A geometric singular characterization of Parallel robots

By: Avshalom ShefferSchool of Mechanical

EngineeringTel Aviv University

Supervised by: Prof. Offer Shai

School of Mechanical Engineering

Tel Aviv University

A GEOMETRIC SINGULAR CHARACTERIZATION OF PARALLEL ROBOTS

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INTRODUCTION

A SCHEMATIC EXPLANATION OF THE GEOMETRIC SINGULAR

CHARACTERIZATION OF THE 6/6 SP

CONCLUSION

TABLE OF CONTENTS

INTRODUCTIONParallel manipulators have a specific mechanical architecture where all the links are connected both at the base.

3/6 Stewart Platform-Spatial Triad

6/6 Stewart Platform Spatial Double Triad

Spatial Tetrad

Hunt’s Singular Configuration

The common line crosses all the leg lines of the SP

Fichter’s Singular Configuration

The moving platform rotates by around the vertical axis

A known singular configurations of 6/6 Stewart Platform

A SCHEMATIC EXPLANATION OF THE GEOMETRIC SINGULAR CHARACTERIZATION OF THE 6/6 SP

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5

6

312

4

31

2

4

In order to calculate the ISA, , we calculate the center axis of the cylindroid of all the possible ISAs of four-leg platform, which is obtained by removal of any two legs from the SP

Let the lines and , be the two lines that cross the three leg lines 1–3 of the remaining four leg lines of the SP. There is an infinite number of such lines

12

4

𝑳𝟐

𝑳𝟏

XX

XX X X

3

We look for the ISA, , as the combination of the lines and

The projection of the platform velocity along each four legs is equal to zero since the legs are rigid bodies. This can be formulated as follows:

1 2

4

𝑳𝟐

𝑳𝟏

XX

XX X X

3

𝑪𝑵𝐿1 ,𝐿2

$𝟏

The line, , obtained by choose another two lines, and , that cross the three legs

Let be the common normal to the lines and and the center axis of the cylindroid of all the possible ISAs of four-leg platform

1

2

43

$𝟏

$𝟐

𝑪𝑵 𝐼

The 6/6 Stewart Platform is in a singular configuration if and only if the three common normal; , and , obtained for each deleted pair of legs have the same common normal.

The common normal, and , obtained by deleted another pair of the three pairs

Due to the generalized Aronhold-Kennedy theorem, this common normal is the ISA of the platform

𝑪𝑵 𝐼

1

243

56

𝑪𝑵 𝐼 𝐼𝐼

𝑪𝑵 𝐼 𝐼

𝑰𝑺𝑨

CONCLUSION

The method presented is consistent with other approaches that appear in the literature

It seems that the method introduced is applicable in finding the singularity of many other types of mechanisms and is not limited to a particular mechanism

The method is based on discrete mathematics thus can be computerized easily

I believe that equimomental line/ screw is a fundamental concept in statics and have a significant potential in characterizing singularity of spatial parallel mechanisms

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