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1. completes the right hand frame 101 Satellite Attitude Dynamics using Euler Angles 3D Rotation Parameterization aig direction x XT VLH Y LVLH Fig 1 View of the orbital plane with the Local Vertical Local Horizon LVLH inertial reference frame Fig 2 shows the CubeSat and its body frame with the origin in O and with Xw Y a and Z as body axes as well as the axes of LVLH which is the fixed inertial frame with respect to whom the body frame will be located For simplification the gravity center G of the satellite was considered as coincident with the geometric center of the cube O 1 e G O The axis z_ of the body frame is oriented on the direction of the sat sat camera objective while x and y axes coincide with the direction of positive torques T and T produced by the two momentum reaction wheels actuating the CubeSat The desired attitude for taking pictures will correspond to Xa Xiv Vea Yivin and Zw Zyypy nadir direction i e null Euler angles for the orientation of the body frame with respect to LVLH see next section for details of Euler angles 3D rotation parameterization m Z LVLH nadir direction Y sat Fig 2 View of the CubeSat with its body axes compared with LVLH axes CubeSat is a standard 10x10x10 cm cube small satellite its mass is around m 1 kg and the following symmetric moment of inertia tensor with respect to the body frame was considered here2. and Y are the only CubeSat axes which are actuated Thus l 0 0 e the roll angle a corresponds to a first x axis elementary rotation matrix R 0 cosa sina sat 0 sina cosa describing the transformation between the orthonormal basis Xi viu Yivtg v y and the intermediate orthonormal basis Xi yvy 1 Z 1 e Vivien R Y and Ziy R Z cosp 0 sinp e the pitch angle B corresponds to a second y axis elementary rotation Rg 0 1 0 describing sin 0 cos the transformation between the intermediate orthonormal basis X Z and another intermediate orthonormal basis X Y Z gt 1 Xivin Rex and Z R Z l 0 0 e finally the roll angle y corresponds to a third x axis elementary rotation R 0 cosy siny 0 siny cosy describing the transformation between the intermediate orthonormal basis X y Z and the body frame orthonormal basis Xat Veat gt Zsar Js 1 y RAY sat and Z R Zoa Based on this x y x roll pitch roll Euler angles parameterization appropriate for the studied case where X and y are the only actuated axes the attitude matrix A is computed as follows 25 1 0 0 cosp 0 sinB 1 0 0 A R R R 0 cosa sina 0 1 0 HO cosy siny 0 sina cosa sin p 0 cosB O siny cosy SY 2 cosp sin Psin y sinBcos y sinasinB cosacosy sinacosBsiny cosasiny sinacosBcosy cosasinB sinacosy cosacosBsiny sinasin y cosa cos cosy 103 Satellite Attitude Dynami
3. and keep it in that attitude The particularity of the considered attitude control problem is that only two axes of the satellite are actuated not all three So one deals with an under actuated system where the difficulty is to control different 3D rotations using only two actuators for three degrees of freedom This two axis moment control problem can occur for a three axis controlled satellite in the accidental case when one moment thruster fails So the study of this two axis only active control case is very important for a majority of satellite missions ACKNOWLEDGEMENT The authors gratefully acknowledge the National Authority for Scientific Research ANCS UEFISCSU for financial support through PN II project nr 106 2007 code ID_247 2007 The authors thank Dr Ing Boris CONONOVICI Dr Ing Ion NITU and the colleagues from the Mechatronics Department at the Institute of Solid Mechanics for useful insights and discussions on the topics of this paper REFERENCES http en wikipedia org wiki Cubesat 2 http www cubesat org KRISHNAMURTHY N Dynamic Modelling of CubeSat Project MOVE Master Thesis Lule University of Technology Sweden 2008 http swisscube epfl ch Swisscube CubeSat project developped by Ecole Polytechnique F d rale de Lausanne http www xatcobeo com Xatcobeo CubeSat project developed by University of Vigo http www leodium ulg ac be cmsms Oufti 1 CubeSat project developed by Unive
4. It is easy to integrate 10 using for example Runge Kutta integration methods In the considered case where the moment of inertia tensor J has the form 1 so T Jo diag tt 1 and for the torque vector T given by 6 then J T ora awe oe J am _ ty J T a oe and XX yy J IT 0 and the differential equations 10 simplify as follows 105 Satellite Attitude Dynamics using Euler Angles 3D Rotation Parameterization g siny aBcosB By Jy sinf a i a B cosy aysinB 11 Ja O T cosp b 7 b Ja t sin i CubeSat direct dynamics results based on the Runge Kutta numerical integration of the differential equations 10 involving the Euler angles will be presented in next subsection 5 SOME ATTITUDE CONTROL RESULTS OBTAINED USING AN INVERSE DYNAMICS TECHNIQUE To show some direct dynamics results obtained by solving the differential equations 11 one needs to use some control law 1 e some evolution for the torques T and T In fact an attitude control technique based on inverse dynamics of 10 11 is already in testing The results presented below are obtained using the control law provided by our attitude control technique which will be presented in a future paper Let s recall that the purpose of the attitude maneuver control is to rotate the satellite from an initial attitude at time tin characterized by the Euler angles a B 7 to the final time tp desire
5. Jn Jy 16 51 0 0 J E _ 2 J J Jy Ja 0 16 51 0 kg cm 1 J yy J J 0 O 16 80 XZ yz The moment of inertia tensor is approximated as constant in time considering the momentum reaction wheels as moving bodies perfectly centered when rotating around their actuation axis The form of J in 1 Dan DUMITRIU Cornel SECARA 102 shows that the satellite considered has a symmetric structure on Xx and y axes 1e J J and there sat yy is no cross term 1 J J J 0 3 EULER ANGLES 3D ROTATION PARAMETERIZATION OF CUBESAT S BODY FRAME WITH RESPECT TO THE INERTIAL LVLH REFERENCE FRAME To parameterize the position of the body frame with respect to the inertial LVLH reference frame Euler angles are used in this paper One can also use quaternions with the corresponding equations of motion 3 17 19 Other 3D rotation parameterizations can be used as well such as Rodrigues parameters angular velocity vector full 3x3 rotation matrix etc 17 22 24 Thus the orientation of the CubeSat can be defined using three Euler angles roll pitch and yaw 3 15 19 22 25 These Euler angles are obtained from the sequence of right hand positive rotations from the CX vin gt Yiviw gt Ziviy LVLH orthonormal basis to the Xats Ysats Zsa body frame orthonormal basis Among the 12 possible sequences of 3D rotations using Euler angles 25 the x y x roll pitch roll sequence was chosen to be used here since X
6. Mod lisation dynamique des syst mes articul s par des vecteurs translation et des matrices rotation Prise en compte des rigidit s par des multiplicateurs de Lagrange Simulations du mouvement laide d un code en C Ph D thesis University of Poitiers France 2003 SPRING K W Euler parameters and the use of quaternion algebra in the manipulation of finite rotations A review Mechanism and Machine Theory 21 5 pp 365 373 1986 http en wikipedia org wiki Euler_angles
7. SISOM 2009 and Session of the Commission of Acoustics Bucharest 28 29 May SATELLITE ATTITUDE DYNAMICS USING EULER ANGLES 3D ROTATION PARAMETERIZATION Dan DUMITRIU Cornel SECARA Institute of Solid Mechanics Romanian Academy Str Constantin Mille nr 15 010141 Bucharest Romania Corresponding author Dr Ing Dan DUMITRIU e mail dumitri04 yahoo com Abstract This paper studies the attitude orientation dynamics of a CubeSat 1 e a 10x10x10 cm cube small satellite with the mass of 1 kg The dynamics considerations presented here remain valid for bigger satellites and for satellites of non cubic forms No atmospheric drag solar radiation pressure or other perturbations were considered in this paper The attitude dynamics of the satellite with respect to its center of mass is formulated using Euler angles 3D rotation parameterization Based on the direct dynamics formulation an inverse dynamics approach has been developed as satellite orientation guidance amp control strategy and will be presented in a future paper The difficulty of the case considered is the fact that the attitude control is performed using a two axis momentum reaction wheels system so not all three axes are controlled but only the x and y axes If a three axis moment control based on inverse dynamics is quite trivial our two axis moment control is not obvious at all one can find always unfavorable cases when the proposed control strategy fails The two axis mom
8. craft Attitude Estimations Using Phase Information of GPS Signals Thammasat Int J Sc Tech 8 pp 44 53 January March 2003 BOLANDI H BADPA A NASIRI SARVI M Application of passive gg aero stabilization for near Earth small satellites Iranian Journal of Information Science amp Technology 3 1 pp 57 71 January June 2005 RODRIGUES D S S ZANARDI M C Spacecraft Attitude Propagation with Different Representations In Advances in Space Dynamics 4 Celestial Mechanics and Astronautics editor H K Kuga pp 143 150 Instituto Nacional de Pesquisas Espaciais INPE S o Jos dos Campos SP Brazil 2004 KANG J Y SHIN K K Analysis of the Antenna Pointing Instability of a Satellite in Spin Stabilized Injection Mode ETRI Journal 16 2 pp 27 41 July 1994 IZZO D BEVILACQUA R VALENTE C Optimal large reorientation manoeuvre of a spinning gyrostat 6th Cranfield Conference on Dynamics and Control of Systems and Structures in Space 2004 Cranfield University Press Bedfordshire UK 2004 pp 607 616 STUCK B W Solar Pressure Three Axis Attitude Control Journal of Guidance and Control 3 2 pp 132 139 1980 WIE B Solar Sail Attitude Control and Dynamics Part 1 amp Part 2 Journal of Guidance Control and Dynamics 27 4 pp 526 535 Part 1 pp 536 544 Part 2 July August 2004 GOGU G COIFFET Ph Repr sentation du mouvement des corps solides Editions Herm s Paris 1996 DUMITRIU D
9. cs using Euler Angles 3D Rotation Parameterization This attitude matrix A transforms an arbitrary vector v from the satellite body frame coordinates to the orbit defined LVLH coordinates V 4 AV That s why it is also called 3x3 direction cosine matrix from body frame coordinates to inertial LVLH coordinates In particular A transforms the moving axes of the body frame into the fixed axes of LVLH i e Xiv ZAX Vovig ZAY and Ziy A Zea sat The singularities associated to Euler angles parameterizations are well known e g in our x y x case if B is a multiple of n then X y and xX coincide and the rotations of angles a and y are getting superposed sat in a single rotation of angle a y around the same axis X X yy In this singular case only the sum of a and y or their difference matters thus there is an infinity of solutions a and y for obtaining the concerned rotation 23 So when working with Euler angles parameterizations one has always to deal with these singularities and be careful to avoid their negative effect on solving the equations of motion From 2 it comes easily the derivative of the attitude matrix HsinB BoosBsiny 7sinBcosy BoosBcosy vsinBsiny l l a csinacos y ycosasiny cicosarcosBsiny amp sinasiny ycosacosy acosacos ceosy A acosasinB Bsinacosp i of fis Se ee S l BsinasinBsiny ysinacos cosy Bsinasin cosy ysinacos siny a CLCOSCLCOSy ysinasiny csinarcosBs
10. d attitude in characterized by a B y The satellite is considered already stabilized before this maneuver i e 0 B 0 y O At the end tr of the maneuver the satellite should have null velocity i e amp 0 Be 0 yp 0 and should be oriented towards the Earth in order to take pictures i e a 0 B 0 Ye O Z must coincide with Z and we will also impose X a Xiv and Ya Yiyip To perform this rotation maneuver in At maneuyv in t t seconds the satellite considered in this paper disposes of active control torques only on two axes more precisely of Ty on xX axis and of T on Y axis Table 1 shows two cases of attitude control maneuver Case and Case 2 successfully performed both for At 100 s maneuv Table 1 Satellite final state tp for Case 1 and 2 of attitude control Qin Bin Yin Case 1 82 29 Case 2 131 62 96 35 109 76 Fig 3 shows the direct dynamics results as well as the torques which produced Case 1 direct dynamics results Fig 4 shows Case 2 direct dynamics results with the corresponding torques 180 Alfa 0 015 torque Tx mn M j Beta genes torque Ty E l amp j 120 Gama 0 01 4 iy 04 SS a Ea 60 by E 0 005 ors a a m7 i Uo o 3 _ La timef s 0 015 4 _ time s Fig 3 Case 1 results in what concerns the Euler angles and the corres
11. e for each axis needed for a correct propagation of the attitude data 4 9 11 a nanospacecraft sized star tracker 14 etc The currents produced by the solar cells can also provide information about the solar angle on the corresponding side of the cube 8 13 Once the attitude information available one must extract determine from this information the attitude of the satellite more precisely to compute the direction of the magnetic field and of the sun according to the position along the orbit This attitude determination is quite a complex estimation it needs different algorithms and models such as a propagator an Earth s magnetic field model a sun vector model etc 4 For example an Extended Kalman Filter EKF estimator can be used for attitude determination using a time series of measurements e g GPS signals and dynamic and or kinematic models 15 In what concerns the actuators implementing the control part of ACDS many CubeSat projects use magnetic torque coils magneto torquers which generate the magnetic fields necessary to orient the satellite 3 4 7 8 More precisely the magnetic field provided by current in the coil interacts with the Earth s natural magnetic field thus producing the resultant torque for orienting the satellite Magneto torquers are not really active actuators they rather realize efficient passive stabilization and control provided that the torque generated by the permanent magnets should be larger than th
12. e other perturbing torques acting on the CubeSat Another passive solution for satellite stabilization is to take some profit from a perturbation 1 e the atmospheric drag 16 These passive stabilization solutions are very frequent for low altitude satellite missions because of their reduced cost and power consumption Several actuator system choices can be made following the attitude control payload a to stabilize the satellite using three magneto torquers one for each of the three orthogonal axes of the CubeSat 3 b to use only two magneto torquers to passively stabilize two of the CubeSat axes 7 and for the third axis one or two momentum wheels can possibly be used as active actuators 9 12 14 c or to use three active actuators e g Momentum reaction wheels one active actuator per each satellite axis 14 Other choices are possible e g Goliat uses only a two momentum wheels actuation system to control its 3D attitude motion 11 This case will be studied here The handicap of controlling only two axes for a 3D rotation motion must be bypassed using an appropriate control This two axis control case of an underactuated satellite 1s very interesting since this special situation can occur even for three axis controlled satellites in the accidental case when one moment thruster fails 2 SATELLITE ATTITUDE DYNAMICS PROBLEM This paper concerns only the attitude maneuver control 1 e the process of controlling t
13. ent control problem can occur also for a three axis controlled satellite in the accidental case when one moment thruster fails Keywords CubeSat Attitude Determination and Control Direct Dynamics 3D Rotation Parameterization Euler Angles 1 INTRODUCTION CubeSat is a standard 10x10x10 cm cube small satellite with a mass of 1 kg maximum proposed since 1999 jointly by Stanford University and California Polytechnic State University CalPoly 1 3 Briefly CubeSat can be defined as 1 liter 1 kilogram 1 Watt 6 due to its small size associated with very low energy consumption The standardization of this miniaturized satellite called also pico satellite is aimed to reduce the launch costs and to induce a better collaboration between universities working on the same topics and model The first CubeSats were launched in 2003 and nowadays it is quite a popular tool for space science technology demonstration with pedagogical purposes Thus many worldwide universities have developed CubeSat projects involving teams of students supervised by their professors Let s cite here just a few most recent CubeSat projects 12 developed by different European universities and research centers Ecole Polytechnique F d rale de Lausanne Switzerland 4 University of Vigo Spain 5 Universit de Liege Belgium 6 University of Trieste Italy 7 Narvik University College Norway 8 Wurzburg University Germany 9 Romanian Space Agency in c
14. erns the attitude dynamics and control of the satellite which is part of the Attitude Determination and Control Subsystem ADCS payload where position velocity and orientation are determined and controlled In terms of control the ADCS shall perform 4 8 1 detumbling after launch or after some free flying period 1 e to reduce the angular velocity spinning rate of the satellite to a minimum 2 attitude stabilization 1 e maintaining an existing orientation 3 attitude maneuver control i e to orientate the satellite from a current initial attitude to the attitude desired for taking pictures or for some other action The ADCS determines and actively controls the attitude of the satellite using a appropriate sensors to provide feedback on satellite orientation b an actuator mechanism to implement the control c the on board computer microcontroller unit which receives the sensor measurements then determines the attitude and finally computes the control signals The control can also be commanded from the ground station Dan DUMITRIU Cornel SECARA 100 There are various sensors providing information for attitude determination sun sensors e g Six photodiodes positioned on the sides of the cube to find the direction of the Sun 8 11 12 three axis magnetometers measuring the Earth magnetic field vector intensity and direction that will be compared with the model 3 4 8 9 11 14 three axis gyroscope to measure the spinning rat
15. he reorientation rotation of the satellite from one attitude to another The satellite is considered already stabilized before this maneuver Also the attitude maneuver control can be followed by attitude stabilization to maintain the new attitude 4 8 More precisely only the direct dynamics formulation 1s presented here while an inverse dynamics control technique will be presented in a future paper The actuation considered is similar with the one conceived on Goliat CubeSat 11 i e a two axes X and y momentum wheels system where each micro motor is fixed in the structure s hole from the center of a face The maximum torque available is 7 0 12 10 N m the minimum torque being null a Servo amplifiers control the torque and the speed between the limits allowing to perform a full 360 rotation in around 40 s 11 Since no magneto torquers where used as actuators but momentum reaction wheels no interaction between the CubeSat satellite and Earth s magnetic field was considered in this paper Fig 1 shows a view from above the orbital plane indicating the orthogonal axes of the Local Vertical Local Horizon LVLH reference frame used to locate the satellite with respect to the reference orbit The LVLH origin is located on the reference orbit Z points in the nadir direction y 1s normal to the orbital plane opposite the angular momentum vector of the reference orbit finally X
16. iny amp cosasiny ysinacosy sinacos Bcosy csinasinf Bcosacosf l oe l l l l BcosasinBsiny ycosacospcosy BcosasinBcosy ycosacos siny Let us calculate now the angular velocity tensor j as 20 23 T4 T4 TpTpT 4 j ATA R R R R R RIA 0 asinBcosy Bsiny dsinBsiny Pcosy 3 asinBcosy Bsiny 0 acosB y asinBsiny Bcos7 acosB 0 where o O O i is the angular velocity vector referenced with respect to the inertial LVLH frame but expressed in body frame coordinates 15 23 and the linear mapping j is nothing else than the cross product on the left by it has the following skew symmetric form 0 O J 0 0 4 0 By identification between 3 and 4 one obtains the components in the body frame O Xat Y sats Zs Of the angular velocity vector 16 19 21 acosp y O sin Bsin y Bcos y 5 3 sin Bcosy sin y Dan DUMITRIU Cornel SECARA 104 4 CUBESAT ATTITUDE DYNAMICS USING EULER S EQUATION OF MOTION The rotation motion of a rigid body is generated by the torques applied on this body For CubeSat the only torques considered here are the two torques produced by the two momentum reaction wheels used as actuators gathered in the torque vector T expressed in the body frame O Xats Yeats Zsat aS T T T a T Fa T 6 0 In orbit perturbations acting on CubeSat e g atmospheric drag or solar radiation pres
17. n CICLOP camera Goliat satellite Bus consists of a number of different subsystems 11 Electrical Power Supply EPS Attitude Determination and Control Subsystem ADCS Radio Communications COMM on board computer etc Each of these customized components fulfills a specific task in order to support the payloads of Goliat For example the power supply of around 2 Watts is generated by solar cells panels placed on the sides of the cube and there is also a reloading battery as buffer for temporarily power peaks i e if for some reason eclipse failure the solar panels don t supply enough power 7 8 11 12 In what concerns the ground stations which will ensure the radio communications with the satellite for Goliat there is one ground station in Bucharest and the second one with a 4m dish telescope is located in Cluj Napoca 11 The camera mounted on a CubeSat should be able to take pictures of the Earth pictures as clear as possible e g CICLOP camera mounted on Goliat has an expected picture area of 50x70 km with a pixel area of 21x28 m 11 One should be able to discern between land and water for example on the downloaded pictures For that it is a must to have a good stability of the camera so of the CubeSat and the appropriate orientation all this in the hazardous and perturbed orbital environment If the CubeSat is not stable then the pictures will show us a blurred image of the area the picture has taken 4 8 This paper conc
18. ollaboration with University of Bucharest and University Politehnica of Bucharest 10 11 Let s cite also pico satellites XI IV and XI V developed by University of Tokio 13 and CanX 2 Canadian Advanced Nanospace eXperiment 2 developed by the Space Flight Laboratory at the University of Toronto Institute for Aerospace Studies 14 Beside its pedagogical role the main CubeSat space exploration purpose is to collect in orbit scientific information and to send this information to the Earth ground station The scientific issues are numerous weather observation and atmospheric measurements airglow phenomena 4 ionizing radiation 5 solar weather magnetic field and atmospheric density measurements in near Earth space radiation environment 7 greenhouse gases detection using a miniature atmospheric spectrometer 14 in orbit temperature measurements 8 micrometeorite detection 10 11 cosmic radiation dose measurement 11 GPS atmospheric occultation 14 etc 99 Satellite Attitude Dynamics using Euler Angles 3D Rotation Parameterization communication technology testing software defined reconfigurable radio SRAD system 5 communication protocol in space for D STAR Digital Smart Technology for Amateur Radio 6 VHF UHF communication with the ground station 8 9 other amateur radio frequency experiments 13 experimental validation of space navigation technology solar panel deployment system 5 Attitude De
19. ponding torques to obtain Case 1 direct dynamics Dan DUMITRIU Cornel SECARA 106 180 0 015 7 torque Tx 150 q4 O G a torque Ty 120 4 0 01 _ 904 E M 3 ns z 0 005 a0 v gt ei s 1 p 5 30 5 0710 20 30 40 50 60 70 80 90 00 D 60 0 005 90 120 4 0 01 4 150 4 180 _ time s _ 0 015 time s Fig 4 Case 2 results in what concerns the Euler angles and the corresponding torques to obtain Case 2 direct dynamics One can see that in Case and 2 the attitude control maneuver was successfully performed But there are still cases where the maneuver is not converging so there is still work to do before proposing in the next paper a reliable attitude control technique based on inverse dynamics 5 CONCLUSIONS AND FUTURE WORK After an extended introduction trying to present general considerations about different CubeSat projects this paper develops the differential equations which are governing the attitude dynamics of a CubeSat A usual Euler angles parameterization was used to represent the 3D rotation of the satellite Based on the direct dynamics formulation an inverse dynamics approach can be used as control technique for performing attitude control maneuver Improving such a control technique is an ongoing work and will be presented in a future paper The purpose is to have a reliable Attitude Determination and Control Subsystem ADCS able to rotate on command the satellite to a desired attitude
20. rsit de Li ge http www2 units it atmocube AtmoCube Cubesat project developed by University of Trieste http hincube hin no Hincube Cubesat project developed at Narvik University College Norway http Awww 7 informatik uni wuerzburg de UWE 3 CubeSat project developed by Wurzburg University Uo OANA NN fH 107 Satellite Attitude Dynamics using Euler Angles 3D Rotation Parameterization 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 http ro wikipedia org wiki GOLIAT Goliat CubeSat project developed by Romanian Space Agency Mugurel BALAN et al Romanian CubeSat Project members GOLIAT project overview presentation CubeSat Developers Workshop 9 10 August 2008 http forum spacealliance ro Space technology and education portal FUNASE R TAKEI E NAKAMURA Y NAGAI M ENOKUCHI A YULIANG C NAKADA K NOJIRI Y SASAKI F FUNANE T EISHIMA T NAKASUKA S Technology demonstration on University of Tokyo s pico satellite XI V and its effective operation result using ground station network Acta Astronautica 61 7 8 pp 707 711 October 2007 SARDA K EAGLESON S CAILLIBOT E GRANT C KEKEZ D PRANAJAYA F ZEE R E Canadian advanced nanospace experiment 2 Scientific and technological innovation on a three kilogram satellite Acta Astronautica 59 1 5 pp 236 245 July September 2006 PURIVIGRAIPONG S Space
21. sure are neglected in this paper So there is no torque generated by the in orbit perturbation forces Considered with respect to the inertial LVLH frame but expressed in the body frame O Xats Veat gt Zsat gt Euler s equation of motion for the CubeSat relates the time derivative of the angular momentum vector H Jo to the torque vector T 3 15 16 18 21 j eee dt JO 7 where it was considered that J 0 for the considered J given by 1 By derivation of 5 one obtains the time derivative of the angular velocity vector amp cosB aBsinB sin Psin y Bcosy cos siny aysinBcosy Bysiny 8 asinBcosy Bsiny aBcosBcosy caysinBsin y By cosy Taking into account the expression 8 of Euler s equation of motion 7 rewritten as JT provides cos 7 aPsinB JT sin sin y Bcosy amp cos sin y aysinBcosy Bysiny J T 9 cisinBcosy Bsiny aBcosBcosy wysinBsin y Bycosy JTT where J stands for the inverse of the moment of inertia tensor J and where JT 1 1 2 3 is the i th component of J T vector From 9 one obtains easily the explicit expressions of amp B and 7 JT sin y JT cos y amp BcosB By sinf B JT cosy JT sin y aysinB 10 f FD ERIO T siny IT cosy Bi S These expressions 10 are the differential equations characterizing the direct dynamics of CubeSat s attitude
22. termination and Control Subsystem ADCS 8 9 formation flying technology demonstration 14 surface material experiment that will measure the effects of atomic oxygen on advanced materials for satellites in low Earth orbit 14 Earth monitoring taking Earth pictures and downloading them to the ground station 8 10 11 13 and astronomy The new Vega launch vehicle of the European Space Agency ESA is scheduled to deploy in November 2009 a number of 9 CubeSats on an orbit characterized by the following parameters at least for AtmoCube 7 semi major axis a 7153 14 km eccentricity e 0 0594 inclination i 71 It results the Ae following period T of the reference orbit T 27 a 6020 8 s 100 35 min where u G Mpan 3 986 10 so It comes also S number of orbits day 14 3 AQ node shift orbit due to Earth rotation 25 09 deg orbit 358 96 deg day average CubeSat velocity 7 458 km s The average visibility at the ground station will be around 8 min 7 11 Among the 9 CubeSats that will be deployed on this orbit there is Goliat CubeSat developed by the Romanian Space Agency in collaboration with University of Bucharest and University Politehnica of Bucharest 10 12 Goliat is planned to accomplish three scientific payloads 10 11 low Earth orbit micrometeorite detection SAMIS platform based on impact sensors 11 cosmic radiation dose measurement Dose N experiment Earth observatio
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