折り畳み構造物をロボットアームで操作するための汎 url …...origami is an old...
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Meiji University
Title折り畳み構造物をロボットアームで操作するための汎
用的なシミュレーションを使ったアプローチ
Author(s) タイ フォン,タオ
Citation
URL http://hdl.handle.net/10291/19729
Rights
Issue Date 2017
Text version ETD
Type Thesis or Dissertation
DOI
https://m-repo.lib.meiji.ac.jp/
明治大学大学院先端数理科学研究科
2016年度
博士学位請求論文
General simulation-based approach for the manipulation
with foldable objects by the robotic arms
（折り畳み構造物をロボットアームで操作するための
汎用的なシミュレーションを使ったアプローチ）
学位請求者 現象数理学専攻
Thai Phuong Thao
i
General simulation-based approach
for the manipulation
with foldable objects by the robotic arms
The thesis submitted for the degree of
Doctor Philosophy
by
Thai Phuong Thao
Advisor
Professor Dr. Ichiro Hagiwara
Graduate School of Advanced Mathematical Sciences
Meiji University
January 2017
ii
Abstract
Origami is a traditional Japanese papercraft that is based on the folding of the
designed structure and can be widely used in industry. Origami folding is not a
difficult task for the human hands; however, folding paper by the robot hands is such
a challenge.
With all the recent advances in robotics, practical robots become available for
home using. These types of robots have to be portable with the small gabarits and
weights to do some real work at home or at the company offices to exclude the
humans from making the target work. The papercraft production as an art and a
hobby for many people, including children, is the important part of the human
activity and can become a popular application for the origami-performing robot.
This thesis presents a new study in a mechanical and geometrical design of the
origami-performing robot that can be placed in the limited living spaces with an aim
to exclude the humans from the complex and time-consuming operations in
producing the origami models. The suggested approach is based on the finite element
simulation of the manipulation of the robot arms with origami crease patterns, which
are given by the sheets of paper.
The virtual design of a robot, that resembles the behavior of the physical robot, is
developed instead of the traditional robotic prototyping approach. The robotic design
includes three main stages for an investigation: conceptual design, computer
modeling, and design development.
The simulation approach becomes the main option in the real-world related with
a robotic activity. Main geometrical, topological and mechanical parameters of a
robot design are defined by the simulation of the folding origami structures.
The central concept underlying the study can be formulated as: each folding
operation of a crease pattern of origami is considered as a function of the mechanical
systems such as a robot. Following this concept, a simulation-based methodology
for the manipulation with the foldable objects, such as a sheet of paper, by using the
iii
robotic arms is developed. The proposed methodology is considered as a basis for a
design of the origami-performing robot without the series of experimental tests.
The main method for a realization of the proposed approach is the finite element
simulation of the formation of the crease lines in origami models and folding
behavior of a sheet of paper by the robot end-effectors in professional solver LS-
DYNA.
Problems of the numerical simulation, including paper material structure,
simulation origami model, the distribution and values of the applying forces, and
others are considered carefully. The configuration of the designed origami-
performing robot is confirmed by the calculation of the reachability and
manipulability of a robot manipulator. Computing software MATLAB is used for
writing a programming code of a motion of the robotic arms.
Two forms of the origami are considered for an investigation: flexible, if folding
paper is produced by bending, and rigid, when forming the 3D shapes from the flat
patterns without having to bend.
The finite element simulating results of forming origami models from the crease
patterns on a sheet of an ordinary paper (flexible form) by using the designed robot
arms are presented and provided by the illustrations. Cardboard origami patterns
(rigid form) are considered as the kinematic systems in simulating.
The simulation results support the idea of the simulation-based designing a robot
to form the 3D origami shapes according to the virtual behavior of the robot arms.
The proposed methodology is useful for scientists, engineers, specialists, and
programmers that work in robotics and origami design as rules for origami-
performing robot design.
iv
Acknowledgement
This dissertation is completed with zealous support and cooperation of many
people.
First and foremost, it is my honor to express my appreciation to Professor
Hagiwara. He has taught, guided and supported me through all my research time in
Meiji University. The joy and enthusiasm he has for the research was the motivation
for me, even during tough time of my PhD pursuit. He also gave me opportunities to
take part in international conferences where I can improve and have a wider
knowledge about my research topic. It is such a great pleasure to work with Professor
Hagiwara.
Secondly, I am extremely grateful to Doctor Maria Savchenko for all her support
and guidance to my research. She has inspired me and given me precious suggestions
for my thoughts. Additionally, she always cares and encourages me whenever I have
troubles with daily life. We worked together and figured out many ideas and
investigations that are truly precious for my research development.
Moreover, I would like to thank to Professor Masayasu Mimura, Professor
Toshiyuki Ogawa and all MIMS member for their kind assistance and
encouragement; and other people at MIMS helped me with my work at various
stages. Besides, I am grateful to MIMS for their financial support throughout my
PhD years.
I also sincerely appreciate Professor Dinh Van Phong, Associate Professor
Nguyen Quang Hoang and other members of Department of Applied Mechanics,
School of Mechanical Engineering, Hanoi University of Science and Technology that
gave me agreement and support during my oversea study.
Having a chance to work with members in Origami engineering group of
Professor Hagiwara laboratory is my honor and they help me a lot. My special thanks
to Nguyen Thai Tat Hoan for his introduction to Professor Hagiwara, so that I had a
chance to go to Japan to study and research in a developed country like Japan. My
v
sincere thanks are due to Yang Yang and Julian Andres Romero Llano for many
fruitful discussions and support for my daily life with my limit Japanese.
I would also thank to all my friends in Tokyo, especially my beloved groups
Xom Nha La and Vietnamese Dancing Club. They always stay by my side and
encourage me to deal with lots of troubles in my daily life. Special thanks to Huyen
Nguyen, although she lives far away from me, in Germany, she always cares and
listens to me whenever I‟m in need.
Last but not least, I would like to express my appreciation to my parents, my
sister, my niece and my nephew, and all other family members for their support and
their belief. I was not able to complete my PhD dissertation without them.
vi
Contents
Chapter 1 Introduction ........................................................................................... 1
1.1. Background of the research ..................................................................... 3
1.1.1. Origami properties ................................................................................... 3
1.1.2. Robotics for origami ................................................................................ 4
1.1.3. Computational mechanics ....................................................................... 8
1.1.4. Simulation-based engineering ................................................................. 9
1.1.5. Modeling of bending paper in computer graphics ................................. 10
1.2. Problem statement and the goal of the thesis ........................................ 10
1.3. Basic problems in the FE simulation of forming crease lines ............... 13
1.4. A framework of the proposed approach for designing the origami-
performing robot .................................................................................... 14
1.5. Structure of the thesis ............................................................................ 16
Chapter 2 An approach for the design of the origami-performing robot ........ 17
2.1. Conceptual design ................................................................................. 18
2.2. Detailed design of the robot parts .......................................................... 20
2.3. Summary ................................................................................................ 24
Chapter 3 Computer model: Main definitions and simulation folding
conditions for flexible origami ............................................................ 25
3.1. Mechanical properties of paper ............................................................. 26
3.2. Placement of grippers on a sheet of paper ............................................. 28
3.3. Gripper forces ........................................................................................ 31
3.4. A working table design and the condition for folding paper ................. 32
3.5. Estimation of a formation of the crease line .......................................... 33
3.6. Summary ................................................................................................ 34
vii
Chapter 4 FE simulation and analysis of forming creases ................................. 36
4.1. General computational model ................................................................ 36
4.2. Meshing of the sheet of paper with the diagonal crease lines ............... 39
4.3. FE simulation of a formation of the intersecting creases ...................... 40
4.3.1. Simulation model for two-intersecting creases ..................................... 40
4.3.2. Simulation model for multi-intersecting crease pattern ........................ 41
4.3.3. Simulation conditions and the FE analysis ............................................ 44
4.4. Elasto-plastic paper deformation and dynamic analysis........................ 46
4.4.1. Stress-strain analysis ............................................................................. 47
4.4.2. Measure of energy stored in an element ................................................ 50
4.5. Summary ................................................................................................ 55
Chapter 5 Design development ............................................................................. 56
5.1. Modification of a holder design............................................................. 56
5.2. Optimal design of the plane contact portion of a robot gripper ............ 59
5.3. Modification of the conceptual design .................................................. 65
5.4. Summary ................................................................................................ 68
Chapter 6 Robot performance analysis ............................................................... 69
6.1 The reachability of the robot manipulators............................................ 70
6.2 The calculation test of the reachability .................................................. 73
6.3 The inverse kinematics of a robot arm .................................................. 75
6.4 Robot performance improvement .......................................................... 77
6.5 Summary ................................................................................................ 78
Chapter 7 Applying the developed methodology ................................................ 80
7.1 Flexible origami ..................................................................................... 80
7.1.1. The “Star” model ................................................................................... 80
7.1.2. The model “Tetrakis Cube” ................................................................... 85
7.2 Rigid origami ......................................................................................... 87
7.2.1. Kinematics of the folding pattern .......................................................... 87
7.2.2. “Miura-ori “ shape forming by the force application ............................ 88
viii
7.2.3. Cardboard packaging by the designed robot arms................................. 90
7.3 Summary ................................................................................................ 96
Conclusion ............................................................................................................... 97
Bibliography ............................................................................................................ 101
1
Chapter 1
Introduction
Origami is an old art of folding paper. It is dated in the 7th
century and was
developed in Japan with originality in China. Nowadays, origami is an interesting for
engineers in the fields of civil engineering, architecture, biotechnology, medicine,
space engineering and others. Since origami has many advantages, its applications
are now used widely in industry and everyday life.
Origami starts from a two-dimensional layer (crease pattern) and transforms to
the three-dimensional structure through folding. The benefit of origami structures is
their abilities to support weight with enough stiffness and to pack a large surface area
into a compact flat shape. Origami principles have broad and varied applications. For
instance, “Miura-ori” origami structure that is developed and formed using low
energy bending and with only one single movement, can be spread into the 3D shape.
This structure has been applied in solar panel in the space plane (Zirbel et al., 2013)
(Figure 1.1). Folding systems in nature is used for building structures to generate
plane structure surfaces (Trautz and Herkrath, 2009).
Additionally, origami structure also is applied in medical surgery, like
mechanisms for stent grafts (Kuribayashi et al., 2006), DNA-sized boxes (Andersen
et al, 2009), or building facades (Del Grosso and Basso, 2010), and self-folding
robotics (Felton et al., 2014) (Fig. 1.2-1.5).
Figure 1.1 Solar array of space craft from Miura-ori structure (Zirbel et al., 2013)
2
Figure 1.3 Cylindrical tube of the stent graft
(Kuribayashi, 2004)
Figure 1.5 Origami self-folding robot
(Felton et al., 2014)
Figure 1.2 Leaf of Chamaerops humilis (Trautz and Herkrath, 2009)
Figure 1.4 Adaptive skin
(Del Grosso and Basso, 2010)
3
1.1. Background of the research
1.1.1. Origami properties
Mathematical background on origami. The Hizuta-Hatori seven axioms are a
set of rules in paper folding related to the mathematical principles of paper folding.
These axioms describe what can be constructed using a sequence of creases with at
most two point or line alignments at once. (Geretschläger, Robert, 2008).
Flat foldability of a design is defined by a possibility to fold it into a single plane
with a thickness determined by a material. Global foldability for multi-vertex folds is
an NP-hard problem. There are several folds in the origami art: a “mountain”
(convex) fold, a “valley” (concave) fold and “swirled fold”.
Foldable conditions for a single vertex flat must satisfied two theorems:
Kawasaki‟s theorem states that the sum of the odd angles must be equal to the sum
of the even angles; Maekawa‟s theorem states that the number of the mountains
must differ from the number of valleys by 2. These theorems and their extension to
more than one vertex are discussed in (T.C.Hull, 2002).
Origami forms. Origami takes two forms: flexible and rigid.
Definition of a flexible form: if paper folding is produced by bending.
Definition of a rigid form: forming the 3D shape from a flat pattern without
having to bend.
Folding rigid steel and cardboard is a good example of the rigid origami.
Understanding how linkages fold and unfold involves rigidity, a key concept in
origami engineering. A linkage can be defined as a graph consisting of vertices and
edges. A configuration is a linkage that includes coordinates for the vertices that
satisfy each edge length. When a linkage folds or moves, it reaches many
configurations and the complete set defines the configuration space. A linkage
configuration is flexible if it can move from some initial configuration in a nontrivial
way (i.e. a motion that is not just a translation or rotation); otherwise it is rigid. The
materials and methods used for fabricating, actuating, and assembling these products
can vary greatly with a length scale. Large-scale origami structures can be
constructed from the thickened panels connected by hinges and can be actuated with
mechanical forces. With developing the origami structures, the material using for
4
folding currently is not only ordinary paper with a small thickness 0.1mm. Special
paper materials, such as cardboard or coated paper, which thickness is bigger (1-
2mm), are used for forming origami structures to increase the stiffness and still keep
the lightweight structure.
From mechanical point of view origami can be defined as a folded structure. The
correct theory to describe folded plates (a rigid origami) is a six parameter shell
theory (3 displacement, and 3 rotations on the displacement field) (Gilewski and
Pelczynski, 2016).
Finite element method (FEM) is necessary to use for numerical analysis, because
of the complex parameters of the structures. FEM needs to be applied with
professional software. The most important task is to develop an effective technique
for efficient computation of structure with a lot of folds. One of the initial issues is
the type of the connections between folds: fixed, linear hinge or truss structure
(Nguyen, 2015).
1.1.2. Robotics for origami
Nowadays, robots are used everywhere in everyday life. With all the recent
advances in robotics, practical robots become available for home use. These types of
robots have to be portable with the small gabarits and weights to do some real work
at home or at the company offices to exclude the humans from making the target
work. The papercraft production as an art and a hobby for many people, including
children, is the important part of the human activity and can become a popular
application for the origami-performing robot. The aim of the origami-performing
robot is to exclude the humans from the complex and time-consuming operations in
producing the origami models. Papercraft, Computer Aided Design (CAD) or
origami design softwares can be connected to the robot system. Users will receive
possibility to create own origami structures, which will be produced at home directly
by a robot.
The geometrical sizes of the compact robot give the possibility to use a robot in
the limited spaces, such as home room (for instance, on a desk), children facilities,
company offices, and so on, for the production of paper models.
5
Nowadays everyone can create many 3D origami forms from the various patterns.
Using robotics for folding the 2D paper pattern for forming the 3D origami model is
considered in this paper. Manufacturing origami-inspired products requires robots
capable of bending and folding materials, such as Robofold (G.Epps, 2012),
Industrial Origami (see Figure 1.6-1.7), (Koshiba et al., 2011). Class of robots that
does only rigid origami is presented by remarkable engineering solutions.
Robots for flexible origami are not widely used now. However, using machine
for folding the paper origami models is not a brand new idea. The first paper folding
machine was discovered in 19th
century.
American worker Margaret Knight studied machinery by a day, and developed
drawings for a bag-making machine at night (see Figure 1.8) (wikipedia). She
received a patent in 1871.
During the last ten years several realizations of the idea to use robotics for
folding paper has been suggested in some publications, such as (Balkcom, 2004),
(Yokokohji, 2013). The robot design that includes a table, a blade, a clamp, and a
plate attached to a robot arm is demonstrated in (Balkcom and Mason, 2004). This
system was succeeded in folding some models like “Samurai hat” and “Airplane”.
Folding origami by two robotic hands was suggested in the paper by Tanaka et al.
(2007). The authors concentrated on origami patterns, which contain various typical
folds and do not include many folding steps. They chose the “tadpole” origami
model for their target. However, this system did not succeed in making the sharp
creases on the folding patterns: sheets can slip out from the fingertip during the
folding operation. Another approach that is connected to packaging robot is
presented in (Yao and Dai, 2008), (Dai and Caldwell, 2010), (Yao et al., 2010);
nevertheless, these approaches do not focus on making the crease lines on a sheet of
paper. They just concentrated on forming the 3D shapes with some folding
techniques, but do not use the imitating movement of human hands or fingers. In
their work, the folding systems were designed before and the folding patterns were
modified to fit the manipulating structure. Two robot hands are used for folding
paper in (Elbrechter et al., 2012). The authors apply a method for real-time detection
and physical modeling of paper and suggest an approach to recognize the shape of a
sheet of paper. A new robotic origami system is proposed in (Namiki and Yokosawa,
6
2015). System configuration includes the left and right hands with two fingers. To
produce dexterous paper folding the authors extract some dynamic motion primitives,
which contain visual or force information. Integrating dynamic primitives that
consists of sensory feedback control allows paper folding. Simulation of the
deformation of a paper sheet is developed with modeling of a sheet of paper by
triangular mesh elements. All above systems were designed based on an
experimental approach with the existing robotic system to fold the origami sheets.
Figures 1.9-1.13 are the illustrations of research mentioned above.
Figure 1.6 Robofold (Epps, 2012) Figure 1.7 Industrial origami robot
Figure 1.8 Paper-bag making machine
7
Figure 1.13 Folding paper by anthropomorphic robot hands
(Elbrechter et al., 2012)
Figure 1.9 Robotic origami folding
(Balkcom, Mason, 2004)
Figure 1.11 Folding robot for
confectionery industry
(Yao et al., 2010)
Figure 1.12 Robotic Origami folding
by Namiki and Yokosawa (2015)
Figure 1.10 Origami folding by
a robotic hand
(Tanaka et al., 2007)
8
1.1.3. Computational mechanics
Any phenomenon in nature can be explained by means of physical laws or
mathematically by using algebraic equations or by integral and differential equations.
The scientists and engineers explain the phenomenon by analytical descriptions,
it means by mathematical models. It is easier to perform the calculation by using the
mathematical models along with numerical methods (Finite Element Method, Finite
Difference Method, Boundary Element Method and etc.) with using computers.
Computational Mechanics connects the mathematical models with the numerical
simulation procedures to define the physical phenomena (Figure 1.14).
In this thesis, the mechanical engineering problem such as a design, kinematics,
material properties, structural analysis, and a motion of the robot are considered as
the numerical simulation problems, which are going to be solved by using numerical
method FEM.
Figure 1.14 Computational mechanics
Engineering problems
Mathematical models
Differential Equations
Formulations Boundary Integral Equations
(BIE) Formulations
Analytical
solutions
Numerical
solutions
Analytical
solutions Numerical
solutions
Finite
Element
Method
(FEM)
Element-
free method
(EFM)
Boundary
Element
Method
(BEM)
Boundary
Node
Method
(BNM)
9
1.1.4. Simulation-based engineering
Engineers and scientists have become increasingly aware that computer
simulation is an indispensable tool for resolving scientific and technological
problems.
A Report of the USA National Science Foundation Blue Ribbon Panel (2006)
on Simulation-Based Engineering Science gives a definition of a simulation as:
“Simulation refers to the application of computational models to the study and
prediction of physical events or the behavior of engineered systems. The
development of computer simulation has drawn from a deep pool of scientific,
mathematical, computational, and engineering knowledge and methodologies”.
Computer simulation as an extension of theoretical science in that is based on
mathematical models. Such models are characterized the physical predictions or
consequences of scientific theories. Simulation provides a powerful alternative to the
techniques of experimental science when measurements are impractical or too
expensive.
Also in Report one can find a definition of the Simulation-Based Engineering
Science: “as the discipline that provides the scientific and mathematical basis for the
simulation of engineered systems.
For the effective engineering design, optimization methods must be closely
coupled with simulation techniques. Fundamental understanding of what constitutes
an optimal design and how to find it in a complex multi-criteria design environment
can be solved by a simulation.
Many engineering communities use simulation software for well-defined,
specific, and independent areas of application.
Commercial software LS-DYNA is a general-purpose finite element program
capable of simulating complex real world problems. It is used by the automobile,
aerospace, construction, military, manufacturing, and bioengineering industries.
In this thesis, LS-DYNA is used as a powerful tool for a simulation of the
formation of the origami models by the robotic arms.
10
1.1.5. Modeling of bending paper in computer graphics
Shape modeling is of great interest to computer graphics. Paper creasing is
considered in (Kergosien et al., 1994), and (Frey, 2004).
In (Bo and Wang, 2007) the method for modeling a developable surface to
simulate bending in interactive applications is presented. This technique is based on
the concept of rectifying developable and can be applied to shape modeling and
design. The paper models are demonstrated in Figure 1.15.
This approach is suitable for bending paper and demonstrates good results for
designing applications. The main problem is that FE analysis is the necessary step at
each designing stage. Professional simulation software gives opportunities to
produce FE analysis for the best design solution.
1.2. Problem statement and the goal of the thesis
This thesis presents a new study in a mechanical and geometrical design of the
origami-performing robot that is based on an iterative FE simulation of the
manipulation of the robot arms with origami patterns. Instead of a physical robot‟s
prototyping, a virtual model of a robot that resembles the behavior of the physical
product is suggested. The design process itself should be thought of as one that
stretches from conceptual design to actual process design and fabrication. This
research is related to applied researches, because problems formulated in this
research are coming from industry and society.
Figure 1.15 Paper models from (Bo and Wang, 2007) generated by
presented method
11
Motivation of the study: Based on the analysis of the existing researches in
origami-performing areas it can be concluded that applications of origami in
engineering is yet rare. It means that improving folding efficiency in many
engineering operations is an open problem. Improving understanding of origami
folding by designing the origami-performing robot in a simulation environment
defines a research topic of this thesis. Also designing a robot, which can be used at
home or office rooms, is a significant technical challenge for automation engineers.
The goal of the research:
1) To develop a general simulation–based methodology for the manipulation with
the foldable objects, such as a sheet of paper, by the robotic arms;
2) To create a virtual model of the robot that resembles the behavior of the
physical design;
3) To design the practical robots that is available for home using.
The central concept underlying the study: each folding operation of a crease
pattern of origami is considered as a function of the mechanical systems such as a
robot.
Methods: Finite element method (FEM) (in solver LS-DYNA) and mathematical
computing software MATLAB are used for:
1) A simulation and finite element analysis (FEA) of forming origami models
according to a behavior of the robot arms.
2) Kinematic modeling and robot performing analysis.
Expected results: The simulation-based methodology for folding various
origami patterns can be a key step in virtual designing of the origami-performing
robots.
Developing a methodology as the general design strategy for the origami-
performing robot can be very useful for studying the robot arm behavior issues when
focusing on the relationship between folding origami model and a robotic
manipulation.
The suggested methodology‟ steps include a set of operations, methods, ideas and
engineering solutions that are important for understanding and developing
mechanical systems, such as a robot.
12
In this thesis, a simulation is considered as a virtual experiment with dynamic
behavior of a robot and the physically-correct simulation model should sufficiently
represent realistic behavior. It is supposed that a simulation model is created with
varying complexity consisting of many elements, and the simulation model should
work as expected.
During the research, main problems to solve in simulating are investigated:
Folding and contact conditions;
Meshing, FE structural analysis;
Kinematic modeling;
Accuracy of crease lines;
Placement of the robot end-effectors on a sheet of paper;
Pressing and tensile forces and directions of their applications
Robot performance analysis.
The main merit of the proposed method to design a robot based on FE structural
analysis of paper material and robot arm‟s behavior is that the designers can estimate
the real robot design and its performance by using only a virtual design. The other
advantages are presented below:
The designers can estimate the real robot design and its performance by
using virtual design based on FE structural analysis of paper material and
robot arm‟s behaviors;
Mathematical simulation of paper and origami folding is complex;
Low costs to produce a robot from scratch;
Robot design can be modified without costs;
In a complex task, robot can be simulated in stages;
Simulations software are available for designers and engineers;
Elapsed time between the start of the design and its completion can be
shortened.
However, there are also some demerits of the suggested approach:
Simulation difficulties connected with the modeling of materials, origami
structures, and robot manipulation;
13
All simulation results depend on the correctness of the simulation
conditions.
Robot applications:
1) “Home-used”: papercraft, education, packaging;
2) Industrial type: architecture, furniture, storage, manufacturing, and etc.
1.3. Basic problems in the FE simulation of forming crease lines
The design based on experiment only is difficult, expensive, time-consuming and
so on. The simulation approach based on software becomes the main option in the
real-world related with a robotic activity for creating basis for their hardware design.
The main problem that we try to solve is how to form the crease lines by the
robot arms in the simulation model. In related works mentioned above, the crease
making approach was considered in the different ways. The crease lines are produced
by putting a sheet of paper in a slot and formed them by using the clamp shut
(Balkcom and Mason, 2004). In research by Tanaka et al. (2007), a rubber ball slides
on the paper to form the crease line by moving a fingernail attached to the fingertip
on an arm-manipulator after the folding operation. In the paper by Yao et al. (2010),
the object is carton box with paperboard material. Creases are made before bending
panel to form the box shape. Although these above methods are able to form the
crease lines on the origami patterns, the accuracy of the results was not analyzed and
illustrated clearly.
The 3D structural simulation of the origami models is carried out by the finite
element modeling (FEM) in LS-DYNA software Livermore Software Technology
Corporation (2001).
The FEM is the most used method for numerical simulations of physical models
based on partial differential equations (P.D.E.). In this method the P.D.E. model is
replaced by a discrete problem for computing. A mesh (computational domain) is
constructing by geometrical elements such as triangles, quads, tetrahedral, hexahedra,
etc. Any numerical simulation may be impossible because of a failure in mesh
construction (Frey et al., 2000).
In the presented research, the crease lines should appear during bending paper
around the folding lines that defined by the origami pattern. Paper is chosen as a
14
material (hereinafter we call it paper) in the simulation model for making apparent
crease lines during the paper folding process. In this case, it is necessary to
investigate a structure of paper and mechanical operations with it for creating the
crease lines by the simulation process. Additionally, the magnitudes of the force and
directions of their application in simulation are considered carefully to achieve the
best results for creasing.
For rigid origami, kinematic modeling and sequences of folding origami patterns
by the robot arms are the main problems for solution in simulating.
1.4. A framework of the proposed approach for designing the
origami-performing robot
The robotic system is suggested to be designed mostly for papercraft and
packaging applications. Small geometrical sizes of all parts of the robot will allow
people to use it at home or in company or school offices.
This study involves three main stages:
Conceptual robot (schematic) design;
Computer modeling the folding objects by the robot arms (a construction of
the FE model and numerical calculations);
Robot design development.
Figure 1.16 demonstrates an approach for the origami-performing robot design.
In Figure 1.17, a Flowchart of the methodology shows the steps of this work.
Figure 1.16 Scheme of the robot design process
Conceptual
(Schematic)
stage
Simulation of the 3D origami
forming using FEM in
LS-DYNA
Design
Developing
stage
15
Step 1
Step 2
Step 3
Step 4
Step 5
Step 6
Step 7
Step 8
Conceptual design
of a robot (Ch.2)
Simulation-based approach for creasing paper
and forming origami models by the robot
arms (Ch.1)
Robot performance analysis
(Ch.6)
Simulation testing of
the 3D shape
forming from the
thickened paper
(Ch.7)
Definition of the
simulation
parameters and
folding conditions.
Definition of the
force‟s types and
directions. (Ch.3)
Computer model,
Numerical calculation
of crease forming.
FEM in the solver
Ls-Dyna.(Ch .3)
Analysis of the
simulation results: the
strengths and
weaknesses in
Conceptual Design
(Ch.4)
Repeatable Robot
Design Developing
Stage (Ch.5)
Testing the origami models
(Ch.7)
Figure 1.17 Flowchart of the methodology
16
1.5. Structure of the thesis
The thesis is organized according to a Flowchart (Figure 1.13) as follow:
Chapter 1 is the introduction of origami properties and robotics for origami. This
chapter includes the goal and the framework of the origami-performing robot design
process.
Chapter 2 presents an overview of the method and conceptual design of the
robot‟s parts. Geometry and the behavior of the end-effectors of the robot arms in the
FE simulation are discussed.
Chapter 3 presents the origami folding conditions for flexible type of origami,
especially the method of forming sharp creases by the robotic arms.
Chapter 4 is the explanation of the simulation model and the Finite Element
analysis for crease forming.
In chapter 5, developing design stage, which includes the modifications of the
conceptual design according to the simulation results, is explained.
Chapter 6 is the analysis of robot performance. In this chapter, the problems of
robot reachability, kinematics and the manipulability extension with the target of
avoiding singularity are investigated.
Chapter 7 presents the testing the origami-performing robot for forming the 3D
shapes for the flexible and rigid origami forms. With flexible origami, the folding
sequences and motion planning for manipulators are demonstrated for the “Star” and
the “Tetrakis Cube”. In rigid origami, the kinematics of folding pattern and the
modification of crease patterns for robotic application are considered carefully. The
simulation results are demonstrated based on the given origami patterns, such as:
“Miura-Ori” and “Gift box”.
Chapter 8 presents summarizing the suggested simulation-based methodology
for manipulation with the foldable objects, calculation and simulation results,
contributions, limitations, and the future work.
17
Chapter 2
An approach for the design of the origami-
performing robot
Robot should be designed according to a problem, which it is trying to solve.
Robots are manipulated devices that keep a position between automatic machine,
which is used for a single task, and human operators for handling a various kind of
jobs. In the robot design process, many requirement and conditions should be
considered to guarantee a functionality of it. The requirements are presented on the
mechanical structure design and the electronic design. A designer considers practical
functions, such as manipulation, geometry and topology, shape and form, materials
that are suitable for the design, and specific details of the design, which must be
satisfied to requirements. Finding a solution of the complex problem by a simple
solution for mechanical design is very interesting and it is an important step in
designing process.
A study of a number of different situations helps to decide exactly the problem
that a robot should solve.
In this thesis, an origami-performing robot is under consideration. It means that
the main problem for its solution is folding paper and forming the 3D shapes. In this
case, an approach for the robot design is based on possibility to use a robot for
folding origami structures.
Automation of a process of the deformation of the flexible objects, such as paper,
by using a robot is the difficult task for an experimental approach. A simulation
based on computer software becomes the main option in real-world robotic related
activity.
The proposed design process that is demonstrated in Figure 2.1 includes three
main stages: conceptual (schematic design), developing a computational model and
numerical calculations, design development. In the simulation, end-effectors of the
18
robot arms are considered. In this chapter, the conceptual design and mechanisms of
the end-effectors are discussed.
2.1. Conceptual design
The conceptual or schematic design is the first phase of the research. As
mentioned in Introduction, the final goal of the research is to design the compact
origami-performing robot. It means that the first criterion is to design the
individualized robot. With this goal, a robot is required to be light-weight, portable,
and be easy to manage the applications. The second criterion is connected to the
robot manipulations. Robot manipulator consists of the arms with end-effectors
(fingers) and a body. For designing the conceptual model of a robot and its
manipulation it is necessary to observe how human hands operate during the folding
process of a sheet of paper. Human being can apply some tricks to make the crease
lines during the folding process. Human fingers grasp two corners of the origami
paper pattern and bring them into precise alignment. Then the fingers of one of hands
smooth the bulges on a sheet of paper. Since paper does not stretch, the creasing lines
form the extreme region of paper. Crease lines are sharpened by applying strength
from fingertips. This operation is important for folding an origami pattern by the
robot arms. It seems to be a simple problem when a person folds a sheet of paper;
anyway, there are at least 6 states in process: holding, folding, flattening, flipping,
gluing, and checking.
Computer model Conceptual model
Developed model
Figure 2.1 Schematic illustration of the suggested approach
Holding fingers
Folding fingers
19
In some related works, there are two kinds of origami folding robot. The first
type can be called a “folding machine” in the research of (Balkcom and Mason,
2004) and the second one is a “folding manipulator”, as in works of (Tanaka et al.,
2007), (Yao et al., 2010), (Namiki and Yokosawa, 2015). In the presented research a
purpose is to build a system of the “folding manipulators” that basically imitates the
operation of the human hands when forming the origami models. In (Tanaka et al.,
2007), the purpose is to design a robot that can fold a sheet of paper by a crease
pattern. The accuracy of robotic folding should be the same as by the human hands;
and the robot configuration should be as simple as possible. There are four
manipulators in their system, two holding and two folding fingers (see Figure 1.9).
Each finger has the same configuration, where a parallel-driven 2-link mechanism is
used for horizontal motions and a ball screw is used for vertical motions. For forming
the creases, a fingernail is installed on the top of the two holding fingers so that these
fingers can insert their fingertip into a gap of the folded paper.
From studies and requirements mentioned above, the conceptual model of the
developed robot includes a working table and two arms: one folding and one holding
(see Figure 2.2). For the folding arm, the main operation is folding paper. The
holding arm is used to fix a sheet of paper on the working table. The experimental
folding machine presented by Balkcom and Mason (2004) includes a blade press for
forming the creases and a working table. In this thesis, the suggested conceptual
design is closed to the design described by Tanaka et al. (2007) for the “tadpole”
model. The authors of this paper use a rubber ball to form creases and explain that
their robot design should be redesigned for each origami model.
In the presented research, it is assumed that the robot will be not redesigned for
the different origami structures. The robot that can fold various kinds of origami
models is may be more complicated to satisfy that requirement.
The lengths of each robot links in the conceptual design are suggested and given
in Table 2.1.
20
Table 2.1 Robot link‟s lengths
2.2. Detailed design of the robot parts
The object of the robotic system is A4 format of the sheets of paper. In this case,
the design of the robot should guarantee the satisfaction to this requirement.
The folding and holding arms of the robot have the same configuration: 3
revolute joints for each manipulator (Figure 2.1) with the different functions. End-
effectors (fingers) are the most important parts of each robot arm. We call holding
arm‟s end-effector as the holder (holding finger) and folding arm‟s end-effector as
the gripper (folding finger). Grasping a paper sheet is classified as a lateral grasping
(Kosuge et al., 2008). A human generally uses only two fingers: the thumb and the
forefinger supported by others for the lateral grasping. The thumb is used to apply
force to the objects supported by the forefinger. Gripping force is required to have a
small magnitude for grasping a flat object like a sheet of paper. The gripper includes
at least 2 fingers: a thumb and a forefinger to grasp one side of paper and bend it
around a crease line. The mechanism, which we choose for the folding gripper, is the
crab propodus. There are two fingers in the crab propodus: one fixed finger and one
l1(mm) l2(mm) l3(mm)
Folding arm 300 100 80
Holding arm 300 100 80
Figure 2.2 Sketch of the conceptual robot design
cc
Holding finger
Folding
fingers The working
table
A sheet of paper
21
movable finger. The fixed finger produces the movement of dactyl (the movable
finger) by two muscles: muscle attaching on the flexor apodeme for closing and
muscle attaching on the extensor apodeme for gaping. The advantages of this
structure are: the simple lever and mechanism, and an adaptation to particular ways
of feeding. The “crank-slider” mechanism is suitable to design the folding gripper
with a fixed finger and a movable one (Figure 2.2a). There are 4 parts in the
mechanism like a “crank-slider”: a base, crank, slider, and connecting rod
(wikipedia). The base is fixed, the crank is rotated around the fixed axis, the slider
motion is a horizontal translation and the connecting rod movement is a motion in
the plane. When the slider moves to the right, it transfers the motion to the
connecting rod and makes the crank rotation (open up to grasp paper).
For the characteristic of the robot, the gripper should be open and closed as
quickly as possible. As paper is not very thick, the opening angle of the gripper is not
unnecessarily to be a very big. The opening angle can be calculated as below:
sind
l , (2.1)
where d is slider‟s moving distance and l is crank‟s length (Figure 2.2a). Moving
distance d is connected to the paper thickness h. In this case, minimal value of α
should be defined by h: minimum(α) =arcsin(h/l).
The geometrical shape for the grippers is inspired by human actions for folding a
sheet of paper. The shape of the gripper from computer-aided design (CAD) and the
prototype of the gripper developed on 3D printer are illustrated in Figure 2.2b and
2.2c. In this case, the shapes of the holders are rectangular cuboids. The geometrical
parameters of the grippers and holders are described as in Figure 2.3a and 2.3b. The
design of the working table is given in Figure 2.3c. The dimension of the geometrical
parameters of the grippers and holders are shown in Table 2.2.
To make a perfect crease, one side of a sheet of paper is fixed on the working
table while the grippers grip the free side of the sheet for bending it around the crease
line. The purpose of this step of the investigation is to make the crease lines
appearance obviously after bending. A sheet of paper has infinite degrees of
freedom; it is better to fix it along the crease line in order to make paper constraint
and prevent bubbling and wrinkling. Since the crease lines may have different
22
lengths, the design of the holder with only one part in a constraint length will not be
an optimal choice. If the design will not satisfy the requirement of the light-weight
and the crease pattern the holding arm can be designed with 2 holders that can fix a
sheet of paper on the working table. The best positions for the grippers on the sheet
of paper should be found during the simulation. To simplify the robot mechanism (in
comparison with the actions of the human hands, when a sharpness of the crease line
is produced by the additional operations by applying the force from the human hand),
a working table is designed with a sharp edge for accurate forming the crease lines
(see Figure 2.2d). Polypropylene (PP) is proposed as a material of end-effectors and
a working table. The working table‟s geometrical parameters are calculated based on
paper size standards. A3 paper format (297x420 mm) is considered as a maximal size
for the working table design.
The conceptual detailed design and mechanisms of the end-effectors are the basis
for the next steps of the simulation-based robot design.
c. 3D printing prototype of the gripper
Figure 2.3 Gripper of a type “Crab” mechanism
slide
r
crank
Connecting rod
d
α
Movable part
a. “Crank slider” mechanism b. Gripper design by 3D CAD design
software “SolidWorks”
23
Table 2.2 Dimension of the geometrical parameters in mm
Grippers Holders
h1 h2 h3 h4 a b c
40 50 5 10 40 50 5
Figure 2.4 Detailed design of the robot parts
b. Holder geometry
a
b
c
a. Gripper: geometrical parameters
Top
Bottom
h2
h1
h3
h4
h1
h2
h3
Working table‟s leg
Paper sheet
c. The CAD model of a working
table
24
2.3. Summary
The conceptual design includes main necessary geometrical and mechanical parts
of the compact origami-performing robot such as grippers, folders, and the working
table. The next step of the methodology is developing the computer model of the
formation of the 3D origami model based on the conceptual design. It is only the first
step in an iterative design. The next steps of the design are presented in the following
chapters.
25
Chapter 3
Computer model: Main definitions and
simulation folding conditions for flexible
origami
Real paper is approximately non-stretchable and shapes obtained by bending a
sheet of paper can be rolled out onto a plane without stretching or tearing. Explicitly
represent surface irregularities such as creases by simulation leads to any
understanding about the deformation of paper. Accurately simulating the behavior of
paper creases includes several tasks like employing mechanical shell modeling. A
sheet of paper can be folded and unfolded without stretching or tearing through
specified angles when the forces are applied.
Creasing refers to the internal de-lamination of a paper sheet by compression
along the line where folding to occur. During folding a sheet of paper at the crease,
all the outward forces that normally would split or crack the sheet are concentrated to
weaken de-laminated line. The correct simulation results are depends on a
complexity of the origami model and right simulation conditions. Below in this
chapter necessary definitions and expected conditions for modeling are introduced.
Crease pattern analysis. For understanding origami structure for numerical
simulation some terms in origami should be presented. A crease is a fold: convex
(mountain) and concave (valley). Crease pattern is a collection of all creases. A
vertex is a point where creases intersect. The degree of vertex is defined by a number
of creases intersecting at this vertex. Figure 3.1 illustrates a crease pattern, in which
AB1 and A1B are diagonal valley folds, CC1 and GG1 are mountain folds, and E is
the crease intersection vertex.
Material. The material used in origami application plays the very important role
for numerical simulations. Paper as an elastic material that prefers to be flat is
26
A
1
C
A B G
E
G1
B1
C1
Figure 3.1 Crease pattern
commonly used in artistic origami. In the presented simulation, paper is called
ordinary.
3.1. Mechanical properties of paper
Here, a structure of paper and forming creases on paper (Lister) are discussed.
Paper is made up of vegetable fibers, which are felted. The fibers are brittle and
during the crease processing paper fibers are permanently fractured along the line of
the crease. This is a permanent line of tiny fractures that forms the crease. Because
this line is indelible, paper is said to "remember" the crease. Even hot ironing will
not get rid of the crease. Finite Element Analysis (FEA) is used to approximate and
verify component reaction under various loading conditions. Material is a main
component to FEA. Elastic material property has a linear stress-strain relationship
regardless of the load applied while the elastic-plastic has a linear relationship up to
yield point and then becomes nonlinear beyond yield point. Fully elastic deformation
is (Hookes Law):
ζ = Eε, where ζ is stress, E is modulus of elasticity, ε is strain.
Paper displays various unconventional mechanical properties both below and
beyond the plastic limit, all the way up to its failure. As it was well known, there is
approximately linear section at small strains in the load-elongation curve of paper.
Nevertheless, the yield point has no unique definition because the deviation from the
linear portion grows gradually as elongation increases. Thus, one way to define the
27
yield point is to identify stress (and strain) at which the smooth curve begins to
deviate from a straight line beyond a certain percentage.
The linear elastic, orthotropic behavior of paper prior to yielding can be modeled
by the classical model.
1 2 1211 11 22
12 21 12 211 1
E E
, (3.1)
1 21 222 11 22
12 21 12 211 1
E E
, (3.2)
12 12 122G G , (3.3)
where the engineering constants are: E1= Young‟s modulus in x1 - direction (MD),
E2=Young‟s modulus in x2 - cross-machine direction direction (CD), 2212
11
is
the Poisson‟s ratio for strain in x2 - direction (CD) when paper is stressed in x1 –
machine direction (MD) only, 1121
22
is the Poisson‟s ratio for strain x1 – (MD)
when paper is stressed in x2 - direction (CD) only, G=shear modulus in x1x2- plane
(Figure 3.2).
Here the same constant of proportionality between elastic module E1 and E2,
Poisson‟s ratios 12 and 21 is required so as to ensure that the stiffness matrix is
orthotropic:
1 12
2 21
E
E
, (3.4)
The crease line is a permanent deformation (shape changing) of a sheet of paper
and can be considered as a plastic deformation. In the current simulation of the
forming crease line, tensile forces along MD direction and CD direction are under
consideration. That configuration of forces allows the elastic analysis. The elasto-
plasticity model is more complicated for the calculation but it is necessary to produce
this analysis for the verification of the real quality of the formed creases.
In the simulation of the crease forming, elasticity of a model is considered in LS-
DYNA terms. Nonlinear analysis is more complex, but gives the more accurate
28
results. Paper plasticity models are defined in terms of stress and strain. By assuming
a linear elastic law, plasticity can be defined in terms of strain only.
3.2. Placement of grippers on a sheet of paper
The performance of folding the sheet of paper along the crease line can be
analyzed and evaluated by conducting a little experiment. Considering human hand
as a robot hand, one side of a folding paper is fixed at the crease line and bent the
other side around the given crease line. The creases do not appear without a tension
of the sheet of paper. The results of this experiment can be applied to a simulation of
folding paper: paper is bent and stays at a tension during the folding. In this case the
crease lines appear apparently.
With a fiber structure mentioned above, a crease will not appear without external
impacts around the crease area. Then, it is not simple procedure to create the crease
lines by using bending paper by a robot. To fold paper accurately, 3 states for folding
fingers are suggested: gripping, rotating, making tensile of paper. These states are not
the complex problems for robotic arms, and the crease lines appear apparently on
paper after bending. Hence, it is necessary to investigate the simulation conditions
for gripper placement during the bending operation on paper.
Calculation of a radius of gyration. Gyration is defined as rotation of a planar
region about some axis lying in the plane.
x2
MD
CD
x1
F2
F1
F2
F1
Figure 3.2 Illustration of tensile force applications
29
For a bar cross-section: several areas, 1 2 3, , ,...a a a at distances 1 2 3, , ,...y y y from a
fixed axis, may be replaced by a single area A , where 1 2 3 ....A a a a at a
distance k from the axis, such that 2 2Ak ay .
k is a distance from an axis and called the radius of gyration of the area A around
the given axis. Since2 2Ak ay I then the gyration radius is determined:
Ik
A (Bird, 2001). I is a second moment of area of the cross-section of a beam
about a given axis of rotation lying in the plane of the cross-section (usually passing
through its centroid). The second moment of area is a quantity much used in the
theory of bending of beams and is needed for calculating bending stress. The
procedure to determine the second moment of area of the regular sections about a
given axis is to find the second moment of area of a typical element and to sum all
such second moments of area by integrating between appropriate limits.
The second moment of an area of the rectangle about an axis is found by initially
considering an elemental strip of a width x , parallel to and a distance x from axis
PP (see Figure 3.3a). Area A of the shaded strip is b x . Second moment of area of
the shaded strip about PP is calculated as 2I x b x .
The second moment of the area is obtained by summing all such strips between
0x and x l , i.e. 2
0
x l
x
x b x
The second moment of area of the rectangle about PP is calculated such as:
3 32
03 3
lx bl
I b x dx b
, (3.5)
Since the total area of the rectangle, A lb , then 2 2
3 3pp
l AlI lb
2
pp ppI Ak thus
2
3pp
lk .
The radius of gyration about asis PP is calculated such as:
2
3 3pp
l lk , where l is a length of the rectangle.
30
In the case of bending a sheet of paper, the radius of gyration is calculated
according to a theory described above: 3
lr , where l is a distance between the
crease line and the edge of a sheet paper (Figure 3.3b).
The placement of the grippers on a sheet of paper. Theoretically the gripper
can be placed on a sheet of paper according to the r (Figure 3.3b). Additionally, it is
necessary to ensure avoiding the collisions between the working table and grippers
during the rotation of a sheet of paper around the crease line by the rotation angle 90
and more degrees. The thickness of the working table ttable is calculated according to
the gripper location in the current folding stage. The real placement of the grippers is
determined by the distance x as it is shown in Figure 3.3c and should satisfy two
Figure 3.3 Placement of the grippers on a sheet of paper
a. The second moment of
area of rectangle
P
b
P
l
δx
b. Bending a sheet of paper
Bending
moment
Crease line Holder
Gripper
r
l
Fpress
c. Placement configuration with
avoiding collision condition
Working table
Grippers
Holders
ttable x
31
conditions: (i) the radius of gyration, (ii) avoiding the collisions between the bottom
part of the gripper and the working table.
3table
lt x , (3.6)
where ttable is the thickness of the working table, 3
l is the radius of gyration.
The best way to get a high second moment of area is to get as possible the longest
distance from the axis.
3.3. Gripper forces
In this section, the forces that should be applied to the paper object by the
grippers to produce folding are considered. Bending alone may not be sufficient to
provide the desired shape and tension may be required also because paper is made up
of vegetable fibers, which are brittle.
When the paper object is grasped by a human hand, the motion of the object is
constrained by contacts with the fingers. Hence, the grippers grasp a sheet of paper
and pull it. This action is called a transient movement. To prevent being slipped out
of the grippers, pressing force is applied on the paper sheet by grippers. Bending
with a tension of a sheet of paper should be performed for forming the crease line.
Figure 3.3 shows the configuration of the movement of the robot hands and applying
forces during the folding process. Gripper pressing force Fpress is applied in the z-
direction and the gripper tensile force Ftensile are applied in the y-direction directions
in the initial configuration in a global coordinate system. The tensile forces are
applied to a sheet of paper from the grippers to keep the flatness of a sheet of paper
during the contact stages with a working table in folding.
In static case, when a sheet of paper does not slip out from the grippers, the
minimal value of the pressing force Fpress is calculated as below:
mgF W F
, (3.7)
where F is a gripper force pressF , W is a weight of a sheet of paper, µ is a static
friction coefficient between the gripper contact portion and the working part of the
32
paper sheet, m is a mass of the sheet, g is the gravitational acceleration. The static
friction coefficient is considered in range [0.1-0.2]; therefore the gripping force is
usually in 10-20 times larger than the weight of a working object. In our case, the
working object is a sheet of paper with a very small weight. For increasing the
sharpness of the crease lines, tensile force is applied permanently during bending
paper (Figure 3.4) in the y-direction in gripper‟s local coordinate system.
Tensile force is applied along the folding sheet in the direction according to the
rotation angle. This is the contact force between the grippers and paper. The contact
force is calculated by Eq. (3.8):
S cf K , (3.8)
where cK is contact stiffness, is penetration.
3.4. A working table design and the condition for folding
paper
The sharp edges of a working table play an important role for a crease forming.
As it is described in chapter 2, the geometrical parameters of the working table are
based on A3 paper format and include in the conceptual design. Here, the working
table design is considered from the paper folding point of view. The working table is
designed with the sharp edges with the aim to crease paper by contacting paper with
the edge of a working table. As paper is fixed by the holders on the working table
and bent by the grippers, paper are strongly deformed at the crease line. If the
working table has each face is a parallelogram with the angles measure 900, to form a
Figure 3.4 Sketch of the simulation of the crease forming by the robot fingers
33
sharp crease line is not possible because an angle of the rotation of a sheet of paper
around the crease line (a working table‟s edge) is limited (Figure 3.5a). In the
presented approach, the working table is suggested to be designed with the faces that
have an isosceles trapezoid shapes with two acute angles α. In this case, a “blade”
edge of the working table allows a formation of the crease line without limit angle
for a concentration of the applied forces at the sharp edge (Figure 3.5b).
3.5. Estimation of a formation of the crease line
As it was shown in this chapter above the crease forming and its accuracy depend
on many factors. The simulation results of folding origami by the robot arms should
be analyzed and tested.
The most important result is the appearance of a crease line. It is supposed that
during the producing paper folding by the grippers around the crease line by 90o, the
dihedral angle between a sheet part on the working table and the folded sheet part is
measured as 107othat is closed to90o
(Figure 3.6). Fold angles are defined in time
intervals and decreasing from 1800 to 100
0. It means that paper is deformed and a
sharp crease is formed. After that, paper remembers a deformation as the crease line.
To evaluate the accuracy of the folding operation circle interpolation approach can
be applied. A circle in 2D coordinate space can be constructed through three points.
In Figure 3.7 this analytic geometry approach is illustrated related with FE modeling.
Figure 3.5 Paper folding conditions related to a working table design
a. A Parallelogram working table
b. An isosceles trapezoid working table
Fixed part of a
sheet of paper
Rotating part of a
sheet of paper
α α
34
In simulating, three mesh nodes can be considered as the given points for
interpolation. In the result of the rotation of a sheet of paper around the crease line
by a small fold angle (Figure 3.7a), a smooth folding allows to construct a circle
through three mesh nodes with a radius R1. All of three mesh nodes out of the crease
area and a node on the generated crease will be presented as a triangle with obtuse
angle inside a circle with radius R1. When a fold is produced by a big fold angle the
circumscribed circles with radius R2 or R3 will be constructed. It means that three
given nodes define a triangle with right or cute angles.
3.6. Summary
This approach can be used for the estimation of the sharpness of the creases
(Figure 3.7b) as well as a FE analysis that is presented in chapter 4.
Figure 3.6 Fold angles with respect to time
35
Figure 3.7 Illustration of the estimation of the crease sharpness
by circle interpolation
R3
c. Folding angle > 90o
a. Small folding angle
R1
b. 90o folding angle
R
36
Chapter 4
FE simulation and analysis of forming creases
4.1. General computational model
FE model for forming the one crease line, which is parallel to the boundary of a
sheet of paper, is prepared by LS-DYNA to achieve a local reduction of the bending
stiffness and thus simplify the folding operations to make the creases more clearly.
The basic assumption is that a sheet of paper is thin in the sense that the thickness t
(prior deformation) << l – a length of the sheet of paper. Paper model is considered
as an elastic shell with 2t thickness. The FEM is the technique that is applied to solve
all simulation problems. A mesh for numerical simulation should be defined. Mesh
for numerical simulation is constructed by LS-DYNA tools. The implementation of
the thin shell with through a thickness is based on the formulation of the Belytschko-
Tsay shell with a relaxation of the thickness variable (Belytschko, Tsay, 1981). The
midsurface of the quadrilateral shell element is defined by the location of the
element‟s four corners. The Belytschko-Tsay shell element is one of the fastest and
popular in finite element codes for thin shell simulations. The kinematics that
includes position and velocity of the shell with through a thickness stretch can be
written in local coordinates as:
3 1 2
3 3 1 2
( ) ( , )
( ) ( , )
i iI I i I
i iI I ij jI I i I
x x s N
v v s e s N
, (4.1)
where 233(1 )
2I I Is t q
.
The kinematics is based on the Belytschko-Tsay shell includes the additional
feature that the thickness is variable. The thickness variable is represented by It and
an additional strain variable Iq to allow for a linear strain through the thickness. The
latter is important to avoid "Poisson locking" that means: for certain finite element
37
schemes, discretization error can become very large when the thickness of the plate
is close to 0. The other variables and parameters are:
:iIx i th component of coordinate of node I,
:iIv i th component of translational velocity of node I,
:jI j th component of rotational velocity of node I,
ijke permutation tensor,
IN shape function localized at node I,
3i Kronecker delta,
:i i th component of the parental coordinate ranging from -1 to 1.
For the fully integrated shell element, the kinematics is adjusted appropriately in
order to avoid spurious locking phenomena. The approach taken is to suitably
modifying the fully integrated shell element (type 16) in LS-DYNA, which has
turned out to be successful. For a single point integrated shell element, the thickness
is constant in the element whereas for the fully integrated element the thickness is
bilinear in the element. For stamping problems, where the reference geometry is just
a flat sheet, any of the two options are applicable.
Belytschko-Tsay shell element (type 2 one point integrated) is used for modeling
a sheet of paper (Hallquist, 1998). The reason to use type 2 is a small value of the
thickness of a sheet of paper.
As paper is a special material with different Young modulus and shear modulus
in different directions, forming creases is investigated based on elasto-plastic
analysis. As paper is orthotropic material, with fiber structure,
ENHANCED_COMPOSITE_DAMAGE (orthogonal material) is chosen for
modeling paper in the LS-DYNA simulation (Chang F.K, Chang K.Y). Arbitrary
orthotropic materials, unidirectional layers in shell structure can be defined. In
addition, special measures are taken for failure under compression, which LS-DYNA
supports. For all shells, laminated shell theory can be activated to properly model the
transverse shear deformation. Lamination theory is applied to correct the assumption
of a uniform constant shear strain through the thickness of the shell. Otherwise, for
comparison, an isotropic material in our simulation is considered in section 4.3.
38
0.1mm
Figure 4.1. Uniform meshing
Size of the finite element for modeling a sheet of paper. Thickness of the
ordinary paper is in the range 0.08÷0.1 mm. For the calculations, the thickness of the
sheet of paper is taken as 0.1mm. In the FE simulation, the origami pattern with one
crease line is modeled by the quadrilateral shell elements uniformly distributed
within the material. Because of a small value of the thickness, the general size of
each element‟s edge is 0.1mm. To save calculation time of the calculations, in some
simulations of folding paper in the real origami models, an adaptive mesh is
preferable. It is a combination of fine (dense) mesh that is generating around the
crease area and a coarse mesh (Figure 4.1 and 4.2).
Size of the finite element for modeling the robot’s parts. In the simulation
model, the robot includes the following parts: the working table, the grippers, and
holders. They are modeled by using 8-nodes hexahedron solid mesh; the size of each
edge of the element is 0.2mm (Figure 4.3).
0.5m
m
0.1m
m Figure 4.2. Adaptive meshing
39
4.2. Meshing of the sheet of paper with the diagonal crease lines
Correctness of the FE computation results of bending a sheet of paper depends on
mesh quality in the bending area (Bathe et al., 2000).
To model a sheet of paper, a suitable mesh for accurate and quick calculations
should be generated. In the simple crease patterns, the crease lines are often parallel
to the edge of a sheet of paper. Therefore, the uniform quadrilateral mesh is
generated for modeling a sheet of paper with non-intersecting lines (Section 4.1).
However, the crease lines that are located in the diagonal directions on a sheet of
paper can be seen in every origami pattern. For this type of the creases, a mesh
generation is not a trivial task. In this case, a mesh is suggested to be generated on
each sheet‟s part separated by the diagonal crease. Mesh is defined as a combination
of the mesh segments, which are generated according to the crease pattern, with the
quadrilateral and triangle elements. After that, a merging of the duplicated nodes
between two meshes is used to construct the single FE model (Figure 4.4).
Grippers
Holder
The working table
Figure 4.3. FEM model for folding a sheet of paper
Sheet of paper
40
4.3. FE simulation of a formation of the intersecting creases
4.3.1. Simulation model for two-intersecting creases
In some origami folding patterns, there are intersecting creases, for instance the
crease pattern in Figure 3.1. For forming the crease line, which intersects already
formed one, meshing and the folding conditions are the main problems for
considerations.
To solve this problem we produce a simulation of an experimental model: a
square shape of a sheet of paper (40x40 mm) and thickness 0.1 mm. The geometrical
sizes for holders are 5x10x0.5 mm; the plane contact area of the gripper is 6x5 mm.
By the calculation, a value of the sheet‟s thickness at the deformed area (the
crease area) is in 10 times less than the value of the initial thickness of the sheet of
paper. It means that the sheet of paper was deformed and it must be considered as a
multi-thickness shell.
For meshing, the sheet of paper is separated into three parts: a deformed area (in
red color), a 2 non-deformed parts (yellow and pink colors) as it is shown in Figure
4.5. Mesh is generated as uniform mesh with the quadrilateral elements 0.1 mm. The
value of the thickness of the deformed part is considered as 0.01mm and of the non-
deformed parts as 0.1 mm.
Figure 4.4 Meshing for a diagonal fold and a zoom-view
41
Forming creases, which are perpendicular to already existing creases, is modeled
with the same conditions and the configuration of the robot arms as for folding an
ordinary crease. The simulation results are demonstrated in section 4.3.3.
4.3.2. Simulation model for multi-intersecting crease pattern
A crease pattern that is presented in Figure 3.1 (Chapter 3) is a good example of
the multi-intersecting crease pattern. There are 4 intersecting creases in this pattern: 2
mountains and 2 valleys in the diagonal directions.
In the simulation model folding the mountain is considered in assuming that all
other creases have already formed. In this case, there are many the deformed parts of
the sheet of paper. In this case, a sheet of paper are considered as a multi-thickness
shell structure. It means that friction condition between a solid structure of the
gripper‟s contact area and the multi-thickness shell structure of paper chosen for the
two-intersecting crease pattern can not be applied.
The friction problem in the FEM can be solved via a simplification of the shell
structure of paper that is called meshing template:
to separate the a sheet of paper into 2 parts: deformed that is fixed on the
working table by the holders and non-deformed folding part;
consider the deformed part of a sheet of paper as a multi-thickness plate with
the thickness values 0.1 mm and 0.01mm;
Figure 4.5 FE model for a simulation of forming the two-intersecting crease lines
The crease area
42
the folding part is considered as the uniformly deformed paper with an
average thickness between deformed and non-deformed parts of the sheet of
paper.
Meshing for the deformed part is done by the mixed mesh elements (quadrilateral
and triangles of 0.1 mm size) in 6 segments: 4 with a thickness 0.1 mm and 3 with a
thickness 0.01mm. Because of the problem with the simulation of the multi-thickness
shell structure (Figure 4.6) in the FEM, the deformed fixing part is considered as a
non-deformed part of the sheet of paper with thickness value equal to 0.06 mm that is
an average value of the sheet‟s thickness after deformation.
In the folding part, the mesh is generating as an uniform quadrilateral mesh with
the element size of 0.1mm. Additionally, this part is modeled as a non-deformed
sheet of paper with the thickness value equal to 0.06 mm as it was explaned above.
The crease line area for the mountain folding is meshing by the triangle and
quadrilateral elements with a thickness value 0.1 mm. The proposed meshing
template ensures a strong contact between paper and the grippers.
Another reason to use this template for a simulation is the FEM problem: there
are many sectors in the simulated paper model with a different mesh element
orientations, which cannot satisfy to the gripper force configuration (MD and CD
directions) (Figure 4.6). Figure 4.7 demonstrates the suggested meshing approach for
the multi-intersecting crease pattern for avoiding the FEM problem. The simulation
model for the forming the crease line for the case of the multi-intersecting creases is
given in Figure 4.8.
Figure 4.6 FEM problem: the complicated mesh for
multi-thickness shell structure
MD
CD
43
Grippers
Holders
Figure 4.8 FE model for a simulation of forming the multi-intersecting creases
Figure 4.7 Meshing template for the multi-intersecting crease pattern
New crease
line
(mixed mesh)
Folding part (uniform
mesh)
Fixing part
(the mixed
mesh)
The formed creases
(mixed mesh)
44
4.3.3. Simulation conditions and the FE analysis
The FE model of the two-intersecting crease lines, which is described in Section
4.3.1 is tested in two separated simulation steps of the formation of the intersecting
creases: fixing and flattening the deformed sheet of paper on the working table by the
holders; forming the crease line on the deformed paper by the grippers. Fixing and
flattening problems are related to kinetic friction between the shell structure of paper
and solid structure of the grippers. The constant coulomb friction model is given in
Equation (4.2):
τ =μσn, (4.2)
where η is the shear stress, μ is the coefficient of friction and σn is the normal stress.
The FE simulation is performed with available μ. Optimal fixing and flattening
conditions can be defined by changing μ value.
AUTOMATIC_SURFACE_TO_SURFACE_TIEBREAK contact was chosen for
a strong friction to fix paper on a working table in the non-deformed paper case (a
single crease line). In the case of the deformed part of the sheet of paper (the
intersecting crease lines), the strong contact may cause a strong deformation in the
existing crease area (the holder force Fpress = 40N, the cofficient of friction μ=0.2)
(Figure 4.9a). As the crease area has the considerable small value of the thickness,
the simulation conditions of fixing and flattening a sheet of paper on the working
table should be changed.
The FE analysis that is demonstrated in Figure 4.9b shows that a normal contact
AUTOMATIC_SURFACE_TO_SURFACE between the holders and a sheet of
paper allows the holders to move on a sheet of paper without deforming or breaking
the structure of paper during flattening process (the holder force Fpress = 30N, the
cofficient of friction μ =0.15).
Second problem is related to bending a sheet of paper by the grippers. Under the
same simulation conditions as for the non-deformed model of paper, bending is
demonstrated not satisfied results in the case with the deformed paper parts. The
reason for it is the crease area with a small thickness affects other parts. In bending,
if the grippers are placed according to the gyration radius, paper is deformed and
tearing is appeared after some bending steps. In Figure 4.10a, the sheet of paper is
45
torn at the its boundaries. In this case, the gripper placements are changed: to the
boundaries of a sheet of paper that are parallel to a crease line. Gripper force‟s
applications are the same as for a single crease case. The FE analysis is presented in
Figure 4.10b. According to the Von-Mises stress analysis, there is the area around
the fold in the sheet of paper with a large deformation (red fringe level). It means that
a new sharp crease line is formed without fractures of the paper material in the crease
area that is formed in the previous step.
In the case of the simulation of forming the multi-intersecting creases, the same
simulation conditions as for the two-intersecting creases can be used, but with a
smaller magnitude of the holder pressing force Fpress = 10N (Figure 4.8). The holder
placements are defined in the middle of the length of the boundaries of a sheet of
paper, which are perpendicular to the fold for avoiding a fracture of the paper
material. FE analysis shows that the intersecting crease (mountain) is formed (green
and red fringe levels) as it is illustrated in Figure 4.11.
Figure 4.9 Flattening the deformed mesh
a. Strong friction and Zoom view of the
existing deformed crease area
Holders
Moving direction
b. Normal friction and Zoom view of the
non-deformed existing crease area
Holders
46
4.4. Elasto-plastic paper deformation and dynamic analysis
The performance of the simulation system by the fold angles and the deformation
of paper during the 3 processing phases: bending, translating, and flattening is an
object of a study in this Section. Paper deformation that is considering as a crease
line on a sheet of paper is evaluated by the finite element stress analysis (FEA). In
the FEM, a computer is unable to distinguish between a ductile and a brittle material.
We use the Von-Misses criteria for checking failure in structures, regardless the
applicable theory for the material (Budynas, 1999).
Figure 4.11 FE analysis of the forming the crease line in the case of the
multi-intersecting creases
Figure 4.10 FE analysis of the forming the two-intersecting crease lines
b. Grippers placement at the boundary of
a sheet of paper
Holders
Grippers
Holders
a. Gripper placements according to the gyration radius
Grippers
Existing creases
creases
Holder 1 Holder 2
Gripper 1 Gripper 2
Fixing part
Folding part
47
Distortion energy failure theorem (Von-Misses failure theorem): Yielding is
predicted to occur when the distortion energy in a part equal or exceeded the
distortion energy in a uniaxial loaded tensile bar at the onset of yielding:
, ,distortion part distortion uniaxialU U (4.3)
Yielding is a purely shear deformation process, which occurs when the effective
shear stress e reaches a critical value.
Material behavior is described by Von-Misses anisotropic yield criterion and
with isotropic hardening. Yielding occurs when the Von-Misses in a part becomes
greater than the yield strength. As paper is brittle material, values of the Von-Misses
stress are greater in the crease area than in other parts of the sheet of paper. Figure
4.5 illustrates the Von-Misses stress analysis by two folding conditions: bending
paper by a rotation it around the crease line and bending paper with an application of
the gripper tensile forces. As it can be seen in Figure 4.12a, when paper is bending
only via rotating, the Von-Misses stress at the crease area does not have a large value
of a stress (green fringe level); it means the crease has not yet appeared. In Figure
4.12b, the Von-Misses stress at the crease area peaks at 533.7 kg/mm2
(red fringe
level); it means a large paper deformation at the bending area and a formation of a
crease line.
4.4.1. Stress-strain analysis
Additionally, the appearance of a crease line is illustrated through the stress-
strain diagram. As paper is considered as a brittle material, the yield point cannot be
a. Bending paper via rotating b. Bending paper via rotating and gripper tensile forces
Figure 4.12 Von-Misses stress concentration at the crease area
48
determined. For analysis, a quadrilateral sheet of paper with the geometrical size
40x40 mm and the thickness of 0.1mm is used. Paper is modeled as elastic material;
when a crease appears the paper structure is deformed. The stress-strain diagram in
MD-CD (x, y) direction is given in Figure 4.13 for forming the mountain crease.
Because of noisy data, diagram is considered as 2 parts (a, b) to approximate a linear
curve (Figure 4.14c). As one can see from Figure 4.14c and Figure 4.14d the crease
is formed at the moment of 0.015sec when strain 0.2 and stress 2178 /N mm .
After crease forming, the stress-strain curve shows nonlinear behavior; it means an
existence of the crease.
Based on the elastic stress-strain analysis, the thickness of the sheet of paper at
the crease area can be calculated as below.
The magnitudes of the tensile forces loaded to the sheet of paper from the
grippers in MD and CD directions (see Figure 4.15) and summarize forces are:
2 2
1 1 2 2 12 12 1 240 , 10 41.2F F N F F N F F F F N
From left and right grippers, the combination force is calculated as follows:
2 2
12 12 60tensileF F F N
The thickness of paper tpaper on the crease area after deformation can be
calculated by the formula:
600.01( )
40 150
tensile tensile tensiletensile paper
paper tensile
F F Ft mm
A a t a
where A – the cross section area of the sheet of paper, a – the length of the crease
line.
Figure 4.13 Stress-strain diagram
a
b
49
a. Stress-strain diagram part 1 (a) b. Stress-strain diagram part 2 (b)
c. Stress-strain diagram after linear approximation
d. Strain diagram with respect to time
Figure 4.14 Elastic-stress analysis
50
4.4.2. Measure of energy stored in an element
The simultaneous description of yield and fracture is regarded essential in the
crease formation. These two mechanisms of material damage interplay and determine
the final state of the product. It is therefore the objective of this work to assess the
changes in local distortion and dilatation assuming that they could be used to forecast
yielding and fracture.
The effective stress, e takes the expression:
2 2 21( ) ( ) ( )
2e x y y z z x , (4.4)
with , ,x y z being the principal stresses (Hwu and Hsu, 1990).
If the effective stress e equal to Von-Misses stress, the yielding will occur.
The strain energy density is a measure of energy that is stored in small volume
elements throughout a material. It is defined as the strain energy per unit volume:
2
2
xxuE
, (4.5)
where E is Young‟s modulus of the material.
The total strain energy in the plane is expressed as this quantity integrated over
the whole volume:
Figure 4.15 Scheme of tensile forces applied on a sheet of paper
F1
F2 F
12 F
12
Bending radius r
Crease line
F1
F2
CD
MD
51
V
U udV (4.6)
which, for a constant cross-section A and length L is:
0
L
U A udx
From Hooke‟s law, the strain energy density of equation (4.5) can be expressed as:
1
2xx xxu (4.7)
As can be seen from Figure 4.16, this is the area under the uniaxial stress-strain
curve.
Consider a shear stress xy acting on the volume element to produce a shear
strain xy as illustrated in Figure 4.17. The element deforms with small angles and
. Only the stresses on the upper and right hand surfaces are shown, since the
stresses on the other two surfaces do not work. The force acting on the upper surface
is xydxdz and moves through a displacement dy . The force acting on the right-
hand surface is xydydz and moves through a displacement dx .
The work done when the element moves through angles d and d is then,
using the definition of shear strain:
( )( ) ( )( ) ( ) (2 )xy xy xy xydW dxdz d dy dydz d dx dxdydz d (4.8)
with shear stress proportional to shear strain, the strain energy density is:
2 xy xy xy xyu d (4.9)
The strain energy density function, /dW dV , for the plane is given by:
2 2 21 1( )
2 2x y x y xy
dWu
dV
, (4.10)
where is the shear modulus of elasticity, is Poisson‟s ratio.
Since the concentrations of stress, strain and energy density are localized in a
crease area, it suffices to isolate them for the presentation of numerical data. Three
sets of numerical results, corresponding to rotation angles 33 ,54 ,90o o o are
obtained and summarized in Table 4.1. For each rotation angles, data are collected in
10 steps from angle 0o to the determined angles. The principle stresses ,x y and
52
xy , as shown on the element in Figure 4.18 are given and the effective stress e is
calculated by equation (4.4).
In Table 4.1, the analysis for Enhanced Damage Composite (Orthotropic
material) is done. The energy density increased with almost linearly up to 90o .
This implies that the energy accumulation process almost ceased for 90o . This is
the range where most of the energy is used in forming the bend in the sheet. As a
result, the crease appears when rotation angle reaches 90o.
In Table 4.2, the analysis is done for Elastic material with isotropic properties,
there is no plastic area and the deformation of the sheet of paper is strong in both
directions. The energy density increases significantly after each step, especially when
the rotating angle 15o .
Figure 4.18 Scheme of a simulation model and principal stresses
y
Figure 4.17 A volume element
under shear stress
x
ζxy
ζxy λ
θ
dy
λdy
θdx
dx
A volume element under stress
θ ζxy
ζx
Holders
ζy
The working table
y
x 20mm
t=0.1mm
A sheet of paper
σ
Figure 4.16 Stress-strain curve
ε
53
The analysis results confirm the correctness of the chosen material and bending
method as a rotation the paper sheet by at least 90o around the crease line by the
robot arms.
Table 4.1 Constant stress and energy density contours in units of kg/mm2 with
Orthotropic material (Enhanced Damage Composite)
Rotating angle x y xy
e /dW dV
33o
1 0 0 0 0 0
2 -2.775 5.557 0.7817 8.959 0.0246
3 -9.091 31.83 0.9029 44.22 0.722468
4 -3.445 87.82 2.716 49.08 5.121259
5 7.98 146.4 6.953 89.01 14.06145
6 8.84 171.5 9.692 149.1 19.32034
7 18.59 290.3 22.17 225.6 55.4382
8 57.01 354.5 36.25 343.9 82.69803
9 97.44 455.9 46.26 426.6 136.9408
10 133.7 537.9 50.61 525.4 190.8045
54o
1 0 0 0 0 0
2 -3.455 7.659 0.9803 13.38 0.045608
3 22.26 117.7 4.55 87.81 9.038172
4 39.48 171.5 5.525 149.1 19.21787
5 74.29 308.6 12.74 275.4 62.31459
6 95.21 395.8 21.66 402.8 102.6378
7 221.2 520.5 35.72 538.1 183.1709
8 254.7 665 64.61 650.3 297.5576
9 445.1 785.5 69.29 791.2 436.633
10 511.9 894.8 82.58 808 568.0937
90o
1 0 0 0 0 0
2 -12.03 58.75 7.246 49.35 2.424064
3 68.59 229.6 17.04 204.8 34.83489
4 83.7 354.5 36.25 343.9 82.92646
5 221.2 454.9 47.42 538.1 143.0765
6 403.2 669 61.86 692.7 321.1983
7 490.7 795.6 73.02 705.4 456.6998
8 553 876 87.77 905.6 557.6909
9 662.8 1048 92.3 927.5 796.9155
10 710.4 1226 98.42 1081 1067.272
54
Table 4.2 Constant stress and energy density contours in units of kg/mm2
with Isotropic (Elastic) material
Rotating angle x y
xy e
/dW dV 18o
1 0 0 0 0 0
2 -2.24229 12.83 -0.4715 10.01 0.112884
3 -2.43329 21.51 -6.241 41.75 0.337431
4 10.73 97.71 -1.252 87.73 6.221729
5 18.14 193.7 -1.502 188.9 24.47358
6 21.21 316 -5.661 311 65.29445
7 70.09 504.9 12.71 427.9 165.9912
8 84.55 682.4 31.73 603.7 303.8608
9 140.2 923.1 35.37 846.9 555.1199
10 167.5 1257 38.33 1000 1029.341
33o
1 0 0 0 0 0
2 -3.98442 2.203 -4.54 14.77 0.022097
3 21.54 171.68 -5.829 168.1 19.21157
4 21.21 316 -5.661 279.9 65.29445
5 85.52 515.2 28.35 493.2 173.1544
6 113.9 782.5 34.19 696.5 399.1326
7 149.6 1104 37.24 988.4 794.1138
8 237.6 1401 39.6 1284 1277.519
9 258.7 1680 36.92 1612 1836.776
10 298.1 2025 60.02 2142 2670.098
90o
1 0 0 0 0 0
2 -2.594 45.95 -4.772 62.23 1.420192
3 46.35 381.7 3.654 334.3 94.88151
4 84.55 682.4 13.74 603.7 303.3217
5 149.6 1104 37.24 988.4 794.1138
6 257.4 1491 37.76 1533 1446.675
7 298.4 1905 46.36 2024 2361.863
8 432.8 2130 61.58 2251 2953.868
9 538.2 2536 63.47 2816 4187.393
10 634.4 2540 110.3 3189 4213.365
55
4.5. Summary
Elastic analysis of paper and dynamic analysis of bending a sheet of paper is
done for testing correctness of the simulation conditions.
Meshing the complicated models with multi-thickness shell is problematic for the
FEM. FEM shell elements are difficult to use for a simulation of the structures with
non-uniform thickness. To model such shell structure the boundary element method
(BEM) can be applied (Wagdy and Rashed, 2014). The reason of it is that the FEM
modelling uses shell elements, but for the BEM only a surface mesh of the structure
is needed.
Because of these problems, the solution for folding deformed paper by using the
FEM is found by meshing a sheet of paper with non-uniform thickness according to
the described template in Section 4.3.2. The results of the FE analysis for the
intersecting creases show that the suggested simulation conditions are suitable for
forming the intersecting crease lines by the robot arms without unwilling defects on
paper.
56
Chapter 5
Design development
Design Developing Stage includes the modification of the conceptual design.
Modification is considered as an interactive procedure to find design solution
according to the simulation results to satisfy the requirements for a robot
performance. The modification can include: changing the number and shapes of the
holding and folding fingers, according to the complexity of the origami models;
configuration of the components of the robotics system; a working table
construction; choosing the material for the robot fingers and other parts, and so on.
For instance, adding number and modification of the shapes of the folding fingertips
(grippers) for grasping paper; designing mechanisms and geometrical parameters for
the robotic arms. In this chapter, the new mechanisms for holding fingertip (holder)
that allows moving them from the center position on the crease line in the direction
to its ends as well as the geometrical design for the grippers are presented.
5.1. Modification of a holder design
As discussed in Chapter 1, the application of the robot is for home or office use,
we would like to modify the robot with portable, light weight properties. In the
conceptual design, the holder has a rectangular parallelepiped shape for the crease
length that is located along the whole size of a sheet of paper. This design is not
optimal because there are many folding lines with the different lengths in the origami
crease pattern. Then the conceptual holder design cannot satisfy to the origami crease
pattern.
The idea behind the gripper modification is to imitate the operation of the human
hand when folding a piece of paper. Human often use 2 fingers to flatten and fix the
sheet of paper on a table when folding. As a result, 2 or 3 holders instead of one in
57
the conceptual design are considered to fix a sheet of paper on the working table.
Polypropylene (PP) is used as a holder material.
The friction of the holders is a problem in the folding simulation. As it is noticed
in the paper by Namiki and Yokosawa (2015) the holders with a higher friction
coefficient are desired. In simulation, the friction between the contact portions of the
holders and a sheet of paper is considered as the contact function between the surface
areas in LS-DYNA solver. This type of the contact allows transmitting a tangential
load if relative sliding occurs, when contact friction between the holder material and
a sheet of paper is active. Additionally, during the folding process without a tension
with using 2 grippers (Figure 5.1a), paper is bubbling, and creases cannot be formed.
Bubble is regularly spaced, protruding air-filled hemispheres. In the case of creating
the origami model by the human hands, one finger is used to flatten the bubble on
paper. By using 3 holders instead of 2, paper is bent without a tension; however the
accuracy of the simulation results still does not increase (Figure 5.1b). Based on
these simulation results, we decided to use the holders that can adjust the length of a
crease line and translate along it to flatten the bubbled sheet of paper in the crease
area (Figure 5.2a). With this idea, the type of contact mentioned above does not work
because the sheet of paper is significantly deformed. Therefore, we have to use
another LS-DYNA supported a contact function that allows translating the holders
along a sheet of paper without a paper deformation. We choose TIEBREAK contact
function because it transmits both compressive and tensile forces to the paper sheet
from the holders. Therefore, it creates the stronger contact than the previous one. The
holders are designed for moving along the crease line: from the center part of the
crease area in both directions to the sheet‟s boundaries. To perform the translational
movement the holders include two plates connecting with a translation screw
mechanisms. The screw moves to adjust the length of the crease lines according to
the folding pattern. Finally, two grippers are located on the boundaries of the sheet
under the pressing holder force (Figure 5.b). As the result of the improving folding
conditions and adding a translation mechanisms, the sheet of paper is fixed on the
working table, bubbling is avoided, and the holders can be designed with the smaller
geometrical parameters (Figure 5.2c).The screw mechanism is given in Figure 5.3.
Based on the analysis of the contact between the holders and paper we assume that a
58
magnitude of the pressing holder force should be enough to keep the sheet of paper
on the working table.
Figure 5.1. Bubbling problem
a. Two holders b. Three holders
a. Sketch of the modified holder design
l
The holders
The grippers
r Crease line
b. Holder „s location
Fpress
The working
table
The holder
A sheet of paper
c. Simulation results without bubbling
Figure 5.2. Paper flattening by the moving holders
59
Figure 5.3 Scheme of a screw mechanism for the holders
Screw
Holders
5.2. Optimal design of the plane contact portion of a robot gripper
Gripping is a key task for the robotic arms. A recent survey on the robotic
grippers is presented in (Tai et al., 2016). A length of the crease line in forming
process by the robot arms can be limited by the gripper sizes. Based on this, we
consider an optimization of the geometry of the plane contact portion of the gripper
for forming the crease line with a given length. Required forces are applied to the
object surface via a contact portion of the gripper. The aim of this investigation is to
define the tensile force configurations and geometrical parameters of the plane
contact portion of the robotic gripper for forming the sharp crease lines in the
origami crease pattern without wrinkles on a sheet of paper.
Wrinkling problem on a sheet of paper. Wrinkles appear on a thin sheet of
paper as the result of the tensile force application from the grippers (Figure 5.4).
In the last few years the wrinkling and folding of thin elastic sheets has attracted
a lot of attention in both mathematics and physics communities (Audoly and
Pomeau), (Kumar et al., 2015).
60
To solve this problem the pure plane geometrical algorithm is developed. Here,
the proposed algorithm is described and illustrated with drawings. The condition,
underlying algorithm, is that the length of crease line should be equal to the length of
a side of A4 paper format (210x294 mm). The optimal geometrical parameters of the
contact part of a gripper can be defined by applying this algorithm. The plane contact
portion of the gripper in an x-y coordinate system is a rectangle with the sides a = 40
mm and b=30mm. Tensile forces should be applied by the grippers to a sheet of
paper to minimize wrinkling during folding. Tensile forces are considered in two
directions x and y as 1F and 2F and apply at a centre of the contact portion of the
grippers. Therefore the summarized tensile force 12F should be applied along one of
two diagonals of the rectangle that is considered as a contact portion. To achieve the
target of the investigation, we construct the intersection points M and T between
diagonals of right and left rectangles and the crease line. Flatness of a sheet of paper
in the crease line area depends on the distance d between these points. From the
mechanical point of view, the distance d can be used as a criterion of the wrinkle
appearance on a sheet of paper in the crease line area. Minimal value of d means the
minimal deformations of a sheet of paper. From designing point of view, d can be
considered as a function of geometrical parameters of the contact portion of the
Figure 5.4 Wrinkles on a sheet of paper
61
grippers, a length of the crease line, and a bending radius: where L is a length of a
crease line, r is a bending radius. We assume that, if the minimal d is defined as d=0,
in this case, diagonals of right and left rectangles intersect the crease line at the same
point M (see Figure 5.5). It means that the summarized tensile forces 12F applied
along these diagonal directions can provide the flatness of a sheet of paper. There are
3 orientations of the sheet of paper on the robot working table: portrait, landscape,
and diagonal. This algorithm is a general for all orientations. Detailed explanation of
the algorithm is provided below for portrait orientation.
Portrait orientation. Main steps of the algorithm:
(1) Constructing the line segment HP based on bending radius r, which can be
defined according to the inequality:
3
lt r .
If t = 5mm, then the value of r is in a range from 5mm to 117mm. The final value
of r is decided based on the simulation results.
(2) Designing the rectangle with two dimensions: a and b. A value of aopt is defining
as one quarter of the crease line L and it is a line segment is HA=L/4.
Figure 5.5 Scheme of force configuration and
geometrical parameters of the grippers
62
(3) The continuation of the line segment MA until the intersection with the longest
edge of a sheet of paper gives a point B. We consider the line segment AB as a
diagonal of the designed rectangle. A value of bopt is calculated based on a
property of similarity of the triangles:
/ 2 / 2
BH AH b a arBMK BAH b
BK MK b r L L a
,
4
La .
(4) Constructing the rectangle AHBE as a contact part of the gripper. The diagonal
AB of the rectangle AHBE is considered as a direction of the summarized tensile
force at the center of the rectangle.
As a result of applying the algorithm the optimal geometrical parameters of the
contact portion of the gripper are defined. For the portrait orientation the values of
these parameters are: a=52mm, b=70mm. Bending radius r is chosen as 70 mm based
on the simulation results. L is equal to 210 mm (Figure 5.6).
This algorithm is also applied for landscape and diagonal orientations of a sheet
of paper. Repeat these steps mentioned above, we have optimized dimensions for
landscape orientation: 73 , 50 , 50a mm b mm r mm (Figure 5.7).
Repeat these steps mentioned above, we have optimized dimensions for diagonal
orientation: 90 , 40 , 40a mm b mm r mm (Figure 5.8).
F1
F2 F
12
F12
aopt
bopt
r
A sheet of paper
Crease line
F
1
F2
Figure 5.6 Algorithm for the optimal design for portrait orientation
aopt
bopt
105
52.5 M N
A
B
K
H
E
P
x
y
63
F11
F2 F
12 F
12
aopt
bopt
r
Crease line
F1
F2
297
148.5
bopt
aopt
73 x
y
Figure 5.7 Algorithm for the optimal design for landscape orientation
Figure 5.8 Algorithm for the optimal design for diagonal orientation
F1
F2 F
12
F12
aopt
bopt
F1
F2
364
90
182
bopt
aopt
x
y
64
Calculation results. The optimal geometrical parameters of the contact portion
of the grippers and magnitudes of tensile forces for three orientations are presented in
Table 5.1.
Table 5.1 The optimal geometrical parameters
To exclude changing the grippers for folding each orientation of a sheet of paper,
designing the unified optimized geometrical shape of the contact portion of the
grippers is a necessary step of our approach. Averaging the values is the simplest
way to solve this problem. We use three types of averaging: arithmetic, geometric,
and root mean square (RMS) methods to find the unified geometry. Based on the
averaged results we decide to design the contact portions of the grippers as the
rectangles with sides: a=72mm and b=53mm, bending radius r is 53mm for the length
of crease lines L= 210, 297, 364 mm. In FE modelling we use shell structure with a
thickness 0.1mm. Shell element size is 0.5 mm. Results of FE simulations can be
seen in Figure 5.9.
Sheet‟s
Orientati-
on
Bending
radius
r(mm)
Geometrical parameters Tensile forces
a (mm) b (mm) F1(N)-x-dir F2(N)-y-dir
Pre- limin-
ary
Opt Pre- limi- nary
Opt Pre- limi- nary
Opt Pre- limi- nary
Opt
Portrait 70 40 52 30 70 88 110 117 117
Landscape 50 50 74 30 50 110 189.2 117 117
Diagonal 40 86 90 30 40 189.2 154 117 117
65
5.3. Modification of the conceptual design
A stage subsequent to schematic design, where the conceptual design decisions
are worked out in greater detail, we call the Design Developing Stage. The Design
Developing Stage includes the modification of the conceptual design. According to
the problems formulated above, the simulation solutions are done, such as:
Paper wrinkling and tearing: by editing the holders that can move from the
center of the crease line in the direction to its ends in order to exclude
wrinkles on paper and strong fix paper strongly on the working table;
(a) Portrait
(b) Landscape
(c) Diagonal
Figure 5.9 FE simulations of unwrinkled forming crease line by the
grippers in 3 orientations of A4 paper format
66
Making crease: by the force values and directions; the locations of the
grippers.
The conceptual robot design is modified for forming the crease lines, which
satisfies the origami pattern. The modified design is different from the schematic
design by changing the number and shapes of both holders and grippers. The holders
will be able to move along the crease line by the screw mechanism, which is
included in the design system. The grippers are placed at the edges of the sheet of
paper instead of one single folding finger located along the whole crease lines as
shown in Chapter 2, Figure 2.1. Additionally, the placements of grippers on a sheet
of paper are changed, and the working table with two parts - a base and a rotating
part - to make it easier to place the origami sheet of paper at the current position for
folding is constructed. Figure 5.10 is the illustration of the final robot design as a
result of a modification in the developing stage. The designed robot can be placed on
the desk in home or office room because of its small dimensions. Table 5.2
demonstrates the difference between the conceptual and modified robot designs.
Design Developing Stage can be repeated several times according to the simulation
results for making a good design decision. The configuration of the robot arms with 3
links, universal joints, and the working table design are illustrated in Figure 5.11.
Figure 5.10 Sketch of the final design of the robot
67
Figure 5.11 Configuration of the robot (CAD models)
a. Holding arm b. Folding arm c. Working table
Rotating Base
Table 5.2 Differences between Conceptual Design and Modified Design
Conceptual Design Modified Design
Number of the
grippers 1 2
Number of the
holders 1 2
Gripper position not defined At the boundaries of a sheet
of paper and according the
radius of gyration
Robot arm‟s
action
The holders: fixing paper on
the working table
The holders: translating
along the crease line
The grippers: rotation of
paper
The grippers: tensile and
rotation of a sheet of paper
The working table 1 part: a base 2 parts: a base and the
rotating part
68
5.4. Summary
The realistic robot design meets requirements such as mechanical, geometrical,
topological. During the simulation of crease forming, some problems are probably
determined, for instance: paper wrinkling and tearing, no forming the creases, no
possibilities to change a sheet‟s position on the working table for producing the next
folding. The modification of the conceptual design is a necessary step in the
proposed methodology. After the modification, the final robot design is more
complicated than the schematic one. It means that the complex origami structures can
not be folded by a simple robot system.
69
Chapter 6
Robot performance analysis
In this chapter, the kinematic analysis of the designed robot arms is investigated.
Robot kinematics is connected to the study of the motion of robots and aspects of
redundancy, collision and singularity avoidance. The kinematic analysis is defined as
the relationships between the positions, velocities, and acceleration of the links of a
robot arm. In the robot kinematics, the grippers can move using rotation of links and
joints.
There are two types of the kinematics: direct kinematics and inverse kinematics.
Direct kinematics involves solving the forward transformation equations to
determine the location of the end-effectors by the angles and displacement between
the links. Inverse kinematics involves solving the inverse transformation equations to
find the relationships between the links from the location of the end-effectors in the
3D space.
For understanding a performance of the presented robot, 3 main problems should
be considered: the calculation test of the reachability, the kinematics, and motion
planning for robot performance.
Robotic arm
with moving
angles q1,q
2,…q
n
of the joint
Direct
kinematics
(forward
kinematics) Cartesian
coordinates of robot
end-effector
x, y, z
Inverse
kinematic
x, y, z Figure 6.1 Scheme of robot performance analysis
70
6.1 The reachability of the robot manipulators
The reachability of a robot manipulator to a target is defined as its ability to move
joints and links in free space in order for the arm to reach the given target.
Thus, calculation test of the reachability is an important step in the robot
designing. When the target is a point, the definition is quite clear. When the target is
a spatial object then the points on the object are considered. If there is at least one
point on the object, which can be reached by the robot arm, the object is said to be
reachable by the robot.
In (Ying and Iyengar, 1995), the mathematical framework of the reachability is
discussed. In order to formulate the problem, a general case should be considered.
Assume that there is a robot manipulator, which has n links and n joints angles,
and moves in the three dimensional workspace. The relationship between the
manipulator‟s joint coordinates and end-effector‟s Cartesian coordinates is given as:
1 2( , ,..., )x x nP f q q q
1 2( , ,..., )y y nP f q q q, (6.1)
1 2( , ,..., )z z nP f q q q
where ( , , )x y zP P P is the Cartesian coordinates of the robot end-effector, 1 2, ,..., nq q q
are its n joint variables, that is, joint angles. When the joint angles change, , ,x y zP P P
will have different values; a set of 1 2( , ,..., )nq q q corresponds to a unique set of
( , , )x y zP P P . , ,x y zf f f are continuous single value function mapping 1 2( , ,..., )nq q q to
( , , )x y zP P P .
The problem whether a given point ( , , )X Y Z is within or out of the
manipulator‟s reachable workspace may be solved by a set of nonlinear equations:
1 2( , ,..., )x nX f q q q
1 2( , ,..., )y nY f q q q, (6.2)
1 2( , ,..., )z nf q q qz
where , ,x y zf f f are identical to the functions given by Equations (6.1)
subject to: min maxi i iq q q , (6.3)
71
where miniq and maxiq , 1...i n are the lower and upper bounds of the manipulator
joint variables.
The left parts of the equations (6.2) are the coordinates of a given point, while the
right parts are determine the manipulator joint variables of the robot arms to reach
the point appropriate to the design parameters. Problem is to find a set of
1 2( , ,... )nq q q , which satisfies equations (6.2) and the constraint conditions (6.3). If
there is at least one solution for the above problem, then the given point ( , , )X Y Z is
located within the reachable workspace of the manipulator; if no solution exists,
( , , )X Y Z is out of the reachable workspace of the manipulator. This is similar to the
inverse kinematic problem, but without an information about orientation of joint
angles.
The reachable workspace for the robot is given by Equation (6.4):
1 2
1 2
1 2
( , ,..., )
: ( , ,..., )
( , ,..., )
x x n
y y n
z z n
P f q q q
W P f q q q
P f q q q
(6.4)
where 1 2( , , ), ( , ,..., ), , ,x y z n x y zP P P q q q f f f have the same definition as in equations
(6.1).
According to the Theorem 8.6.3 from (J.O‟Rourke, 1988): “The reachability
region for n-link arm is an origin-centered annulus with outer radius 0
1
n
i
i
r l
and
inner radius ri =0 if the longest link length lM is less than or equal to half the total
length of the links, and i M i
i M
r l l
otherwise”.
Find Ml and compute 0r then point P is reachable if 0ir p r .
Finding a robot arm configuration. Given point P to reach, first determine if P
is reachable; if so, find configuration recursively. To construct the reachable
workspace of the robot arm recursive, linear algorithm for n -link reachability is
applied:
Annulus radius 0r represents 1n links of n -link arm with circle C of
radius nl centered on P.
72
Recursively find configuration for 1 1 1( ,..., )n nA l l
Append last link nl to this solution to connect to P.
Given point P to reach, first determine if P is reachable; if so, find
configuration recursively.
This method is illustrated in Figure 6.2, and Figure 6.3 is an example of the
reachability of the 2-link robot arm.
l2 l
3 l
4
l1
ri=l
1-(l
2+l
3+l
4)
Figure 6.2 Illustration of Theorem 8.6.3
l
1 can reach
all points on
this circle
l2 can reach all
points on each such
circle centered on a
point of l1’s circle
l1
l2
Figure 6.3 Illustration of a reachability region for 2 links of a robot arm
73
6.2 The calculation test of the reachability
The configuration of the origami-performing robot is given in Figure 6.4. For
testing the reachability, the landscape orientation of a sheet of A4 paper format
(210x297 mm) is considered.
The goal of this testing is to confirm that the designed robot arm can reach the
given point E (0,100,100) on the sheet of paper during its bending around the crease
line to achieve 180 degree in rotation (Figure 6.5).
Robot kinematics. For testing the reachability, it is necessary to consider the
kinematic problem to ensure the reachability of the designed robot. The robotic arm
is designed with the 6-Degree-of-Freedom (6-DOF) for folding the origami models.
Figure 6.4 Schematic model of the 6-DOF robotic manipulator arm
Figure 6.5 Scheme of the reachable region
74
The robot manipulator, which moves in a 3D workspace, has 3 links and 6 joint
angles.
The relationship between the manipulator joint angles and end-effector's
Cartesian coordinates are considered as the coordinates of manipulator end-effector
with the relation with the manipulator joint angles and geometrical parameters of the
robot arm.
These coordinates are calculated by equation (6.5) (Thai, 2012):
2 1 2 3 3 1 4 5 3 1 2 3 4 5
2 1 2 3 3 1 4 5 3 1 2 3 4 5
1 2 2 3 3 2 3 4 5
2 3 4 5
cos cos( )+ sin sin( ) cos cos( )cos( )
sin cos( ) cos sin( ) sin cos( )cos( )
sin( ) sin( )cos( )
sin( )cos( )cos
x
y
z
R l q q q l q q q l q q q q q
R l q q q l q q q l q q q q q
R l l q q l q q q q
R q q q q q
6 2 3 6cos( )sinq q q ,
(6.5)
where [ , , ]x y zR R R are the end-effector‟s coordinates; R is the orientation joint angle
of the end-effector (for bending process, only one orientation of the end-effector is
considered); 1 6[ ,...., ]q q are the joint angles; l1,l2,l3 are the link‟s lengths of the robot
arm.
By the calculation, the working space is a sphere with the radius R=160 mm
(l2+l3) in 3D space. Figure 6.5 shows that the given point E (0,100,100) is located
within the reachable region of the robot manipulator. The joint angle‟s limits for the
robot arm and its geometrical parameters are presented in Table 6.1 and Table 6.2.
Table 6.1 Motion limits of the joint
Table 6.2 Geometry and topology of the robot arm
1 3 5, ,q q q 170o to 170o
2 4 6, ,q q q 90o to 90o
Links Link 1 Link 2 Link 3
Length (mm) 100 80 80
Type of joints Universal Universal Universal
75
6.3 The inverse kinematics of a robot arm
Here, the angles of the joints are discussed in a connection with the desired
location of the end-effector. Inverse kinematics transforms the motion plan into joint
actuator trajectories for the robot arm. The geometrical relation of the joint angle
vector q and the manipulation vector r is defined as ( )r f q .
The inverse kinematics is solved by Equation (6.6):
( )d d
dt dt
r qJ q ; ( )
d
d
rJ q
q, (6.6)
where ( ), ( )tr r J q is the Jacobian matrix, , ,m n m nR R R r q J .
Jacobian matrix is an ( )n m matrix relating differential changes of q to
differential changes to r . Jacobian maps velocities in joint space to velocities in
Cartesian space.
The linear equations (6.6) can be solved but in the calculation case the number of
equations m (the dimension of r) is equal to 4 according to the equation (6.5), and the
number of unknowns n (DOF) is equal to 6, so m<n. Hence, it is impossible to find
the inverse matrix of J, pseudo-inverse matrix is applied to solve equations (6.1) as
follows:
† ( )q J q r , (6.7)
where † 1( )T T J J JJ .
Equation (6.7) may have no solutions, a single solution or an infinite number of
solutions. Instantaneously minimization of the joint velocities does not guarantee that
kinematic singularities are avoided.
As a result, the joint angular velocities are calculated by Equation (6.8):
† †( ) ( ) q J q r I J J , (6.8)
where I denotes the ( )n n identity matrix and is an n-dimensional arbitrary joint
velocity vector. In Equation (6.8), a homogenous term †( )I J J , which is obtained
by filtering the null-space velocity components of , is added to the minimum-norm
solution.
76
Null-space joint velocities cause changes in the configuration of the manipulator
without affecting its velocity at the end-effector. can be exploited to achieve
additional goals - like obstacle or singularity avoidance - besides the desired task of
end-effector (see below).
To exclude the singularity the objective function for optimization of the
reachable region is obtained by using the Jacobian matrix, as follows:
( ) det ( ) ( )T q J q J q (6.9)
Manipulability measure. Manipulability measure plays a key role in robot‟s
behaviour tasks such as grasping, pushing, or pulling objects with satisfactory
dexterity.
A scalar det ( ) ( )T J J at a state of joint variable with respect to
manipulation vector r was proposed in (Yoshikawa, 1985) as a quality measure for
manipulator that describes the distance to singular configuration (see Appendix 1 for
more details).
In thesis notations this measure is calculated by equation (6.9). When reaches
the extreme value, the robot arm avoids a singular configuration. As is a scalar
(Equation (6.8)), it can be represented as a vector [ 1 1 1 1 1]T .
Mechanical singularities. The singularities are caused by the inverse kinematics
of the robot. At a singularity, there is an infinite number of ways for the kinematics
to achieve the same position of the robot end-effector in terms of the manipulator‟s
joint variables. Singularities play a significant role in the design and control of robot
manipulators. Singularities of the kinematic mapping determine the position of the
end–effector in terms of the manipulator‟s joint variables (Donelan, 2007).
Calculation results. The end-effector is considered to move along a circle during
the folding process, where we consider the end-effector motion planning in 2D
space. Robot arms are designed for folding A4 paper format; the motion of the robot
arm is calculated by mathematical computing software MATLAB and is illustrated in
Figure 6.6. As observed in Figure 6.6, the reachability is satisfied to the geometrical
parameters of the robot links and the largest object in folding process. Red points are
the positions of end-effectors during a work time.
77
6.4 Robot performance improvement
By conventional manipulability (Equation 6.9) an undesirable posture sometimes
cannot be predicted because of the joint motion„s limitations.
In this Section, the improvement of performance of the robot manipulator is
considered. The method from (Aomura et al., 2008) for the calculation of “extended
of manipulability” with considering the motion limitations of the robot arm in the
workspace is adapted for the origami-performing robot.
In thesis notations, manipulability ex is calculated by Equation (6.10):
6
1
ex i
i
, (6.10)
where ,i and are given as follows:
2
2
( )exp ,
2
,2 6
i i i i
ii
t t t t
q
LL HL HL LL
, (6.11),
Although the authors of (Aomura et al., 2008) do not describe in details their
method, it can be assumed that the background of it is a probability theory namely
the positive skew normal distribution. It is logically to use this theory for robotics
Figure 6.6 Calculation results of the planar robot arm motion
during the folding process.
a. The robot arm position
0 50 100 150 200 250 300 350-50
0
50
100
150
200
250
300
350
400
t [s]
x(mm)
y(mm)
theta(rad)
b. Coordinates and orientation of
end-effectors with respect to time
78
because as it is well-known, in nature there are many processes, which can be
described by the normal distributions. The probability density function of a normal
distribution with the variable x is represented as:
2
2
( )
21
( )2
x
P x e
, (6.12)
where μ is a mean of x and ζ is a standard deviation of x.
As one can see, the expression for a distribution function i in Equation (6.10) is
a modification of Equation (6.12) with a parameter qi as a variable. The standard
deviation is calculated based on considering the difference between the maximum
(HLti) and minimum (LLti) joint angle‟s limits in terms of 3-sigma. The value of αi is
changed from maximum to minimum in a range of 1.0-0.0. ex does not take a large
value if the posture formed by current joint angles is closed to the singularity, even if
a large value of i . How it is noticed in (Aomura et al., 2008) “the extended
manipulability evaluates the actual mobility by taking into consideration the joint
motion limits of every joint”. When the ex value is large it means that the following
conditions are satisfied: every joint angle has a large scope of movement and the
current posture is far from the singularity. The conventional manipulability, as shown
in Figure 6.7, reaches its maximum value after 130 seconds. The extended
manipulability is decreasing significantly to 0-value after 110 seconds (see Figure
6.8, positive curve‟s part); it means the joint angle‟s motion limit (MATLAB
calculations). Values of the manipulabilities are normalized by their maximum
values. As one can see from Figure 6.7 by applying the extended manipulability
metric, the best posture of the manipulator can be predicted with avoiding a singular
stage at q.
6.5 Summary
The calculation testing of the robot performance produced by using the
mathematical computing software MATLAB shows that geometry and topology of
the designed robot arms can be applied for folding paper with the standard size
formats such as A4 and A3 (see Appendix 2).
79
Figure 6.7 Results of the calculation of conventional manipulability
0 50 100 150 200 250 300 3500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t [s]
Conventional m
anip
ula
bili
ty
Figure 6.8 Results of the calculation of extended manipulability
-200 -150 -100 -50 0 50 100 150 2000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t [s]
Exte
nded m
anip
ula
bility
80
Chapter 7
Applying the developed methodology
In this Chapter, the developed methodology is applied for the forming the given
crease patterns of the origami models by the robot arms in the 3D virtual space.
The origami models, such as the “Star” and the “Tetrakis Cube”, are chosen as
the examples of a flexible form of origami. The “Miura Ori” folding pattern and the
“Gift Box” models are related to a rigid origami form.
7.1 Flexible origami
7.1.1. The “Star” model
The 2D pattern of the origami model “Star” (Figure 7.1b) is chosen to
demonstrate the simulation results of the folding procedure (Figure 7.1a). This
pattern includes 2 mountain lines and 2 diagonal valley lines.
Figure 7.1 The origami model “Star”.
a. The 3D shape
(4)
(3)
(1)
(2)
b. The FEM model of the 2D pattern
81
For the simulation of this model the folding sequence is decided to define the
best order for the robot operations. Folding steps are defined by the robot functions:
rotation, translation, or flipping the sheet of paper.
In this pattern, one can find a special folding condition for the simulation: all
crease lines are intersecting at the center of the folding pattern. It means that paper is
strongly deformed and the sheet of paper has lost flatness. Hence, the flattening as
the robot function should be included as the additional folding step.
Five main stages in the simulation methodology of the formation of this model
are proposed (Figure 7.2):
(1) Mountain fold: Bending the sheet of paper by 90o to make the mountain
fold (1) by the grippers (Figure 7.2a);
(2) The second mountain fold: Flattening the folding sheet by the holders
on the working table; 90o rotating the working table; moving a sheet of
paper by the grippers to the position for the next mountain folding
(Figure 7.2b);
(3) Diagonal fold: Flattening the sheet of paper after the folding of the
second mountain; rotating the working table by 45o;
(4) A sheet of paper is turned over by the grippers;
(5) Moving the sheet of paper on the working table by the gripper to locate
the crease line at the edge of the working table (Figure 7.2c); bending the
sheet of paper for diagonal fold.
For forming the second the diagonal fold the steps 3 and 5 should be produced.
FEM - based modeling. Due to the limitation of the calculation time and the
repetition of the operations only one mountain and one valley folds in the origami
pattern are formed.
The numerical simulation of the model “Star” is produced by using the FEM. In
the simulation model we consider only the robot end-effectors and a sheet of paper as
the origami 2D pattern.
In Chapter 4 meshing problem for the crease pattern as the same with the “Star”
model is discussed.
82
“MAT54/55-Enhanced composite damage” material is chosen as paper material
in LS-DYNA simulation, because the mechanical properties of this material are
closed to paper.
A sheet of paper is described as a shell structure with the mixed elements
(triangular and quadrilateral) with a size of 5 mm (Liedberg, 2014). Using the small
mesh element‟s size allows us to receive the accurate calculation results. In this
simulation, the thickness of a sheet of paper is decided as 0.1 mm (for the ordinary
paper thickness is measured in the range: 0.08 – 0.1 mm). The total number of mesh
elements in the full simulation model is 22003.
The grippers and holders are presented as solid models with the hexagonal
elements. The size of each element is 0.5 mm. The holders (purple and yellow color)
constraint the paper sheet model by pressing, while the grippers (red and brown
color) bend it around the crease line (Figure 7.2a). The upper parts of the grippers are
forced to rotate the paper sheet around the crease line up to 90o or -90
o from the
Figure 7.2 Folding process
b. Flattening
a. Mountain fold
c. Diagonal valley fold
grippers
holder
s
Folding
sheet
Table
83
initial horizontal position according to the valley or mountain lines. Polypropylene
(PP) is decided as a material for the grippers and the holders. Plastic material
provides decreasing the weights of the robotic parts.
The grippers are set up as above mentioned method and the holders are placed in
the crease area. The distance from the grippers to the crease lines is calculated as in
Chapter 3 and has to satisfy equation (7.1):
20010 ( )3
x mm , (7.1)
The length of the mountain fold is 200mm, the thickness of the working table is
10mm.
In this case, the location x equals to 50mm for the grippers. The gripping force,
which is used in this simulation, is the human finger pressing force of 40 N.
Motion planning. The motion of the robotic arms based on a simulation of the
folding sequence for the given model (Figure 7.3) such as:
(1) Start position is the mountain fold (1-1). Positions of the grippers and the
holders are decided according to the conditions described in Section 5.2.
The grippers produce up 90o bending the sheet of paper along the crease
line by applying tensile forces in MD and CD directions. The holders fix
the sheet on the working table under the pressing force.
(2) After forming a crease line, translating and flattening the sheet are
produced by the holders. The working table is rotated to the new start
position; the grippers and the holders are located for bending mountain
fold (2-2) with the same conditions as shown in the step 1.
(Repeat 2 beyond steps to make the second fold)
(3) Flattening the sheet on the working table; turning the folding paper over
by the grippers; rotating the working table up 45o; moving the valley fold
(3-3) to the crease position by the holders.
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Step 1 Step 2 2nd
mountain
fold
Step 4 Step 5
Figure 7.3 Folding sequence for robot motion
1-1 1-1
1-1
2-2
1-1
2-
2
1-1
2-2
1-1
2-2
3-3
85
7.1.2. The model “Tetrakis Cube”
The 2D pattern of the origami model “Tetrakis Cube” (Figure 7.4b) is chosen for
the simulation of the folding procedure (Figure 7.4a). This pattern includes 5
mountain lines and 17 diagonal valley lines. For the simulation of this model we
have to consider the folding sequence to choose the best order for the robot
operations. Folding steps are defined by the robot functions: rotation, translation, or
flipping a paper sheet. The thickness of a sheet of paper is decided as 0.1 mm.
Folding sequence. There are 6 sides of the “Tetrakis cube”, the folding sequence
of each face of the model is similar to each other. The folding sequence of one side
of the cube is presented (Figure 7.4c):
(1) Start position is the valley fold (1), 90o bending down the sheet of paper
by the grippers. Release the holders and grippers to the initial position;
(2) Rotate the working table of 175o, put the holders and grippers at the next
folding position. 90o bending down the sheet of paper by the grippers
(valley fold (2));
(3) Rotate the working table of 88o, put the holders and grippers at the next
folding position. 90o bending down the sheet of paper by the grippers
(valley fold (3));
(4) Rotate the working table of 47o, turning over the paper sheet by grippers,
put the holders and grippers at the next folding position. 90o bending
down the sheet of paper by the grippers (mountain fold (4))
For the other sides of the model, these beyond steps are repeated until the final
forming of the “Tetrakis cube”.
Folding conditions. During bending process, gripper‟s compression force
Fcompress and tensile force Ftensile are applied on a sheet of paper (0.1 mm thickness) in
order to make sharp creases. The magnitudes of these forces are: Fcompress = 40N and
Ftensile = 10N.
86
Figure 7.4 The “Tetrakis Cube” model
a. 3D shape of the “Tetrakis Cube” b. The 2D pattern of model 2
Model 1
Model 2
(1)
(2) (3)
(4) (5) (6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14) (15)
(16)
(17)
(18) (19)
(20) (21)
(22) (23)
c. Folding steps of one side of a cube
Holders Grippers
87
7.2 Rigid origami
Origami model with a thickness of material 1 mm or more is considered as
“rigid” in the origami simulation model. Rigid-foldable origami or a rigid origami is
a piecewise linear origami that is continuously transformable without the
deformation of each facet. Therefore, the rigid origami realizes a deployment
mechanism with stiff panels and hinges, which has advantages for various
engineering purposes, especially for designs in architecture (Tachi, 2011). Facets of
the models are considered as rigid thick-panels that are connected by hinge joints. In
this Chapter the kinematic modeling, the simulation models of the rigid origami
structures, and the simulation results are discussed.
7.2.1. Kinematics of the folding pattern
In engineering applications of origami-inspired design, accommodation of
material thickness is frequently necessary to achieve the design‟s objective. There are
some thickness accommodation methods, such as: the axis method (Tachi, 2011), the
offset joint method (Hoberman, 2010), the membrane folds method (Zirbel et al.,
2013), the offset panel technique (Edmondson et al., 2014).
In this section, in order to enable to construct the generalized rigid-foldable
structure with the thick- panels the kinematic structure is considered that precisely
follows the motion of the rigid origami by locating the rotational axes to lie on the
top or bottom of the panel due to the folding lines as a valley or mountain (Tachi,
2011).
The procedure of preparing the thick panels for folding is as follow. First, a zero-
thickness ideal origami in the developed state is first thickened by offsetting the
surface by constant distance in two directions. In this state, the planes of adjacent
facets collide during the origami folding. Then the surface of each facet is trimmed
by cutting the top or bottom layer to create the hinge joints between the panels.
“Miura Ori” pattern is based on 3D tessellation of parallelograms (Figure 7.5a)
that is considered as a mechanism composed by rigid facets and can be represented
by spherical bar linkages (Beatini and Korkmaz, 2013). In this model, a set of
88
spherical mechanisms is repeated several times in two directions, indicated by the
arrows in Figure 7.5b.
7.2.2. “Miura-ori “ shape forming by the force application
The holders are located on a facet of the rigid structure that is strongly contacted
with the working table (2 valley facet‟s edges).
The folding process is defined as follows:
For beginning the process of shape forming fix one part on the working
table and pick up the moved part (Figure 7.6).
At the same time the gripper under the applied compress force to make
the model movement. The forces for forming the model are greater than
compress forces Fcompress and the initial grasping forces Fgrasp (Figure 7.7).
At the same time the gripper under the applied compress force to make
the model movement. The forces for forming the model are greater than
compress forces Fcompress and the initial grasping forces Fgrasp (Figure 7.7).
Figure 7.5 “Miura Ori” folding pattern
a) 2D pattern of “Miura Ori” model
with mountain and valley crease lines
c) Simulation model
b) Kinematic model
89
Figure 7.7 Shape forming by the force application
Ffix
Fcompress
Fcompress
Ffix
Figure 7.6 Location of holders and grippers
Holders
Grippers
90
7.2.3. Cardboard packaging by the designed robot arms
Here, the implementation of the designed robot for packaging in simulating.
Cardboard is commonly use material in the packaging industry. In (Liu and Dai,
2003) a method to carton-folding trajectory planning by dual robotic fingers is
proposed based on using a simple carton.
In the presented research, we consider more complicated origami-based
structures. The models “Gift box” are designed and demonstrated with the 2D crease
patterns (see Figure 7.8). These origami patterns are created assuming zero (or near
zero) thickness. Material thickness plays the important role, and for calculation a
cardboard sheet of a thickness 1mm is used. These two rigid models have fixed
number of the creases that are placed at fixed positions on a cardboard sheet. Each
facet of the models is considered as a rigid thick-panel.
Analysis of crease patterns. The presented origami patterns are difficult for a
simulation of the robotic origami folding in the 3D virtual space. The reason of it is
the high degrees of freedom (DOF) at the intersection of creases. For instance, points
M, N, P are the intersections in the model‟s variant-1. According to Maekawa‟s
theorem in the mathematics of origami, at every pattern‟s vertex, where creases are
intersected in a flat origami pattern, the difference between the number of mountain
and valley folds is always two (see Figure 7.8a). The total number of folds at each
vertex must be an even number. For instance, five mountains and one valley are
intersected at the vertices M, N, P (Variant-1) as one can see from Figure 7.8b.
Number of folds at each vertex in the crease pattern of the Variant-2 is odd. It means
that special origami properties are not satisfied. With a purpose to form the 3D shape
from the thick cardboard sheet by the robot arms without occurring collisions
between the thick-panels, it is necessary to create the equivalent mechanism of the
cardboard folding for the simulation.
91
Crease pattern modification. Rigid origami folding with a thick-sheet material
is more problematic than for the traditional origami that uses an ordinary paper as a
thin material. To solve this problem in a simulation, the initial crease patterns of the
“Gift box” model should be modified. To avoid the crease intersections, each vertex
of the cardboard pattern can be replaced by the hexagonal (Variant-1) or pentagonal
(Variant-2) vertex-holes on a cardboard sheet by cutting holes out of the thick-sheet.
It can be assumed that after that creases cannot collide. For the sake of simplicity, a
rectangular shape of the vertex-holes in the simulation is decided. The size of a
vertex-hole depends on a thickness of the cardboard sheets to exclude the collisions
between the thick-panels during the formation of the 3D model. These vertex-holes
are working as spherical 3R linkages in the simulation. Spherical joint (see Figure
7.9b) is used for allowing free rotations of 6 facets around the intersection point at
the same time. Hinge joints are located on the crease lines and allow the facets to
rotate around creases as the axes. Notching of creases on cardboard sheet is done on
Figure 7.8 The cardboard model “Gift box”
b. Variant 2
M M N N P P
M N P
a. Variant 1 : Mountain
: Valley
92
the top or bottom surfaces according to mountain and valley folds on the origami
pattern to create hinge joints between the facets.
Simulation models. For the simulation of the origami folding 2 kinds of the
modified “Gift box” patterns are developed: with a single vertex-hole and a double
vertex-hole at each original pattern vertex (see Figure 7.9a and Figure 7.10a). The
thick-panels are modelled by a solid mesh with 4-mm and 2-mm hexagonal elements
for two patterns respectively. Figure 7.9c and Figure 7.10b demonstrate the finite
element (FE) models, which are designed based on the modified crease patterns.
Prediction of the robot end-effectors behaviour in the formation of the 3D shape is a
necessary step in the simulation to decide the folding parameters, force applications,
and others for the correct motions of the robot. In Figure 7.9d and Figure 7.10c the
schemes of the expected behaviour of the robot-arms are presented for two crease
patterns, where θ is a rotation angle of valley crease around y-axis.
Kinematic modeling. The kinematic model of the crease structure with the
double vertex-holes is presented here. A mechanism that includes the facets as the
rigid links and creases as the hinge joints is considered (see Figure 7.11a). Hinge
joint‟s locations are defined based on the axis-shift method (Tachi, 2011), which
shifts each rotational axis to either the top or bottom of the thick origami pattern (see
Figure 7.11b). It is related to the notching the cardboard surface described above. A
chain of the spherical 5R linkages are replaced by spherical 3R linkages. The
subsequent crease folding of the manipulated object is produced according to the
presented kinematic model of the crease motions. A similar mechanism is applied for
the single vertex-hole pattern.
Initialization of the 3D shape forming process. We consider two simulation
cases: (a) the FE model from Figure 7.9c and (b) the FE model shown in Figure
7.10b. In the case (a), the robot manipulator begins to form the model by applying
the pressing force Fpress = 500N by the grippers sequentially at the vertices A, B, C of
the crease pattern (Figure 7.9a). As a result of the force applications, rotational
inertia M appears (the rotational axis y) for forming the valley creases (see Figure
7.9d). The rotation is 120º for this motion.
In the both cases the holders are located on one facet to fix the cardboard sheet
on the working table.
93
Figure 7.9 Simulation approach for the pattern with the single vertex-holes
Figure 7.10 Simulation approach for the pattern with the double vertex-holes
94
The simulation results. The simulation results are demonstrated for the half-size
crease patterns in Figure 7.12 (a, b) and show the successful formation of the 3D
shape by the robot arms.
In the model (Figure 7.10c) there are 3 single vertex-holes. The mechanism at
each one has one DOF in a planar structure. Hence, for the robotic operations, the
whole 3D shape can be formed step by step according to the planning sequence.
Figure 7.13 shows the forming process of the 3D model (Variant-1) according to the
decided sequence movement of the robotic arms. These simulation results show that
the developed mechanism is suitable for robotics to form the shapes from the thick
rigid panels.
Holders
Grippers Grippers
Holders
Figure 7.12 Simulation results for the half-size “Gift box” models
a. Variant 1 b. Variant 2
Figure 7.11 Kinematic modeling
95
Figure 7.13 Sequence steps of forming the 3D “Gift box” model
96
7.3 Summary
The FE simulation results demonstrate that the paper origami models can be
folded from their crease patterns by the robot arms. The limitation of using the
robotics for a formation of the real models is: after the forming simulation, the 3D
shape should be finalized by using the additional folding operations automatically or
by the human hands.
The origami-like cardboard models can be folded by the robot arms in industrial
applications. The limitation of the suggested simulation approach is a folding of the
thick rigid origami models with the curved creases. The curve creases can be
represented as the discrete line segments. In the simulation model, each line segment
can be represented by the hinge joint. In this case, synchronous rotation of the hinge
joints is a main difficulty in the simulation of the pattern. This problem will be
considered in future phase of the research. In Appendix 3, sequence order for folding
crease patterns of other origami structures can be found.
97
Chapter 8
Conclusion
This thesis presents a new study in a mechanical and geometrical design of the
origami-performing robot for using in the limited living spaces such as: an apartment,
an office, and a school classroom. The virtual (simulation-based) design of the robot
arms that resembles the behavior of the physical robot is proposed instead of the
traditional robotic prototyping approach.
A simulation-based methodology for the manipulation with the foldable objects,
such a sheet of paper, by using the robotic arms is developed according to the
solution of many engineering problems, which are related to the modeling of the
design process and the numerical calculations of a formation of the flexible and rigid
origami forms. The FE simulation results are illustrated by the application of the
developed methodology with the real origami models.
The main advantage of a virtual design based on the FE structural analysis of
paper material and robot arm‟s behavior are that the designers can estimate the real
robot design and its performance in each developing stage and low costs to produce a
robot from scratch.
The iterative FE simulation of the manipulation of the robot arms with origami
crease patterns is suggested to use as a tool for the robot design.
In this thesis the FEM (in solver LS-DYNA) and mathematical computing
software MATLAB are used for:
(1) Numerical calculations and the Finite Element Analysis of forming origami
models under a behavior of the robot arms;
(2) Kinematic modeling and robot performing analysis.
The presented thesis is related to the research area that is based on the fusion of
science and technology in many engineering research efforts, including robotics.
That type of the research was defined in (Polanyi, 1990) as: “Systematic technologies
98
are those that are deeply informed by current scientific knowledge, thereby
benefiting from science”.
Simulation-based designing of the origami-performing robot can be considered as
a process that incorporates scientific methods and the new engineering solutions.
In a software LS-DYNA, sufficient mathematical theory is presented for each
technique to provide the user with adequate knowledge to confidently apply the
appropriate analysis technique. LS-DYNA is a general-purpose finite element
program capable of simulating complex real world problems.
Linear elastic theory for orthotropic material demonstrates a correctness of the
simulation results.
In the FEM shell elements are suitable to simulate the folding an ordinary paper
for the simple origami patterns (one crease or two intersecting folds).
In the case of the strong paper deformations, such as the multi-intersecting
origami pattern), shell elements are destroyed when the gripper forces are applied
during bending and so an approximation of properties is poor. This FEM problem as
an object for the consideration is solved by the simplification of the computational
model of the sheet of paper.
In the simulation process, the following problems are considered and solved:
Wrinkling and buckling the sheet of paper during the bending process;
Formation of the crease lines and an estimation of crease sharpness;
Meshing and forming the intersected creases avoiding paper fracture;
Force configurations and their magnitudes, folding and contact conditions;
Optimal geometrical design of the robot end-effectors and their placements on
a sheet of paper for the correct simulation;
Kinematic mechanism of the cardboard patterns includes the facets as the rigid
links and creases as the hinge joints;
Robot performance analysis.
The FE simulation results demonstrate:
Folding an ordinary paper by the crease lines can be performed by the robot
arms;
99
The origami-based cardboard models can be folded by the robot arms in the
industrial applications.
Calculation test of the robot performance:
The robot performance analysis as the kinematic analysis of the designed robot
arms is produces. The reachability calculation test based on MATLAB programming
shows the correct working area that is satisfied to geometrical and topological
parameters of the designed robot. For improvement Yoshikawa measure of
manipulability, a method based on using the probability density function of a normal
distribution of joint angles of robot links is applied and realized by MATLAB
programming. It allows estimating the effect of the limits of the joint angles on
final arm posture during its movement. The calculation results of this metrics of
manipulability are very important for controlling robotic performance to avoid
singularities.
Main contributions in this thesis:
(1) The method to design the origami-performing robot based on the FE
simulation of the formation of the 3D origami shapes by the robotic arms
allowing making an effective decision about the universal robot design for
folding the various origami crease patterns;
(2) Developing an operational methodology in the simulation by providing a
large coverage of the design solutions according to the main requirements
for the production of the origami models;
(3) New mechanical and geometrical solutions for the design of the robot arms
for avoiding paper buckling and wrinkling during the folding process;
(4) An approach based on the meshing template to solve meshing problem
related to the strong deformation of a sheet of paper: multi-thickness
computational model (the origami multi-intersecting crease pattern);
(5) Computer testing of the robot performance by the developing MATLAB
programming code.
Merits in the usage of this methodology:
It shows a way to design the mechanical parts and motion planning of the
robot manipulators by the relationship with forming origami models in FE
simulations without the series of experiments.
100
The proposed methodology is considered as a basis for a design of the
origami-performing robot without the series of experimental tests.
The formulated FE simulation rules and recommendations can be used by
scientists, engineers, specialists, and programmers that work in robotics and
origami design as a manual.
The main demerit of the developed simulation-based robot design:
calculation time consuming and difficulties in finite element analysis may occur for a
simulation of the complex origami models.
Application area: Because of its small dimensions, the robot can be useful for
papercraft, education, packaging at home and school rooms or company offices.
With the large dimensions it can be used in various industrial applications.
Limitations and Future work:
There are some limitations in our methodological approach for forming the
creases. Currently, the robot is not able to form origami patterns with line segments
like “zig-zag” fold (the flexible “Miura-ori” model) represented on an ordinary paper
(0.1 mm thickness). With this kind of the folding structure the best solution for
forming crease lines without redesigning the robot system should be found.
The limitation of the suggested simulation approach for a rigid origami is a
folding of the thick rigid origami models with the curved creases. The curve creases
can be represented as the discrete line segments. In the simulation model, each line
segment can be represented by the hinge joint. In this case, synchronous rotation of
the hinge joints is a main difficulty in the simulation of the pattern. This problem will
be considered in future phase of our research.
A simulation of the whole robot motion to solve the mechanical problems and to
make the perfect design decision is a next step of the research. In the simulation of a
motion of the whole robot (not only the robot arms) some limitations and the
additional mechanical problems may be encountered, such as: a robot workspace for
determining the volume, in which the robot's end-effectors may act, limitations of
joints, singularity, collisions between the robot arms. These kinds of limitations and
others cannot be determined in the current simulation study.
To develop a software for a realistic design of the proposed robotic system can be
a future research.
101
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robotic fingers”, Journal of Robotics and Autonomous Systems 42, pp. 47-63.
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manual-Volume II, version 960.
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Primitieves”, Proceeding of IEEE/RSJ International Conference on Intelligent
Robots and Systems (IROS), pp. 5523-5628.
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Simulation”, PhD thesis, Graduate School of Advanced Mathematical
Sciences, Meiji University.
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editing, p.326
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University of Chicago Press.
Tachi,T. (2011) “Rigid-Foldable Thick Origami”, Origami 5, pp. 253-263.
Tai, K., El-Sayed, A.R., Shahriari, M., Biglarbegian, M., Mahmud, S. (2016) „State
of the Art Robotic Grippers and Applications‟, Journal Robotics , 5(2), 11;
doi:10.3390/robotics5020011.
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Hand”, Proc. IEEE/RSJ International Conference on Intelligent Robots and
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folding system for confectionery industry”, Industrial Robot: An International
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Fingers Based on the Interactive Configuration Space”, Transaction of ASME:
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106
Publications
1. Simulation-based approach for paper folding with the aim to design the origami
performing robotic system
Phuong Thao THAI, Maria SAVCHENKO, Hoan Thai Tat NGUYEN, Ichiro
Hagiwara.
Mechanical Engineering Journal, JSME, Vol. 3 (December 15, 2016) No. 6 p.
15- 00668.
2. Development of a Manufacturing Method for Truss Core Panels Based on
Origami-Forming
Hoan Thai Tat Nguyen, Phuong Thao Thai, Bo Yu and Ichiro Hagiwara
J. Mechanisms Robotics 8(3), ASME, 031011 (March 07, 2016) (8 pages), Paper
No: JMR15-1177; doi: 10.1115/1.4032208.
1
Appendix 1
Manipulability measure (Yoshikawa, 1985)
A manipulator is considered with n degrees of freedom whose joint variables are
denoted by , 1,...,iq i n . Assume that the position and/or orientation of the end-
effector can be described by m variables , 1,..., ( )jr j m m n with respect to a
reference orthogonal coordinate frame and that the kinematic relation between iq and
jr is assumed to be given by:
( )r f q (1.1)
where 1 2[ , ,...., ]T n
nq q q q ( n -dimensional Euclidian space),
1 2[ , ,..., ]T m
mr r r r and the superscript T denotes the transpose. The end-effector
velocity mr corresponding to r , is related to joint velocity q by:
( )r J q q (1.2)
where / nd dt q q , and ( ) m nJ q (the set of all m n real matrices). The
matrix ( )J q is called the Jacobian. A scalar value given by:
det ( )T JJ q (1.3)
is defined to be the manipulability measure at q with respect to r .
Some properties of this manipulability measure will be given in the following.
(1) The set of all end-effector velocity r which is realizable by a joint velocity
q such that: 2 2 2 2
1 2 ... 1nq q q q ( is the Euclidian norm) forms an
ellipsoid in m . This is called the manipulability ellipsoid. Its volume is
given by /2{ / [( / 2) 1]}m m , where ( ) is the gamma function. Therefore,
is proportional to the volume of the manipulability ellipsoid.
(2) When m n , the manipulability measure is simply given by:
2
det J (1.4)
(3) Letting mF denote the force and torque applied to an object by the end-
effector and letting nτ denote the necessary joint driving force and torque,
we have Tτ J F . Hence the set of all manipulating force F which is
realizable by a joint driving force τ such that 1τ , is an ellipsoid in m .
This is called the manipulating force ellipsoid. Its volume is given by
/2{ / [( / 2) 1]}/mn m and is inversely proportional to the manipulability
measure . Also the principal axes of the manipulability ellipsoid and the
manipulating force ellipsoid are the same and their radii in each principal axis
direction are inversely proportional. This means that the direction in which a
large manipulating force can be generated is the one in which the
manipulability is poor and vice versa.
3
Appendix 2
The calculation testing of the robot
performance MATLAB CODING
%robot_parameters
clc
close all
clear all
global robot;
global g;
global n;
n=3;
% body 1
robot.m01=4.1;
robot.J01=0.15;
robot.L01=300;
robot.a01=150;
robot.q01min=-3.22;
robot.q01max=3.22;
% body 2
robot.m02=0;
robot.J02=0;
robot.L02=100;
robot.a02=50;
robot.q02min=-2.7;
robot.q02max=0.61;
% body 3
robot.m03=2.4;
robot.J03=0.06;
robot.L03=70;
robot.L031=50;
robot.a03=35;
robot.q03min=-2.26;
robot.q03max=2.68;
% body 4
robot.m04=0;
robot.J04=0;
robot.L04=50;
robot.a04=25;
4
robot.q04min=-2.96;
robot.q04max=2.96;
% body 5
robot.m05=1.2;
robot.J05=0.005;
robot.L05=80;
robot.a05=40;
robot.q05min=-2.96;
robot.q05max=2.96;
% body 6
robot.m06=0;
robot.J06=0;
robot.L06=80;
robot.a06=40;
robot.q06min=-2.96;
robot.q06max=2.96;
g=9.81;
%forward kinematics
function x = for_Kinematic5dof(q)
% robot_parameter
global robot;
%
l1=robot.L01;
l2=robot.L02;
l3=robot.L03;
l4=robot.L04;
l5=robot.L05;
l6=robot.L06;
q1=q(1);
q2=q(2);
q3=q(3);
q1=q(4);
q2=q(5);
q3=q(6);
x1=((cos(q1)*cos(q2)*cos(q3)-cos(q1)*sin(q2)*sin(q3))*cos(q4)+sin(q1)
*sin(q4))*cos(q5)*l5+(-(cos(q1)*cos(q2)*cos(q3)-cos(q1)*sin(q2)*sin(q3))
*sin(q4)+sin(q1)*cos(q4))*sin(q5)*l5+cos(q1)*cos(q2)*cos(q3)*l3-cos(q1)
*sin(q2)*sin(q3)*l3;
x2=cos(q5)*l5*cos(q4)*sin(q1)*cos(q2)*cos(q3)-cos(q5)*l5*cos(q4)*sin(q1)
*sin(q2)*sin(q3) -cos(q5)*l5*cos(q1)*sin(q4)-sin(q5)*l5*sin(q4)*sin(q1)
*cos(q2)*cos(q3)+sin(q5)*l5*sin(q4)*sin(q1)*sin(q2)*sin(q3)-sin(q5)*l5*
cos(q1)*cos(q4)+sin(q1)*cos(q2)*cos(q3)*l3-sin(q1)*sin(q2)*sin(q3)*l3;
x3=cos(q4)*cos(q5)*l5*sin(q2)*cos(q3)+cos(q4)*cos(q5)*l5*cos(q2)*sin(q3)-
sin(q4)*sin(q5)*l5*sin(q2)*cos(q3)-sin(q4)*sin(q5)*l5*cos(q2)*sin(q3)
+sin(q2)*cos(q3)*l3+cos(q2)*sin(q3)*l3+l1;
5
x4=((sin(q2)*cos(q3)+cos(q2)*sin(q3))*cos(q4)*cos(q5)-(sin(q2)*cos(q3)+
cos(q2)*sin(q3))*sin(q4)*sin(q5))*cos(q6)+(sin(q2)*sin(q3)-cos(q2)*
cos(q3))*sin(q6);
x=[x1; x2; x3; x4];
%Jacobian matrix
function Jacob=Jacob_matr(q)
global robot;
l1=robot.L01;
l2=robot.L02;
l3=robot.L03;
l4=robot.L04;
l5=robot.L05;
l6=robot.L06;
q1=q(1);
q2=q(2);
q3=q(3);
q4=q(4);
q5=q(5);
q6=q(6);
t1 = cos(q5);
t2 = t1 * l5;
t3 = cos(q4);
t4 = t2 * t3;
t5 = sin(q1);
t6 = cos(q2);
t7 = t5 * t6;
t8 = cos(q3);
t9 = t7 * t8;
t11 = sin(q2);
t12 = t5 * t11;
t13 = sin(q3);
t14 = t12 * t13;
t16 = cos(q1);
t17 = sin(q4);
t18 = t16 * t17;
t20 = sin(q5);
t21 = t20 * l5;
t22 = t21 * t17;
t25 = t16 * t3;
t27 = t8 * l3;
t29 = t13 * l3;
t32 = t16 * t11;
t33 = t32 * t8;
t35 = t16 * t6;
t36 = t35 * t13;
t42 = -t4*t33 - t4 * t36 + t22 * t33 + t22 * t36 - t32 * t27 - t35 * t29;
6
t43 = t2 * t17;
t44 = t35 * t8;
t46 = t32 * t13;
t48 = t5 * t3;
t50 = t21 * t3;
t53 = t5 * t17;
t55 = -t43*t44 +t43 * t46 + t2 * t48 - t50 * t44 + t50 * t46 - t21 * t53;
t65 = t12 * t8;
t67 = t7 * t13;
t73 = -t4 *t65 - t4 * t67 + t22 * t65 + t22 * t67 - t12 * t27 - t7 * t29;
t80 = -t43*t9 + t43 * t14 - t2 * t25 - t50 * t9 + t50 * t14 + t21 * t18;
t81 = t3 * t1;
t82 = l5 * t6;
t83 = t82 * t8;
t85 = l5 * t11;
t86 = t85 * t13;
t88 = t17 * t20;
t91 = t6 * t8;
t93 = t11 * t13;
t95 = t81* t83 - t81 * t86 - t88 * t83 + t88 * t86 + t91 * l3 - t93 * l3;
t96 = t17 * t1;
t97 = t85 * t8;
t99 = t82 * t13;
t101 = t3 * t20;
t104 = -t96 * t97 - t96 * t99 - t101 * t97 - t101 * t99;
t105 = t91 - t93;
t111 = cos(q6);
t115 = t11 * t8 + t6 * t13;
t116 = sin(q6);
t118 = (t105 * t3 * t1 - t105 * t17 * t20) * t111 + t115 * t116;
t119 = t115 * t17;
t121 = t115 * t3;
t124 = (-t119 * t1 - t121 * t20) * t111;
Jacob(1,1)=-t4*t9+t4*t14+t2*t18+t22*t9-t22*t14+t21*t25-t7*t27+t12*t29;
Jacob(1,2) = t42;
Jacob(1,3) = t42;
Jacob(1,4) = t55;
Jacob(1,5) = t55;
Jacob(1,6) = 0.0e0;
Jacob(2,1) =t4*t44-t4*t46+t2*t53-t22*t44+t22*t46+t21*t48+t35*t27-t32*t29;
Jacob(2,2) = t73;
Jacob(2,3) = t73;
Jacob(2,4) = t80;
Jacob(2,5) = t80;
Jacob(2,6) = 0.0e0;
Jacob(3,1) = 0.0e0;
Jacob(3,2) = t95;
7
Jacob(3,3) = t95;
Jacob(3,4) = t104;
Jacob(3,5) = t104;
Jacob(3,6) = 0.0e0;
Jacob(4,1) = 0.0e0;
Jacob(4,2) = t118;
Jacob(4,3) = t118;
Jacob(4,4) = t124;
Jacob(4,5) = t124;
Jacob(4,6) = -(t121 * t1 - t119 * t20) * t116 - t105 * t111;
%joint velocity
function output = jointvelocity(input)
global n;
global robot;
q = input(1:n);
xvat = input(n+1:n+12);
xt = xvat(1:4);
xdot = xvat(5:8);
xddot = xvat(9:12);
q1m = robot.q01min;
q1M = robot.q01max;
q2m = robot.q02min;
q2M = robot.q02max;
q3m = robot.q03min;
q3M = robot.q03max;
q4m = robot.q04min;
q4M = robot.q04max;
q5m = robot.q05min;
q5M = robot.q05max;
q6m = robot.q06min;
q6M = robot.q06max;
E = eye(n,n);
W = E;
iW = inv(W);
J = Jacob_matr(q);
invJ = iW*J'*inv(J*iW*J');
S= invJ*J;
muy1=(q1m+q1M)/2;
muy2=(q2m+q2M)/2;
muy3=(q3m+q3M)/2;
muy4=(q4m+q4M)/2;
muy5=(q5m+q5M)/2;
muy6=(q6m+q6M)/2;
s1=(q1M-q1m)/6;
8
s2=(q2M-q2m)/6;
s3=(q3M-q3m)/6;
s4=(q4M-q4m)/6;
s5=(q5M-q5m)/6;
s6=(q6M-q6m)/6;
%
alpha(1)=exp(-((q(1)-muy1)^2))/(2*s1^2);
alpha(2)=exp(-((q(2)-muy2)^2))/(2*s2^2);
alpha(3)=exp(-((q(3)-muy3)^2))/(2*s3^2);
alpha(4)=exp(-((q(4)-muy1)^2))/(2*s4^2);
alpha(5)=exp(-((q(5)-muy2)^2))/(2*s5^2);
alpha(6)=exp(-((q(6)-muy3)^2))/(2*s6^2);
%===========================================================
=========
% tranh va cham tai cac khop
% cac he so c
c = [1 1 1 1 1 1];
qM = 1*[1 1 1 1 1 1];
q_ = [0 0 0 0 0 0];
qm = -1*[1 1 1 1 1 1];
k = 1;
z0 = -k*[(2*c(1)*(q(1)-q_(1))/(qM(1)-qm(1))^2), (2*c(2)*(q(2)- q_(2))/ (qM(2)-
qm(2))^2), (2*c(3)*(q(3)-q_(3))/(qM(3)-qm(3))^2),(2*c(4)*(q(4)-q_(4))/(qM(4)-
qm(4))^2), (2*c(5)*(q(5)-q_(5))/(qM(5)-qm(5))^2),(2*c(6)* (q(6)-q_(6))/(qM(1)-
qm(6))^2)]';
%===========================================================
=========
e = xt - for_Kinematic5dof(q);
Kp = 1*diag([1 1 1 1]);
z = det(J);
wext = alpha*z;
% (E - invJ*J)*z0
q_dot = invJ*(xdot + Kp * e)+(E - invJ*J)*z0;
output=[q_dot',wext,z];
%motion planning
function outputs=taoqdtron(in)
% this function return a circular trajectory of end-effector
t=in(1);
v_max=0.5;
Ta=1;
xO=50;
yO=-37.1052;
zO=351.6025;
% diem xuat phat for A4 landscape
goc= -0.5236;
x0=0;
9
y0=382.1194;
z0=351.6025;
% huong cua ban kep
%ban kinh
R=sqrt((x0-xO)^2+(z0-zO)^2);
phi0=atan2(z0-zO, x0-xO);
% cho ban kep cd tu trang thai dung yen
a=pi/2*v_max/Ta;
if(t<Ta)
at=a*sin(pi/Ta*t);
vt=a*Ta/pi*(1-cos(pi*t/Ta));
st=a*Ta/pi*t-a*Ta^2/pi^2*sin(pi/Ta*t);
epsi= at/R;
ome = vt/R;
phi = phi0+st/R;
else
at=0;
vt=2*a*Ta/pi;
st=-a*Ta^2/pi+2*a*Ta/pi*t;
epsi= at/R;
ome = vt/R;
phi = phi0+st/R;
end
xE=xO+R*cos(phi);
yE=yO;
zE=zO+R*sin(phi);
gocE=goc;
xEv=-R*ome*sin(phi);
yEv= 0;
zEv=R*ome*cos(phi);
gocE_dot= 0;
xEa=-R*ome^2*cos(phi)-R*epsi*sin(phi);
yEa=0;
zEa=-R*ome^2*sin(phi)+R*epsi*cos(phi);
gocE_2dot=0;
outputs=[xE, yE, zE, gocE, xEv, yEv, zEv, gocE_dot, xEa, yEa, zEa, gocE_2dot];
10
Appendix figure 2.1. Simulink model
11
Appendix 3
Folding sequence of freeform Origami model
3.1. The “Bunny” model
The “Bunny” model is illustrated in A.Figure 3.1. The folding pattern and the
paper model are given in A.Figure 3.1a and A.Figure 3.1b. There 4 parts in this
folding model: 0, 1, 2, 3, 4. Each part is folding separately. Folding begins from the
simple parts, such as: part 3 and part 4, part 2 and then part 1 and part 0. The folding
sequence of each part is ordered by the numbers (see A.Figure 3.2). After each step,
the robot arm and working table have to rotate and move to the new folding position.
a. Folding pattern b. The 3D shape
A. Figure 3.1 The “Bunny” model
12
Part 4 Part 3 Part 2
Part 1
Part 0
A.Figure 3.2 Folding sequence of each part of the “Bunny” model
13
3.2. The “Polyhedrons” model
The “Polyhedrons” model is illustrated in A.Figure 3.3. The folding pattern and
the 3D shape are given in A.Figure 3.3a and A.Figure 3.3b. There 2 parts in this
folding model: 0, 1. As these parts are separated, the folding sequence is defined for
of each part and ordering. After each folding step, the robot arms and the working
table have to rotate and move to the new folding position.
a. Folding pattern b. The paper model
A.Figure 3.3 The “Polyhedrons” model
14
A.Figure 3.4 Folding sequence of the “Polyhedrons” model
Part 0
1 2 3
4
5 6
7 8
9 10
11 12 13
14
15 16
17 18
19 20 21
22 23
24 25
26 27
29
Part 1
1 2
3 4 5 6
7 8
9 10
11
12 13
14
15
16 17
18
19
20 21 22
23 24 25
26
15
List of Figures
Figure 1.1 Solar array of space craft from Miura-ori structure (Zirbel et al., 2013) 1
Figure 1.2 Leaf of Chamaerops humilis (Trautz and Herkrath, 2009) ..................... 2
Figure 1.3 Cylindrical tube of the stent graft (Kuribayashi, 2004) ........................... 2
Figure 1.4 Adaptive skin (Del Grosso and Basso, 2010) .......................................... 2
Figure 1.5 Origami self-folding robot (Felton et al., 2014) ...................................... 2
Figure 1.6 Robofold (Epps, 2012) ........................................................................... 6
Figure 1.7 Industrial origami robot ........................................................................... 6
Figure 1.8 Paper-bag making machine ..................................................................... 6
Figure 1.9 Robotic Robotic origami folding (Balkcom, Mason, 2004) .................... 7
Figure 1.10 Origami folding by a robotic hand (Tanaka et al., 2007) ........................ 7
Figure 1.11 Folding robot for confectionery industry(Yao et al., 2010) .................... 7
Figure 1.12 Robotic Origami folding by Namiki and Yokosawa (2015) ................... 7
Figure 1.13 Folding paper by anthropomorphic robot hands (Elbrechter et al., 2012)
................................................................................................................. 7
Figure 1.14 Computational mechanics ....................................................................... 8
Figure 1.15 Paper models from (Bo and Wang, 2007) generated by presented
method ................................................................................................... 10
Figure 1.16 Scheme of the robot design ................................................................... 14
Figure 1.17 Flowchart of the methodology ............................................................... 15
Figure 2.1 Schematic illustration of the suggested approach.................................. 18
Figure 2.2 Sketch of the conceptual robot design ................................................... 20
Figure 2.3 Gripper of a type “Crab” mechanism .................................................... 22
Figure 2.4 Detailed design of the robot parts .......................................................... 23
Figure 3.1 Crease pattern ........................................................................................ 26
Figure 3.2 Illustration of tensile force applications ................................................ 28
Figure 3.3 Placement of the grippers on a sheet of paper ....................................... 30
Figure 3.4 Sketch of the simulation of the crease forming by the robot fingers ..... 32
Figure 3.5 Paper folding conditions related to a working table design................... 33
16
Figure 3.6 Fold angles with respect to time ............................................................ 34
Figure 3.7 Illustration of the estimation of the crease sharpness by circle
interpolation ........................................................................................... 35
Figure 4.1 Uniform meshing ................................................................................... 38
Figure 4.2 Adaptive meshing .................................................................................. 38
Figure 4.3 FEM model for folding a sheet of paper................................................ 39
Figure 4.4 Meshing for a diagonal fold and a zoom-view ...................................... 40
Figure 4.5 FE model for a simulation of forming the two-intersecting crease lines
............................................................................................................... 41
Figure 4.6 FEM problem: the complicated mesh for multi-thickness shell structure
............................................................................................................... 42
Figure 4.7 Meshing template for the multi-intersecting crease pattern .................. 43
Figure 4.8 FE model for a simulation of forming the multi-intersecting creases ... 43
Figure 4.9 Flattening the deformed mesh ............................................................... 45
Figure 4.10 FE analysis of forming the two-intersecting crease lines ...................... 46
Figure 4.11 FE analysis of forming the crease line in the case of the multi-
intersecting creases by the robot ............................................................ 46
Figure 4.12 Von-Misses stress concentration at the crease area ............................... 47
Figure 4.13 Stress-strain diagram ............................................................................. 48
Figure 4.14 Elastic-stress analysis ............................................................................ 49
Figure 4.15 Scheme of tensile forces applied on a sheet of paper ............................ 50
Figure 4.16 Stress-strain curve ................................................................................. 52
Figure 4.17 A volume element under shear stress .................................................... 52
Figure 4.18 Scheme of a simulation model and principal stresses .......................... 52
Figure 5.1 Bubbling problem .................................................................................. 58
Figure 5.2 Paper flattening by the moving holders ................................................. 58
Figure 5.3 Scheme of a screw mechanism for the holders...................................... 59
Figure 5.4 Wrinkles on a sheet of paper ................................................................. 60
Figure 5.5 Scheme of force configuration and geometrical parameters of the
grippers .................................................................................................. 61
Figure 5.6 Algorithm for the optimal design for portrait orientation...................... 62
Figure 5.7 Algorithm for the optimal design for landscape orientation.................. 63
17
Figure 5.8 Algorithm for the optimal design for diagonal orientation.................... 63
Figure 5.9 FE simulations of unwrinkled forming crease line by the grippers in 3
orientations of A4 paper format............................................................. 65
Figure 5.10 Sketch of the final design of the robot ................................................... 66
Figure 5.11 Configuration of the robot (CAD models) ............................................ 67
Figure 6.1 Scheme of robot performance analysis .................................................. 69
Figure 6.2 Illustration of Theorem 8.6.3 ................................................................. 72
Figure 6.3 Illustration of a reachability region for 2 links of a robot arm .............. 72
Figure 6.4 Schematic model of the 6-DOF robotic manipulator arm ..................... 73
Figure 6.5 Scheme of the reachable region ............................................................. 73
Figure 6.6 Calculation results of the planar robot arm motion during the folding
process ................................................................................................... 77
Figure 6.7 Results of the calculation of conventional manipulability .................... 79
Figure 6.8 Results of the calculation of extended manipulability .......................... 79
Figure 7.1 The origami model “Star”...................................................................... 80
Figure 7.2 Folding process ...................................................................................... 82
Figure 7.3 Folding sequence for robot motion ........................................................ 84
Figure 7.4 The “Tetrakis Cube” model .................................................................. 86
Figure 7.5 “Miura Ori” folding pattern ................................................................... 88
Figure 7.6 Location of holders and grippers .......................................................... 89
Figure 7.7 Shape forming by the force application ................................................. 89
Figure 7.8 The cardboard model “Gift box” ........................................................... 91
Figure 7.9 Simulation approach for the pattern with the single vertex-holes ......... 93
Figure 7.10 Simulation approach for the pattern with the double vertex-holes ........ 93
Figure 7.11 Kinematic modeling .............................................................................. 94
Figure 7.12 Simulation results for the half-size “Gift box” models ......................... 94
Figure 7.13 Sequence steps of forming the 3D “Gift box” model ............................ 94
18
List of Tables
Table 2.1 Robot link‟s lengths ............................................................................... 20
Table 2.2 Dimension of the geometrical parameters in mm .................................. 23
Table 4.1 Constant stress and energy density contours in units of kg/mm2 with
Orthotropic material (Enhanced Damage Composite) .......................... 53
Table 4.2 Constant stress and energy density contours in units of kg/mm2 with
Isotropic material (Elastic) .................................................................... 54
Table 5.1 The optimal geometrical parameters ..................................................... 64
Table 5.2 Differences between Conceptual Design and Modified Design ............ 67
Table 6.1 Motion limits of the joint ....................................................................... 74
Table 6.2 Geometry and topology of the robot arm .............................................. 74