A new approach to steepest-ascent trajectory optimization

Abstract

HE primary difference between the present method and others described in the literature concerns the determination of the change in control (angle of attack) in the optimizing direction. The determination of the change in control required to correct errors in specified end conditions is the same as in other versions of the gradient method. In the case where the final velocity is to be maximized, for example, the optimizing angle-of-attack increment is usually determined by specifying a desired increase in velocity. In the method of this paper, we first determine approximate partial derivatives of final velocity with respect to the final values of the other variables. We then proceed to find the angle of attack as if we were using the classical calculus of variations technique, i.e., from the partial derivatives we determine final values of the Lagrangian multipliers. These final values determine the values of the multipliers through the entire trajectory, and the angle of attack can be solved for from the appropriate Euler or adjoint equation. The usual gradient techniques are used to determine the variations in final velocity which would occur as a result of specifying changes in the final values of the other variables, and these variations determine the partial derivatives mentioned previously. Another factor that has been found useful in stabilizing the iteration in particularly sensitive cases is the use of attitude angle 6 as the controlling parameter rather than angle of attack a. In order to use 0 control, it is necessary to estimate the change in path angle 7 which will occur at the next trajectory, because experience has shown that, when the 6 control is determined by adding the new a. to the current value of 7 (0 = 7 + a), the iteration will almost always not converge. Good convergence properties are obtained by determining the value of 6 from 6 = 7 + ^7 + anew The techniques just described and others that are discussed in the paper result in a method for determining trajectories which is believed to combine the best features of the steepest ascent, or gradient method, and the classical methods of the calculus of variations. A computer program using these new techniques has been developed for solving the rocket boost vehicle mission optimization problems that were analyzed in Ref. 1 (using the classical methods of the calculus of variations). As in Ref. 1, a variable coasting time may be included, and the optimum value is determined by the procedure. Effects of planet rotation, aerodynamic forces, variable atmosphere, time-variant thrust, and vehicle design or physiological limitations are included. A numerical example is shown to illustrate the convergence properties of the current method in a somewhat difficult (sensitive) case.