MFGE
404
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
links
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
addition
or
r
1
and
r
2
(and
taking
account
of
link
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
made
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
about
Z

axis
.
•
Joint
2
(R
type)
provides
rotation
about
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
about
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
link
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
links
of
the
manipulator
.
•
Let
A
1
=
a
4
x
4
matrix
that
defines
the
position
and
orientation
of
link
1
with
respect
to
the
world
coordinate
axis
.
•
Similarly,
A
2
=
a
4
x
4
matrix
that
defines
the
position
and
orientation
of
the
link
2
with
respect
to
link
1
.
•
Then
the
position
and
orientation
of
link
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
link
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
links
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

link
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
.
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