Optimization Control SK Box Manoeuvres for GEO Satellites

Global Journal of Science Frontier Research: A Physics and Space Science Volume 15 Issue 7 Version 1.0 Year 2015 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: 2249-4626 & Print ISSN: 0975-5896

Optimization Control SK Box Manoeuvres for GEO Satellites using Electric Thrusters (OCSKBOX) By Louardi Beroual & Djamel Benatia University of Batna, Algeria Abstract- The study presented in this paper deals with an optimization control station keeping box manoeuvers for geostationary satellites equipped with electric propulsion. The station keeping box (SKBOX) represented the maximum permitted values of the excursion of the satellite in longitude and latitude. It can be represented as a pyramidal solid angle, whose vertex is at the centre of the earth, within which the satellite must remain at all times. In this work, the station keeping box is defined by the two half angles at the vertex, one within the plan of the equator (E-W width), and the other in the plan of the satellite meridian (N-S width). A number of different techniques are available for the numerical solution of the station keeping box problem. In this work we will consider the direct method for solution of continuous optimal control problem. Simulation results have demonstrated that the spacecraft can be tightly controlled within station keeping box. Keywords: geostationary satellites, SKBOX, box-limit, electric propulsion, specific impulse, quadratic programming. GJSFR-A Classification : FOR Code: 290207

Optimi zationControlSKBoxManoeuvresforGEOSatellitesusingElectricThrustersOCSKBOX

Strictly as per the compliance and regulations of :

© 2015. Louardi Beroual & Djamel Benatia. This is a research/review paper, distributed under the terms of the Creative Commons Attribution-Noncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/), permitting all non commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Optimization Control SK Box Manoeuvres for GEO Satellites using Electric Thrusters (OCSKBOX)

Louardi Beroual α & Djamel Benatia σ

Abstract- The study presented in this paper deals with an eccentricity must be kept sufficiently small for a

optimization control station keeping box manoeuvers for spacecraft to be tracked with a non-steerable antenna. 201 geostationary satellites equipped with electric propulsion. Also for Earth observation spacecraft for which a very r

The station keeping box (SKBOX) represented the ea

repetitive orbit with a fixed ground track is desirable, the Y maximum permitted values of the excursion of the satellite in eccentricity vector should be kept as fixed as possible. longitude and latitude. It can be represented as a pyramidal A large part of this compensation can be done by using 7 solid angle, whose vertex is at the centre of the earth, within which the satellite must remain at all times. In this work, the a frozen orbit design, but for the fine control station keeping box is defined by the two half angles at the manoeuvres with thrusters are needed [4]. vertex, one within the plan of the equator (E-W width), and the Electric propulsion engines are more efficient

other in the plan of the satellite meridian (N-S width). then chemical ones: they require significantly less V A number of different techniques are available for the propellant to produce the same overall effect, for VII numerical solution of the station keeping box problem. In this example a specific increase in spacecraft velocity. The ue ersion I work we will consider the direct method for solution of s propellant is ejected up to 20 times faster than from s continuous optimal control problem. Simulation results have I chemically-based thrusters, and, thus the same demonstrated that the spacecraft can be tightly controlled within station keeping box. propelling force is obtained with a log less propellant, XV Keywords: geostationary satellites, SKBOX, box-limit, the forces EP produces can be applied continuously for electric propulsion, specific impulse, quadratic very long periods-months or even years. programming. To maintain the satellite within box, orbit

corrections are achieved by applying velocity impulses )

A I. Introduction to the satellite at a point in the orbit. These impulses are ) generated by activating the thrusters that are mounted n astrodynamics orbital station-keeping is the on the satellite as part of the propulsion subsystem. orbital maneuvers made by thruster burns that are needed to keep a spacecraft in a particular The tool developed in the frame of this paper is A based on numerical optimization techniques and uses a Research Volume assigned orbit. For many Earth satellites the effects of the non- thrusters-based model of the satellite to take directly into Keplerian forces, i.e. the deviations of the gravitational account the activity of each thruster used for the control force of the Earth from that of a homogeneous sphere, of the satellite on the optimization process. This paper Frontier gravitational forces from Sun/Moon, solar radiation presents the tool design and the main principle of the pressure and air-drag must be counteracted. optimization algorithm. Usually, control strategies

The deviation of Earth's gravity field from that of consider satellites as a point. The present work includes Science the mathematical definition and the satellite model that a homogeneous sphere and gravitational forces from of Sun/Moon will in general perturb the orbital plane [1]. allow considering it as a system. The results of some For geostationary spacecraft the inclination change simulations and their practical applications are presented. caused by the gravitational forces of Sun/Moon must be Journal counteracted to a rather large expense of fuel, as the II. Mathematical Modeling inclination should be kept sufficiently small for the spacecraft to be tracked by a non-steerable antenna a) Coordinate Frames Global [1,2]. The coordinate system used in this work for Solar radiation pressure will in general perturb describing the perturbing forces is the satellite based the eccentricity (i.e. the eccentricity vector) [3]. For Radial Tangent Normal (RTN) coordinate system with some missions this must be actively counter-acted orthonormal basis , The axis is defined as always with manoeuvers, for geostationary spacecraft the pointing from the Earth’s center along the radius vector 𝑅𝑅�⃗𝑇𝑇�⃗𝑁𝑁��⃗ 𝑅𝑅�⃗ toward the satellite, The axis is normal to the orbit Author α σ: Department of Electrical Engineering, University of Batna, plane with direction of the satellite angular momentum Rue Chahid Boukhlouf, CUBI, Batna, 05000, Algeria. ��⃗ e-mail: [email protected] vector, The axis is perpendicular𝑁𝑁 to R in the orbit plane

and with the direction toward the satellite movement. It and eccentricity e describe the form of the orbit, one completes, with the unit vectors R and N, a right-handed element the mean anomaly M defines the position of orthogonal basis (see Figure 1). satellite along the orbit, the three others, the right ascension Ω, inclination i and argument of perigee ω define the orientation of the orbit in space. Given these

NORTH six elements, it is always possible to uniquely calculate ��⃗ 𝑻𝑻��⃗ the position and velocity vector (see figure. 2). 𝒁𝒁 SATELLITE��⃗ 𝑵𝑵��⃗ 𝑹𝑹 In many application, satellite orbits are chosen to be near-circular, to provide a constant distance from EQUATO the surface of the Earth or a constant relative velocity. RIAL Typical examples are low-altitude remote sensing satellite or geostationary satellite.

201 While there is no inherent difficulty in calculating position and velocity from known orbital elements with e

Year ORBITAL and i close to zero, the reverse task may cause practical ��⃗ ��⃗ 𝑿𝑿 PLAN LINE OF 𝒀𝒀 and numerical problems. These problems are due to NODES 8 singularities arising from the definition of some of the classical orbital elements. The argument of perigee ω, Figure 1 : Coordinate frames for example, is not a meaningful orbital element for small In the flowing, a generic acceleration vector eccentricities, since the perigee itself is not well defined

V induced from propulsive force acting on the satellite will for an almost circular orbit. Similar consideration apply be expressed as 𝑢𝑢�⃗ to small inclination i where the line of node is no longer VII well defined and where the equations for Ω become ue ersion I

s = + + (1) s singular. Several attempts have therefore been made to I substitute other parameter for the classical keplerian Where 𝑢𝑢�⃗ , 𝑢𝑢𝑅𝑅,𝑅𝑅�⃗ 𝑢𝑢𝑇𝑇are𝑇𝑇�⃗ 𝑢𝑢𝑁𝑁the𝑁𝑁��⃗ acceleration XV components along the radial, tangential and normal elements. These elements are usually referred to as 𝑹𝑹 𝑻𝑻 𝑵𝑵 directions. 𝒖𝒖 𝒖𝒖 𝒖𝒖 non-singular, regular or equinoctial elements [4]. The satellite orbit plane is defined thanks to the b) Orbit elements component of the inclination vectors (with modulus A total of six independent parameters are ) tan(i⁄2)) direct alone the line of nodes and pointing

A required to describe the motion of a satellite around the ) towards the ascending node. earth [3,4].Two of these elements, semi-major axis a

= [ ] = 2tan ( )[ ] [ ] , 0 (2) 2 𝑇𝑇 𝑖𝑖 𝑇𝑇 𝑇𝑇 Research Volume The satellite trajectory𝑖𝑖⃗ 𝓅𝓅 𝓆𝓆on its orbit is⁡ defined𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 to 𝑠𝑠𝑠𝑠𝑠𝑠eccentricity𝛺𝛺 ≅ 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 vector𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠directed𝑖𝑖 → alone the line of apsis and

the semi major axis a, and supposing the parameters Ω pointing towards the perigee.

and ω in the same plane, to the component of the Frontier == [ ] = [ ( + ) ( + )] = [ ] (3) 𝑇𝑇 𝑇𝑇 𝑇𝑇

Science 𝑒𝑒⃗ 𝒽𝒽 𝓀𝓀 𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐 𝛺𝛺 𝜔𝜔 𝑠𝑠𝑠𝑠𝑠𝑠 𝛺𝛺 𝜔𝜔 𝑒𝑒 𝑐𝑐𝑐𝑐𝑐𝑐𝜔𝜔� 𝑠𝑠𝑠𝑠𝑠𝑠𝜔𝜔�

of Y

Journal λ Ω Vernal Equinox M

X Global i ω

Ascending e node Perigee

Figure 2 : Orbit elements

Finally, the position of the satellite along its orbit Where is represented by the mean longitude µ v0 station keeping velocity equal to , µ is the earth 0 = + ( ) (4) gravitational coefficient. �𝑎𝑎 And Where Θ is the Greenwich𝑙𝑙 𝜔𝜔� sidereal𝑀𝑀 angle. −𝛩𝛩 𝑡𝑡 c) Dynamics for a GEO satellite 0 = 0 + ( ) (13) The motion of GEO satellite can be described The GEO orbits𝜓𝜓 are𝑙𝑙 characterized𝛩𝛩 𝑡𝑡 by very small by the rat of change of the equinoctial orbital values of eccentricity e and inclination i. the longitude parameters under the influence of the forces acting on and latitude can be defined the satellite. The geostationary dynamics results in the flowing nonlinear time varying system = + 2 ( + ) 2 ( +Θ (14)

( ) = ( ( )) + ( , ( )) + ( , ( )) ( ) (5) 201 𝜆𝜆 =𝑙𝑙 2 𝒽𝒽(𝑠𝑠𝑠𝑠𝑠𝑠+ 𝑙𝑙 ) 𝛩𝛩 2− 𝓀𝓀 𝑐𝑐𝑐𝑐𝑐𝑐( +Θ𝑙𝑙 (15) r

ea Where𝑥𝑥̇ 𝑡𝑡 𝒦𝒦 𝑥𝑥 𝑡𝑡 𝔏𝔏 𝑡𝑡 𝑥𝑥 𝑡𝑡 𝒢𝒢 𝑡𝑡 𝑥𝑥 𝑡𝑡 𝑢𝑢 𝑡𝑡 We denote y the spacecraft position vector, 𝜑𝜑 𝓅𝓅 𝑠𝑠𝑠𝑠𝑠𝑠 𝜆𝜆 𝛩𝛩 − 𝓆𝓆 𝑐𝑐𝑐𝑐𝑐𝑐 𝜆𝜆 Y = [ ] (6) which can be considered as the output variable of the

𝑇𝑇 nonlinear model 9 𝑥𝑥 𝑎𝑎= 𝓅𝓅[ 𝓆𝓆 𝒽𝒽 𝓀𝓀 𝑙𝑙 ] (7) 𝑇𝑇 = ( , ) (16) And the functions,𝑢𝑢 𝑢𝑢𝑅𝑅 𝑢𝑢𝑇𝑇, 𝑢𝑢 and𝑁𝑁 are the variation contribution to the equinoctial elements coming The output equation 𝑦𝑦 𝑓𝑓into𝑥𝑥 𝑡𝑡its Taylor series up to respectively from the Kipler's,𝓚𝓚 𝕷𝕷 Lagrange's𝓖𝓖 and Gausse V the first order around x0 [8], we get the output equation planetary equations [6] and [7]. of the linear time varying system (eq.10) VII

: Describes the satellite motion under the effect of ue ersion I s the gravitational attraction of the earth considered with = ( ) (17) s 𝓚𝓚 I takes into account the effect of the natural perturbing Where 𝑦𝑦 𝐶𝐶 𝑡𝑡 𝔵𝔵 XV forces. = [ ] (18) : Take into account the effect of the natural 0 𝑇𝑇 perturbing forces. And 𝑦𝑦 𝜆𝜆 −𝑙𝑙 𝜑𝜑 𝕷𝕷 : Given by the acceleration by thrusts. )

 −a cos0 ψ −a ψ 00sin 0  A The translation of nonlinear model (equation 5) 0 000 ) 𝓖𝓖   (19) into the linear model we use the Taylor series up to first ( )=  − ψ 0 ψ 00sin2cos21 1  order around the nominal operating points    00 − cos200 ψ 0 2sinψ 0  𝐶𝐶 𝑡𝑡 0 = [ 0 0 0 0 0 0 ] (8) Research Volume 𝑇𝑇 III. Station Keeping Box Problem 𝑥𝑥 0 𝑎𝑎= [0 0 0 ] 𝑙𝑙 (9) We obtain 𝑇𝑇 Formulation

𝑢𝑢 Frontier ( ) = ( ) ( ) + ( ) ( ) + ( ) (10) The station keeping box represents the maximum permitted values of the excursion of satellite in Where 𝔵𝔵̇ 𝑡𝑡 𝒜𝒜 𝑡𝑡 𝔵𝔵 𝑡𝑡 ℬ 𝑡𝑡 𝓊𝓊 𝑡𝑡 𝒟𝒟 𝑡𝑡 longitude λ and latitude . The SK box can be Science = 0 and = 0 (11) considered as pyramidal solid angle, whose vertex is at centre of the earth. 𝜑𝜑 of ( ) ( ) The 𝔵𝔵 and𝑥𝑥 −𝑥𝑥 matrices𝓊𝓊 𝑢𝑢turn− 𝑢𝑢 out to be time Is defined by the two half angles of vertex, one varying because of the presence of periodic terms with within plan of equator E-W width 2λmax and other in the periods equal𝓐𝓐 to𝑡𝑡 multiples𝓓𝓓 𝑡𝑡 of the periodicities of the Journal plan satellite meridian N-S width 2 (see Figure. 3). earth, sun and Moon motion relative to the satellite. max The matrix ( ) is a periodic function with 𝜑𝜑 period equal to 24 hours. Global 𝓑𝓑 𝑡𝑡  20 a0 0   1   00 sinψ 0   2  1 1  00 cosψ  ( ) =  0  (12) v 2 0 − ψ sin2cos ψ 0   0 0  ℬ 𝑡𝑡 ψ ψ  0 cos2sin 0 0     − 2 0 0 

1 = ( ( ) ( ) ( ) + ( ) ( ) ( ) ) (20) 2 𝑡𝑡𝑡𝑡 max 𝑇𝑇 𝑇𝑇 Subj𝐽𝐽 ect∫ 𝑡𝑡to𝑡𝑡 the𝑦𝑦 conditions𝑡𝑡 𝑄𝑄 𝑡𝑡 𝑦𝑦 𝑡𝑡 𝓊𝓊 𝑡𝑡 𝑅𝑅 𝑡𝑡 𝓊𝓊 𝑡𝑡 𝑑𝑑𝑑𝑑

2λ (l0,0) (21)

𝑍𝑍⃗ Where 𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚 −𝑦𝑦 ≤ 𝑦𝑦 ≤ 𝑦𝑦 2 max = [ ] (22)

𝜑𝜑 The thruster𝑦𝑦𝑚𝑚𝑚𝑚𝑚𝑚 accelerations𝜆𝜆𝑚𝑚𝑚𝑚𝑚𝑚 𝜑𝜑𝑚𝑚𝑚𝑚𝑚𝑚 are defined as

⃗ �⃗ control laws in the optimization problems. These control 𝑋𝑋 𝑌𝑌 variables can thus take at any time any value comprised between zero and the maximum thruster acceleration, in 201 Figure 3 : Station keeping box general with j thruster, we can write the control vectors in The station keeping problem can be formulated RTN fram as

Year as constrained linear quadratic continuous time optimal 1 = (23) control problem [9]. Given the linear model equation 10 (10) and equation (17) with initial condition ( ) = , Where m is the𝓊𝓊 s𝑗𝑗pacecraft𝑚𝑚 𝛤𝛤𝐹𝐹𝑗𝑗 and is the thruster the problem is to find the control optimal ( ) over a system configuration matrix can be defined for a satellite 𝑿𝑿� 𝑡𝑡𝑖𝑖 𝑿𝑿�𝑖𝑖 𝛤𝛤 finite time horizon tf-ti the minimize the criterion equipped with four electric thrusters as 𝒖𝒖�𝑜𝑜𝑜𝑜𝑜𝑜 𝑡𝑡 V

VII = (24)

ue ersion I −𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 −𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 −𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 −𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

s 𝛤𝛤 � 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 � I And F is the thrust vector of the thruster−𝑐𝑐𝑐𝑐𝑐𝑐 system𝑐𝑐 −𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐𝑐𝑐In𝑐𝑐 this technique,𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 the control inputs have to be XV written explicitly as function of the stat and its rate of 0 (25) change so that bounds on the control variables have We defined the constraints≤ 𝐹𝐹 on ≤ the𝐹𝐹𝑚𝑚𝑚𝑚𝑚𝑚 control variable by translated in bunds on the attainable rates of change of the state variable [12]. 1 ) ( ) (26)

A 𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚 The linear model (equation 10) can be written in ) −𝐹𝐹 𝑇𝑇 − 𝐹𝐹 different form, characterized by matrix B. to this IV. Numerical𝑚𝑚 ≤Solut𝛤𝛤 𝛤𝛤𝛤𝛤ion𝓊𝓊 of≤ the𝑚𝑚 Problem purpose, we can use the Lyapunov transformation A number of different techniques are available [9,13], in the stat space defined as for the numerical solution of the station keeping box Research Volume = ( ) (27) problem. In this work we will consider the direct method

for solution of continuous optimal control problem, the Where 𝔵𝔵� ℒ 𝑡𝑡 𝔵𝔵 idea behind direct method is to discrete the control time Frontier history and/or stat variable history [10,11].

 00 0 cos2 ψ − sin2 ψ −1   0 0  Science 00 0 3  cos2 ψ 0 − sin2 ψ 0 −  − 3 2  of 0 0 ψ ψ ( ) =  2a0 sin2 0 cos2 0 0  (28) v0  ψ 0 − sin2cos20 ψ 0 0 0 0    ψ ψ ℒ 𝑡𝑡  0 cos2sin20 0 0 − 20 sinψ 0 

Journal  − 2 0 0  a sin2 ψ 0 2cosψ 0  0 

The linear system (equation 10) can be written 1 1 Global = [ ( ) ( )] (31) in the new state variables as − − We obtained 𝐵𝐵� ℒ 𝑡𝑡 𝐵𝐵 𝑡𝑡 ( ) = ( ) ( ) + ( ) ( ) + ( ) (29) 𝖝𝖝� ( ) = ( ) ( ) ( ) ( ) (32) And𝔵𝔵�̇ 𝑡𝑡 we𝐴𝐴̃ can𝑡𝑡 𝔵𝔵� 𝑡𝑡write𝐵𝐵� the𝑡𝑡 𝓊𝓊 control𝑡𝑡 𝐷𝐷� variable𝑡𝑡 of the linear dynamics as a function of the state variable, so Where 𝓊𝓊 𝑡𝑡 𝑀𝑀𝔵𝔵�̇ 𝑡𝑡 − 𝑀𝑀𝐴𝐴̃ 𝑡𝑡 𝔵𝔵� 𝑡𝑡 − 𝑀𝑀𝐷𝐷� 𝑡𝑡

1 1 1 0 0 1 0 0 0 ( ) = ( ) ( ) ( ) ( ) (30) = 1 = (33) − − − 0 0 0 1 0 0 Where 𝓊𝓊 𝑡𝑡 𝐵𝐵� 𝔵𝔵�̇ 𝑡𝑡 − 𝐵𝐵� 𝐴𝐴̃ 𝑡𝑡 𝔵𝔵� 𝑡𝑡 − 𝐵𝐵� 𝐷𝐷� 𝑡𝑡 − 0 0 0 0 1 0 𝑀𝑀 𝐵𝐵� � � ©2015 Global Journals Inc. (US) Optimization Control SK Box Manoeuvres for GEO Satellites using Electric Thrusters (OCSKBOX)

And can be write the state variable as V. Numerical Simulation ( ) = ( ) ( ) + ( ) (34) The initial orbital elements for this simulation can be found in Table 1. And the output equation𝔵𝔵�̇ 𝑡𝑡 𝐴𝐴 ̃given𝑡𝑡 𝔵𝔵� 𝑡𝑡by 𝐷𝐷� 𝑡𝑡 Table I : Initial Orbital Elements ( ) = ( ) ( ) (35)

Orbital parameter Value Where ̃ 𝑦𝑦 𝑡𝑡 𝐶𝐶 𝑡𝑡 𝔵𝔵� 𝑡𝑡 Semi major axis(km) 42166.279 1 ( ) = ( ) ( ) (36) Right ascension of A.N ( °) 82 − Eccentricity 0.0000778 The station keeping problem formulated in the 𝐶𝐶̃ 𝑡𝑡 𝐶𝐶 𝑡𝑡 ℒ 𝑡𝑡 Inclination ( °) 0.002044 previous section as a constrained continuous time Argument of perigee ( °) 315.67725 optimal control problem can be translated in a quadratic Mean anomaly ( °) 324.34109 programming problem with constraint only on the state 201

r The satellite is considered in this simulation is variables. ea

Y Above a finite time horizon tf-ti discretized in N equipped with four thrusters mounted on its anti nadir intervals of length equal h each. The control optimal face (see Figure 4).the configuration of the force vector 11 ( ) is taken constant equal to is 𝑜𝑜𝑜𝑜𝑜𝑜 = [ ] (42) 𝓾𝓾 𝑡𝑡 With k=0,1,…,N-1 (37) 1 2 3 4 𝑜𝑜𝑜𝑜𝑜𝑜 𝑘𝑘 𝐹𝐹 𝐹𝐹 𝐹𝐹 𝐹𝐹 The problem is consist in finding the optimal 𝐹𝐹 V 𝓊𝓊 sequence that minimized the criterion N VII 𝑜𝑜𝑜𝑜𝑜𝑜 1 1 Thruster ue ersion I

𝑘𝑘 1 Thruster s 𝖝𝖝� = + (38) s 2 =1 2 =0 4 γ 3 𝑁𝑁 𝑇𝑇 𝑁𝑁− 𝑇𝑇 I With 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑘𝑘

𝐽𝐽 ∑ 𝑦𝑦 𝑄𝑄 𝑦𝑦 ∑ 𝓊𝓊 𝑅𝑅 𝓊𝓊 XV = (39) σ T Thruster Thruster 𝑘𝑘 𝑘𝑘 𝑘𝑘 2 = +1 𝑦𝑦 𝐶𝐶+̃1 𝔵𝔵� (40) 1 2 𝔵𝔵�𝑘𝑘 −𝔵𝔵�𝑘𝑘 𝔵𝔵�𝑘𝑘 −𝔵𝔵�𝑘𝑘 𝑘𝑘 ℎ ̃𝑘𝑘 𝑘𝑘 R ) 𝓊𝓊 𝑀𝑀 − 𝑀𝑀𝐴𝐴 − 𝑀𝑀𝐷𝐷� Subject A ) Figure 4 : Configuration force vector • The output variable y(t) The characteristics for this satellite can be (41) found in Table 2.

• The control variable𝑚𝑚 𝑚𝑚𝑚𝑚 (t) 𝑘𝑘 𝑚𝑚𝑚𝑚𝑚𝑚 Research Volume −𝑦𝑦 ≤ 𝑦𝑦 ≤ 𝑦𝑦 Table II : Characteristics of Satellite 𝓊𝓊 ( ) 1 (42)

−𝐹𝐹𝑚𝑚𝑚𝑚𝑚𝑚 𝑇𝑇 − 𝐹𝐹𝑚𝑚𝑚𝑚𝑚𝑚 Characteristics of satellite Value

Frontier • The auxiliary state𝑚𝑚 ≤ variable𝛤𝛤 𝛤𝛤𝛤𝛤 ( )𝓊𝓊𝑘𝑘 ≤ 𝑚𝑚 Spacecraft masse (kg) 4500 + Can angles of thruster ( °) 50 +1 = �𝔵𝔵 +𝑡𝑡1 + (43) 2 Slaw angles of thruster ( °) 15 𝔵𝔵�𝑘𝑘 −𝔵𝔵�𝑘𝑘 𝔵𝔵�𝑘𝑘 𝔵𝔵�𝑘𝑘 Science Maximum force modulus (N) 0.17 ℎ ̃𝑘𝑘 �𝑘𝑘 A step-by-step walkthrough𝐴𝐴 of the𝐷𝐷 algorithm is Specific impulse (s). 3800s of as follows: The objective is to determine the set of Step1: Formulation of SKBOX problem manoeuvers to be executed in order to keep the satellite Journal Fixed the finite time horizon tf-ti=1 day. in a latitude and longitude box centered at the station longitude l =10deg with =0.01deg and The weighting matrices are equal to: R=I2×2 and 0 λmax max=0.01deg. Q =I2×2. Global Figure 5, Figure 6 and Figure 7 illustrates the Fixed the initial orbital elements vector x(ti). 𝜑𝜑historical time of the optimal acceleration control Step2: Finding the optimal solution with minimized the components in RTN frame for one year. criterion J with a discretization step of length h=0.01 Negative and positive values of optimal day. acceleration radial allow maintain the satellite in a Step3: Obtained the optimal control with equation (40). latitude window equal to ]-0.01°, 0.01°[, positive values Step4: Finding the output variable with equation (39). of optimal tangential acceleration fix the satellite into a Step5: Repeat the previous steps for 1yaer. longitude interval ]-0.01°, 0.01°[, and the variation of

optimal acceleration normal centered the satellite in the 200 box. 100 (mN)

0 T

F 0

) -100 2 -5 0 50 100 150 200 250 300 350 s Time(day) (km/

R -10 u 200

-15 0 0 50 100 150 200 250 300 350 (mN) N Time(day) F 201 -200 Figure 5 : Optimal acceleration radial control for one 0 50 100 150 200 250 300 350 Year year Time(day) 12 8 Figure 8 : Propulsive force for one year in RTN frame

) 6 Figure 9 Shows the controlled and uncontrolled 2

s manoeuvers time histories of the latitude for the station 4 keeping box in condition ±0.01deg. In SKBox no-

V (km/ T u 2 controlled, the variation in latitude value is not fixing in VII the box. For this problem the SKBox controlled is used ue ersion I

s for fixing the variation in latitude into the box.

s 0

0 50 100 150 200 250 300 350 I Time(day) 0.1

XV Uncontrolled SKB Controlled SKB Figure 6 : Optimal acceleration tangential control for one 0.05 Latitude box year 0 ude (deg) t 40 i ) Lat

) -0.05

) 20 2 s -0.1 0 0 50 100 150 200 250 300 350

(km/ Time(day) N u -20 Research Volume Figure 9 : Time histories of the latitude for one year -40 0 50 100 150 200 250 300 350 Figure 10 illustrates the controlled and Time(day)

Frontier uncontrolled manoeuvers time histories of the true

Figure 7 : Optimal acceleration normal control for one longitude for the station keeping box in condition year ±0.01deg. In SKBox no-controlled, the variation in longitude value is not fixing in the box. For this problem Science Figure 8 illustrates the evolution of the three the SKBox controlled is used for fixing the variation in of components of propulsive force in RTN frame for one longitude into the box. year with the maximum force modulus Fmax=0.17N. The variation in propulsive Force allows Unconttroled manoeuvre Journal 10.0625 Conttroled manoeuvre (λ =0.01°) producing the optimal acceleration to control the max satellite in latitude and longitude box. The thruster forces 10.04 values are lower them maximum propulsive forces. 20 Global tude (deg) 10.0215

200 ongi

100 True l 10 (mN) R 9.98

F 0 50 100 150 200 250 300 350 Time(day) -100 0 50 100 150 200 250 300 350 Time(day) Figure 10 : Time histories of true longitude for one year

VI. Conclusion 13. WANG, T. T.–XIE,W. F.–LIU, G. D.–ZHAO, Y. M. : Quasi-min-max model predictive control for image- In this paper a new method for station keeping based visual servoing with tensor product model box of geostationary satellite equipped with electric transformation, Asian Journal of Control 17 No. propulsion has been developed, we considered a novel 2(2015). 402-416. approach based on direct method for solution of continues optimal control. Using this method, satellite position can be directly controlled based on the optimal acceleration for thruster, simulation results have demonstrated that the satellite can be tightly controlled in the station keeping box.

References Références Referencias 201

1. ECKSTEIN, M. C.: Geostationary Orbit Control r

Considering Deterministic Cross Coupling Effects, ea

Y 41st Congress of the International Astronautical Federation. IAF-90-326, Dresden (1990). 13

2. SOOP, E. M.: Geostationary Orbit Inclination Strategy, ESA Journal, 9 No.1(1985),65-74. 3. CARLINI, S.- GRAZIANI, F.: An Eccentricity Control

Strategy for Coordinated Station Keeping, ESA V Symposium on Spacecraft Flight Dynamics (ESA VII SP-326). Darmstadt, Germany, (1991). ue ersion I s

4. Vallado, D.A.: Fundamentals of Astrodynamics and s Applications, Microcosm Press, Fourth Edition, I

(2015). XV 5. LYSZYK, M. -GRAMERO, P. :Electric Propulsion th System on Bus Platform, In ESA SP-555, 4 International Spacecraft Propulsion Conference,

(2004). )

A 6. BETTS JOHN ,T. : Survey of Numerical Methods for ) Trajectory Optimization , AIAA Journal of Guidance,

Control, and Dynamics, 21 No. 2 (1998), 193-207. 7. CHO, S. B.– MCCLAMRICH, N. H. : Attitude control

of a spacecraft, KSAS international journal 8 No.2 Research Volume (2007),67-75.

8. POURTAKDOUST, S. H. –JALALI, M.: Thrust-limited Optimal Three-Dimensional Spacecraft Trajectories, Frontier AIAA Guidance, Navigation, and Control Conference

and Exhibit, 1995-3325, (2005). 9. CAMACHO, E. F.–BORDONS A.C.: Model Predictive Science Control, Advanced Textbooks in Control and Signal of Processing. Springer-Verlag London Limited, (1999). 10. IGOR, S.– BLAZIC S.–MATKO, D.: Direct fuzzy Journal model-reference adaptive control, International Journal of Intelligent Systems 17 No. 10 (2002), 943- 963. Global 11. PRECUP, R.E.–RADAC, M.B.–PRIETL,S. - PETRIU E.M. – DRAGOS, C.A. :Iterative feedback tuning in fuzzy control systems. Theory and applications, Acta Polytechnica Hungarica 3 No. 3(2006), 81-96. 12. YACOUB,R. R.: functional link neural network for active noise control , International Journal of Artificial Intelligence 12 No. 1, (2014), 6-47.

201 Year

V VII ue ersion I s

s I

XV This page is intentionally left blank )

A ) Research Volume Frontier Science of Journal Global