Linear Equations of Relative Motion

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.

A state-transition matrix (STM) for the relative motion of artificial satellites is presented in this work that is valid in the presence of the gravitational spherical harmonic perturbations. The perturbed relative motion is modeled as a first-order variation of the equinoctial orbital elements describing the reference orbit. A satellite theory is derived for the analytical propagation of the reference orbit that includes the secular and periodic effects due to the zonal, sectorial, and tesseral harmonic perturbations. In case of the relative motion STM, the two main goals achieved are as follows. The analytic expressions for the elements of the STM are derived in a generalized form valid for an arbitrary spherical harmonic, and these expressions are in closed-form without using any series expansion in either the eccentricity or the ratio of the mean motion to the angular velocity of the gravitational body. The STM includes the short-period effects due to all the tesseral harmonics in addition to the secular, long-period, and short-period effects due to all the zonal harmonics. Avoidance of the series expansions in the eccentricity ensures that the accuracy of the STM does not degrade for satellite formations in medium to highly eccentric orbits. Additionally, the STM is nonsingular in case of the reference orbits that are near the resonant region; however, the long-period effects due to the resonant tesseral harmonics are ignored in this work. The proposed relative motion STM is implemented in MATLAB, and its accuracy is validated against numerical propagation with a Earth gravity field in NASA’s General Mission Analysis Tool.

With the recent flurry of research on satellite formation flying, a need has become apparent for a set of linearized equations of relative motion that capture the effect of the J2 geopotential disturbance force. Typically, Hill's linearized equations of relative motion have been used for this analysis, but they fail to capture the effect of the J2 disturbance force on a satellite cluster. In this paper, a new set of constant coefficient, linearized differential equations of motion is derived. These equations are similar in form to Hill's equations, but they capture the effects of the J2 disturbance force. A numerical simulator is employed to check the fidelity of the equations. It is shown that with the appropriate initial conditions, the new lin earized equations of motion have periodic errors (on the order of centimeters) that do not grow in time. The new linearized equations of motion also allow for insight into the effects of the J2 disturbance on a satellite cluster. This includes 'tumbling', the period of the relative orbit, and satellite separation due to differential J2 effects. Overall, a new high fidelity set of linearized equations are produced that are well suited to model satellite relative motion in the presence of the J2 disturbance force.

This paper studies the relative orbit geometry of a leader–follower spacecraft formation flying in unperturbed elliptical reference orbits. The first-order relative position equations, derived using the reference orbital element approach under the condition that the follower and leader spacecraft have equal semimajor axes, are transformed fromtrigonometricformstoparametricandalgebraicforms.Theconditionsforandthenumberofself-intersections oftherelativeorbitprojectedontothethreecoordinateplanesoftheleaderlocal-vertical–local-horizontal frameare obtained. The relative orbit proves to be three-dimensional instead of planar in most cases, and may self-intersect spatially at most once. The collision between the follower and leader spacecraft corresponds to the case where the solution curve to the first-order relative motion equations passes through the origin. The conditions for collision are subsequently determined. For anondegenerate case(in which none of the relative motion in the radial, in-track, and cross-trackdirectionsvanish),threetypesofrelativeorbitarepossible.Mostfrequently,therelativeorbitisonaonesheet hyperboloid. Otherwise, when the relative orbit has a real or finite imaginary self-intersection, it rests on an elliptic cone. In the rest of the cases, including that with an imaginary self-intersection at infinity, the relative orbit is on an elliptic cylinder. The criteria for these three types are given, respectively, followed by examples.

Abstract

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.

Rotational Motion Control by Feedback with Minimum L1-Norm

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.

J2 perturbation and air drag are derived exactly. The dierential equations are composed of two parts, the inertial motion of the reference spacecraft and the relative motion of the chaser with respect to the reference spacecraft. Some useful properties are uncovered. The inertial motion can be described by six variables, the rst ve variables are independent of the sixth variable, namely the the right ascension of the ascending node; The relative motion are also found to be independent of the right ascension of the ascending node of the reference orbit; The air drag has impact on the inertial motion, and the air drag dierence has impact on the relative motion. After the derivation, four potential application areas are discussed: A1-long-time relative motion prediction; A2-eld of view analysis for relative measurement sensor; A3-impact analysis of the coordinate frame precession on reference spacecraft attitude control; A4-impact analysis of the coordinate frame precession on attitude synchronization mission. These application areas show a good and prospective utility of the dynamics dierential equations in formation ying missions.

Linear structural dynamics, unsteady aerodynamics, control system, and actuator models are combined for linear aeroservoelastic equations of motion that are augmented with nonlinear feedback loops based on the increased-order modeling approach. Although the linear equations are formulated in the frequency domain for best combined efficiency, accuracy, and robustness in industrial environment, the nonlinear feedback loops are modeled in the time domain to provide maximal flexibility in adding nonlinear effects in all the involved disciplines. The linear equations are solved first to provide a baseline response to deterministic or stochastic gusts, maneuver commands, or direct-force excitations using fast Fourier transform techniques. Nonlinear effects are then added in a time-marching process that modifies the linear solution using convolution integrals. The numerical process was used in the Dynresp code that was recently developed as a framework for industrial applications and research in the area of nonlinear structural dynamics. The procedure is outlined with emphasis on structural nonlinearities using a fictitious-mass technique. The numerical example exhibits limit-cycle oscillations due to actuator nonlinearities.

Orbital dispersion and Earth-impact probability analysis for fragmented asteroids

This paper extends Hodges–Dowell's partial differential equations of blade motion, by including the effects from gravity, pitch action and varying rotor speed. New equations describing the pitch action and rotor speeds are also derived. The physical interpretation of the individual terms in the equations is discussed. The partial differential equations of motion are approximated by ordinary differential equations of motion using an assumed mode method. The ordinary differential equations are used to simulate a sudden pitch change of a rotating blade. This work is a part of a project on pitch blade interaction, and the model will be extended to include an aerodynamic model and be used for analysis of basic properties of pitch blade interaction. Copyright © 2007 John Wiley & Sons, Ltd.

This paper aims at studying theoretically synchronization of two exciters in a nonlinear vibrating system (NVS), in which the behavior of NVS is mainly reflected at nonlinear restoring forces of springs with piecewise linear characters. The differential motion equations of two rigid frames in the horizontal direction are combined into their relative motion equation. Based on the asymptotic method, the nonlinear stiffness of springs is linearized equivalently as a function of the amplitude of the relative motion. Using Lagrange’s equations, the differential motion equations of the total system are deduced. The criterion of synchronization for two exciters in the analytical form is derived, by the average method. According to the principle that the stable solution of the synchronous states corresponds to a minimum point of Hamilton’s average action amplitude of the system, the criterion of stability of the synchronous states is achieved analytically. States of the system versus the operating frequency under consideration of multiple resonant types are presented. For the synchronous vibrating machines with two rigid frames used in engineering, the condition of the minimum of excitation to foundation is discussed, as well as the ideal working regions of the vibrating machines.

This paper addresses energy-optimal guidance and control of satellite relative motion for formation flying and rendezvous using impulsive maneuvers. To meet the requirements of future multisatellite missions, closed-form solutions of the inverse relative dynamics are sought in arbitrary orbits. Time constraints dictated by mission operations and relevant perturbations acting on the formation are accounted for by splitting the optimal reconfiguration into guidance (long-term) and control (short-term) layers. Both problems are cast in relative orbit element space, which allows the simple inclusion of secular and long-periodic perturbations through a state-transition matrix and the translation of the energy-optimal optimization into a minimum-length path-planning problem. From the choice of state variables, both guidance and control problems can be solved in closed form, leading to optimal, predictable maneuvering schemes for simple onboard implementation. Besides generalizing previous work, this paper finds five new in-plane and out-of-plane closed-form solutions to the relative motion control problem in unperturbed eccentric and perturbed near-circular orbits. A delta-v lower bound is formulated, which provides insight into the optimality of the control solutions. Finally, the functionality and performance of the new maneuvering schemes are rigorously assessed through numerical integration of the equations of motion and systematic comparison with primer vector optimal control.