ADCS Review – Attitude Determination Prof. Der-Ming Ma, Ph.D. Dept. of Aerospace Engineering Tamkang University

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.


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


Spacecraft Coordinate Systems (Cont.) - Earth Centered Inertial (ECI) Coordinate System

ZECI: the rotation axis of the EarthECI is a inertial fixed coordinate system


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


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


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


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


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


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


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.


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



R


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


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


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)



Documents

ADCS Review – Attitude Determination Prof. Der-Ming Ma, Ph.D. Dept. of Aerospace Engineering Tamkang University