7/25/2019 Control of a Quadruped Trot
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C o n t r o lo f a Q u a d r u p e dTro t
J S
Glower nd U. Ozguner
..
..
..
D e p a r t m e n t o f E l e c t r i c a l E n g i n e e r i n g
The Oh ioS t a t eU n i v e r s i t y
Columbus, Ohio 43210
ABSTRACT
A s i m p l ec o n t r o l a w sp r o p o s e d om a i n t a i n
t h e m ov em en t o f a t r o t t i n g q ua dr up ed .he
a s s u m p t i o ne h i n dh i so n t r o la wshat one
s h o u l d b e a b l eoe l y u po n t h eu a d r u p e d ' s
m om entum f o r t a b i l i t y . The c o n t r o la wh o u l d
o n l ye r v eoe d i r e c tra i n t a i nh e
quad rupe d 's momentum. The r e s u l ts a c o n t r o l
l a wh a tw o r k swe l l so l on g as the uad ruped i s
movi ng
INTRODUCTION
Thereas ee n a l o t f e c e n t n t e r e s t n
wa lk ingm a c h i n e s a nd r e s u l t s h av ee e n r e p o r t e d
on b o t hs i x [1,2] a n d o u r [33 l e g g e d o c o m o t i o n .
I n
t h i s p ap er , a c o n t r o la wh a t t a b i l i z e s a
q u a d r u p e d t r o ti t h o u te l y i n g upon s t a t i c
s t a b i l i t ys p resen ted . I n o r d e rot a b i l i z e
t h e q u ad ru p ed , f i v ee p a r a t en p u t sr eo n -
s i d e r e dh a tf f e c th eu a d r u p e d ' su t p u t s :
v e l o c i t y , e i g h t , o l l , e l e v a t i o n , a n dzimuth.
T h r e eu t p u t sf o r w a r de a n ,a t e r a lean , and
1a t e r a l e l o c i t y ) r e o t i e c t l y o n t r o l e d
s i n c e h e ya r e b ou nd ed o r i m p l e r a n s f o r m a t i o n s
o fo t h e ro u t p u t s .
T h e a s s u m p t i o ne h i n dh i so n t r o la ws
t h a t o ne s h o u l d b e a b l e o e l y u po n t h e q ua d-
r u p e d ' s m omentum f o r t a b i l i t y . T he c o n t r o la w
s h o u l d n l ye r v eoe d i r e c t rm a i n t a i nh e
quad rupe d 's momentum. The r e s u l ts a c o n t r o l
l a wh a tw o r k sw e l l
so
l o n g as t h e u a d r u pe d i s
moving.
QUADRUPED MODEL
Them od el f o r a q u ad ru p ed s e d i n h i s p a p e r
i s a s i n g l e i g i d mass i n 3 -s pa ce f o r a b ody w i t h
f o u rx t e n s i b l ee g st t a c h e doa c ho r n e r .
Each l e g sa b l e og e n e r a t e w oh i p o r q u e s
and
o n eo r c el o n gh ee n g t hfh eeg . The
r e s u l t a n t o r c e s a nd t o r q ue s
on
t h e b o d yproduced
b y e g
j
i n body oo rd ina tes re as fo l l o ws [43:
Top View
Side
View
t Y l
Front iew
F i g u r e 1. QuadrupedModel
1496
CH2282-2/86/0000/1496 01.00986BEE
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I f one ssumes ma ss les s e g s , h e n h e e s u l t a n t
a c c e l e r a t i o n so f h e body i n b o dyc oo r d i na t esc an
b e o u n d r o m h e o l l o w in ge q u a t i o n s :
1-
T =
where ~ j ds h e o t a t i o n a l n e r t i a a b o u t h e t h
a x i s , g s h ea c c e l e r a t i o n d ue t og r a v i t y , a n d
T 1 s a 3 x 3 r a n s f o rm a t i o n r o me a r t hc o o r d i n a t e s
t o b od ycoord ina tes .
I nb t a i n i n gq u a t i o n s 3 and
4,
mass less
l egsw e r e as sum ed so t h a t h e e r i v a t i v e
a~
i s
s i m p l y
Q W I f
onea s l e g sw i th mass, th en h e
r o t a t i o n a ln e r t i a i s no longeron sta nt . As a
r e s u l t , a mo re c o m p l i c a t e d o r mo f h ee q u a t i o n s
3 and
4
a r e equ i r ed . As a f i r s to r d e ra p p r o x i -
mat ion, owever, one can assume t h a t h e change
i nh e r o d u c t fo t a t i o n a ln e r t i a and r o t a -
t i o n a le l o c i t ysm a l l .i n c e a t r o t t i n g
quad r uped i se i n go n s i d e r e d ,h eo t a t i o n a l
v e l o c i t yh o u l d be n e g l i g i b l e
i f
t h er o ts
s tab le . Thus ,ne may assume t h a td d i n ge g s
w i t h mass will n o tf f e c tq u a t i o n s 3 and 4
ex ces s iv e ly . However , th e max imum speed th a t a
quad r upedanr av e l wi b e a f f e c t e d .na c t ,
o n e h o u l di n dh a t a s t a b l e r o t x i s t s n l y
f o r i m i t e d speeds.
GAIT COORDINATION
T he u rp os e o f h i s w or k i s o i n d a c on -
t r o l a w o r h e e g o r c e s and t o rq u ess u c h h a t
a quadr uped i s a b l e o r o t a t a s peed f o rw h i c h
c o n c e p t so f t a t i c t a b i l i t y may n o t b e u s e f u l .
T he i n p u t s o h e s y s t e m r e h ee g o r c es and
t o r ques . As a r e s u l t , one m us t av eegs nder -
ne ath he body some o f he im e op r o v i d e h e s e
c o n t r o lo r qu es and f o r c es . The pu r po s e f a i t
c o o r d i n a t i o n s o see t h a t u p p o r t i n g e g s r e
a v a i l a b l e o s u p p o r t h e b od y.
T h e g a i to n s i d e r e dnh i s a p e rsh e
s i n g u l a r r o t 151 I n a s i n g u l a r r o t , one p a i r
o fd i a g o n a l l yo p p o s i t e e g sa r ea l w a y ss u p p o r t i n g
t h e body. When one p a i r f e g s s i f t e d , h e
o t h e r a i rse t down. The g ai to o r d i n a t i o n
a l g o r i t h m ,h e r e f o r e ,u s te t e r m i n e when t h e
s u p p o r t i n g e g sa r e o be s w i t chedandwhere he
new pai
r
o fs u p p o r t i n g e g sa re o be s et down.
S in c e a s i n g u l a r r o t s a s y m m et r ic a i t, a i t
c o o r d i n a t i o ns i m p l e :h e u p p o r t i n ge g s r e
s w i t c hed when t heu p p o r t i n gegs move h a l f a
s t r o k eb e h i n d h e i r e s p e c t i v ehi ps . The new se t
o fu p p o r t i n ge g sr ee t down h a l f a s t r o k e
ahead o f he i
r
r e s p e c t i v e h i p s .
It
h as t o b e m e n t i o n e dh a t ,w i t hm a s s l e s s
l e g s ,s u p p o r t i n g e g sca n be se t down anywhere a t
anyime . When t h eegs havemass,owever , the
l e g d y n a m i cs r ec l u d em m e d i a t ee g o s i t i o n i n g .
When a l e g s n t s w i n g p ha se , it shou ld be
d r i v e n o h e o r r e c t t a r t i n gp o s i t i o n np r e -
p a r a t i o no r e c o m i n g a s u p p o r t i n ge g .nh e
s i m u l a t i o n , h i s sa c c o m p l i s h e db yd e f i n i n g h e
f o r e - a f t o r q u e o f e g j t o be:
where A i s h e e g a n g l e h a t p l a c e s o o t j h a l f
a s t r o k e a he ad o f t s h i p p o s i t i o n .
ATTITUDE CONTROL
G i v e n h a t h e r ea r e u p p o r t i n g e e tu n d e r -
n e a t h h e b od y, t h e u p p o r t i n g e g ' sh i p o r q u e s
an d f o r c e s h a t t a b i l i z e h eq u a d r u p e d ' sm o t i o n
h av e t o b e d e f in e d .F o r h e a k eo f onv en i enc e ,
t h e i po r q u e s an d f o r c e s
w
11erouped as
f o l l ows:
s
= s 1 s 2 + s3 + s4
M =
M 1
+ M2
+
M3 + M4
F = F 1 + F2
+
F3 + F4
As =
+
s2
- s3 - s4
AF
=
F 1
+
F2
-
F3
-
F4
Assuming
= 1 s i n ( a )
Y
=
1 s i n ( @ )
c o s ( p )
= 1 6 )
equa t i ons
1
and
2
c a n b e w r i t t e n a s :
1497
7/25/2019 Control of a Quadruped Trot
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Themodel g iven i n Eq. (7 ) imp1 esw e l v e
degrees fr e ed o m t o b e c o n t r o l l e d .This number
canbe educed t o n i n e
i f
t h e qu ad ru pe d i s o r c e d
t o f a c e h e i r e c t io n fm o t i o n . Assume t h a t X,
t h e o r w a r d e a no f h eq u a d r u p e d , s i m i t e d o
f
1 / 2t r ok e . If h e q u a d r u p e de a n so oa r
f o r w a r d , hen ae w p a i r
pf
s u p p o r t i n g e g s
wi
be et down. Fur the rmor e
Y,
( t h e a t e r a lv e l o c i t y
o fh eu a r u p e d )so r c e doe r o b y
0.
I f
t h e r es some l a t e r a le l o c i t y ,h e nh e
quadruped will t u r n o c a nc e l u th i sa t e r a l
v e l o c i t y .i k e w i s e ,
Y
( t h ea t e r a le a n fh e
quadruped)
will
be smal lue t o
Y
be i ngm a l l .
s i n c eh eu a d r u p e ds o v i n g ,h eu p p o r t i n g
f e e t
will
behanged f a i r l yr e q u e n t l y . As a
r e s u l t , h e m a l l r r o r n
Y will
no t av eime
t o n t e g r a t e o a s ub s ta n t i a l a l ueb e f o r e aew
p a i ro f e g sa r ep l a c e d ahead o f h e i r e s p e c t i v e
h i p s .
S i nc e n l y i ne eg r ees fr eedom eedo
b e o n t r o l l e d , q u a t i o n7 ) may be r e w r i t t e n n
s t a t e s pa ceo r m as f o l l ow s :
+
1498
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - -
* F
AS
0 0 0
F
n n
- x / 1
/cos
B
)
0
- ~ C O S ( U ) C O S ( B )
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Y I C O S a / 1 0
0
_ _
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - - - -
0
0 0
0
- 5 C O S ( p ) / i5 Y C O S ( a ) / l
-
8 )
The fo l low inga r a m e t e r sr eo n s i d e r e dor a
quadruped:
A
= 5 f e e t , B = 2 f e e t , I:
=
1 f o o t ,
oxx =
166 .66 s l ug ee t , @yy
=
866 .66 s l u g ee t ,
aZz =
966.66 s l ug ee t ,
m = 1
s l ugs ,
g = 32 . 2 f t / s ? ,
1 =
6 f e e t
I J s i n gh e s e a r a m e t e r s ,h ei n e a r i z e do r m f
equa t i on
( 8 )
becomes:
x
= 5.360
x +
5.3608
-
0.001667M 0.004629AF
8 =
.6192
X t
3.7158
-
0.001154M
+
0.005769~F
..
=
3.220
Y +
19.32
@ +
0.012000
S
Z
= 32.2 - Xoa
+
0.0092F
. .
4 =
3.181 4
+
0.000862
A S 9 )
CONTROLAW
The purposefh eon t r o l 1 aw i so
s t a b i l i z e a q u ad ru pe d t r o t . o rh e i m u l a t i o n
r uns ,he uadrupe d was moved a t a speed o f
20
f e e tp e r s ec on d w i t h a s t r o k e e n g t ho f 10 f e e t .
T h i s m p l i e s h a t a ew p a i r o f e g s
w i
11 support
t h e body ev ery 0.5 seconds. As a r e su l t ,h e
quadruped 's o rward ean,
X
v a r i e s a s a saw too th
s i g n a l i t h an a m p l i t u d e o f 5 f e e t and a f r e -
quency o f 2Hz.
I f
t heq u a dr u p ed i s om a i n t a i n a
s t a b l er o t ,h e nh eo i s en p u t
X
m us to t
c a u s ex c e s s i v eit ch in g. One way o fs s u r i n g
t h i s s o e q u i r e h a t h eg a i n r o m
X
t o
e
i n
e q u a t i o n
9 )
be smal l .ssum ing
10
degree
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o s c i l l a t i o n s o be t h ea r g e s t c c e p t a b l e ,h e n
t h eg a i n r o m X t o e s hou l d ee s shan -25dB:
The i npu t s
M y S,
F,
A S a n dA Fa r es p e c i f i e d
a s o l l o w s :
. .
M =
-924(X
-
X
-
26240
e
10496
e
S
=
-12000
Q
-
2400
Q
F = .-2OOO(Z - Z o )
-
4000
?
AS = -200000 I, 4000
. .
F
= -26230(X
-
X
-
131200 e
-
52480
e
( 1 0 )
theneach ubsys tem i n e q u a t io n 9 ) i s s t a b l e a n d
t h e a i nr o m X t o
e
i s -28dB .hus, t h i s c on-
t r o la w a pp ea rs t o b e a c a n d i d a t eo r
s t a b i 1i i n g a q u a d r u p e d t r o t .
A
second way o f hec k ingw h e t h e rh e o n -
t r o la w will s t a b i l i z e
a
q ua dru pe d t r o t s o
use a wors t ase pproac h ia yapuno v 's method.
The f o l l ow i ng pp r ox i m a t i onswere made i n equa-
t i o n ( 8 ) :
0.76
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