NASA Technical Reports Server (NTRS) 19630002716: A STUDY OF THE EFFECT OF A DEADBAND ABOUT A DESIRED PERIGEE ON THE GUIDANCE OF A SPACE VEHICLE APPROACHING THE EARTH

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