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Research Papers

Solving the Kinematics of the Planar Mechanism Using Data Structures of Assur Groups

[+] Author and Article Information
Yuanxi Sun

Department of Mechanical
Engineering and Automation,
School of Mechanical Engineering,
Northwestern Polytechnical University,
127 West Youyi Road,
Xi'an, Shaanxi 710072, China
e-mail: sunyuanxi@mail.nwpu.edu.cn

Wenjie Ge

Department of Mechanical
Engineering and Automation,
School of Mechanical Engineering,
Northwestern Polytechnical University,
127 West Youyi Road,
Xi'an, Shaanxi 710072, China
e-mail: gwj@nwpu.edu.cn

Jia Zheng

Key Laboratory of Contemporary Design and
Integrated Manufacturing Technology,
Ministry of Education,
School of Mechanical Engineering,
Northwestern Polytechnical University,
127 West Youyi Road,
Xi'an, Shaanxi 710072, China
e-mail: jiazheng@mail.nwpu.edu.cn

Dianbiao Dong

Department of Mechanical
Engineering and Automation,
School of Mechanical Engineering,
Northwestern Polytechnical University,
127 West Youyi Road,
Xi'an, Shaanxi 710072, China
e-mail: dongdianbiao@mail.nwpu.edu.cn

1Corresponding author.

Manuscript received December 9, 2015; final manuscript received May 5, 2016; published online June 10, 2016. Assoc. Editor: Jian S. Dai.

J. Mechanisms Robotics 8(6), 061002 (Jun 10, 2016) (13 pages) Paper No: JMR-15-1334; doi: 10.1115/1.4033666 History: Received December 09, 2015; Revised May 05, 2016

This paper presents a systematic solution of the kinematics of the planar mechanism from the aspect of Assur groups. When the planar mechanism is decomposed into Assur groups, the detailed calculating order of Assur groups is unknown. To solve this problem, first, the decomposed Assur groups are classified into three types according to their calculability, which lays the foundation for the establishment of the automatic solving algorithm for decomposed Assur groups. Second, the data structure for the Assur group is presented, which enables the automatic solving algorithm with the input and output parameters of each Assur group. All decomposed Assur groups are stored in the component stack, and all parameters of which are stored in the parameter stacks. The automatic algorithm will detect identification flags of each Assur group in the component stack and their corresponding parameters in the parameter stacks in order to decide which Assur group is calculable and which one can be solved afterward. The proposed systematic solution is able to generate an automatic solving order for all Assur groups in the planar mechanism and allows the adding, modifying, and removing of Assur groups at any time. Two planar mechanisms are given as examples to show the detailed process of the proposed systematic solution.

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Figures

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Fig. 1

(a) The decomposition example of Assur groups. (b) The decomposed Assur groups.

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Fig. 2

(a) The topology graph of the planar mechanism in Fig.1(a). (b) The spanning tree of the topology graph.

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Fig. 3

(a) The decomposition example of (augmented) Assur groups. (b) The decomposed (augmented) Assur groups which satisfy rules (a)–(c). (c) The decomposed (augmented) Assur groups which satisfy only rule (a).

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Fig. 4

The topology graph of the planar mechanism in Fig. 3(a)

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Fig. 5

An example of the planar mechanism (a) and its topology graphs with different fundamental loop situations ((b)–(d))

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Fig. 6

Relationships between spanning trees, fundamental trees, and union loops

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Fig. 7

(a) An example of the planar mechanism with Assur groups of different external connecting situations. (b) Its topology graph.

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Fig. 8

Relationships between three types of Assur groups

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Fig. 9

The example of conversions of three types of Assur groups

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Fig. 10

The schematic diagram of the Assur group component, the component stack, the point structure stack, and the angle structure stack

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Fig. 11

The flowchart of solving planar mechanisms using data structures of Assur groups

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Fig. 12

(a) The example planar mechanism 1. (b) The topology graph of the planar mechanism 1.

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Fig. 13

(a) The example planar mechanism 2. (b) The topology graph of the planar mechanism 2.

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Fig. 14

The extended planar mechanism of Fig. 12(a)

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Fig. 15

The status of the component stack and the point/angle structure stack when new Assur groups are added

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Fig. 16

The modified planar mechanism of Fig. 12(a)

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Fig. 17

The removing of an Assur group in the planar mechanism of Fig. 12(a)

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