leads to ambiguities in the case of tree and closed- loop structure robots. The given method has all the advantages of D-H notation in the case of open-loop.
A NEW GEOMETRIC NOTATION FOR OPEN AND CLOSED-LOOP ROBOTS
W. KHALIL - J.F.
KLEINFINGER
Laboratoired'AutomatiquedeNantes UA C.N.R.S. 04/823 1 r u e d e l a N o & 44072 NANTES CEDEX - FRANCE E.N.S.M. ABSTRACT T h i s p a p e r p r e s e n t s a new g e o m e t r i c n o t a t i o n f o r t h e d e s c r i p t i o no ft h ek i n e m a t i co fo p e n - l o o p .t r e ea n d closed-loopstructurerobots. The mechod i s d e r i v e d (D-H) from t h e well-known DenavitandHartenberg notation,which i s powerfulfor serial r o b o t s b u t t r e e andclosedleadstoambiguitiesinthecaseof loop s t r u c t u r e r o b o t s . The g i v e n method h a s a l l the advantagesof D-H n o t a t i o n i n t h e c a s e o f o p e n - l o o p robots. 1
INTRODUCTION
Many methods are a v a i l a b l e f o r t h e d e s c r i p t i o n o f w i t h open-chain mechanism [ I ] . thegeometryofrobots The most common use i s t h e e l e g a n t D-H method [ 2 ] . The D-H method i s d e a l i n g w i t h l i n k s w i t h o n l y two j o i n t s . The d e f i n i t i o n of a j o i n t w i t h r e s p e c t t o the preceeding one i s c a r r i e d o u t b y means of 4 parameters. The u s e o f D-H n o t a t i o n i n r o b o t i c s h a s (geomef a c i l i t e d g r e a t l y a l l themodelingproblems t r i c kinematics,anddynamics) [ 3 ] . The D-H n o t a t i o n , as it is, however, i s s t i l l powerfulanduseful hamperedby c e r t a i n d i f f i c u l t i e s . I n f a c t , the a p p l i D-H n o t a t i o n t o r o b o t s w i t h l i n k s cation of the having more t h a n two j o i n t s i s d i f f i c u l t a n d l e a d s to ambiguities [ 4 ] . ShethandUicker (S-U) [ 4 ] h a sd e v e l o p e da n o t h e r n o t a t i o n which d e s c r i b e s e a c h l i n k by 7 parameters. The S-U methodcanbeused t o d e s c r i b e a n y mechanism, b u t owing t o i t s complexity it h a s b e e n a p p l i e d o n i y [5]. f o rt h ec l o s e d - l o o pr o b o t s In this paper we propose a new g e o m e t r i c n o t a t i o n which can be used for both the closed and the openD-H notalooprobots. It has a l l theadvantagesof t i o n when u s e d f o r o p e n - c h a i n r o b o t s , a n d c a n e a s i l y robots. I n t h e case of b eu s e df o rt h ec l o s e d - l o o p l i n k s w i t h 2 j o i n t s , 4 parameters are needed t o d e s c r i b e a j o i n t w i t h r e s p e c t t o the p r e c e e d i n g o n e , while 2 a d d i t i o n a l p a r a m e t e r s may be needed i n t h e case o f l i n k s w i t h more t h a n t w o j o i n t s . Inthefollowing two s e c t i o n s we w i l l p r e s e n t the D-H and the S-H n o t a t i o n s . The p r o p o s e d n o t a t i o n w i l l be presented in section 4. Two examples w i l l be 5 to illustrate the given notation. giveninsection
2. DENAVIT AND HARTENBERG NOTATION [ 1 ] T h i s method i s t h e m o s t p o p u l a r i n t h e r o b o t i c s world. I t can be u s e d o n l y i n t h e c a s e o f s e r i a l 0 is r o b o t s . A r o b o t i s composed o f n + l l i n k s , l i n k thefixedbase,andlink n is theterminallink, ( i ) .A c o o r d i n a t e j o i n t ( i )c o n n e c t sl i n k s( i - 1 )a n d
frame R . i s a s s i g n e d f i x e d w i t h r e s p e c t t o l i n k
(i).
( i )i s supposedalong
The a x i so fj o i n t .
Z. while -1- 1 common p e r p e n d i c u l a r
t h e X. a x i s i s d e f i n e d a s t h e -1
t o L i - l and Z . ( F i g . 1-a) -1
.
The 4x4 t r a n s f o r m a t i o n m a t r i x which definesframe (i) w i t hr e s p e c tt o frame(i-1) i s o b t a i n e d as f u n c t i o n o f 4 parameters(f3.pr.,di,ai)(Fig. l a ) . T h i sm a t r i x 1
denotedby '-'T.
1
is e q u a l t o
i-lTi
:
= R o t ( 2 , e . )T r a n s ( 2, r . )T r a n s ( X , d i )R o t ( X , a i )
cos 8 . 1-sin 8 . c o s a.1 s i n 8 = sin
€I
B
'
il I
cos 8
i
cos a . '-cos
i
l
; di
cos
I
d .s i n
I
r.
'
1
oi 8 . (1)
11
_ _s i _n _a .
cos a .
I
1--..+--1--
1---'----
D
s i n ai
8. s i n a.
I
n
0
0
Ifjoint
( i )is r o t a t i o n a l , t h e j o i n t v a r i a b l e
e q u a lt o
€Ii, while q1. =
r. if joint 1
Hence q.= (1-0.)€I.+ u , r . where 0 . = 8 i f j o i n t 1
is r o t a t i o n a l a n d
1
1 1
u,=
1 ifjoint
q . is
(i)i s prismatic. (i)
( i )i s p r i s m a t i c .
The geometricmodelof a serial robotcanthusbe o b t a i n e d by t h e s u c c e s s i v e m u l t i p l i c a t i o n s of t h e t r a n s f o r m a t i o n matrices : n- 1 i (2) 'T = OT. T2 Tn I t i s t o b en o t e dt h a tt h ef r a m e( n )c a nb ea l w a y s D-H c o n s t a n t p a r a m e t e r s of definedsuchthatthe frame ( n ) are e q u a lt oz e r o . D-H n o t a t i o n : Two remarks are t o begivenaboutthe
...
i ) The d e f i n i t i o n o f t h e a x i s o f j o i n t
( i )a s lZ -.
i s s o m e t i m e sc o n f u s i n g ,f o rt h i sr e a s o n some people [6-71 f i n d more c o n v e n i e n t t o d e f i n e t h e a x i s o f l i n k ( i )as Z , b u t as a r e s u l t o f D-H n o t a t i o n t h e ( i )w i l l be Ri+l c o o r d i n a t ef r a m ef i x e dw i t hl i n k (Fig.1-b)which,inouropinion, is more confusing thanthefirstcase. i i ) I t i s impossible t o use D-H n o t a t i o n as it is i n t h ec a s eo fc l o s e d - l o o ps t r u c t u r e ,a n dn o te v e ni n tree s t r u c t u r e . F o r example c o n s i d e r t h e t h ec a s eo f s i t u a t i o n shown i n F i g . 2 w h i c h s h o w s 3 r o t a t i o n a l j o i n t s on a tree s t r u c t u r e . Owing t o D-H n o t a t i o n :
.R .
0
is d e f i n e ds u c ht h a t
5 is
t h ea x i s
of j o i n t (1).
T r a v e r s i n gf r o mj o i n t 1 to joint 2 w i l l lead to d e f i n e a c o o r d i n a t e f r a m e R1 f i x e d w i t h r e s p e c t t o l i n k (I), where 2 -1
is the a x i so fj o i n t
2. The v a r i a -
I
x-x+.
1
2 Fig. 1-b
X
-i
F i g u r e 1. DenavitandHartenbergNotation
F i g u r e 2. Ambiguitiesof
D-H
notation
1 i s 81 and it is t h e a n g l e between bleofjoint can be d e f i n e d as u s u a l . a n d x-l . a l , d l , r l
.
Now t r a v e r s i n g from j o i n t 1 t o j o i n t 3 a n o t h e r is t h e j o i n t 3 a x i s frame i s t o b e d e f i n e d w i t h and i s f i x e d a l s o w i t h r e s p e c t t o l i n k (1). T h i s frame i s d e f i n e d by some ( 8 , r , a , d )p a r a m e t e r sb u t parawhat s u b s c r i p t s d o w e have t o a s s i g n f o r t h e s e meters ?
z
of double subscripts s u c h t h a t when t r a v e r s i n g f r o m j o i n t 1 to (81ar12, j o i n t 2 t h e p a r a m e t e r s w i l l be denoted by a.lz,d12)and by ( 8 1 3 , ~ 1 3 , ~ 1 1 3 , d 1 3when ) traversing ql w i l l from j o i n t 1 t o j o i n t 3 , t h e j o i n t v a r i a b l e be 0 1 2 or 8 1 3 , a n a d d i t i o n a l c o n s t a n t p a r a m e t e r which s p e c i f i e s the r e l a t i o n b e t w e e n 012 and 8 1 3 i s t o b ed e f i n e d , i.e. A s o l u t i o n may proposed by the use
e12 = e 1 3 +
Y13
Figure 3. Sheth and Uicker parameters
20
(3)
Anotherconfusion is s t i l l t a k i n g p l a c e b e c a u s e we havealways two f r a m e s f i x e d w i t h r e s p e c t t o l i n k 1. How t o c a l l them R 1 2 and R13 ? A t any case t h e f r a m e ( i )i s nomore f i x e d w i t h r e s p e c t t o l i n k i. W e s e e from t h a t s i m p l e example that D-H n o t a t i o n w i l l looseoneof i t s best advantage which is i t s s i m p l i intheroc i t y . And the mathematicalformulasused b o tm o d e l i n g( g e o m e t r i c - k i n e m a t i c sa n de s p e c i a l l y dynamics) w i l l n o t b e handy.
3. SHETH AND UICKER NOTATION [ 4 ] Owing t o the i n e f f i c i e n c y of t h e D-H n o t a - t i o n i n r e p r e s e n t i n gt h ec l o s e dl o o ps t r u c t u r e ,S h e t ha n d Uicker have developed another notation system, i s composed o f two whereeachtransformationmatrix parts : i)
a c o n s t a n tp a r ts p e c i f y i n gt h es h a p eo f
ii) a d i s t i n c t v a r i a b l e p a r t r e p r e s e n t i n g motion.
Consider(Fig.3)which e a c h3 o i n tc o n t a i n s The f i r s t denoted
the l i n k . the j o i n t
shows two s u c c e s s i v el i n k s , two c o o r d i n a t es y s r e m s . gj i s an a r b i t r a r i l y c h o s e n
sj xj
systemwith
2 , is t h e j o i n t
-1
axis, f i x e d w i t h r e s p e c t
t o l i n k ( j ) and may b e t h o u g h t o f as L o c a t i n g t h e R.-. The o t h e r c o o r position of the joint element
1
D . V . W , is a l s o d e f i n e d f i x e d i n t h e -7 -7 7 R . + . I t i s chosensuch that W. matingjointelement I -7 l i e s along the j o i n t a x i s 2 . b u t U . and V . are -1' -7 -1 arbitrarily oriented.
d i n a t es y s t e m
( j ) i s designatedby9..
The m o t i o n o f t h e j o i n t
3.1.
Hence,
-
IT.+ = V . ( q . ) = 7 3 3
ShapeMatrix
o
.
lk = d i s t a n c ef r o m
8.
= anglefrom
lk
t.
to X
3k
%
4.1.
maThe p r o p o s e d m e t h o d d e f i n e s t h e t r a n s f o r m a t i o n t r i x i n the c a s e o f t w o - j o i n t l i n k by t h e u s e Of 4 parameters as i n t h e D-H n o t a t i o n . I n t h e c a s e Of l i n k s w i t h more t h a n t w o j o i n t s two a d d i t i o n a l p a r a meters may beneeded.
= F. 3k
. The a x i s o f j o i n t . The
coordinateframe
w i t hr e s p e c t
V . (q , ) where q
7
7
is d e f i n e d s u c h t h a t
gi (oi,Xi,yi,zi)
:
( i )w i l l be Ri
is f i x e d
t o l i n k (i)
. The parameterswhichlead
Joint Matrix
The j o i n t m a t r i x will be denoted by
j Tk between t h e c o o r d i n a t e
Introduction
The p r o p o s e d n o t a t i o n
3.2.
j Tk
The aim of t h e new n o t a t i o n is t o d e f i n e a method which can b e u s e d e a s i l y a n d w i t h o u t a m b i g u i t y i n theclosed-looprobots. W e t h i n k t h a t t h i s aim h a s w e w i l l show t h a t t h e g i v e n n o t a been fulfiled and the o p e n - l o o p r o b o t s a s tion can be used also for easyandgeneral as t h a t of D-H n o t a t i o n .
