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Research Papers

A Geometric Approach to Obtain the Closed-Form Forward Kinematics of H4 Parallel Robot

[+] Author and Article Information
Yujiong Liu

Robotics and Mechatronics Laboratory,
Department of Mechanical Engineering,
Virginia Polytechnic Institute
and State University,
Blacksburg, VA 24060
e-mail: yjliu@vt.edu

Minxiu Kong

Robotics Institute School of
Mechatronics Engineering,
Harbin Institute of Technology,
Harbin 150001, China
e-mail: exk@hit.edu.cn

Neng Wan

Advanced Controls Research Laboratory,
Department of Mechanical Science
and Engineering,
University of Illinois at Urbana-Champaign,
Urbana, IL 61801
e-mail: nengwan2@illinois.edu

Pinhas Ben-Tzvi

Mem. ASME
Robotics and Mechatronics Laboratory,
Department of Mechanical Engineering,
Virginia Polytechnic Institute
and State University,
Blacksburg, VA 24060
e-mail: bentzvi@vt.edu

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received March 16, 2018; final manuscript received June 22, 2018; published online July 18, 2018. Assoc. Editor: Damien Chablat.

J. Mechanisms Robotics 10(5), 051013 (Jul 18, 2018) (9 pages) Paper No: JMR-18-1071; doi: 10.1115/1.4040703 History: Received March 16, 2018; Revised June 22, 2018

To obtain the closed-form forward kinematics of parallel robots, researchers use algebra-based method to transform and simplify the constraint equations. However, this method requires a complicated derivation that leads to high-order univariate variable equations. In fact, some particular mechanisms, such as Delta, or H4 possess many invariant geometric properties during movement. This suggests that one might be able to transform and reduce the problem using geometric approaches. Therefore, a simpler and more efficient solution might be found. Based on this idea, we developed a new geometric approach called geometric forward kinematics (GFK) to obtain the closed-form solutions of H4 forward kinematics in this paper. The result shows that the forward kinematics of H4 yields an eighth degree univariate polynomial, compared with earlier reported 16th degree. Thanks to its clear physical meaning, an intensive discussion about the solutions is presented. Results indicate that a general H4 robot can have up to eight nonrepeated real solutions for its forward kinematics. For a specific configuration of H4, the nonrepeated number of real roots could be restricted to only two, four, or six. Two traveling plate configurations are discussed in this paper as two typical categories of H4. A numerical analysis was also performed for this new method.

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Copyright © 2018 by ASME
Topics: Kinematics , Robots
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Figures

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Fig. 1

A typical H4 configuration: the first two P/R-U-U chains connect to the first lateral bar. The third and fourth chain connects to the second lateral bar. The two lateral bars are connected to the central bar by two revolute joints.

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Fig. 2

One practical H4 kinematics configuration. In this configuration, the U-U chain is replaced by a restricted (S-S)2 mechanism and the four chains are designed to have identical properties.

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Fig. 3

Two typical traveling plate structures of the H4 robots. Left belongs to Par4 and the right belongs to I4.

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Fig. 4

Geometric model of the H4 forward kinematics

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Fig. 5

Simplification of the original geometric problem: since the lateral bars keep their direction during movement, each side BiCi is translated from Ci to the corresponding lateral bar center

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Fig. 6

For the case of an articulated traveling plate, the constraints between D1 and D2 are ||D1D2||=e and D1D2·[0 0 1]T=0. The circles with centers O1 and O2 are the possible positions for D1 and D2, respectively, before adding D1D2 constraints.

Grahic Jump Location
Fig. 7

For the case of a prismatic traveling plate, the constraint ||D1D′||=f (D′ is the projection of D1 on C3C4) allows us to make a further simplification that involves the translation of Cir1 from D1 to D′

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Fig. 8

The existence of nonrepeated real root number is illustrated by intentionally letting Cir1 and Cir2 be in the same plane, which is not very hard to find in practice. In these figures, the line segments connecting two circles are the compatible D1D2s with Eq. (12). The unillustrated five root case and seven root case can be achieved from the six root case by slightly enlarging or shrinking Cir2 so that the mid root on the right side disappears or splits into two. A number and an alphabetical letter are used to label a specific root configuration: (a) As Cir2 moves left, the real root emerges from null to three, (b) six real roots case, (c) if we keep moving Cir2 left from the three roots case, the mid root bifurcates and generates the four real roots case. This figure also shows different D1D2s and their corresponding z coordinate, and (d) eight real roots case.

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Fig. 9

The design parameters of a H4 robot with articulated traveling plate (the right figure shows the top view with the parameters of the traveling plate)

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Fig. 10

Workspace of the H4 robot in Fig. 2: position plot of the central bar: (a) three-dimensional view of the workspace of the H4 robot in Fig. 2 and (b) section view of the workspace of the H4 robot in Fig. 2 (section plane x = 0)

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Fig. 11

Distribution of points with different nonrepeated real roots and their enveloping solid: (a) distribution of points with two nonrepeated real roots, (b) distribution of points with four nonrepeated real roots, (c) distribution of points with six nonrepeated real roots, (d) section view (section plane y = −x) of the enveloping solid of (a). Note that this section view shows a hollow inside (a), (e) enveloping solid of (b), and (f) joint view for (d) and (e). The enveloping solid in (d) was changed to transparent green in order to get a better view.

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Fig. 12

Configurations of the four real roots. Note that the inside planar bar near the traveling plate in the parallelogram was set transparent to get a clear view of each configuration. The traveling plate in subfigures (b) and (d) has a similar configuration as in (a) and (c): (a) z = 919.7 mm, (b) z = −673.2 mm, (c) z = 863.3 mm, and (d) z = −612.4 mm.

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