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Automated modular fixture planning: Accuracy, clamping, and accessibility analyses
Robotics and Computer-Integrated Manufacturing 14 (1998) 17—26
Automated modular fixture planning: Accuracy, clamping, and accessibility analyses Y. Wu!, Y. Rong",*, W. Ma", S.R. LeClair# ! Department of Mechanical and Industrial Engineering, University of Illinois, Urbana, IL 61801, U.S.A. " Manufacturing Systems Program, Southern Illinois University, Carbondale, IL 62901, U.S.A. # Materials Directorate, Wright Laboratory, Wright-Patterson Air Force Base, OH 45433, U.S.A. Received 4 March 1997; accepted 26 August 1997
Abstract Automated fixture design has become a research focus in process planning and CAD/CAM integration. An automated fixture configuration design system has been developed where when fixturing surfaces and points are specified, modular fixture components can be automatically selected to generate fixture units and placed into position with satisfying assemble conditions. A geometric analysis has been conducted to determine the fixturing surfaces and points which provide feasible geometric constraints and assembly relationships with modular fixtures. This paper presents an analysis on fixturing accuracy, clamp planning, fixturing accessibility, and clamping stability. When a fixture planning is determined, the analysis results can be applied to verify the performance of the fixture design. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Computer-aided fixture design; CAD/CAM; Fixturing accuracy; Clamp planning; Accessibility; Fixturing stability
1. Introduction Fixturing is an important aspect of manufacturing processes. With the development of flexible manufacturing systems (FMS) and computer-integrated manufacturing systems (CIMS), fixturing has become one of the bottlenecks in implementation of FMS and CIMS. Flexible fixtures are developed to meet the increasing demand on rapid and flexible fixtures. Currently, modular fixtures are the most widely used flexible fixtures in industry. Computer-aided modular fixture design (CAMFD) is the research focus that is necessary to make manufacturing systems truly flexible. Some research has been conducted on CAMFD. Relatively less literature can be found on theoretical study of fixturing principles, including locating accuracy evaluation, clamp planning, and accessibility analysis. An analytical tool on kinematic modeling and characterization of workpiece fixturing was discussed where the condition was derived for a fixture layout to locate a given workpiece uniquely at a desired location [1]. Loading and
* Corresponding author. Tel.: 618453 7857; Fax: 618 453 7455; e-mail: [email protected]. 0736-5845/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved. PII: S0736-5845(97)00025-2
unloading problem was also studied preliminarily on side locating based on an object constraint reasoning for fixture design [2,3]. A model for constraint reasoning was presented for the synthesis of fixtures. However, the discussion was restricted to 3-2-1 locating situation with perpendicular plane surfaces. A geometric assembly analysis-based algorithm was presented for synthesizing 2-D modular fixtures for polygonal parts [4]. Some limitations and improvements on the geometric analysis of modular fixture design have been pointed out [5]. An automated fixture configuration design system has been developed where when fixturing surfaces and points are specified, modular fixture components are automatically selected and placed into position with satisfying assembly relationships of modular fixtures [6]. In this paper, several technical problems related to modular fixture design are discussed, including repeatable locating error analysis, clamp planning and analysis as well as accessibility analysis. A comprehensive fixture planning system is under development in which the fixture planning procedure and relationships of the above mentioned analyses will be presented. The basic assumption in this research is that the primary locating surface is perpendicular to the second and third locating surface. The second and third locating surfaces need not be
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Y. Wu et al. / Robotics and Computer-Integrated Manufacturing 14 (1998) 17—26
perpendicular as required in 3-2-1 locating scheme. Locating accuracy analysis in the latter situation has been presented in Ref. [7]. Although generally, workpiece geometry and fixture configuration may be complicated, it holds true that in most fixture designs, planar and cylindrical surfaces are selected as locating surfaces, especially with modular fixture applications [8]. It is also the fact that the primary locating surface is perpendicular to the secondary and tertiary locating surfaces. Therefore the bottom locating and side locating, as well as top clamping and side clamping, can be considered separately. In this research, only side locating and clamping are considered.
2. Locating accuracy analysis Locating accuracy of a fixture design is related to the repeatability of the workpiece position and orientation relative to the machining tool. The variations of workpiece dimensions and the locator positions are the major source of locating errors. As stated in Ref. [5], the position and orientation of a workpiece can be decided by using a group of equations. To facilitate the calculation of workpiece position and orientation in fixtures, the projected geometry of a workpiece on the primary locating plane is transformed by growing the part by the radius of the locators so that the locators can be treated as ideal points [4,5]. Then, the workpiece is assumed to be translated by (x, y) and rotated by h relative to the base plate, while maintaining contacts between the workpiece and locators. In this paper, the translation of x and y as well as rotation angle h are defined as the workpiece positional configuration. The variation of the workpiece positional configuration is regarded as locating errors. The previous geometric analysis is applicable for the workpiece with ideal geometry as well as perfect locator position. In real production, the workpieces in a batch are always made with a certain tolerance. The variation of the workpiece dimensions may be generated during manufacturing processes. Therefore, the workpiece geometry is expected to vary slightly one by one. Since the workpiece positional configuration is determined by the three locating edges (including line segments and arcs), it will vary slightly corresponding to the workpiece geometry variation. When a fixture is designed and constructed, the locator positions may have positional errors. It also introduces a variation on the workpiece positional configuration. Thus, locating accuracy needs to be studied to ensure the variations of the workpiece positional configuration in fixture within a certain range. In this paper, locating accuracy is represented by the difference between actual workpiece positional configuration and ideal workpiece positional configuration. Let the locating edges triplet be e , e , e which are 1 2 3 designed to contact with three points P , P , P on the 1 2 3
Fig. 1. Workpiece and base plate coordinate systems.
workpiece. The positional configuration of the workpiece is then specified by (x , y , h ) which denotes a relative 0 0 0 transformation of the workpiece position and orientation (i.e., the workpiece coordinate system) to the fixture base plate where the assembly relationship equations should be satisfied as described in Ref. [5]. Fig. 1 shows a sketch of workpiece and base plate coordinate systems (X O ½ and X O ½ ). 8 8 8 " " " When the workpiece geometry varies and/or the locator positions vary, the boundary edge equation parameters (e.g., a , b and c , i"1, 2, 3 for linear edges) will i i i vary within a certain scope. Therefore, x, y, h will have different values corresponding to different edge locating variance. Thus, x, y and h will be the functions of a , i b and c described as: i i x"x (a , a , a , b , b , b , c , c , c ), 1 2 3 1 2 3 1 2 3 y"y (a , a , a , b , b , b , c , c , c ), 1 2 3 1 2 3 1 2 3 h"h (a , a , a , b , b , b , c , c , c ). (1) 1 2 3 1 2 3 1 2 3 When the above equations are differentiated relative to the boundary edge equation parameters, the variation of the workpiece positional configuration, i.e., x, y and h, can be estimated by using 3 Lx 3 Lx 3 Lx x"x # + Da # + Db # + Dc , 0 i i i La Lb Lc i/1 i i/1 i i/1 i 3 Ly 3 Ly 3 Ly y"y # + Da # + Db # + Dc , 0 i i i La Lb Lc i/1 i i/1 i i/1 i 3 Lh 3 Lh 3 Lh Da # + Db # + Dc . (2) h"h # + i i i 0 Lb Lc La i/1 i i/1 i i/1 i In general, it is true that for any geometry, such as arc or curve edges, the workpiece positional configuration error can be analyzed by using the set of equations as shown above. To show the solution procedure, a simple example is provided below (Fig. 2). If assume only the first locating edge varies by Dc , the first locating edge 1 equation will be changed to a x#b y#c #Dc "0. 1 1 1 1
(3)
Y. Wu et al. / Robotics and Computer-Integrated Manufacturing 14 (1998) 17—26
19
The three equations can be re-written as
C
A
B (p !x )B #(q !y )A 1 1 0 1 1 0 1 B (p !x )B #(q !y )A 2 2 2 0 2 21 0 2 A B (p !x )B #(q !y )A 3 3 3 0 3 3 0 3 A
1
CD 1
" 0
dx x"x # Dc , 0 dc 1 1 dy Dc , y"y # 1 0 dc 1 dh Dc . (4) h"h # 1 0 dc 1 When the assembly relationship condition is considered, the line equations can be described as [5]. a [(p !x) cos h#(q !y) sin h] i i i #b [(q !y) cos h!(p !x) sin h]#c "0, (5) i i i i where (p , q ) are the locating hole position coordinates i i on the fixture base plate. The differentiation relative to c becomes 1 dh dx a !(p !x ) sin h ! cos h i i 0 0 dc 0 dc 1 1 dh dy #(q !y ) cos h # sin h i 0 0 dc 0 dc 1 1 dh dy #b !(q !y ) sin h ! cos h i i 0 0 dc 0 dc 1 1 dh dx !(p !x ) cos h # sin h #d "0, (6) i 0 0 dc 0 dc i1 1 1 where i"1, 2, 3 and
C
D
C
D
G
1, d " i1 0,
(7)
where A "!a cos h#b sin h, B "!a sin h!b cos h, i i i i i i and i"1, 2, 3. From the three equations, the first-order derivative at (x , y , h ) dx/dc , dy/dc , and dh/dc can be calculated. 0 0 0 1 1 1 When the determinant of the matrix in Eq. (7) is not equal to 0, there is one unique solution for the three derivatives. Therefore, x, y, and h which defines the workpiece positional configuration can be estimated using the above equation. However, if the determinant equals to zero, there will be no unique solution for the equation set, which is not a valid location of the workpiece. Below, a numerical example is given on locating repeatability analysis. As shown in Fig. 3, one simple triangular workpiece is shown with three locating points defined. For such specific situation, it can be found that x "!120, y "!120, h "0, 0 0 0 when pitch T"30, and a "1, b "1, c "240; a "1, 1 1 1 2 a "0, b "1, c "0, p "0, 3 3 3 1 p "!120, q "0; p "!60, 2 2 3
b "0, c "0; 2 2 q "0; 1 q "!120. 3
Following the above solution procedure, we have
C
D
dx dh a ! #(q !y ) 1 1 0 dc dc 1 1
C
D
dy dh #b ! !(p !x ) #1"0, 1 1 0 dc dc 1 1
C
D
dx dh a ! #(q !y ) 2 2 0 dc dc 1 1
C
D
dy dh #b ! !(p !x ) "0, 2 2 0 dc dc 1 1
C
D
dx dh a ! #(q !y ) 3 3 0 dc dc 1 1
i"1, iO1.
CD
0
Fig. 2. An example of locating edge position variation.
All other parameters remain constant. Therefore, x, y, h can be conceived as the functions of c as 1 x"x(c ), y"y(c ) and h"h(c ). 1 1 1 The workpiece positional configuration with the variation can be estimated by
,
D
dx dc 1 dy dc 1 dh dc 1
#b
C
3
D
dy dh ! !(p !x ) "0. 3 0 dc dc 1 1
(8)
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Y. Wu et al. / Robotics and Computer-Integrated Manufacturing 14 (1998) 17—26
Fig. 3. An example for locating repeatability analysis.
With the specifications in this example, the equation set is simplified to dx dy ! ! #1"0, dc dc 1 1 dh dx ! #120 "0, dc dc 1 1 dy dh ! !60 "0. dc dc 1 1 The solution becomes
(9)
dx "2, dc 1 dy "!1, dc 1 dh 1 " . dc 60 1 In this way, the sensitivity of variance Dc can be 1 evaluated. Although this is a simple example, it illustrates the procedure of the fixturing accuracy analysis which verifies the fixture design results. 3. Clamp planning When locating positions are determined and workpiece positional configuration is calculated in locating planning, clamp positions need to be selected and verified
to secure the workpiece positional configuration. In this section, a step-by-step algorithm is developed to find the possible clamp position. Planar object constraint analysis is performed to evaluate the clamping stability for a given clamping edge. Generally, the possible clamping points can be found by following the three steps: (A) All the possible clamping edges are enumerated as clamping edge candidates after discarding the selected locating edges (or portions of the edges selected for placing locators) and considering the machining envelope. When one clamping edge candidate intersects with the machining envelope, the intersection should be cut off from the clamping edge. (B) All the clamping edge candidates are tested using clamping constraint analysis to find the possible clamping points set corresponding to the locating plans. This step is the major portion of the algorithm and will be discussed in more details. (C) Geometric analysis for modular fixture assembly is finally performed to get possible clamping points. The position of the workpiece has been fully constrained in locating planning. When a clamp unit is designed for one clamping edge, the discrete assembly positions of the clamp are enumerated by the requirement of modular fixture assembly relationships, as described in reference [5]. The actual clamping point on the workpiece will then be calculated. If it falls within the pre-calculated clamping points set, it is a reasonable clamping design. Otherwise, it needs to be re-designed.
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Fig. 4. 2-D object constraint anyalysis [3].
The problem of restraining the planar motion of twodimensional (2-D) object was analyzed [9]. As shown in Fig. 4a, when an object is restrained by a point contact A, it is free to rotate in both counter clockwise (CCW) or clockwise (CW) directions. Let the line perpendicular to the contact surface be L. Line L divides the 2-D space into two regions: “a” and “b”. Any point in region “a” can be the center of instantaneous CCW rotations, and any point in region “b” can be the center of CW rotations. The approach can be also implemented into multiple-point constraints. In Fig. 4b, the 2-D space is divided into six regions by the contacting lines in 3-2-1 locating scheme [3]. Points in region “b3” cannot be the center of rotation, as CCW and CW rotation are restrained by points C and B, respectively. Similarly, points in “b1”, “b2” and “a2” cannot be the center of rotation. The regions of rotation centers are reduced to “a1” and “b4”. Therefore, the instant rotational center (IRC), either CCW or CW, may only fall into the labeled regions “a1” and “b4”, which are referred as CCW IRC and CW IRC regions. The main purpose of clamping is to constrain possible rotations of the workpiece from the locating position. Therefore, the clamping position should be selected such that the CCW IRC and CW IRC regions can be eliminated. The clamping edges can be any geometric entity, such as arc or other curves. Only the normals of the locating edges on the locating points are considered. The position of the clamp should completely reduce the IRC regions to the non-IRC regions, or null regions, thus eliminate rotational freedom of the object. In Ref. [3], only simple workpiece which employs 2-1 side locating principle for 2-D situation is considered. In this research, it is expanded to complex workpiece geometry with linear and arc side locating surfaces which are not necessary to be perpendicular to each other. An analysis procedure is developed to find the possible clamping points satisfying clamping constraint conditions. Assume the workpiece is located by three locators contacting three locating edges e , e and e at P , P , 1 2 3 1 2 and P , respectively. In order to constrain the three 3 degrees of freedom (DOF), e , e and e cannot be all 1 2 3
parallel. Therefore, the contacting force directions at e , 1 e and e , say n , n , n , cannot be all parallel. In general, 2 3 1 2 3 it is assumed that n and n are not parallel, and intersect 1 2 at point O as shown in Fig. 5a. The 2-D space is divided into 4 parts by n and n . From the above approach, 1 2 region B and D are the null regions, A is the CCW IRC region, and C is the CW IRC region. When the third contacting force is applied, the situation is more complicated if considering the different possible directions of the third one. Generally, three cases are discussed below. 1. n is within !n and !n . 3 1 2 In this situation, n divides the regions A and C into 3 four regions and form a new region E. E is the triangle enclosed by n , n , and n , which could be degenerated 1 2 3 into a point if n pass O. It can be found that both 3 A and C regions become the null regions. Region E remains as an IRC with the same rotational direction as A or C (CW in Fig. 5b). 2. n is between n and !n , or n and !n . 3 1 2 2 1 In this situation, both A and C regions are divided into two smaller regions by n (Fig. 5c). One of them 3 becomes the null region and the other will still be the IRC with the same direction as the original region. The IRC regions are enclosed by either two lines or three lines, and may stretch to the infinity. 3. n is between n and n . 3 1 2 In this case, n divides either A or C, say C, into two 3 regions C and E. E is the enclosed triangle by n , n , 1 2 n and becomes the null region. A and C remain the 3 IRC regions, as shown in Fig. 5d. It should be noted that the main purpose of clamping is to constrain possible rotations of workpiece from the locating position, i.e. erase all the possible IRC region. In the above analysis, the IRC regions have been discussed for different situations of contacting directions. All the possible clamping edges have been enumerated in step A for the clamping constraint analysis. The procedure of finding the possible clamping point set with a specific clamping edge is shown below and it is assumed that only one clamp is needed in the fixture configuration.
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Fig. 5. General 2-D constraint analysis.
Step 1: Consider one available clamping edge from the clamping edge candidates. Get its line equation, end points, say AB, and contact force direction (internal normal direction). Step 2: Consider one of the IRC regions. Project the IRC region onto the line segment of the clamping edge. One of its ends may lie in the infinity because the IRC region may stretch to the infinity. Keep record of the end points of the projected line segment, say CD, and its rotational type. Step 3: Find the portion of the projected line segment which falls within the original clamping edge. Let AC and DB be the remaining parts after the intersection of AB and CD has been cut off. Step 4: If CD is within a CW IRC region, the portion AC will be the suitable clamping region and if CD is within a CCW IRC region, the BD should be selected. Step 5: If all the IRC regions have been considered, the selected portion of the clamping edge is returned as the
final possible clamping point set. If there is another IRC region to be processed, let the selected portion of the clamping edge replace the original clamping edge and go to step 1. Step 6: Once the clamping point set is obtained, the fixture configuration design module can be used to generate clamping units with satisfactions of modular fixture assembly relationships [6]. One example on the side clamp planning is given as shown in the Fig. 6. The three side locating points are shown in the figure as well as the IRC regions. Following the above procedure, a derived feasible area (LC) is derived on the clamping edge which can eliminate the two IRCs at the same time. The clamping algorithm is based on the assumption that only one clamp is needed. However, for some workpieces, one clamp may not be sufficient to eliminate all the IRC regions. For example, in Fig. 7a, it is not possible to use one clamp to eliminate the entire IRC region
Y. Wu et al. / Robotics and Computer-Integrated Manufacturing 14 (1998) 17—26
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Fig. 6. One example for side clamping planning.
for the triangular workpiece with three locating points shown in the figure. Two or even more clamps should be used to fully erase the IRC regions. Therefore, additional discussions are necessary. f The condition for using only one side clamp is that such an available clamping edge E exists which satis# fies that the projection of CCW and CW regions are separated spatially, and the CW region projection is on the left side (and/or CCW region projection is on the right side) of the selected clamping point, if looking in the direction of clamp acting direction or the internal normal direction of the selected edge. f If it is found that one clamp is not enough to constrain the possible rotation, two clamps may be used. The design can be carried out by using each clamp to eliminate one IRC region if there are two IRC regions. If only one IRC exists, the first clamp should decrease the IRC region as great as possible and the second one should erase the rest of the IRC region where the clamping sequence needs to be considered carefully. Fig. 7b and Fig. 8 shows examples of using two clamps for a total constraint. It should be noted that the friction effect is not considered in the clamp planning. This is because the friction force effect is always positive on the clamping stability since the friction forces are against any possible motions between the workpiece and fixture components. When the clamp planning is conducted without considering the friction effect, the result is more conservative.
4. Discussion on fixturing accessibility Fixturing accessibility is an important aspect in selecting fixturing (especially locating) surfaces and points. In fixture planning, two types of accessibility should be considered. The first one is the reachability of an individual workpiece surface, which is an important measure in locating and clamping surface selection. The second one is the easiness of loading and unloading the workpiece into a fixture, which refers to the configuration of three locating points on workpiece surfaces. In this section, the loading/unloading accessibility is discussed. It is assumed that clamps have been removed from the fixture configuration. Thus, only three locators are presented in the fixture when the workpiece is loaded or unloaded. There are three kinds of IRC regions as discussed in Section 3. In case A, only one CW or CCW rotational triangle exists and all other regions are null regions. The loading process can be conceived as the workpiece is first placed in the vicinity of the final position, then rotated to contact with the three locators for a final position (Fig. 5b). The rotational direction to make the contact is on the opposite direction of the CW or CCW rotation indicated in the triangular region. The unloading process can be discussed, similar to the loading process. In case B, two IRC regions are found. In this situation, the workpiece is loaded easier into the fixture than the situation in case A. It allows the workpiece to move linearly in one direction (Fig. 5c) to contact with one of the locators, and rotate slightly to contact with the other two locators.
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Y. Wu et al. / Robotics and Computer-Integrated Manufacturing 14 (1998) 17—26
In case C, the workpiece has the best accessibility since it has a wider open scope than the other two. Generally, the workpiece may move in two perpendicular directions to contact with two locators, and finally rotate to contact with the last locator. Usually, the larger the IRC region is, the better the accessibility of the fixture configuration will be. The standard 3-2-1 locating scheme with three perpendicular plane surfaces is an indication of good fixturing accessibility, which should be considered with a priority in locating surface selection. It can be further simplified to consider the directions of the contact forces instead. Assume that two contact forces f and f intersect at 1 2 point O and line ¸ divides the angle between f and 1 f equally. Then the third contact force f maintains an 2 3 angle h with ¸. The accessibility can be generally evaluated as the larger h is, the worse the accessibility is. Fig. 9 is a graphic sketch of the three force vectors. The above discussion only provides one criterion to evaluate the loading/unloading accessibility of the fixture. A complete accessibility analysis could be very complicated that involves more technical problems such as the geometric complexity of the workpiece and reasoning of geometric entities in a CAD model of the workpiece. 5. Clamping stability evaluation
Fig. 7. (a) One workpiece not possible to use one clamp. (b) Using two clamps to eliminate one IRC.
When a modular fixture design is conducted by the computers, the equilibrium between clamping forces and locating responses needs to be evaluated for a reliable locating. The clamping forces can be regarded as active and known input forces. The locating responses are passive forces and need to be solved. When all of the locating
Fig. 8. Using two clamps to eliminate two IRCs.
Y. Wu et al. / Robotics and Computer-Integrated Manufacturing 14 (1998) 17—26
25
Fig. 9. Accessibility analysis. Fig. 10. Clamping stability analysis.
responses are positive in the normal direction, the fixture design is claimed to be clamping stable. Otherwise, the locating and clamping positions needs to be redesigned [10]. In clamping constraint analysis, friction forces are not considered. Generally, frictional forces are presented in fixture performance. However, it is very difficult to solve the locating responses if frictional forces are involved, especially in the 3-D space. Many papers in this area do not consider the frictional forces. In this research, a 2-D analysis approach is applied to calculate the locating responses, which is similar to the method presented in reference [10]. As shown in Fig. 10, basically the equilibrium equations include the force equilibrium in X, ½ directions and a moment equation about Z-axis. RF "0 RF "0 and RM "0. (10) x y z Assume the workpiece has a tendency of rotating in CCW direction, the frictional force at each locating and clamping point is resisting the rotational tendency. Thus, the directions of the frictional forces can be determined as shown in Fig. 10. The magnitudes of the frictional forces should also satisfy f )kN , (11) i i where k is the coefficient of friction, and N is the normal i force at position i. There are more than three unknown variables: N , N , 1 2 N , and all the friction forces in the three equations. If the 3 workpiece is assumed to be a rigid body, one assumption can be made: frictional forces will increase at the same rate at all locating and clamping points, or the frictional forces at all locating points are f "kkN , i i where k is an acting factor between 0 and 1.
(12)
Then there are four unknowns left: N , N , N , and k. 1 2 3 To solve the equations, the coefficient k is increased from 0 to 1 with small increments. When k is increased to a certain value, Eq. (10) may become true with zeros in the right sides. The solved locating responses at this specific k value within the range of 0—1 are the actual normal locating responses. Frictional forces at all points can be also derived. If the solution cannot be found within the range of k"0—1, that means an invalid fixture design which is not stable when the clamping force is applied. Numerical examples can be found in Ref. [10]. It should be mentioned that this calculation only provides a conservative, static evaluation on clamping stability. The friction forces in the primary locating surface and top clamping surfaces as well as the machining force are not included in the calculation.
6. Summary This paper presents analytical discussions of fixturing accuracy, clamping planning, and fixturing accessibility. Together with geometric analysis, these analyses may provide a scientific foundation of automated fixture planning. Although fixture design is a complex task and usually it needs to involve human expertise, applying computer technology to generate and verify feasible solutions with alternatives is possible and greatly beneficial, especially in FMS and CIMS. The necessary conditions of fixture planning can be identified in the analyses, which may make the automated fixture design become possible and applicable in production.
Acknowledgements This research is partially funded by National Science Foundation (NSF), Air Force Office of Scientific
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Research (AFOSR), and Ingersoll Milling Machine Company. Supports from Machine Tool Agile Manufacturing Research Institute (MT-AMRI), Technologies Enabling Agile Manufacturing (TEAM) program of Department of Energy, TechnoSoft Inc., Bluco Corporation, and Qo-Co Companies are appreciated. Technical discussions within the fixturing research group at Southern Illinois University at Carbondale are also acknowledged. References [1] Asada H, By A. Kinematic analysis of workpart fixturing for flexible assembly with automatically reconfiguration fixtures. IEEE Trans Robotics Autom 1985;RA-1(2):86—93. [2] Chou YC, Chandru V, Barash MM. A mathematical approach to automatic configuration of machining fixtures: analysis and synthesis. ASME Trans Engng. Industry 1989;111:299—306. [3] Chou YC, Barash MM. Automatic configuration of machining fixtures: object constraint reasoning. Manufacturing International, Atlanta, GA. 1990.
[4] Brost RC, Goldberg KY. A complete algorithm for synthesizing modular fixtures for polygonal parts. IEEE Trans Robots Autom 1996;12(1):31—46. [5] Wu Y, Rong Y, Ma W, LeClair SR, Automated modular fixture planning: geometric analysis. Robotics Comp. Integrated Manufacturing 1998;14:1—15. [6] Rong Y, Bai Y. Automated generation of modular fixture configuration design. ASME Trans. Manufacturing Sci. Engng 1997;119:208—19. [7] Rong Y, Li W, Bai Y, Locating error analysis for fixture design and verification. ASME Computers in Engineering, Boston, MA, 17—21 Sept. 1995; 825—32. [8] Rong Y, Zhu J, Li S, Fixturing feature analysis for computeraided fixture design. In: Proceeding of ASME International Mechanical Engineering Congress and Exposition, PEDVol. 64, New Orleans, LA, 28 November—3 December 1993; 267—271. [9] Reuleaux F, Kinematics of machinery, London. Macmillan, 1963. [10] Rong Y, Wu S, Chu T. Automatic verification of clamping stability in computer-aided fixture design In: Advances in Manufacturing Systems: Design, Modeling and Analysis, Amsterdam: Elsevier, 1994; 353—58.
Automated modular fixture planning: Accuracy, clamping, and accessibility analyses Y. Wu!, Y. Rong",*, W. Ma", S.R. LeClair# ! Department of Mechanical and Industrial Engineering, University of Illinois, Urbana, IL 61801, U.S.A. " Manufacturing Systems Program, Southern Illinois University, Carbondale, IL 62901, U.S.A. # Materials Directorate, Wright Laboratory, Wright-Patterson Air Force Base, OH 45433, U.S.A. Received 4 March 1997; accepted 26 August 1997
Abstract Automated fixture design has become a research focus in process planning and CAD/CAM integration. An automated fixture configuration design system has been developed where when fixturing surfaces and points are specified, modular fixture components can be automatically selected to generate fixture units and placed into position with satisfying assemble conditions. A geometric analysis has been conducted to determine the fixturing surfaces and points which provide feasible geometric constraints and assembly relationships with modular fixtures. This paper presents an analysis on fixturing accuracy, clamp planning, fixturing accessibility, and clamping stability. When a fixture planning is determined, the analysis results can be applied to verify the performance of the fixture design. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Computer-aided fixture design; CAD/CAM; Fixturing accuracy; Clamp planning; Accessibility; Fixturing stability
1. Introduction Fixturing is an important aspect of manufacturing processes. With the development of flexible manufacturing systems (FMS) and computer-integrated manufacturing systems (CIMS), fixturing has become one of the bottlenecks in implementation of FMS and CIMS. Flexible fixtures are developed to meet the increasing demand on rapid and flexible fixtures. Currently, modular fixtures are the most widely used flexible fixtures in industry. Computer-aided modular fixture design (CAMFD) is the research focus that is necessary to make manufacturing systems truly flexible. Some research has been conducted on CAMFD. Relatively less literature can be found on theoretical study of fixturing principles, including locating accuracy evaluation, clamp planning, and accessibility analysis. An analytical tool on kinematic modeling and characterization of workpiece fixturing was discussed where the condition was derived for a fixture layout to locate a given workpiece uniquely at a desired location [1]. Loading and
* Corresponding author. Tel.: 618453 7857; Fax: 618 453 7455; e-mail: [email protected]. 0736-5845/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved. PII: S0736-5845(97)00025-2
unloading problem was also studied preliminarily on side locating based on an object constraint reasoning for fixture design [2,3]. A model for constraint reasoning was presented for the synthesis of fixtures. However, the discussion was restricted to 3-2-1 locating situation with perpendicular plane surfaces. A geometric assembly analysis-based algorithm was presented for synthesizing 2-D modular fixtures for polygonal parts [4]. Some limitations and improvements on the geometric analysis of modular fixture design have been pointed out [5]. An automated fixture configuration design system has been developed where when fixturing surfaces and points are specified, modular fixture components are automatically selected and placed into position with satisfying assembly relationships of modular fixtures [6]. In this paper, several technical problems related to modular fixture design are discussed, including repeatable locating error analysis, clamp planning and analysis as well as accessibility analysis. A comprehensive fixture planning system is under development in which the fixture planning procedure and relationships of the above mentioned analyses will be presented. The basic assumption in this research is that the primary locating surface is perpendicular to the second and third locating surface. The second and third locating surfaces need not be
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Y. Wu et al. / Robotics and Computer-Integrated Manufacturing 14 (1998) 17—26
perpendicular as required in 3-2-1 locating scheme. Locating accuracy analysis in the latter situation has been presented in Ref. [7]. Although generally, workpiece geometry and fixture configuration may be complicated, it holds true that in most fixture designs, planar and cylindrical surfaces are selected as locating surfaces, especially with modular fixture applications [8]. It is also the fact that the primary locating surface is perpendicular to the secondary and tertiary locating surfaces. Therefore the bottom locating and side locating, as well as top clamping and side clamping, can be considered separately. In this research, only side locating and clamping are considered.
2. Locating accuracy analysis Locating accuracy of a fixture design is related to the repeatability of the workpiece position and orientation relative to the machining tool. The variations of workpiece dimensions and the locator positions are the major source of locating errors. As stated in Ref. [5], the position and orientation of a workpiece can be decided by using a group of equations. To facilitate the calculation of workpiece position and orientation in fixtures, the projected geometry of a workpiece on the primary locating plane is transformed by growing the part by the radius of the locators so that the locators can be treated as ideal points [4,5]. Then, the workpiece is assumed to be translated by (x, y) and rotated by h relative to the base plate, while maintaining contacts between the workpiece and locators. In this paper, the translation of x and y as well as rotation angle h are defined as the workpiece positional configuration. The variation of the workpiece positional configuration is regarded as locating errors. The previous geometric analysis is applicable for the workpiece with ideal geometry as well as perfect locator position. In real production, the workpieces in a batch are always made with a certain tolerance. The variation of the workpiece dimensions may be generated during manufacturing processes. Therefore, the workpiece geometry is expected to vary slightly one by one. Since the workpiece positional configuration is determined by the three locating edges (including line segments and arcs), it will vary slightly corresponding to the workpiece geometry variation. When a fixture is designed and constructed, the locator positions may have positional errors. It also introduces a variation on the workpiece positional configuration. Thus, locating accuracy needs to be studied to ensure the variations of the workpiece positional configuration in fixture within a certain range. In this paper, locating accuracy is represented by the difference between actual workpiece positional configuration and ideal workpiece positional configuration. Let the locating edges triplet be e , e , e which are 1 2 3 designed to contact with three points P , P , P on the 1 2 3
Fig. 1. Workpiece and base plate coordinate systems.
workpiece. The positional configuration of the workpiece is then specified by (x , y , h ) which denotes a relative 0 0 0 transformation of the workpiece position and orientation (i.e., the workpiece coordinate system) to the fixture base plate where the assembly relationship equations should be satisfied as described in Ref. [5]. Fig. 1 shows a sketch of workpiece and base plate coordinate systems (X O ½ and X O ½ ). 8 8 8 " " " When the workpiece geometry varies and/or the locator positions vary, the boundary edge equation parameters (e.g., a , b and c , i"1, 2, 3 for linear edges) will i i i vary within a certain scope. Therefore, x, y, h will have different values corresponding to different edge locating variance. Thus, x, y and h will be the functions of a , i b and c described as: i i x"x (a , a , a , b , b , b , c , c , c ), 1 2 3 1 2 3 1 2 3 y"y (a , a , a , b , b , b , c , c , c ), 1 2 3 1 2 3 1 2 3 h"h (a , a , a , b , b , b , c , c , c ). (1) 1 2 3 1 2 3 1 2 3 When the above equations are differentiated relative to the boundary edge equation parameters, the variation of the workpiece positional configuration, i.e., x, y and h, can be estimated by using 3 Lx 3 Lx 3 Lx x"x # + Da # + Db # + Dc , 0 i i i La Lb Lc i/1 i i/1 i i/1 i 3 Ly 3 Ly 3 Ly y"y # + Da # + Db # + Dc , 0 i i i La Lb Lc i/1 i i/1 i i/1 i 3 Lh 3 Lh 3 Lh Da # + Db # + Dc . (2) h"h # + i i i 0 Lb Lc La i/1 i i/1 i i/1 i In general, it is true that for any geometry, such as arc or curve edges, the workpiece positional configuration error can be analyzed by using the set of equations as shown above. To show the solution procedure, a simple example is provided below (Fig. 2). If assume only the first locating edge varies by Dc , the first locating edge 1 equation will be changed to a x#b y#c #Dc "0. 1 1 1 1
(3)
Y. Wu et al. / Robotics and Computer-Integrated Manufacturing 14 (1998) 17—26
19
The three equations can be re-written as
C
A
B (p !x )B #(q !y )A 1 1 0 1 1 0 1 B (p !x )B #(q !y )A 2 2 2 0 2 21 0 2 A B (p !x )B #(q !y )A 3 3 3 0 3 3 0 3 A
1
CD 1
" 0
dx x"x # Dc , 0 dc 1 1 dy Dc , y"y # 1 0 dc 1 dh Dc . (4) h"h # 1 0 dc 1 When the assembly relationship condition is considered, the line equations can be described as [5]. a [(p !x) cos h#(q !y) sin h] i i i #b [(q !y) cos h!(p !x) sin h]#c "0, (5) i i i i where (p , q ) are the locating hole position coordinates i i on the fixture base plate. The differentiation relative to c becomes 1 dh dx a !(p !x ) sin h ! cos h i i 0 0 dc 0 dc 1 1 dh dy #(q !y ) cos h # sin h i 0 0 dc 0 dc 1 1 dh dy #b !(q !y ) sin h ! cos h i i 0 0 dc 0 dc 1 1 dh dx !(p !x ) cos h # sin h #d "0, (6) i 0 0 dc 0 dc i1 1 1 where i"1, 2, 3 and
C
D
C
D
G
1, d " i1 0,
(7)
where A "!a cos h#b sin h, B "!a sin h!b cos h, i i i i i i and i"1, 2, 3. From the three equations, the first-order derivative at (x , y , h ) dx/dc , dy/dc , and dh/dc can be calculated. 0 0 0 1 1 1 When the determinant of the matrix in Eq. (7) is not equal to 0, there is one unique solution for the three derivatives. Therefore, x, y, and h which defines the workpiece positional configuration can be estimated using the above equation. However, if the determinant equals to zero, there will be no unique solution for the equation set, which is not a valid location of the workpiece. Below, a numerical example is given on locating repeatability analysis. As shown in Fig. 3, one simple triangular workpiece is shown with three locating points defined. For such specific situation, it can be found that x "!120, y "!120, h "0, 0 0 0 when pitch T"30, and a "1, b "1, c "240; a "1, 1 1 1 2 a "0, b "1, c "0, p "0, 3 3 3 1 p "!120, q "0; p "!60, 2 2 3
b "0, c "0; 2 2 q "0; 1 q "!120. 3
Following the above solution procedure, we have
C
D
dx dh a ! #(q !y ) 1 1 0 dc dc 1 1
C
D
dy dh #b ! !(p !x ) #1"0, 1 1 0 dc dc 1 1
C
D
dx dh a ! #(q !y ) 2 2 0 dc dc 1 1
C
D
dy dh #b ! !(p !x ) "0, 2 2 0 dc dc 1 1
C
D
dx dh a ! #(q !y ) 3 3 0 dc dc 1 1
i"1, iO1.
CD
0
Fig. 2. An example of locating edge position variation.
All other parameters remain constant. Therefore, x, y, h can be conceived as the functions of c as 1 x"x(c ), y"y(c ) and h"h(c ). 1 1 1 The workpiece positional configuration with the variation can be estimated by
,
D
dx dc 1 dy dc 1 dh dc 1
#b
C
3
D
dy dh ! !(p !x ) "0. 3 0 dc dc 1 1
(8)
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Fig. 3. An example for locating repeatability analysis.
With the specifications in this example, the equation set is simplified to dx dy ! ! #1"0, dc dc 1 1 dh dx ! #120 "0, dc dc 1 1 dy dh ! !60 "0. dc dc 1 1 The solution becomes
(9)
dx "2, dc 1 dy "!1, dc 1 dh 1 " . dc 60 1 In this way, the sensitivity of variance Dc can be 1 evaluated. Although this is a simple example, it illustrates the procedure of the fixturing accuracy analysis which verifies the fixture design results. 3. Clamp planning When locating positions are determined and workpiece positional configuration is calculated in locating planning, clamp positions need to be selected and verified
to secure the workpiece positional configuration. In this section, a step-by-step algorithm is developed to find the possible clamp position. Planar object constraint analysis is performed to evaluate the clamping stability for a given clamping edge. Generally, the possible clamping points can be found by following the three steps: (A) All the possible clamping edges are enumerated as clamping edge candidates after discarding the selected locating edges (or portions of the edges selected for placing locators) and considering the machining envelope. When one clamping edge candidate intersects with the machining envelope, the intersection should be cut off from the clamping edge. (B) All the clamping edge candidates are tested using clamping constraint analysis to find the possible clamping points set corresponding to the locating plans. This step is the major portion of the algorithm and will be discussed in more details. (C) Geometric analysis for modular fixture assembly is finally performed to get possible clamping points. The position of the workpiece has been fully constrained in locating planning. When a clamp unit is designed for one clamping edge, the discrete assembly positions of the clamp are enumerated by the requirement of modular fixture assembly relationships, as described in reference [5]. The actual clamping point on the workpiece will then be calculated. If it falls within the pre-calculated clamping points set, it is a reasonable clamping design. Otherwise, it needs to be re-designed.
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Fig. 4. 2-D object constraint anyalysis [3].
The problem of restraining the planar motion of twodimensional (2-D) object was analyzed [9]. As shown in Fig. 4a, when an object is restrained by a point contact A, it is free to rotate in both counter clockwise (CCW) or clockwise (CW) directions. Let the line perpendicular to the contact surface be L. Line L divides the 2-D space into two regions: “a” and “b”. Any point in region “a” can be the center of instantaneous CCW rotations, and any point in region “b” can be the center of CW rotations. The approach can be also implemented into multiple-point constraints. In Fig. 4b, the 2-D space is divided into six regions by the contacting lines in 3-2-1 locating scheme [3]. Points in region “b3” cannot be the center of rotation, as CCW and CW rotation are restrained by points C and B, respectively. Similarly, points in “b1”, “b2” and “a2” cannot be the center of rotation. The regions of rotation centers are reduced to “a1” and “b4”. Therefore, the instant rotational center (IRC), either CCW or CW, may only fall into the labeled regions “a1” and “b4”, which are referred as CCW IRC and CW IRC regions. The main purpose of clamping is to constrain possible rotations of the workpiece from the locating position. Therefore, the clamping position should be selected such that the CCW IRC and CW IRC regions can be eliminated. The clamping edges can be any geometric entity, such as arc or other curves. Only the normals of the locating edges on the locating points are considered. The position of the clamp should completely reduce the IRC regions to the non-IRC regions, or null regions, thus eliminate rotational freedom of the object. In Ref. [3], only simple workpiece which employs 2-1 side locating principle for 2-D situation is considered. In this research, it is expanded to complex workpiece geometry with linear and arc side locating surfaces which are not necessary to be perpendicular to each other. An analysis procedure is developed to find the possible clamping points satisfying clamping constraint conditions. Assume the workpiece is located by three locators contacting three locating edges e , e and e at P , P , 1 2 3 1 2 and P , respectively. In order to constrain the three 3 degrees of freedom (DOF), e , e and e cannot be all 1 2 3
parallel. Therefore, the contacting force directions at e , 1 e and e , say n , n , n , cannot be all parallel. In general, 2 3 1 2 3 it is assumed that n and n are not parallel, and intersect 1 2 at point O as shown in Fig. 5a. The 2-D space is divided into 4 parts by n and n . From the above approach, 1 2 region B and D are the null regions, A is the CCW IRC region, and C is the CW IRC region. When the third contacting force is applied, the situation is more complicated if considering the different possible directions of the third one. Generally, three cases are discussed below. 1. n is within !n and !n . 3 1 2 In this situation, n divides the regions A and C into 3 four regions and form a new region E. E is the triangle enclosed by n , n , and n , which could be degenerated 1 2 3 into a point if n pass O. It can be found that both 3 A and C regions become the null regions. Region E remains as an IRC with the same rotational direction as A or C (CW in Fig. 5b). 2. n is between n and !n , or n and !n . 3 1 2 2 1 In this situation, both A and C regions are divided into two smaller regions by n (Fig. 5c). One of them 3 becomes the null region and the other will still be the IRC with the same direction as the original region. The IRC regions are enclosed by either two lines or three lines, and may stretch to the infinity. 3. n is between n and n . 3 1 2 In this case, n divides either A or C, say C, into two 3 regions C and E. E is the enclosed triangle by n , n , 1 2 n and becomes the null region. A and C remain the 3 IRC regions, as shown in Fig. 5d. It should be noted that the main purpose of clamping is to constrain possible rotations of workpiece from the locating position, i.e. erase all the possible IRC region. In the above analysis, the IRC regions have been discussed for different situations of contacting directions. All the possible clamping edges have been enumerated in step A for the clamping constraint analysis. The procedure of finding the possible clamping point set with a specific clamping edge is shown below and it is assumed that only one clamp is needed in the fixture configuration.
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Fig. 5. General 2-D constraint analysis.
Step 1: Consider one available clamping edge from the clamping edge candidates. Get its line equation, end points, say AB, and contact force direction (internal normal direction). Step 2: Consider one of the IRC regions. Project the IRC region onto the line segment of the clamping edge. One of its ends may lie in the infinity because the IRC region may stretch to the infinity. Keep record of the end points of the projected line segment, say CD, and its rotational type. Step 3: Find the portion of the projected line segment which falls within the original clamping edge. Let AC and DB be the remaining parts after the intersection of AB and CD has been cut off. Step 4: If CD is within a CW IRC region, the portion AC will be the suitable clamping region and if CD is within a CCW IRC region, the BD should be selected. Step 5: If all the IRC regions have been considered, the selected portion of the clamping edge is returned as the
final possible clamping point set. If there is another IRC region to be processed, let the selected portion of the clamping edge replace the original clamping edge and go to step 1. Step 6: Once the clamping point set is obtained, the fixture configuration design module can be used to generate clamping units with satisfactions of modular fixture assembly relationships [6]. One example on the side clamp planning is given as shown in the Fig. 6. The three side locating points are shown in the figure as well as the IRC regions. Following the above procedure, a derived feasible area (LC) is derived on the clamping edge which can eliminate the two IRCs at the same time. The clamping algorithm is based on the assumption that only one clamp is needed. However, for some workpieces, one clamp may not be sufficient to eliminate all the IRC regions. For example, in Fig. 7a, it is not possible to use one clamp to eliminate the entire IRC region
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Fig. 6. One example for side clamping planning.
for the triangular workpiece with three locating points shown in the figure. Two or even more clamps should be used to fully erase the IRC regions. Therefore, additional discussions are necessary. f The condition for using only one side clamp is that such an available clamping edge E exists which satis# fies that the projection of CCW and CW regions are separated spatially, and the CW region projection is on the left side (and/or CCW region projection is on the right side) of the selected clamping point, if looking in the direction of clamp acting direction or the internal normal direction of the selected edge. f If it is found that one clamp is not enough to constrain the possible rotation, two clamps may be used. The design can be carried out by using each clamp to eliminate one IRC region if there are two IRC regions. If only one IRC exists, the first clamp should decrease the IRC region as great as possible and the second one should erase the rest of the IRC region where the clamping sequence needs to be considered carefully. Fig. 7b and Fig. 8 shows examples of using two clamps for a total constraint. It should be noted that the friction effect is not considered in the clamp planning. This is because the friction force effect is always positive on the clamping stability since the friction forces are against any possible motions between the workpiece and fixture components. When the clamp planning is conducted without considering the friction effect, the result is more conservative.
4. Discussion on fixturing accessibility Fixturing accessibility is an important aspect in selecting fixturing (especially locating) surfaces and points. In fixture planning, two types of accessibility should be considered. The first one is the reachability of an individual workpiece surface, which is an important measure in locating and clamping surface selection. The second one is the easiness of loading and unloading the workpiece into a fixture, which refers to the configuration of three locating points on workpiece surfaces. In this section, the loading/unloading accessibility is discussed. It is assumed that clamps have been removed from the fixture configuration. Thus, only three locators are presented in the fixture when the workpiece is loaded or unloaded. There are three kinds of IRC regions as discussed in Section 3. In case A, only one CW or CCW rotational triangle exists and all other regions are null regions. The loading process can be conceived as the workpiece is first placed in the vicinity of the final position, then rotated to contact with the three locators for a final position (Fig. 5b). The rotational direction to make the contact is on the opposite direction of the CW or CCW rotation indicated in the triangular region. The unloading process can be discussed, similar to the loading process. In case B, two IRC regions are found. In this situation, the workpiece is loaded easier into the fixture than the situation in case A. It allows the workpiece to move linearly in one direction (Fig. 5c) to contact with one of the locators, and rotate slightly to contact with the other two locators.
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In case C, the workpiece has the best accessibility since it has a wider open scope than the other two. Generally, the workpiece may move in two perpendicular directions to contact with two locators, and finally rotate to contact with the last locator. Usually, the larger the IRC region is, the better the accessibility of the fixture configuration will be. The standard 3-2-1 locating scheme with three perpendicular plane surfaces is an indication of good fixturing accessibility, which should be considered with a priority in locating surface selection. It can be further simplified to consider the directions of the contact forces instead. Assume that two contact forces f and f intersect at 1 2 point O and line ¸ divides the angle between f and 1 f equally. Then the third contact force f maintains an 2 3 angle h with ¸. The accessibility can be generally evaluated as the larger h is, the worse the accessibility is. Fig. 9 is a graphic sketch of the three force vectors. The above discussion only provides one criterion to evaluate the loading/unloading accessibility of the fixture. A complete accessibility analysis could be very complicated that involves more technical problems such as the geometric complexity of the workpiece and reasoning of geometric entities in a CAD model of the workpiece. 5. Clamping stability evaluation
Fig. 7. (a) One workpiece not possible to use one clamp. (b) Using two clamps to eliminate one IRC.
When a modular fixture design is conducted by the computers, the equilibrium between clamping forces and locating responses needs to be evaluated for a reliable locating. The clamping forces can be regarded as active and known input forces. The locating responses are passive forces and need to be solved. When all of the locating
Fig. 8. Using two clamps to eliminate two IRCs.
Y. Wu et al. / Robotics and Computer-Integrated Manufacturing 14 (1998) 17—26
25
Fig. 9. Accessibility analysis. Fig. 10. Clamping stability analysis.
responses are positive in the normal direction, the fixture design is claimed to be clamping stable. Otherwise, the locating and clamping positions needs to be redesigned [10]. In clamping constraint analysis, friction forces are not considered. Generally, frictional forces are presented in fixture performance. However, it is very difficult to solve the locating responses if frictional forces are involved, especially in the 3-D space. Many papers in this area do not consider the frictional forces. In this research, a 2-D analysis approach is applied to calculate the locating responses, which is similar to the method presented in reference [10]. As shown in Fig. 10, basically the equilibrium equations include the force equilibrium in X, ½ directions and a moment equation about Z-axis. RF "0 RF "0 and RM "0. (10) x y z Assume the workpiece has a tendency of rotating in CCW direction, the frictional force at each locating and clamping point is resisting the rotational tendency. Thus, the directions of the frictional forces can be determined as shown in Fig. 10. The magnitudes of the frictional forces should also satisfy f )kN , (11) i i where k is the coefficient of friction, and N is the normal i force at position i. There are more than three unknown variables: N , N , 1 2 N , and all the friction forces in the three equations. If the 3 workpiece is assumed to be a rigid body, one assumption can be made: frictional forces will increase at the same rate at all locating and clamping points, or the frictional forces at all locating points are f "kkN , i i where k is an acting factor between 0 and 1.
(12)
Then there are four unknowns left: N , N , N , and k. 1 2 3 To solve the equations, the coefficient k is increased from 0 to 1 with small increments. When k is increased to a certain value, Eq. (10) may become true with zeros in the right sides. The solved locating responses at this specific k value within the range of 0—1 are the actual normal locating responses. Frictional forces at all points can be also derived. If the solution cannot be found within the range of k"0—1, that means an invalid fixture design which is not stable when the clamping force is applied. Numerical examples can be found in Ref. [10]. It should be mentioned that this calculation only provides a conservative, static evaluation on clamping stability. The friction forces in the primary locating surface and top clamping surfaces as well as the machining force are not included in the calculation.
6. Summary This paper presents analytical discussions of fixturing accuracy, clamping planning, and fixturing accessibility. Together with geometric analysis, these analyses may provide a scientific foundation of automated fixture planning. Although fixture design is a complex task and usually it needs to involve human expertise, applying computer technology to generate and verify feasible solutions with alternatives is possible and greatly beneficial, especially in FMS and CIMS. The necessary conditions of fixture planning can be identified in the analyses, which may make the automated fixture design become possible and applicable in production.
Acknowledgements This research is partially funded by National Science Foundation (NSF), Air Force Office of Scientific
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Research (AFOSR), and Ingersoll Milling Machine Company. Supports from Machine Tool Agile Manufacturing Research Institute (MT-AMRI), Technologies Enabling Agile Manufacturing (TEAM) program of Department of Energy, TechnoSoft Inc., Bluco Corporation, and Qo-Co Companies are appreciated. Technical discussions within the fixturing research group at Southern Illinois University at Carbondale are also acknowledged. References [1] Asada H, By A. Kinematic analysis of workpart fixturing for flexible assembly with automatically reconfiguration fixtures. IEEE Trans Robotics Autom 1985;RA-1(2):86—93. [2] Chou YC, Chandru V, Barash MM. A mathematical approach to automatic configuration of machining fixtures: analysis and synthesis. ASME Trans Engng. Industry 1989;111:299—306. [3] Chou YC, Barash MM. Automatic configuration of machining fixtures: object constraint reasoning. Manufacturing International, Atlanta, GA. 1990.
[4] Brost RC, Goldberg KY. A complete algorithm for synthesizing modular fixtures for polygonal parts. IEEE Trans Robots Autom 1996;12(1):31—46. [5] Wu Y, Rong Y, Ma W, LeClair SR, Automated modular fixture planning: geometric analysis. Robotics Comp. Integrated Manufacturing 1998;14:1—15. [6] Rong Y, Bai Y. Automated generation of modular fixture configuration design. ASME Trans. Manufacturing Sci. Engng 1997;119:208—19. [7] Rong Y, Li W, Bai Y, Locating error analysis for fixture design and verification. ASME Computers in Engineering, Boston, MA, 17—21 Sept. 1995; 825—32. [8] Rong Y, Zhu J, Li S, Fixturing feature analysis for computeraided fixture design. In: Proceeding of ASME International Mechanical Engineering Congress and Exposition, PEDVol. 64, New Orleans, LA, 28 November—3 December 1993; 267—271. [9] Reuleaux F, Kinematics of machinery, London. Macmillan, 1963. [10] Rong Y, Wu S, Chu T. Automatic verification of clamping stability in computer-aided fixture design In: Advances in Manufacturing Systems: Design, Modeling and Analysis, Amsterdam: Elsevier, 1994; 353—58.