Astrodynamics Vol. 5, No. 3, 263–278, 2021 https://doi.org/10.1007/s42064-021-0105-1 1 1 2 1 1 Federico De Grossi (B), Paolo Marzioli , Mengu Cho , Fabio Santoni , and Christian Circi 1. Sapienza University of Rome, 00138 Rome, Italy 2. Kyushu Institute of Technology, Kitakyushu 804-8550, Japan ABSTRACT KEYWORDS The Horyu-VI nano-satellite is an international lunar mission with the purpose of studying trajectory optimization the lunar horizon glow (LHG)—a still unclear phenomenon caused by electrostatically Artemis mission charged lunar dust particles. This study analyzes the mission trajectory with the lunar mission hypothesis that it is launched as a secondary payload of the NASA ARTEMIS-II mission. lunar horizon glow (LHG) In particular, the effect of the solar gravity gradient is studied; in fact, depending on weak stability boundary the starting relative position of the Moon, the Earth, and the Sun, the solar gradient acts differently on the trajectory—changing it significantly. Therefore, the transfer and lunar capture problem is solved in several cases with the initial Sun–Earth–Moon angle as the key parameter. Furthermore, the inclination with respect to the Moon at capture Research Article is constrained to be equatorial. Finally, the problem of stabilization and circularization Received: 30 April 2021 of the lunar orbit is addressed in a specific case, providing an estimate of the total Accepted: 2 July 2021 propellant cost to reach the final orbit around the Moon. © The Author(s) 2021 1 Introduction low a 1-km altitude. A small satellite platform can be used to continuously monitor the forward-scattering The Horyu-VI mission is a nano-satellite lunar mission of sunlight above the lunar terminator region to de- developed by the Kyushu Institute of Technology, in col- tect the column density of the lunar dust cloud. The laboration with the Nanyang Technological University of most recent attempt to detect the lunar horizon glow Singapore, Sapienza University of Rome, and the Cali- came from the lunar orbiter laser altimeter and the fornia Polytechnic State University [1]. The mission was laser ranging telescope onboard the Lunar Reconnais- proposed as a secondary payload of the ARTEMIS-II sance Orbiter [2]. These measurements aimed to test NASA mission. the hypothesis of a major meteor stream producing The purpose of Horyu-VI is to study the lunar horizon sufficient ejecta for the LHG. The outcome is most glow (LHG) phenomenon. It appears as a slim bright- likely that the LHG is a rare occurrence at altitudes ness above the lunar surface, and was observed by the below 20 km during meteor streams. These measure- astronauts of the Apollo missions and pictures of it were ments could not put any constraint for the LHG near a taken by Surveyor 1, 5, 6, and 7. It is believed that the 1-km altitude and below—where most of the exploration LHG is caused by lunar dust particles scattering light at activities are present. The study of the LHG could reveal sunrise and sunset; and since the brightness was stronger critical information about the dust particles near the than the level that could be produced by micrometeorite surface and at low altitudes—information which could be ejecta, a mobilized dust population by electrostatic forces important for future human and robotic missions since was predicted to be responsible for the forward scattering of sunlight [1–12]. the regolith dust is a significant risk factor in the lu- To date, there has been no dedicated mission to ob- nar environment [13–18]. A low lunar orbit is the most serve the lunar horizon glow, especially near and be- promising candidate for studying the LHG, from where B federico.degrossi@uniroma1.it 264 F. De Grossi, P. Marzioli, M. Cho, et al. Nomenclature a semi-major axis (km) e eccentricity i inclination ( ) Ω RAAN ( ) ω argument of pericenter ( ) θ true anomaly ( ) r spacecraft position vector (km) v spacecraft velocity vector (km/s) r position vector from the Moon to the spacecraft (km) R Moon position vector (km) r position vector from the Sun to the spacecraft (km) R Sun position vector (km) F thrust of the spacecraft (N) m mass of the spacecraft (kg) I specific impulse (s) sp 3 2 µ Earth gravitational constant (km /s ) 3 2 µ Moon gravitational constant (km /s ) 3 2 µ Sun gravitational constant (km /s ) θ Sun–Earth–Moon angle at the initial time ( ) sm0 J cost function t time of flight (day) m mass of propellant (kg) α , β angles of the thrust—variables of optimization ( ) N number of arcs of a trajectory arc h lunar pericenter height (km) h height of lunar flyby (km) flyby a solar radiation pressure acceleration vector (m/s ) SRP A spacecraft area (m ) P solar radiation pressure at 1 AU (N/m ) it is also possible, in principle, to extend the scientific In Section 4, the obtained results are presented and objectives by releasing femto-satellites—such as smart analyzed, and a constraint on the final inclination at the dust from the satellite [19]. arrival at the Moon is considered, as well as its influence The present work focuses on the trajectory analysis of on the trajectory. Section 5 presents further analysis on the Horyu mission: the transfer trajectory, in particular the sensitivity of the trajectory to small changes in the the effect that the solar gravity gradient has on it, and initial position of the Moon, affecting the flyby at the the stabilization of the orbit around the Moon. beginning of the trajectory, as well as an estimate of the The spacecraft is considered equipped with low-thrust effect of the solar radiation pressure. Hall effect thrusters, whose acceleration is smaller than In Section 6, the problem of stabilization and circular- the Sun gravity gradient for most of the time during the ization of the orbit is considered for one of the obtained transfer trajectory to the Moon; therefore, the effect of trajectories. The conclusions are nally fi presented in the Sun is of great importance and, according to the Section 7. initial position of the Earth and the Moon with respect to the Sun, the trajectory changes significantly. The 2 Horyu-VI spacecraft architecture and trajectory is computed in several cases, covering all the design features variation range of the Sun–Earth–Moon angle, which is the main parameter that changes the effect of the solar The Horyu-VI nano-satellite is a 12U CubeSat (approx- imately 200 mm × 200 mm × 300 mm) conceived for gradient on the trajectory. LHG observations in a lunar trajectory [1]. The space- The paper is composed of the following sections: Section 2 presents the spacecraft architecture and main craft architecture is illustrated in Fig. 1. The mission was design features. In Section 3, the strategy employed for conceived for NASA’s ARTEMIS-II call for secondary the design and optimization of the trajectory is described. payload proposals issued in 2019. The CubeSat payload Trajectory optimization for the Horyu-VI international lunar mission 265 36,500, 70,000, and 93,500 km [20]. In this mission anal- ysis, it is assumed that the spacecraft is released at an altitude of 70,000 km on the upper stage disposal trajec- tory. If no thrust is applied, the spacecraft encounters the Moon on a flyby—which increases its energy; there- fore, without control, the spacecraft will abandon the Earth–Moon system. The initial conditions for our simulation are in reference Fig. 1 CAD image of Horyu-VI. to December 15, 2017–14:56:42.2 TDB date; and the is composed of imaging sensors at different wavelengths trajectory is propagated considering the Earth–Moon– (monochrome, RGB, near-infrared (NIR) and ultraviolet Sun restricted four-body problem dynamics. The initial (UV) to detect and investigate LHG in lunar orbit. The orbital elements, referred to as the Earth, are given in spacecraft navigation will be made possible by ground- Eq. (1); and the equations of motion of the spacecraft in based range and range rate determination. The on-board a Cartesian Earth-centered inertial reference frame are transponder uses a chip-sized atomic clock (CSAC) for given in Eq. (2). time reference. The cis-lunar space characterization of the behavior of such a time-keeping device, whose oper- a = 206076.92 km, e = 0.9667, i = 28.61 (1) ◦ ◦ ◦ ations were already in-orbit demonstrated through the Ω = 65.96 , ω = 47.92 , θ = 148.41 SPATIUM-I mission in low Earth orbit (LEO), can be r˙ = v considered a secondary objective for spacecraft missions. v˙ = g + g + g + a = − r E pM pS c The spacecraft propulsion system relies on four xenon r R r R F m m s s gas Hall effect micro-thrusters capable of providing a − µ + − µ + + M S 3 3 3 3 r R r R m m m s s thrust of 150 µ N each. The thrusters are located on the (2) same side of the spacecraft, providing a total thrust of where r,v are the position and velocity vectors respec- 600 µ N. The required electrical power for the maneuvers, tively, g is the gravitational acceleration of the Earth, approximately 60 W for the entire system, will be gener- g is the gravity effect from the Moon, g is the grav- pM pS ated by the spacecraft deployable solar panels and stored ity effect from the Sun; a is the thrust acceleration, by the on-board batteries. The total amount of stored µ , µ , and µ are the Earth, the Moon, and the Sun E M S gas propellant is approximately 3 kg; and the dedicated gravitational parameters, respectively; r is the vector CubeSat volume to the propulsion and propellant storage from the Moon to the spacecraft; R is the vector from subsystems was approximately 8 units. The spacecraft the Earth to the Moon; r is the vector from the Sun to propulsion system does not involve any thrust vectoring the spacecraft; R is the vector from the Earth to the systems, so the thrust vector direction is controlled by the Sun; F is the thrust vector; and finally, m is the mass. attitude determination and control subsystem. This will The spacecraft is supposed to have an initial mass rely on reaction wheels for fine attitude control and on m = 20 kg, a total thrust of F = 600 µ N, and a specific cold-gas thrusters, using xenon and sharing the propel- impulse of I = 1000 s. sp lant tanks with the propulsion system for de-saturation. The final orbit around the Moon is a circular, equatorial The main attitude determination sensors are optical star low lunar orbit with a height of 100 km, and there are no trackers, and inertial measurement units. The entire further constraints on the right ascension or the argument system will assure a 0.1 degree pointing accuracy. of the pericenter. For the purpose of this study, the orbit to be achieved can be prograde or retrograde, with a 3 Trajectory design margin of 30 from the conditions of the equatorial orbit. As stated above, the mission begins as a secondary pay- In the following section, only the transfer trajectory is load of the SLS rocket in an Artemis mission. The SLS addressed, with the objective of obtaining an orbit with upper stage can release the secondary payloads on three eccentricity less than one, and pericenter height equal bus stops along its disposal trajectory at altitudes of to 200 km. Initially, the inclination of the arrival orbit 266 F. De Grossi, P. Marzioli, M. Cho, et al. around the Moon was left as free, and then the difference Equation (4) is the thrust vector in the inertial geocen- was shown when it was constrained instead. tric frame during the ith arc; Eq. (5) shows the vector of Different launching dates for the mission imply differ- optimization variables; Eq. (6) is the cost function, where ent Sun–Moon relative initial positions; therefore, the t = ∆t is the duration of the trajectory, and C f i i=1 solar gradient affects the trajectory in different ways— is the constraint on the final state as penalty functions. changing the cost of the trajectory in terms of time and No coasting arcs are considered; therefore, if the t is propellant needed. To analyze these effects, starting from minimum, the propellant mass is also minimum. the configuration of the date to which the initial condi- The design method of the trajectory is as follows: a tions are referred to, the position of the Sun is moved first starting guess is obtained by manually choosing the by intervals of 10 of the true anomaly along its ap- variables, and then an optimization is run with C in parent orbit around the Earth, until a complete turn the cost function in the form of Eq. (7)—where r and is completed. This generates new configurations of the R are the components of the position vector of the Sun–Earth–Moon system, which can be described by the spacecraft and of the Moon in the geocentric frame re- Sun–Earth–Moon angle at the initial time θ ; and the spectively. v and V are the components of the velocity sm0 j m trajectory is therefore computed in all 36 configurations vector of the spacecraft and of the Moon in the geocentric of θ . sm0 frame respectively, as in common rendezvous trajectory approaches [21]. The first optimization aims to find a b b θ = ± arccos(R · R ) (3) sm0 m0 s0 trajectory in which the spacecraft encounters the Moon. The Sun–Earth–Moon angle is computed as in Eq. (3), However, it is not sufficient to obtain the orbit with the and the plus sign is taken for the anti-clockwise rotation of required pericenter height as it often yields final condi- b b R from R , and the minus otherwise. Figure 2 shows s0 m0 tions with non-optimal semi-axis and eccentricity—which the angle, showing the orbital planes of the Moon and can be improved. Therefore, another optimization is run Sun in the Earth-centered frame. with C as shown in Eq. (8), where h is the pericenter The problem is approached as an optimization problem. height over the Moon, h = 200 km, and e and a t m m The trajectory is divided in N arcs of duration ∆ t , are the final eccentricity and semi-axis with respect to during which the thrust direction is fixed in the inertial the Moon respectively. The optimization is repeated, frame and defined by two angles α and β , and the i i progressively reducing a and e , the target semi-axis and t t number of optimization variables is therefore 3N. eccentricity around the Moon respectively, until a and T e no longer decrease, with e < 1 and h = h . m m p t F = F (cos α cos β sin α cos β sin β ) (4) i nom i i i i i 3 3 X X u = (∆t ∆t ··· ∆t , α β α β ··· α β ) (5) opt 1 2 N 1 1 2 2 N N C = c |r − R | + c |v − V | (7) R j m V j m m j m j J = t + C (6) j=1 j=1 Fig. 2 Visualization of the Sun–Earth–Moon angle θ . sm0 Trajectory optimization for the Horyu-VI international lunar mission 267 C = c |h − h | + c |e − e | + c |a − a | (8) later by the NASA GRAIL mission (2011). A similar h p t e m t a m t strategy can be exploited in our case where we are trying The employed optimizers are the interior-point al- to obtain the best effect from the Sun and the Earth to gorithm of the MATLAB function fmincon; the con- achieve a capture similar to the one typical of WSB. In straints considered are only the boundaries: ∆ t = i,min contrast to the classical ballistic WSB trajectory in our ◦ ◦ 0, ∆ t = 400 days, α = − 180 , α = 180 , i,max i,min i,max problem, there is also a flyby, and low-thrust propulsion ◦ ◦ β = − 90 , β = 90 ; the remaining options were i,min i,max is used. set as default, except for the maximum function evalu- After the lunar flyby, the spacecraft needs to be de- ations, which were set at 4000. The Nelder–Mead algo- celerated—but the thrust alone is not sufficient to do rithm of the function fminsearch was used as well, with this, especially if the solar gradient accelerates instead. default options except the maximum number of func- The analysis showed that the most efficient trajectories tion evaluations, which were set at 3000. The MATLAB have an initial phase before the apogee in which the solar version used was R2019a. gradient decelerates the spacecraft, and a phase after the apogee where it accelerates, easing the increase of the perigee. This class of trajectories are the best in terms of 4 Discussion on the results time of flight and propellant among the ones found, and The exploitation of the solar gradient to obtain low-cost appear to be possible for values approximately between ◦ ◦ ◦ ◦ Earth–Moon transfers is employed in the weak stabil- θ ∈ [− 30 , 0 ], and between θ ∈ [160 , 180 ]—with sm0 sm0 ity boundary (WSB) or Belbruno lunar trajectories. In a duration between 81 and 141 days. Figure 3 shows these trajectories, the gravity gradients of the Sun and one such trajectory in the geocentric reference frame, the the Earth are used to modify the trajectory, raising the thrust (blue arrows) and the gradient (red arrows) are perigee in the intermediate phase and decelerating in shown; the red dashed line represents the scaled path of the final phase to obtain a ballistic capture around the the Sun during the trajectory. The trajectory showed is Moon [22–24]. WSB trajectories were used in the Earth– the best among the ones found; it has a t = 81.5 days, a Moon transfers by the JAXA Hiten mission (1990) for mass of propellant required of 0.43 kg, and it achieves a the first time, as suggested by Edward Belbruno and lunar orbit with an eccentricity of 0.93, and the value of Trajectory Fig. 3 WSB-like trajectory in the Earth-centred reference frame, θ = −5.78 . sm0 y 268 F. De Grossi, P. Marzioli, M. Cho, et al. the Sun–Earth–Moon angle is equal to θ = −5.78 . motion of the Earth with respect to the Moon. sm0 Figure 4 shows the same trajectory in the Sun–Earth Figure 5 shows a comparison plot of the norm of the co-rotating frame in the left plot. In this reference frame, thrust acceleration and the norm of the Sun gradient the direction of the Sun is constant; thus, the gradient acceleration during the trajectory of Figs. 3 and 4. It acceleration field does not change—it is represented by can be seen that the solar gradient is a dominating ac- the red arrows in the plot. The right plot shows the celeration in the central part of the trajectory, which is arrival at the Moon. It can be seen that the Earth’s the longest part of it. gravity gradient decelerates the spacecraft in this final For values of θ farther from the most favorable ones, sm0 phase, helping during the capture. The expressions of the trajectories have a farther apogee because the Sun the gradient accelerations of the Sun and Earth plotted gradient decelerates less after the flyby; consequently, the in Figs. 3 and 4 are given in Eq. (9): trajectory requires more time and more thrust effort to r R r R avoid escaping the Earth at the apogee. Trajectories like s s e e g = −µ + , g = −µ + (9) S S E E 3 3 3 3 r R r R Fig. 6 can be found, presenting an unstable behavior at s s e e the apogee, and the angles of these solutions are the limit As before, r and r are the vectors from the Sun to s e values for which WSB-like trajectories are found, after the spacecraft and from the Earth to the spacecraft, which the resemblance with WSB trajectories disappears. respectively; and R , R are the vectors from the Earth s e In the cases where the Sun gradient accelerates the to the Sun and from the Moon to the Earth, respectively. spacecraft after the yb fl y, the thrust is not sufficient to In Fig. 3, the gradient is given by the expression on the avoid the geocentric energy becoming greater than zero; left in Eq. (9) in the Earth-centered frame, where R in these cases, the duration of the trajectory increases varies according to the motion of the Sun with respect to the Earth. In Fig. 4 on the left, g is computed in significantly, and the thrust can be used to allow the a reference frame with the x-axis parallel to the Earth– spacecraft to re-encounter the Earth in a favorable con- Sun direction; therefore, R is constant in this frame. dition for the Moon rendezvous. An example is shown In Fig. 4 on the left, the Earth gradient is given by the in Fig. 7, where the entire trajectory is shown in the left expression on the right in Eq. (9) in a Moon-centered plot, and it is zoomed on the Earth in the right plot to non-rotating frame, where R rotates according to the better show the departure and arrival. Fig. 4 WSB-like trajectory in the Sun–Earth rotating frame and at the arrival at the Moon. Trajectory optimization for the Horyu-VI international lunar mission 269 Fig. 5 Norm of the solar gradient acceleration and norm of the thrust acceleration during the trajectory of Fig. 3. Fig. 6 Trajectory with θ = 27.5 , t = 165.9 days. Earth-centered inertial reference frame. sm0 f The values of the optimization variables for the three possible to find a trajectory that achieved a close orbit trajectories presented above are listed in Tables 1, 2, around the Moon; the fastest trajectory is the one in and 3. Fig. 1 for θ = − 5.78 with t = 81.5 days and m = sm0 f p Overall, for all the values of θ considered, it was 0.43 kg. The longest trajectory has t = 1038.8 days, and sm0 f 270 F. De Grossi, P. Marzioli, M. Cho, et al. Fig. 7 Trajectory with θ = −63 , t = 468.1 days. Earth-centered inertial reference frame. sm0 f Table 1 Design variable values for the trajectory with Table 3 Design variable values for the trajectory with ◦ ◦ θ = −5.78 θ = −63 sm0 sm0 Number of arcs ∆t (day) α (deg) β (deg) Number of arcs ∆t (day) α (deg) β (deg) 1 2.849 112.106 11.059 1 4.112 −169.676 57.919 2 18.899 129.573 0.563 2 14.176 −71.686 −22.700 3 19.989 124.637 0.009 3 27.985 −69.839 −27.865 4 22.069 131.268 48.833 4 7.112 −48.758 −8.150 5 10.607 172.478 69.616 5 9.365 −43.015 −24.340 6 7.126 159.338 0.099 6 23.736 −14.619 −10.417 7 23.755 7.479 −5.122 8 64.520 −11.906 −54.167 Table 2 Design variable values for the trajectory with 9 65.219 170.868 30.101 θ = 27.5 sm0 10 14.117 137.646 16.012 11 54.853 −163.998 −35.248 Number of arcs ∆t (day) α (deg) β (deg) 12 47.469 −160.880 −17.451 1 16.886 129.248 21.155 13 42.336 −103.778 −7.758 2 11.009 113.292 27.056 14 38.809 −55.258 −12.083 3 21.104 −104.280 18.406 15 25.368 −54.266 −21.555 4 42.540 51.244 48.687 16 5.182 43.908 14.801 5 49.601 73.611 0.435 6 6.459 −53.544 7.504 resulted in variations between 829 and 3200 km. 7 10.650 −77.522 −4.959 8 7.699 −176.076 5.307 Figure 9 shows the behavior of the ∆ V with respect to θ . The ∆ V is the effective ∆ V exerted during the sm0 traj m = 5.47 kg, and it is found for θ = 67.21 . The p sm0 trajectory, and its behavior is similar to that of m /m p 0 trajectories have a number of arcs N between 6 and 25. and t , with the same maxima and minima. The ∆ V f circ The best trajectories generally needed a lesser number is the impulsive ∆ V that should be exerted to circularize of arcs, while the longest trajectories required more arcs. the lunar orbit at an altitude of 200 km, and it does not The lunar orbits at arrival have eccentricities between represent the true circularization effort—which will need e = 0.927 and e = 0.963. Figure 8 shows the plots of the to be carried out with low-thrust propulsion. The only time of flight, propellant to initial mass ratio m /m , thing worth noting from it is that the orbits achieved p 0 and the height of the flyby with the Moon for all the are quite similar as the difference in ∆ V between one circ values of the angle θ considered. Minima can be seen another is just some tens of m/s. The ∆ V in the sm0 tot ◦ ◦ around θ = 0 , 180 ; and it can be noted that they final plot is the sum of the previous two, and includes sm0 roughly correspond to the maxima of the h , which the departure ∆ V given by the launcher to put the flyby dep Trajectory optimization for the Horyu-VI international lunar mission 271 Fig. 8 Time of flight ( t ), propellant mass ratio (m /m ), and flyby height (h ). f p 0 flyby Fig. 9 Trajectory ∆V , Circularization impulsive ∆ V , and total ∆ V . payload in lunar transfer from a circular orbit around after 180 of variation of the angle parameter. This be- the Earth—which is estimated to be 3068 m/s. havior is due to the symmetry of the solar gradient around In both Fig. 8 and the first and last plots of Fig. 9, it the Earth for variations of 180 of the Sun’s position. can be seen that the solutions exhibit some periodicity If the orbit of the Earth had been considered perfectly 272 F. De Grossi, P. Marzioli, M. Cho, et al. circular and the perturbation of the Sun approximated the unconstrained trajectories. It can be seen that for for small distances from the Earth, then every trajectory most of the trajectories the difference in duration is less would remain unchanged if the position of the Sun was than 10 days, and the maximum positive difference is shifted by 180 . Because there is a small eccentricity of 32.98 days—corresponding to 173.7 g of extra propellant the Earth orbit and the complete perturbation expression and a ∆ V of 93.22 m/s. In one case a notable difference is considered, there are differences that require calculat- is found: −24.85 days, −130.9 g, and −66.69 m/s. ing the trajectory even for the almost-symmetric cases; To provide an outline of the trajectories not shown however, it was possible to employ the solutions already before, and to visualize how they vary with the variation computed as a starting guess for the corresponding 180 of the angle parameter, twelve trajectories are plotted shifted cases. in Fig. 12 (one every 30 )—around a visualization of As stated before, the inclinations of the obtained lunar the Sun–Earth–Moon angle in which the direction of the orbits are free, and they result in a uniform distribution Moon is visualized as a blue arrow and the direction ◦ ◦ between 30 and 165 . The Horyu mission needs an equa- of the Sun for all the angles as red arrows. With this torial orbit around the Moon in order to properly achieve disposition, it is easy to see the zones in which typical its objective; therefore, a constraint on the inclination trajectories appear. The trajectories in this figure include of the lunar orbit is considered here. All the trajectories the constraint on the inclination. are forced to arrive in the equatorial orbit around the Moon; both prograde and retrograde equatorial orbits are 5 Flyby sensitivity and effect of the solar considered valid, and the interval of inclinations allowed radiation pressure ◦ ◦ ◦ ◦ is i ∈ [0 , 30 ] and i ∈ [150 , 180 ]. It was possible to achieve the orbits in this range for all trajectories—often The initial flyby of the Moon has a significant influence at the cost of a higher t . Figure 10 shows the inclina- on the rendezvous trajectory; further, the initial position tions before the constraint (upper plot) and after the of the Moon is of great importance in WSB transfers. constraint (lower plot). Figure 11 shows the difference Here, a brief analysis of the sensitivity of the problem in time of flight, propellant, and ∆ V with respect to to small variations in the initial position of the Moon is Fig. 10 Lunar inclination with and without equatorial constraint. Trajectory optimization for the Horyu-VI international lunar mission 273 Fig. 11 Differences in time of flight, propellant mass, and total ∆ V between trajectories with equatorial inclination of the final orbit and free trajectories. 152.5 122.7 92.9 33.2 5.8 Z Z Y Y Fig. 12 Outline of the transfer trajectories with the θ parameter. The blue arrow represents the direction of the Moon sm0 and the red arrows represent the direction of the Sun at twelve different angles; one angle is highlighted as an example. Trajectories are plotted in the Earth-centered frame. 274 F. De Grossi, P. Marzioli, M. Cho, et al. Table 4 Results of the variation of the initial position of presented. The best trajectory among the ones found, the Moon in the case where the constraint on the inclination at δt δm δh δr f p flyby apo the arrival at the Moon is considered as a case study; Moon displ. (day) (g) (km) (10 km) the Moon is moved along its orbit by steps of 1 both −1 +28.16 +148.4 −979.7 +4.181 ahead and behind the nominal initial position—then +1 −4.64 −24.5 +891.7 −0.706 the trajectory is re-optimized to achieve the rendezvous +2 −4.08 −21.5 +1242.1 −0.430 again. With a displacement of − 1 , the performance +3 +2.40 +12.6 +1490.6 −0.481 worsens; the flyby has a lower altitude, and the duration +4 +37.84 +19.9 +3211.0 −4.175 of the trajectory is increased by 28.16 days. With a displacement of +1 , the performance improved, the Full specular reflection is considered in such a way that flyby height increased, and the duration was reduced the acceleration given by the solar radiation pressure by 4.64 days. Continuing the analysis in the favorable is always in the opposite direction of the Sun. These direction shows that the initial position displaced by +1 conditions allow an estimate of the maximum possible is a minimum around the nominal trajectory; in fact, effect of the SRP on the trajectory. moving to +2 , the performance is still better—but of a a = P rb (10) SRP E s smaller amount. At +3 instead, the duration increases, and also at +4 even when the shape of the trajectory is Equation (10) shows the expression of the acceleration changed. of the SRP in the considered conditions; P is the solar Table 4 summarizes what is said above, showing the dif- radiation pressure at 1 AU, A is the area of the spacecraft, ference in duration, propellant, yb fl y height, and apogee m is the mass, and rb is the unit vector from the Sun to with respect to the nominal trajectory. Figure 13 shows the spacecraft. The numerical results show that the solar the trajectories found for every position variation and radiation pressure has a negligible effect and the nominal also the nominal trajectory (0 ) for comparison. trajectory has its duration changed by 2600 s. In the analysis in the previous section, the effect of the solar radiation pressure was not taken into account; here, we consider its effect on the best trajectory in a simplified 6 Stabilization and circularization way. The area of the spacecraft is considered constant during the trajectory, equal to the maximum area of In this section, the problem of obtaining a stable orbit the satellite and always facing the direction of the Sun. around the Moon is addressed. The trajectory with the Fig. 13 Transfer trajectory with θ = − 5.78 subject to small variations of the initial position of the Moon. Plots in the sm0 Earth-centred inertial reference frame. Trajectory optimization for the Horyu-VI international lunar mission 275 Table 5 Cost of the trajectory from the Earth to low lunar least time of flight among the previous trajectories was orbit considered as a case study. Transfer trajectory Stabilization and The final semi-axis and eccentricity of the lunar ren- Total (θ = −5.78 ) circularization sm0 dezvous trajectory are a = 29,571 km, e = 0.934. If left t (day) 81.54 684.23 765.77 free, the spacecraft will quickly escape from the Moon; m (kg) 0.4297 2.0775 2.5072 instead, the spacecraft must achieve a circular orbit at an altitude of 100 km. For this purpose, the trajectory is continued with some arcs in which the direction of the thrust is constrained to vary in a cone around the anti-velocity direction; an optimization is run with the objective of reducing the semi-axis below 20,000 km and the eccentricity below 0.7, in order to reach a stable orbit. Then, the orbit is circularized, and the height is reduced by employing the following strategy: • First, the thrust is in the anti-velocity direction during the orbit when the true anomaly is inside an interval centered around the pericenter, here taken ◦ ◦ as [− 130 , 130 ]. When this condition is not met, the thruster is shut off. This reduces the semi-axis and eccentricity of the orbit. • After the eccentricity is reduced to almost zero, the Fig. 14 Trajectory to low lunar orbit, plotted in the Moon- thrust returns continuously during all the orbits in centric inertial frame. the anti-velocity direction until the required height is reached. Table 5 shows the duration and propellant consumption of the considered trajectory; in the first two columns, the time and propellant are divided between the transfer trajectory (without the added arcs used for stabilization) and the stabilization and circularization maneuver until the final orbit is reached. The last column presents the total cost of the mission. The propellant needed is inside the 3 kg limit with a margin of 0.4928 kg, and the total duration is 765.77 days, of which 484.75 days are of thrusting. Figure 14 shows the trajectory from the arrival around the Moon to the end of the maneuver on the final orbit; and Fig. 15 shows the evolution of the orbital elements during stabilization. To be more readable, the plot starts from 81.54 days after departure Fig. 15 Orbital elements during the stabilization and cir- from the Earth, when the original transfer trajectory cularization trajectory. is ended. The semi-axis is gradually reduced and the spacecraft in a stable orbit, then clearly the total cost eccentricity is subject to some oscillations during the would have been notably less. The trajectory presented in initial phase, before becoming more stable and starting this section can be taken as an example; if it is truncated to decrease until it reaches zero. The inclination of the at 37.23 days after the end of the Earth–Moon transfer, orbit reached is 11 —it is therefore acceptable for the it yields an orbit with a semi-axis of about 19,000 km imposed constraints. As a final remark, if instead of a low orbit at an altitude (having a pericenter height of 6400 km, and an apocenter of 100 km, the requirements had been to simply place the of 28,800 km) that remains stable for at least 500 days. 276 F. De Grossi, P. Marzioli, M. Cho, et al. The propellant cost would be 0.2 kg over the transfer mass, and considering the best transfer trajectory, the trajectory cost, so the total cost would be 0.626 kg— mission results are feasible. about a quarter of the mass to reach the 100 km circular orbit. Funding note Open Access funding provided by Universit`a degli Studi 7 Conclusions di Roma La Sapienza. This work presented the trajectory analysis of the Horyu- VI nano-satellite, an international lunar mission with the References goal of studying the lunar horizon glow. [1] Orger, N. C., Cho, M., Iskender, O. B., Lim, W. S., In particular, the effect of the solar gravity gradient on Chandran, A., Ling, K. V., Holden, K. H., Chow, C. the trajectory was analyzed, covering the entire range of L., Bellardo, J., Faure, P., et al. Horyu-VI: Interna- values of the Sun–Earth–Moon angle at departure. With tional CubeSat mission to investigate lunar horizon glow. the given low-thrust propulsion system, a transfer trajec- In: Proceedings of the 71st International Astronautical tory was found for all configurations, achieving a lunar Congress, 2020: IAC-20,B4,2,7,x55547. [2] Barker, M. K., Mazarico, E., McClanahan, T. P., Sun, orbit with eccentricity less than one, and a pericenter at X., Neumann, G. A., Smith, D. E., Zuber, M. T., Head, an altitude of 200 km. The range of propellant required J. W. Searching for lunar horizon glow with the lunar varies from 490 g to 5.47 kg, although the most expensive orbiter laser altimeter. Journal of Geophysical Research: cases require a greater quantity of propellant than the Planets, 2019, 124(11): 2728–2744. expected 3 kg. It was found that the best trajectories [3] Colwell, J. E., Batiste, S., Horanyi, M., Robertson, resemble the WSB Earth–Moon transfers in the way they S., Sture, S. Lunar surface: Dust dynamics and re- exploit the Sun gradient to decelerate and accelerate golith mechanics. Reviews of Geophysics, 2007, 45(2): where most needed; therefore, these types of trajectories 2005RG000184. should be considered the most feasible for the considered [4] Criswell, D. R. Horizon-glow and the motion of lunar problem. dust. In: Photon and Particle Interactions with Surfaces A constraint on the inclination of the arrival orbit at in Space. Grard, R. J. L. Ed. Dordrecht: Springer, 1973: 545–556. the Moon is also considered, and the difference in cost [5] Feldman, P. D., Glenar, D. A., Stubbs, T. J., Retherford, and time is computed with respect to the unconstrained K. D., Randall Gladstone, G., Miles, P. F., Greathouse, case. Reaching an equatorial orbit is possible in all T. K., Kaufmann, D. E., Parker, J. W., Alan Stern, configurations, and the differences in time and propellant S. Upper limits for a lunar dust exosphere from far- are small. ultraviolet spectroscopy by LRO/LAMP. Icarus, 2014, Among the transfer trajectories, some further analysis 233: 106–113. was performed on the best one considering its sensitivity [6] Glenar, D. A., Stubbs, T. J., Hahn, J. M., Wang, Y. to small changes in the Moon’s initial position—which Search for a high-altitude lunar dust exosphere using affect the flyby. It was found that a few degrees of clementine navigational star tracker measurements. Jour- variation in the angular position are admissible without nal of Geophysical Research: Planets, 2014, 119(12): drastic changes in the trajectory—and in some cases, they 2548–2567. [7] Glenar, D. A., Stubbs, T. J., McCoy, J. E., Vondrak, R. might even produce an improvement in the performance. R. A reanalysis of the apollo light scattering observations, It was also assessed that the solar radiation pressure and implications for lunar exospheric dust. Planetary should be of very little importance for this mission. and Space Science, 2011, 59(14): 1695–1707. The problem of the stabilization of the lunar orbit was [8] McCoy, J. E. Photometric studies of light scattering also considered. In the specific case of the best transfer above the lunar terminator from apollo solar corona trajectory, the cost of this maneuver is higher than the photography. In: Proceedings of the Lunar and Planetary transfer trajectory cost. Reaching the final low lunar Science Conference, 1976, 7: 1087–1112. orbit of the Horyu-VI mission required a total of 2.5 kg [9] McCoy, J. E., Criswell, D. R. Evidence for a high altitude of propellant—equal to 12.5% of the initial mass. These distribution of lunar dust. In: Proceedings of Lunar and results show that with the given allocated propellant Planetary Science Conference, 1974, 5: 2991–3005. Trajectory optimization for the Horyu-VI international lunar mission 277 [10] Rennilson, J. J., Criswell, D. R. Surveyor observations Federico De Grossi received his M.S. of lunar horizon-glow. The Moon, 1974, 10(2): 121–142. degree in space and astronautical engi- [11] Severny, A. B., Terez, E. I., Zvereva, A. M. The mea- neering at Sapienza University of Rome surements of sky brightness on lunokhod-2. The Moon, in 2019, with a thesis on rendezvous tra- 1975, 14(1): 123–128. jectories with orbits around asteroids. He [12] Zook, H. A., McCoy, J. E. Large scale lunar horizon glow is currently a Ph.D. student, and his fields and a high altitude lunar dust exosphere. Geophysical of study are space mission trajectory anal- Research Letters, 1991, 18(11): 2117–2120. ysis and optimization, optimization meth- [13] Gaier, J. R. The effects of lunar dust on EVA systems ods, and quantum information and computation. E-mail: during the Apollo missions. Report No. E-15071. NASA federico.degrossi@uniroma1.it. Glenn Research Center, Cleveland OH, 2005. [14] Harris, R. S. Jr. Apollo experience report: Thermal design of Apollo lunar surface experiments package. Re- port No. TN D-6738. National Aeronautics and Space Administration, WDC, 1972. [15] James, J. T., Lam, C. W., Quan, C., Wallace, W. Paolo Marzioli is a post-doctoral re- T., Taylor, L. Pulmonary toxicity of lunar high- search fellow at Sapienza University of land dust. SAE Technical Paper 2009-01-2379, 2009, Rome. He received his Ph.D. degree https://doi.org/10.4271/2009-01-2379. in aeronautical and space engineering in [16] Linnarsson, D., Carpenter, J., Fubini, B., Gerde, P., 2021, with a thesis on the implementation Karlsson, L. L, Loftus, D. J., Prisk, G. K., Staufer, U., of navigation systems for nano-satellites. Tranfield, E. M., van Westrenen, W. Toxicity of lunar His main research topics deal with nano- dust. Planetary and Space Science, 2012, 74(1): 57–71. satellite missions, stratospheric experi- [17] O’Brien, B. J. Review of measurements of dust move- ment development and tracking, and navigation systems ments on the Moon during Apollo. Planetary and Space evaluation for different aerospace mission profiles. E-mail: Science, 2011, 59(14): 1708–1726. paolo.marzioli@uniroma1.it. [18] O’Brien, B. J. Paradigm shifts about dust on the Moon: From Apollo 11 to Chang’e-4. Planetary and Space Sci- ence, 2018, 156: 47–56. [19] Niccolai, L., Bassetto, M., Quarta, A., Mengali, G. A re- view of Smart Dust architecture, dynamics, and mission application. Progress in Aerospace Sciences, 2019, 106: Mengu Cho received his B.S. and M.S. 1–14. degrees from the University of Tokyo, [20] Smith, D. A. Space Launch System (SLS) Mission Plan- and his Ph.D. degree from Massachusetts ner’s Guide. NASA M19-7163, 2018. Institute of Technology, in 1992. After [21] Del Monte, M., Meis, R., Circi, C. Optimization of working at Kobe University and Interna- interplanetary trajectories using the colliding bodies op- tional Space University, he joined Kyushu timization algorithm. International Journal of Aerospace Institute of Technology (Kyutech) in Engineering, 2020, 2020: 9437378. 1996. Since 2004, he has been a pro- [22] Belbruno, E., Miller, J. Sun-perturbed Earth-to-Moon fessor. Currently, he is the director of Laboratory of Lean transfers with ballistic capture. Journal of Guidance, Satellite Enterprises and In-Orbit Experiments. His research Control, and Dynamics, 1993, 16: 770–775. interests include spacecraft environmental interaction and [23] Kawaguchi, J., Yamakawa, H., Uesugi, T., Matsuo, H. satellite systems. He has supervised 11 university satellite On making use of lunar and solar gravity assist for Lunar projects, among which 9 projects and 16 satellites have al- A and planet B missions. Acta Astronautica, 1995, 35: ready been launched. He has authored or co-authored more 633–642. than 180 papers in peer reviewed journals. His research [24] Romagnoli, D., Circi, C. Earth–Moon weak stability interest is nano-/micro-satellite development and applica- boundaries in the restricted three and four body problem. tions. In 2019, he received Frank J. Malina Astronautics Celestial Mechanics and Dynamical Astronomy, 2009, Medal from International Astronautical Federation. E-mail: 103(1): 79–103. cho.mengu801@gmail.kyutech.jp. 278 F. De Grossi, P. Marzioli, M. Cho, et al. Fabio Santoni is an associate pro- lar sails, orbits for planetary observation, and the as- fessor of aerospace systems at Sapienza cent trajectory of Launcher. He is the associate editor University of Rome. His main research for Aerospace Science and Technology and the Interna- topics are nano-satellite mission design, tional Journal of Aerospace Engineering. E-mail: chris- development and operations, space de- tian.circi@uniroma1.it. bris observations and mitigation, and sustainable space exploration. E-mail: Open Access This article is licensed under a Creative fabio.santoni@uniroma1.it. Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and re- Christian Circi is currently an asso- production in any medium or format, as long as you ciate professor in flight mechanics at the give appropriate credit to the original author(s) and the Department of Astronautical, Electrical, and Energy Engineering, Sapienza source, provide a link to the Creative Commons licence, University of Rome. He received his M.S. and indicate if changes were made. degree in aeronautical engineering and The images or other third party material in this article aerospace engineering and his Ph.D. de- are included in the article’s Creative Commons licence, gree in aerospace engineering at Sapienza unless indicated otherwise in a credit line to the mate- University of Rome. He worked as a researcher at the rial. If material is not included in the article’s Creative Grupo de Mecanica of Vuelo-Madrid (GMV) and a research Commons licence and your intended use is not permitted assistant at the Department of Aerospace Engineering. He by statutory regulation or exceeds the permitted use, you is a lecturer in “Interplanetary Trajectories” and “Flight will need to obtain permission directly from the copyright Mechanics of Launcher” in the master degree course of holder. space and astronautical engineering at Sapienza University To view a copy of this licence, visit http://creative- of Rome. His principal research fields are third-body and solar perturbations, interplanetary and lunar trajectories, so- commons.org/licenses/by/4.0/.
Astrodynamics – Springer Journals
Published: Sep 1, 2021
Keywords: trajectory optimization; Artemis mission; lunar mission; lunar horizon glow (LHG); weak stability boundary
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