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Lecture Module 1
Topics: Introduction to Spacecraft Attitude Dynamics; Space environment;
Environmental Torques
Lectures 1 - 4
Introduction
A satellite is an object artificially placed in space for various mission objectives, for example,
remote sensing, telecommunication, weather monitoring, space exploration, space
surveillance, etc.. A majority or all of mission objectives require that a satellite be oriented
(post injection into an intended orbit from a launch vehicle) with respect to an object to capture
maximum information about the object continually over a long period of time. For example, a
communication satellite network requires a satellite to gather (receive) information from one
source and transmit information to the end user. On-board solar panels facing Sun to collect
maximum radiation energy to power a satellite may require orientation of the satellite in a
particular fixed direction. In a fixed orbit, at a fixed location, a satellite may be required to be
re-orientated towards another object of interest thus calling for an attitude maneuver (shifting
from one attitude to another) with the help of on-board control systems. Several mission
objectives thus require attitude (or orientation) of the satellite to be either kept fixed or
manipulated in a controlled manner. This calls for continuous attitude measurement of a
satellite. Attitude measurement sensors are especially designed for satellites with high
precision and accuracy, else we might end up facing a complete loss of signal (or blackout) for
our Television antenna or experience poor transmission leading to blurred images and/or
broken sound and disrupted data transfer. Any deviation from intended attitude read by sensors
has to be corrected either in a passive or in an active manner.
The attitude of a satellite in space can be disturbed by external torques present in space
environment where the satellite is placed. Torques offered by moving parts inside a satellite
can also change its attitude. While small attitude changes are related to stability properties and
small disturbance torques, large attitude changes, during a maneuver, for example, require large
disturbance torques. A satellite is usually equipped with various mechanisms to counter
undesired small and large disturbance torques. Space environment offering various torques can
either be seen as disturbance torques and detrimental to functioning of satellite or they can be
usefully manipulated for various purposes.
Prof. Nandan K Sinha
Aerospace Engineering
IIT Madras
2
Geometric and inertia properties of a satellite are factors which define its stability
characteristics. Through, rigid body motion analysis of a satellite using equations of attitude
dynamics and kinematics, a configuration with desired stability properties can be arrived.
Simulation via numerical integration of the equations of motion, eigenvalue analysis, and
energy based methods for motion analysis are important tools adopted for satellite motion
analysis.
Major goal of this lecture series is to introduce readers to various aspects related to satellite
attitude technology. The lectures are broadly categorised into five modules, covering:
1. Space environment
2. Spherical geometry, introduction to axes systems
3. Satellite attitude dynamics
4. Satellite attitude determination techniques
5. Satellite attitude stabilization and control
Following books were referred to while preparing this lecture series:
1. Spacecraft Attitude Determination and Control, Edited by James R Wertz, Kluwer
Academic Press, Boston, 1978.
2. Modern Spacecraft Dynamics and Control by Marshall H. Kaplan, John Wiley and
Sons, NY, 1976.
3. Spacecraft Dynamics and Control an Introduction by Anton H.J. De Ruiter,
Christopher J. Damaren, and James R. Forbes, John Wiley and Sons, NY, 2013.
Space Environment
Space environment offers many challenges to smooth operation of satellites. Environments
offer sometimes known and at other times unknown disturbances to satellite in term of torques
which changes satellite’s attitude from the desired orientation. Knowledge and modelling of
these attitude disturbance torques are important for accurate prediction of a satellite attitude (or
orientation) using mathematical models (or governing equations of motion). Such predictions
further help in designing control systems for attitude stabilization and maneuvering.
Prof. Nandan K Sinha
Aerospace Engineering
IIT Madras
3
Major sources of attitude disturbance torques are:
a. Earth’s magnetic and gravitational fields,
b. Solar radiation pressure,
c. Aerodynamic drag, and
d. Magnetic disturbance.
1.1 Gravity-gradient torque:
Gravity gradient torque is particularly significant for non-symmetrical objects due to variation
in Earth’s gravitational force over the object.
Geometric center
Center of mass

r'i
ri
dmi
RG
Ri
satellite of arbitrary shape
Earth
Figure 1.1: An arbitrary shaped satellite with distinct center of mass and geometric center
experiencing gravitational force of Earth.
The gravitational force dFi acting on the elemental mass dmi of the satellite is given by
dFi 
 dm i R i
Ri3
Torque about geometric center due this force is:
d N i  r i  d F i  (  r i )  d F i
'
Prof. Nandan K Sinha
(1.1)
Aerospace Engineering
IIT Madras
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The gravity gradient torque on the entire spacecraft is obtained by integrating Eq. (1.1). Thus,
N GG   (   r i ) 
'
 dm i R i
R
3
i



M ˆ
3
RG    3
2
RG
RG
 r
i


 Rˆ G r i .Rˆ G dm i
(1.2)
Where   GM E ; G is the gravitational constant of Earth and ME is the mass of Earth. M is
the mass of the satellite. When the center of mass and the geometric center coincide, i.e.,   0
, the expression for gravity gradient torque simplifies to
N GG 
3
RG3
 r
i




3
 Rˆ G r i .Rˆ G dm i  3 Rˆ G  I .Rˆ G
RG

(1.3)
Where I is the inertia tensor.
Assuming satellite body fixed axis system to be the principal axis system, so that,
I  IP
 I1
 0

 0
0
I2
0
0
0

I 3 
and R0 be the absolute distance between the satellite center of mass and, following Eq. (1.3),
components of Gravity Gradient Torque about satellite axes can be determined to be
G1 
3
( I 3  I 2 ) sin 2 cos 2 
2 R 03
G2 
3
( I 3  I 1 ) sin 2 cos 
2 R 03
G3 
3
( I 1  I 2 ) sin 2 sin 
2 R 03
(1.4)
 ,  in Eq. (1.4) are Euler angles.
Example 1.1: For a spherical body with double symmetry, I 1  I 2  I 3 , gravity gradient
torque G1  G 2  G 3  0 .
Prof. Nandan K Sinha
Aerospace Engineering
IIT Madras
5
Example 1.2: Assuming small angles  ,  , such that, sin( angle)  angle, cos( angle)  1 and
product of angles being of negligible value and insignificant,
G1 
3
3
3
( I 3  I 2 ) sin 2 cos 2  
( I 3  I 2 ).2 .1  3 ( I 3  I 2 )
3
3
2 R0
2 R0
R0
G2 
3
3
3
( I 3  I 1 ) sin 2 cos  
( I 3  I 1 ).2 .1  3 ( I 3  I 1 )
3
3
2 R0
2 R0
R0
G3 
3
3
( I 1  I 2 ) sin 2 sin  
( I 1  I 2 ).2 .  0
3
2 R0
2 R03
(1.5)
Example 1.3: For a body in circular orbit of radius R0, lateral velocity of the satellite
 R0 , and its angular orbital velocity also called orbital rate or frequency,
v
 0  v / R0   R03 , Eq. (x.5) can be re-written as
G1 
3
3
( I 3  I 2 ) sin 2 cos 2  
( I 3  I 2 ).2 .1  3 02 ( I 3  I 2 )
3
3
2 R0
2 R0
G2 
3
3
( I 3  I 1 ) sin 2 cos  
( I 3  I 1 ).2 .1  3 02 ( I 3  I 1 )
3
3
2 R0
2 R0
G3 
3
3
( I 1  I 2 ) sin 2 sin  
( I 1  I 2 ).2 .  0
3
2 R0
2 R03
(1.6)
Homework Exercise 1: For a cylindrical object with two plane of symmetry determine the
Gravity-gradient torque.
Some observations (from Eq. (1.3) with the approximation, that is,   0 ):

The torque is normal to the local vertical,

The torque is inversely proportional to the cube of the geometric distance, and

The torque vanishes for a spherically symmetric spacecraft.
For a spin stabilized satellite or a satellite with a composite of inertial and moving components,
orbital parameters also need to be considered to arrive at gravity-gradient torque [see Ref. 1 for
more details.].
Prof. Nandan K Sinha
Aerospace Engineering
IIT Madras
6
3.1 Solar radiation torque:
Satellite’s surface is subjected to solar radiation pressure (radiation force per unit area equal to
the vector difference between the incident and reflected momentum flux). Near Earth,
magnitude of this pressure is around 4.5  10 6 N / m 2 . Solar radiation pressure on a satellite or
spacecraft in Earth orbit is independent of the altitude of the satellite above Earth because of
the large distance from Sun. Three dominant factors determining the solar radiation torque on
a satellite are:

The intensity and spectral distribution of the incident solar radiation,

The geometry and optical properties of the satellite surface, and

The intensity of the Sun vector relative to the satellite.
In the following, the effect due to direct Solar radiation is considered.
Mean momentum flux P that is also the solar radiation pressure acting on satellite surface
normal to solar radiation is given by, P 
Fe
c
, where Fe is the solar constant which is
wavelength dependent and c is the speed of light.

Solar radiation incident on satellite surface
Figure 1.2: Solar radiation incident upon surface of a satellite at an angle  .
For the part of incident radiation that is absorbed by the surface, the differential radiation force
(for elemental area dA) which is momentum transferred per unit time is given by
d F absorbed   PC a cos SˆdA
Prof. Nandan K Sinha
(0    90 o )
Aerospace Engineering
(1.7)
IIT Madras
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Where Sˆ is the unit vector from satellite to Sun and C a is the absorption coefficient. Part of
the radiation which is specularly reflected (in the direction ( Sˆ  2 Nˆ cos  ) ) or diffused (in all
directions) from the satellite surface results in following differential forces,
d F specular  2 PC s cos 2 Nˆ dA
(0    90 o )
 2

d F diffuse  PC d   cos Nˆ  cos Sˆ  dA
 3

(1.8)
(0    90 o )
(1.9)
Where C s and C d are coefficients of specular and diffusion reflections, respectively. Thus,
total differential force is given by

1

F total   P  1  C s Sˆ  2 C s cos   C d
3


 ˆ
 N  cos dA
 
(0    90 o )
(1.10)
Where C a  C s  C d  1 . The solar radiation torque acting on the spacecraft is given by
N solar   R d F total
(1.11)
In Eq. (1.11), R is the distance between the center of mass of the satellite and the point at
which resultant of force due to solar radiation (integrated over the exposed satellite surface
area) act.
The other two major sources of external torques that a satellite can experience as
disturbance torques in space are Aerodynamic torque (in Low Earth orbit) and Magnetic torque
due to interaction between a magnetic component placed anywhere on the satellite and Earth’s
magnetic field. Some details about these torques follow.
4.1 Aerodynamic Torque
For spacecraft in low Earth orbit (below 400km altitude), the aerodynamic torque is a dominant
environmental disturbance torque acting on the spacecraft/satellite. The aerodynamic force
acting on the satellite is not due to relative wind hitting the satellite surface, but due to
momentum exchange due to molecules arriving at the surface. Therefore, continuum model of
atmosphere do not apply here.
Prof. Nandan K Sinha
Aerospace Engineering
IIT Madras
8
V
dA
C.M.
Figure 1.3: Aerodynamic force acting on a small elemental area of the satellite.
The force, d F aero , on a surface element dA with outward normal Nˆ is given by,
1
d F aero   C D V 2 ( Nˆ .Vˆ )VˆdA .
2
(1.12)
Vˆ is the unit vector in the direction of the relative velocity of the incident airstream (stream of
air molecules),  is the atmospheric density, and C D is the drag coefficient which is a function
of the local angle of attack. An expression for the aerodynamic torque acting on the spacecraft
thus can be arrived at as
N aero   r c  d F aero
(1.13)
where r c is the distance between the center of mass of the spacecraft and the satellite surface
element dA . For a spinning spacecraft the total velocity of the element dA with respect to the
airstream is given by
V  V c   rc
(1.14)
Where V c is the translational velocity of the center of mass of the spacecraft relative to the
airstream, and  is the angular velocity of the spacecraft. The expression for aerodynamic
torque including the spin motion of spacecraft can be obtained to be
N aero 


 Nˆ .  r c  Vˆ c  r c

1
1
C D Vc2  Nˆ .Vˆ c Vˆ c  r c dA  C D Vc  
dA
2
2
 Nˆ .Vˆ c   r c   r c 

Prof. Nandan K Sinha



Aerospace Engineering

(1.15)
IIT Madras
9
A satellite consisting of different parts of different shapes can be decomposed into some basic
shapes. Aerodynamic force on various basic shapes can be thus found easily from empirical
relations and aerodynamic torques due to each individual component integrated over the whole
body of satellite to arrive at the final expression for aerodynamic torque. Some errors due to
interference of different parts are expected in this way and must be accounted for. Shadowing
of one part due to another is another source of error, which must be accounted for.
Expressions for aerodynamic force for some simple geometric shapes are given below.
1
2

Sphere of radius R: F aero   C D V 2R 2Vˆ

1
Plane with surface area A: F aero   C D V 2 A Nˆ .Vˆ Vˆ , where Nˆ is the normal unit
2
 
vector.

Right circular cylinder of length L and diameter D:
 
2
1
F aero   C D V 2 DL 1  lˆ.Vˆ Vˆ
2
,
(1.16)
where lˆ is the unit vector along the length of the cylinder.
4.2 Magnetic Disturbance Torque
Magnetic disturbance torques results from interaction between spacecraft’s residual magnetic
field and geomagnetic field. Sources of spacecraft magnetic field are:

Eddy currents

Hysteresis

Spacecraft’s magnetic moments
The magnetic disturbance torque due to spacecraft magnetic moment is given by
N mag  M  B
(1.17)
Where M is the total magnetic moment (in A.m2) due to permanent and induced magnetism
and spacecraft generated current looks, B is the geocentric magnetic flux density (Wb/m2).
Prof. Nandan K Sinha
Aerospace Engineering
IIT Madras
10
Torques created due to eddy currents and hysteresis are attributed to spacecraft’s spinning
motion in the geomagnetic field. Expression for this torque is given by
N Eddy  k e   B   B
(1.18)
Where  is the spacecraft’s angular velocity vector and ke is a constant coefficient which
depends upon spacecraft geometry and conductivity. ke for some geometric figure of satellite
(or its parts) with conductivity  are:
2 4
r t
3

Thin spherical shell of radius r, thickness t: k e 

Circular loop of radius r and cross-sectional area A located in a plane containing the
spin axis: k e 

Thin

4
walled
r 3 A
cylinder
with
length
L,
radius
r,
and
thickness
t:
2t
L

k e  r 3 Lt 1  tanh 
L
2t 

Magnetic torque due to hysteresis only appreciable for very elongated soft magnetic material
is given by
N Hysterisis 
 E H
 2 t
(1.19)
Where  t is the time over which the torque is being evaluated and E H is the energy loss over
one rotation period given by
E H  V  H .d B
(1.20)
V is the volume of the permeable material, H is the magnetic field of the surrounding medium,
and dB is the induced magnetic induction flux in the material.
Prof. Nandan K Sinha
Aerospace Engineering
IIT Madras

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