Study on Relative Orbit Geometry of Spacecraft Formations in Elliptical Reference Orbits

Revisiting the general periodic relative motion in elliptic reference orbits

T HE relative motion of a satellite (follower) with respect to the reference satellite (leader) in a circular orbit is described by autonomous nonlinear differential equations. The linearized equations at the origin are known as Hill–Clohessy–Wiltshire (HCW) equations [1]. The in-plane motion and the out-of plane motion are independent. The latter is a simple sinusoidal motion. The former has also periodic solutions under the so-called Clohessy–Wiltshire condition. Periodic solutions are given by sinusoidal functions, and the relative orbits are ellipses. These orbits are parametrized by two parameters representing size and phase. Because of this simple nature, relative periodic orbits are used for formation flying [2]. Periodic orbits of the nonlinear relative dynamics are also characterized by the parameters of the follower’s elliptic orbit and its inclination angle in an inertial reference frame, and initial conditions for periodic orbits are explicitly given [3]. If the reference orbit is elliptic, the equations of relative motion involve the true anomaly and the radius of the orbit, which are periodic functions. The linearized equations of motion at the origin are known as Tschauner–Hempel (TH) equations. The in-plane motion and the out-of plane motion remain independent. However, the derivation of the state transition matrix in this case is not immediate. In earlier studies [4–6], the true anomaly is used as independent variable, because this simplifies the resulting equations and the transition matrix can be obtained for them. Rendezvous problems have been extensively studied using this representation [7,8]. The out-of-plane motion is a sinusoidal motion, and the inplane motion has periodic solutions [2,9]. The condition, which generalizes the CW condition, for periodic solutions is given in [9]. Periodic orbits of the TH equations are also used for formation flying [2,9,10]. However, they are more complicated compared to ellipses of the HCW equations, and their shape changes with eccentricity. Some of them are very much distorted. Hence, it is not intuitively clear how to design relative orbits. The direct study of the relative dynamics using time as independent variable is given in [11], and a series expansion of the state transition matrix in the eccentricity is obtained in [12]. A closed form of the state transition matrix is recently derived in [13],where a brief history of elliptic rendezvous is also found. The characterization of periodic orbits of the nonlinear relative dynamics as functions of time is extended to the elliptic case, and initial conditions for periodic orbits are explicitly given [14]. The general solution of the relative dynamics based on the orbital elements is obtained in [15], and the relative motion invariant manifold, where the solution lies, is determined. This technical note is concernedwith the generation of all periodic solutions of the in-plane motion of the TH equations. For this purpose, the state transition matrix of [16], which is given as a function of the true anomaly, is used. The solution is parametrized by four constants, and the condition for periodic solution follows from this. As in the case of the HCWequations, size and phase parameters are introduced. The main contributions of this note are summarized as follows. First, the region where all periodic solutions lie is determined by two ellipses, whose size depends on the eccentricity of the leader orbit. Given size and phase parameters, a graphical method to determine the initial point of a periodic orbit at a given true anomaly is proposed. Varying the value of the true anomaly, the whole family of periodic orbits is generated. The time domain versions of the region and periodic orbits are determined by multiplying the radius of the leader orbit. This clarifies how the shape of a periodic orbit changes with eccentricity and why the shapes of some orbits are distorted. As the eccentricity goes to zero, the region of periodic motion shrinks to a single ellipse, which is a periodic orbit of the HCW equations. The size parameter gives the distance between the leader and the follower, and the phase determines the position of the follower. As in [15], the maximum and minimum distances to the leader can be calculated. Hence, our method is useful to design relative orbits with size and position specifications. This note is organized as follows. Section II reviews the equations of relativemotion along an elliptic orbit. Section III gives a geometric method to draw periodic orbits of the TH equations. Finally, Section IV gives conclusions.

T HE relative motion of a follower satellite with respect to the leader in a given circular orbit is described by autonomous nonlinear differential equations. The linearized equations around the null solution are known as Hill–Clohessy–Wiltshire (HCW) equations [1–3]. The HCWequations were used by many authors to study rendezvous problems (see [4] and references therein). The Tschauner–Hempel (TH) equations replace the HCW equations, when the orbit of the leader is eccentric [2,5]. Rendezvous problems along an eccentric orbit were studied in [4,5]. The HCW equations possess periodic solutions, which are useful as temporary orbits before mission and for proximity operations such as inspection and repair. The leader–follower formation and reconfiguration problems based on the periodic solutions were studied bymany authors [6–10]. The TH equations also have periodic solutions. They are characterized by Inalhan et al. [11], and the initialization procedure to periodic motion is given. Periodic solutions also follow from the transition matrix of the TH system given byYamanaka and Ankersen [12]. The effects of eccentricity on the shape and size of relative orbits are studied by Sengupta andVadali [13]. Periodic solutions of the TH equations are used for formation flying [14,15] because no control efforts are needed to maintain them. However, their period is fixed and is equal to that of the leader orbit, which would be inconvenient for a quick inspection of the leader. The shape of periodic solutions is irregular compared with that of the HCWequations, which would be undesirable for some missions. In this Note, active formation flying for the TH system is considered, in which the desired relative orbit of the follower is generated by an exosystem. This allows for flexibility of the shape and period of the reference orbit. Typical examples are elliptic relative orbits of the HCWequations with higher frequencies. Formation flying for the TH system with generated reference orbits has not been studied in the literature. To realize such a formation flying, the output regulation theory for linear periodic systems recently given by Ichikawa and Katayama [16] is employed. Output regulation covers tracking and disturbance rejection [17], but the tracking aspect of the theory is used. The regulator equation, which is necessary to achieve output regulation, is a differential equationwith periodic coefficients. The main contribution of this Note is to show that the regulator equation can be solved algebraically if the system is in the controllable canonical form and the observation matrix is of a special form. To achieve asymptotic tracking, stabilizing feedback controls are necessary. They are designed by the differential Riccati equation (DRE) of the linear quadratic regulator theory [18]. To show the effectiveness of this approach, two examples are given. In the first example, the orbit of the leader is assumed to be circular and the HCW equations are considered. The reference orbit of the follower for formation flying is circularwith arbitrary frequency. Starting from a periodic solution of the HCW equations, the follower satellite is asymptotically steered to the reference orbit. In this case, both the regulator equation and theRiccati equation become algebraic. TheL1 norm of the input ΔV and the settling time are given as functions of the weight parameter in the Riccati equation. The L1 norm of the input to maintain the reference orbit for one period is also calculated. These performance indices are further examined by varying the frequency of the reference orbit. The second example is concerned with the TH equations for the elliptic leader orbit. The reference relative orbit of the follower is circular, as in the first example. The regulator equation has periodic coefficients, but it is algebraically solved and explicitly given. The DRE is solved backward and the stabilizing periodic solution is obtained. TheL1 normof the input and the settling time are computed by varying eccentricity and frequency of the reference orbit.

This paper deals with the maneuver method for the spacecraft formation reconfiguration in the elliptic reference orbit considering two control input types. During the satellite operation, the formation size and/or geometry should be modified according to the mission requirements and space environment. The follower spacecraft can change the formation radius and/or geometry with respect to the leader spacecraft, while minimizing the control effort during the maneuver. For the continuous control input, the optimal control problem in the relative motion is solved based on the nonlinear programming method, where the initial and final relative positions and velocities are specified in the local frame. For the impulsive control input, on the other hand, the Lambert’s problem is modified to construct the transfer problem in the relative motion, given two position vectors at the initial and final time and the flight time. Furthermore, the minimum velocity change is investigated through the grid search for transferring orbit using the impulsive control input. As a result, the transfer trajectories to resize the formation radius and change the formation geometry are presented by performing the numerical simulation.

Linear Equations of Relative Motion

This paper presents a perturbation approach for determining relative motion initial conditions for periodic motion in the vicinity of a Keplerian elliptic orbit of arbitrary eccentricity, as well as an analytical solution for the relative orbit that accounts for quadratic nonlinearities in the differential gravitational acceleration. The analytical solution is obtained in the phase space of the rotating coordinate system, centered at the reference satellite, and is developed in terms of a small parameter relating relative orbit size, and semimajor axis and eccentricity of the reference orbit. The results derived are applicable for arbitrary epoch of the reference satellite. Relative orbits generated using the methodology of this paper remain bounded over much longer periods in comparison to the results obtained using other approximations found in the literature, because the semimajor axes of the satellites are shown to be matched to the second order in the small parameter. The derived expressions thus serve as excellent guesses for initiating a numerical procedure for matching the semimajor axes of the two satellites. Several examples support the claims in this paper.

PurposeHill equations have definite limitation in the design of multiple spacecraft formation flying in eccentric orbits. To solve the problem, the design method of spacecraft formation flying in a circular reference orbit based on Hill equation can be generalized and applied to spacecraft formation flying in eccentric orbits.Design/methodology/approachIn this paper, T‐H equation is expressed as the explicit function form of reference orbit true anomaly, and the state transition matrix of relative motion of spacecraft formation flying in eccentric orbits is derived. According to the requirement that relative dynamics equation of spacecraft formation flying in eccentric orbits has periodicity solution, the paper theoretically gives the initial condition needed by the long‐term close‐distance spacecraft formation flying including the relationship between relative position and relative velocity. Without perturbation the spacecraft formation, which satisfies the initial periodicity restriction, can keep long‐term close‐distance flying without the need of active control.FindingsBased on the theoretical analysis, some numerical simulations are carried out. The results demonstrate that each spacecraft in eccentric orbits can run in a periodic motion surrounding the center spacecraft under some conditions. And spacecraft formation reconfiguration is implementing according to missions.Originality/valueCombined with the periodicity restriction primary condition a new method about spacecraft formation reconfiguration is put forward. The method given by this paper can be applied to eccentric orbits of arbitrary eccentricity, and provides theoretical reference for orbit design of spacecraft formation flying in eccentric orbits.

I N RECENT years, many proposed satellite missions have undergone a paradigm shift away from large, expensive spacecraft toward the use of smaller satellites flying in formation. Although formation flying offers a number of advantages, such as robustness of the mission to failure and potential cost savings, the benefits are partially offset by the increased complexity of the guidance, navigation, and control systems and by V-limited mission durations. The development of advantageous relative dynamics between the constituent satellites could help extend mission durations by minimizing fuel usage. In particular, periodic or quasiperiodic relative orbits would reduce the control effort necessary to maintain the formation and significantly prolong the mission’s lifetime. Attempts to identify naturally periodic relative orbits can be found throughout the current literature. Vaddi et al. [1] modified initial conditions from the Hill–Clohessy–Wiltshire (HCW) equations to enforce bounded relative motion in the presence of small eccentricity and second-order differential gravity terms. Kasdin and Kolemen [2] solved the Hamilton–Jacobi equation in terms of epicyclic orbital elements to derive bounded, periodic orbits in the presence of higher-order gravity and certain perturbations. Schaub and Alfriend [3] identified the conditions for J2-invariant relative orbits bymatching the secular drift rates of themean orbital elements of the chief and deputy satellites. In most of these analytical cases, however, the authors established precise periodic motion using approximated dynamical models. When higher fidelity dynamics and realistic orbital parameters are used, these methods break down and exhibit relative orbit drift or a high sensitivity to initial condition errors. Several numerical approaches to the problem of relative orbit periodicity have also been attempted. Sabatini et al. [4] and Izzo and Sabatini [5] used a refined genetic algorithm to optimize relative trajectories for maximum periodicity in the presence of J2 perturbations. Quasi-periodic orbits were obtained for two sets of “magic” inclinations (49.11 and 63.43 deg), at which resonance between the formation’s inand out-of-plane motion results in projected circular orbit (PCO) formations with very small orbital drift. Damaren [6] formulated an iterative shooting approach based on the Newton method to close the relative orbit and achieve “almost” periodic initial conditions. This technique requires a low-level state-feedback control loop to track a trajectory based on the HCW equations, but is robust to initial condition errors and exhibits very low drift characteristics in circular orbits. A similar approach was taken by Becerra et al. [7], who applied a nonlinear Hamiltonian model to the Newton method to obtain quasi-periodic relative initial conditions of the deputy in the presence of the J2 perturbation. A linear quadratic regulator (LQR) controller was used to track reference trajectories that were developed from sinusoidal functions of time and based on a Keplerian orbit. As yet, a method of numerically designing reference trajectories that match or closely match the natural perturbed relative motion of the deputy satellite in a generic orbit, and that are robust to initial condition errors, has not been developed. Such trajectories, with a properly designed low-authority controller, could be tracked for minimal V requirements, thus enabling formation flying missions to be significantly extended. The intent of this study is to design such trajectories by continuing the development of the methods used to search for almost-periodic orbits, begun in [6], and applying them to the J2-invariant trajectories in [3]. The resulting relative orbits will approach true periodicity by capitalizing on advantageous initial conditions, numerically designed reference trajectories, and optimal control strategies.

Micro/nano satellites have become the protagonists in most formation flying missions currently, there is an urgent demand to find an approach to quickly calculate multiple-impulsive scheme on board for relative orbital transfer, while considering the optimality of total impulsive consumption. This paper presents an optimal control approach based on multiple impulses for in-plane satellite formation reconfiguration in near circular orbit. A new type of relative orbit motion equation is established based on the initial small deviation in the cylindrical coordinates system in this paper, which is suitable to calculate the solution of relative orbital maneuver by means of multiple impulses. The formation reconfiguration problem is investigated on account of the relative orbit motion, and the orbital motion equations with initial deviations are presented. The formation reconfiguration problem in orbital plane is discussed and analyzed selectively. An optimal three-impulsive approach for relative orbital transfer in plane is obtained by analysis solution based on graphical and numerical way, which is achieved by the combination method from four-impulsive solution. And the method is proved to be effectivity and optimality.

The solution of the initial relative orbit determination problem using angles-only measurements is important for orbital proximity operations, satellite inspection and servicing, and the identification of unknown space objects in similar orbits. In this paper, a preliminary relative orbit determination performance analysis is conducted utilizing the linearized relative orbital equations of motion in cylindrical coordinates. The relative orbital equations of motion in cylindrical coordinates are rigorously derived in several forms included the exact nonlinear two-body differential equations of motion, the linear-time-varying differential equations of motion for an elliptical orbit chief, and the linear-time-invariant differential equations of motion for a circular orbit chief. Using the nonlinear angles-only measurement equation in cylindrical coordinates, evidence of full-relative-state observability is found, contrary to the range observability problem exhibited in Cartesian coordinates. Based on these results, a geometric approach to assess initial relative orbit determination performance is formulated. To facilitate a better understanding of the problem, the focus is on the 2-dimensional initial orbit determination problem. The results clearly show the dependence of the relative orbit determination performance on the geometry of the relative motion and on the time-interval between observations. Analysis is conducted for leader-follower orbits and flyby orbits where the deputy passes directly above or below the chief.

Not all micro-vessels (MV) are traditionally circular and there are examples of elliptic cylindrical MVs in life sciences, particularly if projected with a slant. Similarly, certain biological structures, ferroelectric liquid crystals, aluminum oxide clusters and witherite crystallites’ cross-section appear to be elliptical cones. We studied the mean curvature (H) distribution of these elliptic morphological structures with geometric parameter such as eccentricity; e (ratio of semi-minor to semi-major axes) and a measure of how much diagonal section deviates from circularity and height (h) in case of cones. By means of topographical cues, we defined the curvature-curvature autocorrelation function (gk) and applied this notion to mean curvature (H) of circular and elliptical cylinders and cones. The Fourier transform of correlation function, i.e. “curvature factor” is analogous to “structure factor (or Patterson function)” in X-ray and neutron scattering intensity. It elucidates critically important information related to surface curvature fluctuation relevant to shape (geometry), network and phase transformation. The latter is induced by cells under mechanical stress, occurring in many soft systems (polymeric liquid crystals, foams, bubbles) and biological tissues, particularly cell walls of primary and branched vessels bed in microvasculature that distributes blood within tissue during hypertension and migraines. This perspective is useful in a sustained release of angiogenic/vasculogenic factors and relevant for precision medicine and engineered microvessels and tissues in vitro and in vivo extended cellular processes. The quantitative analysis carried out in this work facilitates our understanding of the mechanical mechanisms associated with thrombosis during surgery that typically occur in bent or stretched MVs due to microenvironment such as localized shear stresses and biochemical factors.

An impulsive feedback control is developed to establish specie c relative orbits for spacecraft formation e ying. The relative orbit tracking errors are expressed in terms of mean orbit elements. The feedback control, based on Gauss’ s variational equations of motion, allows specie c orbit elements or orbit element sets to be controlled with minimal impact on the remaining osculating orbit elements. This is advantageous when J2-invariant orbits are to be controlled, where only the argument of perigee and mean anomaly will drift apart at equal and opposite rates. The advantage of this impulsive feedback control, compared to optimal control solutions, is that it can operate with little computational effort and in a near-optimal manner, whilerequiring only a marginal penalty in fuel cost. When applied to the spacecraft formation e ying problem, this control could also be used to perform general orbit corrections. Formulas are developed providing accurate estimates of the sensitivities of the mean semimajor axis and mean eccentricity with respect to the osculating inclination angle. With these sensitivities, the tracking error in semimajor axis, eccentricity, and inclination angle can be canceled within one orbit.

Relative states autonomous determination is very important for formation flying. In this paper, based on AFF technology, a relative states autonomous determination algorithm for formation flying is brought forward. Relative orbit and attitude motion equation of double satellites formation are taken as states equation. Measurement model is set up based on intersatellite measurement by GPS-like sensors and the measurement equation is established accordingly. A kind of double SKF sub-filters capable of calculating parallelly is put forward to estimate the orbit and attitude parameters at the same time. Comparing the mathematical simulation results of double SKF sub-filters with that of EKF shows the feasibility and rapidity of the proposed algorithm

This paper investigates the problem of relative control for spacecraft formation flying on an arbitrary elliptical orbit. The nonlinear dynamic model describing the relative motion between the leader spacecraft and the follower spacecraft is established in the presence of system model uncertainty and external disturbance. A continuous finite-time control scheme is designed by using a new form of terminal sliding modes. Theoretical analysis and simulation results demonstrate the robustness and fast convergence rate of the proposed method.

A new second order sliding mode (2nd-SM) control algorithm of MIMO nonlinear systems for low thrust multiple picosatellites formation flying is proposed in this paper. The problem is tackled by applying chatteringfree 2nd-SM control law as a control scheme for the case where the leader satellite is in circular and elliptical reference orbits under J2 perturbation. The asymptotic convergence to the 2nd-SM and the robust stability of the asymptotic 2nd-SM control system are for the first time proved with Lyapunov functions. Several numerical results are presented to demonstrate the efficacy of the proposed controllers in maintaining the satellite formation in presence of initial formation errors and J2 perturbation effects that tend to disperse the formation. Simulation results confirm that the suggested methodologies yield high formation keeping precision and effectiveness in ensuring formation maneuvering with thrusters faults of formation flying systems for picosatellites. Using this 2nd-SM method the fuel consumption is lower than in most other works.