2011 IEEE/RSJ International Conference on
Intelligent Robots and Systems
September 25-30, 2011. San Francisco, CA, USA
Experimental verification of hysteresis in gait transition of a quadruped
robot driven by nonlinear oscillators with phase resetting
Shinya Aoi1,3 , Soichiro Fujiki1 , Daiki Katayama1, Tsuyoshi Yamashita1
Takehisa Kohda1, Kei Senda1, and Kazuo Tsuchiya2,3
1
2
Dept. of Aeronautics and Astronautics, Graduate School of Engineering, Kyoto University
Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501, Japan
Dept. of Energy and Mechanical Engineering, Faculty of Science and Engineering, Doshisha University
1-3 Tatara, Miyakodani, Kyotanabe, Kyoto 610-0394, Japan
3
JST, CREST, 5, Sanbancho, Chiyoda-ku, Tokyo 102-0075, Japan
Email: shinya [email protected]
Abstract— In this paper, we investigated the locomotion of
a quadruped robot, whose legs are controlled by an oscillator
network system. In our previous work, simulation studies revealed that a quadruped robot produces walk and trot patterns
through dynamic interactions among the robot’s mechanical
system, the oscillator network system, and the environment.
These studies also showed that a walk-trot transition is induced
by changing the walking speed. In addition, the gait-pattern
transition exhibited a hysteresis similar to that observed in the
locomotion of humans and animals. The aim of the present
study is to verify such dynamic characteristics in the gait
generation of quadrupedal locomotion in the real world by
developing and evaluating a quadruped robot.
I. INTRODUCTION
Humans and animals are endowed with the ability to produce adaptive walking in diverse environments by skillfully
manipulating their complicated and redundant musculoskeletal systems. As a characteristic feature of adaptive locomotor
behaviors, they change their gait patterns depending on the
locomotion speed, e.g., humans walk and run and quadrupeds
walk, trot, and gallop. It has revealed that a hysteresis appears
in their gait transitions, that is, the gait patterns change at
different locomotion speeds depending on the speed change
direction [6], [9], [10], [13], [17], [20], [21], [23]. Although
such gait transitions have been investigated from mechanical,
energetic, kinematic, and kinetic viewpoints [7], [9], [12],
[20], [23], it remains unclear what determines the gait
transitions.
Investigations of configurations and activities of neural
systems have revealed the mechanisms used to perform the
adaptive movements of animals. However, it is difficult to
fully analyze locomotion mechanisms solely in terms of the
nervous system, since locomotion is a well-organized motion
generated through dynamic interactions among the body, the
nervous system, and the environment. In addition to analyzing the nervous system, elucidating the inherent dynamic
characteristics of the body’s mechanical system is crucial. To
overcome the limitations of the nervous system, constructive
approaches using robots and computer simulations have
recently attracted attention. Robots are effective and powerful
tools for testing hypotheses of real-world phenomena, and
978-1-61284-455-8/11/$26.00 ©2011 IEEE
hardware investigations are useful for demonstrating realworld dynamic characteristics. In contrast, simulation studies
allow us to perform thorough investigations, such as parameter dependence, under ideal situations by excluding various
uncertainties of robot systems and environments.
So far, quadruped robots have been investigated based
on biological concepts [8], [14–16], [22]. In our previous
works [1], [24], we constructed a quadruped robot model and
the locomotion control system using an oscillator network
based on the physiological concept of a central pattern
generator. Furthermore, we performed computer simulations,
which revealed that the quadruped robot establishes its walk
and trot patterns through dynamic interactions among the
robot’s mechanical system, the oscillator network system,
and the environment. Our results also showed a walk-trot
transition induced by changing the walking speed. In addition, the gait-pattern transition exhibited a hysteresis similar
to that observed in the locomotion of humans and animals.
Since simulation studies are based on a mathematically
ideal model of a robot, discrepancies between simulation
results and experimental results are inevitable. In addition,
hysteresis is a characteristic property of nonlinear dynamic
systems. These findings suggest that it is important to verify
our simulation results using an actual robot. The aim of the
present study is to verify the obtained dynamic characteristics
in the gait generation of quadrupedal locomotion in the real
world by developing and evaluating a quadruped robot.
II. QUADRUPED ROBOT
We developed a quadruped robot (Fig. 1) composed of a
body and four legs (Legs 1 · · · 4). Each leg consists of two
links connected by pitch joints (Joints 1 and 2) and each
joint is manipulated by a motor. A touch sensor is attached
to the tip of each leg. Table I shows the physical parameters
of the robot. Note that although the body of our robot is
separated to the front and rear bodies as in Fig. 1A, they
are fixed and our robot does not have a roll joint different
from our quadruped robot model in our previous simulation
studies [1], [24].
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Locomotion
speed Gait pattern
Locomotion
Control System
Motion
Generator
∧
Rhythm ∆
Gait
Generator
Generator
A
B
Joint 1
(pitch)
Leg 3
Joint 2
(pitch)
Leg 1
φ
Leg 4
Trajectory Generator
Leg 2
Fig. 1.
Quadruped robot (A) and schematic model (B)
∧
θ
Motion
Controller
TABLE I
.
θθ
Motor Controller
P HYSICAL PARAMETERS OF THE QUADRUPED ROBOT
Link
Body
Upper Leg
Lower Leg
Parameter
Mass [kg]
Length [cm]
Width [cm]
Mass [kg]
Length [cm]
Mass [kg]
Length [cm]
Value
1.50
28
20
0.27
11.5
0.06
11.5
u
Touch
sensor
signal
B
The robot walks on a flat floor with no elevation. The electric power is externally supplied and the robot is controlled
by an external host computer (Intel Pentium 4 2.8 GHz, RTLinux), which calculates the desired joint motions and solves
the oscillator phase dynamics in the locomotion control
system. It receives the command signals at intervals of 1 ms.
The robot is connected with the electric power unit and
the host computer by cables that are held up during the
experiment to avoid influencing the walking behavior.
A. Trajectory generator
We introduce φi (i = 1, . . . , 4) for the phase of Leg i
oscillator, which generates the desired joint motions of Leg
i.
Touch sensor signal Touch sensor signal
Leg 1
oscillator
Leg 2
oscillator
∆ 12
φ1
φ2
∆ 14
Leg 3
oscillator
III. LOCOMOTION CONTROL SYSTEM
Since we used the same locomotion control system as our
previous studies [1], [24], here we briefly explain it. It consists of a motion generator and a motion controller (Fig. 2A).
The motion generator is composed of a gait generator, a
rhythm generator, and a trajectory generator to produce the
desired leg motions based on the desired locomotion speed
and gait pattern. The gait generator creates desired gait
pattern by the phase relationship between the leg movements.
The rhythm generator produces basic locomotor rhythm and
phase for the leg movements using four oscillators (Leg
1 · · · 4 oscillators) and touch sensor signals (Fig. 2B). The
trajectory generator produces the desired leg joint movements
from the oscillator phases. The motion controller consists of
motor controllers to control the joint angles by motors based
on the desired movements. We explain the details of each
control system below.
Mechanical System
∆ 23
∆ 13
∆ 24
φ3
φ4
∆34
Leg 4
oscillator
Touch sensor signal Touch sensor signal
Fig. 2. Locomotion control system [1], [24] (A: architecture of control
system, B: rhythm generator with four oscillators)
The desired leg movements are designed by the desired
trajectory of the leg tip relative to the body in the pitch plane,
which consists of the swing and stance phases (Fig. 3). The
former is composed of a simple closed curve that includes
an anterior extreme position (AEP) and a posterior extreme
position (PEP). It starts from point PEP and continues until
the leg tip touches the ground. The latter consists of a straight
line from the landing position (LP) to point PEP. Distance
between points AEP and PEP is given by D. Nominal duty
factor β is given by the ratio between the stance phase and
step cycle durations when the leg tip touches the ground at
AEP (LP = AEP). These two desired trajectories provide
desired motion θ̂ji (i = 1, . . . , 4, j = 1, 2) of Joint j of Leg
i by the function of phase φi of Leg i oscillator, where we
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Joint 1
Joint 1
Joint 2
Joint 2
AEP
PEP AEP
PEP
LP
LP
D
Stance phase
Swing phase
Fig. 3. Desired trajectories of the forelegs composed of the swing and
stance phases [1], [24]. When the leg tip lands on the ground, the trajectory
changes from the swing to stance phase. When the leg tip reaches point
PEP, the trajectory moves into the swing phase. Bending direction of Joint
2 of the hindlegs is different from the front legs.
where ω is the basic oscillator frequency that uses the same
value among the oscillators, g1i (i = 1, . . . , 4) is the function
regarding the gait pattern shown below, and g2i (i = 1, . . . , 4)
is the function arising from phase resetting by touch sensor
signals given below.
We defined the swing and stance phase durations as Tsw
and Tst , respectively, for the case when the leg tip contacts
the ground at the AEP (LP = AEP). The nominal duty factor
β, the nominal basic frequency ω, the nominal stride length
S, and the nominal locomotion speed v are respectively then
given by
Tst
Tsw + Tst
2π
ω =
Tsw + Tst
Tsw + Tst
D
S =
Tst
D
v =
Tst
β =
Walk
Leg 1
Leg 3 π /2
Leg 2
Leg 4
Trot
Leg 1
Leg 3
Leg 2
Leg 4
π
Fig. 4. Footprint diagram for walk and trot patterns, where right and left
legs move out of phase in each body (forelegs in red and hindlegs in blue)
(3)
These values are satisfied regardless of the gait pattern. In
the present study, we used D = 1.0 cm and Tsw = 0.15 s
and we varied the locomotion speed v by changing the stance
phase duration Tst .
1) Gait pattern control: Function g1i in (2) manipulates
the phase difference between the oscillators regarding the
gait pattern. It is given by
g1i = −
4
ˆ ij ),
Kij sin(φi − φj − ∆
i = 1, . . . , 4
(4)
j=1
used φi = 0 at point PEP and φi = φAEP (= 2π(1 − β)) at
point AEP.
B. Gait generator
Since the desired leg movements are designed by the
oscillator phases, the gait pattern is determined by the phase
difference between the oscillators, which is given by matrix
∆ij (0 ≤ ∆ij < 2π) as follows;
∆ij = φi − φj ,
i, j = 1, . . . , 4
(1)
Since ∆ij = −∆ji , ∆ij = ∆ik + ∆kj , and ∆ii = 0
(i, j, k = 1, . . . , 4) are satisfied, the gait pattern is determined
by three parameters, such as [ ∆12 , ∆13 , ∆34 ]. For example,
[ ∆12 , ∆13 , ∆34 ] = [ π, π/2, π ] is satisfied for the walk
pattern and [ ∆12 , ∆13 , ∆34 ] = [ π, π, π ] is satisfied for the
trot pattern (Fig. 4). The gait generator produces the desired
ˆ ij based on the desired
relationship between the oscillators ∆
gait pattern.
C. Rhythm generator
The rhythm generator produces the basic locomotor
rhythm by using Leg 1 · · · 4 oscillators. The oscillator phase
φi (i = 1, . . . , 4) follow the dynamics
φ̇i = ω + g1i + g2i ,
i = 1, . . . , 4
(2)
where Kij (i, j = 1, . . . , 4) is gain constant (Kij ≥ 0). When
ˆ ij (∆ij = ∆
ˆ ij )
we use a large value for Kij , φi − φj = ∆
will be satisfied.
2) Phase resetting: Phase resetting modulates motor commands based on sensory information to create adaptive walking in biological systems [2], [19] and the phase resetting
mechanism has been used for legged robots [3], [4], [18],
[19], [22]. To produce adaptive walking through dynamic
interactions among the robot mechanical system, the oscillator control system, and the environment, we modulated the
locomotor phase and rhythm by phase resetting based on
touch sensor signals. Function g2i in (2) corresponds to this
regulation. When the tip of Leg i lands on the ground, phase
φi of Leg i oscillator is reset to φAEP from φiland at the landing
(i = 1, . . . , 4). Therefore, functions g2i is written by
g2i = (φAEP − φiland )δ(t − tiland ),
i = 1, . . . , 4
(5)
where tiland is the time when the tip of Leg i touches the
ground (i = 1, 2) and δ(·) denotes Dirac’s delta function.
D. Motor controller
The motor controller manipulates the joint angle based on
the desired joint movement generated by the oscillator phase.
The input torque uij (i = 1, . . . , 4, j = 1, 2) for Joint j of
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uij = −κij (θji − θ̂ji (φi )) − σji θ̇ji
i = 1, . . . , 4, j = 1, 2
Phase difference ∆13 [rad]
Leg i is given by
(6)
where κij and σji (i = 1, . . . , 4, j = 1, 2) are gain constants
and we used adequately large values to establish the desired
leg movements.
E. Conditions for gait pattern
As explained in Section III-B, the phase relationship
between the oscillators determines the gait pattern of our
quadruped robot. This is produced by the interactions among
the oscillators (4) and the phase regulation by phase resetting
(5). When we use neither (4) nor (5), the phase relationship
remains in the initial state, and the gait pattern does not
change. When all elements of matrix ∆ij are determined
based on the desired gait pattern and large values are used for
gain constants Kij in (4), our robot will establish the desired
gait pattern when the gait pattern is stable. In contrast, when
small values are used for gain constants Kij , our robot can
produce a different gait pattern from the desired one due to
the phase regulation by phase resetting (5).
In the same way as our previous studies [1], [24], we
focused on the gait pattern, where the right and left legs
move out of phase. That is, we used
ˆ 12 = π
∆
ˆ 34 = π
∆
(7)
and a large value for gain constants K12 , K21 , K34 , and
K43 .
In our previous studies [1], [24], we did not use the
desired phase relationship between the forelegs and hindlegs
by using zero for the other gain constants Kij . In this
study, however, we determined the desired phase relationship
between the forelegs and hindlegs so that the ipsilateral legs
move out of phase, designating this approach as
ˆ 13 = π
∆
(8)
This means that the desired gait pattern is the trot pattern
in our robot experiment. We used this because the basin of
attraction for the trot pattern is smaller than that for the walk
pattern [1]. However, we used as small a value as possible
for K13 , K31 , K24 , and K42 , and we used zero for the
other gain constants to allow our robot to change the gait
pattern from the desired gait pattern (trot pattern) through
locomotion dynamics due to the phase regulation by phase
resetting.
IV. ROBOT EXPERIMENT
A. Generation of gaits
In our robot experiment, we first increased the stride length
slowly from 0 (standing posture) to S in (3) to generate gaits
for β = 0.54 (Tst = 0.18 s) or β = 0.66 (Tst = 0.29 s),
where we used the following parameters: K12 = K21 =
K34 = K43 = 20.0 and K13 = K31 = K24 = K42 =
0.57. Our robot produced stable locomotion. Figure 5 shows
β=0.54
β=0.66
3
2.5
Trot
2
1.5
Walk
1
0
1
2
3
4
5
Time [s]
6
7
8
Fig. 5. Phase difference ∆13 plotted at foot contact of right foreleg for
various initial values of ∆13 with β = 0.54 and β = 0.66
the phase difference between right foreleg and right hindleg
∆13 , plotted when the right foreleg touches the ground. This
illustrates that the phase difference ∆13 converged to around
2.6 rad from various initial values of ∆13 for β = 0.54,
showing that our robot established the trot pattern at a high
speed (although ∆13 = 2.6 rad is slightly different from
π rad, we considered it the trot pattern in order to distinguish
it from the walk pattern shown below). On the other hand,
the phase difference ∆13 converged to about 1.6 rad for β =
0.66, indicating that our robot achieved the walk pattern at
a slow speed. That is, our robot established a different gait
pattern from the desired gait pattern (trot pattern) at a slow
speed due to the phase regulation by phase resetting.
B. Appearance of hysteresis in the gait transition
After our robot established stable gait pattern as described
in the previous section, we slowly increased the locomotion
speed by changing β from 0.66 to 0.54 or slowly decreased
the locomotion speed by varying β from 0.54 to 0.66.
We investigated what gait pattern, represented by phase
difference ∆13 , emerges and how the gait pattern changes
through locomotion dynamics.
Figure 6 shows the results. We plotted three experiments
for the increase and decrease of locomotion speed, and
they show similar results. When we changed the locomotion
speed, phase difference ∆13 varies between 2.6 and 1.6 rad,
indicating that the gait pattern changes between the walk and
trot patterns (see supplemental movie for gait transition from
walk to trot pattern). In addition, when we decreased the
locomotion speed, the trot pattern transitioned to the walk
pattern at around β = 0.63. On the other hand, when we
increased the locomotion speed, the walk pattern changed to
the trot pattern at around β = 0.58. This means that the gait
transition occurs at different locomotion speeds depending
on the direction of speed change, that is, a hysteresis with
respect to gait pattern appears.
C. Stability characteristics in hysteresis
The hysteresis in the walk-trot transition obtained in the
previous section suggests the coexistence of two different
gait patterns over a range of the locomotion speed [1].
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3
A
3
Phase difference ∆13 n +1 [rad]
Phase difference ∆13 [rad]
Trot
2.5
2
1.5
Walk
1
0.54
0.56
high speed
0.58
0.6
0.62
Duty factor β
β =0.55
2.5
Trot
2
1.5
1
0.64
0.66
low speed
1
1.5
2
2.5
Phase difference ∆13n [rad]
3
B
Fig. 6. Gait transition by changing the locomotion speed using duty factor
β, showing the phase difference ∆13 plotted at the foot contact of the right
foreleg. The gait transition occurs between the walk and trot patterns at
different locomotion speeds, depending on the direction of speed change,
and hysteresis appears.
V. CONCLUSION
In this study, we developed a quadruped robot and showed
that it establishes walk and trot patterns through dynamic
interactions among the robot’s mechanical system, the oscillator network system, and the environment. Evaluation of the
robot also showed a walk-trot transition induced by changing
the walking speed. In addition, the gait-pattern transition
Phase difference ∆13 n +1 [rad]
β =0.60
2.5
Trot
2
Walk
1.5
1
1
1.5
2
2.5
Phase difference ∆13n [rad]
3
C
3
Phase difference ∆13 n +1 [rad]
That is, there is more than one attractor. Since the phase
difference ∆13 determines the gait pattern, stability analysis
for ∆13 will explain the existence of attractors. Therefore,
to investigate the dynamic characteristics in the hysteresis,
we calculated the one-dimensional first return map of the
phase difference ∆13 by plotting the relationship between
the phase difference ∆13n for the nth step and the phase
difference ∆13n+1 for the next step during locomotion. We
can determine possible gait patterns and their stabilities from
the intersection of the return map and the diagonal line
(∆13n+1 = ∆13n ).
Figure 7 shows the results. When β = 0.55 (A), the return
map shows that the trot pattern is the only attractor in this
range of phase difference ∆13 . However, for β = 0.60 (B),
when the initial value of ∆13 is larger than 2.05 rad, it
converges to about 2.5 rad (trot pattern). When the initial
value is smaller than 2.05 rad, it converges to around 1.7 rad
(walk pattern), and this convergence is oscillatory rather than
monotonic. That is, there are two stable gait patterns. In
addition, this return map suggests that there is one unstable
gait pattern between the trot and walk patterns, although it
is difficult to determine whether this instability is oscillatory.
When β = 0.65 (C), the trot pattern disappears due to
the loss of the intersections, and the walk pattern is the
only attractor. These results suggest that the changes in the
stability structure due to the locomotion speed induce the
hysteresis in the walk-trot transition (Fig. 6).
3
β =0.65
2.5
2
Walk
1.5
1
1
1.5
2
2.5
Phase difference ∆13n [rad]
3
Fig. 7. Return map of phase difference ∆13 for (A) β = 0.55, (B)
β = 0.60, and (C) β = 0.65
exhibited a hysteresis as observed in our previous studies [1],
[24].
Different from the robot model in our previous simulation
studies [1], [24], the current robot does not have a compliant
waist joint and uses the desired phase relationship between
the forelegs and hindlegs. In addition, our robot uses different
physical and control parameters. These differences induced
differences in the results, such as the equilibrium points
for phase difference ∆13 and their stability. However, our
robot produced walk and trot patterns and showed a walktrot transition with hysteresis, as in our previous simulation
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studies [1], [24]. Furthermore, stability analysis based on
the return maps revealed a mechanical explanation for the
hysteresis produced in the walk-trot transition, suggesting
that this study verified the dynamic characteristics in the
gait generation of quadrupedal locomotion obtained in our
previous simulation studies [1], [24] using a quadruped robot
in the real world.
The hysteresis in the walk-trot transition of quadrupeds
has also been investigated using a simple body mechanical
model [5]. Hysteresis is a typical characteristic of nonlinear
dynamic systems, as observed in the periodically forced
Duffing equation [11]. To better understand the transition
mechanisms in locomotion dynamics, in the future we should
develop a more sophisticated mathematical model as well as
a more biologically plausible robot.
ACKNOWLEDGMENTS
This paper is supported in part by a Grant-in-Aid for
Scientific Research (B) No. 23360111 and a Grant-in-Aid for
Creative Scientific Research No. 19GS0208 from the Ministry of Education, Culture, Sports, Science, and Technology
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