Solar sail trajectories with piecewise-constant steering laws

Aerospace Science and Technology 13 (2009) 431–441

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Solar sail trajectories with piecewise-constant steering laws Giovanni Mengali ∗ , Alessandro A. Quarta University of Pisa, Dipartimento di Ingegneria Aerospaziale, Via G. Caruso, I-56122 Pisa, Italy

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 14 February 2008 Received in revised form 10 June 2009 Accepted 30 June 2009 Available online 10 July 2009 Keywords: Solar sail Mission analysis Trajectory optimization

Solar sails have much attracted the interest of the scientific community as an advanced low-thrust propulsion means capable of promoting the reduction of mission costs and the feasibility of missions that are not practically accessible via conventional propulsion because of their large  V requirements. To reduce the overall flight time, a given mission is usually analyzed in the framework of a minimum time control problem, with the employment of a continuous steering law. The aim of this paper is to investigate the performance achievable with a piecewise-constant steering law whose aim is to substantially reduce the complex task of reorienting the sail over the whole mission. Unlike previous studies based on direct approaches, here we use an indirect method to optimally select the sail angle within a set of prescribed values. The corresponding steering law translates the results available for continuous controls to the discrete case, and is able of producing trajectories that are competitive in performance with the optimum variable direction program. © 2009 Elsevier Masson SAS. All rights reserved.

1. Introduction Solar sails provide unique capabilities for space missions, due to their inherent characteristics of supplying continuous thrust over the mission life without any propellant expenditure [8,14,27]. This is a particularly important feature for promoting the feasibility of missions that are not practically accessible via conventional propulsion systems due to their large  V requirements [11–13,18, 20]. However, the employment of a solar sail as a primary propulsion means calls for the solution of a number of challenging problems that could jeopardize the mission success if not taken into account properly. Among these, a complex issue is that of orienting and continuously controlling a structure, the solar sail, whose characteristic dimension is of the order of some tens of meters or even more. Therefore, one of the primary goals during the mission analysis phase consists in the development of steering laws capable of combining effectiveness (in terms of required mission time) with simplicity to implement. As the objective of minimizing the mission time typically conflicts with that of using simple steering laws, it is often necessary to look for compromise solutions. In essence, the solar sail attitude control system must be able of providing, at each time instant, the optimal values of two independent control variables, the pitch and clock angles, that define the sail orientation in space [14]. Assuming for simplicity a flat sail

*

Corresponding author. Tel.: +39 050 2217220; fax: +39 050 2217244. E-mail addresses: [email protected] (G. Mengali), [email protected] (A.A. Quarta). 1270-9638/$ – see front matter doi:10.1016/j.ast.2009.06.007

© 2009 Elsevier Masson

SAS. All rights reserved.

and denoting with nˆ a unit vector normal to the sail plane (always directed away from the Sun), the pitch angle defines the direction of the incoming radiation with respect to nˆ and the clock angle characterizes the sunlight direction measured on the sail plane. Optimal (that is, minimum time) solutions usually require the two control angles to be varied with continuity over the whole mission phase. This may be a very demanding task which needs some kind of simplification to make it feasible. In this paper we cope with this problem by using a piecewise-constant steering law, where the number of different sail orientation maneuvers during the mission is a measure of the steering law complexity. The problem of trajectory optimization with a prescribed and discrete number of thrust directions has been studied by Melbourne and Sauer for solar electric propulsion spacecraft [15]. They have shown that a small number of thrust directions (less than three) allows an Earth–Mars rendezvous transfer to be performed with a propellant consumption close (the differences being on the order of 2%) to that attainable with a globally optimum continuous thrust-program. As long as solar sails are concerned, the only few studies dedicated to this subject use a direct approach to discretize the continuous control law with the aim to minimize the differences with respect to the true minimum-time trajectory, see Ref. [19] and the references therein. However, for mathematical tractability, a direct approach usually involves the division of the total mission time in a prescribed number of trajectory arcs of equal length [1,2]. Therefore this technique introduces various additional parameters, such as the number of arcs and their length, whose impact on the mission time can only be calculated with the aid of parametric studies.

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Nomenclature A ac b1 , b2 , b3 fg fp F SRP H Hp J m n nˆ P r rˆ

T

t

U u v

x

sail area characteristic acceleration force coefficients gravitational acceleration vector propulsive acceleration vector sail propulsive force Hamiltonian function reduced Hamiltonian performance index sailcraft total mass number of sail reorientation maneuvers unit vector normal to the sail plane solar radiation pressure ( P ⊕  4.563 μN/m2 ) Sun-sailcraft distance sailcraft position unit vector (r = r rˆ ) polar inertial frame time set of admissible values of α radial velocity circumferential velocity

θ λ λi λv Subscripts

⊕   

Superscripts

·  T

At a distance r = r  from the Sun, the magnitude of the solarlight pressure P exerted on a perfectly absorbing surface is [14]

r⊕

2 (1)

r

where P ⊕  4.563 μN/m2 is the solar radiation pressure at r⊕  1 AU. Assuming a perfectly reflecting (ideal) solar sail of area A, the corresponding solar radiation pressure force is [14,27]

F SRP = 2P A cos α nˆ 2

(2)

where α ∈ [−π /2, π /2] is the sail pitch angle, that is, the angle ˆ Therefore between the direction of rˆ and n.

cos α = rˆ · nˆ

time derivative optimal value transpose integral mean value of 2

2

2. Mathematical model



Earth Mars Sun Venus initial final

0 f

In this paper, instead, the problem is tackled from a sharply different viewpoint. In particular, the maximum number of different orientation angles allowed by the control law is now an input parameter, while an indirect method is employed to optimally select the control angles in the admissible set and generate the piecewise-constant steering law. Accordingly, the admissible controls can be reduced to an arbitrary small number of values. The corresponding impact of the number of orientation angles on the overall mission performance can be easily translated into tradeoff studies. Note also that, unlike a direct approach, the proposed technique allows one to avoid an a-priori discretization of the mission into sub arcs. Indeed, the actual number of arcs with constant pitch angle is an output of the optimization procedure. The paper is organized as follows. First, the ideal and optical solar sail force models are briefly summarized along with the equations of motion. Then, the minimum time rendezvous problem between circular and coplanar orbits is introduced, where the controls are chosen to belong to a discrete set of admissible values. Using an indirect approach, the optimal steering law is found by maximizing, at any time, the Hamiltonian of the system. The new procedure has been used to simulate a classical circle-to-circle rendezvous mission for both Earth–Mars and Earth–Venus transfers.

P = P⊕

state vector sail pitch angle polar angle adjoint vector adjoint variable, with i = (r , θ, u , v ) primer vector

α

(3)

Note that the sail mathematical model used in this paper does not take into account torque disturbances during the trajectory, as for example the offset between the center of mass and the center of pressure. This simplification is consistent with a preliminary interplanetary mission analysis. If one relaxes the assumption of perfect reflectivity, a more accurate model (usually referred to as optical force model) is obtained. The result is [6]



F SRP = 2P A cos α b1 rˆ + (b2 cos α + b3 )nˆ



(4)

where b1 , b2 and b3 (with b1 + b2 + b3  1) are the dimensionless force coefficients related to the thermo-optical properties of the reflective film. Note that Eq. (2) can be recovered from Eq. (4) by simply setting b1 = b3 = 0 and b2 = 1. The performance of a solar sail is usually quantified through the values taken by the characteristic acceleration ac , that is, the acceleration experienced by an ideal solar sail oriented perpendicular to the Sun line (nˆ ≡ rˆ ) at r = r⊕ . From Eq. (2) one has

ac =

2P ⊕ A

(5)

m

where m is the sailcraft mass. The value of ac depends on the technology employed to build the sail, especially that of the reflecting film. Currently admissible values [7,21] are on the order of ac ∈ [0.1, 0.5] mm/s2 , even if near-term technology [4,5] will hopefully allow values of ac ∈ [1, 2] mm/s2 within the next twenty years. Consider a polar inertial frame T (r , θ), where θ is the polar angle measured anticlockwise from some reference position, and introduce the state vector

x  [r

θ

u

v ]T

(6)

where u and v are the radial and circumferential sailcraft velocities, see Fig. 1. The heliocentric equations of motion in vector form can be written as:

x˙ = f g + f p

(7)

where f g and f p represent the gravitational and propulsive accelerations, see Eqs. (8) and (9).

 fg u

v r



v2 r

−

μ r2

 −

uv r

T (8)

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433

Finally, the minimum flight time t f is obtained by enforcing the transversality condition [3] H (t f ) = 1. Invoking Pontryagin’s maximum principle, the optimal sail pitch angle α , to be selected in the domain of feasible controls U , is such that, at any time, the Hamiltonian H is an absolute maximum. Therefore

α = arg max H

(19)

α ∈U

If one introduces the primer vector [10]

λv  [λu λ v ]T

(20)

then Eq. (19) reduces to

α = arg max H p

(21)

α ∈U

where

Fig. 1. Reference frame and state variables.

 f p  ac

r⊕

2



cos α 0 0

r



b1 + b2 cos2 α + b3 cos α

H p  F SRP · λv



T sin α (b2 cos α + b3 )

(9)

Note that Eq. (9) follows from (4) after having substituted here ac of Eq. (5). Consider an interplanetary circle-to-circle transfer problem between an initial (t 0 = 0) orbit of radius r (t 0 ) = r⊕  1AU and a target coplanar orbit of radius r (t f ) = r f , where t f is the final time. Note that a planar problem entails the presence of a single control variable, the sail pitch angle, which is sufficient for fully characterizing the sail orientation. Our aim is to solve the minimum-time problem, that is, to find the optimal control law α = α (t ) that maximizes the performance index J defined as

J  −t f

(10)

Using an indirect approach, we introduce the Hamiltonian function

H  ( f g + f p) · λ

with λ  [λr

λθ

λu λ v ]T

  ∂fg ∂fp ˙λ = − ·λ + ∂x ∂x

(12)

Substituting Eqs. (8) and (9) into Eq. (12) the following four scalar differential equations are obtained:



λθ v r2

+

v2

+ λu

2ac r



− λv



r2

r3





cos α b1 + b2 cos2 α + b3 cos α

r r2

2μ

2

r⊕

uv

−

−

2ac



r⊕

r

r



2

 sin α cos α (b2 cos α + b3 )

λ˙ θ = 0

λ˙ v = −

(13) (14)

λ˙ u = −λr + λ v λθ r

−2

v

(15)

r

λu v r

+

λv u

(16)

r

The problem setup, constituted by the differential problem (7)– (9) and (13)–(16), is completed by the boundary conditions at t 0 and t f :

r (t 0 ) = r⊕ ,

θ(t 0 ) ≡ u (t 0 ) = 0,

r (t f ) = r f ,

u (t f ) ≡ λθ (t f ) = 0,

v (t 0 ) =



v (t f ) =

μ /r⊕



coincides with that portion of the Hamiltonian that explicitly depends on the pitch angle, see Eq. (11). Unlike a classical approach in which the feasible controls coincide with an interval of R (for example, U = {α | −π /2  α  π /2}), in this paper U is a set of admissible values, that is

U = {α1 , α2 , . . . , αk },

k∈N

(23)

From a practical standpoint the maximization of H p can be performed numerically with a limited amount of calculation time. In fact, at a generic time instant t, the admissible values of H p belong to the set { H p (α1 ), H p (α2 ), . . . , H p (αk )} whose maximum value can be found easily with a sorting procedure, such as the quick sort algorithm by Hoare [9]. From Eq. (22), the optimum thrust direction may be thought of as that direction, taken from U , which maximizes the projection of the sail thrust along the primer vector. This result is the discrete version of the optimal continuous steering law found in Refs. [6,16,17].

(11)

where λ is the adjoint vector. The time derivatives of the adjoint variables are provided by the Euler–Lagrange equations:

λ˙ r =

(22)

μ /r f

(17) (18)

3. Trajectory simulation The approach described in the previous section has been applied to study the heliocentric phases of Earth–Mars (r f ≡ r  1.52368 AU) and Earth–Venus (r f ≡ r  0.723331 AU) rendezvous missions with ac = (1, 2) mm/s2 . Because no ephemeris constraint is used, the simulation results are independent of planet phasing and represent the theoretical minimum transfer time between Earth and the target planet. As long as the sail film reflective properties are concerned, both ideal and optical force models have been considered. In the latter case we used the optical force coefficients for a sail with a highly reflective aluminum-coated front side and a highly emissive chromium-coated back side, whose values are [16,27] b1 = 0.0864, b2 = 0.8272 and b3 = −5.45 × 10−3 . A set of canonical units have been used in the integration of the differential equations to reduce their numerical sensitivity. The differential equations were integrated in double precision using a Runge–Kutta fifth-order scheme with absolute and relative errors of 10−8 . The final boundary constraints (18) were set to 1000 km for the position error and to 0.1 m/s for the velocity error. A number of simulations have been performed using different admissible control sets, the differences being in terms of k, see Eq. (23). Irrespective of the chosen value of k, each admissible set U is such that U  ⊂ U , where U   {−90, 0, 90} deg. With such a choice the sailcraft is allowed to generate the maximum available thrust (when α = 0), as well as to set it to zero (when α = ±90 deg), thus introducing coast arcs in the spacecraft trajectory.

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For a given set of admissible controls, the trajectory is characterized through the number n of sail reorientation maneuvers required by the piecewise-constant steering law. For example, a steering law in which α = 0 for 0  t < t 1 , α = 30 deg for t 1  t < t 2 and α = 0 for t 2  t  t f corresponds to n = 2. Note that the value of n is different, in general, from the number of arcs in which the trajectory is divided when a direct optimization method is employed [2]. Therefore, the value of n may be thought of as an index of the complexity of the interplanetary rendezvous mission (the greater being n, the more demanding the corresponding maneuver). Also, the proposed approach allows one to establish

a tradeoff between mission performance (in terms of total transfer time t f ) and attitude performance, which, as stated, depends on n. The optimal solutions obtained solving the problem (21)–(23) have been compared with the globally optimal values, referred to as t f , calculated when the pitch angle is allowed to vary with continuity in the interval α ∈ [−90, 90] deg. In the latter case, as stated, the optimal control law α  (t ) prescribes the thrust vector to maximize its projection along the primer vector [16]. The minimum times t f for the examples studied have been summarized in Table 1. The table also shows the integral mean

¯ . Fig. 2. Pitch angle time history for an optimal Earth–Mars rendezvous mission and corresponding integral mean value α

¯ . Fig. 3. Pitch angle time history for an optimal Earth–Venus rendezvous mission and corresponding integral mean value α

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435

Fig. 4. Heliocentric minimum time trajectories using a continuous steering law.

Table 1 Minimum flight times t f using a continuous steering law for different characteristic accelerations and force models. Mission Earth–Mars

ac [mm/s2 ]

t f [days]

[deg]

ideal

1 2 1 2 1 2 1 2

407.7 323.9 447.7 357.1 204.8 163.7 224.2 180.6

44.94 50.94 43.34 49.91 −46.21 −51.44 −44.79 −50.47

optical Earth–Venus

α¯ 

Reflective model

ideal optical

¯  of the sail pitch angle, defined as value α

t f

t

α¯  

s t f

f

|α | dt with s  sign 

0



α dt 0

where sign(·) is the signum function. The control law time histories α  = α  (t ) corresponding to ac = (1, 2) mm/s2 for Earth–Mars and Earth–Venus rendezvous missions are shown in Figs. 2 and 3, respectively. Note that the analyzed cases cover all of the possible mission typologies, that is, cases in which the pitch angle is always positive (Fig. 2, case a), always negative (Fig. 3, case a) or takes both positive and negative values (Fig. 2, case b and Fig. 3, case b). The corresponding spacecraft trajectories are shown in Fig. 4. 3.1. Set of admissible controls A number of simulations have been performed by varying the complexity of the set of admissible controls U . In this study the least admissible number of elements is k = 5, or

U = {−90, −α f , 0, α f , 90} deg (24)

(25)

where α f > 0 is an optimization parameter. Note, however, that three only of the five pitch angles (that is, α = 0 and α = ±α f ),

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Fig. 5. Simulation results for an Earth–Mars rendezvous mission with piecewise-constant steering laws and U = {−90, −α f , 0, α f , 90} deg. Each point corresponds to a different value of α f , while t f = t f is the minimum time that can be found with a continuous steering law.

correspond to situations in which the solar sail is able to produce thrust, the other two (α = ±90 deg) being representative of coasting behavior, see Eq. (9). For each value of α f ranging in the interval [30, 60] deg, with a step-size α f = 1 deg, the optimization problem (22), (23) has been solved for two different values of the characteristic acceleration, ac = 1 mm/s2 and ac = 2 mm/s2 , with both ideal and optical solar sail force models. The results for Earth–Mars and Earth–Venus rendezvous missions have been summarized in Figs. 5 and 6. For comparative purposes the global minimum time t f = t f (that can be found with a continuous steering law) is also shown. Figs. 5 and 6 show that an optimal value for α f exists, that is

α f  arg min t f

(26)

αf

which depends on the value of the characteristic acceleration and on the solar sail force model. With the aid of Figs. 5, 6 and Table 1 ¯  |, the differences being negligible, less one concludes that α f ≈ |α

than some tenth of degree. Therefore, the integral mean value of the sail pitch angle provides an interesting physical interpretation for α f . Note also that, assuming α f = α f , the difference between the optimal flight time t f (α f ) and the corresponding global minimum time t f decreases as the value of the characteristic acceleration is increased and is less than 4% for both the missions. This means that the use of a piecewise-constant steering law is able of providing performance close to that globally optimal even under the assumption of a limited number of pitch angles belonging to U . Another point worth noting is that for all of the simulations the value of n is very small, ranging between 2 and 4, which results in a considerable simplification of the mission planning. Nevertheless, a small value of n may require large pitch angle variations, as is shown in Figs. 7 and 8 for the two previous missions. For an Earth– Mars rendezvous with an optical force model and ac = 2 mm/s2 Fig. 7 shows that, after approximately 90 days from the departure, a sail reorientation maneuver of 140 deg would be required. In principle, such a maneuver should occur instantaneously, even if,

G. Mengali, A.A. Quarta / Aerospace Science and Technology 13 (2009) 431–441

437

Fig. 6. Simulation results for an Earth–Venus rendezvous mission with piecewise-constant steering laws and U = {−90, −α f , 0, α f , 90} deg. Each point corresponds to a different value of α f , while t f = t f is the minimum time that can be found with a continuous steering law.

in practice, it needs a time interval whose actual length depends on the attitude control system [22–26] mounted on the sailcraft. The attitude maneuvers require the employment of a suitable control system, while usually, during an interplanetary trajectory, the sail attitude changes slowly and its rate is of some degrees per day only. In addition, if a propellantless attitude control system is chosen, as is obtained by shifting and tilting the sail panels [23,24], the problem of guarantying rapid sail reorientation maneuvers is a demanding task especially for trajectories towards outer planets. In fact for these mission typologies the solar radiation pressure decreases significantly due to the considerable distance from the Sun. The same problem is much simpler when missions towards inner planets are investigated and, in these cases, the attitude control system can usually complete the maneuver in some tens of minutes [24]. Such a time interval is so small when compared to the whole mission length, that an instantaneous pitch variation may safely be assumed without making appreciable errors in the estimate of the mission time. For example, using the results from Ref. [24], a small square solar sail (having a side length of 40 m, a total mass of 156 kg and ac  0.081 mm/s2 ) in orbit around the

Earth, is able to perform a pitch reorientation maneuver of 50 deg in about 2 hours. 3.2. Sensitivity to the number of admissible controls The problem of reorientation maneuvers with large solar sail pitch angles can be alleviated by increasing the number of admissible controls in the set U . In particular, it is interesting to investigate the solution sensitivity (both in terms of mission time and total number of reorientation maneuvers) as a function of the dimension of U . To this end, the interval [−90, 90] deg has been discretized, using a finite number of equispaced admissible values, as

U = {−90 : α : 90}

(27)

where α > 0 (in degrees) is the angular step size. For example, α = 45 corresponds to U = {−90, −45, 0, 45, 90} deg. Of course, the number of elements of U increases as long as the value of α is decreased. In the limit as α → 0 a continuous interval [−90, 90] deg is recovered. The simulation

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Fig. 7. Pitch angle time history for an Earth–Mars rendezvous mission with piecewise-constant steering laws and U = {−90, −α f , 0, α f , 90} deg.

results have been summarized in Figs. 9 and 10 for α = (1, 3, 5, 10, 15, 22.5, 30, 45) deg. Note that α = 15 deg is sufficiently small to guarantying that the transfer times are nearly coincident with that globally optimal t f . Clearly, an increase of the number of elements of U tends to increase also the number n of reorientation maneuvers. For example, α = 15 deg corresponds to 10  n  15 for all of the simulations. This is a very satisfactory result as it states that in a mission whose length is on the order of some hundreds of days, about ten reorientation maneuvers are sufficient for a nearly optimal transfer. The piecewiseconstant steering laws corresponding to α = (5, 15, 30, 45) deg have been compared in Fig. 11. Note that, as expected, as α is decreased, the discrete steering law tends to closely follow the shape of the optimal continuous control law. The reason why the case α = 30 deg has a worse performance than α = 45 deg can be explained by observing (see Table 1) that 45 is closer than 30 ¯  |. to a submultiple of |α The differences obtainable using the proposed method with respect to a classical direct optimization approach have been summarized in Table 2 for an Earth–Mars mission using an ideal sail with ac = 1 mm/s2 . Although for a given value of n the two methods provide similar values for the mission times, the new approach requires smaller values of the maximum angular step size. This behavior is clearly shown in Fig. 12 in which the two piecewise-

constant steering laws for the case n = 10 of Table 2 are compared. 4. Conclusions The minimum time problem of a solar sail using piecewiseconstant steering laws has been formulated and solved using an indirect approach. The new procedure has been used to study a classical circle-to-circle orbit transfer problem, even if the same approach can effectively be employed to solve three dimensional problems with minor modifications. Assuming that a unique value of sail orientation must be maintained for the whole mission, the optimal choice consists in choosing the integral mean value of the pitch angle calculated with respect to the continuous steering law. For a prescribed set U of admissible pitch angles, the optimum thrust direction is that direction, taken from U , which maximizes the projection of the sail thrust along the primer vector. This result is the discrete counterpart of the same control logic valid for the continuous case. It has been shown by simulation, for both Earth–Mars and Earth–Venus transfers, that with a few values of admissible pitch angles the optimal mission time is close to that found with a continuous steering law. Therefore, a substantial reduction of the reorientation maneuver complexity is possible with results competitive in performance with the optimal variable direction program.

G. Mengali, A.A. Quarta / Aerospace Science and Technology 13 (2009) 431–441

Fig. 8. Pitch angle time history for an Earth–Venus rendezvous mission with piecewise-constant steering laws and U = {−90, −α f , 0, α f , 90} deg.

Fig. 9. Earth–Mars mission performance with piecewise-constant steering laws and U = {−90 : α : 90}, where α is the maximum value of the sail pitch angle.

439

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Fig. 10. Earth–Venus mission performance with piecewise-constant steering laws and U = {−90 : α : 90}, where α is the maximum value of the sail pitch angle.

Fig. 11. Simulation results for an Earth–Mars rendezvous with piecewise-constant steering laws and U = {−90 : α : 90} for different values of α . The simulations correspond to an optical force model and a characteristic acceleration ac = 2 mm/s2 .

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Table 2 Performance comparison between the new approach with a direct method (data taken from Ref. [19] and displayed between round brackets) as a function of the maximum number of reorientation maneuvers n. Mission

n

Earth–Mars (ideal model)

4 5 7 10 20 35 103

∞

time [days]

α (max) [deg]

penalty [%]

423.9 (418.0) 411.4 409.9 409.2 (409.0) 408.1 (408.8) 407.9 407.8 407.7

45.0 (52.0) 22.5 15.0 10.0 (49.0) 5.0

3.96 (2.40) 0.90 0.55 0.36 (0.20) 0.10 (0.15) 0.04 0.015 0

3.0 1.0 0

[7]

[8] [9] [10] [11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

Fig. 12. Comparison between the proposed method (dash-dot line) with a direct approach (dashed line) for an Earth–Mars mission using an ideal solar sail with ac = 1 mm/s2 . The optimal continuous control law (solid line) is also shown for comparative purposes.

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[21]

[22]

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[24]

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[25]

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[26]

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