paper.dviof a Quadruped Robot
Hiroshi Kimura∗ Yasuhiro Fukuoka∗ Avis H. Cohen∗∗ ∗Graduate School
of Information Systems
Univ. of Electro-Communications 1-5-1 Chofugaoka, Chofu Tokyo
182-8585, Japan
{hiroshi, fukuoka}@kimura.is.uec.ac.jp
Univ. of Maryland, College Park MD 20742, USA
[email protected]
Abstract
We have been trying to induce a quadruped robot to walk with medium
walking speed on ir- regular terrain based on biological concepts.
We propose the necessary conditions for stable dy- namic walking on
irregular terrain in general, and we design the mechanical system
and the neu- ral system by comparing biological concepts with those
necessary conditions described in physical terms. PD-controller at
joints constructs the vir- tual spring-damper system as the
visco-elasticity model of a muscle. The neural system model con-
sists of a CPG (central pattern generator), re- flexes and
responses. A CPG generates rhythmic motion for walking. A CPG
receives sensory in- put and changes the period of its own active
phase as responses. The virtual spring-damper system also receives
sensory input and outputs torque as reflexes. The states in the
virtual spring-damper system are switched based on the phase signal
of the CPG. The physical oscillations such as the motion of the
virtual spring-damper system of each leg and the rolling motion of
the body are mutually entrained with the neural oscillations of
CPGs. Consequently, the adaptive walking is generated by the
interaction with environment. In this paper, we report our
experimental results of dynamic walking on terrains of medium
degrees of irregularity in order to verify the effectiveness of the
designed neuro-mechanical system.
1. Introduction
Many previous studies of legged robots have been per- formed,
including studies on running (Hodgins and Raib- ert 1991) and
dynamic walking (Yamaguchi, Takanishi and Kato 1994; Kajita and
Tani 1996; Chew, Pratt and Pratt 1999; Yoneda, Iiyama and Hirose
1994; Buehler et al. 1998) on irregular terrain. However, all of
those studies assumed that the structure of terrain was
known,
even though the height of the step or the inclination of the slope
was unknown.
On the other hand, studies of autonomous dynamic adaptation
allowing a robot to walk over irregular ter- rain with less
knowledge of it have been started only re- cently and by only a few
research groups. One example is the recent achievement of
high-speed mobility of a hexa- pod over irregular terrain, with
appropriate mechanical compliance of the legs (Saranli, Buehler and
Koditschek 2001; Cham et al. 2002). The purpose of this study is to
realize high-speed mobility on irregular terrain us- ing a
mammal-like quadruped robot, the dynamic walk- ing of which is less
stable than that of hexapod robots, by referring to the marvelous
abilities of animals to au- tonomously adapt to their
environment.
As many biological studies of motion control pro- gressed, it has
become generally accepted that animals’ walking is mainly generated
at the spinal cord by a combination of a CPG (central pattern
generator) and reflexes receiving adjustment signals from a
cerebrum, cerebellum and brain stem (Grillner 1981; Cohen and
Boothe 1999). A great deal of the previous research on this
attempted to generate walking using a neural system model,
including studies on dynamic walking in simulation (Taga, Yamaguchi
and Shimizu 1991; Taga 1995; Miyakoshi et al. 1998; Ijspeert 1998,
2001), and real robots (Ilg et al. 1999; Tsujita, Tsuchiya et al.
2001; Lewis et al. 2003). But autonomously adaptive dynamic walking
on irregular terrain was rarely realized in those earlier studies
except for our studies (Kimura et al. 1999, 2001; Fukuoka et al.
2003). This paper reports on our progress in the past couple of
years using a newly de- veloped quadruped called “Tekken1&2,”
which contains a mechanism designed for walking in 3D space (pitch,
roll and yaw planes) on irregular terrain. Especially, the
self-contained (power autonomous) system was realized in Tekken2
(Fig.1) and we succeeded in outdoor experi- ments using it.
Table 1: Biological concepts of legged locomotion control.
Limit-Cycle-based Control ZMP-based Control by Neural System by
Mechanism
(CPG and reflexes) (spring and damper) good for posture and low
medium speed walking high speed running control of speed
walking
main upper neural system lower neural system musculoskeletal system
controller acquired by learning (at spinal cord, brain stem, etc.)
through self stabilization
battery
controller
battery
radio receiver
Figure 1: Self-contained quadruped robot: Tekken2. The
length of the body and a leg in standing are 30 [cm] and
20 [cm]. The weight including batteries is 4.3 [kg]. The hip
pitch joint, knee pitch joint and hip yaw joint are activated
by DC motors of 23 [W], 23 [W] and 8 [W] through gear ratio
of 20, 28 and 18, respectively.
2. Design concepts for adaptive walking
2.1 Legged locomotion control
Methods for legged locomotion control are classified into ZMP-based
control and limit-cycle-based control (Table 1). ZMP (zero moment
point) is the extension of the cen- ter of gravity considering
inertia force and so on. It was shown that ZMP-based control is
effective for control- ling posture and low-speed walking of a
biped (Takan- ishi et al. 1990) and a quadruped (Yoneda, Iiyama and
Hirose 1994). However, ZMP-based control is not good for medium or
high-speed walking from the standpoint of energy consumption, since
a body with a large mass needs to be accelerated and decelerated by
actuators in every step cycle.
In contrast, motion generated by the limit-cycle-based control has
superior energy efficiency. But there exists the upper bound of the
period of the walking cycle, in which stable dynamic walking can be
realized (Kimura et al. 1990). It should be noted that control by a
neural system consisting of CPGs and reflexes is dominant for
various kinds of adjustments in medium-speed walking of animals
(Grillner 1981). Full and Koditschek (1999)
also pointed out that, in high-speed running, kinetic en- ergy is
dominant, and self-stabilization by a mechanism with a spring and a
damper is more important than ad- justments by the neural system.
Our study is aimed at medium-speed walking controlled by neural
system con- sisting of CPGs and reflexes (Table 1).
2.2 Necessary conditions for stable dynamic walking on irregular
terrain
We propose the necessary conditions for stable dynamic walking on
irregular terrain, which can be itemized in physical terms:
(a) the period of the walking cycle should be shorter enough than
the upper bound of it, in which stable dynamic walking can be
realized,
(b) the swinging legs should be free to move forward during the
first period of the swing phase,
(c) the swinging legs should land reliably on the ground during the
second period of the swing phase,
(d) the angular velocity of the supporting legs relative to the
ground should be kept constant during their pitching motion or
rolling motion around the contact points at the moment of landing
or leaving,
(e) the phase difference between rolling motion of the body and
pitching motion of legs should be main- tained regardless of a
disturbance from irregular ter- rain, and
(f) the phase differences between the legs should be maintained
regardless of delay in the pitching mo- tion of a leg receiving a
disturbance from irregular terrain.
We design the neural system for these necessary condi- tions to be
satisfied in order to realize adaptive walking.
2.3 Mutual entrainment between neural system and mechanical
system
Motion generation based on biological concepts is illus- trated as
Fig.2, where a neural system and a mechanical system have their own
non-linear dynamics. The char- acteristic of this method is that
there is no adaptation through motion planning. These two dynamic
systems
are coupled to each other, generating motion by inter- acting with
the environment emergingly and adaptively (Taga, Yamaguchi and
Shimizu 1991; Taga 1995). We call this method
“coupled-dynamics-based motion gen- eration.”
The coupled dynamic system can induce autonomous adaptation
according to its own dynamics, under changes in the environment
(e.g., adaptive walking on ir- regular terrain) and under
adjustment of the neural sys- tem parameters by an upper-level
controller (e.g., gait transition in change of walking speed).
Therefore, we can avoid such serious problems in robotics as model-
ing of mechanical system and environment, autonomous planning,
conflict between planned motion and actual motion and so on.
neural system
2.4 Design of neural system
We construct the neural system centering a neural oscil- lator as a
model of a CPG, since the exchange between the swing and stance
phases in the short term and the quick adjustment of these phases
on irregular terrain are essential in the dynamic walking of a
quadruped where the unstable two-legged stance phase appears. On
the other hand, Espenschied et al. (1996) constructed the gait
pattern generator proposed by Cruse (1990) refer- ring to a stick
insect. They also employed the sway- ing, stepping, elevator and
searching reflexes observed by Pearson et al. (1984) in a stick
insect. Consequently, their neural system is more sensor dependent
and more decentralized by using a non-oscillator type CPG.
The characteristics of our neural system in comparison with the one
used by Espenschied et al. (1996) are:
(1) The cyclic period is mainly determined by the time constant of
CPGs. This makes it easy for the nec- essary condition (a)
described in Section 2.2 to be satisfied.
(2) The gait is mainly determined by the connecting weights of the
CPGs network.
(3) As sensor feedback for adaptation on irregular ter- rain, a
“response” directly and quickly modulating
the CPG phase is employed in parallel with a “re- flex” directly
generating joint torque (Fig.2).
About the characteristic (3), it is well known in physiol- ogy
that
• some sensory stimuli modify CPG activity and reflex- ive
responses to sensory stimuli are phase dependent under CPG activity
(Cohen and Boothe 1999).
Such interaction between CPG activity and a sensory stimulus is
very important for adaptation, and corre- sponds to the necessary
conditions described in physical terms in Section 2.2.
2.5 Design of mechanical system
Each leg of Tekken has a hip pitch joint, a hip yaw joint, a knee
pitch joint, and an ankle pitch joint (Fig.1). The direction in
which Tekken walks can be changed by using the hip yaw joints. Two
rate gyro sensors and two incli- nometers are mounted on the body
in order to measure the body pitch and roll angles.
In order to obtain appropriate mutual entrainment be- tween neural
system and mechanical system, mechanical system should be well
designed to have the good dy- namic properties. In addition,
performance of dynamic walking such as adaptability on irregular
terrain, energy efficiency, maximum speed and so on highly depends
on the mechanical design. The design concepts of Tekken are:
[a] high power actuators and small inertia moment of legs for quick
motion and response,
[b] small gear reduction ratio for high backdrivability to increase
passive compliance of joints,
[c] small mass of the lowest link of legs to decrease im- pact
force at collision,
[d] small contacting area at toes to increase adaptability on
irregular terrain,
[e] passive ankle joint mechanism to prevent a swinging leg from
stumble on an obstacle quickly (Fig.3).
A
Figure 3: The ankle joint can be passively rotated:(A) if
the toe contacts with an obstacle in a swing phase, and is
locked:(B) while the leg is in a stance phase.
3. Implementation of neural system for adaptive walking
3.1 Rhythmic motion by CPG
Although actual neurons as a CPG in higher animals have not yet
become well known, features of a CPG have been actively studied in
biology, physiology, and so on. Several mathematical models were
also proposed, and it was pointed out that a CPG has the capability
to gener- ate and modulate walking patterns and to be mutually
entrained with a rhythmic joint motion (Grillner 1981; Cohen and
Boothe 1999; Taga, Yamaguchi and Shimizu 1991; Taga 1995). As a
model of a CPG, we used a neu- ral oscillator proposed by Matsuoka
(1987), and applied to the biped simulation by Taga et al. (1991;
1995). A single neural oscillator consists of two mutually inhibit-
ing neurons (Fig.4-(a)). Each neuron in this model is represented
by the following nonlinear differential equa- tions:
τu{e,f}i = −u{e,f}i + wfey{f,e}i − βv{e,f}i
+u0 + Feed{e,f}i + n∑
j=1
wijy{e,f}j
y{e,f}i = max (u{e,f}i, 0) (1) τ ′v{e,f}i = −v{e,f}i +
y{e,f}i
where the suffix e, f , and i mean an extensor neuron, a flexor
neuron, and the i-th neural oscillator, respectively. u{e,f}i is
uei or ufi, that is, the inner state of an extensor neuron or a
flexor neuron of the i-th neural oscillator; v{e,f}i is a variable
representing the degree of the self- inhibition effect of the
neuron; yei and yfi are the output of extensor and flexor neurons;
u0 is an external input with a constant rate; Feed{e,f}i is a
feedback signal from the robot, that is, a joint angle, angular
velocity and so on; and β is a constant representing the degree of
the self- inhibition influence on the inner state. The quantities τ
and τ ′ are time constants of u{e,f}i and v{e,f}i; wfe is a
connecting weight between flexor and extensor neurons; wij is a
connecting weight between neurons of the i-th and j-th neural
oscillator.
In Fig.4-(a), the output of a CPG is a phase signal: yi
. yi = −yei + yfi (2)
The positive or negative value of yi corresponds to ac- tivity of a
flexor or extensor neuron, respectively.
We use the following hip joint angle feedback as a basic sensory
input to a CPG called a “tonic stretch response” in all experiments
of this study. This negative feedback makes a CPG be entrained with
a rhythmic hip joint motion.
Feede·tsr = ktsr(θ − θ0), F eedf ·tsr = −Feede·tsr (3)
Feed{e,f} = Feed{e,f}·tsr (4)
Neural Oscillator(a)
LF : left fore leg
LH : left hind leg RF : right fore leg RH : right hind leg
(b) Neural Oscillator Network
extensor neuron of other N.O.’s
flexor neuron of other N.O.’s
Figure 4: Neural oscillator as a model of a CPG. The suffix
i, j = 1, 2, 3, 4 corresponds to LF, LH, RF, RH. L, R, F or H
means the left, right, fore or hind leg, respectively.
where θ is the measured hip joint angle, θ0 is the origin of the
hip joint angle in standing and ktsr is the feedback gain. We
eliminate the suffix i when we consider a single neural
oscillator.
By connecting the CPG of each leg (Fig.4-(b)), CPGs are mutually
entrained and oscillate in the same period and with a fixed phase
difference. This mutual entrain- ment between the CPGs of the legs
results in a gait. The gait is a walking pattern, and can be
defined by phase differences between the legs during their pitching
motion. The typical symmetric gaits are a trot and a pace. Diagonal
legs and lateral legs are paired and move together in a trot gait
and a pace gait, respectively. A walk gait is the transversal gait
between the trot and pace gaits. We used a trot gait for most of
experiments. But the autonomous gait transition in changing walking
speed was discussed in our former study (Fukuoka et al.
2003).
3.2 Virtual spring-damper system
We employ the model of the muscle stiffness, which is generated by
the stretch reflex and variable according to the stance/swing
phases, adjusted by the neural system. The muscle stiffness is high
in a stance phase for sup- porting a body against the gravity and
low in a swing phase for compliance against the disturbance
(Akazawa et al. 1982). All joints of Tekken are PD controlled to
move to their desired angles in each of three states (A, B, C) in
Fig.5 in order to generate each motion such as swinging up (A),
swinging forward (B) and pulling down/back of a supporting leg (C).
The timing for all joints of a leg to switch to the next state
are:
A→ B: when the hip joint angle of the leg reaches the desired angle
of the state (A)
B → C: when the CPG extensor neuron of the leg becomes active (yi ≤
0)
C → A: when the CPG flexor neuron of the leg becomes active (yi
> 0)
A
Figure 5: State transition in the virtual spring-damper sys-
tem. The desired joint angles in each state are shown by the
broken lines.
The desired angles and P-gain of each joint in each state are shown
in Table 2, where constant values of the desired joint angles and
constant P-gains were de- termined through experiments. Since
Tekken has high backdrivability with small gear ratio in each
joint, PD- controller can construct the virtual spring-damper sys-
tem with relatively low stiffness coupled with the me- chanical
system. Such compliant joints of legs can im- prove the passive
adaptability on irregular terrain.
3.3 CPGs and pitching motion of legs
The diagram of the pitching motion control consisting of CPGs and
the virtual spring-damper system is shown in the middle part of
Fig.6. Joint torque of all joints is determined by the PD
controller, corresponding to a stretch reflex at an α motor neuron
in animals. The desired angle and P-gain of each joint is switched
based on the phase of the CPG output: yi in Eq.(2) as described in
Section 3.2. As a result of the switching of the virtual
spring-damper system and the joint angle feedback signal to the CPG
in Eq.(4), the CPG and the pitching motion of the leg are mutually
entrained.
3.4 Reflexes and responses
It is known in physiology that the reflex to a stimu- lus on the
paw dorsum in the walking of a cat depends on whether flexor or
extensor muscles are active (as re- viewed in Cohen and Boothe
1999). That is,
• when flexor muscles are active, the leg is flexed in order to
escape from the stimulus,
Table 2: Desired value of the joint angles and P-gains at the
joints used in the PD-controller for the virtual
spring-damper
system in each state shown in Fig.5. θ, φ and ψ are the hip
pitch joint angle, the knee pitch joint angle and the hip yaw
joint angle, respectively.
P control angle in state desired value[rad] P-gain[Nm/rad]
θ in A 1.2θC→A G1
θ in B −0.17 G2v+G3
θ in C θstance+ −G4v+G5
body pitch angle
ψ in all states 0 G8
θC→A: the hip joint angle measured at the instance when the state
changes from (C) to (A).
θstance: variable to change the walking speed. body pitch angle:
the measured pitching angle of
the body used for the vestibulospinal reflex. ∗ means that the
desired angle is calculated
on-line for the height from the toe to the hip joint to be
constant.
v [m/s]: the measured walking speed of Tekken.
• when extensor muscles are active, the leg is strongly extended in
order to prevent the cat from falling down.
We call these a “flexor reflex” and a “extensor reflex,”
respectively, and we assume that the phase signal from the CPG of
the leg switches such reflexes.
The following biological concepts are also known (Ogawa et al.
1998):
• when the vestibule in a head detects an inclination in pitch or
roll plane, a downward-inclined leg is ex- tended while an
upward-inclined leg is flexed (Fig.7).
We call this a “vestibulospinal reflex.”
Referring to such biological knowledge, we employed the several
reflexes and responses (Fig.6) to satisfy the necessary conditions
(b)∼(e) described in physical terms in Section 2.2 in addition to
the stretch reflex and re- sponse described in Section 3.2 and 3.1.
The necessary condition (f) can be satisfied by the mutual
entrainment between CPGs and the pitching motion of legs, and the
mutual entrainment among CPGs (Kimura et al. 2001).
3.4.1 Flexor and extensor reflexes
The flexor and extensor reflexes contribute to satisfy the
conditions (b) and (c), respectively. In our former stud- ies
(Kimura et al. 1999, 2001), the stumble of a swinging
- θ swing
α
Figure 6: Control diagram for Tekken. PD-control at the hip
yaw and knee pitch joints are eliminated in this figure.
leg on an obstacle was detected by force sensor, and the flexor or
extensor reflex was activated afterwards. How- ever, the problem
was that the robot typically fell down due to the delayed flexing
motion caused by the delay of sensing and large inertia of the leg
while walking with short cyclic period. Therefore, we substitute
the flexor reflex for the passive ankle joint mechanism (Fig.3)
uti- lizing the fact that the collision with a forward obstacle
occurs in the first half of a swing phase. The extensor reflex has
not yet been implemented in Tekken.
3.4.2 Vestibulospinal reflex and response for pitching
We call the reflex/response for an inclination in the pitch plane a
“vestibulospinal reflex/response for pitching,” the role of which
corresponds to the condition (d) and (f). In Tekken, hip joint
torque in the stance phase is adjusted by the vestibulospinal
reflex (Fig.7-(a)), since the body pitch angle is added to θstance
in Table 2. For the vestibulospinal response, the following
equations are used rather than Eq.(3), (4).
θvsr = θ − (body pitch angle) Feed{e,f}·tsr·vsr = ±ktsr(θvsr −
θ0).
(5)
roll plane
upward -inclined
3.4.3 Vestibulospinal reflex and response for rolling
We call the reflex/response for an inclination in the roll plane a
“tonic labyrinthine reflex/response1 .”
• Tonic labyrinthine reflex
The tonic labyrinthine reflex (TLRF) is employed as the adjustment
of P-gain of the knee joint of the sup- porting legs (G7 in Table
2) according to Eq.(7).
G7 = δ(leg) ktlrf × (body roll angle) +G7 (7)
δ(leg) = {
1, if leg is a right leg; −1, otherwise
When an inclination of a body in roll plane (body roll an- gle) is
detected, the knee joint P-gain of the downward- inclined legs is
increased to extend those legs. In addi- tion, the knee joint
P-gain of the upward-inclined legs is decreased to flex those lges.
As a result, the inclination of the body in roll plane is decreased
(Fig.7-(b)).
• Tonic labyrinthine response
On the other hand, the tonic labyrinthine response (TLRS) is
employed as rolling motion feedback to CPGs (upper left part of
Fig.6). In Tekken, we made the body roll angle be inputted to the
CPGs as a feedback signal expressed by Eq.(8), (9).
Feede·tlrs = δ(leg) ktlrs × (body roll angle) Feedf ·tlrs =
−Feede·tlrs
(8)
Feedf = Feedf ·tsr·vsr + Feedf ·tlrs (9)
1The “tonic labyrinthine reflex” is defined in Ogawa et al. (1998).
The same reflex is called “vestibular reflex” in Ghez (1991).
The rolling motion feedback to CPGs: Eq.(8), (9) con- tributes to
an appropriate adjustment of the periods of the stance and swing
phases while walking on irregular terrain (Fukuoka et al.
2003).
The rolling motion feedback to CPGs is important also in order to
satisfy the condition (e). CPGs, the pitch- ing motions of the
legs, and the rolling motion of the body are mutually entrained
through the rolling mo- tion feedback to CPGs (Fig.8). This means
that the rolling motion can be the standard oscillation for whole
oscillations, in order to compensate for the weak connec- tion
between the fore and hind legs in the CPG network (Fig.4-(b)). As a
result, the phase difference between the fore and hind legs is
fixed, and the gait becomes stable.
Excitatory Connection Inhibitory Connection
angle
Figure 8: Relation among CPGs, pitching motion of a leg
and rolling motion of a body
3.4.4 Stepping reflex to stabilize rolling motion
It is known that the adjustment of the sideway touch- down angle of
a swinging leg is effective in stabiliz- ing rolling motion against
disturbances (Miura and Shi- moyama 1994). We call this a “sideway
stepping reflex,” which helps to satisfy the condition (d) during
rolling motion. The sideway stepping reflex is effective also in
walking on a sideway inclined slope.
For examples, when Tekken walks on a right-inclined slope (Fig.9),
Tekken continues to walk while keeping the phase differences
between left and right lges with the help of the tonic labyrinthine
response. But Tekken can- not walk straight and shifts its walking
direction to the right due to the difference of the gravity load
between left and right legs. In addition, Tekken typically falls
down to the right for the perturbation from the left in the case of
Fig.9-(a), since the wide stability margin: WSM 2 is small. The
sideway stepping reflex helps to stabilize the
2the shortest distance from the projected point of the center of
gravity to the edges of the polygon constructed by the projected
points of legs independent of their stance or swing phases (Fukuoka
et al. 2003)
walking direction and to prevent the robot from falling down while
keepingWSM large on such sideway inclined slope (Fig.9-(b)).
front view
WSM WSM
(a) (b)
a sideway stepping reflex, (b):With.
Since Tekken has no joint round the roll axis, the side- way
stepping reflex is implemented as changing the de- sired angle of
the hip yaw joint from 0 (Table 2) to ψ∗
according to Eq.(10).
ψ∗ = δ(leg) kstpr × (body roll angle) (10)
3.4.5 Stepping reflex to stabilize forward speed
In Table 2, P-gain of hip joints in the state (B) and (C) is
adjusted using the measured walking speed: v. For examples, when v
is increased in the state (B) as the swing phase, the swinging legs
land on the forwarder position because P-gain is increased and the
swinging legs are more strongly pulled forward. As a result, the
increase of v is depressed since the forward motion of the inverted
pendulum in the next stance phase is depressed by the gravity
(Miura and Shimoyama 1994). On the other hand, when v is decreased
in the state (C) as the stance phase while walking up a slope, the
increase of P-gain generates additional torque against the gravity
load at hip joints of the supporting legs. As a result, the
decrease of v is depressed. These reflexes contribute to satisfy
the condition (d).
3.4.6 Re-stepping reflex and response for walk- ing down a
step
When loss of ground contact is detected in a swing phase while
walking over a ditch, a cat activates re-stepping to extend the
swing phase and make the leg land on the forwarder position
(Hiebert et al. 1994). We call this “re-stepping reflex/response,”
which is effective for the necessary condition (c) and (d) to be
satisfied also in walking down a large step (Fig.10).
re-stepping
4. Experiments
4.1 Experiments under instantaneous distur- bances
We made Tekken1 walk on several terrains of medium de- gree of
irregularity in indoor environment (Fig.11) with all responses and
five reflexes, those are, the stretch reflex, flexor reflex,
vestibulospinal reflex for pitching, stepping reflex for forward
speed and re-stepping re- flex, among seven reflexes described in
Section 3.4. This means that Tekken can cope with the instantaneous
dis- turbances while walking over irregular terrain consist- ing of
series of steps and short slopes. Without a tonic labyrinthine
response, the gait was greatly disturbed on irregular terrain, even
if Tekken1 didn’t fall down.
(a) (b)
(c) (d)
Figure 11: Walking over a step 4 [cm] in height: (a), walking
up and down a slope of 10 [deg] in a forward direction: (b),
walking over pebbles: (c) and walking over sideway slopes of
3 and 5 [deg]: (d)
Especially noteworthy, Tekken1 successfully walked down a large
step with approx. 0.5 [m/s] speed using the re-stepping
reflex/response (Fig.12). Without the re-stepping reflex/response,
Tekken1 typically fell down
6.0 6.5 7.0 7.55.5
flexor
extensor
0.14[s]
re-stepping response
Figure 12: Walking down a step of 7 [cm] in height. A re-
stepping response was activated when the contact of the right
fore leg had not been detected for 0.14 [s] after the
activity
of the flexor neuron became zero.
forward because fore legs landed on the backwarder po- sition
excessively and could not depress the increased forward
speed.
4.2 Experiments under long-lasting distur- bances
We made Tekken2 walk on a right-inclined slope of 4 [deg] (0.07
[rad]) in indoor environment with all re- sponses and reflexes
described in Section 3.4 in order to confirm the effectiveness of a
tonic labyrinthine re- flex and a sideway stepping reflex
(Fig.9-(b)) under long- lasting disturbances. As a result of the
experiment, the body roll angle and the hip yaw angle ψ of the
right hindleg and left hindleg are shown in Fig.13. Tekken2 had
walked on the right-inclined slope for 2 [sec], and walked on flat
terrain afterwards.
In Fig.13, we can see that the body roll angle was posi- tive
(0.05∼0.14 [rad]) while walking on the right-inclined slope. The
hip yaw joint of the right hindleg moved to the outside of the body
(right) by approx. 0.04 [rad] due to the sideway stepping reflex in
the swing phase and moved to the inside of the body by approx.
-0.04 [rad] due to the gravity load in the stance phase. On the
other hand, the hip yaw joint of the left hindleg moved to the
inside of the body (right) by approx. -0.04 [rad] due to the
sideway stepping reflex in the swing phase and moved to the inside
of the body furthermore by approx. -0.1 [rad] due to the gravity
load in the stance phase. Consequently, Tekken2 succeeded in
straight walking on the sideway inclined slope.
4.3 Outdoor experiments using Tekken2
Even on a paved road in outdoor environment, there ex- ist a slope
of 3 [deg] at most, bumps of 1 [cm] in height and small pebbles
everywhere. With all responses and reflexes described in Section
3.4, Tekken2 successfully maintained a stable gait on the paved
road for 60 [sec]
-0.1
-0.05
0.05
0.1
0.15
[rad]
[sec]
body roll angle ψ of right hindleg ψ of left hindleg
inclination of a slope (0.07 [rad])
swing phase
stance phase
swing phase
stance phase
Figure 13: Walking on a right-inclined slope of 4 [deg]
Figure 14: Photos of walking in outdoor environment
with approx. 0.5 [m/s] speed while changing its walk- ing speed and
direction by receiving the operation com- mands from the radio
controller (Fig.14).
5. Discussion : how to determine the val- ues of parameters
Values of all parameters in the neural system including the virtual
spring-damper system except for θstance were determined
experimentally. But it should be noted that those values in each
robot were fixed in all experiments independent of kinds of
terrain. Although the size and weight of Tekken2 are different from
those of Tekken1, the values of the parameters of CPGs used for
Tekken2 were same with those used for Tekken1.
The most important subject in coupled-dynamics- based motion
generation is to design and construct a neural system carefully,
while taking into account the dy- namics of a mechanical system and
its interaction with the environment. In this study, we designed a
neural system consisting of CPGs, responses and reflexes, while
taking into account the characteristics of dynamic walk- ing and
utilizing knowledge and concepts in physics, bi- ology, physiology
and so on described in Section 2. and 3.. The relationship between
parameters of CPGs and
the mechanical system was previously analyzed to some extent
(Kimura et al. 2001). Since the relationship be- tween parameters
of reflexes/responses and the mechan- ical system has not yet been
investigated, we manually determined values of those parameters
through experi- ments. The issue of how to construct a neural
system suitable for a mechanical system corresponds to the is- sue
of “embodiment” at the lowest level of sensorimotor
coordination.
In addition, we pointed out the trade-off problem be- tween the
stability and the energy consumption in deter- mining the cyclic
period of walking on irregular terrain, and showed one example to
solve this problem (Fukuoka et al. 2003).
6. Conclusion
In the neural system model proposed in this study, the
relationships among CPGs, sensory input, reflexes and the
mechanical system are simply defined, and motion generation and
adaptation are emergingly induced by the coupled dynamics of a
neural system and a mechanical system by interacting with the
environment. To gener- ate appropriate adaptation, it is necessary
to design both the neural system and the mechanical system
carefully. In this study, we designed the neural system consisting
of CPGs, responses, and reflexes referring to biological concepts
while taking the necessary conditions for adap- tive walking into
account.
We newly employed a tonic labyrinthine reflex, a side- way stepping
reflex, and a re-stepping reflex/response to make the
self-contained quadruped robot (Tekken2) walk in outdoor
environment. In order to increase the degrees of terrain
irregularity which Tekken2 can cope with, we should employ
additonal reflexes and responses, and also navigation ability at
the high level using vision.
MPEG footage of these experiments can be seen at:
http://www.kimura.is.uec.ac.jp.
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