Halo orbit dynamics and properties for a lunar global positioning system design

Transcription

1 doi: /mnras/stu1085 Halo orbit dynamics and properties for a lunar global positioning system design Christian Circi, 1 Daniele Romagnoli 2 and Federico Fumenti 3 1 Department of Astronautic, Electric and Energetic Engineering, Sapienza University of Rome, Via Salaria 851, I Rome, Italy 2 C.I.R.A., Centro Italiano Ricerche Aerospaziali, Via Maiorise, I Capua CE), Italy 3 DLR Institute of Space Systems, Robert-Hooke-Str. 7, D Bremen, Germany Accepted 2014 May 29. Received 2014 May 14; in original form 2014 January 27 ABSTRACT In this paper, the use of north and south families of halo orbits, around the L 1 and L 2 collinear libration points of the Earth Moon system, to realize a lunar global positioning system LGPS) is proposed. The computation of the reference trajectories and the required station-keeping manoeuvres with the associated V are described, as well as different configurations for the ellites constellation. The combination of the north and south families of halo orbits results in an X-shaped configuration allowing optimal performances. A visibility study, from different locations on the lunar surface, has been performed together with a performance analysis in terms of availability of the LGPS signal and quality of the position solution, resulting in a candidate architecture able to guarantee the availability of the LGPS service at the scientifically interesting lunar poles. In addition, the proposed constellations are also interesting from a connection point of view, assuring continuous communication capabilities between the Earth and every location on the surface of the Moon, as well as between any of the points on the ground. Key words: space vehicles celestial mechanics Moon. 1 INTRODUCTION In the last years, the renewed interest in robotic lunar exploration reopened a debate on whether or not having a permanent human outpost on the lunar surface. The importance and the advantages of human exploration are well known, even though there are many difficulties involved in human spaceflight. In particular, the most interesting locations on the lunar surface from the scientific and engineering points of view are situated in very complicated areas, such as the far side of the Moon or the lunar polar regions NASA 2006; Committee for a Decadal Survey of Astronomy and Astrophysics, National Research Council 2010; Committee on the Planetary Science Decadal Survey, National Research Council 2011). Hence, particular effort has to be made in guaranteeing a continuous navigation and communication service. The first studies in this field date back in the early 1970s, with Farquhar proposing the use of a ellite on a periodic orbit about the L 2 libration point of the Earth Moon system to act as a communication relay ellite with the far side of the Moon Farquhar 1971; Farquhar & Kamel 1973). Nonetheless, no big advances took place until the last decade, when the new interest in returning on the Moon dawned. Noteworthy are the works of Li et al. 2010), Romagnoli & Circi 2010), Carretero & Fantino 2012)andZhang&Xu2014). In all of these works, configurations of ellites along with methods and performances of a navigation system are explored. Starting from the idea of Romagnoli & Circi 2010) to place a lunar global positioning system LPGS) around libration points and exploit their equilibrium conditions, this work extends their results to other libration points orbits LPOs), different from the Lissajous ones considered by them. In the last decades, a great number of studies have been carried out on libration points and LPOs regarding dynamics, reaching trajectories and required station keeping. Breakwell & Brown 1979) extended Farquhar s studies of halo orbits around both L 1 and L 2 libration points of the Earth Moon system. They dealt with the whole families of halo orbits, hence considering several dimensions and distances to the Moon. With time, several studies started to include additional complexities, like Howell & Spencer 1986) andcirci2012), who investigate the restricted four-body problem, or Hou & Liu 2011), who focused on elliptic restricted three-body problem. christian.circi@uniroma1.it C 2014 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society

2 3512 C. Circi, D. Romagnoli and F. Fumenti Gòmez et al. 1993) and Alessi, Gòmez & Masdemont 2009, 2010) focused on transfer trajectories to reach LPOs. Mingotti, Topputo & Bernelli-Zazzera 2009, 2010, 2012) discussed about the possibility to execute low-thrust transfers to the Moon exploiting the invariant manifolds associated with LPOs. Correa et al. 2007) compared transfer orbits in the restricted three- and four-body problem. Grebow, Ozimek & Howell 2010) used the restricted three-body problem to study optimal trajectories for lunar pole coverage through low-thrust spacecraft. Using the restricted three-body model, Howell & Pernicka 1993) explored a station-keeping strategy for a ellite in the vicinity of the L 1 point of the Sun Earth/Moon system, while Breakwell, Kamel & Ratner 1974), Simo & McInnes 2009) and Bai & Junkins 2012) focused their attention on the Earth Moon system. Introducing real ephemerides, Gòmez et al. 1998) presented two strategies for the station keeping of a ellite on a halo orbit of L 2 for the Earth Moon system. Grebow et al. 2008) investigated the possibility of using halo orbits to provide coverage to the lunar South Pole region, while Pergola & Alessi 2012) characterized several LPOs in the Earth Moon system partially addressing their visibility and the coverage of the lunar surface. With reference to the realization of a feasible LGPS, the exploitation of LPOs for such a system has been investigated by Romagnoli & Circi 2010), who tackled the realization of lunar global positioning and communication systems by means of ellites on Lissajous orbits around the collinear L 1 and L 2 Lagrangian points of the Earth Moon system. They proposed and compared some lunar global communication system LGCS) and LGPS configurations, proving that the use of LPOs in projecting such kind of systems is nothing but beneficial and showed for the Lissajous-based constellation even better performances with respect to a Walker one. The only drawback of using Lissajous orbits can be found in the lack of continuous line of sight between ellites around L 2 and the Earth, since they can be eclipsed by the Moon. In such an event, if all the ellites are envisaged to communicate with the Earth, ellites around L 2 must use ellites around L 1,which consequently must be projected with greater capabilities. In order to solve the line-of-sight problem, in this paper, the authors move the constellations of ellites from Lissajous orbits to modified halo orbits MHOs), i.e. obtained modifying halo orbits, inasmuch as some halo orbits present the advantage of never being eclipsed by the Moon when seen from the Earth. Nonetheless, the imposed mission requirements and the use of the Sun, the Earth and the Moon real ephemerides do not allow us to directly use halo orbits and MHO are determined. Given the particular shape of halo orbits and due to symmetry reasons, for each Lagrangian point both north and south halo orbits are used to obtain MHO, distributing ellites on an X-shaped configuration. An optimal control problem OCP) is used to determine the MHOs path as well as the station-keeping V required to follow them. In addition, particular effort has been placed in taking the geometric dilution of precision GDOP) parameter into consideration when designing the constellations and evaluating their performances. This paper is organized as follows. In Section 2, the equations of motion are defined along with a short description of the used optimization method. In Section 3, the station-keeping results and the Vs required by MHOs are presented. Section 4 summarizes the fundamentals of GPS with a recap on the pseudo-range measurements and on the DOP parameters to evaluate the accuracy of the obtained solution. Section 5 presents several constellation designs with related discussions. Section 6 closes the dissertation summarizing the results, both the manoeuvring-related ones and both the LGPS -performances-related ones. 2 PROBLEM DEFINITION AND ASSUMPTIONS Let us describe the motion of a spacecraft in an Earth-centred inertial frame Oxyz with Cartesian coordinates: the origin O is placed at the centre of the Earth, the x-axis points to the vernal point, the z-axis coincides with the Earth s rotation axis and the y-axis is placed consequently to obtain a right-hand-oriented frame. The motion can be obtained by solving an n-body problem Aarseth 2003) with n = 4, since the gravity actions of the Moon, the Earth and the Sun are included. The dynamics of the spacecraft is then expressed by the following equations: dx = v x dt dy dt dz dt dv x dt dv y dt dv z dt = v y = v z = μ s xs r 3 s = μ s ys r 3 s = μ s zs r 3 s + x x ) s xm rs 3 μ m + y y s r 3 s + z z s r 3 s r 3 m ) μ m ym r 3 m + x x ) m x rm 3 μ e r 3 + y y m r 3 m ) zm μ m + z z m rm 3 rm 3 ) μ e y r 3 ) μ e z r 3 ) + a x ) + a y ) + a z, 1) [ x, y, z, v x,v y,v z ] T being the ellite s state vector, [ x s,y s,z s ] T the position vector of the Sun, [ x m,y m,z m ] T the position vector of the Moon, [ a x,a y,a z ] T the control acceleration vector, μ s, μ m and μ e the gravitational constants of the Sun, the Moon and Earth, respectively, r s, r m and r the distance between the centre of the Earth and the Sun, the Moon and the ellite, respectively and, finally, r s and r m the distances between the Sun and the ellite and the Moon and the ellite. All the quantities in equation 1) are continuous functions of time.

3 Halo orbits for a lunar GPS design 3513 Positions of the Sun and the Moon to compute distances r s, r m, r s and r m are obtained from real ephemerides. The thrust acceleration, that is, the control acceleration required to follow the desired trajectory, is given by a x = Acos ψ) cosϕ) a y = Acos ψ) sinϕ) a z = Asinψ), where both the norm A and the direction through the standard polar angles ϕ and ψ) are continuous time functions to be determined in order to isfy the mission requirements. The angles ϕ and are allowed to vary in the intervals [ π, π]and[ π 2, π ], respectively. The norm A of 2 the control acceleration is assumed to vary between null that is no thrust is provided) and a maximum value A max = 0, 02 mm obtained from s 2 a literature investigation Farquhar 1971; Simo & McInnes 2010) and preliminary tests. Time and distances are expressed in non-dimensional units TU and DU, respectively) based on the radius and the gravitational parameter of the Earth. An additional reference frame needs now to be introduced. The inertial frame is not convenient to represent the solution for the trajectory around the libration points or the Moon, because the motion of a spacecraft would not be easy to visualize with respect to them. Fig. 1 shows then three Cartesian rotating frames centred on the Moon, on L 1 and on L 2 and denoted asmxyz, L 1 xyz and L 2 xyz, respectively. In each of these frames, the xy-plane is coincident with the Moon s orbital plane and the x-axis overlaps the Earth Moon line. Since the LGPS studied in this paper should be used by users on and around the Moon, it is important to understand the motion of the spacecraft with respect to our ellite; hence, the synodic frame Mxyz is the only one which is going to be used in the following. When the motion is approximated with a circular restricted three-body problem CRTBP), a particular solution is represented by halo orbits, which are closed orbits placed in the proximity of the collinear libration points. The main property of such trajectories is the equal frequencies between the in-plane and the out-of-plane motion, when the motion is studied in the synodic frame of reference. For both L 1 and L 2 points, there exist two families of halo orbits, the north one and the south one, depending on the sign of the out-of-plane displacement. The two families of orbits are symmetric with respect to the xy plane; hence, their projections on this plane are the same. Northern family orbits are also referred to as Class I orbits, while southern family orbits are also denoted Class II orbits, according to Farquhar & Kamel 1973). Fig. 2 shows a set of Class I and II halo trajectories for the L 1 and L 2 collinear libration points of the Earth Moon system. As stated above, in this paper, no simplifying assumptions are imposed on the motion, making the considered dynamics different from the CRTBP. Nevertheless, one expects the real orbits to be not too different from the solution of the approximated problem; then halo orbits turn out to be useful, as it will be shown in the next sections. 2) Figure 1. Rotating Cartesian coordinates systems. Figure 2. Halo orbits in the Earth Moon system.

4 3514 C. Circi, D. Romagnoli and F. Fumenti 2.1 Mission requirements Two mission requirements are considered in this paper. The first one is about the communication between the Earth and the ellites. In order to have continuous communication, ellites need to be seen constantly from the Earth; hence, a ellite around L 2 must not be eclipsed by the Moon. In addition, assuring continuous communication between the Earth and the ellites, continuous communication between the Earth and every point of the Moon s surface can be assured as well, with advantages especially for locations on its far side. The second requirement is about the LGPS. The aim of a global navigation system is to provide the navigation information to the maximum number of possible observers, that is, the system has to guarantee that the most of the surface is provided with the positioning signal. 2.2 The optimal control problem The station-keeping analysis is set as an optimization problem, solved using a direct collocation with non-linear programming DCNLP) approach Hargraves & Paris 1987; Enright & Conway 1991; Herman & Conway 1996; Ozimek, Grebow & Howell 2010; Fumenti, Circi & Romagnoli 2013). In its most generic form, an OCP can be defined through a system of first-order differential equations ẋ = f x t), u t),t) 3) describing the dynamics of the system, the state of the system at the initial and final times xt I ) = x I xt F ) = x F and the performance index to be optimized J = φ [x t), u t),t], 4) φ being a scalar function. In equations 3) and 4), t [t I,t F ]represents the simulation time, x is the n 1 vector of states, u is the m 1 vector of controls and f is the vector of algebraic equations of the states and controls with dimensions n 1. A series of path constraints may be added to the OCP through a vector g of algebraic equations which is bounded by a lower limit g L and an upper limit g U : g L g [x t), u t),t] g U. 5) The DCNLP method performs a discretization of the time interval and an approximation of the solution with piecewise polynomials, so that the stated OCP turns into an NLP problem Fumenti et al. 2013). The time interval is divided into ns subintervals using ns+1 points, referred to as nodes. The solution is approximated using piecewise polynomials to substitute the continuous functions xt) andut) in each subinterval. To give a better insight of the method, let us now consider a generic subinterval [t i,t i+1 ]. Since the method does not require any a priori information on the control, a simple linear trend for the controls ut) is assumed. Hence, u i = ut i )andu i+1 = ut i+1 ) being the control values at the nodes, the control value at the generic time t k [t i,t i+1 ]isgivenbyu k = ut k ) = t k u i+1 u i ) + u i. The approximation of the states, on the other hand, is not as easy: d points taken within the current subinterval are used to build a dth degree polynomial st). The building process of st) represents the main feature of the applied method, inasmuch as some points are used to find the coefficients of the polynomial, some points are used to force the polynomial to respect the equations of motion through constraints called defects. Considering for example the case d = 5, the nodes t i and t i+1 and three internal points t i,1, t i,2 and t i,3 are used. In particular, the points t i,t i,2 and t i+1 are used to build the polynomial, while the points t i,1 and t i,3 are used to define the defects, according to the relations i,1 = ṡ ) t i,1 f sti,1 ), u ) ) t i,1,ti,1 = 0, 6) i,3 = ṡ t i,3 ) f sti,3 ), ut i,3 ),t i,3 ) = 0. 7) Extending the process to all subintervals, the NLP problem is defined. The new unknowns become X T = x T 1, xt 1,2, xt 2,, xt ns+1, ut 1, ut 2,, ut ns+1,) and their values will be determined isfying the defects T = T 1,1, T 1,3, T 2,1, T 2,3,, T ns,1, T ns,3). In this work, the authors set the OCP to find the spacecraft s control time histories and trajectories over a pre-defined timespan while minimizing the total fuel consumption. The performance index has then been chosen to be J = t F ns+1 t I u i t) dt = i=1 t F ns+1 t I A i dt, i=1 where A i denotes the norm A of the control acceleration at the ith node.

5 Halo orbits for a lunar GPS design 3515 Requirements from Section 2.1 have been included in form of path constraints g. The first one is implemented imposing that the trajectory s projection in the yz plane must be outside of a circle whose radius is at least equal to the Moon s radius. Concerning the second requirement instead, the ideal setting would be a circular orbit lying in the yz plane, but from Fig. 2 one can see that natural orbits are quite far from having these features and too much effort should be used to alter them. Constraints too restrictive would be needed to meet the requests and the resulting station keeping would be too expensive. The compromise considered by the authors consisted in imposing path constraints to get trajectories with the same order of magnitude for the maximum displacements along the y- and the z-axes. The last feature required to start the optimization process is the initial guess, then halo orbits are used. Since halo orbits are solution of the CRTBP, they represent an abstraction and it would be meaningless to force a ellite to follow them. Rather they could be used as a starting point to obtain the real trajectory followed by the spacecraft in the real world. A numerical approach for determining a third-orderapproximated halo orbit has been provided by Farquhar & Kamel 1973) and Richardson 1980), and may be successfully applied to obtain the initial guess for the optimization procedure. Actually, there exist several Lagrangian points orbits, but due to their shape and the LGPS requirements, halo orbits seem to be the more appropriate ones. In this paper, for each Lagrangian point, several halo trajectories with different reference amplitudes and belonging to both families have been considered using Richardson s notation, the reference amplitude is denoted as A z and represents the maximum displacement along the z-axis). However, it is important to note that these trajectories act only as a starting point to find a proper solution to the problem: the ellites are not constrained to follow any halo orbit, rather they must isfy well-defined mission requirements, as already stated. 3 STATION-KEEPING ANALYSES During the simulation campaign, V required for the station keeping has been estimated for a time interval of 4 T H, with T H denoting the orbital period of the halo orbit used as initial guess; the total timeframe is then approximately 60 d. Longer periods have not been considered because preliminary tests have been performed for different timeframes 4, 8, 12T H ) and their results comparison showed the same trends without significant differences from each other. Indeed, as the time increases, it has been possible to observe an almost linear V time history. On the other side, the increasing time involved an increase in the computational effort more variables are required for solving the OCP), leading to extremely long simulation durations. For these reasons, the station-keeping analysis has been carried on only for the 4T H timeframe, while results for longer timeframes can be simply extrapolated. In dividing the time interval into subintervals, several values for the ns parameter have been considered, while for approximating the solution time history xt) a polynomial st) ofdegreed = 7 was used. The collocation points used to build the st) polynomial were derived from a Legendre polynomial, in particular from the fifth-degree Legendre polynomial as a consequence of choosing d = 7. In accordance with the halo classification, northern MHOs or of Class I) derive from northern haloes, while southern MHOs or of Class II) derive from southern haloes. Results are presented in Fig. 3 for L 1 point and Fig. 4 for L 2 point with left subfigures for northern MHOs and right subfigures for southern MHOs; in each subfigure, the V values are shown as a function of A z, which is used as an index to mark the orbit dimension. Each mark stands for a single simulation. Lowest Vs are emphasized with a square mark and connected with a solid line. It can be noted that for each A z, several V values are available. This can be explained considering that for each A z value several tests have been executed using different initial conditions; as a matter of fact, for a given subfigure, even if for each A z value the same halo orbit has been used as initial guess, the position along the orbit occupied by the ellite at the initial time was always different. Averaging the entire V set of results, one can get to the value of 3.29 m s 1, which leads to a yearly average consumption of about m s 1. The possibility to extrapolate the yearly consumption from the short period consumption is supported both by theoretical arguments the periodicity of the orbits) and actual results preliminary tests, as mentioned before). At first sight, the yearly consumption of m s 1 could seem high, nevertheless lower V are possible to reach, in fact looking at all the results obtained, it is possible to find cases Figure 3. V required for station keeping of MHOs around L 1 point for approximately 60 d. a) northern family-class I. b) southern family-class II.

6 3516 C. Circi, D. Romagnoli and F. Fumenti Figure 4. V required for station keeping of MHOs around L 2 point for approximately 60 d. a) northern family-class I. b) southern family-class II. in which the requested V is very low. This behaviour can be understood when considering the different initial conditions used inasmuch as the problem treated presents a high level of complexity. As a matter of fact, it must be remembered that the more complex the problem is, the stronger the influence of the initial condition is in finding a solution. The best results, i.e. the lowest Vs for each A z, show that the average V drops to 0.72 m s 1 with an yearly consumption under 4.32 m s 1 these results agree with those obtained by Gòmez et al. 1998). That said, the optimum A z value is not easy to find, because the station keeping is not the only concern for this work. In the main problem of studying an LGPS configuration, considerations about the LGPS performances of the ellite must be taken into account; hence, Figs 3 and 4 can only provide help in finding the less expensive solution but they cannot be considered as stand-alone results. 4 SATELLITE NAVIGATION AND GDOP COMPUTATION The ability of determining the position of an observer using a constellation of ellites comes from the measurement of the so-called pseudo-range between the observer and the ellites of the constellation. Let us introduce the vector X i = [ xi,yi,zi ] T, which defines the vector of the ith ellite, and the location of the observer X observer = [ x observer,y observer,z observer ] T. The distance between the observer and the generic ellite of the constellation can be expressed in terms of the time of travel that the GPS signal requires to spread from the ellite s antenna to the user s receiver. That distance can be expressed as d = x i x observer ) 2 + y i y observer ) 2 + z i z observer ) 2 = ct, 8) c being the speed of light and T the time travel of the GPS signal from the source on board the spacecraft to the user receiver. Equation 8) shows that a distance measurement translates into a time measurement, which is given by the difference between the time of arrival of the GPS signal at the observer and its departure time at the spacecraft. As a consequence, measurement of time is of crucial importance in order to achieve a good approximation of the location of the receiver. Even though the clocks on board the spacecraft are extremely precise, the clock in the receiver is generally less performing and this leads to the need of an additional unknown to the system, which is the offset between the ellites clocks and the receiver s one. Hence, in order to be able to solve the system of four algebraic equations in four unknowns and evaluate the observer s position, a set of minimum four ellites is required. The equation for the generic pseudo-range of the ith ellite is given by p,i R = x i x observer ) 2 + y i y observer ) 2 + z i z observer ) 2 + c t, 9) where t is the offset between the clocks and p,i R is the measured pseudo-range between the ith ellite and the current location of the observer. The pseudo-range equation shows that in order to achieve an accurate result for the user s location, the position of the ellites involved in the computation must be known with a great accuracy. Constellations around the collinear libration points of the Earth Moon system are well suited for this purpose, since range and range-rate measurements can be made from either the lunar surface or the Earth with cm accuracy using laser ranging techniques Dickey et al. 1994). The accuracy of the observer s position obtained using the pseudo-range does not depend only on the precision of the measurement of the ellites positions, but also on the geometrical configuration of the constellation with respect to the observer. As a matter of fact, the larger the separation of the ellites in the observer s sky, the better is the accuracy of the obtained solution. The quantity used to represent this dependence is the DOP, which is defined as the ratio between the position accuracy σ ) and the measurement accuracy σ 0 ), that is, Kaplan & Hegarty 2005) σ = DOP σ 0. Given the distance R i = xi x observer ) 2 + yi y observer ) 2 + zi z observer ) 2 between the ith ellite and the observer, the unit vectors from the receiver to the generic ith visible ellite of the constellation can be written as [ x i x observer ) R i, y i y observer ) R i, z i z observer ) ]and R i

7 Halo orbits for a lunar GPS design 3517 used to compute a matrix where the first three columns of each row are the components of the unit vector from the observer to each ellite and the fourth column is the speed of light. The matrix can be written as ) ) ) x 1 x observer y 1 y observer z 1 z observer c R 1 R 1 R 1 ) ) ) x 2 x observer y 2 y observer z 2 z observer c R 2 R 2 R 2 A = ) ) ) x 3 x observer y 3 y observer z 3 z observer. c R 3 R 3 R 3 ) ) ) x 4 x observer y 4 y observer z 4 z observer c R 4 The matrix A is then used to form a second matrix Q defined as σx 2 σxy 2 σxz 2 σ 2 xt σxy 2 σy 2 σyz 2 σ 2 yt Q =. σxz 2 σyz 2 σz 2 σzt 2 σ 2 xt σ 2 yt σ 2 zt σ 2 t R 4 R 4 Among the different definitions of DOP factors available in the literature, the GDOP is the most commonly used one and it is defined by the following relations: PDOP = σx 2 + σ y 2 + σ z 2, TDOP = σt 2, GDOP = PDOP 2 + TDOP 2, where the elements σ x, σ y, σ z and σ t are those defined in the matrix Q. Note that these elements do not represent variances and covariances as usually defined in statistics and probability, but are purely geometric terms. Small GDOP values represent good geometric configurations of the constellations resulting in better accuracies of the computed location of the observer. The bigger the GDOP value becomes, the worse is the accuracy of the solution provided by the GPS system. In addition, when more than four ellites are visible at the same time, two strategies are available: the first one extracts the solution from the four ellites that provide the minimum GDOP value, the second one uses all the available ellites and computes the solution of the overdetermined system with no unique solution by using the least-squares or similar techniques. 5 CONSTELLATION DESIGNS Two of the critical aspects in defining a global navigation system are the geographical and the time availability of the navigation service. These ensure that a potential user has a continuous and valid navigation message available at every possible location on the surface. The availability of the ellites from the ground and the quality of the navigation signal are mostly related to the geometry of the constellation used, while the time availability of a correct navigation signal relates to the number of ellites within the constellation. Hence, these two aspects have to be properly addressed when designing the LGPS. All the LGPS architectures proposed in this paper use the orbits obtained in the previous sections, i.e. MHOs around both the L 1 and the L 2 Lagrangian points of the Earth Moon system. Hence, due to symmetry reasons, only one Lagrangian point is used to introduce each constellation s configuration, while the complete scenario with both constellations around L 1 and L 2 is used to study different architectures performance. When multiple ellites are located on the same trajectory, they are initially phased so that they are uniformly distributed along the orbit. This criterion helps in spatially spreading the ellites in the sky above the observer s location. The visibility analysis has been performed by sampling the entire Moon s surface in both longitude and latitude using a grid with 5 and 10 resolution, respectively, and the motion of the spacecraft within the constellation has been propagated for four periods, corresponding to roughly 60 d. This value is basically prescribed by the optimization section, but it actually does not represent a limit, because the periodicity of the orbits involves also the repetition of the GDOP trend, allowing then the extension of the results to even longer timeframes. In addition, an elevation mask of 5 has been considered. Limitations of this procedure reside in the Moon libration and nutation motions as well as in the Moon s surface orography, which have not been taken into consideration. 5.1 Single MHO trajectory configuration Let us consider a single northern MHO trajectory around the L 2 Lagrangian point in the Earth Moon system. Since a user can compute his position only if at least four ellites are in his field of view, the first scenario to be tested has four ellites on a single MHO trajectory. A

8 3518 C. Circi, D. Romagnoli and F. Fumenti Figure 5. Initial configuration of the km amplitude L 2 constellation with a single MHO. set A z = [ ] km of out-of-plane reference amplitudes for the northern halo used as initial guess has been tested in order to unveil the relationship between the out-of-plane amplitude of the MHO orbit and the visibility of the four ellites from a location on ground. As an example, Fig. 5 shows the initial positions and the initial trajectory of the four ellites of the constellation around L 2 with out-of-plane reference amplitude of km. The initial conditions used to start the optimization process have been selected to have an even distribution of the ellites along the reference orbit, i.e. a phase separation of 90 for the four-ellite scenario, in order to maximize the spatial distribution of the ellites in the observer s sky and improve the GDOP to gain the best navigation accuracy possible. However, Fig. 5 shows that the solution obtained by the optimizer provides different initial conditions that better isfy the optimization constraints, characterized by a different initial phase separation than the initial guess. Nevertheless, the propagation showed that during the four periods the ellites maintain a reasonable space separation resulting in very few moments where the constellation is concentrated in a small fraction of the observer s sky. The visibility of the ellites from a location is computed using a standard geometrical approach: when the elevation of a ellite over the local horizon at a specific location is bigger than a pre-defined threshold, then the ellite is visible from that location. As examples, the visibility over the four periods for the different amplitude MHO trajectories are shown for a subset of locations on the Moon s surface, in order to provide an overlook of the performance of the different constellation designs at representative locations on ground. In particular, a few locations on the Moon s equator longitude =+25, latitude = 0 ), one at medium latitude longitude = 55, latitude = 45 ), one at high latitude longitude =+35, latitude = 80 ) and one at both high longitude and high latitude longitude =+85, latitude =+85 ) have been considered. Note that this analysis concerned only the visibility and that the GDOP value has not been taken into consideration. Note also that the origin of the reference frame used to define the lunar longitude and latitude is located at the intersection between the x-axis of the rotating synodic frame and the surface of the Moon towards the L 2 libration point, that is, on the far side of the Moon. Table 1 summarizes the results during the whole four-period timeframe for both the visibility analysis, through the minimum and maximum number of visible ellites given in columns 4 and 5, and the GDOP analysis, through an average value given in column 6. Note that the average GDOP values are reported only for those ellites configurations that isfy the visibility constraint from the selected location. In such cases where the number of visible ellites is less than four at some steps of the simulation time, the coverage is not continuous but the average GDOP value can still be computed. Conversely, in such cases where the number of visible ellites is less than four for the whole simulation time, the GDOP cannot be computed at all. As it can be seen from the values in Table 1, a single trajectory with only four ellites does not guarantee the total coverage of the surface nor the condition on the minimum number of visible ellites continuously in sight from the observer s location Fig. 6). As a matter of fact, higher values of latitude imply less visible ellites for increasing values of the reference amplitude. The surface of the Moon, in fact, acts as a shield blocking the line of sight of the ellites moving in the opposite hemisphere with respect to the current location of the receiver. It is, however, noticeable that there are numerous isolated configurations with only four ellites available where the GDOP average value is in the range [2 20], that is, a range usable for proper navigation purposes. This suggests that the spatial distribution provided by an orbit of MHO class is promising for the LGPS, granted the constraints on continuous and full coverage are isfied. Nevertheless, the architecture with four ellites distributed along one single MHO trajectory is not suitable for the LGPS, due to its intrinsic limitations in terms of coverage and number of ellites constantly in sight from any location on ground.

9 Table 1. Summary of the visibility and GDOP results for a single MHO around L 2. Halo orbits for a lunar GPS design 3519 Amplitude km) Longitude deg) Latitude deg) Sat. # Min) Sat. # Max) GDOP avg Figure 6. Visible ellites for a km north MHO around L Two MHO trajectories configuration In order to reduce the problems related to a single MHO configuration, a constellation made of eight ellites distributed over two MHOs a north and a south orbit) around the L 2 Lagrangian point is introduced. Recall that, due to the symmetry of the problem, the constellation can be studied only around one of the two libration points without lack of generality. Due to the characteristic shape of the north and south haloes, the resulting constellation has a peculiar X shape, which is beneficial for users at high values of latitude. The improvement is only slight though, because the availability and the quality of the LGPS signal close to the polar region remains limited due to the geometry of the problem Table 2).

10 3520 C. Circi, D. Romagnoli and F. Fumenti Table 2. Summary of the visibility and GDOP results for a two-mhos configuration around L 2. Amplitude km) Longitude deg) Latitude deg) Sat. # Min) Sat. # Max) GDOP avg From Table 2, it is noticeable that the km configuration is the one that offers the best GDOP performances at high values of both longitude and latitude, with values well below 20, indicating a good navigation solution when the four ellites are available. However, observers located at high latitude may not have continuous access to four ellites of the constellations, resulting in the impossibility of computing the position. 5.3 Four MHO trajectories configuration The continuous access problem is addressed using the complete configuration, i.e. the one exploiting four MHOs: a northern and a southern one around L 1 and a northern and a southern one around L 2 Fig. 7). In this case, a user located near the poles but the same applies for any position near the yz plane of the synodic frame) can see ellites around both Lagrangian points and compene for those moving in the opposite hemisphere and shielded by the lunar surface. The use of four MHOs allows for most of the configurations to have at least four ellites in sight, even if in Table 2 they had the minimum number less than four. For example, the previously cited configuration with reference amplitude of km does not have enough ellites around Figure 7. The complete LGPS constellation with two north and two south MHO trajectories.

11 Halo orbits for a lunar GPS design 3521 Figure 8. GDOP history for a user located at the lunar North Pole, for the km amplitude configuration with 16 ellites a single Lagrangian point Table 2 showed a minimum number of three for the user near the North Pole) but when MHOs around both Lagrangian points are considered this lack vanishes. For this case, the GDOP history is shown in Fig. 8, from which it can be seen that it is always included in the range [2 20], meaning that a good position solution can always be estimated. It is important to note that the requirement of a time continuous availability of the LGPS signal at high latitudes is quite challenging. Moreover, the availability of at least four ellites does not imply that a good position solution can be evaluated, because the ellites disposition in the observer s sky still plays an important role. The overall result is that even when the four-mho structure is envisaged, most of the proposed constellations from Table 2 should still be rejected. As a matter of fact, when these constraints are slightly relaxed, i.e. short timeframes with less than 4 ellites in sight or with the GDOP > 20 are allowed, one can see an increase of the number of constellations with 16 ellites in total spread over 4 different MHOs that can be successfully used. For example, Fig. 9 shows the variation of the GDOP over time with the km reference amplitude constellation for an observer located close to the lunar North Pole. It can be seen that this configuration does not meet the requirements in terms of acceptable performances. However, looking at the GDOP history it is possible to easily recognize that the LGPS signal would be degraded only for very short periods of time, i.e. corresponding to the Figure 9. GDOP history for a user located at the lunar North Pole, for the km amplitude configuration with 16 ellites.

12 3522 C. Circi, D. Romagnoli and F. Fumenti Table 3. Summary of the visibility and GDOP results for a two-mhos configuration with 24 ellites. Amplitude km) Longitude deg) Latitude deg) Sat. # Min) Sat. # Max) GDOP avg spikes in Fig. 9. This suggests that if one could relax the requirement on the continuous availability of a good LGPS signal, the constellation would accomplish the mission. From the performed tests, it emerged that configurations based on the smallest MHOs should be avoided, because the ellites appear condensed in a restricted region of the sky above the location of the observer, resulting in high GDOP values and, as a consequence, bad position solution. On the other hand, too big amplitudes may cause the observer on ground not to have enough ellites in its visibility cone, especially with an elevation mask applied. Hence, a balance between small size and big size MHOs has to be done. Besides, the selection of the reference amplitude of the orbits is crucial also in terms of the required V for the station keeping, as previously mentioned. Given these general constraints on the constellation design, the overall system s performances may be increased by either increasing the number of ellites within the constellation or reducing the elevation mask value currently set at 5, i.e. one may adopt the value ε min = 0.Inorder to maintain the scenario the most realistic possible, the reduction of the elevation mask has not been taken into consideration because it is not compatible with an observer on ground. Therefore, the same performance analysis has been repeated for the complete scenario including both L 1 and L 2 increasing the number of ellites on the two MHOs using six ellites for each MHO trajectory for a total of 12 ellites within the constellation around L 2 and other 12 ellites around L 1. The adoption of a more populated constellation with six ellites on each trajectory and the use of orbits around both libration points dramatically improve the performances of the LGPS in terms of minimum number of visible ellites and GDOP at high latitude and longitude values. In particular, Table 3 clearly shows that starting with the km reference amplitude the performances at high values of both longitude and latitude are able to provide the LGPS functionality at the polar regions. As a further proof of the performances improvement deriving from the 24 ellites utilization, once again the GDOP history for a user located near the lunar North Pole with the km reference amplitude configuration is considered and shown in Fig. 10. As it can be seen, the GDOP is always way below 20, denoting the chance to continuously compute a good estimate of the user s position. Fig. 11 shows the lines of sight from an observer located at the North Pole of the Moon to the 24 ellites of the constellation: the solid lines represent the available lines of sight, i.e. the visible ellites, while the dotted ones indicate interrupted lines of sight, that is, ellites not visible from the selected location. Note that the receiver sees ellites from both the L 1 and the L 2 orbits, allowing a better GDOP value deriving from a better spatial distribution of the ellites in the observer s cone of view. It is important to point out that the advantage of having ellites from both L 1 and L 2 is not only present when the observer is located at the polar regions, but for high longitude values around +90 and 90 as well. Based on the visibility and on the performance analysis, a candidate configuration for the LGPS involves four constellations of mixed north and south MHO orbits with six ellites on each of them. From the GDOP and coverage analysis, it emerged in fact that the best performances are ensured using six ellites on each MHO and reference amplitudes A z above km. From the station-keeping analysis, it came out instead that the cheapest options use MHOs with the highest reference amplitudes. It all led to two final options, both with six ellites on each MHO but with different reference amplitudes A z, i.e and km. Nevertheless, none of them is picked over the other, since they present quite similar performances and before coming out with the final choice further analyses should be performed, for example including features related to the development of the communication subsystem. These configurations guarantee to a ground-based observer the continuous availability of the LGPS service even at both poles of the Moon with a 5 minimum elevation over the local horizon. However, note that the availability of the LGPS increases for observer not on ground, i.e. spacecraft orbiting the Moon, which can benefit of smaller elevation mask angles: as a matter of fact, flying observers may also

13 Halo orbits for a lunar GPS design 3523 Figure 10. GDOP history for a user located at the lunar North Pole, for the km amplitude configuration with 24 ellites. Figure 11. Visibility from the northern lunar polar region. The solid lines represent visible ellites, while the dotted lines indicate not visible ellites. The reference frame is centred on the Moon and aligned with the synodic one. have visible ellites below the local horizon, resulting in negative elevation. In addition, having six ellites on each orbit can successfully cope with possible failures of a ellite: in such an event, in fact, the LGPS service is not interrupted but only degraded. 5.4 MHOs versus Lissajous and polar orbits In order to complete the analysis of the proposed LGPS system, it is important to compare the obtained results with other configurations, like the ones proposed by Romagnoli & Circi 2010) and Carretero & Fantino 2012). As already mentioned throughout the paper, Romagnoli & Circi 2010) proposed to use Lissajous orbits around both L 1 and L 2 points to deploy a constellation of ellites for LGPS and LGCS. Similar assumptions have been made in this paper, which could be seen as an extension of the cited work; hence, a comparison could be useful to spread some light on the behaviour of different orbits in the context of LPOs. Carretero & Fantino 2012) instead envisaged to place the constellation on polar orbits, distributing the ellites on three orbital planes. The interest in the comparison derives directly from the different design between the cited work and the current one, which can be easily perceived by the reader and attributed mainly to the exploited gravitational environment planetocentric motion versus Lagrangian point motion). Before starting with the real comparison though, it is worth saying that not all the constellations using MHOs are considered in the following. The reader can remember that both configurations with 16 and 24 ellites have been proved to successfully isfy the imposed requirements, but since the second ones can offer the best performances, the attention will be focused on them. Note also that the candidate configuration has six ellites on each MHO with reference amplitude A z of either or km, but no specific reference to one or the other is going to be made.

14 3524 C. Circi, D. Romagnoli and F. Fumenti For clearness reasons, let us now denote with CH the proposed configurations with six ellites on each MHO, CL the configuration from Romagnoli & Circi 2010) using Lissajous orbits, CP the configuration from Carretero & Fantino 2012) using polar orbits. From this comparison, one can see that the adoption of MHO trajectories presents important advantages but also some disadvantages over Lissajous and polar orbits. Starting with the disadvantages, they lie mainly in the size of the orbits and in the number of required ellites. Due to the big size and the characteristic shape of a halo orbit, the distance of a ellite from the surface can vary significantly. The use of MHOs tried to reduce this effect but it was not completely solved, neither this was the intention of the authors, as pointed out in the definition of the requirements and of the OCP. The compromise made allowed us to limit the fuel consumption at the expense of a wide range of variations for the ellite surface distances, which in turn affect the complexity of the communication system. Conversely, in CL and CP, this feature is not an issue, since the ellite surface distances vary in a very narrow range. The second drawback of using MHOs lies in the high number of required ellites, with heavy influence in the overall costs. In the CH configuration, 24 ellites are used, while the CL and CP are both below this value with only 8 and 15 ellites, respectively. It is worth noticing that CL does not take into account spare ellites, but even if 1 or 2 ellites were added for each Lagrangian point, the total number would end up being 10 or 12, still far below from the CH configuration. Nevertheless, it must be remembered that also CH configurations with 16 ellites have been found for being successfully used in this case the difference in the number of ellites between CH and CL or CP is not so heavy anymore). The configurations with 24 ellites have just been preferred because of their improved performances, like the redundancy to face failures without interrupting the service, which conversely is not included in the 16 ellites configurations. To see the advantages deriving from the use of MHOs, one should consider the required station keeping, the GDOP parameter evaluation and the requirement of communication with the Earth. The fuel consumption can be related directly to the choice of using LPOs, since the ellites can exploit the equilibrium of the Lagrangian points without heavy penalties on the manoeuvres required to counteract perturbations. The delicate equilibrium among gravitational forces plays the double role of a good/bad character, because even small perturbations can derive rapid drifts but at the same time small perturbations can be counteracted with tiny control actions. For this reason, fuel consumption from CH is far lower than CP, but still similar to CL. For a 10 yr mission, each ellite from CP, CL and CH would require a V of 1 km s 1, ms 1 and m s 1, respectively it must be noted though that a safety margin of 33% is included). An additional comparison parameter related to the fuel consumption could be the V budget required to reach the constellations, i.e. to move the ellites from Low Earth Orbit LEO) to MHO, Lissajous and lunar polar orbits. A first rough literature analysis showed that the V values for insertion are quite similar to each other Circi & Teofilatto 2006; Cadenas, van Damme & Centuori 2007; Romagnoli & Circi 2009). Nevertheless this feature requires a deeper investigation in the field of the transfer trajectories, which analysis was far beyond the intentions of the authors. The second advantage lies in the GDOP value. For what concerns the CL case, no data are available for this purpose; one can only suppose that this configuration offers the worst GDOP performances, since CH and CP rely on a better spatial distribution of the ellites in the observer s sky. Comparing the CH and CP configurations instead, the first one turns out to offer the best performances, since for each of the selected locations, the average value of the GDOP is always less than 20 it is worth noticing that this holds also for configurations with 16 ellites). On the other side, in the CP option there are some ranges of longitude where the GDOP is above the limit, making those regions unusable. Finally, when the line-of-sight feature is taken into consideration, CH is definitely the best option since none of the CL or CP constellations can address this requirement. As a matter of fact, the size and the characteristic shape of an MHO can ensure that none of the ellites, including those orbiting the cislunar point, are eclipsed by the Moon and a continuous communication link with the Earth can be established. Different scenarios could not request for a continuous direct communication with the Earth because such a requirement would ask for a highly complex communication subsystem and most probably, if an LGPS will ever be developed, it will be structured similarly to the GPS with a ground segment consisting of ground stations of the surface of the Moon this assumption has been made for example by Carretero & Fantino 2012)). Nevertheless, continuous direct communication with the Earth is an additional service that cannot be underestimated and for this reason it played a main role in the constellations comparison. As it is common for all engineering activities, a trade-off study has to be carried out before choosing the best configuration, but the possibilities opened by the adoption of the MHO-based LGPS represent an important asset for future Moon exploration missions. 5.5 Communications One of the key aspects when designing the LGPS is the communications between the spacecrafts and the surface of the Moon. Given that this paper is not intended as an in-depth analysis of the communications challenges typical of such a scenario, it is essential to try characterizing the radio-link involved. Knowing what typical the characteristics of GPS requirements are, it has been possible to realize a simplified analysis based on GPS-like requirements, such as: 1. frequency of Transmission f): MHz, the same as GPS L 1 ; 2. transmission bandwidth: 20 MHz similar to the GPS bandwidth); 3. nominal power guaranteed at reception P r ): -160 db W, the same as GPS L 1. Note that a 4.5 db W margin must be added to take uncertainties into account, resulting in a final power at reception of db W; 4. the ellite range is of the order of km; 5. the transmission and reception losses L r and L t ) are both set to 3 db.