# The relationship between different frames

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14 Νοε 2013 (πριν από 4 χρόνια και 6 μήνες)

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11/14/2013

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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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3

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

:

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

Degrees of freedom of joint: the number of relative motions between two

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8

Fig.2
-
22

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The geometrical parameters of the general link are:

-

The
mutual perpendicular distance
:,
a
i
-
1

-

The
,
i
-
1

Fig. 2
-
the following geometrical parameters:

-

d
i :

-

:
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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10

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

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12

Labeling of links: towards a unified representation

ground

0.

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

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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17

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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18

First

and

last

in

the

chain:

Attach

a

frame

to

the

base

of

the

robot,

or

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
.

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

parameters

in

terms

of

the

frames

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

i = the angle between X
i
-
1

Zi;

We usually choose
a
i

> 0 since it corresponds to a distance;
however, other three are signed quantities.

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21

A

final

note

on

uniqueness

is

warranted
.

The

convention

outlined

above

does

not

result

in

a

unique

attachment

of

Frames

to

.

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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22

Fig.2
-
25

Example 1

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Fig.2
-
26

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Example 2

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Parameter table to be given in the classroom

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Summary

1.

2.
D
-

3.
Assign frames to links based on D
-
H.

4.
Benefit of D
-
H: a minimum number of parameters to