Using. Momentum Exchange Devices DISSERTATION. Kevin Anthony Ford. Major, United States Air Force

Transcription

1 AFI/DS/ENY/9- Reorientations of Flexible Spaeraft Using Momentum Exhange Devies DISSERAION Kevin Anthony Ford Major, United States Air Fore AFI/DS/ENY/9- Approved for publi release; distribution unlimited

2 he views expressed in this dissertation are those of the author and do not neessarily reet the oial poliy or position of the Department of Defense or the United States Government.

3 AFI/DS/ENY/9- Reorientations of Flexible Spaeraft Using Momentum Exhange Devies DISSERAION Presented to the Faulty of the Shool of Engineering of the Air Fore Institute of ehnology Air University In Partial Fulllment of the Requirements for the Degree of Dotor of Philosophy Kevin Anthony Ford, B.S., M.S., M.S. Major, United States Air Fore September, 99 Approved for publi release; distribution unlimited

4 AFI/DS/ENY/9- Reorientations of Flexible Spaeraft Using Momentum Exhange Devies Kevin Anthony Ford, B.S., M.S., M.S. Major, United States Air Fore Approved: Dr. Christopher D. Hall Researh Advisor Date Dr. Bradley S. Liebst Committee Member Date Dr. William P. Baker Committee Member Date Dr. Dennis W. Quinn Dean's Representative Date Dr. Robert A. Calio, Jr. Dean, Graduate Shool of Engineering

5 Aknowledgements I annot thank my advisor, Dr. Chris Hall, enough for the time and eort that he has put in to providing me with a rst rate eduation. His talents as a professor and engineer ome from an exeptional blend of tehnial, verbal, and interpersonal skills, along with an unbelievable work ethi. His generosity in sharing his talents with me are greatly appreiated. Dr. Brad Liebst and Dr. William Baker both provided me with an exellent foundation through formal oursework, as well as hours and hours of enlightening ounsel on this projet. Both are experts in their elds and both provided superb guidane. hey were perfet hoies for ommittee members. With regard to the suess I have enjoyed throughout my areer, however, the greatest thanks must go to my wonderful wife, Kelly. Her enouragement, support, and patiene are unparalleled. he saries she makes annot be overstated. I also thank my parents, Barbara and Clayton, for providing a loving and positive environment in whih toenter the world, not only for me, but for my ve outstanding sisters and brothers as well. Finally, I would like to express my appreiation to my hildren, Anthony and Heidi, for so generously sharing CPU time with me at home, forgiving my oasional absene from their lives, and bravely enduring the Air Fore lifestyle. heir saries are appreiated. Kevin Anthony Ford iii

6 able of Contents Page Aknowledgements... iii List of Figures... vii List of ables ix Abstrat x. Introdution.... Bakground Researh Overview Outline of the Dissertation.... Review of the Literature Momentum Exhange Devies Momentum Wheels Control Moment Gyrosopes Spaeraft Reorientations Flexible Bodies Momentum Exhange Devies Gimbaled Momentum Wheel Kinematis.... Equations of Motion System Momenta..... Gimbal Angles..... Spin Axis Angular Momentum Gimbal Axis Angular Momentum Kinematis... iv

7 Page.. GMW Equation Summary Momentum Wheel Equation Summary Control Moment Gyro Equation Summary.... System Energy Summary Euler-Bernoulli Appendages Equations of Motion via the Lagrangian System Energy he Assumed Modes Method Equation Summary for a Spaeraft with GMWs and Euler- Bernoulli Appendages Summary Spaeraft Reorientations Reorientations Using a Momentum Exhange Cluster ALyapunov Feedbak Control Law he CMG Singularity Problem he Singularity-Robust Steering Law he Singular Value Deomposition Singular Diretion Avoidane he Singularity Avoidane Parameter Maneuver Examples Singular Diretion Avoidane Example Eet of the Singularity Avoidane Parameter he Stationary Platform Maneuver Stationary Condition for the GMW Stationary Condition for the Momentum Wheel Stationary Condition for the Control Moment Gyro 9 v

8 Page 5. Lyapunov Control with Stationary Platform Weighting Summary Simulation Results Reorientations of a Small Flexible Satellite he Small Satellite Model Assumed Mode Shapes Momentum Wheels Control Moment Gyros Reorientations of a Flexible Hubble Spae elesope Summary Summary and Conlusions... Appendix A. Denitions, Kinematis, and Fundamentals A. Notation... A. ransformation Matries... 4 A. Eigenaxis Rotations... A.4 Quaternions... 8 A.5 Rigid Body Equations of Motion Appendix B. Simultaneous Eigenaxis and Stationary Platform Rotations. Appendix C. Singularity Attration Using Pseudoinverse Steering Appendix D. Orthogonal CMG Cluster Envelope... 8 Bibliography... Vita vi

9 List of Figures Figure Page. A Gimbaled Momentum Wheel.... A Body with Multiple Gimbaled Momentum Wheels.... A Spaeraft with an Euler-Bernoulli Appendage orque Output Near a Singularity for the SDA and SR Steering Laws 5. orque Produed and orque Desired for the Singularity Avoidane Parameter Example Reorientation Parameters for SDA and SR Control Laws A CMG Stationary Platform Surfae ( F = = ) Cantilevered Beam Mode Shapes Reorientation Parameters for Stationary Platform and Diret Maneuvers Using Momentum Wheels Cluster Norms for the Stationary Platform Maneuver and Diret Maneuver Using MWs Appendage Exitations for Stationary Platform and Diret Maneuvers Using MWs Reorientation Parameters for Lyapunov Maneuver With and Without Stationary Platform Weighting Using MWs Appendage Exitations for Lyapunov Maneuver With and Without Stationary Platform Weighting Using MWs Reorientation Parameters for Stationary Platform and Diret Maneuvers Using CMGs Cluster Norms and Singular Values for the Stationary Platform and Diret Maneuvers Using CMGs.... Reorientation Parameters for Four CMG Control Laws.... Modal Responses for Four CMG Control Laws Singular Values and Potential Energy for Four CMG Control Laws 5 9. Hubble Reorientation Parameters for Lyapunov Maneuver With and Without Stationary Platform Weighting... vii

10 Figure Page. Hubble Appendage Exitations for Lyapunov Maneuver With and Without Stationary Platform Weighting Gimbal Angle Flow for Pseudoinverse Steering Law.... External Momentum Envelope for a hree CMG Orthogonal Cluster viii

11 List of ables able Page. Spaeraft Physial Data for Singularity Avoidane Example SDA Example Simulation Data.... Singularity Avoidane Parameter Comparison Data Physial Data for the Small Flexible Satellite Cluster Data for the Small Satellite with Momentum Wheels Cluster Data for Small Satellite with CMGs Flexible Hubble Spae elesope Physial Data Hubble Spae elesope CMG Cluster Data... 9 ix

12 AFI/DS/ENY/9- Abstrat We study rest-to-rest reorientations of exible spaeraft using momentum exhange devies. A new and onise form of the equations of motion for a rigid body ontaining a luster of gimbaled momentum wheels is developed using the Euler-Newton approah. Speial restritions of the gimbaled momentum wheel equations yield equations of motion for the momentum wheel luster and the ontrol moment gyrosope luster. hough ontrol laws for reorienting rigid bodies using momentum wheels and ontrol moment gyros were previously available, the osillatory nature of a body ontaining a momentum luster presents a hallenge for a spaeraft with exible appendages. In addition, reorientations whih all for high angular aelerations naturally tend to exite osillations of the appendages. A mathematial model of a free spaeraft with Euler- Bernoulli appendages is developed using the Lagrangian approah. Using the assumed modes method, a omplete set of vetor nonlinear dierential equations is developed whih desribes the dynamis of a spaeraft with exible appendages and a luster of gimbaled momentum wheels. his model is useful in omparing the merits of andidate spaeraft reorientations. Speial attention is paid to singularity problems in ontrol moment gyro lusters. he singularity-robust ontrol law ommonly used to avoid singular luster ongurations an ause abrupt hanges in torque output. An improved law based on the singular value deomposition is developed whih avoids torque output ommands in the nearly singular diretion. he stationary platform maneuver, a maneuver along the set of equilibrium solutions of a zero angular veloity spaeraft, is extended to the ontrol moment gyro luster. For a luster of momentum wheels, the set of equilibria is a hyper-ellipsoid in rotor momenta spae. he set of equilibria for a ontrol moment gyro luster is a unique surfae in gimbal angle spae. A ontrol law whih reorients the spaeraft while remaining lose to this surfae is developed using a Lyapunov method. x

13 Reorientations of Flexible Spaeraft Using Momentum Exhange Devies. Introdution. Bakground he spaeraft of tomorrow are no doubt still beyond even the most vivid imaginations. he missions of future spaeraft will likely require that they beome more eient and versatile. While the designs of today seem to be meeting our immediate needs, the maneuverability and attitude ontrol requirements of future raft will likely inrease. he attitude ontrol of spaeraft to date has been largely a problem of maintaining orientation and stability. One notable exeption is the Hubble Spae elesope, whih must be reoriented on ommand to a very preise orientation in inertial spae and maintained there for a period of time. While the telesope's reation wheel ontrol system has proven worthy of the required task, the time required of suh reorientations is measured in large frations of an hour. It is ertainly possible that future missions might require suh reorientations in seonds. We should also antiipate more variation in the sizes of future spaeraft, ranging from mirosats to spaestations. A manned spae station would likely be more eient with inreased size. Perhaps one day, a ommerial station might exist whih would permit ompanies to add on a module to an already existing onguration. Depending on the mission, the ompany might have a requirement for their module or equipment to periodially attain and then maintain a ertain orientation in spae (earthward or sunward for example). his requirement translates to a need for maneuverability. Some sheme to provide a maneuver torque to the struture must be adopted. Of ourse, even to maintain a stationary orientation in spae requires a method to ounterat the ubiquitous environmental torques due to aerodynamis, gravity gradient, and radiation pressure.

14 wo ommon devies for applying ontrol torques today are external thrusters and magneti dipoles. Both have drawbaks and limitations. External thrusters need some type of expendable, hemial fuel for operation. Launhing fuel into spae is ostly. hrusters have the additional fault of expelling propellants. his may reate problems for sensitive sensor equipment. his drawbak apparently ruled out external thrusters for use on the Hubble Spae elesope. Just as importantly, one the thruster onguration is established, there are now limitations on where new modules may be plaed, as the thruster exhaust must be avoided. Magneti torquers are popular as they operate eletrially (a replenishable supply of energy in spae). Magneti torquers take advantage of the earth's magneti eld and an provide a sustained external torque to the platform, thereby hanging the system's angular momentum markedly over a period of time. he magnitude of the torque whih an be ahieved, however, is generally inadequate for eeting any sort of rapid reorientation. Additionally, the strength and diretion of the magneti eld varies with orbital position. Magneti torquers would be useless on spaeraft operating out of the inuene of a magneti eld, suh asinterplanetary spaeraft. One solution to these problems is the use of appropriately mounted internal rotors (ywheels) to store angular momentum. his angular momentum an then be transferred to the vehile struture to eet a desired angular veloity, and subsequently a reorientation. he mehanization of these wheels varies, and the two of partiular interest here are the momentum wheel and the ontrol moment gyro. he momentum wheel is mounted along a xed axis in the platform. Angular momentum is then exhanged between the body and the rotor by applying a torque to the rotor via an eletri motor. he dynamis involve only a relative hange in speed between the rotor and spaeraft. It is ommon to mount at least three momentum wheels in an orthogonal arrangement to permit exhange of angular momentum about any axis in the body. he rate of momentum exhange with the spaeraft (the output torque) is restrited by the rate at whih the wheel spin speeds an be hanged.

15 he ontrol moment gyro (CMG), onversely, operates at a onstant rotor speed. he exhange in momentum is generated by varying the spin axis orientation with respet to the spaeraft. he dynamis of the gyrosopi eets an lead to rather ompliated responses, requiring ompliated algorithms to ompute the required input for a desired response. Also, whereas momentum wheels maintain their design orientation, CMGs an ahieve ertain orientations whih allow no torque apability in a partiular diretion, resulting in a singularity in the ontrol law. his ondition must be guarded against. he rate of momentum exhange with the spaeraft in the ase of the CMG is dependent on the gimbal angle rate. he reorientation problem is one of determining the urrent orientation and angular veloity in inertial spae, determining the desired orientation and angular veloity at some future time (the nal state onditions), and nding the best path to take in reahing the desired nal state onditions. here are an innite number of paths whih an be followed, but ertainly some are better than others. Paths whih might beonvenient for reorientations have been the subjet of many studies. Some of these paths are obvious, whereas others are not. he eigenaxis rotation is the most diret path between two orientations. For any rotation, there exists an axis (the eigenaxis) whih remains xed in spae. If the rotation matrix is known, the eigenaxis an be omputed. Furthermore, the angle of rotation about the eigenaxis is also easily omputed. Reorientation is simply a matter of aelerating to a onstant angular veloity about the eigenaxis, maintaining the veloity, and deelerating to the new orientation. Optimal ontrol theory suggests that, assuming we had the apability to provide a given torque about any axis, the minimum time reorientation would be the result of bangbang ontrol. hat is, aelerate at maximum torque about the eigenaxis until halfway to the desired orientation, and then deelerate at maximum torque into the new orientation. However, it is unlikely that any real ontrol system ould provide the maximum torque in the eigenaxis diretion. Moreover, if the eigenaxis is other than the diretion of a prinipal moment of inertia, a torque perpendiular to the rotation will be required to overome gyrosopi eets.

16 Another path might be ditated by the mehanization of the momentum wheel or CMGs. Supposing that the angular momentum at the beginning of the maneuver is ontained solely in the wheels, then the vehile's total angular momentum is determined and is xed in spae. For a desired nal orientation, the neessary angular momentum relative to the body an be alulated. We might suspet that by adjusting the momentum in the momentum wheels to the required nal values, the desired nal orientation would be ahieved. Ahieving the proper luster momentum in the body frame, however, is neessary but not suient to ensure the desired nal orientation. he variation of the wheel speeds in a linear fashion was investigated by Shultz [4] and was referred to as the diret path. If in fat the body angular veloity an be redued to zero at the nal orientation, then the diret path will ahieve the objetive (to within a rotation about the angular momentum vetor). Unfortunately, for a spei set of nal ondition wheel speeds, the same total angular momentum vetor an be ahieved with an innite number of body rates. he diret method does not neessarily keep body rates low, and most likely will not ahieve the nal orientation with zero body rates. Osillatory motion of the body is a ommon phenomenon assoiated with spaeraft ontaining rotors. his motion presents a serious problem to exible spaeraft, as it an easily exite exible modes of vibration ausing loss of mission eetiveness or even strutural damage. A ontrol law whih keeps angular veloities under ontrol is required for these types of strutures. A tehnique investigated by Hall [] proved promising in the reorientation of exible spae strutures by keeping body rates low throughout the maneuver. hey were termed \stationary platform maneuvers" by Hall, beause they onsist of following paths in rotor momenta spae whih are the equilibrium solutions for a stationary platform. Of ourse, while maneuvering, the platform is not stationary, but by remaining lose to these branhes of equilibria, the dynami eets of the body are minimized and the spaeraft tends to arrive at the nal orientation with low body angular veloity. he purpose of this researh istoinvestigate and ompare various reorientation shemes of exible spae strutures using momentum exhange devies suh as momentum 4

17 wheels and ontrol moment gyros. Insight into the utility of partiular maneuvers should be gained by omparing time required for the maneuvers, ontrol use required for the maneuvers, the possibility ofahieving the desired nal state onditions, osillations of exible appendages during the maneuvers, and the residual osillations at termination.. Researh Overview he objetive of this researh is to develop improved ontrol laws for momentum exhange devies imbedded in exible spaeraft. Sine the available bakground literature on ontrol moment gyros is limited, we develop herein a onise vetorial form of the equations of motion for a body with a luster of gimbaled momentum wheels. he gimbaled momentum wheel is a generalization of the momentum wheel and of the ontrol moment gyro { essentially a variable speed single gimbal ontrol moment gyro. his new development allows for easy speialization to momentum wheels or ontrol moment gyros if desired. Next, ontrol laws whih reorient the spaeraft are investigated. he stationary platform maneuver developed by Hall [8] shows great promise for appliation to exible spaeraft due to its nature of keeping angular veloities low during a maneuver. We examine the maneuver's extension to the gimbaled momentum wheel and the ontrol moment gyro. We show that, beause a momentum luster, regardless of type, possesses a luster momentum whose rate of hange is atually the ontrol torque, the onept is valid for all momentum exhange devies. A drawbak of the stationary platform maneuver is that kinematis are not aounted for, and the nal orientation is only orret to within a rotation about the angular momentum vetor. A method is sought to eet a reorientation to the target attitude while maintaining the stationary platform ondition. he Lyapunov approah is used to produe a orretion to the luster momentum rate whih keeps the luster arbitrarily lose to the stationary platform ondition. he ontrol moment gyro luster has several harateristis whih make its use in attitude ontrol a hallenge. While momentum may be exhanged rapidly with the spaeraft, 5

18 the amount of momentum whih may be exhanged is nite. Control moment gyro lusters also admit singular ongurations whih allow torque output only in a plane. orque ommands normal to this plane near the singular ondition an ause unreasonably high gimbal rate ommands. A singularity-robust ontrol law developed by Oh and Vadali [4] avoids these singular onditions, but an ause abrupt hanges in torque diretion near the singularity. A new singular diretion avoidane law isdeveloped herein. his modiation to the singularity-robust law is based on the singular value deomposition and smooths the gimbal rates while only altering the output torque in the singular diretion. In order to evaluate the eetiveness of the new ontrol laws, a mathematial model of a exible spaeraft is developed. Equations of motion for a free ying spaeraft with Euler-Bernoulli appendages (allowed exure only in a plane) are developed using the Lagrangian approah. he assumed modes method is used to redue the innite degrees of freedom to a nite number suitable for numerial solution. he equations of motion for the exible model an be oupled diretly to the spaeraft/gimbaled momentum wheel equations for numerial evaluation.. Outline of the Dissertation In Chapter we review the literature relevant to this researh. Speially, some of the relevant works on the use of momentum wheels and ontrol moment gyros in the ontrol of spae vehiles are summarized. A review of literature relevant to reorientations and the exible modeling of spae strutures is also presented. he unique qualities of momentum wheels and ontrol moment gyrosopes are addressed in Chapter. We develop the equations of motion for the gimbaled momentum wheel, a generalization ombining the qualities of the momentum wheel and the ontrol moment gyrosope. his hapter inludes some original work regarding the kinematis of the orthogonal CMG luster. he equations are developed in vetor form, with spin axis and gimbal axis torques as the ontrol inputs. A ompat expression for the kineti energy of the system is also given.

19 In Chapter 4, the equations of motion for a body with exible appendages are developed using the Lagrangian approah. he set of equations developed are mixed partial and ordinary dierential equations. hey are then disretized using the assumed modes method, and a nite set of ordinary dierential equations are produed. his set is then integrated with the equations of Chapter to produe a mathematial model of a exible spaeraft with a luster of gimbaled momentum wheels as the attitude ontrol devie. Methods of ahieving reorientations of the body are disussed in Chapter 5. We begin with a general development whih takes the \blak box" approah, assuming that the momentum of the luster relative to the spaeraft an be ontrolled as desired. A general Lyapunov feedbak ontrol law using the luster momentum as the ontrol input is derived. We then relate the ontrol law to the speial ases of momentum wheels and ontrol moment gyros. Some speial onsideration is given to the CMG singularity problem in Chapter 5, and a new approah toavoiding the singularity is presented. We onlude the hapter with a disussion of the stationary platform ondition. he numerial results of the study are presented in Chapter. he qualitative and quantitative results of several proposed ontrol laws are presented. A summary and onlusions of the dissertation researh are in Chapter.

20 . Review of the Literature In this hapter, we review relevant literature in the disiplines whih form the ornerstone of this researh. Many ontributions have been made whih provide an exellent foundation for the study. o the best of our knowledge, the appliation of momentum storage devies to the ontrol of exible strutures has not been onsolidated into a single researh eort.. Momentum Exhange Devies.. Momentum Wheels. he development of equations of motion for a momentum wheel imbedded in a rigid body has been approahed in a number of ways. Any general body to whih is attahed an axisymmetri spinning body is lassied as a gyrostat. he axisymmetri portion of the gyrostat is a momentum wheel. In the ase where the momentum wheels spin axis is aligned with a prinipal axis, the system is alled an axial gyrostat, and a losed-form solution for the angular momentum in terms of Jaobi's ellipti funtions has been given (see Cohran et al. [] for referenes). Hughes [] provided an exellent development of the equations of motion for a rigid body ontaining a single wheel in terms of the absolute angular momentum of the system and the absolute angular momentum of the wheel. Hall [8] extended the development to a system of multiple wheels and derived a torque ontrol law whih maintains his so-alled stationary platform maneuver. Hall [] also redued the equations of motion from the normal N + rst order dierential equations (for an N-rotor gyrostat) to one equation involving a Hamiltonian by applying the method of averaging for the ase of small spinup torques. Hall [8] and Hall and Rand [9] used this averaging approah to show that the Hamiltonian approah an provide insight into the qualitative nature of the dynamis during spinup of the rotors. Shultz [4]investigated and ompared rest-to-rest reorientations of rigid bodies using a luster of momentum wheels. His investigation inluded maneuvers along the stationary platform ondition developed by Hall as well as maneuvers resulting from linear variation of the momentum wheels from initial to the required nal values (onstant torque applied to eah wheel). He found that intermediate angular veloities, as well as nal angular veloi- 8

21 ties and attitude errors were signiantly smaller using the stationary platform maneuver. hese results motivate the investigation of this ontrol law for exible spae strutures. he gyrostat with exible appendages has been investigated mostly in the ontext of stability of motion. Likins et al. [] and Mingori et al. [] were both onerned with the limit yles possible for dual-spin spaeraft whose setions ontain nonlinear exible elements. heir approah was based primarily on the energy sink approah, but was reinfored by omparison with the results of numerial integration. Stabb and Shlak [5] developed a model whih onsisted of a gyrostat with an attahed torsionally exible appendage. hey developed the equations of motion for the system, but the primary thrust was on the appliation of the Krylov-Bogoliubov-Mitropolsky (KBM) method to aid in analyzing the motion of the system for small perturbations from the undeformed state. Mazzoleni et al. [] integrated the work of Stabb [5] and Hall [9] via a double averaging approah to develop a single equation whih allows the projetion of solutions onto a bifuration diagram relating the rotor angular momentum to the system Hamiltonian. Atual appliation of momentum wheel ontrol to a spaeraft is desribed by Creamer et al. [4] in the ase of the Clementine spaeraft. Control laws were developed to maneuver the spaeraft around the eigenaxis using a bang-oast-bang prole. hey simplied development of the ontrol laws by assuming that the nonlinear dynamis were negligible. he omputed open-loop momentum wheel input torques for the maneuver were sent to a feedbak ontrol law for implementation of the atual maneuver, however. Flexible strutural modes were not onsidered... Control Moment Gyrosopes. he torque ampliation phenomenon assoiated with single gimbal ontrol moment gyros make them an appealing alternative to momentum wheels, but the theory for CMGs is not quite as mature. here apparently are no texts whih address the full nonlinear dynamis of the CMG. Jaot and Liska [4] were among the rst to see the usefulness of the CMG for spae vehile attitude ontrol. hough they attempted a areful development utilizing a onservation of momentum approah, they omitted the inertia about the gimbal axis from the very rst equation. hey also linearized early in the development. 9

22 O'Connor and Morine [4] expounded on the merits of various CMG ongurations for spae vehiles and ompared the utility of CMG ontrol laws. hey dismissed torque as a ommand input to the single-gimbal CMG as unusable due to the appreiable frition on the gimbal bearings. Instead they reommended a gimbal rate (feedbak) ontrol law. Singularities were disussed in an early work by Margulies and Aubrun []. hey n showed that for a luster of n CMGs, there exist for any diretion in spae ombinations of gimbal angles for whih the system annot produe any torque in that diretion. In addition, they demonstrated that for n CMGs in a luster, there is an n parameter family of null motions whih produe no output torque. As suh, these null motions an be ombined with torque-produing motions without aeting the dynamis of the system. A useful development of the omplete equations of motion (inluding the gimbal inertia terms) was given in Oh and Vadali [4]. hey presented some andidate feedbak ontrol laws and laimed that, with the gimbal inertias inluded, the ontrol laws must provide gimbal aeleration ommands (instead of gimbal rate ommands). hey also gave some onsiderations for avoiding singularities in the ase of a redundant set of CMGs. Hoelsher and Vadali [] further onsidered open-loop and feedbak ontrol laws whih not only minimized a mix of ontrol eort and maneuver time, but also avoided singular CMG ongurations. Vadali and Krishnan [5] narrowed the fous exlusively to avoiding these singularities by parameterizing gimbal rates as polynomial funtions of time and optimizing the parameters with respet to a singularity avoidane objetive funtion. Going one step further, Vadali et al. [5] developed a method for determining a family of initial gimbal angles whih would avoid singularities during a maneuver. Bedrossian et al. [, 4] developed a way of instituting the addition of null-motion into the ontrol algorithm to avoid singularities. Paradiso [4] developed a omputer algorithm whih was apable of globally avoiding singular states in a feed-forward steering law. An appliation of a highly linearized axis-deoupled CMG steering law for the spae station was presented by Singh and Bossart [48] and another by Bishop et al. []. Neither of these studies extrapolate their apability to a ontrol sheme for large angle reorientations or high angular veloities.

23 . Spaeraft Reorientations Perhaps the best ompendium of notation and methods for dealing with reorientations of spaeraft is the text by Junkins and urner []. Inluded are methods for optimizing the reorientations based on several dierent types of ost funtions, along with some onsiderations for exible modes of vibration. Carrington and Junkins [] developed a solution for the nonlinear spaeraft slew maneuver by assuming a polynomial feedbak form. Kinematis were formulated in terms of Euler parameters (also known as quaternions). One of the early works on feedbak ontrol for reorientations was written by Wie and Barba [55]. hey proposed three distint quaternion feedbak laws apable of general three-axis reorientations. Wie et al. [5] devised a quaternion feedbak regulator to perform eigenaxis rotations (whih they onsidered \optimal") and inluded the ontrol torque required to deouple the gyrosopi oupling torques. Wie and Lu [5] further developed this nonlinear feedbak ontroller to perform eigenaxis rotations under slew rate and ontrol torque onstraints. Cristi et al. [5] provided a quaternion feedbak regulator whih is evidently globally stable and needs no knowledge of the spaeraft inertia matrix, an exellent property for ontrol of a modular spae station. Long [] developed an equivalent axis oordinate frame for rest-to-rest reorientations whih transforms the nonlinear spaeraft ontrol problem into a linear problem. Understandably, the problem of optimum time reorientations has also reeived onsiderable attention. A survey of ontributions in this area was provided by Srivener and hompson [4]. An extremely important ontribution was made by Bilimoria and Wie [] when they demonstrated that with speied ontrol onstraints in all three axes, the eigenaxis rotation is in general not time optimal. he optimal ontrol is bang-bang in all three axes and results in a signiant nutational omponent. he task is only to nd the swithing funtions for eah axis. Byers and Vadali [8] applied these ideas to the development of a method for omputing solutions for the time-optimal ontrol swith times in the reorientation of a rigid spaeraft. hey intentionally omitted the gyrosopi oupling terms from the dynamial equations. Vadali et al. [5] used a parameter optimization sheme to develop an open-loop ontrol law for the reorientation of a partiular ground-

24 based test artile. he ontrol law was then utilized in a feedbak ontroller. heoretial and experimental results were then ompared with apparently exellent agreement. We should point out that studies in the robotis eld have a great deal in ommon with this idea of internal momentum exhange devies to eet reorientation of the primary body. Many works have borrowed the singularity robust inverse presented by Nakamura [9] as a method of avoiding singularities in ontrol laws. An intriguing paper detailing a strategy for planar reorientation of a system of pinned bodies using only internal ontrols is provided by Reyhanoglu and MClamroh [45].. Flexible Bodies An entire text devoted to the dynamis and ontrol of exible spae strutures was prepared by Junkins and Kim [5]. heir treatment of exible struture models and the appliation of the Lagrangian approah for generation of the equations of motion are partiularly noteworthy. A text by Craig [] also provided an overview of various methods useful in modeling a exible body, along with limitations of the nite-dimensional representations of the body whih must be employed for numerial evalution of the problem. Many of the papers onerned with the problem of exible appendages attahed to a satellite were summarized by Modi [8]. An early attempt at onstruting the equations of motion for a exible spaeraft was given by Grote et al. []. he equations were onstruted for a model onsisting of rigid bodies attahed with springs and dampers. It was assumed that the motion of the entral rigid body is presribed, and the exible appendages eet a deviation from this nominal motion. Keat [] desribed an analogous method appliable when the exibility is modeled by generalized oordinates. Most studies investigating reorientation maneuvers with exible appendages attak the problem with simplifying assumptions. Barbieri and Ozguner [] were able to onstrut a minimum-time ontrol law to slew an undamped, one-mode model of a exible struture using the single-axis bang-bang ontrol with multiple swithings. he single exible mode was suppressed at the nal orientation. Byers et al. [9] used smoothed bang-bang ontrol

25 inputs to ahieve a near-minimum time planar reorientation of a xed rigid hub with four idential exible appendages. he ontrol inputs were simply smoothed approximations to the optimal bang-bang inputs for a rigid body. Small antisymmetri deformations of the appendages were assumed. he three-dimensional ase was pursued by Vadali et al. [54] using a parameter optimization approah as a exible extrapolation to [5]. he only onsideration given here to the exible modes was in the smoothing of the ontrol inputs using a multiplier funtion during the initial and nal phases of a maneuver. Bell and Junkins [5] tookthe same approah but used ontrollably sharp spline swithes to redue exible exitations. A momentum exhange feedbak ontrol onept was investigated by Li and Bainum [8], but was restrited to a pair of symmetri appendages axed to a rigid hub. he ontrol torques inluded not only feedbak of the rigid body motion, but also the time rate of hange of momentum resulting from the exible motion. An early introdution to the use of hybrid oordinates in the design of attitude ontrol systems was given by Likins and Fleisher [9]. hey were primarily interested in the stability of an attitude ontroller with the exible modes inluded. Barbera and Likins [] presented a method for testing the stability of a system with an arbitrary disretized appendage, and losed form stability riteria were developed for a very restrited model. Meirovith and Calio [4] ompared three essentially dierent methods for investigating the stability of exible spaeraft. he more spei problem of attitude stability in dualspin systems bearing exible members was addressed by Gale and Likins [] aswell as Cherhas and Hughes []. Not all work has been theoretial in nature. A summary of some experimental investigations into the ontrol of exible strutures was given by Sparks and Juang [49]. hough the aforementioned works provide a rm foundation for this dissertation researh, the study of momentum exhange devies imbedded in a general three-dimensional exible spaeraft remains largely unexplored. We attempt to retain the general nonlinear and -D nature of the problem by avoiding suh simpliations as symmetri appendages, an inertially-xed enter of mass, or the disarding of inertia terms assoiated with the

26 momentum devies. We now turn to the development of a new set of equations of motion for a rigid spaeraft ontaining a momentum exhange luster. 4

27 . Momentum Exhange Devies wo types of momentum exhange devies reeive most of the attention in the literature. he momentum wheel (MW) is a variable speed ywheel mounted on an axis xed in the body. Momentum exhange is eeted by hanging the speed of the wheel relative tothe body using an eletri motor. A luster of at least three MWs with non-oplanar axes an be used to exhange momentum with the body about any axis. he seond type of momentum exhange devie is the ontrol moment gyrosope (CMG). he ywheel of a typial CMG spins at a onstant speed and a gimbal arrangement allows variation of the spin axis in the body referene frame. A double gimbal arrangement an permit the spin axis of the CMG to assume any diretion in the body. he single gimbal CMG (SGCMG) allows reorientation of the spin axis only in a plane whih is perpendiular to the gimbal axis. he advantage of the SGCMG is the well-known torque ampliation property. Essentially, a rate about the gimbal axis an produe an output torque orthogonal to both the gimbal and spin axes whih ismuh greater than the gimbal axis torque. A referene to the CMG in this dissertation implies the single gimbal variety. A gimbaled momentum wheel (GMW) is a generalization of the momentum wheel and single gimbal ontrol moment gyrosope. he GMW allows for variation in wheel speed and reorientation about a gimbal axis (see Figure ). A new form of the equations of motion for the GMW is developed using a momentum approah. he development leads to a system of + N ordinary dierential equations for the system angular momentum, system linear momentum, the angular momenta of the GMWs about the gimbal axes, the angular momenta of the GMWs about the spin axes, and the gimbal angles. One the equations of motion for the GMW are available, speialization to the momentum wheel and CMG follows naturally. When linear momentum is assumed to be zero (and onstant) the equations of motion for the momentum wheel ase redue to a + N order set, whereas the CMG ase redues to a + N order set. Notation and a review of the basi onepts relevant to the subsequent development an be found in Appendix A. 5

28 a s a t Motor a g Figure A Gimbaled Momentum Wheel. Gimbaled Momentum Wheel Kinematis Consider a rigid body in whih N gimbaled momentum wheels are imbedded (Figure ). Eah GMW is omposed of a ywheel mounted in a gimbal frame and inorporates the variable speed of a momentum wheel and the gimbal arrangement of a typial single gimbal CMG. he GMWs are designated W ;W ;:::;W N and the rigid platform is identi- e e b e W b W N W b B Figure A Body with Multiple Gimbaled Momentum Wheels ed as B. he platform is in general not symmetri. A referene frame, F, is established in the body whih has basis ( ~ b ; ~ b ; ~ b ). he body is free to translate and rotate with respet to the inertially xed referene frame, F, with basis ( ~ e ;~ e ;~ e ). i b

29 he wheels spin about their individual axes of symmetry whih are expressed as the unit vetors ~ a ;~ a ;:::;~ a s s sn by the unit vetors ~ a ;~ a ;:::;~ a. A third set of unit vetors given by ~ a ;~ a ;:::;~ a (subsript representing transverse), where ~ a = ~ a ~ a, will prove useful in the derivation. in the vehile body frame F g g gn t t tn We dene a matrix A suh that the olumns of A are the olumn matries a ( j = :::N) whih speify the orientations of the spin axes of the wheels, W ( j = :::N), b. he diretions of the spin axis unit vetors vary with the gimbal angles. he gimbal axes are always orthogonal to the spin axes and are denoted tj sj gj s s sj. hat is A = a a a () s s s sn j he matries A and A are dened similarly. Whereas A is a onstant matrix, the g t g matries A and A depend on the gimbal angles. s t he moment of inertia for the spaeraft is assumed onstant exept for the hange aused by variation in the gimbal angles. It is also assumed that the enter of mass of the spaeraft is xed in the body and does not vary with gimbal angles. he inertia dyadi ~ I is formed from the body inertia dyadi plus the parallel axis ontributions of the wheels. It is given by N X ~ I = ~ I + m ( ~ r ~ r ~ ~~ rr) () B j = j j j j j where m is the mass and ~ r the xed loation of the enter of mass of the j-th GMW. j j We dene the terms I, I, and I to be the total spin axis inertia, the total gimbal sj gj tj axis inertia, and the total transverse axis inertia of the j-th GMW (inluding the gimbal frame). he total spin axis inertia of the GMW is the sum of the gimbal frame inertia and wheel inertia. We split it into the terms I and I so that swj sgj I = I + I () sj swj sgj We form I sw GMW wheels: as a diagonal matrix omposed of the spin axis moments of inertia of the I = diag( I ;I ;:::;I ) (4) sw sw sw swn

30 Four other N N inertia matries, I, I, and I, and I are dened in a similar manner. s g t sg he linear momentum of the system is given by ~ p = m~ v +! ~ ~ (5) where v~ is the veloity of the origin of F and ~ is the rst mass moment of the body/gmw b! ~ b system about the origin of F. he angular veloity of the body (and body referene frame) with respet to inertial spae is. We write Equation (5) in the body oordinate frame as p = mv +! () where we use the ross notation dened by Equation (4). he system angular momentum an be expressed as N X ~ ~! ~ ~ ~ ~ h = I + v + h () j = aj where ~ is the absolute angular momentum of the j-th GMW about its own enter of h aj mass. Instead of grouping the GMW ontributions to the angular momentum by GMW (as did Oh and Vadali [4]), we deompose the GMW ontributions to angular momentum into omponents in the spin, gimbal, and transverse diretions. his is expressed as (using body frame omponents) h I! v Ah Ah Ah = (8) s sa g ga t ta he new terms h, h, and h are N olumn matries whih represent the omponents sa ga ta of absolute angular momentum of the GMWs about their spin axes, gimbal axes, and spin axes respetively. From Equation (8), we see that if we know the gimbal angles (and therefore the N basis matries) along with h, h, h, h, and v, then we an ompute the urrent body angular veloity!. sa ga ta One term in Equation (8) deserves speial attention. he angular momentum of a GMW about its spin axis is a ombination of angular momentum due to the ywheel itself, 8

31 plus a ontribution due to the gimbal frame. We simply split h sa into two terms as h = h + h (9) sa swa sga where h is the N olumn matrix of absolute angular momenta of the wheels about swa their spin axes, and h is the N olumn matrix of absolute angular momenta of the sga gimbal frames about the GMW spin axes. he absolute angular momentum omponents may be expressed in terms of the platform angular veloity and relative angular momenta. he relationships are h I A! h = + () swa sw s swr h I A! h = + () sga sg s sgr h IA! h = + () ga g g gr h IA! h = + () ta t t tr In the ase of mehanial gimbals, the motion of the GMW gimbal is onstrained to rotation about the gimbal axis, so there an be no motion of the GMW relative tothe platform in the transverse diretion, nor an the gimbal rotate relative to the platform about the spin axis. herefore h = h = (4) tr sgr whih implies that we an rewrite Equation (8) as h I AIA AI A! v Ah Ah =( + + ) (5) t t t s sg s s swa g ga Note that the inertia-like matrix multiplying! in Equation (5) is not neessarily onstant. he denitions of this setion allow for a onise expression of the equations of motion, whih wenow address. 9

32 . Equations of Motion In this setion, we develop the equations of motion for a rigid body with GMWs in terms of the system linear and angular momenta, as well as the angular momenta omponents of eah GMW about the spin and gimbal axes. Beause the orientation of the GMW is important (as it gives the diretion of the spin momentum vetor), an equation of motion is also developed for the gimbal angle. A summary of the GMW equations and speializations to the momentum wheel and CMG ases are also presented... System Momenta. related to the time derivative of that vetor in F d~ x dt he absolute time derivative ofanyvetor ~ x in F is i x b by d~ = +! ~ ~ x () dt b i he equation of motion relating linear momentum to the total external fore is ~ p_ = ~ f () e herefore, the dierential equation for linear momentum in the body referene frame is given by! p_ = p + f (8) e he absolute time derivative of angular momentum (in F ) an be shown to be i ~ _ h = ~ v ~ p + ~ g (9) e so that we an write the dierential equation for the total angular momentum in body oordinates as _ =! + () h h v p g e

33 .. Gimbal Angles. We note from Equation () that the relative angular momentum of the GMW about the gimbal axis is h h IA I =! =! () gr ga g g g gr so that we nd the gimbal angle rate from the expression _ = = I h A ()!! gr g ga g whih must be integrated to produe the gimbal angles. he gimbal angles are required to ompute the matries A and A. We assign the gimbal angles to be zero when the spin s t axes point in a set of arbitrary diretions given by a ; a ;:::; a. s s sn he gimbal axis diretion and gimbal angle for a partiular GMW denes a rotation matrix relating the omponents of the the zero gimbal angle unit vetors ( a, a, and a) s t g to the omponents of the urrent unit vetors ( a, a, and a). For an eigenaxis rotation s t g (in this ase about a) through an eigenangle given by, we an express the rotation g matrix relating omponents in the two frames as C= os +(os ) aa sin a () g g g hus, the spin axis unit vetor for the j-th GMW is given by a = C a = os a +( os ) a a a + sin a a (4) sj j sj j sj j gj gj sj j gj sj Sine the gimbal axis is perpendiular to the spin axis ( a a gj sj = ), and if we assume that the gimbal axis is xed in the body frame, then we an further redue Equation (4) to a = os a sin a (5) sj j sj j tj In a similar manner, we an show that the j-th transverse unit vetor is given by a = os a + sin a () tj j tj j sj

34 A s We dene some new terms to aid in the numerial omputation of the N matries and A. From the N matrix,, of gimbal angles, we ompute the N N matries t s and where s = diag(os ) () = diag(sin ) (8) and os and sin are olumn matries of the osines and sines taken term by term of the olumn matrix. By dening the matries A and A as the values of A and A when the gimbal s t s t angles are all zero, then we ompute A and A as funtions of gimbal angles from the expressions s t s A = A A (9) s s t s At At As = + () For single-gimbal GMWs, A = A is xed, so A _ =. he rates of hange of A and A,however, an be shown to be t g g g s A_ = A diag( _ ) () s t _ = diag( _ ) () A t A s.. Spin Axis Angular Momentum. Wenowinvestigate the equations of motion for h and h sine these terms are aeted by the spin axis torque and gimbal torque swa ga respetively. oward this end, we will for the time drop the subsript, j, referring to the j-th GMW, and onsider only a single GMW. We begin by dening a salar omponent of the torque applied to the spinning wheel as g = ~ a ~ g () b w s w

35 a where ~ is a unit vetor in the spin diretion (independent of the referene frame). he term ~ g s bw is the total torque vetor applied by the body/gimbal system on the spinning wheel, and in general inludes omponents orthogonal to the spin axis. he omponent of the absolute angular momentum of the wheel about the GMW enter of mass in the ~ a s diretion is h = ~ a ~ h (4) swa s wa Note that we have inluded only the wheel momentum in the denition, sine we will be applying a spin axis torque to the wheel only. (he total angular momentum of the GMW is ~ h = ~ h + ~ h where ~ h is the gimbal frame momentum about the spin axis.) For a wa ga ga eah GMW, then, we have h _ = ~ a _ ~ h ~ a ~ _ + h swa s wa s wa (5) Sine ~ h _ = ~ g, Equation (5) beomes wa b w h _ = ~ a _ ~ h + g () swa s wa w a Beause ~ is xed in the gimbal referene frame, whih has angular veloity! ~ + ~ _,we have s _ =( + _ ) () ~ a! ~ ~ a ~ a s g s a g Sine the wheel is axisymmetri, the inertia dyadi an be expressed as ~ I = I ~ +( I I ) ~~ aa (8) w tw sw tw s s tw sw ~ ~! ~! ~ where I and I represent the wheel transverse axis inertia and spin axis inertia respetively, and h = I (where is the absolute angular veloity of the wheel with wa w wa wa respet to inertial spae), then Equation () beomes _ =(( + _ a) a) ( +( ) aa) + (9) h! ~ ~ ~ I ~ I I ~~! ~ g swa g s tw sw tw s s wa w

36 he total angular veloity of the wheel is made up of the body angular veloity, plus relative veloities in the spin axis diretion! ~ and gimbal axis diretion! ~. Sowe swr gwr use = + + = a + _ a + (4)! ~! ~! ~! ~! ~ ~! ~ wa swr gwr swr s g and also make use of the fat that ~ a ~ a = ~ a ~ a = whereas ~ a ~ a = to reast Equation (9) as s g s t g g _ =( a _ a ) ( _ a + + a +( ) aa )+ (4) h! ~ ~ ~ I ~ I! ~! I ~ I I ~~! ~ g swa s t tw g tw swr sw s sw tw s s w Eliminating terms based on orthogonality arguments, we obtain _ =( _ ( a ) a ) _ a ( )+ (4) h I ~! ~ ~ ~ I! ~ g swa tw s g t tw w a a a a a whereupon we use the vetor identity (! ~ ~ ) ~ =! ~ ( ~ ~ ) =! ~ ~, giving the result s g s g t h_ = g (4) swa w for the single GMW. his simply states that the rate of hange of the absolute angular momentum omponent in the spin axis diretion is equal to the torque transmitted to the wheel in that diretion. he simpliity of this expression is a result of the axisymmetry of the wheel, and is idential to the symmetry axis equation of motion for an axisymmetri rigid body. Sine we haven GMWs, we form the N olumn matrix h _ for all N GMWs and write swa h_ = g (44) swa he simpliity of this matrix dierential equation motivates a similar approah to deriving the equations of motion for the GMWs about their gimbal axes. w..4 Gimbal Axis Angular Momentum. We now engage in a similar development for the olumn matrix of gimbal axes momenta. o this end, we dene the gimbal axis torque for a single GMW as g = ~ a ~ g (45) g g bgmw 4