14 th AAS/AIAA Space Flight Mechanics Conference

Transcription

1 Paer AAS ASPHERCAL FORMATONS NEAR THE LBRATON PONTS N THE SUN-EARTH/MOON EPHEMERS SYSTEM B. G. Marchand and K. C. Howell School of Aeronautics and Astronautics, Purdue University West Lafayette, ndiana th AAS/AAA Sace Flight Mechanics Conference Maui, Hawaii February 8-12, 2004 AAS Publications Office, P.O. Box 28130, San Diego, CA 92198

2 AAS ASPHERCAL FORMATONS NEAR THE LBRATON PONTS N THE SUN-EARTH/MOON EPHEMERS SYSTEM B.G. Marchand * and K.C. Howell Multi-sacecraft formations, evolving near the vicinity of the libration oints of the Sun-Earth/Moon system, have drawn increased interest for a variety of alications. This is articularly true for sace based interferometry missions such as TPF and MAXM. The resent study considers both continuous and discrete control methods as alied to non-natural formation configurations. Continuous outut feedback linearization methods are emloyed to enforce nonnatural configurations. The general focus is around multi-sacecraft formations that are constrained to evolve along an asherical surface, such as a araboloid, such that the orientation of the formation is fixed in inertial sace. Also, a discrete Floquet controller, reviously develoed for the determination of natural formations, is further investigated. ts alication to inertially fixed formations is of articular interest. The general develoment is resented in the Sun-Earth/Moon ehemeris model. NTRODUCTON The dynamical sensitivity that is characteristic of the region near the libration oints, combined with the ath constraints usually imosed on the envisioned nominal formation configurations, makes formation keeing and deloyment an interesting and challenging roblem. Although some reliminary analyses have already been comleted [1-15], a better understanding of natural and controlled formations in this region of sace is still necessary. n revious work, Howell and Marchand [13-14] consider both continuous and imulsive control to enforce a variety of non-natural formations. Among these, inertially fixed configurations are of articular interest for sace based interferometry. * Graduate Student, School of Aeronautics and Astronautics, Purdue University, West Lafayette, N Professor, School of Aeronautics and Astronautics, Purdue University, West Lafayette, N

3 Continuous control techniques, such as Linear Quadratic Regulator (LQR), nonlinear otimal control, and feedback linearization methods, are successfully alied to this tye of formation [13-14]. Although the mathematical imlementation of these control laws is effective, the resulting control accelerations can be rohibitively small, deending on the desired configuration. For instance, near the libration oints, formations that require the deuty vehicle to remain at a constant distance and orientation, relative to the chief sacecraft, are associated with thrust levels on the order of nn, for a 10 meter searation and a 700 kg vehicle. ncreasing the nominal searation by one order of magnitude also increases the cost by one order of magnitude. This indicates that a 10,000 km searation is required, for this tye of formation, in order for the thrust level to exceed 1 µn of nominal thrust. The resent state of roulsion technology allows for oerational thrust levels on the order of µn via ulsed lasma thrusters, such as those available for attitude control onboard EO-1 [16]. ncreased interest in micro- and nanosatellites continues to motivate theoretical and exerimental studies to further lower these thrust levels, as discussed by Mueller [17], Gonzalez [18] and Phis [19]. Gonzalez [18] estimates that a lower bound of 0.3 nn is ossible via laser induced ablation of aluminum. Aside from their immediate alication to micro- and nano-satellite missions, these concets are also otentially romising for formation flight near the libration oints. However, the resently available technology may be sufficient to ursue other tyes of formation flight configurations, such as those resented here and in reviously considered samle formations [14]. n articular, configurations that require the deuty vehicle to sin, relative to the chief, at some fixed rate, such as those resently and reviously considered by Marchand and Howell [14], can quickly drive the thrust levels into the mn range. Of course, continuous thruster oeration may not always reresent a feasible otion. n such a case, the formation keeing task must rely on a discretized aroach. Mathematically, a standard targeter aroach [14] can accomlish the goal to within a reasonable degree of accuracy, rovided the nominal vehicle searations are on the order of meters. However, because these configurations are not consistent with the natural dynamics near the libration oints, the error incurred between maneuvers grows raidly as this nominal searation increases. Furthermore, the size of the maneuvers can still be rohibitively small. The continuous control laws reviously resented [13-14] and the discrete targeter aroach [14] have something in common. Both of these methods try to force configurations that are not consistent with the natural dynamics and they can both require unreasonably small thrust levels, deending on the constraints on the configuration. This is not surrising given the highly sensitive nature of the dynamics. To circumvent this difficulty, it is advantageous to develo a better understanding of the natural formation 2

4 dynamics near the libration oints and incororate such knowledge in the design of both the nominal configurations and the associated control laws. n one such examle, Howell and Marchand [14] successfully identify a series of naturally existing formations, as well as deloyment into these configurations, through the use of a modified Floquet controller. These naturally existing formations evolve in the vicinity of a reference halo orbit near the L 1 and L 2 libration oints of the Sun-Earth/Moon system. The resulting control law is based on knowledge of the linear stability of the reference orbit. n articular, the controller is designed to remove the unstable mode as well as two of the four center modes associated with the halo orbit. n the resent effort, the natural formations reviously identified in the CR3BP [14], via the Floquet aroach, are transitioned into the more comlete ehemeris model. A constrained two-level differential corrector is then alied to these solutions to enforce eriodicity in the ehemeris model via small imulsive maneuvers alied once a year. Further alication of this methodology as a design tool to identify inertially fixed formations is of articular interest. Also, continuous outut feedback linearization methods, reviously considered in [14], are similarly emloyed. The focus is multi-sacecraft formations that are constrained to evolve along an asherical surface, such as a araboloid, fixed in inertial sace. The develoment is resented in the Sun-Earth/Moon ehemeris model. Potentially, these two aroaches can be combined into one formation keeing strategy, articularly for formations that do not require the nominal configuration to be enforced at all times. For instance, the Floquet controller can be used to drive the vehicles onto a naturally existing formation via a single imulsive maneuver until a reconfiguration, that requires continuous control, is necessary. 3

5 DYNAMCAL MODEL Background n this investigation, the standard form of the relative equations of motion for the n-body roblem, as formulated in the inertial frame ( ), is emloyed. The effects of solar radiation ressure (SRP) are also incororated. Hence, the dynamical evolution of each vehicle in the formation is governed by PP 2 s ( Ps) ( Ps) grav sr () r = f + f + u t. (1.1) For notational uroses, let P 2 denote the central body of integration, in this case the Earth. Then, P s ( P ) identifies the sacecraft, s ( P ) f grav reresents the sum of all gravitational forces acting on P s, f s sr is the force exerted on the vehicle due to solar radiation ressure, and u () t is the control inut vector. From this general exression, the equations of motion for both the chief and deuty sacecraft may be exressed in the following form, () r f f u t f u t, (1.2) PC 2 ( C) ( C) ( C) = grav + sr + C = + C () r f f u t f u t, (1.3) PD 2 i ( Di) ( Di) ( Di) = grav + sr + D = + D i i () () where uc () t and u () t denote the control accelerations required to maintain the desired nominal configuration, and D i ( C) f and ( ) D i f reresent the net force acting on the chief sacecraft and the i th deuty vehicle, resectively. The numerical integration for all vehicles in the formation is erformed in terms of 2 inertial coordinates. Define the measure numbers of a osition vector such that r PD ˆ ˆ ˆ i = x X + yy + z Z. i i i Hence, the vehicle velocities and accelerations are associated with the inertial frame () defined in terms of the unit vectors ˆX, Y ˆ, and Ẑ. The chief sacecraft is assumed to evolve along a quasi-eriodic Lissajous trajectory. Since this is a naturally existing solution in this regime, the baseline control acceleration u () t is zero. The relative equations of motion for the i th deuty, then, are easily determined by subtracting Equation (1.2) from (1.3), C CDi ( Di) ( C) ( Di) ( C) ( Di) = ( grav grav ) + ( sr sr ) + D () = + i Di r f f f f u t f u t. (1.4) () The vector CDi r denotes the osition of the i th deuty relative to the chief sacecraft while f ( D i ) reresents the relative net force vector. 4

6 Let ρ reresent the desired nominal ath of the deuty sacecraft, then, ρ designates the nominal velocity vector, and u () t denotes the associated nominal control effort such that, D i () ρ = +. (1.5) Di f ud t i n this exression, the suerscrit ( ) imlies evaluation on the nominal ath defined by ρ and ρ. SPHERCAL FORMATONS FXED RADAL DSTANCE AND ROTATON RATE n revious studies [14-15] the alication of outut feedback linearization techniques to sherical formation configurations is considered. n the earlier formulation, only the radial distance between the chief and deuty vehicles is enforced while the orientation is left unconstrained. Hence, as long as the deuty sacecraft evolves along the surface of a shere, whose radius is equal to the desired nominal searation, the goal for the controller is satisfied. Since only one variable is tracked, and there are three control inuts, an infinite number of solutions are available. Subsequently, the formation keeing costs can vary dramatically deending on the articular solution that is selected. The difference between each solution is reflected in the rotation rate and the orbital lane corresonding to the converged solution. For instance, the second column of Table 1 summarizes the controller formulations from the earlier analysis. All four controllers in Table 1 accomlish the goal in terms of formation keeing, but the dynamical resonses and the associated costs are significantly different. To better understand these differences, as identified by Howell and Marchand [14-15], consider, briefly, the instantaneous rotating frame defined by i i the radial vector from the chief sacecraft to the deuty ( ˆ CD CD r = r r ), the angular momentum direction ˆ CDi CDi CDi CDi h= r r r r ), and the corresonding tangential unit vector ( ˆ θ = hˆ rˆ ). ( ( ) Table 1- OFL Control of Sherical Formations 5

7 At any given time, the angular momentum vector rovides information about the instantaneous relative orbital lane of the deuty vehicle ( ĥ ) as well as its rotation rate (θ ) about ĥ. The rate of change of the relative angular momentum vector can be exressed in terms of this rotating coordinate system as where f θ, CDi CDi h = r r = r f + u h r f + u ˆ, (1.6) ( ) ˆ ( ) θ θ h h θ D f h, u θ, and u h denote the comonents of f i and ˆ T h h h h resectively. The associated unit normal vector, ( ) 1/ 2 ( ) u D i along the ˆ θ and ĥ directions, =, varies with time as follows, ( + ) dhˆ r fh uh = ˆ θ. (1.7) dt h Thus, the lane of motion is reserved, and comletely determined by the relative initial state, if uh = fh. The last column in Table 1 summarizes the ˆ θ and ĥ comonents of the control inuts listed in the second column. t is simle to show that, for the last three entries in Table 1, the lane of motion is reserved. 2 Also, since h = h = r θ, the rotation rate θ is further determined by u h. For instance, for the second case listed in Table 1, u + f = r. While the radial distance converges to its final value, as commanded by θ θ θ the controller, the quantity u f 0 θ + θ. Hence, h h( tf ) rfθ f = and, thus, the rotation rate will converge on some limiting value. A similar statement can be made about the fourth entry in Table 1. Since the rotation rate is inversely roortional to the square of the radial distance, the formation keeing cost increases significantly for formations that seek to achieve nominal searations on the order of meters. The third entry in Table 1, however, is slightly different. n this case, it is determined that u + f = 0. This leads to h() t = r 2 () t θ () t = h( 0) and, consequently, the rotation rate along the surface of the shere is comletely determined by the initial relative state of the deuty at the time the controller is activated. For the case that emloys radial axis inuts, that is, the first entry in Table 1, the ˆ θ and ĥ comonents of ud i are zero which leads to h = rf hˆ rf ˆ θ h θ. This indicates that, in general, and for an arbitrary set of initial conditions, neither the rotation rate nor the relative orbit normal are reserved with this tye of control aroach. However, if the nominal relative searation is small, it may aear that the solution converges onto a lane, though small deviations are resent. These deviations are most visible at larger searations, on the order of hundreds or thousands of kilometers. These deviations are better visualized from the samle cases in Figure 1. θ θ 6

8 3-Axis Control Radial Control Only 3-Axis Control (a) (b) The blue shere in each subfigure reresents the nominal surface. n Figure 1(a), the nominal distance is 5 km (the radius of the nominal shere) as comared to 5000 km for the illustration in Figure 1(b). Each CD trajectory in Figure 1(a) is associated with an initial state characterized by i ( 0) [ ] ( 0) [ 1 1 1] CD r i = Figure 1 - mact of Relative Orbit Size on Controlled Path of Deuty S/C Radial Control Only/Free Orientation r = km and m/sec. The trajectories in Figure 1(b), are associated with the initial state CD r i ( 0) = [ ] km and CD r i ( 0) = [ 1 1 1] m/sec. Both figures deict two searate aths evolving onto the nominal shere. The green ath is associated with the controller in the fourth entry of Table 1, while the red ath is associated with the controller defined by radial axis inuts only, that is, the first entry in Table 1. Though numerical evidence may at times aear to suggest that each controller is converging onto the same orbital lane, as is aarent from Figure 1(a), the theoretical roof illustrates that radial axis inuts alone do not allow for this tye of solution. This is evident from Figure 1(b). Although reservation of the orbiting lane may be a desirable feature, the numerical and theoretical evidence clearly indicates the need to force the controller to track a articular rotation rate. Forcing a redetermined rotation rate revents the dynamics from inducing a sin rate that may lead to rohibitively high formation keeing costs. To constrain the rate, consider an augmented formulation of the original control law. n the initial develoment, the control inut is introduced into the radial dynamics through r r r r r r r = + r. (1.8) 2 r r r 7

9 t is straightforward to demonstrate that Equation (1.8) can also be written in the more familiar form,. (1.9) 2 r rθ = fr + ur The desired radial error dynamics are defined to follow a critically damed resonse, of natural frequency ω, such that the nominal radial resonse, g (, ) n r rr, is determined as, * * 2 * (, ) 2ω ( ) ω ( ) r = g r r = r r r r r. (1.10) r n n Here, the suerscrit * imlies evaluation on the nominal ath (i.e. the desired vehicle searation). The equation of motion relating the rotation rate to the control inut is determined by differentiating the angular momentum vector exression. This oeration leads to one equation of constraint, u + f = 0, as well as one additional equation of motion, r θ + 2r θ = f + u. (1.11) θ θ h h n this articular examle, the error dynamics for the rotation rate are secified as a decaying exonential, δθ k nt = δθ ( 0) e ω. The desired resonse, g θ ( ) θ, is then determined as ( ) * k ( * n ) θ = g θ = θ ω θ θ, (1.12) θ where k is an arbitrary scale factor and ω n is the same natural frequency as in Equation (1.10). Substitution of Equation (1.12) into (1.11) and of (1.10) into (1.9) leads to the following scalar control laws 2 () = (, ) u t g r r f rθ, (1.13) r r r () = ( θ) + 2 u t rg f r θ, (1.14) θ θ θ h () ( constraint) u t = f. (1.15) h The control inut that is actually alied to the deuty equations of motion, as listed in Equation (1.4), is then determined by transforming the inut vector described by Equations (1.13)-(1.15) back into the ehemeris inertial frame, [ ] E T Di r θ h u = C u u u, (1.16) where E C = rˆ ˆ θ hˆ. A samle imlementation of this control law is illustrated in Figure 2 for a 700 kg sacecraft targeting a 5 km radial searation and rotation rates of one revolution every six or twentyfour hours. The initial state is the same as that in Figure 1(a). 8

10 θ = 1 rev/6 hrs θ = 1 rev/day θ = 1 rev/6 hrs θ = 1 rev/day Figure 2 - OFL Control of Radial Distance + Rotation Rate and Associated Thrust Profile Using this tye of control law, it is ossible to determine how the formation keeing cost varies both as a function of the commanded relative searation and the commanded rotation rate. As before, the equation of constraint guarantees that the lane of motion is entirely determined by the relative state of the deuty before the controller is activated. The associated trends in the cost are illustrated in Figure 3 in terms of the mean thrust required to enforce the formation. Note that the cost increases quadratically with increasing rotation rate and linearly with the searation that is commanded between the chief and deuty vehicles. Figure 3 mact of Commanded Radial Distance and Sin Rate on Formation Keeing Costs 9

11 For a nominal searation of 50 meters, commanding the deuty to sin at one revolution er hour, about the chief sacecraft, requires over 100 mn of thrust, for a 700 kg vehicle. The mean thrust dros to 6.7 mn if the deuty vehicle is to nominally orbit the chief sacecraft once every 4 hours. The thrust levels continue to dro down to 0.19 mn for one revolution a day. f one revolution er day is required, a 500 meter searation raises the mean thrust to 18 mn. Though the lane of motion is not affected by this tye of control aroach, activating the controller at the aroriate time, or biasing the initial velocity, may be sufficient to achieve the desired orbital lane. Overall, the control strategy is concetually simle and numerically efficient for imlementation in the ehemeris model. A more ambitious goal involving OFL control, then, is to target asherical formations. ASPHERCAL FORMATONS Consider a multi-sacecraft formation where the chief vehicle evolves along a quasi-eriodic Lissajous trajectory near L 1 or L 2, as determined in the ehemeris model. The deuty vehicles in the formation are to follow the chief such that their relative motion evolves along the surface of a araboloid that is inertially fixed in orientation. The chief sacecraft and the nadir of the araboloid define the focal line of the formation. Maintenance of a constant distance between the chief vehicle and the nadir of the araboloid is the first objective of the controller. The second requirement is that the motion of all deuties, even during reconfigurations, is constrained to evolve along the surface of the araboloid. To accomlish these goals, it is first necessary to define a suitable arameterization for the formation surface. To that end, consider the illustration in Figure 4. eˆ 3 ( focal line) a Zenith ν ê 1 h u Nadir Figure 4 - Parameterization of a Paraboloid 10

12 The araboloid in Figure 4 is defined in an inertial frame (E) described in terms of the unit vectors ê 1, ê 2, and ê 3. The orientation of this frame, relative to the ehemeris inertial frame (), is defined by the azimuth (α ) and elevation (δ ) of the focal line relative to the inertial frame () unit vectors ˆX, Y ˆ, and Ẑ. This is better visualized from Figure 5, where the unit vector ê 3 is directed from the chief vehicle to the nadir of the nominal araboloid. Ẑ ê 1 eˆ 3 ( focal line ) δ α ˆX Ŷ Xˆ Yˆ Zˆ: Ehemeris nertial Frame ( ) eˆ eˆ eˆ : Paraboloid nertial Frame ( E) Figure 5 - Formation Focal Line Orientation This direction is reresentative of the focal line of the formation. The height of any given deuty vehicle along this surface, measured relative to the nadir oint, is defined by u where 0 u h and h denotes the maximum allowable height along the araboloid. The radius at the zenith of the araboloid ( u = h) is denoted by the variable a. The unit vector ê 1, defined as eˆ ˆ ˆ 1 = Z eˆ 3 Z eˆ 3, is simly a reference direction for the measurement of the angular osition (ν ) along the surface in Figure 1. Of course, eˆ2 = eˆ ˆ 3 e1 comletes the right handed inertial triad. The variables ν and u comletely secify the osition of a deuty vehicle along the formation surface in Figure 4. n the focal frame (E), the osition vector from the nadir to any given deuty vehicle along the surface of the araboloid is defined as (, ν) ( cosν) ( sinν) u = a u h eˆ + a u h eˆ + u eˆ. (1.17) The desired osition of the nadir, relative to the chief sacecraft, is secified as q = qeˆ 3. The equation of motion in (1.5) is associated with the inertial frame (). The nominal relative osition vector can also be written in terms of E-frame coordinates as ρe q ( u, ν ) relative acceleration vector, E ρ, can be exressed as E E = +. n this alternate inertial frame, the nominal ( ) ( ) ρ = q + u + v u + u + u + νν + ν, (1.18) E E u u u ν u νu νν ν 11

13 where u,, uu u ν, ν u, ν, and νν denote the first and second artial derivatives of with resect to u and ν, resectively. The scalar rates u and ν reresent the climb rate and the rotation rate along the nominal surface. Since both the E and frames are inertial, the accelerations in Equations (1.5) and (1.18) are related through the orientation matrix C E such that ρ { E } E = C matrix C E corresonds to the unit vector e ˆ j, for j = 1 3, defined in Figure 5. ρ. The j th column of the E n order for the nominal motion to satisfy Equation (1.18) recisely, the control law must be alied continuously and is determined as i e e 2 2 () = { } + ( + ν) + ( + 2 ν + ν ) ( ) ud t C q i u u uu u u u ν ν νν f r. (1.19) Consider a formation characterized by q ( 10 km) e 3 = ˆ, h = 500 m, a = 500 m. Let the focal line be oriented such that α = 0, and δ = 45. Each vehicle in the formation is to comlete one revolution along the surface once a day, ν = 1 rev/day. The nominal motion of one of the vehicles is initially described by u = 200 meters such that u = u = 0 and ν = 0. After 5 days, the vehicle s nominal ath must be reconfigured, along the araboloid, such that ν and u remain constant. The climb rate, u, is secified such that, after 1 day, the vehicle has raised it s height to 500 meters relative to the nadir of the araboloid, a 300 meter climb relative to the initial orbit. The nominal control law in Equation (1.19) leads to the desired motion as illustrated in Figure 6. Figure 6 Nominal Geometry and Thrust Requirements for a Samle Parabolic Formation 12

14 n this Figure, the off-center circular orbit closest to the nadir (red) reresents the initial nominal motion over a san of 5 days. The green siral defines the reconfiguration segment over the next day while the highest circular orbit (blue) denotes the converged solution over the remaining 5 days. The associated nominal thrust rofile reveals that a range of 1.4 to 2.0 mn of thrust is required to enforce the formation, including the reconfiguration, in the absence of external erturbations. Outut Feedback Linearization: Alication to Parabolic Formations The control design resented above is associated with nominal formation keeing. That is, assuming the vehicles are already in the desired configuration, Equation (1.19) establishes the thrust rofile necessary to enforce the desired formation continuously. n the examle illustrated in Figure 6, each hase is comuted searately and indeendently from the other. For instance, the end-state of Phase, lus the necessary imulsive V, is used as the initial state to comute the trajectory associated with Phase. A similar aroach is emloyed to determine the trajectory associated with Phase. Hence, the above solution addresses neither the deloyment nor the reconfiguration of an actual continuous solution. To address this asect, outut feedback linearization techniques are alied. n rior investigations, inut/outut feedback linearization (FL/OFL) is successfully alied to achieve the desired formation keeing goals. The earlier OFL examles are based on a tracking scheme involving relative distance and rotation rate. Hence, the number of variables to track is less than the number of available control inuts. As such, there are an infinite number of solutions available that satisfy the goals of the controller. n the resent case, a arabolic configuration requires the tracking of three variables: u () t, q() t, and ν () t. That is, the distance to the nadir, the height of the vehicle along the araboloid surface and the orientation time history along the surface. To illustrate how an OFL controller may be alied to this tye of configuration, it is necessary to establish a set of exressions relating the state variables (and control inuts) to the tracked quantities. For instance, recall that ρ [ ] T E = C ρe = x y z, (1.20) where the symbol ~ above each variable indicates that the measure numbers are associated with the focal frame (E) of the araboloid. 13

15 These measure numbers are related to the araboloid arameters in Figure 4 as follows, x = a u / h cosν, (1.21) y = a u / h sinν, (1.22) z = q+ u. (1.23) Squaring and then adding Equations (1.21)-(1.22) reveals that ( / ) u = h a x + y, (1.24) while dividing Equation (1.22) by Equation (1.21) yields Furthermore, Equation (1.23) indicates that ( y x) tan ν = /. (1.25) ( / ) q = z h a x + y (1.26) The associated rates are defined by differentiating Equations (1.24)-(1.26) with resect to time, i.e., 2h 2 a ( ) u = xx + yy, (1.27) 2h 2 a ( ) q = z xx + yy, (1.28) ν = xy yx 2 2 ( x + y ). (1.29) With these exressions, it is ossible to identify relationshis between the control inuts and the tracked variables. These relationshis are obtained by differentiating Equations (1.24)-(1.26) twice with resect to time. Differentiation is straightforward and ultimately suggests that, 2h 2h 2h g u u x + y + x f + y f = xu + yu. (1.30) 2 2 (, ) ( ) u 2 x y 2 x 2 y a a a 2h 2h 2h g qq + x + y + x f + y f f = xu yu + u (1.31) 2 2 (, ) ( ) q 2 x y z 2 x 2 y z a a a ( ν ) ( + )( ) 2 2 ( x + y ) ( y f x x f y) y ( x + y ) ( x + y ) ( x + y ) xx yy xy yx x y g = u u ν x (1.32) 14

16 n Equations (1.30)-(1.32), recall that the ~ reresents quantities associated with the focal frame (E) of the araboloid. Hence, E u C u ux uy uz T = = reresents the transformed control inut vector. Similarly, T E ( Di ) f C f f x f y f and E [ ] T z r = C r = x y z = = denote the differential force and the radial distance to the chief sacecraft, in terms of focal frame coordinates, resectively. The scalar functions u (, ) g u u, g ( qq, ) and ( ν ) q g ν reflect the desired dynamical resonse for the variables u, q, and ν. As an examle, critically damed error dynamics are desired for the distance elements u and q while an exonentially decaying error resonse is sought for ν. An exact solution is available for Equations (1.30)-(1.32). To simlify the form of the solution, let α = 2 hx / a, 2 x α y = 2 hy / a, β x = x / ( x + y ) side of Equations (1.30)-(1.32) be summarized by control inut is exressed 2 2, and β y = y/ ( x + y ) x. Furthermore, let the left hand G u, G q, and G ν, resectively. Then, the commanded ( β G α G ) x u y ν u =, (1.33) y ( α xβx + α yβ y) ( β G + α G ) y u x ν u =, (1.34) ( βα x x + βα y y) u = G + G. (1.35) z u q Note, the above control law is singular if the deuty crosses the focal line ( ê 3 ), x = y = 0. This does not resent a significant issue, however, because once the deuty is on the surface of the araboloid this condition is never met. This singularity can only occur while the deuty is being driven onto the surface during the injection hase. To circumvent this difficulty, it is only necessary to allow the vehicle to coast away from this oint before reactivating the controller. For imlementation of this aroach in the ehemeris model, the integration of each vehicle roceeds searately in an Earth centered inertial frame (). The relative state of the deuty with resect to the chief is comuted from these Earth centered states. These quantities are then transformed into the focal frame (E) via E r = C r and E r = C r. The results of this transformation are substituted into Equations (1.24)- (1.29) to determine the values of the quantities to be tracked as well as the necessary control accelerations, as comuted from Equations (1.33)-(1.35). 15

17 To illustrate the alication of this aroach, consider the nominal samle scenario reviously introduced and deicted in Figure 6. An injection error is introduced such that the initial relative state of the deuty, with resect to the chief sacecraft, is characterized by r ( 4.6X ˆ 4.3Y ˆ 6.934Z ˆ ) = + + km with a relative velocity defined by r = 0.05Xˆ 0.05Yˆ 0.05Zˆ m/sec. This articular initial state is arbitrarily selected to facilitate the visualization rocess. For any arbitrary initial state, the controller should drive the vehicles in the formation to the initial configuration, then reconfigure at the aroriate time. Once deloyed, this evolution is to roceed along the surface of the araboloid. Alication of this controller results in the trajectory illustrated in Figure 7. t days t days Phase 1 4 days + 15 hrs. Phase 2 1 day Phase 3 5 day ê 3 Ẑ t hrs ˆX t 0 ˆ Y t hrs t hrs Figure 7 OFL Controlled Parabolic Formation The resulting ath is divided into three segments. The segment highlighted in orange reresents the injection hase as well as the initial orbit hase, for u = 200 m. The green segment denotes the reconfiguration hase, characterized by u = 300 meters/day. The last segment, in blue, is the final hase associated with u = 500 m. The thrust rofile associated with this solution aears in Figure 8. Note that, at the resent stage of develoment, each vehicle is controlled searately. Future studies must incororate relative information between deuties to assess the robability of collisions. 16

18 Figure 8 - Thrust Profile for OFL Controlled Parabolic Formation Comaring Figure 6 to Figure 8, it is clear that the OFL controller converges to the nominal control excet for the initial correction that is necessary for injection into the nominal configuration. The thrust levels range from 25mN during injection to 1-2 mn for orbit maintenance. Note that the maximum thrust amlitude will vary according to the resonse frequency that is secified. Also, though both the nominal and actual thrust rofiles, aear to converge onto constant segments, there is an oscillation on the order of mn during the orbit hase at u = 200 (last art of Phase 1) m and u = 500 m (Phase 3). Unlike the configurations reviously considered, the chief sacecraft, in this case, is not the center of the formation. NATURAL AND NON-NATURAL FORMATONS N THE EPHEMERS MODEL Besides the non-natural formations identified in the revious discussion, there exist some natural relative behaviors that can be exosed and otentially exloited for formations. For examle, a variety of natural formations can be isolated through knowledge of the dynamical flow near a halo orbit. Howell and Barden [6-7] identify a lanar formation in which individual vehicles evolve along the surface of a twodimensional hollow torus. The vehicles always remain in a lane although the formation exands and contracts and the lane changes orientation. 17

19 This torus is known to envelo the reference halo orbit in the CR3BP. This tye of solution also exists in the ehemeris model and can be identified, numerically, via a two-level differential correction rocess, such as that develoed by Howell and Pernicka [20]. The relative dynamics associated with the torus are determined from a center manifold analysis, based on the linear stability roerties of the reference solution, that is, the reference halo orbit. The reference halo orbits that are of interest for sacecraft missions are usually unstable. The stability of such an orbit is characterized by one stable and one unstable mode, as well as four center modes. The torus formation identified by Howell and Barden [6-7], aearing in Figure 9(a), is associated with two of these center modes. ŷ ˆx ŷ ˆx ẑ ẑ ˆx ŷ ẑ ˆx ẑ ŷ (a) (b) Figure 9 - Natural Quasi-eriodic Formation Associated with Reference Halo Orbit The reference halo orbit evolves along the interior (center) of this two-dimensional hollow torus. Relative to the chief sacecraft, evolving along this reference halo orbit, the deuty motion is constrained to evolve along the surface deicted in Figure 9(b). The surfaces in Figure 9(a-b) are reresentative of the same motion as seen by different observers. Figure 9(a), for instance, is indicative of the motion of the deuty vehicle (D) as seen by an observer fixed at the libration oint (L i ), r Li D. Figure 9(b), then, is reresentative of the motion of the deuty vehicle (D), as observed by the chief sacecraft (C) as it evolves along the LiD LC i reference halo orbit, r r. 18

20 The remaining two center modes lead to nearby eriodic halo orbits. The relative motion associated with these two center modes reveals the existence of eriodic and slowly exanding relative orbits, deending on the target state within the new subsace, as demonstrated by Marchand and Howell [14]. The aroximate eriod of these relative solutions is close to that of the reference halo (~180 days). Figure 10 illustrates the rojection onto the rotating yz-lane of a samle set of these solutions, as determined in the CR3BP. Note, the blue shere in Figure 10 reresents the chief sacecraft but is greatly enlarged for ease of visualization. Figure 10 - Periodic and Slowly Exanding Relative Orbits in the CR3BP n Figure 9(b) and Figure 10, the chief sacecraft is located at the origin. The methodology imlemented by Marchand and Howell [14], based on Floquet theory, is used to numerically identify these relative orbits in the CR3BP. The same rocedure is used as an imulsive control scheme to deloy sacecraft into these natural solutions through a single injection maneuver. n the current work, these solutions are transitioned into the ehemeris model using a two-level corrector [20]. For the nearly eriodic solutions, the resulting motion is essentially unchanged by the transition, as seen from Figure 11(a). Even the addition of solar radiation ressure does not aear to have a significant imact, deending on the sacecraft mass and effective area, of course. However, as the orientation of the relative orbit is shifted towards the vertical orbits, Figure 11(b-c), the imact of SRP is much more visible. These naturally existing motions can be used as initial guesses to comute non-natural formations. n this initial investigation, assume that only imulsive control is available to achieve the objective. For instance, consider the orbit in Figure 11(c). The relative ath is clearly not eriodic but, the initial guess is sufficiently close to eriodic if the effects of SRP are small. 19

21 Figure 11 Natural Formations in the Ehemeris Model mact of Solar Radiation Pressure (SRP) n this case, a differential corrector can be alied to enforce eriodicity if imulsive maneuvers are allowed. The rocess is similar to a method commonly used to transition halo orbits into the ehemeris model. Consider the first two revolutions of the ath (without SRP) in Figure 11(b). A two-level differential corrections rocess, in this case with initial- and end-oint constraints, determines the maneuver necessary to close the orbit over this time eriod. The atch oints associated with the converged solution are then shifted forward in time to add N additional revolutions. The comlete solution is then differentially corrected while allowing maneuvers at the intersections between revolutions. A samle solution, over six revolutions, is illustrated in Figure 0 and is obtained by alying two imulsive maneuvers ranging in size from 2.5 m/sec to 5 m/sec at the secified locations. A similar aroach can be alied to the nearly vertical trajectory in Figure 11(c) to obtain vertical eriodic relative orbits, as illustrated in Figure 13. This articular aroach works very well if eriodicity is enforced in the rotating frame, as oosed to the inertial frame. Relative to an observer fixed in the rotating frame, these solutions aear to be sufficiently close to eriodic and are, subsequently, a suitable initial guess for the differential corrector. However, the associated inertial ersectives, illustrated in Figure 14, are quite different. Thus far, these do not aear to rovide a sufficiently accurate initial guess if eriodicity is required. That is because of the natural geometry of the solution and the fact that the Earth is at a different location every time a revolution is comleted, as oosed to a ersective originating in the rotating frame. At the resent time, however, no conclusive statements can be made since this is still a subject of ongoing study. 20

22 V Figure 12 Controlled Periodic Orbit in the Ehemeris Model (w/o SRP) V Figure 13 Controlled Vertical Orbits in the Ehemeris Model (w/o SRP) 21

23 Figure 14 Natural Formations in the Ehemeris Model nertial Frame Persective of Figures 11(a-c) (w/ SRP) CONCLUSONS n the resent investigation, outut feedback linearization (OFL) techniques are successfully alied to deloy, enforce, and reconfigure sherical and asherical formations near the vicinity of the libration oints of the Sun-Earth/Moon system. All the control laws considered are develoed and imlemented in the full ehemeris model including solar radiation ressure and any number of desired gravitational erturbations. Deending on the constraints imosed on the formation, the results resented in this study often require thrust levels on the order of milli-newton s for a 700 kg sacecraft. n contrast, the deloyment hase can require thrust levels in the Newton range, deending on when the controller is activated. This range of thrust levels aears to be achievable with resently available technology. However, currently, these technologies are mostly devoted to station keeing and attitude control, which does not often require long term oeration. Hence, the lifetime of the roulsion system, within the context of the configurations defined here, may require further enhancements. Even if the formation constraints can be met, theoretically, the sensitivity of these methods to modeling and thruster imlementation errors must be addressed. This is of articular imortance for libration oint missions given the associated dynamical sensitivities to small erturbations. Certainly, on any of the examles resented here, discontinuing the thrust inut at any time, or lacing bounds on its amlitude, leads to divergence from the desired nominal, esecially with the higher rates of rotation that induce higher tangential seeds. Another area of interest in this study is the use of naturally existing configurations to construct non-natural formations via imulsive control. The methods associated with the determination of these natural solutions are determined from center manifold analysis and the alication of a reviously develoed Floquet controller to deloy into these configurations. Thus far, the method resented is only alied to establish eriodic vertical relative orbits in the ehemeris model. These and other techniques are still under study regarding configurations that are, in some general sense, fixed relative to the inertial frame. 22

24 The knowledge gained from the alication of the continuous (OFL Controller) and imulsive techniques (Floquet Controller) can otentially, be alied as a combined formation keeing strategy. That is, if the formation goals only require the nominal configuration to be maintained over a redetermined time interval, the Floquet controller can be used to lace the vehicles into natural formations until the next reconfiguration is necessary. At this oint, the continuous controller can once again be activated to return the vehicles into the desired configuration. ACKNOWLEDGEMENTS This research was carried out at Purdue University with suort from the Clare Boothe Luce Foundation and the National Aeronautics and Sace Administration, Contract Number NCC REFERENCES 1 D.J. Scheeres and N.X. Vinh, Dynamics and control of relative motion in an unstable orbit. AAA/AAS Astrodynamics Secialist Conference, Denver, Colorado, Aug , AAA Paer No P. Gurfil and N. J. Kasdin, Dynamics and Control of Sacecraft Formation Flying in Three-Body Trajectories. AAA Guidance, Navigation, and Control Conference and Exhibit, Montreal, Canada, Aug. 6-9, AAA Paer No P. Gurfil, M. dan, and N. J. Kasdin, Adative Neural Control of Dee-Sace Formation Flying. American Control Conference (ACC), Proceedings. Vol. 4, Anchorage, Alaska, May 8-10, 2002, N.H. Hamilton, Formation Flying Satellite Control Around the L2 Sun-Earth Libration Point. M.S. Thesis, George Washington University, Washington, DC, December D. Folta, J.R. Carenter, and C. Wagner, Formation Flying with Decentralized Control in Libration Point Orbits. nternational Symosium: Saceflight Dynamics, Biarritz, France, June, B.T. Barden and K.C. Howell, Fundamental Motions Near Collinear Libration Points and Their Transitions. The Journal of the Astronautical Sciences, Vol. 46, No. 4, 1998, B.T. Barden, and K.C. Howell, Formation Flying in the Vicinity of Libration Point Orbits. Advances in Astronautical Sciences, Vol. 99, Pt. 2, 1998,

25 8 B.T. Barden and K.C. Howell, Dynamical ssues Associated with Relative Configurations of Multile Sacecraft Near the Sun-Earth/Moon L 1 Point. AAS/AAA Astrodynamics Secialists Conference, Girdwood, Alaska, August 16-19, 1999, AAS Paer No. AAS K.C. Howell and B.T. Barden, Trajectory Design and Stationkeeing for Multile Sacecraft in Formation Near the Sun-Earth L 1 Point. AF 50th nternational Astronautical Congress, Amsterdam, Netherlands, Oct. 4-8, AF/AA Paer No. 99-A G. Gómez, M. Lo, J. Masdemont, and K. Museth, Simulation of Formation Flight Near Lagrange Points for the TPF Mission. AAS/AAA Astrodynamics Conference, Quebec, Canada, July 30-Aug. 2, AAS Paer No K.C. Howell and T. Keeter, Station-Keeing Strategies for Libration Point Orbits - Target Point and Floquet Mode Aroaches. Advances in the Astronautical Sciences, Vol. 89, t. 2, 1995, G. Gómez, K.C. Howell, J. Masdemont, and C. Simó, Station-keeing Strategies for Translunar Libration Point Orbits. Advances in Astronautical Sciences, Vol. 99, Pt. 2, 1998, K.C. Howell and B.G. Marchand, Control Strategies for Formation Flight in the Vicinity of the Libration Points. AAA/AAS Sace Flight Mechanics Conference, Ponce, Puerto Rico, Feb. 9-13, AAS Paer No B.G. Marchand and K.C. Howell, Formation Flight Near L 1 and L 2 in the Sun-Earth/Moon Ehemeris System ncluding Solar Radiation Pressure. AAS/AAA Astrodynamics Secialists Conference, Big Sky, Montana, August 3-8, AAS Paer No K.C. Howell and B.G. Marchand, Design and Control of Formations Near the Libration Points of the Sun-Earth/Moon Ehemeris System Sace Flight Mechanics Symosium Goddard Sace Flight Center, Greenbelt, MD. 16 EO-1 Pulsed Plasma Thruster, htt://sace-ower.grc.nasa.gov/o/rojects/eo1/eo1-t.html. 17 J. Mueller, Thruster Otions for Microsacecraft: A Review and Evaluation of Existing Hardware and Emerging Technologies. AAA/ASME/SAE/ASEE Joint Proulsion Conference & Exhibit, 33rd, Seattle, WA, July 6-9, AAA Paer No

26 18 A. D. Gonzales and R. P. Baker, Microchi Laser Proulsion for Small Satellites. AAA/ ASME/ SAE/ ASEE Joint Proulsion Conference and Exhibit, 37th, Salt Lake City, UT, July 8-11, AAA Paer No C. Phis and J. Luke, Diode Laser-Driven Microthrusters - A New Dearture for Microroulsion. AAA Journal, Vol. 40, No. 2, Feb. 2002, K.C. Howell and H. J. Pernicka, Numerical Determination of Lissajous Trajectories in the Restricted Three Body Problem. Celestial Mechanics, Vol. 41, ,