Minimum Time Orbit Raising of Geostationary Spacecraft by Optimizing Feedback Gain of Steering Law

Trans. JSASS Aerospace Tech. Japan Vol. 17, No. 3, pp. 363-370, 2019

DOI: 10.2322/tastj.17.363

By Kenji KITAMURA,1) Katsuhiko YAMADA,2) and Takeya SHIMA1)

1) Advanced Technology R&D Center, Mitsubishi Electric Corporation, Amagasaki, Japan 2) Graduate School of Engineering, Osaka University, Osaka, Japan

(Received June 23rd, 2017)

This study considers the minimum time orbit-raising problem of geostationary spacecraft with low-propulsion thrusters. This problem is equivalent to determining an appropriate thrust direction during orbit raising. This study proposes a closed- loop thruster steering law that determines the thrust direction based on the optimal feedback gains and control errors of each orbital element. The feedback gains of the steering law are assumed to be the functions of orbital elements and are optimized by a meta-heuristic method. The orbital semi-major axis, eccentricity, and inclination are considered as independent variables for expressing the gains. Numerical simulations show that whichever orbital elements is selected as an independent variable, the same performances are obtained. Therefore, regardless of the initial orbital elements, by selecting the independent variable appropriately, the proposed method can always solve the minimum time orbit-transfer problem.

Key Words: Orbit Raising, Geostationary Orbit, Low-Propulsion Thruster, Optimal Control

Nomenclature Subscripts 0 : initial : semi-major axis : final : eccentric anomaly : normal direction in Local Vertical Local � : eccentricity � Horizontal (LVLH) coordinates � : eccentricity vector � : radial direction in LVLH coordinates � : magnitude of thruster force : transversal direction in LVLH coordinates �� g : gravitational acceleration of Earth at sea � � � level 1. Introduction : orbit angular momentum : specific impulse of thruster In recent years, low-propulsion thrusters, such as ion thruster � �� : inclination � and hall thrusters, have become hopeful propulsion systems for �� : inclination vector spacecraft transferring from a low Earth orbit (LEO), a � : true longitude geostationary transfer orbit (GTO), or a supersynchronous orbit �� 1-3) : spacecraft mass (SSTO) to a geostationary orbit (GEO). Conventionally, a � ��Ω�� geostationary spacecraft is placed in a GTO by a launcher. : mean angular motion � However, to complete the orbit transfer, the spacecraft employs : semi-latus rectum � � �� an on-board high-chemical propulsion thruster, called the : equatorial radius of Earth � � �� apogee kick motor, at the apogee of the GTO so that the semi- : distance from center of Earth to spacecraft �� major axis, eccentricity, and inclination of the orbit correspond : orbital period 4) � with those of the GEO. Although GEO insertion by a chemical : time � �� propulsion thruster is completed in a short period (generally x : vector, consisting of slow less than one week, including initial check-out operation time), � equinoctial elements and mass the low specific impulse of the thruster results in significant �� fuel consumption (approximately half of the wet mass is : in-plane thruster steering angle� �� occupied by the fuel). To overcome this disadvantage, a spiral : ��out-of-plane� thruster steering �angle� � orbit raising using a low-propulsion electric thruster, which has : gravitational constant of Earth � a specific impulse almost ten times larger than that of a : true anomaly � chemical propulsion thruster, is attracting attention as an : right ascension of ascending node alternative orbit transfer method.5) In the case of spiral orbit � : argument of perigee raising, several months are required to complete the orbit Ω transfer, yet it can save a considerable amount of fuel. � In the past, many studies aiming to optimize orbit transfers

Copyright© 2019 by the Japan Society for Aeronautical and Space Sciences and ISTS. All rights reserved. J-STAGE Advance published date: January 31st, 2019 363 Trans. JSASS Aerospace Tech. Japan Vol. 17, No. 3 (2019)

with low propulsion have been conducted. The methods can be swarm optimization (PSO). As independent variables for categorized into three types: analytical methods, numerical expressing the gains, the orbital semi-major axis, eccentricity, methods, and a combination of the two. or inclination are chosen. Numerical simulations show that In the analytical approach, the optimal control theory and the whichever orbital elements are selected as the independent averaging method are often used. Edelbaum6) and Kechichain7) variable, almost identical performances are obtained. solved the minimum-time transfer problem between inclined circular orbits. Bonnard et al.8) derived the explicit form of the 2. Dynamics of Osculating and Mean Orbital Elements Hamiltonian for the minimum energy transfer problem between coplanar orbits. Although the analytical solutions are very In this study, the equinoctial orbital elements defined by Eqs. powerful and useful for trajectory design, the application of the (1) – (5) are utilized for describing the orbital motion:21) analytical approach is limited to specific trajectories, and it is (1) difficult to obtain the minimum-time or minimum-propellant � solutions for the problem with general boundary conditions. � � � cos�Ω � ��, (2) In the numerical approach, direct or indirect optimization �� sin�Ω � �� , (3) methods are widely used. Sackett et al.9) utilized the averaging � (4) method and solved the problem using classical indirect � � ��n��2� cos Ω , optimization. Kluever et al.10) solved the optimal control �� n��2� sin Ω , (5) problem using a direct optimization approach, where the where ( ( and , are the eccentricity vector, � � Ω � � � �, costate variables or the weighting variables were the inclination� vector,� �and� true longitude, respectively. optimization variables of the nonlinear programming problem. 2.1. Dynamics� ,� �, of� ,�osculating�, � equinoctial elements Graham et al.13) proposed a minimum-time trajectory The equinoctial elements are singularity-free. The equations optimization method as a multiple-phase optimal control of motion are written as follows: problem and solved it by utilizing a pseudo-spectral method. In general, numerical optimization explores the search space by (6) using the gradient information of the objective function and �� 2 � � � �� sin � �� , there is a possibility of converging to the local optima. �� √�� Moreover, the computational cost and difficulty increases as �� (7) � � sin � �� cos � � �� the number of revolutions becomes larger. �� The combination of analytical and numerical methods is � ��, � another effective approach that can obtain a sub-optimal � (8) 14- �� solution with an affordable computational cost. Petropoulos � �� cos � �� sin � � �� 16) proposed a Lyapunov feedback control method, called Q- �� law, where the instant thrust direction is determined so that the � � � , (9) � � time derivative of the Lyapunov function (Q function) was � �� minimized. In their work, the multi-objective evolutionary � cos � ��, �� 2� (10) algorithm was utilized to optimize the control parameters and � � find the Pareto-front of the transfer time versus propellant mass. �� 17) � sin � � , Varga et al. modified the Q-law with equinoctial orbital �� 2� � (11) elements to avoid the singularities. As another combining �� 18) where � �� are� defined� , as follows: method, Ruggiero et al. proposed the closed loop steering law, �� (12) which determines the thrust direction by blending the optimal �, �, �, �n� � directions for maximizing the rate of change of each orbital (13) ��, element with the self-adaptive weights. In their work, the (14) �� cos � � �� sin � , weights were obtained based on the difference between current (15) �� sin � � �� cos � , orbital elements and targeted ones. In Eqs. (6) – � (11), � � represent the perturbing In this study, the third approach (combination of analytical � �� . acceleration of spacecraft� � expressed� in LVLH coordinate and numerical method) is utilized to solve the minimum time system, where the �-axis,� points, �n� � to the spacecraft from the orbit-transfer problem with low-propulsion thrusters. In center of the Earth, the -axis is perpendicular to the orbital general, the greatest demerit of the low-thrust orbit transfer is plane, and the -axis� is chosen to complete the right-hand its extra-long transfer time. To suppress this disadvantage, the system. The spacecraft mass� flow rate, owing to the thruster current study focuses on the minimum time problem. This operation, is expressed� as follows: problem is equivalent to determining the appropriate thrust direction during orbit raising. To obtain the optimal direction, (16) the closed-loop thruster steering law given in Ref. 18) is �� 2.2. Dynamics of mean�� equinoctial. elements improved, and the feedback gains for the steering law are �� Generally, orbit �� transfer �� by a low-propulsion thruster introduced. The feedback gains are expressed as functions of a requires several months. It requires a significant amount of time monotonically increasing or decreasing orbital element and are to numerically integrate the osculating equinoctial elements. optimized using a meta-heuristic method, such as the particle

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This can lead to a long convergence time in the numerical following equations are obtained: optimization process, where iterative integration is required. To reduce the integration time, we utilized the following averaged (21) dynamics of equinoctial elements, as performed by Kluever10, �� 2 cos � � � � �� sin � sin � � cos �� , 20) and Utashima11): �� √1�� (22) �� √1�� cos � � �sin � sin � �� (17) �� 1 �� 1 �� cos � � � � �� 1��cos��, � �cos � � �cos�� , where�� is � a� vector�� composed2� �� of the slow variables of the 1��cos� (23) equinoctial elements and the spacecraft mass. In Eq. (17), the �� By taking the� first� derivative� cos�� of Eqs. � �� (21) sin – � (23) � . with respect to relation� is used. The eccentric �� √1�� and and setting them equal to zero, the optimal pitch and anomalies, and , represent the beginning and ending of yaw angles required to maximize the rate of change of these the thruster�� arc in � a ��1 one-orbit � � cos revolution, �� respectively. In the � � three� orbital� elements can be obtained, as given in Table 1. Earth eclipse� (shadow),� it is assumed that power cannot be generated and that the thruster is turned off. Therefore, and

correspond to the exit and entry points of the Earth eclipse� in a one-orbit revolution, respectively. If there is no coastin� g � Earth arc,� and can be set to 0 and , respectively. The

model � of the Earth� eclipse is briefly described in the next section.� � 2� � 2.3. Model of Earth eclipse The Earth shadow in relation to the satellite orbiting the Earth is composed of two regions. One is a penumbra and the other is an umbra. In the penumbra, sunlight is partially Fig. 2. Geometry of LVLH coordinates and thruster direction. obscured by the Earth, while in the umbra, light is completely

blocked. In this study, and are determined based on Table 1. Optimal steering angle for maximum rate of change of semi- the exit and entry point of the penumbra (using a method � � major axis, eccentricity, and inclination. similar to that in Ref. 12).� � Pitch angle Yaw angle

semi-major

Exit axis Penumbra �� sin � tan � � 0 eccentricity 1 � � cos � Sun Earth Umbra �� sin � tan � � 0 inclination cosany � � cos � � sign�cos�� Entry 2 Let us define the optimal pitch and yaw angle for a changing Fig. 1. Geometry of Earth shadow. semi-major axis, eccentricity, and inclination as ∗ ∗ . The associated optimal thrust direction, � � 3. Steering Law , is expressed as follows: � ,�∗ �� , �, �� (24) In this section, we describe the proposed thruster steering �, �, �� ∗ 18) ∗ ∗ ∗ ∗ ∗ � Ruggiero proposed� calculating� � the thrusting� � vector as a law for a low-thrust orbit transfer. This is an improvement from �� sin � cos � cos � cos � sin � � . weighted linear sum of where the weights are the method proposed by Ruggiero.18) We briefly review the determined by the difference∗ between∗ the∗ osculating value of a conventional method and then describe the improved steering � � � specific orbital element �and,� �a target, an� � one� , of the same element. law developed in this study. This calculation is given as follows: 3.1. Conventional steering law In this study, we assume that the thruster force, , remains (25) constant and that the thruster direction is controlled. By � | | � � � �� , (26) introducing a pitch angle, , and a yaw angle, , (Fig. 2), the � � � � �� ∗ � �� ∗ � �� ∗ thruster force vector can be expressed in the LVLH coordinate where�� subscripts� �� represent� �� the initial� � � , and final �� system as follows: � � (target) values of each orbital element, respectively. 0 an� � (18) 3.2. Proposed steering law (19) The conventional method described in the previous section � � � �� sin � cos � , introduces self-adaptive weights and blends each optimal � � (20) � �� cos � cos � , direction, , in accordance with the weights. By using this where is the thruster� � acceleration, defined by . � �� sin �, steering law,∗ it can be expected that all the orbital elements will Substituting Eqs. (18) – (20) into the Gauss planetary equations � � � � of the � semi-major axis, eccentricity, and inclination,� �� the converge to the desired values. However, these self-adaptive weights are not necessarily optimal. We propose to improve the

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steering law as follows: Therefore, the objective function of the minimum-time transfer (27) to the GEO is defined by the following min-max problem: ∗ ∗ ∗ (31) where the improved� weights,� � , � ,� and � �, are given as � � , �� , 4.1.2. Optimization variable follows: |�| min. � � m��, ��, ��. �� Our goal is to obtain the optimal feedback gains , (28) , and , that minimize the objective function . To � �� achieve this, the feedback gains are discretized within the � �� , � � �� (29) �interval�� of � �� , and the nodal values are optimized. � � � � � �� By dividing the interval into segments, � �� , � � � � (30) �� ,� � �� the -th node, , is given as follows� � : �� ,� � �� In the above � equations,�� ∙ cos��and � . are the � � �� ,�,�� (32)� feedback gains of each self-adaptive�� weight, and they are � � � The arbitrary � � between� the� nodes,� and , is functions of the increasing or� decreasing��, � ��, orbital� element�� . In � �� . calculated by a linear interpolation�� of and the case of a coplanar GTO to GEO transfer, the semi-major �� as follows : � � � �� axis or eccentricity can be chosen to be . However, if the � �� initial semi-major axis is equal to that of the GEO (the case of �� ,�,�� (33) � SSTO to GEO), the eccentricity or inclination can be chosen. �� . As the relative values of the weights are important, one of the �� gains can be set to one ( in the GTO to GEO case,

for example). In the following� section, the method to obtain the optimal gains, � ��and�� is described in detail.

In addition, we � explain� the reason� why the monotonically increasing or decreasing� ��, � �� orbital�, element,� ��, , is used as an independent variable instead of time. � 4. Optimization of Feedback Gain for Minimum-time 0 Orbit Transfer Fig. 4. Discretization of feedback gain through . In this section, we describe the method used to calculate the � optimal feedback gains for the minimum-time orbit transfer. In Because and are given from the initial and final

Section 4.1., the minimum time orbit transfer problem (the condition, the� objective� is to obtain the optimal nodal values objective function and search variables) is defined, and in the � � that minimize the

following section, the process of solving the optimization objective� � function. If all the nodal values are given, the thruster problem is described. �vector�� can� ��,�,�� ,�,��be calculated from Eqs. (27)�, – (30) and it is then possible to propagate the orbital elements forward. The 4.1. Problem of minimum-time transfer to GEO propagation ends when the semi-major axis, eccentricity, and 4.1.1. Objective function inclination reach their target values. Therefore, the times, To complete the orbit transfer to the GEO, the semi-major and , as well as the objective function , are easily axis, eccentricity, and inclination values must correspond to obtained by simply forward-propagating the equation of ��,��, �� those of the GEO. Since the steering law, given by Eqs. (27) – motion. Note that if time is chosen to be an independent (30), does not assure that the three orbital elements will variable instead of , it is impossible to propagate the orbital simultaneously achieve the target values, the times when each elements because the final time is an unknown variable, � orbital element reaches its target value should be explicitly and the feedback gain cannot be interpolated within the time �� distinguishable as and (Fig. 3). domain. 4.2. Solving optimal feedback gain by particle swarm ��,��, ��, respectively Orbital Elements optimization The objective function given in Eq. (31) is a min-max function, and it is difficult to define its gradient. This means that gradient-based solution methods for a nonlinear programming problem such as the interior-point method or the sequential quadratic programming method are not suitable for this min-max problem. To solve this problem and achieve optimal nodal values, we use a meta-heuristic optimization time method; the most famous methods are the genetic algorithm (GA) introduced by Holland22) and the PSO method used by Fig. 3. Conceptual diagram for the time difference of each orbital Kennedy and Eberhart.19) These methods do not require any element achieving the target value. gradient information and are suitable for parallel computing. In

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this study, considering the simplicity of implementation, the Table 2. Independent variable for each case PSO is selected as an optimizer for the optimization problem. case1 case2 case3 � In the following section, the PSO algorithm used in this study semi-major axis eccentricity inclination is briefly described. Let us assume that we have a total of particles in the Table 3. Comparison of transfer time (in days) search space and that the position of the -th particle at the - case1 case2 case3 � conventional th generation is defined as , where denotes a ( ) ( ) ( ) � � � 68.7 69.6 69.8 73.0 vector composed of all the optimization variables (nodal values �� 1 of feedback gains). The personal best of the -th particle at �� 70.2 70.3 70.2 77.5 the -th generation, , is defined as follows: �� 69.9 70.3 70.2 83.6 � � �� (34) � 70.2 70.3 70.2 83.6 � �∗ ∗ � � � � � where � �� represents, � the� �� objective�� function�, at . � Furthermore,� the optimal particle at the -th generation� is The number of nodes is set at three for each case. With respect � � defined �as� �the �global best, , as follows: � to the PSO parameters, the following values are chosen: a � particle population of an initial inertia of � (35) ∗ � a final inertia of a final generation of a � ∗ � � � ��, �� .�, By using the�� personal�� ,� best and �the �� global �best,�� the�. state of each particle best attraction of , and a global best attraction �� .�, � � 1��, particle at the -th generation is updated as follows: of . In these numerical� examples, no perturbations except thruster force are considered.� � �.� In the evaluation of the (36) � �� 1� � � � score� for�1.�� each particle, the averaged equation of motion, Eq. �� (17), is utilized for accelerating the propagation speed of the � � � � (37) �� , orbital elements. � � � where is� a�� positive�� parameter��, that represents the inertia of A summary of the transfer time of the three cases and the the particles at the -th generation. The parameters and conventional method is presented in Table 3. In the three cases, �� determine how strongly each particle is attracted to its� personal� and have almost the same 70.3 day transfer time. best and the global best� of the whole particles, respectively.� The� However,�� for ��the result obtained using the conventional method, parameters and are ✕ diagonal matrices whose � ,� , and � differ from each other by several days, with a components are randomly sampled from an interval at transfer time of 83.6 days, which is approximately 13 days �� every generation. In this study,� it is � assumed that the attracting longer� ,� , than that� of the proposed method. factors and are constant for every generation, ��,1�and that Figures 5 through 7 compare the time histories of the

the inertia� parameter� monotonically decreases. By osculating semi-major axis, eccentricity, and inclination defining� the number� of the final generation as , is given obtained by the conventional and proposed steering laws. Note �� as follows : � that these results are obtained by numerically propagating the � � osculating orbital elements by the optimized feedback gains. ��,�,�� (38) These figures show that no matter which orbital element � where and�� represent� �� the� � inertia, parameter at the among and is chosen as the independent variable, � almost the same history of orbital elements is obtained. initial and� final � generation, respectively. By gradually decreasing� the � inertia parameter, each particle can Figure�, 11�, shows the� 3D view of the transfer orbit of case1 in independently move in the search space at the initial generation the inertial coordinate system. Figure 12 compares the transfer and easily converge to the global best at the final generation. orbit of case1 with that of the conventional method. It is clear that the maximum apogee altitude of case1 is much higher than 5. Numerical Examples that of the conventional method. This means that the proposed method prioritizes an increase in the semi-major axis over a In this section, numerical examples of the proposed decrease in the eccentricity at the initial phase of the transfer, optimization method are presented. By assuming GTO which is also indicated in Figs. 5 and 6. Almost the same parameters, the initial orbital elements and mass are set as tendency can be seen in other literature.10,13) This tendency can follows: be roughly explained by using the Gauss planetary equations (21)-(23). These equations indicate that and are approximately proportional to and that � � � � � ��1 ��, � � �.��, � � ��.��, to . This means that if the eccentricity��/�� is larger,��/�� the � � � The thrusterΩ � force ��, � and� specific ��, � impulse� 1�� . are set as control efficiency� for the semi-major1/√ 1�� axis and inclination��/�� is higher,√1�� while the efficiency for the eccentricity is lower. 623.2 mN and 1800� sec, respectively. Here,�� and � � Therefore, it seems that at the initial phase when the represent a perigee altitude of 1000 km and an apogee� of 35786� km, respectively. The target orbital elements of GEO� are � eccentricity is larger, the increase in the semi-major axis and inclination is prioritized over the decrease in the eccentricity. and . The departure date is set �for � � Figures 8 through 10 show the feedback gains obtained as a June 21, 2017.� As for� the independent variable, the following��1�� , three � � cases �, are� considered:� �� function of the semi-major axis, eccentricity, and inclination, �,

367 Trans. JSASS Aerospace Tech. Japan Vol. 17, No. 3 (2019) respectively. As the relative value of the gains is important, is set at one for all cases. Note that in Figs. 9 and 10, the௘ horizontal axes are reversed for comparison. These figuresܭ show the same trend of and decreasing as time proceeds. Although the three௔ cases have௜ different feedback gains, they share the same transferܭ time.ܭ This implies that the shape of curve is more important than the absolute value of . ௝ ߛ െ ܭ In order ௝to investigate how the nodal values of the gain will affectܭ the transfer time, the nodal values of and ܭ௝ are randomly and independently deviated ௔ from௝ the ܭ ൫ߛ ൯ Fig. 7. Comparison of inclination versus time. nominal௜ ௝ ones of case 1 ( ) by . The regions of the deviatedܭ ሺߛ ሻ nodal values of and are shown in Fig. ߛൌܽ േͷͲΨ 13. The distribution of the௔ transfer௝ time௜ ௝ obtained by 3000 Monte-Calro trials are shownܭ ൫ߛ in൯ Fig. ܭ14.൫ߛ Figure൯ 14 indicates that 95 % of the total distribution achieves the transfer time less than 72.3 day (3 % error from the optimal value (70.2 day)) and that 99 % achieves less than 73.7 day (5 % error from the optimal value). This result shows that as long as the shape of the curve is conserved, the suboptimal transfer time could be achieved௝ by the proposed method. ߛെܭ

Fig. 8. Optimal gain as a function of semi-major axis (case1).

a[km]

Fig. 5. Comparison of semi-major axis versus time (The dashed line indicates the semi-major axis of GEO (42,164 km)).

Fig. 9. Optimal gain as a function of eccentricity (case2).

Fig. 6. Comparison of eccentricity versus time.

Fig. 10. Optimal gain as a function of inclination (case3).

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that determined the thrust direction based on the difference between current and target orbital elements. The feedback gains GTO Earth of the steering law were assumed to be functions of the orbital Eclipse elements and were optimized by using the PSO method. To prove the validity of the proposed method, numerical simulations of the orbit transfer from GTO to GEO were GEO conducted. The results indicated that the proposed method Fig. 11. 3D view of GTO to GEO transfer trajectory obtained by reduced the transfer time by 15% compared to the conventional proposed method (case1) (green area represents eclipse). steering law. The results also showed that regardless of which initial orbital element (that monotonically increases or decreases) was selected as an independent variable, similar performances were obtained. Therefore, by selecting the independent variable appropriately, the minimum time orbit- transfer problem can always be solved by the proposed method. Although this study focused on the minimum-time transfer problem to suppress the disadvantage of the low-thrust orbit transfer, it would be possible to introduce the optimal coasting mechanism in the proposed steering law by introducing

(a) Proposed method (case1) (b) Conventional method variable efficiency parameters similar to those in the work by Fig. 12. Comparison of GTO to GEO transfer trajectory obtained by Petropoulos et al. This will be a future study. (a) proposed method and (b) conventional method. Green area represents eclipse. The view point is above Earth’s north pole. References

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