Automatic Sketching of Kinematic Chain Topology Graphs Based on Optimized Branch-chain Matrices

doi:10.3969/j.issn.1004-132X.2026.08.014

Abstract

Abstract:

This paper presents a novel method for the automatic plotting of kinematic chain topological graphs， based on an optimally arranged branch-chain matrix. The structural information of a kinematic chain is first represented using a dendrogram. Branch chains are then extracted according to the connectivity of components and multiple joints， forming a branch-chain matrix. To minimize or eliminate crossings in the final graph， the row order of this matrix is optimized using the concept of intimacy between branch chains and the corresponding crossover-determination theorem， yielding an optimized branch-chain matrix with no or minimal crossings. Subsequently， a bicolored topological graph with minimal or no crossings is generated through a series of operations， including deletion of duplicate elements， element repositioning， insertion of new columns， and element translation. Case studies and comparative analyses demonstrate that the proposed method provides clear procedural rules and preserves distinct loop information. The method effectively resolves line-crossing issues and enables direct generation of topological graphs from the optimized branch-chain matrix.

Key words: topological graph sketching, optimized branch-chain matrix, intimacy, crossover determination

摘要：

提出了一种基于优化排列的支链矩阵进行运动链拓扑图连线绘制的新方法。利用树状结构图描述运动链结构信息，根据各构件及复铰的连接关系分离出各支链，生成支链矩阵。运用支链间的亲密度概念和对应的图的交叉判定定理进行支链矩阵的行排列优化，获得无交叉或最少交叉的优化支链矩阵。在此基础上，通过删除重复元素、调整元素位置、新增列和元素平移等操作，生成无交叉或最少交叉的双色拓扑图。案例分析和对比结果表明，所提绘制方法规则简单、环路信息清晰，能有效解决线条交叉问题，并可直接在优化支链矩阵的基础上生成运动链拓扑图。

关键词: 拓扑图绘制, 优化支链矩阵, 亲密度, 交叉判定

CLC Number:

TH112

XUE Yu, SUN Liangbo, ZHANG Bohou, ZHOU Huaxi, WANG Xiaoyi. Automatic Sketching of Kinematic Chain Topology Graphs Based on Optimized Branch-chain Matrices[J]. China Mechanical Engineering, 2026, 37(8): 1947-1955.

薛钰, 孙亮波, 张博厚, 周华西, 王晓艺. 基于优化支链矩阵的运动链拓扑图自动绘制[J]. 中国机械工程, 2026, 37(8): 1947-1955.

Figures/Tables 8

Fig.1 Structure graph and dendrogram graph of the 11-bar kinematic chain 1

Fig.2 Non-crossing dendrogram graph of kinematic chain 1

Fig.3 Plot of the changes in the position of the main chain

Fig.4 Topological graph of kinematic chain 1

Fig.5 Flowchart for topological graph sketching

Fig.6 Structural graph and dendrogram graph of the 14-bar kinematic chain 2

Fig.7 Deletion and adjustment process for optimized branched-chain matrix M14

Fig.8 Topological graph of kinematic chain 2

References 26

[1]	徐世福. 基于平面运动链的类型综合及创新设计方法的研究［D］. 杭州：浙江理工大学， 2012.
	XU Shifu. The Research on Type Synthesis Based on Planar Kinematic Chains and Innovation Design［D］. Hangzhou： Zhejiang Sci-Tech University， 2012.
[2]	孙良，郑广辉，叶治政，等. 基于图论的多杆式栽植机构构型综合与方法研究［J］. 农业机械学报， 2022， 53（5）： 67-74.
	SUN Liang， ZHENG Guanghui， YE Zhizheng， et al. Structural Synthesis and Innovative Design of Multi-link Planting Mechanism Based on Graph Theory［J］. Transactions of the Chinese Society for Agricultural Machinery， 2022， 53（5）： 67-74.
[3]	YANG Wenjian， DING Huafeng， KECSKEMÉTHY A. Structural Synthesis towards Intelligent Design of Plane Mechanisms： Current Status and Future Research Trend［J］. Mechanism and Machine Theory， 2022， 171： 104715.
[4]	YE Wei， LI Qinchuan. Type Synthesis of Lower Mobility Parallel Mechanisms： a Review［J］. Chinese Journal of Mechanical Engineering， 2019， 32（1）： 38.
[5]	MANIKANDAN H， KAUP V， BABU H. Structural Synthesis and Analysis of 12-link， 3–Freedom Simple Jointed Kinematic Chain［J］. Materials， Mechanics & Modeling （NCMMM-2020）， 2021， 2341： 020031.
[6]	DING Huafeng， YANG Wenjian， KECSKEMÉTHY A. Automatic Synthesis of Planar Multiple Joint Kinematic Chains［M］∥Automatic Structural Synthesis and Creative Design of Mechanisms. Singapore： Springer Nature Singapore， 2022： 199-210.
[7]	VOLKAN YAZıCı M， KOCAK M， USLU T， et al. Structural Synthesis of Platform Type Manipulators via Connection-based Adjacency Matrix： Topology Generation［J］. Journal of Mechanical Design， 2025， 147： 013302.
[8]	DING Huafeng， YANG Wenjian， KECSKEMÉTHY A. Automatic Structural Synthesis and Creative Design of Mechanisms［M］. Singapore： Springer Nature Singapore， 2022： 131-139.
[9]	OLSON D G， THOMPSON T R， RILEY D R， et al. An Algorithm for Automatic Sketching of Planar Kinematic Chains［J］. Journal of Mechanisms， Transmissions， and Automation in Design， 1985， 107（1）： 106-111.
[10]	BELFIORE N P， PENNESTRI E. Automatic Sketching of Planar Kinematic Chains［J］. Mechanism and Machine Theory， 1994， 29（1）： 177-193.
[11]	MAUSKAR S， KRISHNAMURTY S. A Loop Configuration Approach to Automatic Sketching of Mechanisms［J］. Mechanism and Machine Theory， 1996， 31（4）： 423-437.
[12]	XIAO Renbin， LIU Yong， WANG Deshi. An Improved Approach to Automatic Sketching of Mechanism Based on Loop Configuration and Its Implementation［C］∥Volume 5： 27th Biennial Mechanisms and Robotics Conference. American Society of Mechanical Engineers， 2002： 655-660.
[13]	NIE Songhui， LIU Hongzhao， QIU Aihong. A Maximal Loop Approach to Automatic Sketching of Mechanisms［C］∥Volume 9： Mechanical Systems and Control， Parts A， B， and C. ASMEDC， 2007： 501-509.
[14]	NIE Songhui， LIAO Aihong， QIU Aihong， et al. Addition Method with 2 Links and 3 Pairs of Type Synthesis to Planar Closed Kinematic Chains［J］. Mechanism and Machine Theory， 2012， 58： 179-191.
[15]	PUCHETA M A， ULRICH N E， CARDONA A. Combined Graph Layout Algorithms for Automated Sketching of Kinematic Chains［C］∥Volume 4： 36th Mechanisms and Robotics Conference， Parts A and B. American Society of Mechanical Engineers， 2012： 513-523.
[16]	PUCHETA M A， ULRICH N E， CARDONA A. Automated Sketching of Non-fractionated Kinematic Chains［J］. Mechanism and Machine Theory， 2013， 68： 67-82.
[17]	YAN Hongsen， CHIU Y T. An Algorithm for the Construction of Generalized Kinematic Chains［J］. Mechanism and Machine Theory， 2013， 62： 75-98.
[18]	CHU Jinkui， ZOU Yanhuo. An Algorithm for Structural Synthesis of Planar Simple and Multiple Joint Kinematic Chains［J］. Proceedings of the Institution of Mechanical Engineers， Part C： Journal of Mechanical Engineering Science， 2014， 228（12）： 2178-2192.
[19]	DING Huafeng， ZHAO Jing， HUANG Zhen. Unified Structural Synthesis of Planar Simple and Multiple Joint Kinematic Chains［J］. Mechanism and Machine Theory， 2010， 45（4）： 555-568.
[20]	丁华锋，黄真，曹毅. 运动链拓扑图的自动生成及图谱库的建立［J］. 机械工程学报， 2006， 42（4）： 32-36.
	DING Huafeng， HUANG Zhen， CAO Yi. The Topological Graphs Creation Automatically of Kinematic Chains and Atlas Database Establishment［J］. Journal of Mechanical Engineering， 2006， 42（4）： 32-36.
[21]	YANG Wenjian， DING Huafeng， HE Yong， et al. Studying the Optimal Layout of Topological Graphs to Facilitate the Automatic Sketching of Kinematic Chains［J］. Journal of Mechanical Design， 2017， 139（8）： 082301.
[22]	YANG Wenjian， DING Huafeng， KECSKEMÉTHY A. A New Method for the Automatic Sketching of Planar Kinematic Chains［J］. Mechanism and Machine Theory， 2018， 121： 755-768.
[23]	SUN Wei， KONG Jianyi， SUN Liangbo. A Joint–Joint Matrix Representation of Planar Kinematic Chains with Multiple Joints and Isomorphism Identification［J］. Advances in Mechanical Engineering， 2018， 10（6）： 1687814018778404.
[24]	SUN Wei， KONG Jianyi， SUN Liangbo. A Novel Graphical Joint-joint Adjacent Matrix Method for the Automatic Sketching of Kinematic Chains with Multiple Joints［J］. Mechanism and Machine Theory， 2020， 150： 103847.
[25]	YU Luchuan， CHAI Wenhao. Automatic Sketching of Planar Kinematic Chains Based on the Optimal Layout Algorithm［C］∥Proceedings of the 2024 3rd International Symposium on Control Engineering and Robotics. ACM， 2024： 216-222.
[26]	LOHUMI M K， MOHAMMAD A， AHMAD KHAN I. Hierarchical Clustering Approach for Determination of Isomorphism among Planar Kinematic Chains and Their Derived Mechanisms［J］. Journal of Mechanical Science and Technology， 2012， 26（12）： 4041-4046.