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Technical Brief

A Computational Geometric Approach for Motion Generation of Spatial Linkages With Sphere and Plane Constraints

[+] Author and Article Information
Xiangyun Li

School of Mechanical Engineering,
Southwest Jiaotong University,
Chengdu 610031, China
e-mail: xiangyun.app@gmail.com

Q. J. Ge

Department of Mechanical Engineering,
Stony Brook University,
Stony Brook, NY 11794
e-mail: qiaode.ge@stonybrook.edu

Feng Gao

School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: fengg@sjtu.edu.cn

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received June 6, 2018; final manuscript received October 11, 2018; published online December 10, 2018. Assoc. Editor: Andrew P. Murray.

J. Mechanisms Robotics 11(1), 014504 (Dec 10, 2018) (7 pages) Paper No: JMR-18-1168; doi: 10.1115/1.4041788 History: Received June 06, 2018; Revised October 11, 2018

This paper studies the problem of spatial linkage synthesis for motion generation from the perspective of extracting geometric constraints from a set of specified spatial displacements. In previous work, we have developed a computational geometric framework for integrated type and dimensional synthesis of planar and spherical linkages, the main feature of which is to extract the mechanically realizable geometric constraints from task positions, and thus reduce the motion synthesis problem to that of identifying kinematic dyads and triads associated with the resulting geometric constraints. The proposed approach herein extends this data-driven paradigm to spatial cases, with the focus on acquiring the point-on-a-sphere and point-on-a-plane geometric constraints which are associated with those spatial kinematic chains commonly encountered in spatial mechanism design. Using the theory of kinematic mapping and dual quaternions, we develop a unified version of design equations that represents both types of geometric constraints, and present a simple and efficient algorithm for uncovering them from the given motion.

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Figures

Grahic Jump Location
Fig. 1

A spatial displacement

Grahic Jump Location
Fig. 2

Spherical RR-S Leg

Grahic Jump Location
Fig. 8

Solution 1: a sphere with center at (11.2473, 5.3352, 2.4121) and radius of 14.1004

Grahic Jump Location
Fig. 9

Solution 2: a plane defined by the homogeneous equation 0.7736X1 – 0.4464a2X2 + 0.0006X3 + 0.0075X4 = 0

Grahic Jump Location
Fig. 10

Solution 3: a sphere with center at (2.6928, 2.1003, 2.7649) and radius of 3.9451

Grahic Jump Location
Fig. 11

Solution 4: a plane constraint defined by the homogeneous equation –0.7806X1 + 0.4528X2 + 0.0070 * X3 + 0.0794 X4 = 0

Grahic Jump Location
Fig. 12

Solution 5: a sphere with center at (–0.4067, 1.1529, 2.8930) and radius of 4.8645

Grahic Jump Location
Fig. 13

Solution 6: a sphere with center at (3.0556, –2.0322, 2.4240) and radius of 2.1526

Grahic Jump Location
Fig. 14

Solution 7: a plane defined by the homogeneous equation 0.7746X1 + 0.4472X2 = 0

Grahic Jump Location
Fig. 15

A spatial parallel manipulator constructed by a spherical RR-S leg associated with the sphere constraint given by solution 5, a SS leg with the sphere constraint by solution 3, and a RPS leg with the plane constraint by solution 4

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