Iac-10-C1.2.3 Trajectory Control for a Solar Sail Spacecraft in an Offset Lunar Orbit

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 re-targeted toward a specific destination. Spacecraft trajectory design typically begins with a In addition to gravity, multiple small forces act on reference path, whether the design is for a low-Earth a spacecraft in flight. Mission designers typically in- orbiter, a mission to Mars, or an orbit near a La- corporate solar radiation pressure (SRP) as a pertur- grange point. As the mission design process proceeds, bation. For a solar sail, SRP is exploited as a primary the reference trajectory is refined in system models of means of propulsion and, thus, may significantly affect higher fidelity that include planetary ephemerides and the trajectory, enabling new concepts in trajectory de- more accurate representations of spacecraft character- sign. Solar sailing dates back to Tsiolkovsky, Tsander, istics (e.g., reflectivity properties for solar radiation and Oberth in the 1920s.1 If the definition of a so- pressure modeling). Frequently, the reference path lar sail includes any spacecraft that exploits SRP, an includes scheduled, deterministic maneuvers to shift early “sail” mission involved the Mariner 10 space- from one natural arc to another. However, a model is craft that flew by Mercury three times in 1974–75 never perfect. Maneuvers (e.g., insertion maneuvers), and exploited SRP for attitude control.2 More re- ephemerides and the space environment, the behavior cently, MESSENGER employed SRP for trajectory of the spacecraft (e.g., gas leaks, reflectivity), as well control during the 2008–09 Mercury flybys and for an- as the control devices, each introduce uncertainty and gular momentum management.3 These two missions error. “sailed” using their solar panels and not with a highly To maintain a desired orbit or to transfer to an reflective, lightweight sheet of large dimensions con- arc en route to a specified target requires strategies ventionally defined as a solar sail. However, small sails for scheduled station-keeping maneuvers (SMKs) or attached to traditional spacecraft have been proposed trajectory correction maneuvers (TCMs), respectively. for attitude and trajectory control4–10 over the last 50 The processes depend upon precise orbit estimation, years. Only recently has a spacecraft flown with a so- that is, assessing the spacecraft’s current path in rela- lar sail as its only means of propulsion. In the summer tion to its nominal path, and then supplying an input of 2010, the Japanese Space Agency, JAXA, launched for the design of an appropriate maneuver to return a solar sail spacecraft named IKAROS in tandem with the spacecraft to the desired path (SKM) or to cor- another mission to Venus. The sailcraft is the first rect the current trajectory such that the spacecraft is in-flight demonstration of solar sailing.11 One of the more frequently suggested missions that ∗PhD Candidate, School of Aeronautics and Astronautics †Hsu Lo Professor, School of Aeronautics and Astronautics would employ a large solar sails as the primary propulsion device is the Heliostorm (a.k.a. Geostorm) Warn-

IAC-10-C1.2.3 1 of 10 Page 1 of 10 61st International Astronautical Congress, Prague, CZ. Copyright c 2010 by the International Astronautical Federation. All rights reserved. ing Mission. In this mission, a sailcraft is placed at these controllers could be used for other solar sail ap- (or is made to orbit) an artificial libration point lo- plications as well. In the LSP coverage problem, a sin- cated approximately at 0.98 AU, or nearly twice the gle spacecraft must be in continuous view of the LSP distance from the Earth to the natural Earth–Sun L1 at all times. Because of lunar libration and surface point. A spacecraft closer to the Sun would provide topology, the elevation of the spacecraft is constrained advance warning of impending solar storms (i.e., coro- to be at least 15◦. nal mass ejections) that can disrupt electronic systems Solar sails supply an additional force that enables a on Earth.12 Both Yen13 and Sauer14 suggest control- spacecraft orbit to be offset from a central body. It is ling the spacecraft to the sub-L1 point as opposed to advantageous to formulate the LSP problem within the an orbit near that point as suggested by Lisano et context of the Earth–Moon circular restricted three- al.15 Lawrence and Piggott compare a linear quadratic body (CR3B) system, accompanied by the sail’s SRP control scheme and a Gramian controller, both in- force; the Moon is tidally locked to the Earth, and a corporating a sail to maintain the trajectory about base on the Moon is essentially stationary in an Earth– 16 the sub-L1 point. In contrast, Farrés and Jorba Moon CR3B system. Ignoring gravitational pertur- exploit dynamical systems theory to design station- bations originating with the Sun and other bodies is keeping maneuvers with a solar sail to maintain orbits sufficient for initial modeling of a reference trajectory, about a sub-L1 point and to move to other points on but such perturbations can be included in higher fi- the equilibrium surface.17–19 Waters and McInnes lin- delity models and, consequently, in expanded control earize relative to fixed-points used to define a solar-sail schemes. Trajectories in the Earth–Moon CR3B sys- libration-point orbit near the sub-L1 point, then pro- tem do take advantage of sail-modified Earth–Moon duce an optimal control that delivers the trajectory to Lagrange points. Motion in these orbits is offset be- the selected fixed points.20 low the Moon and co-periodic with the Sun’s motion This investigation is focused on an examination of in this reference frame, that is, the orbital period is two strategies for control to a solar sail reference tra- 29.5 days, just as the synodic period of the Sun about jectory. In both schemes, a controller, based on sub- the Earth–Moon system is 29.5 days. matrices of the state-transition matrix along segments The non-dimensional vector equation of motion for of the trajectory arc, is used to correct the sail at- this idealized, yet representative, system are formu- titude profile, thus re-targeting the spacecraft to the lated in terms of a frame, R, rotating relative an reference trajectory or its vicinity. Analysis by How- inertial frame, I, that is, ell and Pernicka for station-keeping along halo orbits R I R R employed ∆v’s in a similar scheme.21 For this analy- a + 2 ω × v + ∇U(r) = as(t) (1) sis, arc segments, are established at constant intervals where the first term is the acceleration as observed in (approximately 1 to 3 days). In the first approach, R 2 the rotating frame (more precisely expressed as d r , a least-squares formulation is employed to solve for dt2 the angles that best target multiple points along an where the left superscript R indicates a derivative arc segment. In the second approach, a collocation- relative to the rotating frame). The second term is based two-point boundary-value problem (TPBVP) the corresponding Coriolis acceleration, evaluated in solver delivers a sequence of three orientations along terms of the velocity relative to the rotating frame, Rdr the arc segment such that the spacecraft trajectory v (more precisely dt ). The angular-velocity vector, I R matches six-dimensional the reference trajectory at the ω , relates the orientation of the rotating frame to end points of each segment. Both control schemes as- the inertial frame. The applied acceleration, from a sume that the sail attitude is inertially fixed along an solar sail in this case, is indicated on the right side by arc segment and that the sail can be instantaneously as(t). The pseudo-gravity gradient, ∇U(r), combines re-oriented at the end of the segment. the centripetal and the gravitational accelerations An ideal sailing spacecraft assumes no reactive mass, (1 − µ) µ and control techniques that are based on only the two ∇U(r) = IωR ×IωR × r+ r + r (2) r3 1 r3 2 degrees of freedom representing sail orientation are 1 2 beneficial for understanding the capabilities of a sail- where µ represents the mass fraction of the smaller craft. To explore the two control strategies in more de- body, or m2/(m1+m2), and r1 and r2 are the distances tail, the dynamical regime is initially described. Then, from the larger and smaller bodies, respectively, that the control techniques are developed, followed by re- is, sults and a discussion of the techniques. p 2 2 2 r1 = (µ + x) + y + z SYSTEM MODEL p 2 2 2 r2 = (µ + x − 1) + y + z The two control schemes examined here are applied to solar sail reference trajectories that address the lu- The system model is illustrated in Fig. 1. The Sun nar south pole (LSP) coverage problem,22–24 although is assumed to be sufficiently far from the system such

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.

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 Fig. 1 Earth–Moon system model O(∆t12) to solve the LSP coverage problem with a sin- that solar gravity is negligible and the rays of sunlight gle sailcraft. The scheme is applied in both a CR3B are parallel. The Sun moves clockwise about the fixed framework and a full ephemeris model. The results are Earth and Moon, and the sunlight vector, `ˆ(t), is a sufficiently precise that a state can be extracted and function of time, and can be represented as explicitly propagated (using a Runge-Kutta or similar integrator) such that the propagated solution closely 24 `ˆ(t) = cos Ωtxˆ − sin Ωtyˆ + 0ˆz (3) matches the collocation solution. The precision of a collocation scheme depends on the degree of its col- where t is non-dimensional time and Ω is the ratio of locating polynomial;27 for example, a method based the sidereal period (27.3 days) to the synodic period on Hermite-Simpson integration rules incorporates a and is approximately equal to 0.9252. third-degree polynomial and has local errors on the The sail is modeled as a perfectly reflecting, flat order of O(∆t5). Other techniques for generating ref- plate.1 The performance of an ideal solar sail can be erence solutions exist as well.28 In reality, because described by one parameter, the characteristic accler- of the sensitivity of the regime, a reference trajec- ation, that is, the acceleration a sail can impart on a tory extracted from a collocation scheme, and based space vehicle at one AU from the Sun. With an ideal on an ephemeris model, still requires a trajectory con- sail, the sail acceleration at 1 AU, as(t), in Eq. (1) trol strategy. Therefore, to develop and test trajectory is directed along a unit vector parallel to the sailface control schemes, a less precise solution is sufficient and normal, u, and is a function of the characteristic ac- warranted. celeration of the sail and the sail attitude, that is In this analysis, a finite-difference method (FDM) is employed to generate reference trajectories. Finite- a 2 c ˆ difference methods possess local errors on the order as(t) = ∗ `(t) · u u, or (4) a of O(∆t2), which are sufficient to assume that the so- ac = cos2 α u (5) lutions are realistic and a sound control strategy is a∗ required to actually follow these solutions. The details where ac is the sail dimensional characteristic accel- of an FDM adapted to the LSP coverage problem are eration (in mm/s2), which is non-dimensionalized by discussed in Wawrzyniak and Howell.22 It is presumed the system acceleration, a∗ (2.73 mm/s2 in the Earth– that if a controller maintains an actual trajectory to Moon system), and α is the sail cone angle, which is follow a reference path that is generated by an FDM the angle between the incoming sunlight and the re- in a CR3B regime, it may lend insight into designing a sultant acceleration due to the solar sail. controller for a more realistic, highly accurate trajec- Higher fidelity models include optical models,1 para- tory modeled in a higher-fidelity ephemeris regime or metric models that incorporate billowing in addition to when other errors are incorporated as well. optical effects,1, 25 and realistic models based on finite- Three reference trajectories generated via finite- element analysis that incorporates optical properties difference methods appear in Fig. 2. In the figure, the and manufacturing flaws.26 Optical effects can repre- Moon appears to scale and the two Earth–Moon La- sent a non-perfectly reflecting solar sail; some energy grange points near the Moon are included for reference. is absorbed, and some is reflected diffusely as well as The three orbits in Fig. 2 possess different trajectory specularly. An ideal sail reflects only specularly. In and attitude profiles. The dark-blue and red orbits all of these models, the resulting acceleration from a are centered under the Moon, while the light-blue or- solar sail is not perfectly parallel to the sailface nor- bit is centered near L2. The sailcraft orbit includes a mal but, instead, is increasingly offset from the sailface minimum elevation as viewed from the LSP of slightly normal as the sail is pitched further from the sun- more than 15◦ near the extreme y values along each or- light direction.1 Fully accounting for realistic solar bit. The initial state of the spacecraft corresponding sail properties attenuates the sail characteristic accel- to each trajectory is defined to occur when the Sun eration by nearly 25% and places an upper limit on is along the −x axis, per Eq. (3). At this time, the the cone angle between 50◦ and 60◦, depending on the spacecraft is in opposition to the Sun with respect to

i.e., kIωRk = 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 θˆı1 + cos φ sin θˆı2 + sin φˆı3 (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 pe- b) xz view

Fig. 2 Reference trajectories below the Moon. the Moon (along the +x axis); in the case of the light- blue 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 same direction as the sunlight vector, `ˆ(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 a a = c (`ˆ· u)2u (6) s a∗ In terms of coordinates defined for the inertial frame, Fig. 4 Inertially defined latitude, φ, and longitude, θ, angles for the three reference orbits. ˆı, the sunlight vector can be written as riod for each orbit, consistent with the motion of the `ˆ= cos(1 − Ω)tˆı + sin(1 − Ω)ˆı + 0ˆı (7) 1 2 3 Earth–Moon system about the Sun over one month. where the normalized rotation rate of the Earth–Moon Of the attitude profiles appearing in Fig. 4, the dark- frame with respect to the inertial frame is equal to one, blue orbit exhibits the least complexity in that the φ

IAC-10-C1.2.3 4 of 10 Page 4 of 10 61st International Astronautical Congress, Prague, CZ. Copyright c 2010 by the International Astronautical Federation. All rights reserved. angle appears to be generally sinusoidal and the θ an- and the length of time spent at a particular attitude gle appears to be generally sinusoidal (plus a secular supplies the third option in a control scheme that tar- term) as compared to the angle evolution correspond- gets the sub-L1 point. In the current problem, it is ing to the other two profiles. The regularity of the desired to control to a specified reference trajectory, dark-blue angle profile indicates that continuous turn not a point. However, a similar turn-and-hold strategy rates and rotational accelerations of the sail itself are is developed and implemented in this CR3B model. likely to be small. As the turns are to be discretized For the turn-and-hold schemes in this investigation, as part of the control schemes, the turns associated the trajectory is decomposed into segments and the with the dark-blue orbit are likely to be small between latitude, φ, and longitude, θ, angles are one or more epochs, relative to the other two orbits. The red and discrete inertial values over each segment i (as a con- light-blue orbits require more complex attitude pro- sequence, the angles are time-varying along the entire files as part of the baseline sailcraft trajectory, in part, arc segment in the Earth–Moon frame). For con- to maintain the elevation constraint on the trajectory venience, φi ≡ φ(ti) and θi ≡ θ(ti). Because the near the respective extreme values in the y direction. reference path from the FDM solution is represented Additionally, these attitude profiles result in large cone by position and velocity states, as well as the required angles (α = cos−1(`ˆ· u)), which may not be allowed attitude, at discrete time intervals, target states at under a more realistic sail model. epochs within the reported intervals are interpolated Each trajectory in Fig. 2 requires a characteristic ac- via an Akima cubic-spline. The initial guesses for 2 celeration of 1.70 mm/s . Recent explorations of the the latitude, φi, and the longitude, θi are the aver- design space conclude that a characteristic accelera- age values of the continuous values of φ and θ over arc tion of approximately 1.5 to 1.7 mm/s2 is required for segment i from the reference solution. The first imple- an ideal sail to consistently maintain the 15◦ elevation mentation of the turn-and-hold controller employs a constraint in the LSP coverage problem.22–24 Only one least-squares fit such that the optimal values of φ and spacecraft to date, IKAROS, has achieved orbit and θ are selected to minimize the difference between the employed a solar sail as its sole source of propulsion to propagated path and the reference path. The second date.11 However, a sailcraft that possesses a net char- implementation employs a collocation-based two-point 2 acteristic acceleration of 0.58 mm/s was designed and boundary-value problem (TPBVP) solver that delivers ground tested by L’Garde for the New Millennium Pro- three turns within a segment such that the positions gram’s Space Technology 9 competition at NASA;29 and velocities at the boundaries of the segment match the characteristic acceleration of the L’Garde sail and the reference trajectory. Path constraints (e.g., ele- 2 structure alone is 1.70 mm/s . In reality, solar sail- vation) are not incorporated into the controllers since ing technology will likely evolve to complement some it is assumed that tracking a reference trajectory, one other type of propulsive device.30 Nevertheless, if the designed with these restrictions, obviates the need to LSP coverage problem is addressed solely with a sail- further constrain the behavior of the controller. craft, some sail technology advancements are required The control algorithms in this analysis are limited to deliver an appropriate characteristic acceleration. controllers in the sense that they do not currently compensate for errors from orbit determination or control CONTROL SCHEMES execution. A study of such error sources is beyond the If a state from each of the reference trajectories in scope of the present investigation. Rather, as an ini- Fig. 2 is used to initialize an explicit propagation algo- tial step, it is necessary to develop a control strategy rithm, such as a classical Runge-Kutta approach, along that can re-target a sailcraft to a path that varies with with the control histories that result from the gen- time. It is presumed that errors arising from inaccu- erating numerical technique, it is not surprising that racies in the reference path and deviations from the the simulated path diverges from the reference prior targeted states at the end of a trajectory segment are to introducing any errors. Spacecraft state knowledge sufficient to develop insight and initiate a controller from statistical orbit determination techniques, ma- design. neuver executions from the attitude control system, and an understanding of the dynamical environment Least-Squares Implementation are all imperfect and possess errors. Therefore, trajec- The least-squares method is a classic approach for tory control schemes are required for the spacecraft to over-determined, or over-constrained, systems by min- follow any reference path. imizing the sum of the squares of the residuals between Two trajectory control schemes are examined; both some observed value and its modeled value. With two assume that the sailcraft can instantaneously turn degrees of freedom for the control (φ and θ at the be- from one orientation to another. Similar “turn-and- ginning of the segment) and at least six constraints hold” schemes are employed by Yen13 and Sauer14 to along the segment (positions and velocities at future control sailcraft to sub-L1 points. In Sauer’s formula- times), a sail cannot supply sufficient control with a tion, the sail angles represent two degrees of freedom single, fixed attitude over the entire the segment to

IAC-10-C1.2.3 5 of 10 Page 5 of 10 61st International Astronautical Congress, Prague, CZ. Copyright c 2010 by the International Astronautical Federation. All rights reserved. target a future state. However, by employing a least- for an extended set of segments. The relationships in squares solution, the sailcraft can get close to its target Eq. (9) are now written as state at some future time along the path.  δx  Given an initial guess for φi and θi, a trajectory  i+1/n   .  δφi is propagated along with a state-transition matrix, . = K6n×2 (11) δθi Φ(ti+1, ti), from time ti to ti+1. The state-transition  δz˙  2×1 i+n/n 6n×1 matrix maps deviations in φi and θi to deviations in the state vector at the end of the segment. This infor- where mation is reflected in the following relationship,  ∂xi+1/n ∂xi+1/n  ∂φi ∂θi  . .   δx  K =  . .  (12)  i+1      δφ ∂z˙i+n/n ∂z˙i+n/n . = K i (9) . 6×2 δθ ∂φi ∂θi   i 2×1  δz˙  T i+1 6×1 As mentioned, a solution to Eq. (11) for {δφi δθi} arises from a weighted least-squares implementation, where that is,  ∂xi+1 ∂xi+1  ∂φi ∂θi    . .  δxi+1/n K =  . .  (10) δφ     i = (KTWK)−1KTW . (13) ∂z˙i+1 ∂z˙i+1 . δθi ∂φi ∂θi    δz˙i+n/n  ∂xi+1 and is the element of Φ(ti+1, ti) that maps a ∂φi where W is a diagonal weighting matrix that bal- deviation in φ to a deviation in x . The size of each i i+1 ances the corrections between the position and velocity matrix or vector is indicated by a subscript. components. The solution from Eq. (13) is used to up- Because the formulation in Eq. (9) does not directly date φ and θ , whereby a new path is simulated and target acceleration, it is sometimes advantageous to i i compared to the reference trajectory. The process is target multiple positions and velocities at future times T iterated until the values of {δφ δθ } longer change, (i.e., t , t , . . . , t , where a fractional sub- i i i+1/n i+2/n i+n/n to within some tolerance. Because the correction is script indicates the time of a turn and the intervals based on a least-squares solution, the elements on the need not be evenly spaced). With a fixed attitude at left sides of either Eqs. (9) or (11) (or, alternatively, t , acceleration is implicitly targeted at t , where i i+n/n within the braces on the right side in Eq. (13)) may t is equivalent to t and n is the number of fu- i+n/n i+1 never converge to zero, but remain close to the ref- ture states in the expanded segment, as illustrated in erence values. Selection of n and the length of time Fig. 5. In the figure, the reference trajectory is repre- between ti and ti+1 affects the resulting controlled trajectory. Three-Turn Two-Point Boundary Value Problem 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 Fig. 5 A “turn-and-hold” segment (red) fit implemented at ti:1/3, ti:2/3, and ti:3/3, where a frac- through n target points along a reference path tional subscript indicates turn one, two, or three of the (black). The attitude is held from ti to ti+n/n. three total turns in an arc segment, supply sufficient control such that the positions and velocities along the sented with a black curve and the path resulting from compensated trajectory match the values on the refer- a simulation incorporating the initial guesses for φi ence trajectory at ti and ti+1, as illustrated in Fig. 6. and θi is in red. The residuals between the simulated An arc segment along the reference trajectory, which trajectory and the reference path at the future epochs supplies an initial guess for the TPBVP solver is rep- are indicated with gray arrows. No residual exists at ti resented in black, while the arc segment of the path because the controller cannot compensate for initial er- that is associated with the three turns that solve the rors in the path, only future errors. A best-fit solution equations of motion appears in red in the figure. for φi and θi that is determined by incorporating these A variety of numerical algorithms are available to extra, intermediate constraints may then more closely solve boundary value problems. A popular family of track the reference trajectory at ti+1 (or ti+n/n) and algorithms are based on collocation methods. Suppose

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. Conveniently, MATLAB R supplies a suite of func- tions for solving TPBVPs based on collocation Fig. 6 A three-turn solution to target position and schemes: BVP4C (most similar to the example illus- velocity at ti+1 given a position and velocity at ti. trated by Fig. 7),31 BVP5C,31 and BVP6C.32 These that, for a one-dimensional case,x ˙(t) = f(x(t)), where three algorithms differ in the degree of their interpo- f(x(t)) is an ordinary differential equation, or an equa- lating polynomial and the resulting accuracy. Both tion of motion, and x(t) solves the equation of motion. BVP4C and BVP5C are based on three- or four-stage th Lobatto IIIa integration rules, respectively, and are Also suppose that Sk(t) is an N -degree polynomial that approximates x(t) in the interval between mesh fourth- or fifth-order accurate uniformly between τk and τ , respectively.31 For BVP6C, a quintic inter- points at τk and τk+1. If, at a time τa ∈ [τk, τk+1], k+1 polant is fit to a mono-implicit Runge-Kutta sixth- Sk(τa) is equivalent to x(τa), then the approximating polynomial is said to be collocated to the solution of order formula for sixth-order uniform accuracy between τ and τ .32 All three methods employ mesh the equation of motion at τa. k k+1 As an example of a collocation method, a third- refinement to minimize the error along the arc such degree polynomial (in red) based on state and deriva- that the sub-arcs need not be uniformly spaced; node points may be added or eliminated as well. tive information at the mesh points (black) at τk and 27 τk+1 appears in Fig. 7. For the example in the Implementing any one of the three collocation func- tions 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 iden- tical, making it simple for the user to switch between methods. For this analysis, BVP6C is employed since Fig. 7 Third-degree collocation this algorithm possesses the best theoretical accuracy figure, only states and derivative information from of the three methods. Fewer mesh points to represent their associated equations of motion are required at the initial guess corresponding to the trajectory seg- the boundaries to generate a third-degree polynomial. ment are required, and convergence is typically faster. However, internal points between the mesh points may Nevertheless, a large number (m ≈ 151) of mesh points be required, depending on the degree of the collocating within the arc bounded by [ti, ti+1] are initially required, in practice, for convergence and to maintain polynomial. At one or more epochs (τa in the figure) the solution path near to the respective initial guesses between τk and τk+1, the derivative of the polynomial (red in the figure) is compared to the derivative for this problem. from the ordinary differential equation (green). The RESULTS residual between these two values is a defect (∆a in the figure). Generally, x(τk) and x(τk+1) are not pre- Both the least-squares implementation and the cisely known initially, and any defect will be non-zero. collocation-based TPBVP strategy (BVP6C) are suc- Therefore, values for x(τk) and x(τk+1) are updated cessful in controlling the sailcraft to the three different through an iterative process, thereby changing the reference trajectories appearing in Fig. 2, within speci- interpolating polynomial and derivative information fied tolerances. The turn-and-hold methods are essen- from the equations of motion, until the defects are tially linear corrections to a non-linear problem, and zero. each orbit possesses different sensitivities. As a conse- To begin the collocation process, the arc within the quence, slight differences in the implementation of the

IAC-10-C1.2.3 7 of 10 Page 7 of 10 61st International Astronautical Congress, Prague, CZ. Copyright c 2010 by the International Astronautical Federation. All rights reserved. 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 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 Fig. 8 Three revolutions of orbits controlled with meet the above convergence requirements for the dark- the least-squares algorithm. and light-blue trajectories. When the absolute and rel- path is generated by scheduling a turn every two solar ative tolerances are tightened, the BVP6C algorithm days using six target points along a two-day segment reports that it “cannot converge without exceeding the of the reference trajectory. The light-blue path re- maximum number of allowable mesh points.” With quires a turn every solar day and is targeted to two the exception of a four-day segment, the conditions points along the one-day segment. After three revolu- leading to convergence are the same for the red trajec- tions, the dark- and light-blue solutions diverge from tory. All three trajectories from the collocation-based the reference trajectory with these targeting schemes. TPBVP control scheme appear in Fig. 10 and the re- The red trajectory in Fig. 8 is generated with turns spective attitude profiles appear in Fig. 11. every two solar days and is targeted to three points Because of the loose convergence tolerances, incor- along the segment. This trajectory tracks its reference porating the turns resulting from the BVP6C algo- path for four-and-a-half revolutions before diverging. rithm into an explicit integration scheme results in Note that the time between turns and the number of position and velocity at the end of the first segment target points is fixed for each orbit. Allowing the time that is not consistent with the boundary conditions es- between turns and the number of target points within tablished for the TPBVP algorithm. Not surprisingly, a segment to fluctuate may result in longer tracking employing this new state as a boundary condition times, especially since the dynamical sensitivities vary in the BVP6C algorithm for the subsequent segment throughout each orbit. The control profiles for the leads to divergence; the solution is highly sensitive to three orbits from Fig. 8 appear in Fig. 9. Note that the initial guess for the path. However, while the suc- the plot for θ includes a secular component because of cess of this collocation-based implementation of the the motion of the Earth–Moon system in inertial space. TPBVP controller is subjective at best (due to the A trade-off between longer hold times and larger atti- errors in the solution), another implementation of a

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 1C. R. McInnes, Solar Sailing: Technology, Dynamics and Mission Applications. Space Science and Technology, New York: Springer-Praxis, 1999. 2D. L. Shirley, “The Mariner 10 mission to Venus and Mercury,” Acta Astronautica, vol. 53, pp. 375–385, August– November 2003. Fig. 10 One revolution of the orbits controlled 3D. J. O’Shaughnessy, J. V. McAdams, K. E. Williams, and with the collocation algorithm. B. R. Page, “Fire Sail: MESSENGER’s use of solar radiation pressure for accurate Mercury flybys,” in Advances in the Astro- nautical Sciences, vol. 133, pp. 61–76, 2009. Paper AAS 09-014. 4G. Colombo, “The stabilization of an artificial satellite at the inferior conjunction point of the Earth–Moon system,” Tech. Rep. 80, Smithsonian Astrophysical Observatory, Novem- ber 1961. 5R. H. Laprade, J. A. Miller, and S. J. Worley, “Satellite stationkeeping by solar radiation pressure,” Space/Aeronautics, vol. 47, pp. 114–117, April 1967. 6W. L. Black, M. C. Crocker, and E. H. Swenson, “Sta- tionkeeping a 24-hour satellite using solar radiation pressure,” Journal of Spacecraft, vol. 5, pp. 335–337, March 1968. 7R. W. Farquhar, “The control and use of libration point satellites,” Tech. Rep. TR-R-346, National Aeronautics and Space Administration, February 1970. 8M. C. Crocker, “Attitude control of a sun-pointing spin- ning spacecraft by means of solar radiation pressure,” Journal of Spacecraft and Rockets, vol. 7, pp. 757–759, March 1970. 9V. 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. Fig. 11 Latitude, φ, and longitude, θ, angles over 10B. W. Stuck, “Solar pressure three-axis attitude control,” one revolution from the collocation formulation of Journal of Guidance and Control, vol. 3, pp. 132–139, February the turn-and-hold controller. 1980. 11O. Mori et al., “Worlds first demonstration of solar power three-turn TPBVP controller may produce a more fa- sailing by IKAROS,” in 2nd International Symposium on Solar vorable result. Sailing, New York City College of Technology, City University of New York, (Brooklyn, New York), July 2010. 12 CONCLUSIONS AND FUTURE WORK R. L. Young, “Updated Heliostorm warning mission: Enhancements based on new technology,” in 48th In this investigation, the authors have developed a AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dy- rudimentary pair of trajectory control schemes that namics, and Materials Conference, (Honolulu, Hawaii), April employ a turn-and-hold strategy for a solar sail space- 2007. Paper AIAA-2007-2249. 13C. L. Yen, “Solar sail geostorm warning mission design,” craft. The least-squares implementation can be im- in Advances in the Astronautical Sciences, vol. 119, pp. 69–82, proved by varying the length of hold times and target February 2004. Paper AAS 04-107. points, while the three-turn TPBVP scheme may re- 14C. G. Sauer, “The L1 diamond affair,” in Advances in the quire a different TPBVP solver. Astronautical Sciences, vol. 119, pp. 2791–2808, February 2004. Paper AAS 04-278. In addition to other implementations of TPBVP 15M. Lisano, D. Lawrence, and S. Piggott, “Solar sail transfer solvers, future efforts will incorporate finite turn times trajectory design and stationkeeping control for missions to the and spacecraft attitude control capabilities. Addition- sub-L1 equilibrium region,” in Advances in the Astronautical ally, a controller should be able to compensate for Sciences, vol. 120, pp. 1837–1854, 2005. 16 errors in the orbit-determination knowledge and turn D. Lawrence and S. Piggott, “Solar sailing trajectory control for sub-L1 stationkeeping,” in AIAA Guidance, Navigation, execution. The control schemes to date focus on dis- and Control Conference and Exhibit, (Providence, Rhode Is- crete orientations. Future research will also examine land), August 2004. Paper AIAA-2004-5014.

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