Attitude Control System Design & Verification for CNUSAIL-1 with Solar

Home , CubeSail

Paper Int’l J. of Aeronautical & Space Sci. 17(4), 579–592 (2016) DOI: http://dx.doi.org/10.5139/IJASS.2016.17.4.579 Attitude Control System Design & Verification for CNUSAIL-1 with Solar/ Drag Sail

Yeona Yoo* Satrec Initiative, Daejeon 34054, Republic of Korea

Seungkeun Kim** and Jinyoung Suk*** Chungnam National University, Daejeon 34134, Republic of Korea

Jongrae Kim**** University of Leeds, Leeds LS2 9JT, United Kingdom

Abstract CNUSAIL-1, to be launched into low-earth orbit, is a cubesat-class satellite equipped with a 2 m × 2 m solar sail. One of CNUSAIL’s missions is to deploy its solar sail system, thereby deorbiting the satellite, at the end of the satellite’s life. This paper presents the design results of the attitude control system for CNUSAIL-1, which maintains the normal vector of the sail by a 3-axis active attitude stabilization approach. The normal vector can be aligned in two orientations: i) along the anti- nadir direction, which minimizes the aerodynamic drag during the nadir-pointing mode, or ii) along the satellite velocity vector, which maximizes the drag during the deorbiting mode. The attitude control system also includes a B-dot controller for detumbling and an eigen-axis maneuver algorithm. The actuators for the attitude control are magnetic torquers and reaction wheels. The feasibility and performance of the design are verified in high-fidelity nonlinear simulations.

Key words: Attitude control system, Cube satellite, Solar sail, Drag sail

1. Introduction Small-scale solar sail and drag sail spacecraft include Nanosail-D, Cubesail, Lightsail-1, and Deorbitsail. In recent years, interest in solar and drag sails has been Nanosail-D, launched by NASA in 2010, was operated in LEO revived. Solar sails use solar radiation pressure (SRP) or at approximately 600 km altitude. Having experienced orbital aerodynamic drag force for propulsion, which provides decay, its sail was deployed and the satellite re-entered the continuous acceleration without requiring chemical atmosphere eight months later [7]. Lightsail-1, developed by propellants. Solar sails may realize a novel spacecraft- the Planetary Society and launched in 2015 [8], successfully propelling method for long-duration missions or deep space deployed a 32 m2 sail, as confirmed by an on-board camera. explorations [1-6]. Meanwhile, drag sails are expected to The orbital decay period of Lightsail-1 was 7 days after sail resolve the debris issue by forcing the satellite to re-enter the deployment [9]. Cubesail by Surrey Space Center (SSC) atmosphere at the end of its life. Some large-sized sail projects utilized the relative change between the center of mass and have been suspended because of their high technology costs the center of solar pressure, and controlled its attitude with and failure risks. Instead, small-scaled sail studies have been magnetic torquers (MTQs) [10]. Cubesail’s missions are conducted in low Earth orbit (LEO), as they are significantly threefold: to deploy a 25 m2 solar sail, to operate solar sailing less costly to develop and verify. over one year, and to increase the aerodynamic drag force

This is an Open Access article distributed under the terms of the Creative Com- * Researcher mons Attribution Non-Commercial License (http://creativecommons.org/licenses/by- ** Associate Professor, Corresponding author: [email protected] nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduc- *** Professor tion in any medium, provided the original work is properly cited. **** Associate Professor

Received: September 14, 2015 Revised: November 23, 2016 Accepted: December 22, 2016 Copyright ⓒ The Korean Society for Aeronautical & Space Sciences 579 http://ijass.org pISSN: 2093-274x eISSN: 2093-2480

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by operating the sail during the deorbiting period. The SSC which have low slew rate limits. Therefore, torque saturation has also developed DeorbitSail, which is destined for rapid becomes problematic in large-angle slewing. Especially, deorbiting from sub-600 km altitudes using a 16 m^2 sail the CNUSAIL-1 mission requires large-angle maneuvers [11]. such as nadir and velocity-vector pointing for minimizing LEO satellites are easily tumbled by external disturbances. and maximizing the aerodynamic drag, respectively. Therefore, the attitude control system is vital for maintaining During sail deployment, CNUSAIL-1 will experience higher the communication link to a ground station and requires aerodynamic drag than normal cubic satellites. Thus, a high battery power. Cube satellite attitude control has feedback control algorithm with slew rate constraints for been implemented by various control approaches, such CNUSAIL-1, which accounts for the realistic saturation limit as magnetic attitude control [12-15], gravity gradient of the micro-reaction wheel is proposed. The reaction-wheel stabilization [16], and zero-bias momentum using reaction saturation is relieved by an MTQ manufactured in-house, wheels or control moment gyros [17]. Passive magnetic which plays a momentum dumping role. The proposed stabilization systems include permanent magnet and scheme is also applicable to future missions of cube satellites hysteresis rods [18-19], spin-control algorithms [20-21], and requiring large-angle maneuvers by a micro-reaction wheel. momentum-biased control combined with MTQs and a The remainder of this paper is organized as follows. momentum wheel. Section 2 introduces the overall and control systems of Attitude control strategies using a solar sail with gimbaled CNUSAIL-1, and the attitude dynamics. Section 3 presents thruster vector control booms, control vanes, and shifting/ the controller design for the operational modes (nadir- tilting sail panels have also been studied [22]. However, these and velocity-vector pointing). The controller performance strategies cannot be easily applied to cube satellites because is verified by nonlinear simulations in Section 4. Section 5 of the mass and volume constraints. Nanosail-D uses only a discusses the performance and limitations of the proposed permanent magnet with no active control approaches [23]. control scheme, and suggests ideas for future works. Three-axis stabilization is achieved by magnetic torque rods and a translation unit in Cubesail Small Satellite Conference, 2008. [24], and by a momentum wheel and MTQs in Lightsail-1 2. Introduction to CNUSAIL-1 system [25]. Some solar sail satellites also implement panel translation with magnetic torqueing [26] or reaction wheel/ 2.1 System description and mission purpose gravity gradient boom stabilization. The CNUSAIL-1 project, proposed by Chungnam National The LEO altitude of the CNUSAIL-1 operation was University in Korea, aims to develop and operate a 4-kg decided from the Cubesat Contest and Developing Program. cubic satellite with a small (2 m × 2 m) solar sail. The sail A relatively high orbital inclination is required. The satellite size is determined by the tradeoff between the attitude/orbit is built in a 3 U cubesat standard configuration (100 mm× dynamics and the control power of the reaction wheel and 100 mm× 300 mm), as shown in Fig. 1. Half of this capacity MTQs. CNUSAIL-1 will be operated in LEO and is targeted is occupied by the bus system, which includes an attitude for solar sail deployment, stabilized 3-axis attitude control, determination and control system, an electronic power and deorbiting at the end-of-life. system, a command and data handling system, and a The main aim of this study is to design an attitude communication system. The remaining 1.5 U are assigned to control system for CNUSAIL-1. The design includes 3-axis the payload (the solar sail deployment system and a camera stabilization using MTQs and a reaction wheel. Two system). The deployment system is divided into two parts: attitude controllers are implemented: i) a B-dot controller boom storage and deployment, and membrane storage. The for detumbling, and ii) a feedback control algorithm for housing volumes of the four quadrant membranes and four eigen-axis maneuver under slew rate constraints. The eigen- booms are 0.5 U and 0.5 U, respectively. Between the payload axis maneuver controller is used during nadir pointing and bus system, there are two cameras with opposite and deorbiting. In most attitude control schemes for cube orientations, which monitor the statuses of sail deployment satellites, a linear quadratic regulator (LQR) or proportional- and health. The satellite base (below the sail membrane) derivative (PD) controller is considered adequate for small- is installed with UHF/VHF antennas, which communicate angle slew maneuvers. However, these schemes cannot with a ground station. guarantee the large-angle slew maneuvers required for The three missions of CNUSAIL-1 are illustrated in Fig. 2. attitude pointing, because they are typically implemented The primary mission is to deploy the solar sail system and by micro-reaction wheel systems in cube-class satellites, operate the satellite in a LEO environment. The secondary

DOI: http://dx.doi.org/10.5139/IJASS.2016.17.4.579 580

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missions are to maneuver the attitude by a spacecraft and heritage must all be considered. The sensors and operation sequence and to deorbit the satellite at the end actuators were selected from commercial on-the-shelf of its operation life. The purposes of CNUSAIL-1 are to (COTS) options, which are conventionally used in small demonstrate the solar sail deployment mechanism, study satellite developments. As the attitude sensors, a sun sensor the orbit and attitude changes, and verify the feasibility of and a magnetometer are selected, and an MEMS gyroscope satellite operations with the solar and drag sail. To this end, a was chosen as the inertial sensor. The actuators are 3-axis 3-axis active attitude stabilization approach is considered, in reaction wheels and three MTQs. which the controllers align and maintain the normal vector The attitude control system operates in three modes: of the sail in the anti-nadir and velocity vector directions, detumbling, nadir-pointing maneuver, and deorbiting respectively. The former maneuver (nadir-pointing mode) maneuver. An ejection from the P-POD triggers antenna minimizes the aerodynamic drag and the latter (deorbiting deployment and the satellite enters the stabilization mode. mode) maximizes it. When the attitude control system receives the detumbling mode command, it damps the initial angular velocity from 2.2 Attitude determination and control system its tip-off rate using the MTQs; this action uses only the magnetometer measurements. In the nadir-pointing mode, The Attitude Determination & Control System (ADCS) in the aerodynamic drag is parallel to the sail plane, which CNUSAIL-1 directs and maintains the satellite orientation maximizes the solar radiation force on the satellite while at the desired attitude throughout the solar sail mission. The minimizing the drag force. In the deorbiting mode, the ADCS controls the attitude such that the normal vector of aerodynamic force is normal to the sail plane. Hence, the the solar sail directly opposes the nadir or the velocity vector. constant aerodynamic force decreases the kinetic energy ADCS utilizes a 3-axis stabilization system and must operate of the satellite, while solar radiation generates a time- autonomously without any ground command in its initial varying force. In both maneuvering modes, the attitude mode. Furthermore, the ADCS must maintain its pointing determination system provides the angular rate and attitude accuracy within 5°, its pointing stability within 0.1°/s, and its estimates using Quaternion ESTimation (QUEST) or pointing knowledge within 3°. extended Kalman filter (EKF) [27]. The ADCS comprises an attitude determination system and an attitude control system (enclosed by the dashed and solid rectangles, respectively, in Fig. 3). It includes two 3. System Dynamic modeling types of actuators, four types of sensors, and an attitude determination board. The size of the cubesat-class satellite 3.1 Coordinate frames imposes numerous constraints on the sensorsFig. and 1. Schematic actuators; of overall configuration of CNUSAIL-1 The body frame fixed at the satellite, the orbital reference size, mass, cost, operational environment, ease of handling,

The three missions of CNUSAIL-1 are illustrated in Fig. 2. The primary mission is to deploy the

solar sail system and operate the satellite in a LEO environment. The secondary missions are to

maneuver the attitude by a spacecraft operation sequence and to deorbit the satellite at the end of its

operation life. The purposes of CNUSAIL-1 are to demonstrate the solar sail deployment mechanism,

study the orbit and attitude changes, and verify the feasibility of satellite operations with the solar and

drag sail. To this end, a 3-axis active attitude stabilization approach is considered, in which the

controllers align and maintain the normal vector of the sail in the anti-nadir and velocity vector

directions, respectively. The former maneuver (nadir-pointing mode) minimizes the aerodynamic drag Fig. 1. Schematic of overall configuration of CNUSAIL-1Fig. 1. Schematic of overall configuration of CNUSAIL-1 and the latter (deorbiting mode) maximizes it.

The three missions of CNUSAIL-1 are illustrated in Fig. 2. The primary mission is to deploy the

solar sail system and operate the satellite in a LEO environment. The secondary missions are to

maneuver the attitude by a spacecraft operation sequence and to deorbit the satellite at the end of its

operation life. The purposes of CNUSAIL-1 are to demonstrate the solar sail deployment mechanism,

study the orbit and attitude changes, and verify the feasibility of satellite operations with the solar and Fig. 2. Operation modes in the CNUSAIL-1 missions Fig. 2. Operation modes in the CNUSAIL-1drag sail. To missions this end, a 3-axis active attitude stabilization approach is considered, in which the

controllers align and maintain the normal vector of the sail in the anti-nadir and velocity vector

directions, respectively. The former maneuver (nadir-pointing mode) minimizes the aerodynamic drag

581 http://ijass.org and the latter (deorbiting mode) maximizes it. 5

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Fig. 2. Operation modes in the CNUSAIL-1 missions

Fig.Fig. 4. 4. Illustrations Illustrations of of coordinate coordinate frames. frames.

3.23.2 Attitude Attitude dynamics dynamics and and kinematics kinematics

TheThe rigid rigid body body dynamics dynamics of of the the satellite satellite are are given given by by the the follo followingwing Euler Euler equation: equation:

Fig. 4. Illustrations of coordinate frames. (1) (1) � � � � � � �� where�� I is the�� moment�� of� inertia�� matrix,�� is the�� angular velocity of the body frame with respect to where3.2 Attitude I is dynamics the moment and kinematics of inertia matrix, is the angular velocity of the body frame with respect to � � � thethe The inertial inertial rigid body frame, frame, dynamics of theisis thesatellite the sum sum are ofgiven of the the �by� disturbancethe� disturbance following Euler torques torques equation: (namely (namely the the aerodynamic, aerodynamic, SRPSRP, , and and Int’l J. of Aeronautical & Space Sci. 17(4), 579–592 (2016) �� (1) gravity-gradientgravity-gradient� torques), torques), � � is is the the magnetic magnetic control control torque, torque, and and is is the the angular angular momentum momentum � �� where�� I is the�� moment�� of inertia�� matrix,� �� is the� angular velocity of the body frame with respect to �� storedstored in in the the reaction reaction wheel. wheel.� � �� frame, and the Earth-centered inertial frame are denoted storedthe inertial in the frame, reaction is wheel. the sum of the� � disturbance torques (namely the aerodynamic, SRP, and

by B,R,and E,respectively. The origin of the body frame To avoid kinematic�� singularities, the attitude between gravity-gradientToTo avoid avoid kinematic kinematictorques),� singularities, singularities, is the magnetic the the control attitude attitude torque, between betweenand two is two the coorangular coordinatedinate momentum systems systems is is expressed expressed as as a a is the satellite’s center of mass. The z axis of the B frame is two coordinate systems is expressed as a quaternion, and is �� aligned along the normal vector of the sail; the x and y axes governedquaternion,quaternion,stored in the by reaction and theand is followingis wheel. governed governed system by by the the followingof following differential system system equations: of of �differen differentialtial equations: equations: are aligned along the principal axes of inertia following the To avoid kinematic singularities, the attitude between two coordinate systems is expressed as a

right-hand rule. The orbital reference frame is centered at quaternion, and is governed by the following system of differential equations: �� (2) the satellite’s center of mass and moves along the satellite �� 00 �� (() ) � �� 00 �� orbit. The z and y axes of the R frame are directed opposite �� 22 �� 0�� 00 �� () �� to the Earth’s center and anti-normal to the orbit’s direction, �� 0 �� 0 �� 2 wherewherewhere�� the the the � bodybody ��body�� rates, rates,0 �� 0 . . respectively, and the x axis is given by the right-hand rule. � �� where�� the ��body rates,�� 0� � . The direction cosine�� matrix�� from�� the ECI to the body During nadir-pointing maneuvers, the satellite maintains its TheThe direction direction cosine cosine� ω matrixω matrix�� from from � � the the � � ECI ECI�� to to the the body body frame frame is is e expressedxpressed as as follows: follows: frame is expressedω as� � follows: �� desired orientation, in which the body frame coincides with The direction cosine matrix from the ECI to the body frame is expressed as follows: the orbit frame. In the Earth-centered inertial frame, the � �� (3) () () origin is the center of the Earth, the z axis is perpendicular �� 2�� 2�� () �� 2��2�� 2�� 2�� 2�� 2�� 3 to the Earth’s equatorial plane, the x axis points to the vernal �� 2��2�� 2��2�� 33 The direction2�� cosine� �� matrix� � from 2�� the� ECI�� to� the� orbital �� referen�� ce frame�� is calculated�� from� the�� unit The direction�� cosine�� matrix from�� the �ECI� �� to the �orbital� �� equinox, and the y axis is defined by the right-hand rule. TheThe direction direction2��2�� cosine �cosine� �� matrix matrix�� from from 2�� 2��the the� �ECI ECI�� to to� �the the�� orbital orbital �� referen ��referen��ce��ce frame frame�� is is calculated calculated from from the the unit unit vectors of the satellite’s position and velocity (denoted and , respectively): These three coordinate systems are shown in Fig. 4 reference frame is calculated from the unit vectors of vectors of the satellite’s position and velocity (denoted and , respectively): Fig.Fig. 4. 4. Illustrations Illustrations of of coordinate coordinate frames. frames.thevectors satellite’s of the satellite’sposition posandition velocity and velocity8 (denoted� (denoted� r and and v , , respectively): 3.2 Attitude dynamics and kinematics respectively): � � 88 � � 3.23.2 Attitude Attitude dynamics dynamics and and kinematics kinematics The rigid body dynamics of the satellite are given by the � (4) (4) followingTheThe rigid Euler rigid body bodyequation: dynamics dynamics of of the the satellite satellite are are given given by by the the follo followingwing Euler Euler equation: equation:� � �� (1) � (1) (1) �� wherewherewhere�� I is��I isIthe isthe the moment ��moment moment�� ofof of inertiainertia inertia�� matrix, matrix,matrix,� �� is is isthe thethe� angular angularangular velocity velocity 3.3of of 3.3the theActuator Actuatorbody body frame frame modelingmodeling with with respect respect to to � � velocity of the body frame with respect to the� inertial frame, thethe inertial inertial frame, frame, is is the the sum sum of of the � the � disturbance � disturbance torques torques (namely (namely3.3.13.3.1 the Magnetic the Magnetic aerodynamic, aerodynamic, torquers torquers SRP SRP , and, and Tdis is the sum of the disturbance torques (namely the �� MTQs generate torque as the magnetic dipole moment aerodynamic,gravity-gradientgravity-gradient SRP, torques), andtorques),� gravity-gradient is isthe the magnetic torques),magnetic control controlTmag is torque, thetorque, and and MTQs is isthe the generate angular angular torquemomentum momentum as the magnetic dipole moment interacts with the geomagnetic field. The

magnetic control torque, and hrw is the angular momentum interacts with the geomagnetic field. The MTQs of �� storedstored in in the the reaction reaction wheel. wheel. � �MTQs� of CNUSAIL-1 are designed to satisfy the size constraints of the bus system. Two torquer rods

ToTo avoid avoid kinematic kinematic singularities, singularities, the the attitude attitude between between two two coor coordinatedinateare mounted systems systems isparallel is expressed expressed to the as x as aand a y axes in the body frame. The rods are made of copper wire and a

quaternion,quaternion, and and is isgoverned governed by by the the following following system system of of differen differentialtial equations: equations:nickel–iron magnetic alloy called permalloy. The generated magnetic dipole is 0.02 at 0.5 . � An air coil (constructed from an aluminum frame wound with copper wire) is mounted Am perpendicular W

� �� 00 �� to the z axis in the body frame.(() The) air coil provides a dipole moment of 0.01 at 0.5 . � � � � �� 00 �� 22 �� 00 �� The magnetic torque generates a control torque perpendicular to the Am Earth’s magnetic W field. � � � �� where�where� � the the ��body ��body rates, rates,�� 00 . . Fig. 3. Block diagram of the Attitude Determination and Control System �� Fig. 3. Block diagram of the Attitude� � Determination and Control System � � �� ωω�� TheThe direction direction cosine cosine matrix matrix from from the the ECI ECI to to the the body body frame frame is ise xpressedexpressed�� as as follows: follows: � is calculated as () 3. System Dynamic modeling �� 5 � � � � � � �3.1� Coordinate frames where is the geomagnetic ( (field) ) vector and is the magnetic dipole generated by each rod type is �� 2��2�� 2�� 2�� The body� � frame� � fixed� � at the� �satellite, the orbital referencegiven fr ame,by [28,and the29]: Earth-centered inertial ��2�� 2�� 33 � 2�� 2�� frame are denoted� �� by � �� . The origin� of �the body frame is the satellite’s center of TheThe direction direction2��2�� cosine� cosine�� matrix �matrix�� from from 2�� 2��the the� ECI� ECI�� to to� the� �the� orbital orbital �� referen� ��referen��ce��ce frame �frame�� is� is calculated calculated from from the the unit unit (6) mass. The z axis of the B, R,B and E,frame respectivelyis aligned along the normal vector of the sail; the x and y axes are �� vectors of the satellite’s position and velocity (denoted and , respectively):� � �� (7) vectors of the satellite’s posalignedition alongand thevelocity principal (denoted axes of inertia following and , respectively):the right-hand rule. The orbital reference frame is � �� centered at the satellite’s center of mass� � and moves�� along �the satellite� �� orbit.�� The � z �� and � y ��axes�� of�� the�� R� � 8 8 In Eqs. (6) and (7), i is the current, is the number of turns, A is the area of the coil, is the

frame are directed opposite to theFig. Earth’s 4. Illustrations center and of anti-norm coordinateal frames.to the orbit’s direction, respectively, � Fig. 4. Illustrations of coordinate frames. relative permeability of the core material� of the rod, and is the demagnetizing factor. Magnetic� and the x axis is given by the right-hand rule. During nadir-pointing maneuvers, the satellite maintains 3.2 Attitude dynamics and kinematics � its desired orientation, in which the body frame coincidescoils with the impose orbit frame. saturation In the Earth-centered limits on the maximum magnetic� moment and coil current. In the present The rigid body dynamics of the satellite are given by the following Euler equation: DOI: http://dx.doi.org/10.5139/IJASS.2016.17.4.579inertial frame, the origin is the center of the Earth, the z axis is perpendicular to the Earth’s equatorial 582 system, the maximum current is (1)0.1 A, and the maximum command dipole moment is 0.2 along plane,� the x axis points to the vernal� equinox,� and the y axis is defined by the right-hand rule. These � � �� where�� I is the�� moment�� of inertia�� matrix,� �� is the� angular velocity of the body frame with respect to Am three coordinate systems are shown in Fig. 4 the x and y axes and 0.1 along the z axis. � � the inertial frame, is the sum of the� � disturbance torques (namely the aerodynamic, SRP, and During magnetic field Am measurements by the magnetometer, the MTQ should be deactivated to �� gravity-gradient torques),� is the magnetic control torque, and is the angular momentum

�� dissipate its�� residual dipole moment [30]. Thus, the MTQs are briefly turned off prior to the (579~592)15-145.indd 582 stored in the reaction wheel.� � 2016-12-27 오후 2:00:33

To avoid kinematic singularities, the attitude betweenmagnetometer two coordinate systemsreading. is expressed as a

quaternion, and is governed by the following system of differen tial equations:

�� 7 �� 0 � �� () 9 �� 0 �� 2 �� 0 �� where�� the ��body rates,�� 0 . � � �� The direction cosineω matrix� �� from � the � ECI� to the body frame is expressed as follows:

� � � � � � � � � � � � � � � � () � �� 2�� 2�� 3 2�� 2�� The direction2�� cosine� �� matrix� � from 2�� the� ECI�� to� the� orbital �� referen�� ce �frame�� is calculated from the unit

vectors of the satellite’s position and velocity (denoted and , respectively):

8 � � 3.3.2 Reaction wheel 3.3.2 Reaction wheel 3.3.2The Reaction reaction wheel comprises DC motors and a flywheel, which are assembled on the flywheel axis. The reaction wheel comprises DC motors and a flywheel, which are assembled on the flywheel axis. TheThe wheel reaction provides wheel an comprises additional DC moment motors ofand inertia a flywheel, and awhich contro arl e torque, assembled enabling on the high-precision flywheel axis. The wheel provides an additional moment of inertia and a control torque, enabling high-precision attitudeThe wheel control provides and fast an maneuvers. additional momentThe cubesat-class of inertia 3-axis and a reaction control torque,wheel installed enabling in high-precision CNUSAIL-1 attitude control and fast maneuvers. The cubesat-class 3-axis reaction wheel installed in CNUSAIL-1 wasattitude developed control andand builtfast maneuvers.by Clyde-space The cubesat-classLtd in Scotland. 3-axis The reaction wheel motors wheel installedare mounted in CNUSAIL-1 on a cube- was developed and built by Clyde-space Ltd in Scotland. The wheel motors are mounted on a cube- was developed and built by Clyde-space Ltd in Scotland. The wheel motors are mounted on a cube- � sized PCB board and aligned along the principal axes of the satellite body. Each reaction wheel � � � � (4)(4) �� sized PCB board and aligned(4) along the principal axes of the satellite body. Each reaction wheel �� −3 −3 �� producessized PCB a torque board up and to aligned0.196 × 10 alongNm the and principal stores up axes to 3.534 of the × 10 satellite body.s of angular Each momentum. reaction wheel �� −3 −3 �� produces a torque up to 0.196 × 10 Nm and stores up to 3.534 × 10 s of angular momentum. �� produces a torque up to 0.196 × 10−3 Nm and stores up to 3.534 × 10−3 s of angular momentum. � The governing differential equation of the DC motor inside the reactionNm wheel, including the back 3.3 Actuator modeling Nm 3.3 Actuator modeling The governing differential equation of the DC motor inside the reactionNm wheel, including the back electromotiveThe governing force differential (emf) voltage equation , is of the DC motor inside the reaction wheel, including the back 3.3.1 Magnetic torquers electromotive force (emf) voltage , is 3.3.1 Magnetic torquers Yeona Yoo Attitude Control Systemelectromotive Design & Verification force (emf) voltagefor CNUSAIL-1 � , is with Solar/Drag Sail V (8) � MTQs generategenerate torquetorque asas thethe magneticmagnetic dipoledipole momentmoment interactsinteracts withwith tthe geomagnetic field.�� The� V� �� V (8) V � V � R i � L �� (8) CNUSAIL-1 are designed to satisfy the size constraints of the wherewhere V is isthe the input input voltage voltage�� to to the the motor, motor, R and and L are are the resistance and inductance of the MTQs ofof CNUSAIL-1CNUSAIL-1 areare designeddesigned toto satisfysatisfy thethe sizesize constraintsconstraints ofof tthe ��bus system.in � Two� �torquer rods�� M MTQs of CNUSAIL-1 are designed to satisfy the size constraints of theVwhere bus�� V system.� � is Two the R � i input torquer� � L voltage rods to the motor, and are the resistance and inductance of the bus system. Two torquer rods are mounted parallel to the x theV resistance � V�� Randi inductance � L �� of the motor� armature, motorwhere V armature, is the respectively, input voltage and to the is motor the , motor R and current L are. The the back resistance emf voltage and inductance of the motor of the is andare y mountedaxes in the parallel body to frame. the x andandThe y rodsaxesaxes ininare thethe made bodybody of frame.frame. copper TheThe rodsrodsrespectively, areare mademade�� ofof andcoppercopper i wirewireis the andand motor aa current. The� back emf motor V armature,�� respectively,M and is the motorR � currentL . The back emf voltage of the motor is V � R L wire and a nickel–iron magnetic alloy called permalloy. The voltagecalculatedmotor of armature, the as motor respectively, is calculated and as i is the motor current. The back emf voltage of the motor is nickel–iron magnetic alloy called permalloy. The generated magnetic dipole is 0.02 at at 0.50.5 .. � generated magnetic dipole is 0.02 Am2 at 0.5 W. An air coil calculated as i � calculated as � i Am� W (9) (9) (constructedAn air coil (constructedfrom an aluminum from an aluminumframe wound frame with wound copper with copper wire) is mounted Am Am perpendicular W W � � � � �� (9) wire) is mounted perpendicular to the z axis in the body where V � K ω is the � K back�ω emf � constant, ω� and are the angular velocities of the wheel(9) to the z axis in the body frame. The air coil provides a dipole momentwhere of 0.01 Kb is the at back 0.5 emf. constant, and ωM, ωrw,and ω are the toto thethe z axisaxis inin thethe bodybody frame.frame. TheThe airair coilcoil providesprovides aa dipoledipole2 momentmoment of 0.01� � at at� 0.50.5 ..� �� frame. The air coil provides a dipole moment of 0.01 Am at where V� � K � ω is � the � K back��ω emf�� constant, ω� and are the angular velocities of the wheel angular V � velocities K�� ω � K of the�ω wheel � ω�relative to the� satellite,�� the relativewhere K to� the is thesatellite, back the emf wheel constant, motor, and and ω the, ω satellite, and ω in the are corresponding the angular velocitiesdirection, ofrespectively. the wheel 0.5 W.The magnetic torque generates a control torque perpendicular to the Am Am Earth’s magnetic W W field. The magnetic torque generates a control torque perpendicularwheel to the motor, Am KEarth’s� and magnetic the W satellite field. in the correspondingω�, ω�� direction,, and ω relative Kto� the satellite, the wheel motor, andω the�, ω satellite��, and ω in the corresponding direction, respectively. The magnetic torque generates a control torque�� Tmag respectively.Therelative torque to theoutputThe satellite, torque of the themotoroutput wheel is ofcalculated motor,the motor and by theis Newton’s calculatedsatellite Law:in theby corresponding direction, respectively. �� perpendicular to the Earth’s magnetic field.T mag is calculated as Newton’sThe torque Law: output of the motor is calculated by Newton’s Law: The torque output of the motor is calculated by Newton’s Law: �� ( ) �� is calculated asis calculated as � is calculated as (5) (() (10) ( ) I��ω� � K�ω� � T� (10 ) �� where is the viscous friction coefficient. The generated torque is the product of the armature �where�� is the geomagnetic field vector and is the magnetic dipole�� generated� by� each rod� type5 is wherewhere� B is�� the is is geomagneticthethe geomagneticgeomagnetic field fieldfield vector vectorvector and andandM is the is is magneticthethe magneticmagnetic dipoledipolewherewhereI ��generatedgeneratedω � K � K is �the isbybyω the � eacheachviscous � T viscous rodrod� typetypefriction friction 5 isis coefficient.coefficient. The generatedThe generated torque is the product of the armature10 whereI ω� � K BK� isω the � T viscous friction coefficient. The generated torque T� is the product of the armature10 dipolegiven generated by [28, 29]: by each rod type is given by [28, 29]: current and the torque coefficient of the motor : given by� [28, 29]: � torque TM is� the product of the armature current iM and the � � � current K � and the torque coefficient of the motor : T� current K� and the torque coefficient of the motor � : T (6) torque coefficienti of the motor(6)(6) KM: K ( ) � (6) � i � K � ( ) �� i� � K (11) ( ) �� T3.4 � KExternal i disturbance torques (7) 11 (7) � � � (7) (7) �� T � K i 11 � �� 3.4� External� � disturbance torques �� T3.4 � KExternali disturbance torques 11 �� In designing a LEO cube satellite with a solar sail, the disturbance torques by the orbital �In Eqs.� ��(6) and�� (7), �� i �� is �� the�� current,�� is the number of turns, A is the area of the coil, is the In Eqs. Eqs. (6) (6) and and (7), (7), i isii is isthe the the current, current, current, n is the is is the thenumber number number of turns, of of turns, turns, A 3.4 is is the the ExternalIn area area designing of of the thedisturbance coil, coil, a LEO isis cube the thetorques satellite with a solar sail, the disturbance torques by the orbital In designing a LEO cube satellite with a solar sail, the disturbance torques by the orbital environment must be considered.� The disturbance torques are sourced from the gravity gradient, A isrelative the area permeability of the coil, of μ ther is thecore relative material� permeability of the rod, and of the is the demagnetizing factor. Magnetic�� relative permeability of the core material� of the rod, and is is thethe demagnetizingdemagnetizingenvironmentIn designing must factor.factor.a LEO be MagneticMagnetic � considered.cube satellite The with disturbance a solar torquessail, the are sourced from the gravity gradient, core material of the rod, and Nd is the demagnetizing factor. aerodynamicenvironment drag, must SRP, be considered. and the residual The disturbancemagnetic dipole torques moment. are sou Inrced the fromabsence the of gravity proper gradient,control, � disturbance torques by the orbital environment must be Magneticcoils impose coils saturationimpose saturation limits on thelimits maximum on the magnetic maximum� moment� and coil current. In the present coils impose saturation limits on the maximum magnetic momentconsidered. andaerodynamic coil current. The drag, Indisturbance SRP, the present and the torques residual are magnetic sourced dipole from moment.the In the absence of proper control, magnetic moment and coil current. In the present system, theseaerodynamic disturbances drag, will SRP, change and the the residual attitude magneticof the satellite dipole or moment.its orbital In elements. the absence of proper control, system, the maximum current is 0.1 A, and the maximum command gravitydipoleipolethese moment moment gradient,disturbances isis 0.20.2aerodynamic will change along along the drag, attitude SRP, of and the satellitethe residual or its orbital elements. the maximum current is 0.1 A, and the maximum command these disturbances will �change the attitude of the satellite or its orbital elements. magnetic dipole moment.� In the absence of proper control, dipolethe x momentand y axes is and0.2 Am0.1 2 along along the xthe and z axis. y axes and 0.1 Am2 Am Am thethe x and y axesaxes andand 0.10.1 along along thethe z axis.axis. these disturbances will Am change the attitude of the satellite or � 10 along the z axis. � During magnetic field Am Am measurements by the magnetometer, theits MTQ orbital should should elements. be be deactivated deactivated to to 10 DuringDuring magneticmagnetic field field measurements measurements by the magnetometer, by the the MTQ should be deactivated to 10 magnetometer, the MTQ should be deactivated to dissipate dissipate its residual dipole moment [30]. Thus, the MTQs are are 3.4.1 briefly briefly Gravity turned turned gradient off off prior prior torque to to the the its residual dipole moment [30]. Thus, the MTQs are briefly 3.4.1 Gravity gradient torque turnedmagnetometer off prior reading.to the magnetometer reading. Gravity-gradient torques result from the Earth’s gravitational force, which varies with distance and position Gravity-gradient torques result from the Earth’s gravitational force, which varies with distance and from the Earth’s center. The gravity-gradient torques on the 3.3.2 Reaction wheel bodyposition frame from are theexpressed Earth’s ascenter. [31] The gravity-gradient torques on the body frame are expressed as [31] 9 The reaction wheel comprises DC motors and9 a flywheel, 3.3.2 Reaction wheel which are assembled on the flywheel axis. The wheel (12) () �� provides an additional moment of inertia and a control �� The reaction wheel comprises DC motors and a flywheel, which are assembledT � �� on �the �� flywheel axis. 12 14 3 2 torque, enabling high-precision attitude control and fast wherewhere μ =3.986∙10 m /s is the Earth’sis the Earth’s gravity gravity constant, constant, ro is the distance from the Earth’s The wheel provides an additional moment of inertia and a control torque, enabling high-precision�� maneuvers. The cubesat-class 3-axis reaction wheel installed is the distance from the Earth’s center, ue is the unit vector � center, � � is 3�� the unit � 1� vector m /s in the nadir direction of the fixed bodyr frame, and I is the moment of in attitudeCNUSAIL-1 control was and developed fast maneuvers. and built The bycubesat-class Clyde-space 3-axis Ltd reaction in wheel the nadir installed direction in CNUSAIL-1 of the fixed body frame, and I is the � in Scotland. The wheel motors are mounted on a cube- momentinertia matrix.of� inertia The matrix. gravity-gradi The entgravity-gradient torques are influenced torques byare the satellite’s moment of inertia. Thus, sizedwas PCB developed board and built aligned by Clyde-space along the principal Ltd in Scotland. axes of theThe wheelinfluenced motors are mountedby the satellite’s on a cube- moment of inertia. Thus, if the if the moment of inertia increases under solar sail deployment, the gradient torques will also increase. moment of inertia increases under solar sail deployment, the satellitesized PCBbody. board Each andreaction aligned wheel along produces the principal a torque axes up of to the satellite body. Each reaction wheel 0.196 × 10−3 Nm and stores up to 3.534 × 10−3 Nms of angular gradient torques will also increase. −3 −3 momentum.produces a torque up to 0.196 × 10 Nm and stores up to 3.534 × 10 s of angular momentum. 3.4.2 Aerodynamic drag The governing differential equation of the DC motor 3.4.2 Aerodynamic drag The governing differential equation of the DC motor inside the reactionNm wheel, including the back inside the reaction wheel, including the back electromotive LEOsLEOs contain contain residual residual atmosphere atmosphere fromfrom the the Earth, Earth, which which imposes a drag force on the satellite. The

forceelectromotive (emf) voltage force V b(emf), is voltage , is imposes a drag force on the satellite. The aerodynamic drag aerodynamic drag acting on the solar sail depends on the atmospheric density, and is given as � acting on the solar sail depends on the atmospheric density, V (8) and is given as (8) �� ( ) �� Vwhere � V � is the R i input � L voltage to the motor, and are the resistance and inductance of the F�� |�|�� 13 where is the aerodynamic force, is the drag coefficient, is the magnitude of the �� motor V armature, respectively, and is the motorR currentL . The back emf voltage of the motor is http://ijass.org 583 �� satelliteF velocity, is the projected sail area,� and is the velocity vector|�| in the fixed body frame. calculated as i� The atmospheric density� is calculated by the NRLMSISE-�� 00 model or an exponential model: (9) � V� � K� ω� � K��ω�� ω� ( ) where is the back emf constant, and are the angular velocities�� of� the wheel (579~592)15-145.indd 583 2016-12-27 오후 2:00:34 � � �� 14 relativeK to� the satellite, the wheel motor, andω �the, ω satellite��, and ω in the correspondingThe aerodynamic direction, respectively.torque is generated by the offset between the center of pressure and the center of

The torque output of the motor is calculated by Newton’s Law: mass . Specifically, it can be represented �� ( ) ( )

�� whereI ω� � K isω the � T viscous friction coefficient. The generated torque T is�� the� r product�� F of�� the armature10 15 Because the aerodynamic drag is influenced by the projected sail area and the satellite altitude, the � � current K and the torque coefficient of the motor : T attitude of the satellite with respect to its velocity vector is considered. � � i K ( )

� � � T3.4 � KExternali disturbance torques 11

In designing a LEO cube satellite with a solar sail, the disturbance torques by the orbital

environment must be considered. The disturbance torques are sourced from the gravity gradient, 11

aerodynamic drag, SRP, and the residual magnetic dipole moment. In the absence of proper control,

these disturbances will change the attitude of the satellite or its orbital elements.

3.4.3 Solar radiation pressure 3.4.3 Solar radiation pressure

Fig. 5. Solar radiation pressure

Fig. 5. Solar radiation pressure The SRP force is sourced from photons striking the satellite and sail surface in space. The SRP

The SRP force is sourced from photons striking the satellite and sail surface in space. The SRP The SRPacting force on is a sourcedflat sail surface from photons with the striking optical theproperties satellite of anthde sail material surface inis given space. by The [1] SRP 3.4.1 Gravity gradient torque 3.4.1 Gravity gradient3.4.1 torque Gravity gradient torque acting on a flat sail surface with the optical properties of the sail material is given by [1] 3.4.1 Gravity gradient torque acting on a flat sail surface with the optical properties of the sail material is given by [1] Gravity-gradientGravity-gradient torques torques result result from from the the Earth’s Earth’s gravitational gravitational force, force, which which varies varies with with distance distance and and ( ) Gravity-gradient torques result from the Earth’s gravitational force, which varies with distance and � � � � Gravity-gradient torques result from the Earth’s gravitational force, which varies with distance and � � �� ( ) �� ( ) � � �� 1 � �� 1�� 1�� 16 positionposition from from the the Earth’s Earth’s center. center. The The gravity-gradient gravity-gradient torques torques on on the the body body frame frame are are expressed expressed as as [31] [31] �� position from the Earth’s center. The gravity-gradient torques on the body frame are expressed as�� [31] � ( ) position from the Earth’s center. The gravity-gradient torques on the body�� frame are expressed as [31] � �� 1 � �� 1�� 1�� 16 ( ) �� () ( ) () �where � �� is 1 the �( �� SRP,) � �� and � �� is the nominal SRP constant at 1 AU from17 the �� () �� 1 � �� 17 �� TT � � �� where� �� is1 the � �� SRP,� �� and12 �12 �� is the nominal SRP constant at 1 AU from17 the T � �� where12 is the� SRP, and � � ��6� � 10 is N/mthe nominal SRP constant at 1 AU from the wherewhere �� is is the the Earth’s Earth’s gravity gravity constant, constant, is is the the distance distancesun. from from the and the Earth’s Earth’s denote the normal �� and tangential� vectors of the sail, respectively, is surface where T � � is � the �� Earth’s gravity constant, is the distance from the Earth’s�� 12 �� where is the Earth’s gravity constant, is the distance�� from the Earth’s �� 6� � 10 N/m �� sun. and denote the normal� � ��6� and � tangential 10 N/m vectors of the sail, respectively, is surface center,center, � � � �is 3��is 3��the the unit� unit 1�� 1� vector ��vector m m/s� /sin in �the the nadir nadir direction direction� of of the the fixed fixed body rbodyr fra frame,me, and and reflectivity,I Iis is the �the moment moment and� of of is the specular reflection coefficient. and are the emission coefficients� of center, � � is 3�� the unit � 1� vector m /s �in � the 3�� nadir � 1�direction m /s of the fixed bodyr frame, and I is ther� moment of center, is the unit vector in the nadir direction of the fixed body frame,� and I is� the moment of � � � reflectivity,� and� is the specular reflection coefficient. and are the emission coefficients� of � inertiainertia matrix. �matrix.� The The gravity-gradi gravity-gradientent torques torques are are influenced influenced by by the the satellite’s satellite’sreflectivity, moment moment andof of inertia. inertia. is the Thus, Thus, specular� reflection coefficient. and � � are the� �emission coefficients of inertia matrix.� The gravity-gradi�ent torques are influenced by the satellite’s moment of inertia. Thus, the front and back surfaces, respectively, and are the corresponding non-Lambertian inertia matrix.� The gravity-gradient torques are influenced by the satellite’s moment of inertia. Thus, � � the front and back� surfaces, respectively, and �� are the�� corresponding non-Lambertian if the moment of inertia increases under solar sail deployment, the gradientthe torques front will and also back� increase. surfaces, respectively, and � � are the� � corresponding non-Lambertian if the moment of inertiaif theincreases moment under of inertia solar sailincreases deployment, under solarthe gradient sail deployment, torques will the also gradient increase. torques coefficients. will also increase. The sun angle is defined as � �� if the moment of inertia increases under solar sail deployment, the gradient torques will also increase. � � coefficients. The sun angle is defined as �� coefficients. The sun angle is defined� as ( ) � 3.4.23.4.2 Aerodynamic Aerodynamic drag drag �� ( ) 3.4.2 Aerodynamic drag � � �� ( ) 18 3.4.2 Aerodynamic drag �� where� is the sun unit vector with respect to the body frame. Furthermore, the LEOsLEOs contain contain residual residual atmosphere atmosphere from from the the Earth, Earth, which which imposes imposes a adrag drag� � force ��force on on�� the� the� ��satellite. satellite. The The 18 LEOs contain residualInt’l J.LEOs ofatmosphere Aeronautical contain from residual & Space the Earth, atmosphereSci. 17(4), which 579–592 fromimposes the(2016) Earth,a drag whichforce onimposes the satellite. a drag Theforce on the satellite. The � where � �� is�� the�� sun unit vector with respect to the body frame. Furthermore, the � � �� aerodynamicaerodynamic drag drag acting acting on on the the solar solar sail sail depends depends on on the the atmosp atmosphericheric density, density, and and issun is given given unit as vectoras will� affect either the front or back surface of the solar sail, depending on the sign of aerodynamic drag acting on the solar sail depends on the atmospheric density, and is given as � �� aerodynamic drag acting on the solar sail depends on the atmospheric density, and�� is ��given�� as�� sun unit� vector� �� will�� affect�� either� the front or back surface of the solar sail, depending on the sign of (13) the sign of cos(α). Thus, . Thus, the(the sail) normalnormal vectorvector is is redefinedredefined as follows: ( ) ( ) � � ( ) � as follows:. Thus, the sail normal vector is redefined as follows: �� . Thus,�� the sail normal vector is redefined�� as follows: �� FF � �� |�|�|��|�� 1313 ( ) F � �� |�where|��where�� isis � thethe aerodynamicaerodynamic force, force, C is theis the drag drag coefficient, is13 the magnitude of the where is the aerodynamicwhereF � �� force, is � the|� | �� aerodynamic is the drag force, coefficient, isd the drag is the coefficient, magnitude �� of is the the magnitude of 13 the �� (19) where is the aerodynamic force, is the drag coefficient, �� is the magnitude of the �� ( ) coefficient, |V| is the magnitude of the satellite velocity, A �� 0� �� 0 0 1� ( 19 ) F�� |�| � F�� satellitesatellite velocity,F velocity, is � is �the the projected projected sail sail area, area,� and and |� | is is the the velocity velocity vector vector|�| in in the the fixed fixed body body frame. frame. � satellite velocity, is isthe the projected projectedF�� sailsail area,area, and is is thethe velocity�� vectorvector inin thethe fixed body��| �frame.| � �� 0� �� 0 0 1� � (20) ( 19 ) satellite velocity, is the projected sail area, and is the velocity vector in the�� fixed � �� body frame. � 0� �� 0 0 � 1� ( ) 20 � fixed body frame. � The atmospheric density ρ is calculated�� by � TheThe atmospheric atmospheric density density� is is calculated �calculated� by by the the NRLMSISE- NRLMSISE-� 0000 model model or or�� an an exponential exponential � �� model model 0� �� : : �� 0 0 � 1� 20 The atmospheric density� is calculated� by the NRLMSISE-� 00 model or an�� exponential model��: � �� 0� �� 0 0 � 1� 20 the3.4.3The NRLMSISE-00 atmospheric Solar radiation modeldensity pressure or an is calculatedexponential by model:the NRLMSISE-00 model or an exponential model: 12 3.4.3 Solar radiation pressure� � � ( ) � (21) ( ) � ( ) ( ) 12 �� (14) �� ( ) � �� ( ) � � �� 1414 0� 21 � � � �� TheThe aerodynamic aerodynamic� torque torque is is generated generated by by the the offset offset between between the the c entercenter of of pressure pressure14�� and�� and the the�� center center� �� of of The aerodynamic torque� � � is generated�� by� �the offset between the center of pressure and the center of� � � � 14 � ( ) The aerodynamic aerodynamic torque torque is generated is generated by the offset by the between offset the centerThe�� of pressureSRP �SRP�� torque � �torque�� and is isthe� calculated� ��calculated center� �� 0� ofsimilarly similarly to theto theaerodynamic aerodynamic torqu e: 21 �� massmass . Specifically,. Specifically, it itcan can be be represented represented mass . Specifically,between it can bethe represented center of pressure and the center of mass rcp. torque:The SRP� torque� is calculated� � similarly to the aerodynamic torque: 3.4.3mass Solar radiation. Specifically, pressure it can be represented �� 0� 21 ( ) Specifically,�� it can be represented �� The SRP torque is calculated similarly to the aerodynamic torque: � �� ( ) (22) ( ) � � �� (� �) �� ( ) 22 (15) 3.4.4 Residual magnetic dipole( ) moment �� ( ) T�� r��F�� 3.4.4� � �Residual�� magnetic dipole15 moment 22 T�� r�� F�� T � r �F 15 15 BecauseBecauseT�� r the the�� aerodynamic �Faerodynamic�� drag drag is is influenced influenced by by the the projected projected sai sail larea area and andThe the the residual satellite satellite magnetic altitude, altitude, dipolethe 15the moment generates a torque by interacting with the geomagnetic field. Because the aerodynamicBecause drag isthe influenced aerodynamic by the projecteddrag is influencedsail area and theby satellitethe altitude,3.4.4�� Residual � �the �� magnetic�� dipole moment 22 Because the aerodynamic drag is influenced by the projected sail area3.4.4 andThe Residualthe residual satellite magneticmagnetic altitude, dipoledipole the momentmoment generates a torque by interacting with the geomagnetic field. projectedattitudeattitude of ofsail the the areasatellite satellite and with withthe respect satelliterespect to to itsaltitude, its velocity velocity the vector vector attitude is i sconsidered. considered. of The resultant torque on the attitude of the satellite is given by attitude of the satelliteattitude with respect of the to satellite its velocity with vectorrespect i sto considered. its velocity vector is considered. TheThe residual residual magneticmagnetic dipoledipole momentmoment generates generates a torquea torque by interacting with the geomagnetic field. the satellite with respect to its velocity vector is considered. The resultant torque on the attitude of the satellite is given by Fig. 5. Solar radiation pressure by interacting with the geomagnetic field. The resultant ( ) The resultant torque on the attitude of the satellite is given by Fig. 5. Solar radiation pressure torque on the attitude of the satellite is given by ( ) 3.4.3 Solar radiation pressure �� 23 The SRP force is sourced from photons striking the satellite and sail surface in space. The SRP The The SRP SRP force force is sourced is sourced fromFig. from 5. Solarphotons photons radiation striking pressure striking the the satellite satellite and sailThe�� surface residual� �� in space.dipole The magnetic SRP moment of the satellite, (23) denoted by , was estimated as (23)

acting on a flat sail surface with the optical properties of the sail materialThe is given residual by [1] dipole magnetic moment of the satellite, denoted by , was estimated as and Thesail SRPsurface force isin sourced space. from The photons SRP strikingacting the on satellite a flat an dsail sail surface in space. The�� SRP� �� 23 � acting on a flat sail surface with the optical properties11 of the sail materialThe is given residual by by [1] PACE dipole CubeSa magnetict [32]. Themoment geomagnetic of the satellite,field vector is determined from a model such with the optical properties of11 the sail material is given11 by [1] acting on a flat sail surface with the optical properties of the sail material11 is given by [1] The�� residual� by dipolePACE CubeSa magnetict [32]. moment The geomagnetic of the satellite, field denotedvector bisy� determined, was estimated from a model as � � such � � denoted10 Am by D, was estimated as D=5×10-4Am2 by PACE � as IGRF�� or� WMM. ( ) �� (16) CubeSat 10as (IGRF Am) [32]. or WMM.byThe PACE geomagnetic CubeSa ( t) [32]. field The vector geomagnetic B is determined field vector � is determined from a model such �� 1 � �� 1�� 1�� 16 �� from a model such as IGRF or WMM. �� 1 � �� 1�� 1�� 16 ( ) � � �� 1 � �� 1�� 1�� 10as �� (IGRF Am) or WMM. 16 � (17) ( ) �� 4. Controller design �where� � �� is1 the ��1 �� SRP, �� and �� is the nominal SRP constant at 1 AU from17 the 17 where�� is the SRP, and is the nominal SRP constant at 1 AU from the � � ��1 � �� -6 � 2 4. Controller design 17 wherewhere F �� is the is theSRP, SRP, and and P=4.563×10 N/m �� is the nominal� is the nominal SRP constant at 1 AU from the sun. �srp and�� denote the normal� � ��6� and � tangential 10 N/m vectors of the sail, respectively, 4. is Controller surfaceThe attitude design control algorithms for CNUSAIL-1 are designed to stabilize the angular rate and � � � ��6� � 10�� N/m� sun. and�� denote the normal and tangential vectors of the sail,4. Controller respectively, design is surface SRP constant� at 1 AU from the sun. n and t denote the normal The attitude control algorithms for CNUSAIL-1 are designed to stabilize the angular rate and sun.reflectivity, � and and� isdenote the specular the reflection normal� � ��6�coefficient. and � tangential 10 and N/m vectorsare the emission of the coefficients� sail, respecti of vely, is surface and tangential vectors of the sail, respectively, r is surface achieve the desired attitude. There are three attitude control modes: a detumbling mode, a nadir- � � The attitude control� algorithms for CNUSAIL-1 are reflectivity, and� is the specular reflection� �coefficient.�� and areachieve theThe emission attitude the desired coefficients control attitude. algorithms of There for are CNUSAIL-1 three attitude are control designed modes: to s atabilize detumbling the angular mode, rate a nadir- and reflectivity,the front� and backand� ssurfaces, is the respectively, specular and reflection are coefficient. the corresponding e non-Lambertian � reflectivity, and is the specular reflection coefficient. andf designed arepointing the emission to maneuver stabilize coefficients themode, angular and of a rate deorbiting and achieve maneuver the desiredmode. The detumbling mode is activated after � � � �� andthecoefficients. e b front are theThe and sun emission back angle surfaces, is coefficientsdefined respectively,as � �� of the and front and back are the correspondingpointingachieve themaneuver desired non-Lambertian mode, attitude. and Therea deorbiting are three maneuver attitude mode. control The modes: detumbling a detumbling mode is mode,activated a nadir- after � �� attitude. There are three attitude control modes: a detumbling surfaces,the front respectively, and back surfaces,and B and respectively, B are the and corresponding are the correspondingseparation from non-Lambertian the P-POD and antenna deployment, and when disturbances increase the angular rate � f b � � mode, a nadir-pointing maneuver mode, and a deorbiting coefficients. The sun angle is defined as � �� separationpointing ( ) maneuver from the mode,P-POD and and a antennadeorbiting deployment, maneuver and mode. when The dist deturbancesumbling increase mode is the activated angular after rate non-Lambertian�� coefficients. The sun angleα is defined�� as� coefficients.� � �� The sun angle is defined as maneuverof the18 satellite. mode. In The this mode,detumbling the MTQs mode reduce is activated the angular after rate of the satellite until the satellite re- where is the sun� unit vector with respect to the body frame. Furthermseparationofseparationore, the the satellite. from from the Inthe this P-PODP-POD mode, and the antennaantenna MTQs deployment,reduce deployment, the angular and andwhen rate dist of urbancesthe satellite increase until thethe angularsatellite rate re- � (18) ( ) � �� stabilizes. The controller follows the B-dot control law and uses only the magnetometer and MTQ as sun unit� vector�� will�� affect�� either� the front or back surface of the solar sail, depending onwhen the sign disturbances of increase ( the) angular rate of the satellite. � of the satellite. In this mode, the MTQs reduce the angular rate of the satellite until the satellite re- � � �� T stabilizes. The controller follows18 the B-dot control law and uses only the magnetometer and MTQ as where Sb=[Sbx, Sby, Sbz] is the sun unit vector with respect to In this mode, the MTQs reduce the angular rate of the � � ��. Thus, ��the� sail� �� normal vector is redefined as follows: sensor and actuator, respectively.18 Once the sail and solar panel are deployed, the operation mode thewhere body frame. Furthermore, isthe the sun sun unit unit vector vector willwith affect respect to the stabilizes.body frame. The Furtherm controllerore, follows the the B-dot control law and uses only the magnetometer and MTQ as where � is the sun unit vector with respect to thesatellite sensorbody untilframe. and the actuator, Furtherm satellite respectively.ore, re-stabilizes. the Once The the controller sail and solarfollows pane l are deployed, the operation mode �� changes ( ) to the nadir-pointing mode. The 3-axis stabilization performed by the ACS is applied to the either the� front� � �� or�� back�� surface�� of the solar sail, depending on the B-dot control law and uses only the magnetometer and 3.4.3 Solar radiationsun unit pressure vector will affect either� the front or back surface of the solar sail, depending on the sign of � �� changessensor and to the actuator, nadir-pointing respectively. mode. Once The the3-axis sail stabilization and solar paneperformedl are deployed, by the ACS the is operation applied to mode the sun�� unit � �� vector� � �� 0� will �� affect�� 0 0 1 either� the front or back surface of the solar MTQ sail, ( 19as depending) sensor and on actuator,the sign ofrespectively. Once the sail and � feedback control logic, as proposed in [33], This control points the antenna toward the nadir for �� . Thus, � 0� the �� sail�� normal� � 0 0 �vector 1� is redefined as follows: solarchanges panel20 to are the deployed, nadir-pointing the operation mode. The mode 3-axis changes stabilization to the pe rformed by the ACS is applied to the . Thus, the sail normal vector is redefined as follows: feedback control logic, as proposed in [33], This control points the antenna toward the nadir for nadir-pointingcommunication mode. with Thea ground 3-axis station, stabilization and for testing performed the distur by bance effect without the aerodynamic �� 12 feedback control logic, as( proposed) in [33], This control points the antenna toward the nadir for �� thecommunication ACS is applied with to the a ground feedback station, control and logic, for testing as proposed the distur bance effect without the aerodynamic � drag torque. At the end of ( the) solar sail mission, the operation mode changes to the deorbiting mode. �� 0� �� 0 0 1�� in [33],communication This control with points a ground ( 19the) station,antenna and toward for testing the nadirthe distur for bance effect without the aerodynamic �� 0� �� 0 0 1� drag torque. At the end of 19the solar sail mission, the operation mode changes to the deorbiting mode. � communicationOrbit decay is handled with a byground ( a fee) dback station, control and logic for testingsimilar tothe the nadir-pointing mode. �� 0� �� 0 0 � 1�� drag torque. At the end of 20the solar sail mission, the operation mode changes to the deorbiting mode. disturbanceOrbit decay effect is handled without by a thefee dbackaerodynamic control logic drag similar torque. to tAthe nadir-pointing mode. �� 0� �� 0 0 � 1� 20 the end of the solar sail mission, the operation mode changes 12 Orbit decay is handled by a feedback control logic similar to the nadir-pointing mode. 12 Fig. 5. Solar radiation pressure to the deorbiting mode. Orbit decay is handled by a feedback 13 Fig. 5. Solar radiation pressure control logic similar to the nadir-pointing mode. The SRP force is sourced from photons striking the satellite and sail surface in space. The SRP 13 acting on a flat sail surface with the optical properties of the sail material is given by [1] 13

DOI: http://dx.doi.org/10.5139/IJASS.2016.17.4.579 ( ) �� 584 �� 1 � �� 1�� 1�� 16 ( )

�� where� �� is1 the � �� SRP,� �� and � �� is the nominal SRP constant at 1 AU from17 the �� sun. � and denote the normal� � ��6� and � tangential 10 N/m vectors of the sail, respectively, is surface

(579~592)15-145.inddreflectivity,� and� 584 is the specular reflection coefficient. and are the emission coefficients� of 2016-12-27 오후 2:00:35

� � the front and back� surfaces, respectively, and � are the� corresponding non-Lambertian

� � coefficients. The sun angle is defined as � ��

� ( ) �� 18 where is the sun unit vector with respect to the body frame. Furthermore, the � � �� sun unit� vector� �� will�� affect�� either� the front or back surface of the solar sail, depending on the sign of

. Thus, the sail normal vector is redefined as follows:

�� ( ) � �� 0� �� 0 0 1� ( 19 ) � �� 0� �� 0 0 � 1� 20

12 communicationcommunication dependsdepends onon thethe satellitesatellite attitude.attitude. ForFor thisthis reasoreason,n, thethe satellitesatellite attitudeattitude isis alignedaligned withwith

thethe orbitorbit referencereference frame.frame. InIn thisthis alignment,alignment, thethe sailsail planeplane isis perpendicularperpendicular toto thethe orbitalorbital planeplane andand thethe

aerodynamicaerodynamic drag drag is is minimized; minimized; moreover, moreover, the the antenna antenna located located at at the the base base of of the the satellite satellite points points

towardtoward the the nadir, nadir, where where it it can can communicate communicate with with the the ground ground stat station.ion. Thus, Thus, in in the the nadir-pointing nadir-pointing Yeona Yoo Attitude Control System Design & Verification for CNUSAIL-1 with Solar/Drag Sail maneuver mode, the satellite maintains the nadir pointing orientation within 5° normal to the solar sail, maneuver mode, the satellite maintains the nadir pointing orientation within 5° normal to the solar sail, which minimizes the aerodynamic drag. In the deorbiting maneuver mode, the satellite rotates 90° 4.14.1 Detumbling Detumbling mode mode with with B-dot B-dot control control 4.2which Nadir minimizes pointing/deorbiting the aerodynamic based drag. In on the eigen-axis deorbiting maneuve r mode, the satellite rotates 90° maneuvers under slew rate constraints 4.1 Detumbling mode with B-dot control from the nadir pointing attitude to align with the velocity vector normal to the solar sail. This 4.1WhenWhen Detumbling the the satellite satellite mode separates separates with B-dot from from control the the P-POD P-POD or or its its angular angular velocity fromincreases, the nadirthe ACS pointing enters attitude the to align with the velocity vector normal to the solar sail. This velocity4.1 Detumbling increases, mode the ACSwith entersB-dot control the detumbling mode as In LEO environments, solar sail missions require that the When the satellite separates from the P-POD or its angular4.1 velo Detumblingcity increases, mode the with ACS B-dot enters control the orientationorientation maximizes maximizes the the aerodynamic aerodynamic drag. drag. The The ACS ACS will will use use th thee 3-axis 3-axis reaction reaction wheel wheel for for showndetumblingWhen in Fig. the mode6.satellite In thisas separatesshown mode, in the fromFig. ACS 6the. In implementsP-POD this mode, or its the aangular B-dot ACS veloimplementscityaerodynamic increases, a B-dot the drag controller, ACS be entersminimized. which the The radio signals from the detumbling mode as shown in Fig. 6. In this mode, the ACS implementsWhen the satellitea B-dot separates controller, from which the P-POD or its angular velocity increases, the ACS enters the controller,detumblingusesWhen the ratethewhich mode satelliteof changeuses as shown separatesthe of magneticrate in fromFig.of change field6the. In P-POD measuredthis of magneticmode, or itsfrom the angular fieldtheACS mag veloimplementsnetometercityantennamaneuvering increases, adata. B-dotcannot Employingthe controller,and ACSpenetrate the entersmagnetic only which the torquesail membranes; for momentum therefore, dumping. 4.1 Detumbling mode with B-dot control measureddetumbling from mode the magnetometeras shown in Fig. data. 6. In Employing this mode, onlythe ACSthe implementssatellite–ground a B-dot controller, communication which depends on the satellite uses the rate of change of magnetic field measured from the magdetumblingnetometer mode data. as Employing shown in onlyFig. 6the. In this mode, the ACS implements a B-dot controller, which magnetometerusesmagnetometer the rate of and andchange MTQ, MTQ, of themagnetic the controller controller field dampsmeasured damps the from angular the mag velocitynetometerattitude. from data. its For initial Employing this reason, tip-off only ratethe the satellite or attitude is aligned with 4.1 Detumbling modeWhen with the B-dot satellite control separates from the P-POD or its angular velocity increases, the ACS enters the magnetometer and MTQ, the controller damps the angularvelocity uses velocity the from rate from its of initial itschange initial tip-off of tip-off magnetic rate rate or field increased or measured body from rate. the Themag netometerthe orbit data. reference Employing frame. only In the this alignment, the sail plane magnetometerincreaseduses the rate body of and changerate. MTQ, The of magneticmagnetic the controller dipolefield measured damps moment the fromgenerated angular the mag velocityalongnetometer axis from i of data. its the initial Employing B-dot tip-off controller only rate the oris When the satellitedetumbling separates mode frommagnetic as theshown P-POD dipole in Fig. or moment 6its. In angular this generated mode, velo thecity ACSalong increases, implements axis ithe of the ACSa B-dot B-dot enters controller, the is whichperpendicular to the orbital plane and the aerodynamic increased body rate. The magnetic dipole moment generatedmagnetometer along axis i of and the MTQ,B-dot thecontroller controller is damps the angular velocity from its initial tip-off rate or controllerincreasedgivenmagnetometer by is body given andrate. by MTQ, The magnetic the controller dipole damps moment the generated angular velocityalong dragaxis from iisof itsminimized; the initial B-dot tip-off controller moreover, rate oris the antenna located at the detumbling modeuses as shownthe rate in of Fig.change 6. ofIn magneticthis mode, field the measured ACS implements from the mag a netometerB-dot controller, data. Employing which only the increased body rate. The magnetic dipole moment generated along axis i of the B-dot controller is given by increased body rate. The magnetic dipole moment generated alongbase axis ofi of the the satellite B-dot controller points toward is the nadir, where it can magnetometer andgiven MTQ, by the controller damps the angular velocity from its initial(24) tip-off rate or ( ) uses the rate of change of magnetic field measured from the magnetometer data. Employing only the communicate with the ground station. Thus, in the nadir- given by given � by � ( ) increased body whererate.�where The� K ��K magneticis is a � apositive positive �dipole� � �� gain gainmoment 2� � and�� generatedM M � is is �� the the alongmagnetic magnetic axis dipole.i of dipole. the B-dot is thecontroller pointingrate of is change maneuver of the mode, body-fixed ( 24the) satellite maintains the nadir magnetometer and MTQ, the controller damps the angular velocity from its initial tip-off rate or � �� pointing orientation within ( 5° normal) to the solar sail, which �where� ��K is a positive�� gain 2� � and�� M � is �� the magneticgiven bydipole. isis� thethe rate rate �of� ofchange change of the of body-fixedthe body-fixed24 geomagnetic� ( ) �wheregeomagnetic� ��K is a fieldpositive�� along � �� gain 2�axis � and� � �i, estimatedM � is �� the bymagnetic the finite dipole. difference �� is methodthe rate asof change of the body-fixed24 increased body rate. The magneticfield dipole� along moment axis� igenerated, estimated along by axis the i offinite the B-dot difference controller is minimizes the aerodynamic drag. In the deorbiting maneuver � �� 2� �� 24 geomagnetic field along axis i, estimated by the finite difference�� where method� ��K is as � a positive�� gain 2� � and�� M � is �� the magnetic dipole. � is the rate ( of ) change of the body-fixed24 methodgeomagneticwhere Kas is a fieldpositive along gain axis and i, estimated M is the bymagnetic the finite dipole. difference �� is methodthe mode,rate as of changethe satellite of the body-fixedrotates 90° from the nadir pointing given by ( ) � � �� attitude to align with the velocity vector normal to the solar where� � ��K is �a positive��geomagnetic � �� gain 2�� and�� M� �� field � is �� the along magnetic axis i ,dipole. estimated is by the the rate finite of change difference �of�� the method body-fixed as24 geomagnetic � field along axis i, estimated ( ) by the finite difference� method as �� (25) sail. This orientation maximizes (25) the aerodynamic drag. The �� with sampling time . can also be expressed as the transfer ( function) of a continuous filter [34] geomagnetic field along axis�� i��, estimated�� by the finite difference�� method as �� 25 ACS will use the 3-axis reaction ( ) wheel for maneuvering and �� ( ) with sampling time . �can� � also �� be ��expressed �� as 2�the � �transfer� � �� function � �� of� �� a continuous �� filter [34] 24 25Fig. 7. Nadir pointing/ Deorbiting maneuver mode where K is a positive gain and withM withis thesampling sampling� magnetic�� time dipole. ��Δt.. � can iscan the also alsorate be ofexpressedbe change expressed asof thethe as transferbody-fixed the function the magneticof a continuous torque filter for [34momentum] dumping. �� ( ) (25) � � �� transfer�with � sampling �function� time of� a continuous. �� can also filter be expressed[34] as the transfer function Generally, of a continuous satellite filter maneuvers [3425] adopt a linear controller with sampling time�� . �� can also ( be )expressed as the transfer function of a continuous filter [34] geomagnetic field� along�� axis�� i, estimated�� by� the� ��finite�� difference method as 25 (26) �� with sampling timewhere . can is also the be cutoff expressed frequency.� as the transfer The resultantfunction ofcontrol a continuous torque filter bysuch [34 the] as MTQs, PD or which LQR. However, is generated these controllers are unsuitable �� (26) Generally, satellite maneuvers ( ) adopt a linear controller such ass PD PD or or LQR. LQR. However, However, these these � � �� 26 for large-angle maneuvers ( because) they tend to saturate the where is the cutoff frequency. The resultant control torque�� by�� the� � MTQs,�� which is generated ( ) 26 �� whereperpendicular�� is theto� the cutoff�� Earth’s frequency. magnetic The field, resultant is then givencontrol by torque by the ( MTQs,) which is generated � �� reaction wheel momentum or torque. Thus, in this study, the � � � � � �� controllerscontrollers areare unsuitableunsuitable forfor26 large-anglelarge-angle maneuversmaneuvers becausebecause ththeyey tendtend toto saturatesaturate thethe reactionreaction wheelwheel � � �� 26 perpendicular� to the Earth’s �ma� �gnetic field,�� is then given bywhere where�� ω c is� the is cutoff the cutoff frequency. frequency. The The resultant resultant control control torque torque 25 by the MTQs, which is generated with sampling time�� . � �can�� also��perpendicularwhere be expressed � is theto as the the cutoff Earth’s transfer frequency. ma functiongnetic The field, of a resultant continuousis then givencontrol filter by torque [34 ] bynadir the 26 MTQs,pointing which and isdeorbiting generated ( ) maneuvers are implemented where is theby cutoffthe MTQs, frequency. which The is generated resultant control perpendicular torque by the to the MTQs, Earth’s which is generated � by momentuma feedback or control torque. logic Thus, for eigen-axis in this study, maneuver the nadir under pointing and deorbiting deorbiting maneuvers maneuvers are are � perpendicular�� to the Earth’s magnetic field, is then given by �� magneticperpendicular �� field, tois thethen Earth’s given ma by gnetic ( field,) is then given by perpendicular � to the� Earth’s� �ma �gnetic � field, communicationis then given by depends on the satellite attitude. ( For )this slew reaso raten, the constraints satellite attitude [34]. is aligned(27) with implementedimplemented byby aa feedbackfeedback controlcontrol logiclogic forfor eigen-axiseigen-axis maneuvermaneuver underunder slewslew raterate constraintsconstraints [34].[34]. �� ( ) � � � � � �� the orbit reference27 frame. In this alignment, (27) the sail plane isConsidering perpendicular to the the maximum orbital plane( andslew) the rate about the eigen-axis, �� 26 ( ) 27 where is the cutoff frequency. �� The resultant control torque by the MTQs, which is generated theConsidering control torque the maximum is given as slew [33]. rat ee aboutabout thethe eigen-axis,eigen-axis, thethe concontroltrol torquetorque isis givengiven asas [33].[33]. �� aerodynamic drag is minimized; moreover, the antenna located at the base of the satellite27 points �� 27 27 �� perpendicular to the Earth’s magnetic field, is then given bytoward the nadir, where it can communicate with the ground station. Thus, in the nadir-pointing (28) (( )) �� maneuver mode, the satellite maintains the nadir ( pointing) where orien�where �tation � �� within�� 5° normal�� to the�� solar sail, 28 �� which minimizes the aerodynamic drag. In the deorbiting maneuve r mode, the satellite rotates 90° � � � � � Fig. 6. Detumbling mode 27 (29) (( ))

from the nadir pointing attitude to align with the velocity vector normal to the solar sail. This Fig. 6. Detumbling mode �� Fig. 6. Detumbling mode � � ��k �� 15 29 (30) (( )) orientation maximizes the aerodynamic drag. The ACS will use the 3-axis reaction wheel for 4.2 Nadir pointing/deorbitingFig. 6. Detumbling based modeon eigen-axisFig. 6. Detumbling maneuvers mode under slew rate constraints Fig. 6. Detumbling mode �� Fig. 6. Detumbling mode ��a� �� (31) (30) 4.2 Nadir pointing/deorbiting based on eigen-axis maneuvers under slew rate constraintsmaneuvering and the magnetic torque for momentum dumping. ( ) 4.2 In Nadir LEO pointing/deorbiting environments, solar based sail on missions eigen-axis require maneuvers that the under aerod slewynamic rate dragconstraints be minimized. The 4.2 Nadir pointing/deorbiting based on eigen-axis maneuvers under slew rate constraints �� 31 In LEO environments, solar sail missions require that the 4.2 aerod Nadirynamic pointing/deorbiting drag be minimized. based Theon eigen-axis maneuvers under slewThe rate last constraints term in Eq.(28) is a kind of nonlinear control to cancelel ω×Iωω×Iω inin Eq.(1)Eq.(1) toto linearizelinearize thethe closed-closed- radio4.2In Nadir LEO signals pointing/deorbiting environments, from the antenna solar based sail cannot on missions eigen-axis penetrate require maneuvers the that sail the membraunder aerod synamiclewnes; rate therefore, dragconstraints be minimized.satellite–ground The In LEO environments, solar sail missions require that the aerodynamic drag be minimized. The radio signals from the antenna cannot penetrate the sail membraIn LEOnes; environments, therefore, satellite–ground solar sail missions require that the aerodynamiclooploop dynamics. dynamics. drag be minimized. TheThe saturationsaturation The functionfunction isis defineddefined as:as: radioIn Fig. LEO signals 6. environments,Detumbling from the mode antenna solar sail cannot missions penetrate require the that sail the membra aerodynamicnes; therefore, drag be satellite–groundminimized. The radio signals from the antenna cannot penetrate the sail membranes; therefore, satellite–ground radio signals from the antenna cannot penetrate14 the sail membranes; therefore, satellite–ground radio signals from the antenna cannot penetrate the sail membranes; therefore, satellite–ground (( )) 4.2 Nadir pointing/deorbiting14 based on eigen-axis maneuvers under slew rate constraints 1 � �� 1 14 14 �� 14 satsat��Pq �� 1 �1 � � �� 1 32 In LEO environments, solar sail missions require that the aerodynamic drag be14 minimized. The �� The positive scalar�1 gain � �� ,, can� can �1 defineddefined as:as: radio signals from the antenna cannot penetrate the sail membranes; therefore, satellite–ground �� (( )) �� k�� 33 14 (( )) Fig. 7. Nadir pointing/ Deorbiting maneuver mode Fig. 7. Nadir pointing/ Deorbiting maneuver mode �� 34 where�� is is thethe maximummaximum slewslew rate.rate. FollowingFollowing [33],[33], itit isis assumedassumed thatthat and and 34 .. �� Generally, satellite maneuvers adopt a linear controller andsuch and � a�s are are PD thenthen or LQR. chosenchosen However, asas these � �0 ��0� � 0 http://ijass.org controllers are unsuitable for large-angle maneuvers585 because they tend to saturate the reaction wheel � �� (( )) momentum or torque. Thus, in this study, the nadir pointing �� and deorbiting maneuvers are �� 2ω ((35 )) implemented by a feedback control logic for eigen-axis maneuver under slew rate constraints [34]. �� Considering the maximum slew rate about the eigen-axis, the�where� � �con 2�� 2��trol k torqueand c isare given the as desired [33]. damping ratio and the natural frequency, respectively. The quaternion36 (579~592)15-145.indd 585 2016-12-27 오후 2:00:36 error,error, denoteddenoted byby ( ) is is defineddefined � � � �� 28 �� where q�� q�� q q�� q q��

�� ( )�� (( )) �� k�� 15 �� 29 �� 37 � � �� where� qccii areare� thethe commandcommand� � quaternionsquaternions� andand qiiareare thethe satellitesatellite attitudeattitude quaternions.quaternions. TheThe errorerror inin thethe

angularangular velocity,velocity, denoteddenoted is is defineddefined as:as: �� ω ω ω ω ω �� (( )) �� 0 � �� 0 0�� 38

4.3 Unloading of the reaction wheel momentum by the MTQ

The momentum of the reaction wheeleel increases increases under under constant constant ext externalernal disturbance. disturbance. If If the the

momentum exceeds the storage limits, the reaction wheel becomes saturatedsaturated andand unableunable toto produceproduce

thethe control control torque. torque. Hence, Hence, the the excess excess momentum momentum should should be be unload unloadeded by by MTQs.. The The momentum momentum

unloading logic is defined as [35]

16 (( ) Int’l J. of Aeronautical & Space Sci. 17(4), 579–592 (2016) ( (( ) ) � � � ( ) ��a�� 30 ��a� �� ((30) ��a� � �� (30) ��a��a� � �� ( 30()30 ) �� (31) ��The last last term term in inEq.(28) Eq.(28) is a is kind a kind of nonlinear of nonlinear control control to canc toel ω×Iω in Eq.(1) to linearize the closed-31 ��The��The last last term term in inEq.(28) Eq.(28) is isa kinda kind of ofnonlinear nonlinear control control to tocanc cancel elω×Iω ω×Iω in5. inEq.(1) NumericalEq.(1) to tolinearize linearize simulation the the closed- closed-3131 cancel��The lastω×I termω in Eq.(1)in Eq.(28) to linearize is a kind theof nonlinear closed-loop control dynamics. to cancel ω×Iω in Eq.(1) to linearize the closed-31 loop dynamics. The saturation function is defined as: Theloop saturation dynamics. function The saturation is defined function as: is defined as: looploop dynamics. dynamics. The The saturation saturation function function is isdefined defined as: as: 5.1 Simulation Conditions ( ) �� The aerodynamic drag, ((SRP,) gravity gradient disturbances, 1 � �� 1 � ��(32)� �� ) 39 1 � �� 1 where is the unloading control gain, and ( (( ) ) is the error between the current and desired wheel � � 1 �� 1 ) � �� 1 ��1 �� 1 � �� 1 �� 1 � 1 and the effect of the residual ( magnetic dipole moment were sat�Pq � � ��1 �� 1 � �� 1 � 1 32 � � � �� 1 �� 1 �� satsat�Pq�Pq�� 1 �� 1 � �� 1 � � �� 1 �� 1 1 momentum� vectors.evaluated However, in removingnumerical the simulations. stored32��32 momentum The by orbit a ma gneticparameters torque compromisesof the Thesat� Pqpositive� � scalar��1 gain � �� 1 ,, �� can� can �1 defineddefined� 1 as:as: 32 ( ) The positive scalar�1�1 gain � �� ,�� can� �1 �1 defined as: TheThe positive positive positive scalar scalar scalar�1 gain gain gain � �� , k,i,, p ��i can can can can� defined �1 defined defineddefined as: as:as: as: �� CNUSAIL-1 are listed in Table 1. The orbital propagation is The positive scalar gain �, � can defined as: power of� the satellite. Thus, the unloading logic is operated when the momentum exceeds a specified �� ) 39 � �� where is thesubjected unloading to control a J2 perturbation. gain, and (( ) The is the geomagnetic error between field the current model and desired wheel �� ) �� limit. (33) ( (( ) ) � �� is implemented in International) Geomagnetic Reference � �� ( k �� 33 k� �� momentum� vectors. However, removing the stored33�� momentum by a magnetic torque compromises the � � �� Field 2011 (IGRF-11). The (sun) location is calculated from the k k�� 33(33) k �� (34) (()33 ) Julian date. ( ( ) ��where � �� is the maximum slew rate. Following power[33], it of is the assumed satellite. that Thus, the unloading and logic34 is. operated when the momentum exceeds a specified where�� is the maximum slew rate. Following [33],5. Numerical it is assumed simulation Thethat moment andof inertia 34is represented. in the body frame. where��where��where � � �� is is is isthethe thethe maximummaximum maximummaximum slew slew slewslew rate. rate.rate.rate. Following FollowingFollowingFollowing [33], [33],[33],[33], it itisitit isisassumedis assumedassumed that thatthat and and and 3434. .. ��where � �� is the maximum slew rate. Following [33], it is assumed that �� and 34 . �� limit. The optical�� 0 properties��0� are �those 0 of metalized kapton sheet, assumed and �that�� are q then≠0 chosenand ω(0)=0. as k and c are then chosen5.1 Simulationas Conditions� �� 0 ��0� � 0 and �� are thene,0 chosen as �� 0 ��0� � 0 and and and � � are�� are are then thenthen chosen chosenchosen as asas from �which��0�0 the solar��0��0� sail � �is 0 0constructed. The satellite and and � are then chosen as � �0 ��0� � 0 � � The (35) aerodynamic drag, SRP, gravity gradient ( disturbances,) and the effect of the residual magnetic � � optical properties before and( ) after the sail deployment are � � � �� ( (() ) � �� dipole moment were evaluated in numerical simulations.( ) The orbit parameters of CNUSAIL-1 are � � 2ω� 5. Numerical simulationlisted in Tables 2 and 3Table35 3, respectively. � � 2ω� � (36) (35 ) � � 2ω� � (35 ) � � � 2ω 2ω � Table 4 and Table 5 list (35 ((35the) ) specifications of the reaction � � 2ω � 5.1listed Simulation in Table Conditions 1. The orbital propagation is subjected(35 ) to a J2 perturbation. The geomagnetic field �where � 2�� k �and c are the desired damping ratio and the natural frequency,wheel respectively. and the TheMTQs, quaternion respectively.36 The reaction wheel wherewhere� � 2��k �andk �and c arec are the the desired desired damping damping ratioratio and the naturalnatural fre quency, respectively. The quaternion36 �where �where� � 2�� 2��k andk �and c arec are the the desired desired damping damping ratio ratio and and the the natural naturalmodel fre isfrequency, implementedquency, respectively. respectively. in Interna The tionalThe quaternion quaternion Geomagnetic3636 Reference Field 2011 (IGRF-11). The sun frequency,�where � 2�� k andrespectively. c are the desired The quaterniondamping ratio error, and thedenoted naturalThe aerodynamic by fre quency,unloading drag,respectively. SRP, logic gravity Theis activated quaterniongradient36 whendisturbances, the angular and the velocity effect of of the residual magnetic error, denoted by is is defineddefined T the reaction wheel reaches 3000 rpm (corresponding to an qerror,e=[qerror,e1 denoted q denotede2 qe3] by is by defined � is is isdefined defineddefined location is calculated from the Julian date. error, denoted by � �� is defined dipole moment were evaluated in numerical simulations.−3 The orbit parameters of CNUSAIL-1 are q� � �q�� q q�� q q�� angular momentum of 1.35 × 10 Nms). � � �� q q�� q�q q�� q q q�� q q�� Table 1. Orbital parameters of CNUSAIL-1 �� q �� q q q�� listed in Table 1. The orbital propagation is( subjected) to a J2 perturbation. The geomagnetic field �� ( ) �� (37) 5.2 Detumbling mode ( ) � �� Element ( ( ) ) Description �� ( ) �� model is implemented in International Geomagnetic37 Reference Field 2011 (IGRF-11). The sun �� Semi-major axis a 37 6963.1 km �� 37 � �� 3737 �� The Inclinationdetumbling mode activates37 under98° two scenarios: the where�� qci are� ��the command�� quaternions�� and qi are the satellite attitude quaternions. The error in the where�� qci are�� the command�� quaternions�� and qi are thelocation satellite is calculated attitude quaternions. from the Julian The date. error in the where�where�� q ciq ciare �areare� ��thethethe� commandcommandcommand�� quaternionsquaternions �quaternions�� andand andqi are qthei are satellite the attitude quaternions.Eccentricity The error in the 0.01939 wherewhere� qc iq areci are the� the command command� �quaternions quaternions� and� and qi areqi are the the satellite satellite attitude attitude quaternions. quaternions. The The error error in inthe the ci i LTDN 10:30 satelliteangular attitude velocity, denotedquaternions. The error in the is is defineddefined angular as:as: Table 1. OrbitalTable parameters 1. Orbital of CNUSAIL-1 parameters of CNUSAIL-1 angularangular velocity, velocity, denoted denoted T � is is isdefined defineddefined as: as:as: Orbit Period 84.50 min velocity,angular denoted velocity, ω denotede=[ωe1 ω e2 ω� e3] is ��defined�� as: � is defined as: �� ω�� ω�� ω�� ω�� ω�� ω �� ( ) �� ω�� ω�� ω�� Element ( ) Description �� ω�ω ω�� ω ω�� ω �� ( ( ) ) � �� ω ω ω (38) Semi-major axis a ( ) 6963.1 km �� 0 � �� 0 0�� The moment of inertia is represented in the38 body frame. The optical properties are those of � � ��0 �� 0�� 38 �� 0 �0 � � � � 0� 0 0� � Inclination 3838 98° 4.3� Unloading�� of� 0 the � reaction � 0� wheel momentum by the MTQ 38 4.3 Unloading of the reaction wheel momentum by themetalized MTQ kapton sheet, Eccentricityfrom which the solar sail is constructed.0.01939 The satellite and optical properties 4.34.3 Unloading Unloading of ofthe the reaction reaction wheel wheel momentum momentum by by the the MTQ MTQ 4.3 Unloading of the reaction wheel momentum by the MTQ LTDN 10:30 4.3 UnloadingThe momentum of the of thereaction reaction wheel wheel momentum increasesbefore under by and constant after the sail ext ernaldeployme disturbance.nt are listed If in Tables the 2 and 3Table 3, respectively. The momentum of the reaction wheel increases under constant externalOrbit disturbance.Period If the 84.50 min ThetheThe momentum MTQ momentum of of the the reaction reaction wh wheeleel increases increases under under constant constant ext externalernal disturbance. disturbance. If If the the momentum exceeds the storage limits, the reaction wheel becomes saturatedsaturated andand unableunable toto produceproduce momentummomentum exceeds exceeds the the storage storage limits, limits, the the reaction reaction wheel wheel becomes becomes saturated saturatedsaturated and andand unable unableunable to totoproduce produceproduce momentumThe momentum exceeds ofthe the storage reaction limits, wheel the reaction increases wheel under becomes Tablesaturated 2. Satellite and unable propertiesTable to produce 2. Satellite properties the control torque. Hence, the excess momentum shouldThe moment be unload ofed inertia by MTQs is represented.. The The momentum momentum in the body frame. The optical properties are those of constantthethe control control external torque. torque. disturbance. Hence, Hence, the the excess If excess the momentum momentum momentum should shouldexceeds be be unload unloadeded by by MTQs MTQs. The.. The The momentum momentum momentum the control torque. Hence, the excess momentum should be unloaded Element by MTQs . The momentum Description theunloading storage logiclimits, is definedthe reaction as [35] wheel becomes saturatedmetalized kapton sheet, from which the solar sail is constructed. The satellite and optical properties unloadingunloading logic logic is isdefined defined as as[35] [35] mass 4 kg andunloading unable tologic produce is defined the controlas [35] torque. Hence, the excess Sail Area 4 m2 before and after the sail deployment are listed in Tables 2 and 3Table 3, respectively. momentum should be unloaded by MTQs. The 16momentum Moment of Inertia Pre-deployed (0.0506, 0.0506, 0.010) 1616 2 unloading logic is defined as [35] 16 (Ix, Iy, Iz) [kg m ] Post-deployed (0.3104, 0.3121, 0.5757) 17 (39) Table 3. Optical properties Table of the 2.( Satellitesolar) sail properties[37] �� Table 3. Optical properties of the solar sail [37] � � �� Element Element 39 DescriptionDescription wherewhere k is the is theunloading unloading control control gain, gain, and and Δh is is the the error error between the current and desired wheel rw Frontmass emission coefficient 4 kg 0.79 between the�� current and desired wheel momentum�� vectors. 2 momentum vectors. However, removing the stored momentum by a maBackgneticSail emission Area torque compromisescoefficient the௙ 4 m 0.55 However, removing the stored momentum by a magnetic Front non-Lambertian coefficientܤ 0.05 Moment of Inertia Pre-deployed஻ (0.0506, 0.0506, 0.010) torquepower compromises of the satellite. the Thus, power the unloading of the satellite. logic is operatedThus, the when the momentum exceeds2 a specifiedܤ (BackIx, Iy, Inon-Lambertianz) [kg m ] coefficientPost-deployed ௙ (0.3104, 0.3121,0.55 0.5757) unloading logic is operated when the momentum exceeds a Reflectivity r ݁ 0.88 limit. ௕ specified limit. Specular reflection coefficient 17 s ݁ 0.94

Table 4 and Table 5 list the specifications of the reaction wheel and the MTQs, respectively. The 5. Numerical simulation DOI: http://dx.doi.org/10.5139/IJASS.2016.17.4.579 586 5.1 Simulation Conditions reaction wheel unloading logic is activated when the angular velocity of the reaction wheel reaches −3 The aerodynamic drag, SRP, gravity gradient disturbances,3000 rpm and (corresponding the effect of theto an residual angular magnetic momentum of 1.35 × 10 Nms).

dipole moment were evaluated in numerical simulations. The orbit parameters of CNUSAIL-1Table 4. Specifications are of reaction wheel

(579~592)15-145.indd 586 2016-12-27 오후 2:00:38 listed in Table 1. The orbital propagation is subjected to a J2 perturbation.Element The geomagnetic fieldDescription Manufacturer Clyde-space Ltd model is implemented in International GeomagneticMass Reference Field 2011 (IGRF-11). The0.275 sun kg Inertia 4.5e-6 kg m2 location is calculated from the Julian date. Maximum torque 1.9 × 10-3Nm Table 1. Orbital parametersMaximum of CNUSAIL-1 Momentum 3.4 × 10-3Nms Maximum Angular rate 7000 RPM Element Description

Semi-major axis a 6963.1 km Inclination Table 5. 98°Specifications of magnetic torquer Eccentricity 0.01939 LTDN 10:30Element Description Orbit Period 84.50 min Length (x, y) 60 mm Torquer rod Radius 4 mm Dipole moment 0.2 Am2 The moment of inertia is represented in the body frame. The optical propertiesArea (z) are those of65mm x 75mm Air Coil metalized kapton sheet, from which the solar sail is constructed. The satelliteDipole and optical moment properties 0.1 Am2

before and after the sail deployment are listed in Tables 2 and 3Table 3, respectively. 5.2 Detumbling mode

The detumbling mode activates under two scenarios: the initial tip-off after the satellite separates Table 2. Satellite properties from the P-POD, and when the angular rate of the satellite increases above 1/s. The solar sail is Element Description deployed by a spiral spring torque, increasing the moment of inertia and reducing the angular rate of mass 4 kg 2 Sail Area the satellite4 m to almost zero [36]. When simulating the post-deployment detumbling mode, the initial Moment of Inertia Pre-deployed (0.0506, 0.0506, 0.010) 2 (Ix, Iy, Iz) [kg m ] Post-deployed (0.3104, 0.3121, 0.5757) 18

Table 3. Optical properties of the solar sail [37] Table 3. Optical properties of the solar sail [37] Element Description Element Yeona Yoo Attitude ControlDescription System Design & Verification for CNUSAIL-1 with Solar/Drag Sail Front emission coefficient 0.79 Front emission coefficient 0.79 Back emission coefficient 0.55 Back emission coefficientܤ ௙ 0.55 initial tip-offFront after non-Lambertian the satellite separates coefficient ܤfrom ௙ the P-POD, 0.05The detumbling gain factor K is set to 30000 and 50000 Front non-Lambertian coefficientܤ஻ 0.05 and when theBack angular non-Lambertian rate of the coefficient satelliteܤ increases஻ above in0.55 the pre- and post-deployment cases, respectively. The Back non-Lambertian coefficient݁௙ 0.55 Reflectivity r ݁௙ 0.88 1°/s. The solar Reflectivitysail is deployed r by a spiral spring௕ torque, sampling0.88 period of the magnetometer and MTQ is 2 s. The Specular reflection coefficient s ݁ ௕ 0.94 increasing the momentSpecular of reflection inertia andcoefficient reducing s the݁ angular simulated0.94 angular rates during pre- and post-deployment are

rate of the satellite to almost zero [36]. When simulating the plotted in Figs. 8 and 9, respectively. In the pre-deployment post-deploymentTable 4 and Table detumbling 5 list the specifications mode, the initialof the reactionangular wherateel and case,the MTQs the angular, respectively. rate reduces The below the desired level for the Table 4 and Table 5 list the specifications of the reaction wheel and the MTQs, respectively. The is assumed lower than the pre-deployment rate; that is, nadir-pointing maneuver after approximately three orbits. reaction wheel unloading logic is activatedT when the angular velocity of the reaction wheel reaches reactionPre-deployment: wheel unloading ω=[ 6 logic-7 3] is °/s activated when the angular velocityIn ofthe the post-deployment reaction wheel reaches case, the reaction wheels reduce Post-deployment: ω=[ 2 -7 1]T °/s −3 the angular rates to below 1°/s. However, the satellite sways 3000 rpm (corresponding to an angular momentum of 1.35 × 10 Nms).−3 3000 rpm (corresponding to an angular momentum of 1.35 × 10 Nms).periodically under the larger aerodynamic drag torque than Table 4. Specifications of reaction wheel Table 4. Specifications of reaction Tablewheel 4. Specifications of reaction wheelin the pre-deployment case. Element Description Element Description 5.3 Nadir-pointing and deorbiting maneuver modes Manufacturer Clyde-space Ltd Manufacturer Clyde-space Ltd 5.3 Nadir-pointingunder and deorbiting slew rate maneuver constraints modes under slew rate constraints Mass 0.275 kg Mass 0.275 kg 2 5.3.1 Nadir-pointing maneuver Inertia 4.5e-6 kg m 2 Inertia 4.5e-6 kg m 5.3.1 Nadir-pointing maneuver -3 Maximum torque 1.9 × 10 Nm-3 The initial error quaternion in the large-angle maneuver simulation is set to Maximum torque 1.9 × 10 Nm The initial error quaternion in the large-angle maneuver angular rate is assumed lower than the pre-deployment rate; that is, -3 Maximum Momentum 3.4 × 10 Nms-3 Maximum Momentum 3.4 × 10 Nms simulation is set to � angular rate is assumed lower than the pre-deployment rate; that is, �� Pre-deployment:Maximum Angular rate /s 7000 RPM �where� the� �� initial error in the � angular �� velocity is� zero. The unloadingT logic will be activated when the Maximum Angular rate୘ 7000 RPM qe,0=[ 0.6460 0.2876 -0.6460 0.2876 ] Pre-deployment: /s Post-deployment: ߱ൌሾ͸ െ ͹͵ሿ /s ୘ angular ratewhere of the reaction the initial wheel exceeds error 3000 in the rpm. angularThe maximu velocitym slew rate is is zero. set to 0.3The/s, and the ߱ൌሾ͸ െ ͹͵ሿ ୘ Post-deployment: ߱ൌሾʹ െ ͹ͳሿ /s The detumblingTableTable 5. 5. SpecificationsgainSpecifications factor K is set ofof tomagnetic magnetic 30000 and torquer torquer 50000 in the pre- and post-deployment cases,unloading logic will be activated when the angular rate of the Table 5. Specifications୘ of magnetic torquer initial eigen-angle is 146.5 . Ideally, an optimal attitude control would require 8.13 min to complete ߱ൌሾʹ െ ͹ͳሿ Therespectively. detumbling The gainsampling factor pe Kriod is of set the to magnetometer 30000 and 50000 and MTQ in the is pre- 2 s. andThe post-deploymentsimulated angular case ratess, reaction wheel exceeds 3000 rpm. The maximum slew rate is Element Description the maneuver. respectively. The sampling Elementperiod of the magnetometer and MTQ is 2 s.Description The simulated angular rates set to 0.3°/s, and the initial eigen-angle is 146.5°. Ideally, an during pre- and post-deployment are plottedLength in Figs.(x, y) 8 and 9, respectively. 60 mm In the pre-deployment case, Length (x, y) 60 mm Fig. 10optimal–13 and Figs.attitude 14–18 control plot the simwouldulation require results in 8.13 the pre- min and to post-deployment complete cases, duringthe angular pre- andTorquerrate post-deployment reduces rod below the are desired plottedRadius level in Figs. for 8the and nadir- 9, respectively.pointing4 mm maneuver In the pre-deployment after approximately case, Torquer rod Radius 4 mm the maneuver. 2 respectively. The nadir-pointing maneuver is completed under the slew rate constraints in both cases. the angular rate reduces below the desiredDipole level moment for the nadir- pointing 0.2 maneuver Am after2 approximately three orbits. In the post-deployment case, Dipolethe reaction moment wheels reduce the 0.2angular Am rates to below 1/s. Fig. 10–13 and Figs. 14–18 plot the simulation results in the pre- In the post-deployment case, momentum unloading is also activated. As shown in Fig. 17, the reaction three orbits. In the post-deployment case,Area the (z) reaction wheels reduce65mm the angular x 75mm rates to below 1/s. However, the Airsatellite Coil sways periodically Areaunder the(z) larger aerodynamic 65mmdrag torque x 75mm than in the pre-and post-deployment cases, respectively. The nadir-pointing Air Coil Dipole moment 0.1 Am2 speeds along the x and z axes are below 3000 rpm, so the magnetic dipole moment command is not However,deployment the case. satellite sways periodically Dipoleunder the moment larger aero dynamic drag 0.1 Am2torque than in the pre-maneuver is completed under the slew rate constraints in both issued around the y axis. deployment case. 5.2 Detumbling mode 5.2 Detumbling mode The detumbling mode activates under two scenarios: the initial tip-off after the satellite separates The detumbling mode activates under two scenarios: the initial tip-off after the satellite separates from the P-POD, and when the angular rate of the satellite increases above 1/s. The solar sail is from the P-POD, and when the angular rate of the satellite increases above 1/s. The solar sail is deployed by a spiral spring torque, increasing the moment of inertia and reducing the angular rate of deployed by a spiral spring torque, increasing the moment of inertia and reducing the angular rate of the satellite to almost zero [36]. When simulating the post-deployment detumbling mode, the initial the satellite to almost zero [36]. When simulating the post-deployment detumbling mode, the initial 18 Fig. 8. B-dot controller verification (pre-deployment)18 Fig. 8. B-dot controller verification (pre-deployment) Fig. 10. Attitude error (pre-deployment) Fig. 8. B-dot controller verification (pre-deployment) Fig. 10. Attitude error (pre-deployment)

Fig. 9. B-dot controller verification (post-deployment)

Fig. 9. B-dot controller verification19 (post-deployment) Fig. 11. Body rate error (pre-deployment) Fig. 9. B-dot controller verification (post-deployment) Fig. 11. Body rate error (pre-deployment) 19

587 http://ijass.org

(579~592)15-145.indd 587 2016-12-27 오후 2:00:38

Fig. 12. Reaction wheel Angular rate (pre-deployment)

Fig. 13. Command torque (pre-deployment)

Fig. 17 Magnetic dipole command for momentum unloading (post-deployment)

Fig. 18. Command torque (post-deployment) Fig. 14. Attitude error (post-deployment)

Int’l J. of Aeronautical & Space Sci. 17(4), 579–592 (2016) 5.3.2 Comparisons between PD control logic and eigen-axis maneuvering under slew rate constraints

This subsection compares the eigen-axis maneuvering algorithm with the PD control logic which is cases. In the post-deployment case, momentum unloading is This subsection compares the eigen-axis maneuvering also activated. As shown in Fig. 17, the reaction speeds along algorithmgiven by with the PD control logic which is given by the x and z axes are below 3000 rpm, so the magnetic dipole (40) ( ) moment command is not issued around the y axis.

� � �Fig. � ��20 plot� ��the attitude errors during a large-angle 40 5.3.2 Comparisons between PD control logic and eigen- maneuverFig. 20 plotcontrolled the attitude by the errors PD during logic aand large-angle the eigen-axis maneuver controlled by the PD logic and the Fig. 15. Body rate error (post-deployment) axis maneuveringFig. 11. Body under rate error slew (pre-deployment) rate constraints

Fig. 11. Body rate error (pre-deployment) eigen-axis algorithm (with eigen-angle = 108), respectively, in the post-deployment case. Unlike the

PD control logic, the eigen-axis maneuvering algorithm achieved the desired nadir pointing.

Fig. 16. Reaction wheel Angular rate (post-deployment) Fig. 12. Reaction wheel Angular rate (pre-deployment) Fig. 16. Reaction wheel Angular rate (post-deployment) Fig. 12. ReactionFig. 12. wheel Reaction Angular wheel Angularrate (pre-deployment) rate (pre-deployment)

Fig. 13. Command torque (pre-deployment) Fig. 17 Magnetic dipole command for momentum unloading (post-de ployment) Fig. 13. Command torque (pre-deployment) Fig. 17. Magnetic dipole command for momentum unloading (post- Fig. 13. Command torque (pre-deployment) Fig. 17deployment) Magnetic dipole command for momentum unloading (post-deployment)

21 21

Fig. 18. Command torque (post-deployment)

Fig. 14. Attitude error (post-deployment) Fig. 18. Command torque (post-deployment)

Fig. 14. Attitude error (post-deployment) 5.3.2 ComparisonsFig. 18. Command between PD torque control (post-deployment)logic and eigen-axis maneuvering under slew rate constraints Fig. 14. Attitude error (post-deployment) 5.3.2This Comparisons subsection comparesbetween PD the control eigen-axis logic maneuvering and eigen-axis algorithm maneuvering with the under PD slewcontrol rate logic constraints which is

givenThis by subsection compares the eigen-axis maneuvering algorithm with the PD control logic which is

given by ( )

� � �� (40) Fig. 20 plot the attitude errors during a large-angle maneuver controlled by the PD logic and the � � �� 40 Fig. 20 plot the attitude errors during a large-angle maneuver controlled by the PD logic and the eigen-axis algorithm (with eigen-angle = 108), respectively, in the post-deployment case. Unlike the

eigen-axisPD control algorithm logic, the (witheigen-axi eigen-angles maneuvering = 108 algorithm), respectively, achieved in the the post-deployment desired nadir pointing. case. Unlike the

Fig. 15. Body rate error (post-deployment) PD control logic, the eigen-axis maneuvering algorithm achieved the desired nadir pointing.

Fig. 15. Body rate error (post-deployment) Fig. 15. Body rate error (post-deployment) Fig. 19. Fig.Errors 19. Errorsin large-angle in large-angle maneuver maneuver controlledcontrolled by by PD PD contro controll logic logic 23

DOI: http://dx.doi.org/10.5139/IJASS.2016.17.4.579 588

Fig. 16. Reaction wheel Angular rate (post-deployment) (579~592)15-145.indd 588 2016-12-27 오후 2:00:39 Fig. 16. Reaction wheel Angular rate (post-deployment)

Fig. 20. Errors in large-angle maneuver controlled by eigen-axis maneuver logic

22 5.3.3 Deorbiting maneuver 22 The deorbiting maneuver signifies the end of the satellite operation in the solar sail mission. During

this operation, the satellite maintains the nadir-pointing position. In the simulation, the initial error

quaternion is set to � �� where the initial error � angular� �� velocities are set to � zero. ��

This case, however, compromises the condition. The control gain matrix P cannot exist

�� for a zero-valued error quaternion. Hence,q when�� , the error quaternions are redefined as

�� q � ��

24 Accordingly, the eigen-angle error in the gain calculation increases by approximately 1.15°.

Fig. 21 –25 plot the angular rate error, attitude angle error, wheel angular rate, and the control

torque, respectively. The satellite is initially aligned with the orbital reference frame. The first phase

completes the deorbiting maneuver. In the second phase, which is in perigee with the maximum

aerodynamic drag torque, the reaction wheels are saturated and the satellite fails to achieve the desired

attitude. The error in the second phase exceeds the initial error and the controller shows poor

Yeona Yoo Attitude Controlperformance, System because Design the gain& Verification was calculated usingfor CNUSAIL-1the initial error. withThe angularSolar/Drag errors are Sail 20 above

the initial angular error and the control gain is updated by the current error quaternion through Eqs.

algorithm (with eigen-angle = 108°), respectively, in the (29)–(34).5.3.4 The Life simulation time results analysis after the gain recalculation are presented in Figs. 26–30. The post-deployment case. Unlike the PD control logic, the deorbiting maneuverThe forces is now imposed partially achieved on the under sail the slewplane rate changeconstraints the and thealtitude satellite nearly eigen-axis maneuvering algorithm achieved the desired maintainsof the the attitude cube that satellite. maximizes Fig. the aerodynamic 31 plots thedrag. altitudeThe satellite decrease periodically at tumbles the and the nadir pointing. reactionsemi-major wheel saturates underaxis the under aerodynamic random drag torque. angular rate rotations, the

5.3.3 Deorbiting maneuver The deorbiting maneuver signifies the end of the satellite operation in the solar sail mission. During this operation, the satellite maintains the nadir-pointing position. In the

simulation, the initial error quaternion is set to qe,0=[-0.0000 0.7071 -0.0000 0.7071 ]T where the initial error angular velocities are set to zero.

This case, however, compromises the qe,0≠0 condition. The

control gain matrix P cannot exist for a zero-valued error Fig. 21. Attitude error (deorbiting, initial gain maintain)

quaternion. Hence, when qe,0<0.1, the error quaternions are Fig. 21. Attitude error (deorbiting, initial gain maintain) redefined as T qe,0=[ -0.1 0.7071 -0.1 0.7071] Accordingly, the eigen-angle error in the gain calculation increases by approximately 1.15°. Fig. 21 –25 plot the angular rate error, attitude angle error, wheel angular rate, and the control torque, respectively. The satellite is initially aligned with the orbital reference frame. The first phase completes the deorbiting maneuver. 25 In the second phase, which is in perigee with the maximum

Fig. 22. Body rate error (deorbiting, initial gain maintenance) aerodynamic drag torque, the reaction wheels are saturated and the satellite fails to achieve the desired attitude. The Fig. 22. BodyFig. rate 22. Body error rate (deorbiting, error (deorb iting,initial initial gain gain maintenance) maintenance) error in the second phase exceeds the initial error and the

controller shows poor performance, because the gain was Fig. 22. Body rate error (deorbiting, initial gain maintenance) calculated using the initial error. The angular errors are 20° above the initial angular error and the control gain is updated by the current error quaternion through Eqs. (29)– (34). The simulation results after the gain recalculation are presented in Figs. 26–30. The deorbiting maneuver is now

partially achieved under the slew rate constraints and the Fig. 23. Reaction wheel angular rate (deorbiting, initial gain maintenance) satellite nearly maintains the attitude that maximizes the Fig. 23. Reaction wheel angular rate (deorbiting, initial gain maintenance) aerodynamic drag. The satellite periodically tumbles and Fig. 23. Reaction wheel angular rate (deorbiting, initial gain mainte- the reaction wheel saturates under the aerodynamic drag nance) torque. Fig. 23. Reaction wheel angular rate (deorbiting, initial gain maintenance) Fig. 19. Errors in large-angle maneuver controlled by PD control logic

Fig. 24. Magnetic dipole command for momentum unloading (deorbiting, initial gain maintenance)

Fig. 20. Errors in large-angle maneuver controlled by eigen-axis maneuver logic Fig. 24. Magnetic dipole command for momentum unloading (deorbiting, initial gain maintenance) 26 Fig. 20. Errors in large-angle maneuver controlled by eigen-axis ma- Fig. 24. Magnetic dipole command for momentum unloading (deor- 26 neuver logic biting, initial gain maintenance) 5.3.3 Deorbiting maneuver

The deorbiting maneuver signifies the end of the satellite operation in the solar sail mission. During 26 this operation, the satellite maintains the nadir-pointing position. In the simulation, the initial589 error http://ijass.org quaternion is set to � �� where the initial error � angular� �� velocities are set to � zero. ��

This case, however, compromises the condition. The control gain matrix P cannot exist

�� for a zero-valued error quaternion. Hence,q when�� , the error quaternions are redefined as (579~592)15-145.indd 589 2016-12-27 오후 2:00:39 �� q � ��

Int’l J. of Aeronautical & Space Sci. 17(4), 579–592 (2016) Fig. 28. Reaction wheel angular rate (deorbiting, gain recalculation)

Fig. 28. Reaction wheel angular rate (deorbiting, gain recalculation) nadir-pointing maneuver, and the deorbiting maneuver. In the case of random angular rate rotations, the satellite orbit decays over 40 days. The orbital decay rate is increased in the deorbiting mode, occurring after approximately 24 days. Fig. 32 compares the lifetime of CNUSAIL-1 with those of existing cube satellites. The lifetime analysis of CNUSAIL-1 compares reasonably with the real data of Lightsail-1 and Nanosail-D.

Fig. 29. Magnetic dipole command for momentum unloading (deorbiting, gain recalculation) Fig. 29. Magnetic dipole command for momentum unloading (deorbiting, gain recalculation) Fig. 29. Magnetic dipole command for momentum unloading (deorbiting, gain recalculation) 5.3.4 Life time analysis

5.3.4The Life forces time imposed analysis on the sail plane change the altitude of the cube satellite. Fig. 31 plots the

altitudeThe forces decrease imposed at the on semi-major the sail plane axis change under therandom altitude angular of the rate cube rotations, satellite. the Fig. nadir-pointing 31 plots the

maneuver,altitude decrease and the atdeorbiting the semi-major maneuver. axis In the under case random of random angular angular rate rate rotations, rotations, the the nadir-pointingsatellite orbit Fig. 25. Command torque (deorbiting, initial gain maintenance) Fig. 25. Command torque (deorbiting, initial gain maintenance) decaysmaneuver, over and 40 the days. deorbiting The orbital maneuver. decay In ratethe case is increased of random in an thegular deorbiting rate rotations, mode, the occurring satellite afterorbit Fig. 25. Command torque (deorbiting, initial gain maintenance) Fig. 25. Command torque (deorbiting, initial gain maintenance) approximatelydecays over 40 24 days. days. The Fig. orbital 32 compares decay rate the lifetimeis increased of CNUSAIL in the deorbiting-1 with those mode, of occurring existing cube after

satellites.approximately The lifetime 24 days. Fig.analysis Fig. 30. 32Command of comparesCNUSAIL-1 torque the compares lifetime(deorbiting, of reasona CNUSAIL gain recalculation)bly with-1 with the real those data of of existing Lightsail-1 cube andsatellites. Nanosail-D. The lifetime In the analysisnadir-poi ofnting CNUSAIL-1 maneuver, compares the satellite reasona will operatebly with for the approximately real data of Lightsail-1 10 years, 28 similarand Nanosail-D. to cube satellites In theFig. nadir-poi without 30. Commandnting a solar maneuver, sail.torque However, (deor the satellitebiting, satel gainlites will recalculation)maintainingoperate for approximatelythe deorbiting 10mode years, or Fig. 30. Command torque (deorbiting, gain recalculation) randomsimilar torotation cube satelliteswill undergo without faster a orbitalsolar sail. decay However, than28 standard satellites cube maintaining satellites. the deorbiting mode or random rotation will undergo faster orbital decay than standard cube satellites.

Fig. 26. Attitude error (deorbiting, gain recalculation) Fig. 26. Attitude error (deorbiting, gain recalculation) Fig. 26. AttitudeFig. error 26. Attitude (deorbiting, error (deorb gainiting, recalculation) gain recalculation)

Fig. 31. Simulated lifetime analysis of CNUSAIL-1 Fig. 31. Simulated lifetime analysis of CNUSAIL-1 Fig. 31. Simulated lifetime analysis of CNUSAIL-1

Fig. 27. Body rate error (deorbiting, gain recalculation) Fig. 27. Body rate error (deorbiting, gain recalculation) Fig. 27. Body rate error (deorbiting, gain recalculation) Fig. 27. Body rate error (deorbiting, gain recalculation)

27 27 27

29 Fig. 32. Lifetimes of CNUSAIL-1 in29 random rotation, deorbiting, and nadir-pointing maneuvers. Lifetimes of real cubesats (Nano- sail-D2 and Lightsail-1) are shown for comparison. Green dots

Fig. 28. Reaction wheel angular rate (deorbiting, gain recalculation) are the estimated lifetimes of the 3U-scaled cube satellites in Fig. 28. Reaction wheel angular rate (deorbiting, gain recalculation) circular orbits.

DOI: http://dx.doi.org/10.5139/IJASS.2016.17.4.579 590

(579~592)15-145.indd 590 2016-12-27 오후 2:00:40

Fig. 29. Magnetic dipole command for momentum unloading (deorbiting, gain recalculation)

Fig. 30. Command torque (deorbiting, gain recalculation)

28 Yeona Yoo Attitude Control System Design & Verification for CNUSAIL-1 with Solar/Drag Sail

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DOI: http://dx.doi.org/10.5139/IJASS.2016.17.4.579 592

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