Reformulation of Edelbaum' s Low-Thrust Transfer Problem Using Optimal Control Theory

The problem of optimal low-thrust transfer between inclined orbits is reformulated within the framework of optimal control theory. The original treatment considered the time-constrained inclination maximization with velocity as the independent variable allowing the use of the theory of maxima. Because the independent variable is double valued for some transfers, two expressions for the inclination change involving inverse-sine functions are needed to describe all possible transfers. The present analysis casts this problem as a minimum-time transfer between given noncoplanar circular orbits and obtains a single analytic expression for the orbital inclination involving a single inverse-tangent function, uniformly valid for all transfers. The D V penalty with respect to the exact transfer solution using the full six-state dynamic equations with optimized thrust proe le during the transfer is shown to be small. I. Introduction A NALYTIC solutions of the low-thrust transfer problem are very useful in preliminary mission analysis as well as spacecraft systems designandoptimization.The overalldesign of asolarelectric transfer vehicle or even an integrated spacecraft requires extensive parametric analyses for optimum sizing of the various power, propulsion, and thermal management systems to maximize delivered payload to the destination orbit. These parametric studies require hundredsofiterations, precluding theuseofthe numerically generated transfer solutions. The analytic solutions are also desirable for future onboard autonomous guidance applications, especially for smaller spacecraft such as in the mini- and microsatellite category where the application of low-thrust technology for orbit maintenance and control is most efe cient. In the early 1960s, Edelbaum 1;2 derived analytic expressions for the maximum change in inclination between two circular orbits of given size with continuous constant acceleration and e xed transfer time. Conversely, he derived an analytic expression for the total 1V needed to carry out the transfer between given inclined circular orbits.ThistheorywaslatergeneralizedbyWieselandAlfano, 3 who allowed for the variation of the out-of-plane or thrust yaw angle during each revolution, unlike Edelbaum, who used the simpler constant yawproe le.Thus, the (a,i)semimajor axisand inclination space was mapped by direct numerical integration of the simplie ed differential equations in a and i, such that the minimum time for a given transfer is read directly from the solution map. InRefs.4and5,theoptimalthrustpitchandyawproe lesrequired for a given transfer were determined in a semianalytic way by also considering discontinuous thrust due to eclipsing. However, these solutions are not analytic and, therefore, are dife cult to implement in systems design optimization software. In Ref. 6, rapid transfer calculations were demonstrated by analytic modeling of the various transfer parameters including shadowing and solar power degradation effects due to the Van Allen radiation belts. The thrust yaw angleisheldconstantthroughoutthetransfer,andtherequiredvalue is determined by iteration. This is not as optimal as the Edelbaum steering solution, which holds the yaw angle constant during each revolution but varies its value from revolution to revolution in an optimal manner. All of these analyses assume that the orbit remains or is forced to be circular after each cycle or revolution.