Attitude Dynamics and Control of a Dual-Body Spacecraft using Variable-Speed Control Moment Gyros

Transcription

1 Attitue Dynamics an Control of a Dual-Boy Spacecraft using Variable-Spee Control Moment Gyros Abstract Marcello Romano, Brij N. Agrawal Department of Mechanical an Astronautical Engineering, US Naval Postgrauate School, Monterey, CA, USA The ynamics equations of a spacecraft consisting of two boies mutually rotating aroun an axis are erive. One boy contains a cluster of single-gimbal variablespee Control Moment Gyros. The equations inclue all the inertia terms an are written in a general form, vali for any cluster configurations an any numbers of actuators in the cluster. A non-linear control law is esigne for the spacecraft attitue an joint rotation. An acceleration-base steering law is use for the variable-spee Control Moment Gyros. The analytical results are teste by numerical simulations. A three-axis test-be was esigne an is in the final set-up phase. Introuction The ynamics an control of multi-boy spacecrafts with gimballe momentum exchange evices is a challenging problem. Control Moment Gyros (CMG s) are unique among attitue control actuators because they can provie high output torque without using expanable fuels an can provie a level of precision an continuity unachievable with jet thrusters. Inee, Control Moment Gyros have been use for ecaes on space stations an on military spacecrafts when fast slewing capability an high pointing accuracy were require. The use of Control Moment Gyros is currently uner consieration also for several future civil spacecrafts requiring high agility [1]. A main rawback to using Control Moment Gyros is the presence of singular gimbal-angle configurations at which the CMG s cluster is unable to prouce the require torque. [], [3] Many previous stuies have consiere the problem of the ynamics an control of spacecrafts using single-gimbal Control Moment Gyros. In particular, Oh an Vaali [4] report the complete equation of motions for the case of a single-boy spacecraft an also consier the CMG s transverse an gimbal inertia; moreover, they introuce a non-linear feeback control law an a singularity robust steering law. Schaub, Vaali an Junkins [5] an For an Hall [6] propose the use of variable-spee Control Moment Gyros (VS-CMG s), which a extra egrees of control to the classical CMG evices an may overcome the gimbal angle singularities while maintaining the output torque equal to the requeste one. In the present paper, the use of variable-spee Control Moment Gyros is analyse for a spacecraft consisting of two rigi boies, which can mutually rotate aroun a common gimbal axis. This high-level moel represents the Bifocal Relay Mirror

2 Spacecraft, which is uner investigation at the Naval Postgrauate School an other institutions. The main mission of the Bifocal Relay Mirror spacecraft, consisting of two mechanically an optically couple telescopes, is to reirect a laser signal between two istant points. The receiver telescope captures the incoming energy from the laser source, while the transmitter telescope irects the laser beam at the target. Previous analytical-numerical stuies have been conucte with the objective of performing preliminary simulations of the ynamics an control of the overall spacecraft using Reaction Wheels (RW s) an incluing a moel of the optical subsystem [7]. The present paper summarizes the analytical research in progress on the use of Variable-Spee Control Moment Gyros [9] an presents the current status of the parallel experimental research effort which intens to valiate the analytical-numerical results through experiments on the groun [8]. Analytical moel of the system ynamics In this section, we report the equations of motion, erive with the Euler-Newton metho, for a ual-boy spacecraft containing a cluster of VS-CMG s. See [9] for the etails on the erivation of the equations of motion. The spacecraft moel use for the erivation of the equations is shown in Figure 1a. The boies C an D, in this conceptual moel, represent the transmitter an the receiver telescopes of a spacecraft for the relay of optical signals. The receiver telescope rotates with respect to the transmitter aroun an axis that contains, as a esign hypothesis, the centre of mass of the receiver telescope itself; therefore, the centre of mass of the overall system oes not change uring the relative rotation. In Figure 1a, O c, O an O b are respectively the centre of mass of the transmitter telescope, the receiver telescope, an the overall spacecraft. A frame with vectrix [10] Fc is fixe w.r. to the transmitter telescope: c 1 is the optical axis of the telescope an c is parallel to the rotation axis between the two telescopes. A frame with vectrix F is fixe with respect to the receiver telescope: 1 is the optical axis of the receiver telescope an is the relative rotation axis. F is rotate with respect to F c of an angle β aroun. Finally F b is parallel to F c but centre in the centre of mass of the overall spacecraft. The transmitter telescope is suppose to contain N single-gimbal VS-CMG s. In summary, our ynamic moel has a total of (N + 7) egrees of freeom. The equation of motions of the system are written in the following compact form, which is suitable for the esign of the control law. [ ] & ( ) J ω J ω ω A B&& D D D & E & F t A & B && β D D & β & tot = tot Ω Ω+ e β + 3 (1) 3 3 T T where Jtot R = JN + m c rc 1 rc r c + m r 1 r r + ( Cb J C b ), m c an m being respectively the mass of the transmitter an of the receiver telescopes an r c = Fb Oc Ob, r = Fb O Ob.

3 Moreover it is efine: with N 3 3 N R = B N + bg i ( r g ) i gb i; + bgi = si gi ti i= 1 J J C J C C a a a () 3 1 T N 1 T N N 1 N ; δ1 δ N ; L & L 1 L N A R = A A R = & & R = δ δ B R = B L B ; D R = D L D, i = 1,, 3 3 N 3 N 1 N i 1i Ni 3 N N 1 T 3 N 1 L N ; 1 L N ; 1 L N E R = E E Ω R = Ω Ω F R = F F T [ ] [ ] A R = C µ J µ = a I + a I ; µ = ; µ = i bgi ( r+ g) i ti ( r+ g) i si ( r+ g) i 1 B R = C J µ = a I + a I + a I i bgi ( r + g ) i si ( r + g ) i gi ( r + g ) i ti ( r + g ) i D R = C µ J µ Ω = a I + a I Ω i1 bgi ri 1 i ti ri si ri i D R = B ω ; D R = C µ J C C J µ C ω i i i 3 bgi ( r + g ) i gbi bgi ( r + g ) i gbi E C J µ a a a F E ω i R = bgi ri 1 = si Iri + gi Iri + ti Iri ; i R = i T (3) a) b) Figure 1: a) High level moel of the Bifocal relay mirror spacecraft b) Four single-gimbal VS-CMG in a pyrami configuration an a si is the unit vector irecte as the rotor spin axis of the i th CMG, a gi is irecte as the gimbal rotation axis, an a ti = a si a gi is irecte as the torque prouce by the VS-CMG. D an D 3 are obtaine from the D 1 an D 13, efine in Eq. 3, by replacing C bg with C b an J ( r+ g) with J. Moreover, the equation of motion of the overall receiver telescope aroun the gimbal axis is given by:

4 ( && T T T T ) ( ) & { } I β + µ & ω + I & ω + I & ω + µ C J C ω ω + D + D β = u (4) b b 3 ij where I inicates the element ( i, j) of the inertia matrix J,, an u, asie the friction, is the torque acting on the receiver telescope ue to the joint motor between the two telescopes. Equations 1 an 4 completely escribe the rotational ynamics of our moel. Noteworthy, the contribution of the receiver telescope to the equations of motion is analogous to the contribution of a single-gimbal CMG with the inertia an mass characteristics of the receiver telescope an fixe rotor. Design of the control law an steering law This section introuces the feeback control law for rotational manoeuvres of the ual-boy spacecraft with N variable-spee Control Moment Gyros, as moelle above. The non-linear feeback law propose here is an extene an improve version of the law propose in [4], an use as a base in [5]. The feeback law is extene in orer to enforce the regulation of the aitional state variable β, which is the angle between the two telescopes. Moreover, the feeback law is improve by exploiting the antisymmetry of some of the matrix factors of the equations of motion, as iscusse in [9]. Lyapunov s irect metho is use for the control law esign. It is assume that estimates of the current state variables of the system ( ω, q, & ββ,, Ω, ) are available in real time. The target state is given by ω f, q f, & β, β f f with free values of an Ω. Let V be the following Lyapunov s function: T 1 T 1 1 V = kq q q+ ω J tot ω+ I ( & β ) + kp β ( β) (5) where k q an k p β are positive constants an we efine the error term q = q q f, an analogously for ω, & ω, β, & β, && β. As it is emonstrate in [9], for V & to be negative semi-efinite an the target state to become asymptotically stable, it is sufficient that the following two conitions are satisfie: where it has been efine B + && D + & EΩ= & t req ; u = u (6) req D3 + D3 f D3 + D3 f D= D1+ D f + ; D = D f + (7) with: ( ) T treq = K ω k q Qq q f J tot & ω f + [ J tot ω] ω f A & F f Ω+ te A & β B && β D & β (8) ( && T T T T β &) & & ( ) f 1 3 ureq = kβ & β kpβ β + I + µ ω + I 1 ω+ I 3ω + µ b b ω ω+ + 3 β C J C D D & These are the esigne control laws. The first control law in Eq. (6) oes not contain the physical control torques of the gimbals an rotors explicitly. Only gimbal an rotor accelerations an gimbal rates appear. In orer to satisfy the control law, a steering law is typically exploite. Starting from the require torque, the steering law etermines the require value of Ω & an the require value of either & (gimbal rate steering law) or && (gimbal acceleration steering law). The rotor

5 an gimbal motors are then commane to track these require values. In our simulations, an acceleration steering law was use. In fact this provies more realistic results because it takes into account the inertia aroun the gimbal axes. In particular, for the case of variable-spee CMG s we use a moifie version of the acceleration steering law introuce in [5], which the reaer is referre to for a etaile explanation. The law is given by: µσ Ω& w 3 s0 e 1 0 T T N N N = ; R [ ]; R µσ 0 k 1 WQ QWQ t req Q = E, D W = && δ & & 0 wg 1 e 1 (9) where 1 is the NbyN ientity matrix, k & δ is a positive T 11 constant, σ = et( DD )/( I r Ω0) is the singularity inex, which is zero at the singular sets of gimbal positions, Ω 0 is the nominal spin rate, an w s0, w g, µ are positive constants. W is the weight matrix for the pseuo-inverse operation. Far from singularity, the factor e µ σ is approximately zero, an it approaches "one" near a singularity. Then, while far from singularity, the require torque is provie by && an &, an when a singularity is approache, the require torque is provie by Ω &, which is otherwise close to zero. Therefore, this steering law can effectively overcome the singularity conition an track the require torque. For the case of simulations with Control Moment Gyros ( Ω & = 0 ) we use the singularity robust steering law introuce in [6] an given by: T ( α ) 1 0 T = && k 1 D DD + e µσ 1 t & & δ req (10) where α 0 is a positive constant. This steering law provies singularity robustness by moifying the output torque with respect to the require torque, near singularities. The Bifocal Three-Axis test-be The Bifocal three-axis test-be has been esigne by the authors with the objective of simulating on-the-groun the attitue ynamics, etermination an control of a two-boy spacecraft like the Bifocal Relay Mirror. The test-be was built by the companies Guiance Dynamics Corporation an Automate Control Environments an is currently being integrate an set-up at the Naval Postgrauate School. The Bifocal Three-Axis test-be is shown in Figures an 3 an its main specifications are reporte in Table 1. The test-be consists of three main parts: the spherical air bearing, the spacecraft bus simulator, an the optical payloa ecks. The zero-g attitue motion is simulate by free floating the spacecraft moel through the spherical air bearing. Inee, a virtually torque-free environment is achieve if the centre of mass of the spacecraft moel coincies with the centre of rotation of the air bearing. The mass of the spacecraft simulator is suitably istribute to coarsely obtain the neee balance. Moreover, three ballast masses are move on mutually-perpenicular linear stages (incluing motors an encoer

6 sensors) for the automatic fine tuning of the balance. The spherical air bearing for the simulator has a 10-inch iameter an requires approximately 4 atm to float the simulator. The main spacecraft bus elements of the simulator are liste here below: 1 three Variable-Spee Control Moment Gyros mounte in a pyrami configuration with changeable base angle; a Northrop-Grumman-Litton LN-00 Inertial Measurement Unit, with three fibre-optic rate gyros; 3 two custom-esigne attitue sensor using lasers to provie high-accuracy vector measurements an initialise the attitue etermination Kalman filter (simulating star-trackers); 4 two Seika inclinometers proviing meium-accuracy attitue measurement aroun pitch an roll axes within a wie range; 5 ata hanling subsystem incluing several signal conitioning an control stations an one PC104 computer with input/output car an wireless Ethernet connection for telemetry an telecommaning; 6 power subsystem supplying regulate voltages to the ifferent on-boar components: the test-be can operate with both external an on-boar power supply. Two ecks are mounte on top of the spacecraft simulator an are available for the mounting of the optical payloa. The two ecks can mutually rotate through a hollow shaft motor. The optical payloa will consist essentially of two telescopes an two fast steering mirrors, an will allow simulation of laser relay operations through the three-axis test-be. Floating platform, mass ~ 600 Kg Floating platform, motion range 360 eg (yaw), +/- 30 eg (pitch an roll) Floating platform, overall imensions Max iameter:.05 m Max height:.75 m VS-CMGs, max torque CMG moe: 1 Nm, RW moe: 1.1 Nm VS-CMGs, max angular momentum RPM IMU, gyroscope bias ~5 eg/hr (1 sigma) IMU, banwith ~500 Hz Laser base attitue sensors, accuracy ~1 arcsec Laser base attitue sensors, banwith ~100 Hz Laser base att. sensors, fiel of view ~1 eg Inclinometers, accuracy ~1 arcmin Inclinometers, banwith ~0 Hz Inclinometers, range +/- 30 eg On-boar computer, real-time OS Xpc target (Mathworks) On-boar power supply Lea-aci batteries Table 1: Main specifications of the Bifocal Three-Axis test-be

7 Figure : Design schematics of the Bifocal Three-Axis test-be Figure 3: Picture of the current status of the Bifocal Three-Axis test-be (the top optical eck, temporarily taken off the simulator, is partially shown sitting on the floor on the left of the picture)

8 J c (transmitter inertia) iag[88, 997, 3164] kg m J (receiver inertia) iag[183, 171, 1560] kg m m c ; m (transmitter & receiver 67 ; 973, kg masses) r c ; r (centre of mass position) [-0.7, -0.49, 0] ; [0.63, 1.15, 0], m β p (VS-CMG pyrami base angle) eg J (r+g) (VS-CMG total inertia) iag[0.7, 0.135, 0.135] kg m J r (VS-CMG rotor inertia) iag[0.45, 0.1, 0.1] kg m k q, k p, k q (control gains) 35, 10, 6.4, Nm K (control gains) iag[616, 705, 881] Nms k & δ, ω g, ω s0, µ, α 0 (steering law gains) 50, s -1, 1, 1,.1,.01 Ω 0 (nominal rotor rates) Table : Characteristic ata of the moel an values of the parameters use for the numerical simulations Simulation results [1, 1, 1, 1] ra/s 0, & 0 (initial gimbals state) π/4 [1, -1, -1, 1] ra, [0, 0, 0, 0] ra/s q 0, ω 0 (initial attitue state) [.31,.54,.64,.46], [.01,.01, -.01] ra/s β 0 (initial joint angle) 0.1 ra q f, ω f, β f (commane state) [0, 0, 0, 1], [0, 0, 0] ra/s, 0 ra The ynamics moel, control laws an steering laws, which have been iscusse above, were coe in Matlab-Simulink in orer to conuct the numerical simulations. The simulation coe was teste by checking the conservation of the angular momentum within the numerical accuracy an by re-obtaining the results of [4] an [5]. The main objective of the numerical simulations presente here was to confirm the asymptotic stability of the propose control law, in the ieal case of no external isturbances an no uncertainties. Moreover, the simulations compare the use of variable-spee Control Moment Gyros (VS-CMG s) to the use of Control Moment Gyros (CMG s). We consiere a cluster of four VS-CMG s in a pyrami configuration mounte on the transmitter-telescope sie of the spacecraft, as shown in Fig. 1b. The characteristic ata of the moel use in our simulations, the use control parameters are presente in Table. An attitue regulation control case was consiere as sample of the attitue slewing performe by the Bifocal Relay Mirror Spacecraft to acquire the laser source an target. For all the actuators, the same control law was use, with zeroe Ω & in case of CMGs.

9 a) Spacecraft angular velocity b) Spacecraft attitue (Euler s param.) c) Spacecraft joint angle an torque ) Actuator spin rates Figure 4: Simulation results, use of VS-CMGs compare to use of CMGs for attitue regulation Figures 4 an 5 report the results of the simulations. As it can be seen in Figures 4a, 4b an 4c, the propose control law is stable an performs satisfactorily well, both with use of the VS-CMG s an the CMG s. Figure 5c shows that a singularity is encountere aroun five secon of the manoeuvre. The VS-CMG s perform better than the CMG s in overcoming the singularity. Inee, the VS-CMG s exploit a variation of the wheel spin rate of about 0 ras/s, as in Figure 4. Therefore, the singularity is overcome while maintaining the total output torque near to that require, as shown in Figure 5. Also the CMG s can overcome the singularity, thanks to the use of the singularity robust steering law, but their output torques significantly fluctuate near the singularity, causing a corresponing fluctuation in the gimbal rates, as can be seen in Figure 5a. These fluctuations, beyon increasing the power consumption, are especially critical for flexible spacecrafts an jittersensitive payloas.

10 Aitional simulations, supporting the results presente here, are reporte in [9]. a) Actuator gimbal rates b) Actuator gimbal angles c) Singularity inex ) Require an obtaine torque Figure 5: Simulation results, use of VS-CMGs compare to use of CMGs for attitue regulation Conclusions The ynamics equations of the motion have been written for a spacecraft moel consisting of two rigi boies connecte by a rotational joint: one of the boies contains a generic number of variable-spee Control Moment Gyros in a generic cluster configuration. All the inertia terms have been taken in account. A new nonlinear control law has been introuce to comman the spacecraft attitue an joint isplacement. A moifie version of an existing acceleration-base steering law has been use. The results obtaine in the simulations with four actuators in a pyrami configuration confirm that the feeback law performs well, in both the regulation an the tracking control.

11 An avance three-axis spacecraft simulator has been esigne an is currently being set-up in our laboratories. This test-be is going to be use to experimentally valiate the results presente here an to simulate on-the-groun other critical aspects relate to the laser relay spacecraft. References [1] Girouart, B., Sebbag, I., an Lachiver, J., Performances of the Pleiaes-HR agile attitue control system, Proceeings of the ESA International Conference on Spacecraft Guiance, Navigation an Control Systems, Frascati, Italy, 00. [] Margulies, G.,an Aubrun, J., Geometric theory of single-gimbal control moment gyro systems, The Journal of the Astronautical Sciences, vol. 6, no., pp , 1978 [3] Berossian, N, Paraiso, J., Bergmann, E. an Rowell, I., Reunant singlegimbal moment gyroscope singularity analysis, Journal of Guiance, Control, an Dynamics, vol. 13, nogimbal control moment gyro systems,. 6, pp , 1990 [4] Oh, H., an Vaali, S., Feeback control an steering laws for spacecraft using single gimbal control moment gyros, The Journal of the Astronautical Sciences, vol. 39, no., pp , [5] Schaub, H., Vaali, S., an Junkins, J., Feeback control law for variable spee control moment gyros, AAS Avances in the Astronautical Sciences: Spaceflight Mechanics, pp , [6] For, K., an Hall, C., Flexible spacecraft reorientations using gimbale momentum wheels, The Journal of the Astronautical Sciences, vol. 49, no.3, pp , 001. [7] Romano, M., an Agrawal, B.N., Acquisition, tracking an pointing control of the Bifocal Relay Mirror Spacecraft, Acta Astronautica, Vol.53, No.4, pp , 003. [8] Agrawal, B.N., Romano, M., an Martinez, T., Three-axis attitue control simulators for bifocal relay mirror spacecraft, Proceeings of John L. Junkins Astroynamics Symposium, College Station, Texas, 003. [9] Romano, M., Agrawal, B.N., Attitue ynamics an control of a ual-boy spacecraft with variable-spee control moment gyros, Journal of Guiance, Control, an Dynamics, to appear in September 004. [10] P.C. Hughes, Spacecraft attitue ynamics, John Wiley & Sons, New York, 1986.

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