# An analytical approach for determining an approximately optimal control and an explicit closed loop guidance strategy

- Paper ID
IAF-69-39

- author
- company
Martin Marietta Corporation Denver Division

- country
U.S.A.

- year
1969

- abstract
A problem of escape from an inverse-square force field by means of a single thrusting interval using minimum fuel is investigated in this paper. The magnitude of the thrust acceleration is assumed to be large and constant during application, and the initial conditions are general, except that neither the initial radius nor the initial speed is zero. In addition, it is assumed that the initial energy is less than the final energy required to escape. Under these conditions, it is desired to determine, analytically, the thrust direction history that requires minimum fuel, pr equivalently, minimum mass ratio. The approximate mass ratios required to escape from circular orbits have been analytically determined by previous investigators using several different thrust histories. For constant thrust acceleration, the case of escape with zero residual speed has been studied by Tsien1 who used radial and circumferentially directed thrust histories, by Benney2 utilizing the tangentially directed thrust program, and by Lawden3 who analytically determined an approximate optimal thrust direction history. The problem was extended to include non-zero residual speeds by Long4 for the cases of tangential and optimally directed thrust programs, and by Anthony5,6 for several different thrust directions and for the optimum thrust direction. Escape maneuvers arising from circular orbits using constant thrust have been studied numerically by Moeckel7 for the case of tangentially directed thrust, and analytically by Anthony and Sasaki8 for optimally directed thrust. In addition to escape maneuvers that start from circular orbits, the problem of escape under more general initial conditions has been investigated by Anthony and Sasaki9 for the case of tangentially directed thrust with constant thrust acceleration. In view of these previous investigations concerning the escape problem, the purpose of the present analysis is three-fold : (1) to present a generalization of the problem for the optimally directed thrust history; (2) to derive an explicit closed loop guidance system that is nearly optimal; and (3) to assess the efficiency of tangentially directed thrust escape maneuvers that originate from arbitrary initial conditions. In Sec. 2 the equations for determining the minimum mass ratio are formulated by utilizing the method of the calculus of variations. Two properties of the exact optimum control, previously known for the case of initially circular orbits, are found to be equally valid for the present case. The equations are solved approximately by using Poisson’s method of expansion in a parameter, s. The impulsive results for the problem are obtained when e vanishes. The dimensionless burning time is determined to second order in the parameter. Certain characteristics of the approximate solution are discussed in Sec. 3. At thrust termination, the approximate optimum control satisfies two properties required by the exact optimum control, and it is shown that, to first order in the parameter, the optimum control rotates uniformly in inertial space. Then, assuming that the state and thrust acceleration are updated continuously, an explicit closed loop guidance scheme is derived. The difference in requirements using optimally directed and tangentially directed thrust is determined analytically. In Sec. 4 the present solution is used to analytically determine the point at which thrust must be initiated in order to escape from a prescribed elliptic orbit with minimum mass ratio. Numerical comparisons arepresented which indicate the differences between finite thrust directed optimally and tangentially and impulsive thrust.