Non-fractionated Kinematic Chains Research Articles

Isomorphism identification algorithm and database generation for planar 2-6 DOFs fractionated kinematic chains combined by two or three non-fractionated kinematic chains

Existing methods for synthesizing fractionated kinematic chains usually avoid generating isomorphic fractionated kinematic chains rather than identifying isomorphic chains. Consequently, no isomorphism identification algorithm for fractionated kinematic chains, especially those with complex structures, has been reported. This study, based on a new isomorphism identification algorithm, proposes a systematic method to synthesize all possible non-isomorphic fractionated kinematic chains with up to 16 links and up to 6 degrees of freedom (DOFs), which are combined by two or three non-fractionated kinematic chains. First, the structural constraint equations are presented to obtain all possible combinations of fractionated kinematic chains. Second, the two concepts of a split-vertex characteristic graph and a combined characteristic graph are developed to acquire the corresponding combined characteristic code, which is the unique representation of planar fractionated kinematic chains and is applied to identify the isomorphism problem. Then, all possible 2-6 DOFs non-isomorphic fractionated kinematic chains combined by two or three non-fractionated kinematic chains are synthesized. Moreover, the corresponding databases of the synthesized fractionated kinematic chains are established. Finally, the synthesis methods and results of this study and the ones in the existing literature are compared and discussed in detail.

Eliminating isomorphism identification method for synthesizing nonfractionated kinematic chains based on graph similarity

A novel method for eliminating isomorphism identification is proposed to improve synthesis efficiency and synthesize planar nonfractionated kinematic chain (KCs) automatically. This method is based on the vertex insertion of contracted graphs. First, similar edges of contracted graphs are divided into groups, in which the similar edges are found and their characteristic matrices are calculated. The edge types are divided based on whether or not isomerism occurs after vertices are inserted. Then, the vertices are inserted into contracted graphs according to the edge condition. In this process, all isomerism caused by the location and number of inserted vertices is reserved, and the property change of similar edges is checked. Lastly, the rigid subchains of remaining isomerism are distinguished. Contracted graphs with four independent loops and some of their inserted vertices are presented in appendix. A complete set of nonfractionated KCs with up to seven independent loops and three degrees of freedom is also provided. The veracity and efficiency of the method are confirmed by conducting a comparative analysis between this synthesis and other literature results.

The family of planar kinematic chains with two multiple joints

This paper proposes an automatic method to synthesize planar non-fractionated kinematic chains with two multiple joints, which are generated from kinematic chains with one multiple joint by transforming a multiple link to a multiple joint. First, the interconversion relationship between kinematic chains with different numbers of multiple joints is revealed. Then, an improved algorithm is developed to obtain the characteristic number string of topological graphs with two multiple joints to enhance the efficiency of isomorphism identification. Finally, based on the atlas database of kinematic chains with one multiple joint, a systematic method is proposed to automatically synthesize kinematic chains with two multiple joints. As a result, the whole family of planar non-fractionated kinematic chains with two multiple joints and up to 15 links is acquired for the first time, and the contrastive analysis with existing research results is also conducted.

Automatic generation of the complete set of planar kinematic chains with up to six independent loops and up to 19 links

The synthesis of all feasible topological (kinematic) structures of kinematic chains still has broad research spaces although various methods have been developed in the past fifty years. In this paper, a systematic method is proposed to automatically synthesize the complete set of planar non-fractionated kinematic chains with up to six independent loops. First, the contracted graphs of kinematic chains are synthesized and classified by their planar and non-planar characteristics. Then, the set of constraint equations is derived from the contracted graphs and used to synthesize topological graphs of kinematic chains. And an improved algorithm is developed to obtain the simplified characteristic codes of topological graphs to increase the efficiency of isomorphism identification. Finally, a systematic synthesis method is proposed and the complete set of planar non-fractionated kinematic chains with up to six independent loops and up to 19 links is achieved for the first time. In addition, a comparative analysis between our synthesis results and the ones in the literature is conducted, illustrating the validity and efficiency of the method.

Automated sketching of non-fractionated kinematic chains

The sketching problem arises frequently in the conceptual design of mechanisms, especially in the enumeration process where a large number of topological solutions automatically generated must be analyzed. This paper presents a new graph layout algorithm to sketch non-fractionated kinematic chains. A combinatorial algorithm based on the independent loops of the graph representation of the kinematic chain, is used to find an adequate initial position of graph vertices with minimal edge crossings; its execution is followed by a force-directed algorithm based on spring repulsion and electrical attraction, including a new concept of edge-to-vertex repulsion to improve esthetics and preserve edge crossings. Both algorithms are used in sequence to generate a representative layout of the graph which optimizes a given quality measure. Finally, standard rules are followed to convert the graph into the sketch, using new heuristic correction rules to avoid newly generated edge crossings. Atlases of complex non-fractionated kinematic chains are used to validate the results. A qualitative comparison with a set of sketches found in the literature is included, showing the advantages of the proposed algorithm.

Erratum: “Complete Atlas Database of 2-DOF Kinematic Chains and Creative Design of Mechanisms” [Journal of Mechanical Design, 2012, 134(3), p. 031006

In page 031006-4, the second paragraph below Eq. (13)“In this way, all the topological graphs of non-fractionated 2-DOF kinematic chains of up to 15 links are synthesized: 3 topological graphs for 7-link kinematic chains, 35 topological graphs for 9-link kinematic chains, 753 topological graphs for 11-link kinematic chains, 27,496 topological graphs for 13-link kinematic chains, and 1,432,732 topological graphs for 15-link kinematic chains.”The correct number is “1,432,730” rather than “1,432,732.”The detailed number information for the atlas database of 15-link 2-DOF non-fractionated kinematic chains is shown in Table 1.

Algorithm to Identify the Type of Degrees of Freedom in Kinematic Chains

An efficient and simple algorithm is developed to classify whether a kinematic chain has total degree of freedom or partial degree of freedom. The developed algorithm uses the sequential removal of binary link strings from the given kinematic chain and can be applied to both planar and nonplanar kinematic chains. The developed algorithm is successfully applied to the enumeration of nonfractionated kinematic chains. Since the algorithm is systematic and efficient, it can be applied to the more complex kinematic chains which have not been reported in the literature yet.

Automatic Method for Enumeration of Complete Set of Kinematic Chains

A fully automatic algorithm is developed to obtain the complete set of nonfractionated kinematic chains for each number of links and degrees of freedom. It is a contracted graph method programmed in LISP. The number synthesis is performed efficiently using the partitioning technique implemented in a recursive procedure. The enumeration of the contracted graphs or bases is computerized by the recurrent application of the contracted graph method. In order to avoid the possible sources of human error, a newly developed automatic algorithm for the detection of the embedded rigid structure is used. The obtained results using the developed algorithm for various numbers of links and degrees of freedom are presented.