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Title

湿潤接着性向上のための形態適応性ソフトコンタクトの機
構

Author(s)NGUYEN, DANG DUYCitationIssue Date

2024-06

Type

Thesis or Dissertation

Text version

ETD

URL

http://hdl.handle.net/10119/19333

RightsDescription

Supervisor: Ho Anh Van, 先端科学技術研究科, 博士

Japan Advanced Institute of Science and Technology

D OCTORAL D ISSERTATION
Mechanics of morphologically adaptive soft contact for
wet adhesion enhancement

N GUYEN Dang Duy

Supervisor: H O Anh Van
Graduate School of Advanced Science and Tehcnology
Japan Advanced Institute of Science and Technology
Materials Science
June, 2024

iii

JAPAN ADVANCED INSTITUTE OF SCIENCE AND TECHNOLOGY

Abstract
Graduate School of Advanced Science and Technology
Doctoral Dissertation
Mechanics of morphologically adaptive soft contact for wet adhesion
enhancement
by N GUYEN Dang Duy
This thesis delves into the concept of animal adaptation to various environments
through the ability to morphologically adapt, often referred to as morphological design. It focuses on the significant role of tribological phenomena in different animal
species, such as human fingers in grasping, gecko toes in dry locomotion, and tree
frog toes in wet gripping, and how these findings can be applied to engineering.
Wet adhesion, a crucial aspect in tribology, is explored, and its potential application in robotics is investigated, particularly in the context of stable robot walking
that requires effective gait planning and control. The mechanical properties of theinteraction between robot feet and the ground surface are considered pivotal.
In this research, we propose a mechanics of morphology changeable soft pad
for robotic foots capable of adapting its morphology to the changing terrains thatrobots encounter. This adaptation includes optimizing various tribological factors
such as friction, adhesion, and particularly wet adhesion when the foot interacts
with wet surfaces.The function of actively changing morphology to adapt to theenvironment plays an important role in embodied robots. The result proposed inthis thesis is promising to use for the application of embodied robots that perceive
their surroundings to manipulate objects or move their bodies, process information,and make decisions.

iv
First, I establish a mathematical model based on energy equations to provide insights into the principles of morphological changes of the robot foot. I then present
two approaches to model the value of wet adhesion in preventing slippage for the
robot’s foot. The first approach simplifies the foot shape to streamline complex parameters for calculating wet adhesion forces. The second approach introduces amore generalized model based on finite element methods to describe wet adhesion
forces on a soft body interacting with various ground surfaces. This generalized
model is validated and simulated using the Simulation Open Framework Architecture (SOFA framework).
Subsequently, we present experimental data that corroborates the accuracy ofthe mathematical models proposed in this research. Finally, we test the application of the morphologically adaptable robot foot in two scenarios using a separate
legged robot and a complete hexapod robot in showcases. These applications aim to
showcase the effectiveness and potential practicality of a morphologically adaptablerobot foot.
In conclusion, this research advances our understanding of animal-inspired robotics,
particularly focusing on the significance of morphological adaptability and wet adhesion in robotic locomotion. The proposed model and its practical application holdpromise for the development of robots capable of efficiently traversing various terrains and adapting to different environments.
Keywords: Tribology, capillary, morphology computation, soft toe pad, animal locomotion, SOFA, finite element, embodied robot

v

Acknowledgements
Completing this doctoral thesis has been a journey of growth, learning, and perseverance. It is with immense gratitude that I extend my thanks to the following
individuals and groups who have supported me on this academic expedition.
First and foremost, I would like to express my deepest appreciation to my mainsupervisor, Assoc. Prof. Ho Anh Van, for his unwavering guidance, invaluable
insights, and unwavering dedication to my research. Your mentorship has been abeacon of wisdom that illuminated my path throughout this journey.
I am equally grateful to my second supervisor, Assoc. Prof. Ji Yonghoon, whose
expertise and constructive feedback have enriched the quality of this work andbroadened my horizons in the field.
I extend my heartfelt appreciation to Prof. Nguyen Le Minh, who served as myAdvisor for Minor Research Project, for providing critical direction and unceasing
support in this endeavor.
My profound thanks also go out to the entire team at the Soft Haptics Lab, Schoolof Materials Science, Japan Advanced Institute of Science and Technology, for creating a vibrant, collaborative research environment and contributing to my academic
and personal growth.
To my family, especially my wife and son, your unwavering support, encouragement, and patience throughout this journey have been my greatest source ofstrength and motivation. I owe this achievement to your understanding and love.
Lastly, my thanks to all my friends and colleagues who have provided inspiration and shared their knowledge.
In this collective effort, I have had the privilege of standing on the shoulders ofgiants, and I am deeply appreciative of each one’s role in making this thesis possible.

vii

Achievements
International Journal• Duy Dang Nguyen, Nhan Huu Nguyen, Van Anh Ho, "Morphology-Changeable
Soft Pads Facilitate Locomotion in Wet Conditions," in IEEE Robotics and Automation Letters, vol. 8, no. 5, pp. 2983-2990, May 2023, doi: 10.1109/LRA.2023.3264731.
(The research was presented orally at The 2023 IEEE/RSJ International Conference on Intelligent Robots and Systems - IROS 2023, Detroit, Michigan, USA.)• Duy Dang Nguyen, Nam Phuong Dam, and Van Anh Ho "Enhancing wet surface adhesion in walking robots: finite element-based analysis and morphologychangeable soft pads." Advanced Robotics (2024): 1-17, doi: 10.1080/01691864.2024.2327649• Harit Pitakjakpipop, Robin Rajan, Kittipong Tantisantisom, Pakorn Opaprakasit, Duy Dang Nguyen, Van Anh Ho, Kazuaki Matsumura, and Paisan Khanchaitit, “Facile Photolithographic Fabrication of Zwitterionic Polymer Microneedles with Protein Aggregation Inhibition for Transdermal Drug Delivery”, Biomacromolecules 2022 23 (1), 365-376, DOI: 10.1021/acs.biomac.1c01325.
IEEE Robotics and Automation Letters metrics: Impact Factor: 5.2, CiteScore:8.0; Ranking: Q1 in Artificial Intelligence, Q1 in Biomedical Engineering, Q1 in
Computer Science Applications, Q1 in Computer Vision and Pattern Recognition,
Q1 in Control and Optimization, Q1 in Control and Systems Engineering, Q1 in
Human-Computer Interaction, Q1 in Mechanical Engineering; #2 journal in top publications of Robotics by Google scholar.
International Conference on Intelligent Robots and Systems (IROS) is flagship international conference of Robotics: Impact score: 2.45; #4 journal in top publicationsof Robotics by Google scholar.

ixContents
Abstractiii

Acknowledgementsv

Achievementsvii

1

Introduction

1

2

Background and Related Works

7

2.1

3

hesion and capillary . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.2

Contact mechanics models for elastic material . . . . . . . . . . . . .

9

2.3

Finite Element method for wet adhesion and capillary . . . . . . . . .

11

2.4

The contribution of this thesis . . . . . . . . . . . . . . . . . . . . . . .

13

Bioinspired robotic foot facilitating wet adhesion

15

3.1

ANALYTICAL MODEL . . . . . . . . . . . . . . . . . . . . . . . . . .

15

3.1.1

Morphology computation model . . . . . . . . . . . . . . . . .

15

3.1.2

Contact model with liquid film . . . . . . . . . . . . . . . . . .

25

3.1.3

Optimal function for design parameters . . . . . . . . . . . . .

28

Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

3.2.1

Inflation and contact-related shape verification . . . . . . . . .

33

3.2.2

Tangential force experiment . . . . . . . . . . . . . . . . . . . .

38

3.2

4

Morphology and bio-inspired structure of animals’ food pad with ad-

The finite element analysis (FEA) for representing wet

43

4.1

The FEA-based solution . . . . . . . . . . . . . . . . . . . . . . . . . .

43

4.1.1

44

The general problem of capillary phenomenon . . . . . . . . .

x
4.1.2
4.2

The FEA-based model for the capillary phenomenon . . . . .

Implementation using SOFA (Simulation Open Framework Architecture) framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1

4.2.2

5

6

48

The FEA-based model for the capillary phenomenon of thepatterned pad . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

Experimental validation . . . . . . . . . . . . . . . . . . . . . . . . . .

58

4.3.1

58

Tangential force experiment . . . . . . . . . . . . . . . . . . . .

Application to slippage resistance of locomotive robot on wet surface

63

5.1

Design and fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

5.1.1

Large scale design . . . . . . . . . . . . . . . . . . . . . . . . .

64

5.1.2

Small scale design . . . . . . . . . . . . . . . . . . . . . . . . .

65

5.2

Case of single legged robot with the proposed foot . . . . . . . . . . .

66

5.3

Case of hexapod robot with the proposed feet . . . . . . . . . . . . . .

70Conclusion and Future Works

75

6.1

Impact of morphology-changeable soft pads . . . . . . . . . . . . . .

75

6.1.1

Design and fabrication . . . . . . . . . . . . . . . . . . . . . . .

75

6.1.2

Analytical model and finite element method model for Morphology changeable soft dome pad . . . . . . . . . . . . . . .

6.1.3

6.2
7

48

The wet adhesion phenomenon simulation for various shape
objects using SOFA . . . . . . . . . . . . . . . . . . . . . . . . .

4.3

44

76

Application for legged robot walking on different terrain without slippage . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

Appendix79

7.1

Appendix 1: Drawing of mold design . . . . . . . . . . . . . . . . . .

79

7.1.1

Appendix 1.1: Drawing of mold design for large scale foot . .

79

7.1.2

Appendix 1.2: Drawing of mold design for small scale foot . .

85

Appendix 2: Matlab code for calculation . . . . . . . . . . . . . . . . .

91

7.2

xi
7.2.1

Morphology changing without pressing force . . . . . . . . .

91

7.2.2

Morphology changing with pressing force . . . . . . . . . . .

95

Bibliography103

xiii

List of Figures
1

Morphology-changeable soft padsfor locomotion on both dry and wet
surfaces. (a) An illustration of a legged robot, highlighting the potential issue of slippage on a wet floor. (b) A conceptual representationof a robot foot equipped with a flexible and adaptable sole. (c) The
manufacturing process of a robotic foot, comprising a sturdy "bone"
enclosed within a soft, flexible skin, and a soft sole featuring adaptable soft pads. (d) A cross-sectional diagram of the foot, offering insights into the adaptable soft pads that are controlled via pneumatic
air pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

2Morphology-changeable soft pads foot and finite element model. (a)The improved version Morphology-changeable soft pads. (b) The
morphology changeable pad foot attached hexapod-robot. (c) Concept of capillary force model on multi-dome pads. (d) finite element
method model for capillary simulation. . . . . . . . . . . . . . . . . .

3The morphology of foot pad’s structure of animals: a- Tree frog toe
pad, b- Gecko toe pad, and c- Starfish tube feet . . . . . . . . . . . . .

4

4

8

Proposed designs and applications inspired from animals’ foot pad.
(a) The design of tree-trog-inspired adhesives in micro-scale [23]. (b)the artificial tube feet of starfish that exhibit robust adhesion and enable effective locomotion [24]. . . . . . . . . . . . . . . . . . . . . . . .

9

xiv
5

A cross-sectional illustration shows the contact between two elastic
spheres using three models: (a) Hertz contact, (b) JKR, and (c) DMT
models. In Hertz theory, no interactions are considered, while the JKR
theory accounts for surface forces only within the contact region. On
the other hand, the DMT theory considers both surface forces within
and attractive forces outside the contact region. Solid lines represent
the contact configuration at zero load, and dashed lines represent thecontact configuration when an applied load, P, is present.[34] . . . . .

6

11Soft Pad Configurations: (a) Depressurized (flat circular soft pad).
(b) Pressurized (dome-shaped soft pad) without ground contact. (c)Pressurized with ground contact during walking. (d) Geometric characteristics (at constant pressures) of the soft pad with and without
contact (represented by dashed lines). . . . . . . . . . . . . . . . . . .

7

15

The spherical surface differential explanation for tension energy: Tension energy is determined by integrating the differentials of sphericalsegments along the z-axis. . . . . . . . . . . . . . . . . . . . . . . . . .

817Capillary action occurs upon contact with the wet floor, induced bya liquid layer surrounding the soft pad with a height of h. In thissituation, the pad exhibits a tendency to slip to the left. . . . . . . . .

925

Optimization solution for design. (a) phenomenon of rising wet liquid. (b) the arrange of inflated dome in the foot and the area caculation. 29

10

The graphical representation illustrates the changes in the liquid’s rising height h versus the initial radius r0 . . . . . . . . . . . . . . . . . .

11

32

Design of the soft pad. (a) Soft pad without air pressure. (b) Illustration of the soft pad design concept. (c) Soft pad under the influence

12of air pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

Experimental design for examining morphological changes. . . . . .

34

xv
13

Dome morphology dimension processed by 400-CAM058 microscope(SANWA SUPPLY): a- without applying pressing force, b- with applying pressing force. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

34

Experiment state 1: Effects of different pressures in the soft pad without floor contact were compared using Matlab (Code matlab in Appendix 7.2.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

15

Experiment state 1: dimension of dome heigh and dome radius. . . .

36

16

Experiment state 2: Response of the soft pad under air pressure of
75 kPa to being pressed at different forces (Code matlab in Appendix
7.2.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

17

Experiment state 2: dimension of dome heigh and dome radius. . . .

37

18

Experiment state 2: dimension of adhesive radius and adhesive deep.

38

19

Experiment to assess the resistance of the soft pad to tangential forces:(a) Setup of the experiment. (b) Soft pad under air pressure without
simulated floor contact, (c) and with simulated wet floor contact. . .

2038(a) Comparison of resulted tangential forces among flat soft pad, anddomed pad under pressures of 65 kPa, 75 kPa, and 85 kPa, respectively; on both wet and dry floors. (b) Zoom-in range with comparison of tangential forces under dry (dashed line) and wet (solid line)
conditions several air pressure values. . . . . . . . . . . . . . . . . . .

21

Comparison of capillary action in wet condition (Fcap = Fwet − Fdry )between experiment (with error bars) and estimation results. . . . . .

22

39

40

Wet process: (a) Soft contact between 2 object. (b) Liquid molecularof wet-adhesion phenomenon in the first time. (c)Wet-adhesion phenomenon in process . (d) Equilibrium state. . . . . . . . . . . . . . . .

2345(a) Wet adhesion phenomenon in general geometry object. (b) Differentials of liquid following polar coordinates. . . . . . . . . . . . . . .

46

xvi
24

(a) The geometric parameters of the differential element of the liquidmass in polar coordinate system. (b) The 2D projection of the differential element of the liquid mass onto its own normal plane. . . . . .

50

25

The algorithm of finite element method for estimation of capillary force. 53

26

SOFA simulation for capillary phenomenon. Note that red dottedboundaries show the uplifted water layer due to adhesion. (a) Dome
shape object. (b) Truncated cone shape object. (c) General geometry
object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

54Tangential capillary force estimation following the wettecd liquid andmovement direction, where Ssection is the projected area of the wetting
liquid on a plane perpendicular to the direction of motion. . . . . . .

28

54

Wet adhesion parameters simulating in case of γl = 0.004 N/m2 withthe rising of γS (simulation using SOFA.) . . . . . . . . . . . . . . . .

54

29

The relationship between surface tension energies and wetted height.

55

30

The relationship between wetted heigh and radius of wetted liquidcurvature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

55

SOFA simulation of wet liquid for soft pads foot. (a) Changeable soft
pads foot meshing in SOFA. (b) Height of rising wet liquid. . . . . . .

57

32

Experimental setup of the tangential force. . . . . . . . . . . . . . . .

59

33

Microscopic images of the experimental soft plastic layer surface: (a)presents the projection of the convex elevation on the rough surface,(b) shows the projection of the roughness density on the rough surface, (c) demonstrates the projection of the smooth surface, (d) displays the projection of the roughness density on the smooth surface.

3460(a) Comparison of resulted tangential forces among flat soft pads foot,and domed pads on both wet and dry floors in two different roughness. (b) Comparison of capillary action in wet condition (Fcap =Fwet − Fdry ) between experiment (with error bars) and estimation results. 61

xvii
35

Fabrication procedure: (a) Arrangement of the foot mold. (b) Pouringof silicone material. (c) Finalizing the foot structure. (d) Soft pad withapplied pressure and dimensions. . . . . . . . . . . . . . . . . . . . . .

36

64

Fabrication process: (a) mold set of foot. (b) Pouring silicone include
two step: pouring Dragon Skin 30 silicone for the body, then pouring
the silicone Ecoflex 00-20 for the thin sheet. (c) Finishing the foot. (d)dome inflation and the robot leg attached the foot. . . . . . . . . . . .

37

65

Design of the robotic leg. (a) The walking system with its components. (b) Mechanical analysis of the linkage system for walking tests
on both (c) dry and (d) wet floors (contour data was tracked to generate the foot paths in Fig. 38). . . . . . . . . . . . . . . . . . . . . . . . .

38

67Tracked foot’s posture (the rectangle in Fig. 37d) during walking action in (a) on a dry floor, (b) on a wet floor with flat foot, and (c) on a
wet floor with domed pad (high inner pressure state). . . . . . . . . .

39

68

Results of the walking robot’s body displacement in five experimental
conditions (tracked by the marker on the robot body as shown in Fig.
12d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

68

3-Degree-of-Freedom Hexapod robot in the experiment: (a) Hexapod
robot with attached markers, (b) leg mechanism illustration. (c) The
concept of hexapod locomotion showcased. . . . . . . . . . . . . . . .

41

70

The positions of the detected markers on leg 2 and leg 5 of the hexapod while walking on the rough surface, illustrating instances of slippage in different scenarios: walking on dry surface with flat foot,walking on wet surface with flat foot, walking on wet surface withdome pad foot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

72

The positions of the detected markers on leg 2 and leg 5 of the hexapod while walking on the smooth surface, illustrating instances ofslippage in different scenarios: walking on dry surface with flat foot,walking on wet surface with flat foot, walking on wet surface withdome pad foot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

xviii43

The positions of the detected markers on the body of the hexapod
while walking on the rough surface in different scenarios: walking
on dry surface with flat foot, walking on wet surface with flat foot,walking on wet surface with dome pad foot. . . . . . . . . . . . . . .

44

73

The positions of the detected markers on the body of the hexapod
while walking on the smooth surface in different scenarios: walking
on dry surface with flat foot, walking on wet surface with flat foot,45

walking on wet surface with dome pad foot. . . . . . . . . . . . . . .

74

Designed mold and completed foot in large and small scale. . . . . .

76

11 Introduction
Recently, biomimetic research has become popular in various engineering applications [1]. This problem-solving approach, often inspired by nature, not only helps
create innovative solutions but also allows exploring biological hypotheses that are
difficult or impossible to test on live organisms [2]. Besides offering practical engineering insights, biomimetic research enhances our understanding of the basic
physics behind different phenomena. This knowledge is particularly valuable insituations where ethical or systematic concerns restrict experimentation on animals.
Among the numerous biomimetic approaches, one area has recently garneredsignificant interest: the concept of morphological adaptation to changing environmental conditions. This approach focuses on creating functional, adaptive structures that can modify their shape, stiffness, or properties in response to different
scenarios. These morphological changes provide a compelling basis for various applications [3]. Researchers are increasingly exploring how animals adapt their morphology to suit their surroundings. This form of adaptation, often referred to as"morphological design," has become a topic of intensive investigation [4].
This research in biomimetics has delved into the details of tribological phenomena, focussing on various aspects of animal adaptation to their environments. For
instance, studies have examined how human fingers achieve exceptional friction
and dexterity when grasping objects, the dry locomotion capabilities of gecko toes[5], and how tree frog toes are optimized for wet gripping [6]. The findings fromthese studies have practical uses in engineering, especially in creating grippers andadhesion systems. As we understand more about how these animals effectively use
their morphology, we gain insights into optimizing mechanisms that enable robots
to interact more effectively with their environment. The ability of actively changing

2

Chapter 1. Introduction

F IGURE 1: Morphology-changeable soft padsfor locomotion on both dry and wet surfaces. (a) An illustration of a legged robot, highlighting the potential issue of slippage
on a wet floor. (b) A conceptual representation of a robot foot equipped with a flexible
and adaptable sole. (c) The manufacturing process of a robotic foot, comprising a sturdy"bone" enclosed within a soft, flexible skin, and a soft sole featuring adaptable soft pads.
(d) A cross-sectional diagram of the foot, offering insights into the adaptable soft pads
that are controlled via pneumatic air pressure.

morphology to adapt to the environment is also an important aspect of embodied
robots, which is an interesting trend. The concept of embodied robotics is groundedin the idea that intelligence and cognition are tightly linked to physical embodiment and interaction with the environment [7]. By having a physical presence, theserobots can engage with the world in a manner similar to living organisms, allowing
them to adapt to changing conditions and navigate complex environments.
One interesting example of this type of adaptation comes from starfish. They canpump water into their tubular limbs through a sieve plate within their skin, thereby
altering the shape, stiffness, and posture of their limbs to enhance their locomotion[8]. This morphological adaptation allows them to harness the principles of tribology effectively and facilitate adhesion between their limbs and the surfaces theynavigate, especially in complex, irregular terrains [9]. This adaptation seems to be a
crucial factor in their ability to adapt to challenging environments, a principle that

Chapter 1. Introduction

3

may hold significance for legged robots as they navigate dynamic, unpredictablesurfaces.
Building on these insights, we propose a novel mechanics of morphology changeable soft pad that takes advantage of adapting its morphology to optimize tribological interactions, such as friction, adhesion, and, notably, wet adhesion. Fig. 1 illustrates the concept of this research. Wet adhesion plays a crucial role when robots
operate on wet surfaces, where slippage can be a challenge (see Fig. 1a). Our robotic
foot incorporates a novel design of soft pad that can adapt its morphology, enhancing the foot’s performance on both dry and wet surfaces. (see Fig. 1b, d).
To provide a theoretical concept for this mechanic, we have developed an analytical model that examines the interaction between the foot’s morphology and theconditions of ground contact. This model takes into account factors such as shape,size, and stiffness of the foot. Furthermore, by actively adjusting the morphologyof the foot’s soft pads to adapt to different surface conditions, we emphasize thecritical role that morphological features play in the locomotion of legged walking
robots.
Besides, slippage is a significant challenge for legged robots, particularly when
they walking on wet surfaces [10]. Various research studies have explored the potential of improving robotic stability by using friction, adhesion, and capillary forces.
Notably, bio-inspired feet with small-scale, soft pads have successfully used capillary forces to enhance robot mobility, mimicking the locomotion of animals such astree frogs on wet surfaces [11]. Additionally, some studies have introduced perspectives on wet adhesion, inspired by the adhesive capabilities of mussels [12].
However, while these bio-inspired foot designs have led to significant improvements in robot stability, there remains a gap in our understanding of the underlying physics. A universal analytical model that reveals not only the performance ofrobot feet but also the mechanisms at play in the natural world is still lacking. The
fundamental principles describing capillary forces in wet adhesion are still a challenge, due to the complicated interaction between the foot’s pads, the liquid, thesurrounding surface, and vapor [13]. Existing analytical models are often limited to

4Chapter 1. Introduction

F IGURE 2: Morphology-changeable soft pads foot and finite element model. (a) The
improved version Morphology-changeable soft pads. (b) The morphology changeable
pad foot attached hexapod-robot. (c) Concept of capillary force model on multi-dome
pads. (d) finite element method model for capillary simulation.

specific geometrical shapes that are readily incorporated into calculations [14], [15].
Furthermore, the arrangement and distribution of these pads are acknowledged as
significant factors in enhancing capillary force [16]. However, this element has notbeen directly incorporated into the capillary force equation. Because of the complicated capillary interactions between objects on wet surfaces, the Finite Element(FE) method has become a useful tool for simulating how liquids behave, especiallywhen it comes to wet adhesion and capillary forces.
To address these challenges, we present an analytical model that employs the
Finite Element method, thoroughly explores the equilibrium state of energy related
to the liquid’s geometric surface. Besides, we also proposed an improved robot footdesign featuring smaller dome pads to maximize the capillary force’s effectivenessand enhance the robot’s locomotion. This research also calculates the capillary force
values, which depend on the geometric characteristics and distribution of the domes
on the foot. Fig. 2 illustrates this purpose, which focuses on another design tooptimize the capillary force (Fig. 2a, c). Besides, Fig. 2b shows the application of thefoot in hexapod robot. Finally, Fig. 2d introduce the result of finite element method

Chapter 1. Introduction

5base model for simulation of capillary.
This study employs the Simulation Open Framework Architecture (SOFA) toconstruct an analytical model, effectively meshing solid objects and considering every element of the liquid’s energy equilibrium state. The analytical model’s accuracy
is validated through force measurement experiments, ensuring its accuracy and reliability. Finally, we evaluate the effectiveness of the robotic foot through a series of
showcase experiments, including scenarios involving locomotion on both dry andwet surfaces.
In summary, this research contributes to our understanding of biomimetic adaptations, morphological design, and the role of tribological phenomena in optimizing
the performance of legged robots. Learning from both nature and engineering helps
us create a solution for robot feet. It highlights the importance of models that connect theoretical understanding with practical application. This result could lead to
improvements in robotics, contributing to the development of robots capable of effective and stable locomotion across a range of surfaces.
In this chapter, I have introduced the overall concept and philosophy of the dissertation, emphasizing the physical meaning, impact, and goals of this research. In
the next chapter, related works that form the concept for our research will be explained in detail.

7

2 Background and Related Works
In this study, a novel mechanics of morphology changeable soft pad were introduced with demonstrated designs and experiments. This mechanics draws its
inspiration from the morphological features and bio-inspired structures found inanimals’ foot pads, which have been extensively explored and employed across diverse applications. Subsequently, a soft contact mechanical model was employed
to reveal the fundamental physical concepts underlying the contact phenomenon
within this mechanics and design. Furthermore, the finite element method was used
to provide a solution for simulating and assessing wet adhesion and capillary effects
within our research-focused contact phenomenon.

2.1

Morphology and bio-inspired structure of animals’
food pad with adhesion and capillary

Researchers have drawn insights from animal footpads, driving advancements
in adhesion mechanisms. Nature’s adaptations have been a source of inspiration,
resulting in practical applications. For example, the tree frog’s toe pads display
a distinctive feature – velocity-dependent shear resistance. This trait has provenvaluable in the development of wet adhesion methods, contributing to bio-inspired
technology [17], [18]. The authors unveiled the significant potential in the conceptof the tree frog’s foot pad, providing a novel approach applicable to a wide array ofcontexts.
In a similar case, gecko toe pads have undergone extensive research due to theirhierarchical structure. These intricate feet have not only been closely studied but

8Chapter 2. Background and Related Works

have also been employed in developing dry adhesion mechanisms [19], [20]. Moreover, geckos can adjust their hydrophobicity in humid environments, enhancing
adhesion and stabilizing friction, highlighting the adaptability of these structures invarious conditions [21]. These examples emphasize the potential of nature-inspired
designs in engineering and technology. Similarly, echinoderm tube feet give another interesting direction for exploration. These remarkable appendages enable
creatures to navigate challenging terrains like sand, stone, or underwater surfaces
with remarkable grace and efficiency [22].
Taking inspiration from the natural world, researchers have proposed bio-inspired
designs for significant applications in various fields, including robotics and medicine.
Authors in [23] proposed design of tree-trog-inspired adhesives in micro-scale toexploit the capillary and adhesion in different applications. Through meticulous
design and fabrication, researchers have created artificial tube feet of starfish thatexhibit robust adhesion and enable effective locomotion, promising innovative solutions in various applications [24]. Soft pad claw robots have successfully integrated
the flexible morphology of soft pads to achieve impressive results. These robots
can climb rough vertical surfaces, showcasing the potential of nature-inspired adhesion designs [25]. Furthermore, a quadruped insect-scale robot uses capillary andF IGURE 3: The morphology of foot pad’s structure of animals: a- Tree frog toe pad, bGecko toe pad, and c- Starfish tube feet

2.2. Contact mechanics models for elastic material

9

F IGURE 4: Proposed designs and applications inspired from animals’ foot pad. (a) The
design of tree-trog-inspired adhesives in micro-scale [23]. (b) the artificial tube feet of
starfish that exhibit robust adhesion and enable effective locomotion [24].

lubrication effects to tackle inclined climbing tasks, underlining the versatility of
bio-inspired strategies in robotics [26].
These examples emphasize the impressive adaptability of animal foot pads, showcasing their significant ability to use adhesion mechanisms in various engineering
applications. This adaptability has evolved over millions of years, with nature finely
honing footpad structures in response to unique environmental challenges.
These examples underline that animal foot pads possess versatile properties capable of reshaping engineering adhesion concepts. Researchers, inspired by naturaldesign, are committed to exploring and applying these insights. Their aim is toincorporate biological lessons into practical solutions that connect biology with advanced technology. As this collaboration progresses, it holds the potential to bringabout significant advancements in various scientific and engineering fields, fromrobotics to material science. This collaboration aims to strengthen the connection
between the insights from nature and human innovation.

2.2

Contact mechanics models for elastic material

The study of soft contact locomotion, seen in the soft pads of animals and evolving robot designs, uncovers a fascinating area marked by a complex interplay offeatures like adhesion, friction, capillary action, and Van der Waals forces. Understanding the roles and physical concepts of these factors in different situations is achallenging scientific puzzle. Adding complexity, many natural foot pads are soft,

10

Chapter 2. Background and Related Works

adaptable, and flexible. These abilities help them move easily on different surfaces,
like tree bark, slippery leaves, or rough rocks. In robotics, this adaptability is a significant tool that helps animals live in their habitat. Researchers want to study fromthese natural abilities, making robots more versatile and innovative. Understanding
how soft contact locomotion works can give us new ideas in fields like robotics andmaterial science.
To solve that problem, one approach is to construct a capillary-action model using elastic materials. Building upon theories such as Hertz theory, Johnson-KendallRoberts, Derjaguin–Muller–Toporov, and Tabor (Fig. 5), we have gained insights
into a multitude of contact conditions that soft bodies can exhibit [27]–[29]. Fig. 5
illustrate the development of contact mechanics theories. In Hertz theory, no interactions are considered, while the JKR theory accounts for surface forces only within
the contact region. On the other hand, the DMT theory considers both surface forces
within and attractive forces outside the contact region. The Maugis-Dugdale theory
has added a crucial analytical perspective by providing valuable data on interactions in the intermediate regions, where Tabor’s result has moderate values [30].
Those results contributed important knowledge into soft contact mechanics.
In our earlier studies, we delved into how soft materials interact in wet conditions, especially concentrating on micro-patterned pads [31]. Taking inspirationfrom how tree frog toes stick, we explored creating and trying out a pad shape thatcan grip thin hemispherical shells using wet adhesion [32], [33]. These inquiries, inspired by nature, have opened up possibilities for robots to adjust their movement
based on the environment.
In the field of soft contact, there’s an interesting aspect related to adhesives inspired by tree frogs. These adhesives provide researchers with an exciting opportunity to investigate the limitless possibilities of technology inspired by nature [35].
They often feature a micropillar pattern connected via a soft homogeneous baselayer to a rigid support. The pioneering works of researchers like Drotlef et al. andLangowski et al. brought forward these designs [23], [36]. These advancements

2.3. Finite Element method for wet adhesion and capillary11

F IGURE 5: A cross-sectional illustration shows the contact between two elastic spheres
using three models: (a) Hertz contact, (b) JKR, and (c) DMT models. In Hertz theory, nointeractions are considered, while the JKR theory accounts for surface forces only within
the contact region. On the other hand, the DMT theory considers both surface forces
within and attractive forces outside the contact region. Solid lines represent the contact
configuration at zero load, and dashed lines represent the contact configuration when an
applied load, P, is present.[34]haven’t just stayed in the scientific contributions but have been used in practical applications, significantly affecting the detection of slipping and enhancing industrial
applications [37]. For example, the legged walking robot introduced by Bloesch etal. [38] changed how slip detection algorithms work, making them more accurate
and efficient by focusing on the robot’s perspective of the situation.

2.3

Finite Element method for wet adhesion and capillary

Wet adhesion and capillary forces represent a fundamental challenge that stems
from the complex interaction between solid and liquid interfaces [39]. Recent research by scholars like Gholamreza and Majid has focussed on the effects of groove
height, width, and contact angles on capillary phenomena [40]. The complexity thatarises when solids interact with liquids, especially in scenarios featuring complex
solid geometries, has led to adopting the Finite Element (FE) method as a powerfultool to study these intricate phenomena [41]. Luo et al. [42] used the FE method
to develop a numerical simulation model for droplet spreading on topologicallyrough surfaces. Moreover, a prior study [43] delved into the deformation of a twodimensional spherical liquid droplet on a horizontal surface. This study presented

12

Chapter 2. Background and Related Worksa mathematical model that described the deformation, using the time-dependent
Navier-Stokes equations in a time-dependent domain.
Over the past several decades, the finite element method (FEM) has proven to bea crucial tool in simulating a wide range of physical phenomena [44], and this holds
true for research into wet adhesion and capillary action in the realm of robotics.
The finite element method involves discretizing complex systems into smaller, moremanageable elements and solving mathematical equations for each element. In thecontext of wet adhesion and capillary effects, this computational technique offers apowerful means of modeling and predicting the behavior of liquid interactions withvarious surfaces.
One of the primary areas where FEM has made significant contributions is inunderstanding how capillary forces play a role in the adhesion of soft materials
to surfaces [45]. This is particularly relevant in the design of robots that need tomaintain stable contact with surfaces in wet environments. FEM simulations allow
researchers to investigate how capillary forces affect the interaction between soft
materials and surfaces under different conditions [46].
This insight is invaluable for designing robots that can effectively navigate andinteract with both dry and wet surfaces. FEM simulations have also been an important method to investigate the mechanics of wet adhesion and capillary forcesin more intricate environments [47]. For example, when designing robots that canadapt to complex terrains, such as rocky landscapes or underwater environments,
researchers employ FEM to model the interactions of fluids with irregular surfaces.
These simulations help engineers optimize the adhesion mechanisms to ensure thatthe robots can securely adhere to the surfaces they encounter [48]. Moreover, FEM
enables the evaluation of the effect of various parameters, such as surface roughness
and liquid properties, on adhesion, leading to more accurate and adaptable robotic
designs [49]. The collaboration between Finite Element Method (FEM) and wet adhesion research contributes to creating novel robotic systems [50]. It facilitates thecreation of soft robots capable of scaling walls, adhering to irregular surfaces, andperforming tasks in challenging environments where traditional rigid robots would

2.4. The contribution of this thesis

13

struggle [51]. The results from FEM simulations help design soft robot parts thatcan effectively use capillary forces to stick, even on wet or uneven surfaces.
In summary, finite element simulations play an important role in researching wet
adhesion and capillary phenomena in the field of robotics. These simulations help
us understand how these forces influence the interaction between robots and surfaces, leading to the development of more adaptable, efficient, and versatile robotic
systems capable of performing tasks in a variety of environments, including those
involving liquids and irregular surfaces. This approach brings together engineering, material science, and computer science, promoting the advancement of robotic
technology.

2.4

The contribution of this thesis

In this research, we focus on exploring both the theory and real-world aspects ofsoft pads with actively morphology adaptations. To bring these concepts to practical applications, we employ both the analytical model and the Finite Element (FE)method, a robust tool, to achieve stable locomotion for our robot foot in both casesof dry and wet surfaces. The main goal is to understand how a flexible shape cansignificantly improve the stability of the foot when it moves on different surfaces.
The important findings of this research are explained below:• We present a mechanics of morphology changeable soft pad that can actively
change the morphology to adapt to the environment. This concept is an important aspect of embodied robot which is grounded in the idea that intelligenceand cognition are tightly linked to physical embodiment and interaction withthe environment.
• We suggest designing a soft pad capable of transitioning from a flat elastic
sheet in the foot sole (suitable for walking in dry conditions) to a raised spherical dome, which enhances walking efficiency on wet surfaces.

14

Chapter 2. Background and Related Works• An analytical model is proposed to investigate the morphology of the soft pad,delving into the complexities of tribological aspects, including friction, adhesion, and capillary action. The findings of our study indicate that, in comparison to the significant size of our robot’s foot, capillary action plays a relatively
minor role when compared to friction and adhesion. Nevertheless, we anticipate that in smaller-scale applications, capillary action could become moresignificant. We plan to explore this aspect further in our future work.
• A mathematical model, based on the FE method, is proposed to delve into
capillary action, and its findings are validated through force measurement experiments.
• Practical experiments involving a robot foot equipped with these soft pads
are presented, showcasing their capacity to enhance walking efficiency acrossdiverse terrains. In this research, we introduce the soft pad foot designs for
robots in both large and small scale, specifically engineered to enhance walking efficiency on wet surfaces.
• We conduct practical experiments involving a hexapod robot equipped withthese adaptive soft pads, thereby highlighting their substantial contribution toenhancing walking efficiency.

In conclusion, this research has explored the aspects of soft contact locomotion, wetadhesion, and capillary forces, revealing novel possibilities. These possibilities may
be used in practical applications to improve the stability and adaptability of robotic
systems in different environments. This work represents the collaboration of biological inspiration with technological applications, providing solutions for solving
recent problems.

15

3 Bioinspired robotic foot facilitating
wet adhesion
3.1

ANALYTICAL MODEL

3.1.1

Morphology computation model

The conceptual mechanics of morphology changeable soft pad , along with aprototype foot equipped with soft pads (illustrated in Fig. 1), represents a important aspect of this study. These soft pads, integrated into the sole of the foot, possess
an adaptability feature that enhances walking performance on wet surfaces. This
adaptability is achieved through the controlled application of compressed air, causing a transformation in the morphology of the soft pads. In this section, we delve
into the theoretical underlying of this transformative process, explaining the fundamental principles of interaction of these soft pads with a wet floor.

F IGURE 6: Soft Pad Configurations: (a) Depressurized (flat circular soft pad). (b) Pressurized (dome-shaped soft pad) without ground contact. (c) Pressurized with groundcontact during walking. (d) Geometric characteristics (at constant pressures) of the soft
pad with and without contact (represented by dashed lines).

16

Chapter 3. Bioinspired robotic foot facilitating wet adhesion
As the robot walks, an inflated soft pad comes into contact with the floor, exert-

ing a contact force represented by F. To comprehensively understand this interaction, we first discuss an analytical model that delves into the relationships between
the morphology of the soft pad and the values of F and air pressure, denoted as p.
The various states of a soft pad are illustrated in Fig. 6(a-c): in its initial state, where
no pressure is applied, it remains a flat circular sheet; upon inflation by air pressure, it transforms into a spherical dome. Next, when this dome pad makes contact
with a floor, the contact force acting on the pad causes the pad’s condition to shift
from a state of high potential equilibrium (characterized by a spherical shape witha resultant volume V1 ) to a lower potential state. In this lower potential state, thepad assumes the form of a thin elastic sheet, as depicted in Fig. 6(d). Examining theenergy equation during this process, the work of the pressing force on the foot (WF )will be equal to the total work of various force components arising throughout theprocess, including: the work to push a portion of the air mass out of the dome (Wp ),the work of the elastic sheet’s tension force (WTen ), and the work of the dry adhesive
force (Wad ).

WF = Wp + WTen + Wad .

(3.1)

The work Wp is equal to the difference of potential energy stored in the compressed air (∆U p ), which can be estimated using the following expression:
Wp = ∆U p = pV1 − pV2 .

(3.2)

Based on the geometric properties displayed in Fig. 6(d), we can calculate the volumes at the two states of the dome as follows:πc1 (3r02 + c21 ),6ππ
V2 = c2 (3r02 + c22 ) − a(3b2 + a2 ).
6
6
V1 =

(3.3)
(3.4)

3.1. ANALYTICAL MODEL

17

X

O
r0
R
Rz

Y
dCRz

R

dv
dv’

c
dl dl’Z

F IGURE 7: The spherical surface differential explanation for tension energy: Tension energy is determined by integrating the differentials of spherical segments along the z-axis.

Thus, ∆U p can be estimated as :ππ
∆U p = pV1 − p c2 (3r02 + c22 ) + p a(3b2 + a2 ).
6
6

(3.5)

Examining the work of tension force which is equal to the difference in elastic tension energy WTen = Ute − Ute0 , where Ute0 is the tension energy of dome withoutpressing force. So, at the energy balancing state, we have Ute0 = pV1 .
Next, we examine the tension energy Ute at the spherical dome surface as thepotential strain energy of an elastic sheet under constraint. The classic model presented by [52] for representing the inflation of an elastic balloon doesn’t directly
apply to the specific portion of the pad, i.e., the spherical dome shape, given the related constraints. In this context, we introduce a new definition of tension energy as
the elastic energy stored within the thin sheet when stretched in all directions (refer
to Fig. 7).
Let’s consider an differential area dv′ dl ′ on the sheet’s surface located at a heightof ( R − c) along the Z-axis within the coordinate system O − XYZ. This area falls
within the region highlighted in red, having a radius of Rz . We can assume that
under low air pressure, this area undergoes stretching to become dvdl at the sameheight (with the value of Rz remaining constant). Consequently, the work required

18

Chapter 3. Bioinspired robotic foot facilitating wet adhesion

for this stretching is confined to the XY-axes, as indicated below:

dUte = dWx + dWy =1
1
Tdu x duy + Tduy du x ,
2
2

(3.6)

where u x , uy is the displacement of a differential element of the sheet following xaxis, y-axis. Here, T represents the elastic tension in the thin sheet. Assuming uniform deformation of the thin sheet at all points and in all directions, the elongation
along the XY-axes can be estimated as follows:

du x = duy ,(3.7)

d ( l − l ′ ) = d ( v − v ′ ),(3.8)

dl = dv.

(3.9)

Taking into account the deformation of the sphere, we divide the sphere into segments denoted as dCRz . Consequently, the tension potential integrates the workrequired for the deformation of each soft pad segment along the Z-axis:

dUte = dWx + dWy =1
1
Tdu x duy + Tduy du x ,22(3.10)

Because du x = duy = d(l − l0 ) = d(v − v0 ) = dl = dv,(3.11)

So dUte = Tdldv,(3.12)

then dWrz =Z
dCRzdUte = TdCRz ,Where:• Ute is the tension energy.
• T is the elastic tension of the thin sheet.
• dv represents the differential of area.
• dl is the differential of length.
• Rz signifies the radius of the red-colored band, which remains constant.

(3.13)

3.1. ANALYTICAL MODEL

19• R denotes the original radius of the sphere.
• c stands for the thickness of the thin sheet.
This approach allows us to quantify the tension energy for this specific constraintdriven deformation of the soft pad segment along the Z-axis. By integrating thework associated with the deformation of the soft pad segment along the z-axis, we
can estimate the tension potential as follows:

Ute =

Z
S

dWrz =

Z

2π Rz Tdl.

S

(3.14)

Subsequently, the differential variable dl is expressed as a function of the differential
variables dx and dz:

dl =

p

dx2 + dz2 , x =dx = √

p

R2 − z2 ,−zdz.
R2 − z2(3.15)(3.16)

So, the differential variable dl is summarized as below:
s
dl =

z2
dz2 + dz2 ,R2 − z2=√R
R2 − z2dz.

(3.17)(3.18)The radius of the round segment of the sphere following the z-axis is represented asa function of the spherical radius and the z-coordinate.

And Rz =q( R2 − z2 ).

(3.19)

20

Chapter 3. Bioinspired robotic foot facilitating wet adhesion

Insert Eq. 3.18, 3.19 into Eq. 3.14, we have:

Ute =

Z R
R−c

2π

=

q

( R2 − z2 ) T √

Z R
R−c

R
R2 − z2dz,

2πTRdz,(3.20)(3.21)

R= 2πTRz,(3.22)

R−c= 2πTRc.

(3.23)

Because 2Rc = r02 + c2 , the tension potential is summarized as below:
Ute = πT (r02 + c2 ).

(3.24)

We consider the tension T when the soft pad inflates:
dl = du + dl ′ , and dv = du + dv′ .

(3.25)The elastic tension that is considered for each element of differential of sphere area:
du
du
=
k
,
dl ′
dv′
dl − dl ′dv − dv′
=k
=k
,
dl0
dv′
dl
dv
= k ( ′ − 1) = k ( ′ − 1),
dl
dv
T
dl
dv
So + 1 = ′ = ′ .
k
dl
dv
T=k

(3.26)(3.27)(3.28)(3.29)

Then, we have:
dl
− 1),
dl ′
dv
= k ( ′ − 1).
dv

2
T
dldv
dS
So
+ 1 = ′ ′ = ′.
k
dl dv
dS
√
√
S − S′
Then, T = k √.
S′
T = k((3.30)(3.31)(3.32)(3.33)

3.1. ANALYTICAL MODEL21

In step 1 when the spherical pad is inflated by inner pressure without any pressing
force, We have:

S=πr02 + c21
q

So T = k, S0 = πr02 ,r02 + c21 − r0
r0.

(3.34)(3.35)(3.36)

Insert into Eq. 3.24:
q
Ute = πkr02 + c21 − r0
r0(r02 + c21 ).

(3.37)In step 2 when the spherical pad is inflated by inner pressure with a pressing force,
We have:
S = π r02 + c22 − a2 , S0 = πr02 ,q
r02 + c22 − a2 − r0.
So T = kr0(3.38)(3.39)

This result leads to the potential energy for state 1 of a spherical soft pad as below:q
Ute = πkr02 + c21 − r0
r0(r02 + c21 ).

(3.40)

Besides, the potential energy for state 2 of the spherical soft pad is built based ongeometrical parameters in this case:q
Ute = πkr02 + c22 − a2 − r0
r0(r02 + c22 − a2 ).

(3.41)

So the work of tension force is estimated as below:q
WTen = ∆Ute = πkr02 + c22 − a2 − r0
r0(r02 + c22 − a2 ) − pV1 .

(3.42)

22

Chapter 3. Bioinspired robotic foot facilitating wet adhesion
The work of adhesive force is defined:
Wad = γ12 πb2 .

(3.43)

Insert Eq. 3.5, 3.41, 3.43 into Eq. 3.1, we have:

WF =

Z

Fdz
3





2 + c2 − a2 2
r
π  2
π
2
= − p c2 3r0 + c22 + p a 3b2 + a2 + π k 0
6
6
r0


2
2
2
−π k r0 + c2 − a + γ 12 π b2 .
Equilibrium state will obtain when:
dWF
= 0,
db
dWF
dWF dWF dz
so:
=
+
= 0,
db
db
dz db

(3.44)(3.45)

with z is a function over c2 and a: z = c1 − c2 + a.
The pressing force is the derivation of total energy over a variant of displacement
following the z-axis.
dWF
= F,
dz
dz
d ( c1 − c2 + a )
da
=
= .
db
db
db

(3.46)(3.47)

Due to the geometrical constraint, the penetration depth a can be expressed as follows:a = R2 −q

R22 − b2 .

(3.48)The result of derivation leads to the equation as below:πpR2 − pq

R22 − b2 + 2γ12qR22 − b2 − F = 0.

(3.49)

3.1. ANALYTICAL MODEL

23
q

Because a << R2 , b << R2 , we have

pπ

R2 −

q

R22 − b2q

R22 − b2 ≈ R2 , so:

R22 − b2 + 2γ12 πq

R22 − b2 − F = 0,(3.50)

pπ R22 − pπR22 + pπb2 + 2γ12 π R2 − F = 0,(3.51)

and: pπ b2 + 2γ 12 π R2 − F = 0.

(3.52)

Finally, the radius of contact area can be expressed as follows:v
 2 2
uu F − 2γ12 π r0 +c2
t
2c2
.
b=
pπ

(3.53)

We consider the force that presses into the surface at the equilibrium of state 2.
dWFdz
dWF da dWF dc2 dWF db
+
+
.
=
da dz
dc2 dzdb dz
F=(3.54)(3.55)As below, we have:z = c1 − c2 + a,

F=

du
= 0,
db

dWF dWF−,dadc2(3.56)(3.57)

By the condition of Eq. 3.44 and Eq. 3.48:F=pπ  2b + a2 + r02 + c222

3kπ−( c2 + a )r0

qr02 + c22 − a2 + 2kπ ( a + c2 ) .

(3.58)

24

Chapter 3. Bioinspired robotic foot facilitating wet adhesion

Here, we found a homogeneous constrain as follows:( R2 − c2 )2 + r02 = R22





2
2


−→ R2 = r0 +c2 ,
2c2




( R2 − a)2 + b2 = R22







−→ R = a2 +b2 .
2

2aThe homogeneous constrain is rewritten as below:
r02 + c22
a2 + b2=.
2c2
2a(3.59)

When F = 0 −→ a = 0, b = 0, Eq. 3.58 and Eq. 3.59 become: 3kπ
pπ  22
F=
r0 + c −(c + a)2
r0

q

r02 + c2 + 2kπ ( a + c) ,r02 + c2
a2 + b2=.
2c2a(3.60)(3.61)The root of the system of equations Eq. 3.60-3.61 is the geometrical parameter c1 ofstep 1. Then we have:

R1 =r02 + c21.
2c1

(3.62)(3.63)The root of the system of equations Eq. 3.53-3.59 is the geometrical parameter ofstep 2 that includes c2 , a, b, and the geometrical constraint is assumed as below:

R2 =r02 + c22.
2c2(3.64)The roots of the system of the equation are solved by the numerical method vpasolveof MATLAB.

3.1. ANALYTICAL MODEL

3.1.2

25

Contact model with liquid film

In the previous section, we computed the specifications of a soft pad operating
in its spherical dome states, which changed by the contact force F and the innerpressure p. In this context, we make the assumption that the dome initially makes
contact with the surface through a surrounding liquid film, as depicted in Fig. 8.
Liquid molecules adhere to the surface of the dome pad along the contact line between the dome pad and the surface of environment. Our aim is to formulate amodel for the resistant force that arises at the contact interface due to this capillaryaction. This model serves to illustrate the role of the dome in improving locomotion.
We calculate the tangential resistant force as follows:

Fre = Ff r + Fad + Fcap ,(3.65)

Ff r = µN,(3.66)

Fad = τA.

(3.67)

where:

Let’s examine the contribution of capillary action to the overall tangential resistantforce. The different pressure between inside and outside of the liquid generates

F IGURE 8: Capillary action occurs upon contact with the wet floor, induced by a liquid
layer surrounding the soft pad with a height of h. In this situation, the pad exhibits atendency to slip to the left.

26

Chapter 3. Bioinspired robotic foot facilitating wet adhesion

resistance when the spherical pad attempts to move, which is referred to as the tangential capillary force. Consequently, the tangential capillary force can be calculated
as follows:

Fcap =ZS∆pl dScosϕ,(3.68)= ∆pl h2ri .

(3.69)

Where γl is the surface tension of liquid, the different pressure between inside andoutside liquid is estimated as:∆pl = γl



1
1
− −
r l ri,and h ≈ 2rl .

(3.70)(3.71)

So, the different pressure between inside and outside liquid is rewritten as:21∆pl = −γl
+
,
h ri


1
2
+
r.
Fcap = −2hγlh ri i(3.72)(3.73)

Finally, the capillary force is summarized as:

Fcap = −2γl (2ri + h) .

(3.74)The phenomenon of liquid rising along a solid surface due to wet adhesion is afascinating interplay of intermolecular forces and gravity. This capillary rise, as it is
known, occurs when a liquid is in contact with a solid substrate. At this interface,
the liquid molecules experience adhesive forces that attract them to the solid. This
adhesive force is often due to the surface tension of the liquid γl , which causes it tospread across the surface.
As the liquid molecules are attracted to the solid surface, they start climbing
upwards. The height to which the liquid can climb is determined by the balance3.1. ANALYTICAL MODEL

27

between this adhesive force and the gravitational force acting on the liquid. Whenthese forces are in equilibrium, the liquid ceases to rise further, and a stable meniscus
or curved shape of the liquid is formed.
The shape of the meniscus is governed by the contact angle, which is the angle
between the tangent to the liquid surface at the solid-liquid interface and the solid
surface. A small contact angle corresponds to a high degree of wetting and results
in a concave meniscus, while a large contact angle leads to a convex meniscus.
Applying Newton’s laws to the system of force for wet adhesion and the weightof amount of wetted liquid, we obtain:
Pl = ∑ Fsl sin(θ + α),(3.75)

ρgVl = 2πγs ri sin(θ + α).

(3.76)In Fig. 8, the radius of the round boundary of the wetted area is determined as:

ri = b + h cot α.

(3.77)

So, the Eq. 3.76 is rewritten as:

ρgVl = 2π (b + h cot α ) γs sin(θ + α),
 i
π
2
2
2
so, hρg π (b + h cot α ) −(b + h cot α ) + b + (b + h cot α b3h= 2π (b + h cot α ) γs sin(θ + α),

hρ g
b
then,
2 (b + h cot α ) −b − b = 2γs sin(θ + α).
3(b + h cot α )Then, the related equation is rewritten as:


b
hρ g
b + 2 h cot α −b = 2γs sin(θ + α).
3(b + h cot α )

with

b
≈ 1.
(b + h cot α )(3.78)(3.79)

28

Chapter 3. Bioinspired robotic foot facilitating wet adhesion

Finally, the height of wetted liquid on the spherical pad is summarized as:sh=3γs sin(θ + α).
ρg cot α(3.80)

Additionally, the contact angle is expressed through the surface tension of the liquidand solid as follows: rγscosθ = 2−1 .
γl(3.81)

Insert these values into Eq. 3.68, resulting in the simplified estimation of Fcap asfollows:"
Fcap = −2γl 2b + (2 cot α + 1)s#
3γs sin(θ + α).
ρg cot α(3.82)As a result, we developed a model to elucidate the alteration in spherical padmorphology induced by air pressure. Subsequently, we demonstrated how thischange provides an advantage in enhancing tangential force through computational
analysis. The analytical model in this chapter is simulated by the Matlab code that
attached in Appendix 7.2.1 and 7.2.2 (Chapter 7: Appendix).

3.1.3

Optimal function for design parameters

Building on the analytical model from the previous section, which formulated
the geometrical shape of the dome and the wet-adhesive force, this section introduces an optimization function. This function helps us choose the optimal parameters for a foot equipped with dome pads. This purpose revolves around the initial
radius, which we call r0 . It’s not just an arbitrary number; it has a particular mission.
We’ve set r0 with the aim of maximizing the number of contact pairs between thesoft sphere pads and the ground surface. This strategy increases the area of contact
and also makes the capillary phenomenon more effective.
In terms of optimizing the dome size on the foot, we start by assuming that the
foot’s area is constant, and it depends on the robot’s size. We also suppose that the3.1. ANALYTICAL MODEL

29

maximum radius R of the dome, starting from the initial radius r0 of a thin sheet,should be about 1.5 times r0 .
This calculation gives us the surface area of the dome, which is represented by
Sdome . Now, Sdome can be computed using this formula:
Sdome = 2πR R +q

R2 − r02√ 
3 3+ 5=2πr02 = 7.8πr02 .

It’s roughly 7.8 times the initial area, which means it elongates by around 180%.

F IGURE 9: Optimization solution for design. (a) phenomenon of rising wet liquid. (b)the arrange of inflated dome in the foot and the area caculation.

This is significantly less than the 845% elongation threshold we observed with silicon Ecoflex 00-20. The number of domes on the foot, indicated as n, approximates
the area Se . Additionally, Stri stands for the area of the triangle formed by threecenters, which you can see in Fig. 9. This aspect is crucial to our design and functionality considerations.
As such, we calculate Sarea as follows:

R = 1.5r0 ,(3.83)

Sarea = n(Sc + Se ) = n(Sc + Stri − 3S p ),


π πR2
2
2
= n πR + 2R cos −
,
3
2
!
2
πr
9n
π
=
πr02 + 2r02 cos − 0 .
432(3.84)(3.85)(3.86)

30

Chapter 3. Bioinspired robotic foot facilitating wet adhesion

The number of pads in a constant area is expressed as a function of the inertial
radius:

n= 

Sarea
√  2.
π
3 + 3 r0

(3.87)In this optimal calculation, we focus on the relationship between the initial radius r0
and various critical performance metrics, such as the robot’s leg’s maximum pressing force and the tangential capillary force. Referring to Eq. 3.60, the pressing force
is determined as follows:q 3kπ
pπ  22
F=
r0 + c −(c + a) r02 + c2 + 2kπ ( a + c) ,
2
r0
√
q
5+3
2
2
r0 .
c = R + R − r0 =2
In the case of an inflated dome with a sphere radius of 1.5r0 , the inner pressure is
calculated as:

F = 0, a = 0,(3.88)

4.3K.
r0(3.89)P=

When the pressing force reaches its limit, the dome is re-pressed to the state of an
approximately flat radius r0 , and the total pressing force on the foot is computed as:

Ftot = nSsh P,=(3.90)

Sarea
4.3K
 πr02
,
√
π
r0
2
+
3
r
0
3

(3.91)

4.3πKSarea
=
√  .
π
3 + 3 r0

(3.92)

The calculated value of Ftot , derived from the equation presented above, is influenced by the radius r0 . A smaller r0 leads to a higher constraint on the pressing

3.1. ANALYTICAL MODEL

31

force. Consequently, we have chosen r0 = 0.5mm as it represents the smallest feasible size within our manufacturing capabilities, thereby optimizing the constraint on
the pressing force.
According to the force balancing equation, the total capillary force a calculated
as below:∑ Fcap = Pli ,(3.93)


so: nγcosθSd = Dl g V0 − nVsph ,then: nγ cosθ 2πa = Dl g

Sbou h −nπh 3a2 + h2
6

(3.94)
!
.

(3.95)

So, the related equation is summarized as below:2γcosθ
a=
Dl g

π
3

√ 
+ 3 Sbou r02 h
Sarea π
3a2 + h2−.
6(3.96)

Using the geometrical constraint of the dome parameters:( R − h )2 + a2 = R2 ,(3.97)

so: a2 = (2R − h) h,(3.98)= (3r0 − h) h,q
then: a = (3r0 − h) h.

(3.99)(3.100)

The relation between the rising liquid heght h and r0 is:
2γ cosθDl g

q(3r0 − h) h =

π
3

+

√  2
3 r0 hπ
9r0 h − 2h2−.
6(3.101)

In investigating the complex interaction of important factors in our soft pad design,
we used the computational tools of MATLAB, particularly the fimplicit function. The
result of this estimation is shown in Fig. 10.
In our computation, we focus on the influence of several influential parameters
including γ = 0.073 N/m (the surface tension), θ = π/3 rad (the contact angle),

32

Chapter 3. Bioinspired robotic foot facilitating wet adhesion

F IGURE 10: The graphical representation illustrates the changes in the liquid’s risingheight h versus the initial radius r0 .

Dl = 1000 kg/m3 (the density of liquid), and g = 9.81 N/m2 (the acceleration due
to gravity). This calculation helps us see the connection between two important
things: h, the height of the liquid, and r0 , the initial size of our soft pad. The curveof h as it moved through changes in r0 showed a clear picture of the optimal range
for this important factor. This result shows that the optimal range is between 0.3 and0.8. Determining the exact optimal value of r0 is challenging. However, considering
our manufacturing capabilities, the range of r0 around 0.5mm aligns well with the
optimal r0 range that maximizes the capillary force represented by the liquid heighth.

3.2

Experimental

In this section, practical experiments were conducted to investigate calculations
and ideas. The concept of the changeable dome pad is illustrated in Fig. 11. This
dome pad can change its morphology by adjusting the inner pressure inside.
The soft pad is made of a silicone material called Dragon Skin 10, provided by
Smooth-On, Inc., USA. Inside it, there’s a rigid, red, 3D-printed structure called the3.2. Experimental

33

F IGURE 11: Design of the soft pad. (a) Soft pad without air pressure. (b) Illustration of
the soft pad design concept. (c) Soft pad under the influence of air pressure.

rigid bone. This part is essential for changing the soft pad’s shape.
A small chamber with a 3 mm diameter is designed as a pathway, guiding compressed air into the soft pad’s container. By controlling the air, we make the soft pad
inflate. Besides, the pad thickness is quite thin with 0.4 mm.
The main goal of choosing r0 was to optimize the contact area between the soft
spherical pads and the ground. This helps increase the friction and adhesion, and it
also makes the capillary effect more effective.

3.2.1

Inflation and contact-related shape verification

Fig. 12 describes the setup used in our experiments to investigate the analytical model. The experimental system comprises an IMADA force gauge attached toan acrylic plate, which is driven by a motorized linear stage. We also have a soft
pad foot that can change its shape by adjusting the inner pressure with a pressure
regulator, a pressure sensor to measure the inner pressure, and a 400-CAM058 microscope (SANWA SUPPLY) to measure the dimensions of the dome. In the dome
shape experiment without applying pressing force, we adjust the inner pressure
at different levels to measure the dome dimensions. Then, we conduct an experiment on the dome shape with pressing force. In this case, the IMADA force gauge
is driven to move into the inflated dome. When the acrylic plate attached to the
IMADA force gauge presses into the dome, the pressing force is displayed on the
IMADA force gauge. Finally, we measure the dome shape parameters using the34

Chapter 3. Bioinspired robotic foot facilitating wet adhesion

F IGURE 12: Experimental design for examining morphological changes.

F IGURE 13: Dome morphology dimension processed by 400-CAM058 microscope(SANWA SUPPLY): a- without applying pressing force, b- with applying pressing force.

400-CAM058 microscope (SANWA SUPPLY). Fig. 13 shows the processed image byusing 400-CAM058 microscope for dimension measuring in 2 case: with applying
pressing force and without applying pressing force. The 400-CAM058 microscopeis used with UM12-CAM software for displaying the scaled dome on the softwareinterface.
In the dome shape experiments without applying pressing force, we manipulated the air pressure, setting it to various levels: 55 kPa, 65 kPa, 75 kPa, and 85 kPa.
Upon completing these experiments, we plotted the results and compared themwith calculated values based on the methodology outlined in the previous section(see Fig. 14, 15). The observed errors typically fell within a range of approximately±3%. Remarkably, our analysis indicated that the magnitude of errors exhibited3.2. Experimental

35

F IGURE 14: Experiment state 1: Effects of different pressures in the soft pad without floor
contact were compared using Matlab (Code matlab in Appendix 7.2.1).

a rising trend in tandem with increased air pressure. This correlation could be attributed to changes in the elasticity coefficient k, which tended to increase as deformation intensified. To assess the pad’s elastic properties, we conducted tests using
an MX2 Motorized test stand. In practice, it’s worth acknowledging that the elasticity coefficient can fluctuate depending on the Young’s modulus, particularly in
scenarios involving significant deformation. However, in this specific study, we assumed that the deformation of the soft pad dome occurred within the linear regionof the Young’s modulus, leading to a constant elasticity coefficient.
Transitioning to the experiment of dome shape with applying pressing force, we
maintained a constant air pressure of 75 kPa. Then, we considered varying levels ofcompressive force, setting them at 0.1 N, 0.2 N, and 0.3 N, respectively. In this, wecollected parameters including b, r0 , and d = (c2 − a), as illustrated in Fig. 6d. The
radius and height of the dome were subsequently calculated using the following

36

Chapter 3. Bioinspired robotic foot facilitating wet adhesion

F IGURE 15: Experiment state 1: dimension of dome heigh and dome radius.

formulate:qc2 =r02 − b2 − d2

2

+ 4r02 d2 − r02 − b2 − d2

2d 2r + c22
R2 = 0.
2c2,Additionally, based on Popov’s research (2010) [39], the relative surface energy√
between the silicone material and acrylic, defined as γ12 = 2 γ1 γ2 , was found tobe approximately 0.006 J/m2 .
Fig. 16, 17, and 18 present the experimental and calculated results. Fig. 16 compares the dome’s geometrical shape under various pressing forces. Fig. 17 and 18
present the dimensions of dome height, radius, adhesive radius, and depth, including error bars.

3.2. Experimental

37

F IGURE 16: Experiment state 2: Response of the soft pad under air pressure of 75 kPa to
being pressed at different forces (Code matlab in Appendix 7.2.2).

F IGURE 17: Experiment state 2: dimension of dome heigh and dome radius.

38

Chapter 3. Bioinspired robotic foot facilitating wet adhesion

F IGURE 18: Experiment state 2: dimension of adhesive radius and adhesive deep.

3.2.2

Tangential force experiment

In this section, we investigate the influence of capillary action on the soft pad
dome’s resistance to slippage. The experimental setup, illustrated in Fig. 19a, incorporates two motorized linear stages responsible for vertical and horizontal movements. Our experiments encompassed both wet and dry conditions, with the soft

F IGURE 19: Experiment to assess the resistance of the soft pad to tangential forces: (a)Setup of the experiment. (b) Soft pad under air pressure without simulated floor contact,(c) and with simulated wet floor contact.

3.2. Experimental

39

pad’s morphology ranging from flat to domed at 65 kPa, 75 kPa, and 85 kPa, respectively. We calculated the resultant capillary force Fcap as the difference between Fwet
and Fdry , the tangential forces measured under wet and dry conditions. In otherwords, Fcap = Fwet − Fdry . These tangential force measurements were recorded prior
to the onset of complete slippage, known as incipient slip (see [53]). Each experimental trial began by pressing the soft pad against a surface, securely affixed to a
3-axis force sensor (DSS300-U, TecGihan, Japan), with varying contact force values.
Subsequently, the horizontal linear stage was set in motion, providing tangential

F IGURE 20: (a) Comparison of resulted tangential forces among flat soft pad, and domed
pad under pressures of 65 kPa, 75 kPa, and 85 kPa, respectively; on both wet and dry
floors. (b) Zoom-in range with comparison of tangential forces under dry (dashed line)
and wet (solid line) conditions several air pressure values.

40

Chapter 3. Bioinspired robotic foot facilitating wet adhesion

displacement to the soft pad at a low speed of 0.2 mm/s to minimize viscosity effects. The outcomes of these experiments are summarized in Fig. 20.
When considering the dry condition, a flat soft pad (i.e., without the sphericaldome induced air pressure) displayed the highest tangential forces across a range of
normal forces (up to 0.4 N). However, under wet conditions, the resulting tangential
force on the flat-surfaced soft pad decreased significantly. This result emphasizes
that a flat soft pad morphology is only suitable for dry environments, as it may experience reduced tangential forces, leading to potential slippage on wet surfaces. In
the case of the domed soft pad morphology, we ensured that slippage did not occur by employing a slow pad movement speed of 0.2 mm/s. Additionally, the air

F IGURE 21: Comparison of capillary action in wet condition (Fcap = Fwet − Fdry ) between
experiment (with error bars) and estimation results.

3.2. Experimental

41

pressure was adjusted to assess the morphological effect. Fig. 20a-b indicate minimal differences in tangential force values between the two conditions: wet (solidlines) and dry (dashed lines). Most notably, the tangential force associated with thedomed soft pad surpassed that of the flat pad in wet conditions, indicating superior
walking stability with a domed soft pad. Therefore, we conclude that a flat soft pad
is suitable for walking on dry surfaces, and a domed pad is preferable for walking on wet
surfaces.
Concerning the domed soft pad, we assessed the effect of its morphology (determined by inner pressure) on the resulting tangential force. The dome cannot endure
high normal force when pressed against the contact surface, as this force could exceed the potential energy stored in the inner pressure. As such, there exists a normal
force limit for each morphological state of the dome. According to our measurements, the limits for normal force at 65 kPa, 75 kPa, and 85 kPa are 0.15 N, 0.3 N, and
0.4 N, respectively (as shown on the horizontal axis of Fig. 9a). Our proposed analytical model (refer to Eqs. (17)-(19)) reveals that a lower normal force results ina reduction of the tangential force during the contact phase. Therefore, increasing
the air pressure to modify the morphology is essential for raising the normal force
limit and, subsequently, enhancing the tangential force. Moreover, substituting thesilicone material, such as switching from Dragon Skin 10 (Shore Hardness 10A) to
Dragon Skin 20 (Shore Hardness 20A) or Dragon Skin 30 (Shore Hardness 30A), canelevate the normal force limit.
In Fig. 21, we compare the resultant capillary force Fcap of a domed pad in wet
conditions across three different air pressure levels (65 kPa, 75 kPa, and 85 kPa) withthe estimated values from our analytical model (see Eq. (28)). Despite the complexity introduced by the liquid film at the contact area, leading to errors ranging from
10 % to 26 %, the experimental results align with the predicted trend.
In this chapter, the design and fabrication process have been explained, and thephysical significance will be clarified through the energy equation. The results obtained from calculations and simulations will be validated through experiments. In
the next chapter, the mathematical model based on Finite Element method that plays

42

Chapter 3. Bioinspired robotic foot facilitating wet adhesion

an important role in this research will be explained.

43

4 The finite element analysis (FEA) for
representing wet
4.1

The FEA-based solution

In the previous chapter, we presented an analytical model to explain the interplay between solid and liquid in wet adhesion phenomena. Nonetheless, this analytical model has constraints when dealing with specific geometrical shapes andis incapable of addressing issues related to general shapes that are non-integrable.
This section introduces a novel Finite Element Method (FEM)-based solution to handle scenarios involving general geometrical shapes. The strategy proposed in thissection aims to serve as a comprehensive solution for understanding wet adhesion
phenomena in cases of solid objects with arbitrary geometrical shapes.
Several studies have employed the FE method to compute geometric parametersin wet adhesion phenomena, such as [54], [55]. Additionally, researchers have published studies utilizing the FE method to describe capillary phenomena [56]. Furthermore, the authors in research [54] provided estimations for calculating capillary
forces in the vertical direction using the FE method. Compared to the mentioned
studies, our FE method model in this study contributes the following novelties:
firstly, our model considers deformation in soft contact, thus accounting for wet
adhesion interaction based on deformed objects; besides, the results we investigate
using the FE method in this study is focussed on the tangential effects of wet adhesion force and capillary force, factors used to assess the sliding mechanism in robot
locomotion; finally, the FE method model used in our study is applied to calculate
in the case of multiple dome pads, which has not been addressed in other studies.

44

Chapter 4. The finite element analysis (FEA) for representing wet

4.1.1

The general problem of capillary phenomenon

The conceptual design and mathematical model for the foot with soft pads arepresented in Fig. 2. In previous chapter, we proposed a method for modeling thecapillary force, which relies on the shape of the dome pad, subject to alterations due
to compressed air pressure and applied force. However, this approach has certain
limitations; it mainly applies to specific pad geometries that are integrable. Whenin motion, these pads can undergo deformations as they interact with the environment. Consequently, when we consider the general geometry of the pad, the curved
surface of the liquid becomes quite complex, leading to certain unidentified parameters affecting the capillary force.
Furthermore, the influence of the distribution of these dome pads on the capillary force remains unclear, especially in cases where the liquid rises between thesepads. In this study, we introduce a novel approach that using the Finite Element(FE) method. This approach is founded on the equilibrium energy equation of theliquid. This approach’s advantage is that it allows us to explore wet adhesion andcapillary forces, not just at the individual pad level but also within the context of the
distributed pattern of pads on the foot. By doing so, we aim to gain a more comprehensive understanding of how these forces operate and interact, enhancing ourknowledge of the interplay between the soft pads and the environment. This could
be a significant step toward improving the design of these pads and enhancing theirperformance in practical applications.

4.1.2

The FEA-based model for the capillary phenomenon

The phenomenon of capillary rise has attracted the attention of researchers in
the history of modern science [57]. The energy balance governing the slow rise of
liquid in a capillary tube is composed of three terms: the work done in lifting theliquid (W), a term accounting for viscous dissipation (K) in the flowing liquid, anda potential energy term (-∆U) representing the energy stored in the liquid column

4.1. The FEA-based solution

45

F IGURE 22: Wet process: (a) Soft contact between 2 object. (b) Liquid molecular of wetadhesion phenomenon in the first time. (c)Wet-adhesion phenomenon in process . (d)Equilibrium state.

[58], [59]:W − K = −∆U,

(4.1)

This dynamic interplay of these components underlies the capillary rise phenomenon,
where the balance between lifting the liquid, energy dissipation, and the stored potential energy in the liquid column ultimately determines the height to which theliquid will ascend within a capillary tube. Understanding this balance is crucial in
the fields, from microfluidics to materials science, offering insights into the capillary
forces that shape the behavior of liquids in confined spaces.
We present Fig. 22, which illustrates the process of liquid rising onto the object
from the perspective of energy balance. Fig. 22a illustrates the interaction between
the soft object and the surface. Following this interaction, the liquid atoms undergo
upliftment, primarily propelled by the action of the wet adhesion force. The work ofthis force is distributed between two significant factors: the potential energy stored
within the liquid and the intermolecular interactions occurring among the liquid

46

Chapter 4. The finite element analysis (FEA) for representing wet

F IGURE 23: (a) Wet adhesion phenomenon in general geometry object. (b) Differentials
of liquid following polar coordinates.

atoms, a phenomenon often referred to as viscous dissipation. These fundamental mechanisms are pivotal in comprehending the process of capillary rise and are
instrumental in revealing the intricacies of liquid wetting on soft objects. This understanding carries implications in various fields where liquid-surface interactions
come into play, from engineering to materials science.
We call the We , Ke , Ue are the differentiation element following polar coordinatesof work of capillary, viscous dissipation, and potential energy, respectively.

W=

Z
S

We ds, K =ZS

Ke ds, ∆U =Z
SUe ds.

(4.2)

In this approach, we consider the balance of energy in each element following polar
coordinates Fig. 23:
We − Ke = −∆Ue .

(4.3)

In the role of capillary forces governing the elevation of liquid on a surface, we
can delve into the components that construct this phenomenon. Here, the capillary
force in each element is denoted as fc , while dl stands for the length of the differential curve. The number of differential layers is represented by n, and ln is the path ofthe object layer. The crucial parameter of liquid’s surface tension energy is called γ.
Additionally, ϕ1 characterizes the angle formed between the capillary force and the4.1. The FEA-based solution

47

curve of the object layer. These variables and their interactions within the context ofcapillary forces define the dynamics of liquid rise and play a significant role in understanding wetting phenomena on various surfaces. In this context, the equation
is formulated as follows:

We =

Z

f c dl(4.4)= ∑ ln γ cos ϕ1 .

(4.5)

Le

n

i =0

In the case of capillary rise on a wet object, the expression for viscous dissipation
can be formulated as follows. Here, Ve = Vn (hi ; ri ) represents the volume of eachliquid layer, dph denotes the hydro-static pressure at each liquid layer, ρ stands forthe liquid’s density, g symbolizes the acceleration due to gravity, and zn signifies theheight of the liquid layer:

Ke =

Z

Ve dph(4.6)

n= ∑ ρgzn Vn (hi ; ri ) .

(4.7)

i =0

The change in the gravitational potential energy (-∆U) can be calculated by integrating the hydro-static force (fh ) as liquid rises from z = 0 to z = h:
∆Ue =

=

Z

Z h
0

Pz dz

(4.8)

ρgVn (hi ; ri ) dz

(4.9)

= ∑ ρ gzn Vn (hi ; ri ) .

(4.10)ni =0Then, the Eq.4.3 become:
n

n

n

i =0

i =0

i =0

∑ ln γ cos θ − ∑ ρgzn Vn (hi ; ri ) − ∑ ln ρ gzn Vn (hi ; ri ) = 0.

(4.11)

48

Chapter 4. The finite element analysis (FEA) for representing wet

4.2

Implementation using SOFA (Simulation Open Framework Architecture) framework

SOFA (Simulation Open Framework Architecture) is a open-source framework
designed for the simulation of interactive, real-time physical phenomena. Developed by a community of researchers and engineers, SOFA has become a promising
platform for simulating complex, dynamic, and real-world scenarios across various domains, including robotics, biomechanics, medical simulation, and computer
graphics. One of SOFA’s distinctive features is its adaptability, allowing users to create highly detailed, customized simulations. This flexibility is achieved through acomponent-based architecture, where each element in the simulation is represented
as an individual module. These components can be easily integrated or replaced,enabling researchers and developers to manage simulations to their specific needs.
SOFA’s open-source encourages collaboration, innovation, and community-driven
development. It provides a large library of pre-built components, models, andsolvers, reducing development time and enabling users to focus on the core aspectsof their simulations. SOFA integrates with other established tools, making it flexible
and powerful. This allows users from diverse fields, like biomechanics, robotics,
and medical training, to create real-time, interactive simulations. SOFA’s powerful framework helps build realistic simulations that run in real-time, pushing theboundaries of virtual reality and computer modeling. Here, the SOFA framework isused to apply the model based on finite element method for capillary simulation.

4.2.1

The wet adhesion phenomenon simulation for various shape
objects using SOFA

To find the parameters of curved surface of liquid and capillary force, we use an
iterative loop, in which the function energy Qe is shown as below:
Qlayer = Wlayer − Klayer + ∆Ulayer .

(4.12)

4.2. Implementation using SOFA (Simulation Open Framework Architecture)49
framework
Base on the equilibrium energy concept in Eq. 4.12, achieving an equilibrium state
Qe = 0. To proceed, we divide the liquid element into layers, aligning them withthe object’s mesh. The layer that meets the equilibrium energy condition Qe = 0
is denoted as the "wet layer." Consequently, layers positioned below the wet layer
exhibit a positive energy function (Qe > 0), while those situated above the wet layer
manifest a negative energy function (Qe < 0). In our finite element method-based
approach, we employ an iterative process to systematically assess each wet layer,commencing from the lowest layer (i.e., the interaction layer between the object andthe surface). The iteration concluded once the condition Qe > 0 is no longer met,
determining the appropriate wet layer.
In each iteration cycle, the energy function Qe is determined by the parametersof the liquid surface formed during the wetting phenomenon on the object’s surface,
as illustrated in Fig. 24. Considering a layer assumed to be the rising liquid level,
the angle θ represents the angle of the surface deformation formed with the verticalaxis z. The ϕ1 angle, obtained from the geometric shape of the pad give us the tangential direction angle of pad surface at differentiation. So, The ϕ2 angle is the angleof the curvature of liquid considered from center. The points A1 , A2 , ..., An are therespective mesh points on the object’s surface. C1 , C2 , ..., Cn are the corresponding
points belonging to each layer situated on the curved surface of the wetting liquid
element in the polar coordinate system of the wetting liquid on the object. Due to theuniformity and symmetry of the liquid molecules, we can construct a circular curve
passing through C1 , C2 , ..., Cn . The differential angle ∆ϕ with respect to the center, as
viewed from each arc C1 C2 , C2 C3 , ..., Cn−1 Cn , is calculated as ∆ϕ = amount o2f layers (seeϕFig. 24b). Besides, α is the angle of considered slice of liquid in polar coordinate. So,
in the iteration cycle of considering each slice of liquid, we need to use the α j foreach cycle.

ai = zi tan(θ − i∆ϕ), with i = 0 : n(4.13)

50

Chapter 4. The finite element analysis (FEA) for representing wet−−−−→
By using the data from the mesh of the object, we determine the vector Ai Ai+1 andthe angle αi . Then, the points of curved surface are determined as below:
−−−→
Ci Ci+1 = ai cosα j ; ai sinα j ; zi ,(4.14)

Ci+1 = ( xci + ai cosα j ; yci + ai sinα j ; zci ).

(4.15)F IGURE 24: (a) The geometric parameters of the differential element of the liquid massin polar coordinate system. (b) The 2D projection of the differential element of the liquidmass onto its own normal plane.

4.2. Implementation using SOFA (Simulation Open Framework Architecture)51
framework
The radius of curvature is determined through the angle ϕ2 and the length of thecurve:n

ϕ2 R = ∑ Ci Ci+1 = C1 C2 + C2 C3 + C3 C4 + ... + Cn−1 Cn .

(4.16)

i =0

Then, the pressure difference between the outside and inside of liquid is calculated:∆p = Υl



1
1
+
R r


.

(4.17)

The capillary force in this case is calculated as follows: (where Ssection is the projectedarea of the wetting liquid on a plane perpendicular to the direction of motion, see
the Fig. 27):
Fcap = ∆pSsection .

(4.18)

The algorithm comprises three main stages: Processing index data, considering
each slice of the surface, and calculating the wet adhesion, executed in sequence.
The processing index data stage begins with the creation of a mesh structure forcontact. The user inputs the elements into SOFA and gets parameters for these elements. The contact layer is identified, and the surface is divided into smaller slices,with their indices arranged from low to high. Next, the considering each slice of thesurface is processed. A loop runs through each layer from 1 to the maximum layer.
Within each layer, coefficients such as C2 , C3 , C4 , C5 are determined. The algorithm
calculates Qe , and based on its sign, the process either returns to the first step orproceeds. Once the loop is broken, the wetted layer is determined.
Finally, the calculating the wet adhesion stage is entered. It starts by calculating
the wetted liquid area, followed by the computation of ∆p, which is important forthe subsequent steps. The algorithm then calculates the section of liquid volume ina defined direction and uses this information to calculate the wet adhesion. In thisstudy, there is another crucial step that we have not fully addressed, which is to investigate the influence of the calculated capillary force on the mesh of the object. The

52

Chapter 4. The finite element analysis (FEA) for representing wet

ultimate result to be examined is the equilibrium state of the object’s deformationand the updated capillary force based on that deformation. For objects with highstiffness, the capillary force is too small to cause significant deformation. However,a comprehensive model based on the finite element method requires addressing thisissue. In this case, a solution has been prepared for future work. We expect to use∆p as a parameter representing the force acting on each element of the mesh object,combined with the normal angle parameter of the element. This information willthen be used to calculate the force vector acting on each element. Subsequently, thisforce parameter will be used to calculate the deformation of the mesh object. We anticipate that this complete algorithm will propose a comprehensive solution in the
future.
This algorithm’s approach ensures a detailed and accurate analysis of wet adhesion for a given surface and liquid interaction. The algorithm of our finite element
method that use to estimate the capillary force is shown as the Fig. 25.
Based on the algorithm explained in this section, we have developed a Python
code in SOFA to simulate the curved surface of wet liquid for various geometricalshapes. Fig. 26 displays the results obtained through our novel approach, and theseresults align well with the observed physical phenomena and related research [60].
Notably, Fig. 26c presents the simulation of wet liquid for a general shape. Utilizing the energy balance equation, we determined the wet layer at each differential
element across the entire surface of the object. Subsequently, the geometrical deformation of the wet liquid was illustrated as shown in Fig. 26. Then, Fig. 27 illustrate
the method to calculate capillary following the direction of movement. (Eq. 4.18).
Next, we provide results calculated in the SOFA environment using the FE methodmodel. As we know, the geometric parameters of the liquid include two factors: theheight of the liquid rising above the solid surface and the radius of the liquid surface. These factors are related to the geometry of the solid, the tension energy ofthe liquid, and the contact angle of wet adhesion forces which depending on theqtension energy of the liquid and solid according to formula cosθ = 2 γγs − 1 [14].
lAccording to the formula, we will see that under wet adhesion conditions, cos theta

4.2. Implementation using SOFA (Simulation Open Framework Architecture)53
framework

F IGURE 25: The algorithm of finite element method for estimation of capillary force.

is in the range [0;1], meaning that the tension is in the range

 γl


;
γ
. Below, we runl4

the model to determine the parameters of the height of the liquid rising above thesolid surface and the radius of the liquid surface for values of liquid tension energy
determined respectively at 0.0016 N/m2, 0.002 N/m2, 0.004 N/m2. The size of thedome pad in the simulation is illustrated in Fig.28. It also shows the trend of theliquid rising on the solid surface in the context of increasing tension energy of the
solid. It is evident that within the range γ4l ; γl , the level of the wet-adhered liquid on the solid surface gradually increases. Fig.29 compares, at increasing levelsof liquid tension energy, the convergence rate to the maximum height of the liquid
rising on the solid surface, which increases with the same considered contact angle.
Finally, Fig.30 shows that the radius of the surface of the wet-adhered liquid block
will be proportional to the height of the rising liquid.

54

Chapter 4. The finite element analysis (FEA) for representing wet

F IGURE 26: SOFA simulation for capillary phenomenon. Note that red dotted boundaries show the uplifted water layer due to adhesion. (a) Dome shape object. (b) Truncated cone shape object. (c) General geometry object

F IGURE 27: Tangential capillary force estimation following the wettecd liquid and movement direction, where Ssection is the projected area of the wetting liquid on a plane perpendicular to the direction of motion.

F IGURE 28: Wet adhesion parameters simulating in case of γl = 0.004 N/m2 with therising of γS (simulation using SOFA.)

4.2. Implementation using SOFA (Simulation Open Framework Architecture)55
framework

F IGURE 29: The relationship between surface tension energies and wetted height.

F IGURE 30: The relationship between wetted heigh and radius of wetted liquid curvature.

This investigation is more convincing if this estimation would be accompanied
by validating experiments. However, evaluating the experimental accuracy of thesesurface parameters in wet adhesion phenomena poses a challenge due to equipment
and technique demands, including imaging technology and the complexity of tension energy in various liquid and solid types. So we just investigated the trends ofthose parameters and compared with other studies [39].

56

Chapter 4. The finite element analysis (FEA) for representing wet

4.2.2

The FEA-based model for the capillary phenomenon of thepatterned pad

In the previous subsection, our approach was employed to simulate wet adhesion, which related to the ability of a liquid to adhere to a solid surface. When aliquid encounters a solid surface, it has the potential to either spread out and forma thin film or cluster together into droplets. Conversely, capillary action is the phenomenon in which a liquid can move against the force of gravity within confined
spaces, such as narrow tubes or amidst small particles. This behavior arises due tothe combined cohesive forces among liquid molecules and adhesive forces between
the liquid and the solid surface. In scenarios with narrower spaces, the liquid isdrawn upward to a greater extent through capillary action.
In this subsection, we propose a FE method to estimate capillary forces in thecase of multiple pads on our changeable-pad foot. The narrow spaces among thepads, which can be controlled by adjusting the inner pressure, are considered asimportant factors affecting the capillary force. Fig. 31 illustrates the distribution ofsoft pads on the foot area. The mesh element parameters are obtained using the tools
GMSH and SOFA. In this scenario, the equilibrium energy equations are formulated
by considering the relationship between the space occupied by the liquid and thework of capillary force. Let n represent the total number of domes, and We that
explicitly formulated as given in Eq.4.4 denote the work of capillary force that lifts
the liquid to its equilibrium height. Subsequently, we use the We formula to calculate
the total work of capillary force as follows:

Wtot = nWe .

(4.19)

Considering the influence of both viscous dissipation and potential energy, as elucidated in Equations Eq. 4.6 and Eq. 4.8, we proceed to compute the volume of eachliquid layer with the following approach. Within this context, dz signifies the heightof the layer element, Sbo denotes the boundary area of the dome, and Vsph represents

4.2. Implementation using SOFA (Simulation Open Framework Architecture)57
framework
the volume of each dome submerged beneath the liquid:

Vn = dzSbo − nSsph(i) ,(4.20)

n

Ktot = ∑ ρgzn (dzSbo − nVsph(i) ),(4.21)∆Utot = ∑ ρgzn (dzSbo − nVsph(i) ).

(4.22)

i =0
n

i =0

The values of dz , Sbo , and Vsph are obtained from the mesh data of GMSH and SOFA.
Gmsh is a powerful open-source finite element mesh generator used for 3D object
discretization. With Gmsh, complex 3D geometries can be divided into smaller elements, simplifying the numerical analysis of various physical processes, including

F IGURE 31: SOFA simulation of wet liquid for soft pads foot. (a) Changeable soft pads
foot meshing in SOFA. (b) Height of rising wet liquid.

58

Chapter 4. The finite element analysis (FEA) for representing wet

structural mechanics, fluid dynamics, and more. It offers a user-friendly interface
and an useful set of tools for mesh creation, making it an important asset in the fieldof computational engineering and science. Whether you’re simulating the behaviorof mechanical components or studying fluid flows within complex structures, Gmsh
provides the means to generate high-quality 3D meshes that are used for accurate
simulations and in-depth analysis.
Therefore, similar to the previous subsection, a loop is utilized to determine theheight of the rising liquid through the energy function Qtot as follows:
Qtot = Wtot − Ktot + ∆Utot .

(4.23)In Fig. 31, we present an overview of our simulation procedure. We utilize abalance energy function to predict the liquid layer’s height. By considering the geometric properties of the liquid’s shape, we calculate the tangential capillary force,as detailed in Eq. 4.18. The subsequent section focuses on the thorough evaluationof our methodology’s accuracy and reliability, including conducting experiments
to validate the computed forces. This experimental assessment is a critical step inverifying the real-world applicability and effectiveness of our simulation.

4.3

Experimental validation

4.3.1

Tangential force experiment

In this section, we assess the impact of capillary action on the morphologicallychangeable foot’s resistance to slippage. The experimental setup is depicted in Fig.
32, which comprises two motorized linear stages facilitating vertical and horizontal
movements. This arrangement involved two key components: a vertical linear stage
motor responsible for vertical motion and a horizontal linear stage motor governing forward and backward movements. The soft pad foot was securely attached to
the vertical linear stage motor, serving as the focal point of the experiment. In thissetup, as the soft pad made contact with the surface, it was subjected to controlled4.3. Experimental validation

59

F IGURE 32: Experimental setup of the tangential force.

conditions under both wet and dry circumstances. The horizontal linear stage motor
played a pivotal role, creating motion and inducing tangential forces that resisted
this motion. The measurement of these tangential resistant forces was a key objective of the experiment. The experimental procedure was as follows: the soft pad footwas gradually pressed onto the surface, with the tests conducted under varying wet
and dry conditions. As the soft pad came into contact with the surface, the horizontal linear stage motor initiated its movement, generating a controlled motion.
During this motion, the tangential resistant forces were measured. This approachallowed us to gather data regarding the pad’s performance and its ability to resist
slippage in response to diverse conditions. The results obtained from this experimental setup investigate the interaction between the soft pad and different surfaces,
focus on its effectiveness and adaptability.
The experimentation was conducted under both wet and dry conditions, varying the soft pad morphology from flat to domed under an air pressure of 50 kPa.
The resultant capillary force Fcap was derived as the difference between Fwet and60

Chapter 4. The finite element analysis (FEA) for representing wet

F IGURE 33: Microscopic images of the experimental soft plastic layer surface: (a)presents the projection of the convex elevation on the rough surface, (b) shows the projection of the roughness density on the rough surface, (c) demonstrates the projection of
the smooth surface, (d) displays the projection of the roughness density on the smoothsurface.

Fdry , corresponding to the tangential forces measured in the respective conditions,specifically Fcap = Fwet − Fdry . The experimental trial initiated with the soft pad being pressed against a surface affixed to a 3-axis force sensor (DSS300-U, TecGihan,
Japan) under varying contact force intensities. Subsequently, the horizontal linear
stage was activated to apply tangential displacement to the soft pad at a velocity of
0.2 mm/s, a rate chosen to minimize any influence from viscosity.
The experimental design included two distinct surface conditions: one with arough texture and the other with a smoother soft plastic surface (as depicted in
the photo within Fig. 33, captured using a 400-CAM058 microscope from SANWA
SUPPLY, JAPAN). The influence of surface roughness on capillary phenomena is amultifaceted aspect that lies beyond the primary focus of this study. However, toensure comprehensive experimentation, we conducted force measurements acrossthese varying surface roughness scenarios. The detailed results can be observed in
the data presented in Fig. 34.
The outcomes presented in Fig. 34 depict a noticeable trend in capillary forces

4.3. Experimental validation

61

F IGURE 34: (a) Comparison of resulted tangential forces among flat soft pads foot, anddomed pads on both wet and dry floors in two different roughness. (b) Comparisonof capillary action in wet condition (Fcap = Fwet − Fdry ) between experiment (with errorbars) and estimation results.

across different surfaces. There is an error attributed to the limitations in measurement accuracy. Simultaneously, the friction forces exhibit a linear relationship withthe applied contact force on the surface. These results closely correspond with thecalculations from the mathematical model detailed in the previous section, with anerror of approximately 15% to 25%.
In the Fig. 34.a, the relationship between tangential force and normal force isaffected by friction and adhesion. Capillary force, although not proportional to thenormal force, doesn’t significantly contribute to the observed trend in tangential
force as its influence is much smaller. Previous research extensively studied models

62

Chapter 4. The finite element analysis (FEA) for representing wet

where tangential force is proportional to the normal force, as cited in [61], [62], [63],[64]. In our study, we specifically focus on capillary force rather than giving muchattention to friction and adhesion.
The wet adhesion and capillary phenomena have been reexamined using the finite element analysis (FEA) approach, contributing to simulation solutions within
the SOFA (Simulation Open Framework Architecture) framework. The results obtained from these solutions have been carefully verified through experiments. In thenext chapter, showcases integrating morphology-changeable soft pads into a legged
robot will be conducted to demonstrate the applicable potential of our design.

63

5 Application to slippage resistance of
locomotive robot on wet surface
5.1

Design and fabrication

In this chapter, we present the design and fabrication processes for two distinct
foot sizes: large-scale and small-scale. Each of these processes has advantages anddisadvantages, making them better suited for specific robotic applications.
In the case of the large-scale fabrication process (refer to Fig. 35), the foot benefits
from enhanced stability, primarily attributed to the presence of a rigid bone encased
within the soft foot structure. However, it’s worth noting that the tangential force
optimization is not optimal, and we delved into this aspect further in the subsequent
Optimization section.
Conversely, in the small-scale fabrication process (as depicted in Fig. 36), we prioritize the optimization of tangential force, significantly improving the foot’s ability
to avoid slippage. Nevertheless, it’s crucial to highlight that under high inner pressure conditions, the set of soft pads experiences deformation, ultimately adopting adome-like shape.
Below, we provide an explanation of both the design and fabrication processes
for these two distinct foot sizes. In particular, the drawing used for fabrication process is present in the Appendix 1: Appendix 1.1: Drawing of mold design for large
scale foot, and Appendix 1.2: Drawing of mold design for small scale foot (Chapter
7: Appendix)

64Chapter 5. Application to slippage resistance of locomotive robot on wet surface

F IGURE 35: Fabrication procedure: (a) Arrangement of the foot mold. (b) Pouring ofsilicone material. (c) Finalizing the foot structure. (d) Soft pad with applied pressure anddimensions.

5.1.1

Large scale design

The design of the foot in the case of large scale is shown in Fig. 35c and 35d,
showcasing the inner "bone" structure, which is printed by ZOTRAZ-300 3D printer.
This inner structure serves as the rigid core of our design. Wrapping around thiscore is a membrane of silicone skin, adding a layer of flexibility and tactile sensitivity
to the design. The pivotal component, the sole, hosts an array of spherical domes,
as illustrated in Fig. 35b.
Initially, in its rest state, the sole of the foot is flat, as exhibited in Fig. 35d. When
providing inner pressure, the soft pad domes emerge in response to the increasing
air pressure. This transformation is central to the foot’s performance, changing its
morphology to adapt to various surface conditions.
The mechanics of this design may apply to robot functions. The external airpressure regulator connects to the rigid bone, and this connection plays a pivotal
role in controlling the inflation and deflation of the soft pad domes. When air pressure is applied, these domes engage, adjusting the sole’s morphology to facilitate
walking on surfaces with different characteristics, particularly those that vary between wet and dry condition by sensing the surface and actively adjusting the innerpressure.
The fabrication process, as illustrated in Fig. 35a-c, is a crucial step in crafting
our soft robotic foot. The fabrication process is initiated by filling the outer mold

5.1. Design and fabrication

65

with silicone material, specifically silicone Ecoflex 00-20 from Smooth-On, Inc., USA.
This silicone forms the outer layer, covering the rigid bone, and giving shape tothe entire foot structure. This step is instrumental in creating the soft, adaptablesurface necessary for our robot to adapt to various terrains. Besides, the inner moldserves the purpose of crafting air hoses. These hoses are important for directing andregulating the flow of air, which is responsible for inflating the thin sheet locatedwithin the foot structure. This controlled inflation allows the soft pad to morph andadapt to the ground, enhancing its efficiency and versatility. The 3D-printed cap,
produced using a ZORTRAX M300 printer and ABS material, plays an importantrole in the assembly process.

5.1.2

Small scale design

In the particular design, a softer sheet material, Silicone Ecoflex 00-20 (SmoothOn, Inc., USA), was employed. This material helps the foot transform into a spherical dome shape when subjected to controlled air pressure adjustments. It is layeredon a firmer substrate, Dragon Skin 30 (Smooth-On, Inc., USA), which provides structural support and stability to the soft pad.
The internal architecture of the foot comprises a network of cylindrical chambers, each with a diameter of 1 mm. These chambers serve a function by acting as

F IGURE 36: Fabrication process: (a) mold set of foot. (b) Pouring silicone include twostep: pouring Dragon Skin 30 silicone for the body, then pouring the silicone Ecoflex 0020 for the thin sheet. (c) Finishing the foot. (d) dome inflation and the robot leg attached
the foot.

66Chapter 5. Application to slippage resistance of locomotive robot on wet surface
conduits for the directed flow of compressed air. The air, when appropriately channeled through these chambers, inflates the soft pad. This pad is designed with athickness of 0.2 mm. An essential design parameter which was considered is theinitial radius of these soft pad domes, denoted as r0 . This value was determined to
optimize the tangential force.
The outer molds were fabricated by ZORTRAX M300 printer with ABS material.
Besides, the inner mold was printed using a FORMLABS FORM 2 3D printer. The
material of choice for this inner mold was grey SLA resin (RS-F2-GPGR-04).

5.2

Case of single legged robot with the proposed footIn this section, we demonstrate the walking efficiency on both wet and dry surfaces using a leg equipped with a foot containing either a flat or a domed soft pad.
While our walking system comprises only a single leg for simplicity, the results holdpromise in validating the potential applicability of this device in more complex systems in the future.
The walking mechanism consists of a body carrying a stepper motor that drives
a one-degree-of-freedom (1-DOF) linkage system, as depicted in Fig. 37a. To assess the effectiveness of preventing foot slippage, we intentionally kept the walking
mechanism simple to avoid distraction associated with a more complex system. We
utilized a one-leg mechanism moving on four wheels constrained by a rail system,
shown in Fig. 37c-d. The basic concept of the linkage mechanism is illustrated in Fig.
37b. The torque τ1 generated by the stepper motor produces external forces F1 , F2 ,F3 , F4 . The hinge joint between the foot and the linkage system, supported by two
springs, is designed to press the foot to the floor, thus increasing the contact time
between the foot and the surface. The external force F4 causes the foot’s backward
motion. When the tangential resistance force Ftot is adequate, slippage between thefoot and the surface is prevented, and the foot remains stationary until the reactive forces N1 , N2 , N3 push the body forward. The motion of the walking bodywas tracked using a green marker on the body and three smaller markers attached5.2. Case of single legged robot with the proposed foot67

to the foot to capture stick/slip motion during walking. A SONY RX10IV(DSCRX10M4) camera was placed to record marker movement (as shown in Fig. 37c-d).
The OpenCV library was used to extract two-dimensional marker movement [65].
By analyzing the contour of the foot based on marker movement, we assessed
walking efficiency in five different walking scenarios: 1,On a dry surface (with aflat soft pad); 2, On a wet surface (with a flat soft pad);3-5, On a wet surface with adomed soft pad pressurized at 40 kPa, 60 kPa, and 75 kPa, respectively. For the firstcase, the foot posture during walking phases is depicted in Fig. 38a. In this case,the foot makes firm contact with the surface (without slippage), propels the bodyforward, and then lifts off from the surface in a stable manner.

F IGURE 37: Design of the robotic leg. (a) The walking system with its components. (b)Mechanical analysis of the linkage system for walking tests on both (c) dry and (d) wet
floors (contour data was tracked to generate the foot paths in Fig. 38).

68Chapter 5. Application to slippage resistance of locomotive robot on wet surface

30

a

20

Z axis(mm)

10
0
30

50

100

150

200

b

20

Move forward

100

Slip backward

30

50

100

150

200

100

150

200

c

20
10
0

50

X axis(mm)F IGURE 38: Tracked foot’s posture (the rectangle in Fig. 37d) during walking action in(a) on a dry floor, (b) on a wet floor with flat foot, and (c) on a wet floor with domed pad(high inner pressure state).
500450

Position(mm)

400
350
300
250
200
150

Walking
Walking on
on drywetsurface
surfacewithout
withoutspherical
sphericalpadpadWalking
spherical
Walking on
on wetwet surface
surface withoutwith spherical
padpad(75kPa)Walking on wet
surface with spherical pad (40kPa)
dry surface
Walking on wet surface with spherical pad (60kPa)(40kPa)(60kPa)Walking on wet surface with spherical pad (75kPa)

100
50
0
0

2

4

6

8

10

12

14

16

18

20

22

Time (s)F IGURE 39: Results of the walking robot’s body displacement in five experimental conditions (tracked by the marker on the robot body as shown in Fig. 12d).

In the second scenario, the foot’s posture is depicted in Fig. 38b. When equippedwith a flat soft pad, the foot exhibited poor performance in wet conditions. Upon
contact with the wet surface (indicated by blue rectangles), the resultant low tangential force caused the foot to slip backward (indicated by red rectangles) rather thanpropelling the body forward. This observation aligns with the earlier experimental
findings indicating a significant reduction in the tangential force of the flat pad in
wet conditions.

5.3. Case of hexapod robot with the proposed feet69

In the third case, walking on a wet surface with a domed soft pad showed alocomotion pattern similar to the first case, where a flat soft-padded foot walked ona dry surface. In both cases, there was minimal slippage upon landing of the foot,as demonstrated in the plot in Fig. 38c. This result underscores the potential of thedomed soft pad in enhancing walking efficiency in a wet environment.
To summarize the time-series positions of the robot body in all five cases, pleaserefer to Fig. 39. Notably, the movements of the body in case 1 (flat pad, dry surface)and case 5 (domed pad, wet surface, 75 kPa) were quite similar. In both scenarios,
the foot executed rapid walking without significant slippage, reaching its destination successfully. However, in case 2 (flat pad, wet surface), reaching the goal tookapproximately five times longer than in case 5, requiring a much higher number ofwalking cycles due to slippage.
Varying the morphology of the domed pad with air pressure (at 40 kPa and
60 kPa, respectively) demonstrated that higher pressure notably improved walking
performance. In a flat state or a domed state with low air pressure, a layer of liquidbecame trapped between the foot and the contact surface, changing dry frictionadhesion into lubrication and reducing the tangential resistance force, ultimately
leading to slippage. In contrast, with sufficient air pressure, the liquid was expelledfrom the contact interface to maintain dry friction-adhesion upon contact. Besides,
the liquid layer was randomly spread across a large area, making it challenging toensure uniform distribution around the soft pad’s domes. Achieving this specificliquid layer condition was not our objective in this experiment, as securing suchconditions in real walking scenarios is difficult. Consequently, the capillary force
in the walking leg experiment is smaller than estimated by our analytical model.
Nonetheless, the uneven liquid height still generates a capillary force, contributing
to tangential capillary effects and aiding the robot in walking on wet surfaces.

70Chapter 5. Application to slippage resistance of locomotive robot on wet surface

5.3

Case of hexapod robot with the proposed feet

In this showcase, we show an experiment that use the Hexapod robot developed by Lynxmotion Inc, Canada. Here, our developed soft pads are attached tothis robot’s feet (see Fig. 40). The robot is a versatile and agile robotic platformcharacterized by three independent rotational axes of motion. This robotic system
mimics the movement of insects with six legs, each of which possesses three degrees of freedom. The three rotational axes typically correspond to the pitch, roll,and yaw orientations, allowing the hexapod to move in a variety of directions andorientations.
In our experimental setup, we employ a straightforward locomotion pattern toguide the robot in a straight path while assessing the performance of its adaptablesoft pad foot. The robot’s movements are tracked using an optical tracking system(NaturalPoint, Inc. DBA OptiTrack), which collects coordinated data from dedicated markers affixed to each leg (refer to Fig. 40). Additionally, three markers

F IGURE 40: 3-Degree-of-Freedom Hexapod robot in the experiment: (a) Hexapod robot
with attached markers, (b) leg mechanism illustration. (c) The concept of hexapod locomotion showcased.

5.3. Case of hexapod robot with the proposed feet

71

placed on the robot’s body provide position and orientation information. Typically,a leg mechanism in a 3 DOF hexapod consists of three articulated joints that provide
rotational freedom. These joints correspond to the hip, knee, and ankle joints (seeFig. 40b).
The hip joint allows the leg to move forward and backward, as well as to swing
laterally to the sides. The knee joint contributes to controlling the leg’s height andstep length. It decides the time when the foot contact into the floor for moving
forward and backward (by swinging the hip joint), or escape from the interaction
with the floor. The ankle joint enables the leg to rotate or tilt, allowing the hexapod
to adjust its foot orientation to the ground. Ankle joint movement is crucial for
maintaining stable contact with the surface. It also aids in adapting the foot’s angle
for optimal traction during movement.
For this experiment, the robot moves directly forward by activating phase differences between two leg groups: group 1 (legs 1, 3, and 5) and group 2 (legs 2, 4, and
6). Fig. 40c illustrates this locomotion concept. As one group swings its hip joint tomove forward, the other group retracts to prepare for its upcoming step. In terms offorce analysis, the forces and moments produced by hip swinging are summarized
as follows: FMov = F1 + F2 + F3 ; M = F1 l − ( F2 + F3 ) l, where F1 , F2 , F3 represent theforces propelling backward leg movement. If the friction force is sufficient, it securesthe foot to the floor, and the reaction force FMov propels the robot forward. As thismovement is linear, M = 0, yielding F1 = 2F2 = 2F3 . Assessing all leg forces, legs 2
and 5 carry a higher risk of slipping during robot locomotion.
Fig. 41 and Fig. 42 provide a comprehensive overview of the experimental results. On dry surfaces, the robot exhibits robust performance in both smooth andrough terrains, with no signs of slippage. However, when a flat soft pad (no inner pressure) contacts into wet surfaces, issues with backward leg slipping become
apparent. Notably, activating the dome pad by increasing the inner pressure significantly mitigates slippage, as observed in Fig. 41 and Fig. 42. These results serve
as a valuable assessment of slippage under diverse conditions, with no predefined
control rules imposed. Consequently, the robot’s trajectory may deviate due to leg

72Chapter 5. Application to slippage resistance of locomotive robot on wet surface

F IGURE 41: The positions of the detected markers on leg 2 and leg 5 of the hexapod while
walking on the rough surface, illustrating instances of slippage in different scenarios:
walking on dry surface with flat foot, walking on wet surface with flat foot, walking on
wet surface with dome pad foot.

F IGURE 42: The positions of the detected markers on leg 2 and leg 5 of the hexapod while
walking on the smooth surface, illustrating instances of slippage in different scenarios:
walking on dry surface with flat foot, walking on wet surface with flat foot, walking on
wet surface with dome pad foot.

slipping. For a deeper understanding of these outcomes, refer to Fig. 43 and Fig. 44,
which depict the paths the robot follows in various experimental scenarios. Whilenavigating wet surfaces with a flat soft pad (absence of inner pressure), the robot

5.3. Case of hexapod robot with the proposed feet73

F IGURE 43: The positions of the detected markers on the body of the hexapod while
walking on the rough surface in different scenarios: walking on dry surface with flatfoot, walking on wet surface with flat foot, walking on wet surface with dome pad foot.

struggles to maintain a straight trajectory, leading to deviations. In contrast, whenequipped with the dome pad, slippage is significantly reduced, allowing the robot
to uphold a straight course, as evidenced by its trajectory on wet surfaces, whichclosely resembles its performance on dry terrain.

74Chapter 5. Application to slippage resistance of locomotive robot on wet surface

F IGURE 44: The positions of the detected markers on the body of the hexapod while
walking on the smooth surface in different scenarios: walking on dry surface with flatfoot, walking on wet surface with flat foot, walking on wet surface with dome pad foot.

75

6 Conclusion and Future Works
In the field of soft robotics, our research introduces the mechanics of morphology changeable soft pad that dynamically adjusts the morphology to maximize performance across varying terrains. This novel solution demonstrates the interesting
potential of biomimetic engineering, as it draws inspiration from nature’s adaptive
solutions. Specifically, our novel soft pad design help the robots change their morphology between two primary states: a flat elastic sheet and a protruding spherical
dome. The former configuration serves for dry surfaces, offering stability, while thelatter enhances efficiency on wet terrains. This feature addresses a crucial challenge
in the field of robotics: adaptability to different environments. The contributions of
this research may impact the field of robotics through practical applications. Furthermore, our approach introduces a novel solution in wet adhesion and capillary
simulation based on the finite element method.

6.1

Impact of morphology-changeable soft pads

6.1.1

Design and fabrication

In this study, we have presented a detailed design and fabrication process aimed
at realizing a biomimetic approach inspired by the mechanisms found in animal footpads. The novel concept lies in the development of a morphology-changeable footpad, carefully designed to optimize friction and adhesion on both dry and wet surfaces. Of significance is the emphasis on exploiting wet adhesion to prevent robot
slippage, an aspect that has been relatively understudied until now. The fabricationof the foot pad is conducted through a specially designed mold, ensuring the physical realization aligns with the conceptual framework. To enhance the efficiency

76

Chapter 6. Conclusion and Future WorksF IGURE 45: Designed mold and completed foot in large and small scale.

of the design and fabrication, we employed optimization algorithms to maximize
forces within specific geometrical constraints. Through this work, we expect to contribute to the field of soft robot, providing a robust design and fabrication processapplicable in diverse applications. Our optimized approach holds promise for thedevelopment of robotics, where the importance of terrain adaptability and wet adhesion has been underscored.

6.1.2

Analytical model and finite element method model for Morphology changeable soft dome pad

In conclusion, our research has introduced a novel approach to the design andanalysis of morphology-changeable soft pads for robotics applications. The main
point of our efforts has been the development of an analytical model based on energy equilibrium equations that examines the morphological properties of the soft
pad. This model encompasses the dynamics of tribological aspects, unraveling theinterplay of friction, adhesion, and capillary action.
The second analytical model, based on the Finite Element (FE) method, specifically addresses capillary action phenomenon. By adopting this approach, we have

6.1. Impact of morphology-changeable soft pads

77

contributed to the understanding of liquid-solid interactions. Moreover, this model
serves as a bridge between theoretical concepts and practical applications in robotics.
To verify our theoretical model, we conducted force measurement experiments toevaluate the accuracy and reliability of our model. This validation process evaluates
our model as a believable approach in the applications of robotics and simulation.
Looking ahead, we expect that our developed model will become an instrumental tool for controlling robots equipped with morphology-changeable soft pads. This
tool is expected to provide a reliable means of simulating the complex behavior of
liquids in contact with solids, enabling the estimation of wet adhesion forces andcapillary forces in real-world scenarios. Our research not only contributes to thetheoretical understanding of soft robotics but also proposed a novel tool for advancing the applications of this technology.

6.1.3

Application for legged robot walking on different terrain without slippage

Moreover, our research has developed a novel morphology-changeable soft pad
for robot feet, showcasing its exceptional adaptability across diverse terrains. Throughexperiments, we have successfully demonstrated the transition of a robot equippedwith these adaptive soft pads between dry and wet surfaces. The outcomes of ourwork provide practical validation, suggesting that these soft pads have the potential
to enhance the locomotion capabilities of walking robots. Importantly, our findings
indicate that the morphology-changeable soft pad foot can offer a useful solution
to improve the performance of walking robots when navigating various terrains.
This research holds promise for overcoming challenges in complex scenarios where
walking robots or humanoid robots need to efficiently complete their tasks. Our research contributes a practical and impactful solution to advance the field of robotics
and address real-world challenges associated with adaptability to varied terrains.

78

Chapter 6. Conclusion and Future Works

6.2

Future work

Moving forward, our research lays the groundwork for several future works.
First and foremost, there is a need for improving the finite element-based mathematical model algorithm. We expect to incorporate the influence of the calculated
capillary force on the mesh of the object, and show the equilibrium state of the object’s deformation and the updated capillary force based on that deformation.
Besides, we expect to develop the finite element-based mathematical model algorithm to expand its capabilities to embrace a more comprehensive range of capillary
phenomena, ensuring a more accurate representation of the complex interactions
between liquids and solids. Delving into the aspects of liquid surface geometry
represents a promising direction. An in-depth examination of how liquid shapes
interact with different surfaces can deepen our insights, potentially uncovering new
information in the behavior of our adaptive soft pads.
Furthermore, the development of adaptive control rules for changing the soft
pad’s morphology is an important content. This involves control rules that enable
the soft pad to dynamically adjust its shape, optimizing the walking robot’s performance across diverse terrains. This adaptive capability could prove instrumental in
addressing the challenges faced by walking robots operating in real-world, unpredictable surfaces (wet or dry).
Lastly, a crucial plan for future work involves the development of sensing technology. Specifically, there’s a need to create a sensing mechanism capable of detecting tangential resistant forces. This development would draw on our evolving
understanding of the interactions that occur in soft contact with the surrounding
liquid layer. Implementing such a sensing technology could enhance the robot’s
ability to respond to its environment, fostering increased adaptability and efficiency
in a variety of scenarios. In sum, these future works aim to push the practical applications of our research, contributing to the evolution of soft robotics in real-world
scenarios.

79

7 Appendix
7.1

Appendix 1: Drawing of mold design

This section shows the drawing of mold design and fabrication discussed in
Chapter 3, Section 3.1: Design and fabrication. Specifically, Appendix 1.1 is utilized
for the fabrication of the large-scale foot (3.1.1), while Appendix 1.2 is employed forthe large-scale foot (3.1.2).

7.1.1

Appendix 1.1: Drawing of mold design for large scale foot

2:1

2:1

2:1

7.1. Appendix 1: Drawing of mold design

7.1.2

Appendix 1.2: Drawing of mold design for small scale foot

85

2:1

2:1

7.2. Appendix 2: Matlab code for calculation

7.2

91

Appendix 2: Matlab code for calculation

This section presents the Matlab code used to simulate the analytical model in
Chapter 3. Appendix 2.1 is employed for simulating morphology changes without applied pressing force, while Appendix 2.2 simulates morphology changes withapplied pressing force. The simulation results are depicted in Fig. 14 and Fig. 16.

7.2.1

Morphology changing without pressing force

clear all
r0=0.0015;
p=5.5*10^4;
k=40;
syms f(c)f(c) = (p*(r0^2+c^2)/(2*c))-(2*k*(1.5*sqrt(r0^2+c^2)-r0)/r0);
sol = vpasolve(f)R=((r0^2+sol^2)/(2*sol))
c=sol;
radius = ((r0^2+c^2)/(2*c));circr = @(radius,rad_ang) [radius*cos(rad_ang);
radius*sin(rad_ang)+sqrt(radius^2-r0^2)];
r_angl = linspace(pi/2+acos((c-radius)/radius),
2*pi+asin((c-radius)/radius));xy_r = circr(radius,r_angl);
c_ex=0.00125;
radius_ex = (r0^2+c_ex^2)/(2*c_ex)
r_anglex = linspace(pi/2+acos((c_ex-radius_ex)/radius_ex),
2*pi+asin((c_ex-radius_ex)/radius_ex));xy_rex = circr(radius_ex,r_anglex);v = VideoWriter(’sphere.avi’);
v.FrameRate = 10;

92

Chapter 7. Appendix

open(v);
figure(1)pause(10);plot([-r0 r0],[0 0],’LineWidth’,2)axis([-1.5*r0

1.5*r00

1.5*r0])axis equal
xlabel(’X axis (mm)’)ylabel(’Z axis (mm)’)set(gca,’FontSize’,30)
hold onpause(3);frame = getframe(gcf);
writeVideo(v,frame);
figure(1)plot(xy_r(1,:), xy_r(2,:),’Color’,’r’,’LineWidth’,2)plot(xy_rex(1,:), xy_rex(2,:),’Color’,’r’,’LineWidth’,2,’LineStyle’,’--’)
axis([-1.5*r0

1.5*r00

1.5*r0])axis equal
xlabel(’X axis (mm)’)ylabel(’Z axis (mm)’)set(gca,’FontSize’,30)
hold onpause(3);frame = getframe(gcf);
writeVideo(v,frame);
p=6.5*10^4;syms f(c)f(c) = (p*(r0^2+c^2)/(2*c))-(2*k*(1.5*sqrt(r0^2+c^2)-r0)/r0);
sol = vpasolve(f)R=(r0^2+sol^2)/(2*sol)

7.2. Appendix 2: Matlab code for calculation

93

c=sol;
radius = (r0^2+c^2)/(2*c);circr = @(radius,rad_ang)[radius*cos(rad_ang);

radius*sin(rad_ang)-sqrt(radius^2-r0^2)];
r_angl = linspace(pi/2+acos((c-radius)/radius),
2*pi+asin((c-radius)/radius));xy_r = circr(radius,r_angl);
c_ex=0.0016;
radius_ex = (r0^2+c_ex^2)/(2*c_ex);r_anglex = linspace(pi/2+acos((c_ex-radius_ex)/radius_ex),
2*pi+asin((c_ex-radius_ex)/radius_ex));xy_rex = circr(radius_ex,r_anglex);figure(1)plot(xy_r(1,:), xy_r(2,:),’Color’,’g’,’LineWidth’,2)plot(xy_rex(1,:),xy_rex(2,:),’Color’,’g’,’LineWidth’,2,’LineStyle’,’--’)
axis([-1.5*r0

1.5*r00

1.5*r0])axis equal
xlabel(’X axis (mm)’)ylabel(’Z axis (mm)’)set(gca,’FontSize’,30)
hold onpause(3);frame = getframe(gcf);
writeVideo(v,frame);
p=7.5*10^4;syms f(c)f(c) = (p*(r0^2+c^2)/(2*c))-(2*k*(1.5*sqrt(r0^2+c^2)-r0)/r0);
sol = vpasolve(f)R=(r0^2+sol^2)/(2*sol)
c=sol;

94

Chapter 7. Appendix

radius = (r0^2+c^2)/(2*c);circr = @(radius,rad_ang)[radius*cos(rad_ang);

radius*sin(rad_ang)-sqrt(radius^2-r0^2)];
r_angl = linspace(pi/2+acos((c-radius)/radius),
2*pi+asin((c-radius)/radius));xy_r = circr(radius,r_angl);
c_ex=0.002;
radius_ex = (r0^2+c_ex^2)/(2*c_ex);r_anglex = linspace(pi/2+acos((c_ex-radius_ex)/radius_ex),
2*pi+asin((c_ex-radius_ex)/radius_ex));xy_rex = circr(radius_ex,r_anglex);figure(1)plot(xy_r(1,:), xy_r(2,:),’Color’,’b’,’LineWidth’,2)plot(xy_rex(1,:), xy_rex(2,:),’Color’,’b’,’LineWidth’,2,’LineStyle’,’--’)
axis([-1.5*r0

1.5*r00

1.5*r0])axis equal
xlabel(’X axis (mm)’)ylabel(’Z axis (mm)’)set(gca,’FontSize’,30)
hold onpause(3);frame = getframe(gcf);
writeVideo(v,frame);
p=8.5*10^4;syms f(c)f(c) = (p*(r0^2+c^2)/(2*c))-(2*k*(1.5*sqrt(r0^2+c^2)-r0)/r0);
sol = vpasolve(f)R=(r0^2+sol^2)/(2*sol)
c=sol;
radius = (r0^2+c^2)/(2*c);7.2. Appendix 2: Matlab code for calculation
circr = @(radius,rad_ang)

95[radius*cos(rad_ang);

radius*sin(rad_ang)-sqrt(radius^2-r0^2)];
r_angl = linspace(pi/2+acos((c-radius)/radius),
2*pi+asin((c-radius)/radius));xy_r = circr(radius,r_angl);
c_ex=0.0024;
radius_ex = (r0^2+c_ex^2)/(2*c_ex);r_anglex = linspace(pi/2+acos((c_ex-radius_ex)/radius_ex),
2*pi+asin((c_ex-radius_ex)/radius_ex));xy_rex = circr(radius_ex,r_anglex);figure(1)plot(xy_r(1,:), xy_r(2,:),’Color’,’k’,’LineWidth’,2)plot(xy_rex(1,:), xy_rex(2,:),’Color’,’k’,’LineWidth’,2,’LineStyle’,’--’)
axis([-1.5*r0

1.5*r00

1.5*r0])axis equal
legend({’$p=0$’,’$p=55kpa$’,’$p=65kpa$’,’$p=75kpa$’,’$p=85kpa$’},’Interpreter’,’latex’,’FontSize’,23)
xlabel(’X axis (mm)’)ylabel(’Z axis (mm)’)set(gca,’FontSize’,30)
hold off
pause(3);frame = getframe(gcf);
writeVideo(v,frame);close(v);7.2.2

Morphology changing with pressing force

syms a c2
fo=0.08;
p=7.5*10^4;

96

Chapter 7. Appendix

pi=3.1415;
E=29.5*10^3;
pois=0.5;
E1=E/(1-pois^2);
ga=0.006;
ga_s=0.0027;
ga_l=0.0073;
theta=1.35;
r0=0.0015;
c1=0.0019412681941617884192468215394673;
k=40;
eq1 = (r0^2+c2^2)/(2*c2)-sqrt(((r0^2+c2^2)/(2*c2))^2-(fo+2*ga*pi*((r0^2+c2^2)/(2*c2)))/(p*pi))==a;eq2 = fo==p*(pi/2)*((fo+2*ga*pi*((r0^2+c2^2)/(2*c2)))/(p*pi)+a^2+r0^2+c2^2)-2*k*pi*(c2+a)*(1.5*sqrt(r0^2+c2^2-a^2)-r0)/r0;[sol_a, sol_c2]= vpasolve([eq1,eq2], [a c2]);b=sqrt((fo+2*ga*pi*((r0^2+sol_c2^2)/(2*sol_c2)))/(p*pi))
a_2=sol_a;c_2=sol_c2;
R1=(r0^2+c1^2)/(2*c1);
R2=(r0^2+c_2^2)/(2*c_2);
phi=asin(b/R2)
cap=2*ga_l*(2*b+(2*cot(phi)+1)*sqrt((3*ga_s*sin(phi+theta))/(9810*cot(phi))))circr = @(R1,rad_ang)[R1*cos(rad_ang);

R1*sin(rad_ang)-sqrt(R1^2-r0^2)];

r_angl = linspace(pi/2+acos((c1-R1)/R1), 2*pi+asin((c1-R1)/R1));
xy_r = circr(R1,r_angl);
if c_2>R2
circr2 = @(R2,rad_ang)[R2*cos(rad_ang);

R2*sin(rad_ang)-sqrt(R2^2-r0^2)];[R2*cos(rad_ang);

R2*sin(rad_ang)+sqrt(R2^2-r0^2)];else
circr2 = @(R2,rad_ang)

7.2. Appendix 2: Matlab code for calculation97

end
c_ex=0.002;
radius_ex = (r0^2+c_ex^2)/(2*c_ex);r_anglex = linspace(pi/2+acos((c_ex-radius_ex)/radius_ex),
2*pi+asin((c_ex-radius_ex)/radius_ex));xy_rex = circr(radius_ex,r_anglex);
r_angl2a = linspace(pi/2+acos((c_2-R2)/R2), pi/2+acos((a_2-R2)/R2));
r_angl2b = linspace(2*pi+asin((a_2-R2)/R2), 2*pi+asin((c_2-R2)/R2));
xy_r2a = circr2(R2,r_angl2a);xy_r2b = circr2(R2,r_angl2b);c_ex=0.0018;
radius_ex = (r0^2+c_ex^2)/(2*c_ex);a_ex=0.000084434;
b_ex=0.0005;
r_angl2a_ex = linspace(pi/2+acos((c_ex-radius_ex)/radius_ex),
pi/2+acos((a_ex-radius_ex)/radius_ex));r_angl2b_ex = linspace(2*pi+asin((a_ex-radius_ex)/radius_ex),
2*pi+asin((c_ex-radius_ex)/radius_ex));xy_r2a_ex = circr2(radius_ex,r_angl2a_ex);xy_r2b_ex = circr2(radius_ex,r_angl2b_ex);v = VideoWriter(’sphere under contact.avi’);
v.FrameRate = 10;
open(v);
figure(1)pause(10);plot(xy_r(1,:), xy_r(2,:),’LineWidth’,2)axis([-1.5*r0

1.5*r00

1.5*r0])hold onplot(xy_rex(1,:), xy_rex(2,:),’Color’,’b’,’LineWidth’,2,’LineStyle’,’--’)
axis([-1.5*r0

1.5*r00

1.5*r0])98

Chapter 7. Appendix

axis equal
xlabel(’X axis (mm)’)ylabel(’Z axis (mm)’)set(gca,’FontSize’,30)
hold onpause(3);frame = getframe(gcf);
writeVideo(v,frame);
figure(1)plot(xy_r2a(1,:), xy_r2a(2,:),’Color’,’r’,’LineWidth’,2)plot(xy_r2b(1,:), xy_r2b(2,:),’Color’,’r’,’LineWidth’,2)axis([-1.5*r0

1.5*r00

1.5*r0])line([-b b],[(a_2-c_2) (a_2-c_2)],’Color’,’r’,’LineWidth’,2)plot(xy_r2a_ex(1,:), xy_r2a_ex(2,:),’Color’,’r’,’LineWidth’,2,’LineStyle’,’--’)
plot(xy_r2b_ex(1,:), xy_r2b_ex(2,:),’Color’,’r’,’LineWidth’,2,’LineStyle’,’--’)line([-b_ex b_ex],[(a_ex-c_ex) (a_ex-c_ex)],’Color’,’r’,’LineWidth’,2,’LineStyle’,’--’)
axis equal
xlabel(’X axis (mm)’)ylabel(’Z axis (mm)’)set(gca,’FontSize’,30)
hold onpause(3);frame = getframe(gcf);
writeVideo(v,frame);
fo=0.19;
eq1 = (r0^2+c2^2)/(2*c2)-sqrt(((r0^2+c2^2)/(2*c2))^2-(fo+2*ga*pi*((r0^2+c2^2)/(2*c2)))/(p*pi))==a;7.2. Appendix 2: Matlab code for calculation99

eq2 = fo==p*(pi/2)*((fo+2*ga*pi*((r0^2+c2^2)/(2*c2)))/(p*pi)+a^2+r0^2+c2^2)-2*k*pi*(c2+a)*(1.5*sqrt(r0^2+c2^2-a^2)-r0)/r0;[sol_a, sol_c2]= vpasolve([eq1,eq2], [a c2]);b=sqrt((fo+2*ga*pi*((r0^2+sol_c2^2)/(2*sol_c2)))/(p*pi))
a_2=sol_a;c_2=sol_c2;
R2=(r0^2+c_2^2)/(2*c_2);
phi=asin(b/R2)
cap=2*ga_l*(2*b+(2*cot(phi)+1)*sqrt((3*ga_s*sin(phi+theta))/(9810*cot(phi))))if c_2>R2
circr2 = @(R2,rad_ang)[R2*cos(rad_ang);

R2*sin(rad_ang)-sqrt(R2^2-r0^2)];[R2*cos(rad_ang);

R2*sin(rad_ang)+sqrt(R2^2-r0^2)];else
circr2 = @(R2,rad_ang)
end
r_angl2a = linspace(pi/2+acos((c_2-R2)/R2), pi/2+acos((a_2-R2)/R2));
r_angl2b = linspace(2*pi+asin((a_2-R2)/R2), 2*pi+asin((c_2-R2)/R2));
xy_r2a = circr2(R2,r_angl2a);xy_r2b = circr2(R2,r_angl2b);c_ex=0.0014;
radius_ex = (r0^2+c_ex^2)/(2*c_ex);a_ex=0.00036217;
b_ex=0.00098;
r_angl2a_ex = linspace(pi/2+acos((c_ex-radius_ex)/radius_ex),
pi/2+acos((a_ex-radius_ex)/radius_ex));r_angl2b_ex = linspace(2*pi+asin((a_ex-radius_ex)/radius_ex),
2*pi+asin((c_ex-radius_ex)/radius_ex));xy_r2a_ex = circr2(radius_ex,r_angl2a_ex);xy_r2b_ex = circr2(radius_ex,r_angl2b_ex);
figure(1)plot(xy_r2a(1,:), xy_r2a(2,:),’Color’,’g’,’LineWidth’,2)

100

Chapter 7. Appendix

plot(xy_r2b(1,:), xy_r2b(2,:),’Color’,’g’,’LineWidth’,2)axis([-1.5*r0

1.5*r00

1.5*r0])line([-b b],[(a_2-c_2) (a_2-c_2)],’Color’,’g’,’LineWidth’,2)plot(xy_r2a_ex(1,:), xy_r2a_ex(2,:),’Color’,’g’,’LineWidth’,2,’LineStyle’,’--’)
plot(xy_r2b_ex(1,:), xy_r2b_ex(2,:),’Color’,’g’,’LineWidth’,2,’LineStyle’,’--’)line([-b_ex b_ex],[(a_ex-c_ex) (a_ex-c_ex)],’Color’,’g’,’LineWidth’,2,’LineStyle’,’--’)
axis equal
xlabel(’X axis (mm)’)ylabel(’Z axis (mm)’)set(gca,’FontSize’,30)
hold onpause(3);frame = getframe(gcf);
writeVideo(v,frame);
fo=0.285;
eq1 = (r0^2+c2^2)/(2*c2)-sqrt(((r0^2+c2^2)/(2*c2))^2-(fo+2*ga*pi*((r0^2+c2^2)/(2*c2)))/(p*pi))==a;eq2 = fo==p*(pi/2)*((fo+2*ga*pi*((r0^2+c2^2)/(2*c2)))/(p*pi)+a^2+r0^2+c2^2)-2*k*pi*(c2+a)*(1.5*sqrt(r0^2+c2^2-a^2)-r0)/r0;[sol_a, sol_c2]= vpasolve([eq1,eq2], [a c2]);b=sqrt((fo+2*ga*pi*((r0^2+sol_c2^2)/(2*sol_c2)))/(p*pi))
a_2=sol_a;c_2=sol_c2;
R2=(r0^2+c_2^2)/(2*c_2);
phi=asin(b/R2)
cap=2*ga_l*(2*b+(2*cot(phi)+1)*sqrt((3*ga_s*sin(phi+theta))/(9810*cot(phi))))if c_2>R2

7.2. Appendix 2: Matlab code for calculation
circr2 = @(R2,rad_ang)

101[R2*cos(rad_ang);

R2*sin(rad_ang)-sqrt(R2^2-r0^2)];[R2*cos(rad_ang);

R2*sin(rad_ang)+sqrt(R2^2-r0^2)];else
circr2 = @(R2,rad_ang)
end
r_angl2a = linspace(pi/2+acos((c_2-R2)/R2), pi/2+acos((a_2-R2)/R2));
r_angl2b = linspace(2*pi+asin((a_2-R2)/R2), 2*pi+asin((c_2-R2)/R2));
xy_r2a = circr2(R2,r_angl2a);xy_r2b = circr2(R2,r_angl2b);c_ex=0.001;
radius_ex = (r0^2+c_ex^2)/(2*c_ex);a_ex=0.0004368;
b_ex=0.0011;
r_angl2a_ex = linspace(pi/2+acos((c_ex-radius_ex)/radius_ex),
pi/2+acos((a_ex-radius_ex)/radius_ex));r_angl2b_ex = linspace(2*pi+asin((a_ex-radius_ex)/radius_ex),
2*pi+asin((c_ex-radius_ex)/radius_ex));xy_r2a_ex = circr2(radius_ex,r_angl2a_ex);xy_r2b_ex = circr2(radius_ex,r_angl2b_ex);
figure(1)plot(xy_r2a(1,:), xy_r2a(2,:),’Color’,’k’,’LineWidth’,2)plot(xy_r2b(1,:), xy_r2b(2,:),’Color’,’k’,’LineWidth’,2)axis([-1.5*r0

1.5*r00

1.5*r0])line([-b b],[(a_2-c_2) (a_2-c_2)],’Color’,’k’,’LineWidth’,2)plot(xy_r2a_ex(1,:), xy_r2a_ex(2,:),’Color’,’k’,’LineWidth’,2,’LineStyle’,’--’)
plot(xy_r2b_ex(1,:), xy_r2b_ex(2,:),’Color’,’k’,’LineWidth’,2,’LineStyle’,’--’)line([-b_ex b_ex],[(a_ex-c_ex) (a_ex-c_ex)],’Color’,’k’,’LineWidth’,2,’LineStyle’,’--’)
axis equal

102xlabel(’X axis (mm)’)ylabel(’Z axis (mm)’)set(gca,’FontSize’,30)
hold off
pause(3);frame = getframe(gcf);
writeVideo(v,frame);close(v);

Chapter 7. Appendix

103

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