ART III: ADVANCED MILITARX SATELLITES

• VOLUME III

EXCLUDED FROM

DECLASSIFICATION JAW E.O. 12958

REFER

TO:

v
ADVANCED WEAPON SYSTEMS STUDY

PART III - ADVANCED MILITARY SATELLITES

VOLUME III

TR-59-0000-00604

DOWNGRADED AT 12 YEAR

;:':TERVALS; NO1 All( OMAT ICAO!

,:),S,11.3StE 

V. DOD D1R 5200.10

1 March 1959

Work Completed

10 September 1958

SPACE TECHNOLOGY LABORATORIES

P.O. Box 95001

Los Angeles 45, California

This doarmint contains I

nfonnetims offomino Me notionol defense of Om United litotes within Ms mooning of Man Emsfoams It U.S.C., Wien 793 and 794, the transmission or 'isolation if which in any wanner to on versotherlisd omen orobilli Laws. MIS hni by law.
11111" TR-59-0000-00604

Page ii

CONTENTS

CHAPTER 1 - ASCENT TRAJECTORIES 

Section

Page

1

1.1 INTRODUCTION 1

1.2 GENERAL CHARACTERISTICS OF ASCENT

TRAJECTORIES 1

1.3 ESTABLISHING A 24-HOUR EQUATORIAL ORBIT • • • • 3

1.3.1 Equatorial. Launch Site 5

1.3.Z Cape Canaveral Launch 7

1.4 ESTABLISHING A 24-HOUR POLAR,ORBIT 19

1.5. TRAJECTORY PHASES 23

CHAPTER 2 - GUIDANCE SYSTEMS 26

2.1 GUIDANCE FUNCTIONS AND ACCURACY . • • • • • • 26

2.1.1 Booster -Powered Flight 30

2.1.2 Low-Altitude Coasting 31

2.1.3 Perigee Burning 32

2.1.4 Ascent Coasting 32

Z.1.5 Apogee Burning 32

2.1.6 Orbital Correction Period 34

2.1.7 Equipment Needed 38

2.2 SYSTEM OPERATION AND COMPONENTS 39

2.2.1 Climb to Low-Altitude Orbit 39

2.2.2 Low-Altitude Coasting . 48

2.2.3 Perigee Burning 51

2.2.4 Ascent Coasting 58

2.2.5 Apogee Burning 63

2.2.6 Orbital Position Correction 65

2.2.7 Weight and Power Summaries 67

CHAPTER 3 - ATTITUDE CONTROL 71

3.1 INTRODUCTION 71

3.2 ATTITUDE CONTROL PRIOR TO ATTAINING THE

24-HOUR ORBIT 72

This document contains information offetting Om notional deem* of Me United Slates within am moaning tho Espi laws, Title

U.S.C., Section 793 end 794. the tronsselsslon or mvoinlinn of ..h/.h in no. ...ono.
TR-59-0000-00604

Page iii

CONTENTS (Continued)

Section Page

3.2 (Continued)

3.2.1 Requirements 72

3.2.2 Mechanization 79

3.3 ATTITUDE CONTROL IN 24-HOUR ORBIT 88

3.3.1 Requirements 88

.3.3.2 Stabilizing Machinery 90

Appendix

A ERROR ANALYSIS FOR HIGH-ALTITUDE CIRCULAR

SATELLITE ORBIT 103

B SATELLITE PERTURBATION RESULTING FROM

LUNAR AND SOLAR EFFECTS 124

ATTITUDE CONTROL 141

TAB &avowal coitteles laffprmeNen affecting am naflonal% Mono of Ow United Stases widths IM ..aiming of Ow Easionsio Laws, TM*
-COWDEN-TR--

Mat

TR-59-0000-00604.

Page III-1

CHAPTER 1

ASCENT TRAJECTORIES

1. 1 INTRODUCTION

In Volume II of this report, a number of applications of advanced

satellites were considered. Out of those considerations came recommendations for two types of satellite orbits--the 24-hour circular equatorial orbit

and the 24-hour circular polar orbit

*

. The purpose of this chapter is to

describe the means of ascent into these recommended orbits. Before

describing the optimum ascent trajectories, we first describe some of the

general features of ascent trajectories in ordet•to give motivation for the

final choice.

1;2 GENERAL CHARACTERISTICS OF ASCENT TRAJECTORIES

It is well known that the moat efficient way, from the standpoint of

propellant conservation, of establiitiinga large radius circular orbit is to

burn propellants near the surface of the earth so as to establish a'horizontal

(or nearly horizontal) velocity vector at burnout just sufficient in magnitude to

cause the rocket to coast in an ellipse to the desired altitude of the circular

orbit, and then, upon reaching the desired altitude, to add an additional vela

city impulse to bring the rocket's speed up to that required for a circular orbit.

The ellipse followed during ascent is called a Hohmann transfer ellipse. Its

perigee is at the burnout point and its apogee at the altitude of the circular

orbit. The rocket, as. seen by an observer in space, coasts halfway around

the earth during its ascent. (Earth's rotation.rriust be taken into account in

determining its position relative to the surface of the earth. )

Figure 1 .. 1 shows the velocities associated with, ascent to a circular

orbit as a function of the altitude of that orbit. In computing these curves it

was assumed that the injection into the Hohmann transfer orbit, i. e., the

location of the low-altitude burnout point, takes place at an altitude of

*We use "24-hour orbits" for brevity. Actually the orbits recommended

would have a period of one siderial day, or 23 hours, 56 minutes,

4.09 seconds, the period of rotation of the earth.

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TR-59-0000-00604

Page 111-2

140 nautical miles. Curve A shows the velocity required at burnout and Curve

B shows the additional velocity impulse which must be added at the apogee of

the Hohmann transfer ellipse to establish the circular orbit. Curve C is the

sum of the velocities in Curve. A and Curve B. It represents the total velocity

impulse which must be provided to the rocket, and consequently gives a good

indication of the payload ratios which can be established in any orbit. These

payload ratios are discussed in Chapter 4 of this volume.

It will be noted that Curve B reaches a maximum at an altitude of about

20,000 nautical miles, but approaches iero at both ends of the curve. This

causes Curve C to have a maximum at about 50,000 nautical miles, and consequently it is more difficult, from the standpoint of propulsion, to establish a

satellite at this altitude than at any other. In Volume II we argued that payload

was relatively insensitive to altitude for altitudes above 10,000 nautical miles.

This observation follows from the relative flatness of Curve C above this

altitude.

The velocity indicated in Curve A is the required burnout velocity with

respect to an inertial frame of reference. A portion of this velocity may be

attained by firing in an easterly direction and taking advantage of the earth's

rotation. For example, an easterly firing from an equatorial launching site

provides about 1526 feet per second toward the velocity requirement shown in

Curve A. An easterly launching from Cape Canaveral, Florida, will provide

about 1341 feet per second.

There is a useful variation from the above method of ascent which costs

nothing in the way of additional velocity impulse. In this variation the rocket

is flown from the launch site and, as before, establishes a horizontal burnout

velocity vector at an altitude of 140 nautical miles. But the speed at burnout

is only sufficient to establish, a circular orbit at 140 nautical miles. After

coasting for some distance in the low-altitude circular orbit, a second burning

period accelerates the rocket up to the speed indicated on Curve . A. After

engine shutdown, the rocket coasts as before in .a Hohmann transfer ellipse,

arriving at the desired altitude at the apogee of the ellipse. There the additional velocity shown in Curve B is applied and the high-altitude circular

orbit is establi.shed. In this method of ascent, the speed shown in Curve A is

This discontent cattalos tolermotion affecting IM national, siolonso of the Unita, States within the moaning of tho Espionage laws, Tills

111, U.S.C., Section 793 and 794, the transmission or revelation el which in any ottootor to an worwthorisod parson is prohibilad by law.
TR-59-0000-00604

Page 111-3

CONFIDENTI7Ai,

gained in two burning periods rather than one, but the total velocity impulse

required has• undergone no change. The low-altitude coast has the effect of

changing the point of entry into the high-altitude circular orbit. Such coast

periods play an important part in the trajectories recommended for establishing the 24-hour orbits.

In the above example we have used 140 nautical miles as the burnout

altitude for entry into the Hohmann transfer ellipse. Had a higher altitude

been chosen, the velocity impulse both on entering and leaving the Hohmann

transfer, ellipse would have been less. But this is more than offset by the

additional velocity impulse required in attaining the initially higher altitude.

The net velocity impulse penalty is shown in Figure 1-2 as a function of the

.altitude of injection into the Hohmann transfer ellipse required for establishing a 24-hour orbit. As can be seen from this figure, the higher the altitude

the greater the penalty. Altitudes lower than 140 nautical miles have not been

recommended because they might lead to some difficulties with atmospheric

drag during the initial burning period and during the low-altitude coast period.

If, for the weight-to-drag ratios of the final vehicle design, the drag is found

to be negligible over the coast periods required for the ascent methods described in this chapter, then it will be possible to assume a somewhat lower

point of injection.

1.3 ESTABLISHING A 24-HOUR EQUATORIAL. ORBIT

The 24-hour circular equatorial orbit with which we are concerned

involves an eastward motion of the satellite in order that the satellite can

remain stationary with respect to the Surface of the earth. An efficient launching requires that the launching take place in a generally easterly direction.

This suggests that Cape Canaveral, Florida, be used as the launch site. Before

discussing the specific trajectories recommended for a Cape Canaveral launch,

it is valuable to consider a more ideal launch site, namely, a hypothetical site

located at the equator. By comparing the total velocity impulse required when

using Cape Canaveral with that required when using the equatorial launch site,

we shall determine the velocity penalty associated with Cape Canaveral.

This sInsuntent contains intsarnation electing the national dolonsa ot the Una within the mooning of the Espiononsi Lows, Till*
TR-59-0000-00604

Page 111-4

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Function of Altitude of Injection into the Hohmann Transfer Ellipse.

Thls demanont conain AlrinrAl Informatios effecting Me nalional slohanso of Om nited States withinYsel 4hisnlis , Tine IS, 11.S.C.. Sectiont

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TR-59-0000-00604

Page III-5

-CONFIDENTIAL1.3.1 Equatorial Launch Site

Figure 1-3 is a schematic drawing of the ascent orbit which would be

used if an equatorial launch site were available. The flight consists of three

powered portions, one in the vicinity of the launch site which ends when the

missile has attained a circular orbit satellite velocity of 25,424.0 feet per

second at an altitude of 140 nautical miles. This is followed by a coast period,

the purpose of which is to ensure that the second burning period of the 24-hour

satellite will be initiated at the proper time so that when in orbit the satellite

shall "hover" over the particular longitude selected for the military application to be performed. The low-altitude coast is followed by the second or

perigee burning period which adds an additional 7996.4 feet per second so that

a horizontal burnout velocity of 33,420.4 feet per second is achieved. The

missile now coasts in .a Hohmann transfer ellipse, proceeding halfway around

the earth during its ascent. A period of 5 hours, 16 minutes, 6.7 seconds is

required for the second coast period. The speed .at apogee has been reduced

to only 5261.3 feet per second. As apogee is reached (or, more exactly, a

minute or two before apogee is reached), the third and final burning period

begins. During this period an additional velocity of 4826.2 feet per second is

added, resulting in .a total velocity of 10,087.5 feet per second, which is just

adequate to ensure that a circular orbit is followed.

There will, of course, be errors in guidance and control during ascent.

and consequently the satellite will not follow precisely the prescribed 24-hour

orbit. The deviations from this orbit can be considerably reduced by observing

and correcting the motion of the satellite over a period of a day or more, an

interval which we shall call the vernier correction period.

During the period from missile lift-off to the establishment of the final

24-hour orbit (a period of 5.4 to 6.8 hours, depending on the length of the lowaltitude coast), the earth has been rotating and the launch site longitude is

some 81 to 102 degrees east of its initial position. This must be taken into

.account in trying to establish the del:fired longitude for the stationary satellite.

There is a direct and simple relationship between the weight of fuel

consumed, the specific impulse of the engine, and the velocity gained during

This Petentent cantata, informolien affecting the notional defense s within the Meaning of the Esoionoto Imws, Titlo 18, U.S.C., Section 703 and 794, the transmission er revelation et which in ony m r to on onautheris•d person is probilsited by law.
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the second and third burning periods. During these periods the drag forces

acting on the missile are completely negligible, and, since the missile is

burning in .a horizontal path, gravity has no effect in reducing the velocity

gained. The velocity gained is given by the simple expression

Mo AV = g I

sp m

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1

where g is the standard acceleration of gravity, Isp is the vacuum specific

impulse, Mo is the mass at the beginning of the burning period, and M I is

the mass at the end of the burning period.

The speed gained during the first burning period is far more complicated to compute. During this time there are drag effects, effects of atmospheric pressure in reducing specific impulse, "gravity losses" due to the

effect of gravity acting back along the path, and, finally, Coriolis "forces"

associated with the.rotation of the earth. In Chapter 2, Volume IV, use is

made of machine computations to obtain the performance of various boosters

during the first burning period. What is important here is a consideration of

the effect of geographical location of the launch site on the velocity gained.

The effects of gravity, atmospheric pressure, and drag are relatively independent of the geographic location of the launch site. We are more concerned

here with the effects of earth rotation and the influence of the location of the

launch site on the velocity gained from the earth's rotation. To a first

approximation, the velocity gained due to the rotation of the earth is merely

the surface speed of the earth at the latitude of the launch site. At the equator this is 1526 feet per second.

1. 3. 2 Cape Canaveral Launch 

The problem of establishing an _equatorial satellite from Cape

Canaveral, which is at a latitude of 28.475°N, is considerably more complex

than when an equatorial launch site is used. If the launch were eastward

from Cape Canaveral directly into a Hohmann transfer ellipse, the plane of

the orbit would be inclined to the equatorial plane by 28.475 degrees and the

apogee of the ascent ellipse would be reached at a latitude of 28, 475°S. At

This document conjoins information affecting the notleaal 4efente ei the United Ste wbitin the meaning el the Elmtempo taws, Title

IS, U.S.C.. Section 793 and 794, the tronsmitsion o, revelation of which In any reenaer to on unevihetized rotten it prohibited b y law.

TR-59-0000-00604

Page 111-7
North

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10,087.5 ft/sec Final 24-Hour Orbit

TR-59-0000-00604

Page 111-8

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apogee, 4826.2 feet per second would have to be added to .obtain a circular.

24-hour orbit (see Section 1.3.1), but the orbit would not be an equatorial one.

As the satellite crosses the equatorial plane„ an additional velocity would have

to be added to convert the orbit into an equatorial one, as illustrated in the

following vector velocity diagram, as viewed from above:

The additional velocity increment of 4966.1 feet per second is very costly in

terms of payload, and consequently it is important to think of an alternative

scheme.

Fortunately, by using a low-altitude coast of the proper length, it is

possible to reduce considerably the penalty associated with not using an equatorial launch site. Launching eastward from Cape Canaveral, the missile is

brought to an altitude of 140 nautical miles and a speed of 25,424.0 feet per

second, the speed required to establish a circular orbit at the burnout altitude.

The missile is then allowed to coast until it crosses the equator, where it

This deamerset contains IntermatIon affecting lite national defense of the United Slates within the massing of the Espionage Lowy, Title

IS. U.S.C., Settle. 793 and 791, the transmission or revelation of which in any manner to on uneetherised person is prohibited by low.
TR-59-0000-00604

Page 111-9

10,087.5 ft sec Final 24- Hour Orbit

DV = 6012.9 ft/sec

enters a second burning period.. During the second-burning period a velocity

increment of 7996.4 feet per second injects the missile into the Hohmann

transfer ellipse. Since the perigee of the Hohmann ellipse is at the equator,

its apogee will be at the equator also. At apogee a velocity increment is

added of such magnitude and in such a direction that the speed is brought up

to 10,087.5 feet per second (the 24-holar orbit speed), and the final direction

of motion is corrected to be in the equatorial plane, as illustrated in the

following diagram:

North

Equator

28. VP

11.v?

It is seen that by combining the operations of increasing the speed and

adjusting its direction into a single operation, the total velocity increment is

only 6012.9 feet per second instead of the 9792.3 feet per second described

earlier (4826.2 feet per second to obtain orbital speed and 4966.1 feet per

aecond to correct the plane of the orbit). The 6012.9 feet per second is only

1186.7 feet per second more than the 4826.2 feet per second which was

required for an equatorial launch site. In addition, there is a penalty paid

during the first building period because the speed of rotation of the earth at

• .. ..... 'oh. ...Noma deltas.. el the United litotes within Um meenIne el Use Isplenereelinr, Thu.
TR-59-0000-00604

Page III-10

-GM-FINN-Rh

Cape Canaveral is only 1341.4 feet per second instead of the 1526.0 feet

per second of an equatorial launch site. This penalty of 184.6 feet per second

added to the 1186.7 feet per second makes a total penalty of 1371.3 feet per

second (or about a 15 per cent decrease in gross payload) associated with the

use of Cape Canaveral.

If the rocket is fired due east from Cape Canaveral, and if its motion

is constrained to the plane which passes through the center of the earth and.

through Cape Canaveral in an easterly direction, then it will cross the equator

at a longitude of 90 degrees east of Cape Canaveral for a nonrotating earth.

The longitude of Cape Canaveral is 80. 55°W; consequently, the missile would

cross the equator at 9. 45°E if the earth were not rotating. Actually, it will

pass the equator west of this point by the angle through which the earth rotates

during the time of flight. The burning period to reach circular satellite velocity at an altitude of 140 nautical miles is approximately 5. 4 minutes, and

during this time the missile will have moved downrange approximately 450 nautical miles,*

which is 7.5 degrees of arc from Cape Canaveral. The time .

required to cover the remaining 82.5 degrees of arc in the low-altitude orbit

is 82.5/360 of the period of the satellite. The satellite period at an altitude

of 140 nautical miles is 88.56 minutes, and consequently the coast time is

20,3 minutes. The total time of flight, therefore, is 25.7 minutes (5. 4 plus

20.3), and during this time the earth will have rotated through 6. 44 degrees.

Consequently, the actual point at which the missile reaches the equator is at

a longitude of 3.01°E.

During the ascent in the Hohmann transfer ellipse, the satellite covers

an arc of 180 degrees. Consequently, if the earth were not rotating during

this period, the satellite would reach apogee at 176.99°W (3.01 o

E of the

International Date Line). However, since the time of ascent is 5 hours,

16 minutes, 6. 7 seconds, the earth will have rotated through 79.24 degrees

* The burning time and downrange distance depend on the details of the

missile design and the powered flight dynamics. The numbers quoted

are approximate, but the results of the arguments given here are relatively insensitive to these quantities.

This desement contains Internagloo effecting $h noti•ael defense of the Untied Stoles within the eteening of the Scalene,. lows, Title

111, U.S.C.„ Settle* 793 end 794, the honsteission it Nveleften of which In any mower to on onovittorlsed porton l• swanshad by low.
Inn TR-59-0000-00604

Page III-11

----GONROOW

during the ascent period, and the apogee is actually attained at a longitude

of 103. 77°E, which will be just above Singapore. It is at this point that

the 24-hour orbit is established, and consequently the satellite would remain

over Singapore.

The question now arises as to what scheme can be used for launching

a 24-hour equatorial satellite which is to achieve a final longitude not equal to

that of Singapore, but still launching from Cape Canaveral. This may be

achieved by using a longer coast period in the circular orbit at 140 nautical

miles and by entering the Hohmann transfer ellipse at some subsequent crossing of the equator. During the low-altitude coast period the rotation of the

earth will bring the desired longitude of injection into the Hohmann transfer

ellipse close to the plane of the 140-nautical-mile orbit. The injection into

the Hohmann transfer ellipse may be introduced at any of the equatorial crossings, whether they be in a southerly or northerly direction, and since these

crossings occur at time intervals of 44.28 minutes apart, the earth will have

rotated through 11.1 degrees between subsequent crossings. We shall

let N be the number of complete half.. revolutions included in the low-altitude

coast.. Therefore, the crossing at 3.01°E described above will be referred

to as the "zeroth crossing." The next crossing, which will be in the northerly

direction, will be referred to as the "first crossing, • " the next southerly

crossing as the "second crossing, "etc. Table 1-1 shows the longitudes of

the successive crossings, the time from. lift -off to the crossing in question,

the coreesponding longitudes of the 24-hour equatorial satellite, and the total

time to establish the satellite (obtained by adding 5 hours, 16.11 minutes for

the ascent time in the Hohmann transfer ellipse). It will be noted that a

satellite can be put up over any point on the earth (to within 11.1 degrees) by

using a low-altitude coast period no longer than 12 hours, 14.2 minutes and

a total time of no longer than 17 hours, 30.3 minutes.

For purposes of this computation we have assumed that perigee and apogee

burning are performed impulsively (with zero burning time). The figures

given here will need refinement when the actual burning programs have

been specified.

'critintepak_

This document contains information effecting the notional defense of Use United States within the meaning of dm Espionage Lows. Title

U.S.C., Section 793 end 794. the transmission or revelation of which in any manner to an oneutimaised person is prohibited by low.
1111.NrIDENTIM,

TA-59-0000-00604

Page 111-12

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Page III-13

The above scheme provides only a discrete set of final longitudes for

the 24-hour satellite unless allowance is made for a coast period longer than

12 hours, 14.2 minutes in the low-altitude orbit. These longitudes could be

shifted somewhat by using an altitude other than 140 nautical miles for the

low-altitude coasting orbit. However, there is another simple scheme for

obtaining any intermediate location with a very small additional velocity

penalty. Instead of firing due east from Cape Canaveral, one could fire

slightly north of east or slightly south of east. For example, if one fires *

6..0 degrees north of east, the plane of the 140-nautical-mile orbit will cross

the equator about 11.6 degrees east of the crossing for a due east launch.

Moreover, the length of the great circle arc from the launching point to the

zeroth crossing of the equator will be about 10.2 degrees greater than 90 degrees. During the extra 2.5 minutes required for the satellite to transverse

this 10.2 degrees, the earth will have rotated through 0.6 degree. When this

is subtracted from the 11.6 degrees, one sees that the zeroth.crossing point

will be 11.0 degrees east of 3.01°E, the crossing point for a due east launch.

If one launches 6. 0 degrees to the south of due east, then the zeroth crossing

will take place 11 degrees west of 3. 01°E. Since the period of the low-altitude

satellite is the same as before, all subsequent crossings are shifted by pre- .

cisely the same increment of longitude.

Figure 1-4 is a plot of the longitude shift given as a function of the

angle between the launching direction and due east. This longitude shift may

be used to adjust the longitudes shown in Table 1-1, both the longitudes of

the equatorial crossing (point of injection into the Hohmann ellipse) and of the

point of establishment of the equatorial satellite.

Figure 1-5 shows the velocity penalty associated with not launching

due east. One of the penalties, shown in Curve A, is associated with not

As seen by observing the path relative to the earth. When the earth's

rotational speed at Cape Canaveral, 1341 feet per second, is added in

the eastward direction, the total velocity is only 5.7 degrees from east

as viewed from a nonrotating frame of reference.

This document contains infermatIon effecting the notional. defense et Om United States within the ramming of the Wiener, Lows, Title

IS, U.S.C.. Section 713 end 704, the ifORSO1181000 or revelation of which In any manner to an unauthorised person is prohibited by law.

--eONFiDENTRL___
TR-59-0000-00604

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U.S.C., Section 793 and 794, the transmission Or revs/knee of which In say miaow to on onaelherised Dotson is swohlbito4 by law.
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TR-59-0000-00604

Page III-15

50

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Deviation of Launch Angle from Due East (degrees)

Velocity penalty due to not fully utilizing earth's rotation.

Velocity penalty incurred during apogee burning because

of increased inclination angle.

C. = A + B = Total velocity penalty.

Figure 1-5. Velocity Penalty Associated With Not Launching Due East.

"CONFIDENTIAr__

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Sealers 793 sod 794. the ttoosreissiee .r einwhotion of which is oily wonnoc to on veeetherised person Is prohibited by low.

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TR-59-0000-00604

Page III- 16

CONFIDENTIA

utilizing fully the earth's rotational velocity, since only the component of the

surface velocity in the direction of launch is effective. Curve B illustrates

the additional penalty incurred at the point of entry into the 24-hour orbit.

This velocity penalty is associated with the fact that when a missile is not

launched due east from Cape Canaveral, the inclination angle, measured from

the equator, of the plane of the orbit is more than. 28.475 degrees. Curve C

is the sum of the two penalties. It will be noted that the maximum penalty

which need be tolerated is when a 6. 05-degree northerly (or southerly) launching is used, and this penalty is only 44 feet per second.

There is one refinement whidi should be mentioned before leaving the

discussion of the 24-hour equatorial orbits launched from Cape Canaveral.

We have assumed heretofore that there was no change in the orbital plane

during the second burning period--that is, at the point of injection into the

Hohmann transfer ellipse. Actually, the velocity added there should be

oriented slightly in the direction to decrease the inclination angle of the plane

Of the orbit. This is illustrated in Figure 1-6. • Decreasing the inclination

angle requires a somewhat greater incremental velocity during the perigee

burning period (Curve B), but it has the effect of reducing the incremental

velocity which must be added at the apogee of the Hohmann transfer ellipse

(Curve .A). The net velocity impulse saved is shown by Curve C. In the • • case of due easterly launch from Cape Canaveral, the optimum plan is* to

decrease the• inclination angle by about 2.2 degrees, This requires an additional 77 feet per second during the perigee burning period but decreases

the velocity increment to be added during the apogee burning by 162 feet

per second, with a net saving of 85 feet per second.

Table 1-2 gives pertinent trajectory information for establishing

equatorial satellites at the particular locations recommended in Volume

of this report.

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TR-59-0000-00604

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Reduction of Inclination Angle During Perigee Burning (degrees)

Figure 1-6. Velocity Saved by Reducing Inclination Angle

During Perigee Burning (Based on East Launch

from Cape Canaveral).

MOM

This document contains infermellen affecting Me national defense of the United States within the meaning of tile Espi Laws Title

II, U.S.C., Settle* 793 and 794, the transmission or revelation of which in any manner in on unauthorized person is prohibited by law.
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CONFIDENTIAL

This decision, contains Information effecting the notional defense of the United Votes wNhin the meaning of the Es

pionage lows, 1$, U.S.C., Section 793 end 794, the transmission or reveielIon of which In any manner to or vneetIwsrlswe person Is 

proltibiled by low.
TR-59-0000-00694

Page III-19

.—CONFZENEAL

1.4 ESTABLISHING A 24-HOUIt ftPLAR ORBIT

Cape Canaveral is not well suittui for the launching of a polar satellite

because firing either to the north or south would involve passage over land

during the initial powered flight. On the other hand, Cooke AFB is quite

ideally situated for the launching of a polar satellite by firing to the south.

In Volume II of this report, it was pointed out that three 24-hour polar

satellites are required to give continuous communications coverage of the

north polar region of the earth. It was there recommended that the polar

satellites be placed in circular orbits which would cross the equator at a

longitude 60°W, and that the satellites be so phased that their northward (or

southward) equatorial crossings would be separated by 8 hours. It will be

noted that in launching the first of the three polar satellites, care need only be

taken that the orbit established for this. satellite .cross the equatorial plane at

60°W. On the other hand, for the second and third satellites we have not only

the above requirement, but also the requirement that the equatorial crossing

take place 8 or 16 hours after the crossing of the first satellite. Therefore,

in launching the second and third polar satellites, it is necessary to time the

countdown so that missile lift-off will take: place at a particular time of the day.

The method of ascent which is here recommended bears a considerable

resemblance to the equatorial launch of an equatorial: satellite described. in

Section 1.3.1. First, we will describe an ascent path which is always applicable to the first polar satellite launched, but which is applicable to the second

and third satellitea only when they are launched on schedule. In the event that

the second or third satellite is launched early or late, a variation of this

ascent method must be used. This variation will be described later.

The " on time" scheme consists of launching due south from Cooke AFB

which is located at longitude 120. 58°W and latitude 34. 75°N. Again we launch

into a circular orbit at an altitude of 140 nautical miles, and, as in the equatorial case, we assume a burning period of 5.4 minutes and a point of injection

into the circular orbit 450 nautical miles downrange from the launching site,

This dementia C0010103 information effecting Ha notional. defense of the United States within the ,meoni,te of the Es pionage Laws, Till,

1111, U.S.C., Section 793 and 794. lint transmission or revelation of which in any manner to on unauthorized person Is prohibited by low.
TR-59-0000-00604

Page 111-20

that is, at a latitude of 27.25°N. The vehicle is allowed to continue in the lowaltitude orbit until it completes a little more than one revolution and arrives • at

a latitude of 14.86°N. Thus, it traverses 372.39 degrees in the 140-nauticalmile orbit. Since the period of the circular orbit is 88.56 minutes, the time

required in this orbit is 91.6 minutes. When this is added to the 5.4 minutes

required for the burning program, we see that the total. time from lift-off is

97.0 minutes, and during this time the earth rotates hrough 24.32 degrees.

Consequently, the point of injection into the Hohmann transfer ellipse is at a

longitude of 144.90°W and a latitude of 14.86°N. If the earth did not rotate

during the ascent in the Hohmann trajectory, the apogee point would be at a

latitude of 14. 86°S and a longitude of 35. 10°E. However, since the earth does

rotate through 79.24 degrees during the . 5 houre,16.11 minutes of ascent *, the

apogee actually will occur at longitude 45. 14°W, latitude 14. 86-S. It is at

this point

*

 that *the.additional velocity is added as required to inject the satellite into a circular orbit. As the satellite, now moving north, 'traverses the

remaining 14.86 degrees to reach the equatorial plane, the.earth also rotates

through 14.86 degrees, and when this is added to the longitude of 45.14°W,

we see that the first equatorial crossing of the satellite takes place at 60°W,

as required.

It will be noted that the entire flight path takes place in one plane in

space. Consequently, we see the similarity to the launching of an equatorial

satellite from an equatorial launch _site, but in this case we gain no advantage

from earth's rotation in establishing the low-altitude orbit. The velocity of an

orbit at 140 nautical miles is 25,424.0 feet per second. As in the equatorial

launch of the equatorial satellite, the velocity which must be added to, the lowaltitude orbital velocity in order to establish the Hohmann transfer ellipse is

Fbr purposes of this computation we have assumed that perigee and apogee

burning are performed impulsively (with zero burning time), The figures

given here will need refinement when the actual burning programs have-been

specified.

This decennia/ contains information allotting the settee', defense of the United Stales within the meaning of the botelette Lows, TM* .•

IS, U.S.C., Section 793 and 791, the frontlet/sloe or revelation of which In any manner to an eneutherlited. person I, ereldbited
TR-59-0000-00604

Page ITI-21

7996.4 feet per second, and the velocity which must be added at the apogee of

the Hohmann transfer ellipse in order to establish the 24-hour circular orbit

is 4826.2 feet per second.

Clearly this scheme can be used to establish the first polar satellite.

If this scheme is also used to establish the second and third polar satellites,

then they too will have their equatorial crossings at 60°W. But if lift-off does

not occur at precisely the right time of day, the crossings will not be spaced

8 hours apart. Necessarily, two of the three satellites will then be spaced

More than .8 hours apart in their successive northward crossings of the equator.

When these two satellites are in the Northern Hemisphere (and the third satellite in the Southern Hemisphere}, there will be a "time when their latitudes are

equal. Nominally, this latitude should be 30°N, but since they are more than

8 hours apart, they will find themselves at the same latitude when they are

somewhat south of 30°N, and consequently the polar communications coverage,

especially that in the Eastern Hemisphere, is decreased. For example, if

the satellites are 8 hours and 40 minutes apart, instead of 8 hours apart, then

they will find themselves at the same latitude when they are at latitude 25°N.

This would have the effect of decreasing communications coverage in Siberia

by approximately 5 degrees of latitude.

Rather than accept this decrease in communications coverage, it is

possible to utilize a late (or early) firing and still establish a pattern of coverage very close to that attained when a precise schedule is maintained. This

is done by establishing an orbit (for the second or third satellite) which crosses

the equator at the right time, though not at the right longitude. For example,

assume that the second satellite is launched an hour late. Instead of starting

the perigee burning when the rocket has coasted to the nominal latitude of

14. 86°N, one should allow it to coast another 16.03 degrees until it reaches

latitude 1.1708. During the extra 3.94 minutes added to the coast period,

the earth will have rotated through 0.99 degree. Consequently, instead of

reaching apogee at a longitude of 45.14°W and a latitude of 14. 86°S, as in the

CONF;DE1'.171AL

1, This decunsoat contains interniallein affecting the national defense of OM ter= I within the Jasmine of the Espionsloo Lows, TIM

IS, U.S.C., Section 793 and 794, the tronsmissim er revelation of which In any manner to on enatitherind person Is prohibited by taw.
TR-59-0000-00604

Page 111-22

GOWEN-TIM.

nominal case for "on-time" firing, the missile would reach apogee at longitude

46. 13°W and latitude 1.17°N. It is at this point that the 24-hour orbit commences. On the nominal firings it takes 63.94 minutes to traverse the 16.03 degrees between 14. 86°S and 1. 17°N. But it will be recalled that 3.94 minutes

were used up by lengthening the coast period. Consequently, exactly 1 hour

has been gained, as required. It will be noted, however, that the equatorial

crossings will now take place at 44. 96°W (1.17 degrees east of apogee longitude). Therefore the longitude of the figure eight traced by the late satellite

is displaced from its correct position by approximately 15 degrees for each

hour of lateness (or earliness). A shift of 15 degrees in longitude has only a

very minor effect on the area covered by the polar communications system.

The guidance system discussed in Chapter 2 of this volume utilizes a measurement of coast time in the low-altitude orbit. In order to effect the correction

which we have just illustrated, it is only necessary to reset the desired coast

time as a function of the actual time of launch.

There is a means of ascent for polar satellites quite analogous to the

scheme suggested in Section 1.3.2 for equatorial satellites launched from nonequatorial bases. This scheme, in principle, is capable of launching a polar

satellite into precisely the correct orbit (correct both in longitude and timing)

regardless of the time of launch. The missile is launched in a generally southerly direction into a low-altitude circular orbit. Twice each period of this

low-altitude orbit it threads through the plane (in space) which contains the

desired 24-hour polar orbit. At one of the crossings the missile is injected

into a Hohmann transfer ellipse which, consequently, has its apogee at the

next point of crossing. It is at this point that the orbital plane is changed,

simultaneously with adding the additional speed' to establish the 24-hour circular orbit. By adjusting the coast time, that is, the number of half-revolutions

in the low-altitude orbit, one can make a gross adjustment of the time of

equatorial crossing of the final 24-hour orbit. As in the equatorial case,

this adjustment allows placement of the satellite in its correct orbit with an

--CONFIDENTiAl

This &moment contain Information allotting the national defense of the United Stoles within the 'meaning of the Espionage LOWS. Till. •ot to a •...t.. •Ifai oto. rA•ashollass. AS whip% Is nom snaRAW 50 GO WIOUlheriZed DIDION is prohibited by low.
TR-59-0000-00604

Page 111-23

angular error of less than 11 degrees. A fine correction is introduced by

choosing the direction of launch so as to slightly change the dihedral angle

between the plane of the original low-altitude orbit and the plane of the desired

polar 24-hour orbit.

Although the method just introduced would seem ideal from the stand-

. point of always establishing the optimum . polar coverage, it suffers from

several practical disadvantages. First, unless the firing time is within an

hour or so of nominal firing : time, the dihedral angle is a large one, and it

is very costly from the standpoint of propulsion to turn through the dihedral

angle at apogee. Second, the location of the apogee point is not necessarily

within the line ...of-sight of conveniently located radio tracking systems, and

consequently the guidance scheme suggested in Chapter 2 might be difficult

to apply. Third, the velocity to be gained by the first stage depends upon the

direction of fire. Because of the rotation of the earth and, unless the fuel

level in .the first stage were changed as a function of the time of firing, it

would be, necessary to carry enough fuel to achieve the low-altitude orbit

under the most unfavorable firing direction. This would penalize the performance unnecessarily. Fourth, a similar argument holds for the amount of fuel

which would have to be carried in the stage which performs apogee burning,

'inasmuch as the dihedral angle, and consequently the velocity increment,

would vary as a function of time of firing. Because of these reasons we do

not recommend this method of ascent.

1.5 TRAJECTORY PHASES , (AND STAGING)

It is important to draw the distinction between burning periods and

vehicle staging. Consider the "first burning period", that which takes the

vehicle from launch up to injection into the low-altitude orbit. In practice

it will be necessary to use several rocket stages during this period, but in

spite of the several stages involved, this will be regarded as one '"burning

period" or one "trajectory phase" because the coast periods between burnout of one stage and ignition of the next can be made as small as practical

considerations (e.g., adequate time for stage separation) will allow. In

Chapter 2, Volume IV, it is shown that one stage added to the Thor, Atlas,

or Titan is capable of achieving injection into the 140-nautical-mile orbit.

This *monies, cenleins intennetlen effecting the nolioni Millie the meaning of the Espionage taws, Thie

IS. U.S.C.. Section 793 tied 794. the trammhslan we irowidattwe 4,1 ot
CONFIDENTIALth T R- 59 - 000 0- 00 604

Page 111-24

Next, although the perigee burning period on entering the Hohmann

orbital transfer ellipse and the apogee burning period on leaving the Hohmann

ellipse may utilize the same rocket stage ignited twice, these are two distinct

burning periods separated by a 5.3-hour coasting period.

Table 1-3 gives a listing of the trajectory phases, applicable to both

equatorial and polar orbits, together with the staging which is recommended

in Chapter 2, Volume IV. The recommended plan is to climb to the lowaltitude orbit using a modification of an existing booster (single-stage Thor,

1-1/2-stage Atlas, or 2-stage Titan) plus a first added stage. Shortly after

this orbit is established, the first added stage is jettisoned. The final stage

Table 1-3

Staging Guidance Phases Time

Booster (1 or 2 stages) Injection into Low •n•5 min

First Added Stage Altitude Orbit

'Low Altitude Coasting

Perigee Burning

Max 10 hr

e-,4 min

Ascent Coasting 5.3 hr Final Stage Apogee Burning min

Orbit. Correction 24 or 48 hr

Attitude Stabilization 6 mo to 1 yr

now coasts in orbit for a period ranging from about 1-1/2 hours for polar

orbits up to as much as about 10 hours for equatorial satellites. The final

stage engine is then ignited for perigee burning. When this burning is completed, the final stage coasts to apogee where an engine (either the same one

that was used at perigee or a different engine, .see Chapter 2, Volume IV) is

ignited for apogee burning. When apogee burning is completed, the vehicle

is reoriented so as to direct its nose downwards. There is no jettisoning of

tanks.or other structure. During the next 24 to 48 hours a series of small

-GMBH-Kik

This eln•smaot caMeine informatlaa al/actin', the notional defense of the United Stales within the mitenine of the Etpienetee Lows, Title
TR-59-0000-00604

Page 111-25

orbital corrections (see next chapter) is effected using gas jets as the means

of obtaining the small impulses needed. Thereafter, for the life of the satellite, the only working components are . the communications (or other payload)

equipment, the auxiliary power supply, the cooling equipment, and the attitude

stabilization equipment. The attitude stabilization problems are discussed in

Chapter 3 of this volume, and electric power and cooling are discussed in

Chapter 1 of Volume' IV.

. n , ....lion .1 *show of Illy Unload Moles within the moaning el the Eugene*, Laws, fills

•

• • c• of which in any 'Nonni', by an unauthorised person It Oftbibited by low.
TR-59-0000-00604

Page 111-26

CONITZUAL

CHAPTER 2

GUIDANCE SYSTEMS

2.1 GUIDANCE FUNCTIONS AND ACCURACY

The guidance system fox satellites of the type described in this report is

more complex than the systems designed for the ICBM or for the IGY

The system to be described in the following pages must be capable of putting a

satellite in an orbit of very slight eccentricity and of extremely accurate known

period. In addition to being in the proper orbit with the proper period, the

satellite must be placed in a particular position in that orbit. When one couples

this required precision with the difficulties of launching from nonideal locations

such as Cape Canaveral, it becomes evident that system complexity is apt to be

greater than in other more well-known guidance systems.

The central problem for the guidance system is to establish and keep the

satellite in the correct orbit. If the satellite, once established in orbit, has an

error in its period, then it will drift off "station" to the east or west and therefore the earth's horizon, as seen by the satellite, will shift (see Figure 2-1).

Since both the high-density communication system and the infrared early-warning

system discussed in Volume U view areas near the horizon, this shift would mean

loss of coverage. It is probably desirable to restrict the drift, say, to about

degrees *

 during the operational life of the satellite. This would restrict the

shift of the boundary of the "covered area" to 120 nautical miles at the equator,

and to only 85 nautical miles at Longitude 45°N. A drift of 1.2 degrees

If, on establishing the orbit, the eccentricity is limited such that the apogee

radius is no more than 200 nautical miles greater than the perigee radius,

then the diurnal oscillation in the east-west direction caused by this eccentricity will be only £112 degree. It is relatively easy to establish an orbit with

this eccentricity. Likewise, the error in establishing the inclination of the

orbital plane is likely to be quite small compared with 2 degrees. The narrow

beamwidth (0.1 degree) ground antennas can track this 2-degree drift by moving

the antenna feed. It is not necessary to move the dish unless considerably

larger drifts are to be accommodated. It is also assumed that as the satellite

drifts, its altitude relative to the earth is maintained. The problems of

maintaining altitude are discussed in Chapter 3.

1 document coot a1 n , information attecting the notional defense of the United States within the meaning of the Espionage Laws, title

9.5.0 So• t • ms 00 • -nn Pan, the transmission or revelation of which in any manner to on unauthorized person is prebibitnel b y law.
TR-59-0000-00604

Page III- 27

. Figure 2-1. Lou of Ground Coverage. if Satellite Changes

Orbital•POsition by 800 NautiCal Miles. • •

.

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TR-59-0000-00604

Page 111-28

corresponds to an east-west drift in orbit of about ±800 nautical miles. * Taking

the operational life between 6 months and 1 year, say 9. months to be definite,

this permits an average drift rate of 0.2 foot per second.

It is beyond the present state-of-the-art to achieve this accuracy when

the satellite is first placed in the high-altitude circular orbit. To achieve such.

accuracy it will be necessary to track the satellite in orbit and then to apply

appropriate small impulses over the next 24 (or at most 48 hours) to correct

the orbit. By such correction schemes the above drift-rate requirement can

be met, and it is even quite likely that the drift rate can be brought down to

0.03 foot per second thus permitting 5-year operational periods if and when

communications, power supply, and attitude subsystem shall have reached a

status of reliable operations over such periods.

It is, of course, possible to relax the orbital drift requirements if

orbital correcting impulses are applied every few weeks or months, but this

would demand repeated operation of the orbital correction guidance and control

subsystem which would introduce additional possible sources of system failure.

However, the repeated corrections will probably be unnecessary since it appears

quite feasible to correct the orbit in 1 or 2'days to the requisite accuracy for

"holding" station for the operational life of the satellite. Thus, some 24 or

48 hours after the orbit is first established, the guidance system will no longer

be required to operate.

The question now arises as to the accuracy required of that portion of

the guidance system which first places the satellite in orbit, the "ascent

guidance." Since any errors which accumulate prior to establishing the

24-hour orbit will be wiped out during the orbital correction period, it would

Therefore such diurnal oscillators east-west or north-south are in any event .

less critical than the steady east-west drift. They too, can be traced by

moving the feed on the ground antenna. They would require, however, a

continuing correction of satellite attitude to compensate the duirnal satellite

displacement. Consequently, it is probably desirable to reduce these' oscillators by the orbital correction. methods described on the following pages,

thereby simplifying the ground tracking and particularly the attitude

stabilization.

01110.4,1114

This dethmenf contains Information affecting the notional defense of the United Stoles within the meaning of the *soionope laws. ilthr

Is. U.S.C.. Bubo. 793 and 794, the transmission or nmetation of which in env Innen., S. .
TR-59-0000-00604

Page 111-29

appear that quite crude guidance equipment can be used. By using a crude

guidance system, guidance equipment weight can be saved. However, if the initial

errors in the orbit are very large, large correcting impulses are required, and

the weight of gas expended in these impulses can exceed the weight saved by

using the crude guidance equipment. No attempt will be made in this report to

arrive at the optimum point in the trade-off between ascent guidance weight and

ascent guidance accuracy. Rather, we have here assumed the use of ascent

guidance equipment based on small additions to and modifications of equipment

which is being developed in any event for ballistic missile systems. Appendix A

is an analysis of how the errors in ascent guidance affect the accuracy of injection into the 24-hour orbit. When the accuracies of the ascent guidance equipment are substituted into the error transformation equations developed in

Appendix A, one finds that the sum of the velocity impulses required of the

orbital correcting gas jets is only about 100 feet per second.

The guidance functions during the various trajectory phases listed in

Table 1-3 are described in Sections 2.1.1 through 2.1.6. However, before getting

into the phase-by-phase description, there are certain general features of the

guidance problem which require mention here inasmuch as they determine the

over-all character of the system.

As mentioned earlier, it is necessary to track the satellite over a period

of 1 or 2 days after it has reached the 24-hour orbit. This tracking provides an

accurate determination of the orbital errors, and therefore provides the information which is required for orbital correction. Radio tracking means are

probably the most dependable means of performing this tracking inasmuch as

optical tracking at such large distances is likely to be quite difficult and is,

moreover, dependent on good weather conditions. A radio transponder beacon

must, therefore, be carried by the satellite in order to provide a sufficiently

strong signal for radio tracking.

Unfortunately, it is not possible to use the same radio command guidance

during all of the guidance phases. In particular, it could not always be used

during perigee burning because perigee burning may take place over open ocean

area which is not within line-of-sight of friendly land areas. For example, as

Gn; I f;

11-4-11

This decvment contains information effecting the national defense of the United States within the meaning of the Ems' Lows Title
*

If the Titan is used, the first stage cutoff signal is derived from level

sensing devices in the first stage fuel and lox tan

. TR-59-0000-00604

Page III- 30

shown in Table 1-2, Chapter 1, the perigee burning for the U. S. -to-Asia

communication satellite, which is to be established at Longitude 170°W takes

place over the Indian Ocean (at 89°E Longitude). For this reason, a selfcontained all-inertial guidance system has been selected for use during the

perigee burning period. An all-inertial system is also well adapted to the

guidance problems encountered in establishing the low-altitude orbit. Modifications of • the current Arma system or certain other systems under development will give satisfactory performance for the climb to the low-altitude orbit

and for perigee burning.

In principle, the inertial system can also be used to carry out guidance

during apogee burning, but this would lead to quite large errors in orbit due,

primarily, to the cumulative effect of the errors introduced in earlier guidance

phases. However, since apogee is always within line-of-sight of friendly territory (it must be in order to establish communications to the U. S. ), the vehicle

can be tracked prior to reaching apogee, and the tracking information may be

used to command the apogee burning, thereby wiping out many of the errors

which accumulated during the earlier blind operation. It turns out, however,

that the smoothing time associated with angular rate tracking at the extreme

ranges of the 24-hour satellite is so long that the inertial guidance system must

be depended upon to carry out the command. Consequently, both radio and

inertial systems are used in the suggested scheme for apogee burning..

2.1.1 Booster-Powered Flight 

Guidance during the booster-powered flight is accomplished with a

gyro-stabilized inertial platform mounting three orthogonal accelerometers.

This platform works in conjunction with a primary guidance digital computer.

During the first-stage booster burning period, the vehicle follows aprogrammed

pitch attitude and is stabilized in roll and yaw using the platform as a reference.

The primary guidance computer calculates position and velocity throughout

this period. Attitude control and steering instructions are provided for subsequent stages in the climb to the low-altitude orbit by the primary computer,

which also issues engine start and cutoff signals.

*

Thisdoceetent contains information affecting the national defense of the United States within the /peening of the Espionage Laws, litho
TR.-:;9-,0000-00604

Page 111-31

A relatively heavy primary computer is required for the preorbital

period because of the complexity of the computations associated with the trajectory during this period. During perigee and apogee burning periods, however,

the computations are sufficiently simple to be carried out by a rudimentary

digital computer, called here the steering and shutoff computer. Therefore, the

primary guidance computer is located in the first added stage and is jettisoned

before perigee burning in order to reduce the guidance weight in the critical

final stage.

Z.1.2 Low-Altitude Coasting.

The inertial guidance system is capable of measuring the velocity and

position of burnout (latitude, longitude, and altitude) with greater accuracy

than it can control these quantities. The resultant control in altitude and speed

errors at the point of injection into the low-altitude orbit effects the low-altitude

coast period and this leads, together with the downrange error at injection into

the low-altitude orbit, to an error in the downrange position of the satellite at

the nominal time of perigee burning. The resulting error, however, in downrange position at perigee burning can be compensated by adjusting the initiation

time for perigee burning. Velocity and altitude errors at the actual time of

perigee burning can also be compensated by adjusting the vector velocity increment to be added at perigee. Computations of the perigee correction biases are

made in the primary guidance computer, and before the first added stage is

separated from the vehicle, the biases are transferred to and stored in the

smaller digital steering and shutoff computer housed in the final stage. This

simple transfer can be avoided if an additional impulse of approximately 30feet

per second is provided at apogee burning to steer out the velocity errors which

result from not biasing the perigee burning program with the transferred data.

The vehicle or final stage is reoriented to the attitude required for

perigee burning after separation of the first added stage. The vehicle is held

at a constant attitude by reference to the stable platform during the coast in

the 140-nautical-mile circular orbit. A simple analog computer commands

the attitude control jets by converting platform gimbal angles into on-off signals

for the jets.

This &cement .0%101111 information affecting the national defense of the United States within the nwpaging el the ES11101100111 Lows. Title
TR-59-0000-00604

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2.1.3 Perigee Burning 

Perigee burning adds approximately 8000 feet per second to the speed.

This nominal desired impulse is somewhat modified by the perigee impulse

correction stored in the steering and shutoff computer. Steering and shutoff

commands are provided by the computer on the basis of accelerometer readings.

Steering commands during this period are fed to the perigee-apogee autopilot

which also make use of platform gimbal•angle measurements and angular rate

data platform from rate gyros.

2.1.4 Ascent Coasting

The guidance system must stabilize the vehicle attitude throughout

ascent in the Hohmann transfer ellipse and the vehilce must be tracked from a

ground station during the final 2 or 3 hours of ascent. By observing a transponding beacon, it will be possible to measure the vehicle's position during

ascent coasting with an angular tracking radar and the vehicle's velocity with

a radio interferometer. The tracking data are then fed into a ground digital

computer, and apogee burning correction biases are sent to the vehicle as

modulations on the tracking waves. These are stored in the steering and

shutoff computer.

Attitude stabilization during ascent coasting is provided by the platform.

During ascent the vehicle must be rotated about 127 degrees to accomplish apogee

burning with the same motor.

2.1.5 Apogee Burning 

Apogee burning must add about 4800 feet per second for polar satellites.

On the other hand, for equatorial satellites launched from Cape Canaveral, since

the plane of motion must be turned through approximately 28.5 degrees about

6000 feet per second are required. The nominal values are adjusted according

to information derived from ground tracking.

Apogee burning is initiated when the vehicle reaches the correct angle

relative to the earth (the correct latitude for the polar satellite and the correct

longitude for the equatorial satellite). The engine start signal is commanded

from the ground on the basis of the angular position tracking information.

01111111 .dirernen1 conteins infemetion effecting the national Mom. .0 th• tin/1.A O.... .
TR-59-0000-00604

Page /II-33

Because burning is initiated on the basis of the correct angle, prior

measurement and control errors will prevent the burning from occurring precisely at apogee. Thus, just before the burning period there will be radial

component of velocity which can be cancelled out by correctly orienting the

velocity increment added during "apogee" burning. Thus, the desired burnout

velocity is horizontal, and this horizontal burnout velocity should be controlled

as a function of altitude to be that which leads to a satellite period of one sidereal

day. It will not be that of the circular one-sidereal-day orbit unless the altitude

at apogee burning is, by chance, exactly the nominal one. Thus, if the altitude

at apogee burning is in error, the satellite is purposely put into an elliptical

(not circular) orbit with a period of one sidereal day (except, of course, for

errors in the command caused by errors in radio tracking and errors in apogee

burning control caused by platform drift and accelerometer errors). The advantages of having a period of very nearly one sidereal day are that secular drift is

very nearly elimintated (at the cost of introducing daily oscillations) and the

orbital correction scheme, discussed in the next section, is simplified.

As in the perigee burning period, cutoff commands are issued by the

steering and shutoff computer on the basis of acceleration measurements obtained

from the platform-mounted accelerometers. Since, at this point in flight, none

of the accelerometer axes will lie along the direction of the desired velocity

increment, some simple computations must be performed in the computer to

resolve the measured velocity increment into a suitable coordinate system.

The vehicle is reoriented by 90 degrees after apogee burning to point

the nose-mounted antennas toward the ground stations. It is. thus oriented with

its long axis radial. This is accomplished with gas jets using data from the

inertial platform to control the orientation.

After rough orientation based on platform information, a more precise

attitude is achieved and unwanted angular rates are eliminated by the attitude

control System described in the next chapter. This attitude control system

must work throughout the life of the satellite to maintain correct orientation.

This deterrent contains information effeding the national defense of the United States within the eseeeing of the Espionage taws. Title

1.• so e vs

ye, -..• nomlalf ma d wohleh In nnv wanner te en unauthorized Porten It orehlblted by low.
-CONFENTIAL TR-59-0000-00604

Page 111-34

2.1.6 Orbital Correction Period 

Assume for the moment that apogee burning is so timed that on the

completion of that burning the satellite is precisely at the correct longitude in

the case of an equatorial satellite (or precisely at the correct latitude in.the

case of the polar satellite). Next, assume that the burning is so controlled

that after apogee burning there is no radial component of velocity. Finally,

assume that the horizontal velocity, chosen as a function of the actual observed

altitude at apogee burning, is that which is required to establish the satellite in

an elliptical orbit with a period of precisely one sidereal day. Under these

assumptions, the satellite is, at the completion of "apogee burning," either at

the apogee or the perigee of an elliptical orbit. There still would remain the

problem of eliminating the eccentricity of this orbit and adjusting the inclination

of the orbital plane.

As an example, let us first consider a satellite which is injected into

an orbit which lies precisely in the equatorial plane, and consider the case

where the actual altitude at.apogee burning is greater than the nominal altitude.

In this case the satellite is at the apogee of an ellipse whose period is one

sidereal day. Since the angular rate of the satellite eastward around the earth

is slowest when it is at apogee, and since its average angular rate is the same

as that of the earth, the satellite recedes slowly to the west as viewed by an

observer on the rotating surface of the earth. As the altitude decreases,' the

apparent angular regression slows up and 6 hours after apogee, as the satellite

crosses the altitude of the nominal circular orbit, the westward regression is

converted into an apparent eastward progression. The eastward rate of pro.-

gression reaches a maximum when the satellite is at perigee.' Thus, 'as illustrated in Figure 2-2, the motion of the satellite relative to an observer on a

rotating earth is, to a first approximation, a small ellipse. An analysis shows

that the satellite traverses this ellipse with harmonic motion, and it can also

be shown that the major axis of the ellipse is horizontal and is always twice the

dimension of the minor axis. In the figure the point C lies on the desired

meridian at the altitude of a circular orbit having a period of one sidereal day.

The point A is the position of the satellite at the end of apogee burning; it lies

111111111

•momaw. i.MrwaNww nifectine She national deform of the United Slates within the meaning of Ni. Espionage Laws, Ohs
Earth

Position at the End

of Apogee Burning

4n••••n•"26,11,•••••••

Meridian

Desired

TR-59-0000-00604

Page III-35

-C-ONHOITA

Figure Z-Z. Apparent Path of Satellite as Viewed

from the Rotating Earth.

This document coracles information offettine Me national defense el tim Unit es within the meaning of the Espionage Lows. Title

U.S.C., Section 793 and 704, the transmission or revelation of which In any manner to an unauthorized person is prohibited by law.
lbls detvw•at nMdn. IninownNao A.I • . •

NMI

TR.-59-0000-00604

Page 111-36

on the same meridian but at an altitude Lih too high. Six hours after apogee

burning the satellite is at the point B; its apparent motion has been to the

west a distance lah. If no orbital correction were made at the point B, the

satellite in its apparent motion would continue around through the points D. and

E. However, at point B, an outward radial impulse is applied which halves the

inward radial velocity component. This has the effect of halving the eccentricity and consequently halving the dimensions of the harmonic ellipse which

represents the path as viewed from the rotating earth. Thus, during the next

12 hours the apparent motion of the satellite is along the elliptical arc BC

represented by the dotted curve in the figure. When the satellite arrives at

the point C, a second impulse is applied equal and opposite to that applied at

point B. This impulse completely arrests all relative motion. From this

time on the satellite is in the desired circular orbit.

In the above correction scheme it was assumed that certain conditions

held at the end of apogee burning. In reality, apogee burning . will not be completed at the correct meridian (in the case of the equatorial satellite) nor will

the radial component of velocity be precisely zero, nor will the period of the

elliptical orbit be precisely one sidereal day. This situation is brought about

because of the unavoidable measurement errors in the tracking radar, which

lead to errors in the apogee burning command, and because of measurement

and control errors in satisfying this command (for example, errors in.the

direction of the velocity increment added at apogee caused by platform drift,

and errors in the magnitude of the velocity increment caused by accelerometer

errors). As a result it will be necessary to use the ground radar for tracking.

the satellite during the orbital correction period. The point B (Figure 2-2)

should be taken as the point where the satellite crosses the altitude of the

nominal circularorbit. Because of errors in apogee burning, the satellite

will not pass through B precisely 6 hours after apogee burning. At point B

the nominal correction, which was described earlier, must be modified to ,

cancel out any apparent horizontal component of velocity, and the vertical

component must be chosen to adjust the eccentricity that the satellite will

pass through the point C. Based on further tracking data collected between

B and C, modifications are made to the nominal correction program at the

point C.

rt‘i
TR-59-0000-00604

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The magnitudes of the impulses which must be added at B and C

depend, of course, on and this, in turn, depends upon the accumulated

guidance errors from launch to apogee. (There is no way to eliminate Ah

during apogee burning.) Although it now appears that accelerometers under

development will be able to measure the velocity increment to better than the

required 0.2 feet per second, should it develop that lower drift rates are

desirable, to provide larger life times without significant departure from

station, there it would be necessary to go through a second correction cycle.

Since the impulses of the second correction cycle will be very small -- of the

order of 0.1 feet per second -- the required increased accuracy could be

established. It should also be noted that any effect of the uncertainty in the

value of GM, the product of the universal gravitational constant and the mass

of the earth, can be wiped out during the one-or two-day orbital correction

period.

To this point only corrections in the equatorial plane have•been discussed. If at the end of apogee burning the plane of the orbit is not precisely

equatorial, then it may be desirable to adjust the inclination of the orbital

plane. A nonequator.ial plane can come about through apogee burning not being

precisely over the equator, or it can come about due to errors in the northsouth component of velocity at apogee burnout. An error analysis indicates

that the total excursion of the north-south oscillation caused by having the

orbital plane inclined to the equatorial plane may not be large enough to warrant correction. However, if it is necessary to reduce this oscillation, this

can easily be accomplished by applying an impulse to the satellite in the

north-south direction as it crosses the equatorial plane.

The above paragraphs discuss primarily the problem of correcting

equatorial orbits. The polar orbital correction problem is somewhat different.

Here it is not necessary to closely control either eccentricity or orbital inclination. The period can be measured by noting the time between two successive

passes over' a given point. A horizontal correcting impulse (or one each on two

successive days to give a vernier control) can then be applied in much the same

way as described above for the equatorial satellite.

Mb doormen. contelos information effecting the noll000l defense 40 the I etas within Pb. losoolno of Pie Espionage Lows, Title

9. 1/.3.C., Wien 793 end 794. The transmission or roveletleo of which in fay manner to in enoetherined person Is prohibited by
WWI

TR-59-0000-00604

Page 111-38

Appendix B is a study of solar and lunar perturbations of a 24-hour

circular orbit. The effect of these perturbations is to cause small oscillations

of the order of three miles or so, and also small steady drifts. The perturbations can be accurately calculated theoretically, and, consequently, the nominal

altitude of the satellite may be so chosen as to compensate for the slow steady

drift rates. It can be concluded, therefore, that lunar and solar perturbations

are of negligible importance so far as their effects on the 24-hour satellite are

concerned.

Orbital correcting impulses can be supplied by a system of gas jets

controlled by orders stored in the steering and shutoff computer which receives

commands through a decoder from the ground tracker and computer. Such a

data link and decoder, associated with the transponder beacon, is currently used

on the Atlas missile. Since the jets cannot be calibrated precisely, a vernier

acceleration sensing system must be used to control the velocity impulses by a

feedback loop through the steering and shutoff computer. The sensitive accelerometers used for the correction period may remain caged prior to the completion

of apogee burning.

Throughout the one- or two-day correction period and during the life of

the satellite, attitude control is maintained by the system described in the next

chapter. The attitude correcting torques can be produced by the same gas jets

which provide orbital correction impulses. This jet system and the gai supply

for it is described in Chapter 3.

2.1.7 Equipment Needed 

The following guidance components need to be carried in the vehicle:

In the booster stage or stages:

Booster autopilots

In the first added stage:

Autopilot of first added stage

3, Primary guidance computer (digital)

TM, deo/meet contains Isfemotlen offestine the notional defense of the United States within the •INOMMO /t

1S, U.S.C., Section 793 and 794, Om ftentoittlen or tovelotten el which in any manner to at enouthetittett
---COARDENTIAL

TR-59-0000-00604

Page 111-39

In the final stage:

Gyro-stabilized inertial platform

Steering and shutoff computer (digital)

Perigee-apogee autopilot

Attitude stabilization computer * (analog)

Tracking beacon with data link and decoder

Vernier acceleration-sensing system

10. Clock for timing various operations aboard the vehicle

(can be part of the steering and shutoff computer).

In addition to these airborne components, the guidance system also relies

upon an accurate tracking radar and radio interferometer and upon a digital computer on the ground. Determinations of vehicle position are made by this ground

system and impulse commands are transmitted to the vehicle over the data link.

Figure 2-3 relates the function of the guidance components to the six

phases of the trajectory.

2.2. SYSTEM OPERATION AND COMPONENTS

2.2.1 Climb to Low-Altitude Orbit 

a. Guidance Scheme 

Guidance during climb to low-altitude orbit (except for the programmed guidance of Stage I) is performed by an inertial system consisting of

a gyro-stabilized platform with three accelerometers and an associated primary

digital guidance computer. This computer controls the powered flight of the

vehicle up .to the time of injection into the low-altitude orbit; it is jettisoned

prior to the'perigee burning period. Figure 2-4 shows a block diagram of the

guidance scheme.

*

This computer works with inputs from the inertial platform,. rate feedback

from the "perigee-apogee autopilot," and controls the vehicle attitude from

launch until the 24-hour orbit is established. Thereafter attitude control

is taken over by the monopulse system, working in conjunction with attitude

computer and other components; the system of attitude control after the

24-hour orbit is reached is the subject of the next chapter.

CONFIDENTIAL

Thh decentent cenhIns inteassallea affecting lb* ealleeal defense of this SnIhd Slates within the meaning of the Es pionage taws, Title

IS, U.S.C., Italian 703 end TH. the tfernalM400 ea. miteelatlea if which In any manner le an yna glherised person is prohibited by law.

, '
Constant

Duration

8 .11

th a

dt

0 fi

Booster and First Added

• Stage Autopilots

Primary Guidance Computer

(in first added stage)

Gyro-Stabilized

Inertial Platform

Steering and Shutoff

Computer

Perigee - Apogee

Autopilot

Tracking Beacon with

Data Link and Decoder

Vernier AcCeleration

Sensing System

TR-59-0000-00604

Page 111-40

Figure 2-3. Airborne Component Operation Timetable.

This decement contains Information attesting the national defense of the United Shari within the meaning of the Esolonolm taws. Title

IS, U.S.C., Section 793 and 794, the transmission or revelation of which in any manna to on eneetherind panne Is prohibited by law.
O

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-a:WEN-PAL

TR-59-0000-00604

Page III-41

.11=•••••nn•14 4••n•n••n11.

This dearnmsal sawfishes Intermation &Udine the notteeel defense of the United States within the meaning of the Espioneoe Laws, Title

UM*. 703 and 791. the transmission or revelation of which in an y manner to an vnetethericed oorsen is prohibited by law.
TR-59-0000-00604

Page 111-42

The stabilization gyros maintain the inner element (on which the

accelerometers are mounted) fixed in an inertial frame. The orthogonally

mounted accelerometers define a rectangular coordinate system in which

the guidance computations are carried out. For reasons of accuracy and ease

of system alignment, one axis (z) is aligned parallel to the local vertical at the

launch point (see Figure 2-5). The y-axis is aligned normal to the poweredflight pitch plane.*

For an eastward firing, the powered-flight pitch plane coincides

with the desired low-altitude orbital plane, that is, the x and z axes of the

computing coordinate system lie in the desired plane of the low-altitude orbit.

The lateral steering during booster-powered flight is then extremely simple,

since it is merely necessary to control and yand jr errors to zero by the time

the booster engines are cut off.

For firing azimuths other than due east, the earth ls velocity has

a component along the y axis at launch, and hence the x-z plane does not coincide with the desired low-altitude orbital plane. It is necessary, then to compute lateral position and velocity errors with respect to the desired orbital

plane and control these errors to zero. If the azimuth of the computing

coordinate system is chosen so that the velocity vector at the final booster

cutoff point for a standard missile (nominal performance parameters, no

disturbances such .as winds) lies in the desired low-altitude orbital plane, then

a standard missile is required to do no yaw maneuvering to compensate for the

earth's rotation. Corrections need be applied only for in-flight disturbances.

Control of the pitch motion of the vehicle is somewhat more complicated. Ideally, it would be desirable to control the pitch motion of the vehicle

and terminate the booster thrust at a time such that the vehicle is established in

a circular orbit at a predetermined altitude with the injection occurring over a

predetermined point on the earth. However, it is impossible to satisfy these

constraints simultaneously without thrust control, although it is possible to

control the booster flight so that a circular orbit at some altitude h is established. It is necessary to control the powered flight by means of controlled

* This is an inertially fixed plane. For a nominal missile executing no yaw

motion, the vector remains parallel to this plane during the booster-powered

flight. Because of the earth's rotation, the missile does not fly in this plane

except in special cases.

This detontont contrens iniermetion •lleoting the pollee& defense et the United Urges within Ow mooning of the Espionege Laws, 71110

U.S.C., Section 793 and 794. Me trentrabstee et reveletion of which in any wa goner to en unauthorised penal Is prohibited by low.
Launch

Point

TR-59-0000-00604

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Figure 2-5. Inertial Computing Coordinate System.

Thb document contains i

nformation affecting the national defense of the United Slabs within the weenies of Me Espionage Laws, Title U.S.C., Settlers 793 and 794, the transmission er revelation of width in 

any manner to an unauthorised person it prohibited by Iwo.
TR-59-0000-00604

Page III-44

thrust termination and pitch steering so that the actual altitude and velocity of

the vehicle is made to approach the required altitude and velocity.

This guidance scheme, using velocity vector control, will lead to

altitude errors as well as latitude, longitude, and time errors at the injection

point. if the missile is nonstandard. At most, altitude errors will amount to a

mile or two. It is possible with the inertial system to measure the position and

time errors at the injection point to about ±600 feet in each direction. Subsequent corrections can be made during later burning periods, so these errors

are of no great consequence. The correction scheme will be described in detail

in Section 2,2.3 and Section 2.2.5.

b. Guidance Accuracy 

Based on the inertial components described later in this section,

the estimated rms guidance measurement errors at injection into the lowaltitude orbit may be stated as follows:

Error in velocity magnitude 12 feet per second

Error in direction of the

velocity vector

in orbital plane 2 x 10 -4 radian

normal to orbital plane 1 x 10 -4 radian

Position errors (all directions) ±600 feet

The control of the actual position and velocity coordinates at the end of the

preorbital burning period will not be as accurate as the measurement accuracy,

but the control errors can be cancelled by subsequent corrections at perigee

burning. It appears that position coordinates will be controlled to about

6000 feet, normal to the flight path and to about 20,000 feet along the flight

path. With no vernier burning period at injection into the low-altitude orbit,

the velocity magnitude can probably be controlled to about 5 feet per second,

and the velocity direction to about 2 x 10 -

4 radian. The measurement errors

specified above are comparable with those required for an ICBM guidance

system with a one- to two-nautical mile cep.

This desetnent contains laleyeetlee effecting the netleool defense of the United Stake within the seeeelne of the Espionage Lows, Title

1$. U.S.C., Seetlee 793 end 794, the trensolsslett er revelation of which la any ~Mr to as etwortilefised pones Is prohibited by low.
TR-59-0000-00604

Page 111-45

c. Computer Requirements 

In order to accomplish the control functions described above, the

primary guidance digital computer must be capable of:

Accepting acceleration data (integrated once) from the stabilized accelerometers, computing the acceleration due to gravity (from position

data), integrating this once to obtain the velocity due to gravity, and combining

this with the accelerometer inputs to obtain the total velocity.

PerforMing another integration to obtain position data.

(3) Computing pitch and. yaw steering commands and engine cutoff

commands based on position, velocity, and time data.

At the low-altitude orbital injection point, the computer must also

calculate corrections to the low-altitude coast time and to the desired velocity

increment to be added during the perigee burning period (see Section 2.2.3).

It is proposed that the desired corrections be calculated shortly after injection

and transmitted to a second (simpler) digital computer, which governs the

flight during all subsequent burning periods. After computing and transmitting

the desired corrections, the primary computer is Jettisoned. The transfer of

this data is optional and could be dispensed with if. the resultant apogee control

errors j600 feet per second can be economically eliminated by judicious steering

during apogee burning, but this study will proceed as though the perigee

corrections are to be made.

In order to provide the necessary computing functions with minimum

weight, size, and power requirements, the primary guidance computer must be

specially designed for the task. Such a computer would be of the fixed program

type and would be similar to the airborne computer being built by Arma for the

WS 107A-2. If building blocks similar to the Arma Computer (diode matrices

for constant storage and magnetostrictive delay lines for dynamic storage) are

used, such a computer would weight about 90 pounds and draw about 250 watts.

Such a computer could be designed and manufactured on a model shop basis

and would be available for flight test about one year after the detailed guidance

equations are frozen. The earliest possible flight test date for this subsystem

would be the middle of 1960.

TUN document COIlieilln Infonoottee Wanes the eollettef Meese of the United Notes within the mooning of IM itolonege lows. Vile .• It • 10..0 .. ►n• ehe 4 mr.4681.111egi I.. Imo.
TR-59-0000-00604

Page III-46

CONFIDENTIAL

A computer having alighter weight (perhaps 40 to 50 pounds) and

drawing less power could be developed using a magnetic drum as the storage

element. Development of a suitable drum to withstand the missile environment

would take a minimum of two. years; the advantages gained by such a computer

do not appear to be very great for this application.

d. Inertial .Measurement Unit 

Table 2-1 indicates some of the major sources of error and approximately the accuracies which can probably be obtained by suitable inertial components within the next two or three years. It is expected that the Arma 10 x 106

gyros currently in development will meet the 0.02 degree per hour drift

specification. It is also likely that one of the gyros now under development at

the MIT Instrumentation Laboratory would be suitable for this application.

The 'precision accelerometers assumed in Table 2-1 are about a

factor of two better than early models of the Arma vibrating-string accelerometer used in the inertial guidance system for the WS 107A-1, but it is

expected that these accelerometers will meet the listed accuracy by the end of

the present development program.

Although it is expected that the component accuracies listed in

Table 2-1 will be met by 1960 or 1961, it should be emphasized that the success

of the ascent guidance system is by no means dependent on such accuracies.

Greater errors will only mean the consumption of slightly greater quantities of

propellant during apogee burning and during the subsequent orbital correction

period.

Table 2-2 summarizes the weight and power requirements of the

inertial measurement unit (platform, accelerometers, and electronic packages)

for two systems, one available in 1960, and the other available in 1961. System I

would use the Arma 4 x106

 gyros and the less accurate accelerometers, System II

would use 10x106 gyros and improved accelerometers. Dimensions of the stable

platform assembly (the largest single unit) and the shock mounts are about

37x 32 x23 inches with a volume of about 5 cubic feet. Total system volume*

including the computer would be about 12 cubic feet.

*

Volume of the early Arma systems used in the WS 107A-1 is about 17 cubic feet.

1111111110 This dievetent contains infosisetlee effecting the notional Mee if Ike United Steles within the weaning if the Esplenotte Sews. Title

U.S.C., Witte 793 end 704. the transmission or revels/lee of which In any menses to se vneethedsed porno I, orehfisited b y low.
TR-59-0000-00604

Page 111-47

Table 2-1. Inertial Measurement Unit Component Accuracies.

Magnitude of Error

(rms) Source of Error

Cyros

Drifts Independent of Acceleration

Drifts Proportional to Acceleration*

Drifts Proportional to Square of **

Acceleratfon (due to vibration)

Accelerometers

Linearity

Cross Coupling

Zero Offset

Orthogonality

Platform Rigidity

Platform Alignment (all axes)

0.02 deg/hr

0.08 deg/hr/g

0.06 deg/hr/g2

2x10 -5 g/g

2x10 -5 g/g

lx10-5 g

6 sec of arc

. 2:. sec of arc/g

6 sec of arc

*

It is assumed that an optimum gyro orientation is used.

**For the optimum gyro orientation, anisoelastic drifts due to steady

accelerations are neglible.

• Table 2-2. Inertial Measurement Unit .(Stable Platform and Accelerometer

System)--Weight and Power Summary.

System

Earliest

Availability

Date

Weight

Power

3(1)ac dc

I

II

Jan 1960

Jan 1961

200

160

350 50

280 20

docomeot iontoins Infotsnotioo ollotting tint noliomol dekilIN of lb. Wiled lldNs wig& IM mooning of Om Isillosogo Laws. This

II U.S.C.. fiction 793 end 794, the tronsmission ., nivolotion of which is goy manner to on onowtherind Derma Is oritisibitod by low.

•
e. Tie-in with the Booster Autopilot 

Figure 2-6 and 2-7 show functional block diagrams of the guidance

c ontrol system for the climb to Jow-altitude orbit. During Stage I the vehicle

follows a programmed pitch attitude and is stabilized in roll and yaw with the

platform as a reference. The primary guidance computer is inactive except

for computing vehicle position and velocity at the appropriate time. If a Titan

booster is used, the first stage cutoff signal is generated by low-level sensors

in fuel and lox tanks. At the beginning of Stage n burning, control of the

vehicle is switched to the primary guidance computer, which issues all subsequent engine start and cutoff signals, and also issues pitch and yaw steering

signals to control the flight path.

As will be noted from the block diagrams (Figures 2-6 and 2-7)

the autopilot (excluding engine actuators and their power supply) is comprised

of the following elements:

Pitch attitude programmer (Stage I)

Three rate gyros

(3) Associated electronic amplifiers and compensation networks.

Weights and power requirements for these components are summarized .in

Section 2.2.7.

2.2.2 Low-Altitude Coasting 

The vehicle moves in a low-altitude (140-nautical-mile) circular

satellite orbit following booster burnout until perigee burning is initiated. The

duration of this orbital coasting is fixed in advance according to the desired

longitude of the vehicle in its high-altitude orbit according to the scheme

described in Chapter 1. Small variations of this coasting time can be exploited

to minimize the control errors accrued at injection into the low-altitude orbit

because of the inability of the vehicle to follow the guidance system commands

exactly and particularly due to the lack of thrust control. Because burnout

position can be measured by the more accurately than the position can be controlled, the position control errors may be in large part compensated by adjusting the coasting time. Velocity control errors at burnout can also be reduced

by appropriate adjustement of the perigee burning and steering program.

TR-59-0000-00604

Page 111-48

SAL moosnmet consoles infermetlea effeming the makes' defame M IM United States Waste Me messing of the EasteneiM kiwi. Thu

U.S.C.. Settlen 793 mei 794. the transmission sr remiletion of whkh in any menses te ea eneelloorited minim is preisildhol 37 lam.
TR-59-0000-00604

Page 111-49

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TR-59-0000-00604

Page III-50

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TR-59-0000-00604

Page 111-51

Referring to Table 1-2, Chapter 1, it can be seen that the low-altitude

coasting may last as long .as 10 hours in order to position certain equatorial

satellites. There are two primary functions to perform.during this time.

Attitude stabilization of the vehicle

Computation of the coasting duration adjustment and the perigee

velocity impulse adjustment to remove control errors imparted

at injection into the low-altitude orbit.

Attitude stabilization is required during the coasting phase to prevent

the gyro-stabilized inertial platform from going into gimbal lock and to orient

the vehicle so that it is aligned approximately in the direction of perigee thrusting.. It is proposed that the vehicle be attitude-stabilized during the coasting

trajectory by measuring vehicle attitude with the stable platform and computing

with.a special analog computer control commands for a gas system. Details of

the atttitude stabilization system are discussed in Chapter 3, Section .3.2.

The bias computations would be performed with the primary digital

guidance computer, using the guidance system measurements of burnout control

errors. (See Section 2.2.3 for details of this computation.) Since the computer

is placed in orbit together with the first added stage in which it is housed, its

utilization during this time requires only that the necessary power be supplied

during the time required to calculate the corrections.

2.2.3 Perigee Burning 

The proposed control scheme for perigee burning is shown in Figure 2-8.

This burning period must add approximately 8000 feet per second to the lowaltitude circular satellite orbit speed of 25,424 feet per second in order to start

the vehicle on its ascent trajectory along the Hohmann transfer ellipse. The

nominal direction for this impulse is tangent to the local earth.surface and in

the original plane of motion.* The computation of coasting duration and velocity

bias corrections by the primary guidance computer for this period is discussed

at the end of this section.

*It may be desirable to reorient the flight plane by a small angle at this point

for propulsion efficiency. (See Chapter 1, Figure 1-6) such maneuvers do

not affect the guidance systems significantly.

Tftls docenent contains bdonnaften aiding

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firth. WI wftld* ftw weaning el doe laplonew Laws, TIM*

Wed. .WilddrIn any Wenner I. ae uneuftwelasd *gram 1. preftIlMnd by
TR-59-0000-00604

Page 111-52

1

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as

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U.S.C.. Sullen 793 end 794, Ow ttansmitsien or reselelle g. ef which In any wanner I. en geguIlterIsed parses Is prehIbited by low.

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TR-59-0000-00604

Page 111-53

The bias corrections are stored in a smaller digital steering and shutoff

computer that remains in the vehicle. This computer controls both the perigee

and apogee burning programs and also regulates the vernier velocity corrections

during the orbital correction phase. It is estimated that such a computer could

be specially designed for a total weight of 20 pounds, and would probably draw

20 watts of 400-cycles per second ac and 20 watts of 20-volt dc power. The computer would be operated only during the burning and orbital correction periods.

Readings from the accelerometers on the inertial platform would be fed

into this computer as velocity data. These measurements would be compared

•with incremental velocity required as adjusted from nominal according to computations made by the primary guidance computer at injection into the low-altitude

orbit. The steering and shutoff computer would compute pitch and yaw steering

signals for the autopilot, and command the engine start and cutoff signals.

The same autopilot would be used for both perigee and apogee burning.

It is estimated that the entire autopilot, including rate gyros, would weigh less

than 15 pounds and need not draw more than .25 watts of 400-cycles per second

ac power and 10 watts of 20-volt dc power. The rate gyros should be good to

10 -4 radian per second over fractions of a second, and would control highfrequency vehicle vibrations and flexures during perigee burning. The inertial

platform accelerometers would control velocity impulse, since they are attitudestabilized and quite accurate (6a/a = 2 x10-5 ). Motor shutdown control errors

will probably limit the speed control to about one foot per second. Since perigee

burning is characterized by low acceleration levels, the g and g 2 platform

errors are neglected and the total velocity orientation control error at the end

of perigee burning is equated to the platforitn drift. The perigee burning errors

are thus

6 y = 6e = ±0.2 degree = 3.5 rnilliradians

6 OV) = ±1 foot per second,

Thu deorment contains Inferootion effecting tiro notional defense of the Uellest.Stefes witble the @waning of Ste Ispionsnot tows, 118le

18, U.S.C., Section 793 one 794, tbe hensmissIon or revelation of which in oily mow to an unauthorized person Is stetablIes1 by low.
TR-59-0000-00604

Page 111-54

e6wifiEHALr

where y is azimuth and c is elevation of the applied velocity increment and

AV is the magnitude of the velocity increment. Since 8000 feet per second i s

added, the angular errors each introduce an unwanted cross component of

velocity equal to Z8 feet per second.

The guidance system is not able to control the missile position and

velocityvectors as accurately it can measure them. This is due primarily to

lack of precise thrust control but also to smoothing, response lags, and motor

shutdown errors. Estimated control and measurement accuracies at the time

of injection into the low-altitude orbit (t o) are compared in Section 2.2.1b. The

error analysis of Appendix A shows that these control errors are transformed

into position and velocity errors at the end of the low-altitude coasting (t1)

according to Equations (2.1) through (2.6). First, however, we define the

symbols used, assuming an equatorial satellite is being launched.

U = circumferential velocity

V = total magnitude of velocity

- transit angle in plane of motion

r distance from vehicle to center of earth

a = climb angle of vehicle above local horizontal

p, = GM, gravitational constant multiplied by mass of earth

= radial velocity =

cr = inclination, 1. e. , dihedral angle between plane of motion

and equatorial plane

= transit angle from initial burnout to "zeroth" crossing of

equatorial plane

= latitude

sgnX0

= +1 for northern latitudes, -1 for southern

= azimuth of velocity vector

N = number of half revolutions from "zeroth" crossing in

low-altitude coast

tONFIDEANt,_

This docomma contains Information affocIlno Wm notional deans* of IM United States within the mooing of IND Ispionop Lows, 7114

U.3.C., Saellon 793 and 794, Iba transmission of revelation of which In any manner so an onavrboriood polo. Is prohlbtlod by law.
TR-5'?-0000-00604

Page III-55

= longitude difference between burnout and "zeroth" crossing

of equator

p = longitude

= angular rate of rotation of earth.

and subscripts indicate the period of occurrence.

Velocity Errors at t1 (end of low-altitude coast)

5Ul

6 r 6V

—7,-- = — (1 — cos cf) o

3. ) 1-7—o

- + (2 cos 04) 1 — 1) v. — sin 4' 6 a. 1 0

o 0 o

6

+ (1 — cos (I)

5

i ) :... , (2.1)

6 W Sr V

. 4) V

r

r

1 = o

sin + 2 sin 4) 1 -72. +cos (1) 1 6 a.0 — sin 4) 1

(2.2)

.o

6 tri = cos (sgn X0 ) 6(30 + sin 9 (sgn X0 ) 6 X0 , ( 2.3)

Position Errors at t1

S rSr 1

Vo

 = (2 — cos 41 )r °

6V

+ 2 (1 cos 4d v o + sin

0

+1 6 a0

— (1 — cos 4) i)

6 x i = [(-1)N cos xo sin ej 6 po + B-oN cos ei 6 xo

Or

+ [(-0N+ I sin o-i (sgn Xi)) [(2 sin — 3 +I) r °

6V

+ (4 sin 4) 1 — 3 4) 1 ) + 2 (cos 4) 1 — 1) 6 a0

4. (2 +1 2 sin cfrl) + —1'0 61 111

O

(2 .4)

(2.5)

This decompose s...oeins infeenetion affecting the natioael deem, a l MO United Stales within the -amine it the Espionage taws, Till.

111. U.S.C., Sactiorr'793 and 794. dm transmission or revelation of which in any mono., to on oneuitteriud person Is prohibited by low.
TR-59-0000-00604

Page 111-56

S r

r

o

°

p i

= sin X.0 6 13 0 + S p0 + n S ti + cos r Z sin 41 —

V

+(4 sin 4) 1 — 3 01 ) --v-2- + 2 (cos 4)1 — 1) 6 o.0

o

U

+ (2 •, — 2 sin 4) 4)1 )

0 

+ -;;;td6 (2.6)

The above equations hold equally well for polar orbits, but the quantities

appearing therein must be redefined: o- is the angle between the plane of the

orbit and the desired plane; p is the azimuthal angle measured in the final

desired plane of the orbit; is the angle from the satellite to the desired

plane; and so on, with the final desired plane replacing the equatorial plane

in the definitions. Of course, in the case of a polar orbit, the term containing

SI in Equation (2.6) does not occur, but a similar term would occur in the

counterpart of Equation (2.5).

If the actual position and velocity could be measured with infinite

precision, the control errors could be computed as the difference between

these and the nominal values stored in the primary guidance computer. The

above equations would allow one to predict the corresponding control variations

in position and velocity which would result at the end of the coasting phase. The

transverse velocity errors [Equations (2.2) and (2.3)] can be compensated by

judicious steering during perigee burning. The speed impulse added during

perigee burning can be altered so as to compensate for too little or too much

speed (6U1 ) at t1 . The altitude error at perigee burning can be corrected

simultaneously with the speed error, since there is an appropriate speed

impulse for transfer takeoff from any altitude. The simultaneous correction

of 6 r 1 and 6 U1 does not depend upon fixing the coasting time.

The latitude and longitude- errors at perigee burning can be minimized

by choosing the coasting time on the basis of the burnout measurements. This

may be seen from Equations (2.5) and (2.6), since if all of the burnout control

errors are measured with infinite precision, the variation of coasting time,

6t 1' can be chosen so as to minimize one or the other, or perhaps both. For

a vehicle fired due east from Cape Canaveral with a maximum transit of seven

This deammat contains helermetion effecting the notional defense el the United States within the meaning it the espienemt tows. Title

111, U.S.C.. Settles 793 end 794. the transmission or revelation which in any manner to an unauthorised person is prohibited by law.
TR-59-0000-00604

Page III-57

corDENTIAL

full revolutions in the low-altitude orbit after the first equatorial crossing,

these last two error expressions assumed the following numerical forms:

6r 6V

6 X1 0.88 613 + 64.3 —la + 63.3 ° 0.95 6 42.5 -6-W V

+ a

r o — o o P.

— 0.56 x 10 -3 6 t 1 ,

6r 6V

6 p 1 = 0.48 6 130 + 6 p0 + 118 ° — 116 — 1.76 6 a V

r °

o

+ 78.2 + 0.95 x 10 -3 6 t1.

Subject to the basic uncertainty in the gravitational product p. = GM,

one may solve for the 6 t 1 , and eliminate either 6 X., or 6 p l . An optimum

strategy seems to be to choose 6 t 1 so that the same bracketed' quantity in

Equations (2.5) and (2.6) vanishes. Actually, the measurements are not

infinitely accurate, so that the propagation of the burnout measurement errors

must be reckoned as control errors when the vehicle arrives at apogee. Correction of a 2-foot-per-second speed-control error at burnout requires that

the coasting time be changed by 9.2 seconds to correct the latitude at perigee

burning, and by 5.4 seconds to correct the longitude. The error coefficients

are not changed significantly during such times, so that the computations are

linear and constant for a fixed nominal trajectory.

Velocity and coasting duration corrections are simply a matter of

transforming the difference between measured and control quantities at burnout

via the above constant matrix. This • could be performed quite easily in the primary

guidance digital computer, which is also brought into the low-altitude coasting

orbit with the vehicle. To take advantage of this computer, however, it is

necessary to postpone separation until the computation is completed and the

correction signals are fed to the steering and shutoff computer which controls

perigee burning. The additional power requirements for this computation are

negligible since it is estimated that the constant-coefficient six-by-six matrix

multiplication could be performed in the computer in less than 1 second.

TONFIDafyit_

fhb Amman/ ceatelas hiletatitless effecting the ittiliwal detente of the UNItid betel within the amino el thet Wenn. Low. TM*

11.S.C.,•/Htlem 713 wed 794. Ow fressalsslea or twolatlei a which Is way saner to ee wwwliwelaed Puma I, ereblbltN by Ws%
-C--ONFIDENNE

• %l • deannont cantatas wrestle'

offecHne the nein& *Wm of Ms United Ste/es within the anierilno el Nue Isolonorte Laws. TM* Id.S, U.S.C., Sidles 793 err rm. the trontedselee or revelation of which In say I. en unetehorked person is 11.d by 'low.

2.2.4 Ascent. Coasting

Perigee burning ends with a total vehicle speed of approximately

33,420 feet per second. This speed is gradually lost as the vehicle exchanges

kinetic energy for potential energy during the slow rise along the elliptical

path to apogee. The combined effect of perigee-burning control errors and

extrapolated booster burnout errors is to prevent precise control of position

and velocity vectors at apogee. There are two primary functions to perform

during this phase:

Attitude stabilization of the vehicle during ascent

Tracking by ground stations to determine the actual position

and velocity vector of the vehicle as it nears apogee.

In the case of an equatorial satellite which enters perigee burning on

a southward crossing of the equator, the angle between the longitudinal axis

of the vehicle and the equatorial plane during perigee burning is about

2865 degrees. (if the plane of the orbit is to be reduced by 2.2 degrees, as

suggested in Chapter 1 for propulsion efficiency, then the angle between the

longitudinal axis and the equator is about 9 degrees less.) Just before apogee.

burning the vehicle is again approaching the equatorial plane at an angle of

28.5 degrees from the south. During apogee burning the longitudinal axis of

the vehicle must be pointed south from the equatorial plane by an angle of

about 24.5 degrees in order to cancel the northerly motion of the vehicle as

it approaches the equatorial plane. The angle between initial and final orientation is about 127 degrees (1.80 degrees - 28.5 degrees • 24.5 degrees). If the

equatorial 'crossing at perigee burning is toward the north, a similar reorientation is required. In addition, it is necessary to control attitude during the

final hours of ascent so that an appropriately designed transponder antenna

pattern will cover the ground tracking station.

The same attitude stabilization system is used during the ascent

coasting trajectory as for the low-altitude coasting phase. Torqueing commands for the gas jets are generated in the analog attitude stabilization

computer using vehicle orientation as read from gimbal angles on the inertial

platform (see Chapter 3, Section 3.2).

TR-59.0000.40604

-CONFIDENTIAL 

Page III-58
TR-59-0000-00604

Page III-59

Although the position and velocity at apogee cannot be controlled with

great precision because of blind navigation errors, it is possible to measure

the vehicle l s position and velocity with considerable accuracy after it rises

above the radar. . horizon of a ground tracker, presumably the same tracker

as would be used 'for the orbital correction phase. The position of a vehicle

launched from an equatorial launch site as a function of time during the ascent

trajectory is shown in Figure 2-9 in a coordinate system which rotates with

the earth. The vehicle spends over 4 of the 5.27 hours of coasting time within

17 degrees of its final Meridian and provides a very good target for position

and velocity. determination. As can be seen from the figure, there is a good

deal of flexibility in the allowable location of the ground station.

Preferably, for accuracy of tracking during the orbital correction phase

(to avoid foreshortening of the base line of the radio interferometer), it should

be close to the nadir point below the satellite. But for logistic simplicity, it

may be desirable to place the tracker at one of the communication stations

which are, of course, within line-of-sight of the satellite l s'terminal position.

Tracking accuracy requirements during ascent coasting are far less

severe than..tracking requirements during the orbital correction phatie. Consequently, it is the' latter phase which sets the specifications and determines

the design of the tracking system.

Since the most critical specification on orbit accuracy is that of meeting

the 0.2-foot per second drift rate, the most critical measurement is angular

rate. The Maui propagation experiments show thatat 10,000 megacycle, I microradian pek second can be achieved with a 20-second smoothing time. (At X-band,

it is probable that the influence of ionospheric diffraction will be less than the

influence of the tropospheric effects.)

Since rate errors are improved in proportion. to the three-halves power

of the smoothing time, a smoothing time of • hour would provide angular rate

accuracies of 0.0004 microradians per second. This would determine horizontal velocity to better than 0.06 foot per second. However, there is some

question about the physical possibility of measuring this accurately over long

Tills dowse* gestates Inteneatlea ettectleg fh. national Ma gee if the UMW Metes within the namin g et the lopleneoe Laws, TM.

U.S.C., Settles 791 sod MC lb. franuallabo or nyvolotkny .f winch In any meant. on onaotherisod warn I Orallbboa by' law.
TR-59-0000-00604

Page 111-60

Figure 2- 9. Position of Vehicle in Equatorial Plane at Various Times

During Ascent Relative to Rotating Earth.

This deco/meet contains Infante.len ettectino the eationel defense et the United Stoles within the meaning et the asp:ones@ Laws, MI*

11, Wit.. Settles 793 sod 794, 1110 tronstelsslon or reeeletion Si which In any manner is on vnewtherised penes is orehlblled by low.
TR-59-0000-00604

Page III-61

periods of time, since diurnal variations in the atmospheric refractive-index

configuration must be recognized together with the random variations whose

effects are being smoothed, and carefully devised experimental tests of

long-term smoothing for tracking data should be initiated.

Equipment for making the above measurements would consist of a

ground interferometer installation possessing three stations spaced approximately 2000 feet apart. Calculations show that adequate signal-to-noise ratio

(approximately 40 db with a 10-cycle bandwidth) can be achieved using 6-foot.

diameter grotmd antennas, one of which is radiating 10 watts at X-band. The

ground antennas must be capable of tracking the vehicle as it rises during the

last few hours. This could be accomplished by slaving them to a single

tracking antenna .which tracks the beacon in the vehicle. 'The three ground

antennas must be connected by phase-coherent links. These could be supplied

either (1.) by waveguide as in the Mod III General Electric Atlas rate system

or (2) by air links.

It seems clear that the various accuracy requirements of the system

(minimizing attitude control problems, easing ground tracking requirement,

etc. )' can be met if angular position is determined to about 1 milliradian. This

requirement can be easily met by the position tracker of the CE system. If

much higher accuracies should ever be required, an alternative for measuring

angular position is to use noise modulation on the X-band radio interferometer

signal sent to the missile transponder. By measuring,the difference of transmission times to the several ground receivers, one determines angular position.

A base line of 2000 feet in each (orthogonal) direction would give a tropospheric

fluctuation accuracy of 10 microradians (rms), according to the Maui tracking

simulation.experiments, but boresighting will limit the absolute accuracy to a

somewhat larger number:

Range rate to the vehicle could be measured by round-trip doppler

techniques to 0.25 foot per second quite readily. The GE radio system has

shown this accuracy in current tests. This measurement could be improved

by a factor of 10 if it were necessary, since propagation noise is not an

important factor.

This decongest contains isfennetien ellestine the notional defense of the United Steles whist* the mooning of the Wit/noes Lows. TM*

IS. U.S.C.. Section 703 end 794. the treessission a revelation et which in any manse. ho an ueoutheruad person is prohibited by low.
TR-59-0000-00604

Page 111-62

Absolute range can be measured to an accuracy of 20 1 feet, which is

limited by the uncertainty of one part in 10 6 for the velocity of light. Propagation effects give a range error of less than 1 foot. It should be noted that relative range can probably be measured to 10 feet at this distance.

The airborne transponder takes the signal from one of the ground

stations, offsets it in frequency and transmits it. The airborne antenna could

be a flush antenna or a fixed horn. The transponder can also be used to receive

commands from the ground tracker. The command channel could be incorporated as modulation on the rate system signals.. The decoder could be built into

the transponder. The average power requirements for the transponder would be

approximately 75 watts. It would radiate 100 milliwatts. The transponder,

including power supply, but not including batteries, might weigh 15 pounds and

occupy three-fourths of a cubic foot. These estimates are based on current

design techniques, using a mixture of currently available vacuu.m tubes and

transistors. It is very likely that the weights and powers of all vehicle-borne

eqUipment could be reduced appreciably if a significant effokt were made in this

direction.

It should also be noted that continuous tracking is not required, since it

is only the low-frequency noise which is critical, for such long smoothing times.

• • It is therefore proposed that the vehicle be tracked for approximately 2 seconds

every 2 minutes to conserve transponder battery power in the vehicle. Since

some of the airborne beacon equipment must remain in operation during the down

times, this probably corresponds to 10 per cent of the nominal total power

requirement.

The tracking data would be fed into a ground digital computer in the

form of round-trip doppler shifts and doppler differences. The computer

would smooth the high-frequency components of the radio tracking data and

then compute the several invariants of a free-fall ellipse (e. g. , eccentricity,

major axis,. inclination of orbital plane, etc. ). These invariants can then be

smoothed to eliminate low-frequency noise. The final results of the computations are -commands transmitted to the vehicle via the transponder data link.

Quantities transmitted . are _the time to commence apogee burning when the

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TR-59-0000-00604

Page 111-63

correct longitude is reached (for the equatorial satellites or the correct

latitude for polar satellites) and adjustments to the incremental vector velocity

(magnitude and direction) to be added during apogee burning.

2.2.5 Apogee Burning 

For the equatorial satellite, the apogee burning period must add

approximately 6000 feet per second to establish the vehicle in the 24-hour orbit,

simultaneously changing the plane of motion by 28.5 degrees. The guidance

scheme proposed for controlling the burning phase is shown in Figure 2-10,

which indicates that the steering and shutoff computer is the central control.

The ground-tracking system measures the actual trajectory during the ascent

coasting phase and transmits velocity bias correction to the vehicle. These

are stored in the steering and shutoff computer. Apogee burning is initiated

when the vehicle reaches the correct position. The engine start signal is

commanded from the ground on the basis of angular position tracking information which establishes the true longitude of the vehicle.

As mentioned in Section 2.1.5 the incremental velocity added during

apogee burning is so chosen that the radial component of velocity and the

north-south component of velocity are both brought to zero, and the eastward

component of velocity is chosen to establish the satellite in an orbit having

a period of one sidereal day. Consequently, the burnout velocity must be

chosen as a function of the observed altitude at apogee burning. The commands to add the correct vector velocity increment are transmitted from the

ground tracker and stored in the steering and shutoff computer.

The steering program during apogee burning is controlled by the

digital steering and shutoff computer using the stored commands from the

ground tracking station and using the platform-mounted accelerometers for

sensing velocity changes of the vehicle. The ground tracker cannot be used

for controlling the velocity increment added during apogee burning nor can

it be used for controlled azimuth steering during apogee burning because the

smoothing times in the angular rate circuits are too long to keep up with the

velocity changes. In principle, the ground tracker could be used for controlling elevation steering since the dopper range rate circuit can have a

This document contains information eitectino the national defense of the United States within the meaning at the Emblems* Laws, Title

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Page III-65

very short smoothing time. However, since the inertial platform must be

used for controlling azimuth steering it is simpler to let it control elevation

steering as well. Since the steering and shutoff computer and the inertial

platform were used during perigee burning, no new components are introduced into the system for apogee burning. The incremental velocity vector

should be controllable to 5 milliradians in angle and about 1 foot per second

in magnitude, including uncertainties in engine shutdown characteristics.

22.6 Orbital Position Correction 

During the orbital correction period, position and velocity data are

derived from the ground tracker and are used to compute commands which

are sent to the vehicle on the data link. The commands consist of orders to

fire the appropriate orbital correction jets together with information on the

incremental velocity to be added, the latter to be stored in the digital steering

and shutoff computer. Since long smoothing times are needed to improve the

latitude and longitude velocity measurements, the tracking can be intermittent.

This will reduce the power consumption by the tracking beacon. It is expected

that only 2 seconds of tracking every 2 minutes are needed during the 24 or

48 hours of vernier control, which calls for an average beacon power of only

7 watts. A block diagram of this system is shown in Figure 2-11.

The gas jets described in Chapter 3 are operated at intervals during.

the first 24 or 48 hOurs. A total velocity impulse of about 100 feet per second

must be supplied for all of these corrections. Since the jets cannot be calibrated precisely, a vernier acceleration sensing system is used to control

velocity impulses by a feedback loop through the computer. Three bodymounted accelerometers plus the associated electronics can probably be

designed for 15 pounds and 30 watts of dc power. Since the impulses will be

supplied at low acceleration levels, it is possible to use sensing schemes

which are more accurate than those used for ICBM flights, where a dynamic

range of 10 g's is required. There is considerable room for invention in this

area of low-thrust accelerometers.

The gas jet system proposed in the next .chapter would produce between

10 to 20 pounds of thrust. This leads to accelerations during the application of

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--CRIFIDENTIAL

MI=

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TR-59-0000-00604

Page 111-67

correcting impulses of about 0.01 g. If we assume that accelerations can be

measured to 10 -5 g's (requiring a null error of 10 -5 g's and a scale factor

error of 10 -3 of full scale), then the 100 feet per second or so which is added

can be measured to 0.1 foot per second which meets the requirements set up

.earlier. If this specification can be met, then the total time required for

corrections should not be greater than about 18 hours. On the other hand, if

the accelerometers do not meet this specification, it may be necessary to go

through a second cycle of corrections as described in Section 2.1.6.

2.2.7 Weight and Power Summaries 

Estimates for subsystem design weights and power requirements are

summarized in Table 2-3. The data in Table 2-3 is combined in Table 2-4

with the airborne component operation time table to produce the estimated

weight and power requirements by component for each phase of the orbitestablishing trajectory. The total weight and power requirements for each

phase of the trajectory is also given, However, the power consumption is.

quite uneven due to the intermittent use of the components. The primary

guidance computer is turned off and jettisoned as part of the first added stage

following the booster burning period to reduce the total weight and power .

required. A single crystal clock accurate to one part in 10 6 , will also be

necessary. However., it will weigh only about 1 pound including circuitry and

will require neglible power.

The analysis of each phase of the guidance sequence given in Section2.2

establishes the probable accuracy and the weight and power requirements. It

is found that 370 pounds of airborne guidance in the last stage could localize the'

vehicle above any point on the equator accurately enough for all of the applications indicated in Volume U. The average power consumption during ascent is

about 450 watts although this requirement varies considerably throughout the

guidance phases.

Analysis of the. accuracy of the system shows that the suggested combinaticin of inertial andradio measurements at appropriate points on the"ascent

trajectory stabilizes for a period of 5 years or more for both the position and

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Page III-68

Table 2-3. Airborne Guidance Components - Final Stage Weight

and Power Requirements.

Component Weight

(pounds)

AC Powerb DC Power

400 cps 3+ 20 volt

(watts) (Watts)

Gyro- stabilized inertial

platform and electronics

Steering and Shutoff

computer (digital)

Perigee - Apogee

autopilota

Tracking beacon with

data link and decorder

Vernier accelerationsensing systemc

Cabling

235 350 50

20 20 20

15 25 • 10

50 50

15 20

25 (per stage)

Total 360 395 150

a Only the weight and power of the measuring instruments and required computing functions are included. Hydraulic power requirements are not included.

These are included in propulsion weights.

b DC inverters must be supplied at additional weight cost to provide alternating

current, since only direct current is supplied by power supply.

c Makes use of the same computer as Item 5. A set of body-mounted accelerometers are used rather than the platform accelerometers.

Min

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TR-59-0000-00604

Page III-70

altitude of the vehicle in the high orbit. The vernier velocity impulses required

to stabilize the orbit are probably about 100 feet per second total.

The guidance system used to establish polar satellites varies slightly

from that for the equatorial, since only broad-beam coverage is required, in

contradistinction to the antenna-bearing problem associated with point-to-point

relaying for equatorial satellites. In essence, this means that the polar satellite

need not be stabilized in yaw after the initial' velocity corrections are made.

TM, Pocoowel Pontotoc infocinollos *Nadine the nolloce1 &foam of IM United Votes wbbio Iba wooeing of gm IseImmo tows, Mb

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TR-59-0000-00604

Page III-71

CHAPTER 3

ATTITUDE CONTROL.

3.1 INTRODUCTION

This chapter is devoted to the problems of establishing and maintaining

the final stage vehicle attitude both before and after the 24-hour orbit has been

attained. The recommended method of attitude control prior to attaining the

24-hour orbit is to use the inertial platform to control a "coarse" gas jet

system. Each jet should produce about 5 pounds of thrust for a Thor-boosted

vehicle or 10 pounds for a Titan- or Atlas-boosted vehicle and when paired

should produce a torque couple of about 25 or 50 foot pounds (assuming Z jets

each, .2-1/2 feet froth the center of rotational inertial). This "coarse" gas jet

system is used to reorient the vehicle during low-altitude coasting, ascent

coasting, and to point the nose of the vehicle toward the earth after apogee

burning. The same gas jet system, with the pairs oriented for adding thrust

rather than for producing a couple, are used to add the small velocity increments needed for orbital correction (Sections 2.1.6 and 2.2.6).

Fine attitude control during low-altitude coasting and ascent coasting is

achieved by again using the inertial platform to control a second set of gas,

jets, each jet producing only 0.005 or 0.01 pound of thrust and when paired

producing torque couples of 0.025 or 0.05 foot pound. The same fine jet

system is used to accurately establish attitude and reduce angular rates after

the 24-hour orbit has been attained.

In the 24-hour orbit a 10- poundflywheel with its axis perpendicular to the

plane of the orbit "pins" the vehicle, stabilizing it against rotations in roll and

yaw. 

* The angular rate of the large wheel is held constant by means of an accurate

electronic (for example, crystal controlled) clock. Although weak pitch stability

might result from the combined effects of gravity and centrifugal force tending to

align the long axis of the vehicle along the vertical; at the altitude of the

24-hour orbit this pitch stability is so weak that even a very small torque such

To represent change in the angle between , the long axis and the local

vertical measured in the plane of the orbit. Roll represents rotation about

the nominal velocity (horizontal). Yaw is rotation about the vertical.

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TR-59-0000-00604

.Page III-72

as might result from a meteor impact or from a slight misal - t of the

large flywheel would upset the balance. Consequently, a small variable speed

wheel with axis perpendicular to the plane of the motion (parallel) to the axis of

the large flywheel) is used to "absorb" small pitch angular momenta. If a

number of meteors should have a cumulative effect causing the small wheel to •

exceed its design speed, then its motion can be stopped as the pitch gas jets

are used to hold attitude; this would leave the pitch wheel free to absorb future

disturbances in pitch. If the assumptions of'AppendixF, Volume IV, on meteor

frequency as a function of size are verified byfuture explorations in space, then

it is quite likely that the small pitch wheel alone will be able to compensate for

meteor impact and the fine gas jets need be used only during the orbital correction period (24 to 48 hours) when relatively large accidental torques may result

from imbalance of the coarse jets used for producing the orbital corrections.

The purpose of attitude control for the communications satellite is to

insure that the communications antennas remain pointed at the ground stations.

For the "stationary" equatorial satellite two small monopulse receiving antennas

with a relatively broad receiving pattern can be used as "finders" for the rela-.

tively narrow beam communications antennas. Error signals derived from the

monopulse antenna system can be used to control the fine gas jet system

and the pitch wheel. For the polar satellite it is only necessary to keep one

16-degree antenna pointed at the earth; there is no requirement for yaw

control. (Pitch and roll measurements of the polar satellite can be made

by an infraredhorizon scanner; these measurements are used to control the

fine gas jet system and the variable speed pitch wheel. The attitude error

signals from either the monopulse system (equatorial) or the horizon scanner

(polar)areprocessed in analogue circuits (the vernier attitude control.

computer) to derive control signals.

3.2 ATTITUDE CONTROL PRIOR TO ATTAINING THE 24-HOUR ORBIT

3.2.1 Requirements

Attitude control of the booster and first added stage during the climb to the

low-altitude orbit is similar to that for an ICBM and will not be treated here.

*

•

* Attitude control during Hohmann accelerations will be regulated either by

using two engines or by using the coarse gas jets. If the latter is used, a

small additional amount of gas will have to be supplied.

-GONFIDENRAt

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U.S.C., Section 793 sod 794, the Inmsoistiott or ortolotion of which in any wormer to so onoolltorizoti wittsoo Is wohlhlearl hit low.
TR-59-0000-00604

Page 111-73

This discussion is limited to the control of the final stage vehicle during coast

periods.

The attitude-control system during the low-altitude coasting phase is

required to:

Prevent the gyro-stabilized inertial platform from going into

gimbal lock, and

Orient the vehicle so that the vehicle is aligned approximately

in the direction of perigee thrusting.

There is no need to align the vehicle so as to direct the beacon transponder

antenna toward the ground since radio tracking will not commence until the

ascent coasting phase. Moreover, since it is recommended in Chapter 1,

Volume IV, that solar surfaces not be deployed until the 24-hour orbit is

reached, solar surfaces will not affect attitude requirement prior to establishing the 244tour orbit.

The vehicle pitch altitude during the low-altitude coast is shown in

Figure 3-1. Figures 3-la and 3-lb show the low-altitude coast of equatorial

satellites launched from Cape Canaveral. These figures show that approximitely a 90-degree attitude reorientation is necessary to correctly align the

vehicle for perigee burning*

whether perigee burning takes place on a southerly

or northerly crossing of the equator. Figure 3-lc shows that less than

20 degrees of attitude reorientation is required when a polar satellite is

launched to the south from Cooke Air Force Base and enters perigee burning

at 14.86°N (see Chapter 1, Section 1.4).

During ascent coasting, the attitude control system is required to

Prevent the gyro-stabilized inertial platform from going into

gimbal lock,

Orient the vehicle so that the beacon transponder's antenna patterrt

illuminates the ground tracker during the final hours of ascent,

(3) Finally, orient the vehicle approximately in the direction of apogee

thrusting.

A net velocity impulse of 85 ft/sec (see Chapter 1, Figure 1-6) may be

saved by reducing the inclination of the orbital plane by 2.2 degrees at

perigee burning. This requires orienting the vehicle about 9 degrees

out of its orbital plane (toward the equatorial plane).

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TR.-59-0000-00604

Page 111-74

Figure 3,1a. Eastward Launch from Cape Canaveral

Perigee Burning on Southerly. Crossing

of Equator.

Figure 3-1b. Eastward Launch from Cape Canaveral

Perigee Burning on Northerly Crossing

of Equator.

Figure 3-1c. Northward Launch, from Cooke Air Force Base

(34-3/4 N) Perigee' Burning at 14-3/4 N.

--CONFIR7NTIAL

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TR-59-0000-00604

Page 111-75

CONFIDENTIAL

As seen in Figure 3-2, in which it is assumed that perigee burning takes

place on a southward crossing of the equator, the vehicle is oriented at perigee

parallel to the orbital plane, some 28.5 degrees from the equatorial plane. (If

the inclination of the orbital plane is reduced by 2. 2 degrees at perigee burning

to save a net of 85 feet per second (see Chapter 1, Figure 1-6), then the vehicle

attitude during perigee burning is about 9 degrees closer to the equatorial

plane..) During apogee burning the thrust must have a southward component to

arrest the northward motion; the vehicle attitude must be about 24.5 degrees

from the equatorial plane (somewhat less if the plane hes been oriented by

2.2 degrees). Altogether a reorientation of 127° (180°-28.5°-24.5° = 127°)

is required. The figure shows this reorientation taking place just after perigee

burning, but clearly it could take place any time during ascent coasting, so far

as meeting the requirement for apogee burning is concerned.

Since the satellite must be tracked during the first 24 or 48 hours after

the 44-hour orbit is reached and since the nose must point toward the earth

during this time (to direct the communications antennas and monopulse attitude

toward the ground stations), the transponder beacon must also have an antenna

which is directed toward the earth from the nose. As seen from Figure 3-2,

if the reorientation were to take place as shown, the nose would be pointed

generally toward the south and nearly parallel to the surface of • the earth during

the last hours of ascent. Consequently, if the satellite is to be tracked during

the final hours of ascent coasting, either the transponder antenna must rotate

with respect to the vehicle frame, or 'the transponder must have a second

antenna on the side of the vehicle, or the vehicle must be reoriented twice.

If the last alternative is chosen, then as shown in Figure 3,3, the

first reorientation through an angle sornewhatrnore than 90 degrees must take

place after perigee burning. This would direct the single transponder.

antenna toward the ground during the last two or three hours of coasting. The

precise orientation should be chosen so as to maximize the tracking time from

a given tracking station. Just before. (possibly one minute or so) the satellite

arrives at the correct longitude (for equatorial satellites) or latitude(for polar

satellites) the. apogee burning commands are transmitted to the, satellite via

the data link and decoder, which are pant of the transponder beacon equipment. This command would actuate the final reorientation through a little

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TR-59-0000-00604

Page 111-78

less than 90 degrees aligning the vehicle along the direction of thrust required

for apogee burning. Then at a certain fixed interval after the instant in which

the command was transmitted, apogee burning is initiated. If a rotatable

antenna or multiple antennas are used, only a single reorientation is required

during ascent coasting. Moreover, there need be no delay to act upon the

command to initiate apogee burning.

After apogee burning a final reorientation through 90 degrees is required

to align the vehicle along the vertical. What then becomes yaw,.i.e., rotation

about the long axis, must then be adjusted to point the antennas at the appropriate

ground stations.

Although the attitude must be controlled during low-altitude and ascent

coasting, there is no need to hold the programmed. attitude with high precision.

Certainly, avoiding gimbal lock imposes no precise attitude control, and

providing transponder beacon coverage imposes an accuracy, requirement of

only a small fraction of the antenna beamwidth. Assuming an antenna beamwidth of 20 or 30 degrees, a reasonable attitude tolerance would be 2 or 3

degrees. Obviously, the orientation of the vehicle at perigee and apogee

burning requires greater accuracy than avoiding gimbal lock and providing

antenna coverage, but even for this case extreme accuracy is not necessary

since residual angular errors can be corrected by the :servoloop controlling

the direction of thrust of the engines during the burnings.

However, at the end of each burning period (or weight jettisoning), the

attitude control system must first arrest residual accidental angular rates

imparted to the vehicle before carrying out anyprogrammed reorientation.

The servoloop used to steer the rocket engines during burning periods has

an appreciable time lag. As a consequence, when the thrust of the rocket

engine is terminated, a small random, residual angular momentum is imparted

to the vehicle.

Based on the estimated characteristics of the rocket engine and the

associated servocontrol loops in the final stage vehicle, it is estimated that

the residual angular rate imparted to the vehicle at rocket engine cutoff will

not exceed a value of 3 degrees per second.

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TR-59-0000-00604

Page 111-79

3.2.2 Mechanization -CORDENTik

Figure 3-4 is a schematic of the "coarse" gas jets whic can be used to

control attitude and to make orbital corrections. Although a number of configurations are possible, the one which is illustrated is a simple configuration

requiring only four gas jets, each of which has single axes of rotation. In

this configuration, no jets are required at the nose or tail of the vehicle and

thereby interference with antennas or the main. engine (or engines) is avoided.

The top three sketches in Figure 3-4 show the jets oriented for produc

a net thrust (but, in principle, no net torque) as required for adding

increment during the orbital correction phase. It will be noted that, whereas

only the top and bottom nozzles can be used for producing east-west thrusting

and only the side nozzles for producing north-south thrusting, either or both

pairs of nozzles may be used for radial thrust. If the required thrust direction

is midway between radial and east-west, then by rotating the top and bottom

•nozzles to an intermediate angle they can be made to produce the required

thrust in one operation. This saves gas When compared with a fixed jet system

which would require thrusting by two sets of jets, one in the radial direction

and the other in the east- west direction. Similarly, if the required thrust is

midway between radial and north-south it can be produced by side jets at an

intermediate angle. However, if the required thrust has both an east-west

and a north-south component then both jets must be used.

During the orbital correction phase a Thor-boosted vehicle weighs about

900 pounds and a Titan- or Atlas.-boosted vehicle weighs about 1900 pounds

(see Volume IV, Section 2.4). A 10-pound thrust (five from each jet) tor a

Thor-boosted vehicle or a 20-pound thrust for a Titan- or Atlas-boosted

vehicle will give accelerations of about 0.01 g's. With such an acceleration,

accelerometers •currently under development can measure the velocity increments (totaling approximately 100 feet per second) to about 0.1 foot per second.

Moreover,. assuming the gas jets can be cut off in 0.05 seconds, the velocity

cutoff error is only 0.016 foot per second.

Assuming the gas jets have an exit velocity of 2000 feet per second

(corresponding to a vacuum specific impulse of 62 pounds-second per pound)

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TR-59-0000-00604

Page 111-81

CONFIDENTIAL

and that the sum of the velocity component increments is 100 feet per second,

the mass of gas expended in orbital corrections must be 5 per cent of the

vehicle mass, i.e., about 45 pounds for the Thor-boosted vehicle and about

95 pounds for the Titan- or Atlas-boosted vehicle.

The bottom three sketches in Figure 3-4 show the same "coarse" gas

jets when oriented to produce torque. The vehicle is shown in the orientation

it would have in the 24-hour orbit. Again by exploiting the single axis of rotation of the jets, torques about axes intermediate between the yaw axis (yaw in

the 24-hour orbit) and either the roll or pitch axis can be achieved in a single

operation using only one pair of jets. However, combinations of roll and pitch

require the use of both pairs of jets.

Because the diameter of the vehicle is about 4 feet (see Volume IV,

Section 2.4) it may be assumed that the jets are each about 2-1/2 feet from

the center line of the vehicle. Thus a pair of jets on the Thor-boosted vehicle

would produce about 25 foot pounds of torque (2 jets x 2-1/2 feet x 5 pounds)

or about 50 foot pou.nds (2 jets x 2-1/2 feet x 10 pounds) on the Titan- or Atlasboosted vehicle.

At the time of this writing the final stage vehicle layout is not sufficiently

solidified to compute exact values for the moments of inertia, but even very

approximate values will suffice to illustrate the general behavior of the

vehicle when acted upon by the torques of the gas jet system. Table 3-1 gives

illustrative numbers for a nominal 90 degree rotation about an axis perpendicular to the long axis of the final stage of a Thor-boosted vehicle. The

rotational characteristics are shown for the vehicle in the low-altitude orbit

when it is completely loaded with fuel, in ascent coasting when the fuel is

partly used up, and in the 24-hour orbit with only payload and structure. The

assumed scheme of reorientation is to use the gas jets until an angular rate

of 4 degrees per second is attained, then "coast" in rotation angle and finally

apply the opposite torque impulse to arrest the angular rate. It will be noted

that the total rotation time (including angular acceleration, coast and deceleration) is in all cases less than a half minute and is only slightly dependent

on the assumed value for the moment of inertia.

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Although the table applies to a Thor-boosted vehicle, the values for

angles, times, angular rates and angular accelerations would be very similar

for the final stage of a Titan- or Atlas-boosted vehicle since both the moments

of inertia and the torques would be approximately doubled.

The total weight of gas used in reorientations may be estimated from

the angular acceleration (and deceleration) times given in. Table 3-1. During

the low-altitude coast the reorientation to align the vehicle for perigee burning

requires about 8.8 seconds of torqueing (4.4 for angular acceleration and 4.4

for deceleration).. Assuming that two approximately 90-degree reorientations

are required during ascent coasting (one to direct the transponder beacon antenna

toward the ground and the second to align the vehicle for apogee burning), an

additional 8.8 secondti\of torqueing is required. To align the long axis of the

vehicle along the vertical just after apogee burning requires another 2.8 seconds

of torqueing. Altogether some 20.4 seconds of torqueing are required.

In the above computation we have neglected the initial torqueing time

required to arrest the accidental angular rates imparted to the vehicle at the

end of each burning period. As stated at the end of Section 3. Z. 1, tlies.e rates

will not exceed 3 degrees per second. Consequently, arresting this angular

rate can increase the torqueing period by 3.3 seconds during the low-altitude

coast, 1.7. seconds during ascent coasting, and 1.0 seconds after reaching the

24-hour orbit. (The change in angle during these periods will have only a slight

effect on the total time to perform the subsequent programmed reorientations.)

When these additional 6. Q seconds, are added to the 20.4 seconds required for

the programmed turns, the total torqueing time is 26.4 seconds.

Since 5 pounds of thrust has been assumed for each nozzle and assuming

that only two nozzles are used in. the rotation, the total impulse is 264 pound

seconds. This requires about 4.3 pounds of gas according to our prior assumption that the specific impulse of the gas jet system is 62 seconds. When the

axis of rotation is such that the four nozzles must be used equally, then this

figure must be increased bgV-r, i.e., about 6.0 pounds of gas are required.

On the assumption that the gas jets can be cut off accurately to within

0.05 second, the last column of Table 3-1 gives the residual angular rates.

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TR-59-0000-00604

Page 111-84

It is not difficult to measure angular rates more accurately t an the numbers

listed, either by using rate gyros (e.g. those in the perigee-apogee autopilot)

or by appropriately differentiating and smoothing the gimbal angles of the

inertial platform.

The most severe requirement for long-term attitude control occurs

during the last 2 or 3 hours of ascent, because the transponder's antenna

coverage requires holding attitude to 2 or 3 degrees. The residual angular

rate of 0.09 degree per second would result in the attitude changing by

3 degrees in only 33 seconds. Consequently, unless throttling is used, a

short impulsive torque would be required every half , minute to keep the attitude within bounds Assuming that each impulse lasts 0.1 second which is

just adequate to reverse the angular rate, the four jets would expend about

8 pounds of gas over the 3-hour period.

However, an alternative is to employ the fine gas jet system which is

necessary in any event to meet the very strict attitude control requirements

of the 24-hour orbit. The fine gas jets can take on a similar configuration to

that shown for the coarse gas jets in Figure 3-4, i.e., there can be four jets

each free to rotate about a single axis of rotation.. Analternative configuration

would use 12 fixed jets mounted in approximately the same positions as the jets

of Figure 3-4, consisting of six pairs, one pair for positive pitch, one for negative pitch, etc. The fine gas jets would each produce only one thousandth the

thrust of the coarse jets, i.e., each would produce 0.005 pound of thrust for

the Thor-boosted vehicle, or 0.010 pound for a Titan- or Atlas-boosted vehicle.

The corresponding torques would be 0.025 foot pound, or 0.050 foot pound.

The fine gas jets would be controlled by the steering and shutoff digital

computer, operating on gimbal angle measurements from the gimbal angle

resolves of the inertial platform. Each gimbal angle will provide information

for controlling a single torque component.

Table 3-2 summarizes the characteristics of the fine gas jet system

in arresting the residual angular rates left by the cutoff error in the coarse

gas jet system. Fifty seconds are required to arrest these residual rates.

During the arresting period an angular change of up to 3.6 degrees (after the

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TR-59-0000-00604

Page 111-86

reorientation in the 24-hour orbit) can occur. As the residual rate is reduced

it can be measured by sequential observations of the gimbal angles of the gyrostabilized platform (monopulse measurements may be used instead of gimbal

angles in the 24-hour orbit, see Section 3.3). The slope of the resolver error.

in measuring these angles is likely to be the limiting factor affecting angular

rate accuracy. Constant errors will not affect rate measurements. Assuming

uncorrelated resolver errors ae high as 0.05 degree, and assuming that an

average angular rate is measured by noting the change in angle each 20 seconds,

one should be able to determine the average angular rate overthe past 20 seconds

to 0.0035 degree per second. This average rate is in effect the angular rate

which existed some 10 seconds (the delay due to smoothing) earlier. To find

the current angular rate it is necessary to add on the change whichhas occurred

during the last 10 seconds. This can be estimated by knowing the calibrated

torque of the fine gas jets. For the case of stabilizing after the satellite is in

the 24-hour orbit, the angular acceleration (see Table 3-2) will be about

0.003 degree per second2

. In 10 seconds this will change the angular rate by

about 0.03 degree per second. Assuming that the calibration is good to 10 per

cent, the change in angular rate can be predicted to about 0.003 degree per

second. The rms current angular rate error due to the combination of resolver

error and jet calibration error is about 0.005 degree per second. Fine gas jet

cutoff errors are negligible when compared to this figure.

When the residual rate, as measured by the above method, is cancelled

out an additional 10 seconds of torqueing is applied to give a nominal angular

rate of 0.03 degree per second (for the case of the 24-hour orbit) in the direction to cancel the angular error which has accrued. About two minutes of

angular coasting are then required to correct out the anaular error of 3.6 degrees

(see Table 3-2). During this angular coast period the constant angular rate can

be measured to about 0.0006 degree per second (again assuming resolver errors

of 0.05 degree). Consequently, as the correct attitude is established, the

required change in angular velocity is known more accurately than it can be

controlled at this time. In braking the approximately 0.03 degree per second

angular rate, the calibration error of 10 per cent will lead to the limiting rate

error of 0.003 degree per second. In 250 seconds more an Angular error..of.

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TR-59-0000-00604

Page 111-87

0.75 degree may have developed, but the angular rate can be measured to

0.0003 degree per second and with a similar error introduced by jet calibration, the new angular rate canbe controlled to about 0.0005 degree per second.

To correct this angular position an angular rate of about 0.0010 degree per

second (controlled to 0.0005 degree per second) can be used to eliminate the

angular error in 750 seconds. This constant angular rate can be measured

to about 0.0001 degree per second and the 10 per cent calibration error in

braking the 0.0010 degree per second is also about 0.0001 degree per second.

However, assuming a jet cutoff uncertainty of 0.05 second, the cutoff 'error

would be 0.00015 and is already the dominant error. The total angular rate

error would have an rms value of 0.0002 degree per second or about

0.7 degree per hour.

The above detailed analysis for the satellite in the 24-hour orbit shows

that attitude can be established to within 0.05 degree of the correct angle as

read from the stable platform (which can have drifted 0,3 degree by this time)

and with a null angular rate to within about 0.7 degree per hour in a matter of

at most 20 minutes -- 30 seconds of coarse jet reorientation, 50 seconds for

five jets to kill coarse jet residual rate, 120 seconds to eliminate accrued

angular error due to coarse jet residual rate, 250 seconds drift away to

measure angular rate, 750 seconds to return and measure angular rate with

accuracy better than fine jet cutoff error.

Of course ,the se numbers are only meant to illustrate the rapid convergence

of the attitude control procedure, even when conservative assumptions are made

concerning angular measurements, jet calibration, and jet cutoff uncertainty.

The 24-hour orbit stabilization problem was chosen for the above numerical

example because the angles and angular rates to be corrected by the fine gas

jet system were greater'than those encountered during low-altitude coasting

or during ascent coasting.. Since the angular accelerations during ascent

coasting are only 60 per cent of those of the vehicle when in the 24-hour orbit,

the cutoff errors will be somewhat less and after no more than 20 minutes of

attitude stabilization the vehicle will probably drift less than 1.5 degrees during

the 3 hours or so in' which the transponder must illuminate the ground tracker.

Therefore, no further attitude corrections are needed.

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TR-59-0000-00604

Page III-88

1..

The total duration of fine gas jet torqueing for a typical eorientation

will not exceed 100 seconds. With two jets each having a thrustof 0. 005 pound,

this only requires 1 pound second of impulse. Assuming this impulse is

required for each of the three axes of rotation and multiplying by four, the

total number of 12 pound seconds is required for the five gas jet system.

Only 0.2 pound of gas are thus consumed.

Figure 3-5 illustrates a possible arrangement for generating gas for

both the coarse and fine jet systems. A liquid monopropellant is stored in a

flexible sack which is sealed to the outlet of the tank. Since the sack is kept

under a constant pressure from a high-pressure gas storage tank, only the

liquid in the sack (and not the pressurizing gas) can escape from the tank,

even in free fall. In decomposition chambers, the monopropellant is catalytically decomposed into its gaseous components and stored in gas storage

tanks. The monopropellant will be fed to the decomposition chamber only

when the computer commands that pressure be maintained and the sensors

on the gas storage tanks also indicate that the pressure is low.

3.3 ATTITUDE CONTROL IN 24-HOUR ORBIT

3. 3. I Requirements

Attitude control requirements, once the vehicle is in the 24-hour orbit,

are set by the beamwidth of the airborne antenna. Of course, this assumes

the vehicle is in the correct orbit. If the eccentricity of the orbit is such that

the difference between perigee and apogee is as large as 250 miles, and if the

attitude of the vehicle remains constant, maximum cyclical loss of coverage

on the ground at the equator would be about 220 miles, and the lass would be,

of course, substantially larger at higher latitudes where, in fact, the actual

coverage is. Moreover, to attempt to continually compensate for the change

in speed of the vehicle during its elliptical orbit by a constant change of attitude in the pitch plane would add considerably to complexity and increase the

power required. Again, if the orbit of the vehicle is inclined to the nominal

24-hour orbit in the equatorial plane by, say, 250 miles, the change in northsouth coverage would be 250 miles. In addition, constantly changing attitude

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TR-59-0000-00604

Page 111-90

COPMENTIAL

to compensate for this out-of-plane error would not only make for additional

complexity, etc, but it would also require that the stabilizing force of the large

flywheel would be constatnly overcome. Therefore, it is important that the

final 24-hour orbit be established as accurately as possible.

As has been shown in Chapter 2, Section 2.6 it is possible to reduce

both eccentricity to 10 -3 and inclination to about 0.1 of a degree, which would

give a daily change in coverage in north-south and east-west directions of

about 50 miles. Such a loss is quite small and can be easily compensated for

by adjusting the airborne • beamwidth to fit the.location of the. particular ground

stations covered by a given antenna..

The attitude should be established initially so accurately that the only

possible change will be in the pitch plane, which is necessaryeince the vehicle

must rotate in the pitch plane one revolution per day. If attitude deviation

angles in roll and yaw are less than ..2 x 10 3

 and the angular rates less than

-

.10 4

 radians per second which the large flywheel must provide, no further

corrections in roll and yaw will be required.

3.3.2 . Stabilizing . Machinery.

The discussion in Sections 3.1 and . 3.2 of this chapter show that the. fine

gas jet system using the platform as a reference and the steering and cutoff

digital computer can supply the appropriate torques to achieve the required

stability.

The system used to provide a reference for maintaining attitude; control

in the 24-hour orbit is illustrated in Figure 3-6. Theanalytia stUdy, Appendix C,

of rigid body motion of a vehicle' at 22,000 miles above the Surface of the earth

shows that it is desirable to accomplish the attitude control in two.periods. In

the first period, immediately following apogee burnout and after the vehicle has

been turned to the generally correct attitude from information given by the platform, the nonlinear terms in the equation of motion are quite important because

of the initial conditions assumed, that is,: an angular error (9) of 2 degrees and

an angular rate error (A) of 5 x 10 -2 radians per second, which are those estimated for ICBM nose cones. Fine gas jets can reduce the angular error to one

rnilliradian and the angular rate error to 10-5 radian per second within minutes.

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TR-59-0000-00604

Page 111-92

CONFIDENTIAL

When the gas jets are turned off, the yaw and roll motions will remain

small (A .4 6 mils and 6 ",10 4

 radians per second) if an internal angular

momentum of 80 slug feet2 radians per second, fixed relative to the body, is

oriented along the pitch axis. But as shown in Appendix C, components of

angular momentum along other axes produce constant driving forces which

tend to disturb yaw and roll stabilization. Therefore, it is normal to the

orbital plane, since this supplies both yaw and roll stabilization. Pitch motion

about the reference vertical is not stabilized and intermittent pitch control-is

required for the life of the vehicle. Since only very small forces are required

for this long-term correction, a small variable speed motor rotating in the

pitch plane can supply all the necessary correcting torques in pitch. However,

if the yaw and roll rates could be reduced below 10 -5 radians per second and

the pitch motion error to 10 -7 radians per second, the very weak torque exercised by the gravitational gradient about the pitch axis may be exploited. In

this event, no control forces will be needed thereafter. However, it appears

that errors in the alignment of the large flywheel or external forces, such as

meteorites will make this 'impossible, and additional forces can be supplied

by the fine gas jets and the small flywheel.

3.3.2.1 Radio Monopulse Attitude-Sensing System for Equatorial Satellites 

Once the vehicle is stabilized using the platform and fine gas jets with

respect to a fixed set of points on the. earth, it is proposed that the vehicle attitude be maintained using a radio direction-finding technique as a reference.

Two ground transmitters will be observed by monopulse receivers in the vehicle

and thereby determine the angle between the antenna axes and the line-of-sight

to the transmitter. These error signals are transformed . in the attitude stabilization computer to give true vehicle orientation about each of its three axes - •

roll, pitch, and yaw. The ground stations would be located as far apart as

possible, since .the yaw measurement (about the reference vertical) is quite

sensitive to the projection difference angle. These ground stations could

readily be combined with the communication transmitters. The airborne

antennas would be fixed on the. vehicle such that they would find the communi

cation ground transmitters (see Figures 3-7 and 3-8) after reorientation in the

final orbit.

•

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TR-59-0000-00604

Page 111-93

CONFIDENTIAL

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TR -59 -0000-00604

Page 111-94

The rigid- body equations of motion for the vehicle are given in Appendix C,

Equation 10. It is necessary to supply both a restoring and damping term for

each angle. It is proposed that the restoring term be computed from the (resolved)

angle measurements of the monopulse channels directly. The angular rate terms

for damping could be obtained by differentiating the angular measurements.

A block diagram of the attitude-sensing arrangement for an antenna in

one plane is shown in Figure 9. The signal at the point e is proportional

to the error in tracking in one plane. The control system in one plane can be

represented approximately as:

7 = torque • = -g1 e - P ; • (3.1)

where I is the moment of inertia of the satellite body, g. 1 is a gain factor, and

p is the magnitude of the damping term. E is the angular position of the tracking antenna, taking zero as the direction to the ground station. The rate term

Is necessary for stability against noise-induced drifts. (See Figure 3-10',) The

slope of the error output as a function of tracking error can be written as S/B,

where S is the amplitude of the signal in * the sum channel at the entrance to the

phase detector and B is the beamwidth of the attitude-control antenna. Equation (2. 20) then becomes

I + p + ; E " n

where n is the noise term at the output of the phase detector.

Propagation irregularities in the troposphere impose a wrinkling

effect on the wavefront received at the vehicle. This produces a zero baseline

tracking error of only 0.1 milliradian, according to the Maui propagation experiments, and shot noise in the receiver is probably the limitation on a system.

which seeks only 1 mil accuracy. There is no stiction noise in the antenna,

since the antennas themselves do not track. On the assumption that the

signal-to-noise ratio in the sum channel is high, the noise has a power spectral density of 2k T F (S2

/5. 2

), where k is Boltzmann ts constant, T is

temperature in degrees Kelvin, F is the receiver noise figure, and Si is

the input signal amplitude in the sum channel. The factor 2 accounts for

111111111it.

This decennial contains intermellen eliectine the nation") debtor al IA. Untied Metes within the ntetenliti el the ftwiennrt low, TUN
TR-59-0000-00604

Page 111-95

05,

s tea

td

Ps U3

bo,d

.s

Ct•

o

Q

0 0

This deataient centoint lidennatien affecting the MON01110 defense al the United States within the meetting el the Esoletansa Laws. Tight

U.S.C., Section 793 end 794. Me hansfalsslea or tevelatien el which In say mamma Is an mouthed:6d person Is prohibited by law.
TR -59-0000-00604

111111113-GO-W-EM Page 111-96

the frequency folding effect of the phase detector in translating the signal at

the IF frequency to zero frequency, and the term.. S ./S i

2

 gives the effect of

the average gain control.

Now, if one assumes that the servosystem is critically darns. , /4nd

writes p for its bandwidth, the mean-square error due to noise perturbation

can be written approximately as

—2

= 2k T F P B2

Si

(2.21)

r

where r is the power signal-to-noise ratio at the antenna input calculation on

the basis of a bandwidth equal to the servobandwidth p. One can calculate what

signal-to-noise ratio and beamwidth are required to obtain a given attitude accuracy from this equation. Taking a 1-foot diameter antenna in the satellite at

X-band and using 2-foot diameter antennae on the ground with 1 0 watts radiated

and a noise figure for the receiver of 10, it is found that at a distance of

22,752 nautical miles the signal-to-noise ratio is approximately 10 6

/4. If one

assumes further that 1-mil tracking is desired, P must be ig 35 cycles. The

equipment to perform the above functions would consist of receivers and would

be similar to that diagrammed in Figure 3-9, but with the addition of two error

channels, one for right-left and one for up-down, for each antenna. The antenna

itself could be a dimple in the .side of the satellite body and thus would weigh

essentially nothing... The receivers themselves would probably weigh about

20 pounds and would use 75-watt dc power.

These elementary calculations show that it should be quite possible

to design a tracking.system for the attitude control which will hold the attitude

to within one milliradian.. Since the complete tracking system is a three-

,

diMensional problem, it is felt that a complete analysis of this system will

require a numerical simulation program.

3.3.2.2. Attitude Control for Polar Satellites

The attitude control systeni proposed for polar satellites is identical

to those used for establishing equatorial satellites up through apogee burning,

(i.e., using the platform as reference). The attitude corrections which are

•

r.

This dosoossof ambles loforsiolloo effecting Ms owsif000f Moose of IM UsINd Saks within tho owsonifig of Um !,plow. LOWS, TM,
"- TR-59-0000-00604

Page 111-97

required .to stabilize the vehicle in a synchronized orbit cannot be accomplished

with the monopulse attitude control system, since the vehicle is continually

changing positions with respect to any pair of ground stations. On the other

hand, for polar orbits only a single communication beam is required to cover

the entire earth with no yaw stabilization above the local vertical through..the

earth's center.

Any attitude sensing system which gives only pitch and roll stability can

be provided by sensing radiation from the earth's surface. Infrared radiation

is given off by the heated earth in sufficient quantities to make an infrared

horizon scanner practical for a 24-hour orbital radius. A preliminary system

study*

 of such a, device shows that the bearing angle to the earth's center can

be considered accurate to approximately one mil for an airborne weight of

7 pounds and .30 watts of dc power ** . This simple device could Lproiride sufficiently accurate data to the computer and roll sets to stabilize the communication

•

3. 3. 3. 3 Fine Constant Stabilization

As discussed in Appendix C, the angular rate of the vehicle in pitch

must be 0.7292116 x 10-4 radians per second if it is to rotate once in .a sidereal.

day. Furthermore, the actual rate must be adjusted .to have an error less than

10 7 radians per second, if the attitude is to freely drift about the equilibrium

position without exceeding the required tolerances. Although the fine gab jets

can produce, angular rates haVing the required accuracy, the cross coupling

terms and minor disturbing torques become important, as shown in Appendix C,

when such fine control is required. These difficulties can, in large part, be

avoided if a stabilizing flywheel pictured in Figure 3-11 is used. The large

flywheel is operated at constant angular momentum. The large constantspeed flywheel normally operates at about 10,000 rpm, has an angular

"Proposal for Development of a Horizon Scanner," RFQ LMSD/36009,

18 October 1957, The Ramo-Wooldridge Corporation.

**

Since the monopulse sybtem will not be used on polar orbits, 17 pounds,

will be gained which can be used to offset the additional power required

for the infrared scanner.

ml. dosomoof confolos Inforosoffoo effecting the nollonol dohoso of IM Wiled Stoles within IM moaning of MO Esol000“ LOW/. Tills

.aerials above.the local vertical adequately.
TR-59 4000 -UUO04

Page 111-98

04J

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v

a

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47:

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This doionsent Infeemetion effecting the notional &tense of the United Steles within the meaning of the belon g/se tows. Tits

IS, U.S.C., Section 793 and 794. the transmission or revelation of which In any stamen to an esouthetised muses Is arebibited by low.
TR-59-0000-00604

Page 111-99

momentum of more than 80 slug feet Z radians per second, and will weigh about

10 pounds. During the ascent trajectory, the flywheel is caged. At the end of

apogee burning, the wheel is brought up to speed by means of a small gas jet.

(Less than one-tenth pound of propellant is required for this purpose. The

angular momentum incurred will be compensated by appropriate use of the

coarse gas jet system.) Once in operation, the angular momentum of this

flywheel is maintained to better than one part in 10 5

by means of temperaturecompensated crystal-controlled electric power furnished to the driving

synchronous motor. The angular momentum of this wheel is so large that

the vehicle may be considered to be "pinned" in roll and yaw by the flywheel

axis; the vehicle is free only in pitch. This flywheel could, in fact, be used

as the inverter.

Fine adjustments in angular speed in pitch may be obtained by varying

the speed of the small control motor and flywheel shown in Figure 3-11. This

. small flywheel has a maximum angular momentum of only 5 x10 -3slug feet2.

Together with its actuating motor, it will weigh less than one pound and will

be about 3 inches in diameter. The controls are so arranged that whenever

the angular momentum of the control flywheel reaches the maximum design

value, corrective torques are applied by the fine gas jets as •shown in Figure 3-4.

The control flywheel is then brought to rest and is thus again free to compensate for any small angular momentum which may develop in the vehicle. For

example, the residual angular rate from the fine gas jet system is 0.7 degree

per hour. To absorb this, the wheel wouldhave to be turned at 0.7 x 10 6 degrees

per -hour, or 33 rpm:

Using information from Appendix F, Volume IV, -Table 3-3, shows that

any velocity changes due to meteoric impact are negligible. For the largest

meteor considered likely to strike the vehicle (and then only once in ten years) the

momentum transfer to the vehicle is 1.69 x 10 -4 slug feet per second which

can give at most a velocity change of 5.5 x 1 0

-6feet per s econd to the 1000-pound

vehicle. The same meteor will produce an angular change in roll or yaw which is

10 ft x 1.69 x 10 -9 slugs x 105

 ft/sec = 0.8 x 104

rad

This doemoone endone, Mformotioo offootioo the ootiond joinno F.; the Ignited Elate/ wi ohln the mooning of dm End/moo' Lows. Tien

80 slug ft2 rad/sec
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TFt- 59 0000 -00604

Page III-100

This down/ant conteles hdennellen affecting the notioael defense of the United Slates within the weaning el the Es pionage Lows Tillie

U.S.C.. Seaton 793 and 701 sh. Ihrmees. G. • •
TR-59-0000-00604

Page III-101

-CON-FIDENTIAL--

Since control to 0.1 degree or 0.01 degree is all that is necessary, this effect

is also negligible.

In pitch the change in angular rate imparted by a similar meteorite is

1.69 x 10-6 radians per second. With such a rate it will take 10 3

 seconds to

drift off by 0.1 degree. The angular momentum imparted to the vehicle by the

meteor is at most 1.69 x 10-3 slug feet2 radians per second, which can easily

be absorbed by the small wheel and motor rotating at 120 rpm.

There are a great many other possible sources of small disturbing

torques in addition to meteorites which may affect the attitude of the vehicle.

For example, the rotation of the heliocentric photovoltaic cells in the pitch

plane will have an effect, since any changes in the moment of inertia will

result in corresponding changes in the angular rate of the vehicle. However,

this effect is small as compared with the effect of the large flywheel and can

easily be controlled by the system here described.

Table 3-4 gives the weight and power required for the attitude-control

system. The dominant factor is the gas required for the velocity control. Of

course, more less gas will be required depending upon the accuracy of the

guidance system used. It should be pointed out that the amount of power

required by the monopulse system is dependent upon the number of measurements required. Probably the misalignment of the large flywheel will be

sufficiently large to require frequent measurements. Other factors, such as

meteorites, will also make frequent observations desirable.

-CONFIDE FIA6

111111110 ThH drovelort centaInn Informal,* DM/an the national defense of the United Stems within the theenins ni thus F snlenn.e. lnw. TM..
TR-59-0000-00604

Page III-102

CONFIM:111,0,1._

Table 3-4. Attitude Control System

Weight

(pounds)

Power

(watts)

Coarse Gas Jets (plumbing and main tank) 15

Gas for Velocity Correction (Thor) 45

(Titan) 90

Gas for Reorientation 6

20*

Fine Gas Jets (plumbing) 5 . Gas for Fine Reorientation 0. 6

Gas for Continuous Attitude Correction in

24-Hour Orbit for Lifetime of One Year 5

Monopulse System 20 75

*

Attitude Stabilization Computer (analog) 15 20

Large Flywheel 12 20

Small Flywheel 2 10

Totals (Thor)

• (Titan)

125.6

170.6

145

Used very intermittently

TAIL skoceseent centeins infernsagien affecting the AMON& &Mesa of 11. .le►t• •u. s-- n••.
TR-59-0000-00604

Page 111-103

APPENDIX A

ERROR ANALYSIS FOR HIGH-ALTITUDE

CIRCULAR SATELLITE ORBIT

1. Introduction 

The flight plane under consideration is designed to place a vehicle in a

specified high-altitude circular satellite orbit. The study is.carried out in a

nonrotating coordinate system with the origin at the center of the earth And

the plane of the desired orbit as one of the principal coordinate planes. With

suitable regard to the rotation of the earth and the inclination of the final

orbit plane to the plane of the equator, these results can be used in the

analysis of either an equatorial orbit or a polar orbit.

The nominal flight consists of four phases:

The powered flight from launch yields burnout conditions suitable for a low-altitude circular orbit. The vehicle is permitted to coast in

this orbit for a given elapsed time. This coast period may extend through

several revolutions of the vehicle about the earth, and nominally terminates

at some particular crossing of the desired final plane. In a nonrotating

coordinate system, the repeated crossings of the desired final plane occur

at the same (two) points, but for a rotating earth the longitudes correspond-.

ing to these crossings vary with time and the duration of the coast period

through multiple crossings provides a means of adjusting the longitude.

After the coast period, a short burning period causes the

vehicle to enter a Hohmann transfer ellipse at perigee. In general, the

plane of this ellipse may be inclined to the plane of the low-altitude circular

satellite.

(c) The vehicle is permitted to coast from perigee to apogee.

Since the burning period at perigee occurred at a crossing of the desired

final plane and was assumed to be short, the apogee position of the vehicle

also occurs at a crossing of the desired final plane.

This document contain. illfArmontine niliseelow P• •
TR. -59 -0000 - 00604

Page 111-104

(d) At apogee, a final short burning period supplies the additional

velocity required to maintain the vehicle in a high-altitude circular orbit and

simultaneously orients the plane of this orbit in the desired final plane.

The purpose of this appendix is to obtain first order estimates of the position and velocity errors at the end of each phase. After summarizing the

notation there is a discussion of the equations of motion of the orbit. This is

followed by an analysis of the perturbations of the orbit due to various error

sources. The results are then applied to the four phases of the flight.

Z. Notation

1111, n 401111 mg

sesired Final Plane

z-direction (determined by "right-hand rule" from

direction of motion in desired final plane)

Vehicle

Plane of Motion

- Direction of Motion in

Desired. Final Plane

Figure A-1.

X = Angular position of vehicle above desired final plane

p = Angular position in desired plane measured from arbitrary fixed reference

(positive sense determined from desired direction of motion in final plane)

r = Radial distance from center of earth

This demesne contains Infoneolion affecting the ,.Bonet defense of Um United Stoles wIthia the .,..ulna of Pb. Espioaaeo Laws, Title

IS, U.S.C.. &Wien 793 and 794, the treasmissise or rovolalIoa of which In any 111011nOt to as unauthorised pen.. Is prohibited by low.
TR -59 -0000 -00604

Page 111-105

X angular position of vehicle above desired final plane

angular position in desired plane measured from arbitrary fixed

reference (positive sense determined from desired direction of

motion in final plane)

radial distance from center of earth

transit angle in plane of motion measured from burnout

V total velocity

elevation angle of velocity vector measured from local horizontal

circumferential velocity, r(i) or V cos a

radial velocity, i. or V sin a

azimuth of velocity vector measured from (-z) direction with

range, 0 - TT

o. dihedral angle between plane of motion and desired final plane

transit angle from initial burnout to first crossing

angular difference between burnout value of p and value of p at

first crossing •

GM where G is the universal gravitational constant and M is the

mass of earth

*

.

T = nominal coast time during first phase

N = number of complete half-revolutions during first phase

T ** = nominal coast time in Hohmann transfer

time measured from initial burnout

t

o=

initial burnout time

t

l=

end of first coast period

t2

=

second burnout time

t

3

.= end of Hohmann transfer

t

4=

final burnout time.

Ttisteriter contain! Inietinatien affecting the notional defense of 11to United Steles within the meaning el the Espiona ge. Lows. Title
TR-59-0000-00604

Page 111-106

0

Direction of

Motion in

Final Plane

b

a

3. Equations of Motion 

The orientation of the plane of motion is determined from the spherical

triangle

Figure A-2

The motion in the plane of the orbit must satisfy the equations

r2 4) = c p

2

I. C

r =

r

3

r

2

and the initial conditions

(r)0 = r

o ,

(1. 4)0 = Vo

 cos ao

(r)o = Vo

 sin u

o

.

NMI

This document contains Inlssonstion olloctino this national dolonso el lise Unitorl Stoles within Oho omsonhop of Oho Isolosonto lows, Title

IS, &Mien 793 end 794. the transmission or revolution of which In any ammo to an onouthorIsod person is orohlbliod by

(A.5)
Mir

nt -59 -0000-00604

Page III-107

CONFIDENTIAL

The solution may be written in the form

2. r = r0

 Vo

 cos a ,

r

o ) = 2

cos 4) — tan a

o

sin 4)

cos ao

cos

ao

where

K _

r V 2

o o

(A.8)

We first verify that Equations (A.6) and (A.7) satisfy the initial conditions

(A.3), (A.4), and (A.5). From (A.?) r = r 0 when 4) = 0. Hence (A.3) is satisfied. Condition (A.4) is satisfied by (A.6) and (A.3). From (A.7)

2

r

o •

r — 1 2

)

sin $— tan a0 cos 4) ,

r cos a 

and using (A.6)

•r = COB2 

ao 

sin 4) vo cos a tan a

o

cos 4)Vo

 cos a

o .

 (A.9)

Thus, when += 0

•

(r)o = Vo sin ao ,

and (A.5) is satisfied. Thus, the initial conditions are all satisfied:

We now verify thats(A.6) and (A.7) satisfy the differential Equations (A.1)

and (A.2). Since r o Vo cos ao is a constant, Equation (A.6) clearly satisfies

(A.1). It only remains to verify that Equation (A,2) is satisfied with the same

constant appearing for c in that equation. Differentiating Equation (A.9),

1 — K

cos 4) — tan ao

 sin 4) cl)Vo

 cos ao • cos ao

This document contains information effecting the notional *donne of nut United States within the moaning of tint Espionono Lows, Title

19. 111.11.C.. Sicilia 793 61141 794. the transmission or rovointion witirir to no. NIIIMONO 16% on .....101.11..4

0

••

r =
TR-59 -0000-00604

Page 111-108

•

r -

Substituting from Equations (A.7) and (A.8)

(r

o K

r

o o

V cos a 2

j o

cos

2

 Q r

2

(r o Vo cosao )2

r

3

r

Thus (A.2) is satisfied.

From Equations (A.6) the time may be expressed by

r

o

 V0 cos ao

t = r d (A.10)

The cases to be studied in this appendix will involve small initial elevation

angles, ao . Writing this small elevation angle as 6a 0 and retaining only first

order terms in

r

Equations (A.6), (A.7), (A.9), and (A.10) yields

= .K + (1 — K) cos cp — 6a sin 4) ,

r

• = r 0 V0

 ,

r

= (1 — K) sin+ + 6 % cos 4 ,

2 6 1

a0 sin 4)

(A.13)

clip . (A.14)

V

ro Vo t = roZ Io{

+ (1 - K) coact]

3 ].

+ (1 - K) cost]

4. Perturbed Equations of Motion

The perturbations of the nominal orbits are due to: (1) initial position

variations, 6r0 , 6X0 , Soo; (2) initial velocity variations, 6Vo , 6a0 ,• 6p0;

(3) variation of the nominal coast time, 6T ; and (4) uncertainty in the value

of GM which will be expressed as Ô .L. It is 'convenient to divide the study of

This deafened eentains Inferesellen effecting the notional defense of the United Vales Vein Pm meaning of tee Espionage Laws, Title

18. V.S.C.. Sidle. 793 nod 791. Ow tranewl•ame ...MA.
R.-59-0000-00604

11. Page 111-109

these variations according to whether they affect the plane of the motion or

cause perturbations of the orbit in the nominal plane of motion.

The in-plane perturbations of the orbit are found from. Equations (A.11),

(A.12), and (A.13) to be

6 r =

r6r 0 22 2

r — r — 6K (1 cos 4)) + -- 6a sin 4' + 1— (1—K) sin+ 6 4 ,

. r o

r

oro

0 r

o

r Vo 6U = 6(r+) ooo r o

r

o () 6r 6V ---- Sr

r

= 6 o + -- — ,

r r = — o

•

r 6 V

8W =

where

6(i) = ° —V06K sin 4) + Vo(1 - K) cos +6 0+ Vo cos +6 Go , Vo

6K _

6r o6Vo — K r

o

Vo

These perturbations are expressed in terms of initial variations, 6p. and

6 . The variation, 6 4,, refers to a peiturbation of the nominal transit angle.

Since the orbit is an ellipse, a variation in the transit angle will cause a

variation in both the position and velocity. The quantity, 6 l,, can be found

from (A.14).

Vo

t = d + 2 6a sin d+

r

o

1:

o

2

+ (1 - K) coq + (1 - K) cos

(A.19') 3

vo t = d 26°'o

r

o o (IC + (1 - K) cosT 1 -

CK + (1 -K)cos4]

dx

x

3

•

V d - K)(1 - cos2 +)+ 2K (1 - cos+)

r

o

+ Sao

t =

0 [K + (1 -K) cos4] [K + (1 - K) cos?

(A.19)

This illechisent someotos InSestotoft*o effecting the motional dolonso of Ma Unitstel Steles within the mooning of lit* Espionage Lows, lift.

IS, U.S.C.. Soctien 793 and 794. the transmission oc rovedolion a! which in onv moon*, to an uontelhotintel nor a an is ssmshIldtod har Inw
TR-59-0000-00604

Page III-110

t

6 -

r

o

6+ 

[K + (1 - K) cos.]

[(1 - K) sing 4,+ 2K (1 - cos 4,)]

+ 6 ao

[K + (1 - K) cos +1 2

3

(A.19

cont)

26K0 (1 - cos+)d4,

[K 4- (1 - K) cos

The two orbits to be studied are the circular satellite And the Hohmann

transfer. For a circular satellite

a

o

0, Wo = 0, Uo

 =_

o

0

Hence, W = 0 and r = r 0 during the entire coast period. From Equation (A.8)

K = 1, and from (A.18)

bK = — μ

6ro2 6 Vo .M•11

r

oVo

These results may be used to simplify Equations (A.15), (A.16), and (A.17),

to yield the variations at the end of the circular coast period (nominally, at

t = T ).

br 6r SV l

° (2—cos + I ) + 2(1 — cos 4)) v ° — (1 — +1) p. + sin 4. 1

Sa r o' o

ro 0

SU br 6V 1

Vo

— 

r: 

(1 — co, +I ) + (2 cos 4), — 1) v ° + (1 — cos 49 — sin + 1 6a0

6W 1

Sr SV o

V = sin +1 + 2 sin cti

v0

0 0

0 — sin +

1

+ cos +1

 Sa

o .

(A.22)

p,

This damming contains toiespatIon a/Iodine the national deers,* of the United 51alos within the weaning of the Espionage laws. Till.

U.S.C.. Section 793 nod Ted. M. • • '• '• • • •

0
TR-59 -0000-00604

Page III-111

Equation (A.19) also may be solved for 64), and the expression simplified by

using K = 1, the above expression for SK,. and the fact that the total transit

angle 4 1 = VoT /ro .

( V T

*

4)1

+ 26K (1 — cos 4)) do— 2 (1 — cos 4)

r

o

V S T

* Sr

°

Sro 2 6 V

o

6 4)1 = 4)1( V0

• 

+ T

*

-

r

o

2(4'1 sin 4. 1 ) p. r

o

v

o

— 2 (1 — cos 4) 1 ) Sao

(A.23)

For a Hohmann transfer starting at t2 and terminating at t 3 = t2 + T**

the conditions are a2 = 0, W2 = 0, 4)3 = Tr. From (A.19)

V.3 T *

6 `= ( _

6 4)3

).

2 6 K

w

(1 - cos 44d4, 

0 [K + (1 - K) cos ci 3 + 6 a 4K 

'

2

(2K - 1)2 r

2 (2x - 1 )2 

(A.24)

where now K = p./r 2 V2

2

. The integral appearing in Equation (A.24) can be

readily evaluated. From any integral table

d$ 1 -1 (07-732 sin 4 a b 0

a+ b cos b+ a cos+

o a - b

d 

a+ b cos 4)

o aZ + 62

d a

+ b cos 2

8a

d ira 

[a + b cos 4] (a2 - b

2 3 2

r:nrINN.TIAL

0 0

71.1. deswmont cantatas infetmetten affecting the natters& defense of the Unilad States within the maaning of the Esoloneste Lows, 91110

111, U.S.C., &Idiom 793 end 794. Ike transmission or revelation of whirls in nay P.101.1. ••••••.110...S...4 m.o..... I. n ooLIG..11... a... •

) o '
TR-59-0000-00604

Page III-1 1 2

( 1 - K) cos cd 3 2 8K 0 [K 4. o [K + (1 - K) cos cli

z

Tr

(1 - cos 41 d 1 8

Tr 

d4

For the particular case, a = K, b = 1 - K

irK 

foir [K + (1 K) cos d q] Z (2K - 1.)312

Therefore, the integral of interest

1 a

= 8K (2K - 1)3/2

w(1 + K)

/2 • 2(2K - 1)5

Substituting, and solving Equation (A.24) for 6 03 gives

**

2 (" 2 1: 6Kw(1 + K) 4 K 6 0. 3

6 4 = (2K — 1) 6 + 2 . r 2 1

This equation may be expressed more conveniently by noting from Equation (A.1.9')•

that the nominal value of V2**/r 2

is given by

V T

d

2 wK 

r

2

[K + (1 - K4) ) cos 4? (2K - 1) 3/2

and from Equation (A.7) with 4 = w

(2K 1)

K =

r 2 + r3

2 r

3

1.

This decimate. contains inferntation effecting the national defense of the United States within the moaning of the Espionage laws, Title

IS, U.S.C., Section 793 and 794, the trontaission or revelation of which in any Monne, to nn 101.6.61••••• 1...1 8. ta- •
6U3

(r2

+ r

3

) 6r2

V3 r2 r

2

6W3

-r3

6 +3

r 3 rz

r

3 e

V3

2—'2 r

2

3 +1. 2

) 6V2

r2 V2

(A.28)

TR-59-0000-00604

Page III-113

CONFIDENTIAL

Thus, the expression for 6 4)3 becomes

6Vz 6 T ** 6r2) 6 4)3 = (2K - 1)2 

wK 

**

(2K - 1)3/

( V2 T rz

w(1 + K)K 6r2 2 6V2)

Vz — 4K6a2

(2K - 1) 1/2 r

2

=

ir(r2 + r 3 ) 3 (rz + r 3 ) 6r2 66V2

2

r

3

r

2 V2

2 3

r

2

 + r

2 3

r

6 Q 3 2

(r + 3r 2 3 ) 1,a

r

3

2r 2 ST

44* r 3 T

(A.25)

Now, from Equations A.15), (A.16), and (A.17), with 4) 3 = it and W3 = 0,

6 2r2 + r 3

6r2

r2 +- r 3

6V2

r

3

(r 2 + r

r

2

r

2

r

2V2

r2 p.

r

3

In the above expressions, W3 = 0 and Equation (A.12) have been used to

replace V2/V3 by r3/r2.

The preceding results provide the in-plane perturbations for the cases of

interest. The perturbations of the plane of motion are obtained from the

spherical triangle (Figure A-2). From Figure A-2a

11111111E

This desement contains Inferteotlen attesting the national defense of IM United States within the mooning of the Espionage LOWS. Title IS, U.S.C., Beetles 793 and 791, the transmission Of revelation of whir A In nnv ......►t‘ss.4
TR -59 -0000 -00604

Page 111-114

• --CONFIDENTIAL--

cos 0. = cos X sin p,

tan 0 .= sin X tan 13,

tan CD = tan X sec 0,

and from Figure A-Zb

cos fr =

tan 0 =

tan =

The variations become

cos X sin (w - 0),

-sin X tan (n p),

-tan X sec (w - p).

sin X sin (3 cos X cos 13 

sin a- 6 X o60, sin

cos X tan 0 sin X sec 2

13 60 6X + 60,

sec A sec t 0

6 sec

2

 X sec 13 6 x ta.n X sec 0 tan 0 613

sec' 2

sect

or

6o- -sin (-X) sin (w -0) + cos X cos (Tr - 0) Sp

sin o' sin osin X sec

2 (

IT in 6p,

sec 0

80 -

cos X tan 0) 6x

sec 0

6 -

sec2

X sec (n - 0) ox tan X sec (w - (3) tan t(n- (3) 6p

sec

2

 0 sec

2

0

Using some identities of spherical trigonometry and the sign convention on

established in Section 2, these results may be written

This &moon. contains leforonoffon &hellos IM NoOlenall defense of PM WOW Moto, wfO►lo the ...dodos of flo. 1.olon000 Lows. THU
Tit -59-0000-00604

Page III-115

CONFIDENTIAL

60- sin 9 (sgn X) 6x — cos t (sgn X) 6(3 ,

60 = cot or cos 13(sgn ).) 6X + csc 0. sin X6 p ,

41) = csc r cos 9 (sgn X) 6X + csc 0• cot a. sin X6 (3

. (A.31)

These equations will be used later in the determination of variations in X and p.

The analysis of the errors introduced during the shortburningperiods does

not require the equations of motion. The short burning periods are assumed to

produce finite increments in the velocity vector in negligible time. In the cases

of interest, the elevation angle a = 0 and the velocity increments take place in

the horizontal plane. The subscripts i and j will be used to denote the states

before and after the velocity change, respectively, and the syMbol A

represent a finite increment. With these assumptions

will

6X.J

= ox. ,

6pi = bpi

,

Or = Sri

.

(A.34)

Consider Figure A-3.

Figure A-3

This determent contains Informatlen attesting the mollooni defense of the is within the mooning el the Espionage Laws, Tide

III„ U.S.C., Section 7V3 and 794, the transmissien or revelation of which In any mama, le en oneetherlsed person is prohibited by low.
TR -59 -0000 -00604

Page 111-116

The initial velocity has magnitude, Vi , and azimuth, withith errors, 6Vi , and

V•1613.

1

• The applied increment has magnitude 

(Aij V); azimuth.; and eleva- , y1

ij' ij ,

13 1.i ij

and the final velocity has magnitude, n, and azimuth, pi . Thus

SW. = 6W. + (A.. V) 6E

1 lj 1.1

(A.35)

and from Figure A-3

6V. = 6V. cos (pi - pi) -

Vi

613i sin — 13.) +6( 3

.. V cos — yij)

J 

+ V) Oyu sin (13i — yij) (A.36) ij

Vi6f3i = 6V1 sin (pi — pi ) + Vi 6 pi cos — V) sin —

+ (Aii V) 6yij cos (pj — yii ) . (A.37)

The results are valid for an arbitrary orientation of the figure. The errors in

the velocity increment can be related to errors in the missile guidance system.

For example, in the case of an all-inertial system, 6yij and 6E ij represent

gyro drifts and 6(il ij V) the error due to an integrating accelerometer.

5. Flight Analysis 

The nominal burnout conditions for the first phase are Wo = 0,

Vo = Uo = 1 77 11. 0, % = 0. Hence at time t 1 = T

*

: r = r o, W1 = 0, U1 = U0,

= ( D + Nn) = Uo -r * /r o . The variations 6r 1 , SU1 , SW1 , 6 41 are given by

Equations (A.20), (A.21), (A.22), and (A.23), respectively, and are summarized in Table A-la. The variation in orbit inclination, 6o-, is given by (A.29)

evaluated at to . In order to express the variations in X and p, it is

convenient to introduce

= angular deficit in transit angle measured to terminal

crossing of desired plane,

8 111 = 6 —5 (1)1,

where 6 4:1) is giVen by Equation (A. 31) evaluated at t

o

 and 6 CI)

1

is given by

Equation (A. 23). Then from Figure A-4,

This deopmeet (*meal inlormetien electing the notional defense of the United Elates *Mule la mooning of the Esslenea Lees. Title

111 11 S r 707 ...I ..1% 4 61.. • • • • .
•

"mtv'llr4-r "-flArletIMM

1

-59-0000-00604

Page 11/-117

04. 1 = (-1)N sin Cr 6 If (sgn x0), (A.38)

bp i = 6p0 + 6 0 - cos cr , (A. 39)

where 6 0 is obtained from (A. 30) evaluated at t o

 and 6po represents the

error in p at initial burnout. Examination of Figure A-4 shows

P I = Tr/2 - (-UN crl ("11 ?1/40) = ir/2 -, (-1)N cro (sgn. k0 ) ,

6 15 1 = (-1)"1 6T1 (sgn ko). (A. 40)

These results are summarized in Table A-lb.

At the end of the circular coast period a short burning period now

supplies the additional velocity required to enter a Hohmann transfer ellipse.

For a given perigee altitude, r o, and a desired apogee altitude, r4 , the velocity required can be obtained from Equation (A. 7),

2- = K + (1 - K) (-1) , r

4

F O

r4

2K -

r4

= ro r4

•

r

o VZ

2

 r4

2+

2 r4

2 r4

+ 0 r 0 r4 ro r4

The variations at the end of this phase are given by Equations (A. 32) through

(A. 37) with i = 1, = 2. The values of AiiV and yid are predetermined for

a given flight and the errors 6(d ij -V), 6-yki , and be can be related to errors

in the inertial guidance equipment. From (A. 34) it may be observed that

r2 r 1 . Nominally r 1 = ro and V1 = Vo• since the first phase is a circular

satellite. These relations and the further relation r 4 = r 3 which also

follows from (A. 34) will be used to simplify the summarizing tables at the

end of this appendix. The results are summarized in Table A-2.

This doormat annals' information affecting the notioaal &fonts Jn within Ow moaning el the Espitmoom lows. Title

la. U.S.C.. &Delimit 793 nnot ?OA -•
TR-59-0000-00604

Page 111-118

Table A-la

S r

o

8 Vo , 8 ao

ba

g

r

Sr*

r

o

V

o T

*

Ori

-=:- 2 - cos 4

'1

2 (1 - cos 01 ) sin +1. - (1 - cos + I ) 0

0

81J 1 -v----

o

-(1 - cos +I ) 2 cos +1 - 1 -sin tpi 1 - cos 4:1. 0

6W1 sin + 1

2 sin 4) 1cos +1- sin 0 10 V;"".

8 4.4 2 sin +1 - 3 CI?, 4 sin 4) 1 - 3+1 2 cos +1 - 1) 2 + 1 - 2 sin 41

Table A-lb

a

0

64A,0 • 6p

o

Scr1

= Scro

=(...e+l (sgn x.0)6131

- cos t(sgn k. ) sin €1(sgn A.0 ) 0 0

6X1

(-1)N cos x

o

 sin 8 0 (-1)N+1 sin o-1(sgnX0)

Sp '

sin X0 0 1 cos 0'1

This tlecomont contains intennotion ofincting the national detente of the United Stains within As swooning of fins Espionage Laws, Title
,0

›•

N

.-I

<I

v

N.N-7n

?"

0

›.

C

......

N

?":1

. I

co

. 

0

.i

to

o

N...... o

7

0

a

4

i

N

In.

to0

u

1 iN1

I

Jo

<1.4

.....N

......

.-•

w

0 0 0 0

••1 4

tO

IN o

:3.-.1

a1

,-,

ta.

ca

0

u

0 o o

0)

0 N

P.

.4

to0

o

ow*.

NI

CO.

I

.-4 S

.4

.

1

o o

..",%1

CCL

I ,-I

CCL

v,

u

oi NI

i.

a.

Lo 0 0 0 0 -1 o

.-4

ed

to

0 0 0 i-• 0 0

rr

o o .-i 0 0 0

o

ev

CCL

1

.-I

CCL

to

o

C.)

o o

*-7.1

c o.

01

11 ...4

10

01 INI

›• I

$.71 1

0

10 k

Pi 0 0 0 °

.

•

40 k

1

,

N

4.0

•

I NI N

,..r

CO

....&

N

to

0

0

00

fitt

••••••••

tO4

II Z+

N ....,

ea. i

‘0 .....

TR -59 -0000 -00604

Page 111-119

TM, &OMNIg contains information affecting the notional defonse of the *Mad States within the weaning of the Elmtense* lows. Till.

IR t 1*, 'Ma . n • . .• . • .
-CONFIDENTIALODD N

6 0

EVEN N

"% Y _ b*

I

6 P- Po)

EVEN N

TR-59-0000-00604

Page 111-1 20

Figure A-4a

-6X

X = Vehicle position at t = T

*

6 T

*

Figure .A-4b

CONFIDENTIAL

This decessent contains intermetien effecting the nallenei defense et the United Dates within the pleasing of the Essienerge Laws, MN

U.S.C., bodies 793 end 794, tire transmission er mwelatien of which in an y manner to an enautherised person Is prohibited by law.
TR -59 -0000 -00604

Page III-121

The vehicle now enters the transfer ellipse at perigee and coasts to apogee.

The nominal conditions at t 2 are a = 0, W 2 2 = 0, and at t 3 = t + 2 T they are

= 0, W3 = 0, 43

 = yr. From (A.6) the nominal velocity at perigee is

r

o V2 V3 = r

4

Furthermore, since the perigee and the apogee positions are nominally on the

desired final plane, X2 = X3 = 0. The variations 6r 3 , 6U3 , 6W3 , and 64)3

are given by Inquations(A.25) through (A.28) and are summarized in Table A-3a.

The variation in inclination angle is facr 3 = 6a.2 where from (A.40) and (A.37)

60. = (-1)N+1

Z

613 2

 (sgn Xo). Since the nominal perigee is at a crossing of the

desired final plane and the nominal trajectory extends through one-half period,

the variations in X and p become

6X3 = — 6X2 + (-1)N (sgn X0 ) sin cr2 54 3 (A.41)

6 13 3 = 6p2 + cos o-2 .6(0 3 . (A.42)

From (A.40) and the 1,:---,wledge that N has increased by unity

p 3 rt/Z 4 ( 1)14 (sgn X0.)

03 1,12 + (-1)N (sgn X0) 0-2

and

6p 3 = (-1)N (sgn Xo) 6a.2 . (A.43)

The results are summarized in Table A-3b.

At apogee a final short burning period is required to maintain the vehicle

in a high-altitude circular orbit and to orient the plane of motion in the desired

plane. The velocity required for the circular orbit is U4 = Equations

(A.32) through (A.37) apply with i = 3, j = 4. The nominal value of p4 is

Tr/2. From the convention in Section 2 regarding p, a positive value for 684

represents a velocity component in the z-direction. The results are summarized

in Table A-4.

CONFIDLN IAL

This dasual•to contains Infer.etiso aff•dino rho notion& dektas. of 1110 WOW Rates within thtt monis, of the Espionage Lows. 7I110

IS. U.S.C. 11..-11... 703 70.1 61.. 1....••• n ••••••• •••••••• n ..1.... .6 •••••,..1. t. . a • • • it • .
TR-59-0000-00604

Page III-122

ra t

Table A- 3a

6r

2

6U

2 6W2 W2 la

11

67.

r

o

V2 vi

T 

**

. _

Sri

( ,_2 ...1r

2

, r4 i + --)

r

o

o

"----

1 0

r

4

-1- ro

)

r

6•U

3

( + 4r

r

o

-(

Zr

+---.1) 0 0 V

3

r

o

(. 1 + -r-i–) r

o

6 W3 3w(r 4 - r o )(r 4 + r 0)2 3ir(r4 - r of ) ro

r4

ir(r42-r: )(ro+3r4 ) 7r(r42'r4)

v

3 8 r 4 r o

o

r

4

r

o r 4

ip. 4 8 r

o r 4

ro

 r 4 4 r

4 r o

r

4

6 +3

3w(r

o

 + r 4 )2 37(ro+r4) ro+r4

2 ---7.--

Tr(r o+r 4)(r 0+3r 4) wro(ro+r4)

2 177—.4 r

4 r 4 zr-7o 2 : 4

.

4r

4

x

o r4 Y .

2 r44 o 4

Table A- 3b

6p2 6X2 6p2 6 +3

6 cr3

=

6 cr2

= (-1)N (sgn X0)6p3

(-1)N+1 (sgn X0 ) 0

6X3 0 -1 0 (-1)N sin crz (sgn Xo)

6p3 0 0 1 CO 8 IT,

z

This stecomont conteins information @fleeing the notional defense of the Wiled Motes within the missingat the Espi Laws, Till.

IS, U.S.C.. &mikes 703 and 714, the transmission or rovelatien of which in any manner to en unauthorised Ironton is rsoblbilod b y law.
TR-59-0000-00604

Page H1-123

.

to

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di M

in

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cel

fil 0

0

0 0 0

14

en

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en I to

to

M

wo to

Nil en

4

en 

0 0 .-4 0 0 0

*I

.0

My

nr en

›. 0

Ven

-

0

•n

COCO

0 0 0

to

en >...

trl 0

U

I do

> 1

en

CM

40

0

—

Mca.

to

0

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,

0 0 0

Mc0.

•

e 144 ›• l

co

en

a.

to

0 0 0 0 ,--1 o

en

tO O 0 0 .-4 a 0

M

1›.

tp

0 0 ei O 0

rn

el 0

CY)

-

0

,-1c0

C0 0

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0

M

Ca.

U)

. 0

U

en I to

1.

I

ker to v ...

.

0 0 0 0 0

J

v14

,o

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14

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to

(11

V°

t<3

ell

›.

di

to

di 0.

to

di t..0.

co •

lish document gee/elms InIermolien effecline the national defense of the Willed Metes within the meanie, of eh. Esol Lows, Title

IS. U.S.C., /*ellen 793 and 794. the Weesmissiee et ,11,14011011 of which in soy wanner to no tannodlwl ...1 ...tem 1. kv
TR-59-0000-00604

• I Page III-124

APPENDIX B

SATELLITE PERTURBATION RESULTING

FROM LUNAR AND SOLAR EFFECTS

1. Introduction .

The motion of an unpowered space vehicle is primarily determined by its

initial motion and the gravitaional field through which it moves. Radiation

pressure, magnetic field effects, collisions with foreign particles, etc. ,, are

neglected 'in the present analysis. For earth satellites computations based

on the gravitational field of the earth alone provide the basic orbits. The

purpose of this appendix is to investigate the perturbations of a 24-hour circular satellite orbit which are due to lunar and solar gravitational forces.

For simplicity, the effects of the sun and the moon are considered

separately. In the solar case, it is assumed that the sun is fixed at the

origin and the earth moves in a circle about the sun. In the lunar case the

origin is taken at the center of mass of the earth-moon system and the earth

and the moon are assumed to move in plane concentric circules about the origin.

After introducing suitable notation and a brief development of the general

equations of motion for a modified three-body problem, the equations are

specialized to the case of a Z4-hour circular satellite orbit about the earth

subject to the influence of either the sun or the moon. A general solution to

these equations is then obtained and the particular effects of the sun and of the

moon are considered separately-. In either case the resulting perturbations

are quite small, which provides post hoc justification for the separation of

the two cases.

jlierote cootoll Wert/ion offeetIno Mo eolleeel defento it !he Ualled Stoles wilt& the "swede? of the Sosionelmeleefs,
TR -59 -0000 -00604

Page 111-125

r = distance from vehicle to earth

R = distance from vehicle to second body

D 1

= distance between earth and center of mass of the system

= distance between second body and center of mass of the system

= angular position of the earth

cr = dihedral angle between "plane of motion" and plane of

gravitating bodies

= transit angle of vehicle in "plane of motion"

= angular position of vehicle measured in plane of gravitating bodies

= elevation angle of vehicle above plane of gravitating bodies

G = universal gravitational constant

M

s

= mass of sun

M

e

= mass of earth

M = mass of moon

m

= GM

s

μe = GM

e

= GMm

A = nominal (constant) value of

B = nominal (constant) value of 4)

D = Di + D2 .

CONFIDENTIAL

Title Iniovi cenloins infroveliew •Hectin. tke Netionsl &Maw' of Ike United Stoles within 1k nmo1159 of tke Espionoo• Lew*, MI*
Second Body

(moon or sun,,

2\

— ---

Center of mass

Plane of Motion of Gravitating Bodies

Instantaneous Plane of Motion of Vehicle

z

TR - 59 -0000 - 00604

Page III-1 L6

2. Notation

Figure B-1. Geometry and Notation for Vehicle and Moon-Sun Interaction.

This doceenset contains infeemation offoctine tho national dolmas of the United Maths within 1h mooning el the Esoloneae Laws. Title

U.S•C.„ Section 793 01111 794, at* transmission or nteeletion NA which io 
TR -59-0000-00604

Page III-127

3. Equations of Motion 

The motion will be determined from the Lagrange equations. The potential

energy is given by

V (B. I)

where 112 equals p.s. or The kinetic energy is

1 2 '2 '2 T = (X + Y + Z ) . (B.2)

It is convenient to convert these equations to earth coordinates:

X = Di

 cos 4) + r cos u. cos y ,

Y = D1

 sin 4 + r cos a sin y , (B. 3)

Z = r sin a .

Consider the spherical triangle

which yields the relations

cos a cos y = cos LI, , (B. 4)

MC:

This deatment contains informeilen ettectine the neNenal defense of the United Slate/ within the mooning of the Eseionotte tows, Title

IS. U.S.C., Section 793 end 794, the tronsmistton or revelation which In any senor to an whouilterised person is prohibited by low.
TR-59-0000-00604

Page III-128

DENTALcos a sin y = cos a tan a Cot o-

= sine sin 4, cote (B.5)

= cos cr sin 4, ,

sin c} = sin o- sin. tp . (B. 6)

From the definitions of 4, and or (Figure B-1) these relations are strictly

true only when the motion is confined to a plane. The motion:is known to

consist of small perturbations from . a plane and consequently these relations

may be used in seeking the perturbations. 'A more rigorous point of view is

to define an "instantaneous plane of motion" determined by the velocity vector

and the center of the earth and then to measure 4, as the great-circle angular

distance in this plane. With this convention, Equations (B.4), (B. 5), and

(B.6) hold for arbitrary motion. These equations may be used in (B. 3) to

obtain

X =

Y =

Z =

Di cos $ + r cos 4, ,

D sin 4, + r cos cr sin 4, , (B. 7)

r sin cr S in 1.11

The distance, R, is given by

R = X + D 2

2

 (cos ) + (Y + D2 sin+) + Z2 Z

. or

R

2 2

= D r + 2 Dr (cos 4) cos 4, + cos cr sin 4) sin 44 (B. 8)

where D = D1 + D. The kinetic energy is obtained by differentiating

Equation (B.7) and is found to be

ml, deseatent modeled inlecateffee effecting lite nellenel detente el Poe United Stoles Wale the weenies el the &Oconee' taws, Title

IS. U.S.C.. Section 753 end TOL the Mmemods.low 'or iroweLotwo .1 NASA. I. •••n•••41..1....1 mamma megaLICS•aal Som.
TR -59 -0000 -00604

Page 111-129

2 T = ;2 + r 2t1, 2 + sin2 4, + B2

+ 2 D Br (cos 4) cos a sin 4, - sing, cos 4,)

+ 2 D i

 Br 4, (cos 4) cos cos ill 4' sin 4) sin4,)

- 2 Di

 Br sin 4, v sin a cos 0, (B. 9)

where B = = constant.

Consider first the Lagrange equation for 4, . Take L = T - V. Then

8L 2 •

4,

a

= r + Di Br (cos 4) cos 0" CO 8 41 + sin 4, sing' ) ,

d (81 d •

--T= Tt- (r 4) 2

 4) ) + Di B; (cos cos cos 4, + sin 4, sin (1,) 84,

- Di Br cr (cos,0 sin a a 080

+ Di

(sin 4) cos 4,- cos 4) cos a sin kl,)

• + Di

B2 r (cos $ sin 4,- sin 4 cos a cos 44 ,

8L• 2

= r2 sin 4, cos tp Cr + Di B ; (cos 4) cos a cos 4,+ sin 4 sin 4')

84,

- Di B r cer cos 4; sin a cos 4, + Di

 B ris (sin 4, cos* - cos 0- sin 4')

1 2 n 8R "'•t 

R°

R 8 = Dr sin 4) cos a cos - cos 4) sin 44 . 84,

CONFIDENT' AL

TMs &moment eentehu IttlemetIon offostIon Hut notIonel dolOnso if the MINI Sales within Net mooning of th. Elnioneno lows, Till.

U.S.C., Sullen 753 end 714. the treososhilen feenIetIon .1 whkb in ear sonener to en enautkerhed MINS k arakikikal
TR-59-0000-00604

Page III-130

Substitution of these quantities in the equation

d (8L

WE a ) 8L = 0

gives

d 2 • 2 . 2 112

rt- (r - r sin +cos* cr + (cos+ sin 4)- sin+ cos+ cos cr) (D1 B

2

r -Dr —3 ) = 0

R

(B. 10)

The Lagrange equation for r is obtained from

8L

•

8r

•

r + Di B. (cos cos a. sin 4, - sin 4) cos+) ,

.

7 + Di B (cos + cos 7 cos 4 + sin 4' sin to

•

D B • (cos + sincr sin 4)) - D1 B2 (sin+ cos cr sin 4,+ cos 4) cos 4,),

8L .2 2 • 2 • = r + r sizx 4, + Di 1341 (cos 4) cos Cr cos + sin + sin 4)) ar

8R Di B it- cos + sin cr sin 4) - Rik -IL2 R Dr ,

r R

8R R = r + D (cos + cos + cos cr sin + sin+) . ar

Substitution of the above in

d (aL) 8L

ar

r t-

.L)
TR-59 -0000-00604

Page 111-131

yields

r

2 .

cr2 'L e P.2 r

r - r sin * + + -r

r R

(p.2 D

+ (cos 4) cos 41+ sin 4) _sin *cos Q)--3— -

R

B

2) = 0 . (B.11)

Similarly, in determining the Lagrange equation for a it is necessary

to. compute

8L 2 2 . r sin 4, - D i Br sin 4, sin a cos 49,

d 8 L ) = d (r 2 sin 2 * cr

.

) - Di sin * sin cr cos 4► - D i r csin* cos crcos

Escr

D 1

 Br *

•

 cos 41 sin cr cos cl) + D 1 B2 r sin 4, sin a sin 4,

8L. .

Di B; cos 4, sin G. sin 4) - "D i Br *cos (I) sin a cos 4,

D i Br O. sin* cos acos /12

-

R 8 R

R8

8 —

cr

R = -Dr sin a sin (1) sin 4).

From

T

d 8L

cr 8.7 = 0

the equation for a is found to be

d 2

(r.

2 • 2 p. D

sin 4, cr) + (r sin 4, sin 4) sin v) D i B --3— = 0 . (B. 12)

CON.F1Drn'''

TlesImatrrMoist Infrostio? offodloo IM notiooti slisfoots Ilor•thollosl Weiss within Ilse mooning of the Esolesesse Lows, TM*
TR-59-0000-00604

page III-132

-CONFIDEKIAL

The basic equations to be•solved are Equations (B. 10), (B.11), and (B. 12).

The quantity [D i B2 - (D p.2 /R3 )] occurs repeatedly. For circular motion of

two gravitating bodies in a plane, D2

= ,J.,/D and D i ci) 2 = p, 2 /D2 . Consequently, B2 = 2 = p.i /D2 /D = 2 . With this observation,

D p.2

 Dμ2 (R2

 - D2) (R2 + D R+ D2)

= —3-3- (R - D) (R + DR + D ) -

D R D

3

 R3 (R+ D)

r B (

131 -35-) .+

(Ry R 15 1 + 75)

[015) + 2 (cos+ costi, + cos cr sincp sin .

(B. 13)

For both the lunar and solar cases, the approximation D in Equation (B. 13)

is sufficiently, accurate for the study of first-order perturbations. In the lunar

.case,

R R

2

A 15- "2)

1. 184 13 1.96;

(C3 i + ';`)

in the solar case,

- 1.500 (.0. 0005.

k

•

.

TInisdaanneof seetaIns Infennallon offotfing gap saliva& Ohm* of Um Unilod Mass ankh, the Ineening .1 the Espionage taws. Till.
TR-59-0000-00604

Page 111-133

CONFIDEr.11

Clearly the approximation is adequate in the solar case; however, in the lunar

case the magnitudes of the predicted first-order perturbations may be in error

by as much as 30 per cent. With this approximation, Equations (B. 10), (B. 11),

and (B. 12) become

— (r 4) = r sin4)cos d 2 2 •2 3r2

2C) [(r tp Cr ---T B -15- 15) + 2 (cos+ cost's+ cos o- sin +sing)] dt

(cos 4) sin 4, - cos 4, sin 4 cos co, (B.14)

r - r4i 2 +- + iz• r sin24, Cr2 r 

7 (

[05.) + 2 (cos4) cos 4, + cos sin +a n4,

r D

cos 4) cos* + sin 4 sin cos Cr), (B.15)

Di (r sin 4,76r) = - Cr (; sin* + r cos 4, if) d

, r132 (

zr -15-)

[() + 2 (cos 4 CPS* + cos cr sin sings sin 4) sin Q. .1 (B.16)

The unperturbed solution is known to be r = 0, o- = 0, = A, where A

is a constant equal to Z w radians per sidereal day. In order to obtain the

first-order perturbations, these unperturbed results are substituted in the

right-hand sides of Equations (B. 14), (B. 15), and (B. 16). Furthermore,

r/D <4 1. in both the lunar and solar cases and, consequently, this term is

dropped in the expression [(r/D) + 2 (cos 4 cos 4, + sin 4, sin 4, cos cr), . With

these further approximations the equations for the perturned motion become

1. 2 2 (Di

(r2 q,) = - 3r B -r) ( COS 4) cos* + cos a- sin 4) sin tp)

(cos 4 sin 41 - cos 4, sin 4 cos co, (8.17)

This doormat cm lints Inl0000lloo a/fading Ms aoh000l &hose 'of do UNIIIIII Stela wlfhlo lho mooning of Ow Esolon000 taws, UM
TR -59-0000-00604

Page 111-134

•• • 2 P'e ij'ar r - + + --- = 3 r 82

r D

cos + cos++ cos sin (I) sin+) Z , (B. 18)

(r sin+ = - 3 r BZ (-DDI) cos + cos+ + cos a sin+ sin+) sin+ sin Q. (B. 19)

4. Solution of Perturbed Equations 

Using elementary trigonometric identities, Equation B. 17) becomes

dt (r2tp) 3 2 E12 ( D1) [sin 2 4 (i cost + sin 24, cos 2 + (1 + cost cr)

- 2 cos a. sin 2 + cos 2 id . (B. 20)

On the right-hand Fide, 4 = A, ;= B, ; 0, and r = 0. Consequently,

Equation (B. 20) may be integrated directly

r2 • r2 A 3r 2

 82

. 

D

1) r 4, = r ".+

(

-if- j cos (2 At + 24, 0) (1 - cos2a.)

2 I cos 2 (At + B t4, 0 + +0) 4. cos 2 (At - Bt ++ 0 - +0) (1 + cos o2 (1 + B/A) (1 - B/A)

cos Cr

[cos 2 (At + Bt ++0 + +0)

(1 + B/A)

cos 2 (At - Bt +4, 0 - +0)1}

(1 - B/A)

w

This docamons soniona ilhOrM eNtta •ilosang the notional defense IM UN'S*/ Soles within M. moaning el the Espionage laws Title se n • n en. ._ on. • •
TR-59 -0000-00604

Page 111-135

3 BZ (D1) = 4, 0 + Ai t + Tr ---ey ( cost Ir) sin 2 (At +14,o)

+

(i + cos a cr)

(1 + B/A) Z (1 B/A) 2

[sin 2 (At + Bt + tko + 00) sin 2 (At - Bt + 1110 - cp0)

cr

]

- cos (B. 22)

(1 + B/A) 2 (1 - B/A) 2

In Equation (B.18), take r = r0 + 6, where r0 is the solution of

A I P2 P•e

.• 1"o

.ro —s + ---2 - 0.

D r

o

(B. 23)

sin 2 (At + Bt + tp o + 4) o) sin 2 (At - Bt +4,0

Equation (B. 18) becomes

- (r +6)(A2 R.2) = 3 r —3 n C% 0

(r

0 -1 U/

(

Di

-15= (cos 4) cos tp+ cos cr sin 4) sin02.

(B. 24)

(1+ 

0

But from Equation (B. 23)

2 R2)_,_ -Re . ..-μe %

( --3 -r --.--- --s = - ( ' -t• 0 j --- 4. ---.-.-.-- -

D (ro+ 6) o

ro

(r0+6) 2

3 • r

o

ll(e

- (ro + 6) A -

ko+

[r o

3

- (ro+ 8)3] ) 

and retaining first-order terms in 8,

P. 3r026% 2) /-1 (ro+6) A2 —DI --73 tro, .0 r0

= - 3μe

(B. 25)

TAI, imater conteles lefernetlee •lletiltle the stelliteel /ekes* et the Unihttl Steles will& the emitting of the Emitting** Lows. THIS
TR -59-0000-00604

Page 111-136

But for the unperturbed circular satellite, = A2

, and Equation (B. Z4)

becomes ro

Di ••

6 - 3 A

Z 

6 r

o

B

2

(cos 4cos4,+ cos crsin.sint1) 2

. (B. Z6)

The solution of the homogeneous equation is 6 = 0, and consequently it is only

necessary to find a particular integral. The equation may be rewritten in the

form

b-3A2 6 = 3r BZ Di o 17

11+ cos 01 2 2

cos 0+ (1 - co sa)Z z cos (4,4-4) 4

(1 - c2s 2 cr) cos 4+40 cos (4, -4) ,

or

2 2 Di t(1 + coso-)2 6-3A 6 = 3r0B -b- 8

[1+cos 24-4

(1 - cosi

 o-) (cos 2tp+ cos 2c1:)} . (1, - cooly)

z

[1+ cos Z (041 + 8 4

(B.27)

Hence,

6 = 6 + Ci cos 2 (,-4) + G2 cos 2 (4)+4)) + C., cos Ztii + C4 cos 24). (B. 28)

Substitution of (B.28) in (B.27) determines the unknown coefficients, C i .

nao ...al.., MO.. • • . ••• • a.
TR-59 -0000-00604

Page 111-137

The result is

ro B2 Di 3 ro 332

6 =

D i (1 + cos 0 2

"2- -2 17 " A 8A2 15- 4(1 - B/A) 2 +

•

cos 2 ( Li) -4) )

2

3ro B (Di (1 -cos o-) 2

8A

D1.4 (1 + B/A)24 2

3r B2 414 (D) (i cos 2 ( ) - 7 - cos 2

 cr) cos 2 4)

3- - (BZ)

:;:2*D (1 - cos2 Q) cos 24) (B. 29)

(3+ 4B2/A2)

Equation (B. 19) for ;J. is written in terms of (r sin 4 is.). This quantity

represents the velocity normal to the nominal plane of motion. Let W = r sin 4,

From Equation (B. 19)

•• _ Bina- Di W —4--- B 3 r 2

[(1 - cos 0") sin (s+ 24)) - (1 + cos cr) sin (+240+ 2 cos Cl" sin 4),

3 r sincr B2)(13 1){(1 - cos o-) cos (4,+ 24)) (1+ cos cr) cos (1)-

4

) + 2 cosa-costd, —A— -17 (i + 2 B/A) (1 - 2 B/A)

(B• 30

= 4

 sin ci)) o- 3 r in . B2 D i [(1 - cos o) sin (4)+ 244 (1 + cos o-) cos (4)- 

——

2 ' ' + 2 cos sin lil . -2 15-

(

A (i+233/A)2 (1 -2 B/A)2

(B.31)

5. Lunar Perturbations

When the moon is the second body, B = 2 radians per month, /80,

and (Di /D)=1 /80. Take r = 26 x 103 miles and B/A = 1/28. The magnitudes

of the various perturbations are then easily estimated.

OM!

deconvent contains tolossooffoo offoeffno Ike notional defense of the United Slates wilful.. Hos Iowans of 1Me Esolenows lows. Title
TR -59-0000-00604

Page 111-138

From Equations (B. 21) and (B.22),

+ 3.75 x 10 -5 (periodic terms) rad/day,

=4, 0 + At + 3 10-6 (periodic terms) rad (t in days).

The periodic terms do not add up to more than 5. With this observation and

recalling that the approximation used in Equation (B. 13) might be 30 per cent

low, the maximum estimated perturbations due to the moon are

6 ()max <1.02 x 10 -5 rad/hr = 1.7 x 10 -7 rad/min = 2.8 x 10 -9 rad/sec,

6 (r )mom 4.0.265 mile/hr = G. 39 ft/sec,

6 itii)max < 1.95 x 10 -5 rad = 0.001 deg,

6 (rtp )max 4 0.51 mile.

From Equation (B. 29),

6 (r)max 4 0.13 mile/hr,

6 (r)in 4 0.26 mile.

The mean value of r is not quite the same as the unperturbed value. From

Equation (B. 23),

3

r

o =

* i B2 Di)

r0 + o 0 3 -2A

Thh document contains Infesmelien effecting the nedismal defense of the Uoiled Slates within the numming of the Espionage Laws. Title ill 11 TO% 1164 . 
TR-59-0000-00604

Page 111-139

3

where r re—

o

*

 = —2 is the unperturbed value of r. The mean value of r is

A

given by

D

r +6 =r o o o*B 15 ro =Yr() - 0.09 mile.

A

From Equations (B. 30) and (B.31),

6 (W)max 0.6 mile/hr = 0.9 ft/sec,

6 (W)max G Z. 4 miles.

6. Solar Perturbations 

(D)

When the sun is the second body, B 2w radians per year and -15-

With r = 26 x 10 3 mile-s and B/A = , the magnitudes of the perturbations

are again easily estimated.

From Equations (B.21) and (B.22),

6 (J)max 44 x 10 -6 rad/hr = 1.1 x 10-9 rad/sec,

•

6 (4)max <0.104 mile/hr = 0.15 ft/sec,

6 (*)max < 7.5 x 10 -6 rad = 4 x 10 -4 deg,

6 (ripmax G. 0.20 mile.

From Equation (B. 29)

(i)max Z. 0.05 mile/hr,

6 (r)m

0.1 mile.

C-04Fitteaukt___

Title . ifentment seateles information affectioe the nefienel defense of the United &Wes within the meowin g el 11s. Es/46mm lows. Title
TR -59 -0000 -00604

Page III-140

The mean radial distance is

r = r - 0.034 mile. 0 0

From Equations (B. 30) and (B. 31)

6 *max 4. 0. 23 mile/hr

6 (W)m <.0.9 mile.

relotaim o►. .11 au. 11.1s.•
TR-59-0000-00604

Page 111-141

APPENDIX C

ATTITUDE CONTROL

The first step in setting up the equations of attitude motion is the choice

of a suitable attitude reference system. We shall choose the geocentric

reference axes xyz with origin at the:center of mass of the vehicle. The

z axis is directed along the outward-pointing geocentric radius through the

vehicle. The x axis is normal to the z axis and lies in the instantaneous

orbital plane. It has the same sense as the vehicle's orbital velocity. The

y axis is chosen normal to the xz plane so as to form a right-handed coordinate system. The angular velocity of this xyz system in inertial space is

—0 w = wX

eX +w y ye +w 

Z Z

e ,

resolved along the xyz system.

There are other more complicated attitude reference systems which are

discussed in detail by R. E. Roberson, a leading authority on attitude motions

of satellite vehicles. We shall use his notation in the following analysis.

Let XYZ form a right-handed orthogonal system of axes imbedded in

the body of the vehicle, such that XYZ coincides with xyz for the correct

attitude. Let us choose the XYZ system along the principal axes of inertia

of the vehicle which has principal moments of inertia I x, 1y, and I. The

angular velocity of XYZ in inertial space resolved along the XYZ system is

.4* —4

= Ls) e e e X X Y Y + w Z Z

Roberson, Robert E., (a) Attitude Control of a Satellite Vehicle--An Outline

of the Problems. Presented at the Eighth International Astronautical

Congress, Barcelona, Spain, October 6 to 12, 1957. (b) Orbital Behavior

of Earth Satellites; (c) Torques on a Satellite Vehicle From Internal Moving

Parts; (d) Gravitational Torque on a Satellite Vehicle; (e) Principles of

Inertial Control of Satellite Attitude.

These papers may be obtained from Autonetics, North American Aviation,

Inc., Downey, California.

This document contains information affecting the notional defense of the United Steles widths the meaning of the Espl laws, TitIe

1111, U.S.C.. Section 793 and 794. the transmission et revelation of which In any manner le an unauthorized person Is prohibited by low.
TR-5.9-0000-00604

-GONFIDERAL-- Page III-142

If LX' LY1 and L are the torques applied about the X, Y, and Z axes

of the body, then Euler's equations, which describe the general motion of

the body about its center of mass, take the form

lex + (lz - Iy) cortoy = Lx

(C. la)

Iy(ay ( Ix Iz) COxWz = Ly (C. 1b)

Izc4 z + (Iy - Ix) w ywx = Lz(C. 1 c)

Now we must express these equations in terms of attitude deviation

angles. Let the angles 9 1 , 92 , and 03 specify the orientation of XYZ

relative to xyz. These angles are measured in the way shown in Figure C-1.

0.

1

 is a rotation about the x axis. 6 2

 is a rotation about the y' axis which

• lies in the yz plane and makes the angle 0 1

 with the y axis. 0

3

 is a

rotation about the z" taxis which lies in the z'x' plane and makes an angle

e

2 with the z' axis. The, unit vectors in the XYZ and xyz systems are

related by

X

= cos 02

 cos 8

31x

+ (sin 0

1

sin 0

2

 cos 93 + sin 9 3

-+ cos 0.) e

y

+ (sin A l sin 0 3 - sin 02 cos 0 1 cos 03) 'Z (C. 2a)

eY = - sin 03 cos 92tx + (cos 0 1 cos 03 - sin Al sin 02 sin 0 3) ly

+ (cos 0 sin 9 3 1 + sin 62 sin 0 3 cos ei)

-043 -0 = sin 02ex - sin 0 1 cos e2ey + cos 0 1 cos Aze z .

(C. 2b)

(C. 2c)

This document conteins latermelien Wooing lb. notional defense at the United Stoles within the eseentne el the Es pIeneoe Lows. Till.

It, U.S.C., Section 793 and 794, the transmission or revelation el which in any manner to an eneetherized person Is prohibited by low.
TR-59-0000-00604

Page III-143

CONFIDENTIAL.

x x'

x, x'

Figure C-1. Relationship of Body Axes to Reference Axes.

Mar

Is decrmenl sagging Inlimm ation Whiting II.. mollersgl Minor of tholed Stags within Ni. mormlno of M. Isnignomp tows, Tills

IS, U.S.C., Sullen 7,3 gad 794, His transmission or roggiallon 01 which in any Nonni, fa an grapollsorised porton Is prehlblled by low.
TR-59-0000-00604

Page 111-144

LILA

Any infinitesimal rotation may be expressed as a. vector 71-..d which is

= de x+ d02 ey, + d0 3

 -gz"

= (dO 1 + d9 3 sin 02 ) ex + (d0 2 cos e i - d.0 3 cos 8 2 sin Oi)

+ (del sin 9 1

 + de cos 82

 cos 1) z •

(C.3)

The angular velocity 2' of the body axes relative to the reference axes is

therefore

-4

= ( 6

1 + 8

3

 sin 92

) ex + (62 cos 1

 - 03 sin 9 1

 cos 92) e y

-I: (0 sin 8 1

 + 93

▪

 cos O f cos 02) z .

The relationship between the angular velocities is

-0* -4 -0

co = co+ .

(C. 4)

(C.5)

and co' may be expressed in terms of ex, ty, and tz by Equations (C. 2).

Then it follows immediately from Equation (C. 5) that, for small attitude.

deviations,

wX =+ + - co 0 +

• 

9

x: y 3 z 2

wY = coy

 + 42

 + co

z

0

1

 - c.ox9

3 - 15 1

9

3

w = co Z z

 + 9 3• + (4)3c8

2

- toye

l

 +81 0 •

(C.6a.)

(C. 6b)

(C.6c)

When these expressions are placed in Equations (C. 1), the equations-of attitude

motion for small values of 0

1' 13 2' and 93 (less than 10 o

) take a useful form:

This detentes,. contains infermation eattling the natlenal defense of the United Slates within the weaning of Me Espionage Lows, Title

11, U.S.C., Section 713 end 7114. the tronswission or revelation of which In any wanner to en eneetherised person is prohibited by low.
—J

TR-59-0000-00604

Page III-145

I (w +6 +0) e - +6e)

X dt x 1 y 3 z 2 z3

+ (1z ." 1y) (0), + 62 + wz6 1 - 0x0 3 - ) (coz

 + A 3 +

x

0

2 - toy

+ 0 10

2

)

= Lx

I 4-- (co + +coe -toe - e Y y 2 z 1 x 3 1 3

+ (Ix - Iz) (wx + 8 1 + 01,9 3 - wze z + 62 ° 3 ) (ta z + 0 3 + wx0 2 - wye i . + 8 i 02)

(C ,.7a)

= L (C. 7b)

1 z 41, (4),_ + 6 3 + cox0 2 - 03, 1 + 8182

+(Iy - Ix) (toy + 62 + taze oxe 3 - 60)3 ) (cox + 8 1 + wy3 coze 2 + 6203)

= LZ (C. 7c)

Now in our application, the terms containing co x and coz in Equations (C. 7) are

extremely small compared to other terms and so may be neglected. If ;1 7 is

the rate of regression of the nodes, y is the angle the orbital plane makes

with the equator, and p is the angle from the ascending node to the satellite,

as shown in Figure C-2, then

co = Â sin y cos p

.

wy = 13 +It cos y

w = z A. 11 sin y sin f3 .

This document contains infoneatiae effecting the national defense of the United Elates within IM meenine of , the Espionage Laws, Title IS. U.S.C.. kellen 703 end 794. the transmission er revelation eV which In any manner to on unauthorised person Is prohibited by low.
TR-59-0000-00604

Page III-146

North

S = Satellite

Figure C-2. Geometry of the Coordinate System

for the Satellite in Orbit.

Thls decemeM sonteins Information *fleeting the notiesal defense of the United Votes with', the mooning of the Eeploneee Lows. Title

IS, U.S.C., Section 7113 sod 794, the transmission Si revelation a which In any manner to on unauthorised person is orohl bbert by
TR-59-0000-00604

Page III-147

CONFIDENTIAL

If the orbit were kept exactly circular and at either y = 0 or w/2, co x

 and

z would be zero. If the orbit is such at y 5 2° or a 88°, the forces

required to overcome the oblateness perturbations are about a gram in size

(assuming a 1000-pound satellite). If these correction forces are not applied,

the values of cox

 and co may be found from the rate of regression of the

nodes given by Blitzer and. Wheelon.

*

[ AS2 4 3 x 10 -9 rad/sec

Y 4 2 -10

co , X Z

As2

For y - 88° -10

cox, (oz .< 10

Thus, it is clear that, in any case, co

x

and caz

are so small that the terms

•

underlined in Equations (C. 7) may be neglected. The terms containing co

may also be neglected and wherever co appears it may be replaced by coo, Y. the angular orbital velocity of the satellite. co o

is approximately 10 -4 rad/sec.

The torque L on the satellite is produced by the gravitational gradient,

by rotating internal parts, by electric and magnetic forces, by radiation

pressure, by a control system, and by many other forces. We shall consider

here only those torques arising from the gravitational field, the control system,

and a rotating part having a constant angular mementum, H, fixed relative to

the body. The other torques are mostly small enough to neglect, though it is

conceivable that in certain satellite designs they could play an important part.

The components of 1,-) are

Blitzer, Leon, Morris Weisfeld, and Albert D. lirheelon, "Perturbations

of a Satellite's Orbit Due to the Earth's Oblateness," J. Appl. Phys., 27,

1141, October 1956.

Blitzer, Leon and Albert D. Wheelon, "Oblateness Perturbation of

Elliptical Satellite Orbits, " J. Appl. Phys., 28, 279, February 1957.

(c) Blitzer, Leon, "Apsidal Motion of an ICY Satellite, " J. Appl. Phys., 28,

1362, November 1957.

This slateMeel contains intemootion affecting the notional deism of the United Pates within the mooning el the Espionage Lows, Title

U.S.C., Section 793 end 794, the transmission or ninelotion whirh In nnw ...Inn.. on. • ••

For

10-10
Y

4

2 + (IX

-

) eY 1 3

+ H(0- 91%) + Lxa

(C. 9a)

w (C. 9b) z

2C - o I ) + LYa

TR-59-0000-00604

Page III-148

CONFIDENTIAL,

= - 3w (1Y

 -I ) 9 +H (6 .+13 12 -91

 w 

y 

)-HZ (wy - e

o 13 +i2 ) (C. 8a)

Ly = - 3cdo

2

 (Ix - Iz)9 +Hz (A +i 29 3 +cay9 3 )- Hx (e 3 + 8 1 92 -toyed+ Lya (C.8b)

LZ HX (62 - 6 1

9

3 + wy) - HY (6 1 + 6 28 3 + wy

8

3) Lie (C. 8c)

-• where La

 is the torque applied; by the control system and

-0 = HL..e +H e +H e. YY ZZ

is an angular mementtun fixed in the body of the satellite. Apparently we

should choose HX= HZ

= 0 in order to eliminate the constant driving torques

- 

in Equations (C. 8a) and (C. 8c). Placing the expressions for Lx, L , and

• Lz

 in Equations IC. 7), replacing w by w y

o

, and dropping small terms, we

obtain

Iz 8 3 + (Iy - Ix + Iz) 8.1 82 + (Iy - Ix - I ) woe i = - Hy (e l + w09 3) + LZa

+- IX) 10 2

Y 9 o 93 (C.9c)

• MN dimming matelot lefetteetten effecting the netlenel defense of the United Steles withle the mennlog the Ispleatiee laws. Title

U.S.C., SWIM 7,3 sad 714, the trensinissicia or teveletilie of whkh le coy 111111111110, S. OR egettthetbeli penes is ptelgtolled by law.
TR-59-0000-00604

Page III-149

Let us now consider the initial motion of the satellite, when it has been

placed. in the desired orbit. To be conservative, let us assume the initial

angles and rates to be around Z

o

and 1/2 rpm (3.5 .x 10 -2 radian and 5 x 10-2

radian per second). These are typical separation angles and tumbling rates

for present nose cones.. Suppose that it is desired to reduce 0 and 6 down

to 10 -3 radians and 10 -4 radians per second, respectively. For this range

of angles and rates, the terms underlined in Equations (C. 9) may be neglected.

• Let us assume that H is of the order of 10 slug ft 2

 radians per second and.

thatIX' IY' and I are of the order of a few hundred slug ft 2. Keeping only

the important terms in Equations (C. 9) we obtain

X1

=HY

O 3

-(x

X 

+I

Z

 -1)66

3

 + L Y Xa

Iy

= - (lx - zz - Iy) 6 1 63 + Lya

= - Hy 6 i - (I - Y X Z 1 2 + ) + LZa

(C. 10a)

(C. 10b)

(C. 10c)

In order to make the solution of these equations as simple as possible, choose

. control torques

(C. ila)

LYa 

= + - IZ IY) 9 j 0 3 (C. 1 b)

k3 0 3

x38 + 3 (Iy Iz) 2 + (C. lic)

Then Equations (C. 10) become

I +Xi +k0. - 0 X i it - i = 1, 1, 3 , (G. 12)

CONillf_NTIAL

This hutment contains IntErneetlas ollecting the neetenel i defense et the. Unbetl Steles within the emceeing et the Esplearige Levi. Till.

IS, U.S.C., Section 713 sod Mt the trenwelstlen or reveletlen of which In say meow kw en uneetherlaed pone. Is psehlbll ynt by tow.

L'Xa = - x1 6 1 + (Ix + 1z - Iy) 6 .9 3 - Hyi 3
TR-59-0000-00604

Page III-150

which have the simple solutions

9 1 (t) = Oi (0) exp -Tr t [cos tti t +,--,--

i i

ri-ro) + 2Ti) sin SE, t.1 i = 1,2;3

600) ki

(C.13)

where

k

-1

- T.1) 2 1/2

1

This solution can be rewritten

(

Xi

9 (t) = C* 

exp mrs- t CO8 ( ni

t+ e

l

)

i i 411

Where

9

1

(0)

C i

cos E

• and

9 1 ( 0 ) + 0, (0)

t * = tan-1 "i 1 

11 1 9 i (0)

(C.14)

(C. 15)

(C. 16)

(C. 17)

If we choose Xi

 80 foot pound seconds per radian and k. tat 90 foot pounds per

radian, El i will have the value 1./3 radian per second and the oscillations will

not exceed 10 degrees in magnitude. The forces needed to apply the torques

X.9 and k9 do not exceed two pounds. The forces required to cancel the terms

containing Hy and Oiej do not exceed one pound. Within two or thrtie minutes

the oscillations should be reduced to the desired values and the control system

turned off. If we choose simpler torques:

This dogwood cordate. inlormetion eNectIne the nelleoel defense of She United States whisks the mooning of rho Isolesege Cows, Me

IS, U.S.C., Section 7►3 end 794. the transmission er reveletion of which In say meaner to en onewthorised person Is oreldirDed by low.
TR-59.-0000-00604

Page III-151

IX

A l= HY

0

3

(C. 18)

I AZ 3

= Hy 0 1 (C. 19)

`

A 3 ( 0) iz + A l ( o) IXIZ sin S2 t - 8 3 (0) Iz cos SI t 20)

equations

•

The solutions are

9 1 (t) = ( 0) +

1"Xa

LYa = - k29 2 - X2

6

2

LZa = - k

3

9

3 - X3

A 3

and solve Equations (C.10) numerically, we find that the oscillations behave

qualitatively in the same way. The angles still remain less than 10 degrees

and in a few minutes they are reduced to the saran values desired.

Now suppose that the oscillations have been reduced so that 9 •4 10 -3 radian

and <10-4 radian per second. Is a control system necessary, or will the

oscillations remain small? The largest terms in Equations (C. 9a) and (C.9c)

are the H terms. The dominant motion of A l and 03

 is then given by the

0 3 (t) = 03 (0) + - A l (0) Ix + A3 (0)11--c-I; sinS1 t+ A i (0) Ix cos f2 t) (C.21)

where

HY Q -

z

(C. 22)

This &Kew**, tootales leterwellen effecting the nolleaol: delouse el the United Motes withle Nor weenie, et the Es pleaoge tows. Title

la, U.S.C., Section 7Ill end MM. the trenunisslee er reveledloo of which In any swum/ N oe enoodworteed person Is prohibited by low.
TR-59-0000-00604

. Page III- 1 52

Therefore, if 0(0) =10 -3and 6 (0) Ix '800 slug it2 and I z =150 slug

ft, then 01 and e 3 will stay less than 1.5 x 10 -3 radians if we choose

HY = 80 slug ft radians per second.

-

The 02

 behavior is given by

+ "X IY) 4143 = 3uct "X IZ) 92 (C. 23)

The gravitational restoring torque on the right-hand side is not large enough

to keep 82 small. To see this, let Ix = Iy + Iz (to remove the nonlinear

term). Then the solution of Equation (C. 23) is

82 (0)

9

2

 (t) = sin w t + 02

(0) cos w t ,

g

where

[ 2 (IX

 — IZ

)11/ 2

3w

o

=10-4

 .

• 

for 0(0) =10- and 8(0) 10-4,

(C. 24)

02 (t)cesin 10-4 t + 10-3 cos 10-4 t ,

from which it is clear that 0 2

 will not remain small. The gravitational

 ,

torgue will not be helpful unless 02 (0) is less than 10 -6 radians per second.

Now let us consider the 0 2 motion when the nonlinear term of

Equation (C.23) is not zero (letting Ix = 800, Ix = 650, and Iz = 150 slug ft 2

).

Iy .42 + (Ix - Iz - Iy) 9183 = 0 (C. 25)

This Ammon. ceabilms Islonmellen offedIng lbw nelkwel•deleas* el lbw WIN Sohn wIlbla lbw asseelog si Ne Espioneye laws. TM.

IL SeclIew 753 wed 7U. lb. Inunills nlso et ameba* el which In soy wwwww le wo Yeeethwripd p.rree k owellblled by low.
TR-59-0000-00604

Page III-153

----0014RDENTik_

(Here we have neglected the gravitational torque because it is small compared

to the nonlinear term.) Using the solutions for 6 1 and 93 given in Equations

(C.20) and (C.21), we obtain

0 (t) = 92

(0) + 2

(0) t

(Ix - IZ - IY)

(o) 93 (o) (1 - cos 212 t) 4 c2.2 I

Y

Z ' X1 (1 + [93

2

 (0) - 9 1

2

(0) 7 sin 2 SIt -

X at.

required are less than a gram in size. If 62 (0) can be reduced to 10 6

 radians

per second, corrections will be needed only about once a day. If H.y were perfectly aligned along the Y principal axis .of.the body,..it might be possible to

reduce all three angular rates to 10 -7 rad/sec, in. which. case it would follow

from Equation (C.23) that control forces would no longer be needed.

This slocumani contains information affecting lito heliostat &has* of Ms 11011ad States within the treenning of Ow balsam's imms, Tills

3crtson 793 and 794, the transmissiaa or revelation of which is any moony to on voodberised Donee is prohibited be low.

Apparently 9 2 will not remain small. Therefore, it will be necessary to use

an applied torque, L

Ya, to control the 0 motion. The control forces
TR-59-0000-00604

Page III-154

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