11/14/2013
Hand out 4
1
Robotics kinematics: Definition,
Motor and End

effector
Each component has a coordinate system or
frame
:
kinematics becomes the relation between frames. Further, if
one frame is set up on the ground called
world frame
, the
“
absolute
” position and orientation of the end

effector is
known.
The relationship between different frames
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2
End

effector
World frame
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How to set up or assign
a
local frame to
each
component
of the robot?
What is called
a component
?
What is called
a joint
?
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4
World frame
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5
Robot Kinematics:
Logics of presentation:
Kinematics: what
Coordinate system: way to describe motion
Relation between two coordinate systems
Definition of component and joint: robot structure
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6
Link
:
Component
with
only
considering
its
joint
line
but
neglecting
its
detailed
shape
.
Next slide (Fig. 2

21) shows
various types of joints
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7
Fig. 2

21
Joint types
Kinematic
pair types
Neglecting the
details of the joint
but relative motions
or relative
constraints between
two connected links
Degrees of freedom of joint: the number of relative motions between two
links that are in connection
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Fig.2

22
General configuration of link
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The geometrical parameters of the general link are:

The
mutual perpendicular distance
:,
a
i

1

The
link twist
,
i

1
Fig. 2

23 shows two links that are connected, which leads to
the following geometrical parameters:

d
i :
link offset

:
joint angle
i
From axis i

1 to axis i
From axis i

1 to
axis i
From axis a(i

1) to axis a(i) along axis i
From axis a(i

1) to axis a(i)
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Fig. 2

23
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11
Denavit

Hartenberg (D

H) notation for
describing robot kinematic geometry. It has the
benefit that only four parameters can describe
completely robot kinematic geometry. The
shortcoming is that they are always across two
links.
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12
Labeling of links: towards a unified representation
The base link or
ground
0.
The last link n.
For other links (
i
=1, 2, …., n

1), their parameters will follow
the D

H rule. However, for the link 0 and link n, there are
some arbitrary situation. For instance, for the example in
the next slide, there is no rule to constrain the definition of
z0 and x3. This will be further discussed later when the
local frame is assigned to each link.
It may also be clear that the geometrical parameters
based on the D

H are dependent on the way of assigning
or defining local frames to links.
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13
All Z axes are all
perpendicular to
the paper plane
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14
Robot Kinematics:
Logics of presentation:
Kinematics: what
Coordinate system: way to describe motion
Relation between two coordinate systems
Definition of component and joint: robot structure
Assign a local frame to each link (D

H notation)
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15
a) If joint 1 is revolute, define the base frame and the first
frame such that d 1 =0.0, and
θ
1=0.0 at the initial time
b) If joint 1 is prismatic, define the base frame and the
first frame such that d
1
=0.0 at the initial time, and
θ
1=0.0.

Regarding d
1
(d
n
) and
θ
1
(
θ
n
), the rule is to make them 0.0
c) If joint n is revolute, define the last frame and the n

1
frame such that d n =0.0, and
θ
n=0.0 at the initial time
d) If joint n is prismatic, define the last frame and the n

1
frame such that d
n
=0.0 at the initial time, and
θ
n=0.0.
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16
Rule to assign a frame to each link (intermediate links)
The
Z

axis
of frame
(
i
),
Zi
, is coincident with the joint
axis
i
.
The origin of frame
(
i
)
is located where the
a
i
perpendicularly intersects with the joint
i
axis.
Xi
points
along
a
i
in the direction from joint
i
to joint
i+
1.
In the case of
a
i
= O,
Xi
is normal to the plane of
Zi
and
Zi+1.
We define
a
i
as being measured in the right

hand
sense about Xi, so we see that the freedom of choosing
the sign of
α
; in this case two choices are available.
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Yi
is
formed
by
the
right

hand
rule
to
complete
the
i

th
frame
.
Fig
.
2

24
shows
the
location
of
frames
{i

1
}
and
{i}
for
a
general
manipulator
.
Fig. 2

24
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Hand out 4
18
First
and
last
links
in
the
chain:
Attach
a
frame
to
the
base
of
the
robot,
or
link
0
,
called
frame
(
0
)
.
This
frame
can
be
treated
as
a
reference
frame
for
measuring
the
position
and
orientation
of
all
other
frames
.
Since
frame
(
0
)
is
arbitrary,
it
always
simplifies
matters
to
choose
Z
0
along
axis
1
and
to
locate
frame
(
0
)
so
that
it
coincides
with
frame
(
1
)
when
joint
variable
1
is
zero
.
Using
this
convention
we
have
:
a
o
=
0
.
0
,
α
o
=
0
.
0
.
Additionally,
this
ensures
that
d
1
=
0
.
0
if
joint
1
is
revolute,
or
θ
1
=
0
.
0
if
joint
1
is
prismatic
.
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19
For joint
n to be a
revolute one, the direction of
X
N
is chosen
so that it aligns with
X
N

1
when
θ
N
= 0.0, and the origin of
frame
(N)
is chosen so that
d
n
= 0.0. For joint
n to be a
prismatic one, the direction of X
N
is chosen so that
θ
N
= 0.0,
and the origin of frame
(N)
is chosen at the intersection of
X
N

1
and joint axis
n
when
d
n
= 0.0.
Summary
of
the
link
parameters
in
terms
of
the
link
frames
If the link frames have been attached to the links according
to our convention, the following definitions of the link
parameters are valid:
ai
= the distance from Zi to Zi+1, measured along Xi;
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20
di
= the distance from Xi_, to Xi

1 measured along Zi; and
i,
= the angle between
Zi,
and
Z
i+1
measured about Xi.
i = the angle between X
i

1
and Xi, measured about
Zi;
We usually choose
a
i
> 0 since it corresponds to a distance;
however, other three are signed quantities.
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Hand out 4
21
A
final
note
on
uniqueness
is
warranted
.
The
convention
outlined
above
does
not
result
in
a
unique
attachment
of
Frames
to
links
.
First
of
all,
when
we
first
align
the
Zi
axis
with
joint
axis
i,
there
are
two
choices
of
direction
in
which
to
point
Zi
.
Furthermore,
in
the
case
of
intersecting
joint
axes
(i
.
e
.
,
a
i
=
O),
there
are
two
choices
for
the
direction
of
Xi,
corresponding
to
the
choice
of
signs
for
the
normal
to
the
plane
containing
Zi
and
Zi+
1
,
.
When
axes
i
and
i+
1
are
parallel,
the
choice
of
origin
location
for
(i)
is
arbitrary
(though
generally
chosen
in
order
to
cause
di
to
be
zero)
.
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Fig.2

25
Example 1
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Fig.2

26
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Example 2
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26
Parameter table to be given in the classroom
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Summary
1.
Link and joint concept.
2.
D

H notation for link.
3.
Assign frames to links based on D

H.
4.
Benefit of D

H: a minimum number of parameters to
describe links and joints.
5.
Shortcoming of D

H: parameters must cross two
consecutively connected links.
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