Nguyen X. Vinh

Stabilizing Motion Relative to an Unstable Orbit: Applications to Spacecraft Formation Flight

Acontrol lawisderivedandanalyzed thatstabilizesa classof unstableperiodicorbitsin theHillrestrictedthreebody problem. The control law is derived by stabilizing the short-time dynamics of motion about a trajectory by the use of a feedback law specie ed by the instantaneous eigenvalue and eigenvector structure of the trajectory. This law naturally generalizes to a continuous control law along an orbit. By applying the control to an unstable periodic orbit, we can explicitly compute the stability of the control over long periods of time by computing the monodromy matrix of the periodic orbit with its neighborhood modie ed via the control law. For the case of an unstable halo periodic orbit in the Hill restricted three-body problem, we e nd that the entire periodic orbit can be stabilized. The resulting stable periodic orbits have three distinct oscillation modes in their center manifold. We discuss how this control can be applied to formation e ight about a halo orbit. Some practical implementation issues of the control are also considered. We show that the control acceleration can be provided by a low-thrust engine and that the total fuel cost of the control can be quite reasonable. I. Introduction T HIS paper studies the stabilization of an unstable periodic orbit in the Hill problem, which can serve as a general model for motion in the Earth‐ sun system. Results of this study will be relevant to the dynamics and control of a constellation of spacecraft in an unstable orbitalenvironment such as found near the Earth‐ sun libration points. It will also shed light on the practical control and computation of a single spacecraft trajectory in an unstable orbital environment over long time spans. We investigate the application of feedback control laws to stabilize a periodic orbit in the sense of Lyapunov (see Ref. 1) (note, not asymptotic stability). Thus, the stabilized trajectory will consist of oscillatory motions about the nominal trajectory, which in this context can be interpreted as motions in the center manifold of the stabilized periodic orbit. We show that an entire class of such control laws can be dee ned and their stability analyzed as a timeperiodic linear system. The fuel expenditure for such control laws is often quite small and scales with the mean distance between the controlled motion and the nominal trajectory (which is a periodic orbit in this application). The problem of spacecraft control in unstable orbits is not new. (See Refs. 2 and 3 for reviews.) However, these previous studies have focused on stationkeeping control for a single spacecraft and have not considered how the relative motion of a formation of spacecraft could be stabilized and their dynamics modie ed, which is what we consider here. Proposed space observatories of the future include ambitious interferometricimagersthatusebaselinesofhundredsorthousandsof kilometers between spacecraft to attain sufe ciently high resolutions to image planets around distant stars. To carry out these imaging procedures requires that the relative motion between spacecraft be known extremely accurately and that the spacecraft “ e ll in” an effective image e eld as they move relative to each other. A periodic

Optimal aeroassisted return from high earth orbit with plane change

(Received 31 January 1984) Abstract--This paper gives a complete analysis of the problem of aeroassisted return from a high Earth orbit to a low Earth orbit with plane change. A discussion of pure propulsive maneuver leads to the necessary change for improvement of the fuel consumption by inserting in the middle of the trajectory an atmospheric phase to obtain all or part of the required plane change. The variational problem is reduced to a parametric optimization problem by using the known results in optimal impulsive transfer and solving the atmospheric turning problem for storage and use in the optimization process. The coupling effect between space maneuver and atmospheric maneuver is discussed. Depending on the values of the plane change i, the ratios of the radii, n = rE/r2 between the orbits and a = r2/R between the low orbit and the atmosphere, and the maximum lift-to-drag ratio E* of the vehicle, the optimal maneuver can be pure propulsive or aeroassisted. For aeroassisted maneuver, the optimal mode can be parabolic, which requires only drag capability of the vehicle, or elliptic. In the elliptic mode, it can be by one-impulse for deorbit and one or two-impulse in postatmospheric flight, or by two-impulse for deorbit with only one impulse for final circularization. It is shown that whenever an impulse is applied, a plane change is made. The necessary conditions for the optimal split of the plane changes are derived and mechanized in a program routine for obtaining the solution.

Minimum-fuel, power-limited transfers between coplanar elliptical orbits

Two methods of computing coplanar, minimum-fuel, power-limited transfers are developed, based on approximate solutions obtained by the averaging method. In the first method, the average solution provides estimates of the initial adjoint variables; in the second, it provides approximations of the optimal controls in feedback form. Both trajectory variables and orbit elements are used in developing these methods. Canonical transformations are derived to convert between these sets of coordinates. The accuracy of the methods for computing coplanar, minimum-fuel, power-limited transfers is assessed for a variety of initial and final orbits. Some initial steps are taken toward the characterization of coplanar, minimum-fuel, power-limited transfers for a wide range of thrust to weight ratios. Circle to ellipse and ellipse to ellipse transfers are considered. Details of the trajectories and thrust profiles for a few illustrative cases are presented. These trajectories and thrust profiles are compared to analytical results obtained using the averaging method and to the analytic solution for infinitesimal transfer. The secular behavior of minimum-fuel transfer is predicted by the averaging results. The shape and orientation of the osculating orbits are predicted quantitatively, while the size is predicted qualitatively. The analytic solution for infinitesimal transfer predicts the qualitative behavior of the thrust during each revolution. Some general principles of minimum-fuel, power-limited transfer are revealed.

General theory of optimal trajectory for rocket flight in a resisting medium

This paper considers the problem of optimizing the flight trajectory of a rocket vehicle moving in a resisting medium and in a general gravitational force field. General control laws for the lift, the bank angle, and the thrusting program are obtained in terms of the primer vector, the adjoint vector associated to the velocity vector. Additional relations for the case of variable thrusting and integrals of motion for flight at maximum lift-to-drag ratio and flight in a constant gravitational field are obtained.

Optimum maneuvers of a skip vehicle with bounded lift constraints

This paper presents the analytical solutions of the problem of optimum maneuvering of a glide vehicle flying in the hypervelocity regime. The investigation is based on the approximation of Allen and Eggers; namely, that along the fundamental part of a reentry or ascent trajectory, the aerodynamic forces greatly exceed the components of the gravitational force in the directions tangent and normal to the flight path.The problem consists of finding an optimal control law for the lift such that the final velocity or the final altitude is maximized. This problem can be viewed as bringing the vehicle to the best condition for interception, penetration, or making an evasive maneuver.If the range is free, the optimal lift control is obtained in closed form. If the lift control is bounded, then bounded control is optimal whenever it is reached. The switching sequences for different cases are discussed, and it is shown that there are at most two switchings. Bounded lift control is always at the ends of the optimal trajectory; for the case of two switchings, the optimal trajectory has an inflection point.

Explicit guidance of drag modulated aeroassisted transfer between elliptical orbits

This paper presents the complete analysis of the problem of minimum-fuel aeroassisted transfer between coplanar elliptical orbits in the case where the orientation of the final orbit is free for selec- tion in the optimization process. The comparison between the optimal pure propulsive transfer and the idealized aeroassisted transfer, by several passages through the atmosphere, is made. In the case where aeroassisted transfer provides fuel saving, a practical scheme for its realization by one passage is proposed. The maneuver consists of three phases: A deorbit phase for non zero entry angle, followed by an atmospheric fly-through withdvariable drag control and completed by a post atmospheric phase. An explicit guidance formula for drag control is derived and it is shown that the required exit speed for ascent to the final orbit can be obtained with a very high degree of accuracy.

Linear stability of a self-gravitating ring

Stability of a self-gravitating ring about a central body is considered. The purpose is to derive a bound on the mass of the ring in order that the system will be linearly stable. Our bound will, in some cases, be the best possible bound. The bound is also expanded as an asymptotic series. Comparisons of our result are made with respect to previous analyses performed by Tisserand, Pendse and Willerding.

Reachable domain for interception at hyperbolic speeds

Abstract The interception of an ICBM at low altitude and in a short time requires hyperbolic speeds. At the acquisition time t0, if updated information about the trajectory of the target dictates a velocity correction for the interception, then it is of interest to assess the potential for interception. The reachable surface is defined as the boundary of the set of all attainable points at a given time t, for a fuel potential, conveniently expressed in terms of the Δv available. Properties of this surface are derived and its mathematical characterization is obtained. As an application of the concept, a necessary condition for a successful interception of a target given in terms of a capture function is explicitly derived.

ANALYTICAL SOLUTION OF MISSILE TERMINAL GUIDANCE

A new guidance law, which combines pursuit guidance and proportional navigation is proposed. This guidance law depends on two parameters that determine the relative importance of pursuit guidance and proportional navigation. Numerical simulations of the nonlinear equations of motion suggest that the parameters of this law can be chosen to reduce the peak value of the missile acceleration. When the engagement ends in a tail chase, and linearization is valid, the linearized equations of motion lead to a confluent hypergeometric equation. This equation is solved in closed form, in the general case where the target performs maneuvers such that its heading angle is a polynomial function of time. The analytic solution based on linearization and the numerical simulation of the nonlinear equations show good agreement.

The restricted P + 2 body problem

Abstract In this paper we study a special case of the restricted n-body problem, called by us the restricted P + 2 body problem. The equilibrium configuration which the P + 1 bodies with mass form consists of one central mass encircled by a ring of P equally spaced particles of equal mass, the ring rotating at a specific angular velocity. We briefly discuss the stability of this configuration. We consider the dynamics of an infinitesimal mass under the influence of such a configuration. First the equilibrium points will be discussed, then the zero-velocity curves. We show that there are 3P, 4P or 5P equilibrium points, depending on the ratio of the ring particle mass to the central body mass. Next motion about the equilibrium points is considered. We show that if the ring particle mass is small enough there will be P stable equilibrium points. Also if the number of particles, P, is large enough and the ratio of the ring particle mass to the central body mass is large enough there will be P different stable equilibrium points. Finally an analysis of the dynamics of the infinitesimal mass will be performed under the restriction that the particle does not cross or come close to the ring and lies in the plane of the ring. Under this restriction an approximate potential can be found which can be made arbitrarily close to the real potential under some circumstances. The dynamics of the particle under the approximate potential are integrable. We find a periodic orbit in this case with the Poincare-Lindstedt method using the mass of the ring as a small parameter. The predictions from this approximate solution of the problem compare well with numerical integrations of the actual system.