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
Attitude Determination
2
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.
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Attitude Determination
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Spacecraft Coordinate Systems
- Spacecraft Body Coordinate System
X-axis
Z-axis
Yaw: rotation around Z-axis
Pitch: rotation around Y-axis
Y-axis
Roll: rotation around X-axis
X-axis
Y-axis
Z-axis (Nadir direction)
2. Euler Angle Definition
1. Spacecraft (ROCSAT-2) Coordinate System
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Attitude Determination
4
Spacecraft Coordinate Systems (Cont.)
- Earth Centered Inertial (ECI) Coordinate
System
ZECI: the rotation axis of the Earth
ECI is a inertial fixed coordinate system
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Attitude Determination
5
Spacecraft Coordinate Systems (Cont.)
- Local Vertical Local Horizontal (LVLH)
Coordinate System
x
z
x
Earth
z
z
x
z
x
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LVLH is not a inertial fixed coordinate system
Attitude Determination
6
Spacecraft Attitude Definition
Spacecraft Attitude: the orientation of the
body coordinate with respect to the ECI
(or LVLH) coordinate system
 Euler angle representation:


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[   y ] : rotate y angle around Z-axis, then
rotate  angle around Y-axis, finally  angle
around X-axis
Attitude Determination
7

Euler Angles  Yaw angle y - 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 ranges
0  y  2




2
2
0    2
Attitude Determination
8
2009/03/05

Referring to the definitions of y, , and , we obtain the
following equations:
 x1   cosy
  
 y1     siny
z   0
 1 
siny
cosy
0
0  x f


0  y f
1   z f





 x2  cos  0  sin    x1 
  
 y 
y

0
1
0
 2 
 1
 z   sin  0 cos    z 
 2 
 1 
0
0   x2 
 xb  1
  
 y 
y

0
cos

sin

 b 
 2 
 z  0  sin  cos    z 
 b 
 2 
Attitude Determination
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2009/03/05

Performing the indicated matrix multiplication, we
obtain the following result:
 xb   cosy cos 
  
   ( siny cos 
 yb     cosy sin  sin  )
  
   (siny sin 
 z    cosy sin  cos  )
 b 
Attitude Determination
siny cos 
(cosy cos 
 siny sin  sin  )
( cosy sin 
 siny sin  cos  )
10
 sin    x f


cos  sin    y f


cos  cos    z f








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
The angular velocity is
  ˆi  ˆj2  kˆ 1y
0
 p  1  1






   q   0   0 cos 
 r  0  0  sin 
0
0  cos 
1


  0 cos  sin   0
 0  sin  cos   sin 


Attitude Determination
11
0  0 
 
sin   1
cos   0
0  sin   0
 
1
0  0y
0 cos   1
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The relationship between the angular velocities in body
frame and the Euler rates can be determined as
0
 sin    
 p  1
 q    0 cos  cos sin    
 
  
 r   0  sin  cos cos   y 
The equations can be solved for the Euler rates in
terms of the body angular velocities and is given by
   1 sin  tan  cos  tan    p 
 
  
cos 
 sin    q 
    0
y   0 sin  sec  cos  sec    r 
 
By integrating the above equations, one can determine
the Euler angles.
Attitude Determination
12
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
Quaternions

The quaternion is a four-element vector q = [q1 q2 q3 q4]T that
can be partitioned as
e sin( / 2) 
q

cos(

/
2)


where e is a unit vector and  is a positive rotation about
e. If the quaternion q represents the rotational
transformation from reference frame a to reference
frame b, then frame a is aligned with frame b when
frame a is rotated by  radians about e. Note that q has
The normality property that ||q||=1.
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Attitude Determination
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
The rotation matrix from a frame to b frame, in terms of
quaternion is
R a 2b
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 q12  q42  q22  q32

  2(q1q2  q3q4 )
 2(q1q3  q2 q4 )

2(q1q2  q3q4 )
q22  q42  q12  q32
2(q2 q3  q1q4 )
Attitude Determination
2(q1q3  q2 q4 ) 

2(q2 q3  q1q4 ) 
q32  q42  q12  q22 
14

Initialization of quaternions from a known direction cosine
matrix is
R (3, 2)  R (2,3)




4q4


R (1,3)  R (3,1)




4q4

q


R (2,1)  R (1, 2)


4
q
4


1

1

R
(1,1)

R
(2,
2)

R
(3,3)


2

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Attitude Determination
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
The Euler angles can be obtained from the of quaternion
  sin 1 (2(q2 q4  q1q3 ))
  arctan 2[2(q2 q3  q1q4 ),1  2(q12  q22 )]
y  arctan 2[2(q1q2  q3q4 ),1  2(q22  q32 )]
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Attitude Determination
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
Quaternion derivatives
 q4
q
1 3
q
2  q2

 q1
or
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q3
q4
q1
q2
q2 
 p

q1   
q

q4 
  r 
q3 
r q
 0
 r 0
p
1
q
2  q p 0

  p q r
Attitude Determination
p
q 
q
r

0
17
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)
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