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Parameterizing the Search Space of Starshade Fuel Costs for Optimal Observation Schedules

2019; American Institute of Aeronautics and Astronautics; Volume: 42; Issue: 12 Linguagem: Inglês

10.2514/1.g003747

1533-3884

Gabriel Soto, Dmitry Savransky, Daniel Garrett, Christian Delacroix,

Astro and Planetary Science

Open AccessEngineering NotesParameterizing the Search Space of Starshade Fuel Costs for Optimal Observation SchedulesGabriel J. Soto, Dmitry Savransky, Daniel Garrett and Christian DelacroixGabriel J. SotoCornell University, Ithaca, New York 14853*Graduate Research Assistant, Sibley School of Mechanical and Aerospace Engineering, 404 Upson Hall.Search for more papers by this author, Dmitry SavranskyCornell University, Ithaca, New York 14853†Assistant Professor, Sibley School of Mechanical and Aerospace Engineering, 451 Upson Hall. Member AIAA.Search for more papers by this author, Daniel GarrettCornell University, Ithaca, New York 14853‡Ph.D. Research Assistant, Sibley School of Mechanical and Aerospace Engineering, 404 Upson Hall.Search for more papers by this author and Christian DelacroixUniversity of Liège, B-4000 Liège, Belgium§Postdoc Associate, Département d’Astrophysique, Géophysique et Océnographie, Allé du Six Août 19c.Search for more papers by this authorPublished Online:5 Aug 2019https://doi.org/10.2514/1.G003747SectionsRead Now ToolsAdd to favoritesDownload citationTrack citations ShareShare onFacebookTwitterLinked InRedditEmail AboutI. IntroductionStarshades are a promising concept for achieving the necessary level of starlight suppression needed to detect habitable zone exoplanets [1–3]. By deploying a separate spacecraft along the line of sight (LOS) of a space telescope to a target star, a shadow is cast onto the telescope pupil, blocking on-axis starlight while passing off-axis exoplanet light. Ensembles of end-to-end mission simulations produce posterior distributions of planet discoveries and the parameters of detected planets for specific mission designs [4,5]. For missions with a starshade, these simulations must incorporate the relevant dynamics, which increases computation time. Of the two main starshade flight modes of station keeping and retargeting between observations, the latter uses the most fuel and is our main focus [6–8]. The retargeting or slewing process creates a vast search space of trajectories: we can choose from any star on a target list and achieve alignment after an arbitrary slew time. At each decision step, the simulation scheduler must select the next best star to observe based on costs and the constraints of required integration times, evolving keepout regions, and mission goals [4,5,7,9,10].Rather than solving a time-constrained optimization problem for each target at each decision step for each simulation, we can approximate the solution via various heuristics. Using simplified fuel cost models as in the work of Glassman et al. [7] or substituting the fuel cost with angular separation between stars as in the work of Savransky et al. [4] reduces the computation time but also accuracy. Kolemen and Kasdin [8] included the full dynamics in their simulations within a cost matrix: they precomputed individual trajectories from star i to star j in their target list and populated a static two-dimensional (2-D) matrix with fuel costs. Their cost matrix, however, was only computed for fixed slew times, whereas the trajectory optimization requires more exploration of the time parameter due to dynamic constraints.We propose a new parameterization of the cost matrix that both captures accurate fuel cost solutions and more adequately explores the constrained search space of possible slews. The parameterization is based on the slew time Δt and angular separation ψ between any two stars; the new cost matrix is sufficiently continuous that a 2-D interpolant can be used globally for any target list with only a small reduction in accuracy. We integrate this new parameterization within the Exoplanet Open-Source Imaging Mission Simulator (EXOSIMS) [5,10], which is a framework for end-to-end mission simulations, and show that examples of science yield results for different simulation ensembles.II. Starshade Dynamical ModelA. Circular Restricted Three-Body ProblemWe assume a space telescope on a nominal halo orbit about the sun–Earth L2 point [11,12]; the starshade trajectories about the halo are governed by the dynamics of the circular restricted three-body problem (CR3BP) [13]. A frame R (with orthogonal unit vectors x^, y^, and z^; Cartesian coordinates (x,y,z), and origin O at the barycenter of the primary masses) is defined to rotate with the two primaries relative to an inertial frame I with unit vectors i^, j^, and k^, as shown in Fig. 1.Fig. 1 Diagram of the sun–Earth rotating frame in the ecliptic plane with locations of L2 and telescope–starshade–star configuration shown. Target stars outside the sun’s keepout region are observable to the telescope.The equations of motion for the third object (which is the starshade in this study) within this rotating frame are x¨−2y˙=∂Ω∂x+fSRP⋅x^(1)y¨+2x˙=∂Ω∂y+fSRP⋅y^(2)z˙=∂Ω∂z+fSRP⋅z^(3)with Ω(x,y,z)=12(x2+y2)+1−μr1+μr2(4)r1=(μ+x)2+y2+z2(5)r2=(1−μ−x)2+y2+z2(6)where Ω(x,y,z) is the effective potential due to the two primaries; r1 and r2 are the distances from the two primaries, respectively; and fSRP is the solar radiation pressure (SRP) force. All units in this formulation are normalized as described by Koon et al. [13]. The reduced mass fraction μ is defined as the smaller primary mass scaled by the mass sum.The SRP force acts on the starshade, which is modeled as an axisymmetric plate of cross-sectional area A with unit vector n^ defined normal to the surface [14]. An intermediate P frame [15] is defined along the sun–starshade vector r1 as shown in Fig. 2, with p^1=r1|r1|(7)p^2=z^×p^1|z^×p^1|(8)and p^3=p^1×p^2(9)The pitch and clock angles, α and δ, are defined as spherical angles relative to P. The normal vector is then n^=cosαp^1+sinαcosδp^2+sinαsinδp^3(10)For the retargeting portions of the starshade mission, we assume no preferential attitude: on average, half of the starshade area faces the sun throughout the trajectory (or, equivalently, α=60°). The clock angle is assumed to be, on average, δ=0°.Fig. 2 Diagram of R and P frames used in equations of motion. P frame is defined perpendicular to the sun–starshade vector. Based on work of Dachwald et al. [15].The total force exerted on the starshade throughout its motion [15] is fSRP=2PAcosα[b1p^1+(b2cosα+b3)n^](11)where b1, b2, and b3 are optical coefficients with values taken from Glassman et al. [7]. Integrating the full equations of motion with initial conditions in the vicinity of the nominal (precomputed) telescope halo orbit provides the motion of the starshade.B. Establishing Line of SightDuring station keeping, the size of the shadow cast onto the telescope pupil plane by the starshade depends on the triangle formed by d, the constant separation distance between the telescope and starshade [8], and the starshade radius RS. This defines a geometric inner working angle θI [1], which is the smallest angle from the telescope-starshade LOS an exoplanet can be situated before its light too gets suppressed by the starshade, given by tanθI=RSd(12)We assume values for the starshade radius and θI that, together, specify the separation distance.The starshade begins the retargeting trajectory at a distance d along the LOS to target star i at time ti; it ends at a distance d along the LOS to target j at time tj=ti+Δt, where Δt is the slew time [7,8]. More details can be found in the work of Soto et al. [9].C. Solving the Boundary Value ProblemThe starshade positions are well defined at the endpoints of the retargeting trajectory but the corresponding velocities are not. This boundary value problem (BVP) is solved using Eqs. (1–3). We use similar methods as Kolemen and Kasdin [8]: the collocation algorithm solve_bvp, which is a Python implementation found in the scipy package [16]. A detailed explanation of our implementation was provided by Soto et al. [9]. The final continuous trajectory from the LOS of star i to the LOS of star j is depicted in Fig. 3. These solutions set the velocities at the start and end of the retargeting trajectory: vRT(ti) and vRT(tj) respectively.Fig. 3 Schematic of two starshade flight modes: station keeping with star i, retargeting to star j, and station keeping with star j. An angle ψ separates LOS vectors to the two stars.D. Calculating Fuel CostChanges in velocity at the transitions between station keeping and retargeting shown in Fig. 3 are modeled as impulsive maneuvers, which instantaneously change the velocity vector without altering the position vector. We quantify the fuel use by these velocity changes Δv. The velocity of the starshade in the inertial frame vISK must match the inertial velocity of the telescope during station keeping [8], which is converted from the rotating to the inertial frames using rotation matrices defined by Koon et al. [13]. The retargeting velocities vRT are similarly converted to inertial velocities (for more details, see the work of Soto et al. [9]). The Δv for each discrete jump is found by Δv(ti)=‖vIRT(ti)−IvSK(ti)‖(13)Δv(tj)=‖IvRT(tj)−IvSK(tj)‖(14)where the superscript I is dropped for conciseness. The total Δv for a given retargeting maneuver from star i to star j for transfer time Δt initiated at t0 is Δv(i,j,Δt,t0)=Δv(ti)+Δv(tj)(15)We calculate fuel use Δm for a nominal chemical propulsion system via Tsiolkovsky’s rocket equation [17] Δm=m0(1−e−(Δv/g0Isp))(16)where m0 is the initial total mass, g0 is the standard gravity constant, and Isp is the specific impulse of the rocket engine. The starshade mass is sequentially decremented throughout the mission simulation as fuel is expended.III. Parameterizing Fuel CostA. Fuel Cost MatrixThe total Δv for retargeting maneuvers, for a telescope on a predefined halo orbit and constant d, is a function of four parameters: the two stars i and j, the time t0 at which the maneuver is initiated (i.e., where the telescope is located on the halo orbit), and the slew time Δt. One may choose to ignore some parameters to create feasible fuel cost heuristics: Kolemen and Kasdin [8] created cost matrices as functions of i and j but kept Δt and t0 fixed. Ignoring the slew time parameter, however, prevents proper application of the time constraints required to properly simulate end-to-end missions.B. Sorting by Star Angular SeparationWe create a new parameterization of the fuel cost matrix using the angular separation between stars. We define a unit vector for the location of each star in the inertial frames r^i/0 and r^j/0. The star angular separation ψ is ψ=sgn(r^j/0⋅k^)arccos(r^i/0⋅r^j/0)(17)The sign of the angular separation is taken with respect to the inertial frame to differentiate alignments with stars ahead of, or behind, the halo motion of the telescope, which require different amounts of fuel.We consolidate the two dimensions of the previous i−j cost matrix into a single dimension: an angular separation from the previously observed star. We populate the second remaining dimension with Δt: all possible slew times to align with a star at some separation ψ. The new 2-D cost matrix in Fig. 4 is generated at a specific t0 relative to some reference star and shows substantially more continuity than previous parameterizations [8]. From the map, it is clear that the relationship between the two parameters is nonlinear. At small values of ψ, fuel costs increase with slew time: quick flights at short distances require less fuel. At large ψ values, fuel costs decrease with slew time: traversing large distances is easier with longer flights.Fig. 4 Heat map of Δv as a function of both star angular separation ψ and slew time Δt. Interpolated values are shown. Color bar is shown in log scale.C. Global Fuel Cost InterpolantWe exploit the continuity of the new fuel cost matrix with a two-dimensional interpolant. We use interp2D, which is a Python implementation found in the scipy package [16], using linear interpolation between points. A single cost matrix can be generated independently of the specific target catalog because only position vectors are needed to calculate angular separations. To reduce interpolation errors, we generate the Δv map using a fake catalog of stars with a compact distribution in ψ, which was described in the appendix of Soto et al. [9].Two parameters are neglected with this approach: the time at which maneuvers are initiated t0, and the location of the previous observed star. We conducted three separate tests to quantify the errors associated with these simplifications; in each, we generated multiple cost matrices where Δv=f(ψ,Δt) and used a target list T⊂Z, which is a subset of integers. We computed each matrix at various t0 taken at 20-day intervals throughout the ∼6 month halo orbit period. In the first test, we selected a random pair of stars i1 and j1∈T and calculated the true retargeting Δv for a range of Δt (ranging from 5 to 80 days in steps of 1 day), all initiated at a single t0. We then compared the true fuel costs to interpolated values from all fuel cost matrices at different t0 for the same star pairing. In the second test, we calculated the true Δv for a different random pair of stars i2∈T and j2∈T using the same Δt range and at every different t0. We then compared this array of true fuel costs to interpolated results using a single fuel cost matrix generated at a single t0. In the third test, we selected 50 random star pairs ∈T at every t0 and compared true and interpolated fuel cost values. The frequencies of the percent errors for each test are shown in Fig. 5. The mean absolute value percent error for interpolation was found to be under 10% for all tests, and it was approximately 2.53% for test 1, meaning that the derived cost matrix can be used as a target selection heuristic, as long as true fuel costs are then evaluated for the selected target. Generating trajectories of 195 realignments for slew times ranging from 5 to 80 days (in steps of 5 days) takes approximately 12 min on a dual-core 2.5 GHz processor. These calculations are now completed, with reasonable errors, within a fraction of a second using the new interpolant and parameterization, allowing for their inclusion in full mission simulations.Fig. 5 Percent errors between true Δv values and interpolated values for three different test cases, shown as solid lines. Mean and median values are calculated from percent error absolute values.IV. Scheduling with Time ConstraintsA. Dynamic Time ConstraintsMission time constraints are imposed on the global optimization problem by defining upper and lower bounds on the Δt axis of the cost matrix. A major time constraint is due to keepout regions [4,7]. A minimum keepout angle must be maintained between the telescope LOS and the direction of any bright object to avoid light contamination in the pupil. Light from a bright object can also reflect from the starshade into the telescope, requiring a maximum keepout angle from the telescope pointing. Solar panel pointing restrictions also limit the permissible telescope look vectors. Figure 1 shows the solar keepout region schematically.The keepout zones evolve through time as the telescope moves along its orbit in an inertial frame. We define an observable window, which is the period of time when a star is continuously observable, via a binary keepout map [4]. We precalculate angles βb between the relative positions of the target star and all bright solar system bodies b (the sun, Earth, moon, etc.) throughout the entire mission and evaluate 45 deg<βSun<90 deg(18)45 deg<βMoon=βEarth(19)1 deg<βothers(20)We populate the binary keepout map with the union of the preceding conditions. Figure 6 shows a sample keepout map over one year of mission time, with different colors representing the cause of a particular keepout. At each decision step in the mission simulation, we impose the observable time windows from the binary keepout map as upper and lower bounds on Δt for every star (at some ψ) within the interpolant. This allows us, for the first time, to combine strict time constraints with full CR3BP fuel cost solutions in end-to-end mission simulations.Fig. 6 Binary keepout map for a subset of stars on a target list.B. Exoplanet Open-Source Imaging Mission Simulator ImplementationWe use EXOSIMS [5] to execute mission simulations: the next best star is sequentially selected for observation while time is still left in the mission and fuel still remains in the starshade. The original code has been modified to accommodate binary keepout maps and 2-D fuel cost interpolants: all prior to running simulations. Other time limits are also implemented, including integration times [10] and other mission-specific constraints. A more detailed review of the code updates is found in the work of Soto et al. [9].We generate an ensemble of simulations using different schedulers [9]: a random walk scheduler that selects the next target at random from currently observable ones, a greedy scheduler choosing the target with the next highest completeness (probability that a star has a single orbiting exoplanet belonging to an assumed population and is observable by the instrument in question) [18,19], and a linear cost scheduler [4]. The last scheduler defines the cost function c=c1Δvmin+c2(1−CO)−c3funv+c4frev(21)where ci are tunable weights; c is an m×1 vector of costs for the m stars in the filtered target list; Δvmin is a vector of the minimum fuel cost values for each m; CO is a similar vector of completeness values; and funv and frev are factors that prioritize targets that have not been observed yet and targets deemed for a revisit, respectively. The linear cost scheduler selects the star corresponding to the root node of the minimum cost path in a truncated search of the tree of all possible future paths (see the work of Savransky et al. [4] for details). Previously, the c1 term used a coarse heuristic for fuel costs given by just the angular separation of two targets, and so the new implementation is significantly better at capturing the true retargeting costs.V. Mission Simulation ResultsThe results of a mission simulation using the linear cost scheduler are shown in Fig. 7; assumptions and mission parameters are listed in Table 1 [20], which are similar to the Wide-Field Infrared Survey Telescope (known as WFIRST) mission [21]. Starshade parameters are taken from studies of the Exo-S mission [6]. Dry and total masses are estimated by assuming the launch vehicle capabilities of a Falcon 9 [22]. In this case, the schedule contains 29 total observations for the three-year mission; six led to positive detections, and three full spectral characterizations were completed.Fig. 7 Sample schedule generated using new observation scheduler for a starshade mission. Each circle represents the location of a different observed star.Figure 7 shows observed target star positions as filled circles (in the equatorial coordinate system), with the circle color representing the completeness value of each target. The first observed target is marked with a bold border. Line colors denote the amount of Δv used for each retargeting slew; line thickness decreases as the mission progresses.We conducted an ensemble of 1000 mission simulations, with each of the different selection schemes outlined in Sec. IV.B. Normalized yield frequencies and total observations are shown in Figs. 8 and 9. The linear cost function produced, on average, the most unique detections and conducted more observations due to more strategic fuel use.Fig. 8 Frequency of unique detections for three ensembles of 1000 mission simulations using a random walk, maximum completeness (Max Comp), and linear cost scheduler.Fig. 9 Frequency of number of observations for three ensembles of 1000 mission simulations using a random walk, maximum completeness, and linear cost scheduler.VI. ConclusionsA fuel cost interpolant based on full solutions to the circular restricted three-body problem starshade trajectories was presented, which effectively explored the slew time tradespace for any pair of targets. These efficient and fast approximations were also implemented within Exoplanet Open-Source Imaging Mission Simulator to perform ensembles of end-to-end exoplanet direct imaging missions for starshade-based imagers. The introduction of these accurate and fast dynamical fuel cost solutions was shown, in simulation, to increase both the total number of observations (because fuel is used more strategically) and the number of unique detections. These improvements have, for the first time, allowed us to realistically treat starshade fuel costs accurately in ensembles of end-to-end mission simulations, thereby greatly increasing our confidence in their results and improving their utility as a mission analysis and design tool. Although the focus was only on the case of chemical propulsion simulated through impulsive maneuvers, a future comparative analysis will be conducted on solar electric propulsion through continuous low-thrust maneuvers.AcknowledgmentsThis work is supported by NASA Grant No. NNG16PJ24C (SIT), NASA Grant No. NNX15AJ67G, and the NASA JPL SURP grant RSA No. 1618976. References [1] Vanderbei R. J., Cady E. and Kasdin N. 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Google Scholar[22] Falcon 9 Launch Vehicle–Payload User’s Guide, Space Exploration Technologies Corp. (SpaceX), Hawthorne, CA, 2015, https://www.spacex.com/sites/spacex/files/falcon_users_guide.pdf [retrieved 21 March 2017]. Google ScholarTablesTable 1 Mission parameters and assumptionsParameterValueMission time3 yearsHalo period179 daysPupil diameter2.37 mΔmag22.5Planet populationKepler-like [21]Total Δv2094.33 m/sInner working angle72 milliarcsecondStarshade diameter26 mSeparation distance37,242.26 kmIsp308 sDry mass1250 kgTotal mass3500 kg Previous article Next article FiguresReferencesRelatedDetailsCited byNext-generation active telescope for space astronomyJournal of Astronomical Telescopes, Instruments, and Systems, Vol. 8, No. 04 What's Popular Volume 42, Number 12December 2019 CrossmarkInformationCopyright © 2019 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the eISSN 1533-3884 to initiate your request. See also AIAA Rights and Permissions www.aiaa.org/randp. TopicsAerospace SciencesAstrodynamicsAstronauticsAstronomical ObservatoryAstronomyCelestial Coordinate SystemCelestial MechanicsOrbital ManeuversPlanetary Science and ExplorationPlanetsSolar PhysicsSpace OrbitSpace Science and TechnologySpace TelescopesTelescopes KeywordsTelescopesEarthHalo OrbitWide Field Infrared Survey TelescopeSolar Electric PropulsionEquatorial Coordinate SystemPythonTsiolkovsky Rocket EquationTrajectory OptimizationBoundary Value ProblemsAcknowledgmentsThis work is supported by NASA Grant No. NNG16PJ24C (SIT), NASA Grant No. NNX15AJ67G, and the NASA JPL SURP grant RSA No. 1618976.PDF Received13 April 2018Accepted26 June 2019Published online5 August 2019