Orbital Relative Motion and Terminal Rendezvous

The first-order solution of the problem of relative motion of a spacecraft in near-circular orbit is known to degrade in accuracy, when compared to the numerically integrated exact solution, at greater distances from the origin of the rotating reference frame. These solutions have been developed to study the problem of the terminal rendezvous guidance where an active spacecraft at several hundred km from its rendezvous target centered at the rotating frame, must maneuver to intercept the target in a given time. In References [1, 2], the relative motion technique was used for a different purpose, namely to describe the future motion of a spacecraft relative to the rotating frame, as it is perturbed by the Earth zonal harmonics J₂ and J₃, and by the luni-solar gravity effects. Although these perturbations have a small effect on the spacecraft motion which would not wander to great distances from the origin of the frame, it is felt that because of the presence of initial non-zero velocities, the subsequent motion may drift to considerable distances from the origin of the frame, thereby, degrading the accuracy of the analytic first-order solution of the equations of motion. The initial velocities exist because the orbit determination-generated osculating orbit is necessarily elliptical in nature with small eccentricity, such that at time zero, or epoch, a reference circular orbit having the same radius as the radial distance of the actual spacecraft is assumed, to describe the future motion of the vehicle itself, which, unlike the frame, experiences the various perturbations just mentioned. Second-order corrections to the linear solution of Reference [3] have been obtained in References [4, 5] to extend the region of accuracy of the analytic solutions at greater distances from the origin. This chapter rederives the second-order solutions, resolving the errors of Reference [4], and the typographical errors of Reference [5], by adopting the nomenclature of these two references in defining the coordinates. The radial coordinate is depicted as y, and the “tangential” coordinate by x, although x is pointing in the opposite direction of motion. The out-of-plane or z coordinate, is along the orbital angular momentum vector. In References [1, 2], x is along the radial direction, y along the “tangential” direction in the direction of motion, and z along the orbital angular momentum vector. When the second-order expressions developed here are added to the first-order solutions of the perturbed motion of References [1, 2], the differences in the coordinates must be properly accounted for.

The analytic first-order solution of the Earth zonal harmomics J₂ and J₃ perturbation effect on the motion of a spacecraft in a near-circular orbit with small eccentricity and arbitrary inclination, energy, node and perigee location is analyzed. The second-order Euler-Hill differential equations are solved after expanding the forcing terms to first-order in the small parameter defining the orbit eccentricity. The construction of the circular reference orbit which defines the rotating Cartesian reference frame with respect to which the relative motion is described, is also shown. For the case of the J₂ harmonic, a position error of 200 m per revolution is sustained when the initial orbit is circular. The equations developed in this chapter can be used to carry out terminal rendezvous in near-circular obit around the oblate Earth.

Following the method of the rendezvous technique used in Reference [1] and Chapter 2 to generate the motion of a spacecraft in near-circular orbit perturbed by Earth’s J₂ and J₃ zonals, and described in the rotating Euler-Hill frame, the same approach is used in this chapter and based on Reference [2] to obtain analytic expression for the spacecraft motion perturbed by the luni-solar gravity.

In Reference [1] and Chapter 3, the effect of solar and lunar gravity perturbations on a spacecraft in near-circular orbit was analyzed. However, in order to simplify the analysis, the lunar as well as the apparent solar orbits were assumed to be circular. In this chapter, this assumption is removed such that the third-body orbits are now allowed to describe Keplerian ellipses. Second and higher-order terms in the eccentricities of the lunar and apparent solar orbits are neglected in this analysis. Analytic solutions for the position and velocity components describing the perturbed motion in the Euler-Hill frame are thus obtained, generalizing thereby the results of [1] and Chapter 3. Once the transformations to the inertial system are carried out, all the orbital elements can be readily obtained and used for example in the maneuver planning function. This theory can be useful for the autonomous navigation of geostationary spacecraft as well as other high near-circular orbit applications such as the GPS spacecraft.

A preliminary attempt is made in this chapter to treat the atmospheric drag perturbation effect on a near-circular orbit with arbitrary inclination, energy, and perigee location and small eccentricity. Following the practice used in References [1, 2], and the preceding chapters, the rotating Euler-Hill frame is used to describe the perturbed motion of the spacecraft by assuming that the atmospheric density obeys an exponential decay law with radial distance from the center of the rotating Earth, that the density scale height is constant, and that the atmosphere is rotating uniformly with the solid Earth at the same angular rate. The Fourier-Bessel series approximation is used to treat the exponential term appearing in the density law, and analytic expressions for the tangential and out-of-plane perturbation acceleration components obtained as the forcing functions driving the linearized Euler-Hill equations of motion. The six position and velocity components are thus derived analytically, from which all the orbital elements can be recovered. More refined derivations considering Earth oblateness and diurnal bulge effects as well as variable density scale height or non-exponential power law density variations are also pursued to obtain a more accurate dynamic model for use in autonomous orbit determination applications for low altitude spacecraft.

The analytic solution of the two-impulse terminal rendezvous problem around the oblate Earth is analyzed in this chapter. Both maneuvering and passive spacecraft are assumed to be in nearby near-circular orbits. The main effect of the oblateness is due to the second zonal harmonic J₂. A highly accurate analytic first-order theory is used to obtain analytic expressions for the first velocity change that initiates the rendezvous. An example is shown to demonstrate the failure of the conic-based rendezvous equations in achieving the required transfer when the subsequent orbits are perturbed by J₂.

The analytic treatment of the problem of impulsive terminal rendezvous in near-circular orbit is presented in this chapter. The velocity change imparted to the active maneuvering vehicle is determined with high accuracy, such that, the error in interceptmg the passive spacecraft in a given time is negligible. This error is evaluated by numerically integrating both spacecraft orbits, by accounting for the second zonal harmonic J₂. A typical large transfer where the two vehicles are initially separated by 2000 km, is achieved with an interception error of 5 km.

The numerical solution of the two-impulse terminal rendezvous for the formation flying of two spacecraft with different ballistic coefficients in general elliptic orbit is presented in this chapter. The motion of the passive vehicle is described by the inverse square gravitational field of the primary body and by atmospheric drag. The center of a rotating reference frame is attached to this vehicle, and the motion of the active maneuvering vehicle is described relative to this accelerating frame. This allows for the solution of the exact, arbitrary duration two-impulse rendezvous problem, with both vehicles continuously subject to drag in their respective orbits during the maneuvering period, meaning between the application of the first or initiating impulse, and the final or terminating impulse which achieves the desired rendezvous. Examples of rendezvous which bring the vehicles to close proximity for various transfer durations are presented to support this analysis.

The full set of second-order nonlinear differential equations describing the exact motion of a spacecraft subject to drag and oblateness perturbations in general elliptic orbit, relative to a rotating reference frame which drags and precesses exactly as a given spacecraft attached to its center is derived. This attached spacecraft is itself flying a general elliptic orbit and can be considered as the passive or non-maneuvering vehicle. The unaveraged form of the J₂ acceleration is used for both vehicles leaving this oblateness perturbation position-dependent for more exacting calculations. These equations can be effectively put to use in calculating by an iterative scheme, the impulsive rendezvous maneuvers in elliptic orbit around the Earth or those planets that are either atmosphere bearing or have a dominant second zonal harmonic, or both.

The algorithm that generates the exact solution of the two-impulse noncoplanar rendezvous in general elliptic orbit is presented in this chapter. The motion of the maneuvering spacecraft is referred to a rotating reference frame attached to the passive spacecraft, and dragging and precessing at the same rate as that spacecraft. An iterative scheme is devised to find the magnitude and orientation of the initiating impulse that brings the active spacecraft to a desired target point in the vicinity of the passive spacecraft in a given time. When the rendezvous point is in the vicinity of the passive spacecraft and not at the passive vehicle location itself, the linear distance between the two vehicles will exhibit variations along their post-rendezvous common orbit which can be of the order of kilometers for highly eccentric orbits. These natural oscillations can be minimized by targeting the active vehicle to the immediate proximity of the passive spacecraft, and be totally eliminated by targeting to the passive vehicle location itself.

The stationkeeping of symmetric Walker constellations is analyzed in this chapter by considering the perturbations arising from a high order and degree Earth gravity field, and the solar radiation pressure. These perturbations act differently on each group of spacecraft flying in a given orbital plane, causing a differential drift effect that would disrupt the initial symmetry of the constellation. The analysis is based on the consideration of a fictitious set of rotating reference frames that move with the spacecraft in the mean sense, but drift at a rate equal to the average drift rate experienced by all the vehicles over an extended period. The frames are also allowed to experience the J₂-precession, such that each vehicle is allowed to drift in 3D relative to its frame. A two-impulse rendezvous maneuver is then constructed to bring each vehicle to the center of its frame as soon as a given tolerance deadband is about to be violated. This chapter illustrates the computations associated with the stationkeeping of a generic Walker constellation, by maneuvering each leading spacecraft within an orbit plane, and calculating the associated velocity changes required for controlling the in-plane motions in an exacting sense, at least for the first series of maneuvers. The analysis can be easily extended to lower flying constellations, which experience additional perturbations due to drag.

This chapter shows the equations used in the JPL computer program ASAP that are specific to the modeling of the central body disturbing function, the third-body disturbing function, as well as the equations of motion due to atmospheric drag. These descriptions are taken directly from Reference [1] by Kwok.