a-new-geometric-notation-for-open-and-closedloop-robots.pdf
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A
NEW
GEOMETRIC
NOTATION
FOR
OPEN
AND
CLOSED-LOOP ROBOTS
W.
KHALIL
-
J.F. KLEINFINGER
Laboratoire d'Automatique de Nantes
UA
C.N.R.S.
04/823
E.N.S.M.
1
rue de
la
No&
44072 NANTES
CEDEX
-
FRANCE
ABSTRACT
This paper presents
a
new geometric notation for the
description of the kinematic of open-loop. tree and
closed-loop structure robots. The mechod
is
derived
from the well-known Denavit and Hartenberg
(D-H)
notation, which
is
powerful for
serial
robots but
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
robots.
1 INTRODUCTION
Many methods
are
available for the description of
the geometry of robots
with
open-chain mechanism
[I].
The most common use
is
the elegant
D-H
method
[2].
The
D-H
method
is
dealing with links
with
only two
joints. The definition of
a
joint with respect to
the
preceeding one
is
carried out by means
of
4 pa-
rameters.
The use of
D-H
notation in robotics has
facilited greatly
all
the modeling problems (geome-
tric
kinematics, and dynamics)
[3].
The
D-H
notation,
powerful and useful
as
it
is,
however,
is
still
hampered by certain difficulties. In fact,
the
appli-
cation of the
D-H
notation
to
robots
with
links
having
more
than two joints
is
difficult and leads
to
ambiguities
[4].
Sheth and Uicker
(S-U)
[4]
has developed another
notation which describes each link by 7 parameters.
The S-U method can be used
to
describe any mechanism,
but owing to
its
complexity
it
has been applied oniy
for the closed-loop robots
[5].
In this paper we propose
a
new geometric notation
which can be used for both the closed and the open-
loop robots.
It
has
all
the advantages of
D-H
nota-
tion when used for open-chain robots, and can easily
be used for the closed-loop
robots.
In the
case
of
links with
2
joints, 4 parameters
are
needed
to
describe
a
joint with respect
to
the
preceeding one,
while
2
additional parameters may be needed in the
case
of links with more than
two
joints.
In the following two sections
we
will
present
the
D-H
and
the
S-H notations. The proposed notation
will
be presented in section
4.
Two
examples
will
be
given in section
5
to illustrate the given notation.
2.
DENAVIT
AND
HARTENBERG
NOTATION
[
1
]
This method
is
the most popular in the robotics
world.
It
can be used only in the case of serial
robots.
A
robot
is
composed of n+l links, link
0
is
the fixed base, and link n
is
the terminal link,
joint
(i)
connects links (i-1) and
(i).
A
coordinate
frame
R.
is
assigned fixed with respect to link
(i).
The axis of joint.
(i)
is
supposed along
Z
.
while
the X. axis
is
defined as the common perpendicular
-1-
1
-1
to
Li-l
and
Z
.
(Fig. 1-a)
.
-1
The 4x4 transformation matrix which defines frame
(i)
with respect to frame (i-1)
is
obtained
as
function
of
4
parameters (f3.pr.,di,ai) (Fig.
la).
This matrix
denoted by
i-lTi
is
equal to
:
'-'T.
=
Rot(2,e.) Trans(2 ,r.) Trans(X,di) Rot(X,ai)
cos
8.
1-sin
8.
cos
a.1
sin
8
sin
ai
;
di cos
oi
11
i
=
(1)
sin
€I
'
cos
8
cos
a.
'-cos
8.
sin
a.
I
d. sin
8.
il
i
11
B
I
sin
a.
I
cos
a.
I
r.
Dl
0
n0
'1
____
1---'----
1--..+--1--
I
If joint
(i)
is
rotational, the joint variable
q.
is
equal to
€Ii,
while
q.=
r. if joint
(i)
is
prismatic.
Hence
q.=
(1-0.)
€I.+
u,r. where
0.=8
if joint
(i)
is
rotational and
u,=
1 if joint
(i)
is
prismatic.
The geometric model of
a
serial
robot can thus be
obtained by the successive multiplications of the
transformation
matrices
:
11
1
1
11
i
n-
1
'T
=
OT.
T2
...
Tn
(2)
It
is
to be noted that the frame (n) can be always
defined such that the
D-H
constant parameters of
frame (n)
are
equal to zero.
Two remarks
are
to be given about the
D-H
notation
:
i)
The definition of the axis of joint
(i)
as
Z.-l
is
sometimes confusing, for this reason
some
people
[6-71
find more convenient to define the axis of
link
(i)
as
Z,
but
as
a
result of
D-H
notation the
coordinate frame fixed with link
(i)
will
be
Ri+l
(Fig. 1-b) which, in our opinion,
is
more
confusing
than the first case.
ii)
It
is
impossible to use
D-H
notation
as
it
is
in
the case of closed-loop structure, and not even in
the case of
tree
structure. For example consider the
situation shown in Fig.
2
which shows3 rotational
joints on
a
tree
structure. Owing
to
D-H
notation
:
.
R
is
defined such that
5
is
the axis of joint
(1).
.
Traversing from joint 1 to joint
2
will
lead to
define
a
coordinate frame
R1
fixed with respect to
link
(I),
where
2
is
the
axis of joint
2.
The varia-
0
-1
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