Huafeng Ding1e-mail: dhf@ysu.edu.cn

Wenao Caoe-mail: cwao1986@163.com

Robotics Research Center,

Yanshan University,

Qinhuangdao 066004, China

Andres KecskemethyChair for Mechanics and Robotics,

University of Duisburg-Essen,

47057 Duisburg,Germany

e-mail: andres.kecskemethy@uni-due.de

Zhen HuangRobotics Research Center,

Yanshan University,

Qinhuangdao 066004, China

e-mail: huangz@ysu.edu.cn

Complete Atlas Database of2-DOF Kinematic Chains andCreative Design of MechanismsThe creation of novel kinematic structures of mechanisms still represents a major challengein the quest for developing new, innovative devices. In this setting, computer models thatcan automatically generate and visualize all possible independent structures in an intuitivemanner prove to be valuable as a support in the creative process of the designer. Thispaper proposes an automatic approach for establishing the complete atlas database of2-DOF kinematic chains and a systematic approach for the creative design of mechanismsbased on such an atlas. First, the transformation of the kinematic structure into a graph-based representation is addressed. Then, an approach for the generation of all nonfractio-nated topological graphs of 2-DOF (degrees of freedom) kinematic chains using contractedgraphs as well as a method for synthesizing all the fractionated topological graphs throughthe combination of corresponding 1-DOF kinematic chains are addressed. Based on thesemethods, the complete atlas database of 2-DOF kinematic chains up to 15 links is estab-lished in this paper for the first time. Using this complete database, a systematic approachfor the creative design of mechanisms can be derived, as illustrated for the example of an11-link 2-DOF rode tractor. [DOI: 10.1115/1.4005866]

Keywords: mechanism, structural synthesis, creative design, topological graph, atlasdatabase

1 Introduction

Multi-DOF mechanisms are widely used in various productsand equipments such as micro-electromechanical systems, highprecision surgical tools, engineering machinery, and robots. Thecreation of novel kinematic structures of mechanisms is an impor-tant issue in the creative design of devices, as it does not only pro-vide a basis for the structural optimization of existing mechanismsbut also to invent novel mechanisms [13]. For a long time, thecreation of a set of candidate mechanisms was mainly realizedthrough the experience and intuition of the researcher. Since1960s, however, many systematic approaches have been proposedfor the synthesis of kinematic chains and creative design of mech-anisms [4,5], which are still under further development.

Crossley [6], Dobrjanskyj and Freudenstein [7], and Woo [8]are among the first to introduce graph theory to the study of thetopological structure of kinematic chains. The dual graphapproach was presented by Sohn and Freudenstein [9] to generateautomatically the topological structures of mechanisms. Yan andHwang [10] proposed the permutation-group approach for thestructural synthesis of kinematic chains. Yan [11] proposed amethodology for the creative design of mechanisms by firstobtaining the general kinematic chain for a specified mechanismand then synthesizing other possible mechanisms based on the ki-nematic chain. Hwang and Hwang [12] proposed a new matrix,the contracted link adjacency matrix, to represent kinematicchains and used it as the representation matrix to synthesize kine-matic chains. Yang [13] and Yang and Yao [14] developed amethod for structural synthesis of kinematic chains with planarand nonplanar graphs on the basis of the single-opened chaintechnique. Rao [15,16] employed the Hamming number techniquefor the structural synthesis of kinematic chains by using F-DOF,(N-2)-links chains as the basic chains to synthesize F-DOF, N-link

chains. The metamorphic mechanism, first proposed by Dai andRees Jones [17], is studied by many researchers and some struc-tural synthesis methods for this novel mechanism were proposed[1820]. By using Polyas theory and the permutation-group, Yanet al. [2123] studied the synthesis of specialized mechanismsfrom given kinematic chains.

However, the progress toward full automation of structural syn-thesis of mechanisms has been slow and hard for quite a long time[2,24,25]. Ding et al. [24] established the unified topological mod-els and corresponding mathematical representations for planarsimple joint, multiple joint and geared (cam) kinematic chains.Moreover, most of the structural synthesis methods reported in lit-erature generate the synthesized structures in an algebraic repre-sentation which is not as intuitive as a direct graphic interactionwith designers and thus makes them impractical for conceptualdesign [25]. A method which would directly display the kinematicstructures in graphs would give designers a better visual under-standing of the linkjoint interrelationship of the generated kine-matic chains and thus make them more suitable in the creativedesign phase [26].

This paper proposes an automatic method to establish the com-plete atlas database (including both nonfractionated and fractio-nated kinematic chains) of 2-DOF kinematic chains up to 15 linksusing highly efficient methods both for isomorphism identification[27] and for rigid subchain detection [28] in the synthesis process.In this setting, the corresponding topological graphs (TG) can beautomatically rendered, giving a visual understanding of the sys-tem at hand to the user. Such an example of creative design ofmechanisms based on the atlas database is presented for an 11-link 2-DOF rode tractor. The overall structure of the paper isorganized as follows. In Sec. 2, the transformation of the kine-matic structure of mechanisms into a graph-based representationis addressed. In Sec. 3, all the kinematic structures of nonfractio-nated 2-DOF kinematic chains with up to 15 links are synthesizedvia their contracted graphs (CG). In Sec. 4, all the fractionated 2-DOF kinematic chains with up to 15 links are synthesized throughthe combination of corresponding 1-DOF kinematic chains. InSec. 5, as an example of the proposed method, the creation design

1Corresponding author.Contributed by the Mechanisms and Robotics Committee of ASME for publica-

tion in the JOURNAL OF MECHANICAL DESIGN. Manuscript received July 4, 2011; finalmanuscript received January 10, 2012; published online February 28, 2012. Assoc.Editor: Pierre M. Larochelle.

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for 11-link 2-DOF rode tractors is illustrated, yielding eight com-pletely novel mechanisms which satisfy the proposed design con-straints as well as the one that has been previously reported in theliterature [29]. This shows the effectiveness and efficiency of themethod.

2 Basic Concepts

2.1 Purpose of the Paper. Multiple DOFs mechanisms arewidely used in robotics and machine equipment. Part of the feasi-ble mechanisms which satisfy a specified design task can be con-ceived on the designers experience and other creative designmethods. However, the creation of all possible mechanisms for agiven task has been for a long time a hard nut to crack. This is aserious drawback, as the selection of a mechanism with the bestperformance for a specified design task is only possible when allthe feasible mechanisms are compared. For example, consider therode tractor in Fig. 1(a). This is a widely used piece of engineer-ing machinery, consisting of 11-links and 2-DOF. The kinematicsketch of the mechanism is shown in Fig. 1(b). Now two impor-tant questions to ask are

(1) Which other possibilities exist for the design of themechanism?

(2) Is the mechanism in Fig. 1(b) the best for an 11-link, 2-DOF rode tractor?

Obviously, only when the first question is answered, one cananswer the second question, after which the fully automatic kine-matic and dynamic analysis can be carried out [3032]. The paperattempts to answer the first question by first establishing a completeatlas database of 2-DOF mechanisms up to 15 links and then usingthe topological graphs in the atlas database to obtain all the feasiblemechanisms for a specified task under specified design constraints.

2.2 Graph-Based Model of Mechanisms. In order to estab-lish the complete atlas database containing all the kinematic struc-tures of mechanisms, the topological graph and the contractedgraph are adopted to represent the topological structure of a mech-anism. It is easy to establish the topological graph of a kinematicchain: vertices of the graph denote the links of the chain and edgesof the graph denote the joints. The TG of a kinematic chain corre-sponds with its structure, i.e., the kinematic structure of a kine-matic chain can be studied through the structure of its topological

graph. For example, Fig. 1(c) depicts the kinematic chain for themechanism in Fig. 1(b), while Fig. 1(d) represents its topologicalgraph. In a topological graph, the degree of a vertex is defined asthe number of edges that are incident with the vertex. A vertexwhose degree is equal to two is defined as a binary vertex and avertex whose degree is greater than two is defined as a multiple-degree vertex.

In a topological graph, a binary path, beginning and endingwith an edge, is a path satisfying the following two conditions:

(1) both the beginning edge and ending edge are incident witha multiple-degree vertex

(2) all vertices on the path are binary vertices

For example, in Fig. 1(d) the path constituted by edge (1,2),vertex 2, edge (2,3), vertex 3, and edge (3,4) is a binary path (anedge is represented by the two vertices it is incident with).

The CG of a topological graph can be obtained by replacingeach binary path in the topological graph with an edge. For exam-ple, after the four binary paths in Fig. 1(d) are replaced by fouredges, the contracted graph is obtained, shown in Fig. 1(e).

Both the topological graph and the contracted graph can be rep-resented by their adjacency matrices. The adjacency matrix for atopological graph is well known in literature and will not beaddressed here. The adjacency matrix for a contracted graph isdefined as

A xij

bb

k; if vertices i and j are adjacent through k edgess; if vertex i has s selfloops0; otherwise

8 1) edges. Thenumber of multiple edges for a contracted graph is defined as thesum of the numbers of multiple edges between every pair of verti-ces. For example, in Fig. 1(e), vertices 1 and 2 have one multipleedge and vertices 3 and 4 have two multiple edges. Thus, thegraph in Fig. 1(e) has three multiple edges.

2.3 Fractionated and Nonfractionated KinematicChains. If a kinematic chain can be separated into two independ-ent kinematic chains at a link or joint, the chain is called a fractio-nated kinematic chain. Fractionation can be divided into two basictypes: link-fractionation and joint-fractionation. Any fractionatedkinematic chain is either one of them or their combination.

For example, the kinematic chain in Fig. 1(c) can be separatedinto two independent 6-link 1-DOF kinematic chains from link 4,so it is a link-fractionated kinematic chain.

If a kinematic chain cannot be separated into two independentkinematic chains at any link or joint, the kinematic chain is a non-fractionated kinematic chain. For example, the kinematic chain inFig. 2 is a nonfractionated kinematic chain.

Based on the classification, the complete atlas database ofN-link F-DOF kinematic chains can be obtained by the followingtwo steps:

(1) Synthesize all the topological graphs of nonfractionated ki-nematic chains with their contracted graphs.

(2) Synthesize all the topological graphs of fractionated kine-matic chains through the combination of appropriate non-fractionated kinematic chains.

Fig. 1 (a) A rode tractor, (b) kinematic sketch of mechanism,(c) kinematic chain, (d) topological graph, and (e) contractedgraph

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3 Atlas Database of Nonfractionated Chains

3.1 Synthesis of Contracted Graphs. The type of links andtheir number that are necessary to form a kinematic chain withspecified number of links (N) and degrees of freedom (F) aredefined as the link assortment. In the paper, the so-called linkassortment array is used to represent link assortment. The linkassortment array of a kinematic chain is represented by

N2;N3;;Np

(3)

where N2, N3, N4, are the numbers of binary, ternary, quaternarylinks, etc., in that order. If the number is greater than nine, alpha-betic letters are used instead in order to avoid multidigital num-bers: A represents 10, B represents 11, C represents 12, and so on.For example, the link assortment array [4, C, 0, 0, 0, 0] denotesthat the number of ternary links (N3) is 12. The highest connectiv-ity p of a link in a kinematic chain is determined by the followingtwo equations [33]:

(1) if F 1, then p N F 1=2(2) if F 2, then p minfN F 1; N F 1=2g

The link assortment array for F-link N-DOF mechanisms isdetermined by [33]

N2 F 3 Ppd4

d 3NdN N2 N3 Np

8>>>>>>:

(5)

where b is the number of vertices of the contracted graph and [d1,d2, , db] (di di1) are the degrees of the vertices correspond-ing to the link assortment array [N2, N3, N4, , Np].

For example, [6,4,1,0] is a possible link assortment array for an11-link 2-DOF kinematic chain. Here N3 4, so there are fourvertices whose degrees are equal to 3 in the corresponding

contracted graphs; N4 1, so there is one vertex whose degree isequal to 4. The synthesis equation set of contracted graphs corre-sponding to the link assortment array [6,4,1,0] is

x12 x13 x14 x15 4x12 x23 x24 x25 3x13 x23 x34 x35 3x14 x24 x34 x45 3x15 x25 x35 x45 3

8>>>>>>>:

(6)

Based on Eq. (5) and an efficient method to detect isomorphism[27], all contracted graphs for 2-DOF nonfractionated kinematicchains up to 15 links can be synthesized automatically. Oneobtains thus only one contracted graph for 7-link kinematicchains, four contracted graphs for 9-link kinematic chains, 17 con-tracted graphs for 11-link kinematic chains, 118 contracted graphsfor 13-link kinematic chains, and 1198 contracted graphs for 15-link kinematic chains.

For example, Table 1 shows all the contracted graphs corre-sponding to 11-link 2-DOF nonfractionated kinematic chains.From the table, it can be seen that all five contracted graphs corre-spond to the link assortment array [5,6,0,0]. The first two havezero multiple edge; the third graph has one multiple edge; thefourth graph has two multiple edges; the last one has three multi-ple edges.

3.2 Synthesis of Topological Graphs. The contracted graphis obtained by replacing every binary path with an edge in a topo-logical graph. Vice versa, the topological graphs can be obtained

Fig. 2 A nonfractionated kinematic chain and its topologicalgraph

Table 1 Contracted graphs of 11-link 2-DOF nonfractionatedkinematic chains

LAA Contracted graphs and their classification

[5,6,0,0]

[6,4,1,0]

[7,2,2,0]

[8,0,3,0]

[7,3,0,1]

[8,1,1,1]

[9,0,0,2]

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if N2 binary vertices are inserted to the edges of a contractedgraph. Although the concept of the contracted graph in graphtheory had been adopted by many researchers to synthesize kine-matic chains, only those consisting of up to 12 links have beenpreviously reported in the literature [33,34]. In this paper, basedon the concept of contracted graphs, topological graphs of 2-DOFnonfractionated kinematic chains with up to 15 links will be syn-thesized and displayed in graph form.

Theorem 1. For a topological graph of nonfractionated F-DOFkinematic chains with more than one loop, the number of binaryvertices on any binary path is less than or equal to F 1.

Proof. Kinematic chains with only one loop, such as 4-link1-DOF kinematic chains, 5-link 2-DOF kinematic chains, etc.,need not to be considered here. For a topological graph of non-fractionated kinematic chains, the subgraph obtained by deletingany binary path is also a valid topological graph with DOF 1. Ifthere exists a binary path with F 2 or more binary vertices on itin a topological graph of nonfractionated kinematic chains, the re-moval of the binary path will reduce the DOFs of the topologicalgraph by at least F. This would lead to a DOF of the remainingsubgraph of less than one. Thus, it is impossible that any binarypath in a topological graph of nonfractionated F-DOF kinematicchains has F 2 or more binary vertices.

In the process of synthesizing topological graphs from con-tracted graphs, we suppose xi binary vertices are to be added toedge ei. In order to make the synthesis of 2-DOF kinematic chainsmore efficient, three rules are proposed.

Rule 1: The scope of xi and the sum of xi satisfy

0 xi 3Pti1

xi N2

8>>>>>>>>>>>>>>>>:

(13)

All 36 valid topological graphs are synthesized after isomorphismidentification and rigid subchain detection, yielding the graphsshown in Fig. 4.

In this way, all the topological graphs of nonfractionated 2-DOFkinematic chains of up to 15 links are synthesized: 3 topologicalgraphs for 7-link kinematic chains, 35 topological graphs for 9-linkkinematic chains, 753 topological graphs for 11-link kinematicchains, 27,496 topological graphs for 13-link kinematic chains, and1,432,732 topological graphs for 15-link kinematic chains.

Table 2 shows the number information for the atlas databasesof 11-link 2-DOF nonfractionated kinematic chains. For examplein the link assortment array [5,6,0,0], the number of topologicalgraphs corresponding to the two contracted graphs with 0 multipleedges are 26 and 80, respectively; the number of topologicalgraphs corresponding to the contracted graph with one multipleedge is 36; the number of topological graphs corresponding to thecontracted graph with two multiple edges is 11. No valid topologi-cal graphs can be synthesized from the contracted graph withthree multiple edges for all the synthesized topological graphscontain rigid subchains.

The number information for the atlas databases of 13-link2-DOF nonfractionated kinematic chains is also shown in Table 3.

4 Atlas Database of Fractionated Chains

4.1 Rules of Combination. Besides nonfractionated 2-DOFkinematic chains, 2-DOF kinematic chains also consist of fractio-nated structures. As kinematic chain with one fractionated joint ortwo or more fractionated links have at least three DOFs, fractionated2-DOF kinematic chains may only contain one fractionated link.

Fig. 3 A contracted graph of an 11-link 2-DOF kinematic chain

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That is, all fractionated 2-DOF kinematic chains can be synthesizedby the combination of two appropriate 1-DOF kinematic chains.

In general, for the synthesis of N-link 2-DOF fractionated kine-matic chains, the number of links of appropriate 1-DOF kinematicchains satisfies

N1 N2 N 1N1 N2Ni 4; 6; 8; 10; i 1; 2

8