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Astrodynamics Vol. 3, No. 3, 231–246, 2019 https://doi.org/10.1007/s42064-019-0061-1 Trajectory design for a solar-sail mission to asteroid 2016 HO 1 2 2 2 Jeannette Heiligers (), Juan M. Fernandez , Olive R. Stohlman , and W. Keats Wilkie 1. Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, the Netherlands 2. Structural Dynamics Branch, Langley Research Center, National Aeronautics and Space Administration, Hampton, Virginia, 23681-2199, USA ABSTRACT KEYWORDS asteroid 2016 HO This paper proposes the use of solar-sail technology currently under development at NASA 3 Langley Research Center for a CubeSat rendezvous mission with asteroid 2016 HO ,a solar sail quasi-satellite of Earth. Time-optimal trajectories are sought for within a 2022–2023 launch solar electric propulsion window, starting from an assumed launcher ejection condition in the Earth–Moon system. trajectory design The optimal control problem is solved through a particular implementation of a direct trajectory optimization pseudo-spectral method for which initial guesses are generated through a relatively simple and straightforward genetic algorithm search on the optimal launch date and sail attitude. The results show that the trajectories take 2.16–4.21 years to complete, depending on the assumed solar-sail reﬂectance model and solar-sail technology. To assess the performance of solar-sail propulsion for this mission, the trajectory is also designed assuming the use of solar Research Article electric propulsion. The resulting fuel-optimal trajectories take longer to complete than the solar-sail trajectories and require a propellant consumption that exceeds the expected Received: 1 May 2019 propellant capacity onboard the CubeSat. This comparison demonstrates the superior Accepted: 6 June 2019 performance of solar-sail technology for this mission. © The Author(s) 2019 to reach. Low-thrust propulsion, either in the form of 1 Introduction solar electric propulsion (SEP) or solar sailing [1,2], has On 27 April 2016, the Pan-STARRS 1 asteroid survey been proven to enable high-energy missions. Examples telescope on Haleakal¯a, Hawaii, USA, detected a for the use of SEP include JAXA’s asteroid sample remarkable asteroid: 2016 HO . Its orbit is extremely return mission Hayabusa [3], NASA’s Dawn mission similar to that of Earth, to the extent that asteroid 2016 that visited the two largest bodies in the asteroid belt HO appears to orbit around our planet. The asteroid is [4], and ESA’s BepiColombo mission to Mercury [5]. therefore considered a near-Earth companion or a quasi- Examples of proposed high-energy solar-sail missions satellite of Earth and is expected to accompany the include NASA’s NEA Scout mission [6], as well as Earth for hundreds of years. This unique characteristic, a range of theoretical mission concepts such as the together with the already signiﬁcant increase in interest Solar Polar Orbiter [7], the Geostorm mission concept in small-body research over recent years, makes 2016 [8, 9], the Interstellar Heliopause Probe [10], asteroid HO an interesting mission target. rendezvous missions [11,12], and, more generally, a wide Even though the orbit of 2016 HO is very similar to range of highly non-Keplerian orbits for novel space that of Earth, its phasing and the 7.8 deg inclination applications [1, 13–18]. with respect to the ecliptic makes it a difficult target This paper investigates the use of solar-sail propulsion to rendezvous with asteroid 2016 HO .In particular, the solar-sail technology currently under development Jet Propulsion Laboratory. Small asteroid is Earth’s constant companion. Available at https://www.jpl.nasa.gov/news/news.php? at NASA Langley Research Center is considered [19]. feature=6537. [Accessed 25 February 2019] The assumed mission conﬁguration is that of a CubeSat m.j.heiligers@tudelft.nl 232 J. Heiligers, J. M. Fernandez, O. R. Stohlman, et al. platform and a launch as secondary payload (e.g., Table 1 Orbital elements of 2016 HO (source: JPL Small-Body Database Browser ) onboard one of the Exploration Missions of the SLS launch vehicle) within a wide 2022–2023 launch window. Orbital element Value Semi-major axis, a 1.0014 AU The assumption of a CubeSat-sized platform drove the Eccentricity, e 0.1040 choice for solar-sail propulsion as mass and dimension Inclination, i 7.7741 deg constraints limit the available space for SEP propellant Right ascension of the ascending node, Ω 66.4066 deg as well as solar arrays to provide power to the SEP Argument of perihelion, ω 306.9337 deg system. To conﬁrm this choice, the trajectory will not only be designed for the use of solar-sail propulsion, but completes the right-handed reference frame. Instead, also for the use of solar electric propulsion. Fig. 1(b) shows the orbit in a synodic frame, B (x , SE SE The objective of the work in this paper is to ﬁnd y , z ), centered at the Sun–Earth barycenter where SE SE time-optimal solar-sail—or alternatively fuel-optimal the x -axis points along the Sun–Earth line, the z - SE SE SEP—trajectories from an assumed launcher ejection axis is oriented perpendicular to the ecliptic plane, and condition in the Earth–Moon circular restricted three- the y -axis completes the right-handed reference frame. SE body problem (CR3BP) to 2016 HO . An initial, This frame thus rotates with the Earth’s motion around ballistic trajectory up to the sphere of inﬂuence of the Sun. The side-view plots in the bottom row of the Earth is assumed to allow spacecraft testing and Fig. 1 clearly show the asteroid’s 7.8 deg inclination with veriﬁcation. Once at the sphere of inﬂuence, the low- respect to the ecliptic. thrust propulsion system is activated and the modelling of the trajectory continues in the Sun–Earth CR3BP. Time- and fuel-optimal trajectories are found through 3 Solar-sail technology the application of a speciﬁc direct pseudospectral The solar-sail architectures assumed for this study optimal control solver, PSOPT [20]. Initial guesses are based upon small satellite solar-sail systems and for the optimal control solver are obtained through a technologies now under development at NASA Langley relatively straightforward genetic algorithm routine that Research Center (NASA LaRC) [19]. These solar-sail ﬁnds the optimal launch date and constant direction of systems are based upon new deployable composite boom the low-thrust acceleration vector with respect to the technologies being developed by LaRC and the German direction of sunlight to minimize the miss-distance and Aerospace Center (DLR) specifically for small satellites . miss-velocity at the asteroid. By subsequently feeding In 2016, NASA LaRC built and ground-tested a 9.2 m× this initial guess to the optimal control solver, where 9.2 m composite-based engineering development unit the control is allowed to vary over time, these errors are (EDU) solar-sail system suitable for 6U CubeSat overcome and the time of ﬂight—or alternatively the spacecraft. This EDU solar-sail system stowed within fuel consumption—is minimized. a20cm×10 cm×15 cm volume inside the 6U CubeSat chassis. This system was initially conceived as a risk- reducing alternative to NASA’s Near Earth Asteroid 2 Orbit of 2016 HO (NEA) Scout solar-sail baseline design, which used open The orbital elements of asteroid 2016 HO are provided cross-section metallic triangular rollable and collapsible in Table 1. With a semi-major axis very close to that of (TRAC) booms [6, 21]. TRAC boom solar-sail designs Earth and only a small eccentricity, its orbit resembles have been used on smaller solar-sail demonstration that of Earth (only inclined at 7.8 deg with respect to ﬂights, most notably with the NASA NanoSail D2 solar the ecliptic). Graphical representations of the asteroid’s sail, and the Planetary Society LightSail 1 (formerly, motion in the timeframe 1960–2020 appear in Fig. 1, where the Earth is assumed to be on a Keplerian, 1 AU Jet Propulsion Laboratory. JPL Small-Body Database Browser. (astronomical unit) circular orbit. Figure 1(a) shows the Available at https://ssd.jpl.nasa.gov/sbdb.cgi. [Accessed 25 February 2019] orbitinaninertialframe, A(X , Y , Z ),where the X - I I I I NASA Game Changing Development Program. Deployable Composite Booms (DCB). Available at https://gameon.nasa.gov/ axis points towards the vernal equinox, the Z -axis is projects/deployable-composite-booms-dcb/. [Accessed 25 February oriented perpendicular to the ecliptic, and the Y -axis 2019] I Trajectory design for a solar-sail mission to asteroid 2016 HO 233 Fig. 1 Orbit of 2016 HO in the 1960–2020 timeframe: (a) in the inertial frame A(X , Y , Z ); (b) in the synodic Sun–Earth frame 3 I I I B (x , y , z ). SE SE SE SE LightSail A) and LightSail 2 solar sails [22]. TRAC following launcher ejection conditions in a synodic booms have been problematic for larger solar sails due to Earth–Moon frame B (x , y , z ) with r EM EM EM EM 0 their high coeﬃcient of thermal expansion (CTE), very the initial position vector (km) and v the initial low torsional stiﬀness, and low deployed precision [23, velocity vector (km/s): ⎡ ⎤ 24]. An improved version of the composite-based EDU solar sail—the Advanced Composites-Based Solar Sail ⎣ ⎦ x = System (ACS3)—is now under development by NASA LaRC and NASA Ames Research Center for a low Earth = [26503.0 −371.3 9134.64.43 2.41 0.85] orbit (LEO) solar-sail technology risk reduction mission (1) in the 2021 timeframe. The 12U ACS3 ﬂight experiment is intended as a technology development pathﬁnder for These launch ejection conditions are derived from a future, larger composite-based small satellite solar-sail early designs of the Exploration Mission-1 (EM-1) system suitable for 12U to 27U CubeSat class spacecraft. mission . Note that frame B (x , y , z ) EM EM EM EM For purposes of this study, a lightness number range is centered at the Earth–Moon barycenter with bounding the anticipated solar-sail performance of a the x -axis pointing to the Moon, the z -axis EM EM notional 12U–27U CubeSat-class spacecraft using the perpendicular to the Earth–Moon orbital plane, ACS3 solar-sail technology is assumed. and the y -axis completing the right-handed EM frame. The state vector in Eq. (1) corresponds to a spacecraft–Earth distance of 32,486 km and an 4 Mission assumptions inertial velocity with respect to Earth of 5.16 km/s To design the trajectory from launch ejection to 2016 (the local escape velocity is 4.95 km/s). HO , a set of assumptions are made: • Launch is assumed to take place in 2022–2023. NASA. Exploration Mission 1. Available at https://www.nasa.gov/ content/exploration-mission-1. [Accessed 25 February 2019] • The trajectory is assumed to start from the 234 J. Heiligers, J. M. Fernandez, O. R. Stohlman, et al. • Starting from the initial state in Eq. (1), a ballistic 5.1 Interplanetary phase arc up to the Earth’s sphere of inﬂuence (SOI) is Considering the relatively close proximity of the asteroid assumed after which the solar sail or SEP system to the Earth and its semi-bounded motion around the is activated to rendezvous with 2016 HO . Earth as shown in Fig. 1(b), the trajectory to 2016 HO • The trajectory propagation up to the Earth’s SOI is is designed in the framework of the Sun–Earth CR3BP. assumed to take place in the Earth–Moon CR3BP, In the CR3BP, the motion of an inﬁnitely small mass, m while the subsequent propelled phase is assumed to (i.e., the spacecraft), is described under the inﬂuence of take place in the Sun–Earth CR3BP. When linking the gravitational attraction of two much larger primary these CR3BPs and when computing the orbit of masses, m (here, the Sun) and m (here, the Earth). 1 2 2016 HO in the Sun–Earth CR3BP the following The gravitational inﬂuence of the small mass on the is assumed for the ephemerides of the Earth and primary masses is neglected and the primary masses are Moon: assumedtomoveincircularorbitsabouttheircommon ◦ For the Earth, a set of analytical ephemerides center-of-mass. The reference frame employed to define is used, but the eccentricity is set to zero. the spacecraft’s dynamics in the Sun–Earth CR3BP ◦ For the Moon, a set of constant Keplerian is that of Fig. 1(b): the synodic Sun–Earth frame elements is used. B (x , y , z ), which rotates at constant angular SE SE SE SE • Regarding the propulsion system, the following velocity, ω,aboutthe z -axis, ω = ωzˆ ; see Fig. 2. SE SE assumptions are made: New units are introduced: the sum of the two primary ◦ Solar sail masses is taken as the unit of mass, i.e., m + m =1. 1 2 The solar-sail lightness number is assumed Then, with the mass ratio μ = m /(m + m ),the 2 1 2 to be in the range β =0.025−0.04 for a masses of these primary bodies become m =1 − μ 12U–27U spacecraft (for a deﬁnition of the and m = μ. Asunitoflength, thedistancebetween lightness number, see below Eq. (7)). the primary bodies is selected, and 1/ω is chosen as the ◦ It is assumed that the solar-sail system can be unit of time, yielding ω =1. Then, one revolution of replaced by a solar electric propulsion system the reference frame (i.e., one year for the Sun–Earth (and power system) with a performance that is CR3BP) is represented by 2π. In this framework, the based on the following assumptions: motion of a low-thrust propelled spacecraft is described An initial spacecraft mass of 14–21 kg. by A speciﬁc impulse of 1600 s [25]. r¨ +2ω × r˙ + ω × (ω × r)= a −∇V (2) Amaximumthrustof0.9mN[25]. with r =[x y z ] the position vector of m. A maximum propellant mass capacity of SE SE SE The terms on the left-hand side of Eq. (2)are 1.5kg[25]. the kinematic, coriolis, and centripetal accelerations, A maximum thruster operation duration of respectively, while the terms on the right-hand side 20,000 hours (2.3 years) [25]. are the low-thrust acceleration and the gravitational acceleration exerted by the primary masses. In frame 5Dynamics B (x , y , z ) the gravitational potential, V,is SE SE SE SE As outlined in the previous section, two diﬀerent sets of dynamical frameworks are adopted to design the trajectory to 2016 HO : the Earth–Moon CR3BP from the initial condition in Eq. (1) up to the Earth’s SOI (hereafter referred to as the Earth–Moon ballistic phase) and the Sun–Earth CR3BP from the Earth’s SOI up to rendezvous with the asteroid (hereafter referred to as the interplanetary phase). The dynamics in both phases will be detailed in the following subsections, starting with the interplanetary phase. Fig. 2 Schematic of circular restricted three-body problem. Trajectory design for a solar-sail mission to asteroid 2016 HO 235 given by 1 − μ μ V = − + (3) r r 1 2 where r and r are the magnitudes of the vectors r = 1 2 1 T T [x +μy z ] and r =[x −(1−μ) y z ] , SE SE SE 2 SE SE SE respectively; see Fig. 2. Finally, the term a is the low-thrust acceleration vector which in this paper is either provided by a solar sail, a = a ,or a solar electric propulsion system, a = a . Its deﬁnition Fig. 3 Side-view schematic of non-ideal solar-sail acceleration therefore depends on the type of low-thrust propulsion components. system employed which will be discussed separately in the following two subsections. Note that, due to the solar sail’s inability to generate 5.1.1 Solar-sail propulsion an acceleration component in the direction of the Sun, To model the solar-sail acceleration, this paper will con- the normal vector always points away from the Sun. sider both an ideal and an optical solar-sail reﬂectance This can be reﬂected through appropriate bounds on model. Considering both these models will allow to the pitch and clock angles: compute not only the theoretically fastest trajectory ◦ ◦ 0 α 90 possible (for the ideal model) but also a more realistic (6) trajectory (for the optical model). While the ideal ⎩ ◦ ◦ −180 δ 180 model assumes the sail to be a perfect, specular reﬂector, the optical model also includes the eﬀects The magnitudes of the solar-sail acceleration com- ponents along the normal and tangential directions in of absorption, diﬀuse reﬂection, and thermal emission. Eq. (4)are givenby[1]: Though diﬀerent in performance, both solar-sail models can be captured in the mathematical deﬁnition provided 1 1 − μ ⎪ a = β (1 + rs ˜ )cos α + B (1 − s)˜ r cos α n f below. ⎪ 2 r The solar-sail acceleration vector can be decomposed ε B − ε B f f b b +(1 − r˜) cos α (7) into a component normal to the sail, a n ˆ,and a ⎪ ε + ε f b ˆ ⎪ component tangential to the sail, a t; see Fig. 3: ⎪ 1 1 − μ a = β (1 − rs ˜ )cos α sin α 2 r a = a n ˆ + a t = a m ˆ (4) s n t s 1 In Eq. (7), β is the solar-sail lightness number, r˜ The normal to the sail, n ˆ, can be deﬁned through is the reﬂectivity coeﬃcient that indicates the fraction two angles, the solar-sail pitch and clock angles, that of reﬂected photons, and s indicates the fraction of deﬁne the solar sail’s orientation with respect to the photons that are specularly reﬂected, while the term direction of sunlight; see Fig. 4. For this, a new reference (1−s) indicates the fraction of photons that are diﬀusely frame S(rˆ , θ , ϕ ˆ ) is deﬁned where the rˆ -unit vector is 1 1 1 1 reﬂected; B and B are the non-Lambertian coeﬃcients f b directed along the Sun–sail line (see also below Eq. (3)) of the front (subscript “f”) and back (subscript “b”) of and the two remaining axes are deﬁned as in Fig. 4 the sail, and ε and ε are the corresponding emissivity f b (left schematic). The pitch angle, α,is then deﬁned coeﬃcients. Values for these optical coeﬃcients for both as the angle between the normal vector, n ˆ,and rˆ ,and an ideal sail and an optical sail model appear in Table 2. the clock angle, δ, isdeﬁnedastheanglebetweenthe The optical sail coeﬃcients have recently been obtained projection of n ˆ onto the (θ , ϕ ˆ )-plane and ϕ ˆ .This 1 1 1 for NASA’s proposed Near Earth Asteroid (NEA) Scout gives the following deﬁnition of the normal vector with mission [27]. Finally, note that, by substituting the respect to frame S(rˆ , θ , ϕ ˆ ): 1 1 1 values for the ideal sail model into Eq. (7), only an ⎡ ⎤ cos α acceleration component normal to the sail remains, i.e., ⎢ ⎥ n = (5) ⎣ sin α sin δ ⎦ ˆ a =0 and therefore a = a n ˆ and m ˆ = n ˆ.From t s n S(rˆ ,θ ,ϕ ˆ ) 1 1 1 sin α cos δ Eq. (7), the resulting acceleration direction, m ˆ ,can be 236 J. Heiligers, J. M. Fernandez, O. R. Stohlman, et al. Fig. 4 Solar-sail pitch and clock angles deﬁned with respect to frame S(rˆ , θ , ϕ ˆ ). Adapted from Ref. [26]. 1 1 1 Table 2 Optical coeﬃcients for an ideal and an optical solar-sail angles, α and δ , respectively: T T reﬂectance model ⎡ ⎤ cos α Reﬂectance model rs ˜ B B ε ε f b f b ⎢ ⎥ T = T sin α sin δ , ⎣ ⎦ ˆ T T Ideal 1 1 — — — — S(rˆ ,θ ,ϕ ˆ ) 1 1 1 sin α cos δ T T Optical [27] 0.91 0.94 0.79 0.67 0.025 0.27 T = R T (13) S→B SE ˆ computed by ﬁrst deﬁning an auxiliary angle, φ;see S(rˆ ,θ ,ϕ ˆ ) 1 1 1 Fig. 3 [1]: with T the SEP thrust magnitude. Note that this time φ =arctan (8) no restrictions need to be imposed on the pitch angle: ◦ ◦ 0 α 180 such that T (14) ◦ ◦ −180 δ 180 θ = α − φ (9) T The direction of m ˆ with respect to frame S(rˆ , θ , ϕ ˆ ) 1 1 1 Finally, due to the consumption of propellant, the canthenbedeﬁnedas spacecraft mass decreases over time according to ⎡ ⎤ cos θ T m ˙ = − (15) ⎢ ⎥ m ˆ = (10) I g ⎣ sin θ sin δ ⎦ sp 0 S(rˆ ,θ ,ϕ ˆ ) 1 1 1 sin θ cos δ with I = 1600 s the assumed SEP thruster’s speciﬁc sp impulse; see Section 4. g is the Earth’s standard free Through a transformation matrix, this normal vector fall constant. The diﬀerential equation in Eq. (15)needs can be transformed to the synodic Sun–Earth frame to be integrated simultaneously with the spacecraft B (x , y , z ) for use in Eq. (4): SE SE SE SE dynamics in Eq. (2). m ˆ = R m ˆ ,R = rˆ θ ϕ ˆ S→B ˆ S→B 1 1 1 SE SE S(rˆ ,θ ,ϕ ˆ ) 1 1 1 5.2 Ballistic Earth–Moon phase (11) As highlighted in the mission assumptions section, the 5.1.2 Solar electric propulsion ﬁrst phase of the trajectory is assumed to be ballistic In the case of employing solar electric propulsion, the (no use of the solar sail or SEP thruster) and is modelled low-thrust acceleration vector in Eq. (2)isdeﬁnedas in the Earth–Moon CR3BP. The dynamics are then as deﬁned in Eq. (2), only now in the B (x , y , a = a = (12) EM EM EM z ) frame with the Earth and Moon as primaries (μ = EM where T =[T T T ] is the Cartesian SEP 0.01215) and a = 0. When integrating the dynamics x,SE y,SE z,SE thrust vector and m is the spacecraft mass. The SEP forward from Eq. (1) up to the sphere of inﬂuence of thrust vector direction can be deﬁned in a similar way the Earth (at a distance of 1,496,513 km), the trajectory as the solar-sail normal vector using the pitch and clock in Fig. 5 is obtained. It takes the spacecraft 9 days to Trajectory design for a solar-sail mission to asteroid 2016 HO 237 t − t , solar sail f 0 J = (16) −m , SEP where t and t are the initial and ﬁnal time in the 0 f interplanetary phase, respectively, and m is the ﬁnal spacecraft mass. The goal then is to ﬁnd the states, x(t), and controls, u(t), that minimize Eq. (16)and satisfy the dynamics in Eq. (2) as well as a set of boundary and path constraints. The states are the position and velocity vectors in the CR3BP. For the SEP conﬁguration, the spacecraft mass is added: Fig. 5 Ballistic trajectory starting from the initial condition ⎨ [r r˙] , solar sail in Eq. (1) depicted in the synodic Earth–Moon frame B (x , EM EM x(t)= (17) ⎩ T y , z ). EM EM [r r˙ m] , SEP where the initial state, x(t )= x , needs to match 0 0 reach the sphere of inﬂuence. Transforming the end of the state vector at the end of the ballistic Earth–Moon the trajectory to the inertial frame A(X , Y , Z ) and the I I I phase at time t and the ﬁnal state, x(t )= x ,needs 0 f f synodic Sun–Earth frame B (x , y , z ),results in SE SE SE SE to coincide with the asteroid’s state vector at time t . the conditions as shown in Fig. 6. In the inertial frame, Furthermore, for the SEP conﬁguration, the initial mass the conditions at the SOI oscillate around the orbit is ﬁxed to a value in the range m =14 − 21 kg (see of the Earth and complete one revolution in one year, Section 4), and the final spacecraft mass is free, i.e., is to whereas in the synodic Sun–Earth frame, these con- be optimized. Note that both the state vectors at the end ditions conduct one revolution per synodic lunar month. of the ballistic phase and of the asteroid are computed in the optimization routine through an interpolation of large state matrices. Furthermore, suitable bounds need 6 Optimal control problem tobeimposedonthe statevectorcomponents: Depending on the low-thrust propulsion system [(1−μ)−0.4 −0.6 −0.4 −0.5 −0.5 −0.50] x(t) employed, the objective in this study is to either minimize the time of ﬂight in the interplanetary part [(1−μ)+0.40.60.40.50.50.5 m ] (18) of the trajectory (for the solar-sail conﬁguration) or the propellant consumption (for the SEP conﬁguration). where the last row in the vectors only applies to the SEP The objective, J, can thus be deﬁned as case. Fig. 6 Conditions at the Earth’s sphere of inﬂuence of the trajectory in Fig. 5 for the year 2022: (a) in the inertial frame A(X , Y , I I Z ); (b) in the synodic Sun–Earth frame B (x , y , z ). I SE SE SE SE 238 J. Heiligers, J. M. Fernandez, O. R. Stohlman, et al. The controls are also deﬁned diﬀerently for the two which are constructed through the following approach: low-thrust propulsion conﬁgurations: • First, the launch date, t , and the pitch and clock angles, (α, δ)or(α , δ ), are ﬁxed. Furthermore, T T T [αδ] , solar sail u(t)= (19) for the solar-sail case, the sail lightness number, [α δ T ] , SEP T T β, is ﬁxed whereas for the SEP case the thrust magnitude is set to its maximum value, T . max where the following bounds are imposed: • Subsequently, the initial condition in Eq. (1)is T ⎨ [0 −π] ππ , solar sail 2 forward integrated from t up to the Earth’s sphere u(t) ⎩ T [0 −π 0] of inﬂuence to construct the ballistic Earth–Moon [ππ T ] , SEP max phase and the end of the trajectory is transformed (20) to the B (x , y , z ) frame. SE SE SE SE In Eq. (20), T = 0.9 mN is the assumed maximum max • Then, the integration is continued to construct thrust magnitude; see Section 4. In addition, a path the interplanetary phase, where the low-thrust constraint is included to avoid close Earth approaches: acceleration is deﬁned by either (α, δ, β)or r (t) 800,000 km (21) (α ,δ ,T ). The integration is truncated after T T max 5 years. In addition, to aid the trajectory Finally, bounds on the initial and ﬁnal time need to be in increasing its inclination to that of the speciﬁed to ensure a launch in the assumed 2022–2023 asteroid, a rudimentary out-of-plane steering law is launch window and to limit the search space on the ﬁnal adopted where the out-of-plane component of the time: acceleration takes the sign of the y -coordinate. SE 1 January 2022 t 31 December 2023 For example, for the solar-sail case: (22) 31 December 2023 t 1 January 2028 [m m |m |],y 0 x,SE y,SE z,SE SE m ˆ = Note that the time in the actual implementation of the [m m −|m |] ,y < 0 x,SE y,SE z,SE SE optimal control problem is deﬁned in non-dimensional (23) units after 1 January 2022, i.e., 1 January 2022 is and similar for the thrust vector, T for the SEP case. represented by t = 0, 1 January 2023 is represented • Subsequently, at each time step in the propagated by t =2π, and so on. trajectory, t, that occurs after 3 years of ﬂight, the The optimal control problem deﬁned in Eqs. (16)–(22) dimensionless error in distance, Δr, and error in is solved with a particular implementation of a direct velocity, ΔV , between the spacecraft’s state-vector ++ pseudospectral method in C ,PSOPT [20]. PSOPT and that of the asteroid is computed. is an open source tool developed by Victor M. Becerra • Finally, the trajectory is truncated at the point of the University of Reading, UK. It can use both where the sum of these errors, Δr+ΔV , is minimal. Legendre and Chebyshev polynomials to approximate Note that, in dimensionless units, a position error and interpolate the dependent variables at the nodes. of 5000 km and a velocity error of 1 m/s are of the However, in this work, only the Legendre pseudospectral same order of magnitude. method is used and PSOPT is interfaced to the NLP For a given performance of the solar-sail or SEP solver IPOPT (Interior Point OPTimizer), an open conﬁguration, i.e., for a given value for β or T the max ++ source C implementation of an interior point method trajectory is fully deﬁned by the following set of three for large scale problems [28]. Furthermore, a consecutive parameters: mesh reﬁnement of [50, 75, 100] nodes is applied, a −6 convergence tolerance of 10 is used, and a maximum [t αδ], solar sail p = (24) number of iterations per mesh reﬁnement of 1000 is [t α δ ], SEP L T T enforced. which deﬁne the objective J(p)=Δr +ΔV (25) 7 Initial guess To ﬁnd the values for the parameters in Eq. (24)that In order to initiate the optimization process, PSOPT requires an initial guess of the states, controls, and time, minimize the objective in Eq. (25), a genetic algorithm Trajectory design for a solar-sail mission to asteroid 2016 HO 239 is employed. In particular, the MATLAB function assumed lightness number range; see Section 4. For each ga.m is used with 1000 individuals and a maximum of lightness number, the genetic algorithm is run ﬁve times 50 generations while enforcing the following bounds on (for the ﬁve diﬀerent seed values), resulting in a total of the optimization parameters: 20 runs per sail model. The best trajectory for each lightness number, i.e., the trajectory with the smallest [1-1-2022 0 − π] objective function value among the ﬁve runs, appears in [1-1-2022 0 − π] Fig. 7 (for an ideal sail model) with numerical values in Table 3 (for both sail models). From these results, it can 31-12-2023 ππ , solar sail be deduced that, the larger the lightness number, the [31-12-2023 ππ], SEP smaller the objective function value (the 8th column in Finally, note that, to account for the inherent random- Table 3), which indicates that the rendezvous conditions ness of the genetic algorithm approach, each simulation at the asteroid can be more easily met with better case is run for ﬁve diﬀerent seed values. solar-sail technology. Furthermore, due to the limited number of controls (especially the constant pitch and 7.1 Solar-sail initial guesses clock angles throughout the trajectory), a mismatch Solar-sail initial guesses are generated for both the ideal in position and signiﬁcant mismatch in velocity still and optical sail models and for four diﬀerent lightness exist between the spacecraft and asteroid at the end numbers, β =[0.025, 0.03, 0.035, 0.04],to cover the ofthetrajectory. Theseerrorscanbeovercomebythe Fig. 7 Solar-sail case—best initial guess trajectories in the synodic Sun–Earth frame B (x , y , z ) for an ideal solar-sail model SE SE SE SE and for (a) β =0.025,(b) β =0.03,(c) β =0.035, and (d) β =0.04. 240 J. Heiligers, J. M. Fernandez, O. R. Stohlman, et al. Table 3 Solar-sail case—details of best initial guesses Time of ﬂight, Pitch angle, Clock angle, β Launch date Arrival date J Δr (km) ΔV (km/s) Δt (y) α (deg) δ (deg) 0.025 10 July 2022 16 Jan 2027 4.51 40.65 −4.17 0.0710 6600 2.114 Ideal sail 0.03 15 Feb 2023 3 June 2027 4.29 31.82 5.76 0.0693 53,357 2.053 model 0.035 17 Jan 2023 19 Jan 2026 3.00 33.38 179.90 0.0580 773,675 1.575 0.04 25 Jan 2022 15 June 2026 4.37 38.97 178.93 0.0451 418,300 1.261 0.025 17 Jan 2023 22 June 2027 4.41 34.67 178.60 0.0863 3,394,765 1.895 Optical sail 0.03 10 July 2022 12 Jan 2027 4.50 28.16 −4.23 0.0695 43,830 2.060 model 0.035 25 Jan 2022 8 July 2025 3.44 33.17 177.26 0.0612 155,470 1.791 0.04 25 Feb 2022 11 June 2026 4.28 39.44 174.38 0.0559 1,105,868 1.444 optimizer, as will be shown later. while this leads to relatively small errors on the position, large errors on the velocity remain. Furthermore, due to 7.2 SEP initial guesses the assumption of continuous thrusting at the maximum thrust magnitude, the propellant consumption is large: For the SEP case, initial guesses are generated for the 6.93 kg and 7.47 kg for m =14 kg and m =21 extremes of the range in initial spacecraft mass: m =14 0 0 kg, respectively. This greatly exceeds the expected kg and m =21 kg; see Section 4. Similar to the propellant budget of 1.5 kg; see Section 4. initial guesses for the solar-sail case, five runs (for the five different seed values) are conducted for each initial mass. The best trajectories for each value for m appear 8Results in Fig. 8 with numerical values in Table 4. From these results, it can be observed that, contrary to the solar- Due to the assumed wide launch window (2022–2023), sail case, out-of-plane thrusting is barely exploited: the the problem at hand contains many local minima. This pitch and clock angles are close to 90 deg, indicating that already became apparent in the results for the initial thrusting takes place entirely in the ecliptic plane and guess in the previous section. It can therefore be along the y -axis. As Fig. 8 and Table 4 clearly show, SE expected that PSOPT will converge to a diﬀerent local Fig. 8 SEP case—best initial guess trajectories in the synodic Sun–Earth frame B (x , y , z ) for (a) m =14 kg and (b) SE SE SE SE 0 m =21 kg. Table 4 SEP case—details of best initial guesses Time of ﬂight, Propellant Pitch angle, Clock angle, m (kg) Launch date Arrival date J Δr (km) ΔV (km/s) Δt (y) mass (kg) α (deg) δ (deg) T T 14 9 Aug 2022 10 June 2026 3.83 6.93 100.89 90.92 0.1467 684,804 4.2334 21 7 Oct 2022 26 Nov 2026 4.12 7.47 97.00 −74.19 0.1568 521,778 4.5662 Trajectory design for a solar-sail mission to asteroid 2016 HO 241 minimum when provided with a diﬀerent initial guess. From the results in Table 5, Fig. 9, and Fig. 10, the To therefore best detect the global minimum, PSOPT following observations can be made: is run for each generated initial guess, i.e., not only for • PSOPT is able to overcome the discontinuities in the best initial guesses that appear in Table 3, but for all the state vector of the initial guess at the end of the 40 (solar-sail case) and 10 (SEP case) generated initial trajectory between the sailcraft and the asteroid, guesses. The results of the runs that generated the best while decreasing the time of ﬂight compared to the trajectory in terms of objective function value (i.e., the initial guess by (on average) 1.4 and 0.7 years for total time of ﬂight) are presented in this section. the ideal and optical sail models, respectively. • ThebestinitialguessasinTable3doesnot 8.1 Solar-sail optimal results necessarily lead to the best optimized result. In fact, only for an ideal sail model and a lightness The results for each considered value for the solar-sail number of β =0.03 did the best initial guess lightness number appear in Table 5 for both the ideal provide the best optimized result. and optical sail models with details for a subset of the • The diﬀerence in launch date between the initial results in Fig. 9 and Fig. 10 for β =0.025 and β =0.04, guess and the corresponding optimized trajectory respectively. Note that the clock angle proﬁles in Fig. 9 is, on average, only 2 days. This implies that and Fig. 10 may appear erratic, but a clock angle switch the reduction in time of ﬂight is only due to an from −π to π does not require an actual physical change advancement of the arrival date, not due to a in the solar-sail attitude. Table 5 Solar-sail case—optimized results Ideal sail model Optical sail model Diﬀerence in Δt between Time of ﬂight, Time of ﬂight, Launch date Arrival date Launch date Arrival date sail models (%) Δt (y) Δt (y) 0.025 7 Oct 2022 10 Apr 2026 3.51 16 Mar 2023 1 June 2027 4.21 19.9 0.03 15 Feb 2023 15 Jan 2026 2.92 24 Feb 2022 3 Aug 2025 3.44 17.8 0.035 15 Feb 2023 19 Aug 2025 2.51 7 Oct 2022 9 Oct 2025 3.01 19.9 0.04 8 Oct 2022 3 Dec 2024 2.16 9 Oct 2022 11 May 2025 2.59 19.9 Fig. 9 Solar-sail case—optimized trajectory and controls for ideal sail model and β =0.025. 242 J. Heiligers, J. M. Fernandez, O. R. Stohlman, et al. Fig. 10 Solar-sail case—optimized trajectory and controls for ideal sail model and β =0.04. change in launch date. For example, for the case discussed later. of an ideal sail model and β =0.03 the launch From the results in the ﬁrst two rows of Table 6 and date of both the initial guess and the optimized Fig. 11, the following observations can be made: trajectory is 15 February 2022, while the time of • PSOPT is again able to overcome the discontinuities ﬂight is 1.4 years shorter; see Table 3 and Table 5. in the state vector of the initial guess at the end The underlying reason can be found in the fact that of the trajectory between the spacecraft and the the launch conditions change much more rapidly asteroid, while this time decreasing the propellant over time than the arrival conditions (see above consumption compared to the initial guess to Fig. 5). The optimizer therefore shies away from 3.60 kg and 5.90 kg for m =14 kg and m = 0 0 altering the launch conditions and prefers to change 21 kg, respectively. However, again, considering the the arrival conditions. expected maximum onboard propellant capacity • The ideal sail model provides the absolute fastest of 1.5 kg (see Section 4), all trajectories appear trajectory possible, which ranges from 2.16 to 3.54 infeasible from that point of view. years for lightness numbers in the range 0.025–0.04. • The decrease in propellant consumption (and The optical sail model presents a more realistic matching of the rendezvous constraints) comes at performance of the sail, but increases the time of a cost in an increase in the time of ﬂight compared ﬂight by 17.8–19.9 percent to 2.59–4.21 years. to the initial guess of approximately 0.5 year: to 4.38 years and 4.62 years for m =14 kg and 8.2 SEP optimal results m =21 kg, respectively. These times of ﬂight exceed the maximum thruster operation duration Following the same approach as for the solar-sail of 20,000 hours or 2.3 years. However, when optimal results, minimum-propellant trajectories can be only considering the time that thrust is actually generated for the nominal SEP case, which appear in produced, this reduces to 15,814 and 25,616 hours the ﬁrst two rows of Table 6 with details for m =14 kg for m =14 kg and m =21 kg, respectively, where in Fig. 11. The table shows that two additional cases 0 0 the former does satisfy the constraint. Finally, also have been considered with larger speciﬁc impulses and note that the times of ﬂight are all longer than maximum thrust magnitudes. These results will be Trajectory design for a solar-sail mission to asteroid 2016 HO 243 Table 6 SEP case—minimum-propellant results Propellant Time of Thruster operating Case m (kg) I (s) T (mN) Launch date Arrival date 0 sp max mass (kg) ﬂight (y) time (h) 1 14 1600 0.9 8 Aug 2022 24 Dec 2026 3.60 4.38 15,814 2 21 1600 0.9 6 Oct 2022 19 May 2027 5.90 4.62 25,616 3 14 2100 1.15 5 Oct 2022 12 Feb 2026 2.91 3.36 13,966 4 21 2100 1.15 5 Oct 2022 16 Dec 2026 4.51 4.20 21,054 Fig. 11 SEP case—minimum-propellant results for case 1: m =14 kg, I = 1600 s, and T =0.9 mN. (a) Trajectory projected 0 sp max onto the ecliptic plane. (b) Trajectory in the out-of-plane direction. (c) Thrust angles. (d) Thrust magnitude. the times of ﬂight obtained for the solar-sail case. • Similar conclusions as for the solar-sail case can be Here, it must be noted that the solar-sail cases were drawn regarding the fact that the best initial guess optimized for the time of ﬂight, while the SEP case as in Table 4 does not necessarily lead to the best is only optimized for the propellant consumption. optimized result and that the launch date does not Therefore, when changing the objective function change much (if at all) with respect to the initial in Eq. (16) for the SEP case to the time of ﬂight, guess. shorter transfers are obtained (2.58 years and 3.87 From the observations listed above, it can be years for m =14 kg and m =21 kg, respectively). concluded that solar electric propulsion (under the 0 0 However, this comes at the cost of a signiﬁcant assumptions of Section 4) does not seem a viable increase in the propellant consumption that further propulsion method for this mission from a propellant violates the constraint on the propellant capacity consumption point of view, thruster operating time (4.65 kg and 6.96 kg for m =14 kg and m =21 kg, and ﬂight time. To further support this conclusion, 0 0 respectively) and thruster operating times that far additional simulations have been conducted for an even exceed the maximum duration (22,520 hours and better performing SEP system (see the results in the 33,745 hours for m =14 kg and m =21 kg, last two rows in Table 6). Here, the speciﬁc impulse has 0 0 respectively). been increased to I = 2100 s and the maximum thrust sp 244 J. Heiligers, J. M. Fernandez, O. R. Stohlman, et al. magnitude to T =1.15 mN, the maximum value as Acknowledgements max speciﬁed in Ref. [25]. The results in Table 6 show that Jeannette Heiligers would like to acknowledge support these improvements lower the propellant consumption from the Marie Skldowska-Curie Individual Fellowship and the thruster operation duration. However, the 658645-S4ILS: Solar Sailing for Space Situational propellant consumption is still larger than the expected Awareness in the Lunar System. budget of 1.5 kg and the thruster operation duration is stilltoolongfor m =21 kg. Finally, while the times of ﬂight are also reduced, they are still longer than those References for the solar-sail trajectories for β> 0.025 (ideal sail) [1] McInnes, C. R. Solar Sailing: Technology, Dynamics and β> 0.03 (optical sail). and Mission Applications. 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GeoSail: An elegant solar Jeannette Heiligers is an assistant sail demonstration mission. Journal of Spacecraft and professor in astrodynamics and space Rockets, 2007, 44(4): 784–796. missions at Delft University of Technology [17] Heiligers, J., Parker, J. S., Macdonald, M. Novel (TU Delft), the Netherlands. She received solar-sail mission concepts for high-latitude earth and her B.Sc. and M.Sc. degrees from TU lunar observation. Journal of Guidance, Control, and Delft in 2008, after which she brieﬂy Dynamics, 2018, 41(1): 212–230. [18] Ozimek,M.T.,Grebow,D.J.,Howell,K.C.Design worked as a junior project manager of solar sail trajectories with applications to lunar in the Dutch space industry. She returned south pole coverage. Journal of Guidance, Control, to academia and completed her Ph.D. and Dynamics, 2009, 32(6): 1884–1897. degree at the Advanced Space Concepts Laboratory [19] Fernandez, J. M., Rose, G., Stohlman, O. R., Younger, (ASCL) of the University of Strathclyde, Glasgow, UK, C. J., Dean, G. D., Warren, J. E., Kang, J. H., in 2012, after which she continued at the ASCL as a Bryant, R.G., Wilkie, K.W.Anadvancedcomposites- research fellow. In 2015 she was awarded a European Marie based solar sail system for interplanetary small satellite Sklodowska-Curie Research Fellowship to conduct research missions. In: Proceedings of the 2018 AIAA Spacecraft Structures Conference, 2018: AIAA 2018-1437. at both the Colorado Center for Astrodynamics Research [20] Becerra, V. M. Solving complex optimal control (CCAR) of the University of Colorado Boulder, USA, problems at no cost with PSOPT. In: Proceedings of and TU Delft. Her research focuses on orbital dynamics, the 2010 IEEE International Symposium on Computer- low-thrust trajectory optimization, and mission design with Aided Control System Design, 2010: 1391–1396. a strong focus on the mission enabling potential of solar [21] Banik, J. A., Murphey, T. W. Performance validation sailing. E-mail: m.j.heiligers@tudelft.nl. of the triangular rollable and collapsible mast. In: Proceedings of the 24th Annual AIAA/USU Juan M. “Johnny” Fernandez is a Conference on Small Satellites, 2010. [22] Betts, B., Spencer, D. A., Nye, B., Munakata, R., research aerospace engineer at NASA Bellardo, J. M., Wond, S. D., Diaz, A., Ridenoure, Langley Research Center. Prior to R. W., Plante, B. A., Foley, J. D., Vaughn, J. joining NASA Langley in 2014, he led LightSail 2: Controlled solar sailing using a CubeSat. several technology development projects In: Proceedings of the 4th International Symposium at the University of Surrey, Surrey Space on Solar Sailing, 2017. Centre related to deployable composite [23] Stohlman, O., Loper, E. Thermal deformation of very structures, solar sails, and satellite deor- slender triangular rollable and collapsible booms. In: biting systems, including the CubeSail, DeorbitSail, and Proceedings of the 2016 AIAA SciTech Conference, 2016. InﬂateSail CubeSat projects, and the Gossamer Deorbiter 246 J. Heiligers, J. M. Fernandez, O. R. Stohlman, et al. Laboratory, and the Jet Propulsion Laboratory, Caltech. project for ESA. He led composites solar sail system Since 2010, he has been head of the Structural Dynamics risk-reduction eﬀorts for the NASA Near Earth Asteroid Branch at NASA Langley, principal investigator for NASA Scout 6U project in 2016, and is currently Principal Langley’s heliogyro advanced solar sail technology development Investigator for the NASA Game Changing Development efforts, and project manager for NEA Scout composite solar Program’s Deployable Composite Booms project. He is sail risk reduction activities, including follow-on composite currently a member of the AIAA Spacecraft Structures solar sail system flight concept development efforts. He Technical Committee, and chair of the AIAA High Strain is currently a member of the AIAA Spacecraft Structures Composites Spacecraft Structures Sub-Committee. E-mail: Technical Committee. E-mail: william.k.wilkie@nasa.gov. juan.m.fernandez@nasa.gov. Olive R. Stohlman is a research Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which aerospace engineer at NASA Langley permits use, sharing, adaptation, distribution and Research Center. Her primary areas of reproduction in any medium or format, as long as you give research are the testing and analysis of appropriate credit to the original author(s) and the source, large deployable structures. Since joining provide a link to the Creative Commons licence, and indicate NASA Langley Research Center in 2014, if changes were made. she has worked on the thermal–structural The images or other third party material in this article are interaction analysis of the Near Earth included in the article’s Creative Commons licence, unless Asteroid Scout solar sail and on sail indicated otherwise in a credit line to the material. If membrane design and packaging for the Advanced material is not included in the article’s Creative Commons Composite Solar Sail Project. She is a member of the licence and your intended use is not permitted by statutory AIAA Spacecraft Structures Technical Committee. E-mail: regulation or exceeds the permitted use, you will need to olive.r.stohlman@nasa.gov. obtain permission directly from the copyright holder. W. Keats Wilkie is the solar sail To view a copy of this licence, visit systems and technology team lead at http://creativecommons.org/licenses/by/4.0/. NASA Langley Research Center and the Principal Investigator for the NASA Small Spacecraft Technology Program Advanced Composite Solar Sail Project. He has over 30 years of technology development experience in multifunctional structures, structural dynamics, and deployable space structures with NASA, the U.S. Army Research
Astrodynamics – Springer Journals
Published: Sep 7, 2019
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