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14 Νοε 2013 (πριν από 3 χρόνια και 7 μήνες)

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Attitude Kinematics

Orbital Dynamics and Attitude Control

Dr. Rafael Ramis Abril

EuMAS
-
European Masters Course in Aeronautics and Space Technology

Departamento de Física Aplicada a la Ingeniería Aeronáutica

Universidad Politécnica de Madrid

OCTOBER 2007

Attitude Kinematics

Attitude Kinematics

i
0

j
0

k
0

k
1

j
1

i
1

“fixed” frame

“moving” frame

Attitude Kinematics

z
0

y
0

x
0

i
0

j
0

k
0

k
1

j
1

i
1

x
1

y
1

z
1

v

“fixed” frame

“moving” frame

The same vector has different components in different frames

Attitude Kinematics

i
0

j
0

k
0

k
1

j
1

i
1

“fixed” frame

“moving” frame

The same vector has different components in different frames

v

Attitude Kinematics

ROTATION MATRIX

Attitude Kinematics

Properties of rotation matrices

Attitude Kinematics

Euler angles

Attitude Kinematics

i
0

j
0

k
0

k
1

j
1

i
1

“fixed” frame

“moving” frame

e



Can we obtain the “moving” frame by rotating the “fixed” frame an angle



around direction defined by unit vector
e

?

Attitude Kinematics

i
0

j
0

k
0

k
1

j
1

i
1

“fixed” frame

“moving” frame

e



Can we obtain the “moving” frame by rotating the “fixed” frame an angle



around direction defined by unit vector
e

?

e
x1
=e
x0

e
y1
=e
y0

e
z1
=e
z0

E
1
=Q
10
E
0

E
1
=E
0
=E

E
T
E=1

(Q
01
-
I)E=0

Yes ! (Euler theorem)

This equation has

non
-
trivial solution

Attitude Kinematics

v
1

v
0

v
0
+
v
1

v
0
-
v
1



e

Gibbs's Vector

Attitude Kinematics

Composition of rotations

Frame 1

Frame 2

Frame 0

g
21

g
02

Some algebra ....

Attitude Kinematics

Quaternions

Attitude Kinematics

Vector rotation with quaternions

Attitude Kinematics

Rotation matrix and quaternion

Choose the biggest one !

Attitude Kinematics

Time derivative of attitude (rotation matrix)

Attitude Kinematics


'


'


'

p

q

r

Time derivative of attitude (Euler angles)

Attitude Kinematics

Time derivative of attitude (Gibbs's vector)

Attitude Kinematics

Time derivative of attitude (Quaternions)

Attitude Kinematics

Rotation

matrix



Direct

and

simple
.

Requires

9

numbers

(can

be

reduced

to

6
)

Euler

angles



Requires

3

numbers
.

Becomes

singular

for


=
0
.

Involves

trigonometric

functions
.

Quaternions



Requires

4

numbers
.

Can

be

normalized

in

a

simple

way
.

Usefull

to

control

systems,

Kalman

filters

...