Analysis and design of hexaslide manipulators for machine tool applications

ANALYSIS AND DESIGN OF HEXASLIDE MANIPULATORS FOR MACHINE TOOL APPLICATIONS

by

ABBARAJU BALA KOTESWARA RAO

Mechanical Engineering Department

Submitted

in fulfillment of the requirements of the degree of Doctor of Philosophy

to the

Indian Institute of Technology, Delhi

July, 2004

CO Indian Institute of Technology New Delhi - 2004

Dedicated- to

914- v beroved Teachers, Parents, and

EamiCy

CERTIFICATE

This is to certify that the thesis entitled, "Analysis and Design of Hexaslide Manipulators for Machine Tool Applications" being submitted by Mr, ABBARAJU BALA KOTESWARA RAO to the Indian Institute of Technology Delhi for the award of the degree of Doctor of Philosophy in Mechanical Engineering is a bonafide record of original research work carried out by him under our supervision i

^CO

Cori iity with the rules and regulations of the Institute.

The results presented in his thesis have not been submitted, in part or full, to any other University or I 'tint - for the award of any degree or diploma.

t.

1)r. P. V. M. Rao

Assistant Professor

Mechanical Engineering Department Indian Institute of Technology Delhi

New Delhi - 110016, India

Dr. S. K. Saha Associate Professor

Mechanical Engineering Department Indian Institute of Technology Delhi

New Delhi - 110016, India

i

ACKNOWLEDGEMENTS

I would like to express my sincere thanks and deep sense of gratitude to my supervisors, Dr. P. V. M. Rao, Asst. Professor, and Dr. S. K. Saha, Assoc. Professor, Mechanical Engineering Department for their constant encouragement and inspiring guidance without which this thesis could have not been completed. Their academic excellence, clear explanations steered me through all the hurdles of my research work. Their critical reviews and constructive comments improved my grasp of the subject and steered to the fruitful completion of the work. Their personal affection, patience, guidance, and encouragement made this thesis possible.

Dr. P. V. M. Rao, despite a very hectic schedule made it a point to spare time for valuable discussions and sharing wealth or his knowledge and experience on the subject. lie has been very kind in his approach and cheered me all through during the course of this work. Dr. S. K. Saha provided excellent help and understanding in every step of my work right from day one. His dedication, punctuality, and elegant presentations motivated me to carry out my research work in a systematic way.

I also take this opportunity to express my heartfelt thanks to the respected SRC members Prof. V. P. Agarwal, Dr. N. 13hatnagar, and Prof. D. K. Sehgal, for their valuable suggestions.

With profound respect, I wish to express my deepest sense of gratitude to Prof P. S. Rao, Prineipal, G. V. P. College or Engineering, Visakhapatnam, Prof. 13, Sarveswara Rao, Chairman, Gayatri Vidya Parishad, and Prof. P. Soma Raju, Secretary, Gayatri Vidya Parishad; because of whose encouragement, by deputing me under Q. I. P. to the

J. T. 1'.

Delhi, this has been achievable. I extend warm respects towards them for their continuous encouragement. I am extremely grateful to the Q. I. P. Coordinator and his team of nienibers, L I. T. Delhi for the opportunity rendered to accomplish my goal.

I would like to express my thanks to Mr. D. Jaitly, Mechatronics Laboratory and Mr.

Ramchandar, N. C. Laboratory for their kind support and co-operation. ^I can not forget Mr. Amit whose timely suggestion made me to feel ICRA Travel Award from IEEE. I wish to acknowledge the strong support and affection from my beloved friends, Mr. V. N. D. Srinivis, Mr. A. Darpe, Mr. A. Mahajan, Mr. T. S. Arvindan, Mr. V. S. Rao, Mr. Prasad, 13., Mr. Naresh K., and Mr. Himanshu Chowdary, Mr. A.S.K.A.V. Prasad Rao, Mr. K. V. Ramana, and others

Gratitude can seldom be expressed in words. I would like to express heartfelt thanks to Dr. K. A. Seetharam and Dr. V. Bhatnagar for their timely treatment to my second son. Without their personal care, I could have not concentrated over this work. Dr. K. A. Seetharan is the One who stands always with me at all instants of time and steers though his cheerful style.

My parents made supreme sacrifices for me throughout my career. No words can express my feelings towards them. My wife, Hanuma, has been a cheerleader to the core and lit, • an uncanny knack of shooting down occasional burst of sadness. Many thanks to her for. showing me silver lining in every dark cloud. And, how can I forget my beloved little sons Tamil and Leela Krishna, who with their innocent smiles showered love that wiped all the traces of tiredness day after day.

I can not forget my uncle, Late Sri K. Subbara Rao, for his strong encourager e lt and motivation; and my aunty for her staying with us at IIT-Delhiby caring my sons sc) affectionately. Thanks to my brothers, sisters, brothers-in-law, and sisters-in-law, for their hearty wishes and blessings.

Last but not the least, I convey my thanks to one and all who have contributed to the realization of this thesis.

43. Kpteswara (Rczo I. I. T. Delhi, India

ABSTRACT

Conventional machine tool structures used for processes like milling, drilling, etc., have serial kinematic chain, with each feed axis is built on the top of another one. The lower axis carries larger mass due to the axes' elements above it, causing high bending due to the Cantilever action. This puts limitations on their accuracy and maximum cutting force to he withstood. Machine tools based on Parallel Kinematics (PKMs) have been found to overcome these limitations due to smaller moving masses. lie::apods are the most common version of PKMs which have been explored for Machine tool applications.

Flexaslides which seem to overcome some of the limitations of hexapods have been of recent interest among PKIVIs.

In this thesis, a comprehensive study of the hexaslides is carried from kinematics, kinetostaties, and dynamics point of view. Each leg of constant length is connected to a slider that moves over a fixed rail. Actuators, usually the heavier part of the machine, that drive We lead-screw on a rail, are mounted on fixed base. So, the moving mass in a hexaslide is drastically reduced, allowing higher cutting forces to be withstood. Due to high rigidity and lightweight structures, hexaslides find applications in high speed machining, precision pleasuring, and others. However, their closed kinematic chains make their design and control difficult. Hence, a comprehensive study on workspace, dexterity, stiffness, and dynamics of hexaslides is presented. Several new performance indices like workspace volume index, workspace shape index, etc. are introduced for design purposes. Optimization is also performed for maximum workspace and dexterity.

Finally, a novel dynamic model based on the Decoupled Natural Orthogonal Complement (DeNOCI) matrices is presented, which is used for the inverse dynamics necessary to estimate actuator power and control.

iv

CONTENTS

Certificate

Acknowledgements Abstract

Contents

List of Figures List of Tables

List of salient Symbols and Abbreviations

Page No.

11

iv

viii xi xii

1 Introduction

1.1 13ackground 1

1.2 Ilexaslide machine tools 5

1.3 Contributions of the present research 6

1.4 Organization of the thesis 8

2 Literature Survey

2.1 Introduction 11

2.2 Kinematics 14

2.3 Workspace and Dexterity 15

2.4 Stiffness 17

2.5 Design optimization 19

2.6 Dynamics 21

3 Kinematic Analysis

3.1 Position analysis 29

3.2 Velocity analysis 31

3.3 Workspace evaluation 33

3.3.1 Workspace of I fexaM 35

3.4 Dexterity evaluation 39

3.4.1 Dexterity of I lexaM 41

contd.

4 Performance Measures and Comparison of Hexasl ides

Page No.

4.1 Workspace shape indices 45

4.2 Workspace volume indices 47

4.3 Workspace indices of HexaM 51

4,4 Global dexterity index 53

4.5 Comparison of hexaslides 54

5 Design Optimization

5.1 Optimization problem 64

5.2 Selection of grid size 66

5.3 Optimization results 67

5.4 Dimension scaling 74

6 Stiffness Analysis

6.1 Effect of

individual

assembly on tool platform 78

6.1.1 Effect of axial leg-deformation 80

6.1.2 Effect of leg bending 81

6.2 Stiffness model for actuator assembly 81

6.3 Stiffness model for leg assembly 84

6.3.1 Effect of axial flexibility 85

6.3.2 Effect of bending flexibility 87

6.4 Stiffness of a hexaslide 88

6.5 Stiffness measures 89

6.5.1 Stiffness along and about the Cartesian coordinate axes 89

6.5.2 Global stiffness index 90

6.6 Stiffness characteristics of hexaslides 90

cont(

vi

7 Dynamic Analysis

Page No.

7.1 Kinematic analysis 99

7.1.1 Angular velocity of legs 102

7.1.2 Angular acceleration of legs 104

7.2 Dynamic modeling 106

7.2.1 Uncoupled Newton-Euler equations 107

7.2.2 1)ecoupled natural orthogonal complement matrices 109

7.2.3 Coupled dynamic equations 114

7.3 Inverse dynamics algorithm 116

7.4 Circular contouring by IlexaM 120

7.4.1 Effect of leg and slider inertias 125

7.5 Dynamics of hexaslides 127

8 Conclusions

8.1 Research summary 133

8.2 Limitations and scope of fUture work 136

References 138

A Appendix

A.1 Direct kinematics 148

A.2 Algorithm to find largest cube 149

A,3 Stiffness formulas 150

A.3,1 Bearing stiffness 150

A.3.2 Nut stiffness 151

A.4 Numerical NOC evaluation 152

vii