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Optimization of Bi-Elliptic transfer with plane change—Part I Osman M. Kamel∗ Astronomy and Space Science Department, Faculty of Sciences, Cairo University, Giza, Egypt Received 11 December 2006; accepted 9 October 2008 Available online 6 December 2008

Abstract We introduce a more advanced analysis than that of Roth concerning the following two points of view: (1) We attempt to find an analytical solution of the optimization problem instead of Roth’s iterative solution. That means a stress is operated on the analytical approach. (2) We consider the terminal orbits to be elliptic and not circular as Roth assumed. This is a generalization of the problem. We shall complete the solution of this Bi-Elliptic transfer with plane change problem in the second part of the present research paper (I). © 2008 Elsevier Ltd. All rights reserved.

1. Introduction Some investigations confirmed that Bi-Elliptic transfer (three impulse transfer) are more economical than the two impulse transfer, for some types of transfer, between coplanar and moreover non-coplanar orbits as well [1]. It is demonstrated that the Bi-Elliptic transfer is the optimal transfer among all three impulse transfers, for the systems of coplanar circular orbits, when the application of the gradient method [2]. When transfer problem is applied to planetary systems, it was shown that eccentricities are of small effects (two dimensional types), whilst inclinations are of very great effects (three dimensional types) [3]. Neustadt, Potter and Stern exposed that minimum consumption of fuel transfer corresponds to a number of impulses which is equal at most to the number of state variables in the final orbit [4].

Prime vector theory is utilized for the special case of coplanar circle to circle optimal rendezvous, time fixed, and multiple impulse solutions [5]. Optimization of space vehicle trajectories, especially the subject of orbital transfer, and rendezvous is considered by Marec as an important application of optimal control. Using very general methods, minimization of fuel expenditure is taken into account, i.e. the maximization of the final mass. A more sophisticated optimization method may be needed, that is functional optimization, more precisely the Contensou–Pontryagin maximum principal. The numerical approach is usually required for fixed time finite transfers [6]. Roth presented a minimized characteristic velocity increment, required for Bi-Elliptic transfer, for the configuration of noncoplanar circular orbits. He reformulated the system of equations to be adequate for an iterative solution [7]. 2. Method and results We refer to Fig. 1 of Roth research Let

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E-mail addresses: [email protected], [email protected] 0094-5765/$ - see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2008.10.002

x = rt /ri ;

y = rt /r f

u, −u−v, v the plane change at ri , rt , rf .

O.M. Kamel / Acta Astronautica 64 (2009) 514 – 517

ri rt rf a1 e1 a2 e2 aT eT

semi-major axis of the second transfer orbit eccentricity of the second transfer orbit first plane change angle second plane change angle third plane change angle total plane change angle ( = 1 +2 +3 = u+2 +v) first velocity impulse at ri second velocity impulse at rt third velocity impulse at rf constant of gravitation

aT eT 1 2 3

Nomenclature peri-apse of the initial elliptic orbit apo-apse of the first transfer orbit peri-apse of the final elliptic orbit semi-major axis of the initial terminal orbit eccentricity of the initial terminal orbit semi-major axis of the final terminal orbit eccentricity of the final terminal orbit semi-major axis of the first transfer orbit eccentricity of the first transfer orbit

V1 V2 V3

∆V3

515

α3 T`

rf

Initial Orbit

∆V1 ri

α1 E

rt α2

Final Orbit

T

ΔV 2

θ

rf

Fig. 1. The general Bi-Elliptic transfer maneuver (ri < rt < r f ).

We have the following relationships according to Fig. 1: rt = xri = xa1 (1 − e1 ) = aT (1 + eT ) = aT (1 + eT ) = yr f = ya2 (1 − e2 ) ri = a1 (1 − e1 ) = aT (1 − eT ) r f = a2 (1 − e2 ) = aT (1 − eT ) aT = 21 (ri + rt ); aT = 21 (r f + rt )

(1)

The total characteristic velocity is given by VT = V1 + V2 + V3 = f (x, y, u, , v)

(2)

V1 , V2 , V3 are the increments in velocity at ri , rt , rf , respectively. x, u, v are considered as the three independent variables, and , y are the dependent variables, since y=

a1 (1 − e1 ) x a2 (1 − e2 )

(3)

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O.M. Kamel / Acta Astronautica 64 (2009) 514 – 517

=−u−v

(4)

is given, accordingly we may write VT = f (x, u, v)

i.e. V2 = (V3 )2 =

(5)

VT is acquired by the trigonometric cosine formula from the following equalities after some reductions: (V1 )2 =

i.e. (V1 )2 =

(1 + e1 ) (1 + eT ) + a1 (1 − e1 ) aT (1 − eT ) (1 + e1 ) (1 + eT ) −2 cos u a1 (1 − e1 ) aT (1 − eT ) (1 + e1 ) 2 rt + a1 (1 − e1 ) (ri + rt ) ri (1+e1 ) 2 rt cos u −2 a1 (1−e1 ) (ri +rt ) ri

(6)

(V2 )2 =

(7)

(8)

(1 − eT ) (1 − eT ) + aT (1 + eT ) aT (1 + eT ) (1 − eT ) (1 − eT ) −2 aT (1 + eT ) aT (1 + eT ) × cos( − u − v)

2

⎡

2xa1 (1 − e1 ) a2 (1 − e2 ){a2 (1 − e2 ) + xa1 (1 − e1 )} 2 2xa1 (1−e1 )(1+e2 ) − cos v a2 (1 − e2 ) a2 (1−e2 )+xa1 (1−e1 )

+

= f 3 (x, v)

(14)

√ i.e. V3 = f 3 (x, v). The first order partial derivative of VT w.r.t. the variables x, u, v should be equal to zero, as a condition of minimization of the consumption of fuel, according to the principles of infinitesimal calculus. Consequently

jVT jV1 jV2 j( − u − v) = + ju ju j( − u − v) ju +

i.e.

i.e.

jV3 jv =0 jv ju

jVT jV1 ju jV2 j( − u − v) = + jv ju jv j( − u − v) jv (10)

+

jV3 =0 jv

a1 (1 − e1 ) a2 (1 − e2 ) + a1 (1 − e1 ) + xa1 (1 − e1 ) a2 (1 − e2 ) + xa1 (1 − e1 )

⎤

⎥ ⎥ ⎦ a1 (1 − e1 )a2 (1 − e2 ) −2 cos( − u − v) {a1 (1 − e1 ) + xa1 (1 − e1 )}{a2 (1 − e2 ) + xa1 (1 − e1 )} = f 2 (x, − u − v) (11)

(V2 )2 =

⎢ 2 ⎢ xa1 (1 − e1 ) ⎣

(12)

(1 + e2 ) a2 (1 − e2 )

jVT jV1 jV2 jV3 = + + =0 jx jx jx jx

(9)

rf ri 2 (V2 ) = + r t ri + r t rt + r f ri r f −2 cos(−u−v) (ri +rt )(r f +rt )

(1 + eT ) (1 + e2 ) + a2 (1 − e2 ) aT (1 − eT ) (1 + e2 ) (1 + eT ) −2 cos v a2 (1 − e2 ) aT (1 − eT )

(1 + e2 ) 2 rt + (V3 ) = a2 (1 − e2 ) (rt + r f ) r f (1 + e2 ) 2 rt cos v −2 a2 (1 − e2 ) rt + r f rf (13) (V3 )2 =

2x (1 + e1 ) + a1 (1 − e1 ) a1 (1 − e1 )(1 + x) (1 + e1 ) 2x −2 cos u a1 (1 − e1 ) a1 (1 − e1 )(1 + x)

= f 1 (x, u) √ i.e. V1 = f 1 (x, u). Similarly

f 2 (x, − u − v) and

2

i.e. (V1 )2 =

O.M. Kamel / Acta Astronautica 64 (2009) 514 – 517

where

517

3. Discussion

j( − u − v) j( − u − v) = = −1; ju jv

ju jv = =0 jv ju

u, v are the independent variables. Whence, we may write jVT jV1 (x, u) jV2 (x, − u − v) = + jx jx jx +

jV3 (u, v) jx

= f (x, u, v) = 0 jVT jV1 (x, u) jV2 (x, u, v) = − ju ju j( − u − v) = g(x, u, v) = 0 jVT jV3 (x, v) jV2 (x, u, v) = − jv jv j( − u − v) = h(x, u, v) = 0

(16a)

or more concisely

Our present analysis is more progressive than that of Roth. In this Section 3, we give an outline of the contents of the whole research paper, Parts I, II. In this Part I, and referring to Fig. 1, we evaluated the expressions for V1 , V2 , V3 the three increments of velocity at the line of nodes. We applied the condition of minimization for (V1 , V2 , V3 ), through the application of the principles of ordinary partial infinitesimal differential calculus. Accordingly we reduced to zero the first order partial derivatives of VT w.r.t. the independent variables x, u, v namely the three functional equations f(x, u, v) = 0, g(x, u, v) = 0 and h(x, u, v) = 0. In Part II, we shall derive the detailed solution of these functional equations, which will yield the optimized values of the independent variables (x, u, v), whence the estimation of (VT ) Min = f 1 (x, u) Min + f 2 (x, − u − v) Min + f 3 (x, v) Min

jVT = f (x, u, v) jx

where 2 = −u−v.

jVT = g(x, u, v) ju

References

jVT = h(x, u, v) jv

(16b)

Since is the total plane variation and is given by = u + 2 + v = 1 + 2 + 3 where u = 1 , v = 3 . Evidently Eqs. (16a) are three first order partial differential equations in the three independent parameters (x, u, v). By solution of Eqs. (16), we get the optimized values of (x, u, v), which renders the total optimized characteristic velocity VT , where VT = f 1 (x, u) + f 2 (x, − u − v) + f 3 (x, v).

[1] R.F. Hoelker, R. Silber, The bi-elliptic transfer between circular co-planar orbits, DA Technical Memo No. 2-59, Army Ballistic Missile Agency, Redstone Arsenal, AL, January 1959. [2] J. Palmore, An elementary proof of the optimality of Hohmann transfers, AIAA Journal, Engineering Notes September–October (1984). [3] J.P. Gravier, C. Marchal, R.D. Culp, Optimal impulsive transfers between real planetary orbits, Journal of Optimization Theory and Applications 15 (5) (1975). [4] T.N. Edelbaum, How many impulses?, Astronautics & Aeronautics November (1967) 64–69. [5] J. Prussing, J.H. Chiu, Optimal-multiple impulse time fixed rendezvous between circular orbits, Journal of Guidance 9 (1) (1986). [6] J.-P. Marec, Space-Vehicle Trajectories: Optimization, System & Control Encyclopedia Theory, Technology, Applications, Pergamon Press, Oxford, 1988, pp. 4481–4490. [7] H.L. Roth, Bi-elliptic transfer with plane change, Aerospace Corp El Segundo Calif Technical Operations, May 1965.