Patent Publication Number: US-11655056-B2

Title: Orientation control device, satellite, orientation control method, and program

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     The present application is based on PCT filing PCT/JP2019/036029, filed Sep. 13, 2019, which claims priority to JP 2018-177919, filed Sep. 21, 2018, the entire contents of each are incorporated herein by reference. 
     TECHNICAL FIELD 
     The present disclosure relates to an attitude control apparatus for controlling the attitude of a satellite, a satellite, a method of controlling attitude, and a program. 
     BACKGROUND ART 
     A geostationary satellite is separated from a rocket and introduced into an orbit, such as low earth orbit (LEO), geostationary transfer orbit (GTO), or super synchronous orbit (SSO). The geostationary satellite then causes firing of a thruster mounted thereon and thus obtains a thrust, thereby shifting from the initial orbit, into which the geostationary satellite has been introduced, to the geostationary earth orbit (GEO). Examples of the thruster include a chemical thruster and an electric propulsion thruster. The chemical thruster can generate a greater thrust, and therefore requires a shorter time for orbit transfer from the GTO to the GEO but requires a higher propellant consumption, in comparison to the electric propulsion thruster. In contrast, the electric propulsion thruster requires a longer time for orbit transfer from the GTO to the GEO but requires a lower propellant consumption, which means a higher specific impulse, in comparison to the chemical thruster. 
     In either case of the chemical thruster and the electric propulsion thruster, in order to shorten the time required for orbit transfer while reducing the propellant consumption during the orbit transfer, the thrust direction of the thruster should extend in a direction appropriate for transfer of the satellite to the target orbit. In addition, in order to cause firing of the thruster, the panel surface of a solar array panel (SAP) for generating electric power to be fed to the thruster should face the direction of the sun. In view of these requirements, some satellites are equipped with attitude control apparatuses that control the attitudes of the satellites during orbit transfer. An example of these attitude control apparatuses is disclosed in Patent Literature 1. The attitude control apparatus disclosed in Patent Literature 1 controls the attitude of a satellite so as to make the panel surface of a SAP orthogonal to the sun vector and make the thrust vector orthogonal to the rotational axis of the SAP. 
     CITATION LIST 
     Patent Literature 
     Patent Literature 1: Unexamined Japanese Patent Application Publication No. 2001-18899 
     SUMMARY OF INVENTION 
     Technical Problem 
     The mechanical attitude control of the satellite is executed by attitude control actuators installed in the satellite. Unfortunately, the attitude control apparatus disclosed in Patent Literature 1 controls the attitude of the satellite regardless of the movement limitation of the attitude control actuators. This configuration may lead to a problem that the actual attitude of the satellite cannot follow the target attitude of the satellite. The panel surface of the SAP therefore fails to face the direction of the sun, resulting in a reduction in power generation efficiency of the SAP. The similar problems arise not only in a geostationary satellite but also in a non-geostationary satellite, which shifts from the initial orbit to a geocentric orbit. 
     An objective of the present disclosure, which has been accomplished in view of the above situations, is to provide an attitude control apparatus, a satellite, a method of controlling attitude, and a program, that can prevent a reduction in power generation efficiency of a SAP during orbit transfer. 
     Solution to Problem 
     In order to achieve the above objective, an attitude control apparatus according to the present disclosure includes an ideal thrust direction calculator, an ideal attitude calculator, a target attitude calculator, and a torque calculator. The ideal thrust direction calculator acquires the position of a satellite including a thruster and a solar panel having a panel surface rotatable about a rotational axis, and calculates an ideal thrust direction that is a thrust direction of the thruster for minimizing a propellant consumption in firing of the thruster during transfer of the satellite to a target orbit. The ideal attitude calculator calculates an ideal attitude that is the attitude of the satellite in which the panel surface faces the sun while the thrust direction aligns with the ideal thrust direction. The target attitude calculator acquires a movement limitation of an attitude control actuator for controlling an attitude of the satellite mechanically, and calculates a target attitude that is an attitude of the satellite in which a deviation from the ideal attitude is minimized within the movement limitation while the panel surface faces the sun. The torque calculator acquires an actual attitude that is an attitude of the satellite, calculates a torque for turning the satellite from the actual attitude to the target attitude, and transmits a torque instruction indicating the calculated torque to the attitude control actuator. 
     Advantageous Effects of Invention 
     The attitude control apparatus according to the present disclosure calculates the target attitude that is the attitude of the satellite in which a deviation from the ideal attitude is minimized within the movement limitation of the attitude control actuator while the panel surface faces the sun. Since the torque instruction indicating the torque for turning the satellite from the actual attitude to the target attitude is transmitted to the attitude control actuator, the configuration can make the attitude of the satellite coincide with the target attitude and thus prevent a reduction in power generation efficiency of the SAP during the orbit transfer. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG.  1    illustrates a geostationary satellite according to Embodiment 1 of the present disclosure; 
         FIG.  2    illustrates a configuration of the geostationary satellite according to Embodiment 1; 
         FIG.  3    is a block diagram illustrating a configuration of an attitude control apparatus according to Embodiment 1; 
         FIG.  4    is a flowchart illustrating an example of attitude control operation executed by the attitude control apparatus according to Embodiment 1; and 
         FIG.  5    illustrates a hardware configuration of the attitude control apparatus according to the embodiments. 
     
    
    
     DESCRIPTION OF EMBODIMENTS 
     An attitude control apparatus according to embodiments of the disclosure is described in detail with reference to the drawings. Components that are the same or equivalent are assigned the same reference signs throughout the drawings. 
     Embodiment 1 
     An attitude control apparatus according to Embodiment 1 is described focusing on an exemplary attitude control apparatus that is installed in a geostationary satellite, which is an exemplary satellite, and controls the attitude of the geostationary satellite during transfer from the initial orbit, into which the geostationary satellite has been introduced, to a geostationary earth orbit (GEO) that is a target orbit. The geostationary satellite  1  illustrated in  FIG.  1    is separated from a rocket and introduced into a geostationary transfer orbit (GTO). The geostationary satellite  1  then causes firing of a thruster  11  that is an electric propulsion thruster to obtain a thrust, and thereby shifts from the GTO to the GEO. The thruster  11  indicates a main thruster, such as ion thruster or plasma thruster. The following description of attitude control for the geostationary satellite  1  defines a geocentric inertial coordinate system having an earth  2  at the center and having the X axis, the Y axis, and the Z axis, and refers to the coordinate system as required, in order to facilitate an understanding. In this geocentric inertial coordinate system, the Z axis extends through the earth  2  from the South Pole to the North Pole, the X axis extends in the direction of vernal equinox, and the Y axis is orthogonal to each of the Z and X axes. 
     As illustrated in  FIG.  2   , the description also defines a satellite coordinate system fixed to the geostationary satellite  1  and having the x B  axis, the y B  axis, and the z B  axis, and refers to the coordinate system as required. In this satellite coordinate system, the z B  axis coincides with the thrust direction of the thruster  11  included in the geostationary satellite  1 , the y B  axis coincides with the rotational axis of a solar panel  12  included in the geostationary satellite  1 , and the x B  axis is orthogonal to each of the z B  and y B  axes. The geostationary satellite  1  includes a housing  10 , the thruster  11  mounted on the housing  10 , the solar panel  12 , a support member  13  mounted on the housing  10  and configured to support the solar panel  12  such that the solar panel  12  is rotatable about the y B  axis, and attitude control actuators  14  that control the attitude of the geostationary satellite  1  mechanically. The thruster  11  is mounted on the housing  10  while the thrust direction is fixed relative to the housing  10 . In the satellite coordinate system defined as illustrated in  FIG.  2   , the firing of the thruster  11  exerts a thrust on the geostationary satellite  1  in the positive z B  axis direction. In order to adjust the direction of the thrust exerted on the geostationary satellite  1 , the attitude of the geostationary satellite  1  is to be controlled, because of the thrust direction of the thruster  11  fixed relative to the housing  10 . The four attitude control actuators  14  that control the attitude of the geostationary satellite  1  mechanically are accommodated in the housing  10 . Examples of the attitude control actuators  14  include reaction wheels and control moment gyros. 
     Although not illustrated in  FIG.  2   , the geostationary satellite  1  also includes an attitude control apparatus accommodated in the housing  10 . The attitude control apparatus electrically controls the attitude control actuators  14 . The attitude control apparatus calculates an ideal attitude that is the attitude of the geostationary satellite  1  in which the thrust direction of the thruster  11  aligns with the optimum direction for minimizing the propellant consumption during the transfer from the GTO to the GEO while the panel surface  12   a  of the solar panel  12  is orthogonal to a unit vector s B  indicating the direction of the sun represented by the dashed arrow in  FIG.  2   , as explained in detail later. The attitude control apparatus then calculates a target attitude having a minimum deviation from the ideal attitude, in view of a movement limitation of the attitude control actuators  14 . The attitude control apparatus then acquires an actual attitude that is the attitude of the geostationary satellite  1 , calculates a torque for turning the geostationary satellite  1  from the actual attitude to the target attitude, and transmits a torque instruction indicating the calculated torque to the attitude control actuators  14 . On the basis of this torque instruction, the attitude control actuators  14  control the attitude of the geostationary satellite  1  mechanically. 
     During the transfer from the GTO to the GEO, anon-illustrated thruster controller causes firing of the thruster  11  at a constant discharge amount. The firing of the thruster  11  at a constant discharge amount generates a constant thrust exerted on the geostationary satellite  1  during the transfer from the GTO to the GEO. 
     The following description is directed to an attitude control apparatus  20 , which directs the thrust direction of the thruster  11  to the optimum direction for minimizing the propellant consumption during the transfer from the GTO to the GEO while a constant thrust is exerted as explained above. With reference to  FIG.  3   , the attitude control apparatus  20  includes an orbit calculator  21  that calculates osculating orbit elements of the current orbit of the geostationary satellite  1 , an ideal thrust direction calculator  22  that calculates an ideal thrust direction of the thruster  11 , a sun direction calculator  23  that calculates a direction of the sun as viewed from the geostationary satellite  1 , an ideal attitude calculator  24  that calculates an ideal attitude of the geostationary satellite  1 , an actual attitude calculator  25  that calculates an actual attitude that is the attitude of the geostationary satellite  1  in real-time, a target attitude calculator  26  that calculates a target attitude that is the attitude of the geostationary satellite  1  to be achieved, a movement limitation determiner  27  that determines conditions of the movement limitation of the attitude control actuators  14 , and a torque calculator  28  that calculates a torque on the basis of the target attitude and transmits a torque instruction to the attitude control actuators  14 . 
     The ideal thrust direction indicates the optimum thrust axis of the thruster  11  for minimizing the propellant consumption during the transfer from the orbit including the geostationary satellite  1  to the GEO and minimizing the time required for the transfer. The ideal attitude is the attitude of the geostationary satellite  1  in which the panel surface  12   a  faces the sun while the z B  axis aligns with the ideal thrust direction. The ideal attitude is preferably the attitude of the geostationary satellite  1  in which the y B  axis extends in the direction orthogonal to the unit vector s B  while the z B  axis aligns with the ideal thrust direction. The target attitude is the attitude of the geostationary satellite  1  in which a deviation from the ideal attitude is minimized within the movement limitation of the attitude control actuators  14  while the panel surface  12   a  faces the sun. The target attitude is preferably the attitude of the geostationary satellite  1  in which a deviation from the ideal attitude is minimized within the movement limitation of the attitude control actuators  14  while the y B  axis extends in the direction orthogonal to the unit vector s B . 
     The orbit calculator  21  calculates an instantaneous position of the geostationary satellite  1  on the basis of the signal acquired from a global positioning system (GPS) receiver installed in the geostationary satellite  1 , and acquires an instantaneous velocity of the geostationary satellite  1  from a speed sensor included in a sensor unit  29 . The speed sensor calculates the velocity of the geostationary satellite  1 , for example, on the basis of the wave from a ground station that communicates with the geostationary satellite  1 . The orbit calculator  21  then calculates osculating orbit elements, which are parameters for specifying the orbit along which the geostationary satellite  1  travels, on the basis of the instantaneous position of the geostationary satellite  1  and the instantaneous velocity of the geostationary satellite  1 . 
     The ideal thrust direction calculator  22  calculates an ideal thrust direction that is the optimum thrust axis of the thruster  11  for minimizing the propellant consumption during the transfer to the GEO from the orbit including the geostationary satellite  1  and specified by the osculating orbit elements calculated by the orbit calculator  21 . In detail, the ideal thrust direction calculator  22  calculates weight coefficients on the basis of the difference of the osculating orbit elements calculated by the orbit calculator  21  from target orbit elements, and calculates the sum of the results of multiplication of the direction vectors providing the maximum change rates of the individual orbit elements by the weight coefficients, thereby calculating the ideal thrust direction in the satellite coordinate system for minimizing the propellant consumption. 
     The sun direction calculator  23  calculates a direction of the sun as viewed from the geostationary satellite  1 . In detail, the sun direction calculator  23  acquires a signal from a sun sensor included in the sensor unit  29 , and calculates a unit vector s B  indicating the direction of the sun in the satellite coordinate system on the basis of the signal acquired from the sun sensor. 
     The ideal attitude calculator  24  calculates an ideal attitude of the geostationary satellite  1  from the ideal thrust direction and the direction of the sun. In detail, the ideal attitude calculator  24  calculates the ideal attitude of the geostationary satellite  1  from the ideal thrust direction and the unit vector s B . 
     The actual attitude calculator  25  acquires a signal from the sensor unit  29 , which includes a magnetic sensor, a gyro sensor, and other sensors installed in the geostationary satellite  1 , and calculates an actual attitude of the geostationary satellite  1  on the basis of the signal acquired from the sensor unit  29 . 
     The movement limitation determiner  27  determines a movement limitation defined by the capacities of the attitude control actuators  14 . In this embodiment, the movement limitation indicates the upper limit ω MAX  of the absolute value of the angular rate of the geostationary satellite  1  that can be achieved by the attitude control actuators  14 . 
     The target attitude calculator  26  acquires the actual attitude of the geostationary satellite  1  from the actual attitude calculator  25 , acquires the ideal attitude of the geostationary satellite  1  from the ideal attitude calculator  24 , and acquires the movement limitation of the attitude control actuators  14  from the movement limitation determiner  27 . On the basis of the actual and ideal attitudes of the geostationary satellite  1  and the movement limitation of the attitude control actuators  14 , the target attitude calculator  26  calculates a target attitude that is the attitude of the geostationary satellite  1  to be achieved. 
     The torque calculator  28  acquires the actual attitude of the geostationary satellite  1  from the actual attitude calculator  25 , acquires the target attitude of the geostationary satellite  1  from the target attitude calculator  26 , and calculates a torque for making the attitude of the geostationary satellite  1  coincide with the target attitude. The torque calculator  28  then transmits a torque instruction indicating the calculated torque to the attitude control actuators  14 . The attitude control actuators  14  control the attitude of the geostationary satellite  1  mechanically in accordance with the torque instruction. 
     A process of controlling the attitude of the geostationary satellite  1  executed by the attitude control apparatus  20  having the above-described configuration is explained with reference to  FIG.  4   . The attitude control apparatus  20  makes the panel surface  12   a  of the solar panel  12  orthogonal to the direction of the sun, that is, the unit vector s B , and then executes the attitude control process illustrated in  FIG.  4   . The attitude control apparatus  20  executes the attitude control process in a time interval T 1  to control the attitude control actuators  14 , so that the attitude control actuators  14  control the attitude of the geostationary satellite  1  mechanically in the constant time interval. The time interval T 1  is a period of several seconds to several minutes, for example. 
     The orbit calculator  21  calculates osculating orbit elements of the geostationary satellite  1  (Step S 11 ). In detail, the orbit calculator  21  calculates the osculating orbit elements of the geostationary satellite  1  at a time t k  and transmits the calculated osculating orbit elements to the ideal thrust direction calculator  22 . The individual components of the attitude control apparatus  20  execute processes in synchronization with the clock signal having the time interval T 1  output from a non-illustrated oscillator circuit. When the osculating orbit elements calculated in Step S 11  are equal to the osculating orbit elements of a geostationary orbit (Step S 12 ; Yes), the attitude control apparatus  20  terminates the attitude control process. 
     When the osculating orbit elements calculated in Step S 11  are not equal to the osculating orbit elements of the geostationary orbit (Step S 12 ; No), the actual attitude calculator  25  calculates an actual attitude of the geostationary satellite  1  in the geocentric inertial coordinate system on the basis of the signal acquired from an attitude sensor included in the sensor unit  29  (Step S 13 ). In detail, the actual attitude calculator  25  calculates a matrix C BkI  representing the actual attitude that is the attitude of the geostationary satellite  1  in the geocentric inertial coordinate system. The actual attitude calculator  25  then transmits the matrix C BkI  to the target attitude calculator  26  and the torque calculator  28 . 
     The sun direction calculator  23  calculates a direction of the sun as viewed from the geostationary satellite  1  in the satellite coordinate system on the basis of the signal acquired from the sun sensor included in the sensor unit  29  (Step S 14 ). In detail, the sun direction calculator  23  calculates a unit vector s B  indicating the direction of the sun in the satellite coordinate system on the basis of the signal acquired from the sun sensor. The sun direction calculator  23  then transmits the unit vector s Bk  indicating the direction of the sun calculated at the time t k  to the ideal attitude calculator  24  and the target attitude calculator  26 . 
     The ideal thrust direction calculator  22  calculates an ideal thrust direction in the satellite coordinate system from the osculating orbit elements calculated in Step S 11  (Step S 15 ). In detail, the ideal thrust direction calculator  22  calculates weight coefficients from the difference of the osculating orbit elements calculated by the orbit calculator  21  at the time t k  from the target orbit elements, and calculates the sum of the results of multiplication of the direction vectors providing the maximum change rates of the individual orbit elements by the weight coefficients, thereby calculating an ideal thrust direction u k+1   d  in the satellite coordinate system at the time t k+1  for minimizing the propellant consumption. The ideal thrust direction calculator  22  then transmits the calculated ideal thrust direction u k+1   d  to the ideal attitude calculator  24 . The time t k+1  is represented by the expression (1) below using the time t k  and the time interval T 1 :
 
 t   k+1   =t   k   +T 1  (1)
 
     The ideal attitude calculator  24  calculates an ideal attitude of the geostationary satellite  1  in the geocentric inertial coordinate system from the ideal thrust direction calculated in Step S 15  and the direction of the sun calculated in Step S 14  (Step S 16 ). In detail, the ideal attitude calculator  24  calculates the ideal attitude of the geostationary satellite  1  from the ideal thrust direction u k+1   d  at the time t k+1  and the unit vector s Bk . The respective unit vectors corresponding to the x B  axis, the y B  axis, and the z B  axis in the satellite coordinate system in the case where the attitude of the geostationary satellite  1  coincides with the ideal attitude at the time t k+1  are represented by x Bk+1   d , y Bk+1   d , and z Bk+1   d . Since the z B  axis aligns with the ideal thrust direction in the ideal attitude as described above, the z B  axis is represented by the expression (2) below:
 
 z   Bk+1   d   u   k+1   d   (2)
 
     In addition, the panel surface  12   a  is orthogonal to the unit vector s B  in the ideal attitude. The direction of the sun at the time t k+1  can be regarded to be identical to the direction of the sun at the time t k  regardless of changes in the position and the attitude of the geostationary satellite  1  during the time interval T 1 , because of the extremely long distance between the geostationary satellite  1  and the sun. That is, the unit vector y Bk+1   d  corresponding to the y B  axis that is the rotational axis of the solar panel  12  can be regarded as orthogonal to the unit vector z Bk+1   d  and the direction of the sun s Bk . Accordingly, the unit vector y Bk+1   d  is represented by the expression (3) below: 
     
       
         
           
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     Because the satellite coordinate system is a right-handed orthogonal coordinate system, the unit vector x Bk+1   d  is represented by the expression (4) below:
 
 x   Bk+1   d   =y   Bk+1   d   ×z   Bk+1   d   (4)
 
     A matrix C Bk+1   d   I  is defined in the expression (5) below, which represents the unit vectors x Bk+1   d , y Bk+1   d , and z Bk+1   d  of the above expressions (2) to (4) in the geocentric inertial coordinate system. The term [x Bk+1   d ] I  in the expression (5) indicates the unit vector x Bk+1   d  represented in the geocentric inertial coordinate system. Also, the term [y Bk+1   d ] I  indicates the unit vector y Bk+1   d  represented in the geocentric inertial coordinate system, and the term [z Bk+1   d ] I  indicates the unit vector z Bk+1   d  represented in the geocentric inertial coordinate system. The ideal attitude calculator  24  transmits the matrix C Bk+1   d   I  to the target attitude calculator  26 .
 
 C   Bk+1   d   I =[[ x   Bk+1   d ] I ,[ y   Bk+1   d ] I ,[ z   Bk+1 ] I ]  (5)
 
     The target attitude calculator  26  calculates a target attitude of the geostationary satellite  1  in the geocentric inertial coordinate system from the actual and ideal attitudes of the geostationary satellite  1  and the movement limitation of the attitude control actuators  14  (Step S 17 ). In detail, the target attitude calculator  26  calculates the target attitude from the matrix C Bk+1   d   I , the matrix C BkI , the unit vector s Bk , and the upper limit ω MAX  of the absolute value of the angular rate. The calculated target attitude preferably has a minimum deviation from the ideal attitude. The target attitude calculator  26  thus calculates a transformation matrix from the ideal attitude of the geostationary satellite  1  at the time t k+1  to the target attitude of the geostationary satellite  1  at the time t k+1 , and minimizes the trace of the transformation matrix. The target attitude calculator  26  thus calculates the target attitude having the minimum deviation from the ideal attitude. The specific calculation process is explained. 
     The transformation matrix C BkBk+1   d  is defined in the expression (6) below, which represents vector transformation from the ideal attitude of the geostationary satellite  1  at the time t k+1  to the actual attitude of the geostationary satellite  1  at the time t k . In the expression (6), C IBk+1   d  indicates a transposed matrix of the matrix C Bk+1   d   I .
 
 C   BkBk+1   d   =C   BkI   C   IBk+1   d   (6)
 
     The transformation matrix C Bk+IBk  from the actual attitude of the geostationary satellite  1  at the time t k  to the target attitude of the geostationary satellite  1  at the time t k+1  is defined in the expression (7) below. The right-hand side of the expression (7) indicates conducting a turn about the unit vector s Bk  by an angle θ and then conducting a turn about the y B  axis by an angle φ, thereby making the attitude of the geostationary satellite  1  coincide with the target attitude. In the expression (7), C 2 (φ) is a coordinate transformation matrix indicating a turn about the y B  axis by the angle φ. In the expression (7), E 3  is a three-dimensional identity matrix, s Bk   T  is a transposed matrix of the s Bk , and s Bk   x  is a cross-product matrix of the s Bk .
 
[Math 2]
 
 C   B     k+1     B     k     =C   2 (ϕ){cos θ E   3 +(1−cos θ) s   B     k     s   B     k     T −sin θ s   B     k     X }  (7)
 
     While the orbital period of the geostationary satellite  1  is approximately 12 to 24 hours, the time interval T 1  is several seconds to several minutes. The angles θ and φ can thus be regarded as extremely small values. The above expression (7) can therefore be approximated by the expression (8) below. In the expression (8), e 2  is a matrix defined by [0 1 0] T . In the expression (8), the transformation matrix C Bk+IBk  is represented by a linear combination of the angles θ and φ.
 
[Math 3]
 
 C   B     k+1     B     k     ≈E   3 −(ϕ e   2   +θs   B     k−1   ) X   (8)
 
     The transformation matrix C Bk+IBk+1   d  from the ideal attitude of the geostationary satellite  1  at the time t k+1  to the target attitude of the geostationary satellite  1  at the time t k+1  is defined in the expression (9) below.
 
[Math 4]
 
 C   B     k+1     B     k+1       d     =C   B     k+1     B     k     C   B     k     B     k+1       d     (9)
 
     The above expression (9) represents a deviation from the ideal attitude of the geostationary satellite  1 . Accordingly, the target attitude closest to the ideal attitude of the geostationary satellite  1  can be obtained by calculating angles θ and φ that provide the minimum trace of the transformation matrix C Bk+IBk+1   d  in the expression (9) while satisfying the expression (10) below based on the upper limit ω MAX  of the absolute value of the angular rate.
 
[Math 5]
 
√{square root over (ϕ 2 +θ 2 )}≤ T 1ω MAX   (10)
 
     On the right-hand side of the above expression (9), the transformation matrix C Bk+IBk  is a linear combination of the angles θ and φ. On the right-hand side of the expression (9), the transformation matrix C BkBk+1   d  is the product of the matrix C BkI  representing actual attitude of the geostationary satellite  1  at the time t k  and the transposed matrix C IBk+1   d  of the matrix C Bk+1   d   I  representing ideal attitude of the geostationary satellite  1  at the time t k+1 , as defined in the above expression (6). The transformation matrix C Bk+IBk+1   d  is therefore a linear combination of the angles θ and φ. That is, the target attitude calculator  26  calculates the target attitude by solving the mathematical programming problem of minimizing the trace of the transformation matrix C Bk+IBk+1   d , which is an evaluation function configured by a linear combination of the angles θ and φ, under the quadratic constraint represented by the above expression (10). This configuration does not require a process of repetitively execute calculations while changing a variable, for example, for solving the mathematical programming problem, and can therefore improve the efficiency of calculating the target attitude. 
     The solutions of the angles θ and φ obtained by solving the above-explained mathematical programming problem are defined as θ* and φ*, respectively. The transformation matrix from the actual attitude of the geostationary satellite  1  at the time t k  to the target attitude of the geostationary satellite  1  at the time t k+1 , obtained by substituting the angles θ* and φ* in the above expression (7), is defined as C B*k+IBk . The matrix C B*k+1I  indicating the target attitude of the geostationary satellite  1  in the geocentric inertial coordinate system at the time t k+1  is represented by the expression (11) below. The target attitude calculator  26  then transmits the matrix C B*k+1I  indicating the calculated target attitude to the torque calculator  28 .
 
[Math 6]
 
 C   B*     k+1     I   =C   B*     k+1     B     k     C   B     k     I   (11)
 
     The attitude control apparatus  20  that executes the above-explained process makes the panel surface  12   a  of the solar panel  12  orthogonal to the direction of the sun at the start of attitude control. As in the above expression (7), the attitude control from the actual attitude of the geostationary satellite  1  at the time t k  to the target attitude of the geostationary satellite  1  at the time t k+1  is achieved by a turn about the unit vector s Bk  and a turn about the y B  axis. That is, the panel surface  12   a  of the solar panel  12  is maintained to be orthogonal to the direction of the sun in the attitude control during the transfer from the GTO to the GEO. This configuration can prevent a reduction in power generation efficiency of the solar panel  12  during the orbit transfer. Furthermore, the attitude of the geostationary satellite  1  is made to coincide with the target attitude having a minimum deviation from the ideal attitude of the geostationary satellite  1  in association with the ideal thrust direction that is an optimum thrust direction of the thruster  11  for minimizing the propellant consumption during the transfer from the GTO to the GEO. This configuration can minimize the propellant consumption during the orbit transfer. 
     The torque calculator  28  calculates a torque required for turning the geostationary satellite  1  from the actual attitude to the target attitude on the basis of the actual attitude of the geostationary satellite  1  calculated in Step S 13  and the target attitude of the geostationary satellite  1  calculated in Step S 17 . The torque calculator  28  then outputs a torque instruction indicating the required torque to the attitude control actuators  14  (Step S 18 ). In detail, the torque calculator  28  calculates the torque for making the attitude of the geostationary satellite  1  coincide with the target attitude at the time t k+1  from the matrix C BkI  representing the actual attitude of the geostationary satellite  1  and the matrix C B*k+1I  indicating the target attitude of the geostationary satellite  1 . The torque calculator  28  transmits the torque instruction indicating the calculated torque to the attitude control actuators  14 . 
     The attitude control actuators  14  control the attitude of the geostationary satellite  1  mechanically in accordance with the torque instruction. The mechanical control of the attitude control actuators  14  over the attitude of the geostationary satellite  1  in accordance with the torque instruction changes the orientation of the geostationary satellite  1 , so that the attitude of the geostationary satellite  1  coincides with the target attitude. The attitude control apparatus  20  executes the above-explained process repetitively in the time interval T 1  until arrival of the geostationary satellite  1  at the GEO. 
     As described above, the attitude control apparatus  20  according to Embodiment 1 calculates a target attitude that is the attitude of the geostationary satellite  1  in which a deviation from the ideal attitude is minimized within the movement limitation of the attitude control actuators  14  while the panel surface  12   a  faces the sun preferably while the y B  axis that is the rotational axis extends in the direction orthogonal to the sun direction s B . Transmitting to the attitude control actuators  14  a torque instruction indicating the torque for turning the geostationary satellite  1  from the actual attitude to the calculated target attitude makes the attitude of the geostationary satellite  1  coincide with the target attitude, thereby preventing a reduction in power generation efficiency of the solar panel  12  during the transfer from the GTO to the GEO. In the above-explained process of calculating the target attitude, calculated is the target attitude having the minimum deviation from the ideal attitude of the geostationary satellite  1  in association with the ideal thrust direction that is an optimum thrust direction of the thruster  11  for minimizing the propellant consumption during the transfer from the GTO to the GEO. This configuration can minimize the propellant consumption during the orbit transfer. 
     Embodiment 2 
     The above-explained process of calculating the target attitude by the target attitude calculator  26  is a mere example. Although the geostationary satellite  1  and the attitude control apparatus  20  according to Embodiment 2 have the same configurations as those according to Embodiment 1, these embodiments have differences in the processes executed by the target attitude calculator  26  and the movement limitation determiner  27 . 
     The movement limitation determiner  27  determines a maximum angular momentum envelope surface, which indicates a collection of maximum angular momenta that can be generated by the control moment gyros or reaction wheels serving as the attitude control actuators  14 . The movement limitation determiner  27  then transmits the maximum angular momentum envelope surface to the target attitude calculator  26 . The following is an additional description about the maximum angular momentum envelope surface. In general, when a certain direction is designated in a satellite coordinate system, the angular momentum that can be generated by each of the attitude control actuators  14  in the designated direction is uniquely determined. The sum of such momenta provides the maximum angular momenta that can be generated by all the attitude control actuators  14  in the designated direction. The surface defined by these maximum angular momenta is called a maximum angular momentum envelope surface. 
     The target attitude calculator  26  calculates a target attitude from the matrix C Bk+1   d   I , the matrix C BkI , the unit vector s Bk , and the maximum angular momentum envelope surface. The calculated target attitude preferably has a minimum deviation from the ideal attitude. The target attitude calculator  26  thus calculates a transformation matrix from the ideal attitude of the geostationary satellite  1  at the time t k+1  to the actual attitude of the geostationary satellite  1  at the time t k , and then solves the inverse kinematics problem of the transformation matrix, thereby obtaining an ideal rotational angle θ d  about the unit vector s Bk  and an ideal rotational angle φ d  about the y B  axis. The target attitude calculator  26  then calculates a target attitude defined by angles θ and φ close to the ideal rotational angles θ d  and φ d . 
     The operation of the target attitude calculator  26  is explained in more detail. The target attitude calculator  26  solves the inverse kinematics problem of the transformation matrix C BkBk+1   d  represented by the above expression (6), and thereby obtains the ideal rotational angle θ d  about the unit vector s Bk  and the ideal rotational angle φ d  about the y B  axis. 
     Regarding a turn from the actual attitude of the geostationary satellite  1  at the time t k  to the ideal attitude of the geostationary satellite  1  at the time t k+1 , a theoretical optimum rotational axis [ρ d ] Bk  regardless of the movement limitation of the attitude control actuators  14  is defined in the expression (12) below. The rotational axis [ρ d ] Bk  is defined in the satellite coordinate system.
 
[ρ d ] Bk =φ d   e2 +θ dSBk   (12)
 
     The target attitude calculator  26  calculates an upper limit h max  of the angular momentum of a turn of the geostationary satellite  1  about the rotational axis [ρ d ] Bk  that can be achieved by the attitude control actuators  14 , on the basis of the maximum angular momentum envelope surface. The target attitude calculator  26  calculates an upper limit ω MAX  of the absolute value of the angular rate of the geostationary satellite  1  about the rotational axis [ρ d ] Bk  that can be achieved by the attitude control actuators  14  as in the expression (13) below. In the expression (13), I B  indicates the inertia matrix of the geostationary satellite  1 . The inertia matrix I B  of the geostationary satellite  1  has three rows and three columns, contains a moment of inertia of the geostationary satellite  1  at the diagonal elements, and a product of inertia at the off-diagonal elements.
 
ω MAX   =h   max   /|I   B [ρ d ] Bk |  (13)
 
     The target attitude calculator  26  then calculates angles θ and φ having minimum deviations from the ideal rotational angle θ d  about the unit vector s Bk  and the ideal rotational angle φ d  about the y B  axis, respectively, while satisfying the above expression (10), and thereby obtains the target attitude closest to the ideal attitude of the geostationary satellite  1 . The angles θ and φ obtained as explained above are defined as θ* and φ*, respectively. The transformation matrix from the actual attitude of the geostationary satellite  1  at the time t k  to the target attitude of the geostationary satellite  1  at the time t k+1 , obtained by substituting the angles θ* and φ* in the above expression (7), is defined as C B*k+IBk . The matrix C B*k+1I  indicating the target attitude of the geostationary satellite  1  in the geocentric inertial coordinate system at the time t k+1  is represented by the above expression (11). The target attitude calculator  26  then transmits the calculated matrix C B*k+1I  indicating the target attitude to the torque calculator  28 . 
     As in Embodiment 1, the attitude control apparatus  20  that executes the above-explained process makes the panel surface  12   a  of the solar panel  12  orthogonal to the direction of the sun at the start of attitude control. As in the above expression (7), the attitude control from the actual attitude of the geostationary satellite  1  at the time t k  to the target attitude of the geostationary satellite  1  at the time t k+1  is achieved by a turn about the unit vector s Bk  and a turn about the y B  axis. That is, the panel surface  12   a  of the solar panel  12  is maintained to be orthogonal to the direction of the sun in the attitude control during the transfer from the GTO to the GEO. This configuration can prevent a reduction in power generation efficiency of the solar panel  12  during the orbit transfer. Furthermore, the attitude of the geostationary satellite  1  is made to coincide with the target attitude having a minimum deviation from the ideal attitude of the geostationary satellite  1  in association with the ideal thrust direction that is an optimum thrust direction of the thruster  11  for minimizing the propellant consumption during the transfer from the GTO to the GEO. This configuration can minimize the propellant consumption during the orbit transfer. 
     As described above, the attitude control apparatus  20  according to Embodiment 2 turns the geostationary satellite  1  about the ideal rotational axis at an angular rate in view of the movement limitation of the attitude control actuators  14 , and thus makes the attitude of the geostationary satellite  1  coincide with the target attitude. This configuration can prevent a reduction in power generation efficiency of the solar panel  12  during the transfer from the GTO to the GEO. Since the movement limitation of the attitude control actuators  14  regarding a turn about the ideal rotational axis is taken into consideration, the angular momenta that can be generated by the attitude control actuators  14  can be maximized even in a geostationary satellite  1  in which the principal axis of inertia providing the maximum second moment of inertia differs from the principal axis of inertia providing the minimum second moment of inertia, for example. In addition, the target attitude having the minimum deviation from the ideal attitude of the geostationary satellite  1  in association with the ideal thrust direction (optimum thrust axis of the thruster  11  for minimizing the propellant consumption during the orbit transfer and minimizing the time required for the transfer). This configuration can minimize the propellant consumption during the orbit transfer. 
       FIG.  5    illustrates a hardware configuration of the attitude control apparatus according to the embodiments. The attitude control apparatus  20  has a hardware configuration for controlling the individual components, which includes a processor  31 , a memory  32 , and an interface  33 . The functions of these components can be performed when the processor  31  executes the program stored in the memory  32 . The interface  33  connects the individual components to each other to establish communication. The interface  33  may be replaced with multiple types of interfaces as required. Although  FIG.  5    illustrates an example in which the attitude control apparatus  20  includes the single processor  31  and the single memory  32 , the attitude control apparatus  20  may include a plurality of processors  31  and a plurality of memories  32 . In this case, the processors  31  and the memories  32  perform the individual functions in cooperation with each other. The attitude control apparatus  20  is connected via the interface  33  to the sensor unit  29  and the attitude control actuators  14 . 
     Furthermore, the above-illustrated hardware configuration and flowchart are a mere example and may be modified and corrected in any manner. 
     The center for executing the control process, which includes the processor  31 , the memory  32 , and the interface  33 , may be configured by a general computer system without a dedicated system. The computer program for executing the above operations may be stored into a non-transitory computer-readable recording medium, such as a flexible disk, a compact disc read-only memory (CD-ROM), or a digital versatile disc read-only memory (DVD-ROM), or may be stored into a storage device on a communication network. In this case, the computer program stored in the non-transitory recording medium or the storage device is installed into a computer, and thereby causes the computer to function as the attitude control apparatus  20  for executing the above process. 
     The above-described embodiments of the disclosure should not be construed as limiting the disclosure. The initial orbit, into which the geostationary satellite  1  separated from the rocket is introduced, may be an orbit other than the GTO. For example, the geostationary satellite  1  may be introduced into a low earth orbit (LEO), a super synchronous orbit (SSO), or other orbits. The attitude control apparatus  20  may conduct attitude control of a satellite other than the geostationary satellite  1 . For example, the attitude control apparatus  20  may control the attitude of a non-geostationary satellite that shifts from the initial orbit to a geocentric orbit. In this case, the target orbit is not the GEO but any geocentric orbit. The geostationary satellite  1  may include a plurality of thrusters  11 . In this case, the z B  axis in the satellite coordinate system indicates a synthetic thrust axis formed by synthetizing the individual thrust axes of the thrusters  11 . The thruster  11  may also be a chemical thruster. 
     The above-described embodiments provide a mere example of the configuration of the attitude control apparatus  20  and the operations of the individual components. The attitude control apparatus  20  may be installed in a ground station. The sun direction calculator  23  may calculate a direction of the sun from the solar calendar. The attitude control apparatus  20  may exclude the movement limitation determiner  27 , and the target attitude calculator  26  may retain the movement limitation of the attitude control actuators  14  in advance. The ideal thrust direction calculator  22  may calculate the optimum thrust direction of the thruster  11  for minimizing the propellant consumption during the transfer from the orbit including the geostationary satellite  1  to the GEO and minimizing the time required for the transfer. This configuration can minimize the propellant consumption during the transfer and minimize the time required for the transfer. 
     The thruster controller may provide the thruster  11  with a thruster instruction value for varying the discharge amount from the thruster  11  during the transfer from the GTO to the GEO. 
     The above-described definition of the satellite coordinate system is a mere example. The satellite coordinate system may be any coordinate system, in which the y B  axis indicating the rotational axis of the solar panel  12  extends in a predetermined direction relative to the z B  axis indicating the ideal thrust direction. 
     The above-described movement limitation of the attitude control actuators  14  is a mere example and may be replaced with the upper limit of the rotational rate of flywheels or reaction wheels serving as the attitude control actuators  14 . In Embodiment 2, the movement limitation of the attitude control actuators  14  may be the upper limit of the absolute value of the angular rate of the geostationary satellite  1  that can be achieved by the attitude control actuators  14 , as in Embodiment 1. 
     The foregoing describes some example embodiments for explanatory purposes. Although the foregoing discussion has presented specific embodiments, persons skilled in the art will recognize that changes may be made in form and detail without departing from the broader spirit and scope of the invention. Accordingly, the specification and drawings are to be regarded in an illustrative rather than a restrictive sense. This detailed description, therefore, is not to be taken in a limiting sense, and the scope of the invention is defined only by the included claims, along with the full range of equivalents to which such claims are entitled. 
     This application claims the benefit of Japanese Patent Application No. 2018-177919, filed on Sep. 21, 2018, the entire disclosure of which is incorporated by reference herein. 
     REFERENCE SIGNS LIST 
     
         
           1  Geostationary satellite 
           2  Earth 
           11  Thruster 
           12  Solar panel 
           12   a  Panel surface 
           13  Support member 
           14  Attitude control actuator 
           20  Attitude control apparatus 
           21  Orbit calculator 
           22  Ideal thrust direction calculator 
           23  Sun direction calculator 
           24  Ideal attitude calculator 
           25  Actual attitude calculator 
           26  Target attitude calculator 
           27  Movement limitation determiner 
           28  Torque calculator 
           29  Sensor unit 
           31  Processor 
           32  Memory 
           33  Interface