Attitude Dynamics and Control

Attitude Determination and Control of Satellites

Introduction

• Attitude Determination and Control Subsystem(ADCS)

- Senses the orientation of spacecraft relative to reference plane usually inertial

- Stabillizes the spacecraft

- Orients the spacecraft in desired direction

Attitude control

Needed because

• Payload requirements

Eg. Focusing the satellite camera to a particular direction

• Communication requirements

Pointing the antenna towards ground

• Power system requirements

Tracking the sun to achieve maximum power generation

Components of ADCS

• Sensors- To determine the orientation of the satellite

• Algorithms-To calculate the deviation from the desired orientation and to generate actuation command to counter the deviation

• Actuators-To act upon the signals given by the control algorithms and to produce the necessary torques

CHANGE IN ATTITUDE

Satellite tends to change its orientation because of environmental torques

– Aerodynamic torque

– Solar radiation pressure

– Gravity gradient torque

– Interaction of Satellite electronics with earth’s magnetic field

Attitude Determination • Earth Sensor (horizon sensor)

– Use IR to detect boundary between deep space & upper atmosphere

– Typically scanning (can also be an actuator) • Sun Sensor • Star Sensor

– Scanner: for spinning S/C or on a rotating mount – Tracker/Mapper: for 3-axis stabilized S/C

• Tracker (one star) / Mapper (multiple stars)

• Inertial Measurement Unit (IMU) – Rate Gyros (may also include accelerometers)

• Magnetometer – Requires magnetic field model stored in computer

Attitude Control

• Actuators come in two types

– Passive

• Gravity Gradient Boom

• Spinning

– Active

• Thrusters

• Wheels

• Gyros

• Torque Rods

Reference frames

• Earth Centered Inertial Frame: Non rotating reference frame denoted by .

Origin : Center of Earth

x –axis : Points towards vernal equinox

y – axis: 90⁰ east in the equatorial plane

z - axis: Extends through North pole

• Earth-Centered Earth Fixed (ECEF) Frame:

x and y axes rotate with the hemisphere relative to ECI frame

origin: center of earth

x-axis: Points toward the intersection of Greenwich meridian and equator

z-axis: Extends through the North pole and rotation is about z axis

y-axis: Completes right handed system.

• Orbit Frame:

Origin: Center of mass of the satellite

x-axis: Points in the direction of motion tangentially to the orbit

z-axis: Points Nadhir (center of Earth)

y-axis: Completes right handed system

• Body frame:

Origin: Coincides with the center of mass of the satellite.

x and y axes: Coincides with the orbit frame axes when the satellite has an attitude of 0⁰ in roll, pitch and yaw.

z-axis: Nadhir side of satellite

Rotation Matrix

Rotation matrix is a description of the rotational relationship between two reference frame.

The rotation matrix R from frame a to b is denoted as Rᵇₐ.

Rotation of a vector from one frame to another frame be given using Rᵇₐ as

• Rotational matrices are denoted by SO(3)

The orientation of satellite is described using rotation matrix

Relation between angular velocity and rotation matrix, and its derivative is

• Attitude Representation:

Most widely used methods:

-Euler angles

-Unit Quaternions

Euler Angles:

-Uses roll (ф), pitch (ϴ), yaw (ψ)

• Transforming the body from the inertial axes to rotated body fixed axes using a rotation matrix:

Where I, j, k are chosen rotation axes and

are transposes of the elementary rotation matrices about their respective axes.

• Unit Quaternions:

-Choose euler axis

- Rotate the object by an angle ф from inertial frame about the euler axis.

Inertia Matrix

Inertia matix Iₒ about origin is defined as

Dynamics of Satellite

Consider satellite as rigid body h = r x p h= IѠ Where h – angular momentum r – position vector p – linear momentum I – moment of inertia Ѡ – angular velocity

Mathematical dynamic model of satellite:

: angular velocity of the body frame with respect to the inertial frame in the body frame

: Torques acting on the satellite in body frame both internal and external.

Modeling of Reaction wheel

• Torque produced by reaction wheel:

: Torque produced by reaction wheel in body frame

is angular momentum of reaction wheels

is friction of reaction wheel

Mathematical Model of spacecraft Attitude System

- (1)

- (2)

- (3)

where is attitude angular velocity expresses in spacecraft’s body fixed reference frame to inertial frame

: rotation inertial matrix

: angular momentum of spacecraft

is attitude angles described by Euler angles and

References

• LI Long, Hou Jianwen. Adaptive Sliding Mode Based Controller Design for Spacecraft Attitude Stabilization, IEEE transactions, 2015

• Attitude determination and control by James R Wertz

• Spacecraft Dynamics and Control by Marcel J. SIDI,1997