adcs review – attitude determination prof. der-ming ma, ph.d. dept. of aerospace engineering...
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ADCS Review – Attitude Determination
Prof. Der-Ming Ma, Ph.D.
Dept. of Aerospace Engineering
Tamkang University
Contents
• Attitude Determination and Control Subsystem
(ADCS) Function
• Spacecraft Coordinate Systems
• Spacecraft Attitude Definition
• Quaternions
• Assignment – Attitude Dynamics Simulation
2009/03/05 2Attitude Determination
ADCS Function The ADCS stabilizes the spacecraft and orients it in
desired directions during the mission despite the external disturbance torques acting on it: To stabilize spacecraft after launcher separation To point solar array to the Sun To point payload (camera, antenna, and scientific
instrument etc.) to desired direction To perform spacecraft attitude maneuver for orbit
maneuver and payloads operation This requires that the spacecraft determine its
attitude, using sensors, and control it, using actuators.
2009/03/05 3Attitude Determination
Spacecraft Coordinate Systems- Spacecraft Body Coordinate System
Z-axis (Nadir direction)
X-axis
Y-axisX-axis
Y-axis
Z-axis
Pitch: rotation around Y-axis
Yaw: rotation around Z-axis
Roll: rotation around X-axis
2. Euler Angle Definition
1. Spacecraft (ROCSAT-2) Coordinate System
2009/03/05 4Attitude Determination
Spacecraft Coordinate Systems (Cont.) - Earth Centered Inertial (ECI) Coordinate System
ZECI: the rotation axis of the EarthECI is a inertial fixed coordinate system
2009/03/05 5Attitude Determination
Spacecraft Coordinate Systems (Cont.) - Local Vertical Local Horizontal (LVLH) Coordinate System
Earth
x
z
x
z
x
z
x
z
LVLH is not a inertial fixed coordinate system
2009/03/05 6Attitude Determination
Spacecraft Attitude Definition Spacecraft Attitude: the orientation of the
body coordinate with respect to the ECI (or LVLH) coordinate system
Euler angle representation: [ ] : rotate angle around Z-axis, then
rotate angle around Y-axis, finally angle around X-axis
2009/03/05 7Attitude Determination
Attitude Determination 8 2009/03/05
Euler Angles - Yaw angle - It is measured in the horizontal
plane and is the angle between the xf and x1 axes.
Pitch angle - It is measured in the vertical plane and is the angle between the x1 and x2 (or xb) axes.
Roll angle - It is measured in the plane which is perpendicular to the xb axes and is the angle between the y2 and yb axes.
The Euler angles are limited to the ranges0 2
2 20 2
Attitude Determination 9 2009/03/05
Referring to the definitions of , , and , we obtain the following equations:
1
1
1
2 1
2 1
2 1
2
2
2
cos sin 0
sin cos 0
0 0 1
cos 0 sin
0 1 0
sin 0 cos
1 0 0
0 cos sin
0 sin cos
f
f
f
b
b
b
x x
y y
z z
x x
y y
z z
x x
y y
z z
Attitude Determination 10 2009/03/05
Performing the indicated matrix multiplication, we obtain the following result:
cos cos sin cos sin
( sin cos (cos cos
cos sin sin ) sin sin sin ) cos sin
(sin sin ( cos sin
cos sin cos ) sin sin cos ) cos cos
fb
fb
fb
xx
yy
zz
Attitude Determination 11 2009/03/05
The angular velocity is
12
ˆˆˆ kji
1
0
0
cos0sin
010
sin0cos
cossin0
sincos0
001
0
1
0
cossin0
sincos0
001
0
0
1
r
q
p
Attitude Determination 12 2009/03/05
The relationship between the angular velocities in body frame and the Euler rates can be determined as
coscossin0
sincoscos0
sin01
r
q
p
The equations can be solved for the Euler rates in terms of the body angular velocities and is given by
r
q
p
seccossecsin0
sincos0
tancostansin1
By integrating the above equations, one can determine the Euler angles.
2009/03/05 Attitude Determination 13
Quaternions The quaternion is a four-element vector q = [q1 q2 q3 q4]T that can be partitioned as
sin( / 2)
cos( / 2)
eq
where e is a unit vector and is a positive rotation aboute. If the quaternion q represents the rotational transformation from reference frame a to reference frame b, then frame a is aligned with frame b whenframe a is rotated by radians about e. Note that q hasThe normality property that ||q||=1.
2009/03/05 Attitude Determination 14
The rotation matrix from a frame to b frame, in terms of quaternion is
2 2 2 21 4 2 3 1 2 3 4 1 3 2 4
2 2 2 22 1 2 3 4 2 4 1 3 2 3 1 4
2 2 2 21 3 2 4 2 3 1 4 3 4 1 2
2( ) 2( )
2( ) 2( )
2( ) 2( )a b
q q q q q q q q q q q q
q q q q q q q q q q q q
q q q q q q q q q q q q
R
2009/03/05 Attitude Determination 15
Initialization of quaternions from a known direction cosine matrix is
4
4
4
(3,2) (2,3)
4
(1,3) (3,1)
4
(2,1) (1,2)
4
11 (1,1) (2,2) (3,3)
2
q
q
q
R R
R R
qR R
R R R
2009/03/05 Attitude Determination 16
The Euler angles can be obtained from the of quaternion
12 4 1 3
2 22 3 1 4 1 2
2 21 2 3 4 2 3
sin ( 2( ))
arctan 2[2( ),1 2( )]
arctan 2[2( ),1 2( )]
q q q q
q q q q q q
q q q q q q
2009/03/05 Attitude Determination 17
Quaternion derivatives
4 3 2
3 4 1
2 1 4
1 2 3
1
2
q q qp
q q qq
q q qr
q q q
q
or 0
01
02
0
r q p
r p q
q p r
p q r
q q
Assignment – Attitude Dynamics Simulation Consider a rectangular box of 10cm X 14 cm X 20cm as
shown in the figure with uniformly distributed mass of 2 Kg. The box has an initial angular velocity of 0.3 rad/sec and 0.05 rad/sec in the positive y and z directions, respectively. The center of mass of the box moves along a 10 m radius orbit with 0.3 rad/sec orbital speed. Neglect gravity effect and any external force or torqu Draw the attitude and the center of mass trajectories of the box
for 10 seconds. Do as much as you can to show the continuous motion of the box
at least for 10 seconds. (You may design an animation routine motion or use on-the-shelf software for the motion)
2009/03/05 18Attitude Determination
2009/03/05 19Attitude Determination