Research Papers

Deterministic Design for a Compliant Parallel Universal Joint With Constant Rotational Stiffness

[+] Author and Article Information
Yan Xie

Robotics Institute,
Beihang University,
Beijing 100191, China
e-mail: xieyan@buaa.edu.cn

Jingjun Yu

Robotics Institute,
Beihang University,
Beijing 100191, China
e-mail: jjyu@buaa.edu.cn

Hongzhe Zhao

Robotics Institute,
Beihang University,
Beijing 100191, China
e-mail: hongzhezhao@gmail.com

1Corresponding author.

Manuscript received June 18, 2017; final manuscript received December 11, 2017; published online March 30, 2018. Assoc. Editor: Guimin Chen.

J. Mechanisms Robotics 10(3), 031006 (Mar 30, 2018) (12 pages) Paper No: JMR-17-1182; doi: 10.1115/1.4039065 History: Received June 18, 2017; Revised December 11, 2017

Compliant universal joints have been widely employed in high-precision fields due to plenty of good performance. However, the stiffness characteristics, as the most important consideration for compliant mechanisms, are rarely involved. In this paper, a deterministic design for a constraint-based compliant parallel universal joint with constant rotational stiffness is presented. First, a constant stiffness realization principle is proposed by combination of the freedom and constraint topology (FACT) method and beam constraint model (BCM) to establish a mapping relationship between stiffness characteristics and topology configurations. A parallel universal joint topology is generated by the constant stiffness realization principle. Then, the analytical stiffness model of the universal joint with some permissible approximations is formulated based on the BCM, and geometrical prerequisites are derived to achieve the desired constant rotational stiffness. After that, finite element analysis (FEA), experimental testing, and detailed stiffness analysis are carried out. It turns out that the rotational stiffness of the universal joint can keep constant with arbitrary azimuth angles even if the rotational angle reaches up to ±5 deg. Meanwhile, the acceptable relative errors of rotational stiffness are within 0.53% compared with the FEA results and 2.6% compared with the experimental results, which indicates the accuracy of the theoretical stiffness model and further implies the feasibility of constant stiffness realization principle on guiding the universal joint design.

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Fig. 1

Visualization of constant stiffness realization principle: (a) a cantilever beam under arbitrary loads within actuation space, (b) a cantilever beam with the free-end restricted to the primary motion, and (c) a serial layout to eliminate the axial load

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Fig. 2

Visualization of constant stiffness in basic flexure modules: (a) parallelogram mechanism, (b) double parallelogram mechanism with mirror symmetry, (c) equivalent double parallelogram mechanism of constant translational stiffness, (d) generalized cross-spring pivot, (e) double cross-spring pivot with mirror symmetry, and (f) equivalent double cross-spring pivot of constant rotational stiffness

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Fig. 5

A generalized blade flexure

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Fig. 4

Constant stiffness realization of the beam-based universal joint: (a) physical embodiment of a constraint line with two noncoplanar blade flexures, (b) physical embodiment of a constraint plane with eight dual blade flexures in parallel, and (c) physical embodiment of a universal joint

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Fig. 3

Conceptual design of constraint-based universal joint: (a) freedom and constraint space of universal joint, and (b) beam-based universal joint design (left) and blade-based universal joint design (right)

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Fig. 8

The rotational deformation of flexure module I: (a) rotational deformation of upper part in flexure module I and (b) partial enlarged view for end-displacements

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Fig. 7

Configuration of flexure module I

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Fig. 6

Load distribution on the moving stage

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Fig. 9

The influence of geometric parameters α1 and λ on rotational stiffness kr

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Fig. 10

The rotational deformation of universal joint around x-axis except flexure modules II and IV

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Fig. 11

The translational deformation of upper part in flexure module I

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Fig. 12

The influence of geometric parameter α1 and external load f on displacement s: (a) isometric view and (b) top view

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Fig. 14

Detailed design of the universal joint: (a) scheme of the flexure module and (b) assembly scheme of the universal joint

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Fig. 15

Proof-of-concept prototypes: (a) flexure module prototype, (b) universal joint prototype, and (c) experimental setup for stiffness test

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Fig. 13

The load-rotation of ACE V: (a) the rotational deformation of ACE V around x-axis and (b) the influence of geometric parameter ξ and rotational angle θ on equivalent load msum

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Fig. 16

Stiffness test of the universal joint with an azimuth angle ψ = 0: (a) moment-rotational angle curves and (b) stiffness-rotational angle and relative error-rotational angle curves

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Fig. 17

Stiffness test of the universal joint with azimuth angles varying from 0 deg to 360 deg: (a) average stiffness under different azimuth angles (0 deg, 15 deg, 30 deg, and 45 deg) and (b) average stiffness and relative errors with azimuth angles varying from 0 to 360 deg



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