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 ! : 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 !˜ : the longitude of periapsis, ! + Ω, deg the body and the shape becomes spiraling. Designing many- P1 : equinoctial element for e, e sin! ˜ revolution orbit, in which acceleration continuously varies, is P2 : equinoctial element for e, e cos! ˜ 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 + !˜ , deg methodology which can solve various many-revolution have L : true longitude, f + !˜ , deg never been developed. K : eccentric longitude, E + !˜ , 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; !; 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 Er 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 periapsis! ˜ . The mean anomaly is trans- anomalies become mathematically undefined value. formed to new element " 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 Z equations with equinoctial elements are averaged via TFCs the- M = ndt + "1 − (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 r theory, which uses Keplerian elements, provides flexibility of d" a p 1 = −2 (1 − e cos E)A + (1 − 1 − e2)(! ˙ + Ω˙ ) maneuvering and transferring for many-revolution. dt µ R ( ) p 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 (#) 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 equa- tion is adequate for describing the dynamics. X1 " ! !# 2πk# 2πk# = v # ∼ + r˙ (1) f ( ) ak cos Λ bk sin Λ (14) µ k=0 v˙ = − 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 Z celerating component in RSW (named radial-transverse-normal mπ 6) 1 or satellite coordinate system) reference frame, a = f (#)d# (15) ( ) ! ! 0 mπ r r × v r × v r Z 0 ! A = A + A + A × (3) 2 mπ 2k# D R r W jr × vj S jr × vj r a = f (#) cos d# (16) k mπ m Z0 ! One of the expressions for Gaussian planetary equations in 2 mπ 2k# 5,7) = # # 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 + f(p + r) cos f + reg AS (5) dt h The Fourier series can be expanded in a time-varying orbital di r cos θ ; ; # = A (6) parameter, such as f E M for . dt h W Ω θ X1 d = r sin ( ) AW (7) = αD # + βD # dt h sin i AD k cos k k sin k (18) d! 1 k=0 = f−p cos f AR + (p + r) sin f AW g = ; ; 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 + f(p cos f − 2re)A − (p + r) sin f A g (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. Z Z 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- ! cess.