An Optimal Working Function Based on the Energetic
Cost for Myriapod Robot Systems: How Many Legs
Are Optimal for a Centipede?
B. T. NOHARA
T. NISHIZAWA
Department of Electric and Computer Engineering, Musashi Institute of Technology,
Tamazutsumi, Setagaya, Tokyo, Japan ([email protected])
Received 28 May 20041 accepted 15 February 2005
Abstract: The objective of this paper is to obtain working functions for the legs of a myriapod robot from a
kinetic energy point of view. The realization of the high performance of energy consumption is indispensable
in the battery-based robot system. We introduce the cost of transport and reduce to the minimum problem of
the cost of transport. The calculus of variations is applied to obtain governing equations and the functions for
legs. We obtain optimal functions for legs in an octarupedal robot.
Key Words: Calculus of variations, gait pattern, myriapod robot, walking robot
NOMENCLATURE
E
JG
JL
JST
JSW
K
KE
KT
M
R
S
Sn
T
U
V
Vn
e
g
kinetic energy consumed by the robot
inertia of the embedded gear of the motor
leg inertia of the lift motion at the swing phase
leg inertia of the swing motion at the stance phase
leg inertia of the swing motion at the swing phase
kinetic energy
induced electromotive force constant
torque constant
weight of the robot
internal electric resistance
stride of the robot
non-dimensional stride
period of the movement of the leg
potential energy
velocity of the body
non-dimensional velocity
kinetic energy consumption of a leg
gravity due to acceleration
Journal of Vibration and Control, 11(10): 1235–1251, 2005
DOI: 10.1177/1077546305054149
1 2005 Sage Publications
1
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1236 B. T. NOHARA and T. NISHIZAWA
i
l
m0
m1
n
r ST
r SW
t
1
2S
2W
3
4
6
7
8
9 S1
9 S2
9 W1
9 W2
9 W3
electric current
radius of the lift motion
mass of each leg including a motor
mass of only a leg
number of legs
radius of the swing motion at the stance phase
radius of the swing motion at the swing phase
time
voltage
angle of the motion of the swing motion
angle of the motion of the lift motion
duty fuctor
K E 5K T
cost of transport
attenuation ratio
Lagrange multiplier
working function of the swing motion at the stance phase
working function of the lift motion at the stance phase
working function of the swing motion at the swing phase
working function of the lift ascent motion at the swing phase
working function of the lift descent motion at the swing phase
torque
angular velocity
1. INTRODUCTION
In the course of studying walking robots, many researchers have focused on the walking mechanism or algorithm (Hirose et al., 19971 Hirai and Hirose, 1999) without tumbling for biped robots, which mimic a human’s walking system. A few institutes (see
http://www.honda.co.jp/robot/ and http://www.sony.co.jp/SonyInfo/QRIO) have developed
biped robot systems to show off their technical might. However, we still have problems in
realizing biped robots in industries, for example, cost/performance, lack of functions, the stability of dynamic walking (Kajita and Tani, 1996), etc. On the other hand, myriapod robots
imitate quadruped animals, insects, spiders, etc. The walking mechanism of myriapod robots
is comparatively easier than biped robots since static walking (Torige, 1993) can be applied
simply. Myriapod robots can even walk in a hard environment in which wheel-installed robots cannot move, since myriapod robots choose safe landing points for legs. The number of
studies of myriapod robots is less than that of biped robots, but we consider myriapod robots
to be more important for realizing industrial robot applications.
Within the study of myriapod robots, we find three perspectives: the stability, the maximum movable speed, and the consumption energy. Past studies have included the gait pattern
of quadruped robots (Kimura et al., 1993) and the development of a hexapod robot from the
point of view of efficient working (Nonami et al., 2001). There have also been some studies
on the generation of the gait pattern of myriapod robots1 these contain the walking control
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ROBOT SYSTEMS: HOW MANY LEGS ARE OPTIMAL FOR A CENTIPEDE? 1237
algorithm of hexapod robots, which mimic the gait pattern of insects (Bartling and Schmitz,
20001 Hirokoshi et al., 20001 Pelletier and Caissie, 2001), the concept design of the gait
pattern by analyzing an animal walking from low to high speed (Pearson, 1976), and the gait
pattern generation for hexapod robots by examining the ecology of spiders (Tanaka et al.,
1999). Moreover, recently studies have been made of the realization of a walking method
that differs in speed (Hirose and Yoneda, 19911 Kimura et al., 1993) and a new generation
method of gait patterns using the autonomous distributed system (Yagi et al., 19991 Hirokawa
et al., 20011 Weangsirna et al., 2001).
In this paper we study myriapod robots and obtain the working function for the legs
from a kinetic efficiency point of view. The realization of the high performance of energy
consumption is indispensable in the battery-based robot system, in which we have to utilize
the battery for the driving force. We introduce the cost of transport as the efficiency of the
kinetic energy and realize the lossless gait of myriapod robots by minimizing the cost of
transport (Hirose and Umetani, 19781 Kaneko et al., 19841 Nishii, 19981 Odasima et al.,
1999). The fact that animals and insects change gait patterns with speed is proof of using
kinetic energy efficiently.
However, in past studies the cost of transport calculated through the working function of
the legs was assumed by the sinusoidal function (Hirose and Umetani, 19781 Nishii, 19981
Suyama et al., 2002a, 2002b, 2002c). So, the problem whether the assumed sinusoidal function is the best function has remained. In this paper, we obtain the optimal working function
for the legs by using the calculus of variations.
In the following section, we describe the myriapod robots dealt with in this paper and
introduce the cost of transport. In Section 3 we obtain a governing equation for the legs and
an optimal working function using the calculus of variations. Optimal working functions of
the swing and lift motion at both the stance and swing phases are obtained using the Euler
equation. In Section 4 we calculate the cost of transport based on the obtained working
functions. Finally, in Section 5 we give concluding remarks and discuss whether a centipede
has an optimal number of legs.
2. LOCOMOTION BASED ON KINETIC ENERGY EFFICIENCY
We consider a myriapod robot, which has some pairs of legs at both sides of a body. Figure 1
shows the cross-sectional view of a pair of legs of the robot. Each leg has one joint with
two degrees of freedom by two DC motors: the swing and lift motor. The swing motor
generates the forward/backward motion of the robot (Figure 2) and the lift motor raises a leg
up (Figure 3). The thrust force of the legs achieves the forward velocity of the body. The
velocity V is obtained by
V 2
S
3T
(1)
where S, T , and 3 indicate the stride (the distance which the leg of the swing phase moves
forward), the period of the movement of the leg (the summation of the duration of the stance
and swing phases), and the duty factor, respectively. The duty factor is defined by the ratio
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1238 B. T. NOHARA and T. NISHIZAWA
Figure 1. Cross-sectional view of a pair of legs.
Figure 2. Swing motion.
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ROBOT SYSTEMS: HOW MANY LEGS ARE OPTIMAL FOR A CENTIPEDE? 1239
Figure 3. Lift motion.
of the number of legs of the stance phase and the total number of legs. From equation (1),
the velocity V is obtained by the flexible combination of the parameters, S, T , and 3. In
order to select the optimal combination of these parameters, the minimization of the kinetic
energy is a reasonable criterion from the engineering point of view.
2.1. Energy Consumption of Each Leg
The electric current, the voltage, the internal electric resistance, the torque, and the angular
velocity of each motor are denoted by i, 1, R, , and , respectively. Then, the kinetic energy
consumption e j of the jth leg is presented by
1
ej 2
1
T
T
2
1tit dt 2
0
4 tt 3
0
3
R 2
t
dt
K T2
(2)
where
4 2
KE
KT
Here, K E and K T indicate the induced electromotive force constant and the torque constant,
respectively. In equation (2), we use the relations:
1t 2
Rit 3 K E t
t 2 K T it
(3)
(4)
The first term of the mostly right-hand side of equation (2) means the mechanical power
consumed by the robot, which is the product of the torque and the angular velocity at each
joint. The second term corresponds to the heat energy loss.
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1240 B. T. NOHARA and T. NISHIZAWA
The total kinetic energy E consumed by the robot is given by the summation of the
kinetic energy consumption of each leg as follows
E2
n
4
ej
(5)
j21
where n is the number of legs.
2.2. Def inition of the Cost of Transport
We introduce the cost of transport as a measure of the kinetic energy efficiency. The cost of
transport 6 is given by
62
E
3E
2
MV T
MS
(6)
where M is the weight of the robot. The cost of transport shows the energy consumption per
unit distance and unit weight of the robot. Therefore, the lower 6 is, the more efficiently the
robot walks.
3. GOVERNING EQUATIONS AND WORKING FUNCTIONS FOR
LEGS
First, we derive the governing equations to determine the working functions for the legs. The
calculus of variations is applied to derive the equations. In this paper, we obtain the optimal
function 9t of the rotational motion, which makes equation (2) minimum. The mass of
each leg including a motor, the mass of only a leg, the attenuation ratio of a motor, and
the inertia of the embedded gear of a motor are m 0 , m 1 , 7, and JG , respectively. Also, the
symbols r ST , r SW , l, 2 S , and 2 W denote the radius of the swing motion at the stance phase,
the radius of the swing motion at the swing phase, the radius of the lift motion, the angle
of the swing motion, and the angle of the lift motion, respectively. Here, the “stance phase”
indicates when a leg supports the weight of the robot. The “swing phase” means the state of
a leg which does not support the robot.
3.1. Stance Phase
The working function of the swing motion at the stance phase is denoted by 9 S1 t. Suppose the initial and boundary conditions are 9 S1 0 2 0, 9 4S1 0 2 0, 9 S1 TS1 2 2 S , and
9 4S1 TS1 2 0. Here, TS1 indicates the swinging time of the stance phase and the prime represents the differentiation with respect to time. The required torque S1 t at the stance phase
is presented by
5
6
S1 2 72 JST 3 JG 9 44S1 t
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ROBOT SYSTEMS: HOW MANY LEGS ARE OPTIMAL FOR A CENTIPEDE? 1241
where JST is the leg inertia of the swing motion at the stance phase and is computed by
JST 2
M 2
r n3 ST
Substituting equation (7) into equation (2) leads to
1
e S1 2
0
TS1
2
3
5
6
62 2
R 5
4 72 JST 3 JG 9 44 S1 t9 4 S1 t 3 2 72 JST 3 JG 9 44 S1 t dt
KT
(8)
In order to minimize equation (8) under the constraint of
1
TS1
9 4S1 t dt 2 2 S (9)
0
using the method of the Lagrange multiplier,
2
3
5 2
6 44
62 44 2
R 5 2
4
4 7 JST 3 JG 9 S1 t9 S1 t 3 2 7 JST 3 JG 9 S1 t dt
J [9 S1 t] 2
KT
0
3
21 TS1
9 4 S1 t dt 5 2 S
(10)
5 8
4
1
TS1
0
must be minimized. Here, S1 t 2 9 4S1 t, and thus equation (10) is rewritten as follows:
1
J [ S1 t] 2
TS1
2
5
6
4 72 JST 3 JG 4 S1 t S1 t
0
62 2
82 S
R 5 2
7 JST 3 JG 4 S1 t 5 8 S1 t 3
2
KT
TS1
3
3
dt
(11)
The integrated function of equation (11) is F S1 t 4S1 t t. Substituting this into the
following Euler equation
F S1 t 4S1 t t
d F S1 t 4 S1 t t
5
20
S1 t
dt
4 S1 t
(12)
62
R 5 2
7 JST 3 JG 44 S1 t 3 8 2 0
2
KT
(13)
yields
2
We obtain the solution of equation (13), which satisfies the initial and boundary conditions,
as follows:
9 S1 t 2
2S 2
t 3TS1 5 2t 3
TS1
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1242 B. T. NOHARA and T. NISHIZAWA
Note that the obtained function 9 S1 t makes the mechanical energy consumption zero.
This fact is easily found by substituting equation (14) into the first term of equation (8).
Accordingly, we find that the amounts of the motoring (
tt 0) and the regeneration
(
tt 0) are balanced in order to minimize the total energy consumption.
The function of the lift motion 9 S2 t is
9 S2 t 2 2 constant
(15)
since this motion simply supports the mass of the robot.
3.2. Swing Phase
The function of the swing motion at the swing phase is 9 W 1 t and we suppose the initial and
boundary conditions to be 9 W 1 0 2 0, 9 4W 1 0 2 0, 9 W 1 TW 1 2 2 S , and 9 4W 1 TW 1 2 0.
Here, TW 1 indicates the swinging time of the swing phase.
In this case, the required torque W 1 t is presented by
6
5
W 1 2 72 JSW 3 JG 9 44W 1 t
(16)
where JSW is the leg inertia of the swing motion at the swing phase and is computed by
2
JSW 2 m 0r SW
Then we obtain the function 9 W 1 t as follows using the same procedure as in Section 3.1:
9 W 1 t 2
2S 2
t 3TW 1 5 2t TW3 1
(17)
Next we find the functions of the lift motion 9 W i t at the swing phase. Here, i 2 2
means the ascent motion, and i 2 3 the descent motion. The required torque W i t is
obtained by the constrained Lagrangian equations of motion
d
dt
7
KWi
9 4 W i t
8
d
5
dt
7
UW i
9 4 W i t
8
5
KWi
UW i
3
2 Wi 9 W i t 9 W i t
(18)
where K W i and UW i denote the kinetic and potential energies, respectively, and are represented by
KWi
UW i
6 2
15 2
7 JL 3 JG 9 4 W i t
2
2 5m 1 gl cos[9 W i t]
2
(19)
(20)
In equation (19), JL indicates the leg inertia of the lift motion at the swing phase and is
computed by
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ROBOT SYSTEMS: HOW MANY LEGS ARE OPTIMAL FOR A CENTIPEDE? 1243
JL 2 m 1 l 2 Using equations (18)–(20), we obtain the required torque as follows:
5
6
W i 2 72 JL 3 JG 9 44 W i t 3 m 1 gl sin[9 W i t]
(21)
Accordingly, we can find the governing equations of 9 W i t by substituting equation (21)
into equation (2). In this case, the problem is
min J [9 4W i t]
(22)
9 4W i t8
under the constraint of
1
TW i
9 4W i t dt 2 2 W (23)
0
where
J [9 4W i t]
1
TWi
2
9
5
6
4 72 JL 3 JG 9 44W i t9 4W i t 3 4 m 1 gl sin[9 W i t]9 4W i t
0
3
3
62 44
6
R 5
R 5 2
7 JL 3 JG 9 W2i t 3 2 2 72 JL 3 JG m 1 gl sin[9 W i t]9 44W i t
2
KT
KT
3
21 TW i
R 2 22 2
4
(24)
m g l sin [9 W i t] dt 5 8
9 W i t dt 5 2 W K T2 1
0
Here
5
6
F[9 W i t 9 4W i t 9 44W i t t] 2 4 72 JL 3 JG 9 44W i t9 4W i t 3 4 m 1 gl sin[9 W i t]9 4W i t
3
62 44
6
R 5 2
R 5
7 JL 3 JG 9 W2i t 3 2 2 72 JL 3 JG
2
KT
KT
6 m 1 gl sin[9 W i t]9 44W i t 3
5 89 4W i t 3
R 2 22 2
m g l sin [9 W i t]
K T2 1
82 W
TW i
(25)
Then, substituting equation (25) into the Euler equation
d F[9 W i t 9 4W i t 9 44W i t t]
F[9 W i t 9 4W i t 9 44W i t t]
5
9 W i t
dt
9 4W i t
9
d2 F[9 W i t 9 4W i t 9 44W i t t]
2 0
3 2
dt
9 44W i t
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1244 B. T. NOHARA and T. NISHIZAWA
Table 1. Major parameters used in the numerical simulation.
M
R5K T2
Sn
g
l
m1
4
72 JL 3 JG
72 JST 3 JG at 3
72 JST 3 JG at 3
72 JST 3 JG at 3
72 JST 3 JG at 3
72 JSW 3 JG
2 758
2 658
2 558
2 458
3.45 kg
0.0001312
0.1167
9.8 m s52
0.105 m
0.02 kg
1
0.00042
0.06161
0.07186
0.08625
0.10781
0.00212
leads to the following equation of motion:
72 JL
3
6
6
5 2
5 2
44
JG 2 9 4
W i t 3 2 7 JL 3 JG m 1 gl cos[9 W i t]9 W i t 5 7 JL 3 JG
6 m 1 gl sin[9 W i t]9 4W i 2 t 3 m 21 g 2l 2 sin[9 W i t] cos[9 W i t] 2 0
i
2 2 3
(27)
3.3. Numerical Examples of the Working Functions
The working functions are obtained using equations (14), (17), and (27) for the swing motion of the stance phase, the swing motion of the swing phase, and the lift motion, respectively. We show numerical examples of these functions. Table 1 shows the major physical
parameters used in the numerical simulation. The period of the movement of a leg T is
calculated using equation (1) based on fixed parameters Sn , Vn , and 3. Sn denotes the nondimensional stride (i.e. the ratio of the actual stride and the length of the robot). Also, Vn
denotes the non-dimensional velocity (i.e. the ratio of the actual velocity and the length of
the robot). Here, Sn 2 01167 unit distance, and 3 2 558. Figure 4 shows functions for
TS1 2 20, 1.4 and 1.0 s (calculated from Vn 2 00583, 0.0833, and 0.1167 unit distance
s51 ) for equation (14). Figure 5 shows functions for TW 1 2 12, 0.84 and 0.6 s (calculated from Vn 2 00583, 0.0833 and 0.1167 unit distance s51 ) for equation (17). Note that
T 2 TS1 3 TW 1 , TS1 2 3T , and TW 1 2 1 5 3T . We must obtain solutions of equation
(27) numerically due to the non-linearity of the differential equation. The initial and boundary conditions are 9 W 2 0 2 0, 9 W 3 0 2 2 W , 9 4W i 0 2 0 (i 2 2 3), 9 W 2 TW 2 2 2 W ,
9 W 3 TW 3 2 0, and 9 4W i TW i 2 0 (i 2 2 3). Figure 6 shows numerical results of the case
2 W 2 518 rad and TW 2 2 TW 3 2 06, 0.42, and 0.3 s. Note that TW 1 2 TW 2 3 TW 3 . Then,
TW 2 and TW 3 are obtained by simple division of TW 1 .
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ROBOT SYSTEMS: HOW MANY LEGS ARE OPTIMAL FOR A CENTIPEDE? 1245
Figure 4. Functions of the swing motion (the stance phase).
Figure 5. Functions of the swing motion (the swing phase).
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1246 B. T. NOHARA and T. NISHIZAWA
Figure 6. Functions of ascent and descent motions.
4. COST OF TRANSPORT
Figures 7 and 8 show the cost of transport of the robot based on the obtained functions of
the legs. These figures present the change of the cost of transport for duty factors when the
robot increases its speed. The non-dimensional stride Sn is set to 0.1167. Figure 7 shows
the case of the hexapod model, and Figure 8 shows the octarupedal model. The locomotion
of minimum energy consumption is performed by selecting the optimal duty factor, which
indicates the minimum cost of transport at each velocity.
In past studies (Hirose and Umetani, 19781 Nishii, 19981 Suyama et al., 2002a, 2002b,
2002c), the cost of transport was obtained by assuming that the working function of the legs
is the sinusoidal function. Thus, the question from the point of the total energy efficiency
has remained. We obtain the optimal working function for the legs by using the calculus of
variations, and we calculate the cost of transport in this study. Figures 9(a) and (b) show the
comparisons of the functions of equations (14) and (27) derived by the calculus of variations
and the pure sine function. Figure 10 presents the cost of transport for these two cases in
the duty factor 3 2 558 of the octarupedal model. We find that the function obtained by the
calculus of variations has smaller cost than the sine function, in spite of a little difference
between the profiles of functions.
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ROBOT SYSTEMS: HOW MANY LEGS ARE OPTIMAL FOR A CENTIPEDE? 1247
Figure 7. Velocity–cost of transport (hexapod model): Sn 2 01167.
Figure 8. Velocity–cost of transport (octarupedal model): Sn 2 01167.
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1248 B. T. NOHARA and T. NISHIZAWA
Figure 9. Comparison of functions: the sine function. (a) Equation (14)1 (b) the solution of equation (27).
Figure 10. Comparison of the cost of transport.
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ROBOT SYSTEMS: HOW MANY LEGS ARE OPTIMAL FOR A CENTIPEDE? 1249
Figure 11. Optimal number of legs.
5. CONCLUDING REMARKS AND FURTHER DISCUSSION
In this paper, we obtain the working functions using the calculus of variations, and we consider the locomotion of the robot based on the maximum energy efficiency. We can obtain
the analytical forms from the Euler equations for the swing working function of the stance
phase and the swing and lift working functions of the swing phase. However, the equation of
motion for the lift function of the swing phase becomes a non-linear form, so we find a solution for this case using the numerical method. The mostly efficient locomotion is presented
through the calculation of the cost of transport.
Furthermore, we consider using this method to study whether a centipede has an optimal
number of legs. The working mechanism of the legs of the robot used in this study is different
from a live centipede, but we try to check how the cost of transport changes when increasing
the number of legs of the robot. Figure 11 shows the change of the cost of transport with the
increase in the number of legs, provided that the total weight of the robot is constant. Each
line indicates the cost of transport with the increasing speed, so the duty factor also changes
(not shown). We formally calculate the cost of transport from six legs to 1000 legs, but only
some of the results are shown in Figure 11 to make it easier to read. We find that the minimum cost of transport occurs for the cases of 48, 26, and 10 legs at low (less than 0.083 unit
distance s51 ), medium (0.083–0.183 unit distance s51 ), and high speed (more than 0.183 unit
distance s51 ), respectively. There are several types of centipede: Scolopendra Subspinipes
Japonica has 42 legs (see http://www.city.nagoya.jp/10eisei/ngyeiken/insect/chilopod/ssj.htm
(in Japanese)] and Bothropolys Rugosus has 30 legs [see http://www.insects.jp/7 tkawabe/konmukadeissun.htm (in Japanese)]. Scolopendra Subspinipes Japonica is a large sized cen-
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1250 B. T. NOHARA and T. NISHIZAWA
tipede and Bothropolys Rugosus is a small centipede. The moving speed of Scolopendra
Subspinipes Japonica is slower than that of Bothropolys Rugosus. The mechanism of the
robot is a very rough model of a centipede, but we obtain nearly the numbers of legs at low
and medium speeds. The attempt to model the legs of a centipede and to estimate the optimal
number of legs may help to solve the process of evolution using an engineering method.
Acknowledgment. The authors are grateful for the useful comments provided by the referees.
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