US 3695554 A Abstract A dual-spin spacecraft having a de-spun platform is arranged to vary the torque on the stabilizing wheel by means of a motor whose speed is varied in accordance with a signal representing nutation motion to damp or attenuate the nutation motion substantially to zero very rapidly. By arranging the mass distribution such that significant cross products of inertia exist, the effects of such products are utilized in a closed loop control system to effect the desired attenuation or damping.
Description (OCR text may contain errors) United States Patent Phillips NUTATION DAMPING IN DUAL-SPIN SPACECRAFT [72] Inventor: Kevin John Phillips, Hightstown, [73] Assignee: RCA Corporation, New York, NY. [22] Filed: Oct. 16, 1970 1211 Appl. No.: 81,450 [52] US. Cl. ..244/15 A 51 lm. Cl. ..B64c 58 Field of Search ..244 15 A, 77 ss, 77 F [56] References Cited UNITED STATES PATENTS 3,350,033 10/1967 Goldberg ..244/15 A 3,547,381 12/1970 Shaw ..244/77 ss Oct. 3, 1972 lorillo ..244/15 A Lannan ..244/15 A Primary Examiner-Richard E. Aegerter Assistant Examiner--Douglas D. Watts Attorney-Edward J. Norton 57 ABSTRACT A dual spin spacecraft having a die-spun platform is arranged to vary the torque on the stabilizing wheel by means of a motor whose speed is varied in accordance with asignal representing nutation motion to damp or attenuate the nutation motion substantially to zero very rapidly. By arranging the mass distribution such that significant cross products of inertia exist, the effects of such products are utilized in a closed loop control system to effect the desired attenuation or damping. 11 Claims, 8 Drawing Figures PATENIEnom m2 3.695.554 sum 3 0r 4 g INVENTOR. KiWA/JPH/L/PJ NUTATION DAMPING IN DUAL-SPIN SPACECRAFI BACKGROUND OF THE INVENTION 1 Field of the Invention This invention relates to orbiting spacecraft, and more particularly to such spacecraft that have a rotating body which is spin stabilized, such spacecraft being of the class known as dual-spin spacecraft. 2. Description of the Prior Art With the advent of the use of satellites as valuable tools for the compilation and transmission of data in the fields of communication, navigation, weather forecasting, and observation stations in general, ithas been found that in conjunction with these varied and multiple uses a need exists to launch larger and larger satellites which can perform such multiple functions. In accordance with these developments there has also been a need to stabilize all satellites of any size within closer tolerances and greateraccuracy than has been achievable in the past. In many applications it is desirable that the spacecraft be maintained in a predetermined orientation or attitude with respect to a given reference direction, such as the direction of the gravity vector, with high accuracy. A particular type of satellite suitable for accurate stabilization with very. small pointing errors is the dualspin spacecraft which consists of a platform, the angular position of which is stabilized relative to the earth by the reaction torque generated by changes in speed of a motor-driven spinning member termed variously, momentum wheel, attitude control wheel or stabilizing wheel. Such a wheel shall be designated herein a momentum wheel and the platform shall be referred to as the de-spun platform as such a portion is now commonly designated in the art. Basically, a spinstabilized satellite exhibits certain types of troublesome motions called wobble, precession, or nutation. All such motions tend to result in a displacement of the statellites geometric axis from its intended mission orientation or attitude. N utation, the coning motion of the pitch or spin axis about the total angular momentum vector, may result from any of the following disturbances: (I) second stage booster nutation, (2) operation of the separation equipment, (3) operation of the magnetic control com ponents, (4) bombardment by micrometeorites, and (5) operation of payload components with uncompensated momentum. In general, and as to be used herein, nutation is rotational motion about either or both of the transverse (non-spinning) axes, which causes a rotational coning motion of the pitch (spinning) axis about the total or resultant momentum vector of the spacecraft. The rate of the coning motion is termed the nutation frequency (w,,). The cone angle of such motion is the amplitude of the nutation which is a measure of the pointing error referred to above. In addition to nutation, the satellite may be rotated about the pitch axis relative to a reference plane or vector such as the gravity vector or the perpendicular to the earth's surface. Such a rotation is termed the pitch error designated 0,,. Nutational stability indicates the manner in which a satellites stabilizing system controls or responds to nutational motions. In certain systems and under certain conditions nutation can increase after the original torque has been removed. In an undamped system, nu- tational motions will continue without increasing or decreasing. When the nutation decreases, it is said to be damped. It is conventionally accepted that nutational stability is critically dependent on various parameters of a spacecraft including the moments of inertia, cross products of inertia, angular momentum, roll (or yaw) angle rotations, and the pitch servo loop system. In conventional design procedures nutation may be reduced by energy absorbing or momentum transfer devices operable on either or both of the transverse axes to attenuate the nutation resulting about the pitch axis. a g Active dampers overcome nutation effects on the pitch axis be developing a torque of opposite phase to that of the nutation torque. Such active dampers may be the form of propulsion jets or a spinning wheel in addition to the stabilizing wheel of the de-spun system such as disclosed in application Ser. No. 612,209, filed Jan. 27, 1967, now U.S. Pat. No; 3,591,108, issued July 6, 1971 by Harold Perkel and William Comerford (assigned to the same assignee as the present application). In conventional designs of dual-spin satellites, the axis of spin of the momentum wheel is selected to be colinear with one of the principal axes of the satellite, the principal axes being defined in this art as the axes about which the products of inertia vanish. In practice some cross products remain in spite of the most careful design and manufacture techniques. The effect of cross products of inertiais the production of a torque about an orthogonal axis relative to a disturbing torque by the coupling effect of the asymmetrical mass distribution. Thus, in a dual-spin spacecraft, any change in speed of the momentum wheel causes a nutation about the axis of the momentum wheel, the amount of which depends on the cross products of inertia of the spacecraft and the amount of speed-changing torque applied to the wheel. Accordingly, good conventional design procedures of both the mass distribution of thesatellite as well as the stabilizing control loop, dictate minimizing if not eliminating, the coupling effect into the spin axis of the cross products of inertia. I have discovered that this coupling effect nevertheless can be utilized to produce a remarkable improvement in attenuating nutation. Accordingly, it is an object of this invention to provide means to rapidly attenuate nutation of the spin axis of a dual-spin spacecraft. It is a further object of the invention to provide means to attenuate nutation about a spin-stabilized axis of a dual-spin spacecraft solely by changing the angular velocity and thus the driving torque of the stabilizing wheel. SUMMARY OF THE INVENTION According to the invention an attitude stabilized dual-spin spacecraft is provided with a spinning wheel whose rotation axis is not parallel to a principal axis so that cross ,products of inertia between the spin axis and an axis at right angles to the spin axis of the wheel, also known as a lateral axis exist. A sensor responsive to lateral motion about a lateral axis develops a signal representing the lateral motion. A closed loop control circuit including a motor for spinning the wheel is arranged to respond to the sensor signal to vary the torque and thus the speed of the motor to attenuate or dampen the lateral motion towards zero at the same time as it controls the motion about the spin axis. IN THE DRAWING FIG. 1 is a diagram in perspective of a spacecraft showing the three axes thereof as related to the momentum and velocity vectors; FIG. 2 is a diagram illustrating the convention of the orbital axes and the angular relationship thereto, of the spacecraft axes; FIG. 3 is a diagram showing the geometric relationship of an orbiting spacecraft and its orientation'relative to the earth illustrating particularly the geometry of a rotating horizon sensor; FIG. 4 is a block diagram illustrating the components that may be incorporated in a spacecraft according to one embodiment of the invention; FIG. 5 is a block diagram of the pitch control loop of a spacecraft embodying the present invention; FIGS. 6 and 7 are Nichols graphical plots of the typical phase-gain characteristics of the pitch control loop showing a separate plot of the open loop responses M(s) and f (s) in FIG. 6, while FIG. 7 is a combined plot thereof; FIG. 8 is a typical plot showing optimum conditions. DETAILED DESCRIPTION According to the invention a dual-spin spacecraft in which a platform usually but not necessarily carrying the system electronics is de-spun to an earth reference or inertial space reference, while a mass in the form of a flywheel or momentum wheel coupled to the platform is rotated at a substantially large angular velocity to provide stabilization about a selected axis of the spacecraft, preferably the pitch axis, colinear with the axis of the flywheel. The mass distribution of the spacecraft is arranged so that cross products of inertia of the platform exist between at least one of the lateral axes and the pitch axis, the pitch axis being colinear with the axis of rotation of the momentum wheel. Thus, the pitch axis is not colinear with any of the principal axes of the spacecraft, it being understood that the principal axes define axes about which no cross products of inertia exist. The magnitude and polarity of the respective cross products of inertia are selected so that changes in torque and thus speed of the momentum wheel in response to a signal representing the nutation induce counter torques on the non-spinning axis to attenuate their motions which cause the nutation of the pitch axis. The control signal is a signal output of a device sensing rotation about a lateral axis representative, thus, of the nutational motion. This signal is used to control the speed of the motor-driven flywheel rotating about the pitch axis. The motor is included in a closedloop pitch axis control system, the gain and phase of which being selected to effect stability for all modes of operation and tolerances of the orbiting satellite including pitch error control. In general, any or all of the three design parameters, i.e. (1) the cross products of inertia, (2) the signal representing motion about a lateral axis, and (3) the phase and gain of the closed-loop pitch axis control system at nutation frequency may be arranged to effect the nutation damping desired. However, for design convenience, it is advantageous to design a system by the selection of the cross products of inertia .and the lateral-axis signal within a given closed-loop pitch control system. According to the invention the method of operating a satellite to effect nutation damping can be optimized by an analysis of the transfer function of the closedloop system in terms of the three design parameters just mentioned above. However, for a given system made in accordance with the invention the cross products only may be varied to develop a wide range of different damping time constants, any of which can be useful even if not of optimum value. Thus, for certain design needs, for example very rapid nutation damping, a satellite can be operative without any other devices of the conventional type heretofore used to control damping. However, for fail-safe purposes, a system utilizing this invention is compatible with such conventional damping devices and in such an arrangement may be considered a cooperative operation wherein the damping time of the damping device is improved by the nutation damping effected by the system of the invention. Referring to FIG. 1, there is shown a body 10 which may be a spacecraft or satellite of any suitable or desired shape. Extending from the center of mass are three mutually perpendicular spacecraft axes designated as 1, 2, and 3 corresponding to conventionally designated yaw, roll, and pitch axes respectively. The pitch (3) axis is defined to be that direction in the spacecraft l0 colinear with the total angular momentum vector H when the spacecraft 10 is operating in its intended mission. The pitch axis is parallel to the axis 12 about which a momentum wheel 14 is rotated. The sign convention is that the pitch axis, shown in FIG. 1, is positive and is in the positive direction of the angular momentum vector H. Thus, according tothe usual convention in this art, the angular momentum possessed by the spacecraft 10 is equivalent to having the body 10 spinning clockwise about the pitch axis as shown along the arrow direction 16 representing the angular velocity (0 about the pitch axis. The yaw and roll axes are mutually perpendicular and orthogonal to the pitch axis. The axis system as defined and used herein according to the usual convention is righthanded in the order 1-2-3. The 3 axis shall at times, it should be noted, be referred to herein as the pitch or spin axis. The 3 axis is parallel to the axis 12 of the spinning wheel 14. According to the invention the pitch axis (3) is selected so that there exist cross products of inertia between at least one of the lateral axes, i.e., the l or 2 axis or both, and the pitch axis. Thus, the pitch axis is selected not to be colinear with the principal axis of the satellite 10. According to the laws of motion, the spacecraft l0 includes a first (translational) motion of the center of passing through the center of mass 20 along the line about which the spacecraft is rotated. The length of the vector on is proportional to the angular speed of the spacecraft 10 about that line. The arrow head 16 of the angular velocity vector (0 indicates the direction of rotation, in this case clockwise about the line as shown. The angular velocity vector 6) can be resolved into the three components (0 m (0 the sense of each of which respectively are indicated by reference numerals 18, 17, and 16. When 6(angular velocity), H (momentum) and the spin (3) axes are not colinear both the mag nitude and direction of m in the l-2-3 coordinate system, that is, the system relative to the spacecrafts body-fixed axes, are functions of time and the spacecraft 10 is said to be nutating. Thus, in other words, as previously mentioned nutation is oscillatory about both of the lateral axes, orthogonal with the spin or pitch axis (3). The problem of eliminating or reducing nutation of a spacecraft is solved with existing techniques in the present state of the art by the use of nutation dampers of either an active or passive form to create essentially cross-coupling torques internal to the spacecraft so that the angular velocity vector 5 and the pitch axis 3 are caused to converge toward coincidence or colinearity with the total momentum vectorfiof the spacecraft 10. According to conventional procedures a satellite is usually designed with a minimum of cross products of inertia relative to the spin axis. For certain satellites, however, the mission equipment such as antenna, solar responsive power generators, and other equipments in or on the spacecraft make it impossible to provide a symmetrical configuration. The stabilization of such configurations both in respect of the pitch error as well as the nutation effects is difficult owing to the cross porducts of inertia that induce nutation. The usual pitch loop control circuits depend upon an input signal that is generated by a sensor of suitable form to determine the pitch error. One form of such sensors, namely horizon sensors, are used to develop a signal of any motion of the platform relative to the earths horizon that represents both the true pitch error as well as nutation. Such a composite signal when applied to the stabilizing closed-loop for pitch control can effect a motor control torque that for certain conditions will render the satellite unstable. To avoid or minimize the unstable modes, orientation of such sensors is made such as to minimize the sensitivity to nutation motions by scanning the horizon portions wherein minimum sensing of the nutation motion is achieved. In this way signals representing nutation motions are minimized to the control loop, thus minimizing the lateral torques demanded through the products of inertia. Nutation is damped by a passive or active damper independently of the pitch closed-loop control system. According to the present invention cross products of inertia are utilized. With such products of inertia, as will be described, a given closed-loop pitch control system will alter the speed of the momentum wheel to attenuate the nutation in response to a signal representing the angular motion about at least one of the transverse or lateral axes. According to the invention the lateral axes angular sensing means may be a horizon sensor having a component output signal representative of both the pitch error and the nutation. Ad- vantageously the larger the signal component representing the nutational motion the better the response characteristic of the pitch loop system to control nutation. Thus, according to the invention, extremely rapid nutation damping is accomplished simply by controlling the torque and thus the speed of the momentum wheel as a function of the sensor signal in a satellite having appropriately selected cross products of inertia and its error sensor oriented to develop a significant signal representing the nutation motion. As will be explained, an optimum damping time is obtained by the combination of optimized cross products of inertia from each or both of the transverse axes, and a sensor oriented to measure lateral or nutation motion. As previously explained, the invention may be utilized in a satellite wherein the only control for stabilization is the momentum wheel. In other applications where a damper is included in the satellite, whether active or passive form, the total damping time of nutation is improved in compatible operation. In such systems, the damper performance requirements may be reduced to thereby simplify the costs and weight of such devices. A better understanding of the manner of selecting the appropriate parameter values according to the invention will now be described in greater detail. Referring now to FIG. 2 there is shown a diagram of several of the parameters that will be used in the description to follow. The 1, 2, and 3 axes described above with respect to FIG. 1 are shown in their relative positions in FIG. 2, it being understood that these axes are identical to the axes heretofore described. The local vertical vector or direction 22 is colinear with the desired reference direction for orienting the spacecraft yaw axis (1). The vector 24 indicates the direction perpendicular to the orbit plane of the spacecraft. The line 26 represents the position of the yaw axis if the spacecraft were to have a yaw angle ll! (psi) and a roll angle qb (phi) but no pitch angle 6,, (theta) relative to the orbital coordinates 22, 24, and 23. The line 28 similarly represents the position of the roll axis, with a yaw angle all and a roll angle but no pitch angle 0,. The angles as shown by the several vectors and directions are defined as follows: 0,, is the spacecraft pitch error, defined as the angle between the yaw axis and the plane formed by the pitch axis and the local vertical 22; (l) is the spacecraft roll angle defined as the angle between the pitch axis and the plane formed by the orbit normal (24) and velocity vector 23; 4; is the spacecraft yaw angle defined as the angle between the orbit normal 24 and the plane defined by the pit ch axis 3, and the local vertical 22. 6,, d; and r1: are the time differentials of the respective angles adopting the convention designation for such differentials. Referring now to FIG. 3 the satellite 10 is shown oriented to scan the horizon 32 of the earth 34 by means of a wheel mounted sensor (not shown). One form of horizon scanner which senses the combination of pitch axis angular error 0,, and a function of the lateral motion, in this case, a function of the roll error will now be described. A sensor of any of the known types is mounted on the spacecraft 10 such that its line of sight is at an angle a from the positive spin vector 3 of the wheel 14. The sensor line of sight is made to rotate around the wheel spin axis at the same rate as the wheel, by either actually mounting the sensor on the wheel or by mounting the sensor on the spacecraft and using a suitably oriented mirror on the wheel. As the wheel rotates, the sensor line of sight 35 traces out a cone about the spin axis, whose half cone angle is equal to (11' a) or a radians depending upon whether a is greater or less than 1r/2 radians. The sensor scan geometry is illustrated in FIG. 3. Once per wheel revolution, the scan line intersects the sky-earth horizon 32. At that instant of time, the change in radiance level detected by the sensor is arranged by electronic means, to generate an electrical pulse usually called the horizon pulse. A device, called an encoder, is mounted on the spacecraft which is so designed to give a second pulse, called the index pulse each time a radial reference line on the wheel passes a given reference point on the despun platform. This spacecraft reference point is so arranged that the horizon pulse and index pulse are time coincident for a correctly oriented spacecraft. Under normal operation the measured pitch error 0,; is determined electronically as a voltage proportional to the ratio of the time lag between two successive horizon pulses, multiplied by 360. It can be shown that is tan 00S tan a sin a sin where p. is half of the angle subtended by the earth at the spacecraft and is a constant for a given altitude. For small disturbances, the expression for G can be rewritten as where dB sec cos t tan sec 5 d sin ,8 tan a sin {3 sin a (4) The expression (3) shows that the sensed error 0,; is equal to the true error 0,, plus the roll error 0, multiplied by the function (LB/dd). Provided that the roll angle variation due to nutation is not large, the value of dB/dd) can be assumed constant, evaluated at the predefined sensor angle a, and the average roll angle, which under normal operation would be zero. Since, depending upon the value of a selected, dfildd) can take any value from minus to plus infinity, as is apparent from an analysis of equation (2), any desired ratio of the lateral motion d) to the pitch axis motion 6,, can be derived for the sensed signal, 6 Referring now to FIG. 4 there is shown in block form the components of spacecraft 10 that are needed to both pitch stabilize the spacecraft and correct as well for nutation motions. Roll sensor 42 responds to angular rotations about the roll axis and generates an electrical signal representing the roll angle. The sensor may be of any of the conventional devices which can be utilized to measure an angular error such as an accelerometer or rate gyroscope suitable calibrated to generate the required electrical signal. Horizon sensor 43, mounted on wheel 14 generates a signal relative to the horizon and a reference on the spacecraft to indicate the pitch angle or pitch error 0 The electrical signal is carried'over lead 52, shaft leads not shown, slip ring 54, lead 55 to a summing and shaping amplifier 44. According to a preferred form of the invention the horizon sensor 43 may be arranged to provide a complex electrical signal representing both the pitch error 0,, and the modified roll dB/dd) d) as explained above, which signal would accordingly be 0 as defined by equation (3). It should be noted that for such an arrangement the sensor 42 is not needed since its function would be redundant. It should thus be appreciated that any form of error sensor may be used to sense the pitch error and the roll error, the roll error being representative of the nutation motion. In the alternative, a yaw error may be used if desired in lieu of the roll error, or the roll and yaw error may be used jointly. The signal output of the sensor 42 is conducted to the shaping and summing amplifier 44 by conductor 56 for correcting the wave shape of the sensor output. The output of the summing and shaping amplifier 44 is conducted via conductor 57 to a power amplifier 46 which develops a voltage to operate motor 15 over conductor 58. Motor 15 may be powered by alternating current or direct current excitation. The power amplifier 46 depends upon its electrical power needs from the power subsystem of the spacecraft converted to whatever form that may be necessary such as a.c. or do for the motor 15, the motion sensor 42, via conductor 42a or sensor 43 via conductors (not shown), and the shaping and summing amplifier 44 via conductor 57. Such power arrangements are well known in the art and accordingly need not be described in detail herein. The input axis of sensor 42 is at right angles to the axis 12 of the attitude momentum wheel 14. The momentum wheel 14 functions to stabilize the platform portion 1 l of the spacecraft 10 in a de-spun relation to the rotating momentum wheel 14. In this manner, as now well known, the platform 11 may be arranged to carry optical viewing devices for a fixed line of sight to a selected spacial body such as the earth, sun, or a star. Devices are provided to correct angular errors about the yaw and roll axis. Such devices, as heretofore known and forming no part of this invention, may include a current carrying wire coil array 48 suitable positioned in the spacecraft to develop external torques on the spacecraft, by developing the necessary magnetic fields to react with the earths magnetic field in response to ground commands. Such external torques will cause the spacecrafts total angular momentum vector H (see FIG. 1) to be moved to any desired position. Thus, the orientation of the spacecraft is controlled to a selected attitude by such ground controlled systems while the speed of the motor 15 is regulated to correct errors from the desired reference attitude. Any of the known forms of nutation damping devices may be utilized in conjunction with the invention to serve as a supplement to the nutation damping system of the invention. Thus, for example, a passive damper 50 of conventional form utilizes a viscous fluid in a closed conduit the actual relative size, it is noted, is much larger than shown. Such dampers, as well known, develop viscous friction in response to nutation movements and thus function to dissipate the nutation energy to dampen the nutation usually over some defined but usually extremely long period of time. An active damper, indicated by block 52, functions in response to signals from a nutation motion sensor 30 of any suitable type to generate a torque in opposition to the nutation torque about the spin axis. Such an active damper and its associated sensing and control apparatus is disclosed in the previously mentioned copending application of Harold Perkel and William Comerford filed Jan. 27, 1967, Ser. No. 612,209 now US. Pat. No. 3,591,108. The closed pitch control loop of the system is shown in FIG. 5, wherein the components are arranged in their respective relationships in the loop. The left-hand portion 60 of the diagram represents the hardware just described with respect to FIG. 4, while the right-hand portion 62 of the diagram represents the dynamics and kinematics of the satellite and their functional relation in the control loop. The motor torque T developed by any change in voltage from the power amplifier 46 will induce in the spacecraft angular motions about all three of the body axes. If there are no cross products of inertia between the spin axis and the transverse axes 2 or 1, no precession torques and thus no motions will be induced about the transverse axes. However with cross products of inertia, a torque T from motor 15 can induce a motion about the transverse axes l and 2 such as to attenuate any angular rotation that may have previously existed. Block 64 represents the spacecraft 10 in its orbiting state and, particularly, the cross products of inertia suitably selected to induce the attenuation effect of the invention. Any torque T will cause the pitch axis 3 to rotate by a certain angular rotation from a desired reference, this rotation being the pitch error 0,, previously described. The component of the sensed pitch error contributed by nutational motion is the roll error (it multiplied by a factor K, plus the yaw angle I11 multiplied by a factor K Alternatively, the component could be measurements of the time derivatives of the roll and yaw errors, or the lateral body rates (0 and ('0 or their time derivatives. The three components are combined at the junction 66 into the pitch error in differential form (6}) and integrated at block 68 (l/s being the conventional form of an integrator) to the sensed pitch error 0, It should be understood that the loop portion 62 is merely an analytical schematic showing the action that occurs when a torque T is developed by the motor 15. In actual fact, the mere change in speed of the motor Causes the rotation of the pitch axis to change the error 6 In order to understand better the invention, an explanation of the mathematical relationships of the system components will now be described. The use of equations defining the spacecraft motion may be helpful in an understanding of the system of this invention. The angular velocities of the spacecraft have been previously described in connection with FIGS. 1 and 2 and the several angles relating the spacecraft or body axes to the orbital axes relative to the earth have also been described in addition with respect to FIG. 3. The total angular momentum (FIG. 1) is related to the component angular momenta according to the following relation: The relationships of the moments of inertia and the cross products of inertia to the momenta are related as follows: H H and H are the magnitudes of the angular momenta about each of the body axes; I 1 and 1 are the moments of inertia about each of the axes 1, 2 and 3 respectively; I, is the moment of inertia of the flywheel; 1 1, and 1 are the cross products of inertia between the respective axes identified by the respective subscripts; w, is the angular speed of the flywheel; (0 m and m and their respective differentials ai 0'1 and (b are the magnitudes of the angular velocities and the angular accelerations (rate of change of velocity) about each of the respective body axes. The sensed pitch error 6 which is the error fed to the control loop is, according to the invention arranged to be either one of the following two [9 a and 9 b relationships: K is any given constant number related to the desired multiplication or cross coupling factor of the roll angle qb and K is a similar factor for the yaw angle. It will be noted that equation (9a) is equal to equation (3) when the general term K is substituted for dflldd) and K is zero. It should be noted also that q) and 11 could be represented in equation (9a) as their first derivatives or double (second) derivatives with appropriate selection of the constants K and K to give the desired result. A still further alternative would be to substitute the time derivatives of (0 and an in equation (9b). The general equations representing the motion of the spacecraft as a function of motor torque (T) are nonlinear and not easily applicable for spacecraft design. For practical applications certain assumptions can be made which will allow the nonlinear equation to be modified into linear relationships that are accurate within the limits of the approximations stated. These assumptions are: l. The angles of true pitch error 6,,, roll error (1), and yaw error ill, are small, typically less than 10; 2. The velocity of orbit (about the earth for example) of the spacecraft is small compared to the angular velocity of the spacecraft body (w,, (a m) such that the effect of the orbit velocity on the equations of motion is negligible; 3. The angle of nutation motion, usually termed the half-cone angle 0,,, is small, less than say 4. The total spin axis momentum H is large compared to the momenta H,, H about either or both of the transverse axes. With these assumptions the momenta H, and H, are small compared with H, which is a substantially constant value. The transfer function relating sensed pitch error 0 to the motor driving torque, T, in linearized form is: where 1,, is (1 1,) the moment of inertia about the spin axis corrected for the moment of inertia effect of the flywheel (l4); s is the LaPlace operator; where f (s) represents the modification to the transfer function due to the cross coupling introduced by the products of inertia. At zero average roll and pitch error, with no nutation damping device on the spacecraft, and with K, set at zero, f (s) can be shown to be approximately: 2 1 a (1131 1) u 22 11, K, IHII3+IHI23J For a system without any existing cross coupling of the products of inertia from one axis into another whereby the products of inertia are zero, i.e., for I 1 0, equation l l reduces tofl(s) l and equation (10) reduces to It can be shown, based on equations (10) and (12) that the open loop transfer function (M[s ]for a pitch control system arranged to respond to cross coupling of the cross products of inertia is where M(s) is the open loop transfer function without cross product effects and f (s) is as defined above. It is known from the principles of control theory that the characteristic equation of the closed loop system is lM(s)f (s)=O (14) Equation (l4).can be factored in general as follows: where 2, and Tare coefficients which are functions of the amplifiers, shaping networks, motors, etc. as well as the products of inertia and in fact the entire dynamics of the spacecraft, as known in this art; r is any positive integer, a is any number including zero, f,(s) is a polynominal in LaPlacian form; j is any position integer less than r. The numerator of equation (15) determines thev stability of the system including the nutational stability 12 provided the f (s) factor is included in its derivation. There will be one quadratic factor in the numerator of the expression which will dominantly determine the spacecraft response to nutational motions or disturbances. This quadratic factor is defined as: l 2;, T s T s and the factor is determined as: where for i a j for all i values from 0 to r, (u being the nutation frequency as described above. The nutational frequency (m of a spacecraft with cross products of inertia can be shown to be approximately equal to The nutation damping time constant T can now be determined from expression l7) by For a system in which a nutating damping device is operative in conjunction with the system of this invention the total or resultant damping time constant 1 can be approximated from the following equation: where 1,, is the damping time constant for the system with the damping device, but, it should be noted, with no cross products of inertia. GENERALIZED DESIGN PROCEDURE FOR DETERMINING SPACECRAFT CONFIGURATION FOR OPTIMUM DAMPING TIME For a given spacecraft configuration with given moments and products of inertia about the l, 2 and 3 axes, and a pitch control loop, designed to give adequate stability to the pitch loop, without cross coupling to the yaw or roll axis, there exists a particular ratio M of K, to K, in equation (9) that will give optimum nutation damping to the spacecraft. Should the ratio of K, to K, be anything other than M, nutation damping may still be derived, but to a lesser and lesser degree as the ratio diverges further and further from M. Alternatively, for predefined values of K, and K (wherein either K, or K, may be zero for example) there exists a particular ratio of 1, to 1 in the spacecraft that will give optimum damping. This ratio can be called N, and should the ratio of 1,, to 1 be anything other than N, nutation damping will be degenerated in the manner just described for the ratio M. The design aim therefore is either a. starting from a spacecraft with given products of inertia, the value of M should be found and the values of K and K chosen such that their ratio is as near as possible to this desired ratio. b. starting with a spacecraft with a given nutational sensing system (such that K, and K are fixed) the value of N should be found, and the values of I and 1 ad justed in the spacecraft such that their ratio is as near as possible to this desired ratio. (a) or (b) can be carried out by means of any computer simulation that can determine the spacecraft nutational time constant for a given set of parameters. For (a) M can be found by making a series of such computer runs with different values of K, and K with the constraint that 1/K +K K where K is held constant for all of these runs. That ratio of K and K in the run that gives the shortest nutation damping time constant will be M. The value of M can be utilized in the design of a spacecraft by either using two sensors measuring nutational motion on the roll (2) and yaw (1) axes with gain factors of K and K as previously described, or by utilizing a single nutation sensor, oriented along an axis in the roll (2) and yaw (1) plane, the axis being at an angle of tan"(K /K from the positive yaw axis and measured in a positive sense towards the positive 2 axis. The gain factor of this single sensor would be K K1 l-K as previously described, and its output is then equal to the combined output of the above described yaw and roll sensors. This single sensor would therefore reduce the amount of equipment necessary in sensing the appropriate nutation motion by a factor of 50 percent. For (b) N can be found by making aseries of computer runs with different values of I and 1 with the constraint that V1 1 I where I is held constant for all these runs. That ratio of 1, to 1 in the run which gives the shortest nutation damping time constant will be N. Having thus optimized the nutation damping system, any given value of nutation damping time constant can be derived for the spacecraft by simply increasing the l4 =tan (123/1 3) (23) Thus, any specified value of 1 and 6 (from 0] to 360) uniquely defines I and 1 as 2:; sin0 25) The spacecraft parameters of an example illustrating an application of these principles are: I 421 lb. in. sec. Yaw axis moment of inertia I 460 lb. in. sec. Roll axis moment ofinertia 1 366 lb. in. sec. Pitch axis moment of inertia l,,,. 5.0 lb. in. sec. Product of inertia l, 4.46 lb. in. sec. Flywheel spin inertia H 140 lb. in. sec. Spacecraft momentum 0.32 radians/sec. Nutation frequency The optimum angle 6 for I,,, is 70. This optimum angle corresponds to the following values of the cross products of inertia: I13= 1.294 lb. in. sec? 1 ,=4.s29 1b. in. sec? These optimum design parameter values give an optimum nutation damping time constant of l 1 minutes. It was also found that a time constant between 11 and 15 minutes is derived or all values of 6 in the range from 30 to 105. Conversion to products of inertia using the yaw (1) and roll (2) shows that a time constant better than 15 now to FIGS. 6 and 7, typical frequency response plots value of K or I (within the practical limits of the system) while still maintaining the correct value of M or N. According to a computerized model, calculations of damping times for various combinations of cross products of inertia of a specific spacecraft were made based upon being held a constant, and K l and K 0. iar ia 23 between that axis and the positive 3 axis. Such an axis is at an angle 0 from the positive yaw (1) axis, measured in the direction of the positive roll (2) axis and is defined as of the general transfer functions M(s) and f (s) are shown. FIG. 6 shows typical plots of M(s) and f (s) for open loop frequency responses showing the phase shift in degrees along the abscissa and the gain or loss in DB (decibels) along the ordinate; The coordinates of the M(s) response at the nutation frequency (ca are the gain -G- DB, at phase (180- The two frequency response plots are shown combined in FIG. 7. It is known from servo mechanism theory that instability occurs if the f, .(s) loop surrounds the critical point where there is a unity gain (0 DB) and a phase shift of -l80. By varying the value of the lateral motion sensor gains K and K as described above, or by varying the relative values of the cross products of inertia I and 1 the size and position of the loop portion f (s) can be varied relative to the curve M(s) about the nutation frequency coordinate w Thus an infinite number of loops can be plotted of possible designs. I have discovered that the optimum damping time in general may be determined when the values of K and K or 1 and 1 are so adjusted to give an f (s) loop on the combined response, such that the line bisecting the f (s) loop and passing through the frequency point to is a straight line continuation of the line joining point 80 (0 DB phase) and thenutation frequency coordinate point ca as is illustrated in FIG. 8, where the op- .15 timized f (s) loop is shown represented. Line 86 represents the bisector of the loop, and is shown as a straight line continuation of line 84. It will be now appreciated that in accordance with this invention a dual-spin spacecraft can be provided with means to attenuate nutation motion substantially to zero very rapidly. By arranging the mass distribution such that significant cross products of inertia exist, the effects of such products are utilized in the closed loop to effect the desired damping by providing an error signal corresponding to the lateral motions representing the nutation. Optimum damping time constants are determined by a unique combination of the cross products for a given nutational motion sensor axis or for a unique sensor axis for a given set of cross products. A feature of the invention is the utilization of the stabilizing wheel of a dual-spin spacecraft whose spin axis is not coincidental with the principal axes, by regulating the speed of the wheel in response to signals representing the nutation to reduce the nutation itself to zero, such signals thus being utilized to couple the effects of the cross products into the closed loop of the pitch control system. What is claimed is: l. A closed loop control system for pitch error control and mutation damping in a dual spin satellite comprising in combination: a platform despun to a selected position reference, a spinning momentum wheel coupled to the platform by means of a motor drive, the common axis of spin of the platform and momentum wheel being the pitch axis of the satellite, the principal axis of the momentum wheel about its center of mass being colinear with the pitch axis, the principal axes of inertia of the platform being noncoincident with respect to said pitch axis, the center of mass of said platform being on the pitch axis, said satellite having two lateral axes perpendicular to each other and orthogonal to said pitch axis, the mass distribution of the satellite thereby being such that cross products of inertia exist between at least one of said lateral axes and said pitch axis, means for sensing angular motion about at least one of said lateral axes, means for sensing the pitch error represented by angular rotation of said pitch axis relative to said selected position reference, means coupled to said angular motion sensor and pitch error sensor for developing a control signal representing the output of both of said sensing means, a closed loop control system for controlling the orientation of the attitude of the platform about the pitch axis by regulating the torque of said motor, and means in said closed loop control system responsive to said control signal for altering the torque of said motor about one of said lateral axes, so that pitch error is reduced sub-stantially to zero and angular motions about either of said lateral axes causing coning of the pitch axis are attenuated rapidly. 2. A control system according to claim 1 including damping means operatively disposed in the satellite to dissipate nutational motion about said spin axis. 3. In an attitude stabilized dual-spin spacecraft comprising a de-spun stable platform anda spinning wheel, he improvement comprising orienting said spinning wheel for spinning about a non-principal axis of the platform whereby cross products of inertia about said axis exist, sensor means responsive to lateral motion about a lateral axis for producing a signal representative of said motion and responsive to errors in pitch angle from the pitch axis for producing a signal representative of pitch error; a closed loop control circuit including a motor for spinning said wheel, and means included in said control circuit responsive to said signal for varying the speed of said motor to attenuate said lateral motion and pitch error towards zero. 4. The spacecraft of claim 3 wherein said sensor means includes one or more rate gyroscopes for sensing said lateral motion. 5. The spacecraft of claim 3 wherein said sensor means includes one or more accelerometers for sensing said lateral motion. 6. The spacecraft of claim 3 adapted for use as an earth orbiting spacecraft wherein said sensor means includes a horizon sensor for producing a complex signal representing both said pitch error and said lateral motion of said spacecraft. 7. The spacecraft of claim 3 wherein the damping time T of nutational motion about the spinning axis is determined by equation (20) as defined and described in the specification, equation (20) being of the form r c/lc- 8. The invention of claim 7 wherein the damping time is approximately 1 1 minutes, the cross products of inertia being as follows: 1 l.3 lb in sec. I 4.8 lb in sec. and the constant K =+l .0, K, 0, said damping time and said parameters being an optimum to satisfy equation l l as set forth and described in the specification, wherein equation l l) is the transfer function modifier f( 9. The invention according to claim 3 including damping means disposed in the spacecraft to dissipate nutational motion about the spinning axis to supplement said speed control means. 10. A method of stabilizing an orbiting spacecraft of the dual-spin type having a wheel de-spun from a stabilized platform said wheel spinning about the spin axis of the spacecraft, comprising the steps of: a. adjusting the mass distribution within the spacecraft so that cross products of inertia exist between at least one of the lateral axes and the spinning axis; b. sensing the lateral motion of the spacecraft about an lateral axis, 0. generating a signal corresponding to the sensed lateral motion for controlling the spinning rate of the spinning wheel. 11. A method according to claim 10 wherein the lateral axis is perpendicular to the spinning axis. . UNITED STATES PATENT OFFICE CERTIFICATE OF CORRECTION PATENT NO. 1 3 695 554 DAT D I October 3 1972 |NVENTOR(S) Kevin John Phillips it is certified that error appears in the above-identified patent and that said Letters Patent are hereby corrected as shown below: In drawing Figure 7, "+360" should read 360 Signed and Sealed this Nineteenth Day of December 1978 [SEAL] RUTH C. MASON DONALD W. BANNER Arresting Oflicer Commissioner of Patents and Trademarks UNITED STATES PATENT OFFICE CERTIFICATE OF CORRECTION Patent No. 3 ,695 .554 Dated October 3 1972 Inventor(S) Kevin John Phillips It is certified that error appears in the above-identified patent and that said Letters Patent are hereby corrected as shown below: Column 2 line 17 change Y'be" to by- Column 5 line 37 correct spelling of "products"; Column 7, line 57 change "6" to Column 8, lines 10 and 61 change respectively "suitable" to --suitably-; Column 10 line 21 the first occurrence of the equals sign, viz, in equation (7) should. be a plus sign, viz Column ll line 30 the term 2 2 S S H H in equation 11 should be H? Column 13, line 27 change "tan (K /K to read ---tan (K /K Signed and sealed this 3rd day of April 1973 (SEAL) Attest: EDWARD M.FLETCHER,JR. ROBERT GOTTSCHALK Attesting Officer Commissioner of Patents FORM POAOSO (10-69) uscoMM-Dc 60376-P69 U.5, GOVERNMENT PRINTING OFF' CE: I969 0-4664 Patent Citations
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