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 Minimum Time Orbit Raising of Geostationary Spacecraft by Optimizing Feedback Gain of Steering Law 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 : ou��t-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 364 Trans. JSASS Aerospace Tech. Japan Vol. 17, No. 3 (2019) This can lead

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