maneuvering operations of the quadruped walking robot on the slope

7/25/2019 Maneuvering Operations of the Quadruped Walking Robot on the Slope

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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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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

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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 .

866



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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.

868


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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

maneuvering operations of the quadruped walking robot on the slope

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