Res Eng Des (I992) 4:75-87 Research in Engineering Design Theory, Applications, and Concurrent Engineering © 1992 Springer-Verlag New York Inc. Conceptual Design of Mechanisms Based on Computational Synthesis and Simulation of Kinematic Building Blocks Sridhar Kota and Shean-Juinn Chiou Department of Mechanical Engineering and Applied Mechanics, The University of Michigan, Ann Arbor, Ann Arbor, Michigan, USA Abstract. Although many ingenious mechanisms have did these designs originate? Addressing these issues been designed, the fundamental task of conceptualizing is like trying to pick the brains of the inventors who these devices is, to a great extent, still an art. While sophis- designed the mechanisms. Two approaches to this ticated computational tools for dynamic analysis of mech- problem have been developed in the course of a long anisms exist, hardly any computational methods exist for history. The first approach is the creation of atlases generalized synthesis. To develop a computational model of mechanisms grouped according to function. for synthesis, a formal foundation for mechanisms design These remain the primary source of ideas. The sec- must be laid by rationalizing the process of mechanical synthesis. Rationalization in synthesis implies that com- ond approach involves an abstract representation of plex mechanical motions can be described in terms of the structure of mechanisms similar in spirit to the primitives or building blocks. In this paper, we present a symbolic representation of chemical compounds in matrix methodology that forms the basis for a computable the field of chemistry. approach to design synthesis. In this methodology, the continuous design space of a mechanisms domain is dis- cretized into functional subspaces, and each subspace is 1.2 Indirect vs. Direct Design Synthesis represented uniquely by a conceptual building block. The matrix scheme serves as a formal means to (a) represent Indirect synthesis of mechanisms using either and reason with the building blocks at different levels atlases or abstract representation of building blocks of abstraction, (b) generate alternate conceptual design is necessary because a design cannot be synthesized configurations, and (c) facilitate rapid simulation of design directly. To perform direct synthesis and optimiza- concepts by connecting a series of building blocks. tion for a given set of design specifications, an ana- lytical solution should exist that provides the joints and the dimensions of all the elements of the system. 1 Introduction The inherent motion characteristics of mechanisms 1.1 Background are too difficult to express analytically, and the same motion can be obtained several different ways, rul- Myriad ingenious mechanisms, including toys, ma- ing out any hope for direct synthesis methods. chine tools, automation equipment, cameras, copy Therefore, a systematic classification of various so- machines, and automobiles, have been designed lution principles through abstractions is necessary over the past 100 years [1, 20]; however, the funda- in order to conceptualize alternate working solutions mental task of conceptualizing how these devices to a given design task. perform the desired motions is, to a great extent, still an art. Once these mechanisms are built, their kinematics are not difficult to understand, but under- 1.3 Early Research standing how they were conceptualized is. One of Reuleaux was among the first to attempt a classifica- the most challenging questions that faces anyone tion scheme and to study machinery design system- who attempts to automate the design process is: atically [32]. He found that a machine consists of a What makes these mechanisms ingenious, and how limited number of parts occurring over and over, and he called these parts constructive elements. Some that he identified are screws and screwed Offprint requests: Design Laboratory, Department of Me- joints, keys, rivets, bearings, pins, shafts, couplings, chanical Engineering and Applied Mechanics, The University of belts, cords and ropes, gears and gear trains, fly- Michigan, Ann Arbor, Ann Arbor, MI 48109-2125, USA wheels, levers, ratchet wheels, and springs. In addi- 76 Kota: Conceptual Design of Mechanisms Based on Computational Synthesis tion to identifying the elements, Reuleaux also as- terized from several different viewpoints, such as a signed them symbols in an attempt to mimic symbols mechanisms's function and its operational con- in chemistry. His symbolism did not include all the straints. The goal of this work, however, is to de- variables necessary for analysis, so his attempts velop an automated synthesis procedure with de- were not successful. The subject of abstract repre- sired behavior as the starting point. sentation of the basic building blocks of machines In spite of its limitations, graph theory is still fell into neglect for almost a hundred years. His a useful and a practical technique for systematic symbolism work was not fully exploited for system- enumeration of alternate kinematic chains, espe- atic synthesis of mechanisms even with the advent cially if an initial mechanism configuration is known of computers. Other researchers, however, did carry a priori. It is particularly useful in patient recogni- on Reuleaux's attempts to identify building blocks tion. Numerous practical applications of graph the- using different names for them, including details, ory have been developed [6, 13, 16, 35]. A detailed elements, and simple-parts. These classifications account of various techniques for creative design consisted of ad hoc labeling and did not provide a and type synthesis of mechanisms is given in [10]. formal definition of why a part should be designated Recently, researchers in computer science have as a basic building block; therefore, they were not investigated qualitative theories for analyzing the useful in systematic synthesis of complex machines. behavior of mechanisms and have qualitatively ana- lyzed the kinematic behavior of mechanisms from the shape and the initial positions of its parts [11, 1.4 Graph Theory 21]. Forbus proposed symbolic place vocabularies Beginning in the mid-1960s, the abstract representa- for qualitative spatial reasoning to capture the geo- tion of kinematic structure was investigated with the metric interactions between physical objects [12]. aid of graph theory first by Freudenstein and Maki His research goal was to develop methods for quali- [14, 15]. This procedure is based on the separation tative descriptions of motion sufficient to understand of the mechanism structure from its function. The a mechanism. His analysis is restricted to two-di- mechanism structure can be enumerated in an essen- mensions. The majority of applications of qualitative tially systematic, unbiased fashion. The method reasoning in kinematics have been in the analysis helps establish the total number of mechanisms in a rather than the synthesis of mechanical motions. given class given the number and type of joints. The The methodology presented in this paper is an ability to enumerate all possible kinematic topolo- alternate approach to type synthesis of mechanisms. gies using graph theory lends itself to development It compliments the graph theory approach by ad- of expert systems [5, 17, 22, 27, 30, 34, 35]. The dressing a higher-level synthesis task. Mechanism kinematic chains that are enumerated using graph solutions generated by the matrix methodology theory are based purely on topological considera- could serve as a starting point for graph theory based tions. The desirable motion characteristics of mech- enumeration of alternate mechanisms. anisms are too complicated to be comprehended by evaluating the kinematic chains. By assigning differ- 1.5 Rational Classification and ent dimensions to individual links, entirely different Matr& Methodology motion characteristics can be obtained by mecha- nisms derived from a single kinematic chain. The work described in this paper provides a system Typically, a designer wishes to specify the desired of indirect design synthesis. It provides a rational behavior in terms of motion characteristics such as: classification scheme for the constructive elements degrees-of-freedom, sequence of output motions, based upon their kinematic nature. In an earlier pa- nature of output motions (translational, rotational, per, we presented a qualitative classification scheme etc.), whether the input-output relationship is linear and computerized catalogs that we developed for a or nonlinear, and continuous or intermittent, unidi- subset of mechanisms [23-25]. This classification rectional or bidirectional, reciprocation or oscilla- scheme is in contrast to the earlier ad hoc methods tion, and so on. One of the limitations of graph the- of labeling that cannot be used for synthesis. Repre- ory is that the enumeration procedure is based sentation methods presented in this paper are based primarily on structural and topological considera- on functions and operating constraints--notions tions. Graph theory accounts mainly for the degrees that are natural to human designers. The work de- of freedom and structural constraints such as scribed in this paper uses computational symbolic whether the input link is connected to frame or not. matrices for synthesis and simulation of a wide vari- Such considerations alone do not reflect the desired ety of mechanisms based on the work of Denavit behavior. Therefore, mechanisms must be charac- and Hartenberg [7]. The matrix methodology, based Kota: