# Accession Number:

## AD0641404

# Title:

## NONLINEAR FEEDBACK SOLUTION FOR MINIMUM-TIME INJECTION INTO CIRCULAR ORBIT WITH CONSTANT THRUST ACCELERATION MAGNITUDE.

# Descriptive Note:

## Technical rept.,

# Corporate Author:

## CRUFT LAB HARVARD UNIV CAMBRIDGE MASS

# Personal Author(s):

# Report Date:

## 1966-07-01

# Pagination or Media Count:

## 25.0

# Abstract:

The instantaneous thrust-direction for a rocket vehicle to perform a minimum-time injection into a circular orbit of prescribed radius is determined as a function of instantaneous distance, and radial and tangential velocity relative to the attracting center. A nonlinear feedback control law for the instantaneous thrust-direction is derived which is based on the approximation that the gravity vector and the vehicle thrust acceleration magnitude during the maneuver are to be constant at values intermediate between their present and expected terminal values. The control law is shown to depend only on two dimensionless functions of the three relevant state variables, so that the solution is, in effect, expressed in a reduced state space of two dimensions. The optimal thrust-direction is defined analytically and graphically as a function on the reduced state space. The open-loop solution to the minimum-time transfer problem is the well-known linear tangent law. The new contributions are 1 showing that the solution depends on only two dimensionless functions of state and 2 putting the solution in the form of a feedback law which depends on these two functions. For a maneuver spanning a considerable arc around the attracting center up to about 40 degrees, the solution may be used directly as a suboptimal control or to give starting values for an iterative solution of true inverse-square gravity problem. Author

# Descriptors:

# Subject Categories:

- Spacecraft Trajectories and Reentry