Introduction to

Flight Dynamics System (FDS)

Pitch Jantawichayasuit

Flight Dynamics and Mission Planning System for THEOS-2 Small Satellite

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Content

• Introduction• Orbit Determination• Orbit Propagation

Introduction

Introduction

• Flight Dynamics System(FDS) is a system that responsible for anything

related to a dynamics (only moving not include rotation) of a satellite.

• FDS usually separate into two different types of operation

➢ Orbit prediction → daily operation

➢ Orbit Maneuver → trajectory calculation (required propulsion system)

• FDS requires GPS telemetry (position, velocity and time) of the satellite

in order to perform either FDS operation.

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Introduction → Orbit Prediction• Unfortunately, the measurement device usually have noise include in the mea

surement data. Therefore, GPS telemetry we receive may not reflect the truth

state (position, velocity and time) of the satellite.

• FDS has a process that deal with this noise in the measurement data in order

to get the truth state of the satellite in the specific time. This process is called

“Orbit Determination”.

• After we get the truth state of the satellite, we can perform a future state predi

ction of the satellite according to this truth state. This process is called “Orbit

Propagation”.

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Introduction → Orbit Prediction (without TLE)

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Introduction → Example of Orbit Propagation result

(Satellite future state prediction)

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Introduction → Orbit Prediction (with TLE)

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Introduction → Example of Two-Line Element (TLE)

A Two-Line Element set (TLE) is a data format encoding a list of orbital elements of an Earth-or

biting object for a given point in time, the epoch.

Therefore, for a ground system which needs satellite future states (position, velocity and time)

information to perform an operation, TLE propagator is required to propagate the TLE to get

satellite future states.

Orbital Element data at epoch

Time (epoch)

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Introduction → Orbit Perturbation

• In space apart from the Earth spherical potential (uniform

distribution), there are many factors that affect the motion

of satellite which also known as “Orbit Perturbation”

• There are 3 major categories of orbit perturbation1. Non-spherical geo-potential (zonal, tesseral, sectorial)

2. Third Body attraction (sun, moon, etc.)

3. Aerodynamic drag (atmospheric drag)

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Orbit Determination

Orbit Determination

• Responsible → determine the truth state of satellite from GPS measurement

data which include noise

• Orbit determination that FDS perform is a post processing (not real time)

• Knowledge required

1. Orbit Propagation

2. STM propagator

3. Least square estimation (non-linear)

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Orbit Determination → Linear Least Square

Linear Function:

y = at + b

Where

a and b are the coefficient in the linear function

y is the fitting value

t is the running value (usually a time)

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Orbit Determination → Linear Least Square

Cost Function :

Where

J is the cost function

ykm is the measurement value

tk is the running value which is time for this case

a and b are the coefficient in the linear function (variable we want to solve)

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Orbit Determination → Linear Least Square

• To obtain a and b that minimise cost function J, we can solve the equation by

0 =𝜕𝐽

𝜕𝑎=𝜕𝐽

𝜕𝑏

𝑘𝑡𝑘

2 𝑘𝑡𝑘

𝑘𝑡𝑘

𝑘1

𝑎𝑏

=

𝑘𝑡𝑘 𝑦𝑘

𝑚

𝑘𝑦𝑘

𝑚

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Orbit Determination → Linear Least Square

Function: y = at+b

Truth: a = 1, b = 2

Noise: mean = 0, SD = 10

Least Square Result:

a = 0.9973

b = 3.5682

Time

y

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Orbit Determination → Linear Least Square

Function: y = at+b

Truth: a = 1, b = 2

Noise: mean = 0, SD = 2

Least Square Result:

a = 1.0006

b = 2.0305

y

Time

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Orbit Determination → Non-Linear Least Square

Non-Linear Function:

y = asin(wt + b)

Where

a and b are the coefficient in the non-linear function

y is the fitting value

t is the running value (usually a time)

w is a known constant (in this case)

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Orbit Determination → Non-Linear Least Square

Cost Function:

Where

J is the cost function

ykm is the measurement value

tk is the running value which is time for this case

a and b are the coefficient in the non-linear function (variable we want to solve)

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Orbit Determination → Non-Linear Least Square

• To obtain a and b that minimise cost function J, we can solve the equation by

0 =𝜕𝐽

𝜕𝑎=𝜕𝐽

𝜕𝑏

• Unfortunately, we cannot easily separate the equation by a and b because

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Orbit Determination → Non-Linear Least Square

Therefore, we need to linearise the problem or “Linearisation”

Linearisation → the linear approximation of a function is the first order Taylor expansion

around the point of interest.

Where

Need to guess Need to guess

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Orbit Determination → Non-Linear Least Square

Therefore,

Notation

The cost function,

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Orbit Determination → Non-Linear Least Square

To obtain Δa and Δb that minimise cost function J, we can solve the equation by

𝑘𝑆𝑘

2 𝑘𝑆𝑘𝐶𝑘

𝑘𝑆𝑘𝐶𝑘

𝑘𝐶𝑘

2

𝛥𝑎𝛥𝑏

=

𝑘𝑆𝑘 𝛥𝑦𝑘

𝑚

𝑘𝐶𝑘 𝛥𝑦𝑘

𝑚

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Orbit Determination → Non-Linear Least SquareAfter we obtain Δa and Δb, we need to make our initial approximate solution a0

and b0 become more reliable approximation a1 and b1 by

Then using a1 and b1, do the same process again to obtain hopefully more relia

ble solution a2 and b2. This process will continue repeat until the solution satisfy

And

For the given sufficiently small value of ε

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Orbit Determination → Non-Linear Least SquareFunction: y = asin(wt+b)

Truth: a = 1, b = 2, w = 0.0025

Noise: mean = 0, SD = 0.25

Guess value:

a0 = 10, b0 = 1.9

Least Square Result:

a = 0.9642

b = 2.0079

Number of iteration: 4

Time

y

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Orbit Determination → Real Problem

• The cost function to minimise is

• Non-linear problem → Linearisation

Orbit Propagation STM Propagator

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Orbit Determination → Real ProblemFunction: F(x0,tk)

Truth: x0 = 20

Noise: mean = 0, SD = 5

Guess value:

x0 = 20.9838

Least Square Result:

x0 = 20.0709

Number of iteration: 5

Time

y

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Orbit Propagation

Orbit Propagation• Responsible → predict the future states of satellite in certain amount of time from

truth state determined from orbit determination

• TLE is also a truth state of the satellite, and any algorithm using a TLE as a data

source must implement one of the SGP models to correctly compute the state at

a time of interest.

• Knowledge required

1. Equation of motion for object orbiting the Earth (Two body)

2. Perturbation (geo-potential, atmospheric drag, third body, etc.)

3. Numerical integrator

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Orbit Propagation → Two bodyTwo body model: σ𝐹 = 𝑚 ሷ𝒓

−(𝐺𝑀𝐸𝑎𝑟𝑡ℎ𝑚

𝑟2)( Ƹ𝑟) = 𝑚 ሷ𝒓

ሷ𝒓 = 𝒂 = −𝜇

𝑟3𝐫

Where

𝒂 is 𝑎1 Ƹ𝑖 + 𝑎2 Ƹ𝑗 + 𝑎3 𝑘

𝒓 is 𝑟1 Ƹ𝑖 + 𝑟2 Ƹ𝑗 + 𝑟3 𝑘

𝑟 is 𝑟12 + 𝑟2

2 + 𝑟32

𝜇 is 398600.4418 𝑘𝑚3

𝑠2

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Orbit Propagation → Two body

We usually make 2nd order differential equation to be 1st order for performing

numerical integration

ቐሶ𝒓 = 𝒗

ሶ𝒗 = −𝜇

𝑟3𝒓

Numerical integrator may effect the accuracy of the propagation result. In

this case, we use RKF45 to be our numerical integrator.

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Orbit Propagation → Two bodyExample: Orbital Element = [7200, 0.001, 98, 30, 20, 30]

Time Semi-axis ecc inc raan omega true ano

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Orbit Propagation → J2 PerturbationJ2 Perturbation model:

ሷ𝒓 = 𝒂 = −𝜇

𝑟3𝐫 + 𝒂𝒑

Where

𝒂 is 𝑎1 Ƹ𝑖 + 𝑎2 Ƹ𝑗 + 𝑎3 𝑘

𝒓 is 𝑟1 Ƹ𝑖 + 𝑟2 Ƹ𝑗 + 𝑟3 𝑘

𝑟 is 𝑟12 + 𝑟2

2 + 𝑟32

𝜇 is 398600.4418 𝑘𝑚3

𝑠2

𝒂𝒑 is the perturbation term

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Orbit Propagation → J2 Perturbation

We usually make 2nd order differential equation to be 1st order for performing

numerical integration

ቐሶ𝒓 = 𝒗

ሶ𝒗 = −𝜇

𝑟3𝒓 + 𝒂𝒑

Numerical integrator may effect the accuracy of the propagation result. In

this case, we use RKF45 to be our numerical integrator.

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Orbit Propagation → J2 PerturbationThe J2 potential function (U2) is

𝑈2 =𝜇

𝑟

𝐽2𝑎𝑒2

2𝑟23𝑟3

𝑟− 1

Where J2 is 1.0826266835532*10-3

ae is 6378.137 km

And, we can get ap by

𝒂𝒑 =𝜕𝑈2𝜕𝒓

Where 𝒂𝒑 = 𝑎1 Ƹ𝑖 + 𝑎2 Ƹ𝑗 + 𝑎3 𝑘

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Orbit Propagation → J2 PerturbationTo calculate a1 a2 a3, it needs to partial derivative the U2 by each r1 r2 r3 respectively.

𝑎1 =𝜕𝑈2𝜕𝑟1

=𝜕𝑈2𝜕𝑟

𝜕𝑟

𝜕𝑟1+𝜕𝑈2𝜕𝑟3

𝜕𝑟3𝜕𝑟1

𝑎2 =𝜕𝑈2𝜕𝑟2

=𝜕𝑈2𝜕𝑟

𝜕𝑟

𝜕𝑟2+

𝜕𝑈2𝜕𝑟3

𝜕𝑟3𝜕𝑟2

𝑎3 =𝜕𝑈2𝜕𝑟3

=𝜕𝑈2𝜕𝑟

𝜕𝑟

𝜕𝑟3+𝜕𝑈2𝜕𝑟3

𝜕𝑟3𝜕𝑟3

After we finish differentiate and get the perturbation term (ap), we can add it to the two-

body model which will result in J2 perturbation model

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Orbit Propagation → J2 PerturbationExample: Orbital Element = [7200, 0.001, 98, 30, 20, 30]

Time Semi-axis ecc inc raan omega true ano

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Orbit Propagation → Comparison0 sec Rx (km) Ry (km) Rz (km) Vx (km/s) Vy (km/s) Vz (km/s)

2-Body 4.388 e+03 1.648 e+03 5.457 e+03 -4.605 -3.428 4.743

J2 4.388 e+03 1.648 e+03 5.457 e+03 -4.605 -3.428 4.743

600 sec Rx (km) Ry (km) Rz (km) Vx (km/s) Vy (km/s) Vz (km/s)

2-Body 9.804 e+02 -5.871 e+02 7.106 e+03 -6.386 -3.780 0.576

J2 9.826 e+02 -5.865 e+02 7.107e+03 -6.380 -3.780 0.582

3600 sec Rx (km) Ry (km) Rz (km) Vx (km/s) Vy (km/s) Vz (km/s)

2-Body -1.258 e+03 4.225 e+02 -7.080 e+03 6.330 3.780 -0.892

J2 -1.309 e+03 3.929 e+02 -7.080 e+03 6.309 3.801 -0.942

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Conclusion

Conclusion• Flight Dynamics System(FDS) is a system that responsible for anything relate

d to a dynamics (only moving not include rotation) of a satellite.

• Normal Operation

➢ Orbit Determination → get the truth state of satellite from GPS telemetry

➢ Orbit Propagation → predict the future state of satellite from the truth state

• Orbit Maneuver

➢ Trajectory calculation → determine when (position) and where (direction) to

firing a propellant

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