An Analytical Solution for Multi-Revolution Transfer Trajectory with Periodic Thrust and Non-Singular Elements

An Analytical Solution for Multi-revolution Transfer Trajectory with Periodic Thrust and Non-Singular Elements

By Yusuke OZAWA1)

1)Department of Advanced Energy, The University of Tokyo, Kashiwa, Japan

(Received June 21st, 2017)

Transfer trajectories from low to high altitude orbits include many revolutions, thus take longer, with low-thrust propulsion systems. Propagating this dynamics with planetary equations by full numerical method has a substantial computational cost because of dividing one revolution into tiny arcs of integration. To reduce computational cost, analytical formulae have been developed previously by many researchers. However, these formulae are restricted to particular cases or low-flexibility thrust control models. In order to achieve the desired flexibility, thrust control can be assumed as Fourier series, whose period synchronizes with one orbit period. Additionally, by averaging it over the course of one revolution, high-frequency terms were found to be reduced to a finite number of coefficients of low-frequency terms by the orthogonality condition. This provides a powerful method to reduce computational cost while bring flexibility for thrust control. However, uncertainties occur when a trajectory is close to a circular orbit and/or inclined by a certain angle. This research offers a new analytical solution to overcome above-mentioned issue by employing the equinoctial elements. This new analytical approach then chases only secular variations of exact solutions, thus can be extended to nearly circular orbit cases.

Key Words: Multi-revolutions, Averaging method, Equinoctial Elements, Analytical solution, Low-Thrust propulsion system

Nomenclature i : initial f : final µ : standard gravitational parameter for earth R : radial component − 398600 km s 2 S : circumferential component a : semi-major axis, km W : normal component e : eccentricity i : inclination, deg 1. Introduction Ω : right ascension of ascending node, deg ω : the argument of periapsis, deg When a spacecraft using a low-thrust propulsion system es- θ : the argument of latitude, deg capes from a central body, the trajectory rotates up around ω˜ : the longitude of periapsis, ω + Ω, deg the body and the shape becomes spiraling. Designing many- P1 : equinoctial element for e, e sinω ˜ revolution orbit, in which acceleration continuously varies, is P2 : equinoctial element for e, e cosω ˜ computationally expensive, because each integration step has Q1 : equinoctial element for i, tan i/2 sin Ω to solve equations of motion which are represented by differ- Q2 : equinoctial element for i, tan i/2 cos Ω ential equations with varying thrust profile. Many studies have M : mean anomaly, deg tackled to reduce the computational cost and analytical solu- f : true anomaly, deg tions have been developed in many special cases, such as a E : eccentric anomaly, deg low-eccentricity spiraling model. However, a general design l : mean longitude, M + ω˜ , deg methodology which can solve various many-revolution have L : true longitude, f + ω˜ , deg never been developed. K : eccentric longitude, E + ω˜ , deg Being adapted to general orbit variance, a design method em- b : semi-minor axis, km ploying perturbation theory have been studied.1–3) A spacecraft h : angular momentum, km2s−1 using low-thrust propulsion have been treated as perturbed Ke- p : semi-latus rectum, km plerian motion. Approximating it with first order perturbation n : mean motion, s−1 expansion on one revolution, and formulating analytical solu- T : period, s tions, we can eventually know secular variations with low com- r : radius, km putational cost. However these methods model manoeuvring A : thrust acceleration, km s−2 pattern as constant control for each circles. In other approach, æ : equinoctial elements [a, l, P1, P2, Q1, Q2] averaged time rate solutions have been found analytically by x : Keplerian elements [a, e, i, Ω, ω, M] treating electric thrust profile as a periodic function with Fourier s : state vector [rx, ry, rz, vx, vy, vz] series. This periodical maneuvering, which is included in vari- α : cos coefficient of thrust Fourier ational equations of classical orbit elements, could be reduced β : sin coefficient of thrust Fourier to finite Fourier coefficients by using trigonometric orthogonal- Subscripts ity conditions,4) i.e. Thrust Fourier Coefficients (TFCs). The Gaussian form Lagrange planetary equations with TFCs are in- tegrated over many-revolution with considerably low computa- Integrating these differential equations with any numerical tional time and can chase secular variations of a transfer. This methods, a detailed time history of osculating orbit elements analytical method gives high thrust flexibility to trajectory de- can be shown. Note that following Kepler’s equations should sign, and effectively finds out trajectory revolutions by reduc- be solved to obtain the identical osculating true anomaly and ing short-term variance. However, this method cannot be used assign it for the next step. for all cases; e.g. dynamical changing inclination angle, which changes inclination to near zero, or transfering from LEO to = − M E√ e sin E (10) GEO and from GTO to GEO, where eccentricity is close to 1 1 + e 1 zero. These trajectories are frequently designed for maneu- tan f = tan E (11) 2 1 − e 2 vering around the Earth, however the periapsis points and the line of ascending node are indefinite. In other words, argument The mean anomaly M is the difference of the mean longitude l of periapsis, right ascension of ascending node and arbitrary and the longitude of periapsisω ˜ . The mean anomaly is trans- anomalies become mathematically undefined value. formed to new element ε with a change of variable as follow- In this paper, the above singularity is eliminated by using new 1 ing. orbital elements called equinoctial elements which have non- singularity on zero eccentricity and inclination.5) The planetary ∫ equations with equinoctial elements are averaged via TFCs the- M = ndt + ε1 − (Ω + ω) (12) ory. Some case studies in this paper prove that this method is suitable to design low eccentricity revolutions with fast com- Then Eq.(9) takes following form. putation. The combination of this method and original TFCs √ theory, which uses Keplerian elements, provides flexibility of dε a √ 1 = −2 (1 − e cos E)A + (1 − 1 − e2)(ω ˙ + Ω˙ ) maneuvering and transferring for many-revolution. dt µ R ( ) √ i 2. Formulation of Many-Revolutions + 2 1 − e2 sin2 Ω˙ (13) 2 2.1. Planetary Equations of Gauss’ form 2.2. Application of Periodic Thrust Model An acceleration of a low-thrust propulsion system is suffi- According to Fourier’s theorem, any piecewise-smooth func- ciently small compared to the central gravitational acceleration. tion f (ϑ) with a finite number of jump discontinuities on the Assuming the direction and magnitude of thrust acceleration interval (0,Λ) can be expressed by Fourier series: can be changed continuously, the following Newtonian equation is adequate for describing the dynamics. ∑∞ [ ( ) ( )] 2πkϑ 2πkϑ = u ϑ ∼ + r˙ (1) f ( ) ak cos Λ bk sin Λ (14) µ k=0 u˙ = − r + A (2) r3 d Time rate of the variational equations formulated by this New- When jump discontinuities exist, Fourier series converge to the Λ = π tonian equations are called the Gaussian form of Lagrange plan- mean of the two limits. For an interval m , the Fourier ffi etary equations. The following formula is the definition of ac- coe cients are found by ∫ celerating component in RSW (named radial-transverse-normal mπ 6) 1 or satellite coordinate system) reference frame, a = f (ϑ)dϑ (15) ( ) ( ) ( ) 0 mπ r r × u r × u r ∫ 0 ( ) A = A + A + A × (3) 2 mπ 2kϑ D R r W |r × u| S |r × u| r a = f (ϑ) cos dϑ (16) k mπ m ∫0 ( ) One of the expressions for Gaussian planetary equations in 2 mπ 2kϑ 5,7) = ϑ ϑ RSW frame as below. bk π f ( ) sin d (17) ( ) m 0 m da 2a2 p = e sin f AR + AS (4) dt h r For any given acceleration component A , each component can [ ] D de 1 then be represented as Fourier series over an arbitrary interval. = p sin f AR + {(p + r) cos f + re} AS (5) dt h The Fourier series can be expanded in a time-varying orbital di r cos θ , , ϑ = A (6) parameter, such as f E M for . dt h W Ω θ ∑∞ d = r sin ( ) AW (7) = αD ϑ + βD ϑ dt h sin i AD k cos k k sin k (18) dω 1 k=0 = {−p cos f AR + (p + r) sin f AW } = , , dt h f D R S W r sin θ cos i − A (8) h sin i W Thrust curve of each component, that are represented as Fourier dM b series, are substituted to Gaussian planetary equations. Eqs.(4)- = n + {(p cos f − 2re)A − (p + r) sin f A } (9) dt ahe R W (9) 2.3. Averaging method These particular remaining coefficients are called as Thrust Averaging the alternation of an open circuit, total amount Fourier Coefficients (TFCs)4) is next presented. of variation for one revolution is solved without calculating αR , βR, αS βS , αW βW sequential change. The orbital elements except for anomaly 0,1,2 1 0,1,2 1,2 0,1,2 1,2 (30) changes slowly for one revolution. Therefore, these elements In this paper, for distinction with TFCs derived from averag- can be considered as constant parameter in the averaging pro- ing with equinoctial elements, these coefficients are named as cess. ∫ ∫ KTFCs (Keplerian TFCs). 1 2π 1 2π x˙¯ = x˙dM = (1 − e cos E)x˙dE (19) 2π 0 2π 0 3. New Formulation with Equinoctial Elements It is notable that this integration takes a processable form owing to the permutation from mean anomaly to eccentric one because 3.1. Planetary Equations with Equinoctial Elements Ω the denominator (1 + e cos f )k emerges in the integration pro- Right ascension of ascending node and argument of pe- ω cess. Some pairs of trigonometric functions appear in the right riapsis are defined upon the line of ascending node, hence, ∼ hand side integrands. Almost orders can be eliminated by the when the orbit inclination is close to zero (i 0), these values Ω ω trigonometric orthogonality condition as follows. ( and ) become undefined ones due to the undetermined line  of ascending node. Furthermore, ω and the anomalies f, M, E ∫ Λ  ,  0 n m are measured from the direction of periapsis, thus as the orbit cos nϑ cos mϑdϑ =  Λ n = m = 0 (20)  shape is approaching to a true circle (e ∼ 0), these values (ω 0 Λ/2 n = m , 0 and anomalies f, M, E) become undefined due to the undeter- ∫ Λ { 0 n , m ω, Ω, i, e sin nϑ sin mϑdϑ = (21) mined periapsis. Accordingly, , and anomalies should Λ/2 n = m , 0 be transformed into new elements which have no singularity 0 ∫ Λ for above cases. In order to find the non-singular elements, new sin nϑ cos mϑdϑ = 0 all cases (22) elements must be defined so as not to depend on the line of 0 ascending node and periapsis. ffi Using this orthogonality condition, all Fourier coe cients ex- For instance, adding both Ω and ω, the singularity about the th st nd cept for 0 , 1 and 2 order are removed and can take a simpler line of ascending node is vanished. The next formulation define 4) form of averaged variational equations. the longitude of periapsisω ˜ . ⟨ ⟩ √ ( ) da a3 1 √ ω ≡ Ω + ω = 2 eβR + 1 − e2αS (23) ˜ (31) dt µ 2 1 0 ⟨ ⟩ √ ( Nevertheless, the singularity related to eccentricity e still exists, de a √ 1 √ = 1 − e2 1 − e2βR + αS in other words, there is a singularity related to periapsis, so the dt µ 2 1 1 ) longitude of periapsisω ˜ itself is not a non-singular element. 3 1 Similarly, adding M toω ˜ , new element is defined as below, and − eαS − eαS (24) 2 0 4 2 is not singular for both the line of ascending node and periapsis. ⟨ ⟩ √ { di a 1 1 = √ + 2 ωαW l ≡ ω˜ + M (32) (1 e ) cos 1 dt µ 1 − e2 2 √ The mean anomaly M can be replaced by the mean longitude 3 W 1 W − e cos ωα − 1 − e2 sin ωβ l, however, it has not yet applied to the uncertainty about true 2 0 2 1 } anomaly f (See the original Gaussian planetary equations as 1 1 √ − e cos ωαW + e 1 − e2 sin ωβW (25) shown in Eqs.(4)-(9)). This singularity can be detected by ex- 4 2 4 2 ⟨ ⟩ √ { amining Kepler equation thoroughly. dΩ a 1 1 √ = √ − 2 ωβW 1 e cos 1 (26) l = ω˜ + M = ω˜ + E − e cos E (33) dt µ 1 − e2 2 1 1 √ + + 2 ωαW − − 2 ωβW Here, the eccentric longitude is defined so as to correspond to (1 e ) sin 1 e 1 e cos 2 2 } 4 the true longitude, 1 W − e sin ωα (27) K ≡ ω + E 4 2 ˜ (34) ⟨ ⟩ √ { L ≡ ω˜ + f (35) dω a 1 1 √ √ = − 1 − e2αR + e 1 − e2αR dt µ e 2 1 0 } Then, Kepler’s equation is rewritten as following 1 1 ⟨ ⟩ + (2 − e2)βS − eβS − cos i Ω˙ (28) l = K + e sinω ˜ cos K − e cosω ˜ sin K (36) 2 1 4 2 ⟨ ⟩ √ { } dε a 1 Moreover, the orbit equation could be transformed to the eccen- 1 = (−2 − e2)αR + 2eαR − e2αR dt µ 0 1 2 2 tric longitude and the true longitude. According to the definition ( √ )( ⟨ ⟩) of the orbit equations. + 1 − 1 − e2 ⟨ω˙ ⟩ + Ω˙ r = a − e E √ i ⟨ ⟩ (1 cos ) +2 1 − e2 sin2 Ω˙ (29) = − ω − ω 2 a(1 e sin ˜ sin K e cos ˜ cos K) (37) and, found in the above process, and there are as below.5) { } da 2a2 p r = p/(1 + e cos f ) = (P sin L − P cos L)A + A (45) dt h 2 1 R r S = p/(1 + e sinω ˜ sin L + e cosω ˜ cos L) (38) [{ ( ) } dl r a p 2b = n − (P sin L + P cos L) + A dt h a + b r 1 2 a R Eccentricity e and longitude of periapsisω ˜ are appeared like ( ) a p e sinω ˜ and e cosω ˜ on both the orbit equations and Kepler’s + + − + 1 (P1 cos L P2 sin L)AS equation. These terms can be replaced to new elements which a b r + (Q cos L − Q sin L)A ] (46) avoid the singularity of periapsis in low eccentricity case. [ 1 2 { (W ) } dP r p p 1 = − cos LF + P + 1 + sin L A dt h r R 1 r S P1 ≡ e sinω ˜ and P2 ≡ e cosω ˜ (39) −P (Q cos L − Q sin L)A ] (47) [ 2 1 { 2 ( W ) } ff dP2 r p p Di erentiating these new elements with respect to time, by sub- = sin LFR + P2 + 1 + cos L AS stituting the variational equations for e andω ˜ as defined in dt h r r +P1(Q1 cos L − Q2 sin L)AW ] (48) Eq.(5),(7) and (8), the new planetary equations for P1 is defined dQ r as below. 1 = (1 + Q2 + Q2) sin LA (49) dt 2h 1 2 W dP1 dω˜ de dQ2 r 2 2 = e cosω ˜ + sinω ˜ = (1 + Q + Q ) cos LAW (50) dt dt dt dt 2h 1 2 [ = −1 − + where, p cos LFW (p r) sin LAS √ h  θ i  b = a 1 − P2 − P2 (51) r sin tan  1 2 −rP A + 2 P A  (40) 1 W h 2 W h = nab (52) p = 1 + P1 sin L + P2 cos L (53) In the case of P2, the same procedure is applied. r r h And, argument of latitude also needs to be expressed by the = (54) true longitude L. h µ(1 + P1 sin L + P2 cos L) For a numerical integration algorithm, to obtain true longitude, θ = ω + = − Ω f L (41) we have to solve two transcendental equations in each iteration steps. thus, l = K + P1 cos K − P2 sin K (55) = − − sin θ = sin L cos Ω − cos L sin Ω (42) r a(1 P1 sin K P2 cos K) (56) The eccentric longitude K and the radius r are obtained by em- Right ascension of ascending node Ω is not a non-singular ele- ploying previous equations. Finally, the true longitude L is de- ment, however, sin θ emerges on the product with tan (i/2). The fined with following relational expressions. terms, tan (i/2) cos Ω and tan(i/2) sin Ω becomes the candidates [( ) ] a a a for replacing to new elements. sin L = 1 − P2 sin K + P P cos K − P r a + b 2 a + b 1 2 1 (57) ≡ i Ω ≡ i Ω [( ) ] Q1 tan sin and Q2 tan cos (43) a a a 2 2 cos L = 1 − P2 cos K + P P sin K − P r a + b 1 a + b 1 2 2 Differentiating the new elements by assigning the classical (58) planetary equations as shown in Eq.(6) and (7), the new ex- 3.2. Averaging with Eccentric Longitude pression is obtained. To reduce the computational cost, it is appropriate to apply an averaging method to design many-revolution. In the case dQ1 i dΩ 1 i di = tan cos Ω + sec2 sin Ω of Gaussian planetary equations with equinoctial elements, the dt 2 dt 2 2 dt true longitude L performs as a time-varying parameter. Accord- r 2 i = sec (sin θ cos Ω + cos θ sin Ω)AW ingly, when the planetary equations are averaged with respect to 2h 2 + + r the true longitude, the factor (1 P1 sin L P2 cos L) remains = (1 + Q2 + Q2) sin LA (44) 2h 1 2 W as denominator in the integrating process. ( ) ∫ ∑∞ The variations of Q2 is also introduced via the same procedure. 2π α + β sin L k=0 k cos kL k sin kL Consequently, the foundation for non-singular elements dL (59) 1 + P1 sin L + P2 cos L a, P1, P2, Q1, Q2, l is established. These elements are defined 0 upon equinoctial plane and the direction of vernal equinox. Solving these integrands numerically, it is found that integral Hence, these elements are called as equinoctial elements. Be- value on the higher order does not converge to zero. For this sides, the new formulation of Gaussian planetary equations is reason, it is impossible to find an analytical form with the true longitude. The original TFCs theory, which uses classical Ke- Only the Fourier coefficients of lower orders remain. The left plerian elements, can obtain analytical solutions by replacing hand side averaged product can be converted to the companion to eccentric anomaly E. Similarly, the eccentric longitude can of averages under the construction of low eccentricity (eq.62). ⟨ ⟩   be converted from the true anomaly with the relations given in ( ) R R da 2a2  α β  Eqs.(57)-(58) and the orbital equation Eq.(56). However, the or- = aP2 + aP2 − a − b P 1 − P 1  + 1 2 1 2 thogonality of trigonometric functions cannot be used because dt h(a b) 2 2 2 ( ) of the existence of factor (1 − P1 sin K − P1 cos K) as denomi- 2a − P2 + P2 − 1 αS (66) nator. h 1 2 0 ( ) ∫ ∑∞ 2π α + β Then, we obtain the averaged planetary equations for semi- sin K k=0 k cos kK k sin kK dK (60) major axis a. For the other elements, the same procedure al- 1 − P sin K − P cos K 0 1 2 lows it to be formulated. This averaging process reveals that This denominator is emerged due to the substitution of re- the averaged mean longitude yields the new Fourier coefficients βR ffi lational expressions, which have the term r at denominator. 2 . In this paper, we do not mention that how the coe cients Hence, these terms can be canceled by multiplying r for both works for the secular variations, yet in spite of that this fact sides. says that, around the singularity, a more detailed thrust model is performed for long-time fluctuations. Note that these remained ( ) ∫ ∑∞ ones are named as Equinoctial Thrust Fourier Coefficients set 2π α + β sin K k=0 k cos kK k sin kK r dK (ETFCs) in this paper. 1 − P sin K − P cos K 0  1 2  αR , βR , αS βS , αW βW ∫ π ∑∞ , , , , , , , , , (67) 2   0 1 2 1 2 0 1 2 1 2 0 1 2 1 2 = a K  α kK + β kK dK = aβ π sin  k cos k sin  1 The following is the vectorized form of this averaged planetary 0 = k 0 equations. The right hand sides of the variational equations are simplified ⟨ ⟩ dæ and can then use the orthogonality conditions. On the other = M f T + M f T + M f T (68) dt R R S S W W hand, the left hand side is represented as following. [ ] ⟨ ⟩ f = αD αD αD βD βD D = {R, S, W} ⟨rx˙⟩ = ⟨r⟩⟨x˙⟩ + r′ x˙′ (61) D 0 1 2 1 2 M , M , M nd R S W are the modulus matrices for each ETFCs as The 2 term on the right hand side shows the averaged product shown in Eqs.69,70,71. of both the deviation of radius and the variances of elements, 3.3. Applying to Integration Algorithm which correspond to the covariance for one revolution. When The appropriate Keplerian elements are decided for the be- orbit is extremely almost a circle, the deviation of radius r′ be- nd ginning as osculating orbit elements. These elements are comes small so as to be able to ignore the 2 term. Conse- converted to the equinoctial elements with the equations as quently, the merit to be able to avoid the singularity problem in shown in Eqs.(32),(39) and (43). Next, these initial osculat- low eccentricity is guaranteed. ing equinoctial elements are determined as the initial condition ⟨ ⟩ ⟨rx˙⟩ ≃ ⟨r⟩⟨x˙⟩ = a ⟨x˙⟩ (62) æi, and calculate the variance of this revolution æ˙ with the ∫ π algebraic expression as shown in Eq.(68). The period of this 1 2 ⟨r⟩ = a(1 − P sin K − P cos K)dK = a (63) revolution is solved with initial osculating semi-major axis. Fi- 2π 1 2 0 nally, the final osculating equinoctial elements are propagated For instance, the formulation process for a is shown. At first, like below form. ⟨ ⟩ both sides of the variational equations as in Eq.(45) are multi- dæ plied by the radius r, and the previous equations (Eqs.56,57 and æ f = T + æi (72) dt 58) are applied to the first and the second term. Each final osculating equinoctial elements are solved similarly 2 da 2a for each revolution; thus, the number of revolutions are 10, and r = (P2r sin L − P1r cos L) FR dt h the number of solutions are 11. The solutions are linearly inter- 2a2 polated in between. To understand instinctively the variations + (r + P r sin L + P r cos L) F h 1 2 S of many-revolution, equinoctial elements should be convert to 2a3 ( ) Keplerian elements. Keplerian is obtained by these equations.5) = aP2 + aP2 − a − b (P K − P K) + 1 2 1 cos 2 sin h(a b) e2 = P2 + P2 (73) 3 ( ) 1 2 −2a 2 + 2 − i P1 P2 1 (64) tan2 = Q2 + Q2 (74) h 2 1 2 averaging this, P1 tanω ˜ = (75) ⟨ ⟩   P2 3 ( )  αR βR  da 2a 2 2  1 1  Q r = aP + aP − a − b P1 − P2  Ω = 1 dt h(a + b) 1 2 2 2 tan (76) Q2 2a3 ( ) − P2 + P2 − 1 αS (65) h 1 2 0   2 δϵ − 2 δϵ −  a ( 1 1)P1 − a ( 1 1)P2   0 h 0 h 0   δ((h−b)ϵ −2b) δ(4b−h)P δbϵ δ(4b−h)P δbϵ   1 2 2 1 − 5   µ 2µ 2µ 2µ 2µ   h(δP2−1) ϵ   hP2 1 ha 5   µ µ 0 − µ 0  MR =  2 4  (69)  ϵ h(1−δP2)   − hP1 ha 5 0 2 0   µ 4µ 2µ   0 0 0 0 0  0 0 0 0 0    0 0 0 0 0   δ λ − δ λ −   −λ P1 1 Q1 − P2 1 Q2   4 2 0 2 0   3aP λ aP (Q −δP λ −P λ ) aP (δϵ λ −λ ) aP (−Q +δP λ −P λ ) aP (δϵ λ −λ )   − 2 4 − 2 1 1 1 1 4 − 2 5 1 2 − 2 2 2 1 1 4 − 2 2 1 3   2hλ −δ 2hλ − λ δϵ 2λh −λ − +δ2hλ − λ δϵ 2λh −λ  MW =  3aP1 4 aP1(Q1 P1 1 P2 4) aP1( 5 1 2) aP1( Q2 P2 1 P1 4) aP1( 2 1 3)  (70)  2h 2h 2h 2h 2h   2 2 2 2 2 2 2   a(1+Q +Q )P1 a(1+Q +Q )δϵ5 a(1+Q +Q )(1−δP )  − 1 2 1 2 1 2 2   2h 8h 0 4h 0   a(1+Q2+Q2)P a(1+Q2+Q2)(1−δP2) a(1+Q2+Q2)δϵ  − 1 2 2 1 2 1 1 2 5 2h 4h 0 8h 0   2 ϵ −  − 2a ( 1 1)   h 0 0 0 0   δ δϵ − µ+ 2 δ δϵ − ϵ µ δ δϵ − µ+ 2 δ δϵ − ϵ µ   0 ( 1 1)(a h )P1 − ( 1 1)a 5 − ( 1 1)(a h )P2 ( 1 1)a 2   2hµ 4hµ 2hµ 2hµ   2µ ϵ + 2 δ µ− 2 2µ ϵ − δµ ϵ + ϵ + 2 δϵ − δµ −ϵ + −ϵ   a ( 1 2h )P1 a (b h ) a P1( 1 1) a(2 (a 1 b 3) 2h ( 4 1)) a P2(a(1 1) b(1 5))   µ − µ − µ − µ − µ  MS =  2h 4h 4h 4h 4h  (71)  µ ϵ − − 2 − δµ ϵ + ϵ − 2 δϵ + + 2µ δµ ϵ − − ϵ − 2 2− µ ϵ aδµP (a(ϵ −1)+b(P2+ϵ )   a(a ( 1 1) 2h )P2 a( 2 (a 1 b 4) 2h ( 3 b) 2a ) a P2(a( 1 1) b( 2 1)) a (h b ) 5 1 1 2 )   2hµ 2hµ 2hµ 4hµ 4µ   0 0 0 0 0  0 0 0 0 0

Where, ϵ = 2 + 2 ϵ = 2 − 2 ϵ = 2 − ϵ = 2 − ϵ = 1 P1 P2 2 P1 P2 3 P1 1 4 P2 1 5 2P1P2 a δ = λ = P Q + P Q λ = P Q + P Q λ = P Q − P Q λ = P Q − P Q a + b 1 1 1 2 2 2 1 2 2 1 3 1 1 2 2 4 1 2 2 1

4. Numerical Verification term fluctuation in the analytical method. Furthermore, note that the computational time with the analytical one is consider- The configuration of a modeled spacecraft for numerical veri- ably reduced. Figure 2 shows the accuracy of the new analytical fication is set as 4 mN on the maximum thrust power, and 500 kg method by comparison with previous analytical methods, which on spacecraft mass. Numerical verification compare the analyt- uses the variational equations with Keplerian. In the previous ical method and numerical integration method with some thrust method, long-term fluctuations could not be followed. Since maneuvering models and initial osculating orbit elements. the value of eccentricity is negative, it will result in more errors in the second iteration. In contrast, the new analytical solution 4.1. Comparison with Benefit of Accuracy on 1 Revolution can follow the long-term variation as well as in the result of the The first verification model is randomly maneuvering thrust osculating equinoctial elements as shown in Figure 1. pattern for one revolution as shown in Figure.1. The black 4.2. Many-Revolution cases line is the virtual thrust pattern as actual maneuvering up to Next, many-revolution case are analyzed under the same con- 10th order Fourier series. The red dashed line represents lower- ditions as the previous one. The result on osculating equinoc- frequency maneuvering which only takes account of ETFCs. tial elements are shown in Figure 4. The analytical method can The Figure shows that the red dashed line follows a trend of follow the secular variation. The analytical solution Q2 gradu- the black line. Nevertheless, it is obvious that the both forms ally differs from the exact one. Since the value of Q2 itself is of detailed thrust are relatively different. The initial osculating relatively small, it does not significantly affect other elements. eccentricity and inclination is set as considerably low as shown Therefore, in order to prevent the order of values from being in Table.1. Other osculating orbit elements are set as insignif- largely different from each other, it is necessary to devise in nu- icant numbers. To observe this comparison easily, the number merical calculations such as nondimensionalization. In Figure of revolutions is set bound to an one revolution. 5, it shows that there is a significant validity of the analytical Table 1. Testing Initial Condition 1 results. a e i Ω ω M Subsequently, the case of some other controls are also veri- 6771 km 0.001 0.001 deg 45.0 deg 22.5 deg 0 deg fied. One is to control in the out-of-plane direction and the other is to control thrust in the in-plane direction. By verifying each Figure 1 shows a comparison of the results of the two meth- of them, it was verified whether the specificity of eccentricity ods. It can be seen that it is possible to calculate the osculat- and inclination can be avoided. ing orbital element after one revolution while ignoring short- Numerical (1.2e+00 sec) -5 Analytical EE (3.6e-03 sec) ×10 Original Thrust 6475 400 Analytical KP (2.5e-03 sec) Only 15 TFCs Thrust ) 2 2 6470 200 0

a (km) 6465 (km/s M (deg) R

A -2 6460 0 0 0.5 1 1.5 0 0.5 1 1.5 0 1 2 3 4 5 6 Time (h) Time (h) -5 ×10-3 ×10 5 400

) 2 2

e 0 200 0 Original Thrust (deg) ω

(km/s Only 15 TFCs Thrust S

A -2 -5 0 0 0.5 1 1.5 0 0.5 1 1.5 0 1 2 3 4 5 6 Time (h) Time (h) -5 ×10 400 0.15 )

2 2 0.1 200 0 (deg) i (deg) Ω

(km/s 0.05 W

A -2 0 0 0 0.5 1 1.5 0 0.5 1 1.5 0 1 2 3 4 5 6 Time (h) Time (h) Eccentric Longitude(rad)

Fig. 1. Thrust acceleration curve of randomly maneuvering pattern: Black Fig. 3. Comparison on osculating Keplerian elements. Black line and Red line is calculated with randomly generated 10th Fourier Orders. Red dashed dashed line are transformed from equinoctial elements (Figure 2). Magenta line is calculated with only ETFCs (2nd Fourier Orders) dash-dotted line is calculated by original TFCs theory with only KTFCs. (Testing thrust curve: Figure 1. Initial Condition: Table 1) Numerical (1.2e+00 sec) 6475 400 Analytical EE (3.6e-03 sec) Numerical (1.3e+01 sec) 6470 6500 400 Analytical EE (3.2e-03 sec) 200 l (deg) a (km) 6465 6450 200 l (deg) 6460 0 a (km) 6400 0 0.5 1 1.5 0 0.5 1 1.5 Time (h) Time (h) 6350 0 ×10-3 ×10-3 0 5 10 15 0 5 10 15 2 0 Time (h) Time (h)

-2 0.04 0 0 P1 P2 -4 0.02 -0.01 P1 P2 -2 -6 0 0 0.5 1 1.5 0 0.5 1 1.5 Time (h) Time (h) -0.02 -0.02 ×10-4 ×10-4 0 5 10 15 0 5 10 15 10 10 Time (h) Time (h) ×10-3 ×10-3 5 5 2 1 Q1 Q2 0 0 1 0 Q1 Q2 -5 -5 0 0 0.5 1 1.5 0 0.5 1 1.5 Time (h) Time (h) -1 -1 0 5 10 15 0 5 10 15 Time (h) Time (h) Fig. 2. Comparison with numerical and analytical method on osculating equinoctial elements; Black line shows the numerical propagation method Fig. 4. Comparison on osculating equinoctial elements with 10 revolu- with 10th Fourier Coeﬃcients, and red dashed line shows the analytical inte- tions. (Testing thrust curve: Figure 1, Initial Condition: Table 1) gration with only KTFCs. (Testing thrust curve: Figure 1, Initial Condition: Table 1) Numerical (1.3e+01 sec) 6500 400 Analytical EE (3.2e-03 sec)

6450 4.2.1. Out-of-Plane Control 200

a (km) 6400 It is assumed that the control is only performed stepwise in M (deg) 6350 0 the out-of-plane direction AW as shown in Figure 6. In this case, 0 5 10 15 0 5 10 15 Time (h) Time (h) the maneuvering interval is for 90deg. 0.03 400

The initial conditions are tested on two cases. The ﬁrst one 0.02

e 200 is the low-eccentricity case (Table 2), and the other is high- (deg) 0.01 ω eccentricity case (Table 3). 0 0 0 5 10 15 0 5 10 15 Table 2. Testing Initial Condition 2 Time (h) Time (h) a e i Ω ω M 400 0.3 0.2 6771 km 0.0001 0.5 deg 45.0 deg 22.5 deg 0 deg 200 (deg) i (deg) Ω 0.1

0 0 Table 3. Testing Initial Condition 3 0 5 10 15 0 5 10 15 Time (h) Time (h) a e i Ω ω M

6771 km 0.3 0.5 deg 45.0 deg 22.5 deg 0 deg Fig. 5. Comparison on osculating Keplerian elements with 10 revolutions. In the first case of initial condition, both the osculating ele- (Testing thrust curve: Figure 1), Initial Condition:Table 1) ments are significantly matched. it is notable that the secular plane. Such dynamical behavior can not be calculated with a fluctuations of inclination are close to zero and increase again. original TFCs method due to the indefinite of the line of as- This says that the orbital plane is crossing over the reference cending node. Numerical (3.6e+01 sec) In the second case of initial condition, the analytical solutions 6472 400 Analytical EE (5.4e-03 sec) are obviously different from exact solutions. This is caused by 6471 200 a (km) the non-negligible product of both the deviation of radius and M (deg) ′ ′ 6470 0 differential elements ⟨r æ˙ ⟩ as shown in Eq.61. This fact proves 0 5 10 15 0 5 10 15 Time (h) Time (h) ×10-4 that this new analytical solutions are can not be use in high- 2 150 eccentricity case. 100

e 1 (deg)

ω 50

-6 ×10 0 0 0 5 10 15 0 5 10 15

) 0 2 Time (h) Time (h) Original Thrust 400 0.6 -5 Only 15 TFCs Thrust (km/s R 0.4 A -10 200 (deg) i (deg) 0 1 2 3 4 5 6 Ω 0.2

×10-6 0 0 0 5 10 15 0 5 10 15

) 0 2 Time (h) Time (h)

-5 Original Thrust

(km/s Only 15 TFCs Thrust S

A -10 Fig. 8. Comparison on osculating Keplerian elements with 10 revolu- 0 1 2 3 4 5 6 tions.(Testing thrust curve: Figure 6 Initial Condition: Table 2) ×10-6

) 0 2 Numerical (3.6e+01 sec) 6472 400 Analytical EE (3.7e-03 sec) -5 (km/s W

A -10 6471 200 l (deg) 0 1 2 3 4 5 6 a (km) Eccentric Longitude (rad) 6470 0 0 5 10 15 0 5 10 15 Time (h) Time (h) Fig. 6. Only maneuvering normal direction: Black line is calculated with 0.278 0.115 randomly generated 1000th Fourier Orders. Dashed Red line is depicted nd 0.2775 0.114 with only ETFCs (2 Fourier Orders). To change inclination angle, normal P1 P2 direction is only maneuvered. 0.277 0.113 0 5 10 15 0 5 10 15 Time (h) Time (h) ×10-3 ×10-3 Numerical (3.6e+01 sec) 4 4 Analytical EE (5.2e-03 sec) 6472 400 2 3 Q1 Q2 6471 200 0 2 l (deg) a (km) -2 1 6470 0 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 Time (h) Time (h) Time (h) Time (h) ×10-5 ×10-5 9.2388 3.8271 Fig. 9. Comparison on osculating equinoctial elements with 10 revolu- 3.827 9.23875 P1 P2 tions. (Testing thrust curve: Figure 6 Initial Condition: Table 3) 3.8269

9.2387 3.8268 0 5 10 15 0 5 10 15 Numerical (3.6e+01 sec) Time (h) Time (h) 7000 400 Analytical EE (3.7e-03 sec) ×10-3 ×10-3 5 4 6500 200 a (km) 0 2 M (deg) Q1 Q2 6000 0 0 5 10 15 0 5 10 15 -5 0 Time (h) Time (h) 0 5 10 15 0 5 10 15 Time (h) Time (h) 0.35 100

e 0.3 50 (deg)

Fig. 7. Comparison on osculating equinoctial elements with 10 revolu- ω tions. (Testing thrust curve: Figure 6 Initial Condition: Table 2) 0.25 0 0 5 10 15 0 5 10 15 Time (h) Time (h) 4.2.2. In-Plane Control 400 0.6

The initial condition of this case is Table 4. To see the avoid- 0.4 200 (deg) i (deg) ing the singularity of eccentricity, in other words, the uncer- Ω 0.2 tainty of priapsis. It is assumed that control is only performed 0 0 0 5 10 15 0 5 10 15 stepwise in the in-plane direction AS as shown in Figure 11. Time (h) Time (h) This thrust curve performs acceleration at apoapsis to raise periapsis and decelerate at periapsis to descendens apasis. Seeing Fig. 10. Comparison on osculating Keplerian elements with 10 revolu- the Figure 13, After eccentricity is close to zero, it raises to be- tions. (Testing thrust curve: Figure 6 Initial Condition: Table 3) come elliptic orbit again. Furthermore, argument of periapsis Table 4. Testing Initial Condition 4 turnovers 180 degrees. This result says that the south-east ma- a e i Ω ω M neuver can be calculated with the new analytical solution while 6771 km 0.03 10 deg 45.0 deg 22.5 deg 0 deg avoiding the singularity of eccentricity. tions of low-eccentricity. The solutions can be used for the ter- ×10-5 1 minal sequence of approaching GEO, and station-keeping op- ) Original Thrust 2 Only 15 TFCs Thrust eration for a geostationary satellite. The method developed is 0 (km/s

R not applicable for high-eccentricity due to the assumption of A -1 0 1 2 3 4 5 6 low-eccentricity while developing process; however, an analyt-

×10-5 ical solutions with the original TFCs theory can solve the high- 1 )

2 eccentricity case. When these methods are combined, design- 0 ing more general transfer could be realized with ﬂexible thrust (km/s S

A -1 direction. The KTFCs and ETFCs can be inputted on a multi- 0 1 2 3 4 5 6 objective optimizer as a design parameter. The resulting thrust ×10-5 1 history could be a good initial estimate solution for a more de- ) 2 tailed trajectory design. 0 (km/s W

A -1 0 1 2 3 4 5 6 References Eccentric Longitude (rad) 1) J.A. Kechichian. Low-Thrust Eccentricity-Constrained Orbit Raising. Fig. 11. Only maneuvering circumferential direction: Black line is cal- Journal of Spacecraft and Rockets, 35 1998. pp327–335. culated with randomly generated 10th Fourier orders. Dashed Red line is 2) J.A. Kechichian. Orbit Raising with Low-Thrust Tangential Accelera- depicted with only ETFCs (2nd Fourier Orders) tion in Presence of Earth Shadow. Journal of Spacecraft and Rockets, 35 1998. pp516–525. 3) F.Zuiani and M.Vasile. Extention of Finite Perturbative Elements For Multi-Revolution, Low-Thrust Transfer Optimization. In 63rd Inter- national Astronautical Congress. Naples, Italy, 43, Oct 2012, pp.661– 673, 4) J.S. Hudson and D.J. Scheeres. Reduction of Low-Thrust Continuous Controls for Trajectory Dynamics. Journal of Guidance, Control, and Dynamics, 32(3) May 2009 pp.780–787 5) R.H. Battin. An Introduction to the Mathmatics and Methods of Astro- dynamics. AIAA Education Series, 1987. 6) Oliver Montenbruck and Eberhard Gill. Satellite Orbits. Springer- Verlag Berlin, 2000. 7) J.M.A. Danby. Fundamentals of Celestial Mechanics. Willmann-Bell, Richmond, VA, 2 edition, 2003.

Fig. 12. Comparison on osculating equinoctial elements with 10 revolutions. (Testing thrust curve: Figure 11 Initial Condition: Table 4)

Fig. 13. Comparison on osculating Keplerian elements with 10 revolutions. (Testing thrust curve: Figure 11 Initial Condition: Table 4)

5. Conclusion

The analytical solutions, which use equinoctial elements and are averaged with TFCs theory, was found under the assump-