View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Chinese Journal of Aeronautics, (2014),27(3): 577–583 Chinese Society of Aeronautics and Astronautics & Beihang University Chinese Journal of Aeronautics [email protected] www.sciencedirect.com Tangent-impulse transfer from elliptic orbit to an excess velocity vector Zhang Gang a,b,*, Zhang Xiangyu a, Cao Xibin a,b a Research Center of Satellite Technology, Harbin Institute of Technology, Harbin 150080, China b National and Local United Engineering Research Center of Small Satellite Technology, Changchun 130033, China Received 6 May 2013; revised 3 July 2013; accepted 14 October 2013 Available online 22 April 2014 KEYWORDS Abstract The two-body orbital transfer problem from an elliptic parking orbit to an excess veloc- Elliptic orbit; ity vector with the tangent impulse is studied. The direction of the impulse is constrained to be Escape trajectory; aligned with the velocity vector, then speed changes are enough to nullify the relative velocity. First, Excess velocity vector; if one tangent impulse is used, the transfer orbit is obtained by solving a single-variable function Orbital transfer; about the true anomaly of the initial orbit. For the initial circular orbit, the closed-form solution Tangent impulse is derived. For the initial elliptic orbit, the discontinuous point is solved, then the initial true anomaly is obtained by a numerical iterative approach; moreover, an alternative method is proposed to avoid the singularity. There is only one solution for one-tangent-impulse escape trajectory. Then, based on the one-tangent-impulse solution, the minimum-energy multi- tangent-impulse escape trajectory is obtained by a numerical optimization algorithm, e.g., the genetic method. Finally, several examples are provided to validate the proposed method. The numerical results show that the minimum-energy multi-tangent-impulse escape trajectory is the same as the one-tangent-impulse trajectory. ª 2014 Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA. Open access under CC BY-NC-ND license. 1. Introduction circular orbit, an approximate analytical solution was obtained for the minimum-energy three- and four-impulse transfers The two-body orbital transfer problem from a parking orbit to between a given circular orbit and a given hyperbolic velocity 1 a given excess velocity vector is a fundamental one in space vector at infinity. For the transfer from the initial circular exploration. The minimum-energy trajectory optimization for orbit to an excess velocity vector, the two-impulse escape is this problem has been studied for many years. For the initial never simultaneously of lower cost than either the one- or three-impulse.2 Recently, Ocampo et al.3,4 studied the one- and three-impulse escape trajectories, which can be con- * Corresponding author. Tel.: +86 451 86413440 8308. structed to serve as initial guesses for determining constrained E-mail addresses: [email protected] (G. Zhang), zxyuhit@ optimal multi-impulsive escape trajectories. For the initial gmail.com (X. Zhang), [email protected] (X. Cao). elliptic orbit, the optimal three-impulse transfer between an Peer review under responsibility of Editorial Committee of CJA. elliptic orbit and an escape asymptote was solved.5 Moreover, a simple numerical technique was proposed to minimize the time-of-flight for the multi-impulse transfer from an arbitrary Production and hosting by Elsevier elliptic orbit to a hyperbolic escape asymptote.6 1000-9361 ª 2014 Production and hosting by Elsevier Ltd. on behalf of CSAA & BUAA. Open access under CC BY-NC-ND license. http://dx.doi.org/10.1016/j.cja.2014.04.006 578 G. Zhang et al. The existing methods for the impulse escape trajectory opti- vf ¼ v0 þ Dv ¼ðv0 þ kÞv0=v0 ð1Þ mization problem do not include any constraint on the impulse 7,8 where rj = krjk, vj = kvjk, j = 0, f, and 0 and f denote the ini- direction. If the impulse direction is aligned with the velocity tial and final orbits, respectively. A relationship between the vector, the impulse is called ‘‘tangent impulse’’ and a speed magnitudes of velocity and position vectors is change will finish the impulse maneuver to nullify the relative sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi velocity. The tangent orbit problem has also existed for many 2 1 years. The classical Hohmann transfer is a cotangent transfer vj ¼ l À ð2Þ rj aj and is the minimum-energy one among all the two-impulse transfers between coplanar circular orbits and between copla- where l is the standard gravitational parameter, and a the nar coaxial elliptic orbits.9 The cotangent transfer problem semimajor axis. Then, the magnitude of the velocity can be between coplanar noncoaxial elliptic orbits has aroused con- written as a function of the initial true anomaly, siderable interest in recent years. The numerical solution was rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10 l 2 obtained based on the orbital hodograph theory. Moreover, v0 ¼ ð1 þ e þ 2e0 cos u Þ ð3Þ p 0 0 the closed-form solution was obtained by using the geometric 0 11 12 þ characteristics and by the flight-direction angle, respec- where u is the true anomaly, u1 the true anomaly at infinity of tively. The latter reference also gave the closed-form solution the excess hyperbolic orbit, e the eccentricity, and p the semil- for the solution-existence condition. In addition, Zhang and atus rectum. Zhou13 studied the tangent orbit technique in 3D based on a From the energy equation, the semimajor axis of the final new definition of orbit ‘‘tangency’’ condition for noncoplanar hyperbolic orbit is orbits. Different from the orbital transfer problem, the orbital l ÀÁ rendezvous problem requires the same time-of-flight for both af ¼ 2 ð4Þ vþ the chaser and the target. Zhang et al. solved the two-impulse 1 rendezvous problem between two coplanar elliptic orbits with þ where v1 is the excess velocity vector. Substituting Eq. (4) into only the second impulse14 and both impulses being tangent,15 Eq. (2) gives respectively. Furthermore, Zhang et al.16 solved the two- sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ÀÁ ÀÁ 2 2l 2 2l impulse rendezvous problem between coplanar elliptic and þ þ vf ¼ v1 þ ¼ v1 þ ð1 þ e0 cos u0Þ ð5Þ hyperbolic orbits. r0 p0 This paper studies the coplanar orbit escape problem from an elliptic orbit to an excess velocity vector only with the tan- Thus, for a given initial true anomaly u0 of the impulse gent impulse. For a given initial orbit and a given excess veloc- point, the magnitude of the tangent impulse is ity vector, the one-tangent-impulse transfer trajectory is k ¼ vf À v0 obtained by solving a single-variable piecewise function. Then sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ÀÁ 2 2l the optimal multi-tangent-impulse escape trajectory is þ ¼ v1 þ ð1 þ e0 cos u0Þ obtained by the genetic method. p0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l 2 2. Orbital elements of transfer orbit À ðÞ1 þ e0 þ 2e0 cos u0 ð6Þ p0 Assume that the spacecraft moves in a given initial orbit, a tan- The eccentricity vector of the final orbit is gent impulse Dv with magnitude k is imposed at P1 (see Fig. 1), 1 2 l where the position vector relative to the Earth’s center F1 is r0 ef ¼ ðvf À Þr0 ðr0 Á vfÞvf ð7Þ l r0 and the velocity vector is v0, then the velocity vector of the transfer orbit (or the final hyperbolic orbit) at P1 is By using the following expression k k l r Á v ¼ 1 þ ðr Á v Þ¼ 1 þ r e sin u 0 f v 0 0 v h 0 0 0 0 0 0 ffiffiffiffiffiffiffi p k e0 sin u0 ¼ lp0 1 þ ð8Þ v0 1 þ e0 cos u0 the eccentricity vector in Eq. (7) can be written as ÀÁ 1 2 l e ¼ vþ þ ð1 þ e cos u Þ r f l 1 p 0 0 0 0 ) ffiffiffiffiffiffiffi 2 p k e0 sin u0 À lp0 1 þ v0 ð9Þ v0 1 þ e0 cos u0 whose magnitude is the eccentricity ef of the final orbit. The normalized eccentricity vector is 2 3 cos xf cos Xf À sin xf sin Xf cos If ef 6 7 ^ef ¼ ¼ 4 cos xf sin Xf þ sin xf cos Xf cos If 5 ð10Þ Fig. 1 Transfer to an excess velocity vector by a tangent impulse, ef sin xf sin If example 1. Tangent-impulse transfer from elliptic orbit to an excess velocity vector 579 where I denotes the orbit inclination, x the argument of peri- Define an XYZ frame, which is obtained by rotating the gee, and X the right ascension of ascending node. With a ECI frame with 3–1 sequence for angles X and i, respectively. coplanar tangent impulse, the orbit plane will not change, The origin of the XYZ frame is also the center of the Earth. which indicates that If = I0, and Xf = X0. In addition, the Then the X and Y axes are in the orbit plane, and Z axis is argument of perigee can be obtained as aligned with the angular momentum vector of the initial (or – final) orbit. atan2ðÞe^fk3 = sin I0; e^fk1 cos X0 þ e^fk2 sin X0 sin I0 0 xf ¼ The eccentricity vector Eq. (9) in the ECI frame can be atan2ðÞe^ ; e^ sin I ¼ 0 fk2 fk1 0 transformed into that in the XYZ frame. Note that the ð11Þ flight-direction angle c0 is the angle from the position vector where e^ ; e^ ; and e^ are three components of ^e in the r0 to the velocity vector v0, then from Eq.