Worldwide, space agencies are increasingly exploiting multibody dynamical structures for their most


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1 Coupled OrbitAttitude Dynamics in the ThreeBody Problem: a Family of OrbitAttitude Periodic Solutions Davide Guzzetti and Kathleen C. Howell Purdue University, Armstrong Hall of Engineering, 71 W. Stadium Ave., West Lafayette, IN Many relatively new techniques are being developed to incorporate the Circular Restricted ThreeBody Problem model into the early stages of the trajectory design. However, the attitude mission profile mostly remains reliant on methods established for Keplerian dynamics. A coupled orbitattitude model might leverage the Circular Restricted Three Body Problem dynamics to explore alternative, possibly more effective, profiles also in terms of the attitude response. The goal of this analysis is a nontrivial solution that is periodic both in its orbital and attitude states, when observed from the rotating frame. In the current investigation, tools largely employed for the orbital analysis (e.g., Floquet theory, differential corrections and continuation schemes) are effectively applied also in the coupled orbitattitude problem. Organized and predictable motion appears to naturally exist under certain conditions in such a model. As an example, a new family of orbitattitude periodic solutions is detailed. I. Introduction Worldwide, space agencies are increasingly exploiting multibody dynamical structures for their most advanced missions, with trajectory designs fundamentally based on an understanding of the Circular 1,, 3 Restricted ThreeBody Problem (CR3BP). These missions are comprised of astronomical observatories, deepspace human habitats or staging infrastructures, 4 as well as repositioned natural bodies. 5, 6 Improving the pointing accuracy of telescopes, safely docking to space stations, or reconstructing the orientation history of captured asteroids are possible challenges in developing the capability to control and predict the attitude motion in more complex dynamical environments. The spacecraft attitude history may also constrain the thrusting direction, thus, limiting the options for maneuvers. 7 There is, therefore, justified interest to better understand the attitude dynamics when it is coupled to the CR3BP regime. The earliest investigations from Kane, Marsh and Robinson consider the attitude stability of different 8, 9, 1 satellite configurations, assuming that the spacecraft is artificially maintained at the equilibrium points. Successive studies introduced Euler parameters, also known as quaternions, and Poncaré maps to explore the dynamics of a single body, that is still fixed to the Lagrangian points. 11, 1 The effects of the gravity torque along libration point orbits are examined by Wong, Patil and Misra for a single rigid vehicle in the Sun Earth system. 13 Wong, Patil and Misra select Lyapunov and halo orbits for their investigation, however, the reference trajectories are expressed in linear form, which limits the applicability of their results to relatively small orbits close to the equilibrium points. Considering another simplification of the CR3BP, the Hill problem, Lara et al. numerically reproduced the orbitattitude coupled dynamics of a large dumbell satellite on halo and vertical orbits in the EarthMoon system. 14 Assuming the spacecraft is in fast rotation, the attitude dynamics can be decoupled from the orbital dynamics by averaging the equations of motion over the fast angle. 14 Under this condition, it is demonstrated that sufficiently elongated structures may affect the stability of halo and vertical L orbits in the Hill problem. 15 Guzzetti et al. numerically solve the nonlinear Ph.D. Student, School of Aeronautics and Astronautics, Purdue University, Armstrong Hall of Engineering, 71 W. Stadium Ave., West Lafayette, IN Hsu Lo Distinguished Professor, School of Aeronautics and Astronautics, Purdue University, Armstrong Hall of Engineering, 71 W. Stadium Ave., West Lafayette, IN Fellow AAS; Fellow AIAA. 1 of 19
2 coupled orbit and attitude equations of motion using the Lyapunov family as reference orbits, but the rotation 16, 17 of the vehicle is limited to the orbital plane. Guzzetti et al. also incorporate solar radiation pressure and simple flexible bodies in the investigation. The full threedimensional coupled motion is explored by Knutson and Howell for a multibody spacecraft in nonlinear Lyapunov and halo reference orbits. 18, 19 Both Knutson and Guzzetti dedicate significant effort to identify conditions that determine bounded attitude solutions relative to the rotating frame in the CR3BP along nonlinear reference trajectories. Attitude maps are proven useful to recognize the orbital characteristics and the body inertia properties that enable the spacecraft to maintain its initial orientation with respect the rotating frame. Most recently, Meng, Hao and Chen analyze the case of a dualspin satellite in halo orbits and, employing a semianalytic expansion of the gravity torque, identify the main frequency components of that motion. 1 Along with stability diagrams at the equilibrium points, mapping techniques and frequency analysis, periodic solutions may contribute to the understanding of the attitude dynamics when it is coupled to the CR3BP. In this paper, solutions are sought that are periodic simultaneously in both the orbital and attitude states, when viewed in the rotating frame in the CR3BP. From the orbital dynamics perspective only, periodic orbits are one of the most successful approaches to the circular restricted three body problem, which lacks a closed form analytical solution. Poincaré first indicated periodic solutions as the primary means of understanding the CR3BP. However, at the time, the search of periodic orbits was significantly limited by the numerical capabilities, to the extent that a prominent investigator, such as Forest Moulton believed that certain periodic solutions are practically impossible to compute. 3 With the advent of artificial calculators, such concern is gone, as numerical procedures grant easy access to several type of periodic solutions. 4, 5 To date, many periodic orbits or their neighbouring dynamical structures have been successfully exploited for space mission applications in both the SunEarth 6 and EarthMoon 7 system. Catalogs of periodic orbits have also been compiled to better understand the dynamical behaviour 7 and to guide the mission design within the context of a given threebody system. 8 Periodic solutions are typically generated by numerically correcting an initial guess to meet specific boundary conditions, which include the continuity between the final and initial states. Physical symmetries or integral of the motion may also be leveraged to enforce periodicity. Even with the current computational capabilities, the convergence of algorithms for periodic orbits depends significantly on the accuracy of the initial guess and the implementation of the targeting scheme. In this investigation, viable methods to obtain precise initial guesses and effectively solve for periodicity are explored in the coupling of orbit and attitude. Several numerical schemes are available to solve boundary values problems. Because of its simplicity and adaptability, single shooting is frequently applied to orbital mechanics in the CR3BP. 9 The TwoPoint Boundary Value Problem (TPBVP) is converted to an Initial Value Problem (IVP), where the selected initial states are iteratively updated, on the basis of a Newton approach, until the constraints at the final time are satisfied within a given tolerance, i.e. differential corrections. If the single shooting is employed in combination with specific symmetry features of the motion, then it is obviously limited to solutions that shares those symmetries. 9 A direct extension of the single shooting scheme is to target multiple states along the path, rather than only the final states. This method, also known as multiple shooting, is introduced by Keller to solve general TPBVP s 3 and is now largely applied to the computation of periodic orbits. 31, 3, 33 A common implementation of the multiple shooting, denoted as parallel shooting, requires all the design variables to be simultaneously corrected to target the complete set of constraints along the path at each iteration. An alternative multiple shooting algorithm is the, so called, two level corrector, that was originally developed in astrodynamics to compute quasiperiodic motions 34 and introduced a nested level of iteration to converge on a subset of the constraint vector by adjusting a subset of the free variables. This method is currently also applied to periodic and nonperiodic trajectories to impose various path constrains to the baseline trajectory. 35 Finally, specific parameters of the periodic solution may be varied to form other periodic solutions that belong to the same dynamical family. This continuation process can be based on the direct modification of the natural parameters or using the direction tangent to the null space of the monodromy matrix associated to the reference periodic solution. The latter algorithm is denoted pseudoarclength continuation and, in some applications, benefits from a more general and robust formulation than the natural parameter continuation. 36 In this investigation the single shooting scheme and the pseudoarclength strategies are adapted to compute and continue periodic orbitattitude solutions. Note that, the procedure can be easily extended to incorporate multiple shooting, after the problem is demonstrated for the single shooting scheme. In Section II, the dynamical model and the equations of motion that describe the orbitattitude evolution of 19
3 for a single rigid spacecraft in three body system are examined. The overall procedure that is adopted to generate orbitattitude periodic solutions is presented in Section III: from defining a good initial guess to continuing the family of solutions. Representative new orbitattitude periodic solutions are illustrated in Section IV for an axisymmetric satellite on L 1 Lyapunov orbits in the EarthMoon system. II. Dynamical Model Consider a single rigid spacecraft in the gravitational field emanated by two massive bodies P 1 and P. Assume that the bodies P 1 and P are moving on circular orbits about their common barycenter, and their motion is unaffected by the presence of the spacecraft (whose mass is negligible compared to the mass of P 1 and P ). The translational motion of the space vehicle is conveniently described by the Cartesian position coordinates (x, y, z) of the spacecraft center of mass relative to the baricenter of the system, as measured in a frame that rotates at the planetary system angular rate Ω. At time t =, the rotating frame ˆx, ŷ, ẑ, is aligned to the inertial frame ˆX, Ŷ, Ẑ. At successive instants of time, the rotating frame is defined such that P 1 and P remain on the ˆxaxis and ẑ is equal to the normal vector Ẑ of the planetary orbiting plane, as depicted in Figure 1. Referring to the figure, the body frame to describe the spacecraft orientation is also depicted, rendered by the tern of unit vectors ˆb 1, ˆb, ˆb 3. In defining kinematics quantities, the notation a c indicates that the motion of a generic c frame is observed from a generic a frame. For convenience, I denotes the inertia frame, r the rotating frame and b the body frame. Figure 1. Frames representation. To reproduce the orbital dynamics of the spacecraft, the gravity force is modelled neglecting the finite extension of the vehicle. Accordingly, the orbital behaviour of the vehicle is equivalent to the response of a pointmass distribution at its center of mass. Perturbations more equally significant as compared to the actual mass distribution, such as the solar radiation pressure, are also neglected in this simplified analysis. The resulting problem is familiar as the Circular Restricted ThreeBody Problem (CR3BP), which 3 of 19
4 is encapsulated in the following equations: ẋ = v x ẏ = v y ż = v z f x = v x = x + v y v y v z = y v x (1 µ)z = d 3 µz r 3 (1 µ)(x + µ) µ(x 1 + µ) d 3 r 3 (1 µ)y d 3 µy r 3, (1) where x, y, z are the position coordinates in the rotating frame; v x, v y, v z are the velocity components of the spacecraft observed from the rotating frame. The distances from P 1 and P are respectively denoted by d = (x + µ) + y + z, r = (x 1 + µ) + y + z. The system of equations (1) is normalized, such that the total mass of the system, the distance between the two attractors, the universal gravitational constant and the angular frequency Ω are unitary. The normalized period of P 1 and P in their orbits about their barycenter is equal to π. After the normalization, the planetary system is dynamically represented by the mass parameter µ only, which is defined as the ratio between the mass of P and the total mass of the system. Assuming the mass of P 1 is greater than mass of P, the location of P 1 along the ˆxaxis in nondimensional units is µ, whereas P is at 1 µ nondimensional units from the barycenter. Particular solutions of Eq. (1) include equilibrium points, periodic orbits and quasiperiodic trajectories. 37 The orientation of the spacecraft is represented through a body reference frame ˆb 1, ˆb, ˆb 3, that is fixed in the center of mass of the spacecraft and aligned with inertia principal directions. Euler equations of motion are used to reproduce the rotational dynamics of the vehicle, incorporating the gravity torque that is exerted by P 1 and P. A secondorder approximation is developed to express the gravitational moment. The resulting dynamical equations for the spacecraft attitude are ω 1 = I ( 3 I 3µ1 I 1 d 5 g g 3 + 3µ ) r 5 h h 3 ω ω 3 f ω = ω = I ( 1 I 3 3µ1 I d 5 g 1g 3 + 3µ ) r 5 h 1h 3 ω 1 ω 3, () ω 3 = I ( I 1 3µ1 I 3 d 5 g 1g + 3µ ) r 5 h 1h ω 1 ω where I ω b = [ω 1 ω ω 3 ] T is the angular velocity vector of the body relative to the inertial frame and expressed using ˆb 1, ˆb, ˆb 3 as the vectorial basis; I 1, I and I 3 denote the principal central moments of inertia in the corresponding directions; µ 1 and µ are the planetary constants of P 1 and P, which satisfy µ 1 = 1 µ and µ = µ in nondimensional units; h i represents the projections of the position vector relative to P 1 into the body frame, while g i are the projection of the position vector relative to P into the same frame. The instantaneous orientation of the body frame (which is the orientation of the vehicle) relative to the inertial frame is defined using the quaternion vector (also known as Euler parameters) I q b = [q 1 q q 3 q 4 ] T : the components [q 1 q q 3 ] T can be interpreted as the axis of rotation expressed in the body frame and scaled as function of the angle of rotation, whereas q 4 contains complementary information for the rotation angle only. The components of the quaternion vector are constrained such that q 1 + q + q 3 + q 4 = 1. (3) 4 of 19
5 The quaternion vector is related to the body angular velocity ω via the kinematic relationship q 1 = 1 (ω 3q ω q 3 + ω 1 q 4 ) f q = q = 1 ( ω 3q 1 + ω 1 q 3 + ω q 4 ) q 3 = 1 (ω q 1 ω 1 q + ω 3 q 4 ). (4) q 4 = 1 (ω 1q 1 + ω q + ω 3 q 3 ) Upon the introduction of quaternions, the projections g i and h i, which determine the gravitational moment in Eq. (), are expressed as functions of the instantaneous orientation of the spacecraft, i.e., g 1 q1 q q3 + q4 (q 1 q + q 3 q 4 ) (q 1 q 3 q q 4 ) g = (q 1 q q 3 q 4 ) q1 + q q3 + q4 cos(ωt) sin(ωt) x + µ (q q 3 + q 1 q 4 ) sin(ωt) cos(ωt) y g 3 (q 1 q 3 + q q 4 ) (q q 3 q 1 q 4 ) q1 q + q3 + q4 1 z (5) and h 1 h h 3 = q1 q q3 + q4 (q 1 q + q 3 q 4 ) (q 1 q 3 q q 4 ) (q 1 q q 3 q 4 ) q1 + q q3 + q4 (q q 3 + q 1 q 4 ) (q 1 q 3 + q q 4 ) (q q 3 q 1 q 4 ) q1 q + q3 + q4 cos(ωt) sin(ωt) x + µ 1 sin(ωt) cos(ωt) y. 1 z (6) Equations (1), () and (4) form the entire set of coupled equations of motion that is necessary to describe the orbitattitude dynamics of a small rigid body within the context of the CR3BP. Given the Eq. () and (4), the attitude response is influenced by the orbital states, but no attitude terms are present in Eq. (1), so that the classic structure of the CR3BP is preserved. Accordingly, this model may not be applicable to spacecraft whose characteristic dimension is large compared to the distance from P 1 and P. External actions, other than gravity, may also introduce a dependency of the orbital path on the body attitude. However, the current model is easily modified to incorporate large spacecraft and external perturbations. Corresponding to the set of orbitattitude equations of motion, the linear differential relationship between initial and final states, or State Transition Matrix (STM), is where A(t) is the timevariant Jacobian of the system A(t) = d Φ(t, ) = A(t)Φ(t, ), (7) dt f x x f q x f ω x f x v f q v f ω v f x q f q q f ω q f x ω f q ω f ω ω. (8) Note that f x q = and f x =, since Eq. (4) does not contain the attitude states; accordingly, the ω Jacobian is a block diagonal matrix. The system to describe the translational and rotational behaviour of the spacecraft consists of 13 equations of motion, but only 1 equations are actually independent. The components of the quaternion vector are, in fact, related by Eq. (3), which implies that one of the kinematic relationships in Eq. (4) is unnecessary for the complete description of the system evolution. One of the quaternion vector components can be considered a function of the remaining components of the vector. Rather than substituting Eq. (3) into the equations of motion, it is more practical to maintain the whole set of equations and reduce only the Jacobian (and the STM, consequently) to a 1 by 1 matrix, which corresponds exclusively to the independent variables. Assume that q 4 is a function of q 1, q, q 3, such that q 4(q 1, q, q 3 ) = 1 q 1 q q 3 ; 5 of 19
6 the infinitesimal variation of q 4 is a function of the independent variations of the remaining quaternions, i.e., The previous relationship between the infinitesimal variations yields q 4 δq 4 = q 1 δq 1 q δq q 3 δq 3. (9) q 4 q i = q i q 4 for i = 1,, 3 (1) which is eventually exploited to compute the partials of the Jacobian matrix relative to q 1, q, q 3 df (q 1, q, q 3, q 4 (q 1, q, q 3 )) = f + f q 4 for i = 1,, 3. (11) dq i q i q 4 q i Since q 4 is not regarded as an independent variable, no partials of the equations of motion with respect to q 4 are necessary to construct the Jacobian matrix. The differential equation for q 4 is then also excluded during the calculation of the Jacobian matrix. Incorporating one less variable and one less equation, the Jacobian is a 1 by 1 matrix. Including the trivial equations, there are total differential equations to simulate the system response and access the linear differential relationship between the initial and final states, which is generally sufficient to identify and precisely compute specific solutions. III. Algorithm In this investigation, periodic solutions are sought in both the orbital and attitude states as viewed from the rotating frame in the CR3BP. As is generally true in the CR3BP, it is difficult to determine a generic position or orientation of the vehicle that is likely to evolve into a bounded motion, particularly one that is likely to remain more controllable than a diverging response. If the spacecraft dynamics are numerically simulated to evaluate the boundedness in the spacecraft orbit and attitude states, it is not typically legitimate to extend any conclusions beyond the observed horizon. However, solutions that are periodic are also bounded over an infinite timespan (assuming no perturbing factors), without the necessity to integrate over that infinite time interval. Additionally, Floquet theory for periodic solutions grants easy access to the stability properties. Presently, periodic solutions must be numerical computed as there not exist an analytical closedform solution for the problem. Numerical methods can produce a periodic response by iteratively adjusting a reference path, but a good initial guess is required. There exist many approaches to retrieve an accurate initial approximation for the periodic solution; these methods include the linear stability analysis of equilibrium points, the scrutiny of bifurcations in the dynamical structure across a family of known periodic motions, resonances and mapping techniques. A straightforward strategy is proposed here to identify complex periodic behavior from the analysis of the dynamical structure across a reference family of elementary solutions. First, the reference orbital path is selected from families of periodic orbits that are already available in the CR3BP. Second, the spacecraft configuration and the initial attitude states are manually assigned to generate an intuitive and elementary periodic response in the rotational dynamics, as observed from the rotating frame, over the reference orbit. Third, the reference orbit is varied across its family and the linear approximation for the dynamical structure is explored such that new complex periodic solutions might emerge. If an adequate approximation for a possible periodic motion is identified, such an initial guess is corrected to accurately produce the corresponding periodic behavior. Lastly, other periodic solutions in the same family may be computed by numerical continuation. The State Transition Matrix (STM) corresponding to a periodic solution is computed over one period and denoted as the monodromy matrix. According to Floquet theory, the monodromy matrix contains information about the behavior in the dynamical neighborhood of the reference periodic solution and may point to alternative periodic motions in its vicinity. Specifically, the eigenvalues of the monodromy matrix are monitored to detect possible dynamical bifurcations that yield a new family of periodic solutions. Since the orbital path is assumed periodic and it is (within the current model) independent from the rotational response, only the eigenvalues corresponding to the attitude dynamics need examination. The STM for the orbitattitude dynamics in Eq. (7) is a 1 by 1 lower triangular block matrix, given that the partials fx q and fx ω equal zero. The two diagonal 6 by 6 blocks are associated to the orbital states and the attitude states, respectively. The 6 by 6 block below the diagonal is due to the coupling of the attitude dynamics to the orbital position and velocity of the spacecraft. The eigenvalues associated with the rotational dynamics 6 of 19
7 are equal to the eigenvalues of the diagonal block corresponding to attitude states. Such eigenvalues are comprised of a unitary trivial pair that represents the existence of the reference periodic solution and its complementary family. The remaining eigenvalues can form either real and reciprocal pairs or complex conjugate pairs. If a nontrivial pair is also equal to 1, the formation of a new family of periodic solutions via tangent bifurcation may be indicated. If the nontrivial pair is unitary but with a negative sign, a periodic solution that has twice the period of the reference may exist (i.e., a perioddoubling bifurcation). Besides tangent and perioddoubling, other types of bifurcations exist, but they are not considered in this analysis, for simplicity. The states along a solution that is periodic, as observed in the rotating frame, do not necessarily preserve periodicity when described in terms of an inertial observer. This statement holds true for both the orbital and attitude variables. In the presence of resonances, the states may be periodic in both the rotating and inertial frames, but that type of motion is not generally the case. The monodromy matrix, i.e., the STM, must reflect the correct choice of the observing frame, to supply accurate information about the periodic motion. Relative to the rotating frame, the spacecraft attitude is described by the quaternion vector r q b = [ q 1 q q 3 q 4 ] T ; by the rule of successive rotations for the quaternion representation, r q b can be transformed into the vector I q b employed in Eq. (3), which describes the orientation of the body frame respect to the inertial frame cos(ωt/) sin(ωt/) I q b sin(ωt/) cos(ωt/) = r q b, (1) cos(ωt/) sin(ωt/) sin(ωt/) cos(ωt/) where the quaternion representing the rotation from the inertial to the rotating frame is, in fact, I q r = [ sin(ωt/) cos(ωt/)] T. To seek periodic solutions relative to the rotating frame, a conversion of the STM to reflect the correct observer seems more practical than the direct substitution of Eq. (1) into the equations of motion and then a reevaluation of the Jacobian. The STM in Eq. (7) linearly relates the variation of the initial states to the variation of the final states relative to the reference solution δx(t f ) δx() Φ xx δx() δ I qr b (t f ) = Φ(tf, ) δ I qr b () = Φ qx Φ qq Φ qw δ I qr b (), (13) δ I ω b (t f ) δ I ω b () Φ wx Φ wq Φ ww δ I ω b () where δ I qr b (t) denotes the independent variations at time t in the quaternion vector that describes the orientation of the body relative to the inertial frame. Recall that the quaternion vector is comprised of four components, which are subjected to the constraint Eq. (3). Thus, only three components can actually describe independent variations. Assume Eq. (9) is employed to define δq 4 as function of the independent variations δq 1, δq, δq 3, such that δ I qr b (t) = [δq 1 δq δq 3 ] T. Using Eq. (1), the variation relative to the inertial frame can be related to the variation in the rotating frame as δ I qr b δq 1 cos(ωt/) sin(ωt/) δ q 1 δ q = δq = sin(ωt/) cos(ωt/) δ q 3 = T (t)δr q b. (14) δq 3 cos(ωt/) sin(ωt/) δ q 4 It is also convenient to rewrite Eq. (9) in a vector form to reduce the variation of the quaternion r q b to its independent components δ q 1 1 δ r q b δ q = δ q 3 = 1 δ q 1 δ q 1 = V ( q 1, q, q 3 )δ r qr b. (15) δ q 3 δ q 4 q 1 / q 4 q / q 4 q 1 /q 3 Equations (14) and (15) combine to yield a linear timevarying relationship between the variations expressed in terms of the rotating and inertial frame, i.e. δ I q b R = T (t)v ( q 1, q, q 3 )δ r q b R = T R δ r q b R, (16) 7 of 19
8 where T R is equal to the identity matrix at the initial time, since the rotational frame is assumed to be initially aligned with the inertial frame. The variation of I q b at the final time t f is computed from the variation at the initial time using Eq. (13); alternatively, δ I q b (t f ) can also be calculated from Eq. (16), if the variation at final time is known relative to the rotating frame rather than the inertial frame. Equating the results from Eq. (16) and Eq. (13) at the final time t f produces δ I q b R(t f ) = T R (t f )δ r q b R(t f ) = Φ qx (t f, )δx() + Φ qq (t f, )δ I q b R() + Φ qω (t f, )δω() (17) which can be rearranged to explicitly express the variation of the spacecraft orientation relative to the reference solution at the final time δ r q b R(t f ) = T R (t f ) 1 Φ qx (t f, )δx() + T R (t f ) 1 Φ qq (t f, )δ r q b R() + T R (t f ) 1 Φ qω (t f, )δω(), (18) where δ I qr b () = δr qr b (). Leveraging Eq. (18), the STM can be transformed to reflect variations of the spacecraft attitude relative to the rotating frame δx(t f ) δx() Φ xx δx() δ r q b R (t f ) δ I ω b (t f ) = Φ(t f, ) δ r q b R () δ I ω b () = T 1 R Φ qx T 1 R Φ qq T 1 R Φ wx Φ wq Φ ww Φ qw δ r q b R () δ I ω b (). (19) The STM Φ in Eq. (19) is the appropriate form to identify and correct solutions that are periodic in the orbital and attitude states relative to the rotating frame. The rotating frame rotates at constant rate relative to the inertial frame, thus, the angular velocity of the spacecraft observed in the rotating frame differs by a constant offset from the angular velocity relative to the inertial observer. Because the offset is constant and it is not an explicit function of time, if a solution is periodic in the rotating frame, the angular velocity is periodic in both the rotating and inertial frames. Therefore, there is no necessity to further modify the STM in Eq. (19). Using the correct form for the STM, if a pair of eigenvalues from the monodromy matrix indicates a possible new periodic behavior nearby the reference solution, the corresponding eigenvectors supply a linear approximation of the direction for varying the initial conditions to trigger the new periodic motion. Given the linear nature of the initial guess, the associated solution is not expected to be precisely periodic. It is, therefore, necessary to introduce a targeting algorithm to produce periodicity of the state variables to the desired level of accuracy. In this investigation, a timevariable single shooting algorithm is employed. Accordingly, the initial states and the final time t f are selected as design/free variables ξ = [x() r q b R() I ω b () t f ] T. Specific cases may require a more restricted set of design variables for convergence, but the general procedure can be formulated including all the independent states. To enforce periodicity, continuity between the final and initial state variables is written as a constraint vector function x(t f ) x() F = qr b (t f ) r qr b () =. () I ω b (t f ) I ω b () Since the reference orbital path is selected a priori to be periodic and it is not directly impacted by the attitude response, it may at first seem redundant to enforce periodicity also in the orbital variables; however, the variation of the final time t f may be required to target periodicity in the rotational components of the motion: if t f is varied, the reference orbit is no longer periodic. Therefore, the orbit must also be adjusted to reestablish periodicity, essentially necessitating a simultaneous correction of both the orbital and the attitude states. Zeros corresponding to the vector constraint function in Eq. () are determined via a NewtonRapson iterative scheme. The constraint function F is expanded about an initial reference in Taylor series to the first order F (ξ) = F (ξ ) + DF (ξ )(ξ ξ ), (1) where DF is the Jacobian of the constraint function with respect to the design variables ξ Φ xx (t f, ) 6x3 6x3 I 6x6 3x3 3x3 ẋ(t f ) DF = Φ qx (t f, ) Φqq (t f, ) Φqω (t f, ) 3x6 I 3x3 I 3x3 q R(t b f ). () Φ ωx (t f, ) Φωq (t f, ) Φωω (t f, ) 3x6 3x3 I I 3x3 ω b (t f ) 8 of 19
9 The linear expansion of F in Eq. (1) is set equal to zero and iteratively solved for ξ. Note that 1 constraint equations are written in terms of 13 design/free variables, which implies infinitely many solutions to the problem. The minimum norm solution is adopted to derive the update equation for the design variables ξ i+1 = ξ i + (DF T (ξ i )DF (ξ i )) 1 DF T (ξ i )F (ξ i ). (3) Accordingly, Eq. (3) is recursively applied to update the free variables until the metric F is below the desired tolerance. Since constraining conditions are only enforced in the final states and the design/free variables are comprised of the initial states plus the total propagation time, the correcting algorithm is basically a timevariable single shooting scheme. After an orbitattitude periodic solution is available, periodic solutions in the same family can be generated using a pseudoarclength continuation procedure. 36 Essentially, the design variables are modified in the direction tangent to their exact nonlinear variation along the family. The tangential direction is computed as the null space of the Jacobian matrix DF for the last converged solution ξ, κ = N (DF (ξ )). (4) Next, an equation is appended to the constraint vector to impose a step of size ds in the tangent direction [ ] F G = (ξ ξ, (5) ) T κ ds such that the derivation of the augmented constraint vector yields a square augmented Jacobian matrix [ ] DF DG =. κ T Finally, a unique solution for the next member of the family is generated via the simple iterative update equation ξ i+1 = ξ i + DG(ξ i ) 1 G(ξ i ). Distinct from the parameter continuation, implementation of the pseudoarclength continuation scheme does not depend upon previous knowledge of the family evolution from one member to the next; additionally, the pseudoarclength approach is reasonably robust and generally prevents the continuation process from jumping to a different family of solutions. IV. A Periodic Solution Considering an axysimmetric satellite moving in the EarthMoon system, a family of solutions that are periodic in both the orbital and attitude variables, as observed from the rotating frame, is constructed. The procedure is initiated with the selection of a reference family of orbitattitude solutions that are known a priori to be periodic in the rotating frame; since such periodic orbitattitude solutions are currently not available in the CR3BP, one possible approach is the assumption of an elementary or intuitive scenario as an initial reference; then, the procedure can be reiterated as more complex solutions are produced. As reference orbital paths, several families of periodic orbits are already accessible in the CR3BP. 8 These families include the L 1 Lyapunov family displayed in Figure, whose members serve as reference periodic orbits to construct an elementary orbitattitude solution in this investigation. In such a way, the orbital states are clearly periodic in the rotating frame. Next, the spacecraft configuration and the initial conditions are selected to generate a simple rotational response that periodically repeats at each revolution along the reference periodic orbit. An axisymmetric mass distribution facilitates the identification of a periodic attitude solution along the reference trajectories and it is a common configuration for space vehicles. There is, therefore, significant interest in commencing the search for orbitattitude periodic solutions from axisymmetric spacecraft. The vehicle is assumed to be axisymmetric about the ˆb 3 axis, such that I t = I 1 = I is the transversal moment of inertia and I a = I 3 is the axial moment of inertia. Accordingly, the spacecraft topology is uniquely described by the inertia parameter k = I a I t I a, 9 of 19
10 which varies in the interval [, 1) to yield a disklike mass distribution with I a I t. If the spacecraft is axisymmetric and I 1 = I, the angular velocity about the axis of symmetry ˆb 3 is constant at all time, since ω 3 = from Eq. (). Then, assume that ˆb 3 is initially orthogonal to the orbiting plane z = for the L 1 Lyapunov orbits, such that q 1 () = q () =, and consider ω 1 () = ω () =. Substituting q 1 = q = and ω 1 = ω = into Eq. () at the initial time yields the result that ω 1 and ω are also equal to zero at any time, if the reference orbit is planar, i.e. z =. Under these conditions, the angular velocity vector is constant throughout the motion and spinning the spacecraft about ˆb 3 at the a rate Ω equal to the rate of the rotating frame produces a periodic solution. In fact, since the angular velocity vector is equal to I ω b = [ Ω] T, and remains constant, the spin axis remains perpendicular to the orbiting plane and the vehicle maintains a rotation rate equal to the rate of the rotating frame relative to the inertial frame. Accordingly, for an observer fixed in the rotating frame the spacecraft never changes its initial orientation, regardless of the x, y location, which is trivially a periodic solution as the vehicle moves along the reference periodic path. Considering this elementary orbitattitude solution, Figure 3 depicts the spacecraft orientation at representative instants of time along a L 1 Lyapunov reference trajectory, both as observed in the rotating frame. The orbitattitude reference solution in this example is comprised of a wellknown periodic trajectory, x y [km] Earth Moon L x [km] x 1 5 Figure. L 1 Lyapunov family of periodic orbits in the EarthMoon rotating frame. (a) 3D perspective. (b) xy plane perspective. Figure 3. Representation in the rotating frame of the elementary orbitattitude solution that is assumed as reference. The colored axes denote the body frame, which remains aligned with the rotating frame at all the time. 1 of 19
11 i.e., an L 1 Lyapunv orbit, and an elementary periodic attitude regime along that trajectory. Varying the reference orbit across the members of the L 1 Lyapunov family, possible bifurcations of the elementary attitude response to various complex periodic solutions may become evident. Generally, one possible type of bifurcation is identified as a local mutation of the linear stability properties of the reference solution. Linear stable modes of the reference periodic solution are associated with eigenvalues of the monodromy matrix λ i, real or complex, that possess a modulus value lower than 1; linear unstable modes correspond to λ i > 1, while marginally linear stable modes to λ i = 1. In the linear approximation, a periodic reference solution is stable (or marginally stable) if the inequality λ i 1 holds true for all the eigenvalues, whereas, it is unstable if any of the eigenvalues possess a modulus greater than one, i.e. λ i > 1 for at least one eigenvalue. When the stability structure of the periodic reference solution changes, one pair of eigenvalues passes through the threshold λ i = 1. If the threshold λ i = 1 is vizualized as a unitary circle on the complex plane, and if the crossing occurs on the real axis at λ i = 1, the change of stability is labelled a tangent bifurcation and may indicate the existence of a new periodic solution in the vicinity of the reference with a similar period. When the stability change along the family occurs on the real axis at λ i = 1, the dynamics may bifurcate to a new periodic solution with twice the period of the reference, denoted a perioddoubling bifurcation. Specifically, tangent and perioddoubling bifurcations at an orbitattitude reference solution can be identified by varying the reference trajectories across the selected family of periodic orbits and monitoring the real component of the eigenvalues for ] [ Φqq Φqω Φ att = Φ ωq Φωω that denotes the diagonal block of the STM in Eq. (19) associated to the rotational dynamics. Considering an inertia ratio k =.8, Figure 4 displays a representative evolution of the real component in the nontrivial eigenvalues of Φ att, as the reference orbit, which is represented by the corresponding orbital period on the xaxis, varies across the L 1 Lyapunov family. A single curve above 1 or below 1 indicates an eigenvalue with modulus certainly greater than 1, such that the reference attitude solution is unstable. When all the curves lie within the range [1 1], the rotational motion may be marginally stable in terms of linear approximation (a more definitive conclusion on the linear stability can be drawn evaluating the modulus of the eigenvalues, not just the real part) a. Two curves simultaneously crossing the line at 1, as depicted in Figure 4, may signal a tangent bifurcation, while a crossing through the line at 1, also indicated in Figure 4, may point to a perioddoubling bifurcation. A large variability in the stability structure is evident in Figure 5 where the bifurcation diagrams appear as a function of the mass distribution, which is represented by sample values of the inertia ratio k within the interval [,1). From the figure, the challenge to identify specific orbitattitude solutions in the CR3BP dynamical regime is evident. The high sensitivity of the stability to system parameters is apparent. Several candidate bifurcations for possible periodic solutions exist across the Lyapunov family as well as the inertia ratio range; a single representative bifurcation is investigated, one that corresponds to an axisymmetric spacecraft with inertia ratio k =.8 travelling along a L 1 Lyapunov reference orbit with approximately a 14day period. The representative bifurcation is indicated by an arrow in Figure 4. A linear approximation of the dynamical behavior nearby the reference solution is employed to initially guess periodic solutions that may emanate from the candidate bifurcation. The linear approximation of a nontrivial attitude periodic solution, associated with the reference periodic orbit, is given by [ ] r q b R I ω b [ ] = r q b R I ω b, + pλ, (6) where [ ] denotes the reference solution, Λ is the normalized eigenvector of Φ att corresponding to the pair of eigenvalues that reflects the stability change, and p represents a scaling factor, one that is sufficiently small to remain in the region of validity for the linear approximation. Although the initial conditions from Eq. (6) do not yield an orbitattitude solution that is precisely periodic in the nonlinear model, a differential corrections scheme, as described in Section III, can be employed to target periodicity to the desired degree a This claim also assumes that the reference orbit is either stable or artificially fixed. Since the attitude response is naturally coupled to the orbital regime, if the reference orbit is unstable, such as L 1 Lyapunov orbits, the instability propagates to the attitude variables, regardless the eigenvalues from the matrix Φ att. 11 of 19
12 Re(λ) Possible tangent bifurcation λ 3 λ 5,λ 6 λ 4 Selected bifurcation 3 Possible period doubling bifurcation Marginally stable attitude motion Figure 4. Dynamical bifurcation diagram of the elementary reference solution for an axysimmetric spacecraft with inertia ratio k =.8 along the L 1 Lyapunov family in the EarthMoon system. of accuracy. Note that, even if the orbital trajectory is known a priori to be periodic, orbital variables may also be incorporated in the corrections process along with the attitude states. In fact, small adjustments in the initial periodic orbit may be required to achieve periodicity in the rotational response. Leveraging specific features of the motion, periodicity in the orbitattitude states can be enforced via a reduced form of the constraints vector in Eq. (). Considering that Lyapunov orbits are symmetric across the xaxis in the rotating frame, periodicity in position and velocity may be achieved accordingly, as described in the mirror theorem, by imposing perpendicular crossings of the xaxis. To meet this criteria, the initial values for y and ẋ are fixed to zero and an identical condition is targeted at the final time. An additional reduction, valid for axisymmetric spacecraft, consists of removing the continuity equation for the angular velocity component about the symmetry axis from the constraint vector F. If the axis of symmetry corresponds to ˆb 3, such that I 1 = I, then Eq. () yields ω 3 =, regardless of any other state variable. Because ω 3 remains constant throughout the motion and its value is not determined by any other free variable except ω 3 () itself, including the periodicity equation for ω 3 in the constraint vector is unnecessary and yields a singular Jacobi constant value. The final constraint vector to target orbitattitude periodic solutions, based on orbits in the L 1 Lyapunov family, is y(t f ) y() ẋ(t f ) ẋ() F = r qr b (t f ) r qr b () =. (7) ω 1 (t f ) ω 1 () ω (t f ) ω () which is comprised of 7 equations of constraint. To target Eq. (7) by iteratively applying Eq. (3), 9 free variables are employed, which are summarized as ξ = [x() ẏ() r q b R() I ω b () t f ] T. The desired output of the corrections algorithm is a solution, distinct from the reference, that is periodic in both translational and rotational states, as observed in the rotating frame in the CR3BP. As the first periodic response is retrieved, the corresponding family of solutions is generated using the pseudoarclength continuation. Note that, if the null space of the Jacobian DF has dimension greater than one, the tangent direction κ in Eq (4) and (5) must be empirically evaluated by testing the multiple directions predicted by the null space to continue the family. 1 of 19
13 4 4 4 Re(λ) Re(λ) Re(λ) (a) k = (b) k = (c) k =.3. Re(λ) Re(λ) Re(λ) (d) k = (e) k = (f) k =.6. Re(λ) Re(λ) Re(λ) (g) k = (h) k = (i) k =.9. Figure 5. Dynamical bifurcation diagrams for the elementary reference solutions for an axysimmetric spacecraft on L 1 Lyapunov orbit in the EarthMoon system; stability as a function of the inertia parameter varies. The results of the corrections and continuation process are members of a family of orbitattitude periodic solutions in the CR3BP, such that the orbital motion describes a L 1 Lyapunov orbit, and the attitude motion follows a nontrivial periodic behavior. The computed family is plotted in Figure 6 as appears in terms of the quaternion subspace. Initial conditions for representative members of this family are reported in Table 1, along with the continuity errors between the initial and final time. From Table 1, different solutions possess different periods, hence, they can not possibly share the same reference periodic orbit, which would be characterized by a unique orbital period. Accordingly, the reference L 1 Lyapunov orbit must be slightly adjusted, as evident in Figure 7, to match the period of the attitude motion as the family of solutions is continued. The adoption of an orbitattitude corrections algorithm is, therefore, warranted to seek rotational periodic behaviors for vehicles on periodic orbits, even if the orbital path is known a priori and does not depend on the spacecraft orientation. Referring to Figure 6, the projection of the solution into the quaternion subspace, which reflects the orientation of the body relative to the rotating frame, appears symmetric with respect to the q 1 axis, i.e. (q 1, q, q 3 ) (q 1, q, q 3 ). That may be linked to a physical symmetry of the motion. Thus, to gain more insight into the physical evolution of the spacecraft orientation, it is useful to monitor the pointing history of the axis of symmetry ˆb 3 = [b 3x b 3y b 3z ] T by plotting its components in the rotating frame as in Figure 8. In the elementary reference configuration, the axis of symmetry is perpendicular to the orbiting plane at any time, such that its components in the rotating frame 13 of 19
14 Table 1. Initial conditions for a nontrivial orbitattitude family of solutions assuming an axisymmetric spacecraft with inertia ratio k =.8 in the EarthMoon system. Given the initial and final time, the continuity error in position and velocity is denoted by orb, while the continuity error in the attitude state variables is represented by att. x ẏ q 1 q q 3 ω 1 ω ω 3 Period orb att e e e e e1 1.7e e e e e e e e e e e e e e e e e e1.176e e e e e e13.36e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e14.74e e e e e1 14 of 19
15 Elementary Reference Solution..1 Elementary Reference Solution q 3.1 q q 1 (a) q 1 q 3 projection (axes not equal) q (b) q q 3 projection (axes not equal). q Elementary Rereference Solution.1.1. q 1 q q.3 Elementary Reference Solution..1 q 1.1. (c) q 1 q projection (axes equal). (d) 3D perspective (axes not equal). Figure 6. Projection of the family of nontrivial orbitattitude solution (in blue) in the quaternion subspace. The quaternion subspace describes the orientation history of the vehicle relative to the rotating frame. The family emanates from an elementary reference solution (in red). are constantly b 3x =, b 3y =, b 3z = 1; in the nearby family of complex periodic solutions, ˆb 3 traces out closed curves identical to those in Figure 8, which are symmetric with respect to the b 3x = plane. It is observed that the ˆb 3 axis of the vehicle periodically oscillates, pointing to directions in the rotating frame that are symmetric relative to a plane orthogonal to the x axis. It is common practice to describe the attitude motion of an axisymmetric satellite in terms of precession and nutation angles. For the selected spacecraft configuration, precession and nutation are defined as the first and second angle of a body33 Euler angle sequence, respectively; the remaining rotation of the sequence is the spin about the ˆb 3 axis, which is the axis of symmetry. These angles describe the orientation of the body relative to the rotating frame. Referring to Figure 9, the family of solutions corresponds to an attitude history that consistently originates from a 9 deg precession and a small nutation angle in the direction opposite to the orbital motion. Essentially, the ˆb 3 axis initially lies in a plane perpendicular to the xaxis, inclined toward y <. During the first half revolution along the L 1 Lyapunov orbit, the axis of symmetry precesses 18 deg, pointing opposite to the initial direction at the next crossing of the xaxis. During the remaining orbital path, the nutation profile symmetrically replicates the first half revolution, while the precession antisymmetrically follows the previous evolution. After a complete L 1 Lyapunov orbit, the vehicle returns to its initial orientation relative to the rotating frame in the CR3BP. Representative instants throughout the motion history for a selected orbitattitude solution are portrayed in Figure 1, which offers an intuitive visualization of the physical response observed from the rotating frame. For completeness, the stability of the solution across the family is investigated, as it may eventually lead to other novel families via dynamical bifurcation. Accordingly, the nontrivial eigenvalues of Φ att are evaluated for the solutions in Table 1 and their real components are 15 of 19
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