Attitude Control System Design & Verification for ...

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 antinadir 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 In recent years, interest in solar and drag sails has been revived. Solar sails use solar radiation pressure (SRP) or aerodynamic drag force for propulsion, which provides continuous acceleration without requiring chemical propellants. Solar sails may realize a novel spacecraftpropelling method for long-duration missions or deep space explorations [1-6]. Meanwhile, drag sails are expected to resolve the debris issue by forcing the satellite to re-enter the atmosphere at the end of its life. Some large-sized sail projects have been suspended because of their high technology costs and failure risks. Instead, small-scaled sail studies have been conducted in low Earth orbit (LEO), as they are significantly less costly to develop and verify.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/bync/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Small-scale solar sail and drag sail spacecraft include Nanosail-D, Cubesail, Lightsail-1, and Deorbitsail. Nanosail-D, launched by NASA in 2010, was operated in LEO at approximately 600 km altitude. Having experienced orbital decay, its sail was deployed and the satellite re-entered the atmosphere eight months later [7]. Lightsail-1, developed by the Planetary Society and launched in 2015 [8], successfully deployed a 32 m2 sail, as confirmed by an on-board camera. The orbital decay period of Lightsail-1 was 7 days after sail deployment [9]. Cubesail by Surrey Space Center (SSC) utilized the relative change between the center of mass and the center of solar pressure, and controlled its attitude with magnetic torquers (MTQs) [10]. Cubesail’s missions are threefold: to deploy a 25 m2 solar sail, to operate solar sailing over one year, and to increase the aerodynamic drag force

* Researcher ** Associate Professor, Corresponding author: [email protected] *** Professor **** Associate Professor

Received: September 14, 2015 Revised: November 23, 2016 Accepted: December 22, 2016 Copyright ⓒ The Korean Society for Aeronautical & Space Sciences 579

(579~592)15-145.indd 579

http://ijass.org pISSN: 2093-274x eISSN: 2093-2480

2016-12-27 오후 2:00:32

Int’l J. of Aeronautical & Space Sci. 17(4), 579–592 (2016)

which have low slew rate limits. Therefore, torque saturation becomes problematic in large-angle slewing. Especially, the CNUSAIL-1 mission requires large-angle maneuvers such as nadir and velocity-vector pointing for minimizing and maximizing the aerodynamic drag, respectively. During sail deployment, CNUSAIL-1 will experience higher aerodynamic drag than normal cubic satellites. Thus, a feedback control algorithm with slew rate constraints for CNUSAIL-1, which accounts for the realistic saturation limit of the micro-reaction wheel is proposed. The reaction-wheel saturation is relieved by an MTQ manufactured in-house, which plays a momentum dumping role. The proposed scheme is also applicable to future missions of cube satellites requiring large-angle maneuvers by a micro-reaction wheel. The remainder of this paper is organized as follows. Section 2 introduces the overall and control systems of CNUSAIL-1, and the attitude dynamics. Section 3 presents the controller design for the operational modes (nadirand velocity-vector pointing). The controller performance is verified by nonlinear simulations in Section 4. Section 5 discusses the performance and limitations of the proposed control scheme, and suggests ideas for future works.

by operating the sail during the deorbiting period. The SSC has also developed DeorbitSail, which is destined for rapid deorbiting from sub-600 km altitudes using a 16 m^2 sail [11]. LEO satellites are easily tumbled by external disturbances. Therefore, the attitude control system is vital for maintaining the communication link to a ground station and requires high battery power. Cube satellite attitude control has been implemented by various control approaches, such as magnetic attitude control [12-15], gravity gradient stabilization [16], and zero-bias momentum using reaction wheels or control moment gyros [17]. Passive magnetic stabilization systems include permanent magnet and hysteresis rods [18-19], spin-control algorithms [20-21], and momentum-biased control combined with MTQs and a momentum wheel. Attitude control strategies using a solar sail with gimbaled thruster vector control booms, control vanes, and shifting/ tilting sail panels have also been studied [22]. However, these strategies cannot be easily applied to cube satellites because of the mass and volume constraints. Nanosail-D uses only a permanent magnet with no active control approaches [23]. 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 [25]. Some solar sail satellites also implement panel translation with magnetic torqueing [26] or reaction wheel/ gravity gradient boom stabilization. The CNUSAIL-1 project, proposed by Chungnam National University in Korea, aims to develop and operate a 4-kg cubic satellite with a small (2 m × 2 m) solar sail. The sail size is determined by the tradeoff between the attitude/orbit dynamics and the control power of the reaction wheel and MTQs. CNUSAIL-1 will be operated in LEO and is targeted for solar sail deployment, stabilized 3-axis attitude control, and deorbiting at the end-of-life. The main aim of this study is to design an attitude control system for CNUSAIL-1. The design includes 3-axis stabilization using MTQs and a reaction wheel. Two attitude controllers are implemented: i) a B-dot controller for detumbling, and ii) a feedback control algorithm for eigen-axis maneuver under slew rate constraints. The eigenaxis maneuver controller is used during nadir pointing and deorbiting. In most attitude control schemes for cube satellites, a linear quadratic regulator (LQR) or proportionalderivative (PD) controller is considered adequate for smallangle slew maneuvers. However, these schemes cannot guarantee the large-angle slew maneuvers required for attitude pointing, because they are typically implemented by micro-reaction wheel systems in cube-class satellites,

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

(579~592)15-145.indd 580

2. Introduction to CNUSAIL-1 system 2.1 System description and mission purpose The LEO altitude of the CNUSAIL-1 operation was decided from the Cubesat Contest and Developing Program. A relatively high orbital inclination is required. The satellite is built in a 3 U cubesat standard configuration (100 mm× 100 mm× 300 mm), as shown in Fig. 1. Half of this capacity is occupied by the bus system, which includes an attitude determination and control system, an electronic power system, a command and data handling system, and a communication system. The remaining 1.5 U are assigned to the payload (the solar sail deployment system and a camera system). The deployment system is divided into two parts: boom storage and deployment, and membrane storage. The housing volumes of the four quadrant membranes and four booms are 0.5 U and 0.5 U, respectively. Between the payload and bus system, there are two cameras with opposite orientations, which monitor the statuses of sail deployment and health. The satellite base (below the sail membrane) is installed with UHF/VHF antennas, which communicate with a ground station. 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

580

2016-12-27 오후 2:00:32

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

and heritage must all be considered. The sensors and actuators were selected from commercial on-the-shelf (COTS) options, which are conventionally used in small satellite developments. As the attitude sensors, a sun sensor and a magnetometer are selected, and an MEMS gyroscope was chosen as the inertial sensor. The actuators are 3-axis reaction wheels and three MTQs. The attitude control system operates in three modes: detumbling, nadir-pointing maneuver, and deorbiting maneuver. An ejection from the P-POD triggers antenna deployment and the satellite enters the stabilization mode. When the attitude control system receives the detumbling mode command, it damps the initial angular velocity from its tip-off rate using the MTQs; this action uses only the magnetometer measurements. In the nadir-pointing mode, the aerodynamic drag is parallel to the sail plane, which maximizes the solar radiation force on the satellite while minimizing the drag force. In the deorbiting mode, the aerodynamic force is normal to the sail plane. Hence, the constant aerodynamic force decreases the kinetic energy of the satellite, while solar radiation generates a timevarying force. In both maneuvering modes, the attitude determination system provides the angular rate and attitude estimates using Quaternion ESTimation (QUEST) or extended Kalman filter (EKF) [27].

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 and the latter (deorbiting mode) maximizes it.

2.2 Attitude determination and control system

The Attitude Determination & Control System (ADCS) in CNUSAIL-1 directs and maintains the satellite orientation at the desired attitude throughout the solar sail mission. The ADCS controls the attitude such that the normal vector of the solar sail directly opposes the nadir or the velocity vector. ADCS utilizes a 3-axis stabilization system and must operate autonomously without any ground command in its initial mode. Furthermore, the ADCS must maintain its pointing accuracy within 5°, its pointing stability within 0.1°/s, and its pointing knowledge within 3°. 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 sensors actuators; Fig.and 1. Schematic 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-1 Fig. 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 drag sail. Tomissions this end, a 3-axis active attitude stabilization approach is considered, in which the Fig. 2. Operation modes in the CNUSAIL-1 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 and the latter (deorbiting mode) maximizes it.

(579~592)15-145.indd 581

581

http://ijass.org

5

2016-12-27 오후 2:00:33

The Therigid rigidbody bodydynamics dynamicsofofthe thesatellite satelliteare aregiven givenby bythe thefollowing followingEuler Eulerequation: equation: �� frames. Fig. 4. Illustrations of��coordinate

where IIisisthe ofofinertia where themoment moment inertiamatrix, matrix, �� isisthe theangular angularvelocity velocityofofthe thebody bodyframe framewit w 3.2 Attitude dynamics and kinematics

rigid body dynamics of the satellite are of given by disturbance the following Euler equation: the inertial frame, �� torques (namely theThe inertial frame, the sum sum of the the disturbance torques (namely the the aerodynamic aerodynami �� isis the


��

��

� ��

� ��

(1)

�

�� gravity-gradient torques), gravity-gradient torques),� �� themagnetic magneticcontrol controltorque, torque,and and �� theangular angular �� isisthe �� isisthe

where I is the moment of inertia matrix, �� is the angular velocity of the body frame with respect to

stored ininthe the reaction storedin thereaction reactionwheel. wheel. frame, and the Earth-centered inertial frame are denoted stored the inertial frame, �� 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,thethe attitude between coordinate systems is exp To singularities, attitude To avoid avoid kinematic kinematic singularities, attitude between coordinate systems is ex gravity-gradient torques), �� is the magneticthe control torque,between and �� two istwo the angular momentum 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 stored in the reaction quaternion, governed by system ofofdifferential quaternion, andis iswheel. governed bythe thefollowing system differentialequations: equations: aligned along the normal vector of the sail; the x and y axes governed byand the following system offollowing differential equations: To avoid kinematic singularities, the attitude between two coordinate systems is expressed as a are aligned along the principal axes of inertia following the �� 00 �� right-hand rule. The orbital reference frame is centered at 00 by��the is��governed of differential equations: �quaternion, �� system �� and �� following �� (2) �� 00 � �� the satellite’s center of mass and moves along the satellite � � �� 0 �� 0 � 0 � � �� 0 orbit. The z and y axes of the R frame are directed opposite � �� (2) 0 �� to the Earth’s center and anti-normal to the orbit’s direction, �� 0 �� where the body rates, where wherethe thebody bodyrates, rates, ωω�� ..�. respectively, and the x axis is given by the right-hand rule. where body rates, cosine ω�� matrix �� .from the ECI to the body The the direction During nadir-pointing maneuvers, the satellite maintains its The matrix Thedirection directioncosine cosine matrixfrom fromthe theECI ECItotothe thebody bodyframe frameisisexpressed expressedasasfollows: follows: frame expressed follows: Theisdirection cosine as matrix from the ECI to the body frame is expressed as follows: desired orientation, in which the body frame coincides with � � �� 2�� 2�� 2�� 2�� 2�� 2�� the orbit frame. In the Earth-centered inertial frame, the �� (3) �� 2�� 2�� 2�� 2�� (3) �� 2�� 2�� origin is the center of the Earth, the z axis is perpendicular 2�� 2�� 2�� 2�� 2�� 2�� to the Earth’s equatorial plane, the x axis points to the vernal The direction cosine matrix from the ECI to the orbital reference frame is calculated from the unit Thedirection direction cosine matrix from the toorbital the orbital equinox, and the y axis is defined by the right-hand rule. The cosine matrix from the totothe reference The direction cosine matrix from theECI ECIECI the orbital referenceframe frameisiscalculated calculatedfrf vectors of the satellite’s and velocity (denoted reference frame isposition calculated from the� and unit�, respectively): vectors of These three coordinate systems are shown in Fig. 4 vectors ofofthe and (denoted Fig. vectors thesatellite’s satellite’s position andvelocity velocity (denoted �� and and �,�,respectively): respectively): Fig.4.4.Illustrations Illustrationsofofcoordinate coordinateframes. frames. the satellite’s positionposition and velocity 8 (denoted r and v, respectively): 3.2 Attitude dynamics and kinematics 3.2 3.2Attitude Attitudedynamics dynamicsand andkinematics kinematics �� The rigid body dynamics of the satellite are given by the �� equation: The dynamics following Euler equation: Therigid rigidbody body dynamicsofofthe thesatellite satelliteare aregiven givenbybythe thefollowing followingEuler Euler equation: �� (1) ��

88

(4) (1)(1)

� � Actuator modeling where I is moment velocity ofof3.3 the body with where I is moment ofofinertia inertia matrix, theangular angular Actuator modeling where Ithe isthe the momentof inertiamatrix, matrix,�� isthe angular velocity3.3 the bodyframe frame withrespect respecttoto � �isis velocity of the body frame with respect to the inertial frame, 3.3.1 Magnetic torquers the the aerodynamic, SRP, theinertial inertialframe, frame,�� isisthe thesum sumofofthe thedisturbance disturbancetorques torques(namely (namely the aerodynamic, SRP,and and 3.3.1 Magnetic torquers Tdis is the sum of the��disturbance torques (namely the generate torque as the magnetic magneticdipole dipole moment aerodynamic, SRP,torques), and gravity-gradient torques), Tmag istorque, the MTQs generate torque as the moment interacts with the geomagnet gravity-gradient �� isisthe angular momentum gravity-gradient torques), ��isisthe themagnetic magneticcontrol control torque,and and�MTQs �� the angular momentum �� interacts with the geomagnetic field. The MTQs of magnetic control torque, and hrw is the angular momentum MTQs of CNUSAIL-1 are designed to satisfy the size constraints of the bus system. Two stored storedininthe thereaction reactionwheel. wheel.

are mounted parallel to theasxasaand ToToavoid systems avoidkinematic kinematicsingularities, singularities,the theattitude attitudebetween betweentwo twocoordinate coordinate systemsis isexpressed expressed a y axes in the body frame. The rods are made of coppe

nickel–iron magnetic alloy called permalloy. The generated magnetic dipole is 0.02 Am quaternion, quaternion,and andisisgoverned governedbybythe thefollowing followingsystem systemofofdifferential differentialequations: equations: �� 0 0 �� 0 �� 0 �� 0 0 �� 0 0 ��

An air coil (constructed from an aluminum frame wound with copper wire) is mounted p

(2) (2)The air coil provides a dipole moment of 0.01 Am� at 0.5 to the z axis in the body frame.

The magnetic torque generates a control torque �� perpendicular to the Earth’s m

� � where �� . �. wherethe thebody bodyrates, rates,ωω ��

Fig. 3. Block the Attitude Determination and Control System Fig. 3. Block diagram of the Attitude Determination and diagram ControlofSystem �� is calculated as

The Thedirection directioncosine cosinematrix matrixfrom fromthe theECI ECItotothe thebody bodyframe frameisisexpressed expressedasasfollows: follows: �� 3. System Dynamic modeling �� 2�� 2�� 2�� 2�� Coordinate �� field vector and � is the magnetic dipole generated by eac � �2�� (3) �� 3.1 �� frames � � � �� 2�� where �� is the geomagnetic (3) �� 2�� 2�� body frame fixed the and the Earth-centered inertial �� The 2�� at��the �� reference 2�� 2�� satellite, �� orbital ��frame, 2�� given by [28, 29]: ��

frame are denoted by B, R, and E, respectively. The origin of the body frame is the satellite’s center of � �� from �� The isiscalculated Thedirection directioncosine cosinematrix matrixfrom fromthe theECI ECItotothe theorbital orbitalreference referenceframe frame calculated fromthe theunit unit

mass. The z axis of the B frame is aligned along the normal vector of the sail; the x and y axes are � ��

� � vectors and (denoted � �and �� rule. ��The ��orbital � ��reference � � vectorsofofthe thesatellite’s satellite’sposition position andvelocity velocity and�,�,respectively): respectively): aligned along the principal(denoted axes of inertia following the right-hand ��frame ��is �

�

centered at the satellite’s center of mass and moves along the satellite orbit. The z and y axes of the R

88

In Eqs. (6) and (7), i is the current, � is the number of turns, A is the area of the c

frame are directed opposite to the Earth’s center and to the orbit’s direction, respectively, Fig. 4. Illustrations ofanti-normal coordinate frames.

Fig. 4. Illustrations of coordinate frames.

relative permeability of the core material of the rod, and �� is the demagnetizing fact 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 orbit frame. In the Earth-centered impose saturation limits on the maximum magnetic moment and coil current. In The rigid body dynamics of the satellite are given by the following Euler equation: inertial frame, the origin is the center of the Earth, the z axis is perpendicular to the Earth’s equatorial DOI: http://dx.doi.org/10.5139/IJASS.2016.17.4.579 0.1 A, and the maximum command dipole moment is 0. 582 system, the maximum current is(1) �� 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 body frame with respect to the z axis. along thevelocity x and ofy the axes and 0.1 Am three coordinate systems are shown in Fig. 4 � the inertial frame, �� is the sum of the disturbance torques (namely the aerodynamic, SRP, and

During magnetic field measurements by the magnetometer, the MTQ should be d

(579~592)15-145.indd 582

gravity-gradient torques), �� is the magnetic control torque, and �� is the angular momentum stored in the reaction wheel.

dissipate its residual dipole moment [30]. Thus, the MTQs are briefly turned off 2016-12-27 오후 2:00:33

was developed andand built by Clyde-space in Scotland. motors areEach mounte sized PCB board aligned along the Ltd principal axes ofThe thewheel satellite body. re (4) (4) sized PCB board and aligned along the principal axes of the satellite body. Each re −3 −3 sized PCB board up and along theand principal of the× 10 satellite Each m re Nm stores upaxes to 3.534 Nmsbody. of angular produces a torque to aligned 0.196 × 10 −3 −3 produces a torque up to 0.196 × 10 −3 Nm and stores up to 3.534 × 10 −3 Nms of angular m up toinside 3.534the × 10 Nmswheel, of angular m produces a torquedifferential up to 0.196equation × 10 Nm The governing of and the stores DC motor reaction includ 3.3 Actuator Actuator modeling modeling The governing differential equation of the DC motor inside the reaction wheel, includ 3.3 The governing equation electromotive forcedifferential (emf) voltage V� , isof the DC motor inside the reaction wheel, includ 3.3.1 Magnetic Magnetic torquers torquers electromotive force (emf) voltage V� , is 3.3.1 Yeona Yoo Attitude Control System Design & Verification for CNUSAIL-1 electromotive force (emf) �� voltage V� , is with Solar/Drag Sail V�� V� � R � i� � L � �� MTQs generate generate torque torque as as the the magnetic magnetic dipole dipole moment moment interacts interacts with with geomagnetic field. The � MTQs geomagnetic V�� the the � V � R i� field. � L ��The � V�� V�� R� � i� � L �� isthe theinput inputvoltage voltage to to the the motor, motor, R R � and and LL are are the resistance and induc where V �� CNUSAIL-1 are designed to satisfy the size constraints of the is where V in M MTQs of of CNUSAIL-1 CNUSAIL-1 are are designed designed to to satisfy satisfy the the size size constraints constraints of of the the bus bus system. system. Two torquer torquer rods rods MTQs Two is the input voltage to the motor, R and L are the resistance and induc where V �� bus system. Two torquer rods are mounted parallel to the x thewhere V resistance andinput inductance motor voltageand toofithethe R � armature, and L areThe theback resistance and induc �� is the respectively, ismotor, the motor current. emf voltage of motor armature, � arey mounted mounted parallel toframe. the xx and and axesare in the the body frame. The rods rods are made made of of and copper wire andmotor are parallel the axes in body are copper wire and aa and axes in the bodyto Theyyrods made of frame. copperThe respectively, i is the current. The back emf M motor armature, respectively, and i� is the motor current. The back emf voltage of motorofarmature, respectively, and i� is the motor current. The back emf voltage of calculated asmotor wire and a nickel–iron alloy called permalloy. The magnetic voltage the calculated at 0.5 W. as nickel–iron magnetic magnetic alloy called called permalloy. The generated generated dipole is is 0.02 Am��is at 0.5 W. nickel–iron magnetic alloy permalloy. The magnetic dipole 0.02 Am calculated as generated magnetic dipole is 0.02 Am2 at 0.5 W. An air coil calculated as � K �ω � ω� V K � ω (9) � � is � perpendicular � �� An air coil (constructed from an aluminum frame wound with copper wire) mounted (constructed from an aluminum frame wound copper An air coil (constructed from an aluminum framewith wound with copper wire) is mounted perpendicular V� � K � ω� � K � �ω�� ω� V� � KK � ω �back K � �ω � ω� �� constant, wire) is mounted perpendicular to the z axis in the body the , ω��ω , and ω where � is � emfemf constant, andand ωM, ω ω� are theare the angular velocities Kb is��the rw,and atback 0.5 W. to the the zz axis axis in in the the body body frame. frame. The The air air coil coil provides provides aa dipole dipole momentwhere of 0.01 Am � at 0.5 W. to 2 moment of 0.01 Am is the back emf constant, and ω , ω , and ω the angular velocities where K frame. The air coil provides a dipole moment of 0.01 Am at � � �� angular of theemf wheel relative to the, ω satellite, theare is the back constant, and arecorresponding the angular velocities where velocities Kto� the � satellite �� , and ω relative satellite, the wheel motor, andωthe in the direction, 0.5 W. perpendicular to the Earth’s magnetic field. The magnetic torque generates a control torque � ��perpendicular to the Earth’s magnetic field. The magnetic torque generates a control torque �� wheel motor, and the satellite in the corresponding direction, �� relative to the satellite, the wheel motor, and the satellite in the corresponding direction, The magnetic torque generates a control torque Tmag relative to the the wheel motor, andbyis the satelliteLaw: inby the corresponding direction, respectively. Thesatellite, torque the motor calculated The torque output of theoutput motor isofcalculated Newton’s �� is calculated as �� is calculated as � perpendicular to the Earth’s magnetic field. Tmag is calculated as The torque output of the motor is calculated by Newton’s Law: �� Newton’s Law: torque output of the motor is calculated by Newton’s Law: IThe �� ω� � K � ω� � T� �� (5) �� (5) �� (5) � I ω � (10) �� K � ω� � T� Iwhere K ω � T �� ω� � � � � K � is the viscous friction coefficient. The generated torque T� is the product of where � is geomagnetic the geomagnetic geomagnetic field vector and � is magnetic the magnetic magnetic dipole dipolewhere generated bythe each rod type type is coefficient. The generated torque T is the product of where is the field vector and is the generated by each rod is B is� field vector and M is� where is viscous friction �the �the � the viscous friction coefficient. Thegenerated generated where KBKKis is the viscous friction coefficient. The torque T� where � and � is the product of the torque coefficient of the motor K : current i � dipole generated by each rod type is given by [28, 29]: given by by [28, [28, 29]: 29]: torque TM is the product of the armature current iM�and the given current i� and the torque coefficient of the motor K � : torque coefficient i� and the torque coefficient of the motor � �� (6)KM: of the motor K � : �� Tcurrent �� K � i� (6) � (6) �� T� � K � i� � K � i�disturbance torques T� External � �� (11) 3.4 �� (7) �� (7) �� (7) �� External disturbance torques 3.4 3.4InExternal disturbance designing a LEO torques cube satellite with a solar sail, the disturbance torques by In Eqs. (6) and (7), i is the current, � is the number of turns, A is the area of the coil, is the the 3.4 External disturbance torques In Eqs. (6) and (7), i is the current, � is the number of turns, A is the area of the coil, �� is In Eqs. (6) and (7), i is the current, n is the number of turns, In designing a LEO cube satellite with a solar sail, the disturbance torques by In designing a be LEO cube satellite with a solar sail, the by environment must considered. The disturbance torques are disturbance sourced fromtorques the grav A isrelative the area of the coil, μ is the relative permeability of the r is the the demagnetizing demagnetizing factor. Magnetic permeability of of the the core core material material of of the the rod, rod, and and � �� is factor. Magnetic relative permeability In designing a LEO cube satellite with a solar sail, the environment must be considered. The disturbance torques are sourced from the grav core material of the rod, and Nd is the demagnetizing factor.� environment drag, must be considered. The disturbance torques are sourced theofgrav aerodynamic and orbital the residual magnetic dipole In the from absence pr disturbance torquesSRP, by the environment must moment. be coils impose impose saturation limits on on the the maximum magnetic moment and and coil coil current. current. In In the the present present coils limits maximum moment Magnetic coils saturation impose saturation limits on the magnetic maximum aerodynamic drag, SRP, and the residual magnetic dipole moment. In the absence of pr considered. Thedrag, disturbance are sourced dipole from the aerodynamic SRP, and torques the the absence of pr these disturbances will change theresidual attitude magnetic of the satellite ormoment. its orbitalInelements. magnetic moment and coil current. In the present system, � � along drag, SRP, and the residual system, the the maximum maximum current current is is 0.1 0.1 A, A, and and the the maximum maximum command command gravity dipole moment is 0.2 Am 0.2 Am gradient, aerodynamic alongthe attitude of the satellite or its orbital elements. system, dipole is thesemoment disturbances will �change the maximum current is 0.1 A, and the maximum command these disturbances will change theabsence attitude of satellite or its orbital elements. magnetic dipole moment. In the of the proper control, 2 2 � �� along dipole is and 0.2 Am along the xthe and y axes and 0.1 Am the axis. the xxmoment and yy axes axes and 0.1 Am along zz axis. the and 0.1 Am these disturbances will change the attitude of the satellite or 10 along the z axis. its orbital elements. During magnetic field measurements by the magnetometer, the MTQ should be deactivated deactivated to to 10 During magnetic field measurements by the magnetometer, the MTQ should be During magnetic field measurements by the 10 magnetometer, the MTQ should be deactivated to dissipate dissipate its residual dipole moment [30]. Thus, the MTQs are briefly turned off prior to the dissipate its residual dipole moment [30]. Thus, the MTQs are 3.4.1 briefly turnedgradient off priortorque to the Gravity its residual dipole moment [30]. Thus, the MTQs are briefly 3.4.1 Gravity gradient torque result from the Earth’s Gravity-gradient torques magnetometer reading. magnetometer reading. turned off prior to the magnetometer reading. gravitational force, which varies with distance and position Gravity-gradient torques result from the Earth’s gravitational force, which varies with from the Earth’s center. The gravity-gradient torques on the 3.3.2 Reaction wheel position Earth’s as center. body framefrom are the expressed [31] The gravity-gradient torques on the body frame are expre 9 The reaction wheel comprises DC motors and 9a9 flywheel, �� 3.3.2 Reaction wheel T�� (12) which are assembled on the flywheel axis. The wheel � ��

�

The reaction wheel comprises a flywheel, which are assembled on the flywheel axis. provides an additional momentDC ofmotors inertiaand and a control � � 3 �� /s Earth’s is the Earth’s constant, whereμ=3.986∙10 � � 3��14m � 1� where /s2 m is the gravitygravity constant, ro r� is the distance from torque, enabling high-precision attitude control and fast The wheel provides an additional moment of inertia and a controlistorque, enabling high-precision the distance from the Earth’s center, u is the unit vector maneuvers. The cubesat-class 3-axis reaction wheel installed center, �� is the unit vector in the nadir edirection of the fixed body frame, and I is th in the nadir direction of the fixed body frame, and I is the in attitude CNUSAIL-1 was developed and built by Clyde-space Ltd control and fast maneuvers. The cubesat-class 3-axis reaction wheel installed in CNUSAIL-1 moment of inertia The gravity-gradient torques are inertia matrix. Thematrix. gravity-gradient torques are influenced by the satellite’s moment of in Scotland. The wheel motors are mounted on a cubewas developed and built by Clyde-space Ltd in Scotland. The wheelinfluenced motors are by mounted on a cubethe satellite’s moment of inertia. Thus, if the sized PCB board and aligned along the principal axes of the if the moment of inertia increases under solar sail deployment, the gradient torques will a moment of inertia increases under solar sail deployment, the satellite body.board Each and reaction wheel produces a torque sized PCB aligned along the principal axesup oftothe satellite body. Each reaction wheel gradient torques will also increase. 0.196 × 10−3 Nm and stores up to 3.534 × 10−3 Nms of angular produces a torque up to 0.196 × 10−3 Nm and stores up to 3.534 × 10−3 Nms of angular momentum. momentum. 3.4.2 Aerodynamic drag drag 3.4.2 Aerodynamic The governing differential equation of the DC motor The governing differential equation of the DC motor inside the reaction wheel, including the back inside the reaction wheel, including the back electromotive LEOs contain fromthe theEarth, Earth,which which LEOs containresidual residualatmosphere atmosphere from imposes a drag force on the electromotive forceVb(emf) force (emf ) voltage , is voltage V� , 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 give acting on the solar sail depends on the atmospheric density, �� (8) V�� V� � R � i� � L (8) � and �� as� �|�|�� F is given ��

� �

where V�� is the input voltage to the motor, R � and L are the resistance and inductance of the �� is the aerodynamic force, �� is the drag coefficient, |�| is the magn where F http://ijass.org back emf voltage of the motor is motor armature, respectively, and i� is the motor current. The 583 �� is the velocity vector in the fixed satellite velocity, � is the projected sail area, and � calculated as The atmospheric density � is calculated by the NRLMSISE-00 model or an exponential V� � K � ω� � K � �ω�� ω� (9) ��

� � � � � �� where K � is the back emf constant, and ω� , ω�� , and ω are the angular �velocities �of the wheel

(579~592)15-145.indd 583

2016-12-27 오후 2:00:34

inertia matrix. The gravity-gradient torques are influenced by the satellite’s moment of inertia. Thus, the front and back surfaces, thetorques front will and back surfaces, respectively, respectively, and and � �� are are the the corresponding corresponding non non ififthe ofofinertia increases under sail the themoment moment inertia increases undersolar solar saildeployment, deployment, thegradient gradient torques willalso alsoincrease. increase. coefficients. The sun angle � is defined as if the moment of inertia under solar sail deployment, the gradient torques will increase. if increases the moment of inertia increases under solar sail deployment, thealso gradient torques will also increase. coefficients. coefficients. The The sun sun angle angle � � is is defined defined as as � � �� 3.4.2 � 3.4.2Aerodynamic Aerodynamicdrag drag �� 3.4.2 Aerodynamic drag 3.4.2 Aerodynamic drag � � �� is the sun unit vector with respect to the body fr where �satellite. � � �� LEOs contain residual atmosphere from the Earth, which imposes a adrag force on the satellite. The LEOs contain residual atmosphere from the Earth, which imposes drag force on the The � LEOs contain residual atmosphere the Earth, which imposes a drag which force on the satellite. Int’l J.LEOs of Aeronautical & Space Sci. 17(4), 579–592 (2016) containfrom residual atmosphere from the Earth, imposes a where drag The force on the �satellite. �� The �� is the sun unit vector with respect to the body frame. Furt � �� where � �� is the sun unit vector with respect to the body frame. Furt sun unit vector will affect either the front or back surface of the solar sail, de aerodynamic drag acting on the solar sail depends on the atmospheric density, and is given as aerodynamic drag on the solar sail depends on theand atmospheric and is given as aerodynamic drag acting on the solar sailacting depends atmospheric density, is given asdensity, aerodynamic drag acting on on thethe solar sail depends on the atmospheric density, and is given asaffect either the front or back surface of the solar sail, depending o sun unit vector will sun unit vector will affect either the front or back surface of the solar sail, depending o �� is redefined redefined as follows: ��.Thus, Thus,the sail normal vector vector �� is the sign of cos(α). sail � (the 13 ) ) normal F�� (13) �� |�|�� |�|�� ( 13 F�� ( 13 ) F�� |�|�� ( 13 ) F�� |�|�� is redefined ��. Thus, the sail normal vector � � as ��. follows:Thus, the sail normal vector �� is redefined as as follows: follows: � �� 0� �� 0 0 1�� |�| where is the aerodynamic force, C is the drag where F is the aerodynamic force, � is the drag coefficient, is the magnitude of the |�| where F is the aerodynamic force, � is the drag coefficient, is the magnitude of the � � d |�| is the magnitude of the �� is the aerodynamic where F coefficient, (19) � 0 0 1�� is the dragforce, �� force, |�| is �� the where F is the the �magnitude aerodynamic �� is the drag coefficient, magnitude of the � � 0 0 1� �� 0� 0� �� coefficient, |V| is of the satellite velocity, A �� 0� �� 0 0 � 1�� isisthe velocity vector ininthe body frame. satellite velocity, � area, isisthe projected sailvelocity area, �� velocity vector thefixed fixed body frame. satellite velocity, the projected area,and and � �� is sail and vector inthe the �� is the thesail velocity vector in the fixed body frame. satellite velocity, �is isthe theprojected projected sail� and � (20) �� is � 0 0 � �� 0� 0�body �� 0 0 � 1� 1�� the velocity vector in�� the fixed frame. satellite velocity, �area, is the projected sail area, and � fixed body frame. The atmospheric density ρ is calculated by The density ��isiscalculated the model ororananexponential Theatmospheric density calculatedbyby theNRLMSISE-00 NRLMSISE-00 model exponentialmodel: model: The atmospheric density �atmospheric is calculated bypressure theor model or an exponential model: � Solar radiation The atmospheric density �NRLMSISE-00 is calculated by the NRLMSISE-00 model or an exponential model: the3.4.3 NRLMSISE-00 model an exponential model: 12 �� 3.4.3 Solar radiation �� pressure � (21) �� ( 14 ) � �� 0� �� ( 14 ) 12 �� ( 14 ) � � �� 12 � � (14) � � � �� ) � � � � � � �� ( 14 0� � � � � � � �� of �� and ��center �� the �� The torque isisgenerated by the offset between the center ofofpressure Theaerodynamic aerodynamic generated offsetof between pressure the of �center �� The aerodynamic torque generated bytorque the offset betweenby thethe center pressurethe andcenter the center of�� and � of � pressure �SRP 0� to to Theisaerodynamic torque is generated by the offset between and center of The aerodynamic torque is generated by the offsetthe centerThe isisthe calculated similarly thethe aerodynamic torque: The SRP torque calculated similarly aerodynamic � torque � � � �� . Specifically, it can be represented mass � . Specifically, it can be represented mass � The SRP torque is calculated similarly to the aerodynamic torque: �� between the center of pressure and the center of mass r . �� torque: cp it can beradiation mass �� . Specifically,3.4.3 Solar pressure it can be represented . represented Specifically, mass �� Specifically, it can be represented The SRP torque is calculated similarly to the aerodynamic torque: � (22) �� TT�� r �� ( 15 r��F�� F�� ( 15) ) T�� r�� F�� ( 15 ) 3.4.4 Residual magnetic dipole T�� r�� F�� ( 15 ) moment (15) �� Residual magnetic dipole moment 3.4.4 The residual magnetic Because the aerodynamic drag is influenced by the projected sail area and the satellite altitude, the Because the aerodynamic drag is influenced by the projected sail area and the satellite altitude,dipole the moment generates a torque by interacting with the geom Because the aerodynamic dragthe isthe influenced by drag the projected sail area andprojected the the satellite the Because aerodynamic drag is influenced by 3.4.4 Residual magnetic dipole moment 3.4.4 magnetic dipole moment Because aerodynamic is influenced by the sail altitude, area and the satellite altitude, the TheResidual residual magnetic dipole moment generates a torque by interacting with the geom projected sail area and the satellite altitude, the attitude of The resultant torque on the attitude of the satellite is given by attitude ofofthe with itsitsvelocity thesatellite satellite withrespect respectto velocityvector vectorisisconsidered. considered. attitude of the satelliteattitude with respect its velocity vector isto residual magnetic dipole moment generates atorque torque The residual magnetic dipole moment ofwith thetosatellite toconsidered. its velocity vector is considered. The theattitude satellite respectwith to itsrespect velocity vector is considered. The resultant torque on the attitude of thegenerates satellite isagiven byby interacting with the geom by �interacting with the geomagnetic field. The resultant Fig. 5. Solar radiation pressure � � � � �� The resultant torque on the attitude of the satellite is given by Fig. 5. Solar radiation pressure torque the of the satellite is given by �� on � � � attitude � 3.4.3 Solar radiation pressure The SRP force is sourced from photons striking the satellite and sail surface in space. The SRP Fig. 5. Solar radiation pressure The residual dipole The magnetic denoted by�, was estimated �� in� space. The SRP force is sourced from photons striking satellite and sail surface SRP moment of the satellite, (23) The SRP force is sourced from photons striking thethe satellite The residual dipole magnetic moment of the satellite, denoted by�, was estimated acting on a flat sail surface with the optical properties of the sail material is given by [1] force isinsourced from photons surface in space. The �� SRP and The sailSRP surface space. The SRP striking actingthe onsatellite a flat and sailsail surface � Am by[1]PACE CubeSat [32].moment The geomagnetic field vector � is determined from 10 acting on a flat sail surface with the optical properties of the sail material is given by 11 The residual dipole magnetic of the satellite, 11 by [1] with theonoptical the properties sail material is material given The�� Am residual magnetic[32]. moment of the satellite, by�, was estimated 11 � acting a flat sail properties surface with theof optical of the sail by [1] by dipole PACE CubeSat The geomagnetic field denoted vector � is determined from 10 11 ��is��given �� , was estimated as D=5×10-4Am2 by PACE denoted byor D �1 �� 1 � �� 1 � �� ( 16 ) as IGRF WMM. � �� 1 �CubeSat �� 1 � �1 �� (16) � �� ( 16 ) [32]. by CubeSat geomagnetic field vector � is determined from 10� ThePACE geomagnetic fieldThe vector B is determined �� 1 � � ��1 �� 1 � � �� (IGRF 16 Am ) [32]. �� as or WMM. � ��

�

�

from a model such as IGRF( 17 or )WMM. � ��1 � �� as(IGRF or WMM. 17 ) �� 1 � �� 1 � �� ( 17 ) �� (17) �� 4. Controller design �� nominal �SRP constant at 1 AU from the where � is the SRP, and � � ��6� � 10 N/m is the where �� is the SRP, and � � ��6� � 10 N/m is the nominal SRP constant at 1 AU from the 4. Controller design -6 � 102�� N/m� is the nominal SRP constant at 1 AU from the whereFsrp�� is the SRP, � � ��6� where is the SRP, andand P=4.563×10 N/m is the nominal The attitude design control algorithms for CNUSAIL-1 are designed to stabilize the angu sun. � and � denote the normal and tangential vectors of the sail, respectively, � 4. is Controller surface 4. The Controller sun. � andat�1 denote and ttangential respectively, � is surface SRP constant AU fromthe thenormal sun. n and denote thevectors normalof the sail, attitudedesign control algorithms for CNUSAIL-1 are designed to stabilize the ang sun. � and thereflection normalcoefficient. and tangential of the sail, respectively, � isattitude. surfaceThere are three attitude control modes: a detumbling m are the emission coefficients of the desired reflectivity, and �� isdenote the specular �� and �� vectors achieve and tangential vectors of the sail, respectively, r is surface Theattitude attitude control algorithms CNUSAIL-1 designed to stabilize the ang The control algorithms CNUSAIL-1 are modes: the emission coefficients ofThere for reflectivity, and � is the specular reflection coefficient. �� and �� are achieve the desired attitude. arefor three attitudeare control a detumbling m the front andand back ssurfaces, respectively, �� are the corresponding non-Lambertian reflectivity, specular reflection coefficient. �� designed are the emission coefficients of arate reflectivity, and �isisthe the specularand reflection coefficient. �� eand f pointing mode, and deorbiting maneuver mode. The detumbling mode is a tomaneuver stabilize the angular and achieve the desired achieve the desired attitude. are three attitude mode. control modes: a detumbling m and ebfront are the emission of theand front back the andsun back ��and �� corresponding non-Lambertian coefficients. The angle surfaces, � is coefficients defined respectively, as pointing maneuver mode, andThere acontrol deorbiting maneuver � are theattitude. There are three attitude modes: a detumblingThe detumbling mode is a separation fromnon-Lambertian the P-POD and antenna deployment, and when disturbances increase the the front and back surfaces, respectively, and � �� are the corresponding � � surfaces, respectively, and B and B are the corresponding f b mode, maneuver mode, and a deorbiting ( 18a) nadir-pointing � � �� pointing maneuver deorbiting maneuver mode. The detumblingincrease mode isthe a separation from the mode, P-PODand andaantenna deployment, and when disturbances coefficients. Thecoefficients. sun angle � The is defined as α is defined as non-Lambertian sun angle of the satellite. thisdetumbling mode, the MTQs the angular maneuver mode.InThe modereduce is activated afterrate of the satellite until th coefficients. The sun angle � is defined as � separation fromthe thethis P-POD and antenna and and when increase frame. Furthermore, the satellite. where �� of the In mode, MTQs deployment, reduce the angular ratedisturbances of the satellite until the th �� is the sun unit vector with respect to the body(18) separation from P-POD deployment, 18and ) theantenna � � �� stabilizes. The controller (follows the B-dot control law and uses only the magnetometer �� ( 18 ) � � �� when disturbances increase the angular rate of the satellite. � affect either the front or back surface of the solar sail, depending on the of sun unit vector will signthe of satellite. In this mode, the MTQs reduce the angular rate of the satellite until th stabilizes. The controller follows the B-dot control law and uses only the magnetometer where Sb=[Sbx, Sby, Sbz]T is the �sun unit vector with respect to In this mode, the MTQs reduce Once the angular of the sensor and actuator, respectively. the sail rate and solar panel are deployed, the ope is redefined asunit follows: ��. Thus, normal vector �� is � � � � � the sun vector with respect to the body frame. Furthermore, the the B-dot control law and uses only the magnetometer where �� the��sail �� the sun unit vector will affect stabilizes. The controller follows the body frame. �� Furthermore, satellite until the satellite re-stabilizes. The controller follows sensor and actuator, respectively. Once the sail and solar panel are deployed, the op � �� 0 0 1� sun unit vector with respect to the body frame. Furthermore, the where �� 0�� is �the ( 19 ) to the nadir-pointing mode. The 3-axis stabilization performed by the ACS is a �� changes either the front or back surface of the solar sail, depending on the B-dot control law and uses only the magnetometer and sun unit vector will affect either the front or back surface of the solar sensor sail, depending on therespectively. sign of 3.4.3 Solar radiation pressure and actuator, sailstabilization and solar panel are deployed, the isopa changes to the nadir-pointing mode.Once The the 3-axis performed by the ACS �� unit � �� 0� �� 0 0 � 1��the front or back surface of the solar ( 20 ) sensor sun vector will affect either sail, onlogic, the sign of MTQ asdepending and actuator, respectively. Once thecontrol sail and feedback control as proposed in [33], This points the antenna toward ��. Thus, the sail normal vector �� is redefined as follows: changes thedeployed, nadir-pointing mode. The 3-axis stabilization performed by the ACS is a solar panelto are operation changes to the feedback control logic, the as proposed inmode [33], This control points the antenna toward ��. Thus, the sail normal vector �� is redefined as follows: communication with a ground station, and for testing the disturbance effect without the nadir-pointing mode. The 3-axis stabilization performed by 12 � ( 19proposed ) station,inand �� 0� �� 0 0 1� feedback controlwith logic, as [33], control points theeffect antenna toward communication a ground for This testing the disturbance without the thedrag ACStorque. is applied to end the feedback logic, as �� 0� �� 0 0 1�� At the of( 19 the) solarcontrol sail mission, theproposed operation mode changes to the deor �� 0� �� 0 0 � 1�� 20 in [33], This control points toward the for mode changes communication with a ground station, for testing the disturbance effect without the drag torque. At the end of( the the) antenna solar sailand mission, thenadir operation to the deo �� 0� �� 0 0 � 1�� ( 20 ) Orbit decay is handled by a feedback control logic similar to communication with a ground station, and for testing thethe nadir-pointing mode. drag torque. the endby of the solar sail mission, the operation mode changes to the deo Orbit decayeffect isAthandled feedback control logic disturbance without athe aerodynamic drag similar torque.toAtthe nadir-pointing mode. theOrbit end of the solar sail mission, the operation mode changes decay is handled by a feedback control logic similar to the nadir-pointing mode. 12 12 to the deorbiting mode. Orbit decay is handled by a feedback Fig. 5. Solar radiation pressure control logic similar to the nadir-pointing mode. 13 Fig. 5. Solar radiation pressure The SRP force is sourced from photons striking the satellite and sail surface in space. The SRP

acting on a flat sail surface with the optical properties of the sail material is given by [1] ��

�1 � �� http://dx.doi.org/10.5139/IJASS.2016.17.4.579 �� 1 � �� 1 � �� DOI: �� 1 � ��

��

13

13 ( 16 ) 584 ( 17 )

where �� is the SRP, and � � ��6� � 10�� N/m� is the nominal SRP constant at 1 AU from the

sun. � and � denote the normal and tangential vectors of the sail, respectively, � is surface

(579~592)15-145.indd reflectivity, and �584 is the specular reflection coefficient. �� and �� are the emission coefficients of

2016-12-27 오후 2:00:35

communication communication depends depends on on the the satellite satellite attitude. attitude. For For this this reason, reason, the the satellite satellite attitude attitude is is

the the orbit orbit reference reference frame. frame. In In this this alignment, alignment, the the sail sail plane plane is is perpendicular perpendicular to to the the orbital orbital p

aerodynamic aerodynamic drag drag is is minimized; minimized; moreover, moreover, the the antenna antenna located located at at the the base base of of the the sa sa

toward toward the the nadir, nadir, where where itit can can communicate communicate with with the the ground ground station. station. Thus, Thus, in in the the nn

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

maneuver maneuver mode, mode, the the satellite satellite maintains maintains the the nadir nadir pointing pointing orientation orientation within within 5° 5° normal normal to to

which minimizes the drag. the which minimizes the aerodynamic aerodynamic based drag. In In on the deorbiting deorbiting maneuver maneuver mode, mode, the the satellit satellit 4.2 Nadir pointing/deorbiting eigen-axis maneuvers under slew rate constraints

4.14.1Detumbling mode with B-dot control Detumbling mode with B-dot control

t control

from the pointing attitude to the nadir nadir pointing to align align with with the the velocity velocity vector vector normal normal to to the the sol so When the separates from the 4.1 Detumbling mode with B-dot control When thesatellite satellite separates from theP-POD P-PODor orits itsangular angular velocity from increases, the ACS entersattitude the velocity increases, thethe ACS enters the In LEO environments, solar sail missions require that the 4.1 Detumbling mode with B-dot control rom the P-POD or its angular velocity increases, ACS enters thedetumbling mode as orientation maximizes the aerodynamic orientation maximizes the aerodynamic drag. drag. The The ACS ACS will will use use the the 3-axis 3-axis reactio reactio 4.1 Detumbling mode with B-dot control When the satellite from theInP-POD or itsthe angular increases, the ACS enters the detumbling as separates shown Fig. 6. this mode, ACS velocity implements a B-dot controller, which shown in Fig.mode 6. In this mode,inthe ACS implements a B-dot aerodynamic drag be minimized. The radio signals from the When the satellite separates from the P-POD or its angular velocitymaneuvering increases, theand ACS enters the torque for Fig. 6. In this mode, the ACS implements a B-dot controller, the magnetic dumping. maneuvering and theenters magnetic for momentum momentum dumping. When the satellite separates or its angular velocity increases, thecontroller, ACS the torque controller, which uses the rate ofwhich change of magnetic field antenna cannot penetrate the sail membranes; therefore, detumbling mode as shown in from Fig. 6.theInP-POD this mode, the ACS implements adata. B-dot which uses the rate of change of magnetic field measured from the magnetometer Employing only the 4.1 Detumblingmeasured mode with B-dot control from the magnetometer the implements satellite–ground communication detumbling mode asEmploying shown in only Fig.data. 6. InEmploying this mode,only the ACS a B-dot controller, which depends on the satellite etic field measured from the magnetometer data.as the detumbling shown Fig. field 6. Inmeasured this mode, the the ACS implements B-dot controller, which uses the ratemode ofand change of the magnetic from magnetometer Employing the magnetometer and MTQ, theincontroller controller damps the angular velocity fromadata. its initial tip-offonly rate or magnetometer MTQ, damps the angular attitude. For this reason, the satellite attitude is aligned with When satellite separates from the P-POD or its angular velocity increases, the ACS enters the Detumbling mode withthe B-dot control uses the rate ofinitial change oftip-off magnetic field measured from the magnetometer data. Employing onlyInthethis alignment, the sail plane ontroller damps the angular velocity from its initial rate or velocity from its tip-off rate or increased body rate. The the orbit reference frame. uses the rate ofand change of magnetic magnetic field measured the magnetometer data. Employing only magnetometer MTQ, the controller damps thefrom angular velocity from itsthe initial rate the or increased body rate. The dipole moment generated along axis i of B-dottip-off controller is detumbling mode as the shown in Fig. 6.itsInangular this mode, the ACS implements a B-dot controller, hen the satellite separates from P-POD ormoment velocity increases, the ACS enters the is which magnetic dipole generated along axis i of the B-dot perpendicular to the orbital magnetometer and MTQ, the controller damps the angular velocity from its initial tip-off rate or plane and the aerodynamic tic dipole moment generatedmagnetometer along axis i ofand theMTQ, B-dot the controller is controller damps thegenerated angular velocity from rate or increased magnetic dipole moment alongdrag axis iisofits theinitial B-dottip-off controller is the antenna located at the givenofbymagnetic isbody givenrate. by The minimized; moreover, the rate of controller change field from the magnetometer data. Employing mbling modeuses as shown in Fig. 6. In this mode, themeasured ACS implements a B-dot controller, which only the increased body rate. The magnetic dipole moment generated alongbase axis i ofthe the B-dot controller is points toward the nadir, where it can increased body rate. The dipole moment generated axis of i of thesatellite B-dot controller (24) along � by �� 2� magnetic �� ( 24is) � magnetometer andgiven MTQ, the � �controller damps the angular velocity from its initial tip-off rate or the rate of change of magnetic �field measured from the magnetometer data. Employing only the communicate with the ground station. Thus, in the nadirgiven by � � �� ( 24 ) along axis i of the B-dot controller given increased body where rate. dipole moment generated maneuver the) satellite maintains the nadir �K by �� 2� �� isis ��the ( 24 � rate ofischange of the mode, body-fixed where Kmagnetic isa� �apositive positive gain and M dipole. �� is the pointing � The is gain and M themagnetic magnetic dipole. netometer 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 � �� 2� �� ( 24 ) � � � � the rate change of the � body-fixed geomagnetic � of rate of change of 2�the body-fixed M is the magnetic dipole. �� is � �� ( 24 ) �is� the given by the rate as of change of the body-fixed where K is a� �� positive gain M is��the by magnetic dipole. �� is method geomagnetic field along axisand i, estimated the finite difference minimizes the aerodynamic drag. In the deorbiting maneuver ased body rate. The magnetic dipole moment generated along axis i of the B-dot controller is field along axis i, estimated by the finite difference� of)change of the body-fixed where K�� isasa positive gain and M is the magnetic dipole. �� is the rate( 24 � � �� where �� finite � �� method �� 2� � �� stimated by the difference mode, the satellite rotates 90° from the nadir pointing is the rate of change of the body-fixed K is a positive gain and M is the magnetic dipole. � �� field �� method geomagnetic along axis i, estimated by the finite difference method as ( 25 ) �� as n by �� attitude geomagnetic field along axis i, estimated finite difference method as to align with the velocity vector normal to the solar � by the rate of change of themethod body-fixed where K is a positive gain and M is the magnetic geomagnetic field along axis i,dipole. estimated finite difference as ( 25��) isbythethe � � � �� (25) sail. This orientation maximizes ( 25 ) the aerodynamic drag. The � � �� 2� �� with �� sampling ( 24 ) function of a continuous filter [34] �� time ��. �� can also be expressed as the transfer �� by the finite difference method as geomagnetic field along i, estimated �� ACS will use the 3-axis reaction ( 25 ) wheel for maneuvering and �� axis � �� ( 25 ) 7. �� function lso be expressed as the transfer of��a continuous filter [34] Fig. Fig. 7. Nadir Nadir pointing/ pointing/ Deorbiting Deorbiting maneuver maneuver mode mode � � � the � �is �magnetic � �� time �dipole. can also be expressed as the transfer function of a continuous filter [34] with sampling time ��. � with sampling Δt. can also be expressed as the � � is the rate of change of the body-fixed e K is a positive gain and M � � the magnetic torque for momentum dumping. � �� ( 26 ) � �� ( 25 ) �� transfer be[34] expressed as the transfer function of a continuous filter [34] with sampling time ��. �� can also functiontime of a continuous filter Generally, satellite maneuvers adopt a linear controller also beas with sampling ��. �� canmethod �� the finite �� difference ( 26 )expressed as the transfer function of a continuous filter [34] magnetic field along axis i, estimated � by � ( 26these ) �� such as PD or LQR. However, controllers are unsuitable �� is the cutoff frequency. The resultant control torque by the MTQs, which is generated � can also be expressed as the transfer function of a continuous filter [34] with sampling time�where ��. � � � � �� Generally, adopt as PD or LQR. Ho (26) �� Generally, satellite satellite maneuvers maneuvers adopt aa linear linear controller controller such (because 26 ) �� for large-angle maneuvers they tend to saturate the such as PD or LQR. Ho �� ( 26 ) � � MTQs, � ��which is generated control torque by the �cy. The resultant ( 25 by ) the MTQs, which is generated �� where � is �� cutoff �� the frequency. The resultant control torque � �� perpendicular to the Earth’s magnetic field, is then given by � � � ( 26 ) wheel momentum or torque. Thus, in this study, the reaction �� controllers are controllers are unsuitable unsuitable for for large-angle large-angle maneuvers maneuvers because because they they tend tend to to saturate saturate the the re re is cutoff the cutoff frequency. The resultant control torque by the MTQs, which is generated where ��the where ωc is frequency. The resultant control torque netic field, is then given by is the cutoff frequency. The resultant control torque by the which generated where � nadirMTQs, pointing and is deorbiting maneuvers are implemented � � � be expressed as the transfer function of a continuous filter [34] sampling time ��. �� can also perpendicular to the Earth’s magnetic field, is then given by � � � ( 27 ) �� frequency. The resultant control torque by the MTQs, which is generated where �� is the bycutoff the MTQs, which is generated perpendicular to the Earth’s torque. Thus, in the pointing momentum or torque. Thus, in this this study, study, the nadir nadir pointing and and deorbiting deorbiting ma m bymomentum a feedback or control logic for eigen-axis maneuver under perpendicular to the Earth’s magnetic field, is then given by ( 27field, ) perpendicular tothen the Earth’s magnetic is then given by �� magnetic field, is given by � � � � � ( 27 ) depends on the satellite attitude.( 26 For )this reason, satellite attitude is aligned with � perpendicular to the�� Earth’s magnetic field, communication is then given by slew ratethe constraints [34]. implemented by control �� implemented by aa feedback feedback control logic logic for for eigen-axis eigen-axis maneuver maneuver under under slew slew rate rate constra constra �� ( 27slew ) Considering the maximum � � � � � ( 27 ) the orbit reference frame. In this alignment,(27) the sail plane is perpendicular to the orbital plane and therate about the eigen-axis, �� ( 27 ) frequency. The resultant control torque by the MTQs, which is generated theConsidering e �� is the cutoff the slew rate Considering the maximum maximum slew rate about about the the eigen-axis, eigen-axis, the the control control torque torque is is given given as as [3 [3 control torque is given as [33]. aerodynamic drag is minimized; moreover, the antenna located at the base of the satellite points

�� endicular to the Earth’s magnetic field, is then given by � � �� the�� toward the nadir, where it can communicate with the ground station. Thus,��in nadir-pointing

�

(28)

maneuver mode, the satellite maintains the nadir(pointing orientation 27 ) where where where within 5° normal to the solar sail,

� � � �

which minimizes the aerodynamic drag. In the deorbiting maneuver mode, the satellite rotates 90°

� �� k ��k��

Fig. 6. Detumbling mode


4.2 Nadir

(29)

from the nadir pointing attitude to align with the velocity vector normal to the solar sail. This


15 15

� � ��a��

orientation maximizes the aerodynamic drag. The ACS will use the 3-axis reaction wheel for pointing/deorbiting based mode on eigen-axis maneuvers under slew rate constraints Fig. 6. Detumbling Fig. 6. Detumbling mode

(30)

Fig. 6. Detumbling mode (31) � � �� Fig.under 6. Detumbling maneuvering and the magnetic torque for momentum dumping. sed on eigen-axis maneuvers slew rate mode constraints 4.2InNadir based eigen-axis maneuvers slew rate drag constraints LEOpointing/deorbiting environments, solar sail on missions require that theunder aerodynamic be minimized. The 4.2 Nadir pointing/deorbiting based on eigen-axis maneuvers under slew rate constraints The term in Eq.(28) is a kind of nonlinear control to cancel ω×Iω in Eq.(1) to lineariz 4.2 Nadir pointing/deorbiting based The on eigen-axis maneuvers under slew ratelast constraints sail missions require that the4.2 aerodynamic drag be minimized. pointing/deorbiting on eigen-axis maneuvers under slew rate constraints InNadir LEO environments, solarbased sail missions require aerodynamic drag be minimized. The radio signals from the antenna cannot penetrate thethatsailthemembranes; therefore, satellite–ground In LEO environments, solar sail missions require that the aerodynamic drag be minimized.loop Thedynamics. The saturation function is defined as: In LEO environments, solar sail missions require that the aerodynamic drag be minimized. The cannot penetrate the sail membranes; therefore, mode satellite–ground 6. environments, Detumbling In Fig. LEO solar sail missions requirethethat aerodynamictherefore, drag be satellite–ground minimized. The radio signals from the antenna cannot penetrate sailthemembranes; radio signals from the antenna cannot penetrate the sail membranes; therefore, satellite–ground 1 � �� 1 14 the sail membranes; therefore, satellite–ground radio signals from the antenna cannot penetrate radio signals from the antenna cannot penetrate the sail membranes; therefore, satellite–ground�� 1 � �� sat�Pq � � �� 1 �� 14 Nadir pointing/deorbiting based on eigen-axis maneuvers under slew rate constraints 14 �1 � �� 1 14 �� 14minimized. The LEO environments, solar sail missions require that the aerodynamic drag be14

signals from the antenna cannot penetrate the sail membranes; therefore, satellite–ground

The positive scalar gain ��, �� can defined as: ��

�� k ��

��

14 �� Fig. 7. Nadir pointing/ Deorbiting maneuver mode Fig. 7. Nadir pointing/ Deorbiting maneuver mode

where �� is the maximum slew rate. Following [33], it is assumed that �� 0 an

Generally, satellite maneuvers adopt a linear controller such �as are PD then or LQR. However, � and chosen as these

585

http://ijass.org

controllers are unsuitable for large-angle maneuvers because they tend to saturate the reaction wheel �

� � 2ω��

momentum or torque. Thus, in this study, the nadir pointing and deorbiting maneuvers are

� � 2��

implemented by a feedback control logic for eigen-axis maneuver under slew rate constraints [34].

where k torque and cisare theasdesired damping ratio and the natural frequency, respectively. Th Considering the maximum slew rate about the eigen-axis, the control given [33]. (579~592)15-145.indd 585

� � � ��

( 28 )

��

2016-12-27 오후 2:00:36

�� a�� a�� 30 )) � (( 30 � � ��a�� 30 (( 30 ( 30 ) ))) ( 30 � � � � � �� (31) � � �� (31) � (31) �� (31) �� (31) The last term in Eq.(28) is a kind of nonlinear control to cancel ω×Iω in Eq.(1) to linearize the closedThe Eq.(28) is aiskind of nonlinear control to cancel in Numerical Eq.(1) to linearize the closedThe last lastterm termin in Eq.(28) a kind of nonlinear control to ω×Iω 5. simulation The last term in Eq.(28) is a kind of nonlinear control to cancel ω×Iω in Eq.(1) to linearize the closedThe lastlast term in in Eq.(28) is is a kind of of nonlinear control to to cancel ω×Iω in in Eq.(1) to to linearize thethe closedThe term Eq.(28) a kind control cancel ω×Iω Eq.(1) linearize closedcancel in Eq.(1) to linearize thenonlinear closed-loop dynamics. loop ω×Iω dynamics. The saturation saturation function is defined defined as: as: loop dynamics. The function is loop dynamics. The saturation function is defined as: The saturation function is defined as: is defined as: loop dynamics. The saturation function 5.1 Simulation Conditions loop dynamics. The saturation function is defined as: � 1 � �� 1 1 � � � � �� ( 39 ) 1 � �� 1 � �� 1 �� 1 ��1 �� 1 � � �� 1 1 �� 1 32 ))gravity gradient disturbances, sat�Pq�� 1 �� 1 � (( 32 sat�Pq The aerodynamic drag, SRP, �� 1 �� (�� 32 sat�Pq � �� (32)�� is the unloading control gain, and �� 1 �� 1 1 � �� ( 32 ) ))is the error between the current and desired wheel sat�Pq �1 � �� 1 where �� 1 � �� 1 � �1 � � 1 ( 32 sat�Pq �� and the effect of the residual magnetic dipole moment were � �� 1 � �� 1 �1 � �� 1 �� 1�1 � �� 1 �� 1 � evaluated in removing numerical Theby orbit parameters momentum vectors. However, thesimulations. stored momentum a magnetic torque of compromises the �� can can defined as: as: The positive positive scalar scalar gain gain � ��,, � � � � � �� ( 39 ) The � can defined �� ican defined as: The positive scalar gain �k��,i,, p� The positive scalar gain defined as: CNUSAIL-1 are listed in Table 1. The orbital propagation is can defined The positive scalar gain � � can , defined as: The positive scalar gain � � � � , � can defined as: The positive scalar gain � power of the satellite. Thus, the unloading logic is operated when the momentum exceeds a specified � � �� 33 subjected a J2 perturbation. geomagnetic field kk�� (( 33 ))The where �� is the unloadingtocontrol gain, and �� is the error between themodel current and desired wheel �� k � � � � � 33 � �� (33) � k � � � � (( 33 )) � k � k�� ( 33 ) �� limit. �� is implemented in International Geomagnetic Reference �� ( 33 ) �� momentum vectors. However, removingThe the sun stored momentum by a magnetic torque compromises the �� (34)location Field 2011 (IGRF-11). is calculated from the �� (34) �� (34) (34) �� (34) JulianThus, date. �� (34)is operated when the momentum exceeds a specified satellite. logic where � �� is is the the maximum maximum slew slew rate. rate. Following Following power [33], itofis isthe assumed that �� the �0 0unloading and ��0� ��0� � 0. 0. 5. Numerical simulation where [33], it assumed that and � �� The moment of inertia in the body frame. � where � is the maximum slew rate. Following [33], it is assumed that � � 0 and ��0� � 0. the maximum slew rate. Following [33], it is is assumed where where�� maximum slew rate. Following [33], that�� and��0� ��0� � 0. �� where thethe maximum slew rate. Following [33], it it is assumed that �� 00 and �is 0.represented ��is is where �� is the maximum slew rate. Following [33], it is assumed that � � 0 and ��0� � 0. �� properties are those limit. The optical of metalized kapton sheet, 5.1 Simulation Conditions � and and �that � are areqthen then chosen as assumed and ω(0)=0. k and c are then chosen as � as e,0≠0 chosen � and � are then chosen as � and � are then chosen as � �and then chosen as as from which the solar sail is constructed. The satellite and and� �are are then chosen The aerodynamic drag, SRP, gravity gradient disturbances, and the effect of the residual magnetic (35) �� 2ω 2ω�� (35) (35) � optical properties before and after the sail deployment are � � � (35) � � 2ω � (35) �� 2ω2ω � � (35) �� moment were evaluated simulations. The orbit parameters of CNUSAIL-1 are 5. dipole Numerical simulation listed in Tablesin2 numerical and 3Table 3, respectively. � (36) � 2�� 2�� (36) �� (36) � (36) � � 2�� Table 4 and Table 5 list the specifications of the reaction � �� 2�� in TableConditions 1. The orbital propagation is(36) subjected to a J2 perturbation. The geomagnetic field (36) �� 2�� 5.1listed Simulation � where k and c are the desired damping ratio and the natural frequency, respectively. The quaternion wheel and the The MTQs, respectively. The reaction wheel where k andc are c are thedesired desireddamping damping ratio ratio and and the respectively. quaternion where k and the the natural naturalfrequency, where k and c are desired damping ratio and natural respectively. The quaternion where k and c are thethe desired damping ratio and thethe natural respectively. The quaternion modelfrequency, isfrequency, implemented in International Geomagnetic Reference Field velocity The sun unloading logic is activated when the angular where k and c are the desired damping ratio and the natural frequency, respectively. The quaternion The aerodynamic drag, SRP, gravity gradient disturbances, and the2011 effect(IGRF-11). ofofthe residual magnetic frequency, respectively. The q quaternion denoted by �q error, denoted denoted by qq� � � �q q�� is iserror, defined �� error, by q q defined � is defined the reaction wheel reaches 3000 rpm (corresponding to an T by q� � �q�� q�� q�� error, denoted qerror, qdenoted isbydefined ��is is �q�q error, q�� defined location is calculated from the Julian date. �� q �� q q q defined e=[q e1denoted e2 qe3] by �� evaluated in numerical simulations. The orbit parameters of CNUSAIL-1 are error, denoted byq�q�� q � � �q �� q �� is defined dipole moment were �� angular momentum of 1.35 × 10−3 Nms). �� Table 1. Orbital parameters of CNUSAIL-1 � � � � �� listed in Table 1. The orbital propagation is(( 37 subjected to a J2 perturbation. The geomagnetic field �� 37 � �� )) � � �� ( 37 37 ) � (37) �� 5.2 Detumbling mode � � ( ) Element Description �� ( 37 ) � �� ( 37 ) � �� model is implemented in International Geomagnetic 6963.1 Reference �� Semi-major axis a km Field 2011 (IGRF-11). The sun � �� The detumbling mode activates under two scenarios: the Inclination 98° are the the command command quaternions quaternions and and qqii are are the the satellite satellite attitude attitude quaternions. quaternions. The error error in in the the where qqcici are The where location is calculated from the Julian date. are the command quaternions and q are the satellite attitude quaternions. The error in the where q are the command quaternions and q are the where q Eccentricity 0.01939 ci the command quaternions and qi are where qciciqare ci ii thei satellite attitude quaternions. The error in the The error in the where ci are the command quaternions and qi are � the satellite attitude quaternions. LTDN 10:30 �ω�� ω angularattitude velocity, denoted denoted � �� The ω ωin is defined defined as: satellite quaternions. error angularas: �� theis �ω angular velocity, ω Table 1. Orbital parameters of CNUSAIL-1 Table 1. Orbital parameters of CNUSAIL-1 � � �� ω�� ω�� angular velocity, denoted � is defined as: � �ω � �ω angular velocity, denoted � � ω ω is defined as: � �� angular velocity, denoted � � ω ω is defined as: Orbit Period 84.50 min T � � �� ω �] �ω �� defined �� ω �� is defined as: velocity, angulardenoted velocity,ωdenoted ω e=[ωe1 ωe2 e3 is �� as: �� 0 � �� 0� �� 0 0�� ( 38 ) � ( 38 ) Element Description � � � � �� 0 � �� 0� � � � �� 38 �0 �0 �� (( 38 � � �� 0� �� ( 38 ) ))) � � (38) � � 0� �� 0 ( 38 � 0� Semi-major axis a km properties are those of The moment of inertia is represented in the body frame.6963.1 The optical Inclination 98° 4.3 Unloading Unloading of of the the reaction reaction wheel wheel momentum momentum by by the themetalized MTQ kapton sheet, from which the solar sail is constructed. The satellite and optical properties 4.3 MTQ Eccentricity 0.01939 4.3 Unloading of the reaction wheel momentum by the MTQ Unloading reaction wheel momentum MTQ 4.34.3 Unloading ofof thethe reaction wheel momentum byby thethe MTQ 4.3 Unloading of the reaction wheel momentum by the MTQ LTDN 10:30 The momentum momentum of the the reactionwheel wheelmomentum increasesbefore under constant external disturbance. If the 2 and 3Table andconstant after the sail deployment are listed If in Tables 3, respectively. 4.3 Unloading of the reaction by The of reaction wheel increases under external disturbance. the Themomentum momentumofof ofthethe thereaction reactionwheel wheelincreases increasesunder underconstant constantexternal external disturbance. Ifthethe the The momentum reaction wheel increases under constant external disturbance. If Orbit Period 84.50 min The disturbance. If the The MTQ momentum of the reaction wheel increases under constant external disturbance. If the momentum exceeds exceeds the the storage storage limits, limits, the the reaction reaction wheel wheel becomes becomes saturated saturated and and unable unable to to produce produce momentum momentum exceeds the storage limits, the reaction wheel becomes saturated and unable to produce momentum exceeds storage limits, reaction wheel becomes saturated and unable produce momentum exceeds thethe storage limits, thethe reaction wheel becomes saturated and unable to to produce momentum exceeds the limits, the reaction wheel becomes saturated and unable to produce 2. Satellite properties Table 2. Satellite properties Thecontrol momentum thestorage reaction wheel increases under the torque.ofHence, Hence, the excess excess momentum should be unloaded unloaded by MTQs. TheTable momentum the control torque. the momentum should be by MTQs. The momentum The moment of inertia is represented in the body frame. The optical properties are those of thecontrol control torque. Hence,thethe theexcess excess momentumshould should beunloaded unloadedbyby byMTQs. MTQs.The Themomentum momentum control torque. Hence, excess momentum should be unloaded MTQs. The momentum thethe torque. Hence, momentum be constant external disturbance. If the momentum exceeds the control torque. Hence, the excess momentum should be unloaded Element by MTQs. The momentum Description unloading logic is defined defined as [35] [35] is as metalized kapton sheet, from which the solar sail is constructed. The satellite and optical properties theunloading storage logic limits, the reaction mass 4 kg unloading logic is defined as [35] wheel becomes saturated unloading logic defined [35] unloading logic is is defined asas [35] unloading logic is defined as [35] torque. Hence, the excess and unable to produce the control Sail Area 4 m2 sail deployment are listed in Tables 2 and 3Table 3, respectively. 16before and after the 16 Moment of Inertia Pre-deployed (0.0506, 0.0506, 0.010) momentum should be unloaded by MTQs. The momentum 16 1616 (Ix, Iy, Iz) [kg m2] Post-deployed (0.3104, 0.3121, 0.5757) 16 unloading logic is defined as [35] � �� J.� ��a�� Int’l of��a�� Aeronautical �� a�� &� �Space Sci. 17(4), 579–592 (2016)

� ��

��

��

17

Table 2. Satellite ( 39 ) sail properties Table 3. Optical properties of the solar [37]

(39)

Table 3. Optical properties of the solar sail [37] Element Description Element Description where is the unloading controlgain, gain,and andΔ�� theerror error between the current and desired wheel ��the where krw�is unloading control h isisthe 0.79 Frontmass emission coefficient ‫ܤ‬௙ 4 kg between the current and desired wheel momentum vectors. 0.55 momentum vectors. However, removing the stored momentum by a magnetic torque compromises Back coefficient ‫ܤ‬the Sailemission Area 4 m2 ஻ However, removing the stored momentum by a magnetic 0.05 0.010) Front non-Lambertian coefficient ݁௙ Moment of Inertia Pre-deployed (0.0506, 0.0506, 2 powercompromises of the satellite.the Thus, the unloading logic is operated when the(IBack momentum exceeds acoefficient specified torque power of the satellite. Thus, the 0.550.5757) ݁௕ Post-deployed (0.3104, 0.3121, x, Iy, Inon-Lambertian z) [kg m ] unloading logic is operated when the momentum exceeds a Reflectivity r 0.88 limit. Specular reflection coefficient 17 s 0.94 specified limit.

5. Numerical simulation

Table 4 and Table 5 list the specifications of the reaction wheel and the MTQs, respectively. The

5.1 Simulation Conditions

reaction wheel unloading logic is activated when the angular velocity of the reaction wheel reaches


586

3000 rpm (corresponding to anresidual angularmagnetic momentum of 1.35 × 10−3 Nms). The aerodynamic drag, SRP, gravity gradient disturbances, and the effect of the Table 4. Specifications of reaction wheel dipole moment were evaluated in numerical simulations. The orbit parameters of CNUSAIL-1 are (579~592)15-145.indd 586

Element

Description

2016-12-27 오후 2:00:38

Table 3. Optical properties of the solar sail [37] Table 3. Optical properties of the solar sail [37] Element Description System Design & Verification for CNUSAIL-1 with Solar/Drag Sail Element Yeona Yoo Attitude Control Description 0.79 Front emission coefficient ‫ܤ‬௙ 0.79 Front emission coefficient ‫ܤ‬௙ 0.55 Back emission coefficient ‫ܤ‬஻ 0.55 Back emission coefficient ‫ܤ‬஻ The detumbling gain factor K is set to 30000 and 50000 initial tip-offFront afternon-Lambertian the satellite separates 0.05 coefficientfrom ݁ the P-POD, 0.05 Front non-Lambertian coefficient௙ ݁௙ in and when the angular rate of thecoefficient satellite increases above 0.55the pre- and post-deployment cases, respectively. The Back non-Lambertian ݁ 0.55 Back non-Lambertian coefficient௕ ݁௕ Reflectivity r 0.88 sampling 1°/s. The solar sail is deployed by a spiral spring torque, Reflectivity r 0.88 period of the magnetometer and MTQ is 2 s. The Specular reflection coefficient s 0.94 simulated increasing the moment inertia and reducing Specularof reflection coefficient s the angular 0.94 angular rates during pre- and post-deployment are plotted in Figs. 8 and 9, respectively. In the pre-deployment rate of the satellite to almost zero [36]. When simulating the the angular rate reduces post-deployment mode, the initial angular wheel rate and case, Table 4 and Tabledetumbling 5 list the specifications of the reaction the MTQs, respectively. The below the desired level for the Table 4 and Table 5 list the specifications of the reaction wheel and the MTQs, respectively. The nadir-pointing maneuver after approximately three orbits. is assumed lower than the pre-deployment rate; that is, reaction wheel unloading activated when the angular velocity of reaction wheel reaches T Inthe thethe post-deployment case, the reaction wheels reduce Pre-deployment: ω=[logic 6 logic -7is 3] °/s reaction wheel unloading is activated when the angular velocity of reaction wheel reaches T −3 the angular rates to below 1°/s. However, the satellite sways Post-deployment: ω=[ 2 -7 1] °/s 3000 rpm (corresponding to an angular momentum of 1.35 × 10 Nms). 3000 rpm (corresponding to an angular momentum of 1.35 × 10−3 Nms). periodically under the larger aerodynamic drag torque than Table 4. Specifications of reaction wheel in the pre-deployment case. Table 4. Specifications of reaction wheel Table 4. Specifications of reaction wheel Element Description Element Description Manufacturer Clyde-space Ltd Manufacturer Clyde-space Ltd Mass 0.275 kg Mass 0.275 kg Inertia 4.5e-6 kg m2 2 Inertia 4.5e-6 kg m Maximum torque 1.9 × 10-3Nm Maximum torque 1.9 × 10-3Nm angular rate is assumed lower than the pre-deployment rate;3.4 that × is, 10-3Nms Maximum Momentum Maximum Momentum 3.4 × 10-3Nms angular rate is assumed lower than the୘pre-deployment rate; that is, Pre-deployment: ൌ ሾ͸ െ ͹͵ሿ Maximum߱Angular rate /s 7000 RPM Maximum Angular rate 7000 RPM ୘

5.3 Nadir-pointing and deorbiting maneuver modes 5.3 Nadir-pointing and deorbiting maneuver modes under slew rate constraints under slew rate constraints 5.3.1 Nadir-pointing maneuver

5.3.1 Nadir-pointing maneuver

The initial error quaternion in the large-angle maneuver simulation is set to

The initial error quaternion in the large-angle maneuver simulation is set to where the initial in the angular velocity is zero. The unloading qe,0=[error 0.6460 0.2876 -0.6460 0.2876 ]T logic will be activated when the Pre-deployment: ߱ ൌ ሾ͸ െ ͹͵ሿ ୘ /s Post-deployment: ߱ ൌ ሾʹ െ ͹ͳሿ /s the initial error3000 in the velocity The and the angular ratewhere of the reaction wheel exceeds rpm.angular The maximum slew rateisiszero. set to 0.3/s, Post-deployment: ߱ ൌ ሾʹ െ ͹ͳሿ୘ /s unloading logic will be activated when the angular rate of the The detumbling gain factor K is set to 30000 and 50000 in the preand post-deployment cases, Table 5. Specifications of magnetic torquer Table 5. Specifications of magnetic torquer initial eigen-angle is 146.5. Ideally, an optimal attitude control would require 8.13 min to complete Table 5. Specifications of magnetic torquer reaction wheel exceeds 3000 rpm. The maximum slew rate is The detumbling gain factor K is set to 30000 and 50000 in the preand post-deployment cases, respectively. The sampling period of the magnetometer and MTQ is 2 s. The simulated angular the rates maneuver. Element Description set to 0.3°/s, and the initial eigen-angle is 146.5°. Ideally, an respectively. Thepost-deployment sampling Element periodare of plotted the magnetometer and MTQ is 2 s.Description The simulated angular rates during pre- and in Figs. In the pre-deployment case, Length (x, y)8 and 9, respectively. 60 mm Fig. 10 –13 and Figs. 14–18control plot the simulation results in 8.13 the preandtopost-deployment Length (x, y) 60 mm optimal attitude would require min complete cases, during preand post-deployment are plotted in Figs. 8 and 9, respectively. In the pre-deployment case, Radius 4 mm the angular Torquer rate reducesrod below the desired level for the nadir-pointing maneuver after approximately Radius 4 mm the maneuver. Torquer rod respectively. The nadir-pointing maneuver is completed under the slew rate constraints in both cases. 2 the angular rate reduces below the desired level for the nadir-pointing maneuver after approximately Dipole moment 2 three orbits. In the post-deployment case, the reaction wheels reduce0.2 theAm angular to below 1/s. Fig. 10–13 and Figs. 14–18 plot the simulation results in the preDipole moment 0.2 Amrates In the post-deployment case, momentum unloading is also activated. As shown in Fig. 17, the reaction Areathe(z) x 75mm three orbits. In the post-deployment case, reaction wheels reduce65mm the angular rates to below 1/s. (z) larger x 75mm However, the Air satellite sways periodically Area under the aerodynamic 65mm drag torque than in the pre-and post-deployment cases, respectively. The nadir-pointing Coil Air Coil speeds the x and z axes are below 3000 rpm, so the magnetic dipole moment command is not Dipole moment 0.1 Am2torque than in the pre-along maneuver is completed under the slew rate constraints in both However, thecase. satellite sways periodically Dipole under themoment larger aerodynamic drag 0.1 Am2 deployment ��

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. 8. B-dot controller verification (pre-deployment)

Fig. 10. Attitude error (pre-deployment)

Fig. 10. Attitude error (pre-deployment)

20

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

Fig. 9. B-dot controller verification (post-deployment) 19

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

Fig. 11. Body rate error (pre-deployment)

19

587

(579~592)15-145.indd 587

http://ijass.org

2016-12-27 오후 2:00:38

Fig. 18. Command torque (post-deployment)

Fig. 14. Attitude error (post-deployment)


5.3.2 Comparisons between PD control logic and eigen-axis maneuvering under slew rate

This subsection compares the eigen-axis maneuvering algorithm with the PD control l This subsection compares the eigen-axis maneuvering given bywith the PD control logic which is given by algorithm

cases. In the post-deployment case, momentum unloading is also activated. As shown in Fig. 17, the reaction speeds along the x and z axes are below 3000 rpm, so the magnetic dipole moment command is not issued around the y axis.

� � ��

(40)

Fig. 20 plot the attitude errors during a large-angle Fig. 20 plot the attitude errors large-angle maneuver controlled by the PD l maneuver controlled by the PDduring logic aand the eigen-axis

5.3.2 Comparisons between PD control logic and eigenaxis maneuvering rate constraints Fig. 11. Bodyunder rate errorslew (pre-deployment)

Fig. 15. Body rate error (post-deployment)

eigen-axis algorithm (with eigen-angle = 108), respectively, in the post-deployment cas

Fig. 11. Body rate error (pre-deployment)

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

23

Fig. 16. Reaction wheel Angular rate (post-deployment)


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

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

22

Fig. 13. Command torque (pre-deployment)

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

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



21 21

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

Fig. 14. Attitude error (post-deployment) Fig. 14. Attitude error (post-deployment)

Fig. 18. Command torque (post-deployment) 5.3.2 Comparisons between PDtorque control (post-deployment) logic and eigen-axis maneuvering under slew rate constraints Fig. 18. Command 5.3.2 Comparisons between PD logicmaneuvering and eigen-axis maneuvering under ratelogic constraints This subsection compares thecontrol eigen-axis algorithm with the PD slew control which is Thisby subsection compares the eigen-axis maneuvering algorithm with the PD control logic which is given given by � � ��

( 40 )

( 40 )

� � ��

Fig. 20 plot the attitude errors during a large-angle maneuver controlled by the PD logic and the

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-axis eigen-angle = 108), respectively, in the case. Unlike the PD controlalgorithm logic, the (with eigen-axis maneuvering algorithm achieved thepost-deployment desired nadir pointing. Fig. 15. Body rate error (post-deployment)

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


19. Errors in large-angle maneuver controlled by PD control logic Fig. 19. Fig. Errors in large-angle maneuver controlled by PD control logic


23

23


(579~592)15-145.indd 588


588

2016-12-27 오후 2:00:39

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 Control Systembecause Design & Verification CNUSAIL-1 with Sail performance, the gain was calculatedfor using the initial error. The Solar/Drag angular errors are 20 above the initial angular error and the control gain is updated by the current error quaternion through Eqs. The Life simulation after the gain recalculation are presented in Figs. 26–30. The 5.3.4 timeresults analysis algorithm (with eigen-angle = 108°), respectively, in the (29)–(34). post-deployment case. Unlike the PD control logic, the deorbiting maneuver is nowimposed partially achieved under theplane slew ratechange constraints andaltitude the satellite nearly The forces on the sail the eigen-axis maneuvering algorithm achieved the desired maintains ofthe the cube satellite. Fig. 31 plots the altitude decrease the and the attitude that maximizes the aerodynamic drag. The satellite periodicallyat tumbles nadir pointing. semi-major axis under random angular rate rotations, the reaction wheel saturates under the aerodynamic drag torque.

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 quaternion. Hence, when qe,0

Attitude Control System Design & Verification for ...

Attitude Control System Design & Verification for ...

Suggest Documents

Design of Attitude Control System for UAV Based on ... - downloads

Robust Control Design for Spacecraft Attitude ...

Design verification for control engineering - CiteSeerX

Attitude control system design of a helicopter ... - Semantic Scholar

Spacecraft drag-free attitude control system design ...

DES Spacecraft drag-free attitude control system design with ...

Fault-Tolerant Attitude Control System for a Spacecraft with Control ...

VEGA Launch Vehicle: Trajectory & Attitude Control System

THE ATTITUDE CONTROL SYSTEM OF THE WILKINSON ...

A New Satellite Attitude Control System

Research Article Design of Attitude Control Systems

Fault-tolerant spacecraft attitude control system

Control System Design

Control System Design

Control System Design Considerations

Control System Design

Vector Based Kinematic Closed-Loop Attitude Control-System for ...

Extensions to the Galileo Attitude Control System for the Probe ...

Design of Sliding Mode Attitude Control for Communication Spacecraft

Robust Attitude Control Systems Design for Solar ...

three axis attitude determination and control system for a ... - CiteSeerX

design of attitude stability system for prolate dual-spin satellite

design of attitude stability system for prolate dual-spin satellite

Control System Design for KEKB Accelerators