AUTOMATIC MINIMUM PRINCIPLE FORMULATION FOR LOW THRUST OPTIMAL CONTROL IN ORBIT TRANSFERS USING COMPLEX NUMBERS

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1 AUOMAIC MINIMUM PRINCIPLE FORMULAION FOR LOW HRUS OPIMAL CONROL IN ORBI RANSFERS USING COMPLEX NUMBERS hierry Dargent hales Alenia Space, 1 Bd du Midi, BP 99, 6322 Cannes La Bocca Cedex France, thierry.dargent@thalesaleniaspace.com ABSRAC In the last recent years a signiicant progress has been made in optimal control orbit transers using low thrust electrical propulsion. Usually 2 dierent techniques o optimisation are used to solve this type o problem: Direct method and Indirect method. Direct method discretizes in parameter the problem and ormulates it as large Non Linear Programming problem to be solved. Indirect methods are using the minimum principle that give necessary condition o optimality and need to solve a 2 or multiple points boundary problem. he advantages o indirect method compare to direct one are a low number o optimisation parameters which will give a rather quick calculation o the solution and the high quality o the solution he main drawback is the need o a good initial guess o the optimisation parameters to allow the convergence o the solver and to be able to have an analytical ormulation o the problem. It restricts the method to analytical model and make diicult or example the introduction o a ull orce perturbation model or multi body environment. In this paper we will describe an innovative eicient method in indirect optimal control to compute numerically in an automatic way all the necessary condition at the same precision like an analytical model. he method allows to use, in a straightorward way, any kind o numerical model o complex dynamic without the need o an analytical model ormulation. he method is base on the utilisation o complex number or calculating sensitivities at machine precision level. It relies on the Cauchy-Riemann equations apply to analytic unction. In an second part we will describe an application o this method to compute low thrust optimal orbit transer using a satellite dynamic model with all the classical perturbation orces describe in: hirdbody perturbation, Solar Radiation Pressure and Spherical Harmonics potential decomposition o the central body Finally, a synthesis o test cases is presented to illustrate the capacities o the method, mixing examples o interplanetary trajectory, and Earth mission. 1 DISCUSSION ON USING MINIMUM PRINCIPLE FORMULAION WHEN USING NON ANALYICAL DYNAMIC FORMULAION Let s introduce J the cost unction or an optimal orbit transer type problem and apply minimal principal [1]. he general orm o the cost unction can be written: xt t xt, t Lx, t x, t t J, x dt (1) Where: x t, t mass maximum problem, x, t is the part o the cost unction at arrival, or example the inal mass o the satellite in Lagrange multipliers, t t are the boundary conditions at arrival and the associated L x, t is the part o the cost unction along the trajectory, or example 1 in

2 , is the dynamic constraint equation along the trajectory and the associated Lagrange multipliers. case o minimum time problem, x t x We deine the Hamiltonian by: H x,, t Lx, t x, t (2) he necessary condition o optimality is J, which ater the variation calculation gives: x x, t (3) n dierential equations x L x (4) n dierential equations o the co-state dynamics he command u(t) being determined by the relation: H u u L u And the boundary conditions: t given at t or t (5) m algebraic relations x k k n boundary conditions on the state at t t t x x t x minimal time problem x t t t (6) n boundary conditions on the co-state at t x L t t, t (8) q boundary conditions on the state at t. (7) 1 boundary condition on the inal time in case o When we want to solve this problem: the classical approach is to ormulate the problem as a 2 points boundary problem where the unknown parameters are a part o the state and the co-state variable at t and the equation to be veriied are the boundary conditions (8) and optimality condition (6)(7) at t. o evaluate the equations (6)(7)(8), we need to compute the state and co-state variables at t by integrating the dynamic o the state and the co-state variables (3)(4) and solving at each step time o the integration process the command equation (5). Integration process is numerically sensitive and numerical error on the dynamic evaluation will create an integrated error that increase exponentially with integration time t -t. he consequence o that is: to be numerically eicient we need to know the co-state dynamic (4) at same level o numerical precision compare to the dynamic (3). In practice that means, we have to know analytically the dynamic expression (3) and derive by ormal calculation an analytical expression o the co-state dynamic (4). his approach is good i the expression o the dynamic is rather simple like or example using a keplerian orce model or the dynamic. It has been successully put in practice by many authors [2][3]. he diiculty arises with this approach when we want to introduce a more complete orce model or the satellite dynamic like introducing hird-body perturbation, Solar Radiation Pressure, Gravitational Fields Perturbation, In this case, describe the perturbation by an analytical model and compute the analytical expression o the co-state dynamic is rather diicult nor possible i or

3 example a numerical model or hird-body ephemeris is use. For this type o dynamic numerical method or automatic dierentiation method as to be use to compute co-state dynamic at the level o machine precision to keep the interest o minimum principle method compare to direct methods. 2 UILISAION OF COMPLEX NUMBER FOR CALCULAING SENSIIVIIES A MACHINE PRECISION LEVEL he most common method to compute numerically the derivatives o a unction deine in subroutine is to use automatic dierentiation program like ADIFOR [4] or Fortran program or to use inite dierence method. he automatic dierentiation method is good rom a numerical precision point o view but have some drawback and limitation one o those are the size o the code generated some limitation in the way o programming, the need to have a ull access o the source code rom and speed eiciency & memory allocation. he inite dierence method is simple and rather strait orward but doesn t provide result with enough precision (typically no more hal o the mantissa is good). he third approach proposes here is to perorm numerical dierentiation using complex number. A comparison o the 3 methods has been study and presented by Joaquim R. R. A. Martins & all [5]. he idea o the numerical dierentiation using complex number is base on property o holomorphic unction (dierentiable unction in complex domain). his properties are describe by the Cauchy- Riemann equations: he Cauchy-Riemann equations on a pair o real-valued unctions u(x,y) and v(x,y) are the two equations: u v x y (9) u v y x ypically the pair u and v are taken to be the real and imaginary parts o a complex-valued unction (x + iy) = u(x,y) + iv(x,y). Suppose that u and v are continuously dierentiable on an open subset o C. hen = u+iv is holomorphic i and only i the partial derivatives o u and v satisy the two Cauchy-Riemann equations (9). hen deining z= x + ih, or small h the irst order development o ((z)=(x + ih) is: ( x) ( x ih) ( x) ih (1) x then rom (1) we can derive the derivative ormula: Im ( x ih) x h (11) he interest o this approach rom a numerical point o view is when computing (1) we never perorm a inite dierence which means we can chose h very small or example 1e-5 and reach the machine precision or the derivative with out any numerical problem and diiculty on the choice o h.

4 For example taken rom [5] we want to evaluate the normalize error on the derivative x e o ( x) (12) when using inite dierence and complex method (11) as a unction 3 3 sin x cos x o h: Reach o the epsilon machine precision point where complex method and analytic calculus give the same result Fig. 1 Comparison o inite dierence and complex method his example shows the capacity o the method to compute derivative up to machine precision compare to analytical derivative. he implementation is rather simple. he main constraint is to be able to program or to have holomorphic unctions in library. his last point is solved i using Fortran or Matlab programming language (see [5] or small restriction and other programming language). 3 APPLICAION OF HIS MEHOD O COMPUE LOW HRUS OPIMAL ORBI Starting rom an existing tool or low thrust optimal control orbit transers [2] we have introduce in the dynamic ormulation classical perturbation orce model: hird-body perturbation [6], All planets & Moon o solar system available with high precision ephemeris Solar Radiation Pressure [7][8] with eclipse model Gravitational Fields [8] with Spherical Harmonics decomposition Model or Earth and Mars he co-state dynamic x is computed using the complex method to evaluate the part x 3.1 EARH ORBIING EXAMPLE he irst example is a minimal time transer or a large Geostationary satellite reaching it operational orbit with electrical propulsion. he satellite model: Initial mass 4415kg Electrical motor thrust: 88mN Electrical motor Isp: 163s

5 Gravitational constant g : he perturbation model is the ollowing: hird body perturbation: Moon and Sun, Solar pressure radiation with a solar arrays surace o 1 m², visible surace absorptivity coeicient. 7 and specular coeicient 1 Earth harmonic spherical decomposition 7*7 using gravity model WGS84-EGM96 rom NASA/NIMA Simulation epoch: 7/6/26 11:56:21. he transer problem is a minimal time transer on Kepler parameters a, e, i,,, l ) : ( geo a initial : km a inal : km e initial : e inal : i initial : 1.5 deg i inal : deg initial : 18 deg inal : 18 deg initial : deg inal : deg Geographic longitude deg Geographic longitude deg Initial Mass: 4415 kg 2 computations have been made one with the non-perturbed dynamical model and one with the perturbed dynamical model. In the irst case minimal time transer is days and in second case the transer time increase to days this increase is mainly due to luni-solar out o plane perturbation which impact inclination evolution. In the igure or this example the blue colour reer to the transer with perturbed dynamical model and the red colour to the non-perturbed dynamical model. he trajectories are quite similar which it was expected. Ones can notice the out o plane command increase about 1.5 to 2.5 along the trajectory x 1 7 Semi major axis evolution F =.88 Nqqqqqqqqqqq Isp = 163 s m = 4415 kg, m = kg DV transert = m/s a initiale = km a inale = km évolution de l'excentricité a in m F =.88 N Isp = 163 s m = 4415 kg, m = kg DV transert = m/s a initiale = km a inale = km ranser in days (total transer= days) Fig. 2 semi major axis evolution excentricité Fig. 3 eccentricity evolution

6 1.5 évolution de l'inclinaison 2 évolution de la longitude vraie( ) 15 1 Inclinaison en Deg 1.5 longitude vraie en deg Fig. 4 Inclination evolution évolution de l'azimut Fig. 5 Satellite Geographic longitude evolution évolution de l'azimut commande en deg 5-5 commande en deg Fig. 6 In plane azimuth command at beginning évolution de l'élévation Fig. 7 In plane azimuth command at the end évolution de l'élévation commande en deg -1 commande en deg Fig. 8 out o plane elevation command at beginning Fig. 9 out o plane elevation command at the end

7 3 azimut onction de l'élévation dans le cas sans pertubation 3 azimut onction de l'élévation dans le cas pertubé 2 2 elevation inertiel en deg 1-1 elevation inertiel en deg azimut inertiel en deg azimut inertiel en deg Fig. 1 elevation / azimuth in inertial rame non perturbed model Fig. 11 elevation / azimuth in inertial rame perturbed model 3.2 INERPLANEARY EXAMPLE he second example is an Earth Mars transer. For this case we leave the Earth the 25/12/21 :: on a hyperbolic branch with Vin = 2 m/s and arrive at Mars the 2/5/213 :: with Vin = ater 877 days o travel. In this test case we will compare the non-perturbed dynamic with one taking into account the Earth perturbation during the escape phase and also the solar pressure radiation. he perturbation o mars at arrival is not taking into account. Satellite parameters: Initial mass 1kg Electrical motor thrust: 1mN Electrical motor Isp: 163s Gravitational constant g : he perturbation model is the ollowing: hird body perturbation: Earth Solar pressure radiation with solar arrays surace o 1 m² visible surace absorptivity coeicient. 7 and specular coeicient 1 he coordinates o the satellite on the hyperbolic branch at departure and at Mars arrival are the ollowing (J2 rame)

8 a initial : m a inal : m e initial : e inal : i initial : deg i inal : deg initial : deg inal : deg initial : deg inal : deg deg deg Initial Mass: 1 kg transer time: 877 days Fig. 12 Satellite trajectory.12.1 Blue pertubed model hrust duration perturbed model 1644 h DV total 4381 m/s hrust Proile Red non perturbed model hrust duration non perturbed model 1514 h DV total m/s hrust on/o in N days rom the 25/12/21 Fig. 13 hrust proile o the perturbed and unperturbed trajectories

9 he overall perormance is quite similar with a inal satellite mass o kg or the nonperturbed model and kg or the perturbed model but the thrust proile present signiicant switching change this test show the need to re-optimize a classical patch-conic approach when using electrical propulsion i we want to catch a precise thrust proile. he radiation pressure perturbation has a negligible impact in the perturbed trajectory compare to the Earth perturbation impact 4 CONCLUSION We have implement and demonstrate an innovative numerical method to solve Optimal control using minimum principle without the usual need o a ormal calculus o the adjoin dynamic and the programming o it. his technique breaks a major constraint and drawback o this type o method and opens the way o creating a ully general solver or optimal control using minimum principle his technique has been also used to compute with accuracy state transition matrix. he implementation o this method has been applied on a high precision orbital dynamic model with success. Some results o simulation show the impact o the perturbation: Around Earth with an example o orbit toping o a GEO satellite On interplanetary Earth Mars mission with the impact on the thrust proile due Earth perturbation on the Escape phase REFERENCES 1. A. E. Bryson et Jr. Yu-Chi-Ho Applied Optimal Control: Optimisation, Estimation and Control Hemisphere Publishing Corporation hierry Dargent, Vincent Martinot, An integrated tool or low thrust optimal control orbit transers in interplanetary trajectories, ISSFD Régis Bertrand, PhD thesis: Optimisation de trajectoires interplanétaires sous hypothèses de aible poussée ADIFOR 2.,Automatic Dierentiation o Fortran, 5. Joaquim R. R. A. Martins, Ilan M. Kroo and Juan J. Alonso, An automated method or sensitivity analysis using complex variable, AIAA J.-P. CARROU, Mécanique Spatiale ome I Cépadues-Edition O. MONENBRUCK, E. GILL Satellite Orbits: Models, Methods, Applications. Spinger 8. C. ROIHMAYR, Contributions o Spherical Harmonics to Magnetic and Gravitational Fields, NASA/M BACK O SESSION DEAILED CONENS BACK O HIGHER LEVEL CONENS