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Research Papers

Planar Linkage Synthesis for Mixed Exact and Approximated Motion Realization Via Kinematic Mapping

[+] Author and Article Information
Ping Zhao

School of Mechanical and
Automotive Engineering,
Hefei University of Technology,
Hefei, Anhui 230009, China
e-mail: ping.zhao@hfut.edu.cn

Xin Ge, Q. J. Ge

Department of Mechanical Engineering,
Stony Brook University,
Stony Brook, NY 11794-2300

Bin Zi

School of Mechanical and
Automotive Engineering,
Hefei University of Technology,
Hefei, Anhui 230009, China

Manuscript received September 12, 2015; final manuscript received November 25, 2015; published online May 4, 2016. Assoc. Editor: Andrew P. Murray.

J. Mechanisms Robotics 8(5), 051004 (May 04, 2016) (8 pages) Paper No: JMR-15-1249; doi: 10.1115/1.4032212 History: Received September 12, 2015; Revised November 25, 2015

It has been well established that kinematic mapping theory could be applied to mechanism synthesis, where discrete motion approximation problem could be converted to a surface fitting problem for a group of discrete points in hyperspace. In this paper, we applied kinematic mapping theory to planar discrete motion synthesis of an arbitrary number of approximated poses as well as up to four exact poses. A simultaneous type and dimensional synthesis approach is presented, aiming at the problem of mixed exact and approximate motion realization with three types of planar dyad chains (RR, RP, and PR). A two-step unified strategy is established: first N given approximated poses are utilized to formulate a general quadratic surface fitting problem in hyperspace, then up to four exact poses could be imposed as pose-constraint equations to this surface fitting system such that they could be strictly satisfied. The former step, the surface fitting problem, is converted to a linear system with two quadratic constraint equations, which could be solved by a null-space analysis technique. On the other hand, the given exact poses in the latter step are formulated as linear pose-constraint equations and added back to the system, where both type and dimensions of the resulting optimal dyads could be determined by the solution. These optimal dyads could then be implemented as different types of four-bar linkages or parallel manipulators. The result is a novel algorithm that is simple and efficient, which allows for N-pose motion approximation of planar dyads containing both revolute and prismatic joints, as well as handling of up to four prescribed poses to be realized precisely.

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References

Figures

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Fig. 1

A planar pose of a moving frame M with respect to the fixed frame F

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Fig. 2

Three types of planar dyad open-chains: RR, PR, and RP

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Fig. 3

Two samples quadratic functions in the form of Eq. (12) plotted in α−β plane with γ set to be 1. These two equations yield up to four intersection points, each denoting one real solution of p in Eq. (11).

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Fig. 4

Eleven-pose example: plot of the 11 poses as listed in Table 1, in which the first, third, eighth, and eleventh poses are critical positions that need to be realized precisely

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Fig. 5

Eleven-pose example: the four-bar linkage is constructed by the two optimal resulting dyads in Table 4. Its coupler motion trajectory is also plotted as comparison to the 11 given poses.

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Fig. 6

Eleven-pose example: plot of the coupler curve and 11 given poses. It can be seen that the first, third, eighth, and eleventh poses are exactly realized.

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Fig. 7

Seven-pose example: seven arbitrary poses in Table 5, in which the third, sixth, and seventh poses require exact realization

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Fig. 8

Seven-pose example: the four-bar linkage is constructed by the first and fourth resulting dyads in Table 7 as well as its coupler motion

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Fig. 9

Seven-pose example: plot of the resulting coupler curve and seven given poses. It can be seen that the third, sixth, and seventh poses are exactly realized while the rest four poses are approximately guided through.

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