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A DERIVATION OF THE THREE DIMENSIONAL THREE DEGREE OF FREEDOM RE-ENTRY EQUATIONS OF MOTION I N TERRESTRIAL NAVIGATIONAL

COORDINATES WITH SUBSEQUENT RESOLUTION INTO INERTIAL PLATFORM COORDINATES

BY

C a r l L C o l w e l l ...-

01s PRICE

MSFC - F ~ r m 523 (Rev. Nwembsr 1960)

https://ntrs.nasa.gov/search.jsp?R=19640005971 2020-05-25T20:25:14+00:00Z

GEORGE C. MARSHALL SPACE FLIGHT CENTER

MTP-ASTR-N-63-10

A DERIVATION OF THE THREE DIMENSIONAL THREE DEGREE OF FREEDOM

COORDINATES WITH SUBSEQUENT RESOLUTION INTO INERTIAL PLATFORM COORDINATES

RE-ENTRY EQUATIONS OF MOTION IN TERRESTRIAL NAVIGATIONAL

By Carl L. Colwell

AB 5 Tuc .r 15- B FA-

The three dimensional three degree of freedom re-entry equations of motion and the input requirements for their solution on the IBM 7090 digital computer are derived. The material presented is the first formulations required to achieve eventually a hardware error analysis capability for three dimensional six degree of freedom closed loop re-entry guidance and control. dimensional three degree of freedom re-entry trajectory expressed in terms of terrestrial navigational coordinates. The derivation of the coordinate transformation equations required to express the solution of the above equations in terms of inertial platform coordinates is also presented.

The presentation consists of the derivation of the three


MTP-ASTR-N-63-10

A DERIVATION OF THE THREE DIMENSIONAL THREE DEGREE OF FREEDOM RE-ENTRY EQUATIONS OF MOTION IN TERRESTRIAL NAVIGATIONAL

COORDINATES WITH SUESEQUENT RESOLUTION INTO INERTIAL PLATFORM COORDINATES

By Carl L. Colwell

PERFORMANCE ANALYSIS SECTION GUIDANCE AND CONTROL SYSTEMS BRANCH

ASTRIONICS DIVISION

TABLE OF CONTENTS Section Page

I . INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 1

I1 . DEFINITION OF COORDINATE SYSTEMS . . . . . . . . . . . . . . 2

I11 . DERIVATION OF SPACE-FIXED ACCELERATIONS . . . . . . . . . . 2

IV . DERIVATION OF GRAVITATIONAL FORCES . . . . . . . . . . . . . . 6

V . DERIVATION OF AERODYNAMIC FORCES . . . . . . . . . . . . . . 7 VI . DERIVATION OF THE COORDINATE TRANSFORMATION EQUATIONS . . . 11 VI1 . ASSEMBLAGE OF THE FINAL EQUATIONS OF MOTION FOR NUMERICAL

SOLUTION . . . . . . . . . . . . . . . . . . . . . . . . . . 15

VI11 . INPUT REQUIREMENTS FOR THE COMPUTATIONAL PROCEDURE . . . . . 18 IX . CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . 19

LIST OF ILLUSTRATIONS

Figure Title Page

1 . RE-ENTRY COORDINATE SYSTEMS . . . . . . . . . . . . . . . . 3

2 . AERODYNAMIC FORCE RESOLUTION . . . . . . . . . . . . . . . . 10

3 . COORDINATE AXES EULER ROTATIONS . . . . . . . . . . . . . . . 13 4 . OBLATE SPHEROID FORMULAE . . . . . . . . . . . . . . . . . . 17

iii

DEFINITION OF SYMBOLS

DEFINITION

Space-fixed azimuth of initial space-fixed velocity

UNITS

rad

rad

SYMBOL

AI

Aw Earth-referenced azimuth of earth-referenced wind velocity

Earth-referenced azimuth of earth-referenced aerodynamic velocity

rad AZ

Aerodynamic drag coefficient for zero angle of attack 0

cD

Total aerodynamic drag coefficient cD

Total aerodynamic lift coefficient cL

D Magnitude of the total aerodynamic drag force

Total aerodynamic force FA Total aerodynamic drag force

Total gravitational force

Total aerodynamic lift force FL

G Universal gravitational constant

Magnitude of the gravitational acceleration in the geocentric radial direction r G

m/s2 Magnitude of the gravitational acceleration in the direction of increasing earth longitude

m/s2 C

Go Magnitude of the gravitational acceleration in the direction of increasing geocentric latitude

J Second order figure constant of the earth rad

kg

kg - s2/m m

m2

L Magnitude of the total aerodynamic lift force

M

R -

Mass of the earth

Displacement from the center of the earth

S Vehicle effective aerodynamic surface area

iv

SYMBOL

DEFINITION OF SYMBOLS (Cont'd)

DEFINITION UNITS

U - vA

vA

vW

D a

aL

a r

A a

a @C

bD

bL

0 h

h

j

m

9

r 0

Gravitational potential of the earth

Aerodynamic velocity

Magnitude of aerodynamic velocity

Magnitude of initial space-fixed velocity

Magnitude of earth-referenced wind velocity

Empirical aerodynamic drag coefficient caused by vehicie angie of attack

Empirical aerodynamic lift coefficient caused by vehicle angle of attack

Magnitude of space-fixed acceleration in instantaneous direction of increasing geocentric displacement

Magnitude of space-fixed acceleration in instantaneous direction of increasing space-fixed longitude

Magnitude of space-fixed acceleration in instantaneous direction of increasing geocentric latitude

Empirical aerodynamic drag coefficient as a function of angle of attack squared

Empirical aerodynamic lift coefficient as a function of angle of attack squared

Ini t ia 1' geocentric a It i tude Geocentric altitude

Inclination of the initial orbit plane

Vehicle mass

Dynamic pressure

Magnitude of initial radial displacement from the center of the earth

m2/s2

m/s

m/s

m/s

m/s

rad

rad

m/s2

m/s2

m/s2

rad

rad

m

m

rad

kg-s2/m

kg/m2

m

V

DEFINITION OF SYMBOLS (Cont'd)

DEFINITION UNITS

m

SYMBOL

Magnitude of radial displacement from the center of the earth

r

Equatorial earth radius m

m

m/s

E r

Polar earth radius

Magnitude of initial radial velocity from the center of the earth

r P

0 ;

Magnitude of radial velocity from the center of the earth

r

.. r m/s2

S

d/S

Magnitude of radial acceleration from the center of the earth

Time

Space-fixed velocity of air molecule in a stationary atmosphere

t - V aE

Space-fixed velocity of air molecule in flowing atmosphere

V . w a

Magnitude of space-fixed velocity in the instantaneous direction of increasing geocentric displacement

V r

Magnitude of space-fixed velocity in the instantaneous direction of increasing space-fixed longitude

Magnitude of space-fixed velocity in the instantaneous direction of increasing geocentric latitude

V @C

Reference space-fixed earth-centered Cartesian coordinate in equatorial plane

X m

Earth-centered Cartesian coordinate generated by first Euler rotation

x1 m

Earth-centered Cartesian coordinate generated by second Euler rotation

X 2

m


m Y

vi

SYMBOLS

y1

y2

z

z1

2 z

V

At

DEFINITION OF SYMBOLS (Cont 'd)

DEFINITION

Earth-centered Cartesian coordinate generated by first Euler rotatJon



Earth-centered Cartesian coordinate generated by first Euler rotation


"Del" vector operator, or "Nabla"

Numerical integration time step

Data printout time step interval

Space-fixed longitude

Space-fixed longitudinal angular velocity

Space-fixed longitudinal angular acceleration

Space-fixed total angular velocity

Angle of attack

Bank angle: defined as the angle between the aerodynamic lift vector and the vertical geocentric plane containing the aerodynamic velocity vector

Space-fixed displacement in the initial cross range direct ion

Space-fixed displacement in the direction of the initial geocentric altitude

Displacement of sea level surface from the center of the earth in the direction of the initial geocentric a 1 t it ude

UNITS

m

m

m

m

m

l / m

S

S

rad

rad/s

rad/s2

rad/s

rad

rad

m

m

m

vii

SYMBOL

?P

@A

@I

0 x x

0 i i .. x 5

P

0 @C

@C

4 0

C i

@C

..

w

DEFINITION OF SYMBOLS (Cont 'd)

DEFINITION

Displacement from sea level surface in the d-rection of the initial geocentric altitude

Earth-referenced dive angle of aerodynamic velocity

Space-fixed dive angle of initial space-fixed velocity

Initial ea r th longitude

Earth longitude

Initial ea r th longitudinal angular velocity

Earth longitudinal angular velocity

Earth longitudinal angular acceleration

Space-fixed displacement in initial downrange direction

Atmospheric mass density

Initial geocentric latitude

Geocentric latitude

Initial geocentric latitudinal angular velocity

Geocentric latitudinal angular velocity

Geocentric latitudinal angular acceleration

Angular velocity of ear th rotation

UNITS

m

rad

rad

rad

rad

rad/s

rad/s

rad/s2

m

kg-s2/m4

rad

rad

rad/s

rad/s

rad/s2

rad/s

viii


MTP-ASTR-N-63-10

A DERIVATION OF THE THREE DIMENSIONAL THREE DEGREE OF FREEDOM RE-ENTRY EQUATIONS OF MOTION IN TERRESTRIAL NAVIGATIONAL COORDINATES WITH SUBSEQUENT RESOLUTION INTO INERTIAL

PLATFORM COORDINATES

By Carl L. Colwell

SUMMARY

The three dimensional three degree of freedom re-entry equations of motion and input requirements for their solution on the IBM 7090 digital computer are derived. The material presented is the first formulations required to achiev,e eventually a hardware error analysis capability for three dimensional six degree of freedom closed loop re-entry guidance and control. The presentation consists of the derivation of the three simultaneous differential equations of motion required to solve for the three dimensional three degree of freedom re-entry trajectory expressed in terms of terrestrial navigational coordinates. The derivation of the coordinate transformation equations required to express the solution of -the above equations in terms of inertial platform coordinates is also presented.

SECTION I. INTRODUCTION

The problem of hardware error analysis requires that accurate simulation models of various types of mission trajectories be available. These trajectories are used to determine the effects of various instrumental errors on the overall accuracy of the guidance systems.

The general approach to obtaining three degree of freedom equations of motion for a three-dimensional re-entry trajectory is a treatment of classical particle dynamics utilizing vector kinematics for the derivation of the space-fixed acceleration. The dynamical model of the re-entry vehicle is a point mass subjected to the forces of the gravitational field of the earth and aerodynamic lift and drag.

First,the space-fixed acceleration, expressed in terms of a moving coordinate system, is derived by vector kinematics for direct substitution into Newton's Second Law. The gravitational and aerodynamic forces are then expressed in terms of that moving coordinate system and equated to the vehicle mass times the space-fixed acceleration. Subsequent cancellation of the unit vectors from the three component equations then yields the three scalar differential equations to be solved simultaneously for the three dimensional re-entry trajectory.

2

A judicious sequence of Euler angle rotations of the moving coordinate axes then transforms the trajectory parameters expressed in the moving coordinate system into equivalent parameters in a coordinate system oriented in the direction of an idealized inertial platform coordinate s y s tem .

SECTION 11. DEFINITION OF COORDINATE SYSTEMS

Two rectangular coordinate systems and one spherical coordinate system are utilized in the treatment of the re-entry problem for which the center of the rotating earth is considered as being space-fixed. These three coordinate systems are illustrated in Figure 1.

- - The &, ly, 1, unit vectors represent a right-handed space-fixed

rectangular coordinate system which has its origin at the center of the earth. The x and y axes are oriented in the equatorial plane with the x axis oriented in a reference space-fixed meridian plane, and the z axis collinear with the North Pole. This coordinate system serves only as a visual aid in deriving the space-fixed acceleration and is not intended to convey any other physical significance.

- - The i@ Y l r Y 1 A unit vectors represent a rotating spherical coordinate

C system which also has its origin at the center of the earth. The unit vectors are oriented in the right-handed sense and represent the instantaneous direction of increasing geocentric latitude, the instantaneous direction of increasing geocentric displacement,and the instantaneous direction of increasing space-fixed longitude, respectively. The three simultaneous differential equations of motion are derived in terms of this coordinate system.

- The i 1 and 1 unit vectors represent a space-fixed coordinate

5 ’ q y c system which has its origin on the surface of the earth directly beneath the initial re-entry point. The unit vectors are oriented in the right- handed sense and represent the initial downrange direction, initial geocentric altitude direction, and initial cross range direction, respectively. The solution of the equations of motion is resolved into this coordinate system.

SECTION 111. DERIVATION OF SPACE-FIXED ACCELERATIONS

To derive the space-fixed accelerations in terms of the - - rotating coordinate system ‘i

vector differential operator on R:

, lr, lA, it is convenient to employ the @C -

3

Vehicle 2. M. /

F1GUR.E 1. RE-ENTRY COORDINATE SYSTEMS

4

where the f irst term i s the der ivat ive of the geocgntrLc displacement l,, the second term r e l a t i v e t o the space-fixed coordinate system lX,

is the derivative Of t h e geocentric displacement re a t i v e t o the ro ta t ing coordinate system 1 , lr, l,, and the th i rd t e r m i s the angular ve loc i ty

vector of the ro ta t ing coordinate system r e l a t i v e t o the space-fixed coordinate system.

l1’

@ C

Referring t o Figure 1, i t i s apparent t h a t the t o t a l angular ve loc i ty vector may be resolved i n t o two cpponent vectors representing angular veloci ty i n the equator ia l plane (A) and angular ve loc i ty i n the polar plane ( Q ~ ) :

Resolving the vector representing the angular ve loc i ty i n the equator ia l plane s t i l l fur ther i n the polar plane:

(3 )

- - Substi tuting equation 3 i n t o equation 1 and expressing R as r l y ie ld the familiar space-fixed veloci ty: r

- . - e . - x ( r l ) = r l + r A c ~ s @ ~ l ~ + r @

C C

r r

Thus, the space-fixed component v e l o c i t i e s are:

v = rQc @ C

v = r r

vA = r A cos Qc

- A subst i tut ion of equation 4 for R i n equation 1 yields the desired space-fixed accelerat ions.

( 4 )

( 6

(7)

5

+ (A cos Q~ + A sin oC lr - 6c 1n> c

.. - . * - " - . - - + r Qc lo + r 1 + r A cos QC 1, r = r QC lQ C C

I. - . . e . - - + r A cos Qc 1, - r Oc A sin QC 1, + r A cos Qc lA

- . . - - r A2 cos2 Qc lr - r @ A sin Qc lA C . -

+ r A2 sin Qc cos 0 - r ic2ir + C @c '.C le i i

.. - = (r oC + 2; i + r ;22 sin oC cos eC)

C C

" - + (r - r ic 2 - r cos2 Q ~ ) lr

.. . . . . - + (r A cos QC + 2r A cos Qc - 2r 0 A sin Qc) lA ( 8 ) C

Thus, the space-fixed component accelerations are: ..

a = r Q c + 2; i + r A2 sin @c cos Q~ C

@C

.. a = r - r Q c 2 - r A2 cos2 aC r .. a = r A cos oC + 2; i cos Q~ - 2r i A sin Q~

A C

It is convenient to make the substitution of earth longitude for space-fixed longitude so that the equations of motion may then be expressed in terms of geocentric latitude, earth longitude, and geocentric altitude, which are the usual terrestrial navigational parameters. This is accomplished by the angular velocity substitution:

6

A = h + w

where X is the earth longitude and w is the angular velocity of the earth. (Note that the new rotating coordinate unit vector i the old rotating coordinate unit vector lA.)

and space-fixed acceleration then become:

is collinear with h - The space-fixed velocity

- 2 .. + [r - r ic 2 - r (i + w)

+ [r 'h' cos ac + 2; (i + w> cos Q~

cos2 ~~1 lr

This acceleration equation is now of the form which can be substituted into Newton's Second Law in order to obtain the equations of motion for a re-entry trajectory.

SECTION IV. DERIVATION OF GRAVITATIONAL FORCES

The gravitational potential external to the surface of an oblate earth is assumed (Ref. 1) to be:

where G is the universal gravitational constant, M is the mass of the earth, r is the geocentric displacement of the point in consideration, J is the second order figure constant for the earth, rE is the equatorial

radius of the earth, and (bc is the geocentric latitude.

The gradient vector operator in spherical coordinates is:

1 A - 1

7

The gravitational force per unit mass may be obtained by taking the negative of the gradient of the gravitational potential:

- - . vu FG - - m

Thus, the gravitational.components of acceleration for the earth are:

G = O h

These-components are the force per unit mass terms caused by the gravitational acceleration. These terms are to be used in Newton's Second Law.

SECTION V . DERIVATION OF AERODYNAMIC FORCES

For the derivation of the aerodynamic forces on the re-entry vehicle, it is necessary to depart from the particle dynamics approach long enough to adopt a rigid body model of the vehicle by which aerodynamic parameters

8

may be defined. conical frustum having a sector of a sphere as its blunt base. vehicle model has an auxiliary, controlled, moveable lateral surface by which an angle of attack with respect to the vehicle symmetry axis may be generated in the vehicle syrmnetry plane. dynamics are assumed not to be coupled with the exterior ballistics of the vehicle center of mass, however, and the aerodynamic lift and drag forces are considered as acting through the vehicle center of mass exclusively. Then, the problem remains essentially one of particle dynamics,with the concept of a rigid body vehicle introduced only for the purpose of defining the direction of the lift and drag forces with respect to the aerodynamic velocity vector.

The vehicle is symbolically represented by a blunted The

The vehicle rigid body

The aerodynamic velocity is the velocity of the vehicle center of mass with respect to the ambient atmosphere. assumed to be stationary with respect to the rotating earth, with any variations being taken into consideration as wind effects. The aerodynamic velocity is computed as the relative velocity of the vehicle center of mass to a coincident air molecule point mass. The space-fixed velocity of the air particle in the stationary atmosphere is assumed to be :

The earth's atmosphere is

Assuming that the wind effects of the air mass are restricted to a mass flow which is laminar with respect to geocentric altitude, the velocity of the air particle then becomes:

45) lA v = v cos + o ir + (r w cos @c + vW sin C

AW w Aw l@

where V

with respect to stationary air at a geocentric displacement r. aerodynamic velocity is then obtained by taking the difference of equation 22 from equation 13 to yield the relative velocity:

and % are the speed and azimuth, respectively, of the air flow W The

- VA = (r @ - V cos i@ + i ir c w

C

For the resolution of the lift and drag vectors into the rotating coordinate system, it is convenient to define two reference angles to describe the orientation of the aerodynamic velocity vector in that coordinate system. Thus, the dive angle of the aerodynamic velocity vector is defined to be:

'A - - cos-1 {k} and the azimuth of the aerodynamic velocity

r X cos $= - Vw sin A = tan

C Z

The aerodynamic drag vector is defined

9

(24)

vector is defined to be:

(25)

to be collinear with the aerodynamic veiocity vecior but oppositely directed. lift vector is defined to be normal to the aerodynamic velocity vector, oriented in the vehicle symmetry plane, and directed oppositely from the vehicle's auxiliary lateral surface.

The aerodynamic

The resolution of the aerodynamic lift and drag vectors into the rotating coordinate axes is illustrated in Figure 2. The bottom view represents the local geocentric vertical plane which contains the aerodynamic velocity vector,and the top view represents the local geocentric horizontal plane. It should be clarified here that the bank angle 7 is defined to be the angle between the lift vector and the vertical geocentric plane containing the aerodynamic velocity vector. The lift and drag resolutions are respectively:

F = L (- sin 7 sin A - cos 7 cos 8 cos AZ) le ' C

L 2 A

+ L (cos 7 sin e,) ir -

+ L (sin 7 cos AZ - cos 7 cos 8 sin A ) lA A Z

FD = D (- sin BA cos AZ) 1@ f D (- COS BA) lr C

- + D (- sin eA sin AZ) lX (27)

With the lift and drag directions resolved into the rotating coordinate system axes, there remains the magnitudes of those aerodynamic forces which are respectively:

1 D = q S CD = 2 p VA2 S (CD + aD a + bD $1 0

10

- L

sin i" - L cos 7 cos BA

' 4 L s in 7

s in BA

y cos AZ

-

D

4 I /-L cos 7 s in eA

! \k

FIGURE 2. AERODYNAMIC FORCE RESOLUTION

11

where p is the atmospheric mass density, S is the vehicle total effective aerodynamic surface area, and aL, bL, CD , aD, and bD are empirical

coefficients which determine the total lift and drag coefficients C,

and CD, respectively, as a function of aerodynamic angle of attack and

Mach number.

0

Equations 28 and 29 yield the aerodynamic forces to be used in Newton's Second Law for the force per vehicle MSS then

- {[-sin 7 FA P 'A'

m - 2 m - - - -

+ [ - sin eA cos Az]

n v.2 s

vehicle equations of motion. The aerodynamic becomes :

sin A - cos 7 cos 8 cos Az] (a, a +bL $1 Z A

<cD + aD a + bD $11 i@ 0 C

P A + - - { [cos 7 sin BA] (aL a + bL $1 2 m

+ E- COS 0 I (cD + a,, a + bD a 9 1 ir A .

{[sin 7 cos AZ - cos 7 cos 0 P 'A' + - - 2 m sin AZ](aL a +bL a2) A

+ E - sin sin Az] (CD + aD a + bD a?)] 1?, 0

SECTION VI. DERIVATION OF THE COORDINATE TRANSFORMATION EQUATIONS

During the course of the re-entry trajectory, the orientation of the rotating coordinate axis system in which the equations of motion are expressed is continuously changing with respect to fixed space. desired to express certain inertial and space-fixed vector quantities of the moving coordinate system in terms of the inertial platform coordinate system defined in Section 11. This resolution may be accomplished by an appropriate sequence of Euler angle rotations of the moving coordinate axes and a subsequent coordinate translation of the origin of the rotated coordinate system from the center of the earth to the surface of the earth.

It is

12

z 2 1

This Euler angle sequence is illustrated - in Figure 3. The first Euler rotation is about the instantaneous 1 h axis through an angle

equa 1 to expressed

he instantaneous geocentric latitude. in matrix form are:

The resolution equations

r COS Q~ sin QC 0

- sin @c COS Qc 0 = I I o 0 1

C

- r 1

- 5

The second Euler rotation is about the newly -

eenerated Ix, axis through I A

I an angle equal to the space-fixed longitude. expressed in matrix form are:


- The third Euler rotation is about the newly generated 1z2 axis through an angle equal to the initial geocentric latitude. equations expressed in matrix form are:

The resolution

O l

30s Q - sin Q C

0 C

0

sin aC cos Qc 0 0 0

0 0 1

(33 1

The fourth Euler rotation is about the initial ir axis through an angle equal to the initial space-fixed azimuth. expressed in matrix form are:


13

FIGURE 3 . COORDINATE AXES EULER ROTATIONS

14

-

1

1

v - 5

cos AI 0 sin AI

0 1 0

- sin AI 0 cos AI

The total vector resolution is then obtained matrix multiplications which yield:

( 3 4 )

by performing the appropriate

cosAI~o~Oc cosAIcosQ sin@c- 0 cO

cosAIsin@ cosAsin@c cosAIsinQ C cosAcosQ C 0

C 0

s inAIs inAs in4 +sinAIsinAcos@c

sin@ sinOc+ C C sin @ cos@

cos@c cosAsinQc cos@c cosAcos@c 0

C 0

0 0

sinAIcos@ cos@ + -sinAIcos@ C sinOc+ 0

C C 0

sinAIsin@ cosAsinOc sinAIsin@ C coshcosOc 0 0 C

+cosAIsinAcos@ C

.cosA sinhsino I C

cosAIsinOc sinA 0

+sinAIcosh

-cos@ sinA C 0

- s inAIs in@ s inll C 0

+cosAIcosA

This combined matrix will then transform the space-fixed displacement, velocity, or acceleration vectors from the instantaneous space-fixed orientation of the moving coordinate system into the idealized inertial platform coordinate system orientation. origin along the 1 platform coordinate system, i.e.,

A subsequent translation of the axis then yields the final idealized inertial

‘I

‘Ip = VE - ro ( 3 6 )

where vp is the platform 7, vE is the resolved space-fixed 7, and r 0

i s the radius of the earth at the re-entry point.

SECTION VII. ASSEMBIAGE OF TKE FINAL EQUATIONS OF MOTION FOR NUMERICAL SOLUTION

The equations of motion may now be obtained by equating the results of equation 8 to the results of equation 17 plus equation 30. Cancellation of the unit vectors then yields the three simultaneous scalar differential equations:

L L D m Z m A m A

- - sin 7 sin A - - cos 7 cos 8 cos AZ - - sin 8 cos AZ (37)

(38) L D m A m A + - cos 7 sin 8 - - COS 8

r X cos aC + 2; <i + w) cos - 2r ic (i + w) sin Q~

L L m Z m A

- - - sin 7 cos A - - cos 7 cos 8 sin AZ

(3 9) D m A - - sin Q sin AZ

Solving each of the differential equations for the highest derivative then puts them in a form solvable by numerical integration on the IBM 7090 digital computer:

2; ic C r C C - 9 J ey sin 2Q C

.. Q = - - - (i + w)2 sin Q cos Q

L m - ${: sin 7 sin AZ + - cos 7 cos eA cos AZ

D m A + - sin e cos AZ) (40)

16

D m A - - cos e

L + r cos Q~ m

1 {i sin y cos AZ - - cos y cos e A sin AZ

D m

To solve this set of differential equations, it is necessary to provide six initial conditions. be\specified by considering the initial position of the vehicle. initial geocentric latitude and initial earth longitude may be specified directly. geocentric altitude and applying the formula derived in Figure 4.

Three of those initial conditions may The

The displacement may be derived by specifying the initial

r =

C

(43 )

The remaining three initial conditions may be derived by considerations of the initial space-fixed velocity of the vehicle:

(44 1

r = VI cos 0

r (i + w) cos Q~ = v sin eI sin I 0

0 0

(45 1

where VI is the initial space-fixed velocity, 8 I is the initial space- fixed dive angle, and A I is the initial space-fixed azimuth. space-fixed azimuth might also be expressed in terms of the inclination (j) of the vehicle's instantaneous orbit plane to the equatorial plane:

The initial

17

North Pole z

z = r sin bc @ C

P2 z2 7 + 7 = 1 rE P

South Pole

P r rE

r - r + h @C

18

The initial.conditions to be obtained from the initial velocity considerations thus become:

VI sin GI COS AI

r - ic -

0 0

I r = VI cos 8 0

I VI sin sin A i = 0 r cos @

0 C 0

SECTION VIII. INPUT REQUIREMENTS FOR TIIE COMPUTATIONAL PROCEDURE

The input requirements for the JBM 7090 re-entry program fall broadly into three categories: (a) geophysical data (b) aerodynamic data, and (c) initial conditions for the equations of motion.

The geophysical data required are the Gaussian constant (GM), second order figure constant (J), the equatorial radius (rE)’ the polar radius (r ), and the angular velocity of rotation (w). > P may be manipulated to produce any oblate spheroid and corresponding gravitational field desired. five items can be found in Reference 2.

The first four parameters

The values used by the author for those

The aerodynamic data required are atmospheric phenomena and vehicle characteristics. The Computation Division provides 1959 ARDC standard atmospheric mass density and speed of sound data, both as a function of altitude. Provisipn is made t o generate any variation from a standard atmosphere by a density modification factor which is constructed as a curve read function of altitude. A wind velocity profile as a function of altitude and a constant wind azimuth are also available. Provision is made for a constant,effective aerodynamic surface area to vehicle mass rates (S/rn), a constant angle of attack ( a ) , and a constant bank angle (7). aD,and bD) which determine

the lift and drag coefficients may be specified as a curve read function of Mach number.

$e empirical constants (aL, bL, cD 0

19

The initial conditions of the equations of motion may be effected via Section VI by specifying the vehicle's initial space-fixed velocity (VI), initial space-fixed dive angle (eI), initial space-fixed azimuth

(AI) or instantaneous orbital inclination (j), initial earth longitude

(Ao), initizl geccentric latitude (Q ), and initial geocentric altitude C

0

The print-out time interval (At) P of the integration step time interval (At) which is also arbitrary accord- ing to the computational accuracy desired.

may be chosen to be any multiple

SECTION IX. CONCLUSION

A system of equations has been derived which can be solved numerically on the IBM 7090 digital computer to describe the motion of a vehi.cle re- entering the atmosphere of the earth. and displacement as a function of time can be obtained in a spare-fixed coordinate system which is parallel to the space-fixed platform on board the vehicle. easily adaptable to hardware accuracy studies.

Components of acceleration, velocity,

Thus, the trajectory simulation given by these equations is

20

REFERENCES

1. Crane, J. A., On the Motion of a Particle about an Oblate Spheroid, British Interplanetary Society Journal, Vol. 17, No. 12, November - December 1960 , pp . 437 -439.

2. Swanson, Conrad D., Geophysical and Astronomical Constants for Geocentric Trajectories, ABMA DS-TN-101, October 17, 1957.

BIBLIOGRAPHY

1. Goldstein, H., Classical Mechanics, Reading, Massachusetts, Addison- Wesley Publishing Company, 1959.

2. MacMillan, W. D., Statics and the Dynamics of a Particle, Dover Publications, Inc. , 1958.

2 1

APPROVAL

A DERIVATION OF THE THREE DIMENSIONAL THREE DEGREE OF FREEDOM

COORDINATES WITH SUBSEQUENT RESOLUTION INTO INERTIAL PLATFORM CQORDINATES

RE-ENTRY EQUATIONS OF MOTION IN TERRESTRIAL NAVIGATIONAL

By Carl L. Colwell

The information in this report has been reviewed for security classification. Defense or Atomic Energy Commission programs has been made by the MSFC Security Classification Officer. This report, in its entirety, has been determined to be unclassified.

Review of any information concerning Department of

MYBROOKS CGief, Aerospace Physics Unit Performance Analysis Section

A? G. G. GASSAWAY Chief, Performance Analysis Section Guidance and Control Systems Branch

3.01 WwU F. B. MOORE Chief, Guidance and Control Systems Branch Astrionics Division

W. WUSSERMANN Director, Astrionics Division

22

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R-ASTR Dr. Haeussermann Mr. Digesu Mr. Mandel Mr. Hosenthien Mr. L. Richard Mr. F. B. Moore Mr. J. Mack Mr. Gassaway Mr . McMahen, I11 Mr. M. Brooks Mr. Carlile Miss Flowers Reference File Branch File (2)

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Scientific and Technical Information

Attn: NASA Representative (S-AK/RKT) P. 0. Box 5700 Bethesda, Maryland

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3' rs:f mtp-astr-n-63-10 >-august 1963 · mtp-astr-n-63-10 a derivation of the three...

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