Postural Balance Strategies for Humanoid Robots in Response toDisturbances in the Frontal Plane

Yuki Yoshida, Kohei Takeuchi, Daisuke Sato and Dragomir Nenchev

Abstract— We examine human body reaction patterns todisturbances in the frontal plane while standing with the aim ofdeveloping balance control strategies for humanoid robots. Fourreaction patterns are identified and related to the magnitude ofthe disturbance. Models for two of them are then developed andimplemented with a humanoid robot HOAP-2. Special attentionis paid to the transitions between the reaction patterns. Theexperimental data show that the models and controllers areappropriate and ensure smooth reaction control under bothimpact-force and continuous-force disturbances.

I. INTRODUCTION

The balance controller plays a central role within the

overall control architecture of a humanoid robot. Similar

to balance control in humans, the control objectives of the

balance controller can be conditionally divided into two large

groups [1]:

• balance control during proactive (preplanned) activities;

• balance control for reactive motion patterns, in response

to unexpected disturbances from the environment.

We focus in this paper on the second type of balance

control objective and more specifically, on the case when

the human/robot is subjected to a disturbance while standing

upright. Most of the research has addressed the problem of

reaction pattern generation and balance recovery for motions

within the sagittal plane. Researchers have paid a lot of at-

tention to balance recovery strategies in humans, in response

to disturbance forces generated by horizontal perturbations

in the support surface while standing upright. It was clarified

that while standing on a normal surface, postural control is

ensured via a reaction pattern called “ankle strategy” that

restores equilibrium by motions in the ankle joints mainly.

On the other hand, with a support surface shortened in

relation to foot length, a different reaction pattern called “hip

strategy” was observed. This pattern produces a horizontal

shear force against the support surface, with little or without

any motion in the ankles, but with a predominant motion

in the hips [1]–[5]. In addition to these two strategies, a

third strategy was identified — the “stepping” or “stumbling”

strategy [4]. This strategy is invoked when certain boundary

values (in position and/or velocity) during the hip strategy

are exceeded.

In the field of humanoid robots, suggestion have been

made about possible ways to adopt the ankle and hip

strategies, so far [6]–[9]. We have succeeded in implementing

The authors are with the Department of Mechanical Systems Engi-neering, Tokyo City University, Tamazutsumi 1-28-1, Setagaya-ku, Tokyo158-8557, Japan. Corresponding author: D. Nenchev (Y. Kanamiya)nenchev@tcu.ac.jp

Fig. 1. Response patterns to disturbances in the frontal plane and thetransitions between them (from motion capture data): (a) Frontal-planeankle strategy; (b) Lift-leg strategy; (c) Cross-leg step strategy; (d) Side-stepstrategy.

the ankle and hip strategies as balance recovery patterns in

response to both impact disturbances [10] and continuous-

force disturbances [11], acting on the back or on the chest of

a small humanoid robot HOAP-2.Thereby, the hip strategy

was implemented as a combined strategy, involving an in-

verted double-pendulum model and the Reaction Null Space

method, developed originally for base disturbance control of

free-flying [12] or flexible-base mounted space robots [13].

We also have shown that the same method is useful for

other types of disturbances within the sagittal plane, such

as sudden motions in the support surface, i.e. slipping [14]

and rotations [15].

In contrast, not much research has been done so far with

regard to postural balance control strategies when a standing

upright human/robot is subjected to a sudden disturbance

within the frontal plane. In [16] a conceptual model for

investigating the role of the central nervous system in the

performance of lateral swaying movements was proposed.

The model suggested was just a simple four-link one, which

enabled only a parallel-leg motion pattern. A more detailed

study [17] came to the conclusion that humans respond either

with an ankle strategy, in response to smaller disturbances,

or with a hip load/unload strategy, in response to larger

disturbances. Also, it was mentioned that humans may need

to make a step to maintain balance, when the disturbance

grows even larger. In the field of humanoid robots, on the

other hand, as far as we know, only [18] has addressed the

problem. The authors proposed a two-stage balance control

1825

Proceedings of the 2011 IEEEInternational Conference on Robotics and Biomimetics

December 7-11, 2011, Phuket, Thailand

strategy implemented with a miniature HOAP-1 robot. The

first stage was the ankle strategy, based on Zero-Moment-

Point (ZMP) [19] feedback control. The second one was a

combined ankle-hip strategy for dealing with larger distur-

bances. The proposed models, however, did not refer to an

in-depth investigation of human postural balance strategies.

The aim of this work is to propose appropriate models

for human balance control in response to disturbances in the

frontal plane, while the human stands upright. We will first

investigate how humans react to such disturbances. Then,

based on the analysis, we will propose the respective models.

Finally, we will implement the models with a humanoid robot

HOAP-2.

II. BACKGROUND: THE REACTION NULL SPACE

A humanoid robot can be described as an underactuated

mechanical system with redundant DOFs. Balance control is

ensured through appropriate link motion and the respective

interaction wrenches between the links in contact with the

environment. A measure for balance stability can be deduced

from the ZMP. Although the ZMP has been widely used as

a means to control balance while standing or walking so far,

we should note that ZMP-based stability measures do not

account for the full state of the feet. Furthermore, the ZMP

can be deduced only for ground contacts on flat surfaces.

Note, on the other hand, that there are many situations in

practice whereby contact conditions cannot be simplified as

in the case with the ZMP. In such situations, the full state

of the links in contact must be accounted for. To deal with

this problem, we transferred the concept of Reaction Null

Space (RNS) from the field of space robotics [12], [13] to

the field of humanoid robots [10]. Through the RNS, it is

easy to obtain all link motions that keep the contact states

unchanged. The RNS of a humanoid robot was defined as

the null space of the inertia submatrix expressing the inertial

coupling between the supporting foot and the rest of the

links. This matrix appears in the equation of motion of an

n-DOF humanoid robot balancing on one of its feet:

⎡⎣Hf Hfl

HTfl H l

⎤⎦⎡⎣Vf

θ

⎤⎦+

⎡⎣Cfcl

⎤⎦+

⎡⎣Gf

gl

⎤⎦ =

⎡⎣0τ

⎤⎦

+

⎡⎣J

Tgf

JTgl

⎤⎦Wg +

⎡⎣J

Tef

JTel

⎤⎦We, (1)

where

H l ∈ �n×n :link inertia matrix

Hf ∈ �6×6 :inertia matrix of the foot

Hfl ∈ �6×n :inertia coupling matrix

cl ∈ �n :link Coriolis and centrifugal forces

Cf ∈ �6 :foot Coriolis and centrifugal forces

gl ∈ �n :link gravity force vector

Gf ∈ �6 :foot gravity force vector

τ ∈ �n :joint torque vector

θ ∈ �n :joint coordinate vector

Vf ∈ �6 :foot twist (spatial velocity)

Wg ∈ �6 :ground reaction wrench

We ∈ �6 :external force wrench

The J{◦}-terms denote proper transforms.

It has been shown [10] that the foot dynamics can be

expressed via the equation:

Hflθ + Hflθ = 0. (2)

The balance can be maintained with joint accelerations

belonging to the set:

{θRL} = −H#flHflθ + {Nfl} (3)

where (◦)# denotes a generalized inverse and {Nfl} stands

for the kernel of the inertia coupling matrix Hfl. The

kernel can be determined e.g. as the set of vectors {(U −H#

flHfl)ζ}, U denoting the unit matrix and ζ standing for

an arbitrary n-vector. The kernel {Nfl} is referred to as the

Reaction Null Space of a humanoid robot [10].

III. HUMAN POSTURAL BALANCE STRATEGIES

IN THE FRONTAL PLANE

Since not many studies could be found on the problem

of human balance control in reaction to unexpected dis-

turbances in the frontal plane while standing upright, we

decided to conduct our own experimental study. The main

goal was to distinguish between possible reactive motion

patterns/balance recovery patterns and to address possible

problems when implementing the motion patterns with a

humanoid robot.

The experimental study was performed with the help of

an optical motion capture system OptiTrack [20]. An impact

force was applied at the right shoulder of several subjects.

We found out that four motion patterns can be clearly dis-

tinguished, depending on the magnitude of the disturbance.

Figure 1 shows a representative motion patterns taken from

one of the subjects (21 years old). First, with a relatively

small disturbance, the subject reacted with movements in

the ankle joints mainly or with a combined ankle/hip/chest

movement. We call this strategy the Frontal-plane ankle

strategy (Fig. 1(a)). Second, with a larger disturbance, the

subject lifted the ipsilateral limb to keep the balance. The

respective balance recovery was to land the limb at or close

to the initial posture. We call this strategy the Lift-leg strategy

(Fig. 1(b)). Or, when the disturbance was stronger than

expected, the lifted ipsilateral limb was moved in front of

the contralateral limb to land across the body for taking

a crossed-leg stable balance posture. We call this strategy

the Cross-leg step strategy (Fig. 1(c)). Sometimes, with a

relatively large disturbance, the subject reacted by directly

taking a step either aside or across to attain a balance posture

resulting in a Side-step strategy (Fig. 1(d)) or Cross-leg step

strategy (Fig. 1(c)), without involving the Lift-leg strategy.

Figure 3 shows data for joint angles and the total CoM y-

axis position components for three strategies: Frontal-plane

ankle (Fig. 3(a)), Lift-leg (Fig. 3(b)) and Side-step (Fig.

3(c)). First, from the ankle strategy data in Fig. 3(a) it

1826

Fig. 2. Coordinate frames of motion capture.

L Hip L Ankle R Hip R Ankle Root Hip Chest

-40

-20

0

20

40

4 4.5 5 5.5 6 6.5 7 7.5 8

Angle

x [

deg

]

Time [s]

(a)

-40

-20

0

20

40

4 4.5 5 5.5 6 6.5 7 7.5 8

Angle

x [

deg

]

Time [s]

(b)

-40

-20

0

20

40

4 4.5 5 5.5 6 6.5 7 7.5 8

Angle

x [

deg

]

Time [s]

(c)

-40

-20

0

20

40

4 4.5 5 5.5 6 6.5 7 7.5 8

CoM

posi

tion [

cm]

Time [s]

(d)

AnkleLift-leg

Side-step

Fig. 3. Representative motion capture data from a subject aged 21, forjoint angles (a)–(c) and the total CoM y-axis position component (d) forthree strategies: Frontal-plane ankle (a), Lift-leg (b) and Side-step (c). “L”and “R” stand for left and right, respectively.

becomes apparent that mainly the ankle joints are used.

Both ankle joints (L Ankle and R Ankle) move in the same

direction. On the other hand, the motion in the two hip joints

(L Hip and R Hip) is subtle. Note that the direction of motion

is opposite to that of the ankle joints. One can conclude from

this data that the motion resembles that of a parallel link

mechanism.

Second, from the lift-leg strategy data in Fig. 3(b) it is

seen that the support leg’s hip joint (L Hip) and the waist

joint (Root Hip) are mainly used. The upper-body inclines

by the large movement in the support leg’s hip joint, and

the swing (right) leg detaches from ground. On the contrary,

no large movement is seen in the swing leg’s hip joint (R

Hip). Note also that the support leg’s hip joint (L Hip) moves

in the opposite direction to the support leg’s ankle joint (L

Ankle). This resembles the ankle-hip strategy known from

previous studies.

Finally, from the side-step strategy data in Fig. 3(c) it

becomes apparent that mainly L Hip and R Ankle joint

movements are employed. There is a large rotation in L Hip,

the foot is opened and makes a step forward. R Hip rotates

in the opposite direction of L Hip. The upper-body rotates

in the negative direction and inclines to the left. Therefore,

θ2

θ4θ1

θ3

m

Kp

pC

τ

pC

Kp

z

f

y

lg

θ

rcy rZMP

rcz

(a)

lg2

LC

KL

m2θ1+θ2

l2

θ2

θ3

m1m3

lg3

z

y

θ1

lg1

l1

m

θ4

f

rcyrZMP(b)

Fig. 4. Model: (a) Frontal-plane ankle strategy, (b) Lift-leg strategy.

R Ankle rotates in the positive direction. L Ankle rapidly

rotates in the negative direction first. It is thought that this

is because the left foot should be parallel to the ground to

prepare for landing. Note also that the CoM position variates

much more than in the previous two strategies (see Fig. 3(d)).

IV. IMPLEMENTATION WITH A SMALL

HUMANOID ROBOT

We have implemented the ankle and the lift-leg strategies

with a small humanoid robot HOAP-2. Thereby, we used the

two planar models shown in Fig. 4. Note that the robot does

not have a joint corresponding to the Root Hip joint of the

human model. Details are given below for each of the two

strategies.

A. Frontal-plane ankle strategy

An inverted pendulum model was used for the ankle

strategy. This was possible because the two legs move in

parallel, similar to the motion of a four-bar linkage. The two

phases of motion, disturbance response and balance recovery,

can be realized with the help of a virtual spring and damper

attached between the vertical and the pendulum (see Fig.

4(a)). The equation of motion can be written as:

Iθ + Cpθ +Kpθ −mglg sin θ = τ, (4)

where m is the total mass, lg is the distance from the ankle

to the CoM, g is the gravity acceleration, I is the moment

of inertia around ankle joint, and the other parameters are

obvious from the model. This equation is simplified by

ignoring the gravity term, which is justified by the high

gear ratios and the internal high-gain feedback controller of

HOAP-2.

We envisioned two types of disturbances: impact force and

continuous force. Consider first an impact force acting on

one of the robot’s shoulders. The moment balance equation

during impact is:

Iθ = malg, (5)

1827

-20-15-10

-5 0 5

10 15 20 25

0 5 10 15 20

Join

t an

gle

[deg

]

Time [s]

R AnkleR Hip

L AnkleL Hip

(a)

-4.0

-2.0

0.0

2.0

4.0

0 5 10 15 20

Acc

eler

atio

n [

m/s

2]

Time [s]

(b)

Fig. 5. Experimental results of Frontal-plane ankle strategy with impactforce.

where a is the acceleration measured by the internal ac-

celeration sensor of the robot located near the total CoM.

Assuming a small impact time interval Δt, and using the

relation θ = Δθ/Δt, we obtain the ankle joint angular speed

step change as

Δθ = malgΔt/I. (6)

The step change determines the initial state for the post-

impact motion. The latter is calculated from the following

joint acceleration derived from (4):

θref = −Cpi

Iθ − Kpi

Iθ, (7)

where a passive joint (τ = 0) is assumed. The acceleration

is then integrated twice to obtain the reference ankle angle

θref (t), which is finally fed to HOAP-2’s internal controller

for execution. With this method, the robot reacts faster to

larger impact forces.

Next, consider a continuous disturbance force acting on

one of the shoulders. In this case, the ankle joint cannot

be assumed to be passive. The ankle joint torque can be

derived from the foot moment balance equation using the

ZMP position:

τ = mg(rZMP − rcy) = frcz, (8)

where f denotes the external force applied at the CoM,

rZMP is the position of the ZMP, and rcy and rcz are the

two components of the CoM position. Incorporating these

relations under simplifying assumption into the equation of

motion (4), we can derive the ankle joint acceleration as:

θrefa = (m/I)g(rZMP − rcy)− (Cp/I)θ − (Kp/I)θ. (9)

This acceleration is integrated twice and fed into the robot’s

controller.

We should note that the spring Kp has to be designed

as a nonlinear spring coefficient in order to ensure smooth

and swift response to the disturbance and also, for balance

recovery. Two components have been included:

Kp(t) = Ksp(θ)−Kg(rZMP − rcy). (10)

Ksp(θ) denotes a component depending on the ankle angle.

It reduces the spring coefficient as the leg inclines. The

coefficient is determined via a 5th-order spline function. The

second component, Kg(rZMP − rcy), is used to improve

the response. It reduces the spring value when there is a

disturbance.

-15-10-5 0 5

10 15 20 25

0 5 10 15 20 25

Join

t an

gle

[d

eg]

Time [s]

R AnkleR Hip

L AnkleL Hip

(a)

-40-30-20-10

0 10 20 30 40 50

0 5 10 15 20 25

-8-6-4-2 0 2 4 6 8 10

r ZM

P -

rcy

[m

m]

f [N

]

Time [s]

(b)

Fig. 6. Experimental results of Frontal-plane ankle strategy with continuousand impact force. Black vertical lines show the timing of impacts.

The robot can distinguish between an impact and a

continuous-force disturbance using the embedded accelera-

tion sensor. We have determined experimentally a suitable

threshold value for the acceleration. Below the threshold, the

continuous-force disturbance controller is activated. When-

ever the acceleration exceeds the threshold, the impact con-

troller is activated. This controller is active until the robot

recovers balance and comes to rest. Thereafter, the con-

trollers are exchanged. Unfortunately, however, the switching

between the controllers may cause an acceleration and the

operation may become unstable. To deal with this problem,

we modify the nonlinear spring variable Kp(t) defined in

(10), as follows. We introduce a new spring component,

Ksm(t), that is used to ensure smooth switching between

the controllers. After the switch, Ksm(t) adjusts Kp to 0.

Thereafter, Ksm(t) becomes 0 by using a fifth-order spline

with a suitably set time interval. Hence, (10) becomes:

Kp(t) = Ksp(θ)−Kg(rZMP − rcy)−Ksm(t). (11)

Experiments: The initial position of the robot was standing

upright with initial joint angles: θi = 0 deg (i = 1, 2, 3, 4).

The virtual spring-damper values were set empirically, as

follows: Cp = 0.5 Nm·s/rad, 8 ≤ Ksp ≤ 25 Nm/rad,

Kg = 0.55 Nm/rad. Cpi = 1.5 Nm·s/rad, Kpi = 1.0 Nm/rad.

The time span for the Ksm-based transition was set to

0.36 s. Impacts and continuous disturbances were applied

to the shoulders of a HOAP-2 robot. The specific type of the

disturbance is determined from the acceleration readings: an

impact has occurred whenever the acceleration exceeded 1.5

m/s2; otherwise the disturbance is regarded as continuous.

First, an experiment with two impact disturbances and

Frontal-plane ankle strategy response was performed (see

Fig. 5). The timing of the two impacts applied to the robot

becomes clear from the acceleration measurement graph (b).

The joint angle graphs (a) shows that the robot reacted

swiftly and then restored the initial posture smoothly.

Next, an experiment with a series of both impact and

continuous force disturbances was performed (see Fig. 6).

The timing of the impacts is shown with vertical lines. The

rest of the deviations in both plots, the joint angle graphs

(a) and the rZMP − rcy graph (b), (a) denote responses to

continuous disturbances, substituting (b) into the (9). The

experimental data show that the robot reacts appropriately to

the two types of disturbances and restores the balance posture

smoothly. Note also that the robot can react to a disturbance

1828

appearing during a balance recovery phase (e.g. around 18

s).

B. Lift-leg Strategy

The lift-leg strategy was realized with the help of the RNS

method described briefly in Section II. Note first that (2) is

integrable:

Hflθ = L, (12)

where the integration constant L denotes angular momentum.

Further on, zero initial conditions are assumed, and hence,

L = 0. Since the last equation is underdetermined, we obtain

the following set of joint velocities:

θ = bn, (13)

where b is an arbitrary scalar and n ∈ �n is a vector from

the kernel of Hfl.

Next, referring to the model in Fig. 4(b), we note that only

the support leg’s ankle θ1 and the support leg’s hip θ2 joints

will be considered. The swing leg’s hip θ3 and the swing

leg’s ankle θ4 joints are fixed. The coupling inertia is then

Hfl ∈ �1×2 and we obtain:

[θ1 θ2]T = b[−Hfl2 Hfl1]

T , (14)

where

Hfl1 = m1lg1C1 +m2(l1C1 − 2l2S12 − lg3C123)

+m3(l1C1 − l2S12 + lg2C12),

Hfl2 = m2(−2l2S12 − lg3C123) +m3(−l2S12 + lg2C12)

and Si = sin θi, Ci = cos θi (i = 1, 2), S12 = sin(θ1 +θ2), C12 = cos(θ1+ θ2), C123 = cos(θ1+ θ2− θ3). Further

on, we set b = 1. Hence, the relation between the two joint

rates becomes:

θ1 = (−Hfl2/Hfl1)θ2. (15)

The reference hip joint rate θ2ref

is calculated from

the inverted pendulum equation for the upper body link.

Referring to (9), we can write:

θref2 =m

Ig(rZMP − rcy)− CL

I(θ1 + θ2)− KL

I(θ1 + θ2).

(16)

C. Integration of Frontal-plane ankle and Lift-leg strategies

We need to ensure smooth transitions between the Frontal-

plane ankle strategy and Lift-leg strategy during balance

control in response to a continuous disturbance.

First, we determine the condition for switching from the

ankle strategy to the lift-leg strategy as a function of the CoM

position. When the CoM position rcy reaches the center of

the supporting foot while moving within the Frontal-plane

ankle strategy, control is switched to the Lift-leg strategy and

the CoM position is maintained with the help of the RNS.

A suitable threshold value for switching is defined as rthcy .

Next, in the opposite case, when switching from the Lift-

leg strategy to the Frontal-plane ankle strategy, we monitor

the value of the supporting leg ankle joint θ1. A suitable

threshold value for switching is defined as θth1 .

When the Frontal-plane ankle strategy was invoked after

the Lift-leg strategy, it was confirmed by simulation that the

hip joint was not corresponding to a negative value of the

ankle joint, as required. This means that a negative value

of the ankle joint will be input to the hip joint during the

Frontal-plane ankle strategy. Note that if the ankle joint angle

doesn’t become equal to the absolute value of the angle of the

hip joint, this will cause a large joint rate during switching.

To cope with this problem, we added an additional transi-

tion phase. Accordingly, we determine each of the joint rates

as:

θref1 = (1−Kw)θ(t)1 +Kw

∫θrefa dt (17)

θrefi = −Kt{θi − (−θ1)} (i = 2, 3) (18)

θref4 = −Kt(θ4 − θ1). (19)

θ(t)1 is the ankle joint rate of the transition, θrefa is the

reference acceleration obtained from (9), Kw and Kt are

variables that change from 0 to 1 and from 0 to 150 1/s,

respectively, with the help of fifth-order splines and suitably

chosen timing. Kt plays the role of a variable feedback gain.

At the end, we obtain four distinct phases of motion. These

are summarized in Table I. The numbers in the brackets

denote equation numbers.

Experiments: The initial conditions were the same as those

in the previous experiment. The time span for the transition

was set to 1 s, within the spline functions for variables Kw

and Kt. The other constants were: rthcy = −39 mm, θth1 =−14.28 deg. Kp = 2 Nm/rad, Cp = 5 Nm·s/rad, KL =0.95 Nm/rad, CL = 2.9 Nm·s/rad.

An arbitrary external disturbance was applied to the left

shoulder of a HOAP-2 robot. The experimental data in-

cluding joint angles (a), ZMP position and CoM ground

projection (b) and the moment of the disturbance rZMP−rcy(c), are shown in Fig. 7. It is seen that the robot is able

to react to the disturbances as desired, switching smoothly

between the strategies and coming finally to rest. Figure 8

shows the respective snapshots.

V. CONCLUSIONS

Four reaction patterns have been identified for human

reaction to disturbances in the frontal plane. There is some

agreement with existing work in the biomechanics literature,

e.g. [17], but further studies are needed. Two of the reaction

patterns, the known ankle strategy and the newly identified

lift-leg strategy, were implemented with a small humanoid

robot HOAP-2. Two types of disturbances have been inves-

tigated: impact forces and continuous forces. The robot was

able to react swiftly and to switch smoothly between the two

strategies, as required to restore the balance.

The models used in this work were simple planar models.

In a future work, we intend to implement the method in

3D in order to incorporate the remaining two strategies

and to explore generation of more complex reactions and

balance recovery patterns via suitable transitions between the

strategies.

1829

TABLE I

MOTION PHASES AND RELATED VARIABLES.

Phase Strategyθ1 θ2 θ3 θ4 Monitored

Support leg’s ankle Support leg’s hip Swing leg’s hip Swing leg’s ankle variable

I Frontal-plane ankle (9) −θ1 −θ1 θ1 -II Transition - - - - rcyIII Lift-leg (15) (16) - - -

IV Transition(17) (18) (18) (19)

θ1Kw: 0 −→spline 1 Kt: 0 −→spline 150 Kt: 0 −→spline 150 Kt: 0 −→spline 150

V (I’) Frontal-plane ankle (9) −θ1 −θ1 θ1

-30-20-10

0 10 20

0 5 10 15 20 25

Join

t an

gle

[deg

]

Time [s]

I II III IVV

R AnkleR Hip

L AnkleL Hip

(a)

-80

-60

-40

-20

0

0 5 10 15 20 25

r ZM

P, r c

y [

mm

]

Time [s]

I II III IV V

Right edge

rcy rZMP

(b)

-15-10

-5 0 5

10 15

0 5 10 15 20 25

-3-2-1 0 1 2 3

r ZM

P -

rcy

[m

m]

f [N

]

Time [s]

I II III IV V

(c)

Fig. 7. Experimental results of Frontal-plane ankle strategy and Lift-legstrategy with continuous force. ”Right edge” denotes the outer boundary ofthe right foot.

Fig. 8. Snapshots of Frontal-plane ankle and Lift-leg strategy. (a): Initialposture; (b)-(c): Frontal-plane ankle strategy; (d)-(h): Lift-leg strategy; (i)and (j): Frontal-plane ankle strategy.

VI. ACKNOWLEDGMENTS

Partial support by Grant-in-Aid for Scientific Research

Kiban (B) 20300072 of JSPS is acknowledged.

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