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Computer Integrated Manufacturing

CIM

A T I L I M U N I V E R S I T Y

Manufacturing Engineering Department

Lecture
8

Industrial Robots Analysis

Fall
2005
/
2006

Dr. Saleh AMAITIK

Industrial Robot Kinematics

Robot

kinematics

is

concerned

with

the

position

and

orientation

of

the

robot

s

end
-
of
-
arm,

or

the

end

effector

attached

to

it,

as

a

function

of

time

but

without

regard

for

the

effects

of

force

or

mass
.

Our

treatment

of

manipulator

kinematics

will

be

limited

to

the

mathematical

representation

of

the

position

and

orientation

of

the

robot

s

end
-
of
-
arm
.

The

kinematics

analysis

involves

two

different

kinds

of

problems
:

1.
Determining

the

coordinates

of

the

end
-
effector

or

end
-
or
-
arm

for

a

given

set

of

joints

coordinates

(Forward

Kinematics)
,

and

2.
Determining

the

joints

coordinates

for

a

given

location

of

the

end
-
effector

or

end
-
of
-
arm

(Backward

Kinematics)
.

Industrial Robot Kinematics

Both

the

joint

space

and

world

space

methods

of

defining

position

in

the

robot

s

space

are

important
.

The

joint

space

method

is

important

because

the

manipulator

positions

its

end
-
of
-
arm

by

moving

its

joints

to

certain

values
.

The

world

space

method

is

important

because

applications

of

the

robot

are

defined

in

terms

of

points

in

space

using

Cartesian

coordinate

system
.

What

is

needed

is

a

means

of

mapping

from

one

space

method

to

the

other
.

Mapping

from

joint

space

to

world

space

is

called

Forward

transformation
,

and

Converting

from

world

space

to

joint

space

is

called

Backward

transformation
.

Forward and Backward Transformations

Forward and Backward Transformation for a Robot with Two Joints

1
-

An

OO

Robot

Forward

Transformation

X =
λ
2

and

Z =
λ
1

Backward

Transformation

λ
1

= Z

and

λ
2

=X

Where

X

and

Z

are

the

coordinate

values

in

the

world

space

λ
1

and

λ
2

are

the

values

in

joint

space

Forward and Backward Transformation for a Robot with Two Joints

2

-

An

RR

Robot

Forward

Transformation

The

forward

transformation

is

calculated

by

noting

that

the

lengths

and

directions

of

the

two

might

be

viewed

as

vectors

in

space
:

)
sin(
),
cos(
sin
,
cos
2
1
2
2
1
2
2
1
1
1
1
1

L
L
r
L
L
r
Forward and Backward Transformation for a Robot with Two Joints

Vector

or

r
1

and

r
2

(and

taking

account

of

L
0
)

yields

the

coordinate

values

of

X

and

Z

at

the

end
-
of
-
arm
:

)
sin(
sin
)
cos(
cos
2
1
2
1
1
0
2
1
2
1
1

L
L
L
Z
L
L
X
Forward and Backward Transformation for a Robot with Two Joints

Backward

Transformation

For

the

backward

transformation,

we

are

given

the

coordinate

positions

X

and

Z

in

world

space,

and

we

must

calculate

the

joint

values

that

will

provide

those

coordinate

values
.

}
sin
)
(
)
cos
(
{
sin
)
cos
)(
(
sin
2
)
(
cos
2
2
0
2
2
1
2
2
2
2
1
0
1
2
1
2
2
2
1
2
0
2
2

L
L
z
L
L
x
xL
L
L
L
z
L
L
L
L
L
z
x

Forward and Backward Transformation for a Robot with Three Joints

Let

us

consider

a

manipulator

with

three

degrees
-
of
-
freedom,

all

rotational,

in

which

the

third

joint

represents

a

simple

wrist
.

The

robot

is

a

RR
:
R

configuration

is

shown

below
:

The robot is limited to the x
-
z plane and the origin of the axis system
at the center of joint 1

Forward and Backward Transformation for a Robot with Three Joints

For

the

forward

transformation,

the

X

and

Z

coordinates

can

be

calculated

as

follows
:

The

arm
-
and
-
body

(RR
:
)

provides

position

of

the

end
-
of
-
arm

The

wrist

(
:
R)

provides

orientation
.

Let

α

the

orientation

angle
.

It

is

the

angle

by

the

wrist

with

the

horizontal
.

It

equals

the

algebraic

sum

of

the

three

joint

angles
:

Forward and Backward Transformation for a Robot with Three Joints

In

the

backward

transformation,

we

are

given

the

world

coordinates

X,

Z,

and

α
,

and

we

want

to

calculate

the

joint

values

θ
1
,

θ
2

and

θ
3

that

will

achieve

those

coordinates
.

This

is

accomplished

by

first

determining

the

coordinates

of

joint

3

as

follows
:

Knowing

the

coordinates

of

joint

3
,

the

problem

of

determining

θ
1

and

θ
2

is

as

follows
:

The

value

of

joint

3

is

then

determined

as

Forward and Backward Transformation for a Robot with Four Joints
in Three Dimensions

Consider

the

four

degree
-
of
-
freedom

robot

shown

below
.

Its

configuration

is

TRL
:
R
.

Joint

1

(T

type)

provides

rotation

Z
-
axis
.

Joint

2

(R

type)

provides

rotation

a

horizontal

axis

whose

direction

is

determined

by

joint

1
.

Joint

3

(L

Type)

is

a

piston

that

allows

linear

motion

in

a

direction

determined

by

joints

1

and

2
.

Joint

4

(R

type)

provides

rotation

an

axis

that

is

parallel

to

the

axis

of

joint

2

Forward and Backward Transformation for a Robot with Four Joints
in Three Dimensions

The

values

of

the

four

joints

are,

respectively,
.
θ
1
,

θ
2
,

λ
3
,

and

θ
4
.

Given

these

values

the

forward

transformation

is

given

by
:

where

Forward and Backward Transformation for a Robot with Four Joints
in Three Dimensions

In

the

backward

transformation,

we

are

given

the

world

coordinates

X,

Y,

Z,

and

α
.

Where

α

specifies

orientation
.

To

find

the

joint

values,

we

define

the

coordinates

of

joint

4

as

follows
:

Homogenous Transformation

The

goal

in

robot

motion

is

often

to

describe

the

effect

of

combined

motions

resulting

from

both

translation

and

rotation
.

Each

of

the

previous

manipulators

required

its

own

individual

analysis,

resulting

in

its

own

set

of

trigonometric

equations,

to

accomplish

the

forward

and

backward

transformations
.

There

is

a

general

approach

for

solving

the

manipulator

kinematics

equations

based

on

homogeneous

transformations
.

The

homogenous

transformation

approach

utilizes

vector

and

matrix

algebra

to

define

the

joint

and

positions

and

orientations

with

respect

to

a

fixed

coordinate

system

(world

space)
.

The

end
-
of
-
arm

is

defined

by

the

following

4

x

4

matrix
:

1
0
0
0
z
z
z
z
y
y
y
y
x
x
x
x
p
a
o
n
p
a
o
n
p
a
o
n
T
Homogenous Transformation

Where

T

consists

of

four

column

vectors

representing

the

position

and

orientation

of

the

end
-
of
-
arm

or

end
-
effector

The

vector

P

defines

the

position

coordinates

of

the

end

effector

relative

to

the

world

x
-
y
-
z

coordinate

system
.

The

vectors

a,

o,

and

n

define

the

orientation

of

the

end

effector
.

The

a

vector,

called

approach

vector,

points

in

the

direction

of

the

end

effector
.

The

o

vector,

or

orientation

vector,

specifies

the

side
-
to
-
side

direction

of

the

end

effector
.

For

a

gripper,

this

is

in

the

direction

from

one

fingertip

to

the

opposite

fingertip

The

n

vector

is

the

normal

vector,

which

is

perpendicular

to

a

and

o
.

Homogenous Transformation

In

manipulator

kinematics,

calculations

based

on

homogeneous

transformations

are

used

to

establish

the

geometric

relationships

among

of

the

manipulator
.

Let

A
1

=

a

4

x

4

matrix

that

defines

the

position

and

orientation

of

1

with

respect

to

the

world

coordinate

axis
.

Similarly,

A
2

=

a

4

x

4

matrix

that

defines

the

position

and

orientation

of

the

2

with

respect

to

1
.

Then

the

position

and

orientation

of

2

with

respect

to

the

world

coordinate

system

(called

T
2
)

is

given

by
:

T
2

= A
1

A
2

Where

T
2

represents

the

position

and

orientation

of

the

end
-
or
-
arm

(end

of

2
)

of

a

manipulator

with

two

joints
.

A
1

and

A
2

define

the

changes

in

position

and

orientation

resulting

from

the

actuations

of

joints

1

and

2

on

1

and

2

respectively
.

Homogenous Transformation

In

general,

the

position

and

orientation

of

the

end
-
of
-
arm

or

end
-
effector

can

be

determined

as

the

product

series

of

homogeneous

transformations,

usually

one

transformation

for

each

joint
-

combination

of

the

manipulator
.

From

this

matrix,

a

five

degree

of

freedom

manipulator

can

yield

a

transformation

matrix

T

=

0
A
5

that

specifies

the

position

and

orientation

of

the

end

point

of

the

manipulator

relative

to

the

world

coordinate

system
.