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UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2008 2008 Frederick A. Leve To my mother and father ACKNOWLEDGMENTS I thank my supervisory committee chair (Dr. Norman FitzCoy) for giving me the chance to embark on this innovative research. I thank him for his patience and advisement. Without his help I would have certainly been lost. I would like to thank my fellow colleagues in my research group were of great assistance to me; Dr. Andrew Tatsch, Shawn Allgeier, Andrew Waldrum, Jaime Bestard, Sharan Asundi, Dante Buckley, Vivek Nagabhushan, Josue Munoz, Nick Martinson, and Will Mackunis. Lastly I would also like to thank my supervisory committee (Dr. Warren Dixon and Dr. Gloria Wiens) for their assistance in validating my master's research. TABLE OF CONTENTS page A CK N O W LED G M EN T S ................................................................. ........... ............. ..... LIST OF TABLES ................... ..................................................... LIST OF FIGURES. .................................................................... 8 L IST O F A B B R E V IA T IO N S ....................... ........................................................... 10 CHAPTER 1 INTRODUCTION ............... .......................................................... 13 R research M motivation .......... ........... ................................ .. .. .... ........ .......... ..... 13 M issio n E x am p le s ............................................................................................................. 14 R e se a rc h F o cu s ..........................................................................................................14 2 ATTITUDE CONTROL ACTUATORS................................................... ..................16 Types of A attitude A ctuators........................................................................... ....................16 T ypes of C M G s .................. .... .............. ...................................................... 19 Single Gimbal Control Momentum Gyroscopes .................................. ...............20 Double Gimbal Control Momentum Gyroscopes................................. ...............20 Variable Speed Control Momentum Gyroscopes...................... ..................21 3 SGCMG DYNAMICS AND PERFORMANCE CHARACTERIZATION ..........................24 D y n a m ic s .......................................................................................................................... 2 4 Torque Am plification ................................... .. ........... .. ............28 S in g u laritie s ............................................................3 0 External Singularities .................................. .. .. ........ .. ............30 Intern al S in gu larities ................. ............................................ ........ .......... .. .... 1 Singular Surfaces................................................ 31 S te e rin g L o g ic s ................................................................................................................. 3 3 M om entum M anagem ent....................................................... .................. ...............38 External Angular M momentum Offloading ............................................ ............... 39 Internal Angular M momentum Offloading ............................................. ............... 39 4 SGCMG ACS CONFIGURATIONS ..............................................................................43 R ooftop C configuration ........ ........................................................................ ..................43 Box Configuration .................................. ... .. .... .... ................. 43 Pyramid Configuration ....................... ................... ...................43 SOBEK Pyram id Configuration ......................................................... .............. 44 SO B EK H ardw are ........................ .. ........................ .. .... ........ ........ 45 SOBEK M mechanical D design .................................................. ........ .... ............... 46 SOBEK Mechanical, Power and Output Specifications.................. .................47 5 SOBEK Attitude Determination...................... ...... .............................. 47 5 R E SU L T S .............. ... ................................................................52 S im u latio n M o d el ...................................................................................................................5 2 Lyapunov Stability Analysis of EMK Attitude Controller.................................. 53 Lyapunov Stability Analysis of Actual Attitude Controller with State Uncertainty .......55 Sim ulation R results and D discussion .............................................................. .....................56 Experim ental R results and D iscussion.......................................................... ............... 58 6 CONCLUSION AND FUTURE RESEARCH ................................ ........................ 70 C o n clu sio n ................... ...................7...................0.......... F utu re R research ................................................................70 APPENDIX OffTheShelf Rw, CMG, And Magnet Torquer Data Specifications................ ..................71 L IST O F R E F E R E N C E S .............................................................................. ...........................74 B IO G R A PH IC A L SK E T C H .............................................................................. .....................78 6 LIST OF TABLES Table page 21 Attitude control actuators and their specifications.................................. ...............21 41 Mechanical and Power Specifications of the SOBEK ACS....................... ...........48 51 Sim ulation param eter values ...................... .. .. ............. ..............................................6 1 Ai Offtheshelf performance specifications for RWs .........................................................71 A2 Offtheshelf performance specifications for CMGs .....................................................72 A3 Offtheshelf performance specifications for Magnet Torquers .............. ...................73 LIST OF FIGURES. Fig. page 11. Onorbit assembly ................................................................... 15 12. E arth m monitoring in L E O .............. ................................................... ......... ......... 15 13 P ostdocking stabilization ........................................................................ ...................15 21. Magnet coil from Tokyo Institute of Technology Cute1.7 CubeSat.............................21 22 D ynacon m miniature R W .......................................................................... ....................22 23. Semilog plot of input power versus output torque for offtheshelf RWs and CMGs ....22 24. Semilog plot of mass versus output torque for offtheshelf RWs and CMGs................22 25. SSTL microsatellite SGCM G flown on Bilsat1 ........................................ ...................23 26. DGCMG developed by L3 Communications ........................................ ............... 23 31. The SGCMG with gimbal coordinate axes.................. .................................... 40 32. A 323 rotation sequence through angles 0,, .................................................. 41 33. Intersecting planes of the spanning of SGCMG angular momentum ..............................41 34. Angular momentum envelope for 0 = 35.26 and e = [1,1,1,1] .................. ...............41 35. Internal singular surface 0 = 35.26 and = [1,1,1,1] .............................................42 36. Z erom om entum configuration .............................................................. .....................42 41. Honeywell rooftop configuration ..............................................................................48 42. H oneyw ell box configuration ........................................ .............................................48 43. External singular surface for box type SGCMG ACS .................................................49 43. H oneyw ell pyram id configuration .......................... ........................................ ............49 44. Pyram idal SG C M G A C S ......................................................................... ...................50 45. R edition of SOBEK tested ......................................... .......... ....... ............... .50 46. A C S w ith hardw are com ponents ............................................. ............................. 51 47. PhaseSpace attitude determination system ............................................ ............... 51 8 51. ACS simulation plots for k = 0.08 and c = 2.5k: A) Quaternion error vector elements versus time, B) Gimbal rates versus time, C) Angular rates versus time, and D) Singularity param eter versus tim e. ............................................ ............................ 62 52. ACS simulation plots for k = 0.08 and c = 2.5k: A) Quaternion error vector elements versus time, B) Gimbal rates versus time, C) Angular rates versus time, and D) Singularity m measure versus tim e. .............................................. ............................. 63 53. ACS simulation plots for k = 4 and c = 3k: A) Quaternion error vector elements versus time, B) Gimbal rates versus time, C) Angular rates versus time, and D) Singularity m measure versus tim e. .............................................. ............................. 64 54. SOBEK tested GUI .............. ....................................... ............... 65 55. ACS experimental plots for k = 0.8 and c = 2.5k: A) Quatemion error vector elements versus time, B) Gimbal rates versus time, C) Angular rates versus Time, and D) Singularity measure versus time. ........................................ ....................... 66 56. ACS experimental plots for k = 0.24 and c = 2.0k: A) Quatemion error vector elements versus time, B) Gimbal rates versus time, C) Angular rates versus time, and D) Singularity m measure versus tim e ...................................................... ............... 67 57. ACS experimental plots for k = 0.32 and c = 3.0k: A) Quatemion error vector elements versus time, B) Gimbal rates versus time, C) Angular rates versus time, D) Singularity m measure versus tim e. .............................................. ............................. 68 58. ACS experiment quaternion error of Moore Penrose solution ..................................69 LIST OF ABBREVIATIONS ACS Attitude control system API Application program interface CMG Control moment gyroscope CM Center of mass DCM Direction cosine matrix DoD Department of Defense DOF Degree of freedom EADS European Aeronautic Defense Space Company EMK Exact model knowledge GUI Graphical user interface FACETS Flywheel Attitude Control and Energy Transmission System IPACS Integrated Power and Attitude Control System LEO Low Earth Orbit MW Momentum Wheel ORS Operationally Responsive Space RW Reaction Wheel R2P2 Rapid Retargeting and Precision Pointing SGCMG SingleGimbal Control Moment Gyroscope SOBEK Spacecraft Orientation Buoyancy Experimental Kiosk SSTL Surrey Space Technology Lab UUB Uniformly Ultimately Bounded Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science DEVELOPMENT OF THE SPACECRAFT ORIENTATION BUOYANCY EXPERIMENTAL KIOSK TESTBED By Frederick Aaron Leve May 2009 Chair: Norman FitzCoy Major: Aerospace Engineering Most satellites are unique and therefore usually take 10 to 15 years to design, fabricate, test, and finally launch. To expedite future advances in space technology, setting standards in terms of "blackboxed" subsystems and interfaces, drastically reduces the time and long term costs needed to complete these tasks. What is meant by "blackboxed" is a stand alone subsystem that is adaptable enough to work with any other subsystems without alteration. Government organizations such as the Department of Defense's (DoD's) Operationally Responsive Space (ORS) office take these tasks into consideration and look to change this pattern by expediting the processes of design, fabrication, test, evaluation, and launch. To accomplish these tasks, smallsats which range from 1 to 1000 kg in mass, are considered for their cheaper platforms and launch costs. With utilization of smallsats of the pico (0.11 kg), nano (1 to 10 kg), and microclasses (10100 kg) for space missions, obstacles occur in terms of power, mass, and volume constraints. Many of these same smallsats obstacles relate to the attitude control system (ACS). Despite these obstacles for many space missions, small satellites must be held to the same pointing requirements as their larger counterparts to be of use. To compensate for these obstacles while maintaining close pointing requirements, innovations in attitude control actuators and determination sensors that satisfy these constraints need to be developed. An excellent choice for attitude control actuators of smallsats are those of the flywheel sort known as zeromomentum and momentum bias actuators. These actuators have the ability to perform precision attitude maneuvers without use of propellant. There are two classes of flywheel attitude control actuators that are used in satellites, reaction/momentum wheels (RWs/MWs) and control moment gyroscopes (CMGs). Due to the scarcity of CMGs available for small satellites below the microclass, RWs are presently utilized. An analysis completed in this thesis shows that although the current stateoftheart CMGs are too massive to fit inside smallsats, the performance relationships in terms of output torque per mass and input power are more favorable than that for RWs. This analysis is an additional motivation for the research in this thesis, which is carried out on the premise that an ACS consisting of singlegimbal control moment gyroscopes (SGCMGs) can meet the performance requirements providing rapid retargeting and precisionpointing (R2P2) for smallsats while enduring their power, mass, and volume constraints. To further test this premise, the topic of this thesis is the development of a ground based SGCMG testbed known as Spacecraft Orientation Buoyancy Experimental Kiosk (SOBEK) was developed to validate the utility of using SGCMGs for smallsat attitude control. The thesis justifies the reasons for using SGCMGs on SOBEK, discusses the spacecraft and actuator dynamics and kinematics associated with an ACS containing CMGs, addresses the geometrical configurations of systems of these actuators and the configuration chosen, and then finally validates the testbed numerically through simulation and experimentation. CHAPTER 1 INTRODUCTION Research Motivation The Department of Defense's (DOD's) Operationally Responsive Space (ORS) office [1] seeks to better the capabilities of development, fabrication, testing, evaluation and launch of space assets. To accomplish the goals of the ORS office, standards such as "blackboxed" systems are considered useful to aid these tasks and reduce the uniqueness factor of every satellite project. To accomplish some of these tasks, it is believed that switching from larger more massive satellites to smallsats of the pico (0.11 kg), nano (110 kg), and microclasses (10100 kg) opens up the doors for a wide variety of missions and provides a quicker and less costly method (i.e., small satellites are cheaper and faster in terms of integration and launch) to develop these blackboxed technologies. The budget of developing smallsats combined with componentsofftheshelf (COTS) makes it possible for some universities and smaller aerospace companies to contribute to the knowledge base and evolution of space technologies. Numerous university satellite projects consisting of missions involving CubeSat and nanosatellite projects have utilized COTS and been flight tested successfully [24]. Standardizations for smallsats in the academic sector are being led by California Polytechnic Institute and Stanford University. These endeavors include CubeSat Kit CubeSat bus architecture (Pumpkin Inc.) and PPod launch vehicle integration [57]. Work is also being done on standardization of satellite bus architecture in the military sector such as the ANGELS, XSS11, and the tactical satellite programs at AFRL [8] and [9]. Although smallsats are a viable choice for testing of new space technologies, they have limited attitude control and propulsive capabilities which leaves some in the aerospace community to regard smallsats as toys or space debris with no significant functionality. This perception of smallsats can be challenged with the development of low power and mass attitude 13 control systems (ACSs) which are near the same pointing and slew requirements of those for larger satellites and hence are still able to accomplish an assortment of many space missions for smallsats where propulsive capabilities can be limited or nonexistent. Mission Examples Some space missions that could possibly utilize smallsats are laser communication, space science, onorbit assembly, satellite servicing, formation flight, blue force tracking, and Earth monitoring. Smallsats that have propulsive capabilities would most likely have a minimum amount of fixed thrusters, therefore their attitude and translation would be coupled. Taking this into account, for a smallsat distribution used for such missions as onorbit assembly shown in Fig. 11, precision in terms of position and orientation need to be maintained for collision avoidance and attitude control where redundant thrusters is most likely not possible. Earth sensing and blue force tracking which have satellite systems in Low Earth Orbit (LEO) require tracking of a specific point on the surface of earth. This becomes nontrivial due to smaller orbital periods (i.e., faster orbital rates). To acquire a longer time of coverage, higher slew rates while maintaining precision are needed. An example of this is shown in Fig. 12. When dealing with missions such as satellite servicing, there may be a situation when the satellite being serviced is noncooperative and is more massive than the satellite which is providing the servicing. In this situation the ACS on board the servicing satellite must be able to provide large amounts of torque responsively to stabilize the system. An illustration of this is shown in Fig. 13. Research Focus The research focus of this thesis is in the development of a ground based testbed to test and evaluate propellantless ACS methodology for smallsats which can endure the volume, mass, and power constraints while still maintaining performance in terms of high torque, slew rates, and precision. The ACS methodology presented in this thesis attempts to validate singlegimbal control moment gyroscopes (SGCMGs) as a viable option of attitude control actuators for smallsats which can contribute high torque, slew rates, and attitude precision while meeting their volume, mass, and power constraints. Examples of other attitude actuators including SGCMGs are discussed in the next chapter for comparison. 0 Z I ,/I Fig. 11. Fig. 12. Onorbit assembly Earth monitoring in LEO Postdocking stabilization Fig. 13. CHAPTER 2 ATTITUDE CONTROL ACTUATORS Types of Attitude Actuators Attitude control actuators for satellites can be passive or active. Passive attitude control actuators such as gravity gradient booms use the gravitational field of the Earth to provide a bounded attitude between 0, = +5 in the orbital plane. Because the gravitational field is conservative, gravity gradient torques ,_, do not dissipate the satellites kinetic energy or dump its angular momentum. The torque is dependent on the distance R between the spacecraft's center of mass (CM) and Earth's CM, the spacecraft principle moments of inertias J,, and the small angles of deviation between the spacecraft's principle axes and the Local verticalLocal horizon coordinate frame 0,. For small angular displacements, the torque is expressed as, (J2 J3 )81 rg = 3 '3( J3 J1 02 Another type of passive attitude control utilizes hysteresis rods or permanent magnets. Typically permanent magnets carry accuracy on the same order of magnitude as gravity gradient booms but cause a satellite to flip at the poles. These actuators like gravity booms are conservative and do not dissipate energy or dump angular momentum. Hysteresis rods on the other hand are passive actuators that may dissipate energy and dump angular momentum. These actuators can be used to null the angular rates of satellites with residual angular momentum after launch [10]. There are three main categories of active attitude control actuators. These categories include magnet torquers, reaction control devices, and flywheel actuators. Magnet torquers are the most common attitude control actuators utilized in smallsats (e.g., magnet coils and torque rods). These actuators are cheaper and lighter than other attitude control 16 actuators used onboard smallsats. Figure 21 shows an example of a magnet coil for attitude control. Magnetic torquers produce a torque zg as a reaction to the change of the local magnetic field of the satellite to that of the Earth. This torque is modeled as an interaction (cross product) between, the magnetic moment of the spacecraft / and Earth's magnetic field direction B as, r =,uxB Since the torque is a function of the cross product between the satellite's magnetic moment and Earth's magnetic field, magnet torquers suffer from a singularity which limits their actuation where no components of torque are available along / and B Due to the change of the local magnetic field of the satellite, sensors for attitude determination such as magnetometers are unable to take readings during use of these actuators until the residual magnetic field of the satellite has decayed. These actuators loose performance as a function of distance squared hence are less useful for satellites at orbits with higher altitudes than LEO. Reaction control devices such as thrusters require propellant which is at this time not available for the majority of nano and picosatellite missions. These devices produce a torque on the satellite via propulsion through jets of fluid, heated or cooled plasma and gas, or charged ions with use of a magnetic field. Thrusters that utilize the chemical energy within the propellant (e.g., hydrazine) produce greater magnitude of output thrust but require large radiators to dump the expended heat created. Thrusters of this sort are not realizable for smallsats which do not have the extra mass and volume to contain large radiators for the thermal protection system. The last type of attitude control actuators known as flywheel actuators can be broken down into two types, reaction and momentum wheels (RWs/MWs) and control moment gyroscopes (CMGs). These are the most accurate sources of attitude control. Reaction and momentum wheels work off the same principle by producing a reaction torque opposite in direction to the acceleration of a flywheel. The difference between MWs and RWs is that, RWs have a zero nominal operating speed (i.e., the speed that the wheels are spun back down to after the maneuver is completed). When spinning RWs/MWs back down to their operating speed, the satellite will revert to its original orientation due to the conservation of angular momentum. For this reason both MWs and RWs must have other actuators onboard such as thrusters or magnet torquers to dump the excess angular momentum, or these actuators will suffer angular momentum saturation of their flywheels. Three RWs/MWs are needed for full threeaxis attitude control of a satellite. An example of a RW used for CubeSat and nano satellite missions is shown below in Fig. 22. The second type of flywheel attitude control actuator known as CMGs provide an instantaneous gyroscopic torque along a torque axis defined as the axis perpendicular to the flywheel and gimbal axes. These actuators rely on shifting the direction of satellites body angular momentum to apply a torque. Three CMGs are needed for threeaxis control in general, although a fourth is typically added for singularity avoidance and/or redundancy. Performance characteristics for typical attitude control actuators were investigated and are shown in Tables 21 and Tables Ai through A3 in the appendix. Table 21 is taken from Space Mission Analysis and Design by Wertz [11]. The tables in Appendix A are for currently available RWs, CMGs, and magnet torquers and supplement that which is shown in Table 21 [1223]. In Table 21, the only attitude control actuators that can meet the performance demands of high precision while maintaining the low power requirement for small satellite systems are flywheel actuators consisting of RWs, MWs, and CMGs. Magnet torquers/coils do not produce sufficient output torque for a given input power and are less accurate due to the uncertainty of Earth's dynamic magnetic field model. A survey was conducted of existing RWs and CMGs. The results of the survey are shown in Figs. 23 and Fig. 24. The supporting data for these figures is provided by Tables Ai and A2 Figures 23 and 24 show that CMGs including that which is developed in the Spacecraft Orientation Buoyancy Experimental Kiosk (SOBEK) have a higher torque output for a given mass and input power than RWs. These figures also illustrate that CMGs are currently nonexistent for classes of satellites equal to or smaller than microsatellites. The scarcity of these actuators is due to their complexity in design and use. If this complexity was removed and they were designed in a "blackboxed" manner (i.e., accepting only an input torque or attitude and producing it), then they could potentially find more use in smaller satellites due to higher torque per mass and power advantage over RWs. There are three CMGs that have been developed for microsatellites by companies such as EADS/Astrium, Honeywell, and Surrey Space Technology Laboratory (SSTL) [2428] Fig. 25 shows one of these three, the SSTL singlegimbal control momentum gyroscope (SGCMG) which flew on the Turkish microsatellite Bilsat1 [29]. Types of CMGs There are two varieties of CMGs: ones with a singe controllable degree of freedom (DOF) known as SGCMGs and those with multiple controllable DOF known as doublegimbal control momentum gyroscopes (DGCMGs) and variablespeed control moment gyroscopes (VSCMGs). The benefits and drawbacks of each type of CMG is discussed next. Single Gimbal Control Momentum Gyroscopes SGCMGs such as the ones shown in Fig. 25 are the least mechanically complex form of CMGs. They utilize a single gimbal axis to produce a gyroscopic torque by rotating a constant speed flywheel about a gimbaled axis perpendicular to the spinaxis. A minimum of three SGCMGs are needed to obtain threeaxis attitude control with these actuators. However an ACS using SGCMGs in some cases utilizes four of these actuators to avoid performance inefficiencies known as internal singularities. The benefit of these actuators among other CMGs lies in the property of torque amplification which is the ratio of output torque from the ACS containing SGCMGs to input torque from the gimbal motors. Discussion of singularities as well as performance characterization of SGCMGs in terms of momentum management and torque amplification is discussed in chapter 3. Double Gimbal Control Momentum Gyroscopes Doublegimbal control momentum gyroscopes (DGCMG) are the most mechanically complex form of CMG actuators. These actuators produce torque by the same method as SGCMGs but possess a redundant mechanical controllable DOF in terms of an extra gimbal axis. This redundancy can be useful when a set of two actuators is used for underactuated attitude control of a satellite but can be harmful when gimbal lock occurs. Gimbal lock occurs when the spin axis of a DGCMG aligns with its axis of freedom and in consequence deprives the DGCMG of one of its controllable DOF thereby eliminating its useful properties. There are however acceptable methods of avoiding these singularities which in turn leaves DGCMGs the benefit of multiple degrees of freedom. Although with the addition of an extra gimbal to these actuators, they are heavier and more mechanically complex than other forms of CMGs which excludes them from being a viable option for smallsats. A figure of a DGCMG developed by L3 Communications for the International Space Station shown in Fig. 26. Variable Speed Control Momentum Gyroscopes VSCMGs are used for their extra DOF added through their reaction wheel mode where flywheel accelerations are nonzero. This extra DOF has its benefits in that it does not require steering logic and is able to produce the required torque when a singularity is encountered [30 32]. Although when in reaction wheel mode the flywheel accelerations require shaft power and therefore input torque thereby increasing their required power and reducing their available torque amplification. Another effect of the reaction wheel modes of VSCMGs is that it becomes near impossible to isolate vibrations from the varying flywheel speeds. In addition to the larger power requirement, less torque amplification, and induced vibration, the motor driver circuitry of VSCMGs is more complicated, and they require more computational and electrical power than standard SGCMGs to control. SGCMGs are considered here as the most viable option for the study of smallsat attitude control and hence were chosen for the SOBEK testbed. Table 21. Attitude control actuators and their specifications Actuator Output Accuracy Mass (Kg) Power (W) (deg) Magnetic Torquers 1 to 4,000 Am2 1 to 5 0.4 to 50 0.6 to 16 Hot Gas Thrusters 0.5 to 9,000 Nm t 5 v 1 to 5 variable N/A Cold Gas Thrusters <5Nm RW & MW 0.005 to 1 Nm 0.001 to 1 1 to 20 10 to 110 CMG 10 to 500 Nm 0.001 to 1 >1 5 to 150 Fig. 21. Magnet coil from Tokyo Institute of Technology Cute1.7 CubeSat 21 Dynacon miniature RW 40tA .n, 120 4 100 80 60 . 40 20 0 ^  * vV MVV r, RA_ SC:EEl/ logio(torque) mNm Semilog plot of input power versus output torque for offtheshelf RWs and CMGs 140 120 100 80 60 40 20  9 0 U U U U U * RV * CMc loglo(torque) mNm Semilog plot of mass versus output torque for offtheshelf RWs and CMGs Fig. 22. Fig. 23. Fig. 24. Fig. 25. SSTL microsatellite SGCMG flown on Bilsat1 Fig. 26. DGCMG developed by L3 Communications CHAPTER 3 SGCMG DYNAMICS AND PERFORMANCE CHARACTERIZATION Dynamics CMGs produce a gyroscopic torque about a torque axis ( orthogonal to both the gimbal and spin axes g, and ,, respectively. An illustration of the CMG gimbal frame 3 and how it relates to the spacecraft body frame 3, is shown in Fig. 31 with gimbal angles, velocities and accelerations, ,, and flywheel velocities and accelerations, Q_, . To analyze the torque generated by CMGs, we first develop the angular momentum expression and coordinatize it in the spacecraft body fixed frame. Application of Euler's equation assuming the external torque is zero yields the governing equation for the CMG system. This development is as follows. The total spacecraftcentroidal angular momentum of the system coordinatized in the body frame is BHC = B c B ( + "hC (31) with the spacecraft centroidal inertia tensor including the CMG components Bj the spacecraft body fixed angular velocity BW)B, and the total CMG angular momentum Bhe about the spacecraft center of mass (CM) coordinatized in the body frame. The spacecraft centroidal inertia BJc in Eq. (32) contains a fixed inertia BJ time n varying inertia from the combined gimbalwheel system CB G T B, and the parallel axis 1=1 components of inertia associated with each CMG m, (&RcT R1 B B~) Rc, with mass m, and position from the spacecraft CM to that of the flywheel, BR . Bc Bj + L[CB C +m, i( ,T& 1B BRc T) (32) 24=1 24 Assuming that BRc is held fairly constant and therefore its time derivative is zero, we differentiate BJC BCB with respect to time as, d(BC BOB) aBJ BB d, ( = + J_ GC = A, + JC G (33) dt d3 dt This CMG contribution of angular momentum B h coordinatized in the body frame is expressed as, 4 Bh =Y (34) h = BG, G h (34) i=1 and is found in by summing the contributions of each individual CMG angular moment 'h , in S which is transformed to the body frame by the DCM CBG Note that for SGCMGs, the DCM CBG varies only with the gimbal angles of the ith CMG S,. The angular momentum of the ith CMG G'h in G is, G,h 0L G hr,= 0 where 0, and I, are the ith flywheel's angular speed and centroidal inertia about its spin axis , and t, and Ig, are the ith gimbal rate and the centroidal inertia of the combined wheelgimbal system about its gimbal axis g,. Taking the time derivative of Bhc and observing that "~h = B h (3,, ,), we get, dBhC O8h dt8 9Bh d8 98h dB M = + + (35) dt 98 dt 85 dt MQ dt with the Jacobian coefficient matrices of the CMG states are defined as, a"h = A2 (37) 98 h a dB =B (38) ahC = C (39) aQ = Combining these terms we get, d B h d B A2 +B3+CO dt =2 = If we also combine A, with A2 which are both multiplied by the gimbal rates d, we have a more compact representation of the output torque from the CMGs in Eq. (310) where the complete Jacobian matrix is D = [(A + A2) B C. com+ (3e S=A, + B =(A+A) B C S =D( 5,,)X (310) dt dt  The form of Eq. (310) is similar to Schaub and Junkins [33] with exception of the different notation. The gyroscopic components of torque are derived from the product of the skew symmetric matrix I B, ] with the total centroidal angular momentum of the system 5 cH as, [B B]X BHc (BB)X Bj BWB +[BB]X Bh +h (311) [B] Cx + B Rc B&rI B B& T)Rc B RTB i=l The complete rotational equation of motion for this system in Eq. (312) is found through the sum of Eqs. (33), (310), and (311) giving, dBH d B C B o B + B +B 0) B ] J o B)B + [ BB] BD + + dt BBBB B[ B ZlCBG, CG, + m, (BRC, Bk Bkc Bkc, )] + (312) S J 2 G, BG _C _c =1 coxI BB CBG G CTG 1 (BkT Bk 1_Bk BkT)J BB If it is assumed that, BJC BJ * Ig << I < , * 1 < I,, and Ig, <<,I, , * Q, 0 and there are no external torques applied to the spacecraft the reduced equation of motion for this system is diH JE Bs B + ) B]x BJs BcoB hoAd+ [coB x B = 0 (313) dt I  with h0 = I,, and B = h0Ad . A 323 DCM through Euler angles ,, 0,, 8, where and 8, are the constant spacing and inclination angles, and 8, is the time varying gimbal angle transforms 3, to 3, for the fourCMG pyramid configuration in Fig. 31. This DCM is expanded as, ~c(8,)c()s(O,)s(6,)s(O) c(9,)s(O)s(9,)c(O)s(0,) c( )c(O,) CBG,= S(S,)C()+C(S,)S ,)S( ,) c(S,)cg ()sS(,)s( )s(O,) s )c(o,) s(8)+c(c,)c(O,) c(8,)s(8)c(O) c (0,) where c(*),s(*) = cos(*),sin () The 323 Euler sequence can be visualized in Fig. 32. Now that we have reduced the general equations of motion to an easily useable form we can address the performance characterization of SGCMGs. There are three areas that describe the performance characteristics of SGCMGs: torque amplification, singularities, and momentum management. Each one of these areas is discussed next. Torque Amplification Torque amplification for a SGCMG is defined as a higher amount of output torque produced than a given gimbal motor input torque required, assuming negligible torque needed by the flywheel motors. This is a reasonable assumption when considering SGCMGs because their flywheels require minimum torque to spin at a constant speed. In the literature (e.g., [34]), torque amplification is defined for a single actuator as, Torque Amplification 2 2 (315) 2 BB ]X Bh 0) If it is assumed that there are no disturbance torques on the system, the equation for internal torque r can be found by decomposing Eq. (313) of a single SGCMG actuator into two separate elements with equal magnitudes and opposite signs as, Bj BO) B _B]X x B)B B whereBL =[X BhC The internal torque can be decomposed to into an output torque t = [Bh ]X x from the CMG and input torque = BB ]X B ~ which is the gyroscopic contribution from the satellite's angular velocity in Eq. (313). If the twonorm of newly defined r, is divided by that of L, we get the same form of the torque amplification equation shown in Eq. (315). 28 When considering pico or nanoclass satellites where the ACS might take up a majority of the mass and volume of the overall satellite and the gimbalwheel system mass may make up a majority of the mass and volume of the ACS, the contributions to torque, and angular momentum from the gimbalwheel system and the gimbal motor friction inefficiencies may not be assumed negligible. Therefore, Eq. (315) is an upper bound for the torque amplification equation. Useful information is gathered from Eq. (315) such as the insight that the spacecraft angular velocity has to be less than the gimbal rate to have a torque amplification greater then one. To understand torque amplification of a single SGCMG for use on smallsats a lower bound was developed [35] and [36]. This lower bound is Irot 211 h0, JTt2 (316) S2 [B w +h + Fi +F, sgn() with gimbal input torques I and motor friction inefficiencies with dynamic and static coefficients F andF,. If the motor friction inefficiencies are of the main concern we get a reduced form of Eq. (316) as, 2 B 1 +F sgnO) + BO + h n2 B  and more compactly as, ,(317) ^l sgn( ) Fd w O+) + d 1 h h where torque from static friction is assumed to be negligibly small when compared to that for dynamic friction. Equation (317), which considers the mechanical inefficiencies of the gimbal motors, is a lower bound in the presence of these motor friction inefficiencies on the torque amplification of an SGCMG. From this equation, it can be seen that choosing the correct gimbal motor to make h the optimization parameter f = as high as possible, will reduce the motor friction effects on Fd the SGCMG torque amplification. It should be noted that increasing the flywheel speed as much as possible will give the largest torque amplification for small satellites where mass, volume and power are limited but will also increase their jitter. The motor inefficiencies of the DC motors used to rotate the flywheels are overlooked for SGCMGs that are not variable speed, due to the fact that they have a very small dynamic friction coefficient and their static friction and residual torques are largest on startup. Now that torque amplification has been discussed, singularities associated with SGCMGs are addressed. Singularities SGCMGs experience two classes of singularities. These singularities are internal and external singularities each of which are instances where the required control torque cannot be produced. External Singularities External singularities occur when the addition of angular momentum from an external source (e.g., disturbances) saturates the SGCMGs of the ACS. Of these disturbance torques, aerodynamic torques are considered to be the most harmful when in LEO due to their size and constant nature in comparison with other disturbance torques. External singularities are usually addressed in mission planning and considered the most harmful at the end of a satellites life cycle when deorbiting. Internal Singularities Internal singularities are encountered on the fly when the Jacobian matrix of the SGCMG ACS becomes rank deficient, whereby the torque vectors lie in a plane and no torque can be produced normal to this plane. Unlike external singularities they must be handled online and cannot simply be designed for a priori. The two types of internal singularities are known as elliptic which are unavoidable/inescapable through null motion (i.e., null motion is motion of the gimbals that produces no net torque on the satellite) and hyperbolic which are avoidable through null motion. Singularity avoidance strategies are discussed later in this chapter Singular Surfaces The CMG torque vector directions f are the columns of the Jacobian matrix. Therefore, singularity occurs when all i lie in the same plane. There are 2n singular configurations (i.e., singular sets of gimbal angles, 3) for any singular direction u normal to this plane. All possible for each CMG span a plane and an example of these planes for a fourCMG pyramid configuration can be visualized clearly in Fig. 33. The CMG torque vector direction is defined as, = torque vector =gk x With a given singular direction u not parallel to a gimbal axis, there are two possibilities for singularity to occur along each singular direction. These possibilities are shown in shown below. u_. =O, uj >Ooru =0, u_ <0 It should be noted that the occurrence of singularity when two gimbal axes are aligned has already addressed as gimbal lock and occurs in DGCMGs. Using the notation found in reference [35] and [32], we can define E, = sign (u ) With these definitions, the spin and torque axis vectors at singular states can be defined as, = (g X")X ,, U g and Therefore the total normalized angular momentum Bh of the CMGs at singular states corresponding to u is, Rh =E= , X )x u g When E, = +1 (or = 1 due to symmetry), the external singular surface known as the angular momentum envelope shown in Fig. 34 is formed as the locus of the maximum projections of the angular momentum of each CMG at the singular directions. External singularities occur on the surface of this envelope when the total internal SGCMG angular momentum is less than that which is external to the spacecraft, and the SGCMGs are then saturated. The internal singular surface is formed in the same way as that of the external singular surface with exception to one E, = 1. An examples of the internal singular surfaces for an ACS containing four SGCMGs in a pyramidal arrangement at an inclination angle 0 = 35.26 is shown in Figs. 35. The inclination angle should not be confused with the skew angle / used to characterize the angular momentum envelope [37] and [38]. The skew angle/? is the minimum angle of the SGCMG planes shown in Fig 33 with that of the horizontal. For a spherical angular momentum envelope, / = 90' 6 = 54.74'. Units of Fig. 34 and Fig. 35 are in terms of the nominal angular momentum magnitude h0 (i.e. diameter of angular momentum envelope involving four SGCMGs is less than or equal to four h0). In Fig. 34, locations that have white circles or holes on the momentum envelope are shown where the gimbal axes are located and there is no angular momentum available. The knowledge of where these singularities occur is known, and there have been methods developed that steer away or escape from these singularities known as steering logics. These methods are discussed in the next section. Steering Logics An ACS utilizing SGCMGs requires an addition of a fourth actuator for minimum redundancy to avoid singularities through null motion (i.e., motion is motion of the gimbals whereby no net torque is produced). This fourth actuator renders the system's Jacobian matrix to be nonsquare which in turn requires a pseudoinverse to map the output torque B, c R3 onto the gimbal rates 3 e R4 Recall, that there are 2n singular configurations for each singular direction of n SGCMGs. Therefore, it is important to note that the addition of actuators does not eliminate the problem of singularities but provides a null space where singularity avoidance may be provided. The MoorePenrose pseudoinverse solution to the gimbal rates is represented as 1 3 =A BA C (318) h, where the MoorePenrose pseudoinverse is, A+ =AT(AAT The MoorePenrose pseudoinverse solution for the gimbal rates in Eq. (318) fails when AA becomes singular. To avoid/escape singularities when mapping the output torque onto the gimbal rates, a steering logic is applied. Steering logics can be broken up into two main groups: those which provide null motion for singularity avoidance such as local gradient methods and global avoidance algorithms, and pseudoinverse solutions which escape singularities through loading of the Jacobian matrix singular values. Common drawbacks to these steering logics are computational complexity, inability to avoid all internal singularities, or loss accuracy by adding torque errors to keep the Jacobian nonsingular with loading of its singular values. Local gradient methods were of the earliest methods used to steer away from internal SGCMG singularities by Margulies and Auburn [39]. This local gradient method, like most steering logics, requires a fourth SGCMG added to provide a 1dimensional null space where null motion can be applied to steer the gimbals away from singularities. The Jacobian matrix null space provides multiple solutions to the gimbal state equation to steer the gimbals away from singularity while maneuvering. An example of local gradient methods is, k1 S= 1(A+ B~ [ A +AAd) (319) where B A is the SGCMG output torque and h, is the nominal magnitude of angular momentum. This equation is similar to Eq. (318) with the addition of a null motion vector d which is mapped to the Jacobian matrix null space by the projection matrix II A+A. The contribution of null motion is scaled by the singularity parameter S= y0 exp(/ m2) (320) which is an explicit function of the singularity measure m, m= det(AA) (321) and design constants p and yo. When using local gradient methods, the null motion vector d can be chosen as the gradient, T d: {8f 8f Of /fY d=\ , as 0a2 3 'a 4 of a objective function f= m This null vector d produces null motion in the direction that maximizes m or the distance away from singularity. It is important to note that steering logics utilizing only null motion are unable to avoid elliptic internal singularities [40] and [41]. Local gradient methods shown here are computationally intensive and cannot avoid all internal singularities such as elliptic internal singularities where null motion is unsuccessful. Global avoidance methods shown in Eq. (322) such as those developed by (Paradiso and Kuhns [42] and [43]) are similar to local gradient methods and differ in the fact that the null motion vector is produced by the difference of the gimbal angle positions from a set of alternate gimbal angle configurations known as preferred trajectories. This difference is shown in Eq. (3 22), where 5* are the preferred trajectories. = (A+ B +7[ A+A](*)) (322) ho0 These preferred trajectories are usually calculated offline. This method is computationally intensive and is not suitable for online use. The most popular pseudo inverse solution is the SR inverse developed first by (Nakamura and Hanafusa [44]), modified later by (Wie [45], and Ford [46]). The SR inverse A" =A (A AA + y) (323) produces the solution to the gimbal rates 3= A#Bhc (324) Because the A# always exists, the maneuver can start out near singularity where local gradient and global avoidance methods are unable. This SRinverse solution works by adding a positive definite matrix of torque errors y! to the positive semidefinite matrix AAT to leave the matrix(AAT + y7) positive definite and hence nonsingular. Rather than avoiding internal singularities as discussed previously, this method which is developed from the MoorePenrose pseudoinverse, approaches singularity and makes a rapid transition to escape. At escape there is a jump in gimbal rates due to the added torque error. As a consequence of these added torque errors, this method is not the preferred method for precision pointing. Under certain circumstances (i.e., when det AA ) = 0 and B e null(A#)), this method can become locked into a singularity, so the generalized SR inverse (GSR) was developed by Wie [45] . The GSRinverse A# = A (AA yE)1 (325) is made up a positive definite symmetric matrix 1 0o sin(03t+ 3) 0 sin (2t +2) E_= o sin (3t +3 ) 1 0 sin (0t + ) >0 (326) S~ sin (2t+2) 2 0 sin(w1lt+ 0) 1 composed of time varying modular parameters E, and scaled by the singularity parameter y in Eq. (320). These modulating parameters ensure that B will not stay locked in the null space of the pseudoinverse, which is not explicitly dependent on time. The GSRinverse has been shown to avoid all internal singularities but adds torque error to all directions. Therefore, the original SRinverse was modified again to minimize the amount of torque error added to the gimbals when avoiding internal singularities. This method developed by Ford is known as Singular Direction Avoidance (SDA) with the pseudoinverse ADA = V StAU and from which the gimbal rates are 3= DA B (343) ho= where V and U are unitary matrices and SDA is, 1 1 3 diag ,,  SIDA 1 02 3 OT The benefit of using this method over the SRinverse methods is that it only adds torque error to the smallest singular value, which in turn requires less null motion and has smoother gimbal rate trajectories than the previous SR and GSRinverse methods. A drawback of this method is that it is required that a SVD be calculated forA every time step, making the steering logic computationally intensive. Also it is developed directly from the original SRinverse and therefore can become locked in singularity when the control torque is in the singular direction of the pseudoinverse. The discussed methods of avoiding or escaping singularities are of the most popular of steering logics developed for SGCMGs [47]. Additional methods have been developed to avoid internal singularities associated with SGCMGs [4244]. An example of one such method includes game theory [48]. Other methods for singularity avoidance have been developed with the addition of an extra degree of freedom while using VSCMGs instead of SGCMGs. In the case of VSCMG, steering logics associated with gimbal velocity and acceleration have been developed by Schaub and Junkins [31] and [49]. These methods use the extra degree of freedom through flywheel acceleration to give an ACS using these actuators the ability to always produce the required output torque needed. Recall it was stated that VSCMGs do not require steering logics due to their reaction wheels modes. Therefore the addition of a fourth VSCMG instead may provide power tracking for systems such as the Flywheel Attitude Control and Energy Transmission System (FACETS) and the Integrated Power and Attitude Control System (IPACS) [50]. These systems are able to use the null space instead to monitor power transmission while converting the extra kinetic energy gathered from spinning down the flywheels after a maneuver to electricity charging the power system on board. Systems such as FACETS require ceramic wheels to make use of the kinetic energy provided from spinning down the flywheels and were deemed unsafe due to the high flywheel speeds of fortythousand rpm and have henceforth not been flight tested. The next section discusses momentum management which deals with the dumping of external and internal angular momentum in order to maintain the performance of an ACS using SGCMGs. Momentum Management There are two situations where the angular momentum of the system must be managed in order to maintain the required ACS performance. These situations occur from external angular momentum added into the system from torque disturbances or internal components of angular momentum remaining post maneuver from gimbal angle saturation. External Angular Momentum Offloading External angular momentum accumulated by the spacecraft from nonzero disturbance torques is considered a priori for each specific mission. If angular momentum is added into the system from the disturbance torque, and it is greater than that from which is allotted from the CMGs, the system encounters an external singularity as discussed in section 3.2. Assuming that a spacecraft is in LEO orbit then the addition of aerodynamic torque is the only source that is considered to saturate the system. This is so because the magnetic disturbance torque is exceptionally small and the disturbance torque from the gravity gradient does not saturate the system with angular momentum due to its periodic nature. Also disturbance torques from solar pressure effects are neglected due to the small amount of solar cell surface area on small satellites. When considering small satellites, aerodynamic torques add a negligible rate of angular momentum into the system in most standard LEO orbits [36]. The addition of angular momentum from disturbance torques is a slower process than that for the internal angular momentum buildup from the SGCMGs. For this reason there is greater concern for offloading of internal angular momentum. Internal Angular Momentum Offloading Internal angular momentum accumulates when components of angular momentum from the SGCMG cannot be cancelled out due to gimbal angle saturation. As previously mentioned these gimbal angle constraints can be reduced by using magnetic bearings and/or slip rings which allow for full range of gimbal rotation. Two cases arise when internal angular momentum needs to be offloaded. The first case occurs when excess components of internal angular momentum is left over from maneuvers due to gimbal angle saturation where unwanted components of angular momentum are unable to be cancelled out. The second case is at startup of the flywheels, where needs to be angular momentum dumped to stop the satellite from tumbling due to the offset in its angular momentum direction. A typical startup configuration of an ACS containing SGCMGs is at zeromomentum. This configuration for a fourpyramid SGCMG cluster has all of the angular momentum vectors of each SGCMG in the body xy plane at a 3 = 0 leaving a zero net momentum for the ACS. An example of this configuration is shown in Fig. 36. In this figure, all of the angular momentum vectors are in the plane although it is not a singular configuration because all of the torque vectors are not in the same plane. For smallsats the offloading of the excess angular momentum on startup as well as that due to gimbal angle saturation can be accomplished by use of magnetic actuators. Now that mathematical and physical aspects of ACSs containing SGCMGs have been discussed, the types of ACS configurations containing these actuators as well is the development and choice of the SOBEK configuration is addressed in the next chapter. Fh Fig. 31. The SGCMG with gimbal coordinate axes A 323 rotation sequence through angles 0, ,, 6, &4~ir~'l~ \I Fig. 33. Intersecting planes spanning of SGCMG torque directions 3 2 1  0i~ 2 .3 .5 5 4 Angular momentum envelope for 0 Fig. 32. Fig. 34. 35.26 and E [1,1,1,1] 5 i . 1.5T2 2 2 = 35. 2 2 Internal singular surface 0 = 35.26" and _s= [1,1,1, 1] t . ft2 II L W It Q34 Zeromomentum configuration Fig. 35. Fig. 36. JA lA. CHAPTER 4 SGCMG ACS CONFIGURATIONS Rooftop Configuration The rooftop design uses two groups of SGCMGs, each with parallel gimbal axes slanted at an angle to each other shown in Fig. 41. A CMG system in this configuration will never have elliptic internal singularities [40]. This means that they will always have continuous momentum trajectories and therefore have continuous gimbal trajectories. This configuration suffers from situations where its Jacobian may approach rank 1. The singular direction where this occurs is on the intersection of the two rows of SGCMGs shown as the red arrow in Fig. 41. Box Configuration The box configuration is a variation of the rooftop configuration containing four SGCMGs with an inclination angle of 0 = 90. This configuration is shown in Fig. 42 where the red arrow indicates the same rank 1 direction where the torque vectors lie. The internal singular surface associated with this configuration covers a large surface of the angular momentum envelope and is seen as the large empty area present inside the external singular surface in Fig. 43. The singularity represented by this singular surface shown at Ih is avoided in the box configuration by utilizing only three SGCMG and keeping the third as a spare while staying within the Ih constraint. Only three of the four SGCMG will be used at any given time so the Jacobian is square and there is no need for a pseudoinverse to map the gimbal rates onto the torque. The major drawbacks of this design are that it is not as compact as other designs and its performance is constrained to Ih of angular momentum, causing the ACS to require larger SGCMGs to meet the missions angular momentum requirements. Pyramid Configuration Common pyramid configurations have numbers of SGCMGs in groups of four or six. ACSs containing these amounts of SGCMGs and have both elliptic and hyperbolic internal 43 singularities. Although an ACS utilizing SGCMGs in a pyramid configuration is susceptible to the elliptic internal singularities, there are methods of escaping these singularities that were discussed in the previous chapter. This configuration has the benefit of giving a somewhat spherical angular momentum envelope for the right choice of inclination angle of the SGCMGs. The SOBEK testbed will utilize a minimal redundancy (i.e., four SGCMGs) pyramid configuration for threeaxis control while applying the GSRinverse steering logic method discussed in chapter 3. SOBEK Pyramid Configuration The SOBEK testbed's ACS is shown in Fig. 44 with motion capture for attitude tracking provided by Phase Space and its visual interpretation of the mapped wireless LEDs in the upper right hand corer. SOBEK's ACS uses four SGCMGs in a pyramid configuration. The testbed was originally designed to float on an air bearing table with multiple ACSs where the footprint of the ACS was limited while trying to minimize weight. For this reason, the inclination was chosen as angle 0 = 54.74'. It should also be mentioned that the formulations in literature consider a zero reference gimbal angle position shifted 90 from where the SOBEK zero reference angle is. The new formulated SGCMG angular momentum vector S s0S()c(81)~ s(2) 2 s(/ c(,3) S(c 4) h,= hk s(5i) + s(0)c(52) + s(53) + s(0)c(54) (41) ScC (0) c() cL C(0c5(2) c(9)c(53) L c(9)c(54), and Jacobian matrix Ls(0)s(8) c(2) s(0)s((3) c(4) A= c (() s (0)s (2) c (3) s(0)s () (42) Sc()s(gl) c(0)s(2) c (0)s(,3) c(0)s(34) takes these changes into account. Other testbeds such as those at Virginia Polytechnical Institute, Lawrence Livermore Laboratory, SSTL, and Honeywell's Line of Site (LOS) testbed are of the few testbeds that comprise the known heritage for this research [5153]. This testbed differs from the ones previously referenced in that it is physically smaller and does not utilize an air bearing but a pivot point to mitigate the forces of gravity. When making the testbed near marginally stable at a point, the only forces and torques considered are the small amount of friction from that point as well as the residual moment from gravity. An illustration of SOBEK that contains the ACS in and stand which holds the ACS on a pivot point is shown in Fig. 45. This illustration was made in ProEngineer, where the SOBEK inertia and mass properties were also estimated. The next section discusses the hardware chosen for the SOBEK and its design shown in Fig. 46 and [54]. SOBEK Hardware The basic electrical hardware for preliminary setup of the SOBEK ACS consists of four Arsape AM2224 twophase Stepper motors, four Faulhaber 1525 BRC brushless DC motors with integrated electronics and four State Electronics 600 Series digital optical encoders. The electrical hardware for the current setup only contains a wireless Phasespace LED controller, a Roving Networks RS232 bluetooth module, and the EZHR17EN stepper driver boards with an RS232 to RS485 adapter. The entire ACS is powered by two lithium polymer batteries. This is possible because there are voltage regulators present to step down the voltage from 14 to 12 volts and 5 volts to meet the specific voltages of the mechanical hardware. Currently the setup has no onboard processing and runs code written in C++ from a Panasonic Toughbook with a Linux operating system. The future setup of this testbed will have everything coded on a single processor board containing the wireless Bluetooth module and motor driver circuitry. There are significant reasons for the choice of the hardware for the testbed. One of these reasons deals with the digital optical encoders. These encoders happen to be more accurate then potentiometers and are dual quadrature which are directly compatible with the Allmotion driver boards chosen. The Allmotion EZHR17EN stepper driver boards have the required output current and voltage necessary for the given Arsape AM2224 stepper motors and so were chosen for this reason. In addition to having the correct current and voltages, these boards also have the ability to daisy chain to other boards. That is for a given set of four motor driver boards controlling a total of four stepper motors two daisy chain boards are needed to send commands to all four motors through the serial port. It is assumed that the gimbal rate commands are sent fast enough serially that they arrive at all four motors at the same time from the daisy chain boards. The nominal speed of the stepper motors needed to be geared down from 134:1. There was a significant loss in efficiency by doing this so the optical rotary encoders are added on the other end of the shaft to reduce the effect of this uncertainty by feedback. The Faulhaber 1525 BRC motors were chosen because they have a high enough stall torque to spin up the flywheels as well as integrated electronics including a frequency output to measure the speed of the motors. This is valuable for use in the testbed, because knowledge of the flywheel angular speed is an integral part of the SGCMG dynamics and control. If the flywheels were spinning at different speeds the angular momentum vector of each SGCMG would have different magnitudes and this would add attitude control error. SOBEK Mechanical Design There are some unique design considerations that were carried out in the design of the SOBEK testbed. One of these design considerations dealt with how the ACS would rotate. To approximate the mass and inertia properties, ProEngineer was used. The inertia properties were measured about the ACS CM. If the CM was made to be below the point of rotation, then the system would be stable (i.e. a pendulum equilibrium point). It should be mentioned that the distance between the point of rotation and the CM cannot be too great or the reaction momentum from gravity will saturate the SGCMGs. For this reason the CM was chosen to be just below the point of rotation to make the system close to neutral stability. To account for the body frame x and y deviation of the CM from the point of rotation, the ACS is connected to a platform that contains a system of weight ballasts. These ballasts allow twodimensional adjustment of the CM. The motor driver and daisy chain boards as well as the batteries and the power terminal are mounted on this platform to conserve space. The zaxis deviation of the CM was accounted for by the vertical adjustment of the ACS center. SOBEK Mechanical, Power and Output Specifications Mechanical specifications of the SOBEK ACS are shown in Table 41 with a total output torque of 267 mNm, mass of 5.489 kg, and power consumption of 8.36 W. SOBEK Attitude Determination The SOBEK ACS is fitted with LEDs that are distinguished by different blinking frequencies where they are read by a system of eight cameras. Shown in Fig. 47 is the testbed setup including the Phasespace attitude determination system. The Phasespace motion capture system is chosen because it is a simple and compatible offtheshelf way to measure attitude accurately. This is possible because Phasespace works on the principle of taking position measurements from LEDs located on the SOBEK robot thereby calculating the attitude of the tracked object. This completes the attitude determination system for SOBEK. The next chapter discusses the simulation and experimental results of this testbed. The simulation results utilize the mass and inertia properties estimated in ProEngineer as the basis for its model. Also discussed in this chapter will be the limitations of the current design iteration of the SOBEK testbed as well as an introduction of the application package interface software (API) that runs it. Table 41 Mechanical and Power Specifications of the SOBEK ACS Current Voltage Weight Micro ACS Power (W) (A) (V) (kg) Output Torque (mNm) Stepper motors 6.000 0.500 12.000 267 DC motors 1.776 0.148 12.000 Encoders 0.600 0.120 5.000 Total 8.376 5.489 6  Fig. 41. Honeywell rooftop configuration Honeywell box configuration Fig. 42. o 1 2 2 External singular surface for box type SGCMG ACS Honeywell pyramid configuration Fig. 43. Fig. 43. Pyramidal SGCMG ACS . t ,..w. . Rendition of SOBEK testbed Fig. 44. Fig. 45. ACS with hardware components Cameras collect rangi measurements from LEDs Motor driver boards send command to the stepper motors T control calculated and sent via bluetooth to ACS q Quaternions sent to command PC PhaseSpace attitude determination system Range data processed Fig. 46. Fig. 47. CHAPTER 5 RESULTS Simulation Model The error quaternion state equation is r L[B"B]' e+le 1 4[e e 2 2 1 G eo) 1Q e e4 B B T 2 e4 2 J= e4 2  where, e and e4, are the vector and scalar elements of the error quaternion and B0B is the spacecraft angular velocity coordinatized in the body frame. The equation for the spacecraft model is S= B B B +B [ BB] Bj B)B where, BjC is the spacecraft centroidal inertia mentioned in Eq. (31) and r is the internal control torque from the SGCMGs mentioned in Eq. (314), both of which are coordinatized in the body frame. The equation for the commanded SGCMG output torque is, B=_Z [BB]j Bh The output torque is mapped onto the gimbal rates with the GSRinverse discussed in chapter 3 giving the gimbal rate solution of o At L A" (51) ho= where, (ct is the actual column matrix of commanded gimbal rates used to generate the given output torque considering the torque error added from the GSRinverse. The internal SGCMG torque r is found in through a nonlinear rest to rest control logic r = Ke CBB + [B B B BB (52) with the symmetric positivedefinite controller stiffness and damping gain matrices K = 2kBJ and C = cBJ The nonlinear exact model knowledge (EMK) controller shown in has been proven to be asymptotically stable [45] and a Lyapunov stability analysis was performed on this controller to understand its behavior when uncertainty is present in the system. Lyapunov Stability Analysis of EMK Attitude Controller A positive definite Lyapunov function is chosen as, V= e'e+(le4)2l[ ] i BBK L>0 (53) 2 53 with time derivative equal to V = 2e +2(1e4 )(,4 )+[B w B K [ B]= 2e ]Xe+ B e4 +2(1e4) B B]Te (54) [B B]) ] K1 [ BB ]X BjBC9B To obtain asymptotic stability, Eq. (54) must be made negative definite. To achieve asymptotic stability the control torque (i.e., internal torque of the SGCMGs) is chosen as Eq. (5 2). Substituting r from Eq. (51) into V that of Eq. (54), we end up with S C [B B]TBB V, = c It)]T c < 0 2k  The final result of this substitution is that the time derivative of the Lyapunov function is negative semidefinite. To prove that the controller in Eq. (52) gives a global asymptotically stable result, we must note that the spacecraft angular velocity goes to zero asymptotically for the following reasons: * Bo) e Lee L ze L, ~ B6B L * BoB E L2 * B )B is uniformly continuous If we apply Barbalat's lemma with these conditions, the spacecraft angular velocity goes to zero asymptotically. If we revisit the control law in Eq. (52), we can see now that it is not possible for the angular velocity to go to zero asymptotically unless the error quaternion vector elements also go to zero, therefore this controller is asymptotically stable. This controller is EMK which assumes full state feedback, therefore the global asymptotic stability does not necessarily hold if accurate full state feedback is not available. The current iteration of SOBEK does not contain gyros for angular rate calculation and encoders for gimbal rate position. This iteration also has no online method of calculating the flywheel speeds, therefore the gimbal angles and angular rates are estimated with uncertainty present and the flywheel speeds are calculated before experimentation. The gimbal angles are estimated through an Euler forward integration of the solution to Eq. (51) as, = + At where Atk is the time step of the control loop. The equation for the calculation of the ACS angular rates is B 0)c B [AS Ae e ,+l e S= L= A += where, Q\ is always invertible. e4k Due to the uncertainty present in this iteration, the best result that the controller can produce on this testbed are shown to be uniformly ultimately bounded (UUB). To account for the many sources of uncertainty in this iteration and obtain the UUB result, another stability analysis of this controller is carried out with uncertainty considered. The control torque is dependent on both the estimated angular rates and gimbal angles. The gimbal angles are integrated from the gimbal rates which are mapped from the control torque. Recall also that the torque utilizes minimal torque errors for singularity avoidance when using the GSRinverse and therefore these torque errors must be considered in addition to the uncertainty. Lyapunov Stability Analysis of Actual Attitude Controller with State Uncertainty For the stability analysis, the actual control torque and angular velocities of the system are defined as, 7ct= + 6 Tact =0 + with the uncertainties and . If we substitute the actual control torque and angular velocities into Eq. (53), we are left with v2 =e +(1e4)2 +I TKB V2e e+(e4 2 oact _ B ) act > 0 2 and its time derivative V )= _K_1 1" c _B_ + 0) .._e (55) V2 actK L 0 act cacte (55) Assuming that the uncertainties and & have negligible time derivatives, the control torque from Eq. (52) give the result V= TK1 K1eC I B e B + O) + a + e 2 cih a simifid r actsct with a simplified results BCB ]' (,I+ B] 1 ( B B 1_ T BB) V, = (56) Examining Eq. (56) while assuming that the uncertainty is less than the actual control torque and angular velocity, the best result possible is UUB on the order of the same magnitude as that of the uncertainty for this control law. Now that the stability analyses have been carried out, the results from simulation and experiment is discussed. Simulation Results and Discussion The simulations were carried out in a Matlab environment with parameters shown in Table 51. Shown in Fig. 51 (A) is the error quaternion vector elements for the first simulation where, el, e2, and e3 are the quaternion error vector elements about the roll, pitch, and yaw directions. This result at first glance seems to have great performance. Recall that this simulation was run assuming exact model knowledge in the absence of uncertainty. The next plot shown in Fig. 51 B) shows the commanded gimbal rates throughout the simulation. In this figure it is clear that the gimbal rates are large in the beginning of the simulation. This is due to the fact that the simulation starts off with the SGCMG gimbals in a singular configuration for a torque needed about the axis. Figure 51 (B) also shows the trend of the gimbal rates going null after the maneuver is completed which is not the case when disturbance torques are added to the system. In this situation, the gimbal rates will need to compensate for the addition of angular momentum and will continue to be nonzero. Figure 51 (C) shows the spacecraft angular rates of the ACS. In this figure, the angular rates are nullified after the maneuver is completed. This is the desired affect for a rest to rest maneuver. The final plot of this simulation is that of the singularity measure m in Fig. 51 (D). Two things can be attributed to Fig. 51 (D). The first is that the singularity measure is zero at the beginning of the simulation (i.e., Jacobian is rank deficient when all gimbal angles are zeroed) and it ramps up quickly to transit from singularity within the first two seconds. Although the ACS is initially at a singular configuration, use of this steering logic enables it to transit away from singularity in a timely manner. The second conclusion for this figure is that m becomes constant after the maneuver is completed. This should be obvious because it is an explicit function of the gimbal angles. Before an experiment was carried out, another simulation was run with the same k and c values from the previous simulation including the addition of a random error signal added to the angular velocity measurement and gimbal angle integration. This random error was added in an attempt to model the uncertainty of the gimbal angles and angular rates as Bck+~ = B cB +O.03rand) 1 = +0.1rand(.) It is clear from the results of this simulation are shown in Figs. 52, while considering an EMK control logic when the controller has uncertainty within its states, the performance is heavily degraded to the point of instability. In addition to this simulation another one including a higher choice of gains k and c was run to compare results and validate the uniformly ultimately bounded result of the stability analysis in Eq. (56). The results in Figs. 53 prove the opposite end of the stability analysis which is with higher choices of a k and c, the steady state error of the UUB solution can be reduced. Experimental Results and Discussion The flywheels onboard the SGCMGs of the ACS are not balanced and are unstable at certain wheel speeds. For safety reasons the wheels are left to spin at a maximum speed of around 6000 rpm to keep them within a stable vibration frequency. This speed does not produce enough angular momentum, and hence, enough torque to overcome the moment from gravity to do large offaxis A or baxis maneuvers and therefore the experimental results are for a yaw maneuvers about the b axis. The GUI associated with this testbed is shown below in Fig. 54. This GUI in addition to rest of the API software was developed by Andrew Waldrum at the University of Florida specifically for SOBEK. The flywheel angular speeds were calculated offline through the output frequency feedback of the DC motors and found to be of different magnitude among the four SGCMGs. A variation of Eq. (51) is used to account for the variation in nominal angular moment for the individual SGCMGs where Ho is the diagonal matrix of nominal angular moment. In this GUI the gains k and c are increased at intervals c = 2.5k to where k varies from .08 and 0.2 in the first experiment. There is no torque to steer away from singularity at startup because the gains are zeroed out initially. All experiments were carried out with the same steering logic parameters and initial conditions as those in the first simulation and only differed by choice of control gain. An initial experiment was run with the controller given in Eq. (52). It is shown in Fig. 55 (A) that at around forty seconds the quaternion error diverges, which can be caused by either the controller's instability, propagation uncertainty from the estimated states, or both. The reason for this error propagation is due to the error rates produced from the uncertainty of the estimates. The maneuvers in these experiments start out with a high initial error about the 3 axis and attempt to align the ACS to a quaternion of q = [0 0 0 1]. The next three plots shown in Figs. 55 (B)(D) show the gimbal rates, angular rates, and singularity measure associated with this maneuver. The next experiment took into consideration the result of the stability analyses and the last two simulations. This experiment was carried out with higher values of the control gains and its results are shown in Figs. 56. In these experiments k was chosen greater than that for the previous experiment keeping the same c/k ratio. The quaternion error vector elements in Fig. 56 (A) contain better results than the previous experiment although they also posses low frequency oscillations. The steady state error in this figure is measured as, b = min[2sin ( e 2,2r 2sin 1( e) = 0.0826 rad= 4.7441V where, the angle e, represents the minimum angle needed to be traced out to align the two frames 3B and 3G about an eigenaxis. This metric is valid because the error quaternion is still a unit quaternion whereby the 2 norm of the vector elements is equal to sin (Oe/2). Noted in red on the plots of Fig. 56 (B) and (D), are the large jumps in gimbal rates when transiting a singularity. This was previously said to occur when the SR or GSRinverse method is applied for singularity escape. The result is still oscillating about zero after sixty seconds and the steady state error S= 0.0826 rad = 4.7441 is still quite large. It is believed through a larger choice of gains, preferably ones with a higher ratio of c/k will reduce the oscillations and steady state error. The next experiment shown in Figs. 57 was carried out with a larger value of control gains with a higher c/k ratio to improve the results from the previous experiment. These values were k = 0.32 and c = 3.0k. The error quaternion vector elements for this experiment converged with a much lower steady state error of = 0.0376 rad = 1.0779 and a smaller amount of steadystate oscillation due to the higher c control gain. The behavior of the system for the chosen control gains is more stable as well as more efficient in terms of less wasted control effort than that for the previous experiment. This is due to the larger ratio of c/k and the fact that the maneuver did not encounter an additional internal singularity within the time of the experiment. With this larger c/k ratio, the steady state error oscillations are reduced by a much larger c gain. The next plot shown is the angular rates versus time for this experiment. Angular rates shown in Fig. 57 (C) show a more desirable behavior than that of Fig. 56 (C) in terms of high slew rates with less oscillation. The last plot of this experiment is that of the singularity measure. It has already been discussed that behavior of this system is more desirable than the previous experiments with the choice of the current gains, and now it also seen that at no time did this system become singular. A final experiment was carried out were the MoorePenrose pseudoinverse which possessed no steering logic (i.e. A = 0). The quaternion error of this experiment is shown in Fig. 58. The SOBEK ACS for all experiments is initially at singularity, therefore without a steering logic large gimbal rates are commanded and the system tumbles out of control as shown in Fig. 58. From this figure, it is clear that without use of steering logics while utilizing SGCMGs, drastic consequences may arise. In the future iterations of the SOBEK testbed, if encoders are provided are for gimbal angle measurement, triaxial gyros provided for angular rate measurements and use of onboard feedback for the flywheel angular speeds will produce more accurate attitude maneuvers with elimination of state uncertainty. Table 51. Simulation parameter values Parameters Values Units 0.04 0.8 J= diag {J1, J22, J33} qc = [q, q2c 3c q4c 0 q(0)= [q q2 q3 4]T S(0) = [0, 02 )3]T E1 = Eorand(1) g2 = orand(2) E3 = Orand(3) A = A0 exp( /m2) 3(0)= [g, g2 3 64 diag {0.0668,0.0756,0.0815} [0 0 0 1]f 54.74 [0.9380 0 0 0.3466]T [0 0 Of 0.0lrand(l) 0.0 lrand(2) 0.0lrand(3) 0.01exp(20m2) [0 0 0 O] Nms none none Nm2 none degrees none rad/s none none none none rad 0 0.05 0.1 o 0.15 0.2 0.25 0.3 0.35 40 50 002  CO 0 08 0 dal/dt da2/dt da3/dt da4/dt 012 10 20 30 time (s) 40 50 60 0.25 0.2 0.15 0.1 . 0.05 S 0 3 0.05 0.1 0.15 202 10 20 30 40 50 60 time (s) E 004 002 0 10 20 30 40 50 60 time (s) ACS simulation plots for k = 0.08 and c = 2.5k: A) elements versus time, B) Gimbal rates versus time, and D) Singularity parameter versus time. Quaternion error vector C) Angular rates versus time, 10 20 30 time (s) e e2 e3 Fig. 51. d5/dt d&Idt 10 23 30 tirm s (f tie (s) ACS simulation plots for k = 0.08 and c = 2.5k: A) Quaternion error vector elements versus time, B) Gimbal rates versus time, C) Angular rates versus time, and D) Singularity measure versus time. times) u) 0.05 0.1 0.10 C Fig. 52. time(s) .1 O i: 0.1 2 001 d. 0) e3 .i .2 .d6dt 0 003 40 10 0 30 40 0 6 40 10 2 3 40 5 60 time (s) time (s) A B Q3, 1.6 0.25 0 1.4 Q2 1.2 1 1 01 E0. 0 a0.6 0.05 0.4 0 \ 0.2 0.2 0 10 23 30 40 50 6 70 00 10 20 30 40 50 60 70 time (s) time (s) C D Fig. 53. ACS simulation plots for k = 4 and c = 3k: A) Quaternion error vector elements versus time, B) Gimbal rates versus time, C) Angular rates versus time, and D) Singularity measure versus time. Euler angles M e U. UU O I w3 I I I I i. uunE I yl 7Lj I 1 Seal wiSTOP Senal wire ~'U fu llla m rIaiumlls p: 1 Y: I0 r" Radians Degrees Kn Kr Kz Ka O7 Draw Range Stop Gimbals Loop is Closed Si re Data Status: BAD Singularity BA  Controller NULL K: J S StepNumber P 1 C: I Commands Sent 77 Rate of Operation  L I SOBEK testbed GUI 1 i :j a Fig. 54. 0.5 0 0 U, 0 d61/dt d62/dt d63/dt d /dt 4 .3 .2 S11 r,/ ,fr 0.1 0 10 20 30 40 50 60 70 80 time (s) 1.4 S1.2 0 1 0.8 E 0.6 0.4 0.2 0 10 20 30 40 50 60 70 80 time (s) Fig. 55. 0 10 20 30 40 50 60 70 80 time (s) ACS experimental plots for k = 0.8 and c = 2.5k: A) Quaternion error vector elements versus time, B) Gimbal rates versus time, C) Angular rates versus Time, and D) Singularity measure versus time. time (s) C I i~v 1.2 1 0.8 0.6 0.4 0.2 0 o0 1 0 20 30 40 50 60 70 time (s) . & I , I .I. d61/dt da /dt ds3/dt ... d /dt " a'w''."." 1 0 10 20 30 40 50 60 70 time (s) B 1 2 3 W  I  0 10 20 30 40 50 60 70 10 11 40 50 60 70 time (s) time (s) C D Fig. 56. ACS experimental plots for k = 0.24 and c = 2.0k: A) Quaternion error vector elements versus time, B) Gimbal rates versus time, C) Angular rates versus time, and D) Singularity measure versus time. e2 e2 e, A P 10 20 30 40 50 60 70 80 time (s) 0.5o 0.5 1.5 1 2/dt .J 3/dt 4J ,/dt 0 10 20 30 40 50 60 70 80 time (s) B 1.5, 10 20 30 40 50 60 70 time (s) D ACS experimental plots for k = 0.32 and c = 3.0k: A) Quaternion error vector elements versus time, B) Gimbal rates versus time, C) Angular rates versus time, D) Singularity measure versus time. S0 0.1 s Fig. 57. time (s) "Awcss  0.1 0. S 2 4 6 8 10 12 14 time (s) Fig. 58. ACS experiment quaternion error of Moore Penrose solution e e CHAPTER 6 CONCLUSION AND FUTURE RESEARCH Conclusion This thesis discusses the development and testing of an attitude control testbed utilizing four SGCMGs in an orthogonal pyramid configuration. SOBEK is of great utility for evaluating attitude control algorithms and has provided insights into the understanding and application of control moment gyrobased actuators. In its current form, SOBEK has only attitude feedback and does not have onboard processing or internal feedback (i.e., operates in an openloop manner). With the addition of onboard processing and sensors for state feedback, it is expected that the testbed will prove to be invaluable for testing control algorithms and steering logics while showing precision in terms of attitude accuracy from SGCMGs. Future Research Future research for this attitude control system will include the changes to the new iteration of SOBEK and testing of different control algorithms and steering logics. A pico satellite CMGbased ACS will be developed using SOBEK as the alphamodel to analyze software that will fly on this miniaturized actuator. APPENDIX A OFFTHESHELF RW, CMG, AND MAGNET TORQUER DATA SPECIFICATIONS Table A1. Offtheshelf performance specifications for RWs RWA Torque (mNm) Mass (kg) Power (W) Bradford Engineering W05 100 3.2 73 W18 200 4.95 63 W45 300 6.95 64 Honeywell HR 0610 55 3.6 80 Dynacon MicroWheel 200 30 0.93 2 MicroWheel 1000 30 1.3 9 Vectronic Aerospace RW1 20 1.8 25 SunSpace SunSpace RW 50 1.98 35 TELDIX RSI 015/15 5 0.6 4 RSI 015/28 5 0.7 4 Goodrich (Ithaco) TW26E300 300 13.9 L3 Space Comm. MARS RWA 15 MWA50 160 10.5 100 Orbital Sciences LEO Star Wheel 140 3.6 55 SSTL MicroWheel5S 5 0.5 8 MicroWheel3S 3 0.75 3 MiniWheel20S 20 3.2 14 MiniWheel20SX 20 2.6 14 NanoWheelm500S 0.5 0.08 0.5 MicroWheel10SPS 10 1.1 5 Table A2. Offtheshelf performance specifications for CMGs CMG Honeywell M50 M95 M160 M225 M325 M325D M715 M600 M1400 M1300 EADS/ASTRIUM CMG 1545S SSTL Bilsat1 CMG SSG SOBEK ACS Torque (mNm) Mass (kg) Power (W) 74570 128803 216931 305059 440641 440641 969410 813491 2E+06 2E+06 45000 95000 66.7 33.1 38.6 44 54 61.2 61.2 89.8 81.6 132 125 2.2 1.38 75 129 217 305 441 441 949 814 1899 1716 12 2.09 Table A3. Strauss Space Micro Magnetic Torque Rods Nano Magnetic Torque Rods Magnetic Torque Coils Vectronic Aerospace MTR5 0.6 0.2 0.04 0.75 3.5 0.36 0.1 6.0 Offtheshelf performance specifications for Magnet Torquers Magnet Torquer Output (Am2) Mass (kg) Power (W) Microcrosm MT21 2.5 0.2 0.5 MT52M 6.0 0.3 0.77 MT61 7.0 0.23 0.25 MT62 8.0 0.3 0.5 MT102H 12 0.35 1.0 MT151M 20 0.43 1.11 MT302CGS 40 1.050 3.6 MT302GRC 35 1.4 1.5 MT701 75 2.6 3.8 MT702 75 2.2 2.6 MT801 100 4.12 3.0 MT802M 90 2.3 4.7 MT1102 120 3.8 2.9 MT1402 170 5.3 1.9 MT2502 300 5.5 4.8 MT4002L 500 7.8 9.0 MT4002 550 11.0 11.4 MT4001 550 9.2 7.7 LIST OF REFERENCES [1] Cebrowski, A. and Raymond, J., Operationally Responsive Space: A New Defense Business Model, Parameters, 2005, pp. 6777. [2] Alminde, L., Bisgaard, M., Vinther, D., Viscor, T., and ostergard, K., The AAU CubeSat Student Satellite Project: Architectural Overview and Lessons Learned, 16th IFAC Symposium on Automatic Control in Aerospace, (Russia), 2004. 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[54] Leve, F., Design of a 3DOF Testbed For Microsatellite Autonomous Operations, AIAA Aerospace Sciences Meeting, 2006. BIOGRAPHICAL SKETCH Frederick Aaron Leve was born in Hollywood, Florida, in 1981. In August 2000 he was accepted into the University of Florida's Department of Aerospace Engineering in the College of Engineering where he pursued his bachelor's degrees in Mechanical and Aerospace Engineering. After completing his bachelor's degrees in May 2005, he was accepted into the master's program in aerospace engineering at the University of Florida. While in the master's program, he received two awards in academia. In January 2007 he received the American Institute of Aeronautics and Astronautics Abe Zarem Award for Distinguished Achievement in Astronautics. For this award he was invited to Valencia, Spain, where he competed in the International Astronautical Federations's International Astronautical Congress Student Competition. Here he received the silver Herman Oberth medal in the graduate category. In May 2006, he was accepted to the Air Force Research Lab Space Scholars Program, where spent his summer conducting space research. His interests include satellite attitude control, satellite pursuit evasion, astrodynamics, and orbit relative motion. PAGE 1 1 DEVELOPEMENT OF THE SPACECRAFT ORIENTATION BUOYANCY EXPERIMENTAL KIOSK By FREDERICK AARON LEVE A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2008 PAGE 2 2 2008 Frederick A. Leve PAGE 3 3 To my mother and father PAGE 4 4 ACKNOWLEDGMENTS I thank m y supervisory committee chair (Dr. No rman FitzCoy) for giving me the chance to embark on this innovative research. I thank him for his patience and ad visement. Without his help I would have certainly been lost. I would like to thank my fellow colleagues in my research group were of great assistance to me; Dr. Andr ew Tatsch, Shawn Allgeier, Andrew Waldrum, Jaime Bestard, Sharan Asundi, Dante Buck ley, Vivek Nagabhushan, Josue Munoz, Nick Martinson, and Will Mackunis. Lastly I would al so like to thank my supervisory committee (Dr. Warren Dixon and Dr. Gloria Wiens) for their assi stance in validating my masters research. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........7 LIST OF FIGURES................................................................................................................ .........8 LIST OF ABBREVIATIONS........................................................................................................ 10 CHAP TER 1 INTRODUCTION..................................................................................................................13 Research Motivation............................................................................................................ ...13 Mission Examples...................................................................................................................14 Research Focus.......................................................................................................................14 2 ATTITUDE CONTROL ACTUATORS................................................................................16 Types of Attitude Actuators.................................................................................................... 16 Types of CMGs.......................................................................................................................19 Single Gimbal Control Momentum Gyroscopes............................................................. 20 Double Gimbal Control Momentum Gyroscopes............................................................ 20 Variable Speed Control Momentum Gyroscopes............................................................ 21 3 SGCMG DYNAMICS AND PERFORMA NCE CHARACTERIZATION.......................... 24 Dynamics................................................................................................................................24 Torque Amplification........................................................................................................... ..28 Singularities............................................................................................................................30 External Singularities......................................................................................................30 Internal Singularities.......................................................................................................31 Singular Surfaces.............................................................................................................31 Steering Logics................................................................................................................ .......33 Momentum Ma nagem ent........................................................................................................38 External Angular Momentum Offloading....................................................................... 39 Internal Angular Momentum Offloading........................................................................ 39 4 SGCMG ACS CONFIGURATIONS..................................................................................... 43 Rooftop Configuration.......................................................................................................... ..43 Box Configuration..................................................................................................................43 Pyramid Configuration.......................................................................................................... .43 SOBEK Pyramid Configuration......................................................................................44 SOBEK Hardware...........................................................................................................45 SOBEK Mechanical Design............................................................................................46 SOBEK Mechanical, Power and Output Specifications .................................................. 47 PAGE 6 6 SOBEK Attitude Determination......................................................................................47 5 RESULTS...............................................................................................................................52 Simulation Model............................................................................................................... ....52 Lyapunov Stability Analysis of EMK Attitude Controller..............................................53 Lyapunov Stability Analysis of Actual Att itude Controller w ith State Uncertainty....... 55 Simulation Results and Discussion......................................................................................... 56 Experimental Results and Discussion..................................................................................... 58 6 CONCLUSION AND FUTURE RESEARCH...................................................................... 70 Conclusion..............................................................................................................................70 Future Research......................................................................................................................70 APPENDIX OffTheShelf Rw, CMG, And Magne t Torquer Data Specifications ........................................... 71 LIST OF REFERENCES...............................................................................................................74 BIOGRAPHICAL SKETCH.........................................................................................................78 PAGE 7 7 LIST OF TABLES Table page 21 Attitude control actuators and their specifications............................................................. 21 41 Mechanical and Power Speci fications of the SOBEK ACS .............................................. 48 51 Simulation parameter values.............................................................................................. 61 A1 Offtheshelf performance specifications for RWs............................................................71 A2 Offtheshelf performance specifications for CMGs......................................................... 72 A3 Offtheshelf performance spec ifications for Magnet Torquers ........................................ 73 PAGE 8 8 LIST OF FIGURES. Fig. page 11. Onorbit assem bly............................................................................................................. 15 12. Earth m onitoring in LEO..................................................................................................15 13. Postdocking stabilization .................................................................................................15 21. Magnet coil from Tokyo Institut e of Technology Cute1.7 CubeSat............................... 21 22. Dynacon miniature RW ....................................................................................................22 23. Semilog plot of input power versus output torque for offtheshelf RWs and CMGs....22 24. Semilog plot of mass versus output to rque for offtheshelf RWs and CMGs................22 25. SSTL m icrosatellite SGCMG flown on Bilsat1..............................................................23 26. DGCM G developed by L3 Communications.................................................................... 23 31. The SGCM G with gimbal coordinate axes.......................................................................40 32. A 323 rotation sequence through angles iii ........................................................41 33. Intersecting planes of the spa nning of SGCMG a ngular momentum............................... 41 34. Angular m omentum envelope for 35.26 and 1,1,1,1.......................................41 35. Intern al singular surface 35.26 and 1,1,1,1 ...................................................42 36. Zerom omentum configuration......................................................................................... 42 41. Honeywell rooftop configuration .................................................................................... 48 42. Honeywell box configuration ........................................................................................... 48 43. External singular surfac e for box type SGCMG ACS ...................................................... 49 43. Honeywell pyram id configuration.................................................................................... 49 44. Pyram idal SGCMG ACS..................................................................................................50 45. Rendition of SOBEK te stbed............................................................................................50 46. ACS with hardware components ......................................................................................51 47. PhaseSpace attitude determ ination system....................................................................... 51 PAGE 9 9 51. ACS si mulation plots for k = 0.08 and c = 2.5k : A) Quaternion error vector elements versus time, B) Gimbal rates versus time, C) Angular rates versus time, and D) Singularity parameter versus time.....................................................................................62 52. ACS si mulation plots for k = 0.08 and c = 2.5k : A) Quaternion error vector elements versus time, B) Gimbal rates versus time, C) Angular rates versus time, and D) Singularity measure versus time........................................................................................ 63 53. ACS si mulation plots for k = 4 and c = 3k : A) Quaternion error vector elements versus time, B) Gimbal rates versus time, C) Angular rates versus time, and D) Singularity measure versus time........................................................................................ 64 54. SOBEK testbed GUI .........................................................................................................65 55. ACS experim ental plots for k = 0.8 and c = 2.5k : A) Quaternion error vector elements versus time, B) Gimbal rates ve rsus time, C) Angular rates versus Time, and D) Singularity measure versus time............................................................................ 66 56. ACS experim ental plots for k = 0.24 and c = 2.0 k : A) Quaternion error vector elements versus time, B) Gimbal rates vers us time, C) Angular ra tes versus time, and D) Singularity measure versus time................................................................................... 67 57. ACS experim ental plots for k = 0.32 and c = 3.0 k : A) Quaternion error vector elements versus time, B) Gimbal rates vers us time, C) Angular ra tes versus time, D) Singularity measure versus time........................................................................................ 68 58. ACS experim ent quaternion error of Moore Penrose solution......................................... 69 PAGE 10 10 LIST OF ABBREVIATIONS ACS Attitude control system API Application program interface CMG Control moment gyroscope CM Center of mass DCM Direction cosine matrix DoD Department of Defense DOF Degree of freedom EADS European Aeronautic Defense Space Company EMK Exact model knowledge GUI Graphical user interface FACETS Flywheel Attitude Control and Energy Transmission System IPACS Integrated Power and Attitude Control System LEO Low Earth Orbit MW Momentum Wheel ORS Operationally Responsive Space RW Reaction Wheel R2P2 Rapid Retargeting and Precision Pointing SGCMG SingleGimbal Control Moment Gyroscope SOBEK Spacecraft Orientation B uoyancy Experimental Kiosk SSTL Surrey Space Technology Lab UUB Uniformly Ultimately Bounded PAGE 11 11 Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science DEVELOPMENT OF THE SPACECRAFT ORIENTATION BUOYANCY EXPERIMENTAL KIOSK TESTBED By Frederick Aaron Leve May 2009 Chair: Norman FitzCoy Major: Aerospace Engineering Most satellites are unique and therefore usuall y take 10 to 15 years to design, fabricate, test, and finally launch. To expedite future advances in space technology, setting standards in terms of blackboxed subsystems and interfac es, drastically reduces the time and long term costs needed to complete these tasks. What is meant by blackboxe d is a stand alone subsystem that is adaptable enough to work w ith any other subsystems without alteration. Government organizations such as the Depa rtment of Defenses (DoDs) Operationally Responsive Space (ORS) office take these tasks in to consideration and look to change this pattern by expediting the proce sses of design, fabric ation, test, evaluation, and launch. To accomplish these tasks, smallsats which range from 1 to 1000 kg in mass, are considered for their cheaper platforms and launch costs. With utiliza tion of smallsats of the pico(0.11 kg), nano(1 to 10 kg), and microclasses (10100 kg) fo r space missions, obstacles occur in terms of power, mass, and volume constraints. Many of these same smallsats obstacles relate to the attitude control system (ACS). Despite these obstacles for many space missions, small satellites must be held to the same pointi ng requirements as their larger counterparts to be of use. To compensate for these obstacles while maintaining close pointing requirements, innovations in attitude control actuators and determination sensors that satisfy these constraints need to be developed. PAGE 12 12 An excellent choice for attitude control actua tors of smallsats are those of the flywheel sort known as zeromomentum and momentum bias actuators. Thes e actuators have the ability to perform precision attitude maneuvers without use of propellant. There are two classes of flywheel attitude control actuators that are used in satellites, reaction/momentum wheels (RWs/MWs) and control moment gyroscopes (CMGs) Due to the scarcity of CMGs available for small satellites below the microclass, RWs are presently utilized. An analysis completed in this thesis shows that although the current stateoftheart CMGs are too massive to fit inside smallsats, the performance relati onships in terms of output tor que per mass and input power are more favorable than that for RWs. This analysis is an additional motivation for the research in this thesis, which is carried out on the premise that an ACS consisting of singlegimbal control moment gyroscopes (SGCMGs) can meet th e performance requirements providing rapidretargeting and precisio npointing (R2P2) for smallsats while enduring their power, mass, and volume constraints. To further test this premise, the topic of th is thesis is the development of a ground based SGCMG testbed known as Spacecraft Orientation Buoyancy Experimental Kiosk (SOBEK) was developed to validate the utility of using SGCMGs for smallsat attitude control. The thesis justifies the reasons for using SGCMGs on SO BEK, discusses the spacecraft and actuator dynamics and kinematics associated with an AC S containing CMGs, addresses the geometrical configurations of systems of these actuators and the configur ation chosen, and then finally validates the testbed numerically th rough simulation and experimentation. PAGE 13 13 CHAPTER 1 INTRODUCTION Research Motivation The Departm ent of Defenses (DODs) Operationally Responsive Space (ORS) office [ 1] seeks to better the capab ilities of developm ent, fabrication, testing, evaluati on and launch of space assets. To accomplish the goals of the ORS office, standards such as blackboxed systems are considered useful to aid these ta sks and reduce the uniquene ss factor of every satellite project. To accomplish some of these task s, it is believed that switching from larger more massive satellites to smalls ats of the pico(0.11 kg), nano(110 kg), and microclasses (10100 kg) opens up the doors for a wide variety of missions a nd provides a quicker and less costly method (i.e., small satellites are cheaper a nd faster in terms of integration and launch) to develop these blackboxed techno logies. The budget of developing smallsats combined with componentsofftheshelf (COTS) makes it possi ble for some universitie s and smaller aerospace companies to contribute to the knowledge base and evolution of space te chnologies. Numerous university satellite projec ts consisting of missions involving CubeSat and nanosatellite projects have utilized COTS and been f light tested successfully [ 24]. Standard izations for smallsats in the acad emic sector are being led by California Polytechnic Institute and Stanford University. These endeavors include CubeSat Kit CubeSat bus architecture (Pumpkin Inc.) and PPod launch vehicle integration [ 57]. Work is also being done on standardization of satellite bus architecture in the military sector such as the ANGELS, XSS11, and the tactical sa tellite programs at AFRL [ 8] and [ 9]. Although sm allsats are a viable choice for tes ting of new space technologies, they have limited attitude control and propulsive capabilit ies which leaves some in the aerospace community to regard smallsats as toys or space debris with no significant functionality. This perception of smallsats can be challenged with the development of low power and mass attitude PAGE 14 14 control systems (ACSs) which are near the same pointing and slew requirements of those for larger satellites and hence are still able to accomplish an assortment of many space missions for smallsats where propulsiv e capabilities can be li mited or nonexistent. Mission Examples Som e space missions that could possibly utilize smallsats are laser communication, space science, onorbit assembly, satellite servicing, formation flight blue force tracking, and Earth monitoring. Smallsats that have propulsive capab ilities would most likely have a minimum amount of fixed thrusters, theref ore their attitude and translati on would be coupled. Taking this into account, for a smallsat distribution used fo r such missions as onorbit assembly shown in Fig. 11, precision in terms of position and orient ation need to be maintained for collision avoidance and attitude control where redundant thrusters is most likely not possible. Earth sensing and blue force tracking which have satellite systems in Low Earth Orbit (LEO) require tracking of a specific point on the su rface of earth. This becomes nontrivial due to smaller orbital periods (i.e., faster orbital rates) To acquire a longer time of coverage, higher slew rates while maintaining precision are neede d. An example of this is shown in Fig. 12. When dealing with missions such as satellite servicing, there may be a situation when the satellite being serv iced is noncooperative and is more ma ssive than the satellite which is providing the servicing. In this si tuation the ACS on board the servic ing satellite must be able to provide large amounts of torque re sponsively to stabilize the system An illustration of this is shown in Fig. 13. Research Focus The research focus of this thesis is in the developm ent of a ground base d testbed to test and evaluate propellantless ACS me thodology for smallsats which can endure the volume, mass, and power constraints while still maintaining performa nce in terms of high torque, slew rates, and PAGE 15 15 precision. The ACS methodology presented in this thesis attempts to validate singlegimbal control moment gyroscopes (SGCMGs) as a viable option of attitude control actuators for smallsats which can contribute high torque, slew rates, and attitude precision while meeting their volume, mass, and power constraints. Examples of other attitude actuators including SGCMGs are discussed in the next chapter for comparison. Fig. 11. Onorbit assembly Fig. 12. Earth monitoring in LEO Fig. 13. Postdocking stabilization PAGE 16 16 CHAPTER 2 ATTITUDE CONTROL ACTUATORS Types of Attitude Actuators Attitud e control actuators for satellites can be passive or active. Passive attitude control actuators such as gravity gradie nt booms use the gravitational fi eld of the Earth to provide a bounded attitude between 5i in the orbital plane. Because the gravitational field is conservative, gravity gradient torques g do not dissipate the satellit es kinetic energy or dump its angular momentum. The tor que is dependent on the distance R between the spacecrafts center of mass (CM) and Eart hs CM, the spacecraft prin ciple moments of inertias iJ, and the small angles of deviation between the spacecrafts principle axes and the Local verticalLocal horizon coordinate frame i For small angular displacements the torque is expressed as, 231 312 33 0gJJ JJ R Another type of passive attitude control util izes hysteresis rods or permanent magnets. Typically permanent magnets carry accuracy on the same order of magnitude as gravity gradient booms but cause a satellite to flip at the poles. These actuators like gravity booms are conservative and do not dissipate energy or dump angular moment um. Hysteresis rods on the other hand are passive actuators that may dissipate energy and du mp angular momentum. These actuators can be used to null the angular rates of satellites with residual angular momentum after launch [ 10]. There are th ree main categories of active at titude control actuators. These categories include magnet torquers, reaction control devices, and flywheel actuators. Magnet torquers are the most co mmon attitude control actuators utilized in smallsats (e.g., magnet coils and torque rods). Th ese actuators are cheaper and lighter than other attitude control PAGE 17 17 actuators used onboard smallsats. Figure 21 show s an example of a magnet coil for attitude control. Magnetic torquers produce a torque B as a reaction to the change of the local magnetic field of the satellite to that of the Earth. This torque is modele d as an interaction (cross product) between, the magnetic moment of the spacecraft and Earths magnetic field direction B as, B B Since the torque is a function of the cross product between the satellites magnetic moment and Earths magnetic field, magnet torquers suffer from a singularity which limits their actuation where no components of torque are available along and B Due to the change of the local magnetic field of the satellite, sensors for att itude determination such as magnetometers are unable to take readings during use of these actuators until the residual magnetic field of the satellite has decayed. These actu ators loose performance as a function of distance squared hence are less useful for satellites at or bits with higher altitudes than LEO. Reaction control devices such as thrusters re quire propellant which is at this time not available for the majority of nanoand picosat ellite missions. These devices produce a torque on the satellite via propulsion through jets of fluid, heated or cooled plasma and gas, or charged ions with use of a magnetic field. Thrusters that utilize the chemical energy within the propellant (e.g., hydrazine) produce greater magnitude of output thrust but require la rge radiators to dump the expended heat created. Thrusters of this so rt are not realizable fo r smallsats which do not have the extra mass and volume to contain larg e radiators for the thermal protection system. The last type of attitude c ontrol actuators known as flywheel actuators can be broken down into two types, reaction and momentum wheel s (RWs/MWs) and control moment gyroscopes (CMGs). These are the most accurate sources of attitude control. PAGE 18 18 Reaction and momentum wheels work off th e same principle by producing a reaction torque opposite in direction to the acceleration of a flywheel. The difference between MWs and RWs is that, RWs have a zero nominal operating sp eed (i.e., the speed that the wheels are spun back down to after the maneuver is completed) When spinning RWs/MWs back down to their operating speed, the satellite will revert to its original orientation due to the conservation of angular momentum. For this reason both MW s and RWs must have other actuators onboard such as thrusters or magnet torquers to dump the excess angular momentum, or these actuators will suffer angular momentum saturation of thei r flywheels. Three RWs/MWs are needed for full threeaxis attitude control of a satellite. An example of a RW used for CubeSat and nanosatellite missions is shown below in Fig. 22. The second type of flywheel attitude c ontrol actuator known as CMGs provide an instantaneous gyroscopic torque al ong a torque axis defined as the axis perpendicular to the flywheel and gimbal axes. These actuators rely on shifting the dire ction of satellites body angular momentum to apply a torque. Three CMGs are needed for threeaxis control in general, although a fourth is typically added for si ngularity avoidance a nd/or redundancy. Performance characteristics for typical attitude control actuators were investigated and are shown in Tables 21 and Tables A1 through A3 in the appendix. Table 21 is taken from Space Mission Analysis and Design by Wertz [ 11]. The tables in Appendix A are for currently available R Ws, CMGs, and magnet torquers and supplement that which is shown in Table 21 [ 1223]. In Table 21, the only attitude control actuato rs th at can meet the performance demands of high precision while maintaining the low power requirement for small satellite systems are flywheel actuators consisting of RWs, MWs, and CMGs. Magnet torquers/coils do not produce PAGE 19 19 sufficient output torque for a gi ven input power and are less accura te due to the uncertainty of Earths dynamic magnetic field model. A survey was conducted of existing RWs a nd CMGs. The results of the survey are shown in Figs. 23 and Fig. 24. The supporting data for these figures is provided by Tables A1 and A2 Figures 23 and 24 show that CMGs includi ng that which is developed in the Spacecraft Orientation Buoyancy Experiment al Kiosk (SOBEK) have a highe r torque output for a given mass and input power than RWs. These figures also illustrate that CMGs are currently nonexistent for classes of satellite s equal to or smaller than micr osatellites. The scarcity of these actuators is due to their complexity in de sign and use. If this complexity was removed and they were designed in a blackboxed manner (i .e., accepting only an input torque or attitude and producing it), then they could potentially find more use in smaller satellites due to higher torque per mass and power advantage over RWs. There are three CMGs that have been developed for microsatellites by companies such as EADS/Astrium, Honeywell, and Surrey Space Technology Laboratory (SSTL) [ 2428] Fig. 25 shows one of these three, the SSTL singlegi m bal control momentum gyroscope (SGCMG) which flew on the Turkish microsatellite Bilsat1 [ 29]. Types of CMGs There are two varieties of CMGs: ones with a singe controllable degree of freedom (DOF) known as SGCMGs and those with multiple cont rollable DOF known as doublegimbal control momentum gyroscopes (DGCMGs) and variables peed control moment gyroscopes (VSCMGs). The benefits and drawbacks of each type of CMG is discussed next. PAGE 20 20 Single Gimbal Control Momentum Gyroscopes SGCMGs such as the ones shown in Fig. 25 are the least mechanica lly complex form of CMGs. They utilize a single gimbal axis to pr oduce a gyroscopic torque by rotating a constant speed flywheel about a gimbaled axis perpendi cular to the spinaxis. A minimum of three SGCMGs are needed to obtain thre eaxis attitude control with th ese actuators. However an ACS using SGCMGs in some cases utilizes four of th ese actuators to avoid performance inefficiencies known as internal singularities. The benefit of these actuators among other CMGs lies in the property of torque amplification which is the ratio of output torque from the ACS containing SGCMGs to input torque from the gimbal moto rs. Discussion of singularities as well as performance characterization of SGCMGs in terms of momentum management and torque amplification is discussed in chapter 3. Double Gimbal Control Momentum Gyroscopes Doublegimbal control momentum gyroscope s (DGCMG) are the most mechanically complex form of CMG actuators. These actu ators produce torque by the same method as SGCMGs but possess a redundant mechan ical controllable DOF in term s of an extra gimbal axis. This redundancy can be useful when a set of two actuators is used for underactuated attitude control of a satellite but can be harmful when gimbal lock occurs. Gimbal lock occurs when the spin axis of a DGCMG aligns with its axis of freedom and in consequence deprives the DGCMG of one of its controllable DOF thereby elimina ting its useful propertie s. There are however acceptable methods of avoiding these singularities which in turn leaves DGCMGs the benefit of multiple degrees of freedom. Although with the addi tion of an extra gimbal to these actuators, they are heavier and more mechanically comp lex than other forms of CMGs which excludes them from being a viable option for smalls ats. A figure of a DGCMG developed by L3 Communications for the International Space Station shown in Fig. 26. PAGE 21 21 Variable Speed Control Momentum Gyroscopes VSCMGs are used for their extra DOF adde d through their reacti on wheel mode where flywheel accelerations are nonzero. This extra DOF has its benefits in that it does not require steering logic and is able to produce the required torque when a singularity is encountered [ 3032]. Although when in reaction wheel mode the fl ywheel acceleration s require shaft power and therefore input torque thereby increasing their re quired power and reducing their available torque amplification. Another effect of the reaction wheel modes of VSCMGs is that it becomes near impossible to isolate vibrations from the varyi ng flywheel speeds. In addition to the larger power requirement, less torque amplification, and i nduced vibration, the moto r driver circuitry of VSCMGs is more complicated, and they require more computational and electrical power than standard SGCMGs to control. SGCMGs are considered here as the most viable option for th e study of smallsat attitude control and hence were chos en for the SOBEK testbed. Table 21. Attitude control actuators and their specifications Actuator Output Accuracy (deg) Mass (Kg) Power (W) Magnetic Torquers 1 to 4,000 Am2 1 to 5 0.4 to 50 0.6 to 16 Hot Gas Thrusters Cold Gas Thrusters 0.5 to 9,000 Nm <5Nm 1 to 5 variable N/A RW & MW 0.005 to 1 Nm 0.001 to 1 1 to 20 10 to 110 CMG 10 to 500 Nm 0.001 to 1 >1 5 to 150 Fig. 21. Magnet coil from Tokyo Inst itute of Technology Cute1.7 CubeSat PAGE 22 22 Fig. 22. Dynacon miniature RW 0 20 40 60 80 100 120 140 202468 log10(torque) mNmPower W RW CMG SOBEK Fig. 23. Semilog plot of input power versus output torque for offtheshelf RWs and CMGs 0 20 40 60 80 100 120 140 202468 log10(torque) mNmMass kg RW CMG SOBEK Fig. 24. Semilog plot of mass versus out put torque for offtheshelf RWs and CMGs PAGE 23 23 Fig. 25. SSTL microsatell ite SGCMG flown on Bilsat1 Fig. 26. DGCMG developed by L3 Communications PAGE 24 24 CHAPTER 3 SGCMG DYNAMICS AND PERFORMA NCE CHARACTERIZATION Dynamics CMGs produce a gyroscopic tor que about a torque axis it orthogonal to both the gimbal and spin axes ig and is, respectively. An illustration of the CMG gimbal frame iG and how it relates to the spacecraft body frame B is shown in Fig. 31 with gimbal angles, velocities and accelerations, iii and flywheel velociti es and accelerations, ii To analyze the torque generated by CMGs, we first develop the angular momentum expression and coordinatize it in the spacecraf t body fixed frame. Application of Eulers equation assuming the external torq ue is zero yields the governing equation for the CMG system. This development is as follows. The total spacecraftcentroidal angul ar momentum of the system coordinatized in the body frame is BBBBB CCCHJh (31) with the spacecraft centro idal inertia tensor incl uding the CMG components B CJ, the spacecraft body fixed angular velocity BB and the total CMG angular momentum B Ch about the spacecraft center of mass (CM) c oordinatized in the body frame. The spacecraft centroidal inertia B CJ in Eq. (32) contai ns a fixed inertia B CJ, time varying inertia from the comb ined gimbalwheel system 1iin GT BGgwBG iCIC, and the parallel axis components of inertia associated with each CMG 1iiiiBTBBBT iCCCCmRRRR with mass im and position from the spacecraft CM to that of the flywheel, iB C R 11iiiiiin BBTBTBBBT CCBGgwBGiCCCC iJJCICmRRRR (32) PAGE 25 25 Assuming that iB C R is held fairly constant and ther efore its time derivative is zero, we differentiate BBB CJ with respect to time as, 1 BBB BBB C C BBBBBB CCdJ J d JAJ dtdt (33) This CMG contribution of angular momentum B Ch coordinatized in the body frame is expressed as, 4 1i iG B CBG ihCh (34) and is found in by summing the contributions of each individual CMG angular momenta iGh iniG which is transformed to the body frame by the DCM iBGC. Note that for SGCMGs, the DCM iBGC varies only with the gimbal angles of the ith CMG i The angular momentum of the ith CMG iGh in G is, 0iwi G g wiI h I where i and w I are the ith flywheels angular speed and centroidal inertia about its spin axis is andi and g w I are the ith gimbal rate and the centroidal inertia of the combined wheelgimbal system about its gimbal axis ig Taking the time derivative of B Ch and observing that ,,BB CChh we get, BBBB CCCCdhhhh ddd dtdtdtdt (35) with the Jacobian coefficient matrices of the CMG states are defined as, PAGE 26 26 2B Ch A (37) B Ch B (38) B Ch C (39) Combining these terms we get, 2 B Cdh ABC dt If we also combine 1A with 2A which are both multiplied by the gimbal rates we have a more compact representation of the output torque from the CMGs in Eq. (310) where the complete Jacobian matrix is 12DAABC 11 2 ,,B B C Cdh d hAAABCDX dtdt (310) The form of Eq. (310) is similar to Schaub and Junkins [ 33] with exception of the different notation. The gyrosco pic components of torque are deri ved from the product of the skew symmetric matrix BB with the total centroidal angular momentum of the system B CH as, 11iiiiiiBBBBBBBBBBB CSC n BBTBTBBBTBB BGgwBGiCCCC iHJh CICmRRRR (311) The complete rotational equati on of motion for this system in Eq. (312) is found through the sum of Eqs. (33), (310), and (311) giving, PAGE 27 27 1 11 1iiiiii iiiiiiB BBBBBBBBBBB C SSC n TBTBBBTBB BGgwBGiCCCC i n BBGTBTBBBTBB BGgwBGiCCCC idH JJDXh dt CICmRRRR CICmRRRR (312) If it is assumed that, BB CSJJ g wiwiII and g wiwigwiwiiIIII 0i and there are no external torques applied to the spacecraft the reduced equation of motion for this system is 00B BBBBBBBBBBB C SSCdH JJhAh dt (313) with 0 whI and 0 B ChhA A 323 DCM through Euler angles iii where i and i are the constant spacing and inclination angles, and i is the time varying gimbal angle transforms iG to B for the fourCMG pyramid configuration in Fi g. 31. This DCM is expanded as, ,cos,siniiiiiiiiiiiii BGiiiiiiiiiiii iiiiiiiccssscsscscc Csccssccssssc scccscc cs where (3.14) The 323 Euler sequence can be visualized in Fig. 32. PAGE 28 28 Now that we have reduced the general equati ons of motion to an easily useable form we can address the performance char acterization of SGCMGs. There are three areas that describe the performance characteristics of SGCMGs: torque amplification, singularities, and momentum management. Each one of these areas is discussed next. Torque Amplification Torque amplification for a SGCMG is defi ned as a higher amount of output torque produced than a given gimbal moto r input torque required, assuming negligible torque needed by the flywheel motors. This is a reasonable a ssumption when considering SGCMGs because their flywheels require minimum torque to spin at a constant speed. In the literature (e.g., [ 34]), torque am plification is defi ned for a single actuator as, 22 2 2Torque Amplification = B Ci out i BBB in Ch h (315) If it is assumed that there are no disturba nce torques on the system, the equation for internal torque can be found by decomposing Eq. (313) of a single SGCMG actuator into two separate elements with equal ma gnitudes and opposite signs as, BBBBBBBBBBBB SSCCJJhh where BB CiChh The internal torque can be decom posed to into an output torque B outCih from the CMG and input torque BBB inCh which is the gyroscopi c contribution from the satellites angular velocity in Eq. (31 3). If the twonorm of newly defined out is divided by that of in we get the same form of the torque am plification equation shown in Eq. (315). PAGE 29 29 When considering picoor nanoclass satellite s where the ACS might take up a majority of the mass and volume of the overall satellite and the gimbalwheel system mass may make up a majority of the mass and volume of the ACS, the contributions to torque and angular momentum from the gimbalwheel system and the gimbal mo tor friction inefficiencies may not be assumed negligible. Therefore, Eq. (315) is an uppe r bound for the torque am plification equation. Useful information is gathered from Eq. (315) such as the insight that the spacecraft angular velocity has to be less than the gimbal rate to have a torque amplification greater then one. To understand torque amplification of a single SGCMG for use on smallsats a lower bound was developed [ 35] and [ 36]. This lower bound is 0 2 2 2sgnout i BBB in Cgwidisih hIFF (316) with gimbal input torques g wi I and motor friction inefficiencies with dynamic and static coefficients dFand s F. If the motor friction inefficiencies are of th e main concern we get a reduced form of Eq. (316) as, 2 2 2 2 2sgnout ii in disi BB B CFF h h and more compactly as, 1 sgnii si d i iF F hh (317) where torque from static friction is assumed to be negligibly small when compared to that for dynamic friction. PAGE 30 30 Equation (317), which considers the mechanical inefficiencies of the gimbal motors, is a lower bound in the presence of these motor friction inefficiencies on the torque amplification of an SGCMG. From this equation, it can be seen that choosing the correct gimbal motor to make the optimization parameter dh F as high as possible, will reduce the motor friction effects on the SGCMG torque amplification. It should be no ted that increasing the flywheel speed as much as possible will give the largest torque amplification for small satellites where mass, volume and power are limited but will also increase their jitt er. The motor inefficiencies of the DC motors used to rotate the flywheels are overlooked for SG CMGs that are not variable speed, due to the fact that they have a very small dynamic friction coefficient and their static friction and residual torques are largest on startup. Now that torque amplification has been discussed, singularities associated with SGCMGs are addressed. Singularities SGCMGs experience two classes of singularities. These singularities are internal and external singularities ea ch of which are instances where th e required control torque cannot be produced. External Singularities External singularities occur when the addition of angular momentum from an external source (e.g., disturbances) saturates the SGCMGs of the ACS. Of these disturbance torques, aerodynamic torques are considered to be the most harmful when in LEO due to their size and constant nature in comparison wi th other disturbance torques. External singularities are usually addressed in mission planning and considered the most harmful at the end of a satellites life cycle when deorbiting. PAGE 31 31 Internal Singularities Internal singularities are encountered on the fly when th e Jacobian matrix of the SGCMG ACS becomes rank deficient, whereby the torque vectors lie in a plane and no torque can be produced normal to this plane. Unlike external singularities they must be handled online and cannot simply be designed for a priori. The tw o types of internal si ngularities are known as elliptic which are unavoidable/inescapable thr ough null motion (i.e., null motion is motion of the gimbals that produces no net torque on the sate llite) and hyperbolic wh ich are avoidable through null motion. Singularity avoidance strategi es are discussed later in this chapter Singular Surfaces The CMG torque vector directions it are the columns of the Jacobian matrix. Therefore, singularity occurs when all it lie in the same plane. There are 2n singular configurations (i.e., singular sets of gimbal angles, ) for any singular direction u normal to this plane. All possible it for each CMG span a plane and an example of these planes for a fourCMG pyramid configuration can be visualized clearly in Fig. 33. The CMG torque vector direction is defined as, torque vector =ii itg s With a given singular direction u not parallel to a gimbal axis, there are two possibilities for singularity to occur along each singular direct ion. These possibilitie s are shown in shown below. 0, 0 or 0, 0iiiiutusutus It should be noted that the occurrence of si ngularity when two gimbal axes are aligned has already addressed as gimbal lock and occurs in DGCMGs. PAGE 32 32 Using the notation found in reference [ 35] and [ 32], we can define iisignus. With these definitions, the spin and torque axis ve ctors at singular states can be defined as, 2 i iiii ig sguug gu and 2 i iii igu tug gu Therefore the total norma lized angular momentum B Ch of the CMGs at singular states corresponding to u is, 2 i B Ciiii ii ig hsguug gu When1i (or 1i due to symmetry), the external singular surface known as the angular momentum envelope shown in Fig. 34 is formed as the locus of the maximum projections of the angular mo mentum of each CMG at the singular directions. External singularities occur on the surface of this envelo pe when the total internal SGCMG angular momentum is less than that which is external to the spacecraft, and the SGCMGs are then saturated. The internal singular surface is formed in the same way as that of the external singular surface with exception to one i = 1. An examples of the inte rnal singular surfaces for an ACS containing four SGCMGs in a pyramidal arrangement at an inclination angle 35.26 is shown in Figs. 35. The inclination angle s hould not be confused with the skew angle used to characterize the angular momentum envelope [ 37] and [ 38]. The skew angle is the minimum angle of the SGCMG planes shown in Fig 33 with that of the horizontal. For a spherical angular PAGE 33 33 momentum envelope,9054.74. Units of Fig. 34 and Fi g. 35 are in terms of the nominal angular momentum magnitude 0h (i.e. diameter of angular momentum envelope involving four SGCMGs is less than or equal to four 0h). In Fig. 34, locations that have white circles or holes on the momentum envelope are shown where the gimbal axes are located and there is no angular momentum available. The knowledge of where these singularities oc cur is known, and there have been methods developed that steer away or escape from th ese singularities known as steering logics. These methods are discussed in the next section. Steering Logics An ACS utilizing SGCMGs requires an addi tion of a fourth actuator for minimum redundancy to avoid singularities through null motion (i.e., moti on is motion of the gimbals whereby no net torque is produced). This fourth actuator renders the systems Jacobian matrix to be nonsquare which in turn requires a pseudoinverse to map the output torque 3B Ch onto the gimbal rates 4 Recall, that there are 2n singular configurations for each singular direction of n SGCMGs. Therefore, it is important to note that the addition of actuators does not eliminate the problem of singularities but provide s a null space where singularity avoidance may be provided. The MoorePenrose pseudoinverse solution to the gimbal rates is represented as 01B CAh h (318) where the MoorePenrose pseudoinverse is, 1TTAAAA PAGE 34 34 The MoorePenrose pseudoinverse solution for th e gimbal rates in Eq. (318) fails when TAA becomes singular. To avoid/escape singularit ies when mapping the output torque onto the gimbal rates, a steering logic is applied. Steering logics can be broken up into two main groups: those which provide null motion for singularity avoidance such as local gradient methods and global avoidance algorithms, and pseudoinverse solutions which escape singularities through load ing of the Jacobian matrix singular values. Common drawbacks to these st eering logics are computational complexity, inability to avoid all internal singularities, or loss accuracy by adding torque errors to keep the Jacobian nonsingular with lo ading of its singular values. Local gradient methods were of the earliest methods used to steer away from internal SGCMG singularities by Margulies and Auburn [ 39]. This local gradient method, like most steering logics, requires a fourth SGCMG added to provide a 1dim ensional null space where null motion can be applied to steer the gimbals away from singularities. The Jacobian matrix null space provides multiple soluti ons to the gimbal state equation to steer the gimbals away from singularity while maneuvering. An ex ample of local gradient methods is, 01B CAhIAAd h (319) where B Ch is the SGCMG output torque and 0h is the nominal magnitude of angular momentum. This equation is similar to Eq. (318) with the addition of a null motion vector d which is mapped to the Jacobian matr ix null space by the projection matrix I AA The contribution of null motion is scal ed by the singularity parameter 2 0exp m (320) which is an explicit functi on of the singularity measure m PAGE 35 35 1 det 2TmAA (321) and design constants and0 When using local gradient methods, the null motion vector dcan be chosen as the gradient, 1234,,,T f fff d of a objective function 1f m This null vector d produces null motion in the direction that maximizes m or the distance away from singularity. It is important to note that steering logics utilizing only null motion are unable to avoid elliptic internal singularities [40] and [ 41]. Local gradient methods shown here are com putationally intensive and ca nnot avoid all internal singulari ties such as elliptic internal singularities where null motion is unsuccessful. Global avoidance methods shown in Eq. (322) such as those developed by (Paradiso and Kuhns [ 42] and [ 43]) are similar to local gradient methods and differ in the fact that the null motion vector is produced by the dif ference of the gimbal angle positions from a set of alternate gimbal angle configurations known as preferred trajectories. This difference is shown in Eq. (322), where are the preferred trajectories. 01B CAhIAA h (322) PAGE 36 36 These preferred trajectories ar e usually calculated offline. This method is computationally intensive and is not suitable for online use. The most popular pseudo inverse soluti on is the SR inverse developed first by (444546 1 # TTAAAAI (323) # 01B CAh h (324) #A This SRinverse solution works by adding a positive definite matrix of torque errors I to the positive semidefinite matrix TAA to leave the matrix TAAI positive definite and hence nonsingular. Rather than avoiding internal singularities as discussed previously, this method which is developed from the MoorePenros e pseudoinverse, approaches singularity and makes a rapid transition to escape. At escape ther e is a jump in gimbal rates due to the added torque error. As a consequen ce of these added torque errors, this method is not the preferred method for precision pointing. Under certain circumstances (i.e., when det0TAA and # B ChnullA), this method can become locked into a singularity, so the generalized SRinverse (GSR) was developed by Wie [ 45] The GSRinverse 1 # TTAAAAE (325) is made up a positive definite symmetric matrix PAGE 37 37 033022 033011 0220111sinsin sin1sin0 sinsin1tt Ett tt (326) composed of time varying modular parametersi and scaled by the singularity parameter in Eq. (320). These modulati ng parameters ensure that B Ch will not stay locked in the null space of the pseudoinverse, which is not explicitly dependent on time. The GSRinverse has been shown to avoid all internal singularities but adds torque error to all directions. Therefore, the original SRinve rse was modified again to minimize the amount of torque error added to the gimbal s when avoiding internal singularities. This method developed by Ford is known as Singular Direction Avoidance (SDA) with the pseudoinverse T SDASDAAVSU and from which the gimbal rates are 01B SDACAh h (343) where V and U are unitary matrices and SDAS is, 3 2 12 311 ,, 0SDA Tdiag S The benefit of using this method ove r the SRinverse methods is that it only adds to rque error to the smallest singular value, which in turn requ ires less null motion and has smoother gimbal rate trajectories than the previous SR and GSRinverse methods. A drawback of this method is that it is required that a SVD be calculated for A every time step, making the steering logic computationally intensive. Also it is developed directly from the original SRinverse and therefore can become locked in si ngularity when the cont rol torque is in the singular direction of the pseudoinverse. PAGE 38 38 The discussed methods of avoiding or escap ing singularities are of the most popular of steering logics developed for SGCMGs [ 47]. Additional methods have been developed to avoid internal singularities associat ed with SGCMGs [4244]. An exam ple of one such method includes game theory [ 48]. Other methods for singularity av oidance have been developed with the addition of an extra d egree of freedo m while using VSCMGs instead of SGCMGs. In the case of VSCMG, steering logics asso ciated with gimbal velocity and acceleration have been developed by Schaub and Junkins [ 31] and [ 49]. These methods use the extra degree of freedom through flywheel accel eration to give an ACS using these actuators the ability to always produce the required output torque needed. Recall it was stated that VSCMGs do not re quire steering logics due to their reaction wheels modes. Therefore the addition of a fourth VSCMG instead may provide power tracking for systems such as the Flywheel Attitude Control and Energy Transmission System (FACETS) and the Integrated Power and Att itude Control System (IPACS) [ 50]. These systems are able to use the nu ll space instead to monitor power tr ansmission while converting the extra kinetic energy gathered from spinning down the flywheel s after a maneuver to electricity charging the power system on board. Systems such as FACETS require ceramic wheels to make use of the kinetic energy provided from spinning down the flywheels and were deemed unsafe due to the high flywheel speeds of fortythousand rpm and have henceforth not been flight tested. The next section discusses momentum ma nagement which deals with the dumping of external and internal angular momentum in order to maintain the performance of an ACS using SGCMGs. Momentum Management There are two situations where the angular mo mentum of the system must be managed in order to maintain the required ACS performance. These situations occur from external angular PAGE 39 39 momentum added into the system from torque di sturbances or internal components of angular momentum remaining post maneuver from gimbal angle saturation. External Angular Momentum Offloading External angular momentum accumulated by the spacecraft from nonzero disturbance torques is considered a priori for each specific mi ssion. If angular momentum is added into the system from the disturbance torque and it is greater than that fr om which is allotted from the CMGs, the system encounters an external singularity as discussed in section 3.2. Assuming that a spacecraft is in LEO orbit then the addition of aerodynamic torque is the only source that is considered to saturate the system. This is so because the magnetic disturbance torque is exceptionally small and the disturbance torque from the gravity gradient does not saturate the system with angular momentum due to its period ic nature. Also disturbance torques from solar pressure effects are neglected due to the small amount of solar cell surface area on small satellites. When considering small satellites, aerodynamic tor ques add a negligible rate of angular momentum into the system in most standard LEO orbits [ 36]. The addition of angular mom entum from disturbance tor ques is a slower process than that for the internal angular momentum buildup from the SGCMGs. For this re ason there is greater concern for offloading of internal angular momentum. Internal Angular Momentum Offloading Internal angular momentum accumulates when components of angular momentum from the SGCMG cannot be cancelled out due to gimbal angle saturation. As previously mentioned these gimbal angle constraints can be reduced by using magnetic bearings a nd/or slip rings which allow for full range of gimbal rotation. Two cases arise when internal angular momentum needs to be offloaded. The first case occurs when ex cess components of internal angular momentum is left over from maneuvers due to gimbal angle saturation where unwanted components of angular PAGE 40 40 momentum are unable to be cancelled out. The s econd case is at startup of the flywheels, where needs to be angular momentum dumped to stop the satellite from tumbling due to the offset in its angular momentum direction. A typical startup confi guration of an ACS c ontaining SGCMGs is at zeromomentum. This configuration for a f ourpyramid SGCMG cluster has all of the angular momentum vectors of each SG CMG in the body xy plane at a 0 leaving a zero net momentum for the ACS. An example of this confi guration is shown in Fig. 36. In this figure, all of the angular momentum vectors are in the plane although it is not a singular configuration because all of the torque vectors are not in the same plane. Fo r smallsats the offloading of the excess angular momentum on startup as well as that due to gimbal angle saturation can be accomplished by use of magnetic actuators. Now that mathematical and physical aspect s of ACSs containing SGCMGs have been discussed, the types of ACS configurations containing these actuat ors as well is the development and choice of the SOBEK configuration is addressed in the next chapter. Fig. 31. The SGCMG with gimbal coordinate axes i 3 b 1 b 2 b i G B ,,iii ,ii PAGE 41 41 Fig. 32. A 323 rotation sequence through angles iii Fig. 33. Intersecting planes sp anning of SGCMG torque directions Fig. 34. Angular momentum envelope for 35.26 and 1,1,1,1 it 1 it 2 it i g is it 1 b 2 b 3 bi i i i i PAGE 42 42 Fig. 35. Internal singular surface 35.26 and 1,1,1,1 Fig. 36. Zeromomentum configuration 1h 2h 3h 4h PAGE 43 43 CHAPTER 4 SGCMG ACS CONFIGURATIONS Rooftop Configuration The roof top design uses two groups of SGCMGs, each with parallel gimbal axes slanted at an angle to each other shown in Fig. 41. A CMG system in this configuration will never have elliptic internal singularities [ 40]. This means that they will always have continuous mom entum trajectories and therefore have c ontinuous gimbal trajectories. Th is configuration suffers from situations where its Jacobian may approach rank 1. The singular direction where this occurs is on the intersection of the two rows of SGCMGs shown as the red arrow in Fig. 41. Box Configuration The box configuration is a variation of the r ooftop configuration containing four SGCMGs with an inclination angle of 90. This configuration is shown in Fig. 42 where the red arrow indicates the same rank 1 dire ction where the torque vectors lie. The internal singular surface associated with this configuration cove rs a large surface of the angular momentum envelope and is seen as the large empty area presen t inside the external singular surface in Fig. 43. The singularity represented by this singular surface shown at 1h is avoided in the box configuration by utilizing only three SGCMG a nd keeping the third as a spare while staying within the 1h constraint. Only thre e of the four SGCMG will be used at any given time so the Jacobian is square and there is no need for a pseudoinverse to map the gimbal rates onto the torque. The major drawback s of this design are that it is not as compact as other designs and its performance is constrained to 1h of angular momentum, causing the ACS to require larger SGCMGs to meet the missions angular momentum requirements. Pyramid Configuration Common pyramid configurations have numbers of SGCMGs in groups of four or six. ACSs containing these amounts of SGCMGs and have both elliptic and hyperbolic internal PAGE 44 44 singularities. Although an ACS utilizing SGCMGs in a pyramid configuration is susceptible to the elliptic internal sing ularities, there are methods of escap ing these singularities that were discussed in the previous chapte r. This configuration has the benefit of giving a somewhat spherical angular momentum envelope for the right choice of inclination angle of the SGCMGs. The SOBEK testbed will utilize a minimal redundancy (i.e., four SGCMGs) pyramid configuration for threeaxis control while applying the GS Rinverse steering logic method discussed in chapter 3. SOBEK Pyramid Configuration The SOBEK testbeds ACS is shown in Fig. 44 with motion capture for attitude tracking provided by Phase Space and its visual interpretati on of the mapped wireless LEDs in the upper right hand corner. SOBEKs ACS uses four SGCM Gs in a pyramid configuration. The testbed was originally designed to float on an air beari ng table with multiple ACSs where the footprint of the ACS was limited while trying to minimize weight. For this reason, the inclination was chosen as angle 54.74. It should also be mentioned that the formulations in literature consider a zero reference gimbal angle position shifted 90 from where the SOBEK zero reference angle is. The new formulat ed SGCMG angular momentum vector 1234 01234 1234 B Cscsscs hhsscssc cccccccc (41) and Jacobian matrix 1234 1234 1234 B Csscssc h Acsscss cscscscs (42) takes these changes into account. PAGE 45 45 Other testbeds such as those at Virginia Polytechnical Institute, Lawrence Livermore Laboratory, SSTL, and Honeywell s Line of Site (LOS) testbed are of the few testbeds that comprise the known heritage for this research [ 5153]. This testbed differs from the ones previous ly referenced in that it is physically smaller and does not utilize an air bearing but a pivot point to mitigate the forces of gravity. Wh en making the testbed near marginally stable at a point, the only forces and torque s considered are the small amount of friction from that point as well as the residual moment from gravity. An illustration of SOBEK that contains the ACS in and stand which holds the ACS on a pivot point is shown in Fig. 45. This illustration was made in ProEngineer, where the SOBEK inertia a nd mass properties were also estimated. The next section discusses the hardware c hosen for the SOBEK and its design shown in Fig. 46 and [ 54]. SOBEK Hardware The basic electrical hardware for preliminary setup of the SOBEK ACS consists of four Arsape AM2224 twophase Stepper motors four Faulhaber 1525 BRC brushless DC motors with integ rated electronics and four State Electr onics 600 Series digital optical encoders. The electrical hardware for the current setup only contains a wireless Phasespace LED controller, a Roving Networks RS232 bluetooth module and the EZHR17EN stepper driver boards with an RS232 to RS485 adapter. The entire ACS is power ed by two lith ium polymer batteries. This is possible because there are voltage regulators presen t to step down the voltage from 14 to 12 volts and 5 volts to meet the specific volta ges of the mechanical hardware. Currently the setup has no onboard processing and runs code written in C++ from a Panasonic Toughbook with a Linux operating system. Th e future setup of th is testbed will have everything coded on a single processor board co ntaining the wireless Bluetooth module and motor driver circuitry. PAGE 46 46 There are significant reasons for the choice of th e hardware for the testbed. One of these reasons deals with the digital opt ical encoders. These encoders happen to be more accurate then potentiometers and are dual quadrat ure which are directly compatible with the Allmotion driver boards chosen. The Allmotion EZHR17EN stepper driver boards have the required output current and voltage necessary for the given Arsa pe AM2224 stepper motors and so were chosen for this reason. In addition to having the correct current and volta ges, these boards also have the ability to daisy chain to other boards. That is for a given set of four motor driver boards controlling a total of four steppe r motors two daisy chain boards are needed to send commands to all four motors through the serial port. It is assumed that the gi mbal rate commands are sent fast enough serially that th ey arrive at all four motors at the same time from the daisy chain boards. The nominal speed of the stepper motors needed to be geared down from 134:1. There was a significant loss in efficiency by doing this so th e optical rotary encoders are added on the other end of the shaft to reduce the effect of this uncertainty by feedback. The Faulhaber 1525 BRC motors were chosen because they have a high enough stall torque to spin up the flywheels as well as integrated electronics includi ng a frequency output to measure the speed of the motors. This is valuable for use in th e testbed, because knowledge of the flywheel angular speed is an integral part of the SGCMG dynamics and control. If the flywheels were spinning at different speeds the angular momentum vector of each SGCMG would have different magnitudes and this would add attitude control error. SOBEK Mechanical Design There are some unique design c onsiderations that were carri ed out in the design of the SOBEK testbed. One of these design consider ations dealt with how the ACS would rotate. To approximate the mass and inertia proper ties, ProEngineer was used. The inertia properties were measured about the ACS CM. If the CM was made to be below the point of PAGE 47 47 rotation, then the system would be stable (i.e a pendulum equilibrium point). It should be mentioned that the distance between the point of rotation and the CM cannot be too great or the reaction momentum from gravity will saturate th e SGCMGs. For this reason the CM was chosen to be just below the point of rotation to make the system close to neutral stability. To account for the body frame x and y deviation of the CM from the point of rotation, the ACS is connected to a platform that contains a sy stem of weight ballasts. These ballasts allow twodimensional adjustment of th e CM. The motor driver and daisy chain boards as well as the batteries and the power terminal are mounted on th is platform to conserve space. The zaxis deviation of the CM was accounted for by the vertical adjustment of the ACS center. SOBEK Mechanical, Power and Output Specifications Mechanical specification s of the SOBEK ACS are shown in Ta ble 41 with a total output torque of 267 mNm, mass of 5.489 kg, and power consumption of 8.36 W. SOBEK Attitude Determination The SOBEK ACS is fitted with LEDs that are distinguished by different blinking frequencies where they are read by a system of ei ght cameras. Shown in Fig. 47 is the testbed setup including the Phasespace attitude determination system. The Phasespace motion capture system is chosen because it is a simple and co mpatible offtheshelf way to measure attitude accurately. This is possible because Phasesp ace works on the principle of taking position measurements from LEDs located on the SOBEK r obot thereby calculating the attitude of the tracked object. This completes the attitude determination system for SOBEK. The next chapter discusses the simulation and e xperimental results of this testbed. The simulation results utilize the mass and inertia properties estimated in ProEngineer as the basis for its model. Also discussed in this chapter will be the limitations of the current design iteration of PAGE 48 48 the SOBEK testbed as well as an introduction of the application package interface software (API) that runs it. Table 41 Mechanical and Power Specifications of the SOBEK ACS Micro ACS Power (W) Current (A) Voltage (V) Weight (kg) Output Torque (mNm) Stepper motors 6.000 0.500 12.000 267 DC motors 1.776 0.148 12.000 Encoders 0.600 0.120 5.000 Total 8.376 5.489 Fig. 41. Honeywell rooftop configuration Fig. 42. Honeywell box configuration PAGE 49 49 Fig. 43. External singular surface for box type SGCMG ACS Fig. 43. Honeywell pyramid configuration PAGE 50 50 Fig. 44. Pyramidal SGCMG ACS Fig. 45. Rendition of SOBEK testbed LEDs Phase Space Graphical User Interface PAGE 51 51 Fig. 46. ACS with hardware components q c Fig. 47. PhaseSpace att itude determination system Phase Space Wireless LED controller Digital Optical Encoders Bluetooth Module Allmotion EZHR17EN Stepper Driver Boards Allmotion Daisy Chain Boards Arsape AM2224 Stepper Motors Faulhaber 1525 BRC DC motors Lithium Polymer Battery PAGE 52 52 CHAPTER 5 RESULTS Simulation Model The error quaternion state equation is 4 44411 11 22 1 22 2BB BB T BBee eee GQ eee e where, e and 4e are the vector and scalar elements of the error quaternion and BB is the spacecraft angular velocity coor dinatized in the body frame. The equation for the spacecraft model is BBBBBBBB CCJJ where, B CJis the spacecraft centroidal inertia mentioned in Eq. (31) and is the internal control torque from the SGCMGs mentioned in Eq. (31 4), both of which are coordinatized in the body frame. The equation for the commanded SGCMG output torque is, BB B B CChh The output torque is mapped onto the gimbal rates with the GSRinverse discussed in chapter 3 giving the gimb al rate solution of # 01B actCAh h (51) where, act is the actual column matrix of commanded gimbal rates used to generate the given output torque considering the torque error added from the GSRinverse. The internal SGCMG torque is found in through a nonlinear rest to rest control logic PAGE 53 53 BBBBBBB CKeC J (52) with the symmetric positivedefinite controller stiffness and damping gain matrices 2B CKkJ and B CCcJ. The nonlinear exact model knowledge (E MK) controller shown in has been proven to be asymptotically stable [ 45] and a Lyapunov stability analysis was performed on this controller to understand its be havior when uncertainty is present in the system Lyapunov Stability Analysis of EMK Attitude Controller A positive definite Lyapunov function is chosen as, 2 1 141 10 2T TBBBBB CVeeeKJ (53) with time derivative equal to 1 144 44 1221 111 221 222T TBBBBB C T TBBBBBB T BBBBBBB CVeeeeKJ eeeee KJ (54) To obtain asymptotic stability, Eq. (54) must be made negative definite. To achieve asymptotic stability the control to rque (i.e., internal torque of the SGCMGs) is chosen as Eq. (52). Substituting from Eq. (51) into1V that of Eq. (54), we end up with 10 2T BBBBc V k The final result of this substitution is that the time derivative of the Lyapunov function is negative semidefinite. To prove that the contro ller in Eq. (52) gives a global asymptotically stable result, we must note that the spacecraft angu lar velocity goes to zero asymptotically for the following reasons: PAGE 54 54 ,BB BBLeLLL 2BB L BB is uniformly continuous If we apply Barbalats lemma with these conditio ns, the spacecraft angular velocity goes to zero asymptotically. If we revisit the control la w in Eq. (52), we can see now that it is not possible for the angular velocity to go to zero asymptotically unless the error quaternion vector elements also go to zero, therefore this controller is asym ptotically stable. This controller is EMK which assumes full st ate feedback, therefore the global asymptotic stability does not necessarily hold if accurate full state f eedback is not available. The current iteration of SOBEK does not contai n gyros for angular ra te calculation and encoders for gimbal rate position. This iter ation also has no online method of calculating the flywheel speeds, therefore the gimbal angles a nd angular rates are estim ated with uncertainty present and the flywheel speeds are calculated be fore experimentation. The gimbal angles are estimated through an Euler forward integra tion of the solution to Eq. (51) as, 1 kkkkt where kt is the time step of the control loop. The equation for the calculation of the ACS angular rates is 11 1 1 1 4442, kk kkk k BB k kk kee e e Qe ee e t where, 4kke Q e is always invertible. PAGE 55 55 Due to the uncertainty present in this iterati on, the best result that the controller can produce on this testbed are shown to be uniformly ultimately bounded (UUB). To account for the many sources of uncertainty in this itera tion and obtain the UUB re sult, another stability analysis of this controller is carried out with uncertainty considered. The control torque is dependent on both the estimated angular rates and gimbal angles. The gimbal angles are integrated from the gimbal rates which are mapped from the control torque. Recall also that the torque utilizes minimal torque errors for singularity avoidance when using the GSRinverse and therefore these torque errors must be cons idered in addition to the uncertainty. Lyapunov Stability Analysis of Actual At titude Controller with State Uncertainty For the stability analysis, the ac tual control torque and angular velocities of the system are defined as, act BB act with the uncertainties and If we substitute the actual contro l torque and angular velocities into Eq. (53), we are left with 2 1 241 10 2TT B act CactVeeeKJ and its time derivative 1 2TBT actactactCactactVKJe (55) Assuming that the uncertainties and have negligible time derivatives, the control torque from Eq. (52) give the result 11 2TB B B B B B BBT act C actCactactVKKeCJ Je with a simplified results PAGE 56 56 1 2 111 2T BBBBBBTBBB CCC TT BBBBBBB CCCcIJJ J V k Jc I JJ (56) Examining Eq. (56) while assuming that the uncertainty is less than the actual control torque and angular veloc ity, the best result possible is UUB on the order of the same magnitude as that of the uncertainty for this control law. Now that the stability analyses have been carried out, the results from simulation and experiment is discussed. Simulation Results and Discussion The simulations were carried out in a Matlab environment with parameters shown in Table 51. Shown in Fig. 51 (A) is th e error quaternion vector elements for the first simulation where, 1e 2e and 3e are the quaternion error vector elemen ts about the roll, pitch, and yaw directions. This result at first glance seems to have great performance. Recall that this simulation was run assuming exact model knowledge in the absence of un certainty. The next plot shown in Fig. 51 B) shows the commanded gimbal rates throughout the simulation. In this figure it is clear that the gimbal ra tes are large in the beginning of the simulation. This is due to the fact that the simulation st arts off with the SGCMG gimbals in a singular configuration for a torque needed about the 3 b axis. Figure 51 (B) also shows the trend of the gimbal rates going null after the maneuver is comp leted which is not the case when disturbance torques are added to the system. In this situati on, the gimbal rates will need to compensate for the addition of angular momentum an d will continue to be nonzero. PAGE 57 57 Figure 51 (C) shows the spacecraft angular rates of the ACS. In this figure, the angular rates are nullified after the maneuver is completed. This is the desired affect for a rest to rest maneuver. The final plot of this simula tion is that of the singularity measure m in Fig. 51 (D) Two things can be attributed to Fig. 51 (D). The first is that the singularity measure is zero at the beginning of the simulation (i.e., Jacobian is rank deficient when all gimbal angles are zeroed) and it ramps up quickly to transit from singularity within the first two seconds. Although the ACS is initially at a singular configuration, use of th is steering logic enables it to transit away from singularity in a timely manner. The second conclusion for this figure is that m becomes constant after the maneuver is completed. This should be obvious because it is an explicit function of the gimbal angles. Before an experiment was carried out, another simulation was run with the same k and c values from the previous simulation including th e addition of a random error signal added to the angular velocity measurement and gimbal angle in tegration. This random error was added in an attempt to model the uncertainty of the gimbal angles and angular rates as 10.03BBBB kkrand 10.1kkrand It is clear from the results of this simulation are shown in Figs. 52, while considering an EMK control logic when the controller has uncerta inty within its states, the performance is heavily degraded to the point of instability. In addition to this simulation another one including a higher choice of gains k and c was run to compare results and validate the uniformly ultimately bounded result of the stability analysis in Eq. (56). The results in Figs. 53 prove the opposite end of the stability analysis PAGE 58 58 which is with higher choices of a k and c, the steady state error of the UUB solution can be reduced. Experimental Results and Discussion The flywheels onboard the SGCMGs of the AC S are not balanced and are unstable at certain wheel speeds. For safety reasons the wh eels are left to spin at a maximum speed of around 6000 rpm to keep them within a stable vibration frequency. This speed does not produce enough angular momentum, and hence, enough torque to overcome the moment from gravity to do large offaxis 1 b or 2 baxis maneuvers and therefore the e xperimental results are for a yaw maneuvers about the 3 b axis. The GUI associated with this testbed is shown below in Fig. 54. This GUI in addition to rest of the API software was developed by Andrew Waldrum at the University of Florida specifically for SOBEK. The flywheel angular speeds were calcul ated offline through the output frequency feedback of the DC motors and found to be of different magnitude among the four SGCMGs. A variation of Eq. (51) 1# 0B actCHAh is used to account for the va riation in nominal angular momenta for the individual SGCMGs where 0H is the diagonal matrix of nominal angular momenta. In this GUI the gains k and c are increased at intervals c = 2.5 k to where k varies from .08 and 0.2 in the first experiment. There is no to rque to steer away from singularity at startup because the gains are zeroed out initially. All experiments we re carried out with the same steering logic parameters and initial conditions as those in the first simulation and only differed by choice of control gain. An initial experiment was run with the controller given in Eq. (52). PAGE 59 59 It is shown in Fig. 55 (A) that at around forty seconds the quaternion error diverges, which can be caused by either the controllers instability, propagation uncertainty from the estimated states, or both. The reason for this error propagation is due to the error rates produced from the uncertainty of the estimates. The mane uvers in these experiment s start out with a high initial error about the 3 b axis and attempt to align the ACS to a quaternion of 0001T cq The next three plots shown in Figs. 55 (B)(D ) show the gimbal rate s, angular rates, and singularity measure associated with this maneuver. The next experiment took into consideration the result of the stability analyses and the last two simulations. This experi ment was carried out with higher values of the c ontrol gains and its results are shown in Figs. 56. In these experiments k was chosen greater than that for the previous experiment keeping the same c/k ratio. The quaternion error vector elements in Fi g. 56 (A) contain be tter results than the previous experiment although they also posses low frequency oscillations. The steady state error in this figure is measured as, 11 22min2sin,22sin0.0826 4.7441eeer a d where, the angle e represents the minimum angle needed to be traced out to align the two frames B and G about an eigenaxis. This metric is valid because the error quaternion is still a unit quaternion whereby the 2norm of the vector elements is equal to sin2e. Noted in red on the plots of Fig. 56 (B) and (D), are the large jumps in gimbal rates when tr ansiting a singularity. This was previously said to occur when the SR or GSRinverse method is applied for singularity escape. PAGE 60 60 The result is still oscillating about zero after sixty seconds and the steady state error 0.0826 4.7441erad is still quite large. It is believed through a la rger choice of gains, preferably ones with a higher ratio of ck will reduce the oscillations and steady state error. The next experiment shown in Figs. 57 was carried out with a larger value of control gains with a higher c/k ratio to improve the results from the pr evious experiment. These values were k = 0.32 and c = 3.0 k. The error quaternion vect or elements for this expe riment converged with a much lower steady state error of 0.0376 1.0779erad and a smaller amount of steadystate oscillation due to the higher c control gain. The behavior of the system for the chosen control gains is more stable as well as more efficient in terms of less wasted control effort than that for the previous experiment. This is due to the larger ratio of c/k and the fact that the maneuver did not encounter an additional internal singularity within the ti me of the experiment. With this larger ck ratio, the steady state error oscillations are reduced by a much larger c gain. The next plot shown is the angular rates versus time for this experiment. Angular rates shown in Fig. 57 (C) show a more desirable behavior than that of Fig. 56 (C) in terms of high slew rates with less oscillation. The last plot of this experiment is that of the singularity measure. It has already been discussed that behavior of this system is more desirable than the previous experiments with the choice of the current gains, and now it also seen that at no time did this system become singular. A final experiment was carried out were the MoorePenrose pseudoinverse which possessed no steering logic (i.e.00 ). The quaternion error of th is experiment is shown in Fig. 58. The SOBEK ACS for all experiments is initially at singularity, theref ore without a steering logic large gimbal rates are commanded and the system tumbles out of control as shown in Fig. PAGE 61 61 58. From this figure, it is clear that without use of steeri ng logics while utilizing SGCMGs, drastic consequences may arise. In the future iterations of the SOBEK testbed, if encoders are provided are for gimbal angle measurement, triaxial gyros provided for a ngular rate measurements and use of onboard feedback for the flywheel angular speeds will produce more accurate attitude maneuvers with elimination of state uncertainty. Table 51. Simulation parameter values Parameters Values Units 0h 0.04 Nms k 0.8 none c 2 none 112233,,JdiagJJJ 0.0668,0.0756,0.0815diag Nm2 1234CCCCT Cqqqqq 0001T none 54.74 degrees 12340Tqqqqq 0.9380000.3466T none 1230T 000T rad/s 10(1)rand 0.01(1)rand none 202 rand 0.01(2)rand none 30(3)rand 0.01(3)rand none 2 0expm 20.01exp20m none 12340T 0000T rad PAGE 62 62 A B C D Fig. 51. ACS simulation plots for k = 0.08 and c = 2.5k: A) Quaternion error vector elements versus time, B) Gimbal rates ve rsus time, C) Angular rates versus time, and D) Singularity parameter versus time. 0 10 20 30 40 50 60 70 0 0.02 0.04 0.06 0.08 0.1 time (s)m 0 10 20 30 40 50 60 70 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 0.25 time (s) (rad/s) 1 2 3 0 10 20 30 40 50 60 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0.02 time (s)d /dt(rad/s) d 1/dt d 2/dt d 3/dt d 4/dt 0 10 20 30 40 50 60 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0.05 time (s)e e1 e2 e3 PAGE 63 63 A B C D Fig. 52. ACS simulation plots for k = 0.08 and c = 2.5k: A) Quaternion error vector elements versus time, B) Gimbal rates ve rsus time, C) Angular rates versus time, and D) Singularity measure versus time. 0 10 20 30 40 50 60 70 0 0.5 1 1.5 2 time (s)m 0 10 20 30 40 50 60 70 0.1 0.05 0 0.05 0.1 0.15 time (s) (rad/s) 1 2 3 0 10 20 30 40 50 60 0.04 0.03 0.02 0.01 0 0.01 0.02 time(s) d /dt(rad/s) d 1/dt d 2/dt d 3/dt d 4/dt 0 10 20 30 40 50 60 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 time (s)e e1 e2 e3 PAGE 64 64 A B C D Fig. 53. ACS simulation plots for k = 4 and c = 3k: A) Quaternion error vector elements versus time, B) Gimbal rates versus time, C) Angular rates versus time, and D) Singularity measure versus time. 0 10 20 30 40 50 60 70 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 time (s)m 0 10 20 30 40 50 60 70 0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 time (s) (rad/s) 1 2 3 0 10 20 30 40 50 60 0.04 0.03 0.02 0.01 0 0.01 time (s)d /dt(rad/s) d 1/dt d 2/dt d 3/dt d 4/dt 0 10 20 30 40 50 60 0.4 0.3 0.2 0.1 0 0.1 time (s)e e1 e2 e3 PAGE 65 65 e e Fig. 54. SOBEK testbed GUI PAGE 66 66 A B C D Fig. 55. ACS expe rimental plots for k = 0.8 and c = 2.5k: A) Quaternion error vector elements versus time, B) Gimbal rates ve rsus time, C) Angular rates versus Time, and D) Singularity measure versus time. 0 10 20 30 40 50 60 70 80 0 0.2 0.4 0.6 0.8 1 1.2 1.4 time (s)m 0 10 20 30 40 50 60 70 80 1 0 1 2 3 4 time ( s ) (rad/s) 1 2 3 0 10 20 30 40 50 60 70 80 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 time (s)d /dt(rad/s) d 1/dt d 2/dt d 3/dt d 4/dt 0 10 20 30 40 50 60 70 80 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 time (s)e e1 e2 e3 PAGE 67 67 A B C D Fig. 56. ACS expe rimental plots for k = 0.24 and c = 2.0k: A) Quaternion error vector elements versus time, B) Gimbal rates ve rsus time, C) Angular rates versus time, and D) Singularity measure versus time. 0 10 20 30 40 50 60 70 0 0.2 0.4 0.6 0.8 1 1.2 1.4 time (s)m 0 10 20 30 40 50 60 70 3 2 1 0 1 2 3 time (s) (rad/s) 1 2 3 0 10 20 30 40 50 60 70 1 0.5 0 0.5 1 1.5 2 2.5 3 time (s)d /dt(rad/s) d 1/dt d 2/dt d 3/dt d 4/dt 0 10 20 30 40 50 60 70 0.2 0 0.2 0.4 0.6 0.8 1 1.2 time (s)e e1 e2 e3 PAGE 68 68 A B C D Fig. 57. ACS expe rimental plots for k = 0.32 and c = 3.0k: A) Quaternion error vector elements versus time, B) Gimbal rates ve rsus time, C) Angular rates versus time, D) Singularity measure versus time. 0 10 20 30 40 50 60 70 80 0 0.5 1 1.5 time (s)m 0 10 20 30 40 50 60 70 80 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 time (s) (rad/s) 1 2 3 0 10 20 30 40 50 60 70 80 2.5 2 1.5 1 0.5 0 0.5 1 time (s)d /dt(rad/s) d 1/dt d 2/dt d 3/dt d 4/dt 0 10 20 30 40 50 60 70 80 0.2 0 0.2 0.4 0.6 0.8 1 1.2 time (s)e e1 e2 e3 PAGE 69 69 Fig. 58. ACS experiment quaterni on error of Moore Penrose solution 0 2 4 6 8 10 12 14 0.2 0.1 0 0.1 0.2 0.3 time (s)e e1 e2 e3 PAGE 70 70 CHAPTER 6 CONCLUSION AND FUTURE RESEARCH Conclusion This thesis discusses the development and tes ting of an attitude control testbed utilizing four SGCMGs in an orthogonal pyramid configurat ion. SOBEK is of great utility for evaluating attitude control algorithms and has provided insights into the unde rstanding and application of control moment gyrobased actuators. In its current form, SOBEK has only attitude feedback and does not have onboard processing or internal feed back (i.e., operates in an openloop manner). With the addition of onboard processing and sensor s for state feedback, it is expected that the testbed will prove to be invaluable for testi ng control algorithms and steering logics while showing precision in terms of attitude accuracy from SGCMGs. Future Research Future research for this attitude control system will include the changes to the new iteration of SOBEK and testing of different cont rol algorithms and steering logics. A picosatellite CMGbased ACS will be developed using SOBEK as the alphamodel to analyze software that will fly on this miniaturized actuator. PAGE 71 71 APPENDIX A OFFTHESHELF RW, CMG, AND MAGNET TORQUER DATA SPECIFICATIONS Table A1. Offthesh elf perfor mance specifications for RWs RWA Torque (mNm)Mass (kg)Power (W) Bradford Engineering W05 100 3.2 73 W18 200 4.95 63 W45 300 6.95 64 Honeywell HR 0610 55 3.6 80 Dynacon MicroWheel 200 30 0.93 2 MicroWheel 1000 30 1.3 9 Vectronic Aerospace RW1 20 1.8 25 SunSpace SunSpace RW 50 1.98 35 TELDIX RSI 015/15 5 0.6 4 RSI 015/28 5 0.7 4 Goodrich (Ithaco) TW26E300 300 13.9 L3 Space Comm. MARS RWA 15 MWA50 160 10.5 100 Orbital Sciences LEO Star Wheel 140 3.6 55 SSTL MicroWheel5S 5 0.5 8 MicroWheel3S 3 0.75 3 MiniWheel20S 20 3.2 14 MiniWheel20SX 20 2.6 14 NanoWheelm500S 0.5 0.08 0.5 MicroWheel10SPS 10 1.1 5 PAGE 72 72 Table A2. Offthesh elf performance specifications for CMGs CMG Torque (mNm)Mass (kg)Power (W) Honeywell M50 74570 33.1 75 M95 128803 38.6 129 M160 216931 44 217 M225 305059 54 305 M325 440641 61.2 441 M325D 440641 61.2 441 M715 969410 89.8 949 M600 813491 81.6 814 M1400 2E+06 132 1899 M1300 2E+06 125 1716 EADS/ASTRIUM CMG 1545S 45000 15 SSTL Bilsat1 CMG 95000 2.2 12 SSG SOBEK ACS 66.7 1.38 2.09 PAGE 73 73 Table A3. Offtheshelf performance specifications for Magnet Torquers Magnet Torquer Output (Am2) Mass (kg)Power (W) Microcrosm MT21 2.5 0.2 0.5 MT52M 6.0 0.3 0.77 MT61 7.0 0.23 0.25 MT62 8.0 0.3 0.5 MT102H 12 0.35 1.0 MT151M 20 0.43 1.11 MT302CGS 40 1.050 3.6 MT302GRC 35 1.4 1.5 MT701 75 2.6 3.8 MT702 75 2.2 2.6 MT801 100 4.12 3.0 MT802M 90 2.3 4.7 MT1102 120 3.8 2.9 MT1402 170 5.3 1.9 MT2502 300 5.5 4.8 MT4002L 500 7.8 9.0 MT4002 550 11.0 11.4 MT4001 550 9.2 7.7 Strauss Space Micro Magnetic Torque Rods15 0.6 3.5 Nano Magnetic Torque Rods 1.5 0.2 0.36 Magnetic Torque Coils 0.1 0.04 0.1 Vectronic Aerospace MTR5 100 0.75 6.0 PAGE 74 74 LIST OF REFERENCES [1] Cebrowski, A. and Raymond, J., Oper ationally Responsive Space: A New Defense Business Model, Parameters, 2005, pp. 67. 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[52] Underhill, B. and Hamilton, B., Momentum Control System and LineofSight Testbed, Advances in the Astronautical Sciences, Vol. 125, 2006, pp. 543. [53] Jung, D. and Tsiotras, P., A 3DoF Expe rimental TestBed for Integrated Attitude Dynamics and Control Research, 2003. [54] Leve, F., Design of a 3DOF Testbed For Microsatellite Autonomous Operations, AIAA Aerospace Sciences Meeting, 2006. PAGE 78 78 BIOGRAPHICAL SKETCH Frederick Aaron Leve w as born in Hollywood, Florida, in 1981. In August 2000 he was accepted into the University of Floridas Departme nt of Aerospace Engineering in the College of Engineering where he pursued his bachelors de grees in Mechanical an d Aerospace Engineering. After completing his bachelors degrees in May 2005, he was accepted into the masters program in aerospace engineering at the University of Florida. While in the masters program, he received two awards in academia. In Janua ry 2007 he received the American Institute of Aeronautics and Astronautics Abe Zarem Award for Distinguished Achievement in Astronautics. For this award he was invited to Valencia, Spain, where he competed in the International Astronautical Federationss International Astronautical Congress Student Competition. Here he received the silver Herman Oberth medal in the graduate category. In May 2006, he was accepted to the Air Force Research Lab Space Scholars Program, where spent his summer conducting space research. His interests include satellite attitude control, satellite pur suit evasion, astrodynamics, and orbit relative motion. 