Proceedings of the 2003 IEEE/RSJ
Intl. Conference on Intelligent Robots and Systems
Las Vegas, Nevada · October 2003

Experimental Study on Energy Efficiency
for Quadruped Walking Vehicles
Ryo Kurazume†

Ahn Byong-won††

Kazuhiko Ohta†

Tsutomu Hasegawa†

Kyushu University†
Mokpo National Maritime University††
Graduate School of Information Science
Division of Marine Engineeringand Electrical Engineering
6-10-1, Hakozaki, Higashi-ku
571-2, Chukyo-dong
Fukuoka, 812-8581
Mokpo, Jeonnam, 530-729
JAPAN
KOREA
kurazume,ahnbw,kazu-o,hasegawa@irvs.is.kyushu-u.ac.jp
VehicleAbstract— Though a legged robot has high terrain adaptability as compared with a wheeled vehicle, its moving
speed is considerably low in general. For attaining a highmoving speed with a legged robot, a dynamically stable gait,such as running for a biped robot and a trot gait or abound gait for a quadruped robot, is a promising solution.
However, the energy efficiency of the dynamically stable gait
is generally lower than the efficiency of the stable gait suchas a crawl gait. In this paper, we present an experimental
study on the energy efficiency of a quaduped walking vehicle.
Energy consumption of two walking patterns for a trot
gait is investigated through experiments using a quadrupedwalking vehicle named TITAN-VIII. The obtained results
show that the 3D sway compensation trajectory proposed in
our previous paper [10] has advantages in view of energy
efficiency as compared with the original sway compensation
trajectory.

Inertia

GravityGroundZMP (Zero Moment Point)
Fig. 1.

classified as a “safety gait” [7] which avoids complete
tumbling by touching a swing leg to ground. However, inthe trot gait, the body of a robot has to be supported bydiagonal legs, and its posture becomes unstable.
For realizing a “stable” trot gait, we have proposed thesway compensation trajectory[16]. This method utilizes its
body motion along a lateral direction to keep its bodybalance. By this body motion, a zero moment point (ZMP)
is controled to move on a desired trajectory. The ZMP is
the projection onto the ground of the force acting on the
body as shown in Fig.1. At this point, no moment exists.
Therefore, by keeping this point on the diagonal line of the
supporting legs, the robot enables to maintain a walking
stability, since there is no moment around this line whichcauses the inclination of the robot body. The proposedsway compensation trajectory uses lateral (along y axis)
body motion to keep the ZMP on a diagonal line between
the supporting legs as shown in Fig.2.
In general, energy efficiency of the dynamically stablegait is lower than the efficiency of a statically stable gaitsuch as a crawl gait. This is mainly due to the fact thatmuch power has to be supplied at each joint to support abody by few legs. This problem is quite critical, especially
if the robot has to carry its own energy source such as aI. I NTRODUCTION
A legged robot has high terrain adaptability as compared with a wheeled vehicle. It enables not only to aviod
an obstacle by going round, but also to stand on or evenstep over an obstacle. Climbing a steep sloop by hooking
its soles, or jumping across a ditch are also possible.
However, a moving speed of a legged robot is considerably
lower than the speed of a wheeled vehicle in general.
For attaining a high moving speed with a legged robot, adynamically stable gait, such as running for a biped robot
and a trot gait or a bound gait for a quadruped robot, isa promising solution. However, since a robot body has tobe supported by few legs in the dynamically stable gait,
special attitude control is requisite for keeping a postureof the robot stable.
For a quadruped walking vehicle, a trot[16],[8],[18],[17], a pace [13], and a bound [12],[3],[4]gaits have been widely noted as fundamental dynamicallystable gaits. Amang these gaits, we have particularly
noticed the trot gait since it has close affinity to a crawl
gait, which is a typical standard statically stable gait. In
addition, the trot gait has an advantege that it can be

0-7803-7860-1/03/$17.00 © 2003 IEEE

Zero moment point(ZMP)

613

The sway compensation trajectory

trajectory.
In this paper, we present an experimental study on theenergy efficiency of a quaduped walking vehicle. Energy
consumption of the sway and 3D sway compensation
trajectories is investigated through experiments using aquadruped walking vehicle named TITAN-VIII, which iscommonly used in many research groups. The obtained
results show that the proposed 3D sway compensation trajectory has advantages in its energy efficiency as comparedwith the original sway compensation trajectory.

ZMP
z (Vertical direction)x (Longitudinal direction)y (Lateral direction)

Fig. 2.

The sway compensation trajectory

II. T HE 3D SWAY COMPENSATION TRAJECTORY
In this section, we give formulation of the conventionalsway compensation trajectory and the 3D sway compensation trajectory which is the expansion of the conventional
one to a sagittal direction.

battery.
The energy efficiency of various legged robots including a monopod[6], a quadruped robot[1], and ahexapod[11],[14], [15] has been investigated in the past.
However, the most of these studies limits their discussionin computer simulation and qualitative analysis.
Gregorio et al.[6] experimentally investigated the energy
consumption of their single legged robot named the ARL
Monopod. They also showed comparison of the specificresistance[5] for several walking vehicles developed in thepast, and human and car.
Kimura[9] examined the energy efficiency of aquadruped vehicle and showed the relation between thewalking cycle and energy consumption for trot and pacegaits. However, a quadruped walking vehicle was modelled as an inverted pendulum and the dynamically stablegait which always maintains the equilibrium state, such as
the gait performed by the sway compensation trajectory,
was not examined.
If the sway compensation trajectory is examined froma viewpoint of energy efficiency, the energy consumption
is expected to be larger than in the case without specificcontrols of body trajectory, since acceleration and deceleration of the body in the lateral direction has to be repeatedperiodically.
However, the position of the ZMP is controllable not
only by the lateral motion but also by the sagittal andvertical motion, respectively. Thus, by combining thesemotions, it may be possible to improve the energy efficiency against the conventional sway compensation trajectory which controls the position of the ZMP only by
the lateral motion.
From these consideration, we proposed the 3D sway
compensation trajectory [10]. This method utilizes the
body motion along not only a lateral direction but alsoa sagittal and a vertical directions. The 3D sway compensation trajectory make it possible to move the robot bodymore smoothly in terms of acceleration. Thus, the 3D sway
compensation trajectory was considered to give higher
energy efficiency than the original sway compensation

A. Conventional sway compensation trajectoryFirst, the formulation of the conventional sway compensation trajectory is presented. Here, we consider a vehicle
as a point mass at xg yg zg . If the ground is flat andthe height of the body from the ground, z, is constant, theposition of the ZMP x z yz 0 is given as



xz
yz



xg
yg



x¨g
y¨g

A



(1)

z

where, A  gg . And, the diagonal line of the support legs
is defined as(2)cos θ x  sin θ y  dThen, in order to stay the ZMP on this line, the center ofgravity has to satisfycos θ xgAx¨g  sin θ yg

Ay¨g   d(3)Assuming that the vehicle moves along the x axis andacceleration of the center of gravity along the moving
direction is constant, the position of the body along the xaxis is expressed as
xg  ax2t 2  ax1t  ax0(4)

By substituting this equation into Eq.(3), we getcos θ ax2t 2  ax1t  ax0

2Aax2   sin θ yg

Ay¨g   d (5)The solution of this differential equation, y g , is given as
the addition of the solutions of the next equationygthat is,Ay¨g  0ttyg  C1y e A  C2y e A

(6)
(7)

and one particular solution that satisfies Eq.(5). Here, we
assume that the general form of the particular solution isexpressed by a polynomial expression of time t; then thesolution of Eq.(5) is derived as
t
t
(8)
yg  C1y e A  C2y e A  ay2t 2  ay1t  ay0

614

From the boundary condition about the continuity ofa trajectory ẏ g t 0  ẏg t T  0,yg t 0  yg t T , each22
coefficient is determined as
T
Tax2  1 e 2 A vy
C1 A cot θ(9)T
T
e2 A e 2 A 
T
Tax2  1 e 2 A vy
A cot θ
(10)
C2 
T
T
e2 A e 2 A ay2 
2ax2 cot θ
(11)
ay1
ay0

ax1 cot θ
ax0 cot θ  d csc θ






T
T
AT e 2 A  e 2 A  2TT
e2 A e 2 A 

T2 x
a
4 2

(15)

By substituting the boundary condition about the continuity of a trajectory, the following equations with twoparameters ax2 and ax1 are derived.



C2x 

C1y 



C2y 

2L

T2

2L



4 AT ax2  2Tax1
T
8 e 2 A 1T
Tax2  1 e 2 A ax1
A cot θT
T
e2 A e 2 A TTax2  1 e 2 A ax1
A cot θT
T
e2 A e 2 A 

(16)

(17)

(18)

(19)

ay2  ax2 cot θ(20)

ay1  ax1 cot θ(21)

ay0  ax0 cot θ  d csc θ

(22)

T2 x
a
4 2

T x
a
4 1
(23)

T
2

0

x¨g 2  y¨g 2  z¨g 2 dt(24)Here, only a regular walk a x2  0 is considered forsimplicity.
First, for the conventional sway compensation trajectoryin which the body is moved only in a sagittal direction, thesum of squared acceleration through the entire trajectory,
ρ , is obtained by substituting a x1  TL into Eq.(24) asTT
L2 A 1  e A  e 2 A  cot2 θ
ρ
(25)
T
A 1  e 2 A 2 T 2
Next, the proposed 3D sway compensation trajectory
including a sway in the sagittal direction is considered.
Since the parameter that can be designed is a x1 , ax1 thatminimizes sum of squared acceleration through the entire
trajectory is given by solving
∂ρ
0
(26)
∂ ax1
as
T
1  e 2 A 2 LT
x
a1 
(27)
T
T
1  e 2 A 2 T 2  16A 1  e 2 A 2 cot2 θAnd the minimum of the sum of squared acceleration ρ
is derived asT
T
L2 A 1  e A  e 2 A  cot2 θ
ρ
(28)
T
T
A 1  e 2 A 2 T 2  16A 1  e 2 A 2 cot2 θ 
By comparing Eqs.(25) and (28), the sum of the squared
acceleration for the 3D sway compensation trajectory issmaller than the sum for the conventional sway compensation trajectory except T  0, θ  π2 , or A  H g  0 ∞.
Therefore, the energy consumption is also expected to be
reduced by the 3D sway compensation trajectory.

The conventional sway compensation trajectory mentioned above is expanded to the form including a swayalong the sagittal direction.
First, Eq.(3) is decomposed into two equations for the x
and y directions, and each solution trajectory is assumed
to be given as Eq.(8) andT 2  4 AT ax2  2Tax1
T
8 e 2 A 1T
T
AT e 2 A  e 2 A  2TT
e2 A e 2 A ρ

B. Expansion for sagittal motion

C1x 



C. Evaluation from a viewpoint of acceleration
The energy consumption of a walking vehicle is affectedby many factors such as mass of the body and legs,configuration of the degrees of freedom, trajectories ofbody and legs, the negative power at each actuator, etc[11],[1]. In this section, the trajectory which minimizes thetotal of the external force applied dynamically, that is, thesum of squared acceleration through the entire trajectory
as defined below, is considered.

(12)
(13)

t
t
xg  C1x e A  C2x e A  ax2t 2  ax1t  ax0d1
cos θ 2where, L is the body stroke in one walking cycle.
We define this expansion of conventional sway compensation trajectory along a sagittal direction as “the 3D sway
compensation trajectory”. Expansion to a vertical directionis also possible[10] in the same way as the formulation toa sagittal direction.

T x
a
4 1
(14)
where, T is a walking cycle. This solution is a trajectoryof the center of gravity that always keeps the ZMP on the
diagonal line of the support legs, and this trajectory is defined as the (conventional) sway compensation trajectory.
ax0 

d
1

cos θ 2

ax0 

615

40

Recovering

35

SupprtingPower [W]

30
25
20
15

Joint 1

10
0Fig. 3.

Joint 2

Joint 0

5
0

0.2

0.4

0.6 0.8
1
Time [s]A simulation model

1.2

1.4

(a) The sway compensation trajectory
40

Recovering

35

III. C OMPUTER SIMULATION AND POWER

SupprtingPower [W]30

MEASUREMENT EXPERIMENTS

A. Computer simulation
To illustrate the advantage of the energy efficiency ofthe 3D sway compensation trajectory, we develop a setof computer simulation. The configuration of degrees of
freedom, weight, etc. of a quadruped walking vehicle
model for computer simulation is the same as the TITANVIII [2] (Fig.3) which is used in following experiments.
Fig.4 shows the power consumption of leg 2 for theconventional sway compensation trajectory and the 3Dsway compensation trajectory. It is clear that the power
consumption of the 3D sway compensation trajectory at
joint 2, which mainly produces the lateral sway motion,
becomes lower than the power consumption of the conventional sway compensation trajectory.
Fig.5 depicts the total energy consumption for a walking
cycle in the cases where walking speed is changed from0.05 to 0.20 [m/s], and γ  a x1 v0 v0  TL  is changed
from 0.2 to 1.0. γ  10 is for the conventional sway
compensation trajectory, and 0  γ  10 is for the 3Dsway compensation trajectory. As γ decreases, the sagittalsway body motion increases. In all the cases, it is verified
that the energy consumption of the 3D sway compensation
trajectory is lower than the energy consumption of theconventional sway compensation trajectory.

25
20
15

Joint 1

10
0

Joint 2

Joint 0

5
0

0.2

0.4

0.6 0.8
1
Time [s]1.2

1.4(b) The 3D sway compensation trajectory
Fig. 4.

Power consumption of leg 2. (cycle time is 1.5 [sec.])

Total energy for one cycle [J]160

Fig. 5.
ax1 v0 )B. Experiments with TITAN-VIII140

γ =1.0γ =0.8120
100
80 γ =0.6
γ =0.460
γ =0.2
40

Solution of Eq(27)20
00.06 0.08 0.1

0.12 0.14 0.16 0.18 0.2
Velocity [m/s]

Total energy consumption for various waking patterns. (γFig.8 shows the power consumption of the leg 2 for thesway compensation trajectory and the 3D sway compensation trajectory In the same way as the simulation results,
the power consumption of the 3D sway compensation
trajectory at joint 2 is lower than the power consumptionof the conventional sway compensation trajectory. Though
the energy consumption at each actuator seems to bedifferent between Fig.4 and Fig.8, this is mainly due
to the difference of joint configuration, that is, TITAN
VIII utilizes the interfere mechanism[2] to improve theefficiency of power consumption at each actuator.
Fig.9 depicts the total energy consumption for a walking

Next, we develop power measurement experiments using a quadruped walking vehicle, TITAN-VIII[2] (Fig.6).
The TITAN-VIII was designed at Tokyo Institute of Technology in 1996, and since it was released, it has been
widely used as the common platform of a quadrupedwaking vehicle in many research groups in Japan.
In order to measure the power consumption at each actuator, we designed an analog circuit (Fig.7), and installed
it in each motor driver. Voltage and current supplied toactuators are multiplied by an analog multiplier via low
pass filters, and stored in a PC via an A/D converter.

616

30

Power measurement circuit

Joint 1

25
Power [W]20

Joint 0
Joint 1

Joint 2

10
5
0
-5

Joint 2

Fig. 6.

Joint 015

Recovering

-10
0

0.2

0.4

Supprting

0.6 0.8 1
Time [s]1.2

1.4(a) The sway compensation trajectory

TITAN VIII with a power measurement circuit

30
Joint 1

25
Power [W]20

Joint 0

15

Joint 210
5
0
-5Recovering-100Fig. 7.

0.2

0.4

Supprting

0.6 0.8 1
Time [s]1.2

1.4(b) The 3D sway compensation trajectory

A power measurement circuit diagram

Fig. 8.

Total energy for one cycle [J]cycle, which is obtained as the average of ten walking
cycles, in the cases where walking speed is changed from
0.067 to 0.20 [m/s], and γ  a x1 v0 v0  TL  is changed
from 0.2 to 1.0. In all the cases, it is verified the energy
consumption of the 3D sway compensation trajectory(γ  02  08) is lower than the energy consumption of
the conventional sway compensation trajectory (γ  10).
As γ decreases, the energy efficiency improves. However,it becomes difficult to keep body balance since large
acceleration and deceleration toward a sagittal direction
is demanded.
The solution of Eq.(27), that is, the parameter which
minimizes the sum of squared acceleration, is also examined, and the energy consumption is plotted in Fig.9.
Though the energy consumption is better than the conventional sway compensation trajectory, it is not an optimum
parameter for low moving velocity. This is due to theactual leg structure, joint friction, or other various factors
which is caused by simplification for analysis.
Next, we compare the energy efficiency between the
proposed trot gait and a crawl gait, which is one of thetypical statically stable gait. The specific resistance[5] hascommonly been used to compare energy efficiency ofvarious moving vehicles. The specific resistance is definedas ε  m Eg L , where E is the energy consumption, m is
the mass of the vehicle, g is the gravitational acceleration,and L is the moving distance. In our case, these values arem  201Kg and L  02m.

Power consumption of leg 2. (cycle time is 1.5 [sec.])

35
30
25
20
15

Solution of Eq(27)γ =1.0γ =0.8γ =0.4γ =0.210
50Fig. 9.
ax1 v0 )γ =0.60.06 0.08 0.1

0.12 0.14 0.16 0.18 0.2
Velocity [m/s]

Total energy consumption for various waking patterns. (γFig.10 dipicts the experimental results for the cases
where the duty factor β is changed from 0.5 to 0.9,
that is, β  05  07 for the trot gait with the 3Dsway compensation trajectory, and β  075  09 for ageneralized trot gait[8],[16].
This figure shows that i) the energy efficiency of thecrawl gait is better than the efficiency of the trot gait forthe same moving velocity, and ii) the energy efficiency of
the trot gait becomes worth as the duty factor increases.

617

Specific resistance ε

1
0.8

trot

0.6

crawl

β=0.7 β=0.65
β=0.6
β=0.75β=0.55β=0.5
β=0.85β=0.8

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β=0.9

0.2
0
0.02

Fig. 10.

0.03

0.04
0.05
Velocity [m/s]

0.06

0.07

Specific resistance for various walking patterns

IV. C ONCLUSION
For a dynamically stable trot gait of a quadruped
robot, we have proposed two control schemes; the sway
compensation trajectory and the 3D sway compensation
trajectory. In this paper, we examined power consumptionof both control schemes experimentally using a quadrupedwalking vehicle, TITAN VIII. The experimental resultsshows that the 3D sway compensation trajectory has
advantages in view of its energy efficiency as comparedwith the original sway compensation trajectory.
V. ACKNOWLEDGMENTS
This research was partly supported by the 21st Century
COE Program ”Reconstruction of Social Infrastructure
Related to Information Science and Electrical Engineering”.
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