machines
Article
Joint Mechanism That Mimics Elastic Characteristics
in Human Running
Takuya Otani 1, *, Kenji Hashimoto 2,3,† , Takaya Isomichi 1,† , Masanori Sakaguchi 4,5,† ,
Yasuo Kawakami 5,† , Hun-Ok Lim 3,6,† and Atsuo Takanishi 3,7,†
1
2
3
4
5
6
7
*
†
Graduate School of Science and Engineering, Waseda University, No. 41-304, 17 Kikui-cho, Shinjuku-ku,
Tokyo 162-0044, Japan; [email protected]
Waseda Institute for Advanced Study, Waseda University, No. 41-304, 17 Kikui-cho, Shinjuku-ku,
Tokyo 162-0044, Japan; [email protected]
Humanoid Robotics Institute (HRI), Waseda University, 2-2 Wakamatsu-cho, Shinjuku-ku,
Tokyo 162-8480, Japan; [email protected] (H.-O.L.); [email protected] (A.T.)
Faculty of Kinesiology, University of Calgary, 2500 University Drive NW, Calgary, AB T2N 1N4, Canada;
[email protected]
Faculty of Sport Sciences, Waseda University, 2-579-15 Mikajima, Tokorozawa-shi, Tokyo 359-1192, Japan;
[email protected]
Faculty of Engineering, Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama 221-8686, Japan
Department of Modern Mechanical Engineering, Waseda University, 2-2 Wakamatsu-cho,
Shinjuku-ku, Tokyo 162-8480, Japan
Correspondence: [email protected]; Tel.: +81-3-3203-4394
These authors contributed equally to this work.
Academic Editors: Marco Ceccarelli and Hui Li
Received: 31 July 2015; Accepted: 19 January 2016; Published: 25 January 2016
Abstract: Analysis of human running has revealed that the motion of the human leg can be modeled
by a compression spring because the joints of the leg behave like a torsion spring in the stance phase.
In this paper, we describe the development of a joint mechanism that mimics the elastic characteristics
of the joints of the stance leg. The knee was equipped with a mechanism comprising two laminated
leaf springs made of carbon fiber-reinforced plastic for adjusting the joint stiffness and a worm gear
in order to achieve active movement. Using this mechanism, we were able to achieve joint stiffness
mimicking that of a human knee joint that can be adjusted by varying the effective length of one of
the laminated leaf springs. The equation proposed for calculating the joint stiffness considers the
difference between the position of the fixed point of the leaf spring and the position of the rotational
center of the joint. We evaluated the performance of the laminated leaf spring and the effectiveness
of the proposed equation for joint stiffness. We were able to make a bipedal robot run with one leg
using pelvic oscillation for storing energy produced by the resonance related to leg elasticity.
Keywords: humanoid; running; joint stiffness; leaf spring
1. Introduction
We previously researched the development of a bipedal humanoid robot that can mimic various
characteristics of human walking; this robot was used for investigating human mechanisms and human
motion control [1–3]. For improving the performance of this robot and using this robot for not only
walking, but also hopping and running, we have initiated the development of a new bipedal humanoid
robot that can run like a human. In human sciences and sports sciences, researchers usually perform
motion capture experiments to realize human motions. In motion capture experiments, however,
motions that pose a risk of injury to human subjects cannot be studied owing to ethical concerns [4],
despite the possibility of improving those motions through coordinated training. If a robot that
Machines 2016, 4, 5; doi:10.3390/machines4010005
www.mdpi.com/journal/machines
Machines 2016, 4, 5
2 of 15
can mimic human hopping and running were developed, it would become possible to mimic other
human motions, as well. Furthermore, if a robot could perform motions that would improve sports
performance, but would be dangerous if a human being were to perform them, it would be possible to
identify more effective human motions.
Relevant studies in human sciences have identified some characteristics of human running,
such as the following:
‚
‚
‚
‚
‚
‚
A stance leg acts like a linear spring, and a human leg can be modeled as a spring-loaded inverted
pendulum (SLIP) [5–7].
The knee and ankle joints of a stance leg act like torsion springs [8,9].
The knee joint stiffness of the stance leg changes with running speed [8,10].
Rapid knee bending occurs in the swing phase to avoid contact of the foot with the ground [11].
Pelvis rotation in the frontal plane increases jumping force [12].
Moment compensation is accomplished using the upper body and arms [13].
In human running, the joints of the leg require more than 1000 W of power [11,14–16], which
is greater than the power of the actuator found in an ordinary life-sized humanoid robot [1,17–20].
To exert a high output during the stance phase, humans utilize joint stiffness for storing the kinetic
energy that the robot has during the flight phase. Therefore, we focused on the leg stiffness and joint
stiffness of the stance leg as the characteristics that are absolutely necessary for attaining jumping
power. There are some studies on running using humanoid robots, but few robots can mimic this
characteristic. Some small humanoids can run, but the dynamics of these robots are different from
those of humans [21]. Life-sized humanoid robots [17,18], such as ASIMO [19] and Toyota’s humanoid
robot [20], do not have human-like leg elasticity. One athletic robot has a human-like musculoskeletal
system in the leg and elastic parts in the foot, as with an artificial leg, but this robot’s ankle joint does
not mimic human ankle joint stiffness [22]. MABEL [23] also has variable-stiffness joints for running
and succeeded in running with axial constraints on the Y-axis, but it cannot vary its joint stiffness
within a range equivalent to that of human joint stiffness. Elastic joint mechanisms were developed for
walking humanoids, such as COMAN [24–26], so that they can interact with the environment safely or
mimic human mechanisms [27,28]. However, the joint mechanisms cannot output enough torque for a
running life-sized humanoid robot. Some artificial legs have been developed for achieving natural
locomotion; however, they mimic only ankle stiffness, not knee joint stiffness, and the amount of joint
stiffness is not equal to that of a human [29–32].
Our intention in this study was to develop a joint mechanism with variable joint stiffness that
mimics human joint stiffness and that can be utilized for running. In addition, the knee joint should
bend in the swing phase to avoid contact between the foot and the ground. The ankle joint should
produce a torque that is the same as that of the knee; however, the joint stiffness of the ankle does
not change according to the running speed, and the ankle joint does not bend like a knee joint during
the flight phase. Thus, the knee joint mechanism should incorporate joint stiffness and should move
actively during the flight phase. We developed a mechanism comprising two laminated leaf springs
made of carbon fiber-reinforced plastic (CFRP) for adjusting joint stiffness and a worm gear that can
change the angle between two laminated leaf springs with an actuator for achieving active movement.
We evaluated the laminated leaf spring and then performed a hopping experiment to evaluate the
effectiveness of the proposed joint mechanism.
This paper is organized as follows. In Section 2, we describe the design of the joint mechanism
that mimics human joint stiffness. In Section 3, we present and discuss experimental results. Finally,
in Section 4, we present conclusions and propose future work.
Machines 2016, 4, 5
3 of 15
2. Joint Mechanism Mimicking the Joint Stiffness of a Human Leg
2.1. Requirements for a Joint Mechanism Based on Human Running
During the stance phase of human running, the knee and ankle joints alternately lengthen and
shrink like a spring, whereas the hip joint, unlike a spring, only lengthens [7,8,14,15]. The leg stiffness
is a result of knee and ankle joint stiffness, and joint stiffness is important for attaining jumping force.
As mentioned above, leg stiffness and joint stiffness vary during the flight phase according to running
speed [8,10]. We began by determining the requirements for the robotic joint based on human running
data; we considered running speeds between 4 m/s and 10 m/s. The requirements for achieving
the joint stiffness of a running human leg were obtained from the results of our preliminary analysis
and published running data (see Table 1) [12,14,16]. From these data, we conclude that the knee joint
should produce a torque of 177 Nm in the stance phase, bend 2.7 rad in the swing phase and be able
to vary its stiffness as needed within the range of human knee joint stiffness, from 300 Nm/rad at a
running speed of 4 m/s to 600 Nm/rad at a running speed of 10 m/s, during the swing phase, which
has a duration of 0.4 s. Based on knee bending, we calculated a required angular velocity of 6.7 rad/s.
Table 1. Joint stiffness for human running.
Knee
Ankle
Min. (Nm/rad)
Max. (Nm/rad)
300
520
600
570
2.2. Design of a Joint Mechanism Mimicking the Joint Stiffness of a Human Leg
There are several methods of mimicking joint stiffness, including using an actuator to control
the joint like a spring, implementing a spring and using a combination of these methods. In human
running, each joint of the leg requires more than 1000 W [11,14–16], whereas the output of the DC
motors used in the legs of some humanoid robots is much lower: approximately 150 W [1,17–20].
If motors with higher power were used, their size and weight would make it difficult to mimic a
human leg. Although hydraulic actuators can be used to produce the required high output, they
require large, heavy pumps. It is therefore very difficult to realize human running using existing
actuators. Consequently, we considered the possibility of mimicking this characteristic by using elastic
bodies, such as a compression coil spring, a torsion spring or a leaf spring. To mimic the variation of
joint stiffness with running speed, we used a leaf spring, with which we can easily adjust the stiffness
by varying the distance between a supporting point and a load point [33]. Figure 1 is a schematic of
the joint stiffness adjustment mechanism. The load point is fixed on Link A, and the leaf spring is fixed
on Link B via the joint axis. When a force is applied to Link A, the force is transmitted to the leaf spring
through the load point. Link A then rotates in accordance with the deformation of the leaf spring.
In this way, we can adjust the joint stiffness by changing the position of the load point. This stiffness
can be calculated as:
M
Kj “
(1)
∆θ
where K j is the joint stiffness, M is the joint torque and ∆θ is the joint displacement. The formula for
the joint displacement is the following:
∆θ “ tan´1 pδ{Lq
(2)
where δ is the deflection of the leaf spring and L is the effective length of the leaf spring. When the
displacement is small, it can be approximated as follows:
∆θ « δ{L
(3)
Machines 2016, 4, 5
4 of 15
Machines 2016, 4, 5
4 of 15
Effective length L
Link B
Rotation center
Roller
Link A
Leaf spring
(a) No load
Load F
Joint torque M
(b) Loaded
Figure 1. Schematic of the joint stiffness adjustment mechanism.
Figure 1. Schematic of the joint stiffness adjustment mechanism.
The deflection is expressed as follows:
The deflection is expressed as follows:
ML2 4 ML2
δ=
= 3
(4)
3EI2
bt
E2
ML
4ML
δ“
“
(4)
where E is the Young’s modulus of the leaf3EI
spring,btI3 Eis the area moment of inertia of the leaf
t is the
spring,
the
width of
the leafof
spring
andspring,
of the leaf
to theseb
where Ebis is
the
Young’s
modulus
the leaf
I isthickness
the area moment
ofspring.
inertia According
of the leaf spring,
equations,
the
joint
stiffness
can
be
approximated
as
follows:
is the width of the leaf spring and t is the thickness of the leaf spring. According to these equations,
the joint stiffness can be approximated as follows:
3EI bt 3 E
(5)
Kj =
=
L
4 L3
3EI
bt E
Kj “
“
(5)
The joint stiffness adjustment mechanism
allows
for4L
changing the stiffness of the joint. However,
L
when we attempted to mimic low joint stiffness, it was difficult to install the leaf spring in a manner
The joint stiffness adjustment mechanism allows for changing the stiffness of the joint. However,
that is consistent with human physical structure. We need to position the load point far from the
when we attempted to mimic low joint stiffness, it was difficult to install the leaf spring in a manner
supporting point to mimic low joint stiffness, and furthermore, the leaf spring cannot withstand the
that is consistent with human physical structure. We need to position the load point far from the
force while in the stance phase if its thickness is reduced to decrease the stiffness. Thus, this leaf
supporting point to mimic low joint stiffness, and furthermore, the leaf spring cannot withstand the
spring joint stiffness adjustment mechanism cannot achieve the required joint stiffness. To resolve
force while in the stance phase if its thickness is reduced to decrease the stiffness. Thus, this leaf
this problem, we incorporated an additional leaf spring into the mechanism. The two leaf springs
spring joint stiffness adjustment mechanism cannot achieve the required joint stiffness. To resolve
were implemented in series through the active actuator in the joint to realize low joint stiffness. This
this problem, we incorporated an additional leaf spring into the mechanism. The two leaf springs
arrangement makes it possible to adjust the stiffness of the joint over a wide range. Moreover, the
were implemented in series through the active actuator in the joint to realize low joint stiffness.
approximation of the displacement becomes more accurate because the deflection of a leaf spring
This arrangement makes it possible to adjust the stiffness of the joint over a wide range. Moreover,
becomes half of the displacement of the joint mechanism. Therefore, we used the above simplification
the approximation of the displacement becomes more accurate because the deflection of a leaf spring
(Equation (3)) for calculating joint stiffness. The theoretical formula for joint stiffness using two leaf
becomes half of the displacement of the joint mechanism. Therefore, we used the above simplification
springs is the following:
(Equation (3)) for calculating joint stiffness. The theoretical formula for joint stiffness using two leaf
K adjustable K fixed
springs is the following:
K'j = K
(6)
K f ixed
Kadjustable
adjustable + K fixed
K1 j “
(6)
Kadjustable ` K f ixed
where K ' j is the joint stiffness, K adjustable is the stiffness of the leaf spring whose effective length we
where K1 j is the joint stiffness, Kadjustable is the stiffness of the leaf spring whose effective length we
can change and K fixed is the stiffness of the leaf spring whose effective length is fixed. Furthermore,
can change and K f ixed is the stiffness of the leaf spring whose effective length is fixed. Furthermore,
we
we devised
devised aa new
new joint
joint mechanism
mechanism in
in which
which the
the angle
angle between
between two
two leaf
leaf springs
springs can
can be
be changed
changed by
by
an
actuator
in
order
to
achieve
active
movement.
The
two
leaf
springs
transmit
the
joint
torque
an actuator in order to achieve active movement. The two leaf springs transmit the joint torque to
to an
an
upper
link
and
a
lower
link
through
the
load
point,
which
can
be
moved
to
change
the
effective
length
upper link and a lower link through the load point, which can be moved to change the effective length
(see
Figure 2a).
2a).When
Whenthe
theactive
active
joint
moves,
rotates
(see Figure
2b). When
the external
(see Figure
joint
moves,
thethe
jointjoint
rotates
(see Figure
2b). When
the external
torque
torque
is applied,
the leaf bend,
springs
andangle
the joint
angle also
(seeIfFigure
2c). If this
is applied,
the leaf springs
andbend,
the joint
also changes
(seechanges
Figure 2c).
this mechanism
is
mechanism
is
to
act
like
a
torsion
spring,
the
angle
between
the
two
leaf
springs
should
be
fixed,
and
to act like a torsion spring, the angle between the two leaf springs should be fixed, and only the leaf
only
theshould
leaf springs
bend
to produce
the joint
torqueis while
the To
robot
is standing.
springs
bend toshould
produce
the joint
torque while
the robot
standing.
accomplish
this, To
we
accomplish
this,
wetoused
a worm
gear from
to which
the torque
an input
shaft
to an outputnot
shaft
used a worm
gear
which
the torque
an input
shaft tofrom
an output
shaft
is transmitted;
all is
of
transmitted;
not
all
of
the
torque
from
the
output
shaft
to
the
input
shaft
is
transmitted
to
the
worm
the torque from the output shaft to the input shaft is transmitted to the worm gear. The transmission
gear.
The transmission
changed
according
to the
leadgear.
angle
of the worm
gear. The
efficiency
was changedefficiency
accordingwas
to the
lead angle
γ of the
worm
Theγ theoretical
formulas
for
theoretical formulas for transmission efficiency from the input shaft to the output shaft ( η1 ) and from
the output shaft to the input shaft ( η2 ) are as follows [34]:
Machines 2016, 4, 5
5 of 15
Machines 2016, 4, 5
5 of 15
transmission
from the input shaft to the output shaft (η1 ) and from the output shaft to
the
Machines 2016, 4,efficiency
5
5 of
15
input shaft (η2 ) are as follows [34]:
tan γ
η1 = tanγ
(7)
tan γ
(7)
η1η1“
= tan(γ + ρ )
(7)
tanpγ
`
ρq
tan(γ + ρ )
tan(γ´- ρq
ρ)
tanpγ
η2 η“2 =tan(γ - ρ )
(8)
tan γ
η 2 = tanγ
(8)
tan γ
where
with
a value
of of
0.14,
determined
byby
the
material
a parameter
with
a value
0.14,
determined
the
materialofofwhich
whichthe
theworm
wormgear
gearisis
whereρ is
ρ aisparameter
is
a
parameter
with
a
value
of
0.14,
determined
by
the
material
of
which
the
worm
gear
is
where
ρ
made
angular velocity
velocityduring
duringrunning.
running.These
These
formulas
plotted
in Figure
3. lead
The angle
lead
madeand
and the
the angular
formulas
areare
plotted
in Figure
3. The
˝
made was
and determined
the angular velocity
during
running.
These
formulas
are plotted in Figure
The lead angle
angle
be 8.73
considering
feasibility
manufacturing
andthe
the3.back-drivability.
back-drivability.
was determined
to beto8.73°
considering
thethe
feasibility
ofofmanufacturing
and
was
determined
to
be
8.73°
considering
the
feasibility
of
manufacturing
and
the
back-drivability.
Using
wewe
designed
the worm
gear gear
so that
would
be possible
to fix thetoangle
between
Usingthese
theseformulas,
formulas,
designed
the worm
soitthat
it would
be possible
fix the
angle
Using
these
formulas,
weactively
designed
the worm
gear so
thatcan
it would
be
possible
tosmall
fix the
angle
the
leaf
springs
and
move
by
using
the
motor,
which
output
150
W
and
is
and
between the leaf springs and move actively by using the motor, which can output 150 W and is light
small
betweento
the leaf
springs andinmove
actively
by using
the motor, efficiency
which canfrom
output
W and
is small
enough
implemented
the leg.
The
torque
the150
input
shaft
the
and light be
enough
to be implemented
in the
leg. transmission
The torque transmission
efficiency
from
thetoinput
and
light
enough
to
be
implemented
in
the
leg.
The
torque
transmission
efficiency
from
the
input
output
the developed
mechanism
50%, and that
fromand
the output
shaft
the input
shaft
is
shaft toshaft
the of
output
shaft of the
developedismechanism
is 50%,
that from
thetooutput
shaft
to the
shaft to
theknee
output
shaft of the
developed
mechanism
is 50%,
and that
from
thesprings
outputinshaft
to the
2.6%.
This
mechanism
(see
Figure
4)
can
fix
the
angle
between
the
two
leaf
the
stance
input shaft is 2.6%. This knee mechanism (see Figure 4) can fix the angle between the two leaf springs
input shaft
is 2.6%. control
This knee
(see Figure 4) can fix the angle between the two leaf springs
phase
actively
themechanism
joint
angle.the
in theand
stance
phase and actively
control
joint angle.
in the stance phase and actively control the joint angle.
Thigh
Thigh
Active axis
Active axis
Leaf spring
Leaf spring
Passive axis
Passive axis
Z
Z
Motor
Motor
Load point
Load point
Shank
Shank
X
X
(b) Rotated by the
(b) Rotated by the
actuator
actuator
(a) Stretched
(a) Stretched
(c) Bent by the external
(c) Bent by the external
torque
torque
Figure 2. Schematic of the knee mechanism comprising two leaf springs.
Figure 2.
Schematic of
of the
the knee
knee mechanism
mechanism comprising
Figure
2. Schematic
comprising two
two leaf
leaf springs.
springs.
Efficiency
Efficiency%%
60
60
η
η1
η11
η1
η
η22
η
η22
40
40
20
20
0
0
0
0
General Developed
General5 Developed
10
5
10
Lead angle deg
Lead angle deg
15
15
Figure 3. Influence of lead angle on torque transmission efficiency.
Figure
Figure 3.
3. Influence
Influence of
of lead
lead angle
angle on
on torque
torque transmission
transmission efficiency.
efficiency.
To vary the joint stiffness within the range of a human leg joint, the load point must move 130 mm
To vary the joint stiffness within the range of a human leg joint, the load point must move 130 mm
in 0.4
In addition,
mechanism
withstand
a load
10,000
in the
direction
perpendicular
Tos.vary
the jointthe
stiffness
withinshould
the range
of a human
legofjoint,
theNload
point
must move
130 mm
in 0.4 s. In addition, the mechanism should withstand a load of 10,000 N in the direction perpendicular
to
the
leaf
spring
and
a
load
of
750
N
in
the
direction
parallel
to
the
leaf
spring.
There
are
several
in 0.4 s. In addition, the mechanism should withstand a load of 10,000 N in the direction perpendicular
to the leaf spring and a load of 750 N in the direction parallel to the leaf spring. There are several
ways
of moving
the load
point,
such
using
a ball parallel
screw ortoathe
rack-and-pinion
system.
When a
to
the leaf
spring and
a load
of 750
N inasthe
direction
leaf spring. There
are several
ways of moving the load point, such as using a ball screw or a rack-and-pinion system. When a
rack-and-pinion
system
is
used,
the
actuator
needs
more
power
to
move
itself
because
it
moves
with
ways of moving the load point, such as using a ball screw or a rack-and-pinion system. When
a
rack-and-pinion system is used, the actuator needs more power to move itself because it moves with
the load point. Therefore, we decided to use a ball screw. When a ball screw is used, the large load in
the load point. Therefore, we decided to use a ball screw. When a ball screw is used, the large load in
the direction parallel to the leaf spring produces a large moment that acts on the actuator. To make it
the direction parallel to the leaf spring produces a large moment that acts on the actuator. To make it
possible for the actuator to withstand this large load and moment, we implemented an electrical
possible for the actuator to withstand this large load and moment, we implemented an electrical
Machines 2016, 4, 5
6 of 15
rack-and-pinion system is used, the actuator needs more power to move itself because it moves with
the load point. Therefore, we decided to use a ball screw. When a ball screw is used, the large load in
Machines
2016, 4,
4,parallel
of 15
15
the
direction
to the leaf spring produces a large moment that acts on the actuator. To make
it
Machines
2016,
55
66 of
possible for the actuator to withstand this large load and moment, we implemented an electrical break
break
to control
control
the moment
moment
and aa linear
linear for
guide
for
theinload
load
in the
the direction
direction
perpendicular
to the
the
leaf
to
control
the moment
and a linear
guide
thefor
load
the direction
perpendicular
to the leaf
spring
break
to
the
and
guide
the
in
perpendicular
to
leaf
spring
(see
Figure
5).
Thanks
to
this
mechanism,
the
load
point
can
be
adjusted
during
the
flight
(see Figure
Thanks
to this mechanism,
the load the
point
canpoint
be adjusted
theduring
flight phase
like
spring
(see 5).
Figure
5). Thanks
to this mechanism,
load
can be during
adjusted
the flight
like aa human.
human.
aphase
human.
phase
like
Figure 4.
4. CAD model
model for the
the knee mechanism
mechanism comprising two
two leaf springs
springs and aa worm
worm gear.
gear.
Figure
Figure 4. CAD
CAD model for
for the knee
knee mechanism comprising
comprising two leaf
leaf springs and
and
Figure 5.
CAD model
model for
for the
the joint
joint stiffness
stiffness adjustment
5. CAD
adjustment mechanism.
mechanism.
Figure
2.3. Joint
Joint
Stiffness
Equation
Considering the
the Fixed
Fixed Point
Point
Joint Stiffness
Stiffness Equation
Equation Considering
2.3.
The position
position
of
the
fixed
point
ofof
the
leaf
spring
is different
different
from
the rotational
rotational
center
of the
theofjoint
joint
The
positionof
ofthe
thefixed
fixedpoint
point
the
leaf
spring
is different
from
the rotational
center
the
The
of
the
leaf
spring
is
from
the
center
of
in
the
developed
joint
mechanism
(see
Figure
6).
Because
the
difference
between
the
rotational
center
joint
the developed
joint mechanism
(see Figure
6). Because
the difference
between
the rotational
in
thein
developed
joint mechanism
(see Figure
6). Because
the difference
between
the rotational
center
and the
the
fixed
point point
of the
theofleaf
leaf
spring
influences
thethe
moment
of of
the
leaf
spring,
we
modified
the
center
and
the point
fixed
the spring
leaf
spring
influences
moment
the
leafspring,
spring,we
wemodified
modified the
the
and
fixed
of
influences
the
moment
of
the
leaf
equation
for
the
stiffness
of
the
mechanism.
equation for the stiffness of the mechanism.
When the
the position
position of
of
the fixed
fixed point
point and
and that
that of
of
the rotational
rotational
center are
are different,
different, the
the positions
positions
position
of the
the
and
that
of the
the
rotational center
center
When
can
be
expressed
in
terms
of
their
two-dimensional
coordinates.
The
coordinates
of
the
rotational
can
be
expressed
in
terms
of
their
two-dimensional
coordinates.
The
coordinates
of
the
rotational
can
two-dimensional
The coordinates of
center are
are set
set as
as the
theorigin
originof
ofthe
thecoordinate
coordinatesystem.
system.The
Thecoordinates
coordinatesofof
ofthe
the
fixed
point
are
fixed
point
are
(a,((h),
center
as
the
origin
of
the
coordinate
system.
The
coordinates
the
fixed
point
are
hh ),),
aa ,, and
M is
and those
those
of load
the load
load
point
are`(( aal,++h).
When
moment
is applied
applied
to
the rotational
rotational
center,
force
those
of the
point
are are
(a
moment
M is M
applied
to theto
rotational
center,
force force
Fls is
and
of
the
point
ll ,, When
hh ).). When
moment
the
center,
F
is
applied
to
the
load
point
of
the
leaf
spring.
This
force
is
expressed
as
follows:
applied
to the to
load
the of
leafthe
spring.
This force
expressed
as follows:
Fls is applied
thepoint
load of
point
leaf spring.
Thisisforce
is expressed
as follows:
ls
M ls
Moment M
Moment
ls
M
M
F == M
FF
lslsls“
L
aaa`
++ L
applied
to
the
load
spring
is
expressed
as follows:
follows:
applied to the load spring is expressed as
M ls == LF
LFls
M
ls
ls
(9)
(9)
(9)
(10)
(10)
Based on
on these
these equations,
equations, this
this moment
moment bending
bending the
the leaf
leaf spring
spring is
is calculated
calculated as
as follows:
follows:
Based
L
M ls == L M
M
M
ls
aa ++ LL
(11)
(11)
Machines 2016, 4, 5
7 of 15
Moment Mls applied to the load spring is expressed as follows:
Mls “ LFls
(10)
Based on these equations, this moment bending the leaf spring is calculated as follows:
Mls “
Machines 2016, 4, 5
L
M
a`L
(11)
7 of 15
According to Equation (11), when the positions of the rotational center and the fixed point of the
According
to Equation
whenzero,
the positions
of the
rotational
center
and theisfixed
pointas
ofthat
the
leaf spring are the
same, i.e.,(11),
a equals
the moment
acting
on the
leaf spring
the same
leaf
spring
are
the
same,
i.e.,
equals
zero,
the
moment
acting
on
the
leaf
spring
is
the
same
as
that
a
acting on the joint. However, when a > 0, the moment acting on the leaf spring is smaller than that
acting
on the
the joint.
joint. This
However,
a >deflection
0, the moment
on the
is smaller
acting on
meanswhen
that the
of theacting
leaf spring
δlsleaf
alsospring
becomes
smaller:than that
acting on the joint. This means that the deflection of the leaf spring δ ls also becomes smaller:
4M L2
4ML3
δls “ 4M3lsls L2 “ 4ML3 3
(12)
δ ls = bt 3E = pa ` Lqbt3 E
(12)
bt E
(a + L)bt E
For calculating the joint stiffness with Equations (1) and (3), the deflection δ perpendicular to the
For calculating the joint stiffness with Equations (1) and (3), the deflection δ perpendicular to
line that passes through the rotational center and the load point is expressed as follows:
the line that passes through the rotational center and the load point is expressed as follows:
a`L
δ “ δls cosθload “ b a + L
δ
(13)
δ = δ ls cos θload =
δ lsls
2
(13)
2 ` 2b2
pa
`
Lq
( a + L) + b
is the angle
anglebetween
betweenthe
theX direction
X direction
that passes
through
the rotational
where θθload
andand
the the
line line
that passes
through
the rotational
center
load is
center
and
the
load
point.
According
to
the
modified
deflection
given
in
Equation
(13),
the
modified
and the load point. According to the modified deflection given in Equation (13), the modified
theoretical formula
formula for
theoretical
for the
the joint
joint stiffness
stiffness is
is the
the following:
following:
b
bt33EE
' bt
K jK1 “
pa
=
(a `
+ LLq
) 22+`bb2 2
j
4L
4 L22
(14)
(14)
1 ' is the
where KK
modified
stiffness.
This indicates
that a difference
greater difference
between
the
modified
jointjoint
stiffness.
This indicates
that a greater
between the
rotational
j jis the
center andcenter
the fixed
point
of point
the leaf
spring
increased
joint stiffness.
We took
this into
rotational
and the
fixed
of the
leaf leads
springtoleads
to increased
joint stiffness.
We took
this
consideration
in
the
design
of
the
leaf
spring.
into consideration in the design of the leaf spring.
Y
Fixed point
( a, b)
Fls
M ls
M
θ load
Load point
( a + L, b )
X
O Rotational center
(0,0 )
Figure 6. Influence of the difference between the rotational center and the fixed point of the leaf spring.
Figure 6. Influence of the difference between the rotational center and the fixed point of the leaf spring.
2.4. Laminated Leaf Spring Made of Carbon Fiber-Reinforced Plastic
2.4. Laminated Leaf Spring Made of Carbon Fiber-Reinforced Plastic
In order to incorporate the leaf springs into the developed joint mechanism, the leaf springs must
In order to incorporate the leaf springs into the developed joint mechanism, the leaf springs must
be able to withstand a large load while the robot is running. One option is to make the leaf spring out
be able to withstand a large load while the robot is running. One option is to make the leaf spring out
of iron. Such a leaf spring could withstand a large load, but it would be very heavy. If an iron leaf
spring were implemented, the joint mechanism would not be able to mimic the mass of a human leg.
To resolve this problem, we used a leaf spring made of CFRP, which is extremely strong, yet also
light. The specific strength of CFRP (2457 kNm/kg) is much higher than that of iron (254 kNm/kg),
and the density of CFRP (1.5 g/cm3) is much lower than that of iron (7.8 g/cm3). On account of these
characteristics, CFRP is used in some prosthetic legs [29]. However, when the CFRP leaf spring was
made small enough to be installed into a robotic leg equivalent in size to a human leg, the stress on
Machines 2016, 4, 5
8 of 15
of iron. Such a leaf spring could withstand a large load, but it would be very heavy. If an iron leaf
spring were implemented, the joint mechanism would not be able to mimic the mass of a human leg.
To resolve this problem, we used a leaf spring made of CFRP, which is extremely strong, yet also
light. The specific strength of CFRP (2457 kNm/kg) is much higher than that of iron (254 kNm/kg),
and the density of CFRP (1.5 g/cm3 ) is much lower than that of iron (7.8 g/cm3 ). On account of these
characteristics, CFRP is used in some prosthetic legs [29]. However, when the CFRP leaf spring was
made small enough to be installed into a robotic leg equivalent in size to a human leg, the stress on the
leaf spring exceeded its strength. To improve the strength, the thickness or width of the leaf spring
should be increased. However, when the width is increased, it becomes difficult to incorporate the
leaf spring into the leg. On the other hand, when the thickness is increased, the deflection of the leaf
spring becomes smaller and the joint stiffness higher than that of a human leg. To resolve this problem,
we stacked two leaf springs, one upon another. The maximum stress σ is expressed as follows:
σ“
6M
bt2
(15)
In contrast, the deflection of the laminated leaf spring and the maximum stress on one leaf spring
in the laminated leaf spring are expressed as follows:
δlaminated “
σlaminated “
6ML2
nbt13 E
(16)
6M
nbt12
(17)
where t1 is the thickness of each leaf spring and n is the number of leaf springs. Thus, t1 is expressed as
follows:
t
t1 “
(18)
n
Considering Equation (18), Equations (16) and (17) can be expressed as follows:
δlaminated “
6n2 ML2
bt3 E
(19)
6nM
bt2
(20)
σlaminated “
Based on Equations (19) and (20), when the number of leaf springs increases, the deflection and
the stress also increase. However, the number of leaf springs influences the deflection more than the
stress. Thus, we can adjust the total thickness to modify the strength and the number of leaf springs to
modify the joint stiffness. To increase the strength of the leaf spring and decrease the joint stiffness,
we used these formulas to design a laminated leaf spring made of two CFRP leaf springs. Compared
to an iron leaf spring whose joint stiffness is the same as that of the laminated CFRP leaf spring, the
laminated leaf spring is thicker and almost the same in length and width, and the mass of the laminated
leaf spring (200 g) is much lower than that of the iron leaf spring (600 g) (see Table 2). Furthermore, the
mass of the joint mechanism using the CFRP laminated leaf springs is 3000 g, compared to 3800 g for
the joint with iron leaf springs. Thus, the use of CFRP laminated leaf springs reduces the mass of the
joint by approximately 21%.
Table 2. Leaf spring characteristics.
Material
Iron
CFRP
Size mm
Mass g
250 ˆ 90 ˆ 3.4
600
220 ˆ 70 ˆ 8.8
200
Machines 2016, 4, 5
9 of 15
2.5. Implementation of the Joint Mechanism
We designed a robotic leg that incorporates the developed joint mechanism. The developed leg
has a knee mechanism comprising two leaf springs, the worm gear and the joint stiffness adjustment
mechanism, as well as an ankle comprising two leaf springs. The joint stiffness of the ankle does not
vary as widely as that of the knee joint (see Table 1), and the ankle’s range of motion during the swing
phase, approximately 30˝ , is more restricted than that of the knee, approximately 60˝ [7]. Therefore,
in order to keep the mass of the ankle more consistent with the mass of a human ankle, we did not
implement the worm gear and the joint stiffness adjustment mechanism in the ankle joint. In the foot,
we implemented a rubber hemisphere at the end of each toe for point grounding.
In addition, the developed leg was implemented with a pelvis mechanism (see Figure 7a). We used
150-W DC motors, timing belts and harmonic drives to actuate the pelvis roll joint and hip joints.
To perform human-like motions, the robot, which weighs 60 kg, must be approximately the same size
as a human [35]. The weight of the robot’s upper body is designed such that the location of the center
of mass and the moment of inertia about the center of mass are consistent with those of the human
Machines 2016, 4, 5
9 of 15
body [36]. The mass of the robot is close to that of a human, and the height is similar to that of a
as a Moreover,
human [35]. The
theeach
robot’s
upper
is designed
such thatto
thethe
location
thethe
center
human’s chest.
theweight
massofof
part
ofbody
the robot
is similar
massofof
corresponding
of mass and
the moment
of inertia
about the
center ofthe
massvariable
are consistent
with those
of the human
part of the human
body.
This was
possible
because
stiffness
actuator
mechanism was
body [36]. The mass of the robot is close to that of a human, and the height is similar to that of a human’s
made lighterchest.
by Moreover,
using the
gearpart
and
therobot
CFRP-laminated
springs.
Table
theworm
mass of each
of the
is similar to the massleaf
of the
corresponding
part3ofdescribes the
body. This
possible
the variablecan
stiffness
actuator
mechanism
madeway a human
configurationtheofhuman
the robot.
Thiswas
robot
has because
nine actuators,
move
its pelvis
in thewas
same
lighter by using the worm gear and the CFRP-laminated leaf springs. Table 3 describes the
does and can jump because it can store energy via leg elasticity and use it efficiently via resonance
configuration of the robot. This robot has nine actuators, can move its pelvis in the same way a human
based on pelvic
was
restricted
to the and
vertical
and horizontal
directions using
doesoscillation.
and can jump Robot
because motion
it can store
energy
via leg elasticity
use it efficiently
via resonance
on pelvic
RobotThe
motion
was restricted
to the
vertical and
horizontal
using
a developedbased
guide
(see oscillation.
Figure 7b).
guide
has two
passive
joints,
anddirections
it was connected
to the
a developed guide (see Figure 7b). The guide has two passive joints, and it was connected to the
robot’s body to ensure that the robot moves around the guide.
robot’s body to ensure that the robot moves around the guide.
Y
R
R
P R
P R
P
P
P
Z
Y
X
P
:active
:passive
(a) Photograph of the robot
(b) Degrees of freedom configuration
Figure 7. Developed robot used in the experiments.
Figure 7. Developed robot used in the experiments.
Table 3. Robot configuration.
Table 3. Robot configuration.
Human [29]
Distance between hip joints (mm)
Thigh length (mm)
Shank length (mm)
Distance between
joints
(mm)
Foothip
length
(mm)
Thigh
length
(mm)
Height of center of the mass (mm)
Shank
length (mm)
Moment of inertia (kg·m2)
Foot length (mm)
Height of center of the mass (mm)
3. Experiments and Discussion
Moment of inertia (kg¨ m2 )
180
374
Human [29]
339
180
170
374
786
339
6.3
170
786
6.3
Robot
180
377Robot
339
170 180
691 377
339
5.8
170
691
5.8
3.1. Verification of the CFRP-Laminated Leaf Spring
verify
the stiffness of the CFRP-laminated leaf spring, we subjected it to a load test. We
3. ExperimentsTo
and
Discussion
developed a test fixture for the leaf spring and applied a load in the vertical direction using a load
testing
machine
(see Figure 8). The
leaf Spring
spring is attached to the lower part of the test fixture, and the
3.1. Verification
of the
CFRP-Laminated
Leaf
upper part of the test fixture can move freely in the vertical direction. When the load testing machine
applies
the leafof
spring
loaded through the upper
of the testwe
fixture.
By changing
To verify
thea load,
stiffness
theis CFRP-laminated
leafpartspring,
subjected
it the
to a load test.
horizontal position of the lower part of the test fixture, we can change the effective length of the leaf
We developed
a
test
fixture
for
the
leaf
spring
and
applied
a
load
in
the
vertical
direction
using
spring. We measured the applied load and the deflection of the leaf spring. To evaluate the loading
a load testing
machine
Figurewe8).
The leaf
is177
attached
to the
oftothe
capacity
of the(see
leaf spring,
applied
loads spring
as high as
Nm, which
is thelower
torque part
applied
a test fixture,
human knee joint during running.
Figure 9 presents experimental results and theoretical values for the deflection of the laminated
leaf spring, a single leaf spring from the laminated leaf spring and a single leaf spring that is as thick
as the laminated leaf spring. The theoretical values are not included if the leaf spring was not able to
withstand the load. As the results show, the laminated leaf spring was able to withstand 177 Nm, and
the mean value of the measured stiffness of the laminated leaf spring 650 Nm/rad was close to the
Machines 2016, 4, 5
10 of 15
and the upper part of the test fixture can move freely in the vertical direction. When the load testing
machine applies a load, the leaf spring is loaded through the upper part of the test fixture. By changing
the horizontal position of the lower part of the test fixture, we can change the effective length of the
leaf spring. We measured the applied load and the deflection of the leaf spring. To evaluate the loading
capacity of the leaf spring, we applied loads as high as 177 Nm, which is the torque applied to a human
knee joint during running.
Figure 9 presents experimental results and theoretical values for the deflection of the laminated
leaf spring, a single leaf spring from the laminated leaf spring and a single leaf spring that is as thick
as the laminated leaf spring. The theoretical values are not included if the leaf spring was not able to
withstand the load. As the results show, the laminated leaf spring was able to withstand 177 Nm, and
the mean value of the measured stiffness of the laminated leaf spring 650 Nm/rad was close to the
theoretical value 610 Nm/rad. It is assumed that the difference between the measured value and the
Machines 2016, 4, 5
10 of 15
theoretical value is caused by approximating the joint displacement.
The stiffness
the 610
laminated
springthat
was
than
that of
singlevalue
leaf and
spring
theoreticalofvalue
Nm/rad. Itleaf
is assumed
thelower
difference
between
thethe
measured
the with the
theoretical
value
is caused of
by the
approximating
joint
displacement.
thickness equal
to the
thickness
laminatedtheleaf
spring.
With the developed joint mechanism, we
The
stiffness
with the
Machines
2016, 4, 5of the laminated leaf spring was lower than that of the single leaf spring
10 of 15
can increase the
joint
stiffness by decreasing the effective length of the leaf spring or
decrease
the joint
thickness equal to the thickness of the laminated leaf spring. With the developed joint mechanism,
stiffness byweincreasing
the
effective
length.
However,
we
cannot
implement
a
leaf
spring
that
is
longer
theoretical
value
610
Nm/rad.
It
is
assumed
that
the
difference
between
the
measured
value
and
the
can increase the joint stiffness by decreasing the effective length of the leaf spring or decrease the
theoretical
value
is caused
by approximating
the
joint stiffness
displacement.
than a human
thigh
or
shank.
Thus,
the
lower
joint
offered
by
the
laminated
leaf
spring
is
joint stiffness by increasing the effective length. However, we cannot implement a leaf spring that is
The stiffness of the laminated leaf spring was lower than that of the single leaf spring with the
longer
than
a
human
thigh
or
shank.
Thus,
the
lower
joint
stiffness
offered
by
the
laminated
leaf
beneficial for the
developed
joint
mechanism.
Theleaf
above
confirms
that the laminated leaf
thickness
equal to the
thickness
of the laminated
spring.discussion
With the developed
joint mechanism,
springweiscan
beneficial
forjoint
thestiffness
joint mechanism.
The above
discussion
that the
increase
by decreasing
the effective
length
the leaf
spring
or confirms
decrease the
spring is advantageous
in the
terms
ofdeveloped
loading
capacity
and size
ofofthe
joint
mechanism.
laminated
leaf spring
is advantageous
in length.
terms of
loadingwe
capacity
and size of
thespring
joint mechanism.
joint stiffness
by increasing
the effective
However,
cannot implement
a leaf
that is
longer than a human thigh or shank. Thus, the lower joint stiffness offered by the laminated leaf
spring is beneficial for the developed joint mechanism. The above discussion confirms that the
laminated leaf spring is advantageous in terms of loading capacity and size of the joint mechanism.
Figure 8. Test fixture used to measure the deflection of the leaf springs under applied vertical loads.
Figure 8. Test fixture used to measure the deflection of the leaf springs under applied vertical loads.
Figure 8. Test fixture used to measure the deflection of the leaf springs under applied vertical loads.
(C)
(B) (A)
(D)
180
160
(B) (A)
(D)
180
(C)
160
140 140
Moment Nm
Moment Nm
120 120
100 100
80
60
80
60
(A) Theoretical
value
layers
Theoritical
value
twotwo
layers
(A) Theoretical
value
layers
Theoritical
value
twotwo
layers
Measured
value
twotwo
layers
(B) Measured
value
layers
Measured
value
two
layers
(B)
Measured
value
two
layers
(C) Theoretical
value
layer (thickness 4.4 mm)
Theoritical
value
oneone
layer(thickness4.4mm)
40
40
20
20
(C)Theoretical
Theoretical
value
one
(thickness
4.4 mm)
Theoritical
value
one
layer(thickness4.4mm)
(D)
value
one
layerlayer
(thickness
8.8 mm)
Theoritical
value
one
layer(thickness8.8mm)
0
0
0
0
0.05
0.05
0.1
0.1
(D) Theoretical
value
layer (thickness 8.8 mm)
Theoritical
value
oneone
layer(thickness8.8mm)
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Angle displacement rad
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Angle of
displacement
rad
Figure 9. Theoretical and measured deflection
leaf springs in
to the magnitude of the
Theoretical
and measured deflection of leaf
springs in relation
relation
to the magnitude
applied vertical load.
Figure 9.
of the applied
vertical load.
Figure 9. Theoretical and measured deflection of leaf springs in relation to the magnitude of the
3.2. Verification
Joint Stiffness
applied
verticalofload.
We conducted an experiment to evaluate the effectiveness of the joint mechanism comprising
two laminated
leaf springs
3.2. Verification
of Joint
Stiffnessin mimicking human knee joint stiffness. In this experiment, we made the
ankle joint passive in order to exclude any influence from the ankle. The robot was lifted then lowered
We
conducted
ana vertical
experiment
to evaluate
the effectiveness
of of
the
mechanism
comprising
to apply
to the leg
downward
force proportional
to the mass
thejoint
robot.
To determine
the
two laminated leaf springs in mimicking human knee joint stiffness. In this experiment, we made the
ankle joint passive in order to exclude any influence from the ankle. The robot was lifted then lowered
to apply to the leg a vertical downward force proportional to the mass of the robot. To determine the
Machines 2016, 4, 5
11 of 15
3.2. Verification of Joint Stiffness
We conducted an experiment to evaluate the effectiveness of the joint mechanism comprising two
laminated leaf springs in mimicking human knee joint stiffness. In this experiment, we made the ankle
joint passive in order to exclude any influence from the ankle. The robot was lifted then lowered to
apply to the leg a vertical downward force proportional to the mass of the robot. To determine the knee
joint stiffness, we measured the knee joint angle using an encoder attached to the knee joint, and we
measured the downward force using a force sensor located on the floor. The joint torque was calculated
from the knee joint angle displacement, thigh length and downward force. In this experiment, the
robotic motion was restricted to the vertical direction using linear guides.
The experimental results are summarized in Table 4. As the results show, the range of the joint
stiffness of the developed joint is wider than that of a human knee joint. In addition, the theoretical
value was almost the same as the measured value by taking into account the difference between
the positions of the fixed point of the leaf spring and the rotational center of the joint mentioned in
Section 2.3.
Table 4. Knee joint stiffness evaluation.
Min. (Nm/rad)
Max. (Nm/rad)
11 of 15
Requirement for knee joint
300
600
Theoretical value
230
690
knee joint stiffness, we measured the knee joint angle using an encoder attached to the knee joint, and
Measured value
230
680
we measured the downward force using a force sensor located on the floor. The joint torque was
Machines 2016, 4, 5
calculated from the knee joint angle displacement, thigh length and downward force. In this
experiment, the robotic motion was restricted to the vertical direction using linear guides.
The experimental results are summarized in Table 4. As the results show, the range of the joint
stiffness of the developed joint is wider than that of a human knee joint. In addition, the theoretical
value was almost the same as the measured value by taking into account the difference between the
positions of the fixed point of the leaf spring and the rotational center of the joint mentioned in Section 2.3.
3.3. Hopping Experiment
To confirm that the developed robot can use its joint elasticity for running, we performed a
hopping experiment and measured the mass displacement and the flight time, in order to verify that
Table 4. Knee joint stiffness evaluation.
the developed joint can be used to attain jumping Min.
power.
We previously developed a running control
(Nm/rad) Max. (Nm/rad)
Requirement
knee jointmovement
300
600 elasticity [37], and we used that
method using the resonance related
toforpelvic
and leg
Theoretical value
230
690
method in this experiment (FigureMeasured
10). value
The robot moved
its pelvis
in the stance phase to attain a
230
680
jumping force with 3.3.
a jumping
height
controller.
Moreover,
the
robot
changed leg joint angles for
Hopping Experiment
stabilization with a ground
reaction
estimation
and
the we
running
with a running
To confirm
that the force
developed
robot can use its
joint controlled
elasticity for running,
performedspeed
a
hopping experiment and measured the mass displacement and the flight time, in order to verify that
speed controller in the
flight
phase.
The
experimental
conditions
are
listed
in
Table
5.
The amplitude
the developed joint can be used to attain jumping power. We previously developed a running control
method
usingthe
the resonance
related to pelvic
movement
and leg elasticity
we used that
of the pelvic oscillation
and
joint stiffness
were
selected
based[37],
onandhuman
running data. In this
method in this experiment (Figure 10). The robot moved its pelvis in the stance phase to attain a
experiment, the robotjumping
initially
stood,
then
started
toMoreover,
move the
itsrobot
pelvis
according
toforthe pelvic oscillation
force with
a jumping
height
controller.
changed
leg joint angles
stabilization with a ground reaction force estimation and controlled the running speed with a running
control method. When
the
robot
was
able
to
jump,
it
moved
its
pelvis
to
the
landing
angle by the next
speed controller in the flight phase. The experimental conditions are listed in Table 5. The amplitude
of
the
pelvic
oscillation
and
the
joint
stiffness
were
selected
based
on
human
running
data.
In
this
landing and moved its hip pitch joint according to the running speed control method, with a reference
experiment, the robot initially stood, then started to move its pelvis according to the pelvic oscillation
running speed of 0.2 control
m/s.method.
In the
running
speed
control,
used
the
value
the
When
the robot was
able to jump,
it movedwe
its pelvis
to the
landing
angle of
by the
nextleg length calculated
and moved its hip pitch joint according to the running speed control method, with a reference
according to the linklanding
length
and
the
joint
angles
of
the
leg
when
the
robot
landed.
The gain for the
running speed of 0.2 m/s. In the running speed control, we used the value of the leg length calculated
according
to
the
link
length
and
the
joint
angles
of
the
leg
when
the
robot
landed.
The
gain
for
the
running speed control was determined experimentally.
running speed control was determined experimentally.
Landing
θ ankle Detector
・Stance
Phase
Jumping Height
Controller
θ pelvis− roll
R
P
・Flight
P
R
P
P
Z
P
Initial joint angle
Ground
x foot
θ ini
Reaction
Running
Joint
Inverse
Force
Speed Control
Kinematics Controller
Estimator x
θ
(Flight phase)
Joint angle
Velocity
Calculator ω yaw
P
R
Y
X
Joints
Sensor
IMU
Ankle
encoder
Figure 10. Block diagram of the control system in the hopping experiment.
Figure 10. Block diagram of the control system in the hopping experiment.
Machines 2016, 4, 5
12 of 15
Table 5. Experimental parameters.
Parameter
Whole mass (kg)
Running speed reference (m/s)
Pelvic oscillation amplitude (˝ )
Knee joint stiffness (Nm/rad)
Ankle joint stiffness (Nm/rad)
Natural period (s)
Hip angle of stance leg at landing (˝ )
Knee angle of stance leg at landing (˝ )
Ankle angle of stance leg at landing (˝ )
Hip angle of swing leg at landing (˝ )
Knee angle of swing leg at landing (˝ )
Ankle angle of swing leg at landing (˝ )
60
0.2
5
220
520
0.3
10
40
90
18
75
90
Figure 11 presents photographs of the running experiment, and Figure 12 depicts the vertical
displacement of the center of mass of the robot. In Figure 12, the orange area means that the robot
took off the ground. The robot started its pelvic oscillation and started to hop and run after a few
oscillations. This indicates that the joint mechanism can withstand the large torque that occurs during
the stance phase of running. During the flight phase, the robot used running speed control. Based
on experimental results, the time of the stance phase was approximately 270 ms, and the time of
the flight phase was approximately 120 ms. The running robot can bend and stretch its knee within
100 ms during the flight phase to achieve alternate bipedal running. The time of the stance phase in
human running is approximately 260 ms, and that of the flight phase is approximately 100 ms [7].
The robot’s gait timing was similar to human gait timing. Moreover, the maximum power that the
joint mechanism output in the hopping experiment was approximately 1000 W, which is similar to the
human data [11,14–16]. It is much greater than the ordinary joint mechanisms for humanoids, which
can output approximately 150 W at most.
The proposed mechanism would be advantageous if the robot could run at a higher speed.
However, this is difficult because of a lack of power in the hip joint. For knee and ankle joints, we
achieved high power output by mimicking human joint stiffness. However, it is difficult to achieve
high power output in a hip joint by using the developed mechanism, because the hip joint of a human
does not move like a spring. We plan to develop a hip joint that can achieve high power output;
nevertheless, in this study, we confirmed that the performance of the developed joint fulfills the
requirements based on human running data and is adequate for higher speed running in terms of
energy storage capacity, maximum output torque, maximum output power, maximum deflection
angle, movable range and angular velocity. The specifications of the developed joint mechanism are
listed in Table 6.
In this experiment, we implemented some control method, such as the foot placement control,
and the ground reaction force estimation, as our previous study [37]. However, to achieve more
stable running or running at higher speed, we consider that the upper body movement is needed
for stabilization. This is derived from humans, which are considered to use their trunk and arms for
stabilization during running [13].
Table 6. Summary of the developed joint mechanism.
Specification
Energy storage capacity (J)
Maximum torque (Nm)
Maximum power (W)
Maximum deflection angle (rad)
Active movement range (rad)
Active movement angular velocity (rad/s)
73
180
1000
0.81
0–1.7
6.7
Machines 2016, 4, 5
Machines
2016,4,4,5 5
Machines
2016,
13 of 15
13 of1315of 15
MassMass
displacement
mmmm
displacement
Figure 11. Photograph of hopping experiment (running speed was 0.2 m/s).
Figure 11. Photograph of hopping experiment (running speed was 0.2 m/s).
Figure 11. Photograph of hopping experiment (running speed was 0.2 m/s).
80
80
60
60
40
40
20
20
0
0
-20
-20
-40
-40
-60
-60
0
1
2
Time s
3
4
5
0
1
2
3
4
5
Figure 12. Mass vertical displacement
Time in
s the hopping experiment.
Figure
12.
Massvertical
verticaldisplacement
displacementin
inthe
thehopping
hoppingexperiment.
experiment.
Mass
4. Conclusions andFigure
Future12.
Work
Our long-term
goal isWork
mimicking human running with a whole body robot. However, this
4. Conclusions
and Future
4. involves
Conclusions
and
Future Work
various
challenging
problems, such as power shortage, stabilization, and so on. Therefore,
Ourpaper,
long-term
goal
is
mimicking
running
a body
wholerobot.
body
robot.
However,
in Our
this
we
reported
the
solution
of human
the running
power
shortage
by mimicking
human
joint stiffness.
Wethis
long-term goal is mimicking
human
with awith
whole
However,
this involves
involves
various
challenging
problems,
such
as incorporates
power
shortage,
and so on.
Therefore,
described
the development
a robotic
joint
that
a jointstabilization,
stiffness
adjustment
various
challenging
problems,ofsuch
as power
shortage,
stabilization,
and
so on.
Therefore,mechanism
in this paper,
in
this
paper,
we
reported
the
solution
of
the
power
shortage
by
mimicking
human
joint
stiffness.
that
uses
two
laminated
leaf
springs
made
of
CFRP
and
a
worm
gear
to
mimic
the
joint
stiffness
of aWe
we reported the solution of the power shortage by mimicking human joint stiffness. We described
described
the
development
of
a
robotic
joint
that
incorporates
a
joint
stiffness
adjustment
mechanism
leg, and we
this
joint
mechanism ainto
thestiffness
leg of a bipedal
robot.mechanism
With the new
thehuman
development
of aincorporated
robotic joint
that
incorporates
joint
adjustment
that
that
uses two laminated
leaftosprings
made
of CFRP
and
worm gear
mimic the
jointofstiffness
mechanism,
it is possible
adjust the
joint
stiffness
bya changing
thetoeffective
length
the leafof a
uses two laminated leaf springs made of CFRP and a worm gear to mimic the joint stiffness of a
springs,
and
thewe
joint
stiffness canthis
be calculated
using a proposed
equation,
which takes
account
human
leg,
and
incorporated
joint mechanism
into the leg
of a bipedal
robot.into
With
the new
human leg, and we incorporated this joint mechanism into the leg of a bipedal robot. With the new
the
difference
between
the
positions
of
the
fixed
point
of
the
leaf
spring
and
the
rotational
center
mechanism, it is possible to adjust the joint stiffness by changing the effective length of theofleaf
mechanism, it is possible to adjust the joint stiffness by changing the effective length of the leaf
the joint.
spring wasusing
lighter
than an equivalent
iron leaf
spring.
We
springs,
andThe
the CFRP-laminated
joint stiffness canleaf
be calculated
a proposed
equation, which
takes
into account
springs, and the joint stiffness can be calculated using a proposed equation, which takes into account
confirmed
thebetween
effectiveness
of the laminated
leaf spring
of loading
capacity
and size, and
we of
the
difference
the positions
of the fixed
point in
ofterms
the leaf
spring and
the rotational
center
the difference between the positions of the fixed point of the leaf spring and the rotational center
verified
that
the
developed
joint
mechanism
can
mimic
the
joint
stiffness
of
a
human
leg.
We
also
the joint. The CFRP-laminated leaf spring was lighter than an equivalent iron leaf spring. We
of confirmed
the joint. the
Theeffectiveness
CFRP-laminated
leaf spring
was for
lighter
than an
equivalent
ironaccording
leaf spring.
ofthe
a proposed
equation
calculating
the
joint stiffness
theWe
confirmed the
effectiveness of
laminated
leaf spring
in terms of
loading
capacity
and size,toand
we
confirmed
the
effectiveness
of
the
laminated
leaf
spring
in
terms
of
loading
capacity
and
size,
and
difference
the positions
the fixed point
of the leaf
and the rotational
center
the
verified
thatbetween
the developed
joint of
mechanism
can mimic
the spring
joint stiffness
of a human
leg. of
We
also
wejoint.
verified
the developed
joint mechanism
mimic the
jointconfirms
stiffnessthat
of athe
human
leg. We
also
The that
developed
robot achieved
hopping viacan
resonance,
which
developed
joint
confirmed the effectiveness of a proposed equation for calculating the joint stiffness according to the
confirmed
the
effectiveness
of
a
proposed
equation
for
calculating
the
joint
stiffness
according
to
mechanism can be used for storing energy for jumping. This energy-storage mechanism is based onthe
difference between the positions of the fixed point of the leaf spring and the rotational center of the
difference
between
the positions
of the the
fixed
point of the
anditthe
rotational
center
human motion
analysis
and improves
performance
ofleaf
the spring
robot, and
could
be applied
alsooftothe
joint. The developed robot achieved hopping via resonance, which confirms that the developed joint
joint.
The
developed
robotorachieved
hopping
viaWe
resonance,
confirms
developed
joint
other
humanoid
robots
new prosthetic
legs.
intend towhich
develop,
in the that
nearthe
future,
an upper
mechanism can be used for storing energy for jumping. This energy-storage mechanism is based on
mechanism
used for
energy
This
energy-storage
based on
body thatcan
willbeallow
us storing
to construct
a for
newjumping.
full-body
running
robot thatmechanism
can mimicisvarious
human
motion analysis
and
improves
the performance
of using
the robot,
and it could be applied also to
characteristics
of human
running,
for example
stabilization
the upper
human
motion analysis
and
improves
the performance
of the
robot,
and itbody.
could be applied also to
other humanoid robots or new prosthetic legs. We intend to develop, in the near future, an upper
other
humanoid
robots
or
new
prosthetic
legs.
We
intend
to
develop,
in
the
near
future, anScience
upperand
body
Acknowledgments: This study was conducted with the support of the Research Institute
body
that will allow us to construct a new full-body running robot that canformimic
various
that
will
allow
us
to
construct
a
new
full-body
running
robot
that
can
mimic
various
characteristics
Engineering, Waseda University, the Institute of Advanced Active Aging Research, Waseda University, and as of
characteristics
of human
running,
for example
stabilization
using the upper body.
part ofrunning,
the humanoid
project atstabilization
the Humanoidusing
Robotics
Waseda University. It was also supported in
human
for example
theInstitute,
upper body.
Acknowledgments: This study was conducted with the support of the Research Institute for Science and
Engineering, Waseda University, the Institute of Advanced Active Aging Research, Waseda University, and as
part of the humanoid project at the Humanoid Robotics Institute, Waseda University. It was also supported in
Machines 2016, 4, 5
14 of 15
Acknowledgments: This study was conducted with the support of the Research Institute for Science and
Engineering, Waseda University, the Institute of Advanced Active Aging Research, Waseda University, and
as part of the humanoid project at the Humanoid Robotics Institute, Waseda University. It was also supported in
part by the MEXT/JSPS KAKENHI Grant Nos. 25220005 and 25709019, the Mizuho Foundation for the Promotion
of Sciences, SolidWorks Japan K.K., DYDEN Corporation, Cybernet Systems Co., Ltd., and TohoTenax Co., Ltd.
We thank all of them for the financial and technical support provided.
Author Contributions: T.O., K.H., T.I., H.-O.L. and A.T. developed the joint mechanism; T.O., K.H., and T.I.
performed the experiments; M.S. and Y.K. analyzed the human motion data; H.-O.L. and A.T. helped to draft the
manuscript. T.O. wrote the paper. All authors read and approved the final manuscript.
Conflicts of Interest: The authors declare no conflicts of interest.
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
Ogura, Y.; Shimomura, K.; Kondo, H.; Morishima, A.; Okubo, T.; Momoki, S.; Lim, H.O.; Takanishi, A.
Human-like Walking with Knee Stretched, Heel-contact and Toe-off Motion by a Humanoid Robot.
In Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems 2006, Beijing,
China, 9–15 October 2006; pp. 3976–3981.
Hashimoto, K.; Hattori, K.; Otani, T.; Lim, H.O.; Takanishi, A. Foot Placement Modification for a Biped
Humanoid Robot with Narrow Feet. Sci. World J. 2014, 2014, 259570. [CrossRef] [PubMed]
Hashimoto, K.; Takezaki, Y.; Lim, H.O.; Takanishi, A. Walking Stabilization Based on Gait Analysis for Biped
Humanoid Robot. Adv. Robot. 2013, 27, 541–551. [CrossRef]
World Medical Association. Declaration of Helsinki; World Medical Association: Helsinki, Finland, 1964.
Blickhan, R. The Spring-mass Model for Running and Hopping. J. Biomech. 1989, 22, 1217–1227. [CrossRef]
McMahon, T.; Cheng, G. The Mechanics of Running: How does Stiffness Couple with Speed? J. Biomech.
1990, 23, 65–78. [CrossRef]
Novacheck, T.F. The biomechanics of running. Gait Posture 1998, 7, 77–95. [CrossRef]
Kuitunen, S.; Komi, P.V.; Kyrolainen, H. Knee and ankle joint stiffness in sprint running. Med. Sci. Sports Exerc.
2002, 34, 166–173. [CrossRef] [PubMed]
Gunther, M.; Blickhan, R. Joint stiffness of the ankle and the knee in running. J. Biomech. 2002, 35, 1459–1474.
[CrossRef]
Ferber, R.; Davis, I.M.; Williams, D.S., III. Gender Differences in Lower Extremity Mechanics during Running.
Clin. Biomech. 2003, 18, 350–357. [CrossRef]
Chapman, A.E.; Caldwell, G.E. Factors determining changes in lower limb energy during swing in treadmill
running. J. Biomech. 1983, 16, 69–77. [CrossRef]
Otani, T.; Hashimoto, K.; Yahara, M.; Miyamae, S.; Isomichi, T.; Hanawa, S.; Sakaguchi, M.; Kawakami, Y.;
Lim, H.; Takanishi, A. Utilization of Human-Like Pelvic Rotation for Running Robot. Front. Robot. 2015.
[CrossRef]
Hinrichs, N.R. Upper Extremity Function in Running. II: Angular Momentum Considerations. Int. J.
Sport Biomech. 1987, 3, 242–263.
Endo, T.; Miyashita, K.; Ogata, M. Kinetics factors of the lower limb joints decelerating running velocity in
the last phase of 100 m race. Res. Phys. Educ. 2008, 53, 477–490. [CrossRef]
Schache, G.A.; Blanch, D.P.; Dorn, W.T.; Brown, A.T.N.; Rosemond, D.; Pandy, G.M. Effect of Running Speed
on Lower Limb Joint Kinetics. Med. Sci. Sports Exerc. 2011, 43, 1260–1271. [CrossRef] [PubMed]
Dalleau, G.; Belli, A.; Bourdin, M.; Lacour, J.R. The Spring-Mass Model and the Energy Cost of Treadmill
Running. Eur. J. Appl. Physiol. 1998, 77, 257–263. [CrossRef] [PubMed]
Nagasaki, T.; Kajita, S.; Kaneko, K.; Yokoi, K.; Tanie, K. A Running Experiment of Humanoid Biped.
In Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, Sendai, Japan,
28 September–2 October 2004; pp. 136–141.
Cho, B.K.; Park, S.S.; Oh, J.H. Controllers for running in the humanoid robot, HUBO. In Proceedings of
the IEEE-RAS International Conference on Humanoid Robots 2009, Paris, France, 7–10 December 2009;
pp. 385–390.
Takenaka, T.; Matsumoto, T.; Yoshiike, T.; Shirokura, S. Running Gait Generation for Biped Robot with
Horizontal Force Limit. JRSJ 2011, 29, 93–100. [CrossRef]
Machines 2016, 4, 5
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
15 of 15
Tajima, R.; Honda, D.; Suga, K. Fast Running Experiments Involving a Humanoid Robot. In Proceedings of
the IEEE International Conference on Robotics and Automation, Kobe, Japan, 12–17 May 2009; pp. 1571–1576.
Tamada, T.; Ikarashi, W.; Yoneyama, D.; Tanaka, K.; Yamakawa, Y.; Senoo, T.; Ishikawa, M. High Speed
Bipedal Robot Running Using High Speed Visual Feedback. In Proceedings of the 14th IEEE-RAS
International Conference on Humanoid Robots, Madrid, Spain, 18–20 November 2014; pp. 140–145.
Niiyama, R.; Nishikawa, S.; Kuniyoshi, Y. Biomechanical Approach to Open-loop Bipedal Running with a
Musculoskeletal Athlete Robot. Adv. Robot. 2012, 26, 383–398. [CrossRef]
Grizzle, J.W.; Hurst, J.; Morris, B.; Park, H.W.; Sreenath, K. MABEL, a new robotic bipedal walker and runner.
In Proceedings of the American Control Conference, St. Louis, MO, USA, 10–12 June 2009; pp. 2030–2036.
Tsagarakis, N.G.; Laffranchi, M.; Vanderborght, B.; Caldwell, D.G. A compact soft actuator unit for small scale
human friendly robots. In Proceedings of the IEEE International Conference on Robotics and Automation,
Kobe, Japan, 12–17 May 2009; pp. 4356–4362.
Ugurlu, B.; Saglia, J.A.; Tsagarakis, N.G.; Caldwell, D.G. Hopping at the resonance frequency: A trajectory
generation technique for bipedal robots with elastic joints. In Proceedings of the IEEE International
Conference on Robotics and Automation, Saint Paul, MN, USA, 14–18 May 2012; pp. 1436–1443.
Moro, F.L.; Tsagarakis, N.G.; Caldwell, D.G. Walking in the Resonance with the COMAN Robot with
Trajectories based on Human Kinematic Motion Primitives (kMPs). Auton. Robots 2014, 36, 331–347.
[CrossRef]
Vu, H.Q.; Yu, X.; Iida, F.; Pfeifer, R. Improving energy efficiency of hopping locomotion by using a variable
stiffness actuator. IEEE/ASME Trans. Mechatron. 2015. [CrossRef]
Renjewski, D.; Seyfarth, A.; Manoonpong, P.; Wörgötter, F. The development of a biomechanical leg system
and its neural control. In Proceedings of the 2009 International Conference on Robotics and Biomimetics,
Guangxi, China, 19–23 December 2009; pp. 1894–1899.
Grimmer, M. Powered Lower Limb Prostheses. Ph.D. Thesis, Technische Universitaet Darmstadt, Darmstadt,
Germany, 2015.
Yuan, K.; Zhu, J.; Wang, Q.; Wang, L. Finite-state control of powered below-knee prosthesis with ankle
and toe. In Proceedings of the 18th IFAC World Congress, Milano, Italy, 28 August–2 September 2011;
pp. 2865–2870.
Waycaster, G.; Wu, S.K.; Xiangrong, S. Design and control of a pneumatic artificial muscle actuated
above-knee prosthesis. J. Med. Devices 2011, 5. [CrossRef]
Hitt, J.; Sugar, T.; Holgate, M.; Bellman, R. An active foot-ankle prosthesis with biomechanical energy
regeneration. J. Med. Devices 2010, 4. [CrossRef]
Morita, T.; Sugano, S. Development of 4-DOF manipulator using mechanical impedance adjuster.
In Proceedings of the IEEE International Conference on Robotics and Automation, Minneapolis, MN,
USA, 22–28 April 1996; pp. 2902–2907.
Merritt, H.E. Worm gear performance. Proc. Inst. Mech. Eng. 1982, 129, 127–194. [CrossRef]
Kouchi, M.; Mochimaru, M. Human Dimension Database; AIST Digital Human Research Center: Tokyo,
Japan, 2005.
Ae, M.; Tang, H.; Yokoi, T. Estimation of inertia properties of the body segments in japanese athletes.
Soc. Biomech. 1992, 11, 23–33.
Otani, T.; Hashimoto, K.; Yahara, M.; Miyamae, S.; Isomichi, T.; Sakaguchi, M.; Kawakami, Y.; Lim, H.O.;
Takanishi, A. Running with Lower-Body Robot That Mimics Joint Stiffness of Humans. In Proceedings of the
IEEE/RSJ International Conference on Intelligent Robots and Systems, Hamburg, Germany, 28 September–2
October 2015.
© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access
article distributed under the terms and conditions of the Creative Commons by Attribution
(CC-BY) license (http://creativecommons.org/licenses/by/4.0/).