Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1967-09 Mass conservative attitude control systems for interplanetary spacecraft. Gilbreath, David Slagle Monterey, California. U.S. Naval Postgraduate School http://hdl.handle.net/10945/12614 I £?^ <£>lu&&fcATH,0. LIBRARY SCHOOL NAVAL POSTGRADUATE 93940 CALIF. MONTEREY, TABLE OF CONTENTS Section I. II. III. IV. V. Page INTRODUCTION 11 ATTITUDE CONTROL REQUIREMENTS 12 Internally Caused Torque 12 Externally Caused Disturbance Torque 13 Dynamic Requirements of ACS 14 ATTITUDE CONTROL SYSTEMS 16 Solar Radiation Torque 16 Active ACS 19 Reaction Wheel 21 Reaction Sphere 24 Control Moment Gyro 26 Momentum Dumping 39 COMPARISON OF ACS 41 Power and Energy Comparison 43 Weight Comparison 47 Angular Impulse Capacity Comparison 47 Pointing Accuracy Comparison 48 Reliability Comparison 48 CONCLUSIONS BIBLIOGRAPHY 50 51 APPENDIX A. Gravity Gradient Torque 52 APPENDIX B. Equations of Angular Motion 54 APPENDIX C. Brushless D. C. Motor Driven Inertia Wheel Built by Sperry Farragut Company for NASA 57 Theoretical, Magnetically Suspended and Torqued Reaction Sphere 59 APPENDIX D. LIST OF FIGURES Page Figure 1. Sun's Radiation vs Distance from the Sun 17 2. Radiation Absorption and Reflection 18 3. Solar Torquers and Energy Collectors 20 4. Electrostatically Supported Magnetically Torqued Reaction Sphere 25 5. Single Degree of Freedom Control Moment Gyro 27 6. Twin Single Degree of Freedom Control Moment Gyro 29 7. Two Degree of Freedom Control Moment Gyro 32 8. Twin Two Degree of Freedom Control Moment Gyro 36 9. Summary of Equations 42 Three Twin Control Moment Gyros 46 10. SYMBOLS A - Angular impulse, ft- lb- sec D - Damping torque per unit relative angular speed, ft-lb-sec/rad E - Energy, ft- lb H - Angular momentum about center of mass, ft- lb- sec I - Mass moment of inertia, ft-lb-sec P - Power, watts S - Absolute angular speed of ACS spin axis, rad/sec T - Torque, ft- lb t - Time, sec X - Vector cross product operator ^ - Vehicle absolute angular speed, rad/sec - Angular speed of ACS rotating element relative to vehicle, rad/sec - Unit vectors along vehicle x, y and spectively St i, j, k 2 z principal axes, re- NOTATION Letters with overbars are vector quantities. are scalar. quantity. Unbarred quantities A dot over a quantity denotes the time derivative of that TERMINOLOGY AND ABBREVIATIONS ACS - Attitude control system/ systems DOF - Degree/degrees of freedom CMG - Control moment gyro /gyros ME - Momentum exchange SYSTEM - Denotes the entire spacecraft VEHICLE - That part of the system separate from a smaller rotating component of the system SUBSCRIPTS control torque c - Control; e.g., T D - Disturbance; e.g., T n M - Torque motor; e.g., T^, torque motor torque R - About the spin axis of the rotating element; e.g., H angular momentum of ACS about the spin axis v - Vehicle; e.g., H x,y,z - Vehicle principal axis; e.g., T x axis , , , disturbance torque , angular momentum of vehicle , control torque about the SECTION I INTRODUCTION Future space travel will involve substantial increases in vehicle size and trip length. Flight paths presently under consideration are low thrust, minimum energy trajectories with long free fall stages. A typical proposed round trip is the Mars flyby which requires about 700 It is for such long duration flights that the attitude control days. systems (ACS) discussed in this paper might be employed. Torque produc- ing jet thrusters have been utilized in all manned space flights to date and are optimum for short duration flights. For the longer voyages con- templated, the amount of fuel required for attitude control thrusters becomes an appreciable percentage of total system weight and storage space. It is this factor that has prompted the search for a substitute for mass expulsion ACS. The mass conservative ACS considered in this thesis are described in Sec. Ill and compared in Sec. IV. Immediately following is a dis- cussion of attitude control requirements. 11 SECTION II ATTITUDE CONTROL REQUIREMENTS Need for attitude control may arise from the following: 1. Vehicle orientation for use of main thrust engines, solar cells, navi- gation and communication equipment, etc. 2. Docking 3. Crew and component limitations on vehicle angular velocity. Orientation for scientific measurements may well be the most demanding in terms of frequency and accuracy. Interplanetary trajectories usually involve midcourse guidance; some programs require several velocity changes with corresponding demands on navigational information. The abil- ity of data gathering instruments to compensate for vehicle attitude error is often quite limited; the Orbiting Astronomical Laboratory concept allows a vehicle pointing error of 0.1 arc second. Torque magni- tudes are mentioned later in this report. Control torque serves to give the vehicle the desired angular velocity and orientation and to counter disturbance or unwanted torque. Dis- turbance torques are conveniently categorized by origin as follows. Internally caused torque . Interaction between components of a spacecraft system does not affect the total angular momentum of the system, but it can change the angular motion of the vehicle; e.g., the stopping or starting of a rotary pump does not change the angular momentum of the system but it does torque the vehicle. Gyroscopic torque results from precession of the momentum of rotating machinery. Sources of these torques include tape recorders and motion of antennas, crew and stored liquid. For the Apollo vehicle, 100 seconds of arc per second represents a typical disturbance from crew movement, 12 [l] Externally caused disturbance torque . Sources of external disturb- ance torque are: 1. Gravity gradient 2. Solar radiation pressure 3. Magnetic field 4. Micrometeorite impingement 5. Thermal radiation from vehicle 6. Gas leaks, out gassing of vehicle material 7. Unwanted torque from ACS 8. Unwanted torque from main thrust engines The facing of the same side of the moon toward earth is due to gravity gradient torque. in Appendix A. The salient features of this torque are described As well as being dependent on system orientation and mass distribution, gravity gradient torque decreases rapidly with in- creasing distance between bodies. The importance of gravity gradient as a disturbance or control torque gives way to solar radiation pressure at about 600 miles from earth. Therefore, although gravity gradient has been successfully employed for earth satellite stabilization, it is of little concern for interplanetary travel. Discussion of solar radiation torque is given in Sec. III. Like gravity gradient, magnetically induced torque is important only in the near vicinity of earth or other bodies with surrounding magnetic fields. Quantitative assessment of micrometeorite induced torque will re- main difficult until more data are accumulated on the sizes, velocities and distribution of these particles. Although the sizes and collision frequency are expected to be low, the engaging speeds are such as to 13 impart considerable momentum to a spacecraft. The extent to which mete- orites produce unbalanced torque is, of course, a function of vehicle geometry. Protruding solar energy collectors and antennas render space- craft more susceptible to this torque. Inefficiencies in spacecraft energy conversion devices necessitate The torque produced by thermal thermal radiation from the vehicle. radiation is minor and readily design controlled. Gas leaks and outgassing of vehicle surface material would normally cause very small torques. ACS non-zero pointing error results from imperfect attitude sensors and/or control systems which cause oscillations or limit cycles about the desired orientation. Momentum exchange (ME) ACS produce unwanted torque due to gyroscopic coupling; this phenomenon is discussed later. During powered flight, directional control of the main engine thrust vector by nozzle gimballing, or other means, affords two axis attitude control; however, supplemental attitude control may be necessary to meet accuracy requirements. Dynamic requirements of ACS . For a given system, pointing accuracy and response time constitute the dynamic capability of the ACS. Since response time is a function of control torque, the latter parameter, being more convenient and general, is used for comparison purposes. The governing general equation of angular motion for a system of masses reduces to simplified equations in limiting cases. and listing of these equations, Nos. The derivation 1-9, are in Appendix B. These equa- tions appear in this paper without lengthy explanation and bear the same identifying number as given in Appendix B. equations are developed in context. Equation (10) and subsequent An assumption made throughout is 14 in response to control or disturbance torque, the vehicle behaves that, as a rigid inelastic body; i.e., the effect on vehicle motion of compo- nent interactions is neglected. These interactions are accompanied by transfer of kinetic to thermal energy and must be considered when dealSpin stabilization is an unlikely ing with spin stabilized vehicles. choice for interplanetary flight due to the need for fixed orientation for the use of solar energy collectors, antennas and other apparatus. The dynamics of rigid body angular motion is described by the following equation. T-flM u/x 4-(I»-Iyy)u;ybUzjr + [Iyyli/y +fl xy where: "Izz)^ [XJZ]1 T = sum of control and disturbance torques I = \JJ mass moment of inertia = vehicle angular speed The coordinate system is fixed in the vehicle and aligned with the vehi- cle's principal axes. The kinematical equations relating angular dis- placement, velocity and acceleration and response time require a coordi- nate transformation to Eulerian Angles or their equivalent. For the purposes of this thesis it suffices to note that response time is a function of acceleration, and this acceleration is torque generated as indicated in Eq. (3). 15 SECTION III ATTITUDE CONTROL SYSTEMS ACS may be termed passive if they do not require energy or mass ex- penditure. In the active category are jet thrusters and momentum ex- Passive control torque is obtained from some of the change devices. same environmental sources that cause external disturbance torque; namely, gravity gradient, magnetic field and solar radiation. For rea- sons previously set forth, only the last of these is significant for interplanetary travel. Solar radiation is also spatially variant but not so unfavorably, as indicated in Fig. 1. Although not the only electromagnetic Solar radiation torque . radiation in our solar system, the sun's emissions dwarf those of all others combined and need be the only radiation considered. Incident radiation is either absorbed or reflected, the reflected part being diffuse or specular. Figure 2 illustrates the three possibilities. The radiation force on a surface is the time derivative of the incident mo- Using Einstein's mass energy equivalency: mentum. Tr 4lmH where E is the - <L(I) , incident energy and is the speed of light. c With P f defined as the pressure of fully absorbed radiation perpendicular to the surface, the expression for solar pressure is: 2 p -fep(\-s)coss -h(/+sf)cos eJpf where 9 and the unit vectors N and /° and S +(i-s<*)pf cos© h "t s/nei are shown in Fig. 2. The symbols are defined as follows: 0<P<\ = fraction of incident radiation reflected 0<S< = fraction of reflected radiation reflected specularly I 16 — / s- /ooo J- \A I-* V 3 /oo i* *>* H /o ' 1 1 1 * 3 1 1 f H Distance f>oyn Sun ^ r~ — ? kvux/o" Figure 1 Sun's Radiation versus Distance from the Sun 17 •/ Absohpt I Specula/- oyi wmiim reflection Diffuse mnrnrm V N Figure 2 Radiation Absorption and Reflection 18 reflection If and p = S=l, the maximum pressure of 2 Pf is realized. = earth's distance from the sun ft- - 9»? X to At an lb/'ft\ Torque generated by solar pressure will, of course, be dependent on moment arm and control surface area as well as the parameters in the pressure equation. The following example illustrates the order of mag- nitude of solar torque. A non-absorbing, specularly reflecting panel, with a moment arm of 10 ft and oriented normal to the 10 ft by 10 ft, sun's radiation at an earth's distance from the sun, will exert a torque of 19.6 x 10" ft-lb. Figure 3 shows some conceptual designs that have appeared in the literature; Figs. 3(b) and 3(c) are self aligning energy It seems likely that if solar torque is utilized for atti- collectors. tude control, it will be in the form of such dual purpose devices. Active ACS . Active ACS are mass expulsion and momentum exchange In the latter group are reaction wheels, reaction spheres (ME) devices. and control moment gyros (CMG) . The three ME ACS provide torque by changing the angular momentum of a mass within the vehicle. Their torque is "internal"; therefore, any change in system angular momentum is due to external disturbance torque. The first two devices effect momentum exchange between vehicle and ACS by changing the angular speed of a wheel or sphere. The CMG maintains constant angular speed and effects momentum exchange by altering the direction of the rotating ele- ment's spin axis. In equation form the ME ACS are described by: Reaction wheel or sphere: otfu+uV) l- CMG : -J c —o s-o (8) olt S-=0 TC = -SX^ =-IR (sx2) (9) 19 Movable Solar vane panels Hinge Vehicle (b) Aligned Reflector^/ Reflector ^Collector Fn5r FORCE L--4 wll tl> Misqliflned Reflector Aligned Boom (c) vehicle Swivel points Collector Parabolic reflectors Figure 3 Solar Torquers and Energy Collectors 20 where: Tc = control torque Hr = ACS angular S = absolute angular Si momentum velocity of the ACS spin axis = angular speed of the wheel or sphere relative to the vehicle W% = component of vehicle angular velocity about the ACS spin axis A more detailed description of the three basic ME ACS follows. Reaction wheel . A reaction wheel is a motor driven wheel. The motor exerts a control torque on the vehicle and an equal and opposite torque on the wheel. cle. The torque axis is fixed with respect to the vehi- Three such wheels with mutually perpendicular spin axes provide complete attitude control; normally the spin axes would be parallel to vehicle principal axes to reduce the coupling evident in Eq. (3). For a reaction wheel parallel to the vehicle x axis, combining Eqs. (3) and (8) yields: Tc=I j? (-d-0; x ) =TP +IxyLU x +(Izi-Iyy)LU y LVz (10) ) where Tp is disturbance torque, and the assumption is made that the moments of inertia of the vehicle approximate those of the system. For the case of steady pointing Eq. (10) reduces to: Tc -lK Sl-TD (11) For attitude changes in interplanetary space, the control torque re- quired for reasonable response time is normally much greater than dis- turbance torque, a possible exception being the disturbance torque resulting from misalignment of the main engine thrust vector. tion, Eq. (10) With this assump- becomes, for the case of commanded reorientation: k=I«(A-lM=£wltf* +(lzz-Iyy)UUylUz 21 (12) : Power required, less motor, bearing and windage losses, is, using Eq. (10): Energy required, less the losses previously noted, is, using Eqs. (10) and (13): I, (-Q-cux )iux Jt jj = 5t [(Ji2 -uy, 2 f-(ii,-u;x/ ^ +[ fc u;x Jt (14) The maximum angular impulse absorbed by the wheel, assuming the wheel initially at rest and using Eq. /W x = fc<# = (10), is: r,f(A-^)<#*i»JW , » UJ*V where the assumption is made that 1 £l^ /T71AX < 15) , Optimization of any ME ACS consists of arriving at the proper blend of the following objectives, commensurate with cost and reliability specifications 1. Maximize control torque 2. Maximize angular impulse capacity 3. Minimize power and energy 4. Minimize weight 5. Minimize space With respect to a reaction wheel the following points evolve: 1. Objective 4 and the need for large wheel moment of inertia dictate a wheel with its mass concentrated in the rim. 2. Objectives 3 and 4 are in direct conflict with one another; 22 for a given control torque, decreasing 1^ necessitates a higher wheel angular acceleration with resulting increase in power and energy. Reference T2~J accounts for this compromise by minimiz- ing an "equivalent weight" which is a function of power as well as weight. This function is a measure of the relative impor- tance of power and weight and is representative of the need for a systems approach to ACS selection and design. The equations that have been derived for a reaction wheel are for a single wheel. The three wheels required for three axis control would, in general, all be spinning. Precession of this momentum by vehicle angular velocity results in gyroscopic torque. set of reaction wheels is, according to Eq. c Lit The torque, then, for a (6): ~SXH R (6) S=rO where, in this case, H R is the net momentum of the three wheels and SrCD in Eq. . Normally the first term would be predominant. (6) is, The second term in general, unwanted since it increases the power re- quired for attitude changes; however, it does afford rate damping for steady pointing. Damping serves to reduce oscillations or limit cycling caused by electromechanical lag in the motor-wheel-vehicle combination. Eddy current, viscous and hysteresis damping have been suggested. Ref- erence [3] claims weight and size advantages for the hysteresis damper. Their design weighs one pound, provides 1.5 ft-lb per rad/sec and consists of permanent magnets attached to the controller housing on either side of a thin vane protruding from the wheel rim. Damping torque re- sults from local magnetization polarity changes in the vane material 23 Appendix C is a brief description as it moves relative to the magnets. of a reaction wheel built for NASA. Reaction sphere . The reaction sphere has the singular capability, among ME devices, of providing torque about any axis. Due to weight and moment of inertia considerations it takes the form of a spherical shell. Lacking mechanical supports it may be suspended by gas bearings, a magnetic field, or an electric field. Gas bearing suspension causes high viscous losses and limited positioning force. If the sphere has equal moments of inertia and the torquing imparts no radial forces, positioning force is necessary only to counter vehicle linear accelerations and centrifugal forces due to vehicle angular speed. Reference [4] suggests magnetic suspension in the form of three orthogonal pairs of servo controlled electromagnets surrounding the sphere. Positioning force results from interaction between the applied magnetic field and eddy currents induced in the spherical shell. of suspension. Some drag torque results from this type Electric field suspension is the only means that does not cause drag torque on the sphere; however, positioning force is re- stricted by the limited voltage gradient maintainable between housing and sphere. Torquing the sphere magnetically has been proposed. Reference [5] suggests an electrostatically suspended, electromagnetically torqued sphere as pictured in Fig. 4. As with magnetic suspension, torque re- sults from interaction between the applied field and eddy currents induced in windings in the sphere's shell. In this design there are three orthogonal stator windings in the shell. rotates, "dragging" along the sphere. The magnetic field Eight electrode areas on the shell and four pairs of electrodes in the housing afford four independent 24 Figure 4 Electrostatically Supported Magnetically Torqued Reaction Sphere 25 positioning forces. The electrodes ideally cover the sphere surface except for motor stators. If a reaction sphere is magnetically torqued and suspended, is unwanted interaction between the magnetic fields. Reference there [4] recom- mends such an arrangement with the claim that this interaction is negligible. A description of this device is in Appendix D. Although the reaction sphere has theoretical advantages, it has not reached the hardware stage because of large scale problems associated with sphere suspension and torquing. Control moment gyro The final and most promising ME ACS is the A CMG may have one or two degrees of freedom; both control moment gyro. will be discussed. . The device produces torque by precessing the angular momentum of a wheel. 7^ S, - -sxh = -l (sxR) r R (9) the absolute angular velocity of the CMG spin axis, is the vector sum of vehicle angular velocity and the angular velocity of the spin axis relative to the vehicle. motor as shown in Fig. The latter motion is effected by a torque 5. The angular speed of the wheel is held constant by a drive motor which, aside from initial speed buildup, needs to overcome only bearing and windage losses. In Fig. 5 the torque motor torque, Tw, acts on the gimbal; an opposite and equal torque acts on the vehicle. tion will be followed in subsequent figures. natural and artificial damping. speed about the output axis; S, This conven- The damper represents S2 is the component of vehicle angular is the gimbal angular speed. angular directions assumed in Fig. 5, 26 For the the gimbal equation of motion is: Cfixec/J C>nba.l Figure 5 Single Degree of Freedom Control Moment Gyro 27 JM -D6 tS£ H R -IS, where I (16) ; is the moment of inertia of the gimbal and rotor about the s torque motor axis. is a gyroscopic torque. H S 2 D6 is the damping R torque, where 6 is the gimbal angular speed relative to the vehicle. Un- wanted torques on the vehicle are those of the damper and the torque and drive motors; normally these would be much less than the output torque S-.H R . A more serious problem arises from the output axis not remaining fixed with respect to the vehicle. As mentioned previously it is desir- able to align the output axis parallel to a vehicle principal axis; the CMG output axis remains perpendicular to the fixed torque motor axis and the wheel spin axis. Therefore, to minimize coupling, such a device must be restricted to small gimbal displacements. Three axis control is provided by a CMG for each vehicle principal axis. As with reaction wheels, such a configuration must contend with unwanted gyroscopic torque due to precession of the wheels vehicle angular velocity. 1 momenta by Since, in general, the CMG possesses a higher angular momentum than the reaction wheel, the problem is more severe with the former. The twin gyro configuration shown in Fig. 6 provides a fixed output axis relative to the vehicle and substantially alleviates the problem of the preceding paragraph. The three axes shown in Fig. 6 are orthogonal. Not shown are the two drive motors that maintain the two wheels at a constant and equal speed. The torquers are coordinated so that both wheels are precessed an equal amount at the same rate. The neutral positions of the angular momentum vectors are along the spin reference axis pointing inward. 28 o u c X o I 4) O • o "2 U c o o vO B o X) u </) L 01 o> J-l 00 •H \ 00 o 00 c •H CO c •H 29 The absolute angular rate about the torque motor axis of each spin axis is the sum of its angular speed relative to the vehicle, 6 , and the component of vehicle angular speed about the torque motor axis. For this configuration it is convenient to consider separately the gyro- scopic torque effected by each of these angular rates. 6 H_ The gyroscopic torque, , is of equal magnitude for both wheels. The components of these torques along the spin reference axis are equal and opposite; the components along the output axis are additive and to- gether equal: =2H^ecos9 Tc (17) The reactance torques on the vehicle from the torquers and dampers are equal and opposite, thus eliminating a minor source of unwanted torque. The output axis is permanently parallel to a vehicle principal axis. With the twin gyro configuration the wheel momenta cancel to some extent; the net momentum is 2H^ sin and is directed along the output axis. Ve- hicle angular speed about the spin reference and torque motor axes acts on the net momentum to produce unwanted gyroscopic torque about the other axis, respectively. An increase in 9 has two detrimental effects: The net momentum increases, causing larger coupling torques, and control torque, Eq. (17), decreases. Therefore a gimbal limit prior to the o 9 = 90 zero torque position is suggested; reference r , [lj limited 9 to ±60°. The equation of motion for an individual gimbal is comparable to Eq. (16). The output axis is again assumed parallel to the vehicle x axis. Ta-De-U^H*cos6-u^H«£wd*Id (18) ; 30 where the signs of the third and fourth terms are dependent on the directions of UJX and 9 , UJy , respectively, and the relative angular acceleration, approximates the absolute gimbal angular acceleration. quantities are the same as in Eq. are observed, Eq. Tm CL 16 -\~UUxHr,COsQ The other If the suggested gimbal limits (16). (18) can be closely approximated by: Power required, less motor losses, for . both torquers is: P~zTh e =z(ie+iux H R cos6)e d9) As with the single CMG, drive motor torque for constant speed operation is necessary only to counter bearing and windage torques. To preclude unnecessarily large drive motors, a long build up period is desirable. The CMG described in reference \l\ was allowed two hours to reach oper- ating speed; yet the power required for this acceleration was twenty times that needed for constant speed operation. Energy required for both torque motors is, using Eqs. (17) and (19): E=2^(l'6mH*cosb)eJt -I(b]-e]) i-^TiLux Jt (20) The maximum angular impulse absorbed, assuming the wheel spin axes are initially aligned with the spin reference axis, is, using Eq. kmn-zH^ecoseJi =2/yR s/A/£^ ax (17): (21) A twin controller for each vehicle principal axis would be necessary for complete attitude control. A two DOF CMG is shown in Fig. ible. 7; the wheel drive motor is not vis- An additional torquer and gimbal freedom enable this CMG to pro- duce gyroscopic control torque about two perpendicular axes. 31 The ' Parallel Z to vehicle axis *-J Damper Parallel to A T M2 vehicle %. Torque motors— 8/ £/;f /0 ^ o/ v Parol lei to vehicle X axis Figure 7 Two Degree of Freedom Control Moment Gyro 32 Y axis Sfi^/ following discussion of the two DOF CMG utilizes the directions of angular motion arbitrarily assumed in Fig. 7. As with the single DOF CMG, control torque is effected by precessing wheel angular momentum. How- ever, with this device the gyroscopic torque is transmitted primarily through the torque motors; the single DOF CMG control torque was conveyed to the vehicle by the gimbals and their supports. For example, control torque about the x axis is obtained by the inner gimbal axis torque motor rotating the inner gimbal. The resulting gyroscopic torque is H S 2. component about the x axis is S sin F H R the ; and is transmitted to the The component S«H R cos F vehicle via the outer gimbal axis torque motor. is perpendicular to the two gimbal axes and is the only control torque not conveyed to the vehicle by a torque motor. bal by its torque motor causes a torque, S..H- Rotation of the outer gimsin that is totally , | about the inner gimbal axis and must be transmitted to the vehicle by the In referring to torque motor induced inner gimbal axis torque motor. gimbal rotation, the assumption is made that $\ —§ *»>Cr S^ — P. The interdependency of the two torquers and gimbal rotations is evident in the gimbal equations of motion. ^-Df-StHrSIMPsI,?! (22) , where I, and I2 are gimbal plus wheel moments of inertia about the outer and inner axes, respectively; I~ is constant and Equation (22) shows that as diminish. When 1 varies with I / 1 departs from 90°, the gyroscopic torques differs from zero, the torque about the inner gimbal axis is no longer parallel to the vehicle y axis. ments are undesirable and indicate a. 33 Both of these develop- need for gimbal limits; reference [6] suggests P both be limited to one radian deflection from their and initial displacements of P 0=0°. = 90° and Remembering that the motor torque, as shown, is acting on the gimbals, the control torque on the vehicle is: f&foCOSPCftstf +TMl SlH(t>)T HRCosrcos^f(DPfS rs2 + (23) yields: (22) and Combining Eqs. (23) 7 / /y /e s/A// An order of magnitude analysis of Eq. damping terms 00 the term S 2 H R cos 7c* and / DP sin fI2 S2 ) s 'N^]k ( 24 > (24) permits the dropping of the If the suggested gimbal limits are observed, . These considerations result in: is negligible. 7 [SiH g SIN/ -I,sll - [(S W R SlN/ , fI 4 l S2 )COS0) J + [StHKCDSrcosb+i^HzSlNP +Iz where the gyroscopic terms SoHpSin/ S l )sm<i\l and S-, H R sin (25) / cos predominate. Determination of the angular impulse capacity of the two DOF CMG is facilitated by considering the components of the absolute gimbal rates, and S x S 2 . S, = +LU* Sl -f' t VUyC0S(f> i-UUz SIN^ (26) ) 34 Wy where and UJz have the proper algebraic sign. Since control torque exists primarily about the vehicle x and y axes, the impulse about only Saturation of the two DOF CMG occurs when these axes is computed. and 0=0°, If the gimbals are initially at are reached. max / r . = 90° and the maximum angular impulses accumulated about the x and y axes are, using Eqs. A*^ (25) and (26): =ftSt//«SWr-I l 4 l )Jt =-^C05^ tH R $(lX)yCOst+LUiSmt>)SlNNt -SlCi+^Jt (27) - Hr ^uux ^inP cost dt -Ti^Cr+d/yCosj-WytsiNt +uuzSiN<l>+lx)z 0cos$)costc/'t A look at the relative magnitudes of the factors comprising Eqs. (28) and in Eq. (28) x average / lOy are small numbers. The term is unintegrable in present form; is assumed. (27) and H R is by far the largest number; 1^, leads to reduction in terms. l>0 I2 , (28) -//* ^ SUvTcosQc/t for illustration purposes an Equations (27) and (28) can now be approximated by: ^Wx" "^ CM/ ih£>, +H« $(Vy«>st +LUzSl»<t)siNrJt (29) where the first term on the right side of each equation usually predominates. A twin two DOF CMG configuration similar to Fig. cancelling of unwanted torques. 6 provides similar Figure 8 is a schematic of such a device showing only the torque motors, momentum vectors and appropriate axes. 35 spin to referee vehicle Figure z ax/*, />*"* //•/ axis 8 Control Moment Gyro Twin Two Degree of Freedom 36 As with the twin single DOF CMG, the deflections of the wheel spin axes from the spin reference axis are mirror images. The directions of torque and angular motion shown in Fig. 8 are for control torque about the vehiAs with the previous twin CMG, it is convenient to examine cle x axis. separately the gyroscopic torque caused by vehicle angular velocity and gimbal angular motion relative to the vehicle. The following analysis applies to Fig. 8; the primed quantities denote the properties of the upper wheel. Mr - HR J"cos RR = rl tSiA/Ps/A/0T +SiNPcos<l>k] H R fcosri Net H R f sinT sin$1 -SinT Qos(b£\ -ZH R [cosrl +SinTsin4 fj (31) _J r - -r\cos<f>7 +s//y0/f] PXHK +D(Hr -2Hk P\sinPI -cosPziN<f>lj (32) Equation (32) describes the net gyroscopic control torque produced by rotation of the inner gimbals by the inner gimbal axes torque motors. If / are each limited to one radian deflection, sin/ and and the y axis component in Eq. (32) is negligible. » cos/ sin The torque compon- ents of the inner gimbal torque motors about the y axis cancel; the com- ponents about the z axis are additive and together equal 2 Tv«2 sin . Presumably control torque from a single controller would be generated about one axis at a time. 0=0, the x axis, and T^ Thus, for the case of control torque about is effecting r ciable gyroscopic torque to the vehicle. small, the z axis component is negligible. 37 and not transferring appre- Since T(^ 2 and sin are both , A similar analysis for control torque about the y axis is accomThe plished by reversing the direction of the upper x axis motor torque. resulting gyroscopic control torque is: $xS*-^X/yR = -2H R ^SINFzos<t> < J 33 > In general, unwanted gyroscopic torque will be caused by vehicle angular velocity precession of the net controller momentum. The twin configura- tion and gimbal limits serve to reduce the net momentum and therefore the unwanted torque. Neglecting damping torques and gimbal accelerations, the angular impulse capacities about the x and y axes are, using Eqs. Awx-sfawJt (33): *2H*$r<;iNrJt = -zH*cosrmLy[ siNtw =-2Af/rsw/; wll5e As expected, Eqs. (34) and <34> < 35 > ; where, again for purposes of illustration, in Eq. been assumed. (32) and (35) (35) an average has / show essentially double the angular impulse capacities of the single wheels described by Eqs. (29) and (30). Torque motor power, less motor losses, for the case of control torque about the x axis, where 0=0 , is: Px=r(TMti-TM Z ) (36) For the directions assumed in Fig. 8, the following holds: Thi =I 2 S 2 + S,HR SINP (22) 38 Si = S2 = rtLUyC0S^ -bUz SW0 + w* = UIK - S2 - r "M^y C05 S2 - F-UJy COS<f) - s, W z -UUz SIN <() SlA/fl Using these relationships Eq. (36) becomes: —2Flz lUz£INf where the term ser order of magnitude of lU2 SIN has been dropped because of the les(p A similar analysis for the case . of control torque about the y axis yields: Py-$(Th t fT«,) -Imfi+WyHfiCosQ sinPJ (38) Torque motor energy, less motor losses, for control about the x axis is, using Eqs. (32) and ^-llzCPPdt + (37): Czt , t \A)*H*SiNf dt =I z (/\-r*)+C Tx UUx dt (39) Similarly, for control about the y axis the torque motor energy is: Ey = I, (&'#,) +[TyWyh (40) Three twin two DOF CMG could be arranged to provide control torque from two controllers about each vehicle principal axis. Momentum dumping . The term, momentum dumping, refers to the removal of accumulated angular impulse from ME ACS. 39 For the reaction wheel or sphere it is a matter of decelerating the rotating element; for any of the CMG discussed it involves aligning the wheel spin axes with the spin reference axis. Saturation is caused by long acting unidirectional dis- turbance torque such as may result from solar pressure, meteorites or gas Presumably any ME ACS would be designed with sufficient impulse leakage. capacity to handle all attitude commands and oscillatory disturbances. Dumping may be effected by solar pressure or jet reaction torque. Since solar torque is very small, it would necessarily be a continuous acting system that would prevent the ME ACS from reaching saturation. Jet reaction appears the more likely candidate; it is more reliable and easier to mechanize. With jet reaction dumping there is a trade off between Amax of the ME device and the fuel required for the reaction jets; as A. ax is increased, the frequency of dumping decreases. \iax is P ro P ortiona l t0 H R = I R ^ Since for all ME ACS, increase in Amax will require additional weight in the form of a larger wheel or bigger drive motor to maintain higher H . 40 SECTION IV COMPARISON OF ACS The following is a discussion of the relative merits of three of the ME ACS. Reaction sphere and solar torque ACS are excluded. reaction sphere is omitted because it has yet to prove feasible. The Its theoretical advantage is that it accomplishes with a single rotating element that which requires three of either of the other type of ME ACS. In addition to suspension and torquing difficulties, disadvantages of the reaction sphere are: The required torquing power is comparable to that of the reaction wheel and therefore suffers the same disadvantage that is noted in the next section; momentary power interruption for a magneti- cally or electrostatically suspended sphere results in total failure. Solar torque is not compared since, with the control surface sizes pres- ently deemed practical, it is insufficient for commanded attitude changes. Bases for comparison are: power and energy, weight, angular impulse capacity, pointing accuracy, reliability, size and cost. Sizes of the ME ACS are not compared since, with the exception of the reaction sphere, they do not vary appreciably. Cost is also not considered. The comparisons, in general, are not numerical; such comparison would require numerical evaluation of the weights, efficiencies and reliabilities of the torque and drive motors. The ACS compared are the reac- tion wheel, twin single DOF CMG and twin two DOF CMG. It is assumed that the superior performance of the twin configurations compared to the single CMG controllers would discourage the selection of the latter. Each ACS is assumed to consist of three controllers oriented parallel to vehicle principal axes. The applicable equations are listed in Fig. for ease of reference. 41 9 1 4. -K •f- 5 nji^» ~T& *+-» / -J v 3 V * O —x m-j o 5' i 1 * * — w + ' 0/ 4. « •<D O o » J? <u 1 m|<m + -i- + •^5? 5 3 O H 1*? * ? I * S s — ' ^ 1 + '?• :<D 2 -v. 3 JE &\ *. U IK M H CM "~"5 CM 5 £ 'a- 5 J* (D X d ~ 3- O U 2 s »/> a: <M «M ? « OS* P R ° CD 2 u O 2 i/» z-4 Xof H _^_ <^ «b ,-*» _,_ 2 o 1^ o t. o •<D U c -H CM R o ^ 3 +-> -2 .5 oo o r q .5 <-» 3 •f 42 Q, ex. o a o u o 3 -h u. us o z Q O Power and energy comparison In comparing ME ACS, power and energy . need not be differentiated if, for a given control torque and vehicle angular velocity, the power is always the same; i.e., the power required Reaction wheels for the ACS is independent of the state of the ACS. violate this condition; power is proportional toil DOF CMG behave identically in this respect; Eqs. fact that the term, 2 ^W^U^COS & . The single and two (17) and (19) and the is predominant show that CMG power , is essentially independent of gimbal position. Maximum power is a criti- cal factor in spacecraft systems; the varying power of a reaction wheel system for identical dynamic response is undesirable. It is illustrative to observe the fraction of total power that goes into torquing the vehicle; the remainder serves to accelerate ACS components. For the reaction wheel this fraction is, using Eqs. IR (ji-w„).a (12) and -ft For the twin single DOF CMG, using Eqs. (17) and (19) and neglecting drive motor power, this fraction is: — TxlUx Hr LUx COS 6 ^ . » » since TO « HrLUx COS 6 Similarly for the twin two DOF CMG the . fractions for the x and y axes are, respectively: LP +H e ujx siA/P ~ l and H F LUySlHPCOs4> ^ . 43 (13): where again drive motor power has been neglected. This omission is jus- tified if drive motor power is negligible compared to the power expended torquing the vehicle, the latter quantity being dependent on the required torque and the vehicle angular speed about the axis in question. Drive motor power is minimized by using high quality bearings, carefully balancing wheels and surrounding the rotating element with a light gas at low pressure. scribed in reference The 50 ft-lb of control torque, two DOF CMG, de[l\ required 18 watts to maintain 12,000 rpnu A twin two DOF CMG producing 50 ft-lb would require 43 watts for torquer and drive motors if the vehicle were rotating at 0.1 rad/sec about the torque axis. A reaction wheel with a 100 per cent efficient drive motor, no windage and bearing losses, and equal wheel size and torque could not exceed 6 rpm if limited to 43 watts; and it would reach that speed in 0.01 seconds from rest. The reaction wheel can compensate to a limited extent for its low efficiency by employing regenerative braking; i.e., when the direction of control torque and wheel speed are such as to require deceleration of the wheel, the drive motor can generate its own armature current down to a certain speed. Although some of the kinetic energy stored in the wheel is regained, the overall efficiency does not approach that of the CMG. To summarize the power and energy comparison of the three ME ACS: 1. The single and two DOF CMG have identical energy conversion effi- ciencies. 2. As evidenced by Eqs. (14), (20), (39) and (40) and the power fractions, the major part of the energy input to a reaction wheel goes into changing the kinetic energy of the wheel; the only appreciable 44 energy loss in a CMG, outside of initial speed buildup, is motor, bearing and windage loss of the drive motor. 3. The reaction wheel is competitive, from a power and energy standpoint, when either infrequent use or small torque and low angular impulse capacity are needed. and Amax dependency on tively. -0. The latter condition results from power as described by Eqs. (13) and (15), respec- The first consideration reflects the constant drive motor power required for the CMG; a reaction wheel, when not in use, idles at a lower speed than an equivalent CMG, thereby requiring less power during This advantage can be negated somewhat by the following: such periods. With regard to the twin single DOF CMG shown in Fig. 6, the spin refer- ence axis of an individual controller can be arbitrarily oriented in a plane perpendicular to the output axis without affecting the net angular momentum or the control torque. Figure 10 represents three identical twin controllers with spin axes aligned along their respective spin ref- erence axes. Vectors A, B, C, A tum vectors of the six wheels. 1 , B 1 and C 1 represent the angular momen- The output axes of the three controllers are parallel to vehicle principal axes. Since the angular momenta of the six wheels are equal in magnitude, the particular orientation of Fig. A', B 10 yields: 1 and C' A+B+C = A and 1 +B +C' 1 = . If A, B and C or are aligned with their respective spin axes as shown and are brought to zero magnitude at equal rates, there is no net torque on the vehicle and the net momentum of the controllers remains zero. procedure halves drive motor power. This The reduced capability would nor- mally be adequate for countering environmental disturbance torque. preceding arrangement is applicable to the twin two DOF CMG as well. 45 The axes are torque X} y ar\d z axes and are parallel To pKucipa.! output ve/ucle axes y afalle/ to axis Figure 10 Three Twin Control Moment Gyros 46 coa's Weight comparison . The rotating elements, torque motors and drive motors are the primary components contributing to the weight of ME ACS. As noted earlier, reaction wheels require larger 1^ than do CMC, for the same torque and power; the larger I_, of course, means increased weight. This difference is offset to a varying degree by the gimbals and extra motors required for CMC Thus, from a weight comparison standpoint, the reaction wheel improves with decreasing control torque and becomes com- petitive when the larger weight of the wheel is offset by the weight of the gimbals and additional motors of the CMG. The two types of CMG compare in much the same fashion as CMG in general compare with reaction wheels. With an ACS composed of three twin two DOF CMG, each axis can be torqued by two controllers. Therefore, an ACS consisting of three twin two DOF CMG requires half as much angular momentum per wheel as a similar system of single DOF CMG; this allows a wheel weight saving for the two DOF system. Although single and two DOF CMG require the same power, the latter lacks the torque multiplication of the single DOF CMG. The gearing necessary to achieve the higher torque results in extra weight for the two DOF system. Thus the two DOF CMG system is lighter than the single DOF system when the wheel weight saving offsets the additional weight of the gearing and the extra gimbal and torque motor. This situation arises with larger vehicles which require greater H^ for larger torques and angular impulse capacities. Angular impulse capacity comparison . The angular impulse capacity of a reaction wheel is equal to the maximum angular momentum of the wheel. The impulse capacity of a CMG would also equal the H^ of the device were it not for gimbal limits imposed. 47 This better "efficiency" v of the reaction wheel is more than offset by J its typically J J low H R max , ' which is a result of power considerations; from a power standpoint the reaction wheel cannot be torqued at the speeds of a comparable CMG. A significant advantage of the two DOF CMG is the fact that its angular impulse capacity is nearly double that of the single DOF CMG. Pointing accuracy comparison . As noted earlier, net momentum af- fords gyroscopic rigidity, thereby improving pointing accuracy. The variable net momentum of a reaction wheel system is comparable to that of either twin CMG system; the individual wheel momenta of the CMG are much greater than those of the reaction wheels, but a large percentage of the CMG momenta is cancelled by the twin configuration. of CMG have identical pointing accuracy. The two types CMG and reaction wheels are subject to similar electromechanical lag in the controllers. In summary, there is no significant inherent difference in pointing accuracy capability between the three ACS. Reliability comparison . The three ME ACS have in common the poten- tially dangerous situation of a rapidly spinning mass supported with a minimum of physical contact for extended time periods. Since actual hardware is not being evaluated, the relative reliability of the ACS may The reaction wheel, single DOF CMG be discussed only in general terms. and two DOF CMG increase in complexity in the order listed. The CMG differ only in the extra gimbal and torquer of the two DOF CMG. Since the motors and bearings of ME ACS are likely failure areas, the addi- tional torquer of the two DOF CMG is a significant disadvantage. The constant and higher speeds of CMG make bearing failure more likely with them than with reaction wheels. The twin CMG configuration affords a limited amount of redundancy, providing failure of one wheel or 48 associated equipment does not incapacitate the other half of the controller. The twin two DOF CMG system provides further redundancy, in that control torque is available about all axes with one controller com- pletely failed. The varying complexity of the three ME devices also appears in the gathering of control logic information from the control- With the reaction wheel, wheel speed is the only parameter moni- lers. tored. For a single two DOF CMG, the control system must maintain con- stant wheel speed and monitor the angular motion of two gimbals. Therefore, despite the ability of the CMG systems to function while partially failed, a rigorous reliability analysis of actual hardware would probably find the reliability varying inversely with the complexity; i.e., the reaction wheel is the most reliable and the two DOF CMG is the least reliable. 49 SECTION V CONCLUSIONS The observations made in Sec. IV lead to the following conclusions: 1. Solar pressure torque is insufficient for attitude control but may be utilized to prevent the saturation of ME ACS. 2. Reaction spheres have theoretical advantages, but there are mechanization problems still to be solved. 3. If power is critical and response time limited, reaction wheels must be relegated to relatively small vehicles; for larger vehicles the torque needed to achieve reasonable response time requires prohibitive power from a reaction wheel ACS. 4. For still larger vehicles the two DOF CMG has significant weight and angular impulse capacity advantages, but, with its increased complexity, it is the least reliable of the three ME ACS compared. 5. Selection of a mass conservative ACS, assuming cost is not per- tinent, therefore hinges on vehicle size, required reliability and the relative priorities placed on ACS weight, power and angular impulse capacity. 50 BIBLIOGRAPHY 1. Simulator Study of Precise Atti Lopez, A. E., and J. W. Ratcliff. tude Stabilization of £i Manned Spacecraft by Twin Gyros and Pulse NASA TND-1645, Modulated Reaction Jets . Ames Research Center. 1964. 2. Yarber, G. W. K. T. Chang, J. Kukel, B. F. McKee, C. S. Smith, Control Moment Gyro A. F. Anderson, C. J. Bertrem and S„ Tarhov. Optimization Study Garrett Airesearch Manufacturing Division. NASA CR-400, 1966. , . 3. Wheeler, P. C. R. G. Nishinaga, J. G. Zaremba and H. Evaluation of a Semi Active Gravity Gradient System NASA CR-594, 1966. , . 4. Williams. TRW Systems. L. Hering, K. W., and R. E. Hufnagel. "Inertial Sphere System for Complete Attitude Control of Earth Satellites," American Rocket Society Journal XXXI (August, 1961), pp. 1074-1078. , 5. and M. H. Smith. "Capabilities and Limitations of Ormsby, R. D. Reaction Spheres for Attitude Control," American Rocket Society Journal XXXI (June, 1961), pp. 808-812. , , 6. "The Control Moment Gyroscope," Dohogne, J. R., and R. F. Morrison. Sperry Engineering Review (Spring, 1965), pp. 33-40. 7. Reaction Wheel with Brushless D. C. Motor Drive . Casaday, W. M. NASA CR-388, 1966. Sperry Farragut Company. 8. Bell, M. W. J. "An Evolutionary Program for Manned Interplanetary Explorations," Journal of Spacecraft and Rockets IV (May, 1967), pp. 625-630. , 9. Purser, P. E., M. A. Faget and N. F. Smith. Manned Spacecraft Engineering and Design . New York: Fairchild Publications Inc., ; 1964. 10. Merrick, V. K., and F. J. Moran. The Highly Coupled System - A Gen eral Approach to the Passive Attitude Stabilization of Space Vehicles . Ames Research Center. NASA TND-3480, 1966. 11. Roberson, R. E. "Torques on a Satellite Vehicle from Internal Moving Parts," Journal of Applied Mechanics XXV (June, 1958), pp. 196-200. , 12. Cannon, R. H. "Gyroscopic Coupling in Space Vehicle Attitude Control Systems," Journal of Basic Engineering LXXXIV (March, 1962), , p. 13. 81. "Control Moment Gyros in Attitude D., and D. J. Liska Control," Journal of Spacecraft and Rockets III (September, 1966), pp. 1313-1320. Jacot, A. u , 51 APPENDIX A GRAVITY GRADIENT TORQUE The important aspects of gravity gradient torque are illustrated by a model consisting of two spheres of equal mass connected by a massless rod and subject to an inverse square gravitational field. Notation: m,= nu = m = mass of rod connected spheres M = mass of attracting body G = universal gravitational constant T = gravity induced torque Newton •s law: 5 hQn ~ P. ^ F - ££x 2 T=4[F s,N fo-'O-F.siN(0**,j * - h. SINfo-Q 1 Jl 2 F2 h SIN0 rsiN<t> > SIN(0+fi,) F,\rSlNJ <IN$ - jGM»*J?rSlN<t 1 y; The gravity gradient torque is zero when orientation, except for the unstable 52 is zero. For any other = 90° position, a torque exists. This example may be generalized for an arbitrary body as follows. A body subject to an inverse square gravitational attraction experiences a torque tending to align its axis of minimum moment of inertia with the gravitational field. 53 I x APPENDIX B EQUATIONS OF ANGULAR MOTION Symbols used only in this appendix: m- /J- = mass of i'th particle = position vector from system center of mass to i'th particle The reference coordinate system is Cartesian with origin at the system center of mass and axes colinear with system principal axes. The system is assumed rigid except for rotation of ACS devices. f>c - Xi + Vl T + I k z-L UJ = LU* I f W y J + LUz k Definitions: Ix y - 1 y x - ~ L Ixz - Izx = - /7yi L X-L yc 4 /m lyz -Izy = - \rrni ff = I /? X /W: L Xi 2 L yi Zj. ?i Equation derivations: <\ = + lux^ it Of H-I/w L j RyJ + zj)u; x -x £ X'l^x f(xj +-f-/£ - UJXPl foe a rigid body =0 +z*)wy -y£ y£ ou y - z u>k] . tf=LxU/xI ilyylUyT + Izz J UUz k 54 £ 2 • u>*j -»-pz. *, ua -2 L yL ujy ^x- fy^Jkj (1) Newton's Second Law as applied to angular motion: Combining Eqs. T= and (2): (1) [ixx LUx + (Izz -Jyy) UUy life] I + [iyyWy *(Iwf-I«)lMiU&] 7 + Jlzz U/ 2 + Clyy "Ixx) bUx UJQ (3) it Equations for system composed of vehicle and ME ACS T^ = = H Hv f H R . (4) ; where T E is torque external to the system as opposed to internal torque generated by interaction of system components, E^ and H^ are the angular momenta of the vehicle and wheel, respectively. In the absence of exter- nal torque: H — — = HV ¥ H r = COHSTMT * * Tc = H v = -HK (5) where T c is control torque on the vehicle generated by the ME ACS. Combining Eqs. Tc=where S from \JJ (2) and (5): " Jt\. SxH * (6) > is the absolute angular velocity of the ACS spin axis; when the spin axis moves relative to the vehicle. S Since moment of inertia of the rotating element about its spin axis, is 55 differs I R , the typically much larger than the element's moment of inertia about its other two principal axes: HP where ^ \JJ K IR (SL+0JK) (7) } is the component of vehicle angular velocity about the spin axis of the ACS. For a reaction wheel or reaction sphere, using Eqs. (6) and (7): = -I, ->s-o (8) Jit 5 For a control moment gyro (CMG) , =0 _Q. » [/Je l using Eqs. (6) and (7) Tc = -SXH* = -I R (s xsl) (9) 56 APPENDIX C BRUSHLESS D.C. MOTOR DRIVEN INERTIA WHEEL BUILT BY SPERRY FARRAGUT COMPANY FOR NASA. General description . is hermetically sealed. The motor is bidirectional. The entire unit Conventional commutation is replaced by photo optical detectors and transistorized switches thus avoiding physical contact between commutator and armature. Energy saving regenerative braking is employed; voltage generated in armature windings affords complete control of wheel when decelerating. When the counter EMF decreases to where it can no longer produce required armature current, the system is automatically switched to a "driving mode." Motor wheel characteristics . Total weight - 13.7 lb Size - 1 Power required - 40 watts max Control torque - 0.65 ft-lb @ 0-250 rpm Friction torque - 0.023 ft-lb HR - 1 I - 1.25 lb-ft - 0-550 R RPM ft x 1 ft x % ft ft-lb-sec @ 250 rpm 2 The contractor offered the following reliability prediction for the motor-wheel combination, considering all components in series and assuming any component failure to be a complete failure: RELIABILITY 1000 HRS 1 YR 3 YRS Motor and wheel actually constructed: 98% 84% 59% Identical device with high reliability parts: 99% 89% 71% 57 The preceding reliability estimate is based on the following power level operation: 5 peak power 1 per cent of the time, half of peak power per cent of the time and 6.9 per cent peak power 94 per cent of the time. 58 APPENDIX D THEORETICAL, MAGNETICALLY SUSPENDED AND TORQUED REACTION SPHERE [4] Spherical shell radius 9.8 in Spherical shell thickness .2 in Spherical shell material aluminum Spherical shell weight 23.4 lb SI @ maximum T 54 rad/sec Maximum torquirtg power 18 watts Maximum T c .103 ft- lb Torquing coils weight 66.2 lb Effective suspension "spring constant" 3 5.72 x 10" lb/in Suspension power 8.4 watts Suspension coils weight 11.1 lb Total weight excluding housing and electronics 101 lb Total power 26.4 watts 59 INITIAL DISTRIBUTION LIST No. Copies 1. Defense Documentation Center Cameron Station Alexandria, Virginia 22314 20 2. Library Naval Postgraduate School Monterey, California 93940 2 3. Commander, Naval Air Systems Command Department of the Navy Washington, D. C. 20360 1 4. Chairman, Department of Aeronautics Naval Postgraduate School Monterey, California 93940 2 5. Dr. Cameron M. Smith Department of Aeronautics 1 Naval Postgraduate School Monterey, California 93940 6. LT David S. Gilbreath, USN Route 1, Box 440 Oak Harbor, Washington 1 7. Dr. Allen E. Fuhs Department of Aeronautics 1 Naval Postgraduate School Monterey, California 93940 60 UNCLASSIFIED Security Classification DOCUMENT CONTROL DATA (Security claeeitlcstlon ol ORIGINATING ACTIVITY I. title, • R&D body ol mmetrmct and indexing annotation mull be entered whan the overall report (Corporate author) 2«. i» cleat 11 Led) REPORT SECURITY CLASSIFICATION UNCLASSIFIED Naval Postgraduate School Monterey, California 26 CROUP REPORT TITLE 3. MASS CONSERVATIVE ATTITUDE CONTROL SYSTEMS FOR INTERPLANETARY SPACECRAFT DESCRIPTIVE NOTES (Type 4- o/ report anrf Inchielve date*) Thesis, M.S. in Aeronautical Engineering, Sept 1967 3- AUTHORfSJ CLmet nam*. Hret name, Initial) GILBREATH, David S. REPORT DATE « 7a. TOTAL NO. OF RACES 58 • a. ORiaiNATOR'S REPORT NUMBERfSJ September 1967 $m. b. CONTRACT OR GRANT PROJECT NO. OTHER REPORT tfiia AVAILABILITY/LIMITATION NOTICES *ni 11. Hi ' -* OP REFS 13 NO. 9b 10. 7*. NO. £ j ii i M in yj^nt^^oj^fm^i m SUPPLEMENTARY NOTES 12. NO(S) (A ny other number* that may be aatigned report) * mM SPONSORING MILITARY ACTIVITY Naval Air Systems Command IS. ABSTRACT Attitude control requirements for interplanetary space are discussed, Spacecraft attitude control systems, excluding mass expulsion devices, are described. Solar pressure, reaction spheres, reaction wheels and control moment gyros are analyzed as sources of control torque. A comparison is made between the reaction wheel and two types of control moment gyros on the basis of weight, power consumption, momentum absorption capability and reliability. The control moment gyros are shown to be the most promising for larger vehicles. DD FORM 1 JAN 64 1473 UNCLASSIFIED 61 Security Classification UNCLASSIFIED Security Classification KEY WO RDS RO L E WT Attitude control systems Control moment gyro Reaction wheel * a* '.: DD AT.. 1473 S/N 0101-807-682 (back) . *i*S4»-,*i(^.. tanujt»-< «*-i-< *•»* w . ,4m-- **» mi UNCLASSIFIED Security Classification 1 62 A-31 409 thesG424 ^Pi! Y KN0X LIBRARY miiiiinii " 3 2768 004148064 Dudley knox library" m mmn Mm m BIBB IIMB mwBR MHiWK Hiii mi - 1HH mm m :: iiH MKwii > '. , NftHffi mn IB m Rljoffi '« IK ill XK» fflM MSHK NHN m 1

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