IAC-12-C1.8.1 COUPLED ORBIT AND ATTITUDE DYNAMICS FOR SPACECRAFT COMPRISED OF MULTIPLE BODIES IN EARTH-MOON HALO ORBITS

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1 63rd International Astronautical Congress, Naples, Italy. Copyright c 1 by the International Astronautical Federation. All rights reserved. IAC-1-C1.8.1 COUPLED ORBIT AND ATTITUDE DYNAMICS FOR SPACECRAFT COMPRISED OF MULTIPLE BODIES IN EARTH-MOON HALO ORBITS Amanda J. Knutson Purdue University, United States of America ajknutso@purdue.edu Kathleen C. Howell Purdue University, United States of America howell@purdue.edu The coupled orbit and attitude motion of a spacecraft in three dimensional halo) reference orbits is explored in this investigation within the context of the Circular Restricted Three Body Problem CR3BP). The motion of a spacecraft, comprised of two rigid bodies connected by a single degree of freedom joint, is examined. The nonlinear variational form of the equations of motion is employed to mitigate numerical effects caused by large discrepancies in the length scales. Several nonlinear, periodic reference orbits in the vicinity of the collinear libration points are selected for case studies and the effects of the orbit on the orientation, as well as the orientation on the orbit, are examined. Additionally, the model is extended to include solar radiation pressure and the motion of a solar sail type spacecraft is considered. I. INTRODUCTION The Circular Restricted Three Body Problem CR3BP) has been examined extensively in recent years; however, considerably fewer studies incorporate the effects of spacecraft attitude in this regime. Numerous missions in the vicinity of the collinear libration points have been completed, are ongoing, or are planned for the near future. 1 3 For example, the James Webb Space Telescope JWST), successor to the Hubble Telescope, is to be deployed in a Sun-Earth L Lissajous orbit within the next decade. 4 Due to the sensitive nature of the dynamics at the libration points, a better understanding of the attitude motion of spacecraft in the vicinity of these points is desired. The ability to predict the attitude motion of a spacecraft as orbital parameters and spacecraft characteristics are varied can be very useful. Once the behavior is investigated, the possibility of exploiting the natural dynamics for passive attitude control in these regimes is an option. A recent investigation examined the motion of a spacecraft constructed as two rigid bodies; the vehicle is located in various planar Lyapunov reference orbits, in the vicinity of the Earth-Moon collinear libration points. 5 One goal of the current investigation is an examination of the effects of incorporating attitude dynamics into the model for the nonlinear three dimensional orbits within the context of the CR3BP, Ph.D. Candidate, School of Aeronautics and Astronautics Hsu Lo Professor of Aeronautical and Astronautical Engineering particularly the impact on vehicles in halo reference orbits. Additionally, the model is extended to include other types of perturbations, for example, a spacecraft model for which solar radiation pressure is significant; the analysis is further expanded to explore the motion of solar sails. Kane and Marsh first consider the attitude stability of a axisymmetric satellite, located exactly at the equilibrium points. 6 In their analysis, the satellite is artificially maintained at each of the equilibrium points and only the attitude motion is considered. Robinson continues this work with an examination of the attitude motion of both a dumbbell satellite 7 and an asymmetric rigid body 8 artificially maintained at the equilibrium points. Abad et al. introduce the use of Euler parameters in the study of a single rigid body located at L 4. 9 More recently, Brucker et al. explore the dynamics and stability of a single rigid body spacecraft in the vicinity of the collinear points in the Sun-Earth system, using Poincaré maps. 1 The effects of the gravity gradient torque are explored by the authors, but the spacecraft is still artificially fixed to the equilibrium point. Wong et al. offer a detailed examination of the motion of a single rigid body near the vicinity of the Sun- Earth libration points, using linear Lyapunov and halo orbit approximations. 11 The response of the spacecraft is detailed by Wong et al. in a meaningful manner, offering orientation results for a number of numerical simulations in terms of a body 3--1 θ, φ, ψ) Euler angle rotation sequence. However, since the Lyapunov IAC-1-C1.8.1 Page 1 of 15

2 63rd International Astronautical Congress, Naples, Italy. Copyright c 1 by the International Astronautical Federation. All rights reserved. and halo orbits are represented in a linear form, the simulations are only valid for relatively small orbits close to the equilibrium points. The main goal of the current research effort is an expansion of the results from Wong et al., that is, to examine the motion of a spacecraft comprised of multiple rigid bodies in three dimensional orbits halos) in the vicinity of the libration points. Knutson and Howell explore the behavior of multibody spacecraft in the circular restricted three body problem, using nonlinear Lyaponov orbits as reference orbits, where the coupled orbit and attitude equations of motion are formulated using Kane s dynamical method. 5 The equations of motion governing the orbit, as well as the spacecraft attitude, are blended in a cohesive manner that minimizes computational difficulties and allows for the propagation of meaningful results. One of the major computational issues in this process is the management of the disparity in length scales in this problem. The distance from the barycenter to the spacecraft is on the order of hundreds of thousands of km depending on the system), and the spacecraft dimensions are typically measured in terms of meters. The differences in the lengths present a computational challenge, as only finite precision is possible. Thus, a nonlinear variational form of the equations of motion is employed to mitigate these numerical effects. The motion of the spacecraft is also monitored relative to some known reference orbit. The equations for the time rate of change of linear momentum first describe the absolute motion of the spacecraft and are then again employed to describe the motion of the point mass reference system. The contributions from the reference are subtracted from the absolute motion, eliminating the difficulties of significantly different lengths coupled within the same differential equations. The same formulation is applied in the current work to analyse motion in three dimensional orbits in the Circular Restricted Three Body Problem. Guzzetti examines the coupled motion of a spacecraft in nonlinear Lyapunov reference orbits, considering the out-of-plane attitude motion. 1 Guzzetti formulates the equations of motion using a Lagrangian approach and also incorporates the effects of solar radiation pressure and flexible bodies. Guzzetti compares the out-of-plane attitude response for several test cases to those generated from the model proposed by Knutson and Howell. 1 The results indicate good correlation between the attitude response using the two models. Another goal of the current work is the assessment of the departure from the vicinity of the reference orbit that is predicted by the coupled orbit and attitude model and the potential to employ this deviation to determine the required attitude configuration to control the orbital motion. A sail is employed to represent the coupled system and it is oriented to remain close to the reference. First, a family of solar sail orbits in the vicinity of the L libration point in the Earth-Moon system is computed using a single shooting algorithm as described by Wawrzyniak. 13 The method is an extension of the vertically shifted linear approximation for elliptical orbits near the libration points originally introduced by Simo and McInnes. 14 Each member in the family of orbits computed numerically by the current authors maintains a fixed sail orientation with respect to the Sun frame in the Earth- Moon system. A multiple shooting scheme is then employed to determine the required corrections to the attitude motion based on an offset in orbital quantities from the desired reference at some instant in time. A turn-and-hold strategy, similar to one proposed by Wawrzyniak, 15 is used to determine attitude corrections. In the turn-and-hold strategy, the spacecraft is considered to maintain a fixed orientation over each multiple shooting segment. Finally, the attitude corrections are applied to the coupled orbit and attitude model. If successful, a satisfactory attitude history maintains the configuration in the vicinity of the reference orbit. The corrections are computed iteratively, as required, based on the departure of the state from the reference path. II. SPACECRAFT MODEL The spacecraft model is comprised of two generally asymmetric rigid bodies, that is, body B and body R. Two coordinate frames are assigned to the spacecraft. The first coordinate frame is fixed in body B, denoted the b-frame. The second set of unit vectors is fixed in body R and is denoted the r-frame. The coordinate systems align with the principal directions of the individual bodies for each center of mass. The mass of body B is m b and the center of mass location is denoted B. The mass of body R is m r and the center of mass location is denoted R. The inertia matrices corresponding to the two bodies are expressed in terms of b-frame and r-frame coordinates, respectively, I B/B = I R/R = I b 1b 1 I bb I b3b 3 I r1r 1 I rr I r3r 3 [1] [] The two bodies are connected by a revolute joint with a single degree of freedom parallel to the ˆb 3 direction. Thus, the ˆb 3 and ˆr 3 directions are aligned for all time. The location of the joint, denoted J, is specified relative to the centers of mass of each body. In the b-frame, the vector from B to J is defined as r J/B = s 1ˆb1 + s ˆb + s 3ˆb3. In the r-frame, the vector from J to R is expressed by r R /J = j 1ˆr 1 +j ˆr +j 3ˆr 3. A schematic representation of the spacecraft model is illustrated in Fig. 1. IAC-1-C1.8.1 Page of 15

3 63rd International Astronautical Congress, Naples, Italy. Copyright c 1 by the International Astronautical Federation. All rights reserved. r 3 duces the rate of change of linear momentum, for any body, S, in a system, r R* r 1 F S nd dt m s n S v ) = [3] B* b 3 J b b 1 The sum of all the external forces on body S is then equal to the resultant F S. The mass for any body, S, is defined by the scalar quantity m s. The single overbar denotes a vector. A similar expression yields the rate of change of angular momentum, M S/S n d ) I S/S n ω s = [4] dt Fig. 1 Spacecraft Model For certain applications, it may be desirable to utilize a single rigid body spacecraft. In this situation, the rotational degree of freedom between body B and body R is fixed, which acts in the same manner as a spacecraft comprised of a single body. III. DYNAMIC FORMULATION The formulation of Kane s equations is explored to model the problem. Kane s method is based on the principles of linear and angular momentum. The method partitions the general formulas for the time rate of change of linear and angular momentum into separate terms in the directions of a set of specified generalized speeds. 16, 17 This approach is employed for several reasons. Kane s method is well suited for deriving equations of motion for systems, such as spacecraft, that are composed of many individual bodies connected by joints in a chain or tree-like structure. Kane s method employs simple matrix multiplication methods to derive the equations of motion. Assuming workless constraints, the contact forces are automatically eliminated. 16 As an alternative to basing the analysis on configuration coordinates position variables), Kane s method uses velocity variables identified as generalized speeds. The generalized speeds, denoted u i, are selected in any manner, but it is usually desirable to select an independent set such that changes to one speed do not affect any other speed. If minimal variables are not assigned, constraint equations are incorporated to reduce the order of the system. If a set of minimal variables is selected, one generalized speed is required for each degree of freedom. The total number of generalized speeds is denoted n. Some configuration coordinates may be required to fully describe the system. The derivatives of the configuration coordinates are formulated by some combination of a subset of the generalized speeds. Kane s equations rely on the principles of linear and angular momentum. The following relationship pro- The sum of all external moments on body B about the center of mass, S, is represented by the resultant sum M S/S. The double overbar represents a dyadic. Then, I S/S is the inertia dyadic relative to S for any three dimensional body. The components of the inertia dyadic are frequently collected in a matrix as, I S/S = I s 1s 1 I ss 1 I s1s I ss I s1s 3 I ss 3 I s3s 1 I s3s I s3s 3 [5] where the vector basis is identified prior to computation. The mass moments of inertia of body S, about the center of mass, S, are described by I s1s 1, I ss, and I s3s 3, and the remaining terms are products of inertia. Equations [3] and [4] are then used to formulate Kane s equations. IV. EQUATIONS OF MOTION The model formulation previously proposed by Knutson and Howell 5 is employed to develop the equations of motion governing the behavior of the spacecraft. An overview of the key concepts that are detailed in Knutson and Howell 5 is presented. To develop the equations of motion governing the behavior of the spacecraft, generalized speeds are first assigned. The two body spacecraft possesses seven degrees of freedom; therefore seven independent generalized speeds, u i, are selected. The angular velocity of the b-frame with respect to the a-frame is expressed, a ω b = u 1ˆb1 + u ˆb + u 3ˆb3 [6] In this application, the relative motion of the two bodies is restricted to be one-dimensional. Thus, an expression for the angular velocity of the r-frame with respect to the b-frame is formulated, b ω r = u 4ˆb3 [7] The remaining generalized speeds are used to describe the translational velocity of B, i.e, the center of mass of body B, relative to the current location in the reference orbit, denoted by O IAC-1-C1.8.1 Page 3 of 15

4 63rd International Astronautical Congress, Naples, Italy. Copyright c 1 by the International Astronautical Federation. All rights reserved. n B v /O = u 5ˆb1 + u 6ˆb + u 7ˆb3 [8] Several other configuration coordinates are also required to fully describe the system translation and orientation. Euler parameters, as elements in a fourvector, are selected to describe the orientation of the b-frame with respect to the frame associated with the reference orbit, denoted the a-frame. The Euler vector is defined, a ɛ b = ɛ 1ˆb1 + ɛ ˆb + ɛ 3ˆb3 [9] with the constraint, ɛ 1 + ɛ + ɛ 3 + ɛ 4 = 1 [1] For visualization purposes, it is sometimes desirable to convert the Euler parameter representation of the orientation to an Euler angle rotation sequence. Two rotation sequences, a body 3--1 θ, φ, ψ) and a body 3--3 α, ζ, γ) are employed to represent the rotation from the a-frame to the b-frame for the various test cases that are examined. Kinematic differential equations are employed to determine the Euler parameter rates and are formulated from the following expression, 17 where and a ɛ b = 1 a ω b E T [11] a ω b = [ a ω1 b a ω b a ω3 b ] [1] E = ɛ 4 ɛ 3 ɛ ɛ 1 ɛ 3 ɛ 4 ɛ 1 ɛ ɛ ɛ 1 ɛ 4 ɛ 3 ɛ 1 ɛ ɛ 3 ɛ 4 [13] The angular velocity of the r-frame, fixed in the second body of the spacecraft, relative to the b-frame is integrated directly to yield the angle between these frames. The direction cosine matrix to convert from the b-frame to the r-frame is described as a rotation about ˆb 3 as follows, C br = cos β sin β sin β cos β 1 [14] The position of R with respect to B is eventually required and is written in b-frame coordinates and evaluated as, δ = g 1 â 1 + g â + g 3 â 3 [16] The current position in the reference orbit is determined by propagating the point mass system forward in time, or by interpolating a known reference path. The motion of the center of mass of the spacecraft system, G, is tracked as the simulation proceeds. This quantity is required to determine the contributions of the gravitational potential from the two primaries. The momentum principles introduced in Equations [3] and [4] are now applied. Equation [3] is first employed to form an equation for the total motion of each of the bodies that comprise the multibody spacecraft, and then to write the equation governing the reference point mass system. To remove the undesirable effects of the discrepancies in the length scales, the point mass equation is subtracted from the total motion of body B, yielding the following variational form, F B F O ) nd dt m b ) n B v /O = [17] Similar expressions are derived for the second body, R. Equation [4] is applied as written since the point mass system includes no inertia properties with respect to the spacecraft center of mass and, therefore, no external moments are applied. External forces and moments are discussed separately below. This model assumes that the net force derived from the gravitational potential due to the two primaries is the only external force applied to the system. The external field force is assumed to be applied to the center of mass of the system, i.e, G. An equivalent system is derived for the two individual bodies, B and R by replacing the force applied at G, with a force and moment pair applied at each of the centers of mass of the individual bodies. The difference between the location of the center of mass of the system, r G, and the point on the reference, r O is represented by δ. Both r G and r O are defined with base points fixed in the inertial frame at the barycenter of the Earth-Moon system. The gravitational force exerted on the center of mass of the system G is represented by F G and the gravitational force exerted on the point mass reference system at O, is represented by F O. The total mass of the spacecraft system of two bodies is defined as m t and the difference between the two forces is denoted, r R /B = r J/B + r R /J C T br [15] To determine the effects of the attitude motion on the orbit, it is assumed that the spacecraft system center of mass, denoted G, is offset from the reference trajectory. The following set of scalar variables, as defined in the rotating orbit frame, is used to track the motion of G with respect to the current position along the reference orbit, O, that is, F = F G F O [18] Since the gravitational force applied at the center of mass of the system, G, is nearly identical to the gravitational force exerted on the point mass reference system at O, a variational representation ensures that the appropriate precision is maintained. A general implementation of Encke s method is employed to achieve this task. 18 The details of this procedure are IAC-1-C1.8.1 Page 4 of 15

5 63rd International Astronautical Congress, Naples, Italy. Copyright c 1 by the International Astronautical Federation. All rights reserved. presented in Knutson and Howell. 5 This approach produces results that are accurate given that the actual spacecraft remains close to the reference path. The reference may require rectification for certain simulations if the spacecraft center of mass is observed to shift sufficiently far from the reference path. The contributions applied to each body individually is determined by partitioning the total mass, m t, into the mass of the two bodies, m b and m r. All force terms are written in the b-frame vector basis. The gravity gradient moment in this problem is delivered by the two primary bodies. For each primary, this moment is derived from the cross product of the distance from the attracting body to the spacecraft and the second-order force expansion. 19 These moments are computed in terms of the b-frame vector basis. n B v n ω b F B n v B M B/B n ω b { n d dt { n d dt m b n B v )} = [19] I B/B )} n ω b = [] These two vector equations for body B are now summed to create one vector equation. Kane splits this equation into two parts, generalized inertia forces and generalized active forces. Generalized inertia forces, denoted Q, combine the contribution of the quantities that depend on the time rate of change of both linear and angular momentum, and the generalized active forces, denoted Q, collect the contributions from external loads. The generalized inertia force and generalized active force for body B are described as follows, The partial velocity matrix is formed for each body from the partial derivative of the center of mass velocity with respect to each generalized speed. The partial derivative of the velocity vector is evaluated for each generalized speed and this vector quantity is stored as a row in the partial velocity matrix, defined for body B as n v B and body R as n v R. The partial velocity matrix and partial angular velocity matrix for both body B and body R are described in terms of the b-frame vector basis. Each matrix has seven rows, one for each generalized speed, and three columns, one for each spatial dimension of the b-frame. The partial angular velocity matrix for each body also originates from the partial derivative of the angular velocity vector with respect to each generalized speed. As described previously, these matrices also have seven rows and three columns, and are described in terms of the b-frame vector basis. The matrices for body B and body R are represented by n ω B and n ω R, respectively. Using the quantities derived previously, the equations of motion are now constructed using Kane s formulation. The partial velocity matrices and the momentum equations are combined to form Kane s equations of motion. The product of the partial velocity matrix and the time rate of change of linear momentum in Equation [3] and the product of the partial angular velocity matrix and the time rate of change of angular momentum in Equation [4] are determined for each body in the system. A coordinate frame fixed in body B is selected as the common vector basis, and the partial velocity and partial angular velocity matrices and Equations [3] and [4] are represented in terms of this frame. For body B, the equations are summarized below, Q B = n v B Q B = n v B n ω b { n d dt { n d dt m b I B/B n B v )} )} [1] n ω b F B + n ω b M B/B [] Similar expression are determined for body R. To apply Kane s method to a multibody system, the contributions from each body must be summed to yield a single equation. A coordinate frame fixed in body B remains as the common vector basis, and all contributions are expressed in terms of this frame. Kane s formulation for a system composed of m bodies is represented by the following simplified expression, m i=1 Q B i + Q Bi = [3] m i=1 The generalized inertia forces and generalized active forces, described in Equations [1] and [], are computed for each body using the partial velocities matrices and the appropriate parts of the momentum equations, from Equations [4] and [17]. These two quantities are combined for each body, as described in Equation [3]. Finally, the contributions from the two separate bodies are added to obtain a single vector equation of motion corresponding to each generalized speed. Thus, the governing equations are produced in the form A ū = w. This system of equations is combined with derivatives for the required configuration coordinates and integrated to determine the motion of the individual bodies in the system. The equations of motion for the reference point mass orbit are also integrated simultaneously and conserved quantities are monitored to ensure accuracy of the solution. IAC-1-C1.8.1 Page 5 of 15

6 63rd International Astronautical Congress, Naples, Italy. Copyright c 1 by the International Astronautical Federation. All rights reserved. V. MOTION IN EARTH-MOON HALOS The reference trajectory for this analysis is a path that reflects the motion of a point mass spacecraft propagated in the CR3BP, where the spacecraft is moving under the gravitational influence of two primary bodies, also modeled as point masses. There are several underlying assumptions that are significant in this model. Since the particle is assumed to be of infinitesimal mass, the primary orbits are modeled as Keplerian. The two primary bodies then move in circular orbits about their common barycenter at a constant rate Ω. A rotating coordinate system is assigned, denoted â 1, â, â 3 ), as depicted in Fig.. The angular velocity of the a-frame relative to an inertial frame, n, is denoted n ω a and is equal to Ωâ 3. a n P 3 points. Three dimensional periodic orbits have been computed by differentially correcting initial conditions in the vicinity of the L 1 collinear libration point. The EOM in Equations [4] -[6] comprise the model for the motion of the point mass reference body. A family of northern Earth-Moon E-M) halo orbits in the vicinity of L 1 is depicted in Fig. 3. a 3 km) x Az km) Fig. P 1 d p r P t a 1 n 1 Circular Restricted Three Body Problem The nondimensional form of the scalar equations of motion for the spacecraft, in the rotating frame â 1, â, â 3 ), are described below. In nondimensional form, the position vector from the barycenter to the spacecraft center of mass is written p = xâ 1 +yâ +zâ 3. The symbol µ represents the mass parameter for the specific system and is defined as the ratio of the mass of the smaller primary body to the total mass of the two primaries. The resulting scalar nondimensional equations of motion are, ẍ = ẏ + x 1 µ)x + µ) µx 1 + µ) d 3 r 3 [4] ÿ = ẋ + y 1 µ)y d 3 µy r 3 [5] 1 µ)z z = d 3 µz r 3 [6] The CR3BP allows five equilibrium solutions as points relative to the rotating system where the gravitational forces and rotating frame dynamical accelerations cancel, assuming a vehicle modeled as a point mass. One of the collinear equilibrium points, L 1, located along the line connecting the two primaries, and located between the two primaries, is of particular interest in this investigation. Both two and three dimensional periodic orbits exist in the vicinity of the libration x 1 4 a km) x 1 5 a 1 km) Fig. 3 Family of Halo Orbits in the Vicinity of L 1 in the E-M System Members of this orbit family are used as references for several test cases. V.I PERTURBED ASYMMETRIC RIGID BODY In a previous analysis, 5 the motion of a perturbed asymmetric rigid body in a set L Earth-Moon Lyapunov orbits is determined and compared to the work by Wong et al., 11 for a similar rigid body moving in a similar set of L Sun-Earth Lyapunov orbits. The results indicate good correlation between the two investigations, especially for smaller orbit sizes. As an expansion of the previous contributions, the motion of an asymmetric rigid body moving in the vicinity of L 1 in the Earth-Moon system is now examined. Two sets of nonlinear halo reference orbits are considered, a small set corresponding to orbits in the blue range of the family, as depicted in Fig 3, and a set of larger halo orbits, which corresponds to orbits in the red range. The small orbit set is displayed below in Fig 4. The green orbit represents a halo orbit corresponding to A z = 5 km, the red orbit is larger such that A z = 1 km, and the blue orbit corresponds to A z = 5 km. Note that A z corresponds to the maximum out-of-plane excursion along the periodic orbit. The spacecraft is modeled as a single asymmetric rigid body. The inertia properties in b-frame coordinates for the rigid body include the moments I b1b 1 =7.83 kg m, I bb = kg m, and I b3b 3 =1.417 kg m, which are consistent with the quantities employed by Wong et al. 11 All products of inertia are equal to zero. A body fixed 3--1 θ, φ, ψ) Euler angle sequence is employed to visualize the orientation response. A perturbation in θ, φ, ψ) is IAC-1-C1.8.1 Page 6 of

7 63rd International Astronautical Congress, Naples, Italy. Copyright c 1 by the International Astronautical Federation. All rights reserved. a 3 km) a 1 4 km) 1 Az=5km Az=1km Az=5km a km) Fig. 4 Small Set of L 1 Halo Reference Orbits in Earth-Moon System explored: 5, 5, 5) deg. The impact on the Euler angles as a result of simulations initialized with the set of perturbations displayed in Figs Az=5km Az=1km Az=5km ψ deg) 4 6 Az=5km Az=1km Az=5km unperturbed perturbed Revs Fig. 7 Asymmetric Orientation Response in ψ for Small Halo Reference Orbits in the E-M System The results that emerge when the vehicle is moving in a halo orbit from the larger set is displayed below in Fig. 8. The green orbit represents a halo orbit corresponding to A z = 1 km, the red orbit possess a maximum out-of-plane excursion equal to A z = 5 km, and the path along the blue orbit reaches A z = 7 km. θ deg) 6 Az=1km Az=5km Az=7km 4 unperturbed perturbed Revs Fig. 5 Asymmetric Orientation Response in θ for Small Halos Reference Orbits in the E-M System a km) 4 φ deg) Az=5km Az=1km Az=5km 4 a 1 4 km) a km) Fig. 8 Large Set of L 1 Halo Reference Orbits in Earth-Moon System 3 unperturbed perturbed Revs Fig. 6 Asymmetric Orientation Response in φ for Small Halo Reference Orbits in the E-M System Then again, a perturbation in θ, φ, ψ) of 5, 5, 5) deg is explored for the large orbit references. The impact on the Euler angles as a result of simulations that are initialized with the perturbations is displayed in Figs In general, the orientation responses are observed to differ depending on the orbit type. For the small reference orbit set, the orientation response in terms of the first Euler angle, θ, appears in Fig. 5 to be oscillatory and relatively periodic over the length of the simulation. The perturbation is noted to shift and increases the response in θ for all three small reference orbits. IAC-1-C1.8.1 Page 7 of 15

8 63rd International Astronautical Congress, Naples, Italy. Copyright c 1 by the International Astronautical Federation. All rights reserved. θ deg) Az=1km Az=5km Az=7km 1 unperturbed perturbed Revs Fig. 9 Asymmetric Orientation Response in θ for Large Halo Reference Orbits in the E-M System 1 5 Az=1km Az=5km Az=7km more quickly than the response when the vehicle is in an orbit from the small orbit set, but overall, generally similar characteristics appear. The exception is the response for the largest orbit, with an A z amplitude of 7 km, where the influence of the close pass to the Moon at.5 rev and 1.5 rev locations is visible. In comparing Figs. 7 and 1, the magnitude in the amplitude response of the third angle ψ, between the small and large reference orbits, is noted to increase much more significantly. The test cases are explored to further expand on the analysis offered in Wong et al. 11 by considering the orientation history resulting from selecting nonlinear, L 1, halo orbits as the reference. While the results presented by Wong et al. 11 display bounded, oscillatory motion for the first two Euler angles, θ and φ, for the length of the simulation, the above sample cases indicate that for larger orbit size, the motion may become unbounded, less oscillatory, and less predictable in general. φ deg) 5 1 unperturbed perturbed Revs Fig. 1 Asymmetric Orientation Response in φ for Large Halo Reference Orbits in the E-M System ψ deg) Az=1km Az=5km Az=7km unperturbed perturbed Revs Fig. 11 Asymmetric Orientation Response in ψ for Large Halo Reference Orbits in the E-M System The response in θ for the large halo orbit set, as apparent in Fig. 9, does not exhibit similar behavior. As the orbit size increases, the motion becomes unbounded and the divergence appears relatively quickly. The motion in the second angle, φ, for the small orbit set is displayed in Fig. 6; the response appears to be oscillatory although the amplitude is increasing with time. The magnitude of the response for a vehicle in one of the orbits from the large set in Fig. 1, diverges V.II PERTURBED AXISYMMETRIC BODY The general model that has been developed is capable of exploring the behavior of a range of vehicles, and is now employed to predict the behavior of an axisymmetric body under a perturbation in the initial orientation. The spacecraft is comprised of two rigid rods, each of length 1 meters and radius.1 meter. The rods are stacked on top of each other, configured so that the axial direction is aligned with ˆb 3. The top rod can spin about ˆb 3 relative to the bottom rod. For visualization purposes, the orientation is expressed in terms of a body fixed 3--3 α, ζ, γ) Euler angle sequence, for a variety of test cases. The angle α represents precession of the b-frame relative to the a frame, the angle ζ is nutation, and spin about the axisymmetic axis is denoted by the angle γ. A 5 degree perturbation in the nutation angle, ζ, is considered for four periodic L 1 halo reference orbits in the Earth-Moon system with A z values of 5 km, 5 km, 1 km, and 5 km. The orbital motion over.5 revolutions of the reference trajectory is first considered for the three smaller A z values. In Fig. 1, the departure of the spacecraft from the reference is evident. It is observed that smaller values of A z actually correspond to larger motion of the spacecraft center of mass, G, away from the isochronous location along the reference path, O. In any family of L 1 halo orbits, the orbits with smaller A z values are unstable. The perturbations in these smaller, more unstable orbits appear to induce greater effect than in larger, less unstable orbits, based on the magnitude of the unstable eigenvalue of the associated monodromy matrix. The attitude response, in terms of the body fixed 3--3 α, ζ, γ) Euler angle sequence is also considered. A perturbation in the nutation angle of ζ = 5 degrees, for the four halo reference orbits is introduced. In Fig. 13, the precession angle of the ˆb-frame with respect IAC-1-C1.8.1 Page 8 of 15

9 63rd International Astronautical Congress, Naples, Italy. Copyright c 1 by the International Astronautical Federation. All rights reserved. a km) a km) Az=5km Az=5km Az=1km a 1 4 km) Fig. 1 Orbital Response to perturbation of ζ = 5 degrees in E-M Halos 3 ζ deg) Az=5km 1 Az=5km Az=1km Az=5km Rev Fig. 14 Orientation Response in ζ to Perturbation ζ = 5 degrees in E-M Halos to the â-frame, α, is observed to be oscillatory and bounded for motion in the orbit with the smallest A z value, irregular for intermediate values of A z, and increasing at a fairly constant rate for the largest A z of 5 km. The nutation angle, ζ, for rods moving in halo orbits of all sizes is observed in Fig. 14 to oscillate around ζ = 9 degrees, which is consistent with knowledge of attitude motion in the two body problem; in terms of the nutation angle, a long thin rod is unstable in the vertical configuration and stable while lying horizontally. The frequency of oscillations appears to increase with the size of the halo reference trajectory. In Fig. 15, the spacecraft spin angle, γ, is noted to be increasing a greater rate for the relatively smaller halo reference orbits than for reference orbits possessing larger values of A z. With the exception of the largest halo orbit however, the spin angle, γ, appears to be fairly irregular. Along the reference trajectory at A z = 5 km, the overall response in γ is observed to be oscillatory and increasing. α deg) Az=5km Az=5km Az=1km Az=5km γ deg) Az=5km Az=5km Az=1km Az=5km Rev Fig. 15 Orientation Response in γ to Perturbation ζ = 5 degrees in E-M Halos rates:.1 rad/s approximately 1 rpm) and 1 rad/s approximately 95 rpm). In Fig. 16 through Fig. 18, the perturbed response is displayed in green, the red line indicates motion corresponding to a spin rate of.1 rad/s, and the blue line reflects to a spin rate of 1 rad/s. 1 1 α deg) Rev Fig. 13 Orientation Response in α to Perturbation ζ = 5 degrees in E-M Halos Spin stabilization of the spacecraft, under the perturbation ζ = 5 degrees, is explored for one E-M halo reference orbit of Az = 5 km and is attempted by spinning body R with respect to body B at two 4 perturbed 5 w spin.1 rad/s) w spin 1 rad/s) Rev Fig. 16 Orientation Response in α to Spin Stabilization in the E-M System It is apparent in Fig. 17 that spin stabilization to IAC-1-C1.8.1 Page 9 of 15

10 63rd International Astronautical Congress, Naples, Italy. Copyright c 1 by the International Astronautical Federation. All rights reserved. ζ deg) perturbed w spin.1 rad/s) w spin 1 rad/s) Rev Fig. 17 Orientation Response in ζ to Spin Stabilization in the E-M System γ deg) perturbed w spin.1 rad/s) w spin 1 rad/s) orbit is considered. The drift in the center of mass location relative to the reference orbit is displayed in Fig. 19. The orbital motion for the perturbed spacecraft is plotted in green, and the spin stabilized spacecraft, with spin rate equal to.1 rad/sec, in red; spin rate of 1 rad/sec is plotted in blue. This figure illustrates that spinning the spacecraft results in only very marginal improvement in maintaining the desired orbital motion relative to the reference orbit. Magnitude of Motion of G* wrt km) perturbed w spin.1 rad/s) w spin 1 rad/s) Rev Fig. 19 Orbital Motion Response to Spin Stabilization in the E-M System Rev Fig. 18 Orientation Response in γ to Spin Stabilization in the E-M System accommodate this particular perturbation depends on the spin rate employed, when focused on the nutation angle ζ. In the simulations of the perturbed motion, the spacecraft exhibits oscillatory behavior in α and unbounded motion in the other two angles which confirms that the spacecraft is not maintaining the initial orientation. Employing a spin rate of.1 rad/sec, the nutation angle response is observed to be damped; however, the spacecraft is still oscillating about ζ = 9 degrees. This response is not unexpected; in addition, recall that this orbit possesses an out-of-plane excursion of 5 km The precession of the ˆb-frame with respect to the â-frame appears in Fig. 16 to be bounded for this spin rate, and the spin angle for body B of the spacecraft is growing less rapidly in comparison to the perturbed case, as apparent in Fig. 18. When the spin rate of body R is increased to 1 rad/s, nearly all effects of the initial perturbation on the nutation angle are eliminated and the body is maintaining the initial orientation in ζ. Not surprisingly, the spacecraft still precesses with respect to the orbit frame at a comparable rate to the perturbed case, but the spin angle response is improved with respect to the perturbed case. Finally, the impact of spin stabilization on the This test case serves two purposes. First, the results confirm that the spacecraft is displaying motion consistent with previous results and serves to verify the model formulation. Secondly, it demonstrates an application of spin stabilization, at least for some orientation parameters, in the coupled orbit and attitude model in the CR3BP. VI. APPLICATION TO SOLAR SAILS One further objective in the development of the coupled orbit and attitude model is the capability to directly influence the orbit with the orientation. A solar sail offers a potentially significant coupling. Thus, it is also desired to consider the departure from the vicinity of the reference orbit, as predicted by the coupled orbit and attitude model, and to determine if the attitude configuration can control the orbital motion of the solar sail. VI.I REFERENCE FAMILY A solar sail immediately introduces the challenge of a more complex reference trajectory. Thus the first step is the construction of a family of periodic solar sail orbits in the vicinity of the L libration point. The computation is accomplished using a single shooting algorithm as described by Wawrzyniak. 13 The scheme is an extension of the vertically shifted linear approximation for elliptical orbits near the libration points that is explored in Simo and McInnes. 14 In the investigation here, an ideal solar sail model is assumed, IAC-1-C1.8.1 Page 1 of 15

11 63rd International Astronautical Congress, Naples, Italy. Copyright c 1 by the International Astronautical Federation. All rights reserved. where the nondimensional acceleration associated with the sail is, a s t) = a c ˆl1 t) û s ) ûs [7] where a c is the characteristic sail acceleration, ˆl 1 is the sunlight direction, t is the nondimensional time and û s is the direction of the sail normal. In the Earth-Moon rotating frame, the sunlight direction frame, denoted the l-frame, is defined, ˆl1 =cosω l t)â 1 sinω l t)â ˆl =sinω l t)â 1 + cosω l t)â [8] ˆl3 =â 3 where Ω l is the rate at which the sunlight direction is changing in the Earth-Moon reference frame, described as the ratio of the sidereal to synodic periods of the Earth-Moon system. 15 The orientation of the solar sail normal direction, û s, with respect to the Earth-Moon rotating frame, the a-frame, can be described by a body 3--1 θ s Ω l t, α s, γ s ) Euler angle rotation sequence. The third angle, γ s, set to zero, as the sail normal can be fully described by solely two angles. The sequence is noted to consistently convey the sail orientation with respect to the Earth-Moon rotating frame. The family of orbits computed by Wawrzyniak is based on the assumption that the vehicle maintains a fixed sail orientation with respect to sunlight direction in the Earth-Moon system. To generate this particular family, the value of θ s is held constant at degrees and the value of α s is held constant at degrees. This value for α s corresponds to the angle that generates the maximum out of plane force, based on the linear equations of motion as described by Simo and McInnes. 14 Using Equations [7] and [8] and the Euler angle rotation sequence, the contributions of the sail are incorporated into the traditional Circular Restricted Three Body Problem equations of motion. Note that the characteristic acceleration is permitted to vary over the family. 1 µ)x + µ) ẍ = ẏ + x d 3 + a c cosα s ) 3 cosω l t) µx 1 + µ) r 3 [9] 1 µ)y ÿ = ẋ+y d 3 µy r 3 a c cosα s ) 3 sinω l t) [3] 1 µ)z z = d 3 µz r 3 a c cosα s ) sinα s ) [31] From these equations of motion, a family of periodic, three dimensional, solar sail reference orbits is computed via a single shooting differential corrections process. The family of orbits is displayed in Fig.. a 3 km) a km) 1 1 a 1 4 km).9 Fig. Family of Solar Sail Reference Orbits in the Vicinity of L in the E-M System The color variation in Fig. indicates the range in values of the nondimensional characteristic acceleration across the family of trajectories. The smallest orbit, displayed in blue possesses a characteristic acceleration of.1.3 nm/s ). The largest orbit, displayed in red, corresponds to a value of nm/s ). VI.II COUPLED MODEL MODIFICATIONS The model framework that was previously developed to formulate the coupled orbit and attitude equations of motion is now modified slightly to incorporate the effects of solar radiation pressure. A particular reference orbit is selected, and accordingly, the characteristic acceleration corresponding to that reference is, thus, available. The rotational degree of freedom between the two bodies in the spacecraft model is assumed to be fixed, creating one rigid body. Realistic parameters are selected to reflect the mass and inertia of an approximate solar sail type spacecraft. The parameters are described such that the sail normal, û s corresponds to the body fixed ˆb 1 direction. The motion of the sunlight direction, as described by ˆl 1 in in Equation [8] is monitored and the force contribution due to the solar sail is computed by incorporating the mass of the solar sail and the acceleration described in Equation [7], dimensionalizing as appropriate. The initial conditions are consistent with the selected solar sail reference orbit. As expected, due to the orbit and attitude coupling, when the system equations of motion are propagated forward in time, the solar sail does not remain in the vicinity of the reference solution. VI.III TURN-AND-HOLD MODEL A multiple shooting scheme is employed in an attempt to control the orbital motion of the spacecraft so it remains in the vicinity of the desired reference. The applied control is not optimal. At some point in time, t i, it is noted that the orbital motion of the coupled orbit and attitude model deviates sufficiently IAC-1-C1.8.1 Page 11 of 15 Characteristic Acceleration nd)

12 63rd International Astronautical Congress, Naples, Italy. Copyright c 1 by the International Astronautical Federation. All rights reserved. from that described by the reference orbit. The position and velocity state of the coupled system at t i is denoted x i. A turn-and-hold strategy, similar to one proposed by Wawrzyniak, 15 is then employed to determine the attitude corrections that are required to guide the solar sail back to the vicinity of the reference orbit at some final time, t f. Intermediate patch points are introduced at times, t i+1, t i+,... t i+np, where n p represents the number of intermediate patch points. The position and velocity states at each intermediate point are described by, x i+1, x i+,... x i+np and the position and velocity at the final state, is denoted x f, and corresponds to a state at t f on the reference trajectory. Initial conditions for the position and velocity states are selected to be close to those associated with the reference trajectory; however, the position and velocity state at each patch point is part of the design vector and vary during the multiple shooting process. In the turn-and-hold strategy, the spacecraft is considered to maintain a fixed orientation with respect to a frame associated with the sunlight direction over each multiple shooting segment. This simplification to the model facilitates the computation of discrete corrections to the spacecraft orientation at a number of intermediate patch points. Finally, if the attitude corrections are successfully designed, their application to the coupled orbit and attitude model produces a configuration that remains in the vicinity of the reference orbit. The corrections are actually determined iteratively as required, based on the departure from the reference orbit. A schematic of the process is displayed in Figs. 1 to 3, using three internal patch points depicted in green). The starting location at t 1 is denoted by the red point and the desired final location at t 5 is the black point. The orientation corrections are applied at t 1 to t 4, computed via the turn-and-hold model, to the coupled orbit and attitude model. These corrections should aid the spacecraft in maintaining an orbital response that remains in the vicinity of the reference path. Coupled Orbit and Attitude Trajectory Multiple Shooting with Turn-and-Hold Model t i t Reference Trajectory t 3 t 4 Fig. 1 Initial Condition for Multiple Shooting Algorithm t 5 Coupled Orbit and Attitude Trajectory The sail normal direction at the initial point and at each intermediate patch point is defined as part of the design vector. Over the multiple shooting segt i t t 3 t 4 t 5 Reference Trajectory Multiple Shooting with Turn-and-Hold Model Fig. Corrections Determined Using Turn-and- Hold Model Coupled Orbit and Attitude Trajectory t i t t 3 t 4 t 5 Reference Trajectory Apply Corrections from Turn-and-Hold Model to Coupled Model Fig. 3 Corrections Applied to Coupled Orbit and Attitude Model ments, the sail normal direction is fixed with respect to a frame associated with the sunlight pointing direction, ˆl 1 as described in Equation [8]. At the initial time,t i, the sail normal direction is expressed, û si = U s1iˆl1 + U siˆl + U s3iˆl3 [3] Since the sail normal direction is a unit vector, the following equality constraint equation is applied at the initial point. U s1i U s1i + U si U si + U s3i U s3i 1 = [33] Similar sail normal directions are described for each patch point, û i+1 to û i+n, and the corresponding constraint equations are applied similarly to Equation [33]. The description of the sail normal from Equation [3] is applied to Equation [7] and the following updated equations of motion are derived that govern the behavior under the turn-and-hold model. Note that in this formulation, the characteristic acceleration is fixed and specified by the selection of the reference orbit. 1 µ)x + µ) µx 1 + µ) ẍ = ẏ + x d 3 r 3 + a c U 3 s1 cosω l t) + Us1U s sinω l t) ) [34] 1 µ)y ÿ = ẋ + y d 3 µy r 3 + a c U s1 U s cosω l t) Us1 3 sinω l t) ) [35] 1 µ)z z = d 3 µz r 3 a cus1u s3 [36] IAC-1-C1.8.1 Page 1 of 15

13 63rd International Astronautical Congress, Naples, Italy. Copyright c 1 by the International Astronautical Federation. All rights reserved. Consistent with the analysis by Wawrzyniak, continuity is enforced in position and velocity at each of the intermediate patch points. The equations of motion in Equations [34-36]are propagated forward using the initial conditions at times t i, t i+1... t i+np until reaching the time associated with the next patch point or the final time in the case of the last patch point, i.e. t i+1, t i+... t f. The position and velocity states at the end of the integration interval are denoted, x i+1 ū si ), x i+ x i+1, ū si+1 ),... x f x i+np, ū si+np ). Therefore, the position and velocity states at the end of the propagation segments are functions of both the state and sail normal vectors delivered at the start of the propagation, except for the first arc where the final states are merely functions of the sail normal vector since the initial position and velocity state are fixed at time t i. To force continuity in position and velocity, the states determined by the propagation must be equal to the states that define the next patch point. Inequality constraints are also implemented at the initial point and each subsequent patch point to ensure that the sail remains oriented with the appropriate side facing the Sun. Slack variables are employed, denoted η i, η i+1... η i+n. 15 Additionally, six inequality constraints must be added for this particular application, to allow for some variation in the position and velocity at the final time, yet remain within some box around the final state along the reference. x flb < x f xi+np, ū si+np ) < xfub [37] where x flb defines the lower bound and x fub specifies the upper bound. Twelve additional scalar slack variables are required to enforce the inequality constraints in Equation [37], six associated with the lower bound constraint, denoted s low and six associated with the upper bound constraint, denoted s upp. After the problem is completely formulated, the variables and constraints are collected in vectors. The design vector for the multiple shooting scheme, X is composed of the sail normal directions, slack variables, and the position and velocity states of the patch points as follows, X = ū si η i x i+1 ū si+1 η i+1... x i+np ū si+np η i+np s low s upp The constraints, F X), applied are follows, [38] F X) = û s i 1 U s1i + η i x i+1 ū si ) x i+1 û s i 1 U s1i+1 + η i+1 x i+ x i+1, ū si+1 ) x i+... x f xi+np, ū si+np ) xflb ) + s low x fub x f xi+np, ū si+np )) + s upp [39] The remainder of the corrections scheme is implemented in the same manner as Wawrzyniak. 15 Once the corrections are determined for the orientation via the turn-and-hold model, the prescribed orientations are applied to the coupled orbit and attitude model. VI.IV APPLICATION The multiple shooting control strategy is now applied to a solar sail type system. As previously noted, a rigid body system representative of a solar sail type spacecraft is created by fixing the degree of freedom between body B and body R. Each body is assigned a sail thickness of 7.5 µm in the ˆb 1 direction, and dimensions of 14 m in the ˆb direction and 7 m in the ˆb 3 direction. This creates a single rigid body with inertia properties on the same order of magnitude as the IKAROS sailcraft. A single reference orbit is selected from the solar sail family displayed in Fig. in blue. This reference orbit is constructed with a nondimensional characteristic acceleration of a c =.1 8. nm/s ). Given the sail reference orbit, the states are then employed in the coupled orbit and attitude solar sail model. The response, in terms of the center of mass motion, appears in red in Fig. 4; the path of G is propagated for.5 rev of the reference path and clearly departs the vicinity of this orbit. Employing spin stabilization, the sail is given a spin rate of.1 rad/s about the ˆb 1 axis in anticipation that spinning the sail improves the orbital motion response. In Fig. 4, the spinning sail response, depicted in black, is observed to remain near the reference for approximately.5 rev, after which it departs the vicinity. The multiple shooting strategy is now employed, in conjunction with the spin stabilized results, to further improve the sail orbital response. The turn-and-hold multiple shooting scheme is now applied. After a time interval corresponding to approximately.35 rev of the reference orbit, the response in the coupled, spinning model response is departed from the vicinity of the reference. The location of the spacecraft at this particular time is depicted by the red dot in Fig. 5. Seven patch points, represented by the green dots in Fig. 5, are assigned. The initial paths of all the intermediate multiple shooting IAC-1-C1.8.1 Page 13 of 15