Novel method of constructing generalized Hoberman sphere mechanisms based on deployment axes

doi:10.1007/s11465-019-0567-5

Front. Mech. Eng.

2020, Vol. 15

Issue (1) : 89-99 https://doi.org/10.1007/s11465-019-0567-5

RESEARCH ARTICLE

Novel method of constructing generalized Hoberman sphere mechanisms based on deployment axes

Xuemin SUN¹, Yan-An YAO¹(

), Ruiming LI²(

)

¹. School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing 100044, China
². Robotics Institute, Beihang University, Beijing 100191, China

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Abstract

This study proposes a method of constructing type II generalized angulated elements (GAEs II) Hoberman sphere mechanisms on the basis of deployment axes that intersect at one point. First, the constraint conditions for inserting n GAEs II into n deployment axes to form a loop are given. The angle constraint conditions of the deployment axes are obtained through a series of linear equations. Second, the connection conditions of two GAEs II loops that share a common deployable center are discussed. Third, a flowchart of constructing the generalized Hoberman sphere mechanism on the basis of deployment axes is provided. Finally, four generalized Hoberman sphere mechanisms based on a fully enclosed regular hexahedron, arithmetic sequence axes, orthonormal arithmetic sequence axes, and spiral-like axes are constructed in accordance with the given arrangement of deployment axes that satisfy the constraint conditions to verify the feasibility of the proposed method.

Keywords deployable mechanism type II generalized angulated elements Hoberman sphere mechanism deployment axes constraint conditions

Corresponding Author(s): Yan-An YAO,Ruiming LI

Just Accepted Date: 04 November 2019 Online First Date: 11 December 2019 Issue Date: 21 February 2020

Cite this article:

Xuemin SUN,Yan-An YAO,Ruiming LI. Novel method of constructing generalized Hoberman sphere mechanisms based on deployment axes[J]. Front. Mech. Eng., 2020, 15(1): 89-99.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-019-0567-5
https://academic.hep.com.cn/fme/EN/Y2020/V15/I1/89

Fig.1 Polyhedra and deployment axes. (a) Regular hexahedron; (b) uniform triangular prism; (c) truncated octahedron; (d) arithmetic sequence axes.

Fig.2 Angulated elements and generalized angulated elements. (a) AEs; (b) GAEs II.

Fig.3 Cartesian coordinate system of two GAEs II.

Fig.4 Loop of n GAEs II.

Fig.5 Parameters of GAEs II.

Fig.6 Connection for two loops. (a) Two loops; (b) three GAEs II of platform A.

Fig.7 Flowchart of constructing a Hoberman sphere mechanism.

Fig.8 Hoberman sphere mechanism based on the regular hexahedron. (a) Deployment axes; (b) configuration I; (c) configuration II; (d) configuration III.

Fig.9 Hoberman sphere mechanism based on a fully enclosed regular hexahedron. (a) Insert deployment axes; (b) configuration I; (c) configuration II; (d) configuration III.

Fig.10 3D printed prototype of a Hoberman sphere mechanism based on a fully enclosed regular hexahedron. (a) Configuration I; (b) configuration II; (c) configuration III.

Fig.11 Hoberman sphere mechanism based on arithmetic sequence axes. (a) Deployment axes; (b) configuration I; (c) configuration II; (d) configuration III.

i	$θ i$ /(° )	$θ ′ i$ /(° )	$θ ″ i$ /(° )	l₂_i_–1/mm	l₂_i/mm
1	15.5	7.25	8.25	37.86	43.05
2	17.5	8.25	9.25	43.05	48.22
3	19.5	9.25	10.25	48.22	53.38
4	21.5	10.25	11.25	53.38	58.53
5	23.5	11.25	12.25	58.53	63.65
6	25.5	12.25	13.25	63.65	68.76
7	27.5	13.25	14.25	68.76	73.85
8	29.5	14.25	15.25	73.85	78.91

Tab.1 Parameters of the Hoberman sphere mechanism based on arithmetic sequence axes

Fig.12 Hoberman sphere mechanism based on orthonormal arithmetic sequence axes. (a) Deployment axes; (b) configuration I; (c) configuration II; (d) configuration III.

j	$ζ j$ /(° )	$ζ ′ j$ /(° )	$ζ ″ j$ /(° )	L₂_j_–1/mm	L₂_j/mm
1	22.65	9.25	13.40	48.22	69.53
2	22.50	11.25	11.25	58.53	58.53
3	20.32	11.25	9.07	58.53	47.30
4	36.37	13.25	23.12	68.76	117.81

Tab.2 Parameters of inserted axes

Fig.13 Hoberman sphere mechanism based on spiral-like axes. (a) Deployment axes; (b) configuration I; (c) configuration II; (d) configuration III.

i	$θ i$ /(° )	$θ ′ i$ /(° )	$θ ″ i$ /(° )	l₂_i_–1/mm	l₂_i/mm
1	45.00	22.50	22.50	45.00	45.00
2	46.65	22.50	24.15	45.00	48.12
3	55.24	24.15	31.09	48.12	60.72
4	61.68	31.09	30.59	60.72	59.85
5	61.68	30.59	31.09	59.85	60.72
6	55.24	31.09	24.15	60.72	48.12
7	46.65	24.15	22.50	48.12	45.00
8	45.00	22.50	22.50	45.00	45.00

Tab.3 Parameters of the Hoberman sphere mechanism based on spiral-like axes

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