Rigid Subchains Research Articles

Structural Synthesis of Planar Kinematic Chains for Non-Redundant Parallel Manipulators

Abstract This work focuses on the synthesis of non-redundant planar kinematic chains applied to parallel manipulators, where the mobility of the kinematic chains is constrained by the planar workspace, allowing for kinematic chains with up to three degrees-of-freedom. The synthesis process consists of two stages. First, a graph generator enumerates non-isomorphic biconnected graphs representing planar kinematic chains with up to three degrees-of-freedom and seven loops. Subsequently, graphs associated to chains with rigid subchains are removed using a degeneracy identification method based on matroid theory. Concise results are presented for chains with up to five loops. For chains with six and seven loops, results are categorized based on link partitions for a direct comparison with previous results. These findings align with prior research, with minor variations in the reported chain numbers. These variations can be attributed to several factors, including graph generation, isomorphism testing, fractionation, and subchain identification. These factors are comprehensively discussed and examined.

A new rigid sub-chain identification and elimination method of kinematic chains with multiple joints

It is a very important process to determine whether there is a rigid sub-chain in the type synthesis for kinematic chains. In this paper, a new method for identifying and eliminating rigid sub-chains of kinematic chains with multiple joints is proposed. Firstly, the properties of the mesh loop set and independent loop set in loop combination were introduced. Properties determine that testing the degrees of freedom (DOF) of all independent loop sets and the combined loops can verify the existence of a rigid sub-chain. Secondly, representation methods and properties of the joint-joint adjacent matrix and loop matrix were explained and the operation method of the loop matrix was explained. A method of obtaining some loop matrices, which represents an independent loop set, by joint-joint adjacency matrix is proposed. Then, the DOF of all independent loop sets and the combined loops of independent loop sets were obtained by the operation method of loop matrix to identify and eliminate the rigid sub-chain. Finally, a large number of cases have proved the feasibility and effectiveness of this method.

Automatic identification method of rigid sub-chains based on branch chain disassembly and replacement

The existing methods of rigid sub-chain identification are mostly based on calculating the degree of freedom (DOF) of the basic loop, the loop combination, and the special branch chain, for which efficiency and accuracy are generally difficult to combine. Therefore, this paper proposes a rigid sub-chain identification method based on the disassembly and replacement of the branch chain, with the core idea of eliminating the non-rigid sub-chain part in the kinematic chain (KC) to highlight the rigid sub-chain part. First, the two-degree chains of length greater than 1 in the KC topology is disassembled. Second, the characteristics of quadrilateral loops are studied, and the quadrilateral loops with a diagonal connection are replaced equivalently to realize further disassembly. Finally, the special edges of the remaining sub-graphs are analyzed and removed. Through the above steps, if topology is eliminated, the structure is reasonable; otherwise, topology contains a rigid sub-chain. This method is used to identify a non-isomorphic KC atlas, and a complete set of planar non-fractionated KCs with simple joints within six loops and 3-DOFs is obtained.

Automatic Identification Method of Rigid Sub-Chains Based on Branch Chain Disassembly and Replacement

Automatic Identification Method of Rigid Sub-Chains Based on Branch Chain Disassembly and Replacement

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.

Reconciling Enumeration Contradictions: Complete List of Baranov Chains With Up to 15 Links With Mathematical Proof

Abstract The identification of Baranov chains is associated with the rigid subchain identification problem, which is a crucial step in several methods of structural synthesis of kinematic chains. In this article, a systematic approach for the detection of rigid subchains, based on matroid theory, is presented and proved. Based on this approach, a novel method for the enumeration of Baranov chains is proposed. A novel algorithm is applied to a database of nonisomorphic graphs of nonfractionated zero-mobility kinematic chains. By means of the proposed algorithm, the previous results for Baranov chains presented in the literature with up to 11 links are compared and validated. Furthermore, discrepancies in the number of Baranov chains with up to 13 links, presented in the literature, are pointed out, discussed, and the proven results are presented. Finally, the complete family of Baranov chains with up to 15 links is obtained. Examples of application of the proposed method are provided.

Structural Synthesis of Baranov Trusses With Up to 13 Links

The structural synthesis of Baranov trusses is still an open problem, although Baranov trusses are widely used in the design and analysis of mechanisms and robots. This paper proposes a systematic method for the structural synthesis of Baranov trusses. First, the definition review and the graph-form representations of Baranov trusses are proposed. Second, seven propositions on structural characteristics of Baranov trusses are concluded. Then, based on the set of constraint equations and a rigid subchain detection algorithm, a systematic method is presented to synthesize the complete set of Baranov trusses with a specified number of links. Finally, the synthesis results of contracted graphs (including valid and rigid contracted graphs) and topological graphs of Baranov trusses with up to 13 links are provided, and the synthesis methods and results between ours and the ones in the existing literature are compared and discussed in detail.

Adding sub-chain method for structural synthesis of planar closed kinematic chains

For at least the past five decades, structural synthesis has been used as a main means of finding better mechanisms with some predefined function. In structural synthesis, isomorphism identification is still a problem unsolved well, and to solve this problem is very significant to the design of new mechanisms. According to the given degree of freedom (DOF) and link connection property of planar closed chain mechanisms, vertex assortment is obtained. For solving the isomorphism problem, a method of the adding sub-chains is proposed with the detailed steps and algorithms in the synthesizing process. Employing this method, the identification code and formation code of every topological structure are achieved, therefore many isomorphic structures could be eliminated in time during structural synthesis by comparing those codes among different topological graphs, resulting in the improvement of synthesizing efficiency and accuracy, and the approach for eliminating rigid sub-chains in and after the synthesizing process is also presented. Some examples are given, including how to add sub-chains, how to detect simple rigid sub-chains and how to obtain identification codes and formulation codes et al. Using the adding sub-chain method, the relative information of some common topological graphs is given in the form of table. The comparison result is coincident with many literatures, so the correctness of the adding sub-chain method is convinced. This method will greatly improve the synthesizing efficiency and accuracy, and has a good potential for application.

A method for identification of isomorphism and structural properties of kinematic chains

The present work deals with the structural analysis and synthesis of kinematic chains of mechanisms and robot manipulators. Four key problems, detection of isomorphism among kinematic chain, identification of isomorphism among inversions of given kinematic chain, identification of rigid sub-chain and type of degree of freedom of kinematic chains are considered. A method based on the flow path matrices of kinematic chains is proposed and two new structural invariants, sum of the absolute values of the characteristic polynomial coefficients (SCPC) and maximum absolute value of the characteristic polynomial coefficient (MCPC) are determined using software MATLAB. These invariants are used as the composite identification number of a kinematic chain and mechanisms. The advantage of this method is that it covers all planar kinematic chains of one, two and three degree of freedom at once for identification of isomorphism and for structural properties.

A new theory for the topological structure analysis of kinematic chains and its applications

Abstract Based on the array representation of loops in topological graphs of kinematic chains, this paper proposes two basic loop operations, “Θ” and “⊕”, for the first time. The existent conditions and properties of “⊕” operation are researched and four laws about the operation are presented. Furthermore, after the important concepts of the independent loop set and its selection theorem are proposed, the loop relationship of kinematic chains is revealed; thus an original theory of loop analysis is established. Finally, some applications are given under the basic theory above, such as the isomorphism identification, the detection of rigid sub-chains, and the freedom type analysis of kinematic chains.

Generation of planar kinematic chains

This work presents a completely automated, mathematically rigorous procedure that generates all planar, non-fractionated, pin-jointed kinematic chains having 2–6 independent loops and 1, 2, or 3 degrees of freedom. It also addresses the isomorphism problem and the elimination of rigid subchains. The computer programs can be run on any personal computer (XT or higher); over 6 million kinematic chains and 100 million inversions have been generated.

An algorithm for the detection of degenerate kinematic chains

A degenerate kinematic chain is a movable kinematic chain with rigid subchains. This paper presents an effective algorithm for the detection of degenerate kinematic chains based on the loop-decreasing method. Some theorems are proved, and the detection algorithm is proposed. A computer program is developed for the automatic detection of degenerate kinematic chains. The result of this work is beneficial to the automation of number synthesis of mechanisms.