LATELY, numerous missions involving satellites flying in

Transcription

1 JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vol. 3, No., January February Secial Inclinations Allowing Minimal Drift Orbits for Formation Flying Satellites M. Sabatini University of Rome La Saienza, Rome, Italy D. Izzo ESA, AZ Noordwijk, The Netherlands and R. Bevilacqua University of Rome La Saienza, 6 Rome, Italy DOI:.54/.334 The ossibility of obtaining a natural eriodic relative motion of formation flying Earth satellites is investigated both numerically and analytically. The numerical algorithm is based on a genetic strategy, refined by means of nonlinear rogramming, that rewards eriodic relative trajectories. First, we test our algorithm using a oint mass gravitational model. In this case the eriod matching between the considered orbits is a necessary and sufficient condition to obtain invariant relative trajectories. Then, the J erturbed case is considered. For this case, the conditions to obtain an invariant relative motion are known only in aroximated closed forms which guarantee a minimal orbit drift, not a motion eriodicity. Using the roosed numerical aroach, we imroved those results and found two coules of inclinations (63.4 and 6.6 deg, the critical inclinations, and 49 and 3 deg, two new secial inclinations) that seemed to be favored by the dynamic system for obtaining eriodic relative motion at small eccentricities. Nomenclature a = orbit semimajor axis a i, b i = eicyclic orbital elements e = orbit eccentricity f = bector function reresenting the satellites relative dynamics f r, f, f z = erturbing forces in r,, z coordinate system i = orbit inclination J = objective function for the genetic otimizer J = Earth flattening term in gravitational field series exansion M = orbit mean motion N gen = number of generations in the genetic otimizer N ind = number of individuals in the genetic otimizer P = erturbing actions = orbit semilatus rectum R = rotation matrix from inertial to LVLH frame R = Earth equatorial radius r X Y Z = satellite absolute osition vector in inertial frame Presented at the AAS/AIAA Sace Flight Mechanics Conference, Tama, FL, 6 January 6; received 7 February 7; revision received 4 May 7; acceted for ublication 4 May 7. Coyright 7 by Marco Sabatini, Dario Izzo, and Riccardo Bevilacqua. Published by the American Institute of Aeronautics and Astronautics, Inc., with ermission. Coies of this aer may be made for ersonal or internal use, on condition that the coier ay the $. er-coy fee to the Coyright Clearance Center, Inc., Rosewood Drive, Danvers, MA 93; include the code 73-59/ $. in corresondence with the CCC. Ph.D. Candidate, School of Aerosace Engineering, Via Eudossiana 6. Advanced Concets Team System Engineer, Euroean Sace Technology and Research Center, Kelerlaan; dario.izzo@esa.int. Ph.D. Candidate, Mathematical Methods and Models for Alied Sciences Deartment, Via Antonio Scara 6; bevilacqua@dmmm. uniroma.it. Student Member AIAA. 94 T = candidate eriod t = time x x y z _x _y _z = satellites relative state vector in Cartesian coordinates i, i = canonical momenta = satellites initial relative hase angle V = velocity instantaneous variation " = erturbation term = on orbit satellite anomaly = decision vector for the genetic otimizer = Earth gravitational constant $ = orbit argument of erigee = satellites relative distance x y z = satellites relative osition vector = orbit right ascension of ascending node! = LVLH frame angular velocity vector, f = value at initial and final time I. Introduction LATELY, numerous missions involving satellites flying in formation have been lanned or studied. A brief review includes ESA missions Proba 3, LISA, XEUS, Darwin, and SMART-3, NASA s missions EO-, ST5, and Terrestrial Planet Finder, the international mission known as A-Train and the JAXA s SCOPE mission. To kee the satellites of the formation in the designed configuration, and therefore to achieve the mission s goals, control actions are needed. The cost of this orbital control in terms of V limits both the mission duration and the exected erformances. Advantageous dynamics could reduce the cost of these oerations, in articular, the ossibility to obtain eriodic or quasi-eriodic natural relative motion would be a significant saving factor as recently argued by Becerra et al. []. Many different aroaches to find a eriodical relative motion are considered in the recent literature on this toic. Inalhan et al. [] found the analytical exression for the initial conditions resulting in eriodic motion based on the classical Tschauner Hemel equations [3]. Kasdin and Koleman [4] used the eicyclic orbital elements theory to derive bounded, eriodic orbits in the resence of various erturbations. Vaddi et al. [5] studied the Hill Clohessy Wiltshire [6] (HCW) modified system to include

2 SABATINI, IZZO, AND BEVILACQUA 95 second-order terms. Finally, Schaub and Alfriend [7] formulated the conditions for invariant J relative orbits introducing relations between the mean orbital elements of the two satellites. The analytical aroaches taken in these works lead to two kinds of findings: ) initial conditions that ensure exact eriodicity in aroximated dynamic models or ) initial conditions resulting in bounded (i.e., with minimum drift, but not eriodic) relative motion in more detailed dynamic models. On the other hand, the numerical aroach taken here is able to reveal reviously unknown features of the minimum drift relative trajectories for satellites in a fully nonlinear, erturbed environment. The aer is structured as follows. In Sec. II, we define our roblem as an otimization roblem and we describe the numerical technique we use to solve it. In Sec. III, we test the behavior of the algorithm by alying it to the well-known and simle unerturbed relative satellite motion case. In Sec. IV we introduce the J erturbation and we aly our algorithm to this case commenting on the unexected results. II. Problem Statement Consider a generic nonautonomous dynamic system _x fx;t, for examle, the relative dynamics of satellites flying in formation. Define x x xt, where x is the system state at the initial time and T is a time variable here called candidate eriod for reasons that will soon be clear. Then, the following otimization roblem is defined: >< find X T T to maximize J Jjxj () >: subject to _x fx;t where the objective function J is constructed in such a way as to have its global maximum at x. The otimization roblem above is equivalent to the task of finding as-eriodic-as-ossible solutions to the system _x fx;t. Exloiting this aroach, we solve the roblem in Eq. () for a number of different dynamic models reresenting the relative motion between satellites under different orbital environments. The as-eriodic-as-ossible solutions corresond, in our case, to minimal relative orbit drift. As we study a number of systems _x fx;t, we face different otimization comlexities and objective function roerties. Thinking about the relative motion between satellites moving on Kelerian orbits, the roblem defined by Eq. () has an infinite number of solutions, corresonding to orbits with equal semimajor axis. A similar structure is also exected when the Kelerian dynamic is erturbed. As a consequence, a genetic aroach, avoiding issues related to domain knowledge and being able to coe with multile local and global minima, has been selected to erform a search in the solution sace. The PIKAIA freely available software [] was used in this work as a genetic otimizer. PIKAIA encodes the decision vector using a decimal alhabet. Table shows the fundamental arameters of a genetic algorithm (GA) used in all the simulations. The best solution returned by the genetic algorithm is then refined locally by means of a nonlinear rogramming numerical solver. In our simulations the decision vector contains the initial relative osition, the initial relative velocity, and the candidate eriod T.We consider the relative motion between two satellites: a chief and a deuty to use a oular terminology connected to formation flying research. The absolute dynamics of both the chief and the deuty are simulated roagating the inertial coordinates of the sacecraft in time, d r dt r 3 r P where P is the erturbing action considered, is the lanetary constant, and r is the orbital radius vector. The relative state is then evaluated by means of Eq. (): ( xyz T RX d Y d Z d T X c Y c Z c T _x _y _z T R _X d _Y d _Z d T _X c _Y c _Z c T! xyz T () where R is the rotation matrix from the inertial coordinate system to the local-vertical/local-horizontal (LVLH) frame in which the relative state is defined. The subscrits c and d stand for chief and deuty satellites. The orbit of the chief is considered known and the initial conditions to roagate the deuty motion are obtained transforming the relative x osition into absolute coordinates inverting Eq. (). As PIKAIA requires the decision vector comonents to satisfy the constraints k ;,wedefine a simle transformation between and a new decision vector that can be used by the otimizer. For the initial relative distances: x y z 3 K (3) This limits the range of variation for the initial relative osition to K; K km. The arameter K allows bounding the dimension of the minimum drift orbit one is interested in finding. Similarly, for the relative initial velocities we set _x _y _z K (4) This limits the initial velocities in the range ; K. We then defined T T ke 7 k, where k is a roerly chosen constant (some tens of seconds) and T ke is the orbital eriod a 3 = of the chief orbit. T is clearly a crucial arameter. At T, the final relative coordinates are comared to the initial relative coordinates, thus determining the quality of the individual. A good individual has a small x and its osition in the individual ranking is high; therefore it has a larger chance to mate and to generate good sons. Its genes will survive in the next generation, and if they will be ranked first in the last generation, they will be further refined by a local otimizer and reresent the set of initial conditions that generate the minimum drift relative orbit. The fitness function we used to rank the individuals is Table Parameters used for the genetic otimizer N ind no. of individuals N gen no. of generations 5 No. of significant digits (no. of genes) 9 Crossover robability.5 Mutation mode Initial mutation rate.5 Minimum mutation rate.5 One-oint, adjustable rate based on fitness Maximum mutation rate Reroduction lan Steady state/relace worst

3 96 SABATINI, IZZO, AND BEVILACQUA J q : x fx x y fy y z fz z _x f _x _x _y f _y _y _z f_z _z (5) A erfect individual (eriodic motion) has a fitness value of, while a ercentage difference of.% between the initial and the final state, brings down the fitness value to 5, a difference of % corresonds to a fitness value of 9, and so on. III. Unerturbed Case To test and tune our aroach, we first consider the dynamic system describing the relative motion around a erfectly sherical Earth with uniform mass, and we comare our results with the semimajor axis matching condition that assures a erfectly eriodic relative motion. We also erform comarisons with several other known aroximate results coming from Taylor exansions of the original nonlinear system. These are described next. A. HCW Case Consider the Hill -Clohessy Wiltshire linear equations, valid for circular unerturbed reference orbits: >< x! _y 3! x y! _x (6) >: z! z 5.4 x Fig. Effects of the nonlinearities on the HCW condition for small (left) and large (right) formations. Drift/orbit (km/orb) Formation initial dimension (km) Fig. Range of validity of the HCW condition. The eriodic motion condition, coming from the suression of the secular term in the equations solution, is _y!x (7) Trivially, as the formation dimension grows, the nonlinearities make this condition (which we refer to as the HCW condition) no longer valid, even when only Kelerian forces are considered. The wellknown relative trajectories in Fig. are obtained by roagating for eriods some initial conditions fulfilling the HCW condition using a nonlinear, nonerturbed model. In Fig. the drift er orbit obtained alying the condition in Eq. (7) is reorted as a function of the formation dimension K. The initial relative osition and velocities considered are K; ; :5; ;!; km. The drift er orbit is measured as the difference between the sacecraft relative distance at the initial time and after one eriod as obtained by roagating the dynamics with a nonlinear model. B. Nonlinear Correction Vaddi et al. [5] develoed a model that takes into account the effects of nonlinearities, both for circular and for ellitical orbits. Following the same aroach of the Taylor series exansion used to derive the HCW equations, but retaining also quadratic terms, one may obtain the following model: >< x! _y 3! x " y! _x "xy >: z! z "xz y z x where " 3=a 4. A condition for eriodic relative orbits is then reached: x ;y ;z sin!t ;cos!t ;sin!t (9)! _x ; _y ; _z cos!t ; _y;!cos!t where is the relative distance and the initial relative hase angle. The only variable influencing the secular growth of the relative motion is _y, which can be written as _y_y h "_y cn () where _y h is the term deriving from the HCW condition and _y cn is the correction for the nonlinearity: _y cn 6 cos 4! () C. Eicyclic Elements A different analytical formulation is found in [4]. Here Kasdin and Koleman use a Hamiltonian aroach to derive the equations of motion for an object relative to a circular or slightly ellitical reference orbit. By solving the Hamilton Jacobi equation in terms of the eicyclic elements they are able to rovide analytical aroximations of the invariance condition. By means of this formalism, they derive bounded, eriodic orbits in the resence of various erturbations, among them the nonlinearities. Here we only reort the conditions found for the circular reference orbit case. Two exressions are given to comute a normalized _y [in Eq. (5) the ()

4 SABATINI, IZZO, AND BEVILACQUA 97 a (m) nonlinear correction second-order eyciclic condition third-order eyciclic condition a (m) ρ (km) Fig. 3 Difference of semimajor axis vs initial dimensions..7 Genetic algorithm Third-order condition ρ (km) Fig. 4 Comarison between GA and third-order eicyclic condition. bars stand for the distances being normalized by the reference orbit semimajor axis a, the time by the angular velocity!, giving a- dimensional quantities]: ) the first one considers second-order terms in the series exansion for the initial conditions, ) the second considers also third-order terms: a 3 5 a a b b 3a b 3 b 3 () a 3 5 a a b b 3a b 3 b 3 3 a b a b b3 (3) In both cases: a cos b sin a cos b (4) sin a 3 3 b 3 3 where i, i are the initial canonical momenta and coordinates, which can be written as functions of the initial conditions. For brevity we omit the subscrit in the following: _x _y 3x _z z 3 _y x 3 x (5) _y z tan tan _x 3 _x y _z Substituting Eq. (5) in Eq. (4), using the conditions in Eq. () or in Eq. (3) (according to the order of the chosen aroximation), and solving for _y, gives the initial _y for bounded orbits. The difference between the semimajor axes of the sacecraft in the formation is a good index of how near the aroximation of the analytical conditions is to the hysical one (i.e., semimajor axis matching); a measure of the drift er orbit can be given as [9] drift =orbit 3a (6) The difference a resulting by using condition () and () or (3) can be lotted for various formation dimensions as shown by Fig. 3. The third-order eicyclic conditions are a very good aroximation of the eriod-matching condition, and indeed the use of a numerical aroach is not necessary in this case. We rather used these analytical results and the eriod-matching condition to test the erformances of the numerical technique and to tune the otimizer arameters. The final comarison between the best analytical (third-order eicyclic) and numerical (genetic algorithm without final local refinement) solutions is showed in Fig. 4. Figure 4 shows the main difference between the analytical and numerical aroach: the a resulting from the genetic algorithm simulations oscillates because of the stochastic nature of the otimizer, while the a resulting from the alication of the thirdorder conditions grows with the formation dimensions. Similar results can be obtained for ellitical unerturbed reference orbits. IV. J Perturbed Case Let us study the solutions of Eq. () in the case where f describes the relative motion between two satellites orbiting around an oblate Earth. The objective function J is again given by Eq. (5). As already mentioned, this corresonds to minimizing the relative orbital drift. Some revious work has been done to determine the ossibility of invariant relative satellite motion when J is considered as a erturbing term. In articular, the aer by Schaub and Alfriend [7] introduces the so-called J invariant relative orbits. In their work, mean orbital elements are used and the secular drifts of the longitude of the ascending node and of the sum of the argument of erigee and mean anomaly are set to be equal between two neighboring orbits. By having both orbits drifting at equal angular rates on the average, they will not searate over time due to the J influence. Two first-order conditions are resented in [7]: a Da e e tani i (7) 4e where 4 tani i e () and D is a arameter deending on i, a, and. The combination of Eqs. (7) and () rovides the two necessary conditions on the mean orbital element differences yielding a J -invariant relative orbit. When designing a relative orbit using the mean orbital element differences, i, e, ora is chosen, the remaining two element differences are then found through the two constraints in Eq. (). The remaining element differences, $, and M can then be chosen at will without affecting the J invariance. Even though called J invariant orbits, these two conditions are only valid in a first-order aroximation. When using these conditions the relative orbit is still exhibiting a relative drift, as Fig. 5 shows for an almost circular 35 deg inclined reference orbit. Proagation is again erformed via a nonlinear model including J as a erturbation. The conditions in Eqs. (7) and () reresent two elegant relations defining relative orbits with a small drift er orbit. We use our numerical aroach based on the solution of the global otimization roblem stated in Eq. () to check to what extent the residual drift obtained with this analytical aroach is an artifact of the use of mean elements. Reeating the calculation for the entire range of inclinations, the results vary sensibly, disclosing a reviously unknown feature of invariant relative motion. In Fig. 6, we reort the final fitness function reached for different inclinations ranging from to deg. The other orbital arameters of the Chief

5 9 SABATINI, IZZO, AND BEVILACQUA Analytical J invariant orbit: Fig. 5 relative orbits for a J erturbed case at 35 deg inclination (using the J invariance condition)..5 6 Fig. 7 relative orbits for a J erturbed case at 35 deg inclination (best individual). 9 GA refined Only GA 4 GA J Invariant Analytical J Invariant Inclination (deg) Fig. 6 Best individual fitness value vs inclination of reference orbit. Fig Time (s) Comarison with a Schaub invariant orbit condition at 35 deg. satellite used for this simulation were a 667 km, e :, $ 9 deg, 7 deg, and deg. The size of the relative orbit is regulated by means of the constant K in Eqs. (3) and (4): in this simulation this is set to. In the figure, we reort both the outut from the genetic algorithm and the final solution obtained refining the solution with a local otimization. For all inclination the minimal drift is not zero, with two remarkable excetions: 49 and 63.4 deg, and their symmetric counterarts (with resect to 9 deg), that is, 3 and 6.6 deg. In the following we will refer to the 49 and 3 deg inclinations as secial inclinations, keeing the term critical inclinations for the 63.4, 6.6 deg case. The heuristic of the genetic algorithms is definitely not resonsible for these eaks, as it turns out by actually roagating the resulting best individuals. At a generic inclination, say 35 deg, the best individual returned by the otimization results in a relative motion that is not eriodic is visualized in Fig. 7. The small residual drift is comarable to the one that results using the Schaub J -invariant orbit condition. To confirm this last statement, we have lotted in Fig. the value of the objective function given by Eq. (5) during one comlete orbit in the case of the Schaub J -invariant orbit and in the case of the condition returned by our genetic algorithm. At critical inclinations (63.4, 6.6 deg) the relative motion turns out to be erfectly eriodical (see Fig. 9). The corresonding otimal decision vector is :6 km; :4 km; :6 km; 5:3 3 km=s; :4 3 km=s; 9:339 3 km=s; 5:45:7 s 6 i= Fig. 9 relative orbits for a J erturbed case at 63.4 deg inclination (best individual). Note that in the J erturbed case, the condition is no longer of eriod matching as a difference in all the six orbital elements is ket, as reorted in Table. The ossibility of obtaining a erfectly eriodical relative motion at these inclinations is robably linked to the cancellation of the secular drift of the erigee argument, which causes the variation of all arameters to haen with the same main frequency. 4 3

6 SABATINI, IZZO, AND BEVILACQUA 99 Table Comaring orbital elements for the J case at the critical inclination Orbital element Chief Deuty Difference a 667 km km :9 km e i deg deg :44 deg $ 9 deg 5.6 deg 39:7 7 deg 9:3 deg.77 deg deg deg deg $ 9 deg 9.46 deg.46 deg Period for Minimum Drift (s) Best Individual Period Kelerian eriod At the two secial inclinations (49, 3 deg), the relative satellite motion resulting from the best individual has only a very small drift, as shown in Fig.. The corresonding decision vector is :65 km; :69 km; :546 km; :69 3 km=s; 3:39 3 km=s; 4: 3 km=s; 5:436:77 s The residual drift does not allow us to conclude that the motion is erfectly eriodical at these inclinations. In fact, we were unable to find a fitness value of (meaning erfect eriodicity) at all the inclinations. A more detailed lot of the objective function achieved around the secial inclination is shown in Fig.. The numeric noise that can be observed in the grah is a consequence of the numerical otimization rocess, amlified by the definition of the objective function given by Eq. (5). At higher values of the fitness very small differences in the residual drift cause significant differences in the Fig. relative orbits for a J erturbed case at 49 deg inclination (best individual) Inclination (deg) Fig. Details around a secial inclination Inclination (deg) Fig. Best individual eriod (s). i= Semimajor axis(km) Fig. 3 Best individual fitness vs increased values for the semi-axis. objective function value. We show in Fig. a lot of the eriod of the found minimum drift orbits. This is clearly quite different from the Kelerian eriod confirming the imortance of having the otimizer to choose it. The existence of the two secial inclinations where the relative motion between satellites can be eriodical is evident from the data resented in Fig. 6 and clearly calls for some exlanation. A number of elegant and interesting exact results have already been established in the years for the eriodicity of absolute satellite motion. Kyner [] deals with the eriodicity of a J erturbed motion. Hughes [] discusses the occurrence and the uniqueness of the critical inclinations. Mortari [] develoed an entire new theory to deal with the eriodicity of satellite constellations with resect to different reference frames. Unfortunately, the case of relative satellite motion has been studied much less. We have already commented on the aroximated minimal drift conditions available in the literature, but to the best knowledge of these authors, there are no exact results on the subject. Recent studies [3,4] try to aroach the henomenon, though not giving a definite answer. One may argue that the existence and the value of the two secial inclinations may be linked to the arameters of the Chief orbit or to the size of the formation (i.e., the relative orbit tyical dimension). By increasing the semimajor axis of the Chief orbit the fitness values reached by formations at both i 49 deg and i 63 deg are not influenced as shown in Fig. 3. The same results aly by changing the value K, related to the formation size as shown if Fig. 4. The validity of the secial inclinations is not affected by the size of the formation or by the semimajor axis of the Chief orbit: they exist for a wide range of formations around circular orbits. For the eccentricity the results are quite different.

7 SABATINI, IZZO, AND BEVILACQUA i=63 excetions: when the formation reference orbit (circular or ellitical) has an inclination of 63.4 or 6.6 deg (which are the classical critical inclinations) and of 49 or 3 deg (which we define as secial inclinations). An extensive simulation camaign is erformed to test the relative motion features at these secial inclinations. While the hysical reasons of this behavior are still under study, a simle conclusion can be drawn. If two satellites have to remain in close formation, the choice of the inclination of the reference orbit is of fundamental imortance, and it results in a smaller control effort to kee the satellites in a bounded formation Formation Dimensions (km) Fig. 4 Best individual fitness vs formation size Eccentricity Fig. 5 Best individual fitness vs eccentricity. i=63 Let us lot the result obtained by increasing the eccentricity value of the Chief orbit. The critical and the secial inclinations are again comared to a general case; the results are reorted in Fig. 5. To allow the study of high eccentricity, the semimajor axis of the reference orbit has been raised to km. The best individual fitness value for the secial and the critical inclinations is quite high for the range of eccentricity analyzed, but surrisingly also at the generic inclination, the fitness may resent remarkable values. Therefore, in the case of eccentric orbits, the drift in the relative motion can be small also outside of the critical and the secial inclinations. V. Conclusions The ossibility to obtain natural eriodic motion of formation flying satellites has been investigated through the use of a numerical global otimization technique such as genetic algorithms, refined by a constrained nonlinear otimization. After validating the aroach on the well-known unerturbed test case, the attention has been focused on erturbations. Although some results obtained are trivial and exected, some others are quite surrising and interesting. In articular, the ossibility to have a eriodic motion is denied, as shown later on, and also when a conservative, symmetric erturbation like J is considered. We find four remarkable Acknowledgments This work has been ossible thanks to a joint research cooeration between the Advanced Concets Team of the Euroean Sace Agency and the deartment Metodi e Modelli Matematici er le Scienze Alicate of the University of Rome. The cooeration was funded under the Ariadna scheme of ESA. References [] Becerra, V. M., Biggs, J. D., Nasuto, S. J., Ruiz, V. F., Holderbaum, W., and Izzo, D., Using Newton s Method to Search for Quasi-Periodic Relative Satellite Motion Based on Nonlinear Hamiltonian Models, Proceedings of the 7th International Conference On Dynamics and Control of Systems and Structures in Sace (DCSSS), Cranfield Univ. Press, London, 6,. 93. [] Inalhan, G., Tillerson, M., and How, J. P., Relative Dynamics and Control of Sacecraft Formations in Eccentric Orbits, Journal of Guidance, Control, and Dynamics, Vol. 5, No.,, [3] Tschauner, J., and Hemel, P., Rendezvous zu Einem in Ellitischer Bahn Umlaufenden Ziel, Acta Astronautica, Vol., March 965, [4] Kasdin, N. J., and Koleman, E., Bounded, Periodic Relative Motion Using Canonical Eicyclic Orbital Elements, AAS Paer 5-6, 5. [5] Vaddi, S. S., Vadali, S. R., and Alfriend, K. T., Formation Flying: Accommodating Nonlinearities and Eccentricity Perturbations, Journal of Guidance, Control, and Dynamics, Vol. 6, No., March Aril 3, [6] Clohessy, W. H., and Wiltshire, R. S., Terminal Guidance System for Satellite Rendezvous, Journal of the Aerosace Sciences, Vol. 7, No. 9, 96, [7] Schaub, H., and Alfriend, K. T., J Invariant Relative Orbits for Sacecraft Formations, Celestial Mechanics and Dynamical Astronomy, Vol. 79, No., Feb., [] Charbonneau, P., and Kna, B., A User s Guide to PIKAIA., NCAR TN 4+A (Boulder: National Centre for Atmosheric Research), 995. [9] Rimrott, Fred P. J., Introductory Orbit Dynamics, Vieweg, Braunschweig, Wiesbaden, 99. [] Kyner, W. T., A Mathematical Theory of the Orbits About an Oblate Planet, Journal of the Society for Industrial and Alied Mathematics, Vol. 3, No., March 965,. 36 7, rinted in the U.S. [] Hughes, S., The Critical Inclination: Another Look, Celestial Mechanics, Vol. 5, No. 3, Nov. 9, [] Mortari, D., and Wilkins, M. P., The Flower Constellation Set Theory Part I: Comatibility and Phasing, IEEE Transactions on Aerosace and Electronic Systems (to be ublished). [3] Sabatini, M., Izzo, D., and Palmerini, G. B., Analysis and Control of Convenient Orbital Configurations for Formation Flying Missions, Advances in Astronautical Sciences, Vol. 4, 6, , Univelt, San Diego, CA. [4] Sabatini, M., Bevilacqua, R., Pantaleoni, M., and Izzo, D., Numerical Search of Bounded Relative Satellite Motion, Nonlinear Dynamics and Systems Theory, Vol. 6, No. 4, 6,