AAS 19-541

INNOVATIVE SOLAR SAIL EARTH-TRAILING TRAJECTORIESENABLING SUSTAINABLE HELIOPHYSICS MISSIONS

James B. Pezent∗, Andrew Heaton† and Rohan Sood‡

The presented trajectory design study demonstrates that near-future solar sails canreach and maintain innovative science orbits about non-traditional Earth-trailingequilibrium points. The proposed configurations can enable long duration stereo-scopic solar imaging while potentially alleviating telemetry requirements con-straining current solar sail studies. Initial guesses for controlled periodic orbitsabout Earth-trailing (1.5◦-15◦) equilibrium points are constructed from a linearizedsolar sail circular restricted three body problem equations of motion. Orbits arethen converged in the full nonlinear model using a high order direct collocationmethod. Transfers to Earth-trailing orbits are then constructed under the assump-tion that the solar sail is a secondary payload on a larger spacecraft en-route to theSun-Earth L1 Lagrange point. Naturally occurring gravitational manifolds flow-ing towards the Lagrange point, as well as navigation data from previous Lagrangepoint missions, are used to generate a set of baseline trajectories for the primaryspacecraft. End-to-end trajectory design and optimization is then carried out toconstruct time optimal solar sail transfers to each target science orbit from ini-tial conditions constrained to lie on the carrier spacecraft’s Lagrange point trajec-tory. Additionally, trade studies are performed to assess how sail performance andEarth-trailing orbit selection affect transfer time of flight and instrument pointingaccuracy. Results indicate that, based on current technological developments, nearfuture solar sails are capable of station keeping about periodic Earth-trailing orbitswhile maintaining near-constant solar observation. Moreover, time of flights onthe order of 1 year are observed for most combinations of sail performance andfeasible initial conditions.

INTRODUCTION

As a successor to the near Earth Asteroid (NEA) Scout spacecraft, a sail-based mission to anear Earth asteroid, there is a growing interest in leveraging solar sails for heliophysics sciencemissions. The presented work is a design study demonstrating the feasibility of a new architectureunder specific operational constraints and initial conditions. Trade studies are carried out to ensurethat a solar sail can reach and maintain innovative science orbits about Earth-trailing equilibriumpoints. Additionally, optimal solutions in terms of time of flight (TOF) and solar observation timeare presented. Preliminary results suggest that the proposed trajectories are robust to varying initialconditions, sail performance levels, and operational constraints.

∗Undergraduate Researcher, Astrodynamics and Space Research Laboratory, The University of Alabama, 245 7th Ave,Tuscaloosa, AL 35401, jbpezent@crimson.ua.edu.†Senior Aerospace Engineer, EV/42, NASA Marshall Space Flight Center, NASA Marshall Space Flight Center,Huntsville, AL 35812, andrew.f.heaton@nasa.gov.‡Assistant Professor, Astrodynamics and Space Research Laboratory, The University of Alabama, 245 7th Ave,Tuscaloosa, AL 35401, rsood@eng.ua.edu.

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OBJECTIVE

Stereoscopic viewing perspectives can provide invaluable insight into the Sun’s structure and dy-namics. This concept has been previously demonstrated by NASA’s STEREO mission, consistingof two spacecraft in heliocentric orbits leading and trailing the Earth.1 The two probes coordinateto provide 3-dimensional images of the Sun as they slowly drift towards its far side. In this work,unique capabilities of solar sails are leveraged to enable long duration stereoscopic solar imaging inconjunction with ground based observations. Previous analysis of non-traditional solar sail equilib-rium points is leveraged to investigate controlled orbits trailing Earth by 1.5◦ to 15◦. The proposedtrajectories could provide sufficient angular separation to construct stereoscopic images while alsopotentially alleviating the telemetry constraints of current solar sail investigations.2

The spacecraft under consideration is assumed to be a secondary payload onboard a spacecrafttraveling to one of the Sun-Earth collinear Lagrange points. Consequently, a variety of possibletrajectories and destination orbits for the carrier spacecraft must be considered in order to properlyassess the feasibility of the secondary solar sail mission as a secondary payload. In this study, halo,Lyapunov, and Lissajous orbits about the L1 Lagrange point are considered as likely destinationsfor the primary spacecraft. The naturally occurring gravitational manifolds flowing towards theLagrange point orbits, as well as navigation data from previous Lagrange point missions are usedto generate initial conditions for the solar sail trajectory. From here, trajectory design is carriedout to ensure that the spacecraft can successfully escape the Earth’s sphere of influence and reachEarth-trailing orbits. Subsequently, the sail is leveraged to counteract the gravitational perturbationsfrom the Earth and maintain the desired configuration. In the process, trade studies are carriedout to assess how sail performance and orbit selection affect transfer TOF and instrument pointingaccuracy.

DYNAMICAL MODELING

Preliminary trajectories are designed within the framework of the Sun-Earth circular restrictedthree-body problem (CR3BP). The spacecraft is modeled as a massless object under the gravitationalinfluence of the Earth and the Sun, which are both assumed to be moving in circular orbits abouttheir barycenter. The equations of motion are expressed in the Sun-Earth rotating frame, and unitsare non-dimensionalized such that the non-dimensional distance between the Earth and Sun is unity,and the period of the system is 2π.3–8 The resulting scalar expressions, shown in Equation 1 , governthe acceleration of the x, y, and z components of the spacecraft’s state relative to the barycenter ofthe system. The additional quantities, r and d, represent the scalar distance of the spacecraft fromthe Sun and Earth respectively, while the mass parameter, µ, is the ratio of the Earth’s mass to thatof the total system.

agx = 2y + x− (1− µ)x+ µ

d3− µx+ µ− 1

r3

agy = −2x+ y − (1− µ)y

d3− µ y

r3

agz = −(1− µ)z

d3− µ z

r3

µ =me

me +ms

~d = [x+ µ, y, z]T

~r = [x+ µ− 1, y, z]T

(1)

2

Though they appear simple, the dynamics are chaotic and exhibit no closed form solutions, butfive equilibrium points, or Lagrange points, are known. The periodic/quasi periodic orbits that existin the vicinity of the Lagrange points can be leveraged to place a spacecraft in orbit about a fixedlocation or to maneuver the craft along the associated manifold for cost-effective transfers.3, 9–11

The small, but constant thrust provided by a solar sail is well suited to exploiting the complexdynamics arising within the CR3BP, as shown by numerous researchers.4–7, 12–16 For an ideal solarsail, the effects of solar radiation pressure (SRP) are modeled by incorporating an additional accel-eration term along the sail-normal direction, n, to Equation 1, and is expressed in vector form inEquation 2 below.

~as = β1− µ

| ~d |2 (cos2(α)) n ; cos(α) = d · n (2)

Figure 1: Visual illustration of solar sail orienta-tion angles and vectors shown in Equation 2.

The solar sail orientation angle, α, measures theangle of incidence/pitch between the solar radi-ation pressure vector, ~rSRP , and the sail’s nor-mal vector, n. The clock angle angle, γ, canbe implicitly defined from Equation 2 as therotation of the pitched normal vector about d.The constant β term, known as the lightnessparameter, is the main performance metric ofthe sail and is a function of the mass and thesail area. In general, higher β will confer in-creased maneuverability and lower TOF. TheNEA Scout spacecraft, currently scheduled tolaunch in June 2020, possesses a β of 0.011.2, 17

With technological advancement, the next gen-eration solar sails are projected to have a β of0.02 to 0.03. As such, values within this rangeare used as a baseline for trajectory and orbitanalysis.

TRAJECTORY TRANSCRIPTION

To design trajectories within the CR3BP, the continuous time optimal control problem is con-verted into a finite dimensional parameter optimization problem using the direct transcription (col-location) method. In the context of the solar sail CR3BP, this process begins by defining the statespace model governing the motion of the solar sail as seen below in Equation 3.

~Xi =

xyzxyz

; ~Ui =

nxnynz

; ~X( ~Xi, ~Ui) =

xyz

agx + asxagy + asyagz + asz

; ~T =

~X1

~U1

...~Xn

~Un∆t1

...∆tn−1

(3)

3

A state along the trajectory, ~Xi, is given by the spacecraft’s position and velocity state and accom-panied with a control vector, ~Ui, which contains the components of the sail’s normal vector. The

instantaneous time derivative, ~X( ~Xi, ~Ui), can then be expressed as a vector function of the statesand control parameters using Equations 1 and 2. The entire trajectory then consists of n states andcontrols arrayed into the design variables vector, ~T , along with the arc times, ∆ti, between eachsequential pair of states. Note that, expressing control directly in terms of the normal vector re-quires an extra variable compared to an α, γ formulation, but is free of coordinate singularities.Furthermore, unitarity of n can be enforced implicitly within the equations of motion.4, 5

Given this discrete representation of the trajectory, Legendre Gauss Lobatto (LGL) collocation isused to represent the system dynamics as a piecewise polynomial function of time. The equationsof motion are considered satisfied if the resulting time-derivative of each piecewise polynomialmatches that of the underlying system dynamics at predefined ‘collocation’ points.6, 7, 18–20 ForLGL methods of order k, the time spacings of states and collocation points on each polynomial’snormalized time interval are restricted to lie at the roots of the kth order Lobatto polynomial. Thiscan be stated succinctly for the lowest order LGL method, LGL-3, otherwise known as the HermiteSimpson method. For every sequential pair of states and controls in the trajectory, a cubic Hermitespline and it’s derivative are constructed and evaluated at the midpoint, ~Xc. The control at themidpoint, ~Uc, is interpolated assuming linear variation over the interval. The constraint equation,~δi, is then evaluated as difference between the interpolated derivative, ~Xc, and the derivative of the

interpolated state, ~X( ~Xc, ~Uc), seen below in Equation 4.

~δi( ~Xi, ~Ui, ~Xi+1, ~Ui+1) = ~X( ~Xc, ~Uc)− ~Xc

where

~Xc =1

2( ~Xi + ~Xi+1) +

∆ti8

( ~X( ~Xi, ~Ui) + ~X( ~Xi+1, ~Ui+1))

~Uc =1

2(~Ui + ~Ui+1)

and

~Xc = − 3

2∆ti( ~Xi − ~Xi+1) +

1

4( ~X( ~Xi, ~Ui) + ~X( ~Xi+1, ~Ui+1))

(4)

The LGL-3, 5, and 7 transcriptions of dynamics and control are used interchangeably throughoutthis work, but regardless of choice, error is assessed and refined using a Runge-Kutta 78 integrator.The reader is referred to the references6, 18, 20, 21 for further description of higher order methods.

For a solar sail, an additional inequality constraint is added to each state and control in the trajec-tory to ensure that the normal vector is directed away from the Sun at all times. This can be formallyincorporated by stating that the dot product of the sail’s position relative to the Sun and its normalvector must be greater than 0, as seen below in Equation 5.

gi( ~Xi, ~Ui) = di · ni > 0 (5)

When Equations 4 and 5 are solved to within some predefined tolerance, the converged trajectoryis considered dynamically feasible. Additional constraints may be applied to compute trajectorieswith required characteristics, such as specific initial or final states. Furthermore, one can define acost function, J(~T ), in terms of discrete trajectory variables, and then minimize/maximize it subject

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to equality, ~F (~T ), and inequality, ~G(~T ), constraints (Equation 6). For example, it may be desirableto minimize the total TOF, which is defined as summation of all arc times, ∆t.

Minimize : J(~T ) =n−1∑i=1

∆ti

Subject To :

~F (~T ) = [~δ1, ~δ2, . . . , ~δn−1]T = ~0

~G(~T ) = [g1, g2, . . . , gn]T > ~0

(6)

Optimal control problems expressed in the above form can be efficiently solved using general pur-pose sequential quadratic programming (SQP) or interior point (IP) methods.22–24 All results pre-sented in this work were formulated and solved using a general purpose SQP solver developedin-house.

EARTH-TRAILING EQUILIBRIUM POINTS

In the Sun-Earth rotating frame, the simplest class of close Earth-trailing trajectories are the solarsail equilibrium points first explored by McInnes.15 If properly positioned, a solar sail can remainstationary in the Sun-Earth rotating frame using a constant α angle. Placing a solar sail at a trailingequilibrium point is an attractive option from the standpoint of pointing on-board scientific instru-ments. By pre-selecting a desired trailing angle, θt, for a mission, solar imaging equipment can bebore-sighted off the sail’s normal vector by an equilibrium α angle, thus allowing the spacecraft toorient itself for constant solar observation. Likewise, communications antennae may be simultane-ously aligned to stay in constant contact with Earth, reducing the need for dedicated communicationslews. Following the analysis of McInnes, the β value and n necessary to maintain an arbitrary equi-librium point within the CR3BP are analytic functions of the sail’s position within the Sun-Earthrotating frame and can be computed using Equation 7.14, 15, 25 However, it is necessary to excludelocations such that cos(α) is less than than zero since the sail cannot provide thrust in the direc-tion of the Sun. Using Equation 7, the planar Earth-trailing configuration space can be sampled toproduce contours of constant β equilibrium points.

n = − ~ag|~ag|

β =|~d|2|~as|

(1− µ)(cos2(α))

(7)

For any given β value, the solutions where θt is equal to 0◦ and 60◦ correspond to the extensivelystudied collinear sub-L1 and triangular sub-L5 points, respectively. However, since this study per-tains only to close Earth-trailing trajectories accessible to near future solar sails, analysis is restrictedto 0◦ to 15◦ for β values ranging from 0.01 to 0.05. A subset of equilibrium contours for these βvalues are plotted in Figure 2(a) . In Figure 2(b), the same contours are represented as labelled whitelines, and the configuration space is colored to show the requisite in-plane α pitch angle necessaryto maintain equilibrium.

Trailing angles are restricted between 0◦-2.5◦, since this range captures the qualitative behaviorof all trailing angles of interest. It is clear from Figure 2(a) that a bifurcation in overall behavioroccurs at a critical lightness number, βcr, between 0.025 and 0.03 (βcr = 0.0281). For β above

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(a) Equilibrium lines for range of β values as given bythe color bar.

(b) Heat-map of α angle as given by the color bar.Constant β contours labelled in white.

Figure 2: Subset of planar Earth-trailing configuration space (θt = 0◦ − 2.5◦).

βcr, an equilibrium contour can be extended smoothly from sub-L1 to sub-L5 . Pitch angles alongthese contours rapidly increase from 0◦ at sub-L1 to a β dependent maximum at approximately0.63◦ θt. A second distinct contour also extends from the vicinity of L1 and hugs the exterior ofEarth’s classical sphere of influence before continuing on towards the classic L5 point. When β isbelow the critical value, contours extending from the corresponding sub-L1 exhibit monotonicallyincreasing α angles as they loop back towards the vicinity of the traditional L1 point. A secondcontour then extends from the sub-L5 point to a β-dependent minimum trailing angle near the Earthbefore turning back towards L5. In this case, pitch angles increase from 0 degrees at sub-L5 to 90◦

when the approaches the classic L5.

Further qualitative information regarding dynamical and stability characteristics can be gainedby examining the linear motion about each equilibrium point. For this case, the state space modeldefined in Equation 3 is restricted to only in-plane components and variations in the control direc-tions are neglected. As noted by McInnes, when α is non-zero, equilibria are not Lyapunov stable,so the eigenvalues of the resulting 4 by 4 linear system lose some of their predictive power when θtis not equal to 0◦ or 60◦ degrees (sub-L1 and sub-L5 respectively). However practical informationregarding the stability and period of nearby motion may still be obtained. An approximate stabilityindex, λr, is defined as the average of the absolute value of the real parts of each eigenvalue. Largervalues of λr indicate a greater tendency to diverge from the equilibrium point. Furthermore, theapproximate period of nearby trajectories is inferred in the standard way from the eigenvalues asso-

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ciated with the most stable subspace. Figure 3 illustrates the evolution of the pseudo-stability indexover the 0◦ to 2.5◦ trailing configuration space and isolates the low α contour for β = 0.03 since itcan be extended smoothly throughout.

(a) Heatmap of stability index λr.

(b) Progression of eigenstructure for β = 0.03.

(c) Progression of period for β = 0.03.

Figure 3: Stability of planar Earth-trailing configuration space (θt = 0◦ − 2.5◦).

Figure 3(b) shows the evolution of the 4 eigenvalues of the indicated contour in the complex plane,while Figure 3(c) illustrates the period associated with pseudo-stable mode (reciprocal eigenvaluetraces with largest complex components). At θt= 0◦, the eigenstructure (Figure 3(b)) is that of thesub-L1 equilibrium point but all modes rapidly decrease in magnitude as the trailing angle increasesuntil the vicinity of the maximum α trailing angle. At this point, the unstable modes converge andthen depart the real axis while stable modes exhibit decreasing real parts. As trailing angle furtherincreases, all eigenvalues reach the unit circle before reversing course and exhibiting decreasingreal parts. If continued beyond what is visible in the plot all the way to θt = 60◦, the eigenstructureasymptotically converges toward the stable configuration of sub-L5. Therefore, it is interpretedthat the sail equilibrium points beyond the 0◦-2.5◦ range will possess a practical degree of longterm stability, and that the surrounding motion mimics the behavior of sub-L5 short period orbits.Furthermore, it can be seen in Figure 3(c) that the period of nearby motion begins to convergetowards one year, as one would expect from a stationary trajectory primarily influenced by the Sun.Examining the stability index for the entire β range in Figure 3(a), the abrupt reduction in the valueof λr below the circular red arc appears to signal the edge of the Earth’s sphere of influence with

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respect to a sail at low α angle. When considered as a whole, it is concluded that a solar sail canbe placed at a trailing angle greater than roughly 2.5◦ to enhance stability characteristics. Note thatthere is a thin strip of comparable stability equililbria (bounded in red) bisecting the trade space ata smaller θt as well, however these are ignored for reasons stated below.

The range of equililbria under consideration as a basis for constructing periodic orbits has beensignificantly narrowed by the above analysis and the desire for stable configurations. Recalling thatthe next generation solar sail will possess a β value on the order of 0.025, it can be seen in Figure3(a) that there is a dead-zone ranging from 0.4◦ to 0.7◦ where it is dynamically impossible to sustainan artificial equilibrium point and that this excludes the sail from being placed in the high-stability,low-trailing angle region bounded by the two red lines in Figure 3(a). The trade space is furtherreduced by combining practical operational constraints from NEA Scout with the stability infor-mation above. Analytic and finite element analysis carried out by Heaton et al. has indicated thatoperating a solar sail at high incidence angles is more challenging from the perspective of attitudecontrol.26–28 Looking back at Figures 2(b) and 3(a), it can be seen that locations with higher inci-dence angles also possess poor stability characteristics. Therefore, wherever two solutions exist at agiven value of θt for a single β, the solutions exhibiting higher α and lying further from the Sun areexcluded. For β values above βcr, this amounts to considering only the equilibrium lines extendingsmoothly from sub-L1 to sub-L5. For β below βcr, points along both contours are considered untilthe point where they reach a lower or upper bound in θt. This reduced subset of the trade space isrepresented in Figure 4(a) below. The dashed red and blue lines denote the cutoff point for β lessthan βcr, whose contour is highlighted in black. Figure 4(b) illustrates the the equilibrium α anglealong each β contour as a function of θt.

(a) Equilibrium lines for range of β val-ues as given by the color bar.

(b) Equilibrium α angle as a function of θt.

Figure 4: Subset of reduced Earth-trailing trade space (θt = 0◦ − 2.5◦).

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EARTH-TRAILING PERIODIC ORBITS

Extensive investigations in the regular CR3BP and in practice have shown that greater long termstability can be achieved by orbiting rather than remaining stationary at an equilibrium point.5, 21, 29

Within the context of a stereoscopic solar imaging mission constrained by Earth communicationsdistance, orbiting also delivers additional benefits. Compared to remaining stationary at an equilib-rium point, an extended orbit allows the spacecraft to observe the Sun from a wider range of trailingangles with respect to Earth-based observatories. Additionally, depending on the trailing angle andorbit size, communication range with Earth is reduced for one half of the orbit’s period.

Subsequently, the motion about Earth-trailing solar sail equilibrium points has been extended intoa new class of planar controlled periodic orbits maintained by a near-optimal station-keeping strat-egy. Starting with a given trailing angle, θt, and β value, the period of a nearby orbit is inferred fromthe frequency of the short period mode of the associated equilibrium point. An initial guess,shownin Equation 8, for the controlled periodic orbit, ~TIG , is constructed as a state history consisting ofmultiple copies of the equilibrium point state, ~Xeq, and control, ~U eq, with all arc times summing tothe estimated period, Peq.

~TIG = [ ~Xeq1 ,

~U eq1 , . . . ,~Xeqn ,

~U eqn ,∆t1, . . . ,∆tn−1]T

n−1∑i=1

∆ti = Peq(8)

The initial guess is then passed to a collocation algorithm constrained to enforce periodic motionand control. An additional constraint is also applied to produce a fixed orbital amplitude, A. Thisis enforced by constraining the initial state to lie a distance, A, along the vector extending from theequilibrium point towards the system’s barycenter. Depending on the trailing angle, this roughlycorresponds to constraining the semi-minor axis of the orbit as viewed in the Sun-Earth rotatingframe, or the eccentricity if viewed in an inertial frame. Equation 9 below illustrates the form andarguments of both the periodicity constraint, ~Cp, and amplitude constraint, ~CA.

~Cp( ~X1, ~U1, ~Xn, ~Un) =

[~X1 − ~Xn

~U1 − ~Un

]= ~0

~CA( ~X1) = (1−A) ~Xeq[1 : 3]− ~X1[1 : 3] = ~0

(9)

When solved with a small non-zero value of A, the solution expands from the equilibrium pointinto a closed orbit. A continuation scheme is applied to produce a family of orbits by increasingthe amplitude of the semi-minor axis and seeding with a previously converged orbit. An exampleof a subset of a fully converged family at a trailing angle of 3◦ and β value of 0.025 is shown inFigure 5. As expected, given the orbital configuration, the period of the final orbits is inheritedfrom the underlying equilibrium point (400 days) and they exhibit characteristics similar to that ofthe sub-L5 short period orbits. As the orbital amplitude increases, small but non-trivial in-planevariations from the reference α angle are necessary to station keep and maintain periodicity, but theclock angle remains constant at 90◦. In this work a number orbits this type have been investigatedfor β ranging from 0.015 to 0.05, and trailing angles, θt, from 1.5◦ to 60◦. In general, it is observedthat the average α angle variation reduces as the trailing angle and β increase. Furthermore, theα control history asymptotically approaches a constant pitch angle of 0◦ for non-zero amplitudesat θt = 60◦ which corresponds to the location of previously studied sub-L5 short period orbits.

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(a) Orbit family in Sun-Earth rotating frame.(b) α, θt history over one period.

Figure 5: Small α variation Earth-trailing orbit family.

When θt << 60◦, this class of trajectories sacrifices the (theoretically) constant pointing directionof an equilibrium point, but α variations are small for next generation β values, and could enableenhanced solar observation. Consider, for example, the largest amplitude orbit (A = 0.01) in Figure5(a). If the imaging instrument is suitably offset by a mean α angle of 2.8◦ (top of Figure 5(b)),any instrument with a field of view greater than 0.5◦ (the average angular diameter of the Sun), willremain in view of the Sun for the majority of the science orbit. Additionally, as observed from theθt vs time plot at the bottom of Figure 5(b), stereoscopic images may be taken at parallaxes of 1.8◦

to 4.5◦, and high volume data transfer with Earth can be concentrated around the 100 day markwhere communications distance is reduced by 40% compared to the equilibrium point. Moreover,in a higher fidelity gravitational model, the station keeping about the equilibrium point may offsetany pointing advantages over the controlled periodic orbits studied in this work; a scenario whichwill be explored in future investigations.

The characterization of the period of motion at the equilibrium point is highly critical to theconstruction of orbits of this type. For the β value (.025) and trailing angle (3.0◦) illustrated above,one might conclude that Earth perturbations will be exceedingly small, and as such that the periodof the underlying periodic orbit will be approximately one year. If the same continuation scheme isthen applied to build a family, orbits are visually almost indistinguishable, but the resulting α controlhistory shown in Figure 6 differs dramatically. Compared to the smooth but irregular control seen inFigure 5, α variation for a one year period orbit appears to be almost a perfect sinusoid. However,for an orbit of comparable size to that analyzed in Figure 5 (A = 0.01), the maximum α variationis 1.75◦, a threefold increase. Without deviating from the periodicity station-keeping strategy, thiswould result in considerably less solar observation time for a statically aligned instrument, thus,these solutions possess less practical utility.

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Figure 6: One year period α control history.

Returning to the low variational solutions presented in Figure 5, it is desirable to assess whethervariations in pitch angle can potentially be eliminated all together. As discussed earlier, assumingthat solar observation equipment is mounted to align with the Sun at some predefined α angle, theclock angle may vary arbitrarily without losing sight of the Sun as long it is accompanied with aroll maneuver about the sun vector. Therefore, it may be possible to maintain both periodicity andcontinuous solar observation at some constant α using only variations in the clock angle. For a giventrailing angle and orbital amplitude, this α angle is not known a priory (and may not exist), but it isclearly close to the minimum-norm solutions produced by the continuation scheme. Therefore, toconstruct a family of orbits with minimal α variation, optimization of a suitable objective functionis employed. To achieve this, the sum of squared deviations between the cosine of the α angle ateach state in the trajectory and a single free pitch angle, αr, is minimized (Equation 10). Recall thatdi and ni are the sun-sail vector and sail-normal vector, respectively.

Jα(~T , αr) =n∑i=1

(di · ni − cos(αr))2 (10)

Minimizing a discrete sum of the squared deviations is selected instead of the integral of squareddeviations to prevent the optimizer from collapsing the trajectory to a single point. It is possible todirectly solve for a trajectory that drives Equation 10 to zero, however such an approach is avoidedsince the resulting problem could be inconsistent with the additional periodicity and amplitudeconstraints. When Equation 10 is incorporated into the continuation scheme using the same initialconditions as before, the resulting orbits (Figure 7(a)) become slightly elongated compared to thenon-optimized case shown in Figure 5(a). By examining the resulting control history of the familyin Figure 7(b), it can be seen that all members are able to converge to a constant α. For smallamplitude orbits, the sail’s normal vector remains almost entirely in-plane. However, as orbitalamplitude increases the optimization routine is able to return constant α trajectories through aninteresting mechanism. It can be seen in Figure 7(c,d) that the sail’s cone angle, γ, begins exhibitingsharp oscillations ±30 days of periapse. Qualitatively, it can be said that the sail is attemptingcompensate for the constant α angle by directly using Earth perturbations to maintain periodicity.When the sail oscillates in clock near periapse (Time = 0, 400), components of thrust counteractingEarth’s pull are momentarily reduced. The maneuver is timed such that the additional impulse fromthe Earth directs the sail back towards the initial state. Clearly, this control history results a smallout-of-plane component to the orbit, however, since clock angle cycles rapidly above and below 90◦

they are essentially planar. Similar to the α variations in non-optimized orbits, the magnitude of theγ variations decreases with both increasing trailing angle, θt, as well as β value.

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(a) Orbit family in Sun-Earth rotating frame.(b) α, γ control history for one period.

(c) Magnified γ control history 0 - 30 days. (d) Magnified γ control history 370 - 400 days.

Figure 7: Constant α Earth-trailing orbit family.

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EARTH-TRAILING TRAJECTORY DESIGN

Given that the next sail-based spacecraft is projected to be a secondary payload on-board a space-craft en-route to an orbit about the L1 Lagrange point, it is desirable to assess whether Earth-trailingorbits are compatible with such a mission architecture. As a first step in the trajectory design pro-cess, it is necessary to compute feasible initial conditions arising from the primary spacecraft’sLagrange point trajectory. A baseline mission is constructed that enables the primary spacecraft toenter into a Sun-Earth L1 Lissajous orbit. To be consistent with the recent Deep Space ClimateObservatory (DSCOVR) and Advanced Composition Explorer (ACE) missions, a Lissajous orbit ofdimensions 280,000 Km (Y) by 160,000 Km (Z) is selected for further analysis.30

Primary Spacecraft Trajectory

Design of the primary spacecraft’s trajectory begins by first finding an unstable manifold ex-tending from a low Earth orbit to the vicinity of the L1 periodic orbit. An unstable manifold ofa northern Halo orbit with a minor Z-amplitude of 160,000 km is selected such that it crosses theXZ plane roughly level with the minimum Z dimension (-160,000 Km) of the target Lissajous.Given this transfer geometry, an initial guess of the target orbit is constructed using a linearizationof motion about L1. Initial in-plane and out-of-plane phase angles of 210◦ and 270◦ are chosensuch that the initial state lies at -160,000 km below the XY plane and in the path of the impingingmanifold. The initial guess is linearly propagated for several revolutions and then converged inthe full CR3BP model.9 The final insertion state is taken to be the orbit’s next 180◦ phase anglecrossing. The primary spacecraft’s transfer from low Earth orbit to the target Lissajous is computedusing a multi-phase direct collocation approach similar to that outlined previously (without the so-lar sail). The trajectory is divided into ballistic arcs whose equations of motion are representedusing LGL collocation. Constraints are applied to enforce position continuity between each sub-sequent ballistic arc, but velocities are left free in order to approximate necessary impulsive ∆V .Additional constraints at the beginning of the first arc and at the end of the last arc are applied toenforce a 200 km LEO departure and Lissajous insertion, respectively. In this case, the trajectoryis modeled using 4 ballistic arcs and allows for 3 impulsive ∆V s to reach the final destination.

Figure 8: Discontinuous initial guess for the baseline L1

Lissajous mission.

The states of the first two arcs aresupplied by the manifold and extendfrom low Earth orbit until its crossingof the XZ plane. A velocity discon-tinuity is placed at 60 days in the as-sumed location of a mid-course cor-rection (MCC) maneuver. The lasttwo arcs are supplied by subdivid-ing the first revolution of the precom-puted Lissajous. The interior veloc-ity discontinuity between these twoarcs approximates a Lissajous orbitinsertion correction (LOIC) maneu-ver to reach the final desired inser-tion state. The eventual intersectionof arcs 2 and 3 represents the locationof the Lissajous orbit insertion (LOI)

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maneuver. This trajectory is then converged to enforce position continuity between LEO and theLissajous insertion state while minimizing the ∆V at each velocity discontinuity.

The final optimized transfer requires a total ∆V of 150 m/s after departure from LEO, and over-all is very similar to the DSCOVR misson. Suitable points along the transfer may then be used asa feasible initial state for the solar sail mission. However, it is advantageous to refrain from de-ploying the sail until the primary spacecraft has reached apoapsis and begun looping back towardsL1. Depending on the size of the sail and constraints on the incidence angle, earlier deploymentsmay result in the sail falling back into Earth’s gravity well. To minimize the mass of the carrierspacecraft, the secondary payload is assumed to separate 2 days prior to the MCC. With this config-uration, sails with lightness numbers ranging from 0.015 to .05 are able to escape the Earth’s SOIand reach Earth-trailing orbits.

Figure 9: Optimized L1 Lissajous trajectory for the primary spacecraft.

Planar Trailing Orbits

Given the previously defined initial conditions, the first class of solar sail missions to be exploredare direct transfers to a single specified trailing orbit. Trailing angles ranging from 1.5◦ to 15◦

will satisfy communications and stereoscopic separation constraints. Therefore, missions to trailingorbits in this range are selected for further study. Likewise, internal NASA studies indicate thatthe next generation solar sail will possess a β value on the order of 0.02 - 0.03; however, β valuesranging from 0.015 to 0.05 are considered in order to thoroughly characterize near future solar sailperformance.

The trade study proceeds by selecting an initial β value and constructing a single transfer tothe 1.5◦ trailing equilibrium point. Due to the proximity of the equilibrium point and the lack ofintervening massive bodies, a simple linear interpolation between the sail deployment state andequilibrium point state suffices as an initial guess for the collocation algorithm. Constraints are

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applied to strictly enforce both boundary values, and the trajectory is optimized to minimize thetotal TOF. This trajectory solution is used to seed a continuation scheme computing and optimizingtransfers to equilibria at incrementally larger trailing angles. Furthermore, once transfers to allequilibrium points are computed for a given β value, they are utilized to seed a continuation schemefor a nearby β value. Since initial guesses for transfer arcs are always very close to a nearby solution,the optimizer is able to converge rapidly, enabling the entire trade space to be explored within just afew minutes. Figure 10 illustrates the progression of the continuation scheme for a constant β valueof 0.03 over the desired range of trailing angles (Figure 10(a)), as well as solutions for all β valuesat a trailing angle of 7.5◦ (Figure 10(b)).

(a) Equilibrium transfers for β= 0.03 over a range of trailingangles.

(b) Equilibrium transfers for all β valuesat a θt = 7.5◦.

Figure 10: Selected transfer trajectories to artificial equilibrium points.

Figure 10(a) provides a qualitative justification for choosing to deploy the sail at the apoapseof the primary spacecraft’s transfer. When the target trailing angles are large (> 3◦), the sail hasample time to adjust the altitude of a powered Earth flyby that minimizes the overall TOF. However,when the target trailing angles are small (< 3◦), higher Earth departure energies would actuallyincrease the overall TOF because of the excess speed that must be bled off before inserting into theequilibrium point. Therefore, in this case, the optimal control strategy uses the sail to make minimalmodifications to the initial L1 halo orbit’s manifold in order to escape the Earth’s SOI.

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Figure 11: Optimized solar sail transfers to 3◦

Earth-trailing orbits.

With this set of baseline solutions, anothercontinuation scheme is carried out to computeoptimal transfers to periodic orbits encirclingeach equilibrium point. For a selected β valueand trailing angle, a family of periodic orbits isfirst computed using the process outlined in pre-vious sections. For this study, the largest mem-ber of the family is limited to a semi-minor axisamplitude of 0.01 in non-dimensional units.Since the optimal insertion location onto the pe-riodic orbit is not known in advance, it must beprovided as a free parameter to the optimizationalgorithm. This is incorporated by represent-ing the target orbit as a piecewise cubic Her-mite spline parameterized by the orbital period.The terminal constraint on the transfer is mod-ified to accept an additional variable represent-ing the time along the target orbit. The splinerepresenting the orbit can then be evaluated toyield a feasible target state. With this modifiedproblem formulation, the continuation schemeis initialized with a converged equilibrium pointtransfer and an estimate of the insertion time,and advances until a transfer to the largest tar-get orbit has been computed. Similar to before,this process can be quickly repeated for all β,θt, and amplitude combinations of interest. Anexample of a β, θt combination of 0.03 and 3◦

is displayed in Figure 11 to the right. Interest-ingly, it can be seen that increasing the size ofthe target orbit can result in considerable TOFbenefits.

The results of the analysis for planar Earth-trailing transfers are succinctly described byFigure 12. In order to smoothly parameterizeoptimal TOFs, the continuation scheme described above sampled the trade space in steps of 0.5◦

in θt and 0.0025 in β. Contours of constant TOF are then interpolated for transfer times to non-traditional equilibrium points and selected periodic trailing orbits, and projected onto the θt vs. βtrade space. For β > 0.02 (shaded in light blue), the TOF varies almost linearly with β and θtfor fully optimized trajectories. Furthermore, it can be observed that transferring to a controlledperiodic orbit confers lower TOF as compared to arriving at the associated equilibrium point, andthe TOFs increase monotonically with trailing angles. However, for β < 0.02 (shaded in light red),smaller sails begin to have difficulty reaching target orbits in a reasonable amount of time as indi-cated by an increase in the slope of constant TOF contour lines. Furthermore, transfer TOFs beginto reach a minimum at roughly θt = 3◦, where the TOF contours become nearly vertical for β valuesless than 0.02.

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Figure 12: Time of flight (days) contours for direct Earth-trailing mission. Contours shown in 30day increments for target orbits of varying size.

CONCLUSION

Results, thus far, have shown that solar sail technology is capable of achieving innovative andsustainable Earth-trailing trajectories. A robust subset of solutions are generated for the given con-straints on the initial conditions associated with the trajectory of a carrier spacecraft traveling tothe vicinity of the Sun-Earth L1. Earth trailing science orbits and trajectories conducive to solarobservation are identified. Baseline transfer solutions are constructed for a variety of sail lightnessparameters and Earth-trailing angles, and are used to thoroughly characterize TOFs for the missionarchitecture. Additional work will be carried out to further assess operational constraints in a higherfidelity dynamical model. Trajectories consisting of multiple destination orbits will also be consid-ered from the perspective of NASA’s heliophysics science goals.

REFERENCES

[1] M. L. Kaiser, T. Kucera, J. Davila, O. S. Cyr, M. Guhathakurta, and E. Christian, “The STEREOmission: An introduction,” Space Science Reviews, Vol. 136, No. 1-4, 2008, pp. 5–16.

[2] J. B. Pezent, R. Sood, and A. Heaton, “Near Earth Asteroid (NEA) Scout Solar Sail Contingency Tra-jectory Design and Analysis,” 2018 Space Flight Mechanics Meeting, 2018, p. 0199.

[3] K. C. Howell, “Three-dimensional, periodic,haloorbits,” Celestial mechanics, Vol. 32, No. 1, 1984,pp. 53–71.

[4] R. Sood and K. Howell, “Solar Sail Transfers and Trajectory Design to Sun-Earth L4, L5: Solar Obser-vations and Potential Earth Trojan Exploration,” The Journal of the Astronautical Sciences, 2019, DOI:10.1007/s40295-018-00141-4.

17

[5] R. Sood, Significance of specific force models in two applications: Solar sails to sun-earth L 4/L 5and grail data analysis suggesting lava tubes and buried craters on the moon. PhD thesis, PurdueUniversity, 2016.

[6] M. T. Ozimek, D. J. Grebow, and K. C. Howell, “A collocation approach for computing solar sail lunarpole-sitter orbits,” Open Aerospace Engineering Journal, Vol. 3, 2010, pp. 65–75.

[7] D. J. Grebow, M. T. Ozimek, and K. C. Howell, “Design of optimal low-thrust lunar pole-sitter mis-sions,” The Journal of the Astronautical Sciences, Vol. 58, No. 1, 2011, pp. 55–79.

[8] V. Szebehely, Theory of orbit: The restricted problem of three Bodies. Elsevier, 2012.[9] K. Howell and H. Pernicka, “Numerical determination of Lissajous trajectories in the restricted three-

body problem,” Celestial Mechanics, Vol. 41, No. 1-4, 1987, pp. 107–124.[10] K. Howell, “Families of orbits in the vicinity of the collinear libration points,” AIAA/AAS Astrodynamics

Specialist Conference and Exhibit, 1998, p. 4465.[11] K. C. Howell and M. Kakoi, “Transfers between the Earth–Moon and Sun–Earth systems using mani-

folds and transit orbits,” Acta Astronautica, Vol. 59, No. 1-5, 2006, pp. 367–380.[12] A. Farres, J. Heiligers, and N. Miguel, “Road Map to L4/L5 with a solar sail,” 2018 Space Flight

Mechanics Meeting, 2018, p. 0211.[13] J. Heiligers, B. Diedrich, W. Derbes, and C. McInnes, “Sunjammer: preliminary end-to-end mission

design,” AIAA/AAS Astrodynamics Specialist Conference, 2014, p. 4127.[14] J. Heiligers and C. McInnes, “Agile solar sailing in three-body problem: motion between artificial

equilibrium points,” 64th International Astronautical Congress 2013, 2013, pp. IAC–13.[15] C. R. McInnes, A. J. McDonald, J. F. Simmons, and E. W. MacDonald, “Solar sail parking in restricted

three-body systems,” Journal of Guidance, Control, and Dynamics, Vol. 17, No. 2, 1994, pp. 399–406.[16] D. Guzzetti, R. Sood, L. Chappaz, and H. Baoyin, “Stationkeeping Analysis for Solar Sailing the L4

Region of Binary Asteroid Systems,” Journal of Guidance, Control, and Dynamics, 2019, pp. 1–13,https://doi.org/10.2514/1.G003994.

[17] L. Johnson, L. McNutt, and J. Castillo-Rogez, “Near Earth Asteroid (NEA) Scout Mission,” 2017.[18] F. Topputo and C. Zhang, “Survey of direct transcription for low-thrust space trajectory optimization

with applications,” Abstract and Applied Analysis, Vol. 2014, Hindawi Publishing Corporation, 2014.[19] J. T. Betts, “Survey of numerical methods for trajectory optimization,” Journal of guidance, control,

and dynamics, Vol. 21, No. 2, 1998, pp. 193–207.[20] A. L. Herman and B. A. Conway, “Direct optimization using collocation based on high-order Gauss-

Lobatto quadrature rules,” Journal of Guidance Control and Dynamics, Vol. 19, No. 3, 1996, pp. 592–599.

[21] D. J. Grebow, Trajectory design in the Earth-Moon system and lunar South Pole coverage. PhD thesis,Purdue University, 2010.

[22] P. E. Gill, W. Murray, and M. A. Saunders, “SNOPT: An SQP algorithm for large-scale constrainedoptimization,” SIAM review, Vol. 47, No. 1, 2005, pp. 99–131.

[23] A. Wachter and L. T. Biegler, “On the implementation of an interior-point filter line-search algorithmfor large-scale nonlinear programming,” Mathematical programming, Vol. 106, No. 1, 2006, pp. 25–57.

[24] A. Barclay, P. E. Gill, and J. B. Rosen, “SQP methods and their application to numerical optimal con-trol,” Report NA, 1997, pp. 97–3.

[25] A. Farres and A. Jorba, “Station Keeping Strategies for a Solar Sail in the Solar System,” Recent Ad-vances in Celestial and Space Mechanics, pp. 83–115, Springer, 2016.

[26] A. Heaton, N. Ahmad, and K. Miller, “Near Earth Asteroid Scout Solar Sail Thrust and Torque Model,”2017.

[27] A. Heaton, B. L. Diedrich, J. Orphee, B. Stiltner, and C. Becker, “Momentum Management for theNASA Near Earth Asteroid Scout Solar Sail Mission,” 2017.

[28] J. Orphee, B. Diedrich, B. Stiltner, C. Becker, and A. Heaton, “Solar Sail Attitude Control System forthe NASA Near Earth Asteroid Scout Mission,” 2017.

[29] M. T. Ozimek, Low-thrust trajectory design and optimization of lunar south pole coverage missions.PhD thesis, Purdue University, 2010.

[30] C. Roberts, S. Case, J. Reagoso, and C. Webster, “Early Mission Maneuver Operations for the DeepSpace Climate Observatory Sun-Earth L1 Libration Point Mission,” 2015.

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