MONTE-CARLO ANALYSIS OF NONIMPULSIVE ORBITAL TRANSFER UNDER THRUST ERRORS, 1.

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1 MONTE-CARLO ANALYSIS OF NONIMPULSIVE ORBITAL TRANSFER UNDER THRUST ERRORS, 1. Antônio Delson Conceição de Jesus Universidade Estadual de Feira de Santana UEFS, Feira de Santana, BA, Brasil Phone/Fax: , Marcelo Lopes Oliveira e Souza Instituto Nacional de Pesquisas Espaciais INPE, São José dos Campos, SP, Brasil Phone: , Fax: , marcelo@dem.inpe.br Antônio Fernando Bertachini de Almeida Prado Instituto Nacional de Pesquisas Espaciais INPE, São José dos Campos, SP, Brasil Phone: , Fax: , prado@dem.inpe.br ABSTRACT In this paper we studied statistically orbital maneuvers with finite propulsion under thrust errors. We studied two transfer maneuvers: a) a high orbit low thrust coplanar transfer; and b) a middle orbit high thrust noncoplanar transfer, that is the first transfer of the satellite EUTELSAT II-F2. This transfer were realized with magnitude and direction ( pitch and yaw ) errors in the thrust applied to the satellite. This errors were modeled as a stochastic process: random-bias and white or pink noise, with zero mean uniform probability function. We studied too the first moment (mathematical expectation) of the standard deviation of each thrust error. The general results suggest one simple nonlinear relation between the causes (in the initial instan and effects (in the final instan, that one progressive deformation of the trajectory distribution along the propulsive arc was verified. 1. INTRODUCTION Most space missions need orbit transfers to reach their goals. These trajectories/orbits are reached sequentially through transfers between them by changing its Keplerian elements, by firing apogee motors or other sources of force. These thrusts have linear and/or angular misalignments that displace the vehicle with respect to its nominal directions. The mathematical treatment for these deviations can be realized under many ways (deterministic, probabilistic, MINMAX, etc of the deviations). Porcelli and Vogel (1980) presented one algorithm that can be used to the determination of the orbit insertion errors in a biimpulsive noncoplanar orbital transfers, using the covariance matrices of the source errors. Adams and Melton (1986) studied transfer with errors analysis to ascent trajectories moved by a finite impulse and low propulsion as one sequence of impulsive burns. They developed one algorithm to compute the navigation and errors propagation 1

2 along of the nominal trajectory, with finite perigee burns. Junkins (1997) considered the precision of the errors covariance matrix method through nonlinear transformations of coordinates. He also found one deformation of the trajectory distribution, due to the initial condition gaussian errors, that was not anticipated by the covariance analysis on linearized models with zero mean errors. Carlton-Wippern (1997) developed one set of differential equations that describes the growth of the position errors variance (in the initial condition or due to drag) of satellites in polar coordinates, by Langevin s equation approximation and one the first order perturbation theory. One deterministic analysis of thrust errors effect, to impulsive assumption, to nonpuntual satellites was done by Rodrigues (1991) and, for the puntual case, by Santos-Paulo (1998). Other papers that studied propulsion errors are Rocco(1999) and Schultz (1997). In this study we present a Monte-Carlo analysis of the nonimpulsive orbital maneuvers under thrust vector errors. The results were obtained for two transfer maneuvers, the first, between high coplanar orbits, the same transfer maneuver used by Biggs (1978 and 1979) and Prado (1989) with low thrust; the second, between middle noncoplanar, a real case transfer (the first part of the transfer of injection of the EUTELSAT II-F2 satellite) by Kuga et alli (1991). The simulations were realized for both cases. The optimization method used by Biggs (1978 and 1979) and Prado (1989) was adapted to the case of the transfers with thrust errors. The angles pitch and yaw were taken as control variables and the minimum fuel consumption is obtained each burn of the thrusters. The sources error that we considered were the magnitude errors, pitch and yaw directions errors of the thrust vector, as causes of the deviations found in the several Keplerian elements of the transfer trajectory. Each deviation was introduced separately along the orbital transfer trajectory. Besides this, we studied two types of errors for each one these causes: the randombias and the white-noise errors. The random-bias errors are even errors during all the transfer arc, while the white-noise errors change along the transfer arc. These errors sources introduced in the orbital transfer dynamic produce effects the final Keplerian elements in the final instant. In this work we present a statistical analysis of these errors for the first moment of the deviations of these Keplerian elements, that is, the arithmetic mean of the Keplerian elements deviations of the final orbit with respect to the reference orbit (final orbit without errors in the thrust vector) to the both maneuvers. The assumption that we used in this work for the treatment of the errors was the probabilistic assumption, being that the errors were assumed having zero mean uniform errors. 2

3 2. MATHEMATICAL FORMULATION AND COORDINATES SISTEM. The orbital transfer problem studied can be formulated in the following way: 1) Global minimization of the performance index: J = m(t 0 ) m(t f ); 2) With respect to α : [t 0,t f ] R ( pitch angle) and β : [t 0,t f ] R ( yaw angle) with α, β C -1 em [t 0,t f ]; 3) Subject to the dynamics: t [t 0,t f ], V r ( = V (.cosϕ ( V (.senϕ( (1) ρ ϕ µ. m m. a r ( = F.cos β(.cosϕ( (2) r 2 ( ( radial direction- X axis of the Figure 1) V T ( = V (.senϕ( V (.cosϕ( (3) ρ + ϕ m T. a ( = F.cos β(.senϕ( (4) ( transversal direction- Y axis of the Figure 1) V z ( = z& ( (5) 3

4 m. a N ( = F.sen β( (6) ( binormal direction- Z axis of the Figure 1) where, a r 2 V (. V ( V ( ( = V& (.cosϕ( V& ρ ϕ ϕ ( ) ρ ϕ t.senϕ(.cosϕ( ρ( ρ( 2. Vρ (. Vϕ (.senϕ( (7) a T 2 Vρ (. Vϕ ( Vϕ ( ( = V & ρ (.senϕ( V& ϕ (.cosϕ(.senϕ( ρ( ρ( (8) a N ( = & z ( (9) cosϕ( = sinα( (10) These equations were obtained by: 1) writing in the dexterous rectangular referential coordinates with inertial directions OX i Y i Z i the Newton s laws for the satellites (S) moving with respect to this referential, centralized in the Earth s center O, with X i axis toward the Vernal point, X i Y i plane coincident with the Earth s Equator, and Z i axis toward approximately to the Polar Star; 2) rewriting them in coordinates of the dexterous rectangular referential with radial, transversal, binormal OXYZ directions, centralized in the Earth s mass center O, X axis toward the satellite S, XY plane coincident with the plane established by the position and velocity vectors of the satellite, and Z axis perpendicular to this plane. And 3) with the aid of the correspondent cylindrical coordinates (ρ, ϕ, z), centralized in the Earth s mass center O, with ρ = r projection in the plane, ϕ = angle between OX and the line that defines ρ in the plane XY, e z = Z (that is, with the goal of obtaing the forces and velocities in the radial X, transversal Y, and binormal Z directions, 4

5 respectively, we write the radial, transversal and binormal accelerations and velocities as functions of that in cylindrical coordinates). Figure 1 - Reference System used in this work. The real thrust vector, with magnitude and directions errors is given by following expressions: r F T r r = F + F (11) r F T r r r = F + F + F (12) T 1 T 2 T3 r F T r r = F, F = F, F = F, i = 1,2,3 T Ti Ti (13) ( F + ).cos( α + α). ( β + β) F T = cos (14) 1 F 5

6 ( F + ).sen( α + α). ( β + β) F T = cos (15) 2 F ( F + ). ( β + β) F T = sen (16) 3 F where: F r, F r r e F r are the thrust vector without errors, the thrust vector, and the error in the thrust vector, respectively; α e β are the errors in the pitch and in the yaw angles, respectively; F T1, F T2 e F T3 are the components of the thrust vector with errors F T in the transversal, radial and normal directions, respectively. These equations were written in the orbit s plane, according with to the Figure 2. 6

7 Figure 2 Thrust Vector applied to the satellite. (traduzir satélite na fig) The magnitude error, F, was computed as a percentage of the nominal force, while the direction errors α e β were computed in units of angle. They are varied inside given ranges, that is, ± DES1 for F, ±DES2 for α and ±DES3 for β. This variation will correspond to the implementation of the random numbers that satisfy a uniform probability distribution into those ranges. In this way, for each implementation of the orbital transfer arc, values of α and β are chosen, whose errors are inside the range, that produce the direction for the minimum fuel consumption. 3. NUMERICAL RESULTS The simulations were performed for system configurations, that is, were realized with random values for each DES1, DES2 and DES3, that the results obtained for the final Keplerian elements represent the arithmetic mean of configurations (mean over the ensemble). The value was chosen to represent the set of simulations because the deviations of all Keplerian elements with respect to the reference go to zero for this number of simulations. The Figures 3 and 4 show graphics for the semi-major axis and eccentricity final deviations versus the simulations number, respectively M ó dulo d o de sv io d o s em i- eix o m aior (k m ) D es v io d a e x ce ntr ic idad e ( x 10 ^ - 4) N (Núme ro de simu la çõ es - amostras) N (Núm ero d e sim ulaçõ es - am ost ra s) 7

8 Figure 3 Semi-major axis Figure 4 - Eccentricity (x10-4 ) deviation vs. N deviation vs. N (Eixos em portugues) These graphics were plotted for direction random-bias errors DES2=1,0 0. The first moment or mathematical expectation computation of the Keplerian elements deviations with respect to the reference can be approximated for the arithmetic mean of them, for simulations as representing of set. So, we can compute the mathematical expectation for the deviation of any Keplerian element, K of the following way, K = E K = N = 1000 Ki ξ. f K ( ξ). dξ (17) N i= 1 It is important to remark that the mean computed em (17) is one mean inside the ensemble and not in the time. In this work we present only the results for the semi-major axis and the eccentricity with respect to the reference for one final instant. The graphics of the Figures 5 to 16 present the behavior of the first moment of these elements as function of the maximum direction errors (random-bias and white noise). For the random-bias errors we found the following results: 1) Uniform random-bias Errors: Semi-major axis (a), pitch errors 8

9 S em i- eixo m aior m édio ( k m ) Semi-eixo maior médio (km) DES2 (grau ) DES2 (grau) Figure 5 First Maneuver: a vs DES2 Figure 6 Second Maneuver: a vs DES2 We observe, in these graphics, behavior very similar for both maneuvers, although they are very different from each other. Easily we observe that the values of the mean semi-major axis present a region of decrease sufficient defined according the growth of the maximum error in pitch, DES2. The Figures 5 and 6 suggest one nonlinear law between these elements for both cases, that is, they suggest one cause vs. effect relation in the orbital transfer phenomenon, not depending of the case studied. 2) Uniform random-bias Errors: Semi-major axis (a), yaw errors S em i- eix o m aior m é dio ( k m ) S em i- eix o m aior m édio ( k m ) DE S3 (grau ) DES3 (grau) 9

10 Figure 7 First Maneuver: a vs DES3 Figure 8 Second Maneuver: a vs DES3 Once more these graphics show behaviors well defined and similar for the semi-major axis as function of the yaw errors, DES3, for both maneuvers studied. That is, there is a region of decrease well defined between the elements a and DES3. In the second case, the curve found seems to turn itself smoothier with respect to the case DES2. This fact is due to the stronger dependence of this element with DES3. So, we can say that this nonlinear relation between the cause in the initial instant and the effect in the final instant also occurs for the out-of-plane maneuvers. 3) Uniform random-bias Errors: Eccentricity (e), pitch and yaw errors 0.56 Excentricidade média Excentricidade média DES2 (grau) Figure 9 First Maneuver: e vs DES2 DES3 (grau) Figure 10 Second Maneuver: e vs DES3 10

11 These graphics show the nonlinear behavior of the mean eccentricity with the thrust direction deviations, respectively, with the pitch and yaw deviations. These graphics were plotted only for the second maneuver because in the first one the change is close to zero of the eccentricity for the usual values of DES2 and DES3. These graphics were plotted with precision of 10-3 for the eccentricity. For the white-noise errors the results were very similar to the results obtained for the random-bias errors but, the curves for the pitch errors present a more defined pattern with respect to those for the yaw errors, where small fluctuations appear in its final form. It is possible to see that the influence of the out-of-plane ( yaw ) errors is so strong in the definition of the orbital transfer trajectory. 4) Uniform white-noise Errors: Semi-major axis (a), pitch errors S em i-eix o m a io r m éd io ( km ) S e m i-e ix o m aior m é dio ( k m ) DE S2 (grau ) Figure 11 First Maneuver: a vs DES DES3 (g rau ) Figure 10 Second Maneuver: a vs DES3 11

12 5) Uniform white-noise Errors: Semi-major axis (a), yaw errors S em i- eix o m a io r m é dio ( k m ) Semi-eixo maior médio (km) DES2 (grau ) Figure 13 First Maneuver: a vs DES2 DES3 (grau) Figure 14 Second Maneuver: a vs DES3 Figures 12 to 14 show clearly the influence of the white-noise errors when the second maneuver is simulated with errors in yaw. The region of decrease still exist, as well as the nonlinear relation, but there are fluctuations in the growth of the maximum yaw error. 6) Uniform white-noise Errors: Eccentricity (e), pitch and yaw errors 12

13 Excentricidade média Excentricidade média DES2 (grau) Figure 15 Second Maneuver: e vs DES2 DES3 (grau) Figure 16 Second Maneuver: e vs DES3 These graphics show that the values of the eccentricity also fluctuate for practical maneuvers with the white-noise errors in yaw, but keeping the region of growth similar to the one verified for the random-bias errors case. So, we can say that all these results suggest and partially characterizes the progressive deformation of the trajectory distribution along the propulsive arc. It occurs due to the loss of optimality of the actual trajectories (with errors) with respect to the nominal trajectories (without errors). The dependence of the final Keplerian elements with the magnitude thrust errors for any of the cases showed that it is null for the first moment of the semi-major axis deviations, that the perturbations occured in this element were small fluctuations of the its mean deviation estimator involved into the numerical error of the experiment and that can be confirmed by Figures 17 and 18, where the mean deviation in the semi-major axis into the cone ± 1 σ (standard deviation of the deviation in the semi-major axis) is shown 13

14 Desvio do semi-eixo maior médio (km) DES1 (%) Figure 17 First case; D a vs DES1 e s (standard deviation) 1σ 1σ De sv io do s e m i-e ixo m a io r m éd io ( k m ) DES1 (%) Figure 18 Second case; D a vs DES1 e s (standard deviation) 1σ 1σ The values for DES1, DES2 and DES3 used to plot these graphics were chosen by considering the actual ranges of the usual errors and also values not usual, with the aim to verify the general behaviors. Obviously it is not practicable values for the pitch errors equal to 30,0 0 or magnitude error equal to 30,0%, for example. 4. CONCLUSION This work presents results of the thrust vector errors implementation for nonimpulsive orbital transfer maneuvers. It was verified that, for any case, the mathematical expectation of the semimajor axis in the final instant presents one nonlinear (approximately parabolic) dependence with the usual thrust direction maxima errors. The same results were verified for the mathematical expectation of the eccentricity in the final instant, for one of the maneuvers simulated. The respective dependences with the thrust magnitude errors were not verified for these expectations. The general results suggest one relation between the cause (in the initial instan and effect (in the final instan, that one progressive deformation of the trajectory distribution along the propulsive arc was verified. This deformation can be associated to the loss of the optimality of the actual trajectory with respect to the nominal trajectory. 14

15 5. REFERENCES 1- Adams, N.J.; Melton, R. G. Orbit Transfer Error Analisys for Multiple, Finite Perigee Burn, Ascent Trajectories. The Journal of Astronautical Sciences, 34(4), p , Biggs, M.C.B. The Optimisation of Spacecraft Orbital Manoeuvres. Part I: Linearly Varying Thrust Angles. The Hattfield Polytechnic Numerical Optimisation Centre, Connecticut, USA, Biggs, M.C.B. The Optimisation of Spacecraft Orbital Manoeuvres. Part II: Using Pontryagin s Maximum Principle. The Hattfield Polytechnic Numerical Optimisation Centre, Connecticut, USA, Carlton-Wippern, K. C. Satellite Position Dilution of Precison (SPDOP). AAS Paper , presented at the 1997 AAS/AIAA Astrodynamics Specialist Conference, Sun Valley, ID, August 4-7, Junkins, J. L. Adventures on the Interface of Dynamics and Control. Journal of guidance, Dynamics, and Control, 20(6),p , Kuga, H. K.; Gill, E.; Montenbruck, O. Orbit Determination and Apogee Boost Maneuver Estimation Using UD Filtering. DLR-GSOC, Wesling, Germany, 1991 (Internal Report DLR-GSOC IB 91-2) 7- Porcelli, G; Vogel, E. Two-Impulse Orbit Transfer error Analysis Via Covariance Matrix. Journal of Spacecraft and Rockets 17(3), p , Prado, A.F.B.A. Análise, Seleção e Implementação de Procedimentos que Visem Manobras Ótimas de Satélites Artificiais. Dissertação de Mestrado. Instituto Nacional de Pesquisas Espaciais (INPE). São José dos Campos, São Paulo, 1989 (INPE TDL/397). 9- Rocco, E. M. Transferências Orbitais Impulsivas Ótimas com Limite de Tempo e Erros nos Propulsores. Tese de Doutorado (em andamento). Instituto Nacional de Pesquisas Espaciais (INPE), DMC. São José dos Campos/SP- Brasil, Rodrigues, D.L.F. Análise Dinâmica da Transferência Orbital. Dissertação de Mestrado. Instituto Nacional de Pesquisas (INPE), São José dos Campos/SP, Brasil, 1991 (INPE TDI/461). 11- Santos-Paulo, M.M.N. Estudo da Influência dos Erros dos Propulsores em Manobras Orbitais Tridimensionais. Dissertação de Mestrado. Instituto Nacional de Pesquisas Espaciais (INPE). São José dos Campos, SP, SCHULTZ, W. Transferências biimpulsivas ente órbitas elípticas não coplanares com consumo mínimo de combustível. Dissertação de Mestrado. DMC. INPE - Instituto Nacional de Pesquisas Espaciais - São José dos Campos/SP - Brasil,