Introduction to ROBOTICS

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14 Νοε 2013 (πριν από 3 χρόνια και 9 μήνες)

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The City College of New York

1

Jizhong Xiao

Department of Electrical Engineering

City College of New York

jxiao@ccny.cuny.edu

Kinematics of Robot Manipulator

Introduction to ROBOTICS


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2

Outline


Review



Robot Manipulators


Robot Configuration


Robot Specification


Number of Axes, DOF


Precision, Repeatability



Kinematics


Preliminary


World frame, joint frame, end
-
effector frame


Rotation Matrix, composite rotation matrix


Homogeneous Matrix


Direct kinematics


Denavit
-
Hartenberg Representation


Examples


Inverse kinematics




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3

Review


What is a robot?


By general agreement a robot is:


A programmable machine that imitates the actions or
appearance of an intelligent creature

usually a human.



To qualify as a robot, a machine must be able to:


1) Sensing and perception: get information from its surroundings

2) Carry out different tasks: Locomotion or manipulation, do
something physical

such as move or manipulate objects

3) Re
-
programmable: can do different things

4) Function autonomously and/or interact with human beings


Why use robots?


4A: Automation, Augmentation, Assistance, Autonomous

4D: Dangerous, Dirty, Dull, Difficult


Perform 4A tasks in 4D environments

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Manipulators


Robot arms, industrial robot


Rigid bodies (links) connected
by joints


Joints: revolute or prismatic


Drive: electric or hydraulic


End
-
effector (tool) mounted
on a flange or plate secured
to the wrist joint of robot

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Manipulators


Robot Configuration:

Cartesian: PPP

Cylindrical: RPP

Spherical: RRP

SCARA: RRP

(Selective Compliance
Assembly Robot Arm)

Articulated: RRR

Hand coordinate:

n:

normal vector;
s
: sliding vector;

a
: approach vector, normal to the

tool mounting plate

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Manipulators


Motion Control Methods


Point to point control


a sequence of discrete points


spot welding, pick
-
and
-
place, loading & unloading


Continuous path control


follow a prescribed path, controlled
-
path motion


Spray painting, Arc welding, Gluing

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Manipulators


Robot Specifications


Number of Axes


Major axes, (1
-
3) => Position the wrist


Minor axes, (4
-
6) => Orient the tool


Redundant, (7
-
n) => reaching around
obstacles, avoiding undesirable
configuration


Degree of Freedom (DOF)


Workspace


Payload (load capacity)


Precision v.s. Repeatability

Which one is more important?

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What is Kinematics


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
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9

What is Kinematics


Inverse kinematics

End effector position

and orientation





Joint variables
-
Formula?

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



)
,
,
,
,
,
(
T
A
O
z
y
x
x
y
z
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10

Example 1

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
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Preliminary


Robot Reference Frames


World frame


Joint frame


Tool frame

x
y
z
x
z
y
W

R

P

T

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12

Preliminary


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




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 ==>

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Preliminary


Mutually perpendicular



Unit vectors

Properties of orthonormal coordinate frame

0
0
0






j
k
k
i
j
i






1
|
|
1
|
|
1
|
|



k
j
i



Properties: Dot Product

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




3
R


cos
y
x
y
x


x
y
x
y
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Preliminary


Coordinate Transformation


Rotation only

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




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?


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Preliminary


Basic Rotation



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



Since


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



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Preliminary


Basic Rotation Matrix





Rotation about x
-
axis with










































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
)
,
(

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Preliminary


Is it True?


Rotation about x axis with










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


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Basic Rotation Matrices


Rotation about x
-
axis with




Rotation about y
-
axis with





Rotation about z
-
axis with




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



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19

Preliminary


Basic Rotation Matrix





Obtain the coordinate of from the coordinate
of

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









































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
Dot products are commutative!

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Example 2


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.




)
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
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21

Example 3


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 degree about OZ axis.

)
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
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Composite Rotation Matrix


A sequence of finite rotations


matrix multiplications do not commute


rules:


if rotating coordinate O
-
U
-
V
-
W is rotating about
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

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Example 4


Find the rotation matrix for the following
operations:




Post
-
multiply if rotate about the OUVW axes

Pre
-
multiply if rotate about the OXYZ axes

...
axis

OU
about

Rotation
axis
OW
about

Rotation
axis

OY
about

Rotation
Answer


































































































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
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24

Coordinate Transformations



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
}

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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



1

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


0
'

o
A
r
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Homogeneous Representation



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

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Homogeneous Transformation


Special cases

1. Translation




2. Rotation










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
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Example 5


Translation along Z
-
axis with h:

















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’

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29

Example 6


Rotation about the X
-
axis by



















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






































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




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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


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31

Example 7


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


:
axis

OZ
about


of
Rotation
axis

OZ

along

d

of
n
Translatio
axis

OX

along

a

of
n
Translatio
axis

OX
about

Rotation
Answer


4
4
,
,
,
,


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





















































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
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32

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’)

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33

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


)
,
,
(
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34

Homogeneous Transformation

2
1
1
0
2
0
A
A
A

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
?

i
i
A
1

Transformation matrix for
adjacent coordinate frames

Chain product of successive
coordinate transformation matrices

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35

Example 8


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
















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?

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36

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

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Orientation Representation


Euler Angles Representation ( , , )


Many different types


Description of Euler angle representations







Euler Angle I Euler Angle II Roll
-
Pitch
-
Yaw

Sequence about OZ axis about OZ axis


about OX axis

of about OU axis about OV axis about OY axis

Rotations about OW axis about OW axis about OZ axis











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38

x

y

z

u
'

v
'





v
"

w
"

w
'
=

=u"

v'"



u
'"

w'"=

Euler Angle I, Animated

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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
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40

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:

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41

Euler Angle II, Animated

x

y

z

u
'

v
'





=v
"

w
"

w
'
=

u"

v
"'



u"'

w"'
=

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

.

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42

Orientation Representation


Matrix with Euler Angle II



























































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 ?

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Orientation Representation


Description of Roll Pitch Yaw

X

Y

Z




Quiz: How to get rotation matrix ?

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44

Thank you!

x
y
z
x
y
z
x
y
z
x
z
y
Homework 1 is posted on the web.

Due: Sept.
16
, 200
8
, before class


Next class: kinematics II