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Research Papers

# A Complete Geometric Singular Characterization of the 6/6 Stewart Platform

[+] Author and Article Information
Michael Slavutin

School of Mechanical Engineering,
Tel Aviv University,
Ramat Aviv 69978, Israel
e-mail: slavutin@post.tau.ac.il

Avshalom Sheffer

School of Mechanical Engineering,
Tel Aviv University,
Ramat Aviv 69978, Israel
e-mail: sheffer@mail.tau.ac.il

Offer Shai

School of Mechanical Engineering,
Tel Aviv University,
Ramat Aviv 69978, Israel

Yoram Reich

School of Mechanical Engineering,
Tel Aviv University,
Ramat Aviv 69978, Israel
e-mail: yoram@eng.tau.ac.il

1Corresponding author.

2Deceased.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received December 12, 2017; final manuscript received April 17, 2018; published online June 1, 2018. Assoc. Editor: Shaoping Bai.

J. Mechanisms Robotics 10(4), 041011 (Jun 01, 2018) (10 pages) Paper No: JMR-17-1414; doi: 10.1115/1.4040133 History: Received December 12, 2017; Revised April 17, 2018

## Abstract

This paper introduces the three-dimensional (3D) dual Kennedy theorem in statics, and demonstrates its application to characterize the singular configuration of the 6/6 Stewart Platform (6/6 SP). The proposed characterization is articulated as a simple geometric relation that is easy to apply and check. We find two lines that cross four of the six legs of the platform. Each one of these two lines has a parallel line that crosses the remaining two legs. Each pair of parallel lines defines a plane. The 6/6 SP is in a singular position if the intersection of these two planes is perpendicular to the common normal of the remaining two legs. The method developed for the singular characterization is also used for the analysis of the mobility and forces of the SP. Finally, the proposed method is compared to some known singular configurations, such as Hunt's and Fichter's singular configurations and the 3/6 Stewart Platform (3/6 SP) singularity. The relation between the reported characterizations of the 6/6 SP and other reported works is highlighted. Moreover, it is shown that the known 3/6 SP characterization is a special case of the results reported in the paper. Finally, a characterization of a platform that does not appear in the literature, 5/6 SP, is developed based on the new approach to demonstrate its utility.

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## Figures

Fig. 1

A schematic explanation of the singular characterization of a general 6/6 SP: (a) the general view of the platform in this example, (b) lines L1 and L2 cross the four leg lines 1–4. Lines L1′ and L2′ are parallel to L1 and L2, respectively, and cross leg lines 5, 6, (c) line m is the intersection of the two planes πL1,L1′ and πL2,L2′, (d) the common normal to leg lines 5, 6 is perpendicular to line m,and (e) the view from above to the perpendicularity.

Fig. 2

The analysis of the mobility of SP 6/6: (a) lines L1 and L2 cross the four leg lines 1–4 and their common normal, CNL1,L2, (b) lines l5′ and l6′ are parallel to l5 and l6, respectively, and cross lines L1 and L2, (c) m′=πl5,l5′∧πl6,l6′ is the line of intersection of planes πl5,l5′ and πl6,l6′, (d) lines l5″ and l6″ lie on plane πl5,l5′, πl6,l6′ and parallel to l5′ and l6′, respectively. Point M′ lies on line m′ and lines l5″ and l6″ pass through this point and construct plane πM′, (e) line m′⊥ lies on the plane πM′ and is perpendicular to line m′, and (f) the ISA $t is perpendicular to the common normal of L1 and L2, i.e., CNL1,L2, and to m′⊥. Fig. 3 Determining the angular velocities ω1 and ω2 and the pitch h of the ISA$t: (a) the angle of lines, L1 and L2, relative to the ISA and (b) the distance of lines, L1 and L2, from the ISA along the k̂ axis

Fig. 4

Case of a single line crossing the six legs of Stewart Platform

Fig. 5

Fichter singular configuration

Fig. 6

A configuration with parallel base and platform

Fig. 7

Case of a robot with collinear joints

Fig. 8

3/6 Stewart platform

Fig. 9

3/6 Stewart platform ISA

Fig. 10

5/6 Stewart platform

Fig. 11

Finding center of hyperboloid drawn on the given three lines

Fig. 12

Determination of the crossing points of the hyperboloid and the fourth given line

## Errata

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