2008 8th IEEE-RAS International Conference on Humanoid Robots
December 1 ~ 3, 2008 / Daejeon, Korea
A Friction Based “Twirl” for Biped Robots
Kanako Miura 1 , Shin’ichiro Nakaoka 1 , Mitsuharu Morisawa 1 , Kensuke Harada 1 , Shuuji Kajita
1
1
National Institute of Advanced Industrial Science and Technology (AIST), Japan
[email protected], [email protected] [email protected]
[email protected], [email protected]
Abstract— This paper presents preliminary results on generating turning motion of a humanoid robot by slipping the feet
on the ground. We start by presenting the fact that such slip
motion is used by humans based on actual human motion capture
data, and show how this is necessary for sophisticated humanlike motion. To generate the slip motion, we predict the amount
of slip using the hypothesis that the turning motion is caused by
the effect of minimizing the power generated by floor friction.
Verification is conducted through both simulation and experiment
with the humanoid robot HRP-2. A ”twirl” utilizing both feet
slip is successfully demonstrated.
I. I NTRODUCTION
In our daily life, we unconsciously exploit the fact that our
feet slip on the ground, when we turn or stop. However, most
current humanoid robot locomotion assumes no slip contacts.
As a result, the robots tend to make many small steps when
turning in place, which usually takes a lot of time to complete.
Taking all these steps does not seem effective in terms of both
energy consumption and stability. Thus we believe the use
of slip is important to apply quick and smooth robot motion.
This paper is a preliminary discussion of the slip phenomenon,
aiming to realize highly sophisticated human like motion on
humanoid robots.
The topics related to our final goal are as follows; elucidating slip phenomena in turning motion, creating a model
that predicts the amount of slip, and the realization of twirl
exploiting slip with a humanoid robot.
In the field of ergonomics and biomechanics, intensive
research has been performed on analyzing slipping and slip
induced in a fall[1] [2] [3]. However, there still exists some
controversy on the conditions which result in falling [4].
Conventional studies on biped locomotion in low friction
environment mainly focus on the prediction, avoidance, or
recovery from the slip. Boone and Hodgins[5][6] simulated
a running biped on a floor with a low friction area, by introducing a reflex control strategy. Park and Kwon[7] designed
a controller to enlarge the frictional force at slipping, and
simulated a 12-DOF biped robot walking on slippery surface.
Kajita[8] described a calculation of ZMP concerning slip.
Though ZMP[9] is useful to check the possibility of falling
down, the conventional way to calculate ZMP cannot be used
in case of slip. It provides a good prediction of falling caused
by slipping. They also successfully demonstrated a walk on
low friction environment using the humanoid robot HRP-2.
There exists some work that addresses the issues regarding the proactive use of slip for humanoid motion.
Nishikawa[10][11] proposed a mechanical method for biped
978-1-4244-2822-9/08/$25.00 ©2008 IEEE
robots to use slip by adding a piston to the heel of the foot to
generate rotational slip. Although the proposed mechanism is
based on the idea of generating friction torque, the physical
model is not numerically revealed. There also exists a study
on the application of slip to the turning motion of a small
humanoid robot HOAP-2 by Koeda[12]. The robot pivots
the foot keeping contact with the floor, instead of lifting
the foot. They claim that the robot is able to keep large
support polygon while turning. The proposed motion used
point contact between one corner of the pivoting foot and the
ground for 90-deg turning motion.
Fig. 1 shows a photograph sequence of a professional
walking model performing a 360 degrees turn. By slipping the
soles of both feet, she turns speedily and smoothly. Aiming
our sights on this motion, we start investigation into the
fundamental properties of rotational slip during a twirl.
Fig. 1.
Rei)
360 degrees turn by a professional walking model (Walking Studio
To realize desired twirling motion, the relation between a
robot motion and the resulting angle is investigated in this
paper. We want to achieve a better understanding of the
phenomenon of rotational slip through this study.
The rest of this paper is organized as follows. After presenting the fact that such slip motion is used by humans based
on actual human motion capture data, we make a hypothesis
that the generated turning motion is caused by the effect of
minimizing the power generated by floor friction in Section
III. According to the hypothesis, we examine pre-calculated
pattern and verify the adequacy of the proposed model in
Section IV. In Section V, 90-deg ”twirl” utilizing both feet
slip is successfully demonstrated by humanoid robot HRP-2.
We conclude this paper and address our future plans in Section
VI.
II. H UMAN T URNING M OTION A NALYSIS
We captured various turning motion performed by a professional walking model using a 3D optical motion capture
system. Fig. 2 is a plot of foot placement during the 180
degrees turn made by the walking model. Fig. 3 shows the
direction of the body (waist) of the model, and relative angles
of both feet. The area with background color signifies the
slipping phase. A twirl procedure obtained from analysis of
the captured motion is described as follows:
a: Landing on the left foot twisting toward the direction
of rotation (Second step in Fig. 2). The body and the right
leg start turning left.
b: Landing on the right foot (Third step in Fig. 2) twisting
and sliding toward the direction of rotation. The body still
remains at a shallow angle.
c: Starting slip of the left foot.
d: Stopping rotation of the right foot. The body and the
left foot are still moving.
e: Stopping rotation of the left foot and the body. It
restarts motion to the next step (Fourth step in Fig. 2).
Both feet rotate during the period between c and d in the
figures. Compared with a 90, or 360 degrees turn, it was
observed that the larger the total angle of rotation becomes, the
larger the angle in this period is. On the other hand, rotational
slip does not occur during b to c when the total angle of
rotation is small. The step just before the turn, the second
step in Fig. 2, also tends to twist in correlation to the total
angle of rotation. The data set was not large enough to supply
the relations with quantitative evidence, thus we can only say
that they have such tendencies.
Human turning motions are realized not only statically
through the trajectory of the feet but also through dynamics
such as the moment of inertia caused by the twisting upper
body or a swinging arm. However we do not go over the
dynamical effects in this paper. In the rest of this paper, we
focus on the rotational slip of the feet and the twirl of the
body caused by it.
Fig. 2. Foot placement during 180[deg] turn by a professional walking model.
The red polygons represent the footsteps of right foot, and blue represents the
left.
III. ROTATION M ODEL FOR E XPLOITING S LIP
Fig. 3.
Waist pose and relative foot angles during 180[deg] turn by a
professional walking model. The red area represents the slipping phase of
the right foot, and blue represents the left. Purple represents simultaneous
slip with the both feet.
In this section, a model of rotation caused by friction force
is proposed. In this proposition, we use the motion as shown
in Fig. 5. The left hand side of Fig. 5 shows the given motion
of feet. The dotted boxes represent the positions of the feet
at the beginning of the motion. The robot stands with its feet
2d apart along the X axis, and 2b apart along the Y axis. The
robot moves them according to the arrows and reaches the
positions represented with solid boxes. The given trajectories
of the feet are both parallel to the x-axis with a distance of
b. Our assumption is that the robots body will rotate from
the reaction force. The expected final positions of the feet
are shown in the right hand side of Fig. 5. In the figure,
the coordinate axis xW yW signifies the world coordinate, and
xB yB signifies the floor projection of the base frame of the
robot which origin is on the pelvis as shown in 4. At the start
of the motion, the two coordinates are exactly the same. We
define theta to be the angle between the world coordinates and
the robot base coordinate. When the robot moves, we expect
theta to increase.
We make a hypothesis that the generated turning motion is
caused by the effect of minimizing the power generated by
floor friction. To simplify, we adapt the following assumption
used by Harada[13] which is based on the method proposed
by Lynch[14]:
1) The model assumed quasi-static motion, in other words,
the motions are slow enough that frictional forces dominate inertial forces.
2) The dynamic and the static coefficient of friction acting
between the feet and the floor are the same.
3) The plane of floor is uniform and horizontal, and the
friction distribution on the sole of the foot does not
change.
According to the assumption 1, we deal in the twirl only by
mechanistic effect of the lower body motion, and the turn built
on the momentum of whole body will be covered in future
work. In this paper we also assume the robot applies the same
weight to the right and left feet, uniformly.
Fig. 6 shows the motion of the right foot for explanation of
the hypothesis. The point P is on the sole of a foot, and it
slides to P while the robot moves its feet. The point P and
P are at a distance of (x, y) from the center of the sole in
the longitudinal and lateral direction, and their positions are
expressed as follows:
Px + x
(1)
P =
Py + y
Cθ −Sθ
x
P =
(2)
Sθ
Cθ
Py + y
Fig. 4.
Model of a robot and its base frame, and world frame.
where, Px and Py are the position of the center of the sole in
the world frame at the beginning, and Sθ = sinθ, Cθ = cosθ.
The loss of the power at P can be calculated by multiplying
the frictional force applied to small area at P by the sliding
distance |P − P |:
WP
f |P − P |
=
(3)
where f is the frictional force per unit area.
Integrating over the area of sole, we obtain expressions for
the total loss of the power at the sole of right foot:
Wright
=
Fig. 5. Motion of both feet: The left shows the given pattern, the right shows
the expected motion. The rotational torque may occur by friction force.
=
ly
2
ly
2
ly
2
lx
2
WP dxdy
− l2x
−
l
− 2y
lx
2
f |P − P | dxdy
− l2x
(4)
where lx and ly are the length and width of the sole, respectively. The frictional force per unit area is given by the
coefficient of friction multiplied by the normal force:
f
μM g 1
2 lx ly
=
(5)
where μ, M , and g are the friction coefficient, the total mass of
the robot, and the gravity coefficient, respectively. In practice,
since the applied normal force f is equal at any point of the
sole, we obtain:
Wright
Fig. 6.
Hypothesis to explain the slip phenomenon
=
μM g
2
ly
2
l
− 2y
lx
2
− l2x
|P − P | dxdy
(6)
The loss of the power on the left sole is the same as the right
since its foot motion is symmetrical to the right about the
origin of the base frame of the robot. Substituting Eq. 1 and
2 into Ew. 6, we obtain the total loss of the power at the both
soles W:
W
=
μM g
|P − P |
−
ly
2
lx
2
|P − P | dxdy
− l2x
(7)
2
T Cθ
x
Px + x −
−Sθ
Py + y
2
Sθ
x
Py + y −
(8)
Cθ
Py + y
2
ly
2
=
+
In order to calculate θ that would minimize W, we do the
following. Differentiating Eq. 7 with respect to θ yields:
∂W
∂θ
∂
μM g
∂θ
=
ly
2
−
ly
2
lx
2
− l2x
|P − P | dxdy
(9)
Since θ that would minimize W requires Eq. 9 to be zero, we
obtain:
l2y l2x
∂
|P − P | dxdy = 0
(10)
∂θ − l2y − l2x
the proposed model. In these simulation and experiment, we
keep to the same motion as shown in the last section.
In the experiment, the robot moves the feet back
and forth using various distances of d[m]. The
start position of the right foot varies over d =
{−0.08, −0.06, −0.04, −0.02, 0.02, 0.04, 0.06, 0.08, 0.10}[m].
The sideway distance of the feet are kept at 2b = 0.19[m].
The vertical axis rotation of the body at the final position is
measured.
For the simulation, we used OpenHRP, which is a
dynamic simulator developed in the Humanoid Robotics
Project(HRP)[15]. Simulated robot model is HRP-2, a 30 DOF
humanoid robot which was also developed in HRP. Fig. 8
illustrates the simulation result at μ = {0.1, 0.2, 0.3, 0.4, 0.5}.
In the simulation we found that the friction between feet and
floor do not affect the slip angle. This result, that the friction
coefficient does not affect the rotation, agrees to our hypothesis
proposed in the last section.
Since this equation has neither friction coefficient, nor the total
mass of the robot, or the gravity acceleration, the shape of the
sole and given motion pattern alone decide the rotation of the
robot. This also means that the friction coefficient does not
affect the rotation.
Eq. 10 can also be described by the square of |P − P | as
follows:
W∗
=
ly
2
ly
2
ly
2
−
=
−
ly
2
lx
2
− l2x
lx
2
− l2x
Fig. 7. Experiment: foot motion of HRP-2 slipping and turning on the low
friction floor
|P − P | dxdy
2
2
{(1 − Cθ )x + Sθ (y + Py ) + Px }
2
=
+ {(1 − Cθ )(y + Py ) − Sθ x} dxdy
4
(1 − Cθ ) 2lx3 ly + lx (ly + Py )3 − (−ly + Py )3
3
+2Sθ Px lx (ly + Py )2 − (−ly + Py )2
+4Px2 lx ly
∂W ∗
∂θ
=
4 3
Sθ lx ly + lx (ly + Py )3 − (−ly + Py )3
3
+8Cθ Px lx Py ly
=
0
(11)
Finally we obtain:
tanθ
=
−
3Px Py
2
lx + ly2 + 3Py2
(12)
IV. S IMULATION BY O PEN HRP AND E XPERIMENT WITH
HRP-2
This section reports the results of a simple slip test with a
humanoid robot by simulation and experiment in order to find
out features of rotational slip, and to verify the adequacy of
Fig. 8. Comparison between different floor friction coefficients. The position
d of the right foot at start position on the x-axis and the rotation at the final
position on the y-axis.
The humanoid robot HRP-2 was used for the experiment.
To obtain low friction, small plastic plates are put on both
side of the foot of HRP-2, where contact with a floor, as shown
in Fig. 9, and a large sheet was set on the floor. The friction
coefficient between the two plastic sheets was 0.23. These
sheets were used because the soles of HRP-2 are covered with
a rubber-like material. The friction coefficient of the rubber
soles were μ > 1.0 on the lab floor, which may cause excessive
motor load when the feet rub against the floor. Thus Eq. 12
is changed as follows:
Fig. 9. The left shows the feet of HRP-2 with plastic plates. The right shows
the figure of the foot. In case of HRP-2, lx = 0.23[m], ly = 0.135[m],
lyc = 0.04[m].
tanθ
= −
3Px Py
lx2 + ly2 + ly lyc + ly2c + 3Py2
(13)
Fig. 10 shows a sequence of photographs of the feet of
HRP-2. Fig. 11 illustrates the results comparing between
the numerical solution of Eq. 13, the simulation and the
experiment. They agree well with each other. It can be said that
the proposed hypothesis is adequate, and that the simulation
agrees rather well with experiment, so the result also shows
accuracy of the OpenHRP simulator. Linear approximation is
also illustrated in Fig. 11. The line for experiment with HRP-2
is not on the origin of the coordinate. The distribution of mass
on both feet of the robot is not completely even. This may have
displaced the robot’s center of mass during the experiment.
Fig. 11. Comparison with numerical result of the proposed model, simulation,
and real experiment . The position d of the right foot at start position on the
x-axis and the rotation at the final position on the y-axis.
are same as the experiment in section above. The pattern of
the motion was designed in reference to a series of human
turning motion, which has been analysed in Section II.
The trajectory of the feet during the turn is shown in Fig.
12. The robot walks straight ahead before and after the 90-deg
turn.The feet are moved in circles so as to fix the stance during
the turn, and it is different from the linear motion used in the
last section. We adopted straight feet path in the last section
in order to deal in the simplest model. The reason for this
change was because we worried that moving the foot twisted
inwards towards the other foot, would cause a collision. We
also expected the motion to look more natural if the distance
between the feet was fixed.
Fig. 12. Motion pattern of the feet. The left shows the second step of Fig.
2. The right shows the twirl with both feet slipping.
Fig. 10. Experiment: foot motion of HRP-2 slipping and turning on the low
friction floor
A sequence of photographs of the demonstration of HRP-2
is shown in Fig. 13, and you can see that a smooth 90-deg turn
has been realized. The 90-deg turn took around 2.5 seconds
to complete. By our conventional way of pattern generation,
the 90-deg turn motion normally spends over 8 seconds with
many small steps.
V. D EMONSTRATION OF T WIRL BY HRP-2
VI. C ONCLUSIONS
This section demonstrates a 90-deg twirl exploiting slip
using HRP-2. The conditions and parameters of the experiment
A preliminary discussion on twirling a humanoid robot
by utilizing the slip between the feet and the ground was
Fig. 13.
Demonstration of HRP-2 turn
described. By using human motion capture data, we figured a
procedure for the twirl by extracting the essence from human
motion. We also conducted both simulation and experiment
with the humanoid robot HRP-2. To generate the slip motion,
we predict the amount of slip under the hypothesis that the
turning motion is caused by the effect of minimizing the
power generated by floor friction. Even though the calculated
trajectory was similar to the experiment and the simulation,
the difference warrants further investigation.
A turning motion utilizing both feet slip is successfully
demonstrated with the humanoid robot HRP-2. Because of the
fewer DOF and limited joint range of HRP-2, it is impossible
to stand on the toes to get smaller friction radius, or to twist
its leg as far as a human. Thus the generated motion was far
from a 360 degrees turn shown by the human. Furthermore,
humans utilize inertial force by twisting the upper body and
swinging the arms. Future work will also address this effect
to achieve highly sophisticated human like motion.
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