A Shape Metric for Design-for-Assembly Gerard J. Kim, George A. Bekey and Kenneth Y. Goldberg Center for Manufacturing and Automation Research Computer Science Department University of Southern California, LA, CA 90089-0782, USA E-mail:
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Abstract \Design-for-Assembly" (DFA) is a process of improving product designs for easy and low cost assembly. DFA often involves an analysis of an existing design according to qualitative criteria that are dicult to analyze. This paper introduces a quantitative shape metric for planar parts, called feedability, based on a stochastic parts feeding algorithm reported in [7][8]. Although feedability is only one possible metric for DFA, it relates one aspect of assembly cost directly to part geometry. Therefore, based on feedability, one can produce a set of geometric redesign strategies. The application of feedability (or Design-for-Feedability) to DFA is illustrated and compared with other DFA methods.
1 Introduction \Design-for-Assembly"(DFA) is a process of improving product designs for easy and low cost assembly. An assembly-conscious design is desirable because it could result in signi cant savings in capital costs and assembly time. For example, the IBM Corporation has successfully applied the concept of DFA to one of its printer models, by redesigning and reducing the number of parts, which drastically lowered the printer's assembly cost, and price [3]. An assembly-conscious design is dicult, in general, because designers tend to focus only on the functionality of the product. Considering \assemblability" concurrently with the functionality of a product seems to be a major source of distraction that most traditional designers would like to avoid. This is especially This project is supported in part by the Institute for Manufacturing and Automation Research (IMAR) Project No. 4 and by the National Science Foundation. Support for Dr. Goldberg is provided in part by a 1991 USC Faculty Research Initiation Fund (FRIF) grant.
apparent when there exist many parts exerting various constraints upon one other. Therefore, there is a desire to automate the DFA process, at least partially, through the use of computers. The purpose of this paper is to introduce a quantitative metric for part shapes, called feedability, and demonstrate its use. Although feedability is only one possible metric for DFA, it relates DFA cost directly to part geometry. Therefore, various redesign strategies are possible by manipulating the geometric parameters of the part and their constraints to improve the feedability. In a previous paper [10], we proposed a model for the DFA process that consists of iterations of three successive stages: analysis, suggestion, and veri cation. The actual implementation revealed some fundamental problems. One of the most important problems was that the current methods of DFA analysis lack a scienti c foundation that can associate DFA cost to a particular set of low level or geometrical design actions. Therefore, current DFA methods can only produce redesign strategies in terms of high level design axioms or design rules: for example, \make part A symmetric", \add a gripping feature to part B", \redesign part C to avoid tangling", etc. The geometric details for achieving above objectives have to made by the humans. One of the goal of this paper is to introduce low level redesign strategies by introducing a metric that can relate assembly cost to part geometry. Then, geometric redesign advice would be possible: for example, \change the length of part A from 3 inches to 5 inches in order to make it symmetric".
2 Other Methods of DFA Currently, there exist two main types of DFA methods. The most popular method is typi ed by the one pioneered by Boothroyd and Dewhurst [4]. In their approach, a product is analyzed according to various
\ease of assembly" criteria (such as symmetry, dimension, mating direction, number of parts, etc), organized with charts of scores, and a tabulated score is used to calculate a \design eciency ratio". The scoring system (or the DFA metric) is empirically based on an engineering time study method [16]. Then, a redesign is proposed manually with the help of the score table which, more or less, pinpoints the source of the design problem. Many other commercial systems or in-house methods [11][14][15] are based on this approach partly because the table lookup and the arithmetic procedure is easily computerizable. The second type of approach is called the \axiomatic method". The axiomatic method is simply a set of design guidelines that have been empirically derived from years of experiences in design and assembly operations. If correctly applied either during the initial design phase or the redesign phase, they should result in a product that has an inherently low assembly cost [9]. Many such DFA axioms have been identi ed by a number of researchers [2][3][9]. In particular, in an e ort to make the axiomatic method more evaluative, Hoekstra has assigned scores to each of the design axioms in a subjective manner (for example, in a scale of 1 to 5). The rst method is suited for a comparative analysis of nished designs, but can not be applied to evolving designs. In practice, the score tables are very useful in analyzing and evaluating designs in terms of DFA. Nevertheless, such tools still have the drawback that they lack a capability of helping designers to explore new possibilities [6]. The second method allows the designer to consider assemblability from the initial design phase. However, in general, the second method lack quantitative measures for analysis and evaluation of nal or evolving designs [16]. Although Hoekstra's scoring system for the design axioms improved upon the evaluating capability of the axiomatic method, it is based on a subjective point of view, and can not relate its scores directly to part geometry. Suh has also cautioned against the handbook approach for the lack of theoretical conception and scienti c basis for the design axioms [17]. A more objective and analytic metric with a scienti c basis, which relates assembly cost directly to the design, could provide a systematic method of making geometric redesigns. Hoekstra best illustrated the need of such a metric by citing part of the following round table discussion on \Design for Assembly" reported in a recent Assembly Engineering Magazine [5][9]. Among the panelists were G. Boothroyd, W. Morill (Manager, Design for Manufacturability, Digi-
tal Equipment Corp.), and B. Branan (Manager, Manufacturing Technology, Motorola, Inc. Boothroyd: ... A typical rule, such as always design parts that are as symmetrical as possible, won't necessarily save anything in manufacturing costs. You simply must quantify these costs at the design stage. Morrill: What's needed is a quantitative measure that can predict what happens when a variable changes and how it effects manufacturability. If there are design alternatives they must be quantifiable. Branan: If you write down a bunch of rules and don't provide a quantitative method for predicting what the results of those rules will be, the designer will say, ``I followed them to the best the situation permitted''
The goal of this paper is to introduce one shape metric for DFA, called feedability, that is quantitative and analytic with a scienti c basis, and can relate the geometry of a design to one type of assembly process as to gain more insights on how DFA redesigns should proceed.
3 Feedability In this section, we introduce the concept of feedability, a new quantitative and analytic measure that can be used as a DFA metric. The concept of feedability is derived from the following stochastic parts feeding framework developed by Goldberg [7][8].
3.1 Stochastic Parts Feeding Goldberg's stochastic parts feeding framework has two components: the feeding mechanism and the planning algorithm [7][8]. The feeding mechanism is a device that can orient parts under software control. Figure 1 shows a picture of a parts feeder which orients parts as they move along on a conveyor belt. The main component of the feeding mechanism is a frictionless parallel jaw gripper which is kinematically yielding on one end of the jaw. While not totally \frictionless", the actual physical gripper has on jaw mounted on ball bearings, so that the friction forces are in fact extremely small. Other components include a conveyor belt for part transportation and a binary lter for distinguishing among orientations whose asymmetry can not be detected by grasping operations alone (See Figure 2). The planning algorithm nds a set of grasping plans in time O(n2) for orienting any n-sided polygonal part up to symmetry using the frictionless gripper. Some of the important assumptions are the following (See [7][8] also).
Parts Feeder
Figure 1: A parts feeder orients parts as they arrive on the left-hand conveyor belt (Reprinted with permission from [8]).
Figure 2: An example of an ambiguous symmetric grasping orientation. The grasping operation itself can not disambiguate between two grasping orientations that are actually asymmetric due to a hole feature. 1. The part is a rigid polygon of known shape. 2. The part's initial position is unconstrained provided it lies somewhere between the two jaws. The part remains between the jaws throughout grasping. 3. There is zero friction between the part and the jaws. 4. We can design a binary lter that accepts a particular orientation of the part and rejects all others.
There are two important functions involved in deriving the grasping plans for a given part. The transfer function provides a mapping from an initial orientation of the part to a nal orientation of the part for each grasping action. A nal orientation of a part is a stable orientation where at least one edge of the part is aligned with the gripper when fully grasped with a frictionless parallel jaw gripper. The set of stable orientations is nite, at most one per edge. Each grasping action corresponds to an orientation of the gripper. Since rotating the gripper is equivalent to rotating the part, the transfer function is de ned in terms of the part's orientation with respect to the gripper. The transfer function depends on the geometry given by the diameter function of the part. The diameter function of a part is a mapping between the part's orientation with respect to the gripper and the distance between parallel jaws of the gripper. The part's initial orientation is uncertain. The rst grasping action in the plan is a random grasp, where gripper orientation is chosen from a uniform probability distribution on the interval [0, 2). After the rst action, the probability that the part is at orientation is related to the probability that the part was initially in some orientation . Accordingly we 0
1
2
1
1
1
2
2
2
Figure 3: Top view of a plan for feeding rectangularparts. Gripper orientations is shown with two parallel lines. Four traces of a two step plan for orienting the part. Each trace runs from top to bottom. Although the part's initial orientation is di erent in each trace, its nal orientation is the same (Reprinted with permission from [8]).
can compute the rst probability distribution by integrating the probability density over the corresponding interval in the transfer function. For a more detailed description, refer to [7][8].
3.2 Example Table 1 and Figure 3 shows an example of orienting a rectangular part. The initial orientation of the part is unknown, but there are only a nite number of stable grasping orientations (in this case, two). Each plan has a probability associated with the possible outcome of the plan. We see that the one-step plan results in orientation 1 (vertical) with probability of 0.705, and in orientation 2 (horizontal) with probability of 0.295. The two-step plan is a guaranteed plan that always orients the part to orientation 1 with probability of 1. Table 1: Plans for orienting a rectangular part. Plan One Step Plan Two Step Plan
Probability Horizontal Vertical 0.705 0.295 1.000 0.000
To compare the one-step plan and the two-step plan, we consider augmenting the one-step plan with a binary lter that rejects parts that are not in the intended correct orientation. The rejected parts loop back into the feeder with random orientations, and the plan is repeated until the part is correctly oriented. On average, the one-step plan must execute 1/0.705 = 1.42 steps until it succeeds. On the other hand, the two step plan must execute 2/1.0 = 2 steps, since it is
guaranteed to succeed. Now, two plans can be compared based on their expected time, assuming that every step in the plan takes one time unit. Under these conditions, the one-step plan has higher expected feed rate.
3.3 Feedability Following Goldberg [8], we de ne the expected feeding cost of a part with an stochastically optimal i-step grasping plan. De nition 1 Expected feeding cost, ci (p), of part, p, with a stochastically optimal i-step grasping plan, is de ned as, ci (p)
= Pi , i
where, (1) i represents the number of plan steps, and (2) Pi represents the probability of most likely orientation the i-step plan will produce.
Then, feedability of a part is de ned by comparing the inverse of the feeding cost of each grasping plan and by choosing the maximum among them. The notion of feedability is de ned within the context of a programmable feeder with a frictionless gripper. However, since feedability is directly associated with both grasp planning and orienting operations, it also addresses the general problem of handling and, therefore, is one example of a metric related to part handling. For example, Boothroyd's score table is based largely on engineering time study methods [16]. Likewise, feedability addresses the handling time in terms of number of grasping steps and expected time to orient the part. The cost of a grasping action includes rotation of the gripper and the grasping, and is constant for any rotation angles or objects. In this paper, the cost (or the time unit) is set to 1. De nition 2 (Feedability) Feedability of a part p, f(p), is de ned as, f (p)
= max[ c (1p) ]. i
i
Based on feedability, we can evaluate di erent parts with respect to their ease of feeding (which is one of the most frequent assembly operations for handling parts), and perhaps suggest a way of improving part designs based on the derivation of their feedability. Note that the de nition of feedability considers neither the time consumed during the possible ltering stage nor the cost of the lter itself.
Figure 4: Feedability (f) vs. Design Parameter (height) of an example part.
4 Design-for-Feedability In this section, we address the problem of modifying an existing design to improve feedability and suggest several heuristic design operators to solve the addressed problem.
4.1 Problem Statement We are concerned with the problem of modifying an existing design to improve its feedability. The problem requires devising operators that transform a given design into a new design with higher feedability. We refer to operators as the feedability-associated design operators. This problem is considered as one of the fundamental problems associated with the development of a system for semi-automatic DFA redesigns, which is our future research goal.
4.2 The Feedability-Associated Design 6Operators Figure 4 illustrates a relationship between the length and theh1feedability for a shape of a common industrial part, taken from the DCLASS Part Fam4 System [1]. Ihe analysis showed that ily Classi cation the sudden change in feedability (the circled portion in Figure 10 4) corresponds to the change in the number of possible stable grasping orientations as the length of the part is changed. To improve the feedability, the design operators can vary the two parameters, i, and P (See De nition 1). These two parameters can be related to the geometry of the part. Here are some examples of the feedabilityassociated design operators that are currently implemented on our system. Reducing Number of Stable Orientation: The ini
creased number of possible stable orientations decreases
Figure 5-1: Four possible heuristic design operators for reducing number of stable orientations. Figure 5-4: Using an orienting feature to increase feedability.
Figure 5-2: Dimensional adjustment to increase Pi .
the parameter Pi because the probability is now distributed across a larger number of possible stable orientations, and therefore, lowers the feedability. Likewise, a decreased number of stable grasping orientation results in a higher feedability. Such design operators include, vertex relocation, edge straightening, parallelization, and edge extension (See Figure 5-1). Dimensional Adjustment: It is also possible to increase Pi by varying the geometric dimension such as distance, angle, etc. One obvious principle is to prefer slender parts with high length-to-diameter (L/D) or length-to-width ratio. For example, use longer screws, if possible, for they are more feedable due to the increased Pi on the longer side of the part (See Figure 5-2). Elimination of Filtering: Goldberg's stochastic parts feeding algorithm nds a set of optimal grasping plans for orienting a polygonal part up to a symmetry. Therefore, even for a guaranteed plan with no planning failure, there may be a need for a lter to disambiguate between two or more symmetric grasping orientations (See Figure 2). There exist two ways of eliminating such ltering (See Figure 5-3, 5-4). One way is to make the part symmetric and, thus, increase Pi . Another way is by introducing a grasping feature to the part.
Relocation 5 Vertex Comparison with Other DFA Methods
It is useful to make a comparison between the concept of feedability for DFA and other DFA methods Edge Straightening
Figure 5-3: Use of symmetry to increase feedability.
introduced earlier in this paper (See section 2). Table 2 shows a list of conventional DFA rules [2][3][4][9] that are relevant for such a comparison 1 . These rules are related to the issue of handling of parts, as is feedability. Table 2: Four DFA rules for part handling. 1. 2. 3. 4.
All parts should be symmetrical If parts are not symmetrical, use increased asymmetry Provide a means to easily grip and hold the part or design particular orienting surfaces such as guide surfaces Avoid parts that tangle, nest or topple.
The comparison was made by applying the above DFA rules to several test examples and observing the change in feedability. The goal of the comparison is to nd if feedability consistently improves when above DFA design rules are applied for redesign. Figure 6 shows some typical examples from the comparison study. The rst three examples show that the feedability consistently improved after applying the corresponding DFA rules from table 4. The rst example in Figure 8 shows the change in feedability, when making a part symmetric (DFA rule 1) by introducing extra features (holes). When a part is perfectly symmetric, there exists a guaranteed plan with one grasping step. Therefore, the feedability is equal Parallelization to one, since only one random grasp is required (P = 1 and i = 1). The second example shows a comparison between an asymmetric part and a part with a symmetric grasping orientation (DFA rule 2). The feedability of the asymmetric part is higher, since the part with the symmetry has lower value of P and requires ltering. The third example shows a case of Feature Manipultion introducing a grasping feature (DFA rule 3). Instead of grasping the object in the left side, grasping the triangular grasping feature can be done in one random grasping operation, thus the feedability of the part with the grasping feature is much higher. i
i
1 Boothroyd also has summarized a list of DFA rules as an emerging principle from his scoring system.
DFA methods. The strength of feedability is that it not only evaluates DFA in terms of both grasping and orienting operations, but also is quantitative metric that can be used for redesigning parts for a particular type of assembly process. The weakness is that it is not applicable for other part handling devices such as dexterous hands, vibratory bowl feeders, and other special end e ectors. Also, it is currently only applicable for two f = 0.232 f = planar 1 dimensional parts. However, since feedability is associated with a speci c assembly process, it encourages a good practice of associating an assembly design with a particular assembly process. For our continuing and future research, we are inDFA Axiom 1: All Parts Should be Symmetrical. terested in developing an interactive knowledge-based system for redesigning mechanical products for DFA. The feedability metric and design operators introduced in this paper will serve an essential role in constructing such a system. f = 0.2
f = 0.25 References
[1] D. Allen and R. Smith, \Part Family Classification", Monograph, No. 3, CAM Software Research Center, Brigham Young University, 1982 [2] M. Andreason, et. al, \Design for Assembly", IFP Publications Ltd. and Springer-Verlag, 1983 Figure 6: Feedability and other DFA rules. DFA Axiom 2: If Parts are not Symmetrical, Use[3]Increased Symmetry. Bailey, \Product Design for Robotic Assembly", Proceedings of the 13th. International Symposium on Industrial Robots and Robots 7, Detroit, Robotics International of SME. [4] G. Boothroyd, and P. Dewhurst, \Product Design for Assembly", The fourth example illustrates a case where DFA Boothroyd and Dewhurst, Inc., 1989 rules do not increase feedability. Extra surfaces were [5] Coleman, \Design for Assembly: Users speak out", Assembly Engineering, pp. 18-24, July, 1988 introduced to prevent parts from toppling on each [6] M. Cutkosky, et. al, \A Methodology and Computational Framework for other (DFA rule 4). Intuitively, introducing extra surConcurrent Product and Process Design", Mechanism and Machine Theory, Vol. 25, No. 3, 1990 faces should decrease the feedability, as there are more [7] K. Goldberg, \Stochastic Plans for Robotic Manipulation", PhD. Thesis, stable grasping orientationsf = possible. However, the exCarnegie Mellon University, August, 1990 0.25 f = 0.5 ample illustrates that the shift in feedability depends [8] K. Goldberg, M. Mason and M. Erdmann, \Generating Stochastic Plans for a Programmable Parts Feeder", Proceedings of the IEEE International on the exact geometry of the part. Thus, our metric Conference on Robotics and Automation, Vol 1., pp 352-359. 67-73, April, 1991 is not always consistent with other DFA rules. [9] Hoekstra, \Design for Automated Assembly: An Axiomatic and Analytical Method", SME Technical Paper, AD89-416, presented at the SME International Conference, Detroit, Michigan, May 1-4, 1989 [10] G. KimHold and G. Bekey, \AEX: A DFA Analysis and Advisory System", DFA Axiom 3: Provide a Means to Easily Grip and the Part. Proceedings of IEEE International Conference on Systems, Man, and Cybernetics, LA, CA, 1990 [11] D. Lee and M. Melkanoff, \Product Design Analysis Using the Assembly Design Evaluation Metric", Proceedings of the ASME Computers in Although, conventional DFA methods can help Engineering 1991 Conference and Exposition, Santa Clara, CA, August 1991. DFA analysis and evaluation, the redesign is totally [12] Light and Gossard, \Modification of Geometric Models through Varialeft to humans. One reason behind this is the lack of tional Geometry", Computer Aided Design, Vol. 14, No. 4, July, 1982 an objective metric that relates assembly cost to geo[13] Miles, \Design for Assembly - A Key Element within Design for Manufacture", Proceedings of Institution of Mechanical Engineers, Vol. 203, metric design. This paper has introduced a new shape 1989 metric, based on Goldberg's stochastic parts feeding [14] Ohashi and Miyakawa, \The Hitachi Assemblability Method (AEM)", International Conference on Product Design for Assembly, Newport, Rhode framework, called feedability, as an analytic approach Island, 15-17 April, 1986 to relate one type of an assembly process to part de[15] Sapphire Design Systems, \User Manual of Assembly View", Sapphire Design Systems, Inc., 1989 sign. We have shown that it can be used to guide [16] H. Stoll, \Design for Manufacture", Manufacturing Engineering, pp. 67the redesign of part geometry subject to geometric 73, Jan., 1988 constraints. We have demonstrated the application of [17] N. f = 0.47 f = 0.38 Suh, \Principles of Design", MIT Press, 1990 0.25 f = 0.30
6 Conclusion and Future Research
f=
feedability to DFA redesigns and compared it to other
DFA Axiom 4: Avoid Parts that Tangle, Nest, or Topple.
