Kinematics of Robot

taupeselectionΜηχανική

14 Νοε 2013 (πριν από 3 χρόνια και 4 μήνες)

61 εμφανίσεις

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
link

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





Rotation about
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?





Rotation about x axis with


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


Rotation about
x
-
axis

with




Rotation about
y
-
axis

with





Rotation about
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
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

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
-
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

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
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
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


Rotation about the X
-
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
about


of
Rotation
axis

OZ

along

d

of
n
Translatio
axis

OX

along

a

of
n
Translatio
axis

OX
about

Rotation
Answer





















































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
adjacent coordinate frames

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

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











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