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湖南工業(yè)大學(xué) 外文翻譯 專 業(yè) 機(jī)械設(shè)計(jì)制造及其自動(dòng)化 學(xué) 生 姓 名 王 曉 雄 班 級(jí) 機(jī)本0303班 學(xué) 號(hào) 26030336 指 導(dǎo) 教 師 黃 開(kāi) 友 MULTI-OBJECTIVE OPTIMAL FIXTURE LAYOUT DESIGN IN A DISCRETE DOMAIN Diana Pelinescu and Michael Yu Wang Department of Mechanical Engineering University of Maryland College Park, MD 20742 USA E-mail: yuwang@eng.umd.edu Abstract This paper addresses a major issue in fixture layout design:to evaluate the acceptable fixture designs based on several quality criteria and to select an optimal fixture appropriate with practical demands. The performance objectives considered are related to the fundamental requirements of kinematic localization and total fixturing (form-closure) and are defined as the workpiece localization accuracy and the norm and distribution of the locator contact forces. An efficient interchange algorithm is uaed in a multiple-criteria optimization process for different practical cases, leading to proper trade-off strategies for performing fixture synthesis. I. INTRODUCTION Proper fixture design is crucial to product quality in terms of precision and accuracy in part fabrication and assembly. Fixturing systems, usually consisting of clamps and locators, must be capable to assure certain quality performances, besides of positioning and holding the workpiece throughout all the machining operations. Although there are a few design guidelines such as 3-2-1 rule, automated systems for designing fixtures based on CAD models have been slow to evolve. This article describes a research approach to automated design of a class of fixtures for 3D workpieces. The parts considered to be fixtured present an arbitrary complex geometry, and the designed fixtures are limited to the minimum number of elements required, i.e. six locators and a clamp. Furthermore, the fixels are modeled as non-frictional point contacts and are restricted to be applied within a given collection of discrete candidate locations. In general, the set of fixture locations available is assumed to be a potentially very large collection; for example, the locations might be generated by discretizing the exterior surfaces of the workpiece. The goal of the fixture design is to determine first, from the proposed discrete domain, the feasible fixture configurations that satisfy the form-closure constraint. Secondly, the sets of acceptable fixture designs are evaluated on several criteria and optimal fixtures are selected. The performance measures considered in this work are the localization accuracy, and the norm and distribution of the locator contact forces. These objectives cover the most critical error sources encountered in a fixture design, the position errors and the unwanted stress in the part-fixture elements due to an overloaded or unbalanced force system. The optimal fixture design approach is based on a concept of optimum experiment design. The algorithm developed evaluates efficiently the admissible designs exploiting the recursive properties in localization and force analysis. The algorithm produces the optimal fixture design that meets a set of multiple performance requirements. II. RELATED WORK Literature on general fixturing techniques is substantial, e.g., [1]. The essential requirement of fixturing is the century-old concept of form closure [2], which has been extensively studied in the field of robotics in recent years [3, 4]. There are several formal methods for analyzing performance of a given fixture based on the popular screw theory, dealing with issues such as kinematic closure [5], contact types and friction effects [6]. A different analysis approach based on the geometric perturbation technique was reported in [7]. An automatic modular fixture design procedure based on this method was developed in [8] to include geometric access constraints in addition to kinematic closure. The problem of designing modular fixtures gained more attention lately [9]. There has also been extensive research in fixture designs, focusing on workpiece and fixture structural rigidity [6], tool accessibility and path clearance [7]. The problem of fixture synthesis has been largely studied for the case of a fixed number of fixture elements (or fixels) [8, 10], particularly in the application to robotic manipulation and grasping for its obvious easons [3, 4]. This article aims to be an extension of the results on the fixture design issues previously reported in [14]. III. FIXTURE MODEL The fundamental performance of a fixture is characterized by the kinematic constraints imposed on the workpiece being held by the fixture. The kinematic conditions are well understood [3, 4, 5, 7, 12]. For a fixture of n locators (i = 1, 2, … , n), the fixture can be represented by: dy=GTdq where define small perturbations in the locator positions and the location of the workpiece respectively. The fixture design is defined by the locator matrixi where and ni and ri denote the surface normal and position at the ith contact point on the workpiece surface. The problem of fixture design requires the synthesis of a fixturing scheme to meet a given set of performance requirements. IV. QUALITY PERFORMANCE CRITERIA FOR A FIXTURE A. Accurate Localization An essential aspect of fixture quality is to position with precision the workpiece into the fixturing system. In general the workpiece positional errors are due to the geometric variability of the part and the locators set-up errors. This paper will focus only on the workpiece positional errors due to the locator positioning errors. As an extension of the fixture model equation (eq.1), the locator positioning errors dy can be related with the workpiece localization error dq as follows: Clearly, for given source errors the workpiece positional accuracy depends only on the locator locations being independent from the clamping system, the Fisher information matrix M = GGT characterizing completely the system errors. It has been shown [12] that a suitable criterion to achieve high localization accuracy is to maximize the determinant of the information matrix (Doptimality), i.e., max(det M). B. Minimal Locator Contact Forces Another objective in planning a fixture layout might be to minimize all support forces at the locator contact regions throughout all the operations with complete kinematic restraint or force-closure. Locator contact forces in response to the clamping action are given as: Normalizing these forces with respect to the clamping intensity we obtain: The force-closure condition requires these forces to be always positive for each locator i of a set of n locators: Computing the norm of the locator contact forces: leads to an appropriate design objective, i.e. min Note that this objective indicates both locator and clamp positions to be determined in the optimization process. C. Balanced Locator Contact Forces Another significant issue in designing a fixture is that the total force acting on the workpiece have to be distributed as uniformly as possible among the locator contact regions. If p represents the mean reactive force in response to the clamp action, then we define the dispersion of the locator contact forces as: Therefore, minimizing the defined dispersion represents an objective for a balanced force-closure: min(d). V. OPTIMAL FIXTURE DESIGN WITH INTERCHANGE ALGORITHMS As mentioned earlier, by generating on the exterior surface of the workpiece to be fixtured a set of discrete locations defined as position and orientation, we create a potential collection for the fixture elements. For example, using the information contained in the part CAD model, a discrete vector collection (unitary, normal vectors) can be generated as uniformly as possible on those surfaces accessible to the fixture components (fig.1). Figure 1: Part CAD model and global collection of candidate locations for the fixture elements. The fixture design layout will select from this collection optimal candidates for locators and clamps with respect to the performance objectives and to the kinematic closure condition. Dealing with a large number of candidate locations the task of selecting an appropriate set of fixels is very complex. As already introduced in [12, 14] an effective method for finding the desired fixture with regard to one of the previous quality objectives is the optimal pursuit method with an interchange algorithm. Due to its own limitations and to the fact that the objectives are functions with many extremes, the exchange procedure may not end up to a unique optimized fixture configuration, but to several improved designs depending on the initial layout. Therefore the solution offered by the multiple interchange with random initialization algorithm is overwhelming favorable, fact that recommends this procedure over the single interchange algorithms. The algorithm can be described as a sequence of three phases: Phase 1: Random generation of initial sets of locators. The starting layout is generated by a random selection of distinct sets, each consisting from 6 locators out of the list of N candidate locations. If the clamp is pre-determined, a valid selection is obtained through a simultaneous check for all kinematic constraints. A big initial set of proposed ocators is preferred, giving the opportunity of finding a convergent optimal solution. However from the efficiency point of view the designer has to balance the algorithm between the accuracy of the final solution and the computation time. Phase 2: Improvement by interchange. The interchange algorithm's goal is to pursue for an improvement of the initial sets of locators with respect to one of the objectives. Basically, this is done iteratively by exchanging one by one the proposed locators with candidate locations from the global collection. It is also essential to consider the form-closure restraint during the exchange procedure. The process will continue as long as an improvement of the objective function is registered. Studying the effect of interchange on the proposed quality measures leads us to some efficient algebraic properties. For example, an interchange between a current locator j (j = 1,2,…,6) and a candidate location k (k = 1,2, … ,N-6) yields changes in the optimized function such that: Thus, at each interchange the pair is selected such that the significant term that controls the function evolution is improving, e.g. max p 2jk and min Δpc , easing the iterative process. Phase 3: Selecting the optimal solution. Applying the interchange algorithm for each initial set of locators we will end up with several distinct solutions on the configuration scheme of the fixture, the best fixture design corresponds evidently to the maximum improvement of the objective function. It should be emphasized that this algorithm can be used sequentially for different objective functions. Depending on the objective pursued the best solution can be evident (for a single objective) or might need the designer's final decision (for multiple objectives). VI. MULTI-OBJECTIVE FIXTURE LOCATOR OPTIMIZATION In many applications the clamp is already fixed given some practical considerations. Then with the clamp predefined, the best fixture with respect to a certain performance criterion is constructed by selecting a suitable set of locators such that a significant improvement of the objective-function is registered. Using the random interchange algorithm we can analyze the impact of the optimization process on the fixture characteristics, as well as we can select the best optimized fixture solution for a specific criterion. In analyzing the effect of random interchange algorithm on several parts, there can be made the following statistical and empirical observations. A. Multi-objective trade-offs In some applications both localization quality and a minimum force dispersion are important. In this case we may have to use a 2-step algorithm: first max(det M) and secondly min(d). The proposed order is a consequence of the above observations. First, maximizing the determinant will automatically decrease the dispersion. Next, a decreasing in dispersion leads in a decreasing in determinant value. Therefore, during the second phase of the algorithm tradeoffs between the two objectives occur. To solve the multi-objective optimization problem the interchange algorithm is applied successively for both objectives. With the clamp pre-defined, a rigorous check for form-closure is needed after each exchange step. A following set of plots present the results when the design requirements of precision localization and uniform contact forces are considered simultaneously. Fig. 2 and Fig. 3 illustrate the global changes of the fixture characteristics during the 2-step algorithm performed on an initial collection of distinct random sets of locators, with the clamp pre-fixed. It can be noticed the advantages of using max(det M) objective as a first step: while the determinant is increasing, the norm and the dispersion of the forces are decreasing, fact benefic for the overall quality of the fixture. Furthermore the solutions are convergent, such that the candidate set of locators for the next step will be significantly reduced. On the other hand, in the second phase, when applying min(d) optimization on sets of locators with a high determinant value the only trend in the determinant evolution is a decreasing one. Therefore, during the second phase of the algorithm tradeoffs between the two objectives occur, fact expressed also through the Pareto-line plot (Fig. 3). In this case the final decision has to be left for the designer to determine the best fixture scheme. Figure 2: Changes upon the fixture characteristics applying the 2-step optimization algorithm on an initial collection of random sets of locators. Figure 3: Behavior during a 2-step random interchange algorithm for a collection of locator sets. As an example, the behavior of a single initial set of locators is studied during the interchange processes of the 2-step algorithm (Fig. 4), confirming the previous remarks. The trade-off zone is decisive in the multiobjective design. The resultant configurations of the fixture after each successive phase are presented in Fig. 5. It can be noticed that the first objective moves the locators close to the boundaries as far as possible from each other, while the second one reorients them to the surfaces' interior. Figure 4: General behavior of a 2-step interchange. Figure 5: Fixture configurations during a 2-step algorithm: (a) initial, (b) after max(det M), and (c) after min(d) respectively. B. Designer decision in finalizing the fixture During the second phase of the algorithm a fairly significant decrease in the determinant value is registered, so few solutions will be acceptable for the multi-objective problem. In order to overcome these problems, an active designer control during min(d) interchange procedure is recommended. Essentially, the modifications consist in controlling the exchange procedure, such that the determinant of the improved locators must be permanently greater than a certain bound, simultaneously with the check for the form-closure condition. Even considering a tight bound for the determinant, more solutions are acceptable for the design than in the uncontrolled min(d) optimization case (fig. 6). As an example, the behavior of a single set of locators is studied during the interchange process of a 2- step algorithm controlled for two different bounds of the determinant value, emphasizing the fact that in the trade off zone the designer decision is decisive in finalizing the fixture configuration (fig. 7). Figure 6: Second phase of a 2-step random interchange algorithm: uncontrolled min(d); controlled min(d). Figure 7: General behavior during a 2-step algorithm applied on a single set of locators. (a) for B1 and (b) for B2. VII. OPTIMAL FIXTURE CLAMPING This section deals with a more complicated problem: to search simultaneously for the optimal clamp and locators in order to achieve a required fixture quality. Varying the clamp, it is obvious that the number of combinations for possible clamp-locators candidates is increasing very much. It will be shown that this problem is manageable for the precise localization objective. For the other objectives we will have to restrain the search of the optimal clamp inside of a small set of proposed locations, such that the optimization procedure could be handled. A. Optimal Clamp from a Set of Clamps In some applications the clamps have certain preferred locations, therefore the need to choose the best clamp from a proposed collection might be raised. For example, let's consider that a collection of preferred clamps is given, and an optimal fixture design with respect to the highly precise localization objective is needed. It is obvious that applying a random interchange procedure successively for each clamp, we find optimal fixture configurations for each specified clamp. Comparing the determinant values offered by these fixture schemes (fig. 8), we end up by selecting an optimal clamp and its corresponding locators, constructing the best- improved fixture design (fig. 9). Figure 8: Clamp selection from a collection of clamps for single-objective design. Figure 9: The initial collection of proposed clamps; the best clamp and the corresponding locators. B. Optimal Clamp from a Set of Clamps Furthermore, by extension, the selection of the optimal clamp from a set of proposed locations with regard to the multi-objective design problem can be considered. It consists of mainly applying the random 2-step interchange algorithm consecutively for each proposed clamp. By collecting the results after applying this procedure for all the clamps, we can compare their different behavior, and select the most appropriate one. It is obvious that an optimal clamp allows only small fluctuations of the determinant while the force dispersion is decreasing significantly (fig. 10). As an example, Fig. 11 illustrates the final fixture design consisting of the best clamp selected from a proposed collection with respect to the multi-objectives and the corresponding optimal locators. Figure 11: The initial collection of proposed clamps; the best clamp and the corresponding locators. VIII. CONCLUSIONS This article focuses on optimal design of fixture layout for 3D workpieces with an optimal random interchange algorithm. The quality objectives considered include accurate workpiece localization, minimal and balanced contact forces. The paper focuses on multi-criteria optimal design with a hierarchical approach and a combined-objective approach. The optimization processes make use of an efficient interchange algorithm. Examples are used to illustrate empirical observations with respect to the design approaches and their effectiveness. The work described here is yet complete. Since the inter-relationship between the locators and the clamps has a determinant role on the fixture quality measures, a more coherent and complete approach to study the influence of the clamp and search of the optimal clamp position is needed in future works. IX. REFERENCES [1] P. D. Campbell, Basic Fixture Design. New York: Industrial Press, 1994. [2] F. Reuleaux, The Kinematics of Machinery. Dover Publications, 1963. [3] B. Mishra, J. T. Schwartz, and M. Sharir, "On the existence and synthesis of multifinger positive grips", Robotics Report 89, Courant Institute of Mathematical Sciences, New York University, 1986. [4] X. Markenscoff, L. Ni, and C. H. Papadimitriou, "The geometry of grasping", International Journal of Robotics Research, vol. 9, no. 1, pp. 61-74, 1990. [5] Y.-C. Chou, V. Chandru, and M. M. Barash, "A mathematical approach to automate