Transcript Rigid motions

Introduction to ROBOTICS
KINEMATICS
POSE (POSITION AND
ORIENTATION)
OF A RIGID BODY
University of Bridgeport
1
Representing Position (2D)
y
5 
p 
2
(“column” vector)
p
2
p  5 xˆ  2 yˆ
xˆ
yˆ
A vector of length one pointing
in the direction of the base
frame x axis
2
A vector of length one pointing
in the direction of the base
frame y axis
5
x
Representing Position: vectors
yˆ b
• The prefix superscript denotes the
reference frame in which the vector
should be understood
yˆ a
p
2
2
3
b
5
b
5 
p  
2
3  8 
  
2 4
a
3
p 
2
3
Same point, two different
reference frames
xˆ a
xˆ b
Representing Position: vectors (3D)
y
y
p
x
2
 
p 5
 
 2 
z
p  2 xˆ  54 yˆ  2 zˆ
x
z
right-handed
coordinate frame
xˆ
A vector of length one pointing in the
direction of the base frame x axis
yˆ
zˆ
A vector of length one pointing in the
direction of the base frame y axis
A vector of length one pointing in the
direction of the base frame z axis
The rotation matrix
A
 cos  
R B  
 sin  
A
A
 sin   

cos   
yˆ B
p
xˆ B

p  RB p
A
yˆ A
B
R B :To specify the coordinate vectors

for the fame B with respect to
frame A
B
R A  RB
A
B
1
 cos  
 
  sin  
5
p RA p
B
A
sin   

cos   
xˆ A
θ: The angle between xˆ A and xˆ B
in anti clockwise direction
USEFUL FORMULAS
B
A
R (B R)
A
RR
B
A
1
1
(B R)
A
T
 I
R .B R  I
A
Det ( R )  1
6
B
10 
p 
10 
Example 1
A
B
yˆ
yˆ
A
find p
B
 cos 30   sin 30  
30

R B  
cos 30  
 sin 30 
30
 3
1 


A
A
B
A
2
2


p  RB p
RB 
 1 7
3
 3
1 




 10   3.6603 
2 
 2
A
2
2


   

P 
 1
3   10   13.6603 


2 
 2
xˆ
A
A
xˆ
BASIC ROTATION MATRIX

Rotation about x-axis with
0
0
1

Rot ( x ,  )  0 C   S 

 0 S 
C
 pu 
 px 
 
 
p y  R ( x, ) pv
 
 
 p w 
 p z 




z
w
P
v

u
y
x
p x  pu
p y  p v cos   p w sin 
p z  p v sin   p w cos 
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BASIC ROTATION MATRICES

Rotation about x-axis with

R x ,

Rotation about y-axis with
Rotation about z-axis with
Pxyz  RP uvw
0 

 S

C  
0
C
S

R y ,

1

 Rot ( x ,  )  0

 0
 C

 Rot ( y ,  ) 
0

  S 
0
1
0
S 

0

C  

R z ,
C 

 Rot ( z ,  )  S 

 0
 S
C
0
0

0

9
1 
EXAMPLE 2
A
point p  ( 4 ,3, 2 ) 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.
uvw
p xyz  Rot ( z , 60 ) p uvw
 0 .5

 0 . 866

 0
 0 . 866
0 .5
0
0   4    0 . 598 
  

0 3  4 . 964
  


1   2  
2
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EXAMPLE 3
A
point a  ( 4 ,3, 2 ) is the coordinate w.r.t. the
reference coordinate system, find the
corresponding point a uvw w.r.t. the rotated
OUVW coordinate system if it has been
rotated 60 degree about OZ axis.
xyz
p uvw  R ot ( z , 60) p xyz
T
O R : p uvw  R ot ( z , 60)
1
p xyz
O R : p uvw  R ot ( z ,  60) p xyz
 0.5

  0.866

 0
0.866
0.5
0
0   4   4.598 
  

0 3   1.964
  


1   2  
2
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COMPOSITE ROTATION MATRIX

A sequence of finite rotations

rules:
if rotating coordinate OUVW is rotating about principal axis of
OXYZ frame, then Pre-multiply the previous (resultant) rotation
matrix with an appropriate basic rotation matrix [rotation about
fixed frame]
 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 [rotation about current
frame]

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ROTATION WITH RESPECT TO CURRENT
FRAME
A
A
P  RB P
B
B
P  RC P
C
C
P  RD P
D
A
P  R D P  R B RC R D P
B
C
A
A
D
A
B
R D  R B RC R D
A
B
C
C
D
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EXAMPLE 4

Find the rotation matrix for the following operations:
R  Rot ( y ,  ) Rot ( z ,  ) Rot ( x ,  )
R otation  about Y axis
R otation  about current Z axis
R otation  about current X axis
 C


0

 - S 
0
1
0
 C C 


S

  S  C 
S   C 

0
S

C    0
 S
C
0
S S  C S C 
C C 
SSC   C S
Pre-multiply if rotate about the fixed frame
Post-multiply if rotate about the current
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frame
0  1

0 0

1   0
0
C
S


 S

C  
0
C S S   S C  

 C S

C  C   S  S  S  
EXAMPLE 5

Find the rotation matrix for the following operations:
R  Rot ( y ,  ) Rot ( z ,  ) Rot ( x ,  )
Rotation
 about X axis
Rotation
 about fixed Z axis
Rotation
 about fixed Y axis
 C


0

 - S 
0
1
0
 C C 


S

  S  C 
S   C 

0
S

C    0
C
0
S S  C S C 
C C 
SSC   C S
Pre-multiply if rotate about the fixed frame
Post-multiply if rotate about the current
frame
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 S
0  1

0 0

1   0
0
C
S


 S

C  
0
C S S   S C  

 C S

C  C   S  S  S  
EXAMPLE 6

Find the rotation matrix for the following operations:
Rotation
 about X axis
Rotation
 about current Z
Rotation
 about fixed Z axis
Rotation
 about current Y axis
Rotation
 about fixed X axis
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axis
Pre-multiply if rotate about the fixed frame
Post-multiply if rotate about the current
frame
EXAMPLE 6

Find the rotation matrix for the following operations:
Rotation
 about X axis
Rotation
 about current Z
Rotation
 about fixed Z axis
Rotation
 about current Y axis
Rotation
 about fixed X axis
axis
R  Rot ( x ,  ) Rot ( z ,  ) Rot ( x ,  ) Rot ( z ,  ) Rot ( y ,  )
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QUIZ

Description of Roll Pitch Yaw

Find the rotation matrix for the following operations:
Z

Rotation  about X axis{ROLL}
Rotation
 about fixed Y axis{PITCH
Rotation
 about fixed Z axis{YAW}
}

X
Y

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Z
ANSWER

Rotation  about X axis
Rotation
 about fixed Y axis
Rotation
 about fixed Z axis

R  Rot ( z ,  ) Rot ( y ,  ) Rot ( x , )
C 

 S

 0
 S
C C 

 SC 

  S 
C
0
0  C

0
0

1   - S 
0
1
0
S   1

0
0

C    0
 SS  C SS
C SS  C S
C S
Y

X
0
C
S


 S

C  
0
C S C   S S  

 C S  S S C 


C C 
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HOMOGENEOUS TRANSFORMATION

Special cases
1. Translation
A
HB
2. Rotation
A
HB
 I 3 3
 
 0 1 3
p 

1 
 ARB
 
 0 1 3
0 3 1 

1 
A
B
20
EXAMPLE 7

Translation along Z-axis with h:
1

0

Trans ( z , h ) 
0

0
0
0
1
0
0
1
0
0
 x  1
  
y
0
  
 z  0
  
 1  0
0

0

h

1
z
0
0
1
0
0
1
0
0
0   p x1   p x1

 
0 p y1
p y1

 
h   p z1   p z1 

 
1 1   1
z



h


P
P
B
y
w
y
z
B
y
v
O
u
x
B
h
O
x
x21
EXAMPLE 7

Translation along Z-axis with h:
0
P H B P
 x  1
  
y
0
  
 z  0
  
 1  0
0
B
0
0
1
0
0
1
0
0
0   p xB

0 p yB

h   p yB

1 1
  p xB
 
p yB
 
  pz 
B
 
  1



h


22
EXAMPLE 8
 Rotation
1

0

Rot ( x ,  ) 
0

0
about the X-axis by
0
0
C
 S
S
C
0
0
0

0

0

1
 x  1
  
y
0
  
 z  0
  
 1  0
0
0
C
 S
S
C
0
0
0   p x1 


0 p y1


0   p z1 


1 1 
z
z1
P

y1

y
x1
x
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HOMOGENEOUS TRANSFORMATION
Composite Homogeneous Transformation Matrix
 Rules:

Transformation (rotation/translation) w.r.t fixed
frame, using pre-multiplication
 Transformation (rotation/translation) w.r.t current
frame, using post-multiplication

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EXAMPLE 9
 Find
the homogeneous transformation
matrix (H) for the following operations:
Rotation
 about OX axis
Translatio
n of a along OX axis
Translatio
n of d along OZ axis
Rotation
of  about OZ axis
H  Rot
Answer :
C 

S
 
 0

 0
z ,
Trans
 S
0
C
0
0
1
0
0
z ,d
Trans
0  1

0 0

0  0

1  0
x ,a
Rot
0
0
1
0
0
1
0
0
x ,
0  1

0 0

d  0

1  0
0
0
1
0
0
1
0
0
a  1

0 0

0  0

1  0
0
0
C
 S
S
C
0
0
0

0

0

25
1
Remember those double-angle formulas…
sin      sin   cos    cos   sin  
cos      cos   cos    sin   sin  
26
Review of matrix transpose
 a 11

A  a 21

 a 31
a 12
a 22
a 32
a 13 

a 23

a 33 
A
 a 11

a
 21
 a 31
a 12
a 22
a 32
T
 a 11

 a 12

 a 13
a 21
a 22
a 23
a 31 

a 32

a 33 
a 13 

a 23

a 33 
27
5 
p 
2
p  5
T
2
T
Important property: A B
T
 BA
T
and matrix multiplication…
 a 11
A  
 a 21
 a 11
AB  
 a 21
a 12 

a 22 
a 12   b11

a 22   b 21
 b11
B  
 b 21
b12 

b 22 
b12   a 11 b11  a 12 b 21
 
b 22   a 21 b11  a 22 b 21
a 11 b12  a 12 b 22 

a 21 b12  a 22 b 22 
Can represent dot product as a matrix multiply:
28

a  b  a xbx  a yb y  a x
bx 
T
ay    a b
b y 

HW


Problems 2.10, 2.11, 2.12, 2.13, 2.14 ,2.15, 2.37,
and 2.39
Quiz next class
29