The authors were introduced last month when we published their article Trajectory optimization by a direct descent process. Preparation of this paper was supported under NASA contract NASr 54(06).

The problem considered is that of trajectory optimization using step-bystep descent to minimum cost along the direction of the cost gradient with respect to the control. Using a hybrid computer, the gradient is computed directly as the response to nearly impulsive control perturbations. A method is presented for computing the gradient when several terminal constraints are enforced. Examples of application of t h e method are presented. It is concluded that the direct gradient computation method has some significant ~ I advantages over other methods.

Extensive investigation of methods of real-time and fast- time computation of earth orbits and reentry trajectories has been conducted for both analog and digital computers. Integrator drift and multiplier errors in the case of analog computers preclude the use of direct computation in in ertial frames or geocentric coordinate systems. Such co ordinate systems can also lead to serious digital computer problems due to truncation and roundoff errors. Introduc tion of a vehicle-centered rotating coordinate system with variables referred to a nominal orbit permits greatly im proved computer scaling which, in turn, very substantially improves the absolute computational errors. In addition, the use of angular momentum and energy constraints is shown to reduce greatly the effect of integrator drift on the in-plane motion. Analog solutions to orbital and re entry problems have been obtained in real-time, fast-time, and high-speed repetitive operation using the above co ordinate system.

continues to serve as a consultant to Link and to Applied Dynamics. His major interests are in flight simulation, space vehicle mechanics, and analog computer applications. He is an Associate Fellow of the AIAA and is a member of SCi.