maneuvering operations of the quadruped walking robot on the slope
TRANSCRIPT
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Maneuvering O peration s of th e Qua dru ped Walking Robot
on th e Slope
Hideyuki TSUKAGOSHI, Shigeo HIROSE, and Iian YONEDA
Dept. of Mechano-Aerospace Eiig. , Tokyo Institute of Technology
2-12-1 Oookayama Meguro-ku Tokyo 152
Japan
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ail: t si1kaBmes
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ac jp
Abstrac t
O n t h e a s s u m p t i o n t h a t a q n a d ru p e d r o b ot wo r k s
o n
a
s l o p e , we d i s c u s s h o w t o ma k e it p r e v e n t t u mb l i n g
over .
Th e larger the d i f f erence becomes be tu ieen the potent ia l
energy of the ce nte r of gravi ty of th,e ini t ial pos i t io n and
that o f the h ighes t po s i t ion a f t er i t s ro ta .t ing , the l ess the
r o b o t t ~ m b l e s . o th i s d i f f erence can be regarded as sta-
b i l i t y margin , and a nove l gai t to obta in larges t s tabi l i t y
ma r g i n i s me n t i o n e d h e r e . It i s n o t h i n g e l s e b u t i n t e r -
mit tent crawl ga.i t whose post l ire
is
i l lust ra ted a f t e r that .
In a d d i t i o n t o it. e n e i y y s t a b i l i ty c o n t o u r , d r awn , b y c o n -
nec t ing equal s tabi l i t y poin t s o n the inc l ined p lan e ,
is
also
explained
in
t h i s p a p e r a n d i t i s h e l p f u l t o d e s i g n s t a n -
dard foot t ra jec tor ies . An o p t i ma l p o s h i r e o n t h e s l o p e
designed in this wa,y, resul ts in inverse trapezoid sha.pe,
which means that upper two l egs are located wider than
l o w e r t w o o n e s . T h i s f o r m w o r ke d for t h e e x p e r i me n -
t al m a c h in e , T I T A N
V I I ,
A i r t h e r m o r e ,
if
t h e s t a n d a r d
t r a j e c to r y f o r o n e d i r e c t i o n i s c o mb i ne d w i t h a n o t h e r d i -
rec t ion t ra jec tory , the quadruped robot can , eas i l y swi tch
i t s
proceeding d i rec t ions , keeping enough s tabi l i t y marg in .
Th i s se q ue n ce i s s h o wn
in
the las t par t .
1.
Intr
oductaon
A quadruped robot is so promising on account
of
its
configuration adaptability that it will be utilized in such
a
rough terrain
as
the construction field. In order to
make the best use of its function, the robot should move
around freely with large stability margin in irregular ter-
rain where conventional vehicles cannot move.
Our robo t is supposed to be used for construction work
on
some inclined terrain. On
a
gentle slope, it
is
expected
to walk around without any support , whi k on a steep one
it would be hoped to climb up by wire towing, shown in
We also assuine the planet exploration by using the
quadruped robot by itself. On this
occasion it will be
strongly desired to walk around craters with no support .
In this stitdy, on t,he assumption tha t th e quadruped
robot walks on a gentle slope without wire support, we
propose a novel gait
and
test it by TITAN VI1 shown in
Photo.1[1].
In section 2, complete consumption of unexpected kine-
matic energy generated by a disturbance is looked upon
import ant to prevent tumbling. Therefore, the concept of
energy st>abilit,ymargin[2] is very useful to evaluate how
stable
t,he
center
of
gravity is. Though paper[Z] also pro-
posed the slope gait, the algorithm there referred to the
optimal position for the center of gravity from the side
of some fixed support legs. But what we want to know
and get is the information about the optimal position for
support legs from the side of the center of gravity in or-
der to realize more stable and active walk for practical
use. To settle the problem, energy stability contour is
proposed here.
In
section 3, intermittent crawl gait is introduced,
wliicli remains t,lie center of gravity in the same place in
swing mode. Th e big difference from existing crawl gait
is featuring zigzag discontinuous movement, for t he center
of gravity. As for discont,inuous walk, paper[3] proposed
straight movement, biit it doesnt take full advantage of
stability possessed in the robot. Compared with paper131
gait, the center of gravity swings to the largest possi-
ble stahilit,y position in the new gai t. Energy stability
contour is used to design the standard foot trajectory.
Fig.1.
Fig.1 Image of a quadruped robot
on
the slope
Proc. IROS 96
0-7803-3213-X/96/ 5.00 O 1996 IEEE
863
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Section 4 shows the way to switch proceeding direc-
tions with the center of gravity kept in
as
stable area as
possible. It reminds s of paper[4] idea which proposes
the sequence from one posture to another posture, fixing
the center of gravity in the same place. But it doesnt
refer
to
stability margin while transforming, so it is neces-
sary to improve the sequence including stability margin.
Apart from it, well show
you
the qudruped robot can
change directions easily by combining two standard feet
trajectories.
2 .
Energy Stubility Contour
2 1.
Energy Stability Margin
While a quaclruped robot traversing some rough ter-
rain, support legs might float in the air because of a dis-
turbance. Supposing the practice use, it would be essen-
tial for the robot to prevent tumbling even in this case.
The simplest way is to make better use of gravitation.
In other words, the center of gravit,y wont reach the high-
est points(Hi(i = 1-4) in Fig.2), if its kinematic energy
generated by the disturbance is completely consumed by
tlie increase of poten tia l energy. .Judging from this point
of view, large difference between the potential energy of
the initi al center of gravity and t hat of its highest po-
sition after rotatin g can lead to the stable gait. This
difference is called energy stabi lity margin[2].
The concept of energy stability margin helps u s
to
achieve the most stable position of the center of grav-
ity in two dimensions when two legs are fixed on an in-
clined plane, as is shown in Fig.2.
On
condition that
tlie distance between the centcr of gravity and the in-
clined plane remains constant, posture(a) put the cent.er
of gravity just in the middle between two feet, but en-
ergy stability margin of the lower side is smaller than
that of the upper one. Posture(b) can divide energy sta-
bility margin equally between upper side and lower side.
Therefore, posture(b) is suitable to work on the slope.
H2
Ib)
Fig.: While
a)
as smaller margin at lower side,
(b) can divide margin into equal a t bo th sides.
2 2 .
How
o
design Energy Stability Contour?
Next, the most stable position of the center of grav-
ity should be mentioned when more than three legs are
touched on the ground. But what we want to know in
practice is the best placement of support legs from the
center of gravity during t he robot walking, not the place-
ment of the center of gravity from arbitrary fixed legs.
This section refers to a new approach to solve the prob-
lem.
First of all, energy stabilit,y maigin in three dimensions
is cleared here. Look a t Fig.3. Whenever
a
quadruped
robot tumbles, the center of gravity rotates around a sup-
port line, a ine coiinecting two support feet. If you take
this phenomenon into account, the most important fac-
tor is not the relation between the center of gravity and
one foot but the relation between the center of gravity
( G ) and the foot of perpendicular line
( P )
on the sup-
port line ( q s ) .Energy stability margin can be defined as
the height difference between the perpendicularly highest
point
of
G
when
P
is rotated around
P
and the initial
point of
G.
z
Fig.3
How
does the quadruped robot tumble in 3
dimensions?
Fig.4 Th e configuration of body coordinate of the
quadruped robot on the slope.
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Second, if you caii know the trajectory connected by
the points P showing the same eiiergy stability margin,
it is helpful for y m to design the placement of support
legs inside t,he kinematic limit. This tr ajecto ry we call
energy stability contour.
On the basis
of
these facts, the equation of energy
stability contour can be derived as follows. To make
the story easier,
we
consider the body coordinate
Ob
X b y b Z b ,
whose 2-b axis is along the maxinmm gradient line
and yb axis is perpendicular to 2-1, 011 the slope. n this
flame, G and
P
are represm ted like these.
OG=
:
OlP = [ ;]
(2)
The unit vector PQ the support line y9 is derived like
this.
If you rotate P around the
P
by 90 you can get a
new vector PR, which is represented
as
follow.
r3=
PQ x P 2
4)
The highest poiiit H can he expressed as follow hy
using two vectors,
P3andP-h.
(5)
PY
= ( c o s c p ) . E J ( P ~ )(sin91
E J ( P ~ )
=
Note that
(coscp).
( ~ 3 )
(sincp) ( ~ k )
tan-'
Y J = P i l ,
co s0
0 i118
[ si:* c,0,0 ]
J =
The inark / on the vect,ors means that those elements
are represented in
a
new coordinate which is fornied by
EJ * rotation
of
o T b Y b Z b flame around yb axis. At
a
result, energy stab ility margin of the quadru ped
robo t, whose weight is expressed as 1 79, is represented
as
follows.
A4 =
m g ( P Y I , - P G q 2 )
=
mg{ ( A os cp + B sin c p -1Yb sill 6 + h cos0)}
( 6 )
A = - 3 - b sin i + 11 cos
B = . -&===g=(hXbsinO+(xX(?
y;)cose)
The set of S b , 16 ) satisfying the equation
6)
s energy
stability contour itself which possesses energy stability
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margin hl. On the basis of this relationship, we'll show
you the exaiiiples of energy stability contour. n this pa-
per, the distance h between the surface
of
the ground and
the center of gravity is fixed to 400[mm], which is the
same colidition as our experiment, and the weight n?g
is 50[kgf].
n
O =
0 case,
energy stability contour be-
comes a concentric circle with center at origin, shown in
Fig.B(a).
On
the other hand, in
6
=
15"
case
(Fig.5(11)),
the curves form the ciicles approximately but they shift
lower and lower along the maxiuium gradient line as M
increases.
40_s00
(a)
Energy stability contour x b [ m r
0 degree
,400
E
x200
200
-
( h ) Energy stability contour x b [ m r
I5
degree
Fig.5
Calculation results of energy stability
contour in the case of
(a)O O
and (b)15
.
Energy Stability
Contour
Crawl Gait
(a ) Crawl
G a i t ( b )
I n t e r m i t t e n t
Fig.G
The gait difference between crawl gait
and i.c. gait
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3. I n t e r m i t t e n t
C r a w l
G a i t
3
1.
S i g n i f i c a n t F e a tu r e
To piit the quadruped robots to pract ical use, we need
to look over the gait again.
As
they have been developed
for walking vehicles to ride in, the continuous movenient
of tlie center of gravity has been regarded as important
and crawl gait is now popular among researchers. How-
ever, Diagonal Transfer Exchanging point in crawl gait,
called DTE(51 in Fig.G(a), decreases energy stab ility mar-
gin sharply, When walking robots are used only for work,
taking enough energy stabilit,y margin
is
much bigger is-
sue rather th an continuity of the center of gravity. In this
sense, the posture represented in Fig.G(b) is the ideal one,
because the center
of
gravity moves only in four legs sup-
port mode and is remained at the most stable position in
swing mode. Thaiiks for this discontinuous movement,
it is easy to switch proceeding direct.ionsas is mentioned
in section 4,
s
well
i l s t o
have large stahilit,y.
These
are
the basic concepts of the novel intermittent crawl gait
proposed here.
The big advantage of this new gait is that it enables
the quadruped robot to avoid the critical points. Fig.7
shows us the transfer of energy stability margin for one
cycle, and you can compare the stability among inter-
mittent crawl gait, duty factor, p = 0.83 crawl gait and
/I = 0.95 crawl gait,. Duty factor means t.he ratio
of
one
leg's support time per one cycle. In
p =
0.83 crawl gait.,
the velocity of the center of gravity is equal t o th e average
velocity in intermittent crawl gait if legs swing in maxi-
mum speed. Both
/I
= 0.83 and
/3
=
0.95
crawl gaits go
through dangerous terms when energy stability margin
per weight is less t.han
loinin,
while intermittent crawl
gait can keep more than 201nm. If you compare Fig.7
result with the foot diagrams shown in Fig.8, energy sta-
bility margin decreases in swing mode in any gates, but
intermittent crawl gait can maintain the decrease as little
as
possible.
time[s]
Fig.7 Comparison
of
energy stability margin among
t,liree gaits.
To make the most
of
the merit of this gait , the optimal
movement of the center of gravity needs to be revealed
from the point of energy stability margin. Considering it
is equivalent to calculate the optimal foot trajectory from
the side
of
body coordinate. Fig.9 shows the movement
of support legs from tlie body for one cycle. Now we
know tlie foot trajectory of intermittent crawl gait forms
approximate V , then all we have to consider is design
V shape inside four legs' kinematic limits.
In
addition
to it, two tiajectories in the same sides with respect to
proceeding direction ar e just the same, while two tra-
jectories put diagonally are a point symmetry about the
center, shown
in
Fig.10.
leg
leg 2
leg
3
leg
4
I
I
5 1 hI
(a ) c raw l ga i t
I 1
5
b l
b) i n t e r m f t t e n t c r a w l g a i t
ez a
sw ing mode
uppor t mode w i th sh i f t
uppor t mode w i thou t sh i f t
Fig.8 Crawl and i.c. gait foot diagrams.
Ybl I
I
I
I
I
I
osture
1
, I
osture
2
nIn I
posture
3
I
I
I
I
posture
4
posture
5
I
posture
6
I
posture
7
posture
8
Fig.9 Standar d foot traject ory of i.c.
gait forms V .
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3 . Designing Foot T r a j e c t o r y
In this section, the way to design the foot trajectory is
shown when a quadruped robot takes intermittent crawl
gait for arbitrary direction. Th e point is that how to de-
cide the form V inside kinematic limit so as to get energy
stability contour
as
large as possible. To get the result,
we assume that we can take advantage of the computers
repeated calculation. For example, we use TITAN
VIPs
kinematic limit shown in Fig.11.
(1) We set large enough stab ility contour
as
is shown in
Fig.l2(1) a t first. Four support lines which cross perpen-
dicularly on th contour with the line from the origin are
drawn, a-a, b-b, c-c, and d-d. But in Fig.lP(1)
case
the contour is too large for
a a
ine to cross kinematic
limit.
So,
it is necessary to reduce the energy stability
contour.
(2) After keeping to reduce energy stability contour
little by little, we can get four support lines which can
cross two kinematic limits (Fig.12(2)).
( 3 )
In order t o decide the form V, we take note of re-
gion I, for example. i. . and
a 2
are the lines wliicli
are shifted in parallel with - and
d
, keeping the
same distance between them, so
as to
make the intersec-
tion between b and i inside the region 1. The
point w in Fig.10 is equivalent to this point. The point
v and
U
should be each situated on the line
U
a and
d
so tha t the line v - u are parallel to the proceeding
direction. But in Fig.12(3),
2-2
cannot cross the region
1,
so
energy stability contour is still too large.
4) o improve the
( 3 ) s
difficulty, the angle of support
lines should be changed or energy stability coiitour should
be reduced again. Fig.12(4)
shows
the first combination
which satisfies the condition
of
the
A
iivw inside region
1.
5 )
The rest of other three trajectoiies should be lo-
cated
so
as to fulfill the Fig.10 regulation. In region 2
case,for example, if A uvw can be put inside kinematic
limit of region 2 after the point v is located on d-d line,
it means that region 2 satisfies the condition. Othe r two
feet location can be derived in the same way (Fig.12(5)).
Fig.10 The feature of standard foot trajec tory
of intermittent crawl gait.
Fig.11 Kinematic limits of TITAN VII.
\
Fig.12 The sequence of designing standard foot
trajectory.
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3
- 3 . Posture
o n
t h e incliened p l a n e
Design of V trajectory can derive the difference be-
tween the level posture and the inclined posture. As the
inclined angle becomes bigger and bigger, energy stabil-
ity contour shifts lower and lower. This effect generates
the inclined posture forming inverse trapezoid, meaning
upper two legs are located more widely than lower two
ones, while left and right support lines are parallel
to
each other in the level posture (Fig.13).
4.Switching Gait
In section 3, standard foot trajectory for only one di-
rection is shown. In this section, the way to switch froin
one direction to anot,her is proposed,
as
the body faces
the same direction.
In practice, it is essential that that the robot should
proceed toward all the directions. When the robot walks
along one standard trajectory, the problem is how
to
switch from one trajectory to another one, maintaining
energy stability margin as large
as
possible.
During some standard trajectory, the quadruped robot
has one support line which enables diagonal two legs to
be swung. In Fig.14 left upper picture, the support line
,connect ing leg1 with leg3 corresponds to it . Both leg2
and leg4 can be swung by using the shift of the center of
gravity. If these two legs are situated on some support
line in the next standard trajectoly, other two legs can
also be shifted on another suppor t line. This switcliing
way carries out only four times steps.
Left downside picture in Fig.14 shows Switching gait
sequence, shifting from a posture along the maximum in-
clined direction to another posture along side direction.
Without consideration of kinematic limit, left picture se-
quence enables this switch. Th e first swung foot is lo-
cated on the point which is some distance far from the
next standard middle point. Number
1
and 3 legs can
be swung keeping energy stability margin in the former
posture, while 4 nd
6
number legs can be swung keeping
the latter posture stability margin.
But on account of kinematic limit, energy stability
margin must be reduced. Then two simple switching
gaits, which can keep the reduction as little as possible,
are explained
as
follows. One is that kinematic limit is
set smaller in advance than the real limit, which allows
legs out a littl e during switching. In this case, though
energy stability margin of standard foot trajectory is de-
creased, switching gait keeps either former gait stability
or latter one. Another way is that stability margin dur-
ing only switching is reduced
so as to
be equal stability
while swinging. In light lower picture in Fig.14, number
1 eg is located in order t o make 3, 4 nd
G
legs indicate
equal stability margin.
This switching gait is fully available by a computer
control, if data bases for standard foot trajectories are
stocked in advance.
Inclined
Fig.13 Level posture and
inclined posture.
Photo.2 Feet are shifted
just like inverse trapezoid.
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x ax is
direction
1
5 . Coizclusions
When a quadruped robot is utilized on
a
slope, it is
inclispensable for i t to preventr tumbling. Th e best
pos-
t.ure for the robot is one which enables it to obtain large
energy stability margin.
For the first step to realize it, the concept of energy
stabilit,y contour on some inclined plane is proposed in
this paper . This means that any suppo rt lines which
cross the contour perpendicnlarly with the line from the
origin, show equal energy stabi lity margin, if th e center
of gravity is rotat,ed aroun d those lines.
By using energy stability contour, it is cleared that
intermittent crawl gait proposed here possesses larger
energy stability margin than crawl gait.
In
this gait,
the center of gravity moves only when four legs support
mode
which contributes to increase stability margin. En-
ergy stability contour
also
helps to design standar d foot
trajec tory i wkinematic limit. Th e design way is also rep-
resented in this paper.
After obtaining stand ard trajectories for each direction
and for each incline, the way t o switch directions is a big
problem. But switching gait proposed here enables this
task easily by combining two t,rajectories simply.
These theories in this paper work
on
15 degrees inclined
plane, by using experimental machine TITAN
VII.
And
we make sure that the concept of energy stability con-
tour and intermittent crawl gait will be available for any
quadruped robot used on sonie gentle slope.
Ackiiourledgnient
We greatly appreciate the cooperation of Tokyu Con-
striiction Company.
References
[l]S.Iiirosc,
I