IAC-10-C1.2.3 TRAJECTORY CONTROL FOR A

c 61st International Astronautical Congress, Prague, CZ. Copyright 2010 by the International Astronautical Federation. All rights reserved.

IAC-10-C1.2.3 TRAJECTORY CONTROL FOR A SOLAR SAIL SPACECRAFT IN AN OFFSET LUNAR ORBIT Geoffrey G. Wawrzyniak∗ Purdue University, United States of America [email protected] Kathleen C. Howell† Purdue University, United States of America [email protected] Solar sailing has the potential to open up design regimes for new mission applications. One such application is the lunar pole sitter, in which a solar sailing spacecraft moves in an orbit that is offset from the center of the Moon and remains in view of the lunar south pole at all times. Trajectory solutions in this configuration are naturally unstable; a sailcraft will eventually diverge from its designated reference path. Two implementations of a “turn-and-hold” control law are developed. The first employs a least-squares solution to deliver the set of angles that will best target a future state along the design path. The second segments the reference orbit into short arcs and examines these segments within the context of two-point boundary-value problems (TPBVP) subject to boundary conditions at either end of the arc. A collocation-based TPBVP solver is then employed to generate three turns at pre-specified times along the segment.

INTRODUCTION Spacecraft trajectory design typically begins with a reference path, whether the design is for a low-Earth orbiter, a mission to Mars, or an orbit near a Lagrange point. As the mission design process proceeds, the reference trajectory is refined in system models of higher fidelity that include planetary ephemerides and more accurate representations of spacecraft characteristics (e.g., reflectivity properties for solar radiation pressure modeling). Frequently, the reference path includes scheduled, deterministic maneuvers to shift from one natural arc to another. However, a model is never perfect. Maneuvers (e.g., insertion maneuvers), ephemerides and the space environment, the behavior of the spacecraft (e.g., gas leaks, reflectivity), as well as the control devices, each introduce uncertainty and error. To maintain a desired orbit or to transfer to an arc en route to a specified target requires strategies for scheduled station-keeping maneuvers (SMKs) or trajectory correction maneuvers (TCMs), respectively. The processes depend upon precise orbit estimation, that is, assessing the spacecraft’s current path in relation to its nominal path, and then supplying an input for the design of an appropriate maneuver to return the spacecraft to the desired path (SKM) or to correct the current trajectory such that the spacecraft is ∗ PhD † Hsu

Candidate, School of Aeronautics and Astronautics Lo Professor, School of Aeronautics and Astronautics

IAC-10-C1.2.3

re-targeted toward a specific destination. In addition to gravity, multiple small forces act on a spacecraft in flight. Mission designers typically incorporate solar radiation pressure (SRP) as a perturbation. For a solar sail, SRP is exploited as a primary means of propulsion and, thus, may significantly affect the trajectory, enabling new concepts in trajectory design. Solar sailing dates back to Tsiolkovsky, Tsander, and Oberth in the 1920s.1 If the definition of a solar sail includes any spacecraft that exploits SRP, an early “sail” mission involved the Mariner 10 spacecraft that flew by Mercury three times in 1974–75 and exploited SRP for attitude control.2 More recently, MESSENGER employed SRP for trajectory control during the 2008–09 Mercury flybys and for angular momentum management.3 These two missions “sailed” using their solar panels and not with a highly reflective, lightweight sheet of large dimensions conventionally defined as a solar sail. However, small sails attached to traditional spacecraft have been proposed for attitude and trajectory control4–10 over the last 50 years. Only recently has a spacecraft flown with a solar sail as its only means of propulsion. In the summer of 2010, the Japanese Space Agency, JAXA, launched a solar sail spacecraft named IKAROS in tandem with another mission to Venus. The sailcraft is the first in-flight demonstration of solar sailing.11 One of the more frequently suggested missions that would employ a large solar sails as the primary propulsion device is the Heliostorm (a.k.a. Geostorm) Warn-

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ing Mission. In this mission, a sailcraft is placed at (or is made to orbit) an artificial libration point located approximately at 0.98 AU, or nearly twice the distance from the Earth to the natural Earth–Sun L1 point. A spacecraft closer to the Sun would provide advance warning of impending solar storms (i.e., coronal mass ejections) that can disrupt electronic systems on Earth.12 Both Yen13 and Sauer14 suggest controlling the spacecraft to the sub-L1 point as opposed to an orbit near that point as suggested by Lisano et al.15 Lawrence and Piggott compare a linear quadratic control scheme and a Gramian controller, both incorporating a sail to maintain the trajectory about the sub-L1 point.16 In contrast, Farr´es and Jorba exploit dynamical systems theory to design stationkeeping maneuvers with a solar sail to maintain orbits about a sub-L1 point and to move to other points on the equilibrium surface.17–19 Waters and McInnes linearize relative to fixed-points used to define a solar-sail libration-point orbit near the sub-L1 point, then produce an optimal control that delivers the trajectory to the selected fixed points.20 This investigation is focused on an examination of two strategies for control to a solar sail reference trajectory. In both schemes, a controller, based on submatrices of the state-transition matrix along segments of the trajectory arc, is used to correct the sail attitude profile, thus re-targeting the spacecraft to the reference trajectory or its vicinity. Analysis by Howell and Pernicka for station-keeping along halo orbits employed ∆v’s in a similar scheme.21 For this analysis, arc segments, are established at constant intervals (approximately 1 to 3 days). In the first approach, a least-squares formulation is employed to solve for the angles that best target multiple points along an arc segment. In the second approach, a collocationbased two-point boundary-value problem (TPBVP) solver delivers a sequence of three orientations along the arc segment such that the spacecraft trajectory matches six-dimensional the reference trajectory at the end points of each segment. Both control schemes assume that the sail attitude is inertially fixed along an arc segment and that the sail can be instantaneously re-oriented at the end of the segment. An ideal sailing spacecraft assumes no reactive mass, and control techniques that are based on only the two degrees of freedom representing sail orientation are beneficial for understanding the capabilities of a sailcraft. To explore the two control strategies in more detail, the dynamical regime is initially described. Then, the control techniques are developed, followed by results and a discussion of the techniques. SYSTEM MODEL The two control schemes examined here are applied to solar sail reference trajectories that address the lunar south pole (LSP) coverage problem,22–24 although IAC-10-C1.2.3

these controllers could be used for other solar sail applications as well. In the LSP coverage problem, a single spacecraft must be in continuous view of the LSP at all times. Because of lunar libration and surface topology, the elevation of the spacecraft is constrained to be at least 15◦ . Solar sails supply an additional force that enables a spacecraft orbit to be offset from a central body. It is advantageous to formulate the LSP problem within the context of the Earth–Moon circular restricted threebody (CR3B) system, accompanied by the sail’s SRP force; the Moon is tidally locked to the Earth, and a base on the Moon is essentially stationary in an Earth– Moon CR3B system. Ignoring gravitational perturbations originating with the Sun and other bodies is sufficient for initial modeling of a reference trajectory, but such perturbations can be included in higher fidelity models and, consequently, in expanded control schemes. Trajectories in the Earth–Moon CR3B system do take advantage of sail-modified Earth–Moon Lagrange points. Motion in these orbits is offset below the Moon and co-periodic with the Sun’s motion in this reference frame, that is, the orbital period is 29.5 days, just as the synodic period of the Sun about the Earth–Moon system is 29.5 days. The non-dimensional vector equation of motion for this idealized, yet representative, system are formulated in terms of a frame, R, rotating relative an inertial frame, I, that is, R a + 2 Iω R × Rv + ∇U (r) = as (t) (1) where the first term is the acceleration as observed in R 2 d r the rotating frame (more precisely expressed as dt 2 , where the left superscript R indicates a derivative relative to the rotating frame). The second term is the corresponding Coriolis acceleration, evaluated in terms of the velocity relative to the rotating frame, R dr v (more precisely dt ). The angular-velocity vector, I R ω , relates the orientation of the rotating frame to the inertial frame. The applied acceleration, from a solar sail in this case, is indicated on the right side by as (t). The pseudo-gravity gradient, ∇U (r), combines the centripetal and the gravitational accelerations µ (1 − µ) r + r ∇U (r) = Iω R × Iω R × r + (2) 1 2 r13 r23 where µ represents the mass fraction of the smaller body, or m2 /(m1 +m2 ), and r1 and r2 are the distances from the larger and smaller bodies, respectively, that is, p r1 = (µ + x)2 + y 2 + z 2 p r2 = (µ + x − 1)2 + y 2 + z 2 The system model is illustrated in Fig. 1. The Sun is assumed to be sufficiently far from the system such

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properties of the sail.26 Nevertheless, this analysis will employ an ideal sail to lend insight into the problem of controlling a sailcraft along a reference trajectory.

Fig. 1

Earth–Moon system model

that solar gravity is negligible and the rays of sunlight are parallel. The Sun moves clockwise about the fixed ˆ Earth and Moon, and the sunlight vector, `(t), is a function of time, and can be represented as ˆ = cos Ωtˆ `(t) x − sin Ωtˆ y + 0ˆ z

(3)

where t is non-dimensional time and Ω is the ratio of the sidereal period (27.3 days) to the synodic period and is approximately equal to 0.9252. The sail is modeled as a perfectly reflecting, flat plate.1 The performance of an ideal solar sail can be described by one parameter, the characteristic accleration, that is, the acceleration a sail can impart on a space vehicle at one AU from the Sun. With an ideal sail, the sail acceleration at 1 AU, as (t), in Eq. (1) is directed along a unit vector parallel to the sailface normal, u, and is a function of the characteristic acceleration of the sail and the sail attitude, that is 2 ac ˆ as (t) = ∗ `(t) · u u, or (4) a ac (5) = ∗ cos2 α u a where ac is the sail dimensional characteristic acceleration (in mm/s2 ), which is non-dimensionalized by 2 the system acceleration, a∗ (2.73 mm/s in the Earth– Moon system), and α is the sail cone angle, which is the angle between the incoming sunlight and the resultant acceleration due to the solar sail. Higher fidelity models include optical models,1 parametric models that incorporate billowing in addition to optical effects,1, 25 and realistic models based on finiteelement analysis that incorporates optical properties and manufacturing flaws.26 Optical effects can represent a non-perfectly reflecting solar sail; some energy is absorbed, and some is reflected diffusely as well as specularly. An ideal sail reflects only specularly. In all of these models, the resulting acceleration from a solar sail is not perfectly parallel to the sailface normal but, instead, is increasingly offset from the sailface normal as the sail is pitched further from the sunlight direction.1 Fully accounting for realistic solar sail properties attenuates the sail characteristic acceleration by nearly 25% and places an upper limit on the cone angle between 50◦ and 60◦ , depending on the IAC-10-C1.2.3

REFERENCE TRAJECTORIES In general, the more accurate a reference trajectory, the less control authority is required. Recent work by Ozimek et al.23 presents a collocation scheme that relies on a seventh-degree polynomial for local errors along the reference trajectory on the order of O(∆t12 ) to solve the LSP coverage problem with a single sailcraft. The scheme is applied in both a CR3B framework and a full ephemeris model. The results are sufficiently precise that a state can be extracted and explicitly propagated (using a Runge-Kutta or similar integrator) such that the propagated solution closely matches the collocation solution.24 The precision of a collocation scheme depends on the degree of its collocating polynomial;27 for example, a method based on Hermite-Simpson integration rules incorporates a third-degree polynomial and has local errors on the order of O(∆t5 ). Other techniques for generating reference solutions exist as well.28 In reality, because of the sensitivity of the regime, a reference trajectory extracted from a collocation scheme, and based on an ephemeris model, still requires a trajectory control strategy. Therefore, to develop and test trajectory control schemes, a less precise solution is sufficient and warranted. In this analysis, a finite-difference method (FDM) is employed to generate reference trajectories. Finitedifference methods possess local errors on the order of O(∆t2 ), which are sufficient to assume that the solutions are realistic and a sound control strategy is required to actually follow these solutions. The details of an FDM adapted to the LSP coverage problem are discussed in Wawrzyniak and Howell.22 It is presumed that if a controller maintains an actual trajectory to follow a reference path that is generated by an FDM in a CR3B regime, it may lend insight into designing a controller for a more realistic, highly accurate trajectory modeled in a higher-fidelity ephemeris regime or when other errors are incorporated as well. Three reference trajectories generated via finitedifference methods appear in Fig. 2. In the figure, the Moon appears to scale and the two Earth–Moon Lagrange points near the Moon are included for reference. The three orbits in Fig. 2 possess different trajectory and attitude profiles. The dark-blue and red orbits are centered under the Moon, while the light-blue orbit is centered near L2 . The sailcraft orbit includes a minimum elevation as viewed from the LSP of slightly more than 15◦ near the extreme y values along each orbit. The initial state of the spacecraft corresponding to each trajectory is defined to occur when the Sun is along the −x axis, per Eq. (3). At this time, the spacecraft is in opposition to the Sun with respect to

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i.e., kIω R k = 1, Ω is the ratio of the sidereal to synodic periods, and time, t, remains nondimensional such that the Earth–Moon system period is 2π. The sail pointing vector is also expanded in terms of inertially fixed unit vectors using spherical coordinates, that is, u = cos φ cos θˆ ı1 + cos φ sin θˆ ı2 + sin φˆ ı3

(8)

where φ and θ are latitude and longitude angles, respectively, defined in terms of the fixed inertial frame as illustrated in Fig. 3. These angle histories for the

a) 3-D view

Fig. 3 Sail pointing vector in terms of inertially defined latitude and longitude angles.

three reference orbits in Fig. 2 are plotted in Fig. 4. Note that θ increases by 29.1205◦ degrees over one peb) xz view Fig. 2

Reference trajectories below the Moon.

the Moon (along the +x axis); in the case of the lightblue orbit below L2 , the spacecraft is on the far side of the orbit relative to the Moon. The arrows along the trajectories represent the direction of the sailface normal, u, at a given epoch and are generally in the ˆ same direction as the sunlight vector, `(t). Since the attitude is to remain constant over a specified time interval, the orientation of the spacecraft is best represented as a set of angles measured with respect to an inertially fixed frame. Recall that the acceleration contributed by the sail is evaluated as ac as = ∗ (`ˆ · u)2 u (6) a In terms of coordinates defined for the inertial frame, ˆ ı, the sunlight vector can be written as `ˆ = cos(1 − Ω)tˆ ı1 + sin(1 − Ω)ˆ ı2 + 0ˆ ı3

(7)

where the normalized rotation rate of the Earth–Moon frame with respect to the inertial frame is equal to one, IAC-10-C1.2.3

Fig. 4 Inertially defined latitude, φ, and longitude, θ, angles for the three reference orbits.

riod for each orbit, consistent with the motion of the Earth–Moon system about the Sun over one month. Of the attitude profiles appearing in Fig. 4, the darkblue orbit exhibits the least complexity in that the φ

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angle appears to be generally sinusoidal and the θ angle appears to be generally sinusoidal (plus a secular term) as compared to the angle evolution corresponding to the other two profiles. The regularity of the dark-blue angle profile indicates that continuous turn rates and rotational accelerations of the sail itself are likely to be small. As the turns are to be discretized as part of the control schemes, the turns associated with the dark-blue orbit are likely to be small between epochs, relative to the other two orbits. The red and light-blue orbits require more complex attitude profiles as part of the baseline sailcraft trajectory, in part, to maintain the elevation constraint on the trajectory near the respective extreme values in the y direction. Additionally, these attitude profiles result in large cone angles (α = cos−1 (`ˆ · u)), which may not be allowed under a more realistic sail model. Each trajectory in Fig. 2 requires a characteristic 2 acceleration of 1.70 mm/s . A recent survey of the design space concludes that a characteristic acceleration of approximately 1.5 to 1.7 mm/s2 is required for an ideal sail to consistently maintain the 15◦ elevation constraint in the LSP coverage problem.29 Only one spacecraft to date, IKAROS, has achieved orbit and employed a solar sail as its sole source of propulsion to date.11 However, a sailcraft that possesses a net char2 acteristic acceleration of 0.58 mm/s was designed and ground tested by L’Garde for the New Millennium Program’s Space Technology 9 competition at NASA;30 the characteristic acceleration of the L’Garde sail and 2 structure alone is 1.70 mm/s . In reality, solar sailing technology will likely evolve to complement some other type of propulsive device.31 Nevertheless, if the LSP coverage problem is addressed solely with a sailcraft, some sail technology advancements are required to deliver an appropriate characteristic acceleration. CONTROL SCHEMES If a state from each of the reference trajectories in Fig. 2 is used to initialize an explicit propagation algorithm, such as a classical Runge-Kutta approach, along with the control histories that result from the generating numerical technique, it is not surprising that the simulated path diverges from the reference prior to introducing any errors. Spacecraft state knowledge from statistical orbit determination techniques, maneuver executions from the attitude control system, and an understanding of the dynamical environment are all imperfect and possess errors. Therefore, trajectory control schemes are required for the spacecraft to follow any reference path. Two trajectory control schemes are examined; both assume that the sailcraft can instantaneously turn from one orientation to another. Similar “turn-andhold” schemes are employed by Yen13 and Sauer14 to control sailcraft to sub-L1 points. In Sauer’s formulation, the sail angles represent two degrees of freedom IAC-10-C1.2.3

and the length of time spent at a particular attitude supplies the third option in a control scheme that targets the sub-L1 point. In the current problem, it is desired to control to a specified reference trajectory, not a point. However, a similar turn-and-hold strategy is developed and implemented in this CR3B model. For the turn-and-hold schemes in this investigation, the trajectory is decomposed into segments and the latitude, φ, and longitude, θ, angles are one or more discrete inertial values over each segment i (as a consequence, the angles are time-varying along the entire arc segment in the Earth–Moon frame). For convenience, φi ≡ φ(ti ) and θi ≡ θ(ti ). Because the reference path from the FDM solution is represented by position and velocity states, as well as the required attitude, at discrete time intervals, target states at epochs within the reported intervals are interpolated via an Akima cubic-spline. The initial guesses for the latitude, φi , and the longitude, θi are the average values of the continuous values of φ and θ over arc segment i from the reference solution. The first implementation of the turn-and-hold controller employs a least-squares fit such that the optimal values of φ and θ are selected to minimize the difference between the propagated path and the reference path. The second implementation employs a collocation-based two-point boundary-value problem (TPBVP) solver that delivers three turns within a segment such that the positions and velocities at the boundaries of the segment match the reference trajectory. Path constraints (e.g., elevation) are not incorporated into the controllers since it is assumed that tracking a reference trajectory, one designed with these restrictions, obviates the need to further constrain the behavior of the controller. The control algorithms in this analysis are limited controllers in the sense that they do not currently compensate for errors from orbit determination or control execution. A study of such error sources is beyond the scope of the present investigation. Rather, as an initial step, it is necessary to develop a control strategy that can re-target a sailcraft to a path that varies with time. It is presumed that errors arising from inaccuracies in the reference path and deviations from the targeted states at the end of a trajectory segment are sufficient to develop insight and initiate a controller design. Least-Squares Implementation

The least-squares method is a classic approach for over-determined, or over-constrained, systems by minimizing the sum of the squares of the residuals between some observed value and its modeled value. With two degrees of freedom for the control (φ and θ at the beginning of the segment) and at least six constraints along the segment (positions and velocities at future times), a sail cannot supply sufficient control with a single, fixed attitude over the entire the segment to

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target a future state. However, by employing a leastsquares solution, the sailcraft can get close to its target state at some future time along the path. Given an initial guess for φi and θi , a trajectory is propagated along with a state-transition matrix, Φ(ti+1 , ti ), from time ti to ti+1 . The state-transition matrix maps deviations in φi and θi to deviations in the state vector at the end of the segment. This information is reflected in the following relationship,     δxi+1   δφi .. = K (9) 6×2 . δθi 2×1     δ z˙i+1 6×1 where   K= 

∂xi+1 ∂φi

.. .

∂ z˙i+1 ∂φi

∂xi+1 ∂θi

.. .

∂ z˙i+1 ∂θi

   

(10)

i+1 is the element of Φ(ti+1 , ti ) that maps a and ∂x ∂φi deviation in φi to a deviation in xi+1 . The size of each matrix or vector is indicated by a subscript. Because the formulation in Eq. (9) does not directly target acceleration, it is sometimes advantageous to target multiple positions and velocities at future times (i.e., ti+1/n , ti+2/n , . . . , ti+n/n , where a fractional subscript indicates the time of a turn and the intervals need not be evenly spaced). With a fixed attitude at ti , acceleration is implicitly targeted at ti+n/n , where ti+n/n is equivalent to ti+1 and n is the number of future states in the expanded segment, as illustrated in Fig. 5. In the figure, the reference trajectory is repre-

for an extended set of segments. The relationships in Eq. (9) are now written as     δxi+1/n   δφi .. = K (11) 6n×2 . δθi 2×1     δ z˙i+n/n 6n×1 where

  K= 

∂xi+1/n ∂φi

.. .

∂ z˙i+n/n ∂φi

∂xi+1/n ∂θi

.. .

∂ z˙i+n/n ∂θi

   

(12)

T

As mentioned, a solution to Eq. (11) for {δφi δθi } arises from a weighted least-squares implementation, that is,     δxi+1/n   δφi .. (13) = (KT WK)−1 KT W . δθi     δ z˙i+n/n where W is a diagonal weighting matrix that balances the corrections between the position and velocity components. The solution from Eq. (13) is used to update φi and θi , whereby a new path is simulated and compared to the reference trajectory. The process is T iterated until the values of {δφi δθi } longer change, to within some tolerance. Because the correction is based on a least-squares solution, the elements on the left sides of either Eqs. (9) or (11) (or, alternatively, within the braces on the right side in Eq. (13)) may never converge to zero, but remain close to the reference values. Selection of n and the length of time between ti and ti+1 affects the resulting controlled trajectory. Three-Turn Two-Point Boundary Value Problem

Fig. 5 A “turn-and-hold” segment (red) fit through n target points along a reference path (black). The attitude is held from ti to ti+n/n .

sented with a black curve and the path resulting from a simulation incorporating the initial guesses for φi and θi is in red. The residuals between the simulated trajectory and the reference path at the future epochs are indicated with gray arrows. No residual exists at ti because the controller cannot compensate for initial errors in the path, only future errors. A best-fit solution for φi and θi that is determined by incorporating these extra, intermediate constraints may then more closely track the reference trajectory at ti+1 (or ti+n/n ) and IAC-10-C1.2.3

The disadvantage in the previous approach involves the fact that the number of boundary conditions (12) may exceed the number of equations of motion (6) plus controls (2). A possible solution is the inclusion of two more turns within a segment, such that the number of controls is 6 and the two-point boundary value problem (TPBVP) is well-posed. In this case, turns implemented at ti:1/3 , ti:2/3 , and ti:3/3 , where a fractional subscript indicates turn one, two, or three of the three total turns in an arc segment, supply sufficient control such that the positions and velocities along the compensated trajectory match the values on the reference trajectory at ti and ti+1 , as illustrated in Fig. 6. An arc segment along the reference trajectory, which supplies an initial guess for the TPBVP solver is represented in black, while the arc segment of the path that is associated with the three turns that solve the equations of motion appears in red in the figure. A variety of numerical algorithms are available to solve boundary value problems. A popular family of algorithms are based on collocation methods. Suppose

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Fig. 6 A three-turn solution to target position and velocity at ti+1 given a position and velocity at ti .

that, for a one-dimensional case, x(t) ˙ = f (x(t)), where f (x(t)) is an ordinary differential equation, or an equation of motion, and x(t) solves the equation of motion. Also suppose that Sk (t) is an N th -degree polynomial that approximates x(t) in the interval between mesh points at τk and τk+1 . If, at a time τa ∈ [τk , τk+1 ], Sk (τa ) is equivalent to x(τa ), then the approximating polynomial is said to be collocated to the solution of the equation of motion at τa . As an example of a collocation method, a thirddegree polynomial (in red) based on state and derivative information at the mesh points (black) at τk and τk+1 appears in Fig. 7.27 For the example in the

Fig. 7

Third-degree collocation

figure, only states and derivative information from their associated equations of motion are required at the boundaries to generate a third-degree polynomial. However, internal points between the mesh points may be required, depending on the degree of the collocating polynomial. At one or more epochs (τa in the figure) between τk and τk+1 , the derivative of the polynomial (red in the figure) is compared to the derivative from the ordinary differential equation (green). The residual between these two values is a defect (∆a in the figure). Generally, x(τk ) and x(τk+1 ) are not precisely known initially, and any defect will be non-zero. Therefore, values for x(τk ) and x(τk+1 ) are updated through an iterative process, thereby changing the interpolating polynomial and derivative information from the equations of motion, until the defects are zero. To begin the collocation process, the arc within the IAC-10-C1.2.3

time block ti and ti+1 is sampled into m mesh points. An initial mesh at node points τ1 , . . . , τk , . . . , τm appears in Fig. 6. Defects from all of the intervals in [ti , ti+1 ] are used to update the states at the collocation points (x(τk ), x(τk+1 ), etc.) such that the states at the boundaries of the arc (x(ti ) and x(ti+1 )) are consistent with the equation of motion and solve the TPBVP. R supplies a suite of funcConveniently, MATLAB tions for solving TPBVPs based on collocation schemes: BVP4C (most similar to the example illustrated by Fig. 7),32 BVP5C,32 and BVP6C.33 These three algorithms differ in the degree of their interpolating polynomial and the resulting accuracy. Both BVP4C and BVP5C are based on three- or four-stage Lobatto IIIa integration rules, respectively, and are fourth- or fifth-order accurate uniformly between τk and τk+1 , respectively.32 For BVP6C, a quintic interpolant is fit to a mono-implicit Runge-Kutta sixthorder formula for sixth-order uniform accuracy between τk and τk+1 .33 All three methods employ mesh refinement to minimize the error along the arc such that the sub-arcs need not be uniformly spaced; node points may be added or eliminated as well. Implementing any one of the three collocation functions requires an initial guess for the trajectory; the discretized reference path is a convenient choice. Also required is an initial guess for the set of controls, that is, some initial sail angle history. In the BVP4C, BVP5C, and BVP6C implementations, these sail angles are collected together as six unknown parameters to be estimated by the algorithm. The attitude of the sailcraft is presumed to change at ti:1/3 , ti:2/3 , ti:3/3 (from Fig. 6), and this set of 3 × 2 angles represent the controls for the TPBVP. The input and output structures for each of the three BVP solvers is identical, making it simple for the user to switch between methods. For this analysis, BVP6C is employed since this algorithm possesses the best theoretical accuracy of the three methods. Fewer mesh points to represent the initial guess corresponding to the trajectory segment are required, and convergence is typically faster. Nevertheless, a large number (m ≈ 151) of mesh points within the arc bounded by [ti , ti+1 ] are initially required, in practice, for convergence and to maintain the solution path near to the respective initial guesses for this problem. RESULTS Both the least-squares implementation and the collocation-based TPBVP strategy (BVP6C) are successful in controlling the sailcraft to the three different reference trajectories appearing in Fig. 2, within specified tolerances. The turn-and-hold methods are essentially linear corrections to a non-linear problem, and each orbit possesses different sensitivities. As a consequence, slight differences in the implementation of the

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two control schemes exist for each reference orbit. Least-Squares Implementation

Each of the three sample orbits is nominally periodic, thus, the success of the over-constrained, or under-controlled, least-squares implementation is measured by a simple evaluation of the controlled solution and the ability to follow the reference path for multiple revolutions. A controller that targets multiple states with only two controls will rarely, if ever, produce a trajectory that coincides with a reference path at the time of the scheduled turns. However, if the controlled trajectory is sufficiently close to the reference path at the points when the angles are changed, it is likely that the controlled solution will follow the reference path. Controlled paths for three periods of the three reference trajectories appear in Fig. 8. The dark-blue Fig. 9 Latitude, φ, and longitude, θ, angles from the least-squares formulation of the turn-and-hold controller over the first revolution.

tude changes is apparent. Future investigations should consider the ability of the spacecraft attitude control system to perform turns in finite amounts of time. Three-Turn TPBVP

Fig. 8 Three revolutions of orbits controlled with the least-squares algorithm.

path is generated by scheduling a turn every two solar days using six target points along a two-day segment of the reference trajectory. The light-blue path requires a turn every solar day and is targeted to two points along the one-day segment. After three revolutions, the dark- and light-blue solutions diverge from the reference trajectory with these targeting schemes. The red trajectory in Fig. 8 is generated with turns every two solar days and is targeted to three points along the segment. This trajectory tracks its reference path for four-and-a-half revolutions before diverging. Note that the time between turns and the number of target points is fixed for each orbit. Allowing the time between turns and the number of target points within a segment to fluctuate may result in longer tracking times, especially since the dynamical sensitivities vary throughout each orbit. The control profiles for the three orbits from Fig. 8 appear in Fig. 9. Note that the plot for θ includes a secular component because of the motion of the Earth–Moon system in inertial space. A trade-off between longer hold times and larger attiIAC-10-C1.2.3

As designed, the orbit from the BVP6C algorithm matches the boundary conditions exactly. However, the path interior to ti and ti+1 is only accurate to within specified tolerances. The tolerances and the initial meshes for the BVP6C algorithm must be “tuned” so that the algorithm converges on a solution to within the specified tolerances. A 151-point initial mesh, with absolute and relative tolerances of 0.001 and one turn per day over a three-day segment result in orbits that meet the above convergence requirements for the darkand light-blue trajectories. When the absolute and relative tolerances are tightened, the BVP6C algorithm reports that it “cannot converge without exceeding the maximum number of allowable mesh points.” With the exception of a four-day segment, the conditions leading to convergence are the same for the red trajectory. All three trajectories from the collocation-based TPBVP control scheme appear in Fig. 10 and the respective attitude profiles appear in Fig. 11. Because of the loose convergence tolerances, incorporating the turns resulting from the BVP6C algorithm into an explicit integration scheme results in position and velocity at the end of the first segment that is not consistent with the boundary conditions established for the TPBVP algorithm. Not surprisingly, employing this new state as a boundary condition in the BVP6C algorithm for the subsequent segment leads to divergence; the solution is highly sensitive to the initial guess for the path. However, while the success of this collocation-based implementation of the TPBVP controller is subjective at best (due to the errors in the solution), another implementation of a

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a continuously re-orienting spacecraft. All controllers should be designed within the limits of sailcraft body rates and accelerations. ACKNOWLEDGEMENTS The first author gratefully acknowledges the National Aeronautics and Space Administration’s (NASA) Office of Education for their sponsorship of his attendance at the 61st International Astronautical Congress in Prague, Czech Republic. Portions of this work were also supported by Purdue University. REFERENCES 1 C.

Fig. 10 One revolution of the orbits controlled with the collocation algorithm.

Fig. 11 Latitude, φ, and longitude, θ, angles over one revolution from the collocation formulation of the turn-and-hold controller.

three-turn TPBVP controller may produce a more favorable result. CONCLUSIONS AND FUTURE WORK In this investigation, the authors have developed a rudimentary pair of trajectory control schemes that employ a turn-and-hold strategy for a solar sail spacecraft. The least-squares implementation can be improved by varying the length of hold times and target points, while the three-turn TPBVP scheme may require a different TPBVP solver. In addition to other implementations of TPBVP solvers, future efforts will incorporate finite turn times and spacecraft attitude control capabilities. Additionally, a controller should be able to compensate for errors in the orbit-determination knowledge and turn execution. The control schemes to date focus on discrete orientations. Future research will also examine IAC-10-C1.2.3

R. McInnes, Solar Sailing: Technology, Dynamics and Mission Applications. Space Science and Technology, New York: Springer-Praxis, 1999. 2 D. L. Shirley, “The Mariner 10 mission to Venus and Mercury,” Acta Astronautica, vol. 53, pp. 375–385, August– November 2003. 3 D. J. O’Shaughnessy, J. V. McAdams, K. E. Williams, and B. R. Page, “Fire Sail: MESSENGER’s use of solar radiation pressure for accurate Mercury flybys,” in Advances in the Astronautical Sciences, vol. 133, pp. 61–76, 2009. Paper AAS 09-014. 4 G. Colombo, “The stabilization of an artificial satellite at the inferior conjunction point of the Earth–Moon system,” Tech. Rep. 80, Smithsonian Astrophysical Observatory, November 1961. 5 R. H. Laprade, J. A. Miller, and S. J. Worley, “Satellite stationkeeping by solar radiation pressure,” Space/Aeronautics, vol. 47, pp. 114–117, April 1967. 6 W. L. Black, M. C. Crocker, and E. H. Swenson, “Stationkeeping a 24-hour satellite using solar radiation pressure,” Journal of Spacecraft, vol. 5, pp. 335–337, March 1968. 7 R. W. Farquhar, “The control and use of libration point satellites,” Tech. Rep. TR-R-346, National Aeronautics and Space Administration, February 1970. 8 M. C. Crocker, “Attitude control of a sun-pointing spinning spacecraft by means of solar radiation pressure,” Journal of Spacecraft and Rockets, vol. 7, pp. 757–759, March 1970. 9 V. J. Modi and K. Kumar, “Attitude control of satellites using the solar radiation pressure,” Journal of Spacecraft and Rockets, vol. 9, pp. 711–713, September 1972. 10 B. W. Stuck, “Solar pressure three-axis attitude control,” Journal of Guidance and Control, vol. 3, pp. 132–139, February 1980. 11 O. Mori et al., “Worlds first demonstration of solar power sailing by IKAROS,” in 2nd International Symposium on Solar Sailing, New York City College of Technology, City University of New York, (Brooklyn, New York), July 2010. 12 R. L. Young, “Updated Heliostorm warning mission: Enhancements based on new technology,” in 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, (Honolulu, Hawaii), April 2007. Paper AIAA-2007-2249. 13 C. L. Yen, “Solar sail geostorm warning mission design,” in Advances in the Astronautical Sciences, vol. 119, pp. 69–82, February 2004. Paper AAS 04-107. 14 C. G. Sauer, “The L1 diamond affair,” in Advances in the Astronautical Sciences, vol. 119, pp. 2791–2808, February 2004. Paper AAS 04-278. 15 M. Lisano, D. Lawrence, and S. Piggott, “Solar sail transfer trajectory design and stationkeeping control for missions to the sub-L1 equilibrium region,” in Advances in the Astronautical Sciences, vol. 120, pp. 1837–1854, 2005. 16 D. Lawrence and S. Piggott, “Solar sailing trajectory control for sub-L1 stationkeeping,” in AIAA Guidance, Navigation, and Control Conference and Exhibit, (Providence, Rhode Island), August 2004. Paper AIAA-2004-5014.

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17 A. Farr´ ` Jorba, “Station keeping close to unstable es and A. equilibrium points with a solar sail,” in AAS/AIAA Astrodynamics Specialist Conference, (Mackinac Island, Michigan), August 2007. Paper AAS 07-347. 18 A. Farr´ ` Jorba, “Solar sail surfing along families es and A. of equilibrium points,” Acta Astronautica, vol. 63, pp. 249–257, July–August 2008. 19 A. Farr´ ` Jorba, “Dynamical system approach for es and A. the station keeping of a solar sail,” Journal of the Astronautical Sciences, vol. 58, pp. 199–230, April–June 2008. 20 T. J. Waters and C. R. McInnes, “Invariant manifolds and orbit control in the solar sail three-body problem,” Journal of Guidance, Control, and Dynamics, vol. 31, pp. 554–562, May– June 2008. 21 K. C. Howell and H. J. Pernicka, “Stationkeeping method for libration point trajectories,” Journal of Guidance, Control, and Dynamics, vol. 16, no. 1, pp. 151–159, 1993. 22 G. G. Wawrzyniak and K. C. Howell, “Accessing the design space for solar sails in the Earth–Moon system,” in AAS/AIAA Astrodynamics Specialists Conference, (Pittsburgh, Pennsylvania), August 2009. Paper AAS 09-348. 23 M. T. Ozimek, D. J. Grebow, and K. C. Howell, “Design of solar sail trajectories with applications to lunar south pole coverage,” Journal of Guidance, Control, and Dynamics, vol. 32, pp. 1884–1897, November–December 2009. 24 M. T. Ozimek, D. J. Grebow, and K. C. Howell, “A collocation approach for computing solar sail lunar pole-sitter orbits,” in AAS/AIAA Astrodynamics Specialists Conference, (Pittsburgh, Pennsylvania), August 2009. Paper AAS 09-378. 25 L. Rios-Reyes and D. J. Scheeres, “Solar-sail navigation: Estimation of force, moments, and optical parameters,” Journal of Guidance, Control, and Dynamics, vol. 30, pp. 660–668, May–June 2007. 26 B. Campbell, An Analysis of Thrust of a Realistic Solar Sail with Focus on a Flight Validation Mission in a Geocentric Orbit. Washington, DC: Ph.D. Thesis, The George Washington University, May 2010. 27 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, pp. 592–599, May–June 1996. 28 G. G. Wawrzyniak and K. C. Howell, “Numerical methods to generate solar sail trajectories,” in 2nd International Symposium on Solar Sailing, New York City College of Technology, City University of New York, (Brooklyn, New York), pp. 195– 200, July 2010. 29 G. G. Wawrzyniak and K. C. Howell, “Investigating the design space for solar sail trajectories in the Earth–Moon system using augmented finite-difference methods,” in preparation, 2010. 30 D. Lichodziejewski, B. Derbes, D. Sleight, and T. Mann, “Vacuum deployment and testing of a 20m solar sail system,” in 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, (Newport, Rhode Island), May 2006. Paper AIAA-2006-1705. 31 M. Macdonald and C. McInnes, “Solar sail mission applications and future advancement,” in 2nd International Symposium on Solar Sailing, New York City College of Technology, City University of New York, (Brooklyn, New York), pp. 1–26, July 2010. 32 The MathWorksTM , Inc., MATLAB , R Version 7.9.0 (R2009b). August 2009. www.mathworks.com. 33 N. Hale and D. R. Moore, “A sixth-order extension to the MATLAB package bvp4c of J. Kierzenka and L. Shampine,” Tech. Rep. 08/04, Oxford University Computing Laboratory, Numerical Analysis Group, Oxford, April 2008.

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