Decelerating effect of atmospheric drag assures minimum fuel consumption and time expenditure during braking for soft landing

Bejczy, A. K.

English

NASA Technical Reports Server

1972
NASA TECH BRIEF 
NASA Pasadena Office
B72-10084 
-4:v 
NASA Tech Briefs announce new technology derived from the U.S. space program. They are issued to encourage 
commercial application. Tech Briefs are available on a subscription basis from the National Technical Information 
Service, Springfield, Virginia 22151. Requests for individual copies or questions relating to the. Tech Brief program may 
be directed to the Technology Utilization Office, NASA, Code KT, Washington, D.C. 20546. 
Optimized Braking of Landing Vehicles

with Atmospheric Drag 
The landing maneuver of a space vehicle on a 
distant planet or on Earth requires in its early stages 
a deceleration so that the terminal velocity will per-
mit a landing. During the final deceleration ma-
neuver, however, propulsive retrothrust devices must 
be used to ensure that the surface is approached with 
a velocity that permits a soft landing. 
The program of application of ret.rothrust must be 
determined carefully and in such a way as to con-
serve fuel, especially when it is necessary to consider 
the cost of transferring one pound of mass from the 
Earth to another planet (and, eventually, back to 
Earth). Accordingly, a study was made of the de-
termination and specification of a control law that 
assures minimum fuel consumption and minimum 
expenditure of time during the propulsive braking 
of a landing vehicle subjected to atmospheric drag 
and following a vertical or a gravity-turn ballistic 
trajectory. 
The distinctive dynamical feature in the solutions 
of the optimal control problems is that the retrothrust 
acts in the same dynamical sense as the atmospheric 
drag. Thus, the drag is a gratis braking force. The drag, 
however, is a strongly state-dependent (velocity-
and altitude-dependent) force. During a propulsive 
soft-landing maneuver on an atmosphere-bearing 
planet, the typical dynamical situation is as follows: 
decreased altitude implies increased drag; decreased 
velocity implies decreased drag; the retrothrust de-
creases the vehicle's velocity and, at the same time, 
retards the vehicle's descent to denser atmospheric 
regions. Hence, the retrothrust reduces the decelerat-
ing effect of the atmospheric drag. Reduction of the
(gratis) decelerating effect of the atmospheric drag, 
however, implies increased integrated retrothrust ef-
fort (= mass expenditure) to achieve a soft landing. 
Therefore, the minimum fuel consumption ( mini-
mum mass expenditure) burning program for the 
retrorocket will necessarily imply the maximization 
of the available decelerating effect of the drag since 
the burning program of the retrorocket implicitly 
also "controls" the time history of the atmospheric 
drag that affects the descending vehicle. 
Solution to the problem assumes that as the vehicle 
descends through the atmosphere, at a certain time 
to the vehicle's altitude, velocity, and path angle 
relative to the surface are known. The vehicle is 
equipped with a retrorocket having a maximum 
thrust level Tm. The program of computation then 
determines ti > to when the retrorocket is turned 
on and maximum thrust is maintained until the ter-
minal time r when the vehicle reaches zero velocity 
at essentially zero altitude and the rocket thrust is 
terminated. The optimization can be performed for 
a vertical or for a ballistic descent in which the 
landing time is minimized. The fuel consumed for 
the maneuver is readily computed. 
Note: 
Requests for further information may be di-
rected to: 
Technology Utilization Officer 
NASA Pasadena Office 
4800 Oak Grove Drive 
Pasadena, California 91103 
Reference: TSP72-10084 
An	 (continued overleaf) 
This document was prepared under the sponsorship of the National 
Aeronautics and Space Administration. Neither the United States 
Government nor any person acting on behalf of the United States
Government assumes any liability resulting from the use of the 
information contained in this document, or warrants that such use 
will be free from privately owned rights.
https://ntrs.nasa.gov/search.jsp?R=19720000084 2020-03-17T03:25:19+00:00Z
Patent status: 
No patent action is contemplated by NASA. 
Source: Antal K. Bejczy of

Caltech/JPL
under contract to

NASA Pasadena Office 
(NPO-11402)
Category 06


Optimized braking of landing vehicles with atmospheric drag

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