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NASA TN B-1607 


NASA TN D-1607 



TECHNICAL NOTE 


D-1607 

A STUDY OF THE EFFECT OF A DEADBAND ABOUT 

A DESIRED PERIGEE ON THE GUIDANCE OF A 

SPACE VEHICLE APPROACHING THE EARTH 

By Jack A. White 

Langley Research Center 
Langley Station, Hampton, Va. 


NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
WASHINGTON February 1963 





NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 


TECHNICAL NOTE D-I 607 

A STUD! OF THE EFFECT OF A DEADBAND ABOUT 
A DESIRED PERIGEE ON THE GUIDANCE OF A 
SPACE VEHICLE APPROACHING THE EARTH 
By Jack A. White 


SUMMARY 


An analysis is made of the guidance of a space vehicle which is approaching 
the earth at supercircular velocities and is attempting to control the trajectory 
to a desired vacuum perigee point. Perigee altitude was controlled hy applying 
a corrective impulse each time the predicted perigee altitude deviated from the 
iesired perigee altitude by a given amount; this given amount is called the dead- 
band. Random errors were assumed in the measurement of velocity and flight-path 
angle and in obtaining the desired magnitude of thrust impulse. The error magni- 
tudes and deadband described in NASA Technical Note D-957 were used for the present 
Investigation. 

It was found that the method of making a correction each time the deadband 
was exceeded yielded poorer perigee-altitude control at approximately the same 
(or slightly less) cost than did the correctional procedure of NASA Technical 
Note D-957 for which corrections were applied at a series of prescheduled points 
along the trajectory and for which no deadband was utilized. 


INTRODUCTION 


In order for a space vehicle approaching the earth at supercircular veloci- 
ties to intercept the earth's atmosphere at desired entry conditions, it may be 
necessary for corrective thrust to be applied during midcourse guidance. Consid- 
erable research has been directed toward the solution of problems associated with 
guidance to a specified perigee altitude and the results of some of these studies 
are presented in references 1 to 6 . The basic problem involves the guidance accu- 
racies required to place the vehicle in a position to enter the earth's atmosphere 
successfully . 

Reference 1 gives the results of a study of three methods of scheduling 
corrective-thrust impulses in the presence of random errors assumed in measuring 
velocity and flight-path angle and in obtaining the desired thrust impulse. Cor- 
rections were applied at preselected correction points along the trajectory. In 
reference 2 , a similar series of scheduled correction points was employed, first, 



by applying a correction only if the predicted perigee altitude deviated from ti 
desired perigee altitude by a certain amount (called the deadband) and, second, 
by applying a correction at each scheduled point (no deadband) . The width of tt 
deadband was decreased as the vehicle approached the earth, as were the errors i 
velocity and flight-path angle. 

The guidance scheme hypothesized in reference 3 is based on a large number 
of prescheduled decision points and employs a constant deadband about the ass um e 
measured value of the trajectory angular rate. The general method of navigation 
used in references 5 and 6 is based on the perturbation theory wherein only devi 
tions in position and velocity from a reference path are utilized. The guidance 
equations which are dependent upon prescheduled decision points use linear predi 
tions of the final deviation to obtain the minimum required corrective velocity 
and were formulated by using optimal filtering of measured data. 

It is of interest to note that each of the guidance schemes presented in 
references i to 6 was found capable of controlling the approach trajectory and 
required a relatively small corrective-velocity increment. One area of agreemeni 
among these studies is the fact that the final deviation from the desired peri get 
point depends on the location of the final correction point. It is also apparenl 
from these results that corrective maneuvers should be executed at the longest 
feasible range inasmuch as the cost of delaying the corrective action becomes sut 
stantial as the range decreases. 

In the present study the effect of a deadband in the guidance scheme upon 
control accuracy and total-corrective-velocity requirements is further investi- 
gated. The same formalization of a deadband as defined in reference 2 Is employe 
and a correction is made whenever the predicted perigee altitude exceeds a bound- 
ary of the deadband. This method differs from reference 2 In that the corrective 
action Is not made at preselected points along the approach trajectory. The mag- 
nitude of each correction was calculated on the basis of assumed measurements of 
position and velocity. According to a conclusion of reference 7, the scheduled 
correction points of reference 2 occur at near optimal frequency along the trajec 
tory. The purpose of this paper is to compare the results of the present study 
with the results of reference 2. 


SYMBOLS 


The English system of units is used In this study. In case conversion to 
metric units is desired, the following relationships apply: 1 foot = 0.3048 mete: 

and 1 statute mile = 5,280 feet = 1,609-344 meters. 

g acceleration due to gravity, 32.2 ft/sec^ 

R radius of earth, 3,960 international statute miles 

r radial distance from center of earth to vehicle, ft, except when used 

In determining the standard deviations of errors in V and 7 and 
then in international statute miles 


2 



radial distance from center of earth to perigee point of flight 
path, ft 

radial distance from center of earth to perigee point of flight path 
after final correction, ft 

desired perigee radial distance, ft 


d 

T 

V 


d 


•7 


f 

a 


V 

r v T 


velocity of space vehicle, ft /sec 

velocity at a given radial distance for the desired trajectory, ft/sec 
magnitude of corrective-velocity vector, ft/sec 

increment of velocity used to establish deadband, ft /sec 
flight -path angle, deg 

flight-path angle at a given radial distance for the desired 
trajectory, deg 

increment of flight-path angle used to establish deadband, deg 

angle between a line from center of earth to space vehicle aind perigee 
radius vector, deg 

magnitude of 9 at final correction point , deg 
change in 9, deg 

standard deviation of normal distribution 
standard deviation of error in V , ft/ sec 

standard deviation of error in V^, percent V^,, ft /sec 
standard deviation of error in 7, deg 


Jub script: 

■> conditions that define initial trajectory 


3 


METHOD OF ANALYSIS 


Approach Conditions and Assumptions 

In all cases investigated in the present study, the space vehicle Is 
approaching the earth on an elliptical path with an eccentricity of almost 1. A 
any point along the approach trajectory, the desired trajectory is that part of 
an ellipse which passes through the point and has a semimajor axis of 100,000 
International statute miles and a perigee radius of R + 250,000 feet. This stu 
Is concerned with the portion of the approach trajectory beginning at 0 O = l6o c 
and ending at the vacuum perigee radius, as shown in the diagram of figure 1. 

The following assumptions, the same as those of references 1 and 2, are mad 
in this study: 

(1) The earth is spherical. 

(2) Motion is considered only in the plane of the orbit for a nonrotating 
earth. 

(3) The space vehicle is close enough to the earth for the gravitation fiel* 
of all other bodies to be neglected (a two-body problem) . 


Correction Technique 

The present study is based on the application of a thrust impulse to control 
the perigee altitude. The basic technique for controlling perigee altitude is tc 
apply corrections throughout the course of the approach trajectory each time the 
predicted perigee altitude exceeds a boundary of a specified deadband about the 
desired perigee altitude. At any point along the approach trajectory the measure 
values of V and 7 (obtained by adding random errors to the true values) are 
used to calculate the orbital characteristics. The calculated (predicted) perige 
radius is compared with a limit (the boundaries of the deadband) of the perigee 
radius to determine if a corrective impulse Is needed. If a correction is indi- 
cated, calculations are then made to determine the optimum direction and magnituc 
of corrective velocity required to correct the trajectory to the center of the 
deadband. An assumed error in corrective velocity is added and the correction ie 
applied in the optimum direction. The procedure given in reference 2 to determir 
the direction in which to apply corrective thrust was used in the present study. 

The standard equations of an ellipse were used (ref. 2) to calculate the 
orbital characteristics of a space vehicle approaching the earth on an elliptical 
path. The following expression for the perigee radius in terms of the trajectory 
variables r, V, and 7 (eq. (l) of ref. 2) was used to calculate the deadband 
within which the space vehicle was to be controlled: 


4 




where v = V d + AV and 7 = 7 ^ - A 7 are used to determine rp on one side of 
the deadband and V = V d - AV and 7 = 7 d + A7 are used to determine r p on 
the other side. 


Range of Initial Conditions 

Two sets of assumed errors, the same as those investigated in reference 2, 
were used in the present study. These errors, which are errors in measuring 
velocity and flight -path angle and in applying corrective thrust, were assumed to 
have a normal distribution. The standard deviations of the errors investigated 
were : 


d v 


a 


7 


First 


— - ft /sec 

10,000 


0 • 012 2 r - degrees 

10,000 



0.013V t ft /sec 


d v = 



Second 

— — — ft/sec 

10,000 

0 . 0379 £ de grees 

10,000 

= 0.039Vijt ft /sec 


Initial conditions were assumed so that without corrections perigee radii 
of 0.75R, 0.99R, 1.01R, 1.25R, 1.5R, and 2. OR would be obtained. The initial 
range was that associated with 0 O = l6o°. 

The deadband width studied for each set of instrumentation errors was that 
obtained when AV = by and A 7 = Oy and is the same as the "0 deadband" in 

reference 2. The r term in by and Oy causes the width of the deadband to 
decrease as the vehicle approaches the earth. 


Method of Control 


For peri gee -altitude control, the present investigation employed a correc- 
tive impulse at every point along the approach trajectory where the predicted 


5 



perigee altitude exceeded a boundary of the deadband. In order to simulate the 
continuous predicted perigee altitude needed in the present study, the digital- 
computer program for the angular method of perigee -altitude control for the study 
reported in reference 2 was modified in the following manner. Small angular 
increments were used to schedule observation points along the approach trajectory. 
At these preselected points, the predicted perigee altitude was determined and 
compared with the deadband. If the predicted perigee altitude was not in the 
deadband, a straight-line approximation was made to determine where the predicted 
perigee altitude exceeded the deadband between the present and previous observa- 
tion points. At the point where the predicted perigee altitude exceeded the dead- 
band, the correctional maneuver was made and the new approach trajectory was 
determined. 

In order to assess the effect of the straight-line approximation used to 
determine the point where the predicted perigee altitude exceeded the deadband, 
angular increments from 2° to 30° were investigated. The perigee altitude and 
total-corrective-velocity probability curves were the same for all angular Incre- 
ments investigated. Therefore angular increments of 30° were used for the study. 


RESULTS AND DISCUSSION 


General Discussion 

The results of this study, presented as solid curves in figures 2 to 7; are 
shown as probability distribution curves. The results are shown for the two sets 
of errors and In the case of the smaller errors for two values of 9^. The 

perigee-altitude probability curves, where each curve Is based on 1,000 runs, are 
presented as the probability of the peri gee -altitude error being less than a 
given value. Likewise, the total-corrective-velocity probability curves, where 
each curve Is based on 100 runs, are presented as the probability of the total 
corrective velocity being less than a given value. For comparison with the pres- 
ent results, data obtained -under the Investigation reported In reference 2 for 
methods with and without a deadband are included in figures 2 to 7* Although all 
results of the present study and those of reference 2 are presented for a desired 
perigee altitude of 250,000 feet, these results are applicable to any desired 
perigee altitude. 


Results of Correcting at Edge of Deadband 
A number of values of r^ were used in the analysis to determine the 

p ,u 

effect of applying a correction each time the predicted perigee altitude exceeded 
the boundary of the deadband about the desired perigee altitude. The results 
presented In figure 2, where the set of measurement errors is represented by 

0 = ft /sec, <t = Q degrees, and cr, r = 0.013Vm ft /sec, and where 

v 10,000 7 10,000 V T x 

= 10°, show a perigee -altitude control within about ±3 >000 feet. For the same 


6 



set of measurement errors, but where no corrective action was taken beyond 
0 = 4o°, the results presented in figure 4 show a perigee-altitude control within 
about ±10,000 feet. The total-corrective-velocity requirements for the perigee- 
altitude control shown In figures 2 and 4 are presented in figures 3 and 5- 

Figures 6 and 7 show the perigee-altitude control and total-corrective- 

velocity requirements for the assumed measurement errors of cr v = — — — ft/sec, 

v 10,000 

a = Q : Q375 , r degrees, and cr v = 0.039Vm ft/sec, and for 9^ = 10°. These results 
7 10,000 V T i 

show a perigee-altitude control within about ±10,000 feet. 


Comparison of Results 

In reference 2 - the results of which are here compared with those of the 
present study - a method of scheduling corrections at different values of the 
angle between perigee and the vehicle* s position vector was investigated with and 
without a deadband. Cases for which errors, Initial conditions, deadband, and 
the final observation point were the same as those of the present study were 
selected from reference 2 and are Included here for comparison with the results 
of the present study. Attention is called to the step or abrupt change in slope 
of the total-corrective-velocity probability-distribution curves of some of the 
data taken from reference 2. As pointed out in reference 2, this step simply 
means that a certain percent of the runs required the same or approximately the 
same value of Vp. 

Figures 2, 4, and 6 show that the perigee- altitude- error band for the correc- 
tion procedure of reference 2 where no deadband was utilized was much smaller 
(approximately 50 percent for most cases) than the band for either of the correc- 
tion procedures utilizing a deadband. 

Figures 3, 5, and 7 show that the probability-distribution curves of total 
corrective velocity are approximately the same for the correction procedure of 
this study and the correction procedure without a deadband. The differences 
between the two methods are not too significant. However, it is Indicated that 
for the runs with small errors and small values of 0p (fig. 3) the Vp required 

was always slightly lower for the present procedure. 

A comparison of the three control procedures leads to the conclusion that 
either the method without a deadband or the method of the present study would be 
the best regarding efficiency. However, if achieving a perigee altitude nearest 
to r p,d t s required, then the method which uses no deadband is the best. In 

just about all cases Investigated, the deadband procedure of reference 2 was poor 
in comparison with the other two methods either for efficient use of Vp or for 

close control of r_ n . 

p,u 


7 



CONCLUDING REMARKS 


A study of the effects of employing a deadband about a desired perigee alti- 
tude on the guidance of a space vehicle approaching the earth was made. A correc- 
tive maneuver was made at any point along the approach trajectory where the pre- 
dicted perigee altitude did not fall within the deadband. A comparison was made 
of these results and the results of the two procedures of perigee altitude control 
of NASA Technical Note D-957. These two procedures were as follows: (l) Correc- 

tive maneuvers were made at scheduled observation points along the trajectory if 
the predicted perigee altitude fell outside a deadband. (2) Corrective maneuvers 
were made at all scheduled observation points (no deadband) . 

By using a correction procedure which omitted the deadband (no deadband of 
NASA Technical Note D-957 ) > the perigee-altitude control about the desired perigee 
value was better under all initial conditions, instrumentation inaccuracies, and 
location of the final correction point than either of the two correction proce- 
dures which included a deadband. The total-corrective-velocity requirements for 
the procedure which omitted the deadband and for the procedure in which a correc- 
tion was made when the predicted perigee altitude exceeded a boundary of the dead- 
band were, in general, approximately the same. The deadband procedure of NASA 
Technical Note D-957 was poor, in comparison with other procedures, either for 
efficient use of corrective velocity or for close control of perigee altitude . 


Langley Research Center, 

National Aeronautics and Space Administration, 

Langley Station, Hampton, Va. , November l 4 , 1962. 


8 



REFERENCES 


1. White, Jack A . : A Study of the Guidance of a Space Vehicle Returning to a 

Braking Ellipse About the Earth. NASA TN D-191, I960. 

2. White, Jack A.: A Study of the Effect of Errors in Measurement of Velocity 

and Flight-Path Angle on the Guidance of a Space Vehicle Approaching the 
Earth. NASA TN D-957, 19^1 • 

5* Harry, David P. , XXI, and Friedlander , Alan L. ; Exploratory Statistical 

Analysis of Planet Approach-Phase Guidance Schemes Using Range, Range-Rate, 
and Angular-Rate Measurements. NASA TN D-268, i960. 

4. Wong, Thomas J. , and Slye, Robert E. : The Effect of Lift on Entry Corridor 

Depth an d Guidance Requirements for the Return Lunar Flight. NASA TR R-oO, 
1961. 

5. McLean, John D. , Schmidt, Stanley F., and McGee, Leonard A.: Optimal Filtering 

and Linear Prediction Applied to a Midcourse Navigation System for the 
Circumlunar Mission. NASA TN D-1208, 1962. 

6. Battin, Richard H. : A Statistical Optimizing Navigation Procedure for Space 

Flight. Rep. R-3Ul, Instrumentation Lab., M.I.T. , Sept. 1961. 

7. Lawden, D. F. : Optimal Program for Correctional Maneuvers. Tech. Rep. 

RR ll86-6o-13 (Contract AF 33(6l6) -5992) , Radiation, Inc., Sept. 27, 19°0. 


9 




Probability;, percent Probability, percent 


Figure 2 



-Correction made when deadband exceeded 


. Corrected at scheduled points if outside deadband (ref. 

Corrected at scheduled points, no deadband (ref. 2) 


2 ) 



Probability-distribution curves of perigee-altitude error. Oy = 


10,000 


ft 


a = °- 01 ?£ r - degrees; = 0 . 013 % ft/sec; e f = 10°. 

y i r\ rir\r\ Vm ■*- 


Probability, percent Probability, percent 



Correction made -when deadband exceeded 

Corrected at scheduled points if outside deadband (ref. 2) 

Corrected at scheduled points, no deadband (ref. 2) 




Probability, percent Probability , percent 



Correction made when deadband exceeded 

Corrected at scheduled points if outside deadband (ref. 2) 

Corrected at scheduled points, no deadband (ref. 2) 














Probability, percent Probability, percent 



Correction made when deadband exceeded 

■ ■ — Corrected at scheduled points if outside deadband (ref. 2 ) 

Corrected at scheduled points, no deadband (ref. 2 ) 



-8 4| 0 It 8 12 16 x 10 

Perigee- altitude error, r , - r , feet 


Figure 4.- Continued. 


Probability , percent probability, percent 


6or~ 


Ijob 



Correction made Tdien deadband exceeded 

Corrected at scheduled points if outside deadband (ref. 2 ) 

Corrected at scheduled points, no deadband (ref. 2) 



Figure U.- Concluded. 





Probability, percent Probability, percent 


Figure 


60 



Correction made iishen deadband exceeded 

Corrected at scheduled points if outside deadband (ref. 2) 

Corrected at scheduled points, no deadband (ref. 2) 



Perigee- altitude error, r^^ ~ r p,a* 


6.- Probability-distribution curves of perigee -altitude error. <jy = 

_ degrees; a v = 0.039Vm ft/sec; 0 f = 10°. 

7 in. non v r L 1 


3r 

10,000 


Probability, percent Probability, percent 


100 


-Correction made when deadband exceeded 

-Corrected at scheduled points if outside deadband (ref. ?.) 
-Corrected at scheduled points, no deadband (ref. 2) 



160 200 2I4O 280 320 360 bOO f 6ho 600 720 760 

Total corrective velocity, V-j., ft/sec 


Figure 7.- Probability-distribution curves of total corrective velocity. Gy - - ft/sec } 

xo ^ 000 

= 2^0575 £ degrees . a = 0.039Vm ft/sec; 0 f = 10°. 

y 10,000 V T 1 


NASA -Langley, 1963