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Semester Project-Thesis
Glass Fiber Reinforced Plastic: New Elastic
Spring Material used in Compliant Legs for
Scalable Energy Efficient Robots
Spring Term 2013
Supervised by: Author:
Xiaoxiang Yu Yafeng Shu
Fumiya Iida
i
Content
Abstract ........................................................................................................................................................ iii
Acknowledgements ....................................................................................................................................... v
1. Introduction ............................................................................................................................................... 1
2. Functions and requirements for springs in legged locomotion robot ........................................................ 4
2.1 Spring mass leg model ........................................................................................................................ 4
2.2 functions of springs in legged locomotion .......................................................................................... 5
2.2.1 Structural component ............................................................................................................ 5
2.2.2 Energy storage component .................................................................................................... 5
2.2.3 Avoid large contact force ...................................................................................................... 5
2.3 properties desired for springs in legged robot ..................................................................................... 6
2.3.1 Load capacity ........................................................................................................................ 7
2.3.2 Energy storage capacity ........................................................................................................ 7
2.3.3 Contact force avoiding .......................................................................................................... 7
3. Curved beam hopping robot ...................................................................................................................... 9
3.1 Mechanical model of curved beam hopper ......................................................................................... 9
3.2 Performance of the curved beam hopping robot ............................................................................... 12
3.3 limitation of metal curved beam ....................................................................................................... 12
4. Properties of Glass fiber reinforced plastic ............................................................................................. 15
5. FEM analysis and experiment test .......................................................................................................... 21
5.1 FEM analyses of GRP and metal springs .......................................................................................... 21
5.2 Comparison between GRP and metal springs ................................................................................... 24
5.2.1 Load capacity ...................................................................................................................... 24
5.2.2 Stiffness ............................................................................................................................... 26
5.2.3 Strain energy ....................................................................................................................... 27
ii
5.2.4 Contact force ....................................................................................................................... 28
5.3 Experiment test ................................................................................................................................. 28
6. Application of GRP in CBH ................................................................................................................... 32
6.1 Experiment platform ......................................................................................................................... 32
6.2 Experimental Setup ........................................................................................................................... 33
6.3 Analysis of Energy Efficiency .......................................................................................................... 34
7. Discussion and conclusion ...................................................................................................................... 37
7.1 Discussion ......................................................................................................................................... 37
7.2 Conclusion ........................................................................................................................................ 38
Symbols, Acronyms and Abbreviations ...................................................................................................... 39
Symbols .................................................................................................................................................. 40
Acronyms and Abbreviations ................................................................................................................. 41
iii
Abstract
Elastic components perform many important functions in legged locomotion.
However limited by material properties, conventional metal spring material cannot
satisfy requirements of legged robot such as light weight, load capacity. From this
perspective, in this thesis a new material, GRP composite, is introduced as spring
material in design of compliant legs for an efficient legged robots. This thesis
systematically compares GRP and conventional metal materials in real application
(curved beam hopping robot) based on simulation and real world test. Results show
that GRP curved beam robot can achieve efficient locomotion (CoT in the range
between 0.4 and 0.6) with scalable loads. While conventional metal curved beam fails
to stand relative large payload.
iv
v
Acknowledgements
I would like to thank my supervisor Xiaoxiang Yu, and Prof. Fumyia Iida for their
support and their helpful suggestion. I would also like to thank Bryan Anastasiades
and Keith Gunura for their help in the workshop.
vi
1
Chapter 1
Introduction
Not like stiff industrial robotics which needs stiffness to improve the precision,
stability and bandwidth of position control, biological system have much more
demands on elasticity and compliance. One example of application of elasticity and
compliance in biology system is legged locomotion. Animals are capable of
autonomously producing a wide range of stable and efficient movements in
environments with unpredictable disturbances. To realize such energy efficient
movements, elasticity of muscle and tendon system has been proved of great
importance in many ways [7].Inspired by biological system, springs are made use in
many legged robot system to provide damping, compliance and the capacity to store
and release mechanical energy. One impressive example is Curved beam hopping
robots (CHBs) as shown in Fig1.1 [3], invented by researchers at the BIRL Ethz. It is
a simple structural compliant legged robot. By exploring resonance of elastic curved
beam, it can achieve a high energy efficient hopping-like locomotion.
Fig1.1 Curved beam hopping robot
Chapter 1. Introduction 2
The behaviors of CHBs are mainly determined by properties of its elastic curved
beam. However limited by material properties, traditional metal spring material can
hardly satisfy its performance requirements, such as light-weight, high load capacity
and high energy store capacity. Other than improving structure design of conventional
metal springs, exploring new elastic spring materials is a prominent solution. The
work present in this paper is an attempt to introduce a new material, GRP composite,
as spring material in design of compliant legs for efficient legged robots with variation
in payload weight. CHB are used as platform to evaluate performance of GRP spring.
The structure of this thesis is as follows: In chapter 2, functions for springs in legged
locomotion and desired spring properties are discussed. In chapter 3, an introduction
to the CBH is made, and limitations of conventional metal spring curved beam are
stated. A possible material solution (GRP material) is come up. Detailed properties of
GRP are discussed in chapter 4. In chapter 5, a systematic comparison between
conventional metal curved beam and the new GRP one is made based on results of
finite element analysis and experiments. Real world experiment is conducted to verify
simulation results. Chapter 6 discusses real application of GRP curved spring beam in
CHB. In the last chapter, chapter 7, the results of this thesis are discussed, and some
conclusions are presented.
3 Chapter 1. Introduction
4
Chapter 2
Functions and requirements for
springs in legged locomotion robot
Springs (the principal springs in animals are muscle-tendons system) perform many
useful functions in legged locomotion of animals. Inspired by biological system,
legged locomotion can generally be modeled as a spring mass system [5]. In this
chapter we will discuss uses and requirements for springs in legged locomotion robot
based on this spring mass model.
2.1 Spring mass leg model
Leg of animal or legged locomotion robot can be modeled as a simplified mass-
spring system [5]. Fig2.1 shows Schematic diagram of the spring mass leg model.
Fig 2.1 Schematic diagram of animal leg and spring mass model
Chapter 2. Functions and requirements for springs in legged locomotion robot 5
In this simplified model, body and foot are represented by two point mass M and m,
which are connected by a leg spring. The stiffness of the leg spring is K. Contact
between foot and ground in modeled by a pad spring k and a rate independent
damping . A rate independent rather than velocity dependent damping is used, as it
has been shown foot of mammal has very similar rate independent damping behavior
[8].
2.2 functions of springs in legged locomotion
Springs perform many useful functions in legged locomotion. Generally they can be
categorized into three classes.
2.2.1 Structural component
It is obvious that springs, as structural component, are parts of leg structure. It helps
support the load of the whole body and bear force when legs interact with ground. As
in the curved beam hopping robot, the whole leg of the robot consists of a curved
beam spring. It bears the whole body weight and ground contact force.
2.2.2 Energy storage component
Reciprocating motion is the major difference distinguishes legged locomotion system
and wheeled vehicle. Due to such reciprocating movement, mechanical energy in the
legged system will fluctuate. While wheeled vehicle moving at constant speed will
have constant kinetic. In order to reduce energy lost due to energy fluctuation, springs
are used as energy storage component to store and recycle energy in system. As in
CHB and ARL monopod, when the leg touch ground, parts of the kinetic energy lost
due to impact will store temporarily as strain energy in the spring and release by the
spring when the leg lift off. By this process, energy is recycled in the system, which
will largely reduce energy expenditure of the whole legged locomotion system.
2.2.3 Avoid large contact force
One important use of spring is to avoid large contact force when the feet touch ground.
Fig2.2 shows typical force response of spring mass leg model when the model’s foot
hit rigid ground [8]. A small peak force is experienced due to rapidly deceleration
of the pad. After decaying in a short time, the force will rise again until a maximum
force is reached. This rise is caused by slow response of the leg spring.
6 Chapter 2. Requirements for springs in legged locomotion robot
Fig2.2 Typical force response of spring mass leg model when hit rigid ground
2.3 properties desired for springs in legged robot
In order to efficiently perform above functions, generally we want the springs used in
legged locomotion robot to have large load capacity, high energy storage capacity,
light weight and suitable stiffness.
2.3.1 Load capacity
Fig 2.3 shows typical response of metal when applied different loads. The spring will
elongate proportional with applied force until a critical point (yield point), where
plastic deformation of metal material happens. If increase force further the structure of
spring will be damaged. And at some tensile point the spring will break. So in order to
get large load capacity, large tensile and yield stress of the spring material are needed.
Fig 2.3 typical stress strain response of metal spring
Chapter 2. Properties desired for springs in legged locomotion robot 7
2.3.2 Energy storage capacity
The strain energy of the spring material is a major factor decides energy storage
capacity of spring. It represents the potential energy that a material of unit weight can
store when it is elastic deformed by external force. The strain energy of a specific
material is expressed as
Where is the strength, is the density and E the Young’s modulus of the spring
material. As the formula shows, the stored elastic strain energy in a spring varies
directly with the square of maximum allowable stress and inversely with the modulus
of elasticity. To store more elastic strain energy, large yield stress, small density and
elastic modulus are general desired
2.3.3 Contact force avoiding
Analysis of force response of the spring mass leg model is taken to get required
conditions for the springs. As shown by Alexander, Bennet, and Ker [8], the
relationship between and are:
⁄
(1)
⁄
(2)
From condition (1) and (2), it can be shown that, small leg weight m and soft spring k
are needed to avoid large contact force. This means we need a light weight spring with
low stiffness k
To conclude above requirements, we need the spring material have low density and E
modulus but large tensile and yield strength. And the spring used in legged robot
should not be too stiff which means compliance is needed. However, these specified
conditions can hardly be satisfied by conventional metal materials, which are
generally with high density and elastic modulus. Exploring new material seems to be a
reasonable solution. In this paper GRP is discussed.
8 Chapter 2. Functions and requirements for springs in legged locomotion robot
9
Chapter 3
Curved beam hopping robot
Curved beam hopping robot is a simple structural compliant legged robot. The main
body of the robot is an elastic curved beam [3].This spring like curved beam will store
mechanical energy when the foot base touch ground and release them when jump off.
By such energy harvest process, energy consumption of the system is largely reduced
[2]. A rotating mass in the head driven by a low-power DC motor is used to induce
resonance vibration of the cured beam. Using resonance of the body dynamics, the
hopping robot can achieve energy efficient hopping-like locomotion [2]. In this
section, we will brief introduce the mechanical model and performance of this type of
legged locomotion system and discuss limitation of conventional metal curved beam.
3.1 Mechanical model of curved beam hopper
The curved beam hopping robot system can be modeled by a simplified spring-mass
system which is shown in Fig3.1 [1]. In this model, the elastic curved beam is
represented by linear and torsional spring-damping and elements. Mass
distribution of the robot is simplified by two point masses A and B. Actuation of the
rotating mass is represented by a constant force F(t) which rotate around the point
mass A. with this mechanical model, the dynamics of the system can be expressed as
follows:
Two featured phases of the hopping movement can be observed, stand phase where
the foot base of the robot is touch on the ground and flight phase when the robot lifts
off in ballistic trajectory [2]. We model dynamics of them separately.
Chapter 3. Curved beam hopping robot 10
(a) Real curved beam (b) mechanical model of curved
hopping robot beam hopper
Fig 3.1 Curved beam hopping robot and its mechanical model
a) Stand phase
During stand phase, the hopping robot can be viewed as a spring-mass inverted
pendulum. By Lagrangian mechanics method, the dynamics equations of the motion
are as follows:
( )
( )
b) Flight phase
During flight phase, the hopping robot fly in a ballistic trajectory, we can get the
dynamics of the system as following:
(
)
[
]
11 Chapter 3. Performance of curved beam hopping robot
[
]
[
]
3.2 Performance of the curved beam hopping robot
In previous research, performance of the curved beam hopping robot has been
systematically evaluated by both simulation and experiment [1] [2] [3]. It has been
shown that, the energy efficiency of such curved beam hopping robot is among the
most efficient legged locomotion systems such as ARL monopod and Cornell Biped
[6]. The specific resistance of curved beam hopper is in the range of 0.2 to 0.6. It has
also been proved theoretically that energy efficient locomotion of such robot can be
scalable to weight of payload [1].
3.3 limitation of metal curved beam
Though has been shown in theory and experiment, curved beam hopping robot is a
prominent energetic efficient legged robot, there are still some limitations need to be
solved. The main performance limitation of the curved beam hopping robot comes
from the conventional metal curved beam.
First, it is necessary to reduce the mass of curved beam to avoid energy loss at ground
collisions. However conventional metal beams are generally heavy in weight, this
limits the possibility to further improve energy efficiency.
Second, besides lightweight, the curved beam is desired to have high strength in order
to take large payload, this is also what conventional metal beam could not satisfy.
Plastic deformation will happen in metal curved beam when relative large load is
applied. As the performance of curved beam hopper is mainly decided by its
morphology [4]. This deformation will permanent change structure of the curved
beam and degrade performance of curved beam hopping robot.
Chapter 3. Curved beam hopping robot 12
Fig3.2 Typical GRP and metals strain stress curve
From this perspective, to get rid of limitations of conventional metal curved beam,
new elastic beam material need to explore. Glass fiber reinforced plastic is a
prominent candidate. Glass fiber-reinforced polymer (GRP) is a composite material
made of a polymeric resin matrix reinforced with glass fiber. Recently it is widely
used in the aerospace, automotive, marine, and civil industries as structural material
due to its high strength-to-weight ratio, excellent corrosion resistance and high tensile
strength. Fig3.2 shows typical stress-strain responses of GRP in contrast with some
conventional metal materials [9]. It is clearly seen in the graph that GRP composite
material exhibit a linear elastic stress-strain behavior right up to brittle failure which
means no plastic deformation happens. And generally GRP composite has much
larger elastic limits and tensile stress than conventional metal materials. Such elastic-
brittle behavior and high elastic strain energy storage capacity of GRP composite
strongly implicate potential application of GRP composite as elastic spring
components in robotic devices. In next section, we will further discuss its properties.
13 Chapter 3. Curved beam hopping robot
Chapter 4. Properties of Glass Fiber Reinforced Plastic 15
14
Chapter 4
Properties of Glass Fiber Reinforced
Plastic
Other than metals like steel and aluminum, GRP does not consist of a polycrystalline
structure. It is formed by continuous glass fibers bonded by a polymeric resin matrix
[9]. Being a material with low density, high tensile strength and excellent corrosion
resistance, it was normally used for structural application to enhance strength and
reduce structure’s weight. In recent years with the improved design, fabrication and
mechanical performance of low-cost composites, GRP composite has been widely
used in the aerospace, automotive, marine, and civil industries. In this chapter
properties of GRP are introduced
Properties of GRP composite are determined by many factors including: amount of
fibers; properties of the fiber; the properties of matrix; and the bonding between the
fibers and matrix.
1) Properties of matrix resin
Glass fiber is hold by matrix resin in GRP. The resin matrix will protect glass fiber,
distribute load among fibers and enhance transverse properties of a laminate. The
most common used resin material is Epoxy.
2) Amount of fibers
Increase in fiber content leads to an increase in tensile strength and elastic modulus.
The maximum volume fraction is about 80%, beyond which fiber can no longer be
completely bonded by resin material. Fig 4.1 shows the effect of glass fiber volume
fraction [10].
Chapter 4. Properties of Glass Fiber Reinforced Plastic 15
Fig4.1 Effect of glass fiber volume fraction
3) Orientation of Fiber
One major different between composite material and metals is that GRP composite is
anisotropic. The strength and stiffness of a composite material is directional
dependent. The orientation of the fiber in resin matrix is an indication of the major
strength direction of the laminate [10].
a) unidirectional (b) cross-plied
Fig4.2 Effect of Fiber Orientation
16 Chapter 4. Properties of Glass Fiber Reinforced Plastic
4) Properties of glass fiber
For different purposes of application, different kinds of glass fibers are used.
Generally they fall into two categories, low-cost general-purpose fibers and special-
purpose fibers. Over 90% of all glass fibers are general-purpose products which are
known by the designation E-glass [14]. The “E” in E-glass stands for electrical as it is
designed for electrical application. However it is also widely used in other general
purpose such as structural application. In this paper, the general-purpose E-glass fiber
is used.
Letter
designation
Property or characteristic
E, electrical Low electrical conductivity
S, strength High strength
C, chemical High chemical durability
M, modulus High stiffness
A, alkali High alkali or soda lime glass
D, dielectric Low dielectric constant
Table4.1 common-used glass-fiber
Typical properties of E-glass and epoxy are listed in table4.2 and table4.3 [14].
Property Units Epoxy
Axial modulus GPa 80-81
Transverse modulus GPa 80-81
Axial Poisson’s ratio -- 0.2
Transverse Poisson’s ratio -- 0.2
Axial shear modulus GPa 35.42
Axial coefficient of thermal
expansion /
m/
5
Axial tensile strength MPa 3100
Filament elongation at break % 4.6
Table4.2 Typical Properties of Glass Fibers (SI System of Units)
Chapter 4. Properties of Glass Fiber Reinforced Plastic 17
Property Units Epoxy
Axial modulus GPa 3.4
Transverse modulus GPa 3.4
Axial Poisson’s ratio -- 0.3
Transverse Poisson’s ratio -- 0.3
Axial shear modulus GPa 1.308
Axial coefficient of thermal
expansion /
m/
63
Axial tensile strength MPa 72
Table4.3 Typical Properties of Matrices (SI System of Units)
For an E-glass/Epoxy GRP lamina with a 70% fiber volume fraction, its
properties could be calculated from that of E-glass and Epoxy listed in table4.2 and
table4.3 [14].
Longitudinal elastic modulus of the unidirectional lamina is calculated by
With Axial modulus of E-glass fiber
Axial modulus of Epoxy resin
Volume fraction of glass fiber
Transverse elastic modulus
Major Poisson’s ratio,
Minor Poisson’s ratio,
18 Chapter 4. Properties of Glass Fiber Reinforced Plastic
In-plane shear modulus
Property Units GRP
Longitudinal E-modulus E1 GPa 57.02
Transverse E-modulus E2 GPa 10.31
Major Poisson’s -- 0.230
Minor Poisson’s -- 0.0394
In-plane shear modulus G12 GPa 4.014
Table 4.4 Properties of GRP
Chapter 4. Properties of Glass Fiber Reinforced Plastic 19
`
20
Chapter 5
FEM analysis and experiment test
Finite element analysis has been widely used to analyze mechanical behavior of
materials and structures. Recently with development of composite materials, a lot of
studies have been conducted on FEA method of composite materials analysis [11]
[12]. In the section, I will introduce an approach using FEM method to analyze GRP
and metal spring. Based on simulation results, a systematic comparison between GRP
and conventional metal material is done. Real world experiments are also conducted
to verify simulation.
5.1 FEM analyses of GRP and metal springs
In this paper ANSYS was used to carry out the finite element analysis. ANSYS is
commercial engineering simulation software for FEM analysis. Here it is used to
predict the deformation and critical buckling load for GRP and metal spring. The steps
of FEM analyses in ANSYS are as following:
a) Define element type
The element used for the laminated GRP spring plates was Shell181, which is a 4-
node 3D shell element. For each node the element has 6 degree of freedom: translation
in X, Y and Z direction and rotation about nodal X, Y and Z axes. It is design to
modeling thin to moderately thick plate or shell structures and allows up to 255
uniform/non-uniform section layers per element. One advantage of this element type
is that it has full nonlinear capabilities including large strain and material models,
which is demanded in this project for analyzing large deformation of the C-Shape
spring.
For isotropic metal materials, simply 3D elastic BEAM4 element was used. BEAM4
is a uniaxial element with tension, compression, torsion, and bending capabilities. The
element has six degrees of freedom at each node: translations in the nodal x, y, and z
Chapter 5. FEM analysis and experiment test 21
directions and rotations about the nodal x, y, and z axes. Stress stiffening and large
deflection capabilities are included.
b) Section definition
For layered composite GRP, the number of layers is defined in the section command
option. For each layer, its thickness and orientation need be specified. The orientation
of the layer is determined by the direction of the fibers.
c) Material properties
For anisotropic composite materials, the material models are not directly built in
ANSYS. We need feed in material properties in the matrix form or layered form. A
code (Appendix A.1) based on the Halpin-Tai equations [13] is used to transfer the
properties listed in table.4 into the layered form parameters (EX, EY,
EZ, ) in ANSYS. Alternatively, the stiffness matrices ([A],
[B], and [D]) can be entered.
For linear isotropic elastic metal materials, Poisson's ratio and Elastic modulus can be
defined in the material properties options.
The analysis of GRP and metal springs are generally the same for the following
procedures.
d) Geometric modeling and meshing
ANSYS provides direct interfaces to all major computer-aided design (CAD) systems.
Existing, native CAD geometry can be used directly, which means the model built in
other CAD software like Solidworks, UG and AutoCAD can be used directly in
ANSYS. This feature largely reduces user’s work in geometric modeling. For the
simple C-shape curved beam spring model, we choose directly model its geometry in
ANSYS. The meshed geometric model of the GRP spring is shown in Fig5.1.
Fig5.1 geometric ANSYS model of GRP spring
22 Chapter 5. FEM analysis
e) Define load and constrain
Full displacement constrain is applied in one end tip of C-Shape curve beam, set the
displacement in all degree of freedom to be zero. Then by applying different loads in
the free-end, we can find the critical load which leads the maximal stress in spring
exceeding the yield stress of the material . This critical load is the
maximal load the spring can stand without failure or plastic deformation. The
boundary condition and load is shown below Fig5.2
Fig5.2 Load and boundary constrain on model
f) Solver and Postprocessor
The deformed shape and Von Mises stress contour plot of the GRP and metal springs
when applied critical load are shown in Fig5.3 Fig5.4
(a) Aluminum spring with (b) steel spring with (c) GRP spring with
maximal payload 100N maximal payload 250N maximal payload 200N
Fig5.3 Deformation of spring with maximal payload
Chapter 5. FEM analysis 23
(a) Aluminum spring with (b) steel spring with (c) GRP spring with
critical stress 95Gpa critical stress 250Gpa critical stress 550Gpa
Fig5.4 Stress of spring with maximal payload
5.2 Comparison between GRP and metal springs
Designing of compliant spring component is a multi-objective and multi-constraints
problem. Geometry structure, Load capacity, stiffness, damping, energy store capacity,
many factors involve. Here a systematic comparison between GRP and metal spring is
conducted. We compare them in two different constraint conditions, with same spring
weight or with same geometry dimension. The results are shown as follows:
5.2.1 Load capacity
Fig5.5 Relation of applied force to displacement for spring beams with the same
dimension
24 Chapter 5. Comparison between GRP and metal curved beam
Fig5.5 shows when apply different load force, vertical displacement of C shape
springs made of GRP, Steel and aluminum. These springs are of the same dimension,
radius 18.5cm, width 5cm, thickness 0.5cm. The maximum load GRP spring can stand
is almost the same as the spring made of steel, however the weight of it is only fifth of
the steel one.
Fig5.6 Relation of applied force to displacement for spring beams with the same
dimension
Fig5.6 shows vertical displacement of C shape springs made of GRP, Steel and
aluminum with different load force. These springs are of identical weight 218g. In this
case, GRP spring can take much higher load than metal ones, which is required for
scalable efficient legged robots.
` Springs with the same dimension Springs with the same weight
Aluminum Steel GRP Aluminum Steel GRP
Maximum
load (N)
100
250
220
43
19
220
Table5.1 Load capacity of GRP, steel and aluminum springs
Chapter 5. Comparison between GRP and metal curved beam 25
5.2.2 Stiffness
Equivalent Stiffness of GRP, aluminum and steel C shape beam are shown below.
Fig5.7 plots equivalent stiffness of C shape springs in different loads when they are of
the same dimensions. And Fig5.8 shows the case when maintain the beam mass the
same. With the same dimension, the GRP spring is much softer than the steel one.
However the stiffness of the GRP spring will increase when its weight rises. As shown
in Fig5.8, with the same weight as steel C beam, it becomes stiffer than the steel one.
In other words, without losing load capacity, GRP spring has much larger stiffness
range when compared with steel C beam under the a constrained maximal spring
weight.
Fig5.7 Relation of applied force to equivalent stiffness for spring beams with the same
dimension
26 Chapter 5. Comparison between GRP and metal curved beam
Fig5.8 Relation of applied force to equivalent stiffness for spring beams with the same
weight
5.2.3 Strain energy
The strain energy of the spring material is a major factor to be considered for
designing of spring. It represents the potential energy that a material of unit weight
can store when it is elastic deformed by external force. The strain energy of a specific
material is expressed as
Where is the strength, is the density and E the Young’s modulus of the spring
material. As the formula shows, the stored elastic strain energy in a spring varies
directly with the square of maximum allowable stress and inversely with the modulus
of elasticity. To store more elastic strain energy, large yield stress, small density and
elastic modulus are general desired. As shown in table5.2 [9], GRP have more elastic
strain energy storage capacity when compared to those of conventional metals.
Chapter 5. Comparison between GRP and metal curved beam 27
Materials Density E-Modulus Yield Stress Tensile Stress
[g/ 3] GPa MPa MPa
Aluminum 2.63 69 95 110
Copper 8.4 117 70 220
Mild Steel 7.81 200 250 400
Glass-reinforced plastic 1.6-2.0 20-55 400-1800 400-1800
(GRP)
Table5.2 general mechanical properties of GRP and common used materials
5.2.4 Contact force
Fig .5.9 Spring mass contact model
Spring mass contact model is used to compare contact force exerted by springs made
of GRP and metals. These springs are with the same dimensions. Dynamics of the
system are formulated as follow:
Fig5.9 shows the simulation results of contact force exerted by different spring, it is
clear that contact force of GRP spring is the lowest one.
28 Chapter 5. Experiment test
5.3 Experiment test
After FEM analysis, two set of experimental tests are conducted to verify the results
of simulation. In the first set of experiment, deformation of GRP C shape beam
laminated by cross-piled glass-fiber fabrics bands is measured when 5Kg payload is
applied. Fig5.10 shows the comparison of simulation result and experiment test. The
error between simulation and experiment is 6-10%.
Fig5.10 comparison between simulation and experiment
Deformation X (m) Y (m)
Simulation 0.094 0.091
Experiment 0.103 0.097
Error (%) 8.7 6.2
Table5.2 Comparison between simulation and experiment
To find out relation between stiffness and thickness of GRP spring beam, stiffness of
GRP beams with different number of fabric layers were measured. In both simulation
and experiment, stiffness raises polynomial with thickness as shown in Fig5.11. The
result of experiment test is fitted by a polynomial of order 3.
Chapter 5. Experiment test 29
Fig5.11 relation between stiffness and number of layers
Fit polynomial:
3
3
With n number of layers
Chapter 5. FEM analysis and experiment test 30
31
Chapter 6
Application of GRP in CBH
In this section performance of the GRP curved beam springs in real robotic
application is examined. Serials of real-world experiments are conducted in a simple
robot platform, curved beam hopping robot. For the purpose of this paper, energy
efficiency in different load condition, especially with large payload, is the key object.
6.1 Experiment platform
As introduced in chapter2, the curved beam hopping robot is able to achieve efficient
hopping locomotion by making use of free vibration of its spring curved beam. Its
hopping behavior is mainly determined by the design and material properties of the
elastic curved beam. In the experiment, metal curved beam is replaced by beams made
of GRP composite. The curved beam is attached to an H shape aluminum foot base. A
non-gear DC motor (Maxon RE) is used to produce periodic actuation. Detailed
mechanical parameters of the hooping robot are shown in table 6.1
Fig6.1 Schematic diagram of Curved beam hopping robot
Chapter 6. Application of GRP in CHB 32
Parameter Unit 10kg hopping robot 5kg hopping robot
M g 8234 4117
m1 g 82 80
L0 cm 37 37
L1 cm 24 14
L2 cm 30 30
e cm 4 4
K1 N/m 5500 1600
Table6.1 parameters of real world GRP curved beam hopping robot
6.2 Experimental Setup
An indoor 2 meter long flat wooden floor is used as track of the hooping robot. The
robot starts from a static posture. A power supply with constant voltage output is used
to drive the motor. Robots with two different payloads, 5kg and 10kg, are used in the
experiment. After the hooping robot fall into stable gaits, power supply’s output
power is measured. The motion of the robot is recorded by a video camera at
30frames/s or with a high-speed camera at 120frames/s. The recorded data were used
to compute the velocity of the robot. Time series photographs of hooping motion for
GRP curved beam hooping robot with different payload are shown in Fig 6.2 and Fig
6.3
Fig.6.2 time series figures of hooping motion for GRP curved beam Hopping robot
with 5kg payload
Fig.6.3 time series figures of hooping motion for GRP curved beam Hopping robot
with 10kg payload
33 Chapter 6. Analysis of energy efficiency
6.3 Analysis of Energy Efficiency
To systematically evaluate energy efficiency of locomotion system, the term” Cost of
Transportation” [15] (CoT) is used, which is defined as follows:
Where P is the power expenditure of the locomotion system, m is total mass, g is the
gravitational acceleration, and v is the locomotion velocity.
By this definition, the CoT of the GRP curved beam hopping robot with different load
is calculated. With 5kg payload, the CoT value of the hopping robot is 0.6. For the 10
kg payload curved beam hopper, 0.485 CoT can be achieved. Compared with other
biological and robotic locomotion systems, the result of our experiment is among the
most energy efficient legged locomotion systems. It is almost the same as human
running. Fig6.4 shows Tucker cost graph of typical locomotion systems [16].
Fig 6.4 Tucker cost graph of typical locomotion system
Chapter 6. Analysis of energy efficiency 34
5 kg payload 10 kg payload
Weight (kg) 5.7 11.8
Speed (m/s) 0.02 0.0589
Power (w) 0.6 3.3
CoT 0.6 0.485
Table 6.2 CoT of GRP curved beam hopping robot
35 Chapter 6. Application of GRP in CHB
36
Chapter7
Discussion and conclusion
This paper introduced a novel material Glass fiber reinforced plastic as spring material
in design of compliant legs for an efficient legged robot. From the experiments by
using the curved beam hopping robot platform and Finite element analysis simulation,
we are able to systematically compare performance of GRP and conventional metals
as elastic spring materials. Based on the comparison results, in the section, we discuss
other potential applications where GRP spring can replace conventional metal spring
and what is the limitation of this new type spring. Finally conclusions are made base
on all the results and discuss.
7.1 Discussion
Light weight is the most prominent feature of GRP spring when compared with
conventional metal spring. With the same load capacity the weight of GRP spring is
general 1/5 of conventional steel spring. This means for those applications where the
weight of spring itself are large, by replacing conventional metal spring with GRP one
can largely reduce the total mass of the system. This is generally needed for energy
efficiency consideration. One example of such applications is the leaf spring used in
suspension system in vehicle. The leaf spring is about 5% to 10% of the total mass of
vehicle. Using GRP spring can largely reduce the weight of the vehicle. By this mean
fuel consumption of the vehicle will be reduced. And energy saving is a major trend
of current world car industry.
High load capacity is another prominent property of GRP spring. With the same
weight the load capacity of GRP spring is about 8 times of steel spring, known from
the simulation result in chapter 5. This feature implies that the GRP can be used to
replace steel as elastic material when large load capacity and strength is needed. Such
as in our curved beam hopping robot, when apply 10kg payload on conventional metal
curved beam, plastic deformation will happen. This will change our CBH’s
Chapter 7. Discussion and Conclusion 37
morphology and degrade performance of it. Other potential application may be as
elastic beam in civil industry such as stay cable in bridge.
Also from analysis in chapter 5, we know that GRP spring has larger stiffness range
when compared with metal spring which is generally stiffer. This means GRP spring
may be used in application where elastic and compliance are more desired such as
robots which interact with human being. For conventional industrial robotics which
stiffness is needed to improve the precision, stability and bandwidth of position
control, GRP spring is not a suitable choice.
Though have many advantages of GRP spring when compared with conventional
metal spring, there are still some limitations which limit application of GRP spring.
Manufacture of GRP spring is somewhat time-consuming especially for the spring
need have complex shape. This will make massive production of GRP spring hard.
Price is another limitation of GRP spring when compared with conventional metal
spring.
7.2 Conclusion
This thesis introduced a new material Glass fiber reinforced plastic as spring material
in design of compliant legs for an efficient legged robot. Performance of it is
evaluated in CHB platform. Finite element analysis and experiments are conducted to
systematically compare GRP with conventional metal elastic material. The results
show that with GRP curved beam the CHB can achieve efficient locomotion scalable
to payload, especially with large load. The Cot of GRP CBH is 0.485, which is among
the most efficient legged locomotion system. Further analysis shows GRP curved
beam has many advantages over conventional metal one. Lighted weighted, high load
and energy storage capacity, large stiffness design range, all these features imply
many applications of GRP spring.
38 Chapter 7. Discussion and Conclusion
39
Appendix A
Symbols, Acronyms and
Abbreviations
Symbols
The symbols are displayed in alphabetical order.
State variable: longitude length of curved beam leg
Initial longitude length of curved beam leg
Length of rotation arm
mass of foot
Mass of payload
Weight of rotating mass
State variable: angle of curved beam leg
Initial angle of curved beam leg
Angle of rotation arm
Angular velocity of rotation arm
Stiffness of longitude linear spring
Stiffness of foot torque spring
Damping coefficiency of longitude linear spring
Appendix A. Symbols, Acronyms and Abbreviation 40
Damping coefficiency of torque spring
State variable: travelling distance of footbase in x axis direction
State variable jumping height of the foot
Longitudinal elastic modulus of the unidirectional GRP lamina
Transverse elastic modulus of the unidirectional GRP lamina
Axial modulus of E-glass fiber
Axial modulus of Epoxy resin
Volume fraction of glass fiber
Major Poisson’s ratio of the unidirectional GRP lamina
Minor Poisson’s ratio of the unidirectional GRP lamina
In-plane shear modulus of the unidirectional GRP lamina
Acronyms and Abbreviations
The acronyms and abbreviations are displayed in alphabetical order.
BIRL Bio Inspired Robotics Lab
CAD Computer Aided Design
CBH Curved Beam Hopping Robot
CoT Cost of Transport
FEM Finite Element Method
GRP Glass Fiber Reinforced Plastic
41 Appendix A. Symbols, Acronyms and Abbreviation
`
42
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