Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects An optimal control approach for minimum-fuel deployment of multiple spacecraft formation flying Richard Epenoy Richard.Epenoy@cnes.fr Centre National d’Etudes Spatiales 18, avenue Edouard Belin 31401 Toulouse Cedex 9 - France CEA-EDF-INRIA School: Optimal control May 30 - June 1 st 2007, INRIA Rocquencourt, France Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment
Problem statement The solution approach Numerical results - A deployment in Low Earth Orbit Conclusion and future prospects Outline Problem statement 1 Space dynamics equations Optimal control formulation The solution approach 2 Pontryagin’s Maximum Principle The continuation-smoothing method Numerical results - A deployment in Low Earth Orbit 3 Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption Conclusion and future prospects 4 Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment
Problem statement The solution approach Space dynamics equations Numerical results - A deployment in Low Earth Orbit Optimal control formulation Conclusion and future prospects Problem statement Problem statement 1 Space dynamics equations Optimal control formulation The solution approach 2 Pontryagin’s Maximum Principle The continuation-smoothing method Numerical results - A deployment in Low Earth Orbit 3 Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption Conclusion and future prospects 4 Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment
Problem statement The solution approach Space dynamics equations Numerical results - A deployment in Low Earth Orbit Optimal control formulation Conclusion and future prospects Problem statement Space dynamics equations Problem statement 1 Space dynamics equations Optimal control formulation The solution approach 2 Pontryagin’s Maximum Principle The continuation-smoothing method Numerical results - A deployment in Low Earth Orbit 3 Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption Conclusion and future prospects 4 Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment
Problem statement The solution approach Space dynamics equations Numerical results - A deployment in Low Earth Orbit Optimal control formulation Conclusion and future prospects Dynamics equations in cartesian coordinates Two-body problem with perturbations and engine thrust − → r − → r � 3 + − γ p 1 + − → → ¨ r = − µ γ p 2 �− → µ : the Earth’s gravitational constant − → γ p 1 : natural perturbative acceleration (geopotential disturbances, lunar and solar third body gravities, atmospheric drag, solar radiation pressure,...) − → − → F γ p 2 = m : perturbative acceleration caused by the thrust − → F : thrust vector of the engine m : mass of the satellite Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment
Problem statement The solution approach Space dynamics equations Numerical results - A deployment in Low Earth Orbit Optimal control formulation Conclusion and future prospects Orbital parameters (1/2) Keplerian osculating elements ω : argument of perigee a : semi-major axis Ω: longitude of the ascending node e : eccentricity v : true anomaly i : inclination Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment
Problem statement The solution approach Space dynamics equations Numerical results - A deployment in Low Earth Orbit Optimal control formulation Conclusion and future prospects Orbital parameters (2/2) - Perturbative acceleration Eccentric anomaly - Mean anomaly cos ( E ) = cos ( v ) + e 1 + e cos ( v ) √ 1 − e 2 sin ( v ) sin ( E ) = 1 + e cos ( v ) M = E − e sin ( E ) Perturbative acceleration in the ( − → T , − → N , − → W ) local orbital frame − → W γ p = T − → T + N − → N + W − → − → → − W T − → − → → − ˙ r ∧ ˙ → − , − → , − → N = − W ∧ − → → r r T = W = T S − → �− → → − � ˙ r ∧ ˙ r � r � − → N Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment
Problem statement The solution approach Space dynamics equations Numerical results - A deployment in Low Earth Orbit Optimal control formulation Conclusion and future prospects Space dynamics equations in orbital parameters Gauss equations 2 V a ( t ) = ˙ T n 2 a 1 r � � e ( t ) = ˙ 2 ( e + cos ( v )) T − sin ( v ) N V a r cos ( ω + v ) ˙ i ( t ) = 1 − e 2 W n a 2 √ r sin ( ω + v ) ˙ Ω( t ) = 1 − e 2 W √ n a 2 sin ( i ) 2 e + (1 + e 2 ) cos ( v ) � � 1 r cos ( i ) sin ( ω + v ) ω ( t ) = ˙ 2 sin ( v ) T + N − 1 − e 2 W √ n a 2 sin ( i ) V e 1 + e cos ( v ) √ e 2 1 − e 2 1 − e 2 � � � � ˙ M ( t ) = n − 2 sin ( v ) 1 + T + cos ( v ) N V e 1 + e cos ( v ) 1 + e cos ( v ) � µ � 2 a (1 − e 2 ) � 1 � with r = , V = µ − and n = a 3 1 + e cos ( v ) r a Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment
Problem statement The solution approach Space dynamics equations Numerical results - A deployment in Low Earth Orbit Optimal control formulation Conclusion and future prospects Natural perturbations taken into account Earth’s oblateness only - Atmospheric drag neglected The non-sphericity of the Earth yields gravitational perturbations: 1 The Earth’s gravity field representation in cartesian coordinates is based on a spherical harmonic expansion 2 The Earth’s oblateness term J 2 is the most important one after the central term 3 The J 2 term corresponds to a certain expression of the perturbative accelerations T , N , and W 4 First effect of J 2 : short-period and long-period oscillations with zero mean on the orbital parameters 5 Second effect: a secular effect, i.e. a linear drift, on Ω (rotation of the orbital plane), ω and M Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment
Problem statement The solution approach Space dynamics equations Numerical results - A deployment in Low Earth Orbit Optimal control formulation Conclusion and future prospects Gauss equations with equinoctial elements and J 2 (1/2) State and control variables Let us consider a formation of n satellites x j : equinoctial orbital parameters for satellite S j ( j = 1 , . . . , n ) a j e x , j = e j cos ( ω j + Ω j ) e y , j = e j sin ( ω j + Ω j ) x j = h x , j = tan ( i j / 2) cos (Ω j ) h y , j = tan ( i j / 2) sin (Ω j ) L j = ω j + Ω j + v j ( a j , e j , i j , ω j , Ω j , v j ): Keplerian osculating elements for S j m j : mass of satellite S j u j : normalized thrust vector for S j in the ( − → T , − → N , − → W ) frame Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment
Problem statement The solution approach Space dynamics equations Numerical results - A deployment in Low Earth Orbit Optimal control formulation Conclusion and future prospects Gauss equations with equinoctial elements and J 2 (2/2) State equations in compact form x j ( t ) = f ( x j ( t )) + F max g ( x j ( t )) u j ( t ) ˙ m j ( t ) � u j ( t ) � ( j = 1 , . . . , n ) m j ( t ) = − F max ˙ g 0 I sp t ∈ [ t 0 , t f ] F max : maximum thrust modulus of the n engines I sp : specific impulse of the n engines g 0 : acceleration due to gravity at sea level t 0 and t f : fixed initial and final dates Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment
Problem statement The solution approach Space dynamics equations Numerical results - A deployment in Low Earth Orbit Optimal control formulation Conclusion and future prospects Problem statement Optimal control formulation Problem statement 1 Space dynamics equations Optimal control formulation The solution approach 2 Pontryagin’s Maximum Principle The continuation-smoothing method Numerical results - A deployment in Low Earth Orbit 3 Statement of the test case Deployment over seven days - Seven local solutions found Balancing the fuel consumption Conclusion and future prospects 4 Richard Epenoy Richard.Epenoy@cnes.fr An optimal control approach for minimum-fuel deployment
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