Dynamical Motion Control for Quadruped Walking With Autonomous Distributed System
Transcript of Dynamical Motion Control for Quadruped Walking With Autonomous Distributed System
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Dynamical Motion Control for Quadruped Walking
with Autonomous Distributed System
Katsuhiko INAGAK I and Hisato KOBAYASHI
Department
of
Electrical Engineering, Hosei University
Koganei, Tokyo
184,
Japan
Abstract his paper discusses dynamical mo-
tion control
for
quadruped walking machine.
Quadruped walking machine requires dynamical
cc..trol to keep its stability in high speed walk-
ing. We make the dynamical compensator by
using a weight oscillator built in the center
of
the body. First we calculate the exact oscil-
lators motion that can neutralize the moment
con.pletely. Next we try to use
a
harmonic os-
cillation for the oscillators motion instead of the
accurate motion. Finally we use a notion of au-
tonomous distributed control to our motion con-
trol and expand it to quasi-dynamical walking.
I. INTRODUCTION
Quadruped walking machine requires dynamic walking
in high speed walking, because the number of standing
legs is less than three. In tr ot gait t hat is known as a
typical dynamic walking pattern, two pairs of diagonal
legs make standing phase, respectively. Thus, there ex-
ists a moment when the body
is
falling down around the
supporting axis. If we leave this moment out of consid-
eration, walking machine may walk with rolling motion,
repeatedly. This fact influences the accuracy of the t ra-
jectory, because each leg makes landing at unexpected
timing. Therefore , we should restrain the moment for
stable walking. For this problem, a method which uses a
notion of ZMP(Zero Moment Point) has been proposed.
In this method, the moment by the gravity can be neu-
tralized by acceleration generated by the trajectory of
the body. However, this method requires vibration
of
the body, and such motion is not desired for some pur-
poses s x h as transportation.
In th is paper, we utilize an oscillation of a small weight
instead of oscillating the body. Our method makes it
possible to neutralize the moment without oscillation of
the body. Usually, horses running accompanies a head
shaking. We think this motion expects similar effect to
1004
the oscillator. By the way, we must not ignore energy
cost problem to discuss the walking machine, because it
has poor energy efficiency compared with other mobile
machines. On this point of view, utilization
of
the os-
cillator causes increase of the body weight. But, we can
settle this problem easily by using mechanical elements
as the oscillator. More important point is to reduce the
energy consumption to drive the oscillator.
In th e first part of this pape r, we make the motion of
the oscillator clear, and show that its motion is similar
to sine wave. Then, we analyze the moment and energy
consumption in case that a simple sine wave is applied
to motion of the oscillator instead of the regular motion.
Finally, we apply this method into our autonomous dis-
tributed control system that has been already proposed.
11.
MOTION L A N N I N G O F O S C I L L A T E R
A .
Standard Gait Rule
First, let us consider gait rules to apply the oscillator
system. In our former report, we have already proposed
a method of gait transition from low speed walking to
high speed walking. Three typical gaits (crawl gait , tro t
gait , gallop gait) are connected smoothly in our method.
The crawl gait is known
as
a typical static gait in slow
speed walking. And, as increase
of
the walking speed,
walking manner changes into the trot gait and the gallop
gait in order.
In this study, we consider only from the crawl gait to
the trot gai t, because the gallop gait requires a jumping
motion. It is difficult to make the gallop gait by only
the oscillator system.
Fig.1
shows a walking pattern of
four legs from the crawl gait to th e trot gait. Black part
means that leg is in the standing phase, and white part
is in the swing phase. We can see that the period of the
standing legs is reduced from three to two
as
increase
of the walking speed . And, during the period of two
standing legs, the standing legs are placed in diagonal
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R-F
R-H
L-F
L-H
0.2
0.1
0.0
Fig. 1 . Gait transition from crawl
gait
to trot gait
Pa1
Pa
2
D i r e c t i o n
Fig.
2 .
Required accelerative direction
position of the body.
B.
M o t i o n
Planning
First of all, we consider the motion planning of the
os-
cillator during the trot gait, because the dynamical con-
trol is always required in this gait . Next, we consider
the motion of the oscillator. There are some directions
of accelerat,ion to generate
a
moment to neutralize the
moment by the gravity. In the trot gait, the pair of
the standing legs is changed at an ins tant. Thus, re-
quired direction of the acceleration is also changed at
an instant as shown in
Fig.2.
At this time, compo-
nent of sideways direction is kept in the same direction,
but that of lengthwise direction is changed into oppo-
site direction. Therefore, it is not wise to generate the
acceleration to the lengthwise direction. Accordingly,
we give the oscillator
a
prismatic motion in sideways di-
rection as degrees of freedom.
Fig.3
shows a walking
situation for our study . The body of the walking ma-
chine goes straight in the y axis direction at
a
constant
speed. Points Pl q,
l),P2 22,~2)
are on the support-
ing diagonal line. At
t
= 0 ,
y
coordinate of t.he body is
/
/
@ C o n t a c t
W a l k i n g
D i r e c t i o n ,/
I
Fig.
3.
Walking situation
on y1, and that is on y2 at
t
=
T .
We can describe the
moment M1 by the gravity around the supporting axis
as follows.
where,
In these equations,
E , )
means
x
coordinate
of
the
point
C
in
Fig.3.
And. other variables are as follows.
mb : massofthe body
m, : mass
of
the oscillator
x,
:
I coordinate
of
the oscillator
g
: acceleration of gravity
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On the other hand, we can make
a
moment to neu-
tralize
M1
by reactive force of the oscillators motion.
This moment
M z
can be described as follows.
Mz
=
-m,hx,
sin 8
(6)
where, h is height of the oscillator
Therefore, the total moment Ma is described as fol-
lows.
To keep the stability of the body, we should solve
following differential equation, because
k f b
must always
keep zero.
where,
a l
= -ag m,+mb) (9)
mbg z l k -
bg m,
+
mb) (10)
2
l =
The solution of this equation is:
where, C1 and
C2
are unknown constants.
C. Boundary Condation
Next, let us consider boundary condition t o find the con-
stants C1,Cz. As stated before, pair
of
the supporting
legs is changed at an instant in the trot gait. Thus, the
motion of the oscillator should be connected smoothly
in this time. Moreover, the total motion of the oscilla-
tor should make gentle oscillatory motion.
From these
reasons, the boundary condition should be described
as
follows.
Sm 0)
=
S , T )
=
0 (12)
By using this condition, the parameters C1,Cz are
derived
as
follows.
(14)
a1
m
1
c2 = --
% I
1+ exp
-m
)
Fig.4 shows a result of a generated trajectory of the
oscillator in the trot gait. Fig.5 shows acceleration of
the oscillator in this time. We can see that the trajectory
is similar to sine wave.
D.
Energy
Consumption
Problem
In the former section, we solved the motion of the oscil-
lator that can neutralize the moment. But, this method
has some problems. First , required acceleration shown
in Fig.5 has discontinuous points at the changing point
of the supporting legs. Such motion may be difficult
to realize and have a bad influence on energy consump-
tion. Even if we can realize
thc
intended motion, walking
motion commonly accompanies other kinds of undesired
acceleration such as vibration of the body. For example,
position error of a standing leg causes such vibration,
because closed mechanical linkage is formed by plural
stand ing legs on the ground. After all, we think
it
is
impossible to neutralize the moment completely. W hat
should be noticed is energy consumption problem ra ther
than precise motion control.
Then, we examine an effect when a single harmonic
oscillation is used insbead of the exact motion in the
former section. It is easy to generate the single harmonic
oscillation. And, th is motion is suitable for economy of
the energy consumption. The motion of the oscillator
can be described as follows.
15)
- = l o cos
ut
+ a )+ l +
2
In thi s equat ion, we must decide the angular veloc-
ity and the phase a suited for the characteristic of
the regular trajectory shown in
Eq.11.
Therefore, we
decided them
as
follows.
x
-
T
w
a = o
(17)
Then, let
us
think about remained undecided value,
amplitude
4.
he amplitude
A
should be decided as the
moment has minimum value. The moment of the body
can be calculated by substituting Eq.15 to Eq.7.
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0 .
U .
x 0 .
- 0 .
- 0 .
4 0 - I
1
I I
e e
e 0
4 0 -
I
1 I
-
0 . 0
0 . 5
1 . 0
1 . 5 2 . 0 2 . 5
y [ml
Fig. 4 . Oscillatory trajectory in trot gait
I I
I I 1
0 . 0 0 . 5
1 . 0
1
?5 2 . 0 2 . 5
y m l
Fig. 5 . Acceleration of oscillation
our former report, we generated a periodic signal based
2 1 + 2 2
E = m A o ( h w 2 + g )
COS
( U t ) + a i t + b i + m g - 2 18)
Therefore, 4 hould satisfy following condition.
on
a
basic motion of a leg. Thus , this method is very
compatible with the oscillator s motion. First p art of
this section, we show basic notion of the autonomous
distributed system and apply it t o the dynamical motion
control
(19)
The solution of this equation is: A . Phase
Szgnal
4a
1
m w 2 h w 2 +
g)T
(20)
Fig.6
shows time response of the moment. The solid
line is the moment when the oscillator makes single har-
monic oscillation with amplitude
A
in Eq.20. And, the
dotted line is the moment when the oscillator is fixed on
the moment compared with the case that the dynamical
motion is not taken.
Our control system consists of one central controller
and four distributed controllers for each leg, as shown
in Fig.7. The central controller makes a basic motion
model of leg by solving following equations.
4 0
=
the center of the body. Our proposed method can reduce
; =
h k1
Ust,
Us,
p,n){q +
p
m ( l , p , 21)
4
= h ( k ,
U s t . h u , P , q ) { - P + q
m ( l , p , q ) } (22)
111. A U T O N O M O U S D I S T R I B U T E D C O N T R O L where
In this section, we adopt the notion of the autonomous
distributed control to our dynamical motion control. In
m ( l , p ,
=
l 2 -
p 2 + q 2 )
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Fig. 6. Time response of moment
Central
Controller
Rhythm
Generator
Left-Fore Left-Hind
Right-Fore Right-Hind
Leg
Distributed Controllers
Fig. 7. Autonomous distributed control system
h ( k , % t ~ ' s W > P ? q )
q+p-:{:p,q)
{ q
+
P 4 l . P Q)l v sw
' .
-usW I
{ q
+ p m l .p . }
I
U,t
u s t