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Euler Angles
Single rotations
X X
X X X 0 S C
0 C S
A
Y Y
Y Y Y S 0 C
C 0 S
A
S C 0
C S 0
A Z Z
z Z Z
Euler sequences – ZXZ (original), ZYZ, YXY, YZY, XYX, XZX
ZXZ sequence ( 1 about global z - 2 about intermediate x - 3 about local z)
2 3 2 3 2
1 2 3 1 3 1 2 3 1 3 1 2
1 2 3 1 3 1 2 3 1 3 1 2 Z X Z S S S C C
C C S S C C C C S S C S
S C S C C S C C C S S S
A A A
2 a cos A 33 1 atan 2 A 13 /A 23 3 atan 2 A 31 /A 32 failsfor 2 0 and
Cardan-Bryant-Tait sequences – XYZ, XZY, YXZ, YZX, ZXY, ZYX
XYZ sequence ( x about global x - y about intermediate y - z about local z)
X Y z X z X Y z X z X Y
X Y z X z X Y z X z X Y
Y z Y z Y X Y Z C S C S S C S S S C C C
S S C C S S S S C C S C
C C C S S
A A A
Y a sin A 13 Zatan 2 A 12 /A 11 Xatan 2 A 23 /A 33 failsforY/ 2 and 3 / 2
ZYX sequence ( z about global z - y about intermediate y - x about local x)
Y Y z Y z
X Y X Y z X z X Y z X z
X Y X Y z X z X Y z X z Z Y X S C S C C
S C S S S C C S S C C S
C C C S S S C C S C S S
A A A
XZY sequence ( x about global x - z about intermediate z - y about local y)
X Y z X z X Y X Y z X z
X Y z X z X Y X Y z X z
Y z Y Y z X Z Y S S C C S S C S S S C C
C S C S S C C C S S S C
C C S C S
A A A
YZX sequence
YXZ sequence
ZXY sequence
A
2 E p^ 2 E p
A T
2 G p^ 2 G p
p 21 E T 21 G T
p T^ p p T p 0
~^ A ^ A T^ 2 E E T 2 E ET
~^ A T^ A ^ 2 G G T 2 G G T
A ^ ~A A~ 2 EG T 2 E GT
Acceleration
^ A
^ 2 E p
^ A T
^ 2 G p
p 21 E T 21 ET 21 E T 41 p T
T 4
T 1 2
T 1 2
T 1 2 p 1 G G G p
41 2
T 41
T 41 p T^ p
E p G p 0
A ^ 2 E G T 2 E G T 2 EG T 2 E G T
E G ^ T E G T
Jerk
^ A A ~
2 E p 2 E p 2 E p 21 ~ 41 T
^ A T A T ~
T 21 41 2 Gp 2 G p 2 G p ~
T 4
T 3 2
1 2
T 1 2
1
T 2
T T 1 2
1 E ~ p
p E E E
^
T 4
T 3 4
1 2
T 1 2
1
T 2
T T 1 2
1
G ~ p
p G G G
G G T 0 G G T 0 G p 0 G p 0 G p G p 0
A ^ 2 EG T 4 E G T 2 E G T 2 E G T 4 E G T 2 E G T
E G ^ T E G T E G T E G T
Three-Dimensional Coordinate Transformations – Haug Notation
Coordinate Frames at the Corners of a Wedge
Adapted from Introduction to Robotics, J.J. Craig, Addison-Wesley, 1989
z
y
x
0 0 1
' b
z
y
x
2
2
2
1
1
1
z
y
x
1 0 0
0 C S
0 S C
a
' b
z
y
x
3
3
3
1
1
1
z
y
x
0 1 0
a
c
z
y
x
5
5
5
4
4
4
z
y
x
0 0 1
S C 0
C S 0
a
c
' b
z
y
x
6
6
6
4
4
4
x (^) 2’
y2’
z2’
x (^) 1’
y 1 ’ z1’
x (^) 3’
y3’ z3’
a
b
c
x (^) 4’
y4’
z4’ x^ 5’
y5’
z 5 x (^) 6’
y6’
z 6 ’ (^) a
b
c
