# Kinematics of Robot

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14 Νοε 2013 (πριν από 4 χρόνια και 8 μήνες)

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Kinematics of
Robot
Manipulator

1

Examples of Kinematics
Calculations

Forward

kinematics

Given joint variables

End
-
effector
position and orientation,

-
Formula
?

)
,
,
,
,
,
,
(
6
5
4
3
2
1
n
q
q
q
q
q
q
q
q

)
,
,
,
,
,
(
T
A
O
z
y
x
Y

x
y
z
Examples of Inverse Kinematics

Inverse kinematics

End effector position

and orientation

Joint variables
-
Formula
?

3

)
,
,
,
,
,
,
(
6
5
4
3
2
1
n
q
q
q
q
q
q
q
q

)
,
,
,
,
,
(
T
A
O
z
y
x
x
y
z
Example
1:
one rotational

4

0
x
0
y
1
x
1
y
)
/
(
cos
kinematics

Inverse
sin
cos
kinematics

Forward
0
1
0
0
l
x
l
y
l
x

l
Robot Reference Frames

World
frame

Joint frame

Tool frame

5

x
y
z
x
z
y
W

R

P

T

Coordinate Transformation

Reference
coordinate
frame OXYZ

Body
-
attached

frame
O’uvw

w
v
u
k
j
i
w
v
u
uvw
p
p
p
P

6

z
y
x
k
j
i
z
y
x
xyz
p
p
p
P

x
y
z
P
u
v
w
O, O’

Point represented in OXYZ:

z
w
y
v
x
u
p
p
p
p
p
p

T
z
y
x
xyz
p
p
p
P
]
,
,
[

Point represented in
O’uvw
:

Two frames coincide ==>

Properties:
Dot Product

Mutually perpendicular

Unit vectors

7

Properties of orthonormal coordinate frame

0
0
0

j
k
k
i
j
i

1
|
|
1
|
|
1
|
|

k
j
i

Let
and be arbitrary vectors in and be
the angle from to , then

3
R

cos
y
x
y
x

x
y
x
y
Coordinate Transformation

Coordinate Transformation

Rotation only

w
v
u
k
j
i
w
v
u
uvw
p
p
p
P

8

x
y
z
P
z
y
x
k
j
i
z
y
x
xyz
p
p
p
P

uvw
xyz
RP
P

u
v
w
How to relate the coordinate in these two frames?

Basic Rotation

Basic Rotation

, , and represent the projections of onto
OX, OY, OZ axes, respectively

Since

9

x
p
P
y
p
z
p
w
v
u
x
p
p
p
P
p
w
x
v
x
u
x
x
k
i
j
i
i
i
i

w
v
u
y
p
p
p
P
p
w
y
v
y
u
y
y
k
j
j
j
i
j
j

w
v
u
z
p
p
p
P
p
w
z
v
z
u
z
z
k
k
j
k
i
k
k

w
v
u
k
j
i
w
v
u
p
p
p
P

Basic Rotation Matrix

Basic Rotation Matrix

x
-
axis

with

10

w
v
u
z
y
x
p
p
p
p
p
p
w
z
v
z
u
z
w
y
v
y
u
y
w
x
v
x
u
x
k
k
j
k
i
k
k
j
j
j
i
j
k
i
j
i
i
i
x
z
y
v
w
P
u

C
S
S
C
x
Rot
0
0
0
0
1
)
,
(

Is it True? Can we check?

11

cos
sin
sin
cos
cos
sin
0
sin
cos
0
0
0
1
w
v
z
w
v
y
u
x
w
v
u
z
y
x
p
p
p
p
p
p
p
p
p
p
p
p
p
p

x
z
y
v
w
P
u

Basic Rotation Matrices

x
-
axis

with

y
-
axis

with

z
-
axis

with

12

uvw
xyz
RP
P

C
S
S
C
x
Rot
0
0
0
0
1
)
,
(

0
0
1
0
0
)
,
(

C
S
S
C
y
Rot

1
0
0
0
0
)
,
(

C
S
S
C
z
Rot

Matrix notation for rotations

Basic Rotation Matrix

Obtain the coordinate of from the coordinate of

z
y
x
w
v
u
p
p
p
p
p
p
z
w
y
w
x
w
z
v
y
v
x
v
z
u
y
u
x
u
k
k
j
k
i
k
k
j
j
j
i
j
k
i
j
i
i
i
13

uvw
xyz
RP
P

w
z
v
z
u
z
w
y
v
y
u
y
w
x
v
x
u
x
k
k
j
k
i
k
k
j
j
j
i
j
k
i
j
i
i
i
R
xyz
uvw
QP
P

T
R
R
Q

1
3
1
I
R
R
R
R
QR
T

uvw
P
xyz
P
<== 3X3 identity matrix

Dot products
are commutative!

Example
2:
rotation of a point in a rotating
frame

A point is attached to a
rotating frame
, the
frame rotates 60 degree about the OZ axis of the
reference frame.

Find
the coordinates
of the point
relative to the
reference frame after the rotation.

14

)
2
,
3
,
4
(

uvw
a

2
964
.
4
598
.
0
2
3
4
1
0
0
0
5
.
0
866
.
0
0
866
.
0
5
.
0
)
60
,
(
uvw
xyz
a
z
Rot
a
Example
3:
find a point of rotated system
for point in coordinate system

A point is the coordinate w.r.t. the
reference coordinate system, find the
corresponding point
w.r.t. the rotated OU
-
V
-
W coordinate system

if it has been rotated 60

)
2
,
3
,
4
(

xyz
a
uvw
a

2
964
.
1
598
.
4
2
3
4
1
0
0
0
5
.
0
866
.
0
0
866
.
0
5
.
0
)
60
,
(
xyz
T
uvw
a
z
Rot
a

Reference coordinate
frame OXYZ

Body
-
attached

frame
O’uvw

Composite

Rotation Matrix

A
sequence of finite rotations

matrix multiplications do not commute

rules
:

if rotating coordinate O
-
U
-
V
-
principal axis of OXYZ frame, then
Pre
-
multiply

the
previous (resultant) rotation matrix with an
appropriate basic rotation
matrix

if rotating coordinate OUVW is rotating about its own
principal axes, then
post
-
multiply

the previous
(resultant) rotation matrix with an appropriate basic
rotation matrix

16

Example
4: finding a rotation matrix

Find the rotation matrix for the following
operations:

17

Post
-
multiply if rotate about the OUVW axes

Pre
-
multiply if rotate about the OXYZ axes

...
axis

OU

Rotation
axis
OW

Rotation
axis

OY

Rotation

S
S
S
C
C
S
C
C
S
S
C
S
S
C
C
C
S
C
S
S
S
C
C
S
C
S
S
C
C
C
S
S
C
C
S
S
C
u
Rot
w
Rot
I
y
Rot
R
0
0
0
0
1
1
0
0
0
0
C
0
S
-
0
1
0
S
0
C
)
,
(
)
,
(
)
,
(
3
Reference coordinate
frame OXYZ

Body
-
attached

frame
O’uvw

Coordinate

Transformations

18

position vector of
P
in {
B
} is transformed
to position vector of
P
in {
A
}

description of {
B
} as
seen from an observer
in {
A
}

Rotation of {
B
} with respect to {
A
}

Translation of the origin of {
B
} with respect to origin of {
A
}

Coordinate Transformations

Two Special Cases

1.
Translation only

Axes of {
B
} and {
A
} are
parallel

2.
Rotation only

Origins of {
B
} and {
A
} are
coincident

19

1

B
A
R
'
o
A
P
B
B
A
P
A
r
r
R
r

0
'

o
A
r
Homogeneous Representation

20

Coordinate transformation
from {
B
} to {
A
}

Homogeneous transformation
matrix

'
o
A
P
B
B
A
P
A
r
r
R
r

1
1
0
1
3
1
'
P
B
o
A
B
A
P
A
r
r
R
r

1
0
1
0
1
3
3
3
3
1
'
P
R
r
R
T
o
A
B
A
B
A
Position
vector

Rotation
matrix

Scaling

Homogeneous Transformation

Special cases

1.
Translation

2.
Rotation

21

1
0
0
3
1
1
3
B
A
B
A
R
T

1
0
3
1
'
3
3
o
A
B
A
r
I
T
Example
5:
translation

Translation along Z
-
axis with h
:

22

1
0
0
0
1
0
0
0
0
1
0
0
0
0
1
)
,
(
h
h
z
Trans

1
1
1
0
0
0
1
0
0
0
0
1
0
0
0
0
1
1
h
p
p
p
p
p
p
h
z
y
x
w
v
u
w
v
u
x
y
z
P
u
v
w
O, O’

h

x
y
z
P
u
v
w
O, O’

Example
6:
rotation

-
axis by

1
1
0
0
0
0
0
0
0
0
0
0
1
1
w
v
u
p
p
p
C
S
S
C
z
y
x

23

1
0
0
0
0
0
0
0
0
0
0
1
)
,
(

C
S
S
C
x
Rot
x
z
y
v
w
P
u
Homogeneous Transformation

Composite Homogeneous Transformation
Matrix

Rules
:

Transformation (
rotation/translation
) w.r.t (
X,Y,Z
)
(
OLD FRAME
), using
pre
-
multiplication

Transformation
(rotation/translation) w.r.t (
U,V,W
)
(
NEW FRAME
), using
post
-
multiplication

24

Example
7:
homogeneous
transformation

Find the homogeneous transformation matrix (T)
for the following operations:

4
4
,
,
,
,

I
T
T
T
T
T
x
a
x
d
z
z

:
axis

OZ

of
Rotation
axis

OZ

along

d

of
n
Translatio
axis

OX

along

a

of
n
Translatio
axis

OX

Rotation

1
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
1
0
0
0
0
1
0
0
0
1
1
0
0
0
1
0
0
0
0
1
0
0
0
0
1
1
0
0
0
0
1
0
0
0
0
0
0

C
S
S
C
a
d
C
S
S
C
Homogeneous Representation

A frame in space (Geometric
Interpretation)

x
y
z
)
,
,
(
z
y
x
p
p
p
P

1
0
0
0
z
z
z
z
y
y
y
y
x
x
x
x
p
a
s
n
p
a
s
n
p
a
s
n
F
n
s
a

1
0
1
3
3
3
P
R
F
Principal axis
n

w.r.t. the reference coordinate system

(X’)

(y’)

(z’)

Homogeneous Transformation

Translation

y
z
n
s
a
n
s
a

1
0
0
0

1
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
z
z
z
z
z
y
y
y
y
y
x
x
x
x
x
z
z
z
z
y
y
y
y
x
x
x
x
z
y
x
new
d
p
a
s
n
d
p
a
s
n
d
p
a
s
n
p
a
s
n
p
a
s
n
p
a
s
n
d
d
d
F
old
z
y
x
new
F
d
d
d
Trans
F

)
,
,
(
Homogeneous Transformation

2
1
1
0
2
0
A
A
A

i
i
A
1

28

Composite Homogeneous Transformation Matrix

0
x
0
z
0
y
1
0
A
2
1
A
1
x
1
z
1
y
2
x
2
z
2
y
?

Transformation matrix for

Chain product of successive
coordinate transformation matrices

Example
8:
homogeneous transformation based on
geometry

For the figure shown below, find the 4x4 homogeneous transformation matrices
and for
i
=1, 2, 3, 4, 5

1
0
0
0
z
z
z
z
y
y
y
y
x
x
x
x
p
a
s
n
p
a
s
n
p
a
s
n
F

1
0
0
0
0
1
0
1
0
0
0
0
0
1
1
0
d
a
c
e
A
i
i
A
1

i
A
0
0
x
0
y
0
z
a
b
c
d
e
1
x
1
y
1
z
2
z
2
x
2
y
3
y
3
x
3
z
4
z
4
y
4
x
5
x
5
y
5
z

1
0
0
0
0
1
0
0
0
0
1
0
1
0
2
0
c
e
b
A

1
0
0
0
0
0
0
1
1
0
0
0
1
0
2
1
d
a
b
A
Can you find the answer by observation
based on the geometric interpretation of
homogeneous transformation matrix?

Orientation Representation

Rotation matrix representation needs 9 elements
to completely describe the orientation of a rotating
rigid body.

Any easy way?

1
0
1
3
3
3
P
R
F
Euler Angles Representation

Orientation Representation

Euler Angles Representation
( , , )

Many different types

Description of Euler angle representations

Euler Angle I Euler Angle II Roll
-
Pitch
-
Yaw

Sequence

of

Rotations

Euler Angle I, Animated

x

y

z

u
'

v
'

v
"

w
"

w
'
=

=u"

v'"

u
'"

w'"=

Orientation Representation

Euler Angle I

1
0
0
0
cos
sin
0
sin
cos
,
cos
sin
0
sin
cos
0
0
0
1
,
1
0
0
0
cos
sin
0
sin
cos
'
'
'

w
u
z
R
R
R
Euler Angle I

cos
sin
cos
sin
sin
sin
cos
cos
cos
cos
sin
sin
cos
sin
cos
cos
sin
sin
sin
cos
cos
sin
sin
cos
cos
sin
sin
cos
cos
'
'
'
w
u
z
R
R
R
R
Resultant
eulerian

rotation
matrix:

Euler Angle II, Animated

35

x

y

z

u
'

v
'

=v
"

w
"

w
'
=

u"

v
"'

u"'

w"'
=

Note the opposite
(clockwise) sense of the
third rotation,

.

Orientation Representation

Matrix with
Euler Angle II

36

cos
sin
sin
sin
cos
sin
sin
cos
cos
sin
cos
cos
cos
cos
sin
sin
cos
sin
cos
cos
cos
sin
cos
sin
cos
cos
cos
sin
sin
Quiz:
How to get this matrix ?

Orientation Representation

Description of Roll Pitch Yaw

X

Y

Z

Quiz:
How to get rotation matrix ?

The City College of New York

38

The City College of New York

Kinematics of Robot Manipulator

Jizhong Xiao

Department of Electrical Engineering

City College of New York

jxiao@ccny.cuny.edu

39

Introduction to ROBOTICS