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

and energy. A parametric Lyapunov differential equation approach is proposed in this paper to solve this constrained regulation problem. After establishing the fact that the Tschauner-Hempel equations are both null controllable with controls of bounded magnitude and energy, this paper proves that the proposed linear periodic controllersemigloballystabilizesthesystem.Equivalently,forany fixedinitialconditions,themagnitudeandenergy of the control can be made as small as desired by tuning some free parameters in the feedback laws. In comparison with the existing quadratic-regulation-based approach, which requires solutions to nonlinear Riccati differential equations,thenewapproachrequiresonlythesolutionoflinearperiodicLyapunovdifferentialequations,whichare investigated inthepaperbyusingtheperiodicgenerator approach.Numericalsimulationsofthe nonlinearmodelof the spacecraft rendezvous instead of a linearized one show that both the magnitude and energy of the control can be reduced to an arbitrarily small level by reducing the values of some parameters in the controller and that the rendezvous mission can be accomplished satisfactorily.

Lyapunov Differential Equation Approach to Elliptical Orbital Rendezvous with Constrained Controls

The optimal control of a spacecraft as it transitions between specified states using continuous thrust in a fixed amount of time is studied using a recently developed technique based on Hamilton-Jacobi theory. Starting from the 1st order necessary conditions for optimality, we derive a Hamiltonian system for the state and adjoints with split boundary conditions. Then, recognizing the two point boundary value problem as a canonical transformation, we employ generating functions to find the optimal feedback control as well as the optimal trajectory. Though we formulate the optimal control problem in the context of the necessary conditions for optimality, our closed-loop solution also formally satisfies the sufficient conditions for optimality via the fundamental connection between the optimal cost function and generating functions. A solution procedure for these generating functions is posed and numerically tested on a non-linear optimal rendezvous problem in the vicinity of a circular orbit. Generating functions are developed as series expansions, and the optimal trajectories obtained from them are compared favorably with those of a numerical solution to the two point boundary value problem using a forward shooting method.

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Solving Optimal Continuous Thrust Rendezvous Problems with Generating Functions

The optimal control of a spacecraft as it transitions between specified states using continuous thrust in a fixed amount of time is studied using a recently developed technique based on Hamilton-Jacobi theory. Started from the first-order necessary conditions for optimality, a Hamiltonian system is derived for the state and adjoints with split boundary conditions. Then, with recognition of the two-point boundary-value problem as a canonical transformation, generating functions are employed to find the optimal feedback control, as well as the optimal trajectory. Although the optimal control problem is formulated in the context of the necessary conditions for optimality, our closed-loop solution also formally satisfies the sufficient conditions for optimality via the fundamental connection between the optimal cost function and generating functions. A solution procedure for these generating functions is posed and numerically tested on a nonlinear optimal rendezvous problem in the vicinity of a circular orbit. Generating functions are developed as series expansions, and the optimal trajectories obtained from them are compared favorably with those of a numerical solution to the two-point boundary-value problem using a forward-shooting method.

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Survey of orbital dynamics and control of space rendezvous

Optimal power-limited rendezvous with variable exhaust velocity is investigated for propulsion systems having both upper and lower bounds on thrust magnitude. In this model the spacecraft thrusters have four admissible states, thrusting at the upper saturation level, unsaturated power-limited thrusting, thrusting at the lower saturation level, and unpowered (i.e., engine off.) A fifth chattering state is also possible. The mathematical structure of the solution of the optimal rendezvous problem associated with this propulsion model is found. Computer simulations of rendezvous with a satellite in circular orbit are presented. For rendezvous near circular orbit four classes of chattering solutions of two types can occur. It is assumed that multiple thrusters can be mounted on the spacecraft and that they can operate independently. Applying the efficiency condition presented in a previous paper, a logic for switching individual thrusters on and off for optimal fuel usage and computational efficiency is presented.

Optimal Power-Limited Rendezvous with Upper and Lower Bounds on Thrust

Very general arguments show that thrust saturation is not helpful in terms of performance in optimal spacecraft trajectories, but these arguments do not provide quantitative information. For optimal power-limited rendezvous problems with thrust saturation and linearized equations of motion, two simple mathematical conditions that are relatively easy to check are presented that may be of value to mission planners. These conditions relate respectively to the problem of reaching a terminal point in a specified time in the presence of thrust saturation and in doing so in a fuel-efficient way. They can be useful in indicating whether or not the flight interval should be lengthened or whether or not additional thrusters should be implemented if possible, and if so how many. For the case of optimal power-limited rendezvous with a satellite in circular orbit, quantitative information is presented for certain boundary conditions. For certain low-thrust saturation levels and boundary conditions, significant improvements in performance and computation can be attained by lengthening the flight time or implementing an additional thruster, if possible. In one example, the use of two thrusters instead of one reduced the mission fuel requirements by nearly 50%.

Optimal power-limited rendezvous with thrust saturation

We consider a second order system of partial differential equations of mixed hyperbolic-elliptic type on a domain bounded by two differentiable curves AC and BC in the hyperbolic region and a non-selfintersecting piecewise differentiabe curve I' connecting A to B in the elliptic region. AC and BC are either characteristic curves or satisfy certain simple inequalities with regard to the leading coefficients. We derive an apriori estimate for the system of differential operators for a certain class of functions with Dirichlet boundary condition on BC and part of and Neuman condition on the other part of r. The corresponding boundary value problem is formulated with uniqueness of a solution following directly from the apriori estimate.

Cyrus J. Pardis

Papers

Optimal Power-Limited Rendezvous with Upper and Lower Bounds on Thrust

Optimal power-limited rendezvous with thrust saturation

A boundary value problem for a system of equations of mixed hyperbolic-elliptic type