Download Euler Angles – Body Fixed

Direction Cosine Matrix (DCM)
b3
C  CBA : B  A
B
A
a3
b2
ˆ1  C11 a
b
ˆ1  C12 a
ˆ2  C13 a
ˆ3
ˆ2  C21 a
b
ˆ1  C22 a
ˆ2  C23 a
ˆ3
a2
ˆ3  C31 a
b
ˆ1  C32 a
ˆ2  C33 a
ˆ3
a1
ˆi  a
where C ij  b
ˆj
b1
Properties :
C 1  C T
ˆ1  C11 C12
b
ˆ  
b2   C21 C22
b
ˆ3  C31 C32
 
C13   a
ˆ1 
ˆ1 
a
C23  a
ˆ2   C BA a
ˆ2 

a
C33  a
ˆ3 
ˆ3 
MMS I, Lecture 2
T
C BA  C AB
CC T  I  C T C
det(C )  1
CCBC BA  C BA CCB
1
Rotation Order
MMS I, Lecture 2
2
Transport Theorem - Example
d r 
ˆ1  r2 b
ˆ2  r3b
ˆ3
   r1b
 d t B
R  R0  r
a3
d r 
ˆ1  r2 b
ˆ2  r3b
ˆ3
 2   r1b
 d t B
2
Acceleration Velocity
P
b3
b2
r
BA
d R0 
d R 
d r 

 
  
d
t
d
t

A 
A d t A
d R 
d r 
  0       BA  r
 d t  A  d t B
R
b1
a2
a1
d2 R0 
d2 R 
d2 r 
  2
 2 
2 
 d t A  d t A d t A
d2 R0 
d2 r 
d r 





r



(


r
)

2



 2
 
BA
BA
BA
BA
2 
d
t
d
t
 d t B
B

A 
Centripetal
MMS I, Lecture 2
Coriolis
3
Euler Angles (Body Fixed 3-2-1)
c3
DC
d3
b3
c3
a3
CB
BA
3
d2
c2
b2
b2
2
1
c1
a1
b1
0
0 
1
C1(1 )  0 c1 s1  C2 ( 2 ) 
0  s1 c1 
CDA  CDC CCB CBA
a2
c2c3

 s1s2c3  c1s3
c1s2c3  s1s3
c1
c 2 0  s 2 
 0 1

0


s 2 0 c 2 
b1
 c 3
C3( 3 )   s 3
 0
s 3 0
c 3 0
0 1
c2 s3
 s2 
s1s2 s3  c1c3 s1c1 
c1s2 s3  s1c3 c1c2 
MMS I, Lecture 2
4
Euler Angle Representations
MMS I, Lecture 2
5
Euler’s Theorem
The rotation of a rigid body
fixed in the point P may
be described as a rotation
about the vector n
through the point P by
the angle 
n

b3
a3
b2
a2
P
a1
MMS I, Lecture 2
b1
6
Euler Eigenaxis rotation
MMS I, Lecture 2
7
Rotation Representations
Representation
Par.
Characteristics
Direction Cosine
matrix
9



Euler Angles
3




Quaternions
4




Nonsingular
Intuitive
Six redundant parameters
Minimal set
Clear physical
representation
Trigonometric functions in
rotation matrix
Singular
Easy orthogonality
Not singular
No clear physical
representation
One redundant parameter
MMS I, Lecture 2
Applications




Analytical
studies
Analytical
studies
Widely used
in simulation
Preferred for
global
rotation
8
Euler Angle Kinematics (3-2-1)
c3
DC
d3
b3
c3
a3
CB
3
.
d2
3
c2
.
b2
.
1
b2
2
2
1
c1
BA
a2
a1
b1
c1
MMS I, Lecture 2
b1
9