A Biologically Inspired Biped Locomotion Strategy
for Humanoid Robots: Modulation of Sinusoidal
Patterns by a Coupled Oscillator Model
Jun Morimoto (corresponding author), Gen Endo, Jun Nakanishi, Member, IEEE,and Gordon Cheng, Senior Member, IEEE

Abstract—Biological systems seem to have a simpler but
more robust locomotion strategy than existing biped walking
controllers for humanoid robots. We show that a humanoid
robot can step and walk using simple sinusoidal desired joint
trajectories with their phase adjusted by a coupled oscillator
model. We use the center of pressure location and velocity
to detect the phase of the lateral robot dynamics. This phase
information is used to modulate the desired joint trajectories.
We do not explicitly use dynamical parameters of the humanoid
robot. We hypothesize that a similar mechanism may exist in
biological systems. We applied the proposed biologically inspired
control strategy to our newly developed human-sized humanoid
robot CB and a small size humanoid robot, enabling them togenerate successful stepping and walking patterns.
Index Terms—Biped Walking, Humanoid Robots, Central
Pattern Generator, Coupled Oscillator, Biologically Inspired Approach

I. INTRODUCTION

B

IOLOGICAL systems seem to have a simpler but morerobust locomotion strategy [1] than existing biped walking controllers for humanoid robots (e.g. [2]). For examples,[2] and [3] showed that the cat locomotion system cangenerate walking pattern without using higher brain function.
An early study of biologically inspired approach to bipedal
locomotion [4] suggested that synchronization property ofneural system with periodic sensor inputs plays an importantrole for robust locomotion control. After this leading study,
there is growing interest in biologically inspired locomotion
control utilizing coupled neural oscillators [5]–[8] or usinga phase oscillator model with phase reset methods [9], [10].
These studies make use of foot contact information, or groundreaction forces in exploiting the entrainment property of theneural or phase oscillator model.

J. Morimoto is with the Japan Science and Technology, 4-1-8 Honcho,
Kawaguchi, Saitama, 332-0012, Japan, and with ATR computational neuroscience labs, 2-2-2 Hikari-dai, Seika-cho, Souraku-gun, Kyoto, 619-0288,
Japan (e-mail: xmorimo@atr.jp)
G. Endo is with Dept. of Mechanical and Aerospace Engineering, Tokyo
Institute of Technology, 4-1-8 Honcho, Kawaguchi, Saitama, 332-0012, Japan.
(e-mail: gendo@sms.titech.ac.jp)
J. Nakanishi is with the Japan Science and Technology, 4-1-8 Honcho,
Kawaguchi, Saitama, 332-0012, Japan, and with ATR computational neuroscience labs, 2-2-2 Hikari-dai, Seika-cho, Souraku-gun, Kyoto, 619-0288,
Japan (e-mail: jun@atr.jp)
G. Cheng is with the Japan Science and Technology, 4-1-8 Honcho,
Kawaguchi, Saitama, 332-0012, Japan, and with ATR computational neuroscience labs, 2-2-2 Hikari-dai, Seika-cho, Souraku-gun, Kyoto, 619-0288,
Japan (e-mail: gordon@atr.jp)

Similarly, since biped walking is a periodic movement, it has
been suggested that methods to synchronize biped controllers
with the robot dynamics are useful to generate stable walking
patterns. Several studies designed walking trajectories as afunction of a physical variable of the robot (e.g. ankle joint
angle) [11]–[13].
However, a neural oscillator model has complex dynamics
and many parameters to be selected [4], [5]. Other approaches
that has synchronization mechanisms usually require proper
gait design [9], [10], [12], [13].
In this study, we undertake the development of a simple
but robust biped controller by means of a coupled oscillator
system, which is said to exist in vertebrates and is widelyreferred as central pattern generator (CPG) [14].
Many biped walking studies have emphasized that humanoid robots have inverted pendulum dynamics, with the top
at the center of mass and the base at the center of pressure,and proposed control strategies to stabilize the dynamics [15]–[19].
We propose using the center of pressure to detect the phaseof the inverted pendulum dynamics. 1) We use simple periodic
functions (sinusoids) as desired joint trajectories. 2) We showthat synchronization of the desired trajectories at each joint
with the inverted pendulum dynamics can generate stepping
and walking. 3) Since our nominal gait patterns are sinusoids,
our approach does not need careful design of desired gait
trajectories. 4) We use smaller numbers of parameters thanexisting neural oscillator approach, and compare to the neuraloscillator model, parameters used in our approach can beeasily selected since the physical meanings of the parameters
are quite simple.
To the best of our knowledge, this study is the first attempt
to apply an oscillator model to a human-sized humanoid robot
CB [20] (Fig.1(a)) for biped walking in a real environment. We
also apply our method to a small humanoid robot (Fig.1(b)).
First, we introduce our biologically inspired biped locomotion
strategy, which use modulated sinusoidal patterns via a coupled oscillator model, described in Section II. In Section III,
we apply our proposed approach to the simulated robot model(see Fig. 1(c)), and also show our experimental results.
II. M ODULATION OF SINUSOIDAL PATTERNS BY A
COUPLED OSCILLATOR MODEL

Our biped control approach uses a coupled phase oscillator
model [21] to modulate sinusoidal patterns. The aim of using

(a)

(b)

(c)

Fig. 1. (a) Our human-sized hydraulic humanoid robot CB developed by
SARCOS [20]. height: 1.59 m, total weight: 95 kg. (b) Small humanoid robot
used in the experiment. (c) Simple 3D biped simulation model. The bipedmodel has ten degrees of freedom. height: 1.59 m, total weight: 95 kg.

We showed the typical time profile of phase difference Φ =φc − φr , angular frequencies ω c and ωr in Fig. 2. Althoughusually the biped dynamics can not represented by the simple
phase dynamics (2), we can still detect the phase from therobot dynamics as described in Section II-B. Then, we use(1) to adjust phase of the controller under assumption thatthe phase dynamics detected from the biped dynamics keeps
similar property to (2).
B. Phase detection from the robot dynamics

Angular frequency ω [rad/s]

Phase difference Φ [rad]As previous studies have pointed out, controlling the inverted pendulum dynamics represented by the center of mass0
and the center of pressure (Fig. 3(a)) is a major issue in2Φcontrolling biped robots. We consider the inverted pendulum
Φ*
−0.2
1.8
dynamics on lateral plane that has a four dimensional state−0.4
1.6
space x = (y, ẏ, ψ r , ψ̇ r ), which depicted in Fig. 3(a).
−0.6
1.4
To detect phase from the inverted pendulum dynamics, we
ω_c
−0.8
1.2
ω_r
project the four-dimensional state space to a two-dimensionalω*state space. Then, convert the two-dimensional state space to
−1
1
0
5
10
0
5
10
Time [s]Time [s]the phase space by using the function arctan, which widelyusedas phase detector for radio wave decoding.
(a) Phase difference(b) Angular frequency
In this study, we consider the center of pressure y and theFig. 2. Typical time profile of the coupled oscillator system. (a) Phase
velocityof the center of pressure ẏ as the variables in the
difference Ψ = φr − φc . The phase difference converged to the analytically
two-dimensional state space since detecting these values byderived value Ψ∗ depicted by dash dot line. (b) Angular frequencies ωr andωc . Angular frequency of each oscillator converged to the analytically derived force sensors on soles is easy for our real robots. Therefore,
compromise frequency ω∗ .
we detect the phase as
 
ẏ
(x)
=
−
arctanφ.
(3)r
the coupled phase oscillator model is to synchronize periodicypatterns generated by the controller with the dynamics of therobot.
We also show a strategy to design nominal desired joint C. Simplified COP detection
angle. One of the simplest way to generate periodic pattern
The center of pressure (COP) depends on a coordinate
at each joint is using only one sinusoidal basis function to system, and we need a kinematic model to detect COP.
represent the desired joint angle. By only using one sinusoidal Alternatively, we use an approximate center of pressure:basis function at each joint, we have smallest numbers ofparameters to represent periodic patterns at each joint. We
yfl oot Fzl + yfr oot Fzr
y
=
,
(4)
introduce our stepping and walking controllers that use theFzr + Fzl
desired joint angle represented by the sinusoidal function.
where Fzl and Fzr represent the left and right ground reaction
force respectively, and y fl oot and yfr oot are the lateral position
A. Coupled oscillator modelof each foot. We assume that feet are symmetrically placedrHere, we consider the behavior of the following dynamics −y lf oot = yf oot . Because we only use this center of pressure toof the phase of a biped controller φ c and the phase of the detect the phase of the robot dynamics φ (x) in equation (3),rrobot dynamics φ r :the scale of the foot position y fl oot and yfr oot can be arbitrary.
lrφ̇c = ωc + Kc sin(φr − φc ),(1) We simply set yf oot = −yf oot = 1.0 m. This simplified COP
detection does not require the kinematic model.
(2)
φ̇r = ωr + Kr sin(φc − φr ),When we derive the center of pressure from the sensor input,
where ωc > 0 and ωr > 0 are natural frequencies of the we use a low-pass filter to eliminate sensor noise from force
controller and the robot dynamics, and K c , Kr are positive sensors on soles.
coupling constants. We can find two fixed points if |ω c −ωr | <Note that when the center of pressure come to an edge of
Kc + Kr . There is no fixed point if |ω c − ωr | > Kc + Kr . A a sole, the detected φr (x) may have discontinuous change
saddle-node bifurcation occurs when |ω c − ωr | = Kc + Kr .
because of discontinuous change of the velocity of the center
If |ωc − ωr | < Kc + Kr , the oscillators run with the phase of pressure (see (3)). However, because we apply the lowdifference: Ψ ∗ = φr − φc = sin−1 ((ωr − ωc )/(Kc + Kr )) and pass filter to the force sensor input, the filtered velocity of thethe compromise frequency: ω ∗ = (Kr ωc + Kc ωr )/(Kc + Kr ) center of pressure do not have such undesirable discontinuous
when they are entrained [21].
change.

z

z

z
y

y

y
COM

COM

z

hip_r

x
hip_p

hip_r

knee_pankle_pankle_r

0
COP

y
COP

0

ankle_r

y

(a)

(b)

where Ap is the amplitude of a sinusoidal function to achievefoot clearance, θ̄p defines the rest posture of the hip, knee,
and ankle joints. We use the oscillator with phase φ c = φc 1
for right limb movement and use the oscillator with phase
φc = φc 2 , which has phase difference φ c 2 = φc 1 + π,
for left limb movement. Two parameters {A p , θ̄p } need to be
selected for the controller for vertical foot movement.
Because the center of pressure is modulated by the joint
angle trajectories, the phase of the controller φ c affects thephase of the robot φ r (x) in (3), through the controllers
introduced in (6)-(10).

Fig. 3. (a) Inverted pendulum model represented by the center of pressure(COP) and the center of mass (COM). (b) Stepping controller.

F. Biped walking controller with additional sinusoids
D. Phase coordination
In this study, we use two oscillators with phases φ c i , wherei = 1, 2. We introduce coupling between the oscillators andthe phase of the robot dynamics φ r (x) to regulate the desired
phase relationship between the oscillators as in (1):φ̇c i = ωc + Kc sin((φr (x) − αi ) − φc i ),(5)

where αi is the desired phase difference. We use two different phase differences, {α 1 , α2 } = {−1/2π, 1/2π}, to makesymmetric patterns for a stepping movement by left and rightlimbs (see section II-E2). Two parameters {ω c , Kc } need to
be selected to define the phase oscillator dynamics.
We empirically found that the natural frequency of a linear
pendulum with the length l can be a good candidate
for the
natural frequency of the controller as ω c = g/l, where gdenotes the acceleration due to the gravity and l denotes theheight of COM when a biped stand straight.
By considering the insight from the oscillator dynamics (1)and (2), we need to use sufficiently large coupling constant
Kc to satisfy |ωc − ωr | < Kc + Kr for keeping fixed points.
E. Stepping controller for lateral movement
1) Side-to-side controller for lateral movement: First, we
introduce a controller to generate side-to-side movement. We
use the hip joints θhip r and the ankle joints θ ankle r (Fig.
3(b)) for the movement. Desired joint angles for each joint
are:
d
θhip
r (φc ) = Ar sin(φc ) + θ̄r ,(6)dθankler (φc ) = −Ar sin(φc ) − θ̄r ,(7)where Ar are the amplitudes of a sinusoidal function for sideto-side movements at the hip and the ankle joints, and we use
an oscillator with the phase φ c = φc 1 . θ̄r defines the rest
posture of the hip, knee, and ankle joints. Two parameters{Ar , θ̄r } need to be selected for the side-to-side controller.
2) Vertical foot movement to make clearance: To achievefoot clearance, we generate vertical movement of the feet (Fig.
3(b)) by using simple sinusoidal trajectories:dθhip
p (φc )= Ap sin(φc ) + θ̄p ,

(8)

d
θknee
p (φc )= −2Ap sin(φc ) − 2θ̄p ,(9)dθanklep (φc )= −Ap sin(φc ) − θ̄p ,(10)

For our biped walking controller, we introduce two additional oscillators φpc i , where i = 1, 2. We then consider phase
dynamics:ωc + Kcp sin((φr (x) − αpi ) − φpc i ) (single support)φ̇pc i =0(double support)(11)
We set the dynamics of the oscillator φ̇pc i = 0 during
double support phase so that we can prevent internal force
generated by friction between the ground and the soles, while
we keep using the phase dynamics (5) . We use two different
phase differences, {α p1 , αp2 } = {0.0, π}, to make symmetricpatterns for a forward movements by left and right limbs. We
empirically figured out that we can generate biped walking by
setting the coupling constant as K cp = Kc /2.
To walk forward, the biped need to make forward step. To
make forward step at proper timing, we introduce an additional
sinusoidal trajectory that has 1/2π phase difference from thesinusoidal trajectories for the stepping movement:
d
p
p
θhip
s (φc ) = Ah s sin(φc ),

(12)

d
p
p
θankles (φc ) = −Aa s sin(φc ),(13)

where Ah s and Aa s are amplitudes of sinusoidal functions
at the hip and the ankle joints for biped walking. We use
the phase φpc = φpc 1 which has 1/2π phase difference withφc 1 , that is phase of the oscillator for lateral movement, forright limb and use the phase φ pc = φpc 2 which has π phasedifference with φpc 1 . The desired nominal trajectories for hipand ankle pitch joints in (8) and (10) become:
d
p
d
d
p
θhip
p (φc , φc ) ← θhip p (φc ) + θhip s (φc ),dpθanklep (φc , φc )

←

(14)

d
d
p
θanklep (φc ) + θankle s (φc ). (15)

Two parameters {Ah s , Aa s } need to be selected for thecontroller for forward movements.

III. SIMULATION AND EXPERIMENTAL RESULT
We applied our proposed method to a simple 3D biped robot
model (Fig. 1(c)), our human-sized humanoid robot CB (Fig.
1(a)), and the small humanoid robot (Fig. 1(b)).

50
5A_r [°]0 2

6

4

A_p [°](a) Pendulum angle

8

3
2
1
0
5
A_r [°]

0 2

6

4

8

A_p [°](b) Stepping period(a) t=0.0 s
Fig. 5.

θ

A. Stepping movement(c) t=0.8 s

12
10
8
6
4
2
0
−2
10

ActualDesired12

14
16Time [s]

18

20

hip_r

[°]

(a) Not using coupled oscillator.

θ1) Application to the simulated biped model: We applied
our proposed method to the biped robot model.
As we proposed in section
 II-D, the natural frequency of thecontroller is set as ωc = g/l = 3.6 rad/s, and the coupling
constant is set as Kc = 10.0. Then, we compared different
parameter settings for the amplitude A r in (6), (7) and A p in(8)-(10).
Figure 4 shows results of the comparison. Using large A p
with small Ar result in falling over. On the other hand, using
small Ap with large Ar can not make single support phase.
By comparing Fig. 4(a) and 4(b), stepping movement thathas smaller pendulum angle ψ r tends to have larger stepping
period.
A proper combination of the parameters A r = 2.5◦ and
Ap = 5.0◦ , which can make stepping movement without
falling over, generated a stepping movement with period 1.4 s.
1Equivalently, average angular frequency φ̇av
c = T (φc (T + t)−
φc (t)) of the stepping movement was φ̇av
c = 4.5 rad/s, where
T is a stepping period. Figure 5 shows successful stepping of
the simulated biped model.
To show how the coupled oscillator model in equation (1)and (2) worked with the biped robot model, we tested adifferent controller with a different natural frequency, ω c = 2.5
rad/s. Although the natural frequency was different, the modulated averaged frequency φ̇av
c = 3.9 rad/s was much closer
to the previous averaged frequency, φ̇av
c = 4.5 rad/s than theselected natural frequency ω c = 2.5 rad/s.
By considering the compromise frequency ω ∗ introduced
in section II-A, this result indicates that the current coupling
constant Kc = 10.0 is large enough to make the controller
frequency close to the natural frequency of the robot dynamics.
Figure 6 shows trajectories of desired and actual hip jointdangles θhipr , θhip r (see Fig. 3(b)). Figure 6(a) represents
the result of without using a coupled oscillator model. The
desired trajectory is the original simple sinusoidal trajectory.
Figure 6(b) represents the result of using a coupled oscillator
model. This modulated trajectory made stepping movement
possible. Large tracking error appeared during single support
phase due to not using very large servo gain. This resultshows that our proposed method does not require accurate(b) t=0.4 s

Stepping movement using coupled oscillator.

hip_rFig. 4. Comparison of using different amplitude parameters. (a) Pendulum
angle ψr in Fig. 3(a). Region has the value ψr = 10 represents that the
robot can not make single support phase with the corresponding parameterselection. Region has the value ψr = 0 represents that the robot falls overwith the corresponding parameter selection. (b) Stepping period. Region has
the value 0 s represents that the robot can not make single support phase withthe corresponding parameter selection. Region has the value 3 s represents
that the robot falls over with the corresponding parameter selection.

[°]

Stepping period [s]ψr [°]

10

12
10
8
6
4
2
0
−2
10

ActualDesired12

14
16Time [s]1820(b) Using coupled oscillator.
Fig. 6. Generated desired and actual trajectory at hip joint. (a) Not using
coupled oscillator, (b) Using coupled oscillator. Large tracking error appeared
during single support phase due to not using very large servo gain. The
result shows that our proposed method does not require accurate tracking
performance.

tracking performance. The desired trajectory is modulated by
the coupling with the phase of the robot dynamics φ r (x).
2) Application to human-sized humanoid robot: The proposed stepping method was applied to the human-sized humanoid robot CB.
The natural frequency of the controller is selected as ω c =3.14 rad/sec and sufficiently large coupling constant is selected
as Kc = 9.4 by following the parameter selection approach
introduced in Section II-D.
We empirically figured out proper amplitude parametersA r = 3.0◦ , and Ap = 3.5◦ , which can generate a stepping
movement.
Figure 7 shows successful stepping of the human-sized
humanoid robot.
3) Application to the small humanoid robot: We applied
the proposed stepping method to the small humanoid robot.
The natural frequency of the controller is selected as ω c =6.28 rad/s and sufficiently large coupling constant is selected
as Kc = 9.4 by following the parameter selection approach
introduced in Section II-D.
We empirically figured out proper amplitude parametersA r = 7.5◦ for hip, A r = 2.5◦ for ankle, and A p = 3.5◦ ,which can generate a stepping movement.
Figure 8 shows successful stepping of the small humanoid

2.3
Walking period [s]Walking velocity [m/s]

0.3
0.2
0.1
0
−0.1
0

5
A_s [°](a) Walking speed

10

2.2
2.1
2
1.9
1.8
0

5
A_s [°]

10(b) Walking period

Fig. 9. Comparison with different amplitude parameters As in (12) and(13). The biped model falls over with the parameter As > 9.0◦ . (a) Walking
speed. (b) Walking period.
Fig. 7.

Fig. 8.

Successful stepping of our human-sized humanoid robot CB.

Successful stepping of the small humanoid robot.

robot.
B. Biped walking
1) Application to simple biped model: We applied our
proposed method to generate walking movements by using
the simulated biped model.
The same parameters as the stepping controller for thenatural frequency of the controller ω c = 3.6 rad/s, and thecoupling constant K c = 10.0 are used. We empirically figuredout proper amplitude parameters A r = 2.5◦ and Ap = 6.0◦
for the walking task.
We compared different amplitude parameters A s by setting
Ah s = As in (12) and A a s = As /2 (13).
Figure 9 shows the results of the comparison. Walking
velocity was linearly increased according to the increase of
the amplitude parameter A s . This is one of good properties ofthe proposed walking controller since we can easily select theamplitude parameter to achieve desired walking velocity.
On the other hand, the walking period did not show monotonic change according to the increase of the amplitude A s
as in Fig. 9(b). Walking velocity can be increased by either
increase walking step or decrease walking period (increase
walking frequency). We can see that walking controller useddifferent strategies to increase walking speed with different
amplitude As .
Figure 10 shows the successful walking pattern generated
by our control approach. We showed that the simulated robot
model could walk by only using the simple sinusoidal trajectory, which composed of at most only two sinusoidal basis
functions at each joint, modulated by the detected phase fromthe center of pressure.

2) Application to human-sized humanoid robot: We applied
our proposed walking method to the humanoid robot CB.
The same parameters as the stepping controller for thenatural frequency of the controller ω c = 3.14 rad/s, thecoupling constant K c = 9.4, Ar = 3.0◦ , and Ap = 3.5◦
are used.
We empirically figured out proper amplitude parameters
Ah s = 4.0◦ and Aa s = 2.0◦ for the walking task.
Figure 11 shows the successful walking pattern of ourhumanoid robot. Our proposed method was able to generate
successful walking patterns even in the real environment.
Note that the black tube from the top of each photo in figure
11 is a hydraulic cable.
3) Application to the small humanoid robot: We applied
our proposed walking method to the small humanoid robot.
The same parameters as the stepping controller for thenatural frequency of the controller ω c = 6.28 rad/s, thecoupling constant K c = 9.4, amplitudes A r = 7.5◦ for hip,A r = 2.5◦ for ankle, and A p = 3.5◦ are used.
We empirically figured out proper amplitude parameters
Ah s = 12.0◦ and Aa s = 8.0◦ for the walking task.
Figure 12 shows the successful walking pattern of thesmall humanoid robot. Our proposed method was able togenerate successful walking patterns for robots of different
sizes. Because our method to design a biped walking controller
does not suffered from singularity problem at knee joints thatcomes from using inverse kinematics, knees can straighten
during walking.
We calculated the ratio between the single support
phase and double support phase. The ratio was around(Tdouble )/(Tdouble + Tsingle ) × 100% = 25%, where Tdouble
is a period of time of double support phase and T single isa period of time of single support phase in a walking cycle.
This ratio is similar to the ratio for adult human biped walking
20%−25% that depends on the age of the human subject [22],[23]. This ratio is achieved by our proposed controller without
using pre-designed trajectories of the center of pressure.
We also tested robustness of our biped controller using four
different ground surfaces with different frictions. Each surface
also has different height. The four surfaces include carpet
with 0.0 mm (base level), plastic sheet with 2.0 mm, rubber
sheet with 3.5 mm, and metal sheet with 3.0 mm. Figure 13
shows successful results of walking over the different surfaces
without changing any parameter of the biped controller.

(a) t=0.00 s(b) t=0.30 s(c) t=0.57 s(d) t=0.93 s(e) t=1.23 s

Fig. 10.

Successful walking pattern using simulated biped robot model. Walking speed is 0.22 m/sec.

Fig. 11.

Successful walking pattern of our human-sized humanoid robot CB.

Fig. 12.

Successful walking pattern of the small humanoid robot. Knees are stretched during walking.

IV. DISCUSSION
In this paper we presented a biologically inspired bipedlocomotion strategy. Our method proposed the utilization of
the center of pressure position and velocity to detect thephase of the lateral robot dynamics. Evidences in biologicallocomotion studies support in part of our work [1], [24].
The detected phase of the robot dynamics was used to modulate sinusoidal joint trajectories. The modulated trajectories
enabled our robots to generate successful stepping and walking
patterns. Because the angular frequency in Equation (5) is
continuously changing during stepping and walking, not only
the frequency of the controller changes toward the resonant
frequency and excites the robot dynamics but also the timecourse of the sinusoidal patterns are modulated.
We applied successfully our proposed control approach to
our newly developed human-sized humanoid robot CB and the(f) t=1.53 s(g) t=1.87 s

small humanoid robot.
In the future, we will consider using optimization methods
such as reinforcement learning or dynamic programming [25],[26] to acquire a nonlinear feedback controller in order toincrease robustness of the walking controller.
A PPENDIX
Here we describe our simulation setups. To follow thedesired trajectories, the torque output at each joint is given
by a PD servo controller:dτ = Kp (θd (φc ) − θ) + Kd (θ̇ (φc ) − θ̇),
10

(16)

where θ (φc ) ∈ R is the target joint angle vector, K pdenotes the position gain matrix, and K d denotes the velocity
gain matrix. Each element of the diagonal position gain matrix
Kp is set to 3000 and each element of the diagonal velocity
gain matrix Kd is set to 100.
dFig. 13. Successful walking pattern of the small humanoid robot over four different surfaces: (1) Gray carpet, (2) Transparent plastic sheet, (3) Black rubber
sheet, (4) Blue metal sheet.

We used the fourth-order Runge-Kutta method with a time
step of Δt = 0.0003 s to numerically integrate the biped
dynamics.
The vertical ground reaction force f z is simulated by aspring-dumper model:
fz = −kpz zcp − kdz żcp ,(17)

where kpz = 30000 is the spring gain, k dz = 1000 is the dumper
gain, and z cp denotes vertical position of a contact point.
The ground reaction force for horizontal directions f x andfy are simulated by a viscose friction:fx
fy= −kdx ẋcp ,= −kdy ẏcp ,

(18)
(19)

where kdx = 2500 and kdy = 2500 are dumper gains. x cp andycp are horizontal position of a contact point.
ACKNOWLEDGMENT
We thank Sony Corp. for allowing us the opportunity to use
the small humanoid robot. We thank Darrin Bentivegna for
helping us to setup initial experiments. We also thank Mitsuo
Kawato, Christopher G. Atkeson, Seiichi Miyakoshi, Sang-Ho
Hyon for helpful discussion.
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