given duration while minimizing the fuel consumption. The fuel consumed is linked to the velocity impulse through the rocket equation derived by Tsiolkovsky [9]:.
JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vol. 39, No. 10, October 2016
Low-Thrust Transfer Between Circular Orbits Using Natural Precession Max Cerf∗ Airbus Defence and Space, 78130 Les Mureaux, France DOI: 10.2514/1.G001331 The minimum-fuel low-thrust transfer between circular orbits is formulated using the Edelbaum averaged dynamics with the addition of the nodal precession due to the first zonal term. The extremal analysis shows that an optimal transfer is composed of three sequences in the regular case. The optimal control problem is solved by a shooting method with a costate guess derived from an approximate solution.
I.
missions raise challenging issues, since they embed optimal control problems (find the minimum fuel transfer between any pair of debris) into a large combinatorial problem (find the optimal visiting order and the optimal rendezvous dates). To efficiently address such mission planning problems, one must be able to solve thousands of transfers between debris pairs in a very fast and reliable way, considering any endpoint conditions. This paper addresses the minimum-fuel low-thrust transfer problem between circular orbits taking into account a right ascension ascending node constraint. The optimal control problem is formulated using the Edelbaum dynamical model, and the solution structure is analyzed. A solution method is proposed using an indirect method and a costate guess derived from a simplified problem. The method proves very fast and reliable, thus opening the way to the optimization of multiple debris-removal missions. It is exemplified in an application case representative of such a mission.
Introduction
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T
HE low-thrust technology for orbit transfers offers promising performance in terms of fuel consumption. This fuel gain is nevertheless offset by a longer transfer duration that may be penalizing for commercial Earth orbit raising missions: for example, to raise a telecommunication satellite from a low Earth orbit to a geostationary Earth orbit. A particular application where time is not an issue is the removal of spent satellites from the near-Earth region. Active debris-removal missions aiming at deorbiting several debris are currently under study. For such missions, the low-thrust technology proves particularly attractive and a quick performance assessment method would be very useful at the vehicle design stage. The targeted debris are mostly old observation satellites evolving on near-circular sun-synchronous orbits. The transfer strategy between two debris must account not only for the altitude and inclination change but also for the ascending node precession due to the Earth flattening (first zonal term J2 ). Several analytical or quasi-analytical solutions exist for the minimum-time low-thrust transfer problem between circular orbits. Some of them rely on perturbation techniques, allowing an explicit solution under specific assumptions [1]. The most famous solution was due to Edelbaum in 1961 [2,3]. It is based on an averaged dynamical model assuming a constantly circular orbit and continuous thrusting. The original work of Edelbaum has been retrieved in the frame of optimal control theory [4], and several extensions have been derived in order to enlarge the application scope of the model. 1) In [5], the Edelbaum model was enhanced to account for the mass variation during the transfer and for a variable specific impulse allowing a reduction of the fuel consumption. 2) In [6], the solution of the Edelbaum problem was enhanced to comply with an altitude upper bound. This is particularly useful when a large inclination change is required, since the Edelbaum solution involves high altitudes to minimize the cost of the inclination change. 3) Still in [6], the dynamics averaging method was adapted to perform a right ascension ascending node (RAAN) change instead of an inclination change. 4) In [7], no-thrust legs due to Earth shadowing were accounted for by shifting the trajectory time at each revolution accordingly. These extensions are devoted to the minimum-time problem. For the debris-removal missions under interest [8], the goal is rather to minimize the fuel consumption within a fixed duration allocated to the mission, assuming that this duration is sufficiently large with respect to the low-thrust engine capabilities. Multiple debris-removal
II.
Problem Formulation and Analysis
This section formulates the low-thrust transfer problem in terms of an optimal control problem (OCP). The solution structure is then analyzed by applying the Pontryagin maximum principle (PMP). A. Dynamics
The dynamics is based on the Edelbaum dynamical model, which assumes constantly circular orbits. At a given date, the orbital plane is defined in the Earth inertial reference frame by the inclination I and the right ascension of the ascending node Ω. The inclination I is the angle of the orbital plane with the Earth equatorial plane. The intersection of the orbital plane with the equator is the line of nodes. The RAAN Ω is the angle between the X axis of the Earth inertial reference frame and the direction of the ascending node (node crossed with a northward motion) (cf. Fig. 1). The circular orbit shape is defined equivalently by the radius a or by the velocity V. The Edelbaum model is derived by an averaging of the Gauss equations, with the following assumptions: 1) The evolution of the orbital parameters is averaged on one period. 2) The averaged orbit is constantly circular throughout the transfer. 3) The acceleration level, denoted f, is constant throughout the transfer. 4) The thrust direction is normal to the radius vector, and it makes a constant angle β with the orbital plane during one period with a sign change at the antinodes. This results in a null RAAN change over the period while maximizing the inclination change for the current value of the out-of-plane angle β. In this paper, the uppercase notations V, I, and Ω represent the averaged orbital parameters. The Edelbaum problem consists of minimizing the transfer duration. The engine is continuously thrusting with the constant acceleration level f, and the transfer is controlled by varying the averaged angle β from one period to the other. We refer to [4] and [9] for the detailed model formulation and the analytical solution of the
Received 25 February 2015; revision received 15 June 2015; accepted for publication 6 July 2015; published online 7 September 2015. Copyright © 2015 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal and internal use, on condition that the copier pay the per-copy fee to the Copyright Clearance Center (CCC). All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the ISSN 07315090 (print) or 1533-3884 (online) to initiate your request. *Mission Analysis Expert, Mission Analysis Department; max.cerf@ astrium.eads.net. 2232
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minimum-time problem. For our purpose, the Edelbaum dynamics is extended by taking into account the first zonal term (denoted J2 ) due to the Earth flattening. This perturbation causes no secular change on the semimajor axis, the eccentricity, and the inclination. The averaged orbit remains circular with radius a and inclination I. On the other hand, a RAAN precession rate is induced, depending on the orbit radius a and the inclination I [9]: 7 μ 2 7 _ − 3 J 2 p cos I −kV 7 cos I (1) μR2E a−2 cos I −k Ω 2 a The constants of the Earth gravitational model are [8] RE 6; 378; 137 m (equatorial radius), μ 3.986005.1014 m3 ∕s2 (gravitational constant), J2 1.08266 (first zonal term), and
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k
3J2 R2E 1.0425 × 10−33 s6 ∕m7 2μ3
C. Extremal Analysis
The dynamics model thus consists of three ordinary differential equations representing the evolution of the averaged orbital parameters (V, I, Ω): 8 > < V_ −f cos β 2 f sin β I_ πV > :Ω _ −kV 7 cos I
(2)
B. Optimal Control Problem
The problem we are interested in consists of transferring the vehicle from a given circular orbit to another given circular orbit in a given duration while minimizing the fuel consumption. The fuel consumed is linked to the velocity impulse through the rocket equation derived by Tsiolkovsky [9]: Zt f M f dt ve ln 0 ΔV M 0 f (3) ⇒ mc M0 − Mf M0 1 − e−ΔV∕ve M0 and Mf are, respectively, the initial and final gross masses; ve is the engine exhaust velocity; and mc is the propellant mass consumed. Minimizing the fuel consumption mc is equivalent to minimizing the velocity impulse ΔV. The optimal control problem is formulated, considering the velocity impulse as a cost function, with completely prescribed endpoints and bounded control. Optimal control problem:
Min J f;β
tf
f dt
subject to
t0
8 > < V_ −f cos β 2 f sin β I_ πV > :Ω _ −kV 7 cos I
(4)
Vt0 V 0 ;
It0 I0 ;
Vtf V f ;
8 2 6 > < p_ V − ∂H ∂V 7pΩ kV cos I πV 2 pI f sin β ∂H 7 p_ I − ∂I −pΩ kV sin I > : p_ − ∂H 0 Ω ∂Ω
(6)
Since the initial and final conditions are completely fixed, there are no transversality conditions on the costate and on the Hamiltonian. Introducing the switching function S, S p0 − pV cos β pI
2 sin β πV
(7)
the Hamiltonian is expressed as _ H fS pΩ Ω
(8)
_ does not depend explicitly on the control variables The term pΩ Ω (f, β). The Hamiltonian is linear with respect to the acceleration level f, and it depends on the thrust direction β through the switching function S. The Hamiltonian maximization yields the optimal acceleration level depending on the switching function sign: When S > 0; then f fmax thrust arc When S < 0; then f 0
coast arc
f cannot be determined directly from the PMP singular arc
Ωt0 Ω0
(9)
In the case of a nonnull acceleration, the Hamiltonian maximization with respect to the thrust direction β yields
Fixed final conditions: tf ;
(5)
2) The costate vector (pV , pI , and pΩ ) is not identically null on an interval of t0 ; tf , and it satisfies the differential system:
When S 0;
Fixed initial conditions: t0 ;
According to the Pontryagin maximum principle [10,11], the necessary conditions for (f, β) to be an optimal control for the OCP are the existence of absolutely continuous functions pV , pI , and pΩ (respective costates of V, I, and Ω), as well as the existence of a nonpositive real p0 (cost multiplier) such that the following is true: 1) The control (f, β) maximizes nearly everywhere the Hamiltonian function is defined as _ H p0 f pV V_ pI I_ pΩ Ω
The control variables are the acceleration level f and its direction β over each period. The acceleration level can be varied between zero and a maximum value fmax .
Z
the endpoint conditions Ωt0 and Ωtf are time dependent due to the natural precession. It can be observed that the velocity and the inclination evolutions do not depend on the RAAN value. The problem is insensitive to a RAAN shift applied identically on Ωt0 and Ωtf . The thrust is used to modify directly the velocity and the inclination, whereas the RAAN change is performed passively by the natural J2 precession. This control strategy based on the Edelbaum model is suboptimal due to the dynamics simplifications. It should be envisioned only if a sufficiently large duration is allocated to the mission allowing a benefit from the J 2 precession. This is normally the case for practical applications such as debrisremoval missions [8]. If not, the Edelbaum control model is no longer suited and the problem should be addressed without control law simplifications.
Itf I f ;
Ωtf Ωf
Control bounds: 0 ≤ f ≤ fmax The problem is autonomous, since the dynamics and the integral cost do not depend explicitly on the time. On the other hand,
( ∂H
∂β 0 ⇒ ∂2 H ≤0⇒ ∂β2
2 pV sin β pI πV cos β 0 2 pV cos β − pI πV sin β ≤ 0
(10)
The costates pv and pI cannot vanish simultaneously on a time interval; otherwise, from Eq. (6), pΩ 0, which would be a contradiction to the PMP (nontrivial costate). System equation (10) therefore defines the thrust direction β without sign ambiguity. Using,
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for example, the function Atan2s; c, where s and c are, respectively, proportional to the angle sine and cosine, we can express β as β Atan2
2 p ; −pV πV I
_ pΩ Ω _ d − fS ≤ pΩ Ω _d pΩ Ω
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2 sin β ≥ p0 πV
(12)
The solution is said to be abnormal when the cost multiplier p0 is null. The switching function is then positive or null, and the trajectory is made of maximum thrust arcs and singular arcs. In the singular case, the switching function S vanishes identically on an interval of t0 ; tf . The acceleration level can no longer be deduced directly from the Hamiltonian maximization. Combining Eq. (7) with Eq. (10) yields the costates pV and pI :
S0 ∂H ∂β 0 2 p0 − pV cos β pI πV sin β 0 ⇒ 2 pV sin β pI πV cos β 0 p0 cos β − pV 0 pV p0 cos β ⇒ ⇒ 2 p0 sin β pI πV 0 pI − πV 2 p0 sin β
8 _ Cte < Ht fS pΩ Ω te p C : Ω fS ≥ 0
(13)
(14)
_ d by H pΩ Ω _ d. Since H and Property 3: If pΩ ≠ 0, we define Ω def
_ d is also constant. pΩ are constant from Eq. (14), Ω
_ is lower than pΩ Ω _ d , and the bound is reached The term pΩ Ω whenever fS 0. All coast arcs (f 0) and all singular arcs (S 0) therefore have _ d. the same precession rate equal to Ω Property 4: If pΩ 0, we have H fS ≥ 0 from Eq. (14). We consider successively the case H ≠ 0 and the case H 0. 1) If H > 0, then S > 0 and, from Eq. (9), the acceleration level is constantly maximum: f fmax . The OCP Eq. (4) is equivalent to the Edelbaum minimum-time problem, for which the solution is analytical [4,9]. 2) If H 0, then either f 0 (coast arc) or S 0 (singular arc). The singular case is not studied in this paper. A regular solution with pΩ 0 is therefore composed of a single arc at either f 0 or f fmax. The control structure is investigated next in the regular case, which is the only one of practical interest. D. Regular Solution
Assuming p0 0 in Eq. (13) leads to pV 0 and pI 0 and, using Eq. (6), to pΩ 0. The costates vanish identically, which is in contradiction with the PMP. A singular solution cannot therefore be abnormal. An abnormal solution is composed of a single maximum thrust arc. In that case, the OCP [Eq. (4)] becomes equivalent to the Edelbaum minimum-time problem, for which the solution is analytical [4,9]. This situation may occur only if the prescribed transfer duration exactly matches the minimum time so that the solution is unique. For practical applications, a sufficiently large duration should be allocated in order to benefit from the natural precession so that the abnormal case can be discarded. Property 2: For the values of H, pΩ , and fS, the OCP [Eq. (4)] is autonomous so that the Hamiltonian is constant. From Eq. (6), pΩ is also constant and, from Eq. (9), the term fS is positive or null:
Fig. 1
(15)
(11)
Before investigating the solution structure, several preliminary results are derived hereafter. Property 1: For the abnormal solution (p0 0), the inequality in Eq. (10) gives a lower bound on the switching function [Eq. (7)]: S p0 − pV cos β pI
Using fS ≥ 0 from Eq. (14), we have
In the regular case, the switching function S does not vanish identically on any interval of t0 ; tf . The acceleration level is either zero or fmax, depending on the sign of S. If pΩ 0, the solution is composed of a single arc (Property 4) and the problem has a direct analytical solution. Assuming pΩ ≠ 0 in the sequel of this section, all _ d (Property 3). During coast phases have the same precession rate Ω the coast phases, the velocity and the inclination remain constant, _ d. whereas the RAAN evolves linearly with the precession rate Ω _ along the transfer, Figure 2 illustrates a possible evolution of pΩ Ω with three propelled sequences separated by two coast phases. This scenario is called scenario 1. The following relationships hold between the endpoints of the coast phases: 8 8 < V2 V1 < V4 V3 and I 4 I 3 I2 I1 : : _ d :t2 − t1 _ d :t4 − t3 Ω2 Ω1 Ω Ω4 Ω3 Ω
(16)
The leg from t2 to t3 is at a maximum acceleration level fmax with a thrust direction law denoted as β23 . This control transfers the state from (V 2 , I2 , Ω2 ) to (V 3 , I 3 , Ω3 ) in a duration of Δt23 t3 − t2 . The RAAN change during this leg is denoted as ΔΩ23 Ω3 − Ω2 . We now consider the scenario obtained by permuting the third and the fourth legs, with each leg keeping the same duration and the same control law as in scenario 1. The dates t2 and t4 are unchanged, whereas the intermediate date t3 is shifted to t30 . The new scenario, called scenario 2, is depicted on the Fig. 3. Scenario 2 is identical to scenario 1 until the date t2 . The coast phase starting at t2 has the duration of Δt34 t4 − t3 and the _ d . It ends at t 0 with the state precession rate Ω 3
Orbital parameters for a circular orbit. a: radius; V: velocity; I: inclination; Ω: right ascension of the ascending node (RAAN).
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Fig. 2
Scenario 1.
Fig. 3 Scenario 2.
8 0 < V3 V2 I 0 I2 : 30 _ d :Δt34 Ω3 Ω2 Ω
(17)
The control (fmax , β23 ) retrieved from scenario 1 is applied from t30 to t4 . Compared to the leg from t2 to t3 of scenario 1, this leg from t30 to t4 has a different starting date (t30 instead of t2 ), the same duration Δt23 , and the same initial state (V 30 V 2 , I 30 I 2 ), except for the RAAN (Ω30 instead of Ω2 ). The shift of the starting date does not change the trajectory, since the problem is autonomous. As observed at the end of Sec. II.B, the solution is also insensitive to a RAAN shift. Applying the control (fmax , β23 ) from the initial state (V 2 , I 2 , Ω30 ) at the date t30 until the date t4 yields a leg identical to scenario 1 with a RAAN change equal to ΔΩ23 . The state at t4 is thus 8 0 < V4 V3 I 0 I3 : 40 _ d :Δt34 ΔΩ23 Ω3 Ω _ d :Δt34 Ω4 Ω30 ΔΩ23 Ω2 Ω (18) Compared with Eq. (16), the final state at t4 is identical to scenario 1: 8 0 < V4 V4 I 0 I4 : 40 Ω4 Ω4
(19)
Scenario 2 reaches the same final conditions as scenario 1. Their costs are identical, since the thrusting sequences at fmax have the same duration. A solution with several coast phases can thus be replaced by an equivalent solution with a single coast phase, and we have the following result. Property 5: In the regular case, a solution of the optimal control problem can be sought by assuming a three-sequence control structure of the form fmax − 0 − fmax .
III.
Solution Method
This section deals with the numerical solution of the optimal control problem in the regular case. A shooting method is applied to the TPBVP issued from the PMP necessary conditions. The shooting method is initiated with a costate guess derived from a simplified problem. A. Shooting Problem
The extremal analysis has shown (Property 5) that an optimal transfer in the regular case could be sought with three sequences and a control structure: fmax − 0 − fmax . The switching dates are denoted as t1 and t2 . The trajectory is obtained by integration of the state and costate [Eqs. (4) and (6)] from the initial date t0 to the final date tf . The acceleration level is set successively to fmax from t0 to t1 , to zero from t1 to t2 , and to fmax from t2 to tf . The thrust direction β is defined by Eq. (11). The problem unknowns are the initial costate components and the switching dates. The final state is prescribed, and the switching
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function must vanish at the switching dates. This shooting problem formulates as follows. Find the three initial costate components pV t0 , pI t0 , and pΩ t0 and the two switching dates t1 and t2 in order to nullify the shooting function F, R5 → R5 , defined as FpV t0 ; pI t0 ; pΩ t0 ; t1 ; t2
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Vtf − V f ; Itf − If ; Ωtf − Ωf ; St1 ; St2
(20)
The shooting problem could be formulated with only three unknowns (initial costate) and three equations (final state), since the acceleration level can be determined from the switching function sign within the trajectory propagation. For the numerical resolution, it proves more efficient to add the two switching dates and the two switching conditions to the nonlinear systems. The shooting method consists of solving the nonlinear system [Eq. (20)] by a Newton method. The main issue lies in the initial costate guess that must be sufficiently close to the solution in order to achieve the convergence of the Newton method. For that purpose, the optimal control problem is approximated by a simplified nonlinear programming (NLP) problem that can be easily solved. The costate guess is then derived from the cost sensitivities of this NLP problem. B. Split Edelbaum Strategy
The control structure of a regular solution is fmax − 0 − fmax . The trajectory is split into a first propelled transfer from the initial orbit (V 0 , I0 ) toward the drift orbit (V d , I d ), a coast phase on the drift orbit, and a second propelled transfer from the drift orbit (V d , I d ) toward the final orbit (V f , I f ). The fuel consumption is directly proportional to the durations of the two propelled transfers. The simplifying approach consists of using the Edelbaum analytical minimum-time solution to assess the two propelled transfers. The problem reduces to the drift orbit parameters (V d , I d ) that should be sought in order to minimize the cost while meeting the final RAAN constraint. This simplified transfer strategy is sub-
Fig. 4
optimal with respect to OCP, since the Edelbaum solution does not account for the RAAN constraint. It should nevertheless be quite close to the OCP solution as long as the propelled duration remains small compared to the drift duration, meaning that the RAAN change is essentially performed during the drift phase. This assumption is the underlying motivation for the formulation of the OCP using the Edelbaum dynamics. The transfer strategy is depicted in Fig. 4, with the spiraling trajectories associated to the initial and final Edelbaum transfers. The simplified NLP problem corresponding to the split Edelbaum strategy (SES) formulates as follows: SES problem: min ΔV
V d; I d
subject to Ωtf Ωf
(21)
For both propelled transfers, respectively, from (V 0 , I 0 ) to (V d , I d ) and from (V d , I d ) to (V f , If ), the Edelbaum formulas [4,9] yield explicit assessments of the minimum duration of the corresponding velocity impulse and of the velocity V and inclination I evolutions with respect to the time. The subscript e in the sequel refers to the Edelbaum analytical solution. These formulas are used to assess, on the one hand, the transfer cost and, on the other hand, the final RAAN value, as explained in the following. The transfer cost is the sum of the velocity impulses of the two propelled transfers, respectively, ΔV e1 and ΔV e2 : ΔV ΔV e1 ΔV e2
(22)
The Edelbaum model considers a thrusting strategy that modifies the inclination at each revolution while leaving the RAAN unchanged. The RAAN evolution throughout the transfer is only due to the J2 precession effect. The final RAAN value is obtained by integration of Eq. (1). During the propelled transfers, the velocity V and the inclination I evolutions are given analytically by the Edelbaum solution. These
Split Edelbaum strategy.
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evolutions along both propelled transfers are denoted, respectively, (V e1 , I e1 ) and (V e2 , I e2 ). The RAAN evolution is assessed by quadrature for the propelled transfers and explicitly for the drift phase, since the precession rate is constant: R Ωt1 Ωt0 tt01 −kV 7e1 cos I e1 dt Ωt2 Ωt1 − kV 7d cos Id t2 − t1 Rt Ωtf Ωt2 t2f −kV 7e2 cos Ie2 dt
(23)
The cost for the SES problem [Eq. (21)] is thus assessed analytically, whereas the constraint is assessed by two quadratures. No integration of differential equations is required.
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C. SES Solution
The SES problem [Eq. (21)] with two unknowns and one equality constraint must be solved numerically using a nonlinear programming software. It can be initiated with a guess based on the Edelbaum analytical costates. For each propelled transfer, the endpoint costates indeed provide the cost derivatives with respect to the endpoint states. The drift orbit parameters represent the final state for the first propelled transfer (date t1 ) and the initial state for the second propelled transfer (date t2 ). Denoting, respectively, (peV1 , peI1 ) and (peV2 , peI2 ) as the Edelbaum costates of the two propelled transfers, we have ( ∂ΔV
e1 ∂V d peV1 t1 and ∂ΔV e1 peI1 t1 ∂I d
( ∂ΔV
e2 ∂V d peV2 t2 ∂ΔV e2 peI2 t2 ∂I d
(24)
The gradient of the SES cost function [Eq. (22)] is thus (
∂ΔV ∂V d peV1 t1 ∂ΔV ∂I d peI1 t1
peV2 t2 peI2 t2
(25)
Assuming that the RAAN change is essentially achieved during the drift phase (which is the underlying motivation of the formulation [Eq. (4)]), the quadratures can be neglected in Eq. (23) and the RAAN constraint becomes analytical: _ d −kV 7 cos I d Ωf − Ω0 Cte Ω d tf − t0
For an optimal control problem with state vector X, costate vector P, cost function J, and Hamiltonian function H, the following relationships hold [11]: Pt0
∂J ; ∂Xt0
Ht0
∂J ∂t0
(28)
where J is the cost value for the extremal starting from X (t0 ) at the date t0 . The initial costate is the partial derivative of the optimal cost with respect to the initial state, whereas the initial Hamiltonian is the partial derivative of the optimal cost with respect to the initial date. These properties are used to build a guess for the shooting problem [Eq. (20)]. For that purpose, the SES problem [Eq. (21)] is first solved for the reference initial conditions yielding the reference cost value. It is then solved again by varying the initial state components one by one and assessing the corresponding costate components by finite differences. An estimate of the Hamiltonian value is also assessed by varying the initial or the final date.
IV.
Application Case
The solution method is illustrated on a typical transfer of a debrisremoval mission. The initial and final orbits are circular and nearly sun synchronous (from 800 km∕98 deg to 900 km∕99 deg). A. Example Data
Table 1 provides the initial and final conditions of the transfer. The transfer duration is bounded to 100 days. The maximal acceleration level is set to 3.5 10−3 m∕s2 . The targeted orbit has a RAAN value of 30 deg at the transfer beginning with a precession rate of 0.982 deg ∕day. The RAAN value that must be targeted at the transfer end is then 128.2 deg (equal to 30 0.982 × 100). The RAAN gap of 30 deg must be caught up within 100 days. It is therefore expected that the drift orbit precession rate will be approximately 0.3 deg ∕day higher than the target precession rate. B. SES Solution
(26)
Equation (26) can be used to eliminate either I d or V d. Choosing, for instance, V d as the free parameter, the cost gradient is dΔV ∂ΔV ∂ΔV dI d dV d ∂V d ∂I d dV d peV1 t1 peV2 t2 peI1 t1 peI2 t2
D. OCP Costate Guess
7 V d tan Id (27)
The assessment of this gradient for any value of V d is completely analytical using, first, Eq. (26) to get I d , and then the Edelbaum formulas to get the endpoint costates peV1 , peI1 , peV2 , and peI2 . The optimal value V d must nullify the gradient [Eq. (27)]. The problem reduces to one unknown nonlinear equation, which can be easily solved. The result is used as a starting point for solving the SES problem [Eq. (21)]. It can be noticed that the difference Ωf − Ω0 in the second member of Eq. (26) is given modulo 2π. The SES problem [Eq. (21)] and, similarly, the OCP [Eq. (4)] therefore have several local minima _ d . In most cases, the corresponding to different drift precession rates Ω best solution can be guessed a priori, depending on the relative configuration of the initial and final orbits in terms of RAAN values and of precession rates. Nevertheless, the existence of local minima must be kept in mind when solving a particular instance of the OCP [Eq. (4)] and, in some cases, different solutions must be compared in order to ensure that the global minimum is found.
The SES problem [Eq. (21)] is solved using a reduced gradient optimizer (Airbus Defence and Space internal software). The initial values of the variables (V d , Id ) are obtained by solving the onevariable equation [Eq. (27)]. The following solution is obtained for the SES problem (in parentheses are the initial values): 1) For the drift orbit velocity, V d 7664.0 m∕s 7564.0 m∕s. 2) For the drift orbit altitude, Zd 404.7 km (585.2 km). 3) For the drift orbit inclination, Id 99.20 deg (100.08 deg). 4) For the total velocity impulse, ΔV 598.1 m∕s. 5) For the drift phase starting date, t1 1.075 days. 6) For the drift phase ending date, t2 99.137 days. The transfer sequences are detailed in the Table 2. The drift precession rate is close to the expected value: 1.284 deg ∕day (≈0.982 0.3). This precession rate is obtained by an optimized combination of altitude decrease and inclination increase. C. Cost Sensitivities
The SES problem is solved again with varying initial conditions in order to assess the cost sensitivities by centered finite differences. The Table 1
Initial and final orbits
Altitude, km Velocity, m∕s Inclination, deg Precession rate, deg ∕day Initial RAAN at t0 0 days, deg Final RAAN at t0 100 days, deg
Initial orbit Final orbit 800.0 900.0 7450.0 7398.6 98.00 99.00 0.917 0.982 0.0 30.0 91.7 128.2
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Table 2
t0 t1 t2 tf
SES sequences
Date, day Altitude, km Velocity, m∕s Inclination, deg RAAN, deg Precession, deg ∕day Impulse, m∕s 0.000 800.0 7450.0 98.00 0.00 0.917 0.0 1.075 404.7 7664.0 99.20 1.18 1.284 328.7 99.137 404.7 7664.0 99.20 127.26 1.284 328.7 100.000 900.0 7398.6 99.00 128.20 0.982 598.1
Table 3
Cost derivatives
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Variation Cost ΔV, m∕s Derivative, deg−1 Reference 0 598.1 — Initial velocity, m∕s −25.950 km 615.5 −0.6300.327 (initial altitude shown in kilometers) 26.0−50 km 582.8 Initial inclination, deg 0.1 583.2 −154.5 −0.1 614.1 Initial RAAN, deg 5 539.3 −13.27 −5 672.0 Final date, day 5 582.3 −3.695 −5 619.3
Derivative, rad−1 — −0.6300.327 −8852.2 −760.31 −3.695
D. OCP Solution Table 4
Velocity Inclination RAAN Switching 1 Switching 2
Initial guess (m∕s, rad, day) pV t0 −0.630 pI t0 −8852.2 pΩ t0 −760.31 t1 1.075 t2 99.137
Table 5
Velocity Inclination RAAN Switching 1 Switching 2
Shooting problem initial guess Constraint value Vtf − V f 5.7 m∕s Itf − I f −0.02 deg Ωtf − Ωf −1.59 deg St1 0.032 St2 0.119
Shooting problem solution
Initial guess (m∕s, rad, day) pV t0 −0.644 pI t0 −9215.9 pΩ t0 −816.97 t1 1.092 t2 99.114
Constraint value Vtf − V f 10−5 m∕s Itf − If 10−8 deg Ωtf − Ωf 10−6 deg St1 10−8 St2 10−8
results are presented in the Table 3 and converted in radians for the angles (inclination, RAAN). The initial velocity variation corresponds to an initial altitude variation of 50 km. The sensitivity to the final date is also assessed. The cost partial derivatives with respect to the initial state provide estimates of the initial costate components (the units are in meters per second and radians): 8 < pV t0 ≈ −0.630 (29) p t ≈ −8852.2 : I 0 pΩ t0 ≈ −760.31 The Hamiltonian is estimated from the cost partial derivative with respect to the final date (the units are in meters per second and days): H−
dΩtf ∂J dJ ∂J dJ _f × − − pΩ Ω dtf ∂tf dtf ∂Ωtf dtf
3.695 −13.27 × 0.982 −9.336
(30) Table 6
t0 t1 t2 tf
The shooting problem [Eq. (20)] is solved by a Newton method (software Hybrd.f) initiated with the SES guess [Eq. (29)]. This guess is already as close to a zero of the shooting function as can be observed on the constraint values in Table 4. The Newton method converges accurately in a few iterations, yielding the shooting problem solution (Table 5). The sequences of the optimal transfer are detailed in Table 6. In this example, the OCP solution is very close to the SES solution. This should be the case, provided that a sufficient duration is allocated to the mission with respect to the engine thrust level so that the propelled durations remain small with respect to the drift duration. If the duration allocated to the transfer becomes small, the Edelbaum thrusting strategy (constant out-of-plane angle β with sign thrust at the antinodes) is no longer adapted to the mission. In such cases, the optimal control problem must be addressed directly without averaging of the dynamics equations and the optimal thrust law should aim at changing simultaneously the inclination and the RAAN.
V.
Conclusions
The minimum-fuel low-thrust transfer problem between circular orbits has been investigated considering a transfer strategy based on the Edelbaum averaged dynamics equations. The thrust is used to change the altitude and the inclination, whereas the RAAN change is realized passively by the natural precession due to the first zonal term. This control strategy requires a sufficient mission duration in order to benefit from the natural precession. It is especially suited for multiple debris-removal missions that are currently under study. The extremal analysis reveals that singular solutions may exist. Ongoing research focuses on the optimality of singular solutions and on the possible junctions between regular and singular legs. For practical applications, only regular solutions are envisioned. Such regular trajectories are composed of three sequences. The first propelled sequence brings the vehicle on a circular drift orbit, the second coast sequence achieves the required RAAN change, and the third propelled sequence completes the transfer to reach the targeted final altitude and inclination.
Optimal transfer sequences
Date, day Altitude, km Velocity, m∕s Inclination, deg RAAN, deg Precession, deg ∕day Impulse, m∕s 0.000 800.0 7450.0 98.00 0.00 0.917 0,00 1.092 407.1 7664.6 99.22 1.19 1.287 330.2 99.114 407.1 7664.6 99.22 127.20 1.287 330.2 100.000 900.0 7398.6 99.00 128.20 0.982 598.1
CERF
A shooting method is applied to solve the optimal control problem. The costate guess is derived from a nonlinear problem that approximates the transfer using the Edelbaum minimum-time solution, with this nonlinear problem itself being initiated with a quasianalytical solution. The guess is quite close to the optimal solution, provided that a sufficient duration is allocated to the mission. The solution method proves fast and reliable enough for an automatic procedure. It opens the way to the optimization of multiple debrisremoval missions. These missions raise combinatorial challenges, since thousands of transfers between debris pairs must be assessed for any endpoint conditions. The solution method proposed in this paper provides such an online assessment, and it has been successfully used for the preliminary planning of future debris-removal missions.
Downloaded by CNES on September 22, 2016 | http://arc.aiaa.org | DOI: 10.2514/1.G001331
References [1] Docherty, S. Y., and Macdonald, M., “Analytical Sun-Synchronous Low-Thrust Orbit Maneuvers,” Journal of Guidance, Control, and Dynamics, Vol. 35, No. 2, 2012, pp. 681–686. doi:10.2514/1.54948 [2] Edelbaum, T. N., “Propulsion Requirements for Controllable Satellites,” ARS Journal, Vol. 31, No. 8, Aug. 1961, pp. 1079–1089. doi:10.2514/8.5723
2239 [3] Leitmann, G., Optimization Technique with Applications to Aerospace System, Academic Press, New York, 1962, pp. 19–30. [4] Kechichian, J. A., “Reformulation of Edelbaum’s Low-Thrust Transfer Problem Using Optimal Control Theory,” Journal of Guidance, Control, and Dynamics, Vol. 20, No. 5, Sept.–Oct. 1997, pp. 988–994. doi:10.2514/2.4145 [5] Casalino, L., and Colasurdo, G., “Improved Edelbaum’s Approach to Optimize LEO-GEO Low-Thrust Transfers,” AIAA/AAS Astrodynamics Specialist Conference, AIAA Paper 2004-5402, Aug. 2004. [6] Conway, B. A., Spacecraft Trajectory Optimization, Cambridge Univ. Press, New York, 2010, pp. 139–177. [7] Kluever, C. A., “Using Edelbaum’s Method to Compute Low-Thrust Transfers with Earth-Shadow Eclipses,” Journal of Guidance, Control, and Dynamics, Vol. 34, No. 1, 2011, pp. 300–303. doi:10.2514/1.51024 [8] Cerf, M., “Multiple Space Debris Collecting Mission—Optimal Mission Planning,” Journal of Optimization Theory and Applications, Jan. 2015. doi:10.1007/s10957-015-0705-0. [9] Chobotov, V., Orbital Mechanics, 3rd ed., AIAA Education Series, AIAA, Reston, VA, 2002, pp. 7–8, 186, 207–208, 335–354. [10] Pontryagin, L., Boltyanskii, V., Gramkrelidze, R., and Mischenko, E., The Mathematical Theory of Optimal Processes, Wiley Interscience, New York, 1962, pp. 19–22. [11] Trélat, E., Contrôle Optimal — Théorie et Applications, Vuibert, Paris, 2005, pp. 103–104.
