Multi-Target Low-Thrust Interplanetary Trajectory Design of DESTINY

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a feasible trajectory. This is particularly more challenging for the PGC members, as their highly eccentric and inclined orbits would impose very narrow time windows of flyby opportunities with very high relative velocities. In DESTINY+’s case, this is combined with low relative velocities at Earth escape and limited thrust capability of ion engines. Additionally, the spacecraft design imposes constraints on the trajectory, making the interplanetary trajectory design of DESTINY+ challenging altogether. The interplanetary trajectory design of DESTINY+ was previously investigated by Sarli et al.10, 11) Sarli et al. (2015) presented ballistic transfer opportunities by solving Lambert’s problem and applying gravity-assist maneuvers by Earth and Mars for multiple asteroid visits to the PGC.10) In a more recent work, Sarli et al. (2017) computed low-thrust optimal trajectories for Earth-Phaethon transfers only and presented potential targets for mission extension regardless of their relation to the PGC.11) (a) xy-plane Building on these works, the present study explores the optimal transfers between Earth and the PGC members by consid- ering low escape energy, limited thrust and mission constraints of DESTINY+. A large database of ballistic transfers for Earth-Phaethon (E1-P2), Phaethon-Earth (P2-E3) and Earth- Secondary Asteroid (2005UD, 1999YC) are generated. Can- didate low-thrust solutions for full mission sequence are identified with a simple “minimum ∆V” criterion and C3-matching algorithm. Transfer opportunities are expanded as compared to Sarli et al. (2015) by the addition of low-thrust propulsion. The chosen initial conditions are then inputted to GALLOP in order to generate optimal solutions. Robustness of a selected trajectory is assessed in a sensitivity study. The paper is structured as follows: In Sec. 2., the Phaethon- Geminid Complex is briefly introduced. In Sec. 3., initial guess generation is explained in more detail. In Sec. 4., optimal solutions are presented. Finally, conclusions are presented in Sec. 5.. (b) xz-plane Fig. 1.: Orbits of the target bodies 2. Phaethon-Geminid Complex

The main scientific target DESTINY+, 3200 Phaethon (for- impose large ∆Vs for a rendezvous mission, therefore only fly- merly known as 1983TB), is an Apollo asteroid with cometary bys are possible. Due to ion engine of DESTINY+, the velocity features.12) It is also the source of Geminids meteroid shower, change that can be given to the spacecraft is limited. This paper which occurs mid-December every year. It was also identified therefore investigates the low-∆V transfers to Phaethon and the to be a potentially hazardous asteroid due to its minimum in- other target bodies. This begins with selection of suitable initial tersection distance with Earth. Extended observations since its conditions that are presented in the next section. discovery allowed researchers to constrain its orbit and properties.13) According to this, Phaethon is 5.1 km ±0.2 km aster- Table 1.: Orbital properties of the Phaethon-Geminid Complex 14) oid with a spin period of ∼3.6 h.13) Phaethon orbits the Sun in Ecliptic J2000 frame (as of April 27, 2019) on a ∼22o inclined, highly eccentric orbit (∼0.89) with respect Property 3200 Phaethon 2005UD 1999YC to ecliptic plane, with 0.14 AU at the closest point to the Sun Eccentricity 0.89 0.87 0.83 Semi-major with an orbit period of approximately 1.4 years.14) Asteroids 1.27 1.27 1.43 axis [AU] (155140) 2005UD and (225416) 1999YC are suggested to orig- Inclination [o] 22.26 28.67 38.23 inate from Phaethon due to their spectroscopic and orbital simi- Arg. of 3, 4) 322.18 207.58 156.40 larity. Both 2005UD and 1999YC are ∼1.5 km-sized objects perihelion [o] on roughly similar orbits Phaethon. 2005UD has an orbit pe- Right ascension of 265.21 19.73 64.78 riod very similar to Phaethon, i.e. 1.4 years, whereas 1999YC the ascending node [o] has longer orbit period (∼1.7 years). Figure 1 shows their or- Period [yr] 1.43 1.44 1.70 bits, and Table 1 tabulates their orbital properties. As can be inferred from the Fig. 1, the orbit characteristics of the PGC 3. Initial guess solutions they show the best patching opportunities, they do not specify whether there exists a suitable low v∞ transfer at Earth depar- Finding a globally optimal solution is a challenging task for ture for a given low patching ∆V. It only specifies that patching mission designers. One deals with multiple local minima for between incoming and outgoing trajectory arcs is possible with given bounds in parameters and for that one needs good ini- low ∆V. We therefore derive a final relation to assess the amount tial guess for a converging optimal solution. One of the most of impulsive ∆V that needs to be provided during E1-P2-E3 se- common methods to generate initial conditions is the solution quence, described as below: of the Lambert problem. The Lambert problem deals with the conic trajectory between a two given position in space for a ∆V = Vdep + V − V (2) given time-of-flight (ToF) between them.15) This is generally total ∞ miss M2M dep a good approximation for high-thrust near-impulsive transfers. where V∞ is the magnitude of the velocity at the Earth escape However, solution to the the low-thrust transfer problem is not and VM2M is the magnitude of velocity gain at the end of mul- as straightforward, because transfer trajectories include large tiple lunar flyby phase. In its current form, Equation 2 assumes dep thrusting arcs in extended periods of time due to limited thrust- that departure V∞ and Vmiss have equal weighting because of ing capability. Nevertheless, the solution to the Lambert trans- the impulsive ∆V assumption. But Eq. 2 is likely to include dep fers can provide hints about the near-ballistic or low-∆V trans- weighting coefficients to scale the contributions from V∞ and fers that can be used in the initial guess generation for the op- Vmiss as a function of, for example, transfer time to Phaethon. timization phase. Therefore, the Lambert transfer problem is This was not considered in the current paper and left as a future solved for each leg of the DESTINY+ mission. The Lambert work. Equation 2 can then be used to compute total ∆V values solver in this paper can account for multiple revolution trajec- for given Phaethon flyby dates and a range of Earth-departure tories around the central body (in this case, the Sun).16) Given and Earth-arrival dates, similar to mismatch ∆V maps. In prin- the limited amount of ∆V that can be provided by DESTINY+, ciple, minimum values in this map should provide converging only prograde solutions are taken into account. Figure 2 shows initial guesses to the optimization process, as shown in Fig. 4 the lowest-∆V transfers between Earth and Phaethon (E1-P2) for the flyby date Jan 4, 2028. for Earth departure dates between Sep, 1 2025 and July 27, Total ∆V up to 3 km/s is shown in the figure. The vertical dep 2029, and Earth arrival dates between July 01, 2026 and June lines in the figure shows departure V∞ values therefore one 05, 2031, in accordance with DESTINY+ schedule. can quickly calculate Vmiss with VM2M = 1.5 km/s assumption. Note that Fig 2 only show transfer ∆Vs lower than 3 km/s. The results show that there exist periodic low-∆V transfer with 3.1. Secondary target solutions about 1.4 years, i.e. Phaethon’s orbit period, in descending of Transfer opportunities to secondary targets are first investi- Phaethon (not shown in the figure, see Ref. [10]). The inset gated individually. Lambert problem is first solved for transfer in Fig 2 provides a more detailed overview of the first flyby opportunities between 2028-2032 Earth departure and 2028- opportunity in Dec 2027 – Jan 2028 time frame with less than 2038 time frame for asteroid flyby. Note that Earth depar- ∆V = 1 km/s, i.e. lower than the upper boundary provided by ture dates are different than what is selected for Earth-Phaethon the multiple lunar flyby phase of the mission. transfers. This is due to the selected mission sequence and po- As mentioned earlier, the Lambert problem is solved each leg tential low-thrust opportunities identified previously.10) Figure of the mission. As the first leg of the interplanetary phase shown 5 and 6 show the Lambert problem solutions for Earth-2005UD in Fig 2, Earth-return leg (P2-E3) of the transfers also have mul- and Earth-1999YC transfers, respectively. tiple solutions. The best combination between E1-P2 and P2-E3 Note that color bars in both figures are scaled such that low- legs can be found when the magnitude of the difference between est value found in the analysis is minimum, and the maximum incoming and outgoing velocity vectors at Phaethon flyby point, value is 5 km/s. Earth-2005UD transfers show plenty of low ∆V defined as Vmiss in Eq. 1: transfer opportunities. The lowest value found is ∼2 km/s. This transfer window was also identified in Sarli et al. (2015) as potential ballistic transfer between Earth and 2005UD.10) The V = |Vout − Vin | (1) miss ∞ ∞ orbital geometry in this time frame between Earth, Phaethon in out where V∞ and V∞ incoming and outgoing V∞ velocities at the and 2005UD is such that it is possible to complete a Phaethon flyby point. Velocity vectors are used in Eq. 1 to ensure the flyby, Earth gravity-assist and 2005 UD flyby in less than a directions of the velocity vectors are also nearly-aligned. Vmiss year. Recall from Fig. 2 that the first flyby opportunity after values can then be generated for a given flyby date and a range DESTINY+’s Earth escape in late 2025 is in December 2027 – of Earth-departure and Earth-arrival dates to determine dates January 2028 time frame. of low-∆V patching between two legs. A number of Vmiss are On the other hand, Earth-1999YC transfers are much more therefore generated for a number of flyby dates selected from limited, with the minimum v∞ value being 4 km/s. The flyby Fig 2 and combined into so-called “Mismatch ∆V maps” to be windows are also very narrow. Sarli et al. (2015) found Mars used for initial conditions generation. An example of a mis- flybys to reach 1999YC with Earth escape v∞ ≈3 km/s. Mars match ∆V map is shown in Fig. 3. The figure shows the mini- flyby is not considered in this study. mum Vmiss for given range of Earth-departure and Earth-arrival Above results demonstrate the opportunities with the direct dates when performing Phaethon flyby on January 04, 2028. Earth-asteroid transfers. However, it is still not clear whether Vmiss values up to 1 km/s in shown in the figure. An im- it is possible to find initial guesses for the multiple-target low- portant point about Mismatch ∆V maps is that, even though thrust optimal solutions. As a result, a v∞-matching procedure Fig. 2.: Lambert solutions found for Earth-Phaethon leg.

Fig. 3.: Mismatch ∆V map for ﬂyby date Jan 04, 2028. Fig. 4.: Total ∆V map for ﬂyby date Jan 04, 2028.

out in V∞ · V∞ φ = arccos out in (3a) |V∞ ||V∞| s !−1 µ φ − rp = in sin 1 (3b) is applied. First, for a given Phaethon flyby date, Lambert V∞ 2 problem is solved Phaethon to Earth leg (i.e., P2-E3). Second, in out for a given range of flyby dates of the selected secondary tar- where φ denote the flyby turn angle, V∞ and V∞ denote in- get, Lambert problem is solved for Earth-asteroid (i.e., E3-A4) coming and outgoing velocity vectors as the output of Lambert transfers. After incoming to Earth and outgoing to asteroid v∞ problem solution. rp is the flyby distance from the center of the vectors are calculated, flyby turn angle and flyby distance is flyby body, µ is the gravitational parameter of the flyby body in found by the formula below: (hence of the Earth’s) and V∞ is the incoming velocity magni- (a) All solutions Fig. 7.

(b) Solution in 2028 Fig. 5.: Transfer opportunities from Earth to 2005UD in 2028- 2038 time frame Fig. 8.

and plotted as straight black line in the figures. In principle, trajectories below that threshold would not require any thrusting, but connect to the target with change in the direction of velocity. As it can be seen, there is a flyby opportunity appear with a flyby altitude below 10 times the Earth’s radius, i.e. o rp <∼63800 km, with a turn angle around 110 . The exact date that matches the threshold value through spline interpolation, as well. The date is found as December 2, 2028. This procedure is repeated for a possible range of Phaethon flyby, Earth arrival and secondary asteroid flyby dates. Follow- ing initial guesses in Table 2 are going to be explored in the low-thrust trajectory optimization process. Fig. 6.: Transfer opportunities from Earth to 1999YC in 2028- During this study, no potential solutions are found for 2038 time frame 1999YC with the techniques applied here. Alternative tech- 17) niques, such as v∞-leveraging, may provide opportunities for tude. During the computations, trajectories with flyby altitudes 1999YC flybys as the secondary target or even the third tar- (i.e. rp − rEarth <) below 500 km are marked as infeasible. get, but this is left as a future work. The next section is there- The output of this computational procedure can then be plot- fore dealing with the optimal low-thrust solutions Phaethon and ted on a so-called C3 − maps where potential dates for the op- 2005UD via an Earth-flyby in between. timization procedure can be seen. Two such maps are shown in Fig. 7 and Fig. 8. In the former, flyby distance is shown in 4. Multiple-target optimal solutions units of Earth radius in color bar. In the latter, the color code represents flyby turn angle. Consistently with the earlier results This section introduces the trajectory optimization tool GAL- shown, the maps depict the flyby case Jan 4, 2028 which return LOP and provides final optimal solutions with the initial condi- to Earth on May 22, 2028. Earth-arrival v∞ is around ∼2.5 km/s tions presented in Table 2. Table 2.: Initial conditions used in this study

# Earth departure (E1) Phaethon ﬂyby (P2) Earth return (E3) Asteroid ﬂyby (A4) v∞ @ E1 [km/s] 1 2025NOV07 2028JAN04 2028MAY14 2028NOV03 2.845 2 2026MAY18 2028JAN04 2028MAY20 2028NOV09 2.647 3 2026MAY16 2028JAN01 2030MAY18 2033FEB17 1.611 4 2026JAN26 2027DEC30 2030AUG03 2033FEB24 1.649

Table 3.: Initial parameters for optimization 4.1. GALLOP Parameter Value Optimal trajectories are computed with GALLOP, a low- Initial mass [kg] 450 thrust trajectory optimization tool based on Sims-Flanagan di- Max. v∞at departure [km/s] 1.5 rect optimization method.18, 19) GALLOP approximates a tra- Thrust [µN] 40 jectory in multiple segments of low thrust arcs. During the op- Duty cycle [-] 0.8 timization, GALLOP takes the whole trajectory in legs and di- Speciﬁc impulse [s] 3000 vide them into segments of low-thrust arcs. In these arcs, thrust 19) magnitude and its direction is assumed to be constant. The In addition, a minimum solar distance of 0.75 AU is also im- optimizer then propagates the equations of motion in Eq. 4 in posed to meet the thermal requirements of the spacecraft. To each leg in backwards and forwards direction to match at a point select the number of nodes in each leg, a number of E1-P2-E3 (usually halfway though the leg). solutions are generated with 30, 50, 70, 100 nodes in each leg. Final mass results are shown in Fig. 9.

x˙ = vx (4a)

y˙ = vy (4b)

z˙ = vz (4c) −µx x¨ = + F /m (4d) r3 x −µy y¨ = + F /m (4e) r3 y −µz z¨ = + F /m (4f) r3 z −F m˙ = (4g) Ispg where x, y and z are the components of the spacecraft state, and p r = x2 + y2 + z2; µ is the gravitational parameter of the attract- ing body, in this case the Sun; Fx, Fy and Fz are the components q Fig. 9.: Final mass results with respect to the number of nodes 2 2 2 of the force provided by the thrust and F = Fx + Fy + Fz ; selected Isp is the specific impulse of the ion engine and g is the gravitational acceleration. If the mismatch between the spacecraft No discernible difference in the final mass is observed, except state after forward and backwards propagation is below toler- one case, with more than 50 nodes. Therefore optimal trajecto- ance, then trajectory becomes feasible. ries are generated by using 50 nodes in the first two legs (i.e., GALLOP uses a solar-electric propulsion model in which the E1-P2-E3) after a trade-off between speed and accuracy of the amount of thrust scales with the distance to the Sun. Specif- computations. As in principle no thrusting is performed in the ically, the thrust model follows an inverse square law if the last leg, number of nodes is selected to be 20. spacecraft further than 1 AU. Specific impulse is assumed to The first initial guess in Table 2 is inputted to GALLOP and be constant. result is in Fig. 10. GALLOP then transcribes the low-thrust trajectory optimiza- In the final optimized trajectory, DESTINY+ escapes the tion problem into nonlinear optimization problem (NLP) where Earth in November 29, 2025, performs a Phaethon flyby on the final mass is the objective. The problem constraints are the its descending node on January 04, 2028. About 5.5 months mismatches in the state variables at the patching points. The later, on May 22, 2028 DESTINY+ performs gravity-assist with NLP is then solved with SNOPT, a software package that im- Earth and reaches 2005UD on November 10, 2028, less than 6 plements sequential quadratic programming (SQP).20) months later. The second (P2-E3) and third legs (E3-A4) are In next subsection, GALLOP will be used to generate opti- ballistic, as depicted in Fig. 11. The trajectory gains a slight mal trajectories for DESTINY+ multiple-target low-thrust opti- inclination before reaching 2005UD in its ascending node, as mization problem. shown in Fig. 10b. Properties of the trajectory is given in Table 4.2. Earth-Phaethon-Earth-Asteroid solutions 4. In the trajectory optimization process, other than the dates As it can be seen from Table 4, an flyby trajectory visiting of initial guesses in Table 2, following inputs are provided to both Phaethon and 2005UD is feasible in given time frame with GALLOP: less than 7.5 kg of propellant. DESTINY+ performs Phaethon Table 4.: Properties of the optimal trajectory Parameter Value Final mass [kg] 442.58 v∞ at E1 [km/s] 1.5 v∞ at P2 [km/s] 33.71 v∞ at E3 [km/s] 2.45 Flyby distance [km] 19225 v∞ at A4 [km/s] 23.99 Minimum solar distance [AU] 0.898 Maximum solar distance [AU] 1.351

timal trajectory is computed for the next grid point by using the previous solution as an initial guess. Now, at this point, it should be noted that a good initial guess is very important for trajectory optimization processes. Particularly for the trajectory optimization problem tackled here, the targeted ﬂyby windows are very narrow due to the properties of the target orbits. It was observed (a) XY-view during this study that the method of moving along the date grid may greatly aﬀect the appearance of the departure window as a mathematical artifact. In this paper, the method illustrated in Fig. 12 was applied and it was found to be robust.

(b) XZ-view Fig. 10.: Optimal trajectory

Fig. 12.: Sensitivity map computational procedure

Fig. 11.: Thrust magnitude along the trajectory In the method illustrated in Fig. 12, the grid is first divided into four quarters. In each quarter, the procedure starts with flyby with 33.71 km/s and the Earth flyby with 2.45 km/s of the nominal solution provided, the dates are first moved along v∞ at around ∼20000 km. Flyby at 2005UD occurs with lower departure axis until the end of the line. Then, after moving v∞, as the spacecraft meets with the target at the maximum so- one point in arrival axis, the procedure is repeated for departure lar distance (1.351 AU) as compared to Phaethon flyby, which axis. The nearest solution is given as initial condition to the next occurs at less than 1 AU. solution. Of course, this method results in redundant computa- In order to assess the sensitivity of the trajectory for the tions at the common axes of between quarters. Those common possible changes in Earth escape dates, a departure sensitivity solutions at each grid are used to compare the solutions in mak- analysis is performed. Earth-departure and Earth-arrival dates ing sure that there is no difference in the final results due to are varied within a given range of days, and the optimization the direction of moving along the grid of dates. The sensitiv- problem is solved for each grid point. Phaethon flyby (P2) ity analysis for the optimal trajectory presented earlier was then and 2005UD flyby (A4) dates are fixed for Earth-departure and performed with the outlined computational procedure and the Earth-arrival at these grid points, i.e. no upper and lower bound- result is presented in Fig.13. ary is given to the dates. the departure date is varied ±20 days, The results show a core region with final masses close to that whereas the arrival date is varied ±10 days. of the nominal trajectory and this value gradually decreases to The computation begins with the nominal solution and an op- lower values as the solution moves away from the best solution. Fig. 13.: Departure sensitivity analysis

The final masses are always above 430 kg, 10 kg above the pro- in February 2033 in this case. One potentially interesting result pellant allocated for the interplanetary phase, except on a small is the Earth-arrival velocity in the third solution, which is on island of lower final masses that can be seen on the top part of the order of v∞ values observed for 1999YC transfers as shown Fig. 13. At this region, the solutions are still feasible, albeit in Fig. 6. The usability, particularly in terms of sensitivity, of with very low final masses. The lowest final mass computed these trajectories for the mission and the 1999YC flyby possi- is 424 kg, which is still above the allocated propellant mass bility with the third solution is not explored in the current paper, of DESTINY+ during the interplanetary phase of the mission. but will be investigated in the future. Figure 13 also shows the flyby v∞ values. The solid black line on the figure depicts the value at P2 whereas the dashed black 5. Conclusions shows the value at A4 phase of the mission. It appears that later Earth-arrival days result in lower v∞ at P2, the difference This paper presented the multiple-target interplanetary tra- is about 0.5 km/s between two extremes of the arrival dates. jectory design of DESTINY+ to Phaethon-Geminid Complex On the other than, earlier Earth-arrival days result in lower v∞ (PGC), which includes asteroid 3200 Phaethon, 2005UD and values at A4, albeit only about 0.2 km/s difference. 1999YC. This results in Fig. 13 suggest a robust optimal trajectory First, a database of transfer opportunities from Earth to in- against earlier and later Earth escapes. It also allows for a wider dividual targets are computed by solving Lambert’s problem. search space for multiple lunar flyby phase to enable Earth es- In order to connect individual trajectory arcs between Earth cape with the maximum v∞ (i.e., 1.5 km/s) considered as an and Phaethon, Phaethon to Earth and Earth to 2005UD (or input for the optimization. 1999YC), mismatch ∆V maps are generated. Finally, suitable Finally, optimal solutions from the other initial conditions in initial guesses for optimization process are established by a sim- Table 2 are also computed. Those were tabulated in Table 5. ple “minimum ∆V” criterion, and applied to the database. It was The first solution is similar to the optimal trajectory discussed found that a suitable Earth-Phaethon-Earth-Asteroid sequence throughout this subsection, but 6 months later, with slightly exist with low Earth-escape energy for Earth departure in the less final mass. The other two solutions also exhibit Phaethon end of 2025. This translates as multiple asteroid flyby opportu- flyby in the same window in December 2027 or January 2028, nities with Phaethon and 2005UD within the same year with but arriving to the Earth gravity assist approximately two years an Earth gravity-assist in between. The optimal trajectories later, as compared to the first solution. 2005UD flyby date is were then generated, and shown that DESTINY+ can indeed Table 5.: Optimal solutions found during the study Final Flyby altitude # E1 P2 E3 A4 Mass [kg] @ E3 [km] 2026MAY05 2028JAN04 2028MAY23 2028NOV09 1 442.3 23802 v∞ = 1.5 km/s v∞ = 33.9 km/s v∞ = 2.35 km/s v∞ = 24.4 km/s 2026DEC04 2028JAN03 2028MAY27 2028NOV09 2 438.1 30392 v∞ = 1.5 km/s v∞ = 33.7 km/s v∞ = 2.40 km/s v∞ = 24.4 km/s 2027APR10 2028JAN03 2028MAY16 2028NOV12 3 439.5 10686 v∞ = 1.5 km/s v∞ = 33.7 km/s v∞ = 2.61 km/s v∞ = 23.6 km/s 2026JAN18 2027DEC30 2028JUL10 2031SEP03 4 438.8 12900 v∞ = 1.5 km/s v∞ = 31.5 km/s v∞ = 2.39 km/s v∞ = 34.02 km/s 2025NOV23 2028JAN02 2030MAY22 2033FEB22 5 443.9 81393 v∞ = 1.5 km/s v∞ = 33.6 km/s v∞ = 1.98 km/s v∞ = 27.9 km/s 2026MAY16 2028JAN01 2030MAY18 2033FEB24 6 444.9 66715 v∞ = 1.5 km/s v∞ = 33.5 km/s v∞ = 2.03 km/s v∞ = 27.9 km/s perform multiple flybys. A departure sensitivity study demon- search, Beijing, July 2006. strated that the generated trajectory is indeed robust. The study 8) Blume, W.H. et al.: Deep Impact Mission Design, Space Science Re- also revealed several other opportunities for optimal Phaethon views, 117 (2005). https://doi.org/10.1007/s11214-005-3386-4 9) Lauretta, D. S., and OSIRIS-REx Team: An Overview of the OSIRIS- and 2005UD transfers for 2028-2033 time frame. No optimal REx Asteroid Sample Return Mission, 43rd Lunar and Planetary Sci- trajectories were found to reach 1999YC with the techniques ence Conference, Paper 2491, Woodlands, TX, (2012). applied here. The future work will improve the existing tra- 10) Sarli, B. 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