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Static balancing of rigid-body

linkages and compliant mechanisms

A THESIS

SUBMITTED FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

IN THE FACULTY OF ENGINEERING

by

Sangamesh Deepak R

Department of Mechanical Engineering

Indian Institute of Science

Bangalore – 560012, INDIA

MAY 2012

c© Copyright by Sangamesh Deepak R 2013

All Rights Reserved

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Abstract

Static balance is the reduction or elimination of the actuating effort in quasi-static

motion of a mechanical system by adding non-dissipative force interactions to the

system. In recent years, there is increasing recognition that static balancing of elastic

forces in compliant mechanisms leads to increased efficiency as well as good force

feedback characteristics. The development of insightful and pragmatic design meth-

ods for statically balanced compliant mechanisms is the motivation for this work.

In our approach, we focus on a class of compliant mechanisms that can be approxi-

mated as spring-loaded rigid-link mechanisms. Instead of developing static balancing

techniques directly for the compliant mechanisms, we seek analytical balancing tech-

niques for the simplified spring–loaded rigid–link approximations. Towards that, we

first provide new static balancing techniques for a spring-loaded four-bar linkage. We

also find relations between static balancing parameters of the cognates of a four-bar

linkage. Later, we develop a new perfect static balancing method for a general n-

degree-of-freedom revolute and spherical jointed rigid-body linkages. This general

method distinguishes itself from the known techniques in the following respects:

1. It adds only springs and not any auxiliary bodies.

2. It is applicable to linkages having any number of links connected in any manner.

3. It is applicable to both constant (i.e., gravity type) and linear spring loads.

4. It works both in planar and spatial cases.

This analytical method is applied on the approximated compliant mechanisms as

well. Expectedly, the compliant mechanisms would only be approximately balanced.

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We study the effectiveness of this approximate balance through simulations and a

prototype. The analytical static balancing technique for rigid-body linkages and the

study of its application to approximated compliant mechanisms are among the main

contributions of this thesis.

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Acknowledgements

The motivating idea for this work is that static balancing techniques for compliant

mechanisms can be developed by extracting insights from static balancing techniques

for rigid-body linkages. The originality of this idea belongs to my advisor, Prof. G.

K. Ananthasuresh. Further, I had regular meetings with my advisor where he offered

his comments and suggestions on my work. Some of his suggestions helped me in

directing this work to this form. I also appreciate his emphasis on prototypes and

the support system he has developed for making them. He has spent significant time

in sharing his experiences and opinions on technical writing. These were valuable in

documenting my work.

In making prototypes presented in the thesis, I had received help from Mr. A.

Ravikumar, Mr. Ramu G., Mr. B. M. Vinod Kumar and Mr. Praveenraj H. K. I

deeply value their knowledge and skills. I also acknowledge the co-operation that I

received from Mr. A. Raja in accessing the workshop.

Almost all the fabrication aspects of the prototype presented in Appendix G was

handled by Mr. Amrith Hansoge. I deeply value his industrial experience, which were

crucial in making a rather impressive prototype.

Prof. Ashitava Ghosal, advisor for my master’s thesis, was very helpful during my

transition from master’s to PhD. I also appreciate the remarkable care he took to see

that I choose a topic of research that suits me.

My course-work in IISc has been one of the most exciting and satisfying aspects

of my life in IISc. I immensely thank all the instructors for their effort in getting

us exposed to challenging concepts. Prof. C. R. Pradeep’s class on Topology ranks

as the best class that I ever had in my academic career. Courses such as these have

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helped me a lot.

Another exciting aspect of my stay in IISc was the weekly group meeting that Prof.

Ananthasuresh conducts. Through these meetings, my advisor and my colleagues in

the lab gave me an exposure to diverse areas of knowledge and research. In an

association spanning a little less than five years, I probably have many things to say

about my advisor. However, I will sum up this association like this – it has been

mostly a pleasure and definitely a privilege.

My lab-mates have been invariably nice, co-operative and understanding. I feel

very privileged to be part of such a good set of people.

A lot of people formed the source for many exciting academic exchanges that I

had. Among them, I would like to make a special mention of Narayana Reddy, Kali-

das, Meenakshi Sundaram, Hariharan and Sreenath. These exchanges have helped

me to put my attention on some of the fundamental principles in mechanics and

mathematics in general.

While my interaction with people in IISc is somewhat less, I must nevertheless

acknowledge that in general people have been kind and understanding towards me. I

would like to specially thank many mess workers who served me with food even if I

was late many times.

The physical training that I received from Master Stephen Kumar, Master Man-

junath and my senior Rahul S., has been one of the best things to happen in my life.

This training helped me to maintain my mental balance even in depressing times.

On personal front, my mother absorbed all the shocks, pulls and pressure to

ensure a free ambience for my growth. It was because of this free ambience that

my understanding of science and mathematics matured over the years. Patience,

perseverance, hope, courage, endurance and wisdom displayed by my mother are

inspirational. I also acknowledge the care shown by my father towards my well-being.

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Contents

Abstract iii

Acknowledgements v

1 Introduction 1

1.1 What is static balance? . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Static balance of rigid-body linkages . . . . . . . . . . . . . . 2

1.1.2 Compliant mechanisms and its static balance . . . . . . . . . . 9

1.1.3 Static balance: rigid-body linkage vs. compliant mechanisms . 11

1.2 Motivation for the thesis . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Scope of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3.1 Three methods to statically balance a zero-free-length spring-

loaded four-bar linkage . . . . . . . . . . . . . . . . . . . . . . 14

1.3.2 Static balancing of cognates . . . . . . . . . . . . . . . . . . . 14

1.3.3 Static balancing of revolute-jointed linkages without auxiliary

links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3.4 Static balancing of spatial and/or revolute jointed linkages

without auxiliary links . . . . . . . . . . . . . . . . . . . . . . 15

1.3.5 Static balancing of flexure-based compliant mechanisms by ad-

dition of springs . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Literature Survey 17

2.1 Rigid-body linkages under gravity loads . . . . . . . . . . . . . . . . . 17

2.1.1 Counter-weight balancing . . . . . . . . . . . . . . . . . . . . 17

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2.1.2 Balancing by addition of springs . . . . . . . . . . . . . . . . . 18

2.2 Rigid linkages under spring loads . . . . . . . . . . . . . . . . . . . . 21

2.3 Static balance of compliant mechanisms . . . . . . . . . . . . . . . . . 22

2.4 Prior Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4.2 Plagiograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4.3 Derivation of the balancing solution in [1] . . . . . . . . . . . 28

3 Static balancing of a four-bar 36

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Static balancing of a given spring-loaded four-bar linkage . . . . . . . 37

3.2.1 Technique 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.2 Technique 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.3 Technique 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3 A prototype with technique 2 . . . . . . . . . . . . . . . . . . . . . . 48

4 Static balance of the cognates 51

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.1.1 Cognates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.1.2 Static balancing parameters and the cognates . . . . . . . . . 52

4.2 A geometric problem and its solution . . . . . . . . . . . . . . . . . . 54

4.2.1 Case (i): a, b and c are parallel. . . . . . . . . . . . . . . . . . 58

4.2.2 Case (ii): a, b and c are concurrent. . . . . . . . . . . . . . . . 59

4.2.3 Case (iii): a, b, and c are neither parallel nor concurrent. . . . 59

4.2.4 Finding the focal pivot . . . . . . . . . . . . . . . . . . . . . . 61

4.3 The result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3.1 A parameterization of the balancing parameters of the three

cognates that has cognate triangle related invariants . . . . . . 63

5 Static balancing of planar linkages 66

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2 Balancing a lever . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

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5.2.1 Potential energy as a function of the configuration variable . . 68

5.2.2 Invariance of potential energy with respect to the configuration

variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.3 Balancing of a rigid body in a plane . . . . . . . . . . . . . . . . . . . 74

5.4 New static balancing techniques for revolute-jointed linkages . . . . . 80

5.4.1 The potential energy of loads on a body transformed as a func-

tion of another body . . . . . . . . . . . . . . . . . . . . . . . 80

5.4.2 Proposition 2 as the recursive relation of an iterative static

balancing algorithm . . . . . . . . . . . . . . . . . . . . . . . . 82

5.4.3 Static balancing of any revolute-jointed linkages with any kind

of zero-free-length spring and constant load interaction within

the linkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.5 A note on prismatic joint . . . . . . . . . . . . . . . . . . . . . . . . . 99

6 Static balancing of spatial linkages 102

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.2 The class of functions in feature 1 . . . . . . . . . . . . . . . . . . . . 103

6.3 Joints that can potentially satisfy feature 2 . . . . . . . . . . . . . . . 104

6.4 Spherical joint has feature 2 . . . . . . . . . . . . . . . . . . . . . . . 107

6.5 Revolute joint has feature 2 . . . . . . . . . . . . . . . . . . . . . . . 109

6.6 Algorithm to synthesize static balancing solution of a spatial revolute/spherical-

jointed tree-structured linkage having zero-free-length spring and/or

gravity loads exerted by a reference link . . . . . . . . . . . . . . . . 111

6.6.1 Illustrative example . . . . . . . . . . . . . . . . . . . . . . . . 113

6.7 Static balance of any kind of spatial revolute and/or spherical jointed

linkage with constant load and zero-free-length spring load interaction 116

6.8 A note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7 Static balance of compliant mechanisms 118

7.1 Balancing a flexure beam . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.1.1 The flexure beam . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.1.2 Rigid-body model for the flexure beam . . . . . . . . . . . . . 120

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7.1.3 Approximation of torsional spring by zero-free-length spring . 122

7.1.4 Static balancing by addition of a zero-free-length spring . . . . 126

7.1.5 Balancing springs on the flexure beam . . . . . . . . . . . . . 130

7.2 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.3 Flexure-based compliant four-bar mechanism . . . . . . . . . . . . . . 136

7.3.1 Description of the mechanism . . . . . . . . . . . . . . . . . . 136

7.3.2 Step 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7.3.3 Step 2 – The effort function . . . . . . . . . . . . . . . . . . . 138

7.3.4 Step 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

7.3.5 Step 4 – Static balance of the linkage under zero-free-length

spring load . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

7.3.6 Step 5 – Approximate static balance of flexure-based four-bar

linkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7.3.7 A way to improve the static balance of the flexure-based four-

bar linkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7.4 Another flexure-based four-bar linkage . . . . . . . . . . . . . . . . . 145

7.4.1 Description of the mechanism . . . . . . . . . . . . . . . . . . 145

7.4.2 Step 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.4.3 Step 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.4.4 Step 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.4.5 Step 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.4.6 Step 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.4.7 First order correction . . . . . . . . . . . . . . . . . . . . . . . 149

7.4.8 Prototype . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7.5 Flexure-based 2R compliant mechanism . . . . . . . . . . . . . . . . . 152

7.5.1 Description of the compliant mechanism . . . . . . . . . . . . 152

7.5.2 Step 1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

7.5.3 Step 2 – Identification of effort function . . . . . . . . . . . . . 153

7.5.4 Step 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

7.5.5 Step 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

7.5.6 Step 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

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7.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7.6.1 Static balancing of compliant mechanisms by individually bal-

ancing flexures . . . . . . . . . . . . . . . . . . . . . . . . . . 159

7.6.2 Static balancing of compliant mechanisms using rigid-body link-

ages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

8 Conclusion 163

8.1 A summary of new static balancing techniques for spring and/or

gravity-loaded rigid-body linkages . . . . . . . . . . . . . . . . . . . . 164

8.1.1 Static balancing of a four-bar linkage loaded by a spring on its

coupler link . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

8.1.2 Static balancing parameters and the cognates of a four-bar linkage164

8.1.3 Static balancing without auxiliary bodies–planar case . . . . . 165

8.1.4 Static balancing without auxiliary bodies–spatial case . . . . . 165

8.2 A framework for designing statically balanced compliant mechanisms 165

8.3 The novelty of the contribution in the context of the current literature 166

8.4 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

A Proofs on finding the focal pivot 169

A.1 If Ia,b and Ic,a circles are coincident, then the given lines a, b and c has

to be concurrent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

A.2 When M and A are distinct, M is the focal pivot. (Refer to section

(4.2.4 and figure (4.11).) . . . . . . . . . . . . . . . . . . . . . . . . . 170

B An elementary theorem 172

C Normal springs 173

D Satisfying Constraints 175

D.1 Satisfying constraints (5.12), (5.13) and (5.14) . . . . . . . . . . . . . 175

D.2 Satisfying constraints (5.19), (5.20) and (5.21) . . . . . . . . . . . . . 177

E Solving balancing constraints – spatial case 178

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F Virtual work calculations 181

F.1 Calculation of stiffness in case 3a based on case 2a . . . . . . . . . . . 186

F.1.1 Obtaining Fx vs. ux in case 2a using virtual work balance . . 186

F.1.2 Obtaining Fx vs. ux in case 3a using virtual work balance . . 191

F.2 Verification of static balance in case 3b through virtual work balance 195

F.3 A first order correction to balancing springs on the flexure-based four-

bar linkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

G Another compliant mechanism balancing 207

H Springs between successive links 212

Bibliography 216

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List of Tables

3.1 Summary of different techniques to statically balance a spring-loaded

four-bar linkage presented in this chapter . . . . . . . . . . . . . . . . 47

4.1 Deduction of various quantities in equations (3.1 – 3.7) from the cog-

nate triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2 Ratio of spring constants of balancing spring 1 for three cognates . . 64

5.1 Potential energy of the weight and the zero-free-length component of

the spring acting on the lever is a linear combination of cos θ, sin θ,

and 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2 Potential energy of weight and spring acting on a link moving in a plane. 75

5.3 Details of the loads in figure (5.5). It may be checked that the loads

satisfy equations (5.12 – 5.14 ) . . . . . . . . . . . . . . . . . . . . . 77

6.1 The potential in the general form shown in figure (6.3), can be ex-

pressed as a linear combination of the basis functions shown in the

table. Each basis function in the table is a function of translational

variable r and the Z-X-Z Euler angle α, β and γ . . . . . . . . . . . . 105

7.1 Relevant quantities to calculate the torsional stiffness of the spring . . 120

7.2 Virtual work calculation for lever in case (2a) . . . . . . . . . . . . . 124

7.3 Virtual work calculation and slope of Fx vs. ux . . . . . . . . . . . . 125

7.4 Verification of equations (5.10–5.11) being satisfied . . . . . . . . . . 129

7.5 Verification of static balance through virtual work calculations . . . . 129

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7.6 The origin and the slope of Fx vs. ux being matched between case 2a

and case 3a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

7.7 Verification of equations (5.19 - 5.21) for the spring-loaded 2R linkage

of figure (7.14) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

7.8 Details of the flexure and calculation of torsional spring constant . . . 147

7.9 Matching value and slopes at the origin of the effort function . . . . . 148

7.10 Details of the original spring and balancing spring . . . . . . . . . . . 149

7.11 First order correction of balancing spring parameters . . . . . . . . . 150

7.12 Details of flexure and calculation of torsional spring constant. . . . . 153

7.13 The value and the first derivative of F vs. u at u = 0 . . . . . . . . 154

7.14 Verification of static balance of springs in case 3b . . . . . . . . . . . 157

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List of Figures

1.1 A lever having two discrete equilibrium configurations, namely (b) and

(c) while (a) is not in equilibrium. . . . . . . . . . . . . . . . . . . . . 3

1.2 A lever having a continuous set of equilibrium configurations; it is in

static balance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 A statically balanced lever as a manually operated road barrier. (Source:

http://www.panoramio.com/photo/42452078) . . . . . . . . . . . . . 4

1.4 Counterweight balancing in a two degree-of-freedom linkage . . . . . 5

1.5 Spring-based balancing in a two degree-of-freedom linkage . . . . . . 5

1.6 Spring-based balancing of a lever . . . . . . . . . . . . . . . . . . . . 6

1.7 Effects of free-length and pre-tension on force-distance plot of a linear

extension spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.8 A compliant crimper . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.9 A spring-loaded four-bar linkage to be statically balanced . . . . . . . 12

1.10 A static balancing solution for the spring-loaded four-bar linkage by

addition of auxiliary links and springs . . . . . . . . . . . . . . . . . . 13

1.11 A compliant four-bar mechanism . . . . . . . . . . . . . . . . . . . . 14

2.1 A counter balancing technique for serial revolute jointed linkages . . . 18

2.2 Use of auxiliary links (colored in grey) along with springs in an existing

technique for balancing a n-degree-of-freedom linkage under constant

load. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 A basic spring force balancer . . . . . . . . . . . . . . . . . . . . . . . 24

2.4 A lever with ordinary springs . . . . . . . . . . . . . . . . . . . . . . 25

2.5 Balanced parallelogram . . . . . . . . . . . . . . . . . . . . . . . . . . 25

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2.6 Balanced two degree-of-freedom parallelogram linkage . . . . . . . . . 26

2.7 Composition of two zero-length-spring into an equivalent one. The virtual,

equivalent spring between E and D is shown in grey color. . . . . . . . . 27

2.8 A plagiograph or a skew pantograph linkage . . . . . . . . . . . . . . . . 28

2.9 A balanced parallelogram with one its spring along a diagonal decom-

posed into two springs. . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.10 The parallelogram linkage with a duplicate that maintains a constant

angle of α from the original . . . . . . . . . . . . . . . . . . . . . . . 30

2.11 Synchronization of motion between two parallelogram linkages . . . . 31

2.12 Scaling the duplicate parallelogram linkage . . . . . . . . . . . . . . . 32

2.13 Removal of a spring and compensating it with increase in stiffness of

another spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.14 Removal of two more links. . . . . . . . . . . . . . . . . . . . . . . . . 33

2.15 The current literature and our contributions . . . . . . . . . . . . . . 35

3.1 A four-bar linkage with a zero-free-length spring anchored from its

coupler to the ground . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Possibility of the four-bar linkage being statically balanced as it is . . 38

3.3 A balanced parallelogram on the load spring . . . . . . . . . . . . . . 39

3.4 Technique 1: Static balancing with two auxiliary links and one balanc-

ing spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.5 Forming a parallelogram using two auxiliary links. . . . . . . . . . . 40

3.6 A virtual spring between opposite vertices of the parallelogram AFED. 41

3.7 Adding balancing spring 2 across opposite vertices of the parallelogram

to balance the load spring and balancing spring 1. . . . . . . . . . . . 41

3.8 Option 2 for balancing the four-bar linkage without requiring auxiliary

links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.9 Technique 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.10 Plagiograph with base pivot at B . . . . . . . . . . . . . . . . . . . . 46

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3.11 A prototype of four-bar linkage that is statically balanced using option

2. The four-bar linkage seen in the above figure is a Watt’s straight-line

mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.12 Realization of a zero-free length spring . . . . . . . . . . . . . . . . . 49

4.1 The cognates of a four-bar mechanism taking a load spring along their

common coupler curve . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2 A cognate triangle and the ground anchor point of a load spring . . . 55

4.3 Technique 3: the base pivot of the plagiograph (U) and the base of the

balanced parallelogram (W) for every cognate are not coincident. . . 55

4.4 Technique 3: the base pivot of the plagiograph (U) and the base of the

balanced parallelogram (W) for every cognate are coincident. . . . . . 56

4.5 Technique 2: the base of the balanced parallelogram (W) at the indi-

cated vertices of the cognate triangle. . . . . . . . . . . . . . . . . . . 57

4.6 A problem in planar geometry . . . . . . . . . . . . . . . . . . . . . . 58

4.7 Finding Sa, Sb, and Sc in case (i) . . . . . . . . . . . . . . . . . . . . 58

4.8 Finding Sa, Sb, and Sc in case (ii) . . . . . . . . . . . . . . . . . . . . 59

4.9 Description of focal pivot . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.10 Points A2, B2 and C2 form a solution to Sa, Sb and Sc . . . . . . . . 60

4.11 Geometric construction to find the focal pivot . . . . . . . . . . . . . 62

5.1 Lack of methods for spring-based n-body linkage balancing without the

usage of auxiliary bodies . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.2 A lever under a constant load and a spring load . . . . . . . . . . . . 69

5.3 Static balancing of a weight by a spring . . . . . . . . . . . . . . . . . 72

5.4 A body that is free to move in a plane . . . . . . . . . . . . . . . . . 74

5.5 A rigid body moving freely in a plane under a constant load is made to

have θ-independent potential energy by addition of two zero-free-length

springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.6 The gravity-loaded serial 4R linkage to be statically balanced. . . . . 86

5.7 Details of Iteration 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 87


xvii



5.11 Statically balanced gravity-loaded serial 4R linkage. . . . . . . . . . . 91

5.12 Statically balanced serial 3R linkage . . . . . . . . . . . . . . . . . . . 93

5.13 Statically balanced serial 2R linkage . . . . . . . . . . . . . . . . . . . 94

5.14 Details of static balance of a 2R linkage under spring load . . . . . . 95

5.15 Details of static balance of a 4R tree-structure linkage under a constant

load and a spring load . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.16 Potential Energy variation of spring loads, constant loads, and their sum 97

5.17 Breaking a problem as a superposition of several problem with each

problem being static balance of revolute-jointed tree-structured linkage

with loads exerted by the root body . . . . . . . . . . . . . . . . . . 98

5.18 Static balance of a tree-structured linkage with inter-body load inter-

actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.19 Balancing the lever lever loads in the second load set of figure (5.18) . 101

6.1 Details of static balancing of six degree-of-freedom spatial balancing

under gravity loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.1 Details of the flexure beam . . . . . . . . . . . . . . . . . . . . . . . . 119

7.2 Our attention is on reducing horizontal force for a range of horizontal

displacements of point P . . . . . . . . . . . . . . . . . . . . . . . . . 121

7.3 Small-length-flexure model applied to the flexure beam . . . . . . . . 122

7.4 Approximation of the torsional spring by a zero-free-length spring . . 123

7.5 The approximate match in Fx vs. ux relation between case 2a and case

3a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

7.6 Static balance of the approximated zero-free-length spring-loaded lever

by addition of a zero-free-length spring . . . . . . . . . . . . . . . . . 128

7.7 All the cases related to the flexure and its approximation by the spring-

loaded lever . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.8 Fx vs. ux relation obtained from finite element analysis of the flexure

beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

xviii

7.9 A consolidated plot of Fx vs. ux for flexure beam and its rigid-body

models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

7.10 A flexure-based four-bar linkage. . . . . . . . . . . . . . . . . . . . . . 137

7.11 The quadrilateral formed by the centers of the flexures . . . . . . . . 137

7.12 Approximation of the flexure-based four-bar linkage as a rigid-body

four-bar linkage with torsional springs. . . . . . . . . . . . . . . . . . 138

7.13 Approximation of torsional springs by zero-free-length springs . . . . 139

7.14 Static balancing by addition of zero-free-length springs . . . . . . . . 141

7.15 A consolidated figure of all the cases . . . . . . . . . . . . . . . . . . 142

7.16 Finite element simulation results for case 1a and case 1b . . . . . . . 143

7.17 Fx vs. ux after first order correction to the stiffness of balancing springs144

7.18 FX vs. ux plot for all the cases shown in figure (7.15) . . . . . . . . . 145

7.19 Consolidated figure containing all the cases . . . . . . . . . . . . . . . 146

7.20 Effort function in all the cases . . . . . . . . . . . . . . . . . . . . . . 148

7.21 A prototype to demonstrate reduction in effort . . . . . . . . . . . . . 151

7.22 A consolidation of all the cases . . . . . . . . . . . . . . . . . . . . . 152

7.23 Details of springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

7.24 Fx vs. u in two views . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7.25 Fy vs. u in two views . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.26 Plot of effort function when flexure length is increased three-fold . . . 159

7.27 Ideal circular arc-like and non-ideal deformation of flexures . . . . . . 160

7.28 Static balancing of each of the flexures, independently of one another 161

8.1 The current literature and our contributions . . . . . . . . . . . . . . 167

A.1 Ia,b and Ic,a circles are coincident . . . . . . . . . . . . . . . . . . . . 170

A.2 Ia,b is at M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

F.1 l, the length of a diagonal of the quadrilateral of four-bar bar linkage

is used as a convenient configuration defining parameter. . . . . . . . 181

G.1 A compliant gripper compensated by a small spring loaded 2R linkage.

(Basement board dimension: 2.5 feet × 2.5 feet) . . . . . . . . . . . 208

xix

G.2 A statically balanced parallelogram and its modification . . . . . . . 209

G.3 A compliant gripper . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

H.1 A tree-structured linkage under gravity load . . . . . . . . . . . . . . 213

xx

Chapter 1

Introduction

Overview

• The concept of static balance.

• The importance of statically balancing a rigid-body linkage.

• Zero-free-length springs, their importance and their practical realization.

• The importance of statically balancing a compliant mechanism.

• The motivation of the thesis.

• The contributions of the thesis.

1.1 What is static balance?

A system is said to be in static balance if it can undergo quasi-static motion without

any external effort when any dissipative force interactions in the system are removed.

In the general motion of a system, at any configuration, inertia forces of the system,

conservative and dissipative force interactions within the system and external forces

acting on the system are in equilibrium. However, in the motion of a statically

1

CHAPTER 1. INTRODUCTION 2

balanced system, the conservative force interactions have to be in equilibrium by

themselves.

Consider the lever shown in figure (1.1a). The conservative force interactions on

it are the gravity between the lever and the ground and the constraint force between

the lever and the pivot-post (fixed to the ground). If, at a configuration, these forces

could be in equilibrium by themselves then we call the configuration as equilibrium

configuration. This system has only two discrete equilibrium configurations shown

in figures (1.1b–c). It is impossible to have quasi-static motion that does not pass

through any configuration other than these two discrete equilibrium configurations.

Hence, the system in not in static balance. In contrast, consider figure (1.2), which is

the same lever with an extra mass added so that the overall centre of gravity is at the

pivot. Here, every configuration is an equilibrium configuration. Hence, one can have

quasi-static motion passing through only equilibrium configurations. Therefore, this

system is in static balance. The effortless motion of this system has been utilized for

a long time in manually operated road barriers such as the one shown in figure (1.3).

If a system is not in static balance, it may be possible to add extra conservative

force interactions to the system such that all the conservative forces are in equilibrium.

This process of addition is called static balancing. An example of static balancing is

the addition of extra mass to the system in figure (1.1) to obtain the system in figure

(1.2).

1.1.1 Static balance of rigid-body linkages

In the example of figure (1.3), static balancing meant nullifying the effect of gravity

forces on the material making up the barrier. In fact, much of the past research on

static balance was concerned with nullifying the effects of gravity on the material

making up a rigid-body linkage. The motivation for such research efforts was that

many practical systems such as leg-orthosis, robots and flight-simulators are made up

of rigid-body linkages and nullifying the gravity effects in them would significantly

reduce the torque or the force requirement from the actuators.

Static balance of rigid-body linkages may be broadly classified into two groups:


g

(a)

g

(b)

g

(c)

Figure 1.1: A lever having two discrete equilibrium configurations, namely (b) and(c) while (a) is not in equilibrium.

ca

ma

co

mo

g

moco +maca = 0

The centres of gravity of the orig-inal lever and that of the extramass are collinear with the pivotco and ca: Coordinates of the cen-tres of gravity of the original leverand the additional mass about thepivotmo and ma: Masses of the originallever and the additional mass

Figure 1.2: A lever having a continuous set of equilibrium configurations; it is instatic balance.


Figure 1.3: A statically balanced lever as a manually operated road barrier. (Source:http://www.panoramio.com/photo/42452078)

counterweight-addition and spring-addition. Figure (1.4) shows the example of a

counterweight balancing technique. The original weight could be that of, say, a lamp

head in a table lamp. Figure (1.5) shows the spring-based balancing of a two degree-

of-freedom system.

Most of the counterweight balancing methods are derived from the lever-balancing

principle shown in figure (1.2). For example, in figure (1.4), the center of mass of the

original weight and that of the counterweight are always collinear with the ground

pivot. Further, the ratio of distances between these points is always the same and the

magnitude of the counterweight added is such that it obeys the same equation as in

figure (1.2). Similarly, the genesis of most of the spring-based balancing techniques

could be traced to the spring-based balancing of a lever shown in figure (1.6).

We now explain how the lever in figure (1.6) is balanced. In figure (1.6), by

geometry, we have

ss = l + hj (1.1)

By denoting the free-length of the spring by s0, the spring force on the lever is given


Counter weight

Original weight

Figure 1.4: Counterweight balancing in a two degree-of-freedom linkage

Original weight

Figure 1.5: Spring-based balancing in a two degree-of-freedom linkage


θ l

kh

wweight

j

s

ss = l + hj

s

Figure 1.6: Spring-based balancing of a lever

by

f =k(s− s0)s

=k(s− s0)1

s

(

l + hj)

, from eqn. (1.1) (1.2)

The moment of the spring force about the lever is given by

M spring =− l× f

=−k (s− s0)

sl×(

l + hj)

, from eqn. (1.2)

=−k (s− s0)

sh(

l× j)

(1.3)

Similarly, the moment of the gravity load is

Mweight = −l×−wj (1.4)

The net moment becomes

M spring +Mweight =

(

−kh (s− s0)

s+ w

)

l× j (1.5)


If we want static balance, then the net moment has to be zero over a continuous range

of θ. Given that l × j is zero at only discrete values of theta, we can only expect its

coefficient to be zero over a continuous set of configurations. However, when s0 6= 0,

the coefficient could become zero, again, at only a discrete set of configurations (Note

that s is a function of θ). Nevertheless, if we can have s0 = 0, i.e., if we can have a

spring of zero free-length, then the coefficient becomes a constant over every θ and,

in particular, zero if kh = w. Thus, the conditions under which the lever shown in

figure (1.6) attains static balance are zero free-length and kh = w. While achieving

kh = w is not hard, it is not popularly known that zero-free-length can also be

achieved in practice. The credit for recognizing the importance of zero free-length in

static balance and also its practical implementation goes to George Carwardine [2]

and Lucien LaCoste [3].

Having understood the necessity of zero-free-length springs for perfect static bal-

ance, we now focus on one of the ways of its practical realization. Figure (1.7) shows

the spring force (f) versus the relative distance (d) of anchor points of a linear ex-

tension spring for various cases. The plot for zero-free-length spring is expected to

be collinear with the origin, as shown in figure (1.7a). A normal extension spring

not only has a finite positive free-length but also what is called as pre-tension. In a

pre-tensioned spring, even if there is no external force, the coils press against each

other. The coils separate and the spring extends only when the external force is more

than the pre-tension. The effect of free-length l0 is to shift the force-distance plot

along l axis by l0 and the effect of pre-tension fp is to shift the plot along the f -axis

by fp. When fp is kl0 where k is slope of the plot (i.e., the stiffness of the spring), the

plot becomes collinear with the origin. Thus, even with unavoidable free-length, by

inducing appropriate pre-tension, one can have force-distance relation to be the same

as that of a zero-free-length spring for l > l0. Therefore, there is no practical hin-

drance in realizing a zero-free-length spring. There are several other ways of realizing

a zero-free-length spring, as discussed in [4] and [5].


l0

fp

l

f

l

f

l0l

f(force)

(distance)

(a) zero free length(b) positive free length without

pre-tension

(c) positive free length with pre-tension

Figure 1.7: Effects of free-length and pre-tension on force-distance plot of a linearextension spring


1.1.2 Compliant mechanisms and its static balance

In recent years, compliant mechanisms have emerged as a plausible design option,

especially in micro mechanical systems. A compliant mechanism, in contrast to a

rigid-body linkage, is made up of a single monolithic piece that transmits force and

displacement through elastic deformation. They are quite amenable to microfabrica-

tion techniques apart from being free of friction and backlash.

Analysis of rigid-body mechanisms can often be carried out analytically. This is

possible since it is the geometry of triangles, quadrilaterals and other polygons that

underlies the kinematics of rigid-body mechanisms. Furthermore, many graphical and

analytical synthesis methods have also been developed for the design of rigid-body

linkages. Accurate analysis of compliant mechanisms, on the other hand, generally

require numerical finite element analysis.

In an attempt to bring the analytical and graphical techniques developed for the

synthesis of rigid-body mechanisms into the realm compliant mechanisms, Midha and

Howell ([6], [7]) developed pseudo-rigid-body model for a class of compliant mecha-

nisms. This model allows a certain class of compliant mechanisms to be approximately

represented by spring-loaded rigid-body mechanisms. Flexure-based compliant mech-

anism form an important subclass of compliant mechanisms that can be represented as

spring-loaded rigid-body mechanisms. Howell and Midha [7] showed that the flexure

can be approximated by a revolute joint with a torsional spring having linear torque–

angle characteristics. This type of approximation is popularly know as small-length

flexure approximation. Thus, with these models, one can bring in several analytical

and graphical rigid-body mechanism design techniques into the realm of compliant

mechanisms.

Static balance of compliant mechanisms

While compliant mechanisms are superior to rigid-body mechanisms in terms of fric-

tion and backlash, they have a feature that could be disadvantageous. Figure (1.8

a) shows a compliant mechanism. Figure (1.8 b) shows the mechanism acting on a

workpiece. Figure (1.8 c) shows the same mechanism requiring effort even though it


is not acting on any workpiece. Mere actuation of the compliant mechanism requires

effort. The source of this effort is due to the spring-like behaviour of the compliant

mechanism arising out of its inherent elasticity.

(a)

(b) (c)

Figure 1.8: A compliant crimper

Statically balancing the elastic forces, i.e., nullifying the spring-like behaviour of

compliant mechanisms is under increasing attention. The reason for that is not just

the reduction in the effort to operate a compliant mechanism but also the prospect


of having tools that offer good force feedback. If a person handling a tool gets a good

sense of the force that the tool is applying on a workpiece, then the tool is said to

offer good force feedback. In certain tools consisting of linkages, such as laparoscopic

grippers, the force feedback is very bad for the reasons attributed to friction in the

joints. Compliant mechanisms on the other hand do not have joint-friction but their

spring-like behaviour can affect the force feedback. If this spring-like behaviour can

be removed, then compliant mechanisms can offer good force feedback.

1.1.3 Static balance: rigid-body linkage vs. compliant mech-

anisms

In rigid-body linkages, for static balance, we often want all possible motion that

the linkage can take to be effortless. This implies that conservative forces in all its

configurations are in equilibrium. As per our definition, “all possible motion” is not

necessary to qualify as static balance. Nevertheless, in this thesis, to comply with the

popular notion, we have struck to this “all possible motion” as far as static balancing

of rigid-body linkages are concerned.

In compliant mechanisms, which have infinite degree-of-freedom, in contrast to

finite degree-of-freedom of rigid-body linkages, it is not feasible to expect all possible

motion to be effortless. Only one or two modes of motion (or deformation) in which

the compliant mechanism normally operates is sought to be made effortless. In other

words, we aim to make configurations only along one or two paths to be in equilibrium.

1.2 Motivation for the thesis

The motivating idea of this thesis is to make use of rigid-body approximations, such

as pseudo-rigid-body model, for static balancing of compliant mechanisms. This mo-

tivation was first proposed in [8]. Pseudo-rigid-body model was successful in the

synthesis of compliant mechanisms because of the existence of simple analytical tech-

niques for the synthesis of rigid-body mechanisms. Similarly, for pseudo-rigid-body

model to be successful in making statically balanced compliant mechanisms, there


has to be simple analytical techniques for static balancing of spring-loaded linkages

rather than gravity-loaded linkages.

Prior to the work of this thesis, there was only one work, carried out by Just Herder

([1]), that recognized the importance of static balancing of spring-loaded rigid-body

linkages. The context of Herder’s work ([1]) was compensating elastic forces of a

cosmetic glove in a hand prosthesis. Herder approximated the motion of the hand

prosthesis as the motion of a four-bar linkage and the elastic forces of the glove as a

zero-free-length spring attached to the coupler link of the four-bar linkage, as shown

in figure (1.9). Herder then derived a static balancing solution to it as shown in figure

(1.10). As can be seen in figure (1.10), the balancing solution involves addition of two

auxiliary bodies and two balancing springs. The result of incorporating this balancing

solution into an actual hand prosthesis was presented in [9], where reduction in the

effort to operate the prosthesis was demonstrated.

Loadingzero-free-lengthspring (kl)

AB

C

D

K

Coupler(Link 2)

Link 3

Link 1

E

Fixed pivots

Anchor point

Figure 1.9: A spring-loaded four-bar linkage to be statically balanced

When we thought of using Herder’s balancing solution ([1]) through a rigid-body

approximation on a compliant four-bar mechanism with flexure pivots, such as the

one shown in (1.11), we noticed that incorporating additional bodies of the balancing

solution of [1] (see figure (1.10)) into the compliant mechanism could be a little


AK

kl

E

DF T

B

H

kb

C

Auxiliary links

G

Balancing spring 1

Balancing spring 2

Loading springkt

Anchor point

Anchor point

Figure 1.10: A static balancing solution for the spring-loaded four-bar linkage byaddition of auxiliary links and springs

cumbersome. This motivated us to take a closer look at the principles of static

balancing to see if there are other ways of statically balancing the same spring-loaded

four-bar linkage (of figure (1.9)), preferably, without using auxiliary bodies. We did

find other ways of balancing the four-bar linkage and one such way does not use

auxiliary bodies. We could also get some new general static balancing techniques

without using auxiliary bodies. We eventually gave a framework for making use of

these static balancing solutions to make statically balanced flexure-based compliant

mechanisms through small-length flexure model. An overview of these contributions,

which make up the bulk of the chapters in the thesis is presented next.


R

Figure 1.11: A compliant four-bar mechanism

1.3 Scope of the thesis

1.3.1 Three methods to statically balance a zero-free-length

spring-loaded four-bar linkage

For the static balancing problem shown in figure (1.9), apart from the method shown

in figure (1.10), which was already in the literature [1], we give two more methods

where the number of additional springs and the number of auxiliary links are less than

or equal to that of the method shown in figure (1.10). Further, we also recognized a

variant of the method given in figure (1.10). Among the methods that we give, one

method does not use any auxiliary link. These results are elaborated in Chapter 3.

1.3.2 Static balancing of cognates

In kinematics, there is a well-known theorem called Roberts-Chebyshev cognate the-

orem (see [10], for example). According to the theorem, for every four-bar linkage

with a specified coupler point on it, one can find two more four-bar linkages and

coupler points on them such that the coupler curves of the three four-bar linkages

are the same. These three four-bar linkages are termed as cognates. The triangle


formed by the ground anchor points of cognates, called the cognate triangle, plays a

central role in a few other elegant relations that the theorem states. With the intent

to extend the theorem to static balancing, we present some relations among static

balancing parameters (spring constants and anchor points of balancing springs) of

different cognates in which the cognate triangle again plays a central role. These

results are elaborated in Chapter 4.

1.3.3 Static balancing of revolute-jointed linkages without

auxiliary links

Building upon a method discussed in Section 1.3.1, we present a general method

that can statically balance any revolute-jointed linkage having zero-free-length spring

force and constant force interactions between the bodies constituting the linkage.

This method adds only zero-free-length springs but not auxiliary links. This result is

elaborated in Chapter 5.

1.3.4 Static balancing of spatial and/or revolute jointed link-

ages without auxiliary links

Further extending the planar method of Section 1.3.3 to spatial linkages, we show that

any spatial revolute and/or spherical jointed linkages with zero-free-length spring and

constant force interactions can be statically balanced. Again, the method of balancing

requires only addition of zero-free-length springs but not auxiliary links. This result

is elaborated in Chapter 6.

1.3.5 Static balancing of flexure-based compliant mecha-

nisms by addition of springs

As pointed in Section 1.2, the motivation to find new methods for static balancing of

spring-loaded rigid-body linkages was to make statically balanced compliant mecha-

nisms through pseudo-rigid-body model. Having found the new methods, we give a


simple step-by-step framework for static balancing of a flexure-based compliant mech-

anisms through small-length flexure model. We give four examples to illustrate the

framework. A prototype of one of the examples is also made. Both simulations and

the fabricated prototype show more than 70 % reduction in the effort. These results

are described in Chapter 7.

Summary

• Static balance implies equilibrium among conservative forces over a continuous

set of configurations.

• Static balancing of rigid-body mechanisms generally focus on compensating

gravity forces.

• Primary focus in static balancing of a compliant mechanism is to compensate

the inherent elastic forces of the compliant mechanism.

• The motivating idea of the thesis is to use rigid-body models of compliant

mechanisms to design statically balanced compliant mechanisms. We recognize

that for the idea to be successful, it is necessary to have new analytical static

balancing methods for spring-loaded rigid-body linkages.

• This thesis presents new static balancing methods for spring-loaded linkages

without using auxiliary bodies. The thesis also presents a simple framework to

use these methods through small-length flexure model for static balancing of

compliant mechanisms.

• The last section of this chapter defined the scope of the thesis and its organi-

zation in the remaining chapters.

Chapter 2

Literature Survey

Overview

• Literature on static balancing of gravity-loaded rigid-body linkages by adding

counter-weights or springs.

• Literature on the importance of static balancing of spring-loaded rigid-body

linkages and the existing methods to handle such problems.

• Literature on various approaches that have been pursued to address the design

of statically balanced compliant mechanisms.

2.1 Rigid-body linkages under gravity loads

There are two dominant ways of balancing a linkage under gravity loads. One is by

addition of counter weights and the other is by addition of springs. The literature in

these fields is described next.

2.1.1 Counter-weight balancing

The simplest of rigid-body linkages is a lever. The conditions for static balance of a

lever under gravity loads was first given by Archimedes ([11]). It relies on making the

17

CHAPTER 2. LITERATURE SURVEY 18

overall center of gravity of the system to be a constant. This lever-balancing principle

has been adapted to pantograph linkages, as in counterweight balanced lamps ([12]).

Further, even though we do not have specific references, counterweight balancing of

a serial revolute jointed linkage such as the one shown in figure (2.1) seems to have

been known for a long time. Such a balancing of serial linkages has been a part of

many balancing schemes, such as in [13].

W

1 2

3 W

W

2W

4W1 2

3

(a) (b)

Figure 2.1: A counter balancing technique for serial revolute jointed linkages

2.1.2 Balancing by addition of springs

George Carwardine, a British engineer, is the pioneer in the area of static balancing

of gravity loaded linkages by addition of springs. In a series of patents he obtained

([14], [15], [16], [17] and [18]) he gave the art of statically balancing a gravity load

supported on a two-revolute jointed linkage. The balancing method involved addition

of two zero-free-length springs and auxiliary links. The patent for statically balanced

Anglepoise lamp ([2]), which is still popular today, is among these patents.

At around the same time as Carwardine, American physicist Lucien LaCoste ([3])

recognized that a pendulum could be in perfect static balance when a zero-free-length

spring is attached to it. LaCoste is credited with first recognizing the role of zero-free-

length springs in perfect static balancing of a gravity-loaded lever. This discovery was

made in the context of devising a pendulum with a very long period of oscillation.

Such a pendulum is apparently useful in seismographs.

Streit and Gilmore [5] made a thorough study of a lever under spring loads. They


discussed achieving a set of discrete as well as a set of continuous equilibrium config-

urations. Some of the ways to realize zero-free-length springs were also discussed.

Nathan [19] gave a way to extend the spring-based lever-balancing to two-degree-

of-freedom linkages using auxiliary parallelogram linkages. Pracht et al. [20] made

a slightly different extension where all the springs are anchored from the ground to

different parts of the linkage. Streit and Shin [21] made another extension using

pantograph linkages. They obtained different design variants by varying input points

and input motion. They also suggested that such a gravity-balancer could be used

in walking machines so that the torque requirement of the motors is reduced. The

attempt of Wongrathanaphisan and Cole [22] to statically balance a load on the

coupler of a four-bar linkage led to a solution that is conceptually not different from

LaCoste’s solution.

Nathan [19] also extended his two-degree-of-freedom linkage-balancing to a n-

degree-of-freedom revolute-jointed linkage. Figure (2.2) illustrates the method in [19]

where, to balance a gravity load on a 3R linkage shown in figure (2.2a), auxiliary

linkages (colored in grey) and extra springs are added. Streit and Shin [23] similarly

extended the work of Pracht et al. [20] to a n-degree-of-freedom revolute-jointed

linkage. Reference [23] termed the extension of Nathan [19] as “vertical link systems”

and their extension of reference [20] as “parallel link systems”. It also presented static

balancing of a linkage having a series of revolute-prismatic pairs of joints.

WW

1 2

3

(a) (b)

Figure 2.2: Use of auxiliary links (colored in grey) along with springs in an existingtechnique for balancing a n-degree-of-freedom linkage under constant load.


Herder’s PhD thesis [4] brought new approaches to the field of spring-based static

balancing. Herder obtained a variety of multi-degree-of-freedom gravity-balancing

linkages using 1) a few modification rules, 2) a few basic statically balanced linkages,

such as the balanced lever of LaCoste, and 3) the properties of auxiliary parallelogram

and pantograph linkages. Herder also introduced the concept of a floating suspension.

Whatever reaction forces that a pivot exerts on a gravity-loaded lever, the same forces

could be exerted by a floating suspension, which is nothing but springs anchored from

the ground to the lever. Thus, under quasi-static conditions, a floating suspension

could form a friction-less replacement for a pivot.

Walsh et al. [24] showed a way to balance a spatial rigid-body attached to the

ground by a two degree-of-freedom joint formed by two revolute joints of intersecting

axes. Streit et al. [25]), in a similar work, dealt with two degree-of-freedom Hooke’s

joint in a somewhat different way. Wongratanaphisan and Chew [26] gave a way to

balance a more general revolute-jointed two-degree-of-freedom serial spatial manipu-

lator using auxiliary links. Agrawal and Fattah [27] provided an interesting method

to balance a spatial gravity-loaded linkage. In the method, by adding auxiliary paral-

lelogram linkages, a physical point that is also the center of mass of the overall system

is first identified. Then, depending on the kind of motion this center of mass under-

goes, springs are added to compensate the gravity. The work of Rahman et al. [28]

gives a straight forward extension of Nathan’s “vertical link systems” to spatial n–

body revolute-jointed linkages. While Rahman et al. [28] used simple parallelograms

in their extension, Lin et al. [29] used spatial RSSR (revolute-spherical-spherical-

revolute) parallelogram linkages to provide a more comprehensive extension. There

is a lot of literature on the static balancing of parallel manipulators such as [13], [30],

[31], [32], [33], [34], [35], [36], [37] and [38].

While the techniques in the above literature use auxiliary bodies in addition to

springs, we (Sangamesh and Ananthasuresh [39]) showed how to balance a n-degree-

of-freedom (n ≥ 1) revolute-jointed spring and/or gravity linkage using only springs

but not auxiliary bodies. Lin et al. [40], using what they called as stiffness matrix

approach, also provided balancing methods without auxiliary bodies for two and three

revolute-jointed serial gravity-loaded linkages. Shieh and Chen [41] showed how to


statically balance revolute-jointed planar one-degree-of-freedom closed-loop linkages

without using auxiliary links. Our work [39] was extended in [42], which gave more

general static balancing conditions.

All the techniques discussed above use zero-free-length springs for perfect static

balance of the gravity loads. While there are a few works that use other kinds of

springs as balancing elements, they are either approximate balancing techniques or use

cams to modulate the spring behaviour. Gopalswamy et al. [43] gave an approximate

static balancing technique where torsional springs are used as balancing elements.

Agrawal and Agrawal [44] presented an approximate static balancing method using

non-zero-free-length springs. The balancing techniques that modulate the behaviour

of springs include the technique in [45], where a pulley of varying radius was used,

and the technique in [46] and [47], where a cam was used.

There are a few works dealing with biomedical applications of static balancing.

The ones dealing with leg include [48], [49], [50], [51], [52], [53], [54], [55], and [56].

The ones that deal with upper limb include [57], [58], [59], [60], [61], [62] and [63].

2.2 Rigid linkages under spring loads

Herder’s work [1] was the first to recognize the importance of a class of problems where

springs forces, which could be an approximation for more complex elastic forces, need

to be compensated. The motivation for the work was to compensate the elastic forces

of the cosmetic glove of a hand prosthesis. In this work, the motion of the fingers of a

hand prosthesis was modelled as the motion of the coupler link of a four-bar linkage.

The elastic forces of the cosmetic glove were lumped into to a zero-free-length spring

attached to a point on the coupler. The work then gives a perfect balancing solution

for statically balancing this spring-loaded four-bar linkage. The obtained solution is

based on extending the balancing of a skew lever to a skew pantograph. Incorporation

of the pantograph results in auxiliary bodies being added to the four-bar linkage.

Visser and Herder [9] used the solution of [1] in an actual hand prosthesis and gave

quantitative data on the reduction of effort in actuation of the fingers.


Our work [8] was the first to recognize that the compensation methods for spring-

loaded linkages are useful in approximately compensating flexure-based compliant

mechanisms as well. The recognition was based on small-length flexure model [64]

where flexure-based compliant mechanisms could be replaced by rigid-body linkages

under torsional-spring loads. The recognition was also the motivation for our work [8]

where we found that a larger set of methods can address the problem of spring-force

compensation enunciated in [1].

One of the methods in [8] led us to a more general class of spring-force compen-

sation methods [39]. Reference [39] showed that any n-rigid-body linkage under zero-

free-length spring and/or constant force can be compensated by addition of springs.

This method does not require addition of auxiliary bodies. The same method was

generalized and also presented in a more systematic manner in [42]. In [42], we ar-

gued that the method is applicable not only for planar linkages but also for spatial

linkages.

The small-length flexure model gives a rigid-body linkage loaded with torsional

springs. Hence, it may seem natural that there be a focus on finding static balancing

methods for rigid-body linkages loaded with torsional springs rather than extension

springs, as in [1], [8], [39] and [42]. However, our own attempts at finding an analyt-

ical solution to static balancing of torsional spring-loaded linkages did not yield any

results. A similar attempt by Radaelli et al. [65] relies on genetic-algorithm-based

numerical optimization or a manual search. To facilitate the manual search, which

could potentially give some insights, they developed an interactive interface called

“interactiveparams”. In, this thesis, we however approximate the torsional springs

by zero-free-length springs through matching a few terms in the Taylor series expan-

sion and then continue to use the analytical methods for balancing zero-free-length

spring-loaded linkages.

2.3 Static balance of compliant mechanisms

One of the earliest papers that recognized the importance of statically balanced com-

pliant mechanisms is [66]. Design of laparoscopic graspers that can give good force


feedback to the operator has been the main motivation in the field of statically bal-

anced compliant mechanisms. The papers that specifically focussed on laparoscopic

graspers include [67], [68] and [69]. Stapel and Herder [67] showed that a fully com-

pliant, statically balanced grasper is feasible. Tolou and Herder [68] separated a

statically balanced compliant grasper into grasper part and balancer part. They fo-

cused on obtaining a negative stiffness balancer that can compensate the positive

stiffness of compliant grasper. De Lange et al. [70] also used the concept of grasper

and balancer. They used topology optimization to match the stiffness characteristics

of a grasper and a balancer.

Various synthesis strategies for statically balanced compliant mechanisms, not re-

stricted to just graspers, has been addressed in [71], [72] and [73]. Morsch and Herder

[71] gave a compliant joint that is statically balanced so that when a linkage is built

out of these compliant joints, the linkage would be in static balance. Building block

approach is one of the known strategies in the synthesis of compliant mechanisms

and Hoetmer et al. [72] explored its extension to static balancing. Rosenberg et al.

[73] made use of the results of Radaelli et al. [65] to design flexure-based statically

balanced compliant mechanisms.

Research in static balance has ventured in tensegrity structures well, as could be

seen in [74] and [75]. The work of Guest et al. [76] where they present a zero-stiff

elastic shell is worth taking note of.

2.4 Prior Art

Prior to our work, the work in [1] was the only work that dealt with analytical solution

to static balancing of a spring-loaded rigid-body linkage. We now describe the way

the solution was arrived at in [1] along with necessary preliminaries that can also be

found in [4].


2.4.1 Preliminaries

Statically balanced lever under spring loads

Consider a lever under the action of two zero-free-length springs, as shown in figure

(2.3). The spring on the left hand side is anchored between points D and P . Let the

spring force on the lever at D be a positive constant times the relative displacement

of point P with respect to D, i.e., k1−−→DP . Similarly, the spring force due to right hand

spring is k2−−→DN . Let these forces be resolved along

−−→DA and

−−→PN . When the constants

and anchor points are chosen such that k1−→AP = −k2

−−→AN , the resolved forces along

−−→PN become zero and the remnant forces along

−−→DA gives zero moment on the lever

about A. This signifies equilibrium. Moreover, this is true for any configuration of

the lever. Hence, we have static balance here.

ANP

k2k1

k2−−→

AN

k2−−→

DAk1−−→

DA

k1−→

AP

Dφ

Figure 2.3: A basic spring force balancer

To appreciate the critical role played by zero-free-length of the springs, consider

a case where the free-length of the spring is not zero. Then, the spring force on D

from A is actuallyl1−l01

l1

−−→DA where l01 is the free-length and l1 is the relative distance

of the anchor points. The resolved components of the forces are as shown in figure

(2.4). Similarly, the resolved components of the right hand side spring are also shown.

It may be noted that no matter what strictly positive values l01 , k1, l02 and k2 take

and where P and N are placed, the resolved forces along−−→PN cannot cancel at every

configuration. The contrast between figures (2.3) and (2.4) in terms of static balance

should highlight the necessity of zero-free-length character for perfect static balance.


ANP

k2k1

l1 − l01

l1k1

−→

AP

Dφ

l1 − l01

l1k1

−−→

DA l2 − l02

l2k2

−−→

DA

l2 − l02

l2k2

−−→

AN

l1 = PD

l2 = ND

Figure 2.4: A lever with ordinary springs

Statically balanced parallelogram linkage

Figure (2.5) shows a parallelogram linkage, ADEN , with two springs attached diago-

nally between the joints. It may be verified that, for the same φ, the potential energy

of this system is the same as that in figure (2.3). If one is statically balanced, so is

the other.

N

D

Aφ

kk

E

Figure 2.5: Balanced parallelogram

Figure (2.6) shows again a parallelogram linkage that now has two degrees of

freedom. For the same φ, the potential energy of this system is the same as that in


figure (2.5). If one is statically balanced, so is the other.

A

N

E

Dφ

k

k

θ

Figure 2.6: Balanced two degree-of-freedom parallelogram linkage

Composition of springs

In [4], a useful concept of composing two zero-free-length springs into an equivalent

one is presented. By referring to figure (2.7), suppose that there is a spring with one

end anchored at A and the other end at E. Also suppose that we desire the spring to

be anchored at point D rather than at A but modification of the spring is not allowed.

In such a case, another spring of spring constant, say k2, is connected between point

E and a point on the ground, say B, such that k1−−→DA = −k2

−−→DB. When the forces

at E are resolved along AD and DE, it can be noticed that forces along AD always

cancel out and the net force is (k1 + k2)−−→ED. The same force is obtained if there

were to be a spring of spring constant (k1 + k2) anchored at D and the other end

at E. Hence, the net effect or composition of the two springs anchored at A and B


is a virtual spring anchored at D. This is an important concept for the techniques

presented in this chapter.

A B

D

E

k1 k2

k1−−→DA = −k2

−−→DB

k2−−→ED

k2−−→DB

k1−−→ED

k1−−→DA

Figure 2.7: Composition of two zero-length-spring into an equivalent one. The virtual,equivalent spring between E and D is shown in grey color.

2.4.2 Plagiograph

In a plagiograph linkage, the path traced at the output point is a scaled and rotated

replica of the path traced at the input point. If the linkage shown in figure (2.8)

satisfies the following conditions:

Condition 1: PQ = SR and PS = QR so that PQRS is a parallelogram,

Condition 2: ∠RQM = ∠NSR (this angle is labelled as α) and RQ

MQ= NS

RS(this ratio is

labelled as m) so that △NSR is similar to △RQM ,

then the linkage is called a plagiograph and it can be proved to have the following

property: output point N follows the input point M through a scaling and rotation

transformation about point P with the scale factor and the rotation angle being m

and α. That is,−−→PN = mRα(

−−→PM) (2.1)


where, Rα is the rotation operator, which operates on a planar vector to rotate it by

angle α.

M

S

α

α

P

α

Q

R

N

Figure 2.8: A plagiograph or a skew pantograph linkage

The point, P , about which the scaling and rotation transformations happen, is

referred to as the base pivot of the plagiograph. A detailed treatment of plagiographs

can be found in [77] and [78].

2.4.3 Derivation of the balancing solution in [1]

In the derivation, Herder starts with a simple statically balanced parallelogram linkage

shown in figure (2.6) and performs a series of modifications to arrive at the statically

balanced four-bar linkage (see figure (1.10)) without losing static balance. We now

discuss those modifications.

In the balanced two degree-of-freedom parallelogram linkage shown in figure (2.6),

the spring between A and E is replaced by two springs as per the principle of spring

composition/decomposition of Section 2.4.1. This replacement is depicted in figure

(2.9). As per the principle of composition, the two springs have to satisfy


AK

E

H

π

k1k2

k3

θφ

N

D

k1−−→

AK

k1

−→

EAk2

−→

EA

k2

−−→

AH

Figure 2.9: A balanced parallelogram with one its spring along a diagonal decomposedinto two springs.

k1−−→AK = −k2

−−→AH (2.2)

so that the net effect of the two springs is along AE. Further, the effective spring

constant (k1 + k2) should equal k3, the spring constant of the spring along DN , in

order to have static balance (see figure (2.6)).

Next, a parallelogram linkage AFGM is formed that is a duplicate of the linkage

ANED. The linkage AFGM undergoes the same motion as the linkage ANED,

except for maintaining a constant angle of α between the two, as shown in figure

(2.10). We show in figure (2.11) how the duplicate linkage synchronizes its motion

with the original linkage. A spring of spring constant k2 is now attached to the

duplicate linkage. The ground anchor point H of the spring is also rotated so that

its length remains the same upon shifting to the duplicate linkage. The spring of

spring constant k3 that was in the original linkage along the diagonal not containing

A, is also duplicated into the duplicate linkage. However, its spring constant in the

original and the duplicate linkage is changed to k4 and k5, such that k4 + k5 = k3.

Since, (k1 + k2) is also k3, for convenience, let us take k1 = k4 and k2 = k5. It may be


noted that the potential energy of the spring in figure (2.10) is the same as that in

figure (2.9). Hence, the static balance in figure (2.9) implies that the linkage of figure

(2.10) is also statically balanced. It may be noted that because of this step, equation

(2.2) gets modified to

k1Rα

(−−→AK

)

= −k2−−→AH (2.3)

where Rα is a rotation operator that rotates a vector by an angle α.

α

φθ K

E

k1

k4

k2

k5

A

H

G

k2

π − α

F N

D

M

Figure 2.10: The parallelogram linkage with a duplicate that maintains a constantangle of α from the original

Figure (2.11) shows how the parallelogram linkages ANED and AFGM can be

made to have the same motion except for maintaining a constant angle between

them. Here, essentially, a plagiograph has been introduced into the parallelogram

linkages. The following conditions have to be satisfied to have the synchronized

motion: ED = DC, ∠EDC = α, CF = FG, ∠CFG = α.

The parallelogram linkage AFGM is scaled by a factor of m, about point A, as

shown in figure (2.12). The plagiograph that maintains the synchronization between


α

αα

K

E

k1

k4

k2

k5

A

H

k2

π − α

N

D

M

F

C

G

Figure 2.11: Synchronization of motion between two parallelogram linkages

the two parallelogram linkages gets modified accordingly, as shown in the figure. The

two springs attached to it are also scaled by the same factor. This necessitates that

the anchor point H be also scaled by the same factor about A. Because of the

spatial scaling, the potential energy of the two springs scales by the square of the

scaling factor m. To restore the potential energy to that of figure (2.11), the spring

constants of the two springs are scaled by a factor of 1/m2. Since the potential energies

in figure (2.11) and (2.12) are the same, the static balance in figure (2.11) implies

the static balance in figure (2.12) too. Because of these modifications, equation (2.3)

gets modified to

k1Rα

(−−→AK

)

= −m2k21

m

−−→AH = −mk2

−−→AH (2.4)

As far as the length is concerned, the spring between M and F is an exact replica

of the spring between D and N , except for a scale factor m. Hence, one can elim-

inate the spring between D and N and increase the stiffness of the spring between

M and F from k2 to (k2 + k1/m2) so that the potential energy remains unaltered.


N

E

D

C

F

M

H

G

α

k1

k2

π − α

KA

α

α

k5 = k2

k4 = k1

Figure 2.12: Scaling the duplicate parallelogram linkage

This alteration is shown in figure (2.13) where two links AN and NE are no longer

necessary. Again, static balance has remained unaltered due to unaltered potential

energy.

As shown in figure (2.14), two more links AM and MG can also be removed by

relocating the spring between M and F to be between A and T where T is a point

on link GFC such that−→GF =

−→FT . Since MF = AT , this relocation does not change

the potential energy. Hence, the static balance remains intact. Again, adding a link

BC, as shown in the same figure does not disturb the static balance. The intent of

this addition is to make the four-bar linkage BCDA. This is how the static balancing

solution given in figure (1.10) was arrived at in [4]. In this thesis, we take a different

approach to arrive at a static balancing solution to the spring loaded four-bar linkage,

leading to many more solutions. Further, we give a static balancing technique that is

applicable to general n-degree-of-freedom revolute and/or spherical jointed linkages.


E

D

C

F

M

H

G

k1

k2

k5 = k2 + k1

m2

KA

π − α

α

α

Figure 2.13: Removal of a spring and compensating it with increase in stiffness ofanother spring

AK

kl

E

DF T

B

H

kb

C

Auxiliary links

G

Balancing spring 1

Balancing spring 2

Loading springkt

Anchor point

Anchor point

Figure 2.14: Removal of two more links.


Summary

• There is a wide array of literature on static balancing of gravity-loaded rigid-

body linkages by addition of counter-weights as well as springs.

• Prior to our work, Herder’s derivation has been the only work in the area of

static balancing of spring-loaded linkages.

• Herder’s derivation is based on cascading modification of simple statically bal-

anced mechanism into the final mechanism without losing static balance in any

of the modifications.

• Literature shows that some of the strategies explored for designing statically bal-

anced compliant mechanisms include topology optimization, designing generic

zero-stiffness compliant joint and the building block approach.

Apart from Chapter 7, the bulk of the thesis is based on our work presented in [8]

and [42]. Hence, it may be apt to pictorially summarize our work and the literature

in a way that brings out the distinct features of our work. Figure (2.15) shows one

such representation.


Rigid linkages

under

gravity load

By

addition

of mass

Rigid linkages

under

spring load

Compliant

mechanism

By

addition

of springs

Approximate

balancing

and

cam using

methods

Perfect

balancing

methods

Gopalswamy et al. (1992)

Agrawal & Agrawal (2005)

Ulrich & Kumar (1991)

Koser (2009)

For

specific

linkages

For

‘n’ - dof

linkages

LaCoste (1934)

Shin & Streit (1991)

Walsh et al. (1991)

Herder (2001)

Lin et al. (2010)

Using

auxiliary

bodies

Without

auxiliary

bodies

Streit & Shin (1993)

Agrawal & Fattah (2004)For planar revolutejointed and spatialrevolute and/orspherical jointedlinkages

Torsion

load

balanced

by

torsion

loads

Extension

spring

balanced

by

extension

springs

Approximate

methods

Perfect

methods

Radaelli et al. (2011)

For

specific

linkages

For

‘n’ -dof

linkages

Basic spring forcebalancer (Herder(2001))

Herder (1998)

Without

auxiliary

bodies

For planar revolutejointed and spatialrevolute and/orspherical jointedlinkages

Deepak & Ananthasuresh (2012)

Limiting case

By

addition

of springs

Other

strategies

Chapter 7 :Extension ofDeepak & Anan-thasuresh (2012)to compliantmechanisms usingpseudo-rigid bodymodel

Topology optimiza-tion (see De Lange etal. (2008))

Building block ap-proach (see Hoet-mer, et al. (2010))

Extension ofRadaelli et al.(2011) to compliantmechanisms as seenin Rosenberg et al(2011)

Original or novel contribu-

tions of this thesisdof: degree of freedom

Figure 2.15: The current literature and our contributions

Chapter 3

Static balancing of a four-bar

linkage loaded by a spring

Overview

• Static balancing of a spring-loaded four-bar linkage by incorporating a parallel-

ogram linkage.

• Static balancing without auxiliary bodies.

• Static balancing with the incorporation of a plagiograph linkage.

• A prototype demonstrating static balancing without auxiliary bodies.

3.1 Introduction

The problem that is addressed in this chapter is static balancing of a four-bar linkage

loaded with a zero-free-length spring connected between its coupler point and a point

on the ground. Figure (3.1) shows one such four-bar linkage. There is already one

method [1] whose derivation was discussed in detail in Section 2.4.3. The motivation

for attempting to find more methods, as discussed in the introduction, is to have a

suitable set of balancing methods that could be used in making statically balanced

36

CHAPTER 3. STATIC BALANCING OF A FOUR-BAR 37

compliant mechanisms using pseudo-rigid-body models. All the results of this chapter

are documented in our paper [8].

Loadingzero-free-lengthspring (kl)

AB

C

D

K

Coupler(Link 2)

Link 3

Link 1

E

Fixed pivots

Anchor point

Figure 3.1: A four-bar linkage with a zero-free-length spring anchored from its couplerto the ground

3.2 Static balancing of a given spring-loaded four-

bar linkage

Consider a four-bar linkage loaded by a zero-free length spring at its coupler point as

shown in figure (3.1). Before adding auxiliary links and balancing springs to balance

it, let us see if there is a possibility of it being in static balance as such. Figure (3.2)

shows the coupler curve traced by point E. If the linkage has to be in equilibrium

at every configuration, then the potential energy of the spring has to be the same at

every configuration. This is possible only when every point of the coupler curve is at

a constant distance from K, i.e., when the coupler curve is a circle centered at K. In


general this is not true except in extreme cases such as when E coincides with D (or

C) and K coincides with A (or B).

AB

C

D

K

E

kl

Figure 3.2: Possibility of the four-bar linkage being statically balanced as it is

Next, various possibilities of static balancing of a four-bar linkage are considered

by adding auxiliary links and springs. Only those possibilities where the number of

additional balancing springs and auxiliary links is less than or equal to that of [1]

are explained. As can be seen in figure (1.10), [1] used two auxiliary links and two

balancing springs.

3.2.1 Technique 1

Motivated by the observation in figure (2.6), the four-bar linkage can be balanced by

creating a balanced parallelogram as shown in figure (3.3). Here, four auxiliary links

and one balancing spring are used. But the number of auxiliary links can be reduced

to two as explained next.

Reducing the number of auxiliary links

As shown in figure (3.4), if T is a point on the (extended) link EP , such that−→EP =

−→PT , then TKQP forms a parallelogram. Relocation of the spring between P and Q

to be between T and K does not change its potential energy and hence static balance

is undisturbed. The advantage of the relocation is that the two auxiliary links KQ


AB

C

E

D

K

kl

kl

Q

P

Figure 3.3: A balanced parallelogram on the load spring

and QE are unnecessary. This way of balancing forms this chapter’s technique 1 to

statically balance a four-bar linkage.

AB

C

E

D kl

kl (balancing spring)

Q

T

P

K

l1 = PE

l2 = PKBalancing parameters

Figure 3.4: Technique 1: Static balancing with two auxiliary links and one balancingspring

The characteristics of technique 1 can be summarized as follows:

1. This way of balancing does not induce any stress in any of the links of the

four-bar linkage.


2. The balancing variables are l1 and l2, i.e., lengths of PE and PK, as shown in

figure (3.4). l1 and l2 can be of any convenient value.

3.2.2 Technique 2

Instead of adding four links to make up a parallelogram as in figure (3.3), if links ED

andDA are used to form the parallelogram as shown in figure (3.5), then two auxiliary

links can be avoided. This parallelogram can balance a spring connected between its

opposite vertices. While the load spring is not between its opposite vertices, another

spring (labelled as balancing spring 1) is added so that the resultant of its composition

with the load spring is between the opposite vertices of the parallelogram, as shown

in figure (3.6). Recall the composition property of springs noted in Section 2.4.1.

B

C

E

kl

F

D

A

K

Figure 3.5: Forming a parallelogram using two auxiliary links.

The virtual equivalent spring between A and E in figure (3.6) is balanced by

adding another spring (labelled as balancing spring 2) across the other two opposite

vertices of the parallelogram as shown in figure (3.7). Thus, the equivalent spring

and balancing spring 2 are in static balance. Equivalently, the load spring, balancing

spring 1 and balancing spring 2 are in static balance. With this, the linkage is in

static balance with two auxiliary links and two balancing springs. The auxiliary links

can be eliminated, as explained next.

As shown in figure (3.8), if T is a point on link DE (extended if necessary)

satisfying−−→ED =

−→DT , then AFDT is always a parallelogram and DF = TA. Hence,


B

kl

F

D

A

K

kb

−−→AH = kl

−−→KA, kt = kl + kb

kb

ktC

E

H

Balancing spring 1

Equivalent spring

Figure 3.6: A virtual spring between opposite vertices of the parallelogram AFED.

B

C

kl

K

kb

−−→AH = kl


kb

E

F

HA

D

kt

Balancing spring 1

Balancing spring 2

Figure 3.7: Adding balancing spring 2 across opposite vertices of the parallelogramto balance the load spring and balancing spring 1.


relocation of the spring between D and F in figure (3.7) to be between T and A as

in figure (3.8), does not change its potential energy. Consequently, the static balance

is also undisturbed. Furthermore, auxiliary links AF and FE are no longer required.

This way of balancing shown in figure (3.8), requiring two balancing spring but no

auxiliary link, constitutes the second technique to balance a four-bar linkage.

B

C

E

kl

kb DK

kt

H

T

kb

−−→AH = kl


−→TD =

−−→DE

A

F

Balancing spring 1

Balancing spring 2

Figure 3.8: Option 2 for balancing the four-bar linkage without requiring auxiliarylinks

This technique has two options. In figure (3.5), a parallelogram was completed

out of links AD and DE. One could have completed the parallelogram out of links

BC and CE as well and proceed in a similar manner. In the former case we say that

the balanced parallelogram is based at A and in the later case, it is based at B.

Among the balancing parameters of this technique, the anchor point of balancing

spring 2 is always one of the fixed pivots of the four-bar linkage. The remaining

balancing parameters: (i) H, the anchor point of balancing spring 1, (ii) kb, the

spring constant of balancing spring 1, and (iii) kt the spring constant of balancing

spring 2, have to be solved from equations included in figure (3.8). Those equations


can be rewritten in a general form so that they are applicable to both the options of

this technique. If W is where the balanced parallelogram is based (i.e., W is A or B)

and K is the anchor point of load spring, then

kb−−→WH = kl

−−→KW (3.1)

kb = klKWWH

(positive if−−→WH is along

−−→KW , and negative otherwise) (3.2)

kt = kb + kl (3.3)

Equation (3.1) says that H has to lie on a line starting fromW and along the direction

of−−→KW . The line is called the straight-line locus of H. Once a point on this line is

chosen as H, kb and kt get determined as per equations (3.2) and (3.3). Thus, the set

of solutions to (H, kb, kt) is a one-parameter family. This choice can be used to fulfill

practical considerations without compromising static balancing.

3.2.3 Technique 3

As mentioned earlier, the only known method in literature to statically balance a

spring-loaded four-bar linkage was described by [1]. In [1], a plagiograph is first

statically balanced and then it is modified to obtain balanced four-bar linkage with two

auxiliary links and two balancing springs as shown in figure (1.10). It is now shown

that the balancing arrangement obtained by [1] can also be obtained by combining

technique 2 of this chapter with the concept of plagiograph. Whereas this later

approach provides four options, the approach of [1] provides only two of these options.

The later approach constitutes the third technique of this chapter. The technique and

its options are described next.

In technique 2, the balancing spring 1 was connected to the same point as that

of the loading spring – the coupler point E. If the coupler point is not accessible to

the balancing spring 1, then another point to anchor the spring has to be found. It is

also desirable that the motion of the other point is related to the coupler point. By

taking a cue from Section 2.4.2, a plagiograph can be completed out of two links of

the given four-bar linkage, as shown in figure (3.9a) so that point G follows a scaled


and rotated locus of E.

}ADCF is a parallelogram

CDED

= GFCF

∠CFG = ∠EDC =⇒ △GFC ∼ △CDE

−→FT =

−→GF

kl

m2

−→LA = kb

−−→AH

kt = kl

m2 + kb

E

D

B

C

A

F

L

H

kb

G

Tkt

K

kl

E

D

B

C

A

F

L

H

kb

G

Tkt

−→FT =

−→GF

kl

m2

−→LA = kb

−−→AH

kt = kl

m2 + kb

−→AL = mRα(

−−→AK)

K

kl

E

D

G

B

C

A

Additional links

−→AG = mRα(

−→AE) (scaling and rotation transformation)

F

m = CDED

is the scale factor

α = ∠EDC (anticlockwise) is the rotation angle

A is the reference point for scaling and rotation

E

D

G

B

C

A

F

L

−→AL = mRα(

−−→AK),

(kl)m2

The fixed point of the plagiograph

(a) (b)

(c)(d)

Technique 2

Final balancing arrangement

Figure 3.9: Technique 3

Whatever scaling and rotation transformation of E that G follows, the same trans-

formation is applied toK to obtain L, i.e.,−→AL = mRα(

−−→AK), as shown in figure (3.9b).

If a spring is attached between L and G, as in figure (3.9b), then the spring is a scaled

(by factorm) and rotated copy of the load spring in figure (3.9a). Further, if the spring

constant of the spring in figure (3.9b) is 1m2 times the spring constant of the load spring

spring, i.e., klm2 , then its potential energy kl

2m2 (LG)2 = kl

2m2 (m(KE))2 = kl2(KE)2 is


the same as that of the load spring. Therefore, the potential energies of the spring

loads in figures (3.9a) and (3.9b) are the same. As a consequence of this, any extra

spring and auxiliary link addition that balances the spring load in figure (3.9b) would

also balance the spring load in figure (3.9a).

In figure (3.9b), one may think of links BC, CF , and FA to constitute a four-bar

linkage loaded with a spring at its coupler point G. Application of technique 2 of

static balancing to this spring-loaded linkage leads to addition of two extra springs,

as shown in figure (3.9c). The same two springs would statically balance the spring

load of figure (3.9a) also.

Balancing the given spring load of figure (3.9a) by adding the balancing spring

loads of figure (3.9c), as shown in figure (3.9d), is the third technique to balance

a spring-loaded four-bar linkage. It may be noted in figure (3.9d) that none of the

balancing springs are connected to the coupler point E.

When the four-bar linkage of figure (3.9b) was balanced using technique 2, there

were two options: balanced parallelogram based at A or based at B (see the section

(3.2.2) describing technique 2). Furthermore, when a plagiograph was completed out

of the four-bar linkage in figure (3.9a), the base pivot of the plagiograph was at A.

We can also complete a plagiograph out of the same four-bar linkage so that the base

pivot is at B, as shown in figure (3.10). Thus, we have 2 × 2 = 4 options – base

pivot of plagiograph at A or B, and balanced parallelogram based at A or B. In the

technique of [1], the base pivot of the plagiograph and the balanced parallelogram

always coincide. Therefore, only two of the above four options are derivable from the

technique of [1].

The following balancing parameters: the anchor point of balancing spring 1 (H),

spring constant of balancing spring 1 (kb) and spring constant of balancing spring

2 (kt), have to be solved from the equations provided in figure (3.9d). If U is the

base pivot of the plagiograph (instead of A) and W is the point where balanced

parallelogram is based (instead of A), then the equations can be generalized as follows,


}=⇒ △DFG ∼ △ECD

A

C

D

E

K

B

F

kl

BFDC is a parallelogram∠ECD = ∠DFGCDCE

= FGFD

Additional links

G

−−→BG = mRα(

−−→BE), where m = CD

CEand α = ∠ECD clockwise

The fixed pivot of the plagiograph

Figure 3.10: Plagiograph with base pivot at B


so that they are applicable to any of the four options of this technique.

−→UL = mRα(

−−→UK) L is the anchor point of the equivalent load spring (3.4)

−−→WH =

klm2kb

−−→LW (3.5)

kb =kl(LW )

m2(WH)(positive if

−−→WH is along

−−→LW , negative otherwise) (3.6)

kt = kb +klm2

(3.7)

m and α in the above equations are found from figure (3.9a) or (3.10) depending and

whether U is A or B. Equation (3.5) indicates that H can lie anywhere on a line

along−−→LW and passing through W . The balancing spring constants kb and kt are

functions of the position of H on this line as per equations (3.6) and (3.7). Therefore,

the set of solutions for (H, kb, kt) is a one-parameter family of solutions.

The main features of the three techniques presented in this section as well as the

Herder’s method are summarized in Table 3.1.

Technique1

Technique2

Technique3

Herder’sTech-nique

Number of auxiliarylinks

2 0 2 2

Number of balancingsprings

1 2 2 2

Variable balancing pa-rameters

l1, l2 H, kb, kt H, kb, kt H, kb, kt

Family of feasible bal-ancing parameters

two-parameter

one-parameter

one-parameter

one-parameter

Number of options one two four two

Table 3.1: Summary of different techniques to statically balance a spring-loaded four-bar linkage presented in this chapter


Figure 3.11: A prototype of four-bar linkage that is statically balanced using option2. The four-bar linkage seen in the above figure is a Watt’s straight-line mechanism.

3.3 A prototype with technique 2

A prototype of a spring-loaded four-bar linkage balanced using technique 2 is shown

in figure (3.11). The zero-free-length springs used in the prototype were realized using

‘pulley and string arrangement’ described in [4]. In the pulley and string arrangement

(see figure (3.12)), a string attached to a spring is passed over a small pulley anchored

to the ground such that when the spring deflection is zero, the end of the string is at

the pulley. Because of this, the distance between the end of the string and the pulley

is the same as the deflection of the spring. Since force exerted by the string-end is

along the line joining the pulley and the string-end, and of magnitude equal to the

deflection of the spring, a zero-free-length spring anchored at the pulley is realized.


x

x

Undeflected spring

Pulley

F = kx

k

Figure 3.12: Realization of a zero-free length spring

The prototype design is such that the ratio of spring constants of loading spring,

balancing spring 1 and balancing spring 2 is 1 : 1 : 2. To realize these three springs,

three identical springs were taken. While two springs were as it is used within spring-

pulley arrangement, only half the length of the spring was used for the remaining

spring, as can be seen in the right hand side of figure (3.11). It may be noted that

the anchor points of loading spring and balancing spring 1 on the coupler link are the

same. Further, the anchor point of loading spring and the anchor point of balancing

spring 2 on the coupler link are located symmetrically with respect to a revolute joint

on the coupler link. It may further be verified that the prototype satisfies equations

(3.1)–(3.3).

To test the prototype, the linkage is first manually constrained to be in a config-

uration. Then the behaviour of the linkage is observed after removing the constraint.

These type of trials were conducted at several configurations in two sets. In the first

set, one or both the balancing springs were removed. In the second set, the original

as well as the two balancing springs were in place. In the first set, it was noted that

the linkage does not sustain its configuration after the manual constraint is removed.

It springs back to one or two specific configurations. In the second set, the linkage

sustains its configuration even after the removal of the manual constraint. Inspite of

the presence of friction, the difference in the behaviour of the two sets of trial strongly

suggest that the prototype with the balancing springs is in static balance.


Summary

• In general, a four-bar linkage loaded by a spring at its coupler point is not in

static balance.

• Three new techniques to static balance the four-bar linkage were recognized.

• In the first technique, a balanced parallelogram linkage was made out of the

loading spring followed by relocation of a balancing spring to reduce the number

of auxiliary bodies.

• In the second technique, composition of springs was used to make a balanced

parallelogram linkage out of the given four-bar linkage itself. Rest of the steps

are the same as those of the first technique.

• In the third technique, by incorporating a plagiograph linkage, a new stati-

cally equivalent virtual-spring-loaded four-bar linkage was obtained. This was

followed by applying the second technique to this equivalent four-bar linkage.

• A prototype demonstrating the second technique was made.

• The number of balancing springs and auxiliary bodies, respectively, for the three

techniques are {one, two}, {two, zero}, and {two, two}.

Chapter 4

Static balance of the cognates of a

four-bar linkage

Overview

• Cognates of a four-bar linkage and its properties

• Quest for a unified parameterization of static balancing variables for the three

cognates.

• A geometric problem and its solution leading to a unified parameterization that

may be considered as an extension of Roberts-Chebyshev cognate theorem.

4.1 Introduction

4.1.1 Cognates

In kinematics of rigid-body linkages, it is well known that for a given four-bar linkage

and a coupler point on it, one can find two more four-bar linkages and corresponding

coupler points such that the coupler curves of all the three four-bar linkages are

the same. The three four-bar linkages are called cognates of each other. Figure

(4.1) shows a set of three cognates: B1C1D1B2, B2C2D2B3 and B3C3D3B1. In the

51

CHAPTER 4. STATIC BALANCE OF THE COGNATES 52

configuration shown in the figure, the coupler points are coincident at E. The triangle

formed by the ground anchor points, △B1B2B3, is called the cognate triangle. There

are a few interesting properties involving this triangle, as could be seen in [10]. One

such property is that △C1D1E, △EC2D2, and △D3EC3 are similar to the cognate

triangle B1B2B3. The cognates and their properties are summarized in what is called

Roberts-Chebyshev Cognate theorem (see [10]).

B3

γ2

γ1

γ2

γ3 γ1

γ2

γ3

γ1

D1

B2

D2

C2

C3

γ3

D3

C1

B1

E

kl

K

Figure 4.1: The cognates of a four-bar mechanism taking a load spring along theircommon coupler curve

4.1.2 Static balancing parameters and the cognates

A designer seeking to statically balance a given spring-loaded four-bar linkage has to

evaluate a family of feasible balancing parameters for each of the options under the

three techniques (see table 3.1) before choosing the one that best meets the design

requirements, including practical considerations. A more general design problem


would be to design a four-bar linkage that would guide the load spring along a specified

path and then statically balance the linkage so that the load spring can be moved

along the path effortlessly. If a four-bar linkage with its coupler curve matching the

specified path is found, then by Roberts-Chebyshev cognate theorem, it follows that

there are two more four-bar linkages whose coupler curves also match the specified

path. In order to make the best design choice, the designer has to evaluate feasible

balancing parameters of options under techniques 2 and 3, described in Chapter 3,

on all the three cognates. This chapter presents a result that will aid the designer to

evaluate these feasible balancing parameters on all the cognates in a unified manner.

As far as technique 1 is concerned, balancing parameters can be evaluated independent

of the four-bar linkage.

Since three four-bar linkages are considered in this chapter, balancing parameters

of techniques 2 and 3, such as H, kb, kt, W , U , and L, are subscripted with 1, 2, or

3, to correspond to the first, the second or the third cognate. Since all the cognates

are evaluated for the same load spring, K and kl are the same for all the cognates.

Among the balancing parameters, it is sufficient to focus only on Hi, kbi , and kti ,

i = 1, 2, or 3, since the remaining parameters are not variables. Nevertheless, to

determine the locus of Hi and the relation between Hi, kbi and kti (see equations (3.1

– 3.3) and equations (3.4 – 3.7)), it is necessary to know the remaining parameters.

While the remaining parameters can be deduced from the details of loaded ith cognate

linkage, table 4.1 shows that the detail is not necessary if the cognate triangle and

the ground anchor point of the loading spring are known.

Figure (4.2) shows the anchor point of a loading spring and a cognate triangle

where each side corresponds to a cognate. Sides corresponding to cognates 1, 2 and

3 are labelled as s1, s2 and s3, respectively. When an option of technique 2 or 3 is

applied on all the cognates with the cyclic symmetry, the straight-line loci of H1, H2

and H3, which are labelled as l1, l2 and l3, could be drawn, as illustrated in figures

(4.3), (4.4) and (4.5). Note that for both techniques, the origin for l1, l2 and l3 is one

of the vertices of the cognate triangle and there is one-one and onto relation between

the origins and the vertices. Because of this feature, it is now shown that H1, H2

and H3, can be varied as if they are at the vertices of the cognate triangle that is


Table 4.1: Deduction of various quantities in equations (3.1 – 3.7) from the cognatetriangle

Quantities other than H, kb and kt in equations (3.1 – 3.7):K, U , W , m, α, L

K Anchor point of the loading springW Base pivot of the balanced parallelogram

⇒ Coincident with a fixed pivot of the four-bar linkage⇒ One of the vertices of the cognate triangle∵ fixed pivots of cognates form the cognate triangle (see figure (4.1))

U Base pivot of the plagiograph⇒ One of the vertices of the cognate triangle (for the same reasons as of W )

m Ratio of two sides of the coupler triangle (see figures (3.9a) and (3.10))⇒ Ratio of two sides of the cognate triangle∵ the cognate triangle is similar to the coupler triangle (see figure (4.1))

α An angle of the coupler triangle (see figures (3.9a) and (3.10))⇒ An angle of the cognate triangle (∵ cognate △ ∼ coupler △)

L Anchor point of the equivalent of the loading spring (see figure (3.9b))

Found using the equation−→UL = mRα(

−−→UK). (U , m, α and K are as above.)

undergoing scaling and rotation transformation about a fixed point, while remaining

at the same time on their respective loci l1, l2 and l3. When H1, H2 and H3 are

varied as noted above, kb1 , kb2 and kb3 also vary, but it would be shown that their

ratio remains constant. This result follows from a solution that we give in this chapter

to a problem in geometry. The way we formulate the problem and find its solution

are described next.

4.2 A geometric problem and its solution

Given a triangle and three straight lines originating from its three vertices, find three

points on the three lines so that they from a triangle that is similar to the given

triangle. With respect to figure (4.6), the problem may be stated as: find points

Sa, Sb and Sc on straight lines a, b and c respectively, so that △SaSbSc is similar to

△ABC.


K

Cognate triangle

Common anchor point of load springs

B3

s2

s3

s1

B2B1

Figure 4.2: A cognate triangle and the ground anchor point of a load spring

K

U1, W1 : B2

U2, W2 : B3

U3, W3 : B1

B2, W1, U1s1

s2

s3

L1

L2

L3

B3, W2, U2l1

l2

l3

B1, W3, U3

Figure 4.3: Technique 3: the base pivot of the plagiograph (U) and the base of thebalanced parallelogram (W) for every cognate are not coincident.


K

U1 : B2 ; W1 : B1U2 : B3 ; W2 : B2U3 : B1 ; W3 : B3

B1

s2

s3

L1

L2

L3

s1

B3

B2

l1

l2

l3

Figure 4.4: Technique 3: the base pivot of the plagiograph (U) and the base of thebalanced parallelogram (W) for every cognate are coincident.


K

W1 : B2

W2 : B3

W3 : B1

B2, W1s1

s2

s3

l1

l2

l3

B1, W3

B3, W2

Figure 4.5: Technique 2: the base of the balanced parallelogram (W) at the indicatedvertices of the cognate triangle.


Ba

c

A

C

b

Figure 4.6: A problem in planar geometry

The method to obtain Sa, Sb and Sc varies for the following three cases, as de-

scribed next.

4.2.1 Case (i): a, b and c are parallel.

Sa, Sb and Sc can be obtained by translating A, B and C along the parallel lines, as

shown in figure (4.7), so that Sa, Sb and Sc stay on lines a, b and c while △SaSbSc is

congruent to △ABC.

b

AC

B

c

a

Sa

Sb

Sc

Figure 4.7: Finding Sa, Sb, and Sc in case(i)


4.2.2 Case (ii): a, b and c are concurrent.

Sa, Sb and Sc can be obtained by scaling A, B and C about the point of concurrence,

as shown in figure (4.8), so that Sa, Sb and Sc stay on lines a, b and c while △SaSbSc

is similar to △ABC.

b

A

C

Bca Sb

Sa

Sc

Figure 4.8: Finding Sa, Sb, and Sc in case(ii)

4.2.3 Case (iii): a, b, and c are neither parallel nor concur-

rent.

If the geometric problem does not fall under case (i) or case (ii), such as the one in

figure (4.6), then it falls under case (iii). In this case, as described later in section

(4.2.4), it is possible to find a non-zero angle η such that when lines a, b and c are

rotated by the same angle η about points A, B, and C, respectively, the lines become

concurrent at a point, as depicted in figure (4.9). The concurrent point is named as

the focal pivot. In figure (4.9) the focal pivot is denoted as P . In order to find a

solution to Sa, Sb and Sc, rotate triangle ABC, about point P , by some angle δ, to

obtain △A1B1C1, as shown in figure (4.10). If the respective intersection points of

lines PA1, PB1, and PC1 with the lines a, b, and c, are A2, B2, and C2, as shown in

figure (4.10), then it is proved in the following paragraph that △A2B2C2 is similar to

△ABC. Points A2, B2 and C2 form a solution to Sa, Sb and Sc, because, in addition

to similarity between △A2B2C2 and △ABC, by definition, A2, B2 and C2 lie on a, b

and c respectively. Different values of angle δ leads to different solutions.


b

AC

B

c

a

P

η

ηη

Figure 4.9: Description of focal pivot

b

AC

B

c

a

P

η

ηη

δ

δ

δ

A1B1

C1

SaSb

Sc

Figure 4.10: Points A2, B2 and C2 form a solution to Sa, Sb and Sc


Proof of similarity of △A2B2C2 and △ABC: △A2PA, △B2PB and △C2PC

are similar to each other since the angles δ and η are common to all of them. Hence,

PA2

PA=PB2

PB=PC2

PC(4.1)

Substitution of PA = PA1, PB = PB1 and PC = PC1 (∵ △A1B2C1 is rotation of

△ABC about P ) in equation (4.1), leads to PA2

PA1= PB2

PB1= PC2

PC1, which implies that

△A2B2C2 is a scale of △A1B1C1 (about P ), which in turn is a rotation of △ABC

(about P ). Hence, △A2B2C2 is similar to △ABC and can be visualized as a rotated

scaling of △ABC about the focal pivot P .

4.2.4 Finding the focal pivot

Let lines a, b and c be rotated by the same angle, say β, about points A, B and C,

respectively. If for some β, the three lines become concurrent, then by definition,

β = η and the point of concurrence is the focal pivot. If Ia,b is the intersection point

of a, b and Ic,a is the intersection point of c, a, then at the concurrence, Ia,b and Ic,a

meet. Ic,a and Ia,b can meet at only the intersection of paths traced by Ia,b and Ic,a,

during rotation.

The path traced by Ia,b is a circle passing through A, B and the original position

of Ia,b (see the theorem in appendix B). Similarly, the locus of Ic,a is a circle passing

through C, A and the original position of Ic,a. The two loci are shown in figure (4.11).

Locating the focal pivots in all the possible types of intersection between the loci is

addressed as follows.

• Ia,b or Ic,a will not exist when the a and b, or c and a are parallel

and hence the Ia,b and Ic,a circles cannot be drawn: Finding the focal pivot is

necessary only in case (iii), where all the three of a, b and c are not parallel to each

other. Hence, it is possible to find at least two pairs among a, b and c, which are not

parallel to each other. Those pairs may be taken as {a, b} and {c, a}, for which Ia,b

and Ic,a exists.

• Ia,b circle and Ic,a circle are coincident: It is shown in appendix (A.1) that

if Ia,b circle and Ic,a circle are coincident then a, b and c before rotation (at β = 0)


have to be concurrent which means that the problem falls under case (ii). Since

the procedure to find the focal pivot is used for only case (iii), the possibility of

coincidence of Ia,b and Ic,a circles does not arise.

• Ia,b circle and Ic,a circle intersect at two distinct points: This is the

generic possibility and is illustrated in figure (4.11). One of the intersection points

is always A. The other intersection point is denoted as M . In appendix (A.2), it is

shown that if rotation angle β is such that Ia,b is at M , then for the same β, Ic,a is

also at M . On the other hand, if β is such that Ia,b is at A, then for the same β, Ic,a

cannot be at A. Hence, it can be concluded that M is the one and only one focal

pivot.

Ia,b

AC

B

c

a

Ic,a

Ib,c

b

M

Figure 4.11: Geometric construction to find the focal pivot

• Ia,b circle and Ic,a circle touch each other at a single point: This is just a

limiting case of two distinct intersection points A and M , of the previous possibility,

merging into one. Here also, M is the only focal pivot but it happens to coincide

with A.

Thus, to find the focal pivot for case (iii), assuming that non-parallel pair of lines

are {a, b} and {c, a}, one should draw two circles: one passing through A, B and Ia,b

(at β = 0) and the other passing through C, A, Ic,a (at β = 0). The two circles either


intersect at two distinct points: M and A, or touch at the point A. In the former

case, the focal pivot is M and in the later case it is A.

4.3 The result

The common properties of solution to Sa, Sb and Sc in all the possible cases presented

in sections 4.2.1, 4.2.2 and 4.2.3, are summarized below.

• Property 1: The set of possibilities for Sa, Sb and Sc constitute a one-parameter

set. A convenient parameter parameterizing the set in case (i) is the translation,

in case (ii) is the scale factor, and in case (iii) is the rotation angle δ.

• Property 2: The ratio ASa : BSb : CSc is the same throughout the one-

parameter set, even though ASa, BSb and CSc themselves vary over the set. The

ratio is 1 : 1 : 1 in case (i), PA : PB : PC in case (ii), and again PA : PB : PC

in case (iii) (see equation (4.1) where A2, B2 and C2 are solutions to Sa, Sb and

Sc).

These properties are now applied to the cognate triangle and loci l1, l2, and l3, as

described next.

4.3.1 A parameterization of the balancing parameters of the

three cognates that has cognate triangle related invari-

ants

The solutions given in section (4.2) are now applied to the cognate triangle and loci

of H shown in figures (4.3–4.5), by taking A, B, C, a, b, c, Sa, Sb, Sc as W1, W2, W3

(the vertices of the cognate triangle), l1, l2, l3, H1, H2, H3, respectively. Then, the

properties at the end of section (4.2) take the following form:

There is a one-parameter parameterization of the balancing parameters of the three

cognates where

• △H1H2H3 is similar to △W1W2W3 (the cognate triangle), and


• the ratio W1H1 : W2H2 : W3H3 = W1P : W2P : W3P or 1 : 1 : 1 is the same for

the entire one parameter set of possibilities.

The second property is used in table (4.2) to rewrite the ratio of spring constants of

the first balancing spring in the three cognates. It is seen that the ratio involves only

constants and hence the ratio itself is invariant for this one-parameter set. It may

further be noted that all these constants are derivable just with the knowledge of the

cognate triangle and the location of anchor point of the loading spring. With this

follows the final result of this section: The balancing parameters of the three cognates,

when an option of technique 2 or 3 is applied on all of them, form a one-parameter

family where

1. The triangle of anchor point of balancing spring 1 (△H1H2H3) of the three

cognates is proportional to the cognate triangle over the entire family. The

two triangles are related by a combination of scaling, rotation and translation

transformations.

2. The ratio of spring constants of balancing spring 1 in the three cognates (kb1 :

kb2 : kb3) is the same throughout the one-parameter family. To calculate this

ratio, the knowledge of the location of the anchor point of the loading spring

and the cognate triangle is enough.

Table 4.2: Ratio of spring constants of balancing spring 1 for three cognates

Technique kb1 : kb2 : kb3 kb1 : kb2 : kb3 rewritten usingW1H1 : W2H2 : W3H3 = W1P :W2P : W3P or 1 : 1 : 1

3 kl(L1W1)

m21(W1H1)

: kl(L2W2)

m22(W2H2)

: kl(L3W3)

m23(W3H3)

(L1W1)

m21(W1P )

: (L2W2)

m22(W2P )

: (L3W3)

m23(W3P )

(see equation (3.6)) or (L1W1)

m21

: (L2W2)

m22

: (L3W3)

m23

2 klKW1

W1H1: kl

KW2

W2H2: kl

KW3

W3H3

KW1

W1K: KW2

W2K: KW3

W3K= 1 : 1 : 1

Case (ii) of sec-tion (4.2) applies

(see equation (3.2)) ∵ K is the same as P in this case

It is believed that the above parameterization would help a designer to visualize

balancing parameters of all the cognates together, when an option of techniques 2 or 3


is being evaluated. This result may be seen as an extension of the Roberts-Chebyshev

cognate theorem to static balancing.

Summary

• Cognates of a four-bar linkage have the same coupler curve.

• When static balancing parameters of the three cognates are considered together,

one can have a unified parameterization with properties related to the cognate

triangle.

• As the unified parameter varies, the triangle formed by the anchor points of the

balancing springs remain similar to the cognate triangle.

• Under the unified parameterization, the ratio of spring constants of a set of

corresponding balancing springs is constant.

• The constant ratio can be geometrically determined from the knowledge of the

cognate triangle and the anchor point of the balancing spring.

• In summary, the results of this chapter may be considered as an extension of

the classical theorem on cognates to static balancing.

Chapter 5

Static balancing of planar linkages

without auxiliary bodies

Overview

• Statically balancing a lever under spring loads and gravity loads.

• Statically balancing the rotational motion of a rigid-body that is free to move

in a plane by addition of springs.

• Transforming potential energy dependence from one body to another body

through a common point of the bodies.

• Recursive application of the transformation to statically balance any revolute-

jointed linkage that is under gravity and or spring loads.

• Examples demonstrating the recursive method.

5.1 Introduction

Figure (5.1a) shows a 4R linkage under a gravity load on its last link. This load

is balanced by counterweights added all through the linkage. Note the pattern of

weights; infact this balancing method extends to a general n-body revolute-jointed

66

CHAPTER 5. STATIC BALANCING OF PLANAR LINKAGES 67

linkage as well. The important fact for us is that there are no auxiliary bodies

added unlike some of the techniques seen in [12] (see figure (1.4)). The fact that

this method of balancing a general n-body linkage is known can be gleaned from the

video [www.youtube.com/watch?v=jJrzIdDUfT4]. Suppose that instead of weights

being the counter balancer, we require springs to be the counter balancer with the

other features – applicability to n-body linkage and non-usage of auxiliary bodies –

remaining. As is evident from the chapter on literature survey, there are no methods

in the literature to fulfil this requirement. Similarly, there are no methods in the

literature on balancing of spring loads through addition of other spring loads. This

chapter fills these gaps in the literature. The relevance of these methods in gravity

balancing as well as balancing elastic forces is already explained in the introductory

chapter of the thesis.

For simplicity, we use constancy of the potential energy as the basis in deriving the

methods. This also helps us to give a rigorous proof on the need for zero-free-length

springs. We consider balancing conditions for (1) a lever, (2) a rigid body in a plane

and (3) eventually for a general n-body revolute-jointed linkage.

5.2 Balancing a lever

Consider a lever pivoted to the ground, as shown in figure 5.2. The configuration of

the lever with respect to the global frame of reference (X−Y ) can be described by θ,

which is the angle from the global frame to the local frame of reference of the lever.

Figure 5.2 also shows two kinds of loads: (1) a spring attached between a point of

the lever and a point of the global frame, and (2) a constant force acting at a point

on the lever. Here, a constant force means that the force has a constant direction

with respect to the global frame and a constant magnitude. A complete specification

of the spring load would involve (1) the spring constant, denoted by k, (2) the local

coordinates of the anchor point on the lever, denoted by a =[

ax ay

]T

, and (3)

the global coordinates of the anchor point on the global reference frame. A complete

specification of the constant force would involve (1) the force components with respect

to the global frame, denoted by f =[

fx fy

]

, and (2) the local coordinates of the


Balancing loads :Springs

Balancing loads :Springs

? ?

(a) (b) (c)

Original load

Original load(black colored)balanced bycounterweights(grey colored)

Original load

Original load

Figure 5.1: Lack of methods for spring-based n-body linkage balancing without theusage of auxiliary bodies

point of action of the force on the lever, denoted by p =[

px py

]

.

5.2.1 Potential energy as a function of the configuration vari-

able

By referring to figure (5.2), the potential energy of the constant load is

PEc = −fT (r +R (θ)p) (5.1)


θ

xy

Xa

k

b

p

f

X

Y r

d

Figure 5.2: A lever under a constant load and a spring load

where r =[

rx ry

]T

is the coordinates of the origin of the local frame on the lever

with respect to the global frame and R is the rotation matrix function given by

R (ψ) =

[

cosψ − sinψ

sinψ cosψ

]

for any angle ψ (5.2)

The potential energy of the spring is

PEs =k

2(l − l0)

2 =k

2l2 − kl0l +

k

2l20 (5.3)

where l0 is the free length of the spring and l is the magnitude of d, the displacement of

one-end point of the spring with respect to the other. This d, referring to figure (5.2),

is:

d = (r +R (θ)a)− b (5.4)

Since l2 = dTd, the potential energy in expression (5.3) may be rewritten as

PEs =k

2dTd− kl0

√

dTd+k

2l20 (5.5)


If the free-length of the spring is zero, then only the first term in equation (5.5)

remains and hence we call it as PEs,zero, i.e.,

PEs,zero =k

2dTd =

k

2((r +R (θ)a)− b)T ((r +R (θ)a)− b)

=k

2(rTr + aTRT (θ)R(θ)a+ bTb− 2rTb+ 2rTR(θ)a

− 2bTR(θ)a)

=k

2

(rTr + aTa+ bTb− 2rTb+ 2rTR(θ)a− 2bTR(θ)a

)(5.6)

∵ RT (θ)R(θ) = I

Since the remaining last two terms of the potential energy in equation (5.5) are non-

zero only if free-length l0 is non-zero, we name these terms as PEs,nonzero, i.e.,

PEs,nonzero = −kl0√

dTd+k

2l20 (5.7)

From equation (5.6), it follows that dTd = 2kPEs,zero. Substituting this in equation

(5.7) leads to the following expression for PEs,nonzero.

PEs,nonzero = −l0√

2kPEs,zero +k

2l20 (5.8)

In the expressions of the potential energy in equations (5.1), (5.6) and (5.8), as

the configuration of the lever varies, f , p, a, b, k, l0 remain constants and r is made

a constant by choosing the origin of the local frame on the lever to coincide with

the pivot point. The dependency of the expressions on the configuration is due the

matrix R(θ), which by examining the definition of R in equation (5.2), can be split

as:

R(θ) =

[

1 0

0 1

]

cos θ +

[

0 −1

1 0

]

sin θ = I cos θ +R(π

2) sin θ (5.9)

This form of R(θ) indicates that PEc in equation (5.1) and PEs,zero in equation

(5.6) can be written as a linear combination of sin θ, cos θ, and 1 (for constants).

The coefficients of sin θ, cos θ and 1 are presented, for clarity, in a tabular form in


Table 5.1: Potential energy of the weight and the zero-free-length component of thespring acting on the lever is a linear combination of cos θ, sin θ, and 1.

Basis CoefficientsWeight Zero-free-length component of spring load

cos θ −fTp

= − (fypy + fxpx)k (r − b)T a

= k (ayry + axrx − ayby − axbx)

sin θ −fTR(π2) p

= (fxpy − fypx)k (r − b)T R(π

2) a

= k(axry − ayrx − axby + aybx)

1 −fTr

= −fyry − fxrx

+ki2

(rTr + aTa+ bTb− 2rTb

)

= +ki2(r2y−2byry+r

2x−2bxrx+b

2y+b

2x+a

2y+a

2x)

table 5.1. Thus, we now have potential energy of constant and spring loads expressed

as functions of configuration variable θ.

5.2.2 Invariance of potential energy with respect to the con-

figuration variable

Trivial conditions

The potential energy of the spring on the lever can have constant potential energy

only under trivial conditions: (1) the spring stiffness is zero (k = 0), (2) the anchor

point on the lever is at the hinge point (a = 0), and (3) the anchor point on the global

frame is at the hinge point (b = r). Similar trivial conditions for the constant loads

are: (1) the load is zero (f = 0), and (2) the load acts at the pivot point (p = 0).

It is only under these trivial conditions that the coefficients of cos θ and sin θ become

zero in table 5.1.

The discovery of Lucien LaCoste

Even though a non-trivial spring and a non-trivial constant load cannot be individu-

ally in static balance, they together can be so, as demonstrated in figure (5.3). This

was first recognized by Lucien LaCoste (see [3]) in the context of having a pendulum

of infinite period. Figure (5.3) shows a lever under the action of a weight W that is

balanced by a zero-free-length spring of spring constant k anchored above the pivot


of the lever at a height of h. As shown in the figure, under the condition W = kh,

the potential energy is invariant with respect to configuration variable θ.

X

xy

Y

θl

kh W

a = p =

[

l

0

]

b =

[

0h

]

f =

[

0−W

]

r =

[

00

]

zero-free-length spring

weight

PEc = (0) cos θ + (Wl) sin θ + constant

PEz = (0) cos θ + (−khl) sin θ + constant+

PEnet = (l (W − kh)) sin θ + constant

0

when W = kh

Figure 5.3: Static balancing of a weight by a spring

Several zero-free-length springs and constant loads

The balancing conditionW = kh of the example in figure (5.3) will now be generalized

to a lever under several constant loads and zero-free-length spring loads. Since several

loads are now being considered, let both constant loads and zero-free-length spring

loads be ordered to allow indexing. The notation ai, bi ki has the same meaning as

a, b and k in figure (5.2) other than that it corresponds to ith spring. f i and pi

also have similar meaning. Further, let the number of constant loads be nc and the


number of zero-free-length spring loads be ns.

Since the potential energy of each of the constant loads and the zero-free-length

spring loads is a linear combination of cos θ, sin θ and 1, their net potential energy is

also a linear combination of cos θ, sin θ and 1. Further, since cos θ and sin θ and 1 are

linearly independent functions of θ, their linear combination is a constant if and only

if the coefficients of non-constant functions, i.e., cos θ and sin θ, are zero. Writing,

with the help of table 5.1, the coefficients of cos θ and sin θ of the net potential energy

of all the loads and equating them to zero lead to the following equations:

−nc∑

i=1

(fy,ipy,i + fx,ipx,i)+

ns∑

i=1

ki (ay,iry + ax,irx − ay,iby,i − ax,ibx,i) = 0 (5.10)

nc∑

i=1

(fx,ipy,i − fy,ipx,i)+

ns∑

i=1

ki(ax,iry − ay,irx − ax,iby,i + ay,ibx,i) = 0 (5.11)

which are the conditions for constant potential energy (or static balance) of several

constant and zero-free-length spring loads on a lever. These conditions are applicable

to all the three categories: 1) balancing weights by weights, 2) balancing weights by

springs, and 3) balancing springs by springs. Further, by choosing appropriate load

parameters, it is possible to satisfy the conditions in practice, as was the case in the

example of figure (5.3).

Normal positive-free-length springs

As far as normally available positive-free-length springs are concerned, the square root

term in equation (5.8) poses a severe restriction on static balancing, as explained in

detail in appendix C. Hence, for the remainder of this chapter, all the spring loads

are of zero free-length with the understanding that a positive-free-length spring can

be brought into the ambit of zero free-length by combining it with an appropriate


θ

xy

Xa1

a2

ai

ans

k1

k2

ki

kns

b1

b2

bns

bi

p1

f1

p2

f2

pnc

fnc

X

Yr

Figure 5.4: A body that is free to move in a plane

negative-free-length spring.

Our next aim is to derive a set of conditions for the static balance of a revolute-

jointed multi-body linkage loaded by constant loads and zero-free-length spring loads.

Before that, it is useful to consider the static balance of a single rigid body moving

freely in a plane.

5.3 Balancing of a rigid body in a plane

Consider the rigid body shown in figure (5.2). An appropriate set of configuration

variables for the body is {r, θ}. It may be noted that r in figure (5.4), in contrast to

figure (5.2), is an independent variable because the body is free to move in the plane.

The loads on the body are a set of zero-free-length spring loads and constant

loads, and both sets of loads are exerted by the global frame of reference as shown

in figure (5.4). The notation nc, ns, ai, bi, ki, f i, and pi has the same meaning as

in Section 5.2. The potential energy of the loads is also the same as in Section 5.2


Table 5.2: Potential energy of weight and spring acting on a link moving in a plane.

Basis Coefficients of the basisWeight Spring load Generalized po-

tential (see eqn.(5.15)

cos θ −(fypy +fxpx)

−k(ayby + axbx) (qywy + qxwx)

sin θ +(fxpy −fypx)

−k(axby − aybx) (qxwy − qywx)

rx cos θ 0 kax vxry cos θ 0 kay vyrx sin θ 0 −kay −vyry sin θ 0 kax vxr2x 0 k

2κ

r2y 0 k2

κ

rx −fx −kbx uxry −fy −kby uy1 0 +k

2(a2x + a2y + b2x +

b2y)c

except that rx and ry are now independent variables. In table 5.1 of Section 5.2, when

linearly independent functions of {r, θ} are pulled out as basis functions, table 5.2

is obtained. As is evident from table 5.2, the potential energy of the loads is now

a linear combination of the following basis functions: cos θ, sin θ, rx cos θ, ry cos θ,

rx sin θ, ry sin θ, r2x, r

2y, rx, ry, and 1.

The net potential energy of nc constant loads and ns spring loads is also a linear

combinations of the same basis functions. Furthermore, cos θ, sin θ, rx cos θ, ry cos θ,

rx sin θ, ry sin θ, r2x, r

2y, rx, ry, and 1 are linearly independent functions of {r, θ}.

Hence, from a reasoning similar to the one in Section 5.2, for the net potential en-

ergy to be independent of the configuration variables, the coefficients of all the basis

functions other than 1 have to be zero. However, it is not practical to make the

coefficients of all these functions as zeros because of the following reasons:

• There are only gravity loads: Gravity is the most important practically seen

instance of a constant load. When all the constant loads are gravity loads,


f i = migi, where mi is the mass and g is the acceleration due to gravity.

Further, the coefficient of rx and ry become

(

−gxng∑

i=1

mi

)

and

(

−gyng∑

i=1

mi

)

.

Since mi > 0, ∀i,ng∑

i=1

mi > 0. Also, since the acceleration due to gravity is

non-zero, both gx and gy cannot be zero. Hence, the coefficients of both rx and

ry cannot be zero.

• There are zero-free-length spring loads, possibly with gravity loads: In

this case the coefficients of both r2x and r2y arens∑

i=1

ki. Since the spring constants

of all the springs considered here are positive (ki > 0, ∀i),ns∑

i=1

ki cannot be zero.

Hence, the coefficients of r2x and r2y cannot be zero.

However, as shown in appendix D.1, there is no such practical difficulty in making

the coefficients of all θ-dependent functions, i.e., cos θ, sin θ, rx cos θ, ry cos θ, rx sin θ,

and ry sin θ, to be zeros. Setting θ-dependent terms to zero amounts to the following

set of independent constraints:

−

nc∑

i=1

((fy,ipy,i + fx,ipx,i))−

ns∑

i=1

(ki(ay,iby,i + ax,ibx,i)) = 0 (5.12)

+

nc∑

i=1

((fx,ipy,i − fy,ipx,i))−

ns∑

i=1

(ki(ax,iby,i − ay,ibx,i)) = 0 (5.13)

ns∑

i=1

(kiai) = 0 (5.14)

It is shown in Appendix D.1 that if these constraints are not satisfied by the loads,

then by adding not more than two zero-free-length springs, these constraints can be

satisfied. A numerical example to demonstrate the same is given in figure (5.5) with

the details of the loads in table 5.3.

Inspite of being able to make the potential energy of the loads on the link inde-

pendent of θ, the dependency on r still remains. In Section 5.4, we show that if the

body is joined to an appropriate linkage, then by adding extra loads to other parts

of the linkage, the r-dependent terms of the potential energy can be balanced out.

Before we proceed to Section 5.4, it may be noted that the potential energy of a


[ 12, 3]T

[ 12, 1]T

Global frameX

Y

[ 12, 0]T

[1, 0]T

Z1

Z2

C1

Xθ

Figure 5.5: A rigid body moving freely in a plane under a constant load is made tohave θ-independent potential energy by addition of two zero-free-length springs

Table 5.3: Details of the loads in figure (5.5). It may be checked that the loads satisfyequations (5.12 – 5.14 )

Spring Loadsa b k

Z1

Z2

[ 12, 0]T [ 1

2, 3]T 1

[− 1

2, 0]T [ 1

2, 1]T 1

Constant Loadsp f

C1 [1, 0]T [0,−1]T

−kbTa −kbTR(

pi

2

)

a ka1

4

−1

4

−fTp

0

−fTR(

pi

2

)

p

3

2

−1

2

−1

[ 12, 0]T

[− 1

2, 0]T


constant load or a zero-free-length spring load fall under the following general form:

Φ = rTu+ κrTr + rTR (θ)v +wTR (θ) q + c (5.15)

where θ and r are the configuration variables of the rigid body on which the load

acts. In the case of constant loads, by comparing equation (5.1) with equation (5.15),

we have

u = −f , κ = 0, v = 0, w = −f , q = p, and c = 0 (5.16)

and in the case of zero-free-length spring loads, by comparing equation (5.6) and

(5.15), we have

u = −kb, κ =k

2, v = ka, w = −b, q = ka, and c =

k

2

(bTb+ aTa

)

(5.17)

Later in the chapter, we encounter potential energy functions that are of the form

given in equation (5.15), but they cannot be attributed to zero-free-length spring loads

or constant loads acting on the body. Hence, there is a need to generalize constraints

(5.10), (5.11), (5.12), (5.13), and (5.14) to the form given in equation (5.15). Such

a generalization is possible because, as can be seen in the last column of table 5.2,

the potential given in equation (5.15) is a linear combination of the basis functions

given in table 5.2 just as in the case of constant and zero-free-length spring loads.

The following proposition states the generalization.

Proposition 1. If there are n functions of the form

Φi = rTui + κirTr + rTR (θ)vi +wTR (θ) qi + ci, i = 1 · · ·n (5.18)

with r and θ as the variables, thenn∑

i=1

Φi is independent of θ if and only if the following


constraints are satisfied:

n∑

i=1

(qy,iwy,i + qx,iwx,i) = 0 (5.19)

n∑

i=1

(qx,iwy,i − qy,iwx,i) = 0 (5.20)

n∑

i=1

vi = 0 (5.21)

When these constraints are satisfied,n∑

i=1

Φi depends only on r in the following form

n∑

i=1

Φi =n∑

i=1

(rTui + κir

Tr + ci)

=(rT)

n∑

i=1

ui +(rTr

)n∑

i=1

κi +n∑

i=1

ci (5.22)

Furthermore, if r happens to be a constant (as in a lever) with only θ being the

variable, thenn∑

i=1

Φi is independent of θ (and hence a constant) if and only if the

following constraints are satisfied:

n∑

i=1

(vy,iry + vx,irx + qy,iwy,i + qx,iwx,i) = 0 (5.23)

n∑

i=1

(vx,iry − vy,irx + qx,iwy,i − qy,iwx,i) = 0 (5.24)

Proof. The proof is along the same lines as the derivation of equations (5.10), (5.11),

(5.12), (5.13), and (5.14).

It may be noted that inspite of considering a general form of potential in equa-

tion (5.18), the inability to make the net potential energy independent of r remains

because of the following reason. In all the cases that we consider next, κi ≥ 0 and

κi > 0 for atleast one value of i. Hence, the r-dependent term,n∑

i=1

κirTr, cannot be

zero in the expression forn∑

i=1

Φi.


5.4 New static balancing techniques for revolute-

jointed linkages

If there is a single rigid body with loads exerted by a reference frame, then the net

potential energy of the loads depends on the configuration of the body with respect

to the reference frame. If there are several such bodies, then the net potential energy

of all the loads on all the bodies depends on the configuration of all the bodies. This

dependency on the configuration of all the bodies can be reduced to that of a single

body provided the bodies are connected by revolute joints (to begin with, say, in a

serial or a tree-structured manner) and the loads are zero-free-length spring loads and

constant loads. If this single body is the reference frame itself, then the net potential

energy is a constant (implying static balance) since the configuration of the reference

frame with respect to itself is always fixed. This result follows as a consequence of

the proposition that is presented next.

5.4.1 The potential energy of loads on a body transformed

as a function of another body

We are now considering several rigid bodies, each of them with its own r, θ, nc, ns,

ai, bi, ki, pi, etc. To distinguish these quantities belonging to different rigid bodies,

we number the rigid bodies and put the number as a superscript to these symbols.

Hence r, θ, nc, ns, ai, bi, ki, pi, and f i of body j are now represented as rj, θj, njc,

njs, a

ji , b

ji , k

ji , pi

j, etc.

Proposition 2. The net sum of a set of functions of the configuration variables of a

body l in the form given in equation (5.15), i.e.,

Φli = rlTul

i + κlirlTrl + rlTR

(θl)vli + wlTR

(θl)qli + cli, i = 1 · · ·nl (5.25)


can be expressed as a function of the same form but of body j, i.e.,

nl∑

i=1

Φli = Φj

i = rjTuji + κjir

jTrj + rjTR(

θj)

vji + wjTR

(

θj)

qji + cji (5.26)

provided the following conditions are satisfied:

• Condition 1: There is a point that is rigidly fixed to both body l and body j.

Such a point is called as a common point of bodies l and j.

• Condition 2: The origin of the local coordinate frame of body l is at the common

point.

• Condition 3: The sum of the set of functions of body l is dependent only on r

in the form given in equation (5.22).

Proof. Let the local coordinates of the common point required by condition 1 in body

l be slj and in body j be sjl . The commonality of the point can be written as follows.

rl +R(θl)slj = rj +R

(

θj)

sjl (5.27)

Condition 2 implies that slj = 0. Substituting slj = 0 into equation (5.27) leads to

rl = rj +R(

θj)

sjl (5.28)

Condition 3 implies that the sum of the set of functions of body l can be written as

nl∑

i=1

Φli = rlT

nl∑

i=1

uli +(

rlTrl

) nl∑

i=1

κli (5.29)

The constant term is omitted in equation (5.29) since it is inconsequential for the

discussion.

Substitution of rl from equation (5.28) into equation (5.29) and simplification


using the fact that RT (θ)R(θ) is identity lead to the following expression fornl∑

i=1

Φli:

rjT

uji

︷︸︸︷

nl∑

i=1

uli

+ rjTrj

κji

︷︸︸︷

nl∑

i=1

κli

+ rjTR(

θj)

vji

︷︸︸︷

2sjl

nl∑

i=1

κli

+

wji

T

︷︸︸︷

nl∑

i=1

uli

T

R(

θj)

qji

︷︸︸︷(

sjl

)

=nl∑

i=1

Φli = Φj

i (5.30)

Again, the constant term is omitted in equation (5.30). It may be readily recognized

that the sum of the set of functions on body l,nl∑

i=1

Φli, as seen in equation (5.30) is

indeed of the form given in equation (5.26).

5.4.2 Proposition 2 as the recursive relation of an iterative

static balancing algorithm

We now show that proposition 2 can be treated as a recursive relation that can be

incorporated into an iterative procedure to achieve static balance of a linkage. For the

purpose of this subsection, we restrict the linkage on which the iterative procedure

can be applied to have the following features:

1. The linkage should be tree-structured (i.e., no closed loops). This feature is

necessary since a recursive relation requires a tree-structure to propagate.

2. All the joints of the tree structure should be revolute joints. This feature is

necessary to satisfy condition 1 of proposition 2.

3. We want all the loads to have potential energy functions of the form given in

equation (5.25) of proposition 2. While we know that zero-free-length springs

and constant loads do have this form (see equations (5.16) and (5.17)) the fact


that there are several bodies involved requires a little attention. The configu-

ration variables(rl, θl

)of different bodies (i.e., of different l) should be with

respect to a common global frame of reference. Hence, constant loads on all

the bodies should be constant with respect to a common global reference frame

and any zero-free-length spring should have its one anchor point on the same

common global frame while the other anchor point can be on any of the bodies

constituting the linkage.

4. The common reference frame should be one of the bodies of the linkage, i.e., it

should join to body/bodies of the linkage by revolute joint/joints.

The iterative static balancing algorithm

We now present the iterative algorithm and prove that it leads to static balance.

Preparatory steps

1. Assign the reference body as the root node of the tree-structure (bodies are rep-

resented as nodes and joints as lines joining the nodes). With this assignment,

for every link/body other than the root, there is a parent body. Further, every

links other than the terminal links has one or more children.

2. On every link, choose a local frame that coincides with the center of the revolute

joint between the link and its parent. For every link k, rk and θk decide the

configuration of its local frame with respect to the frame of the root.

3. Give this tree-structure with the given constant and zero-free-length spring loads

(together referred to as original loads) as an input to the following iterative

procedure.

Iterative procedure

• Entry condition: If the tree-structure contains only the root node, then exit

from the iterative procedure. Otherwise, proceed to step 1.


• Step 1: Any terminal node l, has associated with it the following three kinds

of potential energy functions: 1) due to original loads on body l, 2) due to

association that happened in step 3 of previous iterations, and 3) additional

loads on body l. Let the number of such functions be represented by nlo, n

lc, n

la,

respectively. The first two kinds of functions are known from the given problem

and previous iterations, respectively, and the task in this step is to find the

additional loads so that

– case a: equations (5.19 - 5.21) are satisfied if l is not a child (i.e., not first

generation descendant) of the root.

– case b: equations (5.23 - 5.24) are satisfied if l is a child of the root. Note

that in this case rl is a constant because of the way local frame is chosen

in the preparatory steps.

This task makes sense only if the all kinds of potential energy functions fall

under the form of equation (5.15) with (r, θ) being(rl, θl

). The first and the

third kind of potential energy functions do fall under the form because of the

kind of loads we are considering (see equations (5.16) and (5.17)). The second

kind of potential energy functions conform to the form because of step 2, which

is in accordance with proposition 2. The critical role of proposition 2 in enabling

this iterative procedure may be noted. Further, Appendix D.2 asserts that the

task of this step is always feasible. It may be noted that there are several sets

of additional loads that satisfy these equations. This non-uniqueness calls for

discretion of the designer in choosing a suitable set of additional loads.

• Step 2: In case (a), expressnlo+nl

c+nla∑

i=1

Φli in the form given in equation (5.15)

where r and θ are the configuration variables of the parent of node l. This

is possible since condition 1 (because of revolute joint), condition 2 (because

of preparatory steps) and condition 3 (because of step 1) of proposition 2 are

satisfied. In case (b), recognize thatnlo+nl

c+nla∑

i=1

Φli is a constant as per proposition

1.


• Step 3: Associatenlo+nl

c+nla∑

i=1

Φli with the parent link of l and for energy con-

servation, disassociate Φli, i = 1 · · ·nl

o + nlc + nl

a from node l. Because of this

association, np(l)c (p(l) denotes parent node of l, and nk

c denotes the number

of potential energy functions associated with node k so far at step 3.) gets

incremented by one and the associated function can be written as:

Φp(l)nc

=

nlo+nl

c+nla∑

i=1

Φli (5.31)

• Step 4: With all the potential energy functions disassociated from node l to

its parent, delete this terminal node l.

• Iterator: Once steps 1 to 4 are completed, a new trimmed tree-structure results

where the parents of the nodes deleted in step 4 has additional potential energy

functions associated with them. Follow this iterative procedure again with this

trimmed tree-structure as the input.

With every iteration, the tree-structure shrinks and it eventually gets reduced to

the single root node. Any of the n0c potential energy functions (of the second kind)

associated with this reduced root is from one of the children of the root. As per

step 1, this association is through case (b). Any function associated through case

(b) is a constant as recognized in step 2. Thus, the sum of these n0c potential energy

functions is also constant. Further, the sum of these n0c potential energy functions is

actually the total potential energy of original loads and additional loads on all the

descendants of the root. This can be verified by recursive substitution in equation

(5.31) as exemplified in equation (5.32). Therefore, the original loads are in static

balance with the additional loads.

Illustration of the algorithm on a 4R linkage under constant loads Figure

(5.6) shows a 4R linkage where four revolute joints connect the ground and four other

bodies serially. The ground exerts constant gravitational force on each of the four

bodies. Hence, we take the ground as the root and number the bodies accordingly as


shown in figure (5.6). A local frame of reference is located on each of the bodies as

per the preparatory step 2. The constant loads on bodies 1, 2, 3 and 4 are represented

as C11 , C

21 , C

31 and C4

1 , respectively. Their details (point of action p and force vector

f) are presented in item number 1, 4, 7 and 10 of the tables in figures (5.7 - 5.10).

x0

y0

y1

1

y2

y3

x2

y4

x4

0

2

3

4

C1

1C2

1

C3

1

C4

1

Figure 5.6: The gravity-loaded serial 4R linkage to be statically balanced.

Now, we give the tree-structure to the iterative procedure. The terminal node of

the tree is 4. There is only one original load, C41 , on the node and its potential energy is

represented as Φ41. To emphasize that potential energy function Φ4

1 is because of load

C41 , we write it as Φ4

1 (C41). There is no potential energy function of the second kind

(n4c = 0). Two zero-free-length springs Z4

1 and Z42 are added so that the functions

Φ41 (C

41), Φ

42 (Z

41 ), and Φ4

3 (C42) satisfy equations (5.19 - 5.21) as per case (a) of step 1.

All the details of the springs, constant loads as well as their potential energies in the

standard form (see equation (5.15)) are presented in the table of figure (5.7). Now,


as per step 2, the sum of Φ41 (C

41), Φ

42 (Z

41 ), and Φ4

3 (C42) is transformed as a Φ3

2 (r3, θ3)

in accordance with equation (5.30) of proposition (2). This is followed by making of

a new tree-structure obtained by deleting node 4 and associating Φ32 with node 3 of

the new tree-structure. This completes the first iteration.

4

3

2

10

Φ4

1

(

C4

1

)

, Φ4

2

(

Z4

1

)

, Φ4

3

(

Z4

2

)

n4o

= 1 n4a

= 2

Φ3

1

(

C3

1

)

n3o

= 1

n2o

= 1

Φ2

1

(

C2

1

)

n1o

= 1

Φ1

1

(

C1

1

)

Step 1: add springs/Eqs. (5.19-5.21)

Iteration 13

2

10

n2o

= 1

Φ2

1

(

C2

1

)

n1o

= 1

Φ1

1

(

C1

1

)

Φ3

1

(

C3

1

)

, Φ3

2,

n3o

= 1 n3c

= 1

4

Step 2 and 3

Step 4:deletion

no. Φji a

jk/p

jk b

jk/f

jk k

jk → u

ji κ

ji v

ji w

ji q

ji

1 Φ4

1

(

C4

1

)

[

0.40

] [

0−1

]

eq.(5.16)

[

01

]

0

[

00

] [

01

] [

0.40

]

2 Φ4

2

(

Z4

1

)

[

10

] [

051

32

]

0.8 eq.(5.17)

[

0−

51

40

]

0.4

[

0.80

] [

0−

51

32

] [

0.80

]

3 Φ4

3

(

Z4

2

)

[

−10

] [

035

32

]

0.8 eq.(5.17)

[

0−

7

8

]

0.4

[

−0.80

] [

0−

35

32

] [

−0.80

]

∑

3

i=1u4

i =

[

0−

23

20

]

∑

3

i=1κ4

i = 0.8 eqs. (5.19-5.21) true

j = 3 Φj

i uj

i κj

i vj

i wj

i qj

i

Φ3

2

[

0

−

23

20

]

0.8

[

1.6

0

] [

0

−

23

20

] [

1

0

]

Parameters of potential energy functions associated with node 4 (j = 4)

s34=

[

10

]

From equation (5.30)

Association to node 3

Figure 5.7: Details of Iteration 1

The second iteration acts on the new tree-structure. The tables in figures (5.7-

5.10) give all the details of all the iterations. At the end of four iterations, we are left

with a single root node having constant function Φ01 associated with it. The springs

added in these iterations, along with the original gravity loads are shown in figure


(5.11). By following the dashed arrowed line of figures (5.7-5.10) in the reverse order,

it may be verified that

Φ01 = Φ1

1

(C11

)+Φ2

1

(C21

)+Φ3

1

(C31

)+Φ4

1

(C41

)+Φ4

2

(Z4

1

)+Φ4

3

(Z4

2

)+Φ3

3

(Z3

1

)+Φ2

3

(Z2

1

)

(5.32)

Hence, C41 , C

31 , C

21 , and C1

1 are in static balance with Z41 , Z

42 , Z

31 , and Z2

1 .

3

2

10

n2o

= 1

Φ2

1

(

C2

1

)

n1o

= 1

Φ1

1

(

C1

1

)

Φ3

1

(

C3

1

)

, Φ3

2, Φ3

3

(

Z3

1

)

n3o

= 1 n3a

= 1n3c

= 1

fromiteration 1


2

10

n2o

= 1 n2c

= 1

Φ2

1

(

C2

1

)

, Φ2

2,

n1o

= 1

Φ1

1

(

C1

1

)

Step 2 and 3

Step 4:deletion

3

no. Φji a

jk/p

jk b

jk/f

jk k

jk → u

ji κ

ji v

ji w

ji q

ji

4 Φ3

1

(

C3

1

)

[

0.40

] [

0−1

]

eq.(5.16)

[

01

]

0

[

00

] [

01

] [

0.40

]

5 Φ3

2From iteration 1

[

0−

23

20

]

0.8

[

1.60

] [

0−

23

20

] [

10

]

6 Φ3

3

(

Z3

1

)

[

−10

] [

015

32

]

1.6 eq.(5.17)

[

0−

3

4

]

0.8

[

−1.60

] [

0−

15

32

] [

−1.60

]

∑

3

i=1u3

i =

[

0−

9

10

]

∑

3

i=1κ3

i = 1.6 eqs. (5.19-5.21) true

no. Φji u

ji κ

ji v

ji w

ji q

ji

8 Φ2

2

[

0−

9

10

]

1.6

[

3.20

] [

0−

9

10

] [

10

]

s23=

[

10

]



Iteration 2


To verify the static balance, this linkage along with the loads was modelled in


fromiteration 2


no. Φji a

jk/p

jk b

jk/f

jk k

jk → u

ji κ

ji v

ji w

ji q

ji

7 Φ2

1

(

C2

1

)

[

0.40

] [

0−1

]

eq.(5.16)

[

01

]

0

[

00

] [

01

] [

0.40

]

8 Φ2

2From iteration 2

[

0−

9

10

]

1.6

[

3.20

] [

0−

9

10

] [

10

]

9 Φ2

3

(

Z2

1

)

[

−10

] [

05

32

]

3.2 eq.(5.17)

[

0−

1

2

]

1.6

[

−3.20

] [

0−

5

32

] [

−3.20

]

∑

3

i=1u2

i =

[

0−

2

5

]

∑

3

i=1κ2

i = 3.2 eqs. (5.19-5.21) true

no. Φji u

ji κ

ji v

ji w

ji q

ji

11 Φ1

2

[

0−

2

5

]

3.2

[

6.40

] [

0−

2

5

] [

10

]

s12=

[

10

]



Iteration 3

2

10

n2o

= 1 n2a

= 1n2c

= 1

Φ2

1

(

C2

1

)

, Φ2

2, Φ2

3

(

Z2

1

)

n1o

= 1

Φ1

1

(

C1

1

)

2

10n1o

= 1 n1a

= 0n1c

= 1

Φ1

1

(

C1

1

)

, Φ1

2,

Step 4:deletion

2

Step 2 and 3



fromiteration 3

Iteration 4

Step 4:deletion

Step 2 and 3

1 n1o

= 1 n1a

= 0n1c

= 1

Φ1

1

(

C1

1

)

, Φ1

2,

0

1

0 n0c

= 1

Φ0

1

Step 1:springs unnecessary/eqn. (5.23-5.24)of propn. 1 aresatisfied as it is.

A constant(from propn. 1)Hence it is associated withthe root

no. Φji a

jk/p

jk b

jk/f

jk k

jk → u

ji κ

ji v

ji w

ji q

ji

10 Φ1

1

(

C1

1

)

[

0.40

] [

0−1

]

eq.(5.16)

[

01

]

0

[

00

] [

01

] [

0.40

]

11 Φ1

2s12=

[

10

]

eq.(5.30)

[

0−

2

5

]

3.2

[

6.40

] [

0−

2

5

] [

10

]

eqs. (5.23-5.24) is true with r1 =

[

00

]



x0

y0

y1

1

y2

y3

x2

y4

x4

0

2

3

4

C1

1C2

1

C3

1

C4

1

Z4

1

Z4

2

Z3

1

Z2

1

Figure 5.11: Statically balanced gravity-loaded serial 4R linkage.


ADAMS (www.adams.com). With zero damping, a pulse of energy was initially

introduced to the system. When the dynamic simulation of the system was carried

out it was noticed that the net kinetic energy was constant over time. This implies

that there was no potential gradient along the path that the linkage took in the

dynamic simulation.

In figure (5.12), joint 1 and body 1 are eliminated to modify this 4R example into

a 3R example. Joint 2 now joins body 2 with the ground at r2 =

[

0532

]

. The rest

of the bodies and their numbering remain unchanged. The first two iterations for

this example are identical to the 4R example. The third iteration is the last since

node 2 now is a child of the root. As required at this iteration, it may be verified

that equations (5.23 - 5.24) are satisfied with r2 =

[

0532

]

. One can have a similar

modification of 4R example into a 2R example as shown in figure (5.13).

Illustration of the algorithm on a 2R linkage under a zero-free-length

spring load Just as figures (5.7-5.11) have all the details of the serial 4R example,

figure (5.14) has all the details of this example. The explanation is also along the

same lines as that of the previous example. In this example, the given original load

is Z21 and the balancing loads are Z2

2 and Z23 .

To reaffirm the fact that zero-free-length springs are practical, a prototype of this

example was made, as shown in figure (5.14b). To realize zero-free-length springs,

pulley-string arrangement was used, the details of which were explained in figure

(3.12).

Illustration of the algorithm on a 4R tree-structure linkage under both

constant load and zero-free-length spring load While the previous two exam-

ples had serial architecture, this example has branches emanating form the same node,

as shown in figure (5.15a). The original loads acting on it are C31 and Z4

1 . Instead of

taking original loads to be exclusively constant loads or exclusively zero-free-length

spring loads, here we have taken a combination of both types of loads. These original

loads are balanced by adding springs Z31 , Z

32 , Z

42 , Z

21 , and Z1

1 at various iterations


x0

y0

2

C2

1y2

3

4

Z4

2

Z4

1

C4

1

C3

1

Z3

1

y3

y4

x4

0

r2=

[

05

32

]

Figure 5.12: Statically balanced serial 3R linkage


x0

y0

3

y3

x3

y4

x4

Z4

1

Z4

2

C4

1

C3

1

4

0

r3 =

[

015

32

]

Figure 5.13: Statically balanced serial 2R linkage


Z2

1

Z2

2

Z2

3

2

1

0

Φ21

(

Z21

)

, Φ22

(

Z22

)

, Φ23

(

Z23

)

n2o

= 1 n2a

= 2

Step1: add springsEqs. (5.19-5.21)

2

1

0

Φ11

n1c

= 1

Step 4:Deletion

Steps 2 and 3

1

0Φ01

n0c

= 1

Step 4:Deletion

Step1:Eqs. (5.22-5.23)No extra springs

Steps 2 and 3

Constant fromproposition 1

Iteration 1

Iteration 2

x2

x1y

0

no. Φji a

j

k bj

k kj

k → uji κ

ji v

ji w

ji q

ji

1 Φ2

1

(

Z2

1

)

[

50

] [

1212

]

1 eq.(5.17)

[

−12−12

]

0.5

[

50

] [

−12−12

] [

50

]

2 Φ2

2

(

Z2

2

)

[

50

] [

−12−12

]

1 eq.(5.17)

[

1212

]

0.5

[

50

] [

1212

] [

50

]

3 Φ2

3

(

Z2

3

)

[

−50

] [

00

]

2 eq.(5.17)

[

00

]

1

[

−100

] [

00

] [

−100

]

∑

3

i=1u2

i =

[

00

]

∑

3

i=1κ3

i = 2 eqs. (5.19-5.21) true

4 Φ1

1 s1

2 =

[

10

]

eq.(5.30)

[

00

]

2

[

200

] [

00

] [

50

]


[

00

]

(c)

(a)

(b)

Figure 5.14: Details of static balance of a 2R linkage under spring load

in the iterative algorithm. A pictorial depiction of the iterations on this linkages is

given in figure (5.15b). All the remaining details are given in the table of the same

figure.

To verify the static balance, θs of bodies 1, 2, 3 and 4 are varied in the following

form: θ1 = π4+ π sin (2πt), θ2 = π

12+ π sin (2πt), θ3 = π

1.7+ π sin (2πt), θ4 = π

1.3+

π sin (2πt). The potential energy variation of original loads C31 and Z4

1 as well as the

balancing loads, i.e., Z31 , Z

32 , Z

42 , Z

21 , and Z1

1 , are plotted in figure (5.16). The sum

of all these variations is also plotted and it has turned out to be a constant. This

verifies the static balance.


1

2

34

X

Y

0

Z31

Z32

Z41

Z42

Z21

Z11

no. Φji a

jk/p

jk b

jk/f

jk k

jk → u

ji κ

ji v

ji w

ji q

ji

1 Φ4

1

(

Z4

1

)

[

10

] [

7

23

2

]

1 eq.(5.17)

[

−7

2

−3

2

]

1

2

[

10

] [

−7

2

−3

2

] [

10

]

2 Φ4

2

(

Z4

2

)

[

−10

] [

7

23

2

]

1 eq.(5.17)

[

−7

2

−3

2

]

1

2

[

−10

] [

−7

2

−3

2

] [

−10

]

∑

2

i=1u4

i =

[

−7−3

]

∑

2

i=1κ4

i = 1 eqs. (5.19-5.21) true

3 Φ3

1

(

C3

1

)

[

10

] [

0−1

]

eq.(5.16)

[

01

]

0

[

00

] [

01

] [

10

]

4 Φ3

2

(

Z3

1

)

[

1

2

0

] [

1

2

3

]

1 eq.(5.17)

[

−1

2

−3

]

1

2

[

1

2

0

] [

−1

2

−3

] [

1

2

0

]

5 Φ3

3

(

Z3

2

)

[

−1

2

0

] [

1

2

1

]

1 eq.(5.17)

[

−1

2

−1

]

1

2

[

−1

2

0

] [

−1

2

−1

] [

−1

2

0

]

∑

3

i=1u3

i =

[

−1−3

]

∑

3

i=1κ3

i = 1 eqs. (5.19-5.21) true

7 Φ2

1s23=

[

−13

5

]

eq.(5.30)

[

−1−3

]

1

[

−26

5

] [

−1−3

] [

−13

5

]

8 Φ2

2s24=

[

13

5

]

eq.(5.30)

[

−7−3

]

1

[

26

5

] [

−7−3

] [

13

5

]

9 Φ2

3

(

Z2

1

)

[

0−

6

5

] [

24

]

2 eq.(5.17)

[

−4−8

]

1

[

0−

12

5

] [

−2−4

] [

0−

12

5

]

∑

3

i=1u2

i =

[

−12−14

]

∑

3

i=1κ2

i = 3 eqs. (5.19-5.21) true

10 Φ1

1s12=

[

10

]

eq.(5.30)

[

−12−14

]

3

[

60

] [

−12−14

] [

10

]

11 Φ1

2

(

Z1

1

)

[

1

2

0

] [

5

41

6

]

12 eq.(5.17)

[

−15−2

]

6

[

60

] [

−5

4

−1

6

] [

60

]


[

13

85

4

]

C31

0

1

2

3 4Φ4

1

(

Z4

1

)

, Φ4

2

(

Z4

2

)

n4o

= 1 n4a

= 1

Φ3

1

(

C3

1

)

, Φ3

2

(

Z3

1

)

, Φ3

3

(

Z3

2

)

n3o

= 1 n3a

= 2

0

1

2Φ2

1, Φ2

2, Φ2

3

(

Z2

1

)

n2c

= 2 n2a

= 1

Iteration 1




Step 2 & 3Step 2 & 3

0

1Φ1

1, Φ1

2

(

Z1

1

)

n1c

= 1 n1a

= 1

Iteration 2


Step 2 & 3

0 Φ0

1

A constant

Iteration 2

Step 2 & 3

(a)

(b)

Original loads

Figure 5.15: Details of static balance of a 4R tree-structure linkage under a constantload and a spring load

5.4.3 Static balancing of any revolute-jointed linkages with

any kind of zero-free-length spring and constant load

interaction within the linkage

In the static balancing method for linkages provided in Section 5.4.2, other than

the fact that the linkage to be balanced has to be revolute-jointed and that load

interactions are of zero-free-length spring or constant loads, there were two more

restrictions:


-5

0

5

10

15

20

25

30

35

0 0.2 0.4 0.6 0.8 1

C3

1

Z4

1

Z3

1Z3

2

Z4

2

Z2

1

Z1

1

Net Potential Energy

t −→

PE

Figure 5.16: Potential Energy variation of spring loads, constant loads, and their sum

1. It should be possible to consider that the loads on all the bodies are exerted by

a common reference body (or frame) of the linkage.

2. The linkage should have a tree-structure (i.e., without closed loops).

When the first restriction is violated, as in figure (5.17a), it is always possible

to break the load interactions into a superposition of several load sets with each set

complying with the first restriction. For example, the load interaction in figure (5.17a)

is broken into two load sets in figures (5.17b) and (5.17c). The reference body in

each of these sets is indicated by an asterisk symbol (*) in their respective figures.

Furthermore, in a load set, if there are closed loops, then the closed loops can be

broken by relaxing certain joint constraints. Figures (5.17c) and (5.17d) illustrate

breaking of closed loops in figures (5.17b) and (5.17c), respectively. With closed


Breaking closed loops

Breaking load interactions

+

*

*

(a) (b) (c)

(c) (d)

Joint relaxed

Joint relaxed

Figure 5.17: Breaking a problem as a superposition of several problem with eachproblem being static balance of revolute-jointed tree-structured linkage with loadsexerted by the root body

loops broken, each of the load sets comply with the two restrictions and they can be

statically balanced by adding balancing loads as per Section 5.4.2.

Once each of the load sets is balanced, the joint constraints that were relaxed

for breaking closed loops can be reimposed without disturbing the static balance. In

other words, when the potential energy that is a function of the configuration space is

a constant, it remains as the constant even when the configuration space is restricted

(due to re-attachment of the broken joints).

Once the constraints are reimposed, the linkages in all the load sets are the same

as the original linkage and the loads on all the sets can be superposed. Since each load

set is in static balance, the superposition is also in static balance. In other words, the

sum of several constant potential energy functions due to several load sets is also a


constant. This superposition contains all the original loads on the given linkage. The

remnant loads in this superposition are the additional loads that balance the original

loads. In this way, additional loads that statically balance any revolute-jointed linkage

with zero-free-length spring and constant load interactions between the bodies of the

linkage can always be found. An example is presented next to illustrate this.

A numerical example

Figure (5.18a) shows a linkage which is the same as that in figure (5.15) except for

an additional zero-free-length spring between the three-lobed body and the left-most

body. Unlike in figure (5.15), where the ground can be considered as the reference

body that exerts loads on all the bodies, in figure (5.18a) it is not possible to find a

single reference body that exerts loads on all the bodies. Hence the loads are split

into two sets as in figures (5.18b) and (5.18c). The first set shown in figure (5.18b)

is the same as that in figure (5.15) whose balancing loads are already found using

iterative algorithm detailed in figure (5.15).

In the second set shown in figure (5.18c), the three-lobed body is treated as the

reference frame. The bodies are labelled accordingly with the number 0 assigned to

the reference body. The only spring in the figure is labelled as Z11 . In this figure,

there is only one load interaction, which is between body 1 and the reference frame.

Further, body 1 can be considered as lever on the reference body. An extra spring Z12

between body 1 and 0 is added, the details of which are given figure (5.19), to satisfy

the conditions of static balance of a lever (equations (5.10) – (5.11)). With this the

second load set is balanced.

The balanced first and second load sets are shown in figures (5.18d) and (5.18e),

which are further superposed as in figure (5.18f) to obtain the balancing solution to

the original load set shown in (5.18a).

5.5 A note on prismatic joint

In Section 5.3, it was noted that by merely adding springs, the dependency of a body’s

potential energy on r cannot be annihilated. As shown in the later sections, in the


+

+

0

1

2

34

Set 1 Set 2C31

Z41

C31

Z41

0

1

34Z3

1

Z32

Z42

Z21

Z11

2

0

1

Reference body

The ground

Reference body

2

3

4

Z11

4

3

2

0

1

Z11

Z12

Splitting original loads

Superposition of balanced load sets

Static balancingSee figure (5.15)

Static balancing

See figure (5.19)

(a) (b) (c)

(e)(d)(f)

Figure 5.18: Static balance of a tree-structured linkage with inter-body load interac-tions

case of the body having a revolute joint with another body, this dependency on r

can be transmitted as a dependency on the coordinates of the other body when r

points to the common point of the revolute joint. This allows us to think that the

dependency of the potential energy on r is conceptually removed by transmission.

The static balancing strategy outlined in this chapter is based on carrying out this

transmission in a cascading manner.

If the body has a prismatic joint instead of the revolute joint, there is no scope

for the conceptual removal by transmission since r cannot be expressed in terms of

the coordinates of the other body. This un-annihilated potential energy dependence

of r implies that in a tree-structured chain, if there is a prismatic joint, there is no

way static balance can be achieved by merely adding springs. In such situations,

it is necessary to add revoluted-jointed chain as auxiliary chain and carry out the


4

0

1

Z11

Z12

X

Y

Reference frame

Body 1 is a lever with respect to body 0[

−1

2

3

10

]T

[

3

40]T

[

−3

40]T

Spring details

a

Z12

[

3

40]T

b

[

−1

2

3

10

]T

Z11

[

−3

40]T

[

−1

2

3

10

]T

k

1

1

r =[

−1 35

]T

is a constant

k (r − b)T a k (r − b)T R(

π

2

)

a

3

8

−3

8

−9

40

+ 9

40

Equations (5.10) and (5.11) satisfied

Figure 5.19: Balancing the lever lever loads in the second load set of figure (5.18)

cascading transmission through this auxiliary chain. In closed loop linkages, such

auxiliary chains may be inherently present as in the case of a slider crank linkage

which can be thought as a 3R chain closing its loop with a prismatic joint.

Summary

• We presented a technique to statically balance any planar revolute-jointed link-

age having zero-free-length spring and constant load interactions between the

bodies of the linkage.

• The technique involves only addition of zero-free-length springs but not any

extra link, unlike spring-aided perfect static balancing techniques currently in

the literature.

• The technique relies on a recursive relation to iteratively remove the dependence

of the potential energy on the configuration variables of the bodies of the linkage.

• Recognizing the recursive relation along with the minimal conditions that enable

it constitutes the contribution of this chapter.

Chapter 6

Static balancing of spatial linkages

without auxiliary bodies

Overview

• Just as a recursive relation in the planar case enabled an iterative algorithm,

there is a recursive relation in the spatial case also.

• The recursive relation holds when the loads are constant loads and/or zero-free-

length spring loads and the joints are revolute and/or spherical joints.

• Similar to the planar case, even though the iterative algorithm works on a tree-

structured linkage, extension to an arbitrary linkage is straightforward.

6.1 Introduction

The results of Chapter 5 hinged on Proposition 2, which was recursively applied in

the iterative algorithm of Section (5.4.2). If we can have a similar proposition in the

spatial case, then we can have an iterative static balancing algorithm for spatial case

also. The features of the proposition that allowed its recursive application are:

• Feature 1: The allowable loads on a body, say body j, should have a potential

energy that belongs to a class of functions of configuration variables of the body.

102

CHAPTER 6. STATIC BALANCING OF SPATIAL LINKAGES 103

Let the class of functions be represented by χ(j). In Section 5.4.2, the allowable

loads are zero-free-length spring and constant loads, and the class of functions

are as in equation (5.15).

• Feature 2: A joint between two bodies, say body j and body l, should provide

a relation between configuration variables of j and l that can be used to express

the sum of a set of functions in χ(l) as a function in χ(j). This is essential for

steps 1 to 3 of the iterative procedure in Section 5.4.2.

For spatial linkages, a class of functions and the joints having the two features are

presented next.

6.2 The class of functions in feature 1

If a body in space has its configuration with respect to a reference frame defined by r

and R, the position vector and rotation matrix of the local frame with respect to the

global frame, then the potential energy of a constant load and a positive-stiffness zero-

free-length spring and a general form of potential energy function that can express

potential energies of both constant load and zero-free-length spring loads are:

PEc = (−f)T r + (−f)T Rp (6.1)

PEs = (−kb)T r + (−kb)T Ra+ rTR (ka) +k

2rTr

+k

2

(aTa+ bTb

), k > 0 (6.2)

Φ = uTr +wTRq + rTRv + κrTr + c, κ ≥ 0 (6.3)

The derivation and the notation used in equations (6.1), (6.2), and (6.3) are the same

as those in equations (5.1), (5.6), and (5.15), respectively, except that they are all

spatial quantities now.

The class of functions referred in feature 1 would be the functions of the form

given in equation (6.3). We do not consider a broader class of functions that includes

potential energies of positive zero-free-length springs since the extra terms associated


with positive-free-length spring (see equation (5.8)) cannot be balanced out for the

reasons same as those given in Appendix C.

It may be noted that R, unlike in planar case, is not a function of just a single

variable. Among many possibilities, R may be considered as a function of three Euler

angles. We use body fixed Z −X ′ − Z ′′ system of Euler angles with the three angles

represented as α, β and γ so that R becomes

R =

c1c3 − s1c2s3 −c1s3 − s1c2c3 s1s2

s1c3 + c1c2s3 −s1s3 + c1c2c3 −c1s2

s2s3 s2c3 c2

(6.4)

where c1 = cosα, s1 = sinα, c2 = cos β, s2 = sin β, c3 = cos γ, and s3 = sin γ.

Just as in table 5.1, the function in equation (6.3) can be expressed as a linear

combination a finite number of linearly independent functions where each function is

a function of r and {α, β, γ}. The functions, which are also a basis for the function

in equation (6.3), and the corresponding coefficients are presented in table 6.1.

6.3 Joints that can potentially satisfy feature 2

Consider the following proposition, which is along the same same lines as in Section

5.3. This proposition helps us to narrow down on spatial joints that can potentially

have feature 2, as discussed after the proof of the proposition.

Proposition 3. For a given set of functions of the form

Φi = rTui +wTi Rqi + rTRvi + κir

Tr + ci,

κi ≥ 0 and i = 1 · · ·nf (6.5)

that are associated with a rigid-body defined by r and R,

1. the net sum of these functions along with the potentials of one or more positive-

stiffness zero-free-length springs cannot be independent of r.


Table 6.1: The potential in the general form shown in figure (6.3), can be expressedas a linear combination of the basis functions shown in the table. Each basis functionin the table is a function of translational variable r and the Z-X-Z Euler angle α, βand γ

No. Basis functions Coefficient1 rx ux2 ry uy3 rz uz4 r2x + r2y + r2z κ

5 rx(c1c3 − s1c2s3) + ry(s1c3 + c1c2s3) + rz(s2s3) vx6 rx(−c1s3 − s1c2c3) + ry(−s1s3 + c1c2c3) + rz(s2c3) vy7 rx(s1s2) + ry(−c1s2) + rz(c2) vz8 c1c3 − s1c2s3 wxqx9 s1c3 + c1c2s3 wyqx10 s2s3 wzqx11 −c1s3 − s1c2c3 wxqy12 −s1s3 + c1c2c3 wyqy13 s2c3 wzqy14 s1s2 wxqz15 −c1s2 wyqz16 c2 wzqz17 1 c


2. it is possible to find a finite set of positive-stiffness zero-free-length spring loads

such that the net sum of the functions and potentials of springs is independent

of orientation variable R.

Proof. Let (ki,ai, bi), ki > 0 and i = 1 · · ·na be any set of na ≥ 1 zero-free-length

spring loads on the rigid-body. Their potential energy can be written in the general

form shown in equation (6.3), where the parameters of the general form, corresponding

to na springs, are represented as {ui, wi, qi, vi, κi, ci}, κi > 0, i = nf+1 · · ·nf+na.

The net sum of the given functions and the potential energies of the springs take the

following form.

nf+na∑

i=1

Φi = rT

nf+na∑

i=1

ui +

nf+na∑

i=1

wTi Rqi+

rTR

nf+na∑

i=1

vi + rTr

nf+na∑

i=1

κi +

nf+na∑

i=1

ci

(6.6)

The coefficient of rTr in equation (6.6) is a non-zero positive number since κi ≥ 0 for

i = 1 · · ·nf , κi > 0 for i = nf +1 · · ·nf +na and na ≥ 1. Hence, the net sumnf+na∑

i=1

Φi,

i.e., the net sum of functions along with the potentials cannot be independent of r.

Thus the first part of the proposition is proved.

Now, consider the second part of the proposition. The net sum is also a linear

combination of the basis functions given in table 6.1 with the corresponding coeffi-

cients being the same as in the table except that the coefficients are subscripted with

i and summed over i from 1 to nf + na. The net sum is independent of orientation

if and only if all the coefficients of the basis functions that are dependent on α or β

or γ are zero. The coefficients of the basis function nos. 5, 6 and 7 in table 6.1 being

constrained to zero can be compactly written as

nf+na∑

i=1

vi = 0 (6.7)

The elements in the rows 8 – 16 in the right most column of table 6.1 are actually


nine elements of the matrix qwT . Hence, the coefficients for the net sum of the basis

function nos. 8–16 in table 6.1 being constrained to zero can be written as

[

q1 · · · qnf+na

] [

w1 · · · wnf+na

]T

= 0 (6.8)

It is shown in Appendix E that no matter what the parameters {ui, wi, qi, vi, κi, ci}

for i = 1 · · ·nf are, there are always four zero-free-length springs having such a set

of parameters (ki,ai, bi), ki > 0 and i = 1 · · · 4 that the equations (6.7) and (6.8) are

satisfied. Hence, the second part of the proposition is also true.

If a joint should have feature 2, then the joint should transform the dependence

on r, which cannot be balanced out as per the proposition, to dependence on the

configuration variables of another body between which the joint exists. In other

words, r, i.e., position vector of local origin on the body, should be a function solely

of the configuration variables of the other body. Among the basic joints, such a thing

is possible only if the other body is connected to the body by a revolute or spherical

joint at the origin. Thus, spherical joints and revolute joints can potentially have

feature 2 and we next prove that they indeed have.

From here onwards, we deal with more than one body and each body is assigned a

distinct number. Further, if quantities such as r, R, ui, ki are associated with body

j, then the corresponding symbols are superscripted with j. With this, quantities

associated with different bodies can be distinguished from one another.

6.4 Spherical joint has feature 2

Here is the proposition with its proof that essentially says that for the class of func-

tions given in equation (6.3), the spherical joint has feature 2 under certain conditions.

Proposition 4. For a given set of functions

Φli = rlTul

i +wli

TRlql

i + rlTRlvli + κlir

lTrl + cli,

κli ≥ 0 and i = 1 · · ·nl(6.9)


that is associated with body l, if:

Condition 1: There is a spherical joint between body j and body l such that a point

fixed to body j is coincident with the local origin of body l. Let the local coordi-

nates of the point on body j be sjl .

Condition 2: Equations (6.7) and (6.8) are satisfied with nf + na being nl.

thennl∑

i=1

Φli = rjTu

ji +w

ji

TRjq

ji + rjTRjv

ji + κjir

jTrj (6.10)

where

uji =

nl∑

i=1

uli, w

ji =

nl∑

i=1

uli, q

ji = s

jl ,

vji = 2sjl

nl∑

i=1

κli, κji =nl∑

i=1

κli

(6.11)

Proof. The equation form of condition 1 is

rj +Rjsjl = rl (6.12)

The equations referred in condition 2 are the necessary and sufficient constraints

to make the net sumnl∑

i=1

independent of Rl, as shown in the proof of proposition 3.

Hence, under condition 2, in the expression fornl∑

i=1

Φli, the Rl dependent terms can

be omitted as shown next

nl∑

i=1

Φli = rlT

nl∑

i=1

uji + rlTrl

nl∑

i=1

κli (6.13)

nl∑

i=1

Φli represents potential energy and our interest is in the gradient of the potential

energy with respect to the configuration variables. Hence, the constant terms, which


do not affect gradients, are inconsequential. This is the reason for omitting constant

terms as well in equation (6.13).

On the same lines as in proposition 2, substitution for rl from equation (6.12) into

equation (6.13), simplification and omission of the constant term leads to equations

(6.10) and (6.11).

6.5 Revolute joint has feature 2

Proposition 4 holds for revolute joints as well. However, the fact that in a revolute

joint, a line, rather than a point, fixed to two bodies are coincident allows us to make

condition 2 of proposition 4 less stringent as shown next.

Proposition 5. For given a set of functions

Φli = rlTul

i +wli

TRlql

i + rlTRlvli + κlir

lTrl + cli,

κli ≥ 0 and i = 1 · · ·nl(6.14)

that is associated with body l, if:

Condition 1: There is a revolute joint between body j and body l such that a point

fixed to body j is coincident with the local origin of body l and a unit vector fixed

to body j is always parallel to local z-axis of body l. Let the local coordinates of

the point on body j be sjj and the unit vector fixed to body j in local frame of

reference be kj

l .

Condition 2: Following equations are satisfied.

nl∑

i=1

vli = 0 (6.15)

[

ql1 · · · ql

nl

] [

wl1 · · · wl

nl

]T

= 0 (6.16)

where v = [vx vy]T and q = [qx qy]

T .


thennl∑

i=1

Φli = Φj

i + Φji+1 (6.17)

where φji and Φj

i+1 are functions of the form in equation (6.14) but associated with

body j. uji , w

ji , q

ji , v

ji , κ

ji are as given in equation (6.11), and u

ji+1, w

ji+1, q

ji+1, v

ji+1,

κji+1 are as given next.

uji+1 = 0, w

ji+1 =

nl∑

i=1

qlz,iwli, q

ji+1 = k

j

l

vji+1 = k

j

l

nl∑

i=1

vlz,i, κji = 0

(6.18)

Proof. From condition 1 of the proposition, we have

rj +Rjsjl = rl (6.19)

Rjkj

l = Rl[0 0 1]T = [sl1sl2 − cl1s

l2 cl2]

T (6.20)

What equation (6.20) says is that terms sl1sl2, −c

l1s

l2 and cl2 can be considered as

functions of configuration variables of body j. Hence, for the net sum, it is not

necessary to constrain the coefficients of basis functions involving only these functions

and rl (function nos. 7, 14, 15 and 16 in table 6.1) to zero. Instead, one can transform

the dependence on these function and rl to dependence on the configuration variables

of body j using equations (6.19) and (6.20). This is the reason for truncating v and

q in equations (6.15) and (6.16). This truncation omits coefficients corresponding to

basis function nos. 7, 14, 15 and 16 in table 6.1.

The contribution of basis functions nos. 1, 2, 3, and 4 (as in table 6.1), after

transformation to configuration of body j, is already written in equations (6.10) and

(6.11). Φji in equation (6.17) represent the same. The contribution from basis function

no.7 is

rlT [sl1sl2 − cl1s

l2 cl2]

T

nl∑

i=1

vlz,i (6.21)


The contribution from basis function nos. 14, 15 and 16 is

nl∑

i=1

qlz,iwli

T

[sl1sl2 − cl1s

l2 cl2]

T (6.22)

Substitution for rl and [sl1sl2 − cl1s

l2 cl2] from equations (6.19) and (6.20) into the

sum of expressions (6.21) and (6.22), followed by simplification, yields

rjTRj

kj

l

nl∑

i=1

vlz,i

+

nl∑

i=1

qlz,iwli

T

Rjkj

l (6.23)

This term is represented as Φli+1 in equation (6.17) and is the contribution from basis

function nos. 7, 14, 15 and 16, after transformation to configuration variables of body

j. The reason for not combining two functions on the right hand side of equation

(6.17) into one is that the terms wji

TRjq

ji and wjT

i+1Rjq

ji+1 cannot be combined.

6.6 Algorithm to synthesize static balancing so-

lution of a spatial revolute/spherical-jointed

tree-structured linkage having zero-free-length

spring and/or gravity loads exerted by a ref-

erence link

This is along the same lines as in Section 5.4.2.

Preparatory steps

1. Assign the reference body as the root node of the tree-structure.

2. If the joint between a body and its parent body is a revolute joint, then choose

the local frame of reference of the body such that the z-axis of the local frame


is aligned along the axis of the revolute joint. If the joint between a body and

its parent is spherical joint, then choose the local frame of reference of the body

such that the origin of the local frame coincides with the center of the spherical

joint.

3. Give this tree-structure with the given constant and zero-free-length spring loads

as an input to the following iterative procedure.

Iterative procedure

• Entry condition: If the tree-structure contains only the root node, then exit

from the iterative procedure. Otherwise, proceed to step 1.

• Step 1: For every body/node, there are two kinds of potential energy functions

associated with it. One kind is due to loads acting on the body and the other

kind is due to the association that happened in step 3 of previous iterations.

From equations (5.16-5.17) (which holds for spatial case also), proposition 4

and proposition 5, respectively, both kinds of potential energy are of the form

given in equation (6.3) where r and R are configuration variables of the body.

For every terminal node l of the tree-structure, add extra nla zero-free-length

springs such that the potential energy functions due to 1) nlo original loads on

the body, 2) nlc associations that happened with node l in step 3 of previous

iterations, and 3) nla extra zero-free-length spring loads satisfy condition 2 of

proposition 4 in the case of spherical joint and condition 2 of proposition 5 in

the case of revolute joint. Appendix E shows that such a thing is possible.

• Step 2: Expressnlo+nl

c+nla∑

i=1

Φli as function of configuration variable of the parent

link. This is possible since condition 1 (because of preparatory steps), condi-

tion 2 (because of step 1) of proposition 4 in the case of spherical joint and

proposition 5 in case revolute joint are satisfied.

• Step 3: Associatenlo+nl

c+nla∑

i=1

Φli with the parent link of l and for energy conser-

vation, disassociate Φli, i = 1 · · ·nl

o + nlc + nl

a from node l.


• Step 4: With all the potential energy functions disassociated from node l to

its parent, delete this terminal node l.

• Iterator: Once steps 1 to 4 are completed, a new trimmed tree-structure results

where the parents of the nodes deleted in step 4 has additional potential energy

functions associated with them. Follow this iterative procedure again with this

trimmed tree-structure as the input.

The arguments that this algorithm indeed leads to static balance is also the same as

that in Section 5.4.2. The distinct cases (a) and (b) of Section 5.4.2 exist in spatial

case also. We indicate case (b) in the last iteration of the following example.

6.6.1 Illustrative example

Figure (6.1-(ii)) shows a six degree-of-freedom spatial linkage consisting of four bodies

that are serially connected to the ground. Starting from the ground, the first three

joints are revolute and the last joint is spherical. The architecture is the same as

PUMA robot except that the wrist of the PUMA robot is replaced by a spherical

joint. Starting from 0 for the ground, the bodies are serially numbered as 1, 2, 3

and 4. Bodies 2, 3 and 4 are under the influence of constant loads C41 , C

31 , and C2

1 as

shown in figure (6.1). The constant loads are due to gravitational force on the bodies.

In order to statically balance this spatial linkage, we follow the iterative algorithm

described in Section 6.6.

Local frames of reference are assigned as per the preparatory steps of the algo-

rithm. A D-H table as well as Sjl and k

j

l parameters describing relative position of

the frames are given in tables (c) and (d) of figure (6.1). In the following iterative

procedure, six springs Z41 , Z

42 , Z

43 , Z

31 , Z

21 , and Z1

1 are added to balance the three

constant forces. Tables (a) and (b) of figure (6.1) provide parameters of the constant

forces as well as zero-free-length springs. Further, by comparing equations (6.1) and

(6.2) with equation (6.3), the potential energies in the generalized form for these

constant forces and zero-free-length springs are tabulated in table (e) of figure (6.1).

The generalized form of potential energies that are obtained in steps 2 and 3 of the


0 1

2

3

4 Φ4

1

(

C4

1

)

, Φ4

2

(

Z4

1

)

, Φ4

3

(

Z4

2

)

, Φ4

4

(

Z4

3

)

Φ3

1

(

C3

1

)

Φ2

1

(

C2

1

)

Step 1 of iteration 1:Spring additionCond. 2 of prop. 4

0 1

2

3

4

Φ3

1

(

C3

1

)

, Φ3

2, Φ3

3

(

Z3

1

)

Φ2

1

(

C2

1

)

Step 4 of iteration 1Deletion of node 4

0 1

2

3

Φ2

1

(

C2

1

)

, Φ2

2, Φ2

3, Φ2

4

(

Z2

1

)

0 1

2

Φ1

1, Φ1

2, Φ1

3

(

Z2

1

)

Spring additionEqs. (6.24) and(6.25)

1

x0

y0z

0

2

34

x2

y2

z2

x1

y1z

1

x3

y3

z3

x3

y3

z3

C4

1

C3

1

C2

1

Z4

1

Z4

2

Z4

3

Z3

1

Z2

1

Z1

1

0 (Ground)

D-H parameters

i αi−1 ai−1 di θi

1 0 0 1.2 θ1

2 π

20 −0.175 θ2

3 0 0.6 0.075 θ3

4 −π

20.6 0 ∗ ∗ ∗

Further tranformation offrame 4 is described by 321body fixed euler angles: e

4

1,

e4

2, and e

4

3.

Sjl and k

jl parameters

S0

1

[

0.0 0.0 1.2]T

k0

1

[

0 0 1]T

S1

2

[

0.0 0.175 0.0]T

k1

2

[

0 −1 0]T

S2

3

[

0.6 0.0 0.075]T

k2

3

[

0 0 1]T

S3

4

[

0.6 0.0 0.0]T

Parameters of constant forces

Cji p

ji f

ji

C4

1

[

0.25 0.0 0.0]T [

0.0 0.0 −0.4]T

C3

1

[

0.2 0.0 0.0]T [

0.0 0.0 −0.9]T

C2

1

[

0.2 0.0 0.0]T [

0.0 0.0 −0.9]T

Parameters of zero-free-length springs

Zj

i aj

i bj

i kj

i

Z4

1

[

0.5 0 0]T [

0.7 0.1 0.4]T

1

Z4

2

[

−0.25 0.2 0]T [

0.7 0.1 0.2]T

1

Z4

3

[

−0.25 −0.2 0]T [

0.7 0.1 0.2]T

1

Z3

1

[

−0.2 0.0 0.05]T [

0.7 0.1 1

30

]T9

Z2

1

[

−0.2 0.0 0.025]T [

0.7 0.1 −1

24

]T36

Z1

1

[

0 −41

320−

301

240

]T [

0.7 0.1 −13

240

]T48

Parameters of potentials due to springs, constant forces and transformation cum association in steps 2 and 3

Φj

i uj

i qj

i wj

i vj

i κj

i

Φ4

1

(

C4

1

) [

0.0 0.0 0.4]T [

0.25 0.0 0.0]T [

0.0 0.0 0.4]T [

0.0 0.0 0.0]T

0

Φ4

2

(

Z4

1

) [

−0.7 −0.1 −0.4]T [

0.5 0 0]T [

−0.7 −0.1 −0.4]T [

0.5 0 0]T 1

2

Φ4

3

(

Z4

2

) [

−0.7 −0.1 −0.2]T [

−0.25 0.20 0]T [

−0.7 −0.1 −0.2]T [

−0.25 0.20 0]T 1

2

Φ4

4

(

Z4

3

) [

−0.7 −0.1 −0.2]T [

−0.25 −0.20 0]T [

−0.7 −0.1 −0.2]T [

−0.25 −0.20 0]T 1

2

Φ3

1

(

C3

1

) [

0.0 0.0 0.9]T [

0.2 0.0 0.0]T [

0.0 0.0 0.9]T [

0.0 0.0 0.0]T

0

Φ3

2 =4∑

i=1

Φ4

i

[

−2.1 −0.3 −0.4]T [

0.6 0.0 0.0]T [

−2.1 −0.3 −0.4]T [

1.8 0.0 0.0]T

1.5

Φ4

3

(

Z3

1

) [

−6.3 −0.9 −0.3]T [

−1.8 0.0 0.45]T [

−0.7 −0.1 −1

30

]T [

−1.8 0.0 0.45]T

4.5

Φ2

1

(

C2

1

) [

0.0 0.0 0.9]T [

0.0 0.0 0.9]T [

0.2 0.0 0.0]T [

0.0 0.0 0.0]T

0

Φ2

2 +Φ2

3=3∑

i=1

Φ3

i

[

−8.4 −1.2 0.2]T [

−8.4 −1.2 0.2]T [

0.6 0.0 0.075]T [

7.2 0.0 0.9]T

6[

0.0 0.0 0.0]T [

−0.315 −0.045 −0.015]T [

0.0 0.0 1.0]T [

0.0 0.0 0.45]T

0

Φ2

4

(

Z2

1

) [

−25.2 −3.6 1.5]T [

−0.7 −0.1 1

24

]T [

−7.2 0.0 0.9]T [

−7.2 0.0 0.9]T

18

Φ1

1 +Φ1

2=4∑

i=1

Φ2

i

[

−33.6 −4.8 2.6]T [

−33.6 −4.8 2.6]T [

0.0 0.175 0.0]T [

0.0 8.4 0.0]T

24[

0.0 0.0 0.0]T [

−63

40−

9

40

3

80

]T [

0 −1 0]T [

0 −2.25 0]T

0

Φ1

3

(

Z1

1

) [

−33.6 −4.8 2.6]T [

−0.7 −0.1 13

240

]T [

0.0 −6.15 −60.2]T [

0.0 −6.15 −60.2]T

24

(i) (ii)

(a)

(b)

(c) (d)

(e)

n4o

= 1 n4a

= 3

Step 2 of iteration 1:∑

= Φ3

2(as per eqn. 6.10)

Step 3 of iteration 1:association with node 3


n3o

= 1 n3c

= 1 n3a

= 1


= Φ2

2+Φ2

3(as per eqn. 6.17)


Step 4 of iteration 2Deletion of node 3

n2o

= 1 n2c

= 2 n2a

= 1



= Φ1

1+Φ1

2(as per eqn. 6.17)


Step 4 of iteration 3

Deletion of node 2

a constant

Figure 6.1: Details of static balancing of six degree-of-freedom spatial balancing undergravity loads


iterative procedure are also tabulated in the same table. The table is intended to

facilitate verification of the constraints that we satisfy in the iterative procedure.

The iterative procedure is pictorially depicted in figure (6.1-(i)). The terminal

body of the tree-structure is body 4. n4o = 1 and it is due to load C4

1 . n4c = 0

since this is the first iteration. In the first iteration, three springs Z41 , Z

42 , and Z4

3

are added so that potential energies Φ41 (C

41) (i.e., potential energy due to load C4

1),

Φ42 (Z

41 ), Φ

43 (Z

42 ), and Φ4

4 (Z43 ) satisfy condition 2 of proposition 4. Further, in step

2, the summation of these potential energies is transformed as Φ32, as per equation

(6.10). Steps 3 and 4 involve association of the summed potential with node 3 of a

new tree-structure formed by deleting node 4. All these steps are depicted in figure

(6.1-(i)). Similarly, the details of iteration 2 and 3 are also depicted in the same

figure.

In the fourth iteration, instead of carrying out the same four steps, one can take

advantage of the fact that r1 is a constant and the first two of the Euler angles

describing R1 are zeros. Among the configuration variables describing body 1, only

the last Euler angle, i.e., γ1 is a variable. In this scenario, examination of table 6.1

reveals that any sum of potential energies of the form in equation (6.3) is a linear

combination sin γ1 (i.e., s3) and cos γ1 (i.e., c3) and 1. By setting the coefficients of

sin γ1 and cos γ1 to zero, i.e.,

n1∑

i=1

(r1y,iv

1x,i − r1x,iv

1y,i + w1

y,iq1x,i − w1

x,iq1y,i

)= 0 (6.24)

n1∑

i=1

(r1x,iv

1x,i + r1y,iv

1y,i + w1

x,iq1x,i + w1

y,iq1y,i

)= 0 (6.25)

one gets the condition for the sum of potential energies associated with node 1 to be

a constant. In the fourth iteration, a spring Z11 is added so that potential energies

Φ11, Φ

12 and Φ1

3 (Z11 ) satisfy constraints (6.24) and (6.25). Hence Φ1

1 + Φ12 + Φ1

3 (Z11 )

is a constant. Further Φ11 + Φ1

2 + Φ13 (Z

11 ) is the net sum of potential energies of all

the constant loads and the zero-free-length springs added in the iterative process.

(This may be verified by retracing the arrow lines in figure (6.1-(i))). Since the net


potential is constant, static balance is attained.

6.7 Static balance of any kind of spatial revolute

and/or spherical jointed linkage with constant

load and zero-free-length spring load interac-

tion

The restriction for algorithm of Section 5.4.2 cited in Section 5.4.3 is true for the

spatial algorithm of Section 6.6 also. A general spatial and/or revolute jointed linkage

with constant forces and zero-free-length spring may not fall under these restrictions.

Nevertheless, one can split the loads into load sets followed by breaking of closed loops,

as in figure (5.17), so that the resulting loaded linkages fall within the ambit of the

algorithm. Statically balancing these resulting linkages would balance the original

linkage for the same reasons that are presented in Section 5.4.3. In this way, the

algorithm of Section 6.6 can be used to balance any revolute and/or spherical jointed

linkage with any kind of constant load and zero-free-length spring load interactions

between the bodies.

6.8 A note

Interference is an important issue in a practical implementation of the method pre-

sented in this chapter. In the planar case, the elements that can potentially interfere

can be placed in different parallel planes. Such a leeway is generally not available in

the spatial case. Nevertheless, balancing solutions from the method are not unique.

This non-uniqueness could possibly be exploited to see that the expected motion of

the linkage being balanced falls within the workspace that is devoid of interference.


Summary

• For a set of potential energies of a body that includes constant loads and zero-

free-length springs loads, we derived the conditions under which the sum of

the potential energies can be transformed as potential energies of another body

through revolute joint or spherical joint.

• The conditions for the revolute joint is less stringent in comparison to the spher-

ical joint, reflecting the difference between the joints.

• Recursive application of the transformation of potential energies from one body

to the other leads to static balance.

• Interference could be an issue in a practical implementation of the method

presented in this chapter.

Chapter 7

Towards static balance of

compliant mechanisms

Overview

• Proposal of a framework for analytical, albeit approximate, static balancing

of a flexure-based compliant mechanism that makes use of the analytical static

balancing techniques for spring-loaded rigid-body linkages developed in previous

chapters.

• Static balancing of a flexure-beam as a prelude to understanding the framework.

• Three examples based on the framework with encouraging results.

• A prototype to demonstrate static balancing of a flexure-based compliant mech-

anism.

• A discussion on the limitations of the framework.

118

CHAPTER 7. STATIC BALANCE OF COMPLIANT MECHANISMS 119

7.1 Balancing a flexure beam

7.1.1 The flexure beam

The details of the flexure beam are shown in figure (7.1). It has two portions: a rigid

portion and a flexure. The dimensions of the flexure, geometrically a cuboid, are as

shown in the figure. The flexure beam is considered to have linear isotropic elasticity.

The Young’s Modulus and the Poisson’s ratio are also given in the figure.

l

L−

l

2

h

Material Properties:

Young’s modulus: 205e9 Pa

Poisson’ ratio : 0.28

Dimension of the flexure beam

Rigid

portion

Flexure

b = 20mm

h = 0.5mm

l = 30mm

L = 200mm

hb

Flexure cross-section

Figure 7.1: Details of the flexure beam

The continuous set of configurations over which static balance is sought

The flexure, being an elastic body, can have arbitrary deformation. Figure (7.2) shows

different types of loading and deformation. Among these, the rigid body model of the

flexure beam can accurately model the statics and kinematics of the situation in figure


(7.2a). Here, the interest is to know what horizontal force at point P is necessary for

a range of horizontal displacements of point P . Our static balancing will focus on

considerably reducing this horizontal force over the range of horizontal displacements.

If this force is reduced to zero, it would become perfect static balancing but, as we

will see, it is not the case.

7.1.2 Rigid-body model for the flexure beam

When the flexure beam model [7] is applied to the flexure beam of figure (7.1) a

torsional spring-loaded lever results, as shown in figure (7.3). In this model, the

flexure is replaced by a revolute joint with a linear torsional spring at the joint. The

joint is usually placed at the centre of the flexure’s undeformed configuration. Further,

in the undeformed configuration, the moment exerted by the torsional spring is zero.

To obtain the spring constant, the flexure is modelled as an Euler-Bernoulli beam

(see [79]). An Euler-Bernoulli beam under pure moment will have a constant ratio

between the moment and the relative change in the tangent-angle of the end-points

of the beam. This ratio is evaluated for the flexure and it is taken as the torsional

spring constant. If E is the Young’s modulus of the flexure, I is the area moment of

inertia of the cross-section of the flexure and l is the length of the flexure, then the

ratio is given by

kt =EI

l(7.1)

The numerical values of the relevant quantities are tabulated in table 7.1.

Table 7.1: Relevant quantities to calculate the torsional stiffness of the spring

Numerical value for torsional spring

E Young’s modulus 205× 109[Pa]h Height of the flexure cross-sections 500× 10−6[m]b Width of the flexure cross-sections 2× 10−2[m]

I Area moment of inertia = bh3

122.083× 10−13[m4]

l Length of the flexures 3× 10−2[m]kt Torsional spring constant = EI

l1.424[N −m/rad]

L Lever arm length 0.2[m]


The flexure beamunder bending

P

P

F

F

The flexure beamunder axial tension

The flexure beamunder a loading

(a)

(b) (c)

Figure 7.2: Our attention is on reducing horizontal force for a range of horizontaldisplacements of point P


kt =EI

l

Case 1a Case 2a

Figure 7.3: Small-length-flexure model applied to the flexure beam

7.1.3 Approximation of torsional spring by zero-free-length

spring

We now approximate torsional springs by zero-free-length springs. This approxima-

tion is necessary since we know how to analytically static balance a linkage loaded by

zero-free-length springs but not a linkage with torsional springs.

Our eventual goal is to reduce horizontal force Fx over a range of horizontal dis-

placements ux of point P . So, a zero-free-length spring, shown in case 3a of figure (7.4)

can be considered to approximate the torsional spring of case 2a if the replacement of

the torsional spring by the zero-free-length spring does not significantly perturb Fx

vs. ux relation.

Case (2a) in figure (7.4) shows the lever with the torsional spring. This has to be

approximated by the lever loaded by a zero-free-length spring as shown in case 3a of

the same figure. The task now is to find the parameters of the zero-free-length spring

(spring constant and anchor points) so that Fx vs. ux curve for the two cases match

as closely as possible. Towards that, we find the zero-free-length spring parameters


so that Fx and dFx

duxat ux = 0 match. With this, the two Fx vs. ux curves have the

same Taylor’s expansion upto the first order and we expect them to match atleast in

a range around ux = 0. The details of this are presented next.

Case 2a

PP

x

y

θ

x

y

θ

u

Positive rotational direction: anticlockwise

F

kt

k1 = ?

Case 3a

Figure 7.4: Approximation of the torsional spring by a zero-free-length spring

Fx vs. ux in the torsional spring-loaded lever (case 2a)

There may be ways to find Fx vs. ux relation that are simple in comparison to what

is presented next. However, for the sake of uniformity across the examples, we use

the principle of virtual work to get a balance equation form which Fx vs. ux relation

can be deduced. Let θ and δθ be the variables that parameterize the configuration

and its virtual change. The quantities related to the calculation of the virtual work

are as in table (7.2).

For static equilibrium at configuration θ, the net virtual work should be zero, i.e.,

−δθ (ktθ + FxL cos θ) = 0 (7.2)


Table 7.2: Virtual work calculation for lever in case (2a)

Sl. no. description of quantities at the pivot at point P

1 generalized displacement θ u =

[−L sin θL cos θ

]

2 generalized force −ktθ F =

[Fx

0

]

3 generalized virtual displacement δθ −δθL

[cos θsin θ

]

4 virtual work −δθktθ −δθFxL cos θ

Equation (7.2) can be used to get the following expression for Fx as

Fx = −ktθ

L cos θ(7.3)

In table 7.2 we had already written ux as a function of θ, i.e.,

ux = −L sin θ (7.4)

Equations (7.3 and 7.4) give relation between Fx and ux in the parametric form. The

derivative or the slope of Fx vs. ux is given by

dFx

dux=

dFx

dθduxdθ

=

−ktL

(θ sin θ

cos2 θ+

1

cos θ

)

−L cos θ(7.5)

The reference configuration (corresponding to the undeformed configuration of the

flexure beam) corresponds to θ = 0. At θ = 0, Fx and dFx

duxare given as

Fx = 0 anddFx

dux=ktL2

(7.6)

Fx vs. ux in the lever loaded by zero-free-length spring (case 3a)

The lever loaded by zero-free-length spring is shown as case (3a) in figure (7.4). One

end of the spring is anchored at P and in the reference configuration (θ = 0) the


spring is undeformed. The various quantities for calculating virtual work balance are

tabulated.

Table 7.3: Virtual work calculation and slope of Fx vs. ux

Sl. no. description of quantities at P (spring force) at P (force Fx)

1 generalized displacement u =


]

u =


]

2 generalized force k1L

[sin θ

(1− cos θ)

]

F =

[Fx

0

]

3 generalized virtual displacement −δθL

[cos θsin θ

]

−δθL

[cos θsin θ

]

4 virtual work −δθk1L2 sin θ −δθFxL cos θ

Virtual work balance:

−δθ(k1L2 sin θ + FxL cos θ) = 0 (7.7)

Fx and ux in parametric form

Fx = −k1L sin θ

cos θ(7.8)

ux = −L sin θ (7.9)

Slope of Fx vs. ux

dFx

dux=

dFx

dθduxdθ

=−k1L

(sin2 θcos2 θ

+ 1)

−L cos θ(7.10)

Fx and slope of Fx vs. ux at θ = 0

Fx = 0, anddFx

dux= k1 (7.11)


The parameters of the zero-free-length spring in case 3a

The anchor points of the zero-free-length spring (see figure (7.4)) were chosen such

that in the reference configuration (θ = 0), Fx is zero. Thus, at θ = 0, Fx for case 2a

and case 3a trivially match (see equations (7.6) and (7.11)). The remaining unknown

parameter of the zero-free-length spring is its spring constant k1. This is found by

equatingdFx

duxat θ = 0 for the two cases. From equations (7.6) and (7.11), matching

of the slopes lead to the following expression for k1.

k1 =ktL2

(7.12)

Using the numerical values for kt and L, given in table 7.1, the numerical value for

k1 is 35.600 [N/m]. With this value of k1, we plot Fx vs. ux relation for both case 2a

and case 3a (from equations (7.3), (7.8) and (7.4)) for a range of 5cm displacement

of point P around its reference position, as shown in figure (7.5). The match is not

perfect since there are deviations, which grow towards either end. Nevertheless, for

the range of ux plotted, one can say, by examining the plot, that the deviations are

less than 5%.

7.1.4 Static balancing by addition of a zero-free-length spring

Case 3a of figure (7.4) is the simplest of revolute-jointed linkages and it has only

zero-free-length load. Hence, case 3a in figure (7.4) is apt for application of static

balancing method of Chapter 5. Case 3a of figure (7.4) is a lever. Hence, the loads

on it have to satisfy equations (5.10–5.11) for it to be in static balance. Towards

that, we add a zero-free-length spring of stiffness k2 = k1, as shown in figure (7.6).

It can be verified through table 7.4 that the original zero-free-length spring and the

additional zero-free-length spring together satisfy equations (5.10 – 5.11).

As a cross-check, one may also do virtual work analysis, similar to the way it is in

table 7.3. The virtual work calculation table is the same as that of table 7.3 except

for an additional column corresponding to the balancing spring, as shown in table

7.5.


�0.06 �0.04 �0.02 0.00 0.02 0.04 0.06Horizonatal displacement of point P

�1.5

�1.0

�0.5

0.0

0.5

1.0

1.5

Horiz

onat

l for

ce a

t P (e

ffort

)

Force vs. deflection at the reference point

case 2acase 3a

Figure 7.5: The approximate match in Fx vs. ux relation between case 2a and case3a


Case (3b)

k1

k2 = k1

L

L

x0

y0x1

y1

Balancing spring

Figure 7.6: Static balance of the approximated zero-free-length spring-loaded leverby addition of a zero-free-length spring


Table 7.4: Verification of equations (5.10–5.11) being satisfied

ki ai bi r ki(r − bi)Tai ki(r − bi)

TR(π2)ai

1stspring k1

[L0

] [0L

] [00

]

0 −k1L2

2ndspring k1

[L0

] [0−L

]

0 k1L2

Summation over two springs 0 0

Table 7.5: Verification of static balance through virtual work calculations

no. quantities Original spring balancing spring force Fx

1 displacement u =


]

u =


]

u =


]

2 force k1L

[sin θ

(1− cos θ)

]

k2L

[sin θ

(−1− cos θ)

]

F =

[Fx

0

]

3 virtual displ. −δθL

[cos θsin θ

]

−δθL

[cos θsin θ

]

−δθL

[cos θsin θ

]

4 virtual work −δθk1L2 sin θ +δθk2L

2 sin θ −δθFxL cos θ

Along the same lines as Section 7.1.3, we obtain

Virtual work balance:

−δθ((k1 − k2)L2 sin θ + FxL cos θ) = 0 (7.13)

Fx and its slope at θ = 0

Fx = −(k1 − k2)L sin θ

cos θ(7.14)

dFx

dux= k1 − k2, at θ = 0 (7.15)

From equation (7.14), it may be noted that when k1 = k2, the effort is zero for

any configuration.


7.1.5 Balancing springs on the flexure beam

Case 3a approximates case 2a (see figure (7.4) and case 2a in turn approximates case

1a (see 7.3). Further, a perfectly statically balanced case 3b was obtained by adding

a balancing spring (see figure (7.6)). Case 1b, shown in figure (7.7), obtained by

adding the same balancing spring to the flexure of case 1a is expected to be at least

approximately statically balanced because of the approximation between case 1a and

case 3a. Figure (7.7), by the way, is a consolidation of all the cases.

To check the efficacy of balancing in case 1b, we performed finite element analysis

of the flexure beam using commercially available software COMSOL (www.comsol.com).

The flexure beam was modelled as a plane stress problem with geometric nonlinearity.

Point P was given a horizontal displacement constraint of ux (see figure (7.2)). Fx

was obtained as the reaction force of this constraint. By varying ux, Fx for a range

of values of ux was obtained. This analysis was carried out both with and without

balancing spring force. This data is plotted in figure (7.8).

It is important to note that prior to this, all the calculations to obtain the ap-

proximately balancing spring are analytical. The numerical simulation has been used

only to judge the effectiveness of this approximately balancing spring. In figure (7.8),

while the curve for f1b is not the ideal zero, it has reduced to less than 20% when

compared with unbalanced case 1a.

A way to improve the balance

Having carried out the numerical analysis to judge the effectiveness of the approximate

balance, we are in a position in to find deviation from perfect balance. In particular

we can know what deviations are there for Fx and its slope at the origin. Let us

call these quantities as c and m (these symbols are intended to allude to standard

equation of a straight line: y = mx + c). From the plot, there is no deviation in c

while the deviation in m is 5.4 [N/m]. Further, suppose that we can make changes to

parameters of the balancing spring, say to its stiffness k2. If we can find the sensitivity

of m with respect to k2, then we can make a first order correction to k2. While one

can find this sensitivity from finite element analysis, as a matter curiosity, we have


(1a) (2a) (3a)

(1b) (2b) (3b)

P P P

PPP

Figure 7.7: All the cases related to the flexure and its approximation by the spring-loaded lever


�0.06 �0.04 �0.02 0.00 0.02 0.04 0.06ux (m)

�3

�2

�1

0

1

2

3

Fx (N

)

f1a

f1b

f1c

Fx vs. ux at point P

Figure 7.8: Fx vs. ux relation obtained from finite element analysis of the flexurebeam


used the sensitivity from the rigid-body model, in particular, from equation (7.15).

From the equation, it is evident that the sensitivity is −1. The first order correction

∆k2 should satisfy:

mdesired −mcurrent = (sensitivity of m)×∆k2

0− 5.4 = −1 ∗∆k2

Hence, the correction ∆k2 is 5.4 [N/m] and Fx vs. ux curve for the flexure with the

corrected k2 is shown as f1c in figure (7.8). Without the correction, Fx of case 1b had

decreased to less than 20% of case 1a whereas with the correction Fx has decreased

to less than 2% of case 1a. We end this section with a consolidated fx vs. ux plot for

all cases shown in figure (7.9).

Intermediate nature of cases 2a and 2b

The intermediate nature of cases 2a and 2b is worth noting. One can compose small-

length flexure model and the first order Taylor’s approximation into a single ap-

proximation model to avoid the intermediate cases 2a and 2b. However, by using

small-length flexure model without any modification or composition, we are only

acknowledging the motivational role played by small-length flexure model, which is

already documented in literature. Alternatively, one can use numerical methods such

as Finite Element Methods to directly establish approximations between cases 1a and

3a and between cases 3b and 1b. However, the use of numerical methods defeats our

intent to obtain a simple analytical static-balancing framework that is described next.

7.2 Framework

Based on the example of balancing the flexure beam, we now formally propose a

framework to use the static balancing techniques developed for rigid-body linkages

loaded with zero-free-length springs for static balancing of flexure-based compliant

mechanisms.


�0.06 �0.04 �0.02 0.00 0.02 0.04 0.06ux (m)

�3

�2

�1

0

1

2

3

Fx (N

)

1a

2a, 3a

1b2b, 3b, 1c

Fx vs. ux at point P

case 1acase 2acase 3acase 1bcase 2bcase 3bcase 1c (corrected)

Figure 7.9: A consolidated plot of Fx vs. ux for flexure beam and its rigid-bodymodels


Step 1: Apply small-length-flexure model to the given flexure-based compliant mecha-

nism to obtain a revolute-jointed rigid-body linkage loaded by torsional springs.

The given flexure-based compliant mechanism may be labelled as case 1a and

the linkage loaded by torsional springs as case 2a.

Step 2: Identification of an effort function. Identify a function whose closeness

to zero implies closeness of the system to static balance. The definition of the

function should preferably be such that it is convenient to compute it both in

flexure-based compliant mechanisms and rigid-link mechanisms.

Step 3: Torsional springs to zero-free-length springs. In this step, a new case,

labelled case 3a is formed where the torsional springs are substituted by zero-

free-length springs. The parameters of the zero-free-length springs should be

such that the effort function in case 2a and case 3a should have a close match.

In this thesis, to obtain the match, we ensure that there is a perfect match in

terms of the value of the function and the slope of the function at a reference

configuration. It is known that if two functions have the same value and slope

at a configuration then in a “small” range around the configuration, the two

functions “match closely”. This is the consequence of the match in the Taylor’s

expansion of the two functions upto the first order.

Step 4: Static balancing of the linkage with zero-free-length springs. Statically

balance the spring-loaded linkage of case 3a to obtain case 3b. In this thesis,

we propose the use of static balancing techniques of Chapter 5, where there

is no addition of auxiliary bodies. The previous step was necessary since, till

now, there are analytical static balancing principles only for rigid-body linkages

under zero-free-length spring loads and not under torsional spring loads.

Step 5: Whatever balancing zero-free-length springs were added in case 3b, add the

same springs to the compliant mechanism of case 1a to obtain case 1b. The

assumption here is that with cases 1a, 2a and 3a being approximately close

to each other, whatever additional load balances one case would balance other

cases also. Note that there are instances where this assumption may break


down and so does the whole framework. Breakdown of this step is discussed in

Section 7.6.

A way to improve the static balance

In case 1b, the effort function and its slope are ideally expected to be zero. However,

due to approximations, generally there would be deviations. The deviations could be

found either by finite element analysis or by practical measurements. Based on the

sensitivity of the function value and its slope with respect to the balancing spring

parameters, one can do a first order correction to balancing spring parameters. This

generally brings effort function closer to zero. While the sensitivity should ideally be

evaluated for case 1b using finite element methods, in this thesis, out of curiosity, we

analytically obtain sensitivity based on case 3b. We label this corrected version of

case 1b as case 1c.

We now illustrate this framework on three more examples where the results are

encouraging.

7.3 Flexure-based compliant four-bar mechanism

7.3.1 Description of the mechanism

Figure (7.10) shows a flexure-based four-bar linkage. In the undeformed configuration,

all the flexures are vertical. All the flexures have the same geometrical dimensions

as well as elastic properties. Each flexure is a cuboid. The in-plane dimensions are

0.5 mm × 10 mm and the out-of-plane dimension is 40 mm. In the plane of motion

and in the undeformed configuration, the quadrilateral formed by the centers of the

flexures is a quadrilateral of sides a, b, c, and d with a diagonal of length l0, as shown

in figure (7.11). In the flexure at the vertex between sides c and d (see figure (7.11)),

the bottom-most point is labeled as P (see figure (7.10)). From the figure, it may

also be noted that P is at the centre of the interface between the flexure and the rigid

portion. This point is later used to define the effort function.


Case 1a

P

Flexures

Flexures

Undeformed

configuration

Figure 7.10: A flexure-based four-bar linkage.

a = 150mm

b = 100mm

c = 130mm

d = 120mml0

l0 = 140mm

(drawing not to the scale)

Figure 7.11: The quadrilateral formed by the centers of the flexures


7.3.2 Step 1

Application of small-length flexure model leads to torsional spring-loaded four-bar

linkage, as shown in figure (7.12). All the quantities required to calculate the spring

constant of the torsional spring are shown in the table of the same figure.

Case 2a

kt

kt

ktkt

P

Case 1a

P

E Young’s modulus 205e9[Pa]h Height of the flexure cross-sections 500e-6[m]b Width of the flexure cross-sections 4e-2[m]

I Area moment of inetria = bh3

124.167e-13[m4]

l Length of the flexures 1e-2[m]kt Torsional spring constant 8.5417[N-m/rad]

a

b

c

d

Figure 7.12: Approximation of the flexure-based four-bar linkage as a rigid-bodyfour-bar linkage with torsional springs.

7.3.3 Step 2 – The effort function

When point P is constrained to have a horizontal displacement of ux, there is a

corresponding horizontal constraint force Fx. The function Fx vs. ux over a range

[−0.025m,+0.025m] of ux is taken as the effort function.


7.3.4 Step 3

We next propose to approximate the torsional springs of case 2a by two zero-free-

length springs as in figure (7.13). We further propose to fix the anchor points of the

springs at the coordinates shown in figure (7.13). We find the remaining parameters,

i.e., the spring constants, so that there is a close match in the effort function between

case 2a and case 3a. Towards that, at the reference configuration, the effort function

(Fx at ux = 0) and its slope (dFx/dux at ux = 0) for the two cases are analytically

found and tabulated in table 7.6. The details of this analytical calculation, which is

again based on virtual work balance as in Section 7.1, are presented as WXMAXIMA

(www.sourceforge.net) worksheet in Appendix F.1. To achieve a close match in effort

function between case 2a and case 3a, we equate Fx at ux = 0 and dFx/dux at ux = 0

between the two cases and solve the stiffness from the equations. The stiffness values

so obtained are shown in table 7.6.

Case 2a Case 3a

kt

kt kt

kt

Z2

1

Z1

1

P P

x0

y0

x1

y1

x2

y2

Spring a b klocal global stiffness

m m N/m

Z2

1

[

0.013−

13

300

] [

−0.050.06

]

k21

Z1

1

[

0.080

] [

0.25−0.05

]

k11

Figure 7.13: Approximation of torsional springs by zero-free-length springs


Table 7.6: The origin and the slope of Fx vs. ux being matched between case 2a andcase 3a

at reference configuration case 2a case 3a

Fx 0 0.146k21 − 0.246k11dFx

dux5140.247 2.583k21 − 1.379k11

Matching Fx anddFx

duxleads to

k21 = 2915.88 and k11 = 1735.43

7.3.5 Step 4 – Static balance of the linkage under zero-free-

length spring load

The four-bar linkage with the zero-free-length spring is now statically balanced using

the principles of Chapter 5. As shown in figure (7.14), the unloaded body on the

right side is disregarded. This leads to a 2R linkage with springs Z11 and Z2

1 . Based

on the static balancing algorithm of Chapter 5, two more zero-free-length springs Z22

and Z12 are added to perfectly balance the linkage. The additional springs are added

in two iterations to satisfy equations (5.19 - 5.21). Details of the quantities required

to verify these equations, including spring details, are presented in table 7.7. This

balancing is also cross-checked using virtual work calculations in Appendix F.2.

Table 7.7: Verification of equations (5.19 - 5.21) for the spring-loaded 2R linkage offigure (7.14)

u v w q κ

−kb ka −b ka k

2

Φ2

1(Z2

1) −2915.88

[

−0.050.06

]

2915.88

[

0.013

−13

300

]

−

[

−0.050.06

]

2915.88

[

0.013

−13

300

]

2915.88

2

Φ2

1(Z2

2) −2915.88

[

−0.050.06

]

2915.88

[

−0.01313

300

]

−

[

−0.050.06

]

2915.88

[

−0.01313

300

]

2915.88

2

∑

u = −2915.88

[

−0.100.12

]

s12

=

[

10e − 20

]

∑

κ = 2915.88

Φ1

1(Z2

1,Z2

2) −2915.88

[

−0.100.12

]

2915.88

[

20e − 20

]

−2915.88

[

−0.100.12

] [

10e − 20

]

2915.88

Φ1

2(Z1

1) −1735.43

[

0.25−0.05

]

1735.43

[

8e − 20

]

−

[

0.25−0.05

]

1735.43

[

8e − 20

]

1735.43

2

Φ1

3(Z1

2) −7567.19

[

−0.0092−0.0463

]

7567.19

[

8e − 20

]

−

[

−0.0092−0.0463

]

7567.19

[

8e − 20

]

7567.19

2


Case 3b

0

1

2

x0

y0

x1

y1

Z2

1

Z1

1

x2

y2

Z2

2

Z1

2

Spring a b klocal global

m m N/m

Z2

1

[

0.013−

13

300

] [

−0.050.06

]

2915.88

Z2

2

[

−0.01313

300

] [

−0.050.06

]

2915.88

Z1

1

[

0.080

] [

0.25−0.05

]

1735.43

Z1

2

[

0.080

] [

−0.0092−0.0463

]

7567.19

Figure 7.14: Static balancing by addition of zero-free-length springs


7.3.6 Step 5 – Approximate static balance of flexure-based

four-bar linkage

Because of sucessive approximations from case 1a to case 2a and case 2a to case 3a, we

expect that whatever balances case 3a balances approximately case 1a too. Based on

this, the balancing springs Z22 and Z2

2 are added on to flexure-based four-bar linkage

as shown in figure (7.15–1b). Fx vs. ux plot for case 1a and case 1b, using finite

element simulation, is given in figure (7.16). From the plot, it may be seen that Fx

in case (1b) has reduced to less that 20% of case 1a.

(1a) (2a) (3a)

(1b) (2b) (3b)

P P P

Z2

1

Z1

1

Z2

1

Z1

1

Z2

2

Z1

2Z1

2

Z2

2Z2

2

Z1

2

Figure 7.15: A consolidated figure of all the cases

7.3.7 A way to improve the static balance of the flexure-

based four-bar linkage

However, from the plot of figure (7.16), for case 1b, at ux = 0 there is a deviation

for Fx and dFx/dux from their expected zero values. At ux = 0, let us call Fx and


-100

-50

0

50

100

150

200

-0.02 -0.01 0 0.01 0.02

Fx

[N]

ux [m]

1a1b

Figure 7.16: Finite element simulation results for case 1a and case 1b


dFx/dux as c and m and the deviations as δc and δm. We can make a first order

correction to these deviations by changing k22 and k12 as below:

[

−∆c

−∆m

]

=

∂c

∂k22

∂c

∂k12∂m

∂k22

∂m

∂k12

[

∆k22

∆k12

]

(7.16)

While the partial derivative in the matrix in equation (7.16) should ideally be found

by finite element method based sensitivity analysis, we analytically find them based

on the approximate rigid-body model of case 3b. The details of this calculation

are in Appendix F.3, where it turns out that the corrections are ∆k22 = 309.40 and

∆k12 = 510.51. One more finite element analysis of the flexure-based four-bar linkage

is carried out using the corrected k22 and k12 and Fx vs. ux relation turns out be as in

figure (7.17).

-100

-50

0

50

100

150

200

-0.02 -0.01 0 0.01 0.02

Fx

[N]

ux [m]

Case 1aCase 1b

Corrected

Figure 7.17: Fx vs. ux after first order correction to the stiffness of balancing springs


We end this example with figure (7.18), which is a consolidated plot of all the

cases.

-100

0

100

200

300

400

-0.02 -0.01 0 0.01 0.02

Fx

[N]

ux [m]

1a1b

1c (corrected)2a2b3a3b

Figure 7.18: FX vs. ux plot for all the cases shown in figure (7.15)

7.4 Another flexure-based four-bar linkage

7.4.1 Description of the mechanism

This compliant mechanism is the same as in Section 7.3 except that the flexures are

longer as could be seen in case 1a of figure (7.19). Point P is also defined as in Section

7.3. Its location however is different because of the longer flexure.


R R R

R R R

(1a) (2a) (3a)

(1b) (2b) (3b)

1

x1

y1

x0

y0

Z1

2

Z1

1

Figure 7.19: Consolidated figure containing all the cases


Table 7.8: Details of the flexure and calculation of torsional spring constant

E Young’s modulus 205e9[Pa]h Height of the flexure cross-sections 500e-6[m]b Width of the flexure cross-sections 4e-2[m]

I Area moment of inetria = bh3

124.167e-13[m4]

l Length of the flexures 3e-2[m]kt Torsional spring constant 2.847[Nm/rad]

7.4.2 Step 1

Small-length-flexure model is applied on case 1a to obtain a torsional spring-loaded

rigid-body four-bar linkage (case 2a) as in figure (7.19). The details of the flexures

and the torsional spring constants are given table 7.8.

7.4.3 Step 2

The effort function is again Fx vs. ux where Fx is the horizontal reaction force

corresponding to a horizontal displacement constraint of ux on point P .

7.4.4 Step 3

We propose to approximate the torsional springs of case 2a by a single zero-free-length

spring as shown in case 3a of figure (7.19). Here again, the anchor point of the spring

is fixed and we assign such a spring constant so that there is a close match between

effort functions in case 2a and case 3a. The local coordinates of the anchor point on

body 1 (see case 3a in figure (7.19)) is [0.0632, 0.0301]Tm and the global coordinates

of the anchor point on the ground is [0, 0.07]Tm.

Table 7.9 shows, at ux = 0, Fx and dFx/dux. Fx is zero in both cases. dFx/dux

is a linear multiple of the stiffness in case 3a. By equating dFx/dux between the two

cases, we get the stiffness as shown in the same table. With this stiffness, the extent

of match between case 2a and case 3a could be seen in figure (7.20). The contents of

the table were arrived at analytically (using WXMAXIMA (www.sourceforge.net)) in

a manner similar to Section 7.3.


Table 7.9: Matching value and slopes at the origin of the effort function

at ux = 0 case 2a case 3a

Fx 0 0dFx/dux 2061.10 0.9127kMatching the slope leads to

k = 2258.108

0

50

100

150

-0.02 -0.01 0 0.01 0.02

gene

raliz

ed fo

rce

[N]

generalized displacment [m]

1a1b1c2a2b3a3b

Figure 7.20: Effort function in all the cases


Table 7.10: Details of the original spring and balancing spring

Spring a b klocal global stiffnessm m N/m

Z11

[0.06320.0301

] [0

0.07

]

2258.108

Z12

[0.06320.0301

]

−

[0

0.07

]

2258.108

7.4.5 Step 4

For the purpose of static balancing in case 3a, one only needs to consider the body

labelled as 1 (in figure (7.19)) along with the ground. With such a consideration,

this becomes a spring-loaded lever, similar to case 3a in Section 7.1. We balance the

spring Z11 with additional spring Z1

2 as shown in case 3b of figure (7.19). The details

of each of the springs is given in table 7.10. From the table, it may be verified that

the spring parameters satisfy the lever-balancing equations (5.10 – 5.11).

7.4.6 Step 5

The balancing spring of case 3b is incorporated into case 1a to obtain case 1b, as

shown in figure (7.19). The extent of balance in case 1b can be judged through the

plots in figure (7.20). Just by visual judgement, decrease in Fx to less than 30% of

case 1a can be noted.

7.4.7 First order correction

From the plot for case 1b in figure (7.20) it may be noted that at the origin, i.e.,

ux = 0, the slope of the plot deviates from the ideal zero-slope. The deviation

(calculated using finite element analysis data) is shown table 7.11. Using sensitivity

of the slope with respect to stiffness of the balancing springs (calculated from case 3a),

the correction is found in the same table. The effort function after this correction is

labelled as 1c in figure (7.20) and it may be noted that the effort function has indeed

become closer to zero.


Table 7.11: First order correction of balancing spring parameters

Slope error Slope sensitivity w.r.t. k12 ∆k12N/m N/m

277.53 -0.913 304.055

7.4.8 Prototype

Based on this example, a prototype was made to demonstrate the reduction in effort of

a flexure-based compliant four-bar mechanism, as shown in figure (7.21). The image

in the figure is taken from the backside. Hence correlation between this and case

1b of figure (7.19) has to be done through a flipping transformation. The prototype

deviates from the example in one important respect. It uses normal springs having

finite free-length as balancing springs. In fact four balancing springs are used, which

are mostly in parallel, except for a little offset between anchor points. The balancing

springs were chosen such that the net force of all of them when extended between

anchor points is approximately same as what is exerted by the balancing spring of

case 1c of the example of this section. With further tuning in terms of varying the

number of active coils in springs, we could demonstrate a perceptible difference in the

effort to deflect the prototype mechanism. Almost all people who tried their hand on

this found the difference to be significant. Force measurements showed a reduction

in the effort to less than 25% over a range of 2 cm on either side of the reference

configuration.

What we want to suggest through this prototype is that the theoretical solution

serves as a guideline and one can deviate from the theoretical solution (in this case,

deviation was in free-length) with some compromise on static balance. This compro-

mise may be small enough to be acceptable in many practical situations.


Figure 7.21: A prototype to demonstrate reduction in effort


7.5 Flexure-based 2R compliant mechanism

7.5.1 Description of the compliant mechanism

The flexure-based compliant mechanism shown in case 1a of figure (7.22) is so con-

ceived that when small-length-flexure approximation is applied to it, it becomes a

2R linkage, as shown in case 2a of the same figure. Case 1a of figure (7.22) shows

points P , F1 and F2. F1 is the center of the first flexure and F2 is the center of the

second flexure. P is a point on the rigid portion that comes after the second flexure.

In the undeformed configuration, F1P is horizontal, F1P = 8 cm, PF2 = 6 cm, and

F1F2 = 10 cm.

(1a)

(1b)

(2a)

(2b)

(3a)

(3b)

P P

Z11

Z21F1

F2

Z22

Z12

Figure 7.22: A consolidation of all the cases

7.5.2 Step 1:

The small-length-flexure model is applied to case 1a to obtain torsional spring-loaded

rigid linkage as shown in figure (7.22–2a). The details of the flexures and the calcu-

lation of torsional spring constants from it are given in table 7.12.


Table 7.12: Details of flexure and calculation of torsional spring constant.

E Young’s Modulus 205e9[Pa]h height of the flexure cross-section 500e-6[m]b width of flexure cross-section 4e-2[m]

I area moment of inertia= bh3

124.1666666666666664e-13[m4]

la length of first flexure 1e-2[m]lb length of second flexure 2e-2[m]kta first torsional spring constant=EI

la8.541666666666666 [Nm]

ktb second torsional spring constant=EIlb

4.270833333333333 [Nm]

7.5.3 Step 2 – Identification of effort function

The effort function is F = [Fx Fy]T vs. u = [ux uy]

T where Fx and Fy are

horizontal and vertical reaction forces when point P is constrained to have a hori-

zontal and vertical displacements of ux and uy. We focus our interest in the range of

[−0.015, 0.015]m for both ux and uy.

7.5.4 Step 3

We propose to approximate the torsional springs in case 2a by two zero-free-length

springs, Z21 and Z1

1 , as shown in case 3a of figure (7.22). The details of these springs,

along with the springs that would be added in step 4, are shown in figure (7.23).

Similar to Section 7.3, one can calculate F and its derivative ∇F with respect to u

at u = 0. Note that in case 3a, these quantities are functions of spring constants

that are yet to be determined. In order to have a close match in the effort function

between case 2a and case 3a, we find the stiffness of the spring constants so that at

u = 0, F and ∇F are the same for the two cases. The spring constants turn out be

as in table 7.13. With these springs constants, the extent of match between the effort

function of the two cases could be seen in figures (7.24 – 7.25).

7.5.5 Step 4

The zero-free-length spring-loaded 2R linkage of case 3a is statically balanced by

adding two more zero-free-length springs, Z22 and Z1

2 , as shown in figure (7.22–3b).


(3b)

x0

x1

x2

Z2

1

Z2

2

Z1

1Z1

2

spring a b klocal global stiffnessm m N/m

Z2

1

[

6e− 20

] [

8e− 20

]

1186.34

Z2

2

[

−6e− 20

] [

8e− 20

]

1186.34

coordinates of pivot of (1 to 2) revolute joint

s1

2=

[

10e− 20

]

Z1

1

[

8e− 20

] [

6.4e− 2−4.8e− 2

]

148.28

Z1

2

[

8e− 20

] [

−8e− 20.35e− 2

]

3151.22

Figure 7.23: Details of springs

Table 7.13: The value and the first derivative of F vs. u at u = 0

case 2a case 3a[N/m] [N/m]

F

[00

] [00

]

∇F

[1186.34 0

0 1334.63

] [k21 00 k21 + k11

]

k21 = 1186.34 and k11 = 148.29


Fx-3aFx-3bFx-2aFx-2bFx-1aFx-1b

-0.015 -0.01 -0.005 0 0.005 0.01 0.015ux [m]

-0.015-0.01

-0.005 0

0.005 0.01

0.015

uy [m]

-50-40-30-20-10

0 10 20 30

Fx [N]

Fx-3aFx-3bFx-2aFx-2bFx-1aFx-1b

-50

-40

-30

-20

-10

0

10

20

30

Fx [N]

Figure 7.24: Fx vs. u in two views


Fy-3aFy-3bFy-2aFy-2bFy-1aFy-1b

-0.015-0.01

-0.005 0 0.005

0.01 0.015

ux [m]

-0.015 -0.01 -0.005 0 0.005 0.01 0.015

uy [m]

-30-20-10

0 10 20 30

Fy [N]

Fy-3aFy-3bFy-2aFy-2bFy-1aFy-1b

-30

-20

-10

0

10

20

30

Figure 7.25: Fy vs. u in two views


Table 7.14: Verification of static balance of springs in case 3b

u v w q κ

−kb ka −b ka k2

Φ21(K2,1) −1186.34

[

8e − 20

]

1186.34

[

6e − 20

]

−

[

8e − 20

]

1186.34

[

6e − 20

]

1186.342

Φ21(K2,2) −1186.34

[

8e − 20

]

1186.34

[

−6e − 20

]

−

[

8e − 20

]

1186.34

[

−6e − 20

]

1186.342

∑

u = −1186.34

[

16e − 20

]

s12 =

[

10e − 20

]

∑

κ = 1186.34

Φ11(K2,1,K2,2) −1186.34

[

16e − 20

]

1186.34

[

20e − 20

]

−1186.34

[

16e − 20

] [

10e − 20

]

1186.34

Φ12(K1,1) −148.28

[

6.4e − 2−4.8e − 2

]

148.28

[

8e − 20

]

−

[

6.4e − 2−4.8e − 2

]

148.28

[

8e − 20

]

148.282

Φ13(K1,2) −3084.49

[

−8e − 20.35e − 2

]

3084.49

[

8e − 20

]

−

[

−8e − 20.23e − 2

]

3084.49

[

8e − 20

]

3084.492

The details of all the springs can be seen in figure (7.23). The static balance is done in

two iterations as per the balancing algorithm described in Section 5.4.2. To facilitate

verification of equations that are required to be satisfied by the algorithm of Section

5.4.2, the spring details are written in appropriate form in table 7.14.

7.5.6 Step 5

The balancing springs Z22 and Z1

2 present in case 3b are added on the compliant system

of case 1a to obtain case 1b as shown in figure (7.22). For the reasons discussed in

Section 7.2, we expect the effort function to be closer to zero in case 1b, in comparison

to case 1a. That it is indeed so can be seen in plots of figures (7.24 – 7.25) where

the plots for case 1a and 1b were obtained through a finite element analysis. In most

places, the effort has got reduced to less than 30% of that of case 1a.

7.6 Discussion

The proposed framework of Section 7.2 could fail if


1. Rigid-body model does not very well approximate the actual flexure-based com-

pliant mechanism.

2. The range of configuration changes (in the examples, the range of ux or u that

was in focus) is very large around the undeformed configuration.

The second point is understandable because the rigid-body model of flexure-based

compliant mechanism are themselves valid over a finite but limited range around the

undeformed configuration. Secondly, the approximation between case 2 and 3 is based

on considering the Taylor’s series terms upto first order.

As far as the first point is considered, the deficiencies are the same as that of

the small-length flexure model. The model gets more accurate when the length of

the flexure becomes smaller. Sometimes, it may so happen that for a given flexure

length, the approximation between case 1a and case 2a/3a is acceptable but approx-

imation between case 1b and case 2b/3b is unacceptable. This happens because the

additional load that the flexures experiences due to balancing spring loads could be

high enough to make the flexure deform in way that is deviant from ideal circular

arc like deformation. We noticed this in the example of Section 7.3 when the flexure

length was made longer. In the example, let the length of all the flexures be increased

three times. Inspite of this change, cases 2 and 3 remain the same except that the

spring constants of all the springs change to one third of their values. Figure (7.26)

shows the effort function for cases 1a, 3a, 3b and 1b. It may be noted that the plot

for case 1b is quite different from what was noticed in previous examples. The effort

function of case 1b in some places around the origin is not even less that that of

case 1a. The way we reason is that in case 1a, without any balancing springs, the

deformation of the flexures resembled ideal arc like deformation, as shown in figure

(7.27–a). However, with the balancing springs, the load on the flexure, especially

the one at point O in figure (7.27–b), was high enough to have a non-ideal non-arc

like deformation. Because of this non-ideal deformation, coupled with longer length

of the flexure, the kinematics and statics of the flexure-based compliant mechanism

deviated largely from the rigid-body model of case 3b.


0

50

100

150

-0.02 -0.01 0 0.01 0.02

Fx

[N]

ux [m]

Case 1aCase 1bCase 3aCase 3b

Figure 7.26: Plot of effort function when flexure length is increased three-fold

7.6.1 Static balancing of compliant mechanisms by individ-

ually balancing flexures

In flexure-based compliant systems, the elastic resistance to deformation arises due to

flexures. Hence, it is feasible to statically balance each of the flexures independently.

As an illustration, consider figure (7.28). The given compliant system is shown in

figure (7.28a) which is the same as that of the example in Section 7.5. In figure

(7.28c), the flexure between rigid portions A and B is statically balanced by adding

spring KA based on the principle that is the same as that used in figure (7.7), where

the flexure between the rigid-link and the ground is balanced by adding a spring.

Similarly, spring KB statically balances the flexure between the portion B and the

ground in figure (7.28c). The parameters of the spring KA are determined only by

the flexure between portions A and B and the spring reduces the effort only for the

relative motion between the rigid portions A and B. Similarly comments apply to

spring KB as well. This is what we mean when we say that each of the flexures are

balanced independently of each other. When each of the flexures in a flexure based

compliant system is balanced, the whole compliant system is balanced.

In the above scheme of balancing, the balancing springs were added only between


O

(a)

(b)

Figure 7.27: Ideal circular arc-like and non-ideal deformation of flexures


(a)

PF1

F2

(c)A

B

KA

KB

A

B

Figure 7.28: Static balancing of each of the flexures, independently of one another

successive rigid portions. However, such addition of springs only between successive

rigid links is infeasible for statically balancing gravity-loaded tree-structured rigid-

body linkages, as shown in appendix H.

7.6.2 Static balancing of compliant mechanisms using rigid-

body linkages

In this chapter, we developed a framework for approximately balancing compliant

mechanisms. The balancing elements used were springs, zero-free-length springs in

particular. Since, springs are also compliant elements, the eventual statically balanced

compliant mechanisms consist of only compliant elements. Note that it is not neces-

sary to have a pivot at points where springs anchor with the compliant mechanisms

since springs have negligible bending stiffness.

However, one can also statically balance compliant mechanisms using springs in

conjunction with rigid-body linkages, as shown in Appendix G. We have not empha-

sized this option since some of the advantages that compliant mechanisms have over

rigid-body linkages will be compromised in this option.


Summary

• A way to make use of analytical rigid-body static balancing methods for ap-

proximate analytical static balancing of flexure-based compliant mechanism was

demonstrated through a simple flexure-beam.

• A framework for approximate analytical static balancing of flexure-based com-

pliant mechanisms was proposed.

• Three more examples based on the frame work have shown encouraging static

balance.

• That there is scope for further improvement through simple first order correction

was demonstrated.

• A prototype based on one of the examples was made and it did show perceptible

reduction in operating effort. The prototype also demonstrated that one can

pragmatically deviate from the theoretical solutions with a little compromise

on the extent of static balance.

• The situations where the proposed framework might fail was also discussed.

Chapter 8

Conclusion

Overview

• A summary of the static balancing techniques of rigid-body linkages that this

thesis presented.

• A summary of the framework that this thesis gives for designing statically bal-

anced flexure-based compliant mechanisms.

• Comparison of the contributions of the thesis with the existing literature to

show that the contributions are novel.

Compliant mechanisms are superior to rigid-body mechanisms in terms of lack of

friction and backlash. However they have inherent elastic stiffness that could be un-

desirable in certain situations. Static balancing of the compliant mechanisms would

eliminate or reduce this drawback. Furthermore, because of the elastic deformation

of the body of the compliant mechanism, there is an inherent ‘diversion’ of input

energy into the mechanism. A statically balanced compliant mechanism, which is

suitably pre-loaded, does not suffer from this deficiency in the efficient transmission

of input energy to the output. This thesis proposed using pseudo-rigid-body models,

where a few classes of compliant mechanisms are modelled as spring-loaded rigid-body

linkages, for designing statically balanced compliant mechanisms. Towards that, the

163

CHAPTER 8. CONCLUSION 164

contributions of the thesis are two fold: 1) the development of new analytical tech-

niques for statically balancing spring-loaded rigid-body linkages and 2) a framework

to use the techniques through small-length flexure model for designing statically bal-

anced compliant mechanisms.

8.1 A summary of new static balancing techniques

for spring and/or gravity-loaded rigid-body link-

ages

8.1.1 Static balancing of a four-bar linkage loaded by a spring

on its coupler link

Chapter 3 presented three techniques to statically balance a zero-free-length spring-

loaded four-bar linkage. The first technique requires two auxiliary bodies and one

balancing spring. The second technique requires two balancing springs without using

any auxiliary bodies. The third technique requires two auxiliary bodies and two

balancing springs. The second technique was demonstrated through a prototype.

8.1.2 Static balancing parameters and the cognates of a four-

bar linkage

Chapter 4 presented a unified parameterization for the triplet of the family of static

balancing solutions of the three cognates where the anchor points of the balancing

springs always form a triangle that is similar to the cognate triangle and the ratio of

spring constants of a set of corresponding balancing springs is a constant. This result

came out of a graphical solution that the chapter gave for a problem in geometry.

While the result of this chapter cannot be stated as a new static balancing method,

it helps in exploring various design possibilities as stated in the chapter.


8.1.3 Static balancing without auxiliary bodies–planar case

Chapter 5, through systematic arguments based on making potential energy a con-

stant, gave an algorithm to statically balance any zero-free-length spring-loaded

and/or gravity-loaded revolute jointed rigid-body linkage. In the algorithm, the net

potential energy dependence on the rotational motion of the bodies making up the

linkage are recursively eliminated. The chapter gave several examples to illustrate

the algorithm.

8.1.4 Static balancing without auxiliary bodies–spatial case

Chapter 6 showed that the recursive static balancing algorithm of Chapter 5 also

works for spatial revolute and/or spherical jointed linkages under zero-free-length

spring and/or gravity loads. The chapter highlighted that revolute joints would re-

quire less stringent static balancing conditions. The chapter also gave an example to

illustrate the recursive algorithm in the spatial case.

8.2 A framework for designing statically balanced

compliant mechanisms

Chapter 7 provided a framework for using small-length-flexure model in conjunction

with the new static balancing methods of Chapter 5 for designing approximately

statically balanced compliant mechanisms. Flexure model of compliant mechanisms

gives rise to torsional spring-loaded rigid-body linkages whereas Chapter 5 deals with

static balance of zero-free-length (length extending) spring-loaded rigid-body link-

ages. Therefore, the framework proposed approximating torsional springs by zero-

free-length springs. Instead of using least squares approximation for this approxima-

tion, the framework proposed matching the value and the slope of the effort function

at a reference configuration as a way to approximate the two kinds of springs. This

allowed the framework to avoid numerical optimization. The chapter also presented


four examples along with a prototype of one of the examples to illustrate the frame-

work.

8.3 The novelty of the contribution in the context

of the current literature

The contributions of the thesis covers the following areas of static balance.

1. Rigid linkages under gravity load

2. Rigid linkages under spring load

3. Compliant mechanism under their inherent elastic forces

Figure (8.1), a recapitulation of figure (8.1), pictorially depicts the current literature

as well our contribution (in grey boxes) under these broad areas. Our contribution

for static balancing of gravity-loaded rigid linkages falls within a category where the

methods (1) are perfect (in contrast to approximate methods), (2) add springs for

balancing, and (3) places no restriction on the degree-of-freedom of the linkages.

While there are methods in literature ([23], [27], and [29]) under the category, these

methods use auxiliary links in contrast to our method. Currently, under this category,

our contribution presented in chapters 5 and 6 is the only work that adds only springs

but not auxiliary links. Hence this is a novel contribution.

The literature is relatively sparse for the static balancing techniques of rigid link-

ages under spring loads. Apart from our contribution in this area that is presented

in Chapters 3, 5 and 6, the works in this area are [1], [4] and [65]. While [1] and [4]

deal with balancing zero-free-length spring loads by adding zero-free-length springs,

[65] stands apart as the one that balances, although approximately, torsional spring

loads by adding torsional springs. [4] and [1] give ways to statically balance a lever

and a four-bar linkage under zero-free-length spring loads. In the contribution stated

in Chapter 3, using the concepts not different from the one presented in [4], we give

more methods for the problem addressed in [1] (also see figure (1.9)). As far as the

methods that deal with a n-degree-of-freedom linkage under zero-free-length spring


Rigid linkages

under

gravity load

By

addition

of mass

Rigid linkages

under

spring load

Compliant

mechanism

By

addition

of springs

Approximate

balancing

and

cam using

methods

Perfect

balancing

methods

Gopalswamy et al. (1992)

Agrawal & Agrawal (2005)

Ulrich & Kumar (1991)

Koser (2009)

For

specific

linkages

For

‘n’ - dof

linkages

LaCoste (1934)

Shin & Streit (1991)

Walsh et al. (1991)

Herder (2001)

Lin et al. (2010)

Using

auxiliary

bodies

Without

auxiliary

bodies

Streit & Shin (1993)

Agrawal & Fattah (2004)For planar revolutejointed and spatialrevolute and/orspherical jointedlinkages

Torsion

load

balanced

by

torsion

loads

Extension

spring

balanced

by

extension

springs

Approximate

methods

Perfect

methods

Radaelli et al. (2011)

For

specific

linkages

For

‘n’ -dof

linkages

Basic spring forcebalancer (Herder(2001))

Herder (1998)

Without

auxiliary

bodies

For planar revolutejointed and spatialrevolute and/orspherical jointedlinkages

Deepak & Ananthasuresh (2012)

Limiting case

By

addition

of springs

Other

strategies

Extension ofDeepak & Anan-thasuresh (2012)to compliantmechanisms usingpseudo-rigid bodymodel

Topology optimiza-tion (see De Lange etal. (2008))

Building block ap-proach (see Hoet-mer, et al. (2010))

Extension ofRadaelli et al.(2011) to compliantmechanisms as seenin Rosenberg et al(2011)

Original or novel contribu-

tions of this thesisdof: degree of freedom

Figure 8.1: The current literature and our contributions


loads, we haven’t found any literature. Hence, our contribution stated in Chapters 5

and 6 for a n-degree-of-freedom linkage is new.

A few reported works in the literature on static balance of compliant mechanisms

are listed in figure (8.1). As far as the ones that rely on spring-loaded rigid linkage

balancing techniques, using flexure model as the interface between rigid mechanisms

and compliant mechanisms, there is a work by Rosenberg et al. [73]. However,

they rely on numerical optimization and their method is not completely analytical.

This should signify that our contribution in Chapter 7 is distinct from the existing

literature. Furthermore, that rigid-link balancing techniques could be used for flexure-

based compliant mechanisms was first enunciated in our work [8] (reported in 2009).

8.4 Future work

We now briefly outline the prospects of continuing the work from the thesis.

• In this thesis, though the main algorithm developed in Chapter 5 was for tree-

structured linkages, we showed how to extend it to closed loop linkages. The

concept behind this extension was straightforward and the design obtained

was conservative. An investigation on overcoming this conservative design is

a prospect for future work. The investigation could be based on the work of

Shieh and Chen [41] which exploits closed loop equations.

• Addition of springs during static balancing leads to increase in internal stresses

of the material making up the linkage. This is true in our compliant mechanism

methodology as well. It is useful to investigate ways of reducing this internal

stress without sacrificing the static balance.

Summary

• A summary of previous chapters.

• A comparison with the existing literature to show the distinctness of the con-

tributions of this thesis from the existing literature.

Appendix A

Proofs on finding the focal pivot

A.1 If Ia,b and Ic,a circles are coincident, then the

given lines a, b and c has to be concurrent

In the following proof, a, b, c, Ia,b and Ic,a refer to their original position before

rotation by β. Figure (A.1) shows Ia,b and Ic,a circles that are coincident. Also shown

are lines a and b and their intersection point Ia,b. The point Ic,a lie on this common

circles, since the circles is its locus. By the definition of Ic,a as the intersection of c

and a, it should also line on line a. Hence Ic,a should lie on the intersection of the

circle with the line a.

Since Ic,a and Ia,b circles, by definition pass through A and line a also by definition

pass through A, A is always an intersection point between the line and the circle. The

following types of intersection between the circle and line a are possible:

1. When Ia,b is distinct from A, the line a intersects the circle at two distinct points

:A and Ia,b, as shown in figure (A.1).

2. When Ia,b coincides with A, the line a has to be tangent to the circle at A. (see

the theorem in appendix (B)).

In the first possibility, Ic,a has be either at Ia,b or at A. If it is at A, then line a

has to be tangent to the circle (see the theorem in appendix (B)), which contradicts

169

APPENDIX A. PROOFS ON FINDING THE FOCAL PIVOT 170

b

AC

Ba Ia,b

Coincident Ia,b and Ic,a circles

Before rotation (at β = 0)

Figure A.1: Ia,b and Ic,a circles are coincident

the earlier observation that line a intersects the circle at two distinct points. Hence

it Ic,a cannot be at A and by elimination, has to be at Ic,a. This implies that Ia,b and

Ic,a coincide, which further imply that a, b, and c are concurrent.

In the second possibility, since A is the only intersection between line a and locus of

Ic,a, Ic,a has be at A. Thus, here also both Ia,b and Ic,a coincide implying concurrence

of lines a, b and c.

Hence, if Ia,b and Ic,a circles are coincident, then the given lines a, b and c has to

be concurrent.

A.2 When M and A are distinct, M is the fo-

cal pivot. (Refer to section (4.2.4 and figure

(4.11).)

Let rotation angle β be such that point Ia,b is at A, so that by the theorem in appendix

B, line b aligns along the segment AB and line a becomes tangent to Ia,b circle. If

for the same β, point Ic,a is also at A, then by the theorem in appendix B, line a

becomes tangent to Ic,a circle. This implies that line a is a common tangent to both

Ia,b and Ic,a circles at the A, which is one of the two distinct points of intersection

of the two circles. This contradicts the fact that for two circles intersecting at two

distinct points, there cannot be common tangent at the point of intersection. Hence

APPENDIX A. PROOFS ON FINDING THE FOCAL PIVOT 171

AC

B

ab

M ∼= Ia,b

δ

δ

Figure A.2: Ia,b is at M

there is no β, for which both Ia,b and Ic,a are at A. Therefore A is not a focal pivot.

Let rotation angle β be such that Ia,b is at M as shown in figure (A.2). It may

further be seen that not only circle Ia,b but even line a intersects the Ic,a circle at two

distinct points: A and M . Since point Ic,a has to lie on both its locus Ic,a circle and

on line a (by definition), it should be at one of the intersection points (A or M) of

the circle and the line.

If point Ic,a is at A, then by the theorem in appendix B, line c aligns along CA

and line a becomes tangent to Ic,a circle at A. This contradicts the earlier noted fact

that line a intersects at two distinct points. Hence, point Ic,a cannot be at A and by

elimination of choices, it has to be at M Thus for the angle of rotation β, Ic,a as well

as Ia,b lies on M , implying that M is a focal pivot.

Appendix B

An elementary theorem of

geometry

A chord of a circle subtends the same angle at any point on the circumference of the

circle. The converse of the theorem is as follows: If two non-parallel lines passing

through two ends of a line segment are rotated equally about the respective end-

points of the segment, then the intersection point of the two lines traces a circle. The

circle has the line segment as its chord. During rotation, when the intersection point

reaches one end of the chord, one of the lines will align along the chord while the

other becomes tangent to the circle.

172

Appendix C

Perfect static balance and

positive-free-length springs

This is an appendix to section 5.2. Here the difficulty in achieving perfect static

balance of a lever by using normally available positive free-length springs is discussed.

The dTd term in the potential energy expression of a spring given in equation

(5.5) was seen to be a linear combination of sin θ, cos θ and 1 when expanded as in

equation (5.6). Hence, by writing dTd as α sin θ + β cos θ + γ, the potential energy

expression of the spring becomes:

(k

2(α sin θ + β cos θ + γ)− kl0

√

(α sin θ + β cos θ + γ) +k

2l20

)

(C.1)

The first term in equation (C.1) is the zero-free-length part and the second term is

the free-length part. If the free-length is positive, i.e., l0 > 0, then the free-length

part is negative. The free-length part of the spring is non-constant (i.e., k 6= 0 and

not both α and β is zero) except for trivial situations where spring constant is zero

or the spring is attached to the pivot of the lever. When there are several but finite

positive-free-length and non-trivial springs, the net contribution of the free-length

part is negative, and it is also not known have the possibility of being a constant,

unlike the zero-free-length part. Furthermore, the free-length part is also not known

173

APPENDIX C. NORMAL SPRINGS 174

to be in the function space spanned by sin θ and cos θ. Hence, the possibility of free-

length part cancelling (modulo a constant) with zero-free-length part is also ruled

out. Thus, with several positive-free-length springs, there is no way the net potential

energy could become a constant.

Appendix D

Constraints can be satisfied, if not

as it is, by addition of extra

zero-free-length spring loads

D.1 Satisfying constraints (5.12), (5.13) and (5.14)

This appendix demonstrates how by adding extra zero-free-length spring loads con-

straints (5.12), (5.13) and (5.14) can be satisfied. To differentiate between original

loads and balancing loads, let ns,o and ns,b respectively represent the number of orig-

inal and balancing zero-free-length spring loads with ns,o + ns,b = ns. Also, let the

spring loads be indexed such that the first ns,o loads are original loads with the re-

maining being balancing loads. Similar meaning applies for nc,o and nc,b.

Case 1: Original loads violate the constraint (5.14), and balancing loads are only

zero-free-length springs.

Let us try to satisfy all the constraints by adding a single zero-free-length spring.

As per the notation, this spring gets the index i = ns,o+1. The constraint (5.14)

can be written as follows:

kns,o+1ans,o+1 = −

ns,o∑

i=1

kiai (D.1)

175

APPENDIX D. SATISFYING CONSTRAINTS 176

where the known quantities related to original loads are on the right hand side.

Equation (D.1) gives the unique solution of kiai for i = ns,o+1 to the constraint

(5.14). Furthermore, the constraints (5.12) and (5.13) can be rewritten as

([

kiax,i kiay,i

−kiay,i kiax,i

][

bx,i

by,i

])

i=ns,o+1

=

−nc∑

i=1

((fyipy,i + fx,ipx,i))−ns,o∑

i=1

(ki(ay,iby,i + ax,ibx,i))

+nc∑

i=1

((fx,ipy,i − fy,ipx,i))−ns,o∑

i=1

(ki(ax,iby,i − ay,ibx,i))

(D.2)

The 2× 2 matrix on the left hand side of the equations is known since kiai for

i = ns,o + 1 is already solved in equation (D.1). Furthermore, the matrix is

non-singular since the right hand side of (D.1) that is the same as kns,o+1ans,o+1

is non-zero as per the description this case. We take[

bx,i by,i

]T

for i = ns,o+1

as the inverse of the 2 × 2 matrix times the right hand side of the equation

(D.2) so that the constraints (5.12) and (5.13) can also be satisfied. Thus, the-

oretically, with a single additional zero-free-length spring, all three constraints

(5.12), (5.13) and (5.14) can be satisfied.

Case 2 Original loads satisfy (5.14), but violate atleast one of the constraints (5.12)

and (5.13). Balancing loads are only zero-free-length springs.

If we proceed along the same lines as the previous case, then in equation (D.2)

the 2×2 matrix on the left hand side becomes singular zero-matrix whereas the

right hand side is non-zero by the description of the case. Thus, in this case,

with a single balancing zero-free-length spring, it is not possible to satisfy the

related constraint. However, it may be verified that by adding two balancing

springs, all the constraints can be satisfied.

The cases 1 and 2 cover all possible types of constraint violation. Hence we assert

that if the constraints are not satisfied as it is, then by adding a minimum of one zero-

free-length spring in case (1) (the component related to original loads in constraint

(5.14) is non-zero) and two zero-free-length spring in case (2) (the component related

APPENDIX D. SATISFYING CONSTRAINTS 177

to original loads in constraint (5.14) is zero), the constraints can be satisfied.

D.2 Satisfying constraints (5.19), (5.20) and (5.21)

In appendix B1, if the constraints (5.12), (5.13) and (5.14) are respectively substituted

by constraints (5.19), (5.20) and (5.21), there is going to be no change except for the

right hand side of equation (D.1), which takes the form−no∑

i=1

vi and the right hand side

of equation (D.2), which takes the form−[n∑

i=1

(qy,iwy,i + qx,iwx,i)n∑

i=1

(qx,iwy,i − qy,iwx,i)]T .

There is no change in the left hand side since here also, the additional loads are zero-

free-length springs. Hence, analogous to the conclusion of appendix B1, we conclude

that if the constraints (5.19), (5.20) and (5.21) are not satisfied as it is, then by

addition of a minimum of one zero-free-length spring in case the constraint (5.21) is

originally violated and a minimum of two zero-free-length springs in case the con-

straint (5.21) is not originally violated, the three constraints can be satisfied.

Appendix E

Solving constraints on spatial

orientation independence for the

parameters of additional

zero-free-length springs

This is an appendix to proof of proposition 3. Let the number of zero-free-length

springs, na, be 3. With this equations (6.7) and (6.8) can be split as

[

vnf+1 vnf+2 vnf+3

] [

1 1 1]T

= −

nf∑

i=1

vi (E.1)

[

qnf+1 qnf+2 qnf+3

] [

wnf+1 wnf+2 wnf+3

]T

= −[

q1 · · · qnf

] [

w1 · · · wnf

]T(E.2)

The left hand side of equation are due zero-free-length springs and using relation

(5.17), the parameters ui, wi and qi on the left hand side can be written in terms of

(ki,ai, bi), ki > 0 and i = 1 · · · 3 as

[

k1a1 k2a2 k3a3

] [

1 1 1]T

= −

nf∑

i=1

vi (E.3)

178

APPENDIX E. SOLVING BALANCING CONSTRAINTS – SPATIAL CASE 179

−[

k1a1 k2a2 k3a3

] [

b1 b2 b3

]T

= −[

q1 · · · qnf

] [

w1 · · · wnf

]T(E.4)

Solution to equations (E.3) and (E.4)

Suppose thatnf∑

i=1

vi 6= 0. In the three dimensional space, one can always find a plane

such thatnf∑

i=1

vi 6= 0 is not along the plane. Let m1 and m2 be two two linearly

independent vectors along such a plane. As a result, the matrix[

m1 m2

(

−m1 −m2 −nf∑

i=1

vi

)]

is a full-rank invertible 3 × 3 matrix and also a

solution to the matrix[

k1a1 k2a2 k3a3

]

in equation (E.3). With this as the solution

for[

k1a1 k2a2 k3a3

]

, the matrix[

b1 b2 b3

]T

can be solved by pre multiplying

equation (E.4) with the inverse of[

k1a1 k2a2 k3a3

]

. With this, one has a solution

for k1a1, k2a2, k3a3, b1, b2, and b3. By choosing a convenient positive value for k1,

k2 and k3, one can obtain a1, a2 and a3 as well. Thus, whennf∑

i=1

vi 6= 0 there exist a

non-unique solution for {ki, ai, bi}, i = 1 · · · 3 in equations (6.7) and (6.8).

General case including the possibility ofnf∑

i=1

vi = 0

Whennf∑

i=1

vi = 0 four zero-free-length spring are sufficient to satisfy (6.7) and (6.8),

as shown next. Just as equations (6.7) and (6.8) took the form of equations (E.3)

and (E.4) for three springs, the equations take the following form for four spring.

[

k1a1 k2a2 k3a3 k4a4

] [

1 1 1 1]T

= −

nf∑

i=1

vi (E.5)

−[

k1a1 k2a2 k3a3 k4a4

] [

b1 b2 b3 b4

]T

= −[

q1 · · · qnf

] [

w1 · · · wnf

]T(E.6)

APPENDIX E. SOLVING BALANCING CONSTRAINTS – SPATIAL CASE 180

Let m1, m2 and m3 be any linearly independent vectors in three dimensional space.

The matrix

[

m1 m2 m3

(

−m1 −m2 −m3 −nf∑

i=1

vi

)]

is a solution for[

k1a1 k2a2 k3a3 k4a4

]

in equation (E.5) and at the same time is of full-rank.

With[

k1a1 k2a2 k3a3 k4a4

]

being full rank, there exists solution, though non-

unique, for[

b1 b2 b3 b4

]T

in equation (E.6) not matter what the right hand side

of the equation is. Now we have a solution for {kiai, bi}, i = 1 · · · 4. By choosing a

convenient positive value for ki, one can solve for ai from the known kiai. Thus, there

is always a solution for spring parameters {ki, ai, bi}, i = 1 · · · 4 so that equation

(6.7) and (6.8) are satisfied.

Appendix F

Maxima code on flexure-based

four-bar mechanism - two spring

approximation

In the code below, the length l of a diagonal of the quadrilateral shown in figure (F.1)

is used as configuration defining variable. The choice is motivated by the fact that

the angles of the quadrilateral can be found from l based on cosine rule.

a

b

c

d

O L

M

N

l

Figure F.1: l, the length of a diagonal of the quadrilateral of four-bar bar linkage isused as a convenient configuration defining parameter.

COMMENTS: Angles ∠LON , ∠NLO, ∠LNO, ∠LMN , ∠NLM and ∠MNL are

181

APPENDIX F. VIRTUAL WORK CALCULATIONS 182

written below as functions of l.

(%i1) theta_l_ab : acos( (a^2 + b^2 - l^2)/(2*a*b) );

(%o1) acos

(−l2 + b2 + a2

2 a b

)

(%i2) theta_b : acos( (a^2 + l^2 - b^2)/(2*a*l) );

(%o2) acos

(l2 − b2 + a2

2 a l

)

(%i3) theta_a : acos( (l^2 + b^2 -a^2)/(2*b*l) );

(%o3) acos

(l2 + b2 − a2

2 b l

)

(%i4) theta_l_cd : acos( (c^2 + d^2 - l^2)/(2*c*d) );

(%o4) acos

(−l2 + d2 + c2

2 c d

)

(%i5) theta_c : acos( (l^2 + d^2 - c^2)/(2*l*d) );

(%o5) acos

(l2 + d2 − c2

2 d l

)

(%i6) theta_d : acos( (l^2 + c^2 - d^2)/(2*l*c) );

(%o6) acos

(l2 − d2 + c2

2 c l

)

COMMENTS: The angles and their derivatives with respect to l of the four corners

of the quadrilateral of pivots shown in figure (F.1).

(%i7) psi_ab : theta_l_ab;


(%o7) acos

(−l2 + b2 + a2

2 a b

)

(%i8) d_psi_ab : diff(psi_ab, l);

(%o8)l

a b

√

1− (−l2+b2+a2)2

4 a2 b2

(%i9) psi_bc : theta_a + theta_d;

(%o9) acos

(l2 − d2 + c2

2 c l

)

+ acos

(l2 + b2 − a2

2 b l

)

(%i10) d_psi_bc : diff(psi_bc,l);

(%o10) −1c− l2−d2+c2

2 c l2√

1− (l2−d2+c2)2

4 c2 l2

−1b− l2+b2−a2

2 b l2√

1− (l2+b2−a2)2

4 b2 l2

(%i11) psi_cd : theta_l_cd;

(%o11) acos

(−l2 + d2 + c2

2 c d

)

(%i12) d_psi_cd : diff(psi_cd, l);

(%o12)l

c d

√

1− (−l2+d2+c2)2

4 c2 d2

(%i13) psi_da : theta_c + theta_b;

(%o13) acos

(l2 + d2 − c2

2 d l

)

+ acos

(l2 − b2 + a2

2 a l

)

(%i14) d_psi_da : diff(psi_da, l);


(%o14) −1d− l2+d2−c2

2 d l2√

1− (l2+d2−c2)2

4 d2 l2

−1a− l2−b2+a2

2 a l2√

1− (l2−b2+a2)2

4 a2 l2

COMMENTS: Angle of each side of the quadrilateral with respect to the horizontal

is calculated below.

(%i15) phi_a : 0;

(%o15) 0

(%i16) d_phi_a : diff(phi_a, l);

(%o16) 0

(%i17) phi_b : theta_l_ab;

(%o17) acos

(−l2 + b2 + a2

2 a b

)

(%i18) d_phi_b : diff(phi_b, l);

(%o18)l

a b

√

1− (−l2+b2+a2)2

4 a2 b2

(%i19) phi_c : phi_b - %pi + psi_bc;

(%o19) acos

(l2 − d2 + c2

2 c l

)

+ acos

(l2 + b2 − a2

2 b l

)

+ acos

(−l2 + b2 + a2

2 a b

)

− π

(%i20) d_phi_c : diff(phi_c, l);

(%o20) −1c− l2−d2+c2

2 c l2√

1− (l2−d2+c2)2

4 c2 l2

−1b− l2+b2−a2

2 b l2√

1− (l2+b2−a2)2

4 b2 l2

+l

a b

√

1− (−l2+b2+a2)2

4 a2 b2


(%i21) phi_d : %pi - psi_da;

(%o21) − acos

(l2 + d2 − c2

2 d l

)

− acos

(l2 − b2 + a2

2 a l

)

+ π

(%i22) d_phi_d : diff(phi_d, l);

(%o22)1d− l2+d2−c2

2 d l2√

1− (l2+d2−c2)2

4 d2 l2

+1a− l2−b2+a2

2 a l2√

1− (l2−b2+a2)2

4 a2 l2

COMMENTS: Numerical values for the sides of the quadrilateral.

(%i23) subss(x):= [a=150/1000, b=100/1000,

c=130/1000, d = 120/1000, l=x];

(%o23) subss (x) := [a =150

1000, b =

100

1000, c =

130

1000, d =

120

1000, l = x]

COMMENTS: The value of l corresponding to the reference configuration, i.e., the

undeformed configuration of the flexure-based four-bar mechanism.

(%i24) l_0 : 140/1000;

(%o24)7

50


F.1 Calculation of stiffness in case 3a based on case

2a

F.1.1 Obtaining Fx vs. ux in case 2a using virtual work bal-

ance

COMMENTS: Details of the flexure and calculation of torsional spring constant of

the small-length flexure model.

(%i25) E : 205e9;

(%o25) 2.05 1011

(%i26) flex_width : 500e-6;

(%o26) 5.0000000000000001 10−4

(%i27) flex_thick : 4e-2;

(%o27) 0.04

(%i28) flex_length : 1/100;

(%o28)1

100

(%i29) area_I : 1/12*flex_thick* (flex_width)^3;

(%o29) 4.1666666666666664 10−13

(%i30) kt : E*area_I/flex_length;

(%o30) 8.541666666666666


(%i31) subst(subss(l), psi_ab);

(%o31) acos

(

100(

13400

− l2)

3

)

COMMENTS: Virtual work of the torsional springs.

(%i32) virt_wrk_tor_springs :

subst( subss(l),

-1*(

(psi_ab - subst([l=l_0], psi_ab))*d_psi_ab +

(psi_bc - subst([l=l_0], psi_bc))*d_psi_bc +

(psi_cd - subst([l=l_0], psi_cd))*d_psi_cd +

(psi_da - subst([l=l_0], psi_da))*d_psi_da

)

)$

COMMENTS: Rotation matrix.

(%i33) R_mat(theta) := matrix(

[cos(theta),-sin(theta)],

[sin(theta),cos(theta)]

);

(%o33) R mat (θ) :=

(

cos (θ) −sin (θ)

sin (θ) cos (θ)

)


(%i34) float(

subst( subss(l_0),

matrix([a], [0]) + R_mat( phi_d). matrix( [d], [0]) +

matrix([0], [-0*flex_length/2])

)

);

(%o34)

(

0.16993795176965

0.11833206699467

)

COMMENTS: Local coordinate of point P on the frame attached to the link associ-

ated with side d.

(%i35) ref_coor_d : float(

subst( subss(l_0),

invert(R_mat( phi_d)).

( R_mat( phi_d). matrix( [d], [0]) + matrix([0], [-flex_length/2]) )

)

);

(%o35)

(

0.11506949720856

−8.3074799040210381 10−4

)

COMMENTS: ux as a function of l

(%i36) ref_displ : matrix([1], [0]) .

( subst( subss(l), R_mat(phi_d).ref_coor_d) -

subst( subss(l_0), R_mat(phi_d).ref_coor_d) )$

COMMENTS: The range of parameter l, on which we focus our interest.


(%i37) l_min:94/1000; l_max:170/1000;

(%o37)47

500

(%o38)17

100

(%i39) l_range : [l,l_min,l_max] ;

(%o39) [l,47

500,17

100]

(%i40) plot2d([ref_displ], l_range,

[xlabel, "l (m)"], [ylabel, "u_x (m)"],

[gnuplot_term,ps],[gnuplot_out_file,"./for_thesis_img/ref_displ.eps"])$

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17

u x (

m)

l (m)

COMMENTS: Virtual work contribution of the horizontal force Fx at point P .


(%i41) virt_wrk_ref_force : matrix([1], [0]) .

subst( subss(l) , diff( R_mat(phi_d).ref_coor_d, l) )$

COMMENTS: Explicit expression for Fx in case 2a.

(%i42) f_2a_func_l : kt*virt_wrk_tor_springs/-virt_wrk_ref_force$

(%i43) plot2d([parametric,ref_displ, f_2a_func_l, l_range],

[xlabel, "u_x (m)"], [ylabel, "F_x in case 2a (N)"],

[gnuplot_term,ps],[gnuplot_out_file,"./for_thesis_img/f_2a.eps"] )$

-200

-100

0

100

200

300

400

500

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

Fx

in c

ase

2a (

N)

ux (m)

COMMENTS: Slope of Fx vs. ux at l = l0 in case 2a.

(%i44) f_2a_derivative_l_0 :

float( subst( subss(l_0), diff(f_2a_func_l, l)/diff(ref_displ,l) ) );

(%o44) 5140.246811088702


F.1.2 Obtaining Fx vs. ux in case 3a using virtual work bal-

ance

COMMENTS: Coordinates of anchor points, spring displacement, spring force, and

virtual work contribution of spring K21.

(%i45) spring_c_1_glb_anch : matrix( [-0.05], [0.06]);

(%o45)

(

−0.05

0.06

)

(%i46) spring_c_1_local : subst( subss(l_0), matrix( [c/10], [-c/3]) );

(%o46)

(13

1000

− 13300

)

(%i47) spring_c_1_displ : subst( subss(l),

R_mat(phi_b). matrix([b],[0]) + R_mat(phi_c).spring_c_1_local

)$

(%i48) spring_c_1_force :

-spring_c_1_k*(spring_c_1_displ - spring_c_1_glb_anch)$

(%i49) spring_c_1_virt_wrk :

( spring_c_1_force.diff(spring_c_1_displ, l) )$

COMMENTS: Coordinates of anchor points, spring displacement, spring force, and

virtual work contribution of spring K11.

(%i50) spring_b_1_glb_anch : matrix( [0.25], [-0.05] );


(%o50)

(

0.25

−0.05

)

(%i51) spring_b_1_local : matrix( [80e-3], [ 0] );

(%o51)

(

0.08

0

)

(%i52) spring_b_1_displ :

subst( subss(l), R_mat(phi_b). spring_b_1_local );

(%o52)

2.666666666666667(

13400

− l2)

0.08

√

1−10000 ( 13

400−l2)

2

9

(%i53) spring_b_1_force :

-spring_b_1_k*(spring_b_1_displ - spring_b_1_glb_anch);

(%o53)

−(2.666666666666667

(13400

− l2)− 0.25

)spring b 1 k

−

(

0.08

√

1−10000 ( 13

400−l2)

2

9+ 0.05

)

spring b 1 k

(%i54) spring_b_1_virt_wrk : spring_b_1_force . diff(spring_b_1_displ, l)$

COMMENTS: By virtual work balance, Fx in case 3a is explicitly written as below.

Note that it is a function of l with spring constants of K11 (spring b 1 k) and K2

1

(spring c 1 k) being unknowns.

(%i55) f_3a_func_l_unit_k :

(spring_c_1_virt_wrk + spring_b_1_virt_wrk )/-virt_wrk_ref_force$

COMMENTS: Fx and slope of Fx vs. ux at l = l0.


(%i56) f_3a_func_l_unit_k_origin :

expand( float( subst( subss(l_0), f_3a_func_l_unit_k) ) );

(%o56) 0.14647046533381 spring c 1 k − 0.2461002568181 spring b 1 k

(%i57) f_3a_func_l_unit_k_derivative :

expand( float(

subst(subss(l_0),

diff( f_3a_func_l_unit_k, l) / diff(ref_displ ,l ) )

) );

(%o57) 2.58328234272209 spring c 1 k − 1.37850044099567 spring b 1 k

COMMENTS: Equating Fx and slope of Fx vs. ux at l = l0 between case 2a and case

3a, the stiffness of springs K21 (spring c 1 k) and K1

1 (spring b 1 k) are solved below.

(%i58) spring_cnst_sub: float(

solve(

[f_3a_func_l_unit_k_origin=0,

f_3a_func_l_unit_k_derivative=f_2a_derivative_l_0 ],

[spring_c_1_k, spring_b_1_k])

);

rat : replaced−0.2461002568181by−1625/6603 = −0.2461002574587rat : replaced0.14647046533381

0.14647046501885rat : replaced−5140.2468110887by−395799/77 = −5140.24675324675rat :

replaced−1.37850044099567by−1526/1107 = −1.37850045167118rat : replaced2.58328234272209

2.583282302510717

(%o58) [[spring c 1 k = 2915.879749896249, spring b 1 k = 1735.432003674265]]


(%i59) plot2d([[parametric,ref_displ, f_2a_func_l, l_range] ,

[ parametric,

ref_displ,

subst(spring_cnst_sub , f_3a_func_l_unit_k) ,

l_range]

],

[xlabel, "u_x (m)"], [ylabel, "F_x (N)"],

[legend, "case 2a", "case 3a"],

[gnuplot_term,ps],

[gnuplot_out_file,"./for_thesis_img/f_2a_f_3a.eps"] );

(%o59)

-200

-100

0

100

200

300

400

500

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

Fx

(N

)

ux (m)

case 2acase 3a


F.2 Verification of static balance in case 3b through

virtual work balance

(%i60) spring_c_2_glb_anch : spring_c_1_glb_anch;

(%o60)

(

−0.05

0.06

)

(%i61) spring_c_2_local : -spring_c_1_local$

(%i62) spring_c_2_displ : subst( subss(l),

R_mat(phi_b). matrix([b],[0]) + R_mat(phi_c).spring_c_2_local

)$

(%i63) float( subst( subss(l_0), spring_c_2_displ ) );

(%o63)

(

0.020956478739549

0.12979062150905

)

(%i64) spring_c_2_force :

-spring_c_2_k*(spring_c_2_displ - spring_c_2_glb_anch)$

(%i65) spring_c_2_virt_wrk :

( spring_c_2_force.diff(spring_c_2_displ, l) )$

(%i66) spring_b_2_glb_anch :

matrix( [-0.00916760640646499], [ -0.04633295743741400] );

(%o66)

(

−0.009167606406465

−0.046332957437414

)


(%i67) spring_b_2_local : matrix( [80e-3], [ 0] );

(%o67)

(

0.08

0

)

(%i68) spring_b_2_displ :

subst( subss(l), R_mat(phi_b). spring_b_2_local )$

(%i69) float( subst( subss(l_0), spring_b_2_displ ) );

(%o69)

(

0.0344

0.072226310995371

)

(%i70) spring_b_2_force :

-spring_b_2_k*(spring_b_2_displ - spring_b_2_glb_anch)$

(%i71) spring_b_2_virt_wrk :

spring_b_2_force . diff(spring_b_2_displ, l);

(%o71) 5.333333333333333 l

(

2.666666666666667

(13

400− l2

)

+ 0.009167606406465

)

spring b

177.7777777777778 l(

13400

− l2)

(

0.08

√

1−10000 ( 13

400−l2)

2

9+ 0.046332957437414

)

spring b 2 k

√

1−10000 ( 13

400−l2)

2

9

(%i72) bal_sprng_cnst :

[spring_c_2_k= 2915.87974989625, spring_b_2_k= 7567.19150346676 ];

(%o72) [spring c 2 k = 2915.87974989625, spring b 2 k = 7567.19150346676]


(%i73) f_bal_func_l_unit_k :

(spring_c_2_virt_wrk + spring_b_2_virt_wrk )/-virt_wrk_ref_force$

(%i74) plot2d([[parametric,ref_displ,

( subst(spring_cnst_sub , f_3a_func_l_unit_k)

+ subst( bal_sprng_cnst, f_bal_func_l_unit_k) ) ,

l_range] ,

[ parametric,

ref_displ,


l_range],

[ parametric,

ref_displ,

subst( bal_sprng_cnst, f_bal_func_l_unit_k) ,

l_range]

],

[xlabel, "u_x (m)"], [ylabel, "F_x (N)"],

[legend, "case 3b", "case 3a", "balancing springs"],

[gnuplot_term,ps],

[gnuplot_out_file,"./for_thesis_img/f_2a_f_3a_balnce_contri.eps"]

);

(%o74)

COMMENTS: The following plot shows that virtual work contribution of original

springs and balancing springs in case 3b is zero over the range of l that is plotted.

This signify that static balance has indeed been attained.


-500

-400

-300

-200

-100

0

100

200

300

400

500

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

Fx

(N

)

ux (m)

case 3bcase 3a

balancing springs

F.3 A first order correction to balancing springs

on the flexure-based four-bar linkage

COMMENTS: ux (cms ref displ), Fx (cms f 3a) in case 1a and Fx (cms f 3b) in case

1b are respectively imported from the finite element analysis of the flexure-based

four-bar mechanism.


(%i75) cms_ref_displ : [

-0.025 ,

-0.022619047619047622 ,

-0.02023809523809524 ,

-0.017857142857142856 ,

-0.015476190476190477 ,

-0.013095238095238096 ,

-0.010714285714285714 ,

-0.008333333333333331 ,

-0.005952380952380952 ,

-0.0035714285714285726,

-0.0011904761904761897,

0.0011904761904761932 ,

0.0035714285714285726 ,

0.005952380952380952 ,

0.008333333333333338 ,

0.010714285714285718 ,

0.013095238095238097 ,

0.015476190476190477 ,

0.017857142857142856 ,

0.020238095238095243 ,

0.022619047619047622 ,

0.025 ]$


(%i76) cms_f_3a : [ -104.34594127214568 ,

-95.50949264952318 ,

-86.56303274592234 ,

-77.47330027942202 ,

-68.20345749702217 ,

-58.712126896397784 ,

-48.952203085118825 ,

-38.86932451582366 ,

-28.399875264975144 ,

-17.46832758018575 ,

-5.983643468057328 ,

6.165693524475422 ,

19.118685721280066 ,

33.05194222684286 ,

48.19469250388112 ,

64.8517456852825 ,

83.4398521812106 ,

104.54768638980624 ,

129.03965467289203 ,

158.24607605283924 ,

194.33613042898966 ,

241.11064684981028 ]$


(%i77) cms_f_3b : [ -15.824597608367657 ,

-14.25821375924494 ,

-12.773253920014659 ,

-11.405364973722545 ,

-10.103131463960098 ,

-8.878750274010095 ,

-7.728758343403991 ,

-6.649799654834652 ,

-5.638524800867299 ,

-4.691450134080959 ,

-3.8071963816068135 ,

-2.9739486856122546 ,

-2.1934129950294117 ,

-1.455558594545434 ,

-0.7574982308670888 ,

-0.07118952934501013 ,

0.6255594220880789 ,

1.3732569506544698 ,

2.2493721570413467 ,

3.401967157701034 ,

5.149952707532843 ,

8.172168625874383 ]$

COMMENTS: Deviation in Fx

(%i78) f_3b_orig_error : ( -2.9739486856122546+ -3.8071963816068135 )/2;

(%o78) − 3.390572533609534

COMMENTS: Deviation in slope of Fx vs. ux.


(%i79) f_3b_slope_error :

( -2.9739486856122546- -3.8071963816068135 )/

( 0.0011904761904761932 - -0.0011904761904761932);

(%o79) 349.964032317714

COMMENTS: Two linear equations defining the first order correction on k22 (spring c 2 k)

and k12 (spring b 2 k).

(%i80) correc_eq_1 :float( expand(

subst( subss(l_0), f_bal_func_l_unit_k)

) ) =

-f_3b_orig_error;

(%o80) 0.030088976858302 spring c 2 k−0.011594240624521 spring b 2 k = 3.390572533609534

(%i81) correc_eq_2 : float( expand(

subst( subss(l_0),

diff(f_bal_func_l_unit_k,l)/ diff(ref_displ ,l ) )

) )

=-f_3b_slope_error;

(%o81) −0.028273987618258 spring c 2 k−0.66838579798129 spring b 2 k = −349.964032317714

COMMENTS: The two equations are solved for the correction in stiffness of the

balancing springs.

(%i82) float( solve(

[correc_eq_1, correc_eq_2],

[spring_c_2_k, spring_b_2_k]) );

rat : replaced−3.39057253360953by−79770/23527 = −3.3905725336847rat : replaced−


0.01159424062452by−889/76676 = −0.01159424070113rat : replaced0.030088976858302by443/14723

0.030088976431434rat : replaced349.964032317714by48645/139 = 349.9640287769784rat :

replaced − 0.6683857979813by − 9979/14930 = −0.6683858004019rat : replaced −

0.02827398761826by − 1415/50046 = −0.0282739879311

(%o82) [[spring c 2 k = 309.3997276682762, spring b 2 k = 510.5076505303351]]

COMMENTS: Finite element analysis data of Fx after the correction.

(%i83) cms_3b_post_correc : [

-6.909034235038472 ,

-5.83429860238359 ,

-4.874566950191206 ,

-4.01833402821449 ,

-3.264279702980125 ,

-2.6117285115315623 ,

-2.0606882689860524 ,

-1.6119039235596984 ,

-1.2669288585698968 ,

-1.0282162854171706 ,

-0.9025937472472608 ,

-0.888930983768344 ,

-0.9902145311237647 ,

-1.2313346408838126 ,

-1.6044520551744579 ,

-2.1266404020078657 ,

-2.81101615779267 ,

-3.671581877007283 ,

-4.719935737519259 ,

-5.954192496041346 ,

-7.317875319092065 ,

-8.74382642287233 ]$


(%i84) plot2d([[parametric,ref_displ, f_2a_func_l, l_range] ,

[ parametric,

ref_displ,


l_range],

[ parametric,

ref_displ,

f_2a_func_l+subst( bal_sprng_cnst, f_bal_func_l_unit_k) ,

l_range] ,

[ parametric,

ref_displ,

subst(spring_cnst_sub , f_3a_func_l_unit_k) +


l_range],

[discrete, cms_ref_displ, cms_f_3a],

[discrete, cms_ref_displ, cms_f_3b],

[discrete, cms_ref_displ,cms_3b_post_correc ]],

[legend, "f2a", "f3a", "f2b", "f3b", "f1a", "f1b", "f1c" ],

[xlabel, "generalized displacement [m]"],

[ylabel, "generalized force [N]"],

[gnuplot_term,ps],[gnuplot_out_file,"case_plots.eps"]);

(%o84)

(%i85) load(draw)$

(%i86) case_1a : points(cms_ref_displ, cms_f_3a)$

(%i87) case_1b : points(cms_ref_displ, cms_f_3b)$


(%i88) case_1c : points( cms_ref_displ,cms_3b_post_correc)$

(%i89) case_2a : parametric(ref_displ,

f_2a_func_l, l, l_min, l_max)$

(%i90) case_2b : parametric(ref_displ,

f_2a_func_l+

subst( bal_sprng_cnst, f_bal_func_l_unit_k),

l, l_min, l_max)$

(%i91) case_3a : parametric(ref_displ,


l, l_min, l_max)$

(%i92) case_3b : parametric( ref_displ,

subst(spring_cnst_sub , f_3a_func_l_unit_k) +


l, l_min, l_max)$


(%i93) draw2d(

xlabel="u_x [m]",

ylabel="F_x [N]",

points_joined=true, point_size=1, line_width=1,

color=red, key="1a", point_type=2,case_1a,

color=magenta, key="1b", point_type=1,case_1b,

color=brown, key="1c (corrected)", point_type=3,case_1c,

color=royalblue, key="2a", case_2a,

color=blue, key="2b",case_2b,

color=green,key="3a", case_3a,

color=dark-green, key="3b", case_3b,

terminal = ’eps,

file_name = "case_plots"

);

(%o93) [gr2d (points, points, points, parametric, parametric, parametric, parametric)]

-100

0

100

200

300

400

-0.02 -0.01 0 0.01 0.02

Fx

[N]

ux [m]

1a1b

1c (corrected)2a2b3a3b

Appendix G

Static balancing of compliant

mechanisms using rigid-body

linkages and springs

Figure (G.1) shows a compliant gripper that is statically balanced with a small 2R

linkage loaded by a zero-free-length spring. The zero-free-length spring is realized by

a pulley-string arrangement.

In order to understand the principle behind the static balance in figure (G.1), con-

sider figures (G.2) and (G.3). Figure (G.2a) shows a statically balanced parallelogram

(for details see figure (2.6)). The spring between between B and D in figure (G.2a) is

relocated to be between E and A, as shown in figure (G.2b), where−−→BC =

−−→EB. From

geometry, it follows that this relocation does not change the length of the spring and

hence does not change the potential energy. Therefore the linkage of figure (G.2b) is

also statically balanced.

Figure (G.3) shows a compliant gripper, which is actuated by pulling down the

actuator rod shown in the figure. The actuator rod is guided to maintain symmetric

actuation. Because of the guide and the inherent elasticity of the compliant mecha-

nism, the actuator rod behaves as a one-dimensional spring. We found that for the

range of operation of the compliant mechanism, the force-displacement relation is

linear for practical purposes.

207

APPENDIX G. ANOTHER COMPLIANT MECHANISM BALANCING 208

Figure G.1: A compliant gripper compensated by a small spring loaded 2R linkage.(Basement board dimension: 2.5 feet × 2.5 feet)


l

k

l

l

k

k

(a)

(b)

kA

B

C

D

A

B

C

E

Figure G.2: A statically balanced parallelogram and its modification


Figure G.3: A compliant gripper


If we replace the spring between A and E, in figure (G.2b) by the actuator rod

with the stiffness of the spring between C and A matching that of the actuator rod,

we get the setup shown in figure (G.1). Since the linkage in figure (G.2b) is statically

balanced, the setup in figure (G.1) is also statically balanced. We had demonstrated

this set up in a few places and people invariably felt the difference between the

actuation effort before and after balancing to be dramatic.

Appendix H

Impossibility of statically balancing

a tree-structured gravity-loaded

rigid-body linkage by adding

springs only between successive

links

In the static balancing method of Chapter 5, the balancing springs were always at-

tached from the root to different parts of the linkage. In the case of gravity-loaded

linkages, the root is the ground. A question may arise about the possibility of stat-

ically balancing gravity-loaded tree-structured linkages without using auxiliary links

with springs added only between successive links of the tree-structure. In this ap-

pendix, we show that such a thing is not possible.

Let i be a terminal body of a tree-structured linkage, as shown in figure H.1.

Let p[i] be the center of gravity in the local coordinate frame and f [i] be the net

gravitational force. Then, the potential energy due to the gravity load takes the

form:

PEgravityi = −f [i]TR(θ[i])p[i] − f [i]Tr[i] (H.1)

212

APPENDIX H. SPRINGS BETWEEN SUCCESSIVE LINKS 213

0

i

j

x

y

θ[i]

x[i]

y[i]

r[i]

x

p[i]

f [i]

Figure H.1: A tree-structured linkage under gravity load

The terms dependent on θ[i] in equation H.1 are sinusoidal since f [i] and p[i] are

constants. Hence, equation (H.1) may be rewritten as:

PEgravityi = a cos(θ[i] + α

)− f [i]Tr[i] (H.2)

where a, α and f [i] are constants.

Suppose that static balance is achieved by adding balancing springs, of some type,

attached only between successive links. Two links are successive if they are connected

by a joint. Then, the net potential energy of the springs connected between body i

and its parent j depends on the relative angle between the two, i.e.,

PEspringij = Ψ(θ[i] − θ[j]

)(H.3)

Apart from θ[i], let all the configuration parameters of the linkage be fixed at some

value. In particular, let r[i] be r1 and θ[j] be γ1. Here, we assume that body j is

not the root. It means that the tree topology of linkage should have more than one

generation. Then, for the entire linkage, the only varying potential energy components


are:

a cos(θ[i] + α

)and Ψ

(θ[i] − γ1

)

These components should sum up to a constant since a linkage that is statically bal-

anced continues to be statically balanced even when any of the configuration variables

are fixed or constrained in any manner. Hence, we have have

a cos(θ[i] + α

)+Ψ

(θ[i] − γ1

)= C1 (H.4)

which may be further rewritten as

Ψ(θ[i] − γ1

)= −a cos

(θ[i] + α

)+ C1 (H.5)

Similarly, it is possible to fix the configuration variables, apart from θ[i] at some

different values. In particular, let θ[j] be fixed at γ2 with γ1 6= γ2. In this situation

also, similar to equation (H.5), we can write

Ψ(θ[i] − γ2

)= −a cos

(θ[i] + α

)+ C2 (H.6)

Further, equation (H.5) is true for any θ[i]. In particular, it is true for(θ[i] − (γ2 − γ1)

)

as well. Hence, equation (H.5), may be written as

Ψ(θ[i] − (γ2 − γ1)− γ1

)= Ψ

(θ[i] − γ2

)= −a cos

(θ[i] − (γ2 − γ1) + α

)+ C1 (H.7)

From equations (H.6) and (H.7), we have

a cos(θ[i] + α

)= a cos

(θ[i] + α− (γ2 − γ1)

)+ C2 − C1 (H.8)

What the above equation says is that when there is any non-zero phase shift in the

sinusoidal function, the function changes only by a constant. This is clearly a con-

tradiction. Hence, the statement that static balance is achieved by adding balancing

springs, of some type, connected only between successive links is false. In other

words, it is impossible to statically balance a gravity loaded tree-structured (having


more than one generation) linkage by adding springs, of whatever type, between only

successive links.

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