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 discretized 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 mechanisms 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: Conceptual Design of Mechanisms Based on Computational Synthesis 77 on rationally classified design building blocks, is not map abstract functions to alternate structures or only intuitive but also computable. It also covers a physical artifacts. wide range of mechanisms, such as cams and follow- The concept of mapping functions to structures ers, gear trains, linkages, ratchets, and screw mech- in conceptual design is shared by many researchers anisms. Our method allows for computational syn- [2, 8, 18, I9, 29, 31, 33]. Computational models for thesis of design. configuration design are investigated in [26, 28]. All of these researchers have found that the nature and the granularity of the building blocks are important 2 Computational Synthesis issues in any automated design system. If the granularity of the building blocks is too fine, combinational 2.1 Discretization of Design Space: Basis for explosion occurs. If the granularity is too coarse, Computational Synthesis novel synthesis of building blocks will be hindered. By identifying abstract building blocks, we will Therefore, the challenge in defining building blocks chunk the subspaces inside the continuous design is to determine a proper granularity that precludes space. Each building block represents, in an abstract combinatorial explosion and permits synthesis of way, a subspace of the entire design space. Once new configurations. the most promising subspace is identified for a given Function-to-structure maps are based on physical design problem, we can use mathematical models or working principles. During the function decomposi- design rules (modification rules) to vary the design tion process, it may be possible to discover more parameters within the subspace. We have success- than one way to decompose a given function into fully implemented this methodology in a Dwell- subfunctions or components. These, then, are alter- Expert program for a subset of mechanisms [23-25]. native function-level solutions. Each should lead to Hoeltzel and Chieng [18] have developed a system- a different design configuration. This functional de- atic approach to pattern classification of motion sign methodology has been developed and imple- curves and automated the design process using neu- mented for hydraulic systems [26]. With an under- ral networks. The notion of using building blocks to standing of the nature of the building blocks of describe the domain and using modification rules motion, this methodology can be extended to the to generate variations of the building blocks is like general mechanisms domain. piecing together the design subspaces that were orig- inally discretized. Such discretization and recombi- 2.3 Cam Synthesis as an Example of nation is an essential part of the conceptual design Discretization and Recombination automation process. Consider a classical example of mechanical cam- design process where discretization takes place im- 2.2 Discretization in the Design Process plicitly. The geometry of a cam profile dictates, to a To accomplish computational synthesis in the design great extent, the function of a cam. The function of process, the requirements of a particular design must a cam is described in terms of the displacement of be discretized into functions and constraints. The the mating follower. The follower displacement is a goal of the design is to fulfill first the functional continuous function. However, the desired func- requirements and then satisfy the constraints. So, tional relationship, including first-, second-, and design alternatives based on function are filtered higher-order derivatives, can be approximated by a through the constraints to select a final design. piecewise combination of standard library functions. Not only must the requirements of the design These functions can be, for example, cycloidal, har- be discretized but so must the design knowledge. monic, trapezoidal, or polynomial, using only a finite Design knowledge is discretized into functional and set of standard mathematical functions and follower- physical building blocks [26]. Functional building types. One can systematically synthesize the entire blocks describe only primary functions without con- cam profile by piecing together the individual profile- sideration of constraints, physical descriptions, or segments. Therefore, conceptually, one can create manufacturing processes. Physical building blocks numerous output motions by novel combinations of describe a set of physical artifacts that fulfill a spe- finite building blocks. cific primary function and include all the information omitted for functional building blocks. Functional 3 Computational Synthesis of Motion building blocks can be decomposed into alternatives and components. Physical building blocks are de- Traditionally, kinematic synthesis tasks are classi- composed by alternative alone. This allows us to fied as: motion generation, path generation, and 78 Kota: Conceptual Design of Mechanisms Based on Computational Synthesis

function generation. In practice, motion generation euucaonal Physical Building Blocks

usually is restricted to a finite number of positions Buildi,~Blocks Linear I/O relationship Non-Linear I/O relationship at which the designer desires to control the orienta- r,a,,~atio,, tion of output members. In path generation, a de- ~;~sl,tio, sired path, or discrete points along a desired path, are considered. The synthesis task involves config- Simple wedge Double slider uring mechanism solutions in which the endpoint ~.o.~t~o, traces the desired path. Function generation in- Rotation volves synthesis of mechanisms that satisfy the pre- Cam- Oscillating scribed input-output motion relationships. Com- Gear,,pair Universal Joint Four-bar linkage follower puter-based design tools for these synthesis tasks, Rotation such as LINCAGES [9], perform dimensional syn- ~r;>st,,io,

thesis but not type synthesis. These tools dimension Cam - translating the individual links in a presupposed mechanism Rack and-pinion Screw mechanism Slider-crank follower topology; they do not address the issue of determin- Rotatio, ing the most suitable type of mechanism for given ~,;J'ico~t design task. In the traditional classification of the

phases of a design process, type synthesis can be Spiral gear Bevel-screw gear viewed as conceptual design and dimensional syn - thesis as parameter design. The matrix scheme pre- Fig. 1. A taxonomy of functional and physical building blocks, sented in this paper addresses conceptual design of mechanisms for function-generation tasks. on whether the output and the input must be interchangeable or whether the output must be bidirec- 3.1 Function Generation: Problem Description tional. Regardless of the constraints, the fundamental design requirement of the artifact is often purely • Given: A description of input and output motion kinematic. Therefore, it is important to classify all and a set of constraints, such as kinematic and function-generating systems according to input-out- dynamic operational constraints, cost constraints, put motion relationships (i.e., rotational motion, and reliability. translational motion, and helical motion). This leads • Find: One or more mechanism configurations that to the hierarchical structure shown in Fig. 1. A more fulfill the prescribed functional requirements detailed account of functional and physical building within the constraints. The configuration should blocks must be developed. include descriptions of one or more mechanism The functional design building blocks for the kine- types and a feasible range of parameters that con- matic synthesis of mechanical systems corresponds trol the performance characteristics. to each of the leaf-nodes of the decomposition hierarchy. Figure 1 shows some of the physical building The function-generation task involves synthesis blocks. Corresponding to each functional building of a mechanical device to satisfy a desired motion block, a physical building block has been identified. transformation from the input to the output. For For example, corresponding to the linear rotation instance, the input may be a uniform rotational mo- <-> translation functional building block, a rack- tion, and the output may be an oscillatory motion and-pinion primitive mechanism has been identified. through a certain range with a specified dwell period The form of the rack-and-pinion as shown in Fig. during a portion of the cycle. The input and the 1 should be treated conceptually. Each primitive output are generally considered to have a fixed axis. mechanism has concept variants, physical variations Mappings from functional specifications to mecha- of the same solution principle (see Fig. 2). Although nism types are many-to-many. In a simple function- the devices shown in Fig. 2 have either multiple generation task in which the input is rotational mo- racks or multiple pinions or are used in conjunction tion and the output is translational motion, the possi- with other primitive elements, the underlying design ble candidate mechanism types are slider crank, concept remains the same in all the devices. The rack-and-pinion, screw, rope and pulley, or cam- notion of concept variants permits the grouping of follower systems. The choice of a particular type numerous designs into a single design concept and depends on the constraints imposed by the design thereby helps discretize the design space into build- specifications. For example, the choice may depend ing blocks. Kota: Conceptual Design of Mechanisms Based on Computational Synthesis 79

principle--that is, the concept represented by a building block. The notion of using a limited set of building blocks to describe the entire domain of (a) ~ - (b) " (c) mechanisms does not abandon flexibility. In fact, the notion of modification operators complements the notion of a limited set of building blocks and provides the flexibility to explore physical variations that are typical in the continuous design space of the (d) (e) (0 mechanical world.

Fig, 2. Variants of the rack-and-pinion concept. (a) standard rack-and-pinion, (b) multiple racks, (c) multiple pinion-based transfer mechanism, (d) step-wise motion due to modified inter- 4 Qualitative Matrix Representation of the face, (e) dwell motion due to modified interface, and (f) a clamping Building Blocks device--rack-and-pinion building block in conjunction with a spring and a wedge. Conceptual design is based on an appropriate means of abstracting functionality, whether it be the simple functionality of an individual mechanical widget or 3.2 Concept Variants the complex functionality of an electromechanical The building blocks derived from the function de- system. In our approach to design synthesis, the composition hierarchy represent standard physical functionality of a primitive mechanism is expressed forms. Some of the variations of the standard form as a concatenation of matrices, including a motion of rack-and-pinion are also shown in Fig. 2. It is transformation matrix (MTM) and a sequence of important to note that each of the building blocks constraint matrices. For example, a rack-and-pinion identified represents a "solution-concept" rather building block is expressed as: than any particular physical form. That is, "rack- Output Function matrix and-pinion" is a solution-concept for transformation (MTM) between rotation about a fixed axis and translation. The concept itseff does not preclude, say using multiple racks or multiple pinions. Many other variations may exist. This leads to the notions of standard f } IIi 0°°°°°'00 00 0°0 0il form and modification operators. The modification r tl_i l o o o operators transform the standard form into a suitable Basic constraint matrices variation of the conceptual building block. The standard form and its variations are all essentially differ- 0 0 ~ 0 + f0o0000 0 ~-~ ~-~ ent manifestations of the same working principle. ++ +~ 0 0 The modification operators when applied to the stan- 0 <--> 0 0 0 0 0 Ol dard form change the behavior of the system, re- Input taining, however, the underlying concept. Examples of some modification operators are given here: [o0 0o o0 +o +0 ijl[ ; ,l.R;, + 0 0 0 + + R_~ 1. Duplicate the interacting elements of the building 0 + + 0 0 T.~ ~- 0 + 0 0 r~, block. (Fig. 2b and c) + + 0 0 0 2. Change the form of the interface. The tooth geometry is modified to obtain step-wise motion (Fig. The first 6 x 6 matrix is the MTM which is con- 2d) and dwell motion (Fig. 2e). structed in order to abstract the couplings between 3. Add one or more primitive elements (wedge, the output rotations and translations (the Ro's and spring, lever, etc.). Figure 2f shows a clamping To'S) and the respective input rotations and transla- device based on this modification. tions (the Rs's and Ti's) for x, y, and z coordinates. 4. Apply a kinematic inversion. For instance, this The ones in the matrix represent all the motion trans- rule modifies a standard gear-pair into a planetary formations that are possible with this class of de- gear system. vices. The transformation Ty <-> R z is just one of A generic set of modification operators should be them. The other two 6 x 6 matrices are constraint defined for each building block to enable the com- matrices. The first of these indicates the reversibility puter to create appropriate variations of the solution or irreversibility of the allowed couplings (denoted 80 Kota: Conceptual Design of Mechanisms Based on Computational Synthesis

C : Constant transmission ratio (~) : Continuous motion FBB MTM Physical Building Blocks & ~) : Intermittent ordwell motion V : Variable transmission Their Constraint Matrices <----> : Input & output are interchangeable n : Cycle ratio between the input & output + : Output is in the same direction as input -+ : Reciprocating/Oscillating 000000 3 000000 R~+T 00000O 000000 0000 I 00000'1 ~o o o o'1 Symbolic Algebra : 100000 O+ 000 O_+O 001 o±o oo t 000000 00~0 OOVOO] oovool 0 *© =C) C*c=e ,-----,.,-____,.,...._, O0 " O0 0<-->01 0 *C: = C~ C* v ~v <.-.-.>* <._1.. =< ..... oo o-o" loooo~J oooonj000<--- 0t V * V = V (ratchet)

4 7000000 q

Fig. 3. Operational constraints. R+-~R 0010 4 4 00000 tOoOoo9 oo0 o 00000 - o2_o I}° o°° oo°] Fig. 4. Operational constraints in matrix form. 0000003 22co0, oov o oovool 0 00<-->OI 0 0 O< O00<- - 01 0 On/ 000 l O0 00nJ

00000 O 000000 by the symbol <->), which in this simple case are 000000 T+-~T 000000 all reversible. The other indicates whether the re- 0001 O0 ?, °o oo / - o ooI 000000 v o oi Ioo c 0 0° 00 l O000 0 O" tional (or in the case of rotation, unichiral) or bidirectional (bichiral) outputs, as indicated by the presence of singly or doubly signed matrix elements corre- Fig. 5. A taxonomy of motion building blocks and their matrix representations. FBB, functional building block; MTM, motion sponding to the appropriate couplings. As indicated transformation matrix, by the 1s in the MTM, a generalized rack-and-pinion can be used to convert a translational motion along any direction within the x-y plane into rotational be unique for each building block. Figure 5 shows a motion about the z-axis. The operational constraints taxonomy of motion building blocks and their matrix are expressed symbolically in Fig. 3. The operational representations. As indicated in Fig. 5, although constraints are also expressed in matrix form (Fig. multiple physical building blocks, rack-and-pinion, 4). Concatenation of the constraint matrices is based slider-crank, and screw mechanism in row 1, share a on symbolic manipulation of individual constraints common functional building block (FBB), individual according to the rules of symbolic algebra. constraint matrices are different. Therefore, a func- In a computational design system for conceptual tional matrix, in conjunction with a particular con- design, the MTMs and constraint matrices provide straint matrix, provides a unique representation for a rationalization of the possible functions of mecha- each physical building block. Note that although the nisms. The conceptual design process is initiated function matrix of the slider-crank and the rack-and- with the statement of the design task that describes pinion are the same, their constraint matrices are the input signal and desired output motion. The pre- not. Thus, the initial set of candidate building blocks scribed task is then decomposed into a series of that satisfies the basic motion requirements is fil- subfunction motion transformation matrices. Each tered by imposing the basic motion constraints. mode of decomposition leads to a unique design configuration. The decomposed subfunction matri- 4.1 Different Levels of Representation ces are then matched against a library of matrices (MTMs and constraints) representing existing de- The matrix representation method allows for reason- vices in the knowledge base. Thus, the computable ing with the abstract function of the primitive mecha- process of systematic decomposition and matrix- nisms without cluttering the reasoning process with matching constitute the reasoning processes in- unnecessary details. By investigating the nonzero volved in conceptual design. At the lower levels of terms in the motion transformation matrices, a com- abstraction, discussed in Section 5, the symbolic puter program can select possible candidate mecha- elements of candidate matrices are replaced by the nisms that can fulfill the basic kinematic function actual design parameters before performing qualita- and constraints. The matrix representation allows tive simulation of the designed artifact. The top-level for inclusion of additional matrices to represent the matrix representation of building blocks consisting dynamic and cost constraints. The next tower-level of a string of function and constraint matrices is of abstraction contains design parameters for unique, but the function matrices of building blocks input-output relationships. These matrices are need not be unique. However, in conjunction with called parameter matrices and are used in qualitative constraint matrices, the overall representation will simulation, which is explained in Section 5. Kota: Conceptual Design of Mechanisms Based on Computational Synthesis 81

4.2 Matrix Operations for Task Decomposition other matrix that has its only nonzero element occurring in fh row and i th column as shown A given design task is first formulated as a set of below. system specification matrices. This set consists of desired motion transformation matrix and operational constraint matrix. The first step in our compu- Basic operator Original MTM tational synthesis procedure is to match desired MTM against building block MTMs. If a suitable oooooo ° OOOOOOO match is not found, the desired MTM is then decom- 0 0 0 0 0 , 1 0 0 l 0 0 0 0 0 0 0 0 0 0 0 posed into a series of subfunction matrices. The 0 0 1 0 0 0 0 0 0 0 decomposition process can be automated by per- [°°°°°i;0 0 0 0 0 li!°°°°° 0 0 0 0 0 ] forming matrix manipulations. Based on our prelimi- nary research, we have identified three methods of Tr <-'> Hv T~. <--> Rx decomposition using three different types of matrix manipulation techniques: column shifting, row shifting, and a decomposition matrix. New MTM The goal of these matrix manipulation techniques is to transform or relocate ones in the desired MTM 0 0 0 0 0 0 in such a way that resulting MTMs (or subfunction = E0000001 0 0 0 0 0 0 MTMs) match one or more building block MTMs. 0 0 0 0 0 0 0 1 0 0 1 0 Matrix operators perform these transformations. 0 0 0 0 0 0 1. Column shifting: To shift column i to columnj in Rz<--> Hy the original MTM, postmultiply the original MTM by another matrix that has a nonzero element only in its ith row andf h column as shown below. 3. Decomposition matrix: A given matrix can be decomposed by starting with any of its non-zero Original MTM Basic operator elements. If a non-zero element (i, j) is selected as a starting point, the process of decomposition iooooo,ooo OOO,OOOoooooo takes the following steps: 1 0 0 0 , 0 0 0 0 0 Form a new matrix A with all of its ele- 0 0 0 0 0 0 0 0 0 ments zero except the original nonzero ele- 1 0 0 0 0 0 0 0 0 ments in column j. 1 0 0 0 0 0 0 0 0 Form a decomposition matrix B with Bji as Hy: <--> R z Hy: <--> Ry its only non-zero element Form a matrix C with all of its elements zero except the original non-zero elements New MTM in row i. The original matrix is [A]*[B]*[C]. Matri- 0 1 0 0 0 ces A and C can be recursively decom- (i000000 1 0 0 0 posed, if necessary, using any of the three 0 0 0 0 0 matrix manipulation techniques described 0 1 0 0 0 above. 0 1 0 0 0

Ry <--> R z An example of use of decomposition matrix for decomposing an MTM is shown below with element (4,1) as the starting point Physically, this matrix decomposition implies that to transform a rotational motion about the Z- Original matrix Matrix A axis (Rz) into a helical motion in the x-y plane (Hy~), one could transform the rotational motion 0 0 0 0 0 0 0 0 0 0 0 0 about the z-axis first into a rotational motion 0 0 0 0 0 0 = 0 0 0 0 0 [°°0°0°1 0 1 0 0 0 l I°°°°°°!101 0 0 0 0 about the y-axis (Ry) and then transform Ry into 1 0 l 0 0 0 1 0 0 0 0 Ovz, 0 0 0 0 0 0 0 0 0 0 0 2. Row shifting: To shift row i to rowj in the original MTM, premultiply the original MTM with an- It~.>, <--> H~ Hxy <--> R~. 82 Kota: Conceptual Design of Mechanisms Based on Computational Synthesis

Table I. O 0 0 0 0 0 -+ 0 0 0 Prescribed motion 0 0 C/V 0 0 characteristics Input Output 0 0 0 < .... 0 0 0 0 0 n Type of motion Rotation Translation (reciprocating) Orientation X-axis X-axis Fig. 6. System constraint matrix. Continuity Continuous Continuous Direction + + Linearity Not specified Not specified Reversibility <-- Periodicity Not specified Not specified Step 3: Decompose the desired MTM into a series of subfunction matrices:

Matrix B Matrix C Original matrix Sub function rO 0 0 0 0 0000000 ° 000000 ° 0 0 0 0 0 0 0 0 , 0 0 0 0 0 ;,000001 = °o°°°°° 0 0 0 0 0 li 0 0 0 0 i00 0il [0o0o0;1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 l 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

R.,. <--> Yr- 7 x <--> H~: Subfunction 2

Usually there are several ways in which a matrix Ii i 0 0 0 0 0 0 0 0 0 can be decomposed. Depending on the mode of 0 0 0 0 decomposition, alternate solutions are generated. 0 0 0 0 The design example given in the next section illus- 0 0 0 0 trates this point. Step 4: Check if the new series of MTMs match any existing building block MTMs. 5 Design Example Subfunction 1 Candidate The task is to design a mechanism that converts a building blocks uniform rotational motion about x-axis into a recip- ;000°° ° rocating motion along the same axis. The output and 0 0 0 0 Slider-crank h 00000th 1 0 0 0 Cam-follower the input should not be interchangeable (see Rack-pinion Table I) 0 0 0 0 0 0 0 0 Candidate Subfunction 2 building blocks Step 1: Form the system specification matrices [;0000;o00o J (a) Form or derive the desired MTM from the speci- 0 0 0 0 Worm-gear fications of the type of motion and orientation. 0 0 0 0 ~ Bevel-gear 0 0 0 0 0 0 0 0 [Output matrix] • [Input matrix] ~ = [desired MTM]

° o° oooo Step 5: Check the constraint matrices. • [1 0 0 0 0 0t = 0 0 0 0 0 0 0 0 (a) Write the constraint matrices for each of candi- 0 0 0 0 0 0 0 0 date building blocks (Fig. 7). (b) Concatenate building block constraint matrices in the sequence in which the original MTM was (b) Form the System Constraint Matrix, given that decomposed. Since three building blocks the operational constraints are formulated in a matched the first subfunction MTM and another matrix form (Fig. 6). two building blocks matched the second sub- Step 2: Match the desired MTM against the function, we have a total of six (3 × 2) combina- MTMs of the building blocks. In this case, no match tions leading to six different mechanism config- is found. urations. Constraint matrices for each of the Kota: Conceptual Design of Mechanisms Based on Computational Synthesis 83

Slider-Crank : Rack-Pinion : Cam-Follower : Configuration 4 = [Rack-pinion] * [Bevel-Gear]

0 0 01 o o o o'l o o o o _+ooo +ooo [7°+_oo f°o° o 0 O| o + o o o//o- o o o 0 V 0 0 0 C 0 0 0 0 V 0 0 0 C 0 0|= oo c o o/*/o o c o o 0 0 <---->0 0 0 <---->0 0 0 0 < .... 0 0 0 <---->0 | o o o <---->o[ |o o o <---->o Ii°°°°lIi°°°°lL0 0 0 nJ 0 0 0 nJ 0 0 0 0 L0 0 0 0 nJ 0 0 0 0 nJ k0 0 0 0 n Worm-Gear : Bevel-Gear : Configuration 5 = [Cam-Follower] * [Worm-Gear] C~ 0 0 0 0"] o o 0 - 0 0 0 ooo _+ o o ° ;..'! o o O~ 0 0 0 C 0 o 0 C 0 0 Ii 0°°° V 0 0 ] = 0 0 v 0°!} 0 0 C 0 0 0 < .... 0 0 0 <---->0 0 0 < ..... o o 0 0 < .... 0 0 O< .... 0 0 0 o 0 0 o n 3J o 0 0 n 0 0 0 0 0 0 0 0

Fig. 7. Constraint matrices. Configuration 6 = [Cam-Follower] * [Bevel-Gear]

i0 _+ 0 0 0 [Configuration 1 = [Slider-Crank] * [Worm-Gear] [~":+ 000 0 <00 0ilia00 0 - 0 oo 0 0 0 V 0 0 = 0 0 V 0 0 C 0 0 0 0 < .... 0 0 0 0 ..... 0 0 0 <---.>0 + 0 0 + 0 0 0 [oo o!}o 0 0 0 n 0 0 0 0 nJkO 0 0 0 n ,°ozo 0 V 0 = 0 V 0 0 0 C 0 0 0 < .... 0 0 <--->0 0 0 < .... Fig. 9. Constraint matrices--configurations 4-6. [}°°0 0 °i] 0 0 0 0°° nJ1 0 0 0

Configuration 2 = [Slider-Crank] * [Bevel-Gear]

O 0 0 0 0 Oust Wo~, Input ± 0 0 o + o 0 0 0 - 0 0 0 0 v 0 0 = o v 0 0 * 0 0 C 0 0 0 0 <---->0 0 0 <---->0 [0 0 0 <---->0 ~-- ear [i o°°°° 0 0 nJ1 [i 0°°°° 0 0 n 0 0 0 0 n (a) Configuration[ 00o 3 = [Rack-Pinion]] * [Worm-Gear] ~00 0 + 0 0 0 ~ 0 0 o ol[io o o!1 Fotlower /4 Bevel Gear 0 0 C 0 0 * 0 0 C 0 C 0 = Cam 0 0 < .... 0 0 0 <---->0 0 0 < .... (c) 0 0 0 0 0 0 0 nJ 0 0 0 Fig. 10. Three alternate mechanism configurations generated by Fig. 8. Constraint matrices--configurations 1-3. matrix decomposition. (a) slider-crank and worm-gear; (b) cam- follower and worm-gear; and (c) cam-follower and bevel-gear.

configurations are generated by using symbolic subfunction matrices, we are still left without a way algebra (Figs. 8 and 9). to visualize how the designed system functions. (c) Identify potential solutions--that is, the combi- Therefore, the next task is to enable simulation of nations of building block constraint matrices that the conceptual design using a simplified model. Al- match the system constraint matrix. Of the six though large-scale computer programs such as combinations, only three configurations (I, 5, ADAMS [3], are available and could be used, these and 6) satisfy the requirements after comparing are too complex for our purposes, since a simple and all six constraint matrices with the specified sys- approximate analysis is all that is required during tem constraint matrix. These configurations initial phases of design. Besides, all the pertinent form three alternate solutions to the given task information to perform a thorough dynamic analysis and are shown in Fig. 10. is not usually available at the initial stages. There- fore, a qualitative simulation method, which simply 6 Qualitative Simulation Using combines the building blocks that are selected during Matrix Representation initial phases of design to show the input-output motion relationships, is needed to help evaluate con- While the process of synthesizing building blocks ceptual designs. It is qualitative in the sense that it is accomplished by selecting an appropriate set of hides the details of intermediate joints and links and 84 Kota: Conceptual Design of Mechanisms Based on Computational Synthesis simply illustrates how the output member moves in relation to the motion of the input. The output of the first building block will be automatically treated as the input to the second building block and so on. Note that the sequence in which the building blocks physically combine follows exactly the same sequence in which the original desired motion function is decomposed into subfunctions. Fig. 11. A vector representation of a rack-and-pinion configuration with respect to the global reference frame. The qualitative simulation can be pertbrmed by combining lower-level parametric matrices. The parametric matrix representation for the relative motion between rack-and-pinion is given by the follow- This formulation, which is given only for the x-y ing matrix: plane with pinion-input, is generalized for all cases (rack-input-and-pinion-output, pinion-input-rack- ARI.I,/ output, x-y plane, y-z plane, or z-x plane) and ex- ~R-o = pressed as a single parametric matrix in GRF as: /,.xr,.,,l t s:r.,ij RI ,, 0 0 cos ~ sin ¢- -- 0 0 r /" A Rxi] sin ~ cos,P R~,jTx¢~ 1 0 0 0 0 hR2.,.[ Tvo I" r AR:i~ 0 0 0 cos ,p sin ~ 0 I" g st,,.,-'-'xr,., /f 0 rsin~ r cos ~ 0 0 0 sS, J r cos ~ 0 rsinp 0 0 0 r sin ,p r cos 0 0 0 0 ) 0 0 0 0 0 0 0 O0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 This matrix representation gives the output motion = 0 0 0 0 r sin ,p r cos ,p 0 (e.g., rack AT,,,) given an incremental input (e.g., 0 0 r cos~, 0 rsin~ 0 0 0 r sin ~, 0 r cos g, 0 0 0 AR u to the piMon). To combine the rack-and-pinion 0 0 0 0 0 0 0 building block to the next building block (e.g., slider- li crank), we need to represent each of the building blocks in relation to a Global Reference Frame (GRF). The parametric matrix transformation to 0 GRF is derived below (see Fig. 11) 0 For a pinion input and a rack output in the X-Y 0 0 plane, the vector To is given by 1"sin ~p(Ru/R,.:i) r cos ~(R~.]Rr:i) r sin ~(R~i/R~:i) - r cos g,(R:i/R~:i) 1" sin ~(R,q/R~)i) - r cos ~(RURs~.i) To = T~ + n + (1) 1

All the vectors in Equation (1) are defined respect where to the global coordinate frame X-Y.

Ryzi = Ry i + Rzi _,{o}n = and = {r,;}z, F Rxz i = Rxi + Rzi

The vectors above are defined with respect to the Rxy i = Rxi + Ryi local x-y coordinate. 6.1 Simulation Example To illustrate how individual parameter matrices can + sin ~ cos ~ 3 (2a) be combined to perform qualitative simulation of a mechanical assembly, consider a design task in J T,; + (r cos ~)Rzi + r sin p] which the input is a rotational motion having a con- t T~i + (r sin p)Rzi - r cos ,pj (2b) stant angular velocity (driving shaft) and the desired Kota: Conceptual Design of Mechanisms Based on Computational Synthesis 85 output has a slow cutting stroke (shaping machine) and a quick return stroke. The motor's speed must be reduced by using a gear train. The design function can be decomposed into (a) (b) (e) 1. Speed reduction (R-R), uniform output rotation. 2. Rapid return motion (R-R), nonuniform output rotation. 3. Reciprocating motion (R-T), nonuniform output translation. (e) In this example, the three design building blocks Fig. 12. Building blocks and their assembly. (a) spur gear train are generated by applying modification operators derived from the gear-pair building block; (b) inverted slider- to primitive mechanisms (gear-pair, inverted slider- crank derived from the slider-crank building block; (c) an offset crank, and a slider-crank). An inverted slider-crank slider-crank; and (d) assembly and (e) qualitative simulation of the assembly generated by concatenation of parametric matrices is used to obtain the desired ratio of tbrward to of individual building blocks. return stroke time. The details of design concept generation are not given, as the intention here is to illustrate the application of parameter matrices to simulate the assembly. The individual building blocks are shown in Fig. 12a-c. Figure 12d shows I0 0 0 0 t the physical assembly of the system. Figure 12e R2 = 0 R3 shows the qualitative simulation of the assembly R4 0 obtained by concatenating individual parametric ma- 0 0 0 trices. It is qualitative in the sense that it ignores the details of the individual elements that make up each Inverted slider crank: building block, and it also simply combines the input and output functions of the various building blocks. The result is an approximate analysis showing the 1 0 LIS~ general behavior of the designed artifact. By provid- T3 i 00 = 0 1 ing incremental inputs to the first parametric matrix, 0 0 1 A its output is computed and automatically serves as an input to the second building block. The concatenation of individual parametric matrices is equiva- 0 0 0 lent to the physical assembly of the system. The R3 parametric matrix representation of each building block is given below. 0 0 0 Gear pair 1 and 2: Slider crank:

Ii 0 0 (R~ + ROCO-] ° (L2COtz + \/L~ - (b - L2SOIz)2)C~ - bS'] 1 0 (RI+R2)SO| 1 0 T~ (L 2C O~ z + ~X/-~.~~- ~Ol~ )S tO - b C~ T, = 0 10 O1 J = 0 1 Li0 0 0 0 0 R = [014x4 ; (zero transformation from rotation to rotation) RI 0 0 -RI The output motion matrix can be described as o o [T4][T3][T2]lTi] = [Tout]4 Gear pair 3 and 4:

1 0 0 S~ Ii 0 0 (Rs+ R4)CO~ 0 0 S,, 1 0 (R3 +R4)SO| [To.3~ = o o o" /'2= 001o ~ J 0 0 0 where 86 Kota: Conceptual Design of Mechanisms Based on Computational Synthesis

S 1 = {[L2CO4z + X/L 2- (b - L2SO4z)2]CtO4 - bSq~4} the designer to perform quick simulations of various design concepts. + LIC~3 + (R 3 + R4)Ct~2 + (R t + R 2) The key concept is that we can synthesize complex and novel devices by reasoning with functions 8 2 = {[L2CO4z q- ~v/L 2- (b - L2SO4z)2]gdd4 - bCt~4} and operating constraints of motion building blocks. + L1StO3 + (R 3 + R4)Sth 2 + (RI + R 2) A complete set of building blocks are being identified by analyzing hundreds of ingenious mechanisms and from the literature [1,4]. A library of motion building blocks and their matrix representations is being de- [R4][R3][R2][R1] = [Rout] 4 veloped.

[Rout]4 = [0t4×4 Acknowledgments. The author wishes to thank the National Sci- ence Foundation, Design Engineering Division and the Dynamic 7 Conclusions Systems and Control Program, for sponsoring this research (grant DDM-9103008). The methodology based on matrix representation provides a computable theory for type synthesis of References mechanisms. The computational synthesis process involves manipulation of matrix elements (0s and I. Artobolevski, I.I., Mechanisms in Modern Engineering De- ls). The matrix method is used as a vehicle to repre- sign, v. 1-3, MIR Publishers, Moscow, 1986 sent design building blocks at different levels of ab- 2. Brown, D.C. and Chandrasekaran, B., "An Approach to Expert Systems for Mechanical Design," Proceedings of the straction. At the highest level of abstraction, each IEEE Trends and Applications Conference, 1983~ pp. building block is represented by a string of matrices 173-t80 (function matrix and operational constraint matri- 3. Chase, M.A., "Using DRAM and ADAMS Programs to Sim- ces). A function matrix represents the nature of mo- ulate Machinery, Vehicles," Agricultural Engineering, No- tions that can be accomplished with the mechanism vember 1978 4. Chironis, N., Mechanisms, Linkages and Mechanical Con- building block that it represents. Operational con- trois, McGraw-Hill, 1978 straint matrices reveal inherent limitations of the 5. Datseris, P. and Palm, W., "Principles on the Development building block interms of its capability to provide of Mechanical Hands Which Can Manipulate Objects by reversibility, input-output interchangeability, etc. Means of Active Control," ASME Paper 84-DET-37, ASME, In principle, additional constraint matrices, beyond t 984 6. Datseris, P. and Lee, H.T., "'Development of All Non-iso- operational constraints, can be appended to enable morphic Graphs for Mechanism Design with Emphasis on a comprehensive representation of each building Robot Gripper Design," Proceedings of the Tenth OSU Ap- block. The matrix representation scheme allows pliedMechanisms Conference, New Orleans, December 1987 computational synthesis of building blocks by pro- 7. Denavit, J. and Hartenberg, R.S., "A Kinematic Notation viding a formal means to decompose a given task for Lower-Pair Mechanisms Based on Matrices," Transac- tions of the ASME, 1955, pp. 215-221 into an ordered sequence of subtasks (subfunctions) 8. Dixon, J.R., "Artificial Intelligence and Design: A Mechani- and to match each subfunction matrix with one or cal Engineering View," Proceedings of the AAA1 Fifth Na- more building block function matrices. The repre- tional Conference on Artificial Intelligence, v. 2, American sentation scheme also permits us to generate alter- Association of Artificial Intelligence, 1986, pp. 872-877 nate solutions depending on the mode of decomposi- 9. Erdman, A.G. and Riley, D.R., "Computer Aided Linkage Design Using LINCAGES Package," Paper 81-DET-121, tion. Simple rules of matrix algebra provide ASME, 1981 mathematical operators for task decomposition. 10. Erdman, A.G., "Forty Years of Modern Kinematics," Pro- At lower levels of abstraction, symbols in the ceedings of the Conference on The First Forty Years of Mod- function matrices are replaced by actual design pa- ern Kinematics--A Tribute to Ferdinand Freudenstein, John rameters. By concatenating these matrices, we can Wiley & Son, 1991, Ch. 3 11. Faltings, B., "A Theory of Qualitative Kinematics in Mecha- perform qualitative simulation of the synthesized nisms," Technical Report UIUCDCS-R-86-1274, University mechanism. This allows the designer to verify the of Illinois at Urbana-Champaign, 1987 input-output relationships between each of the con- 12. Forbus, K., Nielsen, P. and Faltings, B., "Qualitative Kine- stituent building blocks as well as the overall motion. matics: A Framework," Proceedings of lJCAt-87, Morgan The simulation is qualitative in the sense that it hides Kaufman Publishers, Milan, 1987 13. Freudenstein, F. and Dobrjansky, L. "On a Theory for the all the details of individual elements of each building Type Synthesis of Mechanisms," Proceedings of the Elev- block and shows only the input-output relation- enth International Conference of Applied Mechanics, 1964, ships. The qualitative simulation capability enables pp. 420-428 Kota: Conceptual Design of Mechanisms Based on Computational Synthesis 87

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