Visual Modeling of the Satellite Dynamics in the

Visual Modeling of the Satellite Dynamics in the LEOSWEEP Project Dmitry A. Khramov Institute of Technical Mechanics NASU & SSAU

Research Subject Task 5.3.3 "Dynamics, Kinematics and Environment models". Given the orbital position and velocity of the shepherd as well as its attitude, we need to compute forces: main gravity force; J2 perturbing force; perturbing gravitational forces from Moon and Sun; aerodynamic drag force; solar radiation force (you will need to establish if the spacecraft is in eclipse or not);

torques: main gravity torque; atmospheric drag torque; solar radiation torque.

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Orbital Motion: Equations The equations of motion of the satellite center of mass ¨rI = −

µrI + fI r3

Perturbations I I f I = fnc + fSI + fM + faI + fpI I fnc – noncentral part of the Earth’s gravitational field; I fS – gravitational force from the Sun; I fM – gravitational force from the Moon; I fa – atmospheric drag; fpI – solar radiation force.

I – Geocentric Inertial Frame (GCI) in J2000.

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Orbital Motion: Simulink model

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Non-Spherical Gravity

The strongest perturbation due to the Earth’s shape arises from J2:

  I fnc

≡

I fJ2

3 µ = − J2 2 2 r



R⊕ r

2    

1−5  1−5  3−5




z 2 r 
z 2 r 
z 2 r

x r y r z r

    

Landis Markley F., Crassidis J. L. Fundamentals of Spacecraft Attitude Determination and Control, Springer, New York, 2014, p. 387

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Perturbing gravitational force from the Sun and the Moon

fiI

 = GMi

rIi − rI rIi − |rIi − rI |3 ri3

 .

GMi – Gravitational coefficient∗mass of i-th body; i = Sun, M oon. We used simple equations for the solar and lunar ephemerides that are accurate to about 0.1–1% from [Montenbruck&Gill], but... Montenbruck O., Gill E. Satellite Orbits – Models, Methods and Applications, Springer-Verlag, Heidelberg, 2005, p. 71–72.

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Issue: Long Formulas

Moon’s gravity

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Issue: Long Formulas

Moon’s position 6

Issue: Long Formulas

Fundamental Arguments of the Moon

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Atmospheric Drag

1 A faI = − CD ρvr vr , 2 m vr = v − ω ⊕ × r; ω ⊕ — the Earth’s angular velocity vector.

Density Models Simple exponential decay. Detail NRLMSISE-00.

Landis Markley F., Crassidis J. L. Fundamentals of Spacecraft Attitude Determination..., Table D.1.

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Solar Radiation Pressure

Solar radiation pressure acceleration (normal to the satellite’s surface coincides with the direction to the Sun): fpI = −νP CR

A r 2 3 AU , m r

Eclipse condition (without penumbra): the spacecraft is in the Earth’s shadow if and only if q 2, rI · eES < − r2 − R⊕ eES — the unit vector in the direction from the Earth to the Sun.

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Verification

Nonperturbed Ceplerian Motion: by direct calculation from the known formulas. The impact of the J2 : from the known formulas for the precession and rotation of the orbit.

Unit testing of perturbations models The MATLAB model of perturbation was created. The results of the calculations for MATLAB model and Simulink block for the random initial data set has been compared.

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Estimation of the Perturbations’ Magnitudes Orbit: a = 7078.0 km, e = 0.001, i = 98.57◦ . Satellite: Area = 5.0 m2 , mass = 1000 kg, CD = 2.3, CR = 1.3. Maximum position error within 100 min. propagation interval, m J2,0 J2,2 J4,4 J10,10 Sun Moon Drag Sol. Rad. 606.2 237.5 162.1 27.5 2.7 5.6 1.2 0.6 Maximum position error within 1440 min. propagation interval, m J2,0 J2,2 J4,4 J10,10 Sun Moon Drag Sol. Rad. 5288.4 3199.3 2237.3 549.3 32.8 75.1 297.5 8.3 The influence of the Moon’s gravity is impractical to consider.

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Attitude Motion: Dynamics Reference Frames Оrbital Frame – Local Vertical/Local Horizontal; Body Frame – axis is directed along the principal central axis of inertia of the satellite. The Equations of Motion B ω˙ B BI = J


−1


B B MB − ω B BI × J ω BI ,

where ω B BI — the angular velocity of the Body Frame relative to the Inertial Frame in projections on the Body Frame’s axis. B B MB = MB g + Ma + Mp

MB g — gravity torque; MB a — atmospheric drag torque; MB p — solar radiation torque. 11

B Attitude Motion: Kinematics and ω B BI → ω BO

The orientation of the Body Frame relative to the Orbital Frame is described by a quaternion qBO = q0 + q1 i1 + q2 i2 + q3 i3 = [ q0 q1−3 ]T , The Kinematic Equations 1 dq = dt 2



ω T q1−3 −ω q0 ω + q1−3 × ω

 .

where q ≡ qBO , ω ≡ ω B BO . The ω B is calculated as BO B O ωB BO = ω BI − ABO ω OI ,

where ω O OI — the angular velocity vector of the Orbital Frame relative to the Inertial Frame.  2  q0 + q12 − q22 − q32 2(q1 q2 + q0 q3 ) 2(q1 q3 − q0 q2 ) q02 + q22 − q12 − q32 2(q2 q3 + q0 q1 )  ABO =  2(q1 q2 − q0 q3 ) 2 2(q1 q3 + q0 q2 ) 2(q2 q3 − q0 q1 ) q0 + q32 − q12 − q22 12

Attitude Dynamics Block

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Gravity Torque

MB g =

3µ n × J B n, r3

n — 3rd column vector from the ABO matrix. Verification: the comparison with the known formula for the frequency of a rigid body oscillation in the orbital plane under the action of gravitational torque was performed.

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Atmospheric Drag Torque and Solar Radiation Torque The spacecraft surface is modelled as a sphere. Aerodynamic drag torque B B MB a = r∆ × Fa

rB ∆ – the distance between the center of mass and center of pressure of the spacecraft; FaB – aerodynamic drag force. Solar radiation pressure torque B B MB p = r∆ × Fp

FaB – solar radiation force. 15

Simulator

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Issues Caution! Long formula Atmosphere Density Earth’s Gravity Field Geomagnetic Field We need ready blocks! Solutions (after the LEOSWEEP project!): C/C++ server

Pro high speed of execution easy modelling

Scilab/Xcos

similar to Simulink has necessary libraries (CelestLab & Aerospace Toolbox) converts the model into C-code

Contra requires high qualification of all developers requires a high centralization of the development ?

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Xcos Simulator

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Conclusions

Some disturbances is not significant for our problem. In particular, the gravitational force from the Moon. Do not forget about the geomagnetic field. The problem of creating new blocks will become more acute as the development of the models. A promising approach is the use of Scilab/Xcos.

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Any Questions?

dkhramov.dp.ua

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