Euler Angles – Body Fixed
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Transcript 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 Cij b
ˆj
b1
Properties :
C 1 C T
ˆ1 C11
b
ˆ
b2 C21
b
ˆ3 C31
C12
C22
C32
C13 a
ˆ1
ˆ1
a
C23 a
ˆ2 C BA a
ˆ2
a
C33 a
ˆ3
ˆ3
T
C BA C AB
CC T I C T C
det(C ) 1
CCB C BA C BA C CB
MMS I, Lecture 2
1
Rotation Order
MMS I, Lecture 2
2
Transport Theorem - Example
d r
ˆ1 r2 b
ˆ2 r3b
ˆ3
r1b
d t B
R R0 r
a3
d r
ˆ1 r2 b
ˆ2 r3b
ˆ3
2 r1b
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
DC
d3
b3
c3
a3
CB
BA
3
d2
c2
b2
b2
2
1
a1
c1
b1
0
0
1
C1(1 ) 0 c1 s1
0 s1 c1
CDA CDC CCB CBA
a2
c1
b1
c 2 0 s 2
C2 ( 2 ) 0 1
0 C3 ( 3 )
s 2 0 c 2
c2c3
s1s2c3 c1s3
c1s2c3 s1s3
c2 s3
s1s2 s3 c1c3
c1s2 s3 s1c3
MMS I, Lecture 2
c 3
s 3
0
s 3 0
c 3 0
0 1
s2
s1c1
c1c2
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
DC
d3
b3
c3
a3
CB
3
.
d2
3
c2
.
b2
.
1
b2
2
2
1
c1
BA
a2
a1
b1
c1
MMS I, Lecture 2
b1
9