Beyond the Hierarchy : System-Wide Rearrangement as a Tool to Eliminate Iteration

A primary tenet of axiomatic design theory is the first axiom, stating that independence of functional requirements should be maintained throughout the design process. As the high level requirements are decomposed into greater detail, and information added to the design with the goal of creating a realizable system, the designer creates subsystems that satisfy the first axiom. While higher level decisions imply an intent that should be maintained as detail is added, this is often not done. When a system is designed that results in some unintended interactions between design elements, it is possible to achieve a non-iterative design process by rearranging the leaf level elements as a collective set. This is shown for a subset of elements from the design of a chemical mechanical polishing (CMP) machine tool. Introduction The reader is assumed to be familiar with the axiomatic design process. All terms used in this paper are consistent with those presented by Suh [1]. Axiomatic design begins with the most general requirements of the system, and decomposes these into subrequirements, which are then mapped to design parameters in the physical domain. The hierarchical collection of functional requirements (FRs) and design parameters (DPs) generated during the zigzagging process is termed the system architecture, and elements which require no further decomposition are leaf level elements. Zigzagging is repeated until it is possible to construct the system from the information contained in the system architecture. When systems are designed with axiomatic design, high-level design equations represent conceptual choices made by the designer, and the intent carried with those choices. In order to realize any system, information must be added. Information is added to the system through the decomposition process, which expands FR/DP pairs into sub requirements which are in turn mapped to the physical domain. As this zigzagging process continues, adding information, the decisions must remain consistent with those at higher levels if the original intent is to be maintained. Although this is the goal of the design process, it is not so easily accomplished. Particularly when designing large systems, which must satisfy a large number of functional requirements, it is likely there will be unconsidered influences, or emergent properties that may not be intended, but can not be avoided. When the full system matrix is created of all leaf level design elements, interactions that fall outside of the original design intent may be uncovered, and serve to contradict the original intent. Elements in a design matrix that fall in the upper triangle represent iteration in the design process. While iteration does involve an increase in the design effort, it is often considered inherent to the design process [2]. The conventional solution in axiomatic design would be to rework the design, and develop a set of design parameters that do not result in a coupled system. Such practice would result in a desirable system that avoids all undesirable interactions. Rather than search for a design solution which altogether avoids any of such small scale interactions, it is proposed that it is possible to rearrange the full system matrix, or any subset of leaf level FR/DP pairs beyond the structure that is defined by the hierarchy of decomposition, to reach a design sequence which does not require iteration. Rearrangement of design elements beyond the structure defined at each level of the decomposition process has not been shown within the axiomatic design methodology, and has potential to reduce iteration to the minimum necessary. Another matrix based analysis method, the design structure matrix (DSM), does demonstrate resequencing of design elements, but does not generally keep the hierarchal structure once the matrix has been formed [3]. This is a strength of the DSM method that may be incorporated into the axiomatic design method as demonstrated in this paper. The DSM method acknowledges that iteration is going to exist in the design process and attempts to manage the iteration as necessary [4]. The DSM and axiomatic design matrix are very similar, and have been considered identical [5]; However, there are differences. The design matrix of axiomatic design often includes design parameters that are not strictly physical components. This ability to utilize features of components rather than components themselves is a particular strength of axiomatic design. Also, axiomatic design preserves the concept of FRs in the design matrix, assigning a DP to each FR. FRs and DPs are paired together, linking the rows and columns in the design matrix just as in the DSM, but the matrix information may be different. The design matrix represents the effects of DPs on functions, as opposed to the effects of physical components on each other, as in the DSM. Case Study: Chemical Mechanical Polishing (CMP) machine As an example for the utility of system wide rearrangement, the design of a chemical mechanical polishing machine will be used. This machine was developed as part of a research program at MIT, and has demonstrated advanced capabilities to polish silicon wafers for semiconductor fabrication. The system was developed within the axiomatic design framework, and provides a full system matrix with approximately 100 leaf level elements. While the full matrix should be investigated as a whole to insure a properly sequenced design, much may be learned by looking at a smaller subset of FR/DP pairs. This may be useful, for instance, as a way to collect elements that are relevant to a particular piece of hardware. For the following example, the design elements that are relevant to the wafer carrier will be presented. The wafer carrier is the physical component of the machine that holds the wafer during polishing. As will be shown, FR/DP elements from various parts of the decomposition are embodied in the hardware of the wafer carrier. Therefore, the elements for this piece of hardware may be clustered together and then investigated as a part of the whole design. The relevant levels of decomposition are presented in Appendix A, along with some description of what the design parameters represent. Here, the collection of elements will be shown in matrix form, and the benefits of rearranging the matrix demonstrated. Wafer carrier design matrix If the leaf elements described in Appendix A are combined into a matrix, the result is shown in Figure 7. These are those elements that are relevant to the wafer carrier hardware. Also included in the matrix, but not discussed in detail is FR/DP 1.4. 6: Provide mechanical support-Mechanical structure. The mechanical structure is part of the machine support systems, those systems which are necessary to enable other systems. As is evident by inspecting the matrix, it is not lower triangular. During each stage of the decomposition, a lower triangular matrix was reached. Therefore, the full matrix shown in Figure 7 should be lower triangular. Unfortunately, it was not possible to maintain the intent of the higher level decisions in the strictest sense. The result is a matrix with some elements in the upper triangle. This will result in iteration during the design process, and therefore added time and expense during the design cycle. One important characteristic is the nature of leaf level design elements. Since the leaf levels may be combined to make the parent (branch) levels, they are the elements of the design which must be individually set. Once this is accomplished, the structure of the hierarchy may be followed from the bottom of the top to realize the system. Because all the leaf levels must be determined, it is reasonable to consider them as the necessary and sufficient set of information to realize a system. In the matrix of Figure 7, only leaf levels are represented. Therefore, they may be reordered to reach an appropriate sequence for design. The result of such reordering is shown in Figure 8. As may be seen, the matrix is now lower triangular, to the extent that it can be. There is a fully coupled block that represents the closed loop control system of the retaining ring vertical motion, as discussed above. This is handled with a real time controller that iterates the solution during operation of the machine, guaranteeing FR satisfaction. Figure 7: Matrix of wafer carrier design elements Figure 8: Rearranged matrix of wafer carrier design elements. Full design matrix Similarly to the subset of elements that make up the matrices in Figures 7 and 8, the entire collection of leaf level elements may be investigated and restructured. The full design matrix created by the decomposition for the CMP machine is shown in Figure 9. As before, the full matrix is reordered, and the result is shown in Figure 10. The matrix in Figure 10 represents an improved sequence for the design elements to be set, in a manner that will reduce the iteration required in the design. As may be seen in the figure, some elements remain in the upper triangle of the design matrix. These represent iterative loops that may not be eliminated. Conclusion As has been shown with the CMP case study, although design intent may be for a purely uncoupled or decoupled system, details of the implementation can lead to unpredicted interactions. Due to these interactions, iteration is required in the design process. If the full system matrix, or even a subset of it, is rearranged to create the desired lower triangular form, iterations in the design process may be reduced or eliminated. It would be useful in this process to use an algorithm that would efficiently structure the matrix. While the method described here does show promise for improving the design process, it does carry with it some potential issues. By redefining the correct sequence for design, iteration is reduced; however by ignoring the structure of the hierarchy, other useful concepts of axiomatic design are cha