Course Appunti - Oral Exam Questions

R



 3Gmt

 (1.38)

−

− (I I )c c

I ω̇ + (I I )ω ω = x z 1 3

y y x z x z 3

R

 3Gmt



 − −

I ω̇ + (I I )ω ω = (I I )c c

 z z y x y x y x 2 1

 3

R

These equations can be manipulated by incorporating terms derived directly from the definitions of relative

rotation between the body and LVLH frames. This accounts for angular velocity as a combination of the

angular velocity in the LVLH frame and the relative velocity of the LVLH frame with respect to the body

frame, equal to the Earth’s rotation rate along the pitch axis. The definition of ”c” derives from the alignment

of the LVLH axes with the radial direction for the yaw axis.  

 

 

 −a

ω 1 a α̇

x

z y

x





 −a

=

ω 1 a α̇

 

   

 y

z x

y

  

   





 −a

ω a 1 α̇ + n

 z

y x

z (1.39)



    

−a

c 1 a 1

1 z y





 −a

=

c 1 a 0

 

    

 2 z x

 

    



 −a

c a 1 0

 3 y x

By inserting these into the linearized equations and simplifying, we obtain the new system:

 2 2

− − − −

I α̈ + (I I I )nα̇ + (I I )n α = α̈ + (K 1)nα̇ + K n α = 0

x x z y x y z y x x y y y x



 2 2

− − − (1.40)

I α̈ + (I + I I )nα̇ + 4n (I I )α = α̈ + (1 K )nα̇ + 4K n α = 0

y y x y z x z x y y r x r y

 2 2

−

I α̈ + 3n (I I )α = α̈ + 3n K α = 0

 z z y x z z p z

From the third, decoupled equation, we derive the pitch stability condition, requiring the y-axis inertia to be

greater than the x-axis inertia. Using the definition of ”K”, this excludes a half-plane from the stability region.

Stability conditions for roll and yaw can be derived from the characteristic equation of the first two equations

of the system, which, once simplified, has the form:

4 2 2 2

s + n s (1 + 3K + K K ) + 4n K K = 0 (1.41)

r r y r y

6

Chapter 1. Attitude Dynamics, Kinematics, Disturbances

Analyzing the characteristic equation provides roll and yaw sta-

bility conditions, ensuring consistent signs between the quadratic

term and the other two terms (conditions one and three) as well

as damping of the linear term (condition two):

 c > 0 √

 ( p

 1 + 3K + K K > 4 K K

2 −

−b ±

 b 4c r r y r y

→

< 0 K , K > 0

2 r y



 2 −

b 4c > 0

 (1.42)

Examining the results on a stability diagram reveals that the

only two stable zones correspond to configurations where the z-

axis has maximum or minimum inertia, constrained by additional

conditions. Figure 1.1: Stability Diagram

Why do the different components of the gravity gradient have different transient

1.3.2 SRP - 2022(x3) - 2023 - 2024

Radiation impacting the external surfaces of the satellite carries its own energy and, according to the principle

of momentum conservation, transfers part of its momentum to the spacecraft during interaction. The primary

source of this radiation is solar, but especially for low orbits, the contribution from radiation emitted or reflected

by the Earth’s surface becomes significant. The average pressure exerted by the radiation is described by a

formula that depends on the power of solar radiation per unit area and the speed of light:

W

F e with F = 1358 (1.43)

P = e 2

c m

The interaction between radiation and surfaces can be classified into three main types, with the forces generated

depending on three coefficients governing the surface behavior. The sum of these coefficients must equal one:

• Specular Reflection: Behavior of a white body, with mirror reflection. The incident radiation transfers

twice its momentum in the direction normal to the surface.

2

−2P ·

dF = k cos ϑ N̂ dA (1.44)

spec

spec

• Absorption: Behavior of a black body, with total absorption of the incident radiation. In this case, all

momentum is transferred in the direction of incidence.

−P ·

dF = k cosϑ ŜdA (1.45)

abs

abs

• Diffisive Reflection: Intermediate behavior where radiation is not specularly reflected but instead diffuses

spherically around the incident point, transferring part of the energy in the normal direction and all energy

in the direction of incidence.

2

−P · ·

dF = k cosϑ N̂ + cosϑ Ŝ dA (1.46)

dif f

dif f 3

By discretizing all satellite surfaces and summing their effects (considering only those exposed to radiation,

where the angle of incidence is between 0 and 90 degrees), the complete formula for the forces acting on the

satellite can be derived. This formula is subsequently used to calculate the torque induced on the spacecraft:

1 X

i i i

−A − → ∧

F = P (1 k )

Ŝ + 2 k cosϑ + k N̂ cosϑ T = r F (1.47)

i i i

spec spec dif f

i i i

3

1.3.3 Atmospheric Drag - 2022(x2) - 2024

The air density at orbital altitudes is not exactly zero; therefore, especially at lower altitudes, satellites ex-

perience aerodynamic drag forces that, in turn, induce disturbance torques on the spacecraft. Modeling this

aerodynamic force assumes the total transfer of the kinetic energy of air particles to the surfaces they en-

counter, with aerodynamic coefficients semi-empirically determined around a value of 2. Consequently, the

general formula for aerodynamic drag on each surface is as follows:

1 X

2 · → ∧

dF = C ρ(h, t)v (v̂ n̂ ) A v̂ T = r F (1.48)

D r i i r

r

i i i

i i

2 i 7

The terms requiring additional attention in this formula are air density, which necessitates an atmospheric

model, and the relative velocity between the surfaces and air particles, derived from considerations of the satel-

lite’s motion. Starting with the atmospheric model, one option is linear interpolation between experimentally

tabulated values every 10 kilometers of altitude. An alternative would be a layered atmospheric model, where

each layer is represented by its own exponential model, offering greater precision. Regarding relative velocity,

it is first expressed in the inertial frame as the relative velocity of the satellite (treated as a point mass) and

the air, accounting for the planet’s rotation (and the atmosphere co-rotating with it):

 

ẋ + ωy

inertial

inertial inertial −

− × ẏ ωx

= v ω R = (1.49)

v ⊕  

r 0 ż

This expression must then be transformed into the body axes through the DCM and further combined with the

intrinsic rotation effects of the satellite around its center of mass for each surface, yielding the value to be used

in the general aerodynamic drag formula.

body inertial G

×

= A v + ω r (1.50)

v r r sat i

B/N

i

2. Sensors and Actuators

2.1 Attitude Sensors

2.1.1 Magnetometer - 2022(x3) - 2024(x2)

A magnetometer is a sensor capable of providing a vector measurement of the magnetic field passing through

it. This information is useful when combined with data from other sensors for determining the attitude of a

satellite. The most commonly used type of sensor is the fluxgate magnetometer, as its measurement is based on

the magnetic field flux through a coil of conductive material. The sensor structure consists of two ferromagnetic

cores, around which two different coils are wound: the primary coil and the secondary coil, each with distinct

structures and functions.

• Secondary: measures the magnetic field flux of the two cores by detecting the voltage induced in the coil:

dφ(B + B )

1 2

−

V = (2.1)

S dt

• Primary: The primary coil generates an induced magnetic field with a triangular variation pattern (using

a triangularly varying current) and opposite polarity in the two cores, so the total sum is always zero.

When the magnetometer is exposed to an external magnetic field, this field adds to the induced fields,

creating an imbalance that can be detected. dH

−

V = (2.2)

p dt

In general, this behavior is nonlinear and subject to hysteresis, which is neglected by imposing a saturation

limit on the system. When exposed to an external magnetic field, the induced magnetic field will appear shifted

by the intensity of the external field, while being constrained by the saturation limits:

(a) (b)

This shift causes an imbalance in the saturation of the two magnetic fields (saturation will occur first in

the positive or negative direction depending on the sign of the external field). This imbalance creates time

8 Chapter 2. Sensors and Actuators

intervals during which one field is saturated while the other varies. The variation is detected by the primary

coil, producing discrete voltage pulses. The current in the primary coil, which generates the induced magnetic

field, must be controlled so that, given the expected value of the magnetic field, the system does not remain

completely saturated in one direction or the other, as this would prevent measurement. The external magnetic

field is determined by identifying the time intervals between the voltage pulses:

−

H = (1 2k)H (2.3)

ext D

To achieve a complete measurement of the magnetic field, three sensors are typically placed along the body axes.

However, the sensor has limited functionality, as it is highly susceptible to errors caused by the variability of the

magnetic field due to external factors (such as electronic activity on the satellite). Additionally, the measurement

relies on a geodetic reference model, which requires rotation to align with the sensor’s body frame. The primary

use of magnetometers is associated with magnetic torquers, which require knowledge of the external magnetic

field to operate. However, the magnetometers and torquers cannot be activated simultaneously, as the magnetic

action of the torquers would interfere with the sensor measurements, rendering them unusable.

2.1.2 Gyroscope - 2022 - 2023 - 2024(x2)

The gyroscope is the sensor used to measure the angular velocity of a satellite and can be built using various

technological approaches. The original design involves a high-speed rotating rotor supported by a mechanism

capable of rotating along an axis perpendicular to the rotor’s spin axis. In this setup, a transducer detects the

precession of the gyroscope’s spin axis. By app