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Dynamical Analysis of Rendezvous and Docking with Very Large Space Infrastructures in Non-Keplerian Orbits

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Dynamical Analysis of Rendezvous and Docking with Very Large Space Infrastructures in Non-Keplerian Orbits DYNAMICAL ANALYSIS OF RENDEZVOUS AND DOCKING WITH VERY LARGE SPACE INFRASTRUCTURES IN NON-KEPLERIAN ORBITS Andrea Colagrossi ∗, Michele` Lavagna y, Simone Flavio Rafano Carna` z Politecnico di Milano Department of Aerospace Science and Technology Via La Masa 34 – 20156 Milan (MI) – Italy ABSTRACT the whole mission scenario is the so called Evolvable Deep Space Habitat: a modular space station in lunar vicinity. A space station in the vicinity of the Moon can be exploited At the current level of study, the optimum location for as a gateway for future human and robotic exploration of the space infrastructure of this kind has not yet been determined, Solar System. The natural location for a space system of this but a favorable solution can be about one of the Earth-Moon kind is about one of the Earth-Moon libration points. libration points, such as in a EML2 (Earth-Moon Lagrangian The study addresses the dynamics during rendezvous and Point no 2) Halo orbit. Moreover, the final configuration of docking operations with a very large space infrastructure in a the entire system is still to be defined, but it is already clear EML2 Halo orbit. The model takes into account the coupling that in order to assemble the structure several rendezvous and effects between the orbital and the attitude motion in a Circular docking activities will be carried out, many of which to be Restricted Three-Body Problem environment. The flexibility completely automated. of the system is included, and the interaction between the Unfortunately, the current knowledge about rendezvous in modes of the structure and those related with the orbital motion cis-lunar orbits is minimal and it is usually limited to point- is investigated. A lumped parameters technique is used to mass dynamics. The aim of this paper is to present some represents the flexible dynamics. preliminary results about a possible rendezvous scenario with The parameters of the space station are maintained as a large space infrastructure in non-Keplerian orbits. The dy- generic as possible, in a way to delineate a global scenario namical analysis is based on a coupled orbit-attitude model of of the mission. However, the developed model can be tuned motion in a Circular Restricted Three-Body Problem (CR3BP) and updated according to the information that will be available environment, and includes the flexibility of the structure with in the future, when the whole system will be defined with a a lumped parameters technique. higher level of precision. The research is started with the definition of a possible Index Terms— Large Space Station, Circular Restricted rendezvous strategy that can be inserted as a part of a broad Three-Body Problem, Halo Orbits, Orbit-Attitude Dynamics, mission framework in cis-lunar space. It exploits invariant Flexible Structure. manifolds associated with unstable periodic orbits, such as EML2 Halo orbits, by finding a heteroclinic connection be- tween two different orbits. A tool to optimize this class of 1. INTRODUCTION transfers is developed and discussed. Moreover, a tool to simu- late the relative dynamics during final approach phases is also In the last two decades humanity achieved amazing goals with presented. The analysed rendezvous strategy is an example of space missions in Low-Earth orbit, creating the base for what application of the simulation tools and the coupled dynamical can be called prolonged human habitation in space. In the model that have been and are being developed from the authors. same time, robots have been targeted throughout the Solar A separate section is dedicated to highlight the effects of the System to explore different planets and numerous celestial extended flexible bodies on the dynamics in non-Keplerian objects. Now, the time for another step forward in space orbits. exploration has come. In fact, the future exploration of Solar This work is intended to pose a preliminary base for fur- System will be driven by a cooperation of astronauts and robots ther developments within the research area on very large and in space missions that will be aimed progressively further flexible structure in Three-Body problem environments. The away from the Earth. The roadmap to drive this ambitious purpose is the definition of a new astrodynamics tool, able to program has been already proposed by the International Space simulate such a class of space systems. At the current stage Exploration Group (ISECG) [1], and one of the key points in of development, the model does not yet include perturbing ef- ∗Ph.D Candidate, [email protected]. fects, such as Solar Radiation Pressure and fourth-body (Sun) yAssociate Professor, [email protected] gravity, and the analysis is still concentrated on a single ele- zM.Sc. Student, [email protected] ment of the structure. However, the model has been founded on a ”Multi-Body-Friendly” approach, and the extension of z the results is of easy implementation. 2. THEORETICAL BACKGROUND The present research is based on Circular Restricted Three- Body Problem modelling approach, which consider the motion m m m m m ; m of three masses 1, 2 and , where 1 2 and y m2 < m1. m1 and m2 are denoted as primaries, and are û assumed to be in circular orbits about their common center of m mass. The motion of m does not affect the trajectories of the r primaries. B1 r The dynamics is written in a rotating reference frame, S, B which is called synodic frame and is represented in figure 1. It m1 r is centered at the center of mass of the system, O; the first axis, O B2 ^x, is aligned with the vector from m1 to m2; the third axis, ^z, is in the direction of the angular velocity of S, ! = !^z; ^y completes the right-handed triad. The system can be defined m2 by the mass parameter, m µ = 2 ; x m1 + m2 the magnitude of the angular velocity of S, s G(m + m ) 1 2 Fig. 1: Synodic Reference Frame. ! = 3 ; r12 and the distance between the primaries r12. The equations gravitational potential, Vg = Vg1 + Vg2 and the generalized of motion are usually normalized with respect to r12, ! and potential of the inertia forces, Vi. the total mass of the system mT = m1 + m2. After the The gravitational action exerted by the i-th spherical pri- normalization, the universal constant of gravitation is G = 1. mary on m, can be derived from: The mass m is an extended body, and the model is currently based on a simple and generic structural element: a rigid rod. Gm m Gm 3 V = − i + i (^r · I · ^r ) − tr(I ) ; In this way, it is possible to have a solid foundation, which gi r r3 2 Bi Bi Bi B Bi Bi can be easily extended to more complex configurations of the (2) space system, with a Multi-Body technique. The body m has where rBi and ^rBi are respectively magnitude and direction of five degrees of freedom: the position of its center of mass B, rBi : position vector of m with respect to the i-th primary. The rB, and two independent parameters to define the orientation previous expression is an expansion up to the second order of of the versor aligned with the rod, ^u. the gravitational potential generated by a spherical attractor on The equations of motion can be derived starting from a a small extended body [2]. Lagrangian Formulation, where the Lagrangian function, L = The generalized potential of the inertia forces is needed to T − G, includes the kinetic energy, T , and the generalized write the equations of motion in S, and it can be expressed as: potential, G. The kinetic energy T of the rigid body can be expressed as m V = r · [! × (! × r ) + 2! × _r ] the kinetic energy of the translational motion of the center of i 2 B B B mass plus the kinetic energy of the rotational motion of m as: 1 − ! · IB · (! + 2Ω): (3) 1 1 2 T = m _r · _r + Ω · I · Ω; (1) 2 B B 2 B It is important to remember that a generic generalized potential V(P; P_ ), where P is the position and P_ the velocity, _r B Ω where B is the velocity of , is the angular velocity of the is defined in a way that the related force can be computed as: body relative to the S frame and IB is the inertia tensor about B. d @V @V The generalized potential G is related with the gravitational F = ( ) − : (4) dt @P_ @P forces and with the inertia forces, since the synodic frame S is a non-inertial reference frame that is rotating with the two In order to write the normalized equation of motion, L primaries. It can be expressed as the sum of the ordinary has to be written in non-dimensional form. In fact, from now on, all the variables will be intended to be non-dimensional: express orientation of m as: lengths will be divided by r12, masses by mT and times by 1=!. In the same way, from now on, the time derivative will ^u = [cos φ2 cos φ3; be taken with respect to the non-dimensional time τ = !t: ◦_ = d ◦ /dτ cos φ1 sin φ3 + sin φ1 sin φ2 cos φ3; At this point, it is possible to derive the equations of motion sin φ1 sin φ3 − cos φ1 sin φ2 cos φ3]: (11) as: d @L @L The equations of motion in terms of Bryant angles can be − = Aj; (5) derived using a Newton-Euler formulation. In fact, the angular dτ @q_j @qj momentum at B, hB, is related to the torque applied to the center of mass, mB, as: where qj; j = 1;:::; 5 are the generalized coordinates, and Aj the generalized non-dimensional contributions due to different d hB external forces, such as the solar radiation pressure or the = mB: (12) dt presence of a fourth-body.
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