Composite Structures 140 (2016) 344–350
Contents lists available at ScienceDirect
Composite Structures
journal homepage: www.elsevier.com/locate/compstruct
Carbon/epoxy composite foot structure for biped robots
Jinyi Lee a, Dongyoung Lee a, Jongwon Park b, Ilbeom Choi c, Jun Woo Lim d, Soohyun Kim a, Dai Gil Lee a,⇑
a
School of Mechanical, Aerospace & Systems Engineering, Korea Advanced Institute of Science and Technology, 291, Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea
Korea Atomic Energy Research Institute, 111, Deadeok-daero 989 beon-gil, Daejeon 34057, Republic of Korea
c
Agency for Defense Development, P.O. Box 35, Yuseong-gu, Daejeon 34060, Republic of Korea
d
LANL-CBNU Engineering Institute Korea, Chonbuk National University, 567 Baekje-daero, Deokjin-gu, Jeonju-si, Jeollabuk-do 54596, Republic of Korea
b
a r t i c l e
i n f o
Article history:
Available online 7 January 2016
Keywords:
Carbon composite
Fast running robot leg
Impact
Damping
Fatigue
a b s t r a c t
Although aluminum structures are generally used for robot structures due to their high specific strength,
aluminum feet for fast running biped robots are vulnerable to fatigue failure due to the low fatigue limit
and low vibration damping of aluminum structures under repeated impact loadings on the feet. On the
other hand, carbon/epoxy composites not only have a much higher specific fatigue limit but also have a
higher material damping than that of aluminum.
In this study, a carbon/epoxy composite foot structure of a biped robot was developed. The composite
foot structure was designed for optimum performances such as weight saving, natural frequency,
damping, and compliance for vibration isolation. Then its performances were analytically and experimentally obtained and compared with those of an aluminum foot structure. Finally, an optimum configuration of the composite foot structure was suggested for the reliable dynamic performance of the
biped robot.
Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction
The ability of fast and agile robotic locomotion is crucial issue in
dangerous situations, such as natural disasters, huge plant accidents, and reconnaissance in war scenarios that consist of environments with various obstacles. A fast legged robotic platform can be
an ideal solution for rescuing or finding enemies on rough and
complex terrain. Over the past several decades, many researchers
have developed various types of legged robots due to their potential for rough terrain mobility compared with wheeled vehicles [1].
There are two approaches to increase the speed of the legged
robot: increasing the actuator capacity and decreasing the leg inertia. Since increasing the actuator capacity causes higher actuator
mass that correspondingly requires the increased leg strength
and leg mass, decreasing the total inertia of the leg structure might
be a more practical solution. However, the design of robot leg with
low inertia but high strength to withstand large ground impacts
during high speed running is coupled.
The distal part of robot leg is more critical due to its larger contribution to rotational inertia. Therefore, there have been some
attempts to reduce leg inertia and apply compliance to distal parts
by decreasing the load transmissibility of the robot body. To
⇑ Corresponding author. Tel.: +82 42 350 3221; fax: +82 42 350 5221.
E-mail address: dglee@kaist.ac.kr (D.G. Lee).
http://dx.doi.org/10.1016/j.compstruct.2016.01.022
0263-8223/Ó 2016 Elsevier Ltd. All rights reserved.
minimize leg inertia, several techniques have been used, such as
under-actuated legs [2,3], locating the actuators closer to the body
[4], cable driven structures [5], or applying high specific strength
materials [6,7]. In addition, series-elastic actuation methods or
mechanical suspensions have been used to obtain compliance
[1]. Despite recent advances in robot technologies, the realization
of agile dynamic movement still remains a difficult problem
because of these design trade-offs.
The raptor robot was developed for agile dynamic locomotion
on irregular terrain at high speed [3]. For fast running, aluminum
structures are generally adopted due to their low density compared with steel. The raptor robot is composed of the underactuated 9-bar linkage structure, which is driven by a single electrical actuator, as shown in Fig. 1(a), whose specifications are
shown in Table 1. An Achilles tendon made of butyl rubber, which
acts as spring and damper, was adopted to reduce ground impact
and achieve efficient movement. The robot has reasonable dynamic
properties, such as being light weight and having a small moment
of inertia, for achieving high speed running at 27 km/h. Due to the
low resilience of aluminum (yield strain 0.2%), a rib structure is
required for the foot, as shown in Fig. 1(b). Because such a stiff foot
structure transmits a large impact load to the robot body, damping
materials were attached at the foot for protection of the body, as
shown in Fig. 1(a). In spite of the damping materials, failure
occurred at the body under repeated impact loading when the
J. Lee et al. / Composite Structures 140 (2016) 344–350
345
Fig. 1. Fast biped raptor robot: (a) raptor robot with a conventional aluminum foot and damping materials; (b) rib structure of the aluminum foot; (c) fractured components
of the robot body during the running test.
Table 1
Specification of the biped robot with the aluminum foot structure.
Specification
Total height (mm)
Mass (kg)
Max. speed (km/h)
Body material
Leg material
500
2.5
27
Aluminum 7075
Aluminum 7075
speed of the robot increased to over 27 km/h, as shown in Fig. 1(c).
Therefore, attaching damping materials was not an appropriate
solution for protection of the body.
Carbon composites have high specific strength, high damping,
high failure strain, and low coefficient of thermal expansion
(CTE) [8]. Because of their excellent properties, carbon composites have been widely used in robot structures as well as in
aircraft and spacecraft structures, machines, automotive structures, and prosthetic structures for amputees [6,8–18]. Therefore,
it might be possible to design a simple and compliant foot
structure to reduce the transmitted load from impact and the
moment of inertia using composite materials. Moreover,
additional damping materials are unnecessary due to the
compliance of the foot.
The purpose of this study is to develop a carbon/epoxy foot
structure for a biped raptor robot to improve the running performance and life cycles of the robot. The composite foot structure
is designed considering its levels of performance, such as weight
saving, natural frequency, and damping and compliance for vibration isolation. For the design of optimum configuration of the composite foot, strain energy method and the maximum strain failure
criterion was selected. Its performances were analytically and
experimentally obtained and compared with those of the aluminum foot structure. Finally, the developed composite foot structure was experimentally verified for the reliable dynamic
performance and life cycles of biped robot.
2. Design of the composite foot structure
2.1. Design of a composite foot model using the strain energy method
The composite foot of the raptor robot has a curvilinear shape,
with a cantilever beam joined to a semicircular beam, as shown
in Fig. 2(a). The cantilever beam of the foot structure is connected
to the aluminum foot connector using screw bolts. To design the
composite foot, a cantilever model and a simple curved beam
model were used, as shown in Fig. 2(b). Table 2 describes the
parameters and values used in the formulations. The subscripts c
and s are indicated curved beam and straight beam (cantilever
beam), respectively. The vertical deflection at the ground contact
point of the foot B is caused by the axial force, shear force and
bending moment. However, the contributions of the axial load
and shear are negligible when the radius to thickness ratio (r=h)
is larger than 10 [19]. Therefore, the strain energy that was due
Fig. 2. Schematic diagram of composite foot structure: (a) curvilinear shape of the
composite foot structure; (b) free body diagram.
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J. Lee et al. / Composite Structures 140 (2016) 344–350
½h ¼ Q
½h
¼Q
E
11
11
Table 2
Design parameters of the composite foot structure.
Radius of the curved beam (mm)
Width of the beam (mm)
Length of the cantilever beam (mm)
Maximum axial load (N)
Thickness of the beam
Stacking angle
Number of symmetry plies
Symbol
Values
r
b
l
F max
h
h
n
75
14
43
88
To be determined
0 6 h 6 90
1 6 n 6 10
ð8Þ
½h is expressed as a function
The transformed reduced stiffness Q
11
of stacking angle h.
½h ¼
Q
11
cos4 h þ 2
E1
1 m12 m21
E2
4
sin h
þ
1 m12 m21
m12 E2
2
þ 2G12 cos2 h sin h
1 m12 m21
ð9Þ
Finally, Eq. (6) can be expressed as follows:
only to bending was used to calculate the deflection of the beam in
this study.
Since the bending moment M s of the straight part of the beam is
the product of the axial force F at the contact point B and the radius
r of the curved beam (Ms ¼ F r), which is constant, the strain
energy of the straight part, U s is expressed as follows:
Us ¼
1 M 2s l 1 ðFrÞ2 l
¼ 2 EI
2 EI
ð1Þ
is the longitudinal Young’s modulus of the composite foot,
where E
and I and l are the moment of inertia and length of the straight part
of the foot, respectively. The bending moment M c of curved beam
part is expressed as follows:
M c ¼ Fr sin /
ð2Þ
where / is the angle between the force and point P as shown in
Fig. 2(b). Thus, the strain energy, U c of the curved beam part is
expressed as follows:
Uc ¼
Z p=2 2 2
2
F r sin /
p F 2 r3
r d/ ¼
8 EI
2EI
0
ð3Þ
Thus, the total strain energy U of the composite foot is
expressed as follows:
2 2
2 3
2 2
F r l pF r
ð4l þ prÞF r
U ¼ Us þ Uc ¼ þ
¼
8 EI
2EI
8EI
ð4Þ
The vertical deflection d of the foot at the contact point B is
expressed as follows:
d¼
@U ð4l þ prÞFr2
¼
@F
4EI
ð5Þ
And the spring constant k of the foot at the contact point B is
expressed as follows:
3
Ebh
F
4EI
k¼ ¼
¼
2
d ð4l þ prÞr
3ð4l þ prÞr 2
ð6Þ
2.2. Optimization of composite foot structure
In Fig. 2(a), neglecting the stress concentration effect due to the
curvature of the semicircle part when r=h > 10, the maximum laminate tensile stress occurs at the outer side of the straight part of
the foot (A) as follows:
r½h
x ¼
F max r h
F max r
¼6
2
I
2
bh
ð7Þ
where b and h represent the width and thickness of the composite
foot, respectively. One of the main objectives of the composite foot
is to reduce the magnitude of k to enable the robot to jump up
higher, stride farther, and reduce the impact from the ground when
the running frequency is lower than the fundamental natural frequency of the foot [20]. The high damping of the composite foot will
decrease the impact load transmissibility. For the ease of manufacturing the composite foot structure, the ply stacking sequence of
½hns was chosen. Then, for the wide rectangular beam, the longitu is expressed as the transformed reduced
dinal Young’s modulus E
stiffness as follows [12]:
ð11Þ
Thus, the ply stresses are expressed as follows [12]:
r1 ¼ cos2 h r½h
x
ð12Þ
r2 ¼ sin2 h r½h
x
ð13Þ
r6 ¼ cos h sin h r½h
x
ð14Þ
To design the composite foot structure, the maximum strain
failure criterion was used because it yielded an analytical solution,
and was generally used for the design of dynamic structures such
as aircraft composite structures [21]. The design condition for the
maximum strain failure criterion for each ply [12], is expressed
as follows:
Xc
cos2 h m12 sin h
Yc
3
bh
12
ð10Þ
Eq. (10) gives the design criteria with respect to thickness h and
stacking angle h when the foot width b is constant; the magnitude
of b is generally determined empirically to avoid the tilting of the
foot from the ground. The formulas can be used to optimize the
foot structure for minimum spring constant and weight.
2
The moment of inertia of the foot structure I is expressed as
follows:
I¼
3
bh
E1
m12 E2
4
cos
h
þ
2
þ
2G
12
3ð4l þ prÞr 2 1 m12 m21
1 m12 m21
E
2
2
4
sin h
cos2 h sin h þ
1 m12 m21
k¼
2
sin h m21 cos2 h
< r½h
x <
< r½h
x <
Xt
2
cos2 h m12 sin h
Yt
2
sin h m21 cos2 h
S
S
< r½h
x <
sin h cos h
sin h cos h
ð15Þ
ð16Þ
ð17Þ
The failure index f , which should be less than 1, is expressed as
follows:
f ¼
8
9
rx ðcos2 hm12 sin2 hÞ >
>
>
>
t
>
>
X
<
=
stress
2
¼ rx ðsin hmt 21 cos2 hÞ < 1
>
Y
strength >
>
>
>
: rx ðcos h sin hÞ >
;
ð18Þ
S
t
t
where X , Y , S and m12 are the longitudinal strength, transverse
strength, shear strength, and Poisson’s ratio of unidirectional composite, respectively. In this study, the design objective is to minimize the magnitude of k in Eq. (10) while satisfying the constraint
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J. Lee et al. / Composite Structures 140 (2016) 344–350
3. Experimental
Failure index
No. symmetry ply (n)
10
5.0
1.0
4.0
8
3.0
6
2.0
4
1.0
2
0.0
0
10
20
30
40
50
60
70
80
90
stacking angle (θ)
(a)
5
Spring constant
Failure index
4
12.1
12
11.1
10.6
9.65
8.68
8
3
6.86
2
3.66
4
1
1.57
Failure index (f)
16
Spring constant (N/mm)
of Eq. (18). The stacking angle and thickness of the composite foot
were determined by calculating the above formulations to optimize
the stacking sequence. The design parameter and its values are
shown in Table 2.
In this work, a high strength carbon/epoxy prepreg (USN150, SK
Chemicals, Korea) was used for the foot structure whose material
properties are shown in Table 3. The maximum axial force on each
leg is known to be approximately 3 times the body weight [22].
Preliminary running test results with aluminum feet also showed
similar trend (F max ¼ 68 N at 27 km/h). A safety factor (SF) of 1.2
was applied for optimization. Therefore, the maximum applied
load was 88 N (F max ¼ 3mg SF).
To solve optimization problems, design parameters of the foot
structures (h, n) were increased from 0 to 90 degree by 5 degree
and 1–10 plies by 1 symmetry plies, respectively. Firstly, searching
the failure index f under 1 with respect to the stacking angles and
number of symmetry plies as shown in Fig. 3(a) with labeled f ¼ 1.
With these results, the local minimum spring constants in the multiple stacking angles with same symmetry plies were organized as
shown in Fig. 3(b), which were selected to satisfy the failure index
less than 1 (f < 1). The results of the 0 degree cases were neglected
due to the splitting problem although it gave the lowest spring
constant. Therefore, the optimum stacking sequence was chosen
as ½54s , whose spring constant and failure index were 3.66 N/
mm, and 0.61, respectively.
0.46
0
[0]2s
[0]3s
[±5]4s
[±10]5s [±15]6s [±30]7s [±45]8s [±50]9s [±60]10s
0
Stacking sequence
3.1. Specimen fabrication
(b)
The composite foot specimens were fabricated using a high
strength unidirectional carbon/epoxy prepreg (USN150, SK Chemicals, Korea) to verify the design optimization results, ½54s . The
configuration and dimensions of the specimens are shown in
Fig. 4. The prepreg were stacked with a stacking sequence of
½54s on a curvilinear shape mold. The mold was polished and
treated with a liquid-type mold release (Safelease30, Air tech, United States) for easy demolding. The stacked composite specimen
was cured by the autoclave vacuum bag degassing method at
125 °C and 0.6 MPa for 2 h.
3.2. Compression test of the composite foot structure
Compression tests were performed using a universal testing
machine (Instron 4469, Instron Corporation, United States) at
25 °C to evaluate the mechanical properties of the fabricated composite foot and the conventional aluminum foot structure. The
experimental setup is shown in Fig. 5(a). The composite foot specimen was clamped by a jig, and then a load was applied by a lubricated flat plate with a crosshead speed of 2 mm/min. The spring
constant of the composite foot structure (½54s ) was 3.75 N/mm,
which was well matched to the analytical model (3.66 N/mm) from
Table 3
Material properties of the high strength carbon/epoxy composite (USN 150).
Material properties
Symbol
Values
Longitudinal modulus
Transverse modulus
Shear modulus
Longitudinal tensile strength
E1 (GPa)
E2 (GPa)
G12 (GPa)
131
10.8
5.65
2000
Transverse tensile strength
Y t (MPa)
S (MPa)
Shear strength
Poisson’s ratio
Density
Thickness
X t (MPa)
m12
q (kg/m3)
tply (mm)
61
70
0.28
1540
0.15
Fig. 3. Analysis results: (a) contour map of the failure index; (b) spring constants of
selected stacking sequences under f < 1.
the Section 2.2. The failure load of the composite feet was 184 N. In
addition, the compression test of the aluminum foot was performed in the load range of 0–88 N. Table 4 shows the compression
test results of both foot structures. The spring constant and mass of
the composite foot structure were 55 times and 3 times lower than
the corresponding conventional aluminum foot, respectively.
Using composite materials, the spring constant, mass and moment
of inertia of the whole leg structure reduced significantly.
3.3. Vibration test of the specimen
Vibration tests were performed with respect to the stacking
sequence to measure the natural frequency and damping ratio of
the composite foot structure. Before the tests, composite foot specimens were fabricated with respect to the stacking sequences of
½54s ; ½104s ; ½204s ; ½304s ; and ½354s to investigate vibration
characteristics of the composite foot with respect to the stacking
angles. The experimental setup is shown in Fig. 5(b). A strain gauge
was attached at the curved part of the composite foot. A dummy
mass, which had the same mass as the robot (2.5 kg), was hung
at the end of the foot. The vibration of the feet was generated by
pulling down 10 mm and then releasing the dummy mass. A low
pass filter was used to remove the high frequency noise. Fast Fourier Transform (FFT) was performed to obtain the natural frequency
of the composite foots and to calculate the damping ratio using the
logarithmic decrement method [23]. The natural frequency and
damping ratio of the composite foot structures with respect to
the stacking sequences were measured and calculated as shown
in Table 5. The natural frequencies of the composite feet were significantly lower than those of the conventional aluminum foot; in
particular, the ½54s foot (5.59 Hz) was 88% lower than aluminum
foot (45.8 Hz). The damping ratio of the ½54s composite foot
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J. Lee et al. / Composite Structures 140 (2016) 344–350
Fig. 4. Configuration of the composite foot specimen.
Fig. 5. Experimental setup (a) compression and dynamic loading test; (b) vibration test.
Table 4
Compression test results.
Foot structure
Materials
Spring constant, k (N/mm)
Mass (g)
Thickness (mm)
Composite foot ½54s
Aluminum foot
Carbon/epoxy composite (USN 150)
Aluminum 7075
3.75
207
8.5
25.0
2.4
N/A
Table 5
Vibration test results.
½104s
½204s
½304s
½354s
Aluminum
3.75
5.59
0.029
3.51
4.97
0.015
2.98
3.94
0.009
2.26
3.18
0.008
1.88
2.90
0.014
207
45.8
0.0016
(0.029) was 18 times higher than aluminum foot (0.0016). Fig. 6
shows the damping ratio and spring constant of the composite foot
structures with respect to the stacking sequence. The highest
damping ratio occurred at ½54s ; which was different from the
trend of tensile specimen where the highest damping occurred at
higher stacking angles. These results may be caused by the low
angled fibers that suppressed the bending motion of the foot structure, which induced the interlaminar shear deformation.
3.4. Life cycles of the composite foot structure
The dynamic loading tests were performed to measure the life
cycles of the developed composite foot structure using a dynamic
materials testing machine (Instron 8801, Instron Corporation, United States) at 25 °C. The experimental setup is shown in Fig. 5(a).
The composite foot was clamped by the jig, and the dynamic load
was applied via a lubricated flat plate using a crosshead frequency
Damping ratio
Stiffness
0.030
0.025
3
0.020
2
0.015
0.010
1
Stiffness (N/mm)
½54s
Damping ratio ( )
Spring constant, k (N/mm)
Natural frequency (Hz)
Damping ratio (f)
0.005
0.000
[±5]4s
[±10]4s
[±20]4s
[±30]4s
[±35]4s
0
Stacking sequence
Fig. 6. Vibration test results with respect to the stacking sequence.
of 3 Hz, which is same as the leg rotating frequency. Load was
applied in one direction only with a stress ratio, R ¼ 0. The dis-
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J. Lee et al. / Composite Structures 140 (2016) 344–350
Table 6
Dynamic test results after 105 cycles.
24.9
Before dynamic test
After dynamic test
3.75
3.7
placement control was used, whose condition was set according to
the static compression test results at 0–90 N (0–20 mm). The maximum stress of the foot was approximately 500 MPa by Eq. (11).
Table 6 presents the spring constant of the composite foot
25
Acceleration (m/s2)
Spring constant (N/mm)
30
20
14.9
15
10
5
5
before and after dynamic loading test of 10 cycles, which was
equivalent of 14 km/h speed for 9 h. The spring constant did not
change without any damage of the composite foot structure. In
conclusion, the newly developed composite foot was very effective
and promising for fast robot running.
0
Al foot
Compostie foot ([±5]4s)
(a)
15
14.1
To verify the developed composite foot structure (½54s ),
dynamic running tests were performed. The raptor robot was
attached to the tethered boom, which realizes quasi-planar motion
on the treadmill, as shown in Fig. 7(a). The robot is composed of
the under-actuated 9-bar linkage structure, which is driven by a
single electrical actuator attached to the crank link. The leg structure of the raptor robot has one degree of freedom, and the foot trajectory is generated by the crank. Fig. 7(b) describes the schematic
diagram of the leg movement as the crank rotates. The leg trajectory of the robot can be determined by the angular position of
Speed (km/h)
3.5. Dynamic running test
11.4
10
5
0
Al foot
Compostie foot ([±5]4s)
(b)
Fig. 8. Experimental results of the running test: (a) acceleration through feet; (b)
speed.
the crank (a). A magnetic rotary encoder (ML 512 CPT, Maxon
Motor Co., Switzerland) was used to measure the rotational speed
of crank. An incremental encoder (E30S, Autonics, Korea) was
installed at each revolute joint of the boom to measure the position
data of the robot. A 3-axis acceleration sensor (EBIMU-9DOFV2,
E2box, Korea) was installed at the center of the robot body to measure inertial force during running. The raptor robot was operated at
3 Hz for the corresponding aluminum and composite foot structures. The acceleration at the body through the composite foot
was 40% lower than that through the aluminum foot, as shown
in Fig. 8(a). This result indicated that the major parts of the robot
were effectively protected by the composite foot. The robot speed
with the composite foot (14.1 km/h) was 23.7% faster than that
with the aluminum foot (11.4 km/h), although the leg rotating frequency was same as 3 Hz, as shown in Fig. 8(b). The composite foot
acted as a series spring linkage and increased the stride length,
which was the distance between two successive placements of
the same foot.
4. Conclusion
Fig. 7. Experimental setup and leg configuration: (a) raptor robot with composite
foot structure; (b) schematic diagram of the leg movement.
In this study, a novel composite foot structure was developed
for a dynamic biped raptor robot using carbon/epoxy composite
material due to its high specific strength, material damping capacity and ease of spring constant control. For the stiffness model, the
strain energy method was used. The design optimization process
was performed with the maximum strain failure criterion to solve
analytically, from which an optimum stacking sequence of ½54s
was obtained.
The spring constant and mass of the composite foot were 55
times and 3 times lower than those of the conventional aluminum
foot, respectively. The natural frequencies of the composite feet
were 88% lower than those of the conventional aluminum foot.
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J. Lee et al. / Composite Structures 140 (2016) 344–350
Moreover, the damping ratio of the composite foot was 18 times
higher than that of the aluminum foot. At a leg rotating frequency
of 3 Hz, the impact load through the fabricated composite foot
reduced by 40% compared with the conventional aluminum foot
due to a lower spring constant and a higher damping ratio. Moreover, the robot speed of the composite foot was 23.7% faster than
that of the aluminum foot under the same rotating frequency of
the leg. As a result, the proposed composite foot structure was very
effective and represents a promising development in the realization of a fast dynamic biped robot.
Acknowledgements
This research was supported by the Climate Change Research
Hub of KAIST – South Korea (Grant No. N01150036) and the UTRC
(Unmanned Technology Research Center) at KAIST, originally
funded by DAPA – South Korea, and ADD – South Korea (Grant
No. N04130041). Their support is greatly appreciated.
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