This paper assumes a requirement for an unmanned multiunit satellite to be assembled in orbit. The requirement to be met is to bring the satellites together so tha t they do not collide but actually rendezvous. The equations of motion of the rendezvous satellite in a relative coordinate system are derived and used to compute a final injection velocity which would effect collision after a time r. The velocity is corrected periodically by a command guidance system and just before impact retrothrust is applied. A terminal infrared homing sj^stem is required to actually accomplish physical contact and joining of the satellites. The first satellite placed in orbit is the "control satellite" and controls all the satellites to be assembled and contains the ccmputer, command guidance equipment, precision orientation equipment, and other features necessary to effect rendezvous. The succeeding satellites contain a propulsion system, a rough at t i tude control system, and a command receiver plus whatever scientific equipment they carry to perform their basic mission. This paper presents the following:

Terminal Guidance System for Satellite Rendezvous

A new state transition matrix is described for the nonlinear problem of relative motion on an arbitrary elliptical orbit. A linearization is performed, leading to a set of linear differential equations with time-dependentcoefficients. A new and simpler solution to those equations is represented in a convenient state transition matrix form. This new state transition matrix is valid for arbitrary elliptical orbits of 0≤e<1. The state propagation using the new state transition matrix shows good agreement with numerical results.

New State Transition Matrix for Relative Motion on an Arbitrary Elliptical Orbit

Formation eying is a key technology for both deep-space and orbital applications that involve multiple spacecraft. Many future space applications will beneet from using formation e ying technologies to perform distributed observations (e.g., synthetic apertures for Earth mapping interferometry) and to provide improved coverage for communication and surveillance. Previous research has focused on designing passive apertures for these formation e ying missions assuming a circular reference orbit. Those design approaches are extended and a complete initialization procedure for a large e eet of vehicles with an eccentric reference orbit is presented. The main result is derived from the homogenous solutions of the linearized relative equations of motion for the spacecraft. These solutions are used to end the necessary conditions on the initial states that produce T-periodic solutions that have the vehicles returning to the initial relative states at the end of each orbit, that is, v(t0)=v(t0+T). This periodicity condition and the resulting initialization procedure are originally given (in compact form) at the reference orbit perigee, butthis is alsogeneralized to enable initialization atanypoint around thereference orbit. In particular, an algorithm is given that minimizes the fuel cost associated with initializingthe vehicle states (primarily the in-track and radial relative velocities) to values that are consistent with periodic relative motion. These algorithms extend and generalize previously published solutions for passive aperture forming with circular orbits. The periodicity condition and the homogenous solutions can also be used to estimate relative motion errors and the approximate fuel cost associated with neglecting the eccentricity in the reference orbit. The nonlinear simulations presented clearlyshowthatignoringthereferenceorbiteccentricitygeneratesanerrorthatiscomparabletothedisturbances caused by differential gravity accelerations.

http://acl.mit.edu/papers/ecc_orb_jgcd.pdf

Relative Dynamics and Control of Spacecraft Formations in Eccentric Orbits

Many dynamical systems have an impulsive dynamical behavior due to abrupt changes at certain instants during the evolution process. The mathematical description of these phenomena leads to impulsive differential equations. In this work we present a new approach via variational methods and critical point theory to obtain the existence of solutions to impulsive problems. We consider a linear Dirichlet problem and the solutions are found as critical points of a functional. We also study the nonlinear Dirichlet impulsive problem.

Variational approach to impulsive differential equations

An analytic method is presented to establish J
2 invariant relative orbits. Working with mean orbit elements, the secular drift of the longitude of the ascending node and the sum of the argument of perigee and mean anomaly are set equal between two neighboring orbits. By having both orbits drift at equal angular rates on the average, they will not separate over time due to the J2 influence. Two first order conditions are established between the differences in momenta elements (semi-major axis, eccentricity and inclination angle) that guarantee that the drift rates of two neighboring orbits are equal on the average. Differences in the longitude of the ascending node, argument of perigee and initial mean anomaly can be set at will, as long as they are setup in mean element space. For near polar orbits, enforcing both momenta element constraints may result in impractically large relative orbits. It this case it is shown that dropping the equal ascending node rate requirement still avoids considerable relative orbit drift and provides substantial fuel savings.

J 2 Invariant Relative Orbits for Spacecraft Formations

Abriefsurvey and classie cation of much of the published material on linearized rendezvous is presented. Thisis followed by a new form of solution of the terminal rendezvous problem that is valid in a general central force e eld. This solution and the solution of the related adjoint system are used to construct a general state transition matrix. Because of the generality of the assumptions, this state transition matrix is very concise and e exible. Finally, the work is applied to the problem of terminal rendezvous near any Keplerian orbit in a Newtonian gravitational e eld using the Tschauner ‐Hempel equations. Because this solution is presented in terms of the true anomaly, considerable care is taken to avoid the types of singularities that are typical in this kind of problem. The result is a state-transition matrix for linearized rendezvous studies that is thought to be simpler and more convenient than other versions found in the literature. I. Introduction L INEARIZED equations of motion are useful in describing the terminal rendezvous phase of a mission or in satellite station keeping. These areas of astrodynamics are rich in the variety of linearizations available to investigators and in the resulting mathematical analysis and computations that follow. This paper presents a brief survey of the types of linear models found in rendezvous studies, followed by a new general model that incorporates much previous work as special cases. The work combines some ideas in 19th century celestial mechanics with some recent discoveries. Because this new model assumes a general central force gravitational e eld,muchofthecomplexitiesfoundinspecie ccasesareavoidedin designingarelativelysimplestatetransitionmatrixthatisapplicable to a variety of problems. The work is then applied to the important special case of linearized rendezvous in a gravitational e eld dee ned by an inverse square law using the Tschauner ‐Hempel equations. In fact, it was this problem that motivated the study. The search for a solution devoid of singularities, valid for any Keplerian orbit, that avoids universal functions can lead to a cluttered set of equations. The new general model avoids much of this clutter and presents the results in a relatively concise form.

State Transition Matrices for Terminal Rendezvous Studies: Brief Survey and New Example

Based on the linearized equations of motion of a spacecraft near a satellite in general Keplerian orbit and the assumptions of a bounded thrust magnitude and constant exhaust velocity, a fixed-duration, fuel-optimal rendezvous problem is formulated and is investigated for the constant-mass case through the solution for the primer vector, which is shown to satisfy the original differential equations that describe the spacecraft motion during unpowered flight. It is shown that there are no singular solutions to this rendezvous problem for noncircular Keplerian orbits and that consequently all optimal solutions for orbit eccentricities greater than zero are constructed from a finite number of intervals of full thrust and coast where the switches are determined from the primer vector.

Fuel-Optimal Rendezvous Near a Point in General Keplerian Orbit

are removable singularities, they may lead to computational instabilities in the solution. Especially bothersome is the fact that computational problems can occur where the true anomaly is near zero. These problems are avoided in the work of solution that does not involve 7(0), but their work is confined to elliptical orbits. The purpose of this Note is to modify the original form by replacing 1(0) by a related integral 7(0), thereby removing all singularities and computational instabilities. The resulting solution, in terms of /(0), is identical for hyperbolic, parabolic, or noncircular elliptic orbits, but the particular case determines the nature of the closed-form evaluation of 7(0). Application of this work to actual problems usually involves the solution of a two-point boundary-value problem and is not considered here. Although we emphasize the case of bounded thrust, the unbounded thrust case can also be investigated through the use of the simpler equations for unpowered flight, which we also present. If the maximum number of impulses is known for an optimal rendezvous in this case, the two-point boundary-value problem is reduced to a problem of parameter optimization on the velocity increments and their locations. We conjecture that the maximum of impulses for this problem is four. If the number of impulses is restricted to two, the problem is solved by methods similar to those of Weiss et al.11'12 For the case of bounded thrust, a method such as that used for the rendezvous problem near circular orbit16 can be applied with starting iteratives obtained from the related unbounded thrust case.

New form for the optimal rendezvous equations near a Keplerian orbit

The problem of minimizing the total characteristic velocity of a spacecraft having linear equations of motion and finitely many instantaneous impulses that result in jump discontinuities in velocity is considered. Fixed time and fixed end conditions are assumed. This formulation is flexible enough to allow some of the impulses to be specifieda priori by the mission planner. Necessary and sufficient conditions for solution of this problem are found without using specialized results from control theory or optimization theory. Solution of the two-point boundary-value problem is reduced to a problem of solving a specific set of equations. If the times of the impulses are specified, these equations are at most quadratic. Although this work is restricted to linear equations, there are situations where it has potential application. Some examples are the computation of the velocity increments of a spacecraft near a real or fictitious satellite or space station in a circular or more general Keplerian orbit. Another example is the computation of maneuvers of a spacecraft near a libration point in the restricted three-body problem.

Optimal impulsive space trajectories based on linear equations

The relative-motion equations of a spacecraft in the vicinity of a satellite in an orbit that is not highly eccentric but decays as a result of drag are investigated. Because the initial orbit is not highly eccentric, theequations of motion of both the satellite and the spacecraft can be approximated by simpler equations. Some transformations are then applied to the equations of relative motion. If the drag is quadratic in the magnitude of the velocity and varies inversely with the distance from the center of attraction, the equations simplify further. There are some interesting consequences if the two objects in orbit have the same drag constant. In these cases, the transformed equations of relative motion generalize the Tschauner-Hempel equations and asymptotically approach an extension of the Clohessy-Wiltshire equations, modified to include the quadratic drag model. If the initial orbit is circular, the relative-motion equations become the new modified Clohessy-Wiltshire equations. The closed-form solution of these equations has similar structure to that of the Clohessy-Wiltshire equations. Among the potential applications of these new equations are fast preliminary studies for terminal rendezvous, station keeping, formation flying, and constellations of satellites.

Thomas Carter

Papers

State Transition Matrices for Terminal Rendezvous Studies: Brief Survey and New Example

Fuel-Optimal Rendezvous Near a Point in General Keplerian Orbit

New form for the optimal rendezvous equations near a Keplerian orbit

Optimal impulsive space trajectories based on linear equations

Clohessy-Wiltshire Equations Modified to Include Quadratic Drag