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Ballistic Misssile Trajectory Equations
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pliriA(I (11ON OF B.ALL I STIC MI SSI LE TRAJ ECTORI ES
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CONTENTS
LIST OF ILLUSTRATIONS................... .. ..... ... .. .. .. .. . .....
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
I INTRODUCTION..................... . .. ..... . . .... .. .. .. .. .....
11 EQUATION OF MOTION .. .. .................. ........ 2
III ENDOATNOSPIIERIC PREDICTION .. .. .................. .... 6A. Characteristics of Lte Ballistic Trajectories
in Endoatmosphere. .. .. .................. ...... 7B. Influence of P~ in Endoatmosphere .. .. ................. 11C. Effect of Nonli..earity in Endoatmosphere .. .. ............. 18
*D. Influence of w' in Endoettrosphere .. .. ................. 19E. Effect of Eccentricity in Endoatmosphere .. .. ............. 23F. Approximation of Nonlinear Term in Endoatmosphere..... ...... 23
IV EXOAT'MOSPHERIC PREDICTION .. .. ......... .............. 27
A. Effect of Nonlinc-rity in Exoltmosphere. .. .. ............. 31B. Approximation of Nonlinear Term in Exoatmosphere. .. .......... 32
C. Influence of w~ in Exoatmosphere. .. .. ................. 33
D. Influence of Eccentricity in Exoatmosphere .. .. ........... 33
V SENSITIVITY OF IMPACT POINTS TO INITIAL VALUES .. .. ............ 33IV~I CO.NCLVSION .. .. .................. ........... 41
APPENDIX .. ... ....... ...... ....... ..... ...... 42
RE.ERENCES .. .. .......... ... ..................... 49
ILLUSTRATIONS
Fig. 1 Coordinate System .. .. ... .... ... .... ... .... ....... 2
Fig. 2 Density of the Atmosphere .. .. .... ... ... .... .... ...... 8
Fig. 3 Ballistic Trajectories on the z-y Plane, Case 1. .. .. ... .... ..... 9
Fig. 4 Ballistic Trajectories on the z-x Plane, Case 1 .. .. . .... ... ..... 10
Fig. S Ballistic Coefficient - Altitude Graph for fio l''8 2. .. .. ... ...... 12
Fig. 6 Ballistic Coefficient - Altitude Graph for 163, 18,1....... ........ 13
Fig. 7 Ballistic Trajectories on the x-y Plane, Case 2 .. .. . ... .... ..... 14
Fig. 8 Ballistic Trajectories on the z-x Plane, Case 2 .. .. . ... .... ...... 1
Fig. 9 Ballistic Trajectories on the z-y Plane, Case 3 .. .. . ... .... ..... 16
Fig. 10 Ballistic Trajectories on the z-x Plane, Case 3 .. .. .... ... ....... 7
Fig. 11 Influences of e, w, and C (Endoatmosphere.,B = 60) z-y Plane. .. .. ...... 20
aFig. 12 Influences of e, w., and C (Endoatmosphere.9 = 0 z-x Plane. .. .. ...... 21
Fig. 13 Influences of e, w, and C (Exoatmosphere), z-y Plane .. .. .. .. .......28
Fig. 14 Influences Of e, W, and C (Exoatmosphere), z-z Plane .. .. .. .. ....... 29
Fig. A-i Illustrations of Unit Vectors. .. .. ... ... .... ... ....... 46
TABLES
Table i Influences of e, w, and C (Endoatmosphere P= PO) . .. ............. 22
Table 11 Influences of e, wv, and C (Endoatmosphere 83 = /34). .. ............. 22
Table III Approximations of Nonlinear Terms (Endoatmosphere,
Hish 18= PO) AX +B .. .. ...... ....... .. ..... .... 24
Table IV Approximations of Nonlinear Terms (Endoatmosphere
Low/3/=4 ) i= AX+B +C ... .. ..... ....... .......... 25
Table V Influences of e, ,~, and C (Exoatmosphere). .. ................. 30
Table VI Sensitivity of Impact Points to Initial Values(Endoatrnosphere), 10% Error. .. .. ....................... 37
Table VII Sensitivity of Impact Points to Initial Values(Endoatmosphere), 20% Error. .. .. ....................... 38
Table VIII Sensitivity of I mpact Points to Initial Values(Exoatm~osphere), 10% Error. .. .. ....................... 39
Table IX Sensitivity of Impact Points to Initial ValuestExoatmosphere), 20% Error. .. .. ....................... 40
Table X Summary of Influences of e, w., C, and /.. .. .. . .. . .. . .. . .... 41
iv
I INTRODUCTION
A critical problem in a missile defense system is that of predicting
the trajectory and impact point for a ballistic reentry vehicle. This
memorandum will describe methods of prediction as well as numerical results
for several representative examples. There are several reports ,2describ-
ing the estimation of the states of a ballistic missile; the ballistic
trajectory and impact point will be predicted by using these estimated
values.
There are several important points to be considered. Fiist is the
choice of the coordinate system to be employed. Either a radar coordinate
system or a rectangular coordinate system centered at the radar site can
be used for the problems being studied. Second is the treatment of
physical parameters in the equation of motion, such as the ballistic co-
efficient of the reentry vehicle and the eccentricity and rotation of the
earth. Since the time required for computation may become significantly
large, it is also very desirable to find a closed-form solution of the
equation of motion, which is a rather complex nonlinear differential
equation. The important point here is how much the accuracy of the solu-
tion is degraded in obtaining a closed-form solution. Third is the propa-
gation of initial errors to the final values in prediction; this is val-
uable in order to trade off the magnitude of errors and the computation
time in estimation. For this purpose, the sensitivity of the initial
values to impact points is briefly investigated.
S (rfreftree are listed at the end of th text.
U
I I EQUATION OF MOTION
The following differential equation in state-variable form describes
Lhe motion of a ballistic missile; the derivation of this equation is
shown in the Appendix and the coordinate system is shown in Fig. 1.
X=AX+ B +C +D
k (z)
RADAR SITE
FIG. 1 COORDINATE SYSTEM
_g
where
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0
L 2- 0 0 0 2wsin /-3,cosu
0
o w 2 sn 2 jL -W.~2 si 4 cos /ju -2,j sin IA& - 0 0r32
0
a'$
0 ,12 sin p cos/A w2 c0 2 g. 2 o Vol 2w~cs ~~Cos A 0 pg V
L0
0
0
0
B -0
-c2fR sin u cos/p
am,w2R cos 2 A -
r-30
0
0
0
(o 1 r30 ~ 0
( r 3 r
00 Ir 1 r3
0
0
0
- (1 - 5 sin 2 ))x
/) r 3 r)
- -t -i - {cos- - -( - -( - -] - -i -, -i ( - - - - --}
*-r)iG- (I (1 Ssinin 0)1
r
_D - 1- $Ssin.t- #l~ J B( , )s-J sinos
& / ,2 sin ;zcos Fz[cas(, - )- 1) sin L. sin( , -
( 0) ( 5 sill 0p[ + at Cos4 s in ,in
ca IL CB scos AcosA( - M1) - sin i, sin(MA -~)
.~~ - 8(coa(M - ,) - I]
~r 3
in which
AT " [ , target state
w L rth rotation rate
GiV ' Product of gravitational constant'and missof earth
* Geodetic latitude of radar
* Geocentric latitude of radar
- Atmospheric mass density At target position
- Ballistic coefficient ( a.et, ight-to-drag ratio)
t * Earth radius to radar site
r Magnitude of posttion vector from earth centerto target
V Yvelocity magnitude or target
g ' Gravitational accelcration
rot ne ad pO are the ioitial talues of r. V and p
I i
C - 'dEt'tntri'ity of re'r-rence ell ipsoidal earth(,"." 0.006bQ45)
J - Iimen.eiplless colistant - 1.624 x 10- 3
Gzeocentr-it latitude of target
a nquatorial radius of earth (20,926,743 ft).
l1 e linear coeitir-sent mat rix A is a 6 x 6 matrix whose elements are
co:stant except for three , lements containing the atmospheric mass density
'llie vector B is constant amd the %ector C is a nolinear term thatbe negligible if , the magnitude of tie target ve locity, and r, the
magnitude of the position vector from the earth to the target, do not
change significantly. The last term D contains the elements describing
the influence of the eccentricity of the reference ellipsoidal earth.
ien e., if eccentricity r is considered to be zero, then the tern, D vanishes.
One of the objectives of this report is to investigate simplifications
of the differential equation des,:ribed above. If the terms C and Q are
negligible, the differential equation will become . v AA - R. It is true
that a linear differential equation with time varying coefficients is no
better for finding an analytical solution than a nonlinear different. al
equation. However, if tle time varying cofficients are approximated as
constant for a certain tame interval, tten piecewise closed-form solutions
'an be obtained.
Raxed on plh. iral considerations, it is helpful for the purpose oi
the following discussion to divide the atmosphere into two regions. One
i- called e.oatsuphere, defined as the space above an altitude of
300.000 ft; the other i4 called endoatsosphere, defined al the space below
an ailtit'jde of 300,000 ft.
III ENDOATMOSPHERIC PREDICTION
For ballistic missiles at an altitude of 300,000 ft with a range
of 200 miles, it will take less than a minute for the high-,. vehicles
to impact and at most several minutes for the low-2 vehicles. There-
fore, the gravity gradient due to the oblateness oi the eatth is negli-
gible, and the term _ can be omitted in the investigaLion ot endoatmos-
pheric trajectory prediction. The effect of the earth's rotation rate
cc is also negligible, except for small deviations that are observed
during the last 10,000 ft before impact. The impact oint is defined
in this report as the point at which a trajectory reaches an altitude
of 10,000 ft.
Theoretical considerations and numerical results obtained indicate
that the term ( in Eq. (1) is negligible for endoatmosphere prediction.
and this is especially true ior high-.- missiles. An approximate differ-
ential equation describing the ballistic trajectory takes the form
X X (2)
Moreover. if w can be considered as zero. then the differential equation
is simplified further and becomes
0 0 0 1 0 0 0
0 0 0 0 1 0 0
0 0 0 0 0 1 0
0ooo 0 -0
0 G 0 0 V o 0
0 0-f 0 0 t- lVN
1)1
06L
Thie aboye equation- geterates predicted trajectories, which are very close
to the actual tiajt'ctories down. to 50.000-ft altitude from an altitude of
'00, 000 ft. Ilencv, i f Ohe initercept. altitude is- higher than 50,000 ft,
the differential Eq. (3) Is a good approximation to the equation of motion
for both highl-,3 and 1ow-Omissiles. For prediction of the trajectory down
to 10,000-ft altitude, the above differential equation is still a good
mathematical model for hl-gb--3 missiles.
The densitv. of the atmosphere changes in a complex manner; an expo-
nenti'al curve was u:sed t o approximate the density-altitude curve. As
Fig. 2 sho~ws, this curve does not match exactly with the U.S. Standard
Atriospherei%92', However, for thc prediction of an impact point, this
exp~one~ntial model is sufficiently accur ate.
Several characteristics of Eq-. (1) in endoatmo~pherc are discussed
tin ;he follo~ing sections, and the sensitivity of impact points to initial
%alues is mentioned in Sec. V.
V. Characteristics of tih Ballistic Trajectoriesin Endoatmosphere
As shown by the numerical results (see Fig., 3), p-roj ec tions' of the
trajectories on the x.-y plane are almost straight' lines.- If the initial
conditions are the samne for trajectories with different co n a an t values
of $, then the x-y projections of their trajeftories will- lie on top of
each other with the high-," missiles flying further than the low-p missiles.
The impact Points lie on a straight line in the- x-.y plane regardless of-
.3 values.
from Fig. 3 it can be se6en that the projections of ballistic trajcc-
tories on the x-y plane do not differ very much for different values of
3. lortwvei, the projections if- ballistic trajectories on the z-x plane
di-fler slightly for different values of (see Fig. 4).
rhe sensiti.'ity of the impact point prediction to the ballistic co-
effici%:nt . 3Is a function of the value of ,B; for low 6 the sensitivity
is large, and for high f' the sensitivity becomes small. Therefore, it is
rather imp.)rtant to detect whether ia target missile has low ,B or high 8
In tit, hIgh-P case, it is possible to predict the impact point wit~h high
Mcuracy. An) inaccuracy can be made smaller by re- estimating the bal-
list ic coefficient. The major effect of P~ on the trajectory occurs at
dit Altitude of less than 150,000 ft. Since 6 comes into the differential
7
10-1
U ..S. STANDARD ATMOSPHERE 1962
EXPONENTIAL APPROXIMATION
10-2
Po0a 7.6474 X 10-2K .OXIO
9 32.174 ft/soc2
.
104
10-5
10,6
0 100 200 300z - kft
TO- 5188-451
FIG. 2 DENSITY OF THE ATMOSPHERE
120
100 INITIAL CONDITIONS:zz9. 20640 x 105 ft:4,51515 x 1O5t
z3.27897 zIO5 ft
80 - 1. 81864 X10 4 ft /SeC1 1. 12315 X 04 ft/SGC
zi-7.01397 110 3 ft/sec0-IMPACT POINT
60
40
20
0 5000,2000
-201100 200 300
x - kftTO- 5166- 452
FIG. 3 BALLISTIC TRAJECTORIES ON THE x-y PLANE, CASE 1
100 3031
INITIAL CONDITIONS: 31
X= 9.20640X10 5 ft 32
90 y= 4.51515 xlO5 ft
3,27897 xlO~ft 33
i=- 1. 81864 X 104 ft/sec 32ic - 1.12315 x 104ft/seC 34
80 i-701397 xiJ 3 ft/sec[r
70 36
37
3635
39
500
40 38t
I
3960
30
t =40 ec, 70 - /3 20
20 sO
a, 5000,2000 4' 400
42 9 0
10 60
bOO
K 30tOO 200200
Iso
K hf100 o ,00! g x kft
TB- Sign- 453
FIG. 4 BALLISTIC TRAJECTORIES ON THE z-x PLANE, CASE 1
i10
7equations in tile form ,/P and p is a verv rtmall value for high altitude,
ti/P is not a significant term unless the altitude is comparatively low
(i.e., less than 150,000 ft).
B. Influence of 13 in Endoatmosphere
In general, the shape of the /3-altitude graph is parabolic-like and
has a maximum value. Several examples are shown in Figs. 5 and 6. Ex-
perimental computations have been conducted by using the minimum, average,
and maximum values for ,. These results were then compared with the exact
solution. The predicted trajectories and impact points for some repre-
sentative cases are shown in Figs. 3, 4, 7, 8, 9, and 10. When the win-
imum value (constant) was used, the numerical results turned out to be
quite different from the exact solution. On the other hand, if the
maximum value (constant) of /3 is used, the deviation from the exact solu-
tion is not too large. For the case of high g3, the maximum value will
give the impact point without a significant error. However, the approxi-
mate value of ' will cause an error in the impact time.
If a target missile is known to have a characteristic of-high g3, it
is permissible to calculate the impact point by using a predicted maximum
value of .2 or a value slightly smaller (by 10 to 20 percent) than the
maximum value of 63. If a target missile has a characteristic of low /,the prediction of the impact point will be more difficult than that for
a high-P missile (refer to Figs. 3 and 4).
Consider the x-y projc:ction of the trajectory. The impact points
for high-B ballistic coefficients are very close to each other even
though the /5 values are different Some examples where /3 ranges from1000 to 5000 lbi'ft 2 show that the deviations are 5000 ft in the x
direction and 3000 ft in the y direction at the impact point (refer to
Figs. 3 and 4).
For low-fr and high-f missiles having the same initial conditions
and impacting on the surface of the earth, the impact points for the
low-h" missiles lie on the x-y projection of a trajectory for a high-P
missile; in other words, the projectionn of low-fr missiles on the x-y
plane are shorter than those of high-9 missiles. Moreover, the x-y pro-
jections of these trajectories are almost straight lines.
II
50000
4000
I 3000
2000
j 1000
0 I I300 200 100 0
(a)
2500 1
2000
-. 1500 B2
1I000
500
030 200 ISO 100 50
ALTITUO - kht
(b)
FIG. 5 BALLISTIC COEFFICIENT - ALTITUDE GRAPH FOR
* Po, PIt, P2
12
1000y
800-
600
e400
2001
200 ISO 120 s0 40 0
ALT ITUDE kft a5134
FIG. 6 BALLISTIC COEFFICIENT - ALTITUDE GRAPH FOR P3, P4- PS
13
0
I *
z
a.'1 00
LU
0o
00
0K
00'no -no v6 % 0z
0
w 3w
a wL
140
iI
ILIImI
IIII0
oa
2 2 *
is
J-
4 , , , I I
2 21
15
200
INITIAL CONDITIONS:
ISO- it a3. 30a I0pf ty a 3.38 a 165za *2.00 aleft
160 i -1.53 a 4ft/uian
120-
* 700-
so-
40-
DuD.. 300
fIG. 9 BALLISTIC TftAJECTORIES ON THE x-y PLANE. CASE 3
I NITIAL CONDITIONS'a3.38 Io 5 ft
100l y 6 3.383 105 it
a2.00 a 15ft I._1-.53 1O4f/t/sec
g0 g- 1-.53 104 ft/,.t
50
tou 10
40
30
20 15
20 0,.00
10
FIG. 10 BALLISTIC TRAJECTORIES ON THE a PLANE.CASE 3
11
Projections of trajectories on the :--x plane for different values
of /3are again almost tht same down to an altitude of 100,000 ft. Below
100,000 ft, the z-x projections start separating and produce Comparatively
significant differences at impact. Some examples show that, because of
different /3, values (from 200 to 5000 lb/ft 2 ), projections of trajectories
on the x-y plane differ almost 50,000 ft in the z direction an(, jOOD00 ft
in the y direction (refer to Figs. 3, 4, 7, and 8).
Naturally the deviations differ according to the different values
of the initial conditions. If the initial altitude is low and the descent
speed 1 is high, then the deviation due to the different values of P3 isnot large. For comparison purposes one example of this kind is shown in
Figs. 9 and 10.
C. Effect of Nonlinearitv in Endoatmosphere
For endoatmospheric prediction, the nonlinear term C is negligibleiiif there is no significant, change in values of r and 1'. lit the endoatmos-
phere case the maximum deviation of r will be
-o rimpact) (a 300,000) -a .3 UY ,
r(impact) a L4 1-
where a is the radius of the earth. Hence, the value i/rO3 - lir 31in the
term C is negligrible compared to Iro for the endbotmospheric missilej
traj ectori es.
On the other hand, the value [Y0 - i in the ttrma r is net neetssarily
negligible compared to Vtt. Since a high-/3 missile doeS not A1o% do*wn Sig-
nificantly, the value [V0 - V is negligible compared to i'o. Thi. telocit~y
of a low-,! missile chantres its velocity m~uch more than that of a high.,8
misaile. As a result, the value IV~ V)h is not negliltible, and for somte
IOW.-0 ami ssiI*$
beezc% 0.5 or grater.
-- -- --- -
In conclusion, for endoatnmospheric prediction the term C is negli-
_ giblc for high-/3 missiles, and the term C should be handled ':arefully for
* low-,? missiles. lit order to illustrate the effects of the term C on
prediction, trajectories of high-Bl and low-,6 missiles are shown. Figures
11 and 12 and Table I show the high-/3 case, and Table 11 shows the low-'l
case.
D. Influence of -, in Endoatmosohere
For endoatmospheric prediction, the influence of the -earth rotation
rate c, is much more significant than that of the eccentricity e. An ex-
ample of a high-,3 missile in the endoatmosphere is shown in Figs. 11 and
12 ad Table I. With and without consideration of w, the flight time
difference is less than 0.5 sec andthe deviation of the impact points is
ab(;ut 6400 ft (5000 ft in the x direction and 4000 ft in the y direction).
In this example. it takes about-43 sec to impact. A fixed point on
the equator moves about 10 nautical miles during these 43 seconds. Then
why ib the deviation of the impact points with end without consideration
of c. about 1 nautical mile rather than 10 nautical miles? The answer to
this q -stion is straightforward; The velocity of a target is measured
with respect to the moving coordinate system, which is. fixed to the
earth at the radar site and rota.es with the earth. Therefore, the
deviation of impact points with and without w is not caused by the motion
*1 of the radar site but is mainly caused by the effect of the coriolis term
in Eq. (1).
Another example is shown in Table II. This is the case of a low-,8
missile in the endoatmosphere. The deviation of the impact points with
and without consideration of c is about 5000 ft (4000 ft in the x direc-
tion and 3000 ft in the y direction). Since this is a low-I3 missile, it
takes about 74 sec to impact, which is about 80 percent longer than the
time required for the high-,3 missile with the same initial conditions.
The low-I3 missile takes a longer time to impact than the high-/3 missile,
vet the deviation of impact po:nts for the low- 3 missile with and without
consideration of u is smaller than that for the high-A missile. This is
because the main contribution 'f o) is the coriolis term, which is pro-
port ional to the vector product u x V of the earth rotation rate and the
yelocity of the nissile. Since _, is constant, the effect due to coriolis
trM depends oni the magnitude of L. The higher the velocity of a missile,
the arger the deviation that will occur.
18
20 111
39
INITIAL CONDITIONS: 3
xa 9.20640 x 105 ft
y a 4.5151 5% IO5 ft
z a3.2T89Tt los5t
1-.8 1864 i 104 ft/Sec
1.I 12315 x IO4 ft0..
10 z a-7.01397 a 1O3 ft/se
40
WITH ALL TERM
(in' a a *0
0 WITHOUT TERM C :0 41 4
42
-t0 4242.26
42AO
42.51.
-20 II60 170 I80 190 200 210 220
a-kit
FIG. 11 INFLUENCES OF w, , AND C (Endloatmo sphere, = x-y PLANE
20
40
IiTIAL CONDITIONS:
9.20640 x 10 fty --4.51515 x 10 5 ft
z-3.27897 %1 o5 ft 19i c-1.81864 x 1 04 ft /See
~~ 10l 421 O ft/seciz7.01~397 x lO0 ft/sec 39
30
f 40
WITHOUT TERM C 40& * :0
WITH ALL TERMS20wO & szQ 41
41
I442
42.4042
42.53 42.26I0L
160 )70 10 O 19 200 210 220 230-kft
FIG. 12 INFLUENCES OF e, a), AND C (Endoatmosphere ; -x PLANE
21
;:W210 10 m
0% I-- Ot 1* C
... ~~~~~ C4.CJ ~ C.. *
-' 0 - - t4 =I
C6 c - I I I I
cr- -t - - M
cor
CIAj d ~ U nIf l t~ f f
oo 0f No- -a..2 c- cc ot- .n
t- C, a, '0 '0 r C' -
in C4 t0 m'C C In II II- II I
ar *.. s
22
E. Effect. of Eccentricity in Endoatmosphere
Eccentricity e comes into the differential equation of motion as the
correction introduced into the gravitational force term due to the oblate-
ness of the earth. The deviations of impact points with and without con_
sideration of the eccentricity e are shown in Figs. I1 and 12 and in
Tables I and II.
Figures l1 and 12 and Table I show the case of a high- rissile in
endoatmosphere. The deviation is about 220 ft (180 ft in the x direction
and 130 ft in the y direction). There is no difference in impact time.
Table II shows the case of a low-i3 missile in endoatmosphere. The
deviation is about 100 ft (100 ft in the x'direction and 22 ft in the y
direction). The difference in impact time is about 0.03 sec.
In conclusion, the effect of the eccentricity e is entirely negligible
for trajectory prediction in the endoatmosphere.
F. Approximation of Nonlinear Term in Endoatmosphere
* As discussed in the previous sections, Eq. (1) can be approximated
as
X = AX + B for high 13 missiles, (4)
and
X AX + B + C for low ,8 missiles (5)
in endoatmospheric trajectory prediction.
If we consider p1/3, r, and V to be piecewise constant, then the
mutrix A and the term C become piecewise constant. Hence, it is possible
to find a piecewise closed-form solution for Eqs. (4) and (5). The accu-
racy of the solution depends upon the integration step size and the time
interval during which pl/3, r, and V are kept constant. Experimental
computations were performed by taking five time intervals, namely, 1, 2,
3, 4, and S sec. The results are shown in lables III and IV.
The purpose of obtaining a closed-form solution is to shorten the
computation time to predict the missile trajectory. The approximation
described above produces s,me inaccuracies. If the inaccuracies can be
23
ell - 0
I-a.
o 0 Lm 0 0
UD - - kr PCP C CP
SC4- -P's
I I4 Ic
PC7
In i 'Ii C -(PP 0 0'. VI C3
.~ (P if~ a' CCnU - ~. PC- -M
- 0 0 0 0 0 0MIi~~~~o inC P C PC P C PLe. 16*H a.. .. - Ci 1 - C Pf' 'C C
"a f4 Gal fog .4. 4
+ k -l 6 A 6
%j- it W o A. i
I24
I-I-
ttL.. cn n 00%C
'.4 %C4 t-1
C4 CN4C4
t4n~ .. 4 t-. .~. Sn - e,fn. ~ - n n S
1. I I I
It -4 -4 -4
-i - 44 4 44 4425
I
tolerated, a closed-form solution should be used in order to reduce the
computation time.
As a reference, it may be helpful to give an approximate computation
time to solve the differential Eq. (1) numerically. If the iteration step
size At is taken as 100 msec and if one iteration does not exceed 0.2 msec,
then the numerical calculation of a missile trajectory for a 50-sec: flight
requires less than 0.1 sec of computatioa time.
I.2i 26
IV EXOATMOSPHERIC PREDICTION
In this reporL, the exoatmosphere is defined as the space above alti-
tude 300,000 ft.. Since the characteristics of the motion of ballistic
missiles in the endontmosphere ane exoatmosphere are significantly dif-
fzrent, it is very meaningful to oLtain schemes of predicting trajectories
separatell'.
A s discussed in Sec. 111, the equation of motion of missiles in endo-
atmosphere is decvribed by Eqs. (2) or (3). The equation of motion of
missiles in exoatmosphere can be approximated as
Projections of the trajectories in exoatmosphere on the x-y plane
are almost straight lines. Projections of trajectories on the z-x plane
have the shape of an ellipse or a parabola. One example is shown in
figs. 13 and 14 and Table V.
The values of q~ is 1.488 x 10-' lb ft-3 at altitude 300,000 ft
(iFg 7. 647 4 '- 10-21b ft-3 at the earth surface). Therefore, the
elIem cn t (A g;, 2"W tins very little influence on the solutions regardless of
the value of g~. The classification of missiles is no longer meaningful
in cxoattnospheric prediction problems. If we consider the term(tg/2')y
to be negligible.. then Eq. (6) is simplified as
X +8+(7)
Whet e
0 U
*I1w
.
0 .
22 *2
ItVuIII0 3
T
-I---
4:
9
0 *a
U g
: : 'U*~ @ '42 . . - -
222222N ~ - U £ 0..
~ tjt. ~ 9~1' 2 Ng
Ee ~2~5 S.
o , Sm
U . u * U S S
8 - M ~*W 1.U
I- I* i
t 'U-U'
iU.0
iA~U0
o IL*U i- 5-S..
iiia* t2
i-US
8 8 8 8II~
I
'S
I_I
- - - - - -
0 0 00 0 0 0 00~ ., ~ ~ P t -~ ~ '0 '0
ea . (~i e~- -
~ ff~00000 0 0' 0 00
- - -
& - - - - - N - -
0 *~ *'~ C 0 0 - - '0 ~i-~ o - a' ~ .'~ ., - a'
- - - C '0 I **' 0
- ~ - - ~ 4. ~ 0 1- I-
0 0 - 0 0' 3' '' ~
(*~ t'. *. C C C C - -
I I I I I I I I
- - -
- *9 *9 .9 .9 .9 .9 .9 .9 .9
0 0 '0 00
* - - a N N - - - = 4
* 0~ ~i -~ ~ -p 0 ~ 0' 0'
* - z~ a~ ~'I ~' 0-- - - - - - -('a .p - ("a ~ -i 0 0
I. .- - -
- I I I I I
- - - - - - - -
- .9 .9 .9 .9 .9 .9 .9 .9 .9 .9
r o 0 0 '0 0 00
-
a N - * a N = -
a -
'a ' 3' '~ ~ ~ 3' ~- 4- '..~ ('4 C 0 ('~ 3' 0'
- 'N - I-.
~? -
I- 0 0 0 0 0 0 0 I" I~- - - - - -
ul ' ' I I I- - - - - - -
~ '- ~ ~ Sn C 4
~ 0 '0 0 0 00
aN -
~: *~ 0 - r- - a't. 4- '-~ - ~ *0 4'- a *-~ 0 - 9.. 9
?a 'a a' t- '0 '00 . .
.9 (44 (44 444 944 - -
- 41; - - - - - - - - - -
* ,,4 ~ Sn Sn Sn .. i C 'a0 0 0 0 '0 0 0 ~
- a - -
* a - a a - - -,
0' ~I
"~ ('~'~'~I,~44 '44 I~4 0 ('4 -b
'a - -
.1 -~ - *4 454 - - -
7'I! ~ I I
4' -5
'a C - -
-~ I A %~ ~ I
- - a
~ 'aI. A
~- &- .9 *' * * a a * * .1a- '
- S
- - - - -
ai
3.
it
II
.1
oo0 0 0 0
o0 0 0 0
0 0 0 0~sn ~t~Op
o20~~ir~- 2uw sillcos tsi, 0 2t 0o
33
A r00
00
Inth oloing cos we will dis 2cus p dere ofte curcof te aove pprxilltios inEqs (b)and(7)
A. rfec of0niertixamshr
Asdicuse bfoeth nnlnert~rm i nglgilefo hgho
rnssle i edotmsper. owve,~th trmC holdbehadld ar0
The cffet of nolinear erm (r 3, is no3clgbefrayms
31
r
approximation allows us to reduce Eqs. (6-) and (7) to linear differential
equations with constant coefficients. As a result., it is possible t, ob-
tain a piecewise dlosed-form solutioii. This appruJih is discussed in the
next section.
B. Approximation of Nonlinear Term in Exoatmosphere
In the Nike-Zeus system, prediction of the-trajectory of a reentry
vehicle is based on an analytical closed-form solution of ai, approximate
set of equations of motion. 4 In order to obtain an analvtic solution to
the equations, the effect of gravity is omitted, and the resulting predic-
tions are corrected for gravity. This is one way of approximating the
original differential equation to find a closed-form solution.
Another approach is to divide the total flight time into several
intervals and to find a closed-form solution for each interval. In other
words, term C is eliminated and the-initial values of r and V [only r in
Eq. (7)] are used for a certain time interval [0, Atl. ihe values of r
and V are recalculated by using the state values at time "It. and these
revised constant values of r and V are used for ca'culating the trajectory
for the next time interval [At, 2At]. This same procedure is continued
until impact is reached.
fhis idea is demonstrated for an exoazmospheric trajectory, and the
results are tabulated in Table V. The deviations of the predicted posi-
tion from the exact value after 240-sec flight are the following:
50,000 ft ------ without term C
20,000 ft ------ 60-sec correction of r and V in C
10,000 ft ------ 30-sec correction of r and V in C
5,000 ft ------ 10-sec correction of r and V in C
The more frequently corrections are made, the more accurately we wi;1l
obtain solutions. The suitable number of intervals for dividing the total
time depends on the error constraints.
If the total time T is divided into N intervals of At, then the
original equation is approximated as
R((t) = An X n(0 + B for nat < t < (+ lAt
Tn = 0, 1.... At
32
and the initial rosiditions are defined as
XQ(O) X_
TX,[(nAt) +- + l)An, 1 1, 2,"'".At
In the case of exoatmospheric trajectories, , is negligible; hence,
An and B,, can be considered to be constant.
C. Influence of in Exoatmosphere
An example of a missile trajectory in exeatmosphere is shown in
Figs. 13, 14. and Table V. The deviation of the impact points (compared
at the same time rather than at the same altitude) with and without con-
sideration of w is about 75,000 ft (8,000 in the x direction, 25,000 ft
in the y direction and 70,000 ft in the z direction) after 240 sec of
flight.
The velocity of the missile in this example is much larger than that
in endoatmosphere examples shown previously. Hlence, the effect of the
coriolis term is greater.
Although the effect of a is negligible for endoatmospheric trajectory
predictions, the effect is very significant for exoatmospheric trajectory
predictions.
D. Influence of Eccentricity in Exodmosphere
Figures 13, 14, and 'Fable V show the case of a missile in exoatmos-
phere. The deviation of impact points with and without consideration of
the eccentricity e is about, 2100 ft (200 ft in the x direction, 500 ft in
the y direction and 2000 ft in the - direction). The eccentricity e is
negligible (with certain reservation) for the cases in exoatosphere.
The main objective of neglecting the term D contain ing the eccentric-
try e is to simplify the differential equation in order to obtain a closed-
form solution. For exoatmospheric trajectories, the deviation can be as
large as 0.5 nautical miles after a 240-see flight. If the tolerance of
the error is several miles, then the eccentricity e can be considered
as zero. If eccentricity e cannot be considered as zero, then it is
c earl\ difficult to find a closed-form solution.
33
One way to overcome this difficulty is to find at efficient numerical
integration technique. It is feasible to obtain a solution of the dif-
ferential equation (I) by using about 0.5 sec of computer time on a
present-day computer (e.g., UNIVAC 1108). The flight time for these exo-
atmospheric cases is on the order of 5 minutes or more; therefore, 0.5 sec
can be well justified for the computer calculations.
It is also possible to neglect the effect of eccentricity e and to
simplify the differential equation so that the closed-form solution can be
found. In order to support the above statement, it is usefL' to tabulate
the state values at 60 sec after the initial time for trajectories with
and without consideration of the eccentricity e. The last two i'ows of
Table V show that the deviation is about 160 ft after 60 sec.
i.
! -
V SENSITIVITY OF IMPACT POINTS TO INITIAL VALUES
When the state values of an incoming missile are estimated, some
errors are inevitable. In order to reduce errors, the computation time
must be increased significantly. The knowledge of the propagation of
initial erirs to the final values in prediction is very meaningful in
order to evaluate the trade-off between the magnitude of prediction errors
and the computation time in estimation. For this purpose, the sensitivity
of the. initial values to the impact points is briefly investigated.
The following simplified equation is used for the sensitivity analysis:
0
0
t 0
0
a
where a and 6 are constant. Then the solution is described as
1 0 0 t 0 0 01t 2
DO 00 10 0 0t 0 -tX(t) 0 0 I 0 0 t (O) + ! bt2 (8)
2
0 0 0 1 00 0
0 0 0 0 1 0 at
0 0 0 0 0 1 bt
If there is a small error AX in X(0), then the state value X(T) at time
T is expressed as
35
1 0 0 T 0 0
o 1 0 0 T 0 a
0 1 0 0 1 0 - 1'"!I(T) Co 0 1 0 0 T IX(O) + AX] + T2
2
0 0 0 1 0 0 T0 o) 0 0 1 0 ,T
1 " 0 0 T 0 O0
0 0 1 0 0 T'- X(T) + (9)-0 0 0 1 0 0 -
0 0 o 0 1 0
The examples considered previously are used again for the sensi-
tivity analysis. In both endoatmosphere and exoatmospihere cases, 10 per-
cent and 20 percent errors are independently introduced into each initial
value. The propagation of each error to the final values is evaluated
by integrating the differential Eq. (1) numericall) and b\ using theI relationship in Eq. (8). These results are shown in Tables VI through
.X. The values in parentheses in these tables are theoretical results
by using the relationship in Eq. (8). According to Bef. 2, the estima-
t tion errors will become about 2 percent after 5 -see of filtering. Hence,
this sensitivity analysis will give better results for more realistic
Fcases.
This coarse sensitivity analysis gives a good indication ot th°, propa-
gation of errors in the initial values. In the ex..mplo of exoatmosphere,
the sensitivity analpsis, and the numerical integratio, agree very well. Inthe case of endoatmospht re, the analysis and the numeri,_al integration match
very well in most rasv , but the cases %here errors esist in :(}0 aid 1-0)
do have signi ficant deh\tations. Errors in x(0), y(t , x0)), a1d 1 0) propla-
gate in the manlier expresstd in Fa1. (9). The psitimit error.' tr1pagate
withiout an)) amplificai tion and iave little inifluence on liht, ,elocit . iThP\ blocity errors plropla t;l without ly ampl I ii icati on ili the yr lo itv Itlf,
it they have a sigriilicant flfect o the posi t ion tI r I'S
36
o ~ ~ c CIA00 0 0 0
- - - - -
.1 ~ ~ ~ o e4 4 = 4 4 l
.44 ~ . 44~~4 -0 LM% 2k4 C 40'. - - e~ 4 C
4 ~4a,
0 03 3 'o .0
r- 0 .0 . 3.- '0 '0 '. at
I 3 3 I S I At
L337
l - -V - -P -P-
*.a~ ~ ~ ~ 1 'o N 0 f E
0't 0" -q t - Lr 0Q - - - 1- 01 0 0
-4 co C30 EN, m' -m N
co- co a, cc 0 ~ o -
I--
.0 M.
EN ~ r lie 0 0 0' c,
n 0 0 0 ~ ~ *nI .
- -i - 0 U38
C4 4 4 4 CI 4
CIS 0 0 04 0
- f. In L5M5 -
co 0 v CIOc en -t 4N eq - t~-
t's en VC' . - C14 '0)
o 4 4 4 4 en 4
-e r.. - 4 9" - Cdo~e do~4 n 4 eqi
- e l l -r-
e l %ne q qe
-~cr r- C 444-i C 1 '7!00 0
'n c- %0 COSf q f~~~-t - - 4 0 SI CIS eli9-
a SoI I It as at lit
- -
* .~ i SR SIC CR sS SRn 4
-~t el 0 0 0~ I- - - - - el- l
4 55 a- 1% too5 - 5
- ~ ~ I tA q 0 I
- *1.A
AAA* 0 0 0 0
Soe 04 C ~4 l CA
* C4 o fn en,
a, 7il - . . .. .
U .- .- - -. -= i m ! l all it AI A
* * In 0% '0_.0
C4 cc _ - cc; .
F 0 0 S.I '
0. 1--'., 'C . . 0,, I i I I III
I~I 9 - - fn - -
lJ ~~' or. W:1.... ..
II -, . = A - U, A:
in 0 A vi 4
-~ ~ ~ S 0 09 -- 0 -
- - - fn-, ~ ~ ~ e fn I II
V. I- .9 a
a~ - a
4. IS IS I S Il A A A
all so to a I '
'C IA 40
VI CONCLUSION
The different'i;l Eq. (1) is a good mathematical model of the bal-
listic motion. Without any approximation, it seems hopeless to find a
closed-form solution of Eq, (1). The only way to find a solution of
Eq (1) is numerical integration. As a result, it requires a signifi-
cant amount of computation time. This memorandum describes a simplifica-
tion of the differential Eq. (1) of the ballistic trajectories. The main
purpose of an approxiirition is to obtain a closed-form solution.
T!,e problems are divided into two domains, namely the endoatmospheric
problem and the exoatmospheric problem. The endoatmospheric problem is
again divided into two. namely, the high-,_ case and the low-f case. In
each case. the influences of the eccentricity e. the rotation rate CV. the
ballistic coefficient &. and the nonlinearities are considered. A summary
of the influences is shown in Table X. In exoatmospheric Frediction prob-
lems, the earth rotation rate c4 and the nonlinear te-i C should be treated
carefully, and in endoatmospheric prediction problem. the ballistic co-
efficient - should be handled properly. 1he effect. of the ballistic
coefficient is very significant on the trajectory at low altitudes (e.g..
for impact point and impact time prediction). further research effort
should be oriented toward improving the estimation and prediction of
ballistic coefficients.
Future work on the prediction problem is to obtain closed-formsolutions of Eqs (3) and (7), One possible was is to find a piecewise
,'losed-form solution over a suitable time interval by taking constant
values of ,. and C in Eqs, (3) and (), respectively.
Table X
ft- .ivell'ift I-,, .4t I,' l l ll lli IitW -'-Ipli llo .0 t . ar,-aI I sll fl~
41
!I
APPENDIX
The derivation of the differential Eq. (I) is discussed in this
appendix.
If NaP is the absolute ac,.eleration of a mi ssile P (it is considered
to be a particle) in a reference frame N and a is the niss of P. then
the inertia force F acting on P in N satisfies
F (A.1)
The reference frame N is a reference frame in which the center C of
the earth and the earth's axis, line NS. are fixed suh that the angular
velocity of the earth E with respect to the reference N. Y. is given by
where w~ 2-r- ad/day and nis a unit vector parallel to line NS. This
reference frame N is a good approximation to a Xvwton a retert:ace frame.
From Fig. 1.
$ • xt •yj • AA A
where t, and k are unit vectors defined in Fig. I and x v, 4and are/. A "
the measure numetra in the directionq k,, and k. resptctivtly. Thtv
velocity of P *ith-reapect to the reference is then expro.tsed as
-dt
The accelerations '9,p at time 0 art related bv
xf, P 4 Ca Mb P*z
- t.'as 1k 0M,~ tw U44 tI +t t.vifdi
42
V
whtre P* is the point fixed in E that coincides with P at time t*, and
a is Illed the coincident-point velocity and the vector 2w x 1VP is
called the coriolis acceleration of P for the reference frames E and N.
The coincident point v'elocity I aP" satisfies
haQ N SaC IWOx r *w x (w x r)
andN dw
dt
The acceleration of P with respect to E is
E d E VE P EEP A IN~
a -i + Yj + zdt
Therefore,
A ANaP + +j + it + > (w x r) + 2& X EVP (A. 3)
.Sincr cA 4nd r are expressed as
A
(c = sin + (." COS /I])Ax A
r = ri + rYj + rk
r = X
r = - R sin (IL- (tc)
r = + R cos (j. - c.) (A.4)
'Ihe last two terms of Eq. (A. 3) are written as
(w ×) - w• rw
A A
-cr, i + u 2 [(rY cos /L + r, sin iz) cos 4- r,]J
+ W'[r cos k4 + r sin AL) sin L - rlk
E. × VP A,( Ao . - si z"COS sin i + x o sin j x- (A cos/
43
Therefore,
* NdP M!+ 4 cos /.L -'sin i) A
A
2 ^j
+ {y 2iu' sinii + c(r cos . r, sin u) cosj i
; - 21cc cob !z t ,2 (r cos + r sin p) sin } (A.5)
Let us now consider the left-Iand side of Eq. (A. 1). The force F
acting on a missile P is divided into two elements. namely the drag
force Fd and the gravitational force F.; hence,
F -- Fd + F% (A.6)
The drag force per unit. mass acting on a body is given approximately
by the equation
Ed - 2 i- E P (A.7)
Next, let us find the expression of the gravitational force. [The gravitational potential at P due to the earth E is expressedb as
';EiP m-r - 2r 2-- [31 - (11 , I. + +
where I is the moment of inertia about the line OP and I,, I, and I
are the principal moments of inertia of the earth E of the mass m.
Since the gravitational force per unit mass is the gradient of thegrav.itational potential, the equation
F8 /i = vuEP
holds, where the operator V is defined as
A A A A
+ M 4 n-
44
and X 1 . X., and X3 are measure numbers of the principal axes of the e "th
E. Let us define mutually perpendicular unit vectors ii and 12 as shown
in Fig. A-I. Then
F1 la /m =VUEIP
GM ~ 3Gh+ - 1(-21, + I + L3)t.r 2 2r 4 2
+ 2112 2 + 2113131
Where I1, 1", 13 are the moments of inertia of the earth about the centerthe(h cio A A A 1
alotg the directions n , n' n 3 , respectively, and I 12 are the momentA Aof inertia of the earth about the center for the pair of directions n, n,
and n , n 3 respectively.
By using algebraic transformation, the above equation becomes
GMA 3C AF /M -I k - (In- I)(1 - 5 sin2 )k 1
r2 r
4
J0 - 1 )2 sin n A
2r4t If
3(1, - I)
2,Ia 2
andIA Ar = k i ,
then
F M A Agrr + gn (A.8)
where
I + .1 (1 5 sin- q')g r
45
X3
0 X
Im
TA- 514S -440
FIG. A-i ILLUSTRATIONS OF UNIT VECTORS
46
6-,
A A A
1lint vectors r anid n are described in terms of i j and k as
r A r A r A_____
AA
A A
An = cos p-j + sin 4k
ilence Eq. (A.7) is expressed as
r + r\yA r' r'n = g rt! i " (g I 'gCos L 1 + g sin k
(A.9)
B. substituting Eqs. (A.5), (A.7) and (A.9) into Eq. (A.1) and comparing
t~e easurA values of . A, elements, it is found that
. + 2 cs -s sin 4) - r - + rr
N ,'" 2- s i n u w w rz sin 2
p r ry
+ .(,2r. sin i- cos IL = - , + g + cos FL (A. 10)
- 2wx cos - r cos2 L
+ ar sil L COS LL , + g r + g, sin 4y 2, 72 o
w he r e
r x
r y - R sin(L - 4L,)
r - + R cos(,L - ,)
r (r2
+ r2 +. 7 2
V' - (j.2 ," + "
g,(), (1 5 sin "k)
47
Li
2 r)
sin cp~ fy cos ju + z sin /I + R sin/k
If a 6 x 1 vector X' is defined as
AT [X, y, z~xj
then Eq. (A.10) is written as Eq. (1) in the main text.
44
4
REFERENCES
H. E. kalmatn. "A New Approach to Linear Filtering and Prediction Problems," Trans. AS.HE,
J. 1j6sjc Engr. (March 1%O).
R. E. Ldrson. B.' %. Pressler. B. S. Patner, "Application of the Extended Kalman Filter
to Bllistic Trajetory "stimation," Final eport, Contract UA-U - O I-0 1 ' 9OOOb(Y),
SRI1 project SlI9-1lU3, St.anford Research Institute, Menlo Park, California (January 1967).
3. US. Standard Atmosphere, 19b2, Prepared by NASA, USAF, USWIH (December 1962).
1. Nike-Zeus TIC Equations, Bell Telephone Laboratories.
T. H. Kane, Analytical Elements of Mechanics, Vol. 2 (Academic Press, 1961).
u. S. . McCuskey, Introduction to Celestial Mechanics (Addison-Wesley, 1963).
9
Sm
44
