New
Notation
for
Serial
Kinematic
Chains
Berthold
K.
P.
Horn
May
1987
Abstract:
The
description
of
a
serial
kinematic
chain
should
be
unique,
unambiguous, simple
to
determine, easy
to
use
and
wellbehaved
when
small
changes
are
made
in
the
arrangement
of
the
elements
of
the
chain.
The
notation
currently
in
use, introduced
by
Denavit
and
Hartenberg, does
not
satisfy
all
of
these
criteria.
It
involves
arbitrary
choices, so
that
more
than
one
description
may
apply
to
a
given
kinematic
chain.
More
impor
tantly, the
parameters
relating
the
links
in
the
chain
can
be
very
sensi
tive
to
small
changes
in
the
physical
arrangement
of
the
chain.
This
is
particularly
true
of
socalled
ideal
chains, ones
that
permit
closedform
solution
of
the
inverse
kinematic
problem, since
these
often
involve
ge
ometries
where
adjacent
axes
are
parallel, perpendicular
or
intersect.
A
new
notation
is
proposed
here
that
does
not
suffer
the
abovementioned
shortcomings.
To
demonstrate
some
of
the
advantages
of
the
new
nota
tion, it
is
applied
to
the
problem
of
fingerprinting
a
robot
arm
and
to
the
solution
of
the
inverse
kinematic
problem
of
nearideal
arms.
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1
Introduction
Let
us
start
right
away
by
defining
the
new
notation:
Let
the
home
position
of
a
serial
kinematic
chain
be
an
arbitrary
posi
tion
specified
in
terms
of
the
joint
variables.
The
home
position
may, for
example, be
chosen
to
be
the
one
where
all
joint
variables
are
zero.
The
base
coordinate
system
is
an
arbitrary
external
coordinate
system
fixed
with
respect
to
the
base
of
the
kinematic
chain.
Erect
a
coordinate
system
in
each
link
of
the
chain
in
such
a
way
that
the
coordinate
axes
are
paral
lel
to
those
of
the
base
coordinate
system
when
the
chain
is
in
the
home
position.
The
kinematic
chain
is
then
fully
specified
when
the
following
are
given
for
the
chain
in
the
home
position:
•
The
set
of
unit
vectors
{
o
∗
}
, parallel
to
the
directions
of
motion
of
ˆ
i
the
prismatic
joints.
•
The
set
of
unit
vectors
{
ω
∗
}
, parallel
to
the
axes
of
the
revolute
joints.
ˆ
i
•
The
set
of
offset
vectors
{
d
∗
}
determined
recursively
with
respect
to
i
the
chosen
base
origin
as
follows:
The
first
reference
point
is
the
origin
of
the
given
base
coordinate
system.
Drop
a
perpendicular
from
this
reference
point
to
the
first
revolute
joint
axis.
This
defines
the
first
offset
vector
as
well
as
a
reference
point
on
the
first
revolute
joint
axis.
Now
drop
a
perpendicular
from
this
new
reference
point
onto
the
second
revolute
joint
axis.
This
defines
the
second
offset
vector
and
a
new
reference
point.
Continue
in
this
fashion
to
the
last
revolute
joint.
Connect
the
last
reference
point
so
found
to
a
chosen
tip
reference
point
in
the
final
link.
This
defines
the
last
offset
vector.
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 Spring '04
 BertholdHorn
 Kinematic Chain, kinematic chains, ﬁnal link

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