INVARIANT RELATIVE SATELLITE MOTION. Marco Sabatini Dip. Ingegneria Aerospaziale, Università di Roma La Sapienza

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1 AC Workshop on Innovative Concepts. ESA-ESEC 8-9 January 8. INVARIAN RELAIVE SAELLIE MOION Marco Sabatini Dip. Ingegneria Aerospaziale, Università di Roma La Sapienza Dario Izzo European Space echnology and Research Center, Keplerlaan, 1 AZ, Noordwijk, he Netherlands Dario.izzo@esa.int Giovanni B. Palmerini Dip. Ingegneria Aerospaziale, Università di Roma La Sapienza giovanni.palmerini@uniroma1.it ABSRAC Formation lying is a key technology enabling a number o missions which a single satellite cannot accomplish: rom remote sensing to astronomy and undamental physics. he design o relative navigation and control systems o the spacecrat is certainly one o the most challenging topic. In order to ease the tasks o these subsystems, a proper reerence trajectory must be conceived, and a relative motion, which shows no drit even in presence o a large disturbance as the J eect, could be a very attractive solution. his paper describes the research activities or inding invariant relative orbits under J eect. Numerical tools as genetic algorithms enabled the discovery o two special inclinations which represent the necessary conditions or periodicity o the motion. hese results generated interest, and analytical explanations or the numerical evidence have been proposed: on-going studies ace this problem rom dierent points o view, and basic results are reported. 1 INRODUCION Formation lying is a key technology enabling a number o missions which a single satellite cannot accomplish: rom remote sensing to astronomy and basic physics. In order to keep the satellites o the ormation in the designed coniguration, and thereore to achieve the mission's goals, control actions are needed. he cost o this orbital control in terms o V limits both the mission duration and the expected perormances. Advantageous dynamics could reduce the cost o these operations, in particular the possibility to obtain periodic or quasiperiodic natural relative motion would be a signiicant saving actor. Many dierent approaches to ind a periodical relative motion are considered in the recent literature. Inalhan, illerson and How [1] ound the analytical expression or the initial conditions resulting in periodic motion based on the classical schauner-hempel equations []. Kasdin and Koleman [3] used the epicyclic orbital elements theory to derive bounded, periodic orbits in presence o various perturbations. Vaddi, Vadali, and Alriend [4] studied the Hill-Clohessy- Wiltshire [5] (HCW) modiied system to include second order terms. Finally, Schaub and Alriend [6] ormulated the 1

2 AC Workshop on Innovative Concepts. ESA-ESEC 8-9 January 8. conditions or invariant J relative orbits introducing relations between the mean orbital elements o the two satellites. he analytical approaches taken in these works lead to initial conditions that ensure exact periodicity in approximated dynamical models or initial conditions resulting in bounded (but not periodic) relative motion in more detailed dynamical models. In the rame o the 4 ESA ARIADNA project, the possibility to obtain natural periodic relative motion o ormation lying LEO (Low Earth Orbits) satellites has been investigated numerically. he algorithm is based on a genetic strategy (GA), reined by means o nonlinear programming, that rewards periodic relative trajectories. Only the J perturbed case is considered, as presence o a dissipative disturbance like drag does not enable a periodic motion. Using the proposed numerical approach, it has been possible to ind two couples o inclinations (63.4 and 116.6, the critical inclinations, and 49 and 131, two new special inclinations) that seemed to be avoured by the dynamical system or obtaining periodic relative motion at small eccentricities. his interesting numerical result still missed a mathematical or physical explanation. Vadali, Sengupta, Yan, and Alriend in [7] ind the values or inclination which enable equal in-plane and out-o-plane undamental requencies, resulting in non-precessing and distortionree relative orbits over the short-run. In this way, however, the results obtained by means o the numerical optimization are only partially validated. In act, according to [7], all the inclinations which all in the interval between the special inclinations, i [ 49.1,63. 4 ], turn out to be valid or periodic motion, while only the range limits were identiied with GA and nonlinear optimization. Following these results, a new kind o study has been started, on mathematical basis, which aims to show that: 1) a truly periodic relative motion in a J perturbed environment is generally not possible; ) two special inclinations exist or which at least the projection on a coordinate plane o the relative motion is periodic HE NUMERICAL APPROACH: DEFINIION OF HE PROBLEM Consider a generic non-autonomous dynamical system x = ( x, t), e.g. the relative dynamics o satellites lying in ormation. Deine x = x x ( ), where x is the system state at the initial time and is a time variable here called candidate period or reasons that will soon be clear. hen, the ollowing optimisation problem is deined: ind : = [ x ] to maximise: J = J ( x ) subject to : x = ( x, t) (1) where the objective unction J is constructed in such a way as to have its global maximum at x =. he optimisation problem above is equivalent to the task o inding as-periodic-aspossible solutions to the system x = ( x, t). hese solutions correspond, in our case, to minimal relative orbit drit. As we study a number o systems x = ( x, t), we ace dierent levels o complexity or the optimisation and or objective unction properties. hink about the relative motion between satellites moving on keplerian orbits, the problem deined by Eq.(1) has an ininite number o solutions, corresponding to orbits with equal semi-major axis. A similar structure is also expected when the keplerian dynamic is perturbed. As a consequence, a genetic approach, avoiding issues related to domain knowledge and able to cope with multiple local and global minima, has been selected to perorm a search in the

3 AC Workshop on Innovative Concepts. ESA-ESEC 8-9 January 8. solution space. he PIKAIA reely available sotware [7] was used in this work as genetic optimiser. able 1 shows the undamental parameters o the genetic algorithm used in all the simulations. able 1 Parameters used or the Genetic Optimizer Number o individuals Number o generations 5 Number o signiicant digits Crossover Probability.85 Initial Mutation Rate.5 Minimum Mutation Rate.5 Maximum Mutation Rate 1 Reproduction Plan 9 Steady-State- Replace-Worst he best solution returned by the genetic algorithm is then reined locally by means o a nonlinear programming solver. In our simulations the decision vector contains the initial relative position, the initial relative velocity, and the candidate period. We consider the relative motion between two satellites: a chie and a deputy to use a popular terminology connected to ormation lying research. he absolute dynamics o both the chie and the deputy are simulated propagating the inertial coordinates o the spacecrat in d r µ time, = r + P where P are the 3 dt r perturbing action considered, µ is the planetary constant and r is the orbital radius vector. he relative state is then evaluated by means o Eq. (): ( d d d c c c ) [ x y z] R [ X Y Z ] [ X Y Z ] = [ x y z ] = R( X Y Z X Y Z ) ω [ x y z] d d d c c c () where R is the rotation matrix rom the inertial coordinate system to the Local Vertical Local Horizontal (LVLH) rame in which the relative state is deined. he subscripts c and d stand or chie and deputy satellites. he orbit o the chie is J ( κ ) 1 3 considered known and the initial conditions to propagate the deputy motion are obtained transorming the relative x position into absolute coordinates inverting Eq.(). We then deined kep 7 = ± κ k, where k is a properly chosen constant (some tens o seconds) and is the orbital period 3 a π µ kep o the chie orbit. is clearly a crucial parameter. At, the inal relative coordinates are compared to the initial relative coordinates, thus determining the quality o the individual. A good individual has a small x and its position in the individual ranking is high, thereore it has a larger chance to mate and to generate good sons. Its genes will survive in the next generation, and i they will be ranked irst in the last generation, they will be urther reined by a local optimiser and represent the set o initial conditions that generate the minimum drit relative orbit. he itness unction we used to rank the individuals is: = 1 x x y y z z x x y y z z x y z x y z (3) A perect individual (periodic motion) has a itness value o 1, while a percentage dierence o.1% between the initial and inal state, brings down the itness value to 5, a dierence o 1% corresponds to a itness value o 9, and so on. 3 HE NUMERICAL APPROACH: RESULS Let us study the solutions o Eq. (1) in the case describes the relative motion between two satellites orbiting around an oblate Earth. As already mentioned, this corresponds to minimising the relative orbital drit. Some previous work has been done to determine the possibility o 3

4 AC Workshop on Innovative Concepts. ESA-ESEC 8-9 January 8. invariant relative satellite motion when J is considered as a perturbing term. In particular, the paper by Schaub and Alriend [6] introduces the so called J invariant relative orbits. In their work mean orbital elements are used and the secular drits o the longitude o the ascending node and o the sum o the argument o perigee and mean anomaly are set to be equal between two neighbouring orbits. Even though called J invariant orbits, these two conditions are only valid in a irst order approximation. We use our numerical approach based on the solution o the global optimisation problem stated in Eq. (1) to check to what extent the residual drit obtained with this analytical approach is an artiact o the use o mean elements. Repeating the calculation or the entire range o inclinations, the results vary sensibly, disclosing a previously unknown eature o invariant relative motion. In Fig. 1 we report the best itness unction reached or dierent inclinations ranging rom to 18 degrees. he other orbital parameters o the Chie satellite used or this simulation were a = 6678 km, e =.118, ϖ = 9, Ω = 7, θ =, with the usual meaning o the symbols. In the Fig. 1 we report both the output rom the genetic algorithm and the inal solution obtained reining the solution with a local optimisation. 131 and In the ollowing we will reer to these as special inclinations. he heuristic o the genetic algorithms is deinitely not responsible or these peaks, as it turns out by actually propagating the resulting best individuals. At a generic inclination, say 35, the best individual returned by the optimisation results in a relative motion that is not periodic, as visualised in Fig.. he small residual drit is comparable to the one that results using Schaub J -invariant orbit condition. At the special inclinations 63.4 and the relative motion turns out to be perectly periodical (see Fig. 3). he possibility o obtaining a perectly periodical relative motion at these inclinations is probably related to the cancellation o the secular drit o the perigee argument, which causes the variation o all parameters to happen with same main requency. At the other two special inclinations (49, 131 ) the relative satellite motion resulting rom the best individual has only a very small drit, as shown in Fig. 4. Fig. 1 Relative Orbits or a Perturbed case at Inclination (Best Individual) Fig. 1 Best Individual Fitness Value vs. Inclination o Reerence Orbit For all inclination the minimal drit is not zero, with two remarkable exceptions: 49 and 63.4, and their symmetric counterparts (with respect to 9 ), i.e. Fig. 3 1 Relative Orbits or a Perturbed case at Inclination (Best Individual) 4

5 AC Workshop on Innovative Concepts. ESA-ESEC 8-9 January 8. Fig. 4 Relative Orbits or a Perturbed case at 49 Inclination (Best Individual) he residual drit does not allow us to conclude that the motion is perectly periodical at these inclinations. We were in act unable to ind a itness value o 1 (meaning perect periodicity, according to Eq. (3)) at any inclination. A more detailed plot o the objective unction achieved around the special inclination is shown in Fig. 5. he clutter that can be observed in the graph is a consequence o the numerical optimisation process, ampliied by the deinition o the objective unction given by Eq.(3). At higher values o the itness very small dierences in the residual drit cause signiicant dierences in the objective unction value. For completeness we also report in Fig. 6 a plot o the period o the ound minimum drit orbits. his is clearly quite dierent rom the keplerian period conirming the importance o having let the optimiser to choose it. Fig. 5 Details around a special inclination Fig. 6 Best Individual Period (sec) he existence o the two special inclinations represented one o the main indings o the 4 ARIADNA project on the search o invariant relative motion o satellites, but it clearly called or some physical or analytical explanation. 4 AN ANALYICAL EXPLANAION Vadali et al. in [7] look or a physical explanation o the special inclinations. he irst step consists in a geometric description o the relative motion using the unit-sphere approach. he expressions or the in-plane and cross-track motion variables are linearized to extract the undamental requencies o motion. A set o classical dierential orbital element initial condition ormulae are derived, valid even or circular orbits. In this way the dierence between the in-plane and cross-track requencies (n xy and n z, respectively), over a short time interval (compared to the period o the dierential nodal precession rate), is evaluated as: sin i Ω i nz nxy = ω 1 i + ( sin i ( )) Ω (4) where i is the reerence orbit inclination, Ω, Ω, i the dierences in RAAN rate, RAAN and inclination between the ormation members. he drit rate or the argument o perigee can be written as 5

6 AC Workshop on Innovative Concepts. ESA-ESEC 8-9 January 8. 5 ω = k sin i n (5) where: RE k = 1.5J n (6) a he dierential nodal precession rate or near-circular orbits is Ω = k sin i i (7) Neglecting the eect o, the requency mismatch is estimated as ollows: 5 i n n = k sin i + z xy i + ( sini Ω( ) ) (8) For the special case o the projected circular orbit (re. [9]), it can be shown that: ( ) ρ Ω ( ) = sin α( ) (9) a sin i ( ) cosα ( ) ρ i = (1) a where α ( ), is the desired initial phase angle and ρ ( ) is the initial radius o the relative orbit in the y-z plane By substituting Eqs. (9) and (1) into Eq. (8), the requency matching condition is satisied by two possible values o the inclination: 1 i * = sin (11).5 + cos α ( ) he inclinations or () = are i*= and and or () = 9, the results are the critical inclination values: i*= and Anyway, α ( ) is a parameter varying between and 36, this means that i * [ 49.11,63. 4 ], while the numerical evidence is or i* equal to the boundary values o the interval. his partially contradictory result led to the research o a dierent approach. 5 AN ALERNAIVE MAHEMAICAL APPROACH In order to try to solve the contradictions between the numerical and the analytical results, a completely dierent approach is adopted. In [1] and [11], a linear model is obtained, describing the relative dynamics o a ormation in a J perturbed environment (circular reerence orbit case). I the GA exploited or the nonlinear ormation dynamics shown in section is used now or the linear model, similar results are obtained, as Fig. 7 conirms. Fig. 7 Results o the application o the numerical method applied to the linear J model It is thereore possible to ocus the attention on the linearized dynamics, since the causes o the existence o special inclinations are in the irst order gradients o gravity and J disturbance. he adoption o a linear system as object o the study allows the use o relevant theorems valid or linear systems, as the one demonstrated by Bose in [1]: Let Y = A( t)y be a n-dimensional homogeneous linear system and Φ ( t) its undamental matrix; let k be a nonnegative integer, k n. here exists a k-dimensional sub-space S k o the solution space S n o the linear homogeneous system such that each member o S k is periodic o period and no member o S n S k is periodic o period i and only i λ = 1 is an eigen value o the scalar matrix Φ( ) 1 Φ( t) o multiplicity k. 6

7 AC Workshop on Innovative Concepts. ESA-ESEC 8-9 January 8. Our case perectly matches the hypothesis o this theorem, since the linear model adopted is homogeneous; however, the period is not known a priori (but clearly in the close proximity o the keplerian period, see again Fig. 6). hereore, the time behaviour o the six eigen values λ o Φ ( t) has been analysed. Φ ( t) is evaluated by means o: dφ( t) = A( t) Φ( t), (1) dt with Φ( ) = I (the identity matrix). he A(t) state matrix is a 6-by-6 matrix: according to the Bose s theorem a relative periodic motion is possible only i a time t* exists or which the eigen values o Φ t * are equal to 1 (i.e. imag( λ) = and ( ) ( λ) = 1 re ) with multiplicity 6. For reerence orbits inclined rom to 49.1, there is not a time instant when this is veriied; as an example, see Fig. 8, reerred to the i=45 case. he imaginary part o the eigen values is never or more than a pair o eigen values at the same time, thereore λ = 1 with multiplicity greater than two is never achieved. projection o the relative motion is periodic (a 4-dimensional subspace necessarily is comprehensive o two coordinates and their derivatives, i.e., a coordinated plane). Fig. 9 ime behaviour o the imaginary part o the 6 Φ t : 49.1 case eigen values o ( ) Fig. 1 Multiplicity o the eigen values having imag ( λ) =, and multiplicity o eigen values having re ( ) = 1 : 49.1 case λ Fig. 8 ime behaviour o the imaginary part o the 6 Φ t : 45 case eigen values o ( ) However the pairs A 1, A and B 1, B have an imaginary part which is zero or time instants which get closer as the inclination increases, inally coinciding or i equal to the special inclination 49.1 (Fig. 9). Fig. 1 conirms that not only 4 eigen values have a zero imaginary part in a certain instant, but that in that instant also the real part o the same eigen values is equal to 1. Following Bose s theorem, this means that there exists a coordinated plane, where the Similar considerations can be done or the other special inclination, 63.1 : at these two special inclinations the motion o the ormation is by ar more stable than at any other inclination, though it is not mathematically periodic in its six components, yet just in our. In act, or the inclinations in the range [49.1, 63.1 ] while there are still 4 eigen values with imaginary part equal to zero in a certain instant, they do not have real part equal to one (Fig. 11 and Fig. 1), and so the Bose s theorem is not veriied, not even or a 4-dimensional subspace. For inclinations greater than 63.1, even i it is possible that 4 eigen values have imaginary part equal to zero, again they do not have real parts equal to one, and a 7

8 AC Workshop on Innovative Concepts. ESA-ESEC 8-9 January 8. periodic motion is not possible, not even a periodic projection o the motion. Fig. 11 ime behaviour o the imaginary part o the Φ t : 55 case 6 eigen values o ( ) though it is true that J is not the only environmental disturbance, it is certainly the largest or a wide range o missions proiles. In addition, the special inclination 49.1 is quite promising since it is not very dierent rom the inclination o GPS, o the uture GALILEO, o the International Space Station. Having an almost ree-ocontrol ormation could suggest a number o solution, dierent rom the usual ormation tasks: it can be useul or occupying the same orbital slot without conlicting, serving as a spare satellite, or or continuous monitoring o the health o the main satellite. Furthermore, once invariant J orbits have been assessed, the eects o other perturbations can be isolated and contrasted in a most eicient way, i needed, or alternatively, they can be ully exploited on purpose. 7 CONCLUSIONS Fig. 1 Multiplicity o the eigen values having imag ( ) =, and multiplicity o eigen λ λ values having re ( ) = 1 : 55 case. 6 FUURE DEVELOPMEN he proposed mathematical demonstration still have a number o drawbacks: irst, it deals with a linearized dynamical model and not with the complete dynamics; moreover, it is not explained why the special inclinations do exist. hereore a physical investigation should be the natural ollowing o these researches. In act, it is not important just to notice that the phenomenon exists, but it is signiicant to understand the orces acting on the two satellites which produce it. Only in this way, the hidden potential in the perturbations o the LEO environment could be ully exploited. A periodic relative orbit represents a very low-cost station keeping solution. Even he possibility o a periodic relative motion o satellites in a J perturbed environment has been numerically conirmed in a ESA ARIADNA project in 4. An invariant motion is ound to be not easible, with two remarkable exception, when the reerence orbit is inclined at 49.1 and hose results generated interest in the scientiic community, and new studies has been ocused on this topic, searching or a physical or analytical explanation o the numerical evidence. Since inevitable approximations are necessary to simpliy the dynamic equations, the results are not matching completely the numerical evidence, and a new approach is here outlined: it shows the uniqueness o the special inclinations 49.1 and 63.4 which enable periodic relative trajectories. REFERENCES 1. Inalhan, G., illerson, M., How, J. P. Relative dynamics and control o spacecrat ormations in eccentric orbits. Journal o Guidance, Control and Dynamics, Vol. 5, No. 1,, pp

9 AC Workshop on Innovative Concepts. ESA-ESEC 8-9 January 8.. schauner, J., and Hempel, P. Rendezvous zu einem in Elliptischer Bahn Umlauenden Ziel. Acta Astronautica, Vol. 11, 1965, pp Kasdin, N. J., Koleman, E. Bounded, Periodic Relative Motion using Canonical Epicyclic Orbital Elements. Paper AAS 5-186, 15th AAS/AIAA Space Flight Mechanics Meeting, Copper Mountain, Colorado, Vaddi, S. S., Vadali, S.R., and Alriend, K.. Formation Flying: Accommodating Nonlinearities and eccentricity Perturbations Journal o Guidance, Control, and Dynamics, Vol. 6, No., March-April 3, pp Clohessy, W. H., and Wiltshire, R. S., erminal Guidance System or Satellite Rendezvous. Journal o the Aerospace Sciences, Vol. 7, No. 9, 196, pp Schaub, H., Alriend, K.. J Invariant Relative Orbits or Spacecrat Formations. Celestial Mechanics and Dynamical Astronomy, Vol. 79, 1, pp Vadali, S.R., Sengupta, P., Yan, H., and Alriend, K.. On the undamental requencies o relative motion and the control o satellite ormations. Proceedings the AAS / AIAA Astrodynamics Specialist Conerence, paper AAS 7-47, Charbonneau, P., Knapp, B., A user s guide to PIKAIA 1.. NCAR echnical note 418+1A (Boulder: National Centre or Atmospheric Research), Vadali, S.R., Vaddi, S.S. and Alriend, K.. An Intelligent Control Concept or Formation Flying Satellite Constellations. International Journal o Nonlinear and Robust Control, dedicated to Formation Flying, ; 1: Izzo, D., Sabatini, M., Valante, C., A New Linear Model Describing Formation Flying Dynamics under J Eects, Proceedings o the 17th AIDAA National Congress, Rome, Italy, Vol. 1, pp , Sabatini, M., Palmerini, G.B. Linearized Formation-Flying Dynamics in a Perturbed Orbital Environment to be presented in 8 IEEE Aerospace Conerence, Bigsky, Montana, 1-8 March 8 1. Bose, A.K. On the periodic solutions o linear homogeneus systems o dierential equations. Internat. J. Math. & Math. Sci., 5():35 39,