 2. A similar approach can be used to test if other assembly criteria are satisfied.
6.3.3
Assembly Relationship Reasoning System and Examples
Figure 17 shows the architecture of automatically reasoning assembly relationship engine. Once the MFEDDB is available, the reasoning engine will examine all element pairs to find their assembly relationship. The results are stored in an MFEARDB, which is based on the MFEARG model discussed in (Mantyla, 1988). This information is used to automatically design the modular-fixture configuration. To illustrate the implementation of the method, an example is given in which a console and a surface/edge block are chosen as El and E2 (Fig. 18). VI' the vector of the supporting face FI of El, is (0, 0, 1, 0) and V 2 , the vector of the supported face F2 of E2 is (0, 0, -1, 0). P ll = (60, 75, 120, 1) and P I2 = (30, 45, 120, 1) are the two locating holes on F I. P 21 = (45, 15, 0, 1) and P 22 = (IS, 45, 0, 1) are the two locating holes on F 2. According to Eq. (IS), one solution can be identified: x
= 75,
Y = 30,
Z
= 120,
a
= 0,
J3
= 0,
'Y
= 90
The solution shows that there is a potential assembly relationship between El and E 2, which satisfied the conditions (I) Fl against F2 and (2) P ll and P l2 align with P 21 and P 22 , respectively. It is obvious, in further checking, that conditions 3 and 4 are also satisfied. Therefore, there is an assembly relationship between El and E2 with a high locating accuracy. The result can be stored in the MFEARDB. When more than two fixture elements are considered, the assembly relationships can be established in pairs. Figure 19 shows three fixture components-a console, a surface/edge block, and a surface locating tower. Table 1 shows the reasoning result between the console and surface/edge
Chapter 6
190
Modular
FlXlure
It-----;&~
(MFEDB)
L -_ _~...:....:...----..J
no
no
no
Modular
FJX1ure
Element
L______====~~~~------~~ Database P yes
yes
Figure 17
Architecture of assembly relationship reasoning.
(MFEARDB)
191
Automated Fixture Configuration Design
'0 (0
s:1
C> \}
Q
C> C)
\)
yl
;,(1
(j
\}
\J
(a)
[2
z2
(b) Figure 18 An example of reasoning assembly relationship between (a) console and (b) surface/edge block.
block when the former serves as a supporting element and the latter a supported element. The assembly criteria can be satisfied with several possible relative assembly positions, which are identified and stored into the MFEARDB. It should be noted that for different assembly positions and orientations, the effective fixturing function (e.g., locating direction) may be different. When the console serves as a supported element and the surface/
Locating tower
Figure 19
Surface/edge block
Examples of fixture elements.
Console
Table 1
Assembly Relationship Reasoning Results of the Console and SurfacelEdge Block
Potential assembly relationship I
2 3 4 5 6
7 8
Relative orientation
Relative position
x
y
Z
i), the final transformation matrix becomes T ii , which is given by (23)
When i = 1 and j = n, where n is the number of elements in a fixture unit, we get the transformation matrix between top element and baseplate. Equation (23) gives the transformation relationship between the acting point of a fixture unit and the global coordinate system of the baseplate in a specific combination of fixture element assemblies. When all possible combinations are considered, a best fixture unit candidate can be selected to approach the desired acting point with accuracy. Assume that (x*, y*, z*) are coordinates of the suggested point of locating or clamping in the baseplate (global) coordinate system and (x a, Ya, za) are coordinates of the contacting point (or acting point) of locator or clamp with the workpiece in its own (local) coordinate system. A set of acting point coordinates of the locator or clamp in the baseplate coordinate system, (x, y, z), can be calculated as
i,
= I, 2,
... , 0,; i2
= I, 2,
... , O2 ;
••• ;
ir
= I. 2,
... ,Or
(24)
For different assembly combinations, the coordinates of the acting point of the locator or clamp may be changed. The combination that makes the acting point of locator or clamp closest to the suggested locating or clamping point are the ones we want to choose; that is, (x* -
X)2
+ (y* - yi + (z* -
Z)2 ~
mioimum
(25)
Once the best combination is found, the position and orientation of the
203
Automated Fixture Configuration Design
fixture elements in the baseplate coordinate system can be calculated based on the bottom-up calculation procedure.
6.4.4
Determination of Spatial Positions of Fixture Elements
In the fixture unit generation algorithm, the fixture unit mounting algorithm, and the interference checking algorithm, we need to transfer the position and orientation of the fixture elements from local coordinate systems into the global coordinate system. Let (x i+I, Yi+ I, Zi+ I) denote the coordinates of the origin Pi+ 1 of fixture element i + 1 and axi+l, ayi+l, azi +1 and bXi+l> bYi+l> b'i+1 be the direction cosines of the coordinate axes in the global coordinate system O(X, Y, Z). Then, the coordinate of Pi(X i, Yi' zJ in O(X, Y, Z) is calculated by applying the transformation matrix:
(Xi' Yi'
zJ =
'
,
,
(Xi' Yi' Zi)
[ a,,,,
a yi + 1
a,,+1
b xi + 1
b Yi + 1
b'i+1
Cxi + 1
Cyi + I
C,i+1
Xi+ 1
Yi+l
Zi+
~l
I
(26)
To determine the orientation of fixture element i in O(X, Y, Z), the direction cosines of the first two axes of the local coordinate system are calculated as a xi
[ ]T = [a~i ay.
[r
a;i
a~J
az.
b xi
b Yi b Zi
= [b~i
b;i b;,J
~'"
]
a xi + I
a yi + 1
b"i + I
b Yi + 1
b,,+1
c xi + I
Cyi + 1
C,.+l
[ a.
60 40 20
0 0
(b)
2
3
4
5
F (xl0e3 N)
Figure 4 (a) Sketch of fixturing deformations; (b) deformation curves of fixture component deformations.
greater fastening force. When the supports separates from the baseplate to a certain degree, the extension of the bolt dominates the deformation. The deformation curve becomes linear again in the third region. Figure 5 shows the deformation and separation process in the interface of constructing blocks and the baseplate. At different stages of the exerted external force, the contributions of the four individual deformations of the total deformation play different roles. Table 1 shows the percentages of the four individual deformations in the total deformation.
370
Chapter II Construct i ng : Blocks I IL ________ II
Interface
II
/-----
III
Figure 5
11.2.2
bolt extension
Deformation process in the interface of fixture components.
Major Operational Effects on Fixturing Deformation
There are several major factors affecting the fixturing deformation. Studying these effects may lead to an optimal design and assembly of T-slot-based modular fixtures in the aspect of improving the fixturing stiffness. (a)
Fastening Force Effect
Increasing the fastening force will enhance the fixturing stiffness and decrease the total deformation. However, as the fastening force is increased, the fixture component wear becomes a problem when it is disassembled for reuse. A too large fastening force may damage the lip of the T-slots. Figure 6a shows the fixture component deformation under different fastening forces. Figure 6b shows the decrease of the total deformation as the fastening force is increased. It is seen that when the fastening force increased to a certain value, the decrease of the total deformation becomes insignificant. Therefore,
Table 1 Region I
11 III
Percentages of Individual Deformations in Different Regions
Ys
y"
Yl
Ye
Yt
100 100 100
34 34-21 21-12
5 5-39 39-62
58 58-35 35-21
3 3-5 5-6
371
Fixturing Stiffness and Clamping Stability D
E 2-
320
•
280
A
+
c
240
~
200
.£ .u
160
.2
e
"0 b{)
)(
0
Q = 800lb Q= 1200lb Q= 1600lb Q= 2000 Ib Q = 2400 Ib Q =2800 Ib
120
.§
a
80
u...
df)
.~
0 0
300
600
2c
1500
1800
F(N)
(a)
E
1200
900
240 0
200
)(
0
'0
C':I
160
+
120
•
El
.£ .u
A
"0
D
b{)
.§
80
~
40
u::
.,
•
..
.,
e
: : e
F= 300 (N) F= 600 (N) F= 900 (N) F= 1200(N) F= 1500(N) F= 1800 (N)
0
0
6CX)
900
1200
(b)
1500
1800
2100
2400
2700
3000
Fastening force (Jb)
Figure 6 (a) Deformation curves affected by the fastening force; (b) fixturing deformation versus fastening force.
the fastening force needs to be optimized based on more understanding of its effect.
(b)
Locating Key Effect
Using locating keys will not only ensure the locating accuracy but also decrease the fixturing deformation by reducing the shift displacement between the baseplate and supports. The experimental results are shown in Fig. 7. It is shown that under a large workload, the total deformation is decreased if the locating keys are applied. Therefore, in the application of T-slot-based
372
Chapter // 25 III
20
•
E
15
:>::
10
without-key with-key
2-
5 0 0
300
600
900
1200
1500
1800
F(N)
Figure 7
Fixturing defonnation affected by locating keys.
modular fixtures, the proper use of locating keys is important for the locating accuracy and fixturing stiffness. (c)
Fixture Configuration Effect
Different geometric shapes (square or rectangular cross section) and orientation of the support will affect the fixturing deformation. Figure 8 shows the deformation curves using different supports. Table 2 shows percentages of four individual deformations with different supports. These results are useful for optimal design and verification of fixture configurations. The method of mounting the baseplate to a machine table also affects the fixturing stiffness. Figure 9 shows deformations of the baseplate under different conditions (i.e., free, corner-edge-strap clamped, and central-strap clamped). It is clear that using central-edge-strap clamps reduces the baseplate deformation, although only two straps are used in comparison to the corner-strap clamping method.
11.2.3
Mathematical Modeling of Fixture Deformation Curve
The fixture deformation curve in Fig. 3 can be expressed in a mathematical model in which the total deformation is a function of the external force; that is, y,
= f(F)
(3)
Based on an adequacy analysis of the model order, y, is represented by a second-order polynomial function of F,
Fixturing Stiffness and Clamping Stability
373 Q
Q
Q
(a) Square supporter
(b) Rectangular supporter I
(c) Rectangular supporter 11
320
8: e c::
0
'::2
280 240
cd
200
t3
160
§ 4)
-0 bL)
c::
c • •
(a) (b) (c)
120
'§
80
)(
u::
40
0
0
300
900
600
12(X)
1800
1500
F(N)
Figure 8
Table 2
Deformation curves for different supporters,
Percentages of Individual Deformations with Different Supports
Supports Square Rectangular I Rectangular II
Ys
Yb
Yi
Ye
Yt
100 100 100
34 44 54
5 6 5
58 47 36
3 4 5
D
80
'8
2c::
.9
60
~
E
..8
.g
40
B t"tS
0.. 4)
20
Vl
t"tS
CO
0 (C)
SOO
750
1000
1250
1500
1750
F(N)
Figure 9 Baseplate deformation with different mounting methods: (a) free clamping; (b) comerstrap clamped; (c) central-edge-strap clamped.
375
Fixturing Stiffness and Clamping Stability
(4)
where do, d 1 and d 2 are model parameters which can be determined from regressions of experimental data. For example, in Fig. 6a, when the fastening force Q = 1200 lbs, the total deformation becomes Y s = 1.2
+
5.638F
+
(5)
0.321 F2
If the fastening force Q is considered in the model, the total fixture deformation Ys becomes a function of F and Q; that is, (6)
Ys = [(F, Q)
Also, based on the adequacy analysis, the multivariable regression model becomes y, = b m + blOF + b 20F2 + b01Q + b02Q2 + b11FQ + b l2 FQ2 + b 2I F2Q + b 22 F2Q2
(7)
Then, Fig. 6a can be represented as y, = 15.682 - 13.712F + 3.591F2 - 18.922Q + 6.754Q2
+
30.036FQ -
1O.790FQ2 - 5.037F2Q
+
1.792F2Q2
(8)
In Eqs. (6) and (8), the first term is nonzero. This is because of the hysteresis in the initial loading and unloading processes, which presents in experiment results and will be explained in Sect. 11.3.2. The mathematical model presents relationships between the typical fixture unit deformation and the external force, as well as the fastening force. It may provide useful information for an optimal determination of fastening forces in fixture design.
11.2.4
Dynamic Fixturing Stiffness
When the modular fixture is used during a machining operation, the dynamic fixturing stiffness is more important because the external force (F) is dynamic. Figure lOa shows the frequency response of the basic assembly unit of T-slot-based modular fixtures. The major natural frequency (948 Hz) represents the first mode bending vibration of the baseplate and supports, as shown in Fig. 11. Another frequency observed is a frequency a little lower than the first one and with a much smaller amplitude (864 Hz), which is
Chapter 11
376 Q
10 Il) Vl
c: 0
8
0. Vl
~ u
6
'E
~
c:
>0
4
2 0
0
200
400
600
800
1000
1200
Frequency (Hz)
(a)
10
~
8
~
6
i
4
•
~
.!::! Cl
2
0 0 (b)
200
400
600
800
1000
1200
Frequency (Hz)
Figure 10 (a) Dynamic response of assembly unit with a square supporter; (b) dynamic response of assembly unit with rectangular supporter I; (c) dynamic response of assembly unit with rectangular supporter 11.
caused by local contact vibration in the interface between baseplate and supports. These frequencies are relatively higher compared with the natural frequencies in machining. If the rectangle cross-section supports are used, the frequency responses are smaller, as shown in Fig. lOb and IOc. Figure 12 shows the frequency response:; in cases of other fixturing configurations.
377
Fixturing Stiffness and Clamping Stability Q
! 0
10
8
en
c::
0 0. en
6
~ U
.~
4
c::
>.
Cl
2
0 0 (c)
200
400
600
800
1000
1200
Frequency (Hz)
Figure 10
Continued
Figure 11
The first mode of basic assembly unit vibration.
n
JCjl{!j" . ~I
14~ . \
1\I I
l()()
. response of mo dular fixtures. Figure 12 J()() DynamIc
a
Fixturing Stiffness and Clamping Stability
11.3
379
FIXTURING STIFFNESS OF DOWEL-PIN-BASED FIXTURES
Dowel-pin modular fixtures are more suitable for applications with computer numerical control (CNC) machines and are widely used in the United States. The stiffness of dowel-pin modular fixtures is obviously improved. This is because holes are used to replace T-slots on the body of fixture components and the open structure in bending deformation is eliminated. However, fixturing stiffness may be still a problem for the following reasons: 1. 2.
3.
4.
There are many holes machined on fixture components in order to satisfy different assembly and adjustment requirements. In the current market, dowel-pin modular fixtures are made of Cr steel, whereas T-slot fixtures are made of CrNi steel. Therefore, the deformation under Hertz contact force becomes more significant compared with the situation of using T-slot modular fixtures. Usually, the baseplate thickness is less than that in T-slot-based fixtures and the baseplate deformation makes a significant contribution to the total fixture deformation. Fixture component selection is often limited by available space and fixture component types.
In our preliminary study of dowel-pin modular-fixture deformation, an experimental analysis method is applied similar to that in the T-slot modularfixture study. The total deformation of a typical assembly unit is tested and decomposed into individual deformations of fixture components and their connections. The effects of major factors are examined and compared with the study of T-slot fixtures.
11.3.1
Typical Assembly Unit and Deformation Curve
A typical fixture unit is built up with a baseplate, console block, and a locating tower, as shown in Fig. 13. The locating tower is connected to console block with an M I 0 screw-bolt without a dowel pin, and the console block is connected to the baseplate with two dowel pins and four screwbolts. Before the total deformation is measured, the hysteresis in initial loading and unloading processes should be considered. Figure 14 shows the remaining deformation in the loading and unloading process under a 3000lb. external force. Generally speaking, the remaining deformation is mainly caused by assembling clearances between fixture components. It is also caused by the plastic deformation and part of the elastic deformation in the contact region, which is resistant to restoration by static friction. This remaining deformation can be reduced by using dowel pins in the connection
380
Chapter J J F
Figure 13
A typical fixture unit of a dowel-pin modular fixture.
of fixture components and applying a certain preload to eliminate or reduce the influence of the remained deformation. Figure 15 shows the total deformation curve of the typical assembly unit, which is obtained after the remaining deformation is removed in the loading/unloading processes. Compared with the deformation curve for Tslot fixtures, the nonlinearity is not significant. This is because in dowelpin-based modular-fixture assemblies, there is usually no long screw-bolt used in fastening. Instead, the fastening is performed between each pair of fixture components. Because dowel-pin fixtures are made of relatively soft steel, the Hertz contact deformation is quite significant. Permanent press marks can be observed as the fastening force increases, which becomes a limit of the fastening-force amplitude. -
D(XI0
Loodir'l9 Unlooding -4
in.>
300
200
100
o 500 1000 1500 2000 2500 3000
Figure 14
re 110)
The hysteresis in initial loading and unloading processes.
Fixturing Stiffness and Clamping Stability D(X]04
381
Ir , )
300
200
100
o 500
Figure 15
1000 1500 2000 2500 3000 FClb)
The total defonnation curve.
Figure 16 shows the individual deformation modes decomposed from the total deformation curve, including the effects of baseplate bending (dP), the contact deformation between the console and baseplate (dB l ), the console bending (dB 2 ), the console shifting (dB 3 ), the contact deformation between the console and locator (dS I ), the locator bending (dS 2 ). and the locator shifting (dS 3 ). Figure 17 shows their contributions to the total deformation and Table 3 indicates the percentages of individual deformations in the total deformation under a load of 3000 lbs. The most significant effect is the baseplate bending (dP). which can be reduced by applying clamping straps when it is mounted to a machine table. The second most significant effect is the contact deformation between the console and baseplate (dB I ), which may be reduced by applying a large fastening force in the connection. The locator shifting (dS 3 ), the contact deformation between the locator and console (dS I ), and the console bending (dB 2 ) also make contributions in the total deformation, where dS 3 is mainly caused by the clearance in the screwbolt connection. Other effects seem insignificant to the total deformation. The principle of analyzing the individual deformations is the same as the method used in T-slot modular-fixture study. It should be mentioned that the preliminary experimental results are obtained from a specific fixture unit configuration. When a different fixture unit is selected as the typical fixture unit, the deformation percentages of different effects may be different, but the general deformation pattern is similar.
382
Chapter 11
-j6P r
I
F - ----. I
1r~B3
11
Figure 16
11.3.2
an
11
Individual deformation modes of a dowel-pin modular-fixture unit.
Major Operational Effects on Fixturing Deformation
Similar to T-slot-based modular fixtures, there are several major factors affecting the fixturing deformation. First, increasing the fastening force will reduce the fixturing deformation, as shown in Fig. 18. Because the material used in dowel-pin modular fixtures is softer than the material used in T-slot modular fixtures, as the fastening force increases, the fixture component wear becomes more significant in dowel-pin modular-fixture applications. The use of locating pins will not only ensure the locating accuracy but also decrease the fixturing deformation by reducing the shift displacement between fixture components. For example, if no pins are used between the console and the baseplate in the fixture unit shown in Fig. 13, the total fixturing deformation will increase under large external forces. The experi-
383
Fixturing Stiffness and Clamping Stability DCX10
-4
in)
75 aB 1 50
25
o 500
Figure 17
1000 1500 2000 2500 3000 rClb)
Contributions of individual deformations to the total deformation.
Table 3 Percentages of Individual Deformations Contributing to Total Deformation ~S,
~P
30.6
21.1
11.7
3.6
13.4
2.3
15.4
~ 210 0
8
190
8. c
170
-F=2000Ib
~
150
-6--F = 1600 Ib - F = 1200 Ib
0
;~
E 130
.E Q) "0
~
~
x u:
110 90
_F=800Ib
70 400
600
800
1000
1200
1400
1600
Fastening force (0)
Figure 18
Deformation reduction as the fastening force increases.
384
Chapter 11
400
-with pin --- without pin
I 08d(1 b)
Figure 19
Effect of locating pins in modular-fixture deformation.
mental results are shown in Fig. 19. Therefore, the proper use of locating pins is important to the locating accuracy and fixturing stiffness. Baseplate deformation makes a significant contribution to the total fixturing deformation. Proper mounting of the baseplate to the machine table will improve the stiffness of the fixturing stiffness. Experimental results show that if a central-strap clamp is used in the baseplate mounting, the total fixturing deformation will be reduced up to 20%. This conclusion is consistent with the results from the T-slot fixture testing.
11.3.3
A Brief Summary
The fixturing stiffness analysis of dowel-pin-based modular fixtures is similar to that of T-slot-based modular fixtures. The following is a brief summary of major findings in the experimental study of dowel-pin modularfixture stiffness. The total fixture deformation is composed of the contact deformation between fixture components and the deformation of the fixture components themselves. The former is the major part and the latter is minor, except the baseplate. Because there is rarely long screw bolts used in dowel-pin fixtures, the nonlinearity of the deformation curve is not significant. When the fixture components are connected with dowel pins in addition to the screw fastening, the shifting deformation is small, whereas without dowel pins, it becomes larger and improper for a precise location. In addition, the remaining deformation is also large without dowel pins in the connection.
Fixturing Stiffness and Clamping Stability
385
Because dowel-pin-based modular fixtures are made of relatively soft steel materials, contact surface damage may occur when the fastening force is large and an insufficient fastening force usually leads to a weak stiffness.
11.4
CLAMPING STABILITY VERIFICATION
This section presents a method for automated verification of clamping stability in computer-aided modular-fixture designs. When a modular-fixture design is conducted by using a CAD system, the equilibrium between clamping forces and locating responses needs to be evaluated for a reliable locating. If the positions and directions of locating and clamping components are not appropriately placed, the clamping action may not secure the locating but destroy it. In this section, two technical problems related to clamping stability verification are studied: automated extraction of positions/directions of clamping forces and locating responses, and evaluation of clamping equilibrium of the fixture design. The automated extraction of the locating/ clamping positions/directions is implemented by adding special attributes into a fixture component CAD database based on an analysis of locating and clamping methods using modular-fixture components. To verify the clamping stability of a fixture design, the clamping forces are assumed active and of known input forces, and the locating responses are passive and variables to be solved. In the solution, all locating responses should be non-negative for a stable clamping and the friction should be in a feasible range for a stable clamping.
11.4.1
Clamping Stability in Computer-Aided Fixture Design
When a workpiece is fixtured, the clamping stability is defined as the equilibrium between clamping forces and locating responses. If the positions and directions of locating and clamping components are improper, the clamping action may not secure the locating; it may destroy the locating. Figure 20 shows an example of unstable fixture designs, where the 3-2-1 locating principle is applied. If the position of the clamping force is much higher than the locating response position in the Y direction (h2 » hi), there is a rotational motion trend of the workpiece under a moment caused by the clamping force at a higher position relative to the locator position. Therefore, the contact of the workpiece and the right locator in the Y direction becomes unreliable or unstable. Once a fixture design is finished by using a CAD system, the clamping stability should be verified so that the fixture design can be used in the workshop.
386
Chapter //
~
________~CLAMPING FORCE
h 2 » h 1 unstable
Locating response
Camping force
Locating response Figure 20
11.4.2
Locating response
An example of unstable fixture design.
Locator and Clamp Analysis
The contacts of workpiece and fixture components are usually between the workpiece and locating/clamping components. Therefore, only the positions and orientations of the locators and clamps need to be identified in a fixture design, where a geometric-modeling-based analysis of the manufactured workpiece is not necessary and may be difficult because the workpiece could be very different and complex. In order to identify the positions and directions of the contact forces, the locating and clamping methods of using fixture components are analyzed based on dowel-pin modular-fixture systems. Typical locating components include round-pin locators (top and/or edge locating), plane surface locators (top and/or edge locating), and V-block locators. Figure 21 shows some examples of locating methods and locating response directions. If there are different methods of using a locator, they are treated as different locating components in the CAFD system (Figs. 21 a and 2Ib). In modular-fixture applications, the major clamping methods include top clamping with straps and side clamping using screw-bolts. Figure 22 shows examples of the clamping components and clamping forces they may pro-
387
Fixturing Stiffness and Clamping Stability Use top surface
@ \1fflt-
Plain locator
1
Pt
LlLJ Use edge surface
2
a)
Use top surface
I
Shoulder locator
1
ij
"Iorkplece
mLOCo.tor
I
I Bcu;e
Plo. tE'
@ _et-pt
CD
Use shoulder
_~rkPlece
1
2
I
}Locotor
I Base 1-
plQte
@ 0
-1 Ct - pt
b)
Figure 21
Locators and their application modes: (a) plain locator; (b) shoulder locator; (c) round locating pin; (d) diamond (relieved) locating pin; (e) locator: (f) edge supports; (g) V-blocks.
Chapter II
388
Round locatinq pin
IJorkplece Bo.se plo.te
c)
Diaaond (Relieved) locatinq pin
d) Figure 21
Continued
Fixturing Stiffness and Clamping Stability
389
Lacator
~
~
I
\rIorkpiece
I I J.-Locotor ~
Bo.se Plo. te
e)
Edge supports
t) Figure 21
Continued
@ CC!?
Chapter J J
390
V blocks
Vblock
g) Figure 21
Continued
vide. The purpose of the extraction of the positions/directions of the clamping forces and locating responses is to catch the positions and orientations of the locators and clamps from a computer-aided modular-fixture design.
11.4.3
Modification of Fixture Component Database
Fixture component CAD databases provided by fixture companies usually contain two-dimensional (2-D) or three-dimensional (3-D) drawing files of each fixture component. The number of fixture components are limited in a fixture system. In order to obtain the information of fixture-workpiece contact positions and directions, special attributes are added to each locator and clamp in the fixture component database based on the analysis of all locators and clamps in modular-fixture systems. In Fig. 23, a contact vector is defined by two points in the fixture component coordinate system for each locating and clamping method. The first point on the locator (or clamp) surface is defined as the contact point, which specifies the position of the locating (clamping) force. The other point is defined inside the locator (clamp) so
Fixturing Stiffness and Clamping Stability
391
Adjustable screw clamp (horizontal)
W'orkplece
a) Figure 22 Clamps and their application modes: (a) adjustable screw clamp (horizontal); (b) screw edge clamps (angle); (c) tapped-heel clamps.
that the direction of the locating response (clamping force) can be determined by connecting the two points. When a specific CAD command is executed, the position of the two points will be automatically recorded to an output file. Therefore, when a fixture design is finished, the fixtureworkpiece contact points (positions) and directions of the clamping forces and locating responses are provided by listing all attributes into an external file.
11.4.4
Verification of Clamping Stability
In order to verify the clamping stability, equilibrium equations are first established where the friction forces are considered. To overcome the difficulty of solving the equations in 3-D where the directions and amplitudes of friction forces are hard to determine, all the forces are projected to 2-D planes, which may lead to a more conservative conclusion of clamping stability in fixture design.
392
Chapter 11
Screw edge clamps (angle)
\lorkplece
b)
Tapped-heel clamps
A
Clo.l"lp \.
I
t
I I
~
I
,. I !(
)
CL:
: I
I
W'orkPlefe .,
I I
j-\
I
Bo.se
pl~te
c) Figure 22
Continued
! I
I
393
Fixturing Stiffness and Clamping Stability
attribute value L811 attribute tag CT-PT X 24.4789 Y = 17.0462 Z = 2.0 attribute value L812 attribute tag DIRECTION X 24.4789 Y 17.0462 Z 0.0
----
I I I I
I
I
I I
CONT AC T POINT}
ATTRIBUTE DIRECTION
I
1..., __r J I
I
I
I
I
I
I
I
Figure 23
(a)
Attributes attached to the CAD model of fixture components.
Equilibrium Equations
Based on the information in the output file from the fixture CAD system, the equilibrium equations can be established and solved for the clamping stability verification by running an external program. Basically, the equilibrium equations include a force and moment equilibrium about the X, Y, and Z axes; that is,
(9)
In the clamping stability verification of fixture designs, it is assumed that the clamping forces are active and known input forces. The locating responses are passive and variables to be solved. If there is no friction considered, the above equations are solvable when the six-point locating principle is applied to the fixture design, where the six locating responses are equal to the number of equations. When one or more locating responses are identified as negative, the fixture design is considered unstable. However, the solution may not be valid in a real situation where the friction between
Chapter //
394
the workpiece and fixture components gives a positive support of the clamping stability.
(b)
Friction Force Discussion
The friction forces bring in some uncertainties in the calculation. The directions of friction forces need to be identified, which should be opposite to the trend of relative motion at the interface of workpiece and fixture components (locators and clamps). Depending on the external force, the amplitudes of the friction forces are in the range (10)
where Ft is the friction force, /J.. is the friction coefficient, and Fn is a normal force at the interface of the workpiece and fixture components. If the workpiece is assumed as a rigid body, a uniform friction acting factor is applied, which allows the friction forces to increase uniformly from o to the maximum value (f.LFn): (11 )
where ex is the friction acting factor, which may vary m a range from 0 to 1.
(c)
Conversion to 2-D Problems
Because of the uncertainty of friction forces, especially their directions, there are no theoretical solutions for Eq. (9). In this research, the 3-D stability problem is converted into three 2-D problems. All clamping forces and locating responses are projected into three orthogonal planes (the X- Y, Y-Z, and X-Z planes). For the clamping stability problem, if a fixture design is verified stable under the three 2-D cases, it is certainly stable in the 3-D situation. During the conversion, three locating responses are maintained in each of the 2-D models according to their effectiveness in the stability problem. Figure 24 shows an example of simplified 2-D models in the X- Y plane, where a locating response in the Y direction is omitted at the position between two other locating responses in the Y direction. Once the stability problem is simplified into 2-D problems, a recursive algorithm can be applied to provide solutions. The procedure of the algorithm is as follows:
395
Fixturing Stiffness and Clamping Stability Clamping Force C
IR
Workpiece
... Negative Locating Force Locator
Locating Force
Figure 24
The 2-D simplification of the stability analysis model.
Solve the equilibrium equations without friction effects If there is any negative locating response is identified (otherwise stable), let it be zero and determine a possible rotation center (e IR ) by considering the other two contacts Determine the directions of all friction forces which contribute to the resistance of the rotation Increase the acting factor of the friction forces from 0 to 1 and find the solution of the equilibrium equations Give a conclusion of clamping stability: stable if all solutions for the locating responses are non-negative for a feasible acting factor of friction forces (less than 1), otherwise unstable.
(d)
Discussion of Underlocating and Overlocating Problems
In practical fixture designs in the workshop, it is possible to have overlocating (using more than one point to restrict one degree of freedom) or underlocating (one or more degrees of freedom are not restricted). For example, four locators may be utilized under a large workpiece to restrict three degrees of freedom and provide an additional support. In this case, the above-mentioned equilibrium equations are still valid for clamping stability verification. When it is simplified to 2-D problems, the extra locating response is omitted according to its effectiveness for the stability. An equivalent six-point locating system is actually formed. In the case of underlocating, the equilibrium equation is omitted in the direction in which the degree of freedom is not restricted. The number of remaining equilibrium equations is equal to the number of variables (locating responses), which can be solved for clamping stability verification.
396
11.4.5
Chapter II
Implementation Example
Figure 25 shows a block diagram of the automated clamping stability verification system. Figure 26 shows two fixture configurations for the workpiece presented in Fig. 20, where (a) is the unstable situation and (b) is the improved design (a clamping angle is applied). Figure 27 shows the outputs with position/direction attributes for all locators and clamps used in the fixture design from the CAD system. The differences between these two designs can be identified through an examination and comparison of the attribute outputs. Based on the information in the output file, the clamping stability can be verified by applying the recursive algorithm. In the first design, the locating responses include a negative value even when the acting factor of friction forces is increased to I. Therefore, it is unstable. Figure 28 shows another example of modular-fixture designs, verified stable in clamping.
Fixture design modification
Figure 25
The clamping stability verification system.
397
Fixturing Stiffness and Clamping Stability 0
0
0
0
0
/;!t~o
0
0
0 0
0
°0 0
o
0
0
0
0
000
0
0
0 0
0
0
0
0
0
0
0
0
0
000
0
0
0
000° 000
0
0
0
0
0
0 00
:~" :
00 0
0
0
Y : 18.0462 Z ~ 3.375
attribute value L192 attribute tag DIRECTION 18.4789 19.0462 3.375
X
Y : Z = attribute value L191 attribute tag eT-PT
X = 24.4789 = 18.0462 Z : ).)75 attribute value L192 attribute tag DIRECTION X = 24.4789 " = 19.0462 Z ~ 3.375 attribute value C022 attr ibute tilq DIRECTION X ~ 21. 4691 Y ~ 8.65945 Z = 4.5 attribute value C021 attribute tag CT-PT X = 21.4691 Y ~ 11.0144 Z -, 4.5 attribute value L051 attribute tag eT-PT x 24.4789 Y
0
0
0
y
~
Z
=
13.0462 2.0
attribute value L052 attribute tag DIRECTION X Y
Figure 26
=
24.4789 1300462 0.0
attribute value L052 attribute tag DIRECTION X '" 24.4789 Y '" 17.0462
Z '" 0.0 attribute value L392 attribute tag DIRECTION
X = 17.4632 Y '" 15.0462 Z '" 0.0 attribute value L391 attribute tag CT-PT X - 17.4632 Y '" 15.0462 2.0
attribute value L382 attribute tag DIRECTION X = 16.4789
=
/ 0
oo~
o 00 000
attribute value LOS1 attribute tag CT-PT
Y
°0
O~"f1
X .. 24.4789 Y '" 17.0462 Z '" 2.0
Z '"
0° °
,/
0
attribute value L191 attribute tag CT-PT X 18.4789
o0 " 0"_,_
,.(~~
o
0
0
0
/':;-",
0
0
0 000
15.0462
Z '" 2.25 attribute value L381 attribute tag eT-PT X '" 17.2914 Y = 15.0462 Z = 2.25 attribute value C021 attribute tag CT-PT X 25.3289 Y = 15.9907 Z = 3.69454 attribute value C022 attribute tag DIRECTION
X
29.6039 Y 15.9907 Z '" 3.69454
An unstable fixture design and the attribute output.
/'
0 0
°.. 1I
Chapter 11
398
1°
)0
I
0 ()
o
o
o
:, ')C'
o
o
0
0
10
°
0
>': :~O 00
l
')
o~oOo o 0
o
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000
0 000
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0 0
L_ _ _ _---' attribute value L391 attribute tag CT-PT X .7.4632 Y = 15.0462 Z = 2.0 attribute value L392 attribute tag DIRECTION X 17.4632 Y = 15.0462 Z = 0.0 attribute value Ll91 attribute tag CT-PT X = 18.4789 Y = 18.0462 Z = 3.375 attribute value L192 attribute tag DIRECTION X 18.4789 Y ., \9.0462 Z = 3.375 attribute value L191 attribut~ tag CT-PT X 24.4789 Y = 18.0462 Z = ).375 attribute value L192 attribute tag DlRECTION X 24.4789 Y ~ 19.0462 Z=3.)75 attribute value C022 attribute tag DIRECTION X 21.4691 Y = 8.65945 Z = 4.5 attribute value C021 attribute tag CT-PT X 21.4691 Y 13.0344 Z - 4.5
Figure 27
attribute value L811 attribute tag CT-PT X = 24.4789 Y = 17.0462 Z = 2.0 attribute value LB12 attribute tag DIRECTION X = 24.4789 'i = 17.0462 Z = 0.0 attribute value L821 attribute tag CT-PT X '" 24.4789 Y '"' 15.0462 Z = 2.0 attribute value L822 attribute tag DIRECTION X =< 24.4789 Y =< 15.0462 Z = 0.0 attribute value COlI attribute tag CT-PT X 25.1787 Y = 16.0612 Z = 2.71674 attribute value C012 attribute tag DIRECTION X 28.4164 Y = 16.0612 Z
=
).0
attribute value LJ81 attribute tag CT-PT X 17.2914 Y = 15.0462 Z ~ 2.25 attribute value L)82 attribute tag DIRECTION X 16.4789 'i 15.0462 Z '" 2.25
The improved fixture design and the attribute output.
Fixturing Stiffness and Clamping Stability
Figure 28
399
Another example of a modular-fixture design.
REFERENCES Berry, D. C. (1982), "Application of CAD/CAM to Fixture Design," in First Biennial International Machine Tool Technolog.\' Conference, Chicago, IL, pp. 45-66. Chou, Y c., V. Chandru, and M. M. Barash (1989), "A Mathematical Approach to Automatic Design of Fixtures: Analysis and Synthesis," Journal (~f Engineering for Industry, Vol. Ill, pp. 299-306. Rong, Y, S. Li, and Y Bai (1994a), "Development of Flexible Fixturing Technique in Manufacturing Industry," in Fifth International Symposium on Robotics and Manufacturing, Maui, HI, pp. 661-666. Rong, Y, S. Wu, and T. Chu (1994b), "Automatic Verification of Clamping Stability in Computer-Aided Fixture Design," in ASME Computers in Engineering, Minneapolis, MN, pp. 421-426. Thompson, B. S., and M. V. Gandhi (1989), "Commentary on Flexible Fixturing," Applied Mechanics Reviel1.', Vol. 39, No. 9, pp. 1365- 1369. Trappey, 1. c., and C. R. Liu (] 989), "An Automatic Workholding Verification System," in 4th International Conference 011 the M(/Ill~f(/ctllrillg Science (/lId Technology (~f the Future, Stockholm, Sweden, pp. 23-34.
400
Chapter 11
Zhang. S. (1981). "Experimental Study on Fixturing Stiffness of Small-size-series Modular Fixture," M.S. Thesis, Tsinghua University, Beijing. Zhu, Y., and S. Zhang (1990), Modular Fixture Systems: Theory and Applications, Machinery Press, Beijing. Zhu, Y., S. Zhang, and Y. Rong (1993), "Experimental Study on Fixturing Stiffness of T-Slot-Based Modular Fixtures," in NAMRC XXI, Still water, OK, pp. 231-235.
12 Fast Interference-Checking Algorithlll for Autolllated Fixture-Design Verification
Fixtures are tooling devices used to locate, support, and hold workpieces during a manufacturing process. The major purpose of a computer-aided fixture design (CAFD) system is to provide a fixture design based on fixturing principle and workpiece information. Interference checking between the machining tool and fixture units, as well as between fixture units, is one of the important performances of fixture design. Once the fixture configuration design is generated with CAFD, interference checking should be employed. Interference checking is an important aspect of fixture-design verification. Although there is an interference-checking function provided in most CAD systems, it may require a significant amount of time because there may be many fixture components involved in a modular-fixture design. In this study, the interference between the workpiece and fixture components is conducted by applying the interference-checking function between solids within CAD systems, as only one workpiece is involved in the iteration process. Because the geometry of modular-fixture components can be simplified into simple shapes and their combinations, a rapid interference-checking algorithm is studied for detecting possible interference between the machining cutter and the fixture components.
12.1
INTRODUCTION
Computer-aided fixture design has been intensively studied in recent years. Once a fixture is designed by using CAFD, its performance needs to be evaluated. Fixture-design performance may include the locating accuracy for 401
402
Chapter 12
ensuring tolerance requirements of a product design, clamping and machining stability and fixturing stiffness to resist fixture component deformations, and tool-path interference (Trappey and Liu, 1990). In previous research of CAFD, a possible interference of the tool-path and fixture components was visually checked (Berry, 1982). It is obviously desired that the interference can be checked automatically. Several techniques may be applied to interference checking.
12.1.1
Related Research on Interference Checking
The detection of collision and interference between moving objects plays an important role and has been studied in the areas of computer graphics, motion simulation, autonomous coordinating planning :If multiple robots, and programming and control of the manufacturing system. Many algorithms have been developed for detecting collisions and interference between two objects represented by boundary representations (B-rep) (Herman, 1986; Canny, 1988; Esterling and Rosendale, 1987). Because the algorithms are realized by a successive intersection check between the surfaces of the objects, the calculation time and cost is directly proportional to a combined number of the faces. If it is a detection between moving objects, additional calculation time, proportional to the number of the vertices of the objects, is required for renewing their coordinates. Constructive solid geometry (CSG) can be used to represent a solid object by a set-theoretic Boolean expression of primitive objects. Many commercial packages of CAD provide the functions for detecting the interference between two solid objects. When the number of objects in a system is large (i.e., if there are many fixture components are involved in a fixture design), the computation time is also significant. In order to increase the efficiency of the collision detection between moving objects, several fast algorithms were studied by representing all objects in hierarchical models such as octrees, a sphere, and an octsphere (Noborio and Tanimoto, 1989; Ahuja and Nash, 1990; Sandberg, 1987; Yang et al., 1994). Swept volume represents the cumulative volume of occupancy of a solid moving in space. It could be applied to represent the moving cutter swept volume and, then, to check the interference between the cutter and fixture components where the Boolean intersection of the swept volume and the fixture component models is calculated. The analytical expressions of swept volumes, generated by a sphere and a cylinder, has been presented (Ganter and Uicker, 1986; Kieffer and Litvin, 1991; and Ling and Chase, 1996). The mathematical expression is complicated and may cause intensive computation (Wang and Wang, 1986).
Fast Interference-Checking Algorithm
403
Spatial representation is another method to represent a solid as a combination of variable orthorhombic cells (Ngoi and Whybrew, 1993), which has been applied to the problem of designing assemblies of T-slot modular fixtures (Ngoi et aI., 1997). It is suited to the shape of the modular-fixturing blocks, but not the fine geometric details of curved or angled surfaces.
12.1.2
Fixture-Design-Related Interference Checking
During machining processes, fixtures are applied to locate the workpiece relative to the cutting tools. Generally, there are four types of interference which may occur related to the fixture design: •
• •
Type A interference: interference between fixture components and the swept volume generated by the cutting tool, as shown in Fig. I Type B interference: interference between the workpiece and moving cutter during the machining process, as shown in Fig. 2 Type C interference: interference between the fixture components and workpiece, as shown in Fig. 3 Type D interference: interference among fixture components, also shown in Fig. 3, where two fixture units, a clamping unit, and a sidelocating unit are positioned on the same side of the workpiece with an insufficient distance; therefore, interference occurs.
Figure 1
Type A interference.
404
Figure 2
Chapter 12
Type B interference.
In fixture-design verification stage, only types A, C, and D interference are considered because no fixture components are involved in type B interference. Interference checking is an important topic in the fields of CAD/ CAM, robotics, and computer simulation or animation. Basically, the methods for interference checking can be divided into two categories: continuous time checking and discrete time checking (also called the step-and-step check approach). The basic problems in the interference checking include
Figure 3
Types C and D interference.
Fast Inteiference-Checking Algorithm
405
(1) the tool-path representation, (2) simplification of fixture component models, and (3) algorithms for interference detection. The basic requirements of an effective and efficient interference checking algorithm include the following: • •
•
Fast: Obviously, the interference checking process has too many iterations, which is time-consuming. Precise: Interference checking has to be as precise as the requirements of tolerance, otherwise the results of interference checking are not reliable. Detailed interference information: Interference location and amount are usually required to be retrievable in the verification stage.
Because standard components with relatively regular shapes are utilized in modular-fixture systems, the geometric models of the fixture components can be much simplified. Therefore, a new method is developed and implemented in this research. Because a CAFD system has been developed (Rong and Bai, 1997), the interference-checking implementation can be integrated with the CAFD system. The major functions developed in this research include the following;
•
12.2
To retrieve the fixture design created by the computer-aided fixturemodular design system, FIX-Des To simplify the fixture component models To generate a tool-path representation in 3-axis and 5-axis numerical control (NC) machining To detect interference between fixture components as well as the workpiece To detect interference between modular-fixture components and tool path To develop a method to check interference between fixture components To report detailed interference information
INTERFERENCE CHECKING BETWEEN FIXTURE COMPONENTS AND TOOL PATH
In order to simplify the algorithm of interference checking, the cutter and fixture components need to be mode led. The modular-fixture design is usually composed of a baseplate, several locators and clamps, as well as other supporting components. The modular-fixture components in commercial systems are usually relatively simple in geometry. Our study on the modular-
406
Chapter 12
fixture components shows that the modular-fixture component can be classified into three types: block type, cylinder type, or block-cylinder type, as shown in Fig. 4 (Rong and Bai, 1997). Some modular-fixture components are assemblies which may be complicated in geometry, but they can be always decomposed into the combination of block type and cylinder type. Even for some complex fixture components, the two basic type models can be used to represent their geometry approximately. Therefore, the modularfixture design can be geometrically represented by a number of blocks and cylinders which are placed in a specific space on the working coordinate systems, as shown in Fig. 5. It is well known that the moving cutter can be mode led as a cylinder whose axis is either perpendicular or parallel (for vertical or horizontal machining operations, respectively) to the machine table in 2.5- and 3-axis Ne machining operations. In this case, the fixture components can be simplified and represented in a two-dimensional (2-0) space. This method of object representation is similar to the method used by Brost and Goldberg (1996) in their fixture design where fixture components were decomposed into blocks or cylinders and projected into 2-D rectangles or circles. For each fixture component, a number of blocks or cylinders may be involved, also as may be several sets of 2-D geometrical contours with respect to certain height values, as shown in Fig. 6. For the purpose of interference checking, the cutter can be simplified as an axis segment if the fixture components boundaries are expanded by the amount of the cutter radius.
~ ~ C
.--.-.-.---.--.-
Figure 4
Some fixture components.
00
-
407
Fast Interference-Checking Algorithm
Figure 5
An example of a modular-fixturing system.
0
0
0
0
0
0
0
0
0
(',0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
lS?~:~~2:g~S)lJ
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1{31 o~
o o 0 0
~ffi\
0
0
0
0
0
0
o I,' ,--I
0
0
0
0
0
0
0
0
~~}~: ~
yJ, start angle aI, and end angle (X2' The only difference between them is that the new one has a larger radius, R + r. In other words, the old
410
Figure 8
Chapter 12
Expanding a line segment.
arc has been expanded along the radius direction with an offset r, which is the cutter radius. The start and end angles, a l and ab respectively, can be obtained by
(Xl
= arctan
Yl -Ye) (Xl Xl'
(3a)
(3b)
The new coordinates of the expanded arc with new start and end points [x;, y;] and [x~, y~] respectively can be calculated:
r
Figure 9
Expanding an arc.
411
Fast Interference-Checking Algorithm
(4a)
(4b)
12.2.2
Cutter Modeling and Tool-Path Representation for Three-Axis NC Machining
The expanding algorithm is fully based on the assumption that the cutter can be simplified as a cylinder and represented by the axis of the cylinder. Therefore, it is important to understand the geometrical relation between the cutter and the fixture in the algorithm. In 3-axis NC machining operations, the milling cutter is simplified as a cylinder with a radius. During the machining operation, the cutter (or the cylinder) is always perpendicular to the machine table where the baseplate holds the workpiece and all the fixture components. After projecting the cylinder on the baseplate, the cutter shape becomes a circle with height information. Once the projecting boundaries of fixture components are expanded, the circle is transformed to a dot. Therefore, the tool path becomes the path of the dot moving through on the plane, as shown in Fig. 10. An NC machine is a piece of manufacturing equipment that performs machining operations automatically to produce parts. It is controlled by a computer which reads in a set of motion commands and other control commands to direct the operation of the machine. This set of commands is called an NC program and is generated by the NC programmer with a CAM system
The cylinder which represents a cutter
The circle center point which represents the cutter in XY plane
Figure 10
Simplified cutter model.
412
Chapter 12
or other tool-path generation methods. In general, NC machine motions usually involve two motion control modes: the linear interpolation mode and the circular interpolation mode. With the linear interpolation mode, the cutter moves relative to the workpiece from point to point on a straight-line path. With the circular interpolation mode, the cutter moves from point to point along a circular arc path. By the simplification of the cutter model, the tool path can be modeled as a moving dot (the tool tip center) in a 3-D space. The tool path can be defined as F(x. y, z) = 0
It is true that the tool path is generated with respect to time, it is well represented by parametric equations
x
(5) 1.
Therefore.
= X(t)}
=
(6)
y y(t) z ::: z(t)
Furthermore, the moving cutter can be modeled as a 2-D continuous curve with certain height values after being projected on a X-Y plane. Usually, the height z, or value on the Z axis, is a variable of time 1; that is, F(x(t), y(t»
z
=0
(7)
= z(t)
If the linear interpolation is used to represent the tool path, the positions of the two end points can be obtained by a given interval .11. If the first point is [XI' YI' zd and, after .1t, the cutter is moved to the second point [X2' Y2' Z21. a 3-D line segment can be defined by
(8)
The first equation in Eq. (8) represents a 2-D line segment after a 3-D line segment is projected on the X- Y plane because a 3-D line segment projection is also a linear segment on a 2-D plane. The second equation provides the height information of the line segment. Circular arc model is also considered as a basic element of tool-path when a projected circular arc on XY plane is recognized. An arc segment can be defined as
Fast Inteiference-Checking Algorithm
413
(9)
where (xC' yJ is the center point of the arc, R is the radius of arc, and h is a constant representing the height of the arc.
12.2.3
Interference Detection Between Tool Path and Fixture Components
After the simplifications, both fixture component and cutter models are 2-D geometric elements with certain heights. Therefore, basically, the 3-D interference checking is degraded into a 2-D interference detection with respect to an additional height detection. Figure 11 shows a diagram of the interference detection procedure. It is important to know all the geometrical elements representing the fixture components and the tool path, which are directional with a start point and an end point, as shown in Fig. 12. Usually, there are three possible conditions of the tool-path element relative to the fixture components: (a) (b) (c)
the tool-path element, as a line segment u, is exterior to the fixture component contour (ABeD in Fig. 12). u is interior to ABeD. u intersects an edge of ABeD.
When the tool-path element is interior to (case b) or intersects with (case c) the fixture component contour, a possible collision presents if the height of the tool overlaps the fixture component. These three conditions can be identified by studying the following two basic problems: I.
2.
If a point is interior to the contour If and where the intersection occurs between a line segment (or arc) and a contour of the fixture component
Each tool-path element always has a starting point. Whether the point lies in the fixture component (Fe) contour must first be determined because, without this information, it is unknown if the next tool-path element lies in or out of the FE contour. Also, sometimes the tool path may be perpendicular to the baseplate, which means the line segment becomes a point after the tool-path element is projected on the baseplate plane. Therefore, an algorithm for the basic problem 1 is developed to detect if the starting point of a tool-path element is in an Fe contour, whereas another algorithm for the basic problem 2 is to identify the intersection between the tool-path element and the Fe contour.
Chapter 12
414
rr-----t--·_·,·-tl______....
...--....L.------.
"
........ _
• • 0-. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
f
"1
I
i
I
:
,
I
I I I i
\ ............................•_.....•....•.....) Height data
Interference data results (Fixture component ID t e - - - - t - - i tool-path ID interference position and volume)
:=~ , ......... _....................._..............
...
If the next Fe is NULL
If the next tool-path is NULL
Figure 11
I
Interference checking diagram.
;
4/5
Fast Interference-Checking Algorithm D
A
D
A
"" '"
./
'" \I
/ Ii'
\I
/
D
A
u
/\
.......
.......
B
C
B
....-
a. Figure 12
C
B
C
c.
b. Three intersection conditions.
(a)
Detection of a Point Within a Contour In fixture component models, all contours are simplified as closed-loop polygons: r is used to denote the given point representing the projected tool path and an arbitrary point P is used to define a ray from r through P, as shown in Fig. 13. P is rotated about r for 360 with a distance to ensure that P is not inside of the FC contour. Then, the intersection is detected for each edge of the contour against the ray where the geometrical intersection detection method can be applied and which will be discussed in a later section. After the detection, the number of intersection points is counted. If the total number of intersection points is n, then the r lies in the contour if n is odd number; on the other hand, r lies out of the contour if n is even number. There are two conditions before the conclusion whether or not r is interior is drawn. First, if the ray has a tangent intersection point with the contour, it cannot be added to the total number n. Second, it is possible that the ray has some intersection points which are exactly the end points of one or more edges of the contour. Each end point of an edge in the contour is a joint point between two consecutive edges. Theoretically, it is the two end points which lie on two edges. Consequently, such a point should be detected twice in the mathematical algorithm. In other words, we have to drop extra points from n. 0
(b)
Detection of Geometric Element Intersection
Because fixture components and tool-path elements are represented by line and arc segments, the interference detection becomes a detection of intersections of these segments. Furthermore, the interference location should be determined also. In order to detect the location of the interference, the direction of the intersection needs to be identified as the tool-path intersects in or out of the fixture component contour. Because the fixture component
416
Figure 13
Chapter 12
Detection of a point r within a contour.
is represented by a closed-loop contour, each element of the contour can be represented by a vector, v. Similarly, the tool-path element can be represented by a vector, u. When there is an intersection between the fixture component and the tool path, the cross-product of the vectors u and v can be used to identify the direction of the intersection. Mathematically, we have
C
=U x
V
U,
j
k
U2
u~
( lOa)
that is,
( lOb)
The vector directions of both the fixture component contour and tool path are defined as positive in the CCW direction. When the tool-path element runs into the fixture component contour, the cross-product of the two vectors gives a vector direction going into the paper «(8». When the toolpath element goes out of the fixture component contour, the cross-product of the two vectors gives a vector direction going out of the paper (8). In Fig. 14a, U represents a line segment of the tool path. ABCD is the fixture component contour where one contour element intersects the tool path, which is represented by v. When U is going into ABCD, the crossproduct of the vectors can be evaluated, C = U X v. The direction of C is (8). In this example, the result is straightforward because every geometrical element is a line segment which can be directly used as a vector. Similarly, in Fig. 14b, a circular arc or full circle can be also represented by a vector.
417
Fast Interference-Checking Algorithm D
A
c
B
a. Figure 14
b. Detection of a vector u in (@) and out «(0 YI), (x 2 , Y2)] and v = [(x~, y~), (x~, y~)]), the cross-product becomes (11 )
If C > 0, the machining tool path is moving out of the fixture component contour; otherwise, the tool path is going inside. This is under the assumption of interference, which is detected as follows: ].
Intersection detection of two line segments. In general, the intersection of two line segments can be determined by examining the line equations. When a line is defined by two points I x h YI] and [X2' Y2], as shown in Fig. 15a, the coordinates of a point on the line can be expressed as ( 12a) ( 12b)
where Tu is a coefficient with a value 0 :::; T" :::; 1. When T. is changed from 0 to 1, the point Ix", Ynl moves on the line segment from one end to the other. Thus, if Tn < 0 or T" > ], x" and Y.. will extend beyond the segment.
418
Chapter 12
~ (x"y,) /
(x.,y.)
(x\, YI)
(a)
(b) Figure 15
(a) A line segment; (b) intersection of two line segments.
If the intersection occurs between two lines and the second line is defined by points [X3, Y3] and [x 4 , Y.d, as shown in Fig. I5b, the intersection point becomes x"
= xj3
and
y"
= YI3
( 13a)
Then (l3b) ( 13c)
Finally, the solution of these simultaneous equations is T
= -(Xl
-
-(x 2
-
"
T _
-(Xl -
13 - -(Xl -
XI)(Y4 - Yl) XI)(Y4 - Y.l)
+ (y, + (Y2 -
YI)(X 4
-
X.l)
YI)(X 4 -
X,)
XI)(Y3 - YI) XI )(Y4 - y,)
+ +
YI)(X 3
-
XI)
(Yl - YI)(X 4
-
x.\)
(Y2 -
( 14a)
( 14b)
419
Fast Interference-Checking Algorithm
If the denominator of the expression defining Tu and T J3 is zero, the lines are parallel. Hence, they do not intersect. If a solution is found when 0 :::; To: :::; 1 and 0 :::; T J3 :::; 1, the segments intersect. Once To. and T J3 are obtained under the intersection condition, the intersection position can be figured out by solving Xu and y Intersection detection of a line segment and an arc. An arbitrary point on an arc with radius Rand center (xc, yJ can be defined as 0:'
2.
x'"
= Xc
YOI. = Ye
e + R sin e + R cos
(15a) ( 15b)
where the angular range of the arc is (e" e2 ), as shown in Fig. 16. If an intersection of the arc occurs with a line segment, we have ( 16a) ( 16b)
Rearranging the terms to cancel
Figure 16
Intersection of line and arc.
e,
420
Chapter 12
and sin 2e + cos 2 e = 1. This equation can be rewritten as a quadratic function, Ae
+
Bt
+
C
=0
(17)
where the constant A, B, and C are defined as A
= (X2
-
XI)2
+
B = 2(x , - xJ(x 2 C
= (XI
-
XJ2
+
(Y2 _ YI)2 -
XI)
(YI -
+
2(y, -
yJ2 -
yJ(yz -
YI)
R2
and ~ = B2 - 4AC. If ~ < 0, there is no intersection between the line and circle. If ~ = 0, the line is tangent to the circle. Then, one solution is obtained from the equation: B T =-" 2A
( ISa)
In this case, whether the intersection occurs can be determined in two general steps. First, the intersection point is checked laying on the line segment under the condition 0 < Ta < 1, which was discussed above. Otherwise, there is no intersection between the line segment and the arc. In the second step, when T et is known, e" can be calculated. If el < e" < e2 , the intersection occurs; otherwise, no intersection point exists on the line segment and arc. If a > 0, the line and the arc have two valid intersection points for which
T"
3.
=
vLi
B :!: 2A
( ISb)
Similarly, each of two points is checked by using the exactly same method discussed when a = O. If 0 ~ T" ~ 1, the intersection occurs and the intersection position can be identified. Intersection detection of two arcs. The equations of two arcs are defined as Arc 1 and Arc 2 •
421
Fast Interference-Checking Algorithm
Arc,:
Y = Yo
+ Ro cos . e} + Ro SIn e
e1< e < e, -
(l9a)
x = XI Y = YI
+ RI cos . e} + RI SIn e
e~ < e < e -
( 19b)
x = Xo
4
where the angle range of Arc, and Arc 2 is [8" 8 2 ] and [8:1. 8 4 ], respectively, as shown in Fig. 17. When e is canceled, these equations can also be written as (X -
XO)2
+
(Y -
Yo)2
= R~
(20a)
(x -
XI )2
+
(Y -
YI)2
= R~
(20b)
Combining these equations, we have (2x -
Xo -
xl)(-X O
+
XI)
+
(2y - Yo - YI)(-Yo
+
YI)
= Ri) -
R~
(21)
By rearranging the equation, it becomes X
=
~ -
R~
+
x~ -
2(xl -
Ar~CI_ __
Figure 17
Intersection of two arcs.
x~ xo)
+
y~ - yz,
YI -
Yo
XI -
Xo
- --- Y
(22a)
422
Chapter 12
If the constants are defined as ~~ - R~ + xi - x~ + yi - y~ A=-----------"-----"2(xl -
XO)
B = YI - Yo XI -
Xo
It becomes
x
=A
- By
(22b)
By substituting Eq. (22b) into Eq. (19a), it becomes (23)
Rearranging the equation, we have
The constants can be defined as
= (B 2 + D = [2B(A C
I)
- xo) + 2YoI
Then, we have the standard quadratic function Cy2
+
Oy
+
E
=0
(25)
Based on ~ = D~ - 4CE, the intersection of two circles can be identified in the following cases: (1) If ~ < 0, there is no intersection occurring between the two circles. (2) If ~ = 0, two circles are ta~ent to each other at the position (x', y'), where y' = (- 0 ::!: V ~)/2C and x can be obtained from Eq. (22b) using the information of point (x', y') and Eq. (19), the angle 8' of each arc corresponding to the point (x', y') can be computed. If the tangent point lies on Arc 1 in the condition if 8 1 < 8' < 8 2 , and also on Arc 2 in the condition H~ < 8' < 8-l, the intersection occurs between two arcs; otherwise, there is no intersection occurring. (3) If ~ > 0, there are two possible intersection points on the two circles. The procedure
423
Fast Inteiference-Checking Algorithm
of determining whether each point lies on both arcs is similar to the calculation when ~ = O. It can be seen that the algorithm discussed earlier is based on the assumption that Xl - Xo 0 in Eq. (22a). If this assumption is not true, another mathematical algorithm should be developed for the case, Xl - Xo = 0 or Xl = Xo (Fig. 18), Actually if XI = Xl = XO, Eq. (20) becomes
*
(x - xli + (Y - YO)2
+
(x - XI )2
= R~
(26a)
Ri
(26b)
(Y - YI)2 =
Then, Eq. (21) becomes (27)
Rearranging the equation, we have (28)
and X
= +- ~..0'()2
_
YI -2 Yo (~2 RI 2+2 -
2(YI - Yo)
(Xo, Yo) ~,,~~~,t I
...
......
I I f
~~~~~~~~~~
...... ...... ' ........
(x(,
......
yd
~.~---------I I I I I
I
Xo
Figure 18
=x(
Intersection of two arcs when
Xo
=
XI'
_
2
Yo
)+
X 0
(29)
424
Chapter 12
Therefore, the two possible intersection points are obtained, and if the possible points lie on two arcs, intersection points can be determined by using the same method presented above. The same problem may be present, as it is possible that Yt - Yo = 0. When Yt = Yo, the two arcs have the same center because, in this case, XI = Xo and YI = YD' Therefore, if two arcs have different radii, it is impossible to have any intersection points. But if their radii are the same, the two arcs are checked to determine if they overlap or partly overlap in the given angular range.
12.2.4
Discussion on Interference Checking in 5-Axis NC Machining
Because the cutter may not be perpendicular (or parallel) to the baseplate in 5-axis Ne machining, the simplified cutter can be considered as a union object of many spheres sitting along the cutter axes, as shown in Fig. 19a. The interference between the cutter and the bounding solids of fixture components can be treated as the interference at one or more of the aligned spheres with the bounding solids. The geometric characteristics of the spheres assure that if the bounding solids are expanded with the sphere radius, the interference between the sphere and the solids can be detected by checking the sphere center with the expanded solids. Therefore, the interference for the cutter against the bounding solids can be detected by determining if there is an interference between the cutter axis and the expanded fixture component models. In Fig. 19b, P,,(t) is a vector representing the position of the cutter tip end and A,,(t) represents the vector of the cutter axis, that is, P/t) AJt)
= xJt)i + yc(t)j + zc(t)k = cos[a(t)]i + cos[(3(t)]j +
(30a)
cos('y(t»k
(30b)
where t is a time index parameter, a, ~, and 'Y are the angles between the cutter axis and the x, y, and z axes, respectively. Therefore, the moving cutter axis can be modeled by an arbitrary point on the cutter axis: (31 )
where s is a parameter to specify the point along the cutter length, which is in the range rO, L], and L is the length of the cutter.
Fast Interference-Checking Algorithm
425
(a)
z
y
(b) Figure 19 (a) Cutter modeling in 5-axis NC machining; (b) geometrical representation of cutter.
Simply, the cutter position can be expressed as
yet, s)
= xc(t) + = yJt) +
s
z(t, s)
= Zc(t) +
s cos["y(t)]
x(t, s)
s cos[a(t)] cos[~(t)]
(32)
426
Chapter 12
Because the tool path is determined in NC programming, xcCt), yc(t), zc(t), a(t), ~(t), and 'Y(t) are known. Therefore, the interference checking can be performed by calculating the possible interference between the tool path specified by the cutter position and all the expanded bounding solids of the fixture components. The fixture component geometry has been simplified into blocks, cylinders, and their combinations, and they are projected into 2-D lines and arcs with height information. Once the tool path is represented by a moving point with parameters t and s, the method of the interference checking becomes the same as presented for the case of 3-axis NC machining, where an additional iteration of s is required.
12.3
INTERFERENCE CHECKING BETWEEN FIXTURE COMPONENTS
In our study, the automated fixture configuration design software is augmented with commercial CAD packages. Although there is an interference detection function in most CAD packages based on solid union operations, the interference detection could be slow in modular-fixture design verification, as there are many fixture units and components involved in a fixture design. Besides, for the purpose of automated fixture-design modification, the information is required about where, in which direction, and how much the interference appears, which is usually not provided by the standard CAD function. It is desired that the interference detection be performed during fixture-design generation. Because the modular-fixture components can be simplified into blocks, cylinder, and their combinations, these prismatic objects can be projected along the z axis into lines and arcs with height information. The interference detection between the fixture components becomes a detection of intersections of two contour loops in two dimensions, which is similar to the method used in interference detection between the tool path and the fixture components. It should be mentioned that there is no need to expand the fixture component boundaries in this case. Basically, the intersection between two loops should be identified. Once it occurs, the new loop presenting the intersection area should be given. In Fig. 20, a rectangular contour ABCD and a circular contour H represent two fixture components. The algorithm of interference checking between the cutter and the fixture components is quite general and can be applied to this case directly. If contour ABCD is considered as a sequence of the tool path starting at point A, the intersection of loop AB CD is calculated with loop H which represents a fixture component projection. The line segments EC
427
Fast Inteiference-Checking Algorithm
A
r-------~~------~
D
B
Figure 20
Intersection of two contours.
and FC are within loop H. Then, loop H is considered a set of tool paths, and an arc, FE, is detected within loop ABCD. The intersection area is bounded by line segments EC and FC, and arc FE. However, it should be noted that if the rectangle and circle do not intersect, it is possible that the rectangle is within the circle or the circle is within the rectangle, as shown in Fig. 21. The algorithm for finding if a point is within a loop can be applied to identify this situation, which has been described in the previous section. A point is selected on a loop (say, ABeD), and it is checked whether this point is within the other loop (e.g., H). If this is true, and no intersection is detected, loop AB CD is entirely inside H. Similarly, no intersection is detected if H is entirely inside ABCD. Figure 22 shows the procedure used for interference detection between fixture components.
A
D
B
Figure 21
Two conditions of two contours without intersection.
c
428
Chapter 12
,........•_...............................................\
I
I ...--+--__.------tl...
Fixture component geometric
L..-_ _ _...,..-_ _ _ _-.J
models
y
Height data with respect to fixture components ~+-----~--------------~y
.......................................................
j
"
Interference data results (Fixture components ID interference position and volume)
IfJ1h component
N
is NULL?
N
If ith component is NULL?
Figure 22
Flowchart of interference detection of fixture components.
429
Fast Interference-Checking Algorithm
12.4
ALGORITHM IMPROVEMENT DISCUSSION
The purpose of the rapid interference detection algorithm is to simplify the computation effort where the geometrical models of fixture components and tool path are greatly simplified. For a given tool-path element, actually it is not necessary to check the interference against all the fixture components in a modular-fixture configuration. If the fixture components far away from the tool path element are filtered out before the interference-checking calculation, the computational efficiency can be even improved. For the purpose of improving the efficiency, grids are defined on the baseplate plane, as shown in Fig. 23. Fixture components projected on the plane may be located in or cross one or more grids. To find out which grids are occupied by a specific fixture components is a simple task. For example, grids B2, B3, C2, and C3 are occupied by fixture components FEl and B3~ B4 is occupied by FE2. After a precomputation, a 2-D array is generated with respect to the grids on the baseplate. Each position of the array records if any fixture component is occupying the corresponding grid, as shown in Fig. 24. Probably, one position of the array has more than one fixture component, and many positions may not have any fixture component occupied. The interference detection needs to be performed only between fixture components which have overlapping occupations of the grids; this will lead to a significant reduction of computation effort. This method is especially suitable for the interference checking between the moving cutter and the fixture components because the cutter is always
2
3
4
A
B
B2
r-- ----
:1 FEl ~------~-----, 1 ,--
C
C2
B3 r-- ---- B4
: :
FE2
1 1
___, C3
D
Figure 23
l
, ______ ~
1
Grids defined on the baseplate.
5
430
Chapter 12 2
3
4
5
None
None
None
FE2
None
A
None
None
B
None
FEl
FEl FE2 C
None
FEl
FEl
None
None
D
None
None
None
None
None
Figure 24
The data structure corresponding to grids.
moving continuously. Therefore, the interference detection is only performed relevant to the fixture components within the grid through which the moving cutter passes. An example is illustrated in Fig. 25. A sequence of the tool path is defined as a, b, c, d, e, f, g, and h. The first few steps, a, b, c, and d, cross four grids, Cl, B I, A I, and A2. No fixture components are located
3
2
4
5
. ----- . - --------------.,.---------------r---------------r---------------"'------, . . I I f
A
.
:
• I ,
• • ,
b: ~
"'"
~:
I I
I
d 1
1
:
:
:. .................... .. ....... }......................... .. .. ............................... !-.............................. .. ........................... . . ,, , ,, ~
,
B
:
:
,
B2
:
B3
CEJ' B4 FE2
~ ........ ~..... ~ ..... ---... !~}............. :............................. .
C
: "
:
:
:
,"
~
,
e:
C2 ~l
~ ......... h .. -. ~ ...g-- ........ ,, ,, ,,
D
,
:,
C3:
+. --.- - --'
,, ,,, , ,
,,
-of -- -.. --- .... -- .............. .. ,, ,
, ,,,
I
l.. .. ______________ •!. __________ .. ____ !..I .. ______________ !.I ____ ... ____________________ .. _ .. __
Figure 25
An example under the improved method.
Fast Interference-Checking Algorithm
431
in those areas. Therefore, no calculation is necessary. The next step, e, crosses grids A2, B2, and C2. FE 1 can be found in grids B2 and C2. The interference detection algorithm is used on the tool-path element e and the fixture component FE 1. After processing step e, element f also lies in grid C2. Therefore, interference detection is performed between e and FE1, although there is no interference between them in this example. This method greatly improves the efficiency of the interference-checking algorithm, especially in 5-axis NC machining processes.
12.5
IMPLEMENTATION
An automated interference checking (AIC) system has been developed (Hu, 1998), which can be integrated with the automated modular-fixture configuration design system. Information derived from the current generated fixture-design layout is required to be an INPUT data set. The algorithm program, PROCESSOR, is applied to find which steps in tool motion along the tool-path are interfering with fixture components in the layout, as well as the interference between fixture components. After all interference checking is finished, the information is stored into an OUTPUT file, which can be read by the CAFD core program to accomplish further design modification. An overview of this system is illustrated in Fig. 26. The input information includes the fixture-design information and the machining-tool information. The first input is from a fixture component database generated in the CAFD system, including all the geometric information about the modular-fixture component. Second, the current fixture configuration design of modular fixtures is also required, which is generated by implementing the CAFD core program. The fixture-design file specifies the fixture components used in the design, as well as the positions and orientations of these components. Third, the radius of a selected cutter and a sequence of tool paths in the machining process can be obtained from NC programming or a CAM package for the geometrical computation purpose. In the processing stage, two major processing steps are used in this method: the contour expanding and interference calculation on preprocessed models. Each fixture component contour is projected to a 2-D X-Y plane and expanded by an offset equal to the cutter radius where the cutter is simplified as an axis and projected as a dot. After the expansion, the interference-checking algorithm is applied to detect possible interference in the fixture design. Finally, if any interference is identified during the interference checking between each tool-path element and any fixture component, the interference position and exact area is determined and reported, including which fixture
Chapter 12
432
rr------------:------------:--.:----------------------:--:--:--:--=--------:----:--:----:------:--.::--------------:----:--------:----:--------:--,~
:: INPUT
::
!! :: !l
""
" "
Fixture component database
Fixture configuration design
Machining cutter radius
",I" ""
!! :: :: "
"
""
Machining tool-path
" "
l
I "t."'.
''..
I,
"~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~4 ~:::::::::::::::::::::::::::::
: : PROCESSOR
-----------" :::::::::::::::::::::::::::::::::::::::: ----------.:
~r
"
" " " " " "
Fixture component expanding
"
" " " " " " " " " "
"
"" "
""
" " "
" " " " "
Expanded contour data
" " "
" " " "" " " " " " " " " " " " " " "
'.'. '.
'. '. '''... "
.'
"" "
Interference checking algorithm
" " ""
'. " " " " " "" "
"
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1
