FORMATION FLYING SATELLITE CONTROL AROUND THE L2 SUN-EARTH LIBRATION POINT By Nicholas H. Hamilton United States Air Force and David Folta and Russell Carpenter NASA/GSFC Abstract This paper discusses the development of a linear control algorithm for formations in the vicinity of the L2 sun-Earth libration point. The development of a simplified extended Kalman filter is included as well. Simulations are created for the analysis of the stationkeeping and various formation maneuvers of the Stellar Imager mission. The simulations provide tracking error, es- timation error, and control effort results. For formation maneuvering, the formation spacecraft track to within 4 meters of their desired position and within 1.5 millimeters per second of their desired zero velocity. The filter, with few exceptions, keeps the estimation errors within their three-sigma values. Without noise, the controller performs extremely well, with the formation spacecraft tracking to within several micrometers. Each spacecraft uses around 1to 2 grams of propellant per maneuver, depending on the circumstances. INTRODUCTION quadratic-regulator framework is applied for control of SI in the vicinity of the L2 Sun-Earth libration point. NASA would like to fly many distributed The development of a simplified extended Kalman fil- spacecraft missions in the next decade and beyond. ter is included as well. Among these concepts, the Magnetospheric Multiscale and Magnetospheric Constellation _will study the mag- SI MISSION DESIGN AND CONTROL netotail of the earth, Stellar Imager 2(SI), MAXIM 3,and REQUIREMENTS Constellation-X 4 will image stars and black holes re- SI is a concept for a space-based, UV-optical spectively, and the Laser Interferometer Space An- interferometer, proposed by Carpenter and Schrijver _1. tenna 5 (LISA) will attempt to detect gravitational The purpose of the mission is to view many stars with a waves. Some of these missions, including SI and sparse aperture telescope in an attempt to better under- MAXIM, plan to operate near the Sun-Earth libration stand the various effects of stars' magnetic fields, the points, and have very precise formation control re- dynamos that generate them, and the internal structures quirements. and dynamics of stars. The leading concept for SI is a Farquhar 6 initially examined control tech- 500-meter diameter Fizeau-type interferometer com- niques for the station-keeping of satellites orbiting a posed of 30 small drone satellites that reflect incoming libration point. Gomez, Masdemont, and Simo 7have light to a hub satellite. The hub will recombine, process, extensively studied dynamics and control problems and transmit the information back to Earth. As Figure 1 involving libration points. Scheeres and Vinh 8 have shows, in this concept, the hub satellite lies halfway researched the dynamics and control of relative motion between the surface of a sphere containing the drones of two spacecraft in unstable orbits. Hoffman 9 and and the sphere origin. Focal lengths of both 0.5 km and Wie _° used the linear quadratic regulator to control 4 km are being considered. This would make the radius problems involving the Earth-Moon cotlinear libration of the sphere either 1km or 8kin. Details of the re- points. Currently, diverse approaches to support these quired formation geometry may be found in the appen- mission types are under study. In this paper, the use of dix. high fidelity dynamics along with a discrete linear- The type of orbit and location in space is an important part of mission design. The best orbit choice 1 American Institute of Aeronautics and Astronautics fortheformatioanfterconsideratioofngravitygradi- The second tier is "intermediate" control with entss,cattereadndstraylight,andelemenretplacement a modulated laser ranging system as sensors and thrust- isaLissajouosrbitarountdheSun-EarLth2point.The ers as actuators. The relative positions will be con- y-amplitudoef the Lissajousorbitwill be about trolled to within 50 microns at this tier. This level will 600,00k0in,butisnotcriticatlothemissionW. iththis drive primary attitude adjustments and small translation orbit,SIwillbeabletocovertheentireskyeveryhalf maneuvers. Twelve 10-100 micro-Newton Indium yearwhilemaintaininagnaimperpendicutloatrhesun. Field Emission Electric Propulsion (FEEP) thrusters ForusefuilmagingS,Imustaimwithin10degreeosf will be used for this level. This propulsion technology perpendicuflraormthesun. is currently available, but by 2015 should be vastly im- Drones proved. Basically, this tier functions to smooth the transition from the rough control of the first-tier to the fine precision control of the third-tier. The third-tier is fine precision control. At this Sphere Origin level, the satellites themselves will not move; instead, the optics will be adjusted by extremely accurate me- chanical devices with an accuracy in the nanometer range. Rather than having a traditional sensor to deter- mine measurements, phase diversity and wave-front \ 500m error (WFE) sensing algorithms, using data from the incoming light rays, will determine the needed control. Figure 1. Satellite formation with focal length 500m Currently, phase diversity and WFE sensing are in their infancy for use with spacecraft and formation flying concepts. To function properly, SI will need to accom- modate a wide range of control functions. In addition Controller Development to maintaining its desired trajectory around L2, the A common approximation in research of this formation must slew about the sky requiring movement type models the dynamics of a satellite in the vicinity of of a few kilometers and attitude adjustments of up to the sun-earth L2 point using the circular restricted 180 degrees. While imaging, though, the drones must three-body assumptions. These assumptions only ac- maintain position within 3 nanometers of accuracy in count for gravitational forces from the sun and Earth. the direction radial from the hub and within 0.5 milli- The moon is also included, but not as an independent meters of accuracy along the sphere surface. The accu- body. The masses of the earth and moon are combined racy required for attitude control while imaging is 5 and assumed to be at the earth-moon barycenter. The milli-arcseconds tip and tilt (rotations out of the surface motion of the sun and the earth-moon barycenter is also of the sphere). The rotation about the axis radial from assumed to be circular around the system barycenter. the hub (rotation within the sphere) is a much less strin- This analysis uses high fidelity dynamics gent 10 degrees. based on a simulation named Generator that Purdue To achieve these requirements, Leitner and University has developed. Generator creates much Schnurr _2propose a three-tiered formation control ap- more realistic Lissajous orbits than those derived from proach. The first tier is "rough" control using radio the circular restricted three-body problem. Using frequency (RF) ranging and modified star trackers for ephemeris files, Generator takes into account the effects sensors and thrusters for actuators. The relative posi- of eccentricity, an independent moon, the other planets tions will be controlled to within a few centimeters. of the solar system, and solar radiation pressure. The This level will drive lost-in-space emergencies, forma- resulting Lissajous orbit can then be used as a more tion initialization, large translations due to formation accurate reference orbit. In addition to providing the slewing, collision avoidance, and maintenance of the reference positions and velocities, Generator also nu- formation's trajectory about L2. The RF ranging sys- merically computes and outputs the linearized dynamics tem will provide range measurements, and the modified matrix, A, for a single satellite at each epoch. This data star trackers will provide azimuth and elevation meas- can be used onboard for autonomous computation by urements. The actuators for this tier will be four low- simple uploads or onboard computation as a back- thrust, high specific impulse (Isp) thrusters. The thrust ground task of the 36 matrix elements and the state level will be on the Newton to milli-Newton order of vector. A Generator reference orbit is shown in Figure magnitude. It is this tier that this paper primarily ad- 2. The origin in Figure 2is the earth. The X coordinate dresses, although these results could be applied at the connects the two primary bodies, the Z coordinate is next tier as well. parallel to their angular velocity of the system, co,and the Y coordinate completes a right-handed system. The 2 American Institute of Aeronautics and Astronautics referencoerbitperiodis359daysandthescaleisin where u is the control vector and B is the matrix that kilometers. maps the control effort to the state-space. The control is modeled as ideally applied acceleration in the x, y, and z directions. Therefore, xi_1_renc! fo;hub rate,lilt 1xIOs -0 0 0- 0 0 0 Iu]' o__._i.....i-i_-)--1 -o. .t.. 0 0 0 and U: U 2 , (10) B= 1 1 0 0 LU3l 0 OS 1 _5 2 0 05 1 15 2 10'_ x x IO6 xI0_ x x 1Q' 0 1 0 .....!. ._'."i.... 0 0 1 " oIl.....k:/-,_--_,,--;I"-":..... I L_.:,i " where u I is the control in the x direction, u2 is the ..i....... 2 control in the y direction, and u3 is the control in the z x _01_' "----.'r-_'""0 x " _O'y direction. Y x 101 Figure 3. Generator based 359 day reference orbit Letting subscript 1 denote the hub, the con- around L2 troller will regulate the hub to follow a desired path around L2, while the drones, denoted by higher sub- The coordinates of a satellite can be written as scripts, will be controlled relative to the hub only. Thus X = X 0 + x (]) the drone's state vectors are actually relative to the hub, not the libration point, and for satellite 2, the tracking Y = Y0 + Y (2) error obeys Z = Z0 + z, (3) ,..L X2 = A_ 2 +Bu 2-Bu 1, (11) where X 0, Yo, and Z 0 are the coordinates of any one For the complete formation, it is convenient to redefine of the libration points. The linearized equations of mo- A and B such that [' 1 tion about a collinear point can be expressed in state- space notation as = Ax, (4) A = " ... B = -B_ B_ where Aj iB "" Bj x=[x y z x y _]r, (5) -- I The system _sdiscretized using a sample-and-hold ap- and A is output from Generator. proach. The state transition matrix is created from the For SI, the overall formation must follow a dynamics partials output from Generator by prescribed path about the libration point, and individual satellites must maintain desired relative configurations; k(t-to) k i.e. the controller must be of the tracking type. Define q)(t -to) = eAt'-'°) = _ , (13) the state error as k=0 truncating the series after the quadratic term. Although X(t) = X(t) -- Xref (t), (6) the A matrix from Generator is slowly time-varying, it SOthat the desired linear control law is is assumed to be constant over the time step, which is in U = -K(t)x, (7) all cases much smaller than the "period" of the Lissa- jous orbit. At each time step, the steady-state LQR The problem is therefore in the form of the linear- control gain is computed from the solution of the dis- quadratic-regulator (LQR), which provides the gain K crete algebraic Riccati equation corresponding to the that minimizes the cost function state transition matrix for that epoch. _t-[ + Since the measurements are critical to the fea- (8) sibility of the SI control concept, and since the types of to measurements under consideration are nonlinear in the where W is a penalty on the mean square tracking er- states of the controller described above, a simplified rors, and Vis apenalty on the control energy. Assum- extended Kalman filter that uses the dynamics de- ing the reference satisfies the state differential equation, scribed above, augmented by zero-mean, white, Gaus- the closed loop dynamics are sian process and measurement noise was also studied. : (A - eK(t))_, (9) The discretized state dynamics for the filter are 3 American Institute of Aeronautics and Astronautics "xk+=t Aa'xk + Bauk + w (14) In terms of the states, the measurements for one satellite are where w is the random process noise vector. The (non- linear) measurement model is Yk = m(xk) +v (15) r = 4x 2+y2 +z 2 (19) The covariances of process and measurement noise are E[wwTJ= Q, E[vvrJ= R, (16) For the hub, we assume Earth-based range, el = sin-'(-Z)'r and az = sin-'(r coX(el) )(20) azimuth, and elevation, are measured. To keep the fil- ter simple, the angles were assumed to be measured in These models are used with standard techniques, e.g. as relative to the ecliptic plane. may be found in Brown and Hwang _4,to construct the For the drone satellites, positions are deter- simplified extended Kalman filter for this study. mined relative to the hub satellite. Range is a scalar, so whether it comes from a sensor on the hub or the drone RESULTS is unimportant. Azimuth and elevation, on the other hand, usually come from sensors on the drone space- Three different scenarios make up the position craft. They are in the frame of reference of the local control problem--maintaining the Lissajous orbit, coordinate system centered on the drone. Because the slewing the formation to aim at another star, and recon- states of the drone spacecraft are represented with re- figuring the formation to take another snapshot of a star spect to the hub spacecraft, and the angles are with re- when necessary. These three scenarios are treated in- spect to the drone spacecraft, a coordinate transforma- dependently. To determine the amount of fuel, the ve- tion is required. However, if the local coordinate sys- tem on the drone is oriented the same as the reference locity change, or AV, is needed. The AV in each coordinate system on the hub (b_ is aligned with x, b2 is direction is found by numerical integration: aligned with y, and b3is aligned with z), the position of 359 359 Z359lu..Iz the drone can be determined from simple trigonometry. AV_=ZIu_IT, AV,,=ZI.,IT,AV: = (21) k=l k=l k=l For convention positive azimuth is defined as where T is the maneuver interval, the upper limit of the counter-clockwise from the b2 direction in the bl-bz sum is the simulation length. The absolute value of the plane. Positive elevation is defined from the bFb2 control is taken because the direction of the maneuver plane upwards in the positive b3direction. The position of the drone relative to the hub is then has no beating on the fuel used. The total A V for one simulation is calculated by x = r cos(e/) sin(az), (17) AV = AV x dc mVy "1- AV z (22) z = -r sin(e/), (18) Lissa[ous Orbit Maintenance Following the Lissajous orbit is not aproblem where r is the range, el is the elevation, and az is the of formation control, but rather aproblem of maintain- azimuth as shown in Figure 3. ing an orbit. Therefore, only the hub satellite needs simulation to determine the amount of control and fuel needed to maintain aLissajous orbit. The results can be extended similarly to other satellites in the formation. The continuous state weighting matrix and the continuous control weighting matrix are chosen to be (23) le6 ] V = I I 1 X tV = le6 1 le3 le3 Dl'C _azr = le3 The values of the process and the measurement noise b2 covariance matrix for the hub satellite are chosen tobe iJ b_ Figure 3. Range, azimuth, and elevation of one spacecraft 4 American Institute of Aeronautics and Astronautics ,0 l l (24) suits. Figure 5 shows the estimation errors of a dozen simulations. 0 0 Ol: hub tracking error 0.3 1= Q"=/ le-6 R= 1500000 j le -6 le 6, o_./_ _ _ ° ........ ax) -05 "_ -1301 The strength of the process noise is set at a 0 200 400 0 200 400 value large enough to be noticed, but not so much as to constrict or destabilize the system. The first term in the measurement noise covariance matrix assumes range g measurements from the earth to the hub within 0.1 km >"-1 _ -002 or 100 meters. The second and third terms assume that 0 200 400 2O0 4O0 the arc lengths corresponding to the azimuth and eleva- 2 ! _ 005 tion angles are three times less accurate than the range _" 0 _',....... " ...... 0 # ® measurements. To find the angular accuracy, simply divide the accuracy of arc length by the range. The L2 N "20 200 400 "4 -005 0 2OO 400 point is approximately 1.5 million kilometers from the time (days) time (days) earth and is used as a roughly constant scaling distance. Figure 4. Hub tracking errors The measurement covariance for the drone satellites is different from that of the hub: The plots on the left are the position estimation errors, and the plots on the right are the velocity esti- 0.0001 " mation errors. The blue lines represent the actual esti- R (0'0003/: mation errors, and the red lines represent the three- = _-; fo.ooo3T, (25) sigma value of the covariance. k 0.59 Here, the range measurement is assumed to be accurate Hub Estimation En-ors (12 simulations) x104 to within 0.1 meters, with three times less accurate arc 05 2 _----_ .... ;...... "..... lengths. The range from the hub to the drones is the focal length of the interferometer (either 0.5 or 4km). 3 41_5 The initial covariances for the hub and drones are 0 100 200 300 400 x 010,4 tO0 200 300 400 05 .... (26) _ - I OOff p,(+) = 1 864: Pj÷I = 0o1: 0864: ) -0. 0" 100 :_00 300 400 _ _ tO-4100 200 300 400 864: 0864: /J 0.5 _ , , , "_ , '-:J--:--';_-,_,-.:._...1 L 864: 0864_ 1 o '_ _ x _i =_o....:'-'"" .... --. respectively. For the hub, this assumes about 1 km -2_- - accuracy for position (10 times greater than measure- -05 "_ 0 100 200 300 400 N o lOO 200 300 400 ment noise covariance) and 1 meter per second for ve- lime (days) time (days) locity. The 86.4 term is the conversion to kilometers Figure 5. Hub estimation errors (12 simulations) per day from meters per second. For drone satellites, the covariance corresponds to a position accuracy of 1 The estimation errors lie within the three- m and velocity accuracy of 1 millimeter per second. sigma values of the covariance with very few excep- Figure 4 shows the tracking errors plotted against time tions. The position estimation errors are within 250 for a simulation one year, with a time step of 1day. The meters, and the velocity estimation errors are within plots on the left are position tracking errors, and the 2e-4 meters per second. Figure 6 shows the control plots on the right are velocity tracking errors. Running effort over the course of one simulation. Averaging the many simulations, it can be seen that the steady-state determined A V from a dozen simulations, the A V position tracking errors are within 0.25 km for each required to keep a satellite in a Lissajous orbit about L2 direction, and the steady-state velocity tracking errors for 359 days is approximately 0.38 meters per second. are within 7.5e-4 meters per second for each direction. Using the rocket equation, and Stellar Imager of initial Because the estimation errors are based on randomness, spacecraft mass of 550 kg for the hub, a 100 kg for a many simulations must be run to determine useful re- drone, and the Isp for the low-thrust, high-Isp thruster of 10000 seconds, the amount of propellant needed to 5 American Institute of Aeronautics and Astronautics maintaianLissajouosrbitfor359daysislessthan2.2 Formation gramsforthehubandlessthan0.4gramsforeach 0.51 drone. I C_,ntml 1 21 r _ F i r i r x I i i _ ! ! i i -0.5 0 0.5 -05 0 0.5 x(kin) x(kin) 0 50 100 if0 200 250 300 350 400 0.5 2 r I I I I 0.5 .... - ; _ ;-. I : ! : i t N -2 0 50 100 150 200 250 300 350 400 5 0.5 -0.5 0 05 yIkm) -05 .O5 x(kin) y(kin) Figure 7. SI formation before and after 90 degree slew 0 50 1O0 150 200 250 300 350 400 (0.5 km focal length) lime Figure 6. Hub control effort The black dots represent drones at the beginning of the simulation, and the open circles represent drones at the Formation Slewin_ end of the simulation. The hub is the black asterisk at A key part of the SI mission is to image many the origin. The upper-right plot illustrates the stars. Following a Lissajous orbit around L2, SI could Golomb 15rectangle formation projected into the x-z view the entire sky approximately every half-year while plane. The lower-left plot clearly shows the drones slewing about just the radial (x) axis. This will also slewing 90 degrees about the hub-centered xaxis. maintain the aiming angle perpendicular to the sun. Figures 8 and 9 show the tracking errors for The formation slewing simulation follows a similar the hub and drone. Although the plots are specific to a algorithm as the Lissajous orbit simulation. 90 degree slew with a 0.5 km focal length, they are For this simulation, the maneuver interval is 1 qualitatively representative of all the different slew an- minute and the length of the simulation is 24 hours, gle and focal length scenarios. using a constant A-matrix from day 2 of the year-long hub Itackirlg elTOt simulation made above. Various slew angles are ex- i i i i. g amined. All other tuning parameters are the same, ex- cept that their discretized values are adjusted by the change in time step. The strength of the process noise -005 _ -0 05L is different from the Lissajous orbit simulation, 500 1000 1500 0 ,500 1000 1500 1 I l g : 0_-_ ,- ' .-- Q_ = le-24 >. -1 10 0 0 te-24 1 (27) 0 500 1000 1500 500 1000 1500 : t 1e-24 _"o2.(,, ! i _ 0.1 As before, the strength of the process noise is set at a 1 value large enough to be noticed, but not so much as to constrict or destabilize the system. 0 500 1000 1500 500 1000 1500 lime time The hub tracking error, estimation error, and Figure 8. Hub tracking error (90 degree slew and 0.5 control are found for focal lengths of both 0.5 and 4 km focal length) km Each drones' tracking error, estimation error, and control are determined as well. For conciseness, only For both 90 degree and 30 degree slews with the results for one drone will be shown (satellite 2). either a 0.5 or 4 km focal length, the hub tracks to Two slewing angles are investigated, 90 degrees and 30 within 50 meters of its reference position and to within degrees. Figure 7 provides an image of the entire SI 5 millimeters per second of its desired velocity. The formation slewing 90 degrees, with a 0.5 km focal drones all track to within 3 meters of their desired ref- length. erence positions and to within 1millimeter per second of their desired zero velocities. 6 American Institute of Aeronautics and Astronautics drone 2tracking error Hub EsUm_ion Error x10.:= o_,,c................ °_ o 0.02t_ , ' ] _ 0.01 × -001 0 500 10(30 1500 500 1000 1500 - ' ' ' 0 A 0x10._ 500 1000 1500 02 i , g °' _ o ...... i ............. _ o I 02t, _ 01 2 L i :_ -02 _ -01 t, -o2 0 5OO 10(30 1500 500 1000 1500 0 500 1000 1500 0x10-3 500 1000 1500 "Co>-......... v_ o.......... 020 , ' I ' E i......; 1f i _ 02 l ._----- N "10 500 10(30 1500 N . 0 500 1000 1500 0 500 1000 1500 N 0 500 1000 1500 time time time (mini time (rain) Figure 9. Drone 2tracking error (90 degree slew and Figure 10. Hub estimation error for 0.5 km focal length 0.5 km focal length) and 90 degree slew Table 1 lists the total position tracking error x104 Drone 2EE x10_ 5 after one day for the different scenarios when noise is turned off. _ o -_-- _ ,_..,_, ._ 3 ......... L.._ Table 1. Formation slewing position tracking errors -2 _ -5 0 104 500 1000 1500 " O10_ 500 1000 1500 with no noise Focal Slew Hub Drone2 Drone 31 length angle position position position ==o "- (km) (deg) tracking tracking tracking error error (m) error (m) 0.5 90 8.33e-6 3.075e-6 3.075e-6 O10" 500 1000 1500 _ 0 10_ 500 1000 1500 B 0.5 30 3.05e-6 1.126e-6 1.126e-6 _ _o 4 90 7.35e-5 2.713e-5 2.713e-5 4 30 2.69e-5 9.93e-6 9.93e-6 00 _ 0 500 10_0 1500 time (mini time (mini For all scenarios, neglecting noise, the velocity Figure 11. Drone 2 estimation errors for 0.5 km focal tracking error is essentially zero after one day. If noise length and 90 degree slew is turned off, the tracking errors go asymptotically to zero, as expected with a linear quadratic regulator con- The three-sigma value change from one simulation to trol strategy. Clearly, the noise and estimation errors another can be seen in the drone estimation error plots. have a significant effect on the tracking errors. The range from the hub to the drone is either 0.5 or 4 Figure 10 shows the estimation error results of kin, whereas the range from the hub to the earth is a dozen simulations for the hub with varying slew an- about 1.5 million km. gles and focal lengths. The red lines are the three-sigma From the control efforts, the directional A V values. The hub's steady-state position three-sigma can be determined as before but the time interval of values are about 50 meters, and the steady-state veloc- 1440 minutes must be taken into account. The total ity three-sigma values are about 1millimeter per second for all scenarios. The estimation errors are within the AV is found from the directional AV 's. The differ- three-sigma values with few exceptions. The three- ences between slewing AV s and orbit maintenance sigma values change for each simulation (because the A V s are the number of maneuvers over the course of noise is random), but the change cannot be seen for the the simulation and the maneuver interval. The forma- hub because the order of magnitude of change is much, tion slewing simulation runs for one day, with one ma- much less than their overall value. neuver per minute (1440 maneuvers), whereas the Lis- Figure 11 shows the estimation errors for a sajous orbit simulation runs for 359 days with one ma- dozen simulations for the first drone satellite. Drone 2's neuver per day. Table 2shows the average A V's for a estimation errors are, with few exceptions, within the dozen simulations for the various scenarios. three-sigma values for all scenarios. 7 American Institute of Aeronautics and Astronautics Table2 AveragfeormatiosnlewingA V's the drones, through the hub, to the desired star, is Focal Slew Hub Drone 2 Drone 31 maintained with such a reorientation. Figure 12 shows Length Angle AV(m/s) AV (m/s) AV (m/s) the first four drones of the formation before and after (km) (deg) reorientation (with a 0.5 km focal length). Only four 0.5 30 1.0705 0.8271 0.8307 0.5 90 1.1355 0.9395 0,9587 drones are pictured for clarity. 4 30 1.2688 1.1189 1.1315 Formation Reodefltation 1 4 90 1.8570 2.1907 2.1932 0=51 • ',o o;: The larger the angle the formation slews N O[iL.......... _ ....... through, the more A V is needed. Also, the larger the .4] focal length, the more A V required. Table 3 shows -0:5 the corresponding propellant masses needed to achieve -1 0 1 -05 0 05 x x the AV's given in Table 2, with the assumptions that 05 the Isp of the low-thrust thrusters is 10000 seconds, the o _.-,.;..... _...... i initial mass of the hub is 550 kg, and the initial mass of °st..... i i ...... each drone is 100 kg. .o.sk........ .. "'_'-. i N -0.i I- ........ Jr"........... Table 3. Avera formation slewing propellant masses 0_05 -0.5 0,5 y -0.5 4]5 x Focal Slew Hub Drone2 Drone 31 Length Angle (kin) (deg) m prop(g) m prop(g) mprop (g) Figure 12. Formation reonentation (0.5 krn focal length) 0.5 30 6.0018 0.8431 0.8468 0.5 90 6.3662 0.9577 0.9773 The scale is in kilometers in all four pictures. 4 30 7.1135 1.1406 1.1534 The plot in the upper-right shows the projection of the 4 90 10.4112 2.2331 2.2357 formation in the x-z plane. The black dots are the drones at their initial conditions, and the open circles When the noise is turned off, the required are the drones after the simulation. The formation ap- AV and propellant mass is reduced significantly. Ta- pears to have rotated clockwise 90 degrees about the y ble 4 shows the AV's and Table 5 shows the corre- axis, but the hub is into the page, so the rotation is sponding propellant masses when noise is removed. counterclockwise when viewed from the hub. Figures 13 and 14 show the tracking errors for the hub and the drone for the 0.5 km focal length case. Table 4. Required AV's for formation slewing cases without noise hub Itacking error 02 0.05 Focal Slew Hub Drone 2 Drone 31 i Length Angle A V (m/s) AV (m/s) AV (m/s) • (km) (deg) 0.5 30 0.0504 0.0853 0.0998 x -02 _ -O.oGL 0.5 90 0.1581 0.2150 0.2315 0 500 1000 1500 500 1000 1500 05 F , _ 02 4 30 0.4420 0.5896 0.6441 4 90 1.3945 1.9446 1.9469 ....... o >"-05 " '_ _,-_-0,2 Table 5. Required propellant masses for formation 0 500 1000 1500 50'0 1000 150j0 slewing cases without noise _ : Focal Slew Hub Drone 2 Drone 31 05[ _ 0.2 Length Angle _-" -'\ _-->+._ _ ', , _ 0 (kin) (deg) m prop (g) m prop (g) m prop (g) 0_5 30 0.2826 0.0870 0.1017 0 500 1000 1500 N -02 500 1000 1500 0.5 90 0.8864 0.2192 0.2360 time time 4 30 2.4781 0.6010 0.6566 Figure 13. Hub tracking error (0.5 km focal length) 4 90 7.8182 1.9822 1.9846 The hub tracks to within 40 meters of its reference po- Formation Reorientation sition, and to within 8 millimeters per second of its ref- erence velocity for both focal lengths. For some stars, one snapshot from the SI for- mation will not provide enough sampling data for suffi- cient resolution. In these cases, the drones must rotate 90 degrees and take another snapshot. The aim from 8 American Institute of Aeronautics and Astronautics drone 2tracking error tion. Table 7 gives the average A V's from a dozen simulations to reorient the formation. o, ] 0 --. /" Table 7. Average formation reorientation AV's x -05v0ir 500 1000 1500 _-O,f 500 1000 1500 Focal Length Hub Drone 2 Drone 31 005 [ ', ', "_ 005 (km) AV (m/s) AV (m/s) AV (m/s) 0.5 1.0126 0.8421 0.8095 4 1.0133 0.8496 0.8190 >, -0 05 = : o>- -005 0 SO0 tO00 1500 500 1000 1500 002 The focal length has no discernible effect on the AV I' :. : ! needed to reorient the formation. This makes sense because the rotation is about the y-axis, and the focal o;-'Li/--i.._...-..._."_; - ' length is assumed to be the measurement along the y -0 05 -002 0 500 1000 1500 SO0 1000 1500 axis from the hub to the drones. Table 8 gives the pro- time time pellant masses that correspond to Table 7. Figure 14. Drone 2 tracking error (0.5 km focal length) Table 8. Average formation reorientation propellant Every drone tracks to within 4 meters of its reference masses position, and to within 1.5 millimeters per second of its Focal Length Hub Drone 2 Drone 31 desired zero velocity for both focal lengths. (km) When noise is turned off, the satellites track mprop (g) mprop (g) mprop (g) much better than when the noise is included. Table 6 0.5 5.6771 0.8584 0.8252 4 5.6811 0.8661 0.8349 shows the total position tracking errors for the various satellites after one day, when noise is eliminated. Without noise, the AV needed to reorient the Table 6. Formation reorientation position tracking formation is much less, as shown in Table 9. errors with no noise Focal length Hub position Drone 2 Drone 31 Table 9. Required A V for formation reorientation (km) tracking position position without noise error (m) tracking tracking error error (m) (m) Focal Length Hub Drone 2 Drone 31 0.5 2.147e-6 0.792e-6 0.793e-6 (km) AV (m/s) AV (m/s) AV (m/s) 4 2.146e-6 0.792e-6 0.793e-6 0.5 0.0408 0.1529 0.1496 4 0.0408 0.1529 0.1495 The tracking errors go asymptotically to zero, and the velocity tracking errors are essentially zero at Table 10 gives the propellant masses that correspond to the end of a day. Just as with the formation slewing Table 9. simulation, the noise, and in turn the estimation errors, are the largest reason for imperfect tracking. Table 10. Required propellant masses for formation Since the hub and drone estimation errors for a reorientation without noise dozen simulations with the different focal lengths is Focal Length Hub Drone 2 Drone 31 similar to the slew simulations, they are not shown (kin) m prop (g) mprop (g) mprop (g) here. The steady-state x three-sigma value is about 30 0.5 0.2287 0.1623 0.1525 meters. The steady-state y and z three-sigma value is 4 0.2287 0.1623 0.1524 about 50 meters. The steady-state velocity three-sigma values are about 1 millimeter per second. The hub es- timation errors stay within the three-sigma values ex- SUMMARY AND CONCLUSION cept for rare occasions. For any drone and either focal The control strategy and Kalman filter pro- length, the steady-state position three-sigma values are vided satisfactory results. The hub satellite tracks to its less than 0.1 meters, and the steady-state velocity three- reference orbit sufficiently for the SI mission require- sigma values are less than le-6 meters per second. ments. The drone satellites, on the other hand, track to Also, the estimation errors stay within the three-sigma only within a few meters. Without noise, though, the values with rare exceptions. drones track to within several micrometers. The first tier of the proposed control scheme for SI requires the drones to track within centimeters. This could be ac- The A V can be determined, for each satellite, complished with better sensors to lessen the effect of from the control effort required to reorient the forma- the process and measurement noise. Tuning the con- 9 American Institute of Aeronautics and Astronautics trollerandvaryingthemaneuveinrtervalsshouldpro- 13. Howell, K.C., and Anderson, J., Generator User's videadditionaslavingasswell.TheKalmanfilterper- Guide. Version3.0.2. 2001. formedsuchthattheestimatioenrrorswereforthemost 14. Brown, R.C., and Hwang, P.Y.C., Introduction to partwithinthethree-sigmvaaluesT.hepropellanmtass Random Signals and Applied Kalman Filtering. andA V results provide a minimum design boundary 2"dEdition. John Wiley & Sons, Inc. 1992. for the SI mission. Additional propellant will be 15. Golomb, S.W., and Taylor, H. IEEE Trans. Info. needed to perform all attitude maneuvers, tighter con- Theo., 28, #4. 1982. trol requirement adjustments, and other mission func- tions. Future studies must examine the attitude dy- Appendix: SI Formation Geometry namics and control problem and integrate this into the The hub spacecraft is assumed to be at the translation control. Other items that should be consid- center of a sphere, on which the drones lie. The x di- ered in the future include; non-ideal thrusters, collision rection of the hub-centered Cartesian coordinate system avoidance, system reliability and fault detection, and is parallel to and in the same direction as the radial axis nonlinear control and estimation. For the nanometer of the earth-centered rotating coordinate system used by level preciseness of the second and third control tiers, Generator. The y direction of the hub-centered system new control strategies and algorithms may be required. is parallel to and in the same direction as the along- track direction of the earth-centered system. The z di- REFERENCES rection of the hub-centered system is parallel to and in 1. "Magnetospheric Constellation," the same direction as the cross-track direction of the htto://s _tp.gsfc.nasa.gov/missions/mc/mc.htm. Oc- earth-centered system. Figure 14 shows the relation- tober 2001 ship between the two coordinate systems. 2. "The GSFC Stellar Imager Homepage," T cross-track htlp://hires.gsfc.nasa.gov/-si/. October, 2001. 3. "Micro Arcsecond X-Ray Imaging Mission," ng-track http://maxim.gsfc.nasa.gov. October, 2001 4. "Constellation-X', http://constellation- X/docs/ga/genaudience.html, May 2001 5. "Laser Interferometer Space Antenna," http://lisa.jpl.nasa.gov. October 2001.. 6. Farquhar, R., The Control and Use of Libration- Point Satellites. NASA Technical Report R-346. radial"N x 1970. Figure 14. Hub-centered Cartesian coordinate system 7. Gomez, G., Masdemont, J., and Simo, C., "Quasi- halo Orbits Associated with Libration Points," The drones' positions are expressed in the spherical Journal of the Astronautical Sciences, Vol. 46. 1998. coordinates, r, 0, and 4, which relate to the hub- 8. Scheeres, D.J., and Vinh, N.X., "Dynamics and centered Cartesian coordinates by Control of Relative Motion in an Unstable Orbit," x = rsin(0) , y = rcos(0)sin(0), (la) AIAA Paper 2000-4135. 2000. and z = r cos(0) cos(_b) (2a) 9. Hoffman, D.A., Station-keeping at the Collinear Equilibrium Points of the Earth-Moon System, The radial coordinate, r, is the distance from JSC-26189. NASA Johnson Space Center. 1993. the hub to the drone and always equals the radius of the 10. Wie, B., Space Vehicle Dynamics and Control. AIAA Education Series. 1998. sphere, which is twice the desired focal length. The _b 11. Carpenter, K.G., Neff, S.G., Schrijver, C.J., Allen, coordinate is measured from the positive z axis toward R.J., and Rajagopal, J., "The Stellar Imager (SI) the positive y axis. Standing on the positive x axis and Mission Concept." In proceedings of the 36th looking back, a positive rotation is clockwise. The /9 Liege Astrophysical Colloquium: From Optical to coordinate is measured from the position in the y-z Millimetric Intefereometry: Scientific and Techni- plane along the sphere in a clockwise direction when cal Challenges. 2001. considered from the positive z axis looking back. 12. Leitner, J., and Schnurr, R., "Stellar Imager (SI): Initially, the center of the formation will be Formation Flying." Presentation from the Inte- placed directly behind the hub in the along-track direc- grated Mission Design Center, Goddard Space tion. Thus, Flight Center, NASA. 2001. 10 American Institute of Aeronautics and Astronautics