Notes_11_02
1 of 10
Three-Dimensional Coordinate Transformations
x i 
x i   a i 11 a i 12
 
  
 y i    y i   a i 21 a i 22
z 
 z  a
 i
 i   i 31 a i 32
P
ri P  ri   A i s i ' P
s i P  A i s i ' P
s i ' P  A i T s i P
[A] matrices are orthonormal
[A] -1 = [A] T
● all columns are unit vectors
● all columns are mutually orthogonal
● all rows are unit vectors
● all rows are mutually orthogonal
● det [A] = +1
Single rotations
0
1
A X   0 C X
0 S X
 C Y
A Y    0
 S Y
C z
A Z   S Z
 0
0 
 S X 
C X 
0 S Y 
1
0 
0 C Y 
 S Z
C Z
0
0
0
1
a i 13  x i 
 
a i 23   y i 
a i 33   z i 
'P
Notes_11_02
2 of 10
Coordinate Frames at the Corners of a Wedge
Adapted from Introduction to Robotics, J.J. Craig, Addison-Wesley, 1989
y3’

z3’
x3’
c
z 1’
y1’
x2’
a
z2’
x1’
y2’
b
x 1  b  1 0 0 x 2  '
    
 
 y1   0   0  1 0  y 2 
 z  0  0 0 1  z 
 1   
 2 
x 1  b  0  S C x 3  '
    
 
 y1   0   0 C S   y 3 
 z  a   1 0
0   z 3 
 1   

z4’
z5
y6’
x5’
x4’
x6’
y5’
c
z6’
a
y4’
b
x 4  ' 0  1 0 0  x 5  '
 
  
 
 y 4   c    0 0  1  y 5 
z 
a   0  1 0   z 
 4
  
 5 
0  x 6  '
x 4  '  b C S
 
  
 
 y 4    c    S  C 0   y 6 
z 
a  0
0
 1  z 6 
 4
  
Notes_11_02
3 of 10
Euler Angles
Euler sequences – ZXZ (original), ZYZ, YXY, YZY, XYX, XZX
ZXZ sequence (1 about global z - 2 about intermediate x - 3 about local z)
C1
A Z A X A Z   S1
 0
 S1
C1
0
0 1
0


0 0 C X
1 0 S X
 S1C 2S 3  C1C 3
A Z A X A Z    C1C 2S3  S1C3

S 2S 3
0  C 3
 S X   S 3
C X   0
 S 3
C 3
0
 S1C 2 C 3  C1S 3
C1C 2 C 3  S1S 3
S 2 C 3
0
0
1
S1S 2 
 C1S 2 
C 2 
 2  a cosa 33  1  a tan 2 a 13 / a 23  3  a tan 2a 31 / a 32  fails for  2  0 and 
ZYZ sequence
YXY sequence
YZY sequence
XYX sequence
XZX sequence
Notes_11_02
4 of 10
Cardan-Bryant-Tait sequences – XYZ, XZY, YXZ, YZX, ZXY, ZYX
XYZ sequence (x about global x - y about intermediate y - z about local z)
C Y C z


A X A Y A Z    S XS Y C z  C XS z
 C X S Y C z  S X S z
 C Y S z
S Y
 S X S Y S z  C X C z
C X S Y S z  S X C z

 S X C Y 
C X C Y 
 Y  a sin a 13   Z  a tan 2 a 12 / a 11   X  a tan 2 a 23 / a 33  fails for  Y   / 2 and 3 / 2
ZYX sequence (z about global z - y about intermediate y - x about local x)
C X C Y
A Z A Y A X   S X C Y
  S Y
C X S Y S z  S X C z
 S X S Y S z  C X C z
C Y S z
C X S Y C z  S X S z 
 S X S Y C z  C X S z 

C Y C z
 Y  a sin  a 31   Z  a tan 2a 32 / a 33   X  a tan 2a 21 / a 11  fails for  Y   / 2 and 3 / 2
XZY sequence (x about global x - z about intermediate z - y about local y)
C Y C Z


A X A Z A Y   C X C YS Z  S XS Y
S X C Y S Z  C X S Y
 S Z
C X C Z
S X C Z
S Y C Z

C X S Y S Z  S X C Y 
S X S Y S Z  C X C Y 
 Z  a sin  a 12   X  a tan 2a 32 / a 23   Y  a tan 2a 13 / a 11  fails for  Z   / 2 and 3 / 2
YZX sequence
YXZ sequence
ZXY sequence
Notes_11_02
5 of 10
Chasles’ Angle and Euler Parameters
 u 2 V + C uvV - wS uwV + vS


 A  uvV + wS v 2 V + C vwV - uS 
 uwV - vS vwV + uS w 2 V + C 
Rotation  about unit direction  u 
C  ( tr A  1) / 2
u 
a 32  a 23 
1 
 

û   v  
 a 13  a 31 
w  2S a  a 
 
 21 12 
fails for   0 and 
Euler parameters (unit quarternion)

e 0   C 2 
e    
uS
p   1    2 
e 2   vS 2 
e 3  wS  
2

 e 02  e12  12
A  2e1e 2  e 0 e 3
e1 e 3  e 0 e 2

e 0  ( tr A  1) / 4
2
pT p  e 02  e12  e 22  e32  1
e1 e 2  e 0 e 3
e 02  e 22  12
e 2 e 3  e 0 e1
e1 e 3  e 0 e 2 

e 2 e 3  e 0 e1 
e 02  e 32  12 
 e1 
a 32  a 23 
1 
 
e  e 2   a 13  a 31 
e  4e1 a  a 
 3
 21 12 
fails for e 0  0 at   
Notes_11_02
  e1
E   e 2
  e 3
A  EG 
T
e0
 e3
e3
 e2
e0
e1
  e1
G   e 2
  e 3
e2 
 e1 
e 0 
2e1e 2  e 0 e 3 
e 02  e12  e 22  e 32
2e 2 e 3  e 0 e1 
e 02  e12  e 22  e 32

 2e1e 2  e 0 e 3 
 2e1e 3  e 0 e 2 

EET  GGT  I 3 
ET E  G T G   I 4   ppT
Ep  Gp  0
Velocity
 A 
  2Ep   2E p
  A T 
  2G p   2G p
p   12 ET   12 GT 
pT p   p T p  0
~   A AT  2EE T  2E ET
~   AT A   2G G T
 
 G T
 2 G
A   ~ A  A~   2EG 
T

T
 2 E G 
6 of 10
e0
e3
- e3
e2
e0
- e1
- e2 
e1 
e 0 
2e1e 3  e 0 e 2  

2e 2 e 3  e 0 e1  
e 02  e12  e 22  e 32 
Notes_11_02
Acceleration
   A 
   2Ep
   AT  
   2G p
p  12 ET   12 E T   12 ET   14 pT 
p  12 GT    12 G T   12 GT    14 p 
T
p T p   14 T   14    14 2
T
E p   G p   0
A   2E G 
T


 2E G
EG T  E GT
T

  
 T
 GT  2 E G
2E
7 of 10
Notes_11_02
8 of 10
Jerk
  A  A~   
~ 
  2Ep 2E p  2Ep 12 
   14 T 

  AT   AT ~  
  2G p  2G p  2G p  12 ~     14   
T
p  12 ET   E T    12 E T 
T
~ 
  12 
   12 T   34 pT 

 12 E 
p  12 G T   G T    12 G T 

G G
T
1
2
G T    12 ~     14      34 p  
T


G G 
T
0
A  2EG
T
T
  
0
G p  0
  

Gp  0
  
G pG p   0
  
 T  2 E
 T  2 E GT  4 E
 T  2 E G
 T
 G
 G
 4 E G
EGT  E G T  EGT  E G T
Notes_11_02
9 of 10
Numerical partial derivatives of rotation matrices with respect to Euler parameters can
produce different results
 e 02  e12  12
A  2e1e 2  e 0 e 3
e1 e 3  e 0 e 2

A  EG 
T
e1 e 2  e 0 e 3
e 02  e 22  12
e 2 e 3  e 0 e1
e 02  e12  e 22  e 32

 2e1e 2  e 0 e 3 
 2e1e 3  e 0 e 2 

e1 e 3  e 0 e 2 

e 2 e 3  e 0 e1 
e 02  e 32  12 
2e1e 2  e 0 e 3 
e  e12  e 22  e 32
2e 2 e 3  e 0 e1 
2
0
numerical
a 11
0
e 3
2e1e 3  e 0 e 2  

2e 2 e 3  e 0 e1  
e 02  e12  e 22  e 32 
numerical
a 11
 2e 3
e 3
Notes_11_02
Rodriguez Parameters
R  tan    û
2
10 of 10