Guidance, Navigation, and Control Conference and Exhibit August 5 8, 2002/Monterey, CA

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1 AIAA Guidance, Navigation, and Control Conerence and Exhibit 5-8 August 22, Monterey, Caliornia AIAA AIAA Tracking and pointing o target by a Biocal Relay Mirror Spacecrat using attitude control and ast steering mirrors tilting Marcello Romano and Brij N. Agrawal Spacecrat Research and Design Center Naval Postgraduate School, Monterey, CA Guidance, Navigation, and Control Conerence and Exhibit August 5 8, 22/Monterey, CA For permission to copy or to republish, contact the copyright owner named on the irst page. For AIAA-held copyright, write to AIAA Permissions Department, 8 Alexander Bell Drive, Suite 5, Reston, VA, Copyright 22 by the author(s). Published by the American Institute o Aeronautics and Astronautics, Inc., with permission.

2 Tracking and pointing o target by a Biocal Relay Mirror Spacecrat using attitude control and ast steering mirrors tilting Marcello Romano and Brij N. Agrawal Spacecrat Research and Design Center Naval Postgraduate School, Monterey, CA The present paper reports the results o numerical simulations carried out on a model o the Biocal Relay Mirror spacecrat. This spacecrat consists o two large mechanically and optically coupled telescopes, used to redirect a laser beam rom a ground-based, airborne or spacecrat based source to a distant target point on the earth or in space. The two telescopes are gimbaled and the spacecrat inertia has a large variation during the angle maneuvers needed to maintain the laser cross link. Moreover the spacecrat has very tight pointing and jitter requirements. The task o the presented simulations was to preliminarily validate and compare two dierent control approaches proposed or the tracking and pointing o the target o the Biocal Relay Mirror. The attitude control system consists o reaction wheels, star trackers and rate gyros. The optical control system consists o two two-axis ast steering mirrors and two optical tracker sensors. In the irst control option considered, eedorward and PD eedback are used or the spacecrat attitude control, while PID eedback is used or the optical subsystem, in order to compensate the pointing error. In the second control approach, the spacecrat and the optical control systems are integrated. Nomenclature h Angular momentum vector T Torque vector I Moment o inertia dyadic ω Angular velocity vector x, y, z Cartesian axes coordinates O Cartesian axes origin i, k Indexes β Relative angle o a mirror with respect to its base ξ Damping ratio ω n Natural requency t Internal torque i Versor j Moment o inertia A Subscripts S, S/C Spacecrat R Receiver portion o the spacecrat Ph.D., US National Research Council Associate Fellow. AIAA member. Proessor and Director. AIAA Associate Fellow. Copyright c 22 by Marcello Romano and Brij N. Agrawal. Published by the American Institute o Aeronautics and Astronautics, Inc. with permission. T w m rel Transmitter portion o the spacecrat Reaction wheel Mirror Relative B Acronyms BRM Biocal Relay Mirror d.o.. Degree o reedom IRU Inertial Reerence Unit FSM Fast Steering Mirror LOS Line o sight FF Feed-Forward I Introduction MANY o the near uture space missions will require high accuracy pointing and tracking o multiple targets at the same time. Applications include optical communications relay link satellites, laser sensors or ormation lying leets o space probes and several other civil and military uses. Optical requencies provide extremely high antenna gain or relatively small antenna size, thereby allowing cross links to be closed with relatively low transmitter power and small terminals. Moreover, in case o relay satellite, the laser sources are on the grounds o American Institute o Aeronautics and Astronautics

3 and that allow or relatively easy interchanging o lasers having dierent power or wavelengths or dierent type o utilization o the cross link. However, the extremely narrow beam-width poses severe pointing, acquisition and tracking requirements. The Spacecrat Research and Design Center o Naval Postgraduate School participated in the Biocal Relay Mirror (BRM) project, aiming to the preliminary study o a laser relay spacecrat or nonweapon military applications o laser links. A Relay Mirror space application is an application o increased diiculty with respect to the typical spacebased telescopes application. In act beyond a high line o sight stabilization capability it requires also the capability o line o site rate tracking. Moreover the oreseen Biocal Relay Mirror Spacecrat application is still more challenging than the Relay Mirror Experiment described in re. 2 In act, while that experiment used both cooperative source and target (that is, both source and target were sending a beacon beam to the relay mirror spacecrat, allowing a closed loop pointing control) the Biocal Relay Mirror Spacecrat is required to work with uncooperative targets. The Biocal Relay Mirror spacecrat is composed o two optically coupled telescopes and is used to redirect the laser light rom ground-based, airborne or spacecrat based laser sources to distant target points on the earth or in space. The receiver telescope captures the incoming energy rom the laser source, while a separate transmitter telescope, movable with respect to the other telescope, directs the laser beam at the desired target. The preliminary design o the BRM Spacecrat was carried out by students o Naval Postgraduate School under a spacecrat design course. The transmitter and receiver telescopes have Cassegrain coniguration with primary mirror o.64 meter diameter, and are gimbaled around a single axis. The transmitter telescope has the majority o the spacecrat bus subsystems, including the attitude control sensors and actuators. The spacecrat mass is 33 kg at launch and the spacecrat orbit altitude is 75 km with an inclination o 4 degrees. The mission requirements are or a 3 meters spot beam on the ground, beam jitter less than 44 nano radians and mean dwell duration per pass o 25 seconds. The spacecrat has three primary operational modes: sun bath, acquisition, and tracking. During the sun bath mode, non-operating period, the solar cell suraces o the telescopes are kept normal to the sun axis to maximize the electric power output. During the acquisition mode, both source and target points are acquired, and then pointed and tracked or the duration o the dwell. The acquisition sequence consists o pitch and roll motion o transmitter telescope to acquire the target point. The yaw motion o the transmitter telescope and the one axis motion o the receiver telescope is used to acquire the source point. The preliminary design eort identiied the need to develop several technologies to meet the mission requirements, especially in the area o ine acquisition, tracking, and pointing, and beam control optics. In particular there are several unique multibody pointing and tracking control problems in the Biocal Relay Mirror spacecrat: in act the motion o the two massive telescopes results in a continual change o the spacecrat inertia during the operation. The present paper ocuses on the problem o pointing and tracking o the target. In particular, the task o the research presented here was to perorm preliminary simulations o the dynamics and control o the overall Biocal Relay Mirror Spacecrat during the engagement operational phase, in order to validate dierent control options and compare their perormances. Beyond the dynamics and control o the spacecrat attitude, the dynamics and control o the two ast steering mirrors, constituting the tertiary mirrors o the transmitter and receiver telescopes, is taken in account. Section two o this paper introduces the dynamic model o the Biocal Relay Mirror spacecrat. Section three describes the studied control approaches. The implementation o the simulation program is described in section our. Finally, the results o the numerical simulations are reported in section ive. A II Dynamics o the Biocal Relay Mirror system Model o the spacecrat The Biocal Relay Mirror Spacecrat consists o two main bodies: transmitter telescope and receiver telescope. The receiver telescope rotates with respect to the transmitter around an axis, that contains, as a design hypothesis, the center o mass o the receiver telescope itsel: then the center o mass o the overall system does not change during the relative rotation o the two telescopes. Looking at igure the relative rotation axis is x R, while the center o mass o the receiver and o the overall system are respectively O R and O S. The other bodies considered in the dynamic model are: our reaction wheels mounted in tetrahedral coniguration on the transmitter telescope sec- 2 o American Institute o Aeronautics and Astronautics

4 z T y T O T Transmitter Telescope x T y S z S OS xs y R Receiver Telescope OR z R xr A. Motion o the center o mass It is derived in the simulation by propagation o the analytical solution or the circular orbit. A.2 Attitude motion The developed simulation program exploits the ormulation o the attitude dynamics o the Biocal Relay Mirror Spacecrat, which has been presented in re. 3 A brie outline o the model is reported here below. The attitude dynamics equations o the system, assumed to be composed o rigid bodies, can be written in the ollowing vectorial orm (see, or instance, re. 4 or the underlying theory): Laser to target Laser rom source Fig. High level model o the Biocal Relay Mirror Spacecrat tion o the system and used as actuators or the spacecrat attitude control; two ast steering mirrors, mounted as tertiary mirrors o the two telescopes. Each ast steering mirror is actuated in such a way it can rotates about two radial axes crossing its center o mass. In summary, the Biocal Relay Mirror spacecrat, as described in our model, has a total o signiicant degrees o reedom: three d.o.. or the position o the center o mass o the system; three d.o.. transmitter; or the attitude position o the one d.o.. or the position o the receiver with respect to the transmitter; two d.o.. or the position o the transmitter ast steering mirror with respect to the transmitter; two d.o.. or the position o the receiver ast steering mirror with respect to the receiver. Let us consider the equations used to simulate the evolution o the dynamics and control o the Biocal Relay Mirror. h S = T S () where h S and T S are respectively the absolute total angular momentum and the total external torque about the center o mass O S o the overall system. The time derivative on the let side o equation is carried out with respect to the inertial rame. In particular, the total angular momentum is given by the ollowing expression: 4 2 h S = I S ω+i R ω rel + I wi ω relwi + I mk ω relmk i= k= (2) where I S and I R are the moments o inertia dyadics respectively o the overall spacecrat and o the receiver about their respective centers o mass; I wi is the moment o inertia dyadic o the i th reaction wheel about its center o mass, I mk is the moment o inertia dyadic o the k th mirrors about its center o mass. Moreover ω is the absolute angular velocity o the transmitter telescope, ω rel = ω R ω is the relative angular velocity o the receiver with respect to the transmitter; ω relwi is the relative angular velocity o the i th reaction wheel with respect to the transmitter telescope and ω relmk is the relative angular velocity o the k th mirrors with respect to its base. In particular, the ourth term on the right o the equation 2 gives the eect o the motion o the ast steering mirrors on the dynamics o the spacecrat. Developing the vectorial equation 2, expressing the vectors and dyadics in the same reerence rame (we used the rame x S, y S, z S in igure ) and choosing a set o attitude parameters (we used the Euler parameters) three dierential scalar equations, describing the attitude motion o the transmitter portion o the spacecrat, are inally obtained. 3 o American Institute o Aeronautics and Astronautics

5 A.3 Motion o the receiver telescope with respect to the transmitter telescope This motion is considered to ollow the precomputed reerence angular motion, designed in order to maintain the optical link between source and target during an orbital passage. A.4 Motion o the ast steering mirrors with respect to their base Typically, a ast steering is actuated by our voicecoil actuators. These are mounted behind the mirror in a cross coniguration parallel to the two rotation axes. The voice coils are utilized in push-pull pairs in order to produce two orthogonal rotations. Then the mirror pivots around its center, as i it was mounted on a gimbal joint. Each ast steering mirror is here modelled as a rigid body connected to the spacecrat by a series o two hinges, located at the center o mass o the mirror and each possessing torsional stiness and torsional damping. With the hypothesis o considering small relative rotation angles, each ast steering mirror is modelled as a set o two decoupled linear torsional oscillators. Then the equations or each o the two ast steering mirrors have the linear orm: β xm + 2 ξ xm ω nx m β xm + ω 2 n xm β xm = t xm j xm ω i xm β ym + 2 ξ ym ω nym βym + ω 2 n ym β ym = t ym j ym ω i ym (3) where β xm and β ym are the relative angles o rotation o the mirror with respect to its base, around the axes x m and y m ; ξ are the damping ratios, ω n the natural requencies, t the torques applied on the mirror, j the moments o inertia o the mirror. Finally, the last terms on the right o equations 3 give the component o the absolute angular acceleration o the base o the mirror along the two mirror axes, which are identiied by their versors i xm and i ym. B Model o the Environmental Torques As external disturbance torques, the gravity gradient and magnetic disturbance are considered in the model. C Model o the Inertial Reerence Unit sensors Rate gyros to determine spacecrat angular rates and star trackers to determine spacecrat attitude Receiver Telescope secondary mirror primary mirror Laser rom source ast steering mirror Light rom source scene Common ocal plane Light rom target scene Laser to target Transmitter Telescope a) High level scheme o the optical subsystem Light rom target scene Light rom source scene target Pointing error laser beam b) Images on the common ocal plane: pointing error c) Compensated pointing error Fig. 2 Basic concepts o the optical subsystem o the Biocal Relay Mirror Spacecrat are considered in the model. The rate gyros s bias errors and star trackers measurement gaps have been simulated. During the measurement gap or the star trackers, rate gyros are used to determine angular rates and angular position. When the star trackers measurements are taken, using a simulated Kalman Filter, the angular position is corrected and rate gyro biases are updated. D Model o the optical subsystem Figure 2(a) gives a high-level description o the optical subsystem o the Biocal Relay Mirror spacecrat, as it has been considered or the present study. This is a basic model o the working concept: the ast steering mirrors o the transmitter and receiver telescopes convey the light respectively rom the target scene and the source scene toward a common ocal plane. Figure 2(b) shows a typical situation at the beginning o the pointing process, looking on the common ocal plane. While the laser beam is approximately in the center o the image rom the source, which is the nominal situation, there is a pointing error between laser beam and target. By suitably moving the transmitter ast steering mirror, the image o the target moves on the common ocal plane with respect to the image o the source and the error be- 4 o American Institute o Aeronautics and Astronautics

6 P?? P P P? tween the laser beam and the target is reduced, as igure 2(c) shows. Several eects are present in reality, which are not considered in our preliminary optical model: in particular the jitter due to presence o atmosphere and to the structural vibrations o the spacecrat, the need o a pointing-ahead angle due to the combination o the eects o the inite propagation time o the light across the link and the relative motion o the spacecrat with respect to source and target. Two optical tracker sensors, supposed ixed with respect to the common ocal plane, process the images rom the two telescopes, and their outputs are available to the controllers o the ast steering mirrors. In particular the source optical tracker senses the motion o the source beam relative to the ocal plane and commands the tilting o the ast steering mirror o the receiver telescope, in order to maintain the laser beam nominally in a ixed point o the ocal plane. That point corresponds to the center o the target sensor. At the same time the target optical tracker senses the motion o the target relative to the ocal plane and commands the ast steering mirror o the transmitter telescope in order to reduce the pointing error. For the study presented in this article, we ocused on the control o the transmitter portion o the optical subsystem, considering the receiver portion always working nominally and maintaining the laser beam in the center o the tracker sensor. This simpliying hypothesis is based on the act that the tracking and pointing o the source is intrinsically less demanding than the tracking and pointing o the target, because the source is supposed to be cooperative. Then, or a irst characterization o the possible system perormance, we concentrated our eort in simulating more realistically the target tracking and pointing. Nevertheless, we included both the receiver and the transmitter ast steering mirrors in our dynamics model. III Control approaches The two control approaches, described in the ollowing subsections, have been considered in the simulations: A Independent control o ast steering mirrors and spacecrat attitude Using this approach, the spacecrat attitude is controlled independently o the ast steering mirrors motion. While the spacecrat attitude is controlled by eed-orward and PD commands, sensing the attitude error with the IRU, the transmitter and receiver ast steering mirrors are controlled by a PID control, " " " "! a) Independent spacecrat-ast steering mirrors control KOK, -/ e e m m N G "&5*6( ;: #%$ &'<( *+ A 4? 324,.-/ "! #%$ &')( *+ Receiver Telescope ? B E F G B C EDC,.&5*H K ML #%$ &')( *+ =!>56 >:? ICJ b) Integrated spacecrat-ast steering mirrors control Fig. 3 The two control approaches considered sensing the target and source errors with the optical sensors. This control approach is described by the block diagram in igure 3(a), with the ollowing signiicance o the symbols: S/C is the true attitude o the spacecrat. In the reality this quantity is not known directly. In the simulations it is given by the integration o the equations o motion; S/C M is the measured attitude o the spacecrat; C S/C is the reerence attitude o the spacecrat; T arget is the attitude o the spacecrat at which the transmitter telescope axis hits the target. In the presented simulations this quantity is considered equal to C S/C ; e transm = T arget S/C is the pointing error; uncertainties, the line o sight uncertainties. They can derive rom dierent causes: motion o the target, imperect knowledge o the reerence attitude, LOS jitter. In some o the presented simulations (deined as simulations with target in uncertain position), we considered uncertainties given by a low requency pseudo-random signal, as later on explained; e LOS is the total pointing error, including the uncertainties and the correction rom the ast steering mirror. 5 o American Institute o Aeronautics and Astronautics

7 B Integrated control o transmitter ast steering mirror and spacecrat attitude Using this control approach, the transmitter ast steering mirror position relative to the structure is measured and used to control the spacecrat attitude. Indeed, the mirror position is proportional to the not corrected target pointing error. The angular position around the yaw axes and the angular rate data, or the attitude control, still comes rom the IRU sensors. As regards the receiver ast steering mirror, it is still considered controlled independently. The concept o this control approach is described by the block diagram in igure 3(b). The dashed line on the output o the target tracker indicates that, in the simulation with ixed ast steering mirror, we considered directly the output o the target as a eedback signal. IV Implementation o the simulation program The system dynamics and control laws described above has been implemented in a simulation program coded in MATLAB/SIMULINK, using as a base the sotware described in re. 3 The main simpliying hypotheses, considered in the development o the model, are summarized here below:. The error between the correct attitude and the actual attitude is considered small, in order to apply the simpliications o small angles in the related attitude kinematics equations and in order to model as two decoupled second order systems the dynamics o each ast steering mirror around its two axes; 2. The relative motion between transmitter and receiver telescope is considered ollowing the reerence relative motion without error; 3. The target and source tracker sensors are considered ideal, giving continuous output; 4. The beam jitter has not been modelled; 5. The reerence attitude proile was chosen similar to a typical slewing maneuver or the Biocal Relay Mirror, without a detailed design o the maneuver. While the irst hypothesis will maintain its validity in the real operational lie o the Biocal Relay Mirror, at least during the acquisition mode phases, the other hypotheses are considered helpul in the preliminary study carried out. The ollowing two dierent sample cases were studied in the simulations: A. Target and source in certain position: in this case the position o target and source is considered perectly known a priori ( uncertainties = ). The design o the reerence attitude motion is based on that knowledge. Thereore, in principle, the spacecrat attitude motion along that reerence motion, guarantees a perect alignment o the laser beam both with the source and target. This is an ideal condition, useul to evaluate the maximum achievable perormance. The target and source trackers sense the relative pointing errors between the target or source and the laser beam spots. In the simulations, the computation o the relative pointing error is based on the knowledge o the exact attitude o the spacecrat, obtained by the integration o the spacecrat dynamics. 2. Target in uncertain position: in this more realistic case the position o the target is not considered perectly known a priori ( uncertainties ). Such uncertainty can derive by dierent causes, as the imperect knowledge o the orbital trajectory o the spacecrat, the motion o the target with respect to the background scene, and the LOS jitter. The uncertainty in the target position has been modelled in the simulation program by adding pseudo-random signals to the target relative pointing error, output by the target sensor. The pseudo-random signals were obtained by low pass iltering the output o a white noise signal. On the contrary, the source position is still considered perectly known. V Simulation results Value o the main simulation parameters A. Integration parameters The simulation time period is 5 seconds. The simulation solver method is ode5 (Dormand-Prince), and the solver ixed step size is.5 seconds. A.2 Dynamics parameters Altitude o the circular orbit: 75 Km. Mass o the transmitter telescope: m = 2267 Kg, mass o the receiver telescope: m 2 = 972 Kg. Transmit telescope inertia: I xt x T = 2997 Kg m 2, I yt y T = 364 Kg m 2, and I zt z T = 882 Kg m 2 ; receiver telescope inertia: I xr x R = 72 Kg m 2, I yr y R = 56 Kg m 2, and I zr z R = 83 Kg m 2. For both the transmitter and receiver ast steering mirrors: j xm = j ym =. Kg m 2. Natural requency around both x and y axes: ω = ω nx m n y = Hz. Damping m ratio around both x and y axes ξ xm = ξ ym =.. 6 o American Institute o Aeronautics and Astronautics

8 A.3 Optical parameters Magniication actor o both telescopes: 8.2. Field o view or both telescopes: 5 e 4 rad. This ield o view corresponds to a circular ootprint area with radius o 375 m on the earth surace, when the telescope axis is perpendicular to the surace. A.4 Disturbance parameters The secular torques magnitude is e 4 Nm. The control law delay or initial determination errors is 3 seconds. A star tracker measurement gap is considered in the period between and 3 seconds. The rate gyros static rate biases are e 4 [,.5, ] rad/sec. The initial errors are set: or quaternion [.8,.2,.8] and or angular rate [.,.,.2] rad/sec. Control gains or the PD control o the spacecrat, k = [5, 35, 225] and k d = [, 2, ]. For the reaction wheels, the maximum allowable torque is Nm and the maximum angular momentum is Nm/sec. Gains or the PID control o the transmitter ast steering mirrors: k = [6, 6], k d = [, ], k i = [5, 5]. B Reerence attitude motion o the spacecrat The proile o the reerence attitude motion is shown in igures 4. The proile resembles the maneuver required to maintain transmitter and receiver telescopes pointing the source and target during an overhead pass to conduct laser relay operations. The majority o the maneuver is perormed in the spacecrat pitch axis, q 2, as both telescopes orient to point at ixed ground sites. The largest relative angle, that in reality is based on the ground site location o target and source, is considered about 3 degrees during a near overhead pass. C Outline o the results The independent and integrated control approaches, described in paragraph III, were applied to both the cases with certain and uncertain knowledge o the target position, described in paragraph IV. This combination o tests has been carried out with the ollowing two conigurations o the system:. Ideal IRU sensors output, continuously available; this constitutes a reerence case; 2. Realistic IRU sensors output and Kalman ilter. Figures 5 report the pseudo-random signals added to the target sensor output, during the simulations with uncertain knowledge o the target position. In particular, Figure 5(b) gives the simulated view o the target track relative to the common ocal plane, looking through the telescope. In other words that q q 2 q 3 q time [s] a) Transmitter telescope reerence rotation in quaternions α [deg] time [s] b) Relative rotation angle o the receiver telescope with respect to the transmitter Fig. 4 The reerence motion o the spacecrat is the simulated view o the target as captured by the target tracker sensor. The laser beam spot rom the source on the common ocal plane corresponds to the center o the square area in the igures and each side o the square corresponds to the ield o view o the telescope. This kind o igure was considered the most eective method o representing the results in the ollowing. Figures 6 and 7 represent the results o the simulations in the case o system with ideal IRU. The results are represented ater the initial 3 seconds, during which the attitude control is switched o, while the initial determination is carried out. The case o uncontrolled target error (i.e. mirror ixed) is compared to the case o target error controlled by tilting the ast steering mirror. Moreover the case o independent spacecrat-ast steering mirrors control 7 o American Institute o Aeronautics and Astronautics

9 is compared to the case o integrated spacecrattransmitter ast steering mirror control. For the simulations o igures 7 the position o the target was considered uncertain, with the meaning speciied in paragraph IV. Figures 8 and 9 represent the results o the simulations in the case o system with realistic IRU sensors and Kalman ilter. For the simulations in igures 9 the position o the target was considered uncertain. Figures represent the time history o the track error, or the case in igures 9. The picks in the error around 3 seconds, in igure (a), are due to the end o the measurement gap in the star tracker. In the simulations with ideal sensors and uncertainties in the target position (igures 7), the integrated control worsens the perormance with respect to the independent control. On the contrary in the case o realistic IRU sensors (igures 9), the integrated control improves the perormance with respect to the independent control. An explanation or these acts is that in both cases the target sensor is considered ideal and giving continuous reading. The uncertainties are added to the ideal output o the target sensor. Then, while in the irst case the target sensor output does not improve the reading o the IRU sensor, but spoils that reading because o the added uncertainties, in the second case it does improve the reading o the IRU sensor, that is aected by the biases o the rate gyros. Aknowledgment This work was carried out while Dr.Marcello Romano was holding a National Research Council Research Associateship Award at the Spacecrat Research and Design Center. The work o Lt.Chris Senenko on a previous version o the simulation sotware is grateully acknowledged. Reerences G.Baister and P.V.Gatenby, Pointing, acquisition and tracking or optical space communications, electronics and communication engineering journal, vol. 6, pp , December J.F.Sullivan, J.E.Anspach, and P.W.Kervin, Relay mirror experiment and wideband angular vibration experiment, program summary, Ball Aerospace Systems Group, B. Agrawal and C. Senenko, Attitude dynamics and control o biocal relay mirror spacecrat, in AAS/AIAA Astrodynamics Specialists Conerence, (AAS Paper -48, Quebec City, Canada), August 2. 4 P.C.Hughes, Spacecrat attitude dynamics. New York: John Wiley & Sons, 986. VI Conclusions Simulations have been carried out on a dynamics model o the Biocal Relay Mirror Spacecrat in order to preliminary validate and compare two dierent proposed control approaches. In particular, an independent control o ast steering mirrors and spacecrat attitude and an integrated control o transmitter ast steering mirror and spacecrat attitude have been investigated. Based on the results obtained in the simulations, it appears that the best solution to control the Biocal Relay Mirror system is probably a trade-o between the independent spacecrat-ast steering mirrors control and the integrated spacecrat-transmitter ast steering mirror control. That is a control that uses a usion o the data rom the target and source tracking sensors and the data rom the IRU sensors, weighted on the base o the knowledge o the characteristics o the speciic sensors. Further studies can be carried out by developing the dynamics model and the simulation program in order to overcome the hypotheses taken in account in the present study and summarized in paragraph IV. 8 o American Institute o Aeronautics and Astronautics

10 ixed mirror ξ u x [rad] x ξ u x [rad] x time [s] a) Pseudo-random signals over time x 3 detail x x 4 detail 2 2 x 4 = ov/2 = 2.5e 4 rad a) Case with independent spacecrat-ast steering mirror control ixed mirror = ov/2 = 2.5e 4 rad b) Simulated view o the target track relative to the common ocal plane Fig. 5 Pseudo-random signals added to the target sensor output, during the simulations with uncertain knowledge o the target position x 3 detail x x 4 detail x 4 = ov/2 = 2.5e 4 rad b) Case with integrated spacecrat-transmitter ast steering mirror control. Fig. 6 Simulated view o the track o the target on the target sensor: spacecrat with ideal IRU and certain knowledge o the target position. 9 o American Institute o Aeronautics and Astronautics

11 ixed mirror ixed mirror = ov/2 = 2.5e 4 rad a) Case with independent spacecrat-ast steering mirror control = ov/2 = 2.5e 4 rad a) Case with independent spacecrat-ast steering mirror control ixed mirror ixed mirror = ov/2 = 2.5e 4 rad b) Case with integrated spacecrat-transmitter ast steering mirror control. Fig. 7 Simulated view o the track o the target on the target sensor: spacecrat with ideal IRU and uncertain knowledge o the target position. = ov/2 = 2.5e 4 rad b) Case with integrated spacecrat-transmitter ast steering mirror control. Fig. 8 Simulated view o the track o the target on the target sensor: spacecrat with realistic IRU and Kalman ilter, and certain knowledge o the target position. o American Institute o Aeronautics and Astronautics

12 x 4 ixed mirror e x [rad] 5 5 x 4 e y [rad] 5 = ov/2 = 2.5e 4 rad a) Case with independent spacecrat-ast steering mirror control time [s] a) Case with independent spacecrat-ast steering mirror control ixed mirror x 4 e x [rad] = ov/2 = 2.5e 4 rad b) Case with integrated spacecrat-transmitter ast steering mirror control. Fig. 9 Simulated view o the track o the target on the target sensor: spacecrat with realistic IRU and Kalman ilter, and uncertain knowledge o the target position. e y [rad] x time [s] b) Case with integrated spacecrat-transmitter ast steering mirror control. Fig. Track error history vs time. Spacecrat with realistic IRU and Kalman ilter, and uncertain knowledge o the target position. o American Institute o Aeronautics and Astronautics