ISSN: ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 3, Issue 11, May 2014

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1 ISSN: ISO 9001:008 ertified International Journal of Engineering and Innovative echnology (IJEI olume 3, Issue 11, May 014 nalytic Solution for the problem of Optimization of the Bi-elliptic ransfer Orbits with Plane hange M.K. mmar Mathematics Dept. Faculty of Science, Helwan University, (11795 airo, Egypt bstract his study gives the optimal total characteristic velocity required to transfer a space-vehicle between non-coplanar elliptic orbits having common center of attraction and collinear major axes (coaxial ellipses. he bi-elliptic transfer considered here consists of two semi-elliptic transfer orbits connecting the initial and final orbits which are considered ellipse also. For the optimal 3-impulse transfer, we have 3-plane changes, with angles α, β, γ respectively with their sum as the total inclination of the final to the initial orbital planes. he method were illustrated though two examples. Index erms Space maneuvers, Bielliptic ransfer, Optimization, Plane hange. I. INRODUION he determination of a specific orbit and the procedure to calculate orbital maneuvers for artificial satellites are problems of extreme importance in the study of orbital mechanics. herefore, the transferring problem of a spaceship from one orbit to another and the attention to this subject has increased during the last years. R. H.Goddard [1] was one of the first researchers to work on the problem of optimal transfers of a spacecraft between two points. He proposed optimal approximate solutions for the problem of sending a rocket to high altitudes with minimum fuel consumption. fter him comes the very important work done by Hohmann []. He solved the problem of minimum Δ transfers between two circular coplanar orbits. His results are largely used nowadays as a first approximation of more complex models and it was considered the final solution of this problem until detailed study of this transfer can be found in Marec [3] and an analytical proof of its optimality can be found in Barrar [4]. he Hohmann transfer would be generalized to the elliptic case (transfer between two coaxial elliptic orbits by Marchal [5]. Smith [6] shows results for some other special cases, like coaxial and quasicoaxial elliptic orbits, circular-elliptic orbits, two quasi-circular orbits. Hohmann type transfers between non-coplanar orbits are discussed in several papers, like Mcue [7], that study a transfer between two elliptic inclined orbits including the possibility of rendezvous; or Eckel and inh [8] that solve the same problem with time or fuel fixed. More recently, the literature studied the problem of a two-impulse transfer where the magnitude of the two impulses are fixed, like in Jin and Melton [9]; Jezewski and Mittleman [10]. Later, the three-impulse concept was introduced in the literature. Using this concept, it is possible to show that a bi-elliptical transfer between two circular orbits has a lower Δ than the Hohmann transfer, for some combinations of initial and final orbits. Roth [11] obtained the minimum Δ solution for a bi-elliptical transfer between two inclined orbits. Following the idea of more than two impulses, we have the work done by Prussing [1] that admits two or three impulses; Prussing [13] that admits four impulses; Eckel [14] that admits N impulses. In this study we shall obtain the optimal total characteristic velocity required to transfer a space-vehicle between non-coplanar elliptic orbits having common center of attraction and collinear major axes (coaxial ellipses. he bi-elliptic transfer considered here consists of two semi-elliptic transfer orbits connecting the initial and final orbits which are considered ellipse also. herefore three impulses are required, and these are assumed to occur only at the apsides of the ellipses. For the optimal 3-impulses, we have 3-plane changes, with angles,, respectively with their sum as the total inclination of the final to the initial orbital planes. he first and the third are partial angles of plane change and are always small. Hence the second plane change, is dominant, which takes place at the coincident apo-centers of the first and the second transfer orbits. II. DESRIPION OF HE MNEUER It is required to transfer a space-vehicle which is moving in an elliptic orbit, with eccentricity e 1, semi-major axis a 1 and peri-centre distance r 1, to another non-coplanar elliptic orbit with parameters e, ( a and peri-centre distance r, he two orbits having a common center of attraction and collinear major axes as shown in Fig.1. We have 4-feasible configurations of this maneuver, discussed in the planar case by Kamel, mmar [15]. Here we shall consider the first configuration in which the transfer orbits O and O connect the peri-centre of the initial orbit O 1 and the peri-centre of the final orbit O. he geometry of the maneuver is described in Fig.1. From Fig. (1, we can obtain the following relations, resulting from the cotangential conditions at the apsides. - t the peri-center : r 1 r ; a 1 e a (1 e (1- a 1( 1 8

2 ISSN: ISO 9001:008 ertified International Journal of Engineering and Innovative echnology (IJEI olume 3, Issue 11, May 014 r r ; 1 a ( 1 e a (1 e (1- b r a - t the apo-center B: B B - t the peri-center : ; a ( 1 e a (1 e r r (1- c lso, we put r E a(1 e, rd a1(1 e1 (1- d r rb r r r r B 1 r a r r E r r r r E (- e (- f Where GM, is the gravitational constant of the celestial body in our problem? Now, we calculate the three required impulses for the maneuver: (i he first impulse is applied at the peri-center 1 of the initial orbit at an angle to the initial plane, which changes the orbital velocity from 1 to, causing a rotation of the orbital plane of the first transfer orbit O by the angle about the line of nodes (co-axial major axis of the orbits. his impulse is given by Fig. 1: Geometry of the maneuver Referring to Fig. 1, and using the vis - visa equation 1 r a We can obtain the velocities at the apsides of the orbits as: 1 B 1 rd r a r r r 1 D 1 r B r a r r rb 1 r r rb a r rb r rb B 1 r B a r r r r B r r B (-a (-b (-c (- d cos rd rb r r r r r D B r D rb cos r rd r rb (ii he second impulse is applied at the apo-center B of the first transfer orbit at an angle to the plane, which changes the orbital velocity from B to B, causing a rotation of the orbital plane of the second transfer orbit by an angle about the line of nodes. his impulse is given by cos B B B B r r r r r r r r r B B B r r cos r rb r rb (3 (4 9

3 ISSN: ISO 9001:008 ertified International Journal of Engineering and Innovative echnology (IJEI olume 3, Issue 11, May 014 (iii Finally, he third impulse 3 is applied at the peri-center of the second transfer orbit O r making an angle a 1e with the orbital plane, which changes the orbital velocity from to, causing a rotation of the final orbit O by an angle about the nodal line and transferring the vehicle to the target orbit. his impulse is given by cos 3 r rb r E r r r r r r B E r r B E cos r rb r r E Such that we use relations (1 and ( to get the equations (3, (4 and (5. (5 1 1 (11 he two variables x, y are not independent, and we can obtain the relation between them as : y r r r r r r B B k x (1 he total characteristic velocity 3 i / i 1 could be written in terms of x and the angles α, β, γ as : x h x h cos 1x 1x cos x 1 x 1 k x (1 x (1 k x III. NLYI SOLUION o simplify the notations, we shall introduce new variables x, y defined as: rb 1 e x, r 1 e r B 1 e y (6 r 1 e lso, introducing the new constants k, h and g, defined as: k r r /, g 1 e, h1 e1. (7 he three impulses in terms of the new notations can be written as: x h x 1x 1x 1 h cos 1 1 x 1 x 1 y 1 cos (1 x(1 y y y 1 3 x 1 y 1 g 1 y cos 1g 1 y (8 (9 (10 Where is the circular velocity at peri- center, of the initial elliptic orbit, given by k x g k x k g cos 1k x 1k x hus, we can write (13 ( x, B( x, k ( x, (14 where x h x ( x, h cos 1x 1x 1 1 B( x, x 1 x 1 k x 1 cos (1 x (1 k x k x g k x ( x, g cos 1k x 1k x (15 (16 (17 We shall use the Lagrange s multiplier technique to obtain the extreme values for the total impulse required for the maneuver. hus define the characteristic function Fx (,,, as: F ( x,,, ( x,,, ( (18 where θ, is the total inclination between the initial and final orbits, and λ is the Lagrange s multiplier. he optimization 10

4 ISSN: ISO 9001:008 ertified International Journal of Engineering and Innovative echnology (IJEI olume 3, Issue 11, May 014 conditions that minimize the total impulse are: F 0 x F F F, 0, 0 and 0 (19 Using (14, we obtain the optimization condition of the characteristic function with respect to x as: F / x B / x / x k 0 x B Using (15-(17, we obtain: F 1 x h 1 x cos x x (1 x k k x g k 1kx cos x (1 kx 1 (1 kx k (1 x B x (1 x (1 kx (1 k kx (1 x (1 kx cos x (1 x (1 kx 1 (1 kx (1 x B x (1 x (1 kx (1 x (1 kx cos 0 x (1 x (1 kx (0 (1 he optimization condition of the characteristic function with respect to the inclinations i.e. the distribution of the plane change among the three maneuvers according to the optimality conditions, are given by: hx sin F 1 x 0 1 sin F x (1 x (1 k x 0 B g k x sin F 1 kx 0 Where we have used the relation: above equations we can obtain the relations: ( (3 (4. From the 1 hx sin 1 x 1 sin B 1 (1 x x k x k g x sin 1 kx (5 Since we have assumed that α and γ are small, then without a loss of generality we can take as a simplification the case where α = γ. In this case we have (6 Equating (, and (4, we obtain after simplifications: k g 1 x h 1 k x (7 Substituting the values of and, squaring, and solving for cos α in terms of x, we obtain: cos hg (1 k [ hk ( g k g ( h] x 1 1 h gkx kx gk hx x o obtain cos β, in terms of x, we put cos ( x, cos ( x, p cos, q sin Knowing that cos cos( pcos q sin hen cos ( x p [ ( x 1] q ( x 1 ( x (8 (9 Substituting cos α, cos β from (8, (9 into (13 and (1, we obtain the total characteristic impulse and d /dx in terms of x as: 11

5 ddxaxis axis 1/ ISSN: ISO 9001:008 ertified International Journal of Engineering and Innovative echnology (IJEI x h x h ( x 1x 1x 1 1 x 1 x 1 k x 1 ( x (1 x (1 k x 1/ 1/ 1/ kx gkx k g ( x 1kx 1kx ( ( x h x x x x (1 x (31 1/ k k x g k 1 kx ( x x (1 kx 1 (1 kx k (1 x B x (1 x (1 kx (1 k kx (1 x (1 kx ( x x (1 x (1 kx 1 (1 kx (1 x B x (1 x (1 kx (1 x (1 kx ( x 0 x (1 x (1 kx I. NUMERIL EXMPLES Example 1: (Earth Pluto: We shall consider the bielliptic transfer from Earth to Pluto. We shall use the following data: a 1 = 1.U, a = U, e 1 = , e = , θ = he value of x, which causes the minimum total characteristic velocity Δ, can be calculated by solving equation (31. he graph of the total characteristic velocity is shown in Fig. olume 3, Issue 11, May xaxis Fig.: otal impulse vs x for Earth-Pluto bielliptic transfer he graph of the derivative of total characteristic velocity with respect to x, is shown in Fig xaxis Fig.3: Derivative of total impulse vs x for Earth-Pluto bielliptic transfer Solving equation (31 numerically and substitution into (8 - (30, we obtain the solution, x = 71.15, β = 16.80, α = γ = Solving Equations (1 and using (6, we can obtain the elements of the transfer orbits action as a function of x: x 1 kx1 e 0.977, e , kx1 x 1 r1 a ( x U. r a ( k x U. he total characteristic velocity is found to be equals to = Example : (ircular Geocentric transfer We shall consider the bi-elliptic transfer from a geocentric circular orbit of radius 7000 km to one of radius km, and the inclination between the two orbits is θ = 8.5 o. he graph of the total characteristic velocity with respect to x, is shown in Fig.4, and the graph of the derivative of total characteristic velocity with respect to x, is shown in Fig.5. Solving equation (31 numerically, we obtain the solution, x = Substitution into (8 and (9, we obtain the required angles for plane changes as: 1

6 ddx axis axis ISSN: ISO 9001:008 ertified International Journal of Engineering and Innovative echnology (IJEI α = γ = 0. 4 o, β = 7.65 o x axis olume 3, Issue 11, May 014 Fig. 4: otal impulse vs x for circular geocentric bielliptic transfer x axis Fig.5:otal impulse d/dx vs x for circular-geocentric bielliptic transfer Hence, we can obtain the elements of the transfer ellipses as: e , e , a km, a km, he total characteristic velocity is found to be equal to ONLUSION n analytical formulation was derived and implemented, to solve the problem of three impulsive orbital transfers between elliptical non-coplanar orbits in a Keplerian dynamics problem with minimum fuel consumption. numerical algorithm was developed for fast practical use to obtain the minimum velocity increment needed to perform this type of maneuver. he paper introduces a simple and straight forward algorithm for calculating the elements of the transfer orbits and change plane angles required for optimization. I. KNOWLEDGEMEN he work completed here was started on suggestions from Professor Osman M.K. he author wishes to thank Professor Osman M.K. for his interest, guidance and suggestions which allowed me to improve considerably this work. lso, I would like to thank Mr. Wael Youssif - El Sherouk cademy for helping me in reviewing the graphs of the paper. REFERENES [1] GODDRD, R.H. Method of Reaching Extreme ltitudes. Smithsonian Inst Publ Misc ollect 71(, [] HOHMNN, W. "Die Erreichbarheit der Himmelskorper", Oldenburg, Munich, 195. [3] MRE, J.P. Optimal Space rajectories, New York, NY, Elsevier, [4] BRRR, R.B. n analytic proof that the Hohmann-type transfer is the true minimum two-impulse transfer. cta stronautica, 9(1:1-11, Jan. /Feb [5] MRHL,. ransferts optimaux entre orbites elliptiques coplanaires (Durée indifférente. cta stronautic, 11(6 : , Nov./Dec [6] SMIH, G.. he calculation of minimal orbits. cta stronautica, 5(5:53-65, May [7] McUE, G.. Optimum two-impulse orbital transfer and rendezvous between inclined elliptical orbits. I Journal, 1(8: , ug [8] EKEL, K.G.; INH, N.X. Optimal switching conditions for minimum fuel fixed time transfer between non- coplanar elliptical orbits. stronautic cta, 11(10/11: , Oct. /Nov [9] JIN, H.; MELON, R.G. ransfers between circular orbits using fixed impulses. S paper In: S/I Spaceflight Mechanics Meeting, Houston, X, Feb [10] JEZEWSKI, D.J.; MILEMN, D. n analytic approach to two-fixed impulse transfers between Keplerian orbits. Journal of Guidance, ontrol, and Dynamics, 5(5: , Sept. /Oct [11] ROH, H.L. Minimization of the velocity increment for a bi-elliptic transfer with plane change. stronautic cta, 13(: , May/pr [1] PRUSSING, J.E. Optimal two- and three-impulse fixed-time rendezvous in the vicinity of a circular orbit. I Journal, 8(7:11-18, July [13] PRUSSING, J.E. Optimal four-impulse fixed-time rendezvous in the vicinity of a circular orbit. I Journal, 7(5:98-935, May [14] EKEL, K.G. Optimum transfer in a central force field with n-impulses. stronautica cta, 9(5/6:30-34, Sept./Dec [15] Osman, M.K.; mmar, M.K. elocity corrections in generalized coplanar coaxial Hohmann and bi-elliptic impulsive transfer orbits. cta stronautica, 58(5: 43-50, March