# Introduction to ROBOTICS

Mechanics

Nov 14, 2013 (5 years and 2 months ago)

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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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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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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

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
-

Continuous path control

-
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

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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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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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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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

-
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?

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

-
axis with

-
axis with

-
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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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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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
-
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

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
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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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Example 6

-
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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Example 7

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

:
axis

OZ

of
Rotation
axis

OZ

along

d

of
n
Translatio
axis

OX

along

a

of
n
Translatio
axis

OX

Rotation

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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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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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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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

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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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

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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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39

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