Interplanetary trajectory design for a hybrid propulsion system

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Interplanetary trajectory design for a hybrid propulsion system

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School of Aerospace, Tsinghua University, Beijing 100084, China

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Shengping Gong 1 , Junfeng Li 2 , Fanghua Jiang 3

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Article history: Received 11 September 2012 Received in revised form 11 August 2014 Accepted 28 April 2015 Available online xxxx Keywords: Hybrid propulsion system Solar electric propulsion Solar sail Interplanetary trajectory design Optimization

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A fuel-optimal interplanetary trajectory design is investigated for a hybrid propulsion system based on solar sailing and solar electric propulsion. The optimal-control problem is formulated using the calculus of variations and Pontryagin’s maximum principle. The solution to the corresponding two-point boundary problem is a bang–bang control for the solar electric propulsion engine that is solved by an indirect method combined with the homotopic technique. Two interplanetary missions, including the Venus and Apophis rendezvous, are used to analyze the fuel consumption for different combinations of the two propulsion systems. The departure time associated with the launch window and the transfer time are two factors that influence the fuel consumption. The optimal departure time can be obtained for a given launch window time range. Additionally, a solar sail used as an auxiliary system to solar electric propulsion can decrease the propellant expense to zero if the transfer time is sufficiently long. Solar electric propulsion used as an auxiliary to a solar sail can significantly shorten the transfer time, while consuming only a small quantity of propellant. Therefore, a hybrid of the two systems can avoid their respective limitations and lead to the development of pure solar sails for space missions. © 2015 Elsevier Masson SAS. All rights reserved.

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1. Introduction

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In deep space, solar radiation pressure (SRP) is a predominant extra force exerted on a solar sail. SRP can be used for orbital maneuvers and interplanetary transfers. JAXA has launched the first solar sail demonstrator, IKAROS [1]. Although IKAROS is mainly propelled by traditional propulsion, SRP provides a velocity increment during the journey of IKAROS to Venus. However, solar sail technology’s readiness level is low at present, and it remains difficult to rely exclusively on a solar sail to complete an interplanetary mission. In contrast, solar electric propulsion (SEP) has been tested in several deep space missions [2,3]. Early in 1972, MacNeal compared the solar sail and electric propulsion systems and noted that both propulsion systems had advantages and disadvantages [4]. A solar sail is superior to electric propulsion for missions of long duration and targets in the inner solar system [5]. The idea of using a solar sail as a propellantless auxiliary system to decrease the propellant expense for a given mission has been proposed. The hybrid propulsion system (HPS) can avoid the respective limitations

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E-mail addresses: [email protected] (S. Gong), [email protected] (J. Li), [email protected] (F. Jiang). 1 Associate professor. 2 Professor. 3 Assistant professor. http://dx.doi.org/10.1016/j.ast.2015.04.020 1270-9638/© 2015 Elsevier Masson SAS. All rights reserved.

of the two systems at the cost of increasing spacecraft complexity. SEP can provide an accurate and reliable full range of the thrust vector, which is particularly important when the spacecraft approaches the target. Furthermore, SEP may be used to decrease the transfer time and increase the reliability of the HPS. As an auxiliary system, a solar sail can provide a portion of the acceleration to save propellant, and the electric thruster can even be shut down to completely rely on the solar sail when the SEP engine propellant is exhausted. Moreover, the hybrid spacecraft can lead to the gradual introduction of pure solar sails for space missions. Several novel HPS applications have been proposed. Baig and McInnes used a hybrid sail to generate artificial equilibria above L1 in the Sun–Earth system for Earth observation [6]. Simo and McInnes exploited hybrid propulsion to search for displaced periodic orbits in the Earth–Moon restricted three-body system [7]. Ceriotti and McInnes discussed an Earth pole-sitter using hybrid propulsion [8]. Additionally, the idea of using an HPS for interplanetary transfers has been proposed. However, the available literature is rare. Mengali and Quarta investigated optimal coplanar interplanetary transfers to Venus and Mars using an indirect optimization method [9]. With respect to solar sail trajectory optimization, a number of studies on solar sailing interplanetary trajectory design have been performed since the 1970s. Sauer [10], Jayaraman [11] and Wood [12] presented detailed trajectory designs based on generalizations of the variational approach and assuming the orbits of the Earth

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Nomenclature

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μ

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β

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m m0 mf t0 tf Fc

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R V T max

λR , λV λm u

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σ

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Gravitational constant of the Sun . . . . . . . . . . . . . Nm /kg Lightness number (ratio of the maximum solar radiation pressure force to the solar gravitational force) Mass of the spacecraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kg Initial mass of the spacecraft . . . . . . . . . . . . . . . . . . . . . . . kg Final mass of the spacecraft . . . . . . . . . . . . . . . . . . . . . . . . kg Time of the sailcraft’s departure from the Earth, date Time of the sailcraft’s arrival at the target body, date Total acceleration of solar radiation pressure and solar electric propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m/s2 Position vector of the sail with respect to the Sun m Velocity vector of the sail with respect to the Sun m/s Maximum thrust of the SEP engine . . . . . . . . . . . . . . . . . N Lagrange multipliers of the position and velocity vector, dimensionless Lagrange multiplier of the mass, dimensionless Positive number between zero and one to describe the thrust magnitude Unitary vector representing the thrust direction

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I sp

Specific impulse of the solar electric propulsion engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . s Gravitational acceleration on the Earth, 9.80665 m/s2 Fuel-optimal performance index . . . . . . . . . . . . . . . . . . . . kg Energy-optimal performance index . . . . . . . . . . . . . . . . . kg Performance index considering fuel and energy . . . . kg Homotopic parameter Unit vector along the sail normal, a unit vector Switch function Pitch angle of the sail normal with respect to the sunlight direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rad Angle measured from −λ V to the sunlight direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rad Hamilton function of the fuel-optimal problem, dimensionless Hamilton function of the new defined problem, dimensionless Unit vector perpendicular to the sunlight in the plane spanned by sunlight and λ V A positive weight constant

g0 J ˜J Jn

ε n

ρ α α˜ H fo Hn e

κ

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and Mars to be circular and coplanar. When an indirect method is employed in the optimization, the long simulation time makes the solution a difficult task particularly because the boundary values are extremely sensitive to the initial guess value of the costates. A direct method is always used to solve the transfer trajectory design problem. The advantage of such control laws lies in the capability to approximate the optimal trajectory through simple numerical simulations [13]. Dachwald has thoroughly examined solar sail trajectory optimization using an evolutionary neurocontrol-based global trajectory optimization [14]. An approach based on locally optimal control laws is particularly useful for planetary escape missions [15,16]. Macdonald used the blended locally optimal control laws to optimize the interplanetary transfer trajectories [17]. Recently, increasing attention has been focused on the fueloptimal trajectory design of low-thrust propulsion. The Global Trajectory Optimization Competition (GTOC) is a worldwide initiative of the European Space Agency (ESA) to solve the low-thrust trajectory design problems [18]. The fuel-optimal problem of an interplanetary trajectory always leads to a discontinuous optimal control derived from the calculus of variations, as the switching function switches at zeros. If the switching function takes the value zero on some intervals, the optimal solution will involve singular arcs. In this case, one has to differentiate the switching function until the control appears explicitly in order to derive an expression of the control along such singular arcs [19]. Previous literatures show that the optimal form of the fuel-optimal problem of the transfer trajectory is a bang–bang control and no singular arc exists, which is verified numerically [20,21]. Even without a singular arc, solving the bang–bang control is not tricky. The discontinuity of the control makes the solution be highly sensitive to the initial guess of the costates. Mantia uses a postulated preceding control strategy to avoid this difficulty [22]. Alternatively, in a compromise approach, Nah employs the energetic optimum as the index to obtain better convergence [23]. The homotopic approach from the energetic optimum to the fuel optimum performs well to solve the sensitivity of the initial guess and results in better numerical stability [20]. This paper discusses the interplanetary transfer trajectory design for an HPS that consists of a solar sail and SEP. The solution to the corresponding two-point boundary problem of the optimal problem is a bang–bang control. An indirect method that employs

a homotopic technique to avoid guessing the sensitive initials of costates is used to obtain the solution. The transfer time and fuel consumption are a conflicting performance index for this system. The lightness number of the solar sail and the maximum acceleration of SEP are two important parameters that influence the transfer time and fuel consumption. The relationships among these parameters and performance indexes are investigated for two rendezvous missions.

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2. Optimal control for the Hybrid Propulsion System

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Consider a spacecraft with an HPS composed of SEP and a solar sail. The total control acceleration exerted on the spacecraft is given by the sum of two terms as follows:

Fc = β

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( R · n)2 n + T max

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(1)

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I sp g 0

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(2)

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The fuel-optimal low-thrust rendezvous can be formulated as an optimal-control problem, and the associated optimal-thrust program can be derived by Pontryagin’s maximum principle (PMP) [21]. The problem addressed in this paper is to find the optimalcontrol law that minimizes the fuel consumption. The performance index J is defined as

J =κ

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A two-body model approximates the ephemeris model well in interplanetary trajectory design, and is widely used in previous literatures [17,21,23]. Similarly, all of the types of perturbation forces are not considered in this paper, and only solar gravity and SRP force exert on the solar sail. In such a model, the dynamic equation of motion in the ecliptic inertial frame can be given by

t

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udt

The Hamiltonian function of the fuel-optimal problem is

(3)

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H fo = κ

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The optimal values of the control variables are obtained in the domain of feasible controls by invoking PMP, that is, by minimizing H fo at any time. By imposing the necessary condition, one has

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[ n(t )

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Fig. 1. The optimal-control law description.

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t f

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u − ε u (1 − u ) dt

(10)

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The definition of this performance index will be explained later. The Hamiltonian function of this new optimal control problem is

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The optimal control can now be derived as

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⎧ if ρ > 0 ⎨u = 0 if ρ < 0 u=1 ⎩ u ∈ [ 0 1 ] if ρ = 0

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ρ =1−

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I sp g 0 λ V  mκ

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Similarly, the optimal values of the control variables are obtained by imposing the necessary condition, one has

(8)

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(9)

The energy-optimal problem is easier to solve because it leads to a continuous control law by invoking PMP. To relate the energyoptimal problem to the fuel-optimal problem, a new performance index is defined to formulate the optimal-control problem, which gives

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[ n(t )

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t

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ρ may be written as

This control law will lead to the singular control if the switching function takes the value zero on some intervals. However, several previous literatures discussed the fuel-optimal problems [20,21]. They usually solve the problem numerically using shooting methods and then the singular arcs can be observed in the profile of the switching function if they exist. The result in the literatures indicated that no singular arcs existed in the fuel-optimal trajectory optimization and the optimal solution was a bang–bang control. However, singular arcs indeed exist in some optimal-control problem. Bonnans et al. discussed the singular control in the Goddard Problem [19]. To conduct a shooting method, the structure of the control must be prescribed by assigning a fixed number of switching times that correspond to junctions between nonsingular and singular arcs. However, these switching times are unknown. Similarly, they used the homotopic method to solve the optimal problem assuming that no singular arcs exist. Then, the history of the switching function is used to provide an indication on the expected structure of the optimal trajectory for the original problem. In this paper, we also use the homotopic method to solve the fueloptimal problem and the results in numerical examples will show that the switching function ρ reaches zero, only at certain finite points. Therefore, the bang–bang control is regarded as the optimal control of the fuel-optimal problem. In fact, this discontinuous bang–bang control is one element that renders the fuel-optimal problem less smooth, and thus more difficult, than the energyoptimal problem. The performance index that corresponds to the energy-optimal problem is defined as

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T max

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where the switch function

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(6)

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For SEP, the thrust direction that minimizes the Hamiltonian function is along the negative direction of the primer vector.

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(12)

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The thrust direction of the SEP system that minimizes the Hamiltonian function is still given by Eq. (6). The optimal control is different from bang–bang control, which is given by

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⎧ u=0 ⎪ ⎪ ⎨ u=1 ⎪ ⎪u = 1 − ρ ⎩

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This optimal control is continuous as long as ε does not take the value zero. Therefore, the new optimal control problem is easier to solve. Most importantly, the solution that results from this new performance index changes from being a solution for the energyoptimal problem to being a solution for the fuel-optimal problem as the parameter ε varies continuously from 0 to 1. This will be explained in ‘homotopic technique’ of the next section. For the solar sail, minimizing the Hamilton function means adjusting the sail attitude to minimize the projection of the SRP force along λ V (t ). In fact, this problem is a locally optimal-type problem [17]. Thus, one knows that the normal vector lies in the plane spanned by the sunlight and λ V (t ). In the plane, the angle between the sunlight and the sail normal is given by

−3 ±

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(14)

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As shown in Fig. 1, the optimal-control law can be written in the vector form of

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⎧ R ⎪ ⎨ n = cos α + sin α e

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 R ⎪ ⎩ − λ V = cos  α + sin α e λV R

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(15)

The time derivatives of the costates are provided by the Euler– Lagrange equations

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⎧ ∂H μ 3μ μ ⎪ = 3 λ V − 5 ( R · λ V ) R − 2β 4 ( R · n)(n · λ V ) λ˙ R = − ⎪ ⎪ ⎪ ∂ R R R ⎪ R ⎪ ⎪ 2( R · n ) R ⎪ ⎪ ⎨ × n− 2 R

m2

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⎪ ∂H ⎪ ⎪ ˙ ⎪ ⎪ λ V = − ∂ V = −λ R ⎪ ⎪ ⎪ ⎪ u ⎩ λ˙ = − T λ v  m max

The state differential equation and costate differential equations are coupled via the control parameters. Thus, the state and costate equations and the various boundary conditions fully define the fuel-optimal rendezvous problem. For a rendezvous mission, the initial conditions are only partially imposed.

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R (t 0 ) = R 0 ,

V (t 0 ) = V 0 ,

m(t 0 ) = m0

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A part of the terminal conditions is

(17)

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R (t f ) = R f ,

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f

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The remaining boundary conditions are supplied by the transversality conditions:

λm (t f ) = 0

(19)

Thus, the optimal rendezvous problem results in a typical twopoint boundary value problem (TPBVP), which is equivalent to solving such nonlinear algebraic equations.

⎡

R (t f ) − R f

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(20)

The problem becomes finding the initial costates λ R (t 0 ), λ V (t 0 ), and λm (t 0 ) to satisfy the nonlinear algebraic equations. Solving the nonlinear algebraic equations efficiently is nontrivial. Typically, one must rely on a shooting method based on certain local algorithms such as Newton’s or Powell’s method.

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3. Numerical methods

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3.1. Normalization

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Because of the local characteristics of the algorithm, the sensibility to the initial guess in solving the above equations by the single shooting method requires a thorough understanding of the underlying physics of the optimal-rendezvous problem. For example, to solve the shooting equation, the initial costates are typically allowed to vary in a broad domain. A reasonable admissible domain is important for obtaining reasonable initial values for the shooting variables by searching algorithms. To decrease the sensibility to interaction in the initial guess, one should first set a reasonable admissible domain for the shooting variables and then employ the global searching algorithm. The admissible domain can be normalized to a bounded unit sphere, which is a significantly smaller and thus a more reasonable searching domain. Jiang [23] noted the homogeneousness of the state equations (3), the costate equations (12) and the boundary conditions with respect to the hybrid multipliers, which include positive factor κ and the initial costates λ R (t 0 ), λ V (t 0 ), and λm (t 0 ). If the performance index is multiplied by a positive factor κ , the ratio of the new costates to the old ones will automatically be κ . However, the optimal control and the optimal trajectory will remain the same. This insight will simplify the initialization of the shooting variables λ R (t 0 ), λ V (t 0 ), and λm (t 0 ) through the parameterization of the high-dimensional abstract unit sphere. That is, after normalization, the multipliers should satisfy

    2 λ R (t 0 )2 + λ V (t 0 )2 + λm (t 0 ) + κ 2 = 1

(21)

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The parameters that define the above unit sphere will be the shooting variables, all of which fall in the bounded domain. With the reasonable admissible domain for shooting variables, Particle Swarm Optimization (PSO) is then adopted to globally search for the preliminary initial interaction value for solving the energyoptimal problem, whose object function is a performance index plus the residual of the shooting equations.

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3.2. Homotopic technique

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The previously mentioned sensibility to the initial guess in solving the above equations by the single shooting method is one of the various difficulties encountered in fuel-optimal trajectory optimization. This difficulty is intrinsic in solving high-dimensional nonlinear equations, a typical topic in applied mathematics. However, a more daunting difficulty results from the different physics that underlies the various performance indexes. In the previously mentioned fuel-optimal performance index in Eq. (3), the integrand u is proportional to the thrust vector’s norm, the smoothness of which is not satisfying. Moreover, because of the linearity of the Hamilton function H fo with respect to the normalized magnitude of thrust vector u, the bang–bang control (7) is discontinuous. These factors will significantly decrease the smoothness of the shooting functions and thus the convergence radius of the associated gradient algorithm. The difficulty can be removed to a certain degree if the so-called homotopic technique is applied. In fact, the linearity of the integrand in the performance index and of the state equations with respect to the normalized thrust magnitude u causes discontinuous bang–bang control. This observation requires a smoother shooting function using a new performance index ˜J , in which the integrand is not linear with respect to u but quadratic form u 2 . It is straightforward to derive the corresponding optimal control associated with the energy-optimal performance index by PMP. Quadratic form u 2 in ˜J leads to quadratic Hamilton function with respect to u, and one can finally obtain a continuous optimal control, which makes shooting function smoother. Initially, it appears that the new continuous optimal control is unrelated to the former discontinuous control. The homotopic technique suggests that one may find continuous mapping, known as homotopic mapping, which gradually transitions from the smoother shooting function to the less smooth function by varying the so-called homotopic parameter continuously, in the present case from the energy-optimal problem to the fuel-optimal problem. A new performance index J n in (10) is defined. The solution that results from this new performance index changes from the energy-optimal problem to fuel-optimal problem as the homotopic parameter ε varies continuously from 0 to 1. The details are omitted. However, the interested reader can consult reference [21] and the related references therein. With the homotopic technique, one can obtain the fuel-optimal solution by starting with the energy-optimal problem. The difficulty in obtaining the energy-optimal solution is managed with techniques introduced in the section on normalization.

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3.3. Algorithm validation

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A typical rendezvous mission with Venus is employed to validate the algorithm. The validation is conducted step by step because the scant literature makes comparison impossible. First, the program is run with the lightness number set to zero (β = 0), which causes the problem to degenerate to a pure low-thrust fueloptimal problem. The program for this problem was developed by our research team during the GTOC [24]. By setting the same parameters for both cases, the result of this hybrid system is identical

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Fig. 2. Hamiltonian functions for different transfer trajectories of the Venus rendezvous mission.

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spacecraft mass of 1000 kg is assumed. To better understand the influence of the solar sail system on the mass consumption, HPSs of different lightness numbers are used to accomplish the transfer missions. Firstly, the result of the energy-optimal problem is compared to that of the fuel-optimal problem. A pure SEP system is employed to achieve the Apophis rendezvous mission in 700 days. Fig. 3(a) gives evolution of the norm of the thrust with respect to the homotopic parameter ε , as it illustrates the evolution from a continuous control (energy-optimal problem) to a discontinuous control (fuel-optimal problem). The final masses of the energyoptimal and fuel-optimal results are 849 and 870 kg, respectively. It means that the fuel-optimal trajectory saves 21 kg fuel. The saving is a considerable for a spacecraft of 1000 kg. Therefore, the fuel-optimal problem is solved in the following simulations.

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4. Numerical simulations The spacecraft must escape the Earth before the spacecraft can journey to an interplanetary trajectory. Either an impulse maneuver is used before the sail deployment or SRP is used to escape the Earth. The interplanetary transfer does not include the geocentric orbit, and the sail is assumed to depart from the heliocentric orbit of the Earth. The departure position and the velocity of the sail are identical to those of the Earth: C ∞ = 0. The fuel-optimal algorithm is applied to study two rendezvous missions, Venus and Apophis, an inner solar planet and a near-Earth asteroid, whose classical orbit elements are provided in Table 1. The maximum thrust of SEP is 0.2 N, and the corresponding specific impulse is 3800 s. An initial

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4.1. Venus rendezvous mission

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A Venus rendezvous mission is considered because such a mission is typical for solar sailing. A solar sail has advantages for missions within the inner planets because the SRP intensity is inversely proportional to the square of the distance from the Sun. First, the transfer time of a pure solar sail and the fuel consumption of pure SEP are provided as references for the HPS. Then, the fuel-optimal transfer trajectories of the HPS are studied for different departure and transfer times. Table 2 provides the transfer time and fuel consumption of pure propulsion systems. The departure time is fixed as February 8, 2015. For pure SEP, a transfer time of 350 days requires 154.6 kilograms of propellant. For a low-performance solar sail, a much longer transfer time is necessary. For example, a transfer time of 1165 days is required for a solar sail of β = 0.02. The transfer time is almost linear with the lightness number and is shortened to only 330 days for β = 0.1, which is close to the time required for SEP. For an approximate estimate, the maximum acceleration of SRP for β = 0.1 is approximately 6 × 10−4 m/s2 and 1.2 × 10−3 m/s2 at the solar distance of the Earth and Venus, respectively. The acceleration caused by the low thrust of 0.2 N is 2 × 10−4 m/s2 and 2.3 × 10−4 m/s2 at the beginning and the end of the mission, respectively. It appears that the solar sail performs better than SEP. However, one must consider that SRP acceleration is significantly decreased for a large sun angle and the acceleration direction cannot be sunward. Fig. 4 provides the fuel consump-

Earth Venus Apophis

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Table 1 Classical orbit elements of planets.

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to that of the pure low-thrust problem. Second, the program is run with the thrust set to zero, (T max = 0), which causes the system to degenerate to a pure solar sail system. In this case, the transfer time is set freely and the transversality condition H (t f ) = 0 is forced. The resulting solution is identical to that of a previous study [25], where a time-optimal interplanetary transfer trajectory for a solar sail is discussed. Finally, a general case is run and the Hamiltonian function is constant over the transfer time. The simulation parameters are the same to those in the section of ‘numerical simulation’. As shown in Fig. 2, the Hamiltonian functions of the transfer trajectories of the Venus rendezvous mission are constant over time as expected. The stepwise validation guarantees that the resulting solution satisfies the first-order necessary condition of PMP.

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Fig. 3. Optimal solutions for different ε .

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Fig. 4. Fuel consumption for different transfer times.

Fig. 5. Fuel consumption for different departure times.

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Table 2 Fuel consumption and transfer time for single propulsion.

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tion of the HPSs for different transfer times and different lightness numbers. The departure time for all of the simulations is identical to that provided in Table 2. The fuel consumption decreases with the transfer time and becomes zero if the transfer time is sufficiently long for the solar sail to accomplish the transfer mission alone. For the case of β = 0.02, the final mass increases approximately 0.07 kilograms for a one-day increment of the transfer time if the transfer time is less than 900 days, and the increment becomes approximately 0.24 kilograms for a transfer time longer than 900 days. Similar results can be observed for the cases of β = 0.03 and β = 0.04. With the SEP engine in operation during the entire transfer journey, the minimum transfer time is calculated to be 300 days for β = 0.02, and the corresponding fuel consumption is approximately 106 kilograms. Compared with pure SEP, the transfer time is shorter and the fuel consumption is less. For a fixed transfer time, the fuel consumption decreases as the capability of the solar sail increases. As shown in Fig. 4, the transfer time is 300 days, and approximately 20 kilograms fuel can be saved for each 0.01 increment of the lightness number. Therefore, a small solar sail may be used as an auxiliary system for SEP. We know that the launch window is important for the mission design. In the present paper, the departure time is associated with the launch window. Fig. 5 provides the fuel consumption for different departure times. The departure time in Table 2 corresponds to zero in the figure, and the transfer time is fixed at 400 days. The fuel consumption increases as the departure time is delayed. The relative position of the Earth with respect to Venus recurs approximately every 19 months. If the optimal departure time is missed, one must wait approximately 19 months for another opportunity. Fig. 6 shows the SEP operation time and switching function for different lightness numbers. The SEP engine switches its state as the switching function changes sign. Fig. 7 provides the pitch angle of the solar sail during the transfer journey. The transfer time is 350 days, and the departure time is February 8, 2015 for all of the cases. The SEP engine starts and shuts off three times for each case. The engine operation time decreases as the lightness number of the solar sail increases. For pure SEP, the thrust should be switched on at the end of the transfer time to achieve the rendezvous. For

the HPS, SRP is used to approach the rendezvous during the final stage because the SRP intensity is the largest during the final stage. The pitch angles in Fig. 7 suggest that the angles must be adjusted to decelerate during the final stage to reach the rendezvous, whereas during the cruise stage, the angle is maintained close to 35 degrees to obtain large energy growth. The time histories of the pitch angles for different lightness numbers are similar. 4.2. Apophis rendezvous mission Asteroid Apophis, also known as 2004 MN4, is an NEA with a diameter of 320 m and mass of approximately 4.6 × 1010 kg. Apophis was predicted to pass approximately 36 350 km above the Earth on April 13, 2029. Recent observations using Doppler radar at the Arecibo radio telescope in Puerto Rico confirmed that Apophis will pass at approximately 32 000 km above the Earth in 2029 with a chance of a resonant return in 2036. Wie studied using a small solar sail to tow Apophis from a 600 m keyhole area in 2029 to eliminate the possibility of the asteroid’s resonant return in 2036 [26]. Gong investigated using a solar sail formation-flying gravitational tractor to tow Apophis [27]. The solar sail gravitational tractor uses SRP instead of low-thrust SEP to counter the gravity of the asteroid, and a rendezvous of the solar sail with the asteroid is necessary before the gravitational tractor starts to operate. In the present paper, we study the Apophis rendezvous mission, which may be regarded as the transfer segment before the solar sail gravitational tractor operates. Similarly, the transfer time and the fuel consumption of the HPS are discussed. Additionally, the transfer trajectory for pure SEP is presented for discussion. Assume that the spacecraft must be launched before 2018 and the departure time from the Earth is July 28, 2017. The transfer time using the pure solar sail propulsion and the fuel consumption using pure SEP are presented in Table 3. The minimum transfer time for pure SEP is 570 days, and the spacecraft is under thrust during the entire journey. The corresponding fuel consumption is approximately 264.3 kilograms. For a low-performance solar sail, a significantly longer transfer time is required than for an inner-planet mission. For example, a transfer time of 1260 days is required for β = 0.02. The transfer time decreases sharply as the lightness number increases and is shortened to 468 days for β = 0.1, which is significantly shorter than the time required for SEP. Obviously, a high-performance solar sail may complete the mission in a short time and consume no fuel. However, designing a high-performance solar sail is a difficult engineering task, particularly because no pure solar sail propulsion has been tested in a real mission. Therefore, an HPS using a low-performance solar sail and SEP presents an acceptable option for the Apophis rendezvous mission. Reliable SEP guarantees that the spacecraft will travel to the target position, where the solar sail will operate to achieve the mission’s goal.

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Fig. 6. Duration of thrust segments of different HPSs.

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Table 3 Fuel consumption and transfer time for single propulsion.

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Fig. 7. Pitch angles of the solar sail for different HPSs.

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Fig. 8 provides the fuel consumption of the HPS for different transfer times and lightness numbers. The departure time for the simulation is provided in Table 2: July 28, 2017. The fuel consumption decreases as the transfer time increases, and the SEP engine will not operate for the entire transfer time if the transfer time is sufficiently long for the solar sail to accomplish the transfer mission alone. The final mass increases quickly with the transfer time when the transfer time is short. For the case of β = 0.02, the final mass increases approximately 0.46 kilograms for each one-day increment of transfer time if the transfer time is less than 750 days, and the increment decreases to approximately 0.125 kilograms for a transfer time longer than 750 days. This outcome means that SEP can be used to significantly decrease the transfer time while consuming a small quantity of propellant. For example, only 80 kilograms of additional extra fuel will decrease the transfer

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Fig. 9. Fuel consumption for different departure times.

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time from 1260 to 670 days. The transfer time may be decreased to approximately 500 days for an HPS of β = 0.02 if the SEP thrust is continuously operated. Similar results can be obtained for large lightness numbers of the HPSs. Compared with pure SEP, the HPS can decrease the transfer time or the fuel consumption, depending on the mission requirements. The HPSs ability to maneuver quickly and save fuel increases as the capability of the solar sail increases. As shown in Fig. 8, the transfer time is fixed at 570 days, and approximately 30 kilograms fuel can be saved for every increment of lightness number of 0.01. This result does not consider the mass increment of the HPS caused by the solar sail.

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Fig. 10. Thrust segment duration of different HPSs for the Apophis mission.

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The influence of the departure time on fuel consumption is shown in Fig. 9. The departure time in Table 3 corresponds to zero in the figure, and the transfer time is fixed at 700 days. The difference in fuel consumption may reach approximately 60 kilograms for different departure times. However, the fuel consumption is insensitive to the departure time compared with the case of the Venus rendezvous. The departure time for the simulations is not a favorable, and the fuel may be saved for a delayed departure time. The relative position of the Earth with respect to the Apophis recurs approximately every 7.7 years because the periods of the Earth and Apophis are similar, which means that the launch must wait more than 7 years for comparable opportunity. However, similar conditions can be found near the original departure time because the relative position changes slowly with time. Fig. 10 shows the duration of the thrust segments of SEP and the switching function for different HPSs. The SEP engine switches its state as the switching function changes sign. Fig. 11 provides the attitude histories of the solar sail. The transfer time is 570 days, and the departure time is July 28, 2017. The thrust operates continuously for pure SEP because this transfer time is the minimum time required to complete this mission. There are three thrust segments for the HPSs, the durations of which shrink as the transfer time increases and reach zero for a sufficiently long transfer time (Fig. 12). The histories of the pitch angle in Figs. 11 and 12 show that the pitch angle remains close to 35 degrees when the SEP engine is in operation. This result means that the solar sail will accelerate to achieve the fastest growth in energy when the SEP engine is working. However, the direction of the acceleration must be considered if the solar sail is to accomplish the rendezvous mission without the help of SEP.

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4.3. Discussion on simulations

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Typically, the velocity increment required to accomplish a specific transfer mission is provided by two sources in the HPS: one source is the SRP, and the other source is SEP. A large velocity increment from the SRP typically results in a small velocity increment from the SEP. The distribution between the two sources depends on several parameters, including the lightness number, the transfer time, the departure time, and the history of the sail attitude. In the present paper, the solution to the fuel-optimal problem has guaranteed the optimality of the attitude history.

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Fig. 11. Pitch angles of the solar sail for different HPSs, Apophis mission.

The lightness number is a parameter that represents the acceleration capability of the solar sail. A spacecraft of a larger lightness number can accumulate a larger velocity increment under the same condition. Therefore, an HPS with a larger lightness number typically consumes less fuel. The transfer time is another import parameter that influences the fuel consumption. Increasing the transfer time decreases the fuel consumption. Therefore, the trajectory design of the HPS seeks a compromise between the transfer time and the fuel consumption. The contradiction between decreasing transfer time and saving fuel also affects traditional trajectory design. However, the contradiction is different for the HPS. The energy difference between the initial and target orbits should be compensated to achieve the transfer mission. There is no extra energy input for a typical chemical propulsion system, and the minimum quantity of fuel is consumed. Typically, transfer time can be sacrificed to reach or approach minimum fuel consumption. For the HPS, the velocity increment from SRP is accumulated over time. It can be expected that no fuel is required as long as the transfer time is sufficiently long for the solar sail to accumulate the total velocity increment for the transfer mission. Therefore, the fuel consumption can be zero for an HPS. The departure time determines the phase difference between the initial and target positions for the rendezvous mission. An

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improper phase relationship may result in particularly high fuel consumption. Therefore, it can be expected that the departure time influences the fuel consumption significantly. If the initial and target orbits have a short resonant period, the transfer trajectory will be sensitive to the departure time (as in the Venus rendezvous mission), and the fuel consumption increases sharply as the departure time varies approximately 100 days. The transfer trajectory is insensitive to the departure time if the two orbits have a large resonant period. For example, in the Apophis rendezvous mission, the influence is not obvious for a departure time variation of more than one year.

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5. Conclusions

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A fuel-optimal problem for interplanetary trajectory design is investigated for a hybrid propulsion system that consists of a solar sail and solar electric propulsion. The optimal-control problem is formulated using the calculus of variations and Pontryagin’s maximum principle. The solution to the problem is a bang–bang control that is solved using an indirect method. There are three factors that influence the fuel consumed during the transfer: the lightness number of the hybrid propulsion system, the transfer time and the departure time from the Earth. The fuel consumption decreases with the lightness number and the transfer time. The lightness number is determined by the solar sail technology and cannot be changed by the trajectory design. The transfer time is an important parameter for the trajectory design of the hybrid propulsion system and can be prolonged to decrease the fuel consumption until no fuel is consumed. The compromise between the transfer time and the fuel consumption is the main task for the trajectory design of the HPS. The departure time from the Earth is another parameter that influences the fuel consumption. The optimal departure can be determined for a given range of departure times.

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Conflict of interest statement

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None declared.

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Acknowledgements

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This work is supported by the National Natural Science Foundation of China (Grants No. 11272004).

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Appendix A. Supplementary material

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Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.ast.2015.04.020.

References

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