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 2 Attitude 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 3 Attitude Determination

Spacecraft Coordinate Systems - Spacecraft Body Coordinate System Z-axis (Nadir direction) X-axis Y-axis X-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 4 Attitude Determination

Spacecraft Coordinate Systems (Cont.) - Earth Centered Inertial (ECI) Coordinate System Z ECI : the rotation axis of the Earth ECI is a inertial fixed coordinate system 2009/03/05 5 Attitude 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 6 Attitude 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 7 Attitude Determination

8 2009/03/05 Euler Angles -  Yaw angle  - It is measured in the horizontal plane and is the angle between the x f and x 1 axes.  Pitch angle  - It is measured in the vertical plane and is the angle between the x 1 and x 2 (or x b ) axes.  Roll angle  - It is measured in the plane which is perpendicular to the x b axes and is the angle between the y 2 and y b axes.  The Euler angles are limited to the ranges

Attitude Determination /03/05 Referring to the definitions of , , and , we obtain the following equations:

Attitude Determination /03/05 Performing the indicated matrix multiplication, we obtain the following result:

Attitude Determination /03/05 The angular velocity is

Attitude Determination /03/05 The relationship between the angular velocities in body frame and the Euler rates can be determined as The equations can be solved for the Euler rates in terms of the body angular velocities and is given by 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 = [q 1 q 2 q 3 q 4 ] T that can be partitioned as 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.

2009/03/05 Attitude Determination 14  The rotation matrix from a frame to b frame, in terms of quaternion is

2009/03/05 Attitude Determination 15  Initialization of quaternions from a known direction cosine matrix is

2009/03/05 Attitude Determination 16  The Euler angles can be obtained from the of quaternion

2009/03/05 Attitude Determination 17  Quaternion derivatives or

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 18 Attitude Determination

2009/03/05 19 Attitude Determination